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1,108,101,566,715 | arxiv | \section{Overview of notation}
\label{sec:notation_overview}
The following notation is used in the paper.
\begin{itemize}
\item $G(n)$ = the number of graphic sequences of length $n$.
\item $\lw=(\lw_k)_{k\geq 0}$=lazy random walk, that is, a random process with independent and identically distributed increments that take the value $0$ with probability $1/2$, $-1$ with probability $1/4$ and $+1$ with probability $1/4$.
\item $\lwa_k=\sum_{i=0}^k \lw_i$, so $(\lwa_k)_{k\geq 0}$ is an integrated random walk.
\item $\lwae_i$ = area of the $i$th excursion of $\lw$.
\item $\lwac_k=\sum_{i=1}^k \lwae_i$. We can see $\lwac$ as a `subsequence' of $\lwa$, where $\lwa\ge 0$ if and only if $\lwac\ge 0$.
\item $\zeta_1=\inf\{k\geq1 : \lw_k=0, \lwa_k\le 0\}$.
\item $\rho = \Prb(\lwa_{\zeta_1}=0)=$ probability that $(\lwa_k)_{k\geq 0}$ hits zero before the first time it goes negative.
\item $\lb=(\lb_k)_{k=0}^n$= lazy random walk bridge, that is, $(\lw_k)_{k=0}^n$ conditioned on $\lb_n=0$.
\item $\lba,\lbae,\lbac$ the counterparts of $\lwa,\lwae,\lwac$ respectively, with $\lw$ replaced by $\lb$ in the definition.
\item $N_n$ = number of times $\lb$ hits 0 after time 0 (so $N_n\in \{1,\dots,n\}$).
\item $M_n$ = the number of times $\lbac$ hits zero.
\item $\pbac,\pbae$ = perturbed variants of $\lbac$ and $\lbae$ respectively.
\item $\xi_i$ = the $i$th time $\lbac$ hits zero.
\item $\probareazero{n}=\Prb(M_n\ge 1\mid \lbac_1,\dots, \lbac_{N_n}\ge 0)$.
\end{itemize}
\section{Proof of Lemma \ref{lem:useful}}
\useful*
We include the proof of Lemma \ref{lem:useful} from Burns \cite{Burns2007TheNO} below for the convenience of the reader.
For $x\in \R_{>0}^n$, let
\[\textstyle
A(x)=\big\{(\sigma,s):\sigma \in S_n,\,s\in \{-1,1\}^n,\,
\sum_{i=1}^k s_ix_{\sigma(i)}\ge 0 \text{ for all }k\in [n]\big\}.
\]
We will show that
\[
|A(x)|\ge (2n-1)!!
\]
and that equality holds if, for all distinct $S,S'\subseteq [n]$, the
corresponding sums are also distinct.
We call a vector $y\in \R_{>0}^n$ \emph{rapidly decreasing} if
\[
y_i > y_{i+1}+\dotsb+y_n
\]
for all $i\in [n]$. We will first use induction on $n$ to show that $|A(y)|=(2n-1)!!$ for all rapidly decreasing
$y\in \R_{>0}^n$. It is clear that $A(y)=\{(\text{Id},1)\}$ when $n = 1$, and the claim holds.
Suppose that we have shown the claim for some $n\ge 1$, and let $y\in \R_{>0}^{n+1}$. Since the sequence is
rapidly decreasing, the pair $(\sigma, s)$ is in $A(y)$ if and only if $s(i) = 1$ for all $i \in [n+1]$
such that $\sigma(i)$ is the lowest number seen so far, i.e. $\sigma(i) < \sigma(j)$ for all $j < i$.
Let $\alpha = \sigma^{-1}(n+1)$. If $\alpha \ne 1$, then the term $s_\alpha y_{n+1}$ has no impact on whether
the sequence is valid, while if $\alpha = 1$, we require $s_1=1$ and then we need the remainder of the
sequence to satisfy the condition. More formally, let $y'$, $\sigma'$ and $s'$ be what is left after
removing $y_{n+1}$, $n+1$ and $s_{\alpha}$ respectively. That is
\[
y'= (y_1,\dots,y_n),\quad \sigma'(i) = \sigma\big(i+\one_{\ge\alpha}(i)\big),
\quad s' = (s_1,\dots,s_{\alpha-1},s_{\alpha+1},\dots,s_{n+1}).
\]
If $\alpha = 1$, then $(\sigma,s)\in A(y)$ if and only if $s_1=1$ and $(\sigma',s') \in A(y')$.
If $\alpha\ne 1$, then $(\sigma,s)\in A(y)$ if and only if $(\sigma',s') \in A(y')$.
Hence, there are $2n+1$ pairs $(\sigma,s)$ in $A(y)$ for each pair $(\sigma',s')$ in $A(y')$.
Clearly, $y'$ is also rapidly decreasing and the result follows by induction.
For $x\in \R^n$ and $S\subseteq [n]$, we write $x_S=\sum_{i\in S}x_i$. We call $x\in \R_{>0}^n$
\emph{sum-distinct} if $x_S\ne x_{S'}$ for all distinct subsets $S,S'\subseteq[n]$.
Let $x\in \R^n_{>0}$ be sum-distinct. We will construct $y\in \R_{>0}^n$ that is rapidly decreasing and for which $|A(x)|=|A(y)|$.
Since $x$ is sum-distinct, we can define a total order $<_x$ on the power set $\mathcal{P}([n])$ by $S <_x S'$ if and only if
$x_S < x_{S'}$. We note that for a given $\sigma$ and~$s$, the condition $\sum_{i=1}^k s_ix_{\sigma(i)}\ge 0$
is equivalent to the condition $x_{S_-} <_x x_{S_+}$ where $S_\pm = \{\sigma(i): i \le k,\,s_i = \pm1\}$,
and hence $A(x)$ only depends on $x$ through $<_x$.
To get to the vector $y$, we will increase $x_1$ until it is larger than $x_2+\dotsb+x_n$, and then increase
$x_1$ and $x_2$ until we also have $x_2>x_3+\dotsb+x_n$ etc. We will do this in a series of steps so that each
increase only changes the total ordering by a ``small" amount, and the following claim shows this
preserves $|A(x)|$. We defer the proof of this claim to the end of this section and first finish the current proof.
\begin{claim}\label{claim:small-change}
Let $x$ and $x'$ be sum-distinct and assume there is a unique pair $\{L,R\}$ of disjoint subsets for which
$L<_{x}R$ yet $R<_{x'}L$. Then $|A(x)| = |A(x')|$.
\end{claim}
Suppose that we already have $x_{j} > x_{j+1}+\dotsb+x_{n}$ for all $j<i$, and we wish to extend this to include
$j=i$ as well. We will slowly increase $x_1$, $x_2$, \dots, $x_i$ so that we only change one disjoint inequality
on the power set at a time and we can use the claim above to show that $|A(x)|$ does not change. We first ensure
that no signed sums are the same by slightly perturbing $x$ by a small amount, which we choose to be small enough to not
change the order $<_x$. Indeed, if
\[
\eps=\min_{S\ne S'}|x_S-x_{S'}|=\min_{S\ne S'}|x_{S\setminus S'}-x_{S'\setminus S}|,
\]
then perturbing each entry of $x$ by less than $\eps/n$ cannot possibly change the order $<_x$.
Let $z$ be a random vector formed by adding a small independent $\Unif[0,\eps/n]$ random variable to each entry
of~$x$, and note that $\mathord{<_x}=\mathord{<_z}$,
so $|A(x)| = |A(z)|$. Almost surely there are no two pairs of disjoint sets
$(A,B)$ and $(C,D)$ such that $z_A-z_B = z_C-z_D$, and we can order the pairs $(A_t,B_t)$ of disjoint subsets
of $[n] \setminus [i]$ such that $z_{A_t}-z_{B_t}$ is increasing in~$t$. Suppose $(A_\tau,B_\tau)$ is the
first pair for which $z_{A_\tau}-z_{B_\tau} > z_i$. If there is no such pair, then we already have
$z_i > z_{i+1}+\dotsb+ z_{n}$ (by choosing $A=[n]\setminus [i]$ and $B=\emptyset$).
Let $\delta$ be any value in the interval $(z_{A_\tau}-z_{B_\tau}-z_i, z_{A_{\tau+1}}-z_{B_{\tau+1}}-z_i)$
(or in $(z_{A_\tau}-z_{B_\tau}-z_i,\infty)$ if there is no pair $(A_{\tau+1},B_{\tau+1}))$, and consider the vector
\[
z^{(2)} =(z_1+2^{i-1}\delta, z_2+2^{i-2}\delta, \dots, z_i+\delta, z_{i+1}, z_{i+2}, \dots z_n).
\]
This is again sum-distinct and there is a unique pair $(L,R)=(B_\tau\cup\{i\},A_\tau)$ such that
$L <_z R$ but $R <_{z^{(2)}} L$, so $|A(z^{(2)})| = |A(z)| = |A(x)|$. It also still has the property
that $z^{(2)}_{j} > z^{(2)}_{j+1}+\dotsb+z^{(2)}_{n}$ for $j<i$. It may not yet have the property
that $z_{i}^{(2)} > z_{i+1}^{(2)}+\dotsb+z_{n}^{(2)}$, but we do have
$z^{(2)}_i>z_{A_\tau}-z_{B_\tau}$, and we can choose $\delta^{(2)}$ in the
interval $(z_{A_{\tau+1}}-z_{B_{\tau+1}}-z^{(2)}_i, z_{A_{\tau+2}}-z_{B_{\tau+2}}-z^{(2)}_i)$
and repeat to get $z^{(3)}$, $z^{(4)}$, $\dots$. The process terminates at
some $k$ when $z^{(k)}_i > z^{(k)}_{i+1}+\dotsb+z^{(k)}_{n}$, and we take this to be our new~$x$.
Repeating this process for $i = 1,2,\dots,n-1$ in turn, gives a rapidly decreasing vector
$y$ with $|A(y)| = |A(x)|$, as required.
We still need to prove the claim for $x$ which are not sum-distinct. As before let
\[
\eps=\min_{x_S\ne x_{S'}}|x_S-x_{S'}|=\min_{x_S\ne x_{S'}}|x_{S\setminus S'}-x_{S'\setminus S}|.
\]
If each entry is perturbed by less than $\eps/n$ to get a vector $z$, then
\[
\sum_{i=1}^k s_ix_{\sigma(i)}\ge 0 \iff \sum_{i=1}^k s_iz_{\sigma(i)} > -\eps
\impliedby \sum_{i=1}^k s_iz_{\sigma(i)} \ge 0.
\]
Therefore, $A(x) \supseteq A(z)$. If we get $z$ by by adding a small $\Unif[0,\eps/n]$
random variable to each entry of~$x$, then $z$ is almost surely sum-distinct and the
result follows since $|A(x)| \ge |A(z)| = (2n - 1)!!$ almost surely.
We now return to the proof of Claim~\ref{claim:small-change}.
\begin{proof}[Proof of Claim \ref{claim:small-change}]
Let $k = |L\cup R|$, and consider the function $f\colon S_n\times\{-1,1\}^n \to S_n\times\{-1,1\}^n$
which acts as follows. For any $(\sigma,s) \in S_n\times\{-1,1\}^n$, the function $f$ maps
$(\sigma,s)$ to $(\sigma',s')$ where $\sigma'$ is the permutation which is reversed on the first $k$ inputs,
and $s'$ is formed by reversing the order of the first $k$ entries in $s$ and negating them.
That is, $\sigma'(i)=\sigma(k+1-i)$ for $i\in [k]$ and $\sigma'(i)=\sigma(i)$ for $i\in [k+1,n]$,
and $s_i'=-s_{k+1-i}$ for $i\in [k]$ and $s_i'=s_i$ for $i\in [k+1,n]$.
The function $f$ is clearly self-inverse and hence bijective, and we will show that
$f(A(x)\setminus A(x')) \subseteq A(x')\setminus A(x)$. By switching the roles of $x$ and $x'$ and of $L$ and $R$, it follows that $f(A(x')\setminus A(x))\subseteq A(x)\setminus A(x')$, and so $|A(x)| = |A(x')|$.
Suppose that $(\sigma,s) \in A(x)\setminus A(x')$ and let $f(\sigma,s) = (\sigma',s')$.
It is obvious that $(\sigma',s')\notin A(x)$ as
\[
\sum_{i=1}^k s_i'x_{\sigma'(i)}=-\sum_{i=1}^k s_ix_{\sigma(i)}<0.
\]
Since $(\sigma,s) \notin A(x')$, there must be at least one $\ell$ for which $\sum_{i=1}^\ell s_ix_{\sigma(i)}'<0$.
We first show that there is exactly one choice for~$\ell$, and that it is~$k$.
Let $S_{\pm}=\{\sigma(i) : i\in [\ell],\,s_i=\pm 1\}$. Then
\begin{align*}
\sum_{i=1}^\ell s_ix_{\sigma(i)} = x_{S_+}-x_{S_-} > 0,\\
\sum_{i=1}^\ell s_ix'_{\sigma(i)} = x'_{S_+}-x'_{S_-} < 0.
\end{align*}
In other words, $S_- <_x S_+$ and $S_+ <_{x'} S_-$. Since $S_+\cap S_-=\emptyset$,
we find that $(S_-,S_+)=(L,R)$ and the only option for $\ell$ is~$k$.
Using this we can check that $(\sigma',s') \in A(x')$. For $j\in[k]$, we have
\begin{align*}
\sum_{i=1}^j s'_i x'_{\sigma'(i)}
&=-\sum_{i=k-j+1}^{k} s_i x'_{\sigma(i)}\\
&=-\sum_{i=1}^k s_i x'_{\sigma(i)}+\sum_{i=1}^{k-j} s_ix'_{\sigma(i)}.
\end{align*}
Both of these terms are non-negative since $\sum_{i=1}^{\ell} s_ix'_{\sigma(i)}< 0$ if and only
if $\ell=k$. Similarly, for $j>k$, we have
\begin{align*}
\sum_{i=1}^j s'_i x'_{\sigma'(i)}
&=\sum_{i=1}^k s'_i x'_{\sigma'(i)}-\sum_{i=1}^k s_i x'_{\sigma(i)}+\sum_{i=1}^j s_i x'_{\sigma(i)}\\
&=-2\sum_{i=1}^k s_ix'_{\sigma(i)}+\sum_{i=1}^js_ix'_{\sigma(i)}\ge 0.
\end{align*}
Hence, $(\sigma',s') \in A(x')\setminus A(x)$.
\end{proof}
\section{Proof of Lemma~\ref{lem:locallimitlazy}}\label{app:locallimitlazy}
\locallimitlazy*
The proof is an adaptation of the proof of Proposition~1 in~\cite{AurzadaDereichLifshits2014}.
Let $a,b\in\Z$. We will use Fourier inversion to estimate $\Prb(\lw_n=a,\lwa_n=b)$. We let $\lmgf_n\colon\R^2\to \mathbb{C}$ given by
\[
(t_1,t_2)\mapsto \E\left[e^{i(t_1\lw_n+t_2\lwa_n)}\right]
\]
be the characteristic function of $(\lw_n,\lwa_n)$, so that by $2$-dimensional Fourier inversion, we have
\begin{equation}
\Prb\left(\lw_n=a,\,\lwa_n=b\right)
=\frac{1}{(2\pi)^2}\int_{-\pi}^\pi\!\int_{-\pi}^\pi \lmgf_{n}(t_1,t_2)e^{-i(t_1a+t_2b)}dt_1dt_2.
\label{eq:fourierinversion}
\end{equation}
We observe that, for $\lws_1,\lws_2,\dots$ i.i.d.\ random variables distributed as the steps
of $\lw$ (i.e. $\Prb(\lws_i=0)=\tfrac{1}{2}$, $\Prb(\lws_i=-1)=\Prb(\lws_i=1)=\tfrac{1}{4}$) we have
\[
\lws_1\overset{d}{=}\tfrac{1}{2}(\uws_1+\uws_2)
\]
for $\uws_1$ and $\uws_2$ two i.i.d.\ Bernoulli random variables with
$\Prb(\uws_1=1)=\Prb(\uws_2=-1)=\tfrac{1}{2}$. Therefore, the characteristic function of $\lws_1$ satisfies
\[
\E\left[e^{it\lws_1}\right]
=\E\left[e^{it\uws_1/2}\right]^2
=\cos^2\left(\tfrac{t}{2}\right).
\]
Moreover, note that
\[\textstyle
(\lw_n,\lwa_n)\overset{d}{=}\left(\sum_{k=1}^n \lws_k,\sum_{k=1}^n (n-k+1)\lws_k\right)
\overset{d}{=} \left(\sum_{k=1}^n \lws_k,\sum_{k=1}^n k\lws_k\right),
\]
so
\[
\lmgf_n(t_1,t_2)=\prod_{k=1}^n\E\left[e^{i(t_1+kt_2)\lws_k}\right]
=\prod_{k=1}^n\cos^2\left(\tfrac{t_1+kt_2}{2}\right).
\]
In particular, we observe that for $n\ge 2$, the absolute value of the integrand in \eqref{eq:fourierinversion}
is equal to $1$ if and only if $t=(t_1,t_2)=(0,0)$ and is strictly smaller otherwise. We will examine
the contribution to the integral of $t$ in a small region around $(0,0)$, and we will show
that the contribution is negligible outside of that region.
Define $T_1=\{(t_1,t_2)\in[-\pi,\pi]^2:|t_1|+n|t_2|\ge\pi\}$. Then, for $n\ge 2$, it
is not too hard to see that there exists a $c<1$ such that $|\cos(\frac{t_1+kt_2}{2})|\le c$
for at least half of the values of $k=1,\dots,n$. Hence, on $T_1$,
\[
|\lmgf_n(t_1,t_2)|\le c^n=\exp(-\Omega(n)).
\]
Now define $T_2=\{(t_1,t_2):n^{-1/3}\le |t_1|+n|t_2|<\pi\}$.
We observe that we have the bound $\cos^2x\le e^{-x^2}$ for $|x|\le\frac{\pi}{2}$ and
\begin{equation}\label{e:sumsq}
\sum_{k=1}^n(t_1+kt_2)^2=nt_1^2+n(n+1)t_1t_2+\tfrac16n(n+1)(2n+1)t_2^2.
\end{equation}
Hence, on $T_2$, we have
$\sum_{k=1}^n(t_1+kt_2)^2=n(t_1+(n+1)t_2/2)^2+n(n^2-1)t_2^2/12=\Omega(n^{1/3})$ as
$\max\{|nt_2|,|t_1+(n+1)t_2/2|\}=\Omega(n^{-1/3})$. Thus
\[
|\lmgf_n(t_1,t_2)|\le\exp\Big(-\tfrac14\sum_{k=1}^n(t_1+kt_2)^2\Big)=\exp(-\Omega(n^{1/3})).
\]
We deduce that
\[
\frac{n^2}{(2\pi)^2}\iint_{T_1\cup T_2}\lmgf(t_1,t_2)e^{-i(t_1a+t_2b)}dt_1dt_2=o(1)
\]
as $n\to\infty$, uniformly in $a$ and $b$.
Finally consider $T_3=\{t:|t_1|+n|t_2|<n^{-1/3}\}$.
Now for small $x$, $\cos^2x=\exp(-x^2+O(x^4))$, and so \eqref{e:sumsq} implies that on $T_3$ we have
\[
\lmgf_n(t_1,t_2)=\exp\left(-\tfrac{1}{4}\left(nt_1^2+n^2t_1t_2+\tfrac{n^3}{3}t_2^2+O(n^{-1/3})\right)\right).
\]
Writing $t_1=n^{-1/2}s_1$, $t_2=n^{-3/2}s_2$, $v_1=n^{-1/2}a$ and $v_2=n^{-3/2}b$, we have
\begin{align*}
\frac{n^2}{(2\pi)^2}&\iint_{T_3}\lmgf(t_1,t_2)e^{-i(t_1a+t_2b)}dt_1dt_2\\
&=\frac{n^2}{(2\pi)^2}\iint_{T_3}
\exp\left(-\tfrac{1}{4}\left(nt_1^2+n^2t_1t_2+\tfrac{n^3}{3}t_2^2+O(n^{-1/3})\right)\right)e^{-i(t_1a+t_2b)}dt_1dt_2\\
&=\frac{1}{(2\pi)^2}\iint_{|s_1|+|s_2|\le n^{1/6}}\!\!
\exp\left(-\tfrac{1}{4}\left(s_1^2+s_1s_2+\tfrac{1}{3}s_2^2+O(n^{-1/3})\right)\right)e^{-i(s_1v_1+s_2v_2)}ds_1ds_2\\
&\to \frac{1}{(2\pi)^2}\iint_{\R^2}
\exp\left(-\tfrac{1}{4}\left(s_1^2+s_1s_2+\tfrac{1}{3}s_2^2\right)\right)e^{-i(s_1v_1+s_2v_2)}ds_1ds_2
\end{align*}
as $n\to\infty$ uniformly over all $v_1$ and $v_2$. Now
\begin{multline*}
\frac{1}{(2\pi)^2}\iint\exp\left(-\tfrac{1}{4}\left(s_1^2+s_1s_2+\tfrac{1}{3}s_2^2\right)\right)
e^{-i(s_1v_1+s_2v_2)}ds_1ds_2\\
=\frac{1}{(2\pi)^2}\iint\exp\left(-\tfrac{1}{2}s^TR^{-1}s\right)e^{-i s^Tv}ds_1ds_2
=\frac{\sqrt{\det R}}{2\pi}\exp\left(-\tfrac{1}{2}v^TRv\right),
\end{multline*}
where
\[
s=\begin{pmatrix}s_1\\s_2\end{pmatrix},\quad
v=\begin{pmatrix}v_1\\v_2\end{pmatrix},\quad
R^{-1}=\frac{1}{2}\begin{pmatrix}1&\tfrac{1}{2}\\\tfrac{1}{2}&\tfrac{1}{3}\end{pmatrix},\quad
R=\begin{pmatrix}8&-12\\-12&24\end{pmatrix}, \quad \det R=48.
\]
Therefore, we deduce that
\[
n^2\Prb\left(\lw_n=a,\lwa_n=b\right)
\to\frac{\sqrt{\det R}}{2\pi}\exp\left(-\tfrac{1}{2}v^TRv)\right)\\
=\frac{2\sqrt{3}}{\pi}\exp\left(-4v_1^2+12v_1v_2-12v_2^2\right)
\]
as $n\to\infty$, uniformly over all $(v_1,v_2)$, as required.
\section{Proof of Lemma \ref{lem:upperboundintegral0}}\label{app:upperboundintegral}
\upperboundintegral*
With Lemma~\ref{lem:locallimitlazy} in hand, the proof of Lemma \ref{lem:upperboundintegral0} is a direct adaptation of the proof of the
upper bound of Theorem~1 of \cite{AurzadaDereichLifshits2014} for simple symmetric random walks.
We include the proof for our case for completeness. Let $\adjlw=\adjlw(\lw,n)$ be the
adjoint process of $\lw$ on $[n]$, i.e. for $i\in [n]$, we set $\adjlw_i:=\lw_n-\lw_{n-i}$
and let $\adjlwa$ be the area process of~$\adjlw$.
Denote $\lw_k=\sum_{i=1}^k\lws_i$ for all $k$, set $b=\lfloor n/4 \rfloor$, and define the events
\begin{align*}
\Omega^+_n&=\{\lwa_1,\dots,\lwa_b\ge 0 \}\in \sigma(\lws_1,\dots,\lws_b), \\
\bar{\Omega}^+_n&=\{\adjlwa_1,\dots,\adjlwa_b\ge 0\}\in \sigma(\lws_{n-b+1},\dots,\lws_n).
\end{align*}
Observe that $\adjlw$ has the same law as~$\lw$, so $\Omega^+_n$ and $\bar{\Omega}^+_n$ are
independent and have equal probability. Moreover, by Theorem 1 and 2 in~\cite{Vysotsky2010},
$\Prb(\Omega^+_n)=\Theta(n^{-1/4})$.
Furthermore, on the event $\{\lw_n=\lwa_n=0\}$, we see that $\adjlwa_k=\lwa_{n-k}$ for any $k\in [n]$, so
\begin{align*}
\Prb\left(\lw_n=\lwa_n=0,\,\lwa_1,\dots,\lwa_n\ge 0\right)
&\le \Prb\left(\Omega^+_n\cap \bar\Omega^+_n\cap \{\lw_n=\lwa_n=0\}\right)\\
&=\Prb(\Omega^+_n)^2\Prb\left(\lw_n=\lwa_n=0 \mid \Omega^+_n\cap\bar\Omega^+_n\right)\\
&=\Theta(n^{-1/2})\Prb\left(\lw_n=\lwa_n=0 \mid \Omega^+_n\cap\bar\Omega^+_n\right).
\end{align*}
Now, we observe that $\Omega^+_n\cap\bar\Omega^+_n$ only depends on $\lws_1,\dots,\lws_b$
and $\lws_{n-b+1},\dots,\lws_n$, so
\begin{multline*}
\Prb\big(\lw_n=\lwa_n=0 \mid \Omega^+_n\cap\bar\Omega^+_n\big)\\
\le\sup_{(\ell_i)}
\Prb\big(\lw_n=\lwa_n=0 \mid \lws_i=\ell_i\text{ for }i\in\{1,\dots,b\}\cup\{n-b+1,\dots,n\}\big),
\end{multline*}
where the supremum is over all choices of $\ell_1,\dots,\ell_b,\ell_{n-b+1},\dots,\ell_n\in\{-1,0,1\}$.
We see that given the values of $\lws_1,\dots,\lws_b$ and $\lws_{n-b+1},\dots,\lws_n$, the processes
$(\lw,\lwa)$ restricted to $b+1,\dots,n-b$ have the same distribution as a lazy random walk and its
integrated counterpart with both processes started at some different point, which implies that
\[
\Prb\big(\lw_n=\lwa_n=0 \mid \Omega^+_n\cap\bar\Omega^+_n\big)
\le \sup_{y,a}\Prb\big(\lw_{n-2b}=y,\,\lwa_{n-2b}=a\big),
\]
which is $\Theta(n^{-2})$ by Lemma~\ref{lem:locallimitlazy}. The result now follows.
\section{Introduction}
Given a graph $G$ and a vertex $v\in V(G)$, the \emph{degree} of $v$ is the number of edges incident to~$v$,
and the \emph{degree sequence} of $G$ is the non-increasing sequence of its vertex degrees. We consider the
following very natural question: over all graphs on $n$ vertices, how many different degree sequences are there?
Since the degree of a vertex is at most $n-1$ and at least~$0$, a simple upper bound follows by bounding the
number of integer sequences $n-1\ge d_1\ge d_2\ge\dotsb\ge d_n\ge 0$. A `stars-and-bars' argument
shows that there are $\binom{2n-1}{n-1}=\Theta(4^n/\sqrt{n})$ such sequences, but not all of them
are degree sequences of graphs.
Sequences which are the degree sequence of some graph are called \emph{graphic sequences}. A famous result of
Erd\H{o}s and Gallai~\cite{ErdosGallai} provides necessary and sufficient conditions for a sequence to be graphical
and various other characterisations are known~\cite{Hakimi,Havel}.
Let $G(n)$ be the number of graphic sequences of length $n$ (or equivalently the number of degree
sequences across graphs on $n$ vertices). The best known bounds on $G(n)$ were given by
Burns~\cite{Burns2007TheNO} who showed that
\[
\frac{c_1 4^n}{n}\le G(n)\le \frac{4^n}{\sqrt{n}\log^{c_2}n}
\]
for some constants $c_1,c_2>0$ and for all $n\in \N$. To the best of our knowledge, these were the best known
asymptotics before our work, and this has been explicitly mentioned as an open problem in several
computational papers (e.g.\ \cite{LuBressan11,Wang19}).
Our main result pinpoints the asymptotics for $G(n)$.
\begin{theorem}\label{thm:main}
The number graphic sequences of length $n$ is $G(n)=(\cdeg+o(1))4^n/n^{3/4}$,
where $\cdeg>0$ is a constant.
\end{theorem}
This also answers in the affirmative a question of Royle~\cite{royal} who asked\footnote{\label{footnote:royle}Royle
actually asked whether $G'(n)/G'(n-1)\to 4$ where $G'(n)$ is the number of degree sequences
of graphs without isolated vertices. However $G'(n)=G(n)-G(n-1)$,
so this question is equivalent to the question of whether $G(n)/G(n-1)$ tends to~4.}
whether the ratio $G(n)/G(n-1)$ tends to~4.
\paragraph{The value of $\cdeg$ and a connection to random walks.}
We can express the value of the constant~$\cdeg$ in terms of the hitting probabilities of a particular random walk,
which arises from our proof strategy. In particular, computational estimates give
$\cdeg\approx 0.099094$ (see Section~\ref{subsec:rho}).
We will prove Theorem~\ref{thm:main} by viewing a sequence $n-1\ge d_1\ge \dotsb \ge d_n\ge 0$
as a path on a grid from $(0,n)$ to $(n-1,0)$, as depicted in Figure~\ref{fig:reformulation}(a).
We then count the number of graphic sequences of length $n$ by sampling such a path uniformly at random,
and computing the probability that the path satisfies the conditions given by
Erd\H{o}s and Gallai for a sequence to be graphical.
To be more precise, a sequence is graphical if and only if the sum of the degrees is even and it satisfies the dominating condition given in (\ref{eq:DC}). Our aim is to show that the probability that a random sequence satisfies the dominating condition is asymptotically $4\sqrt{\pi}\cdeg n^{-1/4}$. Then, we show that, asymptotically, half of the sequences that satisfy the dominating condition have even sum, and the result then follows from the fact that we considered $\binom{2n-1}{n-1}\approx(2\sqrt{\pi})^{-1}4^n n^{-1/2}$ sequences in total.
Via a number of reformulations (see Section \ref{sec:reformulation}), the probability that a uniformly random sequence satisfies the dominating condition turns out to be the probability that a particular integrated random walk bridge stays non-negative.
Let $Y=(\lw_k)_{k\geq 0}$ be a random walk that has increments that take the value $1$ with probability $\frac14$, the value $-1$ with probability $\frac14$ and the value $0$ otherwise, so that $Y$ is a \emph{lazy simple symmetric random walk}. Let $\lwa_k=\sum_{j=1}^k\lw_j$ be its area process. Our probability of interest is \emph{the probability that $\lwa_1,\dots,\lwa_{n-1}\geq 0$, conditional on the event that $\lw_{n-1}\in \{0,-1\}$}. To introduce its asymptotic value, we need some additional notation.
Let $\zeta_1=\inf\{k\geq 1 ~:~ \lw_k=0, \lwa_k\le 0\}$ be the first visit of $\lw$ to $0$ at which $(\lwa_k)_{k=1}^\infty$ hits $(-\infty,0]$ and let $\rho = \Prb(\lwa_{\zeta_1}=0)$.
We prove the following result about lazy simple symmetric random walk bridges which may be of independent interest.
\begin{prop}\label{prop:areanonneg}
We have that
\[
n^{1/4}\Prb(\lwa_1,\dots,\lwa_{n}\ge 0\mid \lw_n=0) \to \frac{\Gamma(3/4)}{\sqrt{2\pi(1-\rho)}}
\]
as $n\to\infty$.
\end{prop}
The probability that a random process does not go negative is also called the \emph{persistence probability}.
The persistence probability of integrated random processes was first studied by Sina\u{\i} \cite{Sinai1992} in 1992, who showed that the persistence probability of an $n$-step simple symmetric random walk (SSRW) is $\Theta(n^{-1/4})$. The sharp asymptotics (including the constant) follow from a result by Vysotsky \cite[Theorem 1]{Vysotsky2014}. His work on random walk bridges implies that the persistence probability of an $n$-step SSRW bridge is $\Theta(n^{-1/4})$ \cite[Proposition 1]{Vysotsky2014}. The sharp asymptotics for SSRW bridges are a natural next question, which we answer in Proposition \ref{prop:persistenceSSRW} for the SSRW bridge and in Proposition \ref{prop:areanonneg} for the lazy variant.
We use Proposition \ref{prop:areanonneg} to show that $G(n)=(\cdeg+o(1))4^n/n^{3/4}$ for
\[
\cdeg = \frac{\Gamma(3/4)}{4\pi\sqrt{2(1-\rho)}}.
\]
The probability generating function of the area of the first excursion of $\lw$ away from $0$ satisfies a recursive equation which allows us to find that
$\rho$ is approximately $0.5158026$, and plugging this into the equation gives $\cdeg\approx 0.099094$.
\paragraph{Related counting problems.}
Much more is known about related counting problems, such as the number of graphs
with a given degree sequence \cite{BarvinokH13,Wormald19}
and a variant of our problem where the sequence does not
need to be non-increasing (e.g.\ \cite{Stanley}).
The number $T(n)$ of out-degree sequences for $n$-vertex tournaments, also called score sequences,
has received particular interest, and the problem of determining $T(n)$ can be traced back to MacMahon in 1920 \cite{macmahon1920american}. Following work of Moser \cite{Moser}, Erd\H{o}s and Moser (see \cite{moon1968topics}), and Kleitman \cite{kleitman1970number},
it was shown that $T(n)=\Theta(4^n/n^{5/2})$ by Winston and Kleitman~\cite{KleitmanWinston} (lower bound) and Kim and Pittel~\cite{KimPittel} (upper bound).
Recently, Kolesnik \cite{Kolesnik22} determined the exact asymptotics,
showing that there is a constant $c\approx 0.392$ such that $T(n)=(c+o(1))4^n/n^{5/2}$.
Another well-studied variant is the fraction $p(N)$ of partitions of an integer $N$ that are graphical.
This corresponds to the variant of our problem where we fix the number of edges of the graph,
rather than the number of vertices.
In 1982, Wilf conjectured that $p(N)\to 0$ as $N\to \infty$.
Pittel \cite{Pittel99} resolved this problem in the affirmative using a Kolmogorov zero-one law,
and Erd\H{o}s and Richmond \cite{ErdosRichmond93} showed a lower bound of $p(N)\ge \pi/\sqrt{6N}$
for sufficiently large even~$N$. The best-known upper bound is $p(N)=O(N^{-\alpha})$ for $\alpha\approx 0.003$
from Melczer, Michelen and Mukherjee~\cite{MelczerMichelenMukherjee20}.
\paragraph{Exact enumeration.}
We also give an improved algorithm for the exact enumeration of graphic sequences and calculate
the number of graphic sequences of length $n$ for all $n$ up to~1651.
There is previous work on enumerating graphic sequences \cite{ivanyi2011,Ivanyi2012,Ivanyi2013,ruskey1994,Wang2019b},
and the numbers were known up to $n=118$~\cite{Wang19} as OEIS sequence \href{https://oeis.org/A004251}{A004251},
although the numbers $G'(n)$ of zero-free graphic sequences were known up to $n=290$~\cite{Wang2019b},
and these imply the values of $G(n)$ up to the same number by the observation in footnote~\ref{footnote:royle}.
\paragraph{Paper overview.}
In Section~\ref{sec:prelim}, we provide the reformulation of our problem in terms of integrated random walks.
In Section~\ref{sec:proof}, we first show that $G(n) = \Theta(4^n/n^{3/4})$, and then prove
Theorem~\ref{thm:main} and Proposition~\ref{prop:areanonneg}, up to technical lemmas that we postpone to Section~\ref{sec:technical}.
In Section~\ref{sec:comp} we introduce an improved algorithm for computing $G(n)$ exactly
for small $n$ and discuss computational results such as the approximation of $\rho$. We conclude with some open problems in
Section~\ref{sec:conclusion}. An overview of the notation used throughout the paper is given in Appendix \ref{sec:notation_overview}.
\section{Reformulation and notation}
\label{sec:prelim}
In order to prove our results, we need a suitable criterion for when a sequence of non-negative integers
is the degree sequence of a simple graph.
For a given sequence of non-negative integers $d_1\ge d_2\ge\dotsb\ge d_n\ge 0$,
let $d_i'$ be the number of $j$ with $d_j\ge i$ and set
\begin{align*}
s_i &= (n-1)-d_i,\\
s_i'&= n-d_i'.
\end{align*}
Let $\ell$ be the largest $j$ such that $d_j\ge j$. Using these definitions we can now
give the version of the Erd\H{o}s--Gallai Theorem that we will use.
\begin{theorem}[Variant of Erd\H{o}s--Gallai]
\label{thm:EG}
A sequence of integers $d_1\ge d_2\ge\dotsb\ge d_n\ge 0$ is the degree sequence of a
simple graph if and only if the sum $\sum_{i=1}^n d_i$ is even and for all\/ $k\le\ell$
\begin{equation}\label{eq:DC}
\sum_{i=1}^k s_i \ge \sum_{i=1}^k s_i'.
\end{equation}
\end{theorem}
We remark that this is just one of many similar characterizations of graphic sequences and that this form follows from the classical statement of the Erd\H{o}s--Gallai Theorem by rearranging terms and observing that only the first $\ell$ conditions need be checked (see e.g. \cite{hasselbarth1984verzweigtheit,nash-williamsValency,rousseau1995note,sierksma1991seven}).
We will call \eqref{eq:DC} the \emph{dominating condition}, so that a sequence of non-negative integers is a graphic
sequence if it satisfies the dominating condition and its sum is even.
As might be expected, it turns out that about half of the sequences that satisfy the
dominating condition have an even sum although, surprisingly, the exact proportion
seems to converge to $1/2$ quite slowly as $n\to\infty$ (see Section~\ref{subsec:eo}).
We will first focus on counting the number of sequences that satisfy the dominating condition, and we will handle the parity condition in Section \ref{subsec:half} (and briefly in the proof of Proposition \ref{prop:firstasymptotics}).
\subsection{It's a walk!}\label{sec:reformulation}
We now describe a way of associating a random walk to a uniformly random graphic sequence
in such a way that we can easily check whether the sequence is graphic using only the walk.
\begin{lemma}\label{lem:to_walk}
Let $(\lw_i)_{i\ge1}$ be a lazy simple symmetric random walk and let $\lwa_k=\sum_{i=1}^k\lw_i$.
Then the probability that a uniformly random sequence $n-1\ge d_1\ge\dotsb\ge d_n\ge 0$
satisfies the dominating condition \eqref{eq:DC} is equal to
\begin{equation}\label{eq:probDC}
\Prb\big(\lwa_1,\dots,\lwa_{n-1}\ge 0 \mid \lw_{n-1}\in\{0,-1\}\big).
\end{equation}
The probability that it is graphic is
\begin{equation}\label{eq:probgraphic}
\Prb\big(\lwa_1,\dots,\lwa_{n-1}\ge 0, \lwa_{n-1}\in 2\N \mid \lw_{n-1}\in\{0,-1\}\big).
\end{equation}
\end{lemma}
This lemma already gives some intuition for why $G(n)=\Theta(4^n/n^{3/4})$. Note that there are
$\binom{2n-1}{n}=\Theta(4^n/\sqrt{n})$ sequences $n-1\ge d_1\ge\dotsb\ge d_n\ge 0$, so we want
\eqref{eq:probDC} to be $\Theta(n^{-1/4})$. Firstly, note that we only need to check that
$\lwa_k\ge 0$ for $k$ such that $\lw_k=0$ (and for $k=n-1$), because
$\lwa_k$ is monotone on excursions of $\lw$ away from~$0$. Up to time~$n$,
the walk $\lw$ visits zero $\Theta(n^{1/2})$ times (in probability) so the area process restricted to
times $k$ when $\lw_k=0$ is a random process with $\Theta(n^{1/2})$ steps. If this process
were a random walk, then a result from Feller's book \cite[Theorem XII.7.1a]{Feller1971} would tell
us that the probability that it stays non-negative is $\Theta((n^{1/2})^{-1/2})=\Theta(n^{-1/4})$. We also require that the total area $A_{n-1}$ is even, but intuitively this should happen with probability roughly $1/2$. Of course, the lengths (and hence the areas) of the excursions are not independent and we cannot apply Feller's result, but this heuristic does at least suggest the right order for~\eqref{eq:probDC}.
\begin{figure}
\centering
\subcaptionbox{Step 1: View a sequence as a lattice path.\label{subfig:step1}}[0.4\textwidth]{%
\input{tikz_path}%
}\hfill%
\subcaptionbox{Step 2: Cut the path in half and reflect the latter half.\label{subfig:step2}}[0.4\textwidth]{%
\input{tikz_reflect_path}%
}\medskip\\
\subcaptionbox{Step 3: The dominating condition corresponds to the signed area between the pair of paths never going negative. \label{subfig:step3}}[0.4\textwidth]{%
\input{tikz_path_distance}%
}\hfill%
\subcaptionbox{Step 4: The cumulative distance process between the two paths gives the area.\label{subfig:step4}}[0.4\textwidth]{%
\input{tikz_diagonals}%
}%
\caption{By using a number of reformulations, we show that the probability that a uniformly random sequence
$n-1\ge d_1\ge d_2\ge\dotsb\ge d_n\ge 0$ satisfies the dominating condition is equal to the probability that
the integral of a lazy simple symmetric random walk with $n-1$ steps, conditioned to end in $0$ or~$-1$,
does not go negative.}
\label{fig:reformulation}
\end{figure}
We will now prove Lemma~\ref{lem:to_walk}. In order to get from a uniformly random non-increasing
sequence to a lazy random walk conditioned to end in $\{0,-1\}$, we need to make a few reformulations.
\paragraph{Step 1: View a sequence as a lattice path.}
We start by viewing the non-increasing sequence as a lattice path $P$ from $(0,n-1)$ to $(n,0)$
which only takes steps to the right and downwards (see Figure~\ref{subfig:step1}).
Informally, we put $n$ stacks of heights $d_1,d_2,\dots,d_n$ respectively next to each other
and let the $P$ be the path from $(0,n-1)$ to $(n,0)$ that traces the outline of the stacks.
To be precise, the path begins with $n-1-d_1$ steps downwards before taking a step right.
For each $2\le i\le n$, the walk takes $d_{i-1}-d_i$ steps downwards on the line $x=i-1$
before taking a step right, and it ends with $d_n$ steps downwards to end at the point $(n,0)$.
There must be a unique $\ell$ such that the walk goes through the point $(\ell,\ell)$,
and it is not hard to see that $\ell$ is the largest $j$ such that $d_j\ge j$.
\paragraph{Step 2: Cut the path in half and reflect the latter half.}
Starting from $(0,n-1)$, the path takes $n-1$ steps to reach the point $(\ell,\ell)$
as each step decreases $y-x$ by~1, and we define a path $W$ which starts with these $n-1$
steps before ending with one final step down. We now define another path $W'$ starting
at $(0,n-1)$ using the other end of the path $P$. Starting at $(n,0)$, walk backwards along
the path $P$ to $(\ell,\ell)$ and, at each step, add the ``reflected" step to $W'$:
if the step on $P$ is vertical, add a right step to~$W'$; otherwise add a down step to~$W'$.
This gives two paths $W$ and $W'$ which take $n$ steps and both end at $(\ell,\ell-1)$
(see Figure~\ref{subfig:step2}, in which the steps on the two paths have been numbered).
\paragraph{Step 3: The dominating condition corresponds to the signed area between the pair of paths never going negative.}
If the walk $W$ (resp.\ $W'$) has a horizontal line from $(i-1,a)$ to $(i,a)$,
then $s_i = n-1-a$ (resp.\ $s_i'=n-1-a$). In particular, if $W$ is at $(i,a)$
after $k$ steps, the sum $\sum_{j=1}^is_j$ is exactly the area enclosed by the walk~$W$,
the line $y=n-1$ and the line $x=i$. Similarly, if $W'$ is at $(i,a)$
after $k$ steps, the sum $\sum_{j=1}^is'_j$ is exactly the area enclosed by the walk~$W'$,
the line $y=n-1$ and the line $x=i$. Therefore, if $W$ and $W'$ are both at $(i,a)$ after
$k$ steps, then the signed area between $W'$ and $W$ is exactly the
sum $\sum_{j=1}^i(s_j-s_j')$ (see Figure~\ref{subfig:step3}).
The dominating condition checks if this sum is non-negative for all~$i$.
\paragraph{Step 4: The cumulative distance process between the two paths gives the area.}
Define $Z_1,\dots,Z_{n}$ by setting $Z_i$ equal to $+1$ if the walk $W$ goes down at
the $i$th step, and $-1$ if it goes right. Similarly, define $Z_1',\dots,Z_{n}'$ by
setting $Z_i'$ equal to $-1$ if the walk $W'$ goes down at the $i$th step, and $+1$
otherwise. Then,
$\lw_i=\tfrac{1}{2}\sum_{j=1}^i (Z_j + Z_j')$ keeps track of the (signed) number of diagonal
right/up steps from the walk $W$ to the walk $W'$. When the two walks coincide at time~$k$,
there will be a diagonal line through every box between the two walks, and hence the number
of diagonal lines is equal to the signed area between $W'$ and~$W$ up to time $k$
(see Figure~\ref{subfig:step4}). We claim that we only need to check the dominating
condition when the two walks coincide. Indeed, between times at which the walks coincide,
the area process is monotone so if the area process first becomes negative at some time $i$ where the walks do not coincide, the process will still be negative when the walks next coincide.
However, the number of diagonal lines used up to time $k$ is the integral of $\lw$ up to
time~$k$, and it suffices to check that condition $\sum_{i=1}^k \lw_i \ge 0$ at all times $k$ where the walks coincide. Since the integral of $Y$ is also monotone on excursions away from zero (i.e. between times when $W$ and $W'$ coincide), the condition (\ref{eq:DC}) is equivalent to $\sum_{i=1}^k \lw_i \ge 0$ for all $k \le n$. But $Y_n=0$, so it is in fact sufficient that
\begin{equation}\label{eq:WDC}
\sum_{i=1}^k \lw_i \ge 0 \quad \text{for all }k \le n-1.
\end{equation}
\paragraph{Step 5: The parity condition corresponds to the integral of $Y$ being even}
We now claim that
\begin{equation}\label{eqn:parity}
\sum_{i=1}^n d_i\equiv \sum_{i=1}^{n-1} \lw_i \mod 2.
\end{equation}
Split the sum $\sum_{i=1}^n d_i$ into $A_{\text{red}}=\sum_{i=1}^{\ell}d_i$ and $A_{\text{green}}=\sum_{i=\ell+1}^{n}d_i$. In Figure \ref{subfig:step1}, $A_\text{red}$ is the area below the red curve (left of $(\ell,\ell)$) and $A_{\text{green}}$ is the area below the green curve (right of $(\ell,\ell)$). Clearly, $A_\text{red}$ is also the area under the walk $W$. The area under the walk $W'$ is given by $A_\text{green} + \ell(\ell - 1)$. Hence, the signed area enclosed by $W'$ and $W$ is given by \[(A_\text{green}+\ell(\ell-1))-A_\text{red}\equiv A_\text{green}+A_\text{red} \mod 2.\]
It was shown in the previous step that the signed area enclosed by $W'$ and $W$ is given by $\sum_{i=1}^{n} \lw_i=\sum_{i=1}^{n-1} \lw_i$, and the claim follows.
\paragraph{Step 6: The distance process between a random pair of paths is a conditioned lazy random walk.}
Finally, we need to understand the distribution of $\lw$ when we sample a sequence
$n-1\ge d_1\ge\dotsb\ge d_n\ge 0$ uniformly at random. First, observe that Step~2 gives a bijection
between such non-increasing sequences and pairs of lattice paths $(W,W')$ of $n$ steps that
start at $(0,n-1)$ and end at the same point, and such that $W$ ends by taking a downwards step. These are in turn in bijection with pairs of lattice paths $(W, W')$ with $n - 1$ steps that start at $(0, n-1)$ and for which the corresponding walk $(Y_k)_{k=0}^{n-1}$ has $Y_{n-1} \in \{0, -1\}$.
Therefore, sampling a uniformly random sequence
corresponds to sampling a uniformly random pair of such paths. If we ignore the
requirement that $Y_{n-1} \in \{0, -1\}$, then for any~$j$, $Z_j$ and
$Z'_j$ have opposite signs with probability $1/2$, and both have value $+1$ (or $-1$)
with probability $1/4$, making $\lw$ a lazy simple symmetric random walk.
To recover the actual distribution of (the first $n-1$ steps of) $\lw$, we need to restrict to the paths for which $Y_{n-1} \in \{0, 1\}$, and we simply condition on this being the case.
\bigskip
We remark that steps 1 through 5 give a deterministic mapping which maps a sequence $n -1 \geq d_1 \geq \dotsb d_n \geq 0$ to a walk $(Y_i)_{i=1}^n$ and we have shown that we can check if the sequence is graphic by checking certain properties of the walk.
In Section \ref{sec:comp}, we use this reformulation (without introducing randomness) to enumerate the number of graphic sequences for small $n$.
\section{Proof of main result}
\label{sec:proof}
In this section, we prove Theorem~\ref{thm:main}, which is a direct consequence of the following two results.
\begin{prop}\label{prop:areanonnegrectangle}
A uniformly random sequence $n-1\ge d_1\ge\dotsb\ge d_n\ge 0$ has probability
\[
(1+o(1))\frac{\Gamma(3/4)}{\sqrt{2\pi(1-\rho)}}n^{-1/4}
\]
of satisfying the dominating condition \eqref{eq:DC}.
\end{prop}
\begin{lemma}\label{lem:parity}
A uniformly random sequence $n-1\ge d_1\ge\dotsb\ge d_n\ge 0$ which satisfies the
dominating condition \eqref{eq:DC}, has probability $1/2+o(1)$ of being a
graphic sequence \textup(equivalently, of having $\sum_{i=1}^nd_i$ even\textup).
\end{lemma}
The remainder of this section is structured as follows. In Section~\ref{subsec:firstasymptotics},
we show that $G(n)=\Theta(4^nn^{-3/4})$. Then, in Section~\ref{subsec:useful}
we introduce a lemma (due to Burns \cite{Burns2007TheNO}) on exchangeable random sequences that turns out to be very
useful in our proofs. Finally, in Section~\ref{subsec:proofoverview}, we use
this lemma to prove Proposition \ref{prop:areanonneg}, which states that as $n\to \infty$,
\[
n^{1/4}\Prb(\lwa_1,\dots,\lwa_{n}\ge 0\mid \lw_n=0) \to \frac{\Gamma(3/4)}{\sqrt{2\pi(1-\rho)}}.
\]
We then deduce Proposition \ref{prop:areanonnegrectangle} from this in Section \ref{subsec:squaretorectangle}.
The proofs of Proposition~\ref{prop:areanonnegrectangle}, Lemma~\ref{lem:parity}, and the lemmas used in the proof of Proposition~\ref{prop:areanonneg} are postponed to Section~\ref{sec:technical}. An overview of the notation is given in Appendix \ref{sec:notation_overview}.
\subsection{First asymptotics}\label{subsec:firstasymptotics}
In this section, we determine the asymptotics of the number of sequences that satisfy
the dominating condition, as well as $G(n)$, up to a constant factor.
\begin{prop}\label{prop:firstasymptotics}
The number of graphic sequences of length $n$ is $G(n)=\Theta(4^n/n^{3/4})$.
\end{prop}
The following proposition is a key ingredient in the proof of the above proposition.
\begin{prop}\label{prop:basic}
For $(\lw_i)_{i=1}^n$ the lazy SSRW and $\lwa_k=\sum_{i=1}^k\lw_i$ its area process,
\[
\Prb\big(\lwa_1,\dots,\lwa_n\ge 0 \mid \lw_{n} =0\big)=\Theta(n^{-1/4}).
\]
\end{prop}
This proposition is a direct consequence of results from Vysotsky \cite{Vysotsky2014}
on positivity of integrated (right-continuous) random walks.
Let $\lw = (\lw_k)_{k\geq 0}$ be a random walk and let $\cD(\lw)$ be the times for
which the walk has positive probability of being at~0, i.e. $\cD(\lw) = \{n : \Prb(\lw_n = 0) > 0\}$.
If the walk $\lw$ is integer valued, we will say it is \emph{right-continuous} (or \emph{upwards skip-free}) if
$Y_{i+1}\le Y_{i} + 1$ almost surely.
\begin{theorem}[Proposition 1 of \cite{Vysotsky2014}]\label{thm:vysotksy}
Let $\lw = (\lw_k)_{k\geq 0}$ be an integer valued random walk starting at $0$ with
$\E[\lw_1] = 0$ and\/ $\Var(\lw_1) < \infty$. Write $\lwa_k=\sum_{i=1}^k\lw_i$.
Then, for $n \in \cD(\lw)$, we have
\[
\Prb\big(\lwa_1,\dots,\lwa_n>0 \mid \lw_n = 0\big) = O(n^{-1/4}).
\]
Furthermore, if $\lw$ is also right-continuous, then for $n \in \cD(\lw)$,
\[
\Prb\big(\lwa_1,\dots,\lwa_n>0 \mid \lw_n = 0\big) = \Theta(n^{-1/4}).
\]
\end{theorem}
Proposition~\ref{prop:basic} follows by combining this theorem with some simple observations. Indeed, the lower bound follows
since the lazy SSRW is clearly right-continuous and the event $\{\lwa_1,\dots,\lwa_n>0\}$
is a subset of the event $\{\lwa_1,\dots,\lwa_n\ge0\}$.
For the upper bound, consider only the walks beginning with an upwards step and then a downwards step.
After these first two steps the walk behaves like a lazy SSRW of length $n-2$ conditioned to end at~0,
but the partial sums are one higher. This means the event $\{\sum_{i=1}^k\lw_i>0,\,k=1,\dots,n\}$ is exactly
the event $\{\sum_{i=3}^k\lw_i\ge 0,\,k=3,\dots,n\}$. A simple calculation shows that the probability
the walk goes up and then down given that $\lw_n = 0$ is at least $1/16$ for all $n\ge 2$. It follows that
\[
\Prb(\lwa_1,\dots,\lwa_n>0 \mid \lw_n=0)
\ge \tfrac{1}{16} \Prb(\lwa_1,\dots,\lwa_{n-2}\ge 0 \mid \lw_{n-2}=0),
\]
which shows the upper bound.
We cannot immediately apply the proposition to calculate the order of growth of the number of graphic sequences as we need to condition on the walk ending at either $0$ or~$-1$, not only $0$. The following lemma shows that this change in conditioning only changes the probability by a constant factor.
\begin{lemma}\label{lem:ignore1}
For all $n \ge 1$,
\[
\tfrac{1}{2} \le
\frac{\Prb(\lwa_1,\dots,\lwa_{n}\ge 0 \mid \lw_{n}\in\{0,-1\})}{\Prb(\lwa_1,\dots,\lwa_{n}\ge 0 \mid \lw_{n}=0)}
\le 1.
\]
\end{lemma}
The proof of Lemma \ref{lem:ignore1} is elementary, but for the sake of brevity we postpone it to Section~\ref{sec:technical}.
\begin{proof}[Proof of Proposition \ref{prop:firstasymptotics}]
By Lemma~\ref{lem:to_walk}, the probability that a uniformly random sequence
$n-1\ge d_1\ge\dotsb\ge d_n\ge 0$ satisfies the dominating condition \eqref{eq:DC} is equal to
$\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0 \mid \lw_{n-1}\in\{0,-1\})$, which is $\Theta(n^{-1/4})$ by Proposition~\ref{prop:basic} and
Lemma~\ref{lem:ignore1}. Hence, the number of sequences which satisfy the dominating condition is $\Theta(4^n/n^{3/4})$.
Clearly, $G(n) = O(4^n/n^{3/4})$ is immediate, and we only need to show a corresponding lower bound.
Let $H(n)$ denote the number of sequences
$(d_i)_{i=1}^n$ that satisfy the dominating condition for which $\sum_{i=1}^nd_i$ is odd. The number of
sequences of length $n$ that satisfy the dominating condition is therefore $G(n)+H(n)=\Theta(4^n/n^{3/4})$.
Each sequence $(d_i)_{i=1}^n$ in $G(n)$ gives rise to a unique sequence in $G(n+1)$ by appending $d_{n+1} = 0$.
Separately, each sequence $(d_i)_{i=1}^n$ in $H(n)$ gives rise to a unique sequence in $G(n+1)$ by adding an extra 1 to the graphic sequence (in the appropriate place). Hence,
\[
G(n+1)\ge \max\{G(n),H(n)\}\ge \tfrac12(G(n)+H(n)).
\]
Hence, $G(n) \geq \frac{1}{2} (G(n - 1)+H(n - 1))=\Theta(4^n/n^{3/4})$ as required.
\end{proof}
\subsection{A useful lemma}\label{subsec:useful}
We make use of the following lemma that appears in Sections 2.3--2.4
of~\cite{Burns2007TheNO}. We include a proof in the appendix for completeness.
\begin{restatable}{lemma}{useful}\label{lem:useful}
Let $x=(x_1,\dots,x_n)\in\R_{>0}^n$, let $\sigma$ be a uniformly random permutation of\/ $[n]$
and let $s=(s_1,\dots,s_n)$ be an independent uniformly random element of $\{-1,1\}^n$. Then
\[
\Prb\left(\sum_{i=1}^k s_i x_{\sigma(i)}\ge 0\text{ for all }k\in [n]\right)\ge \frac{(2n-1)!!}{2^n n!}.
\]
Moreover, equality holds if for all distinct $A,A'\subseteq [n]$, the
corresponding sums are also distinct: $\sum_{i\in A}x_i\ne \sum_{i\in A'}x_i$.
\end{restatable}
We will apply this lemma to sequences of exchangeable random variables of which the law
is invariant under sign changes of the elements, so we use the following equivalent form of the lemma.
\begin{lemma}\label{lem:useful2}
Let $(X_1,\dots,X_n)$ be a random variable in $\R^n$ such that for any $\sigma\in S_n$
and any $s\in\{-1,1\}^n$,
\[
(X_1,\dots,X_n)\overset{d}{=}(s_1X_{\sigma(1)},\dots,s_nX_{\sigma(n)}).
\]
Then,
\[
\Prb\left(\sum_{i=1}^k X_i\ge 0\text{ for all\/ }k\in [n]\right)\ge \frac{(2n-1)!!}{2^n n!},
\]
and equality holds if, almost surely, for all distinct $A,A'\subseteq [n]$, the
corresponding sums are also distinct: $\sum_{i\in A}X_i\ne \sum_{i\in A'}X_i$.
\end{lemma}
To see that Lemma \ref{lem:useful2} indeed implies Lemma \ref{lem:useful}, observe that for $x$, $s$ and $\sigma$ as in the statement of Lemma \ref{lem:useful}, the random variable $(X_1,\dots,X_n):=(s_1x_{\sigma(1)},\dots,s_nx_{\sigma(n)})$ satisfies the conditions of Lemma \ref{lem:useful2}.
\begin{proof}[Proof of Lemma \ref{lem:useful2}]
We observe that, for $\sigma$ a uniformly random permutation of $[n]$ and $s=(s_1,\dots,s_n)$
an independent uniformly random element of $\{-1,1\}^n$,
\begin{align*}
\Prb\left(\sum_{i=1}^k X_i\ge 0\text{ for all }k\in [n]\right)
&=\Prb\left(\sum_{i=1}^k s_i X_{\sigma(i)}\ge 0\text{ for all }k\in [n]\right)\\
&=\E\left[\Prb\left(\sum_{i=1}^k s_i X_{\sigma(i)}\ge 0\text{ for all }k\in [n]\mid X_1,\dots,X_n\right)\right],
\end{align*}
so that the result follows by applying Lemma~\ref{lem:useful} to
\[
\Prb\left(\sum_{i=1}^k s_i X_{\sigma(i)}\ge 0\text{ for all }k\in [n]\mid X_1,\dots,X_n\right).
\qedhere
\]
\end{proof}
We make the following observation about the value of the lower bound in Lemmas~\ref{lem:useful} and~\ref{lem:useful2}. Note that by taking either an even or odd number of terms in Wallis's product formula
\[
\frac{\pi}{2}=\frac{2}{1}\cdot \frac{2}{3}\cdot \frac{4}{3}\cdot \frac{4}{5}\cdot \frac{6}{5}\cdots
\]
one obtains alternately lower and upper bounds for $\frac{\pi}{2}$. Rearranging these inequalities
gives that for all $n\ge1$
\begin{equation}\label{e:ulbounds}
\frac{1}{\sqrt{\pi(n+1/2)}}\le \frac{(2n-1)!!}{2^nn!}=\binom{2n}{n}\frac{1}{4^n} \le \frac{1}{\sqrt{\pi n}}.
\end{equation}
\subsection{Proof of Proposition \ref{prop:areanonneg}}\label{subsec:proofoverview}
Let $\lb = (\lb_k)_{k=1}^{n}$ be the first $n$ steps of the lazy SSRW $ \lw = (\lw_k)_{k \geq 1}$ conditioned on the event $\lw_n=0$, and
let $N_n$ be the number of times that the walk $\lb$ hits~$0$.
Condition on the event that $N_n = N$. Then the walk $\lb$ has $N$ excursions with areas
$\lbae_1,\lbae_2,\dots,\lbae_N$ say. Since the walk $\lba_k = \sum_{i=1}^k \lb_i$ is monotone during the individual excursions, the walk $\lbac=(\lbac_k)_{k=1}^N$, defined by $\lbac_k=\sum_{i=1}^k\lbae_i$,
is never negative if and only if the walk $\lba$ is never negative.
Ideally, we would like to use Lemma~\ref{lem:useful2} to estimate the probability that $\lbac$
never goes negative. Indeed, the summands of $\lbac$ are exchangeable and their law is invariant under sign changes.
However, the probability given by Lemma~\ref{lem:useful2} is only exact when distinct sums of elements
in $(\lbae_i:1\le i \le N_n)$ are distinct almost surely, and this is in general not the case. We therefore
perturb each $\lbae_i$ by a small random amount, so that the sums are distinct almost surely
and the probability from Lemma~\ref{lem:useful2} is exact. To be precise, let
$\eps_i\sim \Unif[-\tfrac{1}{2n},\tfrac{1}{2n}]$ and define $\pbae_i=\lbae_i+\eps_i$ and
$\pbac_k=\sum_{i=1}^k \pbae_i$. Now, $\pbac$ satisfies the conditions of the equality in
Lemma~\ref{lem:useful2} so the probability that it never goes negative
(conditional on the event $N_n = N$) is exactly
\[
\frac{(2N-1)!!}{2^{N}N!}.
\]
The random variable $n^{-1/2}N_n$ converges in distribution to a $\Ray(\sqrt2)$ distribution,
which we write as $2\sqrt{E}$ where $E\sim \Exp(1)$ has a standard exponential distribution,
so one hopes that we can replace $N$ with $2\sqrt{nE}$ in the probability above.
It requires some extra work to ensure that $n^{-1/2}N_n$ does not put too much mass near~$0$, but
Lemma~\ref{lem:probnonnegperturbed} says that we do indeed get what one would expect. That is,
\[
\Prb\big(\pbac_1,\dots,\pbac_{N_n}\ge 0\big)
=\E\left[\frac{(2N_n-1)!!}{2^{N_n}N_n!}\right]
\sim n^{-1/4}\frac{\Gamma(3/4)}{\sqrt{2\pi}}.
\]
Now, note that our perturbations are so small that $\pbac_i < 0$ whenever $\lbac_i < 0$,
so to compute the probability that $\lbac$ never goes negative we can use the following equality
\[
\Prb\big(\pbac_1,\dots,\pbac_{N_n}\ge 0\big)
=\Prb\big(\lbac_1,\dots,\lbac_{N_n}\ge 0\big)
\Prb\big(\pbac_1,\dots,\pbac_{N_n}\ge 0 \mid \lbac_1,\dots,\lbac_{N_n}\ge 0\big).
\]
We already know $\Prb\big(\pbac_1,\dots,\pbac_{N_n}\ge 0\big)$, so our result will follow if we can get a good grasp on the term
$\Prb(\pbac_1,\dots,\pbac_{N_n}\ge 0\mid\lbac_1,\dots,\lbac_{N_n}\ge 0)$.
The key to this is the observation that, provided $\lbac$ does not go negative, the only times where $\pbac$ can go negative are at the points that $\lbac$ is equal to zero.
Suppose that $\lbac$ is equal to zero $M_n$ times, namely at at $\xi_1,\dots,\xi_{M_n}$, and let us also set $\xi_0 = 0$.
By definition, $\pbac_{\xi_k} = \sum_{i=1}^{\xi_k}\eps_i$, and the increment between
times $\xi_{k-1}$ and $\xi_k$ is exactly $\eta_k=\sum_{i=\xi_{k-1}+1}^{\xi_k}\eps_i$.
The sequence $(\eta_k)_{1\le k\le M_n}$
is an exchangeable sequence and its law is invariant under sign changes of the elements.
Distinct sums of elements in the set
are distinct almost surely and we can therefore use Lemma~\ref{lem:useful2} to deduce that
\[
\Prb\big(\pbac_1,\dots,\pbac_{N_n}\ge 0\mid \lbac_1,\dots,\lbac_{N_n}\ge 0,\,M_n=M\big)
=\frac{(2M-1)!!}{2^{M}M!}.
\]
This reduces our task to understanding the distribution of $M_n$ conditional on the event that $\{\lbac_1,\dots,\lbac_{N_n}\ge 0\}$, and we will show that it converges in distribution to $G\sim\Geom(\rho)$.
For this it is enough to show that for $\ell\ge 1$,
\begin{equation}\label{eq:ellconvtorho}
\Prb(M_n\ge \ell+1 \mid \lbac_1,\dots,\lbac_{N_n}\ge 0,\,M_n\ge \ell)\to \rho.
\end{equation}
Firstly, note that
\[
\probareazero{n}:= \Prb(M_n\ge 1\mid \lbac_1,\dots,\lbac_{N_n}\ge 0)
\]
converges to $\rho$ as $n\to \infty$ by Lemma~\ref{lem:pn_to_rho}.
If $M_n\ge\ell$, then let $\zeta_\ell$ be the time at which $\lb$ and $\lba$ hit $0$
simultaneously for the $\ell$th time. Then, $\lb$ and $\lba$ restricted to $[\zeta_\ell,n]$ are distributed as a
lazy simple symmetric random walk bridge with $n-\zeta_\ell$ steps and its area process respectively, so
\[
\Prb\big(M_n\ge\ell+1 \mid \lbac_1,\dots,\lbac_{N_n}\ge 0,\, M_n\ge\ell,\,\zeta_\ell)
=\probareazero{n-\zeta_\ell}.
\]
Lemma~\ref{lem:hittingtimeat0} implies that (conditional on the event
$\{\lbac_1,\dots,\lbac_{N_n}\ge 0\}$) $\zeta_{M_n}=O(n^{-1/2})$ with
high probability, so $n-\zeta_\ell\sim n$ and \eqref{eq:ellconvtorho} follows from
$\probareazero{n}\to \rho$. This implies that
\[
\Prb\big(\pbac_1,\dots,\pbac_{N_n}\ge 0\mid \lbac_1,\dots,\lbac_{N_n}\ge 0\big)
\to \E\left[\frac{(2G-1)!!}{2^G G!}\right],
\]
and a simple calculation shows that this equals $\sqrt{1-\rho}$. Putting this all together gives
\[
\Prb\big(\lbac_1,\dots,\lbac_{N_n}\ge 0\big)
=\frac{\Prb\big(\pbac_1,\dots,\pbac_{N_n}\ge 0\big)}{
\Prb\big(\pbac_1,\dots,\pbac_{N_n}\ge 0\mid \lbac_1,\dots,\lbac_{N_n}\ge 0\big)}\\
\sim \frac{\Gamma(3/4)}{\sqrt{2\pi(1-\rho)}}n^{-1/4}.
\]
\section{The postponed proofs}
\label{sec:technical}
We will first introduce a coupling between simple symmetric random walks (resp.\ bridges)
and lazy simple symmetric random walks (resp.\ bridges) that turns out to be useful in our proofs.
Let $(\uw_i)_{i\ge 0}$ be a simple symmetric random walk. Then, if we set $\lw_i=\frac{1}{2}\uw_{2i}$
for each $i$ we see that $(\lw_i)_{i\ge 0}$ has the law of a lazy simple symmetric random walk.
Moreover, since $\uw$ is zero only at even times, the zeroes of $\uw$ are in one-to-one correspondence
with the zeroes of~$\lw$, and in particular, if $(\ub_i)_{0\le i\le 2n}$ is a simple symmetric
random walk bridge and $\lb_i=\frac{1}{2}\ub_{2i}$, we have that $(\lb_i)_{0\le i\le n}$ is a
lazy simple symmetric random walk bridge.
To prove Lemma~\ref{lem:ignore1} we first show that the probability of event
$\{\lwa_1,\dots,\lwa_{n}\ge 0\}$ is monotone with respect to the value of $\lw_{n}$.
\begin{lemma}\label{lem:monotoneinY_n}
For all $-n\le k \le k' \le n$,
\[\textstyle
\Prb(\lwa_1,\dots,\lwa_{n}\ge 0 \mid \lw_{n} = k)
\le \Prb(\lwa_1,\dots,\lwa_{n}\ge 0 \mid \lw_{n} = k').
\]
\end{lemma}
\begin{proof}
It suffices to prove the claim when $k' = k+1$. We will use the coupling between the SSRW and the lazy SSRW introduced at the beginning of this section. Consider the SSRW $\uw=(\uw_k)_{i=1}^{2n}$ conditioned on the event $\uw_{2n}=2k$, and define $\lw$ by $\lw_i:=\tfrac12\uw_{2i}$. Then $\lw$ is distributed as a lazy SSRW with $n$ steps conditioned on the event $\lw_{n}=k$.
Now, observe that the increments of $\uw$ conditioned on $\uw_{2n}=2k$ are distributed as a uniform ordering of $n+k$ instances of $1$ and $n-k$ instances of $-1$. We consider a modified sequence obtained from the original sequence by picking a uniformly random $-1$ and changing it to a $+1$, so that the modified sequence has the law of the increments of a SSRW of $2n$ steps conditioned to end in $2k+2$, which, again via the coupling, corresponds to a lazy SSRW of $n$ steps conditioned to end in $k+1$. We see that our modification increased one increment of $\lw$ by $1$, and since the event $\{\lwa_1,\dots,\lwa_n\geq 0\}$ is increasing in the increments of $\lw$, the result follows. \end{proof}
We are now ready to prove Lemma~\ref{lem:ignore1}.
\begin{proof}[Proof of Lemma~\ref{lem:ignore1}]
We abbreviate $\event=\{\lwa_1,\dots,\lwa_{n}\ge 0\}$, and start with the lower bound.
\begin{align*}
\Prb(\event \mid \lw_{n}\in\{0,-1\})
&=\frac{\Prb(\event \mid \lw_{n} = 0)\Prb(\lw_{n} = 0)
+ \Prb(\event \mid \lw_{n} = -1)\Prb(\lw_{n}= -1)}{\Prb(\lw_{n} \in \{0,-1\})}\\
&\ge \frac{\Prb(\event \mid \lw_{n} = 0)\binom{2n}{n}}{\binom{2n+1}{n}}\\
&=\tfrac{n+1}{2n+1} \Prb(\event \mid \lw_{n} = 0)\\
&\ge \tfrac{1}{2} \Prb(\event \mid \lw_{n} = 0).
\intertext{We now turn to the upper bound, for which we use Lemma~\ref{lem:monotoneinY_n}. }
\Prb(\event \mid \lw_{n}\in\{0,-1\})
&=\frac{\Prb(\event \mid \lw_{n} = 0)\Prb(\lw_{n} = 0)
+ \Prb(\event \mid \lw_{n} = -1)\Prb(\lw_{n}= -1)}{\Prb(\lw_{n} \in \{0,-1\})}\\
&\le \frac{\Prb(\event \mid \lw_{n} = 0)(\Prb(\lw_{n} = 0) + \Prb(\lw_{n} = -1))}{\Prb(\lw_{n} \in \{0,-1\})}\\
&=\Prb(\event \mid \lw_{n} = 0).\qedhere
\end{align*}
\end{proof}
\subsection{Lemmas for Proposition~\ref{prop:areanonneg}}
As before, let $N_n=|\{1\leq i\leq n:\lb_i=0\}|$ be the number of returns to $0$ of $\lb$ up to time~$n$.
We have the following lemma.
\begin{lemma}\label{lem:convtorayleigh}
As $n\to\infty$, $n^{-1/2}N_n\overset{d}{\to} 2\sqrt{E}$ where $E\sim\Exp(1)$ has a standard
exponential distribution. Moreover, for any $\gamma\ge0$,
\[
\Prb\big(N_n<\gamma n^{1/2}\big)\le \tfrac{\gamma^2}{2}.
\]
\end{lemma}
\begin{proof}
We fix an $n$ and a $k\le n$. We will count the number of Bernoulli bridges with $2n$ steps and at least $k$ returns to $0$ (i.e.\ at least $k$ excursions away from~$0$). For a Bernoulli bridge with $2n$ steps
and at least $k$ excursions, flip the last $k$ excursions so that they are positive. Then remove the
last step of each of the last $k$ excursions. Each of the removed steps was downward, so we now obtain
a path of length $2n-k$ that ends at level~$k$. We can recover the original position of the removed steps:
the $i$th removed step should be included after the last time the path is at level~$i$. Therefore,
each path of length $2n-k$ that ends at level $k$ corresponds to $2^k$ bridges with more than $k$ zeroes,
so the number of bridges with $2n$ steps and more than $k$ zeroes equals $2^k\binom{2n-k}{n}$. Thus
the probability that a simple symmetric random walk bridge with $2n$ steps returns to $0$ at least $k$ times equals
\[
\frac{2^k\binom{2n-k}{n}}{\binom{2n}{n}}
=\frac{2^k n(n-1)\dotsm(n-k+1)}{(2n)(2n-1)\dotsm(2n-k+1)}
=\prod_{i=1}^{k-1}\left(1-\frac{i}{2n-i}\right).
\]
By the coupling between a lazy random walk bridge on $[n]$ and a simple symmetric random walk bridge
on $[2n]$ that preserves the number of zeroes, we then see that for $k=O(\sqrt{n})$,
\[
\log\Prb(N_n\geq k)=\log\prod_{i=1}^{k-1}\left(1-\frac{i}{2n-i}\right)
=-\sum_{i=1}^{k-1}\Big(\frac{i}{2n}+O(i^2/n^2)\Big)=-\frac{k^2}{4n}+O(n^{-1/2}),
\]
so $\Prb(N_n\geq tn^{1/2})\to e^{-t^2/4}=\Prb(E>t^2/4)=\Prb(2\sqrt{E}>t)$ where $E\sim\Exp(1)$ has
an exponential distribution.
We also see that, for $0\le k\le n$,
\[
\Prb(N_n\geq k)=\prod_{i=1}^{k-1}\left(1-\frac{i}{2n-i}\right)
\ge 1-\sum_{i=1}^{k-1}\frac{i}{2n-i}
\ge 1-\frac{k(k-1)/2}{2n-k}\ge 1-\frac{k^2}{2n}.
\]
Hence, as the above inequality is trivially true for $k>n$,
\[
\Prb\big(N_n< \gamma n^{1/2}\big)=1-\Prb\big(N_n\geq \lfloor\gamma n^{1/2}\rfloor\big)
\le \frac{\lfloor\gamma n^{1/2}\rfloor^2}{2n}\le \frac{\gamma^2}{2}.
\qedhere
\]
\end{proof}
We use Lemma~\ref{lem:convtorayleigh} in the following lemma which considers the asymptotic value of
$\Prb(\pbac_1,\dots,\pbac_{N_n}\ge 0)=\E\big[\frac{(2N_n-1)!!}{2^{N_n}N_n!}\big]$.
\begin{lemma}\label{lem:probnonnegperturbed}
As $n\to\infty$,
\[
\E\left[\frac{(2N_n-1)!!}{2^{N_n}N_n!}\right]\sim n^{-1/4} \frac{\Gamma(3/4)}{\sqrt{2\pi}}.
\]
\end{lemma}
\begin{proof}
Fix some $\delta \in (0,1)$. We split the expectation according to the contribution
with $N_n<\delta n^{1/2}$ and $N_n\ge \delta n^{1/2}$.
By Lemma~\ref{lem:convtorayleigh}, $\Prb(N_n<\gamma n^{1/2})\le \gamma^2/2$ for all~$n$ and
we also recall that
\[
\frac{1}{\sqrt{\pi (m+1/2)}}\le \frac{(2m-1)!!}{2^m m!} \le \frac{1}{\sqrt{\pi m}}.
\]
Hence,
\begin{align*}
n^{1/4}\E\left[\frac{(2N_n-1)!!}{2^{N_n}N_n!}\one_{\{N_n<\delta n^{1/2}\}}\right]
&\le n^{1/4}\sum_{i=0}^\infty
\Prb\big(2^{-i-1}\delta n^{1/2}\le N_n<2^{-i}\delta n^{1/2}\big)\frac{1}{\sqrt{\pi 2^{-i-1}\delta n^{1/2}}}\\
&\le \sum_{i=0}^\infty \tfrac12(2^{-i}\delta)^2\cdot (\pi 2^{-i-1}\delta)^{-1/2}=O(\delta^{3/2}).
\end{align*}
Since $n^{1/4}/\sqrt{\pi N_n}$ is bounded for $N_n\ge \delta n^{1/2}$
and $n^{-1/2}N_n$ converges in distribution to $2\sqrt{E}$ where $E\sim\Exp(1)$,
\[
n^{1/4}\E\left[\frac{(2N_n-1)!!}{2^{N_n}N_n!}\one_{\{N_n\ge \delta n^{1/2}\}}\right]
=\E\left[\frac{n^{1/4}}{\sqrt{\pi N_n+O(1)}}\one_{\{n^{-1/2}N_n \ge \delta\}}\right]
\to\E\left[\frac{\one_{\{2\sqrt{E}\ge \delta\}}}{\sqrt{2\pi\sqrt E}}\right]
\]
as $n\to\infty$. Now
\[
\E\left[\frac{\one_{\{2\sqrt{E}\ge \delta\}}}{\sqrt{2\pi\sqrt E}}\right]
=\E\left[\frac{\one_{\{E\ge \delta^2/4\}}}{\sqrt{2\pi}E^{1/4}}\right]
=\frac{1}{\sqrt{2\pi}} \int_{\delta^2/4}^\infty x^{-1/4}e^{-x} dx
=\frac{\Gamma(3/4)}{\sqrt{2\pi}}+O(\delta^{3/2}).
\]
Hence,
\[
n^{1/4}\E\left[\frac{(2N_n-1)!!}{2^{N_n}N_n!}\right]
\to \frac{\Gamma(3/4)}{\sqrt{2\pi}}+O(\delta^{3/2})
\]
as $n\to\infty$. As $\delta$ was arbitrary, the result now follows.
\end{proof}
We now want to show that
\[
\Prb\left(\exists k \in (n^{1/2},n]: \lba_k=\lb_k=0 \mid \lba_1,\dots,\lba_n\ge 0\right)=o(1),
\]
which is the content of Lemma \ref{lem:hittingtimeat0}. For the proof of this, we will need Lemma \ref{lem:locallimitlazy}, which is a local limit theorem for the position and area of a lazy SSRW, and Lemma \ref{lem:upperboundintegral0}, which we will use to control the probability that the integral and position of an unconditioned lazy SSRW hit $0$ simultaneously at a late time.
\begin{restatable}{lemma}{locallimitlazy}\label{lem:locallimitlazy}
We have
\[
\lim_{n\to \infty}\sup_{a,b}\,\left|n^2 \Prb\left(\lw_n=a, \lwa_n=b\right)-\phi(n^{-1/2}a,n^{-3/2}b)\right|=0
\]
where the supremum runs over all $(a,b)\in \Z^2$ and
\[
\phi(x,y)=\frac{2\sqrt{3}}{\pi}\exp\big(-4x^2+12xy-12y^2\big).
\]
\end{restatable}
The proof of this lemma is postponed to Appendix \ref{app:locallimitlazy}.
\begin{restatable}{lemma}{upperboundintegral}\label{lem:upperboundintegral0}
There exists a constant $C$ such that for all\/~$n\ge1$,
\[
\Prb\left(\lwa_n=\lw_n=0,\,\lwa_1,\dots,\lwa_n\ge 0\right)\le Cn^{-5/2}.
\]
\end{restatable}
With Lemma~\ref{lem:locallimitlazy} in hand, the proof of Lemma~\ref{lem:upperboundintegral0} is a direct
adaptation of the proof of the upper bound of Theorem~1 of \cite{AurzadaDereichLifshits2014} on simple
symmetric random walks. For the sake of completeness, we have included a proof in Appendix \ref{app:upperboundintegral}.
\begin{lemma}\label{lem:hittingtimeat0}
We have that
\[
\Prb\left(\exists k \in (n^{1/2},n]\colon \lba_k=\lb_k=0
\mid \lba_1,\dots,\lba_n\ge 0\right)=O(n^{-3/4}).
\]
\end{lemma}
\begin{proof}
Define $\probareapositive{n}=\Prb(\lwa_1,\dots,\lwa_n\ge 0,\,\lw_n=0)$.
Note that for any $k$,
\begin{align*}
\Prb\left(\lba_k=\lb_k=0 \mid \lba_1,\dots,\lba_n\ge 0\right)
&=\frac{1}{\probareapositive{n}}\Prb\left(\lwa_k=\lw_k=0,\,\lwa_1,\dots,\lwa_n\ge 0,\,\lw_n=0\right)\\
&=\frac{1}{\probareapositive{n}}\Prb\left(\lwa_k=\lw_k=0,\,\lwa_1,\dots,\lwa_k\ge 0\right)\\
&\quad \times \Prb\left(\lwa_{k+1},\dots,\lwa_n\ge 0,\,\lw_n=0 \mid \lwa_k=\lw_k=0\right)\\
&=\frac{\probareapositive{n-k}}{\probareapositive{n}}\,\Prb\left(\lwa_k=\lw_k=0,\,\lwa_1,\dots,\lwa_k\ge 0\right).
\end{align*}
By our earlier results (Proposition~\ref{prop:basic}), $\probareapositive{n}=\Theta(n^{-3/4})$ and by
Lemma~\ref{lem:upperboundintegral0}, the final factor is $O(k^{-5/2})$. Therefore, there exists a $C$
such that for each $k\le n$,
\[
\Prb\left(\lba_k=\lb_k=0 \mid \lba_1,\dots,\lba_n\ge 0\right)\le C k^{-5/2}(n-k+1)^{-3/4}n^{3/4}.
\]
Then, the result follows from the union bound by observing that
\[
\sum_{n^{1/2}< k\leq n/2} Ck^{-5/2}(n-k+1)^{-3/4}n^{3/4}
\le 2^{3/4}C\sum_{k> n^{1/2}} k^{-5/2}=O((n^{1/2})^{-3/2})=O(n^{-3/4})
\]
and
\[
\sum_{k=\lfloor n/2\rfloor }^n Ck^{-5/2}(n-k+1)^{-3/4}n^{3/4}
=O(n^{-7/4}\sum_{j=1}^{\lceil n/2\rceil +1} j^{-3/4})=O(n^{-7/4}n^{1/4})=O(n^{-3/2}).
\qedhere
\]
\end{proof}
We now turn to Lemma~\ref{lem:pn_to_rho} which shows that $\probareazero{n}\to\rho$.
For the proof of this we need the following result.
\begin{lemma}\label{lem:ratioq_n}
Let $q_n=\Prb(\lwa_1,\dots,\lwa_n\ge 0\mid \lw_n=0)$.
Then, uniformly over all $m\le n^{1/2}$, we have $q_{n-m}/q_n\to 1$ as $n\to\infty$.
\end{lemma}
\begin{proof}
We define $\varphi_\ell(k)=\Prb(\lw_\ell=-k)$, which is also the probability that a lazy SSRW starting at $k$ is at 0 at time $\ell$.
Let $0<m<b<n$ and let $Z$ be a random variable that depends only on
$\cF_{n-b}=\sigma(\lw_1,\dots,\lw_{n-b})$. We first show that
\begin{equation}\label{eq:req}
\E[Z\mid \lw_n=0] = \E\left[Z\cdot\frac{\varphi_{b}(\lw_{n-b})}{\varphi_{b-m}(\lw_{n-b})} \frac{\varphi_{n-m}(0)}{\varphi_n(0)}\bigmid \lw_{n-m}=0\right].
\end{equation}
For $b'<n'$ and any $Z'$ that depends only on $\cF_{b'}$,
\begin{align*}
\E[Z'\mid \lw_{n'}=0]
&=\frac{\E[Z'\one{\{\lw_{n'}=0\}}]}{\Prb(\lw_{n'}=0)}\\
&=\frac{\E[\E[Z'\one{\{\lw_{n'}=0\}}\mid \cF_{b'}]]}{\Prb(\lw_{n'}=0)}\\
&=\frac{\E[Z'\Prb(\lw_{n'}=0\mid \cF_{b'})]}
{\Prb(\lw_{n'}=0)}
=\E\left[Z'\cdot \frac{\varphi_{n'-b'}(\lw_{b'})}{\varphi_{n'}(0)}\right].
\end{align*}
Applying this to $Z'=Z \frac{\varphi_{b}(\lw_{n-b})}{\varphi_{b-m}(\lw_{n-b})} \frac{\varphi_{n-m}(0)}{\varphi_n(0)}$, $n'=n-m$ and $b'=n-b$, we find that
\begin{align*}
\E[Z'\mid \lw_{n-m}=0]
&=\E\left[Z'\cdot\frac{\varphi_{b-m}(\lw_{n-b})}{\varphi_{n-m}(0)}\right]\\
&=\E\left[Z\cdot\frac{\varphi_{b}(\lw_{n-b})}{\varphi_{b-m}(\lw_{n-b})} \frac{\varphi_{n-m}(0)}{\varphi_n(0)} \cdot \frac{\varphi_{b-m}(\lw_{n-b})}{\varphi_{n-m}(0)}\right]\\
&=\E\left[Z\cdot\frac{\varphi_{b}(\lw_{n-b})}{\varphi_n(0)} \right]\\
&=\E[Z\mid \lw_n=0]
\end{align*}
as desired.
We assume $m\le n^{1/2}$ and let $b = b(n) \in \mathbb{N}$ be such that $b(n) \sim n^{7/9}$.
Next, we provide bounds on
$\frac{\varphi_{b}(\lw_{n-b})}{\varphi_{b-m}(\lw_{n-b})}\frac{\varphi_{n-m}(0)}{\varphi_n(0)}$.
We first note that for $|k|\le\ell$ we have
\begin{align*}
\varphi_\ell(k)
&=\binom{2\ell}{k+\ell}4^{-\ell}=\binom{2\ell}{\ell}4^{-\ell}\cdot\prod_{j=1}^k\Big(1-\frac{2j-1}{\ell+j}\Big)\\
&=\frac{1}{\sqrt{\pi\ell+O(1)}}\cdot\exp\left(\sum_{j=1}^k\log\Big(1-\frac{2j-1}{\ell+j}\Big)\right)\\
&=\frac{1}{\sqrt{\pi\ell}}e^{O(1/\ell)}\cdot\exp\left(-\sum_{j=1}^k\Big(\frac{2j-1}{\ell}+O(j^2/\ell^2)\Big)\right)\\
&=\frac{1}{\sqrt{\pi\ell}}e^{-k^2/\ell+O(k^3/\ell^2)+O(1/\ell)}.
\end{align*}
Hence, $\varphi_\ell(k)/\varphi_{\ell'}(k)\to 1$ provided $\ell\to\infty$,
$\ell/\ell'\to 1$, $k^2(\ell-\ell')/\ell\ell'\to 0$ and $k^3/\ell^2\to0$.
Now define
\begin{align*}
\underline{\delta}_n&=\min_{|k|\le n^{1/2}}
\frac{\varphi_{b}(k)}{\varphi_{b-m}(k)} \frac{\varphi_{n-m}(0)}{\varphi_n(0)}\\
\overline{\delta}_n&=\max_{|k|\le n^{1/2}},
\frac{\varphi_{b}(k)}{\varphi_{b-m}(k)} \frac{\varphi_{n-m}(0)}{\varphi_n(0)}.
\end{align*}
and note that $\underline{\delta}_n\to 1$ and $\overline{\delta}_n\to 1$ as $n\to \infty$
for our choice of $m$ and~$b$. Indeed,
$\log\underline{\delta}_n,\log\overline{\delta}_n=O(m/b+mn/b^2+n^{3/2}/b^2)\to0$.
Applying \eqref{eq:req} with $Z=\one_{\{\lwa_1,\dots,\lwa_{n-b}\ge0,\,|\lw_{n-b}|\le n^{1/2}\}}$
gives
\begin{align*}
\Prb\big(\lwa_1,\dots,\lwa_{n-b}\ge 0\mid \lw_n=0\big)
&\le \Prb\big(\lwa_1,\dots,\lwa_{n-b}\ge 0,\,|\lw_{n-b}|\le n^{1/2}\mid \lw_n=0\big)\\
&\quad+\Prb(|\lw_{n-b}|>n^{1/2}\mid \lw_n=0)\\
&\le \overline{\delta}_n\Prb\big(\lwa_1,\dots,\lwa_{n-b}\ge 0,\,|Y_{n-b}|\le n^{1/2}\mid \lw_{n-m}=0\big)\\
&\quad+\Prb(|\lw_{n-b}|>n^{1/2}\mid \lw_n=0)\\
&\le \overline{\delta}_n\Prb\big(\lwa_1,\dots,\lwa_{n-b}\ge 0\mid \lw_{n-m}=0\big)\\
&\quad+\Prb(|\lw_{n-b}|>n^{1/2}\mid \lw_n=0).
\end{align*}
Similarly,
\begin{align*}
\Prb\big(\lwa_1,\dots,\lwa_{n-b}\ge 0\mid \lw_n=0\big)
&\ge \Prb\big(\lwa_1,\dots,\lwa_{n-b}\ge 0,\,|\lw_{n-b}|\le n^{1/2}\mid \lw_n=0\big)\\
&\ge \underline{\delta}_n\Prb\big(\lwa_1,\dots,\lwa_{n-b}\ge 0,\,|Y_{n-b}|\le n^{1/2}\mid \lw_{n-m}=0\big)\\
&\ge \underline{\delta}_n\Prb\big(\lwa_1,\dots,\lwa_{n-b}\ge 0\mid \lw_{n-m}=0\big)\\
&\quad-\underline{\delta}_n\Prb(|\lw_{n-b}|>n^{1/2}\mid \lw_{n-m}=0).
\end{align*}
We note that
\begin{align*}
&\Prb(|\lw_{n-b}|>n^{1/2}\mid \lw_n=0)=\Prb(|\lw_{b}|>n^{1/2} \mid \lw_n=0)=O(n^{-\omega(1)}),\\
&\Prb(|\lw_{n-b}|>n^{1/2}\mid \lw_{n-m}=0)=\Prb(|\lw_{b-m}|>n^{1/2} \mid \lw_{n-m}=0) =O(n^{-\omega(1)})
\end{align*}
uniformly in all $m\le n^{1/2}$, where we use that on the event $\{\lw_n=0\}$,
it holds that $(\lw_{i})_{i=1}^n$ and $(\lw_{n-i})_{i=1}^n$ have the same law and the fact
that $n^{1/2}\sim n^{\eps}b^{1/2}$ for some $\eps>0$ (and large enough $n$).
Therefore, using that $q_n=\Theta(n^{-1/4})$ by Proposition \ref{prop:basic},
\begin{equation}\label{eq:fromnton-m}
\Prb\big(\lwa_1,\dots,\lwa_{n-b}\ge 0\mid \lw_n=0\big)
=(1+o(1))\Prb\big(\lwa_1,\dots,\lwa_{n-b}\ge 0\mid \lw_{n-m}=0\big).
\end{equation}
To finish the proof, it suffices to show that
\begin{equation}\label{eq:nosignchangeatend}
\frac{\Prb(\lwa_1,\dots,\lwa_{n-b}\ge 0\mid \lw_n=0)}{\Prb( \lwa_1,\dots,\lwa_n\ge0\mid \lw_n=0)}\to 1
\end{equation}
as $n\to \infty$. Indeed, if we set $b = \lfloor n^{7/9} \rfloor$, we see that
\begin{align*}
\frac{q_{n-m}}{q_n}
&=\frac{\Prb(\lwa_1,\dots,\lwa_{n-m}\ge 0\mid \lw_{n-m}=0)}{\Prb(\lwa_1,\dots,\lwa_n\ge 0\mid \lw_n=0)}\\
&=\frac{\Prb(\lwa_1,\dots,\lwa_{n-m}\ge 0\mid \lw_{n-m}=0)}{\Prb(\lwa_1,\dots,\lwa_{n-b}\ge 0\mid \lw_{n-m}=0)}
\frac{\Prb(\lwa_1,\dots,\lwa_{n-b}\ge 0\mid \lw_{n-m}=0)}{\Prb(\lwa_1,\dots,\lwa_{n-b}\ge 0\mid \lw_{n}=0)}\\
&\quad\times\frac{\Prb(\lwa_1,\dots,\lwa_{n-b}\ge 0\mid \lw_{n}=0)}{\Prb(\lwa_1,\dots,\lwa_n\ge 0\mid \lw_n=0)},
\end{align*}
so \eqref{eq:nosignchangeatend} would imply that the first and the third factor in the product tend to~$1$.
The second factor tends to $1$ by~\eqref{eq:fromnton-m}.
To obtain \eqref{eq:nosignchangeatend} we need to show that
\[
\Prb\big(\exists k\in(n-b,n]\colon \lba_k< 0 \mid \lba_1,\dots,\lba_{n-b}\ge 0\big)\to 0
\]
as $n\to \infty$. We know by our earlier bounds (Proposition~\ref{prop:basic}) that there
is a $c>0$ such that for all $n$ large enough
\[
\Prb\left(\lba_1,\dots,\lba_{n}\ge 0 \right)\ge cn^{-1/4},
\]
so it is sufficient to show that
\[
\Prb\left(\exists k\in(n-b,n]\colon \sgn(\lba_{k-1})\ne \sgn(\lba_k) \right)=o(n^{-1/4}).
\]
Note that
\begin{align*}
\Prb\big(\exists k\in(n-b,n]\colon& \sgn(\lba_{k-1})\ne\sgn(\lba_k)\big)\\
&\le \Prb\big(|\lba_{n}|\le n^{6/5}\big)+\Prb\Big(\max_{n-b+1\le i\le n}|\lb_i| > n^{6/5}/b\Big)\\
&\le \Prb\big(|\lba_{n}|\le n^{6/5}\big)+\Prb\Big(\max_{1\le i\le b}|\lb_i| > n^{6/5}/b\Big)
\end{align*}
by the union bound and the fact that $(\lb_{i})_{1\le i\le n}$ and $(\lb_{n+1-i})_{1\le i\le n}$ have the same law.
By Lemma \ref{lem:locallimitlazy},
\[
\Prb\big(|\lba_{n}|\le n^{6/5}\big)=\frac{\Prb(|\lwa_n|\le n^{6/5},\,Y_n=0)}{\Prb(Y_n=0)}
=\frac{O(n^{6/5}/n^2)}{\Theta(n^{-1/2})}=o(n^{-1/4}).
\]
Furthermore, by the reflection principle for the simple symmetric random walk~$\uw$, we have that, for any $k$
\[
\Prb\Big(\max_{1\le i\le 2b}\uw_i\ge k\Big)\le 2\Prb\Big(\uw_{2b}\ge k\Big)
\]
and under the usual coupling between $(\lw_k)_{1\le k\le b}$ and $(\uw_k)_{1\le k\le 2b}$ we have that
\[
\Prb\Big(\max_{1\le i\le b}|\lw_i|>k\Big)=2\Prb\Big(\max_{1\le i\le b}\lw_i>k\Big)
\le 2\Prb\Big(\max_{1\le i\le 2b}\uw_i\ge 2k\Big)\le 4\Prb\Big(\uw_{2b}\ge 2k\Big),
\]
so observing that $n^{6/5}/b=n^{\eps}b^{1/2}$ for some $\eps>0$ implies that
\[
\Prb\Big(\max_{1\le i\le b}|\lw_i| > n^{6/5}/b\Big)=n^{-\omega(1)}.
\]
Then, $\Prb(\lw_n=0)=\Theta(n^{-1/2})$ implies that also
\[
\Prb\Big(\max_{1\le i \le b}|\lb_i| > n^{6/5}/b\Big)=n^{-\omega(1)}.
\]
So it follows that
\[
\Prb\left(\exists k\in(n-b,n]\colon \sgn(\lba_{k-1})\ne \sgn(\lba_k)\right)=o(n^{-1/4})
\]
as claimed.
\end{proof}
Recall that $\probareazero{n}=\Prb(M_n\ge 1\mid \lbac_1,\dots,\lbac_{N_n}\ge 0)$ where $M_n=\#\{i>0:\lbac_i=0\}$.
We will use the preceding lemma to prove Lemma~\ref{lem:pn_to_rho}.
\begin{lemma}\label{lem:pn_to_rho}
We have $\probareazero{n}\to \rho$ as $n\to \infty$.
\end{lemma}
\begin{proof}
By definition,
\[
\probareazero{n}=\Prb(\exists k\in [n]\colon \lba_k=\lb_k=0 \mid \lba_1,\dots,\lba_{n}\ge 0).
\]
Let $\zeta_1=\zeta_1(\lb)=\min\{k>0:\lb_k=0,\,\lba_k\le 0\}$, with the convention that $\min\emptyset=\infty$. On $\{\lba_1,\dots,\lba_n\ge 0\}$ it holds that $\{\exists k\in [n]\colon\lba_k=\lb_k=0\}=\{\zeta_1\le n\}$ and we first show that, under this conditioning, with high probability $\zeta_1$ is either at most $n^{1/2}$ or larger than $n$.
We will then show that we may stop conditioning on $\{\lba_1,\dots,\lba_n\ge 0\}$ if we instead consider the event $\{\zeta_1\le n^{1/2},\,\lba_{\zeta_1}=0\}$ (and accept a $o(1)$ term). We then further show that the fact that we are considering $Y^{br}$ instead of $Y$ makes a negligible difference. After making these changes, the random variable $\zeta_1$ does not depend on $n$ and the result follows.
Observe that
\begin{align*}
\probareazero{n}&=\Prb(\zeta_1\le n \mid \lba_1,\dots,\lba_n\ge 0)\\
&=\Prb(\zeta_1\le n^{1/2} \mid \lba_1,\dots,\lba_n\ge 0)
+\Prb(\zeta_1\in (n^{1/2},n] \mid \lba_1,\dots,\lba_n\ge 0)
\end{align*}
and by Lemma~\ref{lem:hittingtimeat0},
\begin{multline*}
\Prb(\zeta_1\in (n^{1/2},n] \mid \lba_1,\dots,\lba_n\ge 0)\\
\le \Prb\left(\exists k \in (n^{1/2},n]\colon \lba_k=\lb_k=0 \mid \lba_1,\dots,\lba_n\ge 0\right)
=O(n^{-3/4}).
\end{multline*}
It remains to show that $\Prb(\zeta_1\le n^{1/2} \mid \lba_1,\dots,\lba_n\ge 0)\to \rho$.
Recall that $\probbridgeareapositive{n}=\Prb(\lba_1,\dots,\lba_n\ge 0)$. Then, for any $k\in [n]$,
\begin{align*}
\Prb(\lba_1,\dots,\lba_n\ge 0\mid \zeta_1=k)
&=\Prb(\lba_1,\dots,\lba_k\ge 0\mid \zeta_1=k)\\
&\quad\times\Prb(\lba_{k+1},\dots,\lba_{n}\ge 0\mid \zeta_1 =k,\,\lba_{k}=0)\\
&=\probbridgeareapositive{n-k}\Prb(\lba_1,\dots,\lba_k\ge 0\mid \zeta_1=k)\\
&=\probbridgeareapositive{n-k}\Prb(\lba_{\zeta_1}=0\mid \zeta_1=k).
\end{align*}
where we used that, on the event $\{\zeta_1=k,\lba_k=0\}$, the restrictions of $\lb$ and $\lba$ to $[k,n]$
have the joint law of a lazy simple symmetric random walk bridge on $[n-k]$ and its area process respectively.
Therefore,
\begin{align*}
\Prb(\zeta_1\le n^{1/2}\mid \lba_1,\dots,\lba_n\ge 0)
=\sum_{k=1}^{\lfloor n^{1/2} \rfloor}\Prb(\zeta_1=k)
\frac{\probbridgeareapositive{n-k}}{\probbridgeareapositive{n}}\Prb(\lba_{\zeta_1}=0\mid \zeta_1=k).
\end{align*}
Lemma~\ref{lem:ratioq_n} shows that $\frac{\probbridgeareapositive{n-k}}{\probbridgeareapositive{n}}\to 1$ as $n\to \infty$
uniformly over all $k\le n^{1/2}$, so
\begin{align*}
\probareazero{n}
&=\sum_{k=1}^{\lfloor n^{1/2}\rfloor}\Prb(\zeta_1=k)\Prb\left(\lba_{\zeta_1}=0 \mid \zeta_1=k \right)+o(1)\\
&=\Prb\left(\zeta_1\le n^{1/2},\,\lba_{\zeta_1}=0\right)+o(1).
\end{align*}
We now show that removing the condition that $\{\lw_n=0\}$ has only a negligible effect on the probability.
Fix $0<\eps<1/4$ and note that
\begin{align*}
\Prb\big(\zeta_1\le n^{1/2},\ &\lba_{\zeta_1}=0\big)\\
&=\Prb\left(\zeta_1\le n^{1/2},\lba_{\zeta_1}=0,|\lb_{\lfloor n^{1/2} \rfloor}|<n^{1/4+\eps}\right)+o(1)\\
&=\E\left[\one_{\left\{\zeta_1\le n^{1/2}, \lwa_{\zeta_1}=0, |\lw_{\lfloor n^{1/2}\rfloor}|<n^{1/4+\eps}\right\}}
\frac{\varphi_{n-\lfloor n^{1/2}\rfloor}(\lw_{\lfloor n^{1/2} \rfloor})}{\varphi_{n}(0)}\right]+o(1),
\end{align*}
where we have used the function $\varphi$ from the proof of Lemma~\ref{lem:ratioq_n} and that the event
\[
\left\{\zeta_1\le n^{1/2}, \lba_{\zeta_1}=0, |\lb_{\lfloor n^{1/2} \rfloor}|<n^{1/4+\eps} \right\}
\]
is measurable with respect to $\sigma(\lb_1,\dots,\lb_{\lfloor n^{1/2}\rfloor})$.
From the proof of Lemma~\ref{lem:ratioq_n} we have that
\[
\frac{\varphi_{n-\lfloor n^{1/2}\rfloor}(a)}{\varphi_{n}(0)}\to 1
\]
uniformly over all $|a|<n^{1/4+\eps}$, so
\begin{align*}
\Prb\left(\zeta_1\le n^{1/2}, \lba_{\zeta_1}=0 \right)
&=\Prb\left(\zeta_1\le n^{1/2}, \lwa_{\zeta_1}=0, |\lw_{\lfloor n^{1/2} \rfloor}|<n^{1/4+\eps}\right)+o(1)\\
&=\Prb\left(\zeta_1\le n^{1/2}, \lwa_{\zeta_1}=0\right)+o(1).
\end{align*}
Now observe that, under the law of~$\lw$, we have that $\zeta_1$ is a random variable on $\N$
that does not depend on~$n$, so $\Prb\left(\zeta_1\le n^{1/2},\lwa_{\zeta_1}=0\right)\to \rho$
as $n\to \infty$ and the statement follows.
\end{proof}
\subsection{Conditioning on ending in \texorpdfstring{$0$}{0} or \texorpdfstring{$-1$}{-1}}\label{subsec:squaretorectangle}
In this section, we show how Proposition~\ref{prop:areanonnegrectangle} follows
from Proposition~\ref{prop:areanonneg}. By Lemma~\ref{lem:to_walk}, we need to show
that for $(\lw_k)_{k\ge1}$ a lazy SSRW, and $\lwa_k=\sum_{i=1}^k\lw_i$, we have that
\[
\Prb\left(\lwa_1,\dots,\lwa_{n-1}\ge 0 \mid \lw_{n-1} \in \{0, -1\}\right)\sim n^{-1/4}\frac{\Gamma(3/4)}{\sqrt{2\pi(1-\rho)}}.
\]
\begin{proof}[Proof of Proposition~\ref{prop:areanonnegrectangle}]
First, observe that by Proposition~\ref{prop:areanonneg} and the fact that
$\Prb(\lw_n=0)\sim \tfrac{1}{\sqrt{\pi n}}$,
\[
\Prb\left(\lwa_1,\dots,\lwa_{n-1}\ge 0,\,\lw_{n-1}=0\right)
\sim n^{-3/4}\frac{\Gamma(3/4)}{\pi\sqrt{2(1-\rho)}}.
\]
We also need to calculate
\[
\Prb\left(\lwa_1,\dots,\lwa_{n-1}\ge 0,\,\lw_{n-1}=-1\right).
\]
Note that
\begin{align}
\Prb(\lwa_1,\dots,\lwa_n \ge 0,\lw_{n}=0)
&=\tfrac{1}{4}\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0,\lw_{n-1}=-1)\notag\\
&\quad+ \tfrac{1}{4}\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0,\lw_{n-1}=1)\notag\\
&\quad+\tfrac{1}{2}\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0,\lw_{n-1}=0).\label{eq:splitpositionatlaststep}
\end{align}
Both $\Prb(\lwa_1,\dots,\lwa_n\ge 0,\lw_{n}=0)$ and $\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0,\lw_{n-1}=0)$
are asymptotically equal to $n^{-3/4}\frac{\Gamma(3/4)}{\pi\sqrt{2(1-\rho)}}$, so if we show that
\begin{equation}\label{eq:equalprobpositivenegative}
\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0,\lw_{n-1} =-1) \sim
\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0,\lw_{n-1} =1),
\end{equation}
we can deduce that
\[
\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0,\lw_{n-1} =-1)\sim n^{-3/4}\frac{\Gamma(3/4)}{\pi\sqrt{2(1-\rho)}}.
\]
Then, the fact that $\Prb(\lw_{n-1}\in\{0,-1\})\sim\tfrac{2}{\sqrt{\pi n}}$ implies that
\[
\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0 \mid \lw_{n-1}\in\{0, -1\})
\sim n^{-1/4}\frac{\Gamma(3/4)}{\sqrt{2\pi(1-\rho)}},
\]
as required.
We now prove \eqref{eq:equalprobpositivenegative}.
Let $\cY_-$ be the set of paths $(y_0,y_1,\dots,y_{n-1})$ starting at $0$ with steps in
$\{-1,0,1\}$ which satisfy
\[
\left\{\sum_{i=1}^k y_i\ge 0\text{ for all }k\in [n-1],\,y_{n-1} =-1\right\},
\]
and let $\cY_+$ be the set of paths $(y_0,y_1,\dots,y_{n-1})$ starting at $0$ with steps in
$\{-1,0,1\}$ which satisfy
\[
\left\{\sum_{i=1}^k y_i\ge 0\text{ for all }k\in [n-1],\,y_{n-1} =1\right\}.
\]
We have
\[
\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0, \lw_{n-1}=-1)=\Prb(Y\in \cY_-)
\]
and
\[
\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0, \lw_{n-1}=1)=\Prb(Y\in \cY_+).
\]
We will define an injective map $f$ from $\cY_-$ to $\cY_+$
such that $\Prb(Y=y)=\Prb(Y=f(y))$ for all $y\in \cY_-$. Then,
\begin{align*}\Prb\left(Y\in \cY_-\right)&=\sum_{y\in \cY_-} \Prb(Y=y)\\
&=\sum_{y\in \cY_-} \Prb(Y=f(y))\\&=\sum_{y\in f(\cY_-)} \Prb(Y=y)\\&=
\Prb(Y\in \cY_+)-\Prb(Y\in \cY_+\setminus f(\cY_-)).\end{align*}
so we will be done if we can show that $\Prb(Y\in \cY_+\setminus f(\cY_-))=o(n^{-3/4})$.
We let $f$ be the map that changes the sign of the last excursion away from $0$. To be precise, for $y\in \cY_-$, let $\tau_{max}=\tau_{max}(y)=\max\{k\le n-1:y_k=0\}$ and let
\[
f(y)_i=\begin{cases}y_i,&\text{if }i\le\tau_{max};\\-y_i,&\text{if }i>\tau_{max}.\end{cases}
\]
It is immediate that $f$ has the claimed properties. In particular, $f(\cY_-)\subseteq \cY_+$ because for $y\in \cY_-$, it holds that $f(y)_k\ge y_k$ for all $k$.
Moreover, $\cY_+\setminus f(\cY_-)$
consists of the paths $y$ that end at $1$ for which the area at time $n-1$ is smaller than twice the area of the last excursion away from $0$, that is,
\[
\Prb(Y\in \cY_+, Y\notin f(\cY_-))
=\Prb\left(\sum_{i=1}^k \lw_i\ge 0\text{ for }k\in [n-1],\,\lw_{n-1} =1,
\sum_{i=1}^{n-1}\lw_i<2\sum_{i=\tau_{max}}^{n-1}\lw_i\right).
\]
We see that
\begin{multline*}
\Prb\left(\sum_{i=1}^k \lw_i\ge 0\text{ for }k\in [n-1],\,\lw_{n-1} =1,
\sum_{i=1}^{n-1} \lw_i<2\sum_{i=\tau_{max}}^{n-1}\lw_i\right)\\
\le 4\Prb\left(\sum_{i=1}^k \lw_i\ge 0\text{ for }k\in [n],\,\lw_{n} =0,
\sum_{i=1}^{n} \lw_i<2\sum_{i=\tau_{max}}^{n}\lw_i\right),
\end{multline*}
by conditioning on the value of the $n$th increment (similar to the calculation \eqref{eq:splitpositionatlaststep})
and
\begin{multline*}
\Prb\left(\sum_{i=1}^k \lw_i\ge 0\text{ for }k\in [n],\,\lw_{n} =0,
\sum_{i=1}^{n} \lw_i<2\sum_{i=\tau_{max}}^{n}\lw_i\right)\\
\le \Prb\left(0\le \sum_{i=1}^{n} \lw_i<2\sum_{i=\tau_{max}}^{n}\lw_i \bigmid \lw_{n} =0\right)\Prb(\lw_n=0).
\end{multline*}
Since $\Prb(\lw_n=0)=\Theta(n^{-1/2})$, we are done if we can show that
\[
\Prb\left(0\le \sum_{i=1}^{n} \lb_i<2\sum_{i=\tau_{max}}^{n}\lb_i\right)=o(n^{-1/4}).
\]
Pick any $0<\eps<1/4$. We compute
\begin{align*}
\Prb\left(0\le \sum_{i=1}^{n} \lb_i<2\sum_{i=\tau_{max}}^{n}\lb_i\right)
&\le \Prb\left(\sum_{i=\tau_{max}}^{n}\lb_i\ge n^{1+\eps}\right)
+\Prb\left(0\le \sum_{i=1}^{n} \lb_i<2 n^{1+\eps}\right)\\
&=\Prb\left(\sum_{i=1}^{\tau_1}\lb_i\ge n^{1+\eps}\right)
+\Prb\left(0\le \sum_{i=1}^{n} \lb_i<2 n^{1+\eps}\right),
\end{align*}
where $\tau_1=\tau_1(\lb)$ is the first return time of $\lb$ to~$0$.
We trivially have that $\sum_{i=1}^{\tau_1}\lb_i\le (\tau_1)^2$, and it is easy to see that
\[
\Prb(\tau_1=k)=\frac{\tfrac{1}{2k-1}\binom{2k-1}{k}\binom{2n-2k}{n-k} }{\binom{2n}{n} }
\sim \tfrac{1}{4\sqrt{\pi}} k^{-3/2}n^{1/2}(n-k)^{-1/2},
\]
as $k,n \to \infty$. Hence,
\[\Prb\left(\tau_1\ge n^{1/2+\eps/2}\right)=O(n^{-1/4-\eps/4})=o(n^{-1/4}). \]
Moreover, by Lemma \ref{lem:locallimitlazy},
we see that
\[\Prb\left(0\le \sum_{i=1}^{n} \lb_i<2 n^{1+\eps}\right)=O(n^{1+\eps}n^{-3/2})=o(n^{-1/4}),\]
so we conclude that
\[\Prb(Y\in \cY_+, Y\not\in f(\cY_-))=o(n^{-3/4}).\]
This proves \eqref{eq:splitpositionatlaststep} and
\[
\Prb\left(\sum_{i=1}^k \lw_i\ge 0\text{ for all }k\in [n],\,\lw_{n} =0\right)
\sim n^{-3/4}\frac{\Gamma(3/4)}{\pi\sqrt{2(1-\rho)}}. \qedhere
\]
\end{proof}
\subsection{About half of the sums are even}
\label{subsec:half}
So far we have looked at the probability that a sequence $n-1\ge d_1\ge\dotsb\ge d_n\ge 0$
satisfies the dominating condition~\eqref{eq:DC}. In this section, we consider the parity condition that $d_1 + \dotsb + d_n$ is even and prove Lemma~\ref{lem:parity}.
Among the sequences $n-1\ge d_1\ge\dotsb\ge d_n\ge 0$ that satisfy the dominating
condition~\eqref{eq:DC}, let $\mathcal{E}$ denote the set of such sequences for which the sum $d_1 + \dotsb + d_n$ is even, and $\mathcal{O}$ the sequences for which the sum is odd. We will define a partial matching between $\mathcal{E}$ and $\mathcal{O}$
such that the number of unmatched elements of $ \mathcal{E}\cup \mathcal{O}$ is $o(| \mathcal{E}|+| \mathcal{O}|)$.
This will immediately imply Lemma~\ref{lem:parity}.
Each sequence $n-1\ge d_1\ge\dotsb\ge d_n\ge 0$ corresponds to a right/down path from
$(0,n-1)$ to $(n,0)$ taking $2n-1$ steps. This corresponds to a sequence $(B_1,\dots,B_{2n-1})$
where $B_i\in\{\rightarrow,\downarrow\}$ takes the value $\rightarrow$ if the path goes right
(which it does $n$ times) and $\downarrow$ if the path goes down (which it does $n-1$ times).
We say $j\in [2n-2]$ with $j\equiv 0\mod 2$ is a \emph{switch position} for the sequence $(B_1,\dots,B_{2n-1})$ if
$(B_j,B_{j+1})\in
\{(\rightarrow,\downarrow),(\downarrow,\rightarrow)\}$.
Switching the sequence $(B_1,\dots,B_{2n-1})$ at position $j$ refers to replacing
$(B_j,B_{j+1})$ with the unique element in
$\{(\rightarrow,\downarrow),(\downarrow,\rightarrow)\}
\setminus \{(B_j,B_{j+1})\}$, resulting in some new sequence $(B'_1,\dots,B'_{2n-1})$.
We make two observations:
\begin{itemize}
\item First, for any $j\in [2n-2]$ with $j\equiv 0\bmod 2$, the position $j$ is a switch position for the
sequence $B$ if and only if $j$ is a switch position for the sequence $B'$. Moreover, switching at position $j$ is self-inverse:
switching $B'$ at position $j$ gives $B$.
\item If $B$ corresponds to a sequence $n-1\ge d_1\ge \dotsb \ge d_n\ge 0$, then the sequence
$B'$ also corresponds to a sequence $n-1\ge d'_1\ge \dotsb \ge d'_n\ge 0$, where for some $k\in [n]$
\[
d'_i\in\begin{cases}\{d_i\},&\text{if }i\ne k;\\ \{d_i-1,d_i+1\},&\text{if }i=k.\end{cases}
\]
In particular, the parities of $\sum_i d_i$ and $\sum_i d'_i$ are different.
\end{itemize}
It is not necessarily the case that performing a switch on a sequence in $\mathcal{E}$ results in a sequence
in $\mathcal{O}$ (or vice versa) as the switched sequence may violate the dominating condition~\eqref{eq:DC}.
Therefore, we will only define the matching between subsets $ \mathcal{E}' \subseteq \mathcal{E}$ and $ \mathcal{O}' \subseteq \mathcal{O}$, where we choose $ \mathcal{E}'$ and $ \mathcal{O}'$ so that for all sequences in $ \mathcal{E}'\cup \mathcal{O}'$, have some slack
in the dominating condition~\eqref{eq:DC}. We will show that $| \mathcal{E}'\cup \mathcal{O}'|=(1+o(1))| \mathcal{E}\cup \mathcal{O}|$, for which we need the following two lemmas.
\begin{lemma}\label{lem:slack}
As $n\to \infty$,
\[
\Prb(\lwa_{\lfloor n/2 \rfloor},\dots,\lwa_{n-1}\ge 1
\mid \lwa_1,\dots,\lwa_{n-1}\ge 0, \lw_{n-1}\in\{-1,0\})=1-o(1).
\]
\end{lemma}
\begin{proof}
By Proposition \ref{prop:areanonnegrectangle}, $\Prb(\lwa_1,\dots,\lwa_{n-1}\ge 0, \lw_{n-1}\in\{-1,0\})=\Theta(n^{-3/4})$, so we are done if we show that
\[\Prb(\lwa_i=0\text{ for some }i\geq \lfloor n/2\rfloor, \lwa_1,\dots,\lwa_{n-1}\ge 0, \lw_{n-1}\in\{-1,0\})=o(n^{-3/4}).\]
But by conditioning on the value of the $n$th increment, we see that
\begin{align*}&\Prb(\lwa_i=0\text{ for some }i\geq \lfloor n/2\rfloor, \lwa_1,\dots,\lwa_{n-1}\ge 0, \lw_{n-1}\in\{-1,0\})\\&\quad \leq 4 \Prb(\lwa_i=0\text{ for some }i\geq \lfloor n/2\rfloor, \lwa_1,\dots,\lwa_{n-1}\ge 0, \lw_{n}=0),\end{align*}
and Lemma \ref{lem:hittingtimeat0} and Proposition \ref{prop:areanonneg} imply that the right-hand side is $o(n^{-3/4})$ as claimed.
\end{proof}
Informally, this lemma states that switching the sequence `near the last checks' is unlikely to affect whether the dominating condition (\ref{eq:DC}) holds.
We next show that all but a negligible fraction of the sequences have switch positions `near the last checks' (which is the near the middle of $B$).
We say a sequence $(B_1,\dots,B_{2n-1})$ is $k$-\emph{switchable} if it has a switch position at some
even $i\in [n-2k,n-2]$.
\begin{lemma}\label{lem:switches_everywhere}
For any $k\in[n-1]$, the number of sequences $(B_1,\dots,B_{2n-1})\in \{\rightarrow,\downarrow\}^{2n-1}$
that are not $k$-switchable is at most $2^{-k}4^n$.
\end{lemma}
\begin{proof}
We choose a sequence $(B_1,\dots,B_{2n-1}) \in \{\rightarrow,\downarrow\}^{2n-1}$ by independently choosing each $B_i$ uniformly at random. The probability that $B$ has a switch at an even position $j\in [2n-1]$ is $\frac12$, and the events are independent for different even values of $j$. The probability that there are $k$ independent failures is hence $2^{-k}$.
\end{proof}
We now have all the necessary set-up to conclude the proof.
\begin{proof}[Proof of Lemma \ref{lem:parity}]
Set $k=\lfloor n/4\rfloor$. Let $ \mathcal{F}$ denote the set of sequences $d\in \mathcal{E}\cup \mathcal{O}$ for which the right/down-sequence $(B_1,\dots,B_{2n-1})$ that corresponds it (as defined earlier this section) is not $k$-switchable,
or for which the area process takes the value $0$ at some point in the last $2k$ steps.
Then, applying Lemma~\ref{lem:slack} and Lemma~\ref{lem:switches_everywhere}, we find that $| \mathcal{F}|=o(| \mathcal{E}\cup \mathcal{O}|)$.
We set $ \mathcal{O}'= \mathcal{O}\setminus \mathcal{F}$ and $ \mathcal{E}'= \mathcal{E}\setminus \mathcal{F}$ and define a matching between
$ \mathcal{O}'$ and $ \mathcal{E}'$ as follows. Let $d\in \mathcal{E}'\cup \mathcal{O}'$ and let $B\in \{\rightarrow,\downarrow\}^{2n-1}$
be the corresponding right/down-path. With $(\ell,\ell)$ as
the unique grid point at which the path crosses the diagonal, as defined in Section~\ref{sec:prelim}, we note that $(B_1,\dots,B_{n-1})$
is a path from $(0,n-1)$ to $(\ell,\ell)$. We will attempt to change the parity by making a switch as close to the end $(\ell,\ell)$ of this path as possible. By choice of~$ \mathcal{F}$, there is a switch position at some
even $j\in [n-2k,n-2]$. We perform the switch at the last such position.
This only affects the last at most $2k$ positions of the area process,
which can only be increased by~1, reduced by 1 or stay the same. Therefore, the dominating
condition \eqref{eq:DC} is not affected (by our choice of $ \mathcal{F}$). This defines a matching
with the desired properties.
\end{proof}
\section{Computational results}\label{sec:comp}
In this section we give a new recursion which we have used to calculate $G(n)$ for many new values, and a description of how we numerically estimated the value of $\rho$ (and $\cdeg$). We also make the surprising observation that, while roughly half the sequences which satisfy the dominating condition have even sum, the convergence to a half is rather slow.
\subsection{Determining the exact values of \texorpdfstring{$G(n)$}{G(n)} for small \texorpdfstring{$n$}{n}}
\begin{figure}
\centering
\input{graph}
\caption{The graph depicts our numerical estimation of $\cdeg$ (dotted line) via the approximation of $\rho$, and a numerical estimation based on the exact values of $G(n)$ for small~$n$.}
\label{fig:graph}
\end{figure}
We give a simple recursion to count the number of graphic sequences.
In Section \ref{sec:reformulation}, steps 1 through 5 define a deterministic mapping from a sequence $n-1\geq d_1\geq \dots \geq d_n\geq 0$ to a (non-random) walk $Y$ (with steps in $\{-1,0,1\}$ and ending in $0$ or $-1$), such that the sequence $d$ satisfies the dominating condition (\ref{eq:DC}) if and only if $\sum_{j=1}^k Y_j\geq 0$ for all $1\leq k\leq n-1$. Moreover, as explained in Step 5, we have $\sum_{i=1}^n d_i\equiv \sum_{i=1}^{n-1}Y_i\bmod 2$.
We will use this to count the number of graphic sequences.
Note that a single walk $Y$ corresponds to $2^z$ sequences, where $z$ is the number of zero steps that the sample takes (see Figure \ref{fig:reformulation}). We will weight each walk $Y$ accordingly when counting.
Let $F(N, y, a)$ be the weighted number of walks $(Y_i)_{i=1}^N$ which start at $y$, take $N$ steps, end in $\{-1,0\}$ and satisfy $a + \sum_{i=1}^k Y_i \geq 0$ for all $0 \leq k \leq N$ and $a + \sum_{i=1}^{N}Y_i \equiv 0 \bmod 2$. We think of $a$ as the starting value of the area process, or the area from any earlier steps of the walk. We see that by taking $N = n -1$, $y = 0$ and $a = 0$, we count exactly the walks which correspond to sequences which are graphic. Hence, the number of graphic sequences of length $n$ is $F(n-1, 0, 0)$, and we can calculate this using a recursion for $F$ as follows.
First, we have a boundary condition that $F(N, y, a) = 0$ whenever $a < 0$. If this is not the case, we consider the three options for the first step in the walk and how to complete the walk after this step. A completion would need to take $N - 1$ steps in every case, but the starting position and the area will change. The walk could step up, after which a completion starts at $y + 1$ with area $a + y + 1$; the walk could step down, in which case a completion starts at $y - 1$ with area $a + y - 1$; or the walk could take a step of size $0$, in which case a completion starts at $y$ with area $a + y$. Since in the last case the walk takes a step of size 0 and we are counting the weighted number of completions, this last case must be counted twice. This leads to the following recursion:
\[F(N, y, a) = F(N - 1, y + 1, a + y + 1) + F(N - 1, y - 1, a + y -1) + 2F(N-1, y, a + y),\]
with boundary condition $F(N, y, a) = 0$ whenever $a < 0$, and initial condition $F(0, y, a) = 1$ if $y \in \{0, -1\}$, $a \geq 0$ and $a$ even, and $F(0, y, a) = 0$ otherwise.
To calculate the number of graphic sequences, we calculated every value of $F$ for a given value of $N$ using the pre-computed values for $N - 1$, then calculated the values for $N + 1$ using the values for $N$ and continued this until the calculation ran out of memory. At each step, we get one more value of $G(n)$ by reading off the value of $F(n-1, 0, 0)$.
For this approach to work, we need $F$ to take only finitely many values for each value of $N$, and we only need to calculate $F(N, y, a)$ for small (depending on $N$) values of $|y|$ and $a$. First, observe that $F(N, y, a) = 0$ whenever $y > N$ or $y < - N - 1$ as the walk needs to end in $\{0, -1\}$. We now consider for which values of $a$ we need to calculate $F(N, y, a)$. Clearly, the boundary condition means we do not need to consider $a < 0$. Moreover, if $y < 0$, we can ignore all $a < y(y+1)/2$ as the area process will go negative at some point. This just leaves large $a$.
We claim that, given that the walk starts at $y$, in $N$ steps the area can decrease by at most
\[
a' = \frac{N^2 - 2Ny + 2N - y^2 + \one_{\text{odd}}(N-y)}{4}.
\]
In particular, this means that if $a \geq a'$, the area will always stay non-negative (provided it is already) and $F$ only depends on the parity of $a$. Hence, if $c = \max\{0, a'\}$ and $a \geq c$, the value $F(N, y , a)$ is either $F(N, y, c)$ or $F(N, y, c + 1)$.
Let us briefly justify the claim. First, consider the amount the area process can decrease when the walk starts at 0 and returns to 0. The best thing for the walk to do is for it to take $\floor{N/2}$ steps downwards, then $\floor{N/2}$ steps upwards. If $N$ is odd, one extra step of size 0 should be inserted between the downwards and upwards steps. This reduces the area process by
\[ \sum_{i=1}^{\floor{N/2}} i + \one_{\text{odd}}(N) \floor{N/2} + \sum_{i=1}^{\floor{N/2} - 1} i = \frac{N^2 - \one_{\text{odd}}(N)}{4}.\]
If $0 \leq y \leq N - 1$, the biggest reduction comes from taking $y + 1$ steps downwards to $-1$, then doing the above steps with the remaining steps (but starting and ending at $-1$). This reduces the area by
\[ -\sum_{i = -1}^{y-1} i + \frac{(N - y - 1)^2 - \one_{\text{odd}}(N -y - 1)}{4} + (N - y - 1) = a'.\]
When $y = N$, the same equation holds. When $y < 0$, then the best thing to do is to reserve $-y - 1$ steps which will be used as upwards steps at the end. For the first $N + y + 1$ steps, one should aim to minimise the area while returning to $y$. This gives a total of
\[ \frac{(N + y + 1)^2 - \one_{\text{odd}}(N + y + 1)}{4} - y (N + y + 1) + - \sum_{i= y + 1}^{-1} i = a'.\]
The final optimisation we implement is to consider the function $f$ defined by
\[f(N, y, a) = \begin{cases}
F(N, y, a) - F(N, y, a - 1), & a \ne 0;\\
F(N, y, 0), & a = 0.
\end{cases}\]
This satisfies a similar recursion to the function $F$, but the numbers are generally smaller and this helps with the memory usage.
We wrote a program in Go which uses this recursion to calculate the number of graphic sequences and ran this on a node with 512 GiB of RAM until the program ran out of memory. This produced the first 1392 numbers (starting from $n = 0$). Of course, we are only interested in calculating $F(N, 0 ,0)$ and we do not need to calculate all values of $F(N, \cdot, \cdot)$. Even calculating $F(N + 1, 0, 0)$ only needs three values of $F(N, \cdot, \cdot)$ (and one is trivially 0). We therefore ran the program a second time now we had established at what point the program ran out of memory and stopped the recursion just before this point. We then attempted to calculate $F(N + 1, 0, 0), F(N+2, 0, 0), \dots$ by only calculating the values $F$ on demand (and keeping the calculated values in a map). We note that if $F(N, \cdot, \cdot)$ is known entirely, calculating $F(M, 0, 0)$ this way is clearly more efficient when $M$ is not much bigger than $N$, but it is less efficient when $M$ is much larger than $N$, and in particular, doing this from the start is less efficient than calculating the full layers.
This second step added another 260 values, so that we now know $G(n)$ for all $n\le 1651$.
The code (along with the computed values) is attached to the arXiv submission, and the new values have been added to the OEIS entry \href{https://oeis.org/A004251}{A004251}. The sequence \href{https://oeis.org/A095268}{A095268} counts the number $G'(n)$ of zero-free graphic sequences, which is given by $G(n) - G(n - 1)$, and we have also updated this sequence.
\subsection{A surprising observation on the parity condition}
\label{subsec:eo}
By changing the initial condition, the above recursion can be changed to count $H(n)$, the sequences which satisfy the dominating condition and have odd parity. Lemma \ref{lem:parity} shows that $H(n) \sim G(n)$, and one might naturally assume that $|G(n) - H(n)|$ is exponentially smaller than $G(n)$ (and $H(n)$), but this does not appear to be the case.
Surprisingly, numerical estimates suggest that $G(n) - H(n) = \Theta(4^n/n^{5/2})$ which is only a factor of $\Theta(n^{7/4})$ smaller than $G(n)$.
We remark that if we ignore the dominating condition and just count the number of sequences $n-1\ge d_1\ge\dotsb\ge d_n\ge0$ for which $\sum d_i$ is even and for which $\sum d_i$ is odd we get
\[
\frac{1}{2}\left(\binom{2n-1}{n-1}+\binom{n-1}{\lfloor n/2\rfloor}\right)
\qquad\text{and}\qquad
\frac{1}{2}\left(\binom{2n-1}{n-1}-\binom{n-1}{\lfloor n/2\rfloor}\right)
\]
respectively. The difference between these two quantities is only $O(2^n)$, which is exponentially smaller than either.
The partial matching we used to prove Lemma~\ref{lem:parity} suggests a possible explanation for this. In the proof, we switched at the last switchable position before $n-1$. Adapting the proof of the main result in \cite{AurzadaDereichLifshits2014} to \emph{lazy} SSRW bridges yields that $\Prb(\lwa_{n-1}=0\mid \lw_{n-1}=0,\lwa_1,\dots,\lwa_{n-1}\geq 0)=\Theta(n^{-7/4})$, and a sequence with $\lwa_{n-1} = 0$ cannot be `switched down'. Assuming that roughly half of them would be switched down (with the other half switched up), this yields a proportion $\Omega(n^{-7/4})$ of the even sequences which do not have an odd counterpart.
Of course, there are also odd sequences without an even counterpart (e.g. those with $\lwa_{n-1} = 1$ and $\lwa_{n-2} = 0$ cannot be switched down), but this observation makes it at least reasonable that the order of the difference is $\Theta(4^n/n^{5/2})$.
\subsection{Estimating the constant \texorpdfstring{$\rho$}{rho}}
\label{subsec:rho}
We now give a method for estimating~$\rho$. Recall that $(\lw_k)$ is the lazy simple symmetric walk
and $(\lwac_k)$ is the random walk formed by summing the (signed) excursion areas $\lwae_k$ of $(\lw_k)$.
We first need to find the distribution of the excursion areas $\lwae_k$.
We define the following (partial) generating function
\[
g(x,y)=\sum_{i,j>0}\Prb(\text{excursion is of area $i$ and length $j$})x^iy^j.
\]
Note that this is really only part of the full generating function, which is $\frac{y}{2}+g(x,y)+g(x^{-1},y)$, to take into account excursions below the axis, and also the probability $\frac{1}{2}$ event that the lazy
random walk does not move on the first step (and hence the excursion area is zero).
\begin{lemma}
The generating function $g(x,y)$ satisfies the recursive equation
\[
g(x,y)=\frac{xy^2}{16(1-\frac{xy}{2}-g(x,xy))}.
\]
\end{lemma}
We note that as all terms of $g(x,y)$ have a positive power of~$y$, terms in $g(x,xy)$ have higher $x$-degree
than their counterparts in $g(x,y)$, and hence this equation allows one to recursively evaluate $g(x,y)$
to any order. For example,
\[
g(x,y)=\tfrac{x y^2}{16}+
\tfrac{x^2y^3}{32}+\tfrac{x^3y^4}{64}+
\tfrac{x^4(y^4+2y^5)}{256}+
\tfrac{x^5(y^5+y^6)}{256}+
\tfrac{x^6(2y^5+3y^6+2y^7)}{1024}+O(x^7).
\]
\begin{proof}
To have a positive excursion, the first step of the lazy random walk must be up (probability $\frac{1}{4}$).
Then it either goes down (probability $\frac{1}{4}$) giving a term $\frac{1}{16}xy^2$ in $g(x,y)$,
or it follows some non-negative excursion before returning to height~1.
Let the excursion have area $X_1$ and length $T_1$ (so that the walk is now at $1$ with total area $X_1 + T_1 + 1$ and length $T_1 + 1$).
The rest of the excursion is then
equivalent to one starting at $0$ at time $T_1$ with an initial step up, and say this excursion has area $X_2$ and length $T_2$.
The sum $\sum \Prb(X_1,T_1)x^{X_1}z^{T_1}$ is just $\frac{z}{2}+g(x,z)$ (as we allow the
trivial area zero excursion here), and the sum $\sum \Prb(X_2,T_2)x^{X_2}y^{T_2}$is just $4g(x,y)$
(as we automatically can assume the first step is up). Overall we get an
excursion of length $T_1+T_2$ and area $(X_1+T_1)+X_2$. Setting $z=xy$ and multiplying these (together with
a factor of $\frac{1}{4}$ for the initial step up) gives the remaining terms of $g(x,y)$. Hence,
\[
g(x,y)=\tfrac{xy^2}{16}+\tfrac{1}{4}\big(\tfrac{xy}{2}+g(x,xy)\big)\cdot 4g(x,y),
\]
and the lemma follows from rearranging this equation.
\end{proof}
Now by setting $y=1$ we get a generating function for
the (positive) excursion areas, namely
\[
g(x,1)=\tfrac{x}{16}+\tfrac{x^2}{32}+\tfrac{x^3}{64}+
\tfrac{3x^4}{256}+\tfrac{x^5}{128}+\tfrac{7x^6}{1024}+
\tfrac{21x^7}{4096}+\tfrac{37x^8}{8192}+\tfrac{31x^9}{8192}+O(x^{10}).
\]
Note that $g(1,1)=\frac{1}{4}$ is the probability of a positive excursion.
The next task is to estimate or bound~$\rho$, which is the probability that the random walk $\lwac_k$ with step
sizes $\lwae_k$ hits 0 before becoming negative. To do this we construct a finite state Markov chain
with states $\{-,0,1,2,\dots,n-1,\star\}$ where the states $-$ and $\star$ represent the walk going negative and reaching a state at least $n$ respectively. The states $-$ and $\star$ are absorbing, and otherwise we add a random variable distributed like a signed excursion area. If adding this, takes the walk negative or at least $n$, the walk moves to $-$ or $\star$ as necessary. We start the Markov chain at 0 and run until we either hit 0 again, or one of the states $-$ or $\star$.
By a simple coupling argument it is clear that
\[
\Prb(\text{hit }0)\le \rho \le \Prb(\text{hit }0) + \Prb(\text{hit }\star).
\]
Taking $n$ sufficiently large gives us reasonable bounds on~$\rho$.
As an example, taking $n=2$ we have states $\{-,0,1,\star\}$ and transition matrix
\[\begin{pmatrix}
1&0&0&0\\
\frac14&\frac12&\frac{1}{16}&\frac{3}{16}\\
\frac{3}{16}&\frac{1}{16}&\frac12&\frac14\\
0&0&0&1
\end{pmatrix}.\]
Writing $h_{ij}$ for the probability of hitting $j$ starting at $i$, we have
\[
h_{10}=\tfrac{1}{16}+\tfrac{1}{2}h_{10},\qquad
h_{1\star} = \tfrac{1}{4}+\tfrac{1}{2}h_{1\star},
\]
giving $h_{10}=\tfrac{1}{8}$, $h_{1\star}=\tfrac{1}{2}$. Then
\[
h_{00}=\tfrac{1}{2}+\tfrac{1}{16}h_{10}=\tfrac{65}{128},\qquad
h_{0\star}=\tfrac{3}{16}+\tfrac{1}{16}h_{1\star}=\tfrac{7}{32}.
\]
Hence,
\[
\tfrac{65}{128}\le \rho \le \tfrac{65}{128}+\tfrac{7}{32}=\tfrac{93}{128}.
\]
Using this method with $n=2^{18}$ gives a lower bound of $0.51580258$, which appears to be very close to
the true value of~$\rho$. However the upper bound of $0.54543568$ obtained by this method
seems to still be very far from the truth.
If we make the assumption
that hitting 0 before going negative is a decreasing function of the starting point, we
can amalgamate the states $n-1$ and $\star$ in the above model, giving transitions out of
that state as if they were from $n-1$. Unfortunately we do not have a proof that $h_{i0}$ is decreasing in~$i$,
so this does not give a rigorous bound on~$\rho$. Nevertheless, for $n=2^{18}$ we obtain a (non-rigorous)
upper bound of $0.51580289$ by this method, which does seem much closer to the true value.
An even less rigorous estimate can be obtained by applying Richardson extrapolation
to $h_{00}$ in terms of $1/n$, which gives the estimate
\[
\rho\approx 0.515802638089141858504490255841,
\]
and corresponds to a value of
\[
\cdeg\approx 0.099094083237488745361449340935.
\]
These approximations do not appear to correspond to any number with a simple closed form expression.
\section{Concluding remarks}
\label{sec:conclusion}
We have given a precise asymptotic for the number of graphical sequences.
Similar asymptotics are known for tournaments, but not for other natural classes such as uniform hypergraphs and digraphs.
Another interesting direction with open problems is the asymptotics of the number of graphical partitions of an integer $N$.
We outline some more directions for future research below.
\paragraph{An upper bound in Lemmas \ref{lem:useful} and \ref{lem:useful2}}
For $x\in \R_{>0}^n$, let
\[\textstyle
A(x)=\big\{(\sigma,s):\sigma \in S_n,\,s\in \{-1,1\}^n,\,
\sum_{i=1}^k s_ix_{\sigma(i)}\ge 0 \text{ for all }k\in [n]\big\}.
\]
Lemma \ref{lem:useful} states that $|A(x)|\ge (2n-1)!!$, with equality for all $x$ for which all sums $\sum_{i\in S}x_i$ are distinct (for distinct $S\subseteq[n]$).
\begin{conjecture}
\label{conj:comb}
For $x\in \R_{>0}^n$,
$|A(x)|$ is maximized when $x_1=\dots=x_n$.
\end{conjecture}
Rephrased in probabilistic terms, we conjecture that the simple symmetric random walk has the highest probability of staying non-negative, amongst all random processes with exchangeable increments $(X_1,\dots,X_n)\in (\R\backslash\{0\})^n$ of which the law is invariant under sign changes of the elements.
This conjecture would imply (see \eqref{e:ulbounds}) that for all
exchangeable $(X_1,\dots,X_n)\in (\R\backslash\{0\})^n$ of which the law is invariant under sign changes of the elements,
\[ \frac{1}{\sqrt{\pi (n+1/2)}}\le \Prb\left(\sum_{i=1}^k X_i \ge 0\text{ for all }k\in [n]\right)\le \frac{\sqrt{2}}{\sqrt{\pi n}}.\]
Using Lemma \ref{lem:to_walk} and Lemma \ref{lem:convtorayleigh}, this would immediately imply Proposition \ref{prop:basic}, which yields a direct proof for $G(n)=\Theta(n^{-3/4}4^n)$ that does not rely on the results in \cite{Vysotsky2014}. We also think the conjecture is of independent interest, in particular because it would imply that for this very general class of random processes, the order of the probability of staying positive does not depend on the law, or even the tail behaviour of the increments.
\paragraph{Uniformly random graphic sequences}
A natural next question, that we intend to answer in future work, is the (asymptotic) law of a uniformly random graphic sequence of length $n$. We now make some observations using our reformulation and discuss a potential strategy to answer this question. Firstly, note that a uniform lattice path has the law of a simple symmetric random walk bridge with $2n$ steps that straddles the line $y=n-x$, and in particular, for large $n$, its fluctuations away from the line $y=n-x$ are of order $n^{1/2}$. In fact, for any $\delta>0$, it is exponentially unlikely that at some point the lattice path is at distance more than $\delta n$ from the line $y=n-x$. This means that even after conditioning on the lattice path encoding a graphic sequence (i.e.\ conditioning on an event with probability $\Theta(n^{-1/4})$), it is exponentially unlikely that at some point the lattice path is at distance more than $\delta n$ from the line $y=n-x$, and in general, the large deviations of a uniformly random graphic sequence are completely described by the large deviations of a uniformly random lattice path. (Observe that this implies that a uniform graphic sequence is very different from the degree sequence of a uniform graph, of which the corresponding lattice path stays close to the horizontal line $y=n/2$.)
Therefore, to observe the difference between a uniform graphic sequence and a uniform lattice path, we will need to consider their more fine-grained behaviour, for example by studying the scaling limit of their fluctuations around the line $y=n-x$. We conjecture the following.
\begin{conjecture}
Let $D_1\ge\dotsb\ge D_n$ be a uniformly random graphic sequence of length $n$. Then, there exists a random continuous function $D$ from $[0,1]$ to $\R$ such that
\[\left(n^{-1/2}(D_{\lfloor tn\rfloor}-(1-t)n),0\le t\le 1\right)\overset{d}{\to}\left(D_t,0\le t\le 1\right)\]
in the uniform topology.
\end{conjecture}
We expect $D$ to have two characterizations. Firstly, it can be defined as a Brownian bridge conditioned to satisfy a continuous version of the dominating condition \eqref{eq:DC}. Secondly, via a continuous version of the reformulation described in Section \ref{sec:reformulation}, $D$ can be constructed via two paths, for which the distance between them is distributed as a conditioned Brownian bridge and their midpoint is determined by a Brownian motion (the Brownian motion plays the role of the `lazy steps' that can go either right or down). This result would add graphic sequences to the long and varied list of uniformly random combinatorial structures with a `Brownian' scaling limit; examples are numerous models of trees of which the scaling limit can be described by a Brownian excursion (see the survey paper \cite{LeGall}), mappings with a limit described by the Brownian bridge \cite{randommappings}, various classes of maps with limits encoded by the Brownian snake (see the survey paper \cite{surveyRandomMaps}) and pattern-avoiding permutations with limits described by a Brownian excursion (see \cite{randompermutations} and references therein).
Such scaling limits are interesting in their own right, but can also be exploited to answer questions about the corresponding combinatorial class, such as `what proportion of Cayley trees of size $n$ have height exceeding $tn^{1/2}$?' or `in what proportion of maps from $[n]$ to $[n]$ is the average distance to a cycle larger than $tn^{1/2}$?'.
\paragraph{Persistence probabilities of integrated random processes}
The study of the probability that integrated random walks and random walk bridges stay non-negative started with the work of Sina\u{\i} on the SSRW \cite{Sinai1992} and has attracted a lot of attention in the past decade (\cite{AurzadaDereichLifshits2014,Dembo2013,Denisov2015, Gao2014,Vysotsky2010,Vysotsky2014}; see also the survey paper \cite{Aurzada2015}). However, all work on integrated random walk bridges only finds the right order of the persistence probability \cite{AurzadaDereichLifshits2014,Vysotsky2014} and for random walks the sharp asymptotics (including the value of the constant) are only known under a $(2+\delta)$-moment condition on the step distribution.
Our methods completely carry over to the setting of SSRW bridges, yielding the following result of independent interest. Let $\uw$ be a SSRW and let $\uwa$ be its area process.
\begin{prop}\label{prop:persistenceSSRW}
We have that
\[
n^{1/4}\Prb(\uwa_1,\dots,\uwa_{2n}\ge 0\mid \uw_{2n}=0) \to \frac{\Gamma(3/4)}{\sqrt{2\pi(1-\hat{\rho})}}
\]
as $n\to \infty$, for $\hat{\rho}$ the probability that the random walk with steps distributed as the signed area of the excursions of a SSRW hits $0$ before going negative.
\end{prop}
Adapting the method to numerically estimate the value of $\rho$ described in Section~\ref{subsec:rho} shows that $\hat{\rho}\approx 0.0773408571485249705089600725$.
It is possible that our techniques can be generalized to other models of random bridges. However, there are some difficulties to overcome, both conceptual and technical. Our proof relies heavily on Lemma~\ref{lem:useful2}, which requires the areas of different excursions of the bridge to be exchangeable and for their law to be invariant under sign changes. When only considering random walk bridges, these conditions combined restrict the method to (scalings of) symmetric processes with steps in $\{-1,0,1\}$,
so new ideas are needed to adapt the method to other random walk bridges.
\paragraph{Acknowledgements} The exact numbers were computed using the computational facilities of the \href{http://www.bris.ac.uk/acrc/}{Advanced Computing Research Centre, University of Bristol}.
|
1,108,101,566,716 | arxiv | \section{Introduction}
It is now widely accepted that black holes reside in the centre of
nearly all galaxies
\citep[e.g.][]{1996gfa..book.....B,1998Natur.395A..14R,
1998AJ....115.2285M}. Accretion onto these black holes is believed
to be the mechanism responsible for powering Active Galactic Nuclei
\citep[AGN; e.g.][]{1993ARA&A..31..473A,1995PASP..107..803U}.
The black holes in these galaxies can accrete matter, either from the
interstellar medium close to the event horizon or from thick disks of
gas. In order to be accreted, gas needs to fall in with an angular
momentum that does not exceed that needed for disk capture. In the
vicinity of the sphere of influence of the black hole, the Keplerian
motion of the accreting material is exceedingly high, thus processes
that redistribute the angular momentum of the accreting gas determine
the accretion rate. Galactic mergers, gravitational torques from bars
and other structures can efficiently channel gas into the nuclear
regions
\citep[e.g.][]{1989Natur.338...45S,1990Natur.345..679S,1995ApJ...448...41H}. Closer
to the sphere of influence of the black hole, however, other
mechanisms, such as turbulence driven by magnetic instabilities, are
efficient sources of angular momentum transport
\citep[e.g.][]{1995ApJ...445..767M,1996ApJ...463..656S}.
There are different modes of accretion. The `standard' accretion mode
occurs when matter is accreted onto the black hole through a
radiatively efficient process. This is generally associated with
optically thick and geometrically thin accretion disks
\citep[e.g.][]{1973A&A....24..337S}, and thought to be associated with
quasar activity. This process is likely to be prevalent where there is
a plentiful supply of cold gas, which can accrete efficiently towards
the black hole, and as such the fuel, in the form of the cold gas,
should also be able to condense and provide the fuel for star
formation \citep[e.g.][]{2007MNRAS.376.1849H}. Indirect evidence for
such a scenario comes from the similar evolution of the AGN activity
in the Universe and the star-formation rate density, both increasing
rapidly from $z\sim 0 \rightarrow 1$ and steadily turning over at
$z>3$ \citep[e.g.][]{2009ApJ...696..396S}. Furthermore, such a close
connection between AGN and star-formation activity may provide a
relatively straightforward explanation of the local correlation
between galaxy mass and black-hole mass \citep{1998AJ....115.2285M,2000ApJ...539L...9F,2000ApJ...539L..13G,2004ApJ...604L..89H}.
Accretion in an AGN may also be radiatively inefficient. When mass
accretion is too low ($\dot{M}\ll \dot{M}_{\rm Edd}$), the inflowing
gas may not be able to radiate the gravitational potential energy
liberated due to accretion. Magnetohydrodynamic simulations show that
the energy dissipated during accretion could heat up only the ions in
the disk
\citep[e.g.][]{1999ApJ...520..248Q,2000ApJ...541..811M,2002ApJ...577..524Q}. On
the other hand, radiative losses, synchrotron radiation, and
Comptonization of low-energy photons take energy from the electrons,
cooling them. The characteristic time-scale for the hot ions and the
cold electrons to achieve thermal equilibrium through coupling depends
on the gas temperature in the vicinity of the black hole, and is
inversely proportional to the number density of ions. When the
accretion rate is low, the ionic density is also small, and this
time-scale is of the same order as, or larger than, the inflow
time. Thus a two-temperature plasma is expected to form when
$\dot{M}\ll \dot{M}_{\rm Edd}$. In this regime, advection dominated
accretion flows (ADAFs) can be formed. This radiatively inefficient
process is currently taken as the most likely explanation for
accretion flows around black holes at low accretion rates. The
critical accretion rate below which it becomes radiatively inefficient
is estimated to be $\dot{M}_{\rm crit}\simeq \alpha^{2}\dot{M}_{\rm
Edd}$ \citep{1995ApJ...452..710N}, where $\alpha$ is a dimensionless
parameter that measures the efficiency of angular momentum transport
in disks. With the standard value of $\alpha=0.25$
\citep[e.g.][]{1997ApJ...489..865E}, the critical accretion rate is
situated around $\dot{M}_{\rm crit}\sim0.06$ normalised to the
Eddington accretion rate. Due to the inability of the gas to cool
down radiatively in this regime, the gas can be driven to a higher
temperature, and thus cause a vertical thickening of the disk. ADAFs
are thus thought to be characteristic of hot thick disks.
What causes the different modes of radiatively efficient and
inefficient accretion remains uncertain. It has been
suggested that the nature of the gas being accreted might determine
the type of accretion, with cold gas producing a stable accretion
disk and thus a radiatively efficient accretion, and hot gas producing
flows that would result in an ADAF
\citep[e.g.][]{2007MNRAS.376.1849H}. The black hole spin has also been
suggested to play a role in determining the accretion mode
\citep[e.g.][]{2011MNRAS.414.1937M}.
Radio galaxies are a class of AGN that can be divided into two
sub-categories according to the ratio between the intensity of high
and low excitation emission lines in their optical spectra: low
excitation galaxies (LEGs) and high excitation galaxies (HEGs). The
two distinct classes were first noted by \citet{1979MNRAS.188..111H}
and further categorised by \citet{1994ASPC...54..201L} and
\citet{1997MNRAS.286..241J}. \citet{1997MNRAS.286..241J} defined LEGs
as objects that obey the following requirements: have [O{\sc
iii}]-line equivalent widths $< 10$\AA, have a ratio $\rm [O{\sc
II}]/[O{\sc III}]>1$, or both.
It is widely accepted that LEGs could be the result of radiatively
inefficient accretion processes
\citep[e.g.][]{2007MNRAS.376.1849H,2010A&A...509A...6B,2014MNRAS.440..269M}. \citet{2012MNRAS.421.1569B}
compared a large number of local radio-loud AGN and concluded that the
HEG and LEG populations show different accretion rate distributions,
consistent with the idea that radiative efficiency is determined
purely on accretion rate.
This is a subject that impacts on the most
relevant issues for models of AGN and their roles in galaxy evolution. For instance, if the HEG/LEG
distinction is indeed related to the accretion process, then the fact
that their radio properties are similar requires that the
power being channeled into the jets is independent of the accretion rate.
Moreover, given that radiatively efficient accretion is related to the
`quasar mode' form of feedback processes in AGN, and radiatively
inefficient accretion associated with `radio mode' AGN feedback in
semi-analytic models
\citep[e.g.][]{Croton2006}, the HEG and LEG characterisation could be used to
diagnose the different modes at play in each system. LEGs would for
instance be the perfect laboratories to investigate the `radio mode'
form of feedback, currently taken as being the principal mechanism
responsible for shutting down star formation in the most massive
systems at $z<1$.
In this paper we investigate the HEG/LEG division in radio-loud
sources and their different accretion processes, extending the study
of \citet{2012MNRAS.421.1569B} to high-redshift sources
($z\sim1$). Moving to high redshift is not only complementary in terms
of looking back time but also in terms of luminosity as it provides us
with the opportunity to study the rarer more luminous objects in each
class.
Section~\ref{sec:data} describes the data
selection. Section~\ref{sec:sed_method} describes the SED fitting
method used. In Section~\ref{sec:sed_fits} and \ref{sec:ind} the SED
fits are shown and the physical values extracted from them are
presented. The results are discussed in Section~\ref{sec:discuss}, and
conclusions are presented in Section~\ref{sec:conc}. Throughout this
paper we adopt the following values for the cosmological parameters:
$\rm H_0=70\,km\,s^{-1}\,Mpc^{-1}$, $\rm \Omega_M=0.3$ and $\rm
\Omega_\Lambda=0.7$.
\section{Data}\label{sec:data}
We use {\em Spitzer Space Telescope} observations and multiple data
collected from the literature, ranging from mid-infrared to optical
wavelengths, of the sample of radio galaxies presented by
\citet{2011MNRAS.411.1909F}. This sample consists of 27 radio sources
selected from the spectroscopically complete 3CRR
\citep{1983MNRAS.204..151L}, 6CE
\citep{1997MNRAS.291..593E,2001MNRAS.322..523R}, 6C*
\citep{2001MNRAS.326.1563J,2001MNRAS.326.1585J}, 7CRS
\citep{1999MNRAS.308.1096L,2003MNRAS.339..173W} and TOOT
\citep{HillRawlings2003,2010MNRAS.401.1709V} surveys to have narrow
lines and a redshift of $0.9\lesssim z\lesssim 1.1$. The narrow-line
selection was intended to ensure the exclusion of quasars from the
sample (though 3C343 has since been classified as a
quasar and 3C22 is classified as a weak quasar). \\
\subsection{Spitzer data}
The {\em Spitzer} data consist of observations with MIPS $24\,\mu \rm
m$ and IRAC $3.6$, $4.5$, $5.8$ and $8.0\, \mu \rm m$ bands. The MIPS
and IRAC observations took place in August 2006 and August 2007, as
described by \citet{2011MNRAS.411.1909F}. The photometric measurements for the IRAC
data were performed using a 7~arcsec diameter aperture with aperture
corrections of 1.112, 1.113, 1.125 and 1.218 for IRAC channels 1, 2, 3 and
4 respectively, unless there was a nearby source in which case we used
a 3.5~arcsec diameter aperture with appropriate aperture
corrections. For MIPS we extracted the photometry in a 13~arcsec
diameter aperture with aperture correction of 1.167. A summary of
these data is shown in Table\,\ref{table:spitzer}.
\begin{table*}
\caption{{\em Spitzer} photometry for our sample of radio galaxies.\textbf{Column 1} gives the name of the object;
\textbf{Columns 2, 4, 6, 8, 10} give the flux density at
$3.6$, $4.5$, $5.8$, $8.0$ and $24\,\mu \rm m$, respectively;
\textbf{Columns 3, 5, 7, 9, 11} give the respective flux density errors.} \centering
\begin{tabular}{l c c c c c c c c c c}
\noalign{\smallskip}
\hline\hline
\noalign{\smallskip}
Object & $S_{3.6\,\mu \rm m}$ & Err$_{S_{3.6\,\mu \rm m}}$ & $S_{4.5\,\mu \rm m}$ & Err$_{S_{4.5\,\mu \rm m}}$ & $S_{5.8\,\mu \rm m}$ & Err$_{S_{5.8\,\mu \rm m}}$ & $S_{8.0\,\mu \rm m}$ & Err$_{S_{8.0\,\mu \rm m}}$ & $S_{24\,\mu \rm m}$ & Err$_{S_{24\,\mu \rm m}}$ \\[0.5ex]
& $\mu \rm Jy$ & $\mu \rm Jy$ & $\mu \rm Jy$ & $\mu \rm Jy$ & $\mu \rm Jy$ & $\mu \rm Jy$ & $\mu \rm Jy$ & $\mu \rm Jy$ & $\mu \rm Jy$ & $\mu \rm Jy$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
3C280 & 277.26 & 38.69 & 524.69 & 87.58 & 1004.40 & 126.74 & 2200.45 &
204.88 & 9229.46 & 258.67\\
3C268.1 & 35.45 & 13.94 & 78.20 & 33.91 & 108.84 & 43.09 & 209.22 & 64.37 & 940.96 & 123.57\\
3C356 & 108.00 & 11.00 & 110.00 & 11.00 & 122.00 & 14.00 & 434.00 & 47.00 & 4060.00 & 192.00\\
3C184 & 129.88 & 44.40 & 145.48 & 46.09 & 242.00 & 55.00 & 288.00 & 76.00 & 742.00 & 184.00\\
3C175.1 & 119.60 & 25.54 & 109.55 & 40.21 & 92.03 & 42.56 & 124.70 & 53.55 & 836.90 & 166.56\\
3C22 & 1399.41 & 86.86 & 2431.72 & 188.43 & 3647.24 & 239.86 & 5852.97 & 333.61 & 13744.13 & 310.40\\
3C289 & 38.91 & 14.62 & 44.00 & 25.53 & 26.81 & 25.20 & 37.15 & 26.55 & 3650.16 & 168.21\\
3C343 & 151.35 & 28.62 & 142.75 & 45.74 & 211.85 & 60.29 & 683.25 & 114.58 & 7294.40 & 230.19\\
6CE1256+3648 & 62.50 & 18.44 & 61.01 & 29.99 & 55.72 & 32.24 & 133.86 & 52.70 & 1351.94 & 124.20\\
6CE1217+3645 & 118.37 & 25.34 & 120.80 & 42.11 & 125.26 & 51.99 & 150.98 & 53.52 & 313.64 & 104.94\\
6CE1017+3712 & 26.52 & 12.15 & 31.88 & 21.83 & 43.08 & 30.52 & 129.35 & 53.11 & 1136.21 & 131.80\\
6CE0943+3958 & 72.68 & 19.90 & 87.66 & 35.95 & 141.66 & 49.31 & 244.95 & 70.77 & 1996.79 & 153.62\\
6CE1257+3633 & 81.58 & 21.05 & 70.04 & 32.11 & 36.46 & 27.63 & 110.35 & 48.18 & 850.67 & 108.30\\
6CE1019+3924 & 159.69 & 29.41 & 117.47 & 41.57 & 75.93 & 38.14 & 23.26 & 28.80 & 334.56 & 181.18\\
6CE1011+3632 & 74.20 & 20.11 & 79.07 & 34.14 & 89.07 & 40.89 & 200.03 & 64.69 & 1326.94 & 150.39\\
6CE1129+3710 & 81.67 & 21.08 & 62.57 & 30.39 & 40.67 & 35.62 & 72.84 & 41.39 & 856.71 & 119.07\\
6C*0128+394 & 85.50 & 21.58 & 92.74 & 36.95 & 65.62 & 36.02 & 74.97 & 42.26 & 134.85 & 86.14\\
6CE1212+3805 & 74.06 & 20.10 & 61.81 & 30.24 & 30.77 & 31.14 & 55.14 & 37.46 & 257.42 & 92.98\\
6C*0133+486 & 53.85 & 17.20 & 64.35 & 30.86 & 50.99 & 41.43 & 75.81 & 37.92 & 82.15 & 75.92\\
5C6.24 & 111.17 & 24.57 & 102.82 & 38.89 & 86.34 & 41.21 & 118.65 & 50.69 & 728.22 & 119.30\\
5C7.23 & 54.77 & 17.32 & 48.11 & 26.75 & 67.04 & 35.75 & 41.12 & 34.62 & 623.52 & 160.69\\
5C7.82 & 114.37 & 24.92 & 83.15 & 34.99 & 45.29 & 31.63 & 55.92 & 37.71 & 422.33 & 154.47\\
5C7.242 & 123.35 & 25.87 & 104.96 & 39.33 & 69.57 & 36.58 & 41.92 & 34.64 & 1994.72 & 161.08\\
5C7.17 & 30.88 & 13.10 & 22.64 & 18.50 & 56.68 & 33.88 & 5.14 & 22.51 & 1371.30 & 176.80\\
TOOT1267 & 114.48 & 24.91 & 95.55 & 37.47 & 88.35 & 39.65 & 141.41 & 53.90 & 1079.06 & 121.87\\
TOOT1140 & 50.48 & 16.63 & 64.88 & 30.92 & 47.60 & 31.18 & 28.71 & 28.25 & 136.24 & 80.29\\
TOOT1066 & 49.37 & 16.50 & 35.92 & 23.09 & -3.85 & 14.10 & 16.21 & 22.64 & 298.16 & 86.86\\
\noalign{\smallskip}
\hline
\end{tabular}
\label{table:spitzer}
\end{table*}
\subsection{Literature data}
To increase the number of data points in the near-infrared and optical
bands, we used as many photometric values as possible from the
literature. The values available were extracted with a range of
different apertures and, where possible, we chose apertures of $8\, \rm
arcsec$, or larger, which approximately encompass the whole of the
emission from the galaxies in our sample.
\subsubsection{Aperture corrections}
For consistency, when magnitude values were only available with
smaller apertures we applied a correction to transform them into
magnitudes with a $8\, \rm arcsec$ aperture or larger. Given that the
majority of the photometric data in the literature is presented in
apertures of 8 or 9 arcsec diameter, and that this is a good
approximation to the total flux of the source, we preferred to convert
all the smaller aperture magnitudes to their equivalent value at 8 or
9 arcsec.
All the magnitudes measured using smaller apertures that we converted
to 8 or 9~arcsec were measured either in 4 or 5~arcsec apertures, thus
the corrections are small \citep[e.g. ][]{2009MNRAS.394.2197B}. To
find relations between 4 and 5~arcsec and 8 or 9~arcsec, we gathered a
large sample of galaxies ($\sim 40$ galaxies), from the papers
referenced in Tables~1 to 4 of the Appendix, which had photometry
available for several different apertures. With this sample, we
computed linear relations between the magnitudes at 4~arcsec and
9~arcsec and between 5~arcsec and 8~arcsec. We then used these linear
fits to determine all magnitudes in 8 or 9~arcsec apertures.
We also applied a Galactic extinction correction to all the magnitude
values that had not been corrected based on the maps of
\citet{1998ApJ...500..525S}.
\subsubsection{Line emission correction}
Following \cite{2001MNRAS.326.1585J} we also subtracted the flux of
the dominant emission lines $\rm H\alpha$, [O{\sc{II}}], [Ne{\sc{V}}],
Mg{\sc{II}}, [Ne{\sc{IV}}] and C{\sc{II}}], from J, R, F702W, F606W,
V, B, and g bands, as these lines can contribute a significant
percentage to the total flux of the band where they lie. Where the
line fluxes were not available we used the [O{\sc II}] fluxes from
\citet{2011MNRAS.411.1909F} and determined the remaining emission
lines using the line ratios given by \citet{1993ARA&A..31..639M} and
\citet{1997MNRAS.292..758B}. When the flux of H$\alpha$ is not
known, we used the average ratio estimated by
\citet{1993ARA&A..31..639M} for high redshift radio galaxies
($z\lesssim2$) from the 3CR survey.
To correct for emission line contamination we then determined the
location of each emission line for the redshift of the given source
and account for the shape of the filter at that wavelength. Given the
uncertainties affecting the aperture, emission-line contamination and
Galactic extinction corrections, we assume an additional $10$ per cent error
in the flux density for each magnitude value that had any of these
corrections applied. All these corrections are detailed in
Tables~\ref{table:JHK}, \ref{table:HST} \ref{table:UBVRI}, and
\ref{table:ugriz} of the Appendix.
\section{SED fitting}\label{sec:sed_method}
The photometric data span a wavelength range from $0.36\,\mu\rm m$ to
$24\,\mu \rm m$. At optical wavelengths, the emission from radio
galaxies - obscured AGN - is dominated by the stellar emission of the
host galaxy. At mid-infrared wavelengths, dust in the `torus' region
that re-emits the radiation from the central AGN is the main source of
emission, at least for those sources with a relatively powerful and
radiatively efficient AGN. This can be thought of as a composite of
the central emission passing through a screen of dust. We follow the
approach of \citet{2007MNRAS.379L...6M,2009ApJ...706..184M} and model
the radio galaxies with a galaxy template that dominates the emission
in the optical part of the spectrum and a quasar template,
extinguished by a dust extinction law, to fit the emission in the
infrared spectral region.
\subsection{Galaxy template}
To replicate the host galaxy emission, we consider a Bruzual and
Charlot stellar synthesis model \citep[][hereafter
BC03]{2003MNRAS.344.1000B}, as well as the Maraston et
al. ‘fuel-consumption’ stellar population synthesis model
\citep[][hereafter M05]{2005MNRAS.362..799M}. For both of these we
assume a Salpeter initial mass function and solar metallicity.
Since radio galaxies are almost exclusively hosted by elliptical
galaxies or recent merger remnants
\citep[e.g.][]{2003MNRAS.340.1095D,2010ApJ...713...66F}, we expect
them to have a strong short episode of star formation in the beginning
of their formation and for the star formation rate to quickly drop
subsequently. Therefore, we consider that a single simple stellar
population suffices to reproduce the host galaxy's star formation
history, and a synthesis of SSPs is not required. To select the
stellar age of our template we note that our objects all have
$z\sim1$, when, according to the adopted cosmology, the Universe was
$\sim5.7\,\rm Gyr$ old. We, thus, constrain the range of possible ages
with an upper limit for the stellar population age of $6\,\rm Gyr$. As
for a lower limit, previous studies have shown that for
passively-evolving early-type galaxies in cluster environments, the
bulk of stars form at $z\gtrsim3$ and in low-density environments at
$z\gtrsim1.5-2$
\citep[e.g.][]{1992MNRAS.254..601B,1998MNRAS.299.1193B,2006ARA&A..44..141R}. This
yields a stellar age of $\sim5.6\,\rm Gyr$ for cluster environments
and $\sim1.5-2.5\,\rm Gyr$ for field galaxies. Even though our sample
consists of powerful radio galaxies, which tend to inhabit
high-density environments as suggested by various lines of
observations
\citep[e.g.][]{HillLilly1991,Wold2001,Hardcastle2004,Kauffmann2008,Falder2010},
including X-ray observations that show X-ray cavities identified with
clusters of galaxies to be spatially coincident with non-thermal radio
emission from radio-loud AGN (e.g. \citealt{2002MNRAS.331..369F},
\citealt{2004ApJ...607..800B}), we adopt a conservative lower limit
for the stellar age of $\sim0.5\,\rm Gyr$. We thus use templates with
$0.5\,\rm Gyr$, $1\,\rm Gyr$, $2\,\rm Gyr$, $3\,\rm Gyr$, $4\,\rm
Gyr$, $5\,\rm Gyr$, and $6\,\rm Gyr$ of age for the SED fitting of the
galaxies in our sample.
\subsection{Quasar and dust template}\label{sec:torus}
To model the infrared emission of the radio galaxies, we need to
account for the dusty torus absorbing the radiation from the central
source and re-emitting it at longer wavelengths. This can be done, as
an approximation, by assuming that a radio galaxy is equivalent to a
quasar with a layer of obscuring dust in front of it, where this layer
can have diverse column densities depending on how obscured the galaxy
is \citep[e.g.][]{2007MNRAS.379L...6M,2009ApJ...706..184M}. Indeed,
\citet{2008ApJ...688..122H} find that they can reproduce a diversity
of rest-frame $1.6-10\,\rm \mu m$ SEDs of radio galaxies by combining
various amounts of extinction of AGN light with host galaxy
starlight. This representation is consistent with the
orientation-dependent unified scheme and it is intended as a
simplification only, as a more careful approximation should fully
account for radiative transfer effects in a dusty medium \citep[e.g][]{2008ApJ...685..147N}. This
approach would, however, imply a large number of free parameters with
too few data points to constrain them, resulting in a high degeneracy,
making it difficult to infer physically meaningful quantities. Thus, we
build, as a simplification, a composite model of a quasar and a screen
of dust.
As a quasar template, we use the radio-quiet quasar SED by
\citet{1994ApJS...95....1E}, which was constructed based on the mean
energy distribution of a sample of 29 radio-quiet quasars. We
normalise this template by forcing the bolometric luminosity to be 1.
For this, we used the median value of the B-band luminosity, $L_{\rm
B}$, which, according to Table 17 of \citet{1994ApJS...95....1E},
should be \textbf{$L_{\rm bol}=10.7 L_{\rm B}$.}
More recent works such as \cite{2013ApJ...777..164M} have shown a
composite model of a jet, a truncated thin accretion disk and an ADAF
to be a good fit for low-luminosity AGNs, and therefore possibly
better suited for LEGs. \cite{2013ApJ...777..164M} however report an
insufficient IR-emission produced by the truncated thin disk in order
to match the observations. Given these issues, we opt to use the model
by \citet{1994ApJS...95....1E} for all the objects in the sample for
consistency.
For a dust template, we used the extinction laws derived by
\citet{1992ApJ...395..130P}, which replicate how the extinction caused
by the different types of dust varies with wavelength. These templates are described by the following summation:
\begin{equation}\label{eq:pei}
\frac{A_{\lambda}}{A_{\rm B}}(\lambda)=\sum\limits_{i=1}^6\frac{a_{i}}{(\lambda/\lambda_{i})^{n_{i}}+(\lambda/\lambda_{i})^{-n_{i}}+b_{i}} ~~~~~~(Pei, 1992),
\end{equation}
where $a_{i}$, $b_{i}$ and $n_{i}$ vary for each term and for each
dust type (see Table~4 of \citealt{1992ApJ...395..130P} for a full
description). The six terms involved represent the background,
far-ultraviolet and far-infrared extinctions and the $\rm 2175\,\AA$,
$9.7\,\mu$m and $18\mu$m features. For reddened quasars, SMC dust
has been found to be appropriate
\citep[e.g.][]{2004AJ....128.1112H,2005ApJ...627L.101W}, since dust in
the host galaxy of high-redshift galaxies tends to have lower
metallicity and thus can be better approximated by SMC dust
type. However, for more obscured quasars, and in particular for the
majority of the objects in our sample, we find that SMC dust does not
provide as good a fit as MW dust. This could be explained by the fact
that the dust intersecting the line of sight of the central emission
in obscured galaxies comes from the inner region due to the galaxy's
edge-on orientation, and the central regions of galaxies are usually
more metal rich, as there is in general a metallicity gradient in
galaxies
\citep[e.g.][]{1993MNRAS.262..650D,1994MNRAS.270..523C,2009ApJ...691L.138S}. Therefore,
we adopted a MW dust type for all the galaxies in our sample.
\subsection{Method of fitting}
We fit our model to the photometric data points by applying an
extinction curve to the intrinsic quasar light template and then
adding it to the galaxy light model:
\begin{equation}
\rm S_{\nu,model}=S_{\nu,QSOmodel}+S_{\nu,GALmodel},
\end{equation}
with
\begin{equation}
\rm S_{\rm \nu,QSOmodel} = S_{\rm \nu, QSOtemplate}\times 10^{-\frac{A_{\rm V}\times A_{\lambda}(\lambda)}{2.5}},
\end{equation}
where $S_{\rm \nu, QSOtemplate}$ is the quasar light template and
$S_{\rm \nu,QSOmodel}$ is the quasar light model already affected by
extinction.
To convert the extinction law $\frac{A_{\lambda}}{A_{\rm
B}}(\lambda)$ to magnitudes ($A_{\lambda}$) we multiply equation
\eqref{eq:pei} by the term $1/R_{V}+1$, where $R_{V}$ is the ratio
of total-to-selective extinction defined by the equation:
\begin{equation}\label{eq:Rv}
R_{\rm V}=\frac{A_{\rm V}}{E_{\rm B-V}},
\end{equation}
where $E_{\rm B-V}=A_{\rm B}-A_{\rm V}$ is the colour excess.
We multiply each template
by a coefficient which we allow to vary along a range of physically
motivated values.
The extinction law is multiplied by the visual extinction coefficient
$A_{\rm V}$ spanning values of $0 < A_{\rm V} < 400$. The resolution
we use to run through these values varies from galaxy to galaxy
depending on how precise the fit is, and we make it finer where the
dispersion of the fit is smaller, as detailed below.
After multiplying the extinction law in flux units by the transmitted
quasar light, we force it to be as close as possible to the $24\,\rm
\mu m$ flux density value. This means we do not allow the bolometric
luminosity of the model to vary freely. For each given $A_{V}$, it is
fixed by the observed flux density at $24\,\rm \mu m$. The reasoning
for this is to constrain the fit, as we found that without this step,
the higher degeneracy of the fits often provided poorer matches to the
data in terms of reduced-$\chi^{2}$.
The galaxy template, $\rm S_{\nu,GALtemplate}$, is multiplied by a mass
normalisation factor, $M_{\rm gal}$, which we allow to vary between
$10^{10}$ and $10^{13}\,\rm M_{\sun}$, an interval that comprises the
typical values of stellar mass content in early-type galaxies and
certainly the masses of the majority of radio galaxies studied to date
\citep[e.g.][]{2007ApJS..171..353S}:
\begin{equation}
\rm S_{\nu,GALmodel}=\frac{S_{\nu,GALtemplate}}{M_{\odot}}\times M_{gal}
\end{equation}
We then perform a grid search in $A_{\rm V}$ and $M_{\rm
gal}$ to obtain the best fit.
Given that the data from the literature were gathered from several
different instruments, we opted to use a single set of filter
transmission curves: for $J$, $H$ and $K_{s}$ bands we used the
filters from the Visible and Infrared Survey Telescope for Astronomy
\citep[VISTA; see e.g. ][]{2013MNRAS.428.1281J} For the HST F606W,
F702W and F814W passbands, we used the respective HST filter response
profiles; for $U$, $B$, $V$, $R$ and $I$ bands we used the filters
used on the auxiliary-port camera (ACAM) mounted on the William
Herschel Telescope (WHT); finally, for the $u$, $g$, $r$, $i$ and $z$
bands, given that most data at these wavelengths were extracted from
the SDSS public release, we used the SDSS bandpass filters. For the
small number of photometric data points that were not observed with
the exact same filters we use, the errors are smaller or of the same
order of magnitude as the uncertainties on the data.
Using the filter responses, we evaluate the flux density that
the model produces for each band using the following:
\begin{equation}\label{eq:filter}
\nu S_{\nu, \rm model}=\frac{\int_{\lambda_i}^{\lambda_f}\nu S_{\nu, \rm unfiltered}\times T(\lambda) d\lambda}{\int_{\lambda_i}^{\lambda_f}T(\lambda) d\lambda},
\end{equation}
where $S_{\nu, \rm unfiltered}$ is the flux density of the model
before it has been convolved with the filter response; $S_{\nu, \rm
model}$ is the flux density of the model after the filter response
has been taken into account; $\lambda_i$ and $\lambda_f$ are the
wavelength where the filter response starts and ends respectively;
and $T(\lambda)$ is the filter transmission curve.
To determine the flux $\nu S_{\nu,\rm data}$ of the data points, we
multiply the flux density values by the mean frequency of each band. The
mean frequency is determined by computing the effective wavelength of
each band, given by:
\begin{equation}
\lambda_{\rm eff}=\frac{\int_{\lambda i}^{\lambda f} \lambda T(\lambda) d\lambda}{\int_{\lambda i}^{\lambda f} T(\lambda) d\lambda},
\end{equation}
and converting it to $\nu_{\rm eff}$. \\
We determine the $\chi^{2}$ distribution of each model using,
\begin{equation}
\chi^2(A_{\rm V},M_{\rm gal})=\sum_{n}\left(\frac{\nu S_{\nu, \rm model}(A_{\rm V},M_{\rm gal})-\nu S_{\nu, \rm data}}{\sigma}\right)^{2},
\end{equation}
where $n$ is the number of flux/magnitude data points available, and
$\sigma$ is the error associated with each data point.
We repeat this process for all the different stellar models we are
considering (ages 0.5, 1, 2, 3, 4, 5, and 6\,Gyr for both BC03 and M05
models) and choose the model that has the lowest $\chi^{2}$ amongst
these.\\
Sources that are detected below a 2$\sigma$ level in the imaging
data are plotted as upper limits at a 2$\sigma$ level. We use their
measured photometric flux for the SED fitting and place their error
bars between zero and the detection limit for the $\chi^{2}$
evaluation. However, photometric bands for which both the observed
flux density and the model are lower than the flux density limit make
no contribution to the $\chi^{2}$.
\section{SED fits}\label{sec:sed_fits}
\begin{figure*}
\includegraphics[width=0.99\columnwidth]{Figures/CACF01_age8805.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF02_age8805.eps} \\
\includegraphics[width=0.99\columnwidth]{Figures/CACF03_age1.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF04_age885.eps} \\
\includegraphics[width=0.99\columnwidth]{Figures/CACF05_age2.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF06_age6.eps} \\
\includegraphics[width=0.99\columnwidth]{Figures/CACF07_age884.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF08_age0.eps} \\
\end{figure*}
\begin{figure*}
\includegraphics[width=0.99\columnwidth]{Figures/CACF09_age886.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF10_age881.eps} \\
\includegraphics[width=0.99\columnwidth]{Figures/CACF11_age1.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF12_age1.eps} \\
\includegraphics[width=0.99\columnwidth]{Figures/CACF13_age2.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF14_age8805.eps} \\
\includegraphics[width=0.99\columnwidth]{Figures/CACF15_age881.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF16_age881.eps} \\
\end{figure*}
\begin{figure*}
\includegraphics[width=0.99\columnwidth]{Figures/CACF17_age0.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF18_age0.eps} \\
\includegraphics[width=0.99\columnwidth]{Figures/CACF19_age8805.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF20_age3.eps} \\
\includegraphics[width=0.99\columnwidth]{Figures/CACF21_age881.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF22_age885.eps} \\
\includegraphics[width=0.99\columnwidth]{Figures/CACF23_age881.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF24_age0.eps} \\
\end{figure*}
\begin{figure*}
\includegraphics[width=0.99\columnwidth]{Figures/CACF25_age6.eps}
\includegraphics[width=0.99\columnwidth]{Figures/CACF26_age886.eps} \\
\includegraphics[width=0.99\columnwidth]{Figures/CACF27_age882.eps}\\
\caption{SED fitting of the radio galaxies in our sample. The dashed
lines represent both the quasar template affected by dust and the
stellar template, whose sum is the model that best fits the
data. The solid orange line represents the model that best fits the
data. The points represent the data and their error bars. Symbols
are as follows: red circles are for $12\rm \mu m$ MIPS data; violet
circles are for $\rm 8\mu m$, $\rm 5.8\mu m$,$\rm 4.5\mu m$, and
$\rm 3.6\mu m$ IRAC data; purple circles are for K, H, and J band
data; blue squares are for Hubble F814W, F702W, and F606W band data;
blue triangles are for I, R, V, B, and U band data; blue diamonds
are for z, i, r, g, and u band data. The undetections are
represented as downward arrows. }\label{fig:sed}
\end{figure*}
In Table~\ref{table:Av_Mgal} we present the best-fit values for
$A_{\rm V}$, $M_{\rm gal}$ and the stellar age of the SSP template, and in Figure~\ref{fig:sed} we show these best-fit SEDs
overlaid on the data.
\begin{table*}
\caption{Best-fit parameters for our sample of radio galaxies. We note
that the fits to 3C268.1 are unreliable due to a lack of data. \textbf{Column~1} gives the object name; \textbf{Column~2}
gives the value of $A_{\rm V}$ of the best fit model for the SED of
the object; \textbf{Column~3} gives the $1\sigma$ error associated
with the $A_{\rm V}$ value; \textbf{Column~4} gives the value of
$\log_{10}(M_{\rm gal})$ of the best fit model for the SED of the
object; \textbf{Column~5} gives the $1\sigma$ error associated with
the $\log_{10}(M_{\rm gal})$ value; \textbf{Column~6} gives the
stellar age of the stellar synthesis population model that best
fitted the data (BC03 - \citealt{2003MNRAS.344.1000B}; M05 -
\citealt{2005MNRAS.362..799M}) \textbf{Column~7} gives the reduced
$\chi^{2}$ of the best-fit model.} \centering
\begin{tabular}{l l l c c l l}
\hline\hline
\multicolumn{1}{c}{Object} & \multicolumn{1}{c}{$A_{\rm V}$} & \multicolumn{1}{c}{$\sigma_{A_{\rm V}}$} & $\log_{10}(M_{\rm gal}/\rm M_{\sun})$ & $\sigma_{\log_{10}(M_{\rm gal})}$ & SSP model & $\chi_{\rm red}^{2}$ \\[0.5ex]
\multicolumn{1}{c}{(1)} & \multicolumn{1}{c}{(2)} & \multicolumn{1}{c}{(3)} & (4) & (5) & (6) & (7) \\[0.5ex]
\hline
3C280 & 19 & 2 & 11.20 & 0.04 & 0.5\,Gyr (M05) & 2.006\\
3C268.1 & 17 & 6 & 10.33 &0.42 & 0.5\,Gyr (M05) & 0.190\\
3C356 & 127 & 16 & 11.60 &0.04 & 1\,Gyr (BC03) & 3.114\\
3C184 & 1.5 & 0.4 & 11.82 &0.11 & 5\,Gyr (M05) & 2.655\\
3C175.1 & 123 & 117 & 11.58 &0.04 & 2\,Gyr (BC03) & 3.725\\
3C22 & 7 & 1.3 & 12 &0.04 & 6\,Gyr (BC03) & 2.669\\
3C289 & 209 & 48 & 11.95 &0.03 & 4\,Gyr (M05) & 4.943\\
3C343 & 158 & 26 & 11.63 &0.05 & 0.5\,Gyr (BC03) & 2.683\\
6C1256+36 & 180 & 66 & 11.89 &0.03 & 6\,Gyr (M05) & 1.837\\
6C1217+36 & 1 & 0.1 & 11.30 &0.03 & 1\,Gyr (M05) & 1.387\\
6C1017+37 & 141 & 63 & 11.06 &0.03 & 1\,Gyr (BC03) & 1.227\\
6C0943+39 & 99 & 46 & 11.16 &0.03 & 1\,Gyr (BC03) & 1.192\\
6C1257+36 & 112 & 16 & 11.57 &0.03 & 2\,Gyr (BC03) & 1.560\\
6C1019+39 & 110 & 125 & 11.20 &0.02 & 0.5\,Gyr (M05) & 5.025\\
6C1011+36 & 65 & 37 & 11.13 &0.03 & 1\,Gyr (M05) & 3.664\\
6C1129+37 & 164 & 17 & 11.16 &0.03 & 1\,Gyr (M05) & 0.858\\
6C*0128+39 & 4 & 85 & 11.10 &0.08 & 0.5\,Gyr (BC03) & Inf.\\
6C1212+38 & 39 & 34 & 11.20 &0.04 & 0.5\,Gyr (BC03) & 0.161\\
6C*0133+48 & 3 & 42 & 10.70 &0.08 & 0.5\,Gyr (M05) & 0.542\\
5C6.24 & 84.0 & 22 & 11.76 &0.04 & 3\,Gyr (BC03) & 0.351\\
5C7.23 & $>5$ & & 11 & 0.05 & 1\,Gyr (M05) & Inf.\\
5C7.82 & 146 & 134 & 11.83 &0.04 & 5\,Gyr (M05) & 0.012\\
5C7.242 & 355 & 14 & 11.26 &0.04 & 1\,Gyr (M05) & 0.060\\
5C7.17 & 353 & 23 & 10.78 &0.17 & 0.5\,Gyr (BC03) & 0.179\\
TOOT00\_1267 & 103 & 15 & 11.93 &0.03 & 6\,Gyr (BC03) & 4.744\\
TOOT00\_1140 & 20 & 5 & 11.77 &0.03 & 6\,Gyr (M05) & 8.256\\
TOOT00\_1066 & 232 & 46 & 11.29 &0.03 & 2\,Gyr (M05) & 0.570\\
\hline
\label{table:Av_Mgal}
\end{tabular}
\end{table*}
We note that for several objects (3C289, 6C1019+39, 5C7.17,
6C1011+36, TOOT00\_1267, and TOOT00\_1066) there is an excess of emission at
the highest frequencies of the optical region of the SED that rises
above the galaxy template. This is thought to be either due to
scattered light from the central AGN or blue light from young
stars \citep[see e.g. ][ and references therein]{2010MNRAS.406.1841H}. This does not adversely impact our fit for the stellar masses
however, as the bulk of the stellar mass is traced by the
high-mass-to-light-ratio red stars.
To determine the error associated with each normalisation parameter of
the best fitting model we marginalised over the free parameters,
$A_{\rm V}$ and $M_{\rm gal}$,
choosing a uniform (i.e. constant) prior of $0\leq A_{\rm
V}\leq 400$ and $10.0\leq \log_{10}(M_{\rm gal})\leq 13.0$.
We marginalised the free parameters by integrating the
likelihood over the full range of one parameter, for instance $M_{\rm
gal}$, in order to get the dependence of the total likelihood on the
other parameter, $A_{\rm V}$.
This then allows us to determine the uncertainty from the likelihood function in
relation to $M_{\rm gal}$, by determining the minimal width of the
likelihood function that contains $68$ per cent of all the likelihood.
We then repeat the same
procedure to estimate $\sigma_{A_{\rm V}}$.
\section{Notes on individual objects}\label{sec:ind}
In this section we provide notes on the individual sources and their
best fit model.
\noindent
\textbf{3C280} is well fitted with a model with an $A_{\rm V}$ of $\sim19$ and a
stellar mass of $M_{\rm gal}\simeq 1.6\times10^{11}\,\rm M_{\sun}$.\\[-2ex]
\noindent
\textbf{3C268.1} lacks optical or near-IR data in the literature. The
fit of the stellar mass of the galaxy is thus not well-constrained,
and the best-fit value of $M_{\rm gal}\simeq2.1\times 10^{10}\,\rm
M_{\sun}$ is not reliable due to the lack of data points constraining
the fit. The SED fit of this galaxy yields a value for visual dust
extinction of $A_{\rm V}\simeq17$.\\[-2ex]
\noindent
\textbf{3C356} has two bright infrared galaxies at $z=1.079$
coincident with two radio cores, one northern and one southern,
approximately 5\,arcsec apart from each other. The identification of
the nucleus is a matter of debate in the literature
\citep{1990MNRAS.243P...1E,1992ApJ...385...61R,1997MNRAS.292..758B,2000MNRAS.311....1B}. We
assume the northern component to be the identification of the host
galaxy and radio jet. We found a model with a high $A_{\rm V}$ of
$\sim127$ and a stellar mass of $M_{\rm gal}\simeq 4\times
10^{11}\,\rm M_{\sun}$ to be a good fit for the SED of this
galaxy.\\[-2ex]
\noindent
\textbf{3C184} is well fit by a model with a stellar mass of $M_{\rm
gal}\simeq 6.6\times 10^{11}\,\rm M_{\sun}$.\\[-2ex]
\noindent
\textbf{3C175.1} has a best fit model with a high $A_{\rm V}$ of $\sim123$,
with a rather high uncertainty of $\sigma_{A_{\rm V}}=116.8$, and a
stellar mass of $M_{\rm gal}\simeq 3.8\times10^{11}\,\rm M_{\sun}$.\\[-2ex]
\noindent
\textbf{3C22} is a reddened quasar
\citep{1995MNRAS.274..428R,1999MNRAS.306..828S}. As expected for this
type of object, it is well fit by a model with a low dust extinction,
$A_{\rm V}\sim 7$, and the SED is mainly dominated by the quasar light
component. We find that SMC-type dust fits the SED of 3C22 as well
as the MW type. We chose the MW type template for consistency.\\[-2ex]
\noindent
\textbf{3C289} has a best fit model with a very high $A_{\rm V}$ of
$\sim209$, and a stellar mass of $M_{\rm gal}\simeq 8.9\times10^{11}\,\rm
M_{\sun}$.\\[-2ex]
\noindent
\textbf{3C343:} The best fit model for the SED of 3C343 was found with
a visual extinction of $A_{\rm V}\sim 158$ and a stellar mass of $M_{\rm
gal}\simeq 4.3\times10^{11}\,\rm M_{\sun}$. \\[-2ex]
\noindent
\textbf{6C1256+36} has a best fit model with a high $A_{\rm V}$ of
$\sim180$, and a stellar mass of $M_{\rm gal}=7.8\times10^{11}\,\rm
M_{\sun}$.\\[-2ex]
\noindent
\textbf{6C1217+36} is well fit by a model with a very low dust
extinction, $A_{\rm V}\approx1$, and the SED is mainly dominated by the
galaxy light component, with a mass of $M_{\rm
gal}\simeq 2\times10^{11}\,\rm M_{\sun}$.\\[-2ex]
\noindent
\textbf{6C1017+37} has a best fit model with a high $A_{\rm V}$ of
$\sim141$, and a stellar mass of $M_{\rm gal}\simeq 4\times10^{11}\,\rm
M_{\sun}$. The $5.8$ and $4.5\,\rm\mu m$ fluxes are not detected at the
$2\sigma$ level and represent only limits.\\[-2ex]
\noindent
\textbf{6C0943+39} has a best fit model with a visual extinction of
$A_{\rm V}\sim 99$, and a stellar mass of $M_{\rm
gal}\simeq 1.4\times10^{11}\,\rm M_{\sun}$. \\[-2ex]
\noindent
\textbf{6C1257+36} has a best fit model with a visual extinction of
$A_{\rm V}\sim112$, and a stellar mass of $M_{\rm
gal}=3.7\times10^{11}\,\rm M_{\sun}$. \\[-2ex]
\noindent
\textbf{6C1019+39} is not detected at $8.0\,\rm\mu m$ or 24~$\mu$m at the $2\sigma$
level, which makes the fit of the quasar template more problematic. A
visual extinction of $\sim110$ is our preferred model, with a very high associated uncertainty of
$\sim125$. Despite this, the galaxy is well fit by a template with a stellar mass
of $1.6\times10^{11}\,\rm M_{\sun}$. \\[-2ex]
\noindent
\textbf{6C1011+36} has a best fit model with $A_{\rm V}\sim65$, and
a stellar mass of $M_{\rm gal}\simeq1.3\times10^{10}\,\rm M_{\sun}$. The
high flux of the optical data points compared to the galaxy
model suggests that these shorter wavelengths might be contaminated by scattered light as
discussed in Section~\ref{sec:sed_fits}.\\[-2ex]
\noindent
\textbf{6C1129+37} is not detected at $8.0$ and $5.8\,\rm\mu m$ at a
$2\sigma$ level. The best fit model is found for $A_{\rm V}\sim 164$
and a stellar mass of $M_{\rm gal}=1.4\times10^{11}\,\rm
M_{\sun}$.\\[-2ex]
\noindent
\textbf{6C*0128+39} is not detected at $24$, $8.0$ or $5.8\,\rm\mu m$
at the $2\sigma$ level, which makes the fitting of the quasar light
difficult. We find the best fit quasar model to converge to
an $A_{\rm V}=4$, with a likelihood that decreases up to values of
$A_{\rm V}\sim 50$ but then maintains a constant likelihood
for higher values of $A_{\rm V}$. The estimated dispersion for this
parameter is thus extremely high $\sigma_{A_{\rm V}}\sim85$ and the
$\chi^{2}$ for the combined model diverges to infinity. Even though
there are not many data points constraining the stellar emission in
the SED, the best fit model appears to provide a good fit to
the data, with a stellar mass of $M_{\rm
gal}\simeq1.3\times10^{11}\,\rm M_{\sun}$, and thus we do not
exclude this object from our analysis.\\[-2ex]
\noindent
\textbf{6C1212+38} is not detected at $8.0$ and $5.8\,\rm\mu m$ at a
$2\sigma$ level. The best fit model is found for a high $A_{\rm
V}\sim39$, with a standard deviation of the same order of magnitude,
$\sigma_{A_{\rm V}}\sim 34$, and for a stellar mass of $M_{\rm
gal}\simeq1.6\times10^{11}\,\rm M_{\sun}$.\\[-2ex]
\noindent
\textbf{6C*0133+48} is not detected at $24$, $8.0$ or $5.8\,\rm\mu m$
at the $2\sigma$ level, and thus the quasar light component fit is not
reliable, with a likelihood very similar to that for 6C*0128+39. The best fit model was found for $A_{\rm
V}=3.0$ with a much higher standard deviation of $\sigma_{A_{\rm
V}}\sim42$. However, the stellar dominated part of the SED appears
well fitted by a stellar mass of $M_{\rm gal}\simeq 5 \times10^{10}\,\rm
M_{\sun}$.\\[-2ex]
\noindent
\textbf{5C6.24} has a best fit model with $A_{\rm V}\sim 84$, and a
stellar mass of $M_{\rm gal}\simeq5.7\times10^{11}\,\rm M_{\sun}$.\\[-2ex]
\noindent
\textbf{5C7.23} is not detected at $8.0$, $5.8$ and $4.5\,\rm\mu m$ at
a $2\sigma$ level, and the quasar light component fit does not
converge to a specific value of $A_{\rm V}$. We determine a lower
limit for the visual dust extinction of $A_{\rm V}>5$. The stellar
mass is constrained mainly with the $3.6\,\rm\mu m$ and K-band data
points and a best value of $M_{\rm gal}\simeq1.1\times10^{11}\,\rm
M_{\sun}$ is found.\\[-2ex]
\noindent
\textbf{5C7.82} is not detected at $8.0$ and $5.8\,\rm\mu m$ at a
$2\sigma$ level, and the quasar light component fit has a rather high
error associated. The best fit parameter values is $A_{\rm
V}=146$. The stellar dominated part of the SED looks well
constrained by a model with stellar mass $M_{\rm
gal}=6.8\times10^{11}\,\rm M_{\sun}$.\\[-2ex]
\noindent
\textbf{5C7.242} is not detected at $8.0$ and $5.8\,\rm\mu m$ at the
$2\sigma$ level, and thus the quasar light component fit is not
entirely reliable. The best fit model was found for an extremely high value
of visual extinction, $A_{\rm V}=355$, and a stellar mass of $M_{\rm
gal}\simeq1.8\times10^{11}\,\rm M_{\sun}$.\\[-2ex]
\noindent
\textbf{5C7.17} is not detected at $8.0$, $5.8$ and $4.5\,\rm\mu m$ at
the $2\sigma$ level, and the quasar light component fit is quite
problematic. The best model was found with an extremely high value of
visual extinction, $A_{\rm V}=353$. The stellar mass value that
produces the best model is $M_{\rm gal}\simeq6\times10^{10}\,\rm
M_{\sun}$, with a set of data that is also rather hard to fit, and
thus with a large associated error. The high fluxes of the
optical data points compared to the stellar population model suggests
that these might be contaminated by scattered light as discussed in
Section~\ref{sec:sed_fits} or have ongoing star formation.\\[-2ex]
\noindent
\textbf{TOOT00\_1267} has a best fit model with a visual extinction
value of $A_{\rm V}=103$, and a stellar mass of $M_{\rm
gal}\simeq8.5\times10^{11}\,\rm M_{\sun}$. The high fluxes of the optical
data points compared to the galaxy light model suggest that
these might be contaminated by scattered light or ongoing star formation.\\[-2ex]
\noindent
\textbf{TOOT00\_1140} is not detected at $8.0$, $5.8$ and $4.5\,\rm\mu
m$ at the $2\sigma$ level. The best fit model has $A_{\rm V}\sim
20$. The stellar component is well constrained by a model with a
stellar mass of $M_{\rm gal}\simeq 5.9\times10^{11}\,\rm M_{\sun}$
.\\[-2ex]
\noindent
\textbf{TOOT00\_1066:} is not detected at $8.0$ and $5.8\,\rm\mu m$ at
the $2\sigma$ level. The best fit model is found with a high $A_{\rm
V}\sim 232$, and for a stellar mass of $M_{\rm
gal}\simeq1.9\times10^{11}\,\rm M_{\sun}$.\\
\section{Discussion}\label{sec:discuss}
The SED fitting provides a good estimation for important physical
properties of the radio galaxies such as bolometric luminosity,
extinction properties and stellar mass. The stellar mass, in particular,
grants, for elliptical galaxies, an accurate estimation of its
supermassive black hole mass. Together with the bolometric luminosity,
the Eddington weighted accretion rates of the sample can be
inferred. In this section, first the extracted physical properties of
our sample of radio galaxies are discussed and then the classification
of each galaxy into HEG or LEG is used to investigate the
HEG/LEG dichotomy at $z\sim1$.
\subsection{Physical properties of the sample}
\subsubsection{Visual extinction}\label{sec:discussion_av}
Our visual extinction estimations are complicated by the fact that at
$z\sim1$, the $24\,\mu$m data point lies on the edge of the $10\,\mu$m
silicate feature and thus more data points, or spectroscopic data,
would be desirable to better constrain the values of $A_{\rm
V}$. Nonetheless, previous work in the literature, such as
\citet{2007ApJ...660..117C}, agree reasonably well with the values
found. In their study of galaxies and quasars from the 3CRR survey at
$0.4\leq z\leq 1.2$, \citet{2007ApJ...660..117C} fit IRS and MIPS
Spitzer data, and other measurements from the literature, using models
with a synchrotron and a dust component. They consider two variations
of the dust component, one with a screen of cooler dust, and another
with a mixture of warm dust emitting in the MIR and cooler dust. The
visual extinction of their sample ranges from $0<A_{\rm V}<40$ for the
screen dust component model, and $0<A_{\rm V}<150$ for the mixed dust
component model. In particular, for the common objects in our sample
and their sample, we find that the values of $A_{\rm V}$ for 3C268.1,
3C280 and 3C22 agree well with those of \citet{2007ApJ...660..117C}
for a screen dust component. For 3C343 our model is best fit with a
dust component with an $A_{\rm V}=158$, whereas
\citet{2007ApJ...660..117C} find an $A_{\rm V}\sim31$ for a mixed dust
component, and $A_{\rm V}\sim22$ for a screen dust component, both of
them much lower than our value. However, we note that this object does
exhibit a steeply rising slope towards mid-infrared wavelengths in the
{\em Spitzer}-IRS data, similar to what we find with our
photometry. Our estimated high AGN extinction is inconsistent with the
classification of 3C343 as a quasar by
\citet{2007ApJ...660..117C}. However, it is possible that this
difference results from an $A_{\rm V}$ overestimation of our model due
to the proximity of the $24\,\mu m$ data point to the $10\,\mu m$
feature.
\noindent
\subsubsection{Stellar mass and Black hole mass}
Pioneering studies like those of \citet{1995ARA&A..33..581K},
\citet{1997AJ....114.1771F} and \citet{1998AJ....115.2285M}
established that the hot stellar component of galaxies - i.e. the
bulge - is proportional to their black hole mass. This relation became
known as the $M_{\rm BH}$-$M_{\rm bulge}$ relation, and
\citet{1998AJ....115.2285M} found the black hole-to-bulge mass ratio
to be approximately 0.006. More recent studies of this relation
\citep[e.g.][]{2004ApJ...604L..89H} find that the median black hole
mass is $0.14$ per cent of the bulge mass for nearby galaxies, $M_{\rm
BH}=0.0014 M_{\rm bulge}$. We note that including the intrinsic
scatter of the $M_{\rm BH}-M_{\rm bulge}$ relation does not make any
difference to our results, as the uncertainties in our stellar mass
estimates dominate the error budget.
We can use the stellar mass of the galaxy, $M_{\rm gal}$, provided by
the SED fitting, along with the $M_{\rm BH}$-$M_{\rm bulge}$ relation
to estimate the black hole mass, $M_{\rm BH}$. We assume that there is
no significant evolution of the $M_{\rm BH}$-$M_{\rm bulge}$ relation
at $z\sim1$ from the local $M_{\rm BH}/M_{\rm bulge}$. In fact,
\cite{2006MNRAS.368.1395M} shows that for low redshift ($z \lesssim
1$) radio-loud AGN the black hole to spheroid mass ratio lies within
the uncertainties of that found in the local
Universe. \cite{2010A&A...522L...3S} also find that obscured AGNs at
$z \sim 1\--2$ are fully consistent with the local $M_{\rm BH}$-$M_{\rm
bulge}$ relation. We thus use the local Universe relation $M_{\rm
BH}\sim 0.0014 M_{\rm bulge}$. The values found for $M_{\rm gal}$
and $M_{\rm BH}$ are presented in Table~\ref{table:results}.
\begin{table*}
\caption{\textbf{Column~1} gives the object name; \textbf{Column~2}
gives the black hole masses; \textbf{Column~3} gives the
bolometric luminosity; \textbf{Column~4} gives the
accretion rate; \textbf{Column~5} gives the Eddington ratio;
\textbf{Column 6} gives the optical classification of the object
according to the literature. 'HEG' stands for high-excitation radio
galaxy, 'LEG' stands for low-excitation radio galaxy, RQ stands for
reddened quasar; \textbf{Column 7} gives the reference for the
optical classification in Column~6 (for the 3CRR, 6CE and 6C*
objects) or the reference for the optical spectra (for the 7CRS and
TOOT00 objects). References are as follows: Grimes -
www-astro.physics.ox.ac.uk/$\sim$sr/grimes; J01 -
\citealt{2001MNRAS.326.1585J}; JR97 - \citealt{1997MNRAS.286..241J};
REL01 - \citealt{2001MNRAS.322..523R}; W03 -
\citealt{2003MNRAS.339..173W}; V10 - \citealt{2010MNRAS.401.1709V}}
\centering
\begin{tabular}{l c c c c c c}
\hline\hline
\multicolumn{1}{c}{Object} & $\log_{10}(M_{\rm BH}/\rm M_{\sun})$ & $\log_{10}(L_{\rm bol} /\rm W)$ & $\log_{10}(\dot{M}/\rm~M_{\sun}/yr)$ & $\lambda$ & Opt. Class & Ref \\[0.5ex]
\multicolumn{1}{c}{(1)} & \multicolumn{1}{c}{(2)} & (3) & (4) & (5) & (6) & (7)\\[1ex]
\hline
3C280 & 8.346$\pm$0.425 & 39.707$\pm$0.012 & 0.953$\pm$0.012 & 1.765$^{+0.882}_{-0.710}$ & HEG & JR97 \\
3C268.1 & 7.476$\pm$4.204 & 38.689$\pm$0.057 & -0.065$\pm$0.057 & 1.257$^{+104.822}_{-1.245}$ & HEG & JR97 \\
3C356 & 8.746$\pm$0.425 & 39.435$\pm$0.021 & 0.681$\pm$0.021 & 0.375$^{+0.178}_{-0.155}$ & HEG & JR97 \\
3C184 & 8.966$\pm$1.062 & 38.608$\pm$0.110 & -0.146$\pm$0.110 & 0.034$^{+0.065}_{-0.024}$ & HEG & JR97 \\
3C175.1 & 8.726$\pm$0.425 & 38.578$\pm$0.088 & -0.176$\pm$0.088 & 0.055$^{+0.027}_{-0.022}$ & HEG & JR97 \\
3C22 & 9.366$\pm$0.425 & 39.813$\pm$0.010 & 1.059$\pm$0.010 & 0.215$^{+0.103}_{-0.088}$ & RQ & JR97 \\
3C289 & 9.096$\pm$0.255 & 39.271$\pm$0.020 & 0.517$\pm$0.020 & 0.115$^{+0.026}_{-0.034}$ & HEG & JR97 \\
3C343 & 8.776$\pm$0.467 & 39.594$\pm$0.014 & 0.840$\pm$0.014 & 0.506$^{+0.286}_{-0.219}$ & HEG & Grimes \\
6C1256+36 & 9.036$\pm$0.297 & 39.005$\pm$0.040 & 0.251$\pm$0.040 & 0.072$^{+0.020}_{-0.024}$ & HEG? & REL01 \\
6C1217+36 & 8.446$\pm$0.297 & 38.331$\pm$0.151 & -0.423$\pm$0.151 & 0.059$^{+0.022}_{-0.020}$ & HEG? & REL01 \\
6C1017+37 & 8.206$\pm$0.340 & 38.855$\pm$0.051 & 0.101$\pm$0.051 & 0.343$^{+0.127}_{-0.118}$ & HEG & REL01 \\
6C0943+39 & 8.306$\pm$0.297 & 39.082$\pm$0.033 & 0.328$\pm$0.033 & 0.459$^{+0.123}_{-0.154}$ & LEG? & REL01 \\
6C1257+36 & 8.718$\pm$0.263 & 38.678$\pm$0.056 & -0.076$\pm$0.056 & 0.070$^{+0.016}_{-0.021}$ & HEG & REL01 \\
6C1019+39 & 8.346$\pm$0.212 & 38.182$\pm$0.263 & -0.572$\pm$0.263 & 0.053$^{+0.017}_{-0.019}$ & LEG? & REL01 \\
6C1011+36 & 8.276$\pm$0.297 & 38.911$\pm$0.049 & 0.157$\pm$0.049 & 0.332$^{+0.096}_{-0.108}$ & HEG & REL01 \\
6C1129+37 & 8.306$\pm$0.340 & 38.740$\pm$0.061 & -0.014$\pm$0.061 & 0.209$^{+0.072}_{-0.074}$ & HEG? & REL01 \\
6C*0128+39 & 8.246$\pm$0.849 & 37.795$\pm$0.328 & -0.959$\pm$0.328 & 0.027$^{+0.040}_{-0.018}$ & HEG? & J01 \\
6C1212+38 & 8.346$\pm$0.425 & 38.097$\pm$0.164 & -0.657$\pm$0.164 & 0.043$^{+0.023}_{-0.019}$ & LEG & REL01 \\
6C*0133+48 & 7.846$\pm$0.849 & 37.690$\pm$0.702 & -1.064$\pm$0.702 & 0.054$^{+0.108}_{-0.038}$ & LEG? & J01 \\
5C6.24 & 8.906$\pm$0.425 & 38.682$\pm$0.072 & -0.072$\pm$0.072 & 0.046$^{+0.023}_{-0.019}$ & HEG & W03 \\
5C7.23 & 8.196$\pm$0.500 & 38.640$\pm$0.115 & -0.114$\pm$0.115 & 0.214$^{+0.134}_{-0.098}$ & HEG & W03 \\
5C7.82 & 8.976$\pm$0.382 & 38.278$\pm$0.167 & -0.476$\pm$0.167 & 0.015$^{+0.007}_{-0.006}$ & LEG? & W03 \\
5C7.242 & 8.406$\pm$0.382 & 39.036$\pm$0.035 & 0.282$\pm$0.035 & 0.328$^{+0.144}_{-0.122}$ & HEG? & W03 \\
5C7.17 & 7.926$\pm$1.699 & 38.811$\pm$0.056 & 0.057$\pm$0.056 & 0.590$^{+2.798}_{-0.503}$ & HEG & W03 \\
TOOT00\_1267 & 9.076$\pm$0.255 & 38.743$\pm$0.049 & -0.011$\pm$0.049 & 0.036$^{+0.007}_{-0.011}$ & HEG & V10 \\
TOOT00\_1140 & 8.916$\pm$0.255 & 37.779$\pm$0.294 & -0.975$\pm$0.294 & 0.006$^{+0.002}_{-0.002}$ & LEG & V10 \\
TOOT00\_1066 & 8.436$\pm$0.340 & 38.136$\pm$0.130 & -0.618$\pm$0.130 & 0.039$^{+0.016}_{-0.014}$ & LEG? & V10 \\
\hline
\label{table:results}
\end{tabular}
\end{table*}
The SED models of the observed photometry bands provide a very good
fit for the stellar mass of our objects. In fact, in similar studies
to this, \citet{2007ApJS..171..353S} found that, for a sample of 70
radio galaxies at $1<z<5.2$, their broad-band photometry implied
stellar masses in the range $10^{11}-10^{12}\,\rm M_{\odot}$, with a
mean mass of $\sim 10^{11.55}\,\rm M_{\odot}$ up to redshifts of
$z=3$. The radio galaxies in our sample have an average stellar mass
of $10^{11.59}\,\rm M_{\odot}$ (excluding 3C268.1, for which the
stellar mass estimation is poorly constrained), consistent with the
study of \citet{2007ApJS..171..353S}.
\citet{2006MNRAS.368.1395M} estimated the black hole mass of powerful
radio quasars from the 3CRR survey with redshifts between
$0<z<2$ using the virial mass estimator for
Mg{\sc ii} emission line \citep{2002MNRAS.337..109M,2004MNRAS.352.1390M},
and estimated black hole masses in the range $10^{8.3}\,\rm
M_{\odot}-10^{10.1}\,\rm M_{\odot}$. The radio galaxies in our sample at
$z\sim1$ have black hole masses in the range $10^{7.8}\,\rm
M_{\odot}-10^{9.4}\,\rm M_{\odot}$ if we exclude 3C268.1. These are
consistent with the range of values found in the literature for
similar objects. This reinforces the view that powerful radio sources
reside in the most massive galaxies with the most massive black holes.\\
\subsubsection{Accretion rate}\label{sec:accretion}
The bolometric radiative power of an AGN, $L_{\rm bol}$, is proportional to
the accretion rate of the black hole, $\dot{M}$, and to the fraction
of accreted mass that is radiated, i.e. the radiative efficiency,
$\epsilon$, through the expression:
\begin{equation}\label{eq:lbol}
L_{\rm bol}={\epsilon\dot{M}c^{2}}.
\end{equation}
Assuming that $\epsilon$ takes the fiducial value of $0.1$
\citep[e.g.][]{2004MNRAS.351..169M,2004MNRAS.354.1020S,2009ApJ...692..964M},
we can determine the accretion rate of our sources using their
estimated bolometric luminosity, $L_{\rm bol}$. We calculate the values of $L_{\rm bol}$ from the rest-frame 12$\mu$m
luminosity, as in \citet{2011MNRAS.411.1909F}, using a bolometric
correction of 8.5 \citep[e.g.][fig.12]{2006ApJS..166..470R},
i.e. $L_{\rm bol}=8.5\lambda L_{12\mu\rm m}$. Our estimated values for
the accretion rate are shown
in Table~\ref{table:results}.\\
As reviewed by \citet[sec.3.1]{1997iagn.book.....P}, according to the
most widely accepted model, radiatively efficient AGNs are powered by
gravitational infall of material onto a supermassive black hole, with
this material achieving high temperatures in the dissipative accretion
disk (e.g. \citealt{1964ApJ...140..796S}). For the material in the
galaxy to be in hydrostatic equilibrium, the inward gravitational
force needs to be balanced by the outwards radiation pressure. The
Eddington luminosity, $L_{\rm Edd}$, is the maximum luminosity that a
body needs to be radiating to remain in hydrostatic equilibrium in the
case of spherical accretion, assuming a pure ionized hydrogen
plasma. This energy is a function of the mass of the system and is
given by $L_{\rm Edd}=1.3\times 10^{31} \left( \frac{M_{\rm
BH}}{M_{\odot}} \right) \rm \,W$. The Eddington ratio, $\lambda$
is therefore simply,
\begin{equation}\label{eq:edd}
\lambda\equiv \frac{L_{\rm bol}}{L_{\rm Edd}}.
\end{equation}
This gives an estimate of the actual accretion rate of the
AGN, compared to the maximal Eddington accretion rate.
Having the black hole mass and the accretion rate, we use
equations~\ref{eq:lbol} and \ref{eq:edd} to estimate the Eddington ratio of the sources in our sample. The values obtained are shown in
Table~\ref{table:results}. The accretion rate properties of our sample
are detailed in the following section.
\subsection{The HEG/LEG dichotomy}\label{sec:heg_leg}
Table~\ref{table:results} shows how the radio galaxies in our sample
are distributed in terms of their optical/near-IR spectral
classification. This classification was either taken from the
literature, where available (see references in
Table~\ref{table:results}), or determined by inspecting the optical
spectra of the objects. We label the reddened quasar 3C22
\citep{1995MNRAS.274..428R,1999MNRAS.306..828S}, and distinguish it as
a different class (RQ).
\subsubsection{Relation between $L_{\rm 151MHz}$ and $\lambda$}\label{sec:l151_eddr}
\begin{figure}
\begin{center}
\includegraphics[width=0.99\columnwidth]{Figures/L151_eddr.eps}
\end{center}
\caption[Relation between $L_{\rm 151MHz}$ and $\lambda$.]{Relation
between low frequency radio luminosity and Eddington
ratio. Symbols are: circles for the 3CRR survey objects, diamonds
for the 6CE and the 6C* surveys, triangles for the 7CRS, and
squares for the TOOT00 survey objects. Objects coloured in red are
classified as LEGs, objects in dark blue are HEGs, and those in light
blue are reddened quasars. `?' denotes objects whose
classification is uncertain. The two outliers to the general trend
are identified with their names. Both of them have an uncertain
classification.}
\label{fig:l151_eddr}
\end{figure}
First we discuss the relationship between the jet activity and the
accretion rate. In
Figure~\ref{fig:l151_eddr} we show the distribution of the Eddington
ratio versus $\rm 151\,MHz$ radio luminosity, where objects of
distinct optical classifications are displayed in different colours.
The full sample shows a modest positive correlation between $\rm 151\,MHz$
radio luminosity and Eddington ratio. The Spearman coefficient for
this correlation is $\rho=0.42$ and the standard
deviation in the Spearman Rank correlation is $\sigma_{\rho}=0.03$.
Apart from objects 6C0943+39 and 6C*0128+39, which have the most
uncertain classification into HEG/LEGs, there is a clear trend for the
most luminous radio galaxies, with the highest Eddington ratios to be
HEGs, or reddened quasars, and, conversely, for LEGs to have lower
$\lambda$ and $L_{\rm 151MHz}$ values. The complete HEG population in
our sample have a mean Eddington ratio of $\lambda\approx 0.33$
whereas LEGs have a lower mean of $\lambda\approx 0.09$. Excluding the
two outliers that have uncertain classification, 6C0943+39 and
6C*0128+39, though, the mean ratio for HEGs is $\lambda\approx 0.35$
and LEGs is $\lambda\approx 0.03$. We used a Kolmogorov-Smirnov (K-S)
test to evaluate whether the HEGs have a statistically different
distribution in their accretion rate compared to LEGs, and found, for
the full sample, $D=0.66$ with an associated probability of $p=0.01$,
rejecting the null hypothesis that the two populations are drawn from
the same underlying distribution at a 99 per cent level. A
Mann-Whitney test also yields a 97 per cent probability that the null
hypothesis that the two populations HEGs and LEGs come from the same
distribution is not supported by the data. The transition from LEG to
HEG appears to occur around $\lambda\sim 0.04$, which is in excellent
agreement with the theoretical expectations of where accretion rate
becomes radiatively inefficient
\citep[e.g.][]{1982Natur.295...17R,1995ApJ...452..710N,1997ApJ...489..865E}.
The distribution of HEGs and LEGs with respect to radio luminosity
seems indistinguishable up to 151\,MHz luminosities of
$\sim 3\times 10^{27} \rm \,W\,Hz^{-1}\,sr^{-1}$. For higher luminosities
HEGs fully dominate the sample. Even though the literature generally
states a bias for FRI radio galaxies
to be predominantly LEGs, and FRIIs a mix of both HEGs and LEGs
\citep[e.g.][]{1979MNRAS.188..111H,1994ASPC...54..201L}, there have
been recent studies showing that HEGs and LEGs span a similar range of
radio luminosities \citep[e.g.][]{2012MNRAS.421.1569B} in the local
Universe. If the radio properties of HEGs and LEGs are indeed
indistinguishable, and the HEG and LEG distinction is due to the
accretion process, this means that the
mechanism that generates the jets in both HEGs and LEGs is not
solely related to the accretion rate, and other physical processes
must also play an important role, such as black hole spin.
\subsubsection{Relation between $M_{\rm BH}$ and $\dot{M}$}
From equations~\eqref{eq:lbol} and \eqref{eq:edd}, the black hole mass
and accretion rate can be related through:
\begin{equation}
\dot{M}=\frac{\lambda}{\epsilon c^{2}} 1.3\times 10^{31} \left(
\frac{M_{\rm BH}}{M_{\odot}} \right) \rm \,W.
\end{equation}
If we assume the radiative efficiency to be constant, then the slope
of the relationship between black hole mass and accretion rate should
provide information on the Eddington ratio. The Eddington ratio is
thus proportional to the radiative efficiency times $\dot{M}/M_{\rm
BH}$. The distribution of the inferred accretion rate and bolometric
luminosity versus black hole mass is shown in
Figure~\ref{fig:Mbh_Mdot}.
\begin{figure}
\begin{center}
\includegraphics[width=0.99\columnwidth]{Figures/Mbh_Mdot_LEGs.eps}
\caption[HEGs and LEGs $M_{\rm BH}$ and $\dot{M}$
distribution.]{Black hole mass and accretion rate distribution for
HEGs and LEGs. Symbols are as in Figure~\ref{fig:l151_eddr}. The
right axis shows $L_{\rm bol}$ estimated from $\dot{M}$ using
equation \eqref{eq:lbol}. Object 3C268.1 is identified as it has
an unreliable mass estimation (see Section~\ref{sec:ind}).}
\label{fig:Mbh_Mdot}
\end{center}
\end{figure}
We observe a general trend for more massive black holes to have higher
accretion rates. The sources in our sample of galaxies have Eddington
ratios of $0.005 < \lambda < 1.7$ with only 3C280 measured to be
accreting above Eddington, albeit with a large uncertainty.
The distribution seen in Eddington ratios could either be due to the
accretion rate, or to the black hole mass. Figure~\ref{fig:Mbh_Mdot}
shows that accretion onto the black hole is the dominant factor for
the separation between the two classes, given the large overlap in
black hole mass. Therefore, acknowledging HEGs and LEGs as being
powered by different modes of accretion - i.e. radiatively efficient
and radiatively inefficient, respectively - the rate at which matter
is being accreted onto the supermassive black hole is determinant for
the accretion mode. Furthermore, the HEGs in our sample have similar
accretion rates to quasars at $z\sim 1$ selected from the SDSS survey
which have $\log_{10}(\lambda)\sim -0.5$ \citep{2004MNRAS.352.1390M},
compared to a mean of $\log_{10}(\lambda) = -0.47^{+0.009}_{-0.03}$ for the HEG radio
galaxies in our sample. This is consistent with unified models in
which radio-loud quasars and HEGs are similar objects viewed
at different orientations \citep[e.g.][]{1989ApJ...336..606B}.
\subsubsection{Host galaxies of HEGs and LEGs}
\begin{figure}
\begin{center}
\includegraphics[width=0.99\columnwidth]{Figures/L151_mgal_LEGs.eps}
\end{center}
\caption[Host galaxy properties]{Distribution of stellar mass for
the two populations HEGs and LEGs. Symbols are as in
Figure~\ref{fig:l151_eddr}. Object 3C268.1 is identified as it has
an unreliable mass estimation (see Section~\ref{sec:ind}).}
\label{fig:l151_mgal}
\end{figure}
For a large sample of radio galaxies in the local Universe,
\citet{2012MNRAS.421.1569B} find evidence that HEGs are hosted by less
massive galaxies relative to LEGs. At $z\sim 1$, we do not observe any
clear difference between the host galaxy masses of HEGs and LEGs in
our much smaller sample (see Figure~\ref{fig:l151_mgal}). Given that
our sample was selected to include the most powerful radio sources, we
can expect it to be biased towards the largest masses.
\citet{2004MNRAS.351..347M} and \citet{2011MNRAS.410.1360H} have found
that for a similar sample of radio galaxies at $z\sim0.5$, HEGs
demonstrate a significant correlation between $M_{\rm BH}$ and $L_{\rm
151MHz}$. \citet{2004MNRAS.351..347M} also note that the TOOT
objects deviate from the $M_{\rm BH}$ and $L_{\rm 151MHz}$ correlation, and have higher mean galaxy masses than
other subsamples, such as 7CRS and 6CE, which probe higher radio
luminosities. With a larger redshift span, however,
\citet{2007ApJS..171..353S} find only a weak correlation between
stellar mass and radio luminosity.
In our sample we do not find a strong correlation between $M_{\rm
gal}$ and $L_{\rm 151MHz}$ for the full sample of HEGs. Even
dismissing 3C268.1, which has an unreliable mass estimation (see
Section~\ref{sec:ind}), the correlation between $M_{\rm gal}$ and
$L_{\rm 151MHz}$ has a Spearman rank correlation coefficient of
$\rho=0.39$ and a standard deviation of $\sigma_{\rho}=0.09$, not
highly significant.
\subsubsection{Accounting for the jet power in the accretion rate}
Most studies in the literature define the Eddington ratio as in
section \ref{sec:accretion}, however some studies consider a more
physical approach is to include the contribution of the jet mechanical
energy in the output of the accretion energy, i.e. the total energy
produced due to the accretion process should equal the sum of the
radiative luminosity and the jet mechanical luminosity
\citep[e.g.][]{2007MNRAS.376.1849H,2012MNRAS.421.1569B,2014MNRAS.440..269M}. Considering
the Eddington luminosity as a normalisation factor to compare systems
over a wide range of luminosities, the total Eddington ratio is in
fact an Eddington-scaled accretion rate. The radiative output alone
does not account for the total Eddington accretion, hence neglecting
the contribution of the jet kinetic energy would underestimate the
energy output from the accretion process. This may be particularly
important in LEGs, for which the radiative energy is much lower and
the jet power is an important component of the total energy budget.
\begin{figure}
\begin{center}
\includegraphics[width=0.99\columnwidth]{Figures/L151_eddr_wQjet.eps}
\end{center}
\caption[L151 vs Eddington ratio]{Relation between low frequency
radio luminosity and Eddington ratio, including the jet power
contribution in the Eddington ratio calculation as detailed in the
text. Symbols are as in Figure~\ref{fig:l151_eddr}.}
\label{fig:l151_eddr_wQjet}
\end{figure}
Figure~\ref{fig:l151_eddr_wQjet} shows the Eddington ratio
distribution with radio 151\,MHz luminosity, as in
Figure~\ref{fig:l151_eddr}, but including the contribution of $Q_{\rm
jet}$, i.e.:
\begin{equation}\label{eq:edd2}
\lambda_{\rm rad+mec}\equiv \frac{L_{\rm bol}+Q_{\rm jet}}{L_{\rm Edd}},
\end{equation}
where $\lambda_{\rm rad+mec}$ is the Eddington ratio accounting for
both the radiative energy and the jet mechanical energy. We estimate
the jet power using the relation $Q_{\rm jet}\simeq 3\times
10^{38}f^{3/2}( L_{\nu \rm 151MHz} / 10^{28} )^{6/7} \rm W $
\citep{1999MNRAS.309.1017W}, where $1\leq f\leq 20$ represents several
uncertainties associated with estimating $Q_{\rm jet}$ from $L_{\nu
151MHz}$. Following Fernandes et al (2011), we chose $f=10$ as this
is the expectation value of a flat prior in natural space.
The Spearman rank test, with a Spearman coefficient of $\rho \sim
0.59$ and $\sigma_{\rho}=0.001$, reflects a tighter correlation
between 151\,MHz radio luminosity and Eddington ratio with the
inclusion of $Q_{\rm jet}$ than when accounting solely for the $L_{\rm
bol}$.
The separation between HEGs and LEGs is not as clear as when we
consider the Eddington scaled accretion rate to be solely dictated by
the radiated luminosity $L_{\rm bol}$. Indeed, the K-S test now shows
a much lower $64\%$ probability that the two populations come from
distinct distributions ($p=0.36$ and $D=0.38$). Given that our
sample contains the strongest radio galaxies at $z\sim 1$, the jet power
is not negligible when compared to the radiative energy.
\begin{figure}
\begin{center}
\includegraphics[width=0.99\columnwidth]{Figures/Mbh_Mdot_LEGs_wQjet.eps}
\caption[HEGs and LEGs $M_{\rm BH}$ and $\dot{M}$
distribution.]{Black hole mass and accretion rate distribution for
HEGs and LEGs, including the jet power contribution in the
accretion rate calculation. Symbols are as in
Figure~\ref{fig:l151_eddr}. The right axis shows $L_{\rm bol}$
estimated from $\dot{M}$ using equation \eqref{eq:lbol}.}
\label{fig:Mbh_Mdot_wQjet}
\end{center}
\end{figure}
Figure~\ref{fig:Mbh_Mdot_wQjet} shows the distribution of accretion
rate with black hole mass, where the contribution of the jet
mechanical energy is taken into account in the calculation of the accretion
rate. The separation between HEGs and LEGs is not as clear as when
considering only the radiated energy for the calculation of the
accretion rate, however, it is still clear that HEGs have on average
higher accretion rates than LEGs. The inclusion of the contribution of
$Q_{\rm jet}$ has given rise to a transition range between
$0.1\lesssim \log_{10}(\dot{M}/\,M_{\odot}yr^{-1})\lesssim 0.8$, with
both HEGs and LEGs found at these accretion rates. Only LEGs are found
at lower accretion rates $\log_{10}(\dot{M}/\,M_{\odot}yr^{-1})<0.1$,
and only HEGs above $\log_{10}(\dot{M}/\,M_{\odot}yr^{-1})>0.8$.
As expected, an Eddington ratio that only takes into account the
radiated luminosity to balance the gravitational pull shows a more
clear separation between HEGs and LEGs. Due to the fact that the
sources in our sample span a wide range in radio luminosities, and
include some of the strongest radio sources at $z\sim 1$, it is
expected for the jet mechanical energy of these sources to
significantly contribute to the energy balance. This is particularly
true for LEGs, where the radiated emission is weaker.
Figure~\ref{fig:eddr_histo_wQjet} shows the Eddington ratio
distribution of HEGs and LEGs obtained with both methods of calculating
the Eddington ratio. Solid lines are for $\lambda=(L_{\rm bol}+Q_{\rm
jet})/L_{\rm Edd}$ and dashed lines for $\lambda=L_{\rm bol}/L_{\rm
Edd}$. The total accretion energy for both HEGs and LEGs is significantly
increased, however, the trend for HEGs to show higher Eddington ratios than
LEGs remains.
\begin{figure}
\begin{center}
\includegraphics[width=0.99\columnwidth]{Figures/Eddr_histo_wQjet.eps}
\caption[Eddington ratio histogram]{Eddington ratio distribution of
HEGs and LEGs. Solid lines are for the Eddington ratio estimated
using $\lambda=(L_{\rm bol}+Q_{\rm jet})/L_{\rm Edd}$; dashed
lines are for the Eddington ratio estimated using
$\lambda=L_{\rm bol}/L_{\rm Edd}$.}
\label{fig:eddr_histo_wQjet}
\end{center}
\end{figure}
\section{Conclusions}\label{sec:conc}
We have used {\em Spitze}r $24\mu\rm m$ MIPS and $3.6$, $4.5$, $5.8$,
and $8\mu\rm m$ IRAC data, as well as near-IR and optical data from
the literature, to constrain the SEDs of the sample of radio
galaxies previously studied by \citet{2011MNRAS.411.1909F}. We applied
a dust extinction law to a quasar template to approximate the mid-IR
region of the SED, and a template of an elliptical galaxy.
From these fits we are able to determine the black-hole masses through
$M_{\rm bulge}-M_{\rm BH}$ relation, the quasar bolometric luminosity,
the accretion rate and the Eddington ratio. Our analysis revealed
the following:
\begin{enumerate}
\item We find a significant correlation between the Eddington ratio and
the radio luminosity, with more luminous radio sources yielding
higher Eddington ratios, and thus higher accretion rates.
\item We find that HEGs tend to have higher Eddington ratios and radio
luminosities, whereas the LEGs in our sample predominantly gather
towards the lowest values of $\lambda$ and $L_{\rm 151MHz}$. We find
the HEG/LEG division to lie approximately at $\lambda\sim 0.04$,
which is in excellent agreement with theoretical predictions of
where the accretion rate becomes radiatively inefficient and also
with other studies
\citep{2012MNRAS.421.1569B,2014MNRAS.440..269M}. This result further
confirms the suspicion that the differences between HEGs and LEGs
arise due to the different modes of accretion, which result in
different accretion rates. Higher accretion rates are observed with
the radiatively efficient `cold mode' accretion, whereas lower
accretion rates can be explained with the radiatively inefficient
`hot mode' of accretion, such as ADAFs.
\item By including the contribution of the jet power to the Eddington
ratio instead of solely accounting for the radiated power, the HEG/LEG dichotomy becomes less clear. Indeed, the
contribution of the jet power is expected to have a significant
weight for LEGs, which have a weaker radiated power. Moreover, given
that our sample consists of some of the most powerful radio sources,
the jet power contribution appreciably increases the Eddington
ratios of HEGs as well. The trend for HEGs to have higher accretion
rates than LEGs remains.
\item We find that the more massive black holes have higher
accretion rates for the HEGs population. HEGs and LEGs are similarly
distributed in terms of black hole mass. This relation further
displays the dichotomy between HEGs and LEGs in accretion rate,
meaning that the dichotomy seen in Eddington ratio is due to the
accretion rate and not a dichotomy in black hole mass.
\item We do not find a strong correlation between the host galaxy
mass, or equivalently the black hole mass, and the radio luminosity
of the sample. Similarly to \citet{2007ApJS..171..353S}, there is
only a slight trend for the highest mass galaxies to host the most
powerful radio sources.
\item Recent studies in the local Universe find evidence that LEGs are
hosted by more massive galaxies than HEGs. However, at $z\sim 1$ we
do not observe any difference in terms of host galaxy mass
distribution for HEGs and LEGs, although we note that our sample is
much smaller than those used in the local Universe.
\item It is important to ascertain whether the radio properties of
HEGs and LEGs are indistinguishable or not. Our study hints that, at
least up to radio luminosities of $\sim 3\times 10^{27} \rm
\,W\,Hz^{-1}\,sr^{-1}$, HEGs and LEGs span the same range of radio
luminosities. If HEGs and LEGs' radio properties are indeed
indistinguishable, and the HEG and LEG distinction is due to the
accretion process, this means that the power being channeled into
the jets is most likely not solely dependent on the accretion rate and an
additional process must be influencing the jet
formation. Understanding the HEG/LEG dichotomy may thus bring
important new constraints on the relation between accretion process
and jet formation.
\end{enumerate}
\section*{Acknowledgments}
We thank the anonymous referee for providing constructive comments and
raising questions that have helped improving the contents of this
paper and their presentation. CACF is currently supported by CNPq
(through PCI-DA grant 302388/2013-3 associated with the PCI/MCT/ON
program) and gratefully acknowledges past financial support from the
Foundation for Science and Technology (FCT Portugal) through project
grant PTDC/FIS/100170/2008 and doctoral grant SFRH/BD/30486/2006. MJJ
acknowledges the continued support of the South African SKA project
and the NRF. A.M.-S. gratefully acknowledges a Post-Doctoral
Fellowship from the United Kingdom Science and Technology Facilities
Council, reference ST/G004420/1 as well as support from SEPnet. This
work is based (in part) on observations made with the Spitzer Space
Telescope, which is operated by the Jet Propulsion Laboratory,
California Institute of Technology under a contract with NASA. This
research has made use of the NASA/IPAC Extragalactic Database (NED)
which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics
and Space Administration.
|
1,108,101,566,717 | arxiv | \section{Introduction}
Currently, neural networks and deep learning have completely changed the traditional way of signal processing, image processing, image recognition, etc. Many of these successful cases have been applied to the medical field \cite{krishnan2016genome,gulshan2016development,jin2017deep,chen2017low,wang2018image,kang2018deep,yang2018low,zhang2018sparse}.
Also, the authors of a recent article \cite{wang2016perspective} described the vision of using machine learning to create new CT image reconstruction algorithms to improve conventional analysis and iterative methods. Now, more and more academic research hotspots for CT reconstruction are focused on DL (deep learning) methods, including sinogram domain methods \cite{lee2018deep,pelt2013fast}, image domain methods \cite{han2018framing,shan20183} and sinogram domain to image domain methods \cite{wurfl2018deep,zhu2018image}. In particular, the domain transformation method has great potential to remove noise and artifacts simultaneously during graphics reconstruction \cite{zhu2018image}.
However, there is currently no clear mathematical explanation for each part of the domain mapping network. In this article, we use visualization techniques to study the function of the fully connected layer used in the domain mapping network.
\section{Methodology}
\subsection{Prepare and train the simple network}
In order to explain what the fully connected layer of the network, like AUTOMAP \cite{zhu2018image} for CT reconstruction, learned, we built a simple network, as illustrated in Fig.~\ref{fig:simple_network}, with only one fully connected layer followed by five convolutional layers, each convolutional layer with 128 filters, except for the final layer that has only 1 filter. a $3\times3$ filter with a filter stride 1 is used for all convolutional layers. The activation function for all layers is tanh. The loss function was a simple squared loss between the network output and target image intensity values.
Here we only want to explain the role of the fully connected layer, so all used CT images size were downsampled to $64\times64$. A fan-beam CT imaging geometry was simulated for this simple network. The source to detector distance was 1500.00 mm, and the source to rotation center was 1000.00 mm. There were 128 detector elements, each had a dimension of 1.6 mm. To make the CT images fit in this simulated imaging geometry, we further assumed that all CT images have the same pixel dimension of $1.0 mm\times1.0 mm$. Forward projections, i.e., Radon projections, were collected from 90 views with 4.00 degree angular interval. Notice that this is only a simulated fan-beam CT imaging geometry to to explain the role of the fully connected layer. Also be aware that no noise is added in these simulations.
\begin{figure}[h]
\centering
\includegraphics[width=0.75\textwidth] {figure_simple_network.eps}
\caption{The architecture of the simple network. The network with only one fully connected layer followed by five convolutional layers, each convolutional layer with 128 filters, except for the final layer that has only 1 filter. The weight kernels for all convolutional layers are $3\times3$ with a filter stride 1. The three numbers in each box denotes the image column number, image row number and the channel number, respectively. The activation function for all layers is tanh.}
\label{fig:simple_network}
\end{figure}
As illustrated in Fig.~\ref{fig:simple_network}, the shape of the input image, which be fed to fully connected layer, needs to be reshaped from two dimension ($90\times128$) to one dimension($11520$). Also, the output shape of the fully connected layer(FC1), which be fed to CNNs, needs reverse change from one dimension(($4096$)) to two dimension($64\times64$) .
The network was trained by Adam algorithm \cite{kingma2014adam} with starting learning rate of $10^{-5}$. The learning rate was exponentially decayed by a factor of 0.96 after every 1000 steps. The mini-batch had a size of 60, and batch-shuffling was turned on to increase the randomness of the training data. The network was trained for 200 epochs on the Tensorflow deep learning framework using a single graphics processing unit (GPU, NVIDIA GeForce GTX 1080Ti) with 11 GB memory capacity.
The numerical experimental results of this simple neural network are shown in Fig.~\ref{fig:simp_net_rst}. Sinogram image is fed to this network and CT reconstruction image be generated from it. Because we only using this simple neural network to analyze what the fully connected layer has learned, so wo not do quantitative analysis. But the mid result of this network, which was reshaped for human-readable, was shown(FC1 out). As shown in Fig.~\ref{fig:simp_net_rst} FC1 out is somehow close to label.
\begin{figure}[h]
\centering
\includegraphics[width=0.55\textwidth]{figure_simp_net_rst.eps}
\caption{Numerical experimental results of simple neural network.}
\label{fig:simp_net_rst}
\end{figure}
\subsection{Change the shape of the fully connetced layer weights}
In order to observe the physical meaning of the weights learned by the network, we need to recombine the shape of the full connection layer weights. As illustrated in Fig.~\ref{fig:simple_network}, where $S_i$ denote the reshaped projection data. $V_j$ denote the output of fully connected layer. $W_{i,j}$ is refers to one weight of FC1 (fully connected layer), which denote from input end $i$ to output end $j$. There are $11520\times4096$ weights.
The reshape rules from two dimension ($90\times128$) to one dimension ($11520$) of input sinogram image $I$ can be defined by
\begin{equation}
S_i = I_{p,q}, \quad where\ i = (p-1) \times Q+ q , \quad 1\leq p \leq P;1\leq q \leq Q
\end{equation}
where $I_{p,q}$ represent the pixel value of the p-th row and q-th column of the input sinogram image, $P$ and $Q$ represent the total pixels along the row and column directions. And, physically, $P$ and $Q$ represent the the total acquisition angles and detector cells. Here $P=90$ and $Q=128$.
Because of the convolutional layers, which followed FC1, are end-to-end mapping network, So, the output data shape of FC1 layer is the same as the label image. Then the reshape rules of output of FC1 layer from one dimension ($4096$) to two dimension ($64\times64$) can be defined by
\begin{equation}
M_{c,t} = V_j, \quad where\ j=(c-1)\times T+ t, \quad 1\leq c \leq C;1\leq t \leq T
\end{equation}
where $M_{c,t}$ represent the value of the p-th row and q-th column of the input two dimension feature data for convolutional layers, $C$ and $T$ represent the total numbers along the row and column directions of the feature data. Here $C=64$ and $T=64$.
\begin{figure}[h]
\centering
\includegraphics[width=0.75\textwidth] {figure_fully_connected_layer.eps}
\caption{The reshape rules of fully connected layer weights}
\label{fig:fully_connected_layer}
\end{figure}
Following the rules for changing shapes described above, we can transform the shape of the fully connected layer weights to match the rules of back projection in CT reconstruction. Because in this simulated fan-beam CT imaging geometry the sinogram datas were collected from 90 views with 4.00 degree angular interval, the detector has 128 cells and the CT image be projected with two dimension of $64\times 64$. So we can define a cell matix $H$ with dimension of $K\times L$, as shown in Fig.~\ref{fig:fully_connected_layer} where $K$ denotes total columns of the CT image and $L$ denotes total collected views. Here $K=64$ and $L=90$. Each cell has the dimension of $A\times B$, where $A$ denotes total rows of the CT image and $B$ denotes total numbers of detector cells. Here $A=64$ and $B=128$.
We note that $H^{k,l}$ to represent each element of the cell matix $H$ and $H^{k,l}_{a,b}$ to represent each element of cell $H^{k,l}$. So, $H^{k,l}_{a,b}$ can represent each element of the fully connected layer weights:
\begin{equation}
\begin{aligned}
H^{k,l}_{a,b} = W_{i,j}, \quad where\ i &= (k-1)\times A+(a-1) \\
j &= (l-1)\times B+(b-1),\\
1\leq k \leq K; & 1\leq l \leq L; 1\leq a \leq A;1\leq b \leq B
\end{aligned}
\end{equation}
\section{Visualization of weights}
In this section, we only see what weights map (feature map) look like, not do quantitative analysis. So, the display windows of the images shown in this section are not the same but be adjusted comfortable to see feature shape.
First, let's visualize the weights of one given fixed detector unit for different pixel values of the final reconstructed CT image at different acquisition angles. In order to make it more obvious, we have selected the 64th detector unit ($H^{k,l}_{a,b},b=64$). Let ($H^{k,l}_{a,b},b=64;l=1;k=1;2,...,K;a=1,2,...,A$) be the first ($l=1$) back projection view of CT reconstruction weights map, ($H^{k,l}_{a,b},b=64;l=2;k=1;2,...,K;a=1,2,...,A$) be the second ($l=2$) ,...,($H^{k,l}_{a,b},b=64;l=90;k=1;2,...,K;a=1,2,...,A$) be the last ($l=90$) back projection view of CT reconstruction weights map. The displayed of those maps are show on the right image of Fig.~\ref{fig:visl_fix_unit_diff_deg}. The image on the left of Fig.~\ref{fig:visl_fix_unit_diff_deg} is weights map directly calculated by the back projection analytic algorithm at the same rule of above.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\textwidth]{figure_fix_unit_diff_deg.eps}
\caption{Weights map of the 64th detector unit for different pixel values of the final reconstructed CT image at different acquisition angles. Left weights map is calculated by analytic algorithm under the same rules as the right fully connected layer weights map. Both map images from top to bottom and from left to right are the 0th back projection view weights, the 2th, ..., the 90th back projection view weights.}
\label{fig:visl_fix_unit_diff_deg}
\end{figure}
Second, let's visualize the weights of different detector unit for different pixel values of the final reconstructed CT image at a fixed acquisition angle. Here we select the 12th view degree for visualization ($H^{k,l}_{a,b},l=12$), of course, we can also choose other degrees.
Let ($H^{k,l}_{a,b},b=1;l=12;k=1;2,...,K;a=1,2,...,A$) be the first ($b=1$) first detector unit weight map at the 12th back projection view of CT reconstruction, ($H^{k,l}_{a,b},b=2;l=12;k=1;2,...,K;a=1,2,...,A$) be the second ($b=2$) ,...,($H^{k,l}_{a,b},b=128;l=12;k=1;2,...,K;a=1,2,...,A$) be the last ($b=128$) detector unit weight map at the 12th back projection view of CT reconstruction. The displayed of those images are show on the right image of Fig.~\ref{fig:visl_fix_deg_diff_unit}. The image on the left of Fig.~\ref{fig:visl_fix_deg_diff_unit} is weights map directly calculated by the back projection analytic algorithm at the same rule of above.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\textwidth]{figure_fix_deg_diff_unit.eps}
\caption{Weights map of different detector unit for different pixel values of the final reconstructed CT image at a fixed acquisition angle,12th degree. Left weights map is calculated by analytic algorithm under the same rules as the right weights map. Both map images from top to bottom and from left to right are the 0th detector unit weights at the 12th back projection view of CT reconstruction, the 2th, ..., the 128th detector unit weights at the 12th back projection view of CT reconstruction.}
\label{fig:visl_fix_deg_diff_unit}
\end{figure}
From Fig.~\ref{fig:visl_fix_unit_diff_deg} and Fig.~\ref{fig:visl_fix_deg_diff_unit}, we can realize that, the feature map of fully connected layer of this trained network is the same as that of analytic algorithm, they are very sparse data. The fully connected layer implements mapping from sinogram domain to image domain and do some filtering. In Fig.~\ref{fig:simp_net_rst}, FC1 out image appears to be inversely numerical in comparison with the label image or the net output image. This phenomenon can also be seen from Fig.~\ref{fig:visl_fix_unit_diff_deg} and Fig.~\ref{fig:visl_fix_deg_diff_unit}. Because of the back propagation process of deep learning affects all weight parameters, include fully connected layer weights and convolutional layer weights, so, this phenomenon (inversely numerical and some filted) affected weights may be learned out at the fully connected layer .
After the above analysis, we can draw a conclusion that the main role of the fully connected layer is to implement the back projection function in sinogram domain to image domain mapping CT reconstruction networks.
\section{Discussion}
The main problem of using neural networks to realize back-projection is that it consumes huge memory resources. Just like this simple network, it can only be used to reconstruct CT images of $64\times64$ size, the number of weights in fully connected layer was 45M, if we use 4Bytes for one float type data, the consumption of memory resources is 180M Bytes. If we use the clinical CT image size of $512\times512$ at 360 collection views, 768 detector units, the number of weights in one fully connected layer will reach more than 69G. With today's computer graphics storage technology, it is impossible to achieve such a large amount of data storage in high-speed computing. Therefore, it is currently impossible to obtain practical applications.
By visualizing the weights of fully connected layers, it shows that back-projection can be learned through neural networks. This means that we can directly use analytic algorithms to implement back projection, so that the network can learn more complex problems, such as noise reduction and artifact suppression.
\section*{References}
|
1,108,101,566,718 | arxiv | \section{Introduction}\label{sec:intro}
Nash equilibrium \cite{nash1950} and correlated equilibrium \cite{aumann1974} are important solution concepts that have been
extensively studied in both traditional and computational game-theoretic contexts. Coarse correlated equilibrium
\cite{moulin1978} is a closely related concept that was proposed as a generalization of correlated equilibrium,
which can be more powerful in some settings such as potential games.
In this work we construct protocols for mutually distrusting players to implement any coarse correlated equilibrium (and therefore any correlated equilibrium)
of a strategic game without trusted mediation, via cryptographic cheap talk protocols.
Our approach draws upon cryptography in two ways: first, we introduce
an intermediate, ``cryptographically blinded'' game from which the players sample according to the desired equilibrium; and second,
this sampling is achieved using a secure multi-party computation protocol.
Our results address both the computational and perfect (information-theoretic) settings.
\paragraph{Correlated equilibrium}
Suppose a mediator samples an action profile $a$ from a known distribution $\alpha$, and gives as ``advice'' to
each player $i$ his action $a_i$ in $a$. The distribution $\alpha$ is a correlated equilibrium if, \emph{having seen his advice},
and believing that all other players will follow their advice, no player has incentive to unilaterally deviate from the advice profile.
\cite{aumann1974} showed that correlated equilibria can achieve higher expected payoffs than Nash equilibria.
\paragraph{Coarse correlated equilibrium}
Coarse correlated equilibria are a generalization of correlated equilibria which invokes a notion of commitment.
In the mediated scenario described above, $\alpha$ is a
coarse correlated equilibrium if no player has incentive not to ``promise'' or ``commit'' in advance
-- \emph{before seeing his advice} $a_i$ -- to play according to the advice,
as long as he believes that all other players will commit to do the same.
Note that if a player does not commit, then he will not see the advice at all, and must therefore play an independent strategy:
this is in contrast to correlated equilibria, where deviations may depend on the received advice.
\cite{moulin2013} showed that there is a class of potential games in which the Nash equilibrium payoffs
can be improved upon by coarse correlated equilibria but not by correlated equilibria
(e.g. the Cournot duopoly and public good provision games).
\begin{example}
Let us give a brief example to illustrate the gap between the two types of equilibria.
Suppose Alice plays a game $\Gamma$ where she has a ``safe strategy'' for which her payoff is always zero.
Let $\alpha$ be a distribution over action profiles of $\Gamma$,
and suppose Alice's expected payoff from $\alpha$ is very high, say, a million dollars --
however, some action profiles from $\alpha$ will give her negative payoff.
Now, when Alice receives her advice from the mediator, she might be able to deduce that her payoff
in the advised action profile will be negative. If this is the case, she will choose to deviate to her safe strategy,
so $\alpha$ is not a correlated equilibrium. However, $\alpha$ may still be a coarse correlated equilibrium if Alice can commit
before seeing her advice; and importantly, $\alpha$ may be very desirable from Alice's (risk-neutral) point of view,
since expected payoff is high.
\end{example}
\iffalse
Let us illustrate the difference between the two types of equilibria by the simple example game given in Figure~\ref{fig:cceGame}.
\begin{figure}[h!]\label{fig:cceGame}
\bgroup
\def1{2}
\begin{center}
\begin{tabular}{c|c|c|}
& $A$ & $B$ \\ \hline
~~~$A$~~~ & $0,0$ & $\mathbf{-1,101}$ \\ \hline
~~~$B$~~~ & $\mathbf{101,-1}$ & $0,0$ \\ \hline
\end{tabular}
\end{center}
\egroup
\caption{Simple game to illustrate coarse correlated equilibrium}
\end{figure}
Consider the distribution $\alpha$ which outputs action profiles $(A,B)$ and $(B,A)$ each with probability one-half:
each player's expected payoff from this distribution is $50$. This distribution is not a correlated equilibrium,
because once a player sees his advice, he can determine whether he will get payoff $-1$ or $101$, and in the former
case he will be incentivized to deviate from his advice (and thereby obtain a payoff of $0$). However, $\alpha$ is
a \emph{coarse} correlated equilibrium, and indeed achieves higher expected payoff than any correlated equilibrium
of this game. It is thus clear that for the (risk-neutral) players, the distribution $\alpha$ would be a desirable one to achieve.
\fi
\subsection{Our results}
In this work we address the following question:
\begin{center}
\emph{How can the players of a strategic game implement any coarse correlated equilibrium via (cryptographic) pre-play communication
without trusting each other or a mediator?}
\end{center}
In the computational setting, we give an implementation for general strategic games, in the form of an extended game comprising
a \emph{cryptographic protocol} in the pre-play phase, which securely samples an action profile for a ``cryptographically blinded''
version of the original game, followed by play in the original game. The blinded game's action space consists of \emph{encryptions}
of the original game's actions.
Our implementation has the strong property that any computational coarse correlated equilibrium of the original game
corresponds to a payoff-equivalent Nash equilibrium of the extended game. Furthermore, it
achieves \emph{strategic equivalence} to the original game,
in that every computational Nash equilibrium of the extended game corresponds to a computational coarse correlated equilibrium of the original game.
Pre-play communication is via broadcast, as is standard in the cheap talk literature.
In the information-theoretic setting, we give an implementation for strategic games with four or more players, using a similar
format of a cryptographically blinded pre-play phase followed by (simultaneous) play in the original game, given private pairwise communication
channels between players.
As in the computational setting, we achieve strategic equivalence.
Both the restriction to four or more players and the need for a stronger communication model than broadcast
are unavoidable, as shown by impossibility results of \cite{barany1992,aumann2003long}
which will be discussed in more detail in the next section.
None of our constructions require trusted mediation. After the pre-play phase is complete, there is a single step in which the players
invoke a \emph{verifiable proxy} to play the original game according to their instructions.
Verifiable parties were introduced in \cite{ILM11}, and will be
detailed further in Section~\ref{subsec:priorWork}.
No trust need be placed in the verifiable proxy,
because anyone can check whether ir has acted correctly; and we stress that
unlike the usual mediator for coarse correlated equilibria,
the verifiable proxy does not communicate anything to the players which may \emph{affect their strategies} in the game.
Informally, it simply performs a ``translation'' of a player's chosen strategy from one form into another.
Finally, our constructions require \emph{no physical assumptions} and can be executed entirely over a distributed network.
This contrasts with a number of previous works such as \cite{LMP04,ILM11} which require ``physical envelopes''.
\subsection{Relation to prior work}\label{subsec:priorWork}
\paragraph{Cheap talk}
The \emph{pre-play} literature considers the general problem of implementing equilibria without mediation,
as follows: given an abstract game $\Gamma$, the aim is to devise
a concrete communication game $\Gamma'$ having an equilibrium that is payoff-equivalent to a desirable equilibrium
in $\Gamma$, where the concrete game may have a pre-play \emph{cheap talk} phase in which players engage in
communication that is neither costly nor binding, and has no impact on players'
payoffs except insofar as it may influence future actions.
In the literature there has been much focus on implementing correlated equilibria
\cite{barany1992,BenPorath1998,aumann2003long}.
\paragraph{Power of commitment} It has long been recognized that the possibility to \emph{commit} to
strategies in advance can increase the payoffs achievable in a game, starting with the work of \cite{von1934marktform},
who proposed a leader/follower structure to games where the leader moves first (and thereby ``commits'' to his strategy).
\cite{DBLP:journals/geb/StengelZ10} showed that transforming a strategic game into a leader/follower form
allows the leader (i.e. the committer) to do at least as well as in the Nash and correlated equilibria of the strategic game.
Moreover, they show that coarse correlated equilibria, with their arguably stronger notion of commitment,
can yield higher payoffs than the leader/follower transformation.
More recently, \cite{letchford2012value} studied the advantage of commitment from a quantitative perspective
and showed that the extremal ``value of commitment'' is in fact unbounded in many classes of games.
In this work, we achieve the payoffs of coarse correlated equilibria without resorting to the assumption of
binding contracts: instead, we use the power of encryption to hide information that, if known to the
players, could render the situation unstable. We stress that the players are \emph{given the choice}, rather than forced,
to hide information from themselves -- and we find that it is in their rational interest to do so since coarse correlated
equilibria can offer high payoffs.
\paragraph{Cryptographic cheap talk and computational equilibria}
\cite{DHR00} introduced the idea of \emph{cryptographic cheap talk}, in which
players execute a cryptographic protocol during the pre-play phase; and they defined \emph{computational equilibria},
which are solution concepts stable for computationally bounded (probabilistic polynomial time) players who are
indifferent to negligible gains. Their cryptographic cheap talk protocols efficiently implement some computational correlated equilibria
of two-player games. Moreover, their notion of computational equilibria
suffers from \emph{empty threats}
(Definition~\ref{def:emptyThreatInformal}), which cause instability for sequentially rational players in the pre-play game.
This was partially addressed by a new solution concept of \cite{GLR10}; however,
\cite{HNR13} subsequently showed that in general, correlated equilibria cannot be achieved without empty threats by (cryptographic) cheap talk.
Our results in the computational setting use the equilibrium definitions of \cite{DHR00};
however, in our ``cryptographically blinded'' games, empty threats cannot occur.
By converting games into blinded games, our constructions implement all
coarse correlated equilibria without empty threats: this comes at the cost of a single mediated ``translation'' step using a
third party,
discussed in the next paragraph. We consider this step to be a ``necessary'' and mild requirement given that
the impossibility result of \cite{HNR13} renders some additional assumption necessary to achieve all (coarse) correlated equilibria
without empty threats.
\paragraph{Removing trusted mediation}
Removing the need to trust a mediator in the implementation of equilibria and mechanisms has long been a subject of
interest in game theory and cryptography. The notion of \emph{verifiable mediation} was introduced by \cite{ILM11},
who highlighted the difference between the usual concept of a \emph{trusted mediator}, and the weaker
concept of a \emph{verifiable mediator} who performs actions in a publicly verifiable way and without possessing any information that should be kept secret.
Recent applications of verifiable mediation include the strong correlated equilibrium implementation of \cite{ILM11},
and the rational secret sharing scheme of \cite{MS09}.
In this paper, we introduce the new notion of a \emph{verifiable proxy}.
As in verifiable mediation, the actions of a verifiable proxy are publicly verifiable. However, our notion is incomparable to \cite{ILM11}'s
verifiable mediation, because:
\begin{itemize}
\item a verifiable proxy for a strategic game does not give the players any information that affects their strategic choices in the game; and
\item a verifiable proxy may possess information that should be kept secret.
\end{itemize}
More discussion about the merits of these definitions is given in Section~\ref{subsec:whatCanIDo}.
As a simple illustration, consider a sealed-bid auction: much more trust is placed in a mediator who collects
all the players' bids and just announces the winner, than in a mediator who collects the bids, opens them publicly,
and allows everyone to compute the outcome themselves.
In our setting, the verifiable proxy performs a single ``translation'' step on behalf of the players, at the end of the pre-play phase,
in which it takes strategies submitted by the players and ``translates'' them into a different format.
In particular, the proxy acts independently and identically with respect to each player, and therefore is
not implementing the correlation aspect.
\paragraph{Strategic equivalence property}
An important concern in implementation theory is the strategic equivalence of an implementation to the underlying game:
it is desirable that implementations have the ``same'' equilibria as the underlying game, and in particular do not
introduce new ones.
This was first considered by the \emph{full implementation} concept of \cite{maskin1999}, and extended by subsequent works
such as \cite{ILM11} who proposed a stronger notion of \emph{perfect implementation} for certain games.
Although this literature is not directly applicable to the present work (as our results lie in the pre-play realm),
we extend these ideas and find that the cheap talk extensions of our ``cryptographically blinded'' games
achieve full strategic equivalence in that their Nash equilibria correspond exactly to the coarse correlated equilibria of their underlying games.
\iffalse
\paragraph{Bounded rationality}
\Scomment{I want to omit the bounded rationality... since we have info-theoretic results, and we already basically said this thing about leveraging}
Computational games share some ideas with bounded rationality. We do something very specific "all-or-nothingm" that isn't
the usual idea of bounded rationality.
And we *exploit* bounded rationality to improve players' utility!
\fi
\paragraph{Computationally unbounded setting} To our knowledge, existing work in applying cryptographic tools to game theory has
focused overwhelmingly on the setting of computationally bounded players and computational equilibria. In contrast, we consider
the computationally unbounded setting too. Our result for the computational setting is stronger and more efficient than our
information-theoretic solution: in particular, the computational result holds for games with any number of players,
and requires only a broadcast channel for communication between players.
In the computationally unbounded setting it was proven by \cite{barany1992} that correlated equilibria cannot be
achieved by cheap talk between fewer than four players, and indeed, this fits neatly with a more general result of \cite{BGW88,CCD88}
in the context of secure protocols.
Accordingly, our information-theoretic results only apply for games of four or more players; however, improving on the protocols of
\cite{barany1992}, we achieve not only correlated equilibria but coarse correlated equilibria for all games of this type.
Furthermore, in the computationally unbounded setting it has been proven \cite{aumann2003long} that communication by broadcast alone is \emph{insufficient} to achieve
(non-trivial) correlated equilibria by cheap talk, so our result is of interest notwithstanding its stronger requirement of
private communication channels between players. Indeed, the private-channels model has been extensively studied
in both distributed computing (e.g. \cite{FLP85,gossip03}) and multi-party computation (e.g. \cite{BGW88,CCD88}) as an interesting strengthening of the communication model
that allows for much stronger and/or more efficient protocols than the broadcast model.
We therefore consider it natural and compelling to apply this model in the game-theoretical setting.
\iffalse
\paragraph{Impossibility results}
\begin{itemize}
\item \cite{aumann2003long} already mentioned; can't achieve CE with just broadcast
\item \cite{barany1992} apparently says something about non-trivial equilibria being not achievable for less than 4 players. (see DHR00 intro, p114) CHECK THIS!
\item \cite{kar2010difficulty} maybe mention that their model is different
\item \cite{DBLP:conf/crypto/HubacekNR13} what is the loophole here??? I forgot...
\end{itemize}
\fi
\subsection{Organization}
In Sections~\ref{sec:gameTheory} and \ref{sec:crypto} we provide game-theoretical and cryptographic background.
In Section~\ref{sec:blindedGames} we introduce cryptographically blinded games. These are
the essential building block for the cheap talk protocols detailed in Section~\ref{sec:protocols} that
implement all coarse correlated equilibria of general strategic games. At the end of Section~\ref{sec:protocols}
we discuss the efficiency of our protocols.
\subsection{Notation}
For $n\in\mathbb{N}$, let $[n]$ denote the set $\{1,2,\dots,n\}$.
For a set $S$, let $\mathcal{P}(S)$ denote the powerset of $S$, and let $\Delta(S)$ denote the set of all distributions over $S$.
Let $s\leftarrow S$ denote that $s$ is a random element of $S$.
Let $\sqcup$ denote the disjoint union operation.
We write {{\rm\sc ppt}}{} to mean probabilistic polynomial time, and we call distributions that can be sampled in probabilistic polynomial time
``{{\rm\sc ppt}}-samplable''.
Let $\mathsf{negl}$ denote a negligible function (which tends to zero faster than any inverse polynomial).
\section{Game-theoretic background}\label{sec:gameTheory}
\begin{definition}[Finite strategic game]
A finite strategic game $\Gamma=\langle N,(A_i),(u_i)\rangle$ is defined by
a finite set $N$ of players, and
for each player $i\in N$, a non-empty set of possible actions $A_i$ and
a utility function $u_i:\times_{j\in N}A_j\rightarrow\mathbb{R}$.
\end{definition}
We refer to an \emph{action profile}
$a=(a_j)_{j\in N}$ of a game as an \emph{outcome}, and denote by $A$ the set of outcomes $\times_{j\in N}A_j$.
For a given outcome $a$, we write $a_{-i}$ to denote $(a_j)_{j\in N, j\neq i}$, that is, the
profile of actions of all players other than $i$; and we use $(a'_i,a_{-i})$ to denote the action
profile where player $i$'s action is $a'_i$ and all other players' actions are as in $a$.
\subsection{Equilibrium concepts}
\begin{definition}[Nash equilibrium]\label{def:NE}
A \emph{Nash equilibrium} of strategic game $\Gamma=\langle N,(A_i),(u_i)\rangle$
is a product distribution $\alpha\in\times_{j\in N}\Delta(A_j)$ such that
for every player $i\in N$ and for all $a^*_i\in A_i$
\[
\mathop{\mathbf{E}}_{a\leftarrow\alpha}[u_i(a)]
\geq \mathop{\mathbf{E}}_{a\leftarrow\alpha}[u_i(a^*_i,a_{-i})].
\]
\end{definition}
\begin{definition}[Correlated equilibrium]\label{def:CE}
A \emph{correlated equilibrium} of strategic game $\Gamma=\langle N,(A_i),(u_i)\rangle$
is a probability distribution $\alpha\in\Delta(\times_{j\in N}A_j)$ such that
for every player $i\in N$, and for all $b_i,a^*_i\in A_i$ satisfying $\Pr_{a\leftarrow\alpha}[a_i=b_i]>0$,
\[
\mathop{\mathbf{E}}_{a\leftarrow\alpha}[u_i(a)|a_i=b_i]
\geq \mathop{\mathbf{E}}_{a\leftarrow\alpha}[u_i(a^*_i,a_{-i})|a_i=b_i].
\]
\end{definition}
\iffalse
\Scomment{this story was already in the intro}
The concept of correlated equilibrium is captured by the following common illustration: suppose that
a mediator samples an action profile $a$ from the probability distribution $\alpha$ and
gives (privately) to each player $i$ a piece of ``advice'' $a_i$.
The probability distribution $\alpha$ is a correlated equilibrium if no player has an incentive to
unilaterally deviate from the mediator's advice.
\fi
\begin{definition}[Coarse correlated equilibrium]\label{def:CCE}
A \emph{coarse correlated equilibrium} of strategic game $\Gamma=\langle N,(A_i),(u_i)\rangle$
is a probability distribution $\alpha\in\Delta(\times_{j\in N}A_j)$ such that
for every player $i\in N$ and for all $a^*_i\in A_i$
\[
\mathop{\mathbf{E}}_{a\leftarrow\alpha}[u_i(a)]
\geq \mathop{\mathbf{E}}_{a\leftarrow\alpha}[u_i(a^*_i,a_{-i})].
\]
\end{definition}
The model of coarse correlated equilibrium allows the players either to ``commit in advance''
to play according to the mediator's advice (no matter what it turns out to be),
or to play an \emph{independent} strategy without learning the advice.
A probability distribution is a coarse correlated equilibrium if no player has
an incentive to not commit to play according to the mediator's advice.
Because of linearity of expectation, it is sufficient for these
equilibrium definitions to consider only deviations to pure strategies.
Note that any Nash equilibrium is a correlated equilibrium, and any
correlated equilibrium is a coarse correlated equilibrium.
\subsection{Computational equilibrium concepts}\label{sec:computationalEquilibria}
The following definitions of computational equilibria extend those introduced by \cite{DHR00}.
In the computational setting a strategic game induces a family of games parametrized by the security parameter,
i.e. $\Gamma=\{\langle N,(A_{i}^{(k)}),(u_{i}^{(k)})\rangle\}_{k\in\mathbb{N}}$.
Hence, the corresponding solution concepts are ensembles of probability distributions,
and the security parameter captures the intuition that players are limited to efficiently computable
({{\rm\sc ppt}}{}) strategies and indifferent to gains negligible in $k$.
\begin{definition}[Computational Nash equilibrium] \label{def:compNE}
A \emph{computational Nash equilibrium} of computational strategic game
$\Gamma=\{\langle N,(A_{i}^{(k)}),(u_{i}^{(k)})\rangle\}_{k\in\mathbb{N}}$
is a {{\rm\sc ppt}}-samplable ensemble of product distributions $\alpha=\{\alpha^{(k)}=\times_{j\in N}\alpha_j^{(k)}\}_{k\in\mathbb{N}}$
on $\{\times_{j\in N}A_j^{(k)}\}_{k\in\mathbb{N}}$ such that for all players $i\in N$ and every {{\rm\sc ppt}}-samplable ensemble $\hat{\alpha}_i=\{\hat{\alpha}_{i}^{(k)}\}_{k\in\mathbb{N}}$
on $\{A_i^{(k)}\}_{k\in\mathbb{N}}$, there exists a negligible $\varepsilon(\cdot)$ such that for all large enough $k\in\mathbb{N}$ it holds that
\[
\mathop{\mathbf{E}}_{a\leftarrow\alpha^{(k)}}[u_{i}^{(k)}(a)]
\geq \mathop{\mathbf{E}}_{a\leftarrow\alpha^{(k)},\hat{a}_i\leftarrow\hat{\alpha}_i^{(k)}}[u_{i}^{(k)}(\hat{a}_i,a_{-i})]-\varepsilon(k).
\]
\end{definition}
\begin{definition}[Computational correlated equilibrium]\label{def:compCE}
A \emph{computational correlated equilibrium} of computational strategic game $\Gamma=\{\langle N,(A_{i}^{(k)}),(u_{i}^{(k)})\rangle\}_{k\in\mathbb{N}}$
is a {{\rm\sc ppt}}-samplable probability ensemble $\alpha=\{\alpha^{(k)}\}_{k\in\mathbb{N}}$ on $\{\times_{j\in N}A_j^{(k)}\}_{k\in\mathbb{N}}$
such that for all players $i\in N$ and every {{\rm\sc ppt}}-samplable ensemble $\hat{\alpha}_i=\{\hat{\alpha}^{(k)}_{i}\}_{k\in\mathbb{N}}$
on $\{A_i^{(k)}\}_{k\in\mathbb{N}}$ there exists a negligible $\varepsilon(\cdot)$ such that for all large enough $k\in\mathbb{N}$ it holds that
\[
\mathop{\mathbf{E}}_{a\leftarrow\alpha^{(k)}}[u_{i}^{(k)}(a)]
\geq \mathop{\mathbf{E}}_{a\leftarrow\alpha^{(k)},\hat{a}_i\leftarrow\hat{\alpha}_{i}^{(k)}(a_i)}[u_{i}^{(k)}(\hat{a}_i,a_{-i})]-\varepsilon(k).
\]
\end{definition}
\begin{definition}[Computational coarse correlated equilibrium] \label{def:compCCE}
A \emph{computational coarse correlated equilibrium} of computational strategic game
$\Gamma=\{\langle N,(A_{i}^{(k)}),(u_{i}^{(k)})\rangle\}_{k\in\mathbb{N}}$ is a {{\rm\sc ppt}}-samplable probability ensemble $\alpha=\{\alpha^{(k)}\}_{k\in\mathbb{N}}$
on $\{\times_{j\in N}A_j^{(k)}\}_{k\in\mathbb{N}}$ such that for all players $i\in N$ and every {{\rm\sc ppt}}-samplable ensemble $\hat{\alpha}_i=\{\hat{\alpha}_{i}^{(k)}\}_{k\in\mathbb{N}}$
on $\{A_i^{(k)}\}_{k\in\mathbb{N}}$, there exists a negligible $\varepsilon(\cdot)$ such that for all large enough $k\in\mathbb{N}$ it holds that
\[
\mathop{\mathbf{E}}_{a\leftarrow\alpha^{(k)}}[u_{i}^{(k)}(a)]
\geq \mathop{\mathbf{E}}_{a\leftarrow\alpha^{(k)},\hat{a}_i\leftarrow\hat{\alpha}_i^{(k)}}[u_{i}^{(k)}(\hat{a}_i,a_{-i})]-\varepsilon(k).
\]
\end{definition}
Note that in the above definition of computational coarse correlated equilibrium the output of $\hat{\alpha}_{i}^{(k)}$
is independent of $a_i$, unlike in the definition of computational correlated equilibrium.
\begin{remark}
In later sections we apply the above computational solution concepts in a straightforward way to classical strategic games.
For a finite strategic game $\Gamma=\langle N, (A_i),(u_i)\rangle$ we consider the computational version $\{\Gamma^{(k)}\}_{k\in\mathbb{N}}$,
where $\Gamma^{(k)}=\Gamma$ for all $k\in\mathbb{N}$.
The action space and the utility function do not change with the security parameter in this computational version of $\Gamma$;
however, the players are limited to efficient ({{\rm\sc ppt}}{}) strategies.
\end{remark}
\begin{remark}
In the classical setting, it is implicit that the players of a game have oracle access to the utility functions $u_i$,
that is, players can query $u_i$ on any action profile in constant time\footnote{Other parameters of the original game,
such as the correlated equilibrium distribution, are also assumed to be computable in constant time.}.
Our results apply to all strategic games in the classical setting: hence the requirement
that the security parameter be polynomial in the size of the game (i.e. we ensure that players
are able to perform the standard task of reading the payoff matrix).
With computationally bounded players, however, it seems very natural to consider
the case in which computing $u_i$ takes more time.
To our knowledge, this difference has been recognized (e.g., \cite{DHR00}) but
not much analyzed in the literature; however, it is an important underlying idea
of the present work.
\iffalse
From such a game in the classical setting,
our approach is to compile a ``cryptographically blinded'' (but essentially equivalent) game where some utilities
are hard to compute for bounded players, which will be used in the pre-play phase of an extended game.
For our information-theoretic result, we achieve a similar notion of ``blinding'' for computationally unbounded players by
using information-theoretic cryptography.
\fi
\end{remark}
\subsection{Extensive games}
Definitions of extensive form games and subgames are given in Appendix~\ref{appx:extensiveGames}, along with
corresponding equilibrium concepts for the standard and computational settings.
\section{Cryptographic background}\label{sec:crypto}
\subsection{Encryption schemes}
Our constructions will make use of secret-key and public-key encryption schemes, which are defined below.
Note that encryption schemes are parametrized by a security parameter $k$
that determines the ``security level'' of the scheme.
\begin{definition}[Secret-key encryption scheme]
A \emph{secret-key encryption scheme} over a message space $\mathcal{M}$ is a tuple of
{{\rm\sc ppt}}{} algorithms $\Sigma=(\mathsf{SGen},\mathsf{SEnc},\mathsf{SDec})$ satisfying the following.
Let the ciphertext space be the codomain of $\mathsf{SEnc}$ and be denoted by $\mathcal{C}$.
\begin{itemize}
\item The key generation algorithm $\mathsf{SGen}$ takes no input and outputs a secret key $sk$ according to
some distribution (inherent to $\Sigma$). This is denoted by $sk\leftarrow\mathsf{SGen}()$.
\item The encryption algorithm $\mathsf{SEnc}$ takes as input a message $m\in\mathcal{M}$ and a secret key $sk$, and outputs
a ciphertext $c\in\mathcal{C}$. This is denoted by $c\leftarrow\mathsf{SEnc}_{sk}(m)$.
\item The decryption algorithm $\mathsf{SDec}$ is a deterministic algorithm that takes as input a ciphertext $c$
and a secret key $sk$, and outputs a decryption $m'\in\mathcal{M}$. This is denoted by $m'=\mathsf{SDec}_{sk}(c)$.
\item The decryption is always correct, i.e.
for every security parameter $k$, and every $sk\leftarrow\mathsf{SGen}()$
it holds for every $m\in\mathcal{M}$ that $\mathsf{SDec}_{sk}(\mathsf{SEnc}_{sk}(m))=m$.
\end{itemize}
\end{definition}
\begin{definition}[Public-key encryption scheme]
A \emph{public-key encryption scheme} over a message space $\mathcal{M}$ is a tuple of {{\rm\sc ppt}}{} algorithms
$\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ satisfying the following.
Let the ciphertext space be the codomain of $\mathsf{PEnc}$ and be denoted by $\mathcal{C}$.
\begin{itemize}
\item The key generation algorithm $\mathsf{PGen}$ takes input $1^k$, where $k$ is the security parameter, and outputs
a public key and secret key pair $(pk,sk)$.
\item The encryption algorithm $\mathsf{PEnc}$ takes as input a message $m\in\mathcal{M}$
and a public key $pk$ and outputs a ciphertext $c\in\mathcal{C}$.
\item The decryption algorithm $\mathsf{PDec}$ is a deterministic algorithm that takes
as input a ciphertext $c$ and a secret key $sk$, and outputs a decryption $m'\in\mathcal{M}$.
\item The decryption is always correct, i.e.
for every security parameter $k$, and every $(pk,sk)\leftarrow\mathsf{PGen}(1^k)$
it holds for every $m\in\mathcal{M}$ that $\mathsf{PDec}_{sk}(\mathsf{PEnc}_{pk}(m))=m$.
\end{itemize}
\end{definition}
\subsection{Security definitions}
Here we define the following two standard security notions: perfect (information-theoretic) security,
and computational security against chosen-ciphertext attacks. The latter is commonly referred to as $\mathsf{CCA}$-security,
and is the de facto standard for security of public-key encryption; the former is canonical in the information-theoretic setting.
\begin{remark}
Our constructions make use of perfectly secure secret-key encryption and $\mathsf{CCA}$-secure public-key encryption. For convenience,
therefore, the security definitions given below refer to secret- and public-key schemes respectively.
However, both security definitions may be straightforwardly adapted to apply to both types of encryption
(although it is well known that perfect security is impossible in the public-key setting).
\end{remark}
\begin{definition}[Perfectly secure secret-key encryption]\label{def:perfectSecurity}
A secret-key encryption scheme $\Sigma=(\mathsf{SGen},\mathsf{SEnc},\mathsf{SDec})$ is \emph{perfectly secure} if
for all messages $m_0,m_1\in\mathcal{M}$
and ciphertexts $c\in\mathcal{C}$, it holds that
$\Pr[\mathsf{SDec}(\mathsf{SEnc}(m_0))=m_0]=1$
and
$$\Pr_{sk\leftarrow\mathsf{SGen}()}[\mathsf{SDec}_{sk}(c)=m_0] = \Pr_{sk\leftarrow\mathsf{SGen}()}[\mathsf{SDec}_{sk}(c)=m_1].$$
\end{definition}
An alternative and equivalent definition is that a perfectly secure encryption scheme produces ciphertexts that are independent of the messages that they encrypt.
Next, we shall define $\mathsf{CCA}$-security for public-key encryption schemes. The security definition is based on the following experiment,
which may be considered to be a game played between a malicious adversary $\mathcal{A}$ and an honest challenger.
\begin{framed}
\begin{center}
The $\mathsf{CCA}$ indistinguishability experiment $\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{CCA}}(k)$:
{\small \begin{enumerate}
\item The challenger generates a key pair $(pk,sk)\leftarrow\mathsf{PGen}(1^k)$, and sends $(1^k,pk)$ to $\mathcal{A}$.
\item $\mathcal{A}$ has oracle access to $\mathsf{PDec}_{sk}$, and outputs messages $m_0,m_1\in\mathcal{M}$ of the same length.
\item The challenger samples $b\leftarrow \{0,1\}$, then computes $c\leftarrow\mathsf{PEnc}_{pk}(m_b)$, and sends $c$ to $\mathcal{A}$.
\item $\mathcal{A}$ still has oracle access to $\mathsf{PDec}_{sk}$, but cannot query $\mathsf{PDec}_{sk}(c)$. $\mathcal{A}$ now outputs a bit $b'$.
\item The output of the experiment is $1$ if $b' = b$, and $0$ otherwise.
\end{enumerate} }
\end{center}
\end{framed}
Informally, the adversary ``wins the game'' if he guesses correctly
which of the two messages was encrypted. Clearly, he can win with probability $1/2$
by random guessing. The definition of $\mathsf{CCA}$-security formalizes the intuition that he should not be able to do better than that.
\begin{definition}[$\mathsf{CCA}$-secure public-key encryption]\label{def:CCAsec}
A public-key encryption scheme $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ is \emph{$\mathsf{CCA}$-secure}
(i.e. secure against chosen-ciphertext attacks), if for all {{\rm\sc ppt}}{} adversaries $\mathcal{A}$,
$\Pr[\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{CCA}}(k) = 1] \leq 1/2 + \varepsilon(k)$ for some negligible $\varepsilon$.
\end{definition}
\subsection{Non-malleable encryption}
Non-malleable encryption was introduced by \cite{DDN00} in the
computational setting, and extended to the information-theoretic setting by \cite{HSHI02}.
Informally, non-malleability requires that given a ciphertext $c$, an adversary (who does not know the secret key
or the message encrypted by $c$) cannot generate a different ciphertext $c'$ such that the
respective messages are related by some known relation $R$.
We begin with the simpler information-theoretic definition.
Note that \cite{HSHI02} also give a construction of perfectly non-malleable secret-key encryption.
\begin{definition}[Perfect non-malleability]\label{def:perfNML}
A secret-key encryption scheme $\Sigma=(\mathsf{SGen},\mathsf{SEnc},\mathsf{SDec})$ is \emph{perfectly non-malleable}
if for all $c,c',c''\in\mathcal{C}$ such that $c'\neq c\neq c''$ and all relations $R:\mathcal{M}\times\mathcal{M}\rightarrow\{0,1\}$,
$$\Pr_{sk\leftarrow\mathsf{SGen}()}[R(\mathsf{SDec}(c),\mathsf{SDec}(c'))=1] = \Pr_{sk\leftarrow\mathsf{SGen}()}[R(\mathsf{SDec}(c),\mathsf{SDec}(c''))=1].$$
\end{definition}
Observe that perfect non-malleability implies perfect security (but not vice versa).
The computational definition of non-malleability is more involved, using an indistinguishability experiment similar to that of the
$\mathsf{CCA}$-security definition.
It formalizes the same idea, that an attacker
must be unable (with more than negligible advantage) to modify ciphertexts such that the new decryption satisfies a known relation
with the original decryption.
The definition of non-malleability for (public-key) encryption schemes is based on the following experiment.
\begin{framed}
\begin{center}
The $\mathsf{NM}$ indistinguishability experiment $\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{NM}}(k)$:
{\small \begin{enumerate}
\item The challenger generates a key pair $(pk,sk)\leftarrow\mathsf{PGen}(1^k)$ and sends $(1^k,pk)$ to $\mathcal{A}$.
\item $\mathcal{A}$ has oracle access to $\mathsf{PDec}_{sk}$, and outputs (a description of) an efficiently samplable distribution $M$
on the message space $\mathcal{M}$ (which must give non-zero probability only to strings of a given length).
\item The challenger samples a message $m\leftarrow M$, and sends ciphertext $c=\mathsf{PEnc}_{pk}(m)$ to $\mathcal{A}$.
\item $\mathcal{A}$ still has oracle access to $\mathsf{PDec}_{sk}$, but cannot query $\mathsf{PDec}_{sk}(c)$. $\mathcal{A}$ outputs a ciphertext $c'$ and (a description of) an efficiently computable relation $R:\mathcal{M}\times\mathcal{M}\rightarrow\{0,1\}$.
\item The output of the experiment is $1$ if $c'\neq c$ and $R(m,\mathsf{PDec}_{sk}(c'))$ is true, and $0$ otherwise.
\end{enumerate} }
\end{center}
\end{framed}
Define $\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{NM},\$}(k)$ to be identical to $\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{NM}}(k)$, except that item 3 is replaced by:
\begin{framed}
{\small \begin{itemize}
\item[$3'$.] The challenger samples independent messages $m,\tilde{m}\leftarrow M$, and sends $c=\mathsf{PEnc}_{pk}(\tilde{m})$ to $\mathcal{A}$.
\end{itemize}}
\end{framed}
\begin{definition}[Computationally non-malleable encryption] \label{def:compNM}
A public-key encryption scheme $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ is \emph{$\mathsf{NM}$-$\mathsf{CCA}$-secure} (that is, non-malleable
against chosen ciphertext attacks), if for all {{\rm\sc ppt}}{} adversaries $\mathcal{A}$ there exists a
negligible function $\mathsf{negl}$ such that
\[
\left| \Pr[\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{NM}}(k) = 1] - \Pr[\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{NM},\$}(k)=1] \right| \leq \mathsf{negl}(k).
\]
\end{definition}
In our setting\footnote{When considering security notions other than $\mathsf{CCA}$, standard indistinguishability-based security does not imply non-malleability.
In this work we only use $\mathsf{CCA}$-secure schemes.}, $\mathsf{CCA}$-security is equivalent to computational non-malleability, as stated in Claim~\ref{claim:nm}.
For the proof, we refer the reader to \cite{BDPR98}.
\begin{claim}\label{claim:nm}
An encryption scheme is $\mathsf{CCA}$-secure (Definition~\ref{def:CCAsec}) if and only if it satisfies
computational non-malleability (Definition~\ref{def:compNM}).
\end{claim}
\subsection{Secure multi-party computation}\label{subsec:mpc}
Consider $N$ players, each with an input value $x_i$ for $i\in N$, who wish to
jointly compute a function $f$ on their inputs: $f(x_1,\cdots,x_N)=(y_1,\dots,y_N)$.
These players do not trust each other: they want each player $i$ to receive his output value $y_i$ at the end of the computation,
but they also want a guarantee that no player $i$ can learn any information beyond his designated output $y_i$ (even if he ``cheats'').
Multi-party computation gives interactive $N$-party protocols to solve this problem,
with security and correctness guarantees even when some players may maliciously deviate from the protocol.
\begin{definition}[Secure multi-party computation]\label{def:MPC}
An $N$-party computation protocol is said to be \emph{perfectly secure}
(for up to $t<N$ corruptions) if it satisfies the following properties, against any adversary who corrupts
up to $t$ players\footnote{The corrupted players may be thought of as ``dishonest'' players trying to sabotage the protocol.}:
\begin{itemize}
\item \emph{Correctness}: The output of the computation is equal to $f(x_1,\cdots,x_N)$.
\item \emph{Privacy}: No adversary can obtain any information about the honest parties' inputs,
other than what can be deduced from the corrupted players' input and output values $\{x_i,y_i\}_{i\in S}$
(where $S$ denotes the set of corrupt players).
\end{itemize}
The protocol is said to be \emph{computationally secure} if it satisfies the above properties with all but negligible
probability (in a security parameter $k$) against {{\rm\sc ppt}}{} adversaries.
\end{definition}
The following are general possibility results for multi-party computation that are relevant to this work.
For proofs, we refer the reader to the original papers.\footnote{Dodis and Rabin~\cite{dodis2007cryptography} provide and extended summary of the multi-party computation results with emphasis on the use in the game theoretical context.}
\begin{theorem}[\cite{BGW88,CCD88}]\label{thm:perfectMPC}
Any circuit can be evaluated by an $N$-party protocol with perfect security against $t<N/3$ corruptions.
Moreover, the bound of $t<N/3$ is tight.
\end{theorem}
\begin{theorem}[\cite{GMW87,DBLP:journals/iacr/AsharovL11}]\label{thm:compMPC}
Any circuit can be evaluated by an $N$-party protocol with computational security against up to $t=N-1$ corruptions.
\end{theorem}
\iffalse
An additional desirable property of multi-party computation protocols, other than correctness and privacy,
is \emph{fairness}: the property that if \emph{any} player $i$ receives his output $y_i$, then
each honest (non-corrupt) player is guaranteed to receive her correct output,
even in the presence of an adversary. This property is known to be achievable if and only if $t<N/2$
(that is, a majority of the players are honest) \cite{GMW87,Cle86}.
\fi
An additional desirable property of multi-party computation protocols, other than correctness and privacy,
is \emph{guaranteed output delivery}: the property that every honest (non-corrupt) player is guaranteed to receive her correct output,
even in the presence of an adversary. This property is known to be achievable if and only if $t<N/2$
(that is, a majority of the players are honest) \cite{GMW87,Cle86}.
\subsection{Secret Sharing}
A secret sharing scheme specifies a method for a special party (the ``dealer'')
to \emph{share} a secret $s$ among $N$ players so that only large enough subsets of players can reconstruct the secret value $s$.
The dealer gives privately a share $s_i$ to each player $i$, so that
any set of up to $k-1$ shares contains no information about $s$;
however, it can efficiently be reconstructed given any $k$ or more shares.
The formal definition is given below.
\begin{definition}[Secret sharing scheme \cite{Shamir:1979:SS:359168.359176}]
A \emph{$k$-out-of-$N$ secret sharing scheme} is a pair of algorithms $({\sf Share},{\sf Reconstruct})$ as follows.
${\sf Share}$ takes as input a secret value $s$ and outputs a set of \emph{shares} $S=\{s_1,\dots,s_N\}$ such that
the following two properties hold.
\begin{itemize}
\item \emph{Correctness}: For any subset $S'\subseteq S$ of size $|S'|\geq k$, it holds that ${\sf Reconstruct}(S')=s$, and
\item \emph{Privacy}: For any subset $S'\subseteq S$ of size $|S'|< k$, it holds that $H(s)=H(s|S')$, where $H$ denotes the binary entropy function.
\end{itemize}
${\sf Reconstruct}$ takes as input a (sub)set $S'$ of shares and outputs:
$${\sf Reconstruct}(S')=\begin{cases}\bot & \mbox{if} \qquad |S'|<k \\ s & \mbox{if} \qquad\exists S \mbox{ s.t. } S'\subseteq S \mbox{ and } {\sf Share}(s)=S \mbox{ and } |S'|\geq k \end{cases}.$$
\end{definition}
\section{Cryptographically blinded games}\label{sec:blindedGames}
Now we define ``cryptographically blinded'' games $\Gamma'$ whose
actions are encryptions of the actions of an underlying game $\Gamma$.
Payoffs from corresponding action profiles of $\Gamma$ and $\Gamma'$ are the same.
These blinded games will be an essential tool for our pre-play protocols, which will be detailed in Section~\ref{sec:protocols}.
The following supporting definition formalizes the intuitive notion that two strategic games
are equivalent up to renaming of actions or deletion of redundant actions.
\begin{definition} \label{def:equiv_game}
For any strategic game $\Gamma=\langle N,(A_i),(u_i)\rangle$,
a strategic game $\Gamma'=\langle N,(A_i'),(u_i')\rangle$ is said to be \emph{super-equivalent} to $\Gamma$
if there exist surjective \emph{renaming functions} $\rho_i:A_i'\rightarrow A_i$ such that for all $i\in N$, for all $a_1'\in A_1',\dots,a_N'\in A_N'$, it holds that
$u_i'(a_1',\dots,a_N')=u_i(\rho_1(a_1'),\dots,\rho_N(a_N'))$.
In this case, we write $\Gamma'\eqv{\rho}\Gamma$.
\end{definition}
\paragraph{Notation} For a renaming function $\rho$, let $\rho^{-1}_i:A_i\rightarrow\mathcal{P}(A_i')$ be defined by $\rho^{-1}_i(a_i)=\{a'_i|\rho(a'_i)=a_i\}$. To simplify notation, we define $\rho:A_1\times\dots\times A_N\rightarrow A_1'\times\dots\times A_N'$ to be $\rho(a_1,\dots,a_N)=(\rho_1(a_1),\dots,\rho_N(a_N))$, and let $\rho^{-1}$ be defined similarly.
For a distribution $\gamma'$ on action profiles of $\Gamma'$,
$\rho(\gamma')$ denotes the distribution on action profiles of $\Gamma$
that corresponds to sampling $a'\in A'$ according to $\gamma'$
and outputting $\rho(a')$.
\begin{lemma}\label{lem:rhoEquivalenceOfCCE}
Let $\Gamma$ be a strategic game. Then for any $\Gamma'$ with
$\Gamma'\eqv{\rho}\Gamma$ it holds that: (1) for any coarse correlated equilibrium $\alpha$ of $\Gamma$,
there exists a coarse correlated equilibrium $\alpha'$ of $\Gamma'$
such that $\rho(\alpha')=\alpha$; and (2) for any coarse correlated equilibrium $\alpha'$of $\Gamma'$,
$\rho(\alpha')$ is a coarse correlated equilibrium of $\Gamma$.
\end{lemma}
\begin{proof}
To show item (1), consider the distribution $\alpha'$
on action profiles of $\Gamma'$ obtained by
sampling an action profile $a$ from $\alpha$ and outputting a random $a'\in\rho^{-1}(a)$.
Note that $\rho(\alpha')=\alpha$ by construction.
We need to show that for all $i\in N$ and all $a^*_i\in A'_i$,
\[
\mathop{\mathbf{E}}_{a'\leftarrow\alpha'}[u'_i(a')]
\geq \mathop{\mathbf{E}}_{a'\leftarrow\alpha'}[u'_i(a^*_i,a'_{-i})].
\]
The above can be rewritten, due to the construction of $\alpha'$ and definition of $\Gamma'$, as
$
\mathop{\mathbf{E}}_{a\leftarrow\rho(\alpha')}[u_i(a)]
\geq \mathop{\mathbf{E}}_{a\leftarrow\rho(\alpha')}[u_i(\rho_i(a^*_i),a_{-i})]
$.
This holds for every $\rho_i(a^*_i)\in A_i$ since $\alpha=\rho(\alpha')$
is a coarse correlated equilibrium of $\Gamma$.
Item (2) follows similarly, since $\rho(\alpha')$
is a distribution on action profiles of $\Gamma$ and
$\alpha'$ is a coarse correlated equilibrium.
\end{proof}
\iffalse
Next we show that in the special case when the players are not able
to compute $\rho$,
coarse correlated equilibria and correlated equilibria of such
``blinded'' game $\Gamma'$ coincide.
\fi
The interesting case of the seemingly straightforward definition of super-equivalence
arises when the renaming function $\rho$ is not invertible by the players.
We now define cryptographically blinded games.
Let $\Gamma=\langle N,(A_i),(u_i)\rangle$ be a strategic game,
where players have oracle access to the utility functions $u_i$.
\begin{definition}[Secret-key blinded game]\label{def:PKgame}
Let $\Sigma=(\mathsf{SGen},\mathsf{SEnc},\mathsf{SDec})$ be a secret-key encryption scheme, and let $\Gamma=\langle N,(A_i),(u_i)\rangle$ be a strategic game.
Define the \emph{blinded game} $\Gamma^\SKE_{sk}=\langle N,(A_i'),(u_i)'\rangle$ of $\Gamma$ to be the game such that $sk\leftarrow\mathsf{SGen}()$ is generated and
\begin{itemize}
\item for each player $i\in N$ the action space is $A_i'=A_i\sqcup\{\mathsf{SEnc}_{sk}(a_i)|a_i\in A_i\}$
\item for each player $i\in N$ the utility for all $a'\in\times_{j\in N} A'_j$ is $u_i'(a')=u_i(a)$, where for all $j\in N$
\[
a_j=
\begin{cases}
a'_j &\text{if } a'_j\in A_j,\\
\mathsf{SDec}_{sk}(a'_j) &\text{otherwise.}
\end{cases}
\]
\end{itemize}
\end{definition}
\begin{definition}[Public-key blinded game]\label{def:PKgame}
Let $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ be a public-key encryption scheme, and let $\Gamma=\langle N,(A_i),(u_i)\rangle$ be a strategic game.
Define the \emph{computational blinded game} $\Gamma^\PKE_{(pk,sk)}=\{\langle N,(A_{i}^{\prime{}(k)}),(u_{i}^{\prime{}(k)})\rangle\}_{k\in\mathbb{N}}$ of $\Gamma$ to be the computational game such that for every security parameter $k\in\mathbb{N}$ a corresponding key pair $(pk,sk)\leftarrow\mathsf{PGen}(1^k)$ is generated and
\begin{itemize}
\item for each player $i\in N$ the action space is $A_{i}^{\prime{}(k)}=\{\mathsf{PEnc}_{pk}(a_i)|a_i\in A_i\}$
\item for each player $i\in N$ the utility for all $a'\in\times_{i\in N} A_{i}^{\prime{}(k)}$ is $u_{i}^{\prime{}(k)}(a')=u_i(\mathsf{PDec}_{sk}(a'))$.
\end{itemize}
\end{definition}
\iffalse
\begin{definition}[Public-key blinded game]\label{def:PKgame}
Let $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ be a public-key encryption scheme, and let $(pk,sk)\leftarrow\mathsf{PGen}(1^k)$ be a key pair thereof.
Define the \emph{blinded game} $\Gamma^\PKE_{(pk,sk)}=\langle N,(A'_i),(u'_i)\rangle$ of $\Gamma$ to be the game such that for all $i\in N$, it holds that
$A_i'=\{\mathsf{PEnc}_{pk}(a_i)|a_i\in A_i\}$ and for all $a'\in\times_{i\in N} A'_i$, it holds that $u_i'(a')=u_i(\mathsf{PDec}_{sk}(a'))$.
\end{definition}
\begin{definition}
For a public-key encryption scheme $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ and a key pair $(pk,sk)\leftarrow\mathsf{PGen}(1^k)$,
we define $\Gamma^\PKE_{(pk,sk)}=\langle N,(A'_i),(u'_i)\rangle$ to be the game such that
$$\Gamma^\PKE_{(pk,sk)}\eqv{\rho}\Gamma\mbox{\qquad where \qquad}\rho=\mathsf{PDec}_{sk}
\mbox{\qquad and\qquad}A_i'=\{\mathsf{PEnc}_{pk}(a_i)|a_i\in A_i\}.$$
\end{definition}
\begin{definition}
Let $\Sigma=(\mathsf{SGen},\mathsf{SEnc},\mathsf{SDec})$ be a secret-key encryption scheme, and let $sk\leftarrow\mathsf{SGen}()$ be a secret key thereof.
We define $\Gamma^\SKE_{sk}=\langle N,(A_i'),(u_i)'\rangle$ to be the game such that
$$\Gamma^\SKE_{sk}\eqv{\rho}\Gamma\mbox{\qquad where\qquad}\rho=\mathsf{SDec}_{sk}
\mbox{\qquad and\qquad}A_i'=A_i\sqcup\{\mathsf{SEnc}_{sk}(a_i)|a_i\in A_i\},$$
where $\sqcup$ denotes disjoint union.
\end{definition}
\fi
If $\Gamma^\PKE_{(pk,sk)}$ and $\Gamma^\SKE_{sk}$ are blinded games of the game $\Gamma$, then we say that
$\Gamma$ is the \emph{underlying game} of $\Gamma^\PKE_{(pk,sk)}$ and $\Gamma^\SKE_{sk}$.
Observe that the blinded games $\Gamma^\PKE_{(pk,sk)}$ and $\Gamma^\SKE_{sk}$ are super-equivalent to the underlying game $\Gamma$, with respect to
renaming functions $\rho=\mathsf{PDec}_{sk}$ or $\rho=\mathsf{SDec}_{sk}$ (respectively).
\begin{remark}
In these contexts, players do not have knowledge of
the secret key $sk$, as is standard and necessary when employing encryption schemes.
Therefore, expectations ``from the point of view of the player'' are taken over a secret key $sk\leftarrow\mathsf{SGen}()$ or $(pk,sk)\leftarrow\mathsf{PGen}(1^k)$, where
secret- or public-key encryption schemes are used, respectively.
\end{remark}
It is assumed to be infeasible for players of a game $\Gamma$ to efficiently compute
the utility functions $u_i'$ on arbitrary action profiles in $\Gamma^\SKE_{sk}$ or $\Gamma^\PKE_{(pk,sk)}$, since they cannot (efficiently) decrypt ciphertexts
in the corresponding encryption schemes.
However, our applications require players to be able to pick actions in $A_i'$
for which they know the corresponding expected utility. In fact, if the players cannot do this, then the games become
meaningless in that \emph{any} distribution on $A$ is an equilibrium. In the public-key case, this property is achieved
as players can simply compute the encryption of some $a_i\in A_i$ for which the utility is known.
In the secret-key case, $A_i$ is contained in $A_i'$ for exactly this purpose.
\paragraph{Security parameter for public-key games}
Public-key blinded games have an implicit security parameter $k$ due to
the underlying encryption scheme. When applying computational equilibrium concepts (which have a security parameter $k'$ of their own)
to such games, there must be a fixed relation between $k$ and $k'$ in order to have a meaningful definition of security
for a computational equilibrium of a blinded game.
In our setting, both parameters represent the same quantity:
the computational boundedness of the players of a game. Therefore, we let $k=k'$ and refer to a single security parameter $k$.
\subsection{Correspondence of equilibria in blinded games}
\begin{lemma}\label{lem:CCEvsCEinSKE}
Let $\Sigma=(\mathsf{SGen},\mathsf{SEnc},\mathsf{SDec})$ be a perfectly non-malleable and verifiably decryptable secret-key encryption scheme.
Then for any strategic game $\Gamma$,
it holds that for any coarse correlated equilibrium $\alpha$ of $\Gamma$ there exists a
correlated equilibrium $\alpha'$ of $\Gamma^\SKE_{sk}$ that achieves the same utility profile as $\alpha$.
\end{lemma}
\begin{proof}
Let $\alpha'$ be the probability distribution on $\times_{i\in N} A'_i$ that corresponds to
sampling an action profile $a=(a_1,\ldots , a_N)\in\times_{i\in N} A_i$ according to $\alpha$ and outputting an action profile
$a'=(\mathsf{SEnc}_{sk}(a_1),\dots,\mathsf{SEnc}_{sk}(a_N))$, where $sk$ is the secret key generated by $\mathsf{SGen}$.
Note that $\alpha'$ achieves the same utility profile as $\alpha$ by construction.
To show that such $\alpha'$ constitutes a correlated equilibrium of $\Gamma^\SKE_{sk}$,
we need to verify that the conditions from Definition \ref{def:CE} are satisfied, i.e.
for every player $i$ and for all $b'_i,\hat{a}'_i\in A'_i$ it must hold that
\begin{equation}\label{eqn:CCE_SK_cond}
\mathop{\mathbf{E}}_{sk\leftarrow\mathsf{SGen}(),a'\leftarrow\alpha'}[u'_i(a')|a'_i=b'_i]
\geq \mathop{\mathbf{E}}_{sk\leftarrow\mathsf{SGen}(),a'\leftarrow\alpha'}[u'_i(\hat{a}'_i,a'_{-i})|a'_i=b'_i].
\end{equation}
Since $\Sigma$ is perfectly secure, it follows from Definition~\ref{def:perfectSecurity} that for any $a'_0,a'_1\in A_i'$,
\[
\mathop{\mathbf{E}}_{sk\leftarrow\mathsf{SGen}(),a'\leftarrow\alpha'}[u'_i(a')|a'_i=a'_0]=\mathop{\mathbf{E}}_{sk\leftarrow\mathsf{SGen}(),a'\leftarrow\alpha'}[u'_i(a)|a'_i=a'_1].
\]
Thus, for any player $i$, the expected utility from the distribution $\alpha'$ is independent
of the advice $a'_i$.
Moreover, since the underlying encryption scheme is perfectly non-malleable (Definition \ref{def:perfNML}),
no player $i$ can generate (with any advantage\footnote{More precisely, no player can generate
such a deviation $a^*_i$ with more success than by random guessing.}) a deviation $a^*_i$ satisfying
$R(a^*_i,a_i)$ for any known relation $R$.
It follows that we need only to consider deviations $a^*_i$ that are independent of the received advice $a_i$.
Therefore, equation~\ref{eqn:CCE_SK_cond} can be rewritten as the following:
for every player $i$ and for all $\hat{a}'_i\in A'_i$ independent of $a'_i$,
\[
\mathop{\mathbf{E}}_{sk\leftarrow\mathsf{SGen}(),a'\leftarrow\alpha'}[u'_i(a')]
\geq \mathop{\mathbf{E}}_{sk\leftarrow\mathsf{SGen}(),a'\leftarrow\alpha'}[u'_i(\hat{a}'_i,a'_{-i})]
\]
which holds because $\alpha'$ is by Lemma~\ref{lem:rhoEquivalenceOfCCE} a coarse correlated equilibrium of $\Gamma^\SKE_{sk}$.
\end{proof}
\Pcomment{Should say something about encrypting multiple messages with the same key. This is in general not kosher even though the scheme has perfect secrecy - the above proof goes through since each player gets only a SINGLE ciphertext!}
\begin{lemma}\label{lem:CCEvsCEinPKE}
Let $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ be a $\mathsf{CCA}$-secure public-key encryption scheme.
Then for any strategic game $\Gamma$, it holds that for any computational coarse correlated equilibrium $\alpha$ of $\Gamma$
there exists a computational correlated equilibrium $\alpha'$ of $\Gamma^\PKE_{(pk,sk)}$ that achieves the same utility profile as $\alpha$.
\end{lemma}
\begin{proof}
For each security parameter $k\in\mathbb{N}$, let $(pk,sk)$ be the corresponding key pair generated by $\mathsf{PGen}(1^k)$.
Consider the following probability ensemble $\alpha'=\{\alpha^{\prime{}(k)}\}_{k\in\mathbb{N}}$ on $\{\times_{j\in N} A_{j}^{\prime{}(k)}\}_{k\in\mathbb{N}}$
that corresponds for each $k\in\mathbb{N}$ to sampling an action profile $a=(a_1,\ldots , a_N)\in\times_{i\in N} A_i$ according to $\alpha^{(k)}$
and outputting an action profile $a'=(\mathsf{PEnc}_{pk}(a_1),\dots,\mathsf{PEnc}_{pk}(a_N))$.
Note that $\alpha'$ achieves the same utility profile as $\alpha$ by construction.
Assume that $\alpha'$ is not a computational correlated equilibrium of $\Gamma^\PKE_{(pk,sk)}$ (Definition \ref{def:compCE}), i.e.
there exist a player $i\in N$, a {{\rm\sc ppt}}{}-samplable ensemble $\hat{\alpha}_{i}^{\prime{}}=\{\hat{\alpha}_{i}^{\prime{}(k)}\}_{k\in\mathbb{N}}$
on $\{A_{i}^{\prime{}(k)}\}_{k\in\mathbb{N}}$, and a non-negligible function $\delta(\cdot)$ such that for every $k\in\mathbb{N}$
\begin{equation}\label{eqn:CCE_PK_cond}
\mathop{\mathbf{E}}_{\substack{(pk,sk)\leftarrow\mathsf{PGen}(1^k),\\a'\leftarrow\alpha^{\prime{}(k)}}}
[u_{i}^{\prime{}(k)}(a')]
\leq
\mathop{\mathbf{E}}_{\substack{(pk,sk)\leftarrow\mathsf{PGen}(1^k),\\a'\leftarrow\alpha^{\prime{}(k)},\hat{a}_{i}^{\prime{}}\leftarrow\hat{\alpha}_{i}^{\prime{}(k)}(a'_i)}}
[u_{i}^{\prime{}(k)}(\hat{a}_{i}^{\prime{}},a'_{-i})]-\delta(k).
\end{equation}
We show that one can use such a deviation $\hat{\alpha}_{i}^{\prime{}}$ to construct a {{\rm\sc ppt}}{} adversary
that contradicts the computational non-malleability of the encryption scheme $\Pi$ (Definition \ref{def:compNM}).
Let $\mathcal{A}$ be the adversary that for each security parameter $k\in\mathbb{N}$ behaves as follows.
$\mathcal{A}$ receives a public key $pk$ from the challenger and sends back $M=\alpha^{(k)}$ as the message distribution.
Upon receiving the challenge ciphertext $c$ the adversary $\mathcal{A}$ samples $c'\leftarrow\hat{\alpha}_{i}^{\prime{}(k)}(c)$ and sends $c'$ to the challenger
together with the relation
\[
R(b,\hat{b})=
\begin{cases}
1 &\text{w.p. } \frac{1}{2}\cdot(\mathop{\mathbf{E}}_{a\leftarrow\alpha^{(k)}}[u_i(\hat{b},a_{-i})|a_i=b]-\mathop{\mathbf{E}}_{a\leftarrow\alpha^{(k)}}[u_i(a_i,a_{-i})|a_i=b]+1),\\
0 &\text{otherwise.}
\end{cases}
\]
We can assume without loss of generality that all the utilities of all the players in $\Gamma$ are between $0$ and $1$
(the corresponding linear transformation of the game matrix does not change the strategic properties of the game), hence the above expression
defining the probability that $R(b,\hat{b})$ holds is between $0$ and $1$.
Note that $M$ is efficiently samplable and that the relation $R$ is efficiently computable.
Consider the success probability of $\mathcal{A}$ in the experiment $\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{NM}}(k)$, i.e.
\begin{align*}
\Pr[\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{NM}}(k) = 1]
&=
\Pr_{\substack{(pk,sk)\leftarrow\mathsf{PGen}(1^k)\\a\leftarrow\alpha^{(k)},\hat{a}'_{i}\leftarrow\hat{\alpha}_{i}^{\prime{}(k)}(\mathsf{PEnc}_{sk}(a_i))}}
[\hat{a}'_i\neq\mathsf{PEnc}_{sk}(a_i) \land R(a_i,\mathsf{PDec}_{sk}(\hat{a}'_{i}))]\\
{}&=
\frac{1}{2}\bigg(\mathop{\mathbf{E}}_{\substack{(pk,sk)\leftarrow\mathsf{PGen}(1^k),\\a'\leftarrow\alpha^{\prime{}(k)},\hat{a}_{i}^{\prime{}}\leftarrow\hat{\alpha}_{i}^{\prime{}(k)}(a'_i)}}
[u_{i}^{\prime{}(k)}(\hat{a}_{i}^{\prime{}},a'_{-i})]
-
\mathop{\mathbf{E}}_{\substack{(pk,sk)\leftarrow\mathsf{PGen}(1^k),\\a'\leftarrow\alpha^{\prime{}(k)}}}
[u_{i}^{\prime{}(k)}(a')] + 1 \bigg).
\end{align*}
Note that the scaling needed for relation $R$ is done by some finite factor, since the game matrix of $\Gamma$
does not depend on the security parameter $k$.
Therefore, it follows from equation \ref{eqn:CCE_PK_cond} that this probability is larger than $\delta'(k)$ for some non-negligible
function $\delta'(\cdot)$.
On the other hand, the success probability of $\mathcal{A}$ in the experiment $\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{NM},\$}(k)$, i.e.
\begin{align*}
\Pr[\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{NM},\$}(k)=1]
&=
\Pr_{\substack{(pk,sk)\leftarrow\mathsf{PGen}(1^k)\\a,\tilde{a}\leftarrow\alpha^{(k)},\hat{a}'_{i}\leftarrow\hat{\alpha}_{i}^{\prime{}(k)}(\mathsf{PEnc}_{sk}(a_i))}}
[\hat{a}'_i\neq\mathsf{PEnc}_{sk}(a_i) \land R(\tilde{a}_i,\mathsf{PDec}_{sk}(\hat{a}'_{i}))]\\
{}&=
\frac{1}{2}\bigg(\mathop{\mathbf{E}}_{\substack{(pk,sk)\leftarrow\mathsf{PGen}(1^k),\\a'\leftarrow\alpha^{\prime{}(k)},\hat{a}_{i}^{\prime{}}\leftarrow\hat{\alpha}_{i}^{\prime{}(k)}}}
[u_{i}^{\prime{}(k)}(\hat{a}_{i}^{\prime{}},a'_{-i})]
-
\mathop{\mathbf{E}}_{\substack{(pk,sk)\leftarrow\mathsf{PGen}(1^k),\\a'\leftarrow\alpha^{\prime{}(k)}}}
[u_{i}^{\prime{}(k)}(a')] + 1 \bigg),
\end{align*}
can be at most $\epsilon(k)$ for some negligible function $\epsilon$.
This follows from the fact that $\alpha$ is a computational coarse correlated equilibrium,
and no independent deviation can yield a non-negligible improvement in expectation on the utility of any player $i$.
Putting the above two observations together we conclude that for some non-negligible $\delta^{*}(\cdot)$
\[
\left|\Pr[\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{NM}}(k)=1]-\Pr[\mathsf{PubK}_{\mathcal{A},\Pi}^{\mathsf{NM},\$}(k)=1]\right|\geq \delta^{*}(k),
\]
a contradiction to computational non-malleability of $\Pi$.
\end{proof}
\subsection{What can I do with an encrypted action?}\label{subsec:whatCanIDo}
We employ blinded games as a tool to achieve equilibria in the underlying game.
The pre-play protocols in the next section will issue ``advice'' to the players as \emph{encrypted actions},
that is, actions in the blinded game. In this section we address how an action of the blinded game
can be ``used'' to take a corresponding action in the underlying game.
We return to the concept of verifiability of mediation, introduced in Section~\ref{sec:intro}.
Since the players do not know the secret key associated with a blinded game,
they cannot decrypt an encrypted action (and indeed, this is an essential
property upon which the pre-play protocols will depend).
The players therefore invoke a third party who plays the underlying game \emph{on their behalf}.
The third party will act in a way which can be publicly
verified, so no trust need be placed in him to perform actions correctly: if he misbehaves, then the misconduct will be
detected and he can be held accountable.
This is in contrast to the usual idea of trusted mediation for implementation of equilibria.
The importance of reducing the trust placed in mediators has long been recognized in the literature,
and the first formal definition of a verifiable but not trusted form of mediation was given in \cite{ILM11}, which
introduced the concept of \emph{verifiable mediator}.
\begin{definition}[Verifiable mediator \cite{ILM11}]
A \emph{verifiable mediator} is a mediator which performs all actions in a publicly verifiable way,
and does not use any information that must be kept secret.
\end{definition}
We introduce the new concept of a \emph{verifiable proxy}, which is used in our construction.
Note that the new concept is incomparable to the verifiable mediator of \cite{ILM11}.
\begin{definition}[Verifiable proxy]
A \emph{verifiable proxy} is a mediator which performs all actions in a publicly verifiable way,
and does not give the players any information that affects their strategic choices in the underlying strategic game.
\end{definition}
In our setting, the (only) action that the verifiable proxy performs for the players is to \emph{translate}
the action from an encrypted form to the original form. It is well known that decryption can be done verifiably
(see Appendix~\ref{appx:verifiableDecryption} for details). Importantly, the verifiable proxy acts independently for each player:
the correlation between players' strategies is achieved by the players themselves with no external help, and the verifiable proxy acts
simply as a \emph{proxy} or interface so that the players may use encrypted actions to play in the underlying game.
We believe that (in contrast to general trusted mediators), verifiable proxies are a very realistic and mild requirement
in many scenarios, since many games are already ``set up'' by some entity (e.g. the stock exchange or an online games company),
which could easily set up instead a version of the game incorporating encrypted actions.
Moreover, the impossibility result of \cite{HNR13} shows that without any mediation, even correlated equilibria cannot in general
be achieved by cheap talk: so some weak notion of mediation is necessary in order to bypass this result and give useful correlated equilibrium implementations.
\begin{example}
More concretely, we provide a toy example involving the well-known ``Battle of the Sexes'' game (Figure~\ref{fig:battleSexes}),
where two friends are deciding on a joint activity, and they have opposing preferences but would rather be together than apart:
\begin{figure}[h!]
\bgroup
\def1{1}
\begin{center}
\begin{tabular}{r|c|c|}
\multicolumn{1}{r}{}
& \multicolumn{1}{c}{Bach ($B$)}
& \multicolumn{1}{c}{Stravinsky ($S$)} \\
\cline{2-3}
~~~Bach ($B$)~~~ & $\mathbf{2,5}$ & $0,0$ \\
\cline{2-3}
~~~Stravinsky ($S$)~~~ & $0,0$ & $\mathbf{5,2}$ \\
\cline{2-3}
\end{tabular}
\end{center}
\egroup
\caption{``Battle of the Sexes'' game}
\label{fig:battleSexes}
\end{figure}
It is a correlated equilibrium to randomize over $(B,B)$ and $(S,S)$. In this scenario, the ``encrypted advice''
could be an order to an online ticket vendor for either a Bach or Stravinsky concert,
encrypted under the public key of the vendor. The set-up assumption here would be that the online vendor has published a public key
and accepts encrypted orders. Since accepting orders in a variety of formats desirable to customers is in the vendor's interest,
we consider this to be a very feasible scenario.
Note that as this particular example is a correlated equilibrium, it is unnecessary to encrypt advice
(e.g. since the protocol of \cite{DHR00} applies). However, the example serves to illustrate
that verifiable translation can be a highly realistic and mild assumption.
\end{example}
\section{Our Protocols}\label{sec:protocols}
In this section we give cryptographic protocols (in the computational and information-theoretic settings) that
achieve the utility profile of any coarse correlated equilibrium.
\subsection{Cryptographic cheap talk}
\begin{definition}[Cheap talk extension \cite{DHR00,GLR10}]
For a strategic game $\Gamma$, the \emph{cheap talk extension} $\widetilde{\Gamma}$
is defined as an extensive game consisting of a pre-play phase
in which the players exchange messages, followed by the play in
the original strategic game. The communication is non-binding (unlike in signaling games)
in that it does not directly affect players' utilities in the underlying game,
that is, players' utilities in the cheap talk extension depend only on actions
taken in the strategic game.
The \emph{cryptographic cheap talk extension} is defined exactly like the cheap talk extension,
except that the players exchange messages
during a polynomially bounded number of rounds prior to the play in the
original game $\Gamma$.
\end{definition}
We follow the pre-play paradigm of \cite{barany1992}, where the mediator is replaced
by ``cheap talk'' communication prior to game play.
We construct protocols to be run during pre-play, which implement any (computational) coarse correlated equilibrium of blinded games
as a (computational) Nash equilibrium of the (computational) cheap talk extension.
\subsection{Protocol for computationally bounded players}
In this protocol, the players run a computationally secure multi-party computation to sample an action profile from any computational correlated equilibrium of the blinded game.
\begin{framed}
\begin{center}
\textbf{Protocol \refprot{prot:computational}.} Implementing any computational correlated equilibrium $\alpha'$ of $\Gamma^\PKE_{(pk,sk)}$:
\end{center}
Let $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ be a $\mathsf{CCA}$-secure public-key encryption scheme
and let $(pk,sk)\leftarrow\mathsf{PGen}(1^k)$ with $pk$ known to all players.
Communication is via broadcast.
\begin{enumerate}
\item The players run a computationally secure multi-party
computation protocol (secure against $t\leq N-1$ corruptions) to implement
the function that samples an action profile $a'\leftarrow\alpha'$,
and outputs to each player $i$ his action $a'_i$.
\item Every player takes $a'_i$ as its action in $\Gamma^\PKE_{(pk,sk)}$.
\end{enumerate}
\end{framed}
We show that rational computationally bounded players will follow the above protocol,
so they can use it to implement any computational correlated equilibrium.
Then, by combining the above with our results from Section~\ref{sec:blindedGames} about
correspondence of coarse correlated equilibria in the underlying game and correlated equilibria in its blinded version,
we obtain that the protocol can moreover be used to implement any computational \emph{coarse} correlated equilibrium.
Note that it is necessary to treat the two-player case somewhat differently from the case with three or more players,
because of the problem of guaranteed output delivery in the two-player case (which was described in Section~\ref{subsec:mpc}).
We begin by presenting the simpler Theorem~\ref{thm:PKprotocolThree}, which states that Protocol~\ref{prot:computational} can
be \emph{directly} used by three or more players to implement any computational coarse correlated equilibrium.
Then, we give Theorems~\ref{thm:PKprotocol} and \ref{thm:PKprotocolGeneral} which show that by running a \emph{slightly modified}
version of Protocol~\ref{prot:computational}, it is possible for \emph{any} number of players to
implement any computational coarse correlated equilibrium.
\begin{theorem}\label{thm:PKprotocolThree}
Let $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ be a $\mathsf{CCA}$-secure public-key encryption scheme,
and let $\Gamma$ be any finite strategic game with three or more players.
For any computational coarse correlated equilibrium $\alpha$ of $\Gamma$,
there exists a computational Nash equilibrium $\widetilde{\alpha}$ of the computational cheap talk
extension $\widetilde{\Gamma^\PKE_{(pk,sk)}}$ that achieves the same utility profile as $\alpha$.
\end{theorem}
\begin{proof}
Let $\alpha'$ be the computational correlated equilibrium of $\Gamma^\PKE_{(pk,sk)}$ from Lemma~\ref{lem:CCEvsCEinPKE}
that achieves the same utility profile as $\alpha$.
We show that using Protocol~\ref{prot:computational} in order to implement $\alpha'$ constitutes a computational Nash equilibrium
in the cryptographic cheap talk extension $\widetilde{\Gamma^\PKE_{(pk,sk)}}$. Note that it is payoff-equivalent to $\alpha$ by construction.
By the \emph{privacy} guarantee of the secure multi-party computation protocol, we have that no player can learn any
(non-negligible amount of) information that cannot be deduced from his intended output in the first place,
even if he deviates from the protocol arbitrarily. Moreover,
since there are three or more players and we consider only unilateral\footnote{That is, we only consider deviations
from the protocol by a single (malicious) player, rather than by coalitions of multiple colluding players.}
deviations (as implied by the definition of Nash equilibrium), the protocol has the property of
\emph{guaranteed output delivery}\footnote{We remark that in fact, the slightly weaker property of \emph{fairness} is sufficient: that is,
the property that if any player receives his output in the protocol, then every honest player will receive her correct output too.
However, in the settings we consider, the stronger property of \emph{guaranteed output delivery} is known to hold,
hence we refer to the latter property in order to slightly simplify the proof.}:
therefore, the deviation of any player $i$ cannot prevent any other player $j$ from receiving her correct output $a'_j$.
We have shown that for any player, there is no deviation during the protocol phase that is profitable by more than negligible amount.
Hence, we consider only the case where each player $i$ receives his correct output $a'_i$.
Since $\alpha'$ is, by Lemma~\ref{lem:CCEvsCEinPKE}, a computational correlated equilibrium of $\Gamma^\PKE_{(pk,sk)}$,
no player has an incentive to deviate from the prescribed advice,
and thus the players will play according to the sampled action profile $a'$.
Therefore, to follow Protocol~\ref{prot:computational} is the computational Nash equilibrium $\widetilde{\alpha}$
of $\widetilde{\Gamma^\PKE_{(pk,sk)}}$ payoff-equivalent to $\alpha$.
\end{proof}
\subsubsection{Dealing with the two-player case}\label{sec:twoPlayer}
In the two-player case, the additional complication stems from the fact that
in this setting we do not have guaranteed output delivery:
hence, it is necessary to consider that a player may be incentivized to cause a protocol execution to terminate prematurely.
In order to disincentivize such behavior, we introduce an additional ``punishment'' condition to the protocol, as follows.
\begin{framed}
\begin{center}
\textbf{Protocol \refprot{prot:compWithNE}.} Implementing any computational correlated equilibrium $\alpha'$ of $\Gamma^\PKE_{(pk,sk)}$:
\end{center}
Let $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ be a $\mathsf{CCA}$-secure public-key encryption scheme
and let $(pk,sk)\leftarrow\mathsf{PGen}(1^k)$ with $pk$ known to all players.
Communication is via broadcast.
\begin{itemize}
\item The players run Protocol \ref{prot:computational} as long as no player is detected to deviate from the protocol.
\item If any player $i$ is detected to deviate from the protocol, then all (other) players adopt the strategies (in $\Gamma^\PKE_{(pk,sk)}$) corresponding to the worst Nash equilibrium $\sigma^{i}$ for player $i$.
\end{itemize}
\end{framed}
Using Protocol~\ref{prot:compWithNE}, we obtain the following theorem that applies for \emph{any} number of players.
\begin{theorem}\label{thm:PKprotocol}
Let $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ be a $\mathsf{CCA}$-secure public-key encryption scheme,
and let $\Gamma$ be any finite strategic game.
For any computational coarse correlated equilibrium $\alpha$ of $\Gamma$ that for each player
achieves at least as high utility as the worst Nash equilibrium,
there exists a computational Nash equilibrium $\widetilde{\alpha}$ of the computational cheap talk
extension $\widetilde{\Gamma^\PKE_{(pk,sk)}}$ that achieves the same utility profile as $\alpha$.
\end{theorem}
\begin{proof}[Proof]
Let $\alpha'$ be the computational correlated equilibrium of $\Gamma^\PKE_{(pk,sk)}$ from Lemma~\ref{lem:CCEvsCEinPKE}
that achieves the same utility profile as $\alpha$.
We show that using Protocol~\ref{prot:compWithNE} in order to implement $\alpha'$ constitutes a computational Nash equilibrium
in the cryptographic cheap talk extension $\widetilde{\Gamma^\PKE_{(pk,sk)}}$.
For any security parameter $k$, the following events may occur
during the run of the protocol:
\begin{enumerate}
\item \label{itm:learnsEarly} a player learns its advice before the other players;
\item \label{itm:deviatesNoticed} a player deviates from the protocol and the deviation is detected by the other players; or
\item \label{itm:deviatesUnnoticed} a player deviates from the protocol and it is unnoticed.
\end{enumerate}
Addressing (\ref{itm:learnsEarly}): it follows from $\mathsf{CCA}{}$-security of the public-key encryption scheme $\Pi$
(Definition~\ref{def:CCAsec}) that each player is indifferent (up to a negligible improvement in utility)
between any advice he may receive, and thus gains no advantage from learning
his advice first. In particular, he has no incentive to abort the protocol
and prevent others from learning their advice.
Addressing (\ref{itm:deviatesNoticed}): the expectation of any player $i$ in the default Nash equilibrium
$\sigma^i$ is at most the expectation of player $i$ in $\alpha$.
Addressing (\ref{itm:deviatesUnnoticed}): the security of the multi-party computation protocol ensures that players can cheat
without being caught with at most negligible probability.
Thus, the increase in utility from any cheating strategy is at most negligible.
There is no deviation during the protocol phase profitable by more than negligible amount.
Consider the case that every player $i$ received his advice $a'_i$. Since $\alpha'$
is, by Lemma~\ref{lem:CCEvsCEinPKE}, a computational correlated equilibrium of $\Gamma^\PKE_{(pk,sk)}$,
no player has an incentive to deviate from the prescribed advice,
and the players will play according to the sampled action profile $a'$.
Therefore, to follow Protocol~\ref{prot:compWithNE} is the computational Nash equilibrium $\widetilde{\alpha}$
of $\widetilde{\Gamma^\PKE_{(pk,sk)}}$ payoff-equivalent to $\alpha$.
\end{proof}
It is possible to eliminate the condition (from Theorem~\ref{thm:PKprotocol}) that the implemented coarse correlated equilibrium
does at least as well as the respective worst Nash equilibrium for each player, thereby obtaining a yet more general theorem
as follows.
\begin{theorem}\label{thm:PKprotocolGeneral}
Let $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ be a $\mathsf{CCA}$-secure public-key encryption scheme,
and let $\Gamma$ be any finite strategic game.
For any coarse correlated equilibrium $\alpha$ of $\Gamma$,
there exists a computational Nash equilibrium $\widetilde{\alpha}$ of the computational cheap talk
extension $\widetilde{\Gamma^\PKE_{(pk,sk)}}$ that achieves the same utility profile as $\alpha$.
\end{theorem}
The proof of Theorem~\ref{thm:PKprotocolGeneral} makes use of another variant of Protocol~\ref{prot:computational}.
The details of this variant protocol (Protocol~\ref{prot:compWithMinMax}) are given in Appendix~\ref{appx:minMax}
along with the proof of the theorem.
We remark that Protocol~\ref{prot:compWithNE} has certain more desirable properties than Protocol~\ref{prot:compWithMinMax}:
in particular, Protocol~\ref{prot:compWithNE} is \emph{free of empty threats}, which ensures that Nash equilibria
in the protocol are stable even when players may change strategy \emph{adaptively} during protocol execution
(a formal definition of empty threats may be found in Appendix~\ref{appx:emptyThreats}).
Ultimately, notwithstanding the restriction on the class of achieved coarse correlated equilibria,
we consider Theorem~\ref{thm:PKprotocol} to be the much stronger result compared to Theorem~\ref{thm:PKprotocolGeneral},
for the following reasons:
\begin{itemize}
\item all coarse correlated equilibria that players might rationally wish
to implement by cheap talk do dominate all Nash equilibria (otherwise, they could achieve a better payoff from a Nash equilibrium
without the hassle of a pre-play protocol); and
\item unlike Protocol~\ref{prot:compWithMinMax}, Protocol~\ref{prot:compWithNE} is free of empty threats; and
\item the expected payoff even when the protocol is aborted and the default strategy invoked is higher in Protocol~\ref{prot:compWithNE} than in Protocol~\ref{prot:compWithMinMax}.
\end{itemize}
\paragraph{Strategic equivalence}
Lemma~\ref{lem:PKstrategicEquivalence}, below, proves the strategic equivalence of the
cryptographic cheap talk extension $\widetilde{\Gamma^\PKE_{(pk,sk)}}$ to the underlying game $\Gamma$.
\begin{lemma}\label{lem:PKstrategicEquivalence}
Let $\Pi=(\mathsf{PGen},\mathsf{PEnc},\mathsf{PDec})$ be a $\mathsf{CCA}$-secure public-key encryption scheme,
and let $\Gamma$ be any finite strategic game.
For any computational Nash equilibrium $\widetilde{\alpha}$ of the cryptographic cheap talk extension $\widetilde{\Gamma^\PKE_{(pk,sk)}}$,
there exists a computational coarse correlated equilibrium $\alpha$ of $\Gamma$ that achieves the same utility profile as $\widetilde{\alpha}$.
\end{lemma}
\begin{proof}
We show that the probability ensemble $\alpha$ induced by $\widetilde{\alpha}$ on action profiles of $\Gamma$
is a computational coarse correlated equilibrium of $\Gamma$.
Assume that $\alpha$ is not a computational coarse correlated equilibrium, i.e. there exists a player $i$ that has a {{\rm\sc ppt}}{}-samplable unilateral deviation to
$\alpha$ that improves his expectation for every $k\in\mathbb{N}$ by $\delta(k)$ for some non-negligible $\delta(\cdot)$.
However, such deviation can be used by player $i$ also against $\widetilde{\alpha}$ to gain a non-negligible improvement in his expectation in $\widetilde{\Gamma^\PKE_{(pk,sk)}}$,
a contradiction to the fact that $\widetilde{\alpha}$ is a computational Nash equilibrium of $\widetilde{\Gamma^\PKE_{(pk,sk)}}$.
\end{proof}
\begin{corollary}
For any finite strategic game $\Gamma$, the cryptographic cheap talk extension $\widetilde{\Gamma^\PKE_{(pk,sk)}}$ is
strategically equivalent to $\Gamma$, that is, for every Nash equilibrium $\widetilde{\alpha}$ of $\widetilde{\Gamma^\SKE_{sk}}$, there exists
a coarse correlated equilibrium of $\Gamma$ that achieves the same utility profile as $\widetilde{\alpha}$, and vice versa.
\end{corollary}
\begin{proof}
Follows immediately from Lemma~\ref{lem:PKstrategicEquivalence} and Theorem~\ref{thm:PKprotocol} (or Theorem~\ref{thm:PKprotocolThree}
for the case of three or more players).
\end{proof}
\subsection{Protocol for computationally unbounded players}
An alternative protocol using secret-key encryption implements all coarse correlated equilibria --
not just computational ones -- for all strategic games with four or more players.
As discussed in Section~\ref{sec:intro}, the condition of four or more players is unavoidable.
In this (more traditional) setting, the players are computationally unbounded.
\vspace{2em}
\begin{framed}
\begin{center}
\textbf{Protocol \refprot{prot:perfect}.} Implementing any correlated equilibrium $\alpha'$ of $\Gamma^\SKE_{sk}$:
\end{center}
Let $\Sigma=(\mathsf{SGen},\mathsf{SEnc},\mathsf{SDec})$ be a perfectly non-malleable and verifiably decryptable secret-key encryption scheme
and let $sk\leftarrow\mathsf{SGen}$.
Let each player $i$ possess a distinct share $sk_i$ of an $(N-1)$-out-of-$N$ secret-sharing $\{sk_1,\dots,sk_N\}$ of $sk$.
Communication is via pairwise channels.
\begin{enumerate}
\item The players run a perfectly secure multi-party
computation to implement the function that
samples a profile $a'\leftarrow\alpha'$,
and outputs to each $i$ his action $a'_i$.
\item Every player takes $a'_i$ as its action in $\Gamma^\SKE_{sk}$.
\end{enumerate}
\end{framed}
\begin{theorem}\label{thm:SKprotocol}
Let $\Sigma=(\mathsf{SGen},\mathsf{SEnc},\mathsf{SDec})$ be a perfectly non-malleable and verifiably decryptable secret-key encryption scheme,
and let $\Gamma$ be any finite strategic game with four or more players.
For any coarse correlated equilibrium $\alpha$ of $\Gamma$ there exists a Nash equilibrium $\widetilde{\alpha}$ of the cheap talk
extension $\widetilde{\Gamma^\SKE_{sk}}$ that achieves the same utility profile as $\alpha$.
\end{theorem}
\begin{proof}
Let $\alpha'$ be the correlated equilibrium of $\Gamma^\SKE_{sk}$ from Lemma~\ref{lem:CCEvsCEinSKE} that achieves the same utility profile as $\alpha$. We show that to follow Protocol~\ref{prot:perfect} in order to implement $\alpha'$ constitutes the Nash equilibrium $\widetilde{\alpha}$ in the cryptographic cheap talk extension $\widetilde{\Gamma^\SKE_{sk}}$ that achieves the same utility profile as $\alpha$.
First note that since the players are using a perfectly secure protocol with output guarantee (see Section~\ref{subsec:mpc}) to implement sampling from $\alpha'$, no player can prevent the others from learning their advice by a unilateral deviation during the multi-party computation phase. Moreover, even if a single player $i$ withholds its share $sk_i$ the remaining players hold $N-1$ shares of the secret key $sk$ that are sufficient to reconstruct the secret key and sample an action profile from $\alpha'$. Hence, any unilateral deviation does not influence the distribution on actions taken by the other players. Assume that there exists a unilateral deviation for some player $i$ in $\widetilde{\Gamma^\SKE_{sk}}$ that allows him to gain a higher utility than by playing according to $\widetilde{\alpha}$. This contradicts $\alpha'$ being a correlated equilibrium of $\Gamma^\SKE_{sk}$, since it could be used as a unilateral profitable deviation against $\alpha'$ in $\Gamma^\SKE_{sk}$ as well.
\end{proof}
\paragraph{Strategic equivalence}
Lemma~\ref{lem:SKstrategicEquivalence}, below, proves the strategic equivalence of the
cheap talk extension $\widetilde{\Gamma^\SKE_{sk}}$ to the underlying game $\Gamma$.
\begin{lemma}\label{lem:SKstrategicEquivalence}
Let $\Sigma=(\mathsf{SGen},\mathsf{SEnc},\mathsf{SDec})$ be a perfectly non-malleable and verifiably decryptable secret-key encryption scheme,
and let $\Gamma$ be any finite strategic game with four or more players.
For any Nash equilibrium $\widetilde{\alpha}$ of the cheap talk extension $\widetilde{\Gamma^\SKE_{sk}}$,
there exists a coarse correlated equilibrium $\alpha$ of $\Gamma$ that achieves the same utility profile as $\widetilde{\alpha}$.
\end{lemma}
\begin{proof}
We show that the distribution $\alpha$ induced by $\widetilde{\alpha}$
on action profiles of $\Gamma$ is a coarse correlated equilibrium
of $\Gamma$.
Suppose $\alpha$ is not a coarse correlated equilibrium, i.e. there exists a player $i$ that has a deviation to $\alpha$ which improves his expectation.
However, such a deviation contradicts the fact that $\widetilde{\alpha}$ is a Nash equilibrium of $\widetilde{\Gamma^\SKE_{sk}}$,
since it is also a profitable unilateral deviation against $\widetilde{\alpha}$ in $\widetilde{\Gamma^\SKE_{sk}}$.
\end{proof}
\begin{corollary}
For any game $\Gamma$, it holds that the cheap talk extension $\widetilde{\Gamma^\SKE_{sk}}$ is
strategically equivalent to $\Gamma$, that is, for every Nash equilibrium $\widetilde{\alpha}$ of $\widetilde{\Gamma^\SKE_{sk}}$, there exists
a coarse correlated equilibrium of $\Gamma$ that achieves the same utility profile as $\widetilde{\alpha}$, and vice versa.
\end{corollary}
\begin{proof}
Follows immediately from Theorem~\ref{thm:SKprotocol} and Lemma~\ref{lem:SKstrategicEquivalence}.
\end{proof}
\paragraph{Sequential equilibrium}
We also show that the equilibrium from Theorem~\ref{thm:SKprotocol} is a \emph{sequential equilibrium}
(relevant formal definitions are given in Appendix~\ref{appx:extensiveGames}):
informally, we show that by following the prescribed strategy, the players are making optimal decisions at all points in the game tree.
Our proof relies on perfect security for multi-party computation protocols in the presence of one actively corrupted and one passively corrupted party which can be achieved only for six or more players (as shown by Fitzi, Hirt and Maurer~\cite{DBLP:conf/crypto/FitziHM98}, see Section~\ref{subsec:mpc}). Hence, the statement of the following theorem is less general than the statement of Theorem~\ref{thm:SKprotocol}.
\begin{theorem}
Let $\Sigma=(\mathsf{SGen},\mathsf{SEnc},\mathsf{SDec})$ be a perfectly non-malleable and verifiably decryptable secret-key encryption scheme,
and let $\Gamma$ be any finite strategic game with six or more players.
For any coarse correlated equilibrium $\alpha$ of $\Gamma$ there exists a sequential equilibrium $(\widetilde{\alpha},\mu)$ of the cheap talk
extension $\widetilde{\Gamma^\SKE_{sk}}$ that achieves the same utility profile as $\alpha$.
\end{theorem}
\begin{proof}
We assume without loss of generality that the multi-party computation protocol has the canonical structure where at each round a single player receives a message from one of the other players (i.e. the information sets in the extensive game correspond to histories consistent with the received message). Since there is at least six players, we can assume that multi-party computation is secure in the presence of one static and one active corruption.
Consider the behavioral strategy profile $\widetilde{\alpha}$ corresponding to following Protocol~\ref{prot:perfect} at each history where a player receives a message from some other player (in particular this corresponds to ignoring all received messages after termination of the multi-party computation).
First, we specify the belief system $\mu$ of players at any information set. The beliefs at
any information set on the equilibrium path are derived from the behavioral strategy $\widetilde{\alpha}$ by Bayes' rule, and for any information set $I$
that lies off the equilibrium path (i.e. an information set corresponding to receiving a message out of the scope of the protocol), let $\mu(I)$ be the uniform distribution on all histories in $I$.
To show that $(\widetilde{\alpha},\mu)$ is a sequential equilibrium, we must show that $(\widetilde{\alpha},\mu)$ is both sequentially rational and
consistent.
Since $\widetilde{\alpha}$ is a Nash equilibrium (as shown in Theorem~\ref{thm:SKprotocol}), the behavioral strategy to follow $\widetilde{\alpha}$ is optimal for any information set on the equilibrium path.
Hence, to conclude that $(\widetilde{\alpha},\mu)$ is sequentially rational, we just need to show that $\widetilde{\alpha}$ is also optimal off the equilibrium path, given the beliefs of $\mu$. Let $I$ be an information set of player $i$ at some point off the equilibrium path that corresponds to receiving a message from player $j$. Note that even if $j$ sends to $i$ its complete view of the protocol up to this point player $i$ cannot use such information to produce a profitable deviation, since such deviation would imply an adversary corrupting actively player $i$ and statically player $j$ able to break the perfect security of the multi-party computation protocol. Now consider any history off the equilibrium path after the termination of the multi-party computation, and assume that player $i$ receives the private advice of some other player. There cannot exist a profitable deviation of player $i$, since such a deviation would contradict security of the secret key encryption scheme.
To show that $(\beta,\mu)$ is consistent we use the ``trembling-hand'' approach. Consider the sequence of assessments $\{(\beta^{(n)},\mu^{(n)})\}_{n=1}^{\infty}$ where each $\beta^{(n)}$
assigns non-zero probability $\epsilon^{(n)}$ to all actions that are taken with zero probability in $\beta$, such that $\epsilon^{(n)}$ goes to zero as $n\rightarrow\infty$,
and the belief system $\mu^{(n)}$ is derived from $\beta^{(n)}$ using Bayes' rule.
First note that the sequence $\{(\beta^{(n)},\mu^{(n)})\}_{n=1}^{\infty}$ converges to $(\beta,\mu)$.
The sequence of behavioral strategy profiles $\{\beta^{(n)}\}_{n=1}^{\infty}$ converges to $\beta$ by construction.
Since $\mu^{(n)}$ is derived from $\beta^{(n)}$ by the Bayes' rule,
$\mu^{(n)}$ converges to $\mu$ for every information set on the equilibrium path. For every information set $I$ off the equilibrium path, the distribution $\mu^{(n)}(I)$ is equal to $\mu(I)$.
Finally, $\beta^{(n)}$ is completely mixed for all $n$, hence $(\beta,\mu)$ is consistent.
\end{proof}
\subsection{Remarks on efficiency of multi-party computation}
\paragraph{Computational setting}
With recent advances in efficiency, computationally secure multi-party
computation protocols are now being considered for practical use in various settings.
Its first large-scale deployment was to compute market clearing prices for Danish sugar beet contracts in 2008 \cite{sugarbeet}.
Subsequent advances include \cite{IPS09,DO10}.
Indeed, numerous multi-party computation implementations are
available online, such as VIFF (\texttt{viff.dk}) \cite{DGKN09}.
In the common ``pre-processing model'', where pre-processing time is available
before the main computation, yet faster protocols are possible: \cite{SPDZ} achieves secure 3-party
64-bit multiplication in 0.05 ms. This could be a very reasonable
model when the same $N$ players play multiple or repeated games.
\vskip12pt
We note that there has been a line of work starting with \cite{DHR00}, on designing multi-party computation
protocols specifically for sampling from correlated equilibrium distributions.
However, these address the two-party setting, and have not taken into account the most recent advances in
general multi-party computation techniques, so we do not consider them to be of great relevance here.
\paragraph{Perfect setting}
In the perfect setting, known protocols are less efficient;
and perfectly secure encryption is relatively inefficient
due to inherently large key sizes. Nonetheless, substantial progress has been made: the best known protocol \cite{BH08}
achieves $O(N)$ communication complexity per multiplication\footnote{The circuit that the parties want to compute is usually represented
as addition and multiplication gates, and the multiplication gates have been found to be the bottleneck for multi-party computation.},
improving on previous protocols by $\Omega(N^2)$.
\iffalse
\cite{DIK10} gives one of the fastest
protocols known, which allows $N$ parties to evaluate an arithmetic circuit of size $s$, with a computational overhead\footnote{The
definition of computational overhead for multi-party computation in the perfect setting allows for an additive factor of $\mathsf{poly}(n,k,d,\log s)$
where $d$ is the circuit depth and $k$ a security parameter, which is unavoidable because all players are active in each round. See \cite{DIK10} for
further discussion.}
of $O(s\log s\log^2 N)$ arithmetic operations. Though its performance is significantly worse than protocols in the computational
setting,
\fi
We consider our information-theoretic results to be of interest primarily as
proofs of possibility, and a novel application of cryptographic techniques to game theory without computational restrictions.
Certainly, for efficiency in practice and strength of results, our computational protocols are the ones of interest.
\section{Conclusion}\label{sec:conclusion}
In this work we use standard cryptographic tools -- namely, encryption schemes --
to introduce the concept of blinded games: strategic games in which players take encrypted actions,
and in particular have the possibility to take actions they know nothing about.
Moreover, we provide cryptographic protocols that enable the players to not
rely on trusted mediators in order to achieve equilibrium payoffs.
Our approach suggest new interesting uses of cryptographic methods in game theory.
We show that our blinded games offer a viable and appealing alternative to solution
concepts based on commitment, and a particularly promising direction for
future work is to apply the paradigm of leveraging players' lack of knowledge in
order to avoid commitment, in broader settings.
\paragraph{Acknowledgements}
We are grateful to Alessandra Scafuro for raising the question of encrypting advice,
to Silvio Micali for very helpful advice on exposition, and to Jesper Buus Nielsen for detailed technical comments on the final versions.
Pavel Hub\'{a}\v{c}ek acknowledges support from the European Research Commission Starting Grant 279447;
from the Danish National Research Foundation
and The National Science Foundation of China (grant 61061130540) for the Sino-Danish Center for the Theory of Interactive Computation,
within part of this work was performed; and from the CFEM research center, supported by the Danish Strategic Research Council.
\printbibliography
|
1,108,101,566,719 | arxiv | \section{}
\section{Introduction}
The O~{\sc i} triplet lines at 7771--5~$\rm\AA$
(3s~$^{5}$S$^{\rm o}$--3p~$^{5}$P, multiplet 1) are known to play
important roles in stellar spectroscopy, since they are sufficiently strong to be
visible in stars of wide temperature range (spectral types from K to B) and provide
us with possibilities of studying not only oxygen abundances but also line-broadening
parameters (such as turbulences or projected rotational velocities).
Making use of this merit, our group has been exploited these lines for
investigating the behavior of microturbulent velocities in A--F stars (Takeda 1992)
as well as the O abundances of A--F dwarfs (Takeda \& Sadakane 1997;
Takeda et al. 2008), G--K giants (Takeda, Kawanomoto, \& Sadakane 1998;
Takeda et al. 2015), and M~67 stars from the
turn-off point through the red-giant branch (Takeda \& Honda 2015).
This time we pay our attention to B-type stars, while expecting a usefulness of
applying this O~{\sc i} triplet also to this high-temperature regime.
Studying surface oxygen abundances in B-type stars (which have comparatively high
$T_{\rm eff}$ and $M$) is especially significant in connection with rapid
rotators, since a deep mixing caused by the meridional circulation may salvage
ON-cycled material from the interior to produce more or less an underabundance
of O at the surface, similarly to an overabundance of He due to mixing of
H$\rightarrow$He processed product. Since rapidly-rotating B-type stars showing
He-enhancement are actually observed (e.g., Lyubimkov et al. 2004), O-deficiency
may be another indicator for the existence of efficient rotation-induced mixing,
and thus observational information on the oxygen abundances of a large number of B-type
stars with various rotational velocities can make an important touchstone for
the prediction from the stellar evolution theory (e.g., Georgy et al. 2013).
On the observational side, oxygen abundances of B-type stars have so far been studied
mainly by using O~{\sc ii} lines in the blue region (see, e.g., Gies \& Lambert 1992;
Kilian 1992; Cunha \& Lambert 1994; Korotin, Andrievsky, \& Luck 1999; Sim\'{o}n-D\'{\i}az 2010;
Nieva \& Przybilla 2012; Lyubimkov et al. 2013), Although these ionized oxygen lines
are useful because they are numerous in number, their application is exclusively
limited to early B stars, since the strengths of these lines quickly fall off
below $T_{\rm eff} \sim 20000$~K. This is a shortcoming from a viewpoint of
continuation/comparison with lower temperature stars (A, F, and G types),
for which neutral O~{\sc i} lines have to be used. Besides, since these previous
studies using O~{\sc ii} lines are mostly directed to sharp-lined B stars, prospect
for applying them to rapid rotators is not clear.
Meanwhile, spectroscopic investigations on the O abundances of B-type stars
using neutral oxygen lines have been less common and tend to be biased to
either late-B stars or slow rotators.
Takeda et al. (2010; hereinafter referred to as Paper~I) carried out a non-LTE analysis
on the O (and Ne) abundances of sharp-lined B stars ($v_{\rm e}\sin i \sim$~0--30~km~s$^{-1}$
and 10000~K~$\ltsim T_{\rm eff} \ltsim 23000$~K) by using O~{\sc i} 6156--8 lines
(multiplet 10), which however are inapplicable to rapid rotators because of their weakness.
Niemczura, Morel, and Aerts's (2009) LTE study on the abundances of oxygen (along with
other elements) based on weaker O~{\sc i} lines\footnote{Although they did not explicitly
publish the list of used lines, they must have exploited weaker O~{\sc i} lines
(presumably those at 6156--8~$\rm\AA$), since they stated that O~{\sc i} 7771--5 lines
were excluded because of the strong non-LTE effect.} is limited to B6--B9.5 stars.
Similarly, only late-B (B5--B9) stars were targeted in Hempel and Holweger's (2003)
determination of non-LTE oxygen abundances based on O~{\sc i} 7771-5 lines.
Among the previous abundance studies of B-type stars, most notable
in connection with our present interest would be the recent one of Nieva
and Przybilla (2012), who determined the abundance of O (along with
those of He, C, N, Ne, Mg, Si, and Fe) for 20 sharp-lined mid-to-early
B stars (16000~K $\ltsim T_{\rm eff} \ltsim$ 33000~K) by using
the state-of-the-art non-LTE technique based on high-S/N and
high-resolution spectra. While their analysis mainly depends on
a number of O~{\sc ii} lines, they also employed O~{\sc i} lines
(e,g., O~{\sc i} features at 6156--8~$\rm\AA$, 7771--5~$\rm\AA$,
and 8446~$\rm\AA$) for stars with $T_{\rm eff} \ltsim 27000$~K,
in order to check that O~{\sc i}/O~{\sc ii} ionization equilibrium
is accomplished. Although their work is regarded as an important benchmark,
only apparently slow rotators ($v_{\rm e}\sin i \ltsim 30$~km~s$^{-1}$)
were treated and rapid rotators were out of their consideration.
This situation motivated us to investigate a possibility of applying
O~{\sc i} 7771--5 lines to B stars with much wider range stellar properties; i.e.,
$T_{\rm eff} \sim$~10000--30000~K corresponding to late B through early B type,
slow as well as rapid rotators covering $0 \ltsim v_{\rm e}\sin i \ltsim$~200--300~km~$^{-1}$.\\
--- Is this O~{\sc i} triplet feature still usable for early B-type stars
($T_{\rm eff} \sim$~20000--30000~K) of various rotational velocities, despite
that its strength should be considerably weakened as most O atoms are ionized at
such high $T_{\rm eff}$ range? \\
--- How is the behavior of its strength in terms of the stellar parameters;
e.g., $T_{\rm eff}$, surface gravity ($\log g$), microturbulence ($\xi$),
and oxygen abundance? Is it suitable for abundance determination?\\
--- How much important is the non-LTE effect regarding this strong O~{\sc i} triplet?
Does the required non-LTE abundance correction depend on the atmospheric parameters?
Toward answering these questions, we first conducted non-LTE calculations on an extensive
parameter grid in order to understand how the O~{\sc i} 7771--5 strengths of B stars behave
in terms of various key parameters, and then carried out an observational study of
non-LTE oxygen abundance determinations for 34 selected late-B through early-B type stars
with diversified $v_{\rm e}\sin i$ values based on the spectra newly obtained at
Nishi-Harima Astronomical Observatory. The purpose of this paper is to report the
consequence of this investigation.
The remainder of this article is organized as follows:
The observational data of our program stars are described in section~2,
and their spectroscopic parameters are derived in section~3.
We explain our non-LTE calculations in section~4, and the procedures of abundance
determination (including profile fitting, equivalent width evaluation, and error
estimation) in section~5. The resulting behavior of non-LTE correction and the trend
of oxygen abundances of B-type stars are discussed in section~6, followed by
section~7 where the conclusions are summarized.
\section{Observational data}
The list of our 34 targets is presented in table 1, all of which are apparently
bright ($V \le 5$~mag) and selected from the extensive $v_{\rm e}\sin i$ compilation
of 1092 northern B-type stars published by Abt, Levato, and Grosso (2002),
where B supergiants and Be stars were excluded from our selection.
Our sample includes stars of diversified $v_{\rm e}\sin i$ values (from slow
rotators to rapid rotators up to $v_{\rm e}\sin i \sim 200$~km~s$^{-1}$).
These program stars are plotted on the $\log L$ vs. $\log T_{\rm eff}$
diagram (theoretical HR diagram) in figure 1, where Girardi et al.'s (2000)
theoretical evolutionary tracks corresponding to different stellar
masses are also depicted. We can see from this figure that the masses
of our sample stars are in the range between $\sim 2.5 M_{\odot}$ and
$\sim 15 M_{\odot}$.
Spectroscopic observations of these stars were carried out in the summer season
of 2015 (May through September; observation date for each star is given in table~1)
by using the Medium And Low-dispersion Longslit Spectrograph (MALLS; cf.
Ozaki \& Tokimasa 2005) installed on the Nasmyth platform of
the 2~m NAYUTA telescope at Nishi-Harima Astronomical Observatory (NHAO).
Equipped with a 2K$\times$2 K CCD detector (13.5 $\mu$m pixel), MALLS
can record a spectrum covering $\sim$~400~$\rm\AA$ (7600--8000~$\rm\AA$)
in the medium-resolution mode with the resolving power of $R \sim 12000$.
While the exposure time of one frame was typically a few minutes, several
spectral frames were co-added to improve the signal-to-noise ratio.
The reduction of the spectra (bias subtraction, flat-fielding,
spectrum extraction, wavelength calibration, co-adding of frames to improve S/N,
continuum normalization) was performed by using the ``noao.onedspec'' package of
the software IRAF\footnote{IRAF is distributed
by the National Optical Astronomy Observatories,
which is operated by the Association of Universities for Research
in Astronomy, Inc. under cooperative agreement with
the National Science Foundation.}
in a standard manner.
The S/N ratios of the resulting spectra were estimated to be around $\sim$~200--300
in most cases (except for HR~0779, for which S/N is only $\sim$~60--70), which turned out
to be sufficient for our purposes. Our final spectra in the 7700--7850~$\rm\AA$
region for each of the 34 stars are shown in figure~2.
\section{Atmospheric parameters}
The effective temperature and the surface gravity of each program star
were determined from the colors of
Str\"{o}mgren's $uvby\beta$ photometric system with the help of
Napiwotzki, Sc\"{o}nberner, and Wenske's (1993) {\tt uvbybetanew}
program,\footnote{
$\langle$http://www.astro.le.ac.uk/\~{}rn38/uvbybeta.html$\rangle$.}
where the observational data of
$b-y$, $c_{1}$, $m_{1}$, and $\beta$ were taken from Hauck and
Mermilliod (1998) via the SIMBAD database.
The resulting $T_{\rm eff}$ and $\log g$ are summarized in table 1.
The model atmosphere for each star was then constructed
by two-dimensionally interpolating Kurucz's (1993) ATLAS9
model grid in terms of $T_{\rm eff}$ and $\log g$, where
we exclusively applied the solar-metallicity models.
As to typical errors of these parameters, we may estimate
$\sim 3\%$ in $T_{\rm eff}$ and $\sim 0.2$~dex in $\log g$
for most of our sample stars (i.e., 32 stars at
10000~K~$\ltsim T_{\rm eff} \ltsim$~24000~K)
according to Napiwotzki et al. (1993; cf. section 5 therein).
We should note, however, that parameters derived from Napiwotzki et al.'s
code would not be reliable any more for stars at $T_{\rm eff} > 24000$~K,
as pointed out by Nieva (2013) based on the results of Nieva and Przybilla
(2012) who used non-LTE ionization equilibria of various species along
with the profiles of Balmer lines to determine the atmospheric parameters.
Accordingly, the parameters of the relevant two stars (HR~8238 and HR~8797;
cf. table~1) having particularily high $T_{\rm eff}$ (25231~K and 27853~K,
respectively) should be viewed with caution and their errors may be larger
than the typical amounts mentioned above.
Given that HR~779 (HD~16582) and HR~8238 (HD~205021) in our sample
were also studied by Nieva and Przybilla (2012), it is interesting
to compare our color-based $T_{\rm eff}$ and $\log g$ with their
spectroscopically determined values which may be more reliable.
The results of ($T_{\rm eff}^{\rm ours}/T_{\rm eff}^{\rm Nieva}$,
$\log g^{\rm ours}/\log g^{\rm Nieva}$) are (21747/21250, 3.63/3.80)
for HR~779 and (25231/27000, 3,63/3.95) for HR~8238. We can see
a reasonable consistency within the expected errors for the former
B2 star ($T_{\rm eff} < 24000$~K), while the discrepancy is larger
for the latter B0.5 star ($T_{\rm eff} > 24000$~K) which may be
attributed to the reason mentioned above (i.e., outside of the
reliability limit of Napiwotzki et al.'s calibration).
Regarding the microturbulence, which is necessary for abundance
determination, we had to assign an appropriate value since it was not
possible to establish this parameter based on our spectra.
Here, we divide our targets into two classes of $T_{\rm eff} < 15000$~K
and $T_{\rm eff} > 15000$~K, and tentatively assume $\xi = 1(\pm 1)$~km~s$^{-1}$
for the former and $\xi = 3(\pm 2)$~km~s$^{-1}$ for the latter, based on
the discussion in Takeda et al. (2014; cf. section~3 therein) and
Paper~I (cf. section~3 therein), respectively.
Admittedly, this is a rather rough choice; but it appears to be
reasonable as will be discussed in subsection~6.2.
\section{Non-LTE calculations}
In order to evaluate the non-LTE effect on the strength of O~{\sc i} 7771--5
triplet, we carried out non-LTE calculations for oxygen on an extensive grid
of solar-metallicity model atmospheres resulting from combinations of eleven
$T_{\rm eff}$ values (9000, 10000, 12000, 14000, 16000, 18000, 20000, 22000,
24000, 26000, 28000~K) and four $\log g$ values (3.0, 3.5, 4.0, 4.5),\footnote{
Since ATLAS9 model grid does not include $\log g= 3.0$ models at
$T_{\rm eff} \ge 27000$~K presumably because of an instability problem,
models of only three gravities ($\log g$ = 3.5, 4.0, and 4.5)
could be used for $T_{\rm eff}$ = 28000~K.}
which sufficiently cover the parameter ranges of our program stars.
The statistical equilibrium calculations for oxygen were done
in the same manner as in Paper~I (cf. subsection~4.1 therein).
Our non-LTE calculation program is based on the code described in
Takeda (1991). The atomic model of oxygen is the same as that adopted by
Takeda (2003; cf. subsection~2.1 therein), which is based on the atomic
data given in Kurucz and Bell (1995). Other various fixed collisional
and radiative rates were evaluated as described in Takeda (1992).
The non-LTE departure coefficients [$b(\tau)$]
applied to each star (to be used for non-LTE abundance determination in section~5)
were then derived by two-dimensionally interpolating this grid in terms of
$T_{\rm eff}$ and $\log g$, as was done for model atmospheres.
Toward understanding the behavior of the non-LTE effect to be discussed in
subsection~6.1, theoretical non-LTE and LTE equivalent widths ($W_{\rm N}$ and
$W_{\rm L}$) of the whole triplet were computed on this grid of models for
combinations of four oxygen abundances ([O/H]\footnote{[O/H]
[$\equiv A_{\rm star}$(O) $- A_{\odot}$(O)] is the differential
logarithmic oxygen abundance relative to the Sun, where $A$ is
defined in the usual normalization of $A$(H) = 12.00 and we employed
$A_{\odot}$(O) = 8.93 in order to maintain consistency with the
similar non-LTE grid computed by Takeda (2003) for FGK stars.
Note that this reference solar abundance (8.93), which was taken
from Anders and Grevesse (1989) and adopted as the standard value
in the ATLAS9 program, is appreciably higher by Asplund et al.'s
(2009) recently updated value (8.69) based on their 3D hydrodynamical
solar model photosphere. However, it is closer to the solar non-LTE
oxygen abundance (8.81) derived by Takeda and Honda (2005) with
the same O~{\sc i} 7771--5 lines (see the discussion in subsection~5.1
of Paper~I).} = $-0.6$, $-0.3$, 0.0, and +0.3) and three microturbulences
($\xi$ = 1, 3, and 5~km~s$^{-1}$) by using Kurucz's (1993) WIDTH9 program\footnote{
The original WIDTH9 program had been considerably modified in various respects.
In order to enable simulation of line profiles by taking into account the
non-LTE effect, the LTE line opacity and the LTE line source function
usually used in the WIDTH9 program were multiplied by the depth-dependent
NLTE-to-LTE line opacity ratio ($l_{0}^{\rm NLTE}/l_{0}^{\rm LTE}$) and
the NLTE line source function ratio ($S_{\rm L}/B$) resulting from our non-LTE
calculation code, respectively (cf. subsection 4.2 in Takeda 1991 for more details).
Besides, since the original WIDTH9 program can handle only a single isolated line,
a spectrum synthesis technique was incorporated for calculating the total equivalent
width of a multi-component feature (such as O~{\sc i} 7771--5).}
with the atomic data presented in table~2, where $W$ was evaluated by directly
integrating the synthesized profiles of triplet lines over the wavelength region between
7770.44~$\rm\AA$ and 7776.94~$\rm\AA$ (i.e., 1.5~$\rm\AA$ margin both shortward
of the 7771.94 line and longward of the 7775.39 line).
We also evaluated the corresponding non-LTE abundance corrections ($\Delta$)
defined as $A_{\rm N}- A_{\rm L}$, where $A_{\rm N}$ and $A_{\rm L}$
are the abundances derived by inversely analyzing $W_{\rm N}$ in non-LTE
and LTE, respectively. The resulting $W$ and $\Delta$ values are summarized
in the electronic data tables (file name: ncor7773\_xi?.dat) available
as on-line materials. The behaviors of the computed $W$ and $\delta$ in terms of
the atmospheric parameters are shown in figure 3 for selected representative cases,
on which we will focus later in subsection~6.1.
\section{Abundance determination}
As in Paper~I, our strategy of determining the abundance and related quantities
(e.g., non-LTE correction, uncertainties due to ambiguities of atmospheric
parameters) consists of the following consecutive steps:
(i) derivation of provisional abundance solution
by applying the spectrum-fitting technique, (ii) evaluation of
the equivalent width ($W$) by using this fitting-based abundance,
and (iii) analysis of such derived $W$ with the standard parameters
and changing the conditions (perturbed parameters, LTE and non-LTE).
\subsection{Synthetic spectrum fitting}
We first searched for the solutions for the oxygen abundance ($A$), projected
rotational velocity ($v_{\rm e}\sin i$), and radial velocity ($V_{\rm rad}$)
such as those accomplishing the best fit (minimizing $O-C$ residuals)
between the theoretical and observed spectrum in the $\sim$~7765--7780~$\rm\AA$
region, while applying the automatic fitting algorithm (Takeda 1995).
In this preparatory step, we used the model atmosphere (cf. section~3) and
the non-LTE departure coefficients (section~4) prepared for each star
but adopted the same microturbulence of $\xi =3$~km~s$^{-1}$
for computing the non-LTE theoretical spectrum. Regarding the macroscopic
line-broadening function to be convolved with the intrinsic spectrum,
we included the instrumental broadening (assumed to be the Gaussian function)
corresponding to $R\sim 12000$ and the rotational broadening with the limb-darkening
coefficient of $\epsilon = 0.3$ (estimated from Fig.~17.6 of Gray 2005)
How the theoretical spectrum for the converged solutions fits well
with the observed spectrum is demonstrated in figure~4.\footnote{Note that,
in the evaluation of $O-C$ residuals, we sometimes masked some regions
showing features irrelevant to stellar spectra, such as
spurious spectrum defect, as highlighted in green in this figure.}
The resulting $v_{\rm e}\sin i$ values are presented in table 1,
which are also compared with those given in Abt et al.'s (2002)
catalogue in figure~5.
\subsection{Derivation and analysis of equivalent width}
Next, we inversely computed the equivalent width ($W$) for the whole
O~{\sc i} 7771--5 triplet (as done in section~4)
by using the abundance solution (derived from the fitting analysis)
along with the same model atmosphere/parameters.
Based on such evaluated $W$ value, the non-LTE abundance ($A_{\rm N}$)
as well as LTE abundance ($A_{\rm L}$) were freshly computed by applying
the microturbulence assigned to each star (1 or 3~km~s$^{-1}$ depending on
$T_{\rm eff}$; cf. section~3), from which the non-LTE correction ($\Delta$)
was further derived. These $W$, $A_{\rm N}$, and $\Delta$ values are also
given in table 1.
\subsection{Sensitivity to parameter perturbation}
We also estimated the errors in $A_{\rm N}$ caused by ambiguities
involved in the standard atmospheric parameters (presented in table~1).
Since their typical uncertainties are $\pm 3\%$ in $T_{\rm eff}$,
$\pm 0.2$~dex in $\log g$, and $\pm 1$~km~s$^{-1}$ (for $T_{\rm eff} < 15000$~K)
or $\pm 2$~km~s$^{-1}$ (for $T_{\rm eff} > 15000$~K) in $\xi$ (cf. section 3).
six kinds of abundance variations ($\delta_{T+}$, $\delta_{T-}$,
$\delta_{g+}$, $\delta_{g-}$, $\delta_{\xi+}$, and $\delta_{\xi-}$) were
derived by repeating the analysis on the $W$ values while perturbing
the standard atmospheric parameters interchangeably by these amounts.
Finally, the root-sum-square of these perturbations,
$\delta_{Tg\xi} \equiv (\delta_{T}^{2} + \delta_{g}^{2} + \delta_{\xi}^{2})^{1/2}$,
was regarded as the abundance uncertainty (due to combined errors in
$T_{\rm eff}$, $\log g$, and $\xi$),
where $\delta_{T}$, $\delta_{g}$, and $\delta_{\xi}$ are defined as
$\delta_{T} \equiv (|\delta_{T+}| + |\delta_{T-}|)/2$,
$\delta_{g} \equiv (|\delta_{g+}| + |\delta_{g-}|)/2$,
and $\delta_{\xi} \equiv (|\delta_{\xi+}| + |\delta_{\xi-}|)/2$,
respectively.
The resulting abundances ($A_{\rm N}$), equivalent widths ($W$),
non-LTE corrections ($\Delta$), and abundance variations ($\delta$'s)
in response to parameter changes, are graphically displayed as functions
of $T_{\rm eff}$ in figure~6, which we will discuss in subsection~6.2.
\section{Discussion}
\subsection{Behavior of line strength and non-LTE correction}
Before discussing the results of our analysis derived in section~5 for
the program stars, we first review how the strength of O~{\sc i} 7771--5
triplet and its non-LTE effect behaves in terms of the atmospheric parameters.
The following characteristics can be summarized by inspection of figure~3:\\
--- Generally, $W$ declines with an increase in $T_{\rm eff}$.
While its change is gradual at $T_{\rm eff} \ltsim 25000$~K, a rather
abrupt drop of $W$ occurs at $T_{\rm eff} \gtsim 25000$~K where oxygen
is predominantly ionized and very few neutral oxygen atoms remain.\\
--- Naturally, $W$ increases with an increase in [O/H] and $\xi$,
though it is almost $\xi$-independent at $T_{\rm eff} \gtsim 25000$~K
where the line is weak in the linear part of the curve of growth.\\
--- The behavior of $W$ in response to changing $\log g$ is somewhat
complicated (figure~3c). While the line strength slightly grows with a lowering
of $\log g$ at $T_{\rm eff} \ltsim 25000$~K (because of the increased non-LTE effect;
note that this tendency is seen only in $W_{\rm N}$ but not in $W_{\rm L}$),
the trend is reversed at $T_{\rm eff} \gtsim 25000$~K
where $W$ markedly weakens with a decrease in $\log g$ because the effect of
enhanced ionization becomes more important at the condition of lower-density.\\
--- The non-LTE effect always acts in the direction of intensifying
the line strength ($W_{\rm N} > W_{\rm L}$), which makes
the non-LTE abundance correction ($\Delta$) always negative.\\
--- The extent of $|\Delta|$ has a peak at $T_{\rm eff} \sim$~15000--20000~K
and tends to be larger for higher [O/H], smaller $\xi$, and lower $\log g$,
which can be interpreted that the non-LTE effect becomes more significant
when the line-forming depth is higher or in the lower density region.
\subsection{Trend of oxygen abundances}
We now discuss figure~6. The equivalent widths ($W$) derived in
subsection~5.2 show a decreasing trend with an increase in $T_{\rm eff}$
(figure~6a), which is reasonably consistent with the theoretically
predicted trend for $W_{\rm N}$ (cf. figure~3a--c).
The $|\Delta|$ values are in the range of $\sim$~0.6--1.7~dex (figure~6b),
which are so large that these corrections are indispensable for reliable
determination of oxygen abundances of B-type stars based on the
O~{\sc i}~7771--5 lines. Comparing these results with those of Hempel and
Holweger (2003), who determined the non-LTE oxygen abundances of B5--B9 stars
($T_{\rm eff} \sim$~9000--17000~K) based on the same O~{\sc i} triplet,
we see that their $|\Delta|$ values ($\sim$~0.3--1.2; cf. their table~5)
are considerably smaller than ours. We suspect that they underestimated
the importance of the non-LTE effect, which may explain the significantly
supersolar [O/H]$_{\rm NLTE}$ results (by $\sim$~+0.3--0.6~dex)
they obtained for many of their sample stars.
Figure~6c suggests that the oxygen abundances derived for the 34 B-type
stars distribute around $\sim$~8.7--8.8 without any systematic
$T_{\rm eff}$-dependence. Actually the average abundance over all 34 stars
makes $\langle A \rangle = 8.72$ (with standard deviation $\sigma = 0.16$).
If three stars with $A$(O)~$< 8.4$ (HR~6779, 7236, 8238) showing appreciable
deviations are excluded, we have $\langle A \rangle = 8.76$ ($\sigma = 0.10$).
This means that the surface oxygen abundances of these B-type stars
are mostly near to the solar composition (see footnote~5), which is
in fairly good agreement with
the previous studies for B-type stars; e.g., $8.71 \pm 0.06$ derived in
Paper~I based on O~{\sc i} 6156--8 lines or $8.73 \pm 0.13$ by Lyubimkov et al.
(2013) based on O~{\sc ii} lines (see also table~6 and table 7 therein).
Regarding the abundance errors caused by uncertainties in atmospheric parameters,
$\delta_{\xi}$ ($\sim$~0.1--0.2~dex) is predominantly important over
$\delta_{T}$ ($< 0.1$) or $\delta_{g}$ ($< 0.1$) at $T_{\rm eff} \ltsim 25000$~K,
to which most of the program stars are relevant (cf. figure 6d--f). Accordingly,
the extents of $\delta_{tg\xi}$ (error bars attached to the symbols in figure~6c)
are essentially determined by $\delta_{\xi}$ and on the order of $\sim$~0.1-0.2~dex
(except for the two stars of $T_{\rm eff} \gtsim 25000$~K),
which is consistent with the standard deviation of our abundance results.
It may appropriate here to comment on the assigned values of $\xi$
mentioned in section~3. Since the abundances derived for given
microturbulences of $\xi = 1 (\pm 1)$~km~s$^{-1}$ ($T_{\rm eff} < 15000$~K)
and $\xi = 3 (\pm 2)$~km~s$^{-1}$ ($T_{\rm eff} > 15000$~K) do not show
any systematic trend in terms of $T_{\rm eff}$, this may imply that our choice
is reasonably justified. If a constant $\xi$ were assigned
for all stars, we would have obtained a spurious $T_{\rm eff}$-dependence.\footnote{
Actually, the first provisional abundances derived from the spectrum synthesis
by tentatively using $\xi = 3$~km~s$^{-1}$ (cf. subsection~5.1) showed
a tendency of appreciable underabundance in late B-type stars.}
The oxygen abundances of 34 stars are plotted against $v_{\rm e}\sin i$
as well as $\log L$ in figure~7, where we can not see any systematic
dependence. This means that no appreciable O abundance anomaly is detected
in our sample stars at least within the precision of $\ltsim$~0.1--0.2~dex.
According to Georgy et al.'s (2013) calculation, while no change in the
surface composition is produced at all during the main sequence period
for the case of non-rotating stars, rapidly rotating stars may show
some abundance peculiarities caused by the
rotational mixing of nuclear processed product.
For example, regarding the cases of solar-metallicity ($Z = 0.014$) models,
the changes of He$|$O abundances at the end of the main sequence
for 4~$M_{\odot}$ star is $+0.2\%|-0.7\%$ ($\Omega/\Omega_{\rm crit}$ = 0.5)
and $+3.1\%|-6.2\%$ ($\Omega/\Omega_{\rm crit}$ = 0.95), while those
for 9~$M_{\odot}$ star is $+1.9\%|-4.9\%$ ($\Omega/\Omega_{\rm crit}$ = 0.5)
and $+11.4\%|-17.3\%$ ($\Omega/\Omega_{\rm crit}$ = 0.95), where $\Omega_{\rm crit}$
is the angular rotational velocity corresponding to the critical break-up.
However, even if such an abundance change as large as $\sim$~20\% exists, it
corresponds to only $\ltsim 0.1$~dex in the logarithmic scale. Thus,
it is no wonder that we could not detect such a subtle amount, given that
the precision of our abundance determination is $\ltsim 0.2$~dex.
In this sense, we may state that our result is more or less consistent with
the recent stellar evolution calculations based on the canonical mixing theory.
\subsection{Usability of O~I triplet lines for abundance determination}
Finally, we turn to the question which motivated this investigation (cf. section~1):
Is the O~{\sc i} 7771--5 triplet practically useful for oxygen abundance
determination of B-type stars? Our answer is as follows:\\
--- Yes, it is surely useful as far as stars of $T_{\rm eff} \ltsim 25000$~K
are concerned (given that the non-LTE effect is properly taken into account),
since its strength is still sufficiently large ($W \gtsim 200$~m$\rm\AA$)
and abundances are not seriously affected by uncertainties in the atmospheric
parameters.\\
--- For example, if we consider a case of
$v_{\rm e}\sin i \sim 300$~km~s$^{-1}$ and $W \sim 200$~m$\rm\AA$,
the FWHM and the central depth of the merged triplet feature would
be $\sim 10$~$\rm\AA$ and $\sim 2$\%, respectively, which is measurable
without difficulty on a spectrum with S/N ratios of a few hundred.\\
--- In contrast, regarding very early B stars of $T_{\rm eff} \gtsim 25000$~K
(corresponding to $\sim$~B0), the strength of this triplet quickly drops away
and its sensitivity to ambiguities of $T_{\rm eff}$ or $\log g$ becomes
seriously large, which makes it unsuitable as a reliable abundance indicator.\\
--- We should keep in mind, however, even for the well-behaved case of
$T_{\rm eff} \ltsim 25000$~K stars, the precision of abundance derivation would
be on the order of $\ltsim 0.2$~dex, which is mainly restricted by uncertainties
in the choice of microturbulence. Accordingly, very high-precision should not be
expected.\\
--- In this sense, detection of surface oxygen abundance anomaly based on
these O~{\sc i} lines would be feasible only if the extent of peculiarity
is sufficiently large (e.g., by $\sim 0.3$~dex), such as the case where
non-canonical deep mixing is relevant. The slight abundance change
(up to $\ltsim 0.1$~dex) predicted by the current canonical rotational mixing
theory can hardly be detected.
\section{Summary and conclusion}
In order to examine whether the O~{\sc i} triplet lines at 7771--5~$\rm\AA$ can
be used as a reliable oxygen abundance indicator for B-type stars of high
$T_{\rm eff}$ including rapid rotators, we carried out oxygen abundance determinations
for selected 34 B-type stars based on the spectra ($R\sim 12000$) obtained by
the 2~m NAYUTA Telescope at Nishi-Harima Astronomical Observatory.
Regarding the atmospheric parameters, $T_{\rm eff}$ and $\log g$ were
determined from $uvby\beta$ colors and the microturbulence was assumed
to be $\xi = 1 (\pm 1)$~km~s$^{-1}$ ($T_{\rm eff} < 15000$~K)
and $\xi = 3 (\pm 2)$~km~s$^{-1}$ ($T_{\rm eff} > 15000$~K)
by consulting previous studies. The non-LTE effect was properly
taken into consideration based on our non-LTE calculations
carried out for an extensive grid of models.
We first determined the provisional abundance and $v_{\rm e}\sin i$
by applying the synthetic spectrum-fitting analysis to the 7771--5 feature,
and then inversely determined (from the tentative abundance) the
equivalent width of the whole triplet to be used for further
evaluations of non-LTE/LTE abundances, non-LTE correction, and
errors due to ambiguities of atmospheric parameters.
It was found that the extent of the (negative) non-LTE correction is
appreciably large to be $|\Delta| \sim$~0.6--1.7~dex and its
consideration is indispensable. The resulting non-LTE oxygen abundances
turned out to be nearly solar ($\langle A \rangle \sim$~8.7--8.8 with
$\sigma \ltsim 0.2$~dex) without any clear dependence upon rotation
as well as luminosity (or mass), which is consistent with the results
of published observational studies and stellar evolution calculations
with the canonical mixing theory.
We concluded that this triplet is a useful O abundance indicator (with a
precision of $\ltsim 0.2$~dex) up to $T_{\rm eff} \ltsim 25000$~K, since
its total equivalent width is sufficiently large ($\gtsim 200$~m$\rm\AA$).
In contrast, it is not adequate for abundance determination for stars at
$T_{\rm eff} \gtsim 25000$~K, where its strength rapidly turns down and
its sensitivity to $T_{\rm eff}$ or $\log g$ becomes considerably large.
\bigskip
This research has made use of the SIMBAD database, operated by
CDS, Strasbourg, France.
|
1,108,101,566,720 | arxiv | \section{Introduction}
\label{sec:intro}
Dark matter is an enduring mystery. Despite ironclad evidence for the gravitational effects of dark matter in astrophysical and cosmological observations, the particle characteristics of dark matter are still obscure.
In fact, dark matter may be part of an intricate dark sector, and the dark sector may communicate with the Standard Model (SM) through ``mediator'' particles.
The masses and couplings of the dark matter and mediator particles are not constrained by ab-initio principles, and can, in theory, span many orders of magnitude. (For comprehensive reviews of dark matter and many original references, see e.g.~\cite{Bertone:2004pz,Freese:2008cz,Hooper:2009zm,Feng:2010gw,Garrett:2010hd,Bauer:2017qwy,Buckley:2017ijx}.)
In recent years, dark matter (and its mediators) in the \text{MeV}-\text{GeV}\ mass range has received substantial attention~\cite{Essig:2013lka,Alexander:2016aln, Battaglieri:2017aum, Beacham:2019nyx, Alimena:2019zri,Alimena:2021mdu,Bernal:2017kxu,Curtin:2018mvb} as a compelling alternative to more conventional (and increasingly heavily constrained) dark matter at the weak scale.
Dark sector models, with light mediators of various spin-parity assignments and diverse couplings, give rise to distinctive signatures that are potentially detectable in a variety of experiments, spanning a large range of energies. Of particular interest are cases in which the mediators are stable, or characteristically long-lived, and propagate macroscopic scales uninterrupted. In these cases, the mediators can be detected using their displaced decay signatures~\cite{Konaka:1986cb,Riordan:1987aw,Bjorken:1988as,Bross:1989mp,Davier:1989wz,NOMAD:2001eyx,NA64:2018lsq,LHCb:2017trq,Bernardi:1985ny,Blumlein:1990ay,Blumlein:1991xh,CHARM:1985anb,Bjorken:2009mm,Andreas:2012mt,Blumlein:2011mv,Blumlein:2013cua,Gninenko:2012eq,Gninenko:2011uv,CHARM:1985nku,LSND:1997vqj,Essig:2010gu,Williams:2011qb}, or by their recoil on the event (i.e. the missing energy or momentum)~\cite{BaBar:2017tiz,NA64:2017vtt,DELPHI:2003dlq,DELPHI:2008uka,Essig:2013vha,Davoudiasl:2014kua,NA64:2016oww,Fayet:2007ua}.
In this paper, we explore the sensitivity of the proposed MUonE experiment~\cite{Abbiendi:2677471} to long-lived mediators that decay visibly, and manifest as displaced vertex signatures.
MUonE aims to shed light on the discrepancy between the measurement of the muon anomalous magnetic moment, $(g-2)_\mu$~\cite{Muong-2:2006rrc,Muong-2:2021ojo},
and the theoretic prediction~\cite{Aoyama:2020ynm}, a $4.2\sigma$ anomaly.
The experiment's primary purpose is to directly constrain the contribution of hadronic vacuum polarization (HVP) diagrams to $(g-2)_\mu$ through precise differential measurements of elastic $\mu-e$ scattering. The HVP contribution to $(g-2)_\mu$ is currently the largest source of uncertainty, and MUonE would provide a compelling alternative to existing approaches to the non-perturbative HVP calculation (the $R$-ratio method~\cite{Aoyama:2020ynm} and the lattice~\cite{Borsanyi:2020mff}), which currently have some disagreement between them~\cite{Lehner:2020crt}.
The experiment plans to collide high-energy muons from the CERN M2 beamline onto the atomic electrons in an array of thin Beryllium targets, and measure the electron and muon lab-frame scattering angles with high-accuracy~\cite{Marconi:2019rio,Venanzoni:2018ktr,Abbiendi:2020sxw}.
To this end, MUonE employs a tracking system of very good resolution~\cite{Ballerini:2019zkk, Abbiendi:2677471}. Here, we point out that this makes MUonE extremely sensitive to displaced vertex signatures on the order of a few $\rm mm$ to several $\rm cm$.
(See~\cite{Dev:2020drf,Masiero:2020vxk,Asai:2021wzx} for other proposals to use MUonE to search for new physics, and~\cite{Chen:2017awl,Kahn:2018cqs,Chen:2018vkr,Gninenko:2019qiv,Galon:2019owl} for other proposals to search for light mediators using muon beams.)
To be concrete, we demonstrate a proof of concept using the popular ``vanilla'' dark photon model~\cite{Okun:1982xi,Galison:1983pa, Holdom:1985ag, Boehm:2003hm, Pospelov:2008zw}. In this case, the massive vector boson of a new (broken) $U(1)$ gauge symmetry kinetically mixes with the SM photon and inherits attenuated photon-like couplings. These allow the dark photon to be produced at MUonE in the $2\to 3$ process $\mu^\pm +e^-\to \mu^\pm+e^-+A'$, and allow the dark photon to decay to $e^+e^-$ pairs. The final state is therefore
four leptons, $\mu^\pm e^-e^+e^-$ in which one of the $e^+e^-$ pairs forms a vertex displaced from the target.
MUonE tracking capabilities enable a very low-background search for displaced $e^+e^-$ vertices of decay length $\sim 10-100$~mm. In turn, this allows MUonE to have excellent discovery potential to a wide swath of currently uncovered dark photon parameter space, in the range $m_A\sim 10-100$~MeV and $\epsilon e \sim 10^{-5}-10^{-3}$.
The outline of our paper is as follows. \secref{MUonE_exp} gives a brief description of the MUonE experiment. \secref{model} introduces the dark photon model and how it enables dark photon production and decay at MUonE. \secref{4lep_search} describes our proposed search for displaced vertices at MUonE. Results in the dark photon parameter space are presented in \secref{results}, followed by conclusions in \secref{conclusions}. Additional details are delegated to Appendix~\ref{sec:PSconstraints}-~\ref{sec:contours}.
\section{The $\text{MUonE}$ experiment}
\label{sec:MUonE_exp}
The MUonE experiment~\cite{Abbiendi:2677471} aims to measure the HVP contribution to $(g-2)_\mu$ using the techniques outlined in~\cite{Abbiendi:2016xup,Fael:2019nsf, Banerjee:2020tdt} (see also~\cite{CarloniCalame:2015obs}).
An extremely precise measurement of the differential cross-section for the elastic scattering process
\begin{equation}
\label{eq:2TO2_process}
\mu^\pm\, e^- ~\to~ \mu^\pm\, e^- \,,
\end{equation}
is used to extract the scale dependence (running) of the electromagnetic fine-structure constant $\alpha$ in the space-like region.
The measurement of $\alpha(t)$ is subsequently used as input for the evaluation of the HVP contribution~(see~\cite{Banerjee:2020tdt,Fael:2019nsf} for details).
To carry out this measurement, the MUonE experiment proposes to collide $160$~GeV muons from the CERN M2 beamline with the atomic electrons of thin Beryllium targets organized into a series of $40$~consecutive, identical, and aligned modules. Each module consists of one target, followed by three tracking layers, with cross-sectional area designed to be a $10~\mathrm{cm} \times 10~\mathrm{cm}$ square. To achieve the resolutions required for the precision measurement of the running of $\alpha$, MUonE proposes to use a CMS-based tracking apparatus~\cite{Abbiendi:2677471}, with a resolution on the outgoing angle reaching as low as $0.02~\mathrm{mrad}$.
Downstream from this setup, an electromagnetic calorimeter (ECAL) and a muon spectrometer are located and used for particle identification. Possible muon-electron ambiguity is resolved in the ECAL (the muon spectrometer is used for reducing pion contamination)
as long as the energy of the outgoing electron/muon is at least order GeV~\cite{Venanzoni:2018ktr,Abbiendi:2020sxw}. No magnetic field is employed, so charge discrimination is not possible. A schematic of the experimental setup is shown in \figref{experiment}.
A displaced vertex signature is defined as the intersection of reconstructed tracks away from the nearest upstream target. Due to its high resolution tracking apparatus, MUonE is extremely efficient in detecting displaced vertices for opening angles between tracks which are as low as $0.1~\mathrm{mrad}$~\cite{Clara:talk}.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\linewidth]{Figures/experiment_scheme.pdf}
\caption{Structure of the MUonE experiment and the 2 to 3 process $\mu^\pm e^- \rightarrow \mu^\pm e^- A^\prime$. Orange lines: incoming muon beam and outgoing muons. Blue dashed line: outgoing dark photon. Red line: outgoing electron from scattering. Green lines: decay products $e^+ e^-$ from dark photon. Black layers: tracking stations. Gray layers: thin $Be$-targets. Green: ECAL. Blue: muon spectrometer. The distance between two adjacent targets is $100~\mathrm{cm}$. Notice that the interaction may happen at any of the 40 targets.}
\label{fig:experiment}
\end{figure}
\section{The Dark Photon Model}
\label{sec:model}
The ``Vanilla'' dark photon model~\cite{Okun:1982xi,Galison:1983pa, Holdom:1985ag, Boehm:2003hm, Pospelov:2008zw} is a popular simplified model which is used as a benchmark scenario for exploring light, weakly-coupled, long-lived particles~\cite{Essig:2013lka,Alexander:2016aln, Battaglieri:2017aum, Alimena:2019zri}.
In this model, a dark sector Abelian gauge symmetry, $U(1)_D$, is broken, and kinetically mixes with the SM hypercharge gauge factor.
Dark gauge bosons that are much lighter than the weak-scale, predominantly mix with the
unbroken $U(1)_{EM}$ factors, and their induced couplings are photon-like~\cite{Curtin:2014cca, Feng:2016ijc, Izaguirre:2017bqb}.
Focusing on this, and denoting the dark-photon, and its field-strength by $A'$, and $F'$ respectively, the model reads
\begin{equation}
{\cal L}_{A'} = -\frac14 F'_{\mu\nu}F'^{\mu\nu} -\frac12 m_{A'}^2 A'_\mu A'^\mu
-\epsilon e A'_\mu J_{EM}^\mu
\end{equation}
where the dark photon interaction with the electromagnetic current, $J_{EM}$ is suppressed by the kinetic mixing parameter, $\epsilon$, $J_{EM}^\mu = \sum_k Q_k \bar\psi_k \gamma^\mu \psi_k$, and $Q_k$ is the electric charge of the fermion field $\psi_k$ in units of $e$.
In this work, we assume that the dark photon decays predominantly visibly to $e^+e^-$ pairs.
Such a scenario arises naturally for a dark photon mass is in the range $2m_e \le m_{A'} < 2 m_\mu$, when the dark photon decays to the dark sector are suppressed (due to a small coupling or phase-space).
\subsection{Dark Photon Production in $\mu\,e$ collision}
\label{sec:Ap_production}
In the MUonE experiment, dark photons can be produced in the collision of beam muons with the target's atomic electrons in the $2\to3$ process
\begin{equation}
\label{eq:2TO3_process}
\mu^\pm\, e^- ~\to~ \mu^\pm\, e^- A'
\end{equation}
The amplitude for this process is the sum of four Feynman diagrams shown in Fig.~\ref{fig:Feynman_diagrams}.
In this work, we simulate events using \textsc{MadGraph5\_aMC@NLO v3.1.0}~\cite{Mattelaer:2021xdr,Alwall:2011uj,Alwall:2014hca,Frixione:2021zdp}, with UFO file implemented in \textsc{FeynRules 2.3.32}~\cite{Degrande:2011ua,Alloul:2013bka}. For the mass range $[2~\text{MeV},200~\text{MeV}]$ that we are interested in, the cross section $(\epsilon e)^{-2} \sigma$ varies from $10^9~\rm pb$ to $10^5~\rm pb$, decreasing as mass increases. (See also~\cite{Ilten:2016tkc} for a data-driven approach to dark-photon production cross-section estimation.)
\begin{figure}[h!]
\includegraphics[width=0.5\textwidth]{Figures/dark_photon_diags.pdf}
\caption{The four Feynman diagrams (overlaid) for dark photon production in $\mu\,e^-$ collisions. A dark photon can be radiated off each of the incoming or outgoing lepton lines.
}
\label{fig:Feynman_diagrams}
\end{figure}
In $2\to3$ scattering processes, when the projectile (in this case the muon) is heavier than the target,
phase-space constraints dictate that the emission angle of the outgoing projectile are bounded above, while their energies are bounded above and below (see \appref{PSconstraints}).
Similar constraints apply to the emitted dark photon angle and energy. For the case of MUonE, density plots in the energy vs.\ angle plane for the outgoing muon, electron, and the dark photon are shown in \figref{2dhist}.
\begin{figure}[ht]
\includegraphics[width=0.8\linewidth]{Figures/electron_boundary.pdf}
\includegraphics[width=0.8\linewidth]{Figures/muon_boundary.pdf}
\includegraphics[width=0.8\linewidth]{Figures/X_boundary.pdf}
\caption{Density plots for the energy and outgoing angle of the electron, muon, and the dark photon for dark photon mass $m_{A^{\prime}} = 2, 20, 200~\text{MeV}$ and coupling $\epsilon e = 1.0$. Vertical and horizontal lines: limit on the maximal/minimal angle and energies from the phase space constraints. Curves: phase space constraints.}
\label{fig:2dhist}
\end{figure}
\subsection{Dark Photon Decays}
The total decay width of the dark photon is given by (again, we assume that the dark photon mass is such that $A'$ decays only to $e^+e^-$ in this work)
\begin{equation}
\label{eq:Gamma_Ap}
\Gamma_{A'} = \Gamma(A' \to e^+e^-)= \frac{(\epsilon e)^2}{12\pi}m_{A'}\left(
1 - 4 \frac{m_e^2}{m_{A'}^2}
\right)
\sqrt{
1 + 2\frac{m_e^2}{m_{A'}^2}
}
\end{equation}
The dark photon decay length (in the lab frame) is given by
\begin{equation}
\bar d_{A'}
=\frac{\gamma_{A'}\beta_{A'}}{\Gamma_{A'}} = \frac{p_{A'}}{m_{A'} \Gamma_{A'}}
\approx
74~\mathrm{mm}
\left(\frac{10^{-4}}{\epsilon e}\right)^2
\left(\frac{p_{A'} }{ 10~\text{GeV}}\right)
\left(\frac{ 10~\text{MeV}}{m_{A'}}
\right)^2
\label{eq:dbarAp}
\end{equation}
where we use the characteristic dark photon momentum, $p_{A'}$, from \figref{2dhist}.
In the decay $A' \to e^+e^-$, given that $A'$ is sufficiently energetic, the characteristic opening angle between the electron, and positron is given by
\begin{equation}
\label{eq:opening angle}
\theta_{ee} \simeq \frac{2 m_{A^\prime}}{E_{A^\prime}}
\end{equation}
The minimal energy of $E_{A^\prime}$, shown in \figref{2dhist} implies an upper bound on $\theta_{ee}$. This upper bound, together with the upper bound on $\theta_{A^\prime}$, constraints the decay products inside the fiducial volume. We discuss this in \secref{4lep_search}. Analytical expressions of phase space constraints can be found in \appref{PSconstraints}.
\section{The Displaced Vertex Search}
\label{sec:4lep_search}
Beam muons can scatter off the target atomic electrons in any one of the 40 MUonE modules. The apparatus of the MUonE experiment (see \figref{experiment}) can search for dark photons that are produced in $\mu\, e$ scattering (\eqref{2TO3_process}) and subsequently decay to $e^+e^-$ pairs.
The search strategies discussed here leverage MUonE capabilities to identify displaced vertices, determine particle species, and reject various potential background.
\begin{figure}[h]
\centering
\includegraphics[width=0.7\linewidth]{Figures/apparatus_close.pdf}
\caption{Close look at the decay volume and angular acceptance. Gray: target where collision happens. Black: tracking layers. Blue dashed line: dark photon trajectory. Green lines: decay products $e^+ e^-$ pair. Orange line: outgoing muon. Red line: Outgoing electron from scattering at the target. Cyan region: 'in-volume' requirement region. Red dots: hits left by charged leptons in the detectors.}
\label{fig:close}
\end{figure}
\subsection{``In-volume" decay probability}
We define the dark photon decay volume as the region from right after the producing target, until just before the first tracking layer of the same module (see \figref{close}). For MUonE, the distance between the target and the first tracking layer is $\Delta z = 150~\mathrm{mm}$, and the expected resolution along the $z$-direction of the displaced vertex location is $\delta z = 1~\mathrm{mm}$~\cite{Clara:talk}. For a target with thickness of $15~\mathrm{mm}$, requiring a $5\sigma$ tail away from the target, and from the first tracking layer, one has for the edges of the decay volume
\begin{equation}
L_{min} = 20~\mathrm{mm} \,,
\qquad
L_{max} = 145~\mathrm{mm}
\end{equation}
The probability that a dark photon decays within this volume is given by
\begin{equation}
\label{eq:decay_in_volume}
{\cal P}_{\text{in-vol}} =
\exp\left(-L_{min} / \bar d_{A'} \right)
-\exp\left(-L_{max} / \bar d_{A'} \right)
\quad \,,
\end{equation}
where $\bar d_{A'}$ is given by \eqref{dbarAp}
Requiring that the $A'$ decays ``in volume'' implies that the tracking and vertexing of the $\mu\, e\to \mu\, e\, A',~A'\to e\,e$ event is contained within a single module.
This is advantageous because it allows one to point the event back to a single target vertex with high certainty. In contrast, when tracking relies on multiple MUonE modules, downstream from the producing one, the pointing back resolution is degraded due to the distance, and due to possible additional scattering of the particles.
Moreover, tracking based on a single module is an optimal choice given the exponential character of the decay probability, \eqref{decay_in_volume}, and the target parameter-space as characterized by the decay length in~\eqref{dbarAp}.
\subsection{Angular acceptance}
There are several angular acceptance types at MUonE, corresponding to each of the sub-detector systems, and the search strategy.
The angular acceptance of the electromagnetic calorimeter and muon spectrometer system, which are downstream detectors, depends on the module at which the particles were produced. In contrast, particle tracking and vertexing is performed in the producing module and the acceptance of its three tracking layers only depends on the $A'$ decay location relative to the module's target.
In order to estimate the acceptances, we apply cuts event by event in Monte-Carlo where necessary. Nonetheless, it is instructive to point out the following intuitive argument:
because the MUonE experiment is designed to detect the elastic $2\to2$ $e\,\mu$ scattering it is typically efficient in detecting the $2\to 3$ process.
For the leptons ($e,\, \mu$) the energy-angle distribution in the $2\to3$ process is constrained by a curve that corresponds to the distribution in the elastic scattering case (see \figref{2dhist}, and \appref{PSconstraints}). The scattered electron, and muon should therefore fall within the angular acceptance of all detector sub-systems.
For the dark photon (and its $e^+e^-$ decay products) the energy-angle distribution is also constrained, but not necessarily within the angular acceptance of the detector sub-systems. Nonetheless, requiring a displaced vertex implies that the $A'$ is boosted, and its decay products therefore ``inherit'' its propagation direction (see \eqref{opening angle}). As a result, if a displaced $A'$ is within the experiment's angular acceptance, then its decay products are likely within this acceptance as well.
\subsection{Search Strategies}
\label{sec:search_strategies}
We examine events in which the signatures within a module exhibit four charged tracks.
We devise four search strategies, presented here in order of increasing intricacy.
For simplicity, we use the following naming convention:
\begin{itemize}
\item loose: Only require a displaced charged particle pair ($e^+e^-$ pair) to pass through 3 tracking layers.
\item medium-1: require the $e^+ e^-$ pair to pass through the final ECAL.
\item medium-2: require 4 final state leptons to pass through 3 tracking layers (but not necessarily the ECAL).
\item tight: require 4 final state leptons to enter the final ECAL.
\end{itemize}
The ``loose'' strategy only requires a displaced vertex which is identified from two charged tracks, in all 3 tracking layers of the producing module.
This amounts to applying the ``in-volume'' decay requirement.
The ``medium-1'' strategy adds particle identification, by requiring that the particles that created the vertex also reach, and pass through the ECAL. The ``medium-2'' strategy requires tracking of 4 the leptons in the event.
The ``tight'' strategy requires a displaced vertex, as well as tracking and particle identification using the ECAL for all of the four particles in the event.
In all four strategies, a minimum energy threshold on leptons, $E_\ell \ge 5~\text{GeV}$ is applied to the detected final state leptons, i.e. the $e^+ e^-$ for ``loose'' and ``medium-1'', and all the 4 leptons for ``medium-2'' and ``tight''. In addition, we apply a cut on the lab frame opening angle of the $e^+e^-$ pair coming from the reconstructed displaced vertex, $\theta_{ee} \ge 1~\rm mrad$.
We note that the low energy threshold on the lepton energy implies that lepton identification and energy measurement in the ECAL may be degraded for electrons which travel through many modules. For strategies involving the ECAL, only the last few modules may be effective. For this reason, we only include the last 5 modules in the ``medium-1'', and ``tight'' strategies.
\subsection{Backgrounds}
\label{sec:bkgs}
A comprehensive estimation of the rates for background processes at MUonE requires dedicated simulation tools (for example a Geant4~\cite{agostinelli2003geant4,allison2006geant4,allison2016recent} model of MUonE), and is therefore beyond the scope of this work. Nevertheless, it is instructive, as a preliminary estimate, to enumerate and discuss the potential backgrounds of the displaced vertex search. These backgrounds typically fall into two primary categories:
(a) SM processes with an inherently displaced vertex, and (b) SM process in which a prompt vertex fakes a displaced vertex due to tracker inefficiencies.
In the first category the displaced vertex signature arises from the production of long-lived SM particles such as neutral Kaons. At MUonE, such particles would be produced in both coherent, and deep-inelastic $\mu$-nucleus scattering.
In the coherent case, hadronic emissions off the nucleus tend to be soft because the heavy nucleus recoils very little. In the deep-inelastic case, the shattering of the nucleus implies that hadronic emissions (which need not be soft) are accompanied by additional radiation which is identifiable and does not fit the displaced vertex event topology.
Moreover, the long-lived SM hadrons predominantly decay to final states containing hadrons states (hadronic and semi-leptonic decays). For the displaced vertex to register as coming from an $e^+e^-$, those would have to be misidentified as electron.
We expect that the combination of these arguments makes the background rates in this category negligible (see also~\cite{Gligorov:2018vkc,Aielli:2019ivi} for potential muon induced backgrounds)
In the second category the displaced vertex signature arises from SM processes
which are prompt, i.e. the production of the background event occurs at the target (for a recent calculation of background process at NNLO relevant to this proposed search, see~\cite{Budassi:2021twh}). Due to tracker inefficiencies, tracks in the event may reconstruct to a vertex which manifests as displaced. This category includes processes which share the same particle content as the signal, such as $\mu\,e\to \mu\,e\gamma^*\,(\gamma^*\to e^+e^-)$, as well as processes such as $\mu\,e\to \mu\,e\, C^+C^-$, where $C^\pm$ is any charged particle that can fake an electron signature.
The background from these processes can be characterized and scrutinized in a dedicated detector study. We estimate that the scrutinized decay-in-volume requirements combined with particle-identification requirements should substantially reduce such backgrounds to sufficiently low-levels.
In the remainder of this analysis, we assume that background rates are negligible.
\section{Analysis and Results}
\label{sec:results}
The MUonE sensitivity reach to the dark photon signature described in \secref{4lep_search} is estimated in Monte-Carlo. Samples of 100K $\mu e \rightarrow \mu e A^{\prime}$ events are generated for each dark photon mass point, using the methodology described in \secref{Ap_production}. Dark photon decays to $e^+e^-$ are simulated by hand using Eq.~(\ref{eq:Gamma_Ap}).
Event selection is performed following the four prescriptions of \secref{search_strategies}, where the baseline cuts on lepton energies, $\theta_{ee}$, and decay-in-volume are always applied.
The dark photon event yield following these cuts is given by
\begin{align}
\label{eq:numbers}
N = \mathcal{L} \cdot \sigma \cdot \epsilon_{\rm sim} \cdot {\cal P}_{\text{in-vol}} \,,
\end{align}
where $\epsilon_{\rm sim}$ is the fraction of events that pass the phase-space cuts in each simulated sample, and the planned fixed-target luminosity of the MUonE experiment is~\cite{Masiero:2020vxk}
\begin{align}
{\cal L} = 1.5 \times 10^{4} ~\mathrm{pb}^{-1} \,.
\end{align}
For each mass point, the dark photon production cross-section, $\sigma$, is estimated in simulation with $\epsilon e = 1$, and is rescaled by $e^2 \epsilon^2$ for each $(\epsilon,\, m)$ parameter-space point. Similarly, the decay in volume probability, ${\cal P}_{\text{in-vol}}$, \eqref{decay_in_volume}, is estimated for each parameter-space point using the dark photon decay-length, \eqref{dbarAp}.
Assuming zero background events, the MUonE sensitivity reach at $90\%$ C.L. is shown in~\figref{reach plot} for the four different search strategies.
Event number contours for the signal event can be found in~\figref{contours}. The existing limits (from~\cite{Merkel:2014avp,Bodas:2021fsy, Abrahamyan:2011gv,BaBar:2014zli, TheBABAR:2016rlg, Ablikim:2017aab,CHARM:1985anb,Tsai:2019mtm,CMS:2019kiy,Andreas:2012mt,Bjorken:2009mm,Riordan:1987aw,Bross:1989mp,Adrian:2018scb,Konaka:1986cb,Anastasi:2015qla,Anastasi:2016ktq,Anastasi:2018azp,Babusci:2012cr,Babusci:2014sta,LHCb:2017trq,Aaij:2019bvg,NA482:2015wmo,NA64:2018lsq,Banerjee:2019hmi,NOMAD:2001eyx,Blumlein:1990ay,Blumlein:1991xh,Tsai:2019mtm,Davier:1989wz,Bernardi:1985ny}) are reconstructed via \textsc{darkcast}~\cite{Ilten:2018crw}.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{Figures/reach.pdf}
\caption{Exclusion curves at $90\%$ C.L.(red). Shaded regions are excluded by other experiments. Magenta solid: reach curve for ``loose'' selection. Purple small-dashed: reach curve for ``medium-1'' selection. Blue large-dashed: reach curve for ``medium-2'' selection. Orange dot-dashed: reach curve for ``tight'' selection. The various strategies are discussed in Sec.~\ref{sec:search_strategies}.}
\label{fig:reach plot}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
MUonE is an experiment proposed to measure the HVP contribution to muon $g-2$ through elastic $\mu-e$ scattering. We have shown in this work that the design of the MUonE experiment also makes it a promising environment to search for light mediators with visibly displaced decays. Focusing on the vanilla dark photon model for a proof-of-concept, we have identified $\mu+e\to \mu+e+A'$ with $A'\to e^+e^-$ as a promising final state, and we have argued that backgrounds are likely to be negligible. By searching for visibly displaced $e^+e^-$ pairs, possibly in conjunction with $\mu$ and $e$ from the initial $2\to 3$ scattering process, we have demonstrated that MUonE could have world-leading sensitivity to a crucial range of dark photon parameter space ($m_{A'}$ between 10-100~MeV and $\epsilon e$ between $10^{-5}-10^{-3}$).
Of course, this being just a proof-of-concept study, much work remains to accurately estimate background and signal yields; this likely requires detailed GEANT4 simulations that are beyond the scope of this work.
Also it would be interesting to consider other light mediator models that MUonE could be sensitive to besides vanilla dark photons, such as ALPs and leptophilic mediators. Leptophilic models wouId be an especially attractive target for MUonE, since many of the bounds shown in~\figref{reach plot} rely on couplings of dark photons to quarks and hence would disappear for such models. In any case, we hope this work provides further motivation for the MUonE proposal and illustrates how it could have multiple purposes beyond its original goal of measuring HVP for muon $g-2$.
\begin{acknowledgments}
We are especially grateful to Umberto Marconi and Clara Matteuzzi for their encouragement throughout the course of this project and for crucial discussions about the setup of the MUonE experiment, backgrounds and search design. In addition, we thank Simon Knapen, Paride Paradisi, Yotam Soreq, Robert McGehee and Xun-Jie Xu for helpful discussions. Finally, we are extremely grateful to Clara Matteuzzi for detailed feedback on the draft.
This work was supported by DOE grant DE-SC0010008.
The work of IG is supported in part by the Israel Science Foundation (Grant No. 751/19), and by BSF-NSF grant 2020-785.
\end{acknowledgments}
|
1,108,101,566,721 | arxiv | \section{Proof of Proposition~\ref{prop41}}\label{proofpropR_n}\label{ap:A}
\input{proof_prop_Rn}
\section{Introduction}
The many-dimensional Generalized Excited Random Walk ({\rm GERW}) was introduced by Menshikov et al. in~\cite{menshikov2012general} following a series of works on many-dimensional excited random walk \cite{benjamini2003excited,kozma2003excited,berard2007central}. The model considered in ~\cite{menshikov2012general} is a uniformly elliptic random walk with bounded jumps in dimension $d\ge 2$ such that on already visited sites it behaves as a $d$-dimensional martingale with bounded jumps and zero-mean vector and whenever a site is visited for the first time its increment has drift in some fixed direction $\ell$ of the unit sphere in $\mathbb{R}^d$. They show that the GERW with a drift condition in direction $\ell$ is ballistic in that direction. Besides that, they proved a LLG and a CLT (both for dimensions $d \geq 2$) under stronger hypothesis on the definition of GERW; these particular models were called \textit{excited random walk in random environment}.
What makes the GERW an interesting model is the self-interaction encoded in the different behavior the process has on sites visited for the first time as compared to sites already visited. This makes them important toy models of non-Markovian random walks used to understand how much weakly should the random walk be pushed toward a fixed direction to still present a ballistic behavior. Similar works worth mentioning along these lines are~\cite{angel2021balanced,benjamini2011balanced,peres2016martingale}.
A natural question is what happens to {\rm GERW}{} when the strength of the drift on the first visits decreases with time. Would the process still be ballistic in the direction of the drift? What about LLN and CLT? We propose here a variation of {\rm GERW}{} to contemplate this case by assuming that the drift in a fixed direction $\ell$ at time $n$ is of order $n^{-\beta}$ if at this time a site is visited for the first time. If $\{X_n\}_{n\geq 0}$ denotes the {\rm GERW}{} under this weaker condition on the directional drift, we show that $\lim_{n \rightarrow \infty} X_n \cdot \ell = \infty$ (directional transience) with positive probability if $\beta$ is sufficiently small. This shows that our model has an intermediary behavior between a mean-zero random walk (non excited) and the {\rm GERW}{} considered in \cite{menshikov2012general} (see, Remark~\ref{rem:sub-balistic}). In \cite{AIV} we discuss limit theorems for $\beta \ge 1/2$ for a particular class of {\rm GERW}{} in the same spirit of the excited random walk in random environment discussed in
\cite{menshikov2012general}.
Our proof is based on an adaptation of the arguments presented in~\cite{menshikov2012general} for our time in-homogeneous case. It involves the use of a bound on the range of the walk to guarantee that the walk receives enough impulse in the drift direction.
\subsection{Definition of the model and main result}\label{sec:model}
Let $d \ge 2$ be the fixed dimension and $X = \{ X_n \}_{n \geq 0}$ be a $\mathbb{Z}^d$ valued adapted process on a stochastic basis $(\Omega,\mathcal{F},\mathbb{P},\{ \mathcal{F}_n \}_{n \geq 0})$ satisfying the usual conditions. We denote by $\mathbb{E}$ the expectation with respect to $\mathbb{P}$ and by $||\cdot||$ the euclidean norm in $\mathbb{R}^d$.
Now fix $\{\lambda_n\}_{n\ge 0}$ a sequence of positive real numbers, $\ell \in \mathbb{S}^{d-1}$, where $\mathbb{S}^{d-1}$ is the unit sphere of $\mathbb{R}^d$ and a nonempty set $A \subset \mathbb{Z}^d$. We assume that $X_0=0$ and we call $X$ a $\lambda_n$-{\rm GERW}{} in direction $\ell$ with excitation set $A$, if it satisfies the following conditions:
\begin{condition}[Bounded increments]\label{condição1}
There exists a positive constant $K$ such that $\sup_{n \geq 0} || X_{n+1} - X_{n} || < K$ on every realization.
\end{condition}
\begin{condition}\label{condição2} Almost surely
\begin{itemize}
\item on $\{ X_k \neq X_n \, \forall \; k < n \} \cap \{ X_n \in A\}$,
$$
\mathbb{E} [ X_{n+1} - X_n | \mathcal{F}_n] \cdot \ell \geq \lambda_n \, .
$$
\item on $\{ \exists\, k < n \text{ such that } X_k = X_n \}$ or $\{ X_k \neq X_n \, \forall \; k < n \} \cap \{ X_n \notin A\}$,
\[
\mathbb{E} [ X_{n+1} - X_n | \mathcal{F}_n] = 0\, .
\]
\end{itemize}
\end{condition}
\begin{condition}\label{condição3}
There exist $h, r > 0$ such that
\begin{itemize}
\item $X$ is {\rm uniformly elliptic in direction $\ell$}, i.e., for all $n$
\begin{equation} \label{3 1.4}
\tag{UE1}\mathbb{P} \left[ \left( X_{n+1} - X_n \right) \cdot \ell > r | \mathcal{F}_n \right] \geq h\,, \; {a.s..}
\end{equation}
\item $X$ is {\rm uniformly elliptic on the event $\{\mathbb{E} [ X_{n+1} - X_n | \mathcal{F}_n] = 0\}$:} on the event $ \{ \mathbb{E} [ X_{n+1} - X_n | \mathcal{F}_n] = 0 \}$, for all $\ell' \in \mathbb{S}^{d-1}$, with $|| \ell '|| = 1$
\begin{equation} \label{3 1.5}
\tag{UE2}\mathbb{P} \left[ \left( X_{n+1} - X_n \right) \cdot \ell ' > r | \mathcal{F}_n \right] \geq h\,, \; {a.s..}
\end{equation}
\end{itemize}
\end{condition}
When $A = \mathbb{Z}^d$, we call $X$ simply a $\lambda_n$-{\rm GERW}{}.
For every $\ell \in \mathbb{S}^{d-1}$, let $\mathbb{M}_{\ell}$ denote the positive half-space in direction $\ell$, that is, $\mathbb{M}_{\ell} = \{ x \in \mathbb{Z}^d : x \cdot \ell > 0 \}$.
Our main result is stated below:
\begin{theorem}\label{prop43_pnn0}
Let $X$ be a $\lambda_n$-{\rm GERW}{} in direction $\ell$ with excitation set $A \supset \mathbb{M}_{\ell}$. There exists $\beta_0 < 1/6$ such that if for some $n_0 \in \mathbb{N}$, $\lambda >0$ and $\beta<\beta_0$, we have $\lambda_n \ge \lambda (n_0+n)^{-\beta}$ for every $n\ge 1$, then
$$
\mathbb{P} \big( \lim_{n \rightarrow \infty} X_n \cdot \ell = \infty \big) > 0\,.
$$
\end{theorem}
\begin{remark}\label{rem:sub-balistic}
1. If $\lambda_n$ is $O(n^{-\beta})$, then the $\lambda_n$-{\rm GERW}{} is not ballistic since the total mean drift accumulated by time $n$ is bounded by $n^{1-\beta}$. We conjecture that $\lim_{n \rightarrow \infty} X_n \cdot \ell = \infty$ holds almost surely, and even that $\liminf_{n \rightarrow \infty} n^{\beta - 1} X_n \cdot \ell > 0$ almost surely. 2. The condition $\beta < 1/6$ in the statement of Theorem \ref{prop43_pnn0} follows from limitations in our proof. We also conjecture that the result holds for $\beta < 1/2$. For a discussion on the case $\beta \ge 1/2$ see \cite{AIV}.
\end{remark}
\subsection{Proof of Theorem \ref{prop43_pnn0}}\label{resultados_pn}
The strategy to prove Theorem~\ref{prop43_pnn0} is based on obtaining an analogous result to~\cite[Proposition 4.3]{menshikov2012general} where it is proved that the GERW in direction $\ell$ (with $\lambda_n$ constant) never goes below the origin in direction $\ell$ with positive probability. We generalize this proof to our case.
%
First, we need some auxiliary results. Given a stochastic process $\{ X_n\}_{n \geq 0}$ on the lattice $\mathbb{Z}^d$, we denote its range at time $n$ by
\begin{equation*}
\mathcal{R}_n ^X := \{ x \in \mathbb{Z}^d : X_k = x \text{ for some } 0 \leq k \leq n \}\,,
\end{equation*}
i.e., the set of sites visited by the process up to time $n$. Henceforth $|A|$ denotes the number of elements of a set $A$.
\begin{proposition}\label{prop41}
Let $X$ be a $\lambda_n$-{\rm GERW}. Then, there exist positive constants $\alpha \in (0, 1/6)$, $\gamma_1$, $\gamma_2$ , which depend on $K$, $h$, and $r$, such that for every $\lambda_n$
\begin{equation*}
\mathbb{P} [|\mathcal{R}^X _n | < n^{\frac{1}{2} +\alpha} ] < \exp\{- \gamma_1 n^{\gamma_2}\}\,,
\end{equation*}
for all $n \geq 1$.
\end{proposition}
The proof of Proposition~\ref{prop41} is an adaptation of the proof of~\cite[Proposition 4.1]{menshikov2012general} and, for completeness, is provided in Appendix \ref{ap:A}.
For quite general processes, a similar result is presented in ~\cite[Theorem 1.4]{menshikov2014range}. However,
Proposition~\ref{prop41} does not follow directly from~\cite[Theorem 1.4]{menshikov2014range}. In fact, although the $\lambda_n$-{\rm GERW}{} is a submartingale in direction $\ell$, it does not fit in~\cite[Definition 1.1.d.]{menshikov2014range} required in~\cite[Theorem 1.4]{menshikov2014range}.
\medskip
Set $H(a,b) \subset \mathbb{Z}^d$ for $a<b$ as:
\begin{equation*}
H(a,b) := \{ x \in \mathbb{Z}^d : x \cdot \ell \in [a,b] \}\, ,
\end{equation*}
which represents the strip in direction $\ell$ between levels $a$ and $b$. Roughly speaking, the next proposition states that if the number of sites outside the excitation set in a strip with length of order $n^{\frac{1}{2}+\alpha}$ ($\alpha$ from Proposition~\ref{prop41}) containing the origin, is also of order $n^{\frac{1}{2}+\alpha}$, then $X_n \cdot \ell\;$ is at least of order $n^{\frac{1}{2}+\alpha-\beta}$ with high probability.
\begin{proposition}\label{prop42_pnn0}
Let $X$ be a $\lambda_n$-{\rm GERW}{} in direction $\ell$ with excitation set $A \subset \mathbb{Z}^d$. If for some $n_0 \in \mathbb{N}$, $\lambda >0$ and $\beta<\alpha$ (from Proposition~\ref{prop41}), we have $\lambda_n \ge \lambda (n_0+n)^{-\beta}$ for every $n\ge 1$, and if for some $n \geq n_0$ it holds that
\begin{equation}\label{cond44_pnn0}
\left\vert( \mathbb{Z}^d \setminus A) \cap H\left( - n^{\frac{1}{2} + \alpha}, \frac{2 \lambda}{3} n^{\frac{1}{2} + \alpha}\right)\right\vert \leq \frac{1}{3} n^{\frac{1}{2} + \alpha}\,,
\end{equation}
then,
\begin{equation}\label{prop42c_pnn0}
\mathbb{P}\left[ X_n \cdot \ell < \frac{1}{3} \lambda n^{\frac{1}{2}+ \alpha-\beta} \right] < 5n\exp\{-\vartheta_1 n^{\vartheta_2}\}\, ,
\end{equation}
where
\begin{align*}
\vartheta_1 & = \min\left\{ \gamma_1, \frac{1}{2K^2}, \frac{\lambda^{2}}{18K^2}, \frac{((1/3 -2^{1-\beta}/3) \lambda)^2}{2K^2} \right\} \,,
\\
\vartheta_2 & = \min\left\{ \gamma_2, 2(\alpha-\beta) \right\} \,,
\end{align*}
and $\gamma_1$, $\gamma_2$ are the same as in Proposition~\ref{prop41}.
\end{proposition}
Proposition~\ref{prop42_pnn0} is proved in Appendix \ref{ap:B}. The proof of Proposition~\ref{prop42_pnn0} is an adaptation of the proof of~\cite[Proposition 4.2]{menshikov2012general}
\medskip
The next proposition is our main result. It states that the $\lambda_n$-GERW in direction $\ell$ never goes below the origin in that direction with positive probability bounded below by some $\psi$ whose dependence on $\beta$ and $n_0$ is explicitly described.
Set $\{\eta(X_0) = \infty\}$ as the event in which the process $X$ never returns to the origin in the drift direction.
\smallskip
\begin{proposition}\label{prop43_pnn0-1}
Let $X$ be a $\lambda_n$-{\rm GERW}{} in direction $\ell$ with excitation set $A \supset \mathbb{M}_{\ell}$. If for some $n_0 \in \mathbb{N}$, $\lambda >0$ and $\beta < \alpha$ (from Proposition~\ref{prop41}) we have $\lambda_n \ge \lambda (n_0+n)^{-\beta}$ for every $n\ge 1$, then
\begin{equation*}
\mathbb{P}\left[ \eta(X_0) = \infty \right] \geq \mathbb{P}\left[ X_n \cdot \ell > 0 \text{ for all } n\geq 1\right] \geq \psi\, ,
\end{equation*}
where $\psi = h^{\lceil r^{-1} \rceil C \left(\frac{3}{\lambda} \right)^{\frac{1}{\delta -1}}} c$, with $c \in (0, 1)$, $\delta = (2-\alpha+\beta)(1/2 + \alpha-\beta)$,
\begin{align*}
C & = K^{\frac{1}{\delta-1}} \left( \eta + \lceil r^{-1} \rceil^{\frac{1}{\delta-1}}\right) + n_0\,,
\\
\eta & = \left( \frac{ 2-\alpha+\beta}{\vartheta_1 \varphi_1}\right)^{\frac{1}{\varphi_1}} \, , \quad \varphi_1 = \min \left\{ \alpha-\beta, (2-\alpha+\beta)\vartheta_2 \right\}\, , \nonumber
\end{align*}
and $\vartheta_1$, $\vartheta_2$ are as in Proposition~\ref{prop42_pnn0}.
\end{proposition}
Now we use Proposition~\ref{prop43_pnn0-1} to prove Theorem~\ref{prop43_pnn0}, and we prove the proposition just after.
\smallskip
\begin{proof}[Proof of Theorem~\ref{prop43_pnn0}]
Fix $m \in \mathbb{N}$ and set $\mathcal{F}_n$-stopping times $\nu_{m,0} \equiv 0$,
$$
\tau_{m,j} = \inf \{ n > \nu_{m,j-1} : X_n \notin H(0,m) \}\,,
$$
and
$$
\nu_{m,j} = \inf \{ n > \tau_{m,j} : X_n \in H(0,m) \}\,,
$$
for every $j\ge 1$. Also define $\mathcal{G}_j = \mathcal{F}_{\nu_{m,j}}$ and
$$
\Gamma_j = \{\nu_{m,j}<\infty\} \cap \big\{ X_n \cdot \ell < 0 \textrm{ for some } n \in (\nu_{m,j}, \nu_{m,j} + \hat m] \big\} \,,
$$
where $\hat m = \lfloor m/r \rfloor + 1$. The sets $\Gamma_{\hat m j}$ are $\mathcal{G}_{\hat m (j+1)}$-measurable and Condition~\ref{condição3} implies that
$$
\mathbb{P} (\Gamma_{\hat m j}|\mathcal{G}_{\hat m j}) \geq \mathbbm{1}\{\nu_{m,\hat{m}j}<\infty\}h^{\hat m}\,.
$$
By the second Borel-Cantelli lemma II \cite[Theorem 5.3.2]{durrett}, we have that
$$
\mathbb{P} \big( \Gamma_{j} \ i.o. \big| \nu_{m,j} < \infty \, \forall \, j\ge 1 \big) = 1\,.
$$
Thus, almost surely in $\{\eta(X_0) = \infty\}$ we have that $\{ \forall \, m\ge 1 \textrm{ there exists } j=j(m)\ge 1 \textrm{ such that } \nu_{m,j} = \infty \}$. Therefore almost surely in $\{\eta(X_0) = \infty\}$, we have that $\{ \lim_{n \rightarrow \infty} X_n \cdot \ell = \infty \}$. Now the statement follows from Proposition~\ref{prop43_pnn0-1}.
\end{proof}
\smallskip
\begin{proof}[Proof of Proposition~\ref{prop43_pnn0-1}]
Since on $\{ X_n \cdot \ell > 0 \text{ for all } n \geq 1 \}$ the process does not visit $\mathbb{Z}^d/ \mathbb{M}_{\ell}$, it is sufficient to consider the case $A= \mathbb{Z}^d$.
Without loss of generality we consider $r\le 1$ in Condition~\ref{condição3} and $\lambda<1$. Define
\begin{equation*}
U_0 = \left\{ \left( X_{k+1} - X_k \right)\cdot \ell \geq r\;, \text{ for all} \ k = 0, 1, ..., \lceil r^{-1} \rceil m - 1 \right\}.
\end{equation*}
Note that $X_{\lceil r^{-1} \rceil m} \cdot \ell \geq m$ in $U_0$ and by \eqref{3 1.4} in Condition~\ref{condição3}
\begin{equation}\label{u0_pn}
\mathbb{P} \left[ U_0 \right] \geq h^{\lceil r^{-1} \rceil m}\,.
\end{equation}
Consider the following time shift of $X$: $W_k = X_{\lceil r^{-1} \rceil m + k}$, $k \geq 0$. Then $W$ is a $\tilde{\lambda}_{n}$-{\rm GERW}{} (with $\tilde{\lambda}_n=\lambda_{n+\lceil r^{-1} \rceil m}$) adapted to the filtration $\{\mathcal{F}_{k + \lceil r^{-1} \rceil m}\}_{k \ge 0}$ with excitation set
$$
A' = \mathbb{Z}^d \setminus \{ X_0, \dots, X_ {\lceil r^{-1} \rceil m-1} \}\,,
$$
starting at $W_0 = y_0 := X_{\lceil r^{-1} \rceil m}$. Moreover for every $k$ we have
\[
\mathbb{E}[W_{k+1} - W_k | \mathcal{F}_{k +\lceil r^{-1} \rceil m }] \cdot \ell \ge \lambda(n_0 + \lceil r^{-1} \rceil m + k)^{-\beta}\,,
\]
almost surely on
$\{ W_j \neq W_k \text{ for all } \; j < k \} \cap \{ W_k \in A'\}$.
Set $\delta = (2-\theta)(1/2 + \theta)$, where $\theta=\alpha-\beta$ and
$$m = C \Big(\frac{3}{\lambda }\Big)^{\frac{1}{\delta -1}}\,,$$
where
\begin{equation*}
\label{C}
\begin{split}
C & = K^{\frac{1}{\delta-1}} \left( \eta + \lceil r^{-1} \rceil^{\frac{1}{\delta-1}}\right) + q_0 \quad \text{and }
q_0:=n_0 + \lceil r^{-1} \rceil m \\
\eta & = \left( \frac{ 2-\alpha+\beta}{\vartheta_1 \varphi_1}\right)^{\frac{1}{\varphi_1}} \quad \text{with} \quad \varphi_1 = \min \left\{ \alpha-\beta, (2-\alpha+\beta)\vartheta_2 \right\},
\end{split}
\end{equation*}
and $\vartheta_1$, $\vartheta_2$ as in the statement of Proposition~\ref{prop42_pnn0}.
Since $0<\beta <\alpha<1/6$, we have that $\delta > 1$.
The left-hand side of \eqref{cond44_pnn0} with the set $A' - y_0$ is bounded above by $\lceil r^{-1} \rceil m$. Note that, for all $n \geq m^{2- (\alpha-\beta)}$,
\begin{align*
\frac{1}{3}n^{\frac{1}{2} + \alpha} \geq \frac{1}{3}n^{\frac{1}{2} + \alpha-\beta} \geq \frac{1}{3} m^{\delta} \geq \lceil r^{-1}\rceil m\,,
\end{align*}
where the last inequality follows from
\begin{align}\label{nm_pn2}
\frac{m^{\delta-1}}{3\lceil r^{-1} \rceil} = \frac{3C^{\delta-1}}{3\lceil r^{-1} \rceil \lambda} > \frac{K\lceil r^{-1} \rceil}{\lceil r^{-1} \rceil \lambda} = \frac{K}{\lambda} > 1 \,.
\end{align}
The second inequality above follows from the definition of $C$, since $C > (K \lceil r^{-1}\rceil)^{\frac{1}{1-\delta}}$ and $K\geq 1$. Thus~\eqref{cond44_pnn0} with excitation-allowing set $A' - y_0$ is satisfied for all $n\geq m^{2- \alpha+\beta}$.
Denote $m_0 = 0$, $m_1 = m$ and, for $k \geq 1$, $m_{k+1} = \frac{1}{3} \lambda m_{k}^{\delta}$. The sequence $(m_k, k \geq 1)$ is increasing. This can be proved by induction since
\[ \frac{m_2}{m_1} = \frac{\lambda}{3} m^{\delta - 1} = C^{\delta-1} > 1\,,\]
for all $\theta \in (0, 1/6)$, and assuming $m_k/m_{k-1} > 1$ we have
\[ \frac{m_{k+1}}{m_k} = \frac{ \frac{\lambda}{3} m_k ^{\delta}}{\frac{\lambda}{3} m_{k-1} ^{\delta}} = \left( \frac{m_k}{m_{k-1}} \right)^{\delta} > 1\,. \]
For every $k \geq 1$ consider the following events
\begin{align*}
G_k &= \left\{ \min_{\lfloor m_{k-1} ^{2 - \theta}\rfloor < j \leq m_k ^{2 - \theta}} \left( W_j - W_{\lfloor m_{k-1}^{2- \theta}\rfloor} \right) \cdot \ell > - m_k \right\}
\text{and } \,
U_k &= \left\{ W_{\lfloor m_{k} ^{2 - \theta}\rfloor } \cdot \ell \geq m_{k+1} \right\}\,.
\end{align*}
One can see that
\begin{equation}\label{inclusão_pnn0}
\big\{ X_n \cdot \ell > 0, \text{ for all } n \geq 1 \big\} \supset \Big( \bigcap_{k=1}^{\infty} \left(G_k \cap U_k \right) \Big) \cap U_0\;.
\end{equation}
The process $\{X_n \cdot \ell\}_{n\geq 0}$, is a $\mathcal{F}$-submartingale, so $(W - y_0) \cdot \ell$ is also $\mathcal{F}_{\cdot +\lceil r^{-1} \rceil m }$-submartingale. Write
\begin{equation*}
G_k^c = \bigcup_{j= \lfloor m_{k-1}^{2-\theta} \rfloor + 1}^{m_k^{2-\theta}} \left\{ \left( W_j - W_{\lfloor m_{k-1}^{2-\theta} \rfloor}\right) \cdot \ell \leq -m_k \right\}\,.
\end{equation*}
and by Azuma's inequality (for supermartingales with bounded increments)
\begin{align*}
\mathbb{P}&\left[ \left( W_j - W_{\lfloor m_{k-1}^{2-\theta} \rfloor}\right) \cdot \ell \leq -m_k \right] = \mathbb{P}\left[ \left( W_{\lfloor m_{k-1}^{2-\theta} \rfloor} -W_j\right) \cdot \ell \geq m_k \right]
\\
& \leq \exp \Big( - \frac{m_k^2}{2K^2(j - \lfloor m_{k-1}^{2-\theta} \rfloor) } \Big)
\leq \exp \Big( - \frac{m_k^2}{2K^2 m_{k}^{2-\theta}} \Big)
\leq \exp \Big( - \frac{m_k^{\theta}}{2K^2} \Big)\,.
\end{align*}
Thus, we have
\begin{equation*}
\mathbb{P}[G_k|U_0] \geq 1 -\left( m_k ^{2- \theta} -\lfloor m_{k-1}^{2- \theta}\rfloor\right) e^{- \frac{m_k ^{\theta}}{2K^2}} \geq 1 - m_k ^{2- \theta} e^{- \frac{m_k ^{\theta}}{2K^2}}\,.
\end{equation*}
The sequence $\{m_k\}_{k \geq 1}$ is increasing, then if $m^{2-\theta}\geq q_0$ we obtain that $m_k^{2-\theta}\geq q_0$ for all $k \geq 1$. Observe that
\begin{align*}
m^{1-\theta} - \lceil r^{-1} \rceil \geq \left( \frac{3K\lceil r^{-1} \rceil}{\lambda} \right)^{\frac{1-\theta}{\delta-1}} -\lceil r^{-1} \rceil > 1 \,,
\end{align*}
since for all $\theta \in (0, 1/6)$ we have $(1-\theta)/(\delta - 1)> 1$. Thus
\begin{align*}
m^{2-\theta} - \lceil r^{-1} \rceil m & = m \underbrace{( m^{1-\theta} - \lceil r^{-1} \rceil )}_{ > 1} \geq m \geq n_0
\,.
\end{align*}
Since the process $W-y_0$ satisfies Conditions~\ref{condição1},~\ref{condição2},~\ref{condição3}, the set $A'-y_0$ fulfills~\eqref{cond44_pnn0} for all $n \geq m^{2-\theta}$ and $m_k^{2-\theta}\geq n_0 + \lceil r^{-1} \rceil m$ for all $k \ge 1$ by Proposition~\ref{prop42_pnn0}, it holds that
\begin{equation*}
\mathbb{P}[U_k|U_0] = \mathbb{P}\left[W_{\lfloor m_k ^{2-\theta} \rfloor} \cdot \ell \geq \frac{\lambda }{3} m_{k}^{(2-\theta)(\frac{1}{2}+\theta)}
\right] \geq 1 - 5m_k^{2-\theta} e^{- \vartheta_1 m_k ^{(2- \theta)\vartheta_2}}\,.
\end{equation*}
Now, write
\[
\mathbb{P} \left[ \left( \bigcap_{k=1}^{\infty} \left(G_k \cap U_k \right) \right) \cap U_0 \right] = \mathbb{P}[U_0] \Big( 1 - \sum_{k=1}^{\infty} \mathbb{P}[G_k ^c| U_0] + \mathbb{P}[U_k ^c| U_0] \Big)\,,
\]
which is bounded from below by
\begin{equation}\label{psi_pnn0}
\begin{split}
&\mathbb{P}[U_0] \!\left( \!1 -\! \sum_{k=1}^{\infty}\!\left( m_k ^{2- \theta} e^{- \frac{m_k ^{\theta}}{2K^2}} + 5m_k ^{2- \theta} e^{- \vartheta_1 m_k ^{(2- \theta)\vartheta_2}} \right)\!\right)
\!\geq\! h^{\lceil r^{-1} \rceil m} \!\left(\! 1 \!- 6\sum_{k=1}^{\infty} m_k ^{2- \theta} e^{- \vartheta_1 m_k ^{\varphi_1}}\! \right)\,.
\end{split}
\end{equation}
Note that $m$ is large enough so that $\{m_{k}^{2-\theta}e^{- \vartheta_1 m_k^{\varphi_1}}\}_{k \geq 1}$ is decreasing. Indeed, $m$ is bigger than the inflection point $\left( \frac{ 2-\theta}{\vartheta_1 \varphi_1}\right)^{\frac{1}{\varphi_1}}$ of the function $z(x)=x^{2-\theta}e^{-\vartheta_1 x^{\varphi_1}}$, $x>0$:
\begin{align*}
m = C\left( \frac{3}{\lambda }\right)^{\frac{1}{\delta-1}} & > K^{\frac{1}{\delta-1}} \eta \left( \frac{3}{\lambda}\right)^{\frac{1}{\delta-1}}
= \eta \left(\frac{3K}{\lambda} \right)^{\frac{1}{\delta-1}} \geq \left( \frac{ 2-\theta}{\vartheta_1 \varphi_1}\right)^{\frac{1}{\varphi_1}}\,.
\end{align*}
Thus we have,
\begin{align*
\sum_{k=1}^{\infty}& m_{k}^{2-\theta}e^{-\vartheta_1 m_k^{\varphi_1}} \leq \int_{m_{1}}^{\infty} x^{2-\theta} e^{-\vartheta_1 x^{\varphi_1}} dx \, .
\end{align*}
By a change of variables, we write,
\begin{align}\label{integral_pnn0}
&\int_{m_{1}}^{\infty} x^{2-\theta_1} e^{-\vartheta_1 x^{\varphi_1}} dx = \varphi_1^{-1} \vartheta_1^{\frac{\theta-3}{\varphi_1}}\Gamma \left( \frac{3-\theta}{\varphi_1}, \vartheta_1 m_{1}^{\varphi_1} \right)\, ,
\end{align}
where $\Gamma$ is the incomplete gamma function\footnote{$\Gamma(s, x) = \int_{x}^{\infty} t^{s-1}e^{-t} dt$.}.
As mentioned above $m$ is large enough so that the sequence $\{m_{k}^{2-\theta}e^{- \vartheta_1 m_k^{\varphi_1}}\}_{k \geq 1}$ is decreasing. Thus, in order to obtain that ~\eqref{integral_pnn0} is smaller than $1/6$, we may increase $m$ even further by choosing $C$ sufficiently large.
Thus, for a suitable $C>0$ we obtain
\begin{align}\label{eq: 1/7_pnn0}
\sum_{k=1}^{\infty} m_{k}^{2-\theta}e^{-\vartheta_1 m_k^{\varphi_1}} & \leq \int_{m_{1}}^{\infty} x^{2-\theta} e^{-\vartheta_1 x^{\varphi_1}} dx
= \varphi_1^{-1} \vartheta_1^{\frac{\theta-3}{\varphi_1}}\Gamma \left( \frac{3-\theta}{\varphi_1}, \vartheta_1 m_{1}^{\varphi_1} \right) < \frac{1}{6}\, .
\end{align}
Using~\eqref{eq: 1/7_pnn0} in~\eqref{psi_pnn0}, we obtain that,
\begin{align*}
\mathbb{P} \left[ \left( \bigcap_{k=1}^{\infty} \left(G_k \cap U_k \right) \right) \cap U_0 \right]
&\geq h^{\lceil r^{-1} \rceil m} \left( 1 - 6\sum_{k=1}^{\infty} m_k ^{2- \theta} e^{- \vartheta_1 m_k ^{\varphi_1}} \right)
\geq h^{\lceil r^{-1} \rceil C\left(\frac{3}{\lambda } \right)^{\frac{1}{\delta -1}}} c = \psi \, ,
\end{align*}
where $c \in (0,1)$. Theorem~\ref{prop43_pnn0} then follows from $\eqref{inclusão_pnn0}$.
\end{proof}
\subsection{Some auxiliary results for $d$-dimensional martingales}
Using Lemma~\ref{lema51} and~\cite[Lemma 5.4]{menshikov2012general} we now prove Proposition~\ref{prop41}.
\begin{proof}[Proof of Proposition \ref{prop41}]
Consider $b \in (0,1)$ and $\varepsilon > 0$. Recall that when the process is in an already visited site it behaves like a $d$-martingale with bounded jumps and uniform elliptic condition; we denote this process by $\{Y_n\}_{n \ge 0}$. Then by~\cite[Lemma 5.2]{menshikov2012general} there exists a $b' \in (0, 1)$ and a positive constant $c$ (depending of $K$, $h$ and $r$) such that
\[ \mathbb{E}[||Y_{n+1}||^{b'} | \mathcal{F}_n] \ge ||Y_n||^{b'} 1_{\{||Y_n|| \ge c\}}\,.\]
Set $b = b'$ and $0< e_w < 1/6$ and define
\begin{equation*}
H^n_j := H\big( 2(j - 1)n^{e_w}, 2(j + 1)n^{e_w} \big)\, , \ n\ge 1 \,, \ j\ge 1\,,
\end{equation*}
so that $H^n_j$ is strip of width $4n^{e_w}$ in direction $\ell$. The strip $H^n_j$ will be called a trap if $|\mathcal{R}_n ^X \cap H^n_j | \geq n^{e_t}$, where $e_t = 2e_w(1-(b/2)) - 2 \varepsilon$ is the trap exponent.
Set
\begin{equation*}
G = \{ |\mathcal{R}_n ^X | \geq n^{\frac{1}{2} + e_w (1-b)-4\varepsilon} \}\, .
\end{equation*}
We are going to prove that
\begin{equation}\label{probg}
\begin{split}
\mathbb{P}[G] \geq 1 - \left( \left( 2Kn + 1 \right)e^{-\gamma_1' n^{\frac{\varepsilon}{2}}} + \frac{n^{1-2e_w + \varepsilon}}{2} e^{-\gamma_4' n^{\varepsilon}} \right)\,,
\end{split}
\end{equation}
for every $\varepsilon >0$ sufficiently small.
This would establishes Proposition $\ref{prop41}$ since, for $\alpha < e_w(1-b)-4\varepsilon$ we have that $\{|\mathcal{R}_n ^X | < n^{\frac{1}{2} + \alpha} \} \subset G^c$.
Towards proving \eqref{probg}, let us introduce the event
\begin{equation*}
G_1 = \big\{ L_n (k) \leq n^{\frac{1}{2} + \varepsilon} \text{ for all } k \in [-Kn, Kn] \big\}\,.
\end{equation*}
By Lemma~\ref{lema51}, it holds that
\begin{equation}\label{probg1}
\mathbb{P}[G_1] \geq 1 - (2Kn + 1)\exp\{-\gamma_1' n^{\frac{\varepsilon}{2}}\}\, .
\end{equation}
Now, let us define $\sigma_0 = 0$ and inductively
\begin{equation}\label{sigmak}
\sigma_{k+1} = \min \{ j \geq \sigma_k + \lfloor n^{2e_w - \varepsilon} \rfloor : | \mathcal{R}_j ^X \cap B(X_j, n^{e_w}) | \leq n^{e_t} \}\,,
\end{equation}
(formally, if such $j$ does not exist, we put $\sigma_{k+1} = \infty$). Consider the event
\begin{align*}
G_2 = \Big\{ & \text{at least one new point is hit on each of the time intervals }
\\
& [\sigma_{j-1}, \sigma_j), j = 1,..., \frac{1}{2}n^{1-2 e_w + \varepsilon} \Big\}\,,
\end{align*}
where hitting a new point means to visit a not-yet-visited site. Note that on $G^c_2$, the process does not hit a new point in time interval $[\sigma_{j-1},\sigma_j)$ for some $j=1,..., \frac{1}{2}n^{1-2 e_w +\varepsilon}$. When this happens, the process $X$ evolves as a d-dimensional martingale during time interval $[\sigma_{j-1},\sigma_j)$ and for $Y_\cdot = X_{\sigma_{j-1}+\cdot}$
$$
\tau^Y_{\left(\mathcal{R}^X_{\sigma_{j-1}}\right)^c} \ge \sigma_j - \sigma_{j-1} \ge n^{2e_w-\varepsilon}\,.
$$
To control the probability of $G_2^c$, we will apply~\cite[Lemma 5.4]{menshikov2012general}, setting $\delta = \frac{\varepsilon}{2 e_w}$, $m = n^{2 e_w}$ and $U = \left(\mathcal{R}^X_{\sigma_{j-1}}\right)^c$.
By the definition of $\sigma_{j-1}$, we have that $|\mathcal{R}_{\sigma_{j-1}} ^X \cap B(X_{\sigma_{j-1}}, n^{e_w}) | \leq n^{e_t}$, which for our choice of parameters implies
\begin{align*}
| B(X_{\sigma_{j-1}}, n^{e_w}) \setminus \left(\mathcal{R}_{\sigma_{j-1}} ^X \right)^c| &=| B(X_{\sigma_{j-1}}, (n^{2e_w})^{1/2}) \setminus \left(\mathcal{R}_{\sigma_{j-1}} ^X \right)^c| \leq n^{e_t}
\\
&
= n^{2e_w(1- b/2) -2\varepsilon}=\left(n^{2e_w}\right)^{1-b/2 - 2\delta}\,,
\end{align*}
and thus we can use~\cite[Lemma 5.4]{menshikov2012general} to conclude that
$$
\mathbb{P}\left[\tau^Y_{\left(\mathcal{R}^X_{\sigma_{j-1}}\right)^c} \ge \sigma_j - \sigma_{j-1} \ge \left(n^{2 e_w} \right)^{1 - \frac{\varepsilon}{2e_w}}\right] \le e^{-\gamma_4' n^{\delta}}\,,
$$
where $\gamma_4^\prime$ is a positive constant. Thus
\begin{align}\label{probg2}
\mathbb{P}[G_2]
\geq 1 - \frac{1}{2}n^{1-2 e_w+\varepsilon} e^{-\gamma_4' n^{\delta}}\,.
\end{align}
Next, assuming that $n$ is large enough so that $8n^{1-\varepsilon} < n/2$, we will show that $(G_1 \cap G_2) \subset G$. Suppose that both $G_1$ and $G_2$ occur, but $|\mathcal{R}_n ^X | < n^{\frac{1}{2} + e_w (1-b)-4\varepsilon}\}$. Denote by $\hat{L}_j $ the number of visits to $H^n_j$ up to time $n$, i.e.,
\begin{equation*}
\hat{L}_j = \sum_{k = 2(j-1)n^{e_w}}^{2(j+1)n^{e_w} - 1} L_n (k)\,,
\end{equation*}
On $\{|\mathcal{R}_n ^X | < n^{\frac{1}{2} + e_w (1-b)-4\varepsilon}\}$ we have
$$
n^{\frac{1}{2} + e_w (1-b)-4\varepsilon} > |\mathcal{R}_n ^X| \ge n^{e_t} |\{j: H^n_j \text{ is a trap}\}|\,,
$$
thus the number of traps is at most
$$
2n^{\frac{1}{2} + e_w(1-b) - 4\varepsilon -e_t} = 2n^{\frac{1}{2} - e_w - 2\varepsilon}\,.
$$
On $G_1$, we have,
\begin{equation*}
\sum_{j\in \mathbb{Z}} \hat{L}_j 1_{\{H^n_j \text{ is a trap}\}} \leq 4n^{e_w} \times 2n^{\frac{1}{2} - e_w - 2\varepsilon} \times n^{\frac{1}{2} + \varepsilon} = 8n^{1-\varepsilon}\,.
\end{equation*}
Now observe that, since for $j \leq n$ we have $\mathcal{R}_j ^X \subset \mathcal{R}_n ^X$, if $|\mathcal{R}_j ^X \cap B(X_j, n^{e_w})| > n^{e_t}$ then $X_j$ must be in a trap.
Since $n$ is such that $8n^{1-\varepsilon} < n/2$, we obtain that, on the event
\begin{equation*}
\Big\{ \sum_{j \in \mathbb{Z}} \hat{L}_j 1_{\{H^n_j \text{ is a trap}\}} \leq 8n^{1-\varepsilon} \Big\}\,,
\end{equation*}
the total time (up to time $n$) spent in non-traps is at least $n-8n^{1-\varepsilon} > n/2$. From~\eqref{sigmak}, the latter implies that $\sigma_{\frac{n^{1-2 e_w + \varepsilon}}{2}} < n$. Indeed, up to time $\sigma_{\frac{n^{1-2 e_w + \varepsilon}}{2}}$ we can have at most $n/2$ instances $j$ such that $|\mathcal{R}_j ^X \cap B(X_j, n^{e_w})| \leq n^{e_t}$. Therefore, on the event $G_2$ we have that $|\mathcal{R}_n ^X| \geq \frac{1}{2}n^{1-2e_w + \varepsilon}$. Recall that we assumed that $G_1$ and $G_2$ occur, but $|\mathcal{R}_n ^X | < n^{\frac{1}{2} + e_w (1-b)-4\varepsilon}$. Since for $e_w <\frac{1}{6}$ (and $n$ sufficiently large) it holds that $\frac{1}{2}n^{1-2e_w + \varepsilon} > n^{\frac{1}{2} + e_w (1-b)-4\varepsilon}$, for every $b \in (0,1)$ and $\varepsilon>0$, we obtain a contradiction. Then, $(G_1 \cap G_2) \subset G$, and~\eqref{probg} follows from~\eqref{probg1} and ~\eqref{probg2},
\begin{align*}
\mathbb{P}[G] & \geq \mathbb{P}[G_1 \cap G_2]
\geq 1 - (\mathbb{P}[G_1] + \mathbb{P}[G_2])
\geq 1 - \left( \left( 2Kn + 1 \right)e^{-\gamma_1' n^{\frac{\varepsilon}{2}}} + \frac{n^{1-2e_w +\varepsilon }}{2} e^{-\gamma_4' n^{ \varepsilon}} \right)\,.
\end{align*}
To conclude the proof of Proposition $\ref{prop41}$, just note that for every $\alpha<1/6$, we can find $e_w<1/6$ and $b \in (0,1)$ and $\varepsilon$ (sufficiently small), such that $\alpha< e_w(1-b)-4\varepsilon$. \end{proof}
|
1,108,101,566,722 | arxiv | \section{Conclusion \& Limitations}
We present a globally optimal formulation to jointly search for both the topology and geometry of a planar linkage structure. Our formulation relaxes the problem into a MICP, for which optimal solutions can be found efficiently using BB algorithms. Our results show that our formulation can search for complex structures from trivial and intuitive user inputs, i.e. target end-effector trajectories. Additionally, various design constraints can be easily incorporated. For moderately complex structures, the solve time using these formulations falls in the range between minutes and hours.
As a major limitation, the solve time increases quickly with the number of possible rigid bodies in the planar structure ($K$) and the number of samples on the target trajectory ($T$) because the number of decision variables depends on a multiplication of these two parameters. A related issue is that MICP only satisfies the geometric constraints approximately. As illustrated in \prettyref{fig:failcase}c, a predicted target trajectory with approximate constraints satisfaction can be different from a predicted target trajectory with exact constraints satisfaction after solving \prettyref{eq:NLP}. To reduce the approximation error, we have to increase the approximation granularity by using a larger $S$, which in turn increases the number of binary decision variables. Finally, note that our formulation does not generate all possible planar linkage structures but only those that allow sequential forward kinematic processing. This problem is inherited from \cite{kecskemethy1997symbolic,bacher2015linkedit} by using the same representation as these works. Allowing more general planar linkages is also possible under the MICP formulation by using a new formulation of topology constraints.
\begin{figure}[t]
\centering
\vspace{-5px}
\includegraphics[width=0.9\textwidth]{examplesFailedPov-min.pdf}
\put(-280,0){(a)}
\put(-200,0){(b)}
\put(-120,0){(c)}
\put(-50 ,0){(d)}
\put(-320,75){Ambiguity}
\put(-185,82){Colinear}
\put(-80,90){Sub-Optimal}
\caption{\label{fig:failcase} Failure cases and issues with our formulation. (a,b): MICP only returns the single global optimum. But similar target trajectories can lead to two different linkage structures. (c): We only satisfy geometric constraints approximately, so that the linkage structure might not satisfy these constraints exactly. In this example, we have two rigid rods being colinear. (d): Usually, the early feasible solutions found by MICP are of low-quality, and we have to wait for the MICP to find the global optimum.}
\end{figure}
\subsection{Future Work}
Our future research will focus on a balance between global optimality and formulation efficiency. Such a balance could possibly be achieved by using MINLP formulations. In addition, we observe that different planar linkages, as shown in \prettyref{fig:failcase}ab, can generate very similar target trajectories. This indicates that there exist many local optima with objective function close to the global optimum. However, a BB algorithm will only return the single global optimum. In addition, we found that we need to wait until the BB algorithm finds its global optimum; the intermediary solutions are not usually usable, as illustrated in \prettyref{fig:failcase}d. A potential future direction is to use algorithms such as Bayesian optimization that can explore multiple local optima and return many solutions for users to make a choice.
\subsection{\label{sec:geometry}Geometric Correctness}
The main utility of geometric correctness constraints is to compute the exact positions $\E{n}_i=\TWO{x_i}{y_i}$ of each node in the 2D workspace. These positions are functions of time $t$ and we sample a set of $T$ discrete time instances $t^{1,\cdots,T}$. In this section, we will always use superscripts for timestep indices. For example, at time instance $t^d$, the position of $\E{n}_i$ is $\E{n}_i^d$. We want to find a common geometric specification such that all the end-effector positions $\E{n}_K^{1,\cdots,T}$ can be achieved.
The most important geometric variable is the length of each rigid rod. We define these parameters implicitly using a set of constraints such that, if $\E{n}_i$ and $\E{n}_j$ are connected, then the distance between these two nodes is a constant for all time instances. In other words, we need the following set of constraints if $C_{ji}=1$:
\begin{align}
\label{eq:equidistant_nonconvex}
\|\E{n}_j^d-\E{n}_i^d\|^2=\|\E{n}_j^{(d\bmod T)+1}-\E{n}_i^{(d\bmod T)+1}\|^2\quad\forall 1\leq d\leq T,
\end{align}
after which any distance $\|\E{n}_j^d-\E{n}_i^d\|^2$ can be used as the rigid rod length.
However, there are two challenging issues in modeling these constraints that can affect the performance of the MICP solver. A first challenge is to minimize the use of binary variables. Because any pair of nodes $\E{n}_j$ and $\E{n}_i$ might be connected, a naive formulation will require a number of binary variables proportional to $K^2$. Instead, we introduce auxiliary term $\E{d}_{1i}^d=\TWO{dx_{1i}^d}{dy_{1i}^d}$, which indicates the relative position between $\E{n}_i$ and the first other node connected to it at time instance $t^d$. Similarly, $\E{d}_{2i}^d=\TWO{dx_{2i}^d}{dy_{2i}^d}$ indicates the relative position between $\E{n}_i$ and the second other node connected to it. These definitions induce the following big-M constraints:
\begin{equation}
\begin{aligned}
\label{eq:distanceDef}
&\exists \{dx,dy\}_{ki}^d\quad\forall k=1,2\land i=2,\cdots,K\land d=1,\cdots,T \\
&|\{dx,dy\}_{ki}^d-\{x,y\}_j^d+\{x,y\}_i^d|\leq 2B(1-C_{ji}^k)\quad\forall j=1,\cdots,i-1,
\end{aligned}
\end{equation}
where $B$ is the big-M parameter, implying that all the node positions lie in a bounded region $[-B,B]^2$. Note that the first motor node $\E{n}_1$ follows a circular curve (\prettyref{eq:motor}), which requires special definitions of $\E{d}_{11}^d,\E{d}_{21}^d$ as follows:
\begin{align}
\label{eq:distanceDefMotor}
\{dx,dy\}_{11}^d=\{dx,dy\}_{21}^d=\{x_1^d-X_C,y_1^d-Y_C\},
\end{align}
where the center of rotation $\TWO{X_C}{Y_C}$ is used as an additional auxiliary variable. The second challenge is that these constraints are non-convex because they involve quadratic terms. Fortunately, efficient formulations have been developed to relax non-convex functions using piecewise linear approximation \cite{liberti2004reformulation} and a special ordered set of type 2 ($\mathcal{SOS}_2$) \cite{vielma2011modeling}. $\mathcal{SOS}_2$ effects a constraint that at most two of the variables in an ordered set with consecutive indices can take non-zero values. To use these formulations, we decompose the range $[-B,B]$ evenly into $S-1$ pieces with $S$ nodes:
\begin{align*}
\{\alpha_i|-B=\alpha_1<\alpha_2<\cdots<\alpha_S=B\}.
\end{align*}
As a result, for any $\alpha\in[-B,B]$, a piecewise linear upper bound of $\alpha^2$ is $\tilde{\alpha}$, which is defined in \prettyref{eq:upper}.
\begin{figure}[t]
\vspace{-10px}
\begin{minipage}{.4\textwidth}
\begin{equation}
\begin{aligned}
\label{eq:upper}
\TWOC{\alpha}{\tilde{\alpha}}=\sum_{s=1}^S\lambda_s\TWOC{\alpha_s}{\alpha_s^2}& \\
\{\lambda_{1,\cdots,S}\}\in\mathcal{SOS}_2& \\
\sum_{s=1}^S\lambda_s=1&,
\end{aligned}
\end{equation}
\end{minipage}\hfill\hspace{0.01\textwidth}
\begin{minipage}{.5\textwidth}
\includegraphics[width=0.99\textwidth]{UpperBound.pdf}
\put(-165,-5){$\alpha_1$}
\put(-140,-5){$\alpha_2$}
\put(-90 ,-5){$\alpha_3$}
\put(-33 ,-5){$\alpha_4$}
\put(-10 ,-5){$\alpha_5$}
\put(-82 ,82){$\alpha^2$}
\end{minipage}
\vspace{-5px}
\caption{\label{fig:upperBound} An illustration of the piecewise linear upper bound (blue) of the quadratic curve $\alpha^2$ (red) with $S=5$.}
\vspace{-5px}
\end{figure}
As illustrated in \prettyref{fig:upperBound}, $\alpha^2\leq\tilde{\alpha}$ and this upper bound can be arbitrarily tight as $S\to\infty$. This formulation has been used in \cite{dai2017global} to discretize the space of unit vectors. In the rest of the paper, we use a tilde to denote such an upper bound. Using these upper bounds, the equidistant constraints can be approximated using the following conic constraints:
\begin{equation}
\begin{aligned}
\label{eq:equidistant}
&\forall i=1,\cdots,K\land d=1,\cdots,T \\
&\|\E{n}_i^d-\E{n}_i^{(d\bmod T)+1}\|^2\leq(2\sqrt{2}B)^2(1-F_i) \\
&\forall k=1,2\land i=1,\cdots,K\land d=1,\cdots,T \\
&\|\E{d}_{ki}^{(d\bmod T)+1}\|^2\leq \tilde{dx}_{ki}^d+\tilde{dy}_{ki}^d+(2\sqrt{2}B)^2F_i \\
&\|\E{d}_{ki}^d\|^2\leq \tilde{dx}_{ki}^{(d\bmod T)+1}+\tilde{dy}_{ki}^{(d\bmod T)+1}+(2\sqrt{2}B)^2F_i
\end{aligned}
\end{equation}
where the last term on the right-hand sides is the big-M term that excludes fixed nodes. The idea is to require the length of two vectors to be smaller than the upper bound of one another. Note that \prettyref{eq:equidistant} converges to \prettyref{eq:equidistant_nonconvex} as $S\to\infty$. This formulation will require an upper bound for all $\E{d}_{ki}^d$ and each upper bound requires $\lceil\E{log}S\rceil$ binary variables. As a result, our formulation will introduce $\mathcal{O}(4TK\lceil\E{log}S\rceil)$ binary decision variables. We also introduce a last constraint to ensure that rigid rods are not degenerate by ensuring minimal rod length $l_{min}$:
\begin{equation}
\begin{aligned}
\label{eq:miniLength}
&\forall k=1,2\land i=1,\cdots,K\land d=1,\cdots,T \\
&\tilde{dx}_{ki}^d+\tilde{dy}_{ki}^d\geq l_{min}^2-((2\sqrt{2}B)^2+l_{min}^2)F_i.
\end{aligned}
\end{equation}
\begin{figure}[t]
\vspace{-5px}
\centering
\includegraphics[width=0.99\textwidth]{AngleConstraint.pdf}
\put(-270,60){$\E{v}_1^L$}
\put(-273,43){$\E{v}_1^R$}
\GAMMA{0}{292}{55}{45}{$\gamma_1$}
\GAMMA{1}{292}{55}{45}{$\gamma_2$}
\GAMMA{2}{292}{55}{45}{$\gamma_3$}
\GAMMA{3}{292}{55}{45}{$\gamma_4$}
\GAMMA{4}{292}{55}{45}{$\gamma_5$}
\GAMMA{5}{292}{55}{45}{$\gamma_6$}
\GAMMA{6}{292}{55}{45}{$\gamma_7$}
\GAMMA{7}{292}{55}{45}{$\gamma_8$}
\put(-295,-10){(a)}
\put(-178,113){$\epsilon$}
\put(-160, 90){$\E{d}_{1i}^d$}
\put(-210, 65){$\E{d}_{2i}^d$}
\put(-177,-10){(b)}
\GAMMA{0}{60}{55}{45}{$\gamma_1$}
\GAMMA{1}{60}{55}{45}{$\gamma_2$}
\GAMMA{2}{60}{55}{45}{$\gamma_3$}
\GAMMA{3}{60}{55}{45}{$\gamma_4$}
\GAMMA{4}{60}{55}{45}{$\gamma_5$}
\GAMMA{5}{60}{55}{45}{$\gamma_6$}
\GAMMA{6}{60}{55}{45}{$\gamma_7$}
\GAMMA{7}{60}{55}{45}{$\gamma_8$}
\GAMMAI{0}{60}{55}{45}{$\gamma_9$}
\GAMMAI{1}{60}{55}{45}{$\gamma_{10}$}
\GAMMAI{2}{60}{55}{45}{$\gamma_{11}$}
\GAMMAI{3}{60}{55}{45}{$\gamma_{12}$}
\GAMMAI{4}{60}{55}{45}{$\gamma_{13}$}
\GAMMAI{5}{60}{55}{45}{$\gamma_{14}$}
\GAMMAI{6}{60}{55}{45}{$\gamma_{15}$}
\GAMMAI{7}{60}{55}{45}{$\gamma_{16}$}
\put(-60,-10){(c)}
\vspace{-5px}
\caption{\label{fig:angleConstraint} Linear relaxation of angle constraints. (a): We cut $\mathcal{SO}(2)$ into 8 sectors, each of which is selected by a $\gamma$-flag using $\mathcal{SOS}_1$ constraints. A sector, e.g. the sector selected by $\gamma_1$, is bounded by its left/right unit-length plane-normal vectors $\E{v}_1^L$/$\E{v}_1^R$. (b): If $\E{d}_{1i}^d$ falls in the red area, then we restrict $\E{d}_{2i}^d$ to its left half-space (gray), which is at least $\epsilon$-apart (blue). However, note that when $\E{d}_{1i}^d$ moves across sector boundaries, the gray area will jump discontinuously. (c): To avoid discontinuous changes in the restricted region for $\E{d}_{2i}^d$ when $\E{d}_{1i}^d$ undergoes continuous changes, we propose to double cover $\mathcal{SO}(2)$ using $2S=16$ sectors.}
\vspace{-15px}
\end{figure}
By ensuring a fixed rigid rod length across all time instances, we can make sure that all the end-effector positions $\E{n}_K^d$ can be achieved using the same planar linkage structure. In practice, however, we can only change the end-effector position by moving the first motor node $\E{n}_1$, so we still need to ensure that the mechanics system will not glitch or does not have singular configurations. The most intuitive classification of singular configuration is the rank-deficiency of the Jacobian matrix \cite{BOHIGAS20131}. However, this classification cannot be used in an MICP formulation because it is non-convex and the Jacobian matrix cannot be computed under our implicit representation of rigid rods. Instead, we adopt a heuristic proposed by \cite{Thomaszewski:2014:CDL:2601097.2601143}, which avoids singularities by ensuring that, for any movable node $\E{n}_i$, the two vectors $\E{d}_{1i}^d$ and $\E{d}_{2i}^d$ are not colinear. In other words, the triangle area formed by these two vectors is positive. This constraint takes the following bilinear form:
\begin{align*}
\E{d}_{1i}^d\times\E{d}_{2i}^d\geq\epsilon,
\end{align*}
where $\epsilon$ is a small constant. Although this constraint is bilinear, we can use McCormick envelopes \cite{liberti2004reformulation} to relax it as a conic constraint. If the range $[-B,B]$ is cut into $S-1$ segments, then this formulation will introduce $\mathcal{O}(4TK\lceil\E{log}S\rceil)$. However, a critical flaw of this formulation is that a McCormick envelope is an outer-approximation. As a result, the exact linkage structure can still be singular, although its conic relaxation is non-singular. To ensure strict non-singular formulation, we propose a constraint whereby the angle between the two vectors is larger than $\epsilon$, which is equivalent to the positive area constraint when combined with \prettyref{eq:miniLength}:
\begin{equation}
\begin{aligned}
\label{eq:singular_nonconvex}
\measuredangle\E{d}_{1i}^d,\E{d}_{2i}^d\geq\epsilon.
\end{aligned}
\end{equation}
In addition, we propose an inner approximation such that the exact linkage structure is also guaranteed to be non-singular. Concretely, we cut the space of $\mathcal{SO}(2)$ into $S$ sectors, as illustrated in \prettyref{fig:angleConstraint}a, so that $\E{d}_{1i}^d$ will only fall into one of the $S$ sectors. If $\E{d}_{1i}^d$ falls in a particular sector, then we restrict $\E{d}_{2i}^d$ to its left half-space that is at least $\epsilon$-apart, as shown in \prettyref{fig:angleConstraint}b. If we use an $\mathcal{SOS}_1$ constraint to select the sector in which $\E{d}_{1i}^d$ falls, then only $\mathcal{O}(TK\lceil\E{log}S\rceil)$ binary decision variables are needed. A minor issue with this formulation is that the allowed region of $\E{d}_{2i}^d$ jumps discontinuously as $\E{d}_{1i}^d$ changes continuously. We can fix this problem by double-covering the region of $\mathcal{SO}(2)$ using $2S$ sectors, as shown in \prettyref{fig:angleConstraint}c, which will introduce $\mathcal{O}(TK\lceil\E{log}2S\rceil)$ binary decision variables.
To formulate these constraints, we assume that each sector of $\mathcal{SO}(2)$ is flagged by a selector variable $\gamma_l$, which is bounded by its left/right unit-length plane-normal vectors $\E{v}_l^L$/$\E{v}_l^R$. Then the following constraints must be satisfied if $\E{d}_{1i}^d$ falls inside the sector:
\begin{equation}
\begin{aligned}
\label{eq:sector}
<\E{v}_l^L,\E{d}_{1i}^d>\geq 0&\\
<\E{v}_l^R,\E{d}_{1i}^d>\leq 0& \\
<\E{R}(\epsilon)\E{v}_l^L,\E{d}_{2i}^d>\leq 0& \\
<\E{R}(\pi)\E{v}_l^R,\E{d}_{2i}^d>\geq 0&,
\end{aligned}
\end{equation}
where $\E{R}(\bullet)$ is the $2\times2$ counter-clockwise rotation matrix by angle $\bullet$. Combined with the fact that \prettyref{eq:sector} should only be satisfied for one particular sector and that only movable nodes satisfy these constraints, we have the following formulation:
\begin{equation}
\begin{aligned}
\label{eq:sectorFinal}
\forall i=2,\cdots,K\quad d=1,\cdots,T& \\
<\E{v}_l^L,\E{d}_{1i}^d>\geq 2\sqrt{2}B(\gamma_{l,i}^d-1)-2\sqrt{2}BF_i& \\
<\E{v}_l^R,\E{d}_{1i}^d>\leq 2\sqrt{2}B(1-\gamma_{l,i}^d)+2\sqrt{2}BF_i& \\
<\E{R}(\epsilon)\E{v}_l^L,\E{d}_{2i}^d>\leq 2\sqrt{2}B(1-\gamma_{l,i}^d)+2\sqrt{2}BF_i& \\
<\E{R}(\pi)\E{v}_l^R,\E{d}_{2i}^d>\geq 2\sqrt{2}B(\gamma_{l,i}^d-1)-2\sqrt{2}BF_i& \\
\{(\gamma_{1,1i}^d,\cdots,\gamma_{2S,i}^d\}\in\mathcal{SOS}_1& \\
\sum_{l=1}^{2S}\gamma_{l,i}^d=1&.
\end{aligned}
\end{equation}
These constraints will avoid singular configurations.
\section{Introduction}
A planar linkage is a mechanical structure built with a set of rigid bodies connected by hinge joints. This structure typically has one effective degree-of-freedom actuated by a rotational motor. Since they impose a minimal burden on controller design, these structures are widely used as building blocks for low-cost toys and robots, as illustrated in \prettyref{fig:strandbeest}. By combining a series of hinge joints, the end-effector of the planar linkage will trace out a complex curve that can fulfill various requirements of different types of locomotion, including walking and swimming \cite{HERNANDEZ2016AHO,Thomaszewski:2014:CDL:2601097.2601143}.
A challenging problem in mechanics design is to find the linkage structure with an end-effector that will trace out a given curve. This problem is challenging in that it searches over three coupled variables: topology, geometry, and trajectory. The linkage topology determines which rigid bodies are connected and the order of their connections. Clearly, the topology is a non-smooth and non-differentiable decision variable. The linkage geometry determines the shape of each rigid link. Finally, the trajectory determines the pose of the linkage structure at each time instance. The last two variables are smooth and differentiable, but directly optimizing them induces non-convex functions. Previous works \cite{ha2018computational,Zhu:2012:MMT:2366145.2366146} have proposed various solutions to address problems of this kind. These methods rely on random searches, such as $A^*$ \cite{ha2018computational} and covariance matrix adaptation \cite{Zhu:2012:MMT:2366145.2366146}, to try different topologies. Then, for each topology, they perform non-linear programming (NLP) under the given topology to determine the geometry and trajectory. However, these methods are computationally expensive because a huge number of samples are needed for the random search to converge. Moreover, even after determining the topology, these methods can find only sub-optimal solutions due to the non-convex nature of NLP.
\setlength{\columnsep}{10pt}
\begin{wrapfigure}{r}{6cm}
\centering
\vspace{-15px}
\includegraphics[width=0.4\textwidth]{strandbeest.png}
\vspace{-5px}
\caption{\label{fig:strandbeest} An example of planar linkages, used in a strandbeest robot for 2D walking. See \cite{nansai2013dynamic} for more details.}
\vspace{-10px}
\end{wrapfigure}
\TE{Main Results:} Given the input of a target trajectory, we present a new method that can efficiently compute a planar linkage structure with globally optimal topology and geometry configurations and an accurate trajectory reproduction. Based on recent advances in mixed-integer modeling \cite{vielma2015mixed,dai2017global,trespalacios2015improved}, we relax this joint search problem as an MICP problem, the global optimum of which is arbitrarily close to the global optimum of the original problem. The main benefit of MICP relaxation is that the search can be accomplished efficiently using the BB algorithm. BB is more strategic than random search, as used by \cite{Zhu:2012:MMT:2366145.2366146}, because it cuts impossible or sub-optimal search spaces at an early stage, leading to higher efficiency. We have compared MICP with prior methods using different examples. The results show that our proposed MICP approach finds solutions more efficiently and that the resulting structure matches the target trajectory more closely.
In the rest of the paper, we first review related work in \prettyref{sec:related} and then formulate our joint search problem in \prettyref{sec:problem}. The MICP model and various constraints required for the integrity of the planar linkage are presented in \prettyref{sec:method}. Results and the evaluation of the proposed approach are given in \prettyref{sec:results}.
\subsection{The Complete MICP Formulation}
Combining all the constraints, we minimize two objective function terms. First, we want the end-effector trajectory to match the target trajectory specified by users. Second, to minimize manufacturing cost, we want to use as few rigid rods as possible. To formulate the first objective term, we need to replace a trajectory with a discrete number of samples. However, the order of these samples is discarded. In practice, we find that better solutions can be found by preserving the order between these samples. This requirement is formulated by making sure that $\E{n}_K^d$ will be visited by the end-effector sequentially when the motor node rotates by $2\pi$ either clockwise or counter-clockwise. This requirement is formulated using the following MICP constraints:
\begin{equation}
\begin{aligned}
\label{eq:rotation}
&\forall d=1,\cdots,T-1 \\
\|\E{R}(\frac{2\pi}{T})\E{d}_{11}^d-\E{d}_{11}^{d+1}\|^2&\leq(2\sqrt{2}B)^2D& \\
\|\E{R}(-\frac{2\pi}{T})\E{d}_{11}^d-\E{d}_{11}^{d+1}\|^2&\leq(2\sqrt{2}B)^2(1-D),
\end{aligned}
\end{equation}
where $D$ is a binary variable to choose which direction the motor rotates. Putting everything together, we arrive at the following MICP problem:
\begin{equation}
\begin{aligned}
\label{eq:MICP}
\argmin{}&\sum_{d=1}^T\|\E{n}_K^d-\E{n}_K^{d*}\|^2+w\sum_{i=1}^K U_i \\
\E{s.t.}\quad\;\;{}&\text{\prettyref{eq:state}, \ref{eq:connectivity}, \ref{eq:balance}, \ref{eq:balance2},
\ref{eq:distanceDef}, \ref{eq:distanceDefMotor}, \ref{eq:equidistant},
\ref{eq:miniLength}, \ref{eq:sectorFinal}, \ref{eq:rotation},}
\end{aligned}
\end{equation}
where $\E{n}_K^{d*}$ are the sampled points on the target trajectory and $w$ the regularization weight of the cost-efficiency term. Since non-convexity is not accepted by MICP, the solution returned by MICP is only a piecewise linear approximation of the original nonlinear problem. To return a solution with exact constraint satisfaction, we refine the solution by solving an additional NLP locally using the following formulation:
\begin{equation}
\begin{aligned}
\label{eq:NLP}
\argmin{}&\sum_{d=1}^T\|\E{n}_K^d-\E{n}_K^{d*}\|^2 \\
\E{s.t.}\quad\;\;{}&\text{\prettyref{eq:state}, \ref{eq:connectivity}, \ref{eq:balance}, \ref{eq:balance2},
\ref{eq:distanceDef}, \ref{eq:distanceDefMotor}, \ref{eq:equidistant_nonconvex},
\ref{eq:singular_nonconvex}, \ref{eq:rotation},}
\end{aligned}
\end{equation}
where we fix all the binary variables $U_i,F_i,D$. Note that \prettyref{eq:NLP} is a mixed-integer NLP (MINLP) generalization of \prettyref{eq:MICP} and we have the following lemma:
\begin{lemma}
\prettyref{eq:MICP} converges to \prettyref{eq:NLP} as $S\to\infty$, and the BB algorithm can find the global optimum for \prettyref{eq:MICP}.
\end{lemma}
\section{\label{sec:problem}Joint Search for Planar Linkages}
\begin{figure}[ht]
\centering
\vspace{-5px}
\begin{tabular}{@{}c@{}}
\includegraphics[width=0.25\textwidth]{IllusLinkage.pdf}
\put( 2,100){\textcolor{motor}{$\E{n}_1$}}
\put(-43,95 ){\textcolor{fixed}{$\E{n}_2$}}
\put(-37,137){\textcolor{movable}{$\E{n}_3$}}
\put(-97,111){\textcolor{movable}{$\E{n}_4$}}
\put(-27,63 ){\textcolor{movable}{$\E{n}_5$}}
\put(-83,78 ){\textcolor{movable}{$\E{n}_6$}}
\put(-42,16 ){\textcolor{endeffector}{$\E{n}_7$}}
\end{tabular}
\begin{tabular}{@{}c@{}}
\includegraphics[width=0.4\textwidth]{IllusLinkageTriangle.pdf}
\put(-80 ,75){$\CNN{3}{2}{1}$}
\put(-35 ,75){$\CNN{5}{2}{1}$}
\put(-107,10){$\CNN{4}{3}{2}$}
\put(-62,10){$\CNN{6}{5}{4}$}
\put(-17,10){$\CNN{7}{6}{5}$}
\end{tabular}
\caption{\label{fig:linkage} (a): The Jansen's mechanics used in \prettyref{fig:strandbeest} is a planar linkage structure involving 7 nodes. The motor node $\E{n}_1$ is green, the fixed node $\E{n}_2$ is red, the movable nodes $\E{n}_{3,4,5,6}$ are black, and the end-effector node $\E{n}_7$ is blue. Our goal is to find the topology and geometry of the linkage such that the end-effector curve matches the blue target curve. (b): Our MICP formulation is based on the prior symbolic representation \cite{kecskemethy1997symbolic,bacher2015linkedit}. This representation assumes that each node is connected to exactly two other nodes with lower indices: $\CNN{3}{2}{1}$, $\CNN{5}{2}{1}$, $\CNN{4}{3}{2}$, $\CNN{6}{5}{4}$, $\CNN{7}{6}{5}$.}
\vspace{-10px}
\end{figure}
In this section, we introduce the problem of joint searches for planar linkages. Our problem is to search for a structure, as illustrated in \prettyref{fig:linkage}a, where we have a set of rod-like rigid bodies connected with each other using hinge joints. As a result, the end points of these rigid bodies can take at most $N$ distinct positions, denoted as node set: $\E{n}_{1,\cdots,N}$. Of these nodes, $\E{n}_1$ is the rotational motor and $\E{n}_N$ is the end-effector. Within one limit cycle, $\E{n}_1$ follows a circular curve centered at $\TWO{X_C}{Y_C}$ with a radius $R$:
\begin{align}
\label{eq:motor}
\E{n}_1(t)=\TWO{\E{sin}(t)R+X_C}{\E{cos}(t)R+Y_C},
\end{align}
which induces trajectories of other nodes $\E{n}_i(t)$ via forward kinematics. The other $N-2$ nodes can be one of two kinds: fixed or movable. In addition, a rigid body may exist between each pair of nodes $\E{n}_{i,j}$, in which case $\|\E{n}_i(t)-\E{n}_j(t)\|$ is a constant.
Given these definitions, the input to our problem is a target end-effector trajectory $\E{n}_N^*(t)$. The output of our method is the following set of variables defining both the topology and geometry of a planar linkage:
\begin{itemize}
\item An integer vector of size $N$ (the number of nodes), which containing the type of each node: fixed or movable.
\item An $N\times N$ symmetric binary matrix $C^{N\times N}$ where $C_{ij}=1$ means a rigid body connects $\E{n}_{i,j}$.
\item The position of $\E{n}_{1,\cdots,N}(t)$ at a certain, arbitrary time instance $t$.
\end{itemize}
The goal of our method is to find the globally optimal set of variables that minimizes $\int \|\E{n}_N(t)-\E{n}_N^*(t)\|^2 dt$.
\section{\label{sec:related}Related Work}
In this section, we review related work in robot design optimization, mixed-integer modeling, and topology optimization.
\TE{Robot Design Optimization:} Robot design optimization is a superset of conventional topology and truss optimization \cite{LIU2016161} where the decision variables are only topology or geometry. This is because the specification of a robot design is given as a movement pattern \cite{Ha2017JointOO}, leading to a joint search in the space-time domain. The joint search problem greatly expands the search space. As a result, many prior methods do not work since they only optimize a subset of decision variables \cite{Ha2017JointOO,Thomaszewski:2014:CDL:2601097.2601143,bacher2015linkedit,saar2018model,spielberg2017functional}. Recent works \cite{Zhu:2012:MMT:2366145.2366146,ha2018computational,song2017computational} search for all variables simultaneously. However, these methods are based on random search techniques, which usually require a large amount of trial and error and find sub-optimal solutions.
\TE{Mixed-Integer Modeling:} The main benefit of mixed-integer modeling is the use of the well-studied BB algorithm \cite{lawler1966branch}. BB allows us to find the global optimum of non-convex programming problems, while only visiting a small fraction of the search space. Mixed-integer models have been applied to a large variety of problems including motion planning \cite{ding2011mixed}, inverse kinematics \cite{dai2017global}, network flows \cite{conforti2009network}, and mesh generations \cite{bommes2009mixed}. By applying the big-M method \cite{trespalacios2015improved}, McCormick envelopes and piecewise approximations \cite{liberti2004reformulation}, and general non-convex problems can be easily relaxed as MICP problems. Prior works \cite{kanno2013topology,lobato2003mixed} have also formulated topology optimization problems as MICP. However, our work is the first to formulate the planar linkage problem as MICP and we employ MICP to concurrently find the optimal topology, geometry, and trajectory of a linkage.
\TE{Topology Optimization:} Topology optimization of a continuum is a well-studied problem \cite{LIU2016161}. An efficient algorithm can smoothen the problem and use gradient-based method to search for locally optimal structures over a search space of millions of dimensions. This technique has been widely used in the design of soft robots \cite{zhang2017design,zhang2018design,zhu2017two}. However, the optimization of articulated robots is more challenging because the optimized structure must satisfy the joint constraints, making the decision variable non-smooth. Existing techniques use mixed-integer \cite{kanno2013topology,lobato2003mixed} or random search techniques \cite{Zhu:2012:MMT:2366145.2366146,song2017computational} to optimize over these decision variables.
\section{\label{sec:results}Results and Evaluations}
We have implemented our method using Gurobi \cite{gurobi} as our MICP solver for \prettyref{eq:MICP} and Knitro \cite{byrd2006k} as our NLP solver for \prettyref{eq:NLP}. All the experiments are performed on a cluster with 4 CPU cores per process (2.5GHz E5-2680 CPU). Compared with prior work \cite{Thomaszewski:2014:CDL:2601097.2601143}, the main benefit of our formulation is that we can search for planar linkage structures from a target trajectory of the end-effector that requires trivial effort from users. In \prettyref{fig:results_gallary}, we show a list of different target trajectories and the optimized planar linkage structures.
\begin{figure}[t]
\centering
\includegraphics[width=0.99\textwidth]{examplesPov-min.pdf}
\caption{\label{fig:results_gallary} We show 10 different optimized planar linkage structures with the end-effector trajectory in blue and the user-specified target trajectory in yellow. For all these examples, we choose $K=5\sim7$, $S=9$, and $T=10\sim20$. The end-effector trajectory matches closely with the target trajectory.}
\vspace{-10px}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.32\textwidth]{TimeVsK.pdf}
\includegraphics[width=0.32\textwidth]{TimeVsS.pdf}
\includegraphics[width=0.32\textwidth]{TimeVsT.pdf}
\vspace{-5px}
\caption{\label{fig:scalability} We plot the average computational time for solving MICP in 35 example problems using different parameters in \prettyref{fig:results_gallary}. The computational time for solving MICP grows exponentially with $K$, $\E{log}S$, and $T$.}
\vspace{-15px}
\end{figure}
\begin{figure}[h]
\vspace{-10px}
\centering
\includegraphics[width=0.31\textwidth]{Convergence1.pdf}
\includegraphics[width=0.31\textwidth]{Convergence2.pdf}
\includegraphics[width=0.31\textwidth]{Convergence3.pdf}
\vspace{-5px}
\caption{\label{fig:optimal_gap} We plot the convergence history curve for 3 typical optimizations by showing the objective function values plotted against the number of nodes explored in the BB search tree. The BB algorithm spends most of its time exploring infeasible nodes and the first identified feasible solution is usually very close to the optimal solution, so that the optimizer will return the globally optimal solution after refining the solution for $5-10$ times.}
\vspace{-15px}
\end{figure}
The performance and accuracy of our algorithm heavily depend on the three parameters: the max number of rigid rods $K$, the number of pieces for approximating $S$, and the number of samples on the target trajectory $T$. Since the cost of solving MICP grows exponentially with the number of binary decision variables, which is proportional to $K$, $\E{log}S$, and $T$, our method cannot scale to large problems, as illustrated in \prettyref{fig:scalability}. In practice, we find that, given a maximal computational time of $10$ hours, we can compute globally optimal solutions for most benchmarks with $K\leq 7$, $S\leq 9$, and $T\leq 20$. This is enough if we design robots part-by-part, as is done in the Theo Jansen's strandbeest. For other benchmarks, the computational time is longer than $10$ hours, but a feasible solution has been found, although it is sub-optimal. In \prettyref{fig:optimal_gap}, we plot the average convergence history of a typical optimization. Since we express all the topology and geometric requirements as hard mixed-integer constraints, feasible solutions are quite rare in the search space and the optimizer takes most of the computational time pruning infeasible solutions. Once the first feasible solution is found, it is usually very close to the optimal solution and the optimizer refines it for less than 10 times to reach the optimal solution.
\begin{figure}[h]
\vspace{-10px}
\centering
\includegraphics[width=0.9\textwidth]{SAComparison.pdf}
\put(-270,0){(a)}
\put(-220,0){(b)}
\put(-140,0){(c)}
\put(-20 ,0){(d)}
\vspace{-5px}
\caption{\label{fig:SAComparison} SA can find good enough solutions for simple target curves (a). However, for more complex curve shapes, SA failes (b) while MICP succeeds (c). We also plot the objective function values returned by SA and MICP in 10 computational examples in (d), where MICP outperforms SA in 9 instances.}
\vspace{-15px}
\end{figure}
We have also compared our method with conventional global search algorithms such as simulated annealing (SA). We implemented a similar algorithm as proposed in \cite{Zhu:2012:MMT:2366145.2366146}. In this algorithm, we randomly generate $1000000$ samples by random moves and accept these samples according to the simulated annealing rule. Each random move can be of one of three kinds: geometric change, node addition, and node removal. In geometric change, the length of a rigid rod is randomly perturbed. In node addition, a new node is added and the length of the new rigid rods are randomly picked. In node removal, the end-effector node is removed and the last movable node is used as the new end-effector node. We enhance standard SA algorithm by making sure that each random move is valid. In other words, we introduce an inner loop and repeated try random moves until the modified planar linkage structure satisfies all the topological constraints and has no singular configurations. As illustrated in \prettyref{fig:SAComparison}a, SA algorithm can find satisfactory results for simple target curves, but SA usually fails for more complex curve shapes (\prettyref{fig:SAComparison}bc). In \prettyref{fig:SAComparison}d, we also show the objective function values after convergence. The solution of MICP is almost always better than the solution of SA. However, SA outperforms MICP in one example, which is probably due to the inexact constraint satisfaction of MICP.
Usually, the design of a planar linkage structure is not only subject to a target end-effector trajectory, but also to various other user constraints. For example, the user might require certain nodes to be fixed, which can be easily achieved using our MICP formulation. The user may also reserve certain parts of the robot for some functional units that cannot be occupied by the planar linkages. This type of constraint can be expressed as collision avoidance between a planar linkage structure and a specified convex region, which can be formulated as MICP constraints using a prior method \cite{ding2011mixed}. In \prettyref{fig:constraints}, we show results taking these constraints into consideration.
\begin{figure}[h]
\vspace{-10px}
\centering
\scalebox{0.9}{
\includegraphics[width=0.9\textwidth]{constraintsPov.pdf}
\put(-240,10){(a)}
\put(-210,10){(b)}
\put(-50 ,10){(c)}
\put(-185,145){Fixed}
\put(-220,45){Too Close}
\put(-100,165){Bounded Region}}
\vspace{-10px}
\caption{\label{fig:constraints} We show results taking two different user constraints into consideration. (a): Results with no constraints. The optimizer is guided by the regularization term to use as few nodes as possible. (b): We fix the center of rotation and the optimizer finds a more complex structure with 6 nodes. (c): If we do not want the structure to be too close to the target curve, we can add a bounded region and create a constraint that any nodes (other than the end-effector node) should be inside the bounded region.}
\vspace{-15px}
\end{figure}
\section{\label{sec:method}MICP Formulation of Joint Search}
In this section, we present a set of linear constraints and quadratic objective functions for relaxing the joint search as an MICP problem. We first introduce the set of topology constraints to ensure the well-posed nature of the structure in \prettyref{sec:topology} and then present constraints and objective functions for geometric correctness in \prettyref{sec:geometry}.
\subsection{\label{sec:topology}Topology Constraints}
As illustrated in \prettyref{fig:linkage}b, our method is based on the symbolic representation presented in \cite{kecskemethy1997symbolic,Thomaszewski:2014:CDL:2601097.2601143}, which assumes that each movable node is attached to two other nodes. These nodes can be of any type but must have lower node indices. As a result, forward kinematics can be processed sequentially even on linkage structures with closed loops.
Since the number of nodes is unknown, we assume that the maximal number of nodes is $K>N$. For each node other than the first motor node $\E{n}_1$, we need a binary variable $U_i$ such that $U_i=1$ indicates $\E{n}_i$ is used as a part of the planar linkage structure. In addition, we need another binary variable $F_i$ such that $F_i=1$ indicates $\E{n}_i$ is fixed and $F_i=0$ indicates $\E{n}_i$ is movable. These two sets of variables are under the constraint that only a used node can be movable. In addition, we assume that the last node $\E{n}_K$ is the end-effector that must be used. In summary, we introduce the following sets of variables and node-state constraints:
\begin{equation}
\begin{aligned}
\label{eq:state}
\exists U_i,F_i\in\{0,1\} &\quad\forall i=1,\cdots,K \\
1-F_i\leq U_i& \\
U_1=U_K=1& \\
F_1=0&.
\end{aligned}
\end{equation}
Our next set of constraints ensures local topology correctness. It ensures that each movable node is connected to exactly two other nodes with lower indices. As a result, the movable node and the two other nodes will form a triangle and the position of the movable node can then be determined via the Law of Cosine \cite{Ha2017JointOO}. We introduce auxiliary variables $C_{ji}^1$ to indicate whether $\E{n}_j$ is the first node to which $\E{n}_i$ is connected. $C_{ji}^2$ indicates whether $\E{n}_j$ is the second node to which $\E{n}_i$ is connected. In addition, we introduce two verbose variables $C_{0i}^{1,2}=1$ to indicate that $\E{n}_i$ is connected to nothing. The resulting constraint set is:
\begin{equation*}
\begin{aligned}
\exists C_{ji},C_{ji}^1,C_{ji}^2\in\{0,1\}&\quad\forall j,i=1,\cdots,K\land j<i \\
C_{ji}=C_{ji}^1+C_{ji}^2& \\
C_{ji}^1\leq U_j\land C_{ji}^2\leq U_j& \\
\sum_{j=1}^{i-1} C_{ji}=2-2F_i&\quad\forall i=2,\cdots,K \\
\exists C_{0i}^d\in\{0,1\}&\quad \forall d=1,2 \\
\sum_{j=0}^{i-1} C_{ji}^d=1&.
\end{aligned}
\end{equation*}
When $\E{n}_i$ is fixed in the above formulation, then $F_i=1$ in \prettyref{eq:connectivity} and all $C_{ji}$ are zero except for $C_{0i}^{1,2}=1$ due to the sum-to-one constraints. If $\E{n}_i$ is movable, then $F_i=0$ and $C_{ji}$ sums to two. As a result, there must be $j_1,j_2<i$ such that $C_{j_1i}^1=1$ and $C_{j_2i}^2=1$. Note that $j_1$ and $j_2$ must be different because otherwise the constraint that $C_{ji}\in[0,1]$ will be violated. In addition, since the first node $\E{n}_1$ is the motor node, it is excluded from these connectivity constraints. However, this naive formulation will require binary variables for each pair of $\E{n}_j$ and $\E{n}_i$, which requires $\mathcal{O}(K^2)$ binary variables all together. Instead, we adopt the idea of special ordered set of type 1 ($\mathcal{SOS}_1$) \cite{vielma2011modeling} and model these constraints using $\mathcal{O}(K\lceil\E{log}K\rceil)$ binary variables. Intuitively, $\mathcal{SOS}_1$ constrains that only one variable in a set can take a non-zero value and it can be achieved by using a logarithm number of binary variables. The improved constraint set is:
\begin{equation}
\begin{aligned}
\label{eq:connectivity}
\exists C_{ji},C_{ji}^1,C_{ji}^2\in\TR{[0,1]}&\quad\forall j,i=1,\cdots,K\land j<i \\
C_{ji}=C_{ji}^1+C_{ji}^2& \\
C_{ji}^1\leq U_j\land C_{ji}^2\leq U_j& \\
\sum_{j=1}^{i-1} C_{ji}=2-2F_i&\quad\forall i=2,\cdots,K \\
\exists C_{0i}^d\in\TR{[0,1]}&\quad \forall d=1,2 \\
\TR{\{C_{ji}^d|j=0,\cdots,i-1\}\in\mathcal{SOS}_1}& \\
\sum_{j=0}^{i-1} C_{ji}^d=1&.
\end{aligned}
\end{equation}
Finally, we introduce a third set of constraints to ensure global topology correctness. This set of constraints ensures that the linkage structure contains no wasted structures. In other words, each node must have some influence on the trajectory of the end-effector node and the first motor node must be connected to others. We model these constraints using the MICP formulation of network flows \cite{conforti2009network}. Specifically, each node $\E{n}_i$ will generate an outward flux that equals to $U_i$, and we assume that there is a flow edge defined between each pair of nodes with capacity $Q_{ji}$. We require inward-outward flux balance for each node except for the end-effector node:
\begin{equation}
\begin{aligned}
\label{eq:balance}
\exists Q_{ji}\in[0,\infty]&\quad\forall j,i=1,\cdots,K\land j<i \\
Q_{ji}\leq C_{ji}K& \\
U_i+\sum_{j=1}^{i-1} Q_{ji}=\sum_{k=i+1}^{K} Q_{ik} &\quad\forall i=1,\cdots,K-1,
\end{aligned}
\end{equation}
where we adopt the big-M method \cite{trespalacios2015improved} in the second constraint to ensure that only edges between connected nodes can have a capacity up to $K$. Using a similar idea, we also formulate a constraint that a movable node must be connected to at least one other movable node. We assume that each node $\E{n}_i$ generates a reversed outward flux that equals to $1-F_i$, and we assume that there is a flow edge defined between each pair of nodes with capacity $R_{ji}$. We require inward-outward flux balance for each node except for the motor node:
\begin{equation}
\begin{aligned}
\label{eq:balance2}
\exists R_{ji}\in[0,\infty]&\quad\forall j,i=1,\cdots,K\land j<i \\
R_{ji}\leq C_{ji}K\land R_{ji}\leq (1-F_j)K& \\
\sum_{j=1}^{i-1} R_{ji}=1-F_i+\sum_{k=i+1}^{K} R_{ik} &\quad\forall i=2,\cdots,K,
\end{aligned}
\end{equation}
These three constraints ensure that the planar linkage structure is symbolically correct, independent of the concrete geometric shape. |
1,108,101,566,723 | arxiv | \section{Introduction}
\label{introduction}
Indirect searches for Dark Matter, i.e. searches for `anomalous' features in cosmic rays (e.g. gamma-rays, neutrinos, positrons and anti-protons), have been proposed in the late 70's as a powerful way to reveal the existence of Dark Matter annihilations in the Milky Way halo and beyond~\cite{indirectgamma,indirectcharged}.
These techniques are meant to give precious insights about the nature of the Dark Matter particle and its properties, assuming that a signal is seen. Yet there are several limiting factors which weaken their ability to elucidate the dark matter problem. In particular indirect detection requires a detailed knowledge of the astrophysical backgrounds and foregrounds and therefore depends on the present knowledge of astrophysical sources. To make a discovery one either has to carefully remove known (or modelled) background in order to expose the `anomalous' component or hope that the Dark Matter signal is well above the background and exhibits very clear features, which would be difficult to mimic by invoking astrophysical sources only.
\medskip
Currently there are contradicting claims regarding whether indirect detection is giving clues of dark matter or not. On one hand, there are possible anomaly detections which could be explained in terms of Dark Matter annihilation or decay. These include for example the positron excess, as seen in {\sc Pamela}\ (and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT) data~\cite{PAMELApositrons}, a possible feature in the $e^+ + e^-$ spectrum~\cite{epem}, a claimed $\gamma$-ray excess at $\sim$ 10 GeV energies~\cite{Hooper10GeV} \footnote{All these claims have possible drawbacks, cf \cite{Boehm:2002yz,Boehm:2010kg,Bertone:2008xr,Cirelli:2009vg,Cirelli:2009bb,Cirelli:2009dv,Cirelli:2012ut}.} and, most recently, two possible $\gamma$-ray lines at 111 and 129 GeV~\cite{line130GeV,noline130GeV}. On the other hand, a large bulk of present astrophysical data essentially seem to validate the modelling of astrophysical background sources in the GeV-TeV range (disregarding these possible anomalies), and therefore enables one to set powerful constraints on the Dark Matter properties.
\medskip
By measuring the gamma-ray spectrum over a large energy range relevant for Dark Matter physics, the {\sc Fermi}} %{{\sc Fermi-LAT}-LAT\ collaboration has been able to set stringent limits on the Dark Matter pair annihilation cross section into Standard Model particles. For example, using the diffuse $\gamma$-ray emission in dwarf spheroidal (dSph) galaxies~\cite{FermiDwarfs} and also in the Milky Way~\cite{FermiDiffuseMW,Ackermann:2012qk}, the {\sc Fermi}} %{{\sc Fermi-LAT}-LAT\ collaboration has ruled out Dark Matter candidates with a total annihilation cross section of $\langle \sigma v \rangle = 3 \times 10^{-26} \ \rm{cm^3/s}$ if $m_{\rm DM} \lesssim 30$ GeV.
This constituted a remarkable milestone as such a value corresponds to that suggested by the thermal freeze-out scenario, which is generally considered as a strong argument in favour of Weakly Interacting Massive Particles (WIMPs).
\medskip
These limits nevertheless weaken at higher DM masses, therefore allowing for heavier DM candidates with a larger pair annihilation cross section. For example, for $m_{\rm{DM}} =$ 100 GeV the limit relaxes to $\langle \sigma v \rangle \lesssim 10^{-25} \ \rm{cm^3/s}$ while for $m_{\rm DM} =$ 500 GeV, it reads $\langle \sigma v \rangle \lesssim 3 \times 10^{-25} \ \rm{cm^3/s}$, which is one order of magnitude higher than the `thermal' cross section.
\medskip
DM models with such large values of the pair annihilation cross section have actually been proposed over the last five years as a consequence of the excesses in $e^+$ and $e^++e^-$ fluxes. While they may remain hypothetical, discovering such a configuration would invalidate the WIMP `vanilla' model and either point towards the existence of non-thermal process in the Early Universe (possibly opening up an unexpected window on fundamental physics at high energies) or potentially call for more sophisticated mechanisms, such as Freeze-In and regeneration as proposed in \cite{Hall:2009bx,Chu:2011be}. Explaining the observed dark matter relic density may remain nevertheless challenging. For example, in \cite{Williams:2012pz}, it was shown that candidates with a total annihilation cross section exceeding $\langle \sigma v \rangle = 10^{-24} \ \rm{cm^3/s}$ (corresponding to a thermal relic density smaller than $3 \%$) would be ruled out by the {\sc Fermi}} %{{\sc Fermi-LAT}-LAT experiment if they were regenerated at 100$\%$.
\medskip
In addition to measurements of the $e^+$ and $e^+ + e^-$ spectra mentioned above, there is also the measurement of the galactic $\bar{p}$ flux, presented by the {\sc Pamela}\ collaboration~\cite{PAMELApbar1,PAMELApbar2}. While extensive work was done to explain the electron/positron excesses in terms of Dark Matter annihilations (or decays), the implications of the absence of anomalies in the $\bar{p}$ spectrum has remained relatively unexploited. Indeed only a relatively small number of works~\cite{CKRS,salati,boehm,Evoli:2011id,Garny:2011ii,
Asano:2011ik,Garny:2012eb} have dealt with it and shown that large Dark Matter annihilation cross sections can be constrained by the {\sc Pamela}\ data. Among the most interesting conclusions which have been reached let us cite for example that in \cite{salati} constraints on the annihilation cross section into $b \bar{b}$ were given (for the same mass range as is considered in this paper) and limits on the $W^+ W^- $ final state were mentioned for $m_{\rm DM} = 1$ TeV and one specific set of propagation parameters. In \cite{Asano:2011ik}, constraints on the $q\bar{q}g$ were set for bino-like neutralinos.
\medskip
The first aim of this paper is therefore to propose a more systematic analysis of these general anti-proton constraints on the DM annihilation cross section, including paying attention to the uncertainties associated with DM and astrophysical predictions. The second aim of the present analysis is to demonstrate that these measurements can actually constrain the properties of specific DM scenarios, including the mass spectrum in the dark sector. To illustrate this, we will work within a `simplified' version of the phenomenological Minimal Supersymmetric Standard Model (pMSSM)~\cite{Djouadi:1998di} in which all sfermion masses are set to 2 TeV, except for the stop and sbottom masses. The soft masses for the stop are allowed to be much lighter to obtain a Higgs at 125 GeV. In this scenario the only particles with masses below the TeV threshold are therefore the neutralino, chargino, the supersymmetric Higgses and the lightest stop and sbottom. Such a configuration of `light' gauginos and heavy sfermions may actually seem unnatural from a supersymmetric point of view (albeit close to split SUSY~\cite{splitSuSy}) but it is supported by the unfruitful searches for squarks and gluinos at LHC, at least to some extent~\footnote{Even though, admittedly, those negative searches may also be a sign that Supersymmetry is not realised at the TeV scale.}.
\medskip
With this very set up in mind, one can investigate scenarios where the neutralino pair annihilation cross section into $W^+ \, W^-$ gauge bosons is enhanced (due in particular to the chargino exchange diagram). Such a large annihilation cross section gives both a significant anti-proton and diffuse gamma ray flux, together with a gamma ray line, and is therefore potentially constrained by the {\sc Pamela}\ and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT data.
In Supersymmetry, such an enhancement is realised when the LSP neutralino is mass degenerated with the chargino, i.e. when the neutralino has a significant wino component. The combination of both {\sc Fermi}} %{{\sc Fermi-LAT}-LAT and {\sc Pamela}\ data is therefore expected to constrain the wino fraction of the lightest neutralino, thus realizing our second aim. Note that constraints on the neutralino composition are also expected to be obtained in presence of a lower sfermion mass spectrum. However the effect of the chargino-neutralino mass degeneracy on $\gamma-$ray and $\bar{p}$ production would be much harder to characterise. Hence our choice in favour of a heavy sfermion mass spectrum.
\medskip
The paper is organised as follows. In section \ref{constraints} we derive generic constraints on the Dark Matter pair
annihilation cross section into $W^+W^-$ from anti-proton data and recall the {\sc Fermi}} %{{\sc Fermi-LAT}-LAT limits that are obtained from gamma-ray observations in the Milky Way and dwarf Spheroidal galaxies. In Section \ref{SuSy} we present the Supersymmetric model that we shall consider and explain how we perform the scans of the parameter space. Finally in Section \ref{results} we apply the {\sc Pamela}\ and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT limits to our SUSY model and show that the anti-proton data can be more constraining than gamma-ray observations. We conclude in Section \ref{conclusions}.
\section{Anti-proton and $\gamma$-ray bounds on $\sigma_{{\rm DM} \ {\rm DM} \ \rightarrow \ W^+ W^-}$}
\label{constraints}
In this section we discuss how anti-proton and gamma ray data impose generic constraints on the Dark Matter pair annihilation cross section into $W^+ W^-$ as a function of the Dark Matter mass.
\subsection{Generic bounds on $\sigma_{{\rm DM} \ {\rm DM} \ \rightarrow \ W^+ W^-}$ from anti-protons}
\label{pbarconstraints}
$W^\pm$ production in space leads to abundant anti-proton production as the $W^\pm$'s decay products hadronize. The flux of anti-protons thus produced by DM annihilations into a pair of $W^{\pm}$ gauge bosons in the Milky Way and collected at Earth is therefore determined by the Dark Matter pair annihilation cross section into $W^+ W^-$, the Dark Matter mass and the Dark Matter halo profile. It also depends on the anti-proton propagation parameters which are being considered. Hereafter we will assume that the dark matter halo profile is well described by an Einasto profile (we checked that other choices make a small difference) and consider the standard three sets of propagation parameters (`MIN', `MED', `MAX') summarised in table~\ref{tab:proparam}.
In practice, we use the anti-protons fluxes which are given in~\cite{PPPC4DMID}, to which we refer for further details.
\begin{table}[t]
\center
\begin{tabular}{c|cccc}
& \multicolumn{4}{c}{Antiproton parameters} \\
Model & $\delta$ & $\mathcal{K}_0$ [kpc$^2$/Myr] & $V_{\rm conv}$ [km/s] & $L$ [kpc] \\
\hline
MIN & 0.85 & 0.0016 & 13.5 & 1 \\
MED & 0.70 & 0.0112 & 12 & 4 \\
MAX & 0.46 & 0.0765 & 5 & 15
\end{tabular}
\caption{\em \small {\bfseries Propagation parameters} for anti-protons in the galactic halo (from~\cite{FornengoDec2007,DonatoPRD69}). Here $\delta$ and $\mathcal{K}_0$ are the index and the normalization of the diffusion coefficient, $V_{\rm conv}$ is the velocity of the convective wind and $L$ is the thickness of the diffusive cylinder.
\label{tab:proparam}}
\end{table}
\smallskip
In order to constrain the annihilation cross section, we will consider that all present data define the maximal flux in anti-proton that is allowed by the PAMELA~\cite{PAMELApbar2} experiment~\footnote{To avoid the uncertainty related to solar modulation, we restrict ourselves to using the {\sc Pamela}\ data above an anti-proton energy of 10 GeV.}. Both the predicted energy spectrum and the flux depend on the dark matter mass that is being assumed. For each value $m_{\rm{DM}}$, we will therefore compare the sum of the astrophysical background flux and predicted anti-protons flux originating from Dark Matter with the {\sc Pamela}\ data. Given the uncertainties on the astrophysical background, we will apply two different procedures to derive meaningful limits. One can be regarded as aggressive (it assumes a fixed background) while the other one is more conservative (the background can be adjusted within the uncertainties).
\begin{itemize}
\item[$\circ$] For obtaining aggressive limits (referred to as {\it fixed background} in the following), we adopt the standard flux of astrophysical (secondary) anti-protons from~\cite{BringmannSalati} and add it to the DM anti-protons flux. We then compare the result with the {\sc Pamela}\ data and derive a 95\% C.L. limit by imposing that the global $\chi^2$ of the background $+$ DM flux does not exceed by more than 4 units the $\chi^2$ of the null hypothesis (background only).
\item[$\circ$] For obtaining conservative limits (hereafter referred to as {\it marginalized background}), we take again the standard form of the background spectrum predicted in~\cite{BringmannSalati}, except that now we allow for the normalisation of the background spectrum $A$ and the spectral index $p$ to vary within 40$\%$ and $\pm 0.1$ respectively (for each value of the DM mass and pair annihilation cross section into $W^+ \, W^-$).
In practice, we multiply the standard description of the background spectrum by a factor $A \, (T/T_0)^p$, where $T$ is the anti-proton kinetic energy, $T_0 = 30$ GeV is a pivot energy and with $0.6 < A < 1.4$ and $-0.1 < p < +0.1$. These are quite generous intervals, which allow to include the uncertainty predicted in~\cite{BringmannSalati}. We then add up the DM contribution expected for each point in the parameter space defined as $(m_{\rm DM}, \langle \sigma v \rangle)$ and identify the pair of parameters $A$ and $p$ which minimises the global $\chi^2$ with the {\sc Pamela}\ data. This procedure therefore corresponds to marginalising over the parameters of the uncertain astrophysical background point-by-point in the DM parameter space. Again, the 95\% C.L. is then imposed by requiring that the marginalised global $\chi^2$ does not exceed 4 units with respect to the null hypothesis (which has been marginalised consistently).
By considering a variable background spectrum (within the uncertainties) for each value of the DM mass and cross section, we can increase the gap between the expected $\bar{p}$ background and the actual {\sc Pamela}\ data. As a result this leaves more space for a possible DM injection of anti-protons and leads to weaker limits on the DM pair annihilation cross section. A similar approach was used in~\cite{Delahaye:2011jn} but to reduce the gap between the astrophysical background and the data.
\end{itemize}
\begin{figure}[t]
\parbox[b]{.49\linewidth}{
\includegraphics[width=\linewidth]{pbarcontraintesMIN}}
\parbox[b]{.49\linewidth}{
\includegraphics[width=\linewidth]{pbarcontraintesMED}} \\[2mm]
\parbox[b]{.49\linewidth}{
\includegraphics[width=\linewidth]{pbarcontraintesMAX}}\hfill
\parbox[b]{.42\linewidth}{
\caption{\em \small {\bfseries Anti-proton constraints} on DM annihilation into $W^+W^-$. The upper left, upper right and lower panels refer respectively to the `MIN', `MED' and `MAX' propagation parameters. The constraints obtained by the {\sc Fermi}} %{{\sc Fermi-LAT}-LAT collaboration \ from satellite dwarf galaxies are superimposed. We also display five benchmark points. {\color{white} Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling. Filling.}}
\label{antip_crosssection}
}
\end{figure}
\medskip
Our constraints are displayed in Fig.~\ref{antip_crosssection} for the `MIN',`MED', `MAX' set of parameters. As expected, the `conservative' limits are slightly less constraining than the `aggressive' ones.
Also we find that the choice of propagation parameters has a big impact on the type of constraints that can be set: in terms of cross sections, the difference between the `MIN' and `MAX' limits exceeds a factor 10.
To understand more precisely how these constraints work, we defined 5 scenarios (hereafter referred to as `A',`B',`C',`D',`E'), corresponding to different DM masses, cross sections, propagation parameters and constraint procedures. The corresponding fluxes are plotted in Fig.~\ref{examples}. As one can see, benchmark points `A' to `D' correspond to `borderline' scenarios where the total $\bar{p}$ flux (i.e. the sum of the expected flux from DM and astrophysical background) is not significantly exceeding the data. Point `E', on the other hand, displays `how badly' the data is violated inside the excluded region.
\begin{figure}
\parbox[b]{.32\linewidth}{
\includegraphics[width=\linewidth]{plotexamplepbarA-1}}
\parbox[b]{.32\linewidth}{
\includegraphics[width=\linewidth]{plotexamplepbarB-1}}
\parbox[b]{.32\linewidth}{
\includegraphics[width=\linewidth]{plotexamplepbarC}}\\
\parbox[b]{.32\linewidth}{
\includegraphics[width=\linewidth]{plotexamplepbarD}}
\parbox[b]{.32\linewidth}{
\includegraphics[width=\linewidth]{plotexamplepbarE}}\hfill
\parbox[b]{.30\linewidth}{
\caption{\em \small {\bfseries Examples of the fluxes of anti-protons} (astrophysical background and DM-produced) compared with the data from the {\sc Pamela}\, experiment for the sample points A to E as defined in Fig.~\ref{antip_crosssection}. In each panel the assumed parameters (DM mass, annihilation cross section and propagation scheme) are reported.}
\label{examples}
}
\end{figure}
The first apparent feature from Fig.~\ref{examples} is that one can actually exclude a small excess in anti-protons produced by relatively light Dark Matter particles because the {\sc Pamela}\ data set have very small error bars at energies below 100 GeV, hence the strength of the constraints.
It is then instructive to compare case `A' and `B': these two scenarios refer to the same DM mass and constraint procedure; they also predict a very similar flux, as can be seen in Fig.~\ref{examples}, but have a different annihilation cross section. The latter is much larger for `A' than for `B'. This is because the propagation scheme was assumed to be `MIN' for the former and `MED' for the latter. With the `MIN' propagation set, the yield of anti-protons is about one order of magnitude smaller than with `MED' (since the galactic diffusion zone is much smaller in the former case) and therefore the constraint on the annihilation cross section is about one order of magnitude looser than for the `MED' case. On the other hand the constraint obtained for `MAX' (which is not shown here) is stronger than for `MED'.
The comparison between points `B' and `C' shows the impact of the constraint procedures. Although both `B' and `C' have the same DM mass and propagation scheme, we find that the value of the annihilation cross section that is allowed for `C' is larger than for `B'. The reason is that `C' corresponds to the scenario in which the limit is obtained by using the `marginalised background' procedure (i.e. where the background is allowed to retract within the uncertainties) so there is more room for DM while `B' corresponds to a `fixed background' scenario so the associated constraints are stronger.
Finally, the comparison between `C' and `D' enables one to understand why the `margi\-nalised background' constraints are rather independent of the DM mass, despite the fact that the error bars in the {\sc Pamela}\ data become larger at larger energies. For a large DM mass (case `D') the $\bar{p}$ flux is shifted towards larger energies and rather negligible at $\sim$ 10 GeV with respect to the astrophysical background; there is thus little room to reduce the the bakcground (which alone has to fit the data at low energy) and consequently there is little room left for a DM contribution at large energies. As a result, the bound remains stringent.
\subsection{Generic bounds on $\sigma_{{\rm DM} \ {\rm DM} \rightarrow W^+ W^-}$ from gamma-rays}
\label{gammaconstraints}
In DM scenarios, the $W^\pm$ production is associated with gamma-ray emission through (i) the decay and hadronisation of the decay products of the $W^{\pm}$ bosons, (ii) the radiation of a photon from the internal and/or final states associated with ${\rm DM} \ {\rm DM} \rightarrow W^+ W^-$ (iii) DM annihilations into $\gamma\gamma$ and $\gamma Z$ (which can be seen as a higher order process based on ${\rm DM} \ {\rm DM} \rightarrow W^+ W^-$). The first case leads to a {\it continuum} spectrum of $\gamma$-rays (the energy spectra can be e.g. found in~\cite{PPPC4DMID}, for any value of the DM mass); the second leads to {\it sharp features} in the $\gamma$-ray continuum spectrum and the third to {\it $\gamma$-ray lines}. The resulting fluxes from these process have to be compared with the gamma-ray flux measurements from the Milky Way or from other nearby galaxies. Therefore we now review the current $\gamma$-ray constraints derived in the literature (mainly from {\sc Fermi}} %{{\sc Fermi-LAT}-LAT analyses), paying particular attention to that derived from the $W^+ W^-$ channel since this is the main focus of our analysis.
\medskip
\subsubsection{Continuum} The {\sc Fermi}} %{{\sc Fermi-LAT}-LAT collaboration has recently published two different analyses of the continuum diffuse gamma-ray emission from the Milky Way halo~\cite{FermiDiffuseMW,Ackermann:2012qk}. Since no clear DM signal has been found, these have been used to set upper limits on the DM pair annihilation cross-section into various channels: e.g. $b\bar{b},gg,W^+W^-,e^+e^-,\mu^+\mu^-,\tau^+\tau^-$. For relatively light DM ($\sim$ 20 GeV) and e.g. the $b \bar{b}$ channel the limits reach the canonical value of the cross section (namely $\langle \sigma v \rangle =3\times 10^{-26} {\rm cm}^3/{\rm s}$), provided that the most aggressive procedure is used. For DM masses ${\cal O} (100)\ {\rm GeV}$ and for the $W^+ W^-$ channel the limit reads $\langle \sigma v \rangle \lesssim 2 \times 10^{-24} {\rm cm}^3/{\rm s}$. However the most stringent limits on the DM annihilation cross section have actually been obtained from another {\sc Fermi}} %{{\sc Fermi-LAT}-LAT analysis based on the diffuse $\gamma$-ray emission from dSph galaxies; these Dark Matter dominated objects indeed represent a good target for Dark Matter searches.
\medskip
In the present analysis we will use the results from~\cite{FermiDwarfs,GeringerSameth:2011iw} (see also~\cite{Cholis:2012am}). Although they use slightly different sets of targets \footnote{Ref.~\cite{GeringerSameth:2011iw} uses 7 dSphs $-$Bootes I, Draco, Fornax, Sculptor, Sextans, Ursa Minor, and Segue 1 while Ref.~\cite{FermiDwarfs} uses 10 dSphs $-$the same as above plus Carina, Coma Berenices and Ursa Major II$-$.}, slightly different datasets \footnote{Ref.~\cite{FermiDwarfs} uses 24 months between August 2009 and August 2010 while Ref~\cite{GeringerSameth:2011iw}) uses 3 years between August 2009 and August 2011.} and a different analysis procedure (\cite{GeringerSameth:2011iw} introduces a frequentist Neyman construction), they both derive consistent limits for the $b\bar{b}, \, W^+W^-,\mu^+\mu^-,\tau^+\tau^-$ channels. If we apply -- for definiteness -- the constraints from~\cite{FermiDwarfs} and assume a DM mass of $100\ {\rm GeV}$, the limit for the $W^+W^-$ channel reads as $\langle \sigma v \rangle < 8.5 \times 10^{-26} {\rm cm}^3/{\rm s}$. The analysis procedure in~\cite{FermiDwarfs} allows one to incorporate the uncertainties associated with the DM energy density profile of individual dSph galaxies, which was shown to lead to an error band of about an order of magnitude on the constraint in~\cite{GeringerSameth:2011iw}. Here we do not attempt to address these issues; we simply draw the attention of the reader that these constraints have to be taken with care until a better determination of the DM energy density profile in dSph galaxies is available. Consequently, we adopt the rather conservative constraints in this paper.
\medskip
In Fig.~\ref{antip_crosssection} we compare the dSph galaxies limits with the {\sc Pamela}\ anti-proton bounds that were derived in Sec.~\ref{pbarconstraints}. We see that, depending on the propagation scheme that has been chosen for the anti-protons, the dSph galaxies $\gamma$-ray bounds is somewhat more stringent or looser than the constraints from the anti-proton data. For example, for the `MED' case and `marginalized background', the $\bar p$ limits becomes more constraining than the $\gamma$-ray bounds when $m_{\rm DM} \gtrsim 290$ GeV. However they are stronger than the $\gamma$-ray limits whatever the value of $m_{\rm DM}$ (assuming $m_{\rm DM}> 100$ GeV) for a `fixed' background. Since nevertheless the $\bar{p}$ and $\gamma$-ray limits are basically of the same order of magnitude, we will include both constraints in our study.
\medskip
\subsubsection{Internal Bremsstrahlung and Final State Radiation} Gamma rays produced directly as radiation from an internal line or a final state are in general suppressed by the fine structure constant, $\alpha$. However, for a $t$-channel diagram, the associated cross section can be enhanced when the intermediate particle is almost mass degenerated with the DM. Typically the enhancement factor is about $m_{\rm DM}^2/(M_I^2-m_{\rm DM}^2)$ where $M_I$ the mass of the intermediate particle (i.e. a chargino for neutralino pair annihilation into a $W^{\pm}$ pair). These process are model dependent and cannot be constrained generically but they will be included in our $\gamma$-ray estimates when we investigate the neutralino pair annihilations into $W^+ W^-$ in the pMSSM.
\medskip
\subsubsection{Line(s)} Annihilations directly into
$\gamma\gamma$ or $\gamma Z$ occur at one-loop level
(since DM particles do not couple directly to photons) and are therefore generically suppressed. However they lead to a distinctive signature, namely a mono-energetic gamma-ray line at an energy $E=m_{\rm DM}$ or
$E = m_{\rm DM} \, (1- m_Z^2/(4 m_{\rm DM}^2))$ which can be looked for.
With possible evidence for two gamma-ray lines at 129 and 111 GeV
(which have been speculated as originating from DM particles with a mass of about 130 GeV annihilating into $\gamma \gamma $ and $\gamma Z$), indirect detection of DM particles seem promising. Yet the existence of these lines remain to be confirmed by the {\sc Fermi}} %{{\sc Fermi-LAT}-LAT collaboration and their origin to be shown as being exotic. Since the purpose of this study is to set constraints on the DM properties (and owing to these uncertainties on the existence and origin of these lines) we will disregard the results of \cite{line130GeV} and only consider the {\it constraints} which were reported by the {\sc Fermi}} %{{\sc Fermi-LAT}-LAT collaboration on line searches in the Milky Way~\cite{Ackermann:2012qk}, where the upper limits on $\sigma v_{{\rm DM} {\rm DM} \rightarrow \gamma \gamma}$ and $\sigma v_{{\rm DM} {\rm DM} \rightarrow Z \gamma}$ range from $0.03$ to $4.6 \times 10^{-27}{\rm cm}^3{\rm s}^{-1}$ and $1$ to $10 \times 10^{-27}{\rm cm}^3{\rm s}^{-1}$ respectively, for DM masses up to 200 GeV.
Constraints on $\sigma v_{{\rm DM} {\rm DM} \rightarrow \gamma \gamma}$ were also obtained from dSph galaxies~\cite{GeringerSameth:2012sr} but they are not as stringent as those obtained from the Milky Way.
Since the status of these searches is not definite, we made the choice to not include these constraints to perform the scans over the pMSSM parameter space. However we do check that the scenarios which survive the $\bar{p}$ and $\gamma$-ray constraints are not killed by these line searches.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.6]{neut_ann_char.pdf}
\end{center}
\caption{\em \small {\bfseries Dominant neutralino pair annihilation diagrams} into $W^+ \, W^-$, $\gamma \, \gamma$ and $\gamma \, Z$ for this analysis. }
\label{pMSSM_diagrams}
\end{figure}
\section{Chargino-neutralino mass degeneracy}
\label{SuSy}
Now that we have obtained the maximal value of the Dark Matter pair annihilation cross section into $W^+ W^-$ that is observationally allowed as a function of the Dark Matter mass, we can focus on a specific Supersymmetric model and investigate the impact of this generic limit on the neutralino Dark Matter parameter space.
\subsection{ Neutralino pair annihilations into $W^+ W^-$}
In a scenario where all the sfermions are very heavy, the dominant neutralino annihilation channels are expected to be mostly into gauge bosons, more specifically into $W^+W^-$ pairs. All loop-induced $W^{\pm}$ production diagrams which involve sfermions are expected to be suppressed. Hence the process which are expected to lead to a significant $W^+ W^-$ production in the pMSSM only involve charginos and Z boson. The corresponding diagrams are displayed in Fig.~\ref{pMSSM_diagrams}. Since they correspond to $s-$ and $t-$channel diagrams, we typically expect resonant or enhanced annihilations when $m_{\chi^0_1} \sim m_{Z}/2$ or $m_{\chi^0_1} \sim m_{\chi^\pm}$ (i.e. when the neutralino and chargino are mass degenerated). These ultimately enhance the neutralino pair annihilations into $\gamma \gamma$ \cite{Bergstrom:1997fh,Boudjema:2005hb} and $\gamma Z$ \cite{Boudjema:2005hb,Ullio:1997ke} through in particular the two `loop' diagrams displayed in Fig.~\ref{pMSSM_diagrams}.
The questions that we want to address in the next subsections are:
i) which part of the SUSY parameter space is excluded by the $\bar{p}$ limits and do these limits exclude more allowed configurations than the $\gamma$-rays bounds? ii) which values of the neutralino-chargino mass degeneracy are actually constrained by astrophysical data?
\subsection{Exploring the supersymmetric parameter space}
To answer this, we will explore the pMSSM parameter space using the same Markov Chain Monte Carlo method as in \cite{Boehm:2012rh} coupled to the \texttt{micrOMEGAs} code \cite{Belanger} and the \texttt{SOFTSUSY} spectrum calculator \cite{Allanach}.
Our free parameters and their corresponding range are summarised in Table~(\ref{tab:range}). These include the soft mass terms associated with the squarks of the third generation (i.e. $M_{\tilde{Q}_3}$ and $M_{\tilde{u}_3}$) and the trilinear coupling $A_{t}$. To obtain sfermion masses at the TeV scale, we set all the soft masses to 2 TeV. In addition, we set the trilinear couplings to 0 and the CP-odd Higgs mass to 1 TeV. In this framework, the bino mass $M_{1}$ does not exceed 500 GeV; our choice for the other parameters indeed ensures that the neutralinos and charginos are light and the mass splitting between the neutralinos and charginos remains relatively small.
On top of these free parameters, we had to include some nuisance parameters over which we will marginalise~\cite{Dumont:2012ee}. These are related in particular to the quark content of the nucleons (since they have a non-negligible impact on the computation of the Dark Matter-nucleon scattering cross section) and the top mass (since it has an impact on the Higgs sector). All of them are allowed to vary in the range [$N_{exp}$ -3$\sigma$, $N_{exp}$ +3$\sigma$], with $N_{exp}$ ($\sigma$) the corresponding experimental value (error), as shown in Table~\ref{tab:range}.
We also require that the lightest Higgs mass only varies within the range allowed by the ATLAS and CMS experiments \cite{atlas:2012,cms:2012}, namely $m_h = 125.9 \pm 2.0$~GeV. However, by precaution, we checked that the scenarios which seemed allowed were compatible with the latest version of the HiggsBounds code~\cite{HiggsBounds-3.8.0} (even though the most recent LHC results on the Higgs~\cite{atlas:2012,cms:2012} are not included in this version). Note that we did not add any requirement about the Higgs signal strength to perform the scans. Would ATLAS and CMS confirm an `anomalous' Higgs signal strength into $\gamma \, \gamma$ (i.e. larger than SM expectations) with a high confidence level, the pMSSM would be difficult to reconcile with the data. However the principles of our analysis would remain valid and could still be used to constrain small mass degeneracies between the Dark Matter and another (e.g. $t-$channel exchange) intermediate particle.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|}\hline
Free parameter & Range \\ \hline \hline
$M_{1}$ & [10, 500] GeV\\
$M_{2}$ & [100, 1000] GeV\\
$\mu$ & [-2000, 2000] GeV\\
$\tan \beta$ & [2, 75]\\
$M_{\tilde{Q}_3}, M_{\tilde{u}_3}$ & [100, 3000] GeV\\
$A_{t}$ & [-8000, 8000] GeV\\ \hline
\end{tabular}
\begin{tabular}{|c|c|c|}\hline
Nuisance parameter & Experimental value & Likelihood\\ \hline \hline
$m_u$/$m_d$ & 0.553 $\pm$ 0.043 \cite{Leutwyler:1996qg} & $\mathcal{L}_1(m_u/m_d, 0.51, 0.596, 0.043)$\\
$m_s$/$m_d$ & 18.9 $\pm$ 0.8 \cite{Leutwyler:1996qg} & $\mathcal{L}_1(m_s/m_d, 18.1, 19,7, 0.8)$\\
$\sigma_{\pi N}$ & 44 $\pm$ 5 MeV \cite{Thomas:2012tg} & $\mathcal{L}_1(\sigma_{\pi N}, 39, 49, 5)$\\
$\sigma_{s}$ & 21 $\pm$ 7 MeV \cite{Thomas:2012tg} & $\mathcal{L}_1(\sigma_{s}, 14, 28, 7)$\\
$m_t$ & 173.2 $\pm$ 0.9 GeV \cite{TEV} & $\mathcal{L}_1(m_t, 172.3, 174.1, 0.9)$\\ \hline
\end{tabular}
\caption{\em \small Range chosen for the {\bfseries pMSSM free parameters} and {\bfseries nuisance parameters}. \label{tab:range}}
\end{center}
\end{table}
The neutralino relic density is allowed to vary between $\Omega_{\chi^{0}_{1}} h^2 \in [1\%\textrm{ WMAP7}, \textrm{ WMAP7}]$ with $\Omega_{\rm WMAP7} h^2 = 0.1123 \pm 0.0035$, using WMAP 7-year $+$ BAO $+$ $H_0$ and the \texttt{RECFAST 1.4.2} code~\cite{WMAP}. We do not consider smaller values of the relic density as these correspond to DM scenarios with very large values of the annihilation cross section and ultimately overproduce gamma-rays in the galaxy (i.e. are excluded) if their relic density is entirely regenerated, cf \cite{Williams:2012pz}.
For each scenario (corresponding to a point in the pMSSM parameter space), we then calculate the total likelihood function. The latter is a product of likelihood functions associated with each observable, nuisance parameters and free parameters which have been chosen according to the criteria described below.
\subsubsection{ $\mathcal{L}_1(x, x_{min}, x_{max}, \sigma)$}
To $m_h$, $\Omega_{\chi^{0}_{1}} h^2$ and all nuisance parameters, we associate a likelihood function $\mathcal{L}_1$ which decays exponentially at the edges of a well-defined range $[x_{min}, x_{max}]$ with a variance $\sigma$:
\begin{eqnarray}
\mathcal{L}_1(x, x_{min}, x_{max}, \sigma) =
\begin{cases}
e^{-\frac{\left( x - x_{min}\right)^2}{2\sigma ^2}} & \textrm{if} \: x < x_{min}, \\
e^{-\frac{\left( x - x_{max}\right)^2}{2\sigma ^2}} &\textrm{if} \: x > x_{max}, \\
1 & \textrm{for} \: x \in [x_{min}, x_{max}].
\end{cases}
\end{eqnarray}
Here $x$ is either $m_h$, the LSP relic density or the nuisance parameters. Note that we assume flat prior for all nuisance parameters.
For the free parameters, we will consider a slight modification of the above function, namely
\begin{eqnarray}
\mathcal{L}_1(x, x_{min}, x_{max}, \sigma) \, =
\begin{cases}
0 \ \textrm{for} \: x < x_{min} \ \textrm{or} \ x > x_{max}, \\
1 \ \textrm{for} \: x \in [x_{min}, x_{max}]
\end{cases}
\end{eqnarray}
so as to immediately reject all the scenarios in which one of the free parameters falls outside of the specified range. In fact we also immediately reject points where the neutralino is not the LSP, where the LEP limits on chargino, slepton and squark masses are not satisfied or the calculations of the SUSY spectrum fail. We did not implement LHC limits on sfermion masses because our requirement of a sfermion spectrum at the TeV scale should ensure that they are satisfied. However updates on direct searches for relatively light stop and sbottom would be useful to implement to further constrain the parameter space.
\subsubsection{$\mathcal{L}_2(x, x_{min}, x_{max}, \sigma)$}
We will use a Gaussian Likelihood function, $\mathcal{L}_2$, for the ${\cal B}(b \rightarrow X_s^* \gamma)$ observable (one of the B-physics observables that we consider) with experimental mean value $\mu$ and theoretical $+$ experimental error $\sigma$ :
\begin{equation}
\mathcal{L}_2(x, \mu, \sigma) = e^{-\frac{\left(x-\mu\right)^2}{2\sigma ^2}}.
\end{equation}
These observables are important as they receive a potentially large contribution from chargino/stop loops when either one of these particles is light. This contribution can be compensated by the charged Higgs/top diagram but the latter is however suppressed when the charged Higgs mass is at the TeV scale.
\subsubsection{$\mathcal{L}_3(x, x_{min}, x_{max}, \sigma)$}
We also include a Likelihood function $\mathcal{L}_3$ for the 2012 XENON100 limits \cite{Aprile2012} to ensure that the scans do not select too large values of the Dark Matter-nucleon scattering cross section.
In fact we also associate $\mathcal{L}_3(x, \mu, \sigma)$ to regions of the parameter space where ${\sigma v}_{{\chi^{0}_{1}\chi^{0}_{1}\rightarrow W^{+}W^{-}}} $ is greater than $10^{-27}$~${\rm cm}^3/{\rm s}$. The latter is defined as follows:
\begin{equation}
\mathcal{L}_3(x, \mu, \sigma) = \frac{1}{1+e^{-\frac{x-\mu}{\sigma}}}.
\end{equation}
where the lower or upper experimental bound are associated with the positive or negative variance $\sigma$ respectively. Note that some experimental measurements are very discrepant with the SM expectations (namely the anomalous magnetic moment of the muon $\Delta a_\mu$ and the branching ratio ${\cal B}(B^+ \rightarrow \tau^+ \bar \nu_\tau)$). These observables receive additional contributions from particles in the pMSSM but they are too small to explain the observations. Therefore we associate a Likelihood function to them which corresponds to $\mathcal{L}_3(x, \mu, \sigma)$ so that the Likelihood is equal to unity if the predictions are much below the measured value.
The set of constraints that we use is summarised in Table~\ref{tab:constraints}.
\renewcommand{\arraystretch}{1.3}
\begin{table*}[!tb]
\footnotesize{
\begin{tabular*}{1.031\textwidth}{|c|c|c|c|}
\hline
Constraint & Value/Range & Tolerance & Likelihood \\
\hline \hline
$m_h$ (GeV) \cite{atlas:2012,cms:2012} & [123.9, 127.9] & 0.1 & $\mathcal{L}_1(m_h, 123.9, 127.9, 0.1)$ \\ \hline
$\Omega_{\chi^0_1} h^2$ \cite{WMAP} & [0.001123, 0.1123] & 0.0035 & $\mathcal{L}_1(\Omega_{\chi^0_1} h^2, 0.001123, 0.1123, 0.0035)$ \\ \hline
${\cal B}(b \rightarrow X_s^* \gamma)$ $\times$ $10^{4}$ & 3.55 & exp : 0.24, 0.09 & $\mathcal{L}_2(10^{4} {\cal B}(b \rightarrow X_s^* \gamma), 3.55,$ \\
\cite{Asner:2010qj,Misiak:2006zs} & & th : 0.23 & $\sqrt{0.24^2 + 0.09^2 + 0.23^2})$\\ \hline
$\sigma^{SI}_{\chi^0_1 {\rm Xe}}$ (pb) & ($m_{\rm DM}$, $\sigma_N$) plane & $\sigma_N(m_{\rm DM})$/100 & $\mathcal{L}_3(\sigma^{SI}_{\chi^0_1 {\rm Xe}}, \sigma_N(m_{\rm DM}), -\sigma_N(m_{\rm DM})/100)$ \\
& from \cite{Aprile2012} & & \\ \hline
${\sigma v}_{^{\chi^{0}_{1}\chi^{0}_{1}\rightarrow W^{+}W^{-}}}$ & 1 & 0.01 & $\mathcal{L}_3({\sigma v}_{^{\chi^{0}_{1}\chi^{0}_{1}\rightarrow W^{+}W^{-}}}, 1, 0.01)$ \\
($10^{-27}$~${\rm cm}^3/{\rm s}$) & & & \\ \hline
$\Delta a_\mu$ $\times$ $10^{10}$ \cite{Davier:2010nc} & 28.70 & 0.287 & $\mathcal{L}_3(10^{10} \Delta a_\mu, 28.70, -0.287)$ \\ \hline
${\cal B}(B_s \rightarrow \mu^+ \mu^-)$ $\times$ $10^{9}$~\cite{Aaij:2012ac} & 4.5 & 0.045 & $\mathcal{L}_3(10^{9}\ {\cal B}(B_s \rightarrow \mu^+ \mu^-), 4.5, -0.045)$ \\ \hline
$\Delta \rho$ & 0.002 & 0.0001 & $\mathcal{L}_3(\Delta \rho, 0.002, -0.0001)$ \\ \hline
$R_{B^+ \rightarrow \tau^+ \bar \nu_\tau} (\frac{\rm pMSSM}{\rm SM})$ \cite{Charles:2011va} & 2.219 & 2.219$\times10^{-2}$ & $\mathcal{L}_3(R_{B^+ \rightarrow \tau^+ \bar \nu_\tau}, 2.219, -2.219\times10^{-2})$ \\ \hline
$Z \rightarrow \chi^0_1 \chi^0_1$ (MeV) & 1.7 & 0.3 & $\mathcal{L}_3(Z \rightarrow \chi^0_1 \chi^0_1, 1.7, -0.3)$ \\ \hline
$\sigma_{e ^+ e ^- \rightarrow \chi^0_1 \chi^0_{2,3}} \times $ & 1 & 0.01 & $\mathcal{L}_3(\sigma_{e ^+ e ^- \rightarrow \chi^0_1 \chi^0_{2,3}} \times$ \\
$ {\cal B}(\chi^0_{2,3} \rightarrow Z \chi^0_1)$ (pb) \cite{Abbiendi:2003sc} &&& $ {\cal B}(\chi^0_{2,3} \rightarrow Z \chi^0_1),1, -0.01)$ \\
\hline
\end{tabular*}
}
\caption{\label{tab:constraints} \em \small {\bfseries Constraints imposed in the MCMC}, from \cite{Nakamura:2010zzi} unless noted otherwise.}
\end{table*}
\section{Results}
\label{results}
The results of our scans are shown in Fig.~\ref{premierset}. In the upper left panel is displayed the neutralino pair annihilation cross section into $W^+ W^-$ as a function of the mass degeneracy between the neutralino and the chargino and in terms of the neutralino composition. In the upper and lower right panels we show the pair annihilation cross section into $\gamma Z$ and $\gamma \gamma$ respectively as a function of the neutralino-chargino mass degeneracy $\Delta m = m_{\chi_1^+} - m_{\chi_1^0}$ and in the lower left panel we give the forecasted spin-independent elastic scattering cross section as a function of the neutralino mass for a Xenon-based experiment.
The left upper panel indicates the neutralino composition which maximises the $W^{\pm}$ production. As one can see scenarios where $\sigma v_{\chi^0_1 \chi^0_1 \rightarrow W^+ W^-}$ is the largest and the neutralino-chargino mass splitting is the smallest correspond to neutralinos with a very large wino fraction. Large values of both $\sigma v_{\chi^0_1 \chi^0_1 \rightarrow W^+ W^-}$ and the $\chi_1^0-\chi^+$ mass splitting correspond on the other hand to wino-dominated neutralinos but with a non negligible higgsino component. For these two types of wino-dominated configurations the neutralino and chargino mass degeneracy is large enough to make the $t-$channel (chargino) exchange diagram very large. As the wino fraction decreases, the mass splitting becomes larger and the $t-$channel chargino exchange diagram contribution decreases. However it remains large till the higgsino fraction which ensures large values of the $\chi_1^0-\chi^{+}-W^-$ coupling remains significant (i.e. dominates over the bino fraction).
The upper right panel of Fig.~\ref{premierset} shows which values of the neutralino pair annihilation cross section into $Z \gamma$ are excluded by astrophysical data as a function of the neutralino-chargino mass splitting. A similar plot is shown for $\gamma \gamma$ but the colour code now illustrates the relation between the different values of this cross section and the neutralino `thermal' relic density. As one can see the shape of the scenario distribution for $\gamma \gamma$ and $Z \gamma$ is essentially the same in the $(\Delta m,\sigma v)$ plane. However the $Z \gamma$ cross section is approximately 10 times larger than that for $\gamma \gamma$ for every scenario. Hence combining these two figures actually gives an information about the relic density of the scenarios which are excluded by astrophysical data.
In the $Z \gamma$ plot (upper right panel of Fig.\ref{premierset}), the points excluded by the {\sc Fermi}} %{{\sc Fermi-LAT}-LAT dSph continuum $\gamma$-ray data are displayed in yellow (we do not superimpose the constraints from line searches). Those correspond, by construction, to scenarios where there is a very large $W^{\pm}$ production (and thus a large contribution to the continuum $\gamma$-ray spectrum) but also to the few regions in which the LSP is heavy and where the $b\bar{b}$ final state (associated with the $s-$channel pseudo-scalar Higgs exchange and which cannot be discarded as it is significant) overproduces $\gamma$-rays. The regions which are excluded by the {\sc Pamela}\ data are shown in red. The black points correspond to scenarios excluded by both the {\sc Pamela}\ and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT data while those in green represent the points allowed by these two types of constraints.
As one can see from the distribution of black points the largest values of the annihilation cross sections into $Z \gamma$ (and therefore $W^+ W^-$) are excluded by both measurements. Since these scenarios correspond to a small (or relatively small) chargino-neutralino mass splitting and thus large values of the $t-$channel chargino exchange diagram, we can conclude that both {\sc Pamela}\ and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT data are relevant to constrain wino-dominated neutralinos. A small number of these configurations is however constrained by only one of the {\sc Pamela}\ or {\sc Fermi}} %{{\sc Fermi-LAT}-LAT dataset but this does not affect the maximal value of the $\chi_1^0-\chi^+$ mass splitting that can be excluded by using astrophysical considerations.
By inspecting where the neutralino pair annihilations into $Z \gamma$, $\gamma \gamma$ and $W^+ W^-$ are significant in these plots, one also finds that higgsino-dominated scenarios are constrained by both {\sc Pamela}\ and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT data because the box diagram (cf the lower left diagram in Fig.~\ref{pMSSM_diagrams}) still generates a large $W^{\pm}$ production. In fact, for such a LSP, the annihilation cross section into $ZZ$ also becomes non-negligible compared to that into $W^+W^-$. Since the expected $\gamma$-ray and $\bar{p}$ spectra from $W^{\pm}$ and $Z$ production are very similar, we accounted for them both when we made the comparison with the PAMELA\ and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT data.
Finally the green points which pass all the constraints have a non-negligible bino component. This reduces the chargino exchange diagram contribution and thus enables to decrease the $W^{\pm}$ (and therefore anti-proton and $\gamma$-ray) production. For these bino-like configurations one expects the stop and sbottom exchange to be relevant, leading to quarks in the final state and possibly (in particular for $b \bar{b}$) an overproduction of gamma-rays. Note that such process would also compete with the neutralino pair annihilation into SM fermions near pseudo-scalar Higgs resonances for heavy neutralinos.
\begin{figure}[t]
\parbox[b]{.48\linewidth}{
\includegraphics[width=\linewidth]{diff-vcsWW-compoLSP}}
\parbox[b]{.48\linewidth}{
\includegraphics[width=\linewidth]{diff-vcsgz_IDconstraints_cut}}
\parbox[b]{.48\linewidth}{
\includegraphics[width=\linewidth]{LSP-csSI_IDconstraints_cut}}
\parbox[b]{.48\linewidth}{ \quad \
\includegraphics[width=\linewidth]{diff-vcsgg-RD_cut}}
\caption{\em \small Plots of the neutralino pair annihilation cross section into $W^+ W^-$ (left upper panel) and $\gamma Z$ (right upper panel) as a function of the chargino-neutralino mass splitting and the spin-independent DM-nucleon cross section as a function of the Dark Matter mass (lower left panel) together with the XENON 2012 limit. The Freeze-Out relic density is displayed in the lower right panel for the annihilation cross section into $\gamma \gamma$ as a function of the mass splitting. }
\label{premierset}
\end{figure}
The lower left panel of Fig.\ref{premierset} indicates whether the spin-independent elastic scattering cross section is compatible with the latest results from the XENON100 experiment \cite{Aprile2012}. Again in green are the points which are astrophysically allowed, in black the points which are excluded by both {\sc Pamela}\ and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT data and in red or yellow the points which are either excluded by the {\sc Pamela}\ or {\sc Fermi}} %{{\sc Fermi-LAT}-LAT experiments respectively. Clearly one can see that the combination of both the {\sc Pamela}\ and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT astrophysical constraints surpass the latest exclusion limit set by the XENON100 experiment. In fact in general the astrophysical constraints discussed in this paper even have a stronger exclusion power than the forecasted XENON1T limit, illustrating how important adding astrophysical knowledge is in this specific scenario.
Even though many configurations are excluded by the {\sc Pamela}\ and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT data, we do find scenarios which are neither excluded by the XENON100 2012 limit nor by the astrophysical constraints discussed in this paper.
Hence the XENON100 experiment could still discover evidence for relatively light pMSSM neutralinos ($m_{\chi^0_1} < 500$ GeV) if these particles indeed exist. We note nevertheless that in \cite{Deb}, a constraint as strong as the XENON100 2012 limit was obtained by using the XENON100 2011 data and a Bayesian analysis where the full information available in the $(S_1,S_2)$ scintillation plane was exploited. It is therefore likely that the XENON100 experiment can improve its present exclusion limit with the 2012 data and rule out some of the configurations shown here in green.
In these figures we have assumed that the relic density was regenerated at 100 $\%$ for candidates with a total annihilation cross section much larger than the
`thermal' one (i.e. with a suppressed Freeze-Out relic density). This way we could ensure a fair comparison between theoretical expectations and the limits set by the {\sc Fermi}} %{{\sc Fermi-LAT}-LAT and XENON100 experiments. Looking at the $\sigma v_{\chi^0_1 \chi^0_1 \rightarrow \gamma \gamma}$ plot, one sees that invoking regeneration is needed for all scenarios with a chargino-neutralino mass splitting smaller than $\sim$ 20 GeV \footnote{For larger values of the mass splitting, no regeneration assumption is required but the annihilation cross sections into $\gamma \gamma$ and $\gamma Z$ are strongly suppressed. In particular $\sigma v_{\chi^0_1 \chi^0_1 \rightarrow \gamma \gamma}$ is much below $10^{-29} \ \rm{cm^3/s}$.}. Assuming that all these candidates have the correct relic density, we could indeed exclude scenarios with a neutralino-chargino mass splitting up to 20 GeV and values of $\sigma v_{\chi^0_1 \chi^0_1 \rightarrow \gamma Z}$ down to $10^{-28} \ \rm{cm^3/s}$ (see Fig.\ref{premierset}), corresponding to $\sigma v_{\chi^0_1 \chi^0_1 \rightarrow W^+ W^-} > 10^{-25} \ \rm{cm^3/s}$ and $\Omega h^2 \ll 0.06$.
However, relaxing the regeneration assumption would completely relax the exclusion regions and therefore the bound on the mass splitting (apart perhaps from scenarios with extremely small mass splitting).
As a side comment regarding the so-called `130 GeV line': we do find scenarios where $\sigma v_{\chi^0_1 \chi^0_1 \rightarrow \gamma \gamma} \simeq 10^{-27} \ \rm{cm^3/s}$, which is the value of the cross section that is required to explain the feature in the spectrum. These configurations predict a neutralino-chargino mass splitting greater than $\sim 0.2$ GeV. However none of the points corresponding to neutralinos with a mass of about 130 GeV are allowed by the {\sc Pamela}\ data. Hence, our results suggest that one cannot explain the `130 GeV line' in our simplified version of the pMSSM, which is in agreement with~\cite{Cohen:2012me,Buchmuller:2012rc}. Indeed, due to the anti-proton limit, scenarios with $\sigma v_{\chi^0_1 \chi^0_1 \rightarrow \gamma \gamma} \simeq 10^{-27} \ \rm{cm^3/s}$ rather correspond to neutralinos with a mass of about 400-450 GeV. In fact, for the same reason, all the points with $\sigma v_{\chi^0_1 \chi^0_1 \rightarrow \gamma \gamma}>2 \times 10^{-28} \ \rm{cm^3/s}$ correspond to configurations where $m_{\chi^0_1} >$200 GeV. Finally note that in the pMSSM the existence of 130 GeV neutralinos should give rise to a second $\gamma$-ray line at $\sim$ 111 GeV (on top of that at 130 GeV), corresponding to the neutralino pair annihilation into $\gamma Z$. Given our prediction for $\gamma Z$ and $\gamma \gamma$, the flux associated with this 111 GeV line should be about ten times larger than that corresponding to the 130 GeV line, which is in conflict with the observations.
\begin{figure}[t]
\includegraphics[width=\textwidth]{LSP-vcsWW-diff}
\caption{\em \small {\bfseries Annihilation cross section
into $W^+ W^-$} as a function of the neutralino Dark Matter mass. The chargino NLSP-neutralino LSP mass splitting is shown as colour code.}
\label{WW}
\end{figure}
Finally, in Fig.\ref{WW}, we show the annihilation cross section
into $W^+ W^-$ as a function of the neutralino mass and superimpose the {\sc Pamela}\ (for the `MED' set of propagation parameters and marginalised background, i.e. the conservative limits) and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT limits (red and yellow lines respectively). The colour code indicates the different values of the neutralino-chargino mass splitting. As can be seen from this plot, the {\sc Pamela}\ and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT constraints are actually complementary. The {\sc Fermi}} %{{\sc Fermi-LAT}-LAT limit excludes more configurations below 300 GeV than the {\sc Pamela}\ bound but it assumes that the observations are independent of the Dark Matter energy distribution in dSph galaxies, which can be debated~\cite{Hooper:2003sh,Charbonnier:2011ft}. In contrast, the anti-proton limit excludes a bit more configurations than the gamma-rays above 300 GeV. This is reassuring since it is set by observations `within' the galaxy but the drawback is that it relies on a specific choice of propagation parameters and astrophysical knowledge of astrophysical sources. In any case, the fact that both limits exclude similar configurations enables us to validate the exclusion region that we found.
Hence the main information that one can read of from this plot, combined with that displayed in Fig.~\ref{premierset}, is that:
\begin{itemize}
\item one can rule out neutralino-chargino mass splitting up to $\sim$ 20 GeV if $m_{\chi_1^0} \lesssim 150$ GeV and the neutralino is a mixture of wino and higgsino
\item one can exclude all scenarios in which the wino-chargino mass difference is smaller than 0.2 GeV for $m_{\chi^0_1} < 500$ GeV, thanks to both {\sc Pamela}\ and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT data.
\end{itemize}
\section{Conclusions}
\label{conclusions}
\medskip
In this paper we explicitly derived the constraints on the ${\rm DM} {\rm DM} \rightarrow W^+ W^-$ annihilation cross section by using the {\sc Pamela}\ anti-proton data and paying particular attention to the choice of propagation parameters and uncertainties on the astrophysical background. Our results are independent of the so-called {\sc Pamela}\ positron excess and are obtained for two different (fixed vs marginalised) choices of the background spectrum; they are also consistent with the enhancement factor which was derived in \cite{salati} and the detailed analysis of \cite{Evoli:2011id}, for the cases where the propagation parameters overlap.
\medskip
We then compared these bounds with the most stringent gamma-ray limits which have been derived using the {\sc Fermi}} %{{\sc Fermi-LAT}-LAT\ measurements of the gamma-ray continuum spectrum from dSph galaxies, for the same DM annihilation channel and DM mass range. We found that the anti-proton constraints appear to be very competitive with the gamma-ray bounds. More precisely, choosing the `MED' propagation scheme, the $\bar p$ constraints are slightly weaker than the $\gamma$-ray ones when $m_{\rm DM} \lesssim 300$ GeV and slightly stronger for $m_{\rm DM} \gtrsim 300$ GeV. On the other hand, the anti-proton constraints are stronger if we assume the 'MAX' set of propagation parameters and less powerful if we assume the `MIN' set. We also recall that the gamma ray limits themselves may be subject to some uncertainties related to the modelling of the DM profile in dSph galaxies.
\medskip
Finally we applied as fiducial limits the $\bar p$ constraints relative to `MED' and the marginalized astrophysical background to the neutralino LSP in a simplified version of the pMSSM, where we set all the sfermion masses (apart from that of the third generation) to the TeV scale. We found that the fiducial {\sc Pamela}\ anti-proton and {\sc Fermi}} %{{\sc Fermi-LAT}-LAT gamma-ray limits rule out small but non negligible neutralino-chargino mass splittings. In particular for $m_{\chi_1^0} \lesssim 150$ GeV, one can rule out mass splittings up to 20 GeV. Our results also suggest that pure wino or wino-like neutralinos are excluded if they are lighter than 450 GeV. Overall, this limit surpasses the bounds that can be set by using the XENON100 data and even in fact than the projected XENON1T limit.
\medskip
Hence from this work, we conclude that present indirect detection data already enable one to exclude regions of the parameter space where the neutralino-chargino mass splitting is small but non negligible. Since these regions are difficult to probe directly at the LHC, these findings show that {\sc Fermi}} %{{\sc Fermi-LAT}-LAT and {\sc Pamela}\ data constitute modern tools to explore the supersymmetric parameter space and even beat LHC (and also in fact Direct Detection) searches on their own territory, even though -- on the negative side --
they assume a regeneration of the relic density for neutralinos with a very large annihilation cross section.
\section*{Acknowledgment}
We thank the Galileo Galilei Institute for Theoretical Physics in Florence for the hospitality and the INFN for partial support during the completion of this work.
The work of MC is supported in part by the French national research agency ANR under contract ANR 2010 BLANC 041301 and by the EU ITN network UNILHC.
The work of AP was supported by the Russian foundation for Basic Research, grant RFBR-10-02-01443-a and the LIA-TCAP of CNRS.
JDS is supported by the CMIRA 2011 EXPLO'RA DOC grant and would like to thank IPPP for its hospitality.
|
1,108,101,566,724 | arxiv |
\section{Acknowledgment}
We would like to thank Andrew Childs, Omid Etesami, Salman Beigi, and Mohammad Ali Abam for their comments on an earlier version of the paper.
\section{Bootstrapping}\label{bootstraping}
Recall that in Section \ref{bbc}, we described our bootstrap algorithm that uses itself as the oracle of the metric distance algorithm. Here, we compute the time complexity and the approximation factor of the algorithm.
\begin{theorem}\label{thm:bootstrap}
There exists an $\widetilde O(n^{2-(5-\sqrt{17})/4+\epsilon}\mathrm{poly}(1/\epsilon))$ time quantum algorithm that approximates edit distance within a factor $\mathsf{e_{e}}(\epsilon) = O(1/\epsilon)^{O(\log 1/\epsilon)}$.
\end{theorem}
\begin{proof}
The algorithm is presented in Section \ref{bbc}. Here we prove the claimed time complexity and approximation factor. Suppose the time complexity of our algorithm for the edit distance problem is $\mathsf{t_{e}}(\epsilon)=\widetilde O(n^{2-\phi_\epsilon}\mathrm{poly}(1/\epsilon))$ and the time complexity of our algorithm for the bounded edit distance problem is $\widetilde O((1/\delta)^2 n^{2-2\phi_\epsilon}\mathrm{poly}(1/\epsilon))$. Notice that the total number of windows is equal to $O(n/g)$ and thus Step (i) of the algorithm can be easily done in time $O(n/g)$.
In Step (ii), we use the $O(1/\epsilon)$ approximation algorithm of metric estimation to approximate the distances of the windows. Moreover, we use our algorithm of edit distance $\mathcal{A}(2\epsilon)$ recursively for the oracle of metric estimation. Notice that, the length of every window is $l$. Furthermore, the number of points in the metric is equal to the number of windows, namely $O(n/g)$. Note that the running time and query complexity of the $O(1/\epsilon)$ algorithm of metric estimation are $\widetilde O(n^2\mathrm{poly}(1/\epsilon))$ and $\widetilde O(n^{1.5+\epsilon}\mathrm{poly}(1/\epsilon))$, respectively. Therefore, the total running time of this step is
$$\widetilde O((\mathsf{t_{e}}(2\epsilon)(l)\cdot (n/g)^{1.5+\epsilon}+(n/g)^2)\mathrm{poly}(1/\epsilon)) = \widetilde O((n^{(1-\beta_\epsilon)(2-\phi_{2\epsilon})+1.5\beta_\epsilon+\epsilon \beta_\epsilon}\gamma^{1.5+\epsilon} + n^{2\beta_\epsilon}\gamma^2)\mathrm{poly}(1/\epsilon))$$
since $l = O(n^{1-\beta_\epsilon})$ and $n/g = O(\gamma n /l) = O(\gamma n^{\beta_\epsilon})$.
Step (iii) takes time $O(n+|W_1||W_2|)$ due to Lemma \ref{dp} and thus the running time of this step is $O(\gamma^2n^{2\beta})$. Thus, the overall running time of the algorithm is $\widetilde O((n^{(1-\beta)(2-\phi)+1.5\beta+\epsilon \beta}\gamma^{1.5+\epsilon} + n^{2\beta}\gamma^2)\mathrm{poly}(1/\epsilon))$. If we set $\beta_\epsilon=(\sqrt{17}-1)/4+\epsilon$ and $\phi_\epsilon = 1-\beta_\epsilon$, the running time of the algorithm for $\delta$-bounded edit distance would be $\widetilde O((1/\delta)^2 n^{2-(5-\sqrt{17})/2+2\epsilon}\mathrm{poly}(1/\epsilon))$. Hence, the running time of the algorithm for the edit distance problem is $\widetilde O(n^{2-(5-\sqrt{17})/4+\epsilon}\mathrm{poly}(1/\epsilon)) = O(n^{1.781})$.
To compute the approximation ratio, we should first compute the number of nested levels we use the algorithm in itself. In the $i$th recursion, we use $\mathcal{A}(2^{i-1}\epsilon)$, while it works better than an $O(n^2)$ algorithm. Thus we have
$$2-(5-\sqrt{17})/4+2^{i-1}\epsilon < 2 \implies 2^{i-1}\epsilon < (5-\sqrt{17})/4 \implies 2^i < c/\epsilon \implies i<\log_2(1/\epsilon)+o(1)$$
Hence, we have at most $\log(1/\epsilon)+o(1)$ levels of recursion. We know that $\mathsf{e_{e}}(\epsilon) = 2\mathsf{e_{m}}(\epsilon)\mathsf{e_{e}}(2\epsilon)+1$, because if we get an $\alpha$ approximate of the optimal window-compatible transformation, it is a $2\alpha +1$ approximate of the optimal solution as discussed in Lemma \ref{mainbutnotmain}. We also know that $\mathsf{e_{m}}(\epsilon) = O(1/\epsilon)$. Hence, we can compute the edit distance as follows.
\begin{equation*}
\begin{split}
\mathsf{e_{e}}(\epsilon) &= 2\mathsf{e_{m}}(\epsilon)\mathsf{e_{e}}(2\epsilon)+1
\leq (c/\epsilon) \mathsf{e_{e}}(2\epsilon) = \frac{c^i}{1\cdot 2\cdot . . . \cdot 2^{i-1}\cdot \epsilon^i}\\
&=\frac{(1/\epsilon)^{c'}}{(1/\epsilon)^{(i-1)/2} \cdot \epsilon^i} = O(1/\epsilon)^{O(\log 1/\epsilon)}\\
\end{split}
\end{equation*}
This completes the proof.
\end{proof}
At last, we discuss why the exponent of our algorithm converges to $2-(5-\sqrt{17})/4$. Recall the recursive formula of for computing the running time of the algorithm:
$$\widetilde O(n^{2-\phi_\epsilon}\mathrm{poly}(1/\epsilon)) = \widetilde O((n^{(1-\beta_\epsilon)(2-\phi_{2\epsilon})+1.5\beta_\epsilon+\epsilon \beta_\epsilon}\gamma^{1.5+\epsilon} + n^{2\beta_\epsilon}\gamma^2)\mathrm{poly}(1/\epsilon))\enspace.$$
Intuitively, the running is the maximum of two terms, and the best is when these terms are equal. Thus when $\epsilon\to 0$ we can roughly tell:
$2-2\phi_0 = 2\beta_0 = (1-\beta_0)(2-\phi_0)+1.5\beta_0$. This equation has only one positive answer which is $\beta_0 = (\sqrt{17}-1)/4$ and $\phi_0 = 1-\beta_0$; therefore, the exponent is equal to $2-\phi_0 =2-(5-\sqrt{17})/4$.
\section{Edit Distance}\label{editdistance}
In this section, we use the results of Section \ref{metric} to design a quantum approximation algorithm for the edit distance problem. Our algorithm has an approximation factor of $7+\epsilon$ for an arbitrarily small number $\epsilon>0$ and time complexity $\widetilde O(n^{2-1/7}\mathrm{poly}(1/\epsilon))$. The outline of the algorithm is presented in Section \ref{section:ourResults}. Here we provide detailed proofs of the lemmas and theorems that are previously used for edit distance.
\begin{lemma}\label{dp}
Given a matrix of edit distances between the substrings corresponding to every pair of windows of $W_1$ and $W_2$, one can compute the optimal window-compatible transformation of $s_1$ into $s_2$ in time $O(n+|W_1||W_2|)$.
\end{lemma}
\begin{proof}
We take a dynamic programming approach to find the optimal window-compatible transformation of the two strings. Suppose $W_1$ has $k$ and $W_2$ has $k'$ windows. We recall that $W_1 = \langle w_1, w_2, . . ., w_k\rangle$ and $W_2=\langle w'_1, w'_2, . . ., w'_{k'}\rangle$ are collections of windows for $s_1$ and $s_2$, respectively, with window size $l$ and gap size $g$. We note that every window $w$ corresponds to a subinterval of $[1,n]$, thus we can use a linear time sorting algorithm, such as bucket-sort, and sort all the windows in time $O(n)$. Therefore, we can assume that the windows in $W_1$ and $W_2$ are sorted according to their right side.
Now for every $1\leq i\leq k$ and $1\leq j\leq k'$ we define $c_{i,j}$ to denote the optimal window-compatible transformation of the first $l+(i-1)\cdot g$ characters of $s_1$ into the first $l+(j-1)\cdot g$ characters of $s_2$. These suffixes of $s_1$ and $s_2$ correspond to $\langle w_1, w_2, . . ., w_i\rangle$ and $\langle w'_1, w'_2, . . ., w'_j\rangle$, respectively. For the sake of simplicity, we define $c_{i,0}=l+(i-1)\cdot g$, which is the cost of only deleting, and $c_{0,j}=l+(j-1)\cdot g$, which is the cost of only inserting. For every $1\leq i\leq k$ and $1\leq j\leq k'$ the following recursive formula holds:
\begin{align}
c_{i,j} = \min \Big \{ &c_{i-1,j}+g, \enspace c_{i,j-1}+g,
c_{i-\lceil l/g \rceil, j-\lceil l/g \rceil}+d(w_i, w'_j) \Big \}\enspace.
\label{eq:windowdynamic}
\end{align}
To compute $c_{i,j}$ we have three possibilities in an optimal window-compatible transformation. In particular, either $w_i$ is matched with $w'_j$, or at least one of $w_i$ and $w'_j$ is unmatched. If we match $w_i$ to $w'_j$, then there is a cost of $d(w_i, w'_j)$ for transforming $w_i$ to $w'_j$. Also, the other windows that overlap with $w_i$ or $w'_j$ cannot be used. Consequently, the problem reduces to finding window-compatible transformations for the first $l+(i-1)\cdot g-l$ characters of $s_1$ and the first $l+(j-1)\cdot g-l$ characters of $s_2$ with respect to $\langle w_1, w_2, . . ., w_{i-\lceil l/g \rceil}\rangle$ and $\langle w'_1, w'_2, . . ., w'_{j-\lceil l/g \rceil}\rangle$. This subproblem is captured by $c_{i-\lceil l/g \rceil, j-\lceil l/g \rceil}$. For the case that $w_i$ is unmatched, we need $g$ operations to remove every character in range $l+(i-2)\cdot g+1,\ldots,l+(i-1)\cdot g$ from $s_1$. This is because no other window can cover these characters. For the remaining characters the problem reduces to finding window-compatible transformations for the first $l+(i-2)\cdot g$ characters of $s_1$ and the first $l+(j-1)\cdot g$ characters of $s_2$ with respect to $\langle w_1, w_2, . . ., w_{i-1}\rangle$ and $\langle w'_1, w'_2, . . ., w'_j\rangle$, which is captured by $c_{i-1, j}$, thus $c_{i,j}=c_{i-1,j}+g$. Likewise, we can formulate the case that $w'_j$ is unmatched by $c_{i,j}=c_{i,j-1}+g$.
Note that $c_{k,k'}$ is equivalent to an optimal window-compatible transformation from $s_1$ to $s_2$. By iterating through $i$ from $1$ to $k$ and $j$ from $1$ to $k'$ one can simply calculate $c_{i,j}$ in time $O(1)$ according to \eqref{eq:windowdynamic}. Therefore $c_{k,k'}$ can be calculated in time $O(kk')$, and the proof is complete.
\end{proof}
\begin{corollary}
Given an $\alpha$-approximation matrix of edit distances between the substrings corresponding to every pair of windows of $W_1$ and $W_2$, one can compute an $\alpha$-approximation of the optimal window-compatible transformation of $s_1$ into $s_2$ in time $O(n+|W_1||W_2|)$.
\end{corollary}
\begin{lemma}\label{compatibility}
Given that $\mathsf{edit}(s_1,s_2) \leq \delta n$, there exists a window-compatible transformation of $s_1$ into $s_2$ with respect to $W_1$ and $W_2$ that has at most $(3\delta + 1/\gamma)n + 2l$ operations.
\end{lemma}
\begin{proof}
Recall that $l$ is the length of the windows, and $\gamma$ is the number of layers. Let \textsf{opt}\enspace be a minimum size transformation of $s_1$ into $s_2$. The overall idea of the proof is as follows. We first show that there exists a set of non-overlapping windows of length $l$, such that a window-compatible transformation with respect to them approximates \textsf{opt}. Next, by shifting those windows and losing a small fraction on the approximation factor, we fit them to those in $W_1$ and $W_2$.
Consider a pair of characters $x\in s_1$ and $y\in s_2$, such that \textsf{opt}\enspace transforms $x$ into $y$ either with no change or through a substitution. We call such a pair an edge. Note that there is no collision in the set of all edges in \textsf{opt}\enspace (or generally in any transformation), i.e. for edges $(x_1, y_1)$ and $(x_2, y_2)$ if $x_1<x_2$ then $y_1<y_2$. Let $M=\langle (x_1,y_1),\ldots,(x_m,y_m)\rangle$ be the sequence of all edges in \textsf{opt}\enspace in order from left to right.
Now we find the first set of windows as follows. Roughly speaking, we iterate through $M$ and at each step put as many edges as possible in a window of length $l$. In particular, let $\rho(i)$ be the smallest index in $M$ such that $x_{\rho(i)}$ and $y_{\rho(i)}$ are not covered by any window up to step $i$. We create window $v_i$ of length $l$ starting from $x_{\rho(i)}$ and window $v'_i$ of length $l$ starting from $y_{\rho(i)}$. We stop when any such window goes beyond the length of the strings.
In this way, there might be some edges that have one endpoint in $v_i$ and one endpoint beyond $v'_i$ or vice versa. Consider the case in which these edges have one endpoint in $v_i$. Let $h(i)$ be the number of such edges, and let $p(i)$ be the number of characters in $v'_i$ that \textsf{opt}\enspace transforms them through insertion. We claim that $p(i)\geq h(i)$. This is because $v'_i$ has at most $l-h(i)$ edges in \textsf{opt}. In comparison, a transformation with respect to $v_i$ and $v'_i$ can keep all the edges between $v_i$ and $v'_i$ and apply deletion and insertion for those edges that have one endpoint in $v_i$ and one endpoint out of $v'_i$. This costs at most $2h(i)$.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.7]{figs/fig4.pdf}
\end{center}
\caption{\small{An edge from $x\in s_1$ to $y\in s_2$ shows that \textsf{opt} \enspace transforms $x$ into $y$ with either no change or a substitution. Dashed edges represent those with one covered endpoint. Dotted edges represent those that are remained at the end of the iteration.}}
\label{fig:wincomp1}
\end{figure}
Besides, the number of remaining edges at the end of the iteration on $M$ is at most $l$. Such edges can be transformed by at most $2l$ insertions and deletions. Hence, requiring the transformation to be with respect to all $v_i$'s and $v'_i$'s adds at most $2l+2\sum h(i)\leq 2l+2\sum p(i)\leq 2l+2|\textsf{opt}|\leq 2l+2\delta n$ more operations to the optimum solution. Equivalently, the optimum transformation with respect to these windows has at most $3\delta n+2l$ operations.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.7]{figs/fig5.pdf}
\end{center}
\caption{\small{Dotted rectangles represent $v_i$'s and $v'_i$'s. Dashed rectangles represent shifted windows that are in $W_1$ or $W_2$. Dashed lines represent edges that are left outside of windows after shifting.}}
\label{fig:wincomp1}
\end{figure}
Finally, we note that the gap size between the windows in $W_1$ is $g=l/\gamma$, therefore, one can shift $v_i$'s by at most $g/2$ to the right or left in order to map them to non-overlapping windows in $W_1$. Likewise, one can find non-overlapping windows for $v'_i$'s in $W_2$. Every shift of a window leaves at most $g/2$ of the edges outside, which costs an extra $g$ operations. Since there are at most $n/l$ windows, the overall cost of shifting the windows is $n/\gamma$. Therefore, there exists a subset of $W_1$ and $W_2$ such that the optimum window-compatible transformation of $s_1$ into $s_2$ with respect to them has at most $2l+3\delta n+n/\gamma=(3\delta+1/\gamma)n+2l$ operations.
\end{proof}
The next lemma proves the approximation factor and time complexity of our $7+\epsilon$ approximation algorithm for the $\delta$-bounded edit distance problem.
\begin{lemma}\label{mainbutnotmain}
There exists a quantum algorithm that solves the $\delta$-bounded edit distance problem within an approximation factor of $7+\epsilon$ in time $\widetilde O(n^{2-1/7}\mathrm{poly}(1/\epsilon))$.
\end{lemma}
\begin{proof}
First, without loss of generality we can assume that $\delta > n^{-1/14}$, because otherwise one can use the $O(n+d^2)$ algorithm of Landau \textit{et al.}~\cite{landau1998incremental} for strings of distance at most $d$ and find the exact edit distance in time $O(n+\delta^2 n^2)=O(n^{2-1/7})$.
We prove that the algorithm discussed in Section \ref{section:ourResults} leads to an approximation factor of $7+\epsilon$ in quantum running time $\widetilde O(n^{2-1/7}\mathrm{poly}(1/\epsilon))$. To this end, let us go through the algorithm step by step.
Note that the total number of windows is equal to $O(n/g)$ where $g$ is the gap size. Therefore Step (i) of the algorithm takes time $O(n/g)$. In Step (ii), we use the $3+\epsilon$ approximation algorithm for metric estimation to approximate the distances between the windows. The running time of each oracle invocation is $O(l^2)$ since the length of windows are $l$. Also, there are at most $O(n/g)$ points in this metric estimation instance, therefore due to Theorem \ref{thm:metric1} the total running time of Step (ii) is: $$\widetilde O((l^2(n/g)^{5/3}+(n/g)^2)\mathrm{poly}(1/\epsilon))\enspace .$$
Note that $g=l/\gamma$. By assigning $l=n^{1-\beta}$ the overall running time of Step (ii) is:
$$\widetilde O((n^{2-\beta/3}\gamma^{5/3} + n^{2\beta}\gamma^2)\mathrm{poly}(1/\epsilon))\enspace .$$
Step (iii) takes time $O(n+|W_1||W_2|)$ due to Lemma \ref{dp}, and thus the running time of this step is $O(n+\gamma^2n^{2\beta})$. Thus, the overall running time of the algorithm up to Step (iii) is $$\widetilde O((n^{2-\beta/3}\gamma^{5/3})\mathrm{poly}(1/\epsilon)+n+\gamma^2 n^{2\beta})\enspace .$$
By assigning $\beta = 6/7$, the running time of the algorithm becomes $$\widetilde O(n^{2-2/7}(\gamma^{5/3}\mathrm{poly}(1/\epsilon)+\gamma^2))\enspace .$$
Finally, by choosing $\epsilon'=\epsilon/4$ and $\gamma=(\epsilon'\delta)^{-1}$ the overall running time of the algorithm becomes $O(n^{2-1/7}\mathrm{poly}(1/\epsilon))$. What remains is to show that assigning such values for $l$ and $\gamma$ gives the $7+\epsilon$ approximation factor. Due to Lemma \ref{compatibility}, there exists a window-compatible transformation of $s_1$ into $s_2$ with respect to $W_1$ and $W_2$ that has at most $(3\delta+1/\gamma)n+2l$ operations. According to the proof, at most $2\delta n$ of these operations are inside the windows. Therefore, a $(3+\epsilon')$-approximation of the distances between the windows in Step (iii) of the algorithm gives us a transformation with at most $(2\delta n)(3+\epsilon')+(\delta+1/\gamma)n+2l$ operations. This can be simplified as follows:
\begin{align*}
(2\delta n)(3+\epsilon')+(\delta+1/\gamma)n+2l
&\leq 6\delta n + 2 \epsilon'\delta n + \delta n + \frac{n}{\gamma}+2l &\\
&\leq (7\delta +2\epsilon'\delta+ \frac{1}{\gamma}+2n^{-6/7})n &\\
&\leq (7 +2\epsilon'+ \frac{1}{\delta\gamma}+\frac{2n^{-6/7}}{\delta})\delta n &\\
&\leq (7 +2\epsilon'+ \epsilon'+2 n^{-11/14})\delta n & \delta > n^{-1/14}\\
&\leq (7+4\epsilon')\delta n & \text{for every $n > (\frac{2}{\epsilon'})^{14/11}$} \\
&\leq (7+\epsilon)\delta n \enspace .
\end{align*}
Therefore, the algorithm finds a window-compatible transformation of $s_1$ into $s_2$ with respect to $W_1$ and $W_2$ that is $(7+\epsilon)$-approximation and runs in quantum time $O(n^{2-1/7}\mathrm{poly}(1/\epsilon))$.
\end{proof}
\begin{theorem}\label{main}
There exists a quantum algorithm that solves edit distance within an approximation factor of $7+\epsilon$ in time $\widetilde O(n^{2-1/7}\mathrm{poly}(1/\epsilon))$.
\end{theorem}
\begin{proof}
Let \textsf{opt}\enspace be the edit distance between the two strings. We can check if $\textsf{opt}=0$ in time $O(n)$. Assume that $\textsf{opt}\geq 1$. We guess a value $\rho$ for \textsf{opt}\enspace by iterating through different multiplicative ranges from $1$ to $n$. Let $\epsilon'=\epsilon/9$. In particular, in every step $i\geq 0$ we guess a range $[\delta n,(1+\epsilon')\delta n)$ for \textsf{opt}, where $\delta=(1+\epsilon')^i/n$, and run the algorithm of Lemma \ref{mainbutnotmain} with parameters $\epsilon'$ and $(1+\epsilon')\delta$. Note that at each step we can verify whether the output of the algorithm is a valid transformation or not. We get the first valid transformation as soon as \textsf{opt}\enspace lies within the range of our guess. This valid transformation is of size at most $(7+\epsilon')(1+\epsilon')\delta$ which is no more than $(7+\epsilon)\delta n$. Also, there are at most $\log_{1+\epsilon'}(n)\in \widetilde O(1/\epsilon)$ ranges for which we run the algorithm of Lemma \ref{mainbutnotmain}. Hence, the overall time for the search is $\widetilde O(n^{2-1/7}\mathrm{poly}(1/\epsilon))$.
\end{proof}
\section{Introduction}\label{introduction}
The \textit{edit distance} (a.k.a \textit{Levenshtein distance}) is a well-known metric to measure the similarity of two strings. This metric has been extensively used in several fields such as computational biology, natural language processing, and information theory. The algorithmic aspect of the problem is even more fundamental; the problem of computing the edit distance is a textbook example for dynamic programming.
The edit distance between two strings is defined as the smallest number of \textit{insertions}, \textit{deletions}, and \textit{substitutions} that need to be made on one of the strings to transform it to another one. For two strings $s_1$ and $s_2$ with $n$ characters in total ($|s_1| + |s_2| = n$), a classic dynamic program finds the edit distance between them in time $O(n^2)$. The idea is to define auxiliary variables $d_{i,j}$'s which denote the edit distance between the first $i$ characters of $s_1$ and the first $j$ characters of $s_2$. Next, we iteratively determine the values of the auxiliary variables based on the following formula
\begin{equation*}
\small
d_{i,j} =
\begin{cases}
d_{i-1,j-1}, & \text{if }s_1[i] = s_2[j] \\
1+\min\{d_{i-1,j-1},d_{i,j-1},d_{i-1,j}\} & \text{if }s_1[i] \neq s_2[j].
\end{cases}
\end{equation*}
Despite the simplicity of the above solution, it has remained one of the most efficient algorithms from a theoretical standpoint to this day. Since the 1970s, several researchers aimed to improve the quadratic running time of the problem, however, thus far, the best-known algorithm runs in time $O(n^2/\log^2 n)$~\cite{masek1980faster}. The shortcoming of these studies is partly addressed by the work of Backurs and Indyk~\cite{backurs2015edit} wherein the authors show a truly subquadratic time algorithm is impossible to achieve unless a widely believed conjecture (\textsf{SETH}\footnote{The \textit{strong exponential time hypothesis} states that no algorithm can solve the satisfiability problem in time $2^{n(1-\epsilon)}$.}) fails.
Unfortunately, the quadratic dependency of the running time on the size of the input makes it impossible to use such algorithms for large inputs in practice. For example, a human genome consists of almost three billion base pairs that need to be incorporated in similarity measurements. Therefore, several studies were focused on improving the running time of the algorithm by considering approximation solutions. A trivial $\sqrt{n}$ approximation algorithm follows from an $O(n + d^2)$ exact algorithm of Landau \textit{et al.}~\cite{landau1998incremental} where $d$ is the edit distance between the two strings. Subsequent research improved this to $n^{3/7}$~\cite{bar2004approximating}, to $n^{1/3+o(1)}$~\cite{batu2006oblivious}, to $2^{\widetilde O(\sqrt{\log n})}$~\cite{andoni2012approximating}, and the latest of which provides a polylogarithmic approximation guarantee in subquadratic time~\cite{andoni2010polylogarithmic}. Note that although the running times of these algorithms are almost linear, even if one favors the approximation factor over the running time, slowing down the algorithms to barely subquadratic doesn't yield an asymptotically better approximation guarantee. Despite persistent studies, finding a subquadratic algorithm with a constant approximation factor which is the ``holy grail" here is still open (see Section 6 of Indyk~\cite{indyk2001algorithmic}).
Quantum computation provides a strong framework to substantially improve the running time of many algorithmic problems. This includes a long list of problems from algebraic computational problems, to measuring graph properties, to string matching, to searching, to optimizing programs, etc.~\cite{ramesh2003string,farhi1999invariant,krovi2015quantum,jeffery2013nested,belovs2012learning,beals1997quantum,le2014improved,shor1994algorithms}. However, quantum techniques can only be applied to limited structures. For instance, many classic problems such as sorting or even counting the number of 1's in a 0-1 array are still as time-consuming even with quantum computation. Indeed existing quantum techniques offer no immediate improvement to the running time of edit distance, neither to many classic DP-type problems such as finding the $\mathsf{lcs}$\enspace (longest common subsequence), $\mathsf{dtw}$\enspace (dynamic time wrapping) of two strings or determining the Fr\'{e}chet distance between two polylines. To the best of our knowledge, no exact or approximation algorithm is known for edit distance in subquadratic time via quantum computation.
In this work, we provide a framework to approximate the edit distance between two strings within a constant factor. This framework requires as black box a procedure that takes several smaller strings as input and approximates their distances all at once. For quantum computers, we reduce this black box to finding the distances of a metric, namely \textit{metric estimation}. In this problem, we are given a metric space where any distance is available by a query from a distance oracle.
We show that metric estimation cannot be approximated within a factor better than $3$ with a subquadratic number of quantum queries. On the contrary, we provide positive results for approximation factor $3$ and also larger constant factors. We show our bounds are tight up to constant factors by proving lower bounds on the query complexity of metric estimation.
Our metric estimation quantum algorithms are general tools and may find their applications in other distance-related problems as well.
Combining this black box with our framework yields subquadratic quantum algorithms for approximation edit distance within a constant factor.
Our work is similar in spirit to the work of Le Gall \cite{le2014improved} and D\"{u}rr \textit{et al.} \cite{doi:10.1137/050644719} where combinatorial techniques are used to obtain efficient quantum algorithms. We believe that our work opens an avenue to further investigation of edit distance in quantum setting and perhaps achieving near linear time quantum algorithm for edit distance.
As another application of our framework, we design a MapReduce algorithm for approximating edit distance within an approximation factor of $3$.
MapReduce is one of the most recent developments in the area of parallel computing. It has the benefits of both sequential and parallel computation. Many tech companies such as Google, Facebook, Amazon, and Yahoo designed MapReduce frameworks and have used them to implement fast algorithms to analyze their data.
In this paper, we focus on the well-known MapReduce theoretical framework initiated by Karloff, Suri, and Vassilvitskii \cite{karloff2010model} (and later further refined by Andoni, Nikolov, Onak, and Yaroslavtsev \cite{Andonimapreducestoc}).
Designing MapReduce algorithms for simulating sequential dynamic programs for important problems was recently initiated by Im, Moseley, and Sun \cite{mapreducedyn2017}.
They study DP-type problems with two key properties, monotonicity and decomposability. Their framework does not apply here since edit distance is neither monotone nor decomposable.
Our algorithm runs in a logarithmic number of rounds with a sublinear number of machines and sublinear memory of each machine. Moreover, the running time of each machine is subquadratic.
To the best of our knowledge, both our quantum algorithms and our MapReduce algorithm are first to improve upon the trivial $O(n^2)$ classic algorithm beyond subpolynomial factors for approximating edit distance\footnote{within a constant factor} in these settings. We believe that our framework can be useful to better understand edit distance in other models, such as the streaming and the semi-streaming models.
The closest works to our results are ~\cite{andoni2012approximating} and ~\cite{andoni2008smoothed}. In particular, they use a space embedding approach from \cite{Ostrovsky:2005:LDE:1060590.1060623} with dividing the string into blocks of smaller size, but our main observations and structural lemmas are completely different from their approach. We note that to the best of our knowledge, the ideas of our framework are novel and have not been used in any of the previous work. In ~\cite{apostolico1990efficient}, the authors give a parallel algorithm for determining the edit distance between two strings. Their algorithm uses $\widetilde O(n^2)$ processors and a shared memory of $O(n^2)$. Note that their algorithm cannot be used in MapReduce models, since the number of machines and memory of each machine in a MapReduce algorithm should be sublinear, and the number of rounds should be $O(\mathsf{polylog}(n))$ \cite{karloff2010model}. The major advantage of our MapReduce algorithm over the algorithm of ~\cite{apostolico1990efficient} is that both the number of machines and the memory of each machine is sublinear in our algorithm. Moreover, the number of rounds in our algorithm is $O(\log(n))$.
A similar approach is taken in the work of Nayebi \textit{et al.}~\cite{nayebi2014quantum} wherein the authors study the computational complexity of APSP on quantum computers. They give an APSP algorithm for graph instances with small integer weights. They also give a fine-grained reduction from APSP to negative triangle via quantum computing.
\input{our-contribution.tex}
\subsection{MapReduce Basics}\label{basics}
In this section, we give a brief overview of the MapReduce setting and later show how our framework can be used to design a MapReduce algorithm for edit distance.
In the MapReduce model,
an algorithm consists of several rounds. Each round has a mapping phase and a reducing phase. Every unit of information is represented in the form of a $\langle key; value\rangle$ pair in which both key and value are strings. The input, therefore, is a sequence of $\langle key; value\rangle$ pairs specifying the input data and their corresponding positions. For instance, in the case of edit distance, we assume the input pairs are either in the form of $\langle (s_1,i); s_1[i]\rangle$ or $\langle (s_2,i); s_2[i]\rangle$ where the value represents a character, and the key shows the position of this character in either $s_1$ or $s_2$.
Each round of a MapReduce algorithm is performed as follows: every single input pair is given to a mapper separately and depending on the mapping algorithm, a sequence of $\langle key; value\rangle$'s is generated with respect to the input key. Note that the mappers have to be \textit{stateless} in the sense that the output of every mapper is only dependent on the single $\langle key; value\rangle$ pair given to it. Since the mappers are stateless, parallelism in the mapping phase is straightforward; all the inputs are evenly distributed between the machines. Moreover, there is no limit on the types of the $\langle key; value\rangle$ outputs that the mappers generate. Once \textit{all} the mapper jobs are finished, the reducers start to run. Let $\mathcal{K}$ be the set of all keys generated by the mappers in the mapping stage. In the reducing stage, every $key \in \mathcal{K}$ along with all its associated values is given to a single machine. Note that there is no limit on the number of keys generated in the mapping phase as long as all the outputs together fit in the total memory of all machines. However, the values associated with every key should fit in the memory of a single machine since all such values are processed at once by a single reducer. Every reducer, upon receiving a key and a sequence of values associated to it $\langle key; v_1, v_2, v_3, \ldots, v_l\rangle$ runs a reducer-specific algorithm and generates a sequence of output pairs. Unlike the mapping phase, the output keys of a reducer should be identical to the input key given to them. Moreover, the reducers are not stateless since they have access to all values of a key at once, but they can only access their given key and the values associated with it and should be regardless of the other $\langle key; value\rangle$ pairs generated in the mapping phase. Similar to the mapping phase, the total size of the outputs generated by all reducers should no exceed the total memory of all machines together. In addition to this, the total outputs of a reducer should not be more that its memory. Once \textit{all} reducers finished their jobs, the outputs are fed to the mappers for the next round of the algorithm.
For a problem with input length $n$, the goal is to design a MapRuduce algorithm running on $N_p$ machines each having a memory of $N_m$. $N_p$ and $N_m$ have to be sublinear in $n$ since the input is assumed to be huge in this setting. Moreover, since the overhead of a MapReduce round is time-consuming, the number of MapReduce rounds of the algorithms should be small (either constant or polylogarithmic). Many classic computational problems have been studied in the MapReduce setting. For instance, Karloff, Suri, and Vassilvitskii~\cite{karloff2010model} provide a MapReduce algorithm to compute an MST of a graph with a sublinear number of machines and a sublinear memory for every machine. Lattanzi \textit{et al.}~\cite{lattanzi2011filtering} design a filtering method and based on that, provide MapReduce algorithms for fundamental graph problems such as maximal matchings, weighted matchings, vertex cover, edge cover, and minimum cuts.
We show in Section \ref{mapreducealgorithm} that using $O(n^{8/9})$ machines and $O(n^{8/9})$ memory on each machine, one can design a MapReduce algorithm for edit distance that runs in $O(\log n)$ MapReduce rounds. Moreover, the running time of the algorithm is subquadratic.
\section{Approximating Edit Distance in MapReduce}\label{mapreduce}
Edit distance has been studied in parallel and distributed models since the 90s. However, the sequential nature of the dynamic programming solution makes it difficult to parallelize; therefore most of these solutions are slow or require lots of memory/communication. Using our framework, we give a somewhat balanced parallel algorithm for the edit distance problem in MapReduce model. More precisely, we give a ($3+\epsilon$)-approximation algorithm which uses $O(n^{8/9})$ machines, each with a memory of size $O(n^{8/9})$. Moreover, our algorithm runs in a logarithmic number of rounds and has time complexity $O(n^{1.704})$ on one machine which is truly subquadratic.
The overall communication and total memory of our algorithm are also truly subquadratic, due to the sublinearity of the number of machines and the memory of each machine.
Our algorithm is significantly more efficient than previous PRAM algorithms, for instance \cite{apostolico1990efficient} in terms of the number of machines, the overall memory, and the overall communication.
In addition, this is the first result of its kind for edit distance in MapReduce model.
Although this subject has been studied before, previous studies targeted a different aspect of the problem, such as giving a heuristic algorithm, an algorithm for inputs from a particular distribution model, or an algorithm for edit distance between all pairs of several strings \cite{editmapreduce2}.
We begin by stating some of the MapReduce notions and definitions in Section \ref{basics} and next explain our algorithm is Section \ref{mapreducealgorithm}.
\input{mapreduce-intro}
\subsection{Edit Distance in MapReduce}\label{mapreducealgorithm}
Our solution for approximating edit distance in MapReduce uses the same framework explained in Section \ref{editdistance}. Therefore, we solve the problem by solving the $\delta$-bounded edit distance problem several times. The difference is that here we solve all of these subproblems simultaneously. This only imposes a multiplicative factor of $O((1/\epsilon)\log n)$ to the number of machines and a multiplicative factor of $1+\epsilon$ to the approximation factor, hence in the following, we focus on solving the $\delta$-bounded edit distance problem.
We use two different approaches for large $\delta$'s and small $\delta$'s. For large $\delta$'s, we use our framework and compute the edit distance between some pairs of windows of $s_1$ and $s_2$ all at once. For small $\delta$'s though, we use a new method based on $(\mathsf{min}, +)$ matrix multiplication, also known as distance multiplication. We denote it by $\star$.
We separate the large and the small $\delta$'s with a critical value based on the number of machines\footnote{for $n^{8/9}$ machines $\delta^* = n^{-8/27}$.}.
For $(\mathsf{min}, +)$ matrix multiplication in the MapReduce model, we use a parameterized version of the algorithm presented in \cite{saeedmapreduce}.
\begin{theorem}[Proved in \cite{saeedmapreduce}]\label{minplusfirst}
For any two $n\times n$ matrices $A$ and $B$ and $0 < x \leq 2$, $A \star B$ can be computed with $n^{3(1-x/2)}$ machines and memory $O(n^x)$ in $1 + \lceil (1-x/2)/x\rceil$ MapReduce rounds. Moreover, the total running time of the algorithm is $O((1/x)n^3)$.
\end{theorem}
Given that we have a chain of matrices to be multiplied instead of just two matrices, we can use Theorem \ref{minplusfirst} to halve the number of matrices in two rounds; therefore we have the following corollary.
\begin{corollary}[of Theorem \ref{minplusfirst}]
\label{corollary:mm}
The $(\mathsf{min}, +)$ multiplication of $n^a$ matrices of size $n^b\times n^b$ can be computed in $2\lceil a\log_2 n\rceil$ rounds of MapReduce with $n^y$ machines for any $0\leq y\leq a+3b/2$, with a memory of $O(n^{2(a+3b-y)/3})$ for each machine.
Moreover, the running time of the algorithm (for one machine) is $\widetilde O(n^{a+3b-y})$.
\end{corollary}
Notice that for two $n\times n$ matrices in Corollary \ref{corollary:mm}, we have $a=0$ and $b=1$, hence the number of machines is $n^y$ and the memory of each machine is $O(n^{2 - 2y/3})$ which is the same as Theorem \ref{minplusfirst} where $x = 2 - 2y/3$. Also note that for $0\leq y\leq a+3b/2$, we use Theorem \ref{minplusfirst} with $1\leq x\leq 2$, hence all $1/x$ terms are ignored.
In Sections \ref{sec:largedelta} and \ref{sec:smalldelta}, we discuss our approach for large $\delta$'s and small $\delta$'s, respectively. In Section \ref{sec:conclusion}, we discuss the remaining details of the algorithm.
\subsubsection{Our Approach for Large $\delta$'s}
\label{sec:largedelta}
The overall idea of our solution for large $\delta$'s is to use our framework as follows: we first construct some windows for each string, then we find the edit distance between some pairs of windows, and afterward we find a window-compatible transformation, which is a good approximation to the desired edit distance between two input strings.
The first step of our approach is to find the edit distance between some pairs of windows. Previously, we found an approximated edit distance between all pairs of windows using metric estimation. On the contrary, here we can do better than finding the edit distance between all pairs based on the following observation.
\begin{lemma}
\label{lemma:restrict}
Given that $\mathsf{edit}(s_1,s_2) \leq \delta n$, there exists a window-compatible transformation of $s_1$ into $s_2$ with respect to $W_1$ and $W_2$, that for each window $w_i\in W_1$ that matches to a window $w_2\in W_2$, their indices do not differ by more than $\lceil \delta n/g \rceil$, and the number of operations is at most $(3\delta + 1/\gamma)n + 2l$.
\end{lemma}
\begin{proof}
This lemma is similar to Lemma \ref{compatibility} with an additional condition that the indices of any two matching windows do not differ by more than $\lceil \delta n/g \rceil$. The proof is also similar to Lemma \ref{compatibility}.
Let \textsf{opt} be a minimum size transformation of $s_1$ into $s_2$. Consider a pair of characters $x\in s_1$ and $y\in s_2$ such that \textsf{opt} transforms $x$ into $y$ either with no change or through a substitution. As before we call such a pair an edge. Let $M = \langle(x_1,y_1),\dots,(x_m,y_m)\rangle$ be the sequence of all edges in opt in order from left to right.
Similar to the proof of Lemma \ref{compatibility}, we first find a set of non-overlapping windows of length $l$ by iterating over $M$. In particular, let $\rho(i)$ be the smallest index in $M$ such that no window covers $x_{\rho_i}$ and $y_{\rho_i}$ up to step $i$. We create a window $v_i$ of length $l$ starting from $x_{\rho_i}$ and a window $v{i′}$ of length $l$ starting from $y_{\rho_i}$. We stop when any such window goes beyond the range of the strings or all edges of $M$ are covered.
We then shifted these windows to the left to become consistent with the windows of $W_1$ and $W_2$. We proved there exists a window-compatible transformation with respect to those windows with at most $(3\delta + 1/\gamma)n + 2l$ operations. Now we complete the proof using same windows, by showing that the indices of any matching windows do not differ by more than $\lceil \delta n/g \rceil$.
We know that (for example see Corollary $1$ of \cite{ukkonen1985algorithms}) for any edge $(x_{\rho_i}, y_{\rho_i})\in M$, their indices differ by at most $\mathsf{edit}(s_1, s_2)\leq \delta n$. Therefore, for any $v_i$ and $v'_i$, positions of their first characters differ by at most $\delta n$. From this, we can directly conclude that the indices of the corresponding shifted windows in $W_1$ and $W_2$ differ by at most $\lceil \delta n/g \rceil$.
\end{proof}
We find the edit distance between useful pairs of windows in the first round. To do this, we give some pairs of windows to a machine and use the na\"{i}ve DP-based algorithm to find the edit distance between them. In the next round, we combine the results of the first round to find the best window-compatible transformation. The second round is similar to Lemma \ref{dp}; the difference is that the memory and the running time is slightly reduced by Lemma \ref{lemma:restrict}. The second round uses only one machine.
We have the following lemma for large $\delta$'s (or small $\alpha$'s). To simplify the notation, let $\delta = n^{-\alpha}$.
\begin{lemma}\label{largedeltaslemma}
We can solve the $\delta$-bounded edit distance problem for
\begin{itemize}
\item
$0\leq x\leq 13/20$ and $\alpha\leq 3(x+1)/16$ with $n^x$ machines, and $O((1/\epsilon^2)n^{(11-5x)/8+\epsilon'})$ memory for each machine in time $O((1/\epsilon^2)n^{(35-13x)/16})$ (for one machine), and for
\item
$13/20\leq x \leq 7/6$ and $\alpha\leq 2(4-x)/21$ with $n^x$ machines, and $O((1/\epsilon^2)n^{2(4-x)/7+\epsilon'})$ memory for each machine in time $O((1/\epsilon^2)n^{(50-23x)/21})$ (for one machine).
\end{itemize}
in two MapReduce rounds, where $\epsilon'>0$ is an arbitrary constant.
\end{lemma}
\begin{proof}
We already stated the sketch of our algorithm. To analyze the algorithm, we define and set some parameters carefully.
Recall that we used two parameters of $\beta$ and $\gamma$ to construct the windows of length $l=\lfloor n^{1-\beta}\rfloor$ with a gap size $g=\lfloor l/\gamma \rfloor$ for each of the input strings. Lemma \ref{lemma:restrict} states that given $\mathsf{edit}(s_1,s_2)\leq \delta n = n^{1-\alpha}$, there exists a window-compatible transformation with at most $(3\delta + 1/\gamma)n + 2l$ operations. To keep the approximation factor as small as $3+\epsilon$, we should have $n/\gamma \ll \delta n$ and $2l\ll \delta n$. Setting $\gamma = 1/\delta \epsilon$ and $\beta > \alpha$ suffice.
By doing this, the number of windows for each string is at most $$n_{w_1}, n_{w_2} \leq n\gamma/l = O((1/\epsilon)n^{\alpha+\beta}).$$
In the first round, we find the edit distance between all \textit{useful pairs} of windows, which are in fact the pairs with an edit distance of at most $\lceil \delta n/g\rceil$. By Lemma \ref{lemma:restrict}, the number of such useful pairs is at most
$$min(|W_1|,|W_2|)\cdot(2\lceil \delta n/g\rceil+1) = O((1/\epsilon^2)n^{\alpha+2\beta}) $$
Therefore, if we have $n^x$ machines, every machine gets $O((1/\epsilon^2)n^{\alpha+2\beta-x})$ pairs.
The edit distance between a pair of windows can be computed in time $O(l^2)$ and memory $O(l)$ where $l=\lfloor n^{1-\beta} \rfloor$. Hence, the memory of each machine in round 1 is $O((1/\epsilon^2)n^{1+\alpha+\beta-x})$. Moreover, the time complexity of each machine in this round is $O((1/\epsilon^2)n^{2+\alpha-x})$.
In the second round, we only use one machine to combine the results of the first round. This machine has to get edit distance between all pairs from all machine; hence it needs $O((1/\epsilon^2)n^{\alpha+2\beta})$ memory. The time complexity of this round is also $O((1/\epsilon^2)n^{\alpha+2\beta})$.
By setting $\beta = \alpha+\epsilon'/2$ for an arbitrary constant $\epsilon'>0$, and setting $\alpha$ as stated in the lemma, we get the desired result.
\end{proof}
\subsubsection{Our Approach for Small $\delta$'s}
\label{sec:smalldelta}
The other side of the edit distance problem is the case when the two given strings are similar. In this case, if we try to use our framework, we would encounter too many windows, and this exceeds the time and memory given to the algorithm. Previously, in this case, we used the algorithm of Landau \textit{et al.}{}~\cite{landau1998incremental} with time $O(n+d^2)$. This solution cannot (trivially) become parallel. Here, we instead use a novel approach based on $(\mathsf{min}, +)$ matrix multiplication. We again use the fact that a character $c_1$ from $s_1$ can only be transformed (with no change or a substitution) to a character $c_2$ in $s_2$ only if their positions differ by at most $\mathsf{edit}(s_1, s_2)$~(Corollary $1$ of \cite{ukkonen1985algorithms}).
Let $d(i,j+1,i',j'+1)$ be the edit distance between two substrings of $s_1[i,\dots,j]$ and $s_2[i',\dots,j']$. We have the following lemma.
\begin{lemma}\label{smalldeltaslemma}
\label{lem:dij}
For an arbitrary $k$, $i<k\leq j$, we have:
\begin{equation*}
d(i,j+1,i',j'+1) = \substack{min\\i'-1\leq k'\leq j'}\big\{d(i,k+1,i',k'+1) +d(k+1,j+1,k'+1,j+1)\big\}.
\end{equation*}
\end{lemma}
\begin{proof}
We construct a transformation from $s_1[i,\dots,j]$ to $s_2[i',\dots,j']$ using two transformation: one from $s_1[i,\dots,k]$ to $s_2[i',\dots,k']$ and the other from $s_1[k+1,\dots,j]$ to $s_2[k'+1,\dots,j']$, therefore $$d(i,j+1,i',j'+1) \leq \substack{min\\i'-1\leq k'\leq j'}\big\{d(i,k+1,i',k'+1)+d(k+1,j+1,k'+1,j+1)\big\}.$$
Also, if we define $k^*$ as the largest index that a character from $s_1[i,\dots, k]$ transfers into $s_2[k^*]$ in an optimal transformation, or $k^*=i'-1$ if no such index exists, we have $$d(i,j+1,i',j'+1) = d(i,k+1,i',k^*+1)+d(k+1,j+1,k^*+1,j+1).$$
which completes the proof.
\end{proof}
Moreover, computing $d(i,j+1,i',j'+1)$ is useful only when $|i-i'|\leq d$ and $|j-j'|\leq d$~(Corollary $1$ of \cite{ukkonen1985algorithms}), therefore for a fixed $i$ and $j$, all of these \textit{useful values} form a $(2\delta n+1) \times (2\delta n+1)$ matrix, namely $D^{i,j}$. Rewriting Lemma \ref{lem:dij} in matrices, we have the following corollary.
\begin{corollary}[of Lemma \ref{lem:dij}]
\label{cor:matdij}
For an arbitrary $k$, $i\leq k\leq j$, we have $D^{i,j}=D^{i,k}\star D^{k,j}$, where $\star$ is the $(\mathsf{min}, +)$ matrix multiplication operator.
\end{corollary}
Notice that $\mathsf{edit}(s_1,s_2)=d(1, |s_1|+1, 1, |s_2|+1)$, which is an element of $D^{1,|s_1|}$. To compute this matrix, we do as follows:
for a parameter $y$, $0\leq y \leq 1$, which we'll fix later, we partition $s_1$ into $n^y$ substrings of length at most $n^{1-y}$. Each of these substrings has a matching substring in $s_2$ with a length at most $n^{1-y}+2\delta n$. Using the na\"{i}ve DP-based algorithm, we construct a $(2\delta n+1) \times (2\delta n+1)$ matrix for each of these $n^y$ substrings in the first round. The matrices are $D^{1, t}, D^{t+1, 2t}, \dots, D^{(\lceil|s_1|/t\rceil-1)t+1, |s1|}$ where $t=n^{1-y}$. By Corollary \ref{cor:matdij} we have $D^{1,|s_1|} = D^{1, t}\star D^{t+1, 2t} \star \dots \star D^{(\lceil|s_1|/t\rceil-1)t+1, |s1|}$. Therefore, we obtain the result in remaining rounds by the matrix multiplication algorithm of Corollary \ref{corollary:mm}.
\begin{lemma}
We can solve the $\delta$-bounded edit distance problem for
\begin{itemize}
\item
$0\leq x\leq 13/20$ and $\alpha\geq 3(x+1)/16$ with $n^x$ machines, and $O(n^{(11-5x)/8})$ memory of each machine in time $O(n^{(51-29x)/16})$ (for one machine), and for
\item
$13/20\leq x \leq 7/6$ and $\alpha\geq 2(4-x)/21$ with $n^x$ machines, and $O(n^{2(4-x)/7})$ memory of each machine in time $O(n^{(58-25x)/21})$ (for one machine).
\end{itemize}
in at most $O(\log n)$ MapReduce rounds.
\end{lemma}
\begin{proof}
Here, we analyze the described algorithm in more details.
In the first round, constructing the full matrix is a time-consuming process for one machine; therefore we break this job into $n^t$ parts. More precisely, we partition rows of the solution matrix into $n^t$ parts and give the task of computing each part to one machine. Therefore, in the first round, the number of machines is equal to $n^{y+t}=n^x$. The memory of each machine is the maximum of its input size, its running memory, and its output size, which are equal to $2n^{1-y}+2d$, $O(n^{1-y})$, and $n^{2-2\alpha-t}$, respectively. The time complexity of one machine using the DP-based algorithm is $O(n^{1-y}\cdot n^{1-y} \cdot n^{1-\alpha-t}) = O(n^{3-2y-\alpha-t})$.
The second part of the algorithm is analogous to Corollary \ref{corollary:mm} where $a=y$ and $b=1-\alpha$, therefore if the number of machines is $n^x$, the memory of each machine is $n^{2(y+3(1-\alpha)-x)/3}$.
Moreover, the time complexity of one machine is $\widetilde O(n^{3-3\alpha+y-x})$.
Setting $y=(6\alpha+2x-3)/5$ and $t=x-y$ give us the desired result. Also, note that the range of $x$ is consistent with Corollary \ref{corollary:mm}.
\end{proof}
\subsubsection{Conclusion}
\label{sec:conclusion}
We compute edit distance by solving the $\delta$-bounded edit distance problems for several $\delta$'s in parallel. For each $\delta=n^{-\alpha}$ we use the appropriate MapReduce algorithm based on the value of $x$ and $\alpha$. When all subproblems are finished, we also have a final round for combining the results of these subproblems to obtain the final (approximated) edit distance. Therefore, the desired MapReduce ($3+\epsilon$)-approximation algorithm for edit distance is as follows.
\begin{theorem}
We can solve the edit distance problems in MapReduce model in at most $O(\log n)$ MapReduce rounds with $\widetilde O((1/\epsilon)n^x)$ machines and for
\begin{itemize}
\item
$0\leq x\leq 13/20$ with a memory of at most $O((1/\epsilon^2)n^{(11-5x)/8+\epsilon'})$ for one machine in time $O(n^{(51-29x)/16})$ (for one machine), and for
\item
$13/20\leq x \leq 7/6$ with a memory of at most $O((1/\epsilon^2)n^{2(4-x)/7+\epsilon'})$ in time $O(n^{(58-25x)/21})$ (for one machine).
\end{itemize}
\end{theorem}
\begin{proof}
We solve the problem for $\delta=0$ and $\delta=(1+\epsilon/3)^k/n$ for $0\leq k\leq O((1/\epsilon)\log n)$ in parallel machines. For each subproblem, we use $\lceil n^x\rceil$ machines.
In the zero case, we only check whether $s_1=s_2$ or not. This can be done with at most $n^{1-x}$ memory and $O(n^{1-x})$ for each machine. We handle other subproblems by Lemmas \ref{largedeltaslemma} or \ref{smalldeltaslemma}\footnote{in fact, for small $\delta$'s we can run our algorithm just once for the largest $\delta$}. Therefore, the memory of each machine is at most $O((1/\epsilon^2)n^{(11-5x)/8+\epsilon'})$ for $0\leq x\leq 13/20$ and $O((1/\epsilon^2)n^{2(4-x)/7+\epsilon'})$ for $13/20\leq x \leq 7/6$.
The time complexity of each machine is the maximum time of Lemmas \ref{largedeltaslemma} and \ref{smalldeltaslemma} which is at most $O(n^{(51-29x)/16})$ for $0\leq x\leq 13/20$ and $O(n^{(58-25x)/21})$ for $13/20\leq x \leq 7/6$. The number of rounds is also at most $O(\log n)$.
\end{proof}
By setting $x=8/9$, we minimize the maximum of the number of machines and the memory of each machine. This is shown in Figure \ref{mapreducetradeoff}.
\input{figs/mapreducetradeoff}
\begin{corollary}
We can solve the edit distance problems in MapReduce model with an approximation factor of $3+\epsilon$ in $O(\log n)$ rounds with $\widetilde O((1/\epsilon)n^{8/9})$ machines, a memory of $O((1/\epsilon^2)n^{8/9+\epsilon'})$ for each machine, and in time $O(n^{2-8/27})$ (for one machine), where $\epsilon'>0$ is an arbitrary constant.
\end{corollary}
\section{Metric Estimation}\label{metric}
In this section, we discuss the metric estimation problem. Although the results of this section are only auxilary observations to be later used for edit distance, these results are of independent interest and may apply to future work. As defined previously, in this problem, we wish to estimate the distance matrix of a metric space $\langle\metric, d\rangle{}$ with $n$ points. Notice that, an estimation of a distance $d(p_i, p_j)$ with approximation factor $\alpha$ lies in the range $[d(p_i, p_j), \alpha d(p_i, p_j)]$, therefore, the estimated value cannot be less than the actual distance. However, it can be more than the actual distance by a multiplicative factor of $\alpha$. We tend to minimize the query complexity and the approximation factor, however, our algorithm is allowed to run in time $\widetilde O(n^2)$. Throughout this section, we show a tradeoff between the approximation factor and the quantum query complexity of metric estimation. First, we present an impossibility result that shows the approximation factor cannot be less than $3$ unless we make a quadratic number of queries. Next, in Section \ref{sub:alg1}, we present our desired $3+\epsilon$ approximation algorithm for metric estimation with a subquadratic query complexity. Afterward, we adjust our algorithm to make as few as $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$ oracle call for a larger constant approximation $\mathsf{e_{m}}(\epsilon) = O(1/\epsilon)$.
\subsection{Hardness of Approximation for $\alpha < 3$}\label{sub:hardness}
As aforementioned, the purpose of this section is to show an impossibility result for approximating metric estimation within a factor smaller than $3$ with subquadratic query complexity. To this end, we give a reduction from the well-known parity problem to the metric estimation problem. Parity is one of the problems for which quantum computers cannot perform better than classical computers. Recall the definition of the parity problem from Section \ref{contribution:metric}.
\begin{center}
\noindent\framebox{\begin{minipage}{6.3in}
\textsf{Parity}\\[0.25cm]
\textsf{Input}: an integer $n$, and access to an oracle $\mathcal{O}$ that upon receiving an integer $i$ reports the value of $f(i)$. $f$ is defined over $[n]$ and maps each index to either $-1$ or $1$. \\[0.25cm]
\textsf{Output}: $\mathsf{par}(f) = \prod_{i \in [n]}f(i)$.
\end{minipage}}
\end{center}
Note that, $\mathsf{par}(f)$ is either $+1$ or $-1$ for every function $f$. Farhi \textit{et al.}{}~\cite{Farhi:1998bz} proved that at least $\Omega(n)$ oracle queries are necessary to find $\mathsf{par}(f)$. A classic method to show lower bounds on the time/query complexity of problems is via a reduction from parity. This method has been used to show lower bounds on the quantum query complexity of many problems~\cite{doi:10.1137/050644719, Montanaro2015}.
We are now ready to present our reduction.
The idea is to construct a metric space from a given function $f$, and show that any estimation of the metric with an approximation factor smaller than $3$ can be used to compute the parity of $f$. A metric space should satisfy three properties: identity, symmetry and triangle inequality. Keep in mind that our construction should be in such a way that the metric meets all of the mentioned properties. For a function $f:[n^2]\rightarrow\{-1,1\}$, we construct a metric $\mathcal{M} = \{a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n\}$ with $2n$ points. We divide the points into two groups, namely $a_i$'s and $b_i$'s, where the distances of the points within each group are all equal to $1$. Moreover, for every pair of points $(a_i,b_i)$, the distance of $a_i$ from $b_i$ is either $1/2$ or $3/2$, depending on function $f$. We show that, given an $\alpha < 3$ approximation estimation for the distances of $\mathcal{M}$, one can determine $\mathsf{par}(f)$ uniquely.
\begin{theorem}\label{hardness}
Any quantum algorithm that approximates the metric estimation problem with an approximation factor smaller than $3$ needs to make at least $\Omega(n^2)$ oracle calls.
\end{theorem}
\begin{proof}
As promised, we prove this theorem by reducing the parity problem to the metric estimation problem. Suppose we are given an instance $\mathsf{I}$ of the parity problem consisting of $f:[m] \rightarrow \{0,1\}$ and an oracle $\mathcal{O}$ to access $f$. We assume w.l.o.g that $m = n^2$ and construct an instance $\mathsf{Cor}(\mathsf{I})$ of metric estimation as follows:
let $\langle \mathcal{M}, d\rangle$ be a set of $2n$ points where the distance of the points $p_i$ and $p_j$ is denoted by $d(p_i,p_j)$. We divide the points of the metric into two groups $\{a_1,a_2,\ldots,a_n\}$ and $\{b_1,b_2,\ldots,b_n\}$. As mentioned before, the distances within the points of each group are equal to $1$. Moreover, for every pair of points $a_i$ and $b_j$, we set $d(a_i,b_j)$ as follows:
\begin{equation*}
d(a_i,b_j) =
\begin{cases}
3/2 & f((i-1)n+j) = 1, \\
1/2 & \text{otherwise.}
\end{cases}
\end{equation*}
The identity and symmetry conditions are met by definition. We show that the triangle inequality also holds. If all three points of a triangle are in the same group (either $a_i$'s or $b_i$s), then their distances are all $1$. If they are in different groups, the distances are one of these cases, $\langle1, 1/2, 1/2\rangle$, $\langle1, 1/2, 3/2\rangle$ or $\langle1, 3/2, 3/2\rangle$, all of which meet the triangle inequality. Thus, $\langle\metric, d\rangle{}$ is a valid metric space. One can trivially construct an oracle $\mathcal{Q}$ for $\mathsf{Cor}(\mathsf{I})$, that reports the distance of a pair of points with a single query to $\mathcal{O}$.
Now, suppose for the sake of contradiction that there exists a quantum algorithm that estimates the distances within a factor smaller than 3 with $o(n^2)$ query calls of $\mathcal{Q}$.
We show we can use this algorithm to find $\mathsf{par}(f)$ as follows. We first run the algorithm to approximate all of the distances via $o(n^2)$ query calls to $\mathcal{Q}$. This costs us a total of $o(n^2)$ queries to $\mathcal{O}$ since every query of $\mathcal{Q}$ makes a call to $\mathcal{O}$. Next, for every pair of points $(a_i,b_i)$ we determine $f((i-1)n+j)$ as follows:
\begin{equation*}
f((i-1)n+j) =
\begin{cases}
1 & d^*(a_i,b_j) \geq 3/2. \\
-1 & \text{otherwise}
\end{cases}
\end{equation*}
where $d^*(a_i,b_j)$ is the estimated distance of point $a_i$ from point $b_j$. The correctness of our reduction follows from the fact that the approximation factor of the algorithm for metric estimation is smaller than $3$ and thus if $d^*(a_i,b_j) \geq 3/2$ the actual distance $d(a_i,b_j)$ is more than $1/2$. Finally, we take the multiplication of all determined values for $f$ and compute $\mathsf{par}(f)$ with $o(n^2) = o(m)$ queries. This contradicts the observation of Farhi \textit{et al.}{}~\cite{Farhi:1998bz}.
\end{proof}
\subsection{A $3+\epsilon$ Approximation Algorithm with $\widetilde O(n^{5/3}\mathrm{poly}(1/\epsilon))$ Queries}\label{sub:alg1}
In this section, we present a quantum algorithm to estimate the distances of a metric space within an approximation factor of $3+\epsilon$. Our algorithm makes $\widetilde O(n^{5/3}\mathrm{poly}(1/\epsilon))$ oracle calls.
The first idea of our algorithm is to discretize the distances. Recall that, the distances of the metric are non-negative integers in the interval $[l,u]$. We separate the numbers into disjoint intervals. If $l = 0$, we put a separate interval $[0,0]$ for $0$ and continue on with the numbers in $[1,u]$. Every time, we find the smallest number $l \leq x \leq u$ which is not covered in the previous intervals and add a new interval $[x, (1+\epsilon)x]$ to the list. Since $u = \mathrm{poly}(n)$, the number of intervals is $\mathrm{poly}(\log n)\mathrm{poly}(1/\epsilon) = \widetilde O(\mathrm{poly}(1/\epsilon))$. Now, by losing a factor $1+\epsilon$ in the approximation, we can round up all of the numbers within an interval to its highest value and solve the problem for each interval separately. Therefore, the problem boils down to the following: given a threshold $t$, find all pairs of the points with a distance of at most $t$. We call this problem \textsf{threshold estimation}{}. Note that, since we wish to find a $3$ approximation solution for \textsf{threshold estimation}{}, a false positive is also allowed in the solution. More precisely, the solution should contain all pairs of points within a distance of at most $t$, but pairs within distances up to $3t$ are also allowed to be included.
In order to approximate \textsf{threshold estimation}{}, we subsequently make use of Grover's search algorithm~\cite{boyer1996tight}. Think of the metric as a graph $G$ where every point corresponds to a vertex of the graph and two vertices are adjacent if the distance of their corresponding points is at most $t$.
Let $0 < \tau < 1$ be a fixed parameter. We call a vertex $v$ of the graph \textit{low degree} if the number of edges incident to $v$ are bounded by $n^\tau$ and \textit{high degree} otherwise. Our algorithm deals with low degree vertices and high degree vertices differently. We set the value of $\tau$ after the analysis and show it gives us the best bound.
In our algorithm, we iterate over the vertices of the graph and find their neighbors one by one. To this end, fix a vertex $v_i$ and suppose we wish to find all of its neighbors. Due to Grover's search (Theorem \ref{grover}), we can list up to $n^\tau$ neighbors of $v_i$ with $\sqrt{n^\tau n} = n^{(1+\tau)/2}$ queries. Moreover, with an additional Grover's search, we can determine whether the degree of $v_i$ is more $n^\tau$ with $O(\sqrt{n})$ queries.
If $v_i$ is low degree, we already have all its neighbors, and thus we can report those edges and remove $v_i$ from the graph. Otherwise, the degree of $v_i$ is more than $n^\tau$. In this case, we make $O(n)$ oracle calls and find the distances of all other points from the corresponding point of $v_i$, namely $p_i$. Based on these distances, we construct two sets of vertices $N(v_i, t)$ and $N(v_i, 2t)$ where the former contains all vertices corresponding to points within a distance of at most $t$ of $p_i$ and the latter contains all of the vertices corresponding to points within a distance of at most $3t$ from $p_i$. We then proceed by reporting all the edges between $N(v_i, t)$ and $N(v_i, 2t)$ and removing $N(v_i, t)$ from the graph.
A pseudocode for this algorithm is shown in Algorithm \ref{alg:metric0}.
\begin{algorithm2e}
\KwData{
The number of points in the metric space $\mathcal{M} = \{p_1,p_2,\ldots,p_n\}$, oracle access to the distances between points, and a threshold $t$.}
\KwResult{A 0-1 matrix $A$ of size $n\times n$, where for each $d(p_i,p_j)\leq t$ we have $A_{i,j}=1$, and for each $A_{i,j}=1$ we have $d(p_i,p_j)\leq 3t$.}
Initialize a graph $G$ with $n$ vertices\;
\While{$V(G)$ is not empty}{
Select a vertex $v_i$ from $V(G)$\;
List up to $n^\tau$ neighbors of $v_i$ and find out whether $v_i$ is high degree or low degree\;
\If{$v_i$ is low degree}{
Update the matrix $A$ according to the edges of $v_i$\;
Remove $v_i$ from $V(G)$\;
}
\Else{
Find the distances of $p_i$ from all other points\;
Construct $N(v_i, t)$ and $N(v_i, 2t)$ based on the distances\;
For every $x \in N(v_i, t)$ and $y \in N(v_i, 2t)$, set $A_{x,y} = 1$\;
$V(G) \leftarrow V(G) \setminus N(v_i,t)$\;
}
}
Output $A$\;
\caption{\textsf{EstimateWithThreshold}($n, \mathcal{O}, t$)}
\label{alg:metric0}
\end{algorithm2e}
\begin{theorem}\label{thm:metric0}
For $\tau{} = 1/3$, Algorithm \ref{alg:metric0} approximates \textsf{threshold estimation}{} within a factor of $3$ with $O(n^{5/3})$ oracle calls. Moreover, the running time of Algorithm \ref{alg:metric0} is $O(n^2)$.
\end{theorem}
\begin{proof}
The correctness of our algorithm follows from the triangle inequality. We first show that for every pair of points $p_i$ and $p_j$ such that $d(p_i,p_j) \leq t$, $A_{i,j} = 1$ at the end of the algorithm. To this end, consider the first time that we remove either $v_i$ or $v_j$ from the vertices. This could happen in two ways: either one of $v_i$ or $v_j$ is removed from the graph as a low degree vertex or any of them is removed in an iteration of the algorithm for some high degree vertex. In the former case, since we find all neighbors of the low degree vertices, we detect the edge between them thus $A_{i,j} = 1$. Now, suppose that one of these vertices say $v_i$ is removed from the graph in an iteration for a vertex $v_x$ of the graph. Therefore, $d(v_i, v_x) \leq t$. Moreover, due to the triangle inequality, $d(v_j,v_x) \leq d(v_j,v_i) + d(v_i,v_x) \leq 2t$ and thus $v_j \in N(v_x,2t)$. Thus we set $A_{i,j} = 1$. Moreover, it follows from the triangle inequality that if we set $A_{i,j} = 1$ for some $i$ and $j$, then the distance of the points $p_i$ and $p_j$ is bounded by $3t$.
Trivially, the running time of the algorithm is $O(n^2)$. In what follows we show the query complexity of the algorithm is bounded by $O(n^{5/3})$. Let $Q(n)$ denote the query complexity of the algorithm for the case where $|V(G)| = n$. To compute $Q(n)$, we consider two cases separately: (i) when we select a vertex $v_i$ which is low-degree and (ii) when we select a vertex $v_i$ which is high degree. In any case, we make a search to list up to $n^\tau$ neighbors of $v_i$ and we make at least $O(n^{(1+\tau)/2})$ oracle calls. In addition to this, we make $O(\sqrt{n})$ more oracle calls to find out whether $v_i$ is low degree. In case $v_i$ is low degree, we remove $v_i$ from the graph and continue on with an instance with $n-1$ vertices. Otherwise, we make $O(n)$ more oracle calls and then remove $N(v_i,t)$ from the graph which leaves us an instance with at most $n-n^\tau$ vertices. Therefore, we formulate $Q(n)$ as follows:
\begin{equation*}
Q(n) =
\begin{cases}
O(n^{(1+\tau)/2}) + O(\sqrt{n})+ Q(n-1)
&\text{if }v_i\text{ is low degree,} \\
O(n^{(1+\tau)/2}) + O(\sqrt{n})+ O(n) + Q(n-n^\tau)
& \text{otherwise.}
\end{cases}
\end{equation*}
Now we set $\tau = 1/3$ and thus we obtain
\begin{equation*}
Q(n) =
\begin{cases}
O(n^{2/3}) + O(\sqrt{n}) + Q(n-1) = O(n^{2/3}) + Q(n-1) & \text{if }v_i\text{ is low degree,} \\
O(n^{2/3}) + O(\sqrt{n}) + O(n) + Q(n-n^{1/3}) = O(n) + Q(n-n^{1/3}) & \text{otherwise.}
\end{cases}
\end{equation*}
A trivial analysis shows that for every vertex that we remove from $V(G)$, we make $O(n^{2/3})$ amortized query calls and thus the total number of queries is bounded by $n\cdot O(n^{2/3}) = O(n^{5/3})$.
\end{proof}
Now, we are ready to present our $3+\epsilon$ approximation algorithm with query complexity $\widetilde O(n^{5/3}\mathrm{poly}(1/\epsilon))$.
For each $i$, using Algorithm \ref{alg:metric0}, we can find all distances in range $[0, l(1+\epsilon/3)^{i+1}]$ with some false positive distances in range $[l(1+\epsilon/3)^{i+1}, 3l(1+\epsilon/3)^{i+1}]$. By knowing the same information for $i-1$, we have all points in range $[0, l(1+\epsilon/3)^{i}]$ with some false positive distances in range $[l(1+\epsilon/3)^{i}, 3l(1+\epsilon/3)^{i}]$. Thus we can find all points in range $[l(1+\epsilon/3)^{i}], l(1+\epsilon/3)^{i+1}]$, some false positives in range $[l(1+\epsilon/3)^{i+1}, 3l(1+\epsilon/3)^{i+1}]$, and some false negatives that estimated correctly before. All of these distances are in range $[l(1+\epsilon/3)^{i}, 3l(1+\epsilon/3)^{i+1}]$. Therefore we can estimate these distances as $3l(1+\epsilon/3)^{i+1}$ and the approximation factor is $\frac{3l(1+\epsilon/3)^{i+1}}{l(1+\epsilon/3)^{i}} = 3(1+\epsilon/3) = 3+\epsilon$. The time and query complexity of this algorithm is the time and query complexity of Algorithm \ref{alg:metric0} times $\log_{1+\epsilon/3}(u/l) = \widetilde O(1/\epsilon)$. We handle zero distances separately. You can find the pseudocode of this algorithm in the following.
\begin{algorithm2e}
\KwData{The number of points in the metric space $\mathcal{M}=\{p_1,p_2,\ldots,p_n\}$, oracle access to the distances between points, a small number $\epsilon > 0$, a lower bound, and an upper bound for the distances.}
\KwResult{An $n\times n$ matrix $A$, where $A_{i,j}$ is a $3+\epsilon$ approximation of $d(p_i, p_j)$}
Initialize three matrices $A$, $A^\circ$ and $A^\bullet$\;
$A^\circ \gets \textsf{EstimateWithThreshold}(n, \mathcal{O}, 0)$\;
Initialize the threshold: $t \leftarrow \max(1,l)$\;
\While{$t \leq u$}{
$t\leftarrow t\cdot (1+\epsilon/3)$\;
$A^\bullet \gets $\textsf{EstimateWithThreshold}($n, \mathcal{O}, t$)\;
$A\gets A + (A^\bullet - A^\circ)\cdot 3t$\;
$A^\circ\gets A^\circ\vee A^\bullet$
}
output A
\caption{\textsf{EstimateMetric}($n, \mathcal{O}, \epsilon, l, u$)}
\label{alg:metric1}
\end{algorithm2e}
\begin{theorem}\label{thm:metric1}
Algorithm \ref{alg:metric1} solves metric estimation problem with approximation factor $3+\epsilon$, quantum query complexity $\widetilde O(n^{5/3})$ and time complexity of $\widetilde O(n^2)$ for an arbitrary small constant $\epsilon>0$.
\end{theorem}
\begin{proof}
The correctness of Algorithm \ref{alg:metric1} follows from that of Algorithm \ref{alg:metric0}. Moreover, Algorithm \ref{alg:metric1} runs Algorithm \ref{alg:metric0}, $\widetilde O(\mathrm{poly}(1/\epsilon))$ times; therefore, the query complexity of Algorithm \ref{alg:metric1} is $\widetilde O(n^{5/3+\epsilon}\mathrm{poly}(1/\epsilon))$. Furthermore, the running time of Algorithm \ref{alg:metric1} is $\widetilde O(n^2\mathrm{poly}(1/\epsilon))$.
\end{proof}
In this section, we achieved an algorithm with subquadratic query complexity and approximation factor $3+\epsilon$ for any $\epsilon>0$ which is nearly optimal due to Theorem \ref{hardness}. In Section \ref{metric15}, we reduce the quantum query complexity to $O(n^{3/2+\epsilon})$, but the approximation factor grows to larger constants.
\subsection{A Constant Approximation Algorithm with $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$ Queries}\label{sub:alg2}
\label{metric15}
In Sections \ref{sub:hardness} and \ref{sub:alg1}, we showed that the best approximation factor that we can get with subquadratic oracle calls are bounded from below by $3$ and that a $3+\epsilon$ approximation is possible. In this section, we complement this result by showing that the query complexity can be further reduced to $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$, and moreover, we show that the required query complexity is at least $\Omega(n^{3/2})$ for any constant approximation factor.
To this end, we present a quantum algorithm with expected query complexity $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$ where the approximation factor and the expected running time are $\mathsf{e_{m}}(\epsilon) = O(1/\epsilon)$ and $\widetilde O(n^2\mathrm{poly}(1/\epsilon))$, respectively.
As stated before, the problem reduces to \textsf{threshold estimation}. Similar to what we did for Theorem \ref{thm:metric1}, we divide the vertices into two categories low degree and high degree.
Low degree vertices are easy to deal with; we simply list all of their neighbors using Grover's search and report all of them. If a vertex is high degree though, the algorithm needs to be more intelligent.
The overall idea is summarized in the following: we find a small group of vertices, namely \textit{representatives}, that hits at least one vertex from the neighborhood of any large degree vertex.
Using a standard argument of hitting sets, we can show that a subset of $\widetilde O(n/\eta)$ vertices chosen uniformly at random, as \textit{representatives}, hits every neighborhood of size at least $\eta$ with high probability. Notice that these neighborhoods are at most $n$ fixed but unknown subsets. Other vertices outside \textit{representatives} are either low degree vertices, or \textit{followers} which have at least one neighbor in \textit{representatives}, or both.
Next, we run the following procedure: for every vertex $v_i$ which is not in \textit{representatives}, we first check if it is a follower. For a follower vertex which has at least one neighbor in \textit{representatives}, we select one such vertex and call that the leader of $v_i$. Otherwise, if there is no such neighbor, we conclude that $v_i$ is indeed low degree; thus we can find all its neighbors via Grover's search and update the solution. Next, we solve the problem recursively for all of the representatives.
For any $v_i$ and $v_j$ which are connected, we want the leader of $v_i$ and the leader of $v_j$ to become connected in the recursive result. As a consequence of the triangle inequality, we can achieve this by tripling the threshold.
Finally, we construct our solution based on the approximated solution of the representatives and the leader-follower relations, simply by connecting any two vertices, where their leaders are connected. The approximation factor increases with each recursion, but since the number of recursions is a constant, we achieve a constant approximation factor. Furthermore, in each recursion call, we can increase the degree threshold as far as it doesn't increase the query complexity too much. By increasing the degree threshold to its 3rd power, we have this property.
The number of vertices in nested recursions depleted, as soon as the degree threshold become larger that the number of vertices, in which case we treat all vertices as low degree, thus the next time we have zero vertices and the process finishes.
The pseudocode of the algorithm is shown below.
\begin{algorithm2e}
\KwData{The number of points in the metric space $\mathcal{M}=\{p_1,p_2,\ldots,p_n\}$, oracle access to the distances between points, a threshold $t$, a small number $\epsilon$, and a degree threshold $\degreethreshold{}$}
\KwResult{An $n\times n$ matrix $A$, where $A_{i,j}$ is an $\mathsf{e_{m}}(\epsilon)$ approximation of $d(p_i, p_j)$.}
\If {$n=0$} {
Output an empty matrix\;
} \Else {
Sample a hitting set $\mathcal{R}$ with $O((n/\degreethreshold{})\log n)$ points\;
Initialize an $n\times n$ matrix $A$\;
\For {all points in $\mathcal{M}$ as $v_i$} {
Find a neighbor of $v_i$ or $v_i$ itself in $\mathcal{R}$ and save it as $l(v_i)$ (the leader of $v_i$)\;
\If {no such neighbor of $v_i$ exists and $v_i$ is not in $\mathcal{R}$} {
List all neighbors of $v_i$\;
}
}
$A' \gets \mathsf{FastEstimateWithThreshold}(\mathcal{R}, \mathcal{O}, 3t, 3\epsilon, (\degreethreshold{})^3$)\;
\For {all pairs of points in $\mathcal{M}$ as $(v_i, v_j)$ where $l(v_i)\neq\varnothing$ and $l(v_j)\neq\varnothing$} {
\If {$A'(l(v_i), l(v_j)) = 1$} {
$A(v_i, v_j)\gets 1$\;
}
}
$A\gets A \vee A'$\;
Output $A$\;
}
\caption{\textsf{FastEstimateWithThreshold}($\mathcal{M}, \mathcal{O}, t, \epsilon, \degreethreshold{}$)}
\label{alg:metric150}
\end{algorithm2e}
\begin{theorem}
\label{thm:metric150}
Algorithm \ref{alg:metric150} called with the threshold $t$, the parameter $\epsilon$ and the degree threshold $n^{2\epsilon}$ finds all distances less than $t$ with some false positive distances in range $[t, \mathsf{e_{m}}(\epsilon)\cdot t]$ where $\mathsf{e_{m}}(\epsilon) = O(1/\epsilon)$, in expected query complexity $\widetilde O(n^{3/2+\epsilon})$ and expected time complexity $\widetilde O(n^2)$.
\end{theorem}
\begin{proof}
As aforementioned, we deal with three groups of vertices: \textit{representatives}, \textit{followers} and low degree vertices. Low degree vertices may intersect with the other two, but each vertex is at least in one group. Here, for any low degree vertex outside other two groups, we find its neighborhood explicitly. Therefore, to show the correctness of the algorithm, we focus on two groups of followers and representatives. First, we show that we correctly find the group of representatives. A subset $\mathcal{R}$ of size $2(n/\degreethreshold{})\ln n$ chosen uniformly at random, misses one fixed neighborhood of size at least $\degreethreshold{}$ with a probability of at most $(1-\frac{\degreethreshold{}}{n})^{2(n/\degreethreshold{})\ln n}\approx 1/\mathrm{e}^{2\ln n}=1/n^2$. For all neighborhoods, which are at most $n$ fixed subsets of size at least $\degreethreshold{}$, the probability of missing at least one neighboorhood is at most $n\cdot (1 /{n^2}) = 1/n$ by the union bound. If $\mathcal{R}$ misses at least one large neighborhood, we can reset the algorithm. A standard argument of Las Vegas algorithms ensures that the expected query complexity and expected running time is no more that $\frac{n}{n-1}$ times the query complexity and running time of one execution, respectively (Exercise 1.3 of \cite{motwani1995randomized}). Now we can continue assuming we have leader-follower relations.
Recall that for every follower $v_i$ we select one of its neighbors in $\mathcal{R}$ and call that vertex the leader of $v_i$. To simplify the last part of the algorithm, for any $v_i$ in $\mathcal{R}$, we call $v_i$ as the leader of itself. Thus, all followers and representatives have leaders.
Furthermore, we solve the problem for the group of representatives recursively, with different parameters. We triple the threshold in each recursion. Call the leader of two connected vertices $v_i$ and $v_j$ as $r_i$ and $r_j$, respectively. By the triangle inequality we have $d(r_i,r_j) \leq d(r_i,v_i)+d(v_i,v_j)+d(v_j,r_j) \leq 3t$. Thus the leader of any two connected vertices is connected by the new threshold; hence we find all distances less than $t$, perhaps with some false positives.
Before we compute the approximation factor $\mathsf{e_{m}}(\epsilon)$, we determine the number of nested recursion calls. We call the number of vertices in the $i$'th recursion call $n_i$. Note that $n_i$ is the size of representatives group of the $(i-1)$'th recursion. Thus, we have $n_i = O((n_{i-1}/\degreethreshold{i})\log n_{i-1})$. Using induction, we can show that the degree threshold in $i$'th recursion call is $\degreethreshold{i} = n^{(2\cdot 3^{i})\epsilon}$ and therefore, $n_i = O(n^{1-(3^{i}-1)\epsilon}\cdot O(\log^i(n)))$. The number of vertices becomes zero in $i$'th recursion, where $1-(3^{i}-1)\epsilon<0$ or $i > \log_3(1+1/\epsilon)$. Hence, we have at most $k(\epsilon) = \log_3(1/\epsilon)+1$ nested recursion calls, which is independent of $n$. Notice that $k(3\epsilon) = k(\epsilon) - 1$ and $k(3^{k(\epsilon)}\epsilon) = 0$.
What is remained is to compute the approximation factor $\mathsf{e_{m}}(\epsilon)$. The maximum distance of a pair of vertices that we report an edge between them is at most $2(1+3\mathsf{e_{m}}(3\epsilon))$ times the threshold. We know that $\mathsf{e_{m}}(3^{k(\epsilon)}\epsilon) = 1$, therefore $\mathsf{e_{m}}(\epsilon)\leq 9/\epsilon = O(1/\epsilon)$.
The query complexity of Grover's search in the $i$'th recursion is at most $n_i O(\sqrt{|\mathcal{R}|})$ to find the leader of each point, plus $n_i O(\sqrt{n_i\cdot \degreethreshold{i}})$ to find all neighbors of some low degree points. This is equal to $O(n^{3/2-(\frac{5\cdot 3^{i}-3}{2})\epsilon}\mathsf{polylog}(n)) + O(n^{3/2-(\frac{3^{i}-3}{2})\epsilon}\mathsf{polylog}(n))$. Notice that the latter term dominates the former, and the query complexity for $i=0$ dominates all of the recursions; therefore the query complexity of Algorithm \ref{alg:metric150} is at most $O(n^{3/2+\epsilon})$.
The time complexity is at most $O(n^2)$ in each phase. Thus, the time complexity is $O(n^2\mathsf{polylog}(n))$.
\end{proof}
In what follows, we complete our algorithm using Algorithm \ref{alg:metric150} with several thresholds. This is the same as Algorithm \ref{alg:metric1} with minor differences such as line 8 where $3$ has been replaced with $\mathsf{e_{m}}(\epsilon)$.
\begin{algorithm2e}
\KwData{The number of points in the metric space $\mathcal{M}$, oracle access to the distances between points, a small number $\epsilon>0$, a lower bound, and an upper bound for the distances}
\KwResult{An $n\times n$ matrix $A$, where $A_{i,j}$ is a $\mathsf{e_{m}}(\epsilon)$ approximation of $d(p_i, p_j)$ in $\langle\metric, d\rangle$}
Initialize the distance estimation matrix $A$, $A^\circ$ and $A^\bullet$\;
$A^\circ \gets$\textsf{FastEstimateWithThreshold}($n, \mathcal{O}, 0, \epsilon, n^{2\epsilon}$)\;
Initialize the threshold: $t \leftarrow \max(1,l)$\;
\While{$t \leq u$}{
$t\leftarrow t\cdot (1+\epsilon)$\;
$A^\bullet \gets $\textsf{FastEstimateWithThreshold}($n, \mathcal{O}, t, \epsilon, n^{2\epsilon}$)\;
$A\gets A + (A^\bullet - A^\circ)\cdot \mathsf{e_{m}}(\epsilon)$\;
$A^\circ\gets A^\circ\vee A^\bullet$\;
}
Output $A$\;
\caption{\textsf{FastEstimateMetric}($\mathcal{M}, \mathcal{O}, \epsilon, l, u$)}
\label{alg:metric151}
\end{algorithm2e}
\begin{theorem}
\label{thm:metric151}
Algorithm \ref{alg:metric151} solves the metric estimation with approximation factor $\mathsf{e_{m}}(\epsilon) = O(1/\epsilon)$, with query complexity $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$ in time $\widetilde O(n^2 \mathrm{poly}(1/\epsilon))$.
\end{theorem}
\begin{proof}
The correctness of Algorithm \ref{alg:metric151} follows from that of Algorithm \ref{alg:metric150}, the same as we did in Theorem \ref{thm:metric1}. Moreover, Algorithm \ref{alg:metric151} runs Algorithm \ref{alg:metric150}, $\widetilde O(\mathrm{poly}(1/\epsilon))$ times; therefore, the query complexity of Algorithm \ref{alg:metric1} is $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$. Furthermore, the running time of Algorithm \ref{alg:metric1} is $\widetilde O(n^2 \mathrm{poly}(1/\epsilon))$.
\end{proof}
\subsection{An $\Omega(n^{3/2})$ Time Lower Bound}
Last but not least, we show that the query complexity of metric estimation cannot be reduced any further, so long as the approximation factor is constant, i.e., we need at least $\Omega(n^{3/2})$ queries to approximate metric estimation within a constant factor. We use Ambainis's lower bound technique \cite{Ambainis:2000:QLB:335305.335394}.
\begin{theorem}[proven in~\cite{Ambainis:2000:QLB:335305.335394}, Theorem 6]\label{amba}
Let $f(x_1, \ldots, x_n)$ be a function of $n$ variables
with values from some finite set
and $X, Y$ be two sets of inputs such that $f(x)\neq f(y)$
if $x\in X$ and $y\in Y$.
Let $R\subset X \times Y$ be such that
\begin{enumerate}
\item
For every $x\in X$, there exist at least $m$ different $y\in Y$ such that
$(x, y)\in R$.
\item
For every $y\in Y$, there exist at least $m'$ different $x\in X$ such that
$(x, y)\in R$.
\end{enumerate}
Let $l_{x, i}$ be the number of $y\in Y$ such that $(x, y)\in R$ and $x_i\neq y_i$
and $l_{y, i}$ be the number of $x\in X$ such that $(x, y)\in R$ and $x_i\neq y_i$.
Let $l_{max}$ be the maximum of $l_{x, i}l_{y, i}$ over all $(x, y)\in R$
and $i\in\{1, \ldots, N\}$ such that $x_i\neq y_i$.
Then, any quantum algorithm computing $f$ uses
$\Omega(\sqrt{\frac{m m'}{l_{max}}})$ queries.
\end{theorem}
Now we use an intermediate problem to prove the desired lower bound. A permutation matrix is a boolean $n\times n$ matrix, which has exactly one entry $1$ in each row and each column. It corresponds to a permutation $\pi$ where entries of 1 are in the form of $(i, \pi(i))$. The sign of a permutation matrix is defined as the sign of its corresponding permutation. The next lemma about the problem of determining the sign of a permutation matrix is the main part of out lower bound.
\begin{lemma}
\label{perm}
Any quantum algorithm which takes an $n\times n$ permutation matrix as the input and outputs the sign of the permutation matrix has a query complexity of at least $\Omega(n^{3/2})$.
\end{lemma}
\begin{proof}
To apply Theorem \ref{amba}, we use a single index to address an entity instead of two indices. Assume $f(x_1, x_2, . . ., x_{n^2})$ is a function which takes a permutation matrix as input and outputs a value in $\{-1, 1\}$ as the sign of the matrix. Define $X$ as the set of permutation matrices with sign $-1$, $Y$ as the set of permutation matrices with sign $1$ and $R\subset X\times Y$ such that $(x,y)\in R$ iff their corresponding matrices can be transformed to the other with a swap of just two rows. Therefore, we have $m=m'=\binom n 2$. For an $i$ we have $l_{x,i}=n-1$ and $l_{y,i}=1$ if $x_i=1$ and $l_{x,i}=1$ and $l_{y,i}=n-1$ if $x_i=0$, thus $l_{max} = n-1$. Therefore by Theorem \ref{amba}, every quantum algorithm to solve this problem has a query complexity of at least $\Omega\Big(\sqrt{\frac{{\binom n 2}^2}{n-1}}\Big) = \Omega(n^{3/2}).$
\end{proof}
The problem of determining the sign of an $n\times n$ permutation matrix can be easily reduced to our problem, by constructing a bipartite graph with parts $X$ and $Y$, $n$ vertices in each part and $n$ edges that form a complete matching between $X$ and $Y$. Every matching has a corresponding permutations and vice versa. Therefore, we have the following theorem.
\begin{theorem}
\label{thm:lowerquery}
Any quantum algorithm which estimates distances of a metric space of $n$ points with a constant approximation factor has a query complexity of at least $\Omega(n^{3/2})$.
\end{theorem}
\begin{proof}
We simply reduce the problem of determining the sign of a permutation matrix to this problem. Assume $n$ is an even number. For an instance of a $n/2\times n/2$ permutation matrix $\mathcal{A}$, we construct a metric space $\mathcal{M}$ with $n$ points, $r_i$ for row $i$ and $c_j$ for column $j$ of the matrix. Make the distance between $r_i$ and $c_j$ equal to $1$ where $\mathcal{A}_{i,j}=1$ and a distance of $n^2$ otherwise. The distances meet the necessary conditions. Notice that we do not construct the distances, we construct an oracle which invokes the oracle of $\mathcal{A}$ at most one time. Using Lemma \ref{perm}, the query complexity is at least $\Omega((n/2)^{3/2})=\Omega(n^{3/2})$.
\end{proof}
\section{Conclusion and Open Problems}
In the quantum algorithm of Section \ref{editdistance}, we have a fixed length for windows, $l=n^{1/7}$. By allowing windows of different sizes, we can improve the approximation factor to $3+\epsilon$.
Rubinstein, Schramm, and Song independently improved this factor to $3+\epsilon$~\cite{lcdquantum}.
Moreover, by redefining the metric estimation problem in a way that we only output a diagonal band of the distance matrix, meaning the main diagonal and zero or more diagonals on either side, we can improve the running time of our main algorithm to $\widetilde O(n^{38/21}\mathrm{poly}(1/\epsilon)) = \widetilde O(n^{1.810})$.
The same technique improves the running time of our bootstrapping algorithm of Section \ref{bootstraping} to $O(n^{1.708})$.
Indeed the most important open problem concerning edit distance is whether a subquadratic time algorithm can approximate the edit distance of two strings within a constant factor?
In this regard, our paper proposes the following approach. Suppose we want to approximate the pairwise edit distance between $m$ given strings, each of size $n$, within a constant factor. We call this problem \textit{pairwise edit distance}. A na\"{i}ve solution for pairwise edit distance has running time $O(m^2n^2)$. Obviously, any subquadratic time algorithm for approximating edit distance within a constant factor improves upon this running time.
In this paper, we show that an improvement in the running time of pairwise edit distance also leads to a subquadratic time algorithm for approximating edit distance within a constant factor.
We believe this actually opens a new direction for approximating edit distance for classic computers.
In addition to this, our work gives rise to a number of questions that we believe are important to study in future work.
\begin{itemize}
\item \textit{How efficiently can we approximate metric estimation in classic computers with a subquadratic number of queries, when the distance function is edit distance?}
\item \textit{Is there subquadratic quantum algorithms that approximate other similarity measures, \textsf{LCS} in particular?}
\item \textit{Can a quantum algorithm approximate the edit distance of two strings within a constant factor in near-linear time?}
\item \textit{Is it possible to show a non-trivial lower bound on the quantum computational complexity of computing edit distance?}
\end{itemize}
\section{Other Similarity Measures} \label{section:otherMeasures}
Edit distance is one of many similarity measures for comparing two strings. Furthermore, it is one of many problems with a simple two-dimensional DP solution. Other measures and similar problems include longest common subsequence (\textsf{lcs}), Fr\'{e}chet distance (\textsf{fre}) and dynamic time warping (\textsf{dtw}). While the $O(n^2)$ solution for these problems are very analogous, unfortunately, our approach does not directly apply to them. In the following, we discuss some reasons behind this difficulty. The update rule of these measures are defined as follows:
\begin{align*}
\mathsf{edit}(i,j)=\,&min\big\{\mathsf{edit}(i-1,j)+1,\mathsf{edit}(i,j-1)+1,
\mathsf{edit}(i-1,j-1)+(s_1[i]\neq s_2[j])\big\}\\
\mathsf{lcs}(i,j)=\,&max\big\{\mathsf{lcs}(i-1,j),\mathsf{lcs}(i,j-1),
\mathsf{lcs}(i-1,j-1)+(s_1[i]\neq s_2[j])\big\}\\
\mathsf{dtw}(i,j)=\,&min\big\{\mathsf{dtw}(i-1,j), \mathsf{dtw}(i,j-1),
\mathsf{dtw}(i-1,j-1)\big\}+dis(i,j)\\
\mathsf{fre}(i,j)=\,&max\big\{min\{\mathsf{fre}(i-1,j),\mathsf{fre}(i,j-1),
\mathsf{fre}(i-1,j-1)\}, dis(i,j)\big\}
\end{align*}
Our framework for approximating edit distance is based on two assumptions. First, the usability of Lemma \ref{compatibility}, which states that there is a window-compatible solution which is a good approximation to the optimal solution. Second, to use the metric estimation, the desired measure should be a distance function, namely a metric.
Two similarity measures \textsf{dtw} and \textsf{lcs} are not metric, moreover they cannot be approximated by any metric. For example, for \textsf{dtw} consider $s_1=a^{2k+1}$, $s_2=a^kba^k$ and $s_3=ab^{2k-1}a$. We have $\mathsf{dtw}(s_1, s_2)=1$ and $\mathsf{dtw}(s_2, s_3)=0$, but $\mathsf{dtw}(s_1, s_3)=2k-1$. Therefore the triangle inequality does not hold here.
The similarity measure \textsf{lcs} is in fact, the opposite of a metric function, i.e., for two similar strings, their \textsf{lcs} is large, and for two different strings, their \textsf{lcs} is small. The first property of a distance function does not hold here, for a non-empty string s, $\mathsf{lcs}(s,s)\neq 0$.
The other part of our approach where \textsf{lcs} has a drawback is the Lemma \ref{compatibility}. For a window size $l$, one can consider $s_1 = (ab^{l-1}a^{l-1})^t$ and $s_2=(a^lc^{l-1})t$. We have $\mathsf{lcs}(s_1, s_2)=lt$, but $\mathsf{lcs}$ of a windows-compatible transformation is at most $t$.
Likewise, approximating $\mathsf{lcs}$ in classic computers is also harder that $\mathsf{edit}$. None of the results for approximating $\mathsf{edit}$ is shown for $\mathsf{lcs}$, unless when $lcs(s_1, s_2)=\Omega(n)$.
Another way around this is to approximate \textsf{co-lcs} instead of $\mathsf{lcs}$, where \textsf{co-lcs}$(s_1, s_2)=|s_1|+|s_2|-\mathsf{lcs}(s_1, s_2)$. This measure is very similar to edit distance but without the substitution operation. Using our framework, we can approximate \textsf{co-lcs} with the same approximation factor of $7+\epsilon$ in quantum computers and an approximation factor of $3+\epsilon$ in MapReduce.
Fr\'{e}chet distance is rather a similarity measure for curves instead of strings. For strings, the problem becomes trivial, i.e., zero for same strings and one for different strings. However, \textsf{fre} on curves has a similar dynamic programming solution to \textsf{edit}. This similarity in solution leads us to consider this problem, too.
If we study the problem regardless of its geometric properties, i.e. all distances are given as a matrix, we can prove that approximating \textsf{fre} is as hard as computing its exact value.
\begin{theorem}
\label{thm:fre}
If there exists a quantum (or MapReduce) approximation algorithm for Fr\'{e}chet distance with a constant approximation factor in time $O(n^{2-\epsilon})$, which takes distances as a matrix in the input, there also exists a quantum or MapReduce algorithm which computes the exact Fr\'{e}chet distance in time $O(n^{2-\epsilon})$.
\end{theorem}
\begin{proof}
The idea is a gap producing reduction from the problem to itself. We can do a binary search on the actual value of Fr\'{e}chet distance, thus in each step for a threshold $t$, we want to know whether $\mathsf{fre}(a,b)\leq t$ or not. We define a new distance function as:
\begin{equation*}
dis'(a,b) =
\begin{cases}
1 & dis(a,b)\leq t, \\
n^2 & \text{otherwise.}
\end{cases}
\end{equation*}
If $\mathsf{fre}(a,b)\leq t$ then we have $\mathsf{fre}'(a,b)=1$ and $\mathsf{fre}'(a,b)=n^2$ otherwise. Therefore, if we solve the new instance with an approximation algorithm with a constant factor, we can decide whether $\mathsf{fre}'(a,b)=1$ or not. Thus we can decide whether $\mathsf{fre}(a,b)\leq t$ or not. Hence, we can find the exact value of Fr\'{e}chet distance by executing the approximation algorithm in $O(\log n)$ round. This argument works for both quantum algorithms and MapReduce algorithms.
\end{proof}
Theorem \ref{thm:fre} does not rule out the possibility of a subquadratic quantum algorithm or MapReduce algorithm for Fr\'{e}chet distance, but it states that relaxing the problem in this way does not make the problem easier.
\section{Our Results and Techniques} \label{section:ourResults}
In this section, we explain the ideas and techniques of our framework and show how we obtain a subquadratic algorithm for approximating the edit distance on quantum computers. The basis of our MapReduce algorithm is similar to what we explain here, though some details are modified to run the algorithm in a logarithmic number of MapReduce rounds. More details about the MapReduce algorithm can be found in Section \ref{mapreduce}. Our quantum algorithm is based on several known techniques of quantum computing, algorithm design, and approximation algorithms. On the quantum side, we take advantage of \textit{Grover's search}~\cite{grover1996fast} and \textit{amplitude amplification}~\cite{brassard2002quantum} to improve the lookup time on an unordered set. On the algorithmic side, we benefit from classic algorithmic tools such as dynamic programming techniques, divide and conquer, and randomized techniques. In addition to this, we leverage \textit{the bootstrapping technique} to further improve the running time of our algorithm, by allowing the approximation guarantee to grow to larger constant numbers.
Recall that, the edit distance between two strings is defined as the smallest number of insertions, deletions, and substitutions, that one needs to perform on one of the strings to obtain the other one. For two strings $s_1$ and $s_2$, we denote their edit distance by $\mathsf{edit}(s_1,s_2)$. By definition, edit distance meets all of
the \textit{identity of indiscernibles\footnote{$\mathsf{edit}(s_1,s_2) = 0 \Leftrightarrow s_1 = s_2$.}}, \textit{symmetry\footnote{$\mathsf{edit}(s_1,s_2) = \mathsf{edit}(s_2,s_1)$.}}, and \textit{triangle inequality\footnote{$\mathsf{edit}(s_1,s_2) + \mathsf{edit}(s_2,s_3) \geq \mathsf{edit}(s_1,s_3)$.}} properties, thus for any set of strings $\mathcal{M}$, $\langle\mathcal{M},\mathsf{edit}\rangle$ forms a metric space\footnote{A set of points $\mathcal{M}$ and a distance function $d$ form a metric space $\langle \mathcal{M}, d\rangle$, if $d$ meets all of the aforementioned properties.}. Following this intuition, our algorithm is closely related to the study of the metric spaces.
In the following, we outline our algorithm in three steps. First, we define an auxiliary problem, namely \textit{metric estimation} and present efficient approximation algorithms for this problem accompanied by tight bounds on its quantum complexity. Roughly speaking, in this problem, we are given a metric space with $n$ points and oracle access to the distances, and the goal is to output an $n \times n$ matrix which is an estimate to the distances between the points. One may think of the oracle as an ordinary computer program, that we then convert to the corresponding quantum code and unitary operator using a quantum compiler \cite{Farhi:1998bz}. We give two approximation algorithms that solve the metric estimation problem with approximation factors $3+\epsilon$ and $\mathsf{e_{m}}(\epsilon) = O(1/\epsilon)$ with $\widetilde O(n^{5/3}\mathrm{poly}(1/\epsilon))$ and $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$ oracle queries, respectively. Notice that the running times of the algorithms are $O(n^2\mathrm{poly}(1/\epsilon))$, but the query complexities are subquadratic. This allows us to approximate metrics spaces with sublinear points for which answering an oracle query is time-consuming. We emphasize that our metric estimation results are general and can be used for any metric. In the second step, we show that any algorithm that solves the metric estimation problem within an approximation factor $\alpha$ can be used as a black box to obtain a $1+2\alpha+\epsilon$ approximation solution for edit distance. As we show, the reduction takes a subquadratic time and thus using our $3+\epsilon$ approximation algorithm for metric estimation, we obtain a $7+\epsilon$ approximation algorithm for edit distance. Finally, in Section \ref{bootstraping} we devise a bootstrapping technique to further improve the running time of the algorithm by taking a hit on the approximation guarantee. In what follows, we explain each of the steps in more details. Before we delve into the algorithm, we would like to note some comments.
\begin{itemize}
\item The only step of the algorithm where quantum computation plays a role is the first step where we discuss metric estimation. Nevertheless, everywhere we use the term algorithm, we mean a quantum algorithm unless otherwise is stated.
\item In this section, we explain the abstract ideas and steps of the algorithm. Therefore, sometimes we do not provide formal proofs for some of the arguments that we make.
The reader can find a detailed discussion of all statements and proofs in Sections \ref{metric} and \ref{editdistance}.
\item Everywhere we use the word \textit{operation}, we refer to insertion, deletion, or substitution.
\end{itemize}
\subsection{Metric Estimation}\label{contribution:metric}
As mentioned earlier, in the metric estimation problem, we are given a metric space $\langle \mathcal{M}, d\rangle$ and an oracle $\mathcal{O}$ that reports $d(x,y)$ for two points $x$ and $y$ in an invocation. The goal of the problem is to estimate the distance matrix of the points with as few oracle calls as possible. Due to the impossibility results for exact or even solutions with small approximation factors for this problem (see the rest for more details), our aim is to find an approximation solution.
\begin{center}
\noindent\framebox{\begin{minipage}{6.3in}
\textsf{Metric Estimation}\\[0.25cm]
\textsf{Input}: a metric space $\langle \mathcal{M},d \rangle$ with $n$ points where $\mathcal{M} = \{p_1, p_2, \ldots, p_n\}$ and an oracle function $\mathcal{O}$ to access the distances. \\[0.25cm]
\textsf{Guarantee}: all the distances are integer numbers in the interval $[l,u]$. We assume $u$ is $O(\mathrm{poly}(n))$. \\[0.25cm]
\textsf{An output (with approximation factor $\alpha > 1$)}: an $n \times n$ matrix $A$, where $d(p_i,p_j) \leq A[i][j] \leq \alpha d(p_i,p_j)$ holds for every $1 \leq i,j \leq n$.
\end{minipage}}
\end{center}
Before we state the main ideas and results, we briefly explain two key tools that we borrow from previous work and use as black boxes in our algorithms. The first tool is the seminal work of Grover~\cite{grover1996fast} for making fast searches in an unordered database. Suppose we are given a function $f:[n] \rightarrow \{0,1\}$, where $[n] = \{1,2,3,\ldots,n\}$, and we wish to list up to $m$ distinct indices for which the value of the function is equal to $1$. We refer to this problem as \textit{element listing}.
\begin{center}
\noindent\framebox{\begin{minipage}{6.3in}
\textsf{Element Listing}\\[0.25cm]
\textsf{Input}: integers $n$ and $0 \leq m \leq n$, and access to an oracle that upon receiving an integer $i$, reports the value of $f(i)$. $f$ is defined over $[n]$ and maps each index to either $0$ or $1$. \\[0.25cm]
\textsf{Output}: a list of up to $m$ indices for which the value of $f$ is equal to $1$. If the total number of such indices is not more than $m$, the output should contain all of them.
\end{minipage}}
\end{center}
The pioneering work of Grover~\cite{grover1996fast} implies that the element listing problem can be solved with only $O(\sqrt{nm})$ oracle calls via quantum computation. We subsequently make use of this algorithm in this section.
\begin{theorem}[proven in~\cite{boyer1996tight}]\label{grover}
The listing problem can be solved with $O(\sqrt{nm})$ oracle queries via quantum computation.
\end{theorem}
The second quantum technique that we use in this paper is a tool for
proving lower bounds on the quantum complexity of the problems. Let $f:[n] \rightarrow \{-1,1\}$ be a function defined over the numbers $1,2,\ldots,n$
that maps each index to either $-1$ or $1$ and $\mathsf{par}(f) = \prod_{i \in [n]}f(i)$.
In the parity problem, we are given oracle access to $f$ and the goal is to determine $\mathsf{par}(f)$ with as few oracle calls as possible.
\begin{center}
\noindent\framebox{\begin{minipage}{6.3in}
\textsf{Parity}\\[0.25cm]
\textsf{Input}: an integer $n$, and access to an oracle $\mathcal{O}$ that upon receiving an integer $i$ reports the value of $f(i)$. $f$ is defined over $[n]$ and maps each index to either $-1$ or $1$. \\[0.25cm]
\textsf{Output}: $\mathsf{par}(f) = \prod_{i \in [n]}f(i)$.
\end{minipage}}
\end{center}
Of course, if the numbers of $-1$'s or $1$'s are substantially smaller than $n$ ($o(n)$), one can use Grover's search to list all of such indices and compute the parity with fewer than $\Omega(n)$ oracle calls. However, if this is not the case for either $-1$ or $1$, such an approach fails. The seminal work of Farhi \textit{et al.}~\cite{Farhi:1998bz}, showed that at least $\Omega(n)$ queries are necessary for solving the parity problem and thus quantum computation offers no speedup in this case.
\begin{theorem}[proven in~\cite{Farhi:1998bz}]
The parity problem cannot be solved with fewer than $\Omega(n)$ queries with quantum computation.
\end{theorem}
Based on the result of Farhi \textit{et al.}~\cite{Farhi:1998bz}, we begin with showing an impossibility result. Our first result for metric estimation is a hardness of approximation for factors smaller than $3$ using a subquadratic number of queries. More precisely, in Section \ref{metric}, we show that any quantum algorithm that approximates metric estimation within a factor smaller than $3$, needs to make at least $\Omega(n^2)$ oracle queries.
\vspace{0.2cm}
{\noindent \textbf{Theorem} \ref{hardness} [restated]. \textit{Any quantum algorithm for solving the metric estimation problem with an approximation factor smaller than 3 needs to make at least $\Omega(n^2)$ oracle calls.\\}}
The idea is to show a reduction from parity to metric estimation. Suppose we are given an instance $\mathsf{I}$ of the parity problem. Roughly speaking, we construct an instance $\mathsf{Cor}(\mathsf{I})$ of the metric estimation and prove that $\mathsf{Cor}(\mathsf{I})$ has a valid metric as input. Next, we show that any algorithm that approximates metric estimation within a factor smaller than $3$ with $o(n^2)$ queries can be turned into a quantum algorithm for solving parity with $o(n)$ queries which is impossible due to Farhi \textit{et al.}~\cite{Farhi:1998bz}.
Despite this hardness of approximation for factors better than $3$, we show the problem is significantly more tractable when we allow the approximation guarantee to be slightly more than $3$. In Section \ref{metric}, we show that for any $\epsilon > 0$, a $3+\epsilon$ approximation of metric estimation is possible via $\widetilde O(n^{5/3}\mathrm{poly}(1/\epsilon))$ queries.
\vspace{0.2cm}
{\noindent \textbf{Theorem} \ref{thm:metric1} [restated]. \textit{For any $\epsilon > 0$, there exists a quantum algorithm that solves metric estimation with $\widetilde O(n^{5/3}\mathrm{poly}(1/\epsilon))$ queries within an approximation factor of $3+\epsilon$. Moreover, the running time of the algorithm is $\widetilde O(n^2\mathrm{poly}(1/\epsilon))$.\\}}
Our first take on the solution is to discretize the problem at the expense of imposing an additional $1 + \epsilon$ factor to our guarantee. Notice that all of the distances of the metric lie in the interval $[l,u]$. Therefore, one can divide the distances into $\log_{1+\epsilon/3} (u/l) = \widetilde O(\mathrm{poly}(1/\epsilon))$ disjoint intervals where the distances within each interval differ in at most a multiplicative factor of $1+\epsilon/3$. For every interval $[x,(1+\epsilon/3)x]$ we can set a threshold $t = (1+\epsilon/3)x$ and find all pairs within a distance of at most $t$ with an approximation factor of $3$. Then, based on all these solutions, one can find a $3+\epsilon$ approximation distance for every pair of the points.
Now the problem boils down to the following: given a threshold $t$, find all pairs $(p_i,p_j)$ such that $d(p_i,p_j) \leq t$. Of course, an exact solution for this problem is hopeless due to our impossibility result. Therefore we allow some false positive in our solution as well. More precisely, we restrict our solution to contain all pairs $(p_i,p_j)$ such that $(p_i,p_j) \leq d$, but additional pairs are also allowed to appear, if $(p_i,p_j) \leq 3d$. It is easy to show that any solution that solves the above problem via $\widetilde O(n^{5/3}\mathrm{poly}(1/\epsilon))$ queries, yields a $3+\epsilon$ approximation factor algorithm for metric estimation that uses at most $\widetilde O(n^{5/3}\mathrm{poly}(1/\epsilon))$ oracle calls.
In what follows, we describe the ideas to solve the problem for a fixed threshold $t$. The algorithm is explained in details in Section \ref{metric}, therefore, here, we just mention the tools and techniques. For convenience, we construct a graph $G$ with $n$ nodes, and correspond every point $p_i$ of the metric to a vertex $v_i$ of the graph. For a pair of points $(p_i,p_j)$, we add an undirected edge $(v_i,v_j)$ to the graph, if $d(p_i,p_j) \leq t$. Notice that the oracle function $\mathcal{O}$, provides us the exact value of $d(p_i,p_j)$ for any $p_i$ and $p_j$, therefore we can examine whether an edge exists between two vertices $v_i, v_j$ with a single oracle call. Recall that, Grover's search allows us to find as many as $m$ elements with value $1$ of a function of size $n$ via $O(\sqrt{nm})$ oracle calls. Therefore, if the number of the edges of the graph is $O(n^{4/3})$, we can use Grover's search (Theorem \ref{grover}) to list all of the edges with $O(\sqrt{n^2\cdot n^{4/3}}) = O( n^{5/3})$ queries and solve the problem. Therefore, the non-trivial part of the problem is the case where the graph is dense. In this case, the average degree of the vertices is at least $\Omega(n^{1/3})$. Now, suppose we select a vertex $v_i$ whose degree is at least $n^{1/3}$, and with $n-1$ query calls, find the distances of its corresponding point $p_i$ from all other points of the metric. Let set $D^t$, be the set of all points that have a distance of at most $t$ from $p_i$ and $D^{2t}$ be the of points with a distance of at most $2t$ from $p_i$. Trivially, $D^t \subseteq D^{2t}$. Due to the triangle inequality, all of the edges incident to the vertices corresponding to set $D^t$ are from the vertices corresponding to $D^{2t}$. Moreover, the distances of all points of $D^t$ from points of $D^{2t}$ are bounded by $3t$. Therefore, one can report all such pairs in the solution and proceed by removing $D^t$ from the graph (however, some vertices of $D^{2t}$ remain in the graph). Thus, all that remains is to solve the problem for an instance with at most $n-n^{1/3}$ nodes recursively. Since we make at most $O(n)$ query calls for every $n^{1/3}$ vertices (an amortized of $n^{2/3}$ per vertex), the total number of queries is $O(n^{5/3})$. More details about this can be found in Section \ref{metric}.
In addition to Theorem \ref{thm:metric1}, we show in Section \ref{metric} that with a deeper analysis, one can use the same ideas to further improve the query complexity to $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$ by allowing the approximation guarantee to grow up to $\mathsf{e_{m}}(\epsilon) = O(1/\epsilon)$.
\vspace{0.2cm}
{\noindent \textbf{Theorem} \ref{thm:metric151} [restated]. \textit{For any $\epsilon > 0$, there exists a quantum algorithm that solves metric estimation with $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$ queries within an approximation factor of $\mathsf{e_{m}}(\epsilon) = O(1/\epsilon)$. Moreover, the running time of the algorithm is $\widetilde O(n^2\mathrm{poly}(1/\epsilon))$.\\}}
You can find a summary of the results explained in this section in Table \ref{table}.
\begin{table}[t]\centering\small
\caption{Quality of the approximation algorithms for metric estimation}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
\TBstrut
\begin{tabular}{@{}l@{}}Approx. \\ factor\end{tabular} & $\alpha < 3$ & $\alpha = 3+\epsilon$& $\alpha = \mathsf{e_{m}}(\epsilon)$ & $\alpha= $ any constant\\
\hline
\Tstrut
Number of& $\Omega(n^2)$ & $\widetilde O(n^{5/3}\mathrm{poly}(1/\epsilon))$& $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$ & $\Omega(n^{3/2})$ \\
\Bstrut
queries & (Theorem \ref{hardness}) & (Theorem \ref{thm:metric1}) & (Theorem \ref{thm:metric151}) & (Theorem \ref{thm:lowerquery})\\
\hline
\end{tabular}
\label{table}
\end{table}
\subsection{Approximating Edit Distance within a Factor $7+\epsilon$}\label{ghabli}
In the second step, we provide an algorithm to approximate the edit distance between two strings in subquadratic time, based on a reduction to metric estimation. Our approach here is twofold. Suppose we are given a \textit{guess} $d$, on the actual edit distance between the strings, and we want to find an approximation proof to the guess. More precisely, we wish to find out whether $d$ is smaller than the actual distance of the strings, or report a transformation of the strings with at most $\alpha d$ operations\footnote{insertion, deletion, or substitution} where $\alpha$ is given as an approximation factor. If $d$ is substantially smaller than $n$, then the $O(n + d^2)$ exact algorithm of Landau \textit{et al.}~\cite{landau1998incremental} solves the problem in subquadratic time. Therefore, the only hard instances of the problem are when $d$ is asymptotically close to $n$. Therefore, we define a subtask of the edit distance problem, in which we are given two strings $s_1$ and $s_2$ and guaranteed that the edit distance between the strings is at most $\delta (|s_1|+|s_2|)$ where $\delta$ is not too small. The goal is to find a transformation of the strings with at most $(\delta\cdot \alpha) (|s_1|+|s_2|)$ operations, where $\alpha$ is the approximation factor of the algorithm. We refer to this subtask of edit distance as \textit{the $\delta$-bounded edit distance} problem.
\begin{center}
\noindent\framebox{\begin{minipage}{6.3in}
\textsf{$\delta$-bounded edit distance}\\[0.25cm]
\textsf{Input}: two strings $s_1$ and $s_2$, and a real number $0 \leq \delta \leq 1$. \\[0.25cm]
\textsf{Guarantee}: $\mathsf{edit}(s_1,s_2) \leq \delta(|s_1|+|s_2|)$. \\[0.25cm]
\textsf{Output (with an approximation factor $\alpha>1$)}: a sequence of operations with size at most $(\delta\cdot \alpha) (|s_1|+|s_2|)$ that transforms $s_1$ into $s_2$.
\end{minipage}}
\end{center}
We combine a divide and conquer technique with dynamic programming in order to approximate $\delta$-bounded edit distance. In addition to this, we subsequently make use of the quantum techniques mentioned earlier in our solution.
Recall that the total number of characters in the input is equal to $n$, i.e., $|s_1|+|s_2| = n$. For clarity, we define two parameters $0 < \beta < 1$ and $\gamma > 1$. $\gamma$ is an integer number but $\beta$ is a real number between 0 and 1. We use $\beta$ and $\gamma$ as two parameters of our algorithm, and after the analysis, we show which values for $\beta$ and $\gamma$ give us the best guarantee.
We begin by defining the notion of a \textit{window} and construct a set of windows for each string. Let $l = \lfloor n^{1-\beta}\rfloor$ be the \textit{window size} and define a window of $s_1$, as a string of length $l$ over the characters of $s_1$. Moreover, define $g = \lfloor l/\gamma\rfloor = O(n^{1-\beta}/\gamma)$ as \textit{the gap size} and construct a collection $W_1$ of windows for $s_1$ as follows: for every $0 \leq i \leq \lfloor \frac{|s_1|-l}{g} \rfloor$, put a window $[ig+1,ig+l]$ (i.e., a window from index $ig+1$ to index $ig+l$ of $s_1$) in $W_1$. In other words, $W_1$ contains tentatively $\gamma(|s_1|/l) = O(\gamma n^\beta)$ windows of length $l$ where the gap between the neighboring windows is equal to $g$. Figure \ref{figs:windows} illustrates how the windows of $W_1$ span over the characters of $s_1$. Notice that some of the windows overlap.
\input{figs/windows.tex}
Similar to this, we construct a collection $W_2$ of windows for $s_2$, using the same parameters $l$ and $g$. We define a \textit{transformation} of $s_1$ into $s_2$, as a sequence of insertions, deletions, and substitutions that turns $s_1$ into $s_2$. After a transformation of $s_1$ into $s_2$, we call a character of $s_2$ \textit{old} if it is either substituted by a character of $s_1$, or remained intact during the transformation. In other words, if a character is not inserted during a transformation, it is called old. Based on this, we define the notion of \textit{a window-compatible transformation} as follows:
\begin{Definition}
Let $S = \langle w_1,w_2,\ldots,w_{k}\rangle$ and $S' = \langle w'_1,w'_2,\ldots,w'_{k}\rangle$ be two sequences of size $k$ of non-overlapping windows from $W_1$ and $W_2$, respectively. We call a transformation of $s_1$ into $s_2$ window-compatible with respect to $S$ and $S'$, if (i) all old characters of $s_2$ are in the windows of $S'$ and (ii) every old character of $s_2$ which is in a window $w'_i$, was placed in window $w_i$ of $s_1$ prior to the transformation. We call a transformation window-compatible, if it is window-compatible with respect to at least a pair of sequences of non-overlapping windows \textbf{from} $W_1$ and $W_2$, respectively.
\end{Definition}
Intuitively, a window-compatible transformation with respect to two sequences of windows $S$ and $S'$ does not allow the characters to move in between the windows; if a character is initially placed in a window $w_i$, it should either be deleted or placed in window $w'_i$ of $s_2$ and vice versa. We emphasize that in order for a transformation to be window-compatible, the corresponding windows should be selected from $W_1$ and $W_2$, respectively. A few examples of window-compatible and window-incompatible transformations are illustrated in Figure \ref{figs:examples}.
\input{figs/bothfigs}
As we show in the following, window-compatible transformations are well-structured. In fact, we show in Section \ref{editdistance} that if the edit distances of the windows are accessible in time $O(1)$, a dynamic program can find an optimal\footnote{a transformation with the smallest number of operations.} window-compatible transformation of $s_1$ into $s_2$ in time $O(n+|W_1||W_2|)$.
\vspace{0.2cm}
{\noindent \textbf{Lemma} \ref{dp} [restated]. \textit{Given a matrix of edit distances between the substrings corresponding to every pair of windows of $W_1$ and $W_2$, one can compute an optimal window-compatible transformation of $s_1$ into $s_2$ in time $O(n+|W_1||W_2|)$.\\}}
Lemma \ref{dp} shows that window-compatible transformations are easy to find. It also follows from Lemma \ref{dp} that any $\alpha$ approximation matrix for the edit distances of the windows suffices to find an approximately optimal window-compatible transformation (with the same approximation factor) in time $O(n+|W_1||W_2|)$. This makes the connection of edit distance and metric estimation more clear.
We complement this observation by a structural proof. In Section \ref{editdistance}, we show that the length of the shortest window-compatible transformation of $s_1$ into $s_2$ is not far from $\delta(|s_1|+|s_2|)$. This enables us to use the previously mentioned algorithms to find an approximately optimal window-compatible transformation, and show this is in fact a constant approximation away from $\delta(|s_1|+|s_2|)$.
\vspace{0.2cm}
{\noindent \textbf{Lemma} \ref{compatibility} [restated]. Given that $\mathsf{edit}(s_1,s_2) \leq \delta n$, there exists a window-compatible transformation of $s_1$ into $s_2$ with at most $(3\delta + 1/\gamma)n + 2l$ operations. \\}
Now we can put things in perspective. Lemma \ref{dp}, in light of the results of metric estimation, provides us a nice tool for finding an approximately optimal window-compatible transformation, and Lemma \ref{compatibility} argues that such a transformation is to some extent optimal. Based on this, we outline our algorithm for $\delta$-bounded edit distance as follows:
\begin{enumerate}
\item Construct the windows of $W_1$ and $W_2$ for both $s_1$ and $s_2$.
\item Construct a metric $\langle \mathcal{M}, \mathsf{edit}\rangle$, where $\mathcal{M} = W_1 \cup W_2$ and the distance of two points in $\mathcal{M}$ is equal to the edit distance between their corresponding windows. We use the classic algorithm of edit distance to answer every oracle invocation for reporting the edit distance between two windows. Using the quantum approximation algorithm of metric estimation, find a $3+\epsilon$ approximation solution to the edit distances for every pair of windows (Theorem \ref{thm:metric0}).
\item Based on the estimated distances, find a $3+\epsilon$ approximately optimal window-compatible transformation (Lemma \ref{dp}).
\item Report the transformation as an approximation proof for the $\delta$-bounded edit distance problem.
\end{enumerate}
We show in Section \ref{editdistance}, that by setting $\beta = 6/7$ and $\gamma = 1/\epsilon\delta$, the above algorithm runs in time $\widetilde O(n^{2-1/7}\mathrm{poly}(1/\epsilon))$ and has an approximation factor of $7+\epsilon$.
\vspace{0.2cm}
{\noindent \textbf{Lemma} \ref{mainbutnotmain} [restated] There exists a quantum algorithm that solves the $\delta$-bounded edit distance problem within an approximation factor of $7+\epsilon$ in time $\widetilde O(n^{2-1/7}\mathrm{poly}(1/\epsilon))$. \\}
By Lemma \ref{mainbutnotmain}, we can approximate the $\delta$-bounded edit distance problem in truly subquadratic time in case the guarantee holds. Of course, if this algorithm provides a larger or invalid transformation, one can immediately imply that the guarantee $\mathsf{edit}(s_1,s_2) \leq \delta(|s_1|+|s_2|)$ is violated. The rest of the solution for edit distance follows from a simple multiplicative method. In order to solve edit distance, we first check whether the two strings are equal and in that case, we report that their distance is equal to $0$. Otherwise $\mathsf{edit}(s_1,s_2) \geq 1$. Now, we start with $\rho = 1/n$ and every time run our solution for $\delta$-bounded edit distance with parameter $\delta=\rho$, to find an approximation proof for $\mathsf{edit}(s_1,s_2) = \rho n$. If our algorithm finds a proper transformation with at most $(7\rho + \epsilon) n$ operations, then we report that solution. Otherwise, we know that $\mathsf{edit}(s_1,s_2) > \rho n$, and thus multiply $\rho$ by a factor $1+\epsilon$. Of course, this comes at the expense of an additional multiplicative factor of $1+\epsilon$ to the approximation factor; however, the running time remains $\widetilde O(n^{2-1/7}\mathrm{poly}(1/\epsilon))$. We later refer to this technique as \textit{guess and multiply}.
\vspace{0.2cm}
{\noindent \textbf{Theorem} \ref{main} [restated] There exists a quantum algorithm that solves edit distance within an approximation factor of $7+\epsilon$ in time $\widetilde O(n^{2-1/7}\mathrm{poly}(1/\epsilon))$. \\}
\subsection{Improving the Running Time via Bootstrapping}\label{bbc}
So far, we discussed how to use divide and conquer and metric estimation to approximate edit distance in subquadratic time. In this section, we explain the ideas to improve the running time of the algorithm by taking a hit on its approximation factor.
Recall that, in order to approximate the edit distance, we first construct a set of windows. Next, we use metric estimation to estimate the edit distances of the windows, and finally, we use a dynamic programming algorithm to find an almost optimal window-compatible transformation. As discussed before, such a solution approximates the edit distance within a constant factor. The components of this algorithm are illustrated in Figure \ref{7+}.
\input{figs/7+}
Now, we show that we can improve the algorithm at two points. Firstly, instead of using the $3+\epsilon$ approximation algorithm for metric estimation, we can lose a factor of $\mathsf{e_{m}}(\epsilon)$ in the approximation and estimate the distances in time $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$ (Theorem \ref{thm:metric151}). In addition to this, as an oracle function for metric estimation, we do not really need to compute the exact edit distances of the windows; a constant estimation to the distances suffices. Therefore, one can use our algorithm for approximating edit distance to implement the oracle in subquadratic time. Of course, this again comes at the expense of deteriorating the approximation guarantee but the running time improves. In this section, we show how we combine these ideas to achieve an $\widetilde O(n^{2-(5-\sqrt{17})/4+\epsilon}\mathrm{poly}(1/\epsilon))\simeq\widetilde O(n^{1.781})$ time algorithm. As to why the exponent converges to $2-(5-\sqrt{17})/4$, we refer the reader to a discussion in Section \ref{bootstraping}.
To formalize the above ideas, suppose we are given two strings $s_1$ and $s_2$, and would like to approximate the edit distance between the strings in time $\widetilde O(n^{2-(5-\sqrt{17})/4+\epsilon}\mathrm{poly}(1/\epsilon))$. We call our algorithm for this problem $\mathcal{A}(\epsilon)$, and refer to its time complexity and approximation factor with $\mathsf{t_{e}}(\epsilon) $ and $\mathsf{e_{e}}(\epsilon)$, respectively. We inductively show that $$\mathsf{t_{e}}(\epsilon) = \widetilde O(n^{2-(5-\sqrt{17})/4+\epsilon}\mathrm{poly}(1/\epsilon))$$ and $\mathsf{e_{e}}(\epsilon) = O(1/\epsilon)^{O(\log 1/\epsilon)}$. Notice that if $2-(5-\sqrt{17})/4+\epsilon \geq 2$, $\mathcal{A}(\epsilon)$ can be trivially implemented with the classic $O(n^2)$ algorithm and the approximation factor $\mathsf{e_{e}}(\epsilon) = 1$. Now, assume that $2-(5-\sqrt{17})/4+\epsilon < 2$.
An $\widetilde O((1/\delta)^2n^{2-(5-\sqrt{17})/2+2\epsilon}\mathrm{poly}(1/\epsilon))$ time algorithm for $\delta$-bounded edit distance suffices to design $\mathcal{A}(\epsilon)$. If $\delta \leq n^{-(5-\sqrt{17})/8+\epsilon/2}$ we run the $O(n+\delta^2n^2)$ of Landau \textit{et al.}~\cite{landau1998incremental}, otherwise the running time of our algorithm is $\widetilde O(n^{2-(5-\sqrt{17})/4+\epsilon}\mathrm{poly}(1/\epsilon))$. Moreover, a similar guess and multiply method explained in Section \ref{ghabli} extends this solution to edit distance. Therefore, all we need is to approximate the $\delta$-bounded edit distance problem in time $\widetilde O((1/\delta)^2n^{2-(5-\sqrt{17})/2+2\epsilon}\mathrm{poly}(1/\epsilon))$.
To this end, we again define two parameters $\beta$ and $\gamma$ and set the window size equal to $\lfloor n^{1-\beta}\rfloor$ and the gap size equal to $g = \lfloor l/\gamma\rfloor$. Similar to what explained before, we construct two sets of windows $W_1$ and $W_2$ for $s_1$ and $s_2$ based on the windows size and gap size. Now, we use the same algorithm for finding the edit distance between $s_1$ $s_2$, with two modifications.
\begin{enumerate}
\item Construct the windows of $W_1$ and $W_2$ for both $s_1$ and $s_2$.
\item Construct a metric $\langle \mathcal{M}, \mathsf{edit}\rangle$, where $\mathcal{M} = W_1 \cup W_2$ and the distance of two points in $\mathcal{M}$ is equal to the edit distance between their corresponding windows. We use $\mathcal{A}(2\epsilon)$ (a slightly slower version of our algorithm) for estimating the edit distances of the windows in time $\mathsf{t_{e}}(2\epsilon)=\widetilde O(n^{2-(5-\sqrt{17})/4+2\epsilon}\mathrm{poly}(1/\epsilon))$ as on oracle function. Using the approximation algorithm of metric estimation, find an $\mathsf{e_{m}}(\epsilon)\mathsf{e_{e}}(2\epsilon)$ approximation solution to the edit distances for every pair of windows (Theorem \ref{thm:metric151}).
\item Based on the estimated distances, find an $\mathsf{e_{m}}(\epsilon)\mathsf{e_{e}}(2\epsilon)$ approximately optimal window-compatible transformation (Lemma \ref{dp}).
\item Report the transformation as an approximation proof for the $\delta$-bounded edit distance problem.
\end{enumerate}
Notice that there are two modifications to the previous algorithm. First, instead of using the $3+\epsilon$ factor algorithm for metric estimation, here, we use an $\mathsf{e_{m}}(\epsilon)$ approximation factor algorithm that runs in time $\widetilde O(n^{3/2+\epsilon}\mathrm{poly}(1/\epsilon))$. Moreover, instead of implementing the oracle function via the classic $O(n^2)$ algorithm, we use $\mathcal{A}(2\epsilon)$ for approximating the edit distances.
In Section \ref{bootstraping}, we show that by setting the right values for parameters $\beta$ and $\gamma$, the running time and approximation factor of algorithm $\mathcal{A}(\epsilon)$ would be $\widetilde O(n^{2-(5-\sqrt{17})/4+\epsilon}\mathrm{poly}(1/\epsilon))$ and $\mathsf{e_{e}}(\epsilon) = O(1/\epsilon)^{O(\log 1/\epsilon)}$, respectively.
\vspace{0.2cm}
{\noindent \textbf{Theorem} \ref{mainbutnotmain} [restated] There exists an $\widetilde O(n^{2-(5-\sqrt{17})/4+\epsilon})$ time quantum algorithm that approximates edit distance within a factor $\mathsf{e_{e}}(\epsilon) = O(1/\epsilon)^{O(\log 1/\epsilon)}$. \\}
Figure \ref{figs:bootstrapping} shows the components of $\mathcal{A}(\epsilon)$.
\input{figs/bootstrapping}
|
1,108,101,566,725 | arxiv |
\subsection{Outer Loop for Solution Search}
To make this article self-sustained, we briefly explain the search strategies in the outer loop and summarize it in Algorithm~\ref{alg:outer_pf_search}. Interested readers can refer to \cite{lesieutre2015:efficient}.
\begin{algorithm}[tbhp]
\caption{Outer Loop for Locating Power Flow Solutions}
\label{alg:outer_pf_search}
\begin{algorithmic}[1]
\State Solving for a power flow solution $x_1$.
\State Generating elliptical mapping $\mathscr{E}$ by algorithms in \cite{lesieutre2015:efficient,wu2017:algebraic}.
\State $S \gets x_1$ \Comment{Initialize solution set}
\State $N_{\rm{solu}} \gets |S|$ \Comment{Initialize number of solutions}
\State $k \gets 0$ \Comment{Initialize counting number}
\While{$k \neq N_{\rm{solu}}$}
\State $k \gets k+1$ \Comment{Update counting number}
\State $x_0\gets x_k$ \Comment{Update starting solution}
\For{$l=1,~2,~\cdots,~N_{\rm{eqn}}$}
\State Compute $\mathscr{E}^{-1} \mathbf{e}_l$ \Comment{Equivalent curve design}
\State Algorithm~\ref{alg:HEBC} \Comment{HEBC Algorithm}
\State{Return $S_{\rm{new}}$} \Comment{Return newly found solutions}
\If{$S_{new}$ is not in $S$}
\State $S \gets S \cup S_{\rm{new}}$ \Comment{Update the solution set}
\State $N_{\rm{solu}}\gets |S|$ \Comment{Update the number of solutions}
\EndIf
\EndFor
\EndWhile
\end{algorithmic}
\end{algorithm}
We start Algorithm~\ref{alg:outer_pf_search} with a known solution $x_1$ which can be solved by Newton's method or other techniques\footnote{This step relies on the past extensive research of solving a high voltage solution to the power flow problem. Many mature solvers are able to do this job for very large systems.}. After several initialization steps, designing the curve $\{PF-\alpha\mathscr{E}^{-1} \mathbf{e}_l\}$ which is equivalent to $\{EF_{l-}\}$ for $l$. Following the curve from $l=1$ to the last one by Algorithm~\ref{alg:HEBC} (which will be discussed shortly below) and collect new solutions. When finished tracing curves, assigning the starting point $x_0$ to a newly found solution, say, $x_2$, and repeating the procedure. The whole loop terminates upon every solution having been assigned to a starting point.
Algorithm~\ref{alg:outer_pf_search} presents a procedure to follow each curve sequentially. However, the curve designs at the same starting solution are independent with each other, suggesting a parallel computing framework to simultaneously trace these curves. The parallel computing is not performed in this article, but can be done with ease and increase speed drastically.
\subsection{Inner Loop for Curve Tracing}
Instead of tracing a curve by the traditional predictor-corrector algorithm, we apply the holomorphic embedding technique to quickly pass through the regular curve segments. The predictor-corrector algorithm is only executed for traveling across singularities. It is switched back to the holomorphic embedding as soon as current steps leave a singular point.
Figure~\ref{fig:pade1} shows four holomorphic steps on a selected curve from a 5-bus case \cite{salam1989:parallel}. They reach the singular point very quickly. On the other hand, the blue curve in Figure~\ref{fig:pade1} was generated by the traditional predictor-corrector algorithm. It took dozens of steps to reach the same singularity.
\begin{algorithm}[tbhp]
\caption{Holomorphihc Embedding Based Continuation}
\label{alg:HEBC}
\begin{algorithmic}[1]
\State{Input selected curve $\mathscr{E}^{-1}\mathbf{e}_l$.}
\State{Initialize the $1st$ step.}
\For{$k=1:M$}
\For{$k_h=1:N_h$}
\State{Initialize the $1st$ holomorphic step size $\delta_h$.}
\State{Prepare parameters for holomorphic embedding.}
\State{Compute power series of holomorphic embedding.}
\State{Compute Pad\'{e} approximation.}
\parState{Evaluate voltage values from Pad\'{e} and update $\alpha_{k_h+1}$.}
\parState{Evaluate power mismatch $dP_{\rm{mis}}$ from computed voltages.}
\While{minimum pole $p_{\rm{min}}$ is not determined}
\State{Compute roots $\{ \zeta_i \}$ from Pad\'{e} denominator.}
\parState{$p_{\rm{min}} \gets \zeta_{\rm{min}}$ if the minimum real root $\zeta_{\rm{min}}$ has correct sign.}
\EndWhile
\parState{Increase $\delta_h$ while $dP_{\rm{mis}}< dP_{\rm{max}}$ and $|\text{current point}|< |p_{\rm{min}}|$.}
\parState{Decrease $\delta_h$ while $dP_{\rm{mis}}\ge dP_{\rm{max}}$ or $|\text{current point}|\ge |p_{\rm{min}}|$.}
\parState{Correct current holomorphic predicted point by Newton's method.}
\If{Correction succeeds}
\State{Record current point.}
\Else
\parState{Delete current point and compute a starter for switching algorithm.}
\State{Break.}
\EndIf
\If{$\alpha_{k_h+1}\alpha_{k_h}<0$}
\State{Find a solution nearby.}
\If{Fail to locate the solution}
\parState{Delete current point and compute a cold starter for switching algorithm.}
\State{Break.}
\Else
\State{Record solution to solution set $S_{\rm{new}}$.}
\parState{Check completeness of the curve; jump out Algorithm~\ref{alg:HEBC} if completed.}
\EndIf
\EndIf
\If{$|\alpha_{k_h+1}-\alpha_{k_h}|<d\alpha_{h,\rm{min}}$}
\State{Compute a starter for switching algorithm.}
\State{Break.}
\EndIf
\EndFor
\State{Execute predictor-corrector routine.}
\EndFor
\State{Return solution set $S_{\rm{new}}$}
\end{algorithmic}
\end{algorithm}
\begin{figure}[tbhp]
\centering
\subfigure[Holomorphic Steps ]{\label{fig:pade1}\includegraphics[width=0.48\columnwidth]{5salam_trace1_v3_Pade_local_1.eps}}~~
\subfigure[Predictor-Corrector Steps ]{\label{fig:pade2}\includegraphics[width=0.48\columnwidth]{5salam_trace1_v3_Pade_local_2.eps}}
\caption{Holomorphic Steps and Preditor-Corrector Steps} \label{fig:pade}
\end{figure}
\subsubsection{Criterion to Enter Predictor-Corrector Routine}
Two indicators are considered to trigger the switch of the algorithms in Algorithm~\ref{alg:HEBC}. The first indicator appears when the corrector steps fail to make the holomorphic prediction converge within a certain number of iterations.
Another indicator comes when $|\alpha_{k_h+1}-\alpha_{k_h}|$ is smaller than a threshold value $d\alpha_{h,\rm{min}}$. Both suggest that current holomorphic step is close to singular (or at least badly scaled with respect to $\alpha$).
\subsubsection{Using A Warm Starter to Accelerate Predictor-Corrector Steps}
One can initiate the predictor-corrector routine from a minimum step size, and increase it gradually. We refer it to a \emph{cold starter}.
To avoid slow ``warming up'' steps, a warm starter is proposed and implemented. It relies on an estimated distance $d_{\rm{hp}}$ from the singular point to the last holomorphic point.
We specifically choose the initial step interval $S_{\rm{pc}}$ to be $1/5$ of the estimated distance $d_{\rm{hp}}$ and to be no greater than $0.45$ of the last holomorphic step size. Then, using $S_{\rm{pc}}$ to compute two backward steps to initiate a quadratic predictor. For example, the first two green triangles on the upper curve segment in Figure~\ref{fig:pade2} are the backward points evaluated by Pad\'{e} approximation at the step length $S_{\rm{pc}}$. It makes the predictor-corrector routine quickly pass through the singular point as shown by the rest green triangles
\subsubsection{Criterion to Exit Predictor-Corrector Routine}
When travelling across a singular point, the direction of curve changes. Numerically, there exists a particular step $m_c$ such that
$(\alpha_{m_c}-\alpha_{m_c-1})(\alpha_{m_c+1}-\alpha_{m_c})<0$.
After this moment, we continue the predictor-corrector routine for a while until the curve's slope value returns from infinity back to a tractable value. Instead of evaluating the actual slope of the curve, we monitor the maximum variable secant slope $R_{\rm{m}}$.
\begin{equation}
R_{\rm{m}} := max\{ |(\Vv{k,m}-\Vv{k,m-1})/(\alpha_{m}-\alpha_{m-1})|,~\forall k \}
\end{equation}
As long as $R_{\rm{m}}$ drops to a threshold $R_{\rm{max}}$, say, $2 \times 10^4$, we jump out of the predictor-corrector routine and start a new sequence of holomorphic steps.
\subsection{Holomorphic Embedding of Power Flow Equations}
\subsubsection{PQ Bus Embedding}
We start with the basic complex power balance equation for PQ bus in Equation \eqref{eq:complex_pf}. Note that $\Ss{k} = \Pp{k} + j \Qq{k}$
we define
\begin{subequations}
\begin{align}
\Pp{k}(\alpha) &:= (1+K_{\rm{p},\it{k}} \alpha) \Pp{k,0} \\
\Qq{k}(\alpha) &:= (1+K_{\rm{q},\it{k}} \alpha) \Qq{k,0}
\end{align}
\end{subequations}%
where $\alpha \in \mathbb{C}$; $K_{\rm{p},\it{k}}$ and $K_{\rm{q},\it{k}}$ are obtained from $\mathscr{E}^{-1} \mathbf{e}_l$ for some $l$; $\Pp{k,0}$ and $\Qq{k,0}$ are the fixed starting active and reactive power which admit a known solution.
If we define a new variable $\Ww{k} := \Vv{k}^{-1}$ for $\Vv{k} \neq 0$, and restrict parameterized $\Ww{k}(\alpha)$ to be reflective such that $\Ww{k}(\alpha) = \Ww{k}(\alpha^\star)$, then Equation~\eqref{eq:complex_pf} can be written as
\begin{subequations}
\begin{align}
\sum_{n=1}^{N_{\rm{bus}}} \Ybus{n,k} \Vv{n}(\alpha) &= \bigg( (1+K_{\rm{p},\it{k}} \alpha) \Pp{k,0}- j (1 \nonumber
\\ & + K_{\rm{q},\it{k}} \alpha) \Qq{k,0} \bigg) \Ww{k}^\star(\alpha^\star) \label{eq:para_complex_pf} \\
\Vv{k}(\alpha) \Ww{k}(\alpha) &= 1 \label{eq:para_VW}
\end{align} \label{eq:para_PQ_bus}%
\end{subequations}%
Note that on the right hand side of \eqref{eq:para_complex_pf} we use $\Ww{k}^\star(\alpha^\star)$ instead of $\Ww{k}^\star(\alpha)$ since they are equal by the reflective property\footnote{A more detailed discussion on the reflective requirement can be found in \cite{trias2015:reflection}.}.
Since $\Vv{k}(\alpha)$ and $\Ww{k}(\alpha)$ are holomorphic \cite{trias2015:reflection}, we can use power series to represent them.
Then, \eqref{eq:para_PQ_bus} can be re-written as
\begin{subequations}
\begin{align}
\sum_{n=1}^{N_{\rm{bus}}} \bigg( \Ybus{n,k} \sum_{i=0}^{\infty} \scrv{n,i} \alpha^i \bigg) &= \bigg( (1+K_{\rm{p},\it{k}} \alpha) \Pp{k,0} - j (1 \nonumber
\\&+ K_{\rm{q},\it{k}} \alpha) \Qq{k,0} \bigg)\sum_{i=0}^{\infty} \scrw{n,i}^\star \alpha^i \label{eq:para_complex_pf_Taylor} \\
\sum_{i=0}^{\infty} \scrv{n,i} \alpha^i \sum_{i=0}^{\infty} \scrw{n,i} \alpha^i &= 1 \label{eq:para_VW_Taylor}
\end{align} \label{eq:para_PQ_bus_Taylor}%
\end{subequations}%
where $\scrv{n,i}$ and $\scrw{n,i}$ are the power series coefficients.
Matching up coefficients for every monomial of $\alpha$ in \eqref{eq:para_complex_pf_Taylor} and \eqref{eq:para_VW_Taylor} we can solve $(\scrv{k,1}, \scrv{k,2}, \cdots)$ and $(\scrw{k,1}, \scrw{k,2}, \cdots)$ recursively as long as $\scrv{k,0}$ and $\scrw{k,0}$ are provided.
\subsubsection{PV Bus Embedding}
Next, we consider the holomorphic embedding for PV bus equations.
To retain holomorphicity, we need to bring back the reactive power balance equation \eqref{eq:Qld_eqt} to \eqref{eq:PV_bus} and consider reactive power input as a new variable. Again, by defining $\Ww{k} := \Vv{k}^{-1}$ for $\Vv{k} \neq 0$ and restricting parameterized $\Ww{k}(\alpha)$ to be reflective we have the holomorphic embedded equations
\begin{subequations}
\begin{align}
\sum_{n=1}^{N_{\rm{bus}}} \Ybus{n,k} \Vv{n}(\alpha) &= \bigg( (1+K_{\rm{p},\it{k}} \alpha) \Pp{k,0} - j \Qq{k}(\alpha) \bigg) \Ww{k}^\star(\alpha^\star) \label{eq:para_PQ_PV} \\
\Vv{k}(\alpha) \Vv{k}^\star(\alpha^\star) &= \Vv{k,\rm{m}}^2 + K_{\rm{v},\it{k}} \alpha \label{eq:para_V_PV} \\
\Vv{k}(\alpha) \Ww{k}(\alpha) &= 1 \label{eq:para_VW_PV}
\end{align} \label{eq:para_PV_bus}%
\end{subequations}%
where $\Vv{k,\rm{m}} \in \mathbb{R}$ is the fixed voltage magnitude at bus $k$; $K_{\rm{v},\it{k}}$ is obtained from the corresponding entry of $\mathscr{E}^{-1} \mathbf{e}_l$.
By the holomorphic structure, we represent parameterized unknowns $\Vv{n}(\alpha)$, $\Ww{k}(\alpha)$, and $\Qq{k}(\alpha)$ through their power series. Then, \eqref{eq:para_PV_bus} are re-written as
\begin{subequations}
\begin{align}
\sum_{n=1}^{N_{\rm{bus}}} \bigg( \Ybus{n,k} \sum_{i=0}^{\infty} \scrv{n,i} \alpha^i \bigg) &= \bigg( (1+K_{\rm{p},\it{k}} \alpha) \Pp{k,0} \nonumber
\\ & - j \sum_{i=0}^{\infty} q_{k,i} \alpha^i \bigg) \sum_{i=0}^{\infty} \scrw{n,i}^\star \alpha^i \label{eq:para_PQ_PV_Taylor} \\
\sum_{i=0}^{\infty} \scrv{n,i} \alpha^i \sum_{i=0}^{\infty} \scrv{n,i}^\star \alpha^i &= \Vv{k,\rm{m}}^2 + K_{\rm{v},\it{k}} \alpha \label{eq:para_V_PV_Taylor} \\
\sum_{i=0}^{\infty} \scrv{n,i} \alpha^i \sum_{i=0}^{\infty} \scrw{n,i} \alpha^i &= 1 \label{eq:para_VW_PV_Taylor}
\end{align} \label{eq:para_PV_bus_Taylor}%
\end{subequations}%
where $q_{k,i}$'s are the power series coefficients of $\Qq{k}(\alpha)$.
Matching up coefficients for every monomial of $\alpha$ in \eqref{eq:para_PQ_PV_Taylor}, \eqref{eq:para_V_PV_Taylor} and \eqref{eq:para_VW_PV_Taylor} we can solve $u_i$, $w_i$, and $q_i$ as well.
\subsubsection{Slack Bus Embedding}
Consider the slack bus voltage magnitude equation \eqref{eq:Vslack_eqt}. Its holomorphic embedded equation is
\begin{equation}
\Vv{\rm{s}}(\alpha) \Vv{\rm{s}}^\star (\alpha^\star) = \Vv{\rm{s},m}^2+K_{\rm{s}} \alpha \label{eq:slack_embed}
\end{equation}%
where $\Vv{\rm{s},m}$ is the slack bus voltage magnitude, $K_{\rm{s}}$ is the corresponding entry from $\mathscr{E}^{-1} \mathbf{e}_l$.
Substituting the power series of $\Vv{\rm{s}}(\alpha)$ into Equation~\eqref{eq:slack_embed} and matching up each monomial of $\alpha$ we have
\begin{subequations}
\begin{align}
\scrv{\rm{s},\it{i}} &=-\bigg( \sum_{n=1}^{i-1} \scrv{\rm{s},\it{n}} \scrv{\rm{s},\it{i-n}} \bigg)/(2 \scrv{\rm{s},0})~~\text{for}~~i \ge 2
\\ \scrv{\rm{s},1} &=K_{\rm{s}}/(2 \scrv{\rm{s},0}) \label{eq:slack_recursive}
\end{align}
\end{subequations}
Combining the corresponding equations from the PQ bus, PV bus and slack bus equations we finally solve the power series coefficients for each degree-$i$. In practice, every degree requires solving a real-valued linear system (sparse) with its size $(4N_{\rm{bus}}+N_{\rm{gen}}-3) \times (4N_{\rm{bus}}+N_{\rm{gen}}-3)$ where $N_{\rm{gen}}$ is the number of PV nodes. As $i$ goes to infinity, the power series converges to the actual curve in the convergence range. To compromise accuracy and speed, we usually stop at a given maximum degree $i_{\rm{max}}$\footnote{\cite{trias2012:1stHEM} claims that degree $i$ will deplete double precision digits after $60$. How to choose an appropriate $i_{\rm{max}}$ is beyond the scope of this paper. We choose $i_{\rm{max}}=15$ in our numerical experiments by empirical experience considering speed and accuracy.}.
\subsection{Pad\'{e} Approximation}
The above subsection shows that each node voltage (as well as reactive power at PV bus) can be embedded as a holomorphic function, and demonstrates a recursive way to obtain the coefficients. In practice the holomorphic function can only be evaluated by a finite sequence of power series. Thus, the accuracy of the sequence deteriorates when approaching the singularities of the holomorphic function. To achieve a better convergence performance and to predict the location of singular point, we further compute the Pad\'{e} approximation. It approximates the holomorphic function by a rational function in which the numerator and denominator are polynomials.
According to \cite{stahl1989:convergence,stahl1997:convergence}, the Pad\'{e} approximation has the maximum convergent domain if the degrees of its numerator and denominator have the minimum difference. It provides a criterion for determining the best degree(s) that should be chosen.
Consider an embedded voltage variable $\scrv{k}(\alpha)$ for some $k$. Suppose its first $N$ coefficients are known.
\begin{equation}
\scrv{k}(\alpha) = \sum_{n=0}^{\infty} \scrv{k,n} \alpha^n \approx \sum_{n=0}^{N} \scrv{k,n} \alpha^n
\end{equation}
Let its Pad\'{e} approximation be
\begin{equation}
\sum_{n=0}^{N} \scrv{k,n} \alpha^n
= \sum_{n=0}^{N_{\rm{n}}} u_{k,n} \alpha^n / \sum_{n=0}^{N_{\rm{d}}} l_{k,n} \alpha^n%
\end{equation}%
where we specify $N_{\rm{n}}+N_{\rm{d}}=N$, $N_{\rm{n}} \ge N_{\rm{d}}$, and $N_{\rm{n}}-N_{\rm{d}} \le 1$.
To reach a unique coefficient set, let $l_{k,0}=1$. Matching up the coefficients for each monomial we can solve $u_{k,n}$'s and $l_{k,n}$'s in a $(N+1) \times (N+1)$ complex-valued sparse linear system. If we compute the power series to the maximum degree $i_{\rm{max}}$, the system size in the real space is $2(i_{\rm{max}}+1) \times 2(i_{\rm{max}}+1)$.
Once the Pad\'{e} approximation has been calculated, we can move along the parameterized curve by evaluating Pad\'{e} approximated values until a power mismatch threshold\footnote{In our numerical experiments, this threshold is set at $10^{-3}$ p.u.} has been reached. We can also compute the real-valued zeros to the denominator function of Pad\'{e}. These zeros reveal the locations of singularities on the parameterized curve, which can further assist us designing appropriate arc length for passing through these singular points by the traditional predictor-corrector algorithm. Next section will discuss these designs in detail.
\section{Introduction}
\label{sec:intro}
\input{intro}
\section{Description of Power Flow Problem}
\label{sec:pf}
\input{pf}
\section{Holomorphic Embedding Technique}
\label{sec:holo}
\input{he}
\section{Holomorphic Embedding Based Continuation Method}
\label{sec:alg}
\input{alg}
\section{Computational Complexity Comparison}
\label{sec:comp}
\input{comp}
\section{Numerical Experiments}
\label{sec:num}
\input{num}
\section{Conclusions}
\label{sec:conclusions}
\input{concl}
\bibliographystyle{ieeetr}
\subsection{Comparison To Homotopy Continuation Method}
To demonstrate the superiority of computational efficiency in finding multiple power flow solutions, we begin with a comparison of the proposed HEBC method to the homotopy continuation method. The homotopy continuation is performed by the PHCpack \cite{PHCpack}.
The HEBC method finds all the actual power flow solutions in this comparison as well as case14\footnote{No existing literature claims complete solution sets for larger IEEE test cases.}. Figure~\ref{fig:phc} shows execution time (in logarithmic scale) comparison between two methods. For test cases smaller than 5 buses, the PHCpack runs faster than the proposed HEBC method. However, for cases more than 5 buses, the HEBC outperforms the homotopy continuation method substantially. Considering the HEBC method is coded in Matlab and is not optimized to reach the most computational performance, the time reductions from HEBC are impressive. Test cases larger than 9 buses cannot be solved by PHCpack within 24 hours, thus are not considered in this comparison\footnote{A more recent progress in \cite{chen2018:network} successfully reduced the computational time of case14 to 5 minutes, however, the proposed HEBC is still much faster.}.
\begin{figure}[tbhp]
\centering
\includegraphics[width=0.95\columnwidth]{comparison_phc_hebc.eps}
\caption{Comparison Between Homotopy Continuation and HEBC} \label{fig:phc}%
\end{figure}
\subsection{Comparison To Full Predictor-Corrector Algorithm}
In this part, we testify the traditional full predictor-corrector method from \cite{lesieutre2015:efficient} and the proposed HEBC method on the same set of test cases, and compare their numerical performances. Both methods provide the same solution sets for all cases, but the HEBC method is more efficient than the traditional predictor-corrector method.
Some hard\footnote{A curve is hard to follow in the sense that it contains too many singularities or some singular points are very sharp when turning directions.} sample curves are presented in Appendix~\ref{sec:append_1}.
One can see from the left plots of Figure~\ref{fig:curves} that the traditional full predictor-corrector method, though with quadratic predictor and automatic step length adaption, takes very dense points to trace curves. On the other hand, the right plots of Figure~\ref{fig:curves} are primarily sparse. Small dense point periods only occur around singularities when HEBC switches to the predictor-corrector routine for passing through those singularities.
Summaries of the numerical results are collected in Table~\ref{table:1} and \ref{table:2}.
\begin{table}[tbhp]
{\footnotesize
\caption{Numerical Results by Predictor-Corrector Method}\label{table:1}
\begin{center}
\begin{tabular}{l|c|c|c}
\hline
\multicolumn{1}{c|}{Method} & \multicolumn{2}{c|}{Predictor-Corrector} & \multicolumn{1}{l}{} \\ \hline
\multicolumn{1}{l|}{Case} & overall steps & overall time (s) & \# Solutions \\ \hline
3TS & 2962 & 1.257 & 6 \\ \hline
3 & 3349 & 1.143 & 6 \\ \hline
4gs & 4281 & 1.405 & 6 \\ \hline
4BBc & 8838 & 2.660 & 12 \\ \hline
4BB0 & 13791 & 3.719 & 14 \\ \hline
5Salam & 11465 & 3.626 & 10 \\ \hline
5loop & 17049 & 4.568 & 10 \\ \hline
6ww & 9421 & 3.209 & 6 \\ \hline
7Salam & 5195 & 1.978 & 4 \\ \hline
9 & 22264 & 10.945 & 8 \\ \hline
14 & 151423 & 102.401 & 30 \\ \hline
30 & 5358518 & 6054.987 & 472 \\ \hline
33bw & 311957 & 249.736 & 16 \\ \hline
39 & 3009935 & 3758.195 & 176 \\ \hline
57 & 14647351 & 23864.005 & 606 \\ \hline
\end{tabular}
\end{center}
}
\end{table}
\begin{table}[tbhp]
\begin{threeparttable}
{\footnotesize
\caption{Numerical Results by HEBC Method} \label{table:2}
\begin{tabular}{l|c|c|c|c|c}
\hline
\multicolumn{1}{c|}{Method} & \multicolumn{5}{c}{HEBC} \\ \hline
\multicolumn{1}{c|}{Routine} & \multicolumn{2}{c|}{Holomorphic} & \multicolumn{2}{c|}{Predictor-Corrector} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}overall\\ time (s)\end{tabular}} \\ \cline{1-5}
Case & \# steps & time (s) & \# step & time (s) & \\ \hline
3TS & 253 & 0.373 & 228 & 0.133 & 0.859 \\ \hline
3 & 290 & 0.345 & 387 & 0.155 & 0.798 \\ \hline
4gs & 481 & 0.625 & 766 & 0.197 & 1.248 \\ \hline
4BBc & 850 & 1.174 & 1358 & 0.362 & 2.102 \\ \hline
4BB0 & 1212 & 1.489 & 2805 & 0.773 & 2.890 \\ \hline
5Salam & 1128 & 1.754 & 1224 & 0.290 & 2.661 \\ \hline
5loop & 1695 & 2.448 & 1627 & 0.358 & 3.537 \\ \hline
6ww & 995 & 1.402 & 1143 & 0.288 & 2.249 \\ \hline
7Salam & 564 & 1.068 & 362 & 0.132 & 1.676 \\ \hline
9 & 1668 & 3.158 & 4026 & 1.668 & 5.572 \\ \hline
14 & 13350 & 34.443 & 27238 & 12.784 & 50.013 \\ \hline
30 & 403181 & 2077.966 & 910664 & 674.828 & 2813.249 \\ \hline
33bw & 15904 & 81.896 & 65351 & 43.519 & 129.323 \\ \hline
39 & 184458 & 1247.794 & 1044166 & 930.796 & 2204.543 \\ \hline
57 & 835550 & 10565.59 & 3078609 & 3598.361 & 14304.691 \\ \hline
\end{tabular}
\begin{tablenotes}
\small
\item HEBC provides the same solution sets for all the cases as in Table I.
\end{tablenotes}
}
\end{threeparttable}
\end{table}
\begin{figure}[tbhp]
\centering
\includegraphics[width=1\columnwidth]{num_step.eps}
\caption{Steps Needed for Different Cases with Full Predictor-Corrector and HEBC} \label{fig:num_step}
\end{figure}
Comparing the results in Figure~\ref{fig:num_step}, the total number of steps for HEBC is about $1/6$ to $1/3$ of the total number of steps for the full predictor-corrector method. This ratio, not surprisingly, should depend on the problem structure. In general, fewer singularities and longer horizontal curve segments favor the HEBC more.
\begin{figure}[tbhp]
\centering
\includegraphics[width=0.95\columnwidth]{execution_time.eps}
\caption{Execution Times with Full Predictor-Corrector and HEBC}\label{fig:execution_time}
\end{figure}
To reveal the efficiency of HEBC, we compute the equivalent number of predictor-corrector steps $N_{\rm{eqv}}$
\begin{equation}
N_{\rm{eqv}} := (N_{\rm{pc}}-N_{\rm{he,pc}})/N_{\rm{he,holo}}
\end{equation}%
where $N_{\rm{pc}}$ is the number of full predictor-corrector steps; $N_{\rm{he,pc}}$ is the number of predictor-corrector routine steps in HEBC; and $N_{\rm{he,holo}}$ is the number of holomorphic routine steps in HEBC.
From Table~\ref{table:1} and \ref{table:2} we calculate that one holomorphic step on average can represent $8.5$ predictor-corrector steps, with the worst case of $7$ steps and the best case of $15$ steps.
In Figure~\ref{fig:execution_time} the first $9$ small cases up to case7Salam show a limited time saving by HEBC. However, starting at case9 the HEBC method outperforms the full predictor-corrector method by up to $50 \%$ of the execution time. Larger cases also exhibit at least $30 \%$ time saving in the lower plots of Figure~\ref{fig:execution_time}.
\subsection{Average Number of Steps on Each Dimension}
Recall that the HEBC method calls the Newton's method at each step to correct the predicted point.
These predicted points are sequentially determined over the curve tracing process. Thus, the HEBC method can be regarded as a systematic way to choose initial points for solving the power flow equations, where the number of initial points equals the number of steps in Table~\ref{table:2}, i.e. the sum of entries in the second and forth columns for each case.
From this point of view, one can assess the efficiency of HEBC by computing the average number of initial points (steps) allocated in each dimension
\begin{equation}
R_{eq} := N^{1/d}%
\end{equation}%
where $N$ is the total number of initial points, $d$ is the dimensionality of the problem. $R_{eq}$ represents the number of points required in each single dimension such that the total number of initial points composed by their direct combinations achieves the same amount of initial points $N$ for the whole $d$-dimensional problem. Specifically for our problem, $R_{eq}$ is computed as
\begin{equation}
R_{eq} = (N_{he,pc}+N_{he,holo})^{1/(2N_{bus} -1)}
\end{equation}
\begin{figure}[tbhp]
\centering
\includegraphics[width=0.95\columnwidth]{avg_rand_seed.eps}
\caption{Equivalent No. Random Seeds for Each Variable}\label{fig:avg_rand}
\end{figure}
Figure~\ref{fig:avg_rand} depicts the trend of $R_{eq}$ as system size increases. One can clearly see that the average number of steps distributed on each dimension decreases to nearly $1$. Hence, despite the increase of total number of steps, the average number of steps on each dimension seems to decrease in an asymptotic sense.
\subsection{Power Flow Equations in Rectangular Coordinates}
\label{subsec:pf_rectangular}
Consider a connected power grid with $N_{\rm{bus}}$ nodes. Let the node voltage vector be
\begin{equation}
\mathbf{V} := \mathbf{V}_{\rm{d}} + j\mathbf{V}_{\rm{q}} \label{eq:complex_voltage}
\end{equation}
where $\mathbf{V} \in \mathbb{C}^{N_{\rm{bus}}}$; $\mathbf{V}_{\rm{d}} \in \mathbb{R}^{N_{\rm{bus}}}$ and $\mathbf{V}_{\rm{q}} \in \mathbb{R}^{N_{\rm{bus}}}$ are the real and imaginary parts of $\mathbf{V}$, respectively .
For the PQ bus we have
\begin{equation}
\cV{k} \sum_{n=1}^{N_{\rm{bus}}} \Ybus{n,k} \Vv{n} = \cS{k} \label{eq:complex_pf}
\end{equation}%
where $\Vv{k}$ and $\Vv{n}$ are the corresponding entries of $\mathbf{V}$; $\Ybus{n,k}$ is the $(n,k)$-th entry of the bus admittance matrix $\mathbf{Y} \in \mathbb{C}^{N_{\rm{bus}} \times N_{\rm{bus}}}$; $\Ss{k} \in \mathbb{C}$ is the complex power load at bus $k$; superscript star $\star$ represents the conjugate operator.
Separating the real and imaginary parts of Equation \eqref{eq:complex_pf} gives the two equations about a PQ bus
\begin{subequations}
\begin{align}
\Pp{k} &= \Vdq{\rm{d}, \it{k}} \sum_{n=1}^{N_{\rm{bus}}} \big(\Gg{n,k} \Vdq{\rm{d},\it{n}} - \Bb{n,k} \Vdq{\rm{q},\it{n}} \big) \nonumber
\\ &+ \Vdq{\rm{q},\it{k}} \sum_{n=1}^{N_{\rm{bus}}} \big(\Gg{n,k} \Vdq{\rm{q},\it{n}} + \Bb{n,k} \Vdq{\rm{d},\it{n}} \big) \label{eq:Pld_eqt}
\\ \Qq{k} &= \Vdq{\rm{q},\it{k}} \sum_{n=1}^{N_{\rm{bus}}} \big(\Gg{n,k} \Vdq{\rm{d},\it{n}} - \Bb{n,k} \Vdq{\rm{q},\it{n}} \big) \nonumber
\\ &- \Vdq{\rm{d},\it{k}} \sum_{n=1}^{N_{\rm{bus}}} \big(\Gg{n,k} \Vdq{\rm{q},\it{n}} + \Bb{n,k} \Vdq{\rm{d},\it{n}} \big) \label{eq:Qld_eqt}
\end{align} \label{eq:PQ_bus}%
\end{subequations}
where $\Pp{k} \le 0$ and $\Qq{k} \le 0$\footnote{Usually a load absorbs reactive power, but it can possibly generate reactive power. In that case $\Qq{k} \ge 0$.} are the fixed active and reactive power loads at bus $k$; $\Gg{n,k}$ and $\Bb{n,k}$ are the $(n,k)$-th entries of the bus conductance matrix $\mathbf{G}$ and the bus susceptance matrix $\mathbf{B}$\footnote{$\mathbf{Y}=\mathbf{G}+j\mathbf{B}$}; $\Vdq{\rm{d} ,\it{k}}$, $\Vdq{\rm{d},\it{n}}$, $\Vdq{\rm{q},\it{k}}$ and $\Vdq{\rm{q},\it{n}}$ are the corresponding entries of $\mathbf{V}_{\rm{d}}$ and $\mathbf{V}_{\rm{q}}$, which are unknown variables that should be determined.
For the PV bus we have
\begin{subequations}
\begin{align}
\Pp{k} &= \Vdq{\rm{d},\it{k}} \sum_{n=1}^{N_{\rm{bus}}} \big(\Gg{n,k} \Vdq{\rm{d},\it{n}} - \Bb{n,k} \Vdq{\rm{q},\it{n}} \big) \nonumber
\\ &+ \Vdq{\rm{q},\it{k}} \sum_{n=1}^{N_{\rm{bus}}} \big(\Gg{n,k} \Vdq{\rm{q},\it{n}} + \Bb{n,k} \Vdq{\rm{d},\it{n}} \big) \label{eq:Pgen_eqt}
\\ V_{\rm{m},\it{k}}^2 &= \Vdq{\rm{d},\it{k}}^2 + \Vdq{\rm{q},\it{k}}^2 \label{eq:V_eqt}
\end{align} \label{eq:PV_bus}%
\end{subequations}
where $\Pp{k}$ is a fixed active power injection at bus $k$ which is usually positive but can be negative; $V_{\rm{m},\it{k}}$ is the fixed voltage magnitude at bus $k$.
For the slack bus with an angle reference we have
\begin{subequations}
\begin{align}
V_{\rm{m},\it{s}}^2 &= \Vdq{\rm{d},\it{s}}^2 + \Vdq{\rm{q},\it{s}}^2 \label{eq:Vslack_eqt}
\\ 0 &= \Vdq{\rm{q},\it{s}} \label{eq:angle_eqt}
\end{align} \label{eq:slack_bus}%
\end{subequations}
where subscript $s$ is the slack bus number; $\Vv{\rm{m},\it{s}}$ is the slack bus voltage magnitude.
One can further substitute \eqref{eq:angle_eqt} in \eqref{eq:Vslack_eqt}, \eqref{eq:PV_bus} and \eqref{eq:PQ_bus} to eliminate $\Vdq{\rm{q},\it{s}}$. Finally, \eqref{eq:PQ_bus}, \eqref{eq:PV_bus}, and \eqref{eq:slack_bus} together are the power flow equations we will investigate in this paper. Note that they are in quadratic form, thus can be written succinctly as
\begin{equation}
PF(\mathbf{U}) := \{ f_i(\mathbf{U}) = \mathbf{U}^T \M{i} \mathbf{U} - r_i,~ i=1,\cdots,2 N_{\rm{bus}} \} \label{eq:pf}
\end{equation}%
where $\mathbf{U} := [\mathbf{V}_{\rm{d}}^T~~ \mathbf{V}_{\rm{q}}^T]^T$ is the unknown variable vector; $\M{i} \in \mathbb{SR}^{2 N_{\rm{bus}} \times 2 N_{\rm{bus}}}$ is a symmetric constant matrix for the quadratic part; $r_i \in \mathbb{R}$ is the constant scalar part.
\subsection{Equivalent Curve Design of Elliptical Formulation of Power Flow Equations}
As introduced in Section~\ref{sec:intro}, \cite{molzahn2013:counterexample} presented a counter-example that fails the proposed algorithm in \cite{ma1993} for finding all the real-valued power flow solutions. Then, \cite{lesieutre2015:efficient} introduced the concept of elliptical formulation of power flow equations which substantially changes the topology of path following curves and succeeded for that example. Later, \cite{wu2017:algebraic} showed the existence of elliptical formulation under mild conditions and constructed it in a systematical way.
We start our discussion with a given invertible linear map $\mathscr{E} \in \mathbb{R}^{2 N_{\rm{bus}} \times 2 N_{\rm{bus}}}$ that sends Equation \eqref{eq:pf} to a set of high dimensional ellipses $EF(\mathbf{U})$. The construction of $\mathscr{E}$ can be found in \cite{wu2017:algebraic,lesieutre2015:efficient}. Consider
\begin{displaymath}
\mathscr{E}:PF(\mathbf{U}) \to EF(\mathbf{U})
\end{displaymath}
with
\begin{displaymath}
EF(\mathbf{U}) := \{ g_i(\mathbf{U}) = \mathbf{U}^T \Hh{i} \mathbf{U} - \gamma_i,~i=1,\cdots,2 N_{\rm{bus}} \}
\end{displaymath}
where $\Hh{i} \in \mathbb{SR}^{2 N_{\rm{bus}} \times 2 N_{\rm{bus}}}$ and $\Hh{i} \succ 0$; $\gamma_i > 0$.
Let $\mathscr{Z}(h;x)$ be the operator that takes the projection of $\{(x,y) | h(x,y) = 0 \}$ onto $x$; define
\begin{displaymath}
EF_{l-}(\mathbf{U}) := EF(\mathbf{U}) - \{g_l(\mathbf{U})\}
\end{displaymath}
\begin{displaymath}
EF_{l,\alpha}(\mathbf{U},\alpha) := EF_{l-}(\mathbf{U}) \cup \{g_l(\mathbf{U})-\alpha,~\alpha \in \mathbb{R} \}.
\end{displaymath}
Since $EF(\mathbf{U})$ defines a determined algebraic system, its algebraic set is generically 0-dimensional in $\mathbb{R}^{2 N_{\rm{bus}}}$. By removing one equation from $EF(\mathbf{U})$, $EF_{l-}(\mathbf{U})$ acquires one degree of freedom and defines a 1-dimensional algebraic set in $\mathbb{R}^{2 N_{\rm{bus}}}$. On the other hand, adding one extra degree of freedom to $EF(\mathbf{U})$ makes the algebraic set of $EF_{l,\alpha}(\mathbf{U},\alpha)$ 1-dimensional in $\mathbb{R}^{2 N_{\rm{bus}}+1}$.
The following Lemma~\ref{lemma:1} shows an equivalence between these two 1-dimensional algebraic sets.
\begin{lemma}\label{lemma:1}
\begin{displaymath}
\mathscr{Z}(EF_{l-};\mathbf{U}) = \mathscr{Z}(EF_{l,\alpha};\mathbf{U})
\end{displaymath}
\end{lemma}
The proof is trivial and omitted here.
Next, we state the equivalent curve design of elliptical formulation in Theorem~\ref{thm:1}.
\begin{theorem} \label{thm:1}
\begin{displaymath}
\mathscr{Z}(EF_{l-};\mathbf{U}) = \mathscr{Z}\big(\{PF(\mathbf{U}) - \alpha \mathscr{E}^{-1} \mathbf{e}_l\};\mathbf{U}\big)
\end{displaymath}
where $\mathbf{e}_l \in \mathbb{R}^{2 N_{\rm{bus}}}$ is a unit column vector with the $j$-th entry being 1.
\end{theorem}
\begin{proof}
By definition, $EF_{l,\alpha}(\mathbf{U},\alpha)$ can also be expressed as \{$EF(\mathbf{U}) - \alpha \mathbf{e}_l$\}. Then we have
\begin{subequations}
\begin{align}
\mathscr{E}^{-1} \big( EF(\mathbf{U}) - \alpha \mathbf{e}_l \big) &= \mathscr{E}^{-1} \big(EF(\mathbf{U})\big) - \alpha \mathscr{E}^{-1} \mathbf{e}_l \nonumber
\\ &= PF(\mathbf{U}) - \alpha \mathscr{E}^{-1} \mathbf{e}_l. \nonumber
\end{align}
\end{subequations}
Since $\mathscr{E}$ is an invertible linear map, it is a homeomorphism. Hence,
\begin{displaymath}
\mathscr{Z}(EF_{l,\alpha};\mathbf{U}) = \mathscr{Z}\big(\{PF(\mathbf{U}) - \alpha \mathscr{E}^{-1} \mathbf{e}_l\};\mathbf{U}\big).
\end{displaymath}
Finally, by Lemma~\ref{lemma:1} we conclude that
\begin{displaymath}
\mathscr{Z}(EF_{l-};\mathbf{U}) = \mathscr{Z}\big(\{PF(\mathbf{U}) - \alpha \mathscr{E}^{-1} \mathbf{e}_l\};\mathbf{U}\big).
\end{displaymath}
\end{proof}
|
1,108,101,566,726 | arxiv | \section{Introduction\label{sec::Introduction}}
With a simple, purely geometrical definition and complex behavior
that includes a phase transition, percolation
has become an important theoretical model in statistical physics.
It has also applications in various areas of science, like
conductivity in strongly heterogeneous solids \cite{Hunt2001,Besseaguet2019},
fluid flow in porous media \cite{Sahimi1993,Bolandtaba2011},
epidemics \cite{Ziff2021},
and thermal conductivity of composites~\cite{Shtein2015}.
While some critical exponents and crossing probabilities
in 2-dimensional systems
\cite{Stauffer1994,Cardy1992,Lawler2001,Smirnov2001ci,Flores2017},
as well as percolation thresholds in several particular models
\cite{Sykes1964,Kesten1980,Wierman2009} are known rigorously,
many results in this field has been obtained
with computer simulations. Over the years, advanced
numercial methods have been developed, like the Leath
method~\cite{Leath1976,Lorenz1998,Xun2021},
hull-generating walks~\cite{Voss1984,Ziff1984},
gradient percolation~\cite{Rosso1985,Ziff1986,Tencer2021},
toroidal wrapping~\cite{Newman2001,Wang2013,Koza2016},
spanning clusters~\cite{Ziff92_PRL,Oliveira2003},
rescaled particles~\cite{Torquato12b},
frontier tracking~\cite{Quintanilla2000},
parallelized percolation on distributed machines \cite{Pruessner03},
dynamic programming \cite{Yang2013},
and the transfer matrix method \cite{Feng2008,Jacobsen2015}.
These methods, in all their diversity, share one common feature:
they are usually implemented with the assumption
that the system has a square (rectangular, hypercubic) geometry.
This choice is quite natural: it corresponds to
the most elementary and efficient computer data structure: array.
It also facilitates implementation of useful boundary conditions,
including wrapping boundaries. Wrapping boundaries, in turn,
enable one to study percolation on cylinders and tori,
the shapes for which strong theoretical results have been derived
and for which boundary effects are minimized, which results
in a faster convergence rate to the thermodynamic limit~\cite{Newman2001}.
The symmetry of a rectangle is closely related to the required connectedness
of a cluster: in spanning percolation the two sides that are checked for
spanning are geometrically equivalent, and so are the remaining two. One expects
that this configuration will produce quicker convergence to
the thermodynamic limit than, for instance, a trapezoid.
If one uses square systems, it is possible to investigate clusters
that span or wrap in both directions simultaneously \cite{Newman2001}.
But what about spanning in other number of directions than two or four,
for example, three?
It was recently argued that the probability, $p_3$ that,
in the thermodynamic limit, there exists a three-leg cluster
touching the three sides of a triangle at the percolation threshold
has a universal value $1/2$ \cite{Koza2019,Flores2017}.
This result is supposed to be valid for any lattice and even systems
of arbitrary shape, in which case their perimeter must be divided into
three disjoint parts (arcs). For self-matching lattices this property
holds even for finite systems.
The aim of the paper is to investigate whether the property of
$p_3 \to 1/2$ can be used to develop an efficient
method of finding the percolation threshold, $p_\mathrm{c}$,
for planar lattices.
To this end, we investigate the well-known case of the site percolation
on a square lattice, assuming, however, that the system
is in the shape of an equilateral triangle,
the simplest geometry with the three-fold symmetry corresponding
to the three legs of the clusters. The precise value of $p_\mathrm{c}$
in this model~\cite{Jacobsen2015},
\begin{equation}\label{eq::pc_Jacobsen}
p_\mathrm{c} = 0.592\,746\,050\,792\,10(2),
\end{equation}
will be used as a reference value against which the method
will be evaluated.
\section{Method\label{sec::Method}}
We start from placing an equilateral triangle with sides of length $L$ on a square lattice
(Fig.~\ref{fig:geometry}).
\begin{figure}
\includegraphics[width=0.75\columnwidth]{fig1.eps}
\caption{\label{fig:geometry} Geometry of the model. An
equilateral triangle $ABC$ is located on a square lattice in such a way that
its vertex $A$ is placed randomly inside $[0,1)\times[0,1)$ and the angle between
side $AB$ and the direction of the $x$ axis is a random variable between 0 and 15 degrees.
Circles, squares, and diamonds mark the "edge" site corresponding to three sides of the triangle.
A three-leg percolating cluster must contain at least one site from each of these groups.
}
\end{figure}
Incompatibility of the symmetries of the triangle and the lattice
gives rise to specific problems;
for example, the number of lattice sites encompassed by a triangle
is not invariant under translation and rotation, and the number of sites contained
inside a triangle of side length $L$ only approximately scales as $L^2$,
a problem particularly serious for small triangles.
There are also problems of topological nature: if one side of the triangle
is parallel to the $x$ axis, it cuts $\approx L$ lattice bonds, whereas
the remaining two sides cut $\approx \sqrt{3} L$ bonds. Thus, the sides,
even though of equal lengths, are not equivalent,
which may have an adverse effect on the simulation convergence rate.
To mitigate these problems, for each $L$ we consider an ensemble of
equilateral triangles randomly distributed and oriented relative to
the underlying lattice.
Symmetries of the lattice and the triangle allow to restrict the ensemble to
the cases where vertex $A$ is distributed uniformly inside the square
$[0,1)\times[]0,1)$, and the angle $\alpha$ made by side $AB$ and the direction
of the $x$ axis is distributed uniformly between 0 and 15 degrees.
Each simulation starts from randomization of the location
and orientation of the triangle. Then,
a minimum bounding box for the triangle is computed such that its
corners are at lattice sites. This will be the arena of the simulation:
rectangular geometry trivializes the computation of the neighboring sites.
Next, the set of the sites encompassed by the triangle is determined.
We shall call them ``active sites''. Only active sites can be occupied
as the cluster grows. Some of these sites are marked as
edge sites. We define three groups of edge sites, each corresponding
to a triangle side. A site is an edge site to side $AB$ if
a nearest-neighbor bond starting at this site cuts $AB$. Two other
groups are defined similarly for sides $BC$ and $CA$.
A site can be an edge site for more than one side
(for example, the lattice sites nearest to vertices
$A$ and $B$ in Fig.~\ref{fig:geometry}).
We define a three-leg percolating cluster as a cluster that contains
at least one site from each of these groups.
When the geometry has been established, a list of active sites is shuffled
randomly (we used the 64-bit Mersenne twister mt19937
random number generator from the standard C++ library).
During simulation proper, subsequent elements are popped from this list
and the corresponding sites are marked as occupied. With each new site
occupied, the union-find algorithm \cite{Newman2001,Cormen2009}
is used to monitor clusters' growth.
It also updates the information about the edge groups each cluster has reached.
To this end, before the simulation begins,
the union-find assigns to each site three bits that are used to signal
the property of being connected to a given group of edge sites.
This property is updated whenever clusters merge.
A simulation ends when, for some cluster,
all these bits are set to 1, which indicates that a three-leg
percolating cluster has just been formed.
Finally, the number of occupied sites is recorded.
In this way we obtained the distribution of the probability $R(n; L)$ that
for the ensemble of a randomly positioned and oriented
equilateral triangles of side $L$, with $n$ of
their internal sites occupied,
there is a three-leg cluster that spans all of its sides.
We introduce the occupation probability
\begin{equation} \label{eq::def}
p = \frac{n}{S(L)},
\end{equation}
where $S(L) = (\sqrt{3}/4)L^2$ is the expected number of lattice sites contained inside
the triangle, which, due to the randomization of its placement and orientation,
is equal to its area. Clearly, at percolation, $0 < p < 1$.
Note, however, that in principle a triangle can hold more than $S(L)$ lattice
sites within itself, leading to the possibility of $p > 1$ when
every or nearly every site is occupied. This, however, cannot happen
at the onset of percolation; moreover, the upper bound for $p$ as $L\to\infty$ is 1,
as it should.
An important quantity that we want to obtain from simulations is $R_L(p)$,
the probability that, for site occupancy $p$, a triangle of side $L$
contains a cluster spanning all of its sides.
While it is closely related to directly measured $R(n; L)$,
the relation is,
to some extent, unknown. First, we require $R_L(p)$
to be continuous, whereas $R(n; L)$ is discrete; second,
$R(n; L)$ is not known exactly, but is estimated from simulations.
The discrete nature of
$R(n; L)$ is especially problematic for small $L$. For example,
for $L=4$ only for 8 values of $n$ is this function different from 0 and 1.
The usual way of dealing with these problems is to fit the data
to some function (e.g. linear) in a vicinity of $p$.
Here, however, we use the canonical ensemble method \cite{Newman2001}.
Its main idea is to weight the values of a discrete-value function $F$
with coefficients from the corresponding binomial distribution,
\begin{equation} \label{eq::canonical-def}
F(p) = \sum_{n=0}^N {N \choose n}p^n(1-p)^{N-n}F(n),
\end{equation}
where $N$ is the number of sites in the system.
This, however, poses a subtle problem: our simulations are performed
for en ensemble of triangles in which $N$ may take on many values for
fixed $L$, and its mean value, $S(L)$, is non-integer. We solve this by replacing the binomial distribution with its
normal approximation, $\mathcal{N}\left(S(L)p,S(L)p(1-p)\right)$. This leads to
\begin{equation} \label{eq::canonical}
R_L(p) = \frac{1}{\sigma \sqrt{2\pi} }\sum_{n=0}^{\lfloor S(L) \rfloor}
\exp\left({\frac{-(n-\mu)^2}{2\sigma^2}}\right)R(n; L),
\end{equation}
with $\mu = S(L)p$, and $\sigma^2 = S(L)p(1-p)$.
It is expected \cite{Koza2019} that $R_L(p_\mathrm{c}) \to 1/2$ as
$L\to \infty$ (Fig.~\ref{fig:distr}).
\begin{figure}
\includegraphics[width=0.9\columnwidth]{fig2.eps}
\caption{\label{fig:distr} The probability $R_L(p)$ that a triangle
of side $L$ with occupation probability $p$ of its internal sites
contains a three-leg percolating cluster.
Inset: close-up of the intersection region.
}
\end{figure}
We can use this property to define an $L$-dependent approximation
of the percolation threshold, $p^*_L$, defined as the solution to
\begin{equation} \label{eq:p*}
R_L(p^*_L) = 1/2.
\end{equation}
It was suggested \cite{Oliveira2003} that $p^*_L$ can be approximated with
\begin{equation} \label{eq::expansion-in-L}
p^*_L \approx p_\mathrm{c} + L^{-1/\nu} \sum_{k=0}^M A_k L^{-k},
\end{equation}
where $\nu = 4/3$ is a critical exponent \cite{Stauffer1994},
$A_k$ are some nonuniversal parameters, and $M$ is a cut-off.
However, in our case the sites forming "edge groups" are
situated inside the triangle, so the effective side length
may differ from $L$. We therefore introduce another parameter,
$\lambda$, that accounts for this uncertainty and helps to correct
for the truncation of higher-order terms
\cite{Levinshteln1975,Ziff92_PRL},
\begin{equation} \label{eq::expansion-in-L-lambda}
p^*_L \approx p_\mathrm{c} + (L+\lambda)^{-1/\nu} \sum_{k=0}^M A_k (L+\lambda)^{-k},\quad L\ge L_\mathrm{min}.
\end{equation}
where $L_\mathrm{min}$ is the cut-off for the system size below which this
approximation is invalid.
The uncertainties of $p^*_L$ are estimated using
the bootstrap method \cite{nrc}.
Next, the Levenberg–Marquardt algorithm for nonlinear least squares
curve-fitting is applied to estimate the values of
$p_\mathrm{c}$, $\lambda$, and $A_k$
in Eq.~(\ref{eq::expansion-in-L-lambda}).
The cut-offs $L_\mathrm{min}$ and $M$ are determined from the requirement
that they should minimize the error estimate for $p_\mathrm{c}$.
Quality of the fit was monitored with the regression standard error
$s = \sqrt{\chi^2/\mathrm{dof}}$
(the square root of the chi-squared statistic per degree of freedom).
For a good fit, $s$ is close to or smaller than 1.
\section{Results\label{sec::Results}}
We performed simulations of nearest-neighbor site percolation
on a square lattice for triangles of side $L$
ranging from 4 to 1440 (all lengths in lattice units).
The total number of active sites inside the triangles was
$\approx 2\cdot 10^{14}$ for each $L\ge 16$, and between $10^{12}$
and $10^{13}$ for each $L < 16$.
The uncertainties of $p^*_L$ were below $3 \cdot 10^{-7}$ for $L\ge 16$ and below $10^{-6}$ for $L<16$.
The first question we investigated was the convergence rate in
Eq.~(\ref{eq::expansion-in-L-lambda}). Since $R_L(p_\mathrm{c}) \to 1/2$
as $L\to\infty$, and we use this limiting value in Eq.~(\ref{eq:p*}),
$p^*_L - p_\mathrm{c}$ is expected to scale as $L^{-1 - 1/\nu}$ \cite{Ziff92_PRL,Oliveira2003},
which is equivalent to $A_0 = 0$ in (\ref{eq::expansion-in-L-lambda}).
When we set $M = 1$ in (\ref{eq::expansion-in-L-lambda}) and used
the best known value of $p_\mathrm{c}$, Eq.~(\ref{eq::pc_Jacobsen}),
we obtained $A_0 = 3.5\cdot 10^{-6} \pm 2\cdot 10^{-6}$
with the regression standard error $s \approx 0.6$, in agreement with
the convergence rate $\sim L^{-1 - 1/\nu}$ = $L^{-7/4}$.
Setting $A_0 = 0$, we obtained
$A_1 = 0.20707(6)$, $\lambda = 0.9942(25)$, and
\begin{equation} \label{eq::pc}
p_\mathrm{c} = 0.592\,746\,10(4),
\end{equation}
with $L_\mathrm{min} = 16$ and the regression standard error
$s \approx 0.7$. This fit is shown in Fig.~\ref{fig:3}.
\begin{figure}
\includegraphics[width=0.9\columnwidth]{fig3.eps}
\caption{\label{fig:3}
$p^*_L - p_\mathrm{c}$ (symbols) and its approximation
$A_0/(L+\lambda)^{1+1/\nu}$ (line).
Inset: $\delta(L) = |p^*_L - p_\mathrm{c} - A_0/(L+\lambda)^{1+1/\nu}|$,
the error introduced by using (\ref{eq::expansion-in-L-lambda})
with $M=1$ and $L_\mathrm{min}=16$.
}
\end{figure}
The value of $p_\mathrm{c}$ is in agreement with Eq.~(\ref{eq::pc_Jacobsen}).
In terms of measurement precision, our method turns out to be
at least on par with alternative simulation methods based
on Monte Carlo sampling (\cite{Feng2008,Lee2008} and references therein).
The value of $s$ being less than 1 indicates that the fit is
good: actually, the difference between $p^*_L$ and its approximation
$p_\mathrm{c} + A_1/(L+\lambda)^{1+1/\nu}$ is less than
$2\cdot 10^{-7}$ for all $L \ge 16$ (Fig.~\ref{fig:3}, inset).
For $4 \le L < 16$
it behaves roughly as $L^{-\omega}$ with $\omega = 6.0(4)$.
This suggests that $A_2 , A_3, A_4 \approx 0$ in
(\ref{eq::expansion-in-L-lambda}).
However, fits in this region are poor and the hypothesis that three
consecutive terms in (\ref{eq::expansion-in-L-lambda})
vanish should be taken with caution.
The role of $\lambda$ in (\ref{eq::expansion-in-L-lambda}) is to help
reduce both $M$ and $L_\mathrm{min}$. One could do without it
and set $\lambda = 0$, effectively reducing
(\ref{eq::expansion-in-L-lambda}) to (\ref{eq::expansion-in-L}).
This would require taking $M = 3$, with the total of 4 fitting
parameters instead of 3. The uncertainty of $p_\mathrm{c}$ thus
obtained would be higher by about 25\% compared to the
value reported in (\ref{eq::pc}), which is acceptable,
but the uncertainty of $A_1$ would be tripled.
Is randomization of the position and orientation of triangles necessary?
When we fixed vertex $A$ at (x, y) and $B$ at (x + L, y) with $x=y=0.5$,
we found that the left hand side of (\ref{eq::expansion-in-L-lambda})
still exhibits the $\sim L^{-1-1/\nu}$ behavior, but it also contains
a significant contribution of what can be regarded as noise
(Fig.~\ref{fig:4}).
\begin{figure}
\includegraphics[width=0.9\columnwidth]{fig4.eps}
\caption{\label{fig:4}
$p^*_L - p_\mathrm{c}$ for triangles with vertex $A$ fixed at $(0.5,0.5)$
and angle $\alpha = 0$. The dashed line is a guide to the eye with the
slope $-1-1/\nu = -1.75$.
}
\end{figure}
Its magnitude is so large that it makes the method practically useless.
When we set $x=0.5$ and treated $y$ as a random
variable uniformly distributed in $(0,1)$, this ``noise'' was still present,
though its magnitude was much smaller (data not shown). Thus,
full randomization ot triangles' position and orientation
appears necessary for lattice-based systems. However,
for continuous systems (e.g. percolation of
overlapping discs or squares~\cite{Mertens2012})
this step can be omitted.
Using the canonical ensemble method to estimate $R_L(p)$ is crucial,
because it works well even for small system sizes,
where the discrete nature of $R(n; L)$ is quite problematic for
other methods, e.g.\ those based on approximation.
Further studies are necessary to check whether applying
the exact canonical weighting could improve the validity
of Eq.~(\ref{eq::expansion-in-L-lambda}) for small $L$.
The simulations were run in parallel on several computers with varying
computational power.
Simulations depicted in Fig.~\ref{fig:3} would have taken about 1.5 years
if done only on the PC on which this paper was written
(a four-core processor running 8 threads in parallel at 3.4 GHz).
The computational overhead introduced by randomization
of the triangle orientation is about 100\% for very small systems ($L = 4$),
but quickly drops to $\approx 10\%$ for $L \gtrsim 50$.
The highest (multi-threaded) computational efficiency is obtained for $L\approx 64$,
for which one occupied site is processed at $\approx 33$
processor clock cycles. It decreases $\approx 5$ times for the smallest
and largest system sizes considered here, the former due to
the randomization overhead, the latter due to processor cache misses.
\section{Discussion\label{sec::Duscussion}}
The method presented here can be applied to other system shapes (with an arbitrary
division of their boundary into three compact regions), lattices, and
cluster connectedness definitions, all with the universal three-leg crossing
probability $1/2$. In contrast to this, universal spanning ("two-leg") and
wrapping probabilities are known only for some particular system shapes,
e.g.\ rectangles, and for more complicated cases would have to be treated as
an additional unknown. A disadvantage of our method is a rather slow convergence
rate to the thermodynamic limit, most likely related to open
boundary conditions~\cite{Hovi1996}.
The method can be generalized to more complex shapes, e.g., convex polygons.
The bounding box can be determined from the maximum and minimum values of
the vertex coordinates. Then one can iterate over each of the sites
lying inside the bounding box and test
if it lies inside the polygon and if so, if it is adjacent
to any of the polygon's sides. This general method can be used for complex
networks, e.g., Penrose tiling~\cite{Yonezawa1989I}.
A more efficient way is to shoot a small number of ``rays``
in such a way that they cut all internal sites of the polygon.
In the case depicted in Fig.~\ref{fig:geometry}
they could be shot vertically or horizontally along the grid lines.
If a ray hits a convex polygon, it enters the system at some point $q_1$ and leaves it
at some $q_2$ ($q_1$ and $q_2$ can coincide).
Any lattice site between $q_1$ and $q_2$ must be an internal site and all edge sites
must lie in their close proximity.
This reduces the computational complexity of discovering the edge sites
from quadratic to linear in the characteristic system size.
This method should be applicable for
all networks based on regular lattices, including networks
with voids and bottlenecks~\cite{Haji-Akbari2009} and frustrated
lattices~\cite{Haji-Akbari2015}.
Once the geometry of the system has been established, the remaining steps
are essentially the same as in traditional methods.
The angle by which each system is rotated can be arbitrary.
Here we reduced it to the range from 0 o 15 degrees only because
for equilateral triangles some code simplifications are then possible.
For example, no sites adjacent to edge $AB$ in Fig.~\ref{fig:geometry} can have
the same vertical coordinates.
Bond percolation can be treated in a very similar way.
Rather than with active sites, one would deal with active bonds defined
as those with at least one end lying inside the system.
Edge bonds would be then defined as those that cut a given system edge.
Monte Carlo methods, including the one presented here,
for simple planar lattice models
have recently become overshadowed by transfer-matrix methods \cite{Jacobsen2015}.
Still, we hope that our work can prove useful in more complicated cases,
like networks with voids and bottlenecks, continuous models,
or in higher-dimensions \cite{Gori2015}.
\section{Conclusions\label{sec::Conclusions}}
We have shown that Monte-Carlo simulations in systems with
incompatible symmetries of their geometry and the underlying lattice
can be efficient in determining the percolation threshold.
The key step is randomization of the system orientation
and position relative to the lattice.
The computational overhead related to this additional step is acceptable.
Although the convergence rate to the thermodynamic limit
is slower than in some methods based on wrapping,
this is compensated by very small (or perhaps even vanishing)
values of several higher-order terms and the possibility of using
small lattices.
Three-leg clusters have proved useful in determining
the value of the percolation threshold.
Their advantage is that the universal crossing probability
associated with them is geometry-independent, which opens the room
for further improvements of the method.
|
1,108,101,566,727 | arxiv |
\section{Introduction.}
\label{section:introduction} Let $\xi$ be the {\it
standard} contact structure in oriented 3-space
$\hbox{\sl I\kern-.18em R \kern-.3em}^3 = (\rho, \theta, z)$, that is the kernel of
$\alpha = \rho^2 d\theta + dz$. An oriented knot
$K$ in contact $\hbox{\sl I\kern-.18em R \kern-.3em}^3$ is said to be a {\it
transversal} knot if it is transversal to the planes
of this contact structure. In this paper, the term
`transversal' refers to this contact structure only. If
the knot
$K$ is parametrized by $(\rho(t),\theta(t), z(t))$,
then $K$ is transversal if and only if
$\frac {z'(t)}{\theta'(t)} \neq -(\rho(t))^2$ for every
t. We will assume throughout that
$\alpha > 0$ for all $t$, pointing out later how our
arguments adapt to the case
$\alpha < 0$. \\ \bigskip
For the benefit of the reader who may be unfamiliar
with the standard contact structure, Figure
\ref{figure:standard contact structure}(a) illustrates
typical 2-planes in this structure in $\hbox{\sl I\kern-.18em R \kern-.3em}^3$, when
$z$ is fixed, and
$\rho$ and
$\theta$ vary. The structure is radially symmetric. It
is also invariant under translation of $\hbox{\sl I\kern-.18em R \kern-.3em}^3$
parallel to the $z$ axis. Typical 2-planes are
horizontal at points on the $z$ axis and twist
clockwise (if the point of view is out towards
increasing
$\rho$ from the $z$ axis,) as $\rho\to\infty$. \\ \bigskip
There has been some discussion about whether the planes
tend to vertical as
$\rho\to\infty$ or to horizontal. If one looks at the
limit of
$\alpha$, it appears that the limit is a rotation of
$\pi/2$. However, if one derives the standard contact
structure on $S^3$ from the Hopf fibration, as
described below, and wants to have this structure be
consistent with the one defined on $\hbox{\sl I\kern-.18em R \kern-.3em}^3$, it is
necessary to take the limit to be a rotation up to (but
not through, as a rotation of more than $\pi$ results
in an overtwisted structure)
$\pi$. The resulting contact structures on $\hbox{\sl I\kern-.18em R \kern-.3em}^3$
are equivalent, through a contactomorphism that
untwists the planes from $\pi$ to
$\pi/2$. Thus we can work in the standard contact
structure on
$S^3$, which has horizontal planes in the limit, while
using the contact form $\alpha$, which induces vertical
planes in the limit. The details are below.
\begin{figure}[htpb]
\centerline{\BoxedEPSF{polarcontact.eps scaled 800}}
\caption {The standard contact structure on $\hbox{\sl I\kern-.18em R \kern-.3em}^3$
and the Hopf fibration on $S^3$.}
\label{figure:standard contact structure}
\end{figure}
The standard contact structure extends to $S^3$ and has
an interesting interpretation in terms of the geometry
of $S^3$. Let
$$S^3 = \{(z_1,z_2) = (\rho_1e^{i\theta_1},
\rho_2e^{i\theta_2})\in
\complexes^2 \ / \ \rho_1^2 + \rho_2^2 = constant.\}$$
Then $\xi$ is the kernel of $\rho_1^2d\theta_1 +
\rho_2^2d\theta_2$. The field of 2-planes may be
thought of as the field of hyperplanes which are
orthogonal to the fibers of the Hopf fibration
$\pi:S^3\to S^2$. See Figure \ref{figure:standard
contact structure}(b) for a picture of typical fibers.
Identify the $z$ axis in $\hbox{\sl I\kern-.18em R \kern-.3em}^3$ with the core of
one of the solid tori. There is a fiber through each
point in $S^3$, and the 2-plane at a point is
orthogonal to the fiber through the point.
\\ \bigskip
The (topological) {\it type} ${\cal K}$ of a knot
$K\subset\hbox{\sl I\kern-.18em R \kern-.3em}^3$ is its equivalence class under
isotopy of the pair
$(K,\hbox{\sl I\kern-.18em R \kern-.3em}^3)$. A sharper notion of equivalence is
its {\it transversal knot type}
${\cal TK}$, which requires that $\frac {z'(t)}{\theta'(t)}
+ (\rho(t))^2$ be positive at every point of the
deformed knot during every stage of the isotopy. The
difference between these two concepts is the central
problem studied in this paper. \\ \bigskip
A parametrized knot $K\subset\hbox{\sl I\kern-.18em R \kern-.3em}^3$ is said to be
represented as a {\em closed braid} if
$\rho(t)>0$ and
$\theta^\prime(t)>0$ for all $t$. See Figure
\ref{figure:closed braid}(a). It was proved by
Bennequin in $\S$23 of
\cite{Be} that every transversal knot is transversally
isotopic to a transversal closed braid. This result
allows us to apply results obtained in the study of
closed braid representatives of topological knots to
the problem of understanding transversal isotopy. We
carry Bennequin's approach one step further, initiating
a comparative study of the two equivalence relations:
topological equivalence of two closed braid
representatives of the same transversal knot type, via
closed braids, and transversal equivalence of the same
two closed braids. Transversal equivalence is of course
more restrictive than topological equivalence. \\ \bigskip
Topological equivalence of closed braid representatives
of the same knot has been the subject of extensive
investigations by the first author and W. Menasco, who
wrote a series of six papers with the common title {\em
Studying links via closed braids}. See, for example,
\cite{BM3} and
\cite{BM5}. See also the related papers
\cite{BH} and
\cite{BF}. In this paper we will begin to apply what
was learned in the topological setting to the
transversal problem. See also Vassiliev's paper
\cite{V}, where we first learned that closed braid
representations of knots were very natural in analysis;
also our own contributions in
\cite{BW}, where we began to understand that there were
deep connections between the analytic and the
topological-algebraic approaches to knot theory. \\ \bigskip
\begin{figure}[htpb]
\centerline{\BoxedEPSF{braid.eps scaled 800}}
\caption {(a) Closed braid; (b) Example of a closed
braid projection; (c) Positive and negative crossings}
\label{figure:closed braid}
\end{figure}
A well-known invariant of a transversal knot type
${\cal T}{\cal K}$ is its {\it Bennequin} number $\beta({\cal T}{\cal K})$.
It is not an invariant of ${\cal K}$. We now define it in a
way that will allow us to compute it from a closed
braid representative $K$ of ${\cal T}{\cal K}$. The {\em braid
index $n = n(K)$} of a closed braid $K$ is the linking
number of $K$ with the oriented $z$ axis. A generic
projection of $K$ onto the
$(\rho,\theta)$ plane will be referred to as a {\it
closed braid projection}. An example is given in Figure
\ref{figure:closed braid}(b). The origin in the
$(\rho,\theta)$ plane is indicated as a black dot; our
closed braid rotates about the
$z$ axis in the direction of increasing
$\theta$. The {\it algebraic crossing number}
$e=e(K)$ of the closed braid is the sum of the signed
crossings in a closed braid projection, using the sign
conventions given in Figure
\ref{figure:closed braid}(c). If the transversal knot
type
${\cal TK}$ is represented by a closed braid $K$, then its
Bennequin number
$\beta({\cal TK})$ is:
$$ \beta({\cal TK}) = e(K) - n(K).$$ Since $e(K)-n(K)$ can
take on infinitely many different values as $K$ ranges
over the representatives of ${\cal K}$, it follows that
there exist infinitely many transversal knot types for
each topological knot type. It was proved by Bennequin
in
\cite{Be} that $e(K)-n(K)$ is bounded above by
$-\chi({\cal F})$, where
${\cal F}$ is a spanning surface of minimal genus for
${\cal K}$. Fuchs and Tabachnikov gave a different upper
bound in
\cite{FT}. However, sharp upper bounds are elusive
and are only known in a few very special cases. \\ \bigskip
We now explain the geometric meaning of $\beta({\cal TK})$.
Choose a point
$(z_1,z_2) = (x_1+ix_2,x_3+ix_4)\in TK\subset S^3$.
Thinking of
$(z_1,z_2)$ as a point in $\hbox{\sl I\kern-.18em R \kern-.3em}^4$, let
$\vec{p} = (x_1,x_2,x_3,x_4)$. Let $\vec{q} =
(-x_2,x_1,-x_4,x_3)$ and let
$\vec{r} = (-x_3,x_4,x_1,-x_2)$. Then
$\vec{r}\cdot\vec{p} =
\vec{r}\cdot\vec{q} = \vec{p}\cdot\vec{q}=0$. Then
$\vec{q}$ may be interpreted as the outward-drawn
normal to the contact plane at
$\vec{p}$, so that $\vec{r}$ lies in the unique contact
plane at the point $\vec{p}\in S^3$. Noting that a
transversal knot is nowhere tangent to the contact
plane, it follows that for each point
$\vec{p}$ on a transversal knot $TK\subset S^3$ the
vector $\vec{r}$ gives a well-defined direction for
pushing $TK$ off itself to a related simple closed
curve $TK'$. The Bennequin number
$\beta({\cal TK})$ is the linking number
${\cal L}k(TK,TK')$. See $\S$16 of \cite{Be} for a
proof that
$\beta(TK)$ is invariant under transverse isotopy and
that $\beta(TK) = e(K)-n(K)$. \\ \bigskip
We say that a transversal knot is {\em transversally
simple} if it is characterized up to transversal
isotopy by its topological knot type and its Bennequin
number. In
\cite{El} Eliashberg proved that a transversal unknot
is transversally simple. More recently Etnyre
\cite{Et} used Eliashberg's techniques to prove that
transversal positive torus knots are transversally
simple.
\\ \bigskip
Our first main result, Theorem
\ref{theorem:realizing the maximal Bennequin number},
asserts that if a knot type ${\cal K}$ is exchange
reducible, (a condition we define in Section 2,) then
its maximum Bennequin number is realized by any closed
braid representative of minimum braid index. As an
application, we are able to compute the maximum
Bennequin number for all iterated torus knots. See
Corollary
\ref{corollary:maximum Bennequin numbers for iterated
torus knots}. Our second main result, Theorem
\ref{theorem:exchange reducible implies t-simple},
asserts that if
${\cal TK}$ is a transversal knot type with associated
topological knot type ${\cal K}$, and if
${\cal K}$ is exchange reducible, then
${\cal TK}$ is transversally simple. As an application, we
prove in Corollary
\ref{corollary:iterated torus knots are transversally
simple} that transversal iterated torus knots are
transversally simple. The two corollaries use new
results of Menasco
\cite{Me}, who proved (after an early version of this
paper was circulated) that iterated torus knots are
exchange reducible. In Theorem
\ref{theorem:3 braid examples} we establish the
existence of knot types that are {\em not} exchange
reducible.\\ \bigskip
Here is an outline of the paper. Section
\ref{section:exchange reducibility and transversal
simplicity} contains our main results. In it we will
define the concept of an exchange reducible knot and
prove Theorems
\ref{theorem:realizing the maximal Bennequin number} and
\ref{theorem:exchange reducible implies t-simple}. In
Section
\ref{section:examples} we discuss examples,
applications and possible generalizations.
\\ \bigskip
{\bf Acknowlegements} We thank Oliver Dasbach, William
Menasco and John Etnyre for conversations and helpful
suggestions relating to the work in this paper. We are
especially grateful to Menasco for the manuscript
\cite{Me}. In an early version of this paper, we
conjectured that iterated torus knots might be exchange
reducible. We explained our conjecture to him, and a
few days later he had a proof! We also thank Wlodek
Kuperberg for sharing his beautiful sketch of the Hopf
fibration (Figure 1(b)) with us.
Finally, we thank William Gibson, who noticed
our formula for the Bennequin number of iterated torus knots in an earlier
version of this paper and pointed out to us in a private conversation that it
could be related to the upper bound which was given by Bennequin in \cite{Be},
by the formula in Corollary 3, part (2).
\subsection{Remarks on techniques.} This subsection
contains a discussion of the techniques used in the
manuscripts \cite{BM4}, \cite{BM5}, \cite{BM94} and
\cite{Me}, tools which form the foundation on which the
results of this paper are based. We compare these
techniques to those used in the manuscripts \cite{Be},
\cite{El} and \cite{Et}, although Bennequin's paper
rightfully belongs in both sets. This description and
comparison is of great interest, but is not essential
for the reading of this paper or the digestion of its
arguments. \\ \bigskip
We concern ourselves with two known foliations of an
orientable surface
${\cal F}$ associated to $K$: the {\it characteristic}
foliation $\xi_F$ from contact geometry and the {\it
topological} foliation from braid theory. The
characteristic foliation of ${\cal F}$ is the line field
$\xi \cap T{\cal F}$, given by the intersection of the
planes of the contact structure with the planes of the
tangent space of the surface, which is then integrated
to a singular foliation of ${\cal F}$. The {\it
topological foliation} is the foliation of ${\cal F}$ which
is induced by intersecting the foliation of $\hbox{\sl I\kern-.18em R \kern-.3em}^3$
minus the $z$ axis (see Figure \ref{figure:the braid
structure},) with the surface
${\cal F}$. The foliation of $\hbox{\sl I\kern-.18em R \kern-.3em}^3$ minus the $z$-axis
by half-planes is called the {\em standard braid
structure} on $\hbox{\sl I\kern-.18em R \kern-.3em}^3$ $-$ the
$z$-axis. This structure is given by half-planes with
boundaries on the $z$ axis. The surface used in
\cite{El} and
\cite{BM5} was a spanning surface for $K$; in
\cite{BM4} it was a 2-sphere which intersects $K$
twice; in
\cite{BM94} it was a torus in the complement of $K$; in
\cite{Et} and
\cite{Me} it is a torus $T\subset S^3$ on which $K$ is
embedded. Menasco also considers the foliation of a
meridian disc in the solid torus which $T$ bounds. The
characteristic foliation of a surface (associated to a
transversal knot or to another transversal or
Legendrian curve,) is a tool of study in contact
geometry. It was the main tool in the manuscripts
\cite{Be}, \cite{El} and
\cite{Et}. \\ \bigskip
In topological knot theory, one studies the topological
foliation of
${\cal F}$ defined above. The review article \cite{BF} may
be useful to the reader who is unfamiliar with this
area. The study of the topological foliations has
produced many results, for example the classification
of knots that are closed 3-braids \cite{BM3} and a
recognition algorithm for the unknot \cite{BH}. Braid
theory was also a major tool in the work in \cite{Be},
but it appears that Bennequin's detailed study of the
foliation is based entirely on the characteristic
foliation, as it occurs for knots in $\hbox{\sl I\kern-.18em R \kern-.3em}^3$ and
$S^3$. To the best of our knowledge this paper
contains the first application of braid foliations to
the study of transversal knots.
\\ \bigskip
\begin{figure}[htpb]
\centerline{\BoxedEPSF{braidstructure.eps scaled 700}}
\caption {Half-planes in the braid structure on
$\hbox{\sl I\kern-.18em R \kern-.3em}^3$.}
\label{figure:the braid structure}
\end{figure}
We note some similarities between the two foliations:
The characteristic foliation is oriented and the braid
foliation is orientable. (The orientation is ignored,
but a dual orientation, determined by an associated
flow, plays an equivalent role.) The foliations can
be made to agree in the limiting case, as
$\rho\to\infty$ (see the comments above Figure
\ref{figure:standard contact structure}).
\\ \bigskip
After an appropriate isotopy of
${\cal F}$ both foliations have no leaves that are simple
closed curves. Also, their singularities are finite in
number, each being either an elliptic point or a
hyperbolic point (the hyperbolic point corresponding to
a saddle-point tangency of
${\cal F}$ with the 2-planes of the structure). The signs
of the singularities of each foliation are determined
by identical considerations: the surface is naturally
oriented by the assigned orientation on the knot. If at
a singularity the orientation of the surface agrees
(resp. disagrees) with the orientation of the
foliation, then the singularity is positive (resp.
negative). See Figure
\ref{figure:singularity sign}. \\ \bigskip
\begin{figure}[htpb]
\centerline{\BoxedEPSF{hypellsign.eps scaled 800}}
\caption {A positive elliptic singularity (p) and a
positive hyperbolic singularity (h).}
\label{figure:singularity sign}
\end{figure}
In both foliations the hyperbolic singularities are
4-pronged singularities. If $s$ is a hyperbolic
singular point, then the four branches of the singular
leaf through $s$ end at either elliptic points or at a
point on $K$. (The condition that no singular leaves
of the characteristic foliation connect hyperbolic
points is a genericness assumption appearing in the
literature on Legendrian and transversal knots). The
three possible cases are illustrated in Figure
\ref{figure:the three types of hyperbolic
singularities}. In that figure the elliptic points are
depicted as circles surrounding
$\pm$ signs (the sign of the elliptic singularity) and
the hyperbolic singularities are depicted as black
dots. Two of the four branches of the singular leaf
end at positive elliptic points. The other two end at
either two negative elliptic points, or one negative
elliptic point and one point on
$K$, or two points on $K$. \\ \bigskip
\begin{figure}[htpb]
\centerline{\BoxedEPSF{singtypes.eps scaled 800}}
\caption {The three types of hyperbolic singularities.}
\label{figure:the three types of hyperbolic
singularities}
\end{figure}
There are also differences between the two foliations.
In the braid foliation, elliptic points always
correspond to punctures of the surface by the
$z$ axis. In the characteristic foliation, elliptic
points on the surface may or may not correspond to
punctures by the $z$-axis. That is, there may be elliptic
points not corresponding to punctures, and there might be
punctures not corresponding to elliptic points. Here is an
example. In the braid foliation, if there is a piece of the
surface along the boundary, foliated by a single positive pair
of elliptic and hyperbolic singularities, then the only
possible embedding for that piece is shown in Figure
\ref{figure:singularity sign}. On the other hand, in
the characteristic foliation, if there is a piece of
the surface, also along the boundary, also foliated by
a positive elliptic-hyperbolic pair, then the
corresponding embedding may or may not be the one shown
in Figure \ref{figure:singularity sign}. The
embeddings will coincide if the tangent to the surface
at the $z$ axis is horizontal. \\ \bigskip In work on the
braid foliation one uses certain properties that appear
to have been ignored in work based upon the
characteristic foliation. For example, the work on
braid foliations makes much of the distinction between
the three types of hyperbolic singularities which we
just illustrated in Figure
\ref{figure:the three types of hyperbolic
singularities}, calling them {\it types bb, ab} and
{\it aa}. The resulting combinatorics play a major
role in the study of braid foliations. It seems to us
that the distinction between $bb, ab$ and $aa$
singularities can also be made in the situation of the
characteristic foliation, but that this has not been
done.
\\ \bigskip
In the braid foliation the elliptic points have a
natural cyclic order on the $z$ axis, if we are considering the
ambient space as $S^3$ and the braid axis as one of the
core circles of the Hopf fibration, and the hyperbolic points
have a natural cyclic order in
$0\leq\theta\leq 2\pi$. These orderings do not seem
useful in the contact setting. On the other hand,
the characteristic foliation is invariant under
rotation by $\theta$ and translation by $z$, so the
interesting parameter seems to be the coordinate $\rho$.
\\ \bigskip
An essential tool in manipulating and simplifying the
characteristic foliation is the Giroux Elimination
Lemma (\cite{Gi}, \cite{El}), which allows one to
`cancel' pairs of same sign singularities. In
topological knot theory different modifications have
been introduced that are the braid foliation analogue
of isotopies of the Giroux Elimination Lemma, see
\cite{BM5} and also \cite{BF}. They are called {\em
$ab$ exchange moves} and {\em
$bb$ exchange moves}, and they use pairs of Giroux-like
cancellations, but on a much larger scale.
\\ \bigskip
\section{Exchange reducibility and transversal
simplicity.}
\label{section:exchange reducibility and transversal
simplicity}
Our initial goal is to motivate and define the concept
of exchange reducibility. Let
${\cal K}$ be a topological knot type and let $K$ be a
closed
$n$-braid representative of ${\cal K}$. We consider the
following three modifications of $K$
\begin{itemize}
\item Our first modification is {\em braid isotopy},
that is, an isotopy in the complement of the braid
axis. In \cite{Mo} it is proved that isotopy classes
of closed
$n$-braids are in one-to-one correspondence with
conjugacy classes in the braid group
$B_n$. Since the conjugacy problem in the braid group
is a solved problem, each conjugacy class can then be
replaced by a unique representative that can be assumed
to be transversal. Braid isotopy preserves the
Bennequin number since it preserves both braid index
and algebraic crossing number.
\item Our second move is {\em destabilization}. See
Figure
\ref{figure:destabilization and exchange moves}(a). The
box labeled $P$ contains an arbitrary
$(n-1)$-braid, and the label $n-2$ on the braid strand
denotes $n-2$ parallel braid strands. The
destabilization move reduces braid index from
$n$ to
$n-1$ by removing a `trivial loop'. If the trivial
loop contains a positive crossing, the move is called
a {\em positive} or $+$ destabilization. Positive
destabilization reduces algebraic crossing number and
preserves the Bennequin number. Negative ($-$)
destabilization increases the Bennequin number by 2.
\item Our third move is the {\em exchange move}. See
Figure
\ref{figure:destabilization and exchange moves}(b).
\begin{figure}[htpb]
\centerline{\BoxedEPSF{moves.eps scaled 700}}
\caption {(a) positive destabilization and (b) The
exchange move}
\label{figure:destabilization and exchange moves}
\end{figure} In general the exchange move changes
conjugacy class and so cannot be replaced by braid
isotopy. The exchange move preserves both braid index
and algebraic crossing number, hence preserves the
Bennequin number. \\ \bigskip
\end{itemize}
To motivate our definition of exchange reducibility, we
recall the following theorem, proved by the first
author and W. Menasco:
\\ \bigskip
{\bf Theorem A} (\cite{BM5}, with a simplified proof in
\cite{BF}): {\em Let $K$ be a closed
$n$-braid representative of the $m$-component unlink.
Then
$K$ may be simplified to the trivial $m$-braid
representative, i.e. a union of $m$ disjoint round
planar circles, by a finite sequence of the following
three changes: braid isotopies, positive and negative
destabilizations, and exchange moves.} \\ \bigskip
Motivated by Theorem A, we introduce the following
definition:\\ \bigskip
\underline{ Definition:} A knot type ${\cal K}$ is said to
be {\em exchange reducible} if an arbitrary closed
braid representative $K$ of arbitrary braid index
$n$ can be changed to an arbitrary closed braid
representative of minimum braid index
$n_{min}({\cal K})$ by a finite sequence of braid
isotopies, exchange moves and
$\pm$-destabilizations. Note that this implies that
any two minimal braid index representatives are either
identical or are {\em exchange-equivalent}, i.e., are
related by a finite sequence of braid isotopies and
exchange moves. \\ \bigskip
Our first result is:
\begin{theorem}
\label{theorem:realizing the maximal Bennequin number}
If ${\cal K}$ is an exchange reducible knot type, then the
maximum Bennequin number of ${\cal K}$ is realized by any
closed braid representative of braid index
$n_{min}({\cal K})$.
\end{theorem}
The proof of Theorem \ref{theorem:realizing the maximal
Bennequin number}
begins with a lemma. In what follows, we understand
``transversal isotopy" to mean a topological isotopy
that preserves the condition
$\alpha = \rho^2 d\theta + dz > 0$ at every point of
the knot and at every stage of the isotopy.
\begin{lemma}
\label{lemma:transversal isotopies} If a transversal
closed braid is modified by one of the following
isotopies, then the isotopy can be replaced by a
transversal isotopy:
{\rm (1)} Braid isotopy.
{\rm (2)} Positive stabilization or positive
destabilization.
{\rm (3)} An exchange move.
\end{lemma}
{\bf Proof of Lemma \ref{lemma:transversal isotopies}:}
\\ \bigskip
Proof of (1): Since the braid strands involved in the
isotopy will be
$\gg
\epsilon$ away from the z-axis at each stage (so
avoiding $-\rho^2 = 0$), any isotopy will be
transversal if we keep the strands involved "relatively
flat" ($dz/d\theta \sim 0$) at each stage. Since
everything is happening locally there is space to
flatten the strands involved without changing the
braid.\\ \bigskip
Proof of (2): See Figure
\ref{figure:destabilization}(a). Consider a single
trivial loop around the $z$ axis, with a positive
crossing.
\begin{figure}[htpb]
\centerline{\BoxedEPSF{destab.eps scaled 800}}
\caption {Destabilization, with a singularity at s,
where $d\theta = 0$ and $\rho = 0$.}
\label{figure:destabilization}
\end{figure} We have
$d\theta > 0$ along the entire length of the loop since
we are working with a closed braid. For a positive
crossing we have $dz
\geq 0$ throughout the loop as well. Therefore the
inequality
$dz/d\theta > - \rho ^2$ is true for all non-zero real
values of
$\rho$. Crossing the $z$ axis to destabilize the braid
results in at least one singular point, where
$d\theta = 0$, but if we continue to keep $dz
\geq 0$ then in the limit, as $-\rho ^2
\rightarrow 0$ from the negative real numbers,
$dz/d\theta$ goes to
$\infty$ through the positives. Therefore
$dz/d\theta \neq -\rho ^2$ at any stage in the
isotopy. \\ \bigskip
Proof of (3): The sequence of pictures in Figure
\ref{figure:exchange moves} shows that an exchange move
can be replaced by a sequence of the following moves:
isotopy in the complement of the
$z$ axis, positive stabilization, isotopy again, and
finally positive destabilization. Claim 3 then follows
from Claims 1 and 2. $\|$ \bigskip
\begin{figure}[htpb]
\centerline{\BoxedEPSF{exchange.eps scaled 700}}
\caption {An exchange move corresponds to a sequence
consisting of braid isotopies, a single positive
stabilization and a single positive destabilization.}
\label{figure:exchange moves}
\end{figure}
\begin{remark}: {\rm Observe that the argument given to
prove (2) simply doesn't work for negative
destabilization. See figure
\ref{figure:destabilization}(b). The singularity in
this destabilization is a point at which
$dz/d\theta = -\rho ^2$ in the limit. Indeed, a negative
destabilization can't be modified to one which is
transversal, because the Bennequin number (an invariant
of transversal knot type) changes under negative
destabilization.}
\end{remark}
\begin{remark} {\rm If we had chosen $\alpha < 0$ along
the knot, we would consider {\em negative}
stabilizations and destabilizations as transversal
isotopies and would use those instead of positive
stabilizations and destabilizations in the exchange
sequence. All the other moves translate to the
negative setting without change.}
\end{remark}
{\bf Proof of Theorem \ref{theorem:realizing the
maximal Bennequin number}:} Let $K$ be an arbitrary
closed braid representative of the exchange reducible
knot type ${\cal K}$. Let
$K_0$ be a minimum braid index representative of
${\cal K}$, obtained from $K$ by the sequence described in
the definition of exchange reducibility. We must prove
that the transversal knot
$TK_0$ associated to $K_0$ has maximum Bennequin
number for the knot type
${\cal K}$. Note that in general $K_0$ is not unique,
however it will not matter, for if $K_0'$ is a
different closed braid representative of minimal braid
index, then $K_0$ and $K_0'$ are related by a sequence
of braid isotopies and exchange moves, both of which
preserve both braid index and algebraic crossing
number, so $\beta(K_0') =
\beta(K_0)$.
\\ \bigskip
We obtain $K_0$ from
$K$ by a sequence of braid isotopies, exchange moves,
and
$\pm$-destabilizations. Braid isotopy, exchange moves
and positive destabilization preserve
$\beta(TK)$, but negative destabilization increases the
Bennequin number by 2, so the sequence taking $K$ to
$K_0$ changes the Bennequin number from
$\beta(TK) = c$ to
$\beta(TK_0) = c + 2p$, where p is the number of
negative destabilizations in the sequence. The
question then is whether $c + 2p$ is maximal for the
knot type ${\cal K}$. If
$c + 2p$ is less than maximal, then there exists some
other closed braid representative
$K^\prime$ of the knot type ${\cal K}$ with maximum
$\beta(TK^\prime) > \beta(TK_0)$. Since $K_0$ has
minimum braid index for the knot type
${\cal K}$, it must be that $n(K^\prime) \geq n(K_0)$. If
$n(K^\prime) = n(K_0)$, then the two braids are
equivalent by a sequence of Bennequin number preserving
exchange equivalences, so suppose instead that
$n(K^\prime) > n(K_0)$. Then, since $K^\prime$ is a
closed braid representative of the exchange reducible
knot type
${\cal K}$, there must exist a sequence of braid isotopies,
exchange moves, and
$\pm$-destabilizations taking $K^\prime$ to a minimum
braid index braid representative
$K_0 ^\prime$. (As above, we also take any $K_0 ^\prime
\in
{\cal M}_0$ ). Since
$\beta(TK^\prime)$ is assumed to be maximum, and none
of the moves in the sequence taking
$K^\prime$ to $K_0 ^\prime$ reduce Bennequin number, it
must be that
$\beta(TK_0 ^\prime) =
\beta(TK^\prime)$. But since $K_0 ^\prime$ and
$K_0$ are both minimum braid index representatives of
${\cal K}$, they must be equivalent by a sequence of
Bennequin number preserving exchange moves and
isotopies. Thus
$\beta(TK_0)=
\beta(TK_0^\prime)$. $\|$ \bigskip
Our next result is:
\begin{theorem}
\label{theorem:exchange reducible implies t-simple} If
${\cal TK}$ is a transversal knot type with associated
topological knot type
${\cal K}$, where ${\cal K}$ is exchange reducible, then
${\cal TK}$ is transversally simple.
\end{theorem}
The proof of Theorem \ref{theorem:exchange reducible
implies t-simple} begins with two lemmas. Our first
lemma had been noticed long ago by the first author and
Menasco, who have had a long collaboration on the
study of closed braid representatives of knots and
links. However, it had never been used in any of their
papers. It is therefore new to this paper, although we
are indebted to Menasco for his part in its
formulation. A contact-theory analogue of Lemma
\ref{lemma:Oliver's lemma}, below, appears as Lemma 3.8
of \cite{Et}.
\begin{lemma}
\label{lemma:exchange moves are the obstruction} Using
exchange moves and isotopy in the complement of the
braid axis, one may slide a trivial loop on a closed
braid from one location to another on the braid.
\end{lemma}
{\bf Proof of Lemma \ref{lemma:exchange moves are the
obstruction}:} See Figure
\ref{figure:aaexch}. It shows that, using braid
isotopy and exchange moves, we can slide a trivial
negative loop past any crossing to any place we wish
on the braid. The argument for sliding a positive
trivial loop around the braid is identical.
$\|$ \bigskip
\begin{figure}[htpb]
\centerline{\BoxedEPSF{aaexch.eps scaled 700}}
\caption {An exchange move that allows a negative
trivial loop to slide along a braid.}
\label{figure:aaexch}
\end{figure}
\begin{lemma}
\label{lemma:Oliver's lemma} Let $K_1$ and $K_2$ be
closed
$n$-braids that are exchange-equivalent. Let
$L_1$ and $L_2$ be $(n+1)$-braids that are obtained from
$K_1$ and $K_2$ by either negative stabilization on
both or positive stabilization on both. Then
$L_1$ and $L_2$ are exchange-equivalent.
\end{lemma}
{\bf Proof of Lemma \ref{lemma:Oliver's lemma}:} We
already know there is a way to deform $K_1$ to
$K_2$, using exchange equivalence. Each braid isotopy
may be broken up into a sequence of isotopies, each of
which only involves local changes on some well-defined
part of the braid. (For example, the defining
relations in the braid group are appropriate local
moves on cyclic braids). Similarly, exchange moves
have local support. It may happen that the trivial
loop which we added interferes with the support of one
of the isotopies or exchange moves. If so, then by
Lemma
\ref{lemma:exchange moves are the obstruction} we may
use exchange equivalence to slide it out of the way. It
follows that we may deform
$L_1$ to $L_2$ by exchange equivalence. $\|$ \bigskip
{\bf Proof of Theorem \ref{theorem:exchange reducible
implies t-simple}:} We are given an arbitrary
representative of the transversal knot type ${\cal TK}$. Let
${\cal K}$ be the associated topological knot type. By the
transversal Alexander's theorem
\cite{Be} we may modify our representative
transversally to a transversal closed
$n$-braid $K = TK$ that represents the transversal knot
type ${\cal TK}$ and the topological knot type
${\cal K}$. By the definition of exchange reducibility, we
may then find a finite sequence of closed braids
$$K = K_1\to K_2\to\dots \to K_{m-1}\to K_m,$$ all
representing
${\cal K}$, such that each $K_{i+1}$ is obtained from
$K_i$ by braid isotopy, a positive or negative
destabilization or an exchange move, and such that
$K_m$ is a representative of minimum braid index
$n_{min} = n_{min}({\cal K})$ for the knot type
${\cal K}$. The knots
$K_1,\dots,K_m$ in the sequence will all have the
topological knot type
${\cal K}$. \\ \bigskip
In general ${\cal K}$ will have more than one closed braid
representative of minimum braid index. Let
${\cal M}_0({\cal K})=\{M_{0,1},M_{0,2}\dots\}$ be the set of all
minimum braid index representatives of
${\cal K}$, up to braid isotopy. Clearly
$K_m\in{\cal M}_0$. {By \cite{Be}, each
$M_{0,i}\in{\cal M}_0$ may be assumed to be a transversal
closed braid.
\\ \bigskip
By Theorem \ref{theorem:realizing the maximal Bennequin
number} each
$M_{0,i}$ has maximal Bennequin number for all knots
that represent
${\cal K}$. In general this Bennequin number will not be the
same as the Bennequin number of the original
transversal knot type ${\cal TK}$. By Lemma
\ref{lemma:transversal isotopies} the moves that relate
any two
$M_{0,i}, M_{0,j}\in {\cal M}_0$ may be assumed to be
transversal. After all these modifications the closed
braids in the set ${\cal M}_0$ will be characterized, up to
braid isotopy, by their topological knot type
${\cal K}$, their braid index $n_{min}({\cal K})$ and their
Bennequin number
$\beta_{max}({\cal K})$. \\ \bigskip
If the transversal knot type ${\cal TK}$ had Bennequin number
$\beta_{max}({\cal K})$, it would necessarily follow that
${\cal TK}$ is characterized up to transversal isotopy by
its ordinary knot type and its Bennequin number. Thus
we have proved the theorem in the special case of
transversal knots that have maximum Bennequin number.
\\ \bigskip
We next define new sets ${\cal M}_1,{\cal M}_2,\dots$ of
transversal knots, inductively. Each
${\cal M}_s$ is a collection of conjugacy classes of closed
$(n_{min}({\cal K})+s)$-braids. We assume, inductively,
that the braids in
${\cal M}_s$ all have topological knot type
${\cal K}$, braid index
$n_{min}({\cal K})+s$ and Bennequin number
$\beta_{max}({\cal K})-2s$. Also, their conjugacy classes
differ at most by exchange moves. Also, the collection
of conjugacy classes of
$(n_{min}({\cal K})+s)$-braids in the set
${\cal M}_s$ is completely determined by the collection of
conjugacy classes of braids in the set
${\cal M}_0$. We now define the set
${\cal M}_{s+1}$ by choosing an arbitrary closed braid
$M_{i,s}$ in
${\cal M}_s$ and adding a trivial negative loop. Of
course, there is no unique way to do this, but by Lemma
\ref{lemma:exchange moves are the obstruction} we can
choose one such trivial loop and use exchange moves to
slide it completely around the closed braid
$M_{i,s}$. Each time we use the exchange move of Lemma
\ref{lemma:exchange moves are the obstruction}, we will
obtain a new conjugacy class, which we then add to the
collection ${\cal M}_{s+1}$. The set
${\cal M}_{s+1}$ is defined to be the collection of all
conjugacy classes of closed braids obtained by adding
trivial loops in every possible way to each
$M_{i,s}\in{\cal M}_s$. The closed braids in
${\cal M}_{s+1}$ are equivalent under braid isotopy and
exchange moves. They all have topological knot type
${\cal K}$, braid index $n_{min}({\cal K})+s+1$, and Bennequin
number
$\beta_{max}({\cal K}) - 2(s + 1)$. The collection of closed
braids in the set
${\cal M}_{s+1}$ is completely determined by the collection
of closed braids in
${\cal M}_s$, and so by the closed braids in
${\cal M}_0$.
\\ \bigskip
In general negative destabilizations will occur in the
chain $K_1\to K_m$. Our plan is to change the order of
the moves in the sequence
$K_1\to K_m$, pushing all the negative destabilizations
to the right until we obtain a new sequence, made up
of two subsequences:
$$K = K_1^\star\to K_2^\star\to\dots\to K_r^\star=
K_0^\prime\to\dots
\to K_s^\prime = K_p,$$ where $K_p$ has minimum braid
index
$n_{min}({\cal K})$. The first subsequence
${\cal S}_1$, will be
$K = K_1^\star\to K_2^\star\to\dots\to K_r^\star$,
where every
$K_i^\star$ is a transversal representative of
${\cal TK}$ and the connecting moves are braid isotopy,
positive destabilizations and exchange moves. The
second subsequence, ${\cal S}_2$, is
$K_r^\star = K_0^\prime\to\dots \to K_q^\prime$, where
every
$K_{i+1}^\prime$ is obtained from
$K_i^\prime$ by braid isotopy and a single negative
destabilization. Also,
$K_q^\prime$ has minimum braid index
$n_{min}({\cal K})$.
\\ \bigskip
To achieve the modified sequence, assume that
$K_i\to K_{i+1}$ is the first negative destabilization.
If the negative trivial loop does not interfere with
the moves leading from
$K_{i+1}$ to $K_m$, just renumber terms so that the
negative destabilization becomes
$K_m$ and every other
$K_j, j>i$ becomes $K_{j-1}$. But if it does interfere,
we need to slide it out of the way to remove the
obstruction. We use Lemma \ref {lemma:exchange moves
are the obstruction} to do that, adding exchange moves
as required. \\ \bigskip
So we may assume that we have our two subsequences
${\cal S}_1$ and
${\cal S}_2$. The braids in ${\cal S}_1$ are all transversally
isotopic and so they all have the same Bennequin number
and they all represent
${\cal TK}$. The braids $K^\prime_i
\in {\cal S}_2$ all have the same knot type, but
$\beta(K^\prime_{i+1}) = \beta(K^\prime_i) +2$ for each
$i = 1,
\dots, s-1$. With each negative destabilization and
braid isotopy, the Bennequin number increases by 2 and
the braid index decreases by 1. Each braid represents
the same knot type
${\cal K}$ but a different transversal knot type
${\cal T}{\cal K}$. \\ \bigskip
Our concern now is with ${\cal S}_2$, i.e.
$K_r^\star = K_0^\prime\to\dots \to K_s^\prime$, where
every
$K_{i+1}^\prime$ is obtained from $K_i^\prime$ by a
single negative destabilization and braid isotopy.
The number of negative destabilizations in subsequence
${\cal S}_2$ is exactly one-half the difference between the
Bennequin number
$\beta({\cal TK})$ of the original transversal knot
${\cal TK}$ and the Bennequin number
$\beta_{max}({\cal K})$. \\ \bigskip
Let us now fix on any particular minimum braid index
representative of
${\cal K}$ as a minimum braid index closed braid
representative of the transversal knot type that
realizes
$\beta_{max}({\cal K})$. It will not matter which we choose,
because all belong to the set
${\cal M}_0$ and so are exchange-equivalent. We may then
take the final braid
$K_r^\star$ in ${\cal S}_1$, which is the same as the
initial braid
$K_0^\prime$ in ${\cal S}_2$, as our representative of
${\cal TK}$, because it realizes the minimal braid index
for ${\cal TK}$ and by our construction, any other such
representative is related to the one we have chosen by
transversal isotopy. We may also proceed back up the
sequence
${\cal S}_2$ from
$K_s^\prime$ to a new representative that is obtained
from
$K_0^\prime$ by adding $s$ negative trivial loops,
one at a time. By repeated application of Lemma
\ref{lemma:Oliver's lemma} we know that choosing any
other element of
${\cal M}_0$ will take us to an exchange-equivalent element
of ${\cal M}_s$. In this way we arrive in the set
${\cal M}_s$, which also contains $K_r^\star$, and which is
characterized by ${\cal K}$ and
$\beta$. The proof of Theorem
\ref{theorem:exchange reducible implies t-simple} is
complete.
$\|$ \bigskip
\section{Examples, applications and possible
generalizations.}
\label{section:examples} In this section we discuss
examples which illustrate Theorems
\ref{theorem:realizing the maximal Bennequin number}
and
\ref{theorem:exchange reducible implies t-simple}.
\subsection{The unlink and the unknot.} Theorem A,
quoted earlier in this manuscript, asserts that the
$m$-component unlink, for $m \geq 1$, is exchange
reducible. In considering a link transversally, it
should be mentioned that we are assuming each of the
components of the link satisfy the same inequality
$\alpha > 0$. We also need to define the Bennequin
number properly for this transversal link. The natural
way to do so, suggested by Oliver Dasbach, is by the
following method. For a crossing involving two
different components of the link, assign
$\pm 1/2$ to each component depending on the sign of
the crossing. Assign $\pm 1$ to each crossing
consisting of strands from the same component, as in
the case of a knot. Then the Bennequin number of each
component is the difference between the algebraic
crossing number
$e$ (a sum of $\pm 1$'s and
$\pm 1/2$'s) and $n$, the braid index of that
component. Define the Bennequin number of the link to
be the collection of the Bennequin numbers of the
components of the link. The following corollary is an
immediate consequence of Theorems
\ref{theorem:realizing the maximal Bennequin number}
and A:
\begin{corollary}
\label{corollary:the unlink is transversally simple}
The $m$-component unlink, $m\geq 1$, is transversally
simple. In particular, the unknot is transversally
simple.
\end{corollary} Note that Corollary
\ref{corollary:the unlink is transversally simple}
gives a new proof of a theorem of Eliashberg \cite{El}.
\subsection{Torus knots and iterated torus knots} In the manuscript
\cite{Et} J. Etnyre proved that positive torus knots
are transversally simple. His proof failed
for negative torus knots, but he conjectured that the
assertion was true for all torus knots and possibly
also for all iterated torus knots. In an early draft
of this manuscript we conjectured that torus knots and
iterated torus knots ought to be exchange reducible,
and sketched our reasons. Happily, the conjecture is
now a fact, established by W. Menasco in \cite{Me}.
Two corollaries follow. To state and prove our first
corollary, we need to fix our conventions for the description of torus
knots and iterated torus knots.
\\ \bigskip
\underline{Definition:} Let $U$ be the unit circle in the plane $z=0$,
and let
$N(U)$ be a solid torus of revolution with
$U$ as its core circle. Let
$\lambda_0$ be a longitude for $U$, i.e.
$\lambda_0$ is a circle in the
plane $z=0$ which lies on $\partial N(U)$, so
that $U$ and
$\lambda_0$ are concentric circles in the plane $z=0$. See Figure
\ref{figure:the standard solid torus}. A {\em torus knot} of {\em type}
$e(p,q)$, where $e=\pm$, on $\partial N(U)$, denoted
$K_{e(p,q)}$, is the closed $p$-braid
$(\sigma_1\sigma_2\cdots\sigma_{p-1})^{eq}$ on
$\partial N(U)$, where
$\sigma_1,\dots,\sigma_{p-1}$ are elementary braid
generators of the braid group $B_{p}$. Note that
$K_{e(p,q)}$ intersects the curve $\lambda_0$ in
$q$ points, and note that the algebraic crossing
number of its natural closed braid projection on the
plane $z=0$ is $e(p-1)q$. The knot
$K_{e(p,q)}$ also has a second natural closed braid
representation, with the unknotted circle $U$ as braid axis and the closed
$q$-braid $(\sigma_1\sigma_2\cdots\sigma_{q-1})^{ep}$ as closed
braid representative. Since $p$ and $q$ are coprime integers, one of these
closed braids will have smaller braid index than the other, and without loss
of generality we will assume in the pages which follow that we have
chosen $p$ to be smaller than $q$, so that $K_{e(p,q)}$ has braid index $p$.
\begin{figure}[htpb]
\centerline{\BoxedEPSF{iterated0.eps scaled 700}}
\caption {The standard solid torus $N(U)$, with
$K_{+(2,3)}\subset\partial N(K)$}
\label{figure:the standard solid torus}
\end{figure}
\underline{Definition}: We next define what we mean by
an {\em $e(s,t)$-cable} on a knot $X$ in 3-space. Let
$X$ be an arbitrary oriented knot in oriented
$S^3$, and let $N(X)$ be a solid torus neighborhood of
$X$ in 3-space. A {\em longitude} $\lambda$ for $X$ is
a simple closed curve on
$\partial N(X)$ which is homologous to
$X$ in $N(X)$ and null-homologous in
$S^3\setminus X$. Let $f:N(U)\to N(X)$ be a
homeomorphism which maps
$\lambda_0$ to $\lambda$. Then
$f(K_{e(s,t)})$ is an $e(s,t)$-{\em cable} about
$X$.\\ \bigskip
\underline{Definition}: Let $\{e_i(p_i,q_i), i =
1,\dots,r\}$ be a choice of signs
$e_i=\pm$ and coprime positive integers
$(p_i,q_i)$, ordered so that for each
$i$ we have $p_i,q_i>0$. An {\it iterated torus knot}
$K(r)$ of type
$(e_1(p_1,q_1),\dots,e_r(p_r,q_r))$, is defined
inductively by:
\begin{itemize}
\item $K(1)$ a torus knot of type
$e_1(p_1,q_1)$, i.e. a type $e(p_1,q_1)$ cable on the unknot $U$. Note that,
by our conventions, $p_1<q_1$.
\item $K(i)$ is an $e_i(p_i,q_i)$ cable about
$K(i-1)$. We place no restrictions on the relative magnitudes of
$p_i$ and $q_i$ when $i>1$.
\end{itemize}
Here is one of the simplest non-trivial examples of an
iterated torus knot. Let $K(1)$ be the positive
trefoil, a torus knot of type $(2,3)$. See
Figure
\ref{figure:trefoil as a torus knot}(a) and (b). Note
that in the left sketch the core circle is our unit
circle $U$, while in the right sketch the core circle
is the knot $K(1)$. The iterated torus knot
$K(2) = K_{(2,3),-(3,4))}$ is the $-(3,4)$ cable about
$K(1)$. See Figure \ref{figure:(3,4) cabling}. \\ \bigskip
\begin{figure}[htpb]
\centerline{\BoxedEPSF{iterated1.eps scaled 600}}
\caption {(a) the torus knot $K_{+(2,3)}$. (b)the
solid torus neighborhood $N(K(1))$ of
$K(1)$, with core circle $K(1)$ and longitude
$\lambda_1$ marked.}
\label{figure:trefoil as a torus knot}
\end{figure}
\begin{figure}[htpb!]
\centerline{\BoxedEPSF{iterated2.eps scaled 550}}
\caption {(a) the torus knot of type -(3,4) embedded in
$\partial N(U)$, (b) the iterated torus knot
$K(2) = K_{+(2,3), -(3,4)}$}
\label{figure:(3,4) cabling}
\end{figure}
In \cite{Me}, it was shown that the braid
foliation machinery used for the torus in \cite{BM94}
could be adapted to the situation in which the knot is
on the surface of the torus. The main result of that
paper is the following theorem. \\ \bigskip
{\bf Theorem:} (\cite{Me}, Theorem 1) {\it Oriented iterated torus
knots are exchange reducible.}\\ \bigskip
Combining Menasco's Theorem with Theorem
\ref{theorem:exchange reducible implies t-simple}, we
have the following immediate corollary:
\begin{corollary}
\label{corollary:iterated torus knots are transversally
simple} Iterated torus knots are transversally simple.
\end{corollary}
Our next contribution to the theory of iterated torus knots requires that we
know the braid index of an iterated torus knot. The formula is implicit in the
work of Schubert \cite{Sch}, but does not appear explicitly there.
\begin{lemma}
\label{lemma:braid index of iterated torus knots}
Let $K(r) = K_{e_1(p_1,q_1),\dots,e_r(p_r,q_r)}$ be an r-times iterated torus
knot. Then the braid index of
$K(r)$ is $p_1p_2\cdots p_r$.
\end{lemma}
{\bf Proof:} \ We begin with the case $r=1$. By hypothesis $p_1<q_1$, also the torus
knot $K_{e_1(p_1,q_1)}$ is represented by a $p_1$-braid
$(\sigma_1\sigma_2\cdots\sigma_{p_1-1})^{e_1q_1}$. By the formula given in
\cite{Jo} for the HOMFLY polynomial of torus knots, together with the
Morton-Franks-Williams braid index inequality (discussed in detail in
\cite{Jo}), it follows that this knot cannot be represented as a closed
$m$-braid for any $m<p_1$. \\ \bigskip
Passing to the general case, Theorem 21.5 of \cite{Sch} tells us that the
torus knot $K_{e_1(p_1,q_1)}$ and the array of integers
$e_1(p_2,q_2),\dots,e_r(p_r,q_r)$ form a complete system of invariants of the
iterated torus knot $K(r) = K_{e_1(p_1,q_1),\dots,e_r(p_r,q_r)}$. Lemma
23.4 of \cite{Sch} tells us that, having chosen a $p_1$-braid representative
for $K_{e_1(p_1,q_1)}$, there is a natural $p_1p_2\cdots p_r$-braid
representative of $K(r)$. This representative is the only one on this
number of strings, up to isotopy in the complement of the braid axis. Theorem
23.1 of \cite{Sch} then asserts that $K(r)$ also cannot be represented as a
closed braid with fewer strands. That is, its braid index is $p_1p_2\cdots
p_r$. $\|$ \bigskip
{\bf Remark:} The iterated torus knot $K_r$ has two natural closed braid
representatives. The first is a $p_1p_2\cdots p_r$- braid which has
the core circle $U'$ of the unknotted solid torus $S^3\setminus N(U)$ as braid
axis. The second is a $q_1p_2\cdots p_r$-braid which has the core circle
$U$ of the unknotted solid torus $N(U)$ as braid axis. In the case
$r=1$, the second choice gives a closed braid
which is reducible in braid index, i.e. it has $q_1-p_1$ trivial loops. From
this it follows that if $r>1$ it will have $(q_1-p_1)p_2\cdots p_r$ trivial
loops, thus the second closed braid representation is reducible to the first.
\
We are now ready to state our second corollary about iterated torus knots. Let
$\chi$ be the Euler characteristic of an oriented surface of minimum genus
bounded by $K(r)$.
\begin{corollary}
\label{corollary:maximum Bennequin numbers for iterated
torus knots}
Let $K(r) = K_{e_1(p_1,q_1),\dots,e_r(p_r,q_r)}$ be an iterated torus knot,
where $p_1<q_1$. Then the maximum Bennequin number of
$K(r) = K_{e_1(p_1,q_1),\dots,e_r(p_r,q_r)}$ is given by the following two
equivalent formulas:
\begin{enumerate}
\item [{\rm (1)}] $\beta_{max}(K(r)) = a_r-p_1p_2\cdots p_r,$ where
$a_r = \sum_{i=1}^r e_iq_i(p_i - 1)p_{i+1}p_{i+2}\ldots p_r.$
\item [{\rm (2)}] $\beta_{max}(K(r)) = -\chi - d$,
where $d = \sum_{i=1}^r {(1-e_i)(p_i - 1)q_ip_{i+1}p_{i+2}
\ldots p_r}$.
\item [{\rm (3)}] Moreover, the upper bound in the inequality $\beta_{max}(K(r))
\leq -\chi$ is achieved if and only if all of the $e_i's$ are positive.
\end{enumerate}
\end{corollary}
\
{\bf Proof:} \ We begin with the proof of (1). By Lemma \ref{lemma:braid index
of iterated torus knots} the braid index of $K(r)$ is $p_1p_2\cdots p_r$.
Therefore
$\beta_{max}(K(r)) = a_r - p_1p_2\cdots p_r$, where
$a_r$ is the algebraic crossing number of the unique $p_1p_2\cdots p_r$-braid
representative of $K(r)$. To compute $a_r$ we proceed inductively. If
$r=1$ then $K(1)$ is a type $e_1(p_1,q_1)$ torus knot, which is
represented by the closed $p_1$-braid
$(\sigma_1\sigma_2\dots\sigma_{p-1})^{e_1q_1}$. Its
algebraic crossing number is $a_1 = e_1(p_1-1)q_1$. \\ \bigskip
The knot $K(i)$ is an $e_i(p_i,q_i)$ cable on
$K(i-1)$. Note that $K_{e_i(p_i,q_i)} \subset
\partial N(K_0)$, also
$K_{e_i(p_i,q_i)}$ is a $p_i$-braid, also
$K_{e_i(p_i,q_i)}\cap\lambda_0$ consists of
$q_i$ points. We shall think of the projection of the
braided solid torus
$N(K(i-1))$, which is a $p_1p_2\cdots p_{i-1}$-braid,
as being divided into three parts. The reader may find
it helpful to consult Figures
\ref{figure:iterated torus knots} (a), (b), (c) as we
examine the contributions to $a_i$ from each part.
\begin{figure}[htpb]
\centerline{\BoxedEPSF{iterated3.eps scaled 800}}
\caption {Iterated torus knots.}
\label{figure:iterated torus knots}
\end{figure}
\begin{itemize}
\item [(a)] The first part of $N(K(i-1))$ is the
trivial $p_1p_2\cdots p_{i-1}$-braid. The longitude
$\lambda_{i-1}$ is parallel to the core circle of
$N(K(i-1))$ in this part. The surface $\partial
N(K(i-1))$ contains on one of its $p_1p_2\cdots
p_{i-1}$ cylindrical branches the image under $f$ of
the braided part of
$K_{e_r(p_r,q_r)}$. See Figure
\ref{figure:iterated torus knots}(a), which shows the
braid when
$e_i(p_i,q_i) = -(3,4)$. This part of
$K(i)$ contributes
$e_i(p_i-1)q_i$ to $a_i$. Note that there are
$q_i$ points where $f(K_{e_i(p_i,q_i)})$ intersects
$\lambda_{i-1}.$
\item [(b)] The second part of $N(K(i-1))$ contains all
of the braiding in $K(i-1)$, and so also in
$N(K(i-1))$. In Figure
\ref{figure:iterated torus knots}(b) we have
illustrated a single crossing in $K(i-1)$ and the
associated segments of $N(K(i-1))$. We show a single
crossing of
$\lambda_{i-1}$ (as a thick line) over
$K(i-1)$. The single signed crossing contributes
$p_i^2$ crossings to
$K(i)$, so the total contribution from all of the
crossings in
$K(i-1)$ will be $a_{i-1}p_i^2$. The illustration
shows the case
$p_i=3$.
\item [(c)] The third part of $N(K(i-1))$ is again the
trivial
$p_1p_2\cdots p_{i-1}$-braid. It contains corrections
to the linking number of
$\lambda_{i-1}$ with $K(i-1)$ which result from the
fact that a curve which is everywhere `parallel' to the
core circle will have linking number $a_{i-1}$, not 0,
with
$K(i-1)$. To correct for this, we must allow the
projected image of
$\lambda_{i-1}$ to loop around $\partial N(K(i-1))$
exactly
$-a_{i-1}$ times, so that its total linking number with
$K(i-1)$ is zero. See the left sketch in Figure
\ref{figure:iterated torus knots}(c), which shows the 3
positive loops which occur if $a_{i-1}=-3$.
We have already introduced $q_i$ intersections between
$f(K_{e_i(p_i,q_i)})$ and $\lambda_{i-1}$, and
therefore we must avoid any additional intersections
which might arise from the
$-a_{i-1}$ loops. See the right sketch in Figure
\ref{figure:iterated torus knots}(c). When
$\lambda_{i-1}$ wraps around $N(K(i-1))$ the additional
$-a_{i-1}$ times, the
$p_i$-braid $f(K_{e_i(p_i,q_i)}$ must follow. Each loop
in
$\lambda_{i-1}$ introduces $(p_i-1) + (p_i-2) + \cdots
+ 2 + 1 =
\frac{p_i(p_i-1)}{2}$ crossings per half-twist in
$f(K_{e_i(p_i,q_i)})$. Since there is a full twist to
go around the positive loop this number is doubled to
$p_i(p_i-1)$. We have shown the 12 crossings in
$K(i)$ which come from a single loop in
$\lambda_{i-1}$ when $p_i=4$. The total contribution is
$-a_{i-1}p_i(p_i-1)$.
\end{itemize}
Adding up all these contributions we obtain
$$a_i = e_i(p_i-1)q_i +
a_{i-1}(p_i)^2-a_{i-1}(p_i^2 - p_i) = e_i(p_i-1)q_i +
a_{i-1}p_i.$$ Summing the various terms to compute $a_r$, we have proved part
(1) of the Corollary.
\bigskip
The proof of (2) will follow from that of (1) if we can show that:
$$ \chi = p_1p_2\cdots p_r - d - a_r,$$
where $d = \sum_{i=1}^r {(1-e_i)(p_i - 1)q_ip_{i+1}p_{i+2}
\ldots p_r}$. To see this, we must find a natural surface of minimum genus
bounded by $K(r)$ and compute its Euler characteristic. By Theorem 12, Lemma
12.1 and Theorem 22 of \cite{Sch}, a surface of minimum genus bounded by $K(r)$
may be constructed by Seifert's algorithm, explained in Chapter 5 of
\cite{Ro}, from a representative of K(r) which has minimal braid index. We
constructed such a representative in our proof of Part (1). To compute its
Euler characteristic, use the fact that $\chi$ is the number of Seifert circles
minus the number of unsigned crossings (Exercises 2 and 10 on pages 119 and 121
of \cite{Ro}). By a theorem of Yamada
\cite{Ya} the number of Seifert circles is the same as the braid index, i.e.
$p_1p_2\cdots p_r$ in our situation. The number of unsigned crossings is
$b_r$, where $b_i = (p_i-1)q_i + b_{i-1}p_i$ and
$b_1 = (p_1-1)q_1$ and $\chi = p_1p_2\cdots p_r - b_r$.
Adding up the contributions from all the
$b_i's$ we get $b_r = \sum_{i=1}^r {(p_i - 1)q_ip_{i+1}p_{i+2}
\ldots p_r}$ which can be rewritten as $\sum_{i=1}^r
{(1-e_i+e_i)(p_i - 1)q_ip_{i+1}p_{i+2} \ldots p_r}$.
Separating terms:
$$b_r =
\sum_{i=1}^r {(1-e_i)(p_i - 1)q_ip_{i+1}p_{i+2} \ldots p_r}+
\sum_{i=1}^r {(e_i)(p_i - 1)q_ip_{i+1}p_{i+2} \ldots p_r} = d
+ a_r.$$
The claimed formula for $\chi$ follows.\\ \bigskip
To prove (3), observe that the only case when $\beta_{max} = -\chi$ exactly
occurs when $d=0$, i.e. the sum
$\sum_{i=1}^r {(1-e_i)(p_i - 1)q_ip_{i+1}p_{i+2} \ldots p_r}
= 0$, That is, all the
$e_i$'s are $+1$. $\|$ \bigskip
\subsection{Knots that are not exchange reducible}
\label{subsection:knots that are not exchange
reducible} A very naive conjecture would be that all
knots are exchange reducible, however that is far from
the truth. We begin with a simple example. In the
manuscript \cite{BM3} Birman and Menasco studied knots
that are represented by closed 3-braids, up to braid
isotopy, and identified the proper subset of those
knots whose minimum braid index is 3 (i.e. not 2 or 1).
They prove that the knots that have minimum braid index
representatives of braid index 3 fall into two groups:
those that have a unique such representative (up to
braid isotopy) and infinitely many examples that have
exactly two distinct representatives, the two being
related by 3-braid {\em flypes}. A flype is the knot
type preserving isotopy shown in Figure
\ref{figure:the 3-braid flype}. Notice that the flype
is classified as {\em positive} or {\em negative}
depending on the sign of the isolated crossing. After
staring at the figure, it should become clear to the
reader that the closed braids in a `flype pair' have
the same topological knot type. We say that a braid
representative {\em admits a flype} if it is conjugate
to a braid that has the special form illustrated in
Figure
\ref{figure:the 3-braid flype}.
\begin{figure}[htpb]
\centerline{\BoxedEPSF{flype.eps scaled 700}}
\caption {Closed 3-braids that are related by a flype.}
\label{figure:the 3-braid flype}
\end{figure}
\begin{theorem} {\rm \cite{BM3}}
\label{theorem:3 braid examples} The infinite sequence
of knots of braid index 3 in
{\rm \cite{BM3}}, each of which has two closed 3-braid
representatives, related by flypes, are examples of
knots that are {\em not} exchange reducible.
\end{theorem}
{\bf Proof:} \ It is proved in \cite{BM3} that for all but an
exceptional set of
$P,Q,R$ the closed braids in a flype pair are in
distinct conjugacy classes. Assume from now on that a
`flype pair' means one of these non-exceptional pairs.
Since conjugacy classes are in one-to-one
correspondence with braid isotopy equivalence classes,
it follows that the braids in a flype pair are not
related by braid isotopy. On the other hand, it is
proved in
\cite{BM3} that when the braid index is
$\leq 3$ the exchange move can always be replaced by
braid isotopy, so the braids in a flype pair cannot be
exchange equivalent. $\|$ \bigskip
On closer inspection, it turns out that a positive
flype can be replaced by a sequence of braid isotopies
and positive stabilizations and destabilizations, which
shows that it is a transversal isotopy. See Figure
\ref{figure:positive flypes are transversal}.
\begin{figure}[htpb]
\centerline{\BoxedEPSF{flypeiso.eps scaled 700}}
\caption {A positive flype can be replaced by a
sequence of transversal isotopies.}
\label{figure:positive flypes are transversal}
\end{figure}
The figure is a generalization of Figure
\ref{figure:exchange moves} (proving that exchange
moves are transversal), because flypes are
generalizations of exchange moves. We replace one of
the
$\sigma^{\pm1}_n$ with the braid word we label
$R$. A negative flype also has a replacement sequence
similar to the one pictured in Figure
\ref{figure:positive flypes are transversal}, but the
stabilizations and destabilizations required are
negative. Therefore the negative flype sequence
cannot be replaced by a transversal isotopy using
these methods. There may well be some other
transversal isotopy that can replace a negative flype,
but we did not find one. Thus we are lead to the
following conjecture:
\\ \bigskip
{\bf Conjecture:} {\em Any transversal knot type whose
associated topological knot type
${\cal K}$ has a minimum braid index representative that
admits a negative flype is {\it not} transversally
simple.} \\ \bigskip
The simplest example which illustrates our conjecture
is shown in Figure \ref{figure:flype example}. \\ \bigskip
\begin{figure}[htpb]
\centerline{\BoxedEPSF{flype-ex.eps scaled 600}}
\caption {The simplest example which illustrates our
conjecture}
\label{figure:flype example}
\end{figure}
The essential difficulty we encountered in our attempts
to prove or disprove this conjecture is that the only
effective invariants of transversal knot type that are
known to us at this writing are the topological knot
type and the Bennequin number, but they do not
distinguish these examples. \\ \bigskip
\subsection{Knots with infinitely many transversally
equivalent closed braid representatives, all of minimal
braid index:} At this writing the only known examples
of transversally simple knots are iterated torus knots.
By Theorem 24.4 of \cite{Sch} iterated torus knots have
unique closed braid representative of minimum braid
index, and it follows from this and Theorem
\ref{theorem:realizing the maximal Bennequin number}
that they have unique representatives of maximum
Bennequin number. It seems unlikely to us that all
transversally simple knots have unique closed braid
representatives of minimum braid index, and we now
explain our reasons.
\\ \bigskip
The exchange move was defined in Figure
\ref{figure:destabilization and exchange moves}(b) of
$\S$\ref{section:exchange reducibility and transversal
simplicity}. It seems quite harmless, being nothing
more than a special example of a Reidemeister move of
type II. It also seems unlikely to produce infinitely
many examples of anything, however that is exactly what
happens when we combine it with braid isotopy. See
Figure
\ref{figure:infinitely many closed braids} with
$n = 4$.
\begin{figure}[htpb]
\centerline{\BoxedEPSF{exchiso.eps scaled 700}}
\caption {The exchange move and braid isotopy can lead
to infinitely many distinct closed
$n$-braid representatives of a single knot type. }
\label{figure:infinitely many closed braids}
\end{figure}
Proceeding from the right to the left and following
the arrows, we see how braid isotopy and exchange moves
can be used to produce infinitely many examples of
closed braids which are transversally equivalent. It is
not difficult to choose the braids $R$ and
$S$ in Figure \ref{figure:infinitely many closed
braids} so that the resulting closed braids are all
knots, and also so that they actually have braid index
4, and also so that they are in infinite many distinct
braid isotopy classes (using an invariant of Fiedler
\cite{Fi} to distinguish the braid isotopy classes).
We omit details because, at this writing, we do not
know whether the knots in question are exchange
reducible, so we cannot say whether they all realize
the maximum Bennequin number for their associated knot
type.
\subsection{Generalizing the concept of `exchange
reducibility'}
\label{subsection:remarks on exchange reducibility}
Some remarks are in order on the concept of `exchange
reducibility'. Define two closed braids
$A\in B_n$, $A'\in B_m$ to be {\em Markov-equivalent}
if the knot types defined by the closed braids
coincide. Markov's well-known theorem (see
\cite{Ma35}) asserts that
$Markov$-equivalence is generated by braid isotopy,
$\pm$-stabilization, and $\pm$-destabilization.
However, when studying this equivalence relation one
encounters the very difficult matter that
$\pm$-stabilization is sensitive to the exact spot on
the closed braid at which one attaches the trivial
loop. On the other hand, Lemma
\ref{lemma:exchange moves are the obstruction} shows
that exchange moves are the obstruction to moving a
trivial loop from one spot on a knot to another.
Therefore if we allow exchange moves in addition to
braid isotopy, $\pm$-stabilization, and
$\pm$-destabilization, one might hope to avoid the need
for stabilization. That is the idea behind the
definition of exchange reducibility, and behind the
proof of Theorem A. However, that hope is much too
naive, as was shown by the examples in
$\S$\ref{subsection:knots that are not exchange
reducible}.
\\ \bigskip
A way to approach the problem of transversal knots is
to augment the definition of exchange reducible by
allowing additional `moves'. In their series of papers
{\em Studying knots via closed braids I-VI}, the first
author and Menasco have been working on generalizing
the main result in Theorem A to all knots and links. In
the forthcoming manuscript
\cite{BM01}, a general version of the `Markov theorem
without stabilization' is proved. The theorem states
that for each braid index
$n$ a finite set of new moves suffices to reduce any
closed braid representative of any knot or link to
minimum braid index `without stabilization'. These
moves include not only exchange moves and positive and
negative flypes, but more generally {\em handle
moves} and {\em G-flypes}. Handle moves can always be realized
transversally. The simplest example of a G-flype is the 3-braid
flype that is pictured in Figure
\ref{figure:the 3-braid flype}, with weights assigned to
the strands. This will change it to an $m$-braid flype, for any
$m$. But other examples exist, and they are much more
complicated. We note, because it is relevant to the
discussion at hand, that any {\em positive} G-flype can be realized
by a transversal isotopy. The sequence in Figure
\ref{figure:positive flypes are transversal} is a proof
of the simplest case. Awaiting the completion of
\cite{BM01}, we leave these matters for future investigations.
|
1,108,101,566,728 | arxiv | \section{Introduction}
Can cats or dogs do barter? Trade is one of the essential feature of human intelligence. \cite{10.1371/journal.pone.0001518} reported that chimpanzees who are an intelligent species like human have the ability to trade, but are reluctant to trade. This subsequently led to a major divergence in the fates of the two species. The market is nothing but an expression of human intelligence. Intelligence does not arise only in individual brains; it also arises in groups of individuals~(\cite{malone2015handbook}). The securities market, including crypto-assets, is the ultimate expression of human intelligence. As considered by \cite{fama1965behavior}, if the security market is efficient in the strong or semi-strong sense, information on securities instantaneously changes the traders’ subjective equilibrium and the differences in the speed with which they respond to the information decides who are the winners and losers. If the security market is efficient in the weak sense, the market equilibrium trends according to changes in the traders’ subjective equilibrium because of noise. The efficiency of security market can be summarized as follows.
\begin{itemize}
\item Information is incorporated correctly into the price.
\item Information is incorporated rapidly into the price.
\item Arbitrage deals profit if there are errors or delays.
\item Countless traders are always looking for arbitrage opportunities.
\item The arbitrage opportunity is gradually lost and the market becomes more efficient.
\end{itemize}
Although the standard theory draws a story above, transaction is not established if there is no noise and price will stay distorted if bias is strong.
In an efficient market where noise and bias have no effect and information is perfectly symmetrical, security prices should accurately reflect only information. However, if traders had perfectly and simultaneously symmetrical knowledge and information, including the asset valuation model, the transaction will not function because the traders' valuation of the asset would be the same. They must be doing some rational calculations via intelligence in natural and digital computing. In this sense, a rational representative agent in macroeconomics corresponds to the perfect symmetry of information~(\cite{Lucas1976EconometricPE, Kirman1992WhomOW, 10.2307/2138488}).
While \cite{black1986noise} treated this ``symmetry breaking'' as noise, the effect of noise on a security's price is expected to be symmetrical based on its nature. But if there exist so many irrational noise traders synchronizing erroneous stochastic beliefs, both affect prices and earn higher expected returns, the unpredictability of noise traders' beliefs creates a risk in the price of the asset that deters rational arbitrageurs from aggressively betting against them. As a result, prices can diverge significantly from fundamental values even in the absence of fundamental risk. Moreover, bearing a disproportionate amount of risk that they themselves create enables noise traders to earn a higher expected return than rational investors do~(\cite{doi:10.1086/261703}).
On the other hand, \cite{doi:10.1126/science.185.4157.1124} found the effect of bias is asymmetric. They described three heuristics that are employed in making judgments under uncertainty: 1) representativeness; 2) availability of instances or scenarios; and 3) adjustment from an anchor. These heuristics are highly economical and usually effective, but they lead to systematic and predictable errors. But the effect of these heuristics has not been detected as a global bias in the security market. Consequently, identifying the specific effects of noise and bias on security price is challenging.
If attention is paid to any statistical property in any complex system, the log-normal distribution is the most natural and appropriate among the standard or ‘‘normal’’ statistics to overview the whole system~(\cite{doi:10.1143/JPSJ.80.072001}). The log-normality emerges as familiar and typical examples of statistical aspects in various complex systems. Since every member of any complex system has its own history, each member is in the process of growth (or retrogression). The log-normal distribution is realized as a result of Gibrat' law, or Mathew effect. It is applied to cities size and growth rate, where proportionate growth process may give rise to a distribution of city sizes that is log-normal. When considering the entire size distribution, not just the largest cities, then the city size distribution is log-normal~(\cite{10.2307/2296055}). However, it has been argued that it is problematic to define cities through their fairly arbitrary legal boundaries. According to \cite{10.1162/003355399556133},
Zipf's law is a very tight constraint on the class of admissible models of local growth. It says that for most countries the size distribution of cities strikingly fits a power law: the number of cities with populations greater than S is proportional to 1/S. Suppose that, at least in the upper tail, all cities follow some proportional growth process (this appears to be verified empirically). This automatically leads their distribution to converge to Zipf's law.
Gibrat's law of proportionate effect also states that the proportional change in the size of a firm is independent of its absolute size. An implication of this is that large and small firms have the same average proportionate rates of growth. Against this law, \cite{10.2307/2296055} shows large firms are growing faster significantly. \cite{10.1257/mac.20150051} construct a tractable neoclassical growth model that generates Pareto's law of income distribution and Zipf's law of the firm size distribution from idiosyncratic, firm-level productivity shocks. Executives and entrepreneurs invest in risk-free assets, as well as their own firms' risky stocks, through which their wealth and income depend on firm-level shocks. By using the model, they evaluate how changes in tax rates can account for the evolution of top incomes in the United States. The model matches the decline in the Pareto exponent of the income distribution and the trend of the top 1 percent income share in recent decades. In the same research direction, \cite{NIREI201625} construct a neoclassical growth model with heterogeneous households that accounts for the Pareto distributions of income and wealth in the upper tail. In an standard Bewley model~(\cite{BEWLEY1977252}), they feature households' business productivity risks and borrowing constraints, which they find generate the Pareto distributions. Households with low productivity rely on wages and returns from safe assets, while high productivity households choose not to diversify their business risks. Their model can quantitatively account for the observed income distribution in the U.S. under reasonable calibrations. Furthermore, they conduct several comparative statics to examine how changes in parameters affect the Pareto distributions. In particular, they find that the change in the top tax rates in the 1980s potentially accounts for much of the observed increase in top income dispersion in the last decades. Their analytical result provides a coherent interpretation for the numerical comparative statics.
In this article, I present the true fundamentals hypothesis based on rational expectations~(\cite{10.2307/1909635}) and, using a log-normal distribution model, detect global bias components from the price-earnings (P/E), price-to-book (P/B), and price-to-cash flow (P/CF) ratios. The traditional theory of the firm is based on the assumption that the firm acts in the stockholders' interests and that stockholders are interested in profit, so that the object of the firm is to maximize profit. However, in fact, there is a certain range in the profit concept~(\cite{10.2307/2977477}). The analysis results strongly support the true fundamentals hypothesis as the detected biases show similar characteristics. Additionally, the results show that the cash flow indicators contain relatively few bias components and are closer to the true fundamentals. I further demonstrate and examine why the positive P/IC ratio among the indicators analyzed is a proxy for the true fundamentals that does not include bias components.
\section{Hypothesis}
When the true fundamentals of listed companies at time $t$ is denoted as $X_t$ and their growth rate is denoted as $R_t$, the growth of those companies can be expressed by the following Gibrat's process
\begin{equation}
X_t=R_tX_{t-1} \label{1}.
\end{equation}
It is important to note, however, that I assume that the growth rates
$R_t$s are mutually independent random variables that follow the same
distribution with finite variance. The initial value of the fundamentals
set as $X_0$ yields
\begin{equation}
X_T=X_0\prod_{t=1}^T R_t
\label{2}
\end{equation}
at time $T$. Taking the log of both sides of the equation results in
\begin{equation}
\log X_T =\log X_0+\log R_1+\cdots +\log R_T.
\label{3}
\end{equation}
Therefore,
\begin{equation}
\log X_T\sim LN(\mu, \sigma^2)
\label{4}
\end{equation}
would hold true for a sufficiently large $T$ based on the central limit theorem. Essentially, the true fundamentals $X_T$ of listed companies follows the log-normal distribution.
Furthermore, by assuming rational expectations through the future point in time $T$ as of the present point in time 0 on the premise of a going concern, the following equation becomes true:
\begin{equation}
X_0=\mathbb{E}[X_T]\prod_{t=1}^T \mathbb{E}[R_t^{-1}].
\label{5}
\end{equation}
Therefore, the rational expectations $X_0$ for true fundamentals also follow the log-normal distribution. In other words, the true fundamentals $X_t$ at a given point in time $t$ follow the log-normal distribution.
\section{Analysis}
\subsection{Data}
Among the actual fundamental indicators, the price-earnings (P/E), forward price-earnings (P/FE), price-to-book (P/B), price-to-cash flows from operating activities (P/OC), price-to-cash flows from investing activities (P/IC), price-to-cash flows from financing activities (P/FC), and the price-to-cash equivalents at the year-end (P/CE) ratios are analyzed. The cash flow ratios were divided into positive and negative data. The daily data with absolute values less than 1,000 were extracted for all companies listed in Japan for the period spanning 1,817 business days from January 2007 to May 2014.
\subsection{Test}
If the true fundamentals hypothesis is true, the actual fundamental indicators should follow the log-normal distribution. Therefore, the $p$-values were calculated using the Kolmogorov-Smirnov test, Pearson's $\chi^2$ test, and Anderson-Darling test, to test the goodness of fit to the log-normal distribution. It is recognized that the results of these tests are strongly affected by the sample size.
Since the sample size differs by indicator, samples at the 300 quartiles were extracted to calculate the $p$-value for testing the goodness of fit of the overall average of each indicator's samples to the log-normal distribution with variance. In all cases, the null hypothesis is that ``the indicator follows the log-normal distribution,'' or the otherwise worded ``the indicator is a proxy of true fundamentals.'' The indicator reflects the true fundamentals more when the $p$-value is closer to 1. When the $p$-value is closer to 0, the bias is stronger. The lowest $p$-value for each test is shown in Table~\ref{tab1} for the data spanning 1,817 business days.
The test results indicate that bias has a strong effect on the P/E, P/FE, and P/B ratios. It is consistent with our instincts that forward price-earnings ratio (P/FE) is most biased. Interestingly, among all price-to-cash flow ratios, the bias is significant only for P/OC ratio. It can be easier to manipulate than others. I illustrate the time series of $p$-values since the $p$-value is an indicator of the strength of bias. From table~\ref{tab1}, Figure~\ref{bias1} and Figure~\ref{bias2}, it is clear that the positive P/IC ratio is the best proxy for the true fundamentals that does not include bias components. Probably, it should be difficult to manipulate according to the opinions of management and accounting.
\begin{table}[ht]
\caption{The lowest $p$-values for the data spanning 1,817 business days. Samples at the 300 quantile. Significant (bias) levels: 1\%(***), 3\%(**), 5\%(*)\label{tab1}}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\begin{tabularx}{\textwidth}{llll}
\toprule
\textbf{Index} &\qquad \textbf{Kolmogorov-Smirnov} &\qquad\quad \textbf{Peason's $\chi^2$} &\qquad \textbf{Anderson-Darling}\\
\midrule
P/OC$+$ &\qquad\qquad\qquad 0.0986 &\qquad\qquad 0.2843 &\qquad\qquad\qquad 0.0398$^{*}$ \\
P/OC$-$ &\qquad\qquad\qquad 0.0634 &\qquad\qquad 0.0045$^{***}$ &\qquad\qquad\qquad 0.0850 \\
P/IC$+$ &\qquad\qquad\qquad 0.6205 &\qquad\qquad 0.4441 &\qquad\qquad\qquad 0.7201 \\
P/IC$-$ &\qquad\qquad\qquad 0.2787 &\qquad\qquad 0.7012 &\qquad\qquad\qquad 0.1849 \\
P/FC$+$ &\qquad\qquad\qquad 0.2042 &\qquad\qquad 0.2287 &\qquad\qquad\qquad 0.1093 \\
P/FC$-$ &\qquad\qquad\qquad 0.2522 &\qquad\qquad 0.5763 & \qquad\qquad\qquad 0.1269 \\
P/CE &\qquad\qquad\qquad 0.4611 &\qquad\qquad 0.8740 &\qquad\qquad\qquad 0.2376 \\
P/E &\qquad\qquad\qquad 0.0170$^{**}$ &\qquad\qquad 0.0000$^{***}$ &\qquad\qquad\qquad 0.0018$^{***}$ \\
P/FE &\qquad\qquad\qquad 0.0107$^{**}$ &\qquad\qquad 0.0040$^{***}$ &\qquad\qquad\qquad 0.0019$^{***}$ \\
P/B &\qquad\qquad \qquad0.0032$^{***}$ &\qquad\qquad 0.0023$^{***}$ &\qquad\qquad\qquad 0.0005$^{***}$ \\
\bottomrule
\end{tabularx}
\end{table}
\unskip
\begin{figure}[ht]
\begin{center}
\includegraphics[width=11 cm]{bias1.pdf}
\caption{Time series of the goodness of fit test to the log-normal distribution. Blue: Kolmogorov-Smirnov, Yellow: Peason's $\chi^2$, Blue: Anderson-Darling. The higher the value, the less bias there is, vice versa. \label{bias1}}
\end{center}
\end{figure}
\unskip
\subsection{Shape of Bias}
The detected biases exhibit similar characteristics. To facilitate an intuitive understanding of these results, each indicator is shown in figure~\ref{bias2} and is compared to the log-normal distribution. While the figures only show the data for the first day of the period, January 4, 2007, the characteristics of the biases themselves remain essentially unchanged throughout the period although the levels fluctuated. It is worth noting that the shape of the curve, and in particular the strength of the bias, is independent of whether the economy is doing well or not. The stock market, which is clearly artificial, appears to have a natural intelligence.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=10 cm]{bias2.pdf}
\caption{The more the data deviates from a straight line showing a log-normal distribution, the stronger the bias. These graphs are drawn with data from the first day of the time series. The shape of the bias varies proportionally to Figure~\ref{bias1}, but is stable throughout the time. \url{https://figshare.com/projects/Intelligence_and_Global_Bias_in_the_Stock_Market/131711}\label{bias2}}
\end{center}
\end{figure}
\unskip
\section{Discussion and Conclusion}
The results indicate that the major fundamental indicators including the P/E, P/FE, and P/B ratios are strongly affected by bias. Bias also has a significant effect on the positive and negative P/OC ratios. Additionally, there may be a weak effect of bias on the negative P/IC, positive and negative P/FC, and P/CE ratios.
When the test results are compared, the positive P/IC ratio is the stable proxy of true fundamentals among all these indicators, and it is important to consider why this occurs.
Positive cash flows from investing activities represent the realized gain or loss from past investments such as marketable securities, tangible fixed assets, the sales of investment securities, and income from the collection on loans declared at the end of accounting period. In other words, positive cash flow is the indicator that most directly reflects the past business decisions that determine the growth of a company. The year-end cash equivalent ratio and other cash flow ratios can be interpreted to strongly reflect the true fundamentals because they have less bias components due to the strong characteristic of having definite value.
On the other hand, although the P/E ratio and P/B ratio are definite values, investors might not view them as indicators that reflect the true fundamentals since there is a high degree of freedom in accounting. The P/OC ratio might also have a lower credibility compared to other cash flow indicators.
Where do these biases come from? The answer is extremely simple: ``Cash is a fact, profit is an opinion.'' Namely, opinions of management and accountant are added as noise to true fundamentals. As a result, Kesten process~(\cite{10.1007/BF02392040})
\begin{equation}
X_t = R_t X_{t-1}+\epsilon_t,\quad E[\epsilon_t]>0
\end{equation}
is realized and the Pareto distribution is to be obtained. This means their opinions are accompanied by a positive bias.
In fact, these biases fit the Pareto distribution quite well. The generalized Pareto distribution (GPD) is represented by the following functions.
\paragraph{Generalized Pareto Distribution (GPD)}
\[
F(x)=1-\left(1+\left(\frac{x-\mu}{\kappa}\right)^{1/\gamma}\right)^{-\alpha}
\]
\[
f(x)=\frac{\alpha}{\gamma}\kappa^{-1/\gamma}(x-\mu)^{-1+\frac{1}{\gamma}}\left(1+\left(\frac{x-\mu}{\kappa}\right)^{1/\gamma}\right)^{-\alpha-1}
\]
\begin{figure}[ht]
\begin{center}
\includegraphics[width=11cm]{bias3.pdf}
\caption{Comparison of three distributions: Log-Normal(blue), GDP-1(Yellow), GDP-2(Green)\label{bias3}}
\end{center}
\end{figure}
\unskip
\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{bias4.pdf}
\caption{Left: GDP-1 to fit P/FE ($\kappa=18.82,\alpha=1, \gamma=0.385, \mu=0.993$), Right: GDP-2 to fit P/FE ($\kappa=13.70,\alpha=0.515, \gamma=0.238, \mu=0.993$)\label{bias4}. Note that P/FE has the strongest bias.}
\end{center}
\end{figure}
What are the implications of the results? Does the existence of bias negate the efficient market hypothesis? From the view point of intelligence in natural and digital computing, how should we deal with this problem?
The stock market, which is clearly artificial, and where countless traders are using digital computers to trade, appears to have a natural intelligence. Intelligence does not arise only in individual brains; it also arises in groups of individuals~(\cite{malone2015handbook}). If the true fundamentals hypothesis is really true, the major fundamental indicators including the P/E, P/FE, and P/B ratios are strongly affected by bias. This means that investors might not view them as indicators that reflect the true fundamentals since there is a high degree of freedom in accounting. Globally, this implies that the efficient market hypothesis holds~(\cite{fama1965behavior}). The shape of the curve, and in particular the strength of the bias, is stable throughout the time and independent of whether the economy is doing well or not. The curve with bias is the path of the ants~(\cite{kirman1993ants, app12147019}) to avoid obstacles, so to speak, and traders move back and forth along it in response to noise~(\cite{black1986noise, doi:10.1086/261703}). This noise, among other things, plays a role in activating the market system, which knows true fundamentals of firms and represents as a stable global bias in the stock market.
\newpage
\bibliographystyle{unsrtnat}
\section{Introduction}
Can cats or dogs do barter? Trade is one of the essential feature of human intelligence. \cite{10.1371/journal.pone.0001518} reported that chimpanzees who are an intelligent species like human have the ability to trade, but are reluctant to trade. This subsequently led to a major divergence in the fates of the two species. The market is nothing but an expression of human intelligence. Intelligence does not arise only in individual brains; it also arises in groups of individuals~(\cite{malone2015handbook}). The securities market, including crypto-assets, is the ultimate expression of human intelligence. As considered by \cite{fama1965behavior}, if the security market is efficient in the strong or semi-strong sense, information on securities instantaneously changes the traders’ subjective equilibrium and the differences in the speed with which they respond to the information decides who are the winners and losers. If the security market is efficient in the weak sense, the market equilibrium trends according to changes in the traders’ subjective equilibrium because of noise. The efficiency of security market can be summarized as follows.
\begin{itemize}
\item Information is incorporated correctly into the price.
\item Information is incorporated rapidly into the price.
\item Arbitrage deals profit if there are errors or delays.
\item Countless traders are always looking for arbitrage opportunities.
\item The arbitrage opportunity is gradually lost and the market becomes more efficient.
\end{itemize}
Although the standard theory draws a story above, transaction is not established if there is no noise and price will stay distorted if bias is strong.
In an efficient market where noise and bias have no effect and information is perfectly symmetrical, security prices should accurately reflect only information. However, if traders had perfectly and simultaneously symmetrical knowledge and information, including the asset valuation model, the transaction will not function because the traders' valuation of the asset would be the same. They must be doing some rational calculations via intelligence in natural and digital computing. In this sense, a rational representative agent in macroeconomics corresponds to the perfect symmetry of information~(\cite{Lucas1976EconometricPE, Kirman1992WhomOW, 10.2307/2138488}).
While \cite{black1986noise} treated this ``symmetry breaking'' as noise, the effect of noise on a security's price is expected to be symmetrical based on its nature. But if there exist so many irrational noise traders synchronizing erroneous stochastic beliefs, both affect prices and earn higher expected returns, the unpredictability of noise traders' beliefs creates a risk in the price of the asset that deters rational arbitrageurs from aggressively betting against them. As a result, prices can diverge significantly from fundamental values even in the absence of fundamental risk. Moreover, bearing a disproportionate amount of risk that they themselves create enables noise traders to earn a higher expected return than rational investors do~(\cite{doi:10.1086/261703}).
On the other hand, \cite{doi:10.1126/science.185.4157.1124} found the effect of bias is asymmetric. They described three heuristics that are employed in making judgments under uncertainty: 1) representativeness; 2) availability of instances or scenarios; and 3) adjustment from an anchor. These heuristics are highly economical and usually effective, but they lead to systematic and predictable errors. But the effect of these heuristics has not been detected as a global bias in the security market. Consequently, identifying the specific effects of noise and bias on security price is challenging.
If attention is paid to any statistical property in any complex system, the log-normal distribution is the most natural and appropriate among the standard or ‘‘normal’’ statistics to overview the whole system~(\cite{doi:10.1143/JPSJ.80.072001}). The log-normality emerges as familiar and typical examples of statistical aspects in various complex systems. Since every member of any complex system has its own history, each member is in the process of growth (or retrogression). The log-normal distribution is realized as a result of Gibrat' law, or Mathew effect. It is applied to cities size and growth rate, where proportionate growth process may give rise to a distribution of city sizes that is log-normal. When considering the entire size distribution, not just the largest cities, then the city size distribution is log-normal~(\cite{10.2307/2296055}). However, it has been argued that it is problematic to define cities through their fairly arbitrary legal boundaries. According to \cite{10.1162/003355399556133},
Zipf's law is a very tight constraint on the class of admissible models of local growth. It says that for most countries the size distribution of cities strikingly fits a power law: the number of cities with populations greater than S is proportional to 1/S. Suppose that, at least in the upper tail, all cities follow some proportional growth process (this appears to be verified empirically). This automatically leads their distribution to converge to Zipf's law.
Gibrat's law of proportionate effect also states that the proportional change in the size of a firm is independent of its absolute size. An implication of this is that large and small firms have the same average proportionate rates of growth. Against this law, \cite{10.2307/2296055} shows large firms are growing faster significantly. \cite{10.1257/mac.20150051} construct a tractable neoclassical growth model that generates Pareto's law of income distribution and Zipf's law of the firm size distribution from idiosyncratic, firm-level productivity shocks. Executives and entrepreneurs invest in risk-free assets, as well as their own firms' risky stocks, through which their wealth and income depend on firm-level shocks. By using the model, they evaluate how changes in tax rates can account for the evolution of top incomes in the United States. The model matches the decline in the Pareto exponent of the income distribution and the trend of the top 1 percent income share in recent decades. In the same research direction, \cite{NIREI201625} construct a neoclassical growth model with heterogeneous households that accounts for the Pareto distributions of income and wealth in the upper tail. In an standard Bewley model~(\cite{BEWLEY1977252}), they feature households' business productivity risks and borrowing constraints, which they find generate the Pareto distributions. Households with low productivity rely on wages and returns from safe assets, while high productivity households choose not to diversify their business risks. Their model can quantitatively account for the observed income distribution in the U.S. under reasonable calibrations. Furthermore, they conduct several comparative statics to examine how changes in parameters affect the Pareto distributions. In particular, they find that the change in the top tax rates in the 1980s potentially accounts for much of the observed increase in top income dispersion in the last decades. Their analytical result provides a coherent interpretation for the numerical comparative statics.
In this article, I present the true fundamentals hypothesis based on rational expectations~(\cite{10.2307/1909635}) and, using a log-normal distribution model, detect global bias components from the price-earnings (P/E), price-to-book (P/B), and price-to-cash flow (P/CF) ratios. The traditional theory of the firm is based on the assumption that the firm acts in the stockholders' interests and that stockholders are interested in profit, so that the object of the firm is to maximize profit. However, in fact, there is a certain range in the profit concept~(\cite{10.2307/2977477}). The analysis results strongly support the true fundamentals hypothesis as the detected biases show similar characteristics. Additionally, the results show that the cash flow indicators contain relatively few bias components and are closer to the true fundamentals. I further demonstrate and examine why the positive P/IC ratio among the indicators analyzed is a proxy for the true fundamentals that does not include bias components.
\section{Hypothesis}
When the true fundamentals of listed companies at time $t$ is denoted as $X_t$ and their growth rate is denoted as $R_t$, the growth of those companies can be expressed by the following Gibrat's process
\begin{equation}
X_t=R_tX_{t-1} \label{1}.
\end{equation}
It is important to note, however, that I assume that the growth rates
$R_t$s are mutually independent random variables that follow the same
distribution with finite variance. The initial value of the fundamentals
set as $X_0$ yields
\begin{equation}
X_T=X_0\prod_{t=1}^T R_t
\label{2}
\end{equation}
at time $T$. Taking the log of both sides of the equation results in
\begin{equation}
\log X_T =\log X_0+\log R_1+\cdots +\log R_T.
\label{3}
\end{equation}
Therefore,
\begin{equation}
\log X_T\sim LN(\mu, \sigma^2)
\label{4}
\end{equation}
would hold true for a sufficiently large $T$ based on the central limit theorem. Essentially, the true fundamentals $X_T$ of listed companies follows the log-normal distribution.
Furthermore, by assuming rational expectations through the future point in time $T$ as of the present point in time 0 on the premise of a going concern, the following equation becomes true:
\begin{equation}
X_0=\mathbb{E}[X_T]\prod_{t=1}^T \mathbb{E}[R_t^{-1}].
\label{5}
\end{equation}
Therefore, the rational expectations $X_0$ for true fundamentals also follow the log-normal distribution. In other words, the true fundamentals $X_t$ at a given point in time $t$ follow the log-normal distribution.
\section{Analysis}
\subsection{Data}
Among the actual fundamental indicators, the price-earnings (P/E), forward price-earnings (P/FE), price-to-book (P/B), price-to-cash flows from operating activities (P/OC), price-to-cash flows from investing activities (P/IC), price-to-cash flows from financing activities (P/FC), and the price-to-cash equivalents at the year-end (P/CE) ratios are analyzed. The cash flow ratios were divided into positive and negative data. The daily data with absolute values less than 1,000 were extracted for all companies listed in Japan for the period spanning 1,817 business days from January 2007 to May 2014.
\subsection{Test}
If the true fundamentals hypothesis is true, the actual fundamental indicators should follow the log-normal distribution. Therefore, the $p$-values were calculated using the Kolmogorov-Smirnov test, Pearson's $\chi^2$ test, and Anderson-Darling test, to test the goodness of fit to the log-normal distribution. It is recognized that the results of these tests are strongly affected by the sample size.
Since the sample size differs by indicator, samples at the 300 quartiles were extracted to calculate the $p$-value for testing the goodness of fit of the overall average of each indicator's samples to the log-normal distribution with variance. In all cases, the null hypothesis is that ``the indicator follows the log-normal distribution,'' or the otherwise worded ``the indicator is a proxy of true fundamentals.'' The indicator reflects the true fundamentals more when the $p$-value is closer to 1. When the $p$-value is closer to 0, the bias is stronger. The lowest $p$-value for each test is shown in Table~\ref{tab1} for the data spanning 1,817 business days.
The test results indicate that bias has a strong effect on the P/E, P/FE, and P/B ratios. It is consistent with our instincts that forward price-earnings ratio (P/FE) is most biased. Interestingly, among all price-to-cash flow ratios, the bias is significant only for P/OC ratio. It can be easier to manipulate than others. I illustrate the time series of $p$-values since the $p$-value is an indicator of the strength of bias. From table~\ref{tab1}, Figure~\ref{bias1} and Figure~\ref{bias2}, it is clear that the positive P/IC ratio is the best proxy for the true fundamentals that does not include bias components. Probably, it should be difficult to manipulate according to the opinions of management and accounting.
\begin{table}[ht]
\caption{The lowest $p$-values for the data spanning 1,817 business days. Samples at the 300 quantile. Significant (bias) levels: 1\%(***), 3\%(**), 5\%(*)\label{tab1}}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\begin{tabularx}{\textwidth}{llll}
\toprule
\textbf{Index} &\qquad \textbf{Kolmogorov-Smirnov} &\qquad\quad \textbf{Peason's $\chi^2$} &\qquad \textbf{Anderson-Darling}\\
\midrule
P/OC$+$ &\qquad\qquad\qquad 0.0986 &\qquad\qquad 0.2843 &\qquad\qquad\qquad 0.0398$^{*}$ \\
P/OC$-$ &\qquad\qquad\qquad 0.0634 &\qquad\qquad 0.0045$^{***}$ &\qquad\qquad\qquad 0.0850 \\
P/IC$+$ &\qquad\qquad\qquad 0.6205 &\qquad\qquad 0.4441 &\qquad\qquad\qquad 0.7201 \\
P/IC$-$ &\qquad\qquad\qquad 0.2787 &\qquad\qquad 0.7012 &\qquad\qquad\qquad 0.1849 \\
P/FC$+$ &\qquad\qquad\qquad 0.2042 &\qquad\qquad 0.2287 &\qquad\qquad\qquad 0.1093 \\
P/FC$-$ &\qquad\qquad\qquad 0.2522 &\qquad\qquad 0.5763 & \qquad\qquad\qquad 0.1269 \\
P/CE &\qquad\qquad\qquad 0.4611 &\qquad\qquad 0.8740 &\qquad\qquad\qquad 0.2376 \\
P/E &\qquad\qquad\qquad 0.0170$^{**}$ &\qquad\qquad 0.0000$^{***}$ &\qquad\qquad\qquad 0.0018$^{***}$ \\
P/FE &\qquad\qquad\qquad 0.0107$^{**}$ &\qquad\qquad 0.0040$^{***}$ &\qquad\qquad\qquad 0.0019$^{***}$ \\
P/B &\qquad\qquad \qquad0.0032$^{***}$ &\qquad\qquad 0.0023$^{***}$ &\qquad\qquad\qquad 0.0005$^{***}$ \\
\bottomrule
\end{tabularx}
\end{table}
\unskip
\begin{figure}[ht]
\begin{center}
\includegraphics[width=11 cm]{bias1.pdf}
\caption{Time series of the goodness of fit test to the log-normal distribution. Blue: Kolmogorov-Smirnov, Yellow: Peason's $\chi^2$, Blue: Anderson-Darling. The higher the value, the less bias there is, vice versa. \label{bias1}}
\end{center}
\end{figure}
\unskip
\subsection{Shape of Bias}
The detected biases exhibit similar characteristics. To facilitate an intuitive understanding of these results, each indicator is shown in figure~\ref{bias2} and is compared to the log-normal distribution. While the figures only show the data for the first day of the period, January 4, 2007, the characteristics of the biases themselves remain essentially unchanged throughout the period although the levels fluctuated. It is worth noting that the shape of the curve, and in particular the strength of the bias, is independent of whether the economy is doing well or not. The stock market, which is clearly artificial, appears to have a natural intelligence.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=10 cm]{bias2.pdf}
\caption{The more the data deviates from a straight line showing a log-normal distribution, the stronger the bias. These graphs are drawn with data from the first day of the time series. The shape of the bias varies proportionally to Figure~\ref{bias1}, but is stable throughout the time. \url{https://figshare.com/projects/Intelligence_and_Global_Bias_in_the_Stock_Market/131711}\label{bias2}}
\end{center}
\end{figure}
\unskip
\section{Discussion and Conclusion}
The results indicate that the major fundamental indicators including the P/E, P/FE, and P/B ratios are strongly affected by bias. Bias also has a significant effect on the positive and negative P/OC ratios. Additionally, there may be a weak effect of bias on the negative P/IC, positive and negative P/FC, and P/CE ratios.
When the test results are compared, the positive P/IC ratio is the stable proxy of true fundamentals among all these indicators, and it is important to consider why this occurs.
Positive cash flows from investing activities represent the realized gain or loss from past investments such as marketable securities, tangible fixed assets, the sales of investment securities, and income from the collection on loans declared at the end of accounting period. In other words, positive cash flow is the indicator that most directly reflects the past business decisions that determine the growth of a company. The year-end cash equivalent ratio and other cash flow ratios can be interpreted to strongly reflect the true fundamentals because they have less bias components due to the strong characteristic of having definite value.
On the other hand, although the P/E ratio and P/B ratio are definite values, investors might not view them as indicators that reflect the true fundamentals since there is a high degree of freedom in accounting. The P/OC ratio might also have a lower credibility compared to other cash flow indicators.
Where do these biases come from? The answer is extremely simple: ``Cash is a fact, profit is an opinion.'' Namely, opinions of management and accountant are added as noise to true fundamentals. As a result, Kesten process~(\cite{10.1007/BF02392040})
\begin{equation}
X_t = R_t X_{t-1}+\epsilon_t,\quad E[\epsilon_t]>0
\end{equation}
is realized and the Pareto distribution is to be obtained. This means their opinions are accompanied by a positive bias.
In fact, these biases fit the Pareto distribution quite well. The generalized Pareto distribution (GPD) is represented by the following functions.
\paragraph{Generalized Pareto Distribution (GPD)}
\[
F(x)=1-\left(1+\left(\frac{x-\mu}{\kappa}\right)^{1/\gamma}\right)^{-\alpha}
\]
\[
f(x)=\frac{\alpha}{\gamma}\kappa^{-1/\gamma}(x-\mu)^{-1+\frac{1}{\gamma}}\left(1+\left(\frac{x-\mu}{\kappa}\right)^{1/\gamma}\right)^{-\alpha-1}
\]
\begin{figure}[ht]
\begin{center}
\includegraphics[width=11cm]{bias3.pdf}
\caption{Comparison of three distributions: Log-Normal(blue), GDP-1(Yellow), GDP-2(Green)\label{bias3}}
\end{center}
\end{figure}
\unskip
\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{bias4.pdf}
\caption{Left: GDP-1 to fit P/FE ($\kappa=18.82,\alpha=1, \gamma=0.385, \mu=0.993$), Right: GDP-2 to fit P/FE ($\kappa=13.70,\alpha=0.515, \gamma=0.238, \mu=0.993$)\label{bias4}. Note that P/FE has the strongest bias.}
\end{center}
\end{figure}
What are the implications of the results? Does the existence of bias negate the efficient market hypothesis? From the view point of intelligence in natural and digital computing, how should we deal with this problem?
The stock market, which is clearly artificial, and where countless traders are using digital computers to trade, appears to have a natural intelligence. Intelligence does not arise only in individual brains; it also arises in groups of individuals~(\cite{malone2015handbook}). If the true fundamentals hypothesis is really true, the major fundamental indicators including the P/E, P/FE, and P/B ratios are strongly affected by bias. This means that investors might not view them as indicators that reflect the true fundamentals since there is a high degree of freedom in accounting. Globally, this implies that the efficient market hypothesis holds~(\cite{fama1965behavior}). The shape of the curve, and in particular the strength of the bias, is stable throughout the time and independent of whether the economy is doing well or not. The curve with bias is the path of the ants~(\cite{kirman1993ants, app12147019}) to avoid obstacles, so to speak, and traders move back and forth along it in response to noise~(\cite{black1986noise, doi:10.1086/261703}). This noise, among other things, plays a role in activating the market system, which knows true fundamentals of firms and represents as a stable global bias in the stock market.
\newpage
\bibliographystyle{unsrtnat}
|
1,108,101,566,729 | arxiv | \section{Introduction}
The process of stellar mass accretion is an important aspect of star
formation that is still to be fully understood. Not only is accretion
responsible for building up the young star to its final mass, but it
is also responsible for powering the mass outflows observed from such
systems, which in turn remove the excess angular momentum, and prevent
the star from spin-up. In addition, understanding the mass accretion
rate will significantly impact on the understanding of the inner disk,
disk evolution and the eventual formation of planets. Stellar
accretion can be briefly summarised as follows: material passes from
the envelope through the accretion disk -- which is truncated at a
radius $R_{\mathrm{in}}$ due to the strong magnetic field of the central
star. At this inner region of the disk, the material then flows along
the star-disk magnetic field lines (flux tubes) -- at $\sim$free-fall
velocity -- onto the central star, where a strong shock is formed as the
accreting material impacts the stellar surface.
UV veiling, H$\alpha$, Pa$\beta$, Br$\gamma$, \ion{O}{i}, \ion{Ca}{ii}
and \ion{He}{i} emission lines have all previously been used to probe
the accretion scenario in young stars and Brown Dwarfs (e.g. Natta et
al. \cite{natta04,natta06}; Mohanty et al. \cite{subu03,subu05}). Such
studies are based on the hypothesis that the UV veiling arises from
the hot shock region (e.g. Calvet \& Gullbring \cite{calvet}), whereas
the broad permitted emission lines exhibited by CTTSs arise from the
infalling magnetospheric flow (Muzerolle et
al. \cite{muzerolle98}). The \ion{Ca}{ii} lines have been found to be
particularly good infall tracers (Muzerolle et al. \cite{muzerolle98};
Mohanty et al. \cite{subu05}), with the flux strongly correlated to
the accretion rate, although other lines such as H$\alpha$ and
\ion{He}{i} 5876\AA\, also show good correlations to the accretion
rate (Herczeg \& Hillenbrand \cite{herczeg}).
For the last few decades it has been known that pre-main
sequence stars have strong X-ray emission (Feigelson \& DeCampli
\cite{feigelson81}). This X-ray emission was thought to have
the same origin as that of main sequence stars: low-density plasma
($n_{\mathrm e} \sim 10^{10}$ cm$^{-3}$ enclosed in coronal loop
structures and heated to temperatures of $T \sim 10^{6}-10^{7}$
K (Feigelson \& Montmerle \cite{feigelsonmont99}). More recently,
high resolution X-ray spectroscopy has revealed the presence of soft
($E<0.7$ keV) X-ray emission, originating from high density ($n>10^{11}$
cm$^{-3}$) plasma at temperatures of $\sim$3 MK (see Telleschi et
al. \cite{telleschi07}, and references therein). It has been proposed that
this soft X-ray emission is due to mass accretion. This interpretation
is based on a simple model: assuming the accretion flow has free-fall
velocity $\vel \sim 500$ km\,s$^{-1}$, it becomes heated up by the shock
(due to the impact of the accreting material with the stellar surface)
at a temperature of $T \sim (3/16)(\mu m_{\rm H} \vel^{2})\sim 3 \times
10^6$ K, and then cools down radiatively (Gullbring \cite{gullbring94};
Calvet \& Gullbring \cite{calvet}; Lamzin \cite{lamzin}; Gunther
et al. \cite{gunther07}; Brickhouse et al. \cite{brickhouse}) producing
strong X-ray emission.
Supporting this interpretation is the fact that this plasma component
i) has never been observed in non-accreting stars, and ii) is too
dense to have a coronal origin. This interpretation is also supported
by time-dependent models of radiative accretion shocks in CTTSs
(Koldoba et al. \cite{Koldoba2008MNRAS}; Sacco et al. \cite{germano};
\cite{sacco10}; Orlando et al. \cite{Orlando2010A&A}). In particular,
Sacco et al. (\cite{germano}) carried out a detailed hydrodynamic
modeling of the interaction between the accretion flow and the stellar
chromosphere, synthesizing the high resolution X-ray spectrum, as it
would be observed with the Reflection Grating Spectrometers (RGS) on
board the XMM-Newton satellite. They found an excellent agreement
between predicted and observed X-ray spectra, supporting once again
the idea that this X-ray emission originates mainly from accretion
shocks.
However, there are observational results that are difficult to reconcile
with this framework. In particular, the mass accretion rates derived
from X-rays are usually underestimated if compared to accretion rates
derived from optical and UV data. As an example, for TW Hya, X-rays
indicate $\dot{M} \sim 1 \times 10^{-11} M_{\odot}$\,yr$^{-1}$ (Stelzer
et al. \cite{beate}), while H$\alpha$ and UV provide $5 \times 10^{-10}$
M$_{\odot}$\,yr$^{-1}$ (Muzerolle et al. \cite{muzerolle00}). The
same mismatch holds for BP Tau: $\dot{M} \sim 9 \times 10^{-10}$
M$_{\odot}$\,yr$^{-1}$ from X-rays (Schmitt et al. \cite{schmitt}),
and $3 \times 10^{-8}$ M$_{\odot}$\,yr$^{-1}$ from UV (Gullbring et
al. \cite{gullbring}). To date, only 3 such comparisons have been
made in the literature (TW Hya; BP Tau; MP Mus, Argiroffi et al
\cite{costanza09}), moreover, each of these comparisons has used
differing UV/optical accretion measures and differing means of calculating
the X-ray accretion rate. In order to confirm that the soft X-ray emission
arises from the accretion process, and understand the discrepancies
between the optical and X-ray accretion rates, we have homogeneously
analysed optical data for all the CTTSs for which high-resolution X-ray
spectra, with good S/N in the [0.5, 1.0] keV range, have been
gathered. We compare the X-ray derived mass accretion rates to those
measured from the H$\beta$, \ion{He}{i} 5876\AA, \ion{O}{i} 6300\AA,
H$\alpha$ equivalent width and H$\alpha$ full-width at 10\%.
The structure of the paper is as follows. In section 2 we present our
sample of CTTSs previously observed with Chandra and XMM. We briefly
discuss the parameters of each CTTS (summarised in Table \ref{param}),
along with the previously published X-ray data. In section 3 we
discuss the data reduction and analysis of the optical data
(subsection 3.1), and the derivation of mass accretion rates from the
X-ray data (subsection 3.2). In section 4.1 we draw comparisons
between the different optical tracers of accretion and discuss the
mass accretion rates derived from these tracers. In section 4.2 we
analyse and discuss the variability of TW Hya. In section 4.3 we
compare the optical and X-ray mass accretion rates and discuss
possible scenarios to explain our results, and in section 5 we present
our conclusions.
\begin{sidewaystable*}
\vspace{15cm}
\caption{List of sources and their adopted parameters.}
\label{param}
\centering
\begin{tabular}{lcccccccccccl}
\hline \hline
Name & RA & Dec & Dist. & Mass & Radius & Age & Sp Type & Mag. & A$_{\mathrm{v}}$ & Incl. & Binarity\tablefootmark{a} & References \\
~ &(J2000) & (J2000) & (pc) & (M$_{\odot}$) & (R$_{\odot}$)& Myr& ~ & (Johnson V) & (mag) & (\degr) & ~ & ~ \\
\hline
Hen 3-600 & 11 10 28.1 & -37 31 51 & 45 & 0.2 & 0.9 &8 & M4Ve/M4Ve & 12.04 & 0.7 & $\sim0$ & BS & 1, 2, 3, 4, 5, 6 \\
TW Hya & 11 01 51.9 & -34 42 17 & 56 & 0.7 & 1.0 &8 & K7 & 11.27 & 0 & 7 & SS & 1, 6, 7, 8, 9, 10, 11, 12 \\
RU Lup & 15 56 42.3 & -37 49 15 & 140 & 0.8 & 1.7 &0--3 & K7 & 11.55 & 0.1 & 24 & SS & 7, 9, 13, 14, 15, 16 \\
BP Tau & 04 19 15.8 & +29 06 27 & 140 & 0.8 & 2.0 &1.9 & K7 & 12.13 & 0.5 & 45 & SS & 9, 17, 18, 19, 20, 21, 22 \\
V4046 Sgr & 18 14 10.5 & -32 47 34 & 72 & 0.86 & 1.16 &12 & K5Ve/K7Ve & 10.69 & 0 & 35 & BS & 9, 23, 24, 25 \\
MP Mus & 13 22 07.5 & -69 38 12 & 86 & 1.2 & 1.3 &6--7 & K1IVe & 10.44 & 0.17 & 32 & SS & 9, 26, 27, 28 \\
V2129 Oph & 16 27 40.3 & -24 22 03 & 120 & 1.35 & 2.4 &2 & K5 & 12.28 & 0.3 & 45 & SS & 3, 9, 29, 30 \\
T Tau N & 04 21 59.4 & +19 32 06 & 140 & 2.4 & 3.6 &2.7 & K0 & 9.88 & 1 & 13 & TS & 8, 15, 16, 20, 31\\
\hline
\end{tabular}
\tablefoottext{a}{ SS = single star; BS = binary system; TS = triple system.}
\tablebib{
(1) Muzerolle et al. \cite{muzerolle00};
(2) Torres et al. \cite{torres00};
(3) Geoffray \& Monin \cite{geoffray};
(4) Huenemoerder et al. \cite{huenemoerder};
(5) Kastner et al. \cite{kastner97};
(6) Song et al. \cite{song};
(6) Herczeg \& Hillenbrand \cite{herczeg};
(7) Qi et al. \cite{qi04};
(8) Lasker et al. \cite{lasker};
(9) Wichman et al. \cite{wichman};
(10) Raassen \cite{raasan};
(11) Brickhouse et al. \cite{brickhouse};
(12) Robrade \& Schmitt \cite{robrade};
(13) Stempels, Gahm \& Petrov \cite{stempelsgahmpetrov};
(14) Stempels \& Piskunov \cite{stempels};
(15) Comer\'{o}n et al. \cite{comeron};
(15) G\"udel et al. \cite{gudel};
(16) Grankin et al. \cite{grankin};
(17) Costa et al. \cite{costa};
(18) Donati et al. \cite{donati08};
(19) Kenyon et al. \cite{kenyon};
(20) Kenyon \& Hartmann \cite{kenyonhart};
(21) Stempels \& Gahm \cite{stempelsgahm};
(22) Hutchinson et al. \cite{hutchinson};
(23) Kastner et al. \cite{kastner08};
(26) Mamajek et al. \cite{mamajek};
(27) Cortes et al. \cite{cortes09};
(28) Torres et al. \cite{torres08};
(29) Donati et al. \cite{donati};
(30) Wilking et al. \cite{wilking};
(31) Bertout et al. \cite{bertout}.
}
\end{sidewaystable*}
\section{The Sample}
The sample consists of all the CTTSs currently observed with
high-resolution X-ray spectroscopy (either with Chandra, or RGS on
XMM-Newton) and for which \ion{O}{vii} triplet lines have been
measured, and which also have high resolution optical echelle
spectroscopy available in the data archives. The X-ray data have
previously been published, with the published fluxes of the
\ion{O}{vii} used to derive the mass accretion rate for each star. The
parameters for each source, used in our calculations are given in
Table~\ref{param}. Below are brief descriptions of each source.
{\em \object{Hen 3-600}} is a multiple star system and a member of the
TW Hydra Association, at a distance of 45 pc. This is an average
distance taken from Huenemoerder et al. (\cite{huenemoerder}), who
used the photometric distance from Kastner et al. (\cite{kastner97})
and the typical distance adopted for stars belonging to the TW Hya
association. The primary components (A and B) of this system are
separated by 1\farcs4, and component A is surrounded by a dusty disc
(Jayawardhana et al. \cite{rayjay99}). Component A is also inferred
(by Huenemoerder et al. \cite{huenemoerder}) to be almost pole-on,
from a large mid-IR excess and a negligible optical reddening (based
on a B$-$V = 1.52 which is nearly that of an unreddened M3
photosphere; Johnson \cite{johnson66}). Huenemoerder et
al. (\cite{huenemoerder}) discuss the X-ray Chandra observations of
Hen 3-600, and find high density plasma, and a larger `soft excess'
for component A, than B.
{\em \object{TW Hya}} is one of the closest known CTTSs, at a distance
of only $\sim$ 56 pc (Wichmann et al. \cite{wichman}). It has a
mass of 0.7 M$_{\odot}$, a radius of 1 R$_{\odot}$ (Muzerolle et
al. \cite{muzerolle00}) and is orientated so that it is seen nearly
pole-on (Kastner et al. \cite{kastner97}). Previous X-ray data have
been published by Kastner et al. (\cite{kastner02}), Raassen
(\cite{raasan}), Brickhouse et al. (\cite{brickhouse}) and Stelzer
\& Schmitt (\cite{beate}), who found a strong soft X-ray emission
produced by high density plasma.
{\em \object{RU Lup}} is located at a distance of 140 pc (Hughes et
al. \cite{hughes93}). It has a mass and radius of 0.8 M$_{\odot}$ and
1.7 R$_{\odot}$ respectively (Stempels \& Piskunov
\cite{stempels}), it suffers little absorption (Herczeg et
al. \cite{herczeg05}) and is suggested to be viewed almost pole-on
(Stempels \& Piskunov \cite{stempels}). XMM data are discussed by
Robrade \& Schmitt (\cite{robrade}), who find cool, high density
plasma which they conclude indicates an accretion shock origin.
\begin{table*}
\caption{Optical archival data information for our sample of CTTSs.}
\label{obs}
\centering
\begin{tabular}{lccccc}
\hline \hline
Name & Telescope & Instrument & Observation Date & Exposure time \\
~ & ~ & ~ & (yyyymmdd) & (s) \\
\hline
Hen 3-600 & ESO 2.2m & FEROS & 20040512 & 900 \\
TW Hya & ESO 2.2m & FEROS & 20070426 & 900 \\
RU Lup & VLT & UVES & 20050814 & 900 \\
BP Tau & TNG & SARG & 20071220 & 3600 \\
V4046 Sgr & VLT & UVES & 20050815 & 900 \\
MP Mus & ESO 2.2m & FEROS & 20060418 & 3000 \\
V2129 Oph & VLT & UVES & 20020417 & 900 \\
T Tau & TNG & SARG & 20061130 & 3600 \\
\hline
\end{tabular}
\end{table*}
{\em \object{BP Tau}} is a CTTS in the Taurus-Auriga molecular cloud at a
distance of 140 pc (Kenyon et al. \cite{kenyon}). It has a spectral
type of K7, and a mass of 0.8 M$_{\odot}$ (Kenyon \& Hartmann
\cite{kenyonhart}; G\"{u}del et al. \cite{gudel}). The XMM-Newton
spectra were previously discussed by Schmitt et al. (\cite{schmitt})
and Robrade \& Schmitt (\cite{robrade06}). They report the presence
of high density soft X-ray emission from the \ion{O}{vii} triplet,
suggesting this could be due to accretion shocks.
{\em \object{V4046 Sgr}} is a nearby spectroscopic binary CTTS that is
isolated from any dark cloud or molecular cloud, it has negligible
extinction, and there is evidence for a circumstellar disk (Hutchinson et
al. \cite{hutchinson}). The period is well determined to be 2.4213459 days
(Stempels \& Gahm \cite{stempelsgahm}) and the separation is estimated
to be $\sim$ 10 R$_{\odot}$ (Quast et al. \cite{quast}). The observed
spectral energy IR distribution is consistent with the disk having
an inner radius of 1.8AU, therefore the disk is circumbinary. The
inclination of the system is $\sim$35--45\degr. Chandra data of V4046
Sgr is presented by G\"{u}nther et al. (\cite{gunther}), who found the
X-ray emission is due to high density plasma.
{\em \object{MP Mus}} is a K1 IVe type star located in the Lower
Centaurus Crux (LCC) association at a distance of $\sim$ 86 pc. It is
known to have a dusty disk (Mamajek et al. \cite{mamajek}; Silverstone
et al. \cite{silverstone}) with a dust mass of $\sim 5 \times
10^{-5}$ M$_{\odot}$ (Carpenter et al. \cite{carpenter}). Batalha et
al. (\cite{batalha}) studied the variability in B, V, R and I bands,
and found that it has a low variability for a CTTS. XMM data for MP Mus
have previously been published by Argiroffi et al. (\cite{costanza}),
who found evidence for a high density plasma responsible for the soft
X-ray emission.
{\em \object{V2129 Oph}} is the brightest CTTS in the $\rho$ Oph cloud,
at a distance of 120 pc (Lombardi, Lada \& Alves (\cite{lombardi})). It
is a distant binary, with its very low-mass companion (possibly a
brown dwarf) being about 50 times fainter than the CTTS in the V band
(Geoffray \& Monin \cite{geoffray}). The CTTS is inferred to have a mass
of 1.35 M$_{\odot}$ and a radius of 2.4 R$_{\odot}$. Chandra data
have recently been obtained (Argiroffi et al. in prep) but not published
yet. We include the optical analysis in this paper for easy comparison
once the X-ray data is published.
{\em \object{T Tau} N} is another CTTS located in Taurus-Auriga. It is
optically visible part of a triple system, along with T Tau Sa
and Sb which are the ``infrared companions''. It has a spectral
type of K0 (Kenyon \& Hartmann \cite{kenyonhart}) and a mass of
2.7 M$_{\odot}$ (G\"{u}del et al. \cite{gudel}). It is oriented such
that it is seen almost pole-on (Solf \& B\"{o}hm \cite{solfandbohm};
Akeson et al. \cite{akeson}). X-ray data from XMM and Chandra have
been discussed in G\"{u}del et al. (\cite{gudel}) who found a `soft
excess' but no evidence of high density plasma.
\section{Data Analysis}
\subsection{Optical Data Analysis}
We retrieved high resolution echelle spectra spanning the optical
wavelength range from the archives. No single instrument had observed
the entire sample (in part, due to their locations). Data from three
instruments, FEROS, on the 2.2 m ESO telescope in Chile, UVES, on the
VLT in Chile and SARG on the TNG in La Palma were used in this
analysis. Details of the telescope, instrument, observation date and
exposure times for each source can be found in
Table~\ref{obs}. Relevant calibration files were downloaded with each
dataset and used in the data reduction process.
\begin{table*}
\caption{Measured equivalent widths for the accretion tracers\tablefootmark{a}, along
with the measured full width at 10\% for the H$\alpha$
line.}
\label{tab:linewidths}
\centering
\begin{tabular}{lccccccccc}
\hline \hline
\\
Source & H$\gamma$ 4340\AA & \ion{He}{ii} 4686\AA & H$\beta$ 4861\AA & \ion{He}{i} 5016\AA & \ion{He}{i} 5876\AA & \ion{O}{i} 6300\AA & H$\alpha$ 6563\AA & H$\alpha$ FW10\% & \ion{He}{i} 6678\AA \\
& \AA & \AA & \AA & \AA & \AA & \AA & \AA & km\,s$^{-1}$ & \AA \\
\hline
Hen 3-600& -7.51 & ... & -7.93 & -0.32 & -0.96 & -0.37 & -21.81 & 282.29 & -0.37 \\
TW Hya & -18.49 & -0.48 & -31.9 & -0.32 & -3.16 & -0.54 & -141.58 & 407.55 & -0.7 \\
RU Lup & NA & NA & -28.34 & -9.25 & -4.55 & -1.17 & -68.95 & 578.42 & -1.88 \\
BP Tau & NA & -0.6 & -19.82 & -0.27 & -1.08 & -0.37 & -58.63 & 413.41 & -0.31 \\
V4046 Sgr& NA & NA & -5.97 & ... & -0.41 & -0.12 & -32.66 & 515.19 & ... \\
MP Mus & -1.34 & ... & -3.16 & ... & -0.3 & -0.23 & -23.47 & 502.9 & 0.16 \\
V2129 Oph& NA & NA & -2.54 & ... & -0.33 & -0.13 &-12.07 & 274.24 & -0.04 \\
T Tau & NA & -0.1 & -12.05 & -0.6 & -0.55 & -0.7 & -42.45 & 419.71 & 0.08 \\
\hline
\end{tabular}
\tablefoottext{a}{ Negative values represent emission. Note some sources show
lines in absorption. There are no measurements where the
spectra show no indication of a line present. NA indicates where the wavelength
of the line was not covered in the observation.}
\end{table*}
The data from the three instruments were reduced in the standard
manner using IRAF routines to de-bias, flatfield and extract the
spectra. Wavelength calibrations were carried out based on ThArNe arcs
for FEROS, ThAr arcs for UVES and Thorium arcs for SARG.
Line luminosities, $L_{\mathrm{line}}$, have been calculated from the
equivalent width of each emission line (see
table~\ref{tab:linewidths}). The continuum fluxes have been estimated
from the V magnitudes of the stars, following a method similar to that
used by Mohanty et al. (\cite{subu05}) and Dahm (\cite{dahm}).
Specifically, the continuum flux at 5500\,\AA\, was calculated using
the extinction corrected observed V magnitude, the distance, and the
photometric zero point of the Johnson V filter (at the central
wavelength 5500\,\AA) of 3.75$\times$ 10$^{-9}$
\,erg\,cm$^{-2}$\,s$^{-1}$\,\AA$^{-1}$ (Mitchell \& Johnson
\cite{mj}). The continuum flux at the wavelength of each of the
analysed emission lines also required correction for the stellar
spectral response -- which was measured using the Pickles
Spectrophotometric Atlas of Standard Stellar Spectra (Pickles
\cite{pickles}) in conjunction with the observed stellar spectral
type. It is worth noting that using the observed V magnitude we take
into account the continuum excess due to the accretion process.
The mass accretion rates were calculated following the empirical
relation between accretion and line luminosities found and described
by Herczeg \& Hillenbrand (\cite{herczeg}):
\begin{figure*}
\centering
\includegraphics[width=15cm]{weightedmean.ps}
\caption{{\em Top Panel:\em} Plot of the different mass accretion
rates from different accretion tracer emission lines, for each of
the stars in the sample. The stars are in order of increasing
mass, showing no relation between mass and mass accretion
rate. {\em Bottom Panel:\em} Plot showing the ratio of each mass
accretion rate estimate to the optical mean of the accretion rate
for each star. The different symbols/colors represent different
accretion rate tracers (see upper left corner in the top panel).}
\label{opticalcomp}
\end{figure*}
\begin{equation}
\label{line}
\log L_{\rm acc} = a + b \log L_{\rm line}
\end{equation}
\noindent where the coefficients $a$ and $b$ (listed in
Tab.~\ref{coeffs}) have been calculated -- by Herczeg \& Hillenbrand
(\cite{herczeg}) -- by comparing optical emission line fluxes and
accretion luminosities measured from the UV continuum excess. Errors
on the coefficients $a$ and $b$ depend on the scatter of the line
luminosity values around the best fit relation (from a large sample
of sources), and therefore take into account several effects, such as
optical depth and any contribution to the line emission from stellar
outflows.
Once the accretion luminosity, $L_{\mathrm acc}$, has been
calculated, it is possible to calculate the mass accretion rate
via:
\begin{equation}
\label{massacc}
\dot{M} = \left(1-\frac{R_{\mathrm{*}}}{R_{\mathrm{in}}}\right)^{-1} L_{\mathrm{acc}}\frac{R_{\mathrm{*}}}{G M_{\mathrm{*}}} \approx 1.25 L_{\mathrm{acc}} \frac{R_{\mathrm{*}}}{G M_{\mathrm{*}}}
\end{equation}
\noindent where $(1-R_{\mathrm{*}}/R_{\mathrm{in}})^{-1} \approx 1.25$
is estimated by assuming the accreting gas falls onto the star from
the truncation radius of the disk, $R_{\mathrm{in}} \approx
5R_{\mathrm{*}}$ (Gullbring et al. \cite{gullbring}).
We also calculated the mass accretion rate based on the H$\alpha$ full
width at 10\%, using the following equation (Natta et
al. \cite{natta04}):
\begin{equation}
\label{eq:halpha10}
\log \dot{M} \approx -12.9(\pm 0.3) + 9.7(\pm 0.7) \times 10^{-3}\, \mathrm{H}\alpha10\%
\end{equation}
\noindent where $\mathrm{H}\alpha10\%$ is the H$\alpha$ 10\% full width in
km\,s$^{-1}$ and $\dot{M}$ is in M$_{\odot}$\,yr$^{-1}$. The derived
mass accretion rates, along with a (weighted) mean optical accretion
rate, $\langle\dot{M}_{\mathrm{Opt}}\rangle$, are listed in
Table~\ref{tab:massacc}. The mean optical accretion rate was
calculated using all the mass accretion rates calculated via
Eqs.~\ref{line}\, and~\ref{massacc}\, and therefore does not
include the mass accretion rate derived from the H$\alpha$ 10\% width
(see section~\ref{optcompsect} for more details).
\begin{table}
\caption{Coefficients used in Equation~\ref{line}, for each emission line, from Herczeg \& Hillenbrand (\cite{herczeg})}
\label{coeffs}
\centering
\begin{tabular}{lccc}
\hline \hline
Line & Wavelength & $a$ & $b$ \\
& (\AA) & & \\
\hline
H$\gamma$ & 4349 & 3.0 $\pm$0.2 & 1.24 $\pm$ 0.04 \\
\ion{He}{ii} & 4686 & 3.7 $\pm$ 0.5 & 1.04 $\pm$ 0.08 \\
H$\beta$ & 4861 & 2.6 $\pm$0.2 & 1.22 $\pm$ 0.05 \\
\ion{He}{i} & 5016 & 3.3 $\pm$ 0.3 & 1.04 $\pm$ 0.05 \\
\ion{He}{i} & 5876 & 5.3 $\pm$ 0.7 & 1.46 $\pm$ 0.12 \\
\ion{O}{i} & 6300 & 2.8 $\pm$ 0.8 & 0.96 $\pm$ 0.16 \\
H$\alpha$ & 6563 & 2.0 $\pm$ 0.4 & 1.2 $\pm$ 0.11 \\
\ion{He}{i} & 6678 & 5.2 $\pm$ 0.8 & 1.37 $\pm$ 0.13 \\
\hline
\end{tabular}
\end{table}
\begin{sidewaystable*}
\vspace{15cm}
\caption{Derived optical mass accretion rates based on the equivalent widths and H$\alpha$ full width at 10\% measurements listed in Table~\ref{tab:linewidths}. Also listed are the optical weighted mean accretion rates, $\langle\dot{M}_{\rm Opt}\rangle$, based on all the accretion rates except H$\alpha$10\% width.}
\label{tab:massacc}
\centering
\begin{tabular}{lcccccccccc}
\hline \hline
\\
Source & $\log \dot{M}_{\mathrm{H\gamma\ 4340\mbox{\AA}}}$ & $\log \dot{M}_{\ion{He}{ii}\ 4686\mbox{\AA}}$ & $\log \dot{M}_{\mathrm{H\beta\ 4861\mbox{\AA}}}$ & $\log \dot{M}_{\ion{He}{i}\ 5016\mbox{\AA}}$ & $\log \dot{M}_{\ion{He}{i}\ 5876\mbox{\AA}}$ & $\log \dot{M}_{\mathrm{H\alpha\ 6563\mbox{\AA}}}$ & $\log \dot{M}_{\mathrm{H\alpha\ 10\%}}$ & $\log \dot{M}_{\ion{O}{i}\ 6300\mbox{\AA}}$ & $\log \dot{M}_{\ion{He}{i}\ 6678\mbox{\AA}}$ &$\log \langle\dot{M}_{\mathrm{Opt}}\rangle$ \\
& $M_{\odot}\,yr^{-1}$ & $M_{\odot}\,yr^{-1}$ & $M_{\odot}\,yr^{-1}$ & $M_{\odot}\,yr^{-1}$ & $M_{\odot}\,yr^{-1}$ & $M_{\odot}\,yr^{-1}$ & $M_{\odot}\,yr^{-1}$ & $M_{\odot}\,yr^{-1}$ & $M_{\odot}\,yr^{-1}$ & $M_{\odot}\,yr^{-1}$ \\
\hline
Hen 3-600 & $-9.75\pm0.28$ & $...$ & $ -9.61\pm0.30$ & $ -9.52\pm0.42$ & $ -9.02\pm0.94$ & $ -8.89\pm0.55$ & $ -10.15\pm0.61$ & $ -9.27\pm1.19$ & $ -8.89\pm1.06$ & $-9.53\pm0.17$\\
TW Hya & $-9.31\pm0.26$ & $ -9.17\pm0.66$ & $ -9.05\pm0.27$ & $ -9.82\pm0.41$ & $ -8.38\pm0.88$ & $ -8.46\pm0.50$ & $ -8.94\pm0.78$ & $ -9.39\pm1.15$ & $ -8.91\pm1.03$ & $-9.17\pm0.15$\\
RU Lup & ... & ... & $ -8.06\pm0.25$ & $ -7.38\pm0.35$ & $ -6.91\pm0.82$ & $ -7.79\pm0.47$ & $ -7.28\pm1.06$ & $ -8.20\pm1.03$ & $ -7.16\pm0.94$ & $-7.77\pm0.18$\\
BP Tau & ... & $ -8.14\pm0.62$ & $ -8.26\pm0.25$ & $ -8.97\pm0.39$ & $ -7.86\pm0.86$ & $ -7.89\pm0.48$ & $ -8.88\pm0.79$ & $ -8.68\pm1.09$ & $ -8.26\pm1.01$ & $-8.34\pm0.17$\\
V4046 Sgr & ... & ... & $ -9.34\pm0.28$ & ... & $ -9.07\pm0.91$ & $ -8.79\pm0.51$ & $ -7.89\pm0.95$ & $ -9.62\pm1.17$ & ... & $-9.22\pm0.23$\\
MP Mus & $-9.61\pm0.26$ & ... & $ -9.32\pm0.27$ & ... & $ -8.89\pm0.90$ & $ -8.73\pm0.50$ & $ -8.01\pm0.92$ & $ -9.18\pm1.11$ & ... & $-9.35\pm0.17$\\
V2129 Oph & ... & ... & $ -9.76\pm0.29$ & ... & $ -9.19\pm0.92$ & $ -9.28\pm0.55$ & $ -10.23\pm0.60$ & $ -9.56\pm1.18$ & $ -9.99\pm1.11$ & $-9.64\pm0.24$\\
T Tau & ... & $ -7.90\pm0.60$ & $ -7.22\pm0.23$ & $ -7.50\pm0.35$ & $ -6.98\pm0.81$ & $ -7.16\pm0.44$ & $ -8.82\pm0.80$ & $ -7.68\pm0.97$ & ... & $-7.32\pm0.16$\\
\hline
\end{tabular}
\end{sidewaystable*}
\subsection{X-ray Data Analysis}
\begin{figure*}
\centering
\includegraphics[width=16cm]{Halpha_lines.ps}
\caption{The observed H$\alpha$ line profiles for our sample. The
data show many different line morphologies, with some very asymmetric
lines e.g. V2129 Oph.}
\label{halphalines}
\end{figure*}
It is also possible to infer the mass accretion rate $\dot{M}$ of CTTSs
from their soft X-ray emission (e.g. Schmitt et al. \cite{schmitt}). This
method for deriving $\dot{M}$\, is based on the assumption that the
whole soft X-ray emission is due to accreting material and not to coronal
plasma, and that the post-shock zone can be described as a stationary
isothermal slab of plasma at constant velocity and density. Recent
hydrodynamical simulations (Sacco et al. \cite{germano}; \cite{sacco10}) have
demonstrated that, even if the post-shock zone is not stationary (and
characterized by quasi-periodic shock oscillations), the time-averaged
properties of the post-shock zone are, in general, well-described by
the stationary model. Sacco et al. (\cite{sacco10}) have also shown
that the higher the post-shock temperature the larger the discrepancies
between the stationary and the hydrodynamical models. This is mainly
due to the fact that the thermal conduction (which is not taken into
account in the stationary model and that is more efficient for higher
post-shock temperatures) drains energy from the shock-heated plasma to
the chromosphere through a thin transition region, and acts as an additional
cooling mechanism (see also Orlando et al. \cite{Orlando2010A&A}).
Letting $n_{\mathrm{0}}$ and $\vel_{\mathrm{0}}$ represent the
density and velocity of the pre-shock material, and $n_{\mathrm{1}}$
and $\vel_{\mathrm{1}}$ the corresponding post-shock values, then pre-
and post-shock quantities are linked by these relations:
\begin{equation}
\label{eq:shocks}
n_{\mathrm{1}} = 4 n_{\mathrm{0}},\ \
\vel_{\mathrm{1}} = \frac{1}{4}\vel_{\mathrm{0}},\ \
T_{\mathrm{1}} = \frac{3}{16}\frac{\mu m_{\mathrm{H}}}{k}
\vel_{\mathrm{0}}^{2}
\end{equation}
\noindent
where $T_{\mathrm{1}}$ is the post-shock temperature, $\mu$ is the
average mass per free particle ($\mu \approx 0.5$), and we assume a
strong shock regime. The volume occupied by the hot post-shock plasma
is defined by the stream cross-section, $A$, and by the distance,
$l$, covered by the plasma itself before cooling down. Hence, when
$\tau$ is the cooling time, $l$ is given by $\vel_{\mathrm{1}}\tau$,
and the emission measure -- $EM$ -- of the shock-heated plasma is
$n_{\mathrm{1}}^{2}A\vel_{\mathrm{1}}\tau$. Therefore it is possible to
derive the stream cross-section $A$, and hence the mass accretion rate,
from the $EM$ value measured from the soft X-ray emission via:
\begin{equation}
\dot{M} = \mu m_{\mathrm{H}} n_{\mathrm{1}} \vel_{\mathrm{1}} A
= \mu m_{\mathrm{H}} \frac{EM}{n_{\mathrm{1}}\tau}
= \mu m_{\mathrm{H}} \frac{EM\,P(T_{\mathrm{1}})}{3kT_{\mathrm{1}}}
\end{equation}
\noindent
where $\tau=3kT/(n_{\mathrm{1}}P(T))$, with $P(T)$ indicating the
plasma radiative losses per $EM$ unit. We used the above formula to
obtain $\dot{M}$ from the X-ray data. In particular for each CTTS: 1)
we evaluate $\vel_{\mathrm{0}}$, and hence $\vel_{\mathrm{1}}$, assuming
a free fall velocity from a distance of $5R_{\star}$ (to correspond
with the optical analysis); 2) we evaluate the post shock temperature
$T_{\mathrm{1}}$ from $\vel_{\mathrm{0}}$ (see Eq. \ref{eq:shocks});
3) the knowledge of $T_{\mathrm{1}}$ allows us to derive the plasma
$EM$ from the observed flux of the \ion{O}{vii} resonance line located
at 21.60\,\AA\ via:
\begin{equation}
EM=\frac{L_{OVII}}{G(T_1)\,A_{O}}
\end{equation}
\noindent
where $G(T_1)$ is the emissivity function of the \ion{O}{vii} resonance
line at 21.60\,\AA, and $A_{O}$ is the oxygen abundance.
We chose to rely our $\dot{M}$ estimations on the flux of the
\ion{O}{vii} resonance line because it is mostly produced by plasma at
2\,MK, which is the typical expected temperature for the plasma heated
in the accretion shock, and because its flux measurement is available
for almost all CTTSs observed with high resolution X-ray
spectroscopy.
Note that this method allows us to derive $\dot{M}$ independently of
the knowledge of the plasma density $n_{\mathrm{1}}$. On the other
hand this method depends on the plasma metallicity, which is needed to
calculate the radiative losses $P(T_{\mathrm{1}})$ and to link the
\ion{O}{vii} line flux to the corresponding $EM$. It is worth
noting that X-ray observations provide us calibrated spectra, making
the line flux measurements a simple and solid procedure, especially
for the \ion{O}{vii} resonance line at 21.6\,\AA~that is an isolated
line, and hence its flux estimation is not affected by problems due
to line blending or wrong continuum placement. We decided,
therefore, to infer the $\dot{M}$ values, with the above method,
starting from published values of the fluxes of the \ion{O}{vii}
resonance line.
In Table~\ref{tab:mdotx} we list, for each CTTS, the derived accretion
rates together with the main parameters used.
\begin{table*}
\caption{Observed X-ray parameters and derived X-ray mass accretion rates\tablefootmark{a}.}
\label{tab:mdotx}
\begin{center}
\begin{tabular}{lcccccccc}
\hline\hline
Name & $T$ & O, Ne, Fe & $P(T)$ & $\log L_{\ion{O}{vii}}$ & $\log EM$ & $\log \dot{M}$ & References for $L_{OVII}$\\
& (MK) & & $({\rm 10^{-23}cm^{3}\,erg\,s^{-1}})$ & $({\rm erg\,s^{-1}})$ & $({\rm cm^{-3}})$ & $({\rm M_{\odot}\,yr{-1}})$ & and abundances\\
\hline
Hen 3-600 & $1.0$ & $0.4,1.2,0.2$ & $5.7$ & $27.5 \pm 0.2$ & $ 52.7 \pm 0.2$ & $ -9.53 \pm 0.26$ & 1\\
TW Hya & $3.1$ & $0.2,1.8,0.3$ & $2.2$ & $28.9 \pm 0.1$ & $ 53.4 \pm 0.1$ & $ -9.74 \pm 0.05$ & 2 \\
RU Lup & $2.1$ & $0.6,1.4,1.1$ & $7.2$ & $28.9 \pm 0.1$ & $ 52.8 \pm 0.1$ & $ -9.67 \pm 0.13$ & 3 \\
BP Tau & $1.8$ & $0.6,1.5,0.3$ & $4.7$ & $28.9 \pm 0.1$ & $ 52.8 \pm 0.1$ & $ -9.78 \pm 0.08$ & 4, 5 \\
V4046 Sgr & $3.3$ & $0.1,1.0,0.1$ & $1.6$ & $28.5 \pm 0.1$ & $ 53.5 \pm 0.1$ & $ -9.84 \pm 0.11$ & 6, 7\\
MP Mus & $4.1$ & $0.1,0.4,0.1$ & $1.3$ & $28.6 \pm 0.1$ & $ 53.8 \pm 0.1$ & $ -9.70 \pm 0.06$ & 8 \\
V2129 Oph & ... & ... & ... & ... & ... & ... & ...\\
T Tau & $2.9$ & $0.4,0.8,0.3$ & $2.4$ & $29.4 \pm 0.1$ & $ 53.6 \pm 0.1$ & $ -9.47 \pm 0.11$ & 9, 10\\
\hline
\end{tabular}
\end{center}
\tablefoottext{a}{Chandra data for V2129 Oph have recently been obtained
(Argiroffi et al. in prep) but not published yet.}
\tablebib{(1) Huenemoerder et al. \cite{huenemoerder}; (2) Stelzer \& Schmitt, \cite{beate}; (3) Robrade \& Schmitt \cite{robrade}; (4) Schmitt et al. \cite{schmitt}; (5) Robrade \& Schmitt \cite{robrade06}; (6) G\"unther et al. \cite{gunther}; (7) Argiroffi C., private communication; (8) Argiroffi et al. \cite{costanza09}; (9) G\"udel \& Telleschi \cite{gudeltell}; (10) G\"udel et al. \cite{gudel}.}
\end{table*}
\section{Results \& Discussion}
\subsection{Comparison of optical derived accretion rates}
\label{optcompsect}
Table \ref{tab:massacc} shows the calculated mass accretion rates for
each star based on the different emission lines analysed. Although the
sources were observed at different epochs, all the emission lines were
observed simultaneously for each star. Therefore one might na\"{i}vely
expect the different tracers to lead to very similar mass accretion
rates (if each line is tracing the same accretion phenomena). The top
panel of Fig.~\ref{opticalcomp} shows the different mass accretion
rates calculated for each star in the sample (in order of
increasing stellar mass). We find that all the different tracers
agree within the errors, albeit with a spread of typically $\sim$1
order of magnitude. We see no relation between mass accretion
rate and the mass of the stars. Also, it should be noted, that there
is no apparent relation between the mass accretion rate and the age
of the stars (see table~\ref{param}) -- for example, V2129 Oph has
an age of 2 Myrs, whereas T Tau has an age of 2.7 Myrs and their
mass accretion rates are vastly different. V4046 Sgr has an age of
12 Myrs and MP Mus has an age of 6--7 Myrs, and yet they have
similar mass accretion rates. The bottom panel of
Fig.~\ref{opticalcomp} shows the distribution of the difference
between each mass accretion rate estimate and
$\langle\dot{M}_{\mathrm{Opt}}\rangle$, the optical mean mass
accretion rate. In general, H$\alpha$ flux and \ion{He}{i} 5876\AA\,
lead to higher mass accretion rates than H$\beta$, H$\gamma$ and
\ion{O}{i}.
H$\alpha$ 10\% mass accretion rates are often at the extremes of the
range of estimates, but this is not surprising given the H$\alpha$
line profiles are dependent on inclination, rotation effects (see
Mohanty et al. \cite{subu05} for details), winds, and
outflows. Fig.~\ref{halphalines} shows the H$\alpha$ line profiles for
our sample, some of which are very asymmetric. The H$\alpha$ emission
from Hen 3-600 and MP Mus both have narrow peaks, but with asymmetric
broad wing emission. TW Hya, RU Lup and BP Tau all have broader peaked
emission, albeit with some evidence for self-absorption. V4046 Sgr has
an H$\alpha$ line profile which has a broad peak, with very broad wing
emission. V2129 Oph and T Tau both show clear evidence for absorption
in the H$\alpha$ emission, with V2129 Oph having a very asymmetric
profile with strong absorption of the blue-shifted wing/no blueshifted
component. Interestingly, we see that the accretion rates derived from
H$\alpha$ 10\% emission is to the lower part of the range of mass
accretion rates for Hen 3-600, BP Tau, V2129 Oph and T Tau, at the
upper end of the range for RU Lup, V4046 Sgr and MP Mus and in the
middle of the range for TW Hya (see top panel of
Fig.~\ref{opticalcomp}). The asymmetry of the emission line profile
from V2129 Oph would clearly lead to the under-estimation of the mass
accretion rate for that object, but interestingly, the
under-estimation is not extreme -- the difference between the
H$\alpha$ 10\% mass accretion rate and the optical mean mass accretion
rate being only $-1.7 \times 10^{-10}$ M$_{\odot}$\,yr$^{-1}$. In the
cases of V4046 Sgr, MP Mus and T Tau, these estimates are clear
outliers from the rest of the accretion rate estimates (high estimates
for the first two -- as predicted from the line profiles, and lower
for T Tau -- which has a `normal' line profile, albeit with some
absorption, but this does not affect the 10\% intensity level). Due
to the fact that H$\alpha$ 10\% mass accretion rate estimates are
outliers in almost half our sample, along with the fact that
inclination, rotation, winds and outflows may affect these mass
accretion rates, we have excluded the H$\alpha$ 10\% mass accretion
rate estimates from the calculation of the optical mean mass accretion
rate listed in Table~\ref{tab:massacc}.
\subsection{Variability of TW Hya}
\begin{figure}
\centering
\includegraphics[width=8cm]{variability.ps}
\caption{The variability of TW Hya. The different
color/symbols represent different accretion rate tracers (see
upper right corner). Note the X-axis timescale is not linear. The
variability in accretion rate is observed to be small in comparison
to the range of accretion rates calculated from the different tracers.}
\label{variability}
\end{figure}
TW Hya is the only source within the sample to have been observed over
a period of months (from March 2007 through to July 2007) on both
short (nightly) and long (monthly) timescales, using the same
instrumental set-up. We have analysed 14 nights of data over a 5 month
period for the purpose of studying the accretion variability. Our
results are shown in Fig.~\ref{variability}. We find the variability
for a given tracer small compared to the $\sim$ 1 order of magnitude
spread in mass accretion rates from the different tracers. \ion{O}{i},
H$\alpha$ and \ion{He}{i} 5876\AA\, are the tracers with the least
variability, whilst H$\beta$ and H$\gamma$ have the largest
variability. It could be argued that this small variability is
expected given TW Hya is viewed almost pole-on, and as such it should
be possible to observe all active accretion regions about the facing
pole at all times, therefore eliminating large changes in mass
accretion rates as the hotspots rotate out of view. The variability of
mass accretion rates are not necessarily due only to the inclination
angle of the star-disk system, but may be due to time-dependent
accretion rates. The small amount of variability ($\sim$ 0.25 order of
magnitude) we observe in our TW Hya data may be due to time-dependent
accretion rate.
Alencar \& Batalha (\cite{alencar}) studied the TW Hya accretion rate
for 1.5 yrs, and found the mass accretion rate to be variable between
$10^{-9}$ and $10^{-8}$ M$_{\sun}$\,yr$^{-1}$\, using the \ion{Na}{d}
line profile. Whilst these results fall within the range of mass
accretion rates we measured using different tracers, they suggest a
larger variability. This discrepancy may be due to the method used to
estimate the mass accretion rate or due to the longer timeperiod
covered in their analysis. Rucinsky et al. (\cite{rucinsky}) analysed
two datasets -- from 2007 and 2008 -- and found a photometric
variability of $\Delta V < 0.5$ mag on timescales of $\sim$ 3.5 days
in the 2007 data. This however was no longer observable in the 2008
data, which showed variability (again $\Delta V < 0.5$ mag) on
timescales of 2--9 days, which they suggest may be due to the orbital
decay of accreting gas. In addition, they find `spikes' in the
accretion rate lasting only a fraction of a day. They conclude that
whilst accretion contributes to this variability, it is not the sole
cause.
Our optical and X-ray observations are not coeval, and so it is
important to study the variability of these stars due to accretion
to know if the their variability will limit our ability to compare
mass accretion rates derived from non-coeval observations. We find
TW Hya, a pole-on star, to have a variability within our
errors. Furthermore, all the sources in this sample have
inclinations $\lesssim$ 45$\degr$, with TW Hya, Hen 3-600, RU Lup
and T Tau all having inclinations much closer to pole-on, meaning
any accretion hotspots would be visible all/most of the time,
limiting variability due to observability of the hotspot.Whilst we
cannot rule out time dependent mass accretion rates for the stars
other than TW Hya in our sample, we measure a small variability in
our TW Hya data, and even the larger variabilities suggested in the
literature fall within our errors.
\subsection{Comparison of optical/x-ray derived accretion rates}
\begin{figure*}
\centering
\includegraphics[width=16cm]{opticalX_paper.eps}
\caption{Plots of the optically derived mass accretion rates versus
the X-ray derived mass accretion rates. Each panel shows the
optical accretion rate derived from a different tracer, indicated
in the lower right corner of the plot. The CTTSs are colour coded
(see lower left corner of each panel). We plot only those
cases for which we have measured the mass accretion rate for all
the stars in our sample (see table~\ref{tab:massacc}). The bottom
right panel shows a plot of the optical mean mass accretion rate
plotted against the X-ray accretion rate. There is a very small
range, a factor $\approx 2$, in X-ray accretion rate
calculated for the entire sample, whereas the optical mass
accretion rates span a range of $\sim 3$ orders of magnitude.}
\label{opt-xray}
\end{figure*}
We compare the optical derived mass accretion rates $\langle\dot{M}_{\rm
Opt}\rangle$ to the X-ray derived mass accretion rates $\dot{M}_{\rm
X-ray}$ in Fig.~\ref{opt-xray}. The most striking result is
that $\dot{M}_{\rm X-ray}$ for our sample ranges within a factor
$\approx 2$ around $2 \times 10^{-10}$ M$_{\odot}$\,yr$^{-1}$ (see
Fig.~\ref{opt-xray} and Table~\ref{tab:mdotx}), despite the fact that
the range of $\langle\dot{M}_{\rm Opt}\rangle$ spans almost 3 orders of
magnitude. In addition, as suggested by other studies in the literature,
the X-ray mass accretion rates are always lower than the corresponding
optical mass accretion rates with the discrepancy increasing for
increasing $\langle\dot{M}_{\rm Opt}\rangle$.
The above results can be interpreted in the light of the findings of Sacco
et al. (\cite{sacco10}) who used a detailed hydrodynamic model
to investigate the observability of accretion shocks in X-rays in a wide
range of physical conditions of the accretion stream. They considered the
absorption from the optically thick plasma of the stellar chromosphere
on the X-ray emission arising from the shock-heated material. They
found that the absorption strongly depends on the accretion stream
properties and influences the observability of the post-shock accreting
plasma in the X-ray band. In particular, observable X-ray emission from
accretion shocks is expected in streams with densities in the range
$10^{11}\lesssim n_{\rm e}\lesssim 10^{12}$ cm$^{-3}$ and velocities in
the range $400\lesssim \vel\lesssim 600$ km\,s$^{-1}$. In fact,
denser streams produce post-shock zones that are rather narrow and deeply
rooted in the chromosphere, causing their X-ray emission to be strongly
absorbed. Streams with velocities below 400 km\,s$^{-1}$ produce shocked
plasma with temperatures around 1 MK or even lower and, therefore, hardly
observable in X-rays. The obvious consequence is that, whilst accretion
shocks are almost always observable in the optical band, only a small
fraction of them can be revealed in X-rays, leading in general to an
underestimate of the X-ray derived mass accretion rate and, therefore,
to the observed discrepancy with the optical derived mass accretion
rates (see also discussion in Orlando et al. \cite{Orlando2010A&A}
and Sacco et al. \cite{sacco10}).
Note that the above argument is applicable even if only one stream
is present on the star. In fact, for typical densities and velocities
of accretion streams and stellar magnetic field strengths (as derived
from observations of CTTSs; Johns-Krull \cite{johnskrull}), in general
the plasma $\beta$ (i.e. the ratio gas pressure / magnetic pressure)
is expected to be $\ll 1$ (see Sacco et al. \cite{sacco10}). In these
conditions, an accretion stream is a bundle of fibrils each independent
of the others due to the strong magnetic field which prevents mass
and energy exchange across magnetic field lines (e.g. Orlando et
al. \cite{Orlando2010A&A} and Sacco et al. \cite{sacco10}). If the stream
is structured in density (as suggested, for instance, by 3D MHD models of
the star-disk system carried out by Romanova et al. \cite{romanova04}),
only fibrils with density and velocity in the range of observability
defined by Sacco et al. (\cite{sacco10}) are expected to produce
detectable X-rays, unless they are located close to the centre of
the stream cross-section where they suffer strong absorption from the
accretion column itself.
Interestingly, Hen 3-600 is the lowest mass star in our sample and has
a close agreement between the optical and X-ray mass accretion rates,
suggesting it has very few high density streams (or fibrils) --
hardly detectable in X-rays -- or only low density accretion streams
(fibrils) -- observable in X-rays -- meaning almost everything
is seen in the X-ray, as well as in the optical. At the other end of
the scale, RU Lup could be interpreted to have a much
larger number of high density streams (fibrils).
If, on one hand, the discrepancy between optical and X-ray derived mass
accretion rates may be easily explained in the framework of accretion
shock models as discussed above, on the other hand, more puzzling is the
result that $\dot{M}_{\rm X-ray}$ is almost the same for all the stars
of our sample. This result is even more surprising considering that
the determination of $\dot{M}_{\rm X-ray}$ assumes some fixed parameters
(e.g. the distance, the mass and the radius of the star, the abundance
of the emitting plasma, and the temperature of the post-shock region)
whose uncertainties are unknown and therefore not included in the errors
on $\dot{M}_{\rm X-ray}$. Taking into account all these uncertainties,
one would expect a scatter in the values of $\dot{M}_{\rm X-ray}$
possibly larger than one order of magnitude. On the contrary, the
evidence is that the observed scatter is within a factor $\approx 2$
and it can be explained only if the (unknown) uncertainties on the fixed
parameters used in our calculations are smaller than expected.
As for the almost constant value of $\dot{M}_{\rm X-ray}$, we note
that its estimate relies on the flux of the \ion{O}{vii} resonance line
and is based on the assumption of optically thin plasma. However it can
be inferred that optical depth of the strongest emission lines produced
by the plasma located in the post-shock region is non-negligible. In fact
for a post shock region with dimension $\sim$10$^9$ cm, filled with plasma
at T=2 MK and n$_{\rm e}$=10$^{11}$ cm$^{-3}$, the optical depth of the
\ion{O}{vii} resonance line is $\sim 10$. Supporting this scenario is some
evidence of optical depth effects (i.e. the emitting plasma is not totally
optically thin) which has been detected in the X-ray spectrum of MP Mus
(Argiroffi et al. \cite{costanza09}), implying that the \ion{O}{vii}
flux and the derived $\dot{M}_{\rm X-ray}$ could be underestimated.
The optical depth effects can be addressed by comparing mass
accretion rates measured from the resonance line of the \ion{O}{vii}
triplet with the accretion rate measured using the flux of the
intercombination plus the flux of the forbidden line of the same triplet,
which are less affected by optical depth effects. We performed this
comparison in the case of TW Hya and MP Mus. We found no significant
differences in the former case, while the $\dot{M}_{X-ray}$ of MP Mus
measured from the intercombination plus the forbidden line is twice as
large as the $\dot{M}_{X-ray}$ measured from the resonance line. The lack
of high signal-to-noise spectra does not allow us to further investigate
this issue using this approach for the other stars in our sample. However,
this preliminary result suggests that the absorption from the stellar
atmosphere is the main cause of the discrepancy between the optical and
the X-ray mass accretion rates.
Brickhouse et al. (\cite{brickhouse}) analyzed X-ray spectroscopic
data of TW Hya with high signal-to-noise ratio. This kind of analysis
is limited to the case of TW Hya, which is the brightest CTTS in the
X-ray and has been observed with Chandra for $\sim$500 ks. These
authors found that the flux and the densities measured from the
\ion{Ne}{ix} triplet agree with the accretion shock parameters derived
from optical emission lines, while the density and flux derived from
the \ion{O}{vii} triplet disagree. They proposed therefore a model of
``accretion-fed corona'' in which the X-ray emission originates from
three plasma components: a hot ($T_{\rm e} \approx 10$ MK) corona, a
high density ($n_{\rm e} \approx 6.0 \times 10^{12}$ cm$^{-3}$,
$T_{\rm e} \approx 3.0$ MK) post-shock region close to the shock
front, and a cold less dense ($n_{\rm e} \approx 2 \times 10^{11}$
cm$^{-3}$, $T_{\rm e} \approx 2.0$ MK) post-shock cooling region, with
300 times more volume and 30 times more mass than that of the post
shock region itself. On the other hand, it is worth to note that,
recently, Sacco et al. (\cite{sacco10}) have shown that, if accretion
streams are inhomogeneous and the chromospheric absorption is
efficient, their hydrodynamic model of accretion shock predicts that
different He-like triplets measure different densities of the X-ray
emitting plasma. This is explained because the effects of absorption
increases with wavelength and, as a consequence, the \ion{Ne}{ix}
(13.45 {\AA}) emission results to be less absorbed than \ion{O}{vii}
(21.60 {\AA}) emission. In particular, Sacco et al. (\cite{sacco10})
showed that, in an inhomogeneous stream affected by chromospheric
absorption, the average density measured with the \ion{Ne}{ix} triplet
is, in general, larger than that measured with \ion{O}{vii} triplet
although \ion{Ne}{ix} lines form at temperatures higher than
\ion{O}{vii} lines (as noted by Brickhouse et
al. \cite{brickhouse}). Unfortunately, due to the lack of high
signal-to-noise data covering the \ion{Ne}{ix} triplet, we are not
able to test this model for the other stars of our sample. Further
long exposure X-ray spectroscopic observations of the stars in our
sample are required to test this model as well as optical depth
effects on the estimate of $\dot{M}_{X-ray}$.
If the results presented in this paper are confirmed, we suggest
here that the optical depth effects in the \ion{O}{vii} resonance line may
explain the almost constant $\dot{M}_{\rm X-ray}$ derived in our sample if
the effect is larger in stars with larger optical derived mass accretion
rates. Finally, it is worth to note that our analysis does not allow
us to conclude that there is no relation between $\langle\dot{M}_{\rm
Opt}\rangle$ and $\dot{M}_{\rm X-ray}$. From our analysis we can just
conclude that the relation, if exists, is masked by the (small) scatter in
the values of $\dot{M}_{\rm X-ray}$. To this respect, it is interesting
to note that the lower right panel in Fig.~\ref{opt-xray} seems to
suggest that $\dot{M}_{\rm X-ray}$ slightly increases for increasing
$\langle\dot{M}_{\rm Opt}\rangle$ if we consider that the value of
$\dot{M}_{\rm X-ray}$ for Hen 3-600 shows the largest error in X-rays.
\section{Conclusions}
We have carried out the first homogeneous comparison of optical and
X-ray derived mass accretion rates for a sample of CTTSs. We compare
the different optical tracers to each other, and to the X-ray derived
accretion rates. We also analysed the variability of the CTTS TW
Hya. Our findings lead to the following conclusions:
\begin{enumerate}
\item The mass accretion rates derived from the different optical
tracers agree within the errors for each source, albeit with a large
spread of typically $\approx 1$ order of magnitude (see
Fig.~\ref{opticalcomp}).
\item H$\alpha$ 10\% full width, whilst known to be a good {\em
indicator} of accretion, is not good at {\em measuring} mass
accretion rates.
\item For the CTTS TW Hya (the only source within the sample for
which a time analysis was possible), we find little variation in the
mass accretion rates (within each emission line tracer) over a period
of 5 months (see Fig.~\ref{variability}). The variability is
much smaller than the range of accretion rates derived from different
accretion tracers.
\item The X-ray mass accretion rates are always smaller than the
optically derived mass accretion rates for all sources within
the sample (see Fig.~\ref{opt-xray}). This can be explained if the
accretion streams are inhomogeneous and/or multiple streams with
different densities are present simultaneously. In these cases, Sacco
et al. (\cite{sacco10}) have shown that the chromospheric absorption triggers
a selection of the X-ray emitting shocks, absorbing preferentially
the X-ray emission from high density plasma components. As a result,
only the low density plasma component of the post-shock plasma can
be observed in the X-ray band, leading to a systematic underestimate
of the mass accretion rate.
\item We find that the X-ray derived mass accretion rate
ranges within a factor of $\approx 2$ around $2 \times 10^{-10}$
M$_{\odot}$\,yr$^{-1}$ (see Fig.~\ref{opt-xray}), despite the fact
that the range of optical mass accretion rates span almost 3 orders of
magnitude. Some evidence of non-negligible optical depth of emission
lines produced by post-shock accreting plasma (e.g. Argiroffi et al.
\cite{costanza09}) may explain the almost constant $\dot{M}_{\rm X-ray}$
if the effect is larger in stars with larger optical mass accretion
rates. This issue deserves further investigation in future studies to
assess the severity of optical depth effects on the estimate of mass
accretion rates in the X-ray band.
\end{enumerate}
\begin{acknowledgements}
Based on observations made with ESO Telescopes at the La Silla or
Paranal Observatories under programme ID $<069.C-0481>$, $<073.C-0355>$,
$<075.C-0292>$, $<077.D-0478>$, $<078.A-9059>$, $<079.A-9006>$,
$<079.A-9007>$ and $<079.A-9017>$. Based on observations made with the
Italian Telescopio Nazionale Galileo (TNG) operated on the island of
La Palma by the Fundaci\'{o}n Galileo Galilei of the INAF (Istituto
Nazionale di Astrofisica) at the Spanish Observatorio del Roque de
los Muchachos of the Instituto de Astrofisica de Canarias. This
work was supported by the EU Marie Curie Transfer of Knowledge program
PHOENIX under contract No. MTKD-CT-2005-029768 and in part by Agenzia
Spaziale Italiana under contract No. ASI-INAF I/088/06/0.
\end{acknowledgements}
|
1,108,101,566,730 | arxiv | \section{Introduction}
Reconciling low with high gravitational energy scales remains a subtle
issue that is so far not fully-explored by merely looking at current astrophysical data \cite{referenza1}. Thereby, probing general relativity at both short and large distances with arbitrary accuracy is seemly for guaranteeing the validity of general relativity at different energy scales \cite{ref1,ref2}.
\noindent
For instance, at primordial times the increase of gravitational energy could break down general relativity predictability, requiring the existence of quantum gravity \cite{referenza2,referenza2a,referenza2b}. No conclusive approaches to quantum gravity exist, challenging the standard puzzle of unifying fundamental forces into a single scheme \cite{referenza2bis1,referenza2bis1a,referenza2bis2,referenza2bis3}. On the other hand, at infrared energy domains unknown ingredients, namely dark energy and dark matter, dominate over the other species, possibly suggesting Einstein's gravity extensions\footnote{For instance, the Planck satellite has not excluded Starobinsky's inflation, built up in terms of a second order curvature correction to Hilbert-Einstein's action \cite{planck}.} \cite{referenza3,referenza3.1,referenza3bis}. Again, this poses new unexpected caveats jeopardizing Einstein's gravity, likely requiring new physics behind general relativity.
\noindent
These two energy domains are somewhat not fully-matched to each other and as a possible signature of such a problem could be attributed to the recently-stressed convoluted cosmic tensions \cite{referenza4,referenza4bis}. Certainly, the fact that measurements are poorly constrained due to the small number of data points severely limits our matching \cite{referenza5,referenza5bis,referenza5tris}. Though data catalogs are not large enough, constraining Einstein's gravity at different scales may culminate in groundbreaking discoveries and so several examples can be used to combine small and high red shift measures \cite{referenza6,referenza6a,referenza6b,referenza6c}. In this respect, as a possible example of technique that could be used for both low and high energy domains, we face the red and blue shift, based on local position invariance and precision test, along which red and blue shift measures between two identical clocks, regardless the clock structures, plays a crucial role in bounding how time intervals change in presence of gravity \cite{referenza7}.
\noindent
In this work, we consider the red and blue shift in three distinct contexts of low, intermediate and high gravity. Thereby, we employ a Schwarzschild - de Sitter solution, with its effective cosmological constant, and an axially symmetric Zipoy - Voorhees metric, where we constrain the prolate term, $\delta$, that enters the metric itself as signature of spherical symmetry departure. We get feasible spacetime free parameter constraints from suitable red and blue shift regimes and check the intervals of values these parameters may hold within astrophysical frameworks. To do so, we first assume Schwarzschild - de Sitter spacetime, with $\Lambda$ bound got from Planck mission \cite{planck}, for neutron stars first (high gravity), then for white dwarfs (intermediate gravity) and finally for the Earth - Mars system (low gravity). As expected, the overall treatment provides red and blue shift incompatible values with those predicted by ongoing experiments in the solar system, but leads to slight acceptable bounds as neutron stars and white dwarfs are used as benchmarks. In analogy, we build up the same procedure for the Zipoy - Voorhees metric. Here, we check possible evidence for quadrupole corrections on the same astrophysical objects discussed for the Schwarzschild - de Sitter spacetime. Last but not least, we propose novel experimental setups to improve the quality of our outcomes. In addition, we propose the binary system Mars - Phobos, for working out satellite laser ranging to fix refined red and blue shift constraints. Summing up, we demonstrate the incompatibility between red and blue shifts at astrophysical level. However, we presume our technique to be used at a cosmological level to get experimental bounds and then to compare the corresponding expectations with respect to those measures that cause tensions in cosmology.
\noindent
The paper is structured as follows. In Sec. \ref{sezione2}, we describe the main features of our method, i.e. showing how red and blue shift are characterized. In Sec. \ref{sezione3}, we relate our two spacetimes with the red and blue shifts. Thus, in Sec. \ref{sezione4}, we discuss our numerical results and new experimental configurations then, in Sec. \ref{sezione5}, we develop conclusions and perspectives of our work.
\section{The photon red and blue shift}\label{sezione2}
Red and blue shift represent a tool that measures the frequency modifications measured in the exchange of a photon between two observers. This technique does not involve the field equations but only the spacetime symmetries. In this section, we therefore analyze the general method used to determine the red and blue shift, assuming an emitted photon from a massive object, e.g. a planet, a star, a wormhole, and so forth and received by a distant observer. We first highlight the red and blue shift general treatment in a rotating frame. Then, we work out the same for a static configuration and we focus on how to use the underlying strategy for astrophysical objects.
\subsection{General treatment}
\label{subsec1}
Here we follow the mathematical procedure described in \cite{2.1,2.2} and start with the simplest metric describing a rotating axially symmetric spacetime (in spherical coordinates)
\begin{equation}
\label{metric}
ds^2=g_{tt}dt^2+2g_{t\varphi}dtd\varphi+g_{\varphi\varphi}d\varphi^2+g_{rr}dr^2+g_{\theta\theta}d\theta^2,
\end{equation}
where $g_{\mu\nu}=g_{\mu\nu}(r,\theta)$, in the gauge $g_{r\theta}=0$.
\noindent
Let us now indicate with $r_e$ and $r_d$ the photon's emitter and photon's detector positions, respectively\footnote{From now on, to simplify the notation, we will denote these two positions with $r_p$, where the subscript $p$ can be $e$ or $d$ depending on whether it refers to the emitter or the detector respectively.} and so
$ u^{\mu}_{p}=(u^{t}_{p}, u^{r}_{p}, u^{\theta}_{p}, u^{\varphi}_{p})$
is the four-velocity of the photon's emitter (when $p=e$) or of the photon's detector (when $p=d$). Since the above relation refers to a massive object, the following normalization condition holds
\begin{equation}
\label{norm1}
u^{\mu}_{p}u_{p,\mu}=-1\,,
\end{equation}
which explicitly reads
\begin{strip}
\begin{equation}
\left.[g_{tt}(u^t)^2+g_{rr}(u^r)^2+g_{\varphi\varphi}(u^{\varphi})^2+g_{\theta\theta}(u^{\theta})^2+g_{t\varphi}u^t u^{\varphi}]\right|_{r=r_p}=-1\,.
\end{equation}
\end{strip}
\noindent
Similarly, we indicate with
$ k^{\mu}=(k^{t}, k^{r}, k^{\theta}, k^{\varphi})$
the four-velocity of the photon, albeit normalization condition is now
\begin{equation}
\label{norm2}
k^{\mu}k_{\mu}=0\,,
\end{equation}
to explicitly give
\begin{eqnarray}
\nonumber
&&g_{tt}(k^t)^2+g_{rr}(k^r)^2+g_{\varphi\varphi}(k^{\varphi})^2+\\
&&+g_{\theta\theta}(k^{\theta})^2+g_{t\varphi}k^t k^{\varphi}=0.
\end{eqnarray}
Metric components are independent from the variables $t$ and $\varphi$, therefore there are two commuting Killing vector fields, respectively time-like and rotational ones as follow
\begin{eqnarray}
\label{killing1}
&\xi^{\mu}=(1,0,0,0)\,, \\
\label{killing2}
&\psi^{\mu}=(0,0,0,1)\,.
\end{eqnarray}
These two Killing fields imply the existence of two conserved quantities for the massive particle
\begin{eqnarray}
E&\doteq -g_{\mu\nu}\xi^{\mu}u^{\nu}=-g_{tt}u^t-g_{t\varphi}u^{\varphi}\,,\label{E} \\
L&\doteq g_{\mu\nu}\psi^{\mu}u^{\nu}=g_{\varphi\varphi}u^{\varphi}+g_{t\varphi}u^t\,,\label{L}
\end{eqnarray}
that are the total energy, $E$, and angular momentum, $L$.
\noindent
We now evaluate $u^{\varphi}$ and $u^t$ in function of the energy, $E$, and angular momentum, $L$, from Eqs. (\ref{E}) and (\ref{L}), to give
\begin{eqnarray}
u^{\varphi}&=&-\frac{Eg_{t\varphi}+Lg_{tt}}{g^2_{t\varphi}-g_{\varphi\varphi}g_{tt}},\\
u^t&=&\frac{Eg_{\varphi\varphi}+Lg_{t\varphi}}{g^2_{t\varphi}-g_{\varphi\varphi}g_{tt}}\,,
\end{eqnarray}
and plugging the above expressions into Eq. (\ref{norm1}), we get:
\newpage
\begin{strip}
\begin{equation}
\label{norm3}
\left.\left[g_{rr}(u^r)^2+g_{\theta\theta}(u^{\theta})^2+1-\frac{E^2g_{\varphi\varphi}+L^2g_{tt}+2ELg_{t\varphi}}{g^2_{t\varphi}-g_{\varphi\varphi}g_{tt}}\right]\right|_{r=r_p}=0.
\end{equation}
\end{strip}
\noindent
Even though the four-vector components for velocity and momentum do not vanish, rotating the polar coordinate system, the metric, Eq. (\ref{metric}), does not change. Hence, this intrinsic symmetry implies that we can limit to the equatorial plane, where $\theta=\pi/2$, leading to $u^{\theta}=k^{\theta}=0$.
Further, since we hereafter on circular orbits only, we even require $u^r=0$, providing Eq. (\ref{norm3}) becomes
\begin{equation}
\label{potential}
\left.\left[1-\frac{E^2g_{\varphi\varphi}+L^2g_{tt}+2ELg_{t\varphi}}{g^2_{t\varphi}-g_{\varphi\varphi}g_{tt}}\right]\right|_{r=r_p}=0,
\end{equation}
that reduces to
\begin{equation}
\label{Veff}
V_{\rm{eff}}(r_p)=0.
\end{equation}
The former is the energy conservation law, clearly valid for circular orbits. In addition, these orbits require \cite{2.3,2.4}
\begin{equation}
\label{Veff'}
V_{\rm{eff}}'(r_p)=0\,,
\end{equation}
\begin{equation}
\label{Veff''}
V_{\rm{eff}}''(r_p)\geq 0\,,
\end{equation}
guaranteeing orbit stability \cite{2.4} and the existence of the potential minimum\footnote{For the sake of completeness, the equality only holds for spherical symmetry.}.
\noindent
\\Analogously, the two Killing fields, Eqs. (\ref{killing1}) and (\ref{killing2}), imply the existence of two conserved quantities for the photon, the total energy $E_{\gamma}$ and the angular momentum~$L_{\gamma}$
\begin{eqnarray}
\label{Egamma}
E_{\gamma}&\doteq& -g_{\mu\nu}\xi^{\mu}k^{\nu}=-g_{tt}k^t-g_{t\varphi}k^{\varphi}, \\
\label{Lgamma}
L_{\gamma}&\doteq& g_{\mu\nu}\psi^{\mu}k^{\nu}=g_{\varphi\varphi}k^{\varphi}+g_{t\varphi}k^t.
\end{eqnarray}
\subsection{Evaluating the red and blue shift}
Now we have all the ingredients to determine the red and blue shift of the emitted photon. Thus, the photon frequency at given point $p$ is defined as \cite{2.5}
\begin{equation}
\omega_p = -\left(k_{\mu}u^{\mu}\right)\vert_p.
\end{equation}
Since we consider timelike orbits that are both circular and equatorial, depending on whether we use Eqs. (\ref{E})-(\ref{L}) or Eqs. (\ref{Egamma})-(\ref{Lgamma}), we can rewrite $\omega_p$ in two ways, respectively:
\begin{eqnarray}
\omega_p&=&\left(Ek^t-Lk^{\varphi}\right)\vert_p,\\
\omega_p&=&\left(E_{\gamma}u^t-L_{\gamma}u^{\varphi}\right)\vert_p.
\end{eqnarray}
In particular, the frequency of the photon at the emission point is
\begin{eqnarray}
\nonumber
\omega_e &=& -\left(k_{\mu}u^{\mu}\right)\vert_e=\\
&=& \left(Ek^t-Lk^{\varphi}\right)\vert_e= \\
\nonumber
&=&\left(E_{\gamma}u^t-L_{\gamma}u^{\varphi}\right)\vert_e,
\end{eqnarray}
whereas the frequency of the photon at the detection point is
\begin{eqnarray}
\nonumber
\omega_d &=& -\left(k_{\mu}u^{\mu}\right)\vert_d=\\
&=& \left(Ek^t-Lk^{\varphi}\right)\vert_d= \\
\nonumber
&=&\left(E_{\gamma}u^t-L_{\gamma}u^{\varphi}\right)\vert_d.
\end{eqnarray}
Thus, we define the frequency shift associated with the emission and detection of photons as
\begin{eqnarray}
\label{a}
1+z=\frac{\omega_e}{\omega_d}&=&\frac{\left(E_{\gamma}u^t-L_{\gamma}u^{\varphi}\right)\vert_e}{\left(E_{\gamma}u^t-L_{\gamma}u^{\varphi}\right)\vert_d}=\nonumber\\
\label{fs}
&=&\frac{\left(u^t-bu^{\varphi}\right)\vert_e}{\left(u^t-bu^{\varphi}\right)\vert_d},
\end{eqnarray}
where\footnote{Let us observe here that $b$ is the same both at the numerator and the denominator of (\ref{fs}), since $E_\gamma$ and $L_\gamma$ are determined by the same photon path.}
\begin{equation}\label{eq:b}
b\equiv\frac{L_{\gamma}}{E_{\gamma}}\,.
\end{equation}
It will also be convenient to introduce the red shift $z_c$ corresponding to a photon emitted by a particle located at the center observed by a faraway detector, i.e. $b=0$:
\begin{equation}
\label{zc}
z_c=\frac{u_e^t}{u_d^t}-1,
\end{equation}
since
astronomical data are generally collected in terms of the kinematic red shift, defined as
\begin{eqnarray}
\label{z_kin}
\nonumber
z_{\rm{kin}}\doteq z-z_c&=&\frac{(u^t_e\,u^\varphi_d-u^t_d u^\varphi_e)b}{u^t_d(u_t-b u^\varphi)_d}\,=\\
&=& \frac{(u^t_e\Omega_d-u^\varphi_e)b}{u^t_d(1-\Omega_db)}\,,\label{zkin}
\end{eqnarray}
where the \emph{angular velocity} of a detector located far away from the photons source
\begin{equation}
\label{angularvel}
\Omega_d\equiv\frac{u_d^{\varphi}}{u_d^t}
\end{equation}
has been introduced as well.
\noindent
Of course, $b$ varies with the photon path. Then a value of $b$ as a function of the circular orbit of the emitting source (i.e., as a function of $r$) must be determined, in such a way that its absolute value represents the observed radial distance on either side of the observed center ($b=0$) by a faraway observer. The idea is that frequency shifts yielding maximum and minimum values correspond to photons emitted with initial velocities collinear to the source velocity \cite{NuSaSu2001}. This amounts to require $k^r=k^\theta=0$ at $p=e$, and therefore these photons paths, recalling (\ref{Egamma}), (\ref{Lgamma}), are such that
\begin{equation}
-E_\gamma k^t_e+L_\gamma k^\varphi_e=(k_\mu\,k^\mu)_e=0,
\end{equation}
that gives, using (\ref{eq:b}), two possible solutions for
the so called {\sl apparent impact parameter} $b$:
\begin{equation}
\label{bpm}
b_{\pm}=-\frac{g_{t\varphi}\pm\sqrt{g_{t\varphi}^2-g_{\varphi\varphi}g_{tt}}}{g_{tt}},
\end{equation}
depending on whether the photon is emitted by a receding, $b_-$, or an approaching, $b_+$, object with respect to a distant observer. Hence, the $b_-$ and $b_+$ solutions are related to the red shift and the blue shift once we substitute them in Eq. (\ref{fs}), respectively
\begin{equation}
\label{reds}
{\rm Red\,\, shift\,\,\,} z_{\rm{red}}=\frac{u^t_e-b_{-}u^{\varphi}_e}{u^t_d\left(1-b_{-}\Omega_d\right)}-1,
\end{equation}
\begin{equation}
\label{blues}
{\rm Blue\,\, shift\,\,\,} z_{\rm{blue}}=\frac{u^t_e-b_{+}u^{\varphi}_e}{u^t_d\left(1-b_{+}\Omega_d\right)}-1.
\end{equation}
Finally, given $b_+$ and $b_-$, from Eq. (\ref{z_kin}), we get two possible $z_{kin}$ values, namely $z_1$ and $z_2$
\begin{equation}
\label{z1}
z_1=\frac{(u_e^t \Omega_d -u_e^{\varphi})b_{-}}{u_d^t(1-\Omega_d b_{-})},
\end{equation}
\begin{equation}
\label{z2}
z_2=\frac{(u_e^t \Omega_d -u_e^{\varphi})b_{+}}{u_d^t(1-\Omega_d b_{+})},
\end{equation}
that correspond to the cases in which the photon is emitted by a receding or an approaching source, respectively\footnote{It is remarkable to underline the relation between $z_{red}$, or $z_{blue}$, and $z_{kin}$. In particular, from Eq. (\ref{z_kin}), it reads
\begin{eqnarray}
z_{red}&=&\left.(z_{kin}+z_c)\right|_{b=b_{-}},\\
z_{blue}&=&\left.(z_{kin}+z_c)\right|_{b=b_{+}}.
\end{eqnarray}
}
\subsection{Non rotating spacetime}
As special case, we limit to non rotating spacetimes, i.e. the ones for which $g_{t\varphi}=0$. This will be the case of Schwarzschild - de Sitter and Zipoy - Voorhees metrics that we analyze in the next sections. Thus, Eq. (\ref{metric}) simply reduces to
\begin{equation}
\label{metric2}
ds^2=g_{tt}dt^2+g_{\varphi\varphi}d\varphi^2+g_{rr}dr^2+g_{\theta\theta}d\theta^2,
\end{equation}
with gauge condition, $g_{r\theta}=0$. Clearly, all the previous equations before determined are accordingly simplified and so the conserved quantities associated to the massive particles (observes) now become
\begin{eqnarray}
E&=&-g_{tt}u^t,\\
L&=&g_{\varphi\varphi}u^{\varphi},
\end{eqnarray}
and so the velocities are
$u^t=-\frac{E}{g_{tt}}, u^{\varphi}=\frac{L}{g_{\varphi\varphi}}$, while the equation $
V_{\rm{eff}}(r_p)=0$ for the effective potential becomes
\begin{equation}
\label{Veff_nr}
\left.\left[1+\frac{E^2g_{\varphi\varphi}+L^2g_{tt}}{g_{\varphi\varphi}g_{tt}}\right]\right|_{r=r_p}=0.
\end{equation}
Similarly, the conserved quantities associated to the photon are
\begin{eqnarray}
E_{\gamma}&=&-g_{tt}k^t,\\
L_{\gamma}&=&g_{\varphi\varphi}k^{\varphi},
\end{eqnarray}
from which $
k^t=-\frac{E_{\gamma}}{g_{tt}}, k^{\varphi}=\frac{L_{\gamma}}{g_{\varphi\varphi}}$, so that the apparent impact parameter finally reads
\begin{equation}
\label{bnr}
b_{\pm}=\mp\sqrt{-\frac{g_{\varphi\varphi}}{g_{tt}}}.
\end{equation}
The functional forms of $z_1$ and $z_2$ are identical to Eqs. (\ref{z1}) and (\ref{z2}) since assuming $g_{t\varphi}=0$ modifies only the apparent impact parameters rather than $z_{kin}$. We here observe that $b_{+}=- b_{-}$, implying $z_1=-z_2$.
With the above recipe in our hand we are now in condition to handle spacetime symmetries to model astrophysical landscapes. We therefore report below the two metrics involved in our computation.
\section{Spacetime solutions}\label{sezione3}
Our purpose is to assess astrophysical frameworks by means of given spacetimes. Thereby, we first handle the simplest axisymmetric spacetime based on the Zipoy - Voorhees metric. We aim at modelling astrophysical objects, such as neutron stars and white dwarfs by means of such a metric. Afterwards, we switch to the spherical symmetry based on the Schwarzschild - de Sitter metric. In such a case, differently of the astrophysical case, we intend to work out cosmological scenarios and to compute red and blue shifts by fixing the cosmological constant from Planck's measurements \cite{planck}.
\noindent
Clearly, these two regimes, based on two different spacetime symmetries, are profoundly different from each other and, as above stated, we are therefore considering two distinct energy domains. The first is a regime of high gravity, since it deals with neutron stars and white dwarfs. The second is purely cosmological, involving infrared scales of energy. Below we first summarize each metric formalism and then we argue bounds over the free coefficients.
\subsection{The Zipoy - Voorhees metric}
The strategy of getting red and blue shift is here applied to the \emph{Zipoy - Voorhees metric} \cite{2.6}. The metric, in spherical coordinates, reads
\begin{equation}
\label{ZVmetric}
ds^2=-Fdt^2+\frac{1}{F}\left[Gdr^2+Hd\theta^2+(r^2-2kr)\sin^2\theta d\varphi^2\right],
\end{equation}
where
\begin{eqnarray}
\label{F}
F&=&\left(1-\frac{2k}{r}\right)^{\delta},\\
\label{G}
G&=&\left(\frac{r^2-2kr}{r^2-2kr+k^2\sin^2\theta}\right)^{\delta^2-1},\\
\label{H}
H&=&\frac{\left(r^2-2kr\right)^{\delta^2}}{\left(r^2-2kr+k^2\sin^2\theta\right)^{\delta^2-1}}.
\end{eqnarray}
Here, $\delta$ is a free parameter which can vary into three possible ranges
\begin{itemize}
\item $\delta>1$: tidal forces diverge at the singularity, particles are crushed;
\item $\frac{1}{2}<\delta<1$: the singularity is mild, i.e. particles reach it with zero velocity;
\item $\delta<\frac{1}{2}$: the singularity is repulsive, particles are ejected.
\end{itemize}
It is remarkable to notice the limiting case $\delta\rightarrow1$ provides the Schwarzschild metric, whereas $\delta\rightarrow{1/2}$ could show likely critical effects. For example, in Ref. \cite{naked} the authors worked out naked singularity configuration to get regions of repulsive gravity, using eigenvalue method \cite{autovalori1,autovalori2} and showing this interval as critical. However, we here focus on regular objects, such as NS, WD and/or solar system configurations, and so we do not expect any critical region over $\delta$ and/or red or blue shifts, as we effectively get later. Furthermore, $k=m/\delta$ is the ratio between the mass $m$ of the gravitational field and the $\delta$ parameter.
\noindent
As underlined in Sec. \ref{subsec1}, we are limiting to the equatorial plane, {\it i.e.}, $\theta=\pi/2$. The Zipoy - Voorhees metric describes a non rotating spacetime ($g_{t\varphi}=0$), thus we can consider Eq. (\ref{Veff_nr}) that reads
\begin{equation}
\label{VeffZV}
\left.1-\frac{E^2 \left(r^2-2 k r\right) \left(1-\frac{2 k}{r}\right)^{-\delta }-L^2 \left(1-\frac{2 k}{r}\right)^{\delta }}{r^2-2 k r}\right|_{r=r_p}=0
\end{equation}
Its derivative with respect to $r$ gives the condition for circular orbits, say Eq. (\ref{Veff'}):
\begin{strip}
\begin{equation}
\label{Veff'ZV}
\left.\frac{\left(1-\frac{2 k}{r}\right)^{-\delta } \left[2 \delta k r E^2 (r-2 k)+2 L^2 (\delta k+k-r) \left(1-\frac{2 k}{r}\right)^{2 \delta }\right]}{r^2 (r-2 k)^2}\right|_{r=r_p}=0.
\end{equation}
\end{strip}
\noindent
with $r_p=r_e,r_d$, as before.
Solving the system given by the two last relations, we obtain the total energy and the angular momentum
\begin{equation}
\label{E_ZV}
E=\left.\sqrt{\frac{\left(1-\frac{2 k}{r}\right)^{\delta } (\delta k+k-r)}{2 \delta k+k-r}}\right|_{r=r_p},
\end{equation}
\begin{equation}
\label{L_ZV}
L=\pm\left.\sqrt{\frac{\delta k r (2 k-r) \left(1-\frac{2 k}{r}\right)^{-\delta }}{2 \delta k+k-r}}\right|_{r=r_p}.
\end{equation}
\noindent Consequently, we immediately get
\begin{equation}
\label{utZV}
\left.u^t\right|_{r=r_p}=\left.-\sqrt{\frac{\left(1-\frac{2 k}{r}\right)^{-\delta } (\delta k+k-r)}{2 \delta k+k-r}}\right|_{r=r_p},
\end{equation}
\begin{equation}
\label{ufZV}
\left.u^{\varphi}\right|_{r=r_p}=\left.\pm \sqrt{\frac{\delta k (2 k-r) \left(1-\frac{2 k}{r}\right)^{\delta }}{r (k-r)^2 (2 \delta k+k-r)}}\right|_{r=r_p}.
\end{equation}
Furthermore, from Eq. (\ref{bnr}), we have
\begin{equation}
b_{\pm}=\mp\frac{\sqrt{r^2-2kr}}{\left(1-\frac{2 k}{r}\right)^{\delta}}\,,
\end{equation}
Finally, substituting Eqs. (\ref{utZV}) - (\ref{ufZV}) evaluated in $r=r_d$ into Eq. (\ref{angularvel}), we get the angular velocity:
\begin{equation}
\Omega_{d\pm}=\mp\sqrt{\frac{\delta k (2 k-r_d) \left(1-\frac{2 k}{r_d}\right)^{2 \delta }}{r_d (k-r_d)^2 (\delta k+k-r_d)}},
\end{equation}
where $\Omega_{d+}$ and $\Omega_{d-}$ are respectively referred to a co-rotating and to a counter-rotating photons source with respect to the angular velocity of the gravitational field source. In conclusion, substituting all these equations into Eqs. (\ref{z1}) - (\ref{z2}), we get the expressions for $z_1$ and $z_2$ for the Zipoy - Voorhees metric
\begin{strip}
\begin{eqnarray}
\label{z1_ZV}
\nonumber
z_{1\pm}&=&\pm\Biggl\{\left(1-\frac{2 k}{r_d}\right)^{\delta} \left(1-\frac{2 k}{r_e}\right)^{-\delta}\Biggl[-\sqrt{\frac{ \left(1-\frac{2 k}{r_e}\right)^{\delta} \left(1-\frac{2 k}{r_d}\right)^{2\delta}(r_d-2k)^2(k-r_e+k\delta)k\delta}{(r_d-k)^2(k-r_e+2k\delta)(r_d-k-k\delta)}}+\\
\nonumber
&&+\left(1-\frac{2 k}{r_d}\right)^{\delta}\sqrt{\frac{\left(1-\frac{2 k}{r_e}\right)^{\delta}(2k-r_e)^2 k\delta}{(k-r_e)^2(r_e-k-2k\delta)}}\Biggr]\Biggr\}\Biggl\{\sqrt{\frac{\left(1-\frac{2 k}{r_d}\right)^{\delta }(k-r_d+k\delta)}{k-r_d+2k\delta}}\Biggl[\left(1-\frac{2 k}{r_d}\right)^{\delta}+\\
&&\pm\sqrt{\frac{(r_d-2k)^2\left(1-\frac{2 k}{r_d}\right)^{2\delta}k\delta}{(r_d-k)^2(r_d-k-k\delta)}}\Biggr]\Biggr\}^{-1},
\end{eqnarray}
\end{strip}
\begin{strip}
\begin{eqnarray}
\label{z2_ZV}
\nonumber
z_{2\pm}&=&\pm\Biggl\{\left(1-\frac{2 k}{r_d}\right)^{\delta} \left(1-\frac{2 k}{r_e}\right)^{-\delta}\Biggl[\sqrt{\frac{ \left(1-\frac{2 k}{r_e}\right)^{\delta} \left(1-\frac{2 k}{r_d}\right)^{2\delta}(r_d-2k)^2(k-r_e+k\delta)k\delta}{(r_d-k)^2(k-r_e+2k\delta)(r_d-k-k\delta)}}+\\
\nonumber
&&-\left(1-\frac{2 k}{r_d}\right)^{\delta}\sqrt{\frac{\left(1-\frac{2 k}{r_e}\right)^{\delta}(2k-r_e)^2 k\delta}{(k-r_e)^2(r_e-k-2k\delta)}}\Biggr]\Biggr\}\Biggl\{\sqrt{\frac{\left(1-\frac{2 k}{r_d}\right)^{\delta }(k-r_d+k\delta)}{k-r_d+2k\delta}}\Biggl[\left(1-\frac{2 k}{r_d}\right)^{\delta}+\\
&&\mp\sqrt{\frac{(r_d-2k)^2\left(1-\frac{2 k}{r_d}\right)^{2\delta}k\delta}{(r_d-k)^2(r_d-k-k\delta)}}\Biggr]\Biggr\}^{-1},
\end{eqnarray}
\end{strip}
\noindent
where the subscript $\pm$ is again referred to a co-rotating and counter-rotating source with respect to the angular velocity of the gravitational field source.
\noindent
Let us observe that $z_1=-z_2$, in both the co-rotating and counter-rotating cases, regardless of the mass that generates the gravitational field.
Above we put forward that the $z_1$ and $z_2$ variations can be expressed in terms of $r_d$ for both a rotating and counter-rotating configurations. This would help to argue the intervals of validity for the Zipoy - Voorhees free parameters when this metric is applied to astrophysical situations. We describe this approach in detail below.
\subsection{Gravitational field sources for the Zipoy - Voorhees metric}
We analyze the variation of $z_1$ and $z_2$ as function of the position of the detector $r_d$, in the co-rotating and in the counter-rotating configurations. Our analysis is based on different gravitational field sources
\begin{itemize}
\item a neutron star in the maximally - rotating configuration \cite{2.7}, corresponding to a high gravity regime,
\item a white dwarf in the maximally - rotating configuration \cite{2.7}, correspoding to an intemediate gravity regime,
\item Earth and Mars for the Solar System, corresponding to a low gravity regime.
\end{itemize}
\noindent
We report the plots \ref{fig:NSZV}, \ref{fig:WDZV} and \ref{fig:EarthMarsZV} in which we infer the availability intervals for each term.
\\For the neutron star, the variation of $z_1$ and $z_2$ as function of $r_d$ depend stronger on $\delta$. For this reason, we choose three value of $\delta$, one for each range, Fig. \ref{fig:NSZV}
\begin{itemize}
\item[{\bf 1.}] $\delta=1000$, {\it i.e.}, where we take an arbitrary large value to address the condition $\delta\gg 1$ ,
\item[{\bf 2.}] $\delta=\frac{3}{4}$, as arbitrary close value to $\delta=1$, obtained as mean value of the interval $\frac{1}{2}<\delta<1$,
\item[{\bf 3.}] $\delta=\frac{1}{4}$, as arbitrary close value to $\delta =0$, obtained as mean value of the interval $0<\delta<\frac{1}{2}$.
\end{itemize}
However, for WDs, see Fig. \ref{fig:WDZV}, the increase or decrease of $\delta$ do not seem to modify the overall evolution. The same happens for the binary configuration constituted by the Earth and Mars: one can notice from Fig. \ref{fig:EarthMarsZV} that they very weakly depend upon $\delta$ variation. The above configuration is built up assuming the Earth and Mars as distinct gravitational sources as separate cases. For the sake of clearness, the $\delta$ variation is not so evident from our plots since those variations are extremely small and not particularly visible. The corresponding values have been evaluated for WDs, Earth and Mars, noticing a slight difference that permits one to fix $\delta$ to portray the examples we showed in the aforementioned figures.
\noindent
Even though not so evident from our plots, the above occurrence for which $\delta$ is as larger as one approaches higher gravity regimes turns out to be clear even from a theoretical viewpoint. As one approaches regimes of low gravity any quadrupole deviation is negligibly small and so one can approximate with a spherical symmetry those configurations, without losing generality. Furthermore, in the low and intermediate gravity regimes the symmetries $z_{1+}=z_{2-}$ and $z_{1-}=z_{2+}$ emerge, together with $z_{1\pm} =-z_{2\pm}$, being valid for any gravitational sources.
\noindent
The pending caveat to check would be represented by orbit stability, i.e. Eq. (\ref{Veff''}). For the Zipoy - Voorhees metric, the second derivative with respect to $r$ of Eq. (\ref{VeffZV}) is always zero: since this is a quasi-spherical spacetime, we can assert that all orbits are stable.
\\Finally, let us observe that, for $\delta=1$ and, consequently, $k=m$, all these equations reduce to those obtained in the Schwarzschild metric (see Appendix \ref{appendix}). It is now remarkable to stress that for $\delta=1/2$, in all the analyzed gravity regimes, we do not obtain critical values of z, as expected.
\subsection{The Schwarzschild - de Sitter metric}
In this subsection, we apply the method described above to the Schwarzschild - de Sitter metric, corresponding to a spherical symmetric spacetime with an effective cosmological constant, $\Lambda$ \cite{SdSpaper1,SdSpaper2,SdSpaper3}. For this fundamental property, the metric can be used for cosmological applications, to infer bounds on red and blue shifts, fixing $\Lambda$. In spherical coordinates, we have
\newpage
\begin{strip}
\begin{equation}
\label{metric_SdS}
ds^2=-\left(1-\frac{2m}{r}+\frac{\Lambda r^2}{3}\right)dt^2+\frac{1}{\left(1-\frac{2m}{r}+\frac{\Lambda r^2}{3}\right)}dr^2+r^2d\theta^2+r^2\sin^2\theta d\varphi^2.
\end{equation}
\end{strip}
\noindent
We again stress, as we did in Sec. \ref{subsec1}, we study the phenomenon in the equatorial plane, namely $\theta=\pi/2$. Even this metric is clearly non-rotating and so Eq. (\ref{Veff_nr}) becomes
\begin{equation}
\label{VeffSdS}
\left.\left[1-\frac{E^2r^2-L^2\left(1-\frac{2m}{r}+\frac{\Lambda r^2}{3}\right)}{r^2\left(1-\frac{2m}{r}+\frac{\Lambda r^2}{3}\right)}\right]\right|_{r=r_p}=0\,,
\end{equation}
and its derivative with respect to $r$
\begin{equation}
\label{Veff'SdS}
\left.\left[\frac{6(3m+\Lambda r^3)E^2}{(\Lambda r^3+3r-6m)}-\frac{2L^2}{r^3}\right]\right|_{r=r_p}=0.
\end{equation}
with $r_p=\left\{r_e,r_d\right\}$, as before.
Solving the system given by the former two equations, we obtain
\begin{eqnarray}
\label{E_SdS}
E^2&=&\left.\frac{(\Lambda r^3+3r-6m)^2}{9r(r-3m)}\right|_{r=r_p},\\
\label{L_SdS}
L^2&=&\left.\frac{r^2(3m+\Lambda r^3)}{3(r-3m)}\right|_{r=r_p},
\end{eqnarray}
from which we get the total energy and the angular momentum:
\begin{eqnarray}
E&=&\left.\frac{\Lambda r^3+3r-6m}{\sqrt{9r(r-3m)}}\right|_{r=r_p},\\
L&=&\left.\pm r\sqrt{\frac{3m+\Lambda r^3}{3(r-3m)}}\right|_{r=r_p}.
\end{eqnarray}
Thus, we again find
\begin{equation}
\label{ut_SdS}
\left.u^t\right|_{r=r_p}=\left.\sqrt{\frac{r}{r-3m}}\right|_{r=r_p},
\end{equation}
\begin{equation}
\label{uf_SdS}
\left.u^{\varphi}\right|_{r=r_p}=\left.\pm\sqrt{\frac{3m+\Lambda r^3}{3r^2(r-3m)}}\right|_{r=r_p}.
\end{equation}
Furthermore, we have
\begin{equation}
b_{\pm}=\mp\sqrt{\frac{r^2}{\left(1-\frac{2m}{r}+\frac{\Lambda r^2}{3}\right)}},
\end{equation}
\begin{equation}
\Omega_{d\pm}=\pm\sqrt{\frac{3m+\Lambda r_d^3}{3r_d^3}},
\end{equation}
where $\Omega_{d+}$ and $\Omega_{d-}$ are respectively referred to a co-rotating and to a counter-rotating photons source with respect to the angular velocity of the gravitational field source, as before.
\noindent
Hence, plugging all these relations within Eqs. (\ref{z1})-(\ref{z2}), we get the expressions for $z_1$ and $z_2$ for the Schwarzschild - de Sitter metric
\begin{strip}
\begin{eqnarray}
\label{z1_SdS}
z_{1\pm}&=&\pm\frac{\sqrt{\frac{r_e}{r_e-3m}}\left(\sqrt{\frac{3m+\Lambda r_d^3}{3r_d-6m+\Lambda r_d^3}}-\sqrt{\frac{3m+\Lambda r_e^3}{3r_e-6m+\Lambda r_e^3}}\right)}{\sqrt{\frac{r_d}{r_d-3m}}\left(1\mp\sqrt{\frac{3m+\Lambda r_d^3}{3r_d-6m+\Lambda r_d^3}}\right)},\\
\,\nonumber\\
z_{2\pm}&=&\mp\frac{\sqrt{\frac{r_e}{r_e-3m}}\left(\sqrt{\frac{3m+\Lambda r_d^3}{3r_d-6m+\Lambda r_d^3}}-\sqrt{\frac{3m+\Lambda r_e^3}{3r_e-6m+\Lambda r_e^3}}\right)}{\sqrt{\frac{r_d}{r_d-3m}}\left(1\pm\sqrt{\frac{3m+\Lambda r_d^3}{3r_d-6m+\Lambda r_d^3}}\right)},\label{z2_SdS}
\end{eqnarray}
\end{strip}
where the subscript $\pm$ is again referred to a co-rotating and counter-rotating source with respect to the angular velocity of the gravitational field source. Let us observe that $z_1=-z_2$, in both the co-rotating and counter-rotating cases, regardless of the mass that generates the gravitational field.
\subsection{Gravitational field sources for the Schwarzschild - de Sitter metric}
In analogy to our previous treatment, for the Schwarzschild - de Sitter metric we analyze the variation of $z_1$ and $z_2$ as function of the position of the detector $r_d$, in the co-rotating and in the counter-rotating cases. As before, our analysis is based on different gravitational field sources:
\begin{itemize}
\item a neutron star in the maximally - rotating configuration \cite{2.7}, see Fig. \ref{fig:NSSDS}. Here the employed field is strong,
\item a white dwarf in the maximally - rotating configuration \cite{2.7}, see Fig. \ref{fig:WDSDS}. Here we consider an intermediate field,
\item Earth and Mars for the Solar System, see Fig. \ref{fig:EarthMarsSDS}.
\end{itemize}
Furthermore, we consider $\Lambda$ as the cosmological constant, whose value is $\Lambda=1.1056\times 10^{-52}\,m^{-2}$ \cite{planck}.
\noindent
As for the Zipoy - Voorhees metric, in the low and intermediate gravity regimes we again find $z_{1+}=z_{2-}$ and $z_{1-}=z_{2+}$, in addition to $z_{1\pm} =-z_{2\pm}$.
\noindent
The last thing to check is the orbits stability, i.e. Eq. (\ref{Veff''}). For the Schwarzschild - de Sitter metric, the second derivative of Eq. (\ref{VeffSdS}) is
\begin{equation}
\label{V''effSdS}
\left.\left[\frac{8\Lambda r^4+6mr(1-5\Lambda r^2)-36m^2}{r^2(r-3m)(\Lambda r^3+3r-6m)} \right]\right|_{r=r_p}\geq 0.
\end{equation}
Thus, we study Eq. (\ref{V''effSdS}) for every gravitational field source.
\\By considering the photons emitter placed on the neutron star, i.e. $r_e=R$, with $R$ the neutron star radius, we get that the emitter orbit is stable:
\begin{equation}
V''_{\rm{eff}}(r_e)=7.61641\times 10^{-4}>0,
\end{equation}
while the detector orbit is stable for:
\begin{equation}
r_d\geq 9.4374\;\rm{km}.
\end{equation}
Analogously, by considering the photons emitter placed on the white dwarf, i.e. $r_e=R$, with $R$ the white dwarf radius, we get that the emitter orbit is stable:
\begin{equation}
V''_{\rm{eff}}(r_e)=8.62859\times 10^{-14}>0,
\end{equation}
while the detector orbit is stable for:
\begin{equation}
r_d\geq 1.5876\;\rm{km}.
\end{equation}
Finally, by considering the photons emitter placed on the planet, i.e. $r_e=r_{\rm{Earth}}$ and $r_e=r_{\rm{Mars}}$, with $r_{\rm{Earth}}$ and $r_{\rm{Mars}}$ the Earth and Mars radius respectively, we get that the emitters orbits are stable
\begin{eqnarray}
V''_{\rm{eff}}(r_e)&=&3.42303\times 10^{-17}>0\;\;\mbox{for Earth},\\
V''_{\rm{eff}}(r_e)&=&2.43224\times 10^{-17}>0\;\;\mbox{for Mars},
\end{eqnarray}
while the detectors orbits are stable for
\begin{eqnarray}
r_d&\geq&2.65555\times 10^{-5}\;\rm{km}\;\;\mbox{for Earth},\\
r_d&\geq&2.84142\times 10^{-6}\;\rm{km}\;\;\mbox{for Mars}.
\end{eqnarray}
As before, let us observe that, for $\Lambda\rightarrow0$, all these equations reduce to those obtained in the Schwarzschild metric. For the sake of completeness, we briefly report the details in Appendix \ref{appendix}.
\section{Theoretical discussion}
\label{sezione4}
In this section, we describe our findings confronting our predictions with current experimental bounds, got from experiments. Further, we propose how to build up plausible experiments and develop technical configurations to check the validity of our methods. We first discuss our outcomes in the two perspectives that we described above, {\it i.e.}, Zipoy - Voorhees and Schwarzschild - de Sitter metrics. Then, we highlight the basic demands of likely experimental features to check the goodness of our limits.
\subsection{High and intermediate gravity regimes}
The increase or decrease of our red and blue shift bounds depend upon the choice of our free parameters. The possible underlying configuration is crucial in understanding how to single out the most feasible interval of red shifts or blue shifts. We then split the involved two symmetries below, commenting separately our findings and comparing our bounds with previous expectations got from the literature.
\begin{table}[htp]
\begin{tabular}{ccc}
\hline
\hline
& Numerical values got during computation & \\
\hline\hline
& M & R \\
& ($M_\odot$) & ($10^3 m$) \\
\hline
WD & $0.18$ & $18304.5$ \\
\hline
NS & $1.07$ & $13.61$ \\
\hline
\hline
\end{tabular}
\vspace{0.3cm}
\label{table1}
\caption{\emph{Table of astrophysical values adopted during our computation for high and intermediate gravity regimes. We only consider the maximally rotating configurations for NS and WD, where gravitational effects are stringent.}}
\end{table}
\subsubsection{NS case}
In the case of NS, we compute our expectations over $z_1(r)$ and $z_2(r)$ in the maximally - rotating configuration, with the ranges of masses and radii respectively given by $M\in[1.07;1.47]M_\odot$ and $r\simeq 13.61$. The values got by $z_1$ and $z_2$ reach a plateau as $d\gtrsim0$, i.e., as $d$ becomes larger than zero. This indicates the strong gravity regime of NS, as expected, and happens for both the setups of Schwarzschild - de Sitter and Zipoy - Voorhees spacetimes, when the former is computed for very large $\delta$ values. Particularly, the Zipoy - Voorhees metric seems to match the Schwarzschild - de Sitter solution as the quadrupole increases, in agreement with the fact that NS are described as rotating objects. Further, this indicates the Schwarzshild - de Sitter spacetime is a suitable approximation for determining the NS red and blue shifts, although the metric itself does not describe a rotating object. The very impressive fact is that one underlines very small changes within the interval $\delta \in[0.75;10^3]$. This suggests a limiting regime between the above interval, being compatible with the theoretical bounds over $\delta$ that exclude repulsive effects of gravity.
\begin{figure}
\begin{center}
\includegraphics[width=1.07\columnwidth]{NS_z1plot_ZV.pdf}
\includegraphics[width=1.07\columnwidth]{NS_z2plot_ZV.pdf}
\caption{ \emph{$z_1(r)$ and $z_2(r)$ as function of $d=r_d-r_e$ within the Zipoy - Voorhees spacetime. The gravitational field source is a neutron star of mass $M=1.07M_\odot$, with $M_\odot=1.47$ km the solar mass, and radius $R=13.61$ km, in the maximally - rotating configuration \cite{2.7}. Here, $d\in[0;4\cdot 10^5]$ km, whereas $z_1$ and $z_2$ are in power of $10^{-1}$. In the small zoom, we report $z_1$ and $z_2$ up to $d=40$ km.}}
\label{fig:NSZV}
\end{center}
\end{figure}
\begin{figure}
\includegraphics[width=0.9\columnwidth]{NS_z1plot_SdS.pdf}
\includegraphics[width=0.9\columnwidth]{NS_z2plot_SdS.pdf}
\caption{ \emph{$z_1(r)$ and $z_2(r)$ as function of $d=r_d-r_e$ within the Schwarzschild - de Sitter spacetime. The gravitational field source is a neutron star of mass $M=1.07M_\odot$, with $M_\odot=1.47$ km the solar mass, and radius $R=13.61$ km, in the maximally - rotating configuration \cite{2.7}. Here, $d\in[0;4\cdot 10^5]$ km, whereas $z_1$ and $z_2$ are in power of $10^{-1}$. In the small zoom, we report $z_1$ and $z_2$ up to $d=40$ km.}}
\label{fig:NSSDS}
\end{figure}
\subsubsection{WD case}
In the case of WD, we have regimes of intermediate gravity. We therefore compute our expectations over $z_1(r)$ and $z_2(r)$ in the maximally - rotating configuration, with the ranges of masses and radii respectively given by $M\in[0.18;1.47]M_\odot$ and $r\simeq 18304.5$ km. The values got by $z_1$ and $z_2$ reach a plateau as $d\gtrsim3$, {\it i.e.}, as $d$ becomes larger than zero. As well as NS regime for both the setups of Schwarzschild - de Sitter and Zipoy - Voorhees spacetimes, with very large $\delta$, we encounter the same behaviors. As for the NS, we can deduce that the Schwarzshild - de Sitter metric is even a good approximation for WDs in determining the red and the blue shift. Finally, from Figs. (\ref{fig:WDZV}) - (\ref{fig:WDSDS}) we immediately get
\begin{eqnarray}
&z_{1+,\,{\rm ZV}}\,=\,z_{2-,\,{\rm ZV}}\,=\,z_{1-,\,{\rm SdS}}\,=\,z_{2+,\,{\rm SdS}}\,,\label{simmetrie1}\\
&z_{1-,\,{\rm ZV}}\,=\,z_{2+,\,{\rm ZV}}\,=\,z_{1+,\,{\rm SdS}}\,=\,z_{2-,\,{\rm SdS}}\,,\label{simmetrie2}
\end{eqnarray}
where the subscripts ${\rm ZV}$ and ${\rm SdS}$ indicate Zipoy - Voorhees and Schwarzschild - de Sitter spacetimes respectively.
\begin{figure}
\includegraphics[width=0.9\columnwidth]{WD_z1plot_ZV.pdf}
\includegraphics[width=0.9\columnwidth]{WD_z2plot_ZV.pdf}
\caption{ \emph{$z_1(r)$ and $z_2(r)$ as function of $d=r_d-r_e$ within the Zipoy - Voorhees spacetime. The gravitational field source is a white dwarf of mass $M=0.18M_\odot$, with $M_\odot=1.47$ km the solar mass, and radius $R=18304.5$ km, in the maximally - rotating configuration \cite{2.7}. Here, $d\in[0;4\cdot 10^5]$ km, whereas $z_1$ and $z_2$ are in power of $10^{-3}$. In the small zoom, we report $z_1$ and $z_2$ up to $d=10^4$ km.}} \label{fig:WDZV}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{WD_z1plot_SdS.pdf}
\includegraphics[width=0.9\columnwidth]{WD_z2plot_SdS.pdf}
\caption{\emph{$z_1(r)$ and $z_2(r)$ as function of $d=r_d-r_e$ within the Schwarzschild - de Sitter spacetime. The gravitational field source is a white dwarf of mass $M=0.18M_\odot$, with $M_\odot=1.47$ km the solar mass, and radius $R=18304.5$ km, in the maximally - rotating configuration \cite{2.7}. Here, $d\in[0;4\cdot 10^5]$ km, whereas $z_1$ and $z_2$ are in power of $10^{-3}$. In the small zoom, we report $z_1$ and $z_2$ up to $d=10^4$ km.} }
\label{fig:WDSDS}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{EarthMars_z1plot_ZV.pdf}
\includegraphics[width=0.9\columnwidth]{EarthMars_z2plot_ZV.pdf}
\caption{\emph{$z_1(r)$ and $z_2(r)$ as function of $d=r_d-r_e$ within the Zipoy - Voorhees spacetime. We analyzed the two cases, i.e. emitter on Mars and Earth respectively, with $d\in[0;4\cdot 10^5]$ km, whereas $z_1$ and $z_2$ are in power of $10^{-5}$. In the small zoom, we report $z_1$ and $z_2$ up to $d=6\cdot 10^3$ km. This choice enables to get feasible intervals for Phobos, where $d\simeq 5986.5$ km, with Mars as source, and intervals for LARES2 \cite{lares}, where $d\simeq 5899$ km, with Earth as source.}}
\label{fig:EarthMarsZV}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{EarthMars_z1plot_SdS.pdf}
\includegraphics[width=0.9\columnwidth]{EarthMars_z2plot_SdS.pdf}
\caption{ \emph{$z_1(r)$ and $z_2(r)$ as function of $d=r_d-r_e$ within the Schwarzschild - de Sitter spacetime. We analyzed the two cases, i.e. emitter on Mars and Earth respectively, with $d\in[0;4\cdot 10^5]$ km, whereas $z_1$ and $z_2$ are in power of $10^{-5}$. In the small zoom, we report $z_1$ and $z_2$ up to $d=6\cdot 10^3$ km. This choice enables to get feasible intervals for Phobos, where $d\simeq 5986.5$ km, with Mars as source, and intervals for LARES2 \cite{lares}, where $d\simeq 5899$ km, with Earth as source.}}\label{fig:EarthMarsSDS}
\end{center}
\end{figure}
\subsection{Regime of low gravity}
The use of lunar laser ranging is commonly got using powerful pulsed searchlight, from the Earth to lunar corner cube retroreflectors\footnote{Device made up of three perpendicular reflective surfaces which retro-reflect the signal in the same direction in which it arrived.} \cite{LR1,LR2}. More generally, Laser Ranging (LR) is a technique that gets a measure of the round - trip time of a laser fired by a ground station on Earth, received by a cube corner retroreflector on a satellite and reflected back to the station. If the satellite is not the Moon, we refer to as satellite laser ranging \cite{LR3}. We follow here both the lunar and satellite LR approaches to face the low gravity regime.
\\For every LR experiment is performed, it is important to take into account the relative motion between the laser source and the retroreflector. This causes an angular deflection of the laser beam. In fact, in absence of a relative motion, the laser would exactly return back to the station. The angular deflection is also called velocity aberration (VA) and it is defined as
\begin{equation}
\label{VA}
VA=\frac{2}{c}\left[\Delta v_{d}-v_{e}\cos{\phi}\right],
\end{equation}
where $\Delta v_{d}$ is the difference between the orbital velocity and rotational velocity at the equator of the satellite or planet on which is placed the retroreflector, $v_{e}$ the rotational velocity of the satellite or planet on which is placed the laser source at the equator and $\phi$ the laser source latitude. Consequently, the term in the square brackets in Eq. (\ref{VA}) represents the relative velocity between the involved two objects.
\noindent
We are going to use the VA and the laser wavelength to determine the red and blue shifts through relativistic Doppler effect in the equatorial plane, {\it i.e.},
\begin{equation}
\lambda'=\lambda\sqrt{\frac{1+VA/2}{1-VA/2}}
\end{equation}
where, instead of the ratio between the relative velocity and $c$, we used the VA through Eq. (\ref{VA}). Thus, one immediately computes the required $z$ through the standard formula $z=\frac{\lambda'-\lambda}{\lambda}$.
\noindent
We intend to study the theoretical models by considering Earth and Mars as sources of the gravitational field in the solar system. We want to compare the theoretical results with the experimental ones deriving from the LR missions. We thus analyze our numerical technique for three configurations,
\begin{itemize}
\item Earth - Moon system. Here, we model the two astrophysical objects and involve the lunar LR.
\noindent
\item Earth - satellite system. Here, the configuration is analogous to the standard lunar LR, but using an artificial satellite instead of the Moon. This second case is therefore similar to the previous one\footnote{We here refer to the LARES2 expectations. For us, LARES2 is the acronym of Laser RElativity Satellite No. 2 and makes use of the Earth - satellite LR prerogative that we need. }.
\item Mars - Phobos system. Here we propose \emph{ex novo} an architecture to get feasible red and blue shift ranges. So, our computation provides an alternative view to LR, since it involves a laser source placed on Mars, considered the gravitational source modeled through our spacetime approach. The corner cube retroreflectors could be placed on Phobos, i.e. a natural Mars satellite\footnote{This prerogative goes beyond the binary system Earth - Moon, with the advantage of being closer to each other, i.e. guaranteeing a more precise pulse measure.} A similar approach has been developed by \cite{phobos}, where the authors proposed a system constituted by Earth and Phobos. The procedure we here develop permits to extend previous results through the use of alternative technologies of LR.
\end{itemize}
\noindent
Our predicted values, got from the theoretical models under exam for these three configurations, are portrayed in Tabs. \ref{table2}-\ref{table3}. Here, we again observe the same symmetries reported in Eqs. (\ref{simmetrie1},\ref{simmetrie2}).
\begin{table*}[htp]
\begin{tabular}{|c|c|c|c|}
\hline
\hline
$z\pm\delta z$ & Lunar LR ($ 10^{-5}$) & Satellite LR ($10^{-6}$) & Phobos LR ($10^{-6}$) \\
\hline\hline
& & & \\
$z_{1+}\equiv z_{2-}$ & $2.298042$ & $7.36990$ & $4.72646$ \\
& & & \\
$\delta z_{1+}\equiv \delta z_{2-}$ & $\pm 0.000005$ & $\pm 0.01551$ & $\pm 0.00046$ \\
& & &\\
\hline\hline
& & & \\
$z_{1-}\equiv z_{2+}$ & $-2.298057$ & $-7.37018$ & $-4.72653$ \\
& & & \\
$\delta z_{1-}\equiv \delta z_{2+}$ & $\pm 0.000005$ & $\pm 0.01551$ & $\pm 0.00046$ \\
& & &\\
\hline
\hline
\end{tabular}
\vspace{0.3cm}
\caption{\emph{Table of red and blue shift values, predicted by means of the Zipoy - Voorhees metric. We also report the corresponding error bars, namely $\delta z_{1,2\pm}$, evaluated by the standard logarithmic error propagation. Since for low gravity, the predictions over $z$ do not significantly change by varying $\delta$, with an accuracy smaller than one part over $10^6$, we arbitrarily select $\delta=0.5$. Here we consider Lunar LR for the system Earth - Moon, satellite LR for the system Earth - LARES2 \cite{lares} and finally Phobos LR for the proposed experiment that employs Mars and its satellite, Phobos.}}
\label{table2}
\end{table*}
\noindent
\begin{table*}[htp]
\begin{tabular}{|c|c|c|c|}
\hline
\hline
$z\pm\delta z$ & Lunar LR ($ 10^{-5}$) & Satellite LR ($ 10^{-6}$) & Phobos LR ($ 10^{-6}$)\\
\hline\hline
& & & \\
$z_{1+}\equiv z_{2-}$ & $-2.298057$ & $-7.37018$ & $-4.72653$ \\
& & & \\
$\delta z_{1+}\equiv \delta z_{2-}$ & $\pm 0.000005$ & $\pm 0.01551$ & $\pm 0.00046$ \\
& & & \\
\hline\hline
& & & \\
$z_{1-}\equiv z_{2+}$ & $2.298057$ & $7.37018$ & $4.72653$ \\
& & &\\
$\delta z_{1-}\equiv \delta z_{2+}$ & $\pm 0.000005$ & $\pm 0.01551$ & $\pm 0.00046$ \\
& & &\\
\hline
\hline
\end{tabular}
\vspace{0.3cm}
\caption{\emph{Table of red and blue shift values, predicted by means of the Schwarzschild - de Sitter metric. We also report the corresponding error bars, namely $\delta z_{1,2\pm}$, evaluated by the standard logarithmic error propagation. Here we consider Lunar LR for the system Earth - Moon, satellite LR for the system Earth - LARES2 \cite{lares} and finally Phobos LR for the proposed experiment that employs Mars and its satellite, Phobos.}}\label{table3}
\end{table*}
\noindent
On the other hand, the indirect measurements of $z$, obtained starting from the experimental data of the LR, are instead reported in Tabs. \ref{table5}-\ref{table7} in Appendix \ref{AppB}. The slight differences between experimental and predicted outcomes are due to several facts. First, experimentally speaking, we are handling the VA technique only. Second, the accuracy can be refined adopting more than the configurations here investigated. Below, we summarize how to heal such issues adopting direct measure methods, by proposing novel experimental configurations.
\subsubsection{Designs of proposed experimental setups}
In view of the overall results, we summarize the following consequences of our treatments.
\begin{itemize}
\item At high gravity regimes $\delta$ values, using axisymmetric spacetime, agree with the Schwarzschild - de Sitter prediction, making use of the Planck satellite bounds.
\item As well as NS, the regimes of intermediate gravity, here investigated employing WD, behave in analogy.
\item At low gravity, in the solar system, our spacetime metrics are clearly unadequate to fix stringent limits over the free parameters as well as the red and blue shift intervals that slightly disagree with observations, see Appendix B.
\item Analogously LR cannot be used to get indirect measurements for $z$. Moreover, the results coming from the use of Schwarzschild metric, without taking care about the $\Lambda$ value, seem to agree with our scheme, indicating that $\Lambda$ is quite badly constrained within the solar system (see Tab. \ref{table4} in Appendix \ref{appendix}).
\end{itemize}
\noindent
From the above considerations, a direct measure of $z$ would be more predictive than other indirect treatments. So that, one can build up experimental setups based on genuine wavelength measures only.
\noindent
To do so, let us take the simplest configuration we could work with, based on the system Earth - Moon. We can send a laser pulse whose wavelength is known, as well as in the lunar LR technique. The detector is meant to measure the corresponding wavelength and then to get possible hints on red or blue shifts.
\noindent
Alternatively, another possibility is offered by a rotating orbiter around Mars or more away planets, without excluding to take into account other configurations. The orbiter sends signals, whose wavelength is known. The detector, again placed on the given planet we are considering, gets the signals and provides a direct measure of the pulse shift in wavelength that is converted in red and blue shift. Last but not least, even if the astrophysical configurations of NS and WD could in principle be adopted for future missions to better get $z$, the experimental complexity to put on them instruments would be a great limitation for the experiment itself. On the other hand, cosmological red and blue shifts would be the key to fix the cosmological constant value. The strategy would be to take the large scale structure of the universe, switching the spacetime to more complicated metrics that could overcome the likely issues related to Schwarzschild - de Sitter.
\subsection{Error analysis}
In this subsection, we report the values of $z$ and the respective errors obtained when the field sources are the NS and the WD, both in the two employed metrics. We have selected $3$ values of $d = r_d - r_e$ within the range chosen for the plots, $d\in[0; 4\cdot 10^5]$. In particular:
\begin{itemize}
\item $d = 0$ and $d = 4\cdot 10^5$, i.e. the extremes of the interval;
\item $d = 2\cdot 10^5 $, which is the midpoint of the interval.
\end{itemize}
In the case of the Zipoy - Voorhees metric, we considered the $\delta$ values chosen for the plots ($\delta = 1/4$, $\delta = 3/4$ and $\delta = 1000$); while for the Schwarzschild - de Sitter metric, we considered the $\Lambda$ value of the Planck collaboration: $\Lambda=(1.10566\pm0.022703)\cdot 10^{-46}\;{\rm km^{-2}}$.
\\Errors are calculated through the standard logarithmic error propagation, considering that for NS and WD it is hard to get with high accuracy both mass and radii. We therefore work out the following strategy: we consider Refs. \cite{aggiunta1,aggiunta2} and there we got the maximum and minimum bounds associated to mass and radii for both NS and WD. Then, we consider the average constraints and use for our computation, in particular
\begin{itemize}
\item for NS: $M=\left(1.07\pm0.11\right)M_\odot$ and $R=13.61^{+2.18}_{-0.68}$ km,
\item for WD: $M=\left(0.180^{+0.056}_{-0.004}\right)M_\odot$ and $R=18304.5^{+5491.3}_{-823.70}$ km,
\end{itemize}
where $M_\odot=1.47$ km. Computation has been reported in Appendix \ref{errori}.
\section{Conclusions}\label{sezione5}
In this paper, we evaluated the red and blue shifts for distinct astrophysical and cosmological sources. To do so, we arbitrarily split three gravitational regimes, {\it i.e.}, high, intermediate and low gravity, based on the use of NS, WD and solar system constraints. We characterized two spacetimes, the first inspired by astrophysical configuration, exploring the consequences of the Zipoy - Voorhees metric on NS and WD. The second based on cosmological consideration, using the Schwarzschild - de Sitter spacetime. By varying the free parameters that enter the two metrics, we got feasible red and blue shift intervals and interpreted our expectations in view of current experiments and limits.
\noindent
Since the two underlying spacetimes are mostly different, we consider an axisymmetric solution, {\it i.e.}, the Zipoy - Voorhees metric, to characterize NS and WD, showing that at the level of solar system, the $\delta$ free term is unbounded. Analogously, at low gravity, we assumed, instead, the Schwarzschild - de Sitter solution, where we fixed the $\Lambda$ Planck's value, getting acceptable red and blue shift ranges. Further, we considered lunar and satellite LR techniques and showed an overall overestimation on $z_{1,2\pm}$ when using the VA. In turn, we interpreted such a result noticing that at low gravity the general relativistic effects are clearly disfavored. In fact, even the $\Lambda$ value predicted by the Planck satellite cannot fully reproduce the expected intervals of $z_{1,2\pm}$ that LR experiments estimate.
\noindent
On the other side, bearing in mind the maximally - rotating configurations for NS and WD, we got suitable red and blue shift intervals. We therefore concluded the most suitable approximations for NS and WD objects could be performed involving high quadrupole moments. The same happened for the cosmological constant, besides the solar system regime, showing a good agreement with current bounds and indicating the goodness of Planck $\Lambda$ measurements. Error bars have been computed for specific cases got from experiments over NS and WD mass and radii, in full agreement with our predicted bounds.
\noindent
Possible experimental designs for improving the quality of our results have been naively proposed. We discussed coarse - grained approaches to build up likely experimental configurations and set ups with the aim of refining the current accuracy over red and blue shift. To this end, we propose to adopt the binary system composed by Mars - Phobos to improve $z_{1,2\pm}$ measurements using the satellite LR technique.
\noindent
As red and blue shifts can be used to test the equivalence principle and/or sometimes to check the validity of particular classes of models, we intend to contrive new experiments that will be able to construct bounds over $\Lambda$, instead of postulating it. In so doing, as perspective we expect to work out a back - scattering procedure, different from the one here developed. Moreover, we intend to model other astrophysical sources as possible probes to test the red and blue shifts at different regimes of gravity, with the ambitious aim of reconciling several gravity regimes. In this respect, we underline that a more detailed analysis will be expected for getting experimental bounds in cosmology. So, the proposed experimental set up, with additional requirements that will be studied in incoming efforts, could represent a new technique to alleviate tensions between different measurements of $H_0$ in cosmology \cite{tensione}. We believe this can be also generalized for any other cosmological tension, in general.
\section*{Acknowledgments}
OL acknowledges funds from the Ministry of Education and Science of the Republic of Kazakhstan, IRN AP08052311.
LM acknowledges the support of Istituto Nazionale di Fisica Nucleare (INFN), iniziativa specifica MOONLIGHT2.
|
1,108,101,566,731 | arxiv | \section{Introduction}
The discovery of hundreds of extrasolar planets during the last two
decades has enabled us to investigate the physical properties of these
objects, and to compare them to the known planets of our own solar
system. The discoveries of brown dwarfs, on the other hand, have
opened our view for this interesting class of objects placed between
stars and planets and sharing physical properties with both groups.
Particularly interesting questions are whether planets, brown dwarfs,
and stars share the same physical principles with regard to their
magnetic dynamos, and what typical field strengths must be expected at
objects for which a positive field detection is missing so far (brown
dwarfs and exoplanets).
Radioemissions from magnetized planets result from the interaction of
their magnetospheres with the stellar wind. This accelerates electrons
to several keV, which emit radio waves at the local cyclotron
frequency \citep[e.g.,][]{Zarka92, Zarka98, Farrell99, Zarka07}. A
similar mechanism, the electron cyclotron maser, was identified as the
dominant source of radio emission for a number of very-low mass stars
and brown dwarfs \citep{Hallinan08}. Radio emissions from extrasolar
planets are particularly interesting because they can outshine the
radio emission from a quiet host star \citep[e.g.,][]{Farrell99,
Griessmeier05}, and may therefore be utilized to discover extrasolar
planets. So far, none of the searches for radio emission from
extrasolar planets could present a positive detection. This is not too
surprising because currently available facilities are hardly sensitive
enough to observe the expected weak radio flux.
Both the frequency and the total radio flux depend critically on the
magnetic field strength of the planet, the latter primarily because it
controls the cross section of the magnetosphere interacting with the
stellar wind. In addition, the energy flux of the wind, which depends
on the planet's distance from the host star and on stellar activity,
plays an important role for the total radio flux. The spectrum of the
radio emission is expected to show a sharp cutoff at the electron
cyclotron frequency corresponding to the maximum magnetic field
strength close to the planetary surface. Future observations of the
radio spectrum of extrasolar planets can therefore constrain the
field strength rather reliably.
Usually, one assumes that radio emission can be generated at
extrasolar planets in much the same way as they are generated at
Jupiter. Recent estimates of radio fluxes from known extrasolar
planets were presented by \citet{Farrell99}, \citet{Lazio04},
\citet{Stevens05}, \citet{Griessmeier07a}, and \citet{Jardine08}. One
of the large uncertainties in the prediction of radio emission is the
magnetic moment of extrasolar planets. \citet{Griessmeier04}
discussed the different magnetic moment scaling laws that were
available at that time and applied them to estimate the field of
extrasolar planets. \citet{Christensen06} derived a magnetic field
scaling relation for planets based on dynamos simulations that cover a
broad parameter range. Recently, \citet{Christensen09} generalized
the scaling law and showed that its predictions agree with
observations for a wide class of rapidly rotating objects, from Earth
and Jupiter to low-mass main sequence (spectral types K and M) and
T~Tauri stars \citep[see also][]{Christensen09b}. Here, we revisit
the question of magnetic field strength at brown dwarfs and giant
planets and the estimate for the radio flux from extrasolar planets,
using this scaling law, which we believe to be on more solid grounds
both theoretically and observationally than previously suggested
scaling laws.
\section{Magnetic flux estimate}
Here we use this scaling law in the form given by \citet{Reiners09b},
who expressed the magnetic field strength in terms of mass $M$,
luminosity $L$ and radius $R$ (all normalized with solar values):
\begin{equation}
\label{eq:law}
B_{\rm dyn} = 4.8 \times \left(\frac{M L^{2}}{R^7}\right)^{1/6} \rm{[kG]},
\end{equation}
where $B_{\rm dyn}$ is the mean magnetic field strength at the surface
of the dynamo.
In its original form \citep{Christensen09}, the scaling relation
connects the strength of the magnetic field to the one-third power of
the energy flux (luminosity divided by surface area). It is
independent of the rotation rate, provided the latter is sufficiently
high. A weak 1/6-power dependence on the mean density and a factor
describing the thermodynamic efficiency of magnetic field generation
also enter into the scaling law. The latter factor is found to be
close to one for a large class of objects \citep{Christensen09}, and
this value has been used in Eq. \ref{eq:law}. Aside from this
assumption, the value of the numerical prefactor in this equations is
only based on the results of dynamo simulations.
For massive brown dwarfs and stars the top of the dynamo is close to
or at the surface of the object and the value of $B_{\rm dyn}$ is
directly relevant for observations that relate to the magnetic field
strength. In giant planets, with $M < 13 M_{\rm J}$ (13 Jupiter
masses), the surface of the dynamo region is at some depth, for
example in Jupiter at approximately 83\,\% of the planet's radius and
the field at the surface is somewhat smaller \citep{Christensen09}.
Higher multipole components, which are assumed to make up half of
$B_{\rm dyn}$, drop off rapidly with radius and are neglected for
simplicity. The radius of objects with masses between 0.3 and 70
$M_{\rm J}$ is nearly constant within $\pm$ 20\% of one Jupiter radius
for ages larger than 200 Myr \citep{Burrows01}. In this case the depth
to the top of the dynamo, at the pressure of metallization of hydrogen
in a H-He planet, is approximately inversely proportional to the
mass. We relate the dipole field strength at the equator of a giant
planet, $B_{\rm dip}^{\rm eq}$, to the overall field strength at the
top of the dynamo, $B_{\rm dyn}$, by
\begin{equation}
\label{eq:correction}
B_{\rm dip}^{\rm eq} = \frac{B_{dyn}}{2\sqrt{2}} \left(1 - \frac{0.17}{M/{\rm M}_{\rm J}}\right)^3.
\end{equation}
The factor $2\sqrt{2}$ in the denominator results from the assumption
that at the dynamo surface the dipole field strength is half of the
rms field strength and from the fact that the equatorial dipole field
is $1/\sqrt{2}$ of the rms dipole field.
To estimate the field strength of substellar objects, we employ in
Eq.~\ref{eq:law} radii and luminosities from the evolutionary tracks
calculated by \citet{Burrows93, Burrows97} for substellar objects. Our
model predicts a magnetic field of 9\,G for an object of one $M_{\rm
J}$ at an age of 4.5\,Gyr. This value agrees well with the polar
dipole field strength of Jupiter, which is 8.4\,G
\citep[e.g.,][]{Connerney93}.
\subsection{Rotation of the planets}
We implicitly assume that the objects are rotating above a critical
rotation velocity, so that our scaling law can be applied. This seems
to be the case in most of the solar-system giant planets
\citep[][Section\,4]{Christensen09b} and in brown dwarfs
\citep{Reiners08}. Giant planets with semimajor axes less than about
0.1 -- 0.2\,AU are expected to be slowed down to synchronous rotation
by tidal braking on time scales of less than 1\,Gyr \citep{Seager02}.
However, for planets that are very close to their host star the
synchronous rotation rate may still lie above the critical limit for
our scaling law to apply; the value of the latter is somewhat
uncertain. M-dwarfs with rotation periods up to at least 4 days are
generally found to fall into the magnetically saturated regime
\citep{Reiners09a} for which our scaling law applies. We emphasize
that our results can be considered as upper limits and would be lower
if the objects for some reason would rotate substantially slower.
\section{Radio flux calculation}
We estimate the magnetospheric radio flux from extrasolar giant
planets following the work of \citet{Stevens05}. This means we assume
that the input power into the magnetosphere is proportional to the
total kinetic energy flux of the stellar wind. \citet{Griessmeier07a}
discussed different mechanisms depending on the source of available
energy. They also calculate radio emission for the case where the
input power is provided by magnetic energy of the interplanetary
field, for unipolar magnetic interaction, and for stellar flares. In
general, stellar activity of planet host stars is relatively weak so
that we concentrate on the kinetic energy flux as source for the power
input.
\subsection{Radio flux scaling law}
In order to calculate the radio flux of a giant extrasolar planet at
the location of the Earth, we relate it to the radio flux of Jupiter
using Equation (14) from \citep{Stevens05} neglecting the dependence
on the wind-velocity (see below):
\begin{equation}
\label{eq:radioflux}
P \propto \frac{1}{d^{2}} \left(\frac{\dot{M}_{\star} M_{\rm dip}}{a^{2}}\right)^{2/3},
\end{equation}
with $P$ the radio flux, $d$ the distance to the system,
$\dot{M}_{\star}$ the stellar mass-loss rate, $M_{\rm dip}$ the dipole
moment of the planet, and $a$ the star-planet distance. The dipole
moment is
\begin{equation}
M_{\rm dip} = B_{\rm dip}^{\rm eq} R^3,
\end{equation}
with $R$ the planetary radius. We assume Jupiter-like values for the
stellar wind velocity, $V_W = 400$\,km\,s$^{-1}$. As discussed in
\citet{Stevens05}, the latter may be an overestimate in particular for
close-in planets, because wind velocities are lower closer to the
star. We also note that young stars have higher mass-loss ratios
\citep{Wood05}, but this does not necessarily affect the wind
velocities \citep[see also][who assume a fixed stellar wind
velocity]{Wood02}. Nevertheless, our uncertainties in the wind
velocities are probably on the order of a factor of two, which means
about a factor of $\sim 3$ (0.5\,dex) uncertainty for individual
stars. We refer to \citet{Stevens05} for further details. In our
approximation, for a given semi-major axis, mass-loss rate and
distance, the radio flux is only a function of the magnetic moment or
the average magnetic field.
The electron cyclotron frequency near the surface in the polar region
is $f_{\rm ce} [{\rm MHz}] = 2.8 B_{\rm dip}^{\rm pol} [{\rm G}]$,
where $B_{\rm dip}^{\rm pol} = 2 B_{\rm dip}^{\rm eq}$ is the polar
dipole field strength. For Jupiter it corresponds roughly to the
cutoff frequency observed in the radio emission spectrum
\citep[see][]{Zarka92}. The bulk of Jupiter's radio flux occurs in
roughly the frequency range from 0.1~$f_{\rm ce}$ to the cutoff
frequency.
\subsection{Parameter estimates for known host stars}
\subsubsection{Mass-loss rate}
To calculate the mass-loss rate that is required in
Eq.\,(\ref{eq:radioflux}) for host stars of known extrasolar giant
planets (EGPs), we parameterize the stellar mass-loss rates using the
results from \citet{Wood05} and use X-ray luminosities taken from the
NEXXUS
database\footnote{\texttt{http://www.hs.uni-hamburg.de/DE/For/Gal/Xgroup/\\nexxus/nexxus.html}}
\citep{NEXXUS} in analogy with \citet{Stevens05} but with the updated
parameters from \citet{Wood05}
\begin{equation}
\frac{\dot{M}_{\star}}{{\rm \dot{M}}_\odot} = \left(\frac{R_\star}{{\rm R}_{\odot}}\right)^2 \left(\frac{F_X}{{\rm F}_{X,\odot}}\right)^{1.34}.
\end{equation}
With $F_X = L_X/(4 \pi R^2)$, we get the mass-loss rate as a function
of X-ray luminosity times $(R_{\star}/{\rm R}_{\odot})^{-0.68}$. The
latter factor introduces an error on the order of $\sim 10$\,\% if the
radius differs from the solar radius by 20\,\%. A radius offset of
50\,\% (i.e. for example between 0.6 and 0.9\,R$_{\odot}$, which is
the difference in radius between a G-type star and an early-M star)
introduces an error of 24\,\% in the mass-loss rate ($\sim 0.1$\,dex),
which means $\sim 15$\,\% in the radio flux $P$. Compared to the high
variability in X-ray flux and the large systematic uncertainties of
our approach, we can safely ignore this effect and use for the
calculation of the mass-loss rate the equation
\begin{equation}
\label{eqn:Mdot}
\frac{\dot{M}_{\star}}{{\rm \dot{M}}_\odot} \approx \left(\frac{L_X}{{\rm L}_{X,\odot}}\right)^{1.34}.
\end{equation}
\subsubsection{Magnetic Moment: Mass and age estimate}
In order to compute the average surface magnetic field of the EGPs, we
need to estimate mass and age for each system. While their masses are
known except for the projection uncertainty ($M\,\sin{i}$), the age is
more difficult to determine. As a rough estimate, we determine the age
from the X-ray activity seen on the host star. X-ray activity is known
to be related to the rotation of the star, or to the Rossby number $Ro
= P/\tau_{\rm conv}$, with $P$ the rotation period and $\tau_{\rm
conv}$ the convective overturn time \citep[e.g.,][]{Noyes84,
Pizzolato03}. For the host stars of the EGPs that we discuss in
Section\,\ref{sec:EGPs}, we calculate the convective overturn time
from the $(B-V)$ color according to the relation in \citet{Noyes84},
$B-V$ was taken from the Hipparcos catalogue
\citep{Hipparcos}. \citet{Mamajek08} provides useful relations between
the Rossby number and normalized X-ray luminosity (Eq.\,\ref{eqn:Ro};
their Eq.~2.6), and between rotation period and the age (and the
color) of a star \citep[Eq.\,\ref{eqn:t}; their Eq~2.2, see
also][]{MH08}.
\begin{eqnarray}
\label{eqn:Ro}
Ro & = & P/\tau_{\rm conv} = 0.86 - 0.79 ( \log{L_{\rm X}/L_{\rm bol}} + 4.83)\\
\label{eqn:t}
t & = & \left( \frac{P}{0.407 (BV - 0.495)^{0.325} } \right)^\frac{1}{0.566}
\end{eqnarray}
A full discussion of the uncertainties in this activity-age
relationship goes far beyond the scope of our paper and can be found,
e.g., in \citet{Soderblom83}, \citet{Barnes07} and \citet{Mamajek08}.
The derived ages have 1$\sigma$ uncertainties on the order of at least
50\,\%, so these values are really mostly indications of the real age
of the planets. For our scaling relations, however, this large an
error does not significantly affect the results; at the typical age of
the stars, the average magnetic field, radio flux, and peak frequency
that we calculate shrink by approximately 20\,\% if we increase the
age by a factor of two. Note that we use the age only for our
calculation of the average magnetic field and not for an estimate of
the mass-loss rate, which we derive directly from the X-ray flux
according to Eq.\,\ref{eqn:Mdot}.
For the calculation of the Rossby number in Eq.\,\ref{eqn:Ro}, we need
the bolometric luminosity. We calculate the bolometric luminosity from
bolometric corrections $BC_V$ and the distance and $V$-magnitudes
given for each planet's host star in \emph{The Extrasolar Planets
Encyclopaedia}\footnote{\texttt{exoplanet.eu}}. We determine $BC_V$
from the color $B-V$ following the tabulated values in
\citet{Kenyon95}. In order to correct for the binaries contained in
the sample, we finally fitted the relation between $B-V$ and
bolometric luminosity neglecting obvious outliers. We use the
luminosities that result from $B-V$ according to the relation
\begin{equation}
\log{\frac{L_{\rm bol}}{L_{\odot}} = 1.4531 - 2.0395 \, (B-V). }
\end{equation}
\section{Results}
\begin{figure*}
\centering
\mbox{\includegraphics[width=\textwidth]{BfAge.eps}}
\caption{\label{fig:BfAge}Average magnetic field on the surface of
the object, $B_{\rm dyn}$, for $M > 13$\,M$_{\rm J}$, and dipole
field, $B_{\rm dip}^{\rm pol}$, for $M \le 13$\,M$_{\rm J}$, as a
function of age for giant planets, brown dwarfs, and a very-low
mass star with $M = $125\,M$_{\textrm{J}}$. All low-mass objects
are assumed to be rapidly rotating. An estimate of the average
magnetic field of the Sun is overplotted \citep[gray shaded area;
for today's average field see][]{Schrijver87}.}
\end{figure*}
We show the evolution of the magnetic fields in Fig.\,\ref{fig:BfAge}.
For stars and brown dwarfs, i.e., objects with masses higher than
13\,M$_{\rm J}$, we show the average magnetic field of the dynamo,
$B_{\rm dyn}$, which is probably similar to the average surface field.
For giant planets, we plot the dipole field at the surface, $B_{\rm
dip}$, from Eq.\,\ref{eq:correction}. Evolutionary tracks for a few
planetary mass objects and brown dwarfs are shown, as well as one
model of a very-low mass star (125\,M$_{\textrm J}$). In addition, we
overplot an estimate of the magnetic history of the Sun. The
difference between solar evolution and all other cases considered is
that the Sun developes a radiative core and suffers substantial
rotational braking so that it falls below the saturation threshold
velocity for the dynamo. During the first $\sim 10^8$\,yrs of its
lifetime, the Sun was rotating rapidly enough so that its dynamo
operated in the saturated regime captured by our scaling law. After
that time, the solar rotation slowed down \citep[e.g.,][]{Skumanich72,
Barnes07} and magnetic field generation probably weakened in
proportion to angular velocity. Evidence for such a rotation-magnetic
field relation was found in M dwarfs \citep[see][]{Reiners09a}. We
mark this region in grey in Fig.\,\ref{fig:BfAge} because the
uncertainties in the regime of slow rotation are different from our
problem and should not be discussed here. In contrast to sun-like
stars, fully convective stars do not seem to suffer substantial
rotational braking with the result that, even at an age of several
Gyrs, they are rotating in the critical regime for magnetic field
generation \citep{Delfosse98, Barnes07, Reiners08}.
\subsection{Evolution of magnetic fields and radio flux}
The average magnetic fields of giant planets and brown dwarfs
according to our scenario can vary by about an order of magnitude and
more during the lifetime of the objects. The magnetic field strength
is higher when the objects are young and more luminous
(Eq.\,\ref{eq:law}). For example, a one Jupiter-mass planet is
predicted to have a polar dipole field strength on the order of
$100$\,G during the first few Million years; the field weakens over
time and is less than 10\,G after 10\,Gyr. A planet with five Jupiter
masses has a magnetic field at the surface that is consistently
stronger by a factor of four to five over the entire evolutionary
history.
Because of the higher luminosity that is essentially available for
magnetic flux generation, magnetic fields in brown dwarfs are larger
than fields on extrasolar giant planets, varying typically between a
few kG and a hundred G depending on age and mass. Magnetic fields in
brown dwarfs also weaken over time as brown dwarfs cool and loose
luminosity as the power source for magnetic field generation. Low-mass
stars show a generally different behaviour. A low-mass star with $M =
125$\,M$_\textrm{J}$ can produce a magnetic field of about 2\,kG
during the first ten Myr, and the field grows by about a factor of two
until it stays constant from an age of a few hundred Myrs on. For the
solar case, the magnetic field is roughly constant between $5\,10^{7}$
and $10^{8}$\,yrs, which is maintained by the constant luminosity and
rapid rotation.
\subsection{Comparison to other field predictions}
Magnetic field estimates for extrasolar giant planets are available
from a variety of different scaling laws. \citet{Christensen09b}
summarized scaling laws for planetary magnetic fields that were
proposed by different authors. Most of them assume a strong relation
between field strength and rotation rate. As an example,
\citet{Sanchez04} estimated the dipolar magnetic moments of exoplanets
using the ``Elsasser number'' scaling law, which predicts the field to
depend on the square root of the rotation rate but assumes no
dependence on the energy flux. \citet{Sanchez04} predicts average
magnetic fields of $\sim$ 30--60\,G for rapidly rotating planets and
$\sim 1$\,G for slowly rotating ones. The range of values is
comparable to our predictions. If young planets were generally fast
rotators while old planets rotate slowly, which could be the case when
tidal braking plays a role, the results would be similar. However, our
model predicts that energy flux rules the magnetic field strength so
that extrasolar giant planets have high magnetic fields during their
youth and weak magnetic fields at higher ages even if their rotational
evolution is entirely different (given that their are still rotating
fast enough for dynamo saturation).
\citet{Stevens05} used a very simplistic method to scale the magnetic
fields of extrasolar giant planets assuming that the planetary
magnetic moment is proportional to the planetary mass. This implies no
difference between magnetic fields in young and old planets, and no
difference between rapid and slow rotators (but note that slow
rotators were explicitly left out of his analysis). \citet{Stevens05}
also provides radio flux predictions that we compare to our
predictions in the next section.
\subsection{Radio flux and field predictions for known planets}
\label{sec:EGPs}
We have calculated the radio flux and cutoff emission frequency for
the known planets of stars within 20\,pc and with X-ray detections.
Planet parameters are from \emph{The Extrasolar Planets
Encyclopaedia}. The results are given in Table\,\ref{tab:stars} and
are plotted in Fig.\,\ref{fig:radiopeak}. \citet{Stevens05} assumes
the peak radio flux occurs at the electron cyclotron frequency of the
equatorial surface field, i.e. at half the cutoff frequency. We mark
the frequency range, $(0.1-1)f_{ce}$, over which significant radio
emission can be expected for each planet by horizontal lines in
Fig.\,\ref{fig:radiopeak}. The extension of the range to lower
frequencies is rather arbitrary but indicative for the emission from
most of the planets of our solar system \citep[e.g.,][]{Zarka92}. We
include only planets that are more massive than about 0.5\,M$_{\rm
Jup}$ because in Saturn-sized or smaller planets helium separation
may lead to stable stratification at the top of the electrical
conducting region \citep{Stevenson80}. The associated reduction of the
surface field strength is difficult to quantify.
\begin{table*}
\center
\caption{\label{tab:stars}Parameters of known planets around stars
with X-ray detections within 20\,pc}
\begin{tabular}{lccrrccrr}
\hline
\hline
\noalign{\smallskip}
Planet Name & Planet mass & $a$ & d$^1$ & $\dot{M}$ & Age & $B_{\rm dip}^{\rm pol}$ & radio flux$^1$ & $f_{\rm ce}$\\
& [${\rm M_{Jup}}\sin{i}$] & [AU] & [pc] & [$\dot{M_\odot}$] & [Gyr] & [G] & [mJy] & [MHz]\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\input{Planets_table.tex}
\noalign{\smallskip}
\hline
\end{tabular}\\
$^1$The distance to Jupiter was set to 5.2\,AU for the calculation of its radio flux.
\end{table*}
\begin{figure*}
\centering
\mbox{\includegraphics[width=\textwidth]{RadioFlux.eps}}
\caption{\label{fig:radiopeak}Radio flux for the extrasolar planets
of Table\,\ref{tab:stars}. For each planet, the expected radio
flux at Earth is plotted for the frequency range $(0.1-1)f_{\rm
ce}$, which is the expected range of strong emission as in the
case of Jupiter. The hatched area marks the frequency range below
the ionospheric cutoff.}
\end{figure*}
The maximum radio flux predicted for known extrasolar planets is about
700 mJy in the case of $\tau$~Boo\,b. Maximum emission frequencies are
between 7 and 160\,MHz, i.e., in most cases above the ionospheric
cutoff frequency of 10\,MHz. However, when the maximum frequency is
less than 20\,MHz the peak radio emission may fall below the
ionospheric cutoff. The predicted flux for planets other than
$\tau$~Boo\,b is at least an order of magnitude smaller. The fluxes
for $\upsilon$~And\,b, Gl~86\,b, and HD~189733\,b fall into the range
of 40-60 mJy, and their maximum frequencies are well above the
ionospheric cutoff. For GJ~3012\,b we predict 20\,mJy, and all other
planets fall below 10~mJy.
Our predicted radio flux values are similar to those in
\citet{Stevens05}, which is mainly due to the fact that the distance
to the object is an important factor in the observed radio emission,
and that we use the same assumptions for the stellar mass loss rate.
Our Fig.\,\ref{fig:radiopeak} can be compared to Figs.\,1--3 in
\citet{Griessmeier07a}. In general, the range of radio frequencies and
radio flux are comparable but can differ substantially between
individual objects. Note that we only show estimates for the planets
within 20\,pc. These are the most likely candidates for the detection
of radio emissions.
\section{Summary and Discussion}
We applied the energy flux scaling relation from \citet{Christensen09}
to estimate the magnetic field evolution on giant extrasolar planets
and brown dwarfs. This magnetic field scaling is independent of the
rotation of the objects given that they rotate above a critical
rotation limit, which probably is the case for isolated brown dwarfs,
for young exoplanets, and for exoplanets in orbits not too close to
their central star. Close-in planets suffer tidal braking. This
applies to all candidate planets for which we predict a radio flux
above 10\,mJy, except GJ~3021\,b. However, for planets that are very
close to their host star the synchronous rotation rate may still lie
above the critical limit. The critical period is on the order of four
days in M-dwarfs, and if we assume the that the critical limit is
somewhat higher, the top candidate for the detection of radio
emissions, $\tau$~Boo\,b (3.3\,d), as well as $\upsilon$~And\,b
(4.6\,d) and HD~189733\,b (2.2\,d) rotate rapidly enough. At this
point, we cannot say more about the real critical limit. Gl~86\,b
(15.8 d) is probably rotating too slowly and our magnetic field and
radio flux estimates are likely too high.
Because energy flux scales with luminosity, young exoplanets have
magnetic fields about an order of magnitude higher than old
exoplanets. Brown dwarfs go through a similar evolution but may go
through a temporal magnetic field maximum depending on the details of
the luminosity and radius evolution. Very low-mass stars build up
their magnetic fields during the first few $10^6$ years and maintain a
constant magnetic field during their entire lifetime as long as they
are not efficiently braked. The latter is consistent with observations
of Zeeman splitting in FeH lines, which directly yields a measurements
of the average magnetic field \citep{Reiners07}.
So far, no Zeeman measurements were successful in old brown dwarfs. On
the other hand, \citet[][ and references therein]{Hallinan08} confirms
the observation of electron cyclotron masers on three ultra-cool
dwarfs that are probably brown dwarfs older than a few hundred Myr.
This means, these ultra-cool dwarfs emit radio emission by a mechanism
similar to the one discussed for planets in this paper. In particular,
from radio detections at 4.88\,GHz and 8.44\,GHz, Hallinan et al.
conclude that these brown dwarfs have regions of magnetic fields with
$B>1.7$\,kG and $B>3$\,kG, respectively. All three objects are
high-mass brown dwarfs (0.06 -- 0.08\,M$_{\odot}$) older than a few
hundred Million years. The (average) magnetic field prediction for
this class of objects is $\sim 3$\,kG from our models, which is in
good agreement with the observations. Another prediction from our
model is that old brown dwarfs at several Gyr age have weaker fields.
This needs to be tested in other (older) objects, which requires good
estimates of the ages of (old) brown dwarfs. Furthermore,
measurements at other radio frequencies are desirable in order to
better constraint their maximum field strength.
\citet{Reiners09b} failed to detect kG-strength magnetic fields in
four young ($\la 10$\,Myr) \emph{accreting} brown dwarfs that are
rapidly rotating ($v\,\sin{i} > 5$\,km\,s$^{-1}$). According to our
scenario, the objects should have average magnetic fields of 1--2\,kG,
but upper limits of about one kG are found in all of them. This may be
a hint for a deviation from our scaling law suggesting that brown
dwarfs do not follow this rule, at least not at ages below
$\sim10$\,Myr. Another alternative is that (at least in brown dwarfs)
the average magnetic field may be altered by the presence of an
accretion disk \citep[depending on the position of the X-point and the
amount of trapped flux; see ][]{Mohanty08}.
Radio flux predictions for giant exoplanets based on our scaling
relation are not dramatically different from earlier estimates by
\citet{Farrell99}, \citet{Stevens05}, and
\citet{Griessmeier07a}. Still, the uncertainties in the predicted
radio flux are much larger than the actual values. First, our
assumption of a homogenous magnetic field interacting with an
isotropic wind is certainly oversimplified. Quantitative uncertainties
of our flux estimates mainly come from uncertainties in the dipole
moment (we estimate a factor of 3) and in the mass-loss rate (another
factor of 3). Individual values of the radio flux are therefore
uncertain by at least a factor of 5, but differential comparison
between the radio flux predictions are likely to be more trustworthy
because the objects are in general rather comparable.
We estimate that only few extrasolar planets may emit radio flux
larger than 10\,mJy; $\tau$~Boo\,b is the strongest potential emitter
with $P \sim 700$\,mJy. The maximum emission frequencies are well
above the ionospheric cutoff frequency of 10\,MHz in most cases so
that detections of radio emission should not be hampered by the
ionosphere.
The uncertainties of magnetic field and radio emission predictions are
still very large. Nevertheless, several different scaling scenarios
now exist and the one of \citet{Christensen09} was shown to
successfully reproduce magnetic fields of both planets and stars. The
detection of magnetic fields in giant extrasolar planets through other
techniques like the Zeeman effect will probably remain out of reach
for a few decades, at least in old planets with fields that are
probably well below 100\,G. However, very young and massive planets
may harbor magnetic fields up to one kilo-Gauss, which may become
detectable in a few systems in the near future.
Radio emission from extrasolar planets has not been detected so far,
but the technology for a detection may become soon available with
facilities like LOFAR and others. Our scaling relation favors a
different set of planets than that suggested by \citet{Griessmeier07a}
although $\tau$~Boo\,b remains the most obvious choice. Our scaling
relation can easily be applied to extrasolar planets that will be
detected in the future, and target selection for radio observation
campaigns can be based on these predictions.
\begin{acknowledgements}
A.R. acknowledges research funding from the DFG as an Emmy Noether
fellow (RE 1664/4-1).
\end{acknowledgements}
|
1,108,101,566,732 | arxiv | \section{Introduction}
The physical properties of GCSs in galaxies
have long been considered to be ``fossil records''
that contain vital information on galaxy formation
and evolution (e.g., Searle \& Zinn 1978;
Harris 1991; Ashman \& Zepf 1998;
West et al. 2004).
For example, the observed bimodal colour
distributions, higher specific frequencies ($S_{\rm N}$)
and higher fraction of metal-poor GCs (MPCs)
in elliptical galaxies has motivated various
formation scenarios for elliptical galaxies
(e.g., Ashman \& Zepf 1992; Forbes et al. 1997;
C\^ote et al. 1998).
Correlations between physical properties of GCSs
and those of their host galaxies
also have provided clues to the better understanding galaxy formation
For example,
Strader et al. (2004) showed that the
mean colours of MPCs and MRCs
correlate with the luminosity ($L$) of their host galaxy.
From a survey of Virgo cluster early-type galaxies,
Peng et al. (2006, P06) calculated from g--z colours, the relations
${\rm [Fe/H]}_{\rm gc} \propto L^{0.16 \pm 0.04}$ for MPCs and
${\rm [Fe/H]}_{\rm gc} \propto L^{0.26 \pm 0.02}$ for MRCs.
In addition to the above chemical property of GCSs,
the kinematical and structural properties of GCSs have been
investigated both for MPCs and MRCs
(Kissler-Patig \& Gebherdt 1998;
C\^ote et al. 2001;
Rhode \& Zepf 2004; Richtler et al. 2004
Peng et al. 2005;
Bergond et al. 2006;
Bridges et al. 2006;
Pierce et al. 2006;
Romanowsky 2006;
Hwang et al. 2007;
Woodley et al. 2007).
Peng et al. (2005) showed that the GCS in NGC 5128
has a significant
amount of global rotation
whereas Richtler et al. (2004) did not find any significant
rotation in the GCS of NGC 1399,
which suggests a great diversity in the kinematics of GCSs in
early-type galaxies.
The slopes of power-law density profiles of GCSs are also observed
to be diverse and only loosely
correlated with the luminosity of their host galaxy
(e.g., Harris 1986; Ashman \& Zepf 1998).
In comparison with this remarkable progress in observational
studies of GCSs (Brodie \& Strader 2006),
theoretical modeling has not been as
well developed as to provide useful predictions
that can be compared with the wide range of observations on GCS properties.
This is mainly because,
both {\it subpc-scale} formation processes
of GCs in galaxies and {\it kpc- and Mpc-scales}
merging/interaction processes over
a Hubble time
need to be modeled in a fully self-consistent manner
in order that the GCS properties and GCS-host relations can be
examined quantitatively.
Beasley et al. (2002; B02) first investigated the physical
properties of GCSs and their correlations with those of host
galaxies based on a semi-analytic model of hierarchical galaxy
formation in a $\Lambda$CDM universe. B02 however could not discuss
the structural and kinematical properties of GCSs owing to the
limitations of the adopted semi-analytic model. Although previous
collisionless N-body simulations with GCs have provided some
reasonable explanations for the origin of structural and
kinematical properties of GCSs (e.g., Bekki et al. 2005; Bekki \&
Forbes 2006), they did not model the chemical properties of GCSs.
Kravtsov \& Gnedin (2005) carried out a high resolution gas +
N-body simulation of GC formation in a $\Lambda$CDM universe. In
their model of a Milky Way like galaxy, GCs first formed at
redshift $\sim$ 12 with peak formation occurring at z $\sim$
4. However they only simulated MPCs and only from formation to a
redshift of 3. Thus no simulations have yet addressed the
structural, kinematical, and chemical properties of GCSs in a
self-consistent cosmologically motivated simulation from
formation to the current epoch.
The purpose of this paper is to investigate
{\it both the dynamical and chemical properties} of GCSs
based on cosmological N-body simulations
with semi-analytic models of galaxy formation
and thereby compare the results with
the above-mentioned growing number of observational constraints on GCS-host
relations.
We focus particularly on the following physical properties
of GCSs in galaxies : (i) GC metallicity distribution functions (MDFs),
(ii) number fractions of MRCs ($f_{\rm mrc}$),
(iii) number fractions of GCS with bimodal MDFs ($f_{\rm bimo}$),
(iv) specific frequencies of GCs ($S_{\rm N}$),
(v) mean GCS metallicities (${\rm [Fe/H]}$),
(vi) GCS half-number radii ($r_{\rm e}$),
and (vii) GCS velocity dispersions ($\sigma$).
We mainly investigate correlations between these GCS properties
and their host properties such as $M_{\rm B}$
and Hubble type.
Here we do not discuss other important
observational results of GCSs, such
as the blue tilt (e.g., Strader et al. 2006)
or the GC luminosity function dependence on galaxy luminosities
(e.g., Jord\'an et al. 2006).
Some of these aspects have been already discussed in our previous
papers (e.g., Bekki et al. 2007a, B07). Here we also restrict our
analysis to central galaxies and do not consider satellites
in sub-halos.
The plan of this paper is as follows: In the next section
we describe our numerical method of cosmological N-body simulations,
semi-analytic model of galaxy formation,
and formation model for GCs.
In \S 3, we present our numerical results on
GCS properties in galaxies with different luminosities and
Hubble types.
In \S 4, we discuss our results of GCS-host relations in the context of
galaxy formation and evolution.
We summarize our conclusions in \S 5.
\begin{table*}
\centering
\begin{minipage}{185mm}
\caption{Model parameters in the simulation.}
\begin{tabular}{ccccccccccc}
$\Omega$
& $\Lambda$
& $H_{0}$ (km s$^{-1}$ Mpc$^{-1}$)
& ${\sigma}_{8}$
& {$M_{\rm T}$
\footnote{The total mass of a simulation
in units of $10^{14}$ ${\rm M_{\odot}}$.}}
& { $R_{\rm T}$
\footnote{The box size of a simulation in units of
$h^{-1}$ Mpc }}
& { $z_{\rm i}$
\footnote{The redshift ($z$) at which a simulation starts.}}
& { $z_{\rm trun}$
\footnote{The redshift of truncation of GC formation:
GC formation is completely
truncated in halos that are virialized later than the redshift.}}
& { $\alpha$
\footnote{The coefficient in the functional form describing
the dependence of GC formation rate on the mass ratios of merging two galaxies.}}
& { $\beta$
\footnote{The coefficient in the functional form describing
the dependence of GC formation rate on the gas mass fractions of galaxies.}}
& {SNe feedback
\footnote{The details of the strong and weak feedback effects are given
in N05.}} \\
0.3 & 0.7 & 70 & 0.9 & 4.08 & 70 & 41 & 6 & 0.02 & 0.05 & strong \\
\end{tabular}
\end{minipage}
\end{table*}
\section{The model}
We consider that GCs can be formed within any galaxy at any redshift,
if the physical conditions required for GC formation are satisfied
in the galaxies. We therefore investigate GC formation rates in any
virialized halo
in a high-resolution
cosmological simulation based on
a $\Lambda$CDM cosmology model.
The physical properties of GCs (e.g., metallicities)
are determined by their host
galaxies at the epoch of their formation.
GCs formed in low-mass galaxies at high $z$ can be tidally
stripped during the hierarchical merging of galaxies
to finally become GCs within a giant galaxy at $z=0$.
Since numerical methods and techniques of the present cosmological
simulations and those of semi-analytic models of galaxy formation
have been given in our previous papers
(e.g., Yahagi et al. 2004; Nagashima et al. 2005, N05),
we only briefly describe them in the present study.
We here focus on (i) how to identify GCs in virialized dark matter halos
and (ii) how to allocate physical properties
to the identified GCs
based on their host galaxy properties.
\subsection{Simulations}
We simulate the large scale structure of GCs
in a $\Lambda$CDM Universe with ${\Omega} =0.3$,
$\Lambda=0.7$, $H_{0}=70$ km $\rm s^{-1}$ ${\rm Mpc}^{-1}$,
and ${\sigma}_{8}=0.9$
by using the Adaptive Mesh Refinement $N-$body code developed
by Yahagi (2005) and Yahagi et al. (2004),
which is a vectorized and parallelized version
of the code described in Yahagi \& Yoshii (2001).
We use $512^3$ collisionless dark matter (DM) particles in a simulation
with the box size ($R_{\rm T}$) of $70h^{-1}$Mpc and the total mass
($M_{\rm T}$) of $4.08 \times 10^{16} {\rm M}_{\odot}$.
We start simulations at $z_{\rm i} =41$ and follow it until $z=0$
in order to investigate the physical properties
of GCs outside and inside of virialized dark matter halos.
We use COSMICS (Cosmological Initial Conditions and
Microwave Anisotropy Codes), which is a package
of Fortran programs for generating Gaussian random initial
conditions for non-linear structure formation simulations
(Bertschinger 1995, 2001).
Our method of identifying GCs (or ``GC particles'')
and following their evolution
is described as follows.
Firstly, we select virialized dark matter subhalos at a
given redshift
by using the friends-of-friends (FoF) algorithm (Davis et al. 1985)
with a fixed linking length of 0.2 times the mean DM particle separation.
The minimum particle number $N_{\rm min}$ for halos is set to be 10.
For each individual virialized subhalo,
the central
particle is labeled
as a ``GC'' particle.
This procedure for defining GC particles
is based on the assumption that energy dissipation via radiative cooling
allows baryons to fall into the deepest potential well of dark matter halos
and finally to be converted into GCs and stars.
The adopted initial distributions of GCs with respect to
their host halos would be oversimplified, given that possible
candidates of forming GCs are not necessarily in the central
regions of galaxies at $z=0$.
We think that the adopted assumption
can be regarded as reasonable,
because previous observations suggested that
a significant fraction (or even all) of the Galactic GCs
originate from nuclei of the Galactic building blocks
at high $z$
(e.g., Zinnecker et al. 1988; Freeman 1993).
We stress that the predicted spatial distributions
of GCSs in galaxies at $z=0$ might well weakly depend on the adopted
initial distributions of GCs within halos,
though previous simulations suggested that distributions
of GCSs at $z=0$ do not depend strongly on the adopted range
of reasonable initial GC distributions
(Yahagi \& Bekki 2005).
Secondly, we follow GC particles
until $z=0$ and thereby
derive their locations $(x,y,z)$
and velocities $(v_{\rm x},v_{\rm y},v_{\rm z})$.
We then identify virialized halos at $z=0$ with the FoF algorithm
and investigate whether each GC is within the virial radius
($r_{\rm vir}$) of a halo.
If GCs are found to be within a halo, the
physical properties of the GCS
are investigated.
If a GC is not in any halo,
it is regarded as an intergalactic GC.
We don't discuss these intergalactic
(i.e., intra-group and intra-cluster) GCs further in this
current paper (see Yahagi \& Bekki 2005; Bekki \& Yahagi 2006
for details of such GCs),
though a growing number of observations have revealed
physical properties of these intergalactic GCs
(e.g., Bassino et al. 2006; Jones et al. 2006).
Recent theoretical works have suggested that
heating and gas loss resulting from reionization
can severely suppress star formation in low-mass galaxies during
reionization
(e.g., Susa \& Umemura 2004).
It is therefore highly likely that GC formation is truncated
in low-mass galaxies that are virialized after the completion of reionization.
In order to include the effects of
the suppression of GC formation
via reionization on the final properties of the
simulated GCSs,
we adopt the following somewhat idealized
assumption: If a galaxy is virialized after the completion of
reionization ($z_{\rm reion}$), then
GC formation is totally suppressed in the galaxy.
Therefore, GC particles formed
in galaxies with $z_{\rm vir} < z_{\rm reion}$
are not
considered in the physical
properties of the simulated GCSs.
We define the truncation epoch of GC formation as $z_{\rm trun}$
(rather than $z_{\rm reion}$ for convenience)
in the present study.
Recent quasar
absorption-line studies give a lower limit of 6.4 for
$z_{\rm reion}$ (Fan et al. 2003).
Guided by these observations, we investigate the
model with $z_{\rm trun}=6$.
Thus the GCS of a galaxy
is a collection of GCs that are formed within low-mass galaxies
(i.e., galaxy building blocks)
embedded in massive dark matter halos virialized at high redshifts.
The physical properties
of the GCS of a galaxy therefore depends
on star formation histories, chemical evolution, and merging
histories of the building blocks.
Chemical evolution and star formation histories of building blocks
for galaxies are derived from the semi-analytic model
(N05)
which are based on the merging histories of DM halos
derived from our N-body simulation.
\subsection{Semi-analytic model}
The physical properties of a newly formed
GC in a virialized halo at any redshift
are determined by those of its host galaxy at that redshift.
N05 constructed the Numerical Galaxy Catalog ($\nu$GC)
based on a semi-analytic model combined with high-resolution $N-$body
simulations.
In the present study, we adopt the same semi-analytic model as
that used by N05 which includes various physical
processes associated with galaxy formation,
such as galaxy merging, radiative gas cooling,
star formation, supernovae feedback, and extinction by internal
dust. Since the methods and techniques
are given in detail by
N05, we do not describe them in the present study.
The semi-analytic model by N05 can reproduce many observations reasonably well,
such as cold gas mass-to-stellar luminosity ratios of spiral galaxies,
faint galaxy number counts, cosmic star formation rates,
the Tully-Fisher relation
for bright spiral galaxies,
luminosity functions of local galaxies,
and colour-magnitude relations for massive and dwarf elliptical
galaxies. The successes and limitations of this model in
explaining observed galactic properties are given in N05.
Here we present the results for the properties of GCs from our
semi-analytic model and compare them where possible to
observations. We focus on results from the model with the
``strong feedback effects'' (N05) in which galaxies with smaller
rotational velocities are much more strongly influenced by
heating. An important caveat in this work is that only galaxies
located at the centers of halos are analyzed in the present
study, i.e. we do not investigate satellite galaxies. Thus we
effectively focus on brightest cluster, brightest group and
isolated galaxies in this current work. Observational studies are
often dominated by non-central galaxies (satellites in this
context) and this should be born in mind when we compare our
model predictions to the observations below.
\subsection{Formation efficiencies of GCs}
We convert the star formation rate (SFR) in a galaxy into
a GC formation rate (GCFR) based on the physical properties
of the host galaxy at a given redshift.
The vast majority
of stars are observed to form
in star clusters (SCs) embedded within GMCs (Lada \& Lada 2003).
Strongly bound SCs can evolve into GCs or old open clusters
whereas weakly bound, low-mass ones can be disintegrated into
field stars of the host galaxy.
The mass fraction of new SCs that evolve into GCs
relative to all new SCs is denoted as $C_{\rm eff}$
and is assumed to be dependent on the physical properties of their host
galaxy.
Thus we define the GCFR as:
\begin{equation}
{\rm GCFR} = C_{\rm eff} \times {\rm SFR},
\end{equation}
where SFR is an output from the
semi-analytic model.
\begin{figure}
\psfig{file=f1.eps,width=8.0cm}
\caption{
The distribution of GCs
projected onto the $x$-$y$ plane at $z=0$.
Here only GC particles that are at the very centers of GCSs
for galaxies
at $z=0$ are shown for convenience.
Thus the distribution describes
the large-scale distribution of GCSs of galaxies in the universe at $z=0$.
}
\label{Figure. 1}
\end{figure}
\begin{figure}
\psfig{file=f2.eps,width=8.0cm}
\caption{
Cosmic evolution of GC formation rates (GCFRs) as a function of redshift ($z$)
for MPCs (solid) and MRCs (dotted).
The GCFRs are averaged over the volume of the simulation.
}
\label{Figure. 2}
\end{figure}
\begin{figure}
\psfig{file=f3.eps,width=8.0cm}
\caption{
Total numbers of GCs formed at redshift $z$
for MPCs (solid) and MRCs (dotted). The numbers are normalized
by the maximum values
for $0 \le z \le 12$.
}
\label{Figure. 3}
\end{figure}
We consider both observational results by Larsen \& Richtler (2000, LR00)
and simulation ones by Bekki et al. (2002, BFBC02) in order to determine
$C_{\rm eff}$
in a physically reasonably way.
LR00 investigated correlations between
specific $U-$band cluster luminosities $T_{\rm L}(U)$
of GCSs and their host
galaxy properties both for apparently isolated galaxies and
for interacting/merging ones.
LR00 found that (i) $T_{\rm L}(U)$ correlates with
SFR per unit area, stellar surface brightness, and HI surface
density,
and (ii) $T_{\rm L}(U)$ is more than an order of
magnitude higher in strongly starbursting mergers
(e.g., NGC 1705) than apparently normal galaxies
that are forming young SCs (see Table 1 in LR00).
These results by LR00 imply that $C_{\rm eff}$ can be higher
in galaxies with higher surface densities and higher gas mass fractions
for a given galaxy luminosity.
Numerical simulations of GC formation in merging galaxies
(BFBC02)
found that $C_{\rm eff}$ depends on the mass ratios of the merging
spirals ($f_{\rm m}$ or $m_{2}$) and their gas mass fractions ($f_{\rm g}$)
in such a way that $C_{\rm eff}$ is higher
in mergers with larger $m_{2}$ and $f_{\rm g}$.
These results by BFBC02 combined with LR00
imply that $C_{\rm eff}$ is a minimum for isolated galaxies with $m_2=0$
and maximum for equal mass galaxy mergers with $m_2=1$.
BFBC02 also found that $C_{\rm eff}$ is lower in low surface brightness
galaxies for a given galaxy mass.
Galaxies with their halos virialized at higher $z$
have more compact disks
and thus higher stellar densities for a given mass,
baryonic mass fraction, spin parameter, and halo circular velocity
in galaxy formation models based on
$\Lambda$CDM (e.g., Mo et al. 1998).
The above result by BFBC02 combined with that by Mo et al. (1998)
therefore strongly suggests that
$C_{\rm eff}$ is likely to be higher for galaxies
formed at higher $z$. Thus $C_{\rm eff}$ is a function of the gas
fraction, the merger mass ratio and formation redshift.
We estimate $C_{\rm eff}$ as follows:
\begin{equation}
C_{\rm eff} = C_0
F_{\rm g}(f_{\rm g}) F_{\rm m}(f_{\rm m})
F_{\rm z}(z).
\end{equation}
We need to choose the forms of
these three functions ($F_{\rm g}$, $F_{\rm m}$, and $F_{\rm z}$)
so that they are consistent with previous results by LR00 and BFBC02.
The normalization factor $C_0$ is determined as follows:
\begin{equation}
C_0 = {F_{\rm g,max}}^{-1} {F_{\rm m,max}}^{-1}
{F_{\rm z,max}}^{-1},
\end{equation}
where $F_{\rm g,max}$, $F_{\rm m,max}$,
and $F_{\rm z,max}$
are maximum values of the above three functions.
This method of normalization ensures that $C_{\rm eff}$ is always equal to
or less than 1.
If we adopt a value of $C_0$ significantly smaller than the one
given in the above equation,
physical properties of the simulated GCSs (e.g., total numbers of GCs
in galaxies) can be much less consistent with observations.
In the present study, we choose elementary functions
for $C_{\rm eff}$.
Considering that (i) the dependence of $C_{\rm eff}$ on $f_{\rm m}$
appears to be non-linear (BFBC02) and (ii)
$C_{\rm eff}$ is much higher in major mergers (LR02 and BFBC02),
we assume the following:
\begin{equation}
F_{\rm m}(f_{\rm m})=1 - {(1+\alpha)}^{-1} + {(1+\alpha-f_{\rm m})}^{-1},
\end{equation}
where $\alpha$ is a parameter.
This formula ensures that $F_{\rm m}(f_{\rm m})$ has a minimum ($=1$)
at $f_{\rm m}=0$
corresponding to isolated disk galaxies
and a maximum at $f_{\rm m}=1$ corresponding to major mergers.
Furthermore, if a small value of $\alpha$ ($<0.1$) is adopted,
$F_{\rm m}$ is more than 10, which corresponds to a strongly starbursting
major merger in LR00.
We adopt the same functional form for $F_{\rm g}(f_{\rm g})$:
\begin{equation}
F_{\rm g}(f_{\rm g})=1 - {(1+\beta)}^{-1} + {(1+\beta-f_{\rm g})}^{-1}.
\end{equation}
Thus, $\alpha$ and $\beta$ are free parameters which
are chosen so that observations of GCSs can be self-consistently explained.
Mo et al. (1998) showed that the sizes ($R_{\rm d}$) of disk galaxies
formed at redshift $z$ are inversely proportional to $H(z)/H_0$ for given disk and halo
properties (e.g., disk mass fraction and halo circular velocity).
Therefore $R_{\rm d}$ can be more compact for disk galaxies
formed at higher redshifts so that mean stellar densities ${\Sigma}_{\rm s}$
within $R_{\rm d}$ (${\Sigma}_{\rm s} \propto {R_{\rm d}}^{-2}$
for a given stellar disk mass)
are higher.
Considering the above dependences of disk properties on $H(z)/H_0$
(Mo et al. 1998),
we adopt the following functional form:
\begin{equation}
F_{\rm z}(z)={(H(z)/H_0)}^2 .
\end{equation}
We estimate $F_{\rm z}(z)$ for each halo at a given redshift
based on the virialization redshift ($z_{\rm vir}$) of the halo.
Here we adopt a simpler yet qualitatively reasonable
dependence of $F_{\rm z}(z)$: we do not directly derive
the surface mass densities by assuming initial sizes of halos,
spin parameters, and sizes of galaxies based on the
results of the SAM by N05.
Although we can not discuss the importance of
these physical properties (e.g., initial spin parameters)
in GC formation in the present model,
we can show the importance of surface mass (or gas) densities
(i.e., the epoch of virialization) in GC formation of galaxies in a clearer
and more straightforward way thanks to the adopted model.
It should be stressed here that without introduction
of the $F_{\rm z}(z)$, most GCS properties can not be reproduced
well in the present model.
Although the adopted functional forms of $F_{\rm m}$,
$F_{\rm g}$, and $F_z$ are reasonable at least qualitatively,
no observations have been carried out which allow us to determine
whether the adopted functional form is quantitatively
consistent with observations.
Accordingly other functional forms
could be adopted so that observational properties of GCSs can be
reproduced s well.
Here we do not discuss the results of models
with different functional forms of $F_{\rm m}$,
$F_{\rm g}$, and $F_z$.
$T_{L}(U)$ in some merging galaxies
(e.g., NGC 1705) are observed to be as large as $15$,
which is a factor of $\sim20$ larger than
the average value ($=0.7$) of $T_{L}(U)$
for non-merging galaxies with young GCs
(LR00).
Also there is a factor of $\sim 200$ difference
between the minimum (=0.07) and maximum (=15)
values of $T_{L}(U)$ for galaxies with young GC
candidates in LR00.
Therefore $0.005 \le \alpha \le 0.05$ is a reasonable range
of $\alpha$ in $F_{\rm m}$.
We try to determine the most reasonable values of $\alpha$
and $\beta$
for which both
the observed net formation efficiency of GCs (McLaughlin 1999)
and the number fraction of MRCs (Spitler et al. 2007)
can be well reproduced by the present simulation.
The net formation efficiency of GCs ($\epsilon$) in the present study
is defined as follows:
\begin{equation}
\epsilon=\frac{M_{\rm gc}}{M_{\rm star}},
\end{equation}
where $M_{\rm gc}$ and $M_{\rm star}$ are total masses of GCs
and stars, respectively,
which are formed in {\it all} building blocks in the simulation.
We define $f_{\rm mrc}$ as follows:
\begin{equation}
f_{\rm mrc}=\frac{N_{\rm mrc}}{N_{\rm mpc}+N_{\rm mrc}},
\end{equation}
where $N_{\rm mpc}$ and $N_{\rm mrc}$ are numbers of MPCs and MRCs,
respectively.
Since we find that the model with $\alpha=0.02$ and $\beta=0.05$
can best reproduce the observations,
we discuss this model in this paper.
\begin{figure}
\psfig{file=f4.eps,width=8.0cm}
\caption{
Number of gas-rich (``wet'')
major mergers that form MPCs (solid) and MRCs (dotted)
as a function of $z$. Here wet major mergers are those with
$f_{\rm g} > 0.5$ and $f_{\rm m}>0.5$.
The numbers normalized by their maximum values for $ 0\le z \le 12$
are shown.
}
\label{Figure. 4}
\end{figure}
\begin{figure}
\psfig{file=f5.eps,width=8.0cm}
\caption{
The age-metallicity relation for all GCs.
The mean metallicity of GCs in
age bins are shown by filled
circles.
The $1 \sigma$ dispersion in the metallicities
of GCs for each age bin is shown by an error bar.
}
\label{Figure. 5}
\end{figure}
\begin{figure}
\psfig{file=f6.eps,width=7.0cm}
\caption{
Metallicity distribution functions (MDFs) for four representative
massive galaxies with
${\log}_{10} ( \frac{ M_{\rm h} } { {\rm M}_{\odot} }) \ge 13.0$:
(a) $M_{\rm B}=-20.7$ mag and $B/T=1.0$,
(b) $M_{\rm B}=-21.3$ mag and $B/T=1.0$,
(c) $M_{\rm B}=-21.3$ mag and $B/T=1.0$,
and (d) $M_{\rm B}=-20.9$ mag and $B/T=1.0$.
The mean metallicities of MPCs, MRCs, and all GCs are shown
by solid, dotted, and dashed lines, respectively.
Although the shapes of MDFs are different in different galaxies,
they all clearly show bimodal MDFs.
}
\label{Figure. 6}
\end{figure}
\subsection{Main points of analysis}
The total number of virialized halos at $z=0$ is 95139,
among which only 12179
(thus 12.8 \%) have GCs that are formed in their building
blocks virialized before $z=6$ (i.e., reionization).
The total number of {\it GC particles} at $z=0$
is 998529 in the present simulation.
Fig. 1 shows the large-scale structure of GCs that
are in the very central regions of their host galaxies
at $z=0$. Although these central GCs in galaxies are
relatively metal-rich with a mean metallicity of
${\rm [Fe/H] }=-0.18$ for the 12179 galaxies,
some fraction of them (37.5\%) are metal-poor
(${\rm [Fe/H] }<-1$).
Here we focus on correlations between physical properties
of GCSs and those of their host galaxies: we do not
discuss the internal properties of individual GCSs.
In investigating GCS properties,
we divide GCs into MPCs and MRCs according to their metallicities
([Fe/H]): those with ${\rm [Fe/H]} <-1$
and with ${\rm [Fe/H]} \ge -1$ are defined as MPCs
and MRCs, respectively.
We investigate the physical properties of GCSs separately
for these two GC subpopulations.
We also discuss the bimodality in the metallicity distribution
functions (MDFs)
of GCSs.
We show some examples of individual
representative GCSs in Appendix A.
Following the morphological classification scheme by
Simien \& de Vaucouleurs (1986),
we classify simulated galaxies into three different morphological
types: E, S0, and Sp.
We use the simulated
B-band bulge-to-disk luminosity ratio,
B/D in the above morphological classification.
In this paper,
galaxies with B/D $>$ 1.52, 0.68 $<$ B/D $<$ 1.52,
and B/D $<$ 0.68 are
classified as elliptical (E), lenticular (S0),
and spiral galaxies (Sp), respectively.
\begin{figure}
\psfig{file=f7.eps,width=7.0cm}
\caption{
The age-metallicity relations (AMRs) of GCs for the four
galaxies shown in Fig. 6.
The mean
metallicities
of GCs are shown for the five age bins.
In order to show the AMRs more clearly,
only GCs with ages larger than 10 Gyr are shown as few younger
GCs are present.
The $1 \sigma$ dispersion in the metallicities
of GCs for each age bin is shown by an error bar.
}
\label{Figure.7}
\end{figure}
\begin{figure}
\psfig{file=f8.eps,width=7.0cm}
\caption{
The number fractions of GCSs with bimodal MDFs ($f_{\rm bimo}$)
as a function of $M_{\rm B}$ for Es (top), S0s (middle), and
Sp (bottom).
The error bar in each bin is based on Poisson $\sqrt{N}$
statistics.
}
\label{Figure.8}
\end{figure}
\begin{figure}
\psfig{file=f9.eps,width=8.0cm}
\caption{
Distributions of the simulated GCSs in
the $f_{\rm mrc}$-$B/T$ plane (upper)
and the $f_{\rm mrc}$-$M_{\rm B}$ one (lower),
where $f_{\rm mrc}$ is the number fraction of MRCs in a GCS.
Triangles and squares represent the observational results
by Spitler et al. (2007) and by P06, respectively, whereas
circles show the average values of $f_{\rm mrc}$
in $B/T$ and $M_{\rm B}$ bins.
The $1\sigma$ dispersion
for each bin is shown by an error bar.
}
\label{Figure. 9}
\end{figure}
\begin{figure}
\psfig{file=f10.eps,width=8.0cm}
\caption{
Distributions of the simulated GCSs on
the $f_{\rm y}$-$B/T$ plane (upper)
and on the $f_{\rm y}$-$M_{\rm B}$ one (lower),
where $f_{\rm y}$ is the number fraction of young MRCs with ages
less than 8 Gyrs.
Circles show the average values of $f_{\rm y}$
in $B/T$ and $M_{\rm B}$ bins.
The $1\sigma$ dispersion
for each bin is shown by an error bar.
}
\label{Figure. 10}
\end{figure}
\section{Results}
\subsection{Cosmic evolution}
Since our main focus is to discuss the results of correlations between
physical properties of GCSs and those of their host galaxies at $z=0$,
we do not discuss the
cosmic evolution of GCS.
We do however briefly summarize the cosmic evolution of GCS properties
{\it averaged over all GCs formed in the simulation.}
Fig. 2 shows that the peaks of cosmic GCFRs are at $z \sim 7$ for MPCs
and at $z \sim 4$ for MRCs and that the GCFR is much higher
in MPCs than in MRCs at $z>2$, whereas it is higher in MRCs for $z<2$.
Fig. 3 shows that number distributions of MPCs and MRCs normalized by
their maximum values have two different peaks,
which reflect the differences in cosmic evolution of GCFRs
between MPCs and MRCs.
Fig. 4 shows that both MPCs and MRCs
can be formed from gas-rich ($f_{\rm g}>0.5$),
major ($f_{\rm m}>0.5$) mergers for a wide range of redshifts
and that formation rates of MPCs and MRCs from wet major mergers
peak at $z \sim 5$ for MPCs and at $z \sim 3$ for MRCs.
As shown in Fig. 5,
the mean metallicities for all GCs (including both MPCs and MRCs)
steeply increase for ages $ > 10$ Gyr
as a result of rapid chemical enrichment in the building blocks
of galaxies.
The time evolution of the mean metallicities
appears to be much less dramatic for ages $ < 10$ Gyr
owing to the slower chemical enrichment processes in galaxies.
The dispersions in the mean metallicities
($\sigma ({\rm Fe/H})$) are smaller
for larger ages i.e., older GCs.
We find $\sigma ({\rm Fe/H}) \sim 1.6$ dex for $ \sim 13$
Gyr and $\sigma ({\rm Fe/H}) \sim 0.1$ dex for $ \sim 0.8$ Gyr.
It is interesting to derive
the age-metallicity relation of GCs with ages $t$ (Gyr) older than
10 Gyr in the present simulation at $z=0$
in order to compare the simulated relation with
the observed one when sufficient observational results are available.
The $\chi$-square fit to the simulation data gives
the following:
\begin{equation}
{\rm [Fe/H]} = 10.86-10.83 {\log}_{10} t
\end{equation}
The model predicts the presence of young ($<3$ Gyr) GCs
with ${\rm [Fe/H]} > 0$,
though the number fraction of such GCs
is quite small.
Most of these metal-rich GCs are found to be located in the central
regions of galaxies in the simulation.
It should be however stressed
that very old ($>13$ Gyr), metal-poor (${\rm [Fe/H]} < -1.6$)
GCs can also be found in the very central regions of galaxies at $z=0$.
\begin{figure}
\psfig{file=f11.eps,width=7.0cm}
\caption{
Distributions of the simulated GCSs in
the $M_{\rm B}$-${\rm [Fe/H]}$ plane for all GCs (top),
MRCs (middle), and MPCs (bottom).
Here ${\rm [Fe/H]}$ represents the mean metallicity of a GCS in
a galaxy.
Circles show the average values of ${\rm [Fe/H]}$
in a $M_{\rm B}$ bin.
The $1\sigma$ dispersion
for each bin is shown by an error bar.
For comparison, the observed $M_{\rm B}$-${\rm [Fe/H]}$ relations
of P06 are shown by solid lines in the three panels.
}
\label{Figure. 11}
\end{figure}
\begin{figure}
\psfig{file=f12.eps,width=7.0cm}
\caption{
Distributions of the simulated GCSs in
the $S_{\rm N}$-$M_{\rm B}$ plane for E (top),
S0 (middle), and Sp (bottom).
Circles show the average values of $S_{\rm N}$
in a $M_{\rm B}$ bin.
Open squares represent the observational results by P06,
for which we assume that $B-V=0.9$ for all Es to estimate
$S_{\rm N}$.
The $1\sigma$ dispersion
for each bin is shown by an error bar.
}
\label{Figure. 12}
\end{figure}
\begin{figure}
\psfig{file=f13.eps,width=7.0cm}
\caption{
The same as Fig. 12 but for the $V_{\rm N}$-$M_{\rm B}$ plane,
where $V_{\rm N}$ is the number of GCs per halo mass.
}
\label{Figure. 13}
\end{figure}
\begin{figure}
\psfig{file=f14.eps,width=7.0cm}
\caption{
Distributions of the simulated GCSs in
the $S_{\rm N}$-$M_{\rm h}$ plane for E (top),
S0 (middle), and Sp (bottom).
Circles with error bars show the average values of $S_{\rm N}$
in $M_{\rm h}$ bins.
The $1\sigma$ dispersion
for each bin is shown by an error bar.
}
\label{Figure. 14}
\end{figure}
\begin{figure}
\psfig{file=f15.eps,width=7.0cm}
\caption{
The same as Fig. 14 but for the $V_{\rm N}$-$M_{\rm h}$ plane.
}
\label{Figure. 15}
\end{figure}
\begin{table}
\centering
\caption{Metallicity-luminosity relations. The observed (o; Peng
et al. 2006) and
simulated (s) coefficients are given for MPC, MRC and all GCs,
where [Fe/H] = a + b M$_B$.}
\begin{tabular}{ccccc}
& ${\rm a}_{\rm o}$
& ${\rm b}_{\rm o}$
& ${\rm a}_{\rm s}$
& ${\rm b}_{\rm s}$ \\
MPC & -2.771 & -0.064 & -2.414 & -0.052 \\
MRC & -2.452 & -0.105 & -2.056 & -0.081 \\
All & -4.854 & -0.206 & -3.910 & -0.160 \\
\end{tabular}
\end{table}
\subsection{Metallicities and ages}
Fig. 6 shows that the GCSs in four early-type galaxies with $B/T
\approx 1$ at $z=0$ have bimodal MDFs with peaks below and
above ${\rm [Fe/H]} =-1$. The locations of the two peaks and the
number ratios of GCs in each peak are different between
different galaxies, which implies that merging and star formation
histories of their building blocks are quite diverse in the four
galaxies. The peak metallicities of GCSs reflect the gaseous
metallicities of the building blocks at the epoch when strong
starbursts were triggered mostly by gas-rich (``wet''), major
merging at high $z$ ($>3$). Although these four
galaxies located in the centers of massive clusters of galaxies
with $M_{\rm h} > 10^{13} {\rm M}_{\odot}$ clearly show bimodal
MDFs, less luminous galaxies in the field and groups of
galaxies do not necessarily show bimodal MDFs in their GCSs. The
diverse shapes of MDFs of these galaxies (including those
that show unimodal MDFs) are briefly discussed in Appendix A.
Fig. 7 clearly shows that younger GCs are likely to
be more metal-rich for all four galaxies
and that the age-metallicity relations are similar to one another.
Since the colours of GCs depend on both age and metallicity,
these results suggest that the
colour distributions of the simulated GCSs will be
somewhat different from their MDFs. On the other hand, the GCSs
are dominated by very old GCs in each case.
We plan to investigate whether the colour
distributions of GCSs are consistent with observations
by combining the present GC formation model
with a stellar population synthesis code and the
effects of photometric uncertainty in a future paper.
To investigate the number fractions ($f_{\rm bimo}$) of galaxies
that reveal clear bimodal
MDFs
we determine $f_{\rm bimo}$
as follows:
\begin{equation}
f_{\rm bimo}=\frac{ N_{\rm bimo} }{ N_{\rm gal} },
\end{equation}
where $ N_{\rm bimo}$ and $ N_{\rm gal} $ are
the number of galaxies with obviously bimodal GCS MDFs and
that of all galaxies, respectively.
In the present study, an MDF with obvious bimodality
is defined as revealing a clear metallicity peak at both low (${\rm [Fe/H]} <-1$ for MPCs)
and high metallicity (${\rm [Fe/H]} \ge -1$ for MRCs).
Fig. 8 shows that more than 70\% of galaxies with $M_{\rm B}<-19$ mag
have bimodal MDFs in their GCSs
regardless of their Hubble types.
The number fraction of GCSs with bimodal MDFs ($f_{\rm bimo}$)
depends on $M_{\rm B}$ such that
less luminous galaxies have smaller $f_{\rm bimo}$
(i.e., less likely to have bimodal MDFs).
Considering that we do not
include GCSs with no MPCs in estimating $f_{\rm bimo}$,
the above result means that
only MPCs are formed in
the building blocks of these less luminous galaxies
with no bimodal MDFs. Furthermore gas-rich, major merging
between high-density building blocks
does not occur after their
gaseous metallicities becomes higher than ${\rm [Fe/H]} \sim -1$
in the formation histories of these galaxies.
Fig. 9 shows that the number fraction of MRCs ($f_{\rm mrc}$)
does not depend on $B/T$.
Some very late-type spirals with $B/T<0.05$ have high values of $f_{\rm mrc}$
($>0.6$) whereas some ellipticals with $B/T \approx 1$ have low values
of $f_{\rm mrc}$ ($<0.1$), which suggests that
the origin of the Hubble types of galaxies are not
closely associated with the metal-rich fraction
($f_{\rm mrc}$) in their GCSs.
However $f_{\rm mrc}$ does depend on $M_{\rm B}$ such that
$f_{\rm mrc}$ is higher in more luminous galaxies,
though the dispersions in $f_{\rm mrc}$ are quite large over
a wide range of $M_{\rm B}$.
Galaxies with $M_{\rm B}<-20$ mag cover a similar range for
observed galaxies in the $M_{\rm B}-f_{\rm mrc}$ plane (Spitler
et al. 2007). The mean value of $f_{\rm mrc}$ for all GCSs in
the present simulation is 0.36, which is consistent with that
found by Spitler et al. (2007). We note that the Spitler et
al. sample includes non-central galaxies.
The mean formation epochs of MPCs and MRCs
for all GCs formed in the simulation between $z=41$ and $z=0$
are estimated to be 5.7 and 4.3,
respectively.
These results imply typical ages of MPCs and MRCs
at $z=0$ are 12.7 and 12.3 Gyr, respectively. This is consistent
with the current spectroscopic measurements of extragalactic GCs
(e.g. Brodie \& Strader 2006).
The mean
values of the number fractions of young GCs
with ages less than 8 Gyrs ($f_{\rm y}$)
is very small ($\le$4\%) in the present simulation.
Fig. 10 shows that $f_{\rm y}$ does not depend on $B/T$.
The dependence of $f_{\rm y}$ on $M_{\rm B}$
appears to show no/little trend:
although $f_{\rm y}$ appear to peak around
$M_{\rm B} = -19$,
this apparent trend is of little
statistical significance.
owing to the large dispersions.
Dispersions in the locations of galaxies
in the $B/T-f_{\rm y}$ and the $M_{\rm B}-f_{\rm y}$ planes
reflect the diversity in the formation epochs of young MRCs,
which are mostly via gas-rich major merging.
Next we investigate correlations between mean metallicities of GCs (simply
referred to as [Fe/H]) and $M_{\rm B}$ of their host galaxies.
These correlations are called ``metallicity-luminosity relations''
for convenience in the present study.
The observed metallicity-luminosity relations by P06, after
transformation from observed colours,
can be described as follows:
\begin{equation}
{\rm [Fe/H]}=a + b {M_{\rm B}}
\end{equation}
The values of the observed coefficients $a_{\rm o}$ and $b_{\rm o}$
for MPCs, MRCs, and all GCs are summarized in Table 2.
The values of our simulated $a_{\rm s}$ and $b_{\rm s}$
using the same relation are listed also in Table 2
for comparison.
Fig. 11 shows that the mean metallicities of {\it all GCs} correlate well with
$M_{\rm B}$ such that the metallicities are higher in more luminous
galaxies. The simulated slope of the
${\rm [Fe/H]}-{\rm M}_{\rm B}$ correlation for all GCs
is consistent with the observed one by P06 for non-central
galaxies.
The simulated ${\rm [Fe/H]}-{\rm M}_{\rm B}$ correlation
for MRCs is also consistent with the observed one
which suggests that the present formation model for MRCs
is realistic.
The ${\rm [Fe/H]}-{\rm M}_{\rm B}$ correlation for
MPCs is, however, not so consistent with the observed one:
the slope of the correlation is too flat and the mean metallicities
of MPCs are systematically higher than the observed ones for a wide
range of luminosities.
One way to solve this inconsistency is discussed
later in \S 4.3.
\begin{table}
\begin{minipage}{85mm}
\centering
\caption{Mean $S_{\rm N}$ and $V_{\rm N}$ in different Hubble types}
\begin{tabular}{cccc}
& Sp
& S0
& E \\
$<S_{\rm N}>$ & 1.8 & 2.0 & 4.0 \\
$<V_{\rm N}>$ & 8.3 & 11.4 & 11.5 \\
\end{tabular}
\end{minipage}
\end{table}
\subsection{Specific Frequencies $S_{\rm N}$ and $V_{\rm N}$}
We investigate
specific frequencies ($S_{\rm N}$) of GCSs in galaxies
with different Hubble types.
$S_{\rm N}$ is defined as follows (Harris \& van den Bergh 1981):
\begin{equation}
S_{\rm N}=N_{\rm gc} 10^{0.4(M_{\rm v}+15)},
\end{equation}
where $N_{\rm gc}$ and $M_{\rm v}$ are the total number of globular clusters
in a galaxy and the $V-$band absolute magnitude of the galaxy, respectively.
We also investigate the number of GC per unit halo mass $V_{\rm N}$,
which, following Spitler et al. (2007), is defined as follows:
\begin{equation}
V_{\rm N}=N_{\rm gc} {(\frac{M_{\rm h}}{10^{11} {\rm M}_{\odot} })}^{-1}
\end{equation}
where $M_{\rm h}$ is the total mass
in a galaxy halo including dark matter.
The mean values of $S_{\rm N}$ and $V_{\rm N}$
in GCSs of galaxies of different Hubble types
are summarized in Table 3.
We estimate the $1\sigma$ dispersion in $S_{\rm N}$ and $V_{\rm N}$
for $M_{\rm B}$ and $M_{\rm h}$ bins to investigate the statistical
significance in the simulated correlations of
$S_{\rm N}$ and $V_{\rm N}$
with $M_{\rm B}$ and $M_{\rm h}$.
In estimating these dispersions,
we use the simulated galaxies with $S_{\rm N}$ ($V_{\rm N}$)
less than 20 to avoid unreasonably large dispersions
caused by a very small number of galaxies with unusually
large $S_{\rm N}$ ($>50$).
Fig. 12 shows that $S_{\rm N}$ of E/S0s are typically higher
than those of spirals (Sp) for a given $M_{\rm B}$.
Although more luminous Es with $M_{\rm B}<-20$
have higher $S_{\rm N}$ than
intermediate-luminosity Es with $-19$ $<M_{\rm B}<$ $-17$,
there is no such a trend in spirals (Sp).
Faint galaxies with $M_{\rm B}>-15$ show high $S_{\rm N}$ ($>4$)
which for E/S0 is due to high halo mass-to-light-ratios
($M_{\rm h}/L_{\rm B}$).
The mean $S_{\rm N}$ for Sp, S0, and E populations are
1.8, 2.0, and 4.0, respectively.
Fig. 13 shows that the dependences of $V_{\rm N}$
on the Hubble types of galaxies are not as strong
as those of $S_{\rm N}$: only a factor of $2-3$ difference
in $V_{\rm N}$ between E/S0
galaxies and spirals for a given luminosity.
For a given Hubble type,
$V_{\rm N}$ does not depend strongly on $M_{\rm B}$.
These results mean that the
numbers of GCs per unit halo mass ($M_{\rm h}$) does not
depend strongly on luminosity or the Hubble type
of their host galaxies.
The weak dependence of $V_{\rm N}$ on $M_{\rm h}$ was already
pointed by Bekki et al. (2006, B06),
though $V_{\rm N}$ is estimated only for MPCs in B06.
There is a weak tendency for more luminous Es with $M_{\rm B}<-20$ mag
to have higher $V_{\rm N}$,
though the dispersion in $V_{\rm N}$ is large.
S0s have $V_{\rm N}$ values $>8$ that are significantly
higher than those of spirals for a wide range of $M_{B}$,
suggesting that the higher $V_{\rm N}$ in S0s
is not due to the truncation of star formation
in spirals as $V_{\rm N}$ does not
change when a disk fades via truncation of star formation.
These results imply that if the observed $V_{\rm N}$ are typically
higher in S0s than in spirals,
then only a minor fraction of spirals
can be transformed into S0s via truncation of star formation
in spirals.
\begin{figure}
\psfig{file=f16.eps,width=7.0cm}
\caption{
Dependences of
$r_{\rm e, mrc}/r_{\rm e, mpc}$ (top)
$r_{\rm e, mrc}$ (middle), and
$r_{\rm e, mpc}$ (bottom)
with $M_{\rm B}$.
The values averaged for GCSs in
$M_{\rm B}$ bins are shown by filled circles.
The $1\sigma$ dispersion
for each bin is shown by an error bar.
}
\label{Figure. 16}
\end{figure}
\begin{figure}
\psfig{file=f17.eps,width=7.0cm}
\caption{
Dependence of $r_{\rm e, mpc}$ (i.e., the half-number radius of a GCS)
on $M_{\rm h}$ (halo mass of the host).
The values averaged for GCSs in $M_{\rm h}$ bins are shown by
filled circles.
The $1\sigma$ dispersion
for each bin is shown by an error bar.
The $\chi$-square fit to the simulation data is shown by a solid line.
}
\label{Figure. 17}
\end{figure}
\begin{figure}
\psfig{file=f18.eps,width=7.0cm}
\caption{
Dependences of
${\sigma}_{\rm mrc}/{\sigma}_{\rm mpc}$ (top)
${\sigma}_{\rm mrc}$ (middle), and
${\sigma}_{\rm mpc}$ (bottom)
on $M_{\rm B}$ in the simulation.
The values averaged for GCSs at each
of the six $M_{\rm B}$ bins are shown by triangles.
The $1\sigma$ dispersion
for each bin is shown by an error bar.}
\label{Figure. 18}
\end{figure}
\begin{table*}
\centering
\begin{minipage}{185mm}
\caption{Mean properties of the simulated GCSs.}
\begin{tabular}{cccccccccc}
{$<z_{\rm f,mpc}>$
\footnote{The mean formation epoch of all MPCs in the simulation.}}
& { $<z_{\rm f, mrc}>$
\footnote{The mean formation epoch
of all MRCs in the simulation.}}
& { $<f_{\rm bimo}>$
\footnote{The mean number fraction of galaxies with GCSs showing
clear bimodal MDFs.}}
& { $<f_{\rm mrc}>$
\footnote{The mean number fraction of MRCs in GCSs.}}
& { $<f_{\rm y}>$
\footnote{The mean number fraction of young GCs with ages
younger than 8 Gyrs in GCSs.}}
& { $<S_{\rm N}>$
\footnote{The mean $S_{\rm N}$ (GC number per unit luminosity)
in galaxies with GCs.}}
& { $<V_{\rm N}>$
\footnote{The mean $V_{\rm N}$ (GC number per unit halo mass)
in galaxies with GCs.}}
& { $<s_{\rm eff}>$
\footnote{The mean ratio of half-number radii of MRCs
to those of MPCs in GCSs.}}
& { $<s_{\rm dis}>$
\footnote{The mean ratio of velocity dispersions
of MRCs to those of MPCs in GCSs.}}
& { $<{\epsilon}>$
\footnote{The mean net formation efficiency of GCs
in galaxies.}}
\\
5.7 & 4.3 & 0.52 & 0.36 & 0.04 & 2.11 & 11.25 & 0.84 & 0.98 & 0.0013 \\
\end{tabular}
\end{minipage}
\end{table*}
Fig. 14 shows an ``U-shape'' distribution in the $S_{\rm N}-M_{\rm h}$
results for Es, although
error bars are quite large (up to $\sim 5$ for low-mass halos).
This U-shape distribution
means that $S_{\rm N}$ is higher in smaller $M_{\rm h}$
below the threshold halo mass ($M_{\rm h, th}$)
of ${\log}_{10} M_{\rm h,th}=10^{11} {\rm M}_{\odot}$
whereas it is higher in larger $M_{\rm h}$ above
$M_{\rm h, th}$.
This simulated distribution for Es appears to be broadly consistent with
the latest observation by Peng et al. (2008) for GCSs
in the Virgo cluster of galaxies.
This U-shape distribution can be also seen in spirals,
though it is not so clear in S0s.
The $S_{\rm N}-M_{\rm h}$ relation for halo masses smaller than $M_{\rm h, th}$
is due mainly to
the fact that $M_{\rm h}/L_{\rm B}$ depends strongly on $M_{\rm h}$
(or on $M_{\rm B}$).
The $S_{\rm N}-M_{\rm h}$ relation for halo masses larger than $M_{\rm h, th}$
is due mainly to the fact
that numbers of GCs per unit halo mass
are higher in more massive galaxies.
As shown in Fig 15,
the U-shape distributions are less well defined for
the $V_{\rm N}-M_{\rm h}$ results
of galaxies with different Hubble types.
However, $V_{\rm N}$ in galaxies with $M_{\rm h} > M_{\rm h, th}$
are significantly higher in Es, which means higher numbers of GCs per
unit halo mass in more massive Es.
More luminous spirals show higher $V_{\rm N}$ for
${\log} M_{\rm h} < 10^{11.5} {\rm M}_{\odot}$.
It is not so clear why S0s have $V_{\rm N}$ values
higher than those of Es and Sps at low masses
${\log} M_{\rm h} < 10^{10.5} {\rm M}_{\odot}$.
The results in Figs. 14 and 15 thus show that
$S_{\rm N}$ and $V_{\rm N}$ are fairly high
in Es located in the centers of massive halos with
${\log} M_{\rm h} \sim 10^{12.5} {\rm M}_{\odot}$.
These results suggest that giant elliptical galaxies
(and cDs) in the centers of massive groups and clusters of galaxies
have higher $S_{\rm N}$ and $V_{\rm N}$ than those in the field
and small groups of galaxies.
\subsection{Dynamical properties}
\subsubsection{Structures}
We next investigate the half-number radii of MPCs ($r_{\rm e,mpc}$)
and MRCs ($r_{\rm e,mrc}$) for GCs within the virial radii ($r_{\rm vir}$)
of host galaxy dark matter halos: it should be stressed that
the results below might well depend weakly
on the adopted initial distributions of GCs within their host halos
at high $z$.
The spatial distributions of MPCs have valuable information
of dark matter halos of their host galaxies (B07).
In order to avoid contributions
from intracluster and intragroup
GCs in groups and clusters of galaxies
with $r_{\rm vir}>100$ kpc,
we estimate $r_{\rm e,mpc}$ and $r_{\rm e,mrc}$
within 100 kpc for these groups and clusters.
We define $s_{\rm eff}$ to be:
\begin{equation}
s_{\rm eff}=\frac{r_{\rm e, mrc}}{r_{\rm e, mpc}}.
\end{equation}
We also define the power-law slope ($\gamma$) and
the coefficient ($D_{0}$) of the following relation:
\begin{equation}
r_{\rm e, mpc}=D_{0}{M_{\rm h}}^{\gamma} .
\end{equation}
Fig. 16 shows that $r_{\rm e, mpc}$ and $r_{\rm e, mrc}$ depend
on $M_{\rm B}$ such that both are larger for more luminous
galaxies. The mean values of $r_{\rm e, mpc}$ are larger than
10 kpc for $-22$ $<M_{\rm B}<$ $-14$. This is larger than the observed
the half-number radius of the Galactic GC system which is about
5 kpc (e.g., van den Bergh 2000). Although no observational
studies have so far investigated the correlation of $r_{\rm e,
mpc}$ with $M_{\rm B}$, the above results imply that the
simulation significantly overestimates $r_{\rm e, mpc}$: this
may be true for MRCs as well. In the present model, all GCs within
$r_{\rm vir}$ (virial radius) of the dark matter halo of their host
galaxy are used for estimation of $r_{\rm e, mpc}$. This way
of estimating $r_{\rm e, mpc}$ would contribute significantly to
the possible overestimation of $r_{\rm e, mpc}$ of GCSs in the
simulation.
We find $s_{\rm eff}$ is significantly less than 1 for a wide range of
luminosities, which means that the distributions of MRCs
are more compact than those of MPCs for most GCSs.
The mean $s_{\rm eff}$ is 0.84 in the present model,
which can not be currently compared with observations owing to the lack
of observational studies of $s_{\rm eff}$ in galaxies.
The smaller mean value of $s_{\rm eff}$
and the weak dependence of $s_{\rm eff}$ on $M_{\rm B}$
can be tested against future observational studies.
We also suggest that the slopes of $r_{\rm e, mpc}-M_{\rm B}$
and $r_{\rm e, mrc}-M_{\rm B}$ relations can be used to constrain
theoretical models of GC formation based on hierarchical clustering scenarios,
because these depend on merging histories of galactic building
blocks that form GCs.
Fig. 17 shows $r_{\rm e, mpc}$ of GCSs correlate
with their host halo mass such that $r_{\rm e, mpc}$
is larger for larger $M_{\rm h}$.
The $\chi$-square fit to the simulation data is:
\begin{equation}
{\log}_{10} (\frac { r_{\rm e,mpc} } { {\rm kpc} }) =-0.69
+0.18{\log}_{10} (\frac{M_{\rm h}}{ {\rm M}_{\odot} })
\end{equation}
which means $ r_{\rm e,mpc}=
0.20{( \frac{ M_{\rm h}} { {\rm M}_{\odot} } )}^{0.18}$ kpc
(i.e., $\gamma$=0.18).
The following equation would be more useful for
making an estimation of $M_{\rm h}$ for
a galaxy by measuring $r_{\rm e, mrc}$ of the GCS:
\begin{equation}
{\log}_{10} (\frac{M_{\rm h}}{ {\rm M}_{\odot} }) =
4.5+
5.2 {\log}_{10} (\frac { r_{\rm e,mpc} } { {\rm kpc} }).
\end{equation}
Since the present model
based on a dissipationless simulation
can overestimate $r_{\rm e, mpc}$,
it might be better to use observations
in order to determine the zero-point
of the above power-law relation.
If we use observations of the Galactic
GCS (e.g., van den Bergh 2000 and references therein) and the total
mass of the Galaxy (Wilkinson \& Evans 1999),
then the power-law relation is
$ r_{\rm e,mpc}=
5.0 {( \frac{ M_{\rm h}} { 2 \times 10^{12} {\rm M}_{\odot} } )}^{0.18}$ kpc
or
$M_{\rm h} = 2 \times 10^{12}
{(\frac{ r_{\rm e,mpc} } { {\rm 5 kpc} })}^{5.2}
{\rm M}_{\odot}. $
The derived $M_{\rm h}$
from $ r_{\rm e,mpc}-M_{\rm h}$ relations
can be compared with $M_{\rm h}$
derived from kinematics of GCSs and halo field stars
(e.g., Romanowsky 2006).
\subsubsection{Kinematics}
A growing number of observational data sets on the kinematical properties
of GCSs have been recently accumulated (e.g., Romanowsky 2006).
We investigate correlations
between velocity dispersions ($\sigma$)
for MPCs (${\sigma}_{\rm mpc}$) and MRCs (${\sigma}_{\rm mrc}$)
and $M_{\rm B}$ of their host galaxies.
We first estimate ${\sigma}_{\rm mpc}$ and ${\sigma}_{\rm mrc}$
of a GCS for each of the three projections
($x$-$y$, $x$-$z$, and $y$-$z$) by using line-of-sight-velocities
of all GCs within the virial radius of the halo.
We then make an average for the three projections
and determine one-dimensional velocity dispersions of
${\sigma}_{\rm mpc}$ and ${\sigma}_{\rm mrc}$.
We also investigate correlations between $M_{\rm B}$ and $s_{\rm dis}$,
where $s_{\rm dis}$ is defined as:
\begin{equation}
s_{\rm dis}=\frac{{\sigma}_{\rm mrc}}{{\sigma}_{\rm mpc}}.
\end{equation}
Fig. 18 shows that both ${\sigma}_{\rm mpc}$
and ${\sigma}_{\rm mrc}$ are higher in
more luminous galaxies.
The essential reason for this dependence is described as follows.
Since the GCSs investigated in the present study are for galaxies
located in the centers of dark matter halos
(i.e., GCSs in satellite galaxies are not investigated),
more luminous galaxies are more likely to be embedded in
more massive halos (i.e., larger $M_{\rm h}$).
GCs follow structures and kinematics of underlying
dark matter halos so that the mean velocity dispersions of GCSs
are determined mainly by masses of their halos
(and by their spatial distributions).
Therefore the velocity dispersions of GCSs are higher
in more luminous galaxies.
The difference in $s_{\rm dis}$ for different
$M_{\rm B}$ is quite small and the mean value of $s_{\rm dis}$ is 0.98.
These results imply that velocity dispersions are not so different
between MPCs and MRCs in galaxies.
Owing to the collisionless nature of the present simulation,
the ratios of maximum rotational velocities to central velocity
dispersions ($V/\sigma$) in GCSs appear to be
low ($V/\sigma <0.3$) for
most galaxies. These results are in a striking contrast with our
previous simulations (Bekki et al. 2005)
in which some GCSs in the remnants
of disk-disk major mergers show large $V/\sigma$ ($>0.5$).
Owing to the lack of extensive statistical studies on
$V/\sigma$ of GCSs in galaxies,
it is not clear whether the above results are generally consistent
with observations or not.
\subsection{Mean properties}
Lastly, we briefly summarize
the key physical properties of
GCs averaged over all GCSs in the simulation
in Table 4. The columns are the mean properties
of $z_{\rm f, mpc}$ (column 1),
$z_{\rm f, mrc}$ (2),
$f_{\rm bimo}$ (3),
$f_{\rm mrc}$ (4),
$f_{\rm y}$(5),
$S_{\rm N}$ (6),
$V_{\rm N}$ (7),
$s_{\rm eff}$ (8),
$s_{\rm dis}$ (9),
and $\epsilon$ (10).
In estimating these mean values,
we do not include galaxies with no GCs.
If we include galaxies with no GCs,
the mean values would be significantly
changed, in particular, for $S_{\rm N}$ and $V_{\rm N}$.
\begin{figure}
\psfig{file=f19.eps,width=8.0cm}
\caption{
Distributions of the simulated galaxies with GCs in the
$z_{\rm vir,gc}-B/T$ plane (upper) and
$z_{\rm vir,gc}-M_{\rm B}$ plane (lower)
where $z_{\rm vir,gc}$ is the mean $z_{\rm vir}$ of
building blocks hosting GCs in a galaxy.
Circles show mean values in $B/T$ and $M_{\rm B}$ bins.
The $1\sigma$ dispersion
for each bin is shown by an error bar.
No galaxies have $z_{\rm vir,gc} < 6$, because
formation of GCs in the present model is truncated for galaxies
with $z_{\rm vir}<6$.
}
\label{Figure. 19}
\end{figure}
\begin{figure}
\psfig{file=f20.eps,width=7.0cm}
\caption{
Distributions of the simulated galaxies with GCs in the
$M_{\rm h}/L_{\rm B}-M_{\rm B}$ plane,
where $L_{\rm B}$ is the total $B-$band luminosity of a galaxy,
for E (top),
S0 (middle), and
Sp (bottom).
Circles show mean values in $M_{\rm B}$ bins.
The $1\sigma$ dispersion
for each bin is shown by an error bar.
}
\label{Figure. 20}
\end{figure}
\section{Discussion}
\subsection{Bimodal colour distributions in GCSs}
The origin of the observed bimodal colour distributions of GCSs in
elliptical galaxies have long been discussed in the context
of different formation scenarios of elliptical galaxies,
such as two-phase collapse at high redshift (Forbes et al. 1997),
accretion and stripping of low-mass galaxies with GCs (C\^ote et al. 1998),
and gas-rich major mergers (Ashman \& Zepf 1992).
B02 was the first to investigate the bimodal colour distributions
in a quantitative manner based on the results of a semi-analytic
model of galaxy formation. B02 showed that the observed
bimodality can be reproduced, if MRCs are formed during
dissipative merger events at high redshift and that the formation
of MPCs are truncated at $z \sim 5$. B02 however did not
constrain the truncation mechanism.
Since age differences between MPCs and MRCs are rather small ($<1$ Gyr)
in the present simulation,
we can discuss the simulated MDFs of GCSs in terms of the colour bimodality
of GCSs.
In the present model, all GCs (i.e., both MPCs and MRCs) in a galaxy
originate from low-mass building blocks at $z$ $>3$,
whether they are isolated or in merging galaxies.
Therefore the origin of MRCs in a galaxy is not necessarily associated with
the past major merger events of a galaxy.
The peak metallicity in the MDF for MPCs (MRCs) in a galaxy
reflects the highest GCFR in the galaxy's building blocks
that have stellar metallicities lower (higher) than $-1$ at high $z$.
The mean metallicity of MPCs (MRCs) in a galaxy is determined by
the mean stellar metallicity of the more metal-poor building blocks
with ${\rm [Fe/H]} <-1$ (${\rm [Fe/H]} \ge -1$).
Although MRCs are formed later in more metal-rich building blocks of galaxies,
there is only a slight difference in formation redshifts between MPCs and MRCs.
The major merger events in which MRCs are formed are typically $z$ $>3$
whereas the last major merger events
which determine the final morphological types of galaxies
can happen later at lower redshifts.
Therefore the last major merger events responsible for elliptical galaxy formation
are not necessarily associated with the formation of MRCs
, which is in contrast to
the scenario proposed by Ashman \& Zepf (1992) but consistent
with the so-called `damp' merger interpretation of Forbes et
al. (2007).
In order to show the key requirements for the formation of bimodal
MDFs in GCSs,
we have run two comparative models with $\alpha=0.1$ and $\beta=0.05$
and $\alpha=0.02$ and $\beta=0.25$ (see section 2.3).
For these models, the dependences of the GCFR on $f_{\rm m}$ and $f_{\rm
g}$ are assumed to be very weak.
We find that GCSs in these models show much reduced MDF
bimodality in comparison with the present standard model
with $\alpha=0.02$ and $\beta=0.05$ shown in Table 1
(the results of the models are shown in the Appendix B).
These results clearly show that a strong enhancement in the GCFR
during violent merging (in particular, major merging)
and gas-rich phases (i.e., at high redshifts) is important
for the formation of the bimodality.
The origin of the metal-poor GC peaks is due mainly to the strong
enhancement of the GCFR in gas-rich and high gas-density
building blocks at high-$z$ whereas that of the metal-rich
ones is due mainly to the strong enhancement of the GCFR in more metal-rich
and gas-rich building blocks that experience violent merging
at somewhat later epochs.
The truncation of GC formation via reionization is necessary
for the present model in order
to prevent too many GCs from forming in low-mass galaxies.
The epoch of reionization, however, appears not to determine the
presence of bimodality itself. Although differences
in the merging histories of halos
between different galaxies can introduce a
diversity in GCS properties, such differences appear to have
no {\it direct} link with the resulting bimodal MDFs of the GCSs.
As shown in Fig. 8, not all of galaxies have GCSs with bimodal MDFs.
In order to understand why these galaxies have GCSs without
bimodality, we have investigated formation histories of GCs
in these galaxies (the results for some galaxies are shown
in the Appendix B). We find that these galaxies have only either metal-poor
or metal-rich peaks in their MDFs owing to single burst
events of GC formation.
Galaxies with no, or little, enhancement in the GCFR for metal-poor GCs
and strong enhancement of the GCFR for metal-rich GCs
have no clear bimodality in their MDFs.
On the other hand, those with strong enhancement of the GCFR
for metal-poor GCs at high $z$ and
no enhancement of the GCFR for metal-rich GCs
do not show the bimodality either.
The efficient formation of MPCs is highly unlikely for
galaxies with lower $z_{\rm vir}$ whereas
MRCs are highly unlikely to be
formed in gas-poor merging at later epochs.
The galaxies having GCSs without bimodal MDFs thus have
either lower $z_{\rm vir}$ or few events of gas-rich
merging and accretion of more metal-rich galactic building blocks.
The present model has shown that
a significant fraction ($f_{\rm bimo} \sim 0.2$) of very late-type spirals with
small or no bulges ($B/T<0.05$) have bimodal MDFs in their GCSs.
This is because more metal-rich (${\rm [Fe/H]} \ge -1$)
building blocks of late-type spiral galaxies can form GCs at high $z$
which are later accreted during mergers.
Therefore the origin of MRCs (and thus bimodal MDFs)
in late-type spiral galaxies are not necessarily associated with
the formation of galactic bulges via early major merger events
in the present model (see B02 for a brief discussion on this issue).
If most
bulge-less spirals are found to contain very few MRCs (as appears to be the case for
M33; Chandar et al. 2006)
then the model proposed here may require significant alteration.
\begin{table*}
\centering
\begin{minipage}{185mm}
\caption{ Qualitative assessment of model successes and failures.}
\begin{tabular}{cccc}
Items
& {Consistency \footnote{ $\bigcirc$ ($\times$) means
that simulations are broadly consistent (inconsistent) with observations.
$\bigtriangleup$ means that simulations are only partly
consistent with observations.}}
& Comments \\
Ages
& $\bigcirc$
& Data indicate very old ages for both MRCs and MPCs. \\
Bimodal MDFs
& $\bigcirc$
& Requires predicted MDFs in observed colours. \\
$f_{\rm bimo}-M_{\rm B}$
& $\bigcirc$
& Broadly consistent but data very limited. \\
$f_{\rm mrc}-M_{\rm B}$
& $\bigcirc$
& Consistent with data for luminous galaxies. \\
$f_{\rm y}$
& $\bigcirc$
& Broadly consistent but data very limited. \\
$M_{\rm B}-{\rm [Fe/H]}$
& $\bigcirc$
& Less consistent for MPCs. \\
$S_{\rm N}-B/T$
& $\bigcirc$
& Consistent in terms of higher $S_{\rm N}$ in Es. \\
$S_{\rm N}-M_{\rm B}$
& $\bigtriangleup$
& U-shape of data not clearly seen.\\
$V_{\rm N}-M_{\rm h}$
& $\bigtriangleup$
& Data show a flatter slope. \\
$r_{\rm e, mpc}$
& $\bigtriangleup$
& Data for the Galaxy is much smaller.\\
\end{tabular}
\end{minipage}
\end{table*}
\subsection{Specific Frequencies $S_{\rm N}$ and $V_{\rm N}$}
Previous observations have revealed that the $S_{\rm N}$ of GCSs
depends on the Hubble type and luminosity of the host galaxy
(e.g., Harris 1991; but see also Spitler et al. 2007 for an
alternative view). Although the observed trends
of $S_{\rm N}$ with galactic properties (e.g., the bimodality in
the $M_{\rm V}-S_{\rm N}$ relation) are well reproduced by B06,
other key observations of $S_{\rm N}$ have not been discussed.
In the present study, we have shown that (i) $S_{\rm N}$ of
GCSs in luminous galaxies with $M_{\rm B}<-19$ mag are
significantly higher in early-type (E/S0) galaxies than late-type
(Sp) ones, (ii) S0s typically have higher $S_{\rm N}$ than
spirals, and (iii) low-luminous galaxies with $M_{\rm B}> -15$
mag have higher $S_{\rm N}$, regardless of their Hubble types.
These results are qualitatively consistent with previous
observations by Harris (1991), Forbes (2005), and
Arag\'on-Salamanca et al. (2006). So far we have not
discussed the physical reasons for the above dependences of
$S_{\rm N}$ on galactic properties.
Fig. 19 shows that the mean values ($z_{\rm vir,gc}$)
of virialization redshifts ($z_{\rm vir}$)
of galactic
building blocks that form GCs
are significantly higher in early-type galaxies with $B/T>0.8$
than in late-type ones.
This means that GCFRs in building blocks of early-type
galaxies are higher owing to their higher mass-densities
with higher $z_{\rm vir}$.
Because GCFRs depends strongly on $z_{\rm vir}$
through the term of $F_{\rm z}$ (see equation 6) the higher $z_{\rm vir,gc}$
in early-type galaxies is one reason for the observed higher $S_{\rm N}$
of their GCSs.
Furthermore Fig. 19 shows that $z_{\rm vir,gc}$ is higher in more luminous
galaxies, which can explain why more luminous early-type galaxies
with $M_{\rm B}<-19$ mag are more likely to have higher
$S_{\rm N}$ in comparison with less luminous ones with
$-19.0$ $ < M_{\rm B} < -17$ mag seen in Fig. 12.
Fig. 19 shows that low-luminosity galaxies do not have higher
$z_{\rm vir,gc}$, which implies that higher $S_{\rm N}$ in
low-luminosity galaxies seen in Fig. 9 can not be understood in
terms of $z_{\rm vir,gc}$. B06 suggested that the origin of the
observed higher $S_{\rm N}$ in low-luminosity galaxies is due to
higher mass-to-light-ratios ($M/L \approx M_{\rm h}/L_{\rm B}$)
of these galaxies. Fig. 20 shows that $M_{\rm h}/L_{\rm B}$
steeply depends on luminosity in the sense that $M_{\rm
h}/L_{\rm B}$ is higher in less luminous galaxies. Given that GC
numbers per unit halo mass ($V_{\rm N}$) in galaxies only weakly
depend on the halo masses of the galaxy (B06; see also Fig.13
in the present paper), the above result implies that the higher
$S_{\rm N}$ in low-luminosity galaxies are due mainly to their higher
$M_{\rm h}/L_{\rm B}$ ratios owing to the relation $S_{\rm N}=V_{\rm
N} (\frac{ M_{\rm h} }{ L_{\rm B} })$.
\subsection{Metallicity-luminosity relations}
Although B02 predicted that the mean colours of both MPCs and MRCs
only weakly correlate with the total luminosities of their host galaxies,
these predictions are inconsistent
with latest observations by Strader et al. (2004) and P06.
The present model has successfully reproduced reasonably well,
the observed ${\rm [Fe/H]} - M_{\rm B}$ relations
for MRCs and all GCs.
The derived slope in
the ${\rm [Fe/H]} - M_{\rm B}$ relation for MPCs in the present model
is, however, not consistent with the observed one.
The differences between
B02 and the present study are caused by the differences
in the adopted models of GC formation
between these two studies.
Thus more sophisticated models of GC formation in galaxies
at high $z$ need to be developed so that all
three ${\rm [Fe/H]} - M_{\rm B}$ relations (MPCs, MRCs, and all GCs)
can successfully reproduce the observations.
B07 showed that if $z_{\rm trun} \approx 10$, the observed ${\rm
[Fe/H]} - M_{\rm B}$ relation for MPCs is better reproduced.
However the simulated $\epsilon$ (i.e., the formation efficiency
of GCs) in the model with $z_{\rm trun} =10$ in the present study
is too small ($0.00077$) to be consistent with the observed one
by McLaughlin (1999). The present model seems to have
difficulties in explaining self-consistently both the observed
${\rm [Fe/H]} - M_{\rm B}$ relation for MPCs and $\epsilon$. One
possible way to solve this problem would be to investigate models
with higher $z_{\rm trun}$ (=$6-10$) in which $C_{\rm eff}$
depends more strongly on $z$ than in the present model so enough GCs
can be formed at $z=6-10$. It is, however, unclear
whether the models with higher $z_{\rm trun}$ and $C_{\rm eff}$
can reproduce other key observations such as the $S_{\rm N}-M_{\rm
B}$ relation.
\subsection{Success and failures of the present model}
Although we have so far presented variously different physical properties
of GCSs in galaxies,
only some of them can be directly compared with observations: for example,
the simulated
$f_{\rm mrc}-M_{\rm B}$ relation can be compared with observations by P06
whereas the simulated $s_{\rm dis}-M_{\rm B}$ one can not owing to
the lack of observational data.
It is, however, important to check whether the simulated properties
are consistent with the corresponding observations.
In Table 5 we summarise what we regard as the relative successes
and failures of the current model in a qualitative sense. For
any given physical property there are
subtle, and sometimes large, differences between the
observational data and the corresponding property
predicted by the model.
The references of these observational properties are already given
in the Introduction section \S 1 (e.g., Brodie \& Strader 2006)
and some of the observational data are presented within figures
of this paper.
Less consistent results (e.g., the $M_{\rm B}-{\rm [Fe/H]}$
relation for MPCs) imply that a more sophisticated model for
GC formation would be required for more successful modeling.
\section{Conclusions}
We have investigated the structural, kinematical, and chemical properties
of globular cluster systems (GCSs) in galaxies
in a self-consistent manner based on
high-resolution cosmological N-body simulations combined with
a semi-analytic model of galaxy
and globular cluster (GC) formation.
We have adopted a number of assumptions on formation efficiencies
of GCs which depend on the physical properties of their host galaxies
(e.g., gas mass fraction).
We investigated
correlations between
physical properties of GCSs and those of their host galaxies
for $\sim 10^5$
simulated central halo galaxies for MPCs with ${\rm [Fe/H]} <-1$
and MRCs with ${\rm [Fe/H]} \ge -1$.
We find:
(1) The majority ($\sim$ 90\%) of GCs currently
in halos of galaxies are formed in low-mass galaxies at $z > 3$
with mean formation redshifts of MPCs and MRCs being 5.7 and 4.3,
respectively. This corresponds to 12.7 and 12.3 Gyrs in lookback
time.
The majority of MPCs are formed in low-mass galaxies that are virialized
well before reionization ($z>6$) and thus have higher mass densities.
MRCs are formed slightly later not only within high $z$ major mergers between
high-density galaxies but also within
gas-rich isolated ones.
(2) About 50 \% of galaxies with GCs show clear bimodalities in
their MDFs,
though less luminous galaxies with $M_{\rm B}$ fainter than $-17$
are much less likely to show the bimodalities owing to
no or few MRCs. The
age differences between MPCs and MRCs are quite small ($<$ 1 Gyr)
in most galaxies.
The origin of the simulated bimodality in MDFs is due to
strong dependences of GCFE on $f_{\rm g}$
and $f_{\rm m}$ (i.e., higher GCFE in more gas-rich high-z
galaxies and in mergers with larger mass ratios of the merging two
galaxies).
(3) The number fractions of MRCs ($f_{\rm mrc}$)
range from 0 to
almost 1 with an average of $0.4$.
The $f_{\rm mrc}$ in galaxies does not depend on their Hubble type
(i.e., bulge-to-disk-ratios), which implies that
the formation of MRCs is not necessarily associated with
bulge formation.
The $f_{\rm mrc}$ are likely to be smaller for less luminous galaxies.
(4) The $S_{\rm N}$ of GCSs are typically
higher in ellipticals than in spirals,
and in low-luminosity galaxies with $M_{\rm B}>-15$
regardless of their Hubble types.
The mean $S_{\rm N}$ for Sp, S0, and E populations
are 1.8, 2.0, and 4.0, respectively.
(5) The number of GCs per halo mass ($V_{\rm N}$) does
not depend as strongly on
the luminosity or the Hubble type of the host galaxy
as $S_{\rm N}$ does,
which suggests that the GC number per unit mass
is similar between different galaxies.
$V_{\rm N}$ is, however, likely to be higher
in luminous ellipticals with $M_{\rm B}<-20$.
(6) Although there are no significant
differences in $V_{\rm N}$
between spirals and S0s for luminous galaxies
($M_{\rm B} <-20$),
S0s are more likely to have higher
$V_{\rm N}$
than spirals for less luminous galaxies with $M_{\rm B} > -18$.
These results suggest that
only luminous S0s with moderately
high $S_{\rm N}$ can be transformed spirals via truncation
of star formation and the resultant disk fading,
because $V_{\rm N}$ does not change during disk fading.
(7) The mean metallicities of GCs ([Fe/H]) for MPCs and MRCs
depend on $M_{\rm B}$ of their host galaxies
such that they are higher in more luminous galaxies,
though the dependence for MPCs is weak.
Although the observed correlation of $M_{\rm B}-{\rm [Fe/H]}$
for MRCs
can be well reproduced by the present model,
that for MPCs
is not consistent with the simulated one.
The observed correlation between
the mean metallicities of all GCs and $M_{\rm B}$
can be well reproduced by the present model.
(8) Spatial distributions of MRCs are more compact than those of MPCs
with $r_{\rm e,mrc} \sim 0.84 r_{\rm e, mpc}$ .
The $r_{\rm e,mpc}$ depends strongly on halo mass $M_{\rm h}$ such that
$r_{\rm e,mpc} \propto {M_{\rm h}}^{0.18}$
(or $M_{\rm h} \propto {r_{\rm e}}^{5.2}$)
which implies that $r_{\rm e,mpc}$ can be used for estimating
the total masses of dark matter halos.
(9) There is no significant difference in velocity dispersion
between MPCs and MRCs. Velocity dispersions are larger
in more luminous galaxies both for MPCs and MRCs regardless of
their Hubble types.
(10) The physical properties of GCSs such as MDFs, $f_{\rm mrc}$,
$S_{\rm N}$, and $V_{\rm N}$ are quite diverse between different galaxies,
depending on the virialization
redshifts of their building blocks and subsequent merging, star formation,
and chemical evolution histories of the building blocks.
\section*{Acknowledgments}
We are grateful to the anonymous referee for valuable comments,
which contribute to improve the present paper.
K.B. and D.A.F. acknowledge the financial support of the Australian Research
Council throughout the course of this work.
H.Y. acknowledges the support of the research fellowships of the Japan
Society for the Promotion of Science for Young Scientists (17-10511).
MN was supported by the Grant-in-Aid for the Scientific Research Fund
(18749007) of the Ministry of Education, Culture, Sports, Science and
Technology of Japan and by a Nagasaki University president's Fund
grant.
We are grateful to Lee Spitler for providing observational
data and discussing the present results with us.
The numerical simulations reported here were carried out on
Fujitsu-made vector parallel processors VPP5000
kindly made available by the Center for Computational Astrophysics (CfCA)
at National Astronomical Observatory of Japan (NAOJ)
for our research project why36b and uhy09a.
|
1,108,101,566,733 | arxiv | \section{Introduction}
Ads has been contributing to more than $90\%$ of revenue for companies
as Google, Facebook, \textit{etc.},
therefore improving the performance deserves lots of time and efforts,
not only for advertiser themselves, but also for various platform providers as Google.
While advertisers have been spending lots of efforts generating ads with higher quality,
it is never satisfying enough with multiple reasons.
It's subtle to find out the difference of two ads,
especially when the similarity is high,
humans can hardly find out the minor performance difference,
especially without a through understanding of users,
even when they're expressing strong interests in their products.
Also it suffers from the notorious scaling issue,
since ads human writers' efforts do \textit{not} scale up as machines in the modern era of big data.
\section{Methodology}
\subsection{Formulation of the problem}
For each ads group, advertisers could specify a set of keywords to be matched:
\begin{equation}
\label{eq:keywords}
\bw \equiv ( \cdots w_i \cdots), i = 0, 1, \cdots W- 1
\end{equation}
\noindent
and provide a set of creatives:
\begin{equation}
\label{eq:creatives}
\bc \equiv ( \cdots c_j \cdots), c = 0, 1, \cdots N- 1
\end{equation}
With those two as the two dimensions we define the clicks matrix as below:
\begin{eqnarray}
\label{eq:clicks}
\bK \equiv
\left(
\begin{array}{cccc}
\cdots & \cdots & \cdots \\
\cdots & K_{ij} & \cdots \\
\cdots & \cdots & \cdots
\end{array}
\right)
\end{eqnarray}
Similarly we could define the $\pCTR$ matrix.
Given the limit on number of creatives for each ads group,
there could only be $M$ creatives to be the final candidates,
i.e. it is to select a subset of $M$ elements from $\bc$:
\begin{equation}
\bd= (\cdots \bd_{a} \cdots), a = 0, 1, \cdots M-1
\end{equation}
to maximize the number of clicks, with goal function:
\begin{equation}
G(\bd) = \sum\limits_i \mbox{max}(\{K_{ia}, a = 0 \cdots M-1\})
\end{equation}
The shorthand notation of Greedy Algorithm $G(W, N, M)$
and exact solution of $E(W, N, M)$ will be used through this paper.
The main assumption is that
for each keyword request from external user,
there could be only one creative to be selected,
and we require it to be \textit{the} one with the largest $\pCTR$,
so as to maximize revenue.
\subsection{Greedy Algorithm}
Greedy Algorithm has been a popular approach for many practical problems,
in some cases it proves to be the optimal solution,
while in most cases it can only provide a \textit{local} optimal solution,
though it is decent enough and
could be a very good approximation to the exact optimal solution.
Greedy Algorithm with size $M$ is similar to mathematical induction,
starting with the same problem with smallest size, normally $0$ or $1$,
one can go one step further for each iteration,
and after $M$ steps it solves the original problem,
though not guaranteed to be \textit{globally} optimal.
Greedy Algorithm for our problem works as the following:
\begin{enumerate}
\item $a = 1$: select creative $\mbox{argmax}_j \sum\limits_i K_{ij}, j \in [0, N-1]$,
i.e. the column with the largest sum over all keywords;
\item $\cdots$
\item $a = a$: Select the creative $\mbox{argmax}_j G(\bd \cup
\{c_{j}, j \in [0, N - 1], c_j \notin \bd\}
)$;
\item $\cdots$
\item Until $M$ creatives are selected.
\end{enumerate}
\subsection{Greedy-Power Algorithm: $G^n(r, f; W, N, M)$}
Assuming that Greedy Algorithm is a decent approximation to the exact solution $\bdd$,
it is likely that the difference between those two sets are small,
i.e. the number of creatives in the difference set is $m$, which is much smaller than $M$.
In other words, starting with a solution from Greedy Algorithm,
one only needs to perform a minor tuning and to replace a few creatives if ever necessary,
when greedy algorithm is different from that exact solution $\bdd$.
Here we introduce three parameters,
$r$ as the number of creatives removed from the original solution,
$f$ as the number of creative subsets as starting points,
and $n$ as the number of iterations on top of existing solution.
$r$: With $r$ creatives removed from the original Greedy solution,
one gets a new starting point and
can perform another round of greedy algorithm with problem size $M- r \to M$.
It has the same complexity to the original Greedy Algorithm with a factor of $\pfrac{r}{M} < 1$.
Note that the two solutions can be different by at most $r$ creatives.
$f$: Removing $r = 1, 2, 3 \cdots$ creative has
$M, M (M-1)/2, M(M-1)(M-2)/6 \cdots$ options respectively.
When $r = 1$, the cost is essentially the same to original Greedy Algorithm run with $M$ iterations,
since here it is $M$ runs with \textit{one} single iteration for each run.
When $r > 1$, there could be a few options:
\begin{itemize}
\item Systematic approach to enumerate all combinations, with cost increasingly quickly with $r$,
and much more expensive than the original Greedy Algorithm solution;
\item By sampling $f$ unique combinations from all available options,
one can keep the cost comparable to original one.
\end{itemize}
The $r> 1$ option could be better than $r= 1$, since it has the ability to remove multiple elements together,
while the latter option can at most replace $1$ creative.
A simple calculation for a solution with elements $c_x$ and $c_y$ and an even cut,
i.e. $r = M / 2$ and $M$ is even, the probability of $c_x$ and $c_y$ not being in the same half is:
\begin{equation}
P(\{c_x\}, \{c_y\})= \pfrac{C_{M-2}^{r- 1}}{C_{M-1}^{r-1}} = \pfrac{M-r}{M-1} \sim \pfrac{1}{2}
\end{equation}
At the same time, the probability of $c_x$ and $c_y$ in the same half is also roughly $\pfrac{1}{2}$,
so that they could be simultaneously removed when necessary.
Therefore, a larger $r$ has much more flexibility than a smaller $r$.
$n$: With $r$ creatives replaced by the \textit{2nd} round of Greedy Algorithm,
one can ask the same question again,
can we do better than the current solution?
Actually the question is exactly the same to what we have been asking with a Greedy Algorithm in the first step.
In fact, the new solution is from Greedy Algorithm as well,
and essentially no different from its counterpart in the \textit{first} round.
As long as the goal function keeps increasing,
one can continue this process,
until goal function is the same to the previous iteration,
which implies that we're \textit{not} able to go further,
and it is equivalent to $n = \infty$ in this case.
With those $3$ parameters,
we'd use the notation $G^n(r, f; W, N, M)$ for our proposed Greedy-Power Algorithm.
\section{Simulation}
With the Greedy-Power Algorithm $G^n(r, f; W, N, M)$,
we're able to run some simulations comparing against its baseline Greedy Algorithm.
\subsection{Simulation Setup}
\label{subsec:setup}
\begin{itemize}
\item Matrix elements are generated from normal distribution and then take their absolute value;
\item Matrix sizes are at the order of $50 \times 500$ with $M <= 10$;
\item By default, all simulations are ran multiple times with trajectories
$T= 500$ for each simulation, repeating $3$ times.
\end{itemize}
\subsection{Simulation Results and Discussion}
\subsubsection{$G^2(r, \cdot; \cdots)$}
\label{subsec:benchmark}
Note that as $r$ increases,
the computational cost increases by a factor of $r$,
with $r$ iterations compared with one single iteration when $r= 1$.
Strictly speaking, we are supposed to compare performance of
$G^n(1, M; W,N,M)$ vs
$G^n(r, M/r; W,N,M)$,
since the computational cost match.
However, the search space size increases exponentially with $r$,
while a factor of $\pfrac{1}{r}$ would effectively remove more options,
resulting in a quickly decreasing coverage ratio in the search space.
Therefore, we'd remove the $\pfrac{1}{r}$ factor
and keep the default value $f= M$ unless otherwise noted,
i.e.\ run same number of creative subsets from the starting solution obtained from the Greedy Algorithm.
\begin{table}[!ht]
\begin{tabular}{|l|l|l|lll}
\hline
\# & Matched $(\%)$ & Improvement $(\%)$\\
\hline
$1$ & $31.40$ & $1.19$\\
\hline
$2$ & $30.20$ & $1.13$\\
\hline
$3$ & $28.80$ & $1.14$\\
\hline
\end{tabular}
\caption{$G^2(1; 30, 300, 6)$ simulation results.}
\label{tab:base:r1}
\end{table}
\begin{table}[!ht]
\begin{tabular}{|l|l|l|lll}
\hline
\# & Matched $(\%)$ & Improvement $(\%)$\\
\hline
$1$ & $25.60$ & $1.34$\\
\hline
$2$ & $28.60$ & $1.43$\\
\hline
$3$ & $28.80$ & $1.44$\\
\hline
\end{tabular}
\caption{$G^2(2; 30, 300, 6)$ simulation results.}
\label{tab:base:r2}
\end{table}
\begin{table}[!ht]
\begin{tabular}{|l|l|l|lll}
\hline
\# & Matched $(\%)$ & Improvement $(\%)$\\
\hline
$1$ & $26.20$ & $1.71$\\
\hline
$2$ & $24.80$ & $1.61$\\
\hline
$3$ & $23.60$ & $1.51$\\
\hline
\end{tabular}
\caption{$G^2(3; 30, 300, 6)$ simulation results.}
\label{tab:base:r3}
\end{table}
From those $G^2(r; 30, 300, 6) \equiv G^2(r, 6; 30, 300, 6)$ simulation data with $r= 1, 2, 3$ in
\tabRefThree{tab:base:r1}{tab:base:r2}{tab:base:r3},
one can clearly see that as $r$ increases,
the likelihood to go out of the local optimum increases,
as the matched ratio decreases.
Also note that increasing $r$ has another benefit,
among those trajectories going out of the optimum,
the overall improvement compared with the benchmark $G(W, N, M)$ also increases.
Therefore, one is confident that the computational cost of
an extra and simple factor $r$ is worthwhile compared with $r= 1$.
\subsubsection{$G^2(\cdot\ $f$; \cdots)$}
\begin{table}[!ht]
\begin{tabular}{|l|l|l|lll}
\hline
\# & Matched $(\%)$ & Improvement $(\%)$\\
\hline
$1$ & $17.80$ & $1.71$\\
\hline
$1$ & $18.20$ & $1.81$\\
\hline
$1$ & $19.40$ & $1.79$\\
\hline
\end{tabular}
\caption{$G^2(3, 12; 30, 300, 6)$ simulation results.}
\label{tab:f2:r3}
\end{table}
\begin{table}[!ht]
\begin{tabular}{|l|l|l|lll}
\hline
\# & Matched $(\%)$ & Improvement $(\%)$\\
\hline
$1$ & $16.20$ & $1.89$\\
\hline
$2$ & $16.20$ & $1.90$\\
\hline
$3$ & $15.00$ & $1.87$\\
\hline
\end{tabular}
\caption{$G^2(3, 18; 30, 300, 6)$ simulation results.}
\label{tab:f3:r3}
\end{table}
Note that $r = 1$ has only $M$ candidates in the search space,
while they're all covered by the default choice of $f= M$,
to match the computational cost of a conventional Greedy Algorithm.
\tabRefThree{tab:base:r3}{tab:f2:r3}{tab:f3:r3} present the extra value
one can get with $f= M, 2M, 3M$ for $r= 3$.
While $f = M \to 2M$ improves the probability
to get of local optimum significantly by close to $10\%$,
the value added on average is not that impressing.
At the same time, $f= 2M \to 3M$ shows less value,
with a minor improvement of both metrics,
which implies the selection of $f$ is more of an art,
as the trade-off between computational cost and added value.
\subsubsection{$G^2(\cdots; \cdots M)$}
\begin{table}[!ht]
\begin{tabular}{|l|l|l|lll}
\hline
\# & Matched $(\%)$ & Improvement $(\%)$\\
\hline
$1$ & $11.00$ & $0.94$\\
\hline
$2$ & $12.00$ & $0.93$\\
\hline
$3$ & $10.40$ & $0.96$\\
\hline
\end{tabular}
\caption{$G^2(1; 30, 300, 10)$ simulation results.}
\label{tab:M10:r1}
\end{table}
\begin{table}[!ht]
\begin{tabular}{|l|l|l|lll}
\hline
\# & Matched $(\%)$ & Improvement $(\%)$\\
\hline
$1$ & $ 9.80$ & $1.24$\\
\hline
$2$ & $11.00$ & $1.27$\\
\hline
$3$ & $10.40$ & $1.19$\\
\hline
\end{tabular}
\caption{$G^2(3; 30, 300, 10)$ simulation results.}
\label{tab:M10:r3}
\end{table}
Comparing with $M= 10$ vs $M= 6$ data as shown in
\secRef{subsec:benchmark}
with the same $r= 1, 3$,
\tabRef{tab:base:r1} vs \tabRef{tab:M10:r1} and
\tabRef{tab:base:r3} vs \tabRef{tab:M10:r3},
it shows that when $M$ increases,
the $r= 1$ option $3$ times less likely to be trapped at local optimum,
which also implies that Greedy Algorithm is very unlikely to be the global optimum $P^{\mbox{\tiny opt}}_{\mbox{\tiny Greedy}}< 10\%$.
Comparing with $r = 1$ vs $r = 3$ results for $M= 10$ in
\tabRefTwo{tab:M10:r1}{tab:M10:r3},
again it shown that a larger $r$ shows extra value to improve
both the probability of getting out of the local optimum and
the ability to find a better optimum based on the simple Greedy Algorithm.
\subsubsection{$G^2(\cdots; W, N \cdots)$}
\begin{table}[!ht]
\begin{tabular}{|l|l|l|lll}
\hline
\# & Matched $(\%)$ & Improvement $(\%)$\\
\hline
$1$ & $33.00$ & $0.58$\\
\hline
$2$ & $34.00$ & $0.59$\\
\hline
$3$ & $37.80$ & $0.58$\\
\hline
\end{tabular}
\caption{$G^2(1; 100, 300, 6)$ simulation results.}
\label{tab:K100:r1}
\end{table}
\tabRefTwo{tab:base:r1}{tab:K100:r1} are different by $3$ times on row size,
i.e. the number of keywords space.
Simulation suggests that both metrics get worse when $K$ increases.
$P_{\mbox{\tiny trapped}}$ increases only a little bit,
implying a Greedy Square Algorithm is slightly more likely to be trapped locally under a larger keywords space.
Also the gain from the Greedy Square Algorithm is getting worse, reduced by a factor of $2$.
This is kind of expected,
since each minimal iteration to add one more creative is exact in the keyword dimension,
while the uncertainty lies mostly if not all in the creatives dimension.
In this case,
one might consider searching for a larger space,
possibilities are to increase $r$, $f$, i.e. the size of the search space.
\begin{table}[!ht]
\begin{tabular}{|l|l|l|lll}
\hline
\# & Matched $(\%)$ & Improvement $(\%)$\\
\hline
$1$ & $25.60$ & $1.12$\\
\hline
$2$ & $25.60$ & $1.25$\\
\hline
$3$ & $28.00$ & $1.22$\\
\hline
\end{tabular}
\caption{$G^2(1; 30, 1000, 6)$ simulation results.}
\label{tab:N1000:r1}
\end{table}
\tabRefTwo{tab:base:r1}{tab:N1000:r1} are different by $3$ times on column size,
i.e. the number of creatives space.
Simulation suggests that both metrics improve when $N$ increases,
an indication that a Greedy Square Algorithm is more likely to improve under a larger creatives space.
\subsubsection{$G^n(\cdots; \cdots)$}
Given the solution $g^1$ from conventional Greedy Algorithm,
we should be able to run another round of Greedy Algorithm on top it,
resulting in another solution $g^2$ from the Greedy Square Algorithm.
Given the fact that $g^2$ is a solution resulting from the Greedy Algorithm,
there's essentially little or even no difference vs $g^1$.
That's, starting with the Greedy Algorithm solution in iteration $i$, $g^i$,
one can always run another round of Greedy Algorithm to obtain a solution $g^{i+1}$.
There might be two outcomes:
\begin{itemize}
\item $g^{i+ 1}$ is worse than $g^i$, keep $g^i$ and stop here;
\item $g^{i+ 1}$ is equivalent to (starting in a new direction \textit{could} find something new)
or better than $g^i$, keep $g^{i+ 1}$;
\end{itemize}
Note that when the stopping rule of
$g^{i+ 1}$ being worse than $g^i$ is triggered,
$G^n$ is equivalent to $G^{\infty}$,
which would be equivalent to the exact solution ideally,
though it \textit{could} be trapped somewhere in a local optimum in principle.
\section{Conclusions}
We present a Generalized Greedy Algorithm,
i.e.\ the Greedy-Power Algorithm $G^n(r, f; W,N,M)$,
characterized by three parameters:
number of creatives to remove $r$ from a given solution implying a difference of up to $r$ creatives from original solution,
number of branches to take for a given solution $f \sim M$,
and number of iterations for improvement $n$ implying a Greedy Algorithm with a power of $n$.
With $f \sim M$,
we effectively impose that any following improvement step
should have the same complexity of a conventional Greedy Algorithm,
up to a factor of $r$,
and achieve solutions consistently better than the latter.
As such, with twice (and an additional factor of $r$) the cost of a Greedy Algorithm,
one is likely to achieve decent added value from the Greedy-Power Algorithm,
especially when the dimension $M$ is large,
when a Greedy Algorithm is more likely to be trapped at a local optimum,
and corrected by the Greedy-Power Algorithm.
\subsection{Future Directions}
The setup of the Greedy-Power Algorithm $G^n(r, f; W, N, M)$ is all done,
while the effectiveness with real data is still under exploration,
and it'd be interesting to see any significant difference of improvement
from the random standard normal distribution.
Also, it would provide more insights on the performance of Greedy Algorithm,
and how it walks to the global optimum with a big $n$,
which would be equivalent to the $n\to \infty$ when there is no more gain from the goal function.
At the same time,
there could be other approaches for this problem setup,
and one alternative is to find the exact solution is completely from a matrix perspective,
which will be discussed in a separate paper in the near future.
\section{References}
|
1,108,101,566,734 | arxiv | \section{Introduction}
\label{sec:intro}
Time averaging is a basic yet essential tool in fluid dynamics because of the ubiquity of phenomena involving multiple time scales. Atmospheric and oceanic flows, for instance, can often be decomposed usefully into fast and slow components \citep[e.g.][]{vallis2017atmospheric,vanneste2013balance}. Time averaging is then used to interpret numerical-simulation and observational data and to assimilate the latter in ocean and weather models. Arguably, all numerical simulations of these flows rely implicitly on time averaging since they do not resolve phenomena on time scales shorter than the time step.
Time averaging can be performed in different ways. The most straightforward way is to average the time series of flow variables at fixed spatial positions to obtain the so-called \textit{Eulerian mean}. An alternative is to average flow variables along particle trajectories, that is, at fixed particle labels instead of fixed positions, to obtain the \textit{Lagrangian mean}. There are several practical and conceptual reasons to view Lagrangian averaging as superior to Eulerian averaging.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Vorticities_new.eps}
\caption{Vorticity field and its Lagrangian and Eulerian means in the shallow-water simulation of \S\ref{sec:num_strategy2}: (a) instantaneous vorticity, (b) Lagrangian mean vorticity, and (c) Eulerian mean vorticity. The mean fields are averaged over $3.6$ wave periods. Panels (b) and (c) share the same colourbar shown to the right of panel (c).}
\label{fig:vorticities}
\end{figure}
From a practical viewpoint, the advection of fast motion (often consisting of waves) by slowly evolving flow and vice versa adversely affect their Eulerian decomposition. For instance, a strong, slowly evolving flow `Doppler shifts' the frequency of fast waves. This can lead to wave frequencies observed at fixed spatial locations that are much smaller than the intrinsic frequency. It therefore obscures the time-scale separation between waves and flow. Lagrangian averaging resolves this issue, as clearly demonstrated in several studies \citep{nagai2015spontaneous,shakespeare2017spontaneous,bachman2020particle,shakespeare2021new,jones2022separating}. Conversely, the advection of a slow background flow by waves leads to flow features that are blurred by Eulerian averaging but not by Lagrangian averaging.
We illustrate this in figure \ref{fig:vorticities} showing the vorticity field in a simulation of a turbulent rotating shallow-water flow
interacting with a mode-1 Poincar\'e wave. The instantaneous vorticity in panel (a) is dominated by the high-amplitude wave. Both the Lagrangian and Eulerian means (panels (b) and (c) respectively) filter out the wave, but the Eulerian mean blurs out
fine vorticity structures which are well resolved by the
Lagrangian mean (details of this simulation are presented in \S \ref{sec:num_strategy2}).
From a conceptual viewpoint, Lagrangian averaging provides a powerful tool to study wave--mean-flow interactions. This is because the material conservation of key fields (scalar concentrations, vorticity vector, circulation, potential vorticity) is naturally inherited by the corresponding Lagrangian mean fields. As a result, the Lagrangian mean of the dynamical equations is often simpler and more meaningful than the Eulerian mean \citep{sowa72,bretherton1971general,andrews1978exact,buhler2009waves,gilbert2018geometric}. Relatedly, the Lagrangian mean emerges naturally in the asymptotic derivation of wave--averaged models \citep{grimshaw1975nonlinear,wagner2015available}. A striking example of the dynamical relevance of the Lagrangian mean is provided by the observation that geostrophic balance, the dominant balance in rapidly rotating flows, continues to hold in the presence of strong waves provided it is formulated in terms of Lagrangian mean velocity and pressure instead of their instantaneous or Eulerian mean values \citep{moore1970mass,buhler1998non-dissipative,kafiabad2021wave,GBLApaper}.
Despite its advantages, Lagrangian averaging is not widely used as a practical tool, mainly because Lagrangian means are difficult to compute numerically. Most numerical models are intrinsically Eulerian and provide the fields of interest at fixed spatial locations, typically grid points. The standard approach for the computation of Lagrangian means is then to seed a large number of passive particles in the flow, track them (forward or backward in time) using interpolated velocities, and apply time averaging to the resulting Lagrangian time series \citep[e.g.][]{nagai2015spontaneous,shakespeare2017spontaneous,shakespeare2018life,shakespeare2019momentum,shakespeare2021new,bachman2020particle,jones2022separating}. This has a high computational cost, requires a large memory allocation, suffers from possible particle clustering and, as discussed in \cite{GBLApaper}, is difficult to parallelise efficiently (see \citet{shakespeare2021new} for a parallel implementation).
To circumvent the difficulties of particle tracking, \citet{GBLApaper} developed a grid-based method that computes the Lagrangian mean directly on an Eulerian grid, building the mean through a time step iteration carried out over successive averaging intervals. By eliminating the need to compute explicit particle trajectories, the method reduces memory demands and simplifies integration into parallelised numerical models. The present paper starts with the recognition that the algorithm of \citet{GBLApaper} is a particular discretisation of a PDE governing the evolution of what we term \textit{partial Lagrangian mean}, that is, the mean carried out only up to some intermediate time in the averaging interval. We formulate this PDE using the position of particles at the intermediate time as independent spatial variable, as in \citet{GBLApaper}. The (total) Lagrangian mean is then obtained by taking the intermediate time to be the end of the averaging interval.
In this form, the Lagrangian mean does not match \citeauthor{andrews1978exact}'s (\citeyear{andrews1978exact}) definition of the generalised Lagrangian mean (GLM): this requires the mean fields to be expressed as functions of the mean position of fluid particles. To achieve this, it is necessary to relate the mean positions of particles to their positions at the end of averaging interval, and to carry out a remapping of the Lagrangian mean fields. This constitutes our strategy 1 for the computation of generalised Lagrangian means. We show that the algorithm of \citet{GBLApaper} amounts to a semi-Lagrangian discretisation of the PDEs of strategy 1. We propose an alternative strategy, strategy 2, which formulates PDEs directly for the partial Lagrangian means using the mean position as independent spatial variable. The PDEs involved in both strategies can be solved by broad classes of numerical methods: finite differences, finite volumes, finite elements or spectral methods. We illustrate this with a pseudospectral Fourier implementation for a shallow-water flow in a doubly periodic domain.
The paper is structured as follows. We introduce notation and define the Lagrangian means in \S \ref{sec:formulation}. We derive the PDEs of the two strategies in \S\ref{sec:PDEs}. We discuss their numerical implementation and present an application to a shallow-water simulation in \S\ref{sec:num_impl}. The choice of strategy, their advantages and costs are discussed in \S\ref{sec:discussion}. Technical aspects including the averaging of tensorial fields are relegated to appendices.
\section{Formulation}\label{sec:formulation}
We consider fluid motion in a two- or three-dimensional Euclidean space. We denote the flow map by $\boldsymbol{\varphi}$, with $\boldsymbol{\varphi}(\boldsymbol{a},t) \in \mathbb{R}^2$ or $\mathbb{R}^3$ the position at time $t$ of a particle identified by its label $\boldsymbol{a}$ (which can be taken as the position at $t=0$). The flow map and velocity field are related by
\begin{equation}
\partial_t \boldsymbol{\varphi}(\boldsymbol{a},t) = \boldsymbol{u}(\boldsymbol{\varphi}(\boldsymbol{a},t),t).
\label{flowmap}
\end{equation}
Lagrangian averaging is averaging at fixed particle label $\boldsymbol{a}$, in contrast with Eulerian averaging which fixes the spatial position. Both can involve different types of means: temporal, spatial or -- as often used in theoretical work -- ensemble mean. Here we focus on a straigthforward time average, of the form
\begin{equation}
\BAR g(\bar t) = \frac{1}{T} \int_{\bar t}^{\bar t + T} g(s) \, \mathrm{d} s
\label{ave}
\end{equation}
when applied to a function $g(t)$ that depends on time only. Eq.\ \eqref{ave} introduces the notation $\bar t$ for the time at which the averaging is carried out and $T$ for the averaging period. Usually the middle of averaging interval $\bar t + T/2$ is used as an argument for the mean function, but we prefer to adopt $\bar t$ (the beginning of averaging interval) to simplify the notation in the upcoming derivations. A simple shift in time switches from one convention to the other. A weight function could be inserted in the integrand of \eqref{ave} to generalise the definition of the average; this would lead to minimal changes in what follows.
The Lagrangian mean trajectory associated with \eqref{ave} is represented by the Lagrangian mean map $\BARL{\bphi}$ defined by
\begin{equation}
\BARL{\bphi}(\boldsymbol{a},\bar t) \coloneq \frac{1}{T} \int_{\bar t}^{\bar t + T} \boldsymbol{\varphi}(\boldsymbol{a},s) \, \mathrm{d} s.
\label{barLbphi}
\end{equation}
Thus $\BARL{\bphi}(\boldsymbol{a},\bar t)$ returns the mean position from $\bar t$ to $\bar t +T$ of the particle labelled by $\boldsymbol{a}$. The definition \eqref{barLbphi} makes sense in $\mathbb{R}^n$, when $\BARL{\bphi}$ can be interpreted as a vector and averaged component-wise, but not on other manifolds where more complicated definitions are necessary \citep{gilbert2018geometric}.
The (generalised) Lagrangian mean of a scalar function $f(\boldsymbol{x},t)$ is then defined by
\begin{equation}
\BARL f( \BARL{\bphi}(\boldsymbol{a},\bar t), \bar t) \coloneq \frac{1}{T} \int_{\bar t}^{\bar t + T} f(\boldsymbol{\varphi}(\boldsymbol{a},s),s) \, \mathrm{d} s.
\label{barLf}
\end{equation}
Hence $\BARL f(\boldsymbol{x},\bar t)$ is the average of $f$ along the trajectory of the fluid parcel, regarded as a function of the Lagrangian mean position $\boldsymbol{x}$ and time $\bar t$.
Our aim is the development of an efficient numerical method for the computation of $\BARL f$ that relies on solving PDEs, which can be discretised in a variety of ways, rather than on tracking ensembles of particle trajectories. We propose two strategies and derive the corresponding PDEs in the next section.
\section{Two strategies} \label{sec:PDEs}
\begin{figure}
\begin{minipage}{.5\textwidth}
\centerline{(a)}
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\coordinate (C) at (6,2.2);
\draw[->] (-0.3,0) -- (6.9,0) coordinate[label = {below:$t$}] (xmax);
\draw[->] (0,-0.3) -- (0,5.5) coordinate[label = {right:$f$}] (ymax);
\draw[thick, blue] (A) .. controls (2.5, 1) and (4, 2) ..
(B);
\draw[thick, purple] (A) .. controls (2.5, 2.4) and (4, 2.5) ..
(C) node[xshift=0em,yshift=.2em,anchor=south,align=center] {$\TILDL{f}(\boldsymbol{\varphi}(\boldsymbol{a},\bar t+T),\bar t)$ \\ $= \BARL{f}(\BARL{\bphi}(\boldsymbol{a},\bar t),\bar t) $} ;
\draw[thick, blue, fill=blue!20] (A) circle (.8mm);
\draw[thick, blue, fill=blue!20] (B) circle (.8mm);
\draw[thick, purple, fill=purple!20] (C) circle (.8mm);
\draw[dashed] (A) -- (A |- O) node[anchor=north,xshift=.0em] {$\bar t$} ;
\draw[dashed] (C) -- (C |- O) node[anchor=north] {$\bar t + T$} ;
\node[purple] at (3,3.8) {$\tilde{f}(\bm{\varphi}(\bm{a},t),t;\bar t)$};
\node[blue] at (1.6,2.) {$f(\boldsymbol{\varphi}(\boldsymbol{a},t),t)$};
\end{tikzpicture}
\end{minipage}
\begin{minipage}{.5\textwidth}
\centerline{(b)}
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\draw[->] (0,-.3) -- (0,5.5) coordinate[label = {right:$\bm{x}$}] (ymax);
\draw[thick, blue] (A) .. controls (2.5, 1) and (4, 2) ..
(B) node[xshift=.5em, anchor=south] {$\boldsymbol{\varphi}(\boldsymbol{a},\bar t + T)$};
\draw[thick, purple] (A) .. controls (2.5, 1) and (4, 1) ..
(C) node[xshift=.5em, anchor=west] {$\BARL{\bphi}(\boldsymbol{a},\bar t)$} ;
\draw[thick, blue, fill=blue!20] (A) circle (.8mm);
\draw[thick, blue, fill=blue!20] (B) circle (.8mm);
\draw[thick, purple, fill=purple!20] (C) circle (.8mm);
\draw[dashed] (A) -- (A |- O) node[anchor=north,xshift=.0em] {$\bar t$} ;
\draw[dashed] (B) -- (B |- O) node[anchor=north] {$\bar t + T$} ;
\node[blue] at (3.3,2.7) {$\boldsymbol{\varphi}(\bm{a},t)$};
\node[purple] at (4.5,1.25) {$\bar{\boldsymbol{\varphi}}(\boldsymbol{a},t;\bar{t})$};
\end{tikzpicture}
\end{minipage}
\caption{Lagrangian means and partial Lagrangian means in an averaging interval $(\bar t, \bar t + T)$: (a) means of a function $f$ evaluated along the trajectory $\boldsymbol{\varphi}(\boldsymbol{a},t)$ of a fluid parcel labelled by $\boldsymbol{a}$, and (b) means of the position, shown here along a single coordinate axis.}
\label{fig:oned}
\end{figure}
Following \citet{GBLApaper}, we define another form of Lagrangian mean, $\TILDL{f}$, by
\begin{equation}
\TILDL{f}(\boldsymbol{\varphi}(\boldsymbol{a},\bar t + T), \bar t) \coloneq \frac{1}{T} \int_{\bar t}^{\bar t + T} f(\boldsymbol{\varphi}(\boldsymbol{a},s),s) \, \mathrm{d} s,
\label{tildeLf}
\end{equation}
Comparing this definition with \eqref{barLf} shows that the overbar indicates a mean along a trajectory identified by the Lagrangian mean position, whereas the tilde indicates the same mean but with the trajectory identified by the actual position at $t=\bar t + T$, that is, at the end of the averaging interval; explicitly,
\begin{equation}
\TILDL{f}(\boldsymbol{\varphi}(\boldsymbol{a},\bar t+T),\bar t) = \BARL f( \BARL{\bphi}(\boldsymbol{a},\bar t), \bar t).
\label{tildebar}
\end{equation}
This is illustrated in figure \ref{fig:oned}. We also introduce the \textit{partial mean} versions of \eqref{barLbphi}, \eqref{barLf} and \eqref{tildebar}, namely
\begin{subequations}
\label{partial_means}
\begin{align}
\BAR{\boldsymbol{\varphi}}(\boldsymbol{a},t;\bar t) &\coloneq \frac{1}{t-\bar t} \int_{\bar t}^{t} \boldsymbol{\varphi}(\boldsymbol{a},s) \, \mathrm{d} s, \label{barphi} \\
\BAR f( \BAR{\boldsymbol{\varphi}}(\boldsymbol{a},t; \bar t), t; \bar t) &\coloneq \frac{1}{t-\bar t} \int_{\bar t}^{t} f(\boldsymbol{\varphi}(\boldsymbol{a},s),s) \, \mathrm{d} s, \label{fbar} \\
\tilde f(\boldsymbol{\varphi}(\boldsymbol{a},t),t;\bar t) &\coloneq \frac{1}{t-\bar t} \int_{\bar t}^t f(\boldsymbol{\varphi}(\boldsymbol{a},s),s) \, \mathrm{d} s, \label{ftilde}
\end{align}
\end{subequations}
as in \citet{GBLApaper}; see figure \ref{fig:oned}.
Clearly the partial means give the total means when evaluated at $t = \bar t + T$. We emphasise that all means used in the paper are Lagrangian means and that we only indicate this explicitly by a superscript $L$ for the total means, to distinguish them from the partial means which are undecorated. The counterpart of \eqref{tildebar} holds for the partial means:
\begin{equation}
\tilde f(\boldsymbol{\varphi}(\boldsymbol{a},t),t;\bar t) = \BAR f( \BAR{\boldsymbol{\varphi}}(\boldsymbol{a},t; \bar t), t; \bar t) \label{ftilde2fbar}.
\end{equation}
Since time average quantities vary over time scales larger than the averaging period $T$, it is neither necessary nor desirable to compute Lagrangian means at each of the times at which the velocity $\boldsymbol{u}$ and scalar field $f$ are known, typically discrete times separated by a small time step. Rather, we think of the averaging time $\bar t$ as a slow variable and propose to compute the Lagrangian means only at $\bar t = \bar t_n = n T$ for $n=1,2,\cdots$. We can therefore carry out independent computations for each $t_n$, each involving only the fields for $t \in (\bar t_n, \bar t_n+T)$. We now focus on one such interval and, to lighten the notation, drop the parametric dependence on $\bar t$ from the partial means in \eqref{partial_means}. For instance, we use $\BAR f( \BAR{\boldsymbol{\varphi}}(\boldsymbol{a},t), t)$ instead of $\BAR f( \BAR{\boldsymbol{\varphi}}(\boldsymbol{a},t; \bar t), t; \bar t)$ in the following derivations, keeping in mind the now implicit dependence of $\BAR f $ and $\BAR{\boldsymbol{\varphi}}$ on $\bar t$.
We formulate two distinct strategies for the computation of the Lagrangian mean $\BARL f(\boldsymbol{x},\bar t)$.
\subsection{Strategy 1}
Our first strategy parallels that of \cite{GBLApaper} and consists in solving a PDE for $\tilde f (\boldsymbol{x},t)$, evaluating the result at $t=\bar t + T$ to obtain $\TILDL{f}(\boldsymbol{x},\bar t)$, then deducing $\BARL{f}(\boldsymbol{x},\bar t)$ by applying a suitable re-mapping.
To derive the PDE for $\tilde f (\boldsymbol{x},t)$ we take the time derivative of \eqref{ftilde} at fixed label $\boldsymbol{a}$ and use the chain rule and \eqref{flowmap} to find
\begin{align}
\partial_t \tilde f(\boldsymbol{\varphi}(\boldsymbol{a},t),t) + \boldsymbol{u}( \boldsymbol{\varphi}(\boldsymbol{a},t),t) \bcdot \bnabla \tilde f(\boldsymbol{\varphi}(\boldsymbol{a},t),t) = \frac{1}{t - \bar t} \left( f(\boldsymbol{\varphi}(\boldsymbol{a},t),t) - \tilde f(\boldsymbol{\varphi}(\boldsymbol{a},t),t) \right),
\end{align}
Here and in what follows, the gradient $\bnabla$ is taken with respect to the first argument of $\tilde f$. Replacing $\boldsymbol{\varphi}(\boldsymbol{a},t)$ by $\boldsymbol{x}$ as independent variable yields the sought PDE,
\begin{equation}
\partial_t \tilde f(\boldsymbol{x},t) + \boldsymbol{u}(\boldsymbol{x},t) \bcdot \bnabla \tilde f(\boldsymbol{x},t) = \frac{f(\boldsymbol{x},t) - \tilde f(\boldsymbol{x},t)}{t - \bar t},
\label{ftildeevol}
\end{equation}
which can be integrated in from $\bar t$ to $t = \bar t + T$ to find the total mean $\TILDL{f}(\boldsymbol{x}, \bar t) = \tilde f(\boldsymbol{x},\bar t + T;\bar t) $. This is a forced advection equation in which the forcing can be interpreted as a time-dependent relaxation of $\tilde f$ to $f$.
In a bounded domain, the solution of \eqref{ftildeevol} requires no boundary conditions since the differentiation $\boldsymbol{u}(\boldsymbol{x},t) \bcdot \bnabla$ is along the boundary. The initial condition is that $\tilde f(\boldsymbol{x},\bar t) = f(\boldsymbol{x},\bar t)$ so that the right-hand side is finite.
\begin{figure}
\begin{center}
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\coordinate (B) at (7,1);
\coordinate (C) at (6,-1);
\draw[gray, thick, fill=gray!10] (O) ellipse (1.8 and 1.8);
\draw[gray, thick, fill=gray!10] (6,0) ellipse (2 and 2.5);
\node at (0,-2.4) {label space};
\node at (6,-3) {physical space $\subseteq \mathbb{R}^n$};
\draw[shorten >= 2pt,-{Stealth[scale=1]},thick,blue] (A) to[out=50,in=150] node[above] {$\bm{\varphi}$} (B);
\node[anchor=north,yshift=-0.2em,blue] at (A) {$\bm{a}$};
\node[anchor=north,yshift=-0.2em,purple] at (C) {$\bar{\bm{\varphi}}(\bm{a},t;\bar t)$};
\node[anchor=south,yshift=0.2em] at (B) {$\bm{x}$};
\draw[shorten >= 2pt,-{Stealth[scale=1]},purple] (A) to[out=10,in=150] node[above] {$\bar{\bm{\varphi}}$} (C);
\draw[shorten >= 2pt,-{Stealth[scale=1]},thick,teal] (C) to[out=100,in=200] node[above,xshift=-0.5em] {${\bm{\Xi}}$} (B);
\draw[shorten >= 2pt,-{Stealth[scale=1]},thick,teal] (B) to[out=290,in=20] node[above,xshift=1.2em,yshift=-0.9em] {${\bm{\Xi}^{-1}}$} (C);
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\draw[thick, black, fill=black!20] (B) circle (.8mm);
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\end{tikzpicture}
\caption{Flow map $\boldsymbol{\varphi}$, Lagrangian (partial) mean map $\bar \boldsymbol{\varphi}$, lift map $\boldsymbol{\Xi}$ and its inverse $\boldsymbol{\Xi}^{-1}$.}
\label{fig:Xi}
\end{center}
\end{figure}
Computing $\TILDL f$ may be all that is needed for applications in which the spatial distribution of the Lagrangian mean is not important. For example, wave-averaged geostrophy -- the modified form of geostrophic balance expressed in terms of Lagrangian mean quantities --
can be validated by comparing Lagrangian mean velocity and pressure gradient at the same position (equivalently, the same label) regardless of where the position is located in physical space \citep{kafiabad2021wave,GBLApaper}.
However, in many other applications, it is necessary to compute the (generalised) Lagrangian mean field $\BARL f(\boldsymbol{x},t)$ for specified Lagrangian mean positions $\boldsymbol{x}$. This field can be deduced from $\TILDL f(\boldsymbol{x},t)$ by considering the \textit{lift map} $\boldsymbol{\Xi}(\boldsymbol{x},t)$ which returns the position at time $t$ of the particle with $\boldsymbol{x}$ as partial mean position from $\bar{t}$ to $t$, that is,
\begin{equation}
\quad \boldsymbol{\Xi}(\BAR \boldsymbol{\varphi}(\boldsymbol{a}, t),t) = \boldsymbol{\varphi}(\boldsymbol{a},t) \label{Xi-def}
\end{equation}
\citep{andrews1978exact,buhler2009waves}.
Its inverse $\boldsymbol{\Xi}^{-1}(\boldsymbol{x},t)$ returns the partial mean position of the particle that passes through $\boldsymbol{x}$ at $t$:
\begin{equation}
\boldsymbol{\Xi}^{-1}( \boldsymbol{\varphi}(\boldsymbol{a},t) ,t) = \BAR \boldsymbol{\varphi}(\boldsymbol{a},t) = \frac{1}{t-\bar t} \int_{\bar t}^t \boldsymbol{\varphi}(\boldsymbol{a},s) \, \mathrm{d} s, \label{Xi-1-def}
\end{equation}
where we used the definition \eqref{barphi} of the mean position.
The map $\boldsymbol{\Xi}$ and its inverse $\boldsymbol{\Xi}^{-1}$ are depicted in the figure \ref{fig:Xi}.
The relation \eqref{tildebar} between $\TILDL f$ and $\BARL f$ can then be written in terms of $\boldsymbol{\Xi}^{-1}$ as
\begin{equation}
\TILDL{f}(\boldsymbol{x},\bar t) = \BARL{f}(\boldsymbol{\Xi}^{-1}(\boldsymbol{x},\bar t+T),\bar t). \label{remap-strat1}
\end{equation}
Now, comparing \eqref{Xi-1-def} with \eqref{ftilde} makes it clear that the components of $\boldsymbol{\Xi}^{-1}$ can be viewed as instances of functions $\tilde f$ with $f(\boldsymbol{x})=x_i$. Thus, we can rewrite \eqref{ftildeevol} for this special case to obtain
\begin{equation}
\partial_t \boldsymbol{\Xi}^{-1}(\boldsymbol{x},t) + \boldsymbol{u}(\boldsymbol{x},t) \bcdot \bnabla \boldsymbol{\Xi}^{-1}(\boldsymbol{x},t) = \frac{\boldsymbol{x} - \boldsymbol{\Xi}^{-1}(\boldsymbol{x},t)}{t - \bar t}.
\label{Xi-1evol}
\end{equation}
Alternatively, we can take the time derivative of \eqref{Xi-1-def} and replace $\boldsymbol{\varphi}(\boldsymbol{a},t)$ by $\boldsymbol{x}$ to arrive at \eqref{Xi-1evol}.
Integrating \eqref{Xi-1evol} provides the means to effect the remapping \eqref{remap-strat1} between $\TILDL f$ and $\BARL f$.
To recapitulate, our first strategy consists in solving the PDEs \eqref{ftildeevol} and \eqref{Xi-1evol} from $\bar t$ to $\bar t + T$ to obtain $\TILDL{f}(\boldsymbol{x},\bar t)=\tilde f(\boldsymbol{x},\bar t + T; \bar t)$ and $\boldsymbol{\Xi}(\boldsymbol{x},\bar t+T; \bar t)$, then using
\eqref{remap-strat1} to compute $\BARL{f}$ by interpolation. The algorithm proposed by \citet{GBLApaper} turns out to be a particular discretisation of this strategy (see \S\ref{sec:strat1impl} below).
\subsection{Strategy 2}
Our second strategy bypasses the use of $\tilde f$ and is instead based on PDEs for $\BAR f$ and $\boldsymbol{\Xi}$.
To derive these we first note that taking the time derivative of \eqref{barphi} gives
\begin{equation}
\partial_t \BAR \boldsymbol{\varphi}(\boldsymbol{a},t) = \frac{\boldsymbol{\varphi}(\boldsymbol{a},t)-{\BAR \boldsymbol{\varphi}(\boldsymbol{a},t)}}{t-\bar t} \label{dbarphi_dt} \eqcolon \BAR \boldsymbol{u} (\BAR \boldsymbol{\varphi}(\boldsymbol{a},t),t).
\end{equation}
The second equality defines the auxiliary velocity field $\BAR \boldsymbol{u} $ as the time derivative of the partial Lagrangian mean position. Using \eqref{Xi-def}, this velocity field can be written in terms of the lift map as
\begin{equation}
\BAR \boldsymbol{u}(\boldsymbol{x},t) = \frac{\boldsymbol{\Xi}(\boldsymbol{x},t) - \boldsymbol{x}}{t-\bar t},
\label{baru}
\end{equation}
where the dummy variable $\boldsymbol{x}$ can be thought of as the partial mean position.
We emphasise that $\BAR \boldsymbol{u}(\cdot,t)$, like $\boldsymbol{\Xi}(\cdot,t)$, depends implicitly on $\bar t$ and warn that it should not be interpreted
as the partial mean of the Lagrangian velocity: as discussed in appendix \ref{app:lmv}, its value at the end of the averaging interval, for $t=\bar t + T$, differs from the usual Lagrangian mean velocity, that is, the time derivative of $\BARL \boldsymbol{\varphi}(\boldsymbol{a},\bar t)$ with respect to $\bar t$.
Now, differentiating \eqref{fbar} with respect to $t$ at fixed label $\boldsymbol{a}$ and using \eqref{dbarphi_dt} leads to
\begin{equation}
\partial_t \BAR f(\BAR \boldsymbol{\varphi}(\boldsymbol{a},t),t) + \BAR \boldsymbol{u} (\BAR \boldsymbol{\varphi}(\boldsymbol{a},t),t) \bcdot \bnabla \BAR f(\BAR \boldsymbol{\varphi}(\boldsymbol{a},t),t) = \frac{f(\boldsymbol{\varphi}(\boldsymbol{a},t),t) - \BAR f(\BAR \boldsymbol{\varphi}(\boldsymbol{a},t),t)}{t-\bar t}.
\end{equation}
We obtain the desired PDE for $\BAR f(\boldsymbol{x},t)$ by using \eqref{Xi-def} and replacing $\BAR \boldsymbol{\varphi}(\boldsymbol{a},t)$ by the independent variable $\boldsymbol{x}$ to write
\begin{equation}
\partial_t \BAR f(\boldsymbol{x},t) + \BAR \boldsymbol{u}(\boldsymbol{x},t) \bcdot \bnabla \BAR f(\boldsymbol{x},t) = \frac{f(\boldsymbol{\Xi}(\boldsymbol{x},t),t)-\BAR f(\boldsymbol{x},t)}{t-\bar t}.
\label{barfevol}
\end{equation}
This is a forced advection equation, analogous to the PDE \eqref{ftildeevol} governing $\tilde f(\boldsymbol{x},t)$. However, unlike \eqref{ftildeevol} it is not closed since it involves $\boldsymbol{\Xi}(\boldsymbol{x},t)$, explicitly on the right-hand side and implicitly through $\bar \boldsymbol{u}(\boldsymbol{x},t)$ on the left-hand side. It needs to be solved along an equation for $\boldsymbol{\Xi}(\boldsymbol{x},t)$. We derive this equation by taking the time derivative of \eqref{Xi-def} at fixed $\boldsymbol{a}$ and using \eqref{dbarphi_dt} and \eqref{flowmap} to obtain
\begin{equation}
\partial_t \boldsymbol{\Xi}(\bar \boldsymbol{\varphi}(\boldsymbol{a},t),t) + \bar \boldsymbol{u}(\bar \boldsymbol{\varphi}(\boldsymbol{a},t),t) \bcdot \bnabla \boldsymbol{\Xi}(\bar \boldsymbol{\varphi}(\boldsymbol{a},t),t) = \boldsymbol{u}(\boldsymbol{\varphi}(\boldsymbol{a},t),t).
\end{equation}
Hence, replacing $\boldsymbol{\varphi}(\boldsymbol{a},t)$ by $\boldsymbol{x}$ and using \eqref{Xi-def},
\begin{equation}
\partial_t \boldsymbol{\Xi}(\boldsymbol{x},t) +\bar \boldsymbol{u}(\boldsymbol{x},t) \bcdot \bnabla \boldsymbol{\Xi}(\boldsymbol{x},t) = \boldsymbol{u}(\boldsymbol{\Xi}(\boldsymbol{x},t),t).
\label{Xievol}
\end{equation}
Strategy 2 consists in solving \eqref{barfevol} and \eqref{Xievol}, with $\bar \boldsymbol{u}(\boldsymbol{x},t)$ defined by \eqref{baru}, for $t \in (\bar t,\bar t + T)$, then deduce the Lagrangian mean of $f$ as $\BARL f(\boldsymbol{x},\bar t) = \bar f (\boldsymbol{x},\bar t+T; \bar t)$.
The initial conditions for \eqref{barfevol} and \eqref{Xievol} are that $\bar f(\boldsymbol{x},\bar t)=f(\boldsymbol{x},\bar t)$ and $\boldsymbol{\Xi}(\boldsymbol{x},\bar t)=\boldsymbol{x}$.
The boundary conditions are non-trivial: in bounded domains, $\bar f(\boldsymbol{x},t)$ and $\boldsymbol{\Xi}(\boldsymbol{x},t)$ are defined on the image of the label space by the Lagrangian mean map $\bar \boldsymbol{\varphi}$ (equivalently, the image of the fluid domain by $\boldsymbol{\Xi}^{-1}$). Thus, the problem in principle involves a boundary moving with velocity $\bar \boldsymbol{u}$ and can therefore be difficult to discretise. The common situation where the physical domain has boundaries that coincide with constant-coordinate surfaces (curves in two dimensions) is straightforward, however, because the component-wise definition of $\bar \boldsymbol{\varphi}$ in \eqref{barLbphi} ensures that it maps such boundaries to themselves so the domain remains fixed. The case of periodic domains is also straightforward.
\section{Numerical implementation}\label{sec:num_impl}
The set of equations for each strategy of the previous section can be discretised in a variety of ways. Here we focus on a pseudospectral discretisation which we apply to the computation of Lagrangian means in a turbulent shallow-water flow interacting with a Poincar\'e wave. We make general remarks about the choice of strategy and numerical discretisation but leave a more complete analysis of numerical error and convergence for future studies.
\subsection{Strategy 1} \label{sec:strat1impl}
To solve \eqref{ftildeevol} it is convenient to introduce $\tilde g(\boldsymbol{x},t) = (t - \bar t) \tilde f(\boldsymbol{x},t)$, leading to
\begin{equation}
\left(\partial_t + \boldsymbol{u}(\boldsymbol{x},t) \bcdot \bnabla \right) \tilde g(\boldsymbol{x},t) = f(\boldsymbol{x},t). \label{gtilde-evol}
\end{equation}
Integrating this equation over the averaging period then yields the Lagrangian mean
$\TILDL{f}(\boldsymbol{x}, \bar t) = \tilde g(\boldsymbol{x},\bar t + T)/T$. For simple geometries, periodic in particular, standard pseudospectral methods provide efficient solvers for \eqref{gtilde-evol} and, if the remapping to Lagrangian mean positions is desired,
\eqref{Xi-1evol}. This is particularly convenient if the fluid's governing equations are also solved pseudospectrally, because $f$ is then available on spectral grid points to evaluate the right-hand sides of \eqref{gtilde-evol} and \eqref{Xi-1evol}, and on physical grid points to evaluate the nonlinear terms. An alternative is a semi-Lagrangian discretisation, which leads to the algorithm of \cite{GBLApaper} as detailed in appendix \ref{app:GBLA}.
\begin{figure}
\centering
\includegraphics[trim=1cm 0 0 0, width=\linewidth]{NS2D_compareLMs.eps}
\caption{Vorticity field and its Lagrangian mean for a two-dimensional incompressible inviscid flow: (a) instantaneous vorticity $\zeta$ at $t = 25$, and (b--d) $\tilL{\zeta}$ computed by time-integration of \eqref{ftildeevol} from $\bar t = 0$ to $T=25$ using (b) a semi-Lagrangian discretisation with linear interpolation, (c) semi-Lagrangian discretisation with cubic interpolation and (d) pseudospectral discretisation.}
\label{fig:vorticity2Dflow}
\end{figure}
To investigate the validity of numerical solutions of \eqref{ftildeevol} we consider a two-dimensional incompressible inviscid flow for which the vorticity, $\zeta$ say, is conserved materially. This implies that
\begin{equation}
\zeta(\boldsymbol{x},\bar t + T) = \TILDL{\zeta}(\boldsymbol{x},\bar t). \label{vorticity_itsmean}
\end{equation}
Hence, we can calculate $\TILDL{\zeta}$ by integrating \eqref{ftildeevol} (or \eqref{gtilde-evol}) from $\bar t$ to $\bar t + T$ and compare it with the instantaneous vorticity at $\bar t + T$ to study the accuracy of the computed Lagrangian mean. Note that this is simply a test for \eqref{ftildeevol} using the material conservation of $\zeta$ as opposed to an application of the Lagrangian mean. As mentioned earlier, \eqref{ftildeevol} should usually by solved in tandem with \eqref{Xi-1evol} to get a meaningful spatial distribution of Lagrangian mean quantities.
We perform a numerical simulation in a doubly periodic, $\left[0,\ 2 \pi \right]^2$ domain, using a standard Fourier pseudospectral discretisation with $128^2$ grid points. We start the simulation with the vorticity
\begin{equation}
\zeta(x,y,t=0) = \mathrm{e}^{-(x-\pi+0.1)^2 -(y-\pi+\pi/3)^2 } +
\mathrm{e}^{-(x-\pi-0.1)^2 - (y-\pi-\pi/3)^2},
\end{equation}
corresponding to two like-signed vortices which subsequently merge. We use Heun's method for the time integration of the governing vorticity equation and for \eqref{gtilde-evol}, with time step $ \Delta t = 0.005$. Figure \ref{fig:vorticity2Dflow} displays the instantaneous vorticity $\zeta$ at $t = 25$ and $\TILDL{\zeta}$ obtained for $\bar t = 0$ and $T = 25$. As expected from \eqref{vorticity_itsmean}, the Lagrangian mean $\TILDL{\zeta}$ matches the instantaneous vorticity $\zeta$. The pseudospectral solution for $\TILDL{\zeta}$, shown in panel (d), in particular, shows an excellent agreement with $\zeta$ in panel (a). The results of the semi-Lagrangian algorithm of \cite{GBLApaper} with, respectively, linear and cubic interpolations, are shown in panels (b) and (c) (see appendix \ref{app:GBLA}). These show a poorer agreement with panel (a), especially with the linear interpolation, because of an accumulation of interpolation errors. The computation reported in figure \ref{fig:vorticity2Dflow} is however rather extreme in both the coarseness of the resolution and the length of the averaging interval.
We have confirmed that the three numerical solutions for $\TILDL{\zeta}$ converge to each other and to $\zeta$ as the spatial resolution increases or the length of the averaging interval decreases (not shown).
The pseudospectral method leads to the more the accurate results, but it is not as stable as its semi-Lagrangian counterpart and therefore requires smaller time steps. The difference arises because the implicit time integration and numerical smoothing due to interpolation that are inherent to semi-Lagrangian methods (see appendix \ref{app:GBLA}) have a stabilising effect.
\subsection{Strategy 2}\label{sec:num_strategy2}
We now implement strategy 2 which uses the Lagrangian mean position $\boldsymbol{x}$ as independent spatial variable. In the periodic domain we consider, there are no difficulties associated with moving boundaries and a pseudospectral discretisation is straightforward. It is convenient to rewrite the PDEs to be integrated, \eqref{barfevol} and \eqref{Xievol}, in terms of the displacement map
\begin{equation}
\boldsymbol{\xi}(\boldsymbol{x},t) = \boldsymbol{\Xi}(\boldsymbol{x},t) - \boldsymbol{x},
\end{equation}
since $\boldsymbol{\xi}$ is periodic, unlike $\boldsymbol{\Xi}$. (This is the partial-mean analogue of the displacement introduced by \citet{andrews1978exact}.) As in strategy 1, it is also convenient to solve for $\bar g(\boldsymbol{x},t) = (t - \bar t) \bar f(\boldsymbol{x},t)$ instead of $\bar f$. With these transformations, $\bar \boldsymbol{u} = \boldsymbol{\xi}/(t-\bar t)$ and \eqref{barfevol} and \eqref{Xievol} are rewritten as
\begin{subequations}
\begin{align}
\partial_t \boldsymbol{\xi}(\boldsymbol{x},t) + \frac{\boldsymbol{\xi}(\boldsymbol{x},t)}{t-\bar t} \bcdot \bnabla \boldsymbol{\xi}(\boldsymbol{x},t) &= \boldsymbol{u}(\boldsymbol{x}+\boldsymbol{\xi}(\boldsymbol{x},t),t) - \frac{\boldsymbol{\xi}(\boldsymbol{x},t)}{t-\bar t} , \label{xi_evol}\\
\partial_t \bar g(\boldsymbol{x},t) + \frac{\boldsymbol{\xi}(\boldsymbol{x},t)}{t-\bar t} \bcdot \bnabla \bar g(\boldsymbol{x},t) &= f(\boldsymbol{x}+\boldsymbol{\xi}(\boldsymbol{x},t),t). \label{barf_xi}
\end{align}
\label{setofxifbar}
\end{subequations}
These are the PDEs we solve numerically. When discretising in time, we found it beneficial for stability to first update $\boldsymbol{\xi}$ using \eqref{xi_evol}, then use the updated $\boldsymbol{\xi}$ for the time integration of \eqref{barf_xi}.
As an application, we compute Lagrangian means in a simulation of a turbulent flow interacting with a Poincar\'e wave in a rotating shallow-water model. We write the model in a non-dimensional form. We use a characteristic length $L$, characteristic velocity $U$, time $L/U$ and mean height $H$ to scale the variables and obtain
\begin{subequations}\label{sw_eqs_dimless}
\begin{align}
\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \bcdot \bnabla \boldsymbol{u} + \frac{1}{\rm{Ro}} \hat{\boldsymbol{z}} \times \boldsymbol{u} &= - \frac{1}{\rm{Fr}^2}\ \bnabla h + \frac{1}{\rm{Re}} \nabla^2 \boldsymbol{u} \label{sw_momentum_dimless}\\
\frac{\partial h}{\partial t} + \bnabla \bcdot (h \boldsymbol{u}) &= 0\ , \label{sw_mass_dimless}
\end{align}
\end{subequations}
where we introduce the standard dimensionless numbers \citep[e.g.][]{vallis2017atmospheric}
\begin{equation}
{\rm{Ro}} = \frac{U}{f L}\ , \quad {\rm{Fr}} = \frac{U}{\sqrt{g H}}\quad \textrm{and} \quad {\rm{Re}} = \frac{U L}{\nu}.
\end{equation}
In the above, $g$ is the gravitational acceleration, $\nu $ the kinematic viscosity, $f$ the Coriolis parameter and $\hat{\boldsymbol{z}}$ the vertical unit vector.
\begin{figure}
\centering
\includegraphics[width=.75\linewidth]{ICs_new.eps}
\caption{Initial condition of the shallow-water simulation: (a) vertical vorticity of the geostrophic flow and (b) height field of the Poincar\'e wave.}
\label{fig:initial-cond}
\end{figure}
We initialise our simulation with a turbulent flow that is initially in geostrophic balance, with vorticity $\zeta = \partial_x v - \partial_x u$ shown in figure \ref{fig:initial-cond}(a), and superimpose a mode-1 Poincar\'e wave, with the height field shown in figure \ref{fig:initial-cond}(b). We use the root-mean square velocity of the geostrophic flow as the characteristic velocity $U$, and choose the length scale of first Fourier mode as characteristic length $L$. This makes the dimensionless doubly periodic domain $\left[0,\ 2 \pi \right]^2$ which we discretise with $256 \times 256$ grid points. The right-travelling mode-1 Poincar\'e wave has the form
\begin{equation}\label{initial_wave}
u' = a \cos (x-\omega t), \quad
v' = \frac{a}{\omega \rm{Ro}} \sin (x-\omega t), \quad
h' = \frac{a}{\omega} \cos (x-\omega t),
\end{equation}
with the intrinsic frequency $\omega = \left(\rm{Ro}^{-2}+\rm{Fr}^{-2}\right)^{1/2}$ and $a$ is a constant, taken as $a=-1.8$ in our simulation. We set the dimensionless parameters to
\begin{equation}
{\rm{Ro}} = 0.1 \ , \quad {\rm{Fr}} = 0.5 \quad \textrm{and} \quad {\rm{Re}} = 3.84 \times 10^3,
\end{equation}
which results in $\omega = 10.2$.
We evaluate \eqref{initial_wave} at $t=0$ and add the wave fields to the geostrophic field to form the initial condition. We solve the dynamical equations \eqref{sw_eqs_dimless} in tandem with the Lagrangian mean equations \eqref{setofxifbar} over a single averaging time interval taken to be $T=2.2$, corresponding to approximately $3.6$ wave periods.
We use a pseudospectral discretisation and a forward Euler integrator, with the time step of $1.25 \times 10^{-4}$ for \eqref{sw_eqs_dimless} and $2.5 \times 10^{-4}$ for \eqref{setofxifbar}. A bilinear interpolation is used to evaluate $\boldsymbol{u}$ and $f$ at $\boldsymbol{x} + \boldsymbol{\xi}(\boldsymbol{x},t)$ in the right-hand sides of \eqref{setofxifbar}. The link for the scripts and data used to produce the results of this section is provided in the ``Data availability statement" at the end of this paper.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{PVs_new.eps}
\caption{Potential vorticity $q$ and its Lagrangian and Eulerian means in the shallow-water simulation: (a) instantaneous PV $q$ at $t = 1.1$, (b) Lagrangian mean $\barL{q}$, and (c) Eulerian mean PV. The mean fields are averaged from $\bar t=0$ to $T=2.2$. All panels share the same colourbar.}
\label{fig:PVs}
\end{figure}
Figure \ref{fig:vorticities} displays the vorticity field $\zeta$ at $t=T/2$ (panel (a)) and its Lagrangian and Eulerian means (panels (b) and (c)). The strong wave dominates the instantaneous vorticity field. It is filtered successfully by time averaging. Clearly, the Lagrangian mean captures small-scale structures in the vorticity which are blurred by the Eulerian mean. This blurring is the result of the advection of the vorticity by the velocity field associated with the wave which causes the vorticity structures to oscillate with the wave period. By construction, the Lagrangian mean removes these oscillations, leading to a sharper definition of the flow features.
It is interesting to examine the effect of Lagrangian and Eulerian averaging on the potential vorticity (PV) $q=\zeta/h$. In the absence of dissipation ($\rm{Re} \to \infty$), this is a materially conserved field, meaning that $q(\boldsymbol{\varphi}(\boldsymbol{a},t),t)=q_0(\boldsymbol{a})$, with $q_0$ determined by the initial condition. By definition \eqref{barLf}, the Lagrangian mean PV then satisfies $\barL q(\bar \boldsymbol{\varphi}(\boldsymbol{a},t),t)=q_0(\boldsymbol{a})$. Thus both $q$ and $\barL q$ are (smooth) re-arrangements of the initial PV and hence re-arrangements of one another, specifically $\barL q(\boldsymbol{x},t)=q(\boldsymbol{\Xi}(\boldsymbol{x},t),t)$. This imposes constraints such as the two fields sharing the same values for their local extrema.
Because $\bar \boldsymbol{\varphi}$ is not area-preserving, the distribution functions of $q$ and $\barL q$ (measuring the area of regions where the fields are below specified values) do not coincide. Figure \ref{fig:PVs} shows $q$ at $t=T/2$ and $\barL q$ as well as the Eulerian mean PV. The Lagrangian mean PV $\barL q$ appears as a slight deformation of $q$, consistent with it being rearrangement by a map $\boldsymbol{\Xi}$ that is close to the identity. In contrast, the Eulerian mean PV, which is not materially transported, shows blurred features, with in particular extrema that are substantially reduced compared with those of $q$ and $\barL q$. There is a strong argument that the study of wave--mean-flow interactions, in the shallow-water model and more broadly, would benefit from the systematic analysis of
Lagrangian mean fields such as the ones displayed in panels (b) of figures \ref{fig:vorticities} and \ref{fig:PVs}.
\section{Discussion} \label{sec:discussion}
This paper presents a novel approach for the numerical computation of Lagrangian means which relies on solving PDEs rather than tracking particles. We propose two strategies, each leading to a separate set of PDEs. Both strategies are based on the derivation of equations governing the evolution of partial means with respect to $t$. These partial means are defined as averages over a subset $(\bar t,t)$ of each averaging interval $(\bar t, \bar t +T)$ and yield the (total) means for $t=\bar t + T$.
Strategy 1 uses the position of particles at time $t$ as independent spatial variable. Hence, it requires a map from the positions at the final time $t=\bar t + T$ to the Lagrangian mean positions to ultimately present the results in terms of the latter, as is standard in GLM theory.
Strategy 2 directly computes the Lagrangian means in terms of Lagrangian mean positions.
A natural question is which of the two strategies should be preferred. There is no definitive answer: each strategy has pros and cons. If the spatial distribution of Lagrangian means does not matter in an application, it suffices to solve Eq.\ \eqref{ftildeevol} of strategy 1. When the spatial distribution is needed, strategy 1 requires the re-mapping \eqref{remap-strat1} which can be affected by clustering: the mean positions $\boldsymbol{\Xi}^{-1}(\boldsymbol{x},\bar t+T)$ obtained from \eqref{Xi-1evol} for $\boldsymbol{x}$ on a regular grid may have a highly non-uniform distribution. This can lead to large numerical errors in the interpolation required to discretise \eqref{remap-strat1}. Strategy 2 circumvents this issue, as the generalised Lagrangian mean is computed directly on the desired spatial grid points. However, this advantage comes at the computational cost of evaluating more complicated right-hand sides in \eqref{barfevol} and \eqref{Xievol}, which require interpolation at each averaging time step. Furthemore, strategy 2 leads to PDEs posed on a moving domain, unless the domain is periodic or has boundaries that correspond to constant coordinates.
As discussed in \cite{GBLApaper}, the full potential of our approach in saving memory and reducing computational cost is realised when the PDEs for the Lagrangian mean fields are solved on-the-fly, together with the dynamical model (as opposed to offline, using saved model outputs). In this case, it is beneficial to solve the Lagrangian mean PDEs using a numerical scheme that closely matches that of the dynamical model, because the instantaneous values of $f$ and $\boldsymbol{u}$ (required to solve the Lagrangian mean PDEs) are readily available at the same (physical or spectral) grid points. Moreover, the reasons that led to a particular choice of numerical discretisation for the dynamical equations -- such as the type of boundary conditions -- typically also apply to the Lagrangian mean PDEs.
In the main body of the paper, we restrict our attention to the Lagrangian averaging of a scalar function $f(\boldsymbol{x},t)$. The averaging of vectors, differential forms and more general tensors is however of interest in applications. For instance, the Lagrangian mean of the momentum 1-form $\boldsymbol{u} \bcdot \mathrm{d} \boldsymbol{x}$ (the integrand in Kelvin's circulation) and of the magnetic flux 2-form play crucial roles in the theory of wave--mean-flow interactions in fluid dynamics and MHD \citep{sowa72,andrews1978exact,holm2002lagrangian,gilbert2018geometric,gilbert2021geometric}. The derivations in \S\ref{sec:PDEs} generalise straightforwardly to tensors when the language of push-forwards, pull-backs and Lie derivatives is employed. We illustrate this in appendix \ref{app:tensor} by generalising Eq.\ \eqref{barfevol} of strategy 2 for the partial Lagrangian mean of $f(\boldsymbol{x},t)$ to a tensor field $\tau(\boldsymbol{x},t)$.
We conclude by noting that practical averages such as the time average in \eqref{ave}--\eqref{barLf} do not satisfy exactly the axioms of the more abstract averages assumed in the development of GLM and similar theories. In particular, the basic requirement that averaging leaves mean quantities unchanged, that is, $\bar{\bar g} \not= \bar g$, fails for \eqref{ave},
though the difference is small if there is a clear time-scale separation between mean flow and perturbation. Interpreting numerically computed Lagrangian mean fields in light of these theories will therefore require to understand how theoretical predictions are affected by the precise nature of the average.
\section{How to submit to the \emph{Journal of Fluid Mechanics}}
\label{sec:intro}
Authors must submit using the online submission and peer review system \href{https://mc.manuscriptcentral.com/jfm} {Scholar One} (formerly Manuscript Central). If visiting the site for the first time, users must create a new account by clicking on `register here'. Once logged in, authors should click on the `Corresponding Author Centre', from which point a new manuscript can be submitted, with step-by-step instructions provided. Authors must at this stage specify whether the submission is a {\it JFM Paper}, or a {\it JFM Rapids} paper (see \S4 for more details). In addition, authors must specify an editor to whom the paper will be submitted from the drop-down list provided. Note that all editors exclusively deal with either {\it JFM Paper} or {\it JFM Rapids} (clearly indicated on the list), so please ensure that you choose an editor accordingly. Corresponding authors must provide a valid ORCID ID in order to submit a manuscript, either by linking an existing ORCID profile to your ScholarOne account or by creating a new ORCID profile. Once your submission is completed you will receive an email confirmation. Book reviews should not be submitted via the online submission site, but should instead be submitted by email to [email protected].
\section{Rules of submission}\label{sec:rules_submission}
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\section {Authors responsibilites}
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\subsection {Transparency and Openness Promotion (TOP)}
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\begin{itemize}
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\section{Types of paper}\label{sec:types_paper}
\subsection{Standard papers}
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\subsection{JFM Rapids}
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\subsection{JFM Perspectives}
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\section{File types}\label{sec:filetypes}
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\section{Preparing your manuscript}
Authors should write their papers clearly and concisely in English, adhering to JFM's established style for mathematical notation, as provided in Section \ref{notstyle}. We encourage the submission of online supplementary material alongside the manuscript where appropriate (see Section \ref{online}). Metric units should be used throughout and all abbreviations must be defined at first use, even those deemed to be well known to the readership. British spelling must be used, and should follow the \textit{Shorter Oxford English Dictionary}.
\begin{figure}
\centerline{\includegraphics{Fig1}
\caption{Trapped-mode wavenumbers, $kd$, plotted against $a/d$ for
three ellipses:\protect\\
---$\!$---,
$b/a=1$; $\cdots$\,$\cdots$, $b/a=1.5$.}
\label{fig:ka}
\end{figure}
\subsection{Figures}
All authors need to acquire the correct permissions and licences to reproduce figures, which should be submitted with the production files. Further information on applying for permission to reuse figures can be found \href{https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/information/request-permissions}{here}. Images should be submitted in EPS or high-resolution TIFF format (1200 dpi for lines, 300 dpi for halftone and colour in RGB format, and 600 dpi for a mixture of lines and halftone) and all labels should be editable. Unless very large, vector graphics are preferred to ensure image sharpness regardless of sizing. The minimum acceptable width of any line is 0.5pt. Each figure should be accompanied by a single caption, to appear beneath, and must be cited in the text. Figures should appear in the order in which they are first mentioned in the text and figure files must be named accordingly (`Abstract.eps, Fig1.eps', `Fig2a.tiff', etc) to assist the production process (and numbering of figures should continue through any appendices). Words \textit {figure 1, table 1 and movie 1} should be lower case. For example see figures \ref{fig:ka} and \ref{fig:kd}.
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\begin{figure}
\centerline{\includegraphics{Fig2}}
\caption{The features of the four possible modes corresponding to
(\textit{a}) periodic\protect\\ and (\textit{b}) half-periodic solutions.}
\label{fig:kd}
\end{figure}
\subsection{Tables}
Tables, however small, must be numbered sequentially in the order in which they are mentioned in the text. Words \textit {table 1, table 2} should be lower case throughout.
See table \ref{tab:kd} for an example.
\begin{table}
\begin{center}
\def~{\hphantom{0}}
\begin{tabular}{lccc}
$a/d$ & $M=4$ & $M=8$ & Callan \etal \\[3pt]
0.1 & 1.56905 & ~~1.56~ & 1.56904\\
0.3 & 1.50484 & ~~1.504 & 1.50484\\
0.55 & 1.39128 & ~~1.391 & 1.39131\\
0.7 & 1.32281 & ~10.322 & 1.32288\\
0.913 & 1.34479 & 100.351 & 1.35185\\
\end{tabular}
\caption{Values of $kd$ at which trapped modes occur when $\rho(\theta)=a$.}
\label{tab:kd}
\end{center}
\end{table}
\subsection{Online supplementary material}\label{online}
Relevant material which is not suitable for inclusion in the main article, such as movies or numerical simulations/animations, can be uploaded as part of the initial submission. Movies must be submitted in .mp4 format and have the file designation of `Movie'. Each movie must be numbered in the order they are mentioned and titled movie 1, movie 2 etc and accompanied by a separate caption. To ensure maths terms display correctly they should be bounded by \textcolor{red}{ \$\$} and written in TeX, e.g. movie 1. Side view of numerical Schlieren contours from case E1N at {\$$z=Lz/2$\$}. Each movie should be no more than 50MB. Upon publication these materials will then be hosted online alongside the final published article. Likewise, should there be detailed mathematical relations, tables or figures which are likely to be useful only to a few specialists or take up excessive space in the article, these should also be published online as supplementary material [designated as `Other supplementary material']. Note that supplementary material is published `as is', with no further intervention made during the Production process, all `draft' information should be removed.
\section{Editorial decisions}
\subsection{Revision}
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\subsection{Provisional acceptance}
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\subsection{Mathematical notation}
\subsubsection{Setting variables, functions, vectors, matrices etc}
\begin{itemize} \label{sec:MathNot}
\item {\bf Italic font} should be used for denoting variables, with multiple-letter symbols avoided except in the case of dimensionless numbers such as $\Rey$, $\Pran$ and $\Pen$ (Reynolds, Prandtl, and P\'eclet numbers respectively, which are defined as \verb}\Rey}, \verb}\Pran} and \verb}\Pen} in the template).\\
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\begin {enumerate}
\item (vI) label, e.g. T. t (transpose)\\
\item Fixed operators: sin, log, d, $\Delta$, exp etc.\\
\item Constants: i ($\sqrt{-1}$), $\upi$ (defined as \verb}\upi}),e etc.\\
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\end {enumerate}
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\end{itemize}
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\section{Citations and references}
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\backsection[Supplementary data]{\label{SupMat}Supplementary material and movies are available at \\https://doi.org/10.1017/jfm.2019...}
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\section{Appeals process}
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\begin{itemize}
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\end{itemize}
|
1,108,101,566,735 | arxiv | \section{Introduction}
Assessing the quality and/or impact of research of a given institution (university or research center) and assessing the corresponding improvement over time is one of the most difficult tasks in modern quality assurance systems.
On the other hand, the growing development of university rankings shows that research performance is perceived to be related to the reputation of universities and, in some countries, the allocation of funds in the higher education system is linked to the research performance of institutions.
Several methods for assessing research performances of universities have been adopted. The most controversial is the use of bibliometric indicators, such as the number of publications, total citations and/or journals impact factor.
In this regard, it should be mentioned that various scientific associations have signed the so-called DORA declaration \cite{DORA}, which states the contrariety to the automatic use of bibliometrics in order to allocate funding for research and /or evaluate the careers of individual researchers.
Another one, much more expensive, is the method followed by some national evaluation agencies, which assess the quality of a limited number of publications for each university, using peer-review, informed peer-review, bibliometrics methods or both, according to the scientific area. The final aim of these methods is to rate each publication among $n$ classes of quality (usually, $n$ is taken to be $4$ or $5$). For instance, in the last call of UK Research Excellence Framework (REF 2021) and Italian Quality Research Evaluation (VQR 2015-2019) are suggested, respectively, 4 (1*, 2*, 3*, 4*, apart of unclassified) and 5 (A - excellent and extremely relevant, B - excellent, C - standard, D - sufficiently relevant, E - poor or not acceptable) classes.
There is a lively debate on the procedures, criteria and methods used in these exercises (see for instance \cite{Abramo:2016}, \cite{Baccini}, \cite{Demetrescu}, \cite{Franceschini}, \cite{taylor1995}, \cite{varin2016}). However, in this paper we do not want to go into the substance of the methodology and effectiveness of such evaluation exercises. Our aim is to consider the more subtle question of how researchers, policy makers, citizens could / should interpret the output data. Namely, any university and any department receive an evaluation in terms of the percentage of the submitted publications evaluated in each of the classes stated in the call. Here there are two main issues. First, the assessment, given in this way, appears to be \emph{absolute} and there is the dangerous temptation to compare directly the performances of two universities and/or departments whose composition could be very different from each other.
For instance, since the number of required publications is the same for any scientific area, it is likely that a Department of Physics will obtain, on average, better evaluations than, for instance, a Department of Law, just because of the very high average number of articles per researcher in Physics.
Second, even assuming to compare two homogeneous aggregations (for instance the Departments of Physics of two universities, or the same Department of Physics along two or more editions of the evaluation exercise), it could not be a trivial task to understand if one assessment is better or not than the other. Of course, if the two assessments compared are those represented in the following table
\medskip
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Institution & A & B & C & D & E \\
\hline
Department 1 & 100\% & 0\% & 0\% & 0\% & 0\% \\
Department 2 & 0\% & 100\% & 0\% & 0\% & 0\% \\
\hline
\end{tabular}
\end{center}
\medskip
then it is easy to say that the assessment of Department 1 is better than that one of Department 2. But if the two assessments are, for instance, the following ones
\medskip
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Institution & A & B & C & D & E \\
\hline
Department 1 & 20\% & 20\% & 20\% & 20\% & 20\% \\
Department 2 & 15\% & 25\% & 21\% & 21\% & 18\% \\
\hline
\end{tabular}
\end{center}
\medskip
then it is not straightforward to decide which department got the best evaluation. A further complication is the usual presence of departments composed by different scientific areas, which could lead the political decision-maker to misleading analyses of the output data arising from the evaluation exercise: taking inspiration from some real cases in Italian universities, how to compare the performances of a ``Department of Mathematics, Computer Science and Economics'' and a ``Department of Mathematics and Geosciences''? And how to compare the performances of all departments in the same university?
In this paper we try to address all these questions. Of course there is not just \emph{one} answer, since several methods for interpreting data can be defined. However, we show that the above are essentially \emph{geometric} questions and that there is a natural geometric way to treat this topic.
Our first observation is that the outcomes in an evaluation exercise (for instance the data in the above tables) can be geometrically represented as points of the standard simplex $\Delta_n$, where $n+1$ is the number of attributes involved in the call. Then the overall assessment of a department can be ``measured'' as the ``distance'' between the point $P_0$ in $\Delta_n$, representing the evaluation of the department, and the vertex $P_1=(1,0, \ldots, 0)$ of the simplex, which corresponds to the best possible assessment. We define such distance $\delta(P_0)$ as the length of a natural path joining $P_0$ to $P_1$, corresponding to improving the assessment of the department in the slowest smooth possible way (see Section \ref{natural-path} for details). By the application of the beautiful geometric properties of the simplex, we find an iterative method to determine this path, obtaining a general formula for $\delta(P_0)$.
This procedure permits to associate a real number to any assessment expressed by ordinal variables. However, $\delta$ can not be used directly for comparing different departments, unless they are reasonably homogeneous. At least they should have the same size and the same internal structure in terms of research areas. In fact, random variations are larger for small samples, so that evaluation results tend to be ``funnel-shaped": for mega-universities it is difficult to deviate much from the average (narrow part of the funnel) while among the small ones (large part) it is frequent to see exceptional results, both for positive and negative performances. Furthermore, each scientific area has its own peculiarities, citational trends and editorial practices, making meaningless to compare any two different scientific areas. Depending on the availability of data and on the aims of the evaluation, one can consider other possible homogeneity criteria, such as teaching duties of professors, salary, age, gender, and so on. Let $\mathcal{C}$ denote the set of all the homogeneity criteria chosen. We can consider the set of all (ideal) departments, whose members are randomly selected from all the universities participating to the call, so that they have the same size as a given Department $D$ and satisfy the same homogeneity criteria $\mathcal{C}$ when compared to $D$. Then we can define the \emph{geometric score} of $D$ as the proportion ${\mathcal S}_{\mathcal{C}}(D)$ of such ideal departments $D'$ for which $\delta(D')> \delta(D)$. In other words, ${\mathcal S}_{\mathcal{C}}(D)$ represents the probability that an ideal department $D'$, with the same configuration as $D$ (hence comparable with $D$), performs worst than $D$. As we shall show in the article, it has also a nice geometric interpretation.
In this way one compares any department - and, more in general, any ``aggregate'', including a whole university itself - with its similars only (in fact with all their possible similars). This procedure avoids the methodological error, very frequent in several research assessment exercises as well as in many university rankings, of comparing universities, departments and scientific areas which, in principle, cannot be compared directly.
One issue related to the geometric score is its computability. Even in the case of a small department, the cardinality of the set of all ideal departments is a very large number, and the exact calculation of the geometric score is not practicable. However, we can approximate the geometric score using Monte Carlo techniques that guarantee the almost sure convergence of the estimate to the geometric score. In the last part of the paper we use a simple algorithm for the calculation of the geometric score for some aggregates of the Italian VQR 2011-2014. Namely we deal with the areas of ``Mathematics'' and of ``Statistics and Mathematical Methods for Decisions'', which are composed, respectively, of more than 2000 and 1000 professors in Italy. We show an easily implementable way to compute good approximations of the geometric score, and, interestingly, we find that the geometric score ranking is very different from the official ANVUR ranking which is still in use to allocate conspicuous public fundings to Italian universities.
\section{Preliminary notions: the geometry of the $n$-simplex}
Let $P_{1},\ldots ,P_{n+1}\in \mathbb{R} ^{n+1}$ be $n+1$ points of $\mathbb{R} ^{n+1}$ which are affinely independent, i.e. the vectors $P_{2}-P_{1}, \ldots, P_{n+1} - P_{1}$ are linearly independent. Then, the \emph{$n$-simplex} determined by $P_{1},\ldots ,P_{n+1}$ is the subset of $\mathbb{R}^{n+1}$ given by
\begin{equation*}
\Delta_{P_{1},\ldots,P_{n+1}} := \left\{x_{1}P_{1} + \cdots + x_{n+1}P_{n+1} \colon x_i \geq 0 \hbox{ for all $i=1,\ldots,n+1$ and } \sum_{i=1}^{n+1}x_{i} = 1 \right\}.
\end{equation*}
The convex hull of any non-empty subset of cardinality $m+1$ of $\left\{P_{1},\ldots ,P_{n+1}\right\}$ is, in turn, a simplex, called $m$-\emph{face}. In particular $0$-faces, i.e. the defining points $P_{1}, \ldots , P_{n+1}$ of the simplex, are called the \emph{vertices}, $1$-faces are called the \emph{edges}, and $n$-faces are called the \emph{facets}.
If one takes the points $P_{1}=(1,0,\ldots,0), P_{2}=(0,1,\ldots,0), \ldots, P_{n+1}=(0,0,\ldots,1)$ of the canonical basis of $\mathbb{R} ^{n+1}$, the corresponding simplex
\begin{align*}
\Delta_n&:=\Delta_{P_{1},\ldots,P_{n+1}}\\
& = \left\{\left(x_{1}, x_{2}, \ldots, x_{n+1}\right)\in \mathbb{R}^{n+1} \colon x_i \geq 0 \hbox{ for all $i=1,\ldots,n+1$ and } \sum_{i=1}^{n+1}x_{i} = 1 \right\}
\end{align*}
is called the \emph{standard $n$-simplex} and is denoted by $\Delta_{n}$. Any $n$-simplex $\Delta_{P_{1},\ldots,P_{n+1}}$ can be canonically identified with the standard $n$-simplex through the bijective mapping
\begin{equation*}
\left(x_{1},\ldots, x_{n+1}\right) \in \Delta_{n} \mapsto \sum_{i=1}^{n+1}x_{i}P_{i} \in \Delta_{P_{1},\ldots,P_{n+1}}.
\end{equation*}
Thus, from now on we shall deal only with the standard $n$-simplex. Notice that $\Delta_0$ is just the point $1 \in \mathbb{R}$, $\Delta_1$ the line segment in $\mathbb{R}^{2}$ joining $(1,0)$ to $(0,1)$, $\Delta_2$ the
equilateral triangle in $\mathbb{R}^3$ whose vertices are $(1,0,0)$, $(0,1,0)$, $(0,0,1)$, and $\Delta_3$ the regular tetrahedron in $\mathbb{R}^4$ with vertices $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$.
We point out that $\Delta_n$ is bijective to the set of ordered $(n+1)$-tuples
\begin{equation*}
\Delta_{n}^{\ast}:=\left\{\left(s_{1}, \ldots, s_{n}, s_{n+1} \right)\in \mathbb{R}^{n+1} : 0\leq s_{1} \leq \cdots \leq s_{n} \leq s_{n+1}=1 \right\}.
\end{equation*}
Indeed, the map
\begin{gather} \label{bijection}
\varphi : \Delta_{n} \longrightarrow \Delta_{n}^{\ast}\\
\left(x_{1},\ldots,x_{n+1}\right) \mapsto \left(x_{1}, x_{1}+x_{2}, \ldots, x_{1}+x_{2}+\cdots+x_{n}, x_{1}+x_{2}+\cdots+x_{n+1}=1\right) \nonumber
\end{gather}
is clearly injective and surjective. The facets of $\Delta_n$, which are given by the equation $x_{i}=0$, under this bijection correspond to successive coordinates being equal, $s_{i}=s_{i-1}$.
\section{A natural path toward the best assessment}\label{natural-path}
Let us consider a typical evaluation research call, where each institution is due to submit a certain number of publications depending on its size. Let us fix a hypothetical university department which has to submit $N$ products. According to the call's rules at the end of the evaluation each product is assigned to a class of a predefined ordinal qualitative variable, with $n+1$ attributes, ranging between the possible best assessment (usually ``excellent'') and the worst one (usually ``poor'').
Let $x_i$ be the relative frequency of the number of publications assigned to the $i$th class. Since, for each $i\in\left\{1, \ldots, n+1\right\}$, $x_i \geq 0$ and $\sum_{i=1}^{n+1}x_i=1$, the global assessment of the department can be naturally identified with a point $(x_1, \ldots, x_{n+1})$ in the standard $n-$dimensional simplex $\Delta_n$. Notice that the best evaluation that the department can achieve is represented in the simplex by the point $P_1=(1,0, \ldots,0)$, corresponding to the ideal situation in which all the submitted publications are assessed in the best class. The remaining points in $\Delta_n$ represent intermediate assessments, starting from $P_1$ until the worst evaluation represented by $P_{n+1}$. Thus, while, from a geometrical point of view, the simplex is a highly symmetric object, in our context the order of the vertices is very important.
Suppose that the final evaluation of the department is represented by the point $P_0=(x_1^0, \ldots, x_{n+1}^0)$. Then in this geometrical framework it is natural to try to measure how ``far'' is the point $P_0$ in the simplex from the vertex $P_1$ corresponding to the best possible assessment (see Figure~\ref{figura0}).
\begin{figure}[th!]
\begin{tikzpicture}
\begin{axis}[width=0.5\textwidth,
axis lines=middle,
inner axis line style={color=white},
xmin=-1.4,
xmax=1.7,
ymin=-1.4,
ymax=1.7,
zmin=-1.4,
zmax=1.7,
xtick={0,6},
ytick={0,6},
ztick={0,6},
view={105}{-5}]
\fill({1/5},{7/15},{1/3}) circle (2.4pt);
\node at ({1/5},{7/15},{1/3}) [below] {$P_0$};
\node at (-1.25299, -1.25299, 0.582337) [above left] {${P_2}$};
\node at (0.582337, -1.25299, -1.25299) [left] {${P_3}$};
\node at (-1.25299, 0.582337, -1.25299)[right] {${P_4}$};
\fill(1.5,1.5,1.5) circle (2.4pt);
\node at (1.5,1.5,1.5) [right] {$P_1$};
\draw(1.5,1.5,1.5) -- (-1.25299, -1.25299, 0.582337);
\draw[dashed](1.5,1.5,1.5)--(0.582337, -1.25299, -1.25299);
\draw(1.5,1.5,1.5)--(-1.25299, 0.582337, -1.25299);
\draw(-1.25299, -1.25299, 0.582337)--(-1.25299, 0.582337, -1.25299)--(0.582337, -1.25299, -1.25299);
\draw(-1.25299, -1.25299, 0.582337)--(0.582337, -1.25299, -1.25299);
\fill(-1.25299, -1.25299, 0.582337) circle (2.4pt);
\fill(-1.25299, 0.582337, -1.25299) circle (2.4pt);
\fill(0.582337, -1.25299, -1.25299) circle (2.4pt);
\end{axis}
\end{tikzpicture}
\caption{The final evaluation of the department is represented by the point $P_0=(x_1^0, \ldots, x_{n+1}^0)$ of the simplex. In the figure the case $n=3$.}\label{figura0}
\end{figure}
Being the simplex a subset of the Euclidean space $\mathbb{R}^{n+1}$, in principle a natural choice would be to use the Euclidean distance. However in this way it could happen that departments with very different evaluations have the same distance from $P_1$ (for instance two distinct vertices $P_i$ and $P_j$ have the same Euclidean distance from $P_1$). Thus, since we are dealing with ordinal categorical variables, the way in which such measurement is done should take into account the ordering of the vertices of the simplex. Any possible choice of such a ``distance'' between $P_0$ and $P_1$ should be defined in such a way to respect the order described above. Our idea is to construct a path from the point $P_0$ to $P_1$ so that the evaluation changes continuously through this path as slow as possible.
For exposition purposes we begin with the description of the $1$-dimensional case, in which only two categories are considered (the best and the worst assessments). Here the simplex degenerates into the line segment from the point $P_{1}=(1,0)$ to $P_{2}=(0,1)$. We express by a positive real number $a$ the ``effort'' for going from the worst evaluation $P_2$ to the best evaluation $P_1$. From a mathematical point of view we are declaring that the length of the vector $P_{2} - P_{1}$ is $a$. Then, given a point $P_0=(x_1^0, x_2^0)=(x^0_{1},1-x^0_{1})$ in such segment, a measurement $\delta(P_0)$ of the distance of $P_0$ from the best possible outcome $P_1$ is just given by the length of the segment from $P_0$ to $P_1$, that is (see Figure~\ref{figura1}~(a))
\begin{figure}[b]
\begin{tikzpicture}
\begin{axis}[width=0.45*\textwidth,
axis lines=middle,
xmin=-.5,
xmax=1.5,
ymin=-.5,
ymax=1.5,
xtick={0,2},
ytick={0,2},
]
\draw[line width=1.2pt](1,0)--({1/3},{2/3});
\draw[decoration={markings,mark=at position 1 with
{\arrow[scale=1.5,>=stealth]{>}}},postaction={decorate},color=red, line width=1.2pt,->,>=stealth] ({1/3},{2/3})--(0,1);
\fill({1/3},{2/3}) circle (2.4pt);
\fill(1,0) circle (2.4pt);
\fill(0,1) circle (2.4pt);
\node at (1,0) [below] {$P_2=(0,1)$};
\node at (0,1) [above right] {$P_1=(1,0)$};
\node at ({1/3},{2/3}) [above right] {$P_0=(x_1^0,x_2^0)$};
\end{axis}
\node at (3,-1) [above left] {(a)};
\end{tikzpicture}\hspace{10mm}
\begin{tikzpicture}
\begin{axis}[width=0.55*\textwidth,
axis lines=middle,
inner axis line style={dashed},
xmin=0,
xmax=1.5,
ymin=0,
ymax=1.5,
zmin=0,
zmax=1.5,
xtick={0,2},
ytick={0,2},
ztick={0,2},
view={135}{30}]
\addplot3[patch, color=blue, fill opacity=0.1, faceted color=black, line width=0.8pt] coordinates{(1,0,0) (0,1,0) (0,0,1)};
\draw[decoration={markings,mark=at position 1 with
{\arrow[scale=1.5,>=stealth]{>}}},postaction={decorate},color=red, line width=1.2pt,->,>=stealth] ({1/5},{7/15},{1/3})--({2/3},0,{1/3});
\draw[decoration={markings,mark=at position 1 with
{\arrow[scale=1.5,>=stealth]{>}}},postaction={decorate},color=red, line width=1.2pt,->,>=stealth]({2/3},0,{1/3})--(0,0,1);
\draw(1,0,0)--(1.5,0,0);
\draw(0,1,0)--(0,1.5,0);
\draw(0,0,1)--(0,0,1.5);
\fill(1,0,0) circle (2.4pt);
\fill(0,1,0) circle (2.4pt);
\fill(0,0,1) circle (2.4pt);
\fill({1/5},{7/15},{1/3}) circle (2.4pt);
\fill({2/3},0,{1/3}) circle (2.4pt);
\node at (1,0,0) [above left] {$P_2$};
\node at (0,1,0) [above right] {$P_3$};
\node at (0,0,1) [above right] {$P_1$};
\node at ({1/5},{7/15},{1/3}) [above right] {$P_0$};
\node at ({2/3},0,{1/3}) [above left] {$P_0^{'}$};
\end{axis}
\path (2.6,4.3) node(x) {$\ell_{12}$};
\path (3.1,3.2) node(x) {$\ell$};
\node at (4,-1) [above left] {(b)};
\end{tikzpicture}
\caption{Picture~(a) represents the 1-dimensional case: the length of the red segment is $\delta(P_0)$. Picture~(b) represents the 2-dimensional case: here $\delta(P_0)$ is the sum of the length of the two red segments}\label{figura1}
\end{figure}
\begin{equation}\label{delta1}
\delta(P_0)=a(1-x_{1}^{0}).
\end{equation}
In the previous formula the choice $a=\sqrt{2}$ gives the standard Euclidean distance, while for $a=1$ \eqref{delta1} simplifies to $\delta(P_0)=1-x_{1}^{0}$.
Let us consider now the $2$-dimensional case, represented in Figure~\ref{figura1}~(b), corresponding to the case in which each publication can be evaluated in $3$ possible ways: the ``best'', the ``intermediate'' and the ``worst'' ones. Unlike the $1$-dimensional case, here there is no natural order relation which can be used for giving an immediate measurement of how far $\delta(P_0)$ is from the best possible evaluation $P_1$. We proceed in the following way. As before, let us express by two positive real numbers $a$ and $b$ the ``effort'' for going from $P_{2}$ to $P_{1}$ and from $P_{3}$ to $P_{2}$, respectively. If $O$ denotes the origin in ${\mathbb R}^3$, the vectors $\mathbf{v}_{1}:=P_{1}- O = (1,0,0)$, $\mathbf{v}_{2}:=P_{2}- P_{1} = (-1,1,0)$ and $\mathbf{v}_{3}:=P_{3}- P_{2} = (0,-1,1)$ are linearly independent and thus they form a basis of $\mathbb{R}^3$. Let us consider the scalar product $g$ on $\mathbb{R}^3$, represented by the matrix
\begin{equation*}
M_{\mathcal{B}}(g)=
\begin{pmatrix}
1 & 0 & 0 \\
0 & a^2 & 0 \\
0 & 0 & b^2
\end{pmatrix}
\end{equation*}
with respect to the basis $\mathcal{B}=\left\{ \mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3} \right\}$. This scalar product induces a distance function $d_{g}$ on $\mathbb{R}^3$ in the usual way
\begin{equation*}
d_{g}(P,Q)=\left\| P-Q \right\|_{g} = \sqrt{ g\left( P-Q , P-Q \right) }.
\end{equation*}
The restriction of $d_g$ to $\Delta_{2}$ provides the simplex with a distance, that will be still denoted by $d_g$. Let $P'_0$ be the intersection of the edge $\ell_{1 2}$ of the simplex through the points $P_1$ and $P_2$ with the line $\ell$ through $P_0=(x_1^0, x_2^0, x_3^0)$ with direction $\mathbf{v}_3$ (see Figure \ref{figura1}). If we restrict our attention to the line $\ell$, we recover the natural order relation in the $1$-dimensional case previously considered. More precisely, as one moves from $P_0$ to $P'_0$ along $\ell$, the corresponding outcome is ``improving'' in a natural way, as if we were continuously transferring part of publications that received the worst evaluation to the intermediate class. Once arrived at $P'_0$, we recover again the natural order provided by the geometry of the line $\ell_{1 2}$ connecting $P_1$ to $P_2$. Here, moving from $P'_0$ to $P_1$ corresponds to an increase of the frequency of publications in the best class and a consequent decrease of the frequency in the intermediate class. In this way we are able to find a ``natural'' path connecting $P_0$ to $P_1$ (the red path in Figure \ref{figura1}). Thus we can measure the ``distance'' $\delta(P_0)$ from $P_0$ to $P_1$ by the length of this path:
\begin{equation*}
\delta(P_0)=d_{g}(P_{0}, P'_{0}) + d_{g}(P'_{0}, P_{1}).
\end{equation*}
Since the line $\ell$ can be represented by the parametric equations
\begin{equation*}
\begin{cases}
x_1=x_1^0\\
x_2=x_2^0-t\\
x_3=x_3^0-t
\end{cases}
\end{equation*}
and the line $\ell_{12}$ has Cartesian equations
\begin{equation*}
\begin{cases}
x_1 + x_2 = 1\\
x_3 = 0
\end{cases}
\end{equation*}
the point $P'_0$ has coordinate $(x_{1}^0, 1 - x_{1}^0, 0)$ and so we find
\begin{align*}
\delta(P_0)&=\left\| P'_{0}-P_{0} \right\|_g + \left\| P_{1}-P'_{0} \right\|_g\\
&=\left\| -x_{3}^{0} \mathbf{v}_{3} \right\|_g + \left\| -(1-x_{1}^{0}) \mathbf{v}_{2}\right\|_g\\
&=x_{3}^{0} \sqrt{g(\mathbf{v}_{3},\mathbf{v}_{3})} + (1-x_{1}^{0}) \sqrt{g(\mathbf{v}_{2},\mathbf{v}_{2})}\\
&=(1-x_{1}^{0}-x_{2}^{0})b + (1-x_{1}^{0})a\\
&=(a+b) - (a+b)x_{1}^{0} - b x_{2}^{0}.
\end{align*}
If the evaluation call contemplates four different assessment classes, our model is encoded by the geometry of the $3$-dimensional simplex $\Delta_3$. Let $P_{0}=(x_{1}^{0},x_{2}^{0},x_{3}^{0},x_{4}^{0}) \in \Delta_3$ be the evaluation of the department. Then, generalising the above constructions, we are going to define a natural path joining $P_0$ to the best possible evaluation $P_1$. Here the ``naturality'' of such a path means that: (i) for any $i,j \in\left\{2,3,4\right\}$ such that $i > j$ the path from $P_i$ to $P_1$ should be longer than the path from $P_j$ to $P_1$; (ii) this property should be satisfied also by the vertices of any $2$-dimensional simplex given by the intersection of $\Delta_3$ with a plane whose direction is spanned by $P_{3}-P_{2}$ and $P_{4}-P_{2}$. Let $a$, $b$, $c$ denote positive real numbers expressing the ``effort'' for going from $P_{2}$ to $P_{1}$, from $P_{3}$ to $P_{2}$, and from $P_{4}$ to $P_{3}$, respectively.
As in the previous case the vectors $\mathbf{v}_{1}:=P_{1}- O$ and $\mathbf{v}_{i}:=P_{i+1}-P_{i}$, $i=1,2,3$, form a basis $\mathcal{B}$ of $\mathbb{R}^4$ and we consider the scalar product $g$ on $\mathbb{R}^4$, represented by the matrix
\begin{equation*}
M_{\mathcal{B}}(g)=
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & a^2 & 0 & 0\\
0 & 0 & b^2 & 0\\
0 & 0 & 0 & c^2
\end{pmatrix}\,.
\end{equation*}
This scalar product induces a distance function $d_{g}$ on $\mathbb{R}^4$ and $\Delta_{3}$ becomes a metric space with distance the restriction of $d_{g}$ to ${\Delta_3}$. Let $\pi$ denote the plane through $P_0$ and parallel to the plane containing the points $P_{2}, P_{3}, P_{4}$ (i.e. all the vertices of the simplex except the one corresponding to the best evaluation). Notice that $\pi$ has parametric equations
\begin{equation*}
\begin{cases}
x_1=x_1^0\\
x_2=x_2^0-t-s\\
x_3=x_3^0+t\\
x_4=x_4^0 + s
\end{cases}
\end{equation*}
The intersection of $\pi$ with the simplex $ \Delta_3$ is a $2$-dimensional simplex, the equilateral triangle with vertices $P'_{1}, P'_{2}, P'_{3}$. From the triangle obtained we can recover the $2$-dimensional construction. The path from $P_0$ to $P_1$ is now determined by the union of 3 segments (see Figure~\ref{figura2}).
\begin{figure}
\begin{tikzpicture}
\begin{axis}[width=0.6\textwidth,
axis lines=middle,
inner axis line style={color=white},
xmin=-1.4,
xmax=1.7,
ymin=-1.4,
ymax=1.7,
zmin=-1.4,
zmax=1.7,
xtick={0,6},
ytick={0,6},
ztick={0,6},
view={105}{-5}
]
\addplot3[dashed, patch, color=blue, fill opacity=0.1, faceted color=black, line width=0.4pt] coordinates{(1,0,0) (0,1,0) (0,0,1)};
\draw[ decoration={markings,mark=at position 1 with
{\arrow[scale=1.5,>=stealth]{>}}},postaction={decorate},color=red, line width=1.2pt,->,>=stealth] ({1/5},{7/15},{1/3})--({2/3},0,{1/3});
\draw(0,1,0)--(0,0,1);
\draw[ decoration={markings,mark=at position 1 with
{\arrow[scale=1.5,>=stealth]{>}}},postaction={decorate},color=red, line width=1.2pt,->,>=stealth]({2/3},0,{1/3})--(0,0,1);
\draw[decoration={markings,mark=at position 1 with
{\arrow[scale=1.5,>=stealth]{>}}},postaction={decorate},color=red, line width=1.2pt,->,>=stealth](0,0,1)--(1.5,1.5,1.5);
\fill(1,0,0) circle (2.4pt);
\fill(0,1,0) circle (2.4pt);
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\fill({1/5},{7/15},{1/3}) circle (2.4pt);
\fill({2/3},0,{1/3}) circle (2.4pt);
\node at (1,0,0) [below right] {$P_2^{'}$};
\node at (0,1,0) [right] {$P_3^{'}$};
\node at (0,0,1) [above left] {$P_1^{'}$};
\node at ({1/5},{7/15},{1/3}) [below] {$P_0$};
\node at ({2/3},0,{1/3}) [left] {$P_0^{'}$};
\node at (-1.25299, -1.25299, 0.582337) [above left] {${P_2}$};
\node at (0.582337, -1.25299, -1.25299) [left] {${P_3}$};
\node at (-1.25299, 0.582337, -1.25299)[right] {${P_4}$};
\fill(1.5,1.5,1.5) circle (2.4pt);
\node at (1.5,1.5,1.5) [right] {$P_1$};
\draw(0,0,1) -- (-1.25299, -1.25299, 0.582337);
\draw[dashed](1.5,1.5,1.5)--(0.582337, -1.25299, -1.25299);
\draw(1.5,1.5,1.5)--(-1.25299, 0.582337, -1.25299);
\draw(-1.25299, -1.25299, 0.582337)--(-1.25299, 0.582337, -1.25299)--(0.582337, -1.25299, -1.25299);
\draw(-1.25299, -1.25299, 0.582337)--(0.582337, -1.25299, -1.25299);
\fill(-1.25299, -1.25299, 0.582337) circle (2.4pt);
\fill(-1.25299, 0.582337, -1.25299) circle (2.4pt);
\fill(0.582337, -1.25299, -1.25299) circle (2.4pt);
\end{axis}
\path (2.6,4.8) node(x) {$\ell_{12}$};
\path (6.2,1.8) node(x) {$\ell_{14}$};
\path (2.6,2.) node(x) {$\ell_{13}$};
\end{tikzpicture}
\caption{Representation of the 3-dimensional case: here $\delta(P_0)$ is the sum of the length of the three red segments}\label{figura2}
\end{figure}
Since the line $\ell_{i j}$ joining $P_i$ with $P_j$, $i,j\in\left\{1,2,3\right\}$, has equations
\begin{equation*}
\begin{cases}
x_i + x_j = 1\\
x_k = 0 \ \hbox{ for any } k\notin \left\{i,j\right\}
\end{cases}
\end{equation*}
we find that the coordinates of the vertices of the $2$-dimensional simplex are $P'_{1}=(x_{1}^{0}, 1-x_{1}^{0},0,0)$, $P'_{2}=(x_{1}^{0}, 0,1-x_{1}^{0},0)$, $P'_{3}=(x_{1}^{0},0, 0,1-x_{1}^{0})$. In order to find the coordinates of $P'_0$, let $\ell'_{12}$ and $\ell$ denote, respectively, the line joining $P'_{1}$ with $P'_{2}$ and the line through $P_0$ with direction $P'_{3}-P'_{2} $. Such two coplanar lines have equations
\begin{equation*}
\ell'_{12}:
\begin{cases}
x_1 = x_{1}^{0}\\
x_2 + x_3 = 1 - x_{1}^{0} \\
x_4 = 0
\end{cases} \quad \quad \ell: \begin{cases}
x_1=x_1^0\\
x_2=x_2^0\\
x_3=x_3^0-(1-x_{1}^{0}) t\\
x_4=x_4^0 + (1 - x_{1}^{0})t
\end{cases}
\end{equation*}
Thus $P'_{0}= \ell'_{12} \cap \ell= (x_{1}^{0}, x_{2}^{0}, 1 - x_{1}^{0}-x_{2}^{0}, 0)$. Hence
\begin{align*}
\delta(P_0)&=d_{g}(P_{0}, P'_{0}) + d_{g}(P'_{0}, P'_{1}) + d_{g}(P'_{1}, P_{1})\\
&=\left\| P'_{0}-P_{0} \right\|_g + \left\| P'_{1}-P'_{0} \right\|_g + \left\| P_{1}-P'_{1} \right\|_g \\
&=\left\|-x_{4}^{0} \mathbf{v}_{4} \right\|_g + \left\| -(1-x_{1}^{0}-x_{2}^{0})\mathbf{v}_{3} \right\|_g + \left\| -(1-x_{1}^{0})\mathbf{v}_2 \right\|_g \\
&=x_{4}^{0} \sqrt{g(\mathbf{v}_{4},\mathbf{v}_{4})} + (1-x_{1}^{0}-x_{2}^{0}) \sqrt{g(\mathbf{v}_{3},\mathbf{v}_{3})} + (1-x_{1}^{0})\sqrt{g(\mathbf{v}_{2},\mathbf{v}_{2})}\\
&=(1-x_{1}^{0}-x_{2}^{0}-x_{3}^{0}) c + (1-x_{1}^{0}-x_{2}^{0})b + (1-x_{1}^{0})a\\
&=(a+b+c) - (a+b+c)x^{0}_{1} - (b+c)x_{2}^{0} - c x_{3}^{0}\,.
\end{align*}
In the general case, when one has $n+1$ categories, iterating the above constructions, the defined path joining $P_0=(x_{1}^{0},\ldots,x_{n+1}^{0})$ with $P_1$ can be obtained as the union of $n$ line segments, each of which lying in an
$(n-1)$-dimensional simplex. Thus we we finally obtain the following formula
\begin{equation}\label{eq:delta-n}
\delta(P_0)= a_{1}+\cdots + a_{n} - (a_{1}+\cdots + a_{n})x_{1}^{0} - (a_{2}+\cdots + a_{n}) x_{2}^{0} - \cdots - (a_{n-1}+a_{n}) x_{n-1}^{0} - a_{n} x_{n}^{0}
\end{equation}
where, for each $i \in \left\{1,\ldots, n\right\}$, $a_i$ is a positive real number expressing the ``effort'' for going from the vertex $P_{i+1}$ to $P_{i}$.
A rigorous proof of \eqref{eq:delta-n} can be done by induction and it is reported in the Appendix.
\begin{remark}
The constants $a_i$ in \eqref{eq:delta-n}, i.e. the quantification of the effort for going from a category to that one immediately higher, should be explicated in the call. The most frequent situation is when all such efforts are considered equivalent, so that the constants can be taken all equal to $1$. In this case \eqref{eq:delta-n} simplifies to
\begin{equation}\label{eq:delta-n-bis}
\delta(P_0)= n - n x_{1}^{0} - (n-1) x_{2}^{0} - \cdots - 2 x_{n-1}^{0} - x_{n}^{0}.
\end{equation}
In this particular case, $\delta(P_0)$ can be expressed also in terms of the Minkowski distance $d_M$, often applied in measuring dissimilarity of ordinal data - some recent application in this direction can be found in \cite{Weiss2019} and \cite{Weiss2020}. In this regard, using the bijection \eqref{bijection}, we have
\begin{align*}
\delta(P_0)&= n - n x_{1}^{0} - (n-1) x_{2}^{0} - \cdots - 2 x_{n-1}^{0} - x_{n}^{0}\\
&=(1 - x_{1}^{0}) + (1 - x^{0}_{1} - x^{0}_{2}) + \cdots + (1 - x^{0}_{1} - x^{0}_{2} - \cdots - x^{0}_{n})\\
&=|x_{1}^{0} - 1| + | x^{0}_{1} + x^{0}_{2} - 1| + \cdots + | x^{0}_{1} + x^{0}_{2} + \cdots + x^{0}_{n} - 1|\\
&=d_{M}( (x^{0}_{1}, x^{0}_{1} + x^{0}_{2}, \ldots, x^{0}_{1} + x^{0}_{2} + \cdots + x^{0}_{n} , 1), (1, 1, \ldots, 1, 1))\\
&=d_{M}(\varphi(x^{0}_{1}, x^{0}_{2}, \ldots, x^{0}_{n}, x^{0}_{n+1}), \varphi(1, 0, \ldots, 0, 0))\\
&=d_{M}(\varphi({P}_{0}), \varphi({P}_{1}))\,.
\end{align*}
However, there could be - and actually there were - situations when the assumption $a_1=\cdots a_n = 1$ can not be necessarily taken. One example is the VQR 2011-14 which will be discussed in Section \ref{VQR}.
\end{remark}
\section{Geometric score function}
Starting from $\delta(P_0)$, see equation \eqref{eq:delta-n}, we can naturally define a map
\begin{equation*}
d : \Delta_{n} \times \Delta_{n} \longrightarrow \mathbb{R}
\end{equation*}
such that for any $P_{0}, Q_{0} \in \Delta_{n}$
\begin{equation*}
d(P_{0},Q_{0}):=\mid \delta(P_0) - \delta(Q_0) \mid.
\end{equation*}
In other words, we are comparing the evaluations $P_0$ and $Q_0$ of two departments, measuring how ``far'' is each one from $P_1$. Note that $d$ is clearly non-negative and symmetric. Moreover, it satisfies the triangular inequality, since
\begin{align*}
d(P,P'')&=\mid \delta(P) - \delta(P'') \mid \\
&= \mid \delta(P) - \delta(P') + \delta(P')- \delta(P'') \mid \\
& \leq \mid \delta(P) - \delta(P') \mid + \mid \delta(P')- \delta(P'') \mid \\
&=d(P,P') + d(P',P'').
\end{align*}
Note that $d$ is a pseudo-distance, since it does not satisfies the \emph{identity of indiscernibles} condition. In fact $d(P,P')=0$ does not necessarily imply that $P=P'$. For instance, in the $2$-dimensional case, the points $P=(\frac{1}{4},\frac{3}{4},0)$ and $P'=(\frac{1}{2},\frac{1}{4},\frac{1}{4})$ are such that $\delta(P)=\delta(P')=\frac{3 \sqrt{2}}{4}$, so that $d(P,P')=0$.
\medskip
In the applications, in order to compare the assessments of different departments or other aggregates, it is important to consider the locus of points of the simplex $\Delta_{n}$ that are at distance $0$ from each other. Geometrically, in view of \eqref{eq:delta-n}, such a set can be described by the sheaf of parallel hyperplanes of equation
\begin{equation}\label{hyperplane}
(a_{1}+\cdots + a_{n})x_{1}^{0} +(a_{2}+\cdots + a_{n}) x_{2}^{0} + \cdots + (a_{n-1}+a_{n}) x_{n-1}^{0} + a_{n} x_{n}^{0} = \textrm{const}.
\end{equation}
This allows us to divide the aggregates under study in equivalent classes, corresponding to such loci. More formally, one can consider the relation $\sim$ on $\Delta_{n}$ that identifies any two points $P$ and $P'$ such that $d(P,P')=0$. It can be easily proved that $\sim$ is an equivalence relation and then $d$ turns out to be a distance on the quotient set $\Delta_{n} / \sim$. Thus $\Delta_n$ can be partitioned into equivalence classes, which correspond to the points of the simplex belonging to each hypeplane \eqref{hyperplane} (see Figure~\ref{figura4}~(a)).
\begin{figure}[b]
\begin{tikzpicture}
\begin{axis}[width=0.55*\textwidth,
axis lines=middle,
inner axis line style={dashed},
xmin=0,
xmax=1.5,
ymin=0,
ymax=1.5,
zmin=0,
zmax=1.5,
xtick={0,2},
ytick={0,2},
ztick={0,2},
view={135}{30}]
\draw[color=purple, line width=1.2pt](1,0,0)--(0,.5,.5);
\draw[color=purple, line width=1.2pt](0.5, 0., 0.5)--(0., 0.25, 0.75);
\draw[color=purple, line width=1.2pt](0.75, 0., 0.25)--(0., 0.375, 0.625);
\draw[color=purple, line width=1.2pt](0.75, 0.25, 0.)--(0., 0.625, 0.375);
\draw[color=purple, line width=1.2pt](0.5, 0.5, 0.)--(0., 0.75, 0.25);
\addplot3[patch, color=blue, fill opacity=0.1, faceted color=black, line width=0.8pt] coordinates{(1,0,0) (0,1,0) (0,0,1)};
\draw(1,0,0)--(1.5,0,0);
\draw(0,1,0)--(0,1.5,0);
\draw(0,0,1)--(0,0,1.5);
\fill(1,0,0) circle (2.4pt);
\fill(0,1,0) circle (2.4pt);
\fill(0,0,1) circle (2.4pt);
\node at (1,0,0) [above left] {$P_2$};
\node at (0,1,0) [above right] {$P_3$};
\node at (0,0,1) [above right] {$P_1$};
\end{axis}
\node at (3,0) [above right] {(a)};
\end{tikzpicture}\hspace{5mm}
\begin{tikzpicture}
\begin{axis}[width=0.55*\textwidth,
axis lines=middle,
inner axis line style={dashed},
xmin=0,
xmax=1.5,
ymin=0,
ymax=1.5,
zmin=0,
zmax=1.5,
xtick={0,2},
ytick={0,2},
ztick={0,2},
view={135}{30}]
\draw[color=purple, line width=1.2pt](0.75, 0., 0.25)--(0., 0.375, 0.625);
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\draw(1,0,0)--(1.5,0,0);
\draw(0,1,0)--(0,1.5,0);
\draw(0,0,1)--(0,0,1.5);
\fill(1,0,0) circle (2.4pt);
\fill(0,1,0) circle (2.4pt);
\fill(0,0,1) circle (2.4pt);
\fill(0.225, 0.2625, 0.5125) circle (2.4pt);
\fill[color=red](0.175, 0.7875, 0.0375) circle (2.4pt);
\fill[color=red](0.225, 0.6625, 0.1125) circle (2.4pt);
\fill[color=red](0.45, 0.4, 0.15) circle (2.4pt);
\fill[color=red](0.19, 0.43, 0.38) circle (2.4pt);
\fill[color=red](0.72, 0.19, 0.09) circle (2.4pt);
\fill[color=blue](0.4, 0.05, 0.55) circle (2.4pt);
\fill[color=blue](0.12, 0.24, 0.64) circle (2.4pt);
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\node at (1,0,0) [above left] {$P_2$};
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\node at (0,0,1) [above right] {$P_1$};
\end{axis}
\node at (3,0) [above right] {(b)};
\end{tikzpicture}
\caption{In Picture~(a) a line of the pencil represents points with the same score function. In Picture~(b) a geometric interpretation of ${\mathcal S}_{\mathcal{C}}(A)$}\label{figura4}
\end{figure}
\medskip
We introduce a \emph{score function} for the evaluation of an aggregate $A$ (a department, a scientific area, etc.) in the following way: the assessment of $A$ can be realized as a point $P_A$ of $\Delta_n$, to which we can associate the number $\delta(P_A)$. This is, of course, an ``absolute'' index, in the sense that it can be used for comparing evaluations among homogenous aggregates. For instance, it can be used to assign research fundings or to compare the quality of research of different candidates in a competition within the same scientific discipline. More in general, its usage can go beyond the context of this paper, i.e. research evaluation, since in principle it can be used whenever one deals with a situation where there are ordinal assessments.
However, the above index can not in principle be appropriate to compare two inhomogeneous situations, and in particular the evaluations of departments, which are usually composed of researchers working in different scientific areas. In order to overcome this problem, we propose the following general approach. Let us fix an aggregate $A$ within a certain research evaluation call. We denote by $\mathfrak{I}(A,\mathcal{C})$ the set of all ``ideal'' aggregates whose size and configuration (with respect to some prefixed criteria $\mathcal{C}$) are the same of $A$, and whose members are randomly selected from the set of all researchers that satisfy $\mathcal{C}$, working in any other university partecipating to the same assessment call. To any element of $\mathfrak{I}(A,\mathcal{C})$ it can be associated a point in the simplex. Then we define the \emph{geometric score} of $A$ as
\begin{equation}\label{geometric-score}
{\mathcal S}_{\mathcal{C}}(A)=\frac{|\mathfrak{I}^{-}(A,\mathcal{C})|}{|\mathfrak{I}(A,\mathcal{C})|}
\end{equation}
where $|\mathfrak{I}(A,\mathcal{C})|$ denotes the cardinality of the set $\mathfrak{I}(A,\mathcal{C})$ and $|\mathfrak{I}^{-}(A,\mathcal{C})|$ denotes the number of ideal aggregates in $\mathfrak{I}(A,\mathcal{C})$ that are represented geometrically as the points of the simplex which are lying below the hyperplane \eqref{hyperplane} determined by $A$ (see Figure~\ref{figura4}~(b)). ${\mathcal S}_{\mathcal{C}}(A)$ represents the probability that an ``ideal'' aggregate $A'$, with the same configuration as $A$ (hence comparable with $A$), performs worse than $A$.
The choice of the conditions $\mathcal{C}$ depends on the availability of the data and should be aimed to make the evaluation as homogeneous as possibile. For instance, $\mathcal{C}$ may consist in the requirements that an ideal element of $\mathfrak{I}(A,\mathcal{C})$ must be composed of the same number as $A$ of researchers belonging to a given scientific area, and/or by the same number as $A$ of full / associate / assistant professors and/or the same proportion as $A$ of male and female researchers. Other examples for $\mathcal{C}$ may involve further information on researchers working in $A$, like teaching duties, salary, age, as well as description of the social and economic context where $A$ operates. In principle, the more the possible choices for $\mathcal{C}$ are various and
precise, the more the evaluation is accurate and non-misleading. However, even in the extreme case when $\mathcal{C}=\emptyset$ the geometric score \eqref{geometric-score} is still informative, since it avoids comparisons between aggregates of different size.
From the definition \eqref{geometric-score} and the above considerations, it is clear that the geometric score of $A$ strongly depends on $\mathcal{C}$. This reflects mathematically the fact that there does not exist ``the'' evaluation of $A$, but in fact there are many possible evaluations, each of them depending on the contextual aspects and on the possible refinements that the user (the policy maker, the academician, the future student, the citizen, etc.) is interested to.
Operatively, the institution responsible of the evaluation (in most cases the national evaluation agency) should make available a wide range of information regarding the researchers involved in the evaluation as well as the universities where they work. The utopia could consist in having access to a web-site where the user, according to his/her objectives, may select the most appropriate information that form the criteria $\mathcal{C}$, and consequently compute the corresponding geometric score.
\section{A case study: Italian research assessment VQR 2011-14}\label{VQR}
As an application, we consider the outcomes of the Italian Research Assessment VQR 2011-2014 within two scientific areas, the area of \emph{Mathematics} and the area of
\emph{Statistics and Mathematical Methods for Decisions} (coded, respectively, 01/A and 13/D according to the Italian scientific disciplines codification). These two areas are made of a number of smaller aggregates known in the Italian system as Disciplinary Scientific Sectors (SSDs), as described in Table \ref{descrizione-01A} and Table \ref{descrizione-13D}.
\begin{table}[htbp]
\centering
\tiny
\caption{Composition of the area 01/A - Mathematics according to Italian Higher Education legislation}\label{descrizione-01A}
\begin{tabular}{|l|l|}
\hline
{SSD Code} & {Description} \\
\hline
MAT/01 & Mathematical logic \\
MAT/02 & Algebra \\
MAT/03 & Geometry \\
MAT/04 & Mathematics education and history of mathematics \\
MAT/05 & Mathematical analysis \\
MAT/06 & Probability and mathematical statistics \\
MAT/07 & Mathematical physics \\
MAT/08 & Numerical analysis \\
MAT/09 & Operational research \\
\hline
\end{tabular}
\end{table}
\medskip
\begin{table}[htbp]
\centering
\tiny
\caption{Composition of the area 13/D - Statistics and Mathematical Methods for Decisions according to Italian Higher Education legislation}\label{descrizione-13D}
\begin{tabular}{|l|l|}
\hline
{SSD Code} & {Description} \\
\hline
SECS-S/01 & Statistics \\
SECS-S/02 & Statistics for experimental and technological research \\
SECS-S/03 & Economic statistics \\
SECS-S/04 & Demography \\
SECS-S/05 & Social statistics \\
SECS-S/06 & Mathematical methods for economics, actuarial and financial sciences \\
\hline
\end{tabular}
\end{table}
The choice of these two areas is due to the different criteria adopted by the respective committees for the evaluation of the publications.
We recall that in the 2011-2014 VQR edition each researcher was expected to submit a given number of products (in most cases $2$). The submitted scientific products were classified by a committee in one among the following classes, each one associated to a score: Excellent - A (score: 1), Good - B (score: 0.7), Fair - C (score: 0.4), Acceptable - D (score: 0.1), Limited or not assessable - E (score: 0).
Then to each university a so-called ``normalized average score'', denoted by $R$, was associated and was used by ANVUR to draw up a ranking. We recall that the $R$ score was computed as the ratio between the average score of the researchers of a given university in a given area / SSD and the average score of all the Italian researchers in that area / SSD (see \cite{ANVUR:2017-2}). A $R$ score greater than $1$ indicates that, in a given area / SSD, the university under consideration performs better than the average of the all Italian universities. The final ranking, for all scientific areas, is available on the ANVUR's web-site (\cite{ANVUR:2017-2}).
Notice that the value of $R$ for each university, and hence the corresponding ANVUR ranking, is strongly linked to the score associated to each class, which even when reasonable, it is still arbitrary. Moreover, $R$ depends on the internal composition of the aggregate inside each university, which can considerably differ among universities. For instance, as for the 13/D area, the percentage repartition in the $6$ SSDs described in Table \ref{descrizione-13D} is $40$, $4$, $13$, $7$, $4$, $33$ for Roma ``La Sapienza'', while it is $53$, $0$, $0$, $0$, $0$, $47$ for the Milano Politecnico. Last, $R$ is sensible to the size of the aggregate. This can generate some bias due to the so-called ``funnel effect'' (\cite{Spiegelhalter}).
\medskip
We have done a parallel ranking using our geometric score.
In order to apply the methods discussed in the previous sections, we need to understand how to choose, in this circumstance, the constants $a_{i}$ of formula \eqref{eq:delta-n}. Such positive numbers should correspond to the effort for moving from one category to the upper one, and should be known by the evaluating committee / referees. Unfortunately in this case there is not a univocal answer. A first option is to take into account the description of each class in the VQR call\footnote{\url{https://www.anvur.it/attivita/vqr/vqr-2011-2014/riferimenti-normativi-e-regolamentari/}}, which we have summarised in Table~\ref{VQR-call}. Indeed the call states that a publication should be considered \emph{excellent} if, ideally, it falls in the highest 10\% of the distribution of the international scientific research production of the Area in the period 2011-2014, \emph{good} if it is in the 10-30\% segment, \emph{fair} if it is in the 30-50\% segment, \emph{acceptable} if it is in the 50-80\% segment, \emph{limited or not assessable} if it is in the 80-100\% segment.
Then a possible choice for $a_i$ could be to consider the effort for an upgrade to the higher class, measured as the distance between the lowest limit of each segment and the lowest limit of the next upper category.
Namely, if we put $a_{4}=1$, since the length of the interval corresponding to the class D is $1.5$-times longer than the ones of the classes B and C, one should have $a_{1}=a_{2}=a_{4}=1$ and $a_{3}=1.5$.
\begin{table}[htbp]
\centering
\tiny
\caption{Description of each category in the VQR 2011-14 call and scores assigned by ANVUR to each class}\label{VQR-call}
\begin{tabular}{|l|cc|}
\hline
Category & {Percentile in the distribution of worldwide publications in the area } & {ANVUR scores} \\
\hline
A (Excellent) & 90--100 & 1 \\
B (Good) & 70--90 & 0.7 \\
C (Fair) & 50--70 & 0.4 \\
D (Aceptable) & 20--50 & 0.1 \\
E (Limited or not assessable) & 0--20 & 0 \\
\hline
\end{tabular}
\end{table}
On the other hand, one can argue that a referee is aware of the scores assigned by ANVUR to each publication falling in a certain class and used for computing the $R$ score. Thus another way for determining $a_i$ is to consider the difference between any two consecutive of such scores, obtaining $a_{1}=a_{2}=a_{3}=3$, $a_{4}=1$.
Finally, according to the codification and qualitative description used by ANVUR for the assessment of the scientific products, the referee may perceive the classes as ``equidistant''. In this case we can assume $a_{i}=1$ for each $i\in\left\{1,\ldots,4\right\}$.
It is then interesting to test our geometric score in all these three situations and to compare the results with the official ANVUR outcomes. We fix as $\mathcal{C}$ the property ``each ideal aggregate should have the same composition, with respect to each SSD, as the university under consideration'' .
Since for privacy reasons we can not associate the assessment of a research product to its author, we can not proceed by sampling directly from the set of all researchers in the area under consideration. Then, starting from the internal composition of the areas 01/A and 13/D, for a given university $U$ we have simulated an ideal aggregate with the same number of expected products in each SSD as $U$ and we have computed the corresponding score function. In principle one should compute the score function $\delta$ for all possible ideal aggregates that one can construct. However, the number of such ideal aggregates can be quite huge. For instance, for the 01/A Area of Roma ``La Sapienza'' (the biggest Italian university), this number is $43.65\times 10^{401}$.
To avoid to consider the enormous number of all possible combinations, we calculate the geometric score for each university in the two scientific areas by means of Monte Carlo simulations.
The algorithm used follows a simple scheme. Once fixed the university $U$, we denote with $m_i$ the number of product of the SSDs MAT/0i, $i=1, \ldots,9$ (or alternatively, for the 13/D Area, SECS-S/0i, $i=1, \ldots,6$).
We start with simulating $m_i$ products for the SSD MAT/0i belonging to one of the $5$ categories (from A to E) as a sample from a multivariate hypergeometric distribution with parameters $m_{i j}, j=1, \ldots,5$, the number of publications of Italian researchers in the SSD MAT/0i belonging to the category $j$ (see Tables \ref{VQR-math}--\ref{VQR-stat}). We then obtain the configuration of an ``ideal aggregate'' for which we can compute the value of $\delta$. We repeat such procedure $N$ times and we obtain $N$ values of the indexes $\delta_l, l=1, \ldots,N$. We then compare all values obtained with that of the university $U$. This allows us to calculate the proportions of ideal aggregates performing worse than $U$. Since for each $l$ the random variable $I_{\{\delta_l > \delta(U)\}}$ ($I_A$ being the indicator function of the set $A$) follows a Bernoulli distribution with probability of success equal to ${\mathcal S}_{\mathcal{C}}(U)$, by the strong law of large numbers we can then approximate the value of our geometric score with $ \frac{\sum_{l=1}^{N} I_{\{\delta_l > \delta(U)\}}}{N}$. Then we can determine $N$ imposing that $P\Big( \frac{\sum_{l=1}^{N} I_{\{\delta_l > \delta(U)\}}}{N}-{\mathcal S}_{\mathcal{C}}(U) \geq 0.005 \Big)$ is very small (for instance of the order of $10^{-5}$). Using the Hoeffding's inequality (\cite{Hoeffding: 1963}) we have
$$P\Big( \frac{\sum_{l=1}^{N} I_{\{\delta_l > \delta(U)\}}}{N}-{\mathcal S}_{\mathcal{C}}(U) \geq 0.005 \Big) \leq e^{-2N\cdot 0.005^2 }.$$ For $N=200000$, that we fix as the number of simulations in all cases considered, we have $e^{-2N\cdot 0.005^2 }=4.539993 \cdot 10^{-5}$.
We repeat such procedure for all universities in the area 01/A and then in the area 13/D.
The rankings obtained for the areas of Mathematics (01/A) and Statistics and Mathematical Methods for Decisions (13/D) are reported in Table \ref{tab:01Arank} and in Table \ref{tab:13Drank}, respectively. We have reported the official results of VQR 2011-2014 evaluation and compared it with the results arising by the application of the new method proposed considering the calculation of $\delta$ with the aforementioned three ways of choices of the constant $a_i$.
From the tables we can see how the rankings obtained with our score are quite different to that obtained from the $R$ score used by ANVUR for the VQR assessment. In order to verify the magnitude of the differences obtained, we applied a Kruskal-Wallis non-parametric test to globally compare the four rankings. It emerged that the rankings were statistically different (p-value $ =4.627\cdot 10^{-13}$ for the area of Mathematics, and p-value $=1.377 \cdot 10^{-9}$ for the area of Statistics and Mathematical Methods for Decision). A pairwise comparison, through a post-hoc statistical test based on Wilcoxon statistic, confirmed that there is a difference between the ranking based on $R$ score and each of the rankings determined through ${\mathcal S}_{\mathcal{C}}$ (p-values were of order $10^{-10}$ when considering the scientific area 01/A and of order $10^{-7}$ for the area 13/D). On the other hand there are no relevant differences when comparing between them the rankings obtained from the three versions of ${\mathcal S}_{\mathcal{C}}$ (p-values above $0.94$ in both scientific areas).
\begin{table}[htbp]
\centering
\tiny
\begin{tabular}{|l|cc|ccc|ccc|ccc|}\hline
& & & \multicolumn{3}{|c|}{(A)} & \multicolumn{3}{|c|}{(B)} & \multicolumn{3}{|c|}{(C)}\\\hline
& & & & & & & & & & & \\
University & $R$ & Rank & ${\mathcal S}_{\mathcal{C}}(U)$ & Rank & $\delta(U)$ &${\mathcal S}_{\mathcal{C}}(U)$ & Rank & $\delta(U)$ & ${\mathcal S}_{\mathcal{C}}(U)$& Rank & $\delta(U)$ \\
& & & & & & & & & & & \\
\hline
Pisa Normale & 1.4825 & 1 & 0.99995 & 3 & 0.37 & 0.99996 & 3 & 1.11 & 0.99997 & 3 & 0.37 \\
Trieste SISSA & 1.4747 & 2 & 1 & 1 & 0.42 & 1.00000 & 1 & 1.14 & 1.00000 & 1 & 0.47 \\
Pavia & 1.3733 & 3 & 1 & 1 & 0.59 & 1.00000 & 1 & 1.71 & 1.00000 & 1 & 0.61 \\
Bergamo & 1.3077 & 4 & 0.919075 & 10 & 0.75 & 0.91712 & 12 & 2.11 & 0.91378 & 10 & 0.79 \\
Brescia & 1.2778 & 5 & 0.99351 & 6 & 0.83 & 0.99514 & 6 & 2.33 & 0.99372 & 6 & 0.88 \\
Cassino & 1.2576 & 6 & 0.88475 & 13 & 0.81 & 0.86903 & 14 & 2.43 & 0.84009 & 15 & 0.95 \\
Verona & 1.2469 & 7 & 0.965495 & 9 & 0.91 & 0.97032 & 9 & 2.57 & 0.96872 & 9 & 0.95 \\
Roma Tre & 1.2342 & 8 & 0.999885 & 4 & 0.88 & 0.99977 & 5 & 2.56 & 0.99983 & 4 & 0.94 \\
Roma Tor Vergata & 1.1774 & 9 & 0.999605 & 5 & 1.05 & 0.99979 & 4 & 2.93 & 0.99939 & 5 & 1.13 \\
Torino Politecnico & 1.1554 & 10 & 0.976015 & 8 & 1.1 & 0.97803 & 8 & 3.1 & 0.97619 & 7 & 1.19 \\
Bari Politecnico & 1.1520 & 11 & 0.837835 & 15 & 1.1 & 0.83288 & 15 & 3.12 & 0.85786 & 14 & 1.16 \\
Pisa & 1.1204 & 12 & 0.977295 & 7 & 1.17 & 0.98849 & 7 & 3.23 & 0.97460 & 8 & 1.27 \\
Trento & 1.1197 & 13 & 0.91099 & 12 & 1.18 & 0.92043 & 11 & 3.3 & 0.90233 & 12 & 1.28 \\
Napoli II & 1.1156 & 14 & 0.913965 & 11 & 1.2 & 0.92258 & 10 & 3.34 & 0.90291 & 11 & 1.31 \\
Milano Bicocca & 1.1024 & 15 & 0.88101 & 14 & 1.21 & 0.87323 & 13 & 3.43 & 0.90208 & 13 & 1.29 \\
Marche & 1.0952 & 16 & 0.755735 & 18 & 1.19 & 0.73493 & 19 & 3.37 & 0.75044 & 20 & 1.28 \\
Bologna & 1.0680 & 17 & 0.74515 & 19 & 1.31 & 0.76815 & 18 & 3.65 & 0.75085 & 19 & 1.42 \\
Salento & 1.0618 & 18 & 0.66369 & 21 & 1.28 & 0.60666 & 21 & 3.64 & 0.68925 & 21 & 1.36 \\
Milano & 1.0510 & 19 & 0.761015 & 17 & 1.34 & 0.81407 & 16 & 3.7 & 0.75799 & 18 & 1.46 \\
Padova & 1.0483 & 20 & 0.729325 & 20 & 1.31 & 0.73434 & 20 & 3.67 & 0.77247 & 16 & 1.41 \\
Ferrara & 1.0392 & 21 & 0.7651 & 16 & 1.34 & 0.77155 & 17 & 3.74 & 0.77105 & 17 & 1.44 \\
Udine & 1.0301 & 22 & 0.616185 & 22 & 1.31 & 0.53283 & 22 & 3.77 & 0.59436 & 22 & 1.43 \\
della Calabria & 1.0052 & 23 & 0.159515 & 30 & 1.4 & 0.12672 & 33 & 3.94 & 0.17688 & 30 & 1.51 \\
Roma La Sapienza & 1.0047 & 24 & 0.13766 & 31 & 1.46 & 0.32741 & 28 & 3.92 & 0.12026 & 32 & 1.60 \\
Milano Politecnico & 0.9991 & 25 & 0.06818 & 34 & 1.46 & 0.12891 & 32 & 3.98 & 0.06040 & 35 & 1.60 \\
Piemonte Orientale & 0.9889 & 26 & 0.50371 & 23 & 1.38 & 0.43916 & 24 & 4.02 & 0.53354 & 24 & 1.45 \\
Napoli Federico II & 0.9783 & 27 & 0.46423 & 25 & 1.42 & 0.34901 & 26 & 4.04 & 0.45277 & 25 & 1.55 \\
Sannio & 0.9762 & 28 & 0.112775 & 32 & 1.58 & 0.15709 & 31 & 4.16 & 0.12287 & 31 & 1.73 \\
Cagliari & 0.9733 & 29 & 0.480355 & 24 & 1.48 & 0.46175 & 23 & 4.16 & 0.56943 & 23 & 1.57 \\
Firenze & 0.9644 & 30 & 0.404845 & 26 & 1.46 & 0.34262 & 27 & 4.1 & 0.43252 & 26 & 1.58 \\
Parma & 0.9608 & 31 & 0.33844 & 27 & 1.54 & 0.35131 & 25 & 4.26 & 0.30857 & 28 & 1.69 \\
Urbino Carlo Bo & 0.9444 & 32 & 0.218815 & 29 & 1.69 & 0.31418 & 29 & 4.39 & 0.22355 & 29 & 1.86 \\
Salerno & 0.9373 & 33 & 0.24665 & 28 & 1.53 & 0.17767 & 30 & 4.37 & 0.31840 & 27 & 1.63 \\
Torino & 0.9032 & 34 & 0.081775 & 33 & 1.63 & 0.06161 & 35 & 4.57 & 0.09362 & 33 & 1.77 \\
L'Aquila & 0.8862 & 35 & 0.01643 & 45 & 1.72 & 0.01884 & 44 & 4.72 & 0.01172 & 45 & 1.91 \\
Siena & 0.8804 & 36 & 0.06294 & 36 & 1.76 & 0.11009 & 34 & 4.72 & 0.04854 & 37 & 1.96 \\
Perugia & 0.8682 & 37 & 0.028165 & 42 & 1.75 & 0.03985 & 38 & 4.75 & 0.03183 & 41 & 1.91 \\
Modena e Reggio Emilia & 0.8644 & 38 & 0.06435 & 35 & 1.69 & 0.03189 & 40 & 4.81 & 0.06141 & 34 & 1.86 \\
Bari & 0.8514 & 39 & 0.010645 & 46 & 1.73 & 0.00500 & 46 & 4.87 & 0.01020 & 46 & 1.90 \\
Camerino & 0.8380 & 40 & 0.04613 & 38 & 1.75 & 0.03013 & 41 & 4.97 & 0.04830 & 38 & 1.92 \\
Chieti e Pescara & 0.8167 & 41 & 0.039 & 40 & 1.9 & 0.05355 & 36 & 5.1 & 0.04206 & 39 & 2.08 \\
Catania & 0.8158 & 42 & 0.002995 & 47 & 1.88 & 0.00346 & 47 & 5.14 & 0.00370 & 47 & 2.05 \\
Venezia Ca Foscari & 0.8056 & 43 & 0.0433 & 39 & 1.83 & 0.03373 & 39 & 5.15 & 0.02949 & 42 & 2.08 \\
Basilicata & 0.7963 & 44 & 0.024345 & 43 & 1.86 & 0.01782 & 45 & 5.16 & 0.03771 & 40 & 2.00 \\
Genova & 0.7937 & 45 & 4.00E-05 & 50 & 1.91 & 0.00002 & 50 & 5.27 & 0.00005 & 50 & 2.09 \\
Reggio Calabria & 0.7899 & 46 & 0.055945 & 37 & 1.86 & 0.03995 & 37 & 5.24 & 0.05411 & 36 & 2.06 \\
Messina & 0.7480 & 47 & 0.002355 & 48 & 1.95 & 0.00061 & 49 & 5.51 & 0.00201 & 48 & 2.15 \\
Sassari & 0.7143 & 48 & 0.023 & 44 & 2.15 & 0.02680 & 43 & 5.73 & 0.02723 & 43 & 2.33 \\
Napoli Parthenope & 0.7000 & 49 & 0.00131 & 49 & 2.06 & 0.00063 & 48 & 5.78 & 0.00140 & 49 & 2.28 \\
Roma UNINETTUNO & 0.5952 & 50 & 0.029 & 41 & 2.42 & 0.02956 & 42 & 6.4 & 0.02176 & 44 & 2.71 \\
\hline
\end{tabular}%
\vspace{0.2cm}
\caption{Final rankings for the scientific area 01/A - Mathematics. We have denoted with (A) the choice $a_i=1, i=1, \ldots,5$, with (B) $a_1=a_2=a_3=3,a_4=1,a_5=0$, with (C) $a_1=a_2=a_4=1,a_3=1.5,a_5=0$. For each score calculated we report the associated rankings. Observations are ordered according to the VQR ranking based on $R$ values}
\label{tab:01Arank}%
\end{table}%
\begin{table}[htbp]
\centering
\tiny
\begin{tabular}{|l|cc|ccc|ccc|ccc|}
\hline
& & & \multicolumn{3}{|c|}{(A)} & \multicolumn{3}{|c|}{(B)} & \multicolumn{3}{|c|}{(C)}\\\hline
& & & & & & & & & & & \\
University & $R$ & Rank & ${\mathcal S}_{\mathcal{C}}(U)$ & Rank & $\delta(U)$ &${\mathcal S}_{\mathcal{C}}(U)$ & Rank & $\delta(U)$ & ${\mathcal S}_{\mathcal{C}}(U)$& Rank & $\delta(U)$ \\
& & & & & & & & & & & \\
\hline
Milano Politecnico & 1.7647 & 1 & 0.99997 & 2 & 0.33 & 0.99999 & 3 & 1.00 & 0.99999 & 2 & 0.33 \\
Ferrara & 1.6667 & 2 & 0.97731 & 13 & 0.50 & 0.97776 & 13 & 1.50 & 0.97776 & 13 & 0.50 \\
Roma LUISS & 1.6667 & 2 & 0.99373 & 10 & 0.50 & 0.99380 & 9 & 1.50 & 0.99292 & 10 & 0.56 \\
Milano & 1.6373 & 3 & 0.99990 & 4 & 0.55 & 0.99990 & 4 & 1.65 & 0.99993 & 4 & 0.55 \\
Macerata & 1.6176 & 4 & 0.99792 & 8 & 0.58 & 0.99775 & 7 & 1.75 & 0.99821 & 8 & 0.58 \\
Torino Politecnico & 1.4951 & 5 & 0.91082 & 18 & 0.88 & 0.92918 & 18 & 2.38 & 0.90854 & 18 & 0.94 \\
Milano Bocconi & 1.4764 & 6 & 0.99997 & 2 & 0.94 & 1.00000 & 1 & 2.47 & 0.99996 & 3 & 1.03 \\
Padova & 1.4537 & 7 & 1.00000 & 1 & 0.91 & 1.00000 & 1 & 2.59 & 1.00000 & 1 & 0.96 \\
Sassari & 1.4461 & 8 & 0.99870 & 6 & 0.88 & 0.99833 & 6 & 2.63 & 0.99874 & 6 & 0.94 \\
Perugia & 1.3313 & 9 & 0.99818 & 7 & 1.11 & 0.99761 & 8 & 3.21 & 0.99848 & 7 & 1.17 \\
Urbino Carlo Bo & 1.3235 & 10 & 0.95960 & 15 & 1.08 & 0.94741 & 16 & 3.25 & 0.96497 & 15 & 1.12 \\
Venezia Ca' Foscari & 1.3106 & 11 & 0.99974 & 5 & 1.11 & 0.99899 & 5 & 3.32 & 0.99980 & 5 & 1.16 \\
Trento & 1.2717 & 12 & 0.99497 & 9 & 1.23 & 0.99342 & 10 & 3.51 & 0.99640 & 9 & 1.29 \\
Piemonte Orientale & 1.2572 & 13 & 0.88657 & 19 & 1.35 & 0.92947 & 17 & 3.59 & 0.88162 & 19 & 1.47 \\
Parma & 1.2567 & 14 & 0.96847 & 14 & 1.23 & 0.95430 & 15 & 3.59 & 0.97370 & 14 & 1.30 \\
Chieti e Pescara & 1.2561 & 15 & 0.99164 & 11 & 1.19 & 0.99154 & 11 & 3.37 & 0.99207 & 11 & 1.27 \\
Brescia & 1.2255 & 16 & 0.92867 & 17 & 1.25 & 0.87394 & 19 & 3.75 & 0.94654 & 16 & 1.29 \\
Modena e Reggio Emilia & 1.1928 & 17 & 0.84874 & 20 & 1.42 & 0.87120 & 20 & 3.92 & 0.83820 & 20 & 1.54 \\
Bologna & 1.1596 & 18 & 0.98210 & 12 & 1.46 & 0.98041 & 12 & 4.12 & 0.98151 & 12 & 1.60 \\
Firenze & 1.1410 & 19 & 0.94837 & 16 & 1.51 & 0.95464 & 14 & 4.18 & 0.93483 & 17 & 1.66 \\
Napoli II & 1.1111 & 20 & 0.66715 & 25 & 1.44 & 0.54547 & 27 & 4.33 & 0.67541 & 25 & 1.56 \\
Torino & 1.1099 & 21 & 0.83949 & 21 & 1.55 & 0.83738 & 21 & 4.34 & 0.81922 & 21 & 1.70 \\
Genova & 1.0873 & 22 & 0.63533 & 27 & 1.55 & 0.56815 & 26 & 4.45 & 0.61895 & 27 & 1.70 \\
Milano Bicocca & 1.0565 & 23 & 0.77716 & 22 & 1.62 & 0.71746 & 23 & 4.61 & 0.79543 & 22 & 1.77 \\
Bergamo & 1.0407 & 24 & 0.72058 & 24 & 1.62 & 0.60872 & 25 & 4.69 & 0.76130 & 23 & 1.73 \\
Marche & 1.0392 & 25 & 0.73560 & 23 & 1.70 & 0.71927 & 22 & 4.70 & 0.73174 & 24 & 1.83 \\
Salerno & 1.0114 & 26 & 0.42971 & 30 & 1.74 & 0.41217 & 31 & 4.84 & 0.48007 & 30 & 1.87 \\
Roma Tor Vergata & 1.0076 & 27 & 0.66017 & 26 & 1.75 & 0.65762 & 24 & 4.86 & 0.63218 & 26 & 1.93 \\
Udine & 1.0074 & 28 & 0.52801 & 28 & 1.69 & 0.43525 & 28 & 4.86 & 0.53518 & 28 & 1.84 \\
Pisa & 0.9741 & 29 & 0.33328 & 32 & 1.87 & 0.41446 & 29 & 5.03 & 0.30913 & 32 & 2.08 \\
Pavia & 0.9617 & 30 & 0.50671 & 29 & 1.76 & 0.41242 & 30 & 5.10 & 0.50421 & 29 & 1.93 \\
della Calabria & 0.9447 & 31 & 0.16967 & 37 & 1.86 & 0.15948 & 35 & 5.18 & 0.16538 & 36 & 2.06 \\
Cagliari & 0.9276 & 32 & 0.35820 & 31 & 1.81 & 0.23802 & 32 & 5.27 & 0.39905 & 31 & 1.94 \\
Palermo & 0.8554 & 33 & 0.12075 & 40 & 2.01 & 0.09586 & 38 & 5.64 & 0.13415 & 39 & 2.21 \\
Milano Cattolica & 0.8507 & 34 & 0.05308 & 43 & 2.05 & 0.04405 & 42 & 5.66 & 0.06624 & 42 & 2.23 \\
Napoli Federico II & 0.8287 & 35 & 0.02121 & 44 & 2.01 & 0.00506 & 47 & 5.77 & 0.02201 & 44 & 2.22 \\
L'Aquila & 0.8088 & 36 & 0.20405 & 33 & 2.13 & 0.20193 & 33 & 5.88 & 0.19403 & 35 & 2.38 \\
Insubria & 0.7994 & 37 & 0.20072 & 34 & 2.08 & 0.14105 & 36 & 5.92 & 0.20839 & 33 & 2.27 \\
Cassino & 0.7843 & 38 & 0.15338 & 38 & 2.08 & 0.09103 & 40 & 6.00 & 0.14619 & 38 & 2.31 \\
Salento & 0.7608 & 39 & 0.01263 & 45 & 2.28 & 0.01751 & 44 & 6.12 & 0.01532 & 45 & 2.50 \\
Foggia & 0.7549 & 40 & 0.07633 & 41 & 2.15 & 0.04642 & 41 & 6.15 & 0.05961 & 43 & 2.43 \\
Napoli Parthenope & 0.7376 & 41 & 0.17628 & 35 & 2.19 & 0.09225 & 39 & 6.24 & 0.20028 & 34 & 2.40 \\
Roma LUMSA & 0.7190 & 42 & 0.17231 & 36 & 2.33 & 0.18438 & 34 & 6.33 & 0.15361 & 37 & 2.67 \\
Roma La Sapienza & 0.6790 & 43 & 0.00000 & 48 & 2.46 & 0.00000 & 48 & 6.54 & 0.00000 & 48 & 2.75 \\
Sannio & 0.6398 & 44 & 0.01121 & 46 & 2.42 & 0.00724 & 46 & 6.74 & 0.01463 & 46 & 2.66 \\
Roma Europea & 0.5229 & 45 & 0.13016 & 39 & 2.67 & 0.12068 & 37 & 7.33 & 0.12100 & 40 & 3.00 \\
Napoli Orientale & 0.4902 & 46 & 0.01111 & 47 & 2.83 & 0.01714 & 45 & 7.50 & 0.01295 & 47 & 3.17 \\
Messina & 0.4256 & 47 & 0.00000 & 48 & 2.95 & 0.00000 & 48 & 7.83 & 0.00000 & 48 & 3.28 \\
Teramo & 0.3676 & 48 & 0.05957 & 42 & 2.88 & 0.03435 & 43 & 8.13 & 0.06922 & 41 & 3.19 \\
Bari & 0.3650 & 49 & 0.00000 & 48 & 3.03 & 0.00000 & 48 & 8.14 & 0.00000 & 48 & 3.41 \\
\hline
\end{tabular}%
\vspace{0.2cm}
\caption{Final rankings for the scientific area 13/D - Statistics and Mathematical Methods for Decisions. We have denoted with (A) the choice $a_i=1, i=1, \ldots,5$, with (B) $a_1=a_2=a_3=3,a_4=1,a_5=0$, with (C) $a_1=a_2=a_4=1,a_3=1.5,a_5=0$.
For each score calculated we report the associated rankings. Observations are ordered according to the VQR ranking based on $R$ values}
\label{tab:13Drank}%
\end{table}%
\section{Conclusions and remarks}
As shown in Section \ref{VQR}, the results obtained applying the procedures developed in the present paper can be very different from the ranking obtained by ANVUR and used for funding Italian universities. The more one area is heterogeneous, either with respect to the numerousness either relatively to the specific research domains, the more the ANVUR $R$ score becomes rough and the corresponding results differ from ours. For instance, we observe for the area of Mathematics (see Table \ref{tab:01Arank}) that the University of Pisa Normale loses three positions if evaluated according to geometric score instead of considering the ANVUR $R$ score. For the University of Bergamo and for Milano Politecnico, the loss is more evident with the first one losing on average 6 positions and the second losing on average 8 positions, with a small variability depending on the $a_{i}$ constants chosen. On the other hand there are universities for which there is an evident improvement in the ranking, in some cases of even 10 positions, if the geometric score is used.
Similar considerations can be also done for the area of Statistics and Mathematical Methods for Decisions (see Table \ref{tab:13Drank}).
In fact, as pointed out in Section \ref{VQR} one can not in principle compare two aggregates of different size without risking having a funnel effect. In order to overcome this issue, in some areas ANVUR divided universities in three classes according to the number of researchers of each aggregate (big, medium and small) and, for such areas, only the ranking within these three dimensional classes was given (but in any case the computation of the $R$ score, used for funding allocations, was made regardless the dimensional class to which each university was belonging). However, while this remedy could mitigate some perverse effect, it can not prevent the appareance of funnel effects in each dimensional class. Moreover, in this way the analysis is inevitably less informative, since each aggregate is compared with a fewer number of other aggregates. We stress that to calculate the geometric score there is no need to make distinctions on the base of universities' dimensions as, according to the definition of ${\mathcal S}_{\mathcal{C}}(U)$, each university $U$ is compared to all ideal universities having the same size as $U$.
On the other hand, the scientific homogeneity is another important feature which is not considered in the ANVUR analysis but it is incapsulated in our geometric score. The best example in this way is provided by the Mathematics area. Here we have very different outcomes for each scientific discipline, reflecting the diverse publication customs and trends for the various areas of Mathematics, as well as the different methods of assessment which were used (the MCQ score for Pure Mathematics, Impact Factor for Applied Mathematics, peer review for History of Mathematics). From Table \ref{VQR-math} it is clear that departments with higher number of professors dealing with History of Mathematics are unfairly penalized by the ANVUR's analysis methods. With this respect, we observe that the three universities of Pisa Normale, Bergamo and Milano Politecnico, that we have mentioned above among the ones rewarded from the VQR ranking, did not count any professor or reasearcher in that scientific sector. On the contrary departments with higher number of professors dealing with Applied Mathematics obtain, on average, better results.
\begin{table}[htbp]\label{VQR-math}
\centering
\tiny
\caption{Number of total expected products and their repartition in the 5 classes of VQR 2011-2014, for each scientific sector of the area 01/A - Mathematics. The proportion of products in the 5 classes is also reported}
\begin{tabular}{|c|c|ccccc|}
\hline
{scientific sector} & {expected products} & {A} & {B} & {C} & {D} & {E} \\
\hline
MAT/01 & 72 & 25 & 20 &7 &12& 8 \\
& & 0.347 & 0.278 & 0.097 & 0.167 & 0.167 \\
MAT/02 & 319 & 67 & 97& 59& 32& 64 \\
& & 0.21 & 0.304 & 0.185 & 0.10 & 0.345 \\
MAT/03 & 800 & 246 & 190 & 106 & 82 & 176 \\
& & 0.308 & 0.238 & 0.133 & 0.103 & 0.385 \\
MAT/04 & 132 & 18 & 45 & 28 & 20 & 21 \\
& & 0.136 & 0.311 & 0.212 & 0.152 & 0.227 \\
MAT/05 & 1545 & 616& 388& 220& 94 & 227 \\
& & 0.399 & 0.251 & 0.142 & 0.061 & 0.27 \\
MAT/06 & 255 &97 & 71 & 42 & 22 & 23 \\
& & 0.38 & 0.278 & 0.165 & 0.086 & 0.169 \\
MAT/07 & 609 & 197 & 147 & 98 & 88 & 79 \\
& & 0.324 & 0.241 & 0.161 & 0.145 & 0.22 \\
MAT/08 & 563 & 245 & 143 & 83 & 36 & 56 \\
& & 0.435 & 0.254 & 0.147 & 0.064 & 0.17 \\
MAT/09 & 230 &169 & 68 & 30 & 14 &16 \\
& & 0.569 & 0.229 & 0.101 & 0.047 & 0.082 \\
\hline
\end{tabular}
\end{table}
\begin{table}[htbp]\label{VQR-stat}
\centering
\tiny
\caption{Number of total expected products and their repartition in the 5 classes of VQR 2011-2014, for each scientific sector of the area 13/D - Statistics and Mathematical Methods for Decisions. The proportion of products in the 5 classes is also reported}
\begin{tabular}{|c|c|ccccc|}
\hline
{scientific sector} & {expected products} & {A} & {B} & {C} & {D} & {E} \\
\hline
SECS-S/01 & 794 & 241& 209& 94 &91 &109 \\
& &0.325 & 0.281 &0.126 & 0.122 & 0.146\\
SECS-S/02 & 45 & 11 & 10 & 5 &5& 11 \\
& & 0.267 & 0.244 & 0.111 & 0.111 & 0.267 \\
SECS-S/03 & 281 & 44 & 61 & 41 & 45 & 67 \\
& & 0.171 & 0.235 & 0.160 & 0.174 & 0.260 \\
SECS-S/04 & 131 & 28 & 24 &14 &29& 27 \\
& & 0.229 & 0.198 & 0.115 & 0.237 & 0.221 \\
SECS-S/05 & 130 & 20 & 22 & 33 & 27 & 21 \\
& & 0.162 & 0.177 & 0.269 & 0.223 & 0.169 \\
SECS-S/06 & 776 & 152 & 208 & 85 & 89 & 142 \\
& & 0.224 & 0.308 & 0.126 & 0.132 & 0.21 \\
\hline
\end{tabular}
\end{table}
Somehow ANVUR itself was aware of the aforementioned limits of its methods, so that for the program ``Departments of Excellence'' a different methodology, conceptually much similar to ours, was introduced (see \cite{poggi} and \cite{ANVUR:2017-1}). Without entering into details, we point out that such methodology, which uses the Central Limit Theorem, assumes the independence of the assessments received by each publication. Such independence assumption, however, is unrealistic, especially for smaller sectors, as well as for areas with large numbers of coauthors.
|
1,108,101,566,736 | arxiv |
\section{Introduction}
\label{sec:introduction}
We consider the following inverse problem: Assume that $\grtr \in \mathds{R}^n$ denotes a signal of interest in one spatial dimension. It is assumed to be \emph{$s$-gradient-sparse}, i.e., $\cardinality{\supp( \nabla\grtr)} \leq s$, where $\nabla \in \mathds{R}^{n-1 \times n}$ denotes a \emph{discrete gradient operator}.\footnote{We consider a gradient operator that is based on forward differences and von~Neumann boundary conditions. An extension to other choices is expected to be straightforward.} Instead of having direct access to $\grtr$, the signal is observed via a \emph{linear, non-adaptive measurement process}\footnote{For the sake of simplicity, potential distortions in the measurement process are ignored here, but we emphasize that all results of this work can be made robust against (adversarial) noise.}
\begin{equation}\label{eq:meas}
\y = \mathcal{A} \grtr \in \mathds{R}^m,
\end{equation}
where $\mathcal{A} \in \mathds{R}^{m \times n}$ is a known \emph{measurement matrix}.
The methodology of compressed sensing suggests that, under certain conditions, it remains possible to retrieve $\grtr$ from the knowledge of $\y$ and $\mathcal{A}$ even when $m\ll n$.
Indeed, one of the seminal works of this field by \citeauthor*{candes2006cs}~\cite{candes2006cs} shows that for random Fourier measurements, the recovery of $\grtr$ remains feasible with high probability as long as the number of measurements obeys $m \gtrsim s \log(n)$, where the `$\gtrsim$'-notation hides a universal constant.
For the success of this strategy, it is crucial to employ non-linear recovery methods that exploit the a priori knowledge that $\grtr$ is gradient-sparse.
Arguably, the most popular version of 1D total variation (TV) minimization is based on an adaption of the classical basis pursuit, i.e., one solves the convex problem
\begin{equation}\label{eq:intro:tv-1}\tag{$\text{TV-1}$}
\min_{\x \in \mathds{R}^n} \lnorm{\nabla \x}[1] \quad \text{subject to \quad $\y = \mathcal{A} \x $.}
\end{equation}
The research of the past three decades demonstrates that encouraging a small TV norm often efficaciously reflects the inherent structure of real-world signals. Although not as popular as its counterpart in 2D (e.g., see \cite{rudin_nonlinear_1992,Chambolle1997,Chambolle2004}), TV methods in one spatial dimension find application in many practical scenarios, e.g., see \cite{Little2011,Little2010,Sandbichler2015,Wu2014,Perrone2016}. Furthermore, TV in 1D has frequently been subject of mathematical research \cite{Condat2013,Selesnick2015,Selesnick2012,Mammen1997,Briani2011,Grasmair2007}.
The main objective of this work is to study the 1D TV minimization problem for the benchmark case of Gaussian random measurements. In a nutshell, we intend to answer the following question:
\begin{highlight}
Assuming that $\mathcal{A} \in \mathds{R}^{m\times n}$ is a standard Gaussian random matrix, under which conditions is it possible to recover an $s$-gradient-sparse signal $\grtr \in \mathds{R}^n$ via TV minimization~\eqref{eq:intro:tv-1} with the near-optimal rate of $m \gtrsim s\cdot \PolyLog(n)$ measurements?
\end{highlight}
\section{Why Should We Care?}
At first sight, the aforementioned recovery result of \citeauthor*{candes2006cs}~\cite{candes2006cs} seems to deny the relevance of the previous research question. However, we emphasize that their result applies exclusively to random Fourier measurements. Indeed, the TV-Fourier combination allows for a significant simplification of the problem, since the gradient operator is ``compatible'' with the Fourier transform (differentiation is a Fourier multiplier).
In contrast, the more recent work of \citeauthor{cai_guarantees_2015}~\cite{cai_guarantees_2015} addresses the generic case of Gaussian measurements. However, their main result~\cite[Thm.~2.1]{cai_guarantees_2015} seems to imply a negative answer to the question above: in essence, it shows that the \emph{uniform} recovery of every $s$-gradient-sparse signal by solving~\eqref{eq:intro:tv-1} is possible if and only if the number of measurements obeys
\begin{equation}
m \gtrsim \sqrt{sn} \cdot \log (n).
\end{equation}
The conclusion from this result is as surprising as it is discouraging: It suggests that the threshold for successful recovery of $s$-gradient-sparse signals via \eqref{eq:intro:tv-1} is essentially given by $\sqrt{s n}$-many Gaussian measurements. Remarkably, the latter rate does not resemble the desirable standard criterion ${m \gtrsim s \cdot \PolyLog (n,s)}$.
In Table~\ref{tab:intro}, we have summarized some of the existing guarantees for TV minimization in compressed sensing. We refer the interested reader to~\cite[Sec.~1.2]{Genz2020} and~\cite{krahmer_total_2017} for a more detailed overview of the relevant literature.
\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|c||c|c|}
\hline
\diagbox[width=1.5cm,height=1cm]{$\mathcal{A}$}{$d$D} & 1D & $\geq$2D \\
\hline \hline
\multirow{2}{*}{\footnotesize Gaussian} & { \footnotesize \textbf{$s \log^2 (n)$} (non-unif.) \scriptsize [ours]} & \footnotesize $s \cdot \PolyLog(n,s)$ \\
\cdashline{2-2}
& {\footnotesize $\sqrt{s n} \cdot \log (n)$ (unif.) \scriptsize\cite{cai_guarantees_2015}} & \scriptsize\cite{cai_guarantees_2015,Needell2013,Needell2013b} \\
\hline
\multirow{2}{*}{\footnotesize Fourier} & \multicolumn{2}{c|}{\footnotesize $s \cdot \PolyLog(n,s)$} \\
& \multicolumn{2}{c|}{\scriptsize \cite{candes2006cs,Poon2015,Krahmer2014}} \\
\hline
\end{tabular}
\end{center}
\caption{An overview of known asymptotic-order sampling rates for TV minimization in compressed sensing, ignoring universal and model-dependent constants.}
\label{tab:intro}
\end{table}
\section{Our Contribution}
The main contribution of this work consists in breaking the aforementioned $\sqrt{sn}$-complexity barrier. Taking a non-uniform, signal-dependent perspective, we show that a large class of gradient-sparse signals is already recoverable from $m \gtrsim s \cdot \PolyLog (n)$ Gaussian measurements.
Note that such a result does not contradict the findings of \citeauthor{cai_guarantees_2015}~\cite{cai_guarantees_2015}, as these are formulated uniformly across all s-gradient-sparse. Indeed, the $\sqrt{sn}$-rate describes the worst-case performance on the class of all $s$-gradient-sparse signals. We show that a meaningful restriction of this class allows for a significant improvement of the situation, cf.~the numerical experiments of~\cite{cai_guarantees_2015,genzel2017cosparsity}.
With that in mind, our analysis reveals that the separation distance of jump discontinuities of~$\grtr$ is crucial:
\begin{definition}[Separation constant]\label{def:results:msc}
Let $\grtr \in \mathds{R}^n$ be a signal with $s > 0$ \emph{jump discontinuities} such that $\supp(\nabla\grtr) = \{\nu_1,\dots,\nu_{s}\}$ where $0 \eqqcolon \nu_0 < \nu_1 < \dots < \nu_{s} < \nu_{s+1} \coloneqq n$.
We say that $\grtr$ is \emph{$\Delta$-separated} for some \emph{separation constant} $\Delta > 0$ if
\begin{equation}\label{eq:results:msc}
\min_{i \in [s+1]} \frac{\abs{\nu_i - \nu_{i-1}}}{n} \geq \frac{\Delta}{s+1}.
\end{equation}
\end{definition}
It is not hard to see that the separation constant can always be chosen such that $(s+1)/n \leq \Delta \leq 1$, where larger values of $\Delta$ indicate that the gradient support is closer to being equidistant. Indeed, in the (optimal) case of equidistantly distributed singularities, $\Delta = 1$ is a valid choice, independently of $s$. Based on this notion of separation, our main result reads as follows:
\begin{theorem}[Exact recovery via TV minimization]\label{thm:results:exact}
Let $\grtr \in \mathds{R}^n$ be a $\Delta$-separated signal with $s>0$ jump discontinuities and $\Delta \geq 8 s / n$.
Let $\probsuccess > 0$ and assume that $\mathcal{A} \in \mathds{R}^{m \times n}$ is a standard Gaussian random matrix with
\begin{equation}\label{eq:results:exact:meas}
m \gtrsim \Delta^{-1} \cdot s\log^2(n) + \probsuccess^2.
\end{equation}
Then with probability at least $1 - e^{-\probsuccess^2/2}$, TV minimization \eqref{eq:intro:tv-1} with input $\y = \mathcal{A} \grtr \in \mathds{R}^m$ recovers $\grtr$ exactly.
\end{theorem}
The proof of Theorem~\ref{thm:results:exact} relies on a sophisticated upper bound for the associated conic Gaussian mean width, which is based on a
signal-dependent, non-dyadic Haar wavelet transform. As such, the latter result can be extended to sub-Gaussian measurements as well as stable and robust recovery; see~\cite[Sec.~2.4]{Genz2020} for more details.
The significance of Theorem~\ref{thm:results:exact} depends on the size of the separation constant $\Delta$. In particular, we obtain the near-optimal rate of $m \gtrsim s \cdot \PolyLog (n)$ if $\Delta$ can be chosen independently of~$n$ and $s$.
A typical example of such a situation is the discretization of a suitable piecewise constant function $\contgrtr \colon \intvopcl{0}{1} \to \mathds{R}$. Indeed, based on Theorem~\ref{thm:results:exact}, \cite[Cor.~2.6]{Genz2020} shows that $m \gtrsim s \cdot \log^2(n)$ measurements are sufficient for exact recovery when $\contgrtr$ is finely enough discretized; see Figure~\ref{fig:intro} for a visualization of this result.
\begin{figure}[ht!]
\centering
\begin{subfigure}[t]{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{images/signal_examples/n30.png}
\caption{}
\label{fig:intro:n30}
\end{subfigure}%
\qquad
\begin{subfigure}[t]{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{images/signal_examples/n50.png}
\caption{}
\label{fig:intro:n50}
\end{subfigure}
\vspace{1em}
\begin{subfigure}[t]{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{images/scaling_max.png}
\caption{}
\label{fig:intro:max}
\end{subfigure}%
\qquad
\begin{subfigure}[t]{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{images/scaling_dense.png}
\caption{}
\label{fig:intro:dense}
\end{subfigure}%
\caption{\textbf{Numerical simulation.} Subfigure~\subref{fig:intro:n30} and~\subref{fig:intro:n50} show schematic examples of the signal classes that are considered in this experiment at different resolution levels. The orange signal (with circle symbols) is defined as discretization of the piecewise constant function $\contgrtr \colon \intvopcl{0}{1} \to \mathds{R}$ with $s = 5$ jump discontinuities that is plotted in black. The blue plot (with diamond symbols) shows a so-called \emph{dense-jump signal}, which does not match the intuitive notion of a $5$-gradient-sparse signal; note that the spatial location of the jumps is chosen adaptively to the resolution level here, which does not correspond to a discretization of a piecewise constant function. For each signal class we have created phase transition plots: Subfigure~\subref{fig:intro:max} and~\subref{fig:intro:dense} display the empirical probability of successful recovery via TV minimization~\eqref{eq:intro:tv-1} for different pairs of ambient dimension $n$ and number of measurements $m$; note the horizontal axis uses a logarithmic scale. The corresponding grey tones reflect the observed probability of success, reaching from certain failure (black) to certain success (white). Additionally, we have estimated the conic Gaussian mean width of $\lnorm{\nabla(\cdot)}[1]$ at $\grtr$ (denoted by $\effdim[\conic]{\descset{\lnorm{\nabla(\cdot)}[1], \grtr}}$), which is known to precisely capture the phase transition (cf.~\cite{amelunxen2014edge}). The result of Subfigure~\subref{fig:intro:dense} confirms that the class of dense-jump signals suffers from the $\sqrt{s n}$-bottleneck as predicted by~\cite{cai_guarantees_2015}. On the other hand, Subfigure~\subref{fig:intro:max} reveals that this bottleneck can be broken for discretized signals, as predicted by Theorem~\ref{thm:results:exact}.}
\label{fig:intro}
\end{figure}
\vspace{-1.5\baselineskip}
\section{Discussion and Outlook}
We have shown that the $\sqrt{sn}$-bottleneck for 1D TV recovery from Gaussian measurement can be broken for signals with well separated jump discontinuities. The results of Table~\ref{tab:intro} suggest that TV minimization in one spatial dimension plays a special role in this regard. However, we argue that such a phenomenon can also be observed in higher spatial dimensions. In fact, we conjecture that the common rate of $m\gtrsim s \cdot \PolyLog(n,s)$ only reflects worst-case scenarios, while it can be significantly improved for natural signal classes, such as piecewise constant functions with sufficiently smooth boundaries.
\newpage
\renewcommand*{\bibfont}{\smaller}
\begin{refcontext}[sorting=nyt]
\printbibliography
\end{refcontext}
\end{document}
|
1,108,101,566,737 | arxiv | \section{Introduction} \label{sec:intro}
We apply a new computational method to light-front~\cite{LFreview1,LFreview2,LFreview3,LFreview4}
$\phi^4$ theory in two dimensions, in both the
symmetric and broken phases~\cite{RozowskyThorn,Kim,Varyetal1,Varyetal2,Varyetal3,Varyetal4}.
The method is based on an expansion of the Fock-state wave functions
in a basis of multivariate symmetric polynomials~\cite{GenSymPolys1,GenSymPolys2}.
This allows fine tuning of the resolution, Fock sector by
Fock sector, and incorporates small-$x$ behavior that
captures an integrable singularity. Both features represent
an improvement over the traditional discrete light-cone
quantization (DLCQ)~\cite{PauliBrodsky1,PauliBrodsky2} approach, where the resolution
is fixed across all Fock sectors and the integrable singularity
at zero momentum fraction is ignored. The presentation here
extends earlier work in \cite{phi4sympolys}.
The general method and an application to
the broken phase are described here; the
symmetric phase is discussed specifically
by Chabysheva~\cite{ChabyshevaLC16}.
Section~\ref{sec:formulation} details the
structure of the $\phi^4$ eigenvalue problem
and our method of solution, including the
option of a sector-dependent mass. Results
are presented in Sec.~\ref{sec:results},
followed by a brief summary in Sec.~\ref{sec:summary}
\section{Eigenvalue problem} \label{sec:formulation}
The Lagrangian for $\phi^4$ theory is
${\cal L}=\frac12(\partial_\mu\phi)^2-\frac12\mu_0^2\phi^2-\frac{\lambda}{4!}\phi^4$.
From it, one obtains the light-front Hamiltonian density
${\cal H}^-=\pm\frac12 \mu^2 :\phi^2:+\frac{\lambda}{4!}:\phi^4:$
and the Hamiltonian ${\cal P}^-=\int dx^- {\cal H}^-$
that defines the eigenvalue problem ${\cal P}^-|\psi\rangle=\frac{M^2}{P^+}|\psi\rangle$.
Here, $P^+=E+p_z$ is the light-front momentum conjugate to the light-front
spatial coordinate $x^-\equiv t-z$, and ${\cal P}^-$ is the operator that generates
translations in light-front time $x^+\equiv t+z$. The mass of the
eigenstate $|\psi\rangle$ is $M$.
The Hamiltonian is computed with the mode expansion
\begin{equation} \label{eq:mode}
\phi(x^+,x^-)=\int \frac{dp^+}{\sqrt{4\pi p^+}}
\left\{ a(p^+)e^{-ipx} + a^\dagger(p^+)e^{ipx}\right\}.
\end{equation}
The creation operators $a^\dagger(p^+)$ satisfy the
commutation relation
$[a(p^+),a^\dagger(p^{\prime +})]=\delta(p^+-p^{\prime +})$.
However, before completing the construction of ${\cal P}^-$, we
consider an asymmetric form obtained by a shift in the field.
The generator for the shift is
$U=\exp\int dp^+ [f(p^+)a^\dagger(p^+)-f^*(p^+)a(p^+)]$.
This shifts the creation operator
$Ua^\dagger(p^+) U^\dagger=a^\dagger(p^+)+f^*(p^+)$.
Following Harindranath and Vary~\cite{HariVary}, we choose $f$
to correspond to a zero mode
$f(p^+)=f^*(p^+)\equiv\sqrt{\pi p^+}\delta(p^+)\phi_s$. The
field is then shifted by a constant, $U\phi U^\dagger=\phi+\phi_s$ ,
and, with $\phi_s=\pm\sqrt{6\mu^2/\lambda}$, we obtain
\begin{equation}
U:{\cal H}^-\!: U^\dagger=-\frac32\frac{\mu^4}{\lambda}+\frac12(2\mu^2):\phi^2:
+\frac{\lambda\phi_s}{3!}:\phi^3:+\frac{\lambda}{4!}:\phi^4:
\end{equation}
Now substitution of the mode expansion yields
${\cal P}^-={\cal P}^-_{11}+{\cal P}^-_{22}+{\cal P}^-_{13}+{\cal P}^-_{31}+{\cal P}^-_{12}+{\cal P}^-_{21}$,
with
\begin{eqnarray}
{\cal P}^-_{11}&=&\int dp \frac{2\mu^2}{p} a^\dagger(p)a(p), \\
{\cal P}^-_{22}&=&\frac{\lambda}{4}\int\frac{dp_1 dp_2}{4\pi\sqrt{p_1p_2}}
\int\frac{dp'_1 dp'_2}{\sqrt{p'_1 p'_2}}
\delta(p_1 + p_2-p'_1-p'_2) a^\dagger(p_1) a^\dagger(p_2) a(p'_1) a(p'_2), \\
{\cal P}^-_{13}&=&\frac{\lambda}{6}\int \frac{dp_1dp_2dp_3}
{4\pi \sqrt{p_1p_2p_3(p_1+p_2+p_3)}}
a^\dagger(p_1+p_2+p_3)a(p_1)a(p_2)a(p_3), \\
{\cal P}^-_{12}&=& \mu\sqrt{\frac{3\lambda}{2}}
\int \frac{dp_1^+ dp_2^+}{\sqrt{4\pi p_1^+ p_2^+(p_1^++p_2^+)}}
a^\dagger(p_1^++p_2^+)a(p_1^+)a(p_2^+), \\
{\cal P}^-_{31}&=&({\cal P}^-_{13})^\dagger,\;\;{\cal P}^-_{21}=({\cal P}^-_{12})^\dagger.
\end{eqnarray}
The eigenstate of ${\cal P}^-$, with eigenvalue $M^2/P^+$,
can be expressed as an expansion
\begin{equation} \label{eq:FSexpansion}
|\psi(P^+)\rangle=\sum_m P^{+\frac{m-1}{2}}\int\prod_i^m dy_i
\delta(1-\sum_i^m y_i)\psi_m(y_i)|y_iP^+;P^+,m\rangle
\end{equation}
in terms of Fock states
$|y_iP^+;P^+,m\rangle=\frac{1}{\sqrt{m!}}\prod_{i=1}^m a^\dagger(y_iP^+)|0\rangle$
with normalization
$1=\sum_m \int\prod_i^m dy_i \delta(1-\sum_i^m y_i)|\psi_m(y_i)|^2$.
The eigenvalue problem is then reduced to a coupled system of equations
\begin{eqnarray}
\lefteqn{\left\{\begin{array}{l} +\mu^2 \\ -\mu^2 \\ 2\mu^2\end{array}\right\}
\sum_i^m \frac{1}{y_i }\psi_m(y_i)
+\frac{\lambda}{4\pi}\frac{m(m-1)}{4\sqrt{y_1y_2}}
\int\frac{dx_1 \psi_m(x_1,y_1+y_2-x_1,y_3,\ldots,y_m)}{\sqrt{x_1 (y_1+y_2-x_1)}}} && \\
&& +\frac{\lambda}{4\pi}\frac{m\sqrt{(m+2)(m+1)}}{6}
\int \frac{dx_1 dx_2 \psi_{m+2}(x_1,x_2,y_1-x_1-x_2,y_2,\ldots,y_m)}{\sqrt{y_1 x_1 x_2 (y_1-x_1-x_2)}}
\nonumber \\
&&+\frac{\lambda}{4\pi}\frac{(m-2)\sqrt{m(m-1)}}{6}
\frac{\psi_{m-2}(y_1+y_2+y_3,y_4,\ldots,y_m)}{\sqrt{y_1y_2y_3(y_1+y_2+y_3)}} \nonumber \\
&& +\mu\sqrt{\frac{3\lambda}{8\pi}}m\sqrt{m+1}
\int \frac{dx_1 \psi_{m+1}(x_1,y_1-x_1,y_2,\ldots,y_m)}{\sqrt{x_1 (y_1-x_1) y_1}} \nonumber \\
&& +\mu\sqrt{\frac{3\lambda}{8\pi}}(m-1)\sqrt{m}
\frac{\psi_{m-1}(y_1+y_2,y_3,\ldots,y_m)}{\sqrt{y_1 y_2 (y_1+y_2)}}=M^2\psi_m(y_i), \nonumber
\end{eqnarray}
where the last two terms are kept only for bottom option of $2\mu^2$, which represents the
shifted, asymmetric form of the Hamiltonian. In the symmetric phase, we consider only the
odd eigenstate where the number of constituents $m$ is an odd integer.
We solve this system by truncation in Fock space, to $m\leqN_{\rm max}$, and by expansion
of the Fock-state wave functions in terms of multivariate symmetric
polynomials $P_{ni}^{(m)}(y_1,\ldots,y_m)$~\cite{GenSymPolys1,GenSymPolys2}
\begin{equation} \label{eq:expansion}
\psi_m(y_1,\ldots,y_m)=\sqrt{y_1 y_2\cdots y_m}\sum_{ni} c_{ni}^{(m)} P_{ni}^{(m)}(y_1,\ldots,y_m).
\end{equation}
This converts the system into a generalized matrix eigenvalue problem
\begin{eqnarray} \label{eq:matrixequations}
\lefteqn{\sum_{n'i'}\left[\left\{\begin{array}{l} +1 \\ -1 \\ 2\end{array}\right\}
T^{(m)}_{ni,n'i'}+ gV^{(m,m)}_{ni,n'i'}\right]c^{(m)}_{n'i'}
+ g\sum_{n'i'} V^{(m,m+2)}_{ni,n'i'} c^{(m+2)}_{n'i'}
+ g\sum_{n'i'} V^{(m,m-2)}_{ni,n'i'} c^{(m-2)}_{n'i'}} && \\
&& + \sqrt{g}\sum_{n'i'} V^{(m,m+1)}_{ni,n'i'} c^{(m+1)}_{n'i'}
+ \sqrt{g}\sum_{n'i'} V^{(m,m-1)}_{ni,n'i'} c^{(m-1)}_{n'i'}
=\frac{M^2}{\mu^2}\sum_{n'i'}B^{(m)}_{ni,n'i'}c_{n'i'}^{(m)}, \nonumber
\end{eqnarray}
with $g=\lambda/(4\pi\mu^2)$. The matrices $T$, $V$ and $B$ are defined as
\begin{equation}
T^{(m)}_{ni,n'i'}=m\int\left(\prod_j dy_j \right)\delta(1-\sum_j y_j)\left(\prod_{j=2}^m y_j\right) P_{ni}^{(m)}(y_j)P_{n'i'}^{(m)}(y_j),
\end{equation}
\begin{eqnarray}
V^{(m,m)}_{ni,n'i'}&=&\frac{1}{4}m(m-1)\int\left(\prod_j dy_j\right) \delta(1-\sum_j y_j) \\
&& \times \int dx_1 dx_2 \delta(y_1+y_2-x_1-x_2)
\left(\prod_{j=3}^m y_j\right) P_{ni}^{(m)}(y_j)P_{n'i'}^{(m)}(x_1,x_2,y_3,\ldots,y_m),
\nonumber
\end{eqnarray}
\begin{eqnarray}
\lefteqn{V^{(m,m+2)}_{ni,n'i'}=\frac{1}{6}m\sqrt{(m+2)(m+1)}\int\left(\prod_j dy_j\right) \delta(1-\sum_j y_j)}&& \\
&& \times
\int dx_1 dx_2 dx_3 \delta(y_1-x_1-x_2-x_3)
\left(\prod_{j=2}^m y_j\right) P_{ni}^{(m)}(y_j)P_{n'i'}^{(m+2)}(x_1,x_2,x_3,y_2,\ldots,y_m),
\nonumber
\end{eqnarray}
\begin{eqnarray}
V^{(m,m+1)}_{ni,n'i'}&=&\sqrt{\frac32}m\sqrt{m+1}\int\left(\prod_j dy_j\right) \delta(1-\sum_j y_j) \\
&& \times
\int dx_1 dx_2 \delta(y_1-x_1-x_2)
\left(\prod_{j=2}^m y_j\right) P_{ni}^{(m)}(y_j)P_{n'i'}^{(m+1)}(x_1,x_2,y_2,\ldots,y_m),
\nonumber
\end{eqnarray}
and
\begin{equation}
B^{(m)}_{ni,n'i'}=\int\left(\prod_j dy_j \right)\delta(1-\sum_j y_j)\left(\prod_j^m y_j\right) P_{ni}^{(m)}(y_j)P_{n'i'}^{(m)}(y_j),
\end{equation}
with $V^{(m,m-2)}_{ni,n'i'}$ and $V^{(m,m-1)}_{ni,n'i'}$ obtained as the adjoints
of $V^{(m-2,m)}_{ni,n'i'}$ and $V^{(m-1,m)}_{ni,n'i'}$, respectively.
We convert the matrix problem to an ordinary eigenvalue problem by a singular value
decomposition (SVD)~\cite{SVD} of the overlap matrix $B^{(m)}$. This is an implicit
orthogonalization of the basis. The SVD is $B^{(m)}=U^{(m)}D^{(m)}U^{(m)T}$, where the
columns of $U^{(m)}$ are the eigenvectors of $B^{(m)}$ and
$D^{(m)}$ is a diagonal matrix of the eigenvalues.
We keep in $U^{(m)}$ only those columns associated with eigenvalues of $B^{(m)}$
that are above some positive threshold,
in order to eliminate from the basis those combinations of polynomials that are nearly
linearly dependent~\cite{Wilson}.
We then define new vectors of coefficients $\vec c^{\,(m)\prime}=D^{1/2}U^T\vec c^{\,(m)}$
and new matrices, such as $T^{(m)\prime}=D^{-1/2}U^T T^{(m)} UD^{-1/2}$. In terms of
these, the matrix eigenvalue problem is an ordinary one, which we then diagonalize by
standard means.
\section{Results} \label{sec:results}
Figure~\ref{fig:M2vsg} shows the results for the
mass squared of the lowest eigenstate as a function
of the dimensionless coupling $g$ and for both the
symmetric and broken phases.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=7cm]{extrap-odd.eps} &
\includegraphics[width=7cm]{M2vsg.eps} \\
(a) & (b)
\end{tabular}
\caption{Mass squared vs coupling strength for the (a) symmetric phase
and (b) broken phase. The different Fock-space
truncations in (a) are the three-body (triangles), five-body
(squares), and seven-body (diamonds) Fock sectors. Results
for the light-front coupled-cluster method~\protect\cite{LFCCphi4}
(circles) are also included.
In (b), each set of points corresponds
to a different Fock-space truncation to $N_{\rm max}$ constituents.
the different truncations are the four-body (triangles),
six-body (squares), and eight-body (diamonds) Fock sectors.
Error bars are determined by the fits to extrapolation in
the polynomial basis size.}
\label{fig:M2vsg}
\end{figure}
In the symmetric phase, we find a critical coupling of~\cite{phi4sympolys}
$g=2.1\pm0.05$. In the broken phase, the critical coupling values extrapolate
to $g=0.2\pm0.02$.
These critical coupling values can be compared to Chang's duality~\cite{Chang1,Chang2,Kim,HariVary},
which is obtained from the connection between normal orderings with respect to different
masses~\cite{normal}:
\begin{eqnarray}
N_+[\phi^2]&=&N_-[\phi^2]+\frac{1}{4\pi}\ln\frac{\mu_+^2}{\mu_-^2},\\
N_+[\phi^4]&=&N_-[\phi^4]+6\frac{1}{4\pi}\ln\frac{\mu_+^2}{\mu_-^2}N_-[\phi^2]
+3\left(\frac{1}{4\pi}\ln\frac{\mu_+^2}{\mu_-^2}\right)^2.
\end{eqnarray}
The Hamiltonian density is then written as
\begin{equation}
{\cal H}^-=\left(\frac12\mu_+^2+\frac{\lambda}{4}\frac{1}{4\pi}\ln\frac{\mu_+^2}{\mu_-^2}\right)N_-[\phi^2]
+\frac{\lambda}{4!}N_-[\phi^4]
+\frac{1}{4\pi}\ln\frac{\mu_+^2}{\mu_-^2}
\left(2\mu_+^2+\frac{\lambda}{8}\frac{1}{4\pi}\ln\frac{\mu_+^2}{\mu_-^2}\right).
\end{equation}
This is equivalent to ${\cal H}^-$ with negative mass squared if
$\frac12\mu_+^2+\frac{\lambda}{4}\frac{1}{4\pi}\ln\frac{\mu_+^2}{\mu_-^2}=-\frac12\mu_-^2$.
For the dimensionless couplings $g_\pm\equiv \lambda/4\pi\mu_\pm^2$ this becomes
$\frac{1}{g_+}-\frac12\ln g_+=-\frac12\ln (g_-)-\frac{1}{g_-}$.
The comparison is illustrated in Fig.~\ref{fig:dual}
\begin{figure}
\centering
\includegraphics[width=10cm]{dual.eps}
\caption{The duality in couplings between the symmetric and broken phases.
The solid line is the semi-classical duality of Chang~\protect\cite{Chang2}.
The point corresponds to our numerical results.}
\label{fig:dual}
\end{figure}
The relative probabilities of the higher Fock sectors are readily computed
from the Fock-state wave functions obtained in the eigenvector of the
Hamiltonian matrix. A plot can be found in \cite{ChabyshevaLC16}.
There is no indication of critical behavior at $g=2.1$, where one
would expect that the higher Fock sectors would dominate. A
natural assumption for why this might be happening is that the
numerical calculation used the same constituent mass $\mu$ in every
Fock sector, making the invariant mass of the $N$-constituent
Fock sector of order $\sum_i^N\frac{\mu^2}{1/N}=N^2N\mu^2$. The higher
sectors are then suppressed by this large invariant mass.
This can be avoided by the use of a sector-dependent constituent
mass~\cite{SecDep1,SecDep2,SecDep3,SecDep4,SecDep5,SecDep6};
our approach is described in \cite{ChabyshevaLC16}.
The results for the lowest odd eigenstate are shown in Fig.~\ref{fig:extrap}.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=7cm]{extrap.eps} &
\includegraphics[width=7cm]{extrap-all.eps} \\
(a) & (b)
\end{tabular}
\caption{Mass squared for the lowest eigenstate as a function of the
dimensionless coupling $g$ with (a) a sector-dependent constituent mass
and (b) with both sector-dependent (filled symbols) and independent (open symbols).
For (a), the Fock-space truncations are to a number of constituents
$N_{\rm max}=3$ (circles), 5 (triangles), 7 (diamonds), and 9 (hex); the
error bars estimate the range of fits for the $\mu_1$ extrapolations
used to obtain $g$ and $M$. In (b), the sector-independent,
five and seven-constituent results are nearly identical
with the nine-constituent sector-dependent results. Plot (b)
also includes, as open hexagons, the results from a light-front
coupled-cluster calculation~\protect\cite{LFCCphi4}.
}
\label{fig:extrap}
\end{figure}
They are consistent with the sector-independent approach, and the estimate
of the critical coupling remains unchanged at a value of 2.1.
As can be seen in \cite{ChabyshevaLC16}, the relative probabilities
also remain nearly the same.
\section{Summary} \label{sec:summary}
These calculations have revealed two inconsistencies
in the light-front approach to $\phi^4$ theory. One is the
absence of a vacuum expectation value (VEV) above the critical
coupling, where the $\phi\rightarrow-\phi$ symmetry is to
be broken, and the presumably related behavior in
the explicitly broken phase, where the vacuum
expectation value remains nonzero above the critical
coupling. A Gaussian effective
potential analysis~\cite{GEP1,GEP2} can determine a VEV that does
flip between zero and non-zero, but it does so discontinuously,
which is inconsistent with the known second-order nature
of the transition.
Another inconsistency is the smooth behavior of the computed Fock sector
probabilities as the coupling is increased through the critical value,
as presented in more detail in \cite{ChabyshevaLC16}.
The relative probabilities for the sector-dependent and independent
calculations are essentially the same in the three-body Fock sector.
This indicates full convergence with respect to the Fock-space truncation.
In Fock sectors with five and seven constituents, the relative
probability for the sector-dependent case rises above the probability
in the standard case as the critical coupling is approached.
This greater probability is expected; however, the full expectation
was that these probabilities would increase much more rapidly.
The hypothesis, that a sector-dependent mass would reveal the critical
behavior, must be incorrect. It seems likely that a coherent-state
approach is needed, something that the light-front coupled cluster
method can bring~\cite{LFCC,LFCCphi4}.
\acknowledgments
This work was done in collaboration with S.S. Chabysheva and was
supported in part by the Minnesota Supercomputing Institute
of the University of Minnesota with allocations of computing
resources.
|
1,108,101,566,738 | arxiv | \section{Introduction}\label{se:1}
The theoretical study of exciton dynamics in conjugated polymer systems is both a fascinating and complicated subject.
One reason for this is that characterizing excitonic states themselves is a challenging task: conjugated polymers exhibit strong electron-electron interactions and electron-nuclear coupling, and are subject to spatial and temporal disorder.
Another reason is that exciton dynamics is characterised by multiple (and often overlapping) time scales; it is determined by both intrinsic processes (e.g., coupling to nuclear degrees of freedom and electrostatic interactions) and extrinsic processes (e.g., polymer-solvent interactions); and it is both an intrachain and interchain process.
Consequently, to make progress in both characterizing exciton states and correctly describing their dynamics, simplified, but realistic models are needed.
Moreover, as even these simplified models describe many quantized degrees of freedom, sophisticated numerical techniques are required to solve them. Luckily, fundamental theoretical progress in developing numerical techniques means that simplified one-dimensional models of conjugated polymers are now soluble to a high degree of accuracy.
In addition to the application of various theoretical techniques to understand exciton dynamics, a wide range of time-resolved spectroscopic techniques have also been deployed. These include fluorescence depolarization\cite{Grage03,Ruseckas05,Wells07,Dykstra09}, three-pulse photon-echo\cite{Dykstra05,Yang05, Wells08,Sperling08} and coherent electronic two-dimensional spectroscopy\cite{Consani15}. Some of the timescales extracted from these experiments are listed in Table \ref{ta:2}; the purpose of this review is to describe their associated physical processes.
\begin{table*}
{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|p{2.5cm}|p{2.5cm}|p{9cm}|p{1.5cm}|}
\hline
Polymer & State & Timescales & Citation \\
\hline
MEH-PPV & Solution & $\tau_1 = 50 $ fs, $\tau_2 = 1-2 $ ps & Ref\cite{Ruseckas05} \\
\hline
MEH-PPV & Solution & $\tau_1 = 5-10 $ fs, $\tau_2 = 100-200 $ fs & Ref\cite{Collini09} \\
\hline
PDOPT & Film & $\tau = 0.5 - 4 $ ps & Ref\cite{Westenhoff06} \\
\hline
PDOPT & Solution & $\tau_1 < 1 $ ps, $\tau_2 = 15 -23 $ ps & Ref\cite{Westenhoff06} \\
\hline
P3HT & Film & $\tau_1 = 300 $ fs, $\tau_2 = 2.5 $ ps, $\tau_3 = 40 $ ps & Ref\cite{Westenhoff06} \\
\hline
P3HT & Solution & $\tau_1 = 700 $ fs, $\tau_2 = 6 $ ps, $\tau_3 = 41 $ ps, $\tau_4 = 530 $ ps & Ref\cite{Banerji11} \\
\hline
P3HT & Solution & $\tau_1 = 60 - 200 $ fs, $\tau_2 = 1-2 $ ps, $\tau_3 = 14-20 $ ps & Ref\cite{Busby11} \\
\hline
P3HT & Solution & $\tau_1 \lesssim 100$ fs, $\tau_2 \sim 1 - 10 $ ps & Ref\cite{Wells07} \\
\hline
\end{tabular}}
\caption{Some of the dynamical timescales observed in conjugated polymers whose associated physical processes are summarized in Table \ref{ta:1}.}
\label{ta:2}
}
\end{table*}
As well as being of intrinsic interest, the experimental and theoretical activities to understand exciton dynamics in conjugated polymer systems are also motivated by the importance of this process in determining the efficiency of polymer electronic devices. In photovoltaic devices, large exciton diffusion lengths are necessary so that excitons can migrate efficiently to regions where charge separation can occur. However, precisely the opposite is required in light emitting devices, since diffusion leads to non-radiative quenching of the exciton.
Perhaps one of the reasons for the failure to fully exploit polymer electronic devices has been the difficulty in establishing the structure-function relationships which allow the development of rational design strategies. An understanding of the principles of exciton dynamics, relating this to multiscale polymer structures, and interpreting the associated spectroscopic signatures are all key ingredients to developing structure-function relationships. An earlier review explored the connection between structure and spectroscopy\cite{Barford17}. In this review we describe our current understanding of the important dynamical processes in conjugated polymers, beginning with photoexcitation and intrachain relaxation on ultrafast timescales ($\sim 10$ fs) to sub-ns interchain exciton transfer and diffusion. These key processes are summarized in Table \ref{ta:1}.
\begin{table*}
{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|p{7cm}|p{5cm}|p{2.5cm}|p{1.5cm}|}
\hline
Process & Consequences & Timescale & Section \\
\hline
Exciton-polaron self-trapping via coupling to fast C-C bond vibrations. & Exciton-site decoherence; ultrafast fluorescence depolarization. & $\sim 10$ fs & \ref{se:3.1} \\
\hline
Energy relaxation from high-energy quasi-extended exciton states (QEESs) to low-energy local exciton ground states (LEGSs) via coupling to the environment. & Stochastic exciton density localization onto chromophores. & $\sim 100-200$ fs & \ref{se:3.2}\\
\hline
Exciton-polaron self-localization via coupling to slow bond rotations in the under-damped regime. & Exciton density localization on a chromophore; ultrafast fluorescence depolarization. & $\sim 200-600$ fs & \ref{se:3.3}\\
\hline
Exciton-polaron self-localization via coupling to slow bond rotations in the over-damped regime. & Exciton density localization on a chromophore; post-ps fluorescence depolarization. & $\sim 1- 10$ ps & \ref{se:3.3}\\
\hline
Stochastic torsional fluctuations inducing exciton `crawling' and `skipping' motion. & Intrachain exciton diffusion and energy fluctuations. & $\sim 3-30$ ps & \ref{se:4}\\
\hline
Interchromophore F\"orster resonant energy transfer. & Interchromophore exciton diffusion; post-ps spectral diffusion and fluorescence depolarization. & $\sim 10-100$ ps & \ref{se:5}\\
\hline
Radiative decay. & & $\sim 500$ ps & \\
\hline
\end{tabular}}
\caption{The life and times of an exciton: Some of the key exciton dynamical processes, encompassing over four-orders of magnitude, that occur in conjugated polymer systems.}
\label{ta:1}
}
\end{table*}
The plan of this review is the following. We begin by briefly describing some theoretical techniques for simulating exciton dynamics and emphasize the failures of simple methods. As already mentioned, excitons themselves are fascinating quasiparticles, so before describing their dynamics, in Section \ref{se:3} we start by describing their stationary states. We stress the role of low-dimensionality, disorder and electron-phonon coupling, and we discuss the fundamental concept of a chromophore. Next, in Section \ref{se:4}, we describe the sub-ps processes of intrachain exciton decoherence, relaxation and localization, which - starting from an arbitrary photoexcited state - results in an exciton forming a chromophore. We next turn to describe the exciton (and energy) transfer processes occurring on post-ps timescales. First, in Section \ref{se:5}, we describe the primarily adiabatic intrachain motion of excitons caused by stochastic torsional fluctuations, and second, in Section \ref{se:6}, we describe nonadiabatic interchain exciton transfer. We conclude and address outstanding questions in Section \ref{se:7}.
\section{A Brief Critique of Theoretical Techniques}\label{se:2}
A theoretical description of exciton dynamics in conjugated polymers poses considerable challenges, as it requires a rigorous treatment of electronic excited states and their coupling to the nuclear degrees of freedom. Furthermore, conjugated polymers consist of thousands of atoms and tens of thousands of electrons. Thus, as the Hilbert space grows exponentially with the number of degrees of freedom, approximate treatments of excitonic dynamics are therefore inevitable. There are two broad approaches to a theoretical treatment. One approach is to construct \emph{ab initio} Hamiltonians, with an exact as possible representation of the degrees of freedom, and then to solve these Hamiltonians with various degrees of accuracy. Another approach (albeit less common in theoretical chemistry) is to construct effective Hamiltonians with fewer degrees of freedom, such as the Frenkel-Holstein model described in Section \ref{se:4}. These effective Hamiltonians might be parameterized via a direct mapping from \emph{ab initio} Hamiltonians (e.g., see Appendix H in ref\cite{Book}, Appendix A in ref\cite{Barford14a} and various papers by Burghardt and coworkers\cite{Binder14,Binder20b}) or else semiempirically\cite{Barford14b}. A significant advantage of effective Hamiltonians over their \emph{ab initio} counterparts is that they can be solved for larger systems over longer timescales and to a higher level of accuracy.
As the Ehrenfest method is a widely used approximation to study charge and exciton dynamics in conjugated polymers, we briefly explain this method and describe the important ways in which it fails. (For a fuller treatment, see\cite{Horsfield06,Nelson20}.) The Ehrenfest method makes two key approximations. The first approximation is to treat the nuclei classically. This means that nuclear quantum tunneling and zero-point energies are neglected, and that exciton-polarons are not correctly described (see Section \ref{se:3.3}). The second assumption is that the total wavefunction is a product of the electronic and nuclear wavefunctions. This means that there is no entanglement between the electrons and nuclei, and so the nuclei cannot cause decoherence of the electronic degrees of freedom (see Section \ref{se:4.1}). A simple product wavefunction also implies that the nuclei move in a mean field potential determined by the electrons. This means that a splitting of the nuclear wave packet when passing through a conical intersection or an avoided crossing does not occur
(see Section \ref{se:4.2}), and that there is an incorrect description of energy transfer between the electronic and nuclear degrees of freedom (see Section \ref{se:5.4}). As will be discussed in the course of this review, these failures mean that in general the Ehrenfest method is not a reliable one to treat ultrafast excitonic dynamics in conjugated polymers.
Various theoretical techniques have been proposed to rectify the failures of the Ehrenfest method; for example, the surface-hopping technique\cite{Tully90,Tully12}, while still keeping the nuclei classical, partially rectifies the failures at conical interactions. More sophisticated approaches, for example the MC-TDHF and TEBD methods, quantize the nuclear degrees of freedom and do not assume a product wavefunction.
For a given electronic potential energy surface (PES), the multiconfigurational-time dependent Hartree-Fock (MC-TDHF) method\cite{Beck00} is an (in principle) exact treatment of nuclear wavepacket propagation, although in practice exponential scaling of the Hilbert space means that a truncation is required. In addition, this method is only as reliable as the representation of the PES.
In the time-evolving block decimation (TEBD) method\cite{Vidal03,Vidal04} a quantum state, $|\Psi\rangle$, is represented by a matrix product state (MPS)\cite{Schollwock11}. Its time evolution is determined via
\begin{equation}\label{}
|\Psi(t+\delta t\rangle = \exp(-\textrm{i}\hat{H}\delta t/\hbar)|\Psi(t)\rangle,
\end{equation}
where $\hat{H}$ is the system Hamiltonian and the action of the evolution operator is performed via a Trotter decomposition. Since the action of the evolution operator expands the Hilbert space, $|\Psi\rangle$ is subsequently compressed via a singular value decomposition (SVD)\footnote{A related method is time-dependent density matrix renormalization group (TD-DMRG). This has been successfully applied to simulate singlet fission in carotenoids, D. Manawadu, M. Marcus and W. Barford, in preparation.}. Importantly, this approach is `numerically exact' as long as the truncation parameter exceeds $2^S$, where $S$ is the entanglement entropy, defined by $S = -\sum_{\alpha} \omega_{\alpha} \textrm{ln}_2 \omega_{\alpha}$ and $\{\omega\}$ are the singular values obtained at the SVD. The TEBD method permits the electronic and nuclear degrees of freedom to be treated as quantum variables on an equal footing. It thus rectifies all of the failures of the Ehrenfest method described above and, unlike the MC-TDHF method, it is not limited by the representation of the PES. It can, however, only be applied to quantum systems described by one-dimensional lattice Hamiltonians\cite{Mannouch18}. Luckily, as described in Section \ref{se:4}, such model Hamiltonians are readily constructed to describe exciton dynamics in conjugated polymers.
\section{Excitons in Conjugated Polymers}\label{se:3}
Before discussing the dynamics of excitons, we begin by describing exciton stationary states in static conjugated polymers.
\subsection{Two-particle model}\label{se:3.1}
An exciton is a Coulombically bound electron-hole pair formed by the linear combination of electron-hole excitations
(for further details see\cite{Book, Abe93, Barford13}). In a one-dimensional conjugated polymer an exciton is described by the two-particle wavefunction,
$\Phi_{mj}(r,R) = \psi_m(r)\Psi_j(R)$.
$\Psi_j(R)$ is the center-of-mass wavefunction, which will be discussed shortly. Before doing that, we first discuss the relative wavefunction, $\psi_m(r)$, which describes a particle bound to a screened Coulomb potential, where $r$ is the electron-hole separation and $m$ is the principal quantum number. The electron and hole of an exciton in a one-dimensional semiconducting polymer are more strongly bound than in a three-dimensional inorganic semiconductor for two key reasons.\cite{Book, Barford13} First, because of the low dielectric constant and relatively large electronic effective mass in $\pi$-conjugated systems the effective Rydberg is typically $50$ times larger than for inorganic systems. Second, dimensionality plays a role: in particular, the one-dimensional Schr\"odinger equation for the relative particle\cite{Loudon16,Barford02a} predicts a strongly bound state split-off from the Rydberg series. This state is the $m=1$ Frenkel (`$1B_u$') exciton, with a binding energy of $\sim 1$ eV and an electron-hole wavefunction confined to a single monomer. The first exciton in the `Rydberg' series is the $m=2$ charge-transfer (`$2A_g$') exciton.
With the exception of donor-acceptor copolymers, conjugated polymers are generally non-polar, which means that each $p$-orbital has an average occupancy of one electron. This implies an approximate \emph{electron-hole} symmetry. Electron-hole symmetry has a number of consequences for the character and properties of excitons. First, it means that the relative wavefunction exhibits electron-hole parity, i.e., $\psi_m(r) = + \psi_m(-r)$ when $m$ is odd and $\psi_m(r) = - \psi_m(-r)$ when $m$ is even. Second, the transition density, $\langle \textrm{EX}|\hat{N}_i |\textrm{GS}\rangle$, vanishes for odd-parity (i.e., even $m$) excitons. This means that such excitons are not optically active, and importantly for dynamical processes, their F\"orster exciton transfer rate (defined in Section \ref{se:6.1}) vanishes.
Since Frenkel excitons are the primary photoexcited states of conjugated polymers, their dynamics is the subject of this review. Their delocalization along the polymer chain of $N$ monomers is described by the Frenkel Hamiltonian,
\begin{eqnarray}\label{Eq:1}
\hat{H}_{F} &=& \sum_{n=1}^{N}\epsilon_n \hat{N}_{n} +\sum_{n=1}^{N-1}J_n \hat{T}_{n,n+1},
\end{eqnarray}
where $n = (R/d)$ labels a monomer and $d$ is the intermonomer separation.
The energy to excite a Frenkel exciton on monomer $n$ is $\epsilon_{n}$, where
$\hat{N}_{n} = \ket{n}\bra{n}$
is the Frenkel exciton number operator.
In principle, excitons delocalize along the chain via two mechanisms\cite{Barford09a,Barford13}. First, for even-parity (odd $m$) singlet excitons there is a Coulomb-induced (or through space) mechanism. This is the familiar mechanism of F\"orster energy transfer.
The exciton transfer integral for this process is
\begin{eqnarray}\label{Eq:39}
J_{DA} = \sum_{\substack{i\in D \\ j\in A}}V_{ij} \left[
{_D}\langle \textrm{GS} |\hat{N}_i| \textrm{EX} \rangle_D\right]
\left[{_A}\langle \textrm{EX} |\hat{N}_j|\textrm{GS}\rangle_A
\right].
\end{eqnarray}
The sum is over sites $i$ in the donor monomer and $j$ in the acceptor monomer, and $V_{ij}$ is the Coulomb interaction between these sites.
In the point-dipole approximation Eq.\ (\ref{Eq:39}) becomes
\begin{equation}\label{Eq:4}
J_{DA} = \frac{\kappa_{mn} \mu_0^2}{4\pi\varepsilon_{r}\varepsilon_{0} R_{mn}^3},
\end{equation}
where $\mu_0$ is the transition dipole moment of a single monomer and ${R}_{mn}$ is the distance between the monomers $m$ and $n$.
$\kappa_{mn}$ is the orientational factor,
\begin{equation}\label{Eq:40}
\kappa_{mn} = \bs{\hat{r}_m}\cdot\bs{\hat{r}_{n}} - 3(\bs{\hat{R}_{mn}}\cdot\bs{\hat{r}_{m}})(\bs{\hat{R}_{mn}}\cdot\bs{\hat{r}_{n}}),
\end{equation}
where $\bs{\hat{r}_{m}}$ is a unit vector parallel to the dipole on monomer $m$ and $\bs{\hat{R}_{mn}}$ is a unit vector parallel to the vector joining monomers $m$ and $n$. For colinear monomers, the nearest neighbor through space transfer integral is
\begin{equation}
J_{DA} = -\frac{2 \mu_0^2}{4\pi\varepsilon_{r}\varepsilon_{0} d^3}.
\end{equation}
Second, for all excitons there is a super-exchange (or through bond) mechanism, whose origin lies in a virtual fluctuation from a Frenkel exciton on a single monomer to a charge-transfer exciton spanning two monomers back to a Frenkel exciton on a neighboring monomer. The energy scale for this process, obtained from second order perturbation theory\cite{Book}, is
\begin{equation}\label{Eq:5}
J_{SE}(\theta) \propto -\frac{t(\theta)^2}{\Delta E},
\end{equation}
where $ t(\theta)$ (defined in Eq.\ (\ref{Eq:10c})) is proportional to the overlap of $p$-orbitals neighboring a bridging bond, i.e., $t(\phi) \propto \cos \theta$ and $\theta$ is the torsional (or dihedral) angle between neighboring monomers. $\Delta E$ is the difference in energy between a charge-transfer and Frenkel exciton.
The total exciton transfer integral is thus
\begin{equation}\label{Eq:4}
J_{n} = J_{\text{DA}} + J_{\text{SE}}^0\cos^{2}{\theta_{n}}.
\end{equation}
The bond-order operator,
\begin{eqnarray}\label{Eq:7a}
\hat{T}_{n,n+1} = \left( \ket{n}\bra{n+1} + \ket{n+1}\bra{n}\right),
\end{eqnarray}
represents the hopping of the Frenkel exciton between monomers $n$ and $n+1$.
Evidently, $J_{SE}$ vanishes when $\theta = 0$, but $J_{DA}$ will not. Therefore, even if $J_{SE}$ vanishes because of negligible $p$-orbital overlap between neighboring monomers, singlet even-parity excitons can still retain phase coherence over the `conjugation break'\cite{Barford10a}. This observation has important implications for the definition of chromophores, as discussed in Section \ref{se:3.2}.
Eq.\ (\ref{Eq:1}) represents a `coarse-graining' of the exciton degrees of freedom. The key assumption is that we can replace the atomist detail of each monomer (or moiety) and replace it by a `coarse-grained' site, as illustrated in Fig.\ 1. All that remains is to describe how the Frenkel exciton delocalises along the chain, which is controlled by the two sets of parameters, $\{\epsilon\}$ and $\{J\}$.
Since $J$ is negative, a conjugated polymer is equivalent to a molecular J-aggregate.
\begin{figure}\label{Fi:1}
\includegraphics[width=0.5\textwidth]{Fig1.jpeg}
\caption{The mapping of a polymer chain conformation to a coarse-grained linear site model. Each site corresponds to a moiety along the polymer chain, with the connection between sites characterised by the torsional (or dihedral) angle, $\theta$.}
\end{figure}
The eigenfunctions of $\hat{H}_{F}$ are the center-of-mass wavefunctions, $\Psi_j(n)$, where $j$ is the associated quantum number. For a linear, uniform polymer (i.e., $\epsilon_n \equiv \epsilon_0$ and $J_n \equiv J_0$)
\begin{equation}\label{Eq:251}
\Psi_j(n) = \left(\frac{2}{N+1}\right)^{1/2}\sum_{n=1}^N \sin\left(\frac{\pi j n}{N+1}\right),
\end{equation}
forming a band of states with energy
\begin{equation}\label{Eq:252}
E_j = \epsilon_0 +2J_0 \cos\left(\frac{\pi j }{N+1}\right).
\end{equation}
The family of excitons with different $j$ values corresponds to the Frenkel exciton band with different center-of-mass momenta. In emissive polymers the $j=1$ Frenkel exciton is generally labeled the $1^1B_u$ state.
\subsection{Role of static disorder: local exciton ground states and quasiextended exciton states}\label{se:3.2}
Polymers are rarely free from some kind of disorder and thus the form of Eq.\ (\ref{Eq:251}) is not valid for the center-of-mass wavefunction in realistic systems. Polymers in solution are necessarily conformationally disordered as a consequence of thermal fluctuations (as described in Section \ref{se:5}). Polymers in the condensed phase usually exhibit glassy, disordered conformations as consequence of being quenched from solution. Conformational disorder implies that the dihedral angles, $\{ \theta \}$ are disordered, which
by virtue of Eq.\ (\ref{Eq:4}) implies that the exciton transfer integrals are also disordered.
As well as conformational disorder, polymers are also subject to chemical and environmental disorder (arising, for example, from density fluctuations). This type of disorder affects the energy to excite a Frenkel exciton on a monomer (or coarse-grained site).
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{Fig2.jpeg}
\caption{(a) The density of three local exciton ground states (LEGSs, dotted curves) and the three vibrationally relaxed states (VRSs, solid curves) for one particular static conformation of a PPV polymer chain made up of 50 monomers. The exciton center-of-mass quantum number, $j$, for each state is also shown.
(b) The exciton density of a quasiextended exciton state (QEES), with quantum number $j=7$.
Reproduced from J. Chem. Phys. \textbf{148}, 034901 (2018) with the permission of AIP publishing.}\label{Fi:2}
\end{figure}
As first realized by Anderson\cite{Anderson58a}, disorder localizes a quantum particle (in our case, the exciton center-of-mass particle), and determines their energetic and spatial distributions. The origin of this localization is the wave-like nature of a quantum particle and the constructive and destructive interference it experiences as it scatters off a random potential. Malyshev and Malyshev\cite{Malyshev01a,Malyshev01b} further observed that in one-dimensional systems there are a class of states in the low energy tail of the density of states that are superlocalized,
named local exciton ground states (LEGSs\cite{Malyshev01a,Malyshev01b,Makhov10}). LEGSs are essentially nodeless, non-overlapping wavefunctions that together spatially span the entire chain.
They are \emph{local} ground states, because for the individual parts of the chain that they span there are no lower energy states. A consequence of the essentially nodeless quality of LEGSs is that the square of their transition dipole moment scales as their size\cite{Makhov10}. Thus, LEGSs define chromophores (or spectroscopic segments), namely the irreducible parts of a polymer chain that absorb and emit light. Fig.\ \ref{Fi:2}(a) illustrates the three LEGSs for a particular conformation of PPV with 50 monomers.
Some researchers claim that `conjugation-breaks' (or more correctly, minimum thresholds in the $p_z$-orbital overlap) define the boundaries of chromophores\cite{Athanasopoulos08}. In contrast, we suggest that it is the disorder that determines the average chromophore size, but `conjugation-breaks' can `pin' the chromophore boundaries. Thus, if the average distance between conjugation breaks is smaller than the chromophore size, chromophores will span conjugation breaks but they may also be separated by them. Conversely, if average distance between conjugation breaks is larger than the chromophore size the chromophore boundaries are largely unaffected by the breaks. The former scenario occurs in polymers with shallow torsional potentials, e.g., polythiophene\cite{Barford10a}.
Higher energy lying states are also localized, but are nodeful and generally spatially overlap a number of low-lying LEGSs. These states are named quasiextended exciton states (QEESs) and an example is illustrated in Fig.\ \ref{Fi:2}(b).
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{Fig3.jpeg}
\caption{(a) The energy density of states and (b) the optical absorption (neglecting the vibronic progression) of the manifold of Frenkel excitons (where $|\sigma_J/J_0| = 0.1$). The width of the LEGSs density
of states $\sim |J_0||\sigma_J/J_0|^{4/3}$. Similarly, the width of the optical absorption from both the LEGSs and all states $\sim
|J_0||\sigma_J/J_0|^{4/3}$. The band edge for an ordered chain is at $2|J_0|$ (indicated by the dashed lines), so LEGSs generally lie in the Lifshitz (or Urbach) tail of the density of states, i.e., $E < 2|J_0|$.
}
\label{Fi:3}
\end{figure}
When the disorder is Gaussian distributed with a standard deviation $\sigma$, single parameter scaling theory\cite{Kramer93} provides some exact results about the spatial and energetic distribution of the exciton center-of-mass states:
\begin{enumerate}
\item{The localization length
$L_{loc} \sim ({|J_0|}/{\sigma})^{2/3}$
at the band edge and as
$L_{loc} \sim ({|J_0|}/{\sigma})^{4/3}$
at the band center.}
\item{As a consequence of exchange narrowing, the width of the density of states occupied by LEGSs scales as $\sigma/\sqrt{L_{loc}} \sim \sigma^{4/3}$. Similarly, the optical absorption is inhomogeneously narrowed with a line width $\sim \sigma^{4/3}$.}
\item{The fraction of LEGSs scales as $1/L_{loc} \sim \sigma^{2/3}$.}
\end{enumerate}
These points are illustrated in Fig.\ \ref{Fi:3}, which shows the Frenkel exciton density of states and optical absorption for a particular value of disorder. Evidently, although LEGSs are a small fraction of the total number of states, they dominate the optical absorption.
This section has described LEGSs (or chromophores) as static objects defined by static disorder. However, as discussed in Section \ref{se:5}, dynamically torsional fluctuations also render the conformational disorder dynamic causing the LEGSs to evolve adiabatically. As a consequence, the chromophores `crawl' along the polymer chain.
\subsection{Role of electron-nuclear coupling: exciton-polarons}\label{se:3.3}
As well as disorder, another important process in determining exciton dynamics and spectroscopy is the coupling of an exciton to nuclear degrees of freedom; in a conjugated polymer these are fast C-C bond vibrations and slow monomer rotations. In this section we briefly review the origin of this coupling and then discuss exciton-polarons.
\subsubsection{Origin of electron-nuclear coupling}\label{se:3.3.1}
When a nucleus moves, either by a linear displacement or by a rotation about a fixed point, there is a change in the electronic overlap between neighboring atomic orbitals. Assuming that neighboring $p$-orbitals lie in the same plane normal to the bond with a relative twist angle of $\theta$, the resonance integral between a pair of orbitals separated by $r$ is\cite{Mulliken49}
\begin{equation}\label{Eq:10c}
t(\theta)= t(r)\cos \theta = \beta \exp(-\alpha r) \cos \theta,
\end{equation}
where $ t(r) < 0$. The kinetic energy contribution to the Hamiltonian is
\begin{equation}\label{Eq:11}
\hat{H}_{\textrm{ke}} = t(r)\cos \theta \times {\hat T},
\end{equation}
where the bond-order operator, ${\hat T}$, is defined in Eq.\ (\ref{Eq:7a}). Treating $r$ and $\theta$ as dynamical variables, suppose that the $\sigma$-electrons of a conjugated molecule and steric hinderances provide equilibrium values of $r=r_0$ and $\theta = \theta_0$, with corresponding elastic potentials of
\begin{equation}\label{}
V_{\textrm{vib}} = \frac{1}{2}K_{\textrm{vib}}^{\sigma}(r - r_0)^2
\end{equation}
and
\begin{equation}\label{}
V_{\textrm{rot}} = \frac{1}{2}K_{\textrm{rot}}^{\sigma}(\theta - \theta_0)^2.
\end{equation}
The coupling of the $\pi$-electrons to the nuclei changes these equilibrium values and the elastic constants.
To see this, we use the Hellmann-Feynman to determine the force on the bond. The linear displacement force is
\begin{eqnarray}\label{}
\nonumber
f = -\frac{\partial E}{\partial r} && = -\big\langle\frac{\partial\hat{H}_{\textrm{ke}}}{\partial r}\big\rangle \\
&& = \alpha t(r)\cos \theta \langle {\hat T} \rangle - K_{\textrm{vib}}^{\sigma}(r - r_0).
\end{eqnarray}
Thus, to first order in the change of bond length, $\delta r = (r - r_0)$, the equilibrium distortion is
\begin{eqnarray}\label{}
\delta r = \alpha t(r_0)\cos \theta \langle {\hat T} \rangle/K_{\textrm{vib}}^{\sigma},
\end{eqnarray}
which is negative because it is favorable to shorten the bond to increase the electronic overlap.
Similarly, the torque around the bond is
\begin{eqnarray}\label{}
\nonumber
\Gamma = -\frac{\partial E}{\partial \theta} && = -\big\langle\frac{\partial\hat{H}_{ke}}{\partial \theta}\big\rangle \\
&& = t(r)\sin \theta \langle {\hat T} \rangle - K_{\textrm{rot}}^{\sigma}(\theta - \theta_0)
\end{eqnarray}
and the equilibrium change of bond angle, $\delta \theta = (\theta - \theta_0)$, is
\begin{eqnarray}\label{Eq:16a}
\delta \theta = t(r)\sin \theta_0 \langle {\hat T} \rangle/K_{\textrm{rot}}^{\sigma},
\end{eqnarray}
which is also negative, again because it is favorable to increase the electronic overlap. Thus, the $\pi$-electron couplings act to planarize the chain.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig4.jpeg}
\caption{The $\pi$-bond order expectation values, $\langle \hat{T}\rangle$, for (a) the ground state and (b) the excited state, showing the benzenoid-quinoid transition.
As Eq.\ (\ref{Eq:16a}) and Eq.\ (\ref{Eq:17a}) indicate, the larger bond order of the bridging bond in the excited state implies a smaller dihedral angle and a stiffer torsional potential than the ground state.}\label{Fi:40}
\end{figure}
The electron-nuclear coupling also changes the elastic constants. Assuming a harmonic potential, the new rotational spring constant is
\begin{eqnarray}\label{Eq:17a}
\nonumber
K_{\textrm{rot}}^{\pi} = && \frac{\partial^2 E}{\partial \theta^2} \\
= && - t(r_0)\cos \theta_0 \langle {\hat T} \rangle + K_{\textrm{rot}}^{\sigma}
\end{eqnarray}
and thus
$ K_{\textrm{rot}}^{\pi} > K_{\textrm{rot}}^{\sigma}$ (because $ t(r_0) < 0$).
Interestingly, as shown in Fig.\ 4, because $\langle {\hat T} \rangle_{EX} > \langle {\hat T} \rangle_{GS}$ for the bridging bond in phenyl-based systems, the torsional angle is smaller and the potential is stiffer in the excited state (as a result of the benzenoid to quinoid distortion)\cite{Beenken05}.
\subsubsection{Exciton-polarons}\label{se:3.3.2}
An exciton that couples to a set of harmonic oscillators, e.g., bond vibrations or torsional oscillations, becomes `self-trapped'. Self-trapping means that the coupling between the exciton and oscillators causes a local displacement of the oscillator that is proportional to the local exciton density\cite{Rashba57a,Rashba57b,Rashba57c,Holstein59a,Holstein59b} (as illustrated in the next section). Alternatively, it is said that the exciton is dressed by a cloud of oscillators. Such a quasiparticle is named an exciton-polaron.
As there is no barrier to self-trapping in one-dimensional systems\cite{Rashba82}, there is always an associated relaxation energy.
If the exciton and oscillators are all treated quantum mechanically, then in a translationally invariant system the exciton-polaron forms a Bloch state and is \emph{not} localized. However, if the oscillators are treated classically, the non-linear feedback induced by the exciton-oscillator coupling self-localizes the exciton-polaron and `spontaneously' breaks the translational symmetry. This is a self-localized (or auto-localized) `Landau polaron'.\cite{Landau33, Campbell82} Notice that self-trapping is a necessary but not sufficient condition for self-localization. Self-localization always occurs in the limit of vanishing oscillator frequency (i.e., the adiabatic or classical limit) and vanishing disorder.\cite{Tozer14}
Whether or not an exciton-polaron is self-localized in practice, however, depends on the strength of the disorder and the vibrational frequency of the oscillators. Qualitatively, an exciton coupling to fast oscillators (e.g., C-C bond vibrations) forms an exciton-polaron with an effective mass only slightly larger than a bare exciton\cite{Tozer14}. For realistic values of disorder, such an exciton-polaron is not self-localized. This is illustrated in Fig. \ref{Fi:2}(a), which shows the three lowest solutions of the Frenkel-Holstein model (described in Section \ref{se:4.1}), known as vibrationally relaxed states (VRSs). As we see, the density of the VRSs mirrors that of the Anderson-localized LEGSs.
Conversely, an exciton coupling to slow oscillators (e.g., bridging-bond rotations) forms an exciton-polaron with a large effective mass. Such an exction-polaron is self-localized (as described in Section \ref{se:4.3} and shown in Fig.\ \ref{Fi:5}).
\section{Intrachain Decoherence, Relaxation and Localization}\label{se:4}
Having qualitatively described the stationary states of excitons in conjugated polymers, we now turn to a discussion of exciton dynamics.
\subsection{Role of fast C-C bond vibrations}\label{se:4.1}
After photoexcitation or charge combination after injection, the fastest process is the coupling of the exciton to C-C bond stretches. We now describe the resulting exciton-polaron formation and the loss of exciton-site coherence.
As we saw in Section \ref{se:3.3}, bond distortions couple to electrons. Using Eq.\ (\ref{Eq:11}), it can be shown\cite{Barford14b} that the
coupling of local normal modes (e.g., vinyl-unit bond stretches or phenyl-ring symmetric breathing modes) to a Frenkel exciton is conveniently described by the Frenkel-Holstein model\cite{Holstein59a,Barford14b},
\begin{eqnarray}\label{Eq:2}
\nonumber
\hat{H}_{FH} &=& \hat{H}_{F}
-A\hbar\omega_{\textrm{vib}}\sum_{n=1}^{N} \tilde{Q}_n \hat{N}_{n}
+ \frac{\hbar\omega_{\textrm{vib}}}{2}\sum_{n=1}^{N} \left( \tilde{Q}_n^2 +\tilde{P}_n^2 \right).\\
\end{eqnarray}
$\hat{H}_{F}$ is the Frenkel Hamiltonian, defined in Eq.\ (\ref{Eq:1}), while
$\tilde{Q} = (K_{\textrm{vib}}/\hbar \omega_{\textrm{vib}})^{1/2}Q$
and
$\tilde{P} =(\omega_{\textrm{vib}}/\hbar K_{\textrm{vib}})^{1/2}P$
are the dimensionless displacement and momentum of the normal mode.
The second term on the right-hand-side of Eq.\ (\ref{Eq:2}) indicates that the normal mode couples linearly to the local exciton density\footnote{There is also a weaker and less significant coupling of the normal mode to the exciton bond-order operator\cite{Barford14b, Binder14}.}. $A$ is the dimensionless exciton-phonon coupling constant, which introduces the important polaronic parameter, namely the local Huang-Rhys factor
\begin{equation}\label{Eq:112}
S = \frac{A^2}{2}.
\end{equation}
The final term is the sum of the elastic and kinetic energies of the harmonic oscillator, where
$\omega_{\textrm{vib}}$ and $K_{\textrm{vib}}$ are the angular frequency and force constant of the oscillator, respectively.
The Frenkel-Holstein model is another example of a coarse-grained Hamiltonian which, in addition to coarse-graining the exciton motion, assumes that the atomistic motion of the carbon nuclei can be replaced by appropriate local normal modes.
Exciton-nuclear dynamics is often modeled via the Ehrenfest approximation, which treats the nuclear coordinates as classical variables moving in a mean field determined by the exciton. However, as described in Section \ref{se:2}, the Ehrenfest approximation fails to correctly describe ultrafast dynamical processes.
A correct description of the coupled exciton-nuclear dynamics therefore requires a full quantum mechanical treatment of the system. This is achieved by introducing the harmonic oscillator raising and lowering operators, $\hat{b}_n^{\dagger}$ and $\hat{b}_n$, for the normal modes i.e.,
$ \tilde{Q}_n \rightarrow \hat{\tilde{Q}}_n = (\hat{b}_n^{\dagger} + \hat{b}_n)/\sqrt{2}$
and
$ \tilde{P}_n \rightarrow \hat{\tilde{P}}_n = i(\hat{b}_n^{\dagger} - \hat{b}_n)\sqrt{2}$.
The time evolution of the quantum system can then conveniently be simulated via the TEBD method, as briefly described in Section \ref{se:2}.
Since the photoexcited system has a different electronic bond order than the ground state, an instantaneous force is established on the nuclei.
As described in Section \ref{se:3.3}, this force creates an exciton-polaron, whose spatial size is quantified by the exciton-phonon correlation function\cite{Hoffmann02}
\begin{equation}\label{Eq:15}
C_n^{\textrm{ex-ph}}(t) \propto \sum_m \langle\hat{N}_{m} \hat{\tilde{Q}}_{m+n} \rangle.
\end{equation}
$C_n^{\textrm{ex-ph}}$ correlates the local phonon displacement, $Q$, with the instantaneous exciton density, $N$, $n$ monomers away.
$C_n^{\textrm{ex-ph}}(t)$, illustrated in Fig.\ \ref{Fi:40}, shows that the exciton-polaron is established within 10 fs (i.e., within half the period of a C-C bond vibration) of photoexcitation. The temporal oscillations, determined by the C-C bond vibrations, are damped as energy is dissipated into the vibrational degrees of freedom, which acts as a heat bath for the exciton. The exciton-phonon spatial correlations decay exponentially, extending over ca.\ 10 monomers. This short range correlation occurs because the C-C bond can respond relatively quickly to the exciton's motion. \footnote{In contrast, in the classical limit ($\omega \rightarrow 0$) the nuclei respond infinitesimally slowly to the exciton, so that the correlation length and the exciton-polaron mass diverge causing exciton-polaron self-localization.}
\begin{figure}
\includegraphics[width=9cm]{Fig5.jpeg}
\caption{The time-dependence of the exciton-phonon correlation function, Eq.\ (\ref{Eq:15}), after photoexcitation at time $t=0$. It fits the form $C_{n}^{\text{ex-ph}} = C_0\exp(-n/\xi)$ as $t \rightarrow \infty$, where $\xi \sim 10$. $n$ is a monomer index. The vibrational period is $20$ fs.
}
\label{Fi:40}
\end{figure}
The ultrafast establishment of quantum mechanically correlated exciton-phonon motion causes an ultrafast decay of off-diagonal-long-range-order (ODLRO) in the exciton site-basis density matrix. This is quantified via\cite{Kuhn97,Smyth12}
\begin{equation}\label{Eq:17}
C_n^{\textrm{coh}}(t) = \sum_m \left| \rho_{m,m+n} \right|,
\end{equation}
where $\rho_{m,m'}$ is the exciton reduced density matrix obtained by tracing over the vibrational degrees of freedom.
$C_n^{\textrm{coh}}(t)$ is displayed in Fig.\ \ref{Fi:5}, showing that ODLRO is lost within 10 fs.
The loss of ODLRO is further quantified by the coherence number, defined by
\begin{equation}\label{Eq:18b}
N^{\textrm{coh}} = \sum_n C_n^{\textrm{coh}},
\end{equation}
and shown in the inset of Fig.\ \ref{Fi:5}. Again, $N^{\textrm{coh}}$ decays to ca.\ 10 monomers in ca.\ 10 fs, reflecting the localization of exciton coherence resulting from the short range exciton-phonon correlations.
As discussed in Section \ref{se:4.4}, the loss of ODLRO leads to ultrafast fluorescence depolarization\cite{Mannouch18}.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{Fig6.jpeg}
\caption{The time dependence of the exciton coherence correlation function, $ C_n^{\textrm{coh}}$, Eq.\ (\ref{Eq:17}). The time dependence of the associated coherence number, $N^{\textrm{coh}}$ (Eq.\ (\ref{Eq:18b})), is shown in the inset.
$N^{\textrm{coh}}$ decays within $10$ fs, i.e., within half a vibrational period. Reproduced from J. Chem. Phys. \textbf{148}, 034901 (2018) with the permission of AIP publishing.}\label{Fi:5}
\end{figure}
We emphasise that the prediction of an electron-polaron with short range correlations is a consequence of treating the phonons quantum mechanically, while the decay of exciton-site coherences is a consequence of the exciton and phonons being quantum mechanically entangled. Neither of these predictions are possible within the Ehrenfest approximation.
\subsection{Role of system-environment interactions}\label{se:4.2}
For an exciton to dissipate energy it must first couple to fast internal degrees of freedom (as described in the last section) and then these degrees of freedom must couple to the environment to expell heat. For a low-energy exciton (i.e., a LEGS) this process will cause adiabatic relaxation on a single potential energy surface, forming a VRS\cite{Tretiak02,Bittner03,Sterpone08,Leener09}.
As shown in Fig.\ \ref{Fi:3}(a), however, for a kinetically hot exciton (i.e., a QEES) this relaxation is through a dense manifold of states and is necessarily a nonadiabatic interconversion between different potential energy surfaces. As already stated in Section \ref{se:2}, the Ehrenfest approximation fails to correctly describe this process.\footnote{In fact, the Ehrenfest approximation is the cause of the unphysical bifurcation of the exciton density onto separate chromophores found in Ehrenfest simulations of the relaxation dynamics of high energy photoexcited states\cite{Tozer12}.}
Dissipation of energy from an open quantum system arising from system-environment coupling is commonly described by a Lindblad master equation\cite{Breuer02}
\begin{equation}
\label{Eq:master}
\frac{\partial\hat{\rho}}{\partial {t}} = -\frac{i}{\hbar}\left[\hat{H},\hat{\rho}\right]-\frac{{\gamma}}{2}\sum_{n}\left(\hat{L}_{n}^{\dagger}\hat{L}_{n}\hat{\rho}+\hat{\rho}\hat{L}_{n}^{\dagger}\hat{L}_{n}-
2\hat{L}_{n}\hat{\rho}\hat{L}_{n}^{\dagger}\right),
\end{equation}
where $\hat{L}_{n}^{\dagger}$ and $\hat{L}_{n}$ are the Linblad operators, and $\hat{\rho}$ is the system density operator.
In practice, a direct solution of the Lindblad master equation is usually prohibitively expensive, as the size of Liouville space scales as the square of the size of the associated Hilbert space.
Instead, Hilbert space scaling can be maintained by performing ensemble averages over quantum trajectories (evaluated via the TEBD method), where the action of the Linblad dissipator is modeled by quantum jumps.\cite{Daley14}
In this section we assume that the C-C bond vibrations couple directly with the environment\cite{Mannouch18,Bednarz02}, in which case the Linblad operators are the associated raising and lowering operators (i.e., $\hat{L}_{n} \equiv \hat{b}_{n}$, introduced in the last section). In addition,
\begin{equation}
\label{Eq:ham_correc}
\hat{H}=\hat{H}_{\text{FH}}+\frac{{\gamma}\hbar}{4}\sum_{n}\left(\hat{\tilde{Q}}_{n}\hat{\tilde{P}}_{n}+\hat{\tilde{P}}_{n}\hat{\tilde{Q}}_{n}\right).
\end{equation}
(In Section \ref{se:5} we discuss coupling of the torsional modes with the environment\cite{Albu13}.)
The ultrafast localization of exciton ODLRO (or exciton-site decoherence) described in Section \ref{se:4.1} occurs via the coupling of the exciton to internal degrees of freedom, namely the C-C bond vibrations. We showed in Section \ref{se:3.3} (see Fig.\ \ref{Fi:2}(a)) that this coupling does not cause exciton density localization. However, dissipation of energy to the environment causes an exciton in a higher energy QEES to relax onto a lower energy LEGS (i.e., onto a chromophore) and thus the exciton density becomes localized.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{Fig7.jpeg}
\caption{The time dependence of the exciton localization correlation function, $C^{\textrm{loc}}_n$ (Eq.\ (\ref{eq:17c})), for an initial high-energy QEES. The main figure corresponds to the time evolution with the dissipation time $T = \gamma^{-1} = 100$ fs. The time dependence of the exciton density localization number, $N^{\textrm{loc}}$ (Eq.\ (\ref{eq:17d})), is given in the lower inset.
The upper inset corresponds to the time evolution without external dissipation showing that in this case exciton denisty localization does not occur. Reproduced from J. Chem. Phys. \textbf{148}, 034901 (2018) with the permission of AIP publishing.}\label{Fi:6}
\end{figure}
The spatial extent of the exciton density, averaged over an ensemble of quantum trajectories, is quantified by the correlation function\cite{Spano08a}, approximated by
\begin{equation}\label{eq:17c}
C_{n}^{\text{loc}}= \sum_m \left| \Psi_m \Psi_{m+n}^* \right|.
\end{equation}
Fig.\ \ref{Fi:6} shows the time dependence of $C_{n}^{\text{loc}}$ with an external dissipation time $T = \gamma^{-1} = 100$ fs. The time scale for localization is seen from the time dependence of the exciton localization length\cite{Tempelaar14},
\begin{equation}\label{eq:17d}
N_{\text{loc}}={\sum_{n}\left|n\right|C_{n}^{\text{loc}}}/{\sum_{n}C_{n}^{\text{loc}}},
\end{equation}
which corresponds to the average distance between monomers for which the exciton wavefunction overlap remains non-zero, and is given in the lower inset of Fig.~\ref{Fi:6}.
Evidently, the coupling to the environment - and specifically, the damping rate - controls the timescale for energy relaxation and exciton density localization onto chromophores.
In contrast, the upper inset to Fig.\ \ref{Fi:6} shows an absence of localization without external dissipation, indicating that exciton density localization is an extrinsic process.
Figure~\ref{Fi:6} is obtained by averaging over an ensemble of trajectories. To understand the physical process of localization onto a chromophore, Fig.\ 8 illustrates the exciton density of a single quantum trajectory for a photoexcited QEES (shown in Fig.\ 2(b)). At a time ca.\ 20~fs a `quantum jump' caused by the stochastic application of a Lindblad jump operator causes the exciton to localize onto the $j=2$ LEGS, shown in Fig.\ 2(a), i.e., the high-energy extended state has randomly localized onto a chromophore because of a `measurement' by the environment.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig8.jpeg}\label{Fi:17}
\caption{\label{fig:trajectory} The time dependence of the exciton density for a single trajectory of the quantum jump trajectory method.
The discontinuity in the density at ca.\ 20~fs is a `quantum jump' caused by the stochastic application of a Lindblad jump operator.
The dynamics were performed for an initial high energy QEES given in Fig.\ 2(b), showing localization onto the LEGSs (i.e., a chromophore) labeled $j=2$ in Fig.\ 2(a). Reproduced from J. Chem. Phys. \textbf{148}, 034901 (2018) with the permission of AIP publishing.}
\end{figure}
\subsection{Role of slow bond rotations}\label{se:4.3}
By dissipating energy into the environment on sub-ps timescales, hot excitons relax into localized LEGSs, i.e., onto chromophores. The final intrachain relaxation and localization process now takes place, namely exciton-polaron formation via coupling to the torsional degrees of freedom. For this relaxation to occur bond rotations must be allowed, which means that this process is highly dependent on the precise chemical structure of the polymer and its environment.
Assuming that bond rotations are not sterically hindered, their coupling to the excitons is conveniently modeled (via Eq.\ (\ref{Eq:5}) and Eq.\ (\ref{Eq:10c})) by supplementing the Frenkel-Holstein model (i.e., Eq.\ (\ref{Eq:2})) by\cite{Barford18}
\begin{eqnarray}\label{Eq:8}\nonumber
\hat{H}_{\textrm{rot}} = - \sum_{n=1}^{N-1}
B(\theta_n^0)\times ({\phi}_{n+1} - {\phi}_{n})\hat{T}_{n,n+1}
+ \frac{1}{2} \sum_{n=1}^{N} \left( K_{\textrm{rot}}{\phi}_n^2 + {L}_n^2/I \right).\\
\end{eqnarray}
Here, ${\phi}$ is the angular displacement of a monomer from its groundstate equilibrium value and
${L}$ is the associated angular momentum of a monomer around its bridging bonds.
The first term on the right-hand-side of Eq.\ (\ref{Eq:8}) indicates that the change in the dihedral angle,
$\Delta {\theta}_{n}= ( {\phi}_{n+1} - {\phi}_{n})$,
couples linearly to the bond-order operator, $\hat{T}_{n,n+1}$,
where \begin{equation}\label{Eq:24}
B(\theta_n^0) = J_{SE}\sin 2 \theta_n^0
\end{equation}
is the exciton-roton coupling constant and $\theta_n^0$ is the groundstate dihedral angle for the $n$th bridging bond.
The final term is the sum of the elastic and kinetic energies of the rotational harmonic oscillator.
The natural angular frequency of oscillation is $\omega_{\textrm{rot}} = (K_{\textrm{rot}}/I)^{1/2}$, where $K_{\textrm{rot}}$ is the elastic constant of the rotational oscillator and $I$ is the moment of inertia, respectively. As discussed in Section \ref{se:3.3.1}, $K_{\textrm{rot}}$ is larger for the bridging bond in the excited state than the groundstate, because of the increase in bond order. Also notice that both the moment of inertia (and thus $\omega_{\textrm{rot}}$) of a rotating monomer and its viscous damping from a solvent are strongly dependent on the side groups attached to it. As discussed in the next section, this observation has important implications for whether the motion is under or over damped and on its characterstic timescales.
Unlike C-C bond vibrations, being over 10 times slower torsional oscillations can be treated classically\cite{Barford18}. Furthermore, since we are now concerned with adiabatic relaxation on a single potential energy surface, we may employ the Ehrenfest approximation.
Thus, using Eq.\ (\ref{Eq:8}), the torque on each ring is
\begin{eqnarray}\label{Eq:25}
\nonumber
\Gamma_n &=& - \frac{\partial \langle \hat{H}_{rot} \rangle}{\partial {\phi}_n}\\
&=& -K_{\textrm{rot}}{\phi}_n +\lambda_n
\end{eqnarray}
where we define
\begin{equation}\label{Eq:14}
\lambda_n = B(\theta_{n-1}^0) \langle \hat{T}_{n-1,n} \rangle - B(\theta_{n}^0) \langle \hat{T}_{n,n+1} \rangle.
\end{equation}
Setting $\Gamma_n = 0$ gives the equilibrium angular displacements in the excited state as
${\phi}_n^{\textrm{eq}} = \lambda_n/K_{\textrm{rot}}$.
${\phi}_n$ is subject to the Ehrenfest equations of motion,
\begin{equation}\label{Eq:18}
I \frac{d {\phi}_n}{d {t}} = {L}_n,
\end{equation}
and
\begin{eqnarray}\label{Eq:19}
\frac{d {L}_n}{d {t}} && = \Gamma_n- \gamma L_n,
\end{eqnarray}
where the final term represents the damping of the rotational motion by the solvent.
\subsubsection{A single torsional oscillator}\label{se:4.3.1}
Before considering a chain of torsional oscillators, it is instructive to review the dynamics of a single, damped oscillator subject to both restoring and displacement forces. The equation of motion for the angular displacement is
\begin{equation}\label{}
\frac{d^2\phi(t)}{dt^2} = -\omega_{\textrm{rot}}^2(\phi(t) - \phi_{\textrm{eq}}) -\gamma \frac{d \phi(t)}{dt,}
\end{equation}
where $\phi_{\textrm{eq}} = \lambda/K_{rot}$ is proportional to the displacement force.
In the \emph{underdamped regime}\cite{French71}, defined by $\gamma < 2\omega_{\textrm{rot}}$,
\begin{equation}\label{}
\phi(t) = \phi_{\textrm{eq}}\left(1-\cos(\omega t) \exp(-\gamma t/2)\right),
\end{equation}
where $\omega = (\omega_{\textrm{rot}}^2 - \gamma^2/4)^{1/2}$. In this regime, the torsional angle undergoes damped oscillations with a period $T = 2\pi/\omega$ and a decay time $\tau = 2/\gamma$.
Conversely, in the \emph{overdamped regime}\cite{French71}, defined by $\gamma > 2\omega_{\textrm{rot}}$,
\begin{eqnarray}\label{}\nonumber
\phi(t) = \phi_{\textrm{eq}}\left(1- \frac{1}{4\beta}\left(\gamma_1\exp(-\gamma_2 t/2) - \gamma_2\exp(-\gamma_1 t/2)\right) \right),\\
\end{eqnarray}
where $\gamma_1 = \gamma + 2\beta$, $\gamma_2 = \gamma - 2\beta$ and $\beta = (\gamma^2/4 - \omega_{\textrm{rot}}^2)^{1/2}$.
Now, the torsional angle undergoes damped biexponential decay with the decay times $\tau_1 = 2/\gamma_1$ and $\tau_2 = 2/\gamma_2$. In the limit of strong damping, i.e., $\gamma \gg 2\omega_{\textrm{rot}}$, there is a fast relaxation time $\tau_1 = 1/\gamma = \tau/2$ and a slow relaxation time $\tau_2 = \gamma/\omega_{\textrm{rot}}^2 \gg \tau$. In this limit, as the slow relaxation dominates at long times, the torsional angle approaches equilibrium with an effective mono-exponential decay.
For a polymer without alkyl side groups, e.g., PPP and PPV, $\omega_{\textrm{rot}} \sim \gamma \sim 10^{13}$ s$^{-1}$ and are thus in the underdamped regime with sub-ps relaxation. However, polymers with side groups, e.g., P3HT, MEH-PPV and PFO, have a rotational frequency up to ten times smaller and a larger damping rate, and are thus in the overdamped regime\cite{Wells08}.
\subsubsection{A chain of torsional oscillators}\label{se:4.3.2}
An exciton delocalized along a polymer chain in a chromophore couples to multiple rotational oscillators resulting in collective oscillator dynamics.
Eq.\ (\ref{Eq:24}) and Eq.\ (\ref{Eq:14}) indicate that torsional relaxation only occurs if the monomers are in a staggered arrangement in their groundstate, i.e., $\theta_n^0 = (-1)^n\theta^0$. In this case the torque acts to planarize the chain. Furthermore, since the torsional motion is slow, the self-trapped exciton-polaron thus formed is `heavy' and in the under-damped regime becomes self-localized on a timescale of a single torsional period, i.e., $200 - 600$ fs. In this limit the relaxed staggered bond angle displacement mirrors the exciton density. Thus, the exciton is localized precisely as for a `classical' Landau polaron and is spread over $\sim 10$ monomers\cite{Barford18}.
\begin{figure*}
\centering
\includegraphics[width=0.7\textwidth]{Fig9.jpeg}
\caption{The time-evolution of the staggered angular displacement, $\langle{\phi}_n\rangle\times(-1)^n$. The change of dihedral angle is $\Delta \theta_n = (\phi_{n+1}-\phi_n)$, showing local planarization for a PPP chain of 21 monomers.
The inset displays the time-evolution of the exciton density, $\langle N_n \rangle$,
showing exciton density localization after a single torsional period ($\sim 200$ fs). In the long-time limit (i.e., $t \gtrsim 400$ fs) $\langle{\phi}_n\rangle \propto \langle N_n \rangle\times(-1)^n$, illustrating classical (Landau) polaron formation.
Reproduced from J. Chem. Phys. \textbf{149}, 214107 (2018) with the permission of AIP publishing.}\label{Fi:7}
\end{figure*}
The time-evolution of the staggered angular displacement, $\langle{\phi}_n\rangle\times(-1)^n$, is shown in Fig.\ \ref{Fi:7} illustrating that these displacements reach their equilibrated values after two torsional periods (i.e., $t \gtrsim 400$ fs). The inset also displays the time-evolution of the exciton density, $\langle N_n \rangle$, showing exciton density localization after a single torsional period ($\sim 200$ fs).
So far we have described how exciton coupling to torsional modes causes a spatially varying planarization of the monomers that acts as a one-dimensional potential which self-localizes the exciton. The exciton `digs a hole for itself', forming an exciton-polaron\cite{Landau33}. Some researchers\cite{Westenhoff06}, however, argue that torsional relaxation causes an exciton to become more \emph{delocalized}. A mechanism that can cause exciton delocalization occurs if the disorder-induced localization length is shorter than the intrinsic exciton-polaron size.
Then, in this case for freely rotating monomers, the stiffer elastic potential in the excited state causes a decrease both in the variance of the dihedral angular distribution, $\sigma_{\theta}^2 = k_BT/K_{\textrm{rot}}$, and the mean dihedral angle, $\theta_0$.
This, in turn, means that the exciton band width, $|4J|$, increases and the diagonal disorder\cite{Barford14b}, $\sigma_J = J_{SE} \sigma_{\theta}\sin 2\theta_0$,
decreases. Hence, the disorder-induced localization, $L_{\textrm{loc}} \sim (|J|/\sigma_J)^{2/3}$, increases (see Section \ref{se:3.2}).
\subsection{Summary}
The conclusions that we draw from the previous three sections are that a band edge excitation (i.e., a LEGS, which is an exciton spanning a single chromophore) undergoes ultrafast exciton site decoherence via its coupling to fast C-C bond stretches. It subsequently couples to slow torsional modes causing planarization and exciton density localization on the chromophore. A hot exciton (i.e., a QEES) also undergoes ultrafast exciton site decoherence. However, exciton density localization within a chromophore only occurs after localization onto the chromophore via a stochastic interaction with the environment.
\subsection{Time resolved fluorescence anisotropy}\label{se:4.4}
For general polymer conformations, the loss of ODLRO (or the localization of the exciton coherence function) causes a reduction and rotation of the transition dipole moment.
The rotation is quantified by the fluorescence anisotropy, defined by\cite{Lakowicz06}
\begin{equation}
\label{eq:r}
r = \frac{I_{\parallel}-I_{\perp}}{I_{\parallel}+2I_{\perp}},
\end{equation}
where $I_{\parallel}$ and $I_{\perp}$ are the intensities of the fluorescence radiation polarised parallel and perpendicular to the incident radiation, respectively.
For an arbitrary state of a quantum system, $|\Psi\rangle$, the integrated fluorescence intensity polarised along the $x$-axis is related to the $x$ component of the transition dipole operator, $\hat{\mu}_{x}$, by
\begin{eqnarray}\label{Eq:20}
\label{t}
I_{x} \propto \sum_{v}\left|\langle\Psi|\hat{\mu}_{x}|\textrm{GS},v\rangle\right|^{2},
\end{eqnarray}
where $|\textrm{GS},v\rangle$ corresponds to the system in the ground electronic state, with the nuclear degrees of freedom in the state characterised by the quantum number $v$.
The averaged fluorescence anisotropy is defined by
\begin{equation}
\label{eq:intens_av}
\langle r\left(t\right)\rangle = 0.4\times\frac{\sum_{i}I_{i}\left(t\right)r_{i}\left(t\right)}{\sum_{i}I_{i}\left(t\right)},
\end{equation}
where $I_{i}\left(t\right)$ is the total fluorescence intensity and $r_{i}\left(t\right)$ is the fluorescence anisotropy, associated with a particular conformation $i$ at time $t$. The factor of 0.4 is included on the assumption that the polymers are oriented uniformly in the bulk material.\cite{Lakowicz06} Fig.\ \ref{Fi:18} shows the simulated $\langle r\left(t\right)\rangle$ for both a high energy QEES and a low energy LEGS for an ensemble of conformationally disordered polymers.
\begin{figure}[h]
\centering
\includegraphics[width=0.40\textwidth]{Fig10.jpeg}
\caption{The time dependence of the fluorescence anisotropy, $\langle r\left(t\right)\rangle$, for two initial Frenkel excitons coupled to C-C bond stretches. The red curve corresponds to an initial LEGS, while the blue curve corresponds to a QEES. Reproduced from J.\ Chem.\ Phys.\ \textbf{148}, 034901 (2018) with the permission of AIP publishing.}
\label{Fi:18}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.55\textwidth]{Fig11.jpeg}
\caption{The experimental time dependence of the fluorescence anisotropy, $\langle R\left(t\right)\rangle$, in polythiophene in solution.
$\langle R\left(t\right)\rangle$ has decayed from $0.4$ to $\sim 0.25$ within 10 fs, consistent with the theoretical predictions shown in Fig.\ \ref{Fi:18}.
Subsequent fluorescence depolarization is caused by slower torsional relaxation on timescales of $1-10$ ps followed by possible conformational changes\cite{Wells07}.
Reproduced from J.\ Phys.\ Chem.\ C \textbf{111}, 15404 (2007) with the permission of ACS publishing.}\label{Fi:19}
\end{figure}
It is instructive to express Eq.\ (\ref{Eq:20}) as
\begin{eqnarray}\label{Eq:21}
I_{x} \propto \sum_{m,n} s_m^x s_n^x \rho_{mn},
\end{eqnarray}
where $s^x_m$ is the $x$-component of the unit vector for the $m$th monomer and $\rho_{mn}$ is the exciton reduced density matrix.
Then, using Eq.\ (\ref{Eq:17}), Eq.\ (\ref{Eq:18b}), and Eq.\ (\ref{Eq:21}), we observe that the emission intensity, $I_{x}$, is related to the coherence length, $N^{\textrm{coh}}$. Thus, not surprisingly, the dynamics of $\langle r\left(t\right)\rangle$ resembles that of $N^{\textrm{coh}}(t)$ shown in Fig.\ 6. In particular, we observe a loss of fluorescence anisotropy within 10 fs, mirroring the reduction of $N^{\textrm{coh}}$ in the same time.
Furthermore, since there is greater exciton coherence localization for the QEES than for the LEGS, the former exhibits a greater loss of anisotropy.
This predicted loss of fluorescence anisotropy within 10 fs has been observed experimentally, as shown in
Fig.\ \ref{Fi:19}.
Slower sub-ps decay of anisotropy occurs because of exciton density localization via coupling to torsional modes.\footnote{I.\ Gonzalvez Perez and W.\ Barford, in preparation.}
\section{Intrachain Exciton Motion}\label{se:5}
The last section described the relaxation and localization of higher energy excited states onto chromophores, and the subsequent torsional relaxation and localization on the chromophore. We now consider the relaxation and dynamics of these relaxed excitons caused by the stochastic torsional fluctuations experienced by a polymer in a solvent.
Environmentally-induced intrachain exciton relaxation in poly(phenylene ethynylene) was modeled by Albu and Yaron\cite{Albu13} using the Frenkel exciton model supplemented by the torsional degrees of freedom, i.e., $\hat{H} = \hat{H}_F + \hat{H}_{rot}$ (given by Eq.\ (\ref{Eq:1}) and Eq.\ (\ref{Eq:8}), respectively). Fast vibrational modes were neglected because although they cause self-trapping, they do not cause self-localization, and these modes can be assumed to respond instantaneously to the torsional modes. The polymer-solvent interactions were modeled by the Langevin equation. For chains longer than the exciton localization length the excited-state relaxation showed biexponential behavior with a shorter relaxation time of a few ps and a longer relaxation time of tens of ps.
After photoexcitation of the $n=2$ (charge-transfer) exciton in oligofluorenes, Clark \emph{et al.}\cite{Clark12} reported torsional relaxation on sub-100 fs timescales. Since this timescale is faster than the natural rotational period of an undamped monomer, they ascribed it to the electronic energy being rapidly converted to kinetic energy via nonadiabatic transitions. They argue that this is analogous to inertial solvent reorganization.
Tozer and Barford\cite{Tozer15} using the same model as Albu and Yaron to model intrachain exciton motion in PPP where the exciton dynamics were simulated on the assumption that at time $t+\delta t$ the new exciton target state is the eigenstate of $\hat{H}(t+\delta t)$ with the largest overlap with the previous target state at time $t$.\footnote{This latter assumption was shown by Lee and Willard\cite{Lee19} to be problematic for the non-adiabatic transport described in Section \ref{se:5.4}.}
A more sophisticated simulation of exciton motion in poly(p-phenylene vinylene) and oligothiophenes chains was performed by Burghardt and coworkers\cite{Binder20a, Binder20b, Binder20c, Hegger20} where high-frequency C-C bond stretches were also included, the solvent was modeled by a set of harmonic oscillators with an Ohmic spectral density, and the system was evolved via the multilayer-MCTDH method. Their results, however, are in quantitative agreement with those of Tozer and Barford in the `low-temperature' limit (discussed in Section \ref{se:5.3}), namely activationless, linearly temperature-dependent exciton diffusion with exciton diffusion coefficients larger, but close to experimental values.
The Brownian forces excerted by the solvent on the polymer monomers have two consequences. First, as already noted in Section \ref{se:3.2}, the instantaneous spatial dihedral angle fluctuations Anderson localize the Frenkel center-of-mass wavefunction. Second, the temporal dihedral angle fluctuations cause the exciton to migrate via two distinct transport processes.\footnote{This process is sometimes referred to as Environment-Assisted Quantum Transport\cite{Rebentrost09}.}
At low temperatures there is small-displacement adiabatic motion of the exciton-polaron as a whole along the polymer chain, which we will characterize as a `crawling' motion. At higher temperatures the torsional modes fluctuate enough to cause the exciton to be thermally excited out of the self-localized polaron state into a more delocalized LEGS or quasi-band QEES. While in this more delocalized state, the exciton momentarily exhibits quasi-band ballistic transport, before the wavefunction `collapses' into an exciton-polaron in a different region of the polymer chain. We will characterize this large-scale displacement as a non-adiabatic `skipping' motion.
Before describing the details of these types of motion, we first describe a model of solvent dynamics and consider again exciton-polaron formation in a polymer subject to Brownian fluctuations.
\subsection{Solvent dynamics}\label{se:5.1}
If the solvent molecules are subject to spatially and temporally uncorrelated Brownian fluctuations, then the monomer rotational dynamics are controlled by the Langevin equation
\begin{equation} \label{Langevin}
\frac{d L_{n}(t)}{dt} = \Gamma_{n}(t) + R_{n}(t) -\gamma L_{n}(t),
\end{equation}
where $\Gamma_{n}(t)$ is the systematic torque given by Eq.\ (\ref{Eq:25}).
$R_{n}(t)$ is the stochastic torque on the monomer due to the random fluctuations in the solvent and $\gamma$ is the friction coefficient for the specific solvent.
From the fluctuation-dissipation theorem, the distribution of random torques is given by
\begin{equation}\label{fluc_diss}
\langle R_{m}(t)R_{n}(0) \rangle = 2I\gamma k_{B}T\delta_{mn}\delta(t),
\end{equation}
which are typically sampled from a Gaussian distribution with a standard deviation of
$\sigma_{R} = (2 I\gamma k_{B}T)^{\frac{1}{2}}$.
As a consequence of these Brownian fluctuations the monomer rotations are characterized by the autocorrelation function\cite{Nitzan06}
\begin{eqnarray}\nonumber
&& \langle\delta\phi(t)\delta\phi(0)\rangle =\\
&& \langle\delta\phi^{2}\rangle \left( \cos(\omega_{rot} t)+ \left(\frac{\gamma}{2\omega_{rot}}\right)\sin(\omega_{rot} t)\right) \exp(-\gamma t/2),
\end{eqnarray}
where
$\langle\delta\phi^{2}\rangle = {k_{B}T}/{K_{rot}}$,
$K_{rot}$ is the stiffness and $\omega_{rot} = \sqrt{K_{rot}/I}$ is the angular frequency of the torsional mode.
\subsection{Polaron formation}\label{se:5.2}
As we saw in Section \ref{se:4.3}, at zero temperature torsional modes couple to the exciton, forming an exciton-polaron. At finite temperatures, however, a combination of factors affect the localization of the exciton. First, the exciton will still attempt to form a polaron. However, the thermally induced fluctuations in the torsional angles will affect the size of this exciton-polaron, as there is a non-negligible probability that the exciton will be excited out of its polaron potential well into a more delocalized state at high enough temperatures.
Second, the exciton states will be Anderson localized by the instantaneous torsional disorder.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{Fig12.jpeg}
\caption{The exciton localization length as a function of temperature for the `free' (i.e., `untrapped') exciton (red circles) and exciton-polaron (i.e., `self-trapped') (black squares). The untrapped exciton localization length obeys $L_{\textrm{loc}}^{\textrm{free}} \propto T^{-1/3}$. The lengths coincide when $k_BT \sim $ the exciton-polaron binding energy.
Reproduced from J. Chem. Phys. \textbf{143}, 084102 (2015) with the permission of AIP publishing.}\label{Fi:8}
\end{figure}
Fig.\ \ref{Fi:8} shows how the average localization length varies with temperature both with and without coupling between the exciton and the torsional modes (i.e., `self-trapped' and `free' exciton, respectively). As described in Section \ref{se:3.2}, the localization length for the `free' exciton is determined by Anderson localization. For small angular displacements from equilibrium a Gaussian distribution of dihedral angles implies a Gaussian distribution of exciton transfer integrals. Then, as confirmed by the simulation results shown in Fig.\ \ref{Fi:8}, from single-parameter scaling theory,
$L_{\textrm{loc}}^{\textrm{free}} \propto \sigma_{\theta}^{-2/3} = \langle\delta\theta^{2}\rangle^{-1/3} \propto T^{-1/3}$.
In contrast, the localization length of the `self-trapped' exciton slowly increases with temperature because of the thermal excitation of the exciton from the self-localized polaron to a more delocalized LEGS or QEES. The two values coincide when $k_B T$ equals the exciton-polaron binding energy (i.e., $T \sim 1500$ K in PPP).
\subsection{Adiabatic `crawling' motion}\label{se:5.3}
At low temperatures ($\lesssim 100$ K) the exciton has only a small amount of thermal energy, and not enough to regularly break free from its polaronic torsional distortions. Thus, the exciton-polaron migrates quasi-adiabatically and diffusively as a single unit. This is a collective motion of the exciton and the torsional degrees of freedom, as the torsional planarization accompanies the exciton. The random walk motion is illustrated in Fig.\ \ref{Fi:16}, which shows that the mean-square-distance traveled by the exciton-polaron is proportional to time, i.e., $\langle L^{2} \rangle = 2D_A(T) t$, where $D_A$ is the diffusion coefficient.
Since the migration of the exciton-polaron is an activationless process, as the gradients of Fig.\ \ref{Fi:16} indicate, at low temperatures it obeys the Einstein-Smoluchowski equation,
$D_A(T) = \mu k_{B}T$,
where $\mu$ is the mobility of the particle.
\begin{figure}
\centering
\includegraphics[width=0.50\textwidth]{Fig13.jpeg}
\caption{The intrachain mean-square-distance traveled by an exciton-polaron at low temperatures caused by stochastic torsional fluctuations. The motion is diffusive, as shown by the mean-square-distance increasing linearly with time: $\langle L^{2} \rangle = 2D_A(T) t$. The gradients satisfy the Einstein-Smoluchowski equation, $D_A(T) = \mu k_{B}T$.
Reproduced from J. Chem. Phys. \textbf{143}, 084102 (2015) with the permission of AIP publishing.}\label{Fi:16}
\end{figure}
The time taken for an exciton to diffuse a distance $L$ along the chain is determined by the equation for a one-dimensional random walk, i.e.,
$\tau_D = \langle L^2 \rangle/2D$.
As shown in Fig.\ \ref{Fi:8}, the typical exciton-polaron localization length is $\sim 12$ monomers or $\sim 6$ nm in PPP.
This characteristic length scale implies a characteristic timescale, namely the time taken for the exciton to diffuse along a chromophore length.
As shown in Table \ref{Ta:2}, these timescales are typically $3-30$ ps at room temperature depending on the solvent friction coefficient, being shorter at higher temperatures and smaller damping rates. As we show in Section \ref{se:5}, these timescales are an order of magnitude shorter than F\"orster transfer times in the condensed phase.
As the exciton-polaron migrates along the polymer chain it experiences a different potential energy landscape, so its energy will also fluctuate on a timescale $\sim \tau_D$.
Interestingly, these timescales are consistent with the longer timescale found experimentally in biexponential fits of relaxation processes of polymers in solution (see Table I) and correspond to the longer timescale simulated by Albu and Yaron\cite{Albu13} in longer polymers.
\begin{table}
\centering
{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|M{1.5cm}|M{1.5cm}|M{2.5cm}|M{1.5cm}|}
\hline
$\gamma$ (s$^{-1}$) & $T$ (K) & $D_A$ ($\textrm{cm}^2 \textrm{ s}^{-1}$) & $\tau_D$ (ps) \\
\hline
$10^{11}$ & 300 & $6.0\times 10^{-2}$ & 3.0 \\
$10^{11}$ & 100 & $2.0\times 10^{-2}$ & 9.0 \\
\hline
$10^{12}$ & 300 & $2.7\times10^{-2}$ & 6.7 \\
$10^{12}$ & 100 & $ 9.0\times10^{-3}$ & 20 \\
\hline
$10^{13}$ & 300 & $6.0\times10^{-3}$ & 30 \\
$10^{13}$ & 100 & $2.0\times10^{-3}$ & 90 \\
\hline
\end{tabular}}
\caption{Calculated intrachain adiabatic exciton diffusion coefficients, $D_A$, and times, $\tau_D$, in PPP from ref\cite{Tozer15}.
$\tau_D$ is the time taken for an exciton to diffuse along a chromophore of linear size $6$ nm in a solvent at temperature, $T$, with a damping rate $\gamma$.
From simulations\cite{Tozer15}, $\tau_D \sim \gamma^{1/2}/T$.
Hegger \emph{et al.}\cite{Hegger20} obtained $D_A \sim 10^{-2}$ $\textrm{cm}^2 \textrm{ s}^{-1}$ in oligothiophenes at 300 K and $\gamma = 5\times10^{12}$ s$^{-1}$.}
\label{Ta:2}
\end{table}
\subsection{Nonadiabatic `skipping' motion}\label{se:5.4}
At higher temperatures, adiabatic `crawling' migration of the exciton-polaron, as described above, still occurs. However, a second non-adiabatic mechanism for the dynamics plays an important role. This mechanism involves the exciton-polaron being excited to a high enough energy by the thermal fluctuations to be excited out of the polaron potential well, resulting in a breakdown of the polaron and the exciton to enter an untrapped local exciton ground state (LEGS), or a higher energy quasi-extended exciton state (QEES). Once in this more delocalized state the exciton has quasi-band characteristics and travels quasi-ballistically.
As described in Section \ref{se:4}, however, on a sub-ps timescale the hot exciton will shed some of its excess kinetic energy and relax back into an exciton-polaron. As a result, the time-averaged exciton localization length calculations of Fig.\ \ref{Fi:8} show only a slight increase in localization length with increasing temperatures, as the majority of its lifetime is still spent in self-localized exciton-polarons.
The requirement that the exciton is excited out of the polaron potential well means that this process is activated. Thus, from a simple Fermi golden rule analysis it can be shown that\cite{Barford14c}
\begin{equation}\label{High_T_D}
D_{NA}(T) \sim T^{2/3} \exp\Bigg{(}-\frac{\Delta E}{k_{B}T}\Bigg{)},
\end{equation}
where $\Delta E$ is the exciton-polaron binding energy. At 300 K $D_{NA}$ is approximately twice as large as $D_{A}$ and thus the overall diffusion coefficient is considerably enhanced by this skipping motion.
The role of exciton transport in disordered one-dimensional systems via higher-energy quasi-band states has been discussed in ref\cite{Bednarz02}, where in that work phonons in the condensed phase environment induced non-adiabatic transitions.
\section{Interchain Exciton Motion}\label{se:6}
The stochastic, torsionally-induced intrachain exciton diffusion in polymers in solution described in the last section is not expected to be the primary cause of exciton diffusion in polymers in the condensed phase. Instead, owing to restricted monomer rotations and the proximity of neighboring chains, exciton transfer is determined by Coulomb-induced, F\"orster-like processes. Moreover, since
dissipation rates are typically\cite{Sterpone08,Leener09} $10^{12} - 10^{13}$ s$^{-1}$, whereas exciton transfer rates are typically $10^{9} - 10^{11}$ s$^{-1}$, exciton migration is an incoherent or diffusive process\cite{Scholes11a}.
Early models of condensed phase diffusion assumed that the donors and acceptors are point-dipoles whose energy distribution is a Gaussian random variable\cite{Movaghar86a, Movaghar86b, Meskers01}. An advantage of these models is that they allow for analytical analysis, for example predicting how the diffusion length varies with disorder and temperature\cite{Athanasopoulos19}.
They also reproduce some experimental features, such as the time-dependence of spectral diffusion. A disadvantage, however, is that there is no quantitative link between the model and actual polymer conformations and morphology.
More recent approaches have attempted to make the link between random polymer conformations and the energetic and spatial distributions of the donors and acceptors via the concept of extended chromophores\cite{Beljonne02,Beljonne05,Athanasopoulos08,Singh09} and using transition densities to compute transfer integrals. However, the usual practice has been to arbitrarily define chromophores via a minimum threshold in the $p$-orbital overlaps, and then obtain a distribution of energies by assuming that the excitons delocalize freely on the chromophores thus defined as a `particle-in-a-box'.
As discussed in Section \ref{se:3.2}, an unambiguous link between polymer conformations and chromophores may be made by defining chromophores via the spatial extent of local exciton ground states (LEGSs). Using this insight, a more realistic first-principles model that accounts for polymer conformations can be developed\cite{Barford12b,Barford14c}. This is described in Section \ref{se:6.1}, while its predictions and comparisons to experimental observations are described in Section \ref{se:6.2}.
\subsection{Modified F\"orster theory}\label{se:6.1}
The F\"orster exciton transfer rate from a donor (D) to an acceptor (A) has the general Golden rule form
\begin{equation}\label{Eq:34}
k_{DA} = \left(\frac{2\pi}{\hbar}\right)\left|J_{DA}\right|^2 \int D(E)A(E)dE,
\end{equation}
where $J_{DA}$ is the Coulomb-induced donor-acceptor transfer integral defined by Eq.\ (\ref{Eq:39}).
As we remarked in Section \ref{se:3.1}, the transition density vanishes for odd-parity singlet excitons; it also vanishes for all triplet excitons.
$D(E)$ and $A(E)$ are the donor and acceptor spectral functions, respectively,
defined by
\begin{equation}\label{}
D(E) = \sum_{v}F_{0v}^D\delta(E+{E}^{D}_{0v})
\end{equation}
and
\begin{equation}\label{}
A(E) = \sum_{v}F_{0v}^A\delta(E-{E}^{A}_{0v}),
\end{equation}
where $F_{0v}$ is the effective Franck-Condon factor, defined in Eq.\ (\ref{Eq:43}).
$E^A_{0v} = (E^A_{00} + v\hbar\omega_{vib})$ is the excitation energy of the acceptor,
while $E^D_{0v} = -(E^D_{00} - v\hbar\omega_{vib})$ is the de-excitation energy of the donor.
The link between actual polymer conformations and a realistic model of exciton diffusion is made by realising that the donors and acceptors for exciton transfer are LEGSs (i.e., chromophores).
This assumption is based on the observation that exciton transfer occurs at a much slower rate than state interconversion, so the donors are LEGSs, while the spectral overlap between LEGSs and higher energy QEESs is small, so the acceptors are also LEGSs. Moreover, the energetic and spatial distribution of LEGSs is entirely determined by the conformational and site disorder, as described in Section \ref{se:3.2}. Finally, polaronic effects are incorporated by an effective Huang-Rhys factor for each chromophore and the Condon approximation may be assumed as C-C vibrational modes do not cause exciton self-localization.
Then, as proved rigorously in ref\cite{Barford14c}:
\begin{enumerate}
\item{$J_{DA}$ is evaluated by invoking the Condon approximation and using the line-dipole approximation\cite{Book, Barford07}
\begin{equation}\label{Eq:42}
J_{DA} = \left(\frac{1}{4\pi\varepsilon_{r}\varepsilon_{0}}\right)
\sum_{\substack{n\in D \\ n'\in A}}\frac{\kappa_{nn'}}{R_{nn'}^{3}}\mu_{D}\Psi_D(n)\mu_{A}\Psi_A(n'),
\end{equation}
where $\Psi(n)$ is the LEGS center-of-mass wavefunction on monomer $n$ determined from the disordered Frenkel exciton model (Eq.\ (\ref{Eq:1})).
Since the spatial extent of $\Psi(n)$ defines a chromophore, the sum over $n$ and $n'$ is implicitly over monomers of a donor and acceptor chromophore, respectively.
$\mu_{X}$ is the transition dipole moment of a single monomer of the donor ($X=D$) or acceptor ($X=A$) chromophores (so $\mu_{X}\Psi_X(n)$ is the transition dipole moment of monomer $n$ as part of the chromophore).
$\kappa_{nn'}$ is the orientational factor, defined in Eq.\ (\ref{Eq:40}), and $R_{nn'}$ is the separation of monomers on the donor and acceptor chromophores. The line-dipole approximation is valid when the monomer sizes are much smaller than their separation on the donor and acceptor chromophores; it becomes the point-dipole approximation when the chromophore sizes are much smaller than their separation.}
\item{The spectral functions describe `polaronic' effects, by containing effective Franck-Condon factors which describe the chromophores coupling to effective modes with reduced Huang-Rhys parameters:
\begin{equation}\label{Eq:43}
F_{0v} = \frac{S_{\textrm{eff}}^v \exp(-S_{\textrm{eff}})}{v!},
\end{equation}
where
$S_{\textrm{eff}} = S/\textrm{PN}$,
$S$ is the local Huang-Rhys parameter (defined by Eq.\ (\ref{Eq:112})) and
$ \textrm{PN} = \left(\sum_n |\Psi_n|^4\right)^{-1}$
is the participation number (or size) of the chromophore\cite{Barford14a}.
}
\item{Similarly, the $0-0$ transition energy is defined by
$ E_{00} = (E^{\textrm{vert}}- E^{\textrm{relax}})$,
where
$E^{\textrm{vert}}$ is determined from the Frenkel exciton model and
$ E^{\textrm{relax}} = \hbar\omega S_{\textrm{eff}}$
is the effective reorganisation energy for the effective mode.}
\end{enumerate}
\subsection{Condensed-phase exciton diffusion}\label{se:6.2}
We might attempt to anticipate the results of the simulation of exciton diffusion from the properties of the exciton transfer rate, $k_{DA}$. When the chromophore size, $L$, is much smaller than the donor-acceptor separation, $R$, the point-dipole approximation is valid. In this limit $k_{DA} \sim L^2/R^6$ and thus the hopping rate \emph{increases} with increasing chromophore size. Conversely, when the chromophore size is much larger than the donor-acceptor separation, the line-dipole approximation predicts that for straight, parallel or collinear chromophores\cite{Wong04,Das10,Barford10c} $k_{DA} \sim 1/(LR)^2$ and thus the hopping rate \emph{decreases} with increasing chromophore size.
In practice, Monte Carlo simulations assuming a statistical model of polymer conformations find that the exciton hopping rate is essentially independent of disorder and hence of chromophore size. This is presumably because neither the assumption of straight, parallel chromophores nor point dipoles are valid. These simulations also show that the average time taken for the first exciton hop to occur after photoexcitation is $\sim 10$ ps, whereas the time intervals between hops just prior to radiative recombination is over 10 times longer, and indeed becoming so long that a radiative transition is competitive.
This increase in hopping time intervals occurs because as an exciton diffuses through the polymer system it continuously looses energy. Thus, the energetic condition for exciton transfer to occur, namely $E_A \le E_D$, becomes harder to satisfy and in general the spectral overlap between the donor and acceptor decreases. As the excitons diffuse they eventually become trapped in `emissive' chromophores, from which they radiate. As shown in Fig.\ \ref{Fi:14}, these emissive chromophores occupy the low energy tail of the LEGSs density of states. Their quasi-Gaussian distribution explains spectral diffusion\cite{Hayes95,Meskers01}: a time-dependent change in the fluorescence energy, satisfying $E \propto - \log t$.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{Fig14.jpeg}
\caption{The density of states for absorbing LEGSs (solid line) and emitting trap states (dashed line) for an ensemble of PPV chains, with a Gaussian distribution of dihedral angles.
$\langle \theta_0 \rangle = 10^0$, $\sigma_{\theta_0} = 5^0$, and $\sigma_{\theta} = 5^0$.
Reproduced from J. Chem. Phys. \textbf{141}, 164103 (2014) with the permission of AIP publishing.}\label{Fi:14}
\end{figure}
Typically, the average hop distance is between 4 nm (for strong disorder giving an average chromophore length of 8 nm) to 6 nm (for weak disorder giving an average chromophore length of 30 nm). On average, an exciton only makes four hops before radiating, and thus average diffusion lengths are between $\sim 8-12$ nm, being longer for more ordered systems. These theoretical predictions are consistent with experimental values obtained via various techniques\cite{Markov05,Lewis06,Scully06}
(see K\"ohler and B\"assler\cite{Kohler15} for further experimental references).
The diffusion length is remarkably insensitive to disorder, and from simulation satisfies $L_D \sim L_{\textrm{loc}}^{1/4} \sim \sigma^{-1/6}$; a result that can be explained by the spatial distribution of chromophores in randomly coiled polymers\cite{Barford12b}.
An interesting prediction of Anderson localization is that for the same mean dihedral angle lower energy chromophores are shorter than higher energy chromophores. Now, as the intensity ratio of the vibronic peaks in the emission spectrum, ${I_{00}}/{I_{01}}$, is proportional to the chromophore size\cite{Spano11,Spano14,Barford14a,Barford14b}, i.e.,
\begin{equation}
\frac{I_{00}}{I_{01}} \propto \frac{1}{S_{\textrm{eff}}} = \frac{\langle \textrm{PN} \rangle}{S},
\end{equation}
spectral diffusion also implies that ${I_{00}}/{I_{01}}$ reduces in time, as observed in time-resolved photoluminescence spectra in MEH-PPV (see Fig.\ 3 of ref\cite{Hayes95}).
According to simulations\cite{Barford14c}, ${I_{00}}/{I_{01}} \propto - \log t$.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Fig15.jpeg}
\caption{The calculated optical spectra of PPV assuming a statistical model of random polymer conformations. Exciton migration prior to emission causes a red-shift in energy, a narrowing of the inhomogeneous broadening, and a decrease in $I_{00}/I_{01}$.
Reproduced from J. Chem. Phys. \textbf{141}, 164103 (2014) with the permission of AIP publishing.}\label{Fi:9}
\end{figure}
Some of the key features of exciton relaxation and dynamics described in this review are nicely encapsulated by Fig.\ \ref{Fi:9}. This figure shows the simulated absorption to all absorbing states, the fluorescence via emission from all LEGSs (which occurs in the absence of exciton migration), and the time-integrated fluorescence following exciton migration and emission from `trap' chromophores. We observe that:
\begin{itemize}
\item{The absorption spectrum and the emission spectrum assuming no exciton migration are broadly a mirror image. However, the absorption is broader and has a high energy tail as absorption occurs to both LEGSs and QEESs (as also shown in Fig.\ \ref{Fi:3}(b)), whereas, from Kasha's law, emission occurs only from LEGSs following interconversion from QEESs.
}
\item{The emission following exciton migration is red-shifted, because the emissive states are in the low-energy tail of the density of states (as shown in Fig.\ \ref{Fi:14}).}
\item{The inhomogeneous broadening of the post-migration emission is narrowed, because the emissive states have a narrower density of states than LEGSs.}
\item{Similarly the intensity ratio, $I_{00}/I_{01}$, decreases, because on average emissive chromophores have shorter conjugation lengths than LEGSs.}
\end{itemize}
\section{Summary and Concluding Remarks}\label{se:7}
We have reviewed the various exciton dynamical processes in conjugated polymers. In summary, they are:
\begin{itemize}
\item{Following photoexcitation, the initial dynamical process is the correlation of the exciton and phonons associated with high-frequency C-C bond vibrations. This quantum mechanical entanglement causes exciton-site decoherence, which is manifest as sub-10 fs fluorescence depolarization (see Section \ref{se:4.1}).}
\item{Next, the energy that is transferred from the exciton to the nuclei is dissipated into the environment on a timescale determined by the strength of the system-bath interactions. For a hot exciton (i.e., a QEES) the system-bath interactions cause the entangled exciton-nuclear wavefunction to stochastically `collapse' into a particular LEGS, causing the exciton density to be localized on a `chromophore' (see Section \ref{se:4.2}).}
\item{The fate of an exciton on a chromophore is now strongly dependent on the polymer chemical structure and the type of environment. For underdamped, freely rotating monomers, the coupling of the exciton to the low-frequency torsional modes creates an exciton-polaron, with associated planarization and exciton-density localization (see Section \ref{se:4.3}).}
\item{For a polymer in solution, stochastic torsional fluctuations also causes the exciton-polaron to diffuse along the polymer chain; a process known as environment-assisted quantum transport\cite{Rebentrost09}. The diffusion coefficient is linearly proportional to temperature (see Section \ref{se:5}).}
\item{For a polymer in the condensed phase, the dominant post-ps process is F\"orster resonant energy transfer and exciton diffusion. An exciton diffusing in the random energy landscape soon gets trapped in chromophores occupying the low-energy tail of the LEGSs density of states, exhibiting $\log t$ spectral diffusion. An exciton typically diffuses $\sim 10$ nm before radiative decay, with the diffusion length weakly increasing with decreasing disorder (see Section \ref{se:6}).}
\end{itemize}
In this review we have argued that theoretical modeling of exciton dynamics over multiple time and length scales is only realistically possible by employing suitably parametrized coarse-grained exciton-phonon models. Moreover, to correctly account for the ultrafast processes of exciton-site decoherence and the relaxation of hot excitons onto chromophores, the exciton and vibrational modes must be treated on the same quantum mechanical basis and importantly the Ehrenfest approximation must be abandoned. We have also repeatedly noted that spatial and temporal disorder play a key role in exciton spectroscopy and dynamics; and it is for this reason that exciton dynamics is conjugated polymers is essentially an incoherent process.
In a previous review\cite{Barford17} we explained how spectroscopic signatures are highly-dependent on polymer multiscale structures, and how - in principle - good theoretical modeling of excitons and spectroscopy can be used as a tool to predict these polymer structures. This review builds on that prospectus by describing how time-resolved spectroscopy can be understood via a theoretical description of exciton dynamics coupled to information on polymer multiscale structures. Again, the reverse proposition follows: time-resolved spectroscopy coupled to a theoretical description of exciton dynamics can be used to provide insights into polymer multiscale structures.
\begin{acknowledgments}
I thank Isabel Gonzalvez Perez for helping to compile Table 1.
\end{acknowledgments}
|
1,108,101,566,739 | arxiv | \section{Introduction}
Modeling patient prognosis is a challenging but important topic in clinical research, where researchers analyze and predict clinical outcomes including response to certain therapy (\textit{e.g.}, radiotherapy, chemotherapy, surgery, immunotherapy for oncology), patient progression-free survival (PFS) and overall survival (OS). Research efforts have been paid on seeking significant biomarkers, \textit{e.g.}, EGFR mutation for EGFR-TKI therapy \cite{Yu2013AnalysisOT},
PD-L1 expression and tumor mutational burden (TMB) for immunotherapy \cite{Gibney2016PredictiveBF}. However, these biomarkers are generally costly and invasive, and possibly dissatisfactory for novel therapy, \textit{e.g.}, anti-PD-1 and anti-PD-L1 immunotherapy \cite{Sacher2016BiomarkersFT}. With more novel revolutionary therapy (including combination therapy \cite{Jain2001NormalizingTV}) available, a unified analytic framework for modeling patient prognosis is urged.
We address this issue via emerging deep learning technology by mining clinical data, \textit{e.g.}, electronic health records (EHR) \cite{Rajkomar2018ScalableAA}. Specifically, we focus on a unified approach to model patient prognosis under certain therapy. Prior arts are generally developed on a single data modality \cite{Hosny2018DeepLF,Sun2018ARA}. Besides, only a few studies \cite{Xu2019DeepLP} take into account the temporal / serial information. In clinical practice, multi-modal temporal data is continuously generated with numerous kinds of sensors and records. It is remarkably valuable to mine the easily accessible information to develop the prognosis prediction system, \textit{e.g.}, radiographics, laboratory and clinical information. We formalize the prognosis modeling as a \textbf{multi-modal asynchronous} time series classification task, and propose a \emph{MIA}-Prognosis~framework with \textbf{Measurement}, \textbf{Intervention} and \textbf{Assessment} (MIA) information, where Measurement and Intervention information are treated as inputs of multi-modal asynchronous time series to predict the Assessment as ground truth (details in Sec.~\ref{sec:framework}).
An algorithmic challenge is how to effectively and efficiently process multi-modal asynchronous time series like clinical information. Binkowski \emph{et al.} \cite{Binkowski2017AutoregressiveCN} propose a gated CNN for asynchronous time series analysis, where asynchronous time intervals are regarded as input features. This approach might not be suitable for the clinical scenario since it is not essentially asynchronous; data is needed to learn representation for time intervals. What we need for real-world clinical data processing is a natively asynchronous model, which is flexible and light-weight to learn from expensive clinical data. Inspired by recent advances in natural language processing (NLP), \textit{e.g.}, attention transformers \cite{Vaswani2017AttentionIA,yang2019modeling,Devlin2019BERTPO} and relative position encoding \cite{Shaw2018SelfAttentionWR}, we propose a Simple Temporal Attention (SimTA) module to process asynchronous time series, where attention matrix is learned simply from the time intervals of asynchronous time series (details in Sec.~\ref{sec:simta}).
The SimTA module is proven to be superior to standard RNN-based approaches in a synthetic asynchronous time series prediction dataset. Moreover, we experiment the proposed \emph{MIA}-Prognosis~framework on an in-house retrospective dataset of real-world non-small cell lung cancer (NSCLC) under anti-PD-1 immunotherapy. Our predictive model achieves promising performance on predicting the immunotherapy response after 90 days. Notably, this model could further stratify low-risk and high-risk patients in terms of long-term survival.
\section{Methods}
\subsection{MIA Prognosis: The Framework} \label{sec:framework}
\paragraph{\textbf{Categorizing Clinical Information.}}
In clinical practice, data of a numerous variety of modalities is collected. Most medical data is unaligned in time steps, which means that it has varying intervals between adjacent steps in time series. Such limitations call for a unified framework that integrates asynchronous data of different modalities. To address this issue, we first divide clinical information into three categories according to data sources: \textbf{measurement}, \textbf{intervention} and \textbf{assessment}, which defined our \emph{MIA}-Prognosis~framework. Measurement data comes from medical examinations such as imaging data (computed tomography, ultrasound, X-ray), laboratory and genetic tests. Measurement is the main information in our \emph{MIA}-Prognosis~framework. Interventions include actions such as injections and operations. Assessment evaluates the effectiveness of interventions, \textit{e.g.}, Response Evaluation Criteria in Solid Tumors (RECIST) \cite{Eisenhauer2009NewRE}, or 1-year overall survival rates. In this study, we use RECIST to obtain the ``ground truth''\footnote{RECIST is not theoretecally perfect. We refer to ``ground truth'' in a clinical sense.} of therapy response, where complete response (CR), partial response (PR), stable disease (SD) are regarded as response (R), and PD (progressive disease) is regarded as non-response (non-R). Note that measurement, intervention and assessment can also be categoried into either serial or static data, which depends on its status over time. These categorizations are the basis of our framework's capability of integrating heterogeneous multi-modal data.
\paragraph{\textbf{Model Overview.}}
\begin{figure}[tb]
\includegraphics[width=\textwidth]{fig/framework.pdf}
\caption{The \emph{MIA}-Prognosis~framework, with Measurement, Intervention and Assessment information. The asynchronous time series is encoded by the proposed Simple Temporal Attention (SimTA) module into a summary vector. The summary vector is further added with a temporal encoding of time intervals between the assessment time and the last time stamp ($\delta t_1$ and $\delta t_2$). Together with static information, these features predict the therapy response (R / non-R) after an unobserved period.} \label{fig:framework}
\end{figure}
We propose a framework that integrates multi-modal data in asynchronous time series, named \emph{MIA}-Prognosis. Fig.~\ref{fig:framework} gives an overall description of our framework. Due to the fact that clinical data of different modalities is usually unaligned in time, it is impractical to simply concatenate these vectors together and pad zeroes at the time step where a certain modality is missing. Therefore, we process each modality independently in our framework. We pass serial data of each modality through its own SimTA module, which outputs a summary vector. The summary vector is added with a temporal encoding (adapted from position encoding \cite{Vaswani2017AttentionIA}) of time intervals between the assessment time and the last time stamp. Static data goes through a multi-layer perceptron (MLP) that encodes it in high-dimensional embedding. We then concatenate summary vectors of serial data with static embedding, and input the concatenated vector into another MLP to give the final prediction of therapy response (R: response / non-R: non-response) after an unobserved period.
Existing deep sequential models, such as recurrent neural network (RNN), assume that time series data is synchronous in nature. However, this assumption does not hold in the context of clinical practice. Here we introduce a new module to help us process asynchronous data, named SimTA. Inspired by recent advances in natural language processing (NLP), \textit{e.g.}, attention transformers \cite{Vaswani2017AttentionIA,Devlin2019BERTPO} and relative position encoding \cite{Shaw2018SelfAttentionWR}, SimTA utilizes time interval information of asynchronous series to generate attention matrix, capturing temporal relationships between asynchronous time steps. It is worth noting that the latest steps in time series of different modalities are not likely to coincide with each other. In such cases, we use temporal encoding to make use of this information.
\subsection{Simple Temporal Attention for Asynchronous Time Series} \label{sec:simta}
Simple Temporal Attention (SimTA) is the key factor that enables our framework to process asynchronous times series. Let $X \in \mathbb{R}^{T \times C}$ be an asynchronous time series of length $T$, and $\boldsymbol{\tau}=[\tau_{1},\tau_{2},\dots,\tau_{T-1}] \in \mathbb{R}^{T-1}$ be the time interval vector between any adjacent time steps. A general formula of a single SimTA module can be described as:
\begin{equation}
SimTA(X,\boldsymbol{\tau})=softmax(A)\sigma(f(X)),
\end{equation}
where $\sigma$ denotes an activation function of our choice, and $f$ is a fully-connected layer. $A=S(X,\tau) \in \mathbb{R}^{T\times T}$ is the attention matrix that encodes relations between any two time steps. Matrix $A$ can be calculated using any edge-aware attention mechanism, \textit{e.g.}, multi-head self attention \cite{Vaswani2017AttentionIA} with relative position encoding~\cite{Shaw2018SelfAttentionWR}. In this study, we use an extremely simplified version that only encodes linearly time intervals, which is validated effective in our experiments:
\begin{equation}
A=S(\boldsymbol{\tau})=
\begin{bmatrix}
0, & -\infty,& -\infty, & \dots & -\infty \\
-\lambda \tau_1+\beta, & 0, & -\infty, & \dots & -\infty \\
-\lambda (\tau_1+\tau_2)+\beta, & -\lambda \tau_2+\beta, & 0, & \dots & -\infty\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
-\lambda \sum_{1}^{T-1}{\tau_i}+\beta, &-\lambda \sum_{2}^{T-1}{\tau_i}+\beta, & -\lambda \sum_{3}^{T-1}{\tau_i}+\beta, & \dots & 0
\end{bmatrix},
\label{eq:simta_attn}
\end{equation}
where $\lambda\in \mathbb{R}^+$ and $\beta\in \mathbb{R}$ are trainable parameters that apply a linear transformation on $\boldsymbol{\tau}$. The simplicity of this attention mechanism can help us cope with overfitting as well, considering we only have limited amount of data. The formula is based on the assumption that the more recent time steps should have stronger correlations with the current time step than further ones. Complex temporal information can be captured by stacking multiple SimTA modules. The complete SimTA model pipeline is a SimTA block of one or multiple SimTA modules, which outputs a summary vector, followed by an MLP that outputs the final prediction with softmax activation. To the best of our knowledge, the proposed SimTA is the first study to introduce attention mechanism to asynchronous time series analysis with proven effectiveness.
\subsection{Counterpart Approaches}
To demonstrate SimTA's capability of learning asynchronous temporal relations, we bring in LSTM (Long Short-term Memory) \cite{Hochreiter1997LongSM,Greff2017LSTMAS} as comparison. LSTM is a special type of RNN designed to capture relations over extended time intervals in sequences. In our experiments, we use LSTM models with a comparable size and the same training configuration as SimTA. Let $X_{i}$ be the value of the time series at time step $t_{i}$. Three LSTM approaches are tested, which differ in their input: (1) Only $X_{i}$; (2) $X_{i}$ and time intervals $\tau_{i}$; (3) $X_{i}$ and time stamps $t_{i}$. We mark them as LSTM, LSTM(i) and LSTM(s), respectively.
\section{Experiments}
\subsection{Proof of Concept on Synthetic Dataset}
We first validate the superiority of the proposed SimTA over RNN on asynchronous time series using a synthetic dataset.
\paragraph{\textbf{Dataset and Experiment Settings.}}
The synthetic dataset consists of the summation of $N$ trigonometric functions with random periods. Each time series is computed as:
\begin{equation}
X_{t}=\sum_{j=1}^{N}[{\alpha_{j}\sin(\omega_{j}\pi t+b_{j})+\beta_{j}}]+\eta\epsilon
\end{equation}
where $N$ is the number of trigonometric functions involved. $\epsilon$ is a white noise following the standard normal distribution, whose magnitude is controlled by a constant $\eta$. In our experiments, we choose $N=10$ and $\eta=0.5$. We generate 10,000 such asynchronous series, which are split in 80/20 for training/validation, respectively. The training/validation data is sampled once from the predefined distribution and fixed throughout the experiment.
During training, 10 different $X_{t}$ are randomly sampled from each series. The time intervals between any two adjacent points follow a uniform distribution between $0$ and a maximum interval level $I$. The model is tasked to predict the next 3 points ($+1,+2,+3$) following the last one in the input. Fig.~\ref{fig:poc} shows a sample series of the synthetic dataset.
In our experiments, we compare performances of SimTA and three LSTM models, which are LSTM, LSTM(i) and LSTM(s). For LSTM(i) and LSTM(s), $X_{t}$ and the time information are concatenated into one vector. SimTA follows the model described in Sec. \ref{sec:simta}.
\begin{figure}[tb]
\includegraphics[width=\textwidth]{fig/poc.pdf}
\caption{The synthetic time series and MSE loss curves of LSTM, LSTM(i), LSTM(s) and the proposed SimTA. \textbf{Left}: The illustration of a data sample. \textbf{Middle}: Training MSE loss curves. \textbf{Right}: Validation MSE loss curves. We clip the y axis in both loss curves for the sake of better visualization.} \label{fig:poc}
\end{figure}
\paragraph{\textbf{Results.}}
We train all four models for 100 epochs. Fig.~\ref{fig:poc} shows the training and validation mean squared error over the training phase. SimTA outperforms all three LSTM models on both training and validation data. It achieves significantly lower MSE (2.197 on SimTA and 6.427 on all LSTM approaches) compared with LSTM. The time information does help LSTM(i) and LSTM(s) to converge faster than vanilla LSTM, but all three end up with errors at the same level. It is worth noting that the LSTM model variants underfit the training set. From the observations above, we conjecture that the proposed SimTA outperforms existing standard sequential models such as LSTM for asynchronous time series.
\subsection{Predicting Response to Anti-PD-1 Immunotherapy for Non-Small Cell Lung Cancer (NSCLC)}
\paragraph{\textbf{Background.}}
Lung cancer is the most commonly diagnosed cancer worldwide. According to \cite{Bray2018GlobalCS}, lung cancer accounts for 18.4\% of the global cancer deaths in 2018. NSCLC makes up 80\%-85\% of these cases. Deep learning has shown its potential in precision medicine for lung cancer \cite{zhao20183d,zhao2019toward,yang2019development,yang2020relational}. Recently, immunotherapy has been proven to remarkably increase the overall survival and the life quality of patients with a variaty of cancers, including NSCLC. However, only a small percentage of patients benefit from immunotherapy and show lasting responses. There has been research on the identification of response predictors, whereas most of the effort are focused on biopsy analyses and serum biomarkers, \textit{e.g.}, PD-L1 expression and tumor mutation burden (TMB) for first-line immunotherapy. These methods are expensive, invasive, and not always consistently associated with tumor responses. Furthermore, no biomarker is available for predicting second-line NSCLC immunotherapy outcome so far . Such limitations emphasize the necessity for convenient, economical and non-invasive indicators, especially for second-line immunotherapy treatment.
\paragraph{\textbf{Dataset and Experiment Settings.}}
In this retrospective study, 99 patients with advanced or metastatic stage IIIB and IV NSCLC under second-line immunotherapy are included. The dataset includes 793 CT scans, 1335 laboratory blood tests, 99 clinical data, and 320 response evaluations as per RECIST1.1 \cite{Eisenhauer2009NewRE}. All data is further categorized into serial data and static data.
The CT scans are labelled by one radiologist with 8 years of experience, by manually segmenting the volume of interest (VOI) of the target lesion in each scan. An oncologist with 30 years of experience reviewed and confirmed the segmentation. CT volumes and segmentation masks are resampled to uniform spacing ($1mm \times 1mm \times 1mm$), with B-spline interpolation for CT volumes and nearest-neighbor interpolation for VOI masks. We use radiomics features~\cite{Gillies2016RadiomicsIA} to represent the radiological features due to limited number of samples. With large data available, a fully end-to-end CNN could also be used as the feature extractor. 107 radiomics features are extracted from each VOI using PyRadiomics~\cite{Griethuysen2017ComputationalRS}. Radiomics features are treated as serial data unless there is only one CT examination. The serial blood test features are in 22 dimensions and static clinical information features are in 18 dimensions. All categorical features are encoded in one-hot vectors. Numeric features are normalized by removing the mean and scaling to unit variance to ensure stable training and faster convergence. Intervention information is one-hot encoded (\textit{i.e.}, a binary flag at a time step). Serial radiomics, laboratory blood test and intervention is asynchronous in time.
In our experiments, models are tasked to output binary predictions of R (response) or Non-R (non-response) of each response evaluation, with static data and all serial data before 90 days prior to time of response assessment. We use binary cross entropy (BCE) as the loss function. SimTA model is optimized with Adam optimizer~\cite{Kingma2014AdamAM}. We split the 99 patients into 3-fold (33 patients in each fold), and perform 3-fold cross validation for evaluating our method. Hyperparameters of model structure and training configuration are chosen using a grid search with bootstrap on the training dataset in each cross validation fold. To verify the effectiveness of SimTA on asynchronous time series, we include LSTM, LSTM(i) and LSTM(s) in our ablation study as comparison. LSTM models of comparable parameters are trained under similar setting.
We further associate model prediction with clinical survival benefits, specifically, overall survival (OS) and progression-free survival (PFS). A cutoff value of 0.5 is used for stratifying patients into high-risk and low-risk groups. The serial data before 90 days prior to time of response assessment is used as input to the trained model. Kaplan-Meier analysis and log-rank test for the survival analysis validate the effectiveness of our method in terms of patient survival.
\paragraph{\textbf{Results.}}
\begin{table}[tb]
\centering
\caption{Model performance of on predicting immunotherapy response, including standard RNN approaches (LSTM, LSTM(i), LSTM(s)) instead of the proposed SimTA, and our methods with multi-modal and single-modal inputs.} \label{tab:model-performance}
\begin{tabular*}{\hsize}{@{}@{\extracolsep{\fill}}l|ccc|ccc@{}}
\toprule
Methods & LSTM & LSTM(i) & LSTM(s) & Ours & Ours w/o radiomics & Ours w/o lab \\
\midrule
AUC & 0.71&0.71&0.70 & 0.80&0.47&0.58 \\
\bottomrule
\end{tabular*}
\end{table}
\begin{figure}[tb]
\includegraphics[width=\textwidth]{fig/km_plot.pdf}
\caption{Model performance on predicting NSCLC patient survival under anti-PD-1 immunotherapy. \textbf{Left}: Patient survival curve visualized by Kaplan-Meier (K-M) plot of progression-free survival (PFS), p-value of log-rank test to high/low-risk groups is $<0.01$. \textbf{Right}: K-M plot of overall survival (OS), with $p<0.01$.} \label{fig:km_plot}
\end{figure}
As depicted in Table \ref{tab:model-performance}, promising results are observed in predicting immunotherapy outcome using the proposed framework. The area under curve (AUC) of receiver operating characteristic (ROC) curve is 0.80 with our \emph{MIA}-Prognosis~framework, whereas vanilla LSTM, LSTM(i) and LSTM(s) are achieving 0.71, 0.71 and 0.70 AUC respectively. The LSTM counterparts does not totally fail in this case because the patients are taking CT scans and blood tests on a fairly regular schedule, the interval variance is mostly smaller than seven days. Still, SimTA outperforms LSTM by a large margin in this “mildly” asynchronous serial data modelling task. We also validate the necessity of multi-modal input. Without radiomics feature or laboratory blood test results, our framework reaches very low AUC of 0.47 and 0.58 respectively, suggesting the significance of multi-modal model in this task. Moreover, as shown in Fig. \ref{fig:km_plot}, the p-values for Kaplan-Meier analysis are significant in both PFS and OS tests. Therefore, our predictive model could further stratify the low- and high-risk patients in terms of patient survival.
\section{Conclusion and Further Work}
In this paper, we focus on a unified deep learning framework to predict therapy response, with easily accessible clinical data. The proposed framework named \emph{MIA}-Prognosis~utilizes clinical information including Measurement, Intervention and Assessment to model patient prognosis. We also propose a Simple Temporal Attention (SimTA) module to process the asynchronous time series. The proof-of-concept experiments validate the superiority of SimTA over standard RNN approaches in asynchronous time series analysis. Moreover, our method is proven effective on an in-house dataset on predicting response to anti-PD-1 immunotherapy for real-world non-small cell lung cancer (NSCLC) patients. Importantly, our predictive model is associated with long-term patient survival in terms of progression-free survival (PFS) and overall survival (OS).
In future studies, it is valuable to apply the proposed \emph{MIA}-Prognosis~framework on other therapy and diseases. On the other hand, it is also important to design efficient and effective non-linear temporal attention module to enhance temporal relation learning of SimTA. Besides, a fully end-to-end model with CNN-based Radiomics \cite{yang2019probabilistic} to encode the signature of radiographic features is worth exploring. Furthermore, it is interesting to explain what the \emph{MIA}-Prognosis~models from data-driven approaches.
\subsubsection{Acknowledgment.}
This work was supported by National Science Foundation of China (61976137, U1611461). Authors would like to appreciate the Student Innovation Center of SJTU for providing GPUs.
\bibliographystyle{splncs04}
\input{main.bbl}
\end{document} |
1,108,101,566,740 | arxiv | \section{Introduction}
As it is known, the mathematical approach to vehicular traffic modeling is developed on three representation scales: microscopic, macroscopic, and kinetic. However, the critical analysis proposed in \cite{[BDS11]} states that none of those scale approaches is totally satisfactory. Precisely, each one presents technical and conceptual advantages and disadvantages. Consequently, a multiscale approach is a necessity to obtain a detailed description of the dynamics of vehicles on the road. The author in \cite{[Do2]} compliments this point of view, where a hybrid model is proposed with detailed modeling of the dynamics of the micro-systems which is implemented into a macroscopic hyperbolic equation. An important reference to clarify research activity in the field is the survey of \cite{[HEL01]} on the physics and modeling of multi-particle systems. While, the critical paper by \cite{[DAG95]} provides some drawbacks of the driver-vehicle micro-system, where interactions can even modify the behavior of the driver and his/her ability is conditioned by the local flow conditions. Next, this paper has generated various discussions and reactions to account for the aforementioned criticisms~\cite{[BDF12],[HJ09],[PSTV15],[PSTV17]}. We mention that the approach of \cite{[BDF12]} has been further developed by various papers, for instance by a multiscale continuous velocity and activity variables for vehicular traffic flow \cite{[CNZ19]}.
This paper deals with the mathematical modeling of vehicular traffic flow along a multilane road on the basis of the kinetic theory. Precisely, we refer to Reference \cite{[BG09]}, where the authors proposed a discrete kinetic model with the main features:
\begin{enumerate}
\item The approach is developed at the kinetic scale where the modeling of the interacting vehicles is considered at the microscopic scale with a binary interactions;
\item The velocity variable is assumed to be discrete to overtake the drawback of vehicles number might not large enough to assure continuity of the probability distributions over the micro-states;
\item The quality of the road conditions is taken into account by an additional parameter which takes values in the interval $[0, 1]$, where the extremes of the interval correspond to worst and best conditions respectively.
\end{enumerate}
The proposing discrete model can be considered as a deep revisiting of the proposed model in \cite{[BG09]}. Indeed, the aforesaid paper consider only the numerical simulations of the spatially homogeneous problem where the asymptotic property in time has been shown. In this paper, we consider both the spatially homogeneous and inhomogeneous problems. In the first case, the well-posedness of the related Cauchy problem has been proved according to Banach fixed-point theory, and the Kerner's fundamental diagrams has been reproduced. In the second problem, we show the numerical results of the emerging clusters phenomena. In the modeling part, we deal with the following topics: $i)$ nonlinear additive interactions (rather than binary interactions) between vehicles accounting on $ii)$ perceived density (rather than real one), $iii)$ function depending on the space variable modeling the road conditions (rather than a parameter), and $iv)$ dynamics under external actions such as the presence of tollgates. The modeling is inspired by the books of \cite{[KER04],[PH71]} which reports a detailed interpretation of the physics traffic.
Let us comment now on some references in the literature about the modeling of multilane traffic flow. One can find the macroscopic approach in \cite{[MBY84]} while \cite{[MSR10]} reports an interesting survey of different models for lane changing. Also, a multilane model is analyzed in \cite{[GKR03]} focusing on the total vehicle density across all lanes, while a modeling traffic vehicular with conservation laws is proposed in \cite{[GR20]}. On the basis of kinetic theory, microscopic models are studied in \cite{[KW99a],[KW99b]} and a macroscopic model derived from individual behavior is given in \cite{[HG97]}. Recently, the authors in \cite{[HMV18]} extended the analysis to the second-order Aw-Rascle model and a hybrid stochastic kinetic model, respectively. Note that the approach in \cite{[CC06]} is more similar to the analysis presented in \cite{[HN19]} but with a different source term.
The rest of this paper is organized as follows: Section \ref{Sect2} deals with a concise description of multilane traffic flow and derives a new mathematical structure suitable to include the aforementioned features in addition to those already included in \cite{[BDF12]}. Section \ref{Sect3} is devoted to the derivation of a specific model by inserting a possible form of the terms that appeared in the proposed mathematical structure, that is to say, the perceived density, the encounter rate, table of games, the weight function and the external force. Section \ref{Sect4} presents a qualitative analysis of the spatially homogeneous problem. Moreover, Kerner's fundamental diagrams \cite{[KER04]} are reproduced and the trend to equilibrium (asymptotic in time) is shown. Finally, in Section \ref{Sect5} the numerical simulations of the spatially inhomogeneous problem are provided to capture emerging behaviors.
\section{Mathematical representation and structure}\label{Sect2}
This section is organized through two subsections: the first one deals with the description of traffic flow along a multilane and the second one is devoted to the design of mathematical frameworks for nonlinearly additive interactions along with multilane traffic flow.
\subsection{Representation of traffic flow along a multilane}
This subsection deals with a discrete kinetic description of traffic flow along a multilane road and introduces the classical macroscopic quantities.\\
Herein, all the dependent variables describing the traffic flow are dimensionless and normalized to values in the interval $[0,1]$.
\begin{itemize}
\item[\textbullet] $X$ is the length of the road;
\item[\textbullet] $V_M $ is the maximum velocity attained by an isolated fast vehicle when moving in free flow conditions;
\item[\textbullet] $\tilde{\rho}_M$ is the maximum density of vehicles corresponding to bumper to bumper vehicle distance.
\end{itemize}
Using the above quantities, the following dimensionless variables are identified:
\begin{itemize}
\item[\textbullet] $x$ is the position referred to length of road $X$;
\item[\textbullet] $t$ is the time normalized by means of $\frac{X}{V_M}$;
\item[\textbullet] $v=\frac{V}{V_M}$ is the velocity referred to $V_M$;
\item[\textbullet] $\rho=\tilde{\rho}/\tilde{\rho}_M$ is the density of vehicles referred to the maximum one.
\end{itemize}
In this paper, we adopt the discrete kinetic theory method by introducing the uniform grids of the velocity variable taking the following forms
\begin{equation}
0=v_1<v_2<\dots<v_i<\dots<v_n=1, \qquad i=1,\dots,n.
\end{equation}
Under these hypotheses, the evolution in time and space of the multilane traffic flow is described by introducing in each lane a statistical distribution function as a linear combination of Dirac distributions in variables $v$ and $u$ as follows:
$$f^\ell(t,x,v)=\sum_{i=1}^nf_{i}^\ell(t,x)\otimes\delta_i(v-v_i):\mathbb{R}^+\times[0,1]\times[0,1]\rightarrow\mathbb{R}^+,$$
where $f_{i}^\ell(t,x)=f^\ell(t,x,v_i)$ for $\ell=1,\cdots,L$ with $L$ is the total number of lanes. Thus, macroscopic quantities can derived in each lane as follow
\begin{itemize}
\item[\textbullet] The density in the $\ell$-lane is
$$\rho_\ell(t,x)=\sum_{i=1}^nf_{i}^\ell(t,x); $$
\item[\textbullet] The flux in the $\ell$-lane is
$$q^\ell(t,x)=\sum_{i=1}^nv_if_{i}^\ell(t,x);$$
\item[\textbullet] The average velocity in the $\ell$-lane is
$$U^\ell(t,x)=\frac{q^\ell(t,x)}{\rho^\ell(t,x)}=\frac{1}{\rho_\ell(t,x)}\sum_{i=1}^nv_if_{i}^\ell(t,x);$$
\item[\textbullet] The variance of the velocity in the $\ell$-lane is
$$\Theta^\ell(t,x)=\frac{1}{\rho_\ell(t,x)}\sum_{i=1}^n\big(v_i-U^\ell(t,x)\big)^2f_{i}^\ell(t,x).$$
\end{itemize}
The global density and the global flux are obtained by summing the contributions of all lanes
$$\rho(t,x)=\sum_{\ell=1}^L\rho_\ell(t,x),\qquad q(t,x)=\sum_{\ell=1}^Lq^\ell(t,x).$$
Moreover, the global average velocity and variance are given by
$$U(t,x)= \frac{1}{L}\sum_{\ell=1}^LU^\ell(t,x)\qquad \Theta(t,x)=\sum_{\ell=1}^L\Theta^\ell(t,x).$$
\subsection{A mathematical structure toward modeling}
This subsection reports about the derivation of a specific structure in the case of nonlinearly additive interactions involving the micro-systems, generating the evolution of the distribution function $f^\ell_{i}$. In general, the kinetic approach is such that interactions are modeled by table of games. Thus, three types of vehicles are involved:
\noindent \textbullet $\;$ \emph{Test particle}: It represents the whole system. The related distribution function is $f^\ell_{i}=f^\ell(t,x,v_i)$;\\
\noindent \textbullet $\;$ \emph{Candidate particle}: It changes its current state to that of the test particle as a consequence of an interaction. The related distribution function is
$f^r_{h}=f^r(t,x,v_h)$;\\
\noindent \textbullet $\;$ \emph{Field particle}: It interacts with test and candidate vehicles. The related distribution function is $f^r_{p}=f^r(t,x^*,v_p)$.\\
Our structure is obtained by a balance of vehicles in the elementary volume of the space of the microscopic state which includes position and velocity. This balance of vehicles includes the free transport term, the dynamics of nonlinear additive interactions, and the trend to the velocity imposed by the external actions. We mention that the dynamics of nonlinear additive interactions include a gain term of vehicles that enter in the
aforementioned elementary volume and a loss term of vehicles that leave it. Thus, the resulting structure can be written, at a formal level, as follows:
\begin{equation}
\displaystyle\partial_t f_{i}^\ell(t,x)+v_i\,\partial_x f_{i}^\ell(t,x)
= \mathcal{J}_{i}^\ell[\textbf{f},\textbf{f}](t,x)+\mathcal{T}_{i}^\ell[\textbf{f}](t,x),
\end{equation}
where $f^\ell_{i} = f^\ell(t, x, v_i),\;\textbf{f}=(f_1,\cdots,n)$ and $v_i\,\partial_xf^\ell_{i}
$ is the free flow transport term, while $\mathcal{J}_{i}^\ell$ and $\mathcal{T}^\ell_{i}$ correspond, respectively, to nonlinear additive interactions and interaction with external actions. Thus, the evolution of the distribution function in each lane is given by
{\small\begin{eqnarray}\label{Struct}
\begin{array}{l}
\displaystyle\partial_t f_{i}^\ell(t,x)+v_i\,\partial_x f_{i}^\ell(t,x)
= \mathcal{G}_{i}^\ell[\textbf{f},\textbf{f}](t,x)-f_{i}^\ell(t,x)\,\mathcal{L}^\ell[\textbf{f}](t,x)+\mathcal{T}_{i}^\ell[\textbf{f}](t,x)\\
{}\\
\displaystyle\hskip 0.1cm =\sum_{r=1}^L\sum_{h,p=1}^n\int_{\Omega_v}\eta^r[\rho^\star_r(t,x^*)]\mathcal{A}_{hp,r}^{i,\ell}[\rho_1^\star,\dots,\rho_L^\star;\alpha](t,x^*)f_{h}^{r}(t,x)f_{p}^{r}(t,x^*)\omega^r(x,x^*)dx^* \\
{}\\
\displaystyle\hskip.5cm-f_{i}^\ell(t,x)\int_{\Omega_v}\eta^\ell[\rho^\star_\ell(t,x^*)]\rho_\ell(t,x^*)\,\omega^\ell(x,x^*)dx^*+\mu^\ell[\rho^\star_\ell]\Big(f_{e}^\ell(x;v_e(x))-f_{i}^\ell(t,x)\Big).
\end{array}
\end{eqnarray}}
In model \eqref{Struct}, $\Omega_v=[x,x+\xi_v]$ represents the visibility zone where $\xi_v$ is the visibility length on front of the vehicle, that depends on the quality of the road-environment, that is to say on $\alpha = \alpha(x)$; $\eta^\ell[\rho_\ell(t,x^*),x]$ is the encounter rate, which depends on the probability distributions by means of the perceived density $\rho^\star_\ell$ in the $\ell-$lane; $\mathcal{A}_{h,p,r}^{i,j,\ell}$ defines the table of games, which denotes the probability density that the candidate particle falls into the state of the test particle after an interaction with a field particle; $\omega^\ell(x,x^*)$ represents the weight function in each lane $\ell$; and $\mu^\ell[\rho^\star_\ell]$ models the intensity of the action, which increases with $\rho_\ell^\star$, while $v_{e}(x)$ is the speed imposed by the external action in the $\ell-$lane.\\
Herein, we shall consider admissible interactions that generate transitions only in continuous lanes. In details, the sets of admissible lane transitions are as follows
{\small\begin{equation}
I_r=
\left\{
\begin{array}{ll}
I_r=\{1,2\},& r=1;\\
I_r=\{r-1,r,r+1\}, & r\neq 1,L; \\
I_r=\{L-1,L\}, &r=L.
\end{array}
\right.
\end{equation} }
\section{From mathematical structure to model}\label{Sect3}
This section deals with the derivation of a specific model of multilane vehicular traffic by inserting into the aforesaid mathematical structure (\ref{Struct}) models of interactions at the microscopic scale. This goal is pursued by looking at the modeling of the interaction terms that characterize such structure, that is to say, $\eta^\ell, \mathcal{A}_{hp,r}^{i,\ell}, \omega^\ell,\mu^\ell$ and $f_e^\ell$, such that a good agreement with empirical data, concerning both the Kerner's fundamental diagram and the emerging behaviors in unsteady flow conditions, is given.
\subsection{Modeling of interaction domain and perceived density}
The vehicles have a visibility zone denoted by $\Omega_v = \Omega_v(x) = [x, x + \xi]$, where $\xi$ is the visibility length on front of the particle, that depends on the quality of the road-environment conditions modeled by $\alpha = \alpha(x)$. In more detail, we assume that $\xi = \alpha \, X$, where $X << \xi$ is the visibility length in the case of best quality of the road ($\alpha = 1$). Note that one can consider a sensitive zone $\Omega_s=[x,x+\xi_s]$, necessary to perceive the flow conditions in $\Omega_s$. We refer the interested reader to the paper \cite{[CNZ19]} for more details.\\
The concept of perceived density was introduced in~\cite{[DEA99]}, where it was suggested that this quantity is greater (smaller) than the real one whenever positive (negative) density gradients appear. The following expression is considered
{\small\begin{equation}
\rho^\star_\ell[f] = \rho_\ell + \frac{\partial_x \rho_\ell}{\sqrt{1 + (\partial_x \rho_\ell)^2}}\,\Big((\frac{1}{L}- \rho_\ell)\, H(\partial_x \rho_\ell) + \rho_\ell \, H(- \partial_x \rho_\ell)\Big),
\end{equation}}
where $H(\cdot)$ is the Heaviside function $H(\cdot \geq 0) = 1$, while $H(\cdot < 0) = 0$. Thus, the perceived density, positive gradients increase the value of $\rho^\star_\ell$ from $\rho_\ell$ to the maximum admissible value $\rho_\ell = \frac{1}{L}$, while negative gradients decrease it from $\rho_\ell$ to the lowest admissible value $\rho_\ell = 0$ such that
$$
\partial_x \rho_\ell \to +\infty \Rightarrow \rho^\star_\ell \to \frac{1}{L}, \quad \partial_x \rho_\ell = 0 \Rightarrow \rho^\star_\ell = \rho_\ell, \quad \partial_x \rho_\ell \to - \infty \Rightarrow \rho^\star_\ell \to 0.
$$
\subsection{Modeling of the encounter rate and the weight function}
The encounter rate $\eta^\ell[\rho^\star_\ell(t,x)]$ gives the number of interactions per unit time among the micro-systems in each lane. We assume that this term grows with the local perceived density starting from a minimum value corresponding to driving in vacuum conditions $\eta^0$. The following expression is considered
$$\eta^\ell[\rho^\star_\ell(t,x)]=\eta^0\big(1+\gamma_\eta^\ell L\rho^\star_\ell(t,x)\big),$$
where $\gamma_\eta^\ell$ is the growth coefficient and $\rho^\star_\ell$ is the perceived density in each lane.\\
The weight function $\omega^\ell(x,x^*)$ is assumed to be the same in all lanes and it satisfies the following requirement:
$$\omega^\ell(x,x^*)\geq 0, \qquad \int_{\Omega_v}\omega^\ell(x,x^*)dx^*=1, \qquad \forall x^*\in {\Omega_v}.$$
\subsection{Modeling of the table of games}
We present and discuss a possible form for the table of games $A_{h,p,r}^{i,\ell}$ which gives the probability that a micro-system in the lane $r$ with velocity $v_h$ reaches the velocity $v_i$ in the lane $\ell$, after an interaction with a micro-system traveling at velocity $v_p$ in the lane $r$. It satisfies the following
\begin{equation}\label{tablB}
\mathcal{A}_{hp,r}^{i,\ell}\geq 0,\quad \sum_{\ell=1}^L\sum_{i=1}^n\mathcal{A}_{hp,r}^{i,\ell}=1,
\quad \forall h,p=1,\cdots,n,\quad \forall r=1,\cdots,L
\end{equation}
Herein, we improve the table of games proposed by \cite{[BG09]} by taking into account the nonlinear additive interactions rather than binary ones, perceived density rather than the real one, and space function modeling the road conditions rather than a parameter.\\
We consider that the candidate particle with velocity $v_h$ and assume that it interacts with a field particle with velocity $v_p$ in the same lane. Thus, the cases $v_h > v_p$, $v_h < v_p$, $v_h = v_p$, are analysed separately. Herein, we omit the dependence of the road conditions and the perceived density functions on the space variable $x$. \\
\textbf{ I. Interaction with a faster particle $(v_h < v_p)$.}
In this case, the candidate particle is encountering a faster field particle in the $r$-lane. Then the candidate particle either maintains its lane or it changes it to the right continuous one, which means $I_r=\{r-1,r\}$. In the following we distinguish the case $r=\{2,\cdots,L\}$ from the case $r=1$.
\subparagraph*{a) \underline{Interaction in the case $h\neq1,$ $r\neq1$}}
As a result of the above assumptions, it follows that $I_r=\{r-1,r\}$. Therefore,
{\small\begin{equation}\label{3.3}
\mathcal{A}_{hp,r}^{i,\ell=r}=
\left\{
\begin{array}{ll}
0, & i=1,\cdots,h-1, \\
(1-\alpha)\big(1-L\rho^\star_r(1-L\rho^\star_{r-1})\big), & i=h,\\
\displaystyle\alpha \frac{1}{(i-h)}\frac{1}{\sum_{z=h+1}^{p}(z-h)^{-1}}\big(1-L\rho^\star_r(1-L\rho^\star_{r-1})\big),&i=h+1,\cdots,p.
\end{array}
\right.
\end{equation}
\begin{equation}\label{3.4}
\mathcal{A}_{hp,r}^{i,\ell=r-1}=
\left\{
\begin{array}{ll}
\displaystyle (1-\alpha )(h-i)\frac{1}{\sum_{z=1}^{h-1}(h-z)}L\rho^\star_r(1-L\rho^\star_{r-1}), & i=1,\cdots,h-1, \\
\displaystyle\alpha L\rho^\star_r(1-L\rho^\star_{r-1}),&i=h,\\
0,&i=h+1,\cdots,p.
\end{array}
\right.
\end{equation}}
Note that Eq. (\ref{3.3}) gives the probability that the candidate particle maintains its lane. It can maintain its current speeds or accelerate. In the latter case, it can reach a new velocity $v_i \in\{v_{h+1},\cdots,v_p\}$ with a probability that depends not only on the road conditions and on the perceived density but also on the distance between the velocity classes involved. Eq. (\ref{3.4}) yields the probability that the candidate particle changes its lane to the right lane.
\subparagraph*{b) \underline{Interaction in the case $h=1$ and $r\neq1$}}
In this case, $I_r=\{r-1,r\}$ and the field particle stops or the candidate particle accelerates in the same lane. Therefore,
{\small\begin{equation}
\mathcal{A}_{1p,r}^{i,\ell=r}=
\left\{
\begin{array}{ll}
L\rho^\star_r, & i=1, \\
\displaystyle\frac{1}{(i-1)}\frac{1}{\sum_{z=2}^{p}(z-1)^{-1}}(1-L\rho^\star_r), & i=2,\cdots,p.
\end{array}
\right.
\end{equation}}
\subparagraph*{c) \underline{Interaction in the case $h=1,\cdots,p-1$ and $r=1$}}
In this case, the candidate particle is in the slowest lane $(I_r=1)$. We assume that when it interacts with a faster field particle, it can only maintain its current lane with the same velocity or accelerating. In the latter case, it can reach a new velocity $v_i \in\{v_{h+1},\cdots,v_p\}$ with a probability that depends not only on the road conditions and on the perceived density but also on the distance between the velocity classes involved. Consequently,
{\small\begin{equation}
\mathcal{A}_{hp,1}^{i,\ell=1}=
\left\{
\begin{array}{ll}
0,&i=1,\cdots,h-1, \\
\displaystyle1- \alpha (1-L\rho^\star_1), & i=h, \\
\displaystyle\alpha \frac{1}{(i-h)}\frac{1}{\sum_{i=h+1}^{p-1}(i-h)^{-1}}\big(1-L\rho^\star_1\big) ,&i=h+1,\cdots,p.
\end{array}
\right.
\end{equation}}
\textbf{II. Interaction with a faster particle $(v_h > v_p)$.}
In this case, the candidate particle is encountering a slower field particle in the $r-$lane. Thus, the candidate particle either maintains its lane or it changes to the left continuous one, which means $I_r = \{r, r+1\}$. In what follows, we distinguish the case $r = \{1,\cdots,L-1\}$ where the candidate particle can change its lane, from the case $r = L$ in which it can only maintain the current lane.
\subparagraph*{a) \underline{Interaction in the case $h\neq n$ and $r\neq L$}}
In this case $I_c=\{r,r+1\}$. Therefore,
{\small\begin{equation}\label{3.7}
\mathcal{A}_{hp,r}^{i,\ell=r}=
\left\{
\begin{array}{ll}
L\rho^\star_{r+1}, & i=p, \\
0,&otherwise.
\end{array}
\right.
\end{equation}
\begin{equation}\label{3.8}
\mathcal{A}_{hp,r}^{i,\ell=r+1}=
\left\{
\begin{array}{ll}
\displaystyle(1-\alpha )\frac{h-i}{\sum_{i=p+1}^{h}{(h-z)}}(1-L\rho^\star_{r+1}), & i=p+1,\cdots,h, \\
\displaystyle\alpha \frac{1}{(i-h)}\frac{1}{\sum_{z=h+1}^{n-1}(z-h)^{-1}}(1-L\rho^\star_{r+1}),&i=h+1,\cdots,n-1.
\end{array}
\right.
\end{equation}}
Eq. (\ref{3.7}) yields the probability that the candidate particle maintains its lane when interacting with a slower field particle traveling in the same $r-$lane. The probability depends only on the perceived density in the $(r+1)-$lane. Eq. (\ref{3.8}) yields the probability that the candidate particle changes its position to the left lane. Here, the probabilities depend on $\alpha$, $\rho^\star_{r+1}$, the difference between the velocity classes involved.
\subparagraph*{b) \underline{Interaction in the case $h= n$ and $r\neq L$}}
As a consequence of the above assumptions, it follows that $I_r=\{r,r+1\}$. In this case, the velocity ${v_n}$ of the candidate particle is the maximum allowed. It maintains the current lane reducing its velocity to that of the field particle or it changes to the left lane. While in the left lane, the candidate particle maintains its velocity or brakes depending on the local traffic density. Thus,
{\small\begin{equation}
\mathcal{A}_{np,r}^{i,\ell=r}=
\left\{
\begin{array}{ll}
L\rho^\star_{r+1}, & i=p, \\
0,&otherwise.
\end{array}
\right.
\end{equation}
\begin{equation}
\mathcal{A}_{np,r}^{i,\ell=r+1}=
\left\{
\begin{array}{ll}
\displaystyle\frac{n-i}{\sum_{z=p+1}^n(n-z)}(1-L\rho^\star_{r+1}), & i=p+1,\cdots,n, \\
0,& otherwise.
\end{array}
\right.
\end{equation}}
\subparagraph*{c) \underline{Interaction in the case $h=p+1,\cdots,n$ and $r= L$}}
The candidate particle is in the fastest lane. When it interacts with a slower field particle, we assume that it is obligated to travel with the velocity $v_p$. Thus,
{\small\begin{equation}
\mathcal{A}_{hp,L}^{i,\ell=L}=
\left\{
\begin{array}{ll}
1, & i=p, \\
0,&otherwise.
\end{array}
\right.
\end{equation}}
\textbf{III. Interaction with an equally faster particle $(v_h=v_p)$.}
In this case, the candidate particle and the field particle travel with the same speed in the same $r-$lane. We assume that the candidate particle maintains its lane or it can change both to the right lane or in the left one, which means $I_r=\{r-1,r,r 1\}$. In what follows, we distinguish the case in which the candidate particle can change both to the right and the left lanes from those where it can change only to the right or only to the left lane.
\subparagraph*{a) \underline{Interaction in the case $h\neq 1,n$ and $r\neq 1,L$}} As a consequence of the above assumptions, it follows that $I_r=\{r-1,r,r+1\}$. Thus,
{\small\begin{equation}\label{3.12}
\mathcal{A}_{hp,r}^{i,\ell=r}=
\left\{
\begin{array}{ll}
\displaystyle\frac{1}{2}(1-\alpha )(h-i)\frac{1}{\sum_{z=1}^{h-1}(h-z)}L\rho^\star_r(L\rho^\star_{r-1}+L\rho^\star_{r+1}),& i=1,\cdots,h-1, \\
\displaystyle\frac{1}{2}(1-\alpha )(1-L\rho^\star_r)(L\rho^\star_{r-1}+L\rho^\star_{r+1}),&i=h,\\
\displaystyle\frac{1}{2}\alpha \frac{1}{(i-h)}\frac{1}{\sum_{z=h+1}^{n}(z-h)^{-1}}(L\rho^\star_{r-1}+L\rho^\star_{r+1}),&i=h+1,\cdots,n.
\end{array}
\right.
\end{equation}
\begin{equation}\label{3.13}
\mathcal{A}_{hp,r}^{i,\ell=r-1}=
\left\{
\begin{array}{ll}
\displaystyle\frac{1}{2}(1-\alpha )(h-i)\frac{1}{\sum_{z=1}^{h-1}(h-z)}L\rho^\star_r(1-L\rho^\star_{r-1}),& i=1,\cdots,h-1, \\
\displaystyle\frac{1}{2}\alpha (1-L\rho^\star_{r-1}),& i=h,\\
0,&i=h+1,\cdots,n.
\end{array}
\right.
\end{equation}
\begin{equation}\label{3.14}
\mathcal{A}_{hp,r}^{i,\ell=r+1}=
\left\{
\begin{array}{ll}
0,& i=1,\cdots,h-1, \\
\displaystyle\frac{1}{2}(1-\alpha )(1-L\rho^\star_{r+1}),& i=h,\\
\displaystyle\frac{1}{2}\alpha \frac{1}{(i-h)}\frac{1}{\sum_{z=h+1}^{n}(z-h)^{-1}}(1-L\rho^\star_{r+1}),&i=h+1,\cdots,n.
\end{array}
\right.
\end{equation}}
Eq. (\ref{3.12}) yields the probability that the candidate particle maintains its current lane $r$, in this case, the candidate particle can decelerate or accelerate or maintains its current speed. All the above probabilities depends on the road conditions $\alpha$, the densities $\rho_{r-1}$, $\rho_{r}$, $\rho_{r+1}$, the difference between the velocity classes involved. Eq. (\ref{3.13}) yields the probability that the candidate particle changes its lane to the right lane, in this case the candidate particle cannot accelerate. Eq. (\ref{3.14}) yields the probability that the candidate particle changes its lane to the left lane, in this case, the candidate particle cannot brake.
\subparagraph*{b) \underline{Interaction in the case $h\neq 1,n$ and $r=1$}}
In this case, the interacting vehicles are in the slowest lane and $I_r=\{1,2\}$. Consequently,
{\small\begin{equation}\label{3.15}
\mathcal{A}_{hp,1}^{i,\ell=1}=
\left\{
\begin{array}{ll}
\displaystyle(1-\alpha )(h-i)\frac{1}{\sum_{z=1}^{h-1}(h-z)}L\rho^\star_1L\rho^\star_2,& i=1,\cdots,h-1,\\
\displaystyle(1-\alpha )(1-L\rho^\star_{1})L\rho^\star_2,& i=h,\\
\displaystyle\alpha \frac{1}{(i-h)}\frac{1}{\sum_{z=h+1}^{n}(z-h)^{-1}}L\rho^\star_{2},&i=h+1,\cdots,n.
\end{array}
\right.
\end{equation}
\begin{equation}\label{3.16}
\mathcal{A}_{hp,1}^{i,\ell=2}=
\left\{
\begin{array}{ll}
0,& i=1,\cdots,h-1,\\
\displaystyle(1-\alpha )(1-L\rho^\star_{2}),& i=h,\\
\displaystyle\alpha \frac{1}{(i-h)}\frac{1}{\sum_{z=h+1}^{n-1}(z-h)^{-1}}(1-L\rho^\star_{2}),&i=h+1,\cdots,n.
\end{array}
\right.
\end{equation}}
Eq. (\ref{3.15}) yields the probability that the candidate particle maintains its current lane, depending on the two traffic densities, on the parameter $\alpha$, on the difference between the velocity classes involved. Eq. (\ref{3.16}) yields the probability that the candidate particle changes its lane to the left one, in this case it cannot brake.
\subparagraph*{c) \underline{Interaction in the case $h\neq 1,n$ and $r=L$}}
In this case, the interacting vehicles are in the fastest lane and $I_r=\{L-1,L\}$. The candidate particle or it changes to the right one, depending on the two traffic densities, on the parameter $\alpha$, on the difference between the velocity classes involved. If it changes its lane, it cannot accelerate. Thus,
{\small\begin{equation}\label{3.17}
\mathcal{A}_{hp,L}^{i,\ell=L}=
\left\{
\begin{array}{ll}
\displaystyle(1-\alpha )(h-i)\frac{1}{\sum_{z=1}^{h-1}(h-z)}L\rho^\star_{L-1}L\rho^\star_L,& i=1,\cdots,h-1, \\
\displaystyle(1-\alpha )(1-L\rho^\star_{L})L\rho^\star_{L-1},& i=h,\\
\displaystyle\alpha \frac{1}{(i-h)}\frac{1}{\sum_{z=h+1}^{n}(z-h)^{-1}}L\rho^\star_{L-1},&i=h+1,\cdots,n.
\end{array}
\right.
\end{equation}
\begin{equation}\label{3.18}
\mathcal{A}_{hp,L}^{i,\ell=L-1}=
\left\{
\begin{array}{ll}
\displaystyle(1-\alpha )(h-i)\frac{1}{\sum_{z=1}^{h-1}(h-z)}(1-L\rho^\star_{L-1}),&i=1,\cdots,h-1, \\
\displaystyle\alpha (1-L\rho^\star_{L-1}),& i=h,\\
0,&i=h+1,\cdots,n.
\end{array}
\right.
\end{equation}}
Eq. (\ref{3.17}) and (\ref{3.18}) yields the probabilities that the candidate particle maintains its current lane or it changes to the right one, depending on the two traffic densities, on the $\alpha$, on the difference between the velocity classes involved. If it changes its lane it cannot accelerate.
\subparagraph*{d) \underline{Interaction in the case $h=1$ and $r\neq 1,L$}}
As a result of the above assumptions, it follows that $I_r=\{r-1,r,r+1\}$. Therefore
{\small\begin{equation}\label{3.19}
\mathcal{A}_{11,r}^{i,\ell=r}=
\left\{
\begin{array}{ll}
\displaystyle 1-\alpha L\rho_r\big(1-\frac{1}{2}(L\rho^\star_{r-1}+L\rho^\star_{r+1})\big),& i=1, \\
0,& i=2,\cdots,n.
\end{array}
\right.
\end{equation}
\begin{equation}\label{3.20}
\mathcal{A}_{11,r}^{i,\ell=r-1}=
\left\{
\begin{array}{ll}
0,& i=1, \\
\displaystyle\frac{1}{2}\alpha \frac{1}{(i-1)}\frac{1}{\sum_{i=2}^{n}(i-2)^{-1}}L\rho^\star_r(1-L\rho^\star_{r-1}),& i=2,\cdots,n.
\end{array}
\right.
\end{equation}
\begin{equation}\label{3.21}
\mathcal{A}_{11,r}^{i,\ell=r+1}=
\left\{
\begin{array}{ll}
0,& i=1, \\
\displaystyle\frac{1}{2}\alpha \frac{1}{(i-1)}\frac{1}{\sum_{i=2}^{n}(i-2)^{-1}}L\rho^\star_r(1-L\rho^\star_{r+1}),& i=2,\cdots,n.
\end{array}
\right.
\end{equation}}
Eq. (\ref{3.19}) yields the probability that the candidate particle remains in the same lane, and it maintains its velocity. Eqs. (\ref{3.20}) (\ref{3.21}) yields the probabilities that the candidate particle changes its lane to the right lane or the left one, depending on the traffic densities, on the quality of the road, on the difference between the velocity classes involved and on the activity $u_k$. In this case, it can only accelerate and it cannot brake.
\subparagraph*{e) \underline{Interaction in the case $h=1$ and $r=1$}}
In this case, the two interacting vehicles are in the slowest lane and $I_r =\{1,2\}$. Consequently
{\small\begin{equation}\label{3.22}
\mathcal{A}_{11,1}^{i,\ell=1}=
\left\{
\begin{array}{ll}
\displaystyle 1-\alpha L\rho^\star_1(1-L\rho^\star_{2}),& i=1, \\
0,& i=2,\cdots,n.
\end{array}
\right.
\end{equation}
\begin{equation}\label{3.23}
\mathcal{A}_{11,1}^{i,\ell=2}=
\left\{
\begin{array}{ll}
0,& i=1, \\
\displaystyle\alpha \frac{1}{(i-1)}\frac{1}{\sum_{z=2}^{n}{(z-1)^{-1}}}L\rho^\star_1(1-L\rho^\star_{2}),& i=2,\cdots,n.
\end{array}
\right.
\end{equation}}
Eq. (\ref{3.22}) yields the probability that the candidate particle remains in the same lane and only maintains its velocity $v_1$, depending on the traffic densities, on the quality of the road, on the difference between the velocity classes involved. Eq. (\ref{3.23}) yields the probability that the candidate particle changes its lane to right one, in this case it can accelerate.
\subparagraph*{f) \underline{Interaction in the case $h=1$ and $r=L$}} As a consequence of the above assumptions, it follows that $I_r=\{L-1,L\}$. In this case, the two interacting vehicles stop in the fastest lane. Thus,
{\small\begin{equation}\label{3.24}
\mathcal{A}_{11,L}^{i,\ell=L}=
\left\{
\begin{array}{ll}
\displaystyle 1-\alpha L\rho^\star_L(1-L\rho^\star_{L-1}),& i=1, \\
0,& i=2,\cdots,n.
\end{array}
\right.
\end{equation}
\begin{equation}\label{3.25}
\mathcal{A}_{11,L}^{i,\ell=L-1}=
\left\{
\begin{array}{ll}
0,& i=1, \\
\displaystyle\alpha \frac{1}{(i-1)}\frac{1}{\sum_{z=2}^{n}(z-1)^{-1}}L\rho^\star_L(1-L\rho^\star_{L-1}),& i=2,\cdots,n.
\end{array}
\right.
\end{equation}}
Eq. (\ref{3.24}) yields the probabilities that the candidate particle remains in the same lane and maintains its current velocity $v_1$, depending on the traffic densities, on the quality of the road, on the difference between the velocity classes involved. Eq. (\ref{3.25}) yields the probabilities that the candidate particle changes its lane to right one accelerating, depending on the traffic densities, on the quality of the road, on the difference between the velocity classes involved.
\subparagraph*{g) \underline{Interaction in the case $h=n$ and $r\neq 1,L$}} In this case, $I_r=\{r-1,r,r+1\}$ and the two interacting vehicles travel at the fastest allowed velocity $v_n=1$ and they are both in a central lane. It is considered that the candidate particle maintains its current lane with the same or a slower velocity or it can change going to a continuous one. Consequently,
{\small\begin{equation}
\mathcal{A}_{nn,r}^{i,\ell=r}=
\left\{
\begin{array}{ll}
\displaystyle\frac{1}{2}(n-i)\frac{1}{\sum_{z=1}^{n-1}(n-z)}(L\rho^\star_{r-1}+L\rho^\star_{r+1})L\rho_r,& i=1,\cdots,n-1, \\
\displaystyle\frac{1}{2}(L\rho^\star_{r-1}+L\rho_{r+1})(1-L\rho_r),& i=n.
\end{array}
\right.
\end{equation}
\begin{equation}
\mathcal{A}_{nn,r}^{i,\ell=r-1}=
\left\{
\begin{array}{ll}
\displaystyle\frac{1}{2}(1-\alpha )(n-i)\frac{1}{\sum_{z=1}^{n-1}(n-z)}(1-L\rho^\star_{r-1}),& i=1,\cdots,n-1, \\
\displaystyle\frac{1}{2}\alpha (1-L\rho^\star_{r-1}),& i=n.
\end{array}
\right.
\end{equation}
\begin{equation}
\mathcal{A}_{nn,r}^{i,\ell=r+1}=
\left\{
\begin{array}{ll}
0,& i=1,\cdots,n-1, \\
\displaystyle\frac{1}{2}(1-L\rho^\star_{r+1}),& i=n.
\end{array}
\right.
\end{equation}}
\subparagraph*{h) \underline{Interaction in the case $h=n$ and $r=1$}} In this case, $I_r =\{1,2\}$ and the two interacting vehicles are both in the slowest lane with the maximum allowed speed. The candidate particle maintains its current lane traveling with the same or lower velocity independence of the traffic density $\rho_1$ or it changes to the left lane maintaining its velocity. Consequently,
{\small\begin{equation}
\mathcal{A}_{nn,1}^{i,\ell=1}=
\left\{
\begin{array}{ll}
\displaystyle(n-i)\frac{1}{\sum_{z=1}^{n-1}(n-z)}L\rho^\star_1L\rho^\star_2,& i=1,\cdots,n-1,\\
\displaystyle(1-L\rho^\star_1)L\rho^\star_2,& i=n.
\end{array}
\right.
\end{equation}
\begin{equation}
\mathcal{A}_{nn,1}^{i,\ell=2}=
\left\{
\begin{array}{ll}
0,& i=1,\cdots,n-1, \\
1-L\rho^\star_2,& i=n.
\end{array}
\right.
\end{equation}}
\subparagraph*{i) \underline{Interaction in the case $h=n$ and $r=L$}} As a result of the above assumptions, it follows the $I_r=\{L-1,L\}$. In this case, the two interacting vehicles are both in the fastest lane with the maximum allowed speed $v_n$. The candidate particle maintains its current lane traveling with the same or lower velocity independence of the traffic density $\rho_L$ or alternatively it changes to the right lane maintaining its velocity or braking. Consequently,
{\small\begin{equation}
\mathcal{A}_{nn,L}^{i,\ell=L}=
\left\{
\begin{array}{ll}
\displaystyle(n-i)\frac{1}{\sum_{z=1}^{n-1}(n-z)}L\rho_{L-1}L\rho^\star_L,& i=1,\cdots,n-1, \\
\displaystyle(1-L\rho^\star_L)L\rho^\star_{L-1},& i=n.
\end{array}
\right.
\end{equation}
\begin{equation}
\mathcal{A}_{nn,L}^{i,L=L-1}=
\left\{
\begin{array}{ll}
\displaystyle(1-\alpha )(n-i)\frac{1}{\sum_{z=1}^{n-1}(n-z)}(1-L\rho^\star_{L-1}),& i=1,\cdots,n-1, \\
\alpha (1-L\rho^\star_{L-1}),& i=n.
\end{array}
\right.
\end{equation}}
\subsection{Modeling of external actions}
The last term needed to model is the external action which indicates a prescribed speed as it occurs, for instance, in the presence of tollgates and traffic signs.
The structure of this term is reported in Eq.~(\ref{Struct}), where $f_e^\ell$ is a given function of the prescribed velocity $v_e= v_e(x)$. Thus, from Eq. (\ref{Struct}) it only requires to model the intense of the action. We propose the following form
\begin{equation}\label{mu}
\mu^\ell[\rho^\star_\ell]= \eta^0 \, (1 + \gamma_\mu^\ell \, L\rho_\ell^\star),
\end{equation}
where $\gamma_\mu^\ell$ is the growth coefficient and $\rho^\star$ is the perceived density.
\section{The spatially homogeneous problem}\label{Sect4} In this section, we address the theoretical and computational analysis of the spatially homogeneous problem in which the distribution function $f^\ell$ is independent of the space variable $x$. We define it as follows
$$f^\ell=f^\ell(t,v)=\sum_{i=1}^nf_{i}^\ell(t)\otimes\delta_i(v-v_i):\mathbb{R}^+\times[0,1]\rightarrow\mathbb{R}^+.$$
Consequently, it results that
$$\partial_x f_{i}^\ell=0, \qquad \forall i=1,\cdots,n, \quad \forall \ell=1,\cdots,L.$$
This implies $\rho^\star_\ell=\rho_\ell$ for $ \; \ell=1,\cdots,L$.
Taking into account the above hypotheses, the final form of model \eqref{Struct} in the spatially homogeneous case is reduced to the following Cauchy problem
{\small\begin{equation}\label{SptialP}
\left\{
\begin{array}{l}
\displaystyle\frac{df_{i}^\ell}{dt}=\sum_{r\in I_c}\eta^r[\rho_r(t)]\sum_{h,p=1}^n\mathcal{A}_{hp,r}^{i,\ell}f_{h}^rf_{p}^r-\eta^\ell[\rho_\ell(t)]f_{i}^\ell\rho_\ell(t), \\
\displaystyle f_{i}^\ell(0)=f_{i,0}^{\ell},
\end{array}
\right.
\end{equation}}
where $$\mathcal{A}_{h,p,r}^{i,\ell}=\mathcal{A}_{hk,pq,r}^{i,\ell}[v_h\rightarrow v_i,I_r\rightarrow I_\ell\mid,v_h,v_p,I_r,\rho_1(t),\dots,\rho_L(t)].$$ Recalling the probability density propriety \eqref{tablB}, the above model \eqref{SptialP} satisfies the mass conservation hypothesis, i.e. $ \frac{d\rho}{dt}= 0$, as it is required in spatially homogeneous conditions.
\subsection{Well-posedness of the related Cauchy problem }
Let denote
$\mathbb{M}_{Ln}$ a set of matrix endowed with the $1-$norm
$$ \parallel f(t)\parallel _1= \sum_{\ell=1}^L\sum_{i=1}^n \mid f_{i}(t)\mid , \quad f=(f_{i}^\ell) \in \mathbb{M}_{Ln}.$$
We introduce the linear space $X_T=C ([0,T]; \mathbb{M}_{Ln} )$ of the continuous functions $f=f(t):[0,T]\to \mathbb{M}_{Ln}$ for $T>0$ equipped
with the infinity norm
$$\parallel f \parallel _\infty = \displaystyle\sup_{t \in [0,T] } \parallel f(t)\parallel _1.$$ \\
Note that $(X_T,\parallel .\parallel _\infty )$ is a real Banach space.
{Well-posedness of the spatially homogeneous problem means global in time existence and uniqueness of a solution $f = f(t)$ to the Cauchy problem (\ref{SptialP}). These results pass through two steps. Firstly, we prove the local existence and uniqueness in time of the solution $f$ in $X_T$ for a certain $T > 0$. Secondly, we extend the obtained result to a global solution defined for all $t > 0$. }
We consider the following assumptions to prove the well-posedness of the spatially homogeneous problem \eqref{SptialP}:
$$ (\textbf{\textit{H1}})\qquad {\mathcal{A}_{hp,r} ^{i,\ell}\geq0},\qquad \sum_{\ell=1}^L\sum_{i=1}^n \mathcal{A}_{hp,r} ^{i,\ell}=1 \quad \forall r =1, \dots , L\quad \forall h,p =1, \dots , n $$
whenever $0\leq \rho_\ell{\leq}\frac{1}{L}$ and
$$(\textbf{\textit{H2}}) \qquad \exists C_{\eta} > 0 \quad / \quad 0<\eta^\ell[\rho_\ell] \leq C_{\eta}^\ell,\quad \hbox{when} \quad 0 \leq \rho_\ell {\leq} \frac{1}{L},\quad \ell=1,\dots,L. $$
\begin{theorem}\label{Therm1} Assume $(\textbf{H1})$-$(\textbf{H2})$ are satisfied and let $\displaystyle\sum_{\ell=1}^L\sum_{i=1}^n f^\ell_{ij}:=\rho_0\in[0,1]$, then
$\exists \hspace{0.2cm} T>0$ such that problem \eqref{SptialP} admits unique local non-negative solution $f \in X_T$. Moreover, it satisfies the mass conservation
\begin{equation}\label{Mass}
\parallel f(t)\parallel _1=\rho_0. \end{equation}
\end{theorem}
\noindent We start by giving some estimates on the nonlinear operator $\mathcal{J}$ given by:
$$\mathcal{J}=\mathcal{J}_{i}^\ell:=\sum_{r\in I_c}\eta^r[\rho_r(t)]\sum_{h,p=1}^n\mathcal{A}_{hp,r}^{i,\ell}f_{h}^rf_{p}^r-\eta^\ell[\rho^\ell(t)]f_{i}^\ell(t)\rho_\ell(t).$$
\begin{lem}
Assume $(\textbf{\textit{H1}})$ and $(\textbf {\textit{H2}})$ are satisfied, then $\exists\; C_1>0$ such that
$$ \parallel \mathcal{J}(f)\parallel _1 \leq C_1\parallel f\parallel _1 ^2,$$
$$ \parallel \mathcal{J}(f)-\mathcal{J}(g)\parallel _1 \leq C_1(\parallel f\parallel _1 +\parallel g\parallel _1)\parallel f-g\parallel _1.$$
\end{lem}
\noindent Next, we define the following operator $$\phi(f)(t)=\int_0 ^t \mathcal{J}(f)(s) ds. $$ We have the following estimates:
\begin{lem} \label{Lema3} Assume $(\textbf{\textit{H1}})$ and $(\textbf{\textit{H2}})$ are satisfied, then $\exists\, C_2(T)>0$ such that
$$ \parallel \phi(f)\parallel _\infty \leq C_2\parallel f\parallel _{\infty} ^2,$$
$$ \parallel \phi(f)-\phi(g)\parallel _\infty \leq C_2(\parallel f\parallel _\infty +\parallel g\parallel _\infty)\parallel f-g\parallel _\infty.$$
\end{lem}
\noindent\textbf{Proof of Theorem 1} First, we rewrite problem \eqref{SptialP} as an integral form
$$f=\mathcal{N}(f)$$
where the mild equation is as follows
{\small\begin{eqnarray}\label{MS}
\displaystyle (\mathcal{N}(f))_{i}^\ell = f_{i,0}^\ell + \int_0 ^t\Big(\sum_{r=1}^L\eta^r[\rho_r]\sum_{h,p=1}^n \mathcal{A}_{hp,r} ^{i,\ell} f_{h}^r(s)f^r_{p}(s)- f^\ell_{i}(s) \eta^\ell[\rho_{\ell}]\rho_\ell(s)\Big)ds.
\end{eqnarray}}
Proving the uniqueness and existence of solution of the above mild equation in $X_T$ implies to finding a fixed point of $\mathcal{N}$. This is given by the following lemma
\begin{lem}\label{Lema4}
$\mathcal{N}$ maps the ball $B_{X_T}(0,2\,\parallel f_0\parallel_1)$ into itself.
\end{lem}
\noindent Let $f \in X_T, \quad R>0$ such that $\parallel f\parallel _\infty \leq R$. From Lemma \ref{Lema3} we obtain
$$ \parallel \mathcal{N}(f)\parallel _\infty \leq \parallel f_0\parallel _1 +T C_1 \parallel f\parallel_\infty ^2,$$
$$ \parallel \mathcal{N}(f)\parallel _\infty \leq R +T C_1 R^2.$$
We chose $T$ such that
$$T C_1 R^2< \frac{R}{4},$$
this implies
$$T < \frac{1}{4C_1 \parallel f_0\parallel _1}.$$
Then,
$$ \parallel \mathcal{N}(f)\parallel _\infty \leq \parallel f_0\parallel _1 + \frac{1}{4C_1 \parallel f_0\parallel _1} C_1 \parallel f\parallel _\infty ^2,$$
$$\parallel \mathcal{N}(f)\parallel_\infty \leq \parallel f_0\parallel_1 + \frac{1}{4 \parallel f_0\parallel_1}R ^2.$$
where $R$ is a solution of $\frac{1}{4} R^2 - \parallel f_0\parallel _1 R+ \parallel f_0\parallel _1 ^2=0.$
Then, $R=2\parallel f_0\parallel _1$.
\begin{lem}\label{Lema5}
$\mathcal{N}$ is strictly contraction on a ball $B_{X_T} \big(0,2\parallel f_0\parallel _1\big)$.
\end{lem}
\noindent Let $f , g \in X_T, \; R>0$ such that $\parallel f\parallel_\infty \leq R$ and $\parallel g\parallel_\infty \leq R$. Thanks to Lemma \ref{Lema3}, we have the following estimate
$$ \parallel \mathcal{N}(f)-\mathcal{N}(g)\parallel _\infty \leq T C_1\big(\parallel f\parallel _\infty +\parallel g\parallel_\infty\big)\parallel f-g\parallel_\infty.$$
It follows that
$$ \parallel \mathcal{N}(f)-\mathcal{N}(g)\parallel_\infty \leq 2R T C_1\parallel f-g\parallel_\infty.$$
We chose $T$ such that
$$T < \frac{1}{2RC_1}.$$
Then, Lemma \ref{Lema5} is achieved.\\
Now, by summing over $\ell,i,j$ and using Assumption (\textbf{\textit{H1}}), one gets the conservation of mass given by equation \eqref{Mass}. Notice that we have the non-negativity of the solution thanks to Lemma \ref{Lema4} under the non-negativity of the initial condition $f_{i,0}^\ell>0$. {Indeed,
Eq. (\ref{SptialP}) can be rewritten as follows
{\small\begin{equation}\label{*}
\displaystyle\frac{d}{dt}f_{i}^\ell(t)+f_{i}^\ell(t)L(f)(t)=G_i(f,f)(t):=\sum_{r\in I_c}\eta^r[\rho_r(t)]\sum_{h,p=1}^n\mathcal{A}_{hp,r}^{i,\ell}f_{h}^rf_{p}^r,
\end{equation}}
where $L(f)(t)=\eta^\ell[\rho_\ell(t)]\rho_\ell(t)$.
Let denote $\lambda(t)=\int_{0}^{t}L(f)(s)ds$.
If $f_i^{\ell}$ is a solution of Eq. \eqref{*}, then
$$\displaystyle\frac{d}{dt}\Big(\exp(\lambda(t))f_{i}^\ell(t)\Big)=\exp(\lambda(t))G_i(f,f)(t),$$
which implies
\begin{equation}\label{**}
f_i^\ell(t)=\exp(-\lambda(t))f_{i,0}^\ell+\int_{0}^{t}\exp(\lambda(s))G_i(f,f)(s)ds.
\end{equation}
Consequently, eq. \eqref{**} allows to conclude that, given $f_{i,0}^\ell>0$ and the positivity of the integral function, the solution $f_i^{\ell}(t)$ satisfies non-negativity condition in its domain of existence. }\\
$$\hskip15.5cm\square$$
The following theorem guaranties the existence of global solution of the spatially homogeneous problem \eqref{SptialP}:
\begin{theorem}
Under the same hypotheses of Theorem \ref{Therm1}, Problem (\ref{SptialP}) admits a unique non-negative global solution $f \in C(\mathbb{R}^+, \mathbb{M}_{Ln})$ satisfying estimate \eqref{Mass}.
\end{theorem}
\noindent\textbf{Proof.} It suffices to apply the same reasoning developed in the proof of Theorem
\ref{Therm1} on the interval $(T^*, 2T^*]$, taking $f(T^*)$ as new initial condition. Since $f^\ell_{i}(T) \geq 0$ for all
$\ell=1,\dots,L,\;i = 1,\dots, n$, and $\displaystyle\sum_{\ell=1}^L\sum_{i=1}^n f^\ell_{i} = \rho_0 \in[0, 1],$ we are in the same hypotheses of Theorem \ref{Therm1}. Hence, we conclude the existence and uniqueness of a local in time continuous solution on $[T^*, 2T^*]$ satisfying
\begin{equation}
\parallel f(t)\parallel _1=\rho_0,\qquad \forall t\in[T^*, 2T^*]. \end{equation}
Iterating this procedure on all intervals of the form $(kT^*, (k+1)T^*], k \in \mathbb{N},$ we can construct the global solution on real positive axis $\mathbb{R}^+$.
\subsection{Numerical simulations}
We present some numerical simulations obtained from the spatially homogeneous problem \eqref{SptialP} to validate proposed modeling of each terms, namely of the table of games. This can be done according to the existing results in the literature. Concretely, Kerner's fundamental diagrams which relates the flux and the average velocity to the density, and the asymptotic property in time of the vehicles on the lanes starting from different initial distributions.
\subsubsection{Kerner's fundamental diagrams}
Numerical simulations of Eq.(\ref{SptialP}) have been performed to obtain the fundamental diagrams relating to the average velocity and the macroscopic flux to the density of the vehicles. The problem under consideration has been numerically solved by fixing six-velocity classes $\{v_i\}_{i=1}^6$ and three lanes $L=3$. The result is shown in Figures \ref{5.2}-\ref{5.3} corresponding to three different values of the phenomenological parameter $\alpha$, that is to say $\alpha=0.95, \; \alpha=0.6, \; \alpha=0.3$
and by fixing $\rho^0_{\ell}\in \big[0,\frac{1}{L}\big[, \ell=1,2,3,$ at the initial time.\\
Figure \ref{5.2} shows, respectively, the result of fundamental diagrams for three-lane road, namely the slowest lane, the medium lane, and the fastest lane, and Figure \ref{5.3} shows the global ones. The obtained results can be summarized as follows:
\begin{itemize}
\item [\textbullet] For low density the flux exhibits an almost linear behavior, which is in agreement with experimental observation reported by Kerner under free-flow conditions;
\item [\textbullet] The flux becomes markedly nonlinear when the density increases;
\item [\textbullet] The average speed is initially almost close to the maximum possible, then it drops steeply to zero when the density enters the congested flow range;
\item[\textbullet] The free flow phase reduces as the environmental conditions worsen;
\item[\textbullet] The free flow increases as the vehicles are positioned in the fastest lane.
\end{itemize}
\begin{figure}
\centering
\subfigure{ \includegraphics[height=1.5in ,width=4in]{DiagrQ.pdf}}
\subfigure{ ~\includegraphics[height=1.5in ,width=4in]{DiagrU.pdf}}
\caption{The first line shows fundamental diagrams: flux $q^\ell$ vs. $\rho_\ell$. The second line shows velocity diagrams: means velocity $U^\ell$ vs. $\rho_\ell$. These numerical results are obtained under various road conditions ($\alpha = 0.95, \alpha = 0.6, \alpha = 0.3$ respectively).}
\label{5.2}
\end{figure}
\begin{figure}
\centering
\subfigure{ \includegraphics[height=1.5in ,width=4in]{DFT.pdf}}
\caption{Global fundamental diagrams obtained under various road conditions ($\alpha = 0.95, \alpha = 0.6, \alpha = 0.3$ respectively): (Left) flux $q$ vs. density $\rho$; (right) means velocity $U$ vs. density $\rho$ .}
\label{5.3}
\end{figure}
Figures \ref{5.2}-\ref{5.3} demonstrate that our model is able to catch qualitatively the well-known phase transition from free to congested traffic flow.
\subsubsection{Asymptotic property in time}
We aim to show the asymptotic property in time of the vehicles on the three lanes ($L=3$) beginning from different initial conditions as follows
\begin{equation}
\begin{array}{llll}
\rho=0.2:&\rho_{1}\mid_{t=0}=0.2,& \rho_{2}\mid_{t=0}=0,&\rho_{3}\mid_{t=0}=0;\\
\rho=0.4:&\rho_{1}\mid_{t=0}=0.2,& \rho_{2}\mid_{t=0}=0.2,&\rho_{3}\mid_{t=0}=0;\\
\rho=0.6:&\rho_{1}\mid_{t=0}=0.3,& \rho_{2}\mid_{t=0}=0.15,&\rho_{3}\mid_{t=0}=0.15;\\
\end{array}
\end{equation}
Figure \ref{5.4} shows the obtained numerical simulations in the case of a low global density of $\rho=0.2$ where the micro-systems started from the slowest lane. We notice that when the equilibrium is achieved, micro-systems are distributed in all lanes. Specifically, the medium lane is the most occupied in the case of bad road conditions $\alpha=0.2$ while the fasted lane is the busiest in the case of good road conditions $\alpha=0.6$. Next, we increase the global density $\rho=0.4$ and we assume the same initial distributions for the first and the second lanes while the third lane is empty. Figure \ref{5.5} shows the obtained results. We notice that the fastest lane is the most occupied lane at equilibrium and when the road conditions are bad the fastest and the medium lanes are occupied all the most in the same way. Finally, we increase the global density $\rho=0.6$ and we assumed that the slowest lane is the busiest. We notice that the fastest lane is the most occupied at equilibrium, see Figure \ref{5.5}. Precisely, we have to following results
\begin{itemize}
\item Increasing the density increases the rapidity by which the steady density is reached;
\item Micro-systems tend to occupy the fastest lane. This trend is all the more remarkable as the quality of the road increases and the density is low. On the contrary, we note that when the density is important, the micro-systems tend to occupy the entire line and the state of the road parameter is less relevant;
\item The same road conditions and the same overall density show a tendency towards a balance insensitive to the initial distribution on the lanes.
\end{itemize}
\begin{figure}
\centering
\subfigure{ \includegraphics[height=1.5in ,width=4in]{Test1.pdf}}
\caption{Global density $\rho=0.2$ at different values of $\alpha$.}
\label{5.4}
\end{figure}
\begin{figure}
\centering
\subfigure{ \includegraphics[height=1.5in ,width=4in]{Test4.pdf}}
\caption{Global density $\rho=0.4$ at different values of $\alpha$.}
\label{5.5}
\end{figure}
\begin{figure}
\centering
\subfigure{ \includegraphics[height=1.5in ,width=4in]{Test5.pdf}}
\caption{Global density $\rho=0.6$ at different values of $\alpha$.}
\label{5.6}
\end{figure}
\newpage
\section{The spatially inhomogeneous problem}\label{Sect5}
This section is devoted to present the numerical simulations of the spatially inhomogeneous problem, which describes the spatial and temporal evolution of the traffic flow. Precisely, we aim to validate our proposed modeling by showing the emerging clusters. Herein, we neglect the term of the external force $\mathcal{T}_{i}^\ell$, and we consider the periodic conditions. Thus, the final form of model (\ref{Struct}) is as follows
{\small\begin{equation}\label{InhomgP}
\left\{\begin{array}{ll}
\displaystyle\frac{\partial f_{i}^\ell}{\partial t}+v_i\;\partial_xf_{i}^\ell=\mathcal{J}_{i}^\ell[\textbf{f},\textbf{f}](t,x),&\forall (t,x)\in [0,T]\times]0,1[,\\\\
f_{i}^\ell(0,x)=f_{i}^{\ell 0}(x), & \forall x\in[0,1],\\\\
f_{i}^\ell(t,0)=f_{i}^\ell(t,1), &\forall t\in [0,T],
\end{array}\right.
\end{equation}}
for $\ell=1,\dots,L$ and $i=1,\dots,n$,
where
{\small\begin{equation*}
\begin{array}{l}
\displaystyle\mathcal{J}_{i}^\ell[\textbf{f},\textbf{f}](t,x)=\sum_{r\in I_c}\sum_{h,p=1}^{n}\int_{x}^{x+\xi}\eta^r[\rho^\star_r(t,x^*)]\mathcal{A}_{hp,r}^{i,\ell}[\rho_{1}^\star,\dots,\rho_L^\star;\alpha](t,x^*)\\
\hskip 3cm \times f_{h}^r(t,x)f_{p}^r(t,x^*)\omega^r(x,x^*)dx^*
\\
\hskip 3cm \displaystyle-f_{i}^\ell(t,x)\int_{x}^{x+\xi}\eta^\ell[\rho^\star_\ell(t,x^*)]\rho_\ell(t,x^*)\omega^\ell(x,x^*)dx^*.
\end{array}
\end{equation*}}
We show the numerical simulations of the flow dynamics in two lanes: In the slowest lane, we consider two clusters of vehicles traveling with different speeds while the fastest lane is empty. More precisely, we take
$$f_5^1(t,x)=70\sin^2\big(10\pi(x-0.3)(x-0.4)\big), \qquad x\in[0.3,0.4],$$
$$f_4^1(t,x)=50\sin^2\big(10\pi(x-0.5)(x-0.6)\big), \qquad x\in[0.5,0.6].$$
The hyperbolic system (\ref{InhomgP}) has been integrated by the finite volume method based on the slop limiters technique \cite{[leV]}. The simulations were developed, for different values of the road conditions $\alpha$, by fixing six velocity classes. Namely, Figure \ref{1.5} refers to a high quality road environment $\alpha=0.95$, we notice that:
\noindent \textbullet $\;$ In the slowest lane, the fast group of vehicles after having approached the group of the slow ones, as indicated in Figure \ref{1.5}-(c), leave them behind, while a small group of vehicles with the velocity $v_n$ is taking the lead, see Figure \ref{1.5}-(e);\\
\noindent \textbullet $\;$ In the fastest lane, groups of vehicles appeared. The groups of vehicles in the slowest lane have increased their speed when they took the fastest lane (Figure \ref{1.5}-(d)). The group with a greater speed has taken the lead over the other group. The vehicles in the two groups are no longer as united as they used to be in the slowest lane (Figure \ref{1.5}-(f)), and that is due to the fact that some of the vehicles now have the chance to accelerate.\\
Figure \ref{1.6} shows the evolution time $t$ of the micro-systems $f_i^1$ (left) and $f_i^2$ for $i=3,\dots,6$ under the good road conditions ($\alpha=0.95$). We observe that the number of vehicles with velocities $v_{n-2}$ and $v_{n-1}$ have decreased over time as indicated in Figure \ref{1.6}-(c) and \ref{1.6}-(e). Moreover, a group of negligible vehicles has decelerated. Whereas, another group of a relating greater number of vehicles has accelerated (see Figure \ref{1.6}-(a) and \ref{1.6}-(g)). On the other hand, Figure \ref{1.6} shows the number of vehicles of each group that has taken the fastest lane. We notice that the groups with velocities $v_{n-1}$ and $v_n$ (see Figure \ref{1.6}-(f) and \ref{1.6}-(h)) are the only groups that are greatly increasing in number with the time being. Moreover, we observe that the vehicles of groups are no longer in the united groups as mentioned before.
\begin{figure}
\centering
\subfigure[]{ \includegraphics[height=1.5in ,width=2in]{L1T1.pdf}}
~\subfigure[]{ \includegraphics[height=1.5in ,width=2in]{L2T1.pdf}}\\
{Snapshot of $f_{i}^1$ (left) and $f_i^2$ (right) for $i=1,\dots,6$ at time $t=0.01$.}\\
\subfigure[]{ \includegraphics[height=1.5in ,width=2in]{L1T2.pdf}}
~\subfigure[]{ \includegraphics[height=1.5in ,width=2in]{L2T2.pdf}}\\
{Snapshot of $f_{i}^1$ (left) and $f_i^2$ (right) for $i=1,\dots,6$ at at time $t=0.35$.}\\
\subfigure[]{ \includegraphics[height=1.5in ,width=2in]{L1T3.pdf}}
~\subfigure[]{\includegraphics[height=1.5in ,width=2in]{L2T3.pdf}}\\
{Snapshot of $f_{i}^1$ (left) and $f_i^2$ (right) for $i=1,\dots,6$ at at time $t=1.61$.}\\
\caption{Evolution of the micro-system in the slowest $f_i^1$ and the fastest lanes $f_i^2$ for $i=1,\cdots,6$ under good road conditions $(\alpha=0.95)$ at successive time $t=0.001,\,0.35,\, 1.61$.}
\label{1.5}
\end{figure}
\begin{figure}
\centering
\subfigure[]{\includegraphics[height=1.5in ,width=2in]{f_31.pdf}}
~\subfigure[]{\includegraphics[height=1.5in ,width=2in]{f_32.pdf}}\\
{Evolution of $f_{3}^1$ (left) and $f_3^2$ (right).}\\
\subfigure[]{ \includegraphics[height=1.5in ,width=2in]{f_41.pdf}}
~\subfigure[]{ \includegraphics[height=1.5in ,width=2in]{f_42.pdf}}\\
{Evolution of $f_{4}^1$ (left) and $f_4^2$ (right).}\\
\subfigure[]{\includegraphics[height=1.5in ,width=2in]{f_51.pdf}}
~\subfigure[]{\includegraphics[height=1.5in ,width=2in]{f_52.pdf}}\\
{Evolution of $f_{5}^1$ (left) and $f_5^2$ (right).}\\
\subfigure[]{\includegraphics[height=1.5in ,width=2in]{f_61.pdf}}
~\subfigure[]{\includegraphics[height=1.5in ,width=2in]{f_62.pdf}}
\\{Evolution of $f_{6}^1$ (left) and $f_6^2$ (right).}
\caption{Evolution of the micro-system in the in slowest lane (left) and in the fastest lane (right) under good road conditions $\alpha=0.95$.}
\label{1.6}
\end{figure}
Figure \ref{1.7} shows the evolution of groups of vehicles in the case of bad road conditions $\alpha=0.3$. We notice that:
\noindent \textbullet $\;$ The group of vehicles with velocity $v_{n-1}$ has approached the other in the slowest lane, as shown in Figure \ref{1.7}-(c), and then passes the other group of vehicles with the velocity $v_{n-2}$, see Figure \ref{1.7}-(e);\\
\noindent \textbullet $\;$ In the fastest lane, two groups of vehicles and a small group with velocity $v_n$ have appeared (see Figure \ref{1.8}-(d)), almost merging at a certain point. With the time being the group of vehicles with velocity $v_{n-1}$ passes the other group with velocity $v_{n-2}$ and the small group of vehicles takes the lead (see Figure \ref{1.7}-(f)). \\
The fist column in Figure \ref{1.8} shows the evolution time $t$ of the micro-systems $f_i^1$ for $i=3,\dots,6$ under the bad road conditions ($\alpha=0.3$). We notice that a negligible number of vehicles that would either decease (see Figure \ref{1.8}-(c) and \ref{1.8}-(e)) or increase (see Figure \ref{1.8}-(a) and \ref{1.8}-(g)). On the other hand,
second column in Figure \ref{1.8} shows the evolution of the groups of vehicles in the fastest lane. We notice that the number of vehicles in the two groups $f_4^2$ and $f_5^2$ is increasing with the time being (see Figures \ref{1.7}-(d) and \ref{1.7}-(f)), and that a small group of vehicles with velocity $v_n$ have appeared (see Figure \ref{1.8}-(b) and \ref{1.8}-(h)).
\begin{figure}
\centering
\subfigure[]{\includegraphics[height=1.5in,width=2in]{L1T1.pdf}}~\subfigure[]{\includegraphics[height=1.5in ,width=2in]{L2T1.pdf}}
\\{Snapshot of $f_{i}^1$ (left) and $f_i^2$ (right) for $i=1,\dots,6$ at time $t=0.01$.}\\
\subfigure[]{\includegraphics[height=1.5in ,width=2in]{L12.pdf}}
~\subfigure[]{\includegraphics[height=1.5in ,width=2in]{L22.pdf}}
\\{Snapshot of $f_{i}^1$ (left) and $f_i^2$ (right) for $i=1,\dots,6$ at at time $t=0.35$.}\\
\subfigure[]{\includegraphics[height=1.5in ,width=2in]{L13.pdf}}
~\subfigure[]{ \includegraphics[height=1.5in ,width=2in]{L23.pdf}}
\\{Snapshot of $f_{i}^1$ (left) and $f_i^2$ (right) for $i=1,\dots,6$ at at time $t=1.61$.}\\
\caption{Evolution of the micro-system in the slowest $f_i^1$ and the fastest lanes $f_i^2$ for $i=1,\cdots,6$ under bad road conditions $(\alpha=0.3)$ at successive time $t=0.001,\,0.35,\, 1.61$.}
\label{1.7}
\end{figure}
\begin{figure}
\centering
\subfigure[]{\includegraphics[height=1.5in ,width=2in]{ff_31.pdf}}
~\subfigure[]{\includegraphics[height=1.5in ,width=2in]{ff_32.pdf}}
\\{Evolution of $f_{3}^1$ (left) and $f_3^2$ (right).}\\
\subfigure[]{\includegraphics[height=1.5in ,width=2in]{ff_41.pdf}}
~\subfigure[]{\includegraphics[height=1.5in ,width=2in]{ff_42.pdf}}
\\{Evolution of $f_{4}^1$ (left) and $f_4^2$ (right).}\\
\subfigure[]{\includegraphics[height=1.5in ,width=2in]{ff_51.pdf}}
~\subfigure[]{\includegraphics[height=1.5in ,width=2in]{ff_52.pdf}}
\\{Evolution of $f_{5}^1$ (left) and $f_5^2$ (right).}\\
\subfigure[]{\includegraphics[height=1.5in ,width=2in]{ff_61.pdf}}
~\subfigure[]{\includegraphics[height=1.5in ,width=2in]{ff_62.pdf}}
\\{Evolution of $f_{6}^1$ (left) and $f_6^2$ (right).}\\
\caption{Evolution of the micro-system in the in slowest lane (left) and in the fastest lane (right) under bad road conditions $\alpha=0.3$.}
\label{1.8}
\end{figure}
\section{Conclusions and perspectives}
In this paper we have developed a new discrete kinetic model for multilane traffic flow on the basis of the kinetic theory approach. After a detailed description of a multilane road and the discrete method adopted, we have presented our proposed discrete model. Next, we have modeled each term appearing in the proposed model. We have proposed an improved table of games by taking into account further features of the complexity, namely the nonlinear additive interactions, perceived density rather than real one, and the road conditions as a function depending on the space variable which can used to depict the road maintenance areas. Moreover, we have modeled the external actions such as tollgates as well traffic signs. The well-posedness of the related Cauchy problem for the spatially homogeneous case has been proved by using Banach fixed-point theory. In order to validate the proposed improved model, in both the spatially homogeneous and inhomogeneous cases, and in fact to show its ability to reproduce certain realistic phenomena, we have reproduced the numerical simulations for Kerner's fundamental diagrams, and the asymptotic in time property along three lanes. Finally, we have obtained numerical simulations of the emerging of clusters with closed speed along two lanes by using the higher resolution finite volume method. We mention that all numerical simulations are provided with particular focus on the road environment conditions modeled by a parameter, which has been shown to have a great influence on the dynamic of vehicles.
Looking ahead, the obtained numerical simulations in the case of the spatially inhomogeneous problem are just one of the several emerging behaviors that can be depicted by our proposed modeling. One can also reproduce many realistic phenomena such as stop and go waves, and the external action by tollgates (see \cite{[Do1]} for one lane). In addition, it is interesting to develop an approach of asymptotic limits towards macroscopic models known in the literature, for example, the recent models \cite{[GR20],[HMV18]}. One can draw inspiration from the ideas developed in \cite{[BD11]}.
|
1,108,101,566,741 | arxiv | \section{Introduction}
The existence of a fundamental scalar has been essential to understand an important phenomenon in the nature.
The Higgs boson, responsible for the electroweak symmetry breaking, is the only example confirmed by experiment so far.
However, it is well believed that a scalar field (inflaton) drives an early expansion of the universe~\cite{Brout:1977ix, Sato:1980yn, Guth:1980zm, Linde:1981mu, Albrecht:1982wi} to solve the flatness and horizon problems, seeding the primordial fluctuations in cosmic microwave background~\cite{Starobinsky:1979ty, Mukhanov:1981xt}. A scalar field is also required to break Peccei-Quinn (PQ)
symmetry at an intermediate scale through which the Strong CP conservation is enforced dynamically \cite{Kim:2008hd}.
More recently, scalar particles have been considered extensively as a dark matter candidate for various reasons, which can also be
extremely light~\cite{Hu:2000ke}. Furthermore, a light scalar (dilaton) coupling to the gluon fields could lead a fifth-force between the nucleus, motivating a lot of low energy atomic experiments~\cite{Murata:2014nra}. Such a scalar field with an initial displaced vacuum
could induce time-varying gauge couplings through its coupling to the weak and strong gauge fields, providing an first-order phase transition required by the electroweak baryogengesis~\cite{Ipek:2018lhm, Ellis:2019flb, Berger:2019yxb, Danielsson:2019ftq}.
In this paper, we consider a scalar field $\phi$ coupling to a general $SU(N)$ gauge field which confines at a lower scale.
Then, a $\phi$-dependent vacuum energy is induced and generates a new potential for the scalar boson in the confining phase.
In this way, a new energy scale can emerge to provide us some insight on understanding various phenomena mentioned above.
The paper is organized as follows: we will first briefly overview the mechanism, and then discuss various implications in physics of
dark matter and symmetry breaking.
\section{The mechanism}
\subsection{Confinement potential }
Let us consider an $SU(N)$ gauge theory with $n_f$ light fermions, and a singlet scalar $\phi$ which couples
to the gauge field through a high dimension operator,
\begin{eqnarray}
\mathcal{L} \supset -\frac{1}{4 g^2} \left (1- c \frac{\phi}{M} \frac {\beta} {2g} \right ) G^{ \mu\nu} G_{\mu \nu} .
\label{formula1}
\end{eqnarray}
Note that the pre-factor $\beta/2g$ is included to keep renormalization scale invariance of the operator, $g$ is the gauge coupling constant, and the beta function $\beta$ is defined by
\begin{eqnarray}
\frac{d g}{d \ln \mu} \equiv \beta=-\beta_0 \frac{g^3}{16\pi^2}
\end{eqnarray}
with $\beta_0= (\frac{11}{3} N- \frac{2}{3} n_f)$.
If $SU(N)$ confines below $M$, there occurs gauge condensation $\langle \frac{\alpha_s}{\pi} \tilde G^{ \mu \nu} \tilde G_{\mu \nu} \rangle \simeq \Lambda^4$~\cite{Shifman:1978by, Shifman:1978bx, Gubler:2018ctz}. Here $\tilde G^{ \mu \nu} = \frac{1}{g_s} G^{ \mu \nu} $ which is the normalized gauge field and $g_s$ should be understood as the effective gauge coupling satisfying $\frac{1}{g_s^2} = \frac{1}{g^2} \left (1- c \frac{\phi}{M} \frac {\beta} {2g} \right ) $. $\Lambda$ is the confinement scale which can be estimated as
\begin{eqnarray}
\Lambda = M \exp(-\frac{8\pi^2}{g^2\beta_0} -\frac{c}{4} \frac{\phi}{M} ) .
\end{eqnarray}
The contribution of vacuum energy from confinement is~\cite{Collins:1976yq, Pasechnik:2016sbh} \footnote{Note that a similar term is considered for the radion~\cite{vonHarling:2017yew, Fujikura:2019oyi}.},
\begin{eqnarray}
V_{vac} &=& \frac{1}{4} \langle T^\mu_\mu \rangle = \langle \frac{\beta}{8g_s} \tilde G^{ \mu \nu} \tilde G_{\mu \nu} \rangle + \frac{1}{4} (1-\gamma_m)m_f \langle \bar f f \rangle \nonumber \\
&\simeq& - \frac{\beta_0}{32}\Lambda^4 = - \frac{\beta_0}{32} \Lambda_0^4 \exp(- c\frac{\phi}{M} )
\label{potential}
\end{eqnarray}
where $\Lambda_0$ is the confinement scale when $\phi=0$ and $\gamma_m$ is the anomalous dimension of the fermion mass operator. Here we ignore the fermion contribution by assuming $m_f$ is much smaller than $\Lambda_0$.
Since the $\Lambda$ has depends on $\phi$, it induces a potential for $\phi$. Note that this potential is only valid for $ | c \frac{\phi}{M}| \lesssim \mathcal O(1) $. In the following we will discuss the physics implications of the emergent potential~(\ref{potential}) with the pre-factor ${\beta_0}/{32}$ absorbed by the redefined confinement scale.
\subsection{ Emergence of a new energy scale}
As is well-known, a scale could emerge from dimension transmutation although a theory does not contain any mass parameter.
QCD is a specific example where the condensation scale of $\Lambda_{QCD}\sim 0.1$ GeV arises due to
the color confinement. Another example is the Coleman-Weinberg potential which was originally used for electroweak symmetry breaking (EWSB) without introducing a dimensionful parameter~\cite{Coleman:1973jx}. It is also argued that such classical scale invariance may provide a solution of naturalness problem~\cite{Bardeen:1995kv}. Along these lines, the confinement potential~(\ref{potential}),
generating a dimensionful term for the scalar $\phi$ via dimension transmutation,
can be the source of a new scale: the vacuum expectation value $\langle \phi\rangle$.
Consider the following interactions of the scalar field $\phi$,
\begin{eqnarray}
\mathcal L \supset -\frac \lambda 4 \phi^4 -\frac{1}{4 g^2} \left (1- \frac{\phi}{M} \frac{\beta}{2 g} \right ) G^{ \mu\nu} G_{\mu \nu}
\label{eq3}
\end{eqnarray}
which induces a potential,
\begin{eqnarray}
V = \frac \lambda 4 \phi^4 - \Lambda_0^4 \exp(-\frac{\phi}{M} )
\end{eqnarray}
valid for $|\phi| \lesssim M$. From the minimization of the potential~(\ref{potential}), fulfilling the conditions:
$V^{\prime} =0$ and $V^{\prime \prime} > 0$ at $\phi=\langle \phi \rangle$,
we can find the solution:
\begin{eqnarray}
&& \langle \phi \rangle \simeq - \lambda^{-1/3} \Lambda_0 \left ( \frac{\Lambda_0}{ M} \right )^{1/3} \\
&& m_\phi^2 = 3 \lambda \langle \phi \rangle^2 - \frac{\Lambda^4_0}{M^2}
\end{eqnarray}
for $\lambda > \frac{1}{27} (\frac{\Lambda_0}{M})^4$.
For a complex scalar field $\phi$ charged under $U(1)$ or $Z_2$ symmetry, one can consider the Lagrangian:
\begin{eqnarray}
\mathcal L \supset -\frac \lambda 4 (\phi^\dagger \phi)^2 -\frac{1}{4 g^2} \left (1- c \frac{\phi^\dagger \phi}{M^2} \frac{\beta}{2 g} \right ) G^{ \mu\nu} G_{\mu \nu}
\end{eqnarray}
which induces the potential,
\begin{eqnarray}
V &=& \frac \lambda 4 (\phi^\dagger \phi)^2 - \Lambda_0^4 \exp(- c \frac{\phi^\dagger \phi}{M^2} ) \nonumber \\
&=& \frac \lambda 4 (\phi^\dagger \phi)^2 - \Lambda_0^4 + c \frac{\Lambda_0^4}{M^2} \phi^\dagger \phi + \cdots .
\end{eqnarray}
In the case of $c < 0$, it provides a negative mass term and thus induces the symmetry breaking leading to
\begin{eqnarray}
&&\langle \phi \rangle \simeq (-\frac{2 c}{\lambda})^{1/2} \frac{\Lambda_0^2}{M} \\
&& m^2_\phi \simeq -c \frac{\Lambda_0^4}{M^2} .
\end{eqnarray}
Notice that this mechanism can be applied to the PQ symmetry breaking. Taking $M$ to be the Planck scale
$M_{P}$, one can get the axion scale of $\langle \phi \rangle \sim 10^{11}$ GeV for
$\Lambda_0 \sim 10^{15}$ GeV with $|c|$ and $\lambda \sim \mathcal O(1)$.
\section{Dark matter and gauge condensation}
\begin{figure}[ht]
\centering
\includegraphics[width=2.5in]{potential.pdf}
\caption{The parameter space where $\phi$ as a dark matter candidate. We show the limit from fifth-force measurement \cite{Murata:2014nra} (gray region) and stellar cooling limits from HB stars \cite{Hardy:2016kme} (above green curve), RG stars \cite{Hardy:2016kme} (above blue curve) and SN1987A \cite{Knapen:2017xzo} (pink region). The red curve is the parameter space where $\phi$ satisfies the dark matter relic density with a fraction $f$.
}
\label{fig1}
\end{figure}
\subsection{Minimal model from color confinement}
Consider a scalar dark matter $\phi$ which couples only to the gluon field. In this case, its abundance can be produced through the misalignment of the vacuum before and after the QCD phase transition. Assuming again the following interactions,
\begin{eqnarray}
\mathcal L \supset -\frac \lambda 4 \phi^4 -\frac{1}{4 g_s^2} \left (1- \frac{\phi}{M} \frac {\beta_s} {2g_s} \right ) G_s^{a \mu\nu} G_{s\mu\nu}^a
\label{qcd}
\end{eqnarray}
where we used the index $s$ to denote the color $SU(3)_c$ sector.
Before QCD confines, the $\frac 1 4 \lambda \phi^4$ dominate the potential having the minimum at $\langle \phi \rangle =0$. After the confinement of QCD, the confinement potential arises
\begin{eqnarray}
V = \frac \lambda 4 \phi^4 - \Lambda^4_{\rm QCD} \exp(- \frac{\phi}{M} )
\end{eqnarray}
developing a new minimum at
\begin{equation}
\langle \phi \rangle \simeq - \lambda^{-1/3} \Lambda_{\rm QCD}
\left ( \frac{\Lambda_{\rm QCD}}{ M} \right )^{1/3}
\end{equation}
where we define $\Lambda^4_{\rm QCD} \equiv \frac{9}{32} \langle \frac{\alpha}{\pi}G^{ a \mu \nu}G^a_{\mu \nu} \rangle$ $={9\over 32} 0.028$ GeV$^4$~\cite{Horsley:2012ra} \footnote{Note that there is a large uncertainty in the determination of $\langle \frac{\alpha}{\pi}G^{a \mu \nu} G^a_{\mu \nu} \rangle$~\cite{Narison:2018dcr}.}.
At the same time, $\phi$ gets the mass $m_\phi \simeq \sqrt{3\lambda} \langle \phi \rangle $ which is much larger than the Hubble parameter. Thus, $\phi$ starts immediately to oscillate around the new minimum. If $\phi$ contributes all the dark matter component, the relic density of $\phi$ should satisfy
\begin{eqnarray}
\rho_{\phi} \times \left ( \frac{ T_{eq}}{T_{osc}} \right )^3 \frac{ g_{*} (T_{eq})}{g_{*} (T_{osc} ) } \simeq 0.4 ~\rm (eV)^4
\label{darkmatter}
\end{eqnarray}
where $\rho_{\phi} \simeq \frac{3}{4}{\lambda} \langle \phi \rangle^4$ and $T_{eq} \simeq 0.8$ eV is the temperature at the matter-radiation equality. The oscillation temperature is taken to be $T_{osc} = 0.15 $ GeV~\cite{Petreczky:2012rq}.
In our scenario the dark matter coupling to photons arises at two loop level, The decay width of the light scalar can be estimated as,
\begin{eqnarray}
\Gamma_{\phi\to \gamma\gamma} \approx \sum_q \frac{1}{\pi} \left (\frac{\alpha}{4\pi} \right)^2 \alpha_s^2 Q_q^4 \frac{m_\phi^3}{M^2} .
\end{eqnarray}
Where $Q_q$ is the electric charge of the quarks. The minimization conditions and the dark matter relic density relation (\ref{darkmatter}) set the dark matter lifetime as a function of its mass and fraction $f$:
\begin{equation}
\tau_{\phi \to \gamma \gamma} \approx 5 \times 10^{17} \mbox{sec}\,
\left ( \frac{1}{f} \right ) \left( \mbox{3 keV} \over m_\phi \right)^5 .
\end{equation}
Thus, the dark matter mass should not be larger than about 3 keV.
On the other hand, the interaction in Eq.~(\ref{qcd}) also induces a coupling between the scalar and nucleus at low energy:
\begin{eqnarray}
\frac{\phi}{M} \frac{\beta}{2g} G^{a \mu\nu} G^a_{\mu \nu} \Rightarrow \mathcal O(1) \frac{\phi}{M} m_N \bar N N .
\end{eqnarray}
The fifth-force measurement and the astrophysical observations set a strong limit on such couplings. In Fig.~\ref{fig1} we show the dark matter preferred region as well as these limits. The red curve is the parameter space where $\phi$ satisfies the dark matter relic density. It shows that this scalar can hardly provide all the the dark matter component surviving all the constraints.
More parameter space is available if we consider $\phi$ contributes to a fraction of the dark matter.
\subsection{Additional SU(N) potential}
Let us now generalize the previous consideration by adding an additional confining gauge sector of $SU(N)$:
\begin{eqnarray}
\mathcal{L} \supset && - \frac{1}{4} \lambda \phi^4 -\frac{1}{4 g_N^2} \left (1- \frac{\phi}{M} \frac{\beta_N}{2g_N}\right ) G^2_N \nonumber \\
&& -\frac{1}{4 g_s^2} \left (1- \frac{\phi}{M} \frac{\beta_s}{2g_s}\right ) G^2_s
\end{eqnarray}
where we assumed the same coupling of $\phi$ to the $SU(N)$ and $SU(3)_c$ gauge fields for simplicity. If we allow smaller coupling of $\phi$ to the QCD sector, our bounds will be relaxed accordingly.
The above interactions can be obtained after integrating out by a heavy fermion in bi-fundamental representations of $SU(N)$ and $SU(3)_c$. Then the total confinement potential at low energy becomes,
\begin{eqnarray}
V &=& \frac{1}{4} \lambda \phi^4 - \Lambda_{N}^4 \exp(-{\phi \over M})
- \Lambda_{\rm QCD}^4 \exp(-{\phi \over M}) .
\end{eqnarray}
Assuming $\Lambda_{N} \gg \Lambda_{\rm QCD}$ \footnote{Note that the $\Lambda_{\rm QCD}$ here should be changed into $\Lambda^\prime_{\rm QCD}=\Lambda_{\rm QCD} \exp(\frac{1}{4}\frac{\langle \phi \rangle_{\rm min}}{M})$ since the gauge coupling get renormalized for $\phi \ne 0 $. However, for our case $M \gg \langle \phi \rangle_{\rm min}$, this difference is very small and therefore we safely neglect it. Same argument is for the pure QCD case. }, the scalar potential is dominated by the $SU(N)$ confinement leading to the previous relations (7,8) with $\Lambda_0 \to \Lambda_N$.
The requirement for the dark matter relic density (16) is again applicable with $T_{osc}=\Lambda_N$.
To maintain the two sectors in thermal equilibrium before the hidden sector confines we may add more fermions with smaller mass.
In Fig.~{\ref{fig2}} we show the viable parameter space. The dashed curves show the values of $m_\phi$. The gray region is excluded by the X-ray or $\gamma$-ray searches due to the decaying of the scalar into photons~\cite{Essig:2013goa}. These limits generally require the lifetime of scalar to be larger than $10^{27}$ seconds.
In the present case, the dark matter mass can be in the range of (eV, MeV) where the upper limit is set by the condition:
$M < M_P$ as a reference.
\begin{figure}[ht]
\centering
\includegraphics[width=2.5in]{case2.pdf}
\caption{The parameter space where $\phi$ as a dark matter candidate. All of shaded region are excluded by different searches. The dashed curves show $m_\phi =$ 1 eV, 1 keV, 1 MeV, 1 GeV respectively.
}
\label{fig2}
\end{figure}
\section{ Origin of the Electroweak symmetry breaking}
It is known that the electroweak symmetry breaking is triggered by the condensation of Higgs boson. Although the Higgs properties of the standard model are well established, there still to be understood: "What is the origin of the Higgs potential?".
Here we attempt to drive the electroweak symmetry breaking by applying our mechanism of dimensional transmutation.
\begin{figure}[ht]
\centering
\includegraphics[width=2.7in]{higgself.pdf}
\caption{ Deviation of the Higgs trilinear coupling for different $M$. }
\label{fig3}
\end{figure}
Assuming that the Higgs couples to a hidden strong sector through a dimension-6 operator:
\begin{eqnarray}
\mathcal L \supset - \lambda (H^\dagger H)^2 -\frac{1}{4 g^2} \left (1+ \frac{H^\dagger H}{M^2} \frac{\beta}{2 g} \right ) G^{ \mu\nu} G_{\mu \nu} \nonumber \\
\end{eqnarray}
we get the Higgs potential
\begin{eqnarray}
V &=& {\lambda} (H^\dagger H)^2 - \Lambda_0^4 \exp( \frac{H^\dagger H}{M^2} ) \nonumber \\
&\approx& -\frac{\Lambda_0^4}{ M^2} H^\dagger H
+ (\lambda+\frac{1}{2} \frac{H^\dagger H}{M^2} ) (H^\dagger H)^2 \nonumber \\
&& -\frac{1 }{6} \frac{\Lambda_0^4}{ M^6} (H^\dagger H)^3 + \cdots .
\end{eqnarray}
Taking $ H = ( 0 , (h+v)/\sqrt2 )^T $ in the unitary gauge, the minimization conditions give us
\begin{eqnarray}
\Lambda_0 = \sqrt 2 M \left (\frac{m_h^2}{8 M^2- v^4/M^2} \right )^{1/4}
\end{eqnarray}
where $v=246$ GeV and $m_h=125$ GeV.
The higher dimension operators modify the Higgs trilinear coupling $\lambda_{hhh}$. Defining $\delta \equiv (\lambda_{hhh}/\lambda^{SM}_{hhh})-1$, we find
\begin{equation}
\delta = - \frac{4 v^4}{ 24 M^4-3 v^4} \simeq -\frac{1}{6} (\frac{v}{M})^4
\end{equation}
which is highly suppressed.
As can be seen in Fig.~\ref{fig3}, the deviation of Higgs trilinear coupling from the standard model prediction can be sizable for
$M= 300-400$ GeV, which might be probed at future colliders. It drops quickly below 1\% for $M> 500$ GeV.
For this range, we need $\Lambda_0 \gtrsim 200$ GeV.
\section{{Conclusion}}
We studied a new mechanism where a scalar potential emerges from the scalar coupling to a $SU(N)$ gauge field strength which confines at a low energy scale. Such a potential may play an important role to understand the origin of dark matter, or spontaneous breaking of symmetry such as PQ and electroweak symmetry.
It will be an interesting task to see whether the potential for the inflaton or quintessence can emerge in a same way.
\\
\noindent {\bf{Acknowledgements}}
This project has received support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 690575. E.J.C. acknowledges support from InvisiblesPlus RISE No. 690575. C. H thanks Shi Pi for helpful discussions. CH acknowledges support from the Sun Yat-Sen University Science Foundation.
|
1,108,101,566,742 | arxiv | \section{Introduction}\label{intro}
{\em Possibilistic logic} \cite{DuLaPr,DuPr2} is a well-known uncertainty logic to reason with graded (epistemic) beliefs on classical propositions by means of necessity and possiblity measures.
In this setting, epistemic states of an agent are represented by possibility distributions. If $W$ is a set of classical evaluations or possible worlds, for a given propositional language, a {\em normalized possibility distribution} on $W$ is a mapping $\pi: W \to [0, 1]$, with $\sup_{w \in W} \pi(w) = 1$. This map $\pi$ ranks interpretations according to its plausibility level: $\pi(w) = 0$ means that $w$ is rejected, $\pi(w) = 1$ means that $w$ is fully plausible, while $\pi(w) < \pi(w')$ means that $w'$ is more plausible than $w$.
A possibility distribution $\pi$ induces a pair of dual possibility and necessity measures on propositions, defined respectively as:
\begin{center}
$\Pi(\varphi) = \sup\{ \pi(w) \mid w \in W , w(\varphi) = 1 \}$ \\
$N(\varphi) = \inf\{ 1- \pi(w) \mid w \in W , w(\varphi) = 0\}$
\end{center}
$N(\varphi)$ measures to what degree $\varphi$ can be considered certain given the given epistemic, while $\Pi(\varphi)$ measures the degree in which $\varphi$ is plausible or possible. Both measures are dual in the sense that $\Pi(\varphi) = 1 - N(\neg \varphi)$, so that the degree of possibility of a proposition $\varphi$ equates the degree in which $\neg \varphi$ is not certain.
If the normalized condition over possibility distribution is dropped, then we gain the ability to deal with inconsistency. In \cite{DyP2015}, a possibility distribution which satisfies $\sup_{w \in W } \pi(w) < 1$ is called sub-normal. In this case, given a set $W$ of classical interpretations, a degree of inconsistency can be defined in the following way:
$$ inc(W) = 1 - \sup_{w \in W } \pi(w)$$
When the normalised possibility distribution $\pi$ is $\{0, 1\}$-valued, i.e. when $\pi$ is the characteristic function of a subset $\emptyset \neq E \subseteq W$, then the structure $\langle W, \pi \rangle$, or better $\langle W, E \rangle$, can be seen in fact as a K45 frame. Indeed, it is folklore that modal logic K45, which is sound and complete w.r.t. the class of Kripke frames $\langle W, R \rangle$ where $R$ is a euclidean and transitive binary relation, also has a simplified semantics given by the subclass of frames $\langle W, E
\rangle$, where $E$ is
a non-empty subset of $W$ (understanding $E$ as its corresponding binary relation $R_E$ defined as $R_E(w, w')$ iff $w' \in E$).
When we go beyond the classical framework of Boolean algebras of events to many-valued frameworks, one has to come up with appropriate extensions
of the notion of necessity and possibility measures for many-valued events \cite{DeGoMa}.
In the particular context of G\"odel fuzzy logic \cite{Hajek98}, natural generalizations that we will consider in this paper are the following.
A possibility distribution is as before a function $\pi \colon W \to [0, 1]$ but now, $W$ is a set of G\"odel propositional interpretations that induces the
following generalized possibility and necessity measures over G\"odel logic propositions:
$$\Pi(\varphi) = \sup_{w \in W} \{ \min(\pi(w), w(\varphi)) \}$$ \vspace{-4mm}
$$N( \varphi) = \inf_{w \in W} \{\pi(w) \Rightarrow w(\varphi) \},$$
where $\Rightarrow$ is G\"odel implication, that is, for each $x, y \in [0, 1]$, $x \Rightarrow y = 1$ if $x \leq y$, $x \Rightarrow y = y$, otherwise.\footnote{Strictly speaking, the possibility measure is indeed a generalization of the classical one, but the necessity measure is not, since $x \Rightarrow 0 \neq 1-x$.}
These expressions agree with the ones commonly used in many-valued modal Kripke frames $\langle W, R \rangle$ to respectively evaluate modal formulas $\Diamond \varphi$ and $\Box \varphi$ (see for example \cite{BouJLC} and references
therein) when the $[0, 1]$-valued accessibility relation $R: W \times W \to [0, 1]$
is replaced by a possibility distribution $\pi: W \to [0, 1]$ as $R(w,w') = \pi(w')$, for any $w, w' \in W$.
Actually, modal extensions of G\"odel fuzzy logic have been studied by Caicedo and Rodr\'iguez \cite{CaiRod2015}, providing sound and complete axiomatizations for different classes of
$[0, 1]$-valued Kripke models. These structures are triples $\ensuremath{\frm{M}} = \langle W, R,
e \rangle$, where $W$ is a set of worlds, $R $ is a $[0, 1]$-valued accessibility relation, as above, and $e: W \times Var \to [0, 1]$ is such that, for every $w \in W$, $e(w, \cdot)$ is a G\"odel $[0,1]$-valued evaluation of
propositional variables (more details in next section) that extends to modal formulas as follows:
\begin{align*}
e(w,\Diamond \varphi ) &= \sup_{w'\in W}\{\min(R(w,w'), e(w',\varphi ))\} \\
e(w,\Box \varphi ) &= \inf_{w'\in W}\{R(w,w')\Rightarrow e(w',\varphi )\}.
\end{align*}
We will denote by ${\cal K}45(\mathbf{G})$ the class of $[0, 1]$-models $\ensuremath{\frm{M}} = \langle W, R, e \rangle$ where $R$ satisfies the following many-valued counterpart of the classical properties:
\begin{itemize}
\item {\em Transitivity}: $\forall w, w', w'' \in W$, $\min(R(w, w'), R(w', w'')) \leq R(w, w'')$
\item {\em Euclidean}: $\forall w, w', w'' \in W$, $\min(R(w, w'), R(w, w'')) \leq R(w', w'')$
\end{itemize}
In this setting, the class $\Pi\mathcal{G}$ of {\em possibilistic Kripke
models} $\langle W, \pi, e \rangle$, where $\pi \colon W \to [0, 1]$ is a possibility distribution (not necessarily normalized) on the set of worlds $W$, can be considered as the subclass of ${\cal K}45(\mathbf{G})$ models $\langle W, R, e \rangle$ where $R$ is independent of the world in its first argument, in the sense that $R(w, w') = \pi(w')$. Since $\Pi\mathcal{G} \subsetneq {\cal K}45(\mathbf{G})$, it follows that the set $Val({\cal K}45(\mathbf{G}))$ of valid formulas in the class of
${\cal K}45(\mathbf{G})$ is a subset of the set $Val(\Pi\mathcal{G} )$ of valid formulas in the class $\Pi\mathcal{G}$, i.e. $Val({\cal K}45(\mathbf{G})) \subseteq Val(\Pi\mathcal{G} )$. The interesting question is therefore whether the converse inclusion $Val(\Pi\mathcal{G} ) \subseteq Val({\cal K}45(\mathbf{G}))$ holds, and thus whether the class $\Pi\mathcal{G}$ provides a simplified possibilistic semantics for the modal logic $K45(\mathbf{G})$.
In \cite{BEGR16} the authors claim to provide a positive answer, not for the class ${\cal K}45(\mathbf{G}))$ models but for the subclass of ${\cal KD}45(\mathbf{G}))$ models, i.e.\ those ${\cal K}45(\mathbf{G}))$ models $\ensuremath{\frm{M}} = \langle W, R, e \rangle$ further satisfying the many-valued counterpart of seriality:
\begin{itemize}
\item{\em Seriality}: $\forall w \in W$, $\sup_{w' \in W} R(w, w') = 1$
\end{itemize}
Indeed, they prove that the logic $KD45(\mathbf{G})$, complete w.r.t. the
class of ${\cal KD}45(\mathbf{G}))$ models, is also complete w.r.t. the class $\Pi^*\mathcal{G}$ of possibilistic models $\langle W, \pi, e \rangle$, where $\pi: W \to [0, 1]$ is a {\em normalized} possibility distribution on $W$.
Unfortunately, it has to be noted that the completeness proof in \cite{BEGR16} has some flaws.
In this paper we provide a correct proof, not only for the completeness of $KD45(\mathbf{G})$ w.r.t. to its corresponding class of possibilistic frames, but also for the weaker logic $K45(\mathbf{G})$ accounting for partially inconsistent possibilistic Kripke frames.
{This paper is organized as follows: Section~\ref{sec:preliminaries} introduces the main notions about minimum propositional modal G\"odel logic and its relational semantics;
Section~\ref{sec:K45logic} then formally introduces a calculus for the logic $K45(G)$, some of its extensions and several of its main theorems;
Section~\ref{sec:possibilitysem} presents our simplified possibilistic semantics and the results of completeness;
Section ~\ref{sec:decidability} is devoted to analyze the finite model property for the new semantics;
finally, Section~\ref{sec:Disc-concl} provides some conclusions.}
\section{Preliminaries on propositional and modal G\"odel fuzzy logic} \label{sec:preliminaries}
This section is devoted to preliminaries on the G\"odel fuzzy logic G. We present their syntax and semantics, their main logical properties and the notation we use throughout the article.
The language of G\"odel propositional logic $\mathcal{L}(V)$ is built as usual from a countable set of propositional variables $V$, the constant $\bot$ and the binary connectives $\land$ (conjunction) and $\to$ (implication).
Further connectives are defined as follows:
\begin{eqnarray*}
\top & \mbox{is} & \bot \rightarrow \bot \\
\varphi \vee \psi & \mbox{is} & ((\varphi \rightarrow \psi)\rightarrow
\psi)
\wedge ((\psi \rightarrow \varphi )\rightarrow \varphi),\\
\lnot \varphi & \mbox{is} & \varphi \rightarrow \bot,\\
\varphi \equiv \psi & \mbox{is} & (\varphi \rightarrow \psi) \wedge (\psi
\rightarrow
\varphi).
\end{eqnarray*}
The following are the {\em axioms\/} of $G$: \\
\begin{tabular}{l l}
(A1) &$(\varphi \rightarrow \psi) \rightarrow ((\psi \rightarrow
\chi) \rightarrow (\varphi \rightarrow \chi))$
\\
(A2) &$(\varphi \wedge \psi) \rightarrow \varphi$
\\
(A3) &$(\varphi \wedge \psi) \rightarrow (\psi \wedge \varphi)$
\\
(A4a) &$(\varphi \rightarrow (\psi \rightarrow \chi ))
\rightarrow ((\varphi \wedge \psi) \rightarrow \chi)$
\\
(A4b) &$((\varphi \wedge \psi) \rightarrow \chi) \rightarrow
(\varphi\rightarrow (\psi \rightarrow \chi))$
\\
(A6) & $\varphi \to (\varphi \wedge \varphi)$
\\
(A7) & $((\varphi \rightarrow \psi) \rightarrow \chi) \rightarrow
(((\psi \rightarrow \varphi ) \rightarrow
\chi) \rightarrow \chi)$
\\
(A8) &$\bot \rightarrow \varphi$
\\
\end{tabular}
\ \\
\noindent The {\em deduction rule\/} of $G$ is modus ponens. \\
G\"odel logic can be obtained in fact as the axiomatic extension of H\'ajek's Basic Fuzzy Logic BL \cite{Hajek98} (which is the logic of continuous t-norms and their residua) by means of the contraction axiom (A6) $\varphi \to (\varphi \land \varphi)$. Since the unique idempotent continuous t-norm is the minimum, this yields that G\"odel logic is strongly complete with respect to its standard fuzzy semantics that interprets formulas over the structure $[0, 1]_\mathrm{G} := \langle [0, 1], \min, \Rightarrow_\mathrm{G}, 0, 1 \rangle$\footnote{Called standard G\"odel algebra.}, i.e. semantics defined by truth-evaluations $e$ such that $e(\varphi \land \psi) = \min(e(\varphi), e(\psi))$, $e(\varphi \to \psi) = e(\varphi) \Rightarrow_\mathrm{G} e(\psi)$ and $e(\bot) = 0$.
G\"odel logic can also be seen as the axiomatic extension of intuitionistic propositional logic by the prelinearity axiom $(\varphi \to \psi) \lor (\psi \to \varphi)$. Its algebraic semantics is therefore given by the variety of prelinear Heyting algebras, also known as G\"odel algebras. In fact, it is sound and complete in the following stronger sense, see \cite{CaiRod2010}.
\begin{proposition} \label{ordersoundness}
\begin{itemize}
\item[i)] If $T\cup \{\varphi \}\subseteq \mathcal{L}(X),$
then $T\vdash _G \varphi $ implies $\inf v(T)\leq v(\varphi )$
for any valuation $v:X\rightarrow \lbrack 0,1]$.
\item[ii)]If $T$ is countable,
and $T\nvdash _ G \varphi _{i_{1}}\vee ..\vee \varphi _{i_{n}}$
for each finite subset of a countable family $\{\varphi _{i}\}_{i\in I}$ there is
an evaluation $v:\mathcal{L}(X) \rightarrow \lbrack 0,1]$\ such that $v(\theta )=1$\ for all
$\theta \in T$\ and $v(\varphi _{i})<1$ for all $i \in I$.
\end{itemize}
\end{proposition}
We mention in passing that the algebraic semantics of G\"odel algebra is is given by the variety of prelinear Heyting algebras, also known as G\"odel algebras. A G\"odel algebra is a structure ${\bf A} = \langle A, *, \Rightarrow, 0, 1 \rangle$ which is a (bounded, integral, commutative) residuated lattice satisfying the contraction equation $$x * x = x,$$ and pre-linearity equation $$(x \Rightarrow y) \lor (y \Rightarrow x) = 1, $$where $x \lor y = ((x \Rightarrow y) \Rightarrow y) * ((y \Rightarrow x) \Rightarrow x)$). \vspace{5mm}
Let us consider a modal expansion of G\"odel logic with two operators $\Box$ and $\Diamond$. The set of formulas $\mathcal{L}_{\Box \Diamond }(V)$
is built as $\mathcal{L}(V)$ (always assuming countability of the set of propositional variables $V$) but extending the set of operations with two unary symbols $\Box $ and $\Diamond$. Whenever $V$ is clear from the context we will simply write $\mathcal{L}_{\Box \Diamond}$.
In the style introduced by Fitting \cite{Fitting91,Fi92b} and studied in the works mentioned in the introduction, we define the G\"odel Modal Logic as arising from its semantic definition. This is given by enriching usual Kripke models with evaluations over the previous standard algebra, as in \cite{CaiRod2010, CaiRod2015} and others. Formally:
\begin{definition}
A \termDef{G\"odel-Kripke model} is a structure $\langle W, R, e\rangle$ where $W$ is a non-empty set of so-called worlds, and $R \colon W \times W \rightarrow[0,1]$ and $e \colon V \times W \rightarrow [0,1]$ are arbitrary mappings.
\end{definition}
The evaluation $e$ can be uniquely extended to a map with domain $W \times \mathcal{L}_{\Box\Diamond}$ in such a way that it is a propositional G\"odel homomorphism (for the propositional connectives) and where the modal operators are interpreted as infima and suprema, namely:
\begin{itemize}
\item $e(v, \bot) \coloneqq 0$,
\item $e(v, \varphi \star \psi) \coloneqq e(v, \varphi) \star e(v, \psi)$ for $\star \in \{\wedge, \vee, \rightarrow\}$,
\item $e(v,\Box \varphi) \coloneqq \bigwedge_{w \in W}( R(v,w) \rightarrow e(w, \varphi)),$
\item $e(v,\Diamond \varphi) \coloneqq \bigvee_{w \in W}( R(v,w) \wedge e(w, \varphi)).$
\end{itemize}
A formula $\varphi$ is \emph{valid} in a G\"odel-Kripke model $\langle W, R, e\rangle$, if $e(w,\varphi) = 1$ for all $w \in W$. We will denote by $\cal G$ the class of G\"odel Kripke models and will say that a formula $\varphi$ is \emph{$\cal G$-valid}, written $\models_{\mathcal{G}} \varphi$, if $\varphi$ is valid in all G\"odel Kripke models.
In their paper \cite{CaiRod2015} Caicedo and Rodriguez define the logic $K(\mathbf{G})$ as the smallest set of formulas containing some axiomatic version of G\"{o}del-Dummet
propositional calculus, that is, Heyting calculus plus the prelinearity law, and the following additional axioms and rule: \\
\begin{tabular}{rlrl}
$(K_\Box)$ & $\Box(\varphi \to \psi) \to (\Box \varphi \to \Box \psi)$ &
$(K_\Diamond)$ & $\Diamond(\varphi \lor \psi) \to (\Diamond\varphi \lor \Diamond \psi)$\\
$(F_\Box)$ & $\Box \top$ &
$(P)$ & $\Box(\varphi \to \psi) \to (\Diamond \varphi \to \Diamond \psi)$\\
$(FS2)$ & $(\Diamond\varphi \to \Box \psi) \to \Box(\varphi \to \psi)$ \mbox{} \mbox{} \quad &
$(Nec)$ & from $\varphi$ infer $\Box \varphi$ \\
\end{tabular}
\vspace{5mm}
They denote deduction in this system as $\vdash _{K(\mathbf{G})}$ and they show the following result:
\begin{theorem}[{\cite[Theorem 3.1]{CaiRod2015}}]\label{th:compGK}
Let $\varphi \in \mathcal{L}_{\Box\Diamond}$. Then
\[\vdash_{K(\mathbf{G})} \varphi \text{ if and only if } \models_{\mathcal{G}} \varphi.\]
\end{theorem}
\vspace{0.2cm}
\noindent Proofs with assumptions are allowed with the restriction that $(Nec)$ may be applied only when the premise is a
theorem.
This restriction allows for the following convenient reduction (see \cite{CaiRod2010}).\medskip
\begin{lemma}
{\label{reduction} } Let $ThK(\mathbf{G})$ be the set of theorems of $K(\mathbf{G})$ with no assumptions, then for any theory $T$ and formula $\varphi $ in $\mathcal{L}_{\Box \Diamond
}:T\vdash _{K(\mathbf{G})}\varphi $\ if\ and\ only\ if\ $ T\cup ThK(G)\vdash _{\mathbf{G}}\varphi $ where $\vdash_{\mathbf{G}}$ denotes deduction in G\"odel fuzzy logic G.
\end{lemma}
\noindent The following are some theorems of $K(\mathbf{G})$, see \cite{CaiRod2015}: \vspace{2mm}
$%
\begin{array}{rlrl}
T1. & \lnot \Diamond \theta \leftrightarrow \square \lnot \theta & \\
T2. & \lnot \lnot \square \theta \rightarrow \square \lnot \lnot \theta &
\\
T3. & \Diamond \lnot \lnot \varphi \rightarrow \lnot \lnot \Diamond \varphi & \\
T4. & (\square \varphi \rightarrow \Diamond \psi )\vee \square ((\varphi \rightarrow \psi )\rightarrow \psi )\\
T5. & \Diamond (\varphi \to \psi) \to (\square \varphi \to \Diamond \psi)
\\
\end{array}
$ \vspace{2mm}
\noindent The first one is an axiom in Fitting's systems in \cite{Fitting91}, the next two were
introduced in \cite{CaiRod2015}, the fourth one will be useful in our completeness
proof and is the only one depending on prelinearity. The last is known as
the first connecting axiom given by Fischer Servi.
{In addition, it is interesting to notice that the following rule is derivable:\vspace{2mm}
$
\begin{array}{rl}
(Nec_\Diamond) & \mbox{from } \varphi \to \psi \mbox{ infer } \Diamond \varphi \to \Diamond \psi \\
\end{array}
$ \vspace{2mm}
\noindent Indeed, if $\vdash _{K(\mathbf{G})} \varphi \to \psi$, then $\vdash _{K(\mathbf{G})} \Box(\varphi \to \psi)$ by $Nec$, and by using axiom $(P)$ and modus ponens we get $\vdash _{K(\mathbf{G})} \Diamond \varphi \to \Diamond \psi$.
}
In the next section, we will focus on an extension of $K(\mathbf{G})$ for which we are able to simplify the G\"odel Kripke semantics introduced in this section.
\section{The logic $K45(\mathbf{G})$} \label{sec:K45logic}
\noindent The logic $K45(\mathbf{G})$ is defined by adding to $K(\mathbf{G})$ the following axioms: \vspace{0.1cm}
\begin{tabular}{ll@{\qquad}ll}
\hspace{2mm} $(4_\Box)$& $\Box \varphi \to \Box \Box \varphi$ & \hspace{2.3cm} $(4_\Diamond)$& $\Diamond\Diamond \varphi \to \Diamond \varphi$ \\
\hspace{2mm} $(5_\Box)$& $\Diamond\Box \varphi \to \Box \varphi$ &\hspace{2.3cm} $(5_\Diamond)$& $\Diamond \varphi \to \Box \Diamond \varphi$ \\
\end{tabular}
\vspace{0.2cm}
\noindent We define the logic $KD45(\mathbf{G})$ by adding to $K45(\mathbf{G})$ the following axiom:
\begin{tabular}{rl@{\qquad}ll}
\hspace{2mm} $(D)$ & \blue{$\Diamond \top$} & \ \\
\end{tabular}
\vspace{0.2cm}
\noindent and $KT45(\mathbf{G})$ if we add the following axioms to $K45(\mathbf{G})$: \vspace{0.1cm}
\begin{tabular}{ll@{\qquad}lll}
\hspace{2mm} $(T_\Box)$& $\Box \varphi \to \varphi$ & \hspace{2,8cm} $(T_\Diamond)$& $\varphi \to \Diamond \varphi$ \\
\end{tabular}
\vspace{0.2cm}
On the other hand, the following is a theorem of $KD45(\mathbf{G})$: \vspace{2mm}
$
\begin{array}{rl}
(D') & \Box \varphi \to \Diamond \varphi \\
\end{array}
$ \vspace{2mm}
\noindent Indeed, we can replace $\top$ by $\varphi \to \varphi$ in Axiom (D) and then use T5. In fact, (D) and (D') are interderivable in $K(\mathbf{G})$, that is, we have both $ \Diamond \top \vdash_{K(\mathbf{G})}
\Box \varphi \to \Diamond \varphi $ and $\Box \varphi \to \Diamond \varphi \vdash_{K(\mathbf{G})} \Diamond \top$, the latter holding by first instantiating (D') with $\varphi = \top$ and getting $\Box \top \to \Diamond \top$, and then using that $\Box \top$ is an axiom of $K(\mathbf{G})$.
Next we show that in $K45(\mathbf{G})$ some iterated modalities can be simplified. This is in accordance with our intended simplified semantics for $K45(\mathbf{G})$ that will be formally introduced in the next section.
\begin{proposition} \label{simplif}
The logic $K45(\mathbf{G})$ proves the following schemes:
\vspace{0.2cm}
$
\begin{array}{llll}
\hspace{2mm} ({F}_{\Diamond\Box}) & \Diamond\Box\top \leftrightarrow \Diamond \top &
\hspace{2cm} {(G45)} &
{(\ensuremath{\Box} \varphi \to \ensuremath{\Diamond} \psi) \to \ensuremath{\Box} (\ensuremath{\Box} \varphi \to \ensuremath{\Diamond} \psi)} \\
\hspace{2mm} ({U}_\Diamond) & \Diamond\Diamond \varphi \leftrightarrow
\Diamond \varphi & \hspace{2cm} ({U}_\Box) & \square \square\varphi \leftrightarrow \square\varphi \\
\hspace{2mm} (T4_\Box) & (\Box \varphi \to \Diamond \Box\varphi) \vee \Box \varphi & \hspace{2cm} (T4_\Diamond) & (\Box\Diamond \varphi \to \Diamond \varphi) \vee \Box\Diamond \varphi \\
\hspace{2mm} (Sk_\Diamond) & (\Diamond \top \to \Diamond \varphi) \leftrightarrow \Box\Diamond \varphi & \hspace{2cm} (T4'_\Diamond) & (\Box\Diamond \varphi \to \Diamond \varphi) \vee (\Diamond \top \to \Diamond \varphi)
\end{array}$\\
\end{proposition}
\begin{proof2}
$({F}_{\Diamond\Box})$ is an immediate consequence of $F_\Box$ and $Nec_\Diamond$.
As for schemes $U_\Diamond$ and $U_\Box$, axioms $4_\Box$ and $4_\Diamond$ give one direction of them. The opposite directions, together with the rest of schemes, are obtained as follows:\\
\vspace{-2em}
\[
\begin{array}{ll}
\text{Proof $(U_\Diamond)$:} & \text{Proof $(U_\Box)$:} \\[0.4em]
\begin{array}{@{}ll@{}}
\Diamond \varphi \to \Box \Diamond \varphi & \text{ axiom } (5_\Diamond) \\
\Box(\varphi \to \Diamond\varphi) & \text{ by } MP \text{ and } (FS2) \\
\Diamond \varphi \to \Diamond \Diamond \varphi & \text{ by } MP \text{ and } (P)
\end{array}
&
\begin{array}{@{}ll@{}}
\Diamond \Box\varphi \to \Box \varphi & \text{ axiom } (5_\Box) \\
\Box(\Box \varphi \to \varphi) & \text{ by } MP \text{ and } (FS2) \\
\Box \Box \varphi \to \Box \varphi & \text{ by } MP \text{ and } (K_\Box)
\end{array}
\\[2em]
\mbox{Proof $(T4_\Box)$:} & \mbox{Proof $(T4_\Diamond)$:} \\
\begin{array}{@{}ll@{}}
\Box\Box \varphi \to \Diamond \Box \varphi \vee \Box\Box\varphi & \text{(T4) with } \Box\varphi \\
\Box \varphi \to \Diamond \Box \varphi \vee \Box\varphi & \text{by } (U_\Box)
\end{array}
&
\begin{array}{@{}ll@{}}
(\Box\Diamond \varphi \to \Diamond \Diamond \varphi) \vee \Box((\Diamond \varphi \to \Diamond \varphi) \to \Diamond \varphi) & \text{(T4)} \\
(\Box\Diamond \varphi \to \Diamond \Diamond \varphi) \vee \Box(\top \to \Diamond \varphi) & \text{equiv}
\end{array}
\\[2em]
\text{Proof $(G45)$:} & \\[0.4em]
\multicolumn{2}{l}{
\begin{array}{@{}ll@{}}
(\ensuremath{\Box} \varphi \to \ensuremath{\Diamond} \psi) \to (\ensuremath{\Box} \varphi \to \ensuremath{\Box} \ensuremath{\Diamond} \psi) & \text{by applying } (5_\ensuremath{\Diamond}) \\
(\ensuremath{\Box} \varphi \to \ensuremath{\Box} \ensuremath{\Diamond} \psi) \to (\ensuremath{\Diamond} \ensuremath{\Box} \varphi \to \ensuremath{\Box} \ensuremath{\Diamond} \psi) & \text{by applying } (5_\ensuremath{\Box}) \\
(\ensuremath{\Diamond} \ensuremath{\Box} \varphi \to \ensuremath{\Box} \ensuremath{\Diamond} \psi) \to \ensuremath{\Box}(\ensuremath{\Box} \varphi \to \ensuremath{\Diamond} \psi) & \text{by } (FS2)
\end{array}
}
\\[2em]
\text{Proof $(Sk_\Diamond)$:} & \\[0.4em]
\multicolumn{2}{l}{
\begin{array}{@{}ll@{}}
\Box(\top \to \Diamond \varphi) \to (\Diamond \top \to \Diamond\Diamond \varphi) & \text{by (P)} \\
\Box \Diamond \varphi \to (\Diamond \top \to \Diamond \varphi) & \text{by $(U_\Diamond)$ and equivalences} \\
(\Diamond \top \to \Box \Diamond \varphi) \to \Box (\top \to \Diamond \varphi) & \text{by } (FS2) \\
(\Diamond \top \to \Diamond \varphi) \to \Box \Diamond \varphi & \text{by } (5_\Diamond)
\end{array}
}
\\[2.4em]
\text{Proof $(T4'_\Diamond)$: using } (T4_\ensuremath{\Diamond}) \text{ and } (Sk_\Diamond)
\end{array}
\]
\vspace{-3mm}
\end{proof2}
{Moreover, if we restrict ourselves to formulas starting with $\Box$ or $\Diamond$ we can prove the following property.
\begin{lemma} \label{rmk} Let $X = \{\Box \theta, \Diamond \theta : \theta \in \mathcal{L}_{\Box,\Diamond}\}$. If $\varphi \in X$ then the schemas
$$ \varphi \to \Box\varphi \hspace{5mm} \mbox{ and } \hspace{5mm} \Diamond \varphi \to \varphi$$
are theorems of $K45(\mathbf{G})$.
\end{lemma}
\begin{proof} We check that $K45(\mathbf{G})$ derives $\varphi \to \Box\varphi$ if $\varphi \in X$, the other schema is similar. In fact, we have two cases: if $\varphi = \Box \psi$, then $\varphi \to \Box\varphi$ is in fact one direction of $(U_\Box)$; if $\varphi = \Diamond \psi$, then this is axiom $(5_\Diamond)$.
\end{proof}
}
From now on we will use $ThK45(\mathbf{G})$ to denote the set of theorems of $K45(\mathbf{G})$. We close this section with the following observation:
\begin{lemma} \label{reduction2} If $T$ is a finite set of formulas, $T\vdash _{K45(\mathbf{G})} \varphi $ iff $\vdash _{K45(\mathbf{G})} T^{\land} \to \varphi $, where $T^{\land} = \bigwedge\{ \psi \mid \psi \in T \}$.
\end{lemma}
\begin{proof2} By Lemma \ref{reduction}, we have $T\vdash _{K45(\mathbf{G})} \varphi $ iff $T\cup ThK45(\mathbf{G}) \vdash _{G} \varphi $. By the deduction theorem of G\"odel logic, the latter is equivalent to $ ThK45(\mathbf{G}) \vdash _{G} T^{\land} \to \varphi $, and by (i) again, this is equivalent to $\vdash _{K45(\mathbf{G})} T^{\land} \to \varphi $.
\end{proof2}
\begin{remark} \label{emptyzone} It is worth noting that for any valuation $v$ such that $v(ThK45(\mathbf{G})) = 1$ there is no formula $\varphi$ such that $v(\Diamond \top) < v(\nabla \varphi) < 1$ with $\nabla \in
\{\Box , \Diamond\}$ because both formulae $(\Box \varphi \to \Diamond \varphi) \vee \Box \varphi$ and $\Diamond \varphi \to \Diamond \top$ are in
$ThK45(\mathbf{G})$.
\end{remark}
\section{Simplified Kripke semantics and completeness} \label{sec:possibilitysem}
In this section we will show that $K45(\mathbf{G})$ is complete with respect to a class of simplified Kripke G\"odel frames.
\begin{definition}
\label{simplgodelframe} A {\em possibilistic Kripke frame}, or $\Pi$-frame, is a
structure $\langle W,\pi \rangle $ where $W$ \emph{is a non-empty
set of worlds}, and $\pi:W \rightarrow [0,1]$ is a {\em possibility distribution} over $W$.
A {\em possibilistic G\"odel Kripke model}, \emph{$\Pi G$-model} for short, is a triple $ \langle W, \pi, e\rangle$ where $ \langle W, \pi\rangle $ is a $\Pi$-frame and $e: W \times Var \to [0, 1]$ provides a G\"odel evaluation of variables in each world.
For each $w \in W$, $e(w, -)$ extends to arbitrary formulas in the usual way for the propositional connectives and for modal operators in the following way:
\medskip
\hspace{3.5cm} $e(w,\Box \varphi ):=\inf_{w'\in W}\{\pi(w') \Rightarrow
e(w',\varphi )\}$
\hspace{3.5cm} $e(w,\Diamond \varphi ):=\sup_{w'\in W}\{\min(\pi(w'), e(w',\varphi ))\}$.
\medskip
\noindent If $\pi$ is normalised, i.e. if $\sup_{w \in W} \pi(w) = 1$, then $ \langle W, \pi, e\rangle$ will be called a {\em normalised} possibilistic G\"odel Kripke model, or \emph{$\Pi^* G$-model}. A formula $\varphi$ is \emph{valid} in a $\Pi G$-model $\langle W, \pi, e \rangle$ if $e(w,\varphi) = 1$ for all $w \in W$.
We will denote by $\Pi \cal G$ the class of possibilistic G\"odel Kripke models, and by $\Pi^* \cal G$ the subclass of normalised models. We say that a formula $\varphi$ is \emph{$\Pi \cal G$-valid}, written $\models_{\Pi\mathcal{G}} \varphi$, if $\varphi$ is valid in all possibilistic G\"odel Kripke models, and \emph{$\Pi^* \cal G$-valid}, written $\models_{\Pi^*\mathcal{G}} \varphi$, if $\varphi$ is valid in all normalised possibilistic G\"odel Kripke model.
\end{definition}
Observe that the evaluation of formulas beginning with a modal operator is in fact independent from the current world. Also note that the $e(\cdot,\Box \varphi )$ and $e(\cdot,\Diamond \varphi )$ are in fact generalisations for G\"odel logic propositions of the necessity and possibility degrees of $\varphi$ introduced in Section \ref{intro} for classical propositions, although now they are not dual (with respect to G\"odel negation) any longer.
In the rest of this section we are going to show in detail a weak completeness proof of the logic $K45(\mathbf{G})$ (resp.\ $KD45(\mathbf{G})$) with respect to the class $\Pi\mathcal{G}$ (resp.\ the subclass $\Pi^*\mathcal{G}$) of possibilistic G\"odel Kripke models. In fact one can prove a slightly stronger result, namely completeness for deductions from finite theories.
We start with the case of $K45(\mathbf{G})$. In what follows, for any formula $\varphi$ we denote by $Sub(\varphi) \subseteq \mathcal{L}_{\square
\Diamond }$
the set of subformulas of $\varphi$ and containing
the formula $\bot $. Moreover, let $X:=\{\square \theta ,\Diamond \theta :\theta \in \mathcal{L}_{\square
\Diamond }\}$ be the set of formulas in $\mathcal{L}_{\square \Diamond }$
beginning with a modal operator; then $\mathcal{L}_{\square \Diamond }(Var)=%
\mathcal{L(}Var\cup X)$. That is, any formula in $\mathcal{L}_{\square
\Diamond }(Var)$ may be seen as a propositional G\"odel formula built from the extended set
of propositional variables $Var\cup X$. In addition, for a given formula
$\varphi$,
let $\sim_\varphi$ be the equivalence relation on $[0,1]^{Var\cup X} \times \lbrack 0,1]^{Var\cup X}$
defined as follows:
$$ u \sim_\varphi w \mbox{ iff } \forall \psi \in Sub(\varphi): u(\Box \psi) = w(\Box \psi) \mbox{ and } u(\Diamond \psi) = w(\Diamond \psi) .$$
Now, assume that a formula $\varphi$ is not a theorem of $K45(\mathbf{G})$. Hence by completeness of G\"odel calculus and Lemma \ref{reduction}, there exists a G\"odel valuation $v$ such that $v(ThK45(\mathbf{G}))=1$ and $v(\varphi)<1$.
With the valuation $v$ now fixed, we follow the usual canonical model construction, defining a canonical $\Pi G$-model $\ensuremath{\frm{M}}^v_{\varphi}$ in which we will show $\varphi$ is not valid.
The \emph{canonical model} \emph{\ }$%
\ensuremath{\frm{M}}^v_{\varphi}= \langle W^v_{\varphi},\pi^v_{\varphi},e^v_{\varphi} \rangle$ is defined
as follows
\begin{itemize}
\item $W^v_{\varphi}$ is the set $\{u \in \lbrack 0,1]^{Var\cup X} \mid u
\sim_\varphi v \mbox{ and } u(ThK45(\mathbf{G}))=1 \}$.
\item $\pi^v_{\varphi}(u)=\inf_{\psi \in Sub(\varphi)}\{\min(v(\Box \psi)\rightarrow u(\psi ), u(\psi )\rightarrow v(\Diamond \psi ))\}.$
\item $e^v_{\varphi}(u,p)=u(p)$ for any $p\in Var$
\end{itemize}
In this context, we call the elements of $\Delta_\varphi := \{\square \theta ,\Diamond \theta :\theta \in Sub(\varphi) \}$ the {\em fixed points} of the Canonical Model.
Note that having $\nu(ThK45(\mathbf{G}))=1$ does not guarantee that $\nu$ belongs to the canonical model because it may not take the appropriated values for the fixed points, i.e.\ it may be that $u \not\sim_\varphi \nu$. However, the next lemma shows how to, under certain conditions, transform such an evaluation into another belonging to the canonical model.
\begin{lemma}\label{normalization} Let $u \in W^v_\varphi$ and let $\nu: {Var\cup X} \to [0,1]$ be a G\"odel valuation. Define $\delta = \max \{ u(\lambda) : \nu(\lambda) < 1 \mbox{ and } \lambda \in \Delta_\varphi\}$ and $\beta = \min \{ u(\lambda) : \nu(\lambda)= 1 \mbox{ and } \lambda \in \Delta_\varphi\}$. If $\nu$ satisfies the following conditions:
\begin{description}
\item [a.] $\nu(ThK45(\mathbf{G}))=1$.
\item [b.] For all $\lambda \in X$, we have $u(\lambda) > \delta \Rightarrow \nu(\lambda)=1$.
\item [c.] For any $\psi, \phi \in \{\lambda : u(\lambda) \leq \delta \mbox{ and } \lambda \in X \}$: $u(\psi) < u(\phi)$ implies $\nu(\psi) < \nu(\phi)$.
\item [d.] {For any $\psi, \phi \in \Delta_\varphi: u(\psi) \leq u(\phi)$ implies $\nu(\psi) \leq \nu(\phi)$.}
\end{description}
then, there exists a G\"odel valuation $w \in W^v_\varphi$ such that for any $\varepsilon > 0$ with $\delta+\varepsilon < \beta$, and for any formulae $\psi$ and $\phi$, the following conditions hold:
\begin{enumerate}
\item $\nu(\psi) = 1 $ implies $ w(\psi) \geq \delta+\varepsilon$.
\item $\nu(\psi) < 1 $ implies $ w(\psi) < \delta+\varepsilon$.
\item $1 \neq \nu(\psi) \leq \nu(\phi)$ implies $w(\psi) \leq w(\phi)$.
\item $\nu(\psi) < \nu(\phi)$ implies $ w(\psi) < w(\phi)$.
\end{enumerate}
\end{lemma}
\begin{proof2}
First of all,
notice that if $\nu$ satisfies the conditions {\bf c} and {\bf d}, then necessarily $\delta < \beta$. Indeed, suppose $\delta \geq \beta$. Then there are at least two formulas $\theta_1$ and $\theta_2$ in $\Delta_\varphi$ such that $\nu(\theta_1) <1$, $\nu(\theta_2)=1$ and $\delta \geq u(\theta_1) \geq u(\theta_2) \geq \beta$. Note that the case $u(\theta_1) > u(\theta_2)$ is not possible because it would violate condition {\bf c}, and the case $u(\theta_1) = u(\theta_2)$ is also imposible because it would then violate condition {\bf d}.
Let $B= \{ \nu(\lambda) : \lambda \in \Delta_\varphi , \nu(\lambda) < 1\} \cup \{0 \} = \{ b_0 = 0 < b_1 < \ldots < b_N \}$. Obviously, $b_N < 1$. For each $0 \leq i \leq N$, pick $\lambda_i \in \Delta_\varphi$ such that $\nu(\lambda_i) = b_i$.
Define now a continuous strictly function $g:[0,1]\mapsto \lbrack 0,\delta+\varepsilon) \cup \{1\}$ such
that \medskip
$g(1)=1$
$g(b_i)= u(\lambda_i)$ for every $0 \leq i \leq N$
$g[(b_N, 1)]= (\delta, \delta+\varepsilon)$
\medskip
\noindent Notice that $\delta = g(b_N)$. In addition, define another continuous strictly increasing function $h:[0,1] \mapsto [\delta+\varepsilon, 1]$ such that
\medskip
$h(0)=\delta+\varepsilon$
$h[(0,\beta)] =(\delta+\varepsilon, \beta)$
$h(x) = x$, for $x \in [\beta, 1]$
\medskip
\noindent Then we define the valuation $w: Var \cup X \to [0,1]$ as follows:
\medskip
$w(p) = \left \{
\begin{array}{ll}
g(\nu(p)), & \mbox{if } \nu(p) < 1,\\
h(u(p)), & \mbox{if } \nu(p) = 1.
\end{array}
\right .
$
\medskip
Next step is to prove that $w$ satisfies the required Properties 1--4. And we are going to prove it by induction on {the maximum of the complexity of both formulae $\psi, \phi$.}
First, we consider the base case when both $\psi$ and $\phi$ belong to $Var \cup X$. Then
\begin{enumerate}
\item By definition of $w$, $\nu(\psi) = 1$ implies $w(\psi)= h(u(\psi)) \geq \delta+\varepsilon$. The same happens for $\phi$.
\item By definition of $w$, $\nu(\psi) < 1$ implies $w(\psi)= g(\nu(\psi)) < \delta+\varepsilon$. The same happens for $\phi$.
\item Suppose $1 \neq \nu(\psi) \leq \nu(\phi)$. Since $\nu(\psi) < 1$, by definition of $w$, we have $w(\psi)=g(\nu(\psi)) < \delta+\varepsilon$. Now, we analyse different cases.
If $\nu(\phi) < 1$ then $w(\phi) = g(\nu(\phi)) \geq g(\nu(\psi)) =
w(\psi)$ because $g$ is strictly increasing. Otherwise, If $\nu(\phi) =1$ then $w(\phi)= h(u(\phi)) \geq \delta +\varepsilon > w(\psi)=g(\nu(\psi))$.
\item Suppose $\nu(\psi) < \nu(\phi)$. In this case, the proof is similar to the previous one. If $\nu(\psi) < 1$ and $\nu(\phi)=1$ then $w(\phi)= h(u(\phi)) \geq \delta +\varepsilon > g(\nu(\psi)) = w(\psi)$. On
the contrary, if both $\nu(\psi)<1$ and $\nu(\phi)<1$ then $w(\phi) = g(\nu(\phi)) > g(\nu(\psi)) = w(\psi)$ because $g$ is strictly increasing.
This case will become important when we try to prove that $w$ satisfies the axioms of $K45(\mathbf{G})$. \label{extraproperty}
\end{enumerate}
In fact, in the case both formulae $\psi$ and $\phi$ belong to $Var \cup
X$ we can prove that two further conditions hold:
\begin{enumerate}
\item[5.] If $\nu(\psi)=\nu(\phi)=1$ and $u(\psi) \leq u(\phi)$ then
$w(\psi) \leq w(\phi)$. \label{extraproperty1}
\item[6.] If $\nu(\psi)=\nu(\phi)=1$ and $u(\psi) < u(\phi)$ then $w(\psi) < w(\phi)$. \label{extraproperty2}
\end{enumerate}
Indeed, by definition of $w$, we have $w(\psi)= h(u(\psi))$ and $w(\phi)= h(u(\phi))$, and since $h$ is strictly increasing, $u(\psi) \leq u(\phi)$ (resp. $u(\psi) < u(\phi)$) implies $h(u(\psi)) \leq h(u(\phi))$ (resp. $h(u(\psi)) < h(u(\phi))$), as desired. These properties will become important when proving that $w$ satisfies the axioms of $K45(\mathbf{G})$.
Now, we consider the inductive step. \medskip
\noindent {\bf Claim}: {\em Induction hypothesis (IH): if Properties 1--4 are satisfied by formulas with complexity at most $n$, then these properties also hold for formulae with complexity at most $n+1$.} \medskip
\noindent The proof of this claim is quite technical and it is moved to the appendix.
Finally, we prove that $w \in W^v_{\varphi}$, i.e. we prove that both $w \sim_\varphi v$ and $w(ThK45({\cal G})) = 1$.
\begin{itemize}
\item[(i)] By definition, $w \sim_\varphi v$ iff $w(\Box\phi) = v(\Box\phi)$ and $w(\Diamond\phi) = v(\Diamond\phi)$ for all $\phi \in Sub(\varphi)$. Let $A \in \{ \Box \phi, \Diamond \phi\} \subseteq \Delta_\varphi$.
By definition of $w$, $w(A) = g(\nu(A))$ if $\nu(A) < 1$, and $w(A) =
h(u(A))$, otherwise. Since $A \in \Delta_\varphi$, if $\nu(A) < 1$ then, by construction ($u(A) \leq \delta$), we have $g(\nu(A)) = u(A) \leq \delta$, and if $\nu(A) =1$, again by construction ($(u(A) \geq \beta$), we have $ h(u(A)) = u(A)$. So, we have proved that $w \sim_\varphi u$, but by hypothesis $u \sim_\varphi v$ as well, thus $w \sim_\varphi v$ as well.
\item[(ii)] We first prove that all axioms of $K45(\mathbf{G})$ are evaluated to 1 by $w$. The axioms of $\mathcal{G}$ are evaluated to $1$ by any G\"odel valuation. As for the specific axioms of $K45(\mathbf{G})$ (i.e.\ axioms $
(4_\Box), (4_\Diamond), (5_\Box), (5_\Diamond)$), observe that all these
axioms are of the form $\phi \to \psi$ for some $\phi, \psi \in X$. Then it is enough to prove that for any $ \varphi, \psi \in X$, if $ \nu(\phi \to \psi)=1$ then $w(\phi \to \psi)=1$.
Indeed, if $ \nu(\phi \to \psi) = 1$, then $ \nu(\phi) \leq \nu(\psi)$ and we have two possibilities:
\begin{itemize}
\item if $1 \neq \nu(\phi) \leq \nu(\psi)$, then by Property 3, we have $w(\phi) \leq w(\psi)$.
\item if $\nu(\phi) = \nu(\psi) = 1$, then by definition of $w$, $w(\phi) = h(u(\phi))$ and $w(\psi) = h(u(\psi))$. But since $u \in W^v_\varphi$,
we have $u(\phi \to \psi) = 1$, because we assume that $\phi \to \psi \in K45(\mathbf{G})$ with $\phi, \psi \in X$. Thus $u(\phi) \leq u(\psi)$ and, by Property 5, we know $w(\phi) = h(u(\phi)) \leq h(u(\psi)) = w(\psi)$.
\end{itemize}
Finally, let us consider the axioms of $K(\mathbf{G})$ (Section 2). We have to prove that $w(\chi)=1$ for each of such axioms $\chi$, knowing by assumption that $\nu(\chi) = 1$. The case of $(F_\Box)$ is easy since $(F_\Box) \in X$, and then,
by definition, $w(F_\Box)=h(u(F_\Box))=1$.
As for axiom $(K_\Box)$, note that, using propositional reasoning, it can be equivalently expressed first as $(\Box(\phi \to \psi) \land \Box \phi) \to \Box \psi$, and then as $(\Box(\phi \to \psi) \to \Box \psi) \lor (\Box \phi \to \Box \psi)$. Therefore, if $\nu(K_\Box) = 1$, it means either $\nu(\Box(\phi \to \psi) \to \Box \psi) = 1$ or $\nu(\Box \phi \to \Box \psi)
= 1$. But these two cases concern implications of formulas from $X$, and hence we are in the same situation above as with the specific axioms of
$K45(\mathbf{G})$, and hence using the same reasoning, we can conclude that either $w(\Box(\phi \to \psi) \to \Box \psi) = 1$ or $w(\Box \phi \to \Box \psi) = 1$.
The case of axioms $(K_\Diamond)$ and $(P)$ can be dealt in an analogous way, as they can be written as a disjunction of implications of formulas from $X$. So the case left is that of axiom $(FS2)$, $(\Diamond \phi \to \Box\psi) \to \Box(\phi \to \psi)$.
By hypothesis, we know $\nu(FS2) = 1$, that is $\nu(\Diamond \phi) \Rightarrow \nu(\Box \psi) \leq \nu(\Box(\phi \to \psi))$. According to that,
we have to prove $w(FS2) = 1$ is as well.
Notice that, since $\psi \to (\phi \to \psi)$ is a tautology, $u(\Box(\psi \to (\phi \to \psi)))=\nu(\Box(\psi \to (\phi \to \psi)))=1$. Then, by definition, $w(\Box(\psi \to (\phi \to \psi)))=1$ as well and because axiom (K) is valid for $w$, we have $w(\Box \psi \to \Box (\phi \to \psi)) = 1$, i.e. $w(\Box \psi) \leq w(\Box (\phi \to \psi))$.\\
Now, we consider the following cases:
\begin{itemize}
\item Case $u(\Diamond \phi) \leq u(\Box \psi)$. Then $u(\Box(\phi \to \psi))=1$ which implies $\nu(\Box(\phi \to \psi))=1$. Hence, by construction, $w(\Box(\phi \to \psi))= h(u(\Box(\phi \to \psi)))=1$.
\item Case $u(\Diamond \phi) > u(\Box \psi)$. Here we distinguish three subcases:
\begin{itemize}
\item $u(\Box \psi) \leq \delta < u(\Diamond \phi)$: by Conditions {\bf b} and {\bf c}, $\nu(\Diamond \phi) = 1$ and $\nu(\Box \psi) < 1$, respectively. Therefore, by Property 4, $w(\Box \psi) < w(\Diamond \phi)$ and
thus $w(\Diamond \varphi \to \Box \psi) = w(\Box \psi) \leq w(\Box(\phi
\to \psi))$ and hence $w(FS2) = 1$.
\item $\delta < u(\Box \psi) < u(\Diamond \phi)$: by Condition {\bf b}, $1 = \nu(\Diamond \phi) = \nu(\Box \psi)\leq \nu(\Box(\phi \to \psi))$. Thus, by construction, $w(FS2) = (h(u(\Diamond \varphi)) \Rightarrow
h(u(\Box \psi))) \Rightarrow h(u(\Box(\phi \to \psi)) = 1$.
\item $u(\Box \psi) < u(\Diamond \phi) \leq \delta$: by Condition {\bf c}, $\nu(\Box \psi) < \nu(\Diamond \phi)$, and by Property 4, $w(\Box \psi) < w(\Diamond \phi)$, and hence $w(\Diamond \varphi \to \Box \psi) = w(\Box \psi) \leq w(\Box(\phi \to \psi))$, and hence $w(FS2) = 1$.
\end{itemize}
\end{itemize}
\end{itemize}
{So far, we have proved that $w(\phi) = 1$ if $\phi$ is a $K45(\mathbf{G})$ axiom. To conclude the proof, we need to extend this result to the rest of the formulas in $ThK45({\mathbf{G}})$. Recall that a formula $\phi$ in the set $ThK45({\mathbf{G}})$ is either an axiom of $K45(\mathbf{G})$ or $\phi$ can be proved from the axioms and rules of inference $K45(\mathbf{G})$. In the latter case, there is non-empty finite sequence of formulae $\phi_1, \phi_2,\dots, \phi_n$, with $\phi_n = \psi$, and each $\phi_i$ is either an axiom or it has been obtained by application of an inference rule on some of the preceding formulae $\phi_1, \dots, \phi_{i-1}$. We proceed by induction on the length $n$ of the sequence. The unique interesting case is when $\phi_n$ is obtained by applying the (Nec) rule to some $\phi_j$ with $1 \leq j < n$, i.e. $\phi_n = \Box \phi_j$. It is clear that $\phi_j \in ThK45({\mathbf{G}})$, and by the inductive hypothesis, $w(\phi_j) =1$. Of course, $\Box \phi_j \in ThK45({\mathbf{G}})$ as well, and thus $\nu(\Box\xi) =1$ too. But since $\Box \phi_j \in X$, by definition of $w$, we hace $w(\psi) = w(\Box \phi_j) = h(u(\Box \phi_j)) = 1$. }
This finishes the proof of the lemma.
\end{proof2}
\noindent Completeness will follow from the next truth-lemma.
\begin{lemma} [Truth-lemma] \label{equation-joint} $e^v_{\varphi}(u,\psi )=u(\psi )$ for any $\psi \in
Sub(\varphi)$ and any $u\in W^{v}_\varphi$.
\end{lemma}
\begin{proof2}
For simplicity, we will write $W^c$, $\pi^c$ and $e^c$ for $W^v_{\varphi}, \pi^v_{\varphi}$ and $e^v_{\varphi}$, respectively. We prove the identity by
induction on the complexity of the formulas in $Sub(\varphi)$, considered
now as elements
of $\mathcal{L}_{\square \Diamond }(Var)$. For $\bot $ and the propositional
variables in $Sub(\varphi)$ the equation holds by definition. The only non trivial
inductive steps are:\ $e^{c}(u,\Box \psi)=u(\Box \psi)$ and $
e^{c}(u,\Diamond \psi)=u(\Diamond \psi)$ for $\Box \psi,\Diamond
\psi \in Sub(\varphi).$ By the inductive hypothesis we may assume that $
e^{c}(u^{\prime },\psi)=u^{\prime }(\psi)$ for every $u^{\prime }\in
W^c;$ thus we must prove
\begin{eqnarray}
\inf_{u^{\prime }\in W^c}\{\pi^c(u^{\prime })\Rightarrow u^{\prime }(\psi
)\}=u(\Box \psi ) \label{box} \\
\sup_{u^{\prime }\in W^c}\{\min(\pi^c(u^{\prime }), u^{\prime }(\psi
)) \}=u(\Diamond \psi ) \label{Diam}
\end{eqnarray}
By definition, $\pi^c(u^{\prime })\leq (v(\Box \psi )\Rightarrow u^{\prime
}(\psi ))$ and $\pi^c(u^{\prime })\leq (u^{\prime }(\psi )\Rightarrow
v(\Diamond \psi ))$ for any $\psi \in Sub(\varphi)$ and $u^{\prime }\in W;$
therefore, $u(\Box \psi) = v(\Box \psi) \leq (\pi^c(u^{\prime })\Rightarrow u^{\prime
}(\psi))$ and $\min(\pi^c(u^{\prime }), u^{\prime }(\psi)) \leq
v(\Diamond \psi) = u(\Diamond \psi).$ Taking infimum over $u^{\prime }$ in the first
inequality and the supremum in the second we get
\begin{equation*}
u(\Box \psi )\leq \inf_{u^{\prime }\in W^c}\{\pi^c(u^{\prime })\Rightarrow
u^{\prime }(\psi )\}, \ \sup_{u^{\prime }\in W^c}\{\min(\pi^c(u^{\prime
}), u^{\prime }(\psi ))\}\leq u(\Diamond \psi ).
\end{equation*}
Hence, if $u(\Box \psi )=1$ and $u(\Diamond \psi )=0$ we directly obtain (\ref
{box}) and (\ref{Diam}), respectively. Therefore, it only remains\ to prove
the next two claims for $\Box \psi ,\Diamond \psi \in Sub(\varphi)$.\medskip
\noindent \textbf{Claim 1}. \emph{If $u(\Box \psi )=\alpha <1$ then, for every $\varepsilon > 0$, there exists a valuation $w\in W^c$ such that $\pi^c(w) > w(\psi)$ and $w(\psi ) < \alpha + \varepsilon$, and thus $(\pi^c(w) \Rightarrow w(\psi ))< \alpha + \varepsilon$}.\medskip
\noindent \textbf{Claim 2. }\emph{If $u(\Diamond \psi )=\alpha >0$ then, for any small enough $\varepsilon >0,$ there exists a valuation $w'\in W^c$ such that $\min(w'(\psi ), \pi^c(w')) \geq \alpha -\varepsilon$}.
\medskip \noindent
The proofs of these two claims are rather involved and they can be found in the appendix.
\end{proof2}
\begin{theorem} [Weak completeness $K45(\mathbf{G})$]
\label{WeakCompleteness} For any formula $\varphi $ in $\mathcal{L}_{\square \Diamond }$%
, $\models _{\Pi\mathcal{G}}\varphi $ iff $\vdash _{K45(\mathbf{G})}\varphi .$
\end{theorem}
\begin{proof2} One direction is soundness, and it is easy to check that the axioms are valid in the class $\Pi\mathcal{G}$ of models. As for the other direction, assume $\not\vdash _{K45(\mathbf{G})}\varphi .$ Then $ThK45(\mathbf{G})\not\vdash _G\varphi $ by Lemma \ref{reduction}, and thus
there is, by Proposition~\cite[Proposition 3.1]{CaiRod2010}, a G\"odel valuation $v:Var\cup X\rightarrow [0,1]$ such that $v(\varphi )< v(ThK45(\mathbf{G}))=1.$ Then $v$ is a world of the canonical model $\ensuremath{\frm{M}}_\varphi^v$ and by Lemma~\ref{equation-joint}, $e_{\varphi}^v(v,\varphi )=v(\varphi )<1.$ Thus $\not\models_{\Pi \cal G}\varphi$.
\end{proof2}
In addition, it is also easy to generalize last proof for getting completeness for deductions from finite theories as it is shown by the next theorem:
\begin{theorem} [Finite strong completeness $K45(\mathbf{G})$]
\label{JointCompleteness} For any
finite theory $T$ and formula $\varphi $ in $\mathcal{L}_{\square \Diamond }$, we have: $T\models _{\Pi\mathcal{G}}\varphi $ iff $T\vdash _{K45(\mathbf{G})}\varphi $.
\end{theorem}
\begin{proof2} The proof is an easy adaptation of the one of weak completeness. We will only mention the main differences. If $T \not\vdash_{K45(\mathbf{G})} \varphi$, by completeness of G\"odel logic and Lemma \ref{reduction2}, there exists a G\"odel valuation $v$ such that $ v(ThK45(\mathbf{G}))=1$, $v(T) = 1$ and $v(\varphi)<1$. Now, in order to build a canonical model, we need to take into account not only $v$ and $\varphi$ but also $T$. To do that we have to follow the very same steps as before but replacing everywhere the set $Sub(\varphi)$ of subformulas of $\varphi$ by the larger set $Sub(T, \varphi)$ of subformulas of $T
\cup \{\varphi\}$, i.e. $Sub(T, \varphi) = \bigcup_{\psi \in T \cup \{\varphi\}} Sub(\psi)$. Let us denote by $\ensuremath{\frm{M}}^v_{T, \varphi} = \langle W^c, \pi^c, e^c \rangle$ the canonical model built accordingly, where $v \in W^c$. Note that
there is no need of any modification in neither Lemma \ref{normalization} nor the Truth Lemma \ref{equation-joint} (except for replacing $Sub(\varphi)$ by $Sub(T, \varphi)$ in its statement). Then the theorem follows by observing that
Lemma \ref{equation-joint} guarantees that $e^c(v, \psi) = v(\psi) = 1$ for all $\psi \in T$ while $e^c(v, \varphi) = v(\varphi) < 1$. Therefore, $T \not\models _{\Pi\mathcal{G}}\varphi $.
\end{proof2}
Actually, the proofs for weak and finite strong completeness of $K45(\mathbf{G})$ with respect to the class of simplified possibilistic models $\Pi\mathcal{G}$ easily generalize to completeness of $KD45(\mathbf{G})$, the axiomatic extension $K45(\mathbf{G})$ with axiom $D$, with respect to the class of {\em normalized} possibilistic models $\Pi^*\mathcal{G}$.
\begin{corollary} [Finite strong completeness $KD45(\mathbf{G})$]
For any
finite theory $T$ and formula $\varphi $ in $\mathcal{L}_{\square \Diamond }$, we have: $T\models _{\Pi^*\mathcal{G}}\varphi $ iff $T\vdash _{KD45(\mathbf{G})}\varphi $.
\end{corollary}
\begin{proof}
We only consider the proof of weak completeness, its extension to a proof
of finite strong completeness can be then devised as in Theorem \ref{JointCompleteness}. Indeed, the proof only needs small adaptations to the one
for the case of $K45(\mathbf{G})$. {To start with, for technical reasons that will become clear later, we will actually prove that $\not\vdash _{KD45(\mathbf{G})} \varphi'$ implies $\not\models _{\Pi^*\mathcal{G}} \varphi'$, where $\varphi' = \Diamond \top \to \varphi$. Note that since $\Diamond \top$ is an axiom of $KD45(\mathbf{G})$ and is valid in the class of frames $\Pi^*\mathcal{G}$, the former condition is indeed equivalent to prove that $\not\vdash _{KD45(\mathbf{G})} \varphi$ implies $\not\models _{\Pi^*\mathcal{G}} \varphi$. }
Next, in the definition of the set of worlds $W^v_{\varphi'}$ of the canonical model $\ensuremath{\frm{M}}^v_{\varphi'}$, we need to replace the condition $u(ThK45(\mathbf{G}))=1$ by $u(ThKD45(\mathbf{G}))=1$, i.e. we define
$$W^v_{\varphi'} = \{u \in \lbrack 0,1]^{Var\cup X} \mid u \sim_{\varphi'} v \mbox{ and } u(ThKD45(\mathbf{G}))=1 \},$$
and analogously in the condition $\bf a.$ of Lemma \ref{normalization}. Then in the proof of this lemma (item (ii) after the {\bf Claim}), one has to further check that $w(\Diamond \top) = 1$. Observe that $\Diamond \top$ is provably equivalent to $\square \top \to \Diamond \top$, but this formula is of the form $\phi \to \psi$ for some $\phi, \psi \in X$, and hence it falls under the cases already considered in the proof. Thus Lemma \ref{normalization} holds, and the same happens with Lemma \ref{equation-joint}, that holds as well without any modification. Moreover, Lemma \ref{equation-joint} allows us to
prove that the canonical model belongs in fact to the class $\Pi^*\mathcal{G}$ of normalized possibilistic models. Indeed, by \eqref{Diam} it follows that
$$\sup_{u^{\prime }\in W^v_{\varphi'}}\{\min(\pi^c(u^{\prime }), u^{\prime }(\psi)) \}=u(\Diamond \psi )$$ { for every $\Diamond \psi \in Sub(\varphi')$, and since $\Diamond \top \in Sub(\varphi')$ and $u \in W^v_{\varphi'}$ (and hence $u(\Diamond \top) = 1$), we finally have $$ 1 = u(\Diamond \top) = \sup_{u^{\prime }\in W^v_{\varphi'}}\{\min(\pi^c(u^{\prime }), u^{\prime }(\top)) \}= \sup_{u^{\prime }\in W^v_{\varphi'}} \pi^c(u^{\prime }), $$
in other words, $\pi^c$ is normalized and thus $\ensuremath{\frm{M}}^v_{\varphi'} \in \Pi^*\mathcal{G}$.
In summary, we have found a model $\ensuremath{\frm{M}}^v_{\varphi'} = \langle W^v_{\varphi'}, \pi^c, e_{\varphi'}^v \rangle \in \Pi^*\mathcal{G}$ and a world $v \in W^v_{\varphi'}$ such that $e_{\varphi'}^v(v, \varphi') = e_{\varphi'}^v(v, \varphi) < 1$, and therefore $\not\models _{\Pi^*\mathcal{G}} \varphi$. }
\end{proof}
{We would like to finish this section by noticing that the same kind of proof for weak and finite strong completeness for $K45$ can also be easily adapted for the logic $KT45(\mathbf{G})$, that is in fact equivalent to $KT5(\mathbf{G})$, since Axiom (4) is derivable in $KT5(\mathbf{G})$. For that, we need to adapt some details of our original proof. First, we can take the same definition of the canonical model but considering $ThKT5(\mathbf{G})$ instead of $ThK45(\mathbf{G})$. Next, we follow with Lemma \ref{normalization} where we change condition {\bf a.} by $\nu(ThKT5(\mathbf{G})) = 1$. It is easy to check that the same proof goes through except that we need to prove now that $w(\Box \varphi \to \varphi) = w(\varphi \to \Diamond\varphi)=1$. But again this is easy to be verified,
and the rest of the proof is working well. Then, we are are able to prove the following:
\begin{theorem} [Weak completeness $KT45(\mathbf{G})$]
\label{WeakCompletenessT45} For any formula $\varphi $ in $\mathcal{L}_{\square \Diamond }$:
$$\models _{\Pi^5\mathcal{G}}\varphi \mbox{ iff } \vdash _{KT5(\mathbf{G})}\varphi $$
\end{theorem}
\noindent where $\Pi^5\mathcal{G}$ is the class of possibilistic frames $(W, \pi)$ that validate axioms $(T_\Box)$ and $(T_\Diamond)$. However this result is not new, since it turns out that $(W, \pi)$ validates $(T_\Box)$ and $(T_\Diamond)$ iff $\pi$ is such that $\pi(w) = 1$ for every $w \in W$. In other words, $\Pi^5\mathcal{G}$ is in fact the class of universal models, the simplified semantics for $S5(\mathbf G)$, a result that is well-known in the literature \cite{CMRT19}.
}
\section{Decidability} \label{sec:decidability}
So far, we have shown that $Val( \Pi{\mathcal{ G}}) = ThK45(\mathbf{G})$, i.e.\ the set of valid formulas in $\Pi\mathcal {G}$, the class of all $\Pi G$-frames, coincides with the set of theorems of the logic $K45(\mathbf{G})$, or in other words, the logic $K45(\mathbf{G})$ is sound and complete with respect to $\Pi G$-frames. It is natural to ask whether the logic
$K45(\mathbf{G})$ is decidable.
Unfortunately,
it is easy to check that under the possibilistic semantics the logic $K45(\mathbf{G})$ does not satisfy the finite model property. Indeed, consider the formula
$$\square \lnot \lnot p \rightarrow \lnot \lnot \square p $$
where $p$ is a propositional variable. This formula is not valid in the model $\ensuremath{\frm{M}} = \langle \mathbb{N},\pi, e\rangle,$ where for all $n \in \mathbb{N}$,
\begin{equation*}
\pi (n)=1 \mbox{ and } e(p,n)=\tfrac{1}{n+1}.
\end{equation*}%
Then, for all $n \in \mathbb{N}$ we have:
$$e(\lnot \lnot p,n)=(\frac{1}{n+1} \Rightarrow 0) \Rightarrow 0=1,$$
$$e(\Box p,n )=\inf_{n \in \mathbb{N}} \{ \pi(n) \Rightarrow e(p, n)\} = \inf_{n\in \mathbb{N}}\{1{\Rightarrow} \frac{1}{n+1}\}=0,$$
$$e(\Box \neg\neg p,n )=\inf_{n \in \mathbb{N}} \{ \pi(n) \Rightarrow e(\neg\neg p, n)\} = \inf_{n\in \mathbb{N}}\{1{\Rightarrow} 1 \}=1,$$
and hence $e(\square \lnot \lnot p \rightarrow \lnot \lnot \square p, n)
= 0$.
However, $\Box \lnot \lnot p \rightarrow \lnot \lnot \Box p $ is valid in
any $\Pi G$-model $\langle W,\pi,e\rangle $ where $W$ is finite.
Nevertheless, in \cite{CMRR2013}, an alternative semantics was provided for $K(\mathbf{G})$ which admits the finite model property. In this section, we are going to adapt
that semantics in order to obtain an equivalent class of $\Pi G$-models. For that, we are going to use the same strategy used in \cite{CMRR2013}, i.e. by limiting the truth-values of modal formulae to a finite number of possibilities. According to this idea, we propose the following adaptation of the original semantics given in \cite{CMRR2013}.
\begin{definition}
A ${\Pi GF}$-model is a quadruple $\ensuremath{\frm{M}}= \langle W, \pi, T, e \rangle$, where $ \langle W, \pi, e \rangle
$ is a $\Pi G$-model and $T \in \mathcal{P}_{<\omega}(\left[ 0, 1 \right])$ is a finite set of
truth values satisfying $\{0,1\} \subseteq T \subseteq [0,1]$.
The valuation $e$ is extended to formulas using the same clauses for non-modal connectives as for $\Pi G$-models, and using the following revised clauses for modal connectives:
\begin{eqnarray*}
e(\ensuremath{\Box} \ensuremath{\varphi}, x) & = & \max \left \{r \in T: r \leq \inf_{y \in W} \{\pi(y) {\Rightarrow} \ e(\ensuremath{\varphi}, y)\} \right \} \\
e(\ensuremath{\Diamond} \ensuremath{\varphi}, x) & = & \min \left \{r \in T: r \geq \sup_{y \in W} \{\min(\pi(y),e(\ensuremath{\varphi}, y))\} \right \}.
\end{eqnarray*}
A formula $\varphi$ is said to be \emph{valid} in $\ensuremath{\frm{M}}$ if $e(\varphi,x) = 1$ for all $x \in W$. We will denote by $\Pi \mathcal{GF}$ the class of all $\Pi GF$-models.
\end{definition}
Notice that now the formula $\ensuremath{\varphi} = \ensuremath{\Box} \neg \neg p \to \neg \neg \ensuremath{\Box} p$ has a finite
$\Pi GF$-counter-model. Indeed, consider the $\Pi GF$-model $ \ensuremath{\frm{M}}_0 =
\langle W,\pi,T,e \rangle$ with $W = \{a\}$, $\pi(a) = 1 $, $T
= \{0,1\}$, and such that $e(p,a) = \frac{1}{2}$. Then we have:
\begin{itemize}
\item $e(\neg p, a) = 0$, $\pi(a) \Rightarrow e(\neg \neg p, a) = 1$,
and so $e(\ensuremath{\Box} \neg \neg p, a) = 1$.
\item Further, $e(\ensuremath{\Box} p, a) = 0$ (since $\pi(a) {\Rightarrow} \ e(p, a) = \frac{1}{2}$,
and $0$ is the next smallest element of $ T$);
\item Hence, $e(\neg \ensuremath{\Box} p, a) = 1$ and $e(\neg \neg \ensuremath{\Box} p, a) = 0$.
\item Therefore, $1 = e(\ensuremath{\Box} \neg \neg p, a) > e( \neg \neg \ensuremath{\Box} p, a) =
0$ and $\ensuremath{\Box} \neg \neg p \to \neg \neg \ensuremath{\Box} p$ is not valid in $\frm{M}_0$.
\end{itemize}
Next, we are going to prove that both semantics characterize the same logic. First we need the following lemmas to prove the main result.
\begin{lemma} \label{extended}
Let $\ensuremath{\frm{M}} = \langle W, \pi, T, e \rangle$ be an $\Pi GF$-model. Given an
order-embedding $h \colon [0,1] \to [0,1]$ satisfying $h(0) =0$ , $h(1)
= 1$, and for any $t \in T : h(t) = t$, consider $\wh{\ensuremath{\frm{M}}} = \langle \wh{W}, \wh{\pi}, \wh{T}, \wh{e} \rangle $, with $\wh{W} = W_{\ensuremath{\frm{M}}}$, $\wh{\pi}( x) = h(\pi (x))$,
$\wh{T}(x) = T(x)$, and $\wh{e}(p,x) = h(e(p,x))$ for all $x \in W$
and $p \in \rm{Var}$. Then, for all $\ensuremath{\varphi} \in \ensuremath{\rm{Fml}_{\bo\di}}$ and $x \in W$: $$\wh{e}(\ensuremath{\varphi},x) = h(e(\ensuremath{\varphi},x))$$
\end{lemma}
\begin{proof2} It is a special case of part (c) of Lemma 1 in~\cite{CMRR2013}.
\end{proof2}
Now, we provide the key construction of a $\Pi G$-model taking the same truth values for formulae as a given $\Pi GF$-model.
\begin{lemma}\label{rtlD45}
For any $\Pi GF$-model $\ensuremath{\frm{M}} = \langle W,\pi, T, e \rangle$, there is a $\Pi G$-model $\wh{\ensuremath{\frm{M}}}= \langle \wh{W},\wh{\pi},\wh{e} \rangle$ with $W \subseteq \wh{W}$,
such that $\wh{e}(\ensuremath{\varphi}, x) = e(\ensuremath{\varphi},x)$ for all $\ensuremath{\varphi} \in \ensuremath{\rm{Fml}_{\bo\di}}$ and $x \in W_\ensuremath{\frm{M}}$.
\end{lemma}
\begin{proof2}
We proceed similarly to the proof of Lemma 4 in \cite{CMRR2013}. Given a $\Pi GF$-model $\ensuremath{\frm{M}} = \langle W,\pi, T,e \rangle$, we construct its associated $\Pi G$-model $\wh{\ensuremath{\frm{M}}}$
directly by taking infinitely many copies of $\ensuremath{\frm{M}}$. Consider $T = \{\alpha_1, \ldots, \alpha_n\}$ with $0 = \alpha_1 < \ldots < \alpha_n = 1$ and, using Lemma~\ref{extended}, define a family of order-embeddings $\{h_k\}_{k \in \mathbb{Z}^+}$ from $[0,1]$ into $[0,1$]
satisfying $h_k (0) = 0$ and $h_k(1) = 1$, such that
\begin{center}
\begin{tabular}{rcll}
$h_k(\alpha_i)$ & $=$ & $\alpha_i$ & for all $i \leq n-1$ and $k \in \mathbb{Z}^+$\\
$h_k[(\alpha_i, \alpha_{i+1})]$ & $=$ & $(\alpha_i, \min(\alpha_i + \frac{1}{k}, \alpha_{i+1}))$ & for all $i \leq
n-1$ and even $k \in \mathbb{Z}^+$\\
$h_k[(\alpha_i, \alpha_{i+1})]$ & $=$ & $(\max(\alpha_{i}, \alpha_{i+1}- \frac{1}{k}), \alpha_{i+1})$ & for all $i
\leq n-1$ and odd $k \in \mathbb{Z}^+$.\\
\end{tabular}
\end{center}
For all $k \in \mathbb{Z}^+$, we define a $\Pi G$-model
$\wh{\ensuremath{\frm{M}}}_k = \langle \wh{W}_k, \wh{\pi}_k, \wh{e}_k \rangle$ such that
each $\wh{W}_k$ is a copy of $W$ with distinct
worlds, $ \wh{\pi}_k = h_k(\pi)$ and $\wh{e}_k(\ensuremath{\varphi},x^k) = h_k(e(\ensuremath{\varphi},x))$ for each copy $x^k$ of $x \in W$ and $\ensuremath{\varphi} \in \ensuremath{\rm{Fml}_{\bo\di}}$. We also let
$\wh{W}_0 = W$, $ \wh{\pi}_0 = \pi$ and $\wh{e}_0 = e$. Then $\wh{\ensuremath{\frm{M}}} = \langle \wh{W},\wh{\pi},\wh{e} \rangle$ where
\[
\wh{W} = \bigcup_{k \in \mathbb{N}} \wh{W}_k \qquad {\rm and} \ \text{
for } \wh{x} \in \wh{W}_k : \qquad \wh{\pi}(\wh{x}) = \wh{\pi}_k(\wh{x}) \qquad ;\qquad
\wh{e}(p,\wh{x}) = \wh{e}_k(p,\wh{x}) .
\]
Now, It then suffices to prove that $\wh{e}(\ensuremath{\varphi},x) = e(\ensuremath{\varphi},x)$ for all $\ensuremath{\varphi} \in \ensuremath{\rm{Fml}_{\bo\di}}$ and $x \in W$, proceeding by an induction on $\ell(\ensuremath{\varphi})$. The base case $\ell(\ensuremath{\varphi}) = 1$ follows directly from the definition of $\wh{e}$. For the inductive step, the cases for the
non-modal connectives follow easily using the induction hypothesis. Let us just consider the case $\ensuremath{\varphi} = \ensuremath{\Box} \ensuremath{\psi}$, the case $\ensuremath{\varphi} = \ensuremath{\Diamond} \ensuremath{\psi}$ being very similar. There are two possibilities. Suppose first that
\[
e(\ensuremath{\Box} \ensuremath{\psi},x) = \max\{r \in T : r \leq \inf \{\pi(y) \Rightarrow e(\ensuremath{\psi}, y): y \in W\}\} = 1.
\]
That means for all $y \in W \ : \ \pi(y) \leq e(\ensuremath{\psi}, y)$ and hence, for
all $k \in \mathbb{Z}^+$ and $\wh{y} \in \wh{W}$: $h_k(\pi(y)) = \wh{\pi}_k ({y}^k) = \wh{\pi} (\wh{y}) \leq h_k (e(\ensuremath{\psi}, y)) = \wh{e}_k(\ensuremath{\psi},y^k)= \wh{e}(\ensuremath{\psi},\wh{y})$.
It follows that
\[
\wh{e}(\ensuremath{\Box} \ensuremath{\psi}, \wh{x}) = \inf\{\wh{\pi}(\wh{y}) \Rightarrow \wh{e}(\ensuremath{\psi},\wh{y}): \wh{y} \in \wh{W}\} = 1 = e(\ensuremath{\Box} \ensuremath{\psi},x).
\]
Now suppose that $e(\ensuremath{\Box} \ensuremath{\psi},x) = \alpha_i < 1$ for some $i \leq m-1$. Then $\pi(z) \Rightarrow e(\ensuremath{\psi},z) \geq \alpha_i$ for all $z \in W$, and thus, $(\star)$, $\wh{\pi} (\wh{z}) \Rightarrow \wh{e}(\ensuremath{\psi},\wh{z}) \geq \alpha_{i}$ for all $\wh{z}
\in \wh{W}$, by construction using the order-embeddings $\{h_k\}_{k \in \mathbb{Z}^+}$.
There are two subcases. First, suppose that there is at least one $y \in W$ such that $\pi(y) \Rightarrow e(\ensuremath{\psi},y) =
\alpha_i$; call it $y_0$. This means that $\pi(y_0) > e(\ensuremath{\psi},y_0) = \alpha_i$ and for all $k \in \mathbb{Z}^+$, $\wh{e}(
\ensuremath{\psi}, \wh{y}_0) = \wh{e}_k(\ensuremath{\psi},{y}_0^k) = h_k(e(\ensuremath{\psi},y_0))
= h_k(\alpha_i)= \alpha_i$. Since $\pi(y_0) > \alpha_i$, also for all
$k \in \mathbb{Z}^+$, $\wh{\pi}(\wh{y}_0) = \wh{\pi}_k({y}_0^k) =
h_k(\pi(y_0)) > \alpha_i = \wh{e}(\ensuremath{\psi}, \wh{y}_0)$, and hence, using $(\star)$,
\[
\wh{e}(\ensuremath{\Box} \ensuremath{\psi},\wh{x})=\inf\{\wh{\pi} (\wh{z}) \Rightarrow \wh{e}(\ensuremath{\psi},\wh{z}): \wh{z} \in \wh{W}\} = \alpha_i = e(\ensuremath{\Box} \ensuremath{\psi}, x).
\]
Now suppose that $\pi(y) \Rightarrow e(\ensuremath{\psi},y) > \alpha_i$ for all $y \in W$. Since $e(\ensuremath{\Box} \ensuremath{\psi},x) = \max\{r \in T
: r \leq \inf \{\pi(y) \Rightarrow e(\ensuremath{\psi}, y): y \in W\}\} = \alpha_i$, there is at least one $y \in W$ such that $\pi(y) \Rightarrow e(\ensuremath{\psi},y) \in (\alpha_i, \alpha_{i+1})$; call it $y_0$. Then, by construction, for any $\epsilon > 0$ there is a $k \in \mathbb{Z}^+$ such that $\wh{\pi} ({y}^k_0) \Rightarrow \wh{e}(\ensuremath{\psi}, {y}^k_0) \in (\alpha_i, \alpha_i + \epsilon)$.
Using $(\star)$, this ensures that\\
\qquad$\wh{e}(\ensuremath{\Box} \ensuremath{\psi},\wh{x})= \inf\{\wh{\pi}( z) \Rightarrow \wh{e}(\ensuremath{\psi},z): z \in \wh{W}\} = \alpha_i = e(\ensuremath{\Box}\ensuremath{\psi},x)$.
\end{proof2}
As an immediate consequence of Lemma \ref{rtlD45}, we have the next corollary.
\begin{corollary} \label{Decib1} $Val( \Pi{\mathcal{ G}}) \subseteq Val(
\Pi{\mathcal{ GF}})$.
\end{corollary}
The next lemma paves the way to prove in Theorem \ref{Decib2} that the logic $K45(\mathbf{G})$ has the finite model property and offers a bound
on the complexity. We call a set $\Sigma \subseteq \ensuremath{\rm{Fml}_{\bo\di}}$ a \emph{fragment} if it is closed under subformulas.
\begin{lemma} \label{ltrD45}
{Let $\mathrm{\Sigma} \subseteq \ensuremath{\rm{Fml}_{\bo\di}}$ be a finite fragment.} Then, for any $\Pi G$-model $\ensuremath{\frm{M}}$, there is a finite $\Pi GF$-model $\wh{\ensuremath{\frm{M}}}$ with $\wh{W}
\subseteq W$, such that
$\wh{e}(\ensuremath{\varphi}, x) = e(\ensuremath{\varphi},x)$ for all $\ensuremath{\varphi} \in \mathrm{\Sigma}$ and $x \in \wh{W}$. Moreover, $|\wh{W}|
+|\wh{T}| \leq 2|\mathrm{\Sigma}|$.
\end{lemma}
\begin{proof2}
{Let $\mathrm{\Sigma}\subseteq \ensuremath{\rm{Fml}_{\bo\di}}$ be a finite fragment}, $\ensuremath{\frm{M}} = \langle W, \pi, e \rangle$ a $\Pi G$-model. First, define $\mathrm{\Sigma}_\ensuremath{\Box}$ as the set of
all box-formulas in $\mathrm{\Sigma}$, $\mathrm{\Sigma}_\ensuremath{\Diamond}$ as the set of all diamond-formulas in $\mathrm{\Sigma}$, and $\mathrm{\Sigma}_{
\rm{Var}}$ as the set of all variables in $\mathrm{\Sigma}$. Let us also define $e_{x}[\mathrm{\Delta}] = \{e(\ensuremath{\varphi},x): \ensuremath{\varphi} \in \mathrm{\Delta}\}$ for any $x
\in W$ and $\mathrm{\Delta} \subseteq \ensuremath{\rm{Fml}_{\bo\di}}$. In addition, let $e_{x}[\mathrm{\Sigma}_\ensuremath{\Box} \cup \mathrm{\Sigma}_\ensuremath{\Diamond}] \cup \{0,1 \}=\{\alpha_1, \ldots, \alpha_n\}$ with $0 = \alpha_1 < \ldots < \alpha_n = 1$.
Next, we choose a finite number of $y \in W$. For each $\ensuremath{\Box} \ensuremath{\psi} \in \mathrm{\Sigma}_\ensuremath{\Box}$ such that $e(\ensuremath{\Box} \ensuremath{\psi},x) = \alpha_i < 1$, choose a $y = y_{\ensuremath{\Box} \ensuremath{\psi}}
\in W$ such that $e (\ensuremath{\psi},y_{\ensuremath{\Box} \ensuremath{\psi}}) < \alpha_{i+1}$, and for each $\ensuremath{\Diamond} \ensuremath{\psi} \in \mathrm{\Sigma} _\ensuremath{\Diamond}$, such that $e(\ensuremath{\Diamond} \ensuremath{\psi},x) =
\alpha_i > 0$, choose a $y = y_{\ensuremath{\Diamond} \ensuremath{\psi}} \in W$ such that $e( \ensuremath{\psi},y_{\ensuremath{\Diamond} \ensuremath{\psi}}) > \alpha_{i-1}$. Then let $\wh{W} = \{x
\} \cup \{y_{\ensuremath{\Box} \ensuremath{\psi}} \in W: \ensuremath{\Box} \ensuremath{\psi} \in \mathrm{\Sigma}_\ensuremath{\Box}\} \cup \{y_{\ensuremath{\Diamond} \ensuremath{\psi}} \in W:
\ensuremath{\Diamond} \ensuremath{\psi} \in \mathrm{\Sigma}_\ensuremath{\Diamond}\}$. Clearly $\wh{W}
\subseteq W$ is finite. We define $\wh{\ensuremath{\frm{M}}} = \langle \wh{W}, \wh{\pi}, \wh{T}, \wh{e} \rangle$ where $\wh{T} = e_{x}[\mathrm{\Sigma}_\ensuremath{\Box}
\cup \mathrm{\Sigma}_\ensuremath{\Diamond}] \cup \{0,1\}$, and both $\wh{\pi}$ and $\wh{e}$ are equal to $\pi$ and $e$ restricted to $\wh{W}$, respectively. It then follows by
induction on $\ell(\ensuremath{\varphi})$ that $\wh{e}(\ensuremath{\varphi}, x) = e(\ensuremath{\varphi},x)$ for all $x \in \wh{W}$ and $\ensuremath{\varphi} \in\mathrm{\Sigma}$. The base case follows from the definition of $\wh{e}$. For the inductive step, let $\ensuremath{\varphi} \in \mathrm{\Sigma}$ be of the form $\ensuremath{\varphi} = \ensuremath{\Box} \ensuremath{\psi}$ (the non-modal cases follow directly, using the induction hypothesis). We need to consider the same two cases it was considered in previous
Lemma \ref{rtlD45}. First, note that always it is true that:
$$ \inf\{ \pi(y) \Rightarrow e(\ensuremath{\psi},y): {y \in W} \} \leq \inf\{ \wh{\pi}(\wh{y}) \Rightarrow \wh{e}(\ensuremath{\psi},\wh{y}): \wh{y} \in \wh{W}\}$$
because of $\wh{W} \subseteq W$. Now suppose that $1 = e(\ensuremath{\Box} \ensuremath{\psi}, x)$. Then, because $1 \in \wh{T}$,
\begin{align*}
1 = e(\ensuremath{\Box} \ensuremath{\psi}, x) &= \inf\{ \pi(y) \Rightarrow e(\ensuremath{\psi},y): {y \in W} \} \\
&\leq \inf\{ \wh{\pi}(\wh{y}) \Rightarrow \wh{e}(\ensuremath{\psi},\wh{y}): \wh{y} \in \wh{W}\} \\
&\leq \max\{ r \in \wh{T} : r \leq \inf\{ \wh{\pi}(\wh{y}) \Rightarrow \wh{e}(\ensuremath{\psi},\wh{y}): \wh{y} \in \wh{W}\}\} \\
&= \wh{e}(\ensuremath{\Box} \ensuremath{\psi}, x).
\end{align*}
For the second case, $e(\ensuremath{\Box} \ensuremath{\psi}, x) = \inf\{ \pi(y) \Rightarrow e(\ensuremath{\psi},y) : y \in W\} = \alpha_i < 1$ for some $i \in \{1, \ldots, n-1\}$. According to our choise, there exists a $ y_{\ensuremath{\Box} \ensuremath{\psi}} \in \wh{W}$ such that $ \alpha_i \leq \wh{\pi}(y_{\ensuremath{\Box} \ensuremath{\psi}}) \Rightarrow \wh{e}(\ensuremath{\psi}, y_{\ensuremath{\Box} \ensuremath{\psi}}) < \alpha_{i+1}$. Hence, $$ \alpha_i \leq \inf\{ \wh{\pi}(\wh{y}) \Rightarrow \wh{e}(\ensuremath{\psi},\wh{y}): \wh{y} \in \wh{W}\}\} \leq \wh{\pi}(y_{\ensuremath{\Box} \ensuremath{\psi}}) \Rightarrow \wh{e}(\ensuremath{\psi}, y_{\ensuremath{\Box} \ensuremath{\psi}}) < \alpha_{i+1}$$
Thus
$$ \wh{e}(\ensuremath{\Box} \ensuremath{\psi}, x)= \max\{ r \in \wh{T} : r \leq \inf\{ \wh{\pi}(\wh{y}) \Rightarrow \wh{e}(\ensuremath{\psi},\wh{y}): \wh{y} \in \wh{W}\}\} = \alpha_i$$
The diamond-case case follows similarly to the box-case and is therefore omitted.
Finally, we note that by the construction of $\wh{\ensuremath{\frm{M}}}$, $|\wh{W}| \leq |\mathrm{\Sigma}_\ensuremath{\Box} \cup \mathrm{\Sigma}_\ensuremath{\Diamond}| + 1 \leq |\mathrm{\Sigma}|$ and $|\wh{T}| \leq |\mathrm{\Sigma}_\ensuremath{\Box} \cup \mathrm{\Sigma}
_\ensuremath{\Diamond}| + 2 \leq |\mathrm{\Sigma}|$, and therefore $|\wh{W}| + |\wh{T}| \leq 2 |\mathrm{\Sigma}|$.
\end{proof2}
As a direct consequence of the above lemma we have the converse inclusion of Corollary \ref{Decib1}.
\begin{corollary} $Val( \Pi{\mathcal{ GF}}) \subseteq Val( \Pi{\mathcal{
G}}) $.
\end{corollary}
Finally, we can state the main result of this section.
\begin{theorem}[main] \label{Decib2} For each $\ensuremath{\varphi} \in \ensuremath{\rm{Fml}_{\bo\di}}$:
$\mdl{\Pi \mathcal{G}} \ensuremath{\varphi}$ iff $\ensuremath{\varphi}$ is valid in all {(finite)} $\Pi GF$-models $\ensuremath{\frm{M}} = \langle W,\pi,T,e \rangle$ satisfying
$|W|+|T| \leq 2(\ell(\ensuremath{\varphi})+2)$.
\end{theorem}
{ As a direct consequence we get the decidability of $K45({\bf G})$.
\begin{corollary} The logic $K45({\bf G})$ is decidable.
\end{corollary}
Similarly, one can prove that the logic $KD45({\bf G})$ is decidable as well.
\begin{corollary} The logic $KD45({\bf G})$ is decidable.
\end{corollary}
}
\section{Conclusions} \label{sec:Disc-concl}
In this paper we have studied the logic over G\"odel fuzzy logic arising from many-valued G\"odel Kripke models with possibilistic semantics, and have shown that it actually corresponds to a simplified semantics for the
logic $K45({\bf G})$, the extension of Caicedo and Rodriguez's bi-modal G\"odel logic with many-valued versions of the well-known modal axioms $4$ and $5$. We have also considered the extension with the axiom $D$, the logic $KD45({\bf G})$, and have shown to be captured by normalised possibilistic G\"odel Kripke models. In this way, we have obtained many-valued G\"odel generalizations of the results reported by Pietruszczak in \cite{Pietrus09} about simplified semantics for several classical modal logics. We have also shown the decidability of those logics.
It is worth noticing that the truth-value of a formula $\Diamond \varphi$
in a possibilistic Kripke model is indeed a proper generalization of the possibility measure of $\varphi$ when $\varphi$ is a classical proposition, however the semantics of $\Box \varphi$ is not. This is due to the fact that the negation in G\"odel logic is not involutive. Therefore, a first open problem we leave for further research is to consider to extension of the logic $K45({\bf G})$ with an involutive negation and investigate its possibilistic semantics. {Finally, we would like to study the connection between our simplified possiblistic semantics and the pseudomonadic algebras proposed in \cite{BCR2019}}.
\paragraph{Acknowledgments} Rodriguez acknowledges partial support of Argentinean projects: PICT-2019-2019-00882, UBA-CyT-20020190100021BA and PIP 112-2015-0100412 CO.
Tuyt is supported by Swiss National Science Foundation (SNF) grant 200021\_184693. Esteva and Godo acknowledge partial support by the Spanish project
PID2019-111544GB-C21.
|
1,108,101,566,743 | arxiv | \section{Introduction}
Model predictive control (MPC) is a powerful control technique in dynamic systems, power inverters, and dynamic reference trajectories, as shown in \cite{berberich2022linear}. It can predict future system behavior from the current system state by solving an optimal control problem at each sampling instant. As discussed in \cite{baillieul2019perceptual}, the evolution of a control theory for systems that exhibit the kind of resilience seen in neurobiology involves input and output signals generated by the collective activity of vast numbers of simple elements. This is one of the primary motivations for studying is what we have called overcomplete control systems. Our goal is to understand systems whose overall functionality depends on discrete sets of inputs that operate collectively in groups. Quantized input systems have been studied for many decades, such as in \cite{lewis1963optimum} and \cite{liu1990optimal}, but this work was primarily concerned with the digital round-off errors. The work reported in this paper is focused on developing quantization methods merged with MPC using simple inputs inspired by neurobiology to realize emulation.
The quantized system is described by
\begin{equation}
x(k+1)=Ax(k)+Bu(k),
\label{equation:qs}
\end{equation}
where $x\in\mathbb{R}^n$ is the system state, $u\in\mathbb{U}=\{-1,0,1\}^m$ is the set of possible quantized inputs and $m\gg n$.\\
In \cite{baillieul2021neuromimetic}, we formulated two types of emulation problems and only focused on the restricted problem. The general one is further discussed in this paper. Let $H$ be an $n\times n$ Hurwitz matrix and consider solutions to the linear ordinary differential equation (ODE)
\begin{equation}
\dot x = Hx,\ \ x(0)=x_0.\label{equation:LTI}
\end{equation}
The goal of this general emulation problem is to find piecewise constant quantized inputs with sampling interval $h>0$ such that the resulting trajectories of (\ref{equation:qs}) with initial state $x(0)=x_0$ approximate the continuous system $(\ref{equation:LTI})$. Here, we assume the sampling time to be one unit and solve the emulation problem that finds a partition of the state space $\{U_i\,:\, \cup\ U_i = \mathbb{R}^n;\ \ U_i^o\cap U_j^o=\emptyset;\ U_i^o={\rm interior}\ U_i\}$ and a selection rule depending on the current state $x(k)$ for assigning values of the input at the $k$-th time step to be $u(k)\in{\cal U}=\{-1,0,1\}^m$, so that for each $x\in U_i$, $Ax + Bu(k)$ is as close as possible to $Hx$ according to an appropriate metric like the direction angle and distance as studied in \cite{baillieul2021neuromimetic}.
In the present paper, we show how a Lyapunov-like objective function can be used to formulate
a quantized system model predictive control (QS-MPC) theory along the line pursued in \cite{xu2022steady}. The problem of using a quantized system to optimally emulate systems that are
continuous in both time and state variable is considered. Sufficient conditions guaranteeing the
asymptotic stability of solutions to the optimal emulation problem are established. The
computational complexity of the integer optimization problem is addressed by applying tools
from machine learning to an appropriate least squares reformulation.
\section{MPC for Quantized Systems}
We know that the existence of a control Lyapunov function provides sufficient conditions for the existence of a controller that ensures asymptotical stability for a discrete-time system. Therefore, establishing the MPC's stability can be approached by finding a candidate-Lyapunov function as its cost function. In general, the cost function contains the terminal cost and stage cost with the form \[J(x,U)=p(x_N)+\sum_{i=0}^{N-1}q(x_i,u_i),\] where we have a prediction horizon $N$ and the predicted input sequence $U=\{u_0,\dots,u_{N-1}\}$. The commonly used cost functions $p$ and $q$ are quadratic in the states and control inputs. Then, the QS-MPC approach can be formulated as solving the following optimization problem:
\begin{equation}
\label{MPC_optimal}
\begin{split}
\min\limits_{u_{0|k},...,u_{N-1|k} }&\quad J\left(x(k),U(k)\right)=|x_{N|k} -x_{ref}(k,N)|^2_P\\
&\qquad\ \ \ \ \ \ +\sum_{n=0}^{N-1}|x_{n|k}-x_{ref}(k,n)|^2_Q+ |u_{n|k}|^2_R \\
s.t. &\quad u_{n|k}\in\left\{-1,\ 0,\ 1\right\}^m,\\
&\quad x_{n+1|k}=Ax_{n|k}+Bu_{n|k},\\
&\quad x_{ref}(k,n+1)=e^Hx_{ref}(k,n),\\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \forall n= {0,1,...,N-1},
\end{split}
\end{equation}
where $P,Q, R$ are positive definite matrices and the function $|x|^2_P=x^TPx$, with similar definitions for $Q$ and $R$. The reference system here is a linear time-invariant system (\ref{equation:LTI}). Matrix $H$ could be thought of as specifying a target behavior of a closed loop version of a general dynamic $\dot x=Ax+Bu$, say of the form $H = A + BK$ where K is a stabilizing gain chosen as in \cite{sun2022neuromimetic}. Following general MPC procedures, the optimal input at time $k$ for the quantized system is $u^*(k)=u^*_{0|k}$, which is the first element in the predicted optimal input sequence $U^*(k)$.
As we generally establish the stabilizing MPC and finite control set MPC, we explore the convergence performance of our QS-MPC when conducting the emulation task. The following theorem gives a sufficient condition for stability.
\begin{theorem}
\label{MPC:thm1}
Consider emulating a stable LTI system (\ref{equation:LTI}) utilizing the QS-MPC system (\ref{equation:qs}). The asymptotically stable solution for QS-MPC is guaranteed when the following conditions are satisfied:\\
(a) $A=e^H$;\\
(b) the positive definite matrices $P, Q$ satisfy $Q-P+A^TPA\prec0$.
\end{theorem}
The proof of Theorem\ref{MPC:thm1} is given in the appendix.$\hfill\blacksquare$
\begin{remark}
\nonumber
Most quantized systems will not asymptotically converge to an equilibrium but only to a neighborhood of it. However, in this case, once the states approach the origin, the optimal input sequence can be zero, and the state transition matrix has all its eigenvalues inside the unit circle. Therefore, the QS-MPC is asymptotically stable.
\end{remark}
From Theorem\ref{MPC:thm1}, we find there are strict conditions on matrices $A,$ $P$, and $Q$ needed to guarantee asymptotic stability in emulating the performance of a stable LTI system. Next, we provide a relaxed condition to achieve the tracking task with the loss of the asymptotical feature. The main idea is to cancel the influence of $A$ matrix with matrix $B$ in our quantized system. It is the fact that $B$ has a large number of columns, activation patterns formed by $Bu$ can have the dominant influence on $x(k+1)$ so that it can track the stable LTI system as approaches towards the equilibrium point.
\begin{lemma}
\label{tracking:lemma1}
When utilizing the quantized system $(\ref{equation:qs})$ to do the LTI emulation task and without loss of generality, assume the equilibrium of the LTI is the origin. If $A$ is Schur stable, it can track the stable LTI system (\ref{equation:LTI}) by solving the optimization problem (\ref{MPC_optimal}) and the trajectories of both systems asymptotically approach the origin.
\end{lemma}
\begin{pf}
Because $A$ is Schur stable, when there is no input (i.e., $u(t)=\mathbf{0}^m$), the quantized system can converge to the origin. By solving the optimization MPC problem (\ref{MPC_optimal}), it can decrease the error $\epsilon$ between these two systems, and the maximum error bound can be expressed as
\begin{equation}
\begin{split}
\epsilon_{max}\le&\max\{\max\{|A^tx(k)-e^{Ht}x_{ref}(k)|\}\},\\
&\quad\quad\quad\quad \ \ \ \ t\in\{0,1,\dots,N-1\},k\in\mathbb{N}
\end{split}
\end{equation}
where $\max\{|A^tx(k)-e^{Ht}x_{ref}(k)|\}$ represents the maximum error between two systems from time $k$ to the following predicted time $k+N$. Since the dynamic functions of these two systems are Lipschitz, the trajectories are continuous, and error value $\epsilon$ between two systems is also continuous. Because $A$ is Schur stable and $H$ is Hurwitz, when $k\to\infty$, $A^tx(k)\to 0$ and $e^{Ht}x_{ref}(k)\to0$ which implies $\epsilon\to0$. Therefore, we can conclude there exists a constant can be the upper bound of $\epsilon_{max}$.$\hfill\blacksquare$
\end{pf}
\begin{lemma}
\label{tracking:lemma2}
If there always exists a quantized input $\hat u$ to ensure $||Ax(k)+B\hat u||-||x(k)||\le0,\forall k$, the quantized system can track the stable LTI system (\ref{equation:LTI}) by solving the optimization problem (\ref{MPC_optimal}) and eventually converge to the set $\mathscr{X}=Br(0)$.
\end{lemma}
Since if there exists a direction $\hat u(k)$ guiding the system to the origin at any time $k$, even though the quantized system selects other $u^*(k)$ to minimize the objective function, it can finally reach the origin as LTI approaches $\mathbf{0}$. The rigorous proof is omitted here.
From the above discussion, though solving problem (\ref{MPC_optimal}) can have satisfactory emulating performance as shown in the simulation in Section \ref{simulation}, it is a quadratic integer programming, which is NP-hard. Therefore, obtaining the optimal input sequence $U^*(k)$ is time-consuming, especially when the receding horizon $N$ becomes larger. In the following sections, we will show how to avoid this difficulty at the cost of some loss of accuracy.
\section{Efficient Method to Solve Integer Programming}
As stated in previous sections, solving integer programming is a computational challenge, and the direct rounding method may lead to suboptimal solutions, which can influence the emulating performance. In this section, we adapt the sphere decoding algorithm introduced in \cite{fincke1985improved} and utilized in \cite{hassibi2005sphere} and \cite{geyer2014multistep} based on the branch-and-bound method. Compared with the exhaustive enumeration method, the sphere decoding algorithm shrinks the size of the candidate control sequences by pruning the branch to improve efficiency.
\subsection{Integer Least Squares Problem Formulation}
To apply the sphere decoding algorithm, the problem $(\ref{MPC_optimal})$ needs to be rewritten in the vector form and constructed as an integer least squares problem. Define $ X_k=\begin{pmatrix}
x(k|k)\\
x(k+1|k)\\
\vdots \\
x(k+N|k)
\end{pmatrix}\in\mathbb{R}^{(N+1)n}$, $U_k=\begin{pmatrix}
u(k|k)\\
u(k+1|k)\\
\vdots \\
u(k+N-1|k)
\end{pmatrix}\in\{-1,0,1\}^{(N-1)m}$ and $R_k=\begin{pmatrix}
x_{ref}(k,0)\\
x_{ref}(k,1)\\
\vdots \\
x_{ref}(k,N)
\end{pmatrix}\in\mathbb{R}^{(N+1)n}$. Let $\tilde{A}=\begin{pmatrix}
I\\
A\\
\vdots \\
A^N
\end{pmatrix}$, $\tilde{B}=\begin{pmatrix}
0& 0 & \cdots &0 \\
B& 0& \cdots & 0\\
AB& B& \cdots& 0\\
\vdots&\vdots & \ddots & \vdots\\
A^{N-1}B& A^{N-2}B&\cdots &B
\end{pmatrix}$, $\tilde Q=\begin{pmatrix}
Q& 0 & \cdots &0 \\
0& Q& \cdots &0 \\
\vdots& \vdots & \ddots &0 \\
0& 0 & 0 &P
\end{pmatrix} $ and $\tilde R=\begin{pmatrix}
R& 0 & \cdots &0 \\
0& R& \cdots &0 \\
\vdots& \vdots & \ddots &0 \\
0& 0 & 0 &R
\end{pmatrix}$. Then, $\tilde H=\tilde B^T\tilde Q\tilde B+\tilde R$ is positive definite and symmetric. The reconstructed problem $(\ref{MPC_optimal})$ is
\begin{equation}
\label{integer_least_H}
\begin{split}
\min\limits_{U_k}\quad& ||U_k+\tilde H^{-1}(\tilde Ax(k)-R_k)||_{\tilde H}^2\\
s.t. \quad &u_{n|k}\in\left\{-1,\ 0,\ 1\right\}^m,\ \ \ \ \ \ \forall n= {0,1,...,N-1}.
\end{split}
\end{equation}
Detailed transformation steps can be found in the appendix. It is observed that the closed form solution for the unconstrained problem $(\ref{integer_least_H})$ is $U_{uncon K}=-\tilde H^{-1}$ $(\tilde Ax(k)-R_k)$. Since matrix $\tilde H$ is symmetric positive definite, there exists an invertible and lower triangular matrix $W\in \mathbb{R}^{Nm\times Nm}$ by Cholesky decomposition to factor $\tilde H=W^TW$ and $\tilde H^{-1}=W^{-1}W^{-T}$. Denote $\bar U_{uncon K}=WU_{uncon K}$. Then the optimization problem $(\ref{integer_least_H})$ has an integer least square objective function
\begin{equation}
\begin{split}
J\quad&=\quad(WU_k-\bar U_{uncon K})^T(WU_k-\bar U_{uncon K})\\
&=\quad ||WU_k-\bar U_{uncon K}||_2^2.
\end{split}
\end{equation}
The various modified sphere decoding algorithms can be applied to solving this problem as was done in \cite{karamanakos2015computationally}. Though the sphere decoding algorithm can find the optimal solution by traversing a tree instead of applying an exhaustive search, the complexity of this algorithm depends on the radius $d$ of the sphere. Here, we choose $d=\min\{||WU_{k_B}-\bar U_{uncon K}||_2^2,||W\hat U_{k}-\bar U_{uncon K}||_2^2 \}$ in the initial, where $U_{k_B}$ is the direct integer lattice round-off of $U_{uncon K}$.It guarantees that the radius is as small as possible and that there is at least one lattice point on or inside the sphere.
\subsection{Learning Optimal Activation Patterns Using Neural Networks}
\label{sub:machine learning}
While using the sphere decoding algorithm, we also collect data according to possible approximate metrics as in \cite{baillieul2021neuromimetic}, where we compared vectors in terms of both magnitude and direction. In terms of this metric, the difference between the quantized system state (\ref{equation:qs}), as determined by the MPC optimal input $u^*_{0|k}$ and the state of the LTI system can be recorded at each step. Applying our MPC approach to optimal emulation from different initial points provides a large amount of data, which we then use to train a neural network. The trained model can choose the activation pattern directly at each step based on the current metric value of the two systems without solving the MPC problem. As illustrated in Section \ref{simulation}, the model based on classification is trained in a simple way but still can efficiently solve the emulation problem.
\section{Suboptimal QS-MPC Algorithm}
In this section, we propose an algorithm that can compute the suboptimal input sequence for each iteration instead of solving the original quadratic integer programming by relaxing the constraints. The relaxed quadratic programming problem is
\begin{equation}
\label{relax MPC_optimal}
\begin{split}
\min\limits_{u_{0|k},...,u_{N-1|k} }&\quad J\left(x(k),U(k)\right)=|x_{N|k} -x_{ref}(k,N)|^2_P\\
&\qquad\qquad+\sum_{n=0}^{N-1}|x_{n|k}-x_{ref}(k,n)|^2_Q+ |u_{n|k}|^2_R \\
s.t. &\quad u_{n|k}\in[-1,1]^m,\\
&\quad x_{n+1|k}=Ax_{n|k}+Bu_{n|k},,\\
&\quad x_{ref}(k,n+1)=e^Hx_{ref}(k,n),\\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \forall n= {0,1,...,N-1}.
\end{split}
\end{equation}
After solving this problem and following \cite{grotschel2012geometric}, Babai estimation is used to round the solution to the nearest integer input sequence of vectors from the set $\{-1,0,1\}^m$ denoted by $\tilde U$. Then the algorithm estimates the suboptimal solution $\tilde U^*$ by choosing from the shifted optimal sequence $\hat U$ and $\tilde U$. \\
\begin{algorithm}
\caption{Suboptimal QS-MPC algorithm for emulating LTI systems}
\label{Suboptimal QS-MPC}
\begin{algorithmic}[1]
\State Apply MPC with horizon predicted window size $N$ to solve problem (\ref{relax MPC_optimal}) and compute the optimal input sequence $U^*(0)= \{u^*_{0|0},...,u^*_{N-1|0}\}$;
\State Estimate $\tilde u^*_{i|0},\forall i\in\{0,1,...,N-1\}$ to be $[-1,0,1]^m$ by Babai estimation method. The suboptimal solution is denoted by $\tilde U^*(0)=\{\tilde u^*_{0|0},...,\tilde u^*_{N-1|0}\}$ and take the first element $\tilde u^*_{0|0}$ in this suboptimal input sequence to the quantized system;
\State Initialize the number of iterations k=0, and input the total number of iterations $K$;
\While{the number of iterations $k < K$}
\State Define the shifted sequence $\hat U(k+1)=\{\tilde u^*_{1|k},...,\tilde u^*_{N-1|k},0\}$;
\State Solve the problem (\ref{relax MPC_optimal}) using MPC to compute the optimal input sequence $U^*(k+1)= \{u^*_{0|k+1},...,u^*_{N-1|k+1}\}$, and estimate it to be $\tilde U(k+1)=\{\tilde u_{0|k+1},...,\tilde u_{N-1|k+1}\}$;
\If {$J(x(k+1),\hat U(k+1))\le J(x(k+1),\tilde U(k+1))$}
\State $\tilde U^*(k+1)=\hat U(k+1)$;
\Else
\State $\tilde U^*(k+1)=\tilde U(k+1)$;
\EndIf
\State Take the first element $\tilde u^*_{0|k+1}$ to the quantized system;
\State $k\gets k+1$;
\EndWhile
\end{algorithmic}
\end{algorithm}
\begin{theorem}
\label{MPC:thm2}
When conditions in Theorem \ref{MPC:thm1} are satisfied, Algorithm \ref{Suboptimal QS-MPC} solves the optimal emulation problem for any stable LTI continuous time system. The optimal quantized emulation asymptotically approaches the LTI system equilibrium.
\end{theorem}
\begin{pf}
To prove the convergence of this algorithm is the same as proving the cost function $J$ is a candidate Lyapunov function, which means $J(x(k),\tilde U^*(k))$ is strictly decreasing until it becomes 0. From the proof of Theorem \ref{MPC:thm1}, we obtain that when conditions (a) and (b) are satisfied, $J(x(k+1),\tilde U(k+1))\le J(x(k),\tilde U^*(k))$. After sufficiently many iterations, this inequality become an equality, after which $ \forall l\ge 0,J(x(k+l),\tilde U(k+l))=0$. Therefore, similarly, from time $k$, the optimal input sequence of optimization problem (\ref{relax MPC_optimal}) is $\bold{0}$, and at this time $J(x(k+1),\tilde U(k+1))=J(x(k+1),\tilde U^*(k+1))$. From the algorithm, we have
\begin{equation}
\begin{split}
&\quad J(x(k+1),\tilde U^*(k+1))\\
=&\quad \min\{J(x(k+1),\tilde U(k+1)),J(x(k+1),\hat U(k+1))\}\\
\le&\quad J(x(k+1),\tilde U(k+1))\\
\le&\quad J(x(k),\tilde U^*(k)).
\end{split}
\end{equation}
Therefore, the constructed cost function is a qualified Lyapunov function, and the algorithm converges through iterations.
\end{pf}
\section{Simulation and Analysis}
\label{simulation}
In this section, we provide simulations of the emualtions per Theorem $\ref{MPC:thm1}$ (Fig. \ref{fig:thm1}(a)) and per the Suboptimal QS-MPC of Algorithm $\ref{Suboptimal QS-MPC}$ (Fig. \ref{fig:thm1}(b)) with the same initial conditions. The LTI system we try to emulate is $\dot x=Hx=\begin{bmatrix}
0& 1\\
-1&-2
\end{bmatrix}x$ with all its eigenvalues located in the left-half plane. The quantized system we choose is $x(t+h)=Ax(t)+Bu(t)=e^{Hh}x(t)+\begin{bmatrix}
1& 0 & -1 & 0\\
0& 1& 0&-1
\end{bmatrix}u(t),$ where $u(t)\in\{-1,1,0\}^m$ and time step $h=0.2$. In the cost function, we design $P=\begin{bmatrix}
50 &0 \\
0&50
\end{bmatrix}$, $Q=\begin{bmatrix}
0.1 &0 \\
0&0.1
\end{bmatrix}$ to guarantee the condition (b) in Theorem $\ref{MPC:thm2}$ and Algorithm $\ref{Suboptimal QS-MPC}$ hold. To ensure the tracking performance is satisfactory, we choose the weight matrix $R$ in the cost function to be $0.05*\mathbb{I}_n$. Since if $R$ is too large, the optimal input tends to be zero resulting in no emulation process; if $R$ is zero, it may have singular solutions. We adopt Gurobi to solve the integer programming optimization problem ($\ref{MPC_optimal}$) and obtain the results shown in Fig. $\ref{fig:thm1}$. Blue trajectories are the linear system starting from the points $2*[cos\alpha;sin\alpha],\alpha=0:\pi/4:2\pi$, while red ones are the quantized system from $[cos(\alpha);sin(\alpha)],\alpha=0:\pi/4:2\pi$. It can be observed that both quantized systems asymptotically converge to the linear system, although the relaxed constraints emulating algorithm has a larger error bound. Meanwhile, we also find the cost function $J$ is strictly decreasing and converging rapidly to zero, which convincingly validates theorem $\ref{MPC:thm2}$. Fig. $\ref{fig:lemma}$ is an emulation with $A=\mathbb{I}_n$ and all initial points on the unit circle. It illustrates that the emulation trajectory is stable but not asymptotically converging to the equilibrium, which is exhibited in the brown circle.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.285]{thm1.png}
\includegraphics[scale=0.25]{alg_ex.png}
\end{center}
\caption{Figures are emulation results when solving the integer optimization problem ($\ref{MPC_optimal}$) directly and applying the suboptimal algorithm $\ref{Suboptimal QS-MPC}$ of the quantized system to emulate a stable LTI system, respectively. }
\label{fig:thm1}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45]{lemma.png}
\end{center}
\caption{The emulation result when A is an identity matrix.}
\label{fig:lemma}
\end{figure}
Meanwhile, we have collected data from the left top emulation in Fig. $\ref{fig:thm1}$. There is a total of 3300 data points with activation patterns as their labels. Then we constructed a regular densely-connected four-layer neural network with ReLU, Sigmoid, or Linear as their activation functions, and the number of nodes in each layer is 512, 480, 256, and 25, respectively. The loss value is calculated by the sparse categorical cross-entropy, which is commonly used in multi-class classification problems, and the optimizer is Adam. After 20 training epochs, we obtain a model with a training accuracy of 98.1\%. The test dataset comes from another emulation with all initial points in the unit circle, which contains 840 data points, and the accuracy achieved 94.7\%. Though using the machine learning method has a degraded emulation performance, the running time for training and predicting is 5 minutes, much less than solving integer programming directly, which takes around an hour on the same computer.
\section{Conclusion and future work}
The work described above uses concepts of model predictive control (MPC) to extend out
previous research on neuromimetic emulation of finite dimensional linear systems (See \cite{baillieul2021neuromimetic} and \cite{sun2022neuromimetic}). The main results are shown to hold under
assumptions that we expect to relax in the near future. Future research will also describe
neuro-inspired machine learning approaches to these and other classes of neuromimetic
emulation.
\begin{ack}
This work has benefitted from conversations with Anni Li.
\end{ack}
|
1,108,101,566,744 | arxiv | \section{Introduction}
The resolution of scanned interferometric spectroscopy is usually limited by the maximum optical path difference (MOPD) of the interferometer. As part of our development of mass-correlated rotational alignment spectroscopy (CRASY),\cite{Schroter2011} we constructed an interferometer with an effectively infinite MOPD,\cite{Schroter2018} utilizing the discrete pulse train emitted from a femtosecond laser oscillator. Here we present the highest resolution data measured to-date and discuss the resolution-limiting factors in our molecular beam experiments. Discussed concepts are equally applicable to other types of interferometric or time-domain spectroscopy that rely on the scanning of a spatial path difference or a temporal delay range.
CRASY is a type of rotational coherence spectroscopy (RCS)\cite{Frey2011} and measures rotational Raman spectra in the time-domain by scanning the path difference between two interferometer arms. The resulting data is useful for the structural characterization of non-dipolar molecules\cite{Schroter2011,Schroter2015,Heo2022a,Heo2022b} that are inaccessible to Fourier-transform microwave Spectroscopy (FTMW).\cite{Grabow2011} By extending the scanned interferometer length, our work obtained order-of-magnitude improved resolution\cite{Schroter2018,Lee2019} as compared to preceding RCS experiments or Fourier-transform infrared spectroscopy (FTIR).\cite{Albert2011}
\begin{comment}
Rotational coherence spectroscopy (RCS) is an interferometric, impulsive laser measurement that delivers broad-band rotational Raman spectra.\cite{Frey2011} The resulting data is useful for the structural characterization of non-dipolar molecules that are inaccessible to Fourier-transform microwave Spectroscopy (FTMW).\cite{Grabow2011} Data is acquired by scanning the time-delay between picosecond excitation and an ionization pulses, with the former exciting and the latter probing a coherent rotational wave-packet. As part of our development of correlated rotational alignment spectroscopy (CRASY), an RCS method, we developed an experiment with an effectively infinite interferometer size and improved the resolution by orders-of-magnitude as compared to preceding experiments. Here we will present corresponding experimental data and simulations and discuss the attainable spectroscopic resolution and accuracy of interferometric experiments, in general, and RCS / CRASY experiments, in particular.
\end{comment}
The energy resolution ($\Delta E$) of spectroscopic experiments is fundamentally limited by the observation time $\Delta t$, i.e., the time over which the investigating particles (usually photons) and investigated molecules interact. The time-frequency formulation of Heisenberg's uncertainty principle states this fundamental resolution limit as $\Delta E \cdot \Delta t = \hbar/2$. Independent of the experiment, the effective observation time is always limited by the coherence time (or lifetime) of the observed quantum states. This leads to lifetime-broadening of observed spectral lines, either due to an intrinsically limited lifetime of the observed states or due to interactions with an environment, e.g., through molecular collisions. When lifetime broadening is small, the effective resolution will be limited by the constraints of the spectroscopic experiment.
In frequency domain spectroscopy, the coherence length of the interacting photons limits the effective observation time. The situation is fundamentally different in interferometric Fourier-transform spectroscopy, where the observation time is limited by the MOPD \replaced{that can be achieved within}{of} the interferometer. The non-apodized full-width-at-half-maximum (FWHM) resolution limit in FTIR is given as $\Delta \widetilde{\nu}^{FWHM} = 0.61 \cdot \mathrm{d^{-1}_{MOPD}}$.\cite{Albert2011} The highest-resolution interferometer described in the literature features a MOPD of 11.7~m,\cite{Albert2018} which corresponds to a delay range of $t_\textrm{max} = 39$ ns and a non-apodized resolution limit of 15.6 MHz FWHM.
RCS is based on the impulsive excitation and probing of rotational coherence with ultrafast (femtosecond or picosecond) laser pulses. In our CRASY variant of RCS, the rotationally excited molecules are probed by resonant multi-photon photoionization and rotational coherence is observed as interferometric signal modulation \replaced{of}{in} resulting ion signals. CRASY therefore correlates rotational spectra with observed ion masses and thereby facilitates the assignment of signals in heterogeneous samples. As in FTIR, or other scanned interferometric spectroscopies, RCS experiments scan the optical path difference in an interferometer and shows a resolution limited by the MOPD.
\section{Infinite Interferometer Design for CRASY}
To obtain mass-CRASY data, ion signals were detected in a time-of-flight mass spectrometer as function of the scanned delay between alignment and ionization laser pulses. The experimental details for mass-CRASY measurements were described previously \cite{Schroter2015,Schroter2018,Lee2019,Ozer2020,Lee2021} and here we focus on the interferometer design used to scan extended optical path differences. Focussed alignment pulses with 800 nm wavelength, $\leq$2 ps pulse duration and 100 $\mu$J-level pulse power created a coherent rotational wavepacket by impulsive Raman excitation. Alignment, in this context, denotes the transient molecular alignment that is commonly observed upon excitation of a coherent wavepacket.\cite{Stapelfeldt2004} Ionization pulses with 200 nm or 266 nm wavelength, 45 fs pulse duration and few-$\mu$J pulse power photoionized molecules by two-photon resonant photoionization. Ion signals showed temporal signal modulations due to the interference of the coherent rotational states in the probe step, as depicted in Fig\ \ref{CRASY_data_example} (A) and (B). Fourier-transformation of these signal modulations reveal the spectrum of the coherent wavepacket, as shown in Fig.\ \ref{CRASY_data_example} (C).
\begin{figure}[ht]
\includegraphics[width=240pt]{Fig1_CRASY_data_example.png}
\caption{Mass-CRASY data from a 50 ns delay scan of a sample containing benzene (mass 78 u, blue), perdeuterated benzene (mass 84 u, green), carbon disulfide (mass 76 u, orange), and naturally occurring heavy isotopologues (darker colors). (A) Delay dependent ion signals show significant signal modulation due to interference of coherently excited rotational states. (B) A section of the signal modulation trace for mass 84 u. (C) Rotational Raman spectrum obtained by Fourier-transformation of the trace shown in (B).}
\label{CRASY_data_example}
\end{figure}
The interferometer for high-resolution spectroscopy should have the longest possible MOPD, combined with a small step size and high positioning accuracy. As described above, the achievable spectroscopic resolution is directly proportional to the MOPD and, as described by the Shannon-Nyquist theorem,\cite{Shannon1949} the spectroscopic range $\nu_\mathrm{max}$ is inversely proportional to the sampling step size ($\nu_\mathrm{max} = 1/(2 \cdot t_{step\:size}$)). CRASY is performed on cold molecular beams with beam temperatures below 10 K and a 0.5 ps to 5 ps steps size (maximum spectroscopic range of $\nu_\mathrm{max} =$ 0.1 THz to 1 THz) is sufficient to resolve the complete thermally occupied set of rotational states. The positioning accuracy should remain well-below the scanned step size to avoid a degradation of \added{the} spectroscopic resolution.
Interferometers used for FTIR, RCS, and other types of scanned interferometer spectroscopy are based on opto-mechanical delay stages, i.e., moving mirrors in one interferometer arm to change the optical path length. The largest interferometers can be found at national synchrotron facilities but are not practical within the restricted space and budget of University-based laboratory research. Instead, our interferometer combined electronic and opto-mechanical delays to achieve an infinite effective delay range within a compact and affordable interferometer design.
A schematic representation the infinite interferometer is shown in Fig.\ \ref{InterferometerDesign}. The mechanical delay was based on a 30-cm Physik Instrumente, MD-531 motorized stage. The optical beam path across the stage was 16-times folded to obtain a MOPD of 4.8 m (16 ns). Longer delays were achieved by electronic pulse selection of oscillator pulses that were amplified in two separate regenerative Ti:Sa amplifiers, forming the two arms of the interferometer. The repetition rate of the laser oscillator (Coherent Vitara) was a 80 MHz and selection of subsequent pulses added discrete 12.5 ns delays in the second interferometer arm. Note that the timing accuracy of the electronic delays is governed by the stability of the oscillator repetition rate and not the accuracy of the electronic delay generator.
\begin{figure}[ht]
\includegraphics[width=240pt]{Fig2_InterferometerDesign.png}
\caption{Schematic depiction of the infinite interferometer design. Pulses from a femtosecond laser oscillator are split and recombined with beam splitters (BS) and amplified in two separate amplifiers. Electronic selection of different oscillator pulses for amplifier 1 and 2 introduce\replaced{s}{d} discrete pulse delays in multiples of 12.5 ns. An opto-mechanical delay stage add\replaced{s}{ed} additional delays of 0--16 ns with picosecond step size and femtosecond accuracy. }
\label{InterferometerDesign}
\end{figure}
The pulse selection delay was controlled via an electronic delay generator (SRS-DG535) and allowed to extend the delay range to quasi-arbitrary values. The amplifiers were operating at 1-kHz repetition rate and probe pulses delayed by more than 1 ms therefore arrive after a subsequent pump pulse. The molecular beam velocity in our experiments is in the range of 1000 m/s and molecules travel \replaced{decimeter}{meter} distances within milliseconds. Experiments with $>1$ ms delays can therefore rely on spatial discrimination of the excited molecules. Therefore, for all practical purposes, our set-up represents an interferometer with infinite MOPD and the achievable spectroscopic resolution is no longer limited by the size of the interferometer, but \replaced{rather by}{by other experimental limitations, such as }the ability to track the molecular beam.
\section{High-Resolution CRASY Data}
Fig.\ \ref{CS2_16_vs_50_ns} shows rotational Raman spectra for \ce{CS2} and illustrates the progress achieved by combining electronic and opto-mechanical delays. The maximal opto-mechanical delay range of 16 ns was sufficient to obtain an effective resolution of 60~MHz FWHM, as shown in Fig.\ \ref{CS2_16_vs_50_ns} (Top). The effective resolution remained below the non-apodized resolution limit of 38 MHz because the mechanical delay stage was not perfectly flat, leading to a loss of signal when the stage approached either end of the delay range. A combined opto-mechanical and electronic delay range of 16.7 m (50 ns) gave a greatly enhanced effective resolution of 17.5 MHz FWHM, near the resolution limit of 16.7 MHz, as shown in Fig.\ \ref{CS2_16_vs_50_ns} (Bottom). The sample \replaced{for the latter spectrum}{used for the latter measurement} contained only trace amounts of \ce{CS2} and the spectrum therefore had a lower signal-to-noise ratio \added{(SNR)}.
\begin{figure}[ht]
\includegraphics[width=240pt]{Fig3_CS2_16_vs_50_ns.png}
\caption{Rotational Raman spectra for \ce{CS2} from CRASY data-sets. (Top) Spectrum obtained by scanning a 16 ns delay range (4.8 m MOPD) with an opto-mechanical delay stage. (Bottom) Spectrum obtained by scanning a 56 ns delay range (equivalent to 16.7 m MOPD), using combined opto-mechanical and electronic delays. Enlarged insets reveal the effective resolution for selected transition lines.}
\label{CS2_16_vs_50_ns}
\end{figure}
The measurement time required to scan large delays scales linearly with the scan range and the collection of mass spectra for a large number of alignment-ionization delays was time-consuming and created exceedingly large data-sets. The data shown in Fig.\ \ref{CS2_16_vs_50_ns} (Bottom) was obtained with delay scan range of 50 ns and a 2 ps step-size and therefore required the accumulation of 25\,000 mass spectra. Data was acquired with 500 Hz repetition rate and ion signals for 1000 laser shots were accumulated for each mass spectrum. The resulting measurement time was almost 14 hours. Each time-of-flight mass spectrum contained 400\,000 point\added{s} and the acquired raw data quantity corresponds to 10 Gb. It is readily apparent that the brute-force extension of the scanned delay range will lead to impractical requirements in terms of measurement time and \deleted{for} data storage.
We reduced the data quantity by lossless compression and the use of sparse data formats. Because mass spectroscopic data is highly discrete, we routinely achieved $>100$-fold in-memory compression with zlib compression algorithms. Fourier transform analysis is only possible on decompressed data, but downsampling of the mass axis and the conversion into sparse data formats facilitated the signal analysis.
Random sparse sampling was used to accelerate long delay scans, i.e., data was only measured for a randomly selected sub-set of delay\replaced{s along the}{ points along an} extended time axis. Different sparse sampling strategies were explored in the field of multi-dimensional NMR experiments,\cite{Hoch2014,Pelczer1991} and are discussed in more detail, below. Fig.\ \ref{M76_full_and_sparse_sampling} compares spectra from a fully-sampled and a sparsely-sampled measurement, with the latter collecting mass spectra only for 5.5\% of delays along an extended delay axis. The sparsely sampled data was acquired 2.5-times faster than the fully sampled data and improved the spectroscopic resolution by a factor 20. Sparse sampling added noise to the spectra, as readily apparent in the logarithmic representation depicted in Fig.\ \ref{M76_full_and_sparse_sampling}.
\begin{figure}[ht]
\includegraphics[width=240pt]{Fig4_Sparse_sampling.png}
\caption{Rotational Raman spectra for \ce{CS2} obtained with continuous 1-ps sampling of a 15 ns delay (Top) and random sparse sampling of 17\,000 mass spectra along a 312 ns delay \added{axis }(5.5\% sampling rate, Bottom). Note the logarithmic scale of the ordinate.}
\label{M76_full_and_sparse_sampling}
\end{figure}
The highest resolution spectrum measured to-date with the CRASY technique was based on a 10 $\mu$s scan of a benzene sample, containing residual \ce{CS2} in small concentration. Due to the limited size of our spectrometer window, tracking of the molecular beam was only achieved \replaced{up to}{for a delay range} $<3$ $\mu$s, reducing the achieved signal contrast and resolution. Fig.\ \ref{fig:kHz_resolution_Spectrum} show\added{s} signal for the \ce{CS2} mass channel, with an effective resolution of 330 kHz FWHM. This data represents the highest-resolution Fourier-transform interferometric spectrum in the world, representing a 50-fold improvement over the highest-resolution FTIR data in the literature\cite{Albert2011,Albert2015,Albert2018}. To comprehend the scale of this improvement, we invite the reader to visualize the \removed{large} 11.7 m MOPD interferometer used for the latter experiments (see Ref.\ \citenum{Albert2011b} for a photographic image) versus the km-scale MOPD achieved in our laboratory experiment.
\begin{figure}[ht]
\centering
\includegraphics[width=240pt]{Fig5_kHz_resolution_Spectrum.png}
\caption{\small Highest resolution rotational Raman spectrum obtained with the mass-CRASY technique. The inset, with 30\,000-fold enlarged abscissa, reveals the 330 kHz FWHM \added{effective} resolution for the J=6--8 transition in \ce{CS2}. }
\label{fig:kHz_resolution_Spectrum}
\end{figure}
Table \ref{tab:Spectroscopic_resolution} compares the resolution limit of various spectroscopic techniques that are used to characterize rotational spectra at high resolution. CRASY currently represents the highest-resolution method for rotational Raman spectroscopy and \added{, more generally,} for the investigation of non-dipolar molecules. Modern FTMW experiments reach a significantly higher resolution,\cite{Shipman2011} but can only be performed for dipolar species and only cover a spectral range of tens of GHz, more than one order-of-magnitude below the spectral coverage obtained with CRASY. In terms of resolving power ($\frac{\mathrm{spectral\:range}}{\mathrm{resolution}}$), CRASY is at parity with state-of-the-art FTMW experiments.
\begin{table}
\caption{Resolution for common types of rotationally resolved spectroscopies.}
\begin{ruledtabular}
\begin{tabular}{lc}
{Spectroscopic method} & {Resolution limit}\\
\colrule
Raman, single-mode laser\saa & 1500 MHz \cite{Weber2011}\\
Raman, FTIR\saa & 300 MHz \cite{Weber2011}\\
Raman, RCS\sbb & 150 MHz \cite{Frey2011,Weber2011}\\
Raman, low resolution CRASY\saa & 39 MHz \cite{Lee2019}\\
Coherent anti-Stokes Raman\saa & 30 MHz \cite{Weber2011}\\
FTIR\replaced{\saa}{\sbb} & 16 MHz \cite{Albert2011}\\
Raman, high resolution CRASY\saa & 330 kHz \scc\\
FTMW\saa & few kHz \cite{Grabow2011,Shipman2011}\\
\end{tabular}
\end{ruledtabular}\\
\footnotesize{\saa Achieved effective resolution. \sbb Theoretical resolution limit. \scc This work.}
\label{tab:Spectroscopic_resolution}
\end{table}%
Table \ref{tab:Spectroscopic_resolution} omitted frequency comb measurements\cite{Hansch2006,Diddams2020}\added{and related techniques}. Highest-resolution frequency comb measurement cover only very small spectral ranges and a direct comparison is therefore not meaningful. Dual-comb or direct comb spectroscopy (DCS)\cite{Foltynowicz2011,Gambetta2016,Muraviev2020} allow the rapid, broad-band, and high-resolution characterization of molecular spectra. We omit DCS from our table because it requires extended interaction times with significant molecular sample densities and, to our knowledge, the resolution of all broad-band DCS spectra is subject to significant \removed{Doppler} broadening. A comparison with estimated DCS resolution limits is therefore not meaningful \added{because the achieved effective resolution remains far below the theoretical resolution limit}.
\section{Limits to Spectroscopic Resolution and Accuracy}
The use of an infinite interferometer removes resolution limits due to the MOPD. We must therefore consider other factors that limit the resolution or accuracy of interferometric measurements. Three distinct types of uncertainties must be considered: (i) Uncertainties in delay positions accrued over the length of the opto-mechanical delay line. (ii) Uncertainties in the laser oscillator repetition rate, which affect the accuracy of the discrete 12.5 ns pulse-selection delays. (iii) Uncertainties due to Doppler shifts and Doppler broadening. In the following, we discuss each error source separately.
\subsection{Uncertainties in Opto-Mechanical Delays}
The MD-531 motorized stage in our interferometer contains a 100~nm internal encoder. The encoder is mounted on an aluminum rod with correspondingly large thermal instabilities,\footnote{The thermal expansion coefficient of aluminum at 25$^\circ$C is $1.1\cdot10^{-5} \frac{m}{mK}$ \cite{CRC_Aluminum_expansion_coefficient}.} and the calibration of known spectra revealed relative positioning errors up to $\Delta r / r = 10^{-4}$.
We addressed the problems with the internal stage encoder by mounting an external optical encoder (Sony Laserscale BL57-RE) with a thermal expansion coefficient of $0.7 \cdot 10^{-6}$ m/(m$\cdot$K). Comparison of internal and external encoder positions revealed a very linear error for the internal encoder. The internal encoder was then calibrated over a range of 12.5 ns against the oscillator repetition rate, by measuring laser cross-correlation signals displaced by one oscillator pulse jump. With a typical $\pm 0.2\, ^{\circ}$C temperature stability on our laser table, we found that the internal encoder was sufficient to confine positioning uncertainties to $\Delta r / r < 3 \cdot 10^{-6}$ ($<40$ fs uncertainty across the stage). Higher accuracy was available by monitoring the external encoder with resulting positioning uncertainties below $\Delta r/ r < 2 \cdot 10^{-7}$ ($\approx$3 fs uncertainty across the stage).
Additional uncertainties arose due to the variation of the air refractive index $n$ with air pressure, temperature, and composition. For 800 nm light, the air refractive index changes by $\Delta n / n \approx 10^{-7}$ for a 40 Pa change in air pressure, a 0.1 $^{\circ}$C change in temperature or a 10\% change in air humidity.\cite{NIST_air_refractive_index, Ciddor1996}
These uncertainties were readily suppressed by the continuous measurement of, and correction for, changes in air temperature, pressure, and humidity. The NIST shop-floor equation\cite{NIST_air_refractive_index}\footnote{Air index of refraction $n$ based on pressure $P$ (kPa), temperature $T$ ($^{\circ}$C), and relative humidity $RH$ (\%): $n= 1+7.86 \cdot 10^{-4} \cdot P / (273 + T) - 1.5 \cdot 10^{-11} \cdot RH (T^2 + 160)$} was sufficient to approximate $n$ with a relative uncertainty of $ \Delta n/n < 10^{-7}$.
Uncertainties accrued over the range of the opto-mechanical delay line are reset with each \added{electronic} pulse selection delay jump, when the mechanical delay line is re-set to its initial position. The impact of opto-mechanical delay uncertainties therefore scales inversely to the number $N$ of \replaced{oscillator}{electronic delay} jumps and becomes small for large $N$\replaced{ and it is}{. It is therefore} sufficient to suppress relative stage positioning errors into a regime where the accrued phase shift for the highest measured frequencies becomes negligible. Stage errors in the \added{$\Delta r / r = $}$10^{-6}$ regime correspond to a 12.5 fs phase shift across the mechanically scanned 12.5 ns delay range and \replaced{are}{were} negligible for any feasible experiment with our minimal 50 fs laser pulse duration (impulsive Raman excitation possible for $<10$ THz transition frequencies). We should note that \added{when} significant calibration errors between the mechanical stage and the oscillator repetition rate \added{occurred, this} le\removed{a}d to the formation of 80 MHz sidebands\replaced{, which are}{ that were} readily identified in the experimental spectra.
\subsection{Uncertainties in the Oscillator Repetition Rate}
Extended delays are achieved by delayed electronic pulse-selection of subsequent oscillator pulses from a Coherent Vitara laser oscillator. This adds delays in multiples of 12.5 ns to the probe arm of interferometer. Any undetected drift of the laser oscillator repetition rate from the nominal value of 80 MHz will introduce a corresponding uncertainty. We used a frequency counter (Aim-TTI TF930) to monitor the stability of the oscillator against a GPS-stabilized clock (Leo Bodnar GPSDO) with an expected frequency accuracy of \added{$\Delta \nu / \nu = $}$\le10^{-10}$.
Figs.\ \ref{AllanDev} and \ref{AllanMDev} show the Allan deviation and the modified Allan deviation \added{for the oscillator frequency,} measured over a 1-day period. For periods $<100$ s, \replaced{we observed a}{the observed} slope of -1 (-1.5) in the Allan (modified Allan) deviation\removed{, which} is characteristic for random white noise.\cite{Riley2008} This noise is due to the frequency counter digitization noise ($\pm 1$ count over the measurement period) and does not reflect any drift of the clock or oscillator. For periods $>100$ s, the Allan deviation remained $<10^{-10}$, giving an upper limit for the frequency stability of the oscillator. It is quite possible that the GPS clock stability \replaced{is}{was} limiting in this regime and that the oscillator frequency \replaced{is}{was} more stable than our measurement indicates.
\begin{figure}[ht]
\includegraphics[width=240pt]{Fig6_AllanDev.png}
\caption{Allan deviation for the 80 MHz Coherent Vitara-T laser oscillator, measured against a \deleted{Leo Bodnar }GPS-disciplined clock. }
\label{AllanDev}
\end{figure}
\begin{figure}[ht]
\includegraphics[width=240pt]{Fig7_AllanMDev.png}
\caption{Modified Allan deviation for the 80 MHz Coherent Vitara-T laser oscillator, measured against a \deleted{Leo Bodnar }GPS-disciplined clock. }
\label{AllanMDev}
\end{figure}
Slow frequency drifts of the oscillator are readily corrected by a corresponding adjustment of the opto-mechanical delay position. A continuous monitoring of the oscillator rate therefor allows a \emph{feed-forward} correction of delays, which fulfills the same purpose as the feed-back oscillator stabilization in frequency comb spectroscopy,\cite{Hansch2006} albeit with much smaller \replaced{technological}{technical} efforts. With our inexpensive monitoring system and a typical frequency counter period of 1 or 10 seconds, we readily achieved a single-sigma uncertainty (Allan deviation) $\Delta \nu / \nu\ll 10^{-7}$ and a longer frequency counter integration period can reduce this uncertainty \removed{down} towards the $\Delta \nu / \nu \approx 10^{-10}$ noise floor of the frequency measurement. We expect that the uncertainty can be further reduced with the use of a high-fidelity reference clock: similar oscillators \replaced{are}{were} used in frequency comb experiments and were stabilized to \replaced{several orders}{order}-of-magnitude better performance. The feedback stabilization required for the \replaced{latter}{frequency-comb measurements} introduces an additional source of noise that is absent \replaced{with}{in} our feed-forward stabilization \added{scheme}.
\subsection*{Uncertainties Arising from Doppler Effects}
\label{Doppler effects}
Textbooks commonly label skimmed molecular beam spectroscopy as "Doppler-free". But as illustrated in Figs.\ \ref{Doppler_broadening} and \ref{Doppler_shift}, the non-zero collimation angle and imperfect alignment of a molecular beam contributes some Doppler effects. We separately assessed Doppler broadening and Doppler shifts by geometric consideration of the molecular beam velocity components $v_{\parallel}$ in the direction of the alignment and ionization laser beams.
\begin{figure}[ht]
\includegraphics[width=240pt]{Fig8_Doppler_broadening.png}
\caption{Illustration of Doppler broadening. The non-zero molecular beam collimation angle $\epsilon$ leads to a distribution of beam velocity components $v_\parallel$ that are parallel to the laser beam propagation axis: $v_{\parallel} = v\mathrm{_{beam} \cdot tan}(\epsilon)$.}
\label{Doppler_broadening}
\end{figure}
\begin{comment}
Vapor pressure of CS2 at 20C is ~0.4 bar. At 20 bar we have some
\end{comment}
In our experiments, a 1 mm skimmer at 280 mm distance from the pulsed valve led to a molecular beam collimation angle of $\epsilon = 0.10^{\circ}$. We measured the molecular beam velocity for a helium-seeded beam of \ce{CS2} to be $v_\mathrm{beam} \approx 1100$ m/s and calculated a velocity spread $v_{\parallel} = \pm1.96$ m/s and a maximal Doppler broadening of $\Delta \nu_{\textit{Db}} / \nu = v_{\parallel} / c = 6.6 \cdot 10^{-9}$. This estimated Doppler broadening \replaced{is}{was} significantly larger than a value obtained based on the textbook treatment (see Chapter 4 in Demtr\"oder \cite{Demtroder2011}) because we accounted for the similar velocity profiles of the heavier CS$_2$ molecules and the lighter helium atoms in the seeded molecular beam.
Doppler broadening becomes relevant when it approaches or exceeds the spectroscopic resolution. In our best CRASY data, we observe sub-MHz line-width for 100 GHz line frequencies ($\Delta \nu_\mathrm{FWHM} / \nu$ in the $10^{-6}$ regime). The Doppler broadening estimated above is several orders-of-magnitude smaller than our best achieved resolution and will not affect spectroscopic results until we reach sub-kHz level resolution. The use of slower molecular beams and better molecular beam collimation can further reduce Doppler broadening and we expect that sub-100 Hz line widths \removed{for 100 GHz rotational lines} can be observed \added{for 100 GHz lines} before Doppler broadening becomes a limiting factor.
A Doppler shift occurs if the angle between laser and molecular beam deviates from $\alpha = 90^{\circ}$. As illustrated in Fig.\ \ref{Doppler_shift}, tracking of the molecular beam then changes the effective path length of the alignment versus the ionization arm of the interferometer and introduces an additional delay of $\delta t = \frac{\Delta x}{c} = \Delta t \left(1 + \frac{v_\mathrm{beam}}{c} \cdot \mathrm{sin}(\alpha) \right)$. Resulting Fourier transformed spectra show a frequency shift proportional to the delay time errors $\delta t / \Delta t$. The line position for well-resolved lines can be determined with an accuracy that is orders-of-magnitude better than the spectroscopic resolution and our experiment is therefore highly sensitive to Doppler shifts.
To measure the angle $\alpha$ between laser and molecular beam, we propagated a laser pointer through the skimmer onto the pulsed valve orifice and measured the relative angle of laser pointer beam and alignment / ionization laser beams against a reference frame. For our experiments, we determined an angle of $\alpha = 91.6 \pm 0.4 ^{\circ}$ and calculated a Doppler correction factor of $\delta t / \Delta t_{m} = (1.0 \pm0.26) \cdot 10^{-7}$. The Doppler shift was not negligible for our most-accurate measurements and reduced measured rotational frequencies by one part in $10^7$ (some 320 Hz for a 3.2 GHz rotational constant fitted for \ce{CS2}).\cite{Schroter2018} The Doppler shift uncertainty of $2.6 \cdot 10^{-8}$ can be reduced by a careful measurement of the relative angles between the molecular beam and the laser beams. The Doppler shift can be measured, and thereby completely eliminated, by performing complementary experiments with laser beams propagating in opposite directions, as measurements from opposing directions show opposite signs for the Doppler shift.
\begin{figure}[ht]
\includegraphics[width=240pt]{Fig9_Doppler_shift.png}
\caption{Illustration of the Doppler shift: To correct for molecular beam propagation within the delay $\Delta t$ between alignment and ionization pulses, the ionization laser is tracking the molecular beam for a distance of $d = \Delta t \cdot v_\mathrm{beam}$. When the angle between molecular beam and laser beams deviates from $90^{\circ}$, this leads to an additional path $\delta x$ for the ionization beam and an additional delay $\delta t = \delta x / c$.}
\label{Doppler_shift}
\end{figure}
\subsection*{Signal Degradation by Sparse Sampling}
Sparse sampling is an essential tool to extend the optical path difference and thereby the spectroscopic resolution without excessive requirements in terms of measurement time and data storage. A number of sparse sampling approaches were discussed in the context of multi-dimensional NMR spectroscopy.\cite{Pelczer1991,Hoch2008,Hyberts2012,Hoch2014} Sparse sampling methods other than random sampling affect the line shape and are therefore problematic, unless the natural line shape in the investigated spectra is known or negligible. Randomly sparse sampled data merely shows an elevated noise level and correspondingly reduced signal-to-noise ratio\deleted{(SNR)}, without introducing any significant artifacts.\footnote{Random sparse sampling is equivalent to the multiplication of a continuous time-domain trace with a binary [0,1] \replaced{'white noise array'}{masking array}, which masks out the unmeasured data points. The multiplication of traces in the time domain corresponds to a folding of their spectra in the Fourier-domain. A \replaced{white noise}{random binary} array transforms into a flat \added{noise} spectrum and therefore merely adds noise \replaced{to}{without otherwise affecting} the measured spectrum.} The combination of sparse sampling with an infinite interferometer therefore offers a unique spectroscopic tool, where resolution and SNR can be freely traded against one another.
Fig\replaced{ure}{.\ } \ref{Sparse_Sampling_simulation} shows the effect of random sparse sampling in a simulated delay trace for CS$_2$. The Fourier transform of a fully sampled trace show\replaced{s}{ed} negligible noise due to the synthetic nature of the data. To simulate 3\% random sparse sampling, 97\% of all points in the delay trace were set to zero. The random selection was based on the Mersenne-Twister pseudo-random number generator as implemented in the numpy library of the Python programming language. The Fourier transform of the sparsely sampled data show\replaced{s}{ed} significant noise but only a modest degradation of the resolution. An estimate of the noise distribution, using the modified Z-score,\cite{Iglewizc1993} gave a noise level of $\sigma$ = 0.2\% relative to the largest signal peak. Experimentally observed SNRs in sparsely sampled data showed similar signal degradation. Note that the loss of information scales proportionally to $\sqrt{\mathrm{samples}}$ and longer scans can be performed with lower sparse sampling rates.
\begin{figure}[ht]
\includegraphics[width=240pt]{Fig10_Sparse_Sampling_simulation.png}
\caption{Simulated Fourier-transformed rotational Raman power spectrum for CS$_2$ at 8 K. (a) Full sampling of 100 ns time delay with $\Delta t$ = 1 ps steps ($10^5$ samples). (b) Sparse sampling of 3000 randomly selected delays from (a) created $< 1$\% sampling noise. Insets show a 50-fold or 2500-fold enlarged abscissa.}
\label{Sparse_Sampling_simulation}
\end{figure}
\section{Expected Resolution Limits for CRASY Experiments}
For all practical purposes, the coherence lifetime of cold rotational states in small molecules is only limited by collisions. Rotational decoherence in collision-free, pulsed molecular beams therefore only occurs when the molecules hit the spectrometer wall. The resolution limit of CRASY is therefore purely a function of the MOPD. With the infinite interferometer design presented above, we removed all practical limitations to the MOPD and other experimental factors become limiting: (i) The molecular beam travels with supersonic velocities and must be accurately tracked. (ii) Sparse sampling is necessary to achieve large MOPDs within reasonable measurement times, but may degrade the SNR to a point where spectra can no longer be resolved.
The resolution of current CRASY measurements is limited by factor (i): due to a limited window size we can track the molecular beam only over distances of a few mm. The beam velocity for our dilute, helium-seeded molecular beams was measured to be $v_{b} \approx 1100$ m/s and the 330 kHz resolution data shown in Fig.\ \ref{fig:kHz_resolution_Spectrum} therefore required \removed{a} tracking of the beam over a distance of nearly 2 mm. The molecular beam velocity can be significantly reduced by using a heavier seed gas with lower speed-of-sound. With a suitably larger laser window, the tracking distance can be extended, e.g., an extension to 10 cm tracking, would offer a 50-fold increase of the accessible MOPD. We expect that the combination of longer tracking and slower molecular beams will push the resolution limit into the single-kHz regime. Further extensions would require the construction of a spectrometer with a dedicated chamber for decimeter- or meter-scale tracking of the molecular beam, e.g., as depicted in Fig.\ \ref{fig:Extended_tracking_chamber}. Note that the signal collections might be facilitated with the correlation to other spectroscopic observables. E.g., probing of rotational coherence via fluorescence excitation would remove the nonlinearity of our two-photon photoionization probe step and might allow the multiplexed detection of signals along the molecular beam axis.
\begin{figure}[ht]
\centering
\includegraphics[width=240pt]{Fig11_Extended_tracking_chamber.png}
\caption{\small Schematic representation of a photoion-photoelectron spectrometer with extended optical access for molecular beam tracking. }
\label{fig:Extended_tracking_chamber}
\end{figure}
\replaced{To collect spectroscopic data with single-kHz resolution and a 100 GHz spectral range would require the sampling of a time axis containing some $10^8$ points.}{The sampling of a time axis containing some $10^8$ points could allow to collect spectroscopic data with single-kHz resolution and a 100 GHz spectral range.} Clearly, this is only possible with severe sparse sampling and a corresponding degradation of the SNR. Fig.\ \ref{fig:millisecond_scan_simulation} shows that such measurements are feasible: a simulated spectrum based on the sampling of 100\,000 points along a 2 ms \replaced{time-delay axis}{delay range} combines excellent signal-to-noise ratio with sub-kHz resolution.
\begin{figure}[ht]
\centering
\includegraphics[width=240pt]{Fig12_Simulation_of_ms_Scan.png}
\caption{\small Simulation of a \ce{CS2} spectrum with 0.61 kHz non-apodized resolution, based on a 2 ms scan range, a nominal 5 ps step size, and 0.05\% sparse sampling. Simulated signal count rates were in the range of few-hundred counts per data point, corresponding to typical experimental count rates. The inset shows a 100-kHz section of the simulation.}
\label{fig:millisecond_scan_simulation}
\end{figure}
In conclusion, we demonstrated that \removed{the use of} pulse-selection from a stable laser oscillator allows to perform interferometric spectroscopy with an effectively infinite interferometer. This approach removed previous limits to the available interferometric MOPD and we presented rotational spectra with sub-MHz effective resolution over a 500 GHz spectral range range. The achieved resolution \replaced{is}{was} several orders-of-magnitude better than that achieved by any preceding RCS or FTIR measurements and corresponds to the scanning of \removed{almost} km-scale path differences. Further order-of-magnitude improvements are expected and \replaced{are}{progress is} only limited by experimental challenges such as the requirement to track skimmed molecular beams over extended distances.
\begin{acknowledgments}
The authors acknowledge funding support from the National Research Foundation of Korea, grant NRF-2018R1D1A1A02042720 and Samsung Science and Technology Foundation, grant SSTF-BA2001-08.
\end{acknowledgments}
|
1,108,101,566,745 | arxiv | \section{Introduction}
\label{sec:intro}
\newcommand{\figref}[1]{Figure~\ref{fig:#1}}\
\newcommand{\secref}[1]{Section~\ref{sec:#1}}\
\newcommand{\tblref}[1]{Table~\ref{tbl:#1}}\
The modern picture of galaxy assembly is effective at explaining a wide range of galaxy classes. Star formation is fueled by cool gas, and spiral discs generally progress from late to early types as the gas is depleted. Ram-stripping can accelerate this process, while interactions and mergers are responsible for the formation of bulges and ellipticals. However, this picture is heavily slanted toward explaining galaxies which are easy to observe. With central surface brightnesses \sbB{>22}, fainter than typical sky backgrounds, low surface brightness (LSB) galaxies are less studied than their high surface brightness (HSB) counterparts and often chronically underrepresented in surveys.
The term ``Low Surface Brightness Galaxy'' has recently evolved into an umbrella term, encompassing a wide range of physically distinct galaxy classes. For example, some of the most extreme LSB galaxies are the ultra-diffuse cluster galaxies \citep[e.g.,][]{vanDokkum2015}, which are red/dead discs likely ram-stripped of cool gas billions of years ago. The most numerous LSB galaxies are undoubtedly gas-poor dwarf galaxies, which are thought to have expelled their gas in an early burst of star-formation due to their low escape velocities \citep[e.g.,][]{Sawala2010,Hopkins2012}. Dark matter-poor structures such as the Leo Ring \citep{Schneider1983} and the somewhat contested ``ghost galaxy'' \citep{vanDokkum2018}, which are gas rich but late-forming due to long dynamical times, have also joined the ranks of LSB galaxies. For each of these classes there exists a clear and unique explanation as to why their stellar populations are so sparse.
In contrast, our work focuses specifically on gas rich, blue, dark matter-dominated, rotationally supported LSB disc galaxies, henceforth referred to as ``LSB spirals''. To be clear, though, the division between LSB and HSB spirals is somewhat artificial, with no discernible bimodality in surface brightness \cite[e.g.,][]{McGaugh1995distribution}; our targets are simply the faint tail of the same distribution which includes the Milky Way. For this class of objects there is no obvious reason why more of their gas has not yet been turned to stars. Indeed, there may be multiple mechanisms to arrest their star formation.
The abundant gas in the discs of LSB galaxies typically exhibits low densities and low metallicities. A straightforward application of the Schmidt-Kennicutt Law predicts little/no star formation in LSB discs, but this only pushes the question back a step; why does the gas remain at low densities? Aside from exceptional cases, such as NGC~4395 \citep{Heald2008}, it is unlikely that the H{$\,$\footnotesize I}{} gas in most LSB spirals is recently accreted material. More likely, the majority of the gas in the majority of LSB spirals was accreted at early times, and the gas has remained in a stable but diffuse disc since.
One approach is to use the assembly information fossilized into the stellar populations to paint a clearer picture of the formation of LSB spirals. With average colours somewhat bluer than HSB spirals \citep{McGaugh1994,deBlok1995}, LSB spirals would seem to be young, unevolved systems, however early studies quickly showed that this simple explanation was insufficient. For example, \cite{Zackrisson2005} use multi-band photometry to show that LSB galaxies are poorly modeled by only a young stellar component, and have likely been forming stars for quite some time.
A more viable explanation is that most LSB spirals are indeed old, but their star formation has been patchy and sporadic. \cite{Boissier2008} suggest that the red FUV-NUV but blue optical colours exhibited by many LSB spirals are best explained by a low and slowly evolving star formation rate punctuated by intense bursts of star formation. The few red LSB spirals in their sample, then, have simply had a longer quiescent period since their most recent burst. Some issues remain; for example, the bursts required to replicate the observed properties of LSB spirals are rather extreme, up to hundreds of \sfr{} for the most massive LSB spirals and typically tens of \sfr{} for LSBs spirals in the Milky Way mass range. Also, \cite{Boissier2008} point out that it is difficult to rule out the possibility that the red FUV-NUV colours are the result of a truncated IMF. In this alternate scenario where the IMF is truncated, the star formation rate in LSB spirals is high, but there are insufficient high mass stars to produce the expected hard UV photons, resulting in red UV colours and inaccurately low emission line-derived star-formation rates. However, caveats aside, the ``patchy/sporadic'' mode of star formation described in \cite{Boissier2008} remains highly plausible, and effectively explains many of the observed properties of LSB spirals.
Likewise, \cite{Vorobyov2009} use hydrodynamic models to show that LSB spirals are likely many Gyr old and dominated by patchy/sporadic clumps of star formation. \cite{Schombert2014} model the optical and IR colours of LSB galaxies and rule out the possibility that typical LSB galaxies are more the 5~Gyr younger than typical HSB galaxies. They find that the driving difference is a low overall star-formation rate with sporadic bursts to help explain the variation in colours, versus the initially fast but now declining star-formation rates of HSB galaxies.
Metallicity may play a role in regulating the mode of star formation: Using an N-body simulation, \cite{Gerritsen1999} find that star formation is suppressed in LSB spirals primarily as a result of low metallicities. With lower cooling efficiencies, the formation of molecular clouds is slower and rarer. These simulations make predictions in-line with patchy/sporadic star formation. In their simulations the star-formation rate has strong fluctuations, however the amplitude of the fluctuations are lower and the durations of the quiescent periods are longer than those derived from FUV-NUV colours by \cite{Boissier2008}. Likewise, \cite{Gnedin2011} note that, in most environments, the shielding of H$_2$ is partly accomplished by dust, which is rare in the low metallicity environments of LSB spirals.
The concept of sporadic, patchy, disc-wide star-formation is somewhat at odds with the inside-out description of galaxy evolution, wherein spiral galaxies form stars near their centres earlier and faster than in their outer discs. Flatter optical colour gradients \citep[e.g.,][]{Matthews1997,Galaz2006} and flat or inverted metallicity gradients \citep{deBlok1998I,Young2015} also seem to imply that evolution in LSB galaxies does not seem to show an inside-out or outside-in preference. However, \cite{Bresolin2015} find that this may be an effect of the larger scale radii of LSB Spirals; we will revisit this issue in \secref{context}.
There are also limitations to the analyses to-date. For example, \cite{Zackrisson2005} and \cite{Kim2007} are unable to place hard constraints on the ages of LSB galaxies due to the degeneracy between age and star-formation history. Both of these works emphasise the patchy nature of star formation in LSB galaxies, but both rely on whole-galaxy measurements. Even with an accurate whole-galaxy star-formation history, open questions would remain, such as: have LSBs always had patchy star formation, or did they begin in a fashion similar to HSBs and then diverge? If star formation in LSB galaxies is best characterised by sporadic bursts, then what is the duty cycle of these bursts, and what are the visible properties of LSB galaxies over this cycle? Because LSB galaxies seem to have had star-formation histories that are variable both in time and across their discs, both spatial and spectral resolution are needed to resolve this issue.
In addition to providing insight into galaxy assembly, the early star-formation history of LSB galaxies may be a key element in resolving the ``cusp-core'' problem. A growing body of evidence suggests that the dark matter profiles of typical Milky-Way mass LSB galaxies are better fit by isothermal (cored) profiles rather than the cuspy NFW profiles produced by most cold dark matter simulations \citep{Borriello2001,deBlokBosma2002,KuziodeNaray2008}.
Supernova feedback is typically invoked as an explanation. In this scenario, a central star-forming episode expels enough gas from the centre of a galaxy to gravitationally drag a significant amount of dark matter from the centre, flattening a cusp into a core \citep[e.g.,][]{Pontzen2012,Governato2012}. This mechanism can explain the cored dark matter halos of dwarf galaxies, but a very significant amount of energy is required to restructure the halos of massive LSB galaxies; for example, \cite{DiCintio2014} find that the effectiveness of supernova feed back drops off for galaxies more massive than \lmstar{>8.5}. Resolving the star-formation histories of LSB galaxies allows for a direct comparison between the intensity of early central star formation and the energetics needed to reshape dark matter halo. Additionally, a clearer picture of LSB galaxies before and during the halo resculpting processes would provide a key test for this theory by allowing the identification of galaxies at different points in this process.
In order to address these questions, we present the MUSCEL program (MUltiwavelength observations of the Structure, Chemistry, and Evolution of LSB galaxies). In this paper we use optical IFU spectra in tandem with Spitzer IRAC and Swift UVOT observations to derive the spatially resolved star-formation history of the LSB galaxy UGC~628. The techniques presented here will be used to similarly determine the histories of other galaxies in our sample. The structure of this paper is as follows: In \secref{obs} we outline our target selection and observations, including a description of the established physical characteristics of UGC~628. In \secref{analysis} we present our analysis, including star-formation history fitting. The results of our analysis, including a best-fit history and a discussion of its implications, are presented in \secref{results}. Finally, in \secref{summary} we summarise these results and discuss future directions for the MUSCEL program.
\section{Target Selection and Observations}
\label{sec:obs}
\subsection{Target Selection and Program Architecture}
The crux of our project is the comparison between the observed spectral energy distributions (SEDs) of LSB spirals and SEDs generated from synthetic star formation histories (SFHs). Our modeling program, described below in \secref{analysis}, uses the spectral synthesis program \textsc{P\'egase}{} \citep{PEGASE}. In order to provide the maximum constraint on the SFH, our program uses ground-based optical spectra in tandem with space-based Spitzer 3.6\micron{} and Swift UVM2 photometry.
The UVM2 filter is a broad-band filter similar to Galex NUV. Although our modeling program does not derive properties from the UVM2 measurements directly, the UVM2 photometry constrains the current/recent star formation and helps constrain the extinction parameter. Our program was awarded approximately nine hours\footnotemark[1] to collect UV data on LSB spirals of interest to the MUSCEL project, as part of the Swift Cycle 10 Guest Investigator Program.
\footnotetext[1]{Proposal ID:1013267}
The 3.6\micron{} photometry complements the UVM2 data by constraining the integrated star-formation history. As with the relationship between the UMV2 data and the current star-formation rate, this constraint comes from the fact that the 3.6\micron{} photometry is correlated with stellar mass. We do not derive stellar mass directly prior to SFH fitting, nor do we assume any mass-to-light ratio. Our observations are drawn from archival Spitzer data (warm Spitzer data in the case of UGC~628).
Finally, we take advantage of the intermediate age indicators in the optical spectrum, such as the H$\delta$ line, the 4000\ensuremath{\text{\AA}}{} break, and the overall shape of the continuum, by including in our SEDs spatially-resolved optical spectra in the 3600-5700\ensuremath{\text{\AA}}{} range. Our spectral data were collected using the VIRUS\nobrkhyph{}P{} IFU at the 2.7m Harlen J. Smith telescope. VIRUS\nobrkhyph{}P{} is a fiber-fed IFU spectrograph with \arcsecond{4}{16} fibers which we use to construct spectral data cubes with a \arcsecond{0}{5} plate scale and a seeing resolution limit of 2\arcsec{} (see \secref{photovalid}). Again, we do not derive properties directly from the optical spectra, but the presence of these features allows our fitting method to constrain the star formation history at intermediate ages; see \cite{Young2015} (hereafter Paper~I) for a thorough discussion of the reduction methods and physical properties of UGC~628 derived directly from the optical spectra.
Below in \secref{photovalid} we discuss the reduction and analysis of the VIRUS\nobrkhyph{}P{} data, involving the reconstructing the VIRUS\nobrkhyph{}P{} image plane. As a comparison to our reconstructed VIRUS\nobrkhyph{}P{} data, we include in \figref{reconstruct} BVR data taken with the ARCTIC imager on the Apache Point Observatory 3.5m telescope.
Our targets are drawn from the LSB galaxies cataloged in \cite{Kim2007} and \cite{McGaugh1994}. Our initial study includes only galaxies that fit comfortably within the \arcminute{1}{7}$\times$\arcminute{1}{7} VIRUS\nobrkhyph{}P{} field-of-view while still spanning many 2\arcsec{} seeing-limited resolution elements (typical of Mt.~Locke). In practice, that means that the distances to our target galaxies are typically 10-100Mpc. To maximise galaxy coverage, minimise dust obscuration, and minimise the number of distinct stellar populations along each line-of-sight, we avoided galaxies that are edge-on ($i>85^\circ$). Finally, our target list was also restricted to objects with archival Spitzer IRAC 3.6\micron{} data. The first target in our program with completed ground-based spectra is UGC~628, which is the focus of this paper.
With a central B-band surface brightness of $23.1~\rm mag~arcsec^{-2}$ \citep{Kim2007}, UGC~628 falls clearly on the LSB side of the surface brightness continuum, though it is by no means an extreme member of the population. Using broad-band photometry \cite{Kim2007} estimate $\lmstar{=}$ 10.65 or 10.80, depending on the IMF assumed, making it a near match to the $\lmstar{} = 10.7$ for the Milky Way \citep{Flynn2006,McMillan2011}.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig1_images-eps-converted-to.pdf}
\caption{{\bf left:} UGC~628 seen with ARCTIC BVR, Swift UVM2, and Spitzer IRAC 3.6\micron{} filters. {\bf middle: } Weighted average reconstructed images from VIRUS\nobrkhyph{}P{} data convolved with UBV filters ({\bf top}), or from matching synthetic fibers used to sample the corresponding image in the left column. {\bf right:} Maximum probability reconstructed images.}
\label{fig:reconstruct}
\end{figure}
\subsection{Data Reduction and Image Reconstruction}
\label{sec:reduce}
Our program uses data from three instruments: The VIRUS\nobrkhyph{}P{} IFU imaging spectrograph on the 2.7m Harlen J. Smith telescope, the UVOT imager on the Swift Space Telescope, and the IRAC imager on the Spitzer Space Telescope.
We developed a VIRUS\nobrkhyph{}P{} reduction pipeline optimised for low surface brightnesses, detailed in Paper~I. Once reduced, the VIRUS\nobrkhyph{}P{} spectra were mapped onto a spectral data cube with a \arcsecond{0}{5} plate scale and a 20\ensuremath{\text{\AA}}{} spectral resolution. The angular resolution in our spectral data cubes is limited partly by the seeing at Mt.~Locke (typically around 2\arcsec{}), but also by the \arcsecond{4}{16} diameter of the VIRUS\nobrkhyph{}P{} fibers. Our observing strategy uses a 6-point dither pattern which allows us reconstruct the image plane from the fiber data at with an angular resolution of around 2\arcsec{}. We have chosen to reconstruct the image plane at a \arcsecond{0}{5} plate scale since adopting a plate scale equal to our 2\arcsec{} plate scale would significantly blur out resolution elements not aligned with the 2\arcsec{} pixel boundaries. As in Paper~I, we over-sample the angular resolution and recognise that any apparent structures smaller than 2\arcsec{} are not real.
The only modifications from the VIRUS\nobrkhyph{}P{} data reduction procedures in Paper~I are as follows: {\bf 1)} Because our spectral synthesis program, \textsc{P\'egase}{}, has a resolution of 20\ensuremath{\text{\AA}}{} in the optical part of the spectrum, the VIRUS\nobrkhyph{}P{} data are binned at an early stage to 20\ensuremath{\text{\AA}}{}~bins aligned to the \textsc{P\'egase}{} wavelength bins in the rest frame of UGC~628. By binning only once at this step, we avoid further degradation of the spectra which would be incurred in subsequent binning steps. {\bf 2)} Instead of a simple weighted average for image reconstruction, we now use a maximum probability image reconstruction, which minimises the residuals between fiber data and a Gaussian-smoothed image. A comparison of these image reconstruction methods is show in \figref{reconstruct}. {\bf 3)} Applying our improved image reconstruction technique to the standard star data, we improved our photometric precision, resulting in a photometric agreement within 0.07~magnitudes between our VIRUS\nobrkhyph{}P{} data and SDSS-g photometry (see \secref{photovalid}).
In this paper, we add two more wavelength bins to the data cube, one for the Swift UVOT UVM2 filter and one for the Spitzer IRAC 3.6\micron{} filter. Because SED modeling relies on the relative brightness at different wavelengths, it is essential that each spaxel in our final data cube samples the same part of the sky across wavelength and across instruments. Between these instruments, the poorest angular resolution comes from VIRUS\nobrkhyph{}P{}, which is dominated by the effects of reconstructing the image plane from fiber spectra. As a result, it was necessary to degrade the UVM2 and 3.6\micron{} images to match the VIRUS\nobrkhyph{}P{} data cubes to ensure a high fidelity comparison between the VIRUS\nobrkhyph{}P{} spectra and the UVM2 and 3.6\micron{} photometry.
To do this, we sampled the UVM2 and 3.6\micron{} images with synthetic fibers, which were created using guide-star pointing data to match the actual VIRUS\nobrkhyph{}P{} fibers during each exposure. We then reconstructed the UVM2 images and the 3.6\micron{} images using the same technique as with the VIRUS\nobrkhyph{}P{} spectral cube. The results are shown in \figref{reconstruct}. These synthetic UVM2 and 3.6\micron{} images are exactly matched to the VIRUS\nobrkhyph{}P{} spectral cubes in spaxel plate scale and sky alignment, and, since they were processed through the same image reconstruction algorithms as the VIRUS\nobrkhyph{}P{} spectral cubes, they share any artifacts of the reconstruction process.
Finally, it is worth considering if variations in the PSF could affect our analysis. By chance, the PSF FWHM does not vary much between these instruments. More importantly, the blurring effects of the \arcsecond{4}{16} VIRUS\nobrkhyph{}P{} fibers dominate over PSF, and, as described above, all images and spectra are subject to the same blurring due to coarse fiber sampling.
In \secref{analysis} we will discuss the methodology for fitting the star-formation history to each region in UGC~628. Since we have ensured that each spaxel in our data cube samples the same part of the sky, we are able to fit a unique history to each spaxel. Each pixel in the star-formation history maps presented below is individually derived from a spaxel in the data cubes.
\subsection{Photometric Validation}
\label{sec:photovalid}
\begin{table}
\caption{Whole-galaxy magnitudes compared before and after image reconstruction, along with known calibration uncertainties.}
\begin{tabular}{lccc}
Filter & Before & After & Uncertainty\\
\hline
g (VIRUS\nobrkhyph{}P{}) & --- & 15.792 & 0.07\footnotemark[1] \\
g & 15.795 & 15.730 & 0.01\footnotemark[2] \\
UMV2 & 18.059 & 18.068 & 0.03\footnotemark[3] \\
3.6\micron{} & 15.389 & 15.352 & 0.016\footnotemark[4] \\
\multicolumn{3}{l}{ $^1$ \footnotesize adopted as a result of this table}\\
\multicolumn{3}{l}{$^2$ \cite{Doi2010}}\\
\multicolumn{3}{l}{$^3$ Swift UVOT CALDB Release Note}\\
\multicolumn{3}{l}{$^4$ IRAC Data Handbook}\\
\label{tbl:photvalid}
\end{tabular}
\end{table}
As a test of our data reduction and image reconstruction methodology, we performed a series of comparisons between our data before and after reconstruction, and with SDSS DR14 broad-band data \citep{SDSSDR14}. To compare with SDSS data, we convolved the VIRUS\nobrkhyph{}P{} spectral data cube with the SDSS g\nobrkhyph{}band{} filter response function and reconstructed the field of view, as described above. The g\nobrkhyph{}band{} filter is ideal for this comparison because it is entirely contained within the wavelength range of our VIRUS\nobrkhyph{}P{} spectra. The first row in \tblref{photvalid} lists the magnitude of UGC~628 in the g\nobrkhyph{}band{} image reconstructed from the VIRUS\nobrkhyph{}P{} spectra.
We sampled the SDSS g\nobrkhyph{}band{} image of UGC~628 with synthetic fibers and reconstructed the field of view, just as we did with the UVM2 and 3.6\micron{} images. The second row in \tblref{photvalid} lists the magnitude of UGC~628 in the SDSS g\nobrkhyph{}band{} image before reconstruction (left) and after reconstruction (right). Likewise, the third and fourth rows in \tblref{photvalid} compare the magnitude of UGC~628 in the archival and reconstructed UVM2 and 3.6\micron{} images as a check on the reconstruction method and on the method used to sample the broad-band images.
The agreement between the before and after magnitudes is a test of the fidelity with which we can reconstruct the FOV from IFU data and the degree of uncertainty we introduce by doing so. Likewise, the agreement between the the SDSS g\nobrkhyph{}band{} magnitude and the g\nobrkhyph{}band{} magnitude derived from our VIRUS\nobrkhyph{}P{} spectra is a test of our data reduction and calibration techniques.
We see excellent agreement between the magnitudes before and after reconstruction. We conclude that sampling and reconstructing broad-band images does not impact our results.
The agreement between the SDSS image and our reconstructed g\nobrkhyph{}band{} image is approximately 0.07~mag. This disparity is larger than our formal error bars, and larger than the 0.01~mag photometric calibration errors of the SDSS image \citep{Doi2010}. While SDSS images are typically too shallow for surface photometry on LSB galaxies, the galaxy-wide photometry has a much higher S/N, and should be considered reliable. Likewise, our VIRUS\nobrkhyph{}P{} exposure times were calculated for S/N of ten at the VIRUS\nobrkhyph{}P{} resolution of 5.1\AA, but averaging over the entire g\nobrkhyph{}band{} boosts the S/N significantly, roughly $10\times$ excluding systematics.
If this disparity were an artifact of the image reconstruction process, then we would see a similar disparity between the before and after reconstruction magnitudes for the UVM2, 3.6\micron{}, and SDSS g\nobrkhyph{}band{} images. Because these images were sampled with synthetic fibers, and those sampled values were processed in exactly the same manner as the VIRUS\nobrkhyph{}P{} measurements, any photometric offsets introduced by the reconstruction process would be included in the difference between the magnitudes before and after reconstruction. Instead, these differences are all smaller than the 0.07~mag offset between the SDSS g\nobrkhyph{}band{} and the VIRUS\nobrkhyph{}P{} g\nobrkhyph{}band{} photometry. Additionally, we tested the 2\arcsec{} kernel width used in image reconstruction by varying the kernel width, and found that a kernel width of 2\arcsec{} minimised the disparity between the before and after UVM2 and 3.6\micron{} magnitudes and between the VIRUS\nobrkhyph{}P{} and SDSS g\nobrkhyph{}band{} magnitudes. We conclude that this disparity is likely a calibration error, either in our data or SDSS, and, to be conservative, we adopt the 0.07~mag disparity as our calibration uncertainty.
\section{Analysis}
\label{sec:analysis}
The goal of our analysis is to determine the likelihood distribution of past and present star-formation rates for each location within our target galaxy. To accomplish this, we divided the history of each spaxel, from 10~Gyr ago to present, into 15 semi-logarithmically-spaced age ranges, and used our data to determine the average star-formation rate within each timestep. The histories of all the spaxels collectively give a map of the star-formation rate in UGC~628 over cosmic time.
The number of timesteps is arbitrary, but it is limited by the sensitivity of the SEDs to age, our lack of knowledge of other parameters that influence the SED (see \secref{fixed}), and the quality of our data. Our model takes larger timesteps as we go further backward in time because the SED of a stellar population changes more slowly with age, making it more difficult to discriminate between populations of different ages. In this work the terms ``Gya'' and ``Mya'' are used to mean billions and millions of years ago {\it as seen from our perspective today}. Since UGC~628 is \about{78.3}Mpc away , it is necessary to add \about{243}~Myr of lookback time when comparing the timeline presented here to timelines for other galaxies.
For each of these age ranges, a normalised basis observation set (spectra and photometry) was generated from synthetic spectra produced with the \textsc{P\'egase}{} spectral synthesis tool \citep{Fioc2011}. \textsc{P\'egase}{} accepts a star-formation history as an input and produces a synthetic spectrum as an output, covering a wavelength range from the far UV to the far IR. The user is free to specify a parameterized star-formation rate (such as a single burst or an exponential decay), or to specify the exact star-formation rate at each point in time. The \textsc{P\'egase}{} output lists luminosities for the nebular emission lines separately from the stellar continuum, leaving it up to the user to combine the two.
We start by creating basis observation sets by providing \textsc{P\'egase}{} with a star-formation rate which is constant within each time step but zero at all other times. We then sample the output \textsc{P\'egase}{} spectra to match the VIRUS\nobrkhyph{}P{} spectra and convolve the \textsc{P\'egase}{} spectra with the UVM2 and 3.6\micron{} filters to produce synthetic photometry, thereby creating synthetic spaxels identical in format and meaning to the spaxels in the real data cube (described in \secref{reduce}). These are the basis observation sets.
Each of these basis observation sets represents what our observations would be if given a spaxel within UGC~628 had only formed stars during a given window of time and at a rate of \sfr{1}. The process of SFH fitting then reduces to decomposing the real observations into a linear combination of these basis observation sets, convolved with an optimal set of environmental parameters, such as extinction normalization ($A_V$), choice of extinction law, and gas-phase metallicity. The extinction normalization is a free parameter in our fitting method, and is determined from the data on a spaxel-by-spaxel basis. The remaining environmental parameters are fixed; the values for these are discussed in detail in \secref{fixed}.
In addition to the rate of star formation in each epoch, the metallicity of the stars formed during each epoch also affects the observed spectra. One solution would be to adopt the current ISM metallicity that we reported in Paper~I, $\log{\rm O/H}+12=7.8$, or $\zstar = \zsol{13\%}$ \citep{Asplund2009}, however it is likely that the ISM metallicity has fluctuated over time due to enrichment and accretion of low metallicity IGM gas. Instead, we leave the stellar metallicity as a free parameter, and interpolate between four different metallicity tracks. The lowest track is around \zsol{3\%}, chosen to provide a robust lower bracket since it is much lower than expected for a galaxy of this mass, and lower than all but the most extreme dwarf galaxies \citep{Kniazev2018}. The second and third tracks, \zsol{30\%} and \zsol{60\%}, are more typical of what one would expect for an LSB spiral of this mass. In order to generously bracket the current ISM value and account for the possibility of more enriched stellar populations, we also include a fourth solar metallicity track. Our SFH fitting code generates stellar populations with intermediate mediate metallicities by interpolating between any two of these four tracks. Although the effects of metallicity on stellar spectra are, in detail, non-linear, the low-resolution of our VIRUS\nobrkhyph{}P{} spectra makes our program fairly insensitive to these effects, allowing us to extract the star-formation histories. In Sections \ref{sec:uniqueness} and \ref{sec:stability} we present a thorough discussion of the robustness of our fitted histories. Nevertheless, we emphasise that the stellar metallicities presented should be taken as gross estimates only.
It is worth noting that we have left the stellar metallicity as completely free parameter (within the bounds of \zsol{3\%-100\%}). It is not constrained to the gas-phase metallicity, and is allowed to go up or down with time. This choice is reasonable given the growing body of evidence that galaxies may acquire pristine IGM gas after they have formed some fraction of their stellar population \citep[e.g.,][]{Huang2013,Sanchez2014,Elmegreen2016}. Such acquisition has the effect of diluting their ISM and necessarily lowering the metallicity of subsequently formed stars. For example, in their study of the stellar populations in the dwarf galaxy DDO~68, \cite{Annibali2019} find that the typical stellar metallicity is around ten times as great as the gas-phase metallicity. DDO~68 shows signs of a recent merger with several smaller dwarfs, including a gas-rich dwarf, and the conclusion is that the intruder recently diluted the ISM. Although the stellar metallicity in most galaxies, including DDO~68, generally increases over time, any stars formed in DDO~68 in the near future will likely have a much lower metallicity that the older population in DDO~68.
The other factor which significantly motivates us to leave the stellar metallicity free to go up or down is that both the stars and the gas within UGC~628 have likely migrated over time. This is discussed further in \secref{results}, but the oldest populations are more smeared over the disk out than the younger populations. While this effect is primarily azimuthal, it is still possible for old populations with high metallicity to mix with young populations with low metallicity, even if the trend in metallicity over time for any one dynamical unit is purely monotonic. This consideration does not impact our ability to resolve the history of each spaxel, but our analysis cannot tell us exactly where the stars in each spaxel originated, an unavoidable uncertainty when studying the spatially resolved histories of galaxies.
\begin{figure}
\includegraphics[width=\columnwidth]{fig2_basis-eps-converted-to.pdf}
\caption{The synthetic VIRUS\nobrkhyph{}P{} spectra in each timestep, grouped by metallicity. The units are relative luminosity density, normalised to the red end of the spectra. The youngest timesteps are blue, the oldest are red.}
\label{fig:basis}
\end{figure}
For visual comparison, the synthetic VIRUS\nobrkhyph{}P{} spectra from each timestep and each metallicity are shown in \figref{basis}. These are the spectral components of the basis observations. To guide the eye, the spectra are colour coded with the youngest timesteps coded as blue and the oldest coded as red. The spectra show the expected trends --- overall reddening with age, the disappearance of emission lines within the first few Myr, and, later, the strengthening of metal absorption lines with respect to the hydrogen absorption lines. Although less drastic that the effects of age, the differences between the spectra due to metallicity are significant, particularly at the blue end of the spectrum, underscoring the need to anchor SED fits with UV measurements.
Summing over bins in both time and metallicity, the standard $\chi^2$ comparison between model and observation can be expressed as:
\begin{equation}
\chi^2 = \sum_\lambda\left[ \left( \frac{f_\lambda - \sum\limits_b w_b g_{\lambda,b}}{\Delta f_\lambda} \right)^2\right]
\end{equation}
where the $f_\lambda$ are the observations, $g_{\lambda,b}$ are the model basis spectra, and $w_b$ are the relative weights, determined from the star-formation rate and stellar metallicity at each timestep. By dividing each disparity term by $\Delta f$, this classic form has the desirable property that each term is inversely weighted by the uncertainty in the measurements and, assuming that $\Delta f_\lambda$ is of a similar order of magnitude as the disparity between the model and the data, each term represents a relative disparity between model and data that contributes equally to $\chi^2$ even if the $f_\lambda$ values vary strongly with $\lambda$.
Near the solution, this assumption is reasonable. However, any search algorithm must explore parts of parameters space far from the solution. In such cases an asymmetry manifests itself: in the case where the model values are too low, there is a maximum disparity that can be incurred (100\%), whereas for model values that are too large there is no maximum. As an example, consider a candidate solution for a region with bright emission lines in which the lines are fit correctly but the continuum is underestimated. Ideally the algorithm would raise the continuum and refit the lines, but because underfit values are penalized less than overfit values, this candidate solution is at a local $\chi^2$ minimum.
To create this symmetry and still retain the relative disparity aspect, instead of dividing each disparity term by $\Delta f_\lambda$ we divide by the geometric mean of the model and the data. To additionally capture the aspect of the classic form of $\chi^2$ wherein each term is inversely weighted by its relative uncertainty, we multiply each term by its signal-to-noise $f_\lambda/\Delta f_\lambda$. Near the solution, this reduces to the original $\chi^2$. This modification is purely to aid the search algorithm, and has no significant impact on the interpretation of the final $\chi^2$ values.
Additionally, in recognition of the fact that some of our wavelength bins are wide photometric bins ($\Delta\lambda > 1000\rm\AA$) while others are much smaller spectral bins ($\Delta\lambda = 20\rm\AA$), each disparity term is multiplied by the relative size of the bin $N_\lambda$.
\begin{equation}
\chi^2 = \sum_\lambda \left[ N_\lambda \frac{\left(f_\lambda - \sum\limits_b w_b g_{\lambda,b}\right)^2}{f_\lambda \sum\limits_b w_b g_{\lambda,b}} \left(\frac{f_\lambda}{\Delta f_\lambda}\right)^2 \right]
\end{equation}
Finally, for the sake of clarity, we rewrite our model spectra $g_{\lambda,b}$ as the product of model luminosities $L_{\lambda,b}$ and environmental factors $\phi_\lambda$, which include fixed parameters such foreground extinction and ISM metallicity, as well as the fitted model parameter $A_V$:
\begin{equation}
\chi^2 = \sum_\lambda \left[ N_\lambda \frac{\left(f_\lambda - \sum\limits_b w_b L_{\lambda,b}\phi_\lambda\right)^2}{f_\lambda \sum\limits_b w_b L_{\lambda,b}\phi_\lambda} \left(\frac{f_\lambda}{\Delta f_\lambda}\right)^2 \right]
\end{equation}
\subsection{Fixed Model Parameters}
\label{sec:fixed}
\begin{figure*}
\includegraphics[width=0.9\textwidth]{fig3_RegionA-eps-converted-to.pdf}
\caption{The best fitting photometry and residual spectra to the large star-forming complex identified as Region~$A$ using a battery of combinations of IMFs and extinction laws. Since the residual spectra are represented as fractions of the observed uncertainties, the $y$-axis of the top central panel ranges from $-20\%$ to $+20\%$ error. Region~$A$ was selected for this experiment because the bright emission lines, particularly \HB{}, provide a strong secondary diagnostic of the current and recent star-formation rate, and the effects of the choice of IMF and extinction law are more tied the current star-formation rate than the older population. The Calzetti Law is the best-fit extinction law, but the gross characteristics of the fitted history are not strongly affected the choice of IMF. The best overall fit is the Calzetti Extinction Law with the Miller-Scalo IMF.}
\label{fig:perm}
\end{figure*}
The \textsc{P\'egase}{} spectral synthesis tool provides the luminosities of the stellar continuum and emission lines separately. We chose to adopt the emission line luminosities produced by \textsc{P\'egase}{}, with the exception of the \ionl{O}{2}{3727}, \ionl{O}{3}{4959}, and \ionl{O}{3}{5007} lines. Like many of the emission lines, these are sensitive to the ISM metallicity and ionization parameter, however unlike many of the other metal lines, they contribute a significant fraction of the the total luminosity in star-forming regions, especially the \ionl{O}{3}{5007} line. The ratios of these lines to \HB{} are related to the ISM metallicity and ionization parameters. In Paper~I we used observations of these lines in select H$\,${\small II}{} regions in UGC~628 to derive a metallicity and ionization parameter profile with the R23 bright line method. Our findings indicate that UGC~628 has a fairly flat metallicity and ionization profile, with values around $\log{\rm O/H}+12\approx7.8$ and $\rm U=-2.8$. Since there is comparatively little variation in these values (in the H$\,${\small II}{} regions where they can be sampled), in this paper we adopt these metallicity and ionization parameter values globally, even in locations where they cannot be measured with high fidelity. Our modeling program then corrects the fluxes of these oxygen lines so that their ratios against \HB{} are consistent with the adopted metallicity and ionization parameters, essentially the reverse of the process used for the emission-line bright regions in Paper~I.
However, we have fewer direct constraints on the stellar IMF and extinction law. The stellar IMF and choice of extinction law primarily affect measurements of the current/recent star-formation through their influence on UV continuum. Since emission lines, particularly \HB{}, also track the current star-formation rate, we have chosen to use the large star-forming complex on the North-Western limb of UGC~628, identified in Paper~I as Region~$A$, as a laboratory to study our choice of IMF and extinction law. As a case-study, we fit the brightest part of Region~$A$ with a combination of IMFs and extinction laws, permuting the IMFs from \cite{Kroupa1993} (top-heavy) and \cite{Miller1979} (bottom-heavy) with the extinction laws from \citep{Cardelli1989} (Milky Way Law), \cite{Koornneef1981} (LMC Law), \cite{Rayo1982} (SMC Law), and \cite{Calzetti1994} (starburst galaxies). This results in eight combinations. As noted in \secref{analysis}, the extinction normalization ($A_V$) is a free parameter in the fitting processes.
\figref{perm} presents the best fit histories for each of these eight models for Region~$A$, along with observed and model photometry and the residuals between the observed and model spectra. The normalised $\chi^2$ values for each model are also listed in the figure legend. All the models agree well with the observations, as can be seen from the fact that all the residual spectra in \figref{perm} fall entirely within the 20\% error bars and mostly within the 10\% error bars, similar to the 0.07~mag uncertainty we quoted in \secref{photovalid}. Because the residual spectra are also very similar to each other, and we conclude our methodology is not capable of determining which extinction law or which IMF is the appropriate choice. However, our goal is to determine the star-formation history, not the IMF or extinction law.
The fitted histories shown in \figref{perm} are all fairly similar to each other. All the histories have in common that Region~$A$ has an 8-10~Gyr old stellar population, and then a younger population that was formed between 5-50~Mya, with nearly zero current star formation and nearly zero star formation between 8~Gya and 50~Mya. Since the histories are similar, we conclude that the choice of extinction law and IMF does not significantly impact the gross characteristics of our results. However, since the differences between the histories in the 5-50~Mya range are roughly $\sim 2\times$, it is worth considering which extinction law and IMF to choose.
The two models with the lowest $\chi^2$ values are those that use the Calzetti Extinction Law; the Calzetti Law is naturally most appropriate for our project since it is derived from unresolved stellar populations mixed with star forming regions. The choice of IMF seems to have less of an effect, however the bottom-heavy Miller-Scalo IMF is a slightly better fit. Hereafter, we adopt the best fit model, \cite{Miller1979} IMF and the \cite{Calzetti1994} extinction law, and note that this choice does not significantly impact our conclusions.\\
\section{Confidence Limits}
\label{sec:error}
The reconstructed star-formation history of UGC~628 is illustrated in \figref{timeline}. The top left panel shows the star-formation rates only, the top right panel shows the star-formation rates colour-coded by the stellar metallicities, the bottom left panel shows the confidence limits the star-formation rates, and the bottom right panel shows the total mass formed in each epoch. Because UGC~628 is an inclined disc, the overall shape in \figref{timeline} is an ellipse. The edges of the ellipse look tattered in many places; as the light profile tapers off, the S/N falls below the threshold where fitting is possible, leaving an uneven boundary. Several spots are masked out from our analysis due to contamination from foreground stars and a background galaxy. The implications of the star-formation history will be discussed in \secref{results}; here we discuss the calculation of the confidence limits and diagnostics of systematics.
\subsection{Limit Determination}
\label{sec:limit}
\begin{figure*}
\includegraphics[width=.9\textwidth]{fig4_snapshot-eps-converted-to.pdf}
\caption{{\bf top left: } A map of star formation during 15 epochs in the history of our first target LSB galaxy. {\bf top right: } Like above, except points are colour-coded based on the metallicity of the stars formed during each era. {\bf bottom left: }Uncertainties in the star-formation rates shown in the top panel. The units and scale are the same as the top panel. {\bf bottom right: }The total stellar mass contributed to UGC~628 for formation in each epoch.}
\label{fig:timeline}
\end{figure*}
In order to estimate the confidence limits on the star-formation rate in each epoch, we determined, for each pixel, the limits on the star-formation rate in each timestep (and $A_V$) that correspond to $\Delta\chi^2=\chi^2-\chi^2_0=1$. Many of the star-formation rates have a degree of covariance; that is, if the star-formation rate in one bin is changed, it may be possible to mitigate the effects on $\chi^2$ by shuffling that star-formation rate to an adjacent timestep. \secref{stability} and \secref{uniqueness} explore the impacts of this solution degeneracy on our conclusions. While we do not find that solution degeneracy affects our overall picture of the star-formation history of UGC~628, it is the primary source of uncertainty in our star-formation rates. In order to fully capture this effect while exploring the $\Delta\chi^2=1$ limits for each parameter, the remaining parameters are re-optimised at each step. The $\Delta\chi^2=1$ limit for each parameter is adopted as the uncertainty for that parameter.
In \figref{timeline} we see that the uncertainties range from around 0.005~$\rm M_\odot yr^{-1}kpc^{-2}$ near the edge of the galaxy to 0.05~$\rm M_\odot yr^{-1}kpc^{-2}$ near the centre. The amounts to roughly 25\% uncertainties in timesteps with modest levels of star-formation. The notable exception is the low uncertainties in the 0-4~Mya timestep; current star formation is tightly constrained. The uncertainties are highest near the centre in the oldest timestep because most of the star formation near the centre of UGC~628 took place in the earliest timesteps, and, since older stellar populations have less of an effect on the present-day spectrum, the constraints are weaker. In contrast, Region~$A$, which has a very high star-formation rate in a recent timestep, shows very small uncertainties, even in the early timesteps.
\subsection{Linearity Validation}
\label{sec:uniqueness}
\begin{figure*}
\includegraphics[width=0.9\textwidth]{fig5_linearity-eps-converted-to.pdf}
\caption{{\bf left:} Five pairs of regions in UGC~628 (top right), each selected to have identical $A_V$ values but different SF histories. For each pair, we show the fitted SFH for each separate region (coded red and blue), along with the sum of the two separate fits (black dashed line), and the fit of the summed SED of the the two regions (black solid). The similarity between the black dashed and black solid lines indicates our ability to determine the history of any composition stellar populations as a composite of their histories. The discrepancy between the back dashed and solid lines represents a systematic uncertainty in our methodology.}
\label{fig:linearity}
\end{figure*}
Each resolution element in our SEDs will contain the light from different stellar populations with different histories blurred together. Ideally, we should derive the correct composite history from a composite population. A flawed methodology might produce a fitted SFH that disproportionately favors one sub-population, or, worse, produces a history that does not accurately represent any sub-population.
After the initial fitting was complete, we identified pairs of regions with similar best-fit extinction parameters but different histories, shown in \figref{linearity}. We then added their spectra and photometry to create summed SEDs, and fitted those summed SEDs with the same methodology that we applied to individual regions. If our methodology is robust, then the SFH fitted to the summed SED should be the sum of the fits to the individual SEDs. Note that it is important that we chose regions with similar extinction parameters since, unlike starlight, extinction does not add linearly.
We see in \figref{linearity} that the fitted SFHs of the summed SEDs closely resemble the sums of the fits to the individual SEDs. The match is not exact, but the gross characteristics of the SFHs match in all cases. We take this as a validation of our method, and a confirmation that we are able to spectroscopically distinguish multiple histories even when they are combined into a single SED. The small disparity between the sums of the fits to the individual regions and the fits to the summed SEDs can be considered a rough estimate on the confidence of our method, independent (though generally consistent with) the uncertainties presented in \secref{limit}.
\subsection{Numerical Stability Validation}
\label{sec:stability}
With the fitted star-formation histories in hand, we consider one final issue with our analysis: the possibility that the residuals between the model and the data have only a shallow dependence on one or more of the parameters, or perhaps a linear combination of parameters. For example, since the colours of older stellar populations change more slowly than the colours of younger populations, our residuals might depend on the total sum of star formation prior to 1~Gya, but with only a weak preference for the particular age range. In this scenario, a fitting algorithm might consistently converge on a solution for each region, when in fact a radically different solution is nearly as good a fit.
The natural method to test for cases where solutions have a shallow dependence on data is to refit the data with a small variance added; if the solution is robust, the solution will vary slightly, but if the solution has a shallow dependence on the data, the solution will vary wildly. The pixel-to-pixel variation of our best-fit star-formation histories, shown in \figref{timeline}, naturally provides this test.
In \secref{obs} we describe how our data cubes and SFH maps are rendered at a somewhat arbitrary plate scale of \arcsecond{0}{5}, significantly oversampling the PSF (dominated by the 2\arcsec{} seeing at Mt. Locke). Our adopted \arcsecond{0}{5} plate scale gives us the a final check on the stability of our method.
At the 78.3 Mpc distance of UGC~628, \arcsecond{0}{5} corresponds to 190pc. Star-forming regions and groups of stars may be smaller than this scale so, without atmospheric seeing, we might expect neighboring pixels in our SFH maps to have radically different histories. With the 2\arcsec{}, the light from each stellar population is spread across approximately four spaxels. If our algorithm works correctly, fitting each spaxel separately (without knowledge of its neighbors) should produce histories that vary smoothly from pixel-to-pixel, with no rapid variations on scales smaller than four pixels. If our algorithm is numerically unstable (as with parameter degeneracy), we might see significant pixel-to-pixel `speckling' in the SFH maps, caused by minor differences in the observed flux leading to significant changes in the derived history.
Examining the solutions presented in \figref{timeline}, the majority of the frames show very few cases of significant pixel-to-pixel variations in the SFHs. Careful examination of the oldest bins reveals some speckling; these are the timesteps that are the closest to degenerate, since the composite spectra of older stellar populations change more slowly. Aside from this, the solutions vary continuously, gradually transitioning from one to the other, just as the values in the data cubes do. We conclude that our model and methodology are highly unlikely to suffer from serious issues with solution degeneracy, and are stable against small perturbations in the measurement values.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig6_radialsfr-eps-converted-to.pdf}
\caption{Azimuthally averaged star-formation rate densities, derived from the star-formation rate maps in \figref{timeline}. For the sake of clarity, the star-formation rate densities shown here have been smoothed with a Gaussian kernel with a FWHM of 0.5~kpc. These curves clearly illustrate a migration of centrally concentrated star formation early in the history of UGC~628 to disc-wide patchy star formation continuing through present-day.}
\label{fig:radial}
\end{figure}
\begin{table}
\caption{Brief History of UGC~628}
\begin{tabular}{lllllr}
era & SFR & \ensuremath{\rm M_\bigstar} & \ensuremath{\rm\mu_0(B)} & \ensuremath{\overline{R_{SF}}}\\
Mya & \ensuremath{\rm M_\odot\,yr^{-1}} & \ensuremath{\rm 10^{9} M_\odot} & & kpc \\
\hline\\
8000-10000 & 15.18 & 22.2 & 21.2 & 5.0 \\
6000-7999 & 3.29 & 23.7 & 22.4 & 5.9 \\
5000-5999 & 6.95 & 27.9 & 22.1 & 4.1 \\
3000-4999 & 1.01 & 27.1 & 22.6 & 4.8 \\
1500-2999 & 1.51 & 27.6 & 22.2 & 2.9 \\
1000-1499 & 4.65 & 29.2 & 21.9 & 4.1 \\
700-999 & 2.07 & 29.3 & 22.2 & 5.3 \\
500-699 & 0.54 & 29.2 & 22.5 & 7.7 \\
300-499 & 0.38 & 29.1 & 22.6 & 7.3 \\
200-299 & 1.52 & 29.1 & 22.7 & 7.2 \\
100-199 & 8.30 & 29.8 & 22.7 & 5.0 \\
50-99 & 4.67 & 29.9 & 22.7 & 8.3 \\
20-49 & 20.93 & 30.5 & 22.5 & 7.7 \\
5-19 & 5.52 & 30.5 & 22.7 & 9.1 \\
0-4 & 0.42 & 30.5 & 22.7 & 7.3 \\
\label{tbl:timeline}
\end{tabular}
\end{table}
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig7_rgb-eps-converted-to.pdf}
\caption{A UBV recreation of the stellar populations of UGC~628 as they were at the end of each epoch, assuming present-day extinction values. We have not corrected for orbital motion, and the stellar populations are represented at their current locations. The final frame is a match to the present-day appearance.}
\label{fig:ubv}
\end{figure}
\section{Discussion}
\label{sec:results}
As mentioned above, the reconstructed star-formation history of UGC~628 is illustrated in \figref{timeline}. Azimuthally averaged histories are shown in \figref{radial}, and interesting galaxy-wide properties in each summarised in \tblref{timeline}. In order to compare to present-day galaxies, \figref{ubv} shows a recreated UBV image of UGC~628 for each timestep, assuming present-day extinction. Note that because of this assumption, the data in \figref{ubv} can only be used for gross characterization, and even then only with caution. Also note that \figref{ubv} is not a recreation of the actual appearance of UGC~628 during these epochs since we are not correcting for the orbits of the stars, and many dynamical times have passed since the earliest timesteps.
What follows is a discussion of the fitted history of UGC~628 set in the context of what is known about LSB spirals in general, both from other observations and from models. We emphasise that UGC~628 is only one object, and broader conclusions will require a larger sample. Using \figref{timeline} as a guide, we break the history of UGC~628 into several eras, summarised below:\\
\subsection{Timeline of Events}
\label{sec:timeline}
{\bf 8-10 Gya} Intense star formation is concentrated in the central 5kpc, but with noticeable levels of star formation through the disc. 60\% of the stellar mass was formed during this era (see bottom right of \figref{timeline} and \tblref{timeline}). The stars formed in the central region during this era are currently spatially coincident with the bar, and it seems likely that this is the era of bar formation (see discussion in \secref{context}). The distribution of star formation in UGC~628 during this era is grossly in-line with inside-out formation, and the buildup of a now-mature stellar population near the centre explains the current red-to-blue colour gradient reported in Paper~I. We also see the formation of solar-metallicity stars in the outer disk. The buildup of metals is not surprising given the duration of this era and the level of star formation, however the contrast with the low metallicity star formation dominating the central region is interesting, and will be discussed further in \secref{context}. During this era the unextinguished \sbB{} was bright enough that, had UGC~628 been observed, it might have been classified as an HSB spiral.
{\bf 5-8 Gya} In these two timesteps star formation persists through the disc of UGC~628, albeit at a much lower rate. We see one knot of star formation offset from the centre, at about the position where the bar currently ends. Interestingly, \figref{timeline} shows that the stars formed in UGC~628 during this era have a higher metallicity than the stars formed near the centre during the 8-10~Gya peak-bar era. Although, as discussed in \secref{analysis}, our stellar metallicities should be taken as gross estimates only, this difference between these eras makes sense chronologically: the early initial burst locally enriched the central gas.
{\bf 3-5 Gya} UGC~628 is fairly quiet during this period, with an average star-formation rate of around \sfr{1}. We cannot rule out short high amplitude bursts, such as the one seen in the later 20-49~Mya timestep, but we can rule out the possibility that they were common. This was likely a {\it nearly} burst-free era. A possible analog to UGC~628 during this phase is UGC~8839, with a centre that is fairly bright but somewhat red for an LSB spiral, but an extremely faint blue extended outer disc.
{\bf 700 Mya - 3 Gya} These three timesteps show a burst of star formation. In the 1.5-3~Gya timestep we see no activity except in the core. This is the last point in the history of UGC~628 when the core is more active than the outer disk. In the 1-1.5~Gya timestep the core goes quiet and the disk becomes active. The activity in the inner disk is not clearly associated with the bar. During this era the unextinguished value of \sbB{\approx 22} would make UGC~628 a marginal LSB galaxy.
{\bf 200-699 Mya} These three timesteps are very quiet. The average level of galaxy-wide star formation is similar to today, around \sfr{0.4}. Like the 3-5~Gya era, this was likely a nearly burst free era. The scarce star formation during the 200-299~Mya timestep shows a strong preference for the outer disc of the galaxy.
{\bf 100-199 Mya} This timestep shows another burst of star formation, this time predominately in the outer disk. \figref{timeline} shows a ring of star formation around the core. We caution against over interpreting this as a true dynamically driven ring. More likely, this is an extension of the patchy mode of star formation down to smaller galactocentric radii. It is worth noting, though, that star formation now clearly avoids the core. Given the sharp swings in star-formation rate in subsequent timesteps (see below), it is possible that this timestep was mostly quiet with a brief episode of intense star formation; only an average value is possible in our analysis.
{\bf 50-99 Mya} A fairly quiet era. Low-level star formation is now strongly biased toward the outer rim of the galaxy. Because our analysis necessitates that the earlier timesteps be larger, more recent timesteps may give us insight into the short-term variations that may have occurred during earlier timesteps. This timestep may be representative of inter-burst periods during the earlier timesteps.
{\bf 20-49 Mya} A major episode of patchy, nearly disc-wide, low metallicity star formation. Star formation again avoids the core/bar region entirely, and is, instead, spread along the outer rim of the galaxy. This edge-dominated star formation would seem to be the likely cause of the mostly flat and slightly inverted metallicity gradient we reported in Paper~I. This event may have been driven by tidal torques from an interaction, but the seemingly undisturbed morphology of UGC~628 makes a major merger unlikely. During this era UGC~628 was still an LSB spiral, however, had UGC~628 been observed during this epoch, it might have been excluded from LSB samples because of its bright knots. The average star-formation rate was \sfr{\approx 20}, the highest in our reconstructed timeline; UGC~628 would have been considered a starburst galaxy since, at this rate, as it could have assembled its entire stellar population in 1.5~Gyr. However, since this era only lasts 29~Myr, it only accounts for 2\% of the stellar population.
{\bf 5-19 Mya} This post-burst era sees UGC~628 with one knot of younger stars, Region~$A$, located on the Northwestern edge of the disc. Region~$A$ itself shows a significant drop in star-formation rate since the previous era, and also an increase in the metallicity of stars forming now compared to those formed in the previous era; as with the core enrichment seen in the first few timesteps, this makes sense chronologically if Region~$A$ self-enriched during the 20-49~Mya era.
{\bf 0-4 Mya} Present-day UGC~628 sees star formation is largely shutdown, with only low levels in a few remaining areas such as Region~$A$. We see that the current phase in the life of UGC~628 is simply an inter-burst period, with relics such as Region~$A$, the inverted metallicity profile, and the redder bar hinting toward a more active past.
\subsection{Comparison with Other Observed and Simulated LSB Spirals}
\label{sec:context}
One of the most striking aspects seen in our fitted history of UGC~628 is the large swings in star formation. The recent history of UGC~628 seems to be made of long periods of quiescence punctuated by eras of star formation. During the most recent episode of star formation UGC~628's star-formation rate was approximately $10\times$ what would be required to assemble the current stellar mass in a Hubble Time. Conversely, during the period between 300-700~Mya the star-formation rate was fairly low, similar to the current level. This picture is similar to that portrayed in other works. Recall that \cite{Boissier2008} use Galex FUV-NUV colours to show that LSB spirals are likely in a post-burst phase; having lost their most massive stars, LSB spirals have weak emission lines and red UV colours, but are still optically blue. We see the extreme bursts of star formation discussed in \cite{Boissier2008} mirrored in our fitted history of UGC~628.
The metallicity of UGC~628 that we report in Paper~I, $\log{\rm O/H}+12=7.8$, or roughly $\zsol{13\%}$, is in the range that \cite{Gerritsen1999} suggest could suppress the cooling and formation of molecular clouds and trigger the patchy/sporadic behavior that we see. The star-formation rate fluctuations predicted in \cite{Gerritsen1999} are powers of ten lower in amplitude, and the quiescent periods are shorter than our timesteps are capable of resolving, however their galaxy is almost a factor of ten less massive, and \cite{Boissier2008} find that the burst amplitudes and quiescent durations seem to correlate with galaxy mass. Suppressed star formation via chronically low metallicity gas would be in-line with the idea of galactic ``overfeeding'' that we suggested as a possible explanation in Paper~I, a scenario in which a galaxy accretes pristine IGM too quickly for normal enrichment processes and becomes trapped in a low metallicity / low star-formation rate state.
As discussed in \secref{intro}, \cite{Vorobyov2009} use numerical hydrodynamic modeling of an LSB galaxy, and they find that their simulated galaxy exhibits bursts of star formation on timescales less that 20~Myr, similar to what we find for UGC~628. Star formation in the simulated galaxy peaks 2.6~Gyr after assembly (9.8~Gya), declining to a current value of \sfr{0.08}. Star formation in UGC~628 also experienced a peak around a similar time, with about 60\% of its stars formed during the 8-10~Gya timestep (see \tblref{timeline}). This is grossly in keeping with the concept of galactic downsizing, wherein both models and large surveys suggest that most of the star formation in ``typical'' galaxies in the mass range of UGC~628 should be at $z\sim1$ \citep[e.g.][]{Leitner2012,Moster2013,Behroozi2013}. However, since the 8-10~Gya era, the behavior of UGC~628 has departed from the model galaxy in \cite{Vorobyov2009}, with star formation not declining steadily but instead exhibiting wild swings. Indeed, a particularly vigorous episode of star formation occurred fairly recently in the 20-49~Mya time-step.
The bar may figure significantly into the overall history of UGC~628. \cite{Chequers2016} provides a thorough dynamical discussion of the formation of the bar in UGC~628 specifically, and find that their model best matches the current appearance of UGC~628 several Gyr after peak bar strength. However, the evolution in appearance after peak bar strength is fairly slow, and so \cite{Chequers2016} note that the window of time during which the appearance would match is fairly wide. Indeed, \cite{Mihos1997} find that, since LSB spiral discs are so stable, bars may be difficult to form but, once formed, are long lived.
In the earliest era, 8-10~Gya, we see star formation cover the area of the bar, almost entirely and almost exclusively. As we noted above, many dynamical times have passed, and all we can say is that most of the stars currently spatially coincident with the bar formed during this time period, and that comparatively few stars formed during this period which are not now spatially coincident with the bar. \cite{Chequers2016} find that the bar in their simulated galaxy begins to form when their galaxy is 3~Gyr old and peaks in strength at 5~Gyr. If we assume that the events in the 8-10~Gya timestep represent peak bar strength, that would marginally allow the formation of UGC~628's disc within the accepted age of the universe. It is much more likely that star formation in the 8-10~Gya timestep is associated with the era of bar growth.
\begin{figure}
\includegraphics[width=0.4\textwidth]{fig8_analogs-eps-converted-to.pdf}
\caption{SDSS DR14 ugr images of size spiral discs galaxies with an overall low surface brightness but marked by large star-forming complexes in the outer disc. These may be an analogs to UGC~628 during the burst phases of its life cycle.}
\label{fig:analogs}
\end{figure}
Regardless of the bar, early star formation in UGC~628 was much more centrally concentrated than it is now. In \tblref{timeline} and the lower right panel of \figref{timeline} we see that 60\% of the stellar population formed during this era, and within the central few kpc. This period of early formation matches the conclusion in previous works \citep[e.g.,][]{vandenHoek2000,Boissier2008,Schombert2014}, that, despite their overall blue colours, LSB spirals cannot be totally unevolved galaxies with only young stars. It would seem UGC~628 began its life on the inside-out track, and only after the shutdown of the bar/core did star formation follow the patchy/sporadic mode. The magnitude and concentration of early star formation sets the 8-10~Gya era apart from the others, and suggests that a real transition followed. Looking at the azimuthally averaged star-formation rate profiles in \figref{radial}, we see a clear migration of star formation out of the bar/core region. Likewise, in \tblref{timeline} the quantity $R_{SF}$ is the average radius weighted by the star-formation rate; from the earlier timesteps to the more recent ones it increases by almost a factor of two, and almost monotonically. Star formation in the most recent timesteps seems to preferentially avoid the bar/core, often occurring in clumps on the outer edge of the disc.
The evolution of the stellar metallicities shown in \figref{timeline} makes the galactic overfeeding scenario that we discussed in Paper~I seem unlikely, at least in UGC~628's centre. In the 8-10~Gya timestep we see the core dominated by low metallicity star formation while the outer disk hosted higher metallicity star formation. As with all our stellar metallicities, these should be taken with caution (see \secref{analysis}). But, even with significant systematic errors, it would still be true that most of the stars formed around the core had lower metallicities than those in the outer disk. This suggests that during this era the core had an ample supply of metal poor gas, possibly via accretion, while the outer disk was left to self enrich. In the next timestep we see an overall drop in star formation, with only higher metallicity stars forming around the core, as we would expect if the core had exhausted the supply of metal poor gas. If star formation in UGC~628 had been suppressed due to poor cooling or self shielding in low metallicity clouds, then we would expect to see star formation ``seeded'' in areas where higher metallicities are achieved, which is the opposite of our observations. As such, our data are more suggestive of shutdown due to gas depletion, expulsion, or heating.
During the 20-49~Mya era UGC~628 would have met both the technical definition starburst galaxy (star-formation rate high enough to grow its current stellar population in less than a Hubble Time), and of an LSB galaxy (faint central brightness). This statement is robust: even if our estimates for the star-formation rate are incorrect by a factor of nine, UGC~628 would still have been a starburst galaxy during this era. With a small number of extremely active star forming complexes on the outer edge of an otherwise faint disc, such a starburst LSB galaxy would differ significantly from the more traditional nuclear starbust galaxies.
During a starburst phase, especially 20-49~Mya, UGC~628 likely resembled the galaxies shown in \figref{analogs}. These galaxies would not normally be classified as LSB spirals because of their bright blue clumps, despite an otherwise low surface brightness disc. The core region in these galaxies is brighter and redder and the surrounding disk, though still comparatively faint and blue, similar to UGC~628. It is tempting to think of these objects as analogs to the Cartwheel Galaxy, which was likely a giant LSB disc before a face-on interaction with a smaller galaxy which plunged through the centre of the disk. However, like UGC~628, none of the examples in \figref{analogs} is a giant LSB, nor do any show signs of interaction or disturbance or have any obvious companions. In the case of UGC~628, the scenario where a dwarf galaxy plunges through the centre of the disk face-on is clearly ruled out as an explanation for the edge-dominated star formation since, in this scenario, we would expect star formation to start from the centre and work outwards, whereas we see random patches of star formation that simply tend to be found more often near the edge. We can also rule out a massive but distant companion since there are no galaxies near UGC~628 visible in the SDSS images.
If the recent starburst was due to an interaction, it was likely a small companion which has since merged with or been obscured by UGC~628. The upper right panel of \figref{timeline} shows that the stars formed during the starburst had a lower metallicity than most of the stars formed since the early HSB-like burst 8-10~Gya. One interpretation would have a small gas-rich galaxy merging with UGC~628, diluting the ISM metallicity, and triggering a starburst. However, this scenario would have to be reconciled with the flat/inverted gas-phase metallicity profile we reported in Paper~I. Simplistically, the addition of low metallicity gas at the edge of the disk would make the metallicity profile steeper. The enrichment from the starburst mitigates the problem, but it may not solve it entirely. This problem would be solved if the inflow of gas was distributed throughout the disk of UGC~628, but in that case it is unclear why the starburst was localized near the edge.
It is also possible that the hypothetical companion was gas poor and/or did not contribute much to the ISM gas involved in the associated starburst. This scenario is particularly interesting because, if correct, it would lend credence to the concept that minor interactions and mergers with may play a significant role in the star-formation histories of their larger partners through means other than the simply contributing to the supply of cool gas. For example, \cite{Tanaka2017} find low surface brightness features around the Seyfert galaxy M77, which suggest a minor merger on a timescale compatible with the idea that the merger initiated a burst of star formation and potentially AGN activity. Several works \citep[e.g.,][]{MartinezDelgado2012} find compelling evidence that activity in the starburst dwarf irregular NGC4449 was triggered by a merger with an even smaller dwarf. At the extreme, even tiny galaxies such as DDO~68 show signs of starburst triggered by the accretion of faint dwarfs \citep{MartinezDelgado2012,Annibali2016}. In all of these cases, the tell-tail signs come from ultra-low surface brightness features, which would not be visible in any existing images of UGC~628. One interpretation of the star-formation histories presented in this paper is that we are seeing the accretion history of UGC~628, marked by edge-dominated star-forming events.
The bottom right panel of \tblref{timeline} and \figref{timeline} give a sense of the significance of the later bursts of star formation to UGC~628. 60\% of the stellar mass of UGC~628 was formed in an inside-out, HSB-like event. The later starbursts contributed 40\% of the stellar mass, but primarily much further out in the galaxy. Those characteristics are in line with what would be expected from accretion events. However, the accretion of stars normally contributes to the buildup of a spheroid, and these events seem to have contributed to the buildup of the disk. If they were accretion events, it seems likely that either they primarily contributed gas or their primarily influence was to disturb the existing gas disk.
We see evidence for annular morphology reminiscent of that found in UGC~628 in H{$\,$\footnotesize I}{} observations of other LSB spirals. For example, \cite{Pickering1997} find large amounts of diffuse H{$\,$\footnotesize I}{} in their sample of four giant LSB galaxies. The H{$\,$\footnotesize I}{} is below the critical density for star formation \citep{Kennicutt1989} throughout most of the disc, reaching the threshold only in a few locations at large radii. If UGC~628 is analogous, H{$\,$\footnotesize I}{} maps of UGC~628 would likely show a gaseous disc which is subcritical except where we see star formation, near the visible edge of the disc. In a more theory-oriented work looking at multi-component models of Milky Way-mass LSB spirals, \cite{Garg2017} suggest that disc stability in many LSB spirals may reach a local minimum and achieve star formation only at radii several kpc from the galactic centres. It seems plausible, then, that UGC~628 and the galaxies in \figref{analogs} represent different phases of internally driven disc evolution which, in these cases, favors star formation at large radii.
Careful studies of HSB spirals show that edge-dominated star-formation and starbursts centered in the outer disk are, in fact, not unheard of in late-type spirals. \cite{Perez2013} examine the first 105 galaxies in the CALIFA survey, and find that inside-out formation is the dominant mode of galaxy assembly in systems larger than \mstar{=6\times 10^{10}\rm M_\odot}, but, below that limit, star formation is more evenly distributed throughout the disk, and small systems even exhibit outside-in formation. \cite{Huang2013} use spectra of 1000 galaxies to show that edge-dominant starbursts are more common in late-type and/or low surface density galaxies. These findings set a precedent for edge-dominated starbursts, although UGC~628 is more extreme than these examples. The outer-disk starbursts described in \cite{Huang2013} typically only account for 10-20\% of the total stellar mass, but 40\% of the stellar mass in UGC~628 formed in the patch/sporadic, edge-dominant mode.
We can also look to the gas-phase metallicity for clues to past star formation. In a study of star-forming galaxies with \lmstar{>10}, \cite{Moran2012} find that the most gas-rich 10\% of their sample show sudden drops in the gas-phase metallicity in their outer disks, which they interpret as a sign of recent accretion. In Paper~I we reported the opposite in UGC~628, a nearly flat but slightly inverted gradient in the gas-phase metallicity, as sampled by the R23 bright line method. When compared to the findings in \cite{Moran2012}, a simplistic interpretation would favor secular disk instabilities instead of accretion events as the drivers of the outer-disk starbursts in UGC~628. However, \cite{Bresolin2015} find that seemingly flat or shallow metallicity gradients in LSB spirals are actually very similar to the steeper metallicity gradients in HSB spirals when recalculated as gradients with respect to the exponential scale radii instead of physical units (kpc). This makes intuitive sense: if star formation in LSB spirals is spread out over a larger area, the enrichment will be as well. In this view, edge-dominated star formation would cleanly explain the larger scale radii of LSB spirals. If UGC~628 fits into this scheme in a straightforward way, it may be a typical object at a transitional time: The recent and fairly intense burst of star formation 20-49~Mya may have temporarily left it with a slightly inverted gradient, to be flattened out over time by further bursts and, to some extent, by radial migration. These hypotheses could be explored by metallicity measurements of the extended disc, beyond the bright H$\,${\small II}{} regions we report in Paper~I.
\section{Summary and Conclusions}
\label{sec:summary}
By analyzing space-based UV and IR photometry in tandem with ground based IFU spectra we have produced spatially resolved maps of the star-formation history of the LSB spiral UGC~628. We have demonstrated that this technique is stable and reliable, and not particularly sensitive to most assumptions about environmental parameters. The exception is the choice of extinction law; after exploring a battery of extinction laws, we find that the Calzetti Extinction Law best agrees with our data. We find that, although UGC~628 exhibits only low-level star formation today, it has been more active in the past, broadly in agreement with existing numerical and hydrodynamic simulations. Our main findings regarding the history of UGC~628 are summarised below:\\
\begin{enumerate}
\item UGC~628 is \about{10}Gyr old, with \about{60\%} of its stellar mass formed within the first 2~Gyr.
\item Early star formation seems to have followed the inside-out pattern, as in HSB galaxies, and may be associated with the formation of UGC~628's bar.
\item During the earliest era, 8-10~Gya, UGC~628 was likely an HSB spiral. Since that time, it has been an LSB spiral, although at times only marginally.
\item After the first few Gyr, UGC~628 transitioned to a patchy/sporadic mode of star formation.
\item Recent star formation has avoided the central 2kpc and, instead, shows a marked preference for the visible edge of the galaxy. It may be that these edge-dominated bursts mark the accretion of smaller objects.
\item During some eras, such as 20-49~Mya, UGC~628 would have exhibited large star forming complexes on the visible edge of the galaxy. These properties would likely have excluded UGC~628 from most LSB catalogs and surveys, even though the underlying stellar disc would still have had \sbB{> 22}.
\item During the 20-49~Mya era specifically the edge-dominated star-formation rate was high enough to qualify UGC~628 as a starburst galaxy.
\item The current era can best be described as post-burst, with Region~$A$ as a relic from the recent episode.
\end{enumerate}
\subsection{Future Work}
\label{sec:futurework}
The recent burst in UGC~628 20-49Mya suggests the possibility of an interesting class of galaxy, a starburst LSB spiral. Further exploration into galaxies such as those shown in \figref{analogs} may reveal if they are truly analogs to UGC~628 during this period. If this patchy/sporadic mode of star formation is common among LSB spirals, then it may be possible to examine galaxies at different phases in the burst cycle to better explain the physics driving and inhibiting star formation in LSB spirals.
As before, we emphasise that UGC~628 is only one object, and we caution against drawing global conclusions of LSB spirals based on the history of UGC~628. Indeed, since it is a barred LSB spiral it may not even be typical for this class. Following up on our analysis of the star-formation history of UGC~628, the goal of the MUSCEL program is to extend our methodology to other LSB spirals.
Preliminary analysis of LSB spirals in our sample suggests that comparatively flat ISM metallicity gradients are common in our sample, in keeping with earlier findings of other LSB galaxies \citep{deBlok1998I}. As noted in \secref{context}, the analysis in \cite{Bresolin2015} cleanly explains this in the context of enrichment: typical LSB spirals have flatter metallicity profiles because the stellar population have shallower profiles. UGC~628 would seem to be a transitional object, with a slightly inverted metallicity profile as a result of its recent burst of annular star formation. If this is correct, then we can utilise our program's ability to detect past bursts of star formation and see if there is a relationship between the ISM metallicity gradient and the location of and time since the most recent burst.\\
This work was done in collaboration with the Mount Holyoke College GeoProcessing Lab.\\
This work was performed in part using high performance computing equipment obtained under a grant from the Collaborative R\&D Fund managed by the Massachusetts Technology Collaborative.
\bibliographystyle{apj}
|
1,108,101,566,746 | arxiv | \section{Introduction}
The spin-boson model is an important toy model for investigating
the influence of dissipation on quantum tunneling and has a wide
range of applications.\cite{leg,weiss} Over decades the model has
been studied by various methods,\cite{leg} like path
integral,\cite{fey} renormalization group
calculation,\cite{bray,cha} variational
calculation,\cite{em,hew,zw,sh,chen,nie,ct} and the numerical
renormalization group(NRG) calculation,\cite{bulla} etc.. One
important issue is to study the cross-over from the delocalized to
localized phases as the dissipation increases. Most of the studies
are concentrated on the Ohmic dissipation case which is considered
as corresponding to real physical systems and the cross-over
picture is well understood. On the other hand, the situation for
the sub-Ohmic dissipation, which is of less physics interest but
still important for a well understanding of the spin-boson model,
has some confusions. Renormalization group calculation shown that
quantum tunneling is totally suppressed by dissipation for any
non-zero sub-Ohmic coupling at T=0,\cite{cha,leg} while different
conclusion was found by mapped the spin-boson model to an Ising
model\cite{sp} and using the well-known result for Ising
model.\cite{dys} The sub-Ohmic case was also studied by using
infinitesimal unitary transformation and the cross-over was found
to be discontinuous.\cite{keh} Recently, the NRG calculation,
which is considered as a powerful tool for investigation of the
Kondo model and its generalizations, confirmed the delocalized to
localized phase cross-over in sub-Ohmic dissipation case and the
cross-over is identified as continuous.\cite{bulla} Variational
calculation has been used to study the spin-boson model with a
Ohmic bath and the result of cross-over boundary is in good
agreement with the renormalization group
calculation.\cite{zw,sh,chen,nie} The variational calculation for
non-zero temperature\cite{sh} was generalized to sub-Ohmic case
recently and the discontinuous cross-over behavior was found to
exist at non-zero-temperature.\cite{ct} Up to now, the description
for this discontinuous cross-over is just limited to the
discontinuous change of the tunneling splitting at the cross-over
point, while a scenario for such a discontinuous cross-over is
still lacking. According to Ginzburg-Landau theory{\cite{gol,cl},
the evolution of the free energy around the critical point for the
first order(discontinuous) phase transition is rather complicated
and merely a discontinuous change of order parameter at the
cross-over point is certainly no enough for a complete description
of this discontinuous transition. In this paper, we present
further analysis on this discontinuous cross-over by examining the
evolution of the solutions of the self-consistent equation derived
from the variational calculation. It is found that the evolution
of the solutions near the phase boundary is consistent with the
general picture of the first order phase transition. Basing on the
constructed picture, it is shown that the critical points
determined in the general way are not thermodynamically critical
points and the true critical point is calculated. The arrangement
of the paper is as follows. In the next section, the model and a
brief explanation on variational calculation are presented. In
section III we present analysis on the discontinuous phase
transition by comparing the evolution of the solutions of the
self-consistent equation for Ohmic and sub-Ohmic dissipation cases
near the critical point. Conclusions and discussion are given in
the last section.
\section{The model and variational calculation}
The Hamiltonian of the spin-boson model is given
by(setting $\hbar=1$)\cite{leg,weiss}
\begin{equation}
H=\frac{\epsilon}{2}\sigma_z+\frac{\Delta}{2}\sigma_x+\sum_k
b_k^{\dagger}b_k\omega_k+\sigma_z\sum_k c_k(b_k^{\dagger}+b_k),
\end{equation}
where $\sigma_i(i=x,y,z)$ is the Pauli matrix,
$b_k(b_k^{\dagger})$ is the annihilation(creation) operator of the
$k$th phonon mode with energy $\omega_k$ and $c_k$ is the coupling
parameter. The main interest of the present paper will be the zero
temperature so we set the bias $\epsilon=0$ in the following. It
is known that the solution of this model is determined by the
so-called the bath spectral function(density) defined
as\cite{leg,weiss}
\begin{equation}
J(\omega)=\pi \sum_kc_k^2\delta(\omega-\omega_k).
\end{equation}
Generally $J(\omega)$ is characterized by a cut-off frequency
$\omega_c$ and has a power-law form, i.e.,
\begin{equation}
J(\omega)=\frac{\pi}{2}\alpha
\omega^s/\omega_c^{s-1},~~~~~~~0<\omega\le\omega_c,
\end{equation}
where $\alpha$ is a dimensionless coupling strength which
characterizes the dissipation strength. Parameter $s$ specifies
the property of the bath, $s=1$ is the case of Ohmic dissipation
and $0\le s<1$ the sub-Ohmic dissipation case. It should be noted
that $J(\omega)$ can take some different forms,\cite{leg,ct} like
$J_1(\omega)=\frac{\pi}{2}\alpha
\omega^s/\omega_c^{s-1}e^{-\omega/\omega_c}$ and
$J_2(\omega)=\frac{\pi}{2}\alpha \omega^s/\omega_s^{s-1}$ with
$\omega_c \rightarrow \infty$, while we find that the solution is
almost the same in sub-Ohmic case(see below).
As one can see from the Hamiltonian given in Eq.(1), when
$\Delta=0$ the localized phase is stable since in this case we
have $[\sigma_z,H]=0$. This result implies that the coupling to
phonon bath alone cannot lead to tunneling and thus this problem
can be treated approximately in the way without coupling to the
bath as given in the quantum mechanics textbook.\cite{lan} To
ensure the tunneling is small which is a precondition of our
treatment, the following calculation is restricted to the
condition $\Delta/\omega_c\ll 1$. We denote the eigen-states of
spin-up(down)-plus-bath as
$|\uparrow\rangle|\phi_+\rangle(|\downarrow\rangle|\phi_-\rangle)$,
where $|\phi_{\pm}\rangle$ represent the eigen-state of the phonon
bath without tunneling. When the tunneling is taken into account,
the eigen-state of the whole system can be approximately given
by\cite{lan}
\begin{equation}
|\Phi_{\pm}\rangle=(|\uparrow\rangle|\phi_+\rangle\pm
|\downarrow\rangle|\phi_-\rangle)/\sqrt{2},
\end{equation}
then the tunneling splitting in the presence of dissipation is
\begin{equation}
\Delta'=\langle\Phi_+|H|\Phi_+\rangle-\langle\Phi_-|H|\Phi_-\rangle=
\Delta\langle\phi_+|\phi_-\rangle,
\end{equation}
a well known result that $\Delta'$ is determined by the overlap
integral of the phonon ground states.\cite{chen,nie} In the
absence of tunneling(i.e., $\Delta=0$), Hamiltonian (1) can be
diagonalized by a well-known displaced-oscillator-transformation
and the phonon ground states are the so-called
displaced-oscillator-states\cite{mah}
$$
|\phi_{d\pm}\rangle={\rm
exp}\{\pm\sum_k\frac{c_k}{\omega_k}(b_k-b_k^{\dagger})\}|0\rangle,
$$
where $|0\rangle$ is the vacuum state of the phonon. The essence
of the variational calculation is that, in the presence of
tunneling, the phonon ground states are suggested to still have
the same from, i.e.,
\begin{equation}
|\phi_{\pm}\rangle={\rm
exp}\{\pm\sum_kg_k(b_k-b_k^{\dagger})\}|0\rangle,
\end{equation}
but leaving the parameter $g_k$ to be determined from the
condition that the ground state energy of the whole system is a
minimum with respect to $g_k$. Substituting the above equation to
Eq.(4), the ground state energy of the whole system is found to be
\begin{equation}
E[g_k]=\sum_k(\omega_kg_k^2-2c_kg_k)-\frac{1}{2}\Delta\exp\{-2\sum_kg_k^2\},
\end{equation}
which is a functional of $g_k$, then $\frac{\delta E}{\delta
g_k}=0$ leads to
\begin{equation}
g_k=\frac{c_k}{\omega_k+ \Delta\exp\{-2\sum_kg_k^2\}},
\end{equation}
the tunneling splitting, by Eq.(5), is given by
\begin{equation}
\Delta'= K \Delta, ~~~~~~K=F[g_k]\equiv \exp\{-2\sum_kg_k^2\}.
\end{equation}
Using Eq.(8) and the definition of the spectral function, we find
that $K$ is determined by the following self-consistent(or
variational) equation
\begin{equation}
K=f(K),~~~~~~f(K)\equiv {\rm exp}
\{-\alpha\int_0^1\frac{x^s~dx}{[x+(\Delta/\omega_c)K]^2} \}.
\end{equation}
Such a kind of self-consistent equation has been derived in
previous works\cite{hew,zw,sh,chen,nie,ct} and it plays an
important role in dealing with the cross-over from the delocalized
to localized phases. It is easy to see that $K=0$ is the trivial
solution of Eq.(10) and this solution represents the localized
phase. When the coupling strength $\alpha$ is large enough, $K=0$
is the only solution of the self-consistent equation, while as
$\alpha$ decreases to some value $\alpha_c$, the self-consistent
equation begins to have, in addition to the trivial solution,
non-zero solutions, then $\alpha_c$ is identified as the critical
point at where the cross-over from the localized($K=0)$ to
delocalized($K>0$) phases happens. This is the general way to
determine the phase boundary used in previous
works.\cite{zw,sh,chen,nie,ct}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{sb3f1.eps}
\caption{Phase boundary determined by $\alpha_c$ for various
$\Delta/\omega_c$. The inset shows the comparison with the result
by using different spectral function $J_1(\omega)$ and
$J_2(\omega)$(see the text) in the case of
$\Delta/\omega_c=0.01$.}
\end{figure}
In the case of Ohmic dissipation, i.e., $s=1$, the self-consistent
equation can be solved analytically and the phase boundary is
found to be $\alpha_c=1$ in the case of $\Delta/\omega_c\ll 1$, in
agreement with the renormalization group
calculation.\cite{zw,nie,sh} In the case of sub-Ohmic dissipation,
the self-consistent equation can be solved numerically and the
phase boundary between the localized($K=0)$ and delocalized($K>0$)
phases determined in this way is shown in Fig.1. The result by
using different spectral functions, i.e., $J_1(\omega)$ and
$J_2(\omega)$, are also shown in the inset. Our result shows that,
for $\Delta/\omega_c\ll 1$, the phase boundary is almost the same
for all three spectral functions as $s\le 0.7$, while $\alpha_c$
is a little bit lower for $J_2(\omega)$ when $s>0.7$. Also, it is
found that the relation between the critical coupling $\alpha_c$
and $\Delta/\omega_c$ has a simple power-law form $\alpha_c\propto
(\Delta/\omega_c)^{1-s}$ as found by NRG calculation.\cite{bulla}
Notably, such a relation can be deduced from Eq.(16) in
ref.\cite{keh} by using the spectral function given
here.\cite{note} However, as we shall show in the next section,
the $\alpha_c$ determined in this way for sub-Ohmic case is not
thermodynamically the critical point, but just the the limit of
metastability for superheating of the first order phase
transition.\cite{cl}
\section{The discontinuous cross-over in sub-Ohmic
case} Now we turn to present a scenario for such a discontinuous
cross-over from the delocalized to localized phases in sub-Ohmic
case. The key point is to examine the evolution of the solutions
of the self-consistent equation with the coupling strength
$\alpha$ near the phase boundary. For clarity, we first see what
happens in the Ohmic case. Fig.2 shows the evolution of the
solutions of Eq.(10) with the increase of $\alpha$ in Ohmic
dissipation case. When $\alpha>\alpha_c$, we have the trivial
solution only, while a non-zero solution($K_1\not=0$) appears for
$\alpha<\alpha_c$. As one can see from the figure, the non-zero
solution $K_1$ continuously tends to 0 as $\alpha$ approaches
$\alpha_c$. This is consistent with the picture of a
continuous(second order) transition: \cite{gol,cl} above the
critical point($\alpha>\alpha_c$), there is only one stable
phase(one energy minimum located at some $g_{k0}$ satisfying
$F[g_{k0}]=0$ in the present case), below the critical point, this
stable phase becomes unstable($E[g_{k0}]$ becomes the maximum of
the energy) and a second stable phase appears($E[g_{k1}]$ is the
new energy minimum, where $F[g_{k1}]=K_1>0$), the cross-over
behavior is continuous.
The situation for the sub-Ohmic dissipation case is qualitatively
different. As shown in Fig.3, when $\alpha<\alpha_c$,
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{sb3f2.eps}
\caption{Evolution of the solutions of Eq.(10) with the increase
of coupling strength $\alpha$ for $\Delta/\omega_c=0.1$ in the
case of Ohmic dissipation $s=1$.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{sb3f3.eps}
\caption{Evolution of the solutions of Eq.(10) with the increase
of coupling strength $\alpha$ for $\Delta/\omega_c=0.1$ in the
case of sub-Ohmic dissipation $s=0.3$. }
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.23\textwidth]{sb3f41.eps}
\includegraphics[width=0.23\textwidth]{sb3f42.eps}
\caption{The $\alpha$-dependence of the non-zero solution of
Eq.(10) for $s=1$(left) and $s=0.3$(right). As
$\alpha/\alpha_c\rightarrow 1$, the non-zero solution of $s=1$
approaches 0 continuously, while for $s=0.3$, the non-zero
solution jumps from $K_0\not=0$ to 0.} \label{aaa}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{sb3f5.eps}
\caption{Evolution of the K-dependence of the ground state energy E
with coupling strength $\alpha$ for sub-Ohmic case($s=0.3$) when
$\Delta/\omega_c=0.1$. }
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{sb3f6.eps}
\caption{The evolution of the energy extrema with the coupling
strength $\alpha$ for the sub-Ohmic case. Thermodynamically the
cross-over point is $\alpha_1$, while $\alpha_c$ is just the point
where the second minimum begins to develop.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{sb3f7.eps}
\caption{The phase boundary determined by $\alpha_c$ and
$\alpha_1$ in the case of $\Delta/\omega_c=0.1$. }
\end{figure}
there are {\it two} non-zero solutions of Eq.(10)($K_2>K_1\not=0$)
in additional to the trivial solution.\cite{e1} As the coupling
strength $\alpha$ increases, $K_2$ decreases while $K_1$ increases
and tends to meet $K_2$ as $\alpha$ approaches $\alpha_c$. At
$\alpha=\alpha_c-0$, we have $K_1=K_2=K_0\not=0$ and at this point
$K_0$ is the point of tangency for the line $y=x$ and curve
$y=f(x)$. At $\alpha=\alpha_c+0$, the solution $K_0$ disappears
{\it suddenly} and only the trivial solution is found. The
$\alpha$-dependence of the non-zero solutions of Eq.(10) for Ohmic
and sub-Ohmic dissipation cases are shown in Fig.4 where one can
see the qualitatively different cross-over behavior. The result of
sub-Ohmic dissipation case clearly shows that the cross-over is
discontinuous since the non-zero solution of Eq.(10) and thus the
tunneling splitting $\Delta'$ changes discontinuously at the point
$\alpha=\alpha_c$. Such a behavior was found before\cite{keh,ct}
and took as the evidence for a discontinuous transition since the
tunneling splitting has a physics meaning of the order parameter.
What we want to emphasize here is the two non-zero solutions when
$\alpha<\alpha_c$. Physically we need to know which solution is
stable and the meaning of the second non-zero solution. As one can
see from Fig.4(right), the non-zero solution $K_1$ increases with
$\alpha$, one can intuitively conclude that $K_1$ is unstable
since physically the tunneling splitting should decrease with
$\alpha$. Basing on the variational calculation, we cannot make
further analysis on the stability of the solution, so we turn to
energy analysis, i.e., to see which solution is energy preferable.
Typical evolution of the $E-K$ curve with $\alpha$ is shown in
Fig.5. The result shows that, we have $E(K_1)>E(0)$ and
$E(K_1)>E(K_2)$ when $\alpha\ll \alpha_c$, $E(0)$ decreases while
both $E(K_1)$ and $E(K_2)$ increase relatively as $\alpha$
increases but $E(K_1)$ is always the largest, finally
$E(K_1)=E(K_2)=E(K_0)$ as $\alpha=\alpha_c$ and we have
$E(0)<E(K_0)$. This implies that, both $K=0$ and $K_2$ are energy
preferable while $K_1$ is unstable when $\alpha<\alpha_c$. Such a
result is consistent with the scenario of a first order
transition. In the scenario of the first order
transition,\cite{gol,cl} below the critical point, there are two
free energy minima and a maximum lies between, while above the
critical point, only one global free energy minimum survives. In
the sub-Ohmic dissipation case, when $\alpha<\alpha_c$, three
solutions of Eq.(10) represent the two energy minima and one
energy maximum, that is, $E[g_{k0}]$ with $F[g_{k0}]=0$ and
$E[g_{k2}]$ with $F[g_{k2}]=K_2$ are the two energy minima, while
$E[g_{k1}]$ with $F[g_{k1}]=K_1$ is the energy maximum lies
between as shown in Fig.6. As $\alpha$ increases and approaches
$\alpha_c$, $E[g_{k1}]$ tends to meet $E[g_{k2}]$ and at the point
$\alpha=\alpha_c$, these two energy extrema merge into a point of
inflection at $F[g_k]=K_0$, then only one energy minimum
$E[g_{k0}]$ survives when $\alpha>\alpha_c$. Based on the picture
for the discontinuous phase transition, it is now clear that
$\alpha_c$ is {\it not} the critical point for the cross-over to
happen, but just the point where the second energy minimum begins
to develop. $\alpha_c$ can be considered as the limit of
metastability for superheating,\cite{cl} i.e., the limit of
metastability for increasing the dissipation strength in the
present case. Thermodynamically the critical point, as shown in
Fig.6, should be $\alpha_1$ where we have\cite{gol,cl}
\begin{equation}
E(K_2)=E(0),
\end{equation}
from this, $\alpha_1$ can be determined by Eqs.(7), (8) and (10).
Comparison between the phase boundary determined by
$\alpha_c$ and $\alpha_1$ is shown in Fig.7. It is easy to see
that $\alpha_1<\alpha_c$ while the difference between $\alpha_c$
and $\alpha_1$ decreases as $s$ increases and tends to zero as
$s\rightarrow 1$ where the transition becomes continuous. We also
find that the difference between $\alpha_c$ and $\alpha_1$
decreases with $\Delta/\omega_c$.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{sb3f8.eps}
\caption{The phase boundary determined by $\alpha_1$ for various
$\Delta/\omega_c$. The inset shows the comparison with the NRG
calculation in the case of $\Delta/\omega_c=0.001$.}
\end{figure}
The phase boundary deduced in this way is shown in Fig.8 which is
similar to that shown in Fig.1 but with all the critical points
lower. It is found that the phase boundary determined by
$\alpha_1$ is in good agreement with that obtained by NRG
calculation when $\Delta/\omega_c \le 0.01$.
\section{Conclusions and discussion}
In conclusion, we have study the cross-over behavior from
localized to delocalized phases of a spin-boson model with a
sub-Ohmic bath by variational method. By examining the evolution
of the solutions of self-consistent equation (10) with the
coupling strength, we are able to present the scenario of the
discontinuous transition in sub-Ohmic dissipation case. Based on
the constructed picture, it is shown that the $\alpha_c$, at where
the self-consistent equation begins to have non-zero solutions, is
not thermodynamically the critical point, but just the point where
the second energy minimum begins to develop. The true critical
point is determined according to Ginzburg-Landau theory for the
first order phase transition and the obtained phase boundary is
in agreement with the NRG calculation. Our analysis shows that the
cross-over behavior in spin-boson model is directly related to the
evolution of solutions of the self-consistent equation derived
from the variational calculation. The evolution behavior of
solutions for a continuous cross-over(in Ohmic dissipation case)
is qualitatively different from that of a discontinuous
cross-over(in sub-Ohmic dissipation case). The present work, on
one hand, provides convincing evidence for a discontinuous
cross-over in sub-Ohmic case and on the other hand, demonstrates
the new way to deal with the cross-over behavior in spin-boson
model by the variational method.
According to the definition of stable and unstable fixed points
for renormalization group,\cite{jp} geometrically one can see from
Fig.3 that, both $K=0$ and $K_2$ are stable fixed points while
$K_1$ is unstable fixed point as $\alpha<\alpha_c$ in sub-Ohmic
case. On the other hand, we only have one stable fixed point(i.e.,
$K_1$) and one unstable fixed point as $\alpha<\alpha_c$ in Ohmic
case. This result is in agreement with the NRG calculation,
\cite{bulla} where 3 fixed points(2 stable and 1 unstable) were
found in sub-Ohmic case while the third unstable fixed point
disappeared in Ohmic case. However, the cross-over behavior in
sub-Ohmic case was identified as continuous in NRG calculation,
this implies further analysis is needed for seeking a deeper
relation. Although the work by Kehrein and Mielke is not based on
the variational calculation,\cite{keh} the cross-over behavior was
studied by a self-consistent equation and the discontinuous
behavior was judged by the discontinuous change of the tunneling
splitting at the critical point $\alpha_c$, where the
self-consistent equation begins to have non-zero solutions. Some
results, like the $(\Delta/\omega_c$) dependence of critical
coupling $\alpha_c$ and the $s$-dependence of tunneling splitting
at the critical point also show quantitative agreement with our
work determined from Eq.(10) at $\alpha=\alpha_c$. This may lead
to a conclusion the the critical point determined in
ref.\cite{keh} is just $\alpha_c$ given in the present work, i.e.,
not thermodynamically the critical point.
The author(Chen) thanks Dr. N. H. Tong for supplying the data of
phase diagram by NRG calculation. This work was supported by a
grant from the Natural Science Foundation of China under Grant
No. 10575045.
$^*$ Author to whom correspondence should be addressed. E-mail: [email protected]
|
1,108,101,566,747 | arxiv | \section{Introduction}
Computational high-throughput screening ever-increasingly allows the
coverage of larger subsets of chemical space. Extracting a property
of interest across many compounds helps infer structure-property
relationships, of interest both for a better understanding of the
physics and chemistry at hand, as well as for materials
design~\cite{pyzer2015high, jain2016computational, Bereau2016,
von2018quantum}. Recent hardware and algorithmic developments have
enabled a number of applications of high-throughput screening in hard
condensed matter~\cite{curtarolo2013high, ghiringhelli2015big}, while
comparatively slower development in soft
matter~\cite{ferguson2017machine, bereau2018data}.
Soft-matter systems hinge on a delicate balance between enthalpic and
entropic contributions, requiring proper computational methods to
reproduce them faithfully. Physics-based methodologies, in particular
molecular dynamics (MD), provide the means to systematically sample
the conformational ensemble of complex systems. High-throughput
screening using MD presumably involves one simulation per compound.
This remains computationally prohibitive at an atomistic resolution.
As an alternative, we recently proposed the use of coarse-grained (CG)
models to establish a high-throughput scheme. Coarse-grained models
lump several atoms into one bead to decrease the number of degrees of
freedom~\cite{Noid2013}. The computational benefit of a
high-throughput coarse-grained (HTCG) framework is two-fold: ($i$)
faster sampling of conformational space; but most importantly ($ii$) a
significant reduction in the size of chemical space, tied to the
transferable nature of the CG model (i.e., a finite set of bead
types). Effectively the reduction in chemical space due to
coarse-graining still leads to a combinatorial explosion of chemical
space, but with a significantly smaller prefactor. This many-to-one
mapping is empirically probed by coarse-graining large databases of
small molecules---an effort made possible by automated force-field
parametrization schemes~\cite{Bereau2015}. Using the CG Martini force
field~\cite{periole2013martini}, we recently reported the
\emph{exhaustive} characterization of CG compounds made of one and two
beads, corresponding to small organic molecules between 30 and 160~Da.
Running HTCG for 119 CG compounds enabled the predictions of
drug-membrane thermodynamics~\cite{menichetti2017silico} and
permeability~\cite{menichetti2018drug} for more than $500,000$ small
organic molecules.
Pushing further our exploration of chemical space, how do we further
diversify the probed chemistry in a systematic manner? Scaling up to
larger molecular weight using more CG beads will ultimately lead to a
combinatorial explosion: there are already 1,470 linear trimers and
19,306 linear tetramers in Martini. Instead of an exhaustive
enumeration, we propose to explore regions of chemical space that are
of particular interest. Specifically, rather than following the
chemistry we navigate chemical space according to a \emph{target
property}---in this work, the tendency of a small organic molecule to
partition at the interface of a phospholipid bilayer.
Akin to importance sampling used extensively to characterize
conformational space, we first introduce a Markov chain Monte Carlo
(MC) procedure to sample chemical compound space. Our methodology
consists of a sequence of compounds where trial alchemical
transformations are accepted according to a Metropolis criterion
(Fig.~\ref{fig:intro}a). While the acceptance criterion for
conformational sampling typically includes the energy of the system,
compositional sampling (i.e., across chemical space) means averaging
over the environment and thus calls for a free energy. As such, MC
moves will push the exploration towards small molecules that increase
their stability within a specific condensed-phase
environment---algorithmically akin to constant pH
simulations~\cite{mongan2004constant}.
In this work we focus on the free-energy difference of transferring a
small molecule from the aqueous environment to the lipid-membrane
interface (Fig.~\ref{fig:intro}b). The MC acceptance criterion is here
dictated by pharmacokinetic considerations: while hydrophobic
compounds will more easily permeate through the bilayer
\cite{menichetti2018drug}, they may display poor solubility
properties~\cite{dahan2016solubility}. Our MC criterion thus aims at
balancing the delicate interplay between solubility and permeability.
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=\linewidth]{fig1.png}
\caption{(a) Importance sampling across coarse-grained compounds
via a Markov chain Monte Carlo scheme. Only the dark-blue
region is sampled. (b) Background: Simulation setup of a solute
(yellow) partitioning between water (not shown) and the lipid
membrane. Foreground: Potential of mean force along the normal
of the bilayer, $G(z)$, and definition of the three transfer
free energies of interest between the three state points (red
circles): bilayer midplane (``M''), membrane-water interface
(``I''), and bulk water (``W''). (c) The MC-sampled free
energies (dark-blue region) form the training set for a machine
learning model, used to predict a larger subset of compounds
(light-blue region). (d) Each coarse-grained compound
represents a large number of small molecules.}
\label{fig:intro}
\end{center}
\end{figure*}
To further boost the size of chemical space that is probed, we further
predict a more extended subset of compounds that were not sampled by
using machine learning (ML; see
Fig.~\ref{fig:intro}c)~\cite{rasmussen2004gaussian}. Despite known
limited capabilities to extrapolate beyond the training set, we
observe remarkable accuracy for the predicted compounds. This
excellent transferability can be associated to a simplified learning
procedure at the CG resolution: structure-property relationships are
easier to establish~\cite{menichetti2018drug} and compound similarity
is compressed due to the reduction of chemical space. The range of
reliable predictions is made clear by means of the ML model satisfying
linear thermodynamic relations across
compounds~\cite{menichetti2017silico}---a more robust confidence
metric compared to the predictive variance. The CG results are then
systematically backmapped (Fig.~\ref{fig:intro}d) to yield an
unprecedentedly-large database of free energies.
\section{Methods}
\subsection{Coarse-grained simulations}
MD simulations of the Martini force field \cite{periole2013martini}
were performed in {\sc Gromacs} 5.1. The integration time-step was
$\delta t=0.02~\tau$, where $\tau$ is the model's natural unit of
time. Control over the system temperature and pressure ($T=300~K$ and
$P=1$~bar) was obtained by means of a velocity rescaling thermostat
\cite{bussi2007canonical} and a Parrinello-Rahman barostat
\cite{parrinello1981polymorphic}, with coupling constants
$\tau_T=~\tau$ and $\tau_P=12~\tau$. Bulk simulations consisted of
$N_{\textup{W}}=450$ and $N_{\textup{O}}=336$ water and octane
molecules, where the latter was employed as a proxy for the
hydrophobic core of the bilayer \cite{menichetti2017silico}. As for
interfacial simulations, a membrane of $36~\text{nm}^2$ containing
$N_{\textup{L}}=128$ 1,2-dioleoyl-\emph{sn}-glycero-3-phosphocholine (DOPC) lipids (64 per layer) and
$N_{\textup{W}}'=1890$ water molecules was generated by means of the
{\sc Insane} building tool \cite{wassenaar2015computational}, and
subsequently minimized, heated up, and equilibrated. In all
simulations containing water molecules we added an additional $10\%$
of antifreeze particles.
\subsection{Free-energy calculations}
\label{sec:free_en}
Water/interface and interface/membrane transfer free energies $\Delta G_{\textup{W}\rightarrow\textup{I}}$
and $\Delta G_{\textup{I}\rightarrow\textup{M}}$ for all compounds investigated in this work were obtained
from alchemical transformations, in analogy with
Ref.~\cite{menichetti2017silico}. This construction is based on the
relation linking the transfer free energies of two compounds $A$ and
$B$ ($\Delta G^{A}_{\textup{W}\rightarrow\textup{I}}$,
$\Delta G^{B}_{\textup{W}\rightarrow\textup{I}}$ and
$\Delta G^{A}_{\textup{I}\rightarrow\textup{M}}$,
$\Delta G^{B}_{\textup{I}\rightarrow\textup{M}}$) to the free energies
of alchemically transforming $A$ into $B$ in the three fixed
environments, $\Delta G^{A\rightarrow B}_{\textup{I}}$,
$\Delta G^{A\rightarrow B}_{\textup{W}}$ and
$\Delta G^{A\rightarrow B}_{\textup{M}}$
\begin{equation}
\nonumber
\Delta G^{B}_{\textup{W}\rightarrow\textup{I}}=\Delta
G^{A}_{\textup{W}\rightarrow\textup{I}}+(\Delta G^{A\rightarrow
B}_{\textup{I}}-\Delta G^{A\rightarrow B}_{\textup{W}}),
\end{equation}
\begin{equation}
\label{eq:alch_transf}
\Delta G^{B}_{\textup{I}\rightarrow\textup{M}}=\Delta
G^{A}_{\textup{I}\rightarrow\textup{M}}+(\Delta G^{A\rightarrow
B}_{\textup{M}}-\Delta G^{A\rightarrow B}_{\textup{I}}).
\end{equation}
$\Delta G^{A\rightarrow B}_{\textup{I}}$, $\Delta G^{A\rightarrow
B}_{\textup{W}}$, and $\Delta G^{A\rightarrow B}_{\textup{M}}$ were
determined by means of separate MD simulations at the interface, in
bulk water, and in bulk octane. For the calculation of each $\Delta
G^{A\rightarrow B}_i$, $i={\textup{I}},{\textup{W}},{\textup{M}}$ we
relied on the multistate Bennett acceptance ratio (MBAR)
\cite{shirts2008statistically, klimovich2015guidelines}, in which
free-energy differences are obtained by combining together the results
from simulations that sample the statistical ensemble of a set of
interpolating Hamiltonians $H(\lambda)$, $\lambda\in [0,1]$, with
$H(0)=H_A$ and $H(1)=H_B$. We employed 24 evenly spaced
$\lambda$-values for each alchemical transformation and in each
environment (interface, water, octane). The production time for each
$\lambda$ point was $4 \cdot 10^{4}~\tau$ at the interface and $2
\cdot 10^{4}~\tau$ in bulk environments. To calculate $\Delta
G^{A\rightarrow B}_{\textup{I}}$ we added a harmonic potential with
$k=240~{\rm kcal}\,{\rm mol}^{-1}\,{\rm nm}^{-2}$ between the
compound's center of mass and the bilayer midplane at a distance
$\bar{z}=1.5~{\rm nm}$, to account for the spatial localization of the
interface. The value of $\bar{z}$ was fixed by analyzing the potential
of mean force $G(z)$ (see Fig.~\ref{fig:intro}b) for the insertion of
various solutes that preferentially sit near the lipid headgroups in a
DOPC bilayer~\cite{menichetti2017silico}. The minimum of these
profiles was found to be located at $\bar{z}\approx1.8~{\rm nm}$
irrespective of the compound's chemical detail, suggesting that the
location of the dip is largely determined by the membrane environment.
In this work, we corrected $\bar{z}$ to account for the horizontal
shift in the potentials of mean force generated by the additional bead
of the Martini DOPC model originally employed in
Ref.~\onlinecite{menichetti2017silico}~\cite{menichetti2017efficient}.
We further emphasize that we only restrained the compound's center of
mass, while the \emph{orientation} of the linear molecule with respect
to the bilayer normal was left unbiased. Notably, we do expect (and
observed) compounds to display very different preferential
orientations---from parallel to perpendicular with respect to the
bilayer normal. There are two reasons motivating our choice: ($i$) the
CG simulations efficiently explore conformational space anyway, such
that this degree of freedom is relatively easily sampled; and ($ii$)
the information between interpolating Hamiltonians that are simulated
during an alchemical transformation are efficiently exchanged thanks
to the MBAR method. The small corrections operated during the
thermodynamic-cycle optimization we apply a posteriori attest of our
assumptions.
\subsection{Monte Carlo sampling}
We perform a stochastic exploration of the chemical space of CG linear
trimers and tetramers through the generation of Markovian sequences of
compounds. Given the last compound $A$ of a sequence, the new compound
$B$ is proposed by randomly selecting a bead of $A$ and changing its
type. The move from $A$ to $B$ is then accepted with probability
\begin{equation}
\label{eq:metr_mc}
P_{A\rightarrow B} = \min \left\{ 1, \exp \left[-\beta(
\Delta G^{B}_{\textup{W}\rightarrow\textup{I}} -
\Delta G^{A}_{\textup{W} \rightarrow\textup{I}})\right] \right\},
\end{equation}
where $\Delta G^{A}_{\textup{W}\rightarrow\textup{I}}$ and $\Delta
G^{B}_{\textup{W}\rightarrow\textup{I}}$ are the water/interface
transfer free energies of $A$ and $B$, respectively. $P_{A\rightarrow
B}$ aims at driving the Monte Carlo sampling towards compounds that
favor partitioning at the membrane interface (Fig.~\ref{fig:intro}b).
This is significantly different from optimizing $\Delta G_{\textup{W}\rightarrow\textup{M}}$, because those
would likely be poorly soluble in an aqueous
environment~\cite{dahan2016solubility}.
While in this work we set $\beta=1/k_\mathrm{B}T$, we stress
that $\beta$ can in principle be chosen independently of the system
temperature. The free-energy difference in Eq.~\ref{eq:metr_mc} is
derived from the alchemical free-energy differences of transforming
$A$ into $B$ in the three fixed environments
$\Delta G^{A\rightarrow B}_i,~i=\textup{W, I, M}$ (first
relation in Eq.~\ref{eq:alch_transf}), which we compute from MD
simulations.
We generated up to five independent Markovian sequences in parallel,
each starting from a different initial compound. To avoid
recalculating alchemical transformations already visited, we stored
the history of calculations and looked up previously-calculated values
when available.
\subsection{Thermodynamic-cycle optimization}
\label{sec:topt}
By combining together the results of all independent Markovian sequences,
the outcome of our Monte Carlo sampling consists of an alchemical
\emph{network}. Each node of the network represents a
compound, and an edge connecting two nodes $A$ and $B$ corresponds to
an alchemical transformation that was sampled via an MD simulation,
see Fig.~\ref{fig:network}. Each edge is characterized by the
free-energy differences $\Delta G^{A\rightarrow B}_{\textup{i}}$ in
the three fixed environments, $i=\textup{W, I, M}$. The network
representation was created with {\sc
NetworkX}~\cite{hagberg2008exploring} and visualized with {\sc
Gephi}~\cite{bastian2009gephi}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.95\linewidth]{fig2.png}
\caption{Network of CG compounds, each denoted by its set of
Martini bead types. Their size is proportional to the number of
edges. The blue-to-red path illustrates a sequence of accepted
compounds sampled from the Monte Carlo scheme. The dashed, green
triangle denotes a closed path in compound space. Sampling all
of its three branches closes the thermodynamic cycle, used as
constraint to further refine each free energy. }
\label{fig:network}
\end{center}
\end{figure}
For each environment, the net free-energy difference along any closed
cycle in the network must be zero (as shown in green in
Fig.~\ref{fig:network}), by virtue of a free energy being a state
function. We thus enforced this thermodynamic condition to optimize
the set of free-energy differences calculated from MD simulations. We
employed the algorithm proposed by Paton~\cite{paton1969algorithm} to
identify the cycle basis that spans the alchemical network, i.e., each
cycle in the network can be obtained as a sum of the $N_{C}$ basis
cycles. We denote the MD free-energy differences involved in at least
one basis cycle by $\Delta G^{j}_i,~j=1,..,N_{G},~i=\textup{W, I, M}$,
while nodes connected to only a single edge cannot be taken into
account. For each environment, we optimized the set of free energies
$\Delta \hat{G}^{j}_i$ by minimizing the loss function
\begin{equation}
\label{eq:ObjFct}
\mathcal{L}_i
= \sum_{j=1}^{N_G} (\Delta G^j_i - \Delta \hat{G}^j_i)^2 +
\sum_{k=1}^{N_c} \omega \Big(\sum_{j \in k}
(-1)^{s_{j,k}}\Delta \hat{G}^j_i\Big)^2.
\end{equation}
While the first term ensures that the optimized free-energy
differences $\Delta \hat{G}^{j}_{\textup{i}}$ remain close to the MD
simulation results, the second term ($\omega=10.0$) penalizes
deviations from zero for each thermodynamic cycle within a basis
cycle. The exponent $s_{j,k}$ controls the sign of the
free-energy difference in the cycle, taking values of $0$ or $1$. To
minimize the cost functions, we employed the
Broyden-Fletcher-Goldfarb-Shanno method
(BFGS)~\cite{avriel2003nonlinear} (see Figs.~S1 and S4 for trimers and
tetramers, respectively).
\subsection{Machine learning}
We use kernel ridge regression~\cite{rasmussen2004gaussian}, where the
prediction of target property $p({\bf x})$ for sample ${\bf x}$ is
expressed as a linear combination of kernel evaluations across the
training points ${\bf x}^*_i$
\begin{equation}
p({\bf x}) = \sum_i \alpha_i K({\bf x}^*_i, {\bf x}).
\end{equation}
The kernel consists of a similarity measure between two samples
\begin{equation}
K({\bf x}, {\bf x}') = \exp \left( - \frac{||{\bf x}-{\bf
x}'||_1}{\sigma} \right),
\end{equation}
which corresponds to a Laplace kernel with a city-block metric (i.e.,
$L_1$-norm), and $\sigma$ is a hyperparameter. The representation
${\bf x}$ corresponds to the vector of water/octanol partitioning free
energies of each bead---it is described more extensively in the
Results. The optimization of
the weights $\alpha$ consists of solving for the samples in the
training with an additional regularization term $\lambda$: ${\bf
\alpha} = ({\bf K} + \lambda \mathrm{I})^{-1}{\bf p}$. The confidence
of the prediction is estimated using the predictive variance
\begin{equation}
\epsilon = {\bf K}^{**} - ({\bf K}^*)^T ({\bf K} + \lambda
{\bf I})^{-1} {\bf K}^*,
\end{equation}
where ${\bf K}^{**}$ and ${\bf K}^*$ represent the kernel matrix of
training with training and training with test datasets,
respectively~\cite{rasmussen2004gaussian}. The two hyperparameters
$\sigma$ and $\lambda$ were optimized by a grid search, yielding
$\sigma = 100$ and $\lambda = 10^{-4}$. Learning curves are shown in
Figs.~S2 and S5 for trimers and tetramers, respectively.
\section{Results}
We consider the insertion of a small molecule across a
single-component phospholipid membrane made of
1,2-dioleoyl-\emph{sn}-glycero-3-phosphocholine (DOPC) solvated in
water. The insertion of a drug is monitored along the collective
variable, $z$, normal distance to the bilayer midplane
(Fig.~\ref{fig:intro}b). We focus on three thermodynamic state points
of the small molecule: the bilayer midplane (``M''), the
membrane-water interface (``I''), and bulk water (``W''). We link
these quantities in terms of transfer free energies, e.g., $\Delta G_{\textup{W}\rightarrow\textup{M}}$
denotes the transfer free energy of the small molecule from water to
the bilayer midplane.
\subsection{Importance sampling}
We ran MC simulations across CG linear trimers and tetramers (results
for tetramers are shown in the SI), randomly changing a bead type,
calculating the relative free energy difference between old and new
compound in the three different environments, and accepting the trial
compound using a Metropolis criterion on the water/interface transfer
free energy $\Delta G_{\textup{W}\rightarrow\textup{I}}$ (Fig.~\ref{fig:intro}a and Eq.~\ref{eq:metr_mc}).
This criterion aims at selecting compounds that favor partitioning at
the water/membrane interface.
The MC algorithm yielded an acceptance ratio of 0.2. While initially
most trial compounds contributed to expand the database, the sampling
scheme quickly reached a stable regime where roughly half of the
compounds had already been previously visited. Because each
free-energy calculation is expensive, we avoid recalculating identical
alchemical transformations to help efficiently converge the protocol.
To monitor and control the possible accumulation of statistical error
during the MC chain of alchemical transformations, we optimized the
network of sampled free energies---i.e., the one obtained
by combining together the results of all independent MC sequences---according to thermodynamic
constraints (Sec.~\ref{sec:topt}). A short MC sequence of accepted
compounds is shown in Fig.~\ref{fig:network}. We display the sequence
within the network of sampled compounds, each node being represented
by the set of Martini bead types involved. We find a large number of
closed paths within this network: since the free energy is a state
function, the closed path represents a thermodynamic cycle---it must
sum up to zero. We thus enforced this condition on the whole set of
basis cycle in the network and for each of the three environments to determine a set of optimized free-energy
differences, at the same time pushing the optimized values to remain
close to MD simulation results, see Eq.~\ref{eq:ObjFct}. We stress
that many free energies are involved in multiple basis cycles,
enhancing the robustness of the optimization by combining constraints.
The outcome of the optimization for both trimers and tetramers is
presented in the supporting information (Fig.~S1 and S4,
respectively). We found small modifications of the free energies
calculated via MD simulations to be sufficient to virtually enforce a
zero net free energy over the whole set of basis cycles. Aggregate
changes along cycles never exceeded 0.2, 0.1, and 0.01~kcal/mol at the
interface, in water, and in the membrane core.
Any cycle in the network can be written as a combination of basis
cycles. As such, the condition enforced in the optimization offers
free-energy differences between two compounds to be calculated along
\emph{any} path connecting them in the network. This highlights the
robustness of our optimization scheme and hinders a significant
accumulation of errors in the relative free energies along a sequence
of compounds.
By combining together the results for the three thermodynamic environments through Eq.~\ref{eq:alch_transf},
we thereby obtain an \emph{optimized} network of compounds whose edges feature relative
transfer free energies $\Delta G_{\textup{W}\rightarrow\textup{I}}$ and $\Delta G_{\textup{I}\rightarrow\textup{M}}$ between the two connecting
nodes. Relative free energies can be summed up to trace the total
change in free energy during a path. This only leaves us to determine
the \emph{absolute} transfer free energies $\Delta G_{\textup{W}\rightarrow\textup{I}}$ and $\Delta G_{\textup{I}\rightarrow\textup{M}}$ for an
arbitrary starting compound. These transfer free energies were
extracted from the potential of mean force, $G(z)$
(Fig.~\ref{fig:intro}b), calculated following the simulation protocol
described in Ref.~\onlinecite{menichetti2017silico}. A number of
reference PMFs for trimers and tetramers were calculated. The lack of
compounding of the statistical errors, as evidenced by our
thermodynamic-cycle optimizations, and the sufficient number of MC
cycles make the choice of reference compounds insignificant.
\subsection{Machine learning}
The ML models used here infer the relationship between the
CG composition of a compound and its various transfer free energies. A
key component of an efficient ML model is its
representation~\cite{huang2016communication}. It should include
enough information to distinguish a compound's chemical composition
and geometry, as well as encode the physics relevant to the target
property~\cite{faber2018alchemical}. Because the CG compounds all
consist of beads arranged linearly and equidistant, we have found that
encoding the geometry had no benefit to the learning (data not shown).
Instead we simply encode the water/octanol partitioning of each bead,
yielding for linear trimers
$
{\bf x} = \left(\Delta G_{\textup{W}\rightarrow\textup{Ol}}^{(1)}, \Delta G_{\textup{W}\rightarrow\textup{Ol}}^{(2)}, \Delta G_{\textup{W}\rightarrow\textup{Ol}}^{(3)}\right).
$
Reference values for $\Delta G_{\textup{W}\rightarrow\textup{Ol}}^{(i)}$ were extracted from alchemical
transformations of each bead type between the two bulk
environments~\cite{Bereau2015}. Note that while the problem we
consider in this work contains reflection symmetry for the compounds
(i.e., $ABC$ is equivalent to $CBA$), we did not need to encode this
in the representation. Instead we sorted the bead arrangement when
generating compounds for the importance sampling and machine learning.
When trained on most of
the MC-sampled data, we obtained out-of-sample mean absolute errors
(MAE) as low as 0.2~kcal/mol for $\Delta G_{\textup{W}\rightarrow\textup{I}}$ and $\Delta G_{\textup{I}\rightarrow\textup{M}}$, on par with the
statistical error of the alchemical transformations (see Fig.~S2).
Remarkably, the prediction of $\Delta G_{\textup{W}\rightarrow\textup{M}}$ converges to an MAE lower than
0.05~kcal/mol, illustrative of the strong correlation between
water/octanol and water/membrane free energies in
Martini~\cite{menichetti2017silico}. For all three quantities we
monitor a correlation coefficient above 97\%, indicating excellent
performance.
Next, we train our ML model on the entire dataset of MC-sampled
compounds. We use this model to predict all other CG linear
trimers---a similar protocol was applied to tetramers. Because of the
importance-sampling scheme, the predicted compounds will typically
feature different characteristics, e.g., more polar compounds that
would preferably stay in the aqueous phase. As such the ML model is
technically extrapolating outside of the training set. As a measure of
homogeneity between training and validation sets,
Fig.~\ref{fig:pred_var} displays the distributions of confidence
intervals (see Methods) between out-of-sample predictions and the
expansion of the dataset. While we find significant overlap between
the MC and ML distributions for trimers, we observe larger deviations
in the case of tetramers.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\linewidth]{fig3.pdf}
\caption{Distribution functions of confidence intervals, $p(\epsilon)$ (see Methods for definition), for both the out-of-sample predictions within the MC-sampled compounds and the ML-predicted compounds. (a) Trimers and (b) tetramers.}
\label{fig:pred_var}
\end{center}
\end{figure}
The extrapolation can also be seen in the projections of predicted
transfer free energies, highlighting distinct coverages of sampled and
predicted trimer compounds (Fig.~\ref{fig:linear}). However, the main
panels (a) and (b) display notable linear relations between transfer
free energies---similar behavior is found for tetramers (Fig.~S3).
Importantly, similar linear relations had already been observed for CG
unimers and dimers, highlighting thermodynamic relations for the
transfer between different effective bulk
environments~\cite{menichetti2017silico}. We also argue that the ML
models do not simply learn linear features, since we optimize
independent models for the different predicted transfer free energies.
The linear behavior displayed across both sampled and predicted
compounds testifies to the robustness of the ML model, despite the
extrapolation.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.95\linewidth]{fig4.pdf}
\caption{(a) Transfer free energies from water to interface
$\Delta G_{\textup{W}\rightarrow\textup{I}}$ as a function of the compound’s water/membrane
partitioning free energy, $\Delta G_{\textup{W}\rightarrow\textup{M}}$. The dark and light blue
points depict corresponding quantities for trimers estimated
from MC sampling (3B-MC) and the ML predictions (3B-ML),
respectively. Linear fits highlight the molecular-weight
dependence. (b) Transfer free energies from the interface to
the membrane $\Delta G_{\textup{I}\rightarrow\textup{M}}$ as a function of the compound's
water/membrane partitioning free energy, $\Delta G_{\textup{W}\rightarrow\textup{M}}$. The coverages
are projected down along a single variable on the sides. Error
bars for 3B-MC are on par with the datapoint sizes (not shown).}
\label{fig:linear}
\end{center}
\end{figure}
The ML predictions also offer higher accuracy compared to simple
linear fits: We selected a small set of 50 reference compounds
spanning the entire dataset and measured the performance of the ML
predictions and linear regression. The deviation of both predictions
against reference alchemical transformations for each compound is
shown in Fig.~\ref{fig:linearFit_vs_ML}, displaying predictions for
$\Delta G_{\textup{W}\rightarrow\textup{I}}$ and $\Delta G_{\textup{W}\rightarrow\textup{M}}$. We find a mean-absolute error (MAE) of 0.3 and
0.5~kcal/mol for the ML and linear fit, respectively. The linear
regressions display equal but opposite errors between $\Delta G_{\textup{W}\rightarrow\textup{I}}$ and
$\Delta G_{\textup{W}\rightarrow\textup{M}}$, by construction. The compounds are sorted according to the ML
deviation of $\Delta G_{\textup{W}\rightarrow\textup{I}}$. Interestingly, this ranking of compounds shows
no clear pattern: for instance, it only correlates weakly with
hydrophobicity (21\%). On the other hand hydrophobicity correlates
much more strongly with the predictive variance (50\%). While the
latter naturally stems from the representation, the absence of
correlation between ML deviation and hydrophobicity points at a more
complex structure of the interpolation space (i.e., the CG chemical
space)---a feature that will only worsen when learning atomistic
compounds. Complementary information can be further probed from
Fig.~\ref{fig:linearFit_vs_ML} by comparing the ML deviations between
$\Delta G_{\textup{W}\rightarrow\textup{I}}$ and $\Delta G_{\textup{I}\rightarrow\textup{M}}$: we observe a correlation coefficient of 63\%
between the unsigned prediction errors. Training two independent ML
models on identical subsets of chemical space leads to large
correlations, further emphasizing the predominance of the
interpolation space over the target property when learning.
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=0.8\linewidth]{fig5.pdf}
\caption{Deviations of the ML and linear-fit (``Fit'')
predictions from reference alchemical transformations (``MC'')
for (a) $\Delta G_{\textup{W}\rightarrow\textup{I}}$ and (b) $\Delta G_{\textup{I}\rightarrow\textup{M}}$. Error bars for the ML model
display the 95\% confidence intervals form the predictive
variance. Compounds are sorted according to the ML prediction
error for $\Delta G_{\textup{W}\rightarrow\textup{I}}$. Free energies displayed in units of
kcal/mol.}
\label{fig:linearFit_vs_ML}
\end{center}
\end{figure*}
A systematic coarse-graining of compounds in the GDB~\cite{Fink2007}
using {\sc Auto-Martini}~\cite{Bereau2015} was performed to identify
small organic molecules that map to CG linear trimers. The algorithm
is deterministic, such that it leads to a unique mapping from molecule
to CG representation. We identified
1.36~million compounds, for which we can associate all three transfer
free energies, $\Delta G_{\textup{W}\rightarrow\textup{M}}$, $\Delta G_{\textup{W}\rightarrow\textup{I}}$, and $\Delta G_{\textup{I}\rightarrow\textup{M}}$. We note that the
sampled and predicted CG representations amount to similar numbers of
compounds, such that the ML boosting introduced here offers an
additional 0.8~million compounds to the database. The database is
provided as supporting material for further data analysis.
\section{Conclusions}
The overwhelming size of chemical space naturally calls for
statistical techniques to analyze it. A variety of data-driven
methods such as quantitative structure-property relationships (QSPR)
and ML models at large have been applied to chemical
space~\cite{rupp2012fast, faber2016machine, bartok2017machine,
zhang2017machine}. While sparse databases easily lead to
overfitting~\cite{swift2013back}, a dense coverage can offer
unprecedented insight~\cite{ramakrishnan2014quantum}. Here we rely on
tools from statistical physics to ease the exploration of chemical
space: the application of importance sampling guides us toward the
subset of molecules that enhances a desired thermodynamic property.
This approach is similar to recent generative ML
models~\cite{sanchez2018inverse}, but without the a priori requirement
for labeled training data.
In this work, we provide estimates of different transfer free energies (e.g.,
from water to the membrane interface, $\Delta G_{\textup{W}\rightarrow\textup{I}}$) for a large number of
CG compounds. The combination of alchemical transformations with MC
sampling motivates the calculation of free energies \emph{relative} to
the previous compound (e.g., $\Delta G^{A\rightarrow
B}_{\textup{I}}$). Estimating the stability of compound $C_i$, sampled
at the $i$-th step of an MC procedure, requires the summation of all
previous free-energy contributions, all the way to the initial
compound, for which we computed absolute free energies from umbrella
sampling. Because each step in the MC procedure involves a free
energy, there is a statistical error that is compounded during this
reconstruction. We can take advantage of thermodynamic cycles to
measure deviations from a net free energy of zero in a closed loop,
and thus estimate this compounding of errors. Remarkably, we find
deviations that are much smaller than the estimated statistical error
of the alchemical transformations. This illustrates the robustness of
estimating free energies at high throughput using an MC scheme.
A conceptually-appealing strategy to expand the MC-sampled
distribution is through an ML model. Effectively we train an ML model
on the MC samples and further boost the database with additional ML
predictions. Unfortunately, the limited extrapolation behavior of
kernel models means that accurate predictions can only be made for
compounds \emph{similar} to the training set. \emph{How} similar is
often difficult to estimate a priori. Similarity metrics are often
based at the level of the ML's input space---here the molecular
representation. In Fig.~\ref{fig:pred_var} we used the predictive
variance as a metric for the query sample's distance to the training
set~\cite{rasmussen2004gaussian}.
Beyond similarity in the ML's input space via the predictive variance,
we also consider the target properties directly. Our physical
understanding of the problem offers a clear requirement on the
transfer free energies, through the linear relationships shown in
Fig.~\ref{fig:linear}~\cite{menichetti2017silico}. As such, the
thermodynamics of the system impose a physically-motivated constraint
on the predictions. Rather than specific to each prediction, this
constraint is \emph{global} to the ensemble of data points. Satisfying
it grounds our predictions within the physics of the problem, ensuring
that we accurately expand the database.
Remarkably, we find that we can significantly expand our
database---doubling it for trimers and a factor of 10 for tetramers
(see SI)---while retaining accurate transfer free energies. Unlike
conventional atomistic representations~\cite{faber2017prediction}, our
ML model is encoded using a CG representation, such that compounds
need only be similar at the CG level. This CG similarity is strongly
compressed because ($i$) coarse-graining reduces the size of chemical
space~\cite{menichetti2017silico}, but also ($ii$) of a more
straightforward structure-property link~\cite{menichetti2018drug}. The
latter is embodied by the additive contribution of bulk partitioning
free energies for each bead, efficiently learning the molecular
transfer free energy in more complex environments. All in all,
backmapping (Fig.~\ref{fig:intro}d) significantly amplifies the
additional region of chemical space reached by the ML model. Our work
highlights appealing aspects of bridging physics-based methodologies
and coarse-grained modeling together with machine learning, offering
increased robustness and transferability to explore significantly broader
regions of chemical space.
\section{Supporting Information}
The attached supporting information contains additional details on the
optimization of thermodynamic cycles; the learning curves of the
machine learning model; and results on linear tetramers. In addition,
we provide databases for the transfer free energies of all trimers and
tetramers, as well as atomistic-resolution compounds that map to
trimers in a repository~\cite{zenodo}.
\section*{Acknowledgments}
The authors thank Alessia Centi and Clemens Rauer for critical reading
of the manuscript. The authors acknowledge Chemaxon for an academic
research license of the Marvin Suite. This work was supported by the
Emmy Noether program of the Deutsche Forschungsgemeinschaft (DFG) and
the John von Neumann Institute for Computing (NIC) through access to
the supercomputer JURECA at J\"ulich Supercomputing Centre (JSC).
|
1,108,101,566,748 | arxiv | \section{Introduction}
Let $e_\lambda$ denote the $L^2$-normalized eigenfunction on a compact, boundary-less Riemannian surface $(M,g)$, i.e.,
$$-\Delta_g e_\lambda=\lambda^2 e_\lambda, \quad \text{and } \, \, \int_M |e_\lambda|^2 \, dV_g=1.$$
Here $\Delta_g$ is the Laplace-Beltrami operator on $(M,g)$ and $dV_g$ is the volume element associated with metric $g$.
Various questions concerning analytic properties
of eigenfunctions have drawn much attention from number theorists, analysts and physicists in recent years. In particular, it is an area of interest to study quantitative behaviors of eigenfunctions restricted to smooth curves. Recently, there are a lot of interests in studying the period integral of eigenfunctions over smooth closed curves on compact hyperbolic surfaces due to its significance in number theory. See e.g. \cite{Zel}, \cite{Pitt}, \cite{Rez} and the references therein.
Using the Kuznecov formula, Good \cite{Good} and Hejhal \cite{Hej} proved independently that if $\gamma_{per}$ is a periodic geodesic on a compact hyperbolic
surface $M$ parametrized by arc-length, then, uniformly in $\lambda$,
\begin{equation}\label{i.1}
\Bigl|\, \int_{\gamma_{per}} e_\lambda \, ds\, \Bigr| \le C_{\gamma_{per}}.
\end{equation}
A few years later, Zelditch~\cite{ZelK} generalized this result by showing that if $\lambda_j$ are the eigenvalues of $\sqrt{-\Delta_g}$ for an orthonormal basis of eigenfunctions $e_{\lambda_j}$ on
a compact Riemannian surface, and if $p_j(\gamma_{per})$ denote the period integrals of $e_{\lambda_j}$ as in \eqref{i.1},
then
\begin{equation}\label{Zel}\sum_{\lambda_j\le \lambda}|p_j(\gamma_{per})|^2 =c_{\gamma_{per}} \lambda +O(1),\end{equation}
where the remainder being $O(1)$ implies \eqref{i.1}. Further work for hyperbolic surfaces giving more information about the lower order remainder in terms
of geometric data of $\gamma_{per}$ was done by Pitt~\cite{Pitt}. By Weyl's Law, the number of eigenvalues (counting multiplicities) that are smaller than
$\lambda$ is about $\lambda^2$, and thus \eqref{Zel} implies that, on average, one can do much better than \eqref{i.1}. It was pointed out by Chen and Sogge \cite{CSPer} that \eqref{i.1} is sharp on both the sphere $S^2$ and the flat torus $\mathbb T^2$. On $S^2$, \eqref{i.1} is saturated by $L^2$-normalized zonal spherical harmonics of even degree restricted to the equator, while it is trivially sharp on the flat torus ${\mathbb T}^2$. In contrast, as an analogy with the Lindel\"of conjecture for certain $L$-functions, it is conjectured by Reznikov \cite{Rez} that the period integrals over closed geodesics/ geodesic circles on a compact hyperbolic surface satisfy the following:
\begin{conjecture}[\cite{Rez}]Let $\gamma$ be a periodic geodesic or a geodesic circle on a compact hyperbolic surface $(M,g)$. Then given $\varepsilon>0$, there exists a constant $C_\varepsilon$ depending on $\varepsilon$, $M$ and the length of $\gamma_{per}$, such that
\begin{equation} \Bigl|\, \int_{\gamma}\, e_\lambda \, ds\, \Bigr| \le C_\varepsilon\lambda^{-\frac12+\varepsilon}.
\end{equation}
\end{conjecture}
In a paper of Chen and Sogge \cite{CSPer}, the first improvement over \eqref{i.1} for closed geodesics was obtained. Indeed, they proved a stronger statement saying that the period integrals in \eqref{i.1} converge to 0 as $\lambda\to \infty$, if $(M,g)$ has strictly negative
curvature. The proof exploited the simple geometric fact that, due to the presence of negative curvature, there is no non-trivial geodesic rectangle on the universal cover of $M$. This allowed them to show that the period integrals goes to 0 as $\lambda\rightarrow\infty$. In a recent paper of Sogge, the second author and Zhang \cite{Gauss}, this method was further refined, and they managed to show that
\begin{equation}\label{gauss}
\Bigl|\, \int_{\gamma_{per}} e_\lambda \, ds\, \Bigr| = O((\log\lambda)^{-\frac12}),
\end{equation}
under a weaker curvature assumption, where the curvature $K=K_g$ of $(M,g)$ is assumed to be non-positive but allowed to vanish at an averaged
rate of finite type. The key idea of \cite{Gauss} was to use the Gauss-Bonnet Theorem to get a quantitative version of the ideas used in \cite{CSPer}, that is, to quantitatively avoid geodesic rectangles on the universal cover.
Since then, there have been plenty new developments in this area. The first author (\cite{emmett2}, \cite{emmett1}) generalized the results in \cite{CSPer} and \cite{Gauss} to curves which have geodesic curvature bounded away from the curvature of limiting circles. See also the recent work of Canzani, Galkowski \cite{canzani}
for $o(1)$ bounds for averages over hypersurfaces under weaker
assumptions. To describe this result, we shall need to introduce a few notations. If $\gamma$ is a smooth curve in $M$, we denote by $\kappa_\gamma(t)$ the geodesic curvature of $\gamma$ at $t$, i.e.
\[
\kappa_\gamma(t) = \frac{1}{|\gamma'(t)|} \left| \frac{D}{dt} \frac{\gamma'(t)}{|\gamma'(t)|} \right|,
\]
where $D/dt$ is the covariant derivative in the parameter $t$. Fixing a point $p \in M$ and $v \in T_pM$, we denote by $v^\perp$ a choice of vector in $T_pM$ for which $|v^\perp| = |v|$ and $\langle v^\perp, v \rangle = 0$. We also need a function $\mathbf k$ on the unit sphere bundle of $M$ representing the ``curvature of a limiting circle."
\begin{deff}[Curvature of a limiting circle] \label{def k}
For a point $p \in M$ and $v \in S_pM$ and let $r \mapsto \zeta(r)$ the unit speed geodesic with $\zeta(0) = p$ and $\zeta'(0) = v$. Let $J$ be a Jacobi field along $\zeta$ satisfying
\begin{equation} \label{J initial condition}
J(0) = \zeta'(0)^\perp.
\end{equation}
We let $\mathbf k_p(v)$ denote the unique number such that
\begin{equation} \label{J bounded}
|J(r)| = O(1) \quad \text{ for } r \in (-\infty,0]
\end{equation}
if $J$ satisfies the additional initial condition
\begin{equation} \label{J' initial condition}
\frac{D}{dr} J(0) = \mathbf k_p(v) J(0).
\end{equation}
\end{deff}
The name ``curvature of a limiting circle" will be clear after a lift to the universal cover. By the Cartan-Hadamard Theorem, we identify the universal cover of $M$ with $(\mathbb R^2, \tilde g)$, where $\tilde g$ is the pullback of the metric tensor $g$ through the covering map. If $p \in M$ and $v \in T_pM$, denote by $\tilde p$ and $\tilde v$ their respective lifts to $\tilde M$ and $T_{\tilde p}\tilde M$. Then $\mathbf k_p(v)$ denotes the limiting curvature of the circle at $\tilde p$ with center taken to infinity along the geodesic ray in direction $-\tilde v$. In the flat case, $\mathbf k_p(v) = 0$ for all $p\in M$, while if $M$ is a compact hyperbolic surface with sectional curvature $-1$, then $\mathbf k_p(v) = 1$ for all $p\in M$, which equals the curvature of a horocycle in the hyperbolic plane. See Figure 1.
\begin{figure}
\centering
\includegraphics[width=.65\textwidth]{figure1.pdf}
\caption{}
\label{fig1}
\end{figure}
\begin{theorem}[\cite{emmett2}] \label{W}
Let $(M,g)$ have nonpositive sectional curvature and let $\gamma$ be a smooth closed unit-speed curve in $M$. Then we have
\[
\left|\int_\gamma \,e_\lambda \, ds\right| = O((\log \lambda)^{-1/2}),
\]
provided that
\begin{equation} \label{curvature hypotheses}
\kappa_\gamma(s) \neq \mathbf k_{\gamma(s)}(\gamma'^\perp(s)) \quad \text{ and } \quad \kappa_\gamma(s) \neq \mathbf k_{\gamma(s)}(-\gamma'^\perp(s))
\end{equation}
for all points $\gamma(s)\in\gamma$.
\end{theorem}
A natural way to look at the period integral \eqref{i.1} is to regard it as the 0-th order Fourier coefficient of $e_\lambda|_{\gamma_{per}}$. The general $\nu$-th order Fourier coefficients of $e_\lambda|_{\gamma_{per}}$ are called {\it generalized periods} (see \cite{Rez}). Generalized periods for hyperbolic surfaces naturally arise in the theory
of automorphic functions, and are of interest in their own right, thus have been studied considerably by number theorists. In the paper \cite{Rez}, Reznikov showed that on compact hyperbolic surfaces, if $\gamma$ is a periodic geodesic or a geodesic circle, the $\nu$-th order Fourier coefficients of $e_\lambda|_\gamma$ is uniformly bounded if $\nu\le c_\gamma\lambda$ for some constant $c_\gamma$ depending on $\gamma$. In this spirit, the second author \cite{inner} generalized Reznikov's results to arbitrary smooth closed curves over arbitrary compact Riemannian surfaces.
\begin{theorem}[\cite{inner}]\label{inner}
Let $\gamma$ be a smooth, closed, unit-speed curve on $(M,g)$. Let $|\gamma|$ denote its length. Given $0<c<1$, if $\nu$ is an integer multiple of $2\pi|\gamma|^{-1}$ such that $0\le\frac{|\nu|}{\lambda}\leq c<1$, then we have
\begin{equation}\label{period}
\Big|\int_{\gamma}e_\lambda(\gamma(s)) e^{-i\nu s}\, ds\Big|\le C |\gamma|\|e_\lambda\|_{L^1(M)},
\end{equation}
where the constant $C$ only depends on $(M,g)$ and $c$, and will be uniform if $\gamma$ belongs to a class of curves with bounded geodesic curvature.
\end{theorem}
As for the period integrals, the above bounds are sharp for surfaces with constant non-negative curvature. (See \cite[Section 5]{inner}.) \eqref{period} improves \eqref{i.1} for the case $\nu=0$ by having an $L^1(M)$ norm on the right hand side, and it trivially implies these generalized periods are bounded,
\begin{equation}\label{period'}
\Big|\int_{\gamma}e_\lambda(\gamma(s)) e^{-i\nu s}\, ds\Big|\le C |\gamma|.
\end{equation} We also remark that the frequency gap condition $0\le\frac{\nu}{\lambda} \leq c < 1$ is necessary, in the sense that, at the resonant frequency $\nu=\lambda$, \eqref{period} fails to hold on $S^2$. Indeed, on $S^2$, $L^2$-normalized highest weight spherical harmonics with frequency $\lambda$ restricted to the equator have $\nu$-Fourier coefficient $\sim\nu^\frac14$, which represents a big jump from \eqref{period}.
Another key insight provided by
\cite{Gauss} and \cite{emmett2} is that under suitable curvature assumptions (for both $\gamma$ and $M$), the $\nu$-th Fourier coefficients of $e_\lambda|_{
\gamma}$ satisfies
\begin{equation}\label{gauss'}
\Bigl|\, \int_{\gamma} e_\lambda e^{-i\nu s} \, ds\, \Bigr| \le C_\nu|\gamma|(\log\lambda)^{-\frac12},
\end{equation} where $C_\nu$ is a constant times a positive power of $\nu$. On the other hand, it is conjectured in \cite{Rez} that for compact hyperbolic surface, we should expect much better estimates.
\begin{conjecture}[\cite{Rez}]\label{C2}Let $\gamma$ be a closed geodesic or a geodesic circle on a compact hyperbolic surface $(M,g)$. Then given $\varepsilon>0$, $0<c<1$, there exists a constant $C_\varepsilon$ depending on $\varepsilon$, $M$ and the length of $\gamma_{per}$, such that for $0\le\frac{\nu}{\lambda}\leq c <1$, we have
\begin{equation} \Bigl|\, \int_{\gamma_{per}} e_\lambda(\gamma_{per}(s)) e^{-i\nu s} \, ds\, \Bigr| \le C_\varepsilon\lambda^{-\frac12+\varepsilon}.
\end{equation}
\end{conjecture}
The above conjecture, if true, illustrates the huge differences between the sphere/ torus case and the hyperbolic case. The curvature of the surface being negative somehow ``filters" out almost all lower frequency oscillations of eigenfunctions over certain closed curves. In a recent paper of the second author, a first result towards the above conjecture was obtained for the case when $\gamma$ is a periodic geodesic.
\begin{theorem}[\cite{GP}]\label{GP}
Let $\gamma=\gamma_{per}$ be a periodic geodesic on a Riemannian surface $(M,g)$ with curvature $K$ satisfying the averaged vanishing conditions in the sense of \cite{Gauss}. Given $0<c<1$, if $\nu$ is an integer multiple of $2\pi|\gamma|^{-1}$ such that $0\le\frac{|\nu|}{\lambda}=\epsilon\leq c<1$, then we have
\begin{equation}\label{GPs}
\Big|\int_{\gamma}e_\lambda(\gamma(s)) e^{-i\nu s}\, ds\Big|\le C |\gamma|(\log\lambda)^{-\frac12},
\end{equation}
where the constant $C$ only depends on $M$ and $c$.
\end{theorem}
It is clear that Theorem \ref{GP} is an improvement over both \eqref{period'} and \eqref{gauss}, where it is better than \eqref{period'} for negatively curved surfaces, and contains \eqref{gauss'} as the special case when $\nu=0$. This was the first result showing generalized periods converge to 0 uniformly for all $|\nu|<c\lambda$ over closed geodesics on compact hyperbolic surfaces.
The proof of Theorem \ref{GP} followed the strategies developed in \cite{CSPer} and \cite{Gauss}. The Gauss-Bonnet Theorem was used to exploit the defects of geodesic quadrilaterals which arise naturally in these
arguments. Gauss-Bonnet Theorem allows one to quantitatively avoid geodesic parallelograms on negatively curved surfaces, which in turn provides favorable controls over derivatives of the phase functions
which occur in the stationary phase arguments.
The purpose of this paper is to prove log-improved generalized periods bounds for a larger class of curves, which, in particular, includes geodesic circles of arbitrary radius. Since we will not be dealing with geodesics mostly, the Gauss-Bonnet Theorem will not be as handy as in \cite{Gauss} and \cite{GP}. Instead, we shall turn to the strategies developed by the first author in \cite{emmett1} and \cite{emmett2}, and use these to prove an analog of Theorem \ref{W} for generalized periods over curves satisfying assumptions analogous to \eqref{curvature hypotheses}. However, in this case, our curvature assumption on $\gamma$ has to involve the frequency ratio $\epsilon=\nu/\lambda$. Our main result is the following.
\begin{theorem}[Main Theorem]\label{main}
Let $\gamma$ be a smooth, closed, unit-speed curve on a compact Riemannian surface $(M,g)$ with nonpositive curvature. Let $E_\gamma$ denote the set of $\epsilon \in (-1,1)$ for which
\begin{align}
\label{curv hyp +'} \left\langle \frac{D}{dt} \gamma', \gamma'^\perp \right\rangle &\neq - \sqrt{1 - \epsilon^2} \mathbf k_\gamma(\gamma'^\perp), \qquad \text{ and } \\
\label{curv hyp -'} \left\langle \frac{D}{dt} \gamma', \gamma'^\perp \right\rangle &\neq \sqrt{1 - \epsilon^2} \mathbf k_\gamma(-\gamma'^\perp)
\end{align}
at each point along $\gamma$.
If $\nu$ is an integer multiple of $2\pi|\gamma|^{-1}$ such that $\nu/\lambda \in E_\gamma$, then we have
\begin{equation}\label{periods}
\left|\int_{\gamma}e_\lambda(\gamma(s)) e^{-i\nu s}\, ds\right| \le C (\log\lambda)^{-\frac12},
\end{equation}
where $C$ is uniform for $\nu/\lambda$ in a compact subset of $E_\gamma$.
\end{theorem}
\begin{remark} \label{main remark}
Fix $\gamma$ and $E_\gamma$ as in Theorem \ref{main}, and let $K \subset E_\gamma$ be compact. Note $E_\gamma$ still contains $K$ even if we perturb $\gamma$ so that the change in the first two derivatives of $\gamma$ are small. At the same time, careful observation throughout the proof of Theorem \ref{main} shows that the constant $C$ is uniform if we perturb $\gamma$ so that the change in the first $N$ derivatives of $\gamma$ are small, where $N$ is some fixed, finite number. Hence the constant $C$ in \eqref{periods} is uniform if $(\gamma, \nu/\lambda)$ belongs to a compact subset of
\[
\{ (\gamma,\epsilon) : \gamma \in C^\infty(\mathbb R,M), \epsilon \in E_\gamma \} \subset C^\infty(\mathbb R,M) \times (-1,1)
\]
with respect to the subspace topology.
\end{remark}
On the flat torus $\mathbb T^2$, \eqref{curv hyp +'}-\eqref{curv hyp -'} are valid for any closed curves with non-vanishing curvature, since $\mathbf k_p(v)\equiv0$ in this case. On the other hand, for hyperbolic surfaces with curvature $K=-1$, it is known that $\mathbf k_p(v)\equiv1$, thus Theorem \ref{main} implies that any curves with curvature bounded away from $\sqrt{1-\epsilon^2}$ will enjoy log-improved generalized periods bounds for $\nu=\epsilon\lambda$. An direct yet significant corollary of our main theorem gives the first result towards Conjecture \ref{C2} in the case of geodesic circles. It follows from the fact that for any given geodesic circle on a Riemannian surface of nonpositive curvature always satisfies \eqref{curv hyp +'}-\eqref{curv hyp -'}, and therefore, Theorem \ref{main} and Remark \ref{main remark} implies the following.
\begin{corr} \label{corollary}
Let $\gamma$ be a unit-speed geodesic circle on a compact Riemannian surface $(M,g)$ with nonpositive curvature. If $\nu$ is an integer multiple of $2\pi |\gamma|^{-1}$, then given any $0<\delta<1$, we have
\[
\left| \int_\gamma e_\lambda(\gamma(s)) e^{-i\nu s} \, ds \right| \leq C(\log \lambda)^{-1/2},
\]
where the constant $C$ is uniform over the sets of all geodesic circles $\gamma$ with radii in $[\delta, \delta^{-1}]$ and all $\nu$ with $|\nu|/\lambda$ in $[0, 1 - \delta]$.
\end{corr}
Note that the number $\delta$ in Corollary \ref{corollary} can be taken to be arbitrarily close to 0, and thus this log-improved bounds indeed hold for any fixed geodesic circle with arbitrary radius.
Another corollary is about the weak $L^2(\gamma)$ limit of eigenfunctions restricted to curves.
\begin{corr} \label{co} Let $\gamma$ be a curve on a compact Riemannian surface $(M,g)$ with nonpositive curvature that satisfies the assumption of Theorem \ref{main}, with $0$ being an interior point of $E_\gamma$, e.g. a geodesic circle. Then for any orthonormal sequence of eigenfunctions $\{e_{\lambda_j}\}$ with frequency $\lambda_j$, $e_{\lambda_j}|_\gamma\rightarrow 0$ weakly in $L^2(\gamma)$ if and only if it is bounded in $L^2(\gamma)$.
\end{corr}
The quantum ergodic restriction (QER) theorem of Toth and Zelditch \cite{TZ} implies that for a compact hyperbolic surface, any orthonormal sequence of eigenfunctions will have a full density subsequence that has bounded $L^2(\gamma)$ normal over a given geodesic circle $\gamma$. Therefore Corollary \ref{co} implies the following.
\begin{corr} Let $(M,g)$ be a compact hyperbolic surface, $\{e_{\lambda_j}\}$ an orthonormal basis of eigenfunctions of frequency $\lambda_j$. Then given a geodesic circle $\gamma$, there exists a density one subsequence $ \{e_{\lambda_{j_k}}\}$ such that $ e_{\lambda_{j_k}}|_\gamma\rightarrow 0$ in weak $L^2(\gamma)$.
\end{corr}
Our paper is organized as follows. In the next section we shall perform several standard reductions by using the reproducing kernels for eigenfunctions and lifting the calculations to the universal cover $(\mathbb R^2,\widetilde g)$. By microlocally decomposing the measure of $\gamma$ into one tangential and two transversal components, we reduce the proof of Theorem \ref{main} to estimating a few microlocalized
oscillatory integrals over curves in $(\mathbb R^2,\widetilde g)$. In \S 3, we recall a stationary phase technique from \cite{HI}, and then use it to handle the term which is local in time. In \S 4, we gather a few bounds for the non-local kernel which are from \cite{Gauss}, and then use them to handle the tangential term.
In \S 5 we prove a few phase function bounds by employing strategies from \cite{emmett2}, and then finish the proof of Theorem \ref{main} by proving favorable bounds for the two transversal parts.
The proof for the corollaries can be found at the end of \S 5.
In what follows, by possibly rescaling the metric, we shall assume that the injectivity radius of $M$, $\text{Inj}\,M$, is at least 10. We shall always use the letter $\epsilon$ to denote a number in $(-1,1)$ that equals the frequency ratio $\nu/\lambda$. The letter $C$ will be used to denote various positive constants depending on $(M,g)$ and $\delta$, whose value could change from line to line.
\textit{Acknowledgments.}
The authors would like to thank Professor Sogge for his constant support. The second author want to thank Professor Greenleaf and Iosevich for their invaluable mentorship.
\section{Standard Reductions and Microlocal Decompositions }
Since we are taking $\epsilon$ to be in a compact subset of $E_\gamma$, we may assume there exists a small constant $\delta > 0$ such that
\begin{align}
|\epsilon| &\leq 1 - \delta,
\end{align}
\begin{align}
\label{curv hyp +} \left|\left\langle \frac{D}{dt} \gamma', \gamma'^\perp \right\rangle + \sqrt{1 - \epsilon^2} \mathbf k_\gamma(\gamma'^\perp) \right| &\geq \delta, \qquad \text{ and } \\
\label{curv hyp -} \left|\left\langle \frac{D}{dt} \gamma', \gamma'^\perp \right\rangle - \sqrt{1 - \epsilon^2} \mathbf k_\gamma(-\gamma'^\perp) \right| &\geq \delta
\end{align}
By using a partition of unity on $\mathbb R/|\gamma|\mathbb Z$ and the triangle inequality, we can obtain \eqref{periods} by showing
\begin{equation} \label{main2 bound}
\left| \int b(t) e_\lambda(\gamma(t)) e^{-i\nu t} \, dt \right| \leq C (\log \lambda)^{-1/2}
\end{equation}
where $b$ is a smooth function on $\mathbb R$ with small support. To begin we assume the support of $b$ is contained in some unit interval in $\mathbb R$, though we may further restrict the support of $b$ as needed.
Let us choose a function $\rho\in {\mathcal S}(\mathbb R)$ satisfying
$$\rho(0)=1 \quad \text{and } \, \, \Hat \rho(\tau)=0 , \quad |\tau|\ge 1/4,$$
and for any $T>0$ define the Fourier multiplier operator $\rho(T(\lambda-\sqrt{-\Delta_g}))$ by the spectral theorem, i.e.
\[
\rho(T(\lambda - \sqrt{-\Delta_g})) = \sum_j \rho(T(\lambda - \lambda_j)) E_j
\]
where $E_j$ is the orthogonal projection operator onto the eigenspace spanned by $e_j$.
$\rho(T(\lambda - \sqrt{-\Delta_g}))$ reproduces eigenfunctions, in the sense that $\rho(T(\lambda-\sqrt{-\Delta_g}))e_\lambda=e_\lambda$. \eqref{main2 bound} will follow from the stronger\footnote{\eqref{main2 bound'} implies Theorem \ref{main} holds for $L^2$-normalized quasimodes which have spectral support on bands $[\lambda, \lambda + 1/\log \lambda]$ of length $1/\log \lambda$.} bound
\begin{equation} \label{main2 bound'} \tag{\ref{main2 bound}$'$}
\left|\, \int b(t) e^{-i\nu t} \rho(T(\lap g - \lambda))f(\gamma(t))\, dt \, \right|\le C \, (\log\lambda)^{-1/2}\, \|f\|_{L^2(M)},
\end{equation}
where
\begin{equation} \label{T = clog}
T = c\log \lambda
\end{equation}
for some sufficiently small $c$.
Choose Fermi local coordinates $x = (x_1,x_2)$ about $\gamma$, so that $x_1 \mapsto (x_1,0)$ parametrizes $\gamma$ and $x_2 \mapsto (x_1,x_2)$ are geodesics normal to $\gamma$. By construction,
\begin{equation}\label{fermi metric}
g = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \text{ for } x_2 = 0.
\end{equation}
Let $B_1$ be a smooth function on $S^1$ taking values in the range $[0,1]$, and for which
\begin{align*}
B_1(\xi) &= 1 \qquad \text{ for } \xi_2 \geq \delta/2 \text{ and } \\
B_1(\xi) &= 0 \qquad \text{ for } \xi_2 \leq \delta/4
\end{align*}
where here $\xi = (\xi_1,\xi_2) \in S^1$. Set
\[
B_{-1}(\xi) = B_1(-\xi) \qquad \text{ and } \qquad B_0 = 1 - B_1 - B_{-1}
\]
(see Figure \ref{fig2}).
\begin{figure}
\centering
\includegraphics[width=.65\textwidth]{figure2.pdf}
\caption{}
\label{fig2}
\end{figure}
For $i = -1,0,1$, we define
\begin{equation} \label{def B}
B_i(x,y,\xi) = b_*(x) \beta(|x - y|) B_i(\xi/|\xi|) \Upsilon_{c_1}(|\xi|/\lambda)
\end{equation}
where $x$ and $y$ are expressed in our Fermi local coordinates,
and we have taken $b_*$ to be a smooth function supported on a neighborhood of $\gamma$ such that $b_*(\gamma(s))=b(s)$; $\beta \in C^\infty(\mathbb R,[0,1])$ with $\beta \equiv 1$ on a neighborhood of $0$ and $\text{supp } \beta$ small; given constant $0<c_1<1$, $\Upsilon_{c_1}\in C^\infty(\mathbb R)$ satisfies
\begin{equation}
\Upsilon_{c_1}(r)=1, \quad r\in[c_1,c_1^{-1}],\quad \Upsilon_{c_1}(r)=0, \quad r\not\in[c_1/2,2c_1^{-1}]
\end{equation}
We associate operators\footnote{The function $B_i(x,y,\xi)$ will serve the same purpose as a zero-order symbol in $\xi$, and their respective operators will similarly function like zero-order classical pseudodifferential operators. It is important to distinguish these objects from symbols and pseudodifferential operators due to the presence of the cutoff away from $|\xi| = \lambda$, even though they play the same roles.} with $B_i$ by
\[
B_if(x) = \frac{1}{(2\pi)^2} \int_{\mathbb R^2} \int_{\mathbb R^2} e^{i\langle x - y , \xi \rangle} B_i(x,y,\xi) f(y) \, dy \, d\xi
\]
in our local coordinates. First, note that $B_i$ is a bounded operator on $L^\infty$ with norm
\begin{equation} \label{B_i norm}
\| B_i \|_{L^\infty\rightarrow L^\infty} = O(\lambda^2)
\end{equation}
for $i = -1,0,1$.
Secondly, note
\[
B_1 + B_{-1} + B_0 = 1-B_\#,
\]
where $B_\#$ is a psedudodifferential with symbol supported away from the set $\{(x,y,\xi):||\xi|/\lambda\in[c_1,c_1^{-1}]\}$. As in \cite[Page 141]{SFIO}, one can then use a parametrix for the half-wave operator to see that if $c_1$ is small enough, $T=c\log\lambda$, we have
\begin{equation} \label{Bsharp}
\|B_\#\circ \rho(T(\lap g - \lambda))\|_{L^2\rightarrow L^\infty}\le C_N\lambda^{-N}, \quad \text{for any }N\in\mathbb N,
\end{equation}
Indeed, using the H\"ormander parametrix for the half wave operator, we can see that the highest order term of the kernel associated to the above operator is
\begin{multline*}
K(T,\lambda;x,y) =\iiiint e^{i(\varphi(z,y,\xi) +\langle x - z ,\zeta \rangle - \tau( p(z,\xi) - \lambda ))} \\ a(T, \lambda; \tau, z, y, \xi)B_{\#}(x,z,\zeta) \,
dz\,d\xi \, d\zeta\,d\tau.
\end{multline*}
where $p(y,\xi) $
is the principal symbol of $\lap g$; $\varphi\in C^\infty(\mathbb R^n\setminus\{0\})$ is homogeneous of degree $1$ in $\xi$, and satisfies
\begin{equation} \label{local varphi}
|\partial_{\xi}^\alpha (\varphi(x,y,\xi) - \langle x - y, \xi \rangle)| \leq C_\alpha |x - y|^2|\xi|^{1 - |\alpha|},
\end{equation}
for multiindices $\alpha \geq 0$ and for $x$ and $y$ sufficiently close;
the symbol $a$ behaves like a zero order symbol in $\xi$. For a more detailed description of the H\"ormander paramterix, see section 3 or ~\cite{SFIO}.
Notice that $p(y,\xi)\sim|\xi|$, then if $|\xi|\not\in[c\lambda,c^{-1}\lambda]$ for some suitable constant $c$ depending on the metric, we can integrate by parts in $\tau$ to see that
\[\left|\int e^{i \tau( p(y,\xi) - \lambda )} a(T, \lambda; \tau, z, y, \xi) \,d\tau\right|\le C_N(|\xi|+\lambda)^{-N},\]
thus the difference between $K(T,\lambda;x,y)$ and
\begin{multline*}
\widetilde K(T,\lambda;x,y) =\iint e^{i(\varphi(z,y,\xi) +\langle x - z ,\zeta \rangle - \tau( p(y,\xi) - \lambda ))}\\ \Upsilon_{c}(|\xi|/\lambda) q(T, \lambda; \tau, z, y, \xi)B_{\#}(x,z,\zeta) \,
dz\,d\xi \, d\zeta\,d\tau,
\end{multline*}
satisfies the required bound if $c_1$ is sufficiently small. It is then clear $\widetilde K$ gives an operator satisfying bounds in \eqref{Bsharp} if we integrate by parts in $z$ a few times. Therefore, by \eqref{Bsharp}, we have
\begin{equation}
\left|\, \int b(t) e^{-i\nu t} B_\# \rho(T(\lap g - \lambda))f(\gamma(t))\, dt \, \right|\le C_N \, \lambda^{-N} \|f\|_{L^2(M)},\quad \text{for any }N\in\mathbb N.
\end{equation}
To prove \eqref{main2 bound'}, it suffices to show that
\begin{equation} \label{main2 bound''} \tag{\ref{main2 bound}$''$}
\left|\, \int b(t) e^{-i\nu t} B_i \rho(T(\lap g - \lambda))f(\gamma(t))\, dt \, \right|\le C \, (\log\lambda)^{-1/2}\, \|f\|_{L^2(M)},
\end{equation}
for each $i = -1,0,1$.
To set up the proof of \eqref{main2 bound''} we first note that the kernel of the operator thereof is given by
$$
B_i\rho(T(\lap g - \lambda))(x,y)=\sum_j \rho(T(\lambda_j-\lambda)) B_ie_j(x)\overline{e_j(y)}.
$$
Hence, by the Cauchy-Schwarz inequality, we would have \eqref{main2 bound'} if we could show that
$$
\int_M\left|\, \int \,e^{-i\nu t} \sum_j \rho(T(\lambda_j-\lambda)) \, B_i e_j(\gamma(t)) \overline{e_j(y)} \, dt \, \right|^2 \, dV_g(y)
\le C(\log\lambda)^{-1},
$$
By orthogonality, if $\chi(\tau)=|\rho(\tau)|^2$, this is equivalent to showing that
\begin{equation}\label{2.2}
\left|\, \iint e^{i\nu(s-t)}\sum_j \chi(T(\lambda_j-\lambda)) B_i e_j(\gamma(t)) \overline{B_i e_j(\gamma(s))} \, dt ds\, \right|
\le C(\log \lambda)^{-1}.
\end{equation}
By Fourier inversion,
\begin{align*}
\sum_j \chi(T(\lambda_j-\lambda))B_ie_j(x)\overline{B_ie_j(y)} &= \frac{1}{2\pi T} \sum_j \int \hat \chi(\tau/T) e^{-i\tau \lambda} e^{i\tau \lambda_j} B_ie_j(x)\overline{B_ie_j(y)} \, d\tau \\
&= \frac1{2\pi T} \int \hat \chi(\tau/T) e^{-i\tau \lambda}
\bigl(B_i e^{i\tau \sqrt{-\Delta_g}} B_i^*\bigr)(x,y) \, d\tau,
\end{align*}
and \eqref{2.2} is equivalent to
\begin{equation} \label{2.3}
\frac{1}{T} \left| \iiint \hat \chi(\tau/T) e^{i\nu(s-t)} e^{-i\tau \lambda} (B_i e^{i\tau \lap g} B_i^*)(\gamma(t), \gamma(s)) \, ds \, dt \, d\tau \right| \leq C (\log \lambda)^{-1}.
\end{equation}
We cut the integral into $\beta(\tau)$ and $(1 - \beta(\tau))$ components where, as before, $\beta \in \mathbb C_0^\infty(\mathbb R,[0,1])$ is a bump function with $\beta(\tau) = 1$ for $|\tau| \leq 1$ and $\beta(\tau) = 0$ for $|\tau| \geq 2$. We bound the $\beta(\tau)$ component first.
\begin{proposition}\label{beta component}
\begin{equation}\label{local term}
\frac{1}{T} \left| \iiint \beta(\tau) \hat \chi(\tau/T) e^{i\nu(s-t)} e^{-i\tau \lambda} (B_i e^{i\tau \lap g} B_i^*)(\gamma(t), \gamma(s)) \, ds \, dt \, d\tau \right| \leq CT^{-1}.
\end{equation}
for $i = -1,0,1$.
\end{proposition}
The proof of Proposition \ref{beta component} is quite standard yet a bit involved. We will give a detailed proof in the next section with the help of a stationary phase lemma from \cite{HI}.
What remains is to show
\begin{align}
\nonumber \frac{1}{T} \left| \iiint (1 - \beta(\tau)) \hat \chi(\tau/T) e^{i\nu(s-t)} e^{-i\tau \lambda} (B_i e^{i\tau \lap g} B_i^*)(\gamma(t), \gamma(s)) \, ds \, dt \, d\tau \right| &\\
\label{precosine} &\hspace{-4em} \leq C (\log \lambda)^{-1}.
\end{align}
To deal with \eqref{precosine}, we lift to the universal cover.
Before this, however, we want to replace the operator $e^{i\tau \lap g}$ with $\cos(\tau \lap g)$ so we can make use of H\"uygen's principle. By Euler's formula,
\[
e^{i\tau \lap g} = 2\cos(\tau \lap g) - e^{-i\tau \lap g},
\]
and so the integral in \eqref{precosine} is
\begin{align*}
&\frac{2}{T} \iiint (1 - \beta(\tau)) \hat \chi(\tau/T) e^{i\nu(s-t)} e^{-i\tau \lambda} (B_i \cos(\tau \lap g) B_i^*)(\gamma(t), \gamma(s)) \, ds \, dt \, d\tau \\
&+ \frac{1}{T} \iiint (1 - \beta(\tau)) \hat \chi(\tau/T) e^{i\nu(s-t)} e^{-i\tau \lambda} (B_i e^{-i\tau \lap g} B_i^*)(\gamma(t), \gamma(s)) \, ds \, dt \, d\tau.
\end{align*}
By reversing our reduction, the second line is
\begin{equation}\label{+term}
\frac{1}{T} \sum_j X_T(-(\lambda_j + \lambda)) \left| \int e^{-i\nu t} B_i e_j(\gamma(t)) \, dt \right|^2
\end{equation}
where $\hat X_T(\tau) = (1 - \beta(\tau)) \hat \chi(\tau/T)$.
We claim \eqref{+term} is $O(\lambda^{-N})$ uniformly for $T \geq 1$ for each $N = 1,2,\ldots$. Indeed, \eqref{B_i norm} and the general sup-norm estimates
\[
\|e_j\|_{L^\infty(M)} \leq C \lambda_j^{1/2}
\]
(see for example ~\cite{SFIO}) imply the integral in \eqref{+term} is bounded by $C\lambda_j^{1/2}\lambda^{2}$. The $O(\lambda^{-N})$ bound follows since $X_T(-(\lambda_j + \lambda))$ is rapidly decaying in $\lambda_j + \lambda$.
It now suffices to show
\begin{align}
\nonumber \frac{1}{T} \left| \iiint (1 - \beta(\tau)) \hat \chi(\tau/T) e^{i\nu(s-t)} e^{-i\tau \lambda} (B_i \cos(\tau \lap g) B_i^*)(\gamma(t), \gamma(s)) \, ds \, dt \, d\tau \right|&\\
\label{2.4} &\hspace{-5em} \leq C (\log \lambda)^{-1}.
\end{align}
Here $\bigl(\cos \tau\sqrt{-\Delta_g}\bigr)(x,y)$ is the wave kernel for the map $$C^\infty(M)\ni f\to u\in C^\infty(\mathbb R\times M)$$ solving the Cauchy problem
with initial data $(f,0)$, i.e.,
\begin{equation}\label{Cauchy}\bigl(\partial_\tau^2-\Delta_g\bigr)u=0, \quad u(0, \, \cdot \, )=f, \quad \partial_\tau u(0, \, \cdot \, )=0.\end{equation}
To be able to compute the integral in \eqref{2.4} we need to relate the wave kernel on $M$ to the corresponding wave kernel on the universal cover of $M$. By the Cartan-Hadamard Theorem (see e.g. \cite[Chapter 7]{doCarmo}), we can lift the calculations up to the universal cover $(\mathbb{R}^2,\tilde g)$ of $(M,g)$.
Let $\Gamma$ denote the group of deck transformations preserving the associated covering map ${\mathbb{R}^2} \to M$ coming from the exponential map about the point $\gamma(0)$. The metric $\tilde g$ is the pullback of $g$ via the covering map. We shall measure the distances in $(\mathbb{R}^2,\tilde g)$ using its Riemannian distance function $d_{\tilde g}({}\cdot{},{}\cdot{})$. We choose a Dirichlet fundamental domain, $D \simeq M$, centered at the lift $\tilde \gamma(0)$ of $\gamma(0)$, which has the property that ${\Bbb R}^2$ is the disjoint union of the
$\alpha(D)$ as $\alpha$ ranges over $\Gamma$ and $\{\tilde y\in {\Bbb R}^2: \, d_{\tilde g}(0,\tilde y)<10\} \subset D$ since we are assuming that $\text{Inj}\,M\ge10$. It then follows that we can identify every point $x\in M$ with the unique point $\tilde x\in D$ having the property
that $\tilde x \mapsto x$ through the covering map. Let also $\{\tilde \gamma(t)\in\mathbb R^2: |t|\le \tfrac12\}$ similarly denote the set of points in $D$ corresponding to our segment
$\gamma$ in $M$. Derivatives of $\gamma$ correspond to push-forwards of the corresponding derivatives of $\tilde \gamma$ through the covering map. In particular, $\tilde \gamma$ is a unit-speed curve with the same geodesic curvature as $\gamma$.
Moreover, if $p \in M$ and $v \in S_pM$, and $\tilde p$ and $\tilde v$ are their respective lifts to $D$ and $T_{\tilde p}D$, then $\tilde \mathbf k_{\tilde p}(\tilde v)$ as defined on the universal cover coincides with $\mathbf k_p(v)$. Hence, the hypotheses \eqref{curv hyp +} and \eqref{curv hyp -} correspond exactly to
\begin{align*}
\left| \left\langle \frac{D}{dt} \tilde \gamma', \tilde \gamma'^\perp \right\rangle + \sqrt{1 - \epsilon^2} \tilde \mathbf k_{\tilde \gamma}(\tilde \gamma'^\perp) \right| &\geq \delta, \qquad \text{ and } \\
\left| \left\langle \frac{D}{dt} \tilde \gamma', \tilde \gamma'^\perp \right\rangle - \sqrt{1 - \epsilon^2} \tilde \mathbf k_{\tilde \gamma} (-\tilde \gamma'^\perp) \right| &\geq \delta.
\end{align*}
Finally, if $\Delta_{{\tilde g}}$ denotes the Laplace-Beltrami operator associated to ${\tilde g}$, then, since solutions of the Cauchy problem \eqref{Cauchy} for $(M,g)$ correspond exactly
to $\Gamma$-invariant solutions of the corresponding Cauchy problem associated to the lifted wave operator $\partial^2_t-\Delta_{{\tilde g}}$, we have the following
Poisson formula relating the wave kernel on $(M,g)$ to the one for the universal cover $({\Bbb R}^2,{\tilde g})$:
\begin{equation}\label{2.10}
\bigl(\cos \tau \sqrt{-\Delta_g} \big)(x,y)=\sum_{\alpha\in \Gamma}\bigl(\cos \tau \sqrt{-\Delta_{\tilde g}} \bigr)(\tilde x, \alpha(\tilde y)).
\end{equation}
Then,
\begin{align*}
&B_i \cos(\tau \lap g) B_i^*(x,y) \\
&= \frac{1}{(2\pi)^4} \iiiint e^{i\langle x - w, \eta \rangle } B_i(x,w,\eta) \cos(\tau \lap g)(w,z) e^{i\langle z - y, \zeta \rangle} \overline{B_i(z,y,\zeta)}\\
&\hspace{28em} \, dw \, dz \, d\eta \, d\zeta \\
&= \frac{1}{(2\pi)^4} \sum_{\alpha \in \Gamma} \iiiint e^{i\langle \tilde x - \tilde w, \eta \rangle } \widetilde B_i(\tilde x,\tilde w,\eta) \cos(\tau \lap {\tilde g})(\tilde w,\alpha (\tilde z)) e^{i\langle \tilde z - \tilde y, \zeta \rangle} \overline{ \widetilde B_i(\tilde z,\tilde y,\zeta)}\\
&\hspace{29em} \, d\tilde w \, d\tilde z \, d\eta \, d\zeta
\end{align*}
where
\[
\widetilde B_i(\tilde x, \tilde y,\xi) =
\begin{cases}
B_i(x,y,\xi), & {\text{if }}{\tilde x, \tilde y \in D,} \\
0, & \text{otherwise,}
\end{cases}
\]
whereafter we write
\[
B_i \cos(\tau \lap g) B_i^*(x,y) = \sum_{\alpha \in \Gamma} \widetilde B_i \cos(\tau, \lap {\tilde g}) \widetilde B_{i,\alpha}^{*}(\tilde x, \tilde y),
\]
where, in terms of the kernel, $$\widetilde B_{i,\alpha}(\tilde x,\tilde y)=\widetilde B_i(\alpha^{-1}(\tilde x),\tilde y).$$
\eqref{2.4} will follow from
\begin{equation} \label{lifted bound}
\frac{1}{T} \sum_{\alpha \in \Gamma} \left| \iint e^{i\nu(s - t)} K_{i,\alpha} (T,\lambda;\tilde \gamma(t), \tilde \gamma(s)) \, ds \, dt \right| \leq C (\log \lambda)^{-1}
\end{equation}
where
\begin{equation} \label{kernel def}
K_{i,\alpha}(T,\lambda;x,y) = \int {(1 - \beta(\tau))} \hat \chi(\tau/T) e^{-i\tau \lambda} \widetilde B_i \cos(\tau \lap {\tilde g}) \widetilde B_{i,\alpha}^* (x,y) \, d\tau.
\end{equation}
The bound \eqref{lifted bound}, and hence Theorem \ref{main}, follows from the propositions below and \eqref{T = clog} for some sufficiently small $c$. The first proposition treats the identity term in the sum for $i = -1,0,$ and $1$.
\begin{proposition} \label{small time} If $\alpha=I$, the identity deck transformation, we have
\begin{equation}\label{2.17}
\frac{1}{T} \left| \iint e^{i\nu(s - t)} K_{i,\alpha}(T,\lambda;\tilde \gamma(t), \tilde \gamma(s)) \, ds \, dt \right| \leq C T^{-1}.
\end{equation}
for $i = -1,0,1$.
\end{proposition}
The next proposition treats the remaining terms for the $i = 0$ case.
\begin{proposition} \label{nonsmall time}
\begin{equation}\label{2.18}
\frac{1}{T} \sum_{\alpha \in \Gamma \setminus \{I\}} \left| \iint e^{i\nu(s-t)} K_{0,\alpha}(T,\lambda; \tilde \gamma(t), \tilde \gamma(s)) \, ds \, dt \right| \leq Ce^{C'T} \lambda^{-1/2}.
\end{equation}
\end{proposition}
Fix a constant $R \gg 1$ to be determined later and which is independent of $T$, $\lambda$, $\gamma$ and $\nu$ subject to the hypotheses of Theorem \ref{main}. We set
\begin{equation} \label{A def}
A = \{ \alpha \in \Gamma : d_{{\tilde g}}(\tilde \gamma, \alpha (\tilde \gamma)) \leq R \}.
\end{equation}
and treat the contributions of $A \setminus \{I\}$ and $\Gamma \setminus A$ to the sum in \eqref{lifted bound} separately.
\begin{proposition} \label{medium time}
\begin{equation}\label{2.20}
\frac{1}{T} \sum_{\alpha \in A \setminus \{I\}} \left| \iint e^{i\nu(s - t)} K_{i,\alpha}(T,\lambda;\tilde \gamma(t), \tilde \gamma(s)) \, ds \, dt \right| \leq CT^{-1}
\end{equation}
for $i = -1,1$.
\end{proposition}
\begin{proposition} \label{large time}
\begin{equation}\label{2.21}
\frac{1}{T} \sum_{\alpha \in \Gamma \setminus A} \left| \iint e^{i\nu(s - t)} K_{i,\alpha}(T,\lambda;\tilde \gamma(t), \tilde \gamma(s)) \, ds \, dt \right| \leq Ce^{C'T} \lambda^{-1/2}.
\end{equation}
for $i = -1,1$.
\end{proposition}
As argued, Theorem \ref{main} follows from Propositions \ref{beta component} through \ref{large time}.
\section{Stationary Phase Tool and Proof of Proposition \ref{beta component}}
We will be dealing with oscillatory integrals with up to eight variables of integration, so it will be convenient to be able to use the method of stationary phase in stages to avoid having to work with excessively large matrices. To this end, we use ~\cite[Theorem 7.7.6 ]{HI}, summarized below for convenience.
Let $\phi(x,y)$ be a smooth phase function on $\mathbb R^m \times \mathbb R^n$ with
\[
\nabla_y \phi(0,0) = 0 \qquad \text{ and } \qquad \det \nabla_y^2 \phi(0,0) \neq 0,
\]
and let $a(\lambda;x,y)$ be a smooth amplitude with small, adjustable support satisfying
\[
|\partial_\lambda^j \partial_x^\alpha \partial_y^\beta a(\lambda;x,y)| \leq C_{j,\alpha,\beta} \lambda^{-j} \qquad \text{ for } \lambda \geq 1
\]
for $j = 0,1,2,\ldots$ and multiindices $\alpha$ and $\beta$. $\nabla_y^2 \phi \neq 0$ on a neighborhood of $0$ by continuity. There exists locally a smooth map $x \mapsto y(x)$ whose graph in $\mathbb R^m \times \mathbb R^n$ contains all points in a neighborhood of $0$ such that $\nabla_y \phi = 0$, by the implicit function theorem. Let $\sigma$ denote the signature of $\nabla_y^2 \phi$. By continuity, $\sigma$ is constant on a neighborhood of $0$. We adjust the support of $a$ to lie in the intersection of these neighborhoods.
\begin{lemma}[\cite{HI}]\label{st phase}
Let
\[
I(\lambda; x) = \int_{\mathbb R^n} e^{i\lambda \phi(x,y)} a(\lambda; x, y) \, dy
\]
with $\phi$ and $a$ as above. Then for any fixed positive integer $N$, there exists $R_N(\lambda;x)$ such that
\begin{align*}
I(\lambda; x) = (\lambda/2\pi)^{-n/2} &|\det \nabla_y^2 \phi(x,y(x))|^{-1/2} e^{\pi i \sigma/4} e^{i\lambda \phi(x,y(x))} a(\lambda; x, y(x))\\
&\hspace{8em}+ \lambda^{-n/2 - 1} e^{i\lambda \phi(x,y(x))} R_N(\lambda; x) + O(\lambda^{-N})
\end{align*}
where $R_N$ has compact support and satisfies
\[
|\partial_\lambda^j \partial_x^\alpha R(\lambda; x)| \leq C_{j,\alpha} \lambda^{-j} \qquad \text{ for } \lambda \geq 1,
\]
and where the $O(\lambda^{-N})$ term is uniform in $x$.
\end{lemma}
Inspection of the proof of ~\cite[Theorem 7.7.6]{HI} shows that all the bounds, constants, supports, and neighborhoods in Lemma \ref{st phase} are uniform and only depend on finitely many derivatives of $a$ and $\phi$. Now we are ready to prove Proposition \ref{beta component}.
\begin{proof}[Proof of Proposition \ref{beta component}] For the sake of simplicity, we shall only give the proof for the case when $i=0$, the other two cases follow from the same proof.
By the H\"ormander's parametrix for the half-wave operator, (see \cite[Chap. 4]{SFIO}), we can write
\[
e^{i\tau\lap g}(x,y) = \frac{1}{(2\pi)^n} \int_{\mathbb R^n} e^{i(\varphi(x,y,\xi) + \tau p(y,\xi))} q(\tau,x,y,\xi) \, d\xi
\]
modulo a smooth kernel, where $p(y,\xi) $
is the principal symbol of $\lap g$, and where $\varphi\in C^\infty(\mathbb R^n\setminus\{0\})$ is homogeneous of degree $1$ in $\xi$ and satisfies
\begin{equation} \label{local varphi}
|\partial_{\xi}^\alpha (\varphi(x,y,\xi) - \langle x - y, \xi \rangle)| \leq C_\alpha |x - y|^2|\xi|^{1 - |\alpha|}
\end{equation}
for multiindices $\alpha \geq 0$ and for $x$ and $y$ sufficiently close.
Moreover, $q$ satisfies bounds
\begin{equation} \label{q bounds}
|\partial_\xi^\alpha \partial_{\tau,x,y}^\beta q(\tau,x,y,\xi)| \leq C_{\alpha,\beta} (1 + |\xi|)^{-|\alpha|},
\end{equation}
and where for $\tau \in \text{supp } \beta$, $q$ is supported on a small neighborhood of $x = y$. Hence, the main term of \eqref{local term} is
\begin{equation}
\frac{1}{T} \iint K(T,\lambda;\gamma(s),\gamma(t))e^{i\nu(s-t)} \, ds \, dt,
\end{equation}
if the kernel is the following
\begin{multline*}
K(T,\lambda;x,y) =\idotsint e^{i(\varphi(w,z,\xi) +\langle x - w,\eta \rangle +\langle z - y ,\zeta \rangle + \tau( p(z,\xi) - \lambda ))} \\ a(T, \lambda; \tau, x, y, \xi, w, z, \eta, \zeta) \,
dw\,dz\,d\xi \, d\eta \, d\zeta\,d\tau,
\end{multline*}
where the amplitude is given by \[
a(T, \lambda; \tau, x, y, \xi, w, z, \eta, \zeta) = \beta(\tau)\hat \chi(\tau/T) B_0(x,w,\eta) q(\tau, w, z, \xi) \overline{ B_0(y,z,\zeta)},
\]
and satisfies
\begin{equation} \label{q bounds}
|\partial_\xi^{\alpha_1} \partial_\eta^{\alpha_2} \partial_\zeta^{\alpha_3} \partial_{\tau,x,y,w,z}^\beta a| \leq C_{\alpha,\beta} (1 + |\xi|)^{-|\alpha_1|}(1 + |\eta|)^{-|\alpha_2|}(1 + |\zeta|)^{-|\alpha_3|}
\end{equation}
It suffices to show that
\begin{equation}\label{local goal}
\left|\iint K(T,\lambda;\gamma(s),\gamma(t))e^{i\nu(s-t)} \, ds \, dt\right|\le C.
\end{equation}
After a change of coordinates sending $(\xi,\eta,\zeta) \mapsto (\lambda \xi,\lambda \eta,\lambda \zeta)$, we have
\begin{multline*}
K(T,\lambda;x,y) =\lambda^6\idotsint e^{i\lambda\Phi(\tau,x,y,\xi,w,z,\eta,\zeta)} \\ a(T, \lambda; \tau, x, y, \lambda\xi, w, z, \lambda\eta, \lambda\zeta) \,
dw\,dz\,d\xi \, d\eta \, d\zeta\,d\tau,
\end{multline*}
After fixing $\tau,x,y,\xi$, the phase function
\[
\Phi(\tau,x,y,\xi,w,z,\eta,\zeta) = \varphi(w,z,\xi) + \langle x - w,\eta \rangle + \langle z - y ,\zeta \rangle + \tau( p(z,\xi) - 1 )
\]
has a unique critical point in the four variables $(w,z,\eta,\zeta)$ at
\[
(w,z,\eta,\zeta) = (x,y,\nabla_x \varphi(x,y,\xi), -\nabla_y \varphi(x,y,\xi) + \tau \nabla_y p(y,\xi)).
\]
It is easy to see that at this critical point, the Hessian of $\Phi$ is
\[
\nabla_{w,z,\eta,\zeta}^2 \Phi = \begin{bmatrix}
* & * & -I & 0 \\
* & * & 0 & I \\
-I & 0 & 0 & 0 \\
0 & I & 0 & 0
\end{bmatrix},
\]
which has determinant $-1$ and signature $0$. Then by Lemma \ref{st phase}, we see that
\begin{equation*}
K(T,\lambda; x,y) = \lambda^2 \iint \widetilde a(T,\lambda; \tau, x, y, \xi) e^{i\lambda \widetilde\Phi(\tau, x, y, \xi)} \, d\tau \, d\xi,
\end{equation*}
modulo lower order terms. Here
\[
\widetilde a(T,\lambda; \tau, x, y, \xi) = a(T, \lambda; \tau, x, y, \xi, x,y,\nabla_x \varphi(x,y,\xi), -\nabla_y \varphi(x,y,\xi) + \tau \nabla_y p(y,\xi))
\]
and
\[
\widetilde\Phi(\tau,x,y,\xi) = \varphi(x,y,\xi) + \tau( p(y,\xi) - 1 ).
\]
Let
\[
\Psi(\tau,s,t,\xi) = \widetilde\Phi(\tau,\gamma(s),\gamma(t),\xi)+\epsilon(s-t),
\]
then we can see that the main term of \eqref{local goal} is
\begin{equation*}
\iiiint \widetilde a(T,\lambda; \tilde \gamma(s), \tilde \gamma(t)) e^{i\lambda \Psi(\tau,s,t,\xi)} \, ds \, dt \, d\xi \, d\tau.
\end{equation*}
The gradient of the phase function $\Psi$ is
\[
\nabla_{\tau,s,\xi_1,\xi_2} \Psi = \begin{bmatrix}
|\xi| - 1 \\
\epsilon + \xi_1 + O(|s - t||\xi|)\\
s - t + O(|s-t|^2) + \tau \partial_{\xi_1} |\xi|\\
\tau \partial_{\xi_2} |\xi| + O(|s - t|^2)\\
\end{bmatrix},
\] with critical points at $(\tau, s, \xi_1, \xi_2) = (0,t,-\epsilon, \pm \sqrt{1 - \epsilon^2})$. The Hessian at these critical points is:
\[
\nabla_{\tau, s, \xi_1, \xi_2}^2 \Psi = \begin{bmatrix}
0 & 0 & -\epsilon & \pm\sqrt{1-\epsilon^2} \\
0 & * & 1 & 0 \\
-\epsilon & 1 & 0 & 0 \\
\pm\sqrt{1-\epsilon^2} & 0 & 0 & 0
\end{bmatrix}
\]
Since by our assumption, $1 - \epsilon \geq \delta$, $\sqrt{1-\epsilon^2}$ is bounded away from zero, this matrix has determinant uniformly bounded away from zero. Then if we invoke the method of stationary phase again, \eqref{local goal} follows, finishing the proof of Proposition \ref{beta component}.
\end{proof}
\section{Kernel Bounds and Proofs of Propositions \ref{small time} and \ref{nonsmall time}}
We shall need an explicit expression for the kernel
\begin{equation}
K(T,\lambda;x,y)=\int \bigl(1-\beta(\tau)\bigr)\hat \chi(\tau/T) e^{i\tau \lambda} \bigl(\cos \tau \sqrt{-\Delta}\bigr)(x,y) \, d\tau,
\end{equation}
evaluating at $(x,y)=(\tilde \gamma(t),\alpha(\tilde \gamma(s)))$. Note that $K_{i,\alpha}$ is the kernel corresponding to $\widetilde B_i$-conjugation of $K$.
The following proposition is adapted from ~\cite[Proposition 5.1]{Gauss} and characterizes the kernel $K(T,\lambda;x,y)$. In what follows, we let $\Delta_x$ and $\Delta_y$ denote the Laplace-Beltrami operators on $\tilde M$ operating in the $x$ and $y$ variables, respectively.
\begin{proposition}[{\cite[Proposition 5.1]{Gauss}}]\label{prop5.1} Let $T=c\log\lambda$. If $d_{\tilde g}\ge1$ and $\lambda\gg 1$, we have
\begin{equation}\label{kernel1}
K(T,\lambda;x,y)= \lambda^{1/2}\sum_\pm a_\pm(T,\lambda; x,y) e^{\pm i\lambda d_{\tilde g}(x,y)} +R(T,\lambda,x,y),
\end{equation}
where
\begin{equation}\label{kernel2}
|a_\pm(T,\lambda; x,y)|\le C,
\end{equation}
and
if $\ell,m=1,2,3,\dots$ are fixed
\begin{equation}\label{kernel3}
\Delta^\ell_x \Delta_y^m a_\pm(T,\lambda; x,y) = O(\exp(C_{\ell,m} d_{\tilde g}(x,y)))
\end{equation}
and
\begin{equation}\label{kernel5}
|R(T,\lambda,x,y)|\le\lambda^{-5},
\end{equation}
provided the constant $c>0$ is sufficiently small.
Also, in this case we also have
\begin{equation}\label{kernel6}
K(T,\lambda;x,y)=O(\lambda^{-5}), \quad \text{if } \, \, \, d_{\tilde g}(x,y)\le 1.
\end{equation}
\end{proposition}
We remark that the bounds \eqref{kernel3}, \eqref{kernel5}, and \eqref{kernel6} are stronger than those stated in \cite{Gauss}. The first follows from the pure derivative bounds
\begin{align*}
&\Delta_x^\ell a_\pm(T,\lambda;x,y) = O(\exp(C_\ell d_{\tilde g}(x,y))) \qquad \text{ and }\\
&\Delta_y^\ell a_\pm(T,\lambda;x,y) = O(\exp(C_\ell d_{\tilde g}(x,y)))
\end{align*}
given by ~\cite[Proposition 5.1]{Gauss} and Proposition \ref{mixed derivatives prop} in the appendix.
The latter two follow easily from the proof in \cite{Gauss} by increasing the number of terms used in the Hadamard parametrix and doing integration by parts a few more times. Now one can easily see that Proposition \ref{small time} follows directly from \eqref{kernel6} and the fact that each $\widetilde B_i$ is bounded on $L^\infty$ with norm about $\lambda^2$.
Now we can use Lemma \ref{st phase} to compute $K_{i,\alpha}$ for $i=-1,0,1$. Indeed, from Proposition \ref{prop5.1}, we see that
\begin{multline}\label{K_i}
K_{i,\alpha}(T,\lambda;x,y) \\
= \frac{\lambda^{1/2}}{(2\pi)^4} \sum_\pm \iiiint a_\pm(T,\lambda;w,z) e^{ i (\pm\lambda d_{\tilde g}(w,z)+\langle x - w, \eta \rangle+\langle \alpha^{-1}(z) - y, \zeta \rangle)} \\\widetilde B_i(x,w,\eta) \overline{\widetilde B_i(\alpha^{-1}(z),y,\zeta)} \, dw \, dz \, d\eta \, d\zeta
+ B_i RB_i^*(T,\lambda; x,y).
\end{multline}
Here, the $B_iRB_i^*$ term maps $L^\infty\rightarrow L^\infty$ with norm $O(\lambda^{-1})$ thanks to \eqref{kernel5} and \eqref{B_i norm}. It suffices to compute the first term.
After a change of variables sending $\eta \mapsto \lambda \eta$ and $\zeta \mapsto \lambda \zeta$, we can see that the main term above is
\[
\lambda^{9/2} \iiiint a(T,\lambda; x,y,w,z,\eta,\zeta) e^{i\lambda \Phi(x,y,w,z,\eta,\zeta)} \, dw \, dz \, d\eta \, d\zeta
\]
where
\[
\Phi(x,y,w,z,\eta,\zeta) = \pm d_{\tilde g}(w,z) + \langle x - w,\eta \rangle + \langle \alpha^{-1}(z) - y , \zeta \rangle
\]
and
\[
a(T,\lambda; x,y,w,z,\eta,\zeta) = a_\pm(T,\lambda; w,z) \widetilde B_i(x,w,\lambda \eta) \overline{\widetilde B_i(\alpha^{-1}(z),y,\lambda \zeta)}.
\]
Let us first look at the gradient of the phase function $\Phi$ in all variables $(w,z,\eta,\xi)$,
\[
\nabla_{w,z,\eta,\zeta} \Phi = \begin{bmatrix}
-\eta \pm \nabla_w d_{\tilde g}(w,z) \\
\zeta \pm \nabla_z d_{\tilde g}(w,z) \\
x - w \\
\alpha^{-1}(z) - y
\end{bmatrix},
\]
which has a unique critical point at $(w,z,\eta,\zeta) = (x,\alpha (y),\pm \nabla_x d_{\tilde g}(x,\alpha (y)), \mp \nabla_y d_{\tilde g}(x,\alpha (y)))$. In particular, if $\sigma$ is the unit-speed geodesic connecting the two points $x$ and $\alpha (y)$, with $\sigma(0) = x$ and $\sigma(d_{\tilde g}(x,\alpha(y))) = \alpha (y)$, the critical point is $(x,y,\pm \sigma'(0), \pm \alpha_* \sigma'(d_{\tilde g}(x,\alpha (y))))$, where $\alpha_*$ is the map induced on the cotangent bundle by $\alpha$. The Hessian matrix of $\Phi$ is
\[
\nabla_{w,z,\eta,\zeta}^2 \Phi = \begin{bmatrix}
* & * & -I & 0 \\
* & * & 0 & I \\
-I & 0 & 0 & 0 \\
0 & I & 0 & 0
\end{bmatrix},
\]
which has full rank. This matrix has signature $0$ and determinant $-1$ so by Lemma \ref{st phase}, modulo a $O(e^{Cd_{{\tilde g}}(x,\alpha (y))} \lambda^{-1/2})$ error, \eqref{K_i} is equal to
\begin{equation} \label{K_i'}
\lambda^{1/2} \sum_\pm a_\pm(T,\lambda;x,\alpha (y)) [b_*(x)]^2 [b_*(y)]^2 B_i(\mp \sigma'(0)) B_i(\mp \alpha_* \sigma'(d_{\tilde g}(x,\alpha(y))) e^{\pm i \lambda d_{\tilde g}(x,\alpha (y))}.
\end{equation}
Now we are ready to prove Propositions \ref{small time} and \ref{nonsmall time}. By \eqref{kernel6} and the same argument as before, $K_{i,\alpha}(T,\lambda;x,y) = O(\lambda^{-1})$ for $d_{\tilde g}(x,y) \leq 1$, whence follows Proposition \ref{small time}. We prove Proposition \ref{nonsmall time} below.
\begin{proof}[Proof of Proposition \ref{nonsmall time}.]
Notice that by Huygen's Principle, the number of nonzero summands in \eqref{2.18} is at most exponential in $T$, and thus to prove Proposition \ref{nonsmall time}, it suffices to show that for each $\alpha\neq I$, there exists a constant $C$ independent of $\alpha$, such that
\begin{equation} \label{nonsmall goal}
\left| \iint e^{i\nu(s-t)} \widetilde K_{0,\alpha}(T,\lambda; \tilde \gamma(t), \tilde \gamma(s)) \, ds \, dt \right| \leq Ce^{CT} \lambda^{-1/2}.
\end{equation}
In fact, by \eqref{K_i'}, the left hand side of \eqref{nonsmall goal} is equal to
\begin{multline}
\iint \lambda^{1/2} \sum_\pm a_\pm(T,\lambda;\tilde \gamma(t), \alpha(\tilde \gamma(s))) [b(t)]^2 [b(s)]^2 B_i(\mp \sigma'(0))\\ B_i(\mp \alpha_* \sigma'(r_\alpha(t,s))) e^{i\lambda(\epsilon(s-t) \pm r_\alpha(t,s))} \, ds \, dt,
\end{multline}
where we have set $r_\alpha(t,s) = d_{\tilde g}(\tilde \gamma(t), \alpha( \tilde \gamma(s)))$ and where $\sigma$ is the unit-speed geodesic connecting the two points $\tilde \gamma(t)$ and $\alpha (\tilde \gamma(s))$.
The key observation here is that the phase function
$\epsilon(s-t) \pm r_\alpha(t,s)$ has no critical point in the support of the integrand. In fact, the $(t,s)$ gradient of this phase function is
\[
(-\epsilon \pm \partial_t r_\alpha(t,s), \epsilon\pm \partial_s r_\alpha(t,s)),
\]
which vanishes only if the angle made by the geodesic $\sigma$ and $\tilde\gamma$ has cosine value to be equal to $\mp\epsilon$, and at the same time the angle made by $\sigma$ and $\alpha( \tilde\gamma)$ has cosine value equal to $\pm\epsilon$. Since $|\epsilon|\le 1-\delta$ is uniformly bounded away from 1, at a critical point of the phase these two angles will be uniformly bounded away from $0$ and $\pi$ independent of the choice of $\alpha$. However, neither of these can happen in the support of $a_\pm$, due to our choice of the phase support of $B_0$, see Figure \ref{fig2}. Thus, we have an absolute lower bound for the gradient of the phase function which is uniform in $\alpha$. Now we can use \eqref{kernel3} to integrate by parts in $s$ to get \eqref{nonsmall goal}, finishing the proof of Proposition \ref{nonsmall time}.
\end{proof}
\section{Phase Function Bounds and Proofs of Propositions \ref{medium time} and \ref{large time}}
By \eqref{K_i'}, we write
\begin{multline} \label{final osc int}
\iint e^{i\nu(s-t)} K_{i,\alpha}(T,\lambda; \tilde \gamma(t), \tilde \gamma(s)) \, ds \, dt \\
= \lambda^{1/2} \sum_\pm \iint a_\pm(T,\lambda, \alpha;t,s) e^{i \pm \lambda \phi_\alpha(t,s)} \, ds \, dt + O(e^{CT}\lambda^{-1/2})\\
\end{multline}
with phase function\footnote{Strictly speaking, this should be $\pm\epsilon(t - s) + r_\alpha(t,s)$, however, our $\epsilon$ is allowed to be negative, so we omit the $\pm$ sign for simplicity.}
\[
\phi_\alpha(t,s) = \epsilon(t - s) + r_\alpha(t,s)
\]
and amplitude
\[
a_\pm(T,\lambda,\alpha;t,s) = a_\pm(T,\lambda; \tilde \gamma(t), \alpha( \tilde \gamma(s))) [b(s)]^2[b(t)]^2 B_i(\mp \sigma'(0)) B_i(\mp \alpha_* \sigma'(r_\alpha))
\]
where $r_\alpha(t,s) = d_{\tilde g}(\tilde \gamma(t), \alpha( \tilde \gamma(s)))$, $\sigma$ is the geodesic adjoining $\tilde \gamma(t)$ and $\alpha( \tilde \gamma(s))$ as before,
and where by \eqref{kernel3} the amplitude satisfies
\begin{equation}\label{amp bound}
|\partial_t^j \partial_s^k a_\pm(T,\lambda,\alpha;t,s)| = O(e^{C_{j,k} r_\alpha}).
\end{equation}
Notice that we can control the support of the amplitude by controlling the support of $b$.
In what follows, we will only consider the case where $i = +1$. The arguments for when $i = -1$ are similar. Fix a unit normal vector field $v(t) = (0,1)$ in our Fermi coordinates along $\tilde \gamma(t)$. Then, $B_1(\xi)$ is supported in the region $\langle v,\xi \rangle \geq \frac\delta4 |\xi|$. Hence the amplitude is supported only for those $t$ and $s$ for which
\begin{align}
&\label{supp of a} \langle \mp \sigma'(0), v(t) \rangle \geq \delta/4 \qquad \text{ and } \\
&\nonumber \langle \mp \alpha_* \sigma'(r_\alpha(t,s)), v(s) \rangle \geq \delta/4,
\end{align}
where the signs $\pm$ must match. We will use the methods of stationary and nonstationary phase to provide the desired bounds on the right side of \eqref{final osc int}, so we need some information about the first and second derivatives of $\phi_\alpha$.
Firstly, the $(t,s)$ gradient of the phase function is
\[
\nabla_{t,s} \phi_\alpha(t,s) = \begin{bmatrix}
\epsilon + \partial_t r_\alpha(t,s) \\
-\epsilon + \partial_s r_\alpha(t,s)
\end{bmatrix}.
\]
Note $\phi_\alpha$ has a critical point wherever the geodesic $\sigma$ is incident to both $\tilde \gamma$ and $\alpha(\tilde \gamma)$ at an angle with cosine value $\epsilon$. From ~\cite{emmett2} we have the computation
\begin{equation} \label{partial_s^2}
\partial_s^2 \phi_\alpha(t,s) = \cos(\theta) (\pm \kappa_{\tilde \gamma}(s) + \cos(\theta) \kappa_{S(\tilde \gamma, r)}(s))
\end{equation}
where $\kappa_{S(\tilde \gamma,r_\alpha)}(s)$ denotes the geodesic curvature at $\alpha(\tilde \gamma(s))$ of the circle centered at $\tilde \gamma(t)$ with radius $r_\alpha(t,s)$, and $\theta$ is the angle this circle makes with $\alpha(\tilde \gamma)$ (see Figure \ref{fig3}). The sign of $\pm$ agrees with that of $\langle \sigma'(r_\alpha(t,s)), \frac{D}{ds} \alpha(\tilde \gamma(s)) \rangle$, i.e. positive if $\sigma'$ agrees with the direction of curvature of $\alpha(\tilde \gamma)$ and negative otherwise. A similar formula
\begin{equation} \label{partial_t^2}
\partial_t^2 \phi_\alpha(t,s) = \cos(\theta') (\mp \kappa_{\tilde \gamma}(t) + \cos(\theta')\kappa_{S(\alpha(\tilde \gamma),r_\alpha)}(t))
\end{equation}
holds for the second derivative in $t$, where $\theta'$ is the angle the circle centered at $\alpha(\tilde \gamma(s))$ makes with $\tilde \gamma$ at $\tilde \gamma(t)$, and where the sign $\mp$ \emph{disagrees} with the sign of $\langle \sigma'(0), \frac{D}{dt}\tilde \gamma(t) \rangle$.
\begin{figure}
\centering
\includegraphics[width=.65\textwidth]{figure3.pdf}
\caption{}
\label{fig3}
\end{figure}
We are now in a position to prove Proposition \ref{medium time}.
\begin{proof}[Proof of Proposition \ref{medium time}]
Since the number of terms $\alpha \in A \setminus I$ is fixed and finite, it suffices to show
\[
\frac{\lambda^{1/2}}{T} \left| \iint a_\pm(T,\lambda,\alpha;t,s) e^{\pm i \lambda \phi_\alpha(t,s)} \, dt \, ds \right| \leq C_\alpha T^{-1}
\]
for each such $\alpha$, where the constant $C_\alpha$ is allowed to depend on $\alpha$. We claim that the union of open neighborhoods
\[
\{(t,s) : \nabla_{t,s} \phi_\alpha(t,s) \neq 0 \} \cup \{(t,s) : \partial_s^2 \phi_\alpha(t,s) \neq 0 \} \cup \{(t,s) : \partial_t^2 \phi_\alpha(t,s) \neq 0 \}
\]
covers the diagonal $t = s$ of the support of $a_\pm$. We then restrict the support of $a_\pm$ so that it lies entirely within one of these open neighborhoods. The desired bound is obtained by nonstationary phase, in the first case, or by stationary phase (e.g. by ~\cite[Corollary 1.1.8]{SFIO}) in the appropriate variable.
Suppose $(t,s)$ is in the diagonal of $\text{supp } a_\pm$ and that $\nabla_{t,s} \phi_\alpha(t,s) = 0$ and $\partial_s^2 \phi_\alpha(t,s) = 0$. Then the geodesic $\sigma$ adjoining $\tilde \gamma(t)$ to $\alpha(\tilde \gamma(s))$ is incident to both $\tilde \gamma$ and $\alpha(\tilde \gamma)$ at an angle with cosine value $\epsilon$. Moreover, $\cos(\theta) = \sqrt{1 - \epsilon^2}$ since the geodesic $\sigma$ and the circle $S(\tilde \gamma,r_\alpha)$ intersect $\alpha(\tilde \gamma)$ at complimentary angles. By \eqref{partial_s^2},
\[
0 = \pm \kappa_{\tilde \gamma}(s) + \sqrt{1 - \epsilon^2} \kappa_{S(\tilde \gamma,r_\alpha)}(s).
\]
Note this situation requires the sign $\pm$ to be negative, so that $\sigma'$ points in a direction contrary to the direction of the curvature of $\alpha(\tilde \gamma)$. Since $s = t$ and $(t,s) \in \text{supp } a_\pm$, $\sigma'$ must also point in a direction contrary to that of that of the curvature of $\tilde \gamma$ (see ~\eqref{supp of a}). Therefore the sign in \eqref{partial_t^2} is positive and
\[
\partial_t^2 \phi_\alpha(t,s) = \sqrt{1 - \epsilon^2} \left( \kappa_{\tilde \gamma}(t) + \sqrt{1- \epsilon^2} \kappa_{S(\alpha(\tilde \gamma),r_\alpha)}(t) \right) > 0
\]
as desired.
\end{proof}
Proposition \ref{large time} requires us obtain some uniform bounds on \eqref{final osc int}, so we will need some uniform bounds on the second derivatives of $\phi_\alpha$. The bounds which follow are largely adapted from the corresponding bounds in ~\cite{emmett2}. We begin with the mixed derivative.
\begin{lemma}[{~\cite[Lemma 3.1]{emmett2}}] \label{off-diagonal lemma} We have absolute bounds
\[
|\partial_s \partial_t \phi_\alpha(t,s)| \leq 2/r_\alpha(t,s).
\]
\end{lemma}
To bound the pure derivatives \eqref{partial_s^2} and \eqref{partial_t^2}, we need to be able to describe the behavior of the curvature of circles of large radius. For this we have the following lemma.
\begin{lemma}[{~\cite[Lemma 4.1]{emmett2}}] \label{large radius} Using the notation above, we have absolute bounds\footnote{Note while this lemma is stated for circles with centers along $\tilde \gamma$, it holds for circles in general.}
\[
0 < \kappa_{S(\tilde \gamma,r_\alpha)}(s) - \mathbf k(\sigma'(r_\alpha(t,s))) < 1/r_\alpha(t,s).
\]
\end{lemma}
Our final geometric lemma provides bounds on the pure second derivatives of $\phi_\alpha$ and is adapted from ~\cite[Lemma 4.2]{emmett2}.
\begin{lemma} \label{diagonal lemma}
Suppose
\[
|\pm \kappa_{\tilde\gamma}(s) - \sqrt{1-\epsilon^2}\kappa_{S(\tilde \gamma,r_\alpha)}(s)| > \varepsilon_0 \qquad \text{ for all } t,s \in \mathcal I
\]
for some $0<\varepsilon_0\ll \delta $. Here, as before, the $\pm$ sign matches that of $\langle \sigma'(r_\alpha(t,s)), \frac{D}{ds} \alpha(\tilde \gamma(s)) \rangle$. Then there exist positive constants $c_0$ and $\eta< \delta$ independent of $\alpha$ such that if the diameter of $\mathcal I$ is less than $c_0$, and $\partial_s \phi_\alpha$ is nonvanishing on $\mathcal I \times \mathcal I$, then
\[
|\partial_s \phi_\alpha(t,s)| \geq \eta \qquad \text{ for } t,s \in \text{supp } b.
\]
On the other hand if $\partial_s \phi_\alpha(t_0,s_0) = 0$ for some $s_0,t_0 \in \mathcal I$, then
\[
|\partial_s^2 \phi_\alpha(t,s)| \geq \sqrt\delta\varepsilon_0/2 \quad \text{ for } t,s \in \mathcal I.
\]
This result holds similarly for derivatives in $t$.
\end{lemma}
\begin{proof}
The curvature of any geodesic circle in $(\mathbb R^2,\tilde g)$ with radius at least $1$ is bounded uniformly by Lemma \ref{large radius} and the fact that $\mathbf k$ is bounded ($\mathbf k$ is continuous on $M$). Hence, we select a global constant $C$ so that
\[
\sup_{s \in \mathcal I} \kappa_{\tilde\gamma}{(s)} + \sup_{t,s \in \mathcal I} \kappa_{S(\tilde \gamma,r_\alpha)}(s) \leq C \qquad \text{ for all } \alpha \neq I
\]
where, as before, $\kappa_{S(\tilde \gamma,r_\alpha)}(s)$ is the curvature of the geodesic circle at $\alpha(\tilde \gamma(s))$, with center at $\tilde \gamma(t)$ and radius $r=\phi_\alpha(t,s)$.
Set
\[
\eta' = \min\left( \frac{1}{10}\delta, \frac{\sqrt{\delta}\,\varepsilon_0}{4C} \right).
\]
We claim that
\begin{equation}\label{claim}
|\partial_s^2 \phi_\alpha| \geq \sqrt\delta\varepsilon_0/2 \quad \text{ if } \quad |\partial_s \phi_\alpha| \leq \eta'.
\end{equation}
To prove this claim, first note that
\[
|-\epsilon+\sin(\theta)| = |\partial_s \phi_\alpha(t,s)|,
\]
where $\theta$ is as in Figure \ref{fig3},
then if $|\partial_s \phi_\alpha(t,s)| \leq \eta'$,
\[
|\cos (\theta) - \sqrt{1-\epsilon^2}| \leq \frac{|\sin^2 (\theta) - \epsilon^2|}{\cos (\theta) + \sqrt{1 - \epsilon^2}} \leq \frac{2|\sin (\theta) - \epsilon|}{\sqrt{\delta}} \leq \frac{\varepsilon_0}{2C}
\]
Since $\eta' \leq \delta/10$, we have that $\cos(\theta) \geq \sqrt{\delta}$ by default.
Hence,
\begin{align*}
|\partial_s^2 \phi_\alpha| &\geq \sqrt{\delta}\left| \pm \kappa_{\gamma} + \cos\theta \kappa \right|\\
&= \sqrt{\delta}\left| \pm \kappa_{\gamma} + \sqrt{1-\epsilon^2}\kappa - (\sqrt{1-\epsilon^2}- \cos\theta) \kappa \right|\\
&\geq \sqrt{\delta} | \pm\kappa_{\gamma} + \sqrt{1-\epsilon^2}\kappa | - \sqrt{\delta}|\sqrt{1-\epsilon^2} - \cos\theta| |\kappa|\\
&\geq \sqrt{\delta} \varepsilon_0 -\sqrt{\delta} \frac{C\varepsilon_0}{2C}\\
&= \frac{\sqrt{\delta}\varepsilon_0}{2},
\end{align*}
proving \eqref{claim}.
Now set
\[
c_0 = \frac{\eta'}{2\sqrt{2}(1+C^2)^{1/2}}.
\]
By \eqref{partial_s^2},
\[
|\partial_s^2 \phi_\alpha| \leq C.
\]
Moreover by Lemma \ref{off-diagonal lemma}, the fact that $\mathcal I$ has diameter at most $1$, and that the injectivity radius is at least $10$, we have
\[
|\partial_t \partial_s \phi_\alpha(t,s)| \leq 1.
\]
Hence for any $(t,s)$ and $(t_0,s_0)$ in $\mathcal I \times \mathcal I$,
\begin{align*}
|\partial_s \phi_\alpha(t,s) - \partial_s \phi_\alpha(t_0,s_0)| &\leq (1 + C^2)^{1/2}|(t,s) - (t_0,s_0)| \leq \frac{\eta'}{2}
\end{align*}
since the diameter of $\mathcal I \times \mathcal I$ is no greater than $\sqrt 2 c_0$.
In particular if $\partial_s \phi_\alpha(t_0,s_0) = 0$, then
\[
|\partial_s \phi_\alpha(t,s)| \leq \eta'/2 \qquad \text{ for all } t,s \in \mathcal I
\]
and so $|\partial_s^2 \phi_\alpha(t,s)| \geq \sqrt\delta\varepsilon_0/2$ by our claim \eqref{claim}.
Now suppose $|\partial_s \phi_\alpha(t,s)| > 0$ for all $t,s \in \mathcal I$. In the case that $|\partial_s \phi_\alpha(t_0,s_0)| \leq \eta'/2$ for some $t_0,s_0 \in \mathcal I$, $|\partial_s \phi_\alpha(t,s)| \leq \eta'$ for all $t,s \in \mathcal I$, and hence by our claim, $\partial_s \phi_\alpha(t,s)$ is monotonic in $s$, and so $\partial_s \phi_\alpha$ is smallest near the endpoints of $\mathcal I$. Since $\text{supp } b$ is closed and $\mathcal I$ open, the distance $d(\text{supp } b, \mathcal I^c)$ from $\text{supp } b$ to the complement of $\mathcal I$ is positive. Hence,
\[
|\partial_s \phi_\alpha(t,s)| \geq d(\text{supp } b, \mathcal I^c) \sqrt\delta\varepsilon_0/2 > 0.
\]
The proof is complete after setting
\[
\eta = \min(\eta'/2, d(\text{supp } b, \mathcal I^c) \sqrt\delta\varepsilon_0/2).
\]
\end{proof}
Now we are in a position to finish the proof of Proposition \ref{large time} and hence the proof of our main theorem.
\begin{proof}[Proof of Proposition \ref{large time}.]
By our hypothesis \eqref{curv hyp +} and \eqref{curv hyp -} on the curvature of $\gamma$, and since $\mathbf k$ is continuous, we restrict the support of $b$ and the interval $\mathcal I$ so that
\begin{align}\label{ca}
\inf_{t,s \in \mathcal I} \left| \left\langle \frac{D}{dt} \gamma', \gamma'^\perp \right\rangle(t) + \sqrt{1 - \epsilon^2} \mathbf k_\gamma(\gamma'^\perp)(s) \right| &\geq 2\varepsilon_0, \qquad \text{ and } \\\nonumber
\inf_{t,s \in \mathcal I} \left| \left\langle \frac{D}{dt} \gamma', \gamma'^\perp \right\rangle(t) - \sqrt{1 - \epsilon^2} \mathbf k_\gamma(-\gamma'^\perp)(s) \right| &\geq 2\varepsilon_0.
\end{align}
for some small $\varepsilon_0 > 0$. We first require $R$ in \eqref{A def} be at least as large as $16\varepsilon_0^{-1}\delta^{-1/2}$ so that, by Lemma \ref{off-diagonal lemma},
\begin{equation} \label{phase function mixed}
\sup_{t,s \in \mathcal I} |\partial_t \partial_s \phi_\alpha(t,s)| \leq \varepsilon_1/8, \qquad \text{ if } \alpha \in \Gamma \setminus A,\ \varepsilon_1=\varepsilon_0\sqrt\delta.
\end{equation}
By Lemma \ref{large radius} and our requirement that $R > 16 \varepsilon_0^{-1}\delta^{-1/2}$,
\[
\sqrt{1-\epsilon^2}|\kappa_{S(\tilde \gamma,r_\alpha)}(s) - \mathbf k(\sigma'(r_\alpha(t,s)))|< \varepsilon_0.
\]
Hence, if the $\pm$ sign matches that of $\langle \sigma'(r_\alpha(t,s)), \frac{D}{ds} \alpha(\tilde \gamma(s)) \rangle$, it follows from \eqref{ca} that
\[|\pm\kappa_{\tilde\gamma}(t) - \sqrt{1-\epsilon^2}\kappa_{S(\tilde \gamma,r_\alpha)}(s)|> \varepsilon_0, \qquad \text{ for } t,s \in \mathcal I.
\]
By proposition \ref{prop5.1} and Huygens' principle, the number of nonzero summand in \eqref{2.21} is $O(e^{CT})$, it suffices to show
\begin{equation}\label{final integral}
\left| \iint a_\pm(T,\lambda,\alpha;t,s) e^{\pm i \lambda \phi_\alpha(t,s)} \, dt \, ds \right| \leq C e^{CT} \lambda^{-1},
\end{equation}
for all $\alpha\in\Gamma\setminus A$ and $C$ is independent of $\alpha$.
By a partition of unity, we restrict the diameter of $\mathcal I$ to be less than the constant $c_0$ in Lemma \ref{diagonal lemma}. If $|\partial_s \phi_\alpha| > 0$ on $\mathcal I \times \mathcal I$, $|\partial_s \phi_\alpha| \geq \eta$ on $\text{supp } b \times \text{supp } b$ for some $\eta > 0$ independent of $\alpha$. Then we integrate by parts in $s$ to see that much better bounds are satisfied in this case. We obtain \eqref{final integral} similarly if $\partial_t \phi_\alpha$ does not vanish in $\mathcal I \times \mathcal I$.
What is left is the case that $\nabla \phi_\alpha$ vanishes at exactly one point $(t_0,s_0) \in \mathcal I \times \mathcal I$. By a translation, we assume without loss of generality that $(t_0,s_0) = (0,0)$. In this case,
\begin{equation} \label{diagonal bounds}
|\partial_s^2 \phi_\alpha| \geq \varepsilon_1/2 \quad \text{ and } \quad |\partial_t^2 \phi_\alpha| \geq \varepsilon_1/2
\end{equation}
on $\mathcal I \times \mathcal I$. Together with \eqref{phase function mixed}, it is easy to see that
\[
|\nabla^2\phi_\alpha(t,s) \xi| \geq \frac{c}{4} |\xi| \qquad \text{ for all } \xi \in \mathbb R^2.
\]
Hence by the mean value theorem,
\[
|\nabla \phi(t,s)| \geq \frac{c}{4}|(t,s)| \qquad \text{ for } (t,s) \in \mathcal I \times \mathcal I.
\]
\eqref{final integral} then follows by a standard stationary phase argument in both variables $s$ and $t$. Indeed, we gain a $\lambda^{-1}$ factor from the stationary phase, and only lose by a factor of $e^{CT}$ thanks to our bounds on the amplitude \eqref{amp bound}.
\end{proof}
\begin{proof}[Proof of Corollary \ref{corollary}.]
The collection of circles in $M$ with radii in $[r_1,r_2]$ with $0 < r_1 < r_2 < \infty$ have uniformly bounded derivatives by compactness. Let $\gamma'^\perp$ denote the unit vector normal to such a circle $\gamma$ pointing towards its center. Note
\[
\left| \left\langle \frac{D}{dt} \gamma',\gamma'^\perp \right\rangle + \sqrt{1 - \varepsilon^2} \mathbf k_\gamma(\gamma'^\perp) \right| > 0
\]
and by Lemma \ref{large radius},
\[
\left| \left\langle \frac{D}{dt} \gamma',\gamma'^\perp \right\rangle - \sqrt{1 - \varepsilon^2} \mathbf k_\gamma(-\gamma'^\perp) \right| > 0.
\]
Again by compactness, these quantities are uniformly bounded away from $0$, say by $\delta > 0$. Hence, $E_\gamma = (-1 + \delta, 1 - \delta)$. The corollary follows from Theorem \ref{main}.
\end{proof}
\begin{proof}[Proof of Corollary \ref{co}.]
Let $\gamma$ be such a curve. As a consequence of the principle of uniform boundedness, every weakly convergent sequence in a Hilbert space is bounded. It suffices to show that any sequence of eigenfunctions with bounded $L^2(\gamma)$ restriction norms converges to $0$ weakly in $L^2(\gamma)$
For the sake of simplicity, we may now assume that $|\gamma|=2\pi$, and the $L^2(\gamma)$ norm of $e_{\lambda_j}$ is bounded by 1. Let $g\in L^2(\gamma),$ it then suffices to show that given any $\varepsilon>0$, for large enough $j$, we have
\[\left|\int_\gamma e_{\lambda_j}(\gamma(s))\,g(s)\,ds\right|\le \varepsilon,\]
Now if we write $g$ in terms of its Fourier series,
\[g(s)=\sum_ka_ke^{iks}.\]
then $g$ being in $L^2(\gamma)$ implies that there exists a $N>0$ depending on $\varepsilon$, such that
\[\sum_{|k|>N}|a_k|^2\le\frac14\varepsilon^2.\]
If we denote
\[b_{j,k}=\left|\int_\gamma e_{\lambda_j}\,e^{iks}\,ds\right|,\]
then by Theorem \ref{main}, and the fact that $0$ is in the interior of $E_\gamma$,
\[b_{j,k}\le C(\log\lambda_j)^{-\frac12},\]
where $C$ will be an absolute constant provided that $|k|\le\delta\lambda_j,$ for some fixed $\delta>0$ such that $(-\delta,\delta)\subset E_\gamma$.
Now we can see that
\begin{align*}\left|\int_\gamma e_{\lambda_j}\,g\,ds\right|&\le\sum_{|k|\le N}|a_k||b_{k,j}|+\Big[\sum_{|k|>N}|a_k|^2\Big]^\frac12\|e_{\lambda_j}\|_{L^2(\gamma)}\\
&\le C{(\log\lambda_j)^{-\frac12}}\sum_{|k|\le N}|a_k|+\frac12\varepsilon\\&\le C(\log\lambda_j)^{-\frac12}N^\frac12[\sum_{|k|\le N}|a_k|^2]^\frac12+\frac12\varepsilon.\end{align*}
Notice that if we take $j$ large enough, such that $\log{\lambda_j}\ge[{4\varepsilon^{-2}NC^2\|g\|^2_{L^2(\gamma)}}]+1,$ the first term on the last line will be less than $\varepsilon/2$, then we have
\[\left|\int_\gamma e_{\lambda_j}\,g\,ds\right|\le \varepsilon.\]
This choice of $j$ can be justified, since $N/\lambda_j$ would be comparable to $\varepsilon^2\lambda_j^{-1}\log\lambda_j<\delta$ when $\lambda_j$ is large enough, which guarantees the uniformity of $C$.
\end{proof}
\section{Appendix}
We present here a proposition which allows us to obtain the bounds on the mixed derivatives \eqref{kernel3} from the corresponding bounds on the pure derivatives.
\begin{proposition} \label{mixed derivatives prop}
Let $M$ and $\tilde M$ be as above and let $f \in C^\infty(\tilde M, \tilde M)$ satisfy bounds
\begin{equation} \label{mixed derivatives prop 1}
|\Delta_x^\ell f( x, y)| \leq C_\ell e^{C_\ell d_{\tilde g}(x, y)} \qquad \text{ and } \qquad |\Delta_y^\ell f( x, y)| \leq C_\ell e^{C_\ell d_{\tilde g}( x, y)}
\end{equation}
for $\ell = 0,1,2,\ldots$, where $\Delta_x$ and $\Delta_y$ denote the Laplace-Beltrami operators on $\tilde M$ in the $x$ and $y$ variables, respectively. Then,
\[
|\Delta_x^\ell \Delta_y^m f( x, y)| \leq C_{\ell,m} e^{C_{\ell,m} d_{\tilde g}( x, y)} \qquad \text{ for } \ell,m = 0,1,2,\ldots
\]
where each of the constants $C_{\ell,m}$ depends only on $M$ and finitely many of the constants $C_\ell$.
\end{proposition}
\begin{proof}
Let $\beta \in C_0^\infty(\mathbb R,[0,1])$ be equal to $1$ near $0$ and be supported in the interval $(-\text{inj} M, \text{inj} M)$. Fix $x_0,y_0 \in \tilde M$ and set
\[
F(x,y) = \beta(d_{\tilde g}(x,x_0)) \beta(d_{\tilde g}(y,y_0)) f(x,y).
\]
The support of $\beta$ allows us to interpret $F$ has a function on $M \times M$. The distance function $d_{\tilde g}$ satisfies similar bounds as \eqref{mixed derivatives prop 1}, and hence
\begin{equation} \label{mixed bounds prop 3}
|\Delta_x^\ell F( x, y)| \leq C_\ell' e^{C_\ell' d_{\tilde g}(x_0, y_0)} \qquad \text{ and } \qquad |\Delta_y^\ell F( x, y)| \leq C_\ell' e^{C_\ell' d_{\tilde g}( x_0, y_0)}.
\end{equation}
for $\ell = 0,1,2,\ldots$ where the constants $C_\ell'$ depend only on $C_\ell$, $\beta$, and $M$. Moreover, it suffices to show
\[
\| \Delta_x^\ell \Delta_y^m F \|_{L^\infty(M \times M)} \leq C_{\ell,m}' e^{C_{\ell,m}' d_{\tilde g}(x_0,y_0)} \qquad \text{ for } \ell,m = 0,1,2,\ldots
\]
where $C_{\ell,m}'$ only depends only on $M$ and finitely many of the constants $C_\ell'$.
Note $\Delta_x + \Delta_y$ is the Laplace-Beltrami operator on the product manifold $M \times M$ endowed with the product metric. Moreover, $e_p(x)e_q(y)$ for $p,q = 0,1,2,\ldots$ form a Hilbert basis of eigenfunctions of $\Delta_x + \Delta_y$ with
\[
-(\Delta_x + \Delta_y)e_p(x) e_q(y) = (\lambda_p^2 + \lambda_q^2) e_p(x) e_q(y).
\]
Hence if we write
\[
\hat F(p,q) = \iint_{M\times M} F(x,y) \overline{e_p(x) e_q(y)} \, dx \, dy,
\]
we have by Sobolev embedding
\begin{align*}
&\|\Delta_x^\ell \Delta_y^m F \|_{L^\infty(M\times M)}^2\\
&\lesssim \| (1 - \Delta_x - \Delta_y)^3 \Delta_x^\ell \Delta_y^m F \|_{L^2(M\times M)}^2 \\
&= \sum_{p,q} (1 + \lambda_p^2 + \lambda_q^2)^{6} \lambda_p^{4\ell} \lambda_q^{4m} |\hat F(p,q)|^2 \\
&\leq C_{\ell,m} \sum_{p,q} (1 + \lambda_p^{4(3 + \ell + m + 1)} + \lambda_q^{4(3 + \ell + m + 1)}) |\hat F(\ell,m)|^2\\
&\leq C_{\ell,m} \left( \| F \|_{L^2(M\times M)}^2 + \| \Delta_x^{3+\ell+m+1} F \|_{L^2(M\times M)}^2 + \| \Delta_y^{3+\ell+m+1} F \|_{L^2(M\times M)}^2 \right).
\end{align*}
The desired bounds follow from H\"older's inequality and \eqref{mixed bounds prop 3}.
\end{proof}
|
1,108,101,566,749 | arxiv | \section*{Some examples}
We consider two gauge mediation models for a demonstration.
In the following, mass spectrums are calculated by ISAJET 7.72~ \cite{ISAJET} and
we use programs Herwig 6.5~\cite{HERWIG6510} and AcerDET-1.0~\cite{RichterWas:2002ch}
to simulate LHC signatures.
The first example is a strongly interacting gauge mediation (SIGM) model ~\cite{Hamaguchi:2008yu}, in which
the NLSP is a neutralino.
We take the same SIGM parameters as the example in Sec. 4 of Ref.~\cite{Hamaguchi:2008yu}.
The mass spectrum is shown in Fig.~\ref{fig:spectrumSIGM}.
The masses of the lightest neutralino and gravitino are 356 GeV and 10 eV, respectively.
\begin{figure}[t]
\input{spectrumSIGM}
\caption{Mass spectrum of SIGM}
\label{fig:spectrumSIGM}
\end{figure}
We take the events cuts as follows:
\begin{itemize}
\item $\ge 4$ jets with $p_{\rm T}>50$ GeV and $p_{\rm T,1,2}>100$ GeV.
\item $\ge 2$ photons with $p_{\rm T}>20$ GeV.
\item $M_{\rm eff}>500$ GeV, where
\begin{equation}
M_{\rm eff} = \sum_{\rm jets}^{4} p_{{\rm T}j} + p_{\rm T}^{\rm miss}.
\end{equation}
\item $p_{\rm T}^{\rm miss}>0.2 M_{\rm eff}$.
\end{itemize}
Under these cuts, we see that the standard-model backgrounds are almost negligible.
\begin{figure}[t]
\begin{tabular}{cc}
\begin{minipage}{0.55\hsize}
\begin{center}
\epsfig{file=SIGMpart.eps ,scale=.45,clip}
(a)
\end{center}
\end{minipage}
\begin{minipage}{0.55\hsize}
\begin{center}
\epsfig{file=SIGMdet.eps ,scale=.45,clip}
(b)
\end{center}
\end{minipage}
\end{tabular}
\caption{A distribution of $M_{\rm T2}$ for the SIGM example. (a): parton level signature.
(b): detector level signature.
}
\label{fig:SIGM}
\end{figure}
In Fig.~\ref{fig:SIGM}-(a), a parton level distribution of $M_{\rm T2}$ is shown for an integrated luminosity of 10 fb$^{-1}$.
Here, we take the sum of gravitino and neutrino transverse momenta
as the parton level missing $\Vec{p}_{\rm T}$.
As discussed in Ref.~\cite{Hamaguchi:2008yu}, very little number of leptons are produced in the SIGM.
Therefore, missing $\Vec{p}_{\rm T}$ is due to almost only gravitinos and the assumption that $\pT{\rm miss}
= \pT{\rm LSP1}+\pT{\rm LSP2}$ is satisfied.
There is a clear edge at $M_{\rm T2} \simeq m_{\tilde{\chi}^0_1}=356$ GeV.
In Fig.~\ref{fig:SIGM}-(b), we show a distribution of $M_{\rm T2}$ after taking account of detector effects.
In order to extract the point of the edge, we use a simple fitting function;
\begin{equation}
f(x) = (ax+b)\theta(-x+M) + (cx+d)\theta(x-M), \label{eq:fit}
\end{equation}
where $\theta(x)$ is the step function and $a, b, c, d$ and $M$ are fitting parameters.
We fit the data with $f(x)$ over $300 \le M_{\rm T2} \le 500$ GeV and find
\begin{equation}
m_{\tilde{\chi}^0_1} = 357 \pm 3~ {\rm GeV}.
\end{equation}
Here, the estimation of the error is done by `eye' because of lack of information
on the shape of the $M_{\rm T2}$ distribution.
The estimation that $m_{\tilde{\chi}^0_1} = 357 \pm 3$ GeV is very good agreement with the true value $m_{\tilde{\chi}^0_1} = 356$ GeV.
Next we show another example.
We study the Snowmass benchmark point SPS8~\cite{Allanach:2002nj},
which is a minimal gauge mediation model with a neutralino NLSP.
In Fig.~\ref{fig:spectrumSPS}, SPS8 mass spectrum is shown.
The masses of the lightest neutralino and gravitino are 139 GeV and 4.8 eV, respectively.
\begin{figure}[t]
\input{spectrumSPS}
\caption{Mass spectrum of SPS8}
\label{fig:spectrumSPS}
\end{figure}
\begin{figure}[thbp]
\begin{tabular}{cc}
\begin{minipage}{0.55\hsize}
\begin{center}
\epsfig{file=SPSpart.eps ,scale=.45,clip}
(a)
\end{center}
\end{minipage}
\begin{minipage}{0.55\hsize}
\begin{center}
\epsfig{file=SPSdet.eps ,scale=.45,clip}
(b)
\end{center}
\end{minipage}
\end{tabular}
\caption{A distribution of $M_{\rm T2}$ for the SPS8. (a): parton level signature.
The blue and dashed line represents the case that $\pT{\rm miss} = \sum_{\rm gravitino}\Vec{p}_{\rm T}$ and
the red and solid line $\pT{\rm miss} = \sum_{\rm gravitino}\Vec{p}_{\rm T}+\sum_{\rm neutrino}\Vec{p}_{\rm T}$.
(b): detector level signature.
}
\label{fig:SPS}
\end{figure}
In Fig.~\ref{fig:SPS}-(a), a parton level distribution of $M_{\rm T2}$ is shown for an integrated luminosity of 10 fb$^{-1}$.
The event cuts are the same as in the previous SIGM case.
The blue and dashed line represents the case that $\pT{\rm miss} = \sum_{\rm gravitino}\Vec{p}_{\rm T}$ and
the red and solid line $\pT{\rm miss} = \sum_{\rm gravitino}\Vec{p}_{\rm T}+\sum_{\rm neutrino}\Vec{p}_{\rm T}$.
In SPS8, there are many neutrino production sources.
Hence, we cannot see a clear edge as in the SIGM case.
However, there is a cliff at $M_{\rm T2} \simeq m_{\tilde{\chi}^0_1} = 139$ GeV.
In Fig.~\ref{fig:SPS}-(b), detector level distribution of $M_{\rm T2}$ is shown.
To get the value of $m_{\tilde{\chi}^0_1}$, we fit the data with $f(x)$ in Eq.~(\ref{eq:fit}) over
$110\le M_{\rm T2}\le 180$ GeV. Then we get
\begin{equation}
m_{\tilde{\chi}^0_1} = 139 \pm 3~{\rm GeV}.
\end{equation}
The error estimation is done by `eye'.
This value
agrees with the true value ($m_{\tilde{\chi}^0_1} = 139 ~{\rm GeV}$).
In summary, we have presented a determination of the neutralino mass
for the SUSY models with an ultralight gravitino LSP and a neutralino NLSP,
which may work in the early stage of the LHC.
Though we have considered GMSB models with a neutralino NLSP, our method is applicable to any model
in which the signal events will lead to a pair of cascade decays that result in
\begin{equation}
\cdots \to {\rm any~cascade~decay}\to A \to B + X, \label{eq:decay}
\end{equation}
where $B$ is a visible (standard-model) particle and $X$ is a missing particle that is almost massless.
The mass of $A$ is then determined by the two $B$s' momenta and the missing
transverse momentum.
For example, let us consider GMSB models with a slepton NLSP.
In this case, the slepton, lepton and gravitino correspond to $A$, $B$ and $X$ in Eq.~(\ref{eq:decay}), respectively.
In addition to leptons from the sleptons' decays, many other leptons are produced in this scenario.
However, we may see which of observed leptons is produced through the slepton decay by measuring lepton's momentum,
or by detecting a kink of its track for a long-lived slepton.
In such a case, we can measure the slepton mass with the $M_{\rm T2}$ method as discussed above.
Furthermore, the present method may work in an axino LSP scenario.
\section*{Acknowledgements}
We thank Tsutomu Yanagida for useful discussion.
This work was supported by World Premier International Center Initiative
(WPI Program), MEXT, Japan.
The work by KH is supported by JSPS (18840012).
The work of SS is supported in part by JSPS
Research Fellowships for Young Scientists.
\section*{Appendix}
In this appendix we derive Eq.~(\ref{eq:mt2nlsp}).
We start from Eqs.~(\ref{eq:mt2}) and (\ref{eq:mt}).
We assume that $B$ is massless.
\paragraph{(i) $m_X=0$ case:}
First, we consider the case that $X$ is a massless particle.
The $M_{\rm T2}$ variable is defined by Eqs.~(\ref{eq:mt2}) and (\ref{eq:mt}) with $m_{B}=m_{X}=0$.
If a momentum splitting is the correct one, i.e.,
$\pT{{\rm miss},1}=\pT{X,1}$ and $\pT{{\rm miss},2}=\pT{X,2}$,
then each transverse mass is smaller than the mass of $A$, $m_A$:
\begin{eqnarray}
m_A^2=(p^{B,i}+p^{X,i})^2=2\big(|\pT{X,i}||\pT{B,i}|\cosh(\Delta y^i)-\pT{X,i}\cdot
\pT{B,i}\big)\geq\big(M_{\rm T}^{(i)}\big)^2
\end{eqnarray}
for $i=1,2$,
where $\Delta y^i$ is the rapidity difference of $B$ and $X$ in each decay chain.
From this, it is clear that
\begin{eqnarray}\label{eq:mT2_bound}
M_{\rm T2}\leq m_A.
\end{eqnarray}
We do not assume the relation
\begin{eqnarray}\label{eq:BtoB}
\pT{B,1}+\pT{B,2}=-\pT{\rm miss},
\end{eqnarray}
which holds in the case of a ``back-to-back'' pair production of $A\,$s.
We may assume that $\pT{B,1}$ and $\pT{B,2}$ are linearly independent and
$\pT{\rm miss}$ can be expressed as
\begin{eqnarray}
\pT{\rm miss}=c_1\pT{B,1}+c_2\pT{B,2}.
\end{eqnarray}
Here, $c_1$ and $c_2$ are real coefficients and they are given by
\begin{eqnarray}
c_1=\frac{1}{\sin^2\theta}\bigg[\frac{\pT{\rm miss}\cdot\pT{B,1}}{(p_1)^2}
-\frac{\pT{\rm miss}\cdot\pT{B,2}}{p_1p_2}\cos\theta\bigg],
\end{eqnarray}
\begin{eqnarray}
c_2=\frac{1}{\sin^2\theta}\bigg[\frac{\pT{\rm miss}\cdot\pT{B,2}}{(p_2)^2}-
\frac{\pT{\rm miss}\cdot\pT{B,1}}{p_1p_2}\cos\theta\bigg],
\end{eqnarray}
where
\begin{eqnarray}
p_1\equiv|\pT{B,1}|,\quad p_2\equiv|\pT{B,2}|,\quad
\cos\theta\equiv\frac{\pT{B,1}\cdot\pT{B,2}}{p_1p_2}.
\end{eqnarray}
The momentum splitting $\pT{{\rm miss},1}$ and $\pT{{\rm miss},2}$
can also be expressed as
\begin{eqnarray}
\pT{{\rm miss},1}=(c_1-x)\pT{B,1}+y\pT{B,2},
\end{eqnarray}
\begin{eqnarray}
\pT{{\rm miss},2}=x\pT{B,1}+(c_2-y)\pT{B,2},
\end{eqnarray}
where $x$ and $y$ are real variables. We rewrite Eq.~(\ref{eq:mt2}) as
\begin{eqnarray}
(M_{\rm T2})^2=2p_1p_2\mathop{\rm min}_{x,y\in{\bf R}}
\Big[{\rm max}\Big\{z_1(x,y),z_2(x,y)\Big\}\Big],
\end{eqnarray}
where
\begin{eqnarray}\label{eq:z1}
z_1(x,y)\equiv\frac{\big(M_{\rm T}^{(1)}(x,y)\big)^2}{2p_1p_2}=
\sqrt{\Big[\frac{c_1-x}{r}+y\cos\theta\Big]^2+y^2\sin^2\theta}-
\Big[\frac{c_1-x}{r}+y\cos\theta\Big],
\end{eqnarray}
\begin{eqnarray}\label{eq:z2}
z_2(x,y)\equiv\frac{\big(M_{\rm T}^{(2)}(x,y)\big)^2}{2p_1p_2}=\sqrt{\big[x\cos\theta+(c_2-y)r\big]^2+x^2\sin^2\theta}
-\big[x\cos\theta+(c_2-y)r\big],
\end{eqnarray}
and $r\equiv p_2/p_1$. It is clear that
\begin{eqnarray}
z_1(x,y)\geq0,\quad{\rm and}\quad z_1(x,y)=0\ \Leftrightarrow\ y=0\ \&\ x\leq c_1,
\end{eqnarray}
\begin{eqnarray}
z_2(x,y)\geq0,\quad{\rm and}\quad z_2(x,y)=0\ \Leftrightarrow\ x=0\ \&\ y\leq c_2.
\end{eqnarray}
From this, we can infer that
\begin{eqnarray}
\big(M_{\rm T2}\big)^2=0\quad{\rm if}\quad c_1\geq0\ \&\ c_2\geq0,
\end{eqnarray}
and for other values of $c_1$ and $c_2$, $\big(M_{\rm T2}\big)^2$ is given by
$\big(M_{\rm T}^{(1)}(x,y)\big)^2=\big(M_{\rm T}^{(2)}(x,y)\big)^2$ at the point $(x,y)=(x_0,y_0)$
where the contours of $z_1(x,y)$ and $z_2(x,y)$ in the $x$--$y$ plane become tangent to each other.
We denote the corresponding value $z\equiv z_1(x_0,y_0)=z_2(x_0,y_0)$ in the following.
Eqs. (\ref{eq:z1}) and (\ref{eq:z2}) yield
\begin{eqnarray}
x_0=-\frac{r\sin^2\theta}{2z}\bigg(y_0-\frac{z\cos\theta}{\sin^2\theta}\bigg)^2+\frac{rz}{2\sin^2\theta}+c_1,
\end{eqnarray}
\begin{eqnarray}
y_0=-\frac{\sin^2\theta}{2rz}\bigg(x_0-\frac{z\cos\theta}{\sin^2\theta}\bigg)^2+\frac{z}{2r\sin^2\theta}+c_2
\end{eqnarray}
with the tangential condition
\begin{eqnarray}
\frac{\sin^4\theta}{z^2}\bigg(x_0-\frac{z\cos\theta}{\sin^2\theta}\bigg)
\bigg(y_0-\frac{z\cos\theta}{\sin^2\theta}\bigg)=1.
\end{eqnarray}
We can obtain $z$ by solving these three equations.
A straightforward calculation yields that these equations reduce to
\begin{eqnarray}\label{eq:ab}
4(a-b)^2-\big(\sqrt{2(a+b)+3}-1\big)\big(\sqrt{2(a+b)+3}+3\big)^3=0,
\end{eqnarray}
where
\begin{eqnarray}
a=\frac{1}{r^{\frac{1}{3}}}\bigg(\frac{r}{2}-\cos\theta+c_1\sin^2\theta\frac{1}{z}\bigg),\label{eq:app_a}
\end{eqnarray}
\begin{eqnarray}
b=r^{\frac{1}{3}}\bigg(\frac{1}{2r}-\cos\theta+c_2\sin^2\theta\frac{1}{z}\bigg).\label{eq:app_b}
\end{eqnarray}
It can be checked that the above equations have a unique real positive solution of $z$.
Eqs.~(\ref{eq:ab})-(\ref{eq:app_b}) have been used for the analysis in this work.
In the special case of a ``back-to-back'' pair production, in which Eq. (\ref{eq:BtoB}) holds,
we recover the result obtained by taking the massless limit of the formula in Ref.~\cite{MT2_back2back},
\begin{eqnarray}
\big(M_{\rm T2}\big)^2\Big|_{\rm back-to-back}
=2\big(|\pT{B,1}||\pT{B,2}|+\pT{B,1}\cdot \pT{B,2}\big)
=2p_1p_2(1+\cos\theta).
\end{eqnarray}
\paragraph{(ii) $m_X\ne 0$ case:}
Generalization of the above result for the case with massive $X$, i.e., $m_X\ne0$, is straightforward.
In this case, the $M_{\rm T2}$ variable is defined by Eq.~(\ref{eq:mt2}) with $m_{B}=0$.
The same argument as above shows that Eq.~(\ref{eq:mT2_bound}) holds also in this case.
Calculating in the same way as above, it can be shown that
\begin{eqnarray}
\big(M_{\rm T2}\big)^2=m_X^2+2p_1p_2z
\end{eqnarray}
with $z$ being the solution of Eq.~(\ref{eq:ab}) with
\begin{eqnarray}
a=\frac{1}{r^{\frac{1}{3}}}\bigg(\frac{r}{2}-\cos\theta+c_1\sin^2\theta\frac{1}{z}-\frac{r\sin^2\theta}{2}
\frac{m_X^2}{p_2^2}\frac{1}{z^2}\bigg),
\end{eqnarray}
\begin{eqnarray}
b=r^{\frac{1}{3}}\bigg(\frac{1}{2r}-\cos\theta+c_2\sin^2\theta\frac{1}{z}
-\frac{\sin^2\theta}{2r}\frac{m_X^2}{p_1^2}\frac{1}{z^2}\bigg).
\end{eqnarray}
For the case with massive $X$, this expression for $M_{\rm T2}$ is valid for
any values of $c_1$ and $c_2$. The existence of a unique positive real solution of $z$ can also be checked.
|
1,108,101,566,750 | arxiv | \section{Introduction}
{It is important to check whether the covariates of interest contribute to the response, given the other covariates. In linear regression models, this is formulated as testing whether the parametric vector of interest is equal to zero.
This paper studies inference of ultrahigh-dimensional parameter vector of interest with ultrahigh-dimensional nuisance parameter vector. This problem is of great importance in practice. For instance, researchers may aim to test whether a gene pathway, consisting of ultrahigh-dimensional genes for the same biological functions, is important for certain clinical outcome, given the other ultrahigh-dimensional genes.
For this challenging problem, there are several proposals available in the literature. The coordinate-based maximum tests have been proposed recently. See for instance \cite{ning2017general}, \cite{zhang2017simultaneous}, \cite{dezeure2017high}, \cite{ma2021global} and \cite{wu2021model}. These methods are computationally expensive because many penalized optimization implementations with ultrahigh-dimensional parameter vector are involved. Further, these tests require the sparsity assumption on both nuisance parameter vector and parameter vector of interest, otherwise, the max-type statistics would have relatively low power. A Wald-type test was suggested by \cite{guo2021group}, which is computationally low costed. They imposed the boundedness of eigenvalues of covariance matrix and sparse structure on the whole parameter vector. To tackle the problem in the study of the asymptotic properties, brought by too small variance (\cite{guo2021group} pointed out), a positive tuning parameter over the sample size is added to the estimated variance. As the limiting null distribution remains untractable, the critical values determined by their method cause the test conservative in theory (see the discussion on page 11 of \cite{guo2021group}).
But in practice, the test could be either very liberal or very conservative in different models as numerical studies in Section 6 of our paper indicate.
Score function-based testing procedures are also popular. As this method can easily exploit the information contained in the null hypothesis, mild conditions on the whole model are in need. This is particularly useful in ultrahigh dimensional paradigms. When the dimension of the nuisance parameter vector is low or diverging at relatively slow rate, and the parameter vector of interest is high-dimensional, the references include \cite{goeman2006testing}, \cite{zhong2011tests}, \cite{guojrssb2016} (for generalized linear models), \cite{cui2018test}, and \cite{guo2022conditional}. The recent development of score function-based testing procedure is made by \cite{chen2022testing} who extended the score function-based test of \cite{guojrssb2016} to handle ultra-high dimensional nuisance parameter vector. Due to the adoption of score function, only the nuisance parameter vector requires the sparsity assumption, and the dimension of the parameter vector of interest can grow polynomially with the sample size to guarantee nontrivial power. As a result, this test is suitable under dense alternative hypotheses. \cite{chen2022testing} showed some merits of their test in numerical studies. In theory, the eigenvalue boundedness assumption on covariance matrix is still imposed to control the correlations between covariates and the asymptotic limiting distributions under the local alternatives are not established. Although the limiting null distribution was established (Theorem~1, \cite{chen2022testing}), some technical details need further careful checks.
The above observations motivate us to further study the score function-based test for linear models, extend the results in the literature and propose new test to handle high correlation between nuisance and testing covariates. To be specific, we will do the following. First, we need to reanalyze, under weaker conditions, the properties of the test statistic and extend the results to the case where the testing parameter and nuisance parameter vectors are ultrahigh dimensional simultaneously at the rates up to the exponential of the sample size. To this end, we derive the limiting distributions under the null and local alternative hypotheses. Second, when the correlation between the covariates of interest and the nuisance covariates is strong, a non-negligible bias causes that the tests in \cite{chen2022testing} fail to work. Therefore, we propose an orthogonalization procedure to reduce the possible bias. Although this technique has been adopted in the recent high-dimensional inference literature \citep{zhang2014confidence,van2014asymptotically,javanmard2014confidence,belloni2015uniform,chernozhukov2018double}, to the best of our knowledge, it has not been applied to constructing test statistics based on the quadratic norm of the score function for ultra-high dimensional testing parameter vector. Two merits shown in our investigation are as follows. The orthogonalization can debiase the error terms and convert the non-degenerate error terms to degenerate, thus relaxing the correlation assumption between the covariates of interest and nuisance covariates; it can also reduce the variance {of the test statistic} and thus enhance the power performance, which was not observed in the literature.
Technically, we establish the asymptotic normality of the two proposed test statistics in a different way from those used by \cite{zhong2011tests, guojrssb2016, cui2018test} for the quadratic norm-based test statistics. Instead of calculating the relatively complex {spectral} norm of the high dimensional sample matrix, we derive the order of element-max norm of the high-dimensional $U$-statistics with the help of maximal inequalities established in \cite{chernozhukov2015comparison} and \cite{chen2018gaussian}. The technique developed in this paper can be useful for other high-dimensional inference problems.
The rest of the paper is organized as follows. Section \ref{sec2} re-analyzes the test statistic in \cite{chen2022testing} to handle the case with higher dimensional parameter vector of interest and presents the limiting distributions under both null and local alternative hypotheses. The failure of this test statistic in the high correlation case is also discussed. Section \ref{sec3} introduces the orthogonalization procedure. Section \ref{sec4} contains an oracle inference procedure to illustrate the merits of the orthogonalization approach. Further, Section \ref{sec5} develops an orthogonalization-based test in the general case and derives the relevant asymptotic analysis. Section \ref{sec6} presents simulation studies and a real data analysis. Section \ref{sec7} offers some conclusions. The detailed proofs of the theoretical properties are in the Appendix. The Supplementary Materials include some proofs of the theoretical properties, technical lemmas, and additional simulation results.}
Before closing this section, we introduce some necessary notations. For a $d$-dimension vector $\mathbf{U}$, write $\lVert \mathbf{U}\rVert_r = (\sum_{k=1}^{d}|U_{k}|^{r})^{1/r}$ and $\lVert \mathbf{U} \rVert_{\infty} = \max_{1\leq k\leq d}\lvert U_{k}\rvert$ to denote $L_r$ and $L_{\infty}$ norms of $\mathbf{U}$, where $U_{k}$ is the $k$-th element of $\mathbf{U}$. Further define $\|\mathbf{U}\|_{0}=\#\left\{k: U_{k} \neq 0\right\}$. A random variable $X$ is $\operatorname{\mathit{sub-Gaussian}}$ if the moment generating function (MGF) of $X^2$ is bounded at some point, namely $\mathbb{E}\exp(X^2/K^2)\leq2$, where $K$ is a positive constant. A random vector ${\bf{X}}$ in $\mathbb{R}^p$ is called $\operatorname{\mathit{sub-Gaussian}}$ if $x^\top{\bf{X}}$ are sub-gaussian random variables for all $x\in\mathbb{R}^p$. For $a,b\in\mathbb{R}$, write $a\vee b=\max\{a,b\}$. {For $p_1\times p_2$ dimensional matrix $\mathbf{A}$, write $\lambda_{\rm{max}}(\mathbf{A})$ to denote the spectral norm of $\mathbf{A}$.} Further define $\lVert\mathbf{A}\rVert_{0} = \#\{(i,j): A_{ij} \neq 0 \}$ and $\lVert\mathbf{A}\rVert_{F} = \{\mathrm{tr}(\mathbf{A}\bA^\top)\}^{1/2}$, where $A_{ij}$ is the $(i,j)$-th element of $\mathbf{A}$.
\section{The score function-based test and new results}
\label{sec2}
Let $Y\in\mathbb{R}$ be the response variable along with the covariates ${\bf{X}} = (X_{1},\ldots,X_{p_{\bm{\beta}}})^\top\in\mathbb{R}^{p_{\bm{\beta}}}$ and $\mathbf{Z} = (Z_{1},\ldots,Z_{p_{\bm{\gamma}}})^\top\in\mathbb{R}^{p_{\bm{\gamma}}}$. Consider the following linear model:
\begin{align}
\label{linearmodel1}
Y = \bm{\beta}^\top{\bf{X}} + \bm{\gamma}^\top\mathbf{Z} + \epsilon,
\end{align}
where $\epsilon$ is the random error satisfying $\mathbb{E}(\epsilon) = 0$ and $\mathbb{E}(\epsilon^{2}) = \sigma^{2}$. Let $\bm{V} = ({\bf{X}}^\top,\mathbf{Z}^\top)^\top$ and $\bm{\Sigma}$ be the covariance matrix of $\bm{V}$. Without loss of generality, assume that $\mathbb{E}(\bm{V}) = \bm{0}$, $\bm{\Sigma}$ is positive definite and $\epsilon$ is independent of $\bm{V}$. Our primary interest is to detect whether ${\bf{X}}$ contributes to the response $Y$ or not given the other covariates, which is testing the following inference problem:
\begin{align}
\label{question1}
\bm{H}_{0}:\bm{\beta} = \bm{0},\qquad \mathrm{versus}\qquad \bm{H}_{1}:\bm{\beta} \neq \bm{0}.
\end{align}
To test the above hypothesis, we can construct test statistics based on score functions. An advantage of score function-based tests is that we do not need to estimate the parametric vector of interest and thus no sparsity assumption on it is required. To be precise, consider the following $L_2$ loss function:
\begin{align*}
\mathcal{L}(\bm{\beta},\bm{\gamma}) = \mathbb{E} (Y-\bm{\beta}^\top{\bf{X}} - \bm{\gamma}^\top\mathbf{Z})^2,
\end{align*}
and the corresponding score function of $\bm{\beta}$:
\begin{align*}
\partial\mathcal{L}(\bm{\beta},\bm{\gamma})/\partial\bm{\beta}=\nabla_{\bm{\beta}}\mathcal{L}(\bm{\beta},\bm{\gamma}) = \mathbb{E}\{(Y-\bm{\beta}^\top{\bf{X}} - \bm{\gamma}^\top\mathbf{Z}){\bf{X}}\}.
\end{align*}
Then $\nabla_{\bm{\beta}}\mathcal{L}(\bm{0},\bm{\gamma}) = \bm{0}$ corresponds to $\bm{H}_0$; otherwise, to $\bm{H}_{1}$.
A test statistic can be based on the quadratic norm $\nabla_{\bm{\beta}}\mathcal{L}(\bm{0},\bm{\gamma})^\top\nabla_{\bm{\beta}}\mathcal{L}(\bm{0},\bm{\gamma})$. As the nuisance parameter $\bm{\gamma}$ is unknown, we can replace $\bm{\gamma}$ with {an estimator} $\hat{\bm{\gamma}}$.
Now suppose $\{{\bf{X}}_i,\mathbf{Z}_i,Y_i\}_{i=1}^{n}$ is a random sample from the population $({\bf{X}},\mathbf{Z},Y)$. \cite{guojrssb2016} proposed the following test statistic based on the quadratic norm of the score function:
\begin{align}
\label{teststatisticguo}
T_{n} = \frac{1}{n}\sum_{i\neq j}(Y_i - \hat\bm{\gamma}^\top\mathbf{Z}_i)(Y_j - \hat\bm{\gamma}^\top\mathbf{Z}_j){\bf{X}}_i^\top{\bf{X}}_j,
\end{align}
where $\hat\bm{\gamma}$ is the least squares estimator. Clearly $T_n$ can be seen as a $U$-statistic type estimator of $(n-1)\nabla_{\bm{\beta}}\mathcal{L}(\bm{0},\hat\bm{\gamma})^\top\nabla_{\bm{\beta}}\mathcal{L}(\bm{0},\hat\bm{\gamma})$. In the asymptotic analysis, the growth rate of the dimension of $\bm{\gamma}$ is required to be slower than $n^{1/4}$. Recently, \cite{chen2022testing} extended it to handle the ultrahigh dimensional nuisance parameter situation. They obtained the estimator $\hat\bm{\gamma}$ by solving the following penalized problem,
\begin{align}
\label{bgammaestimator}
\hat\bm{\gamma} = \mathop{\mbox{argmin}}\limits_{\bm{\gamma}\in\mathbb{R}^{p_{\bm{\gamma}}}}\frac{1}{2n}\sum_{i=1}^{n}(Y_i - \bm{\gamma}^\top\mathbf{Z}_i)^2 + \lambda_{Y}\lVert\bm{\gamma}\rVert_{1},
\end{align}
where $\lambda_Y$ is the tuning parameter and $\bm{\gamma}$ has a sparse structure. Other penalties such as SCAD and MCP, are also applicable. To deal with the case with ultrahigh dimensional parameter vector of interest, relax some unrealistic conditions, and fill up some leaks in their technical proofs,
we conduct a further investigation for their test.
\subsection{Limiting null distribution}
\label{limitnulldisofsec2}
Let $\bm{\Sigma}_{{\bf{X}}}$ and $\bm{\Sigma}_{\mathbf{Z}}$ be the covariance matrices of the covariates ${\bf{X}}$ and $\mathbf{Z}$ respectively.
{Denote $p_{\bm{\beta}}$, $p_{\bm{\gamma}}$ as the dimension of $\bm{\beta}$ and $\bm{\gamma}$, and let $p = p_{\bm{\beta}} + p_{\bm{\gamma}}$.
Denote $\bm{\gamma}_{\phi} = \bm{\Sigma}_{\mathbf{Z}}^{-1}\mathbb{E}(\mathbf{Z} Y)$, and $\bm{\gamma}_{\phi} = \bm{\gamma}$ under $\bm{H}_{0}$.
Let $s$ be a positive integer and represents the sparsity level of $\bm{\gamma}_{\phi}$. Let $\varrho^2 = \max_{1\leq k\leq p_{\bm{\gamma}}}\lVert\mathbb{E}(Z_{k}{\bf{X}})\rVert_{2}^2$ and $\varpi^2 = s^2\log p_{\bm{\gamma}}\varrho^2$. Here $\varrho^2$ describes the dependence of covariates of interest and nuisance covariates. Next, under some technical assumptions, we study the asymptotic null distribution of the test statistic $T_n$ with $\hat\bm{\gamma}$ in \eqref{bgammaestimator}.
\begin{assumption}
\label{assumptionbmatrix2}
$\mathrm{tr}(\bm{\Sigma}^4_{{\bf{X}}})=o(\mathrm{tr}^2(\bm{\Sigma}_{{\bf{X}}}^2))$ and $\mathrm{tr}(\bm{\Sigma}_{{\bf{X}}}^2)\rightarrow\infty$ as $(n,p_{\bm{\beta}})\rightarrow\infty$.
\end{assumption}
\begin{assumption}
\label{assumptionb5}
$\bm{V}$ can be expressed as
\begin{align*}
\bm{V} = \bm{\Gamma} \bm{\nu},
\end{align*}
where $\bm{\Gamma}$ is a $p \times m$ dimensional matrix with $p\leq m$.
The $L_2$ norms of row vectors in $\bm{\Gamma}$ are uniformly bounded.
$\bm{\nu}$ is an $m$-dimensional sub-Gaussian random vector with mean zero and identity covariance matrix.
\end{assumption}
\begin{assumption}
\label{assumptionb6}
$\lVert \hat{\bm{\gamma}} - \bm{\gamma}_{\phi} \rVert_{1} = O_p(s\sqrt{\log p_{\bm{\gamma}}/n})$.
\end{assumption}
\begin{assumption}
\label{assumptionb8}
$\log p_{\bm{\gamma}} = O(n^b)$ for some constant $0<b<1/3$.
\end{assumption}
\begin{assumption}
\label{assumptionb7}
$\epsilon$ is $\operatorname{\mathit{sub-Gaussian}}$ with bounded $\operatorname{\mathit{sub-Gaussian}}$ norm.
\end{assumption}
Assumption \ref{assumptionbmatrix2} frequently appeared in the literature \citep{chen2010two, guojrssb2016, cui2018test} and is required in applying the martingale central limit theorem. If all the eigenvalues of $\bm{\Sigma}_{{\bf{X}}}$ are bounded, then Assumption \ref{assumptionbmatrix2} holds. Assumption \ref{assumptionb5} says that $\bm{V}$ can be expressed as a linear transformation of an $m$-dimensional sub-Gaussian vector $\bm{\nu}$ with zero mean and unit variance. \cite{tony2020semisupervised} imposed a similar assumption.
This assumption is similar to the pseudo-independence assumption, which is widely used in the literature such as \cite{bai1996effect, chen2009effects, chen2010two, zhong2011tests, cui2018test}. The boundedness assumption of $L_2$ norms of row vectors in $\bm{\Gamma}$ is imposed to ensure that sub-Gaussian norms of the components of $\bm{V}$ are uniformly bounded. Assumption \ref{assumptionb6} requires the $L_1$ error bound of $\hat\bm{\gamma}$ at the order of $s(\log p_{\bm{\gamma}}/n)^{1/2}$. Many estimators, such as Lasso, SCAD, and MCP, can achieve such a rate of convergence.
{See for instance \cite{loh2015regularized}.} Assumption \ref{assumptionb8} allows the dimension of the nuisance parameter in an exponential order of the sample size. Assumption \ref{assumptionb7} is standard in the analysis for high dimensional linear models.
\begin{theorem}
\label{limitnulldisofguosum}
Under $\bm{H}_{0}$ in \eqref{question1} and Assumptions~\ref{assumptionbmatrix2} -- \ref{assumptionb7}, and the following two conditions:
\begin{align}
\label{condition1inguosum}
\varpi^2\vee (\log p_{\bm{\gamma}})^{1/2}\varpi\lambda_{\rm{max}}^{1/2}(\bm{\Sigma}_{{\bf{X}}}) = o(\sqrt{\bm{\Lambda}_{{\bf{X}}}}),
\end{align}
and
\begin{align}
\label{condition2inguosum}
s(\log p_{\bm{\gamma}})^{3/2}/\sqrt{n} = o(1),
\end{align}
we have
\[
\frac{T_n}{\sqrt{2\bm{\Lambda}_{{\bf{X}}}}}\rightarrow N(0,1)\quad \text{in distribution}
\]
as $(n,p_{\bm{\beta}},p_{\bm{\gamma}})\rightarrow\infty$, where $\bm{\Lambda}_{{\bf{X}}} = \sigma^4\mathrm{tr}(\bm{\Sigma}^2_{{\bf{X}}})$.
\end{theorem}
Note that \cite{zhang2017simultaneous} assumed $s^*(\log p)^{3/2}/\sqrt{n} = o(1)$. Here $s^*$ is the sparsity level of the whole parameter vector $(\bm{\beta}^\top, \bm{\gamma}^\top)^\top$. Thus, condition \eqref{condition2inguosum} is weaker when the sparse structure exists.
Condition \eqref{condition1inguosum} restricts the sparsity, the dimensions of the nuisance and testing parameter, and the correlations among the covariates. Generally speaking, if the correlations are weak and the dimension of the testing parameter is high, the sparsity level and the dimension of the nuisance parameter can be very high.
Therefore, when the correlation between ${\bf{X}}$ and $\mathbf{Z}$ is weak,
{$T_{n}$} still has a tractable limiting null distribution.
Particularly, when $\bm{\Sigma}$ has bounded eigenvalues, $\varrho^2$ and $\lambda_{\rm{max}}(\bm{\Sigma}_{{\bf{X}}})$ can be bounded by a constant, condition \eqref{condition1inguosum} can be simplified as
{$s^2\log p_{\bm{\gamma}} = o(\sqrt{p_{\bm{\beta}}}).$ Furthermore, if $p_{\bm{\beta}}\asymp n^2$, condition \eqref{condition1inguosum} can be further simplified as $s^2\log p_{\bm{\gamma}} = o(n)$ which is weaker than condition \eqref{condition2inguosum}. This implies that $T_n$ has tractable null distribution even when both $p_{\bm{\beta}}$ and $p_{\bm{\gamma}}$ are of exponential order of $n$.}
To formulate the testing procedure based on Theorem \ref{limitnulldisofguosum}, we use
\[R_{1n}= \hat{\sigma}^{4}\frac{1}{2\tbinom{n}{4}}\sum_{i< j<k<l}^{n}({\bf{X}}_{i} - {\bf{X}}_{j})^\top({\bf{X}}_{k} - {\bf{X}}_{l})({\bf{X}}_{j} - {\bf{X}}_{k})^\top({\bf{X}}_{l} - {\bf{X}}_{i})\]
to estimate $\bm{\Lambda}_{{\bf{X}}}$, where $\hat{\sigma}^{2}$
is a consistent estimator of the error variance $\sigma^2$, such as the one in \cite{sun2012scaled}.
Under the null hypothesis, $R_{1n}$ is a ratio consistent estimator of $\bm{\Lambda}_{{\bf{X}}}$. The consistency of $R_{1n}$ has been discussed by many authors such as \cite{zhong2011tests,cui2018test,guo2022conditional}.
Combining Theorem \ref{limitnulldisofguosum} and Slutsky Theorem, we reject $\bm{H}_{0}$ at a significance level $\alpha$ if
\[
T_n\geq z_{\alpha}\sqrt{2R_{1n}}.
\]
Here $z_{\alpha}$ is the upper-$\alpha$ quantile of standard normal distribution.
\subsection{Power analysis}
\label{poweranalysisofsec2}
Next, we study the limiting distribution of $T_n$ under a class of alternative hypotheses. Let $\mathbb{E}({\bf{X}}\mathbf{Z}^\top)=:\bm{\Sigma}_{{\bf{X}}\mathbf{Z}}=\bm{\Sigma}_{\mathbf{Z}{\bf{X}}}^\top$ be the covariance matrix between ${\bf{X}}$ and $\mathbf{Z}$ and ${\bm{\eta}} = {\bf{X}} - \bm{W}^\top\mathbf{Z}$, where
\begin{align}
\label{bW}
\bm{W} = \bm{\Sigma}_{\mathbf{Z}}^{-1}\bm{\Sigma}_{\mathbf{Z}{\bf{X}}}.
\end{align}
Let $\bm{\Sigma}_{{\bm{\eta}}}$ be the covariance matrix of ${\bm{\eta}}$.
Define the following family of local alternatives:
\[
\mathscr{L}_1(\bm{\beta}) = \biggl\{\bm{\beta}\in\mathbb{R}^{p_{\bm{\beta}}}\bigg\vert \bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}\bm{\beta} = o(1),\, \bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}^{2}\bm{\beta} = o\biggl(\frac{\bm{\Lambda}_{{\bf{X}}}}{n\varpi^2}\biggr) \,\,\text{and}\,\,\bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}\bm{\Sigma}_{{\bf{X}}}\bm{\Sigma}_{{\bm{\eta}}}\bm{\beta} = o\biggl(\frac{\bm{\Lambda}_{{\bf{X}}}}{n}\biggr)\biggr\}.
\]
We have the following theorem.
\begin{theorem}
\label{limitpowerofguosum}
Under the conditions in Theorem \ref{limitnulldisofguosum}, for $\bm{\beta}\in\mathscr{L}_1(\bm{\beta})$, we have
\[
\frac{T_n - n\bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}^{2}\bm{\beta}}{\sqrt{2\bm{\Lambda}_{{\bf{X}}}}}\rightarrow N(0,1)\quad \text{in distribution}
\]
as $(n,p_{\bm{\beta}},p_{\bm{\gamma}})\rightarrow\infty$.
\end{theorem}
Theorem \ref{limitpowerofguosum} tells that the asymptotic power under the local alternatives $\mathscr{L}_1(\bm{\beta})$ of $T_n$ is
\begin{align}
\label{powerfunctionofguosum}
\phi_{1n} = \Phi\biggl(-z_{\alpha} + \frac{n\bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}^{2}\bm{\beta}}{\sqrt{2\bm{\Lambda}_{{\bf{X}}}}}\biggr),
\end{align}
where $\Phi(\cdot)$ denotes the standard normal cumulative distribution function. \eqref{powerfunctionofguosum} implies that the proposed test has nontrivial power as long as the signal-to-noise ratio $n\bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}^{2}\bm{\beta}/\sqrt{2\bm{\Lambda}_{{\bf{X}}}}$ does not vanish to 0 as $(n,p_{\bm{\beta}})\rightarrow\infty$. Compared with \cite{chen2022testing}, we now establish the asymptotic distribution of $T_n$ under local alternative hypotheses. Both the parameter vector of interest and nuisance parameter vector are allowed to be ultra-high dimensional.
\subsection{The failure of $T_n$}
\label{sec2.3}
As discussed, the asymptotic distribution of $T_n$ can be established when the correlation between ${\bf{X}}$ and $\mathbf{Z}$ is weak. However, in highly correlated cases, condition \eqref{condition1inguosum} fails as $\varrho^2$ can be divergent in this situation.
This may cause the failure of the asymptotic normality. To illustrate this problem, consider the following toy example.
\begin{example}
\label{example2}
Generate the covariates according to the following model:
\begin{align}
\label{eqofexample2}
{\bf{X}} = \bm{W}^\top\mathbf{Z} + {\bm{\eta}},
\end{align}
where ${\bm{\eta}}$ is a $p_{\bm{\beta}}$-dimensional random vector and ${\bm{\eta}}$ is independent of $\mathbf{Z}$. $\bm{W}$ is a $p_{\bm{\gamma}}\times p_{\bm{\beta}}$ dimensional matrix with non-zero corners
\begin{align}
\label{bWexample}
\bm{W}^\top = \begin{pmatrix}
\bm 0 & \bm 0 & \bm 0\\
\bm{Q}^\top & \bm 0 & \bm{Q}^\top
\end{pmatrix},
\end{align}
where $\bm{Q}$ is an $d_{\mathbf{Z}}\times 2d_{{\bf{X}}}$ dimensional matrix and all the elements of $\bm{Q}$ equal $0.5$. In this example, we set $\bm{\beta} = \bm{0}$ and $\bm{\gamma}\neq\bm{0}$. The null hypothesis $\bm{H}_{0}$ then holds. We derive $\varrho^2$ increases with the increase of $d_{{\bf{X}}}$
by some calculation.
Let $d_{\mathbf{Z}} = 3$ and vary $d_{{\bf{X}}}$ from $0$ to $15$.
\end{example}
Left penal of Figure \ref{figureofexample2} plots the empirical sizes of $T_n$. As $d_{{\bf{X}}}$ increases, the empirical size rises rapidly, and the significance level cannot be maintained. Right penal of Figure \ref{figureofexample2} reports the empirical probability density function of $T_n$, which can be well approximated by the standard normal distribution when $d_{{\bf{X}}}=0$. However, the deviation gradually increases with the increase of $d_{{\bf{X}}}$. This toy example illustrates that when ${\bf{X}}$ and $\mathbf{Z}$ are not weakly dependent, $T_n$ is not applicable.
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{Tn_hist.pdf}
\caption{Left panel presents the empirical sizes of $T_n$ with different $d_{{\bf{X}}}$.
Right panel presents the empirical probability density function of $T_n$ with different $d_{{\bf{X}}}$ ($d_{{\bf{X}}}$ = 0,5,10,15). The pink shade represents the probability density function of the standard normal distribution. Set $n=100$, $p = 600$ and $p_{\bm{\beta}} = p_{\bm{\gamma}}$. We generate 500 replications and reject the null hypothesis at the significance level $\alpha = 0.05$. More details can be seen in Scenario 3 in Section \ref{sec6}.}
\label{figureofexample2}
\end{figure}
In the proof of Theorem \ref{limitnulldisofguosum}, the following error term depends on the correlation between ${\bf{X}}$ and $\mathbf{Z}$:
\begin{align}
\label{error1}
\operatorname{ERROR}_1= (\bm{\gamma} - \hat\bm{\gamma})^\top\frac{1}{n}\sum_{i\neq j}\mathbf{Z}_i{\bf{X}}_i^\top{\bf{X}}_j\mathbf{Z}_j^\top(\bm{\gamma} - \hat\bm{\gamma}),
\end{align}
where $n^{-1}\sum_{i\neq j}\mathbf{Z}_i{\bf{X}}_i^\top{\bf{X}}_j\mathbf{Z}_j^\top$ is a $p_{\bm{\gamma}}\times p_{\bm{\gamma}}$ dimensional random matrix, and the $(k,l)$-th element is a non-degenerate U-statistic with non-zero mean $\mathbb{E}(Z_{k}{\bf{X}})^\top\mathbb{E}(Z_{l}{\bf{X}})$.
The order of $\operatorname{ERROR}_1$ is $O_p(s^2\log p_{\bm{\gamma}}\varrho^2)$. Thus condition \eqref{condition1inguosum} is required to reduce the impact of bias term $\operatorname{ERROR}_1$ on the asymptotic behavior of $T_n$.
Clearly the bias is no longer negligible if $s^2\log p_{\bm{\gamma}}\varrho^2\geq C\sqrt{\Lambda_{{\bf{X}}}}$.
In the following, we suggest an orthogonalization-based test to reduce the bias term.
\section{An orthogonal score function-based test}
\label{sec3}
As discussed in subsection \ref{sec2.3}, the estimation error of $\hat\bm{\gamma}$ may make the test statistic $T_n$ fail to work when the correlation of ${\bf{X}}$ and $\mathbf{Z}$ is high. To make the score function immune to the estimation error of $\hat\bm{\gamma}$, we consider orthogonalizing the score function of $\bm{\beta}$,
\begin{align*}
\mathcal{S}(\bm{\beta},\bm{\gamma}) = \nabla_{\bm{\beta}}\mathcal{L} - \bm{W}^\top\nabla_{\bm{\gamma}}\mathcal{L}\quad \mathrm{with}\quad \bm{W}^\top = \nabla_{\bm{\beta}\bm{\gamma}}\mathcal{L}(\nabla_{\bm{\gamma}\bgamma}\mathcal{L})^{-1}.
\end{align*}
Here $\mathcal{L}=\mathcal{L}(\bm{\beta},\bm{\gamma}) = \mathbb{E} (Y-\bm{\beta}^\top{\bf{X}} - \bm{\gamma}^\top\mathbf{Z})^2,
\nabla_{\bm{\beta}}\mathcal{L}=\partial\mathcal{L}/\partial\bm{\beta}, \nabla_{\bm{\beta}\bm{\gamma}}\mathcal{L}=\partial\mathcal{L}^2/\partial\bm{\beta}\partial\bm{\gamma}$. $\nabla_{\bm{\gamma}}\mathcal{L}$ and $\nabla_{\bm{\gamma}\bgamma}\mathcal{L}$ are similarly defined.
The main idea of orthogonalization is to construct a statistic for the target parameter, which is locally insensitive to the nuisance parameter. The orthogonalization plays an important role in high dimensional inference problems and has been successfully applied in the recent literature. See for example, \cite{belloni2015uniform}, \cite{ning2017general} and \cite{belloni2018uniformly}. However, the current adoption of orthogonalization only focuses on low-dimensional parameters. Actually the coordinate-based maximum tests firstly consider orthogonalization for each element of testing parameter vector and then take the maximum of all individual test statistics for each elements. To the best of our knowledge, orthogonalization has not been investigated for test statistics based on the quadratic norm of the score function for ultra-high dimensional testing parameter vector.
In our model setting, $\mathcal{S}(\bm{\beta},\bm{\gamma})$ is equal to
\begin{align*}
\mathcal{S}(\bm{\beta},\bm{\gamma}) = \mathbb{E}\{(Y - \bm{\beta}^\top{\bf{X}} - \bm{\gamma}^\top\mathbf{Z})({\bf{X}} - \bm{W}^\top\mathbf{Z})\}.
\end{align*}
Again $\bm{H}_0$ corresponds to $\mathcal{S}(\bm{0},\bm{\gamma}) = \bm{0}$. Compared with $\mathcal{L}(\bm{\beta},\bm{\gamma})$, we replace ${\bf{X}}$ with ${\bf{X}} - \bm{W}^\top\mathbf{Z}$ now. We can then construct a test statistic based on $\mathcal{S}(\bm{0},\bm{\gamma})^\top\mathcal{S}(\bm{0},\bm{\gamma})$.
For the sample $\{{\bf{X}}_{i},\mathbf{Z}_{i},Y_{i}\}_{i=1}^{n}$,
define:
\begin{align}
\label{teststatisticstar1}
M_{n}^{*} = \frac{1}{n}\sum_{i\neq j}(Y_i - \bm{\gamma}^\top\mathbf{Z}_i)(Y_j - \bm{\gamma}^\top\mathbf{Z}_j)({\bf{X}}_i - \bm{W}^\top\mathbf{Z}_i)^\top({\bf{X}}_j - \bm{W}^\top\mathbf{Z}_j).
\end{align}
We construct a final test statistic in Sections \ref{sec4} and \ref{sec5}.
\section{The oracle inference}
\label{sec4}
To illustrate the merits of the orthogonalization technique, we first consider the case with given $\bm{W}$. Recall $\bm{W}$ is defined in \eqref{bW} and $\bm{W} = \bm{\Sigma}_{\mathbf{Z}}^{-1}\bm{\Sigma}_{\mathbf{Z}{\bf{X}}}$. A sufficient condition for known $\bm{W}$ is that the joint distribution of $({\bf{X}}^\top,\mathbf{Z}^\top)^\top$ is known in advance. This assumption is given in the recent literature on high-dimensional statistics, such as the model-X knockoff procedure in \cite{candes2018panning}.
Based on the term in \eqref{teststatisticstar1}, consider the following test statistic:
\begin{align}
\label{teststatisticoracle1}
M_{n}^{\text{o}} = \frac{1}{n}\sum_{i\neq j}(Y_i - \hat\bm{\gamma}^\top\mathbf{Z}_i)(Y_j - \hat\bm{\gamma}^\top\mathbf{Z}_j)({\bf{X}}_i - \bm{W}^\top\mathbf{Z}_i)^\top({\bf{X}}_j - \bm{W}^\top\mathbf{Z}_j).
\end{align}
We obtain the estimator $\hat\bm{\gamma}$ by solving a penalized least squares problem in \eqref{bgammaestimator}.
\subsection{Limiting null distribution}
Recall ${\bm{\eta}}$ is defined in subsection \ref{poweranalysisofsec2} and ${\bm{\eta}} = {\bf{X}} - \bm{W}^\top\mathbf{Z}$.
$\bm{\Sigma}_{{\bm{\eta}}}$ is the covariance matrix of ${\bm{\eta}}$.
Give the following assumption.
\begin{assumption}
\label{assumptionbmatrix}
$\mathrm{tr}(\bm{\Sigma}^4_{{\bm{\eta}}})=o(\mathrm{tr}^2(\bm{\Sigma}_{{\bm{\eta}}}^2))$ and $\mathrm{tr}(\bm{\Sigma}_{{\bm{\eta}}}^2)\rightarrow\infty$ as $(n,p_{\bm{\beta}})\rightarrow\infty$.
\end{assumption}
Assumption \ref{assumptionbmatrix} is a counterpart of Assumption \ref{assumptionbmatrix2} in the case of the orthogonal score function.
Theorem \ref{limitnulldisoforaclesum} states the limiting null distribution of the oracle test statistic $M_{n}^{\text{o}}$ in \eqref{teststatisticoracle1}.
\begin{theorem}
\label{limitnulldisoforaclesum}
Under $\bm{H}_0$, and Assumptions~\ref{assumptionb5}-\ref{assumptionb7}, \ref{assumptionbmatrix} and condition \eqref{condition2inguosum},
we have
\[
\frac{M_{n}^{o}}{\sqrt{2\bm{\Lambda}_{{\bm{\eta}}} }}\rightarrow N(0,1)\quad \text{in distribution}
\]
as $(n,p_{\bm{\beta}},p_{\bm{\gamma}})\rightarrow\infty$, where $\bm{\Lambda}_{{\bm{\eta}}} =\sigma^4 \mathrm{tr}(\bm{\Sigma}_{{\bm{\eta}}}^2)$.
\end{theorem}
Notably, compared with Theorem \ref{limitnulldisofguosum}, condition \eqref{condition1inguosum} is removed in Theorem \ref{limitnulldisoforaclesum}. Except for Assumption \ref{assumptionbmatrix}, there are no additional restrictions on the relationship between the covariates in Theorem \ref{limitnulldisoforaclesum}. The dependence requirement is greatly relaxed. The proof for this theorem is similar to that for Theorem \ref{limitnulldisofguosum}, but the error term becomes
\[
\operatorname{ERROR}^{\text{o}} = (\bm{\gamma} - \hat{\bm{\gamma}})^\top\frac{1}{n}\sum_{i\neq j}\mathbf{Z}_i{\bm{\eta}}_{i}^\top{\bm{\eta}}_{j}\mathbf{Z}_j^\top(\bm{\gamma} - \hat\bm{\gamma}),
\]
where ${\bm{\eta}}_{i} = {\bf{X}}_i - \bm{W}^\top\mathbf{Z}_i$ and $n^{-1}\sum_{i\neq j}\mathbf{Z}_i{\bm{\eta}}_{i}^\top{\bm{\eta}}_{j}\mathbf{Z}_j^\top$ is a $p_{\bm{\gamma}}\times p_{\bm{\gamma}}$ dimensional matrix with zero-mean degenerate $U$-statistic components. Benefitting from the zero-mean property of $n^{-1}\sum_{i\neq j}\mathbf{Z}_i{\bm{\eta}}_{i}^\top{\bm{\eta}}_{j}\mathbf{Z}_j^\top$, the order of the bias term $\operatorname{ERROR}^{\text{o}}$ is much smaller than that of $\operatorname{ERROR}_1$. From the theory of $U$-statistics (see, e.g., \cite{serfling1980approximation}), the order of a typical degenerate $U$ statistic is $O_p(n^{-1})$. Thus, by the property of degenerate $U$ statistic, $n^{-1}\sum_{i\neq j}\mathbf{Z}_i{\bm{\eta}}_{i}^\top{\bm{\eta}}_{j}\mathbf{Z}_j^\top$ is much easier to handle than $n^{-1}\sum_{i\neq j}\mathbf{Z}_i{\bf{X}}_{i}^\top{\bf{X}}_{j}\mathbf{Z}_j^\top$. In the proof, we show that $\operatorname{ERROR}^{\text{o}} = o_p(\sqrt{2\Lambda_{{\bm{\eta}}}})$. In summary, the orthogonalization technique has two merits: debiasing and converting a non-degenerate $U$-statistic to a degenerate one. The first is frequently observed in the literature, such as \cite{zhu2006empirical}, \cite{zhang2014confidence}, and \cite{van2014asymptotically}.
We have not seen in the literature any study to show the second merit of the orthogonalization technique.
\subsection{Power analysis}
\label{poweranalysisofsec4}
To study the power performance of $M_{n}^{\text{o}}$, consider the following local alternatives:
\begin{align*}
\mathscr{L}^{\text{o}}(\bm{\beta}) = \biggl\{\bm{\beta}\in\mathbb{R}^{p_{\bm{\beta}}}\bigg\vert \bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}\bm{\beta} = o(1)\,\,\text{and}\,\,\bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}^{3}\bm{\beta} = o\biggl(\frac{\Lambda_{{\bm{\eta}}}}{n}\biggr)\biggr\}.
\end{align*}
\begin{theorem}
\label{limitpoweroforaclesum}
Under conditions in Theorem \ref{limitnulldisoforaclesum}, and for $\bm{\beta}\in\mathscr{L}^{\text{o}}(\bm{\beta})$, we derive
\[
\frac{M_{n}^{\text{o}} - n\bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}^{2}\bm{\beta}}{\sqrt{2\bm{\Lambda}_{{\bm{\eta}}} }}\rightarrow N(0,1)\quad \text{in distribution}
\]
as $(n,p_{\bm{\beta}},p_{\bm{\gamma}})\rightarrow\infty$.
\end{theorem}
Similarly, the asymptotic power function of $M_{n}^{\text{o}}$ under the local alternatives is
\begin{align}
\label{powerfunctionoforaclesum}
\phi^{\text{o}}_{n} = \Phi\biggl(-z_{\alpha} + \frac{n\bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}^{2}\bm{\beta}}{\sqrt{2\bm{\Lambda}_{{\bm{\eta}}}}}\biggr).
\end{align}
Comparing with Theorem \ref{limitpowerofguosum}, the Pitman asymptotic relative efficiency (ARE) of $M_{n}^{\text{o}}$ concerning the $T_n$ is
\[
\operatorname{ARE}(M_{n}^{\text{o}},T_n) =
\biggl\{\frac{\mathrm{tr}(\bm{\Sigma}_{{\bf{X}}}^2)}{\mathrm{tr}(\bm{\Sigma}_{{\bm{\eta}}}^2)}\biggr\}^{1/2}.
\]
By the definition of ${\bm{\eta}}$, $\mathrm{tr}(\bm{\Sigma}_{{\bf{X}}}^{2})\geq \mathrm{tr}(\bm{\Sigma}_{{\bm{\eta}}}^{2})$. Thus the power of $M_{n}^{o}$ is higher than $T_n$.
This result is important as the orthogonalization technique can simultaneously reduce bias and variance, thus improving power performance.
\section{The test with unknown \texorpdfstring{$\bm{W}$}{\textbf{W}}}
\label{sec5}
In this case, the test statistic is defined by using a plug-in estimator of $\bm{W}$:
\begin{align}
\label{teststatistic1}
M_{n} = \frac{1}{n}\sum_{i\neq j}(Y_i - \hat\bm{\gamma}^\top\mathbf{Z}_i)(Y_j - \hat\bm{\gamma}^\top\mathbf{Z}_j)({\bf{X}}_i - \hat\bm{W}^\top\mathbf{Z}_i)^\top({\bf{X}}_j - \hat\bm{W}^\top\mathbf{Z}_j).
\end{align}
We obtain the estimator $\hat\bm{\gamma}$ by solving a penalized least squares problem in \eqref{bgammaestimator}.
The $k$-th column of $\bm{W}$ can be estimated by
\begin{align}
\label{bWestiamtor}
\hat\bm{W}_k = \mathop{\mbox{argmin}}\limits_{\bm{W}_k\in\mathbb{R}^{p_{\bm{\gamma}}}}\frac{1}{2n}\sum_{i=1}^{n}(X_{ik} - \bm{W}_k^\top\mathbf{Z}_i)^2 + \lambda_{X_k}\lVert\bm{W}_k\rVert_{1},
\end{align}
where $X_{ik}$ is the $k$th component of ${\bf{X}}_i$, and $\lambda_{X_k}$ is the tuning parameter.
\subsection{Limiting null distribution}
We let $\lVert\bm{W}\rVert_{0}\leq s^{\prime}$, where $s^{\prime}$ is a positive integer and represents the sparsity level of $\bm{W}$.
Let $\varphi^2 = \max_{1\leq k\leq p_{\bm{\gamma}}}\lVert\mathbb{E}(Z_{k}\mathbf{Z})\rVert_{2}^{2}$ and $\vartheta^2 = s^2s^\prime(\log p_{\bm{\gamma}})^{2}\varphi^2/n^{\prime}$.
Next, under the following assumption, we study the asymptotic null distribution of the test statistic $M_{n}$.
\begin{assumption}
\label{assumption8}
The estimator $\hat\bm{W}$ is independent of the data $\{{\bf{X}}_i,\mathbf{Z}_i,Y_i\}_{i=1}^{n}$, and
$\lVert\hat\bm{W} - \bm{W}\rVert_{F} = O_p(\sqrt{s^{\prime}\log p_{\bm{\gamma}}/n^\prime})$
for some positive integer $n^\prime$.
\end{assumption}
Assumption \ref{assumption8} requires the Frobenius norm bound of $\hat{\bm{W}}$ in the order of $(s^\prime\log p_{\bm{\gamma}}/n^\prime)^{1/2}$, which can be satisfied by most of existing high-dimensional estimators. See section 9.7 in \cite{highprobability2019} for instance. Besides, $n^\prime$ represents the sample size used to estimate $\bm{W}$. We can estimate $\bm{W}$ using additional data of ${\bf{X}}$ and $\mathbf{Z}$ if we have, otherwise, applying the sample-splitting approach. See, for instance, \cite{belloni2012sparse} and \cite{chernozhukov2018double}.
We also note that the independence between $\hat\bm{W}$ and $\{{\bf{X}}_i,\mathbf{Z}_i,Y_i\}_{i=1}^{n}$ can make the asymptotic analysis more easily. However, { the simulation study shows that the proposed test statistic $M_n$ still works well
numerically even when we estimate $\bm{W}$ based on the same data set. Therefore, we guess this independence might not be necessary, although we have not yet got rid of it in the technical deduction.}
\begin{theorem}
\label{limitnulldisofsum}
Under $\bm{H}_0$, Assumptions
\ref{assumptionb5}-\ref{assumptionb7},
\ref{assumptionbmatrix}, and \ref{assumption8}, condition \eqref{condition2inguosum}, and
\begin{align}
\label{condition1insum}
\vartheta^2 \vee (\log p_{\bm{\gamma}})^{1/2}\vartheta\biggl(\lambda_{\rm{max}}(\bm{\Sigma}_{{\bm{\eta}}}) + \lambda_{\rm{max}}(\bm{\Sigma}_{\mathbf{Z}})\frac{s^\prime\log p_{\bm{\gamma}}}{n^\prime}\biggr)^{1/2}\vee \lambda_{\rm{max}}(\bm{\Sigma}_{\mathbf{Z}})\frac{s^\prime\log p_{\bm{\gamma}}}{n^\prime} = o(\sqrt{\Lambda_{{\bm{\eta}}}}),
\end{align}
we have
\[
\frac{M_{n}}{\sqrt{2\bm{\Lambda}_{{\bm{\eta}}} }}\rightarrow N(0,1)\quad \text{in distribution}
\]
as $(n,p_{\bm{\beta}},p_{\bm{\gamma}})\rightarrow\infty$, where $\bm{\Lambda}_{{\bm{\eta}}}$ is defined in Theorem \ref{limitnulldisoforaclesum}.
\end{theorem}
Compared with Theorem \ref{limitnulldisofguosum}, condition \eqref{condition1inguosum} is replaced by condition \eqref{condition1insum} that handles the impact caused by the estimation error of $\hat\bm{W}$.
As discussed, the error term caused by the correlation between ${\bf{X}}$ and $\mathbf{Z}$ is reduced by the orthogonalization technique, but the estimation error is brought by $\hat\bm{W}$.
Now we compare condition \eqref{condition1insum} with \eqref{condition1inguosum}. By the formula of these conditions, it suffices to compare $\vartheta^2$ with $\varpi^2$. Note that
\begin{align}
\label{ratiobetweenvarthetaandvarrho}
\frac{\vartheta^2}{\varpi^2} = \frac{\varphi^2}{\varrho^2}\frac{s^\prime\log p_{\bm{\gamma}}}{n^{\prime}}.
\end{align}
If ratio \eqref{ratiobetweenvarthetaandvarrho} is small, condition \eqref{condition1insum} can be weaker than \eqref{condition1inguosum}.
When the relationship between nuisance covariates is weak while the relationship between covariates of interest and nuisance covariates is high, the ratio \eqref{ratiobetweenvarthetaandvarrho} can be small. When the matrix $\bm{W}$ is sparse, or $n^\prime$ is large, the ratio \eqref{ratiobetweenvarthetaandvarrho} can also be small.
Similar to the test construction in Section \ref{limitnulldisofsec2}, we estimate $\bm{\Lambda}_{{\bm{\eta}}}$ by
\[
R_{n} = \hat{\sigma}^{4}\frac{1}{2\tbinom{n}{4}}\sum_{i< j<k<l}^{n}(\hat{{\bm{\eta}}}_{i} - \hat{{\bm{\eta}}}_{j})^\top(\hat{{\bm{\eta}}}_{k} - \hat{{\bm{\eta}}}_{l})(\hat{{\bm{\eta}}}_{j} - \hat{{\bm{\eta}}}_{k})^\top(\hat{{\bm{\eta}}}_{l} - \hat{{\bm{\eta}}}_{i}),
\]
where $\hat{\bm{\eta}}_{i} = {\bf{X}}_{i} - \hat{\bm{W}}^\top\mathbf{Z}_{i}$. Under the null hypothesis and conditions in Theorem \ref{limitnulldisofsum}, $R_{n}$ is a ratio consistent estimator of $\Lambda_{{\bm{\eta}}}$; see the details in Supplementary Material. We reject $\bm{H}_{0}$ at the significance level $\alpha$ if
\[
M_{n}\geq z_{\alpha}\sqrt{2R_{n}}.
\]
Here $z_{\alpha}$ is the upper-$\alpha$ quantile of standard normal distribution.
\subsection{Power analysis}
Consider the class of local alternatives as follows:
\[
\mathscr{L}(\bm{\beta}) = \biggl\{\bm{\beta}\in\mathbb{R}^{p_{\bm{\beta}}}\bigg\vert\bm{\beta}\in\mathscr{L}^{\text{o}}(\bm{\beta})\quad\text{and}\quad \bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}^{2}\bm{\beta} = o\biggl(\frac{\Lambda_{{\bm{\eta}}}}{n(\vartheta^2\vee \lambda_{\max}(\bm{\Sigma}_{\mathbf{Z}})s^\prime\log p_{\bm{\gamma}}/n^\prime)}\biggr)\biggr\}.
\]
Then the power performance is stated in the following theorem.
\begin{theorem}
\label{limitpoweroftruesum}
Assume conditions in Theorem \ref{limitnulldisofsum}, and for $\bm{\beta}\in\mathscr{L}(\bm{\beta})$, we derive
\[
\frac{M_{n} - n\bm{\beta}^\top\bm{\Sigma}_{{\bm{\eta}}}^{2}\bm{\beta}}{\sqrt{2\bm{\Lambda}_{{\bm{\eta}}} }}\rightarrow N(0,1)\quad \text{in distribution}
\]
as $(n,p_{\bm{\beta}},p_{\bm{\gamma}})\rightarrow\infty$.
\end{theorem}
Compared to $\mathscr{L}^{\text{o}}(\bm{\beta})$, this class of alternatives have more restrictions on $\bm{\beta}$. This is because we have to handle the extra error caused by estimating $\bm{W}$. Also, comparing this theorem to Theorem~\ref{limitpoweroforaclesum}, under designed conditions, the test $M_{n}$ has the same power performance as the oracle test $M_{n}^o$ with a known $\bm{W}$ asymptotically.
\section{Numerical studies}
\label{sec6}
\subsection{Simulations}
We first compare the performances among four tests: (1). the score function-based test $T_n$
in Section \ref{sec2}; (2). the orthogonalied score function-based test $M_n$ in Section \ref{sec5}; (3). the studentized bootstrap-assisted test $ST$ in \cite{zhang2017simultaneous}; and
(4). the Wald-type test $WALD$
in \cite{guo2021group}.
Generate data from the following ultrahigh-dimensional linear model:
\begin{align*}
Y &= \bm{\beta}^\top{\bf{X}} + \bm{\gamma}^\top\mathbf{Z} + \epsilon,
\end{align*}
where the covariates $\bm{V}_i=({\bf{X}}_i^\top,\mathbf{Z}_i^\top)^\top$ are generated from the multivariate normal distribution. The details are given later, and the regression error $\epsilon_i \sim N(0,1)$ independent of $\bm{V}_i$. Denote $s_{\bm{\beta}}$ and $s_{\bm{\gamma}}$ as the sparsity levels of $\bm{\beta}$ and $\bm{\gamma}$ respectively. The regression coefficients $\bm{\beta}$ are set as: $\beta_{j}=b_0$ for $1 \leq j \leq s_{\bm{\beta}}$ and $\beta_{j}=0$ otherwise. Similarly let $\gamma_{j}=g_0$ for $1 \leq j \leq s_{\bm{\gamma}}$ and $\gamma_{j}=0$ otherwise. Throughout the simulation study, let $s_{\bm{\gamma}} = \lfloor 5\% p_{\bm{\gamma}} \rfloor$ and $g_0 = 0.5$.
There are three settings for the values of $\bm{\beta}$:
\begin{enumerate}[itemsep=0pt, topsep=0pt, leftmargin=2.2cm, label={\bf Setting \arabic*}:]
\item Consider $b_0 = 0$ to assess the empirical Type-I error.
\item Let $s_{\bm{\beta}} = \lfloor 5\% p_{\bm{\beta}} \rfloor$ and $b_0\neq 0$ to assess the empirical power with sparse alternative.
\item Let $s_{\bm{\beta}} = \lfloor 50\%p_{\bm{\beta}} \rfloor$ and $b_0\neq 0$ to assess the empirical power with dense alternative.
\end{enumerate}
The experiment is repeated $500$ times for each simulation setting to assess the empirical type-I error and power at the significance level $\alpha = 0.05$.
The tuning parameter $\lambda_Y$ in \eqref{bgammaestimator} and $\lambda_{X_k}$ in \eqref{bWestiamtor} are selected by 10-fold cross-validations using the R-package \texttt{glmnet}. Based on these settings, we consider the following three scenarios.
{\bf{Scenario 1.}} We aim to compare our tests with other testing methods in this scenario. The covariates are generated from the multivariate normal distribution $N_p(\bm{0}_p, \bm{\Sigma})$. Here $\bm{\Sigma} = (\sigma_{ij})_{p\times p}$ follows the Toeplitz design, that is, $\sigma_{ij} = 0.5^{|i-j|}, i,j = 1, \ldots, p$. The sample size $n = 100$, the covariate dimension $p = 600$ and $p_{\bm{\beta}} = p_{\bm{\gamma}}=300$. In the sparse alternative (setting 2) and the dense alternative (setting 3), we vary $b_0$ from $0$ to $\sqrt{\lVert\bm{\gamma}\rVert_{2}^{2}/s_{\bm{\beta}}}$.
{\bf{Scenario 2.}} This scenario investigates the performance of our tests thoroughly when the correlation between covariates of interest and nuisance covariates is weak. Generate the covariates from the multivariate normal distribution $N_p(\bm{0}_p, \bm{\Sigma})$, where $\bm{\Sigma} = (\sigma_{ij})_{p\times p}$ with $\sigma_{ij} = \rho^{|i-j|}, i,j = 1, \ldots, p$ and $\rho = 0.3,0.5,0.7$. The sample size $n = 100,200$, the covariate dimension $p = 600,1000,2000,4000$ and $p_{\bm{\beta}} = p_{\bm{\gamma}}$. In the sparse alternative (setting 2) and the dense alternative (setting 3), we set $b_0 = \sqrt{\lVert\bm{\gamma}\rVert_{2}^{2}/s_{\bm{\beta}}}$.
{\bf{Scenario 3.}} This scenario investigates the performance of our tests when ${\bf{X}}$ and $\mathbf{Z}$ are highly correlated.
The covariates are generated according to the following model:
\begin{align}
\label{eq1ofexample2}
{\bf{X}} &= \bm{W}^\top\mathbf{Z} + {\bm{\eta}},
\end{align}
where ${\bm{\eta}}$ is a $p_{\bm{\beta}}$-dimensional random vector and ${\bm{\eta}}$ is independent of $\mathbf{Z}$. ${\bm{\eta}}\sim N_{p_{\bm{\beta}}}(\bm{0}_{p_{\bm{\beta}}},\bm{\Sigma}_{{\bm{\eta}}})$ and $\mathbf{Z}\sim N_{p_{\bm{\gamma}}}(\bm{0}_{p_{\bm{\gamma}}},\bm{\Sigma}_{\mathbf{Z}})$, where $\bm{\Sigma}_{{\bm{\eta}}}$ and $\bm{\Sigma}_{\mathbf{Z}}$ follow the Toeplitz design with $\rho=0.5$ respectively. $\bm{W}$ is defined in \eqref{bWexample} in subsection \ref{sec2.3}.
Throughout the scenario, $d_{\mathbf{Z}} = 3$ and $d_{{\bf{X}}} = 10$.
The sample size $n = 100$, the predictor dimension $p = 600$ and $p_{\bm{\beta}} = p_{\bm{\gamma}} = 300$. In the sparse alternative (setting 2) and the dense alternative (setting 3), we vary $b_0$ from $0$ to $\sqrt{\lVert\bm{\gamma}\rVert_{2}^{2}/s_{\bm{\beta}}}$.
Figure \ref{figure2} displays the empirical size-power curves of the four tests in scenario 1.
It can be observed that $T_{n}$, $M_{n}$ and $ST$ tests control the size well. $T_n$ and $M_n$ are generally more powerful than $ST$ under the sparse and dense alternative hypotheses. Under the dense alternative, the empirical powers of $ST$ can be as low as the significance level. The empirical powers of $T_n$ and $M_n$ increase quickly as the signal strength $b_0$ becomes stronger.
{On the other hand, the $WALD$ test is very liberal to have very large empirical size when we use the tuning parameter $\tau =1$ recommended by \cite{guo2021group}. While the numerical studies in \cite{guo2021group} suggest that the $WALD$ test with $\tau=1$ can be very conservative in their setting. We have also conducted different settings with different dimensions and sample sizes, and found that with different values $\tau= 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0$ the test can be either very liberal or very conservative. Thus selecting a proper tuning parameter is difficult in general.}
It is worth noticing that $T_n$ and $M_n$ have similar performances in this scenario. $T_n$ performs well enough when the correlation between ${\bf{X}}$ and $\mathbf{Z}$ is relatively weak.
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{plotscenario1.pdf}
\caption{The left penal represents empirical sizes and powers of the $T_n, M_n$, $ST$ and $WALD$ in the sparse alternative (setting 2). The right penal corresponds to dense alternative (setting 3). The solid line with circle points, dash line with diamond points, dot-dash line with square points and two-dash line with triangle points represent the empirical sizes and powers of $T_n$, $M_n$, $ST$ and $WALD$, respectively.}
\label{figure2}
\end{figure}
Table \ref{tabletoeplitz} reports the simulation results of scenario 2. We have the following observations. First, $T_n$ and $M_n$ control the type I error well, even when the dimension is $4000$. Second, the empirical powers increase when the dimension decreases and the sample size increases. Third, there is no significant difference in power between sparse and dense alternatives as long as $\lVert\bm{\beta}\rVert_{2}$ stays the same.
{\small
\begin{table}[htbp]
\renewcommand\arraystretch{0.8}
\centering \tabcolsep 4pt \LTcapwidth 6in
\caption{The empirical type-I errors and powers for $T_n$ and $M_n$.}
\label{tabletoeplitz}
\begin{threeparttable}
\begin{tabular}{cccccccccccc}
\toprule
& & & \multicolumn{3}{c}{Type-I Error} & \multicolumn{3}{c}{Power (Sparse)} & \multicolumn{3}{c}{Power (Dense)} \\
\cmidrule(r){4-6} \cmidrule(lr){7-9} \cmidrule(l){10-12}
$n$ & $p$ & $\rho$ & $0.3$ & $0.5$ & $0.7$ & $0.3$ & $0.5$ & $0.7$ & $0.3$ & $0.5$ & $0.7$ \\
\midrule
$100$ & $600$ & $T_n$ & $0.046$ & $0.070$ & $0.060$ & $0.950$ & $0.998$ & $1.000$ & $0.956$ & $1.000$ & $1.000$ \\
& & $M_n$ & $0.050$ & $0.058$ & $0.054$ & $0.964$ & $0.998$ & $1.000$ & $0.950$ & $1.000$ & $1.000$ \\
$100$ & $1000$ & $T_n$ & $0.038$ & $0.050$ & $0.076$ & $0.820$ & $0.972$ & $1.000$ & $0.826$ & $0.980$ & $1.000$ \\
& & $M_n$ & $0.042$ & $0.050$ & $0.076$ & $0.808$ & $0.974$ & $1.000$ & $0.824$ & $0.980$ & $1.000$ \\
$100$ & $2000$ & $T_n$ & $0.052$ & $0.058$ & $0.048$ & $0.552$ & $0.800$ & $0.982$ & $0.522$ & $0.760$ & $0.996$ \\
& & $M_n$ & $0.042$ & $0.062$ & $0.042$ & $0.532$ & $0.796$ & $0.984$ & $0.504$ & $0.780$ & $0.990$ \\
$100$ & $4000$ & $T_n$ & $0.052$ & $0.066$ & $0.068$ & $0.382$ & $0.534$ & $0.816$ & $0.356$ & $0.546$ & $0.848$ \\
& & $M_n$ & $0.046$ & $0.066$ & $0.064$ & $0.376$ & $0.516$ & $0.826$ & $0.348$ & $0.528$ & $0.852$ \\
$200$ & $600$ & $T_n$ & $0.048$ & $0.056$ & $0.060$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ \\
& & $M_n$ & $0.048$ & $0.052$ & $0.052$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ \\
$200$ & $1000$ & $T_n$ & $0.062$ & $0.062$ & $0.054$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ \\
& & $M_n$ & $0.070$ & $0.064$ & $0.060$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ \\
$200$ & $2000$ & $T_n$ & $0.070$ & $0.048$ & $0.068$ & $0.982$ & $1.000$ & $1.000$ & $0.978$ & $0.998$ & $1.000$ \\
& & $M_n$ & $0.070$ & $0.042$ & $0.060$ & $0.986$ & $1,000$ & $1.000$ & $0.976$ & $0.998$ & $1.000$ \\
$200$ & $4000$ & $T_n$ & $0.040$ & $0.050$ & $0.052$ & $0.818$ & $0.956$ & $1,000$ & $0.796$ & $0.986$ & $1.000$ \\
& & $M_n$ & $0.040$ & $0.054$ & $0.052$ & $0.818$ & $0.960$ & $1,000$ & $0.794$ & $0.980$ & $1.000$ \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item ``Type-I error", ``Power (Sparse)," and ``Power (Dense)" correspond to Setting 1, Setting two, and Setting 3.
\end{tablenotes}
\end{threeparttable}
\end{table}
}
Figure \ref{figure3} displays the empirical size-power curves of $T_n$ and $M_n$ in scenario 3. We find that $T_n$ is too liberal to maintain the significance level. On the contrary, $M_n$ maintains the level well. As $b_0$ increases, although the empirical powers of $T_n$ and $M_n$ increase rapidly, the empirical power of $T_n$ does not go to $1$ as the increase of $b_0$. In contrast, the empirical power of $M_n$ can increase to 1 quickly. The results show that $M_n$ can also improve the power compared with $T_n$. This confirms the theory.
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{plotscenario3.pdf}
\caption{The left penal represents empirical sizes and powers of the $T_n$ and $M_n$ in the sparse alternatives (setting 2). The right penal corresponds to dense alternatives (setting 3). The solid line with circle points and dash line with diamond points represent the empirical sizes and powers of $T_n$ and $M_n$, respectively.}
\label{figure3}
\end{figure}
The above simulation studies conclude that our proposed tests perform well even when the testing and nuisance parameters are both ultrahigh dimensional. When a relatively high correlation exists between the covariates of interest and the nuisance covariates, $M_n$ can enhance the power performance over $T_n$. These confirm the merits of the orthogonalization technique.
\subsection{Real data analysis}
We apply our tests to a data set about riboflavin (vitamin B2) production rate with Bacillus Subtilis. This data set was made publicly by \cite{buhlmann2014high} and analyzed by several authors, for instance \cite{van2014asymptotically}, \cite{javanmard2014confidence}, \cite{dezeure2017high}, and \cite{fei2019drawing}. It consists of $71$ observations of strains of Bacillus Subtilis and $4088$ covariates, measuring the log-expression levels of 4088 genes. The response variable is the logarithm of the riboflavin production rate.
First, we screen the covariates using the sure independence screening procedure in \cite{fan2008sure}. 133 genes are picked out using R-package \texttt{SIS}. Table \ref{selectedgenesetofrealdata} in the Supplementary Material reports the names of selected genes. Denote $\mathcal{G}$ as the selected gene set and $\mathcal{G}^c$ as its complement. The set $\mathcal{G}$ contains some detected genes reported in the literature, such as YXLD\_at, YXLE\_at, YCKE\_at, and XHLA\_at. A natural question is whether the selected genes contribute to the response given the other genes? Consider the following regression modeling:
\begin{align*}
Y = \bm{\beta}^\top\bm{V}_{\mathcal{G}} + \bm{\gamma}^\top\bm{V}_{\mathcal{G}^c} + \epsilon,
\end{align*}
where $Y$ is the response variable, $\bm{V}_{\mathcal{G}}$ is the vector of selected genes, $\bm{V}_{\mathcal{G}^c}$ denotes the genes in set $\mathcal{G}^c$, and $\epsilon$ is the regression error. Further $\lVert\bm{\beta}\rVert_{0} = \lvert\mathcal{G}\rvert = 133$, $\lVert\bm{\gamma}\rVert_{0} = \lvert\mathcal{G}^c\rvert = 3955$. The null hypothesis of interest is $\bm{H}_{01}:\bm{\beta} = \bm{0}$. Thus the testing parameter is $\bm{\beta}$, and the nuisance parameter is $\bm{\gamma}$. The dimensions of $\bm{\beta}$ and $\bm{\gamma}$ are higher than the sample size ($n = 71$).
To verify whether some genes in $\mathcal{G}^{c}$ contribute to the response given the other genes, we also consider the null hypothesis $\bm{H}_{02}:\bm{\gamma} = \bm{0}$. In this testing problem $\bm{H}_{02}$, the testing parameter is changed to be $\bm{\gamma}$, and $\bm{\beta}$ is the nuisance parameter correspondingly.
Apply $T_n$, $M_n$, $ST$, and $WALD$. We standardize the data and report the $p$-values in Table \ref{pvalueofrealdata}. For the testing problem $\bm{H}_{01}$, only $T_n$ and $M_n$ reject the null hypothesis at the significance level $\alpha = 0.01$. For the testing problem $\bm{H}_{02}$, all tests do not reject the null hypothesis at the significance level $\alpha = 0.01$. The results suggest that the selected gene set $\mathcal{G}$ contributes to the response, and there is no significant gene in $\mathcal{G}^c$.
{\small
\begin{table}[ht]
\renewcommand\arraystretch{0.8}
\centering \tabcolsep 4pt \LTcapwidth 6in
\caption{The $p$-values for real data example. }
\label{pvalueofrealdata}
\begin{threeparttable}
\begin{tabular}{ccccc}
\toprule
Method & $T_n$ & $M_n$ & $ST$ & $WALD$ \\
\midrule
& \multicolumn{4}{c}{$p$-value} \\
\cmidrule(l){2-5}
$\bm{H}_{01}$ & 0.008 & $<$0.001 & 0.052 & 0.101\\
$\bm{H}_{02}$ & 0.794 & 0.789 & 0.402 & 0.377\\
\bottomrule
\end{tabular}
\end{threeparttable}
\end{table}
}
\section{Conclusions}
\label{sec7}
This paper considers testing the significance of ultrahigh-dimensional parameter vector of interest with ultrahigh-dimensional nuisance parameter vector. We first reanalyze the score function-based test under weaker conditions to show the limiting distributions under the null and local alternative hypotheses. We construct an orthogonalized score function-based test to handle the correlation between the covariates of interest and nuisance covariates. Our investigation shows that the orthogonalization technique can debiase the error term, convert the non-degenerate error terms to degenerate, and reduce the variance to achieve higher power than the non-orthogonalized score function-based test.
{Our procedure is very generic. Extensions to other regression models such as generalized linear regression models and partially linear regression models are possible. We would investigate these extensions in near future.}
|
1,108,101,566,751 | arxiv | \section{Introduction}
Reinforcement
Learning (RL) follows the principle of behaviourist psychology and learns
in a similar way as a child learns to perform a new task. RL has been
repeatedly successful in the past~\cite{singh2002optimizing,tesauro1995temporal}, however, the successes were mostly limited to low-dimensional problems. In recent years, deep learning has significantly advanced the field of RL, with the use of deep learning algorithms within RL giving rise to the field of ``deep reinforcement learning''. Deep learning enables RL to operate in high-dimensional state and action spaces and
can now be used for complex decision-making problems~\cite{arulkumaran2017brief}.
Deep RL algorithms have mostly been applied to video or image processing domains that include playing video games~\cite{silver2016mastering,mnih2015human} to indoor navigation~\cite{zhu2017target}. Only a very limited number of studies have explored the promising aspects of deep RL in the field of audio processing in particular for speech processing.
In this paper, we study this under-researched topic. In particular, we conduct a case study of the feasibility of deep RL for
automatic
speech command classification.
A major challenge of deep RL is that it often requires a prohibitively large amount of training time and data to reach reasonable performance, making it inapplicable in real-world settings~\cite{cruz2017pre}. Leveraging humans to provide demonstrations (known as learning from demonstration (LfD) in RL has recently gained traction as a possible way of speeding up deep RL~\cite{vinyals2017starcraft,hester2018deep,kurin2017atari}. In LfD,
actions demonstrated by the human are considered as the ground truth labels for a given input game/image frame.
An agent closely simulates the demonstrator's policy at the start, and later on
learns to surpass the demonstrator~\cite{cruz2017pre}. However, LfD holds a distinct challenge, in the sense that it often requires the agent to acquire skills from only a few demonstrations and interactions due to the time and expense of acquiring them~\cite{Calinon2018}. Therefore,
LfDs are generally not scalable for especially high-dimensional problems.
Pre-training the underlying deep neural network (in Section~\ref{sec:methods} we discuss the structure of RL in detail) is another approach to speed up training in deep RL. In~\cite{abtahi2011deep}, the
authors combine Deep Belief Networks (DBNs) with RL to take benifit of the unsupervised pre-training phase in DBNs, and then use the DBN as the opening point for a neural network function approximator. Furthermore, in~\cite{anderson2015faster}, the
authors demonstrate
that a pre-trained hidden layer architecture can reduce the time required to solve reinforcement learning problems. While these studies show the promise of using pre-trained deep neural
networks in non-audio domains, the feasibility of pre-training is not well understood for
the audio field in general.
We found very few studies in audio using RL/deep RL. In~\cite{stockholm2009reinforcement}, the
authors describe
an avenue of using RL to classify audio files into several mood classes depending upon listener response during a performance. In~\cite{lakomkin2018emorl},
the authors introduce the `EmoRL' model that triggers an emotion classifier as soon as it gains enough confidence while listening to an emotional speech. The authors cast this problem into a RL problem by training an emotion classification agent to perform two actions: wait and terminate. The agent selects the terminate action to stop processing incoming speech and classify it based on the observation. The objective was to achieve a trade-off between performance (accuracy) and latency by punishing wrong classifications actions as well as too delayed predictions through the reward function. While
the authors in the above studies use RL for audio, they do not have a focus on pre-training to improve the performance of deep RL.
In this paper,
we propose pre-training for improving
the performance and speed of Deep RL while conducting speech classification. Results from the case study with the
recent public Speech Commands Dataset \cite{warden_speech_2018} show that pre-training offers significant improvement in accuracy and helps achieve faster convergence.
\section{Methodology}
\label{sec:methods}
In this study, we investigate the feasibility of pre-training in RL algorithms for speech recognition. We present the details of the proposed model in this section.
\subsection{Speech Command Recognition Model}
The considered policy network model consists of a speech command recognition model as shown in figure~\ref{fig:speech_command_model}.
\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{figures/model-architecture/Model_Architecture-v8.png}
\caption{Speech command recognition model architecture.}
\label{fig:speech_command_model}
\end{figure}
Considering the fact that Convolutional Neural Networks (CNNs) and Long Short Term Memory (LSTM) Recurrent Neural Networks (RNNs) can be combined to improve the performance \cite{latif2019direct},
we assembled CNN layers on top of an LSTM RNN layer. LSTM RNNs are good at learning the temporal structure of a feature map \cite{latif2018phonocardiographic}, and CNNs are strong in diminishing frequency variations \cite{rana2019multi}. The outputs from the LSTM RNN layer are passed on to fully connected layers to learn discriminative features during training \cite{latif2019direct}. In this way, our proposed policy network is empowered by convolutional layers for learning high-level abstraction, an LSTM layer to capture long-term temporal context, and finally fully connected layers for learning discriminative representation.
\subsection{Deep Reinforcement Learning Framework}
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{figures/RL-Architecture-v5.png}
\caption{Framework of the proposed Deep RL. }
\label{fig:rl_framework}
\end{figure}
The reinforcement learning framework mainly consists of two major entities namely ``agent'' and ``environment''. The \textit{action} decided by the agent is executed on the environment and it notifies the agent with the reward and next state in the environment. In this work, we focus on deep RL that involves a Deep Neural Network (DNN) structure in the agent module to resolve the action taken by observing the state which is illustrated in Figure \ref{fig:rl_framework}. We modelled this problem as a Markov decision process (MDP) \cite{tetreault_reinforcement_2008}. This can be considered as a tuple $(S,A,P,R)$, where $S$ is the state space, $A$ is the action space, $P$ is the state transition policy, and $R$ is the reward function. Since the core goal of this problem is classification, we modelled the MDP in such a way that the predicted classes are to be as actions, $A$, and the states, $S$ are the features of each audio segment in a batch of size $\eta$. An action decision is carried out by an RL agent which receives a reward ($r_{t}$) using the following reward function:
\begin{equation}
r_{t}=
\begin{cases}
+1, & \text{if}\ a_{t} = g_{t} \\
-1, & \text{otherwise},
\end{cases}
\end{equation}
where $g_{t}$ is the ground truth value of the specific speech utterance. We modelled the probability of actions using the following equation:
\begin{equation}
a_{t} = \text{Softmax}(W^{a}\cdot S_{t} +b^{a}),
\end{equation}
where $a_{t}$ is the action selection probability, and $W^{a}$ and $b^{a}$ are the weight and bias values. $S_{t}$ is the output from the previous hidden layer. The softmax function is defined as:
\begin{equation}
\text{Softmax}(\alpha_{j}) = \frac{e^{\alpha_{j}}}{\sum_{i=1}^{n} e^{\alpha_{i}} }.
\end{equation}
The target of the RL agent is to maximise the expected return in the policy
\begin{equation}
J_{a} (\theta_a,\theta_s) = \EX_{\pi(a_{t}|s_{t}; \theta_a,\theta_s)}[r_{t}],
\end{equation}
where $\pi(a_{t}|s_{t}; \theta_a,\theta_s)$ is the policy of agent, and $r_{t}$ is the expected reward return at state $t$.
\subsubsection{REINFORCE Training}
The REINFORCE algorithm is used to approximate the gradient to maximise the objective function $J(\theta_{a},\theta_{s})$.
\begin{algorithm}[t!]
\SetAlgoLined
initialise state space\;
initialise policy network model\;
pre-train policy network\;
retrieve initial state $s_{1}$\;
\For{$i \leftarrow 1$ \KwTo $N_{E}$}{
initialise $E_{s}, E_{a}, E_{r}$\;
\While{!d}{
$a_{i} \leftarrow $ get action($s_{i}$)\;
$s_{i+1}$, $r_{i}$, $d \leftarrow execute(a_{i}$)\;
$E_{s} \leftarrow s_{i} + E_{s}$\;
$E_{r} \leftarrow r_{i} + E_{r}$\;
$E_{a} \leftarrow a_{i} + E_{a}$\;
}
$train(E_{s}, E_{a}, E_{r})$\;
}
\caption{REINFORCE algorithm implementation}
\label{alg:rl_algo}
\end{algorithm}
Algorithm~\ref{alg:rl_algo} describes the algorithmic steps followed throughout the RL action prediction process, where $N_{E}$ indicates the maximum number of episodes to run (10,000 experiments), $S_{i}$ is the state at instant $i$, $a_{i}$ is the predicted action for the $s_{i}$ at $i^{th}$ instant, $r_{i}$ is the reward obtained by executing the predicted action $a_{i}$, $d$ is a boolean flag indicating the end of an episode, where the end of the episode is decided when $i$ reaches the step size $\eta$ (50). $E_{s}, E_{a}, E_{r}$ are arrays collecting the values of $s_{i}$, $a_{i}$, $r_{i}$ for each step, which is consumed by the policy model's training method $train$.
For a given set of examples in the state space $S$, initially, the environment sends the $s_{1}$ to the RL agent. The RL agent infers the corresponding action probabilities through the policy network and selects the highest probable action $a_{1}$ and returns it to the environment. The environment then calculates the reward $r_{1}$ for the action-state combination and returns to the agent with the reward and next state $s_{t+1}$. Each $r_{i}$, $a_{i}$, and $s_{i}$ are stored, and then, the policy network is trained with those values in an episode.
\section{Experimental Setup}
\subsection{Datasets}
The Speech Commands Dataset \cite{warden_speech_2018} is an audio corpus of 105,829 utterances containing 30 command keywords spoken by 2,618 speakers. Each utterance of a one-second file is stored in the `.wav' file format with 16\,kHz sampling rate. This dataset contains mainly two subsets of command keywords, namely ``Main Commands'', and ``Sub Commands''. Table~\ref{tab:sc_dataset} shows the distribution of the 30 keywords among the two subsets.
\begin{table}[h]
\centering
\caption{Distribution of keywords in the Speech Commands Dataset}
\begin{tabular}{p{8em}p{13em}}
\hline
Subset & Commands \\
\hline
Main Commands & one, two, three, four, five, six, seven, eight, nine, down, go, left, no, off, on, right, stop, up, yes, zero \\\hline
Sub Commands & bed, bird, cat, dog, happy, house, Marvin, Sheila, tree, wow
\\\hline
\end{tabular}
\label{tab:sc_dataset}
\end{table}
\subsection{Feature Extraction}
In this study, we use Mel Frequency Cepstral Coefficients (MFCC) to represent the speech signal. MFCCs are very popular features and widely used in speech and audio analysis/processing \cite{latif2019direct,davis1980comparison}.
We extract 40 MFCCs from the mel-spectrograms with a frame length of 2,048 and a hop length of 512 using Librosa \cite{mcfee_librosa:_2015}.
\subsection{Model Recipe}
We use the Tensorflow library to implement the model, a combination of CNN and LSTM: The initial layers are 1d convolution layers wrapped in time distributed wrappers with filter sizes of 16 and 8, respectively, followed by a max-pooling layer. The feature maps are then passed to an LSTM layer of 50 cells for learning the temporal features. A dropout layer of dropout rate 0.3 is used for regularisation. Finally, three fully connected layers of 512, 256, and 64 units respectively are added before the softmax layer.
The input to the model is a matrix of $n \times f$, where $n$ is the number of MFCCs (40), and $f$ is the number of frames (87) in the MFCC spectrum.
We use a Stochastic gradient descent optimiser
with a learning rate of $10^{-4}$.
The score value $V_{j}$ is defined as the sum of the rewards $r_{i}$ produced within the $j^{th}$ episode. The score variable can be utilised to infer the overall accuracy ($H$) of the RL Agent within a given episode.
\begin{equation}
H_{i} = \frac{ V_{i} - \eta \times \text{min}(r)}{\eta \times (\text{max}(r)-\text{min}(r))} \times 100 \%.
\end{equation}
\section{Results}
Experiments were carried out focusing on the effect that pre-training has on the accuracy of the RL Agent. Three subsets of Speech Command datasets were selected, namely ``binary'', ``20 class'', and ``30 class''. A binary subset contains only the speech commands of the ``left'' and ``right'' classes. The 20 classes and 30 classes subsets contain "Main" commands and the merge of ``Main'' and ``Sub'' commands, respectively. Each subset was experimented as without pre-training and with pre-training.
\begin{figure*}[t!]
\centering
\includegraphics[trim=0.2cm 0.2cm 0.2cm 0.2cm,clip=true,width=\textwidth]{figures/score-episode/all-class-score-episode-2000dpi-v2.png}
\caption{Performance evaluations of the model on three different scenarios}
\label{fig:performence}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[trim=0.2cm 0.2cm 0.2cm 0.2cm,clip=true,width=\textwidth]{figures/std-episode/all-class-std-episode-2000dpi-v2.png}
\caption{Standard Deviation of the score with episode on three different scenarios}
\label{fig:std_dev_episode}
\end{figure*}
The mean score of a batch of 200 episodes is plotted against the episode number in Figure~\ref{fig:performence}. Interpreting the graphs in Figure~\ref{fig:performence}, one can find that the pre-training increases the overall score. It can be seen that the binary class classification nearly reaches its maximum score at 50 within initial 2,500 episodes. The other 2 subsets show a score over 25 within 10,000 episodes.
The rates of change of score (velocity of score) for the initial 500 and 1,000 episodes were calculated by equation~\ref{eq:velocity} and tabulated in Table~\ref{tab:score_velocity}.
\begin{equation}
\text{Velocity}_x = \frac{\text{mean}(V_x:V_{x+5}) - \text{mean}(V_0:V_5)}{x}.
\label{eq:velocity}
\end{equation}
The results in Table~\ref{tab:score_velocity} convey that the velocity of score increases by pre-training within the initial 1,000 episodes for all 3 subsets of experiments. This lets one conclude that the pre-training can decrease the time taken for the RL agent to converge to better accuracy.
\begin{table}[t!]
\centering
\scriptsize
\caption{Velocity of score change in the initial episodes.}
\label{tab:score_velocity}
\resizebox{\linewidth}{!}{%
\begin{tabular}{|l|r|r|}
\hline
\multirow{2}{*}{\# Classes} & \multicolumn{2}{c|}{Change of Velocity of score (\%)} \\ \cline{2-3}
& Initial 500 episodes & Initial 1000 episodes \\ \hline
2 & -0.9 & \textbf{4.4} \\ \hline
20 & \textbf{15.2} & 8.8 \\ \hline
30 & \textbf{6.4} & 11.2 \\ \hline
\end{tabular}%
}
\end{table}
\begin{table}[]
\scriptsize
\centering
\caption{Improvement of the score with pre-training.}
\resizebox{\linewidth}{!}{%
\begin{tabular}{|l|r|r|r|}
\hline
\multirow{2}{*}{\# Classes} & \multicolumn{2}{c|}{Score after 10000 episodes} & \multirow{2}{*}{Improvement (\%)} \\ \cline{2-3}
& w/o pre-train & w/ pre-train & \\ \hline
2 & 29.4 & 49.0 & 19.6 \\ \hline
20 & -23.8 & 37.0 & 60.8 \\ \hline
30 & -23.4 & 29.0 & 52.4 \\ \hline
\end{tabular}%
}
\label{tab:score_improvement}
\end{table}
Table~\ref{tab:score_improvement} shows the mean score of the latest 5 episodes for the ``with'' (w/) and ``without'' (w/o) pre-train scenarios. The improvement column shows the increment of score of the ``with pre-train'' with respect to ``without pre-train'' scenario, where the improvement is calculated by equation~\ref{eq:improvement}. Each improvement is a positive improvement. This concludes that the overall final score (accuracy) of the RL agent policy network model can be improved by pre-training.
\begin{equation}
\text{Improvement} = \frac{x_{\text{with}} - x_{\text{without}}}{\eta \times (\text{max}(r)-\text{min}(r))} \times 100 \%.
\label{eq:improvement}
\end{equation}
According to Figure~\ref{fig:std_dev_episode}, the standard deviation of the score decreases rapidly with time in the pre-trained scenario. This again shows that predictions of the RL agent are increasing with time and pre-training is accelerating the process.
\section{Conclusions}
In this paper, we propose the use of pre-training in deep reinforcement learning (deep RL) for speech recognition. The proposed model uses pre-training knowledge to achieve a better score while reducing the convergence time. We evaluated the proposed RL model using the speech command dataset for three different classifications scenarios, which include binary (two different speech commands), and 10 and 30 class tasks. Results show that pre-training helps to achieve considerably better results in a lower number of episodes. In future efforts, we want to study the feasibility of using unrelated data to pre-train deep RL to further improve its performance and convergence.
|
1,108,101,566,752 | arxiv | \section{Introduction}
As a class, normal elliptical galaxies are not actively forming stars. Their star formation
rate (SFR) must be significantly less than $1$~M$_\odot$~yr$^{-1}$, or such activity would have been
identified long ago in the form of prominent H$\beta$ lines or an excess of blue light. However, lower levels
of star formation are suggested by small changes in optical line indices, which
can be used to date stellar populations of galaxies. At constant metallicity, the Balmer line indices
are sensitive to age, with inferred ages often surprisingly young. For example, in the sample of
ellipticals in \citet{2000Trager}, 40\% have mean
ages less than 6 Gyr and several are younger than 3 Gyr. \citet{2005Denicolo} obtain similar
results with similar methods. This implies that many elliptical galaxies were forming stars
quite recently ($z < 0.5$), in conflict with other observations that indicate little star
formation in ellipticals since $z \approx 1$ \citep{2005Daddi,2005Labbe}. One explanation for
this discrepancy is the ``frosting'' effect, whereby a small rate of ongoing star formation
contaminates the Balmer lines, making them relatively strong, while contributing little to the
mass of the galaxy. The degree of the frosting could be understood if the rate of ongoing
star formation were known, resulting in more reliable age determinations.
Several mechanisms can lead to small amounts of star formation. One avenue involves stellar
winds from the ensemble of old stars in a galaxy, amounting to $0.1-1$~M$_\odot$~yr$^{-1}$ for
$\sim$L$^*$ ellipticals. Most of this stellar mass loss is believed to collide with the ambient
interstellar medium (ISM) and become heated to X-ray emitting temperatures that are typically
$5\times 10^{6}$~K \citep{2003Mathews}, although
the efficiency of this process is unknown and some gas may remain cool \citep{2008Parriott},
and occasionally may
form into new stars. Most of the hot ISM in a galaxy has a cooling time much less than a
Hubble time, and if not driven out as a galactic wind (through active galactic nuclei or supernovae),
the gas will radiatively cool.
Radiatively cooling gas is found in $30-40$\% of ellipticals, as apparent from the detection of
the \ion{O}{6} line in Far Ultraviolet Spectroscopic Explorer (FUSE) observations \citep{2005Bregman},
implying a cooling rate of $\sim 0.1-0.5$~M$_{\odot}$~yr$^{-1}$.
Gas emitting \ion{O}{6} is at the peak of the cooling curve, so it will cool to the $10-10^{4}$~K
range. This cooled gas, confined to the central region ($< 1$~kpc), is a natural source of
material for star formation. In addition to the internal recycling of galactic gas, there may
be infall of material onto a galaxy from other smaller galaxies or from ambient group material.
Although the gaseous content of elliptical galaxies is generally dominated by hot
($5\times 10^{6}$~K) X-ray emitting material with masses of $10^{8}-10^{10}~$M$_\odot$
\citep{1991Roberts}, some evidence for cool gas ($< 10^{4}$~K) exists in addition to \ion{O}{6}
emission. H$\alpha$-emitting material at $\sim 10^{4}$~K is present in the central kpc
($10-20''$) of most ellipticals \citep{1990Matthews,2000Caon}, although it constitutes a
relatively small amount of gas ($10^{4}-10^{5}$~M$_\odot$). While few elliptical galaxies have
detectable amounts of \ion{H}{1} ($<10^{7}-10^{8}$~M$_\odot$) or H$_2$
\citep{1991Roberts,2007diSeregoAlighieri}, 5-10\% were detected at 60-100~$\mu$m by IRAS
\citep{1998Bregman}, showing that dust emission is occasionally present. Extinction by dust
lanes is also seen in $\sim50$\% of ellipticals, and is generally near the center
\citep{2005Lauer}. Evidently, cool gas exists in ellipticals and is most common in the
central region, but we do not know the fate of this material nor whether its presence is
stable or varies with time.
A good strategy for quantifying the amount of recent star formation in normal ellipticals is to work in the ultraviolet (UV),
where old, red stars that dominate these galaxies contribute little compared to hot
horizontal branch stars, post-Asymptotic Giant Branch (p-AGB) stars, and young stars \citep{1999OConnell}.
Indeed, many near-UV (NUV) observations have been made of early-type galaxies in an attempt to quantify amounts of
recent star formation.
For instance, using the Wide-Field Planetary Camera 2 (WFPC2) on the
Hubble Space Telescope (HST), \citet{2000Ferreras} found that most ellipticals in a $z=0.4$ cluster
must have young stellar mass fractions of at least $0.1\%$, and in many cases $1$ to $10\%$, to account for
their rest frame $2000$\AA\ emission. At lower redshift, \citet{2007Kaviraj} studied a sample of $2100$
early-type galaxies selected systematically from the Sloan Digital Sky Survey and found that $30\%$
of them had Galaxy Evolution Explorer (GALEX) NUV-optical
colors that required a young ($<1$~Gyr) component, amounting to $1$--$3\%$ of their stellar mass
\citep[see also the similar earlier study by][]{2005Yi}.
A handful of UV observations have also been made of individual stars, though these
observations have mostly been limited to the nearest galaxies. Perhaps
the most comprehensive set comes from \citet{2008Brown}, where the UV population of stars
in the nearby elliptical M32 (with a distance, $d=800$~kpc) was studied via near and far-UV data from the
Space Telescope
Imaging Spectrograph (STIS) on HST \citep[see also][]{2000Brown}
and using stellar evolution models. However, because \citet{2008Brown} focused on determining the UV contribution from
p-AGB stars and other stars at late evolutionary stages, and because the
field of view of STIS was small ($25\times 25\arcsec$), no analysis of young stars in M32 was
performed. This technique of imaging individual stars has also been applied to nearby lenticular galaxies; for example,
NGC~5102, a nearby S0 ($d = 4.0$~Mpc), where \citet{1997Deharveng}
used the Faint Object Camera (FOC) on HST to search for individual young stars as point sources
and deduced a limit for the star formation rate of $5\times 10^{-4}$~M$_\odot$~yr$^{-1}$.
\citet{1997Deharveng} also observed NGC~3115, another nearby S0 ($d = 9.7$ Mpc), though they
detected no stars and concluded that the FOC was not useful for detecting these stars at
the distances of most elliptical galaxies.
More recently, \citet{2011Crockett} detected clumps of blue stars in the more distant S0 NGC~4150
($d = 13.7$ Mpc), using HST's Wide Field Camera 3 (WFC3), and determined the star formation
history of NGC~4150 by fitting the UV-optical SED on a pixel-by-pixel basis to a two-burst model.
In this paper we present HST WFC3 UV imaging of four nearby elliptical galaxies, which we used to determine
star formation rates through the direct identification of young stars and star clusters. Star formation rates
were measured to levels of $10^{-5}$~M$_\odot$~yr$^{-1}$, a limit that is much lower than the
$\la 0.01$~M$_\odot$~yr$^{-1}$ limits that have previously been attained for galaxies
at similar distances \citep{1999OConnell}.
Our sample, which we describe in \S \ref{sec:targets}, includes three ellipticals
with potential signposts of star forming conditions, plus one control galaxy with no prior
indication of star formation.
Observations and photometry are also described in \S\ref{sec:data}.
We present the detected UV bright sources in \S\ref{sec:population}, and discuss their physical nature in
\S\ref{sec:ids}. Star formation rates and histories from the UV bright clusters
are presented in \S\ref{sec:sfr}, and we conclude with a discussion in \S\ref{sec:discussion}.
\section{Data}
\label{sec:data}
\subsection{Sample Selection}
\label{sec:targets}
Our sample consists of nearby normal $\sim$L$^*$ elliptical galaxies, which are located between $10.6$~Mpc
and the Virgo Cluster, with the farthest targeted galaxy at $d=18.4$~Mpc \citep[all distances were
determined via surface brightness fluctuations by][]{2001Tonry}. Three normal elliptical galaxies that
demonstrated potential signposts of star formation were included, as well as one typical elliptical
that has shown no indication of having current star formation. The observed targets are summarized
in Table \ref{tab:targets} and are presented below:
\begin{deluxetable*}{lrcccl}
\tablecaption{Target Summary\label{tab:targets}}
\tablehead{\colhead{Galaxy} & \colhead{Type} & \colhead{$m-M$} & \colhead{$d$} & \colhead{$R_{\mathrm{eff}}$} & \colhead{Notes}\\
\colhead{} & \colhead{} & \colhead{} & \colhead{(Mpc)} & \colhead{(arcsec)} & \colhead{} }
\startdata
NGC~3379 & E1 & 30.12 & 10.6 & 29.9 & no prior star formation indicator\\
NGC~4697 & E6 & 30.35 & 11.8 & 42.4 & PAH emission\\
NGC~4636 & E0-1 & 30.83 & 14.7 & 59.3 & \ion{O}{6} emission, cooling flows\\
NGC~4374 & E1 & 31.32 & 18.4 & 34.8 & \ion{O}{6} emission, AGN activity
\enddata
\tablecomments{Galaxy types are from RC3 \citep{1991deVaucouleurs}, distances are from \citet{2001Tonry},
and effective radii are from The 2MASS Large Galaxy Atlas \citep[LGA;][]{2003Jarrett}.}
\end{deluxetable*}
NGC~3379 --- We selected the nearest normal elliptical that is old \citep[$\approx 9$~Gyr;][]{2000Trager}
and has no indication of recent star formation, NGC~3379 (M105). It is the closest galaxy in our sample at
$d=10.6$~Mpc, and was chosen as the control galaxy to allow tests of whether the possible star formation
indicators in the other galaxies within our sample, such as \ion{O}{6} emission,
AGN activity, and PAH emission, are reliable tracers of ongoing star formation.
NGC~4697 --- With a distance of $11.8$~Mpc, NGC~4697 is our second closest target and is usually
designated a typical elliptical galaxy. However, from a sample of 30 nearby normal ellipticals observed
with the Spitzer Space Telescope \citep{2006Bregman_a,2006Bregman_b}, it is the only galaxy that exhibited a
polycyclic aromatic hydrocarbon (PAH) feature. Although PAH emission can be produced without star
formation, e.g., via the winds of intermediate-age stellar populations \citep{2010Vega}, it is often
an indicator of such activity.
NGC~4636 --- A cooling flow elliptical within the Virgo Cluster, NGC~4636 has a distance of $14.7$~Mpc. It
has abundant X-ray emitting
gas that cools on short timescales, and exhibits \ion{O}{6} emission \citep{2001Bregman,2005Bregman}. This \ion{O}{6}
emission is an excellent tracer of cooler gas that could feed star formation and indicates a cooling rate of
$0.3$~M$_\odot$~yr$^{-1}$.
NGC~4374 --- The farthest of our targets, NGC~4374 (M84) is in the Virgo Cluster and has a distance of $18.4$~Mpc.
\ion{O}{6} emission has been detected in this galaxy \citep{2005Bregman}, suggesting the presence of
cooling gas that can feed star formation. Radio jets and AGN activity are also present, which may correlate
with star formation \citep[e.g., see][]{2004Heckman}.
\subsection{Observations and Data Reduction}
\label{sec:observations}
Observations were conducted using HST's WFC3 between 2009 November 23
and 2010 March 29 (Cycle 17, PID 11583). These observations were made using UVIS
through two filters, F225W and F336W, which are spaced far enough in wavelength that stars
on the main sequence would stand out in color from possible contaminants such as UV-bright
globular clusters on a color-magnitude diagram (CMD). Not only are the background
levels in the UV substantially lower than at optical wavelengths, but fluctuations in the
background due to horizontal branch stars \citep{1993Worthey} are much fainter than young stars,
which are expected to emit strongly in the UV.
Each galaxy was observed over two orbits for each of the F225W and F336W filters.
Dithered observations were performed over each two-orbit sequence, using the UVIS box dithering
pattern with four pointings per orbit to aid in the removal of artifacts. Each pointing had an exposure
time of 600s, for a total of 4800s per filter per galaxy. As the exposure times were the same for all
four of our targets, the data of NGC~3379 and NGC~4697, our closest targets,
are deeper than those of NGC~4636 and NGC~4374.
Data were reduced on-the-fly during retrieval from the Multimission Archive at STScI (MAST), resulting in
pipeline-calibrated and flat-fielded individual exposures. We then removed cosmic rays,
corrected geometric distortions, and combined the dithered pointings from each orbit using
Multidrizzle \citep{2002Koekemoer}. The resulting drizzled images were used as the reference frame while the
photometry was performed.
\subsection{Photometry}
\label{sec:photometry}
We performed PSF photometry using {\sc dolphot} \citep[version 2.0;][]{2000Dolphin}, a stellar
photometry package with specific HST instrument modules. In addition to its customized settings for
WFC3 data, {\sc dolphot} can perform localized background subtraction during the
photometering, which was crucial due to the background emission near the center of each galaxy that
needed to be subtracted carefully, particularly in F336W.
For each galaxy, we used the deep multidrizzled F225W image as the reference frame. Initial estimates
for the position shifts, magnification and rotation were made using the {\it wfc3fitdistort} task, and
the individual
dithered exposures were then photometered on a chip by chip basis. All recommended settings for WFC3 data
were used, with the exception of the sky-fitting setting (FitSky). Instead of fitting the sky prior to each photometry
measurement, the sky was fit inside the PSF region of 10 pixels but outside the photometry aperture of 4
pixels. This was done to properly take into the account the varying background levels near the center of
each galaxy, particularly at radii dominated by background levels in F336W.
The output magnitudes from {\sc dolphot} are automatically calibrated using Vega zeropoints, while
aperture corrections were determined manually using the {\it qphot} task in IRAF.
Reddening corrections are based on the dust maps of \citet{1998Schlegel} using
$R_V=3.1$\footnote{http://wwwmacho.mcmaster.ca/JAVA/Acurve.html}.
Completeness limits were determined based on artificial star tests using approximately $7\times 10^5$
artificial stars.
We imposed a $1/r$ density distribution for the artificial stars, avoiding the chip gap, so that there
are approximately equal number of stars in uniformly-spaced radial bins.
Completeness curves were then computed within circular annuli centered on each galaxy.
We imposed magnitude cuts at the 80\% completeness level in both of the F225W and F336W filters, and then
derived the accuracy of the recovery based on distance from the center of the galaxy per chip, as there
is a difference in sensitivity per UVIS chip. The 80\%
completeness limit changes as a function of radius, with sources farther from the center
of the galaxy being easier to detect due to the lower background.
This trend is demonstrated in Figure \ref{fig:completenessNGC4697}, which is the $80\%$ completeness limit as
a function of radius for NGC~4697, per filter and chip. The radial trend is weak outside the inner 200 pixels,
with only a 0.3 magnitude maximum variation, where 200 pixels corresponds to only $3.5\%$ of the area within
the effective radius of NGC~4697.
\begin{figure}
\plotone{f1.eps}
\caption{$80\%$ completeness limits as a function of radius for one of our targeted galaxies, NGC~4697, for
the F225W (blue) and F336W (red) filters. Chips are distinguished by solid (chip 1) and dashed (chip 2)
lines. While there is a radial trend in the ability to detect sources, it is only in the innermost region
of the galaxy that the variation is significant.}
\label{fig:completenessNGC4697}
\end{figure}
After completeness limits were applied, further cuts were made to the detected source list based on
output from {\sc dolphot}. First, elongated objects, objects that are too sharp, and extended objects
were omitted from the list (i.e., sources with object types greater than 2). All sources containing too
many bad or saturated pixels were also omitted (i.e., those with error flags greater than 3).
Last, we applied a sharpness cut of
$|\mathrm{sharpness}_{\mathrm{F225W}}+\mathrm{sharpness}_{\mathrm{F336W}}| \leq 0.4$, which we
determined via testing to be the most appropriate cut for selecting point sources.
Despite performing the photometry carefully, imposing strict completeness limits based on the
artificial star tests, and applying careful cuts based on {\sc dolphot} quality flags, some contamination from
extended and background sources was apparent in all four galaxy fields as well as all eight blank fields that
were used to estimate background contamination (\S\ref{bcf}). We
visually identified these obvious features, which stood out particularly in F336W, and then removed them by hand. We did not remove anything that
was ambiguous, i.e., we only removed sources that were clearly elongated and/or clearly part of a nearby background
galaxy whose bright, star forming regions were detected as individual clumps.
After removing these features, the remaining sources do not appear to correspond to any visible galaxies when
their positions are overlaid on optical images.
\subsection{Background Control Field}
\label{bcf}
While many background galaxies are easily identifiable by their extended structures, it is certain
that unresolved background galaxies and point-like AGNs contaminate our source lists. To estimate
the effect of this contamination in our fields, we searched MAST for WFC3 observations in the
F225W and F336W filters that had similar exposure times to our data, and used all available data
that are suitable as blank fields; eight fields from HST PID 11359 (Cycle 17; HUDFs 1-8). We
retrieved the pipeline-calibrated WFC3 F225W and F336W data for these fields from MAST and
photometered them identically to our four galaxies.
Although the blank fields were observed with the same filters and instrument, the
exposure times differ from those of our observations: each F225W blank field was observed for 5688s
(roughly 800s longer than our exposures) and each F336W blank field was observed for 2778s (roughly
2000s shorter than our exposures). The similar exposure times of the F225W images in the blank fields
and in our fields, which were used as reference frames in all cases, lends to the photometry being
performed quite similarly with {\sc dolphot}. As we are particularly interested in blue sources,
the depth in the redder filter is not as important, though it isn't particularly shallow, being only
0.3 magnitudes more shallow in F336W.
Identical cuts based on output from {\sc dolphot} were applied to the blank fields, including object
type, error flag, and sharpness. Likewise, all obvious extended sources that were not removed by these
cuts were removed by hand, as was done in each of our target fields.
We also imposed a magnitude
cut based on the artificial star tests that were run on our galaxies, to ensure that we did not include
any objects that were only visible due to the deeper F225W data. Aperture and reddening corrections
were applied as described in \S \ref{sec:photometry}.
We assume all remaining sources within the blank fields to be representative of the unresolved
background galaxy and point-like AGN population, as there is no reason to expect other contaminants such
as stars in these fields. We combined the source lists of all eight blank fields and scaled their total
area to the area of one WFC3 field to estimate the magnitudes and quantity of expected background
sources in one WFC3 field. While cosmic variance may play a role in the amount of background contamination,
we cannot account for these effects due to the small area of the sky covered by the eight fields in the HUDF
and the lack of similar data in other blank fields.
\section{UV-bright Sources}
\label{sec:population}
UV-bright sources are detected in all four of our targeted galaxies, though there is variation between the number
and positions of sources per galaxy. An example of several of these sources can be seen in
Figure~\ref{fig:imageszoom}, which displays a small portion of the observed NGC~3379 field
in both the F225W and F336W filters. Figure~\ref{fig:imageszoom} also demonstrates a noticeable
difference in the
surface brightness of NGC~3379 per filter, where the surface brightness in F225W is extremely
low, even compared to that of F336W. This demonstrates the importance of removing the background emission
surrounding individual sources carefully during the photometry process, as discussed
in \S \ref{sec:photometry}. It also demonstrates the usefulness of the F225W and F336W filters in searching
for young stars, as the optical filters would be even more dominated by each galaxy's surface brightness,
rendering this experiment impossible.
\begin{figure*}
\plottwo{f2a.eps}{f2b.eps}
\caption{A small portion of the smoothed HST WFC3 F225W (left) and F336W (right) images of NGC~3379. Several
UV-bright sources are detected in both filters (circles). For scale, the white, horizontal line in the
bottom, right corner of each image indicates $1\arcsec$ (the entire WFC3 field is $162 \times 162\arcsec$).}
\label{fig:imageszoom}
\end{figure*}
\subsection{Color-Magnitude Diagrams}
\label{sec:cmds}
Color-magnitude diagrams for all four targeted galaxies are presented in Figure~\ref{fig:cmds}.
Magnitude limits were determined via artificial star tests, and the deepest magnitude limits are indicated
on the CMDs by dashed lines.
The apparent magnitude limits, which are a function of radius, typically ranged from
$25.4 < \mathrm{F225W} < 25.7$ and $25.5 < \mathrm{F336W} < 26.3$. These
variations were also dependent on chip, as the UVIS chip 2 is more sensitive than chip 1 by 0.2 magnitudes.
The inner radial bin, with a radius of 100 pixels, had a much shallower limiting magnitude than
the other radial bins, being about 1 magnitude shallower than the deepest limit, except in NGC~4374 where
there is a lot of structure in the innermost bin, resulting in the inner bin being about 2 magnitudes
shallower. However, the vast majority of sources are not within the first radial bin as it encloses
very little area ($\sim 0.2\%$) of the entire WFC3 field. The absolute magnitude limit is strongly
dependent on
the distance to each of these galaxies, resulting in much deeper data for NGC~3379 and NGC~4697 than for
our other two targets, which accounts for much of the discrepancy in the number of detected sources for
each galaxy.
\begin{figure*}
\plottwo{f3a.eps}{f3b.eps}
\plottwo{f3c.eps}{f3d.eps}
\caption{CMDs of targeted galaxies, showing all detected sources with $\mathrm{F225W}-\mathrm{F336W} < 1$
(diamonds). Apparent magnitude is on the right y-axis while absolute magnitude is on the left y-axis.
The grayscale Hess diagram indicates the probability of a star being at a given location on the CMD assuming a
Salpeter IMF and constant SFR over the past 100~Myr, in units of $10^{-6}$ stars per bin of width 0.02 mag and
height 0.1 mag, and is based on stellar evolutionary tracks from {\it BaSTI}
\citep{2004Pietrinferni}. The choppy appearance of the Hess diagram is an artifact due to the interpolation between different
mass tracks. All sources with colors $>1$ are omitted from these CMDs as they are
predominantly globular clusters, which are beyond the scope of this study. Error bars indicate the
mean uncertainty, as estimated by {\sc dolphot}, for sources within bins of 0.5 in F225W magnitude.
Sources are detected in all of our targets, though there are more detected in the nearest targets.
Dashed lines show the deepest apparent magnitude limits. }
\label{fig:cmds}
\end{figure*}
Sources are overlaid on a grayscale Hess diagram, which indicates the probability of stars
at a given location on the CMD assuming a Salpeter initial mass function (IMF) and a constant star formation rate (SFR)
over the past 100~Myr. The Hess diagram is based on stellar evolutionary tracks from {\it BaSTI}
\citep{2004Pietrinferni}, assuming solar metallicity, solar alpha abundance, $\eta=0.4$ canonical models,
and a Salpeter IMF from 2 to 30 M$_\odot$ (we are limited at the high mass end by the absence of more
massive {\it BaSTI} tracks, and lower mass stars will not be observable with our data).
The numbers have been rescaled to account for lower-mass stars assuming a \citet{2003Chabrier} IMF.
As globular clusters emit in the UV and are commonly found in ellipticals, we determined the colors of
globular clusters in the F225W and F336W filters using the existing \citet{2005Dirsch} catalogue for NGC~4636,
where we matched the positions of optically identified globular clusters to detected sources in our sample.
Out of $\sim 80$ matched globular clusters, only 1 had a color blueward of $\mathrm{F225W-F336W} < 1$. We
therefore expect very little, if any, contamination from globular clusters within the color range of
$\mathrm{F225W-F336W} < 1$. Redward of $\mathrm{F225W-F336W} = 1$, however, the CMDs are dominated by globular
clusters, and so we limit the CMD color range to $\mathrm{F225W-F336W} < 1$ as the globular cluster
populations of these galaxies are beyond the scope of this study.
\subsection{Source Distribution}
WFC3 F225W images for each galaxy are shown in Figure \ref{fig:images}, where the positions of sources
in our sample (all sources with $\mathrm{F225W-F336W} <1$) are marked by colored dots.
Each color represents a different F225W$-$F336W range: blue represents all sources that are near the
main sequence ($-1.3 < \mathrm{F225W-F336W} < -0.7$), yellow represents all sources with colors to the red of
the main sequence ($\mathrm{F225W-F336W} > -0.7$), and green represents all sources to the blue of the main sequence
($\mathrm{F225W-F336W} < -1.3$); there are only 3 green points, all of which are in NGC~4374.
Many sources are concentrated around the centers of NGC~3379, NGC~4697, and NGC~4374, while there are
fewer sources near the center of NGC~4636. Also, most sources are not within the color range of the main sequence,
and instead have colors within $-0.7 < \mathrm{F225W-F336W} < 1$.
\begin{figure*}
\plottwo{f4a.eps}{f4b.eps}
\plottwo{f4c.eps}{f4d.eps}
\caption{Smoothed HST WFC3 F225W images of observed galaxies. Colored dots represent the
positions of UV-bright sources in our sample, and are color-coded by their $\mathrm{F225W-F336W}$ value:
blue dots represent sources that are near the main sequence ($-1.3 < \mathrm{F225W-F336W} < -0.7$), yellow
dots are sources with colors to the red of the main sequence, and green dots are sources to the blue of
the main sequence. The dot-dashed red annuli show the effective radius. UV-bright sources are detected in
all four galaxies, though there is variation between the number and position of sources per galaxy. For
example, many sources are concentrated around the centers of NGC~3379, NGC~4697, and NGC~4374, while there
are very few sources near the center of NGC~4636. The total WFC3 field is $162\times 162 \arcsec$. For
reference, the white, horizontal line in the bottom, right corner denotes $30~\arcsec$.
The sources that are visible on these images without overlaid dots did not meet our selection criteria.}
\label{fig:images}
\end{figure*}
\subsection{Radial Surface Densities}
Radial surface density profiles of the detected sources are given as a function of effective radius in
Figure~\ref{fig:radprofs}. The dotted, horizontal lines in each plot represent the mean density of
background sources determined from the background control field, the CMD for which is shown in Figure~\ref{fig:cmdhudfs}
and was determined by combining the eight blank HUDF fields. This background has
not been subtracted from the target galaxies CMDs in Figure~\ref{fig:cmds}.
The solid, curved lines represent de Vaucouleurs profiles \citep{1948deVaucouleurs} that are scaled by each galaxy's
corresponding $R_{\mathrm{eff}}$ and are arbitrarily scaled in amplitude for reference. Different F225W$-$F336W
ranges are represented by different colors, where blue represents sources that have colors expected of individual
young stars based on stellar evolutionary tracks, green spans colors expected of background galaxies
and AGNs (see Figure~\ref{fig:cmdhudfs}), as well as other potential objects that are off the main sequence, and red represents the
remaining sources with F225W$-$F336W redder than that expected from the background sample and main
sequence stars. The surface densities account for unobservable regions within each radial bin, such as the corners of
the fields and the chip gap.
\begin{figure*}
\plottwo{f5a.eps}{f5b.eps}
\plottwo{f5c.eps}{f5d.eps}
\caption{Radial surface density profiles of UV-bright sources within targeted galaxies (points). Horizontal dotted
lines are the mean density of background sources and solid curves are de Vaucouleurs profiles
\citep{1948deVaucouleurs} arbitrarily scaled in amplitude for reference. Different colors represent
different F225W$-$F336W ranges, where blue spans colors expected to be dominated by main sequence stars
($-1.3 \leq \mathrm{F225W-F336W} < -0.8$), green spans the range expected to be dominated by open clusters and
background galaxies ($-0.8 \leq \mathrm{F225W-F336W} < 0.3$), and red spans colors greater than these
two ranges ($0.3 \leq \mathrm{F225W-F336W} < 1.0$). Both the red and blue horizontal dotted lines are
at very low levels, while the green dotted lines are elevated, showing the background galaxy level.
}
\label{fig:radprofs}
\end{figure*}
\begin{figure}
\plotone{f6.eps}
\caption{CMD of background control field, determined by combining eight blank HUDF fields.
All sources are confined to colors between $-1.1$ and $0.8$, with most between $-0.8$ and $0.3$, suggesting that
the majority of contamination will be found within this range.}
\label{fig:cmdhudfs}
\end{figure}
The majority of sources are in the intermediate (``background'') color range. In both NGC~3379 and NGC~4697, the
surface density of sources in all three color ranges are above the mean density of background sources and reveal
that these sources are concentrated about the centers of these galaxies. These trends are also seen in NGC~4374, but
at lower significance. There is no clear evidence for an excess above the background in NGC~4636, except possibly for
the reddest sources at intermediate radius. This is in agreement with the impression given from Figure~\ref{fig:images},
where the sources do appear concentrated towards the center of the galaxy, especially those within the intermediate color
range. Some fraction of these UV-bright sources must therefore be associated with
the target galaxies and cannot simply be an overlap in background sources.
\section{Source Identification}
\label{sec:ids}
A variety of sources emit in the UV, including old, horizontal branch and p-AGB
stars, young stars, helium burning stars on the blue loop, open and globular clusters, AGNs, and
background galaxies. It is therefore crucial to determine the expected colors and magnitudes of
each of these sources and address the likelihood of their detection in our targeted galaxies,
which we describe in the following subsections.
As discussed in \S\ref{sec:cmds}, we have already eliminated globular clusters from our sample by restricting
our analysis to sources
with $\mathrm{F225W-F336W}<1$, as sources with F225W-F336W greater than $1$ are much too red to be
the young stars and open clusters that we are searching for. Globular clusters are therefore
omitted from the discussion below.
\subsection{Main Sequence Stars}
\label{sec:stars}
We used stellar evolutionary tracks from {\it BaSTI} \citep{2004Pietrinferni}, which were derived specifically
for WFC3 filters, to construct a Hess diagram assuming a Salpeter IMF and
constant SFR over the past 100~Myr. The tip of the main sequence is clearly visible in Figure~\ref{fig:cmds}, spanning
$-1.1 < \mathrm{F225W}-\mathrm{F336W} < -0.7$,
and is where young, hot stars would be expected on the CMD if
they were present in our targets. In all but one of the targeted galaxies there is at least one detected source
within this color range, making it entirely possible that they are young O or B stars. For reference, main sequence O stars range from $M_\mathrm{F225W}=-8.8$ to $-6.8$, while main sequence B stars rapidly become too faint to be detected (e.g., a B2V star has $M_\mathrm{F225W}=-4.8$).
It is clear in the CMDs, however, that the majority of sources are not located on the main sequence, but
instead are mostly
redder in color. To determine whether main sequence stars could be found at these colors, we estimated the
effect of metallicity on the location of the main sequence. We created Hess diagrams using $\alpha$-enhanced
models with $Z=0.0001$, $0.01$, $0.03$, and $0.04$. Higher metallicities indeed shift the evolutionary tracks
to the red, but the magnitude of the effect is small: e.g., using $Z=0.04$ ($\approx 2.5 Z_\odot$) instead
of solar metallicity results in a shift of the main sequence of +0.1 in color.
Another possibility that could account for the appearance of the sources being redder than that of the expected
location of the main sequence is the presence of dust. It is possible that there is dust at the center of some
or all of our targeted galaxies, and indeed is evident by obvious dust lanes that are visible in our data of NGC~4697 and NGC~4374.
However, these visible dust lanes are limited to the very central regions of the galaxy (within a radius of $7\arcsec$ for
NGC~4374, the galaxy with the most prominent dust features), and most of our targets are much farther from
this region. So, although some sources may appear redder due to dust, it is unlikely that this is the case for
all of the targets between $-0.7 < \mathrm{F225W}-\mathrm{F336W} < 1.0$.
We have also corrected for foreground reddening based on the dust maps of \citet{1998Schlegel}.
We conclude that if the sources lie on the grayscale of the Hess diagram in Figure~\ref{fig:cmds}, then
by definition the sources can be individual stars with masses of $<30$~M$_\odot$, but that none or very few of the
redder sources are individual stars.
It is possible, however, that they represent a population of open star clusters, a theory we investigate
in \S \ref{sec:clusters}.
\subsection{Star Clusters}
\label{sec:clusters}
If stars are forming in these nearby elliptical galaxies, it is likely that $70-90\%$ of the stars form
embedded in clusters instead of individually \citep{2003Lada}.
Although zero-age clusters are dominated by the most massive star, and therefore would lie on the grayscale in
Figure~\ref{fig:cmds}, they quickly evolve to redder colors and become fainter as they lose their bright O and
B stars.
A CMD for each galaxy is shown in Figure~\ref{fig:cmdsclustertracks} with evolutionary tracks of open clusters
overlaid. The evolutionary tracks were determined assuming a Salpeter IMF and using {\it BaSTI} stellar
evolutionary tracks \citep{2004Pietrinferni}, as a funtion of cluster mass, assuming that stars die instantly
once they evolve off of the main sequence.
The main sequence lifetimes, $\tau_{\mathrm{MS}}$, are assumed to follow:
\begin{equation}
\tau_{\mathrm{MS}} = 10^{10}~\left(\frac{M}{M_{\sun}}\right)^{-2.5}~\mathrm{yr}
\end{equation}
which is a good approximation to the lifetimes in Figure 4 of \citet{1992Schaller} over the
$30~\mathrm{Myr} \la \tau_{\mathrm{MS}} \la 2~\mathrm{Gyr}$ range of relevance.
Tracks for clusters with masses ranging from $10^2$~M$_\odot$ to $3\times 10^4$~M$_\odot$ are overlaid.
These CMDs have also
been corrected for background contamination, where points on the CMD were matched to the distribution
of sources in the background CMD, which was scaled to the area of one WFC3 field, have been removed for a clearer
picture of the intrinsic CMD. As this procedure is statistical in nature, it should not be used when
examining individual sources, but it gives an accurate picture of the effect of background contamination on the CMD.
Cluster tracks are the same for all four galaxies.
\begin{figure*}
\plottwo{f7a.eps}{f7b.eps}
\plottwo{f7c.eps}{f7d.eps}
\caption{CMDs with evolutionary tracks of clusters overlaid. Cluster tracks assume a Salpeter IMF and are derived from the
{\it BaSTI} stellar evolutionary tracks \citep{2004Pietrinferni}.
Tick marks indicate ages of 0, 100, 200, 400, 600, 800, 1000, 1200, 1400, 1600, 1800, and 2000~Myr, while dashed lines indicate
100 Myr (left) and 1 Gyr (right).
Magnitude limits restrict the age at which we can detect lower mass clusters.
These CMDs have been corrected for background contamination.}
\label{fig:cmdsclustertracks}
\end{figure*}
If the sources are open clusters, their distribution in the CMD reflects the cluster mass function, the rate
at which clusters traverse the CMD, and the recent star formation history (SFH). For a constant SFH, the
first two factors approximately cancel out at a given absolute magnitude: high mass clusters are produced
less often than low mass clusters, but evolve more slowly (and therefore ``pile up'' on the CMD) at a given
luminosity because they are later in their evolution. For example, if the cluster IMF slope is $-2$
\citep[e.g., see][]{2009Gieles}, clusters with masses between $10^2$~M$_\odot$ to $3\times 10^2$~M$_\odot$ are 10
times more numerous as $10^3$~M$_\odot$ to $3\times 10^3$~M$_\odot$ clusters, but evolve 14 times faster between
$-6 < \mathrm{F225W} < -5$, where most of the sources are located. We therefore expect star clusters to be
approximately uniformly distributed in color at a given magnitude, as observed.
We estimated the disruption timescale of the clusters using equation (9) in \citet{2005Lamers}, which scales
as cluster mass to the 0.62 power and as the inverse square root of the ambient density. This latter quantity,
which is dominated by the baryonic component at these small radii, we estimate using the \citet{1983Jaffe}
model, which provides a good model for elliptical galaxy light profiles. For a mass of $2\times 10^{10}$~M$_\odot$
and an effective radius of 2 kpc, typical of the target galaxies, the ambient density at the effective radius is
$\sim 0.05$~M$_\odot$~pc$^{-3}$, and the cluster disruption timescale is $>1$~Gyr for $10^4$~M$_\odot$ clusters,
and $\ga 100$~Myr for even $10^2$~M$_\odot$ clusters. We would therefore expect that clusters from present-day
star formation have not yet disrupted. Moreover, disruption is not instantaneous, and clusters in the process
of dissolving may well still appear as point sources at these distances.
To determine the radial extent of the sources, it was necessary to stack all sources to increase the
signal-to-noise. A variety of stacks were created spanning different color ranges to ensure that there was
no dependence of the radius on color; color had no effect on the determined radii. Stacks were scaled by
the inverse of the flux, making all sources weighted equally regardless of magnitude. We used Tiny Tim
HST PSF modeling software \citep{2011Krist} to determine the PSF for WFC3 F225W filter, resulting in a PSF
with $FWHM=1.7$. The stacked radius was more extended than the PSF, with a $FWHM=2.5$. The radius of the
stacked sources for the F336W filter was also more extended than the PSF derived using Tiny Tim. While
the minimum resolvable radius is 3.4~pc, we determine that the mean cluster radius is 5.0~pc (based on
the radius of the F225W cluster stack). This is consistent with the sources being open clusters; for example,
the median cluster radius in the Catalogue of Open Cluster Data \citep[COCD;][]{2005Kharchenko} is 4.3~pc.
\subsection{Post-Main Sequence Stars}
\label{subsec:pmss}
Post-main sequence stars, such as hot horizontal branch and p-AGB stars, contribute much of the UV light from
galaxies \citep{1999OConnell}.
Typical p-AGB stars detected in M32 and modeled from stellar evolutionary tracks by \citet{2008Brown} are too
faint by several magnitudes relative to our data to contaminate our sample. Theoretical considerations also make it
unlikely that the sources we detect are old p-AGB stars. The p-AGB evolutionary tracks of \citet{1994Vassiliadis}
have been overlaid on our deepest CMD in Figure~\ref{fig:cmdspagbtracks}, where the bolometric correction has been
computed assuming a blackbody spectrum. Although the colors of our sources are consistent with being p-AGB stars, the
magnitudes are not: only stars with initial masses $\ga 1.5$~M$_\odot$ could be bright enough to be the sources we
detect, which implies that they would have to belong to a reasonably young ($<4$~Gyr) population. Moreover, such
massive stars live very short lifetimes as p-AGB stars, as seen by the length between 100 yr tick marks on
Figure~\ref{fig:cmdspagbtracks}.
As these stars spend more time blueward of the main sequence than redward of it, the complete absence of sources blueward of the main sequence is inexplicable if they are p-AGB stars, but is a natural consequence if they are star clusters that evolve redward from locations slightly to the red of the main sequence. Horizontal branch stars are similar in
temperature to p-AGB stars, but significantly fainter, and so cannot be the sources we detect.
\begin{figure}
\plotone{f8.eps}
\caption{CMD of UV-bright sources in NGC~3379, our closest target and therefore deepest CMD.
Overlaid curves represent p-AGB evolutionary tracks from \citet{1994Vassiliadis}. Tracks are labelled according
to the stellar model initial mass, $M_i$, and the final core mass, $M_c$. Ticks are plotted every 100 yr.
A main sequence turnoff mass of $M_i=1.0$ corresponds to a 10~Gyr population, $M_i=1.5$ corresponds to an age
of $<4$~Gyr and brighter tracks are produced by turnoff stars from even younger populations.
The bottom track is therefore the only place we would expect to see a
population of p-AGB stars from the dominant old population, and our
sources are too bright to be these p-AGB stars. This
CMD has been corrected for background contamination.}
\label{fig:cmdspagbtracks}
\end{figure}
Helium burning stars on the blue loop are expected to have colors redder than the main sequence, though that
number should be low based on the grayscale probability distribution derived from stellar evolutionary tracks,
i.e., the blue loops are the regions seen in Figure~\ref{fig:cmds} between
$-0.5 \lesssim \mathrm{F225W-F336W} \lesssim 0.2$ at the magnitudes we are detecting.
The grayscale probability distribution clearly
illustrates that few stars are expected in this region, i.e., only the helium burners on the blue loop would
be in these color ranges, and even then they are so short-lived that they would be very unlikely. Also, if these
were blue loop stars, many more stars on the main sequence would be expected based on the Hess diagram, i.e., at
magnitudes brighter than $M_\mathrm{F225W}=-4.6$, we would expect to see 19 times more stars on the main sequence
than on the blue loop, though we only detect a few sources on the main sequence while detecting many sources
redward of this.
\subsection{Point-like Background Galaxies and AGNs}
Contamination from point-like background galaxies and AGNs is unavoidable. However, the amount of contamination
can easily be estimated by using a blank control field to get an estimate of the colors, magnitudes, and number
of sources expected in such a field. In section \ref{bcf} we described eight blank fields observed with
WFC3 using the same filters as our observations, which we photometered to derive a background sample.
After scaling the total area of the combined blank fields to the area of one WFC3 field, we determined
how many background sources would be expected on our CMDs and in what position on the CMD they would be in.
We then matched these background sources to the most likely counterpart in our CMDs based on their magnitudes
and colors, and then subtracted them to determine what the CMD corrected for background contamination would
be. The resulting CMDs of this process can be see in Figure~\ref{fig:cmdsclustertracks}, which demonstrates that
although a background sample has been removed from the CMD, there are still numerous sources within the CMD
that are unaccounted for, in each of our galaxies. Although galaxy clustering can change the magnitude of the
background counts, the radial dependence of the source distribution clearly demonstrates that the majority
of detected sources are contained in the target galaxy and are therefore not point-like background galaxies or
AGNs. An estimate of the variance in the background can be obtained by looking at the field-to-field
dispersion in the eight blank fields; the mean number of sources is 11.6, with a standard deviation of 5.8. In
contrast, the number of sources detected within the color range of our study in NGC~3379, NGC~4636, NGC~4697,
and NGC~4374 are 98, 31, 87, and 41, respectively.
\subsection{Surface Brightness Fluctuations}
\label{sec:sbf}
It is important to address the likelihood of false detections due to surface brightness fluctuations (SBFs),
which will not only result in an increased number of detections, but can also mimic a radially
concentrated population. Models by \citet{1993Worthey} determine the characteristic SBF magnitude in
F284W and the optical U band for a variety of old stellar populations. In the case where there is both an extended/blue
horizontal branch and a low-mass p-AGB star population contributing to the SBFs, the characteristic SBF
magnitude will be the luminosity-weighted mean luminosity of the populations. Using their low-mass p-AGB
case as a worst case scenario, $(\overline{\mathrm{F284W}}, \overline{\mathrm{U}}) = (1.2, 2.7)$, which, when converted
to the filters we are interested in, implies $(\overline{\mathrm{F225W}}, \overline{\mathrm{F336W}}) = (-0.3, 2.7)$.
As the deepest limit in our data is -4.5, this corresponds to nearly a $50\sigma$ fluctuation, which is extremely
unlikely, particularly over the WFC3 field of view. We therefore rule out the possibility that these sources
are surface brightness fluctuations.
\section{Results}
\label{sec:sfr}
As the detected sources are most likely individual main sequence stars and star clusters, we can deduce their
individual ages and masses from evolutionary tracks. The properties of our sample of UV-bright sources
for each galaxy are presented in Table~\ref{tab:basic}, where the galaxy ID is in column 1, right
ascension ($\alpha$) and declination ($\delta$) are in columns 2 and 3, apparent F225W and F336W
magnitudes are in columns 4 and 5, and color (F225W$-$F336W) is in column 6. The derived cluster ages
and masses are also presented in Table~\ref{tab:basic} in columns 7 and 8. Errors for the cluster
properties were determined by Monte Carlo sampling the error ellipse in the CMD around each cluster,
and represent the 16th and 84th percentiles of the Monte Carlo distribution ($1 \sigma$).
Because the clusters evolve faster at younger ages, the errors are asymmetric and there is expected
to be a net bias in the age and mass determinations. We estimate the effect of this by examining the
mean of the Monte Carlo error distribution. We find that the mean age after accounting for the
photometric error is $\sim 30$~Myr older than the input age for clusters younger than $1$~Gyr (the
method is not well defined for older clusters because the Monte Carlo points run off of the calculated
evolutionary tracks), and the recovered mean mass is larger than the input mass by a factor falling
from ~3 at $100$~M$_\odot$ to $\sim 1.5$ at $1000$~M$_\odot$ and is negligible at $10^4$~M$_\odot$. This
systematic error should be kept in mind when interpreting the cluster properties.
\begin{deluxetable*}{lrrccrcc}
\tabletypesize{\scriptsize}
\tablecaption{Properties of UV-Bright Clusters \label{tab:basic}}
\tablehead{
\colhead{Galaxy}&\colhead{$\alpha$}&\colhead{$\delta$}&\colhead{F225W}&\colhead{F336W}&\colhead{F225W$-$F336W}&\colhead{Mass}&\colhead{Age}\\
\colhead{}&\colhead{}&\colhead{}&\colhead{}&\colhead{}&\colhead{}&\colhead{($M_{\odot}$)} & \colhead{(yr)}}
\startdata
NGC~3379 & 10:47:43.078 & +12:35:20.59 & $25.04 \pm 0.19$ & $24.94 \pm 0.08$ & $0.10 \pm 0.21$ & $7.04^{+6.56}_{-3.78} \times 10^{3}$ & $7.06^{+3.24}_{-2.66} \times 10^{8}$\\
& 10:47:44.170 & +12:34:42.50 & $25.37 \pm 0.25$ & $26.03 \pm 0.19$ & $-0.67 \pm 0.32$ & \nodata & \nodata\\
& 10:47:44.325 & +12:34:34.47 & $25.03 \pm 0.19$ & $25.16 \pm 0.09$ & $-0.13 \pm 0.21$ & $3.07^{+3.67}_{-1.76} \times 10^{3}$ & $4.00^{+2.87}_{-1.49} \times 10^{8}$\\
& 10:47:44.539 & +12:34:48.99 & $24.98 \pm 0.18$ & $25.37 \pm 0.11$ & $-0.38 \pm 0.21$ & $1.17^{+1.69}_{-0.79} \times 10^{3}$ & $2.19^{+2.52}_{-0.99} \times 10^{8}$\\
& 10:47:44.621 & +12:34:21.47 & $25.48 \pm 0.30$ & $26.15 \pm 0.21$ & $-0.67 \pm 0.37$ & \nodata & \nodata\\
& 10:47:45.475 & +12:35:01.37 & $24.60 \pm 0.14$ & $24.44 \pm 0.05$ & $0.17 \pm 0.15$ & $1.30^{+0.70}_{-0.51} \times 10^{4}$ & $8.23^{+2.02}_{-1.98} \times 10^{8}$
\enddata
\tablecomments{
Table \ref{tab:basic} is published in its entirety in the electronic edition of the {\it Astrophysical Journal}.
A portion is shown here for guidance regarding its form and content.
Sources that lie outside the cluster evolution tracks do not have mass or age listed. Sources where the upper age error lies outside the cluster evolution tracks are listed as lower limits.
}
\end{deluxetable*}
Assuming that all star formation results in an extant cluster, we can add together the mass of clusters
within a given age range to estimate the star formation rate at that time.
Star formation rate as a function of cluster age is shown in Figure \ref{fig:sfrvsage}, where the
individual points represent the star formation rate contributed by clusters as a function of mass and the solid line
represents the total star formation rate per bin. The cluster age errors are typically $\lesssim 100$~Myr for
ages $<200$~Myr, rising to $200-400$~Myr for older clusters. We have therefore adjusted the bin sizes in
Figure~\ref{fig:sfrvsage} to be similar to the typical errors.
There is ongoing star formation in all four of the observed
galaxies, and there is variation between galaxies. As the data get shallower, there is a bias that lower mass
clusters are lost first. We have applied a completeness correction, which assumes that the cluster catalog is
complete to the mass where the cluster age tracks cross the typical magnitude limit for each galaxy, using a
cluster initial mass function that is a power law of slope $-2$ from $10^2$ to $10^6$~M$_\odot$
\footnote{The total mass has a very mild logarithmic dependence on the upper and lower limits
of the mass function, so the choice of these particular values does not introduce a large
uncertainty.}. As can be seen
in Figure~\ref{fig:sfrvsage}, applying the completeness correction does not have a dramatic effect on the implied
star formation rates, and we are therefore directly constraining the star formation rate with the observed clusters.
The derived SFRs take background contamination into account by subtracting those sources that best match
the distribution of the scaled control background field.
At ages $> 10^9$~yr, the colors of star clusters become more sensitive to cluster metallicity
than to cluster age --- changes of $0.1$~dex in metallicity become enough to move the clusters
into different age bins --- and therefore the ages of clusters we list as older than this
should be considered with care.
Similarly, the details of the star formation histories
beyond 1~Gyr are very uncertain.
As noted above, photometric errors cause a mean overestimate in cluster masses at the low mass end, resulting
in a tendency to overestimate the star formation rate contribution by those clusters. However,
only for the lowest-age bin in NGC~4636 do the lowest-mass clusters contribute a significant amount to
the total estimated star formation, so this should have a minimal effect on the derived star
formation rates. The mean age overestimate due to photometric errors is significantly smaller than
the size of the age bins, so it should not have a significant effect.
\begin{figure*}
\plottwo{f9a.eps}{f9b.eps}
\plottwo{f9c.eps}{f9d.eps}
\caption{Star formation rate as a function of cluster age. Individual points represent
the star formation rate contributed by clusters as a function of mass, while the solid line
represents the total star formation rate per bin. There is ongoing star formation in all four
of the observed galaxies, though more in some than in others. These have
been corrected for background contamination. The completeness
correction assumes that the cluster catalog is complete to the mass where the cluster age tracks
cross the typical magnitude limit for each galaxy, using a cluster initial mass function that is a
power law of slope $-2$ from $10^2$ to $10^6$~M$_\odot$.
Error bars are plotted for the ``Total'' line, and are based on bootstrap resampling of a Poisson-distributed
number of clusters within each age bin.
The relative errors on the completeness-corrected SFRs are identical.}
\label{fig:sfrvsage}
\end{figure*}
In three of the four galaxies, we detect current star formation at rates that range from
$1.0\times 10^{-6}$~M$_\odot$~yr$^{-1}$ for NGC~4636 to $8.0\times 10^{-5}$~M$_\odot$~yr$^{-1}$ for
NGC~3379 (note that because the completeness is different between galaxies, we quote completeness-corrected
values here, although the difference is minimal for the present-day SFR); in NGC~4374, the lack of
detected $<100$~Myr clusters corresponds to an upper limit of $1.15\times 10^{-6}$~M$_\odot$~yr$^{-1}$.
Star formation within the past Gyr is detected in all four galaxies,
with average rates somewhat larger, ranging from $2.9\times 10^{-4}$~M$_\odot$~yr$^{-1}$ for NGC~4374 up
to $4.6\times 10^{-4}$~M$_\odot$~yr$^{-1}$ for NGC~4697.
Encouragingly, these numbers are of the same order as the average star formation rates found by
\citet{2010DonovanMeyer}
for early-type galaxies that had no other evidence for star formation, based on total GALEX NUV emission. It
therefore seems likely that the emission they see is the unresolved sum of the sources we detect with HST.
The cluster mass function presents another projection of the cluster data and is shown in
Figure~\ref{fig:clustermassfunction} for the nearest galaxy in our sample, NGC~3379. The
data for the remaining galaxies are too shallow to draw strong conclusions on the mass
function, but they are consistent with what is seen in NGC~3379.
Because the mass function is complete to different masses at different ages, we have plotted
the mass function for clusters in three different age ranges, with the completeness limit shown in each case as
the vertical dotted line. The black diagonal line shows a power law of slope $-2$ for reference.
\begin{figure}
\plotone{f10.eps}
\caption{Cluster mass function for NGC~3379 for three different age ranges, with the completeness limit shown
in each case as the vertical dotted line. The black diagonal line shows a power law of slope $-2$ for reference.
}
\label{fig:clustermassfunction}
\end{figure}
One notable feature in Figures~\ref{fig:sfrvsage} and \ref{fig:clustermassfunction} is the lack of very massive
($> 3000$~M$_\odot$), very young ($< 300$~Myr) clusters. Such clusters would not be too
faint to be detected, and they should not have been disrupted. We can imagine several
potential explanations for this, however most are not viable upon further examination.
For instance, particularly luminous sources could have diffraction spikes that are
artificially broken up by the source finding algorithm; however, the successful
detection of more luminous sources at redder colors refutes this explanation.
Moreover, visual inspection of those few detections that dolphot removed for being
extended revealed all to be truly extended objects rather than diffraction spikes.
The systematic age overestimate from photometric errors also cannot be the explanation,
as this would be more important for low mass clusters than high mass clusters.
Another potential explanation is that there are no high mass clusters at any age and the
apparent old high mass clusters are actually blue loop stars. However, as argued in
Section~\ref{subsec:pmss}, the lack of observed main sequence stars precludes a large population of
blue loop contaminants. Furthermore, such a population would produce a peak in
Figure~\ref{fig:clustermassfunction} in the $3000-10000$~M$_\odot$, $200-600$~Myr bin, where no such
feature is seen. While it is true that the cluster evolution tracks are based on stellar
models that are poorly constrained in the UV, and could potentially be
systematically too blue resulting in systematically high age estimates,
this also cannot be the explanation because it would require the highest
mass stars to be significantly bluer than the lower mass stars.
The most likely explanation is that there is a physical reason why galaxies
in this sample have star formation that is not vigorous enough to
generate clusters more massive than $3000$~M$_\odot$. Given that the galaxies
are all very red ellipticals, this is entirely possible, particularly
as they have current star formation rates that are extremely low
even relative to their average over the past Gyr. This is augmented
by small number statistics -- at these low star formation rates, only
one very massive very young cluster per galaxy would be expected.
Figure~\ref{fig:sfrvsmass} shows how the current and recent star formation of the galaxies vary as a function of
their stellar mass, where the stellar masses were calculated from 2MASS LGA \citep{2003Jarrett} $K_{tot}$ magnitudes and the $M/L$
calibration of \citet{2001Bell} using the $V-K$ color from the RC3 $V_T$ magnitudes. The Gyr-averaged
SFRs are more
or less independent of stellar mass, while the current star formation appears to decline with mass.
Lines of constant specific star formation rate (SSFR) are overplotted. The SSFR values range from
$<8\times10^{-18}$~yr$^{-1}$ to $2\times10^{-15}$~yr$^{-1}$ at the present day, and $2\times10^{-15}$~yr$^{-1}$ to
$10^{-14}$~yr$^{-1}$ averaged over the past Gyr.
\begin{figure}
\plotone{f11.eps}
\caption{Star formation rate as a function of stellar mass for all four galaxies in our sample. Filled
symbols indicate the SFR over the past $100$~Myr, while open symbols are the average SFR over the past
Gyr. Gray lines of constant specific star formation rate are overlaid. The Gyr-aged SFRs are more
or less independent of stellar mass, while the current star formation appears to decline with mass.
}
\label{fig:sfrvsmass}
\end{figure}
\section{Discussion}
\label{sec:discussion}
This work presents the first derivation of star formation rates of true ``red and dead'' ellipticals via
direct detection of point sources in the UV using the WFC3 on HST. Our targets have distances ranging from
10.6 Mpc to the Virgo Cluster at 18.4 Mpc, and three were chosen because they demonstrated physical
evidence of possibly harboring young stars, in the form of PAH emission, OVI emission, and AGN activity,
while the fourth was selected as the control galaxy with no star formation indicator. The detected
sources are best explained as being individual star clusters with masses between $10^2$ and $10^4$~M$_\odot$
and ages less than $1$~Gyr: they are radially concentrated about the center of each galaxy, ruling out
background contamination; they are too bright to be p-AGB stars from the dominant old population and
too red as a population to be p-AGB stars from an intermediate-age population; and the absence of a large
number of bright main sequence turnoff stars precludes a large enough population of blue loop stars to
account for any but a tiny fraction of the observed sources. Our detections of young stars and star
clusters in all four of our targeted ellipticals confirms that low-level star formation is
ongoing in these ``red and dead'' galaxies.
All of the observed galaxies have line index ages \citep{2010Kuntschner,2006Sanchez,2000Trager} that
are uniformly very old. To estimate whether the very small amount of star formation we detect is consistent
with these ages, we compared the total F336W light coming from our detected sources to the total U-band flux
from RC3. For NGC~3379, where we go deepest down the young cluster mass function, only $0.03\%$ of
the U-band light within the effective radius is coming from our sources. It is therefore consistent with the
old ages determined spectroscopically. A current SFR that is $1$--$2$ orders of magnitude larger, which would
still be immeasurable by other techniques but is easily ruled out for the galaxies in this sample, would be
required to provide enough frosting to begin to contribute enough light at the age-sensitive wavelengths
to alter the derived ages. Our data are also consistent with the models of \citet{2010Rogers}, which were
performed to study the star formation history of NGC~4697 (amongst other ellipticals). Their results show that
a simple stellar population model has an equally good fit to the data as their best fit frosting model, meaning
that up to $3\%$ of the stellar mass in a younger component can be added without changing the fit. Our
derived value is consistent with this result, as the young population we detect makes up a much smaller percentage
than this.
The typical star formation rates of these galaxies are $\sim 10^{-5}$~M$_\odot$~yr$^{-1}$.
The two closest galaxies, NGC~3379 and NGC~4697, where we have the best measurements of the SFR, both show current SFR at a
similar level, despite the presence of PAH emission from one and not the other. The errors are much
larger for the more distant galaxies with OVI emission, NGC~4636 and NGC~4374, but they also show consistent results. This perhaps
indicates that such emission is a poor tracer of the small-scale minute amounts of cold gas that must
be present to form stars.
Comparison to the current stellar mass of the galaxies gives specific star formation rates of
$\sim 10^{-16}~\mathrm{yr}^{-1}$ at the present day, which implies that stars younger than $100$~Myr provide
a frosting that consists of $10^{-8}$ of the total stellar mass. The Gyr-averaged specific star formation rate
is somewhat higher, $\sim 10^{-14}~\mathrm{yr}^{-1}$, corresponding to $10^{-5}$ of the stellar mass.
There is no obvious structure to the spatial distribution of clusters beyond their general concentration
towards the center of each galaxy, or any obvious correlation between the locations of the clusters and
any other feature in the galaxy.
The star formation histories of the galaxies reveal a drop in the star formation within the past 300 Myr,
particularly in NGC~3379 and NGC~4697 where the data are the deepest and the sources are most obviously
physically associated with the galaxy. There are several possible origins for this, such as a contaminant
that we have not properly accounted for that pollutes the region of the diagram where older clusters lie,
though this explanation is highly unlikely.
Another possibility is that the cluster evolution tracks evolve too slowly in this region of
the CMD. More sophisticated population synthesis models in the WFC3 UVIS bands would be required to test
this hypothesis, which are not presently available.
Or perhaps our selection of galaxies, which was not random but specifically targeted old red galaxies with no known
star formation, picked out galaxies that are unusual quiescent at the present day. The youngest clusters are the
easiest to see, easier than old clusters,
but they are completely absent in 2 of the 4 galaxies. One of these galaxies, NGC~4374, hosts an AGN,
which may have recently heated the gas and made it unavailable for star formation. This hypothesis can be
tested with a larger and more representative sample of elliptical galaxies, which we hope to obtain.
These results provide a key observable for the amount of residual star formation in quenched early-type galaxies.
By comparing the rate at which stellar mass transitions across the ``green valley'' to the stellar mass function
of the red and blue galaxy populations, \citet{2007Martin} inferred a quenching timescale on the order of hundreds
of Myr. However, because these small levels of star formation have minimal impact on the global galaxy color, the
residual star formation had been unconstrained. These results will therefore provide useful input for models that
attempt to measure the ages of the stellar populations of galaxies based on
their spectral features. In particular, it would be interesting to see whether frosting models such as those of
\citet{2000Trager}, when using these SFHs, are able to reconcile the apparent discrepancy between the young ages
inferred by the spectra compared to the old ages at which the majority of stars must have been formed.
Future work includes a similar study of other nearby ellipticals and also lenticular galaxies,
searching for individual UV-bright young stars and open star clusters, to not only place tight constraints on
their low-level star formation rates and histories, but to also address the role of environment and
galaxy properties on star formation.
\acknowledgments This work has made use of {\it BaSTI} web tools. We thank Adriano Pietrinferni
for providing stellar evolutionary tracks for the relevant WFC3 UVIS filters that were not publicly
available. We also thank Jeremy Bailin, Eric Bell, and Sally Oey for providing valuable comments and
assistance. Support for program \# 11583 was provided by NASA through a grant from the Space Telscope
Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc,.
under NASA contract NAS 5-26555. The National Radio Astronomy Observatory is a facility of the
National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
|
1,108,101,566,753 | arxiv | \section{Introduction}
\label{sec:introduction}
Strong gravitational lenses are rare cases where a distant galaxy or quasar is aligned so closely with a foreground galaxy or cluster of galaxies that the gravitational field of the foreground object creates multiple, highly distorted images of the background object. These take the form of multiple images of a quasar and Einstein rings surrounding a galaxy and of highly distorted arcs seen through galaxy groups and clusters. The first strong lens was discovered in 1979 \citet{1979Natur.279..381W} and since then many valuable scientific uses have been found for them. They have been used to study how dark matter is distributed in galaxies and clusters \citep[e.g.][]{1991ApJ...373..354K,2001ApJ...554.1216C,2002ApJ...568L...5K,2003ApJ...587..143R,2003ApJ...583..606K,2005MNRAS.360.1333W,2005ApJ...623...31D,2009MNRAS.392..945V,2016MNRAS.463.3115T}. They have been used to measure the Hubble constant and other cosmological parameters
\citep[e.g.][]{1964MNRAS.128..307R,1992ARAandA..30..311B,2000ApJ...544...98W,2013ApJ...766...70S,2016AandARv..24...11T}.
Their magnification has been used as a natural telescope to observe otherwise undetectable objects \citep[e.g][]{2007ApJ...671.1196M,2017MNRAS.464.4823B,2016ApJ...833..264S}. They have put limits on the self interaction of dark matter and on the theory of gravitation \citep{0004-637X-606-2-819}. Through microlensing they have been used to study the structure of quasars \citep{2008ApJ...689..755M,2008ApJ...673...34P,2011ApJ...729...34B}. To access this wealth of information the first step is to find the lenses.
The Square Kilometer Array (SKA)\footnote{http://skatelescope.org/}, the Large Synoptic Survey Telescope (LSST)\footnote{https://www.lsst.org/} and the Euclid space telescope\footnote{https://www.euclid-ec.org/} are expected to increase the number of potential lenses by orders of magnitude \citep{collett_15,euclidSLWGwhitepaper,2015aska.confE..84M,2010MNRAS.405.2579O}. For example it is estimated that there will be approximately 200,000 observable galaxy-galaxy lenses in the Euclid data set among tens of billions of potential objects. So far, less than a thousand lenses have been found in total across many heterogeneous data sets. These surveys will bring a new era for strong lensing where large relatively well defined samples of lenses will be possible. It will also require handling much larger quantities of data than has been customary in this field.
Up to this point, the most widely used method for finding lenses in imaging surveys has been by visual inspection of candidates that have been selected on the bases of luminosity and/or colour. This has been done in the radio \citep{2003MNRAS.341...13B} and in the visible with space and ground based data \citep{2008MNRAS.389.1311J,2008ApJS..176...19F,2010AandA...517A..25S,2014MNRAS.439.3392P}. A related method pioneered by the SPACE WARPS project \citep{2016MNRAS.455.1171M,2016MNRAS.455.1191M,2015MNRAS.452..502G} has been to crowd source the visual inspection through an online platform. These efforts have proved very fruitful, but they deal with orders of magnitude fewer candidates and lenses than will be necessary in the future. Dealing with such large quantities of data will not be practical for any visual inspection approach. In addition, the efficiency and detection bias of human inspection methods are difficult to rigorously quantify.
Spectroscopic searches for galaxy scale lenses have also been done by looking for high redshift stellar lines in the spectra of lower redshift large galaxies. Notably this was done in the Sloan Lens ACS (SLACS) survey, producing a relatively well defined and pure sample of Einstein ring lenses \citep{2006MNRAS.369.1521W,2006ApJ...638..703B,2012ApJ...744...41B,2015MNRAS.449.3441S}. However, spectroscopy is telescope-time consuming and for the foreseeable future we are not likely to have spectroscopic surveys that cover anywhere near the number of objects as the planned imaging surveys.
Some automated algorithms have been developed in the past to detect lenses in images. These have been designed to detect arc-like
features \citep{2006astro.ph..6757A,2007AandA...472..341S, 2017AandA...597A.135B} and rings
\citep{2014ApJ...785..144G,2014AandA...566A..63J}. They have been applied to survey data and found of order 200 lenses \citep{2007AandA...461..813C,2012ApJ...749...38M,2016AandA...592A..75P}. Another approach has been to use a lens modeling code to fit a model to all the candidates and pick out the ones that fit the model well \citep{2009ApJ...694..924M,2017arXiv170401585S}.
More recently, machine learning techniques that have become widely used in the fields of computer image processing and artificial intelligence have been applied to this problem and others in astronomy. In particular, artificial neural networks (ANNs), support vector machines (SVM), and logistic regression. SVMs and logistic regression methods belong to the family of reproducing kernel Hilbert Space methods. They learn from a training set how to classify objects using features given by predefined kernel functions. ANNs (and a popular variant convolutional neural networks (CNNs) ) are even more flexible in learn directly from a training set which features are the most important for distinguishing categories of objects. These have been used widely for such tasks as handwriting and facial recognition. In astronomy, these families of algorithms are beginning to be used for categorizing galaxy morphologies \citep{2015MNRAS.450.1441D}, photometric redshifts \citep{2017MNRAS.465.1959C,2016PASP..128j4502S,2017NewA...51..169S}, supernova classification \citep{2016ApJS..225...31L} and the lens finding problem \citep{2017MNRAS.471..167J,2017arXiv170207675P,2017MNRAS.465.4325O,2017AandA...597A.135B,hartley2017support}.
Given the future of this field with large amounts of data coming soon and many new ideas emerging, it is timely to stage a series of challenges to stimulate new work, determine what can realistically be done in lens finding and get a better idea of the strengths and weaknesses to different methods. The long term goal is to get a set of algorithms that can handle Euclid, LSST or SKA data sets and produce high purity and high completeness lens samples with well defined efficiency or selection. We anticipate further challenges in the future in which the realism of the data simulations will become progressively better. Here we have chosen to concentrate on galaxy/small group scale lenses where the background source is a galaxy because we feel that this is where the most progress can be made and the scientific return is the highest, but QSO lenses and cluster/group lenses challenges may follow.
The paper is organised as follows. The form of the challenge and its rules are described in the next section. The methods used to simulate mock images of galaxies and gravitational lenses are described in section~\ref{sec:simulation}. In section~\ref{sec:entries}, each of the methods that were used to solve the challenge are briefly described. We discuss the metrics used to evaluate entries in section~\ref{sec:figure_of_merit}. The performance of each of the methods broken down in different ways is presented in section~\ref{sec:performance}. Finally, in section~\ref{sec:conclusion}, we conclude with a discussion of what was learned and how methods can be improved in the future.
\section{The Challenge}
\label{sec:challenge}
The challenge was in fact two separate challenges that could be entered independently. One was designed to mimic a single band of a future imaging data set from a satellite survey such as Euclid. The other was designed to mimic ground based data with multiple bands, roughly modeled on the Kilo-Degree Survey (KiDS)\footnote{http://kids.strw.leidenuniv.nl/} \citep{2013ExA....35...25D}. In neither case were the simulated images meant to precisely mock these surveys, but the surveys were used as guides to set noise levels, pixel sizes, sensitivities, and other parameters.
In each case, a training set of 20,000 images in each band was provided for download at any time along with a key giving some properties of the object including whether it was a gravitational lens. Each image was $101\times101$ pixels. The participants were free to download these sets and train their algorithms on them. To enter the contest, the participants needed to register with a team name at which point they would be given a unique key and the address of a test data set. These data sets contained 100,000 candidates. In the case of the multi-band ground-based set this was 400,000 images. The participants had 48 hours to upload a classification of all candidates consisting of a score between 0 and 1 -- 0 signifying the lowest confidence that it is a lens and 1 signifying the highest confidence that it is a lens. This ranking could have been a simple binary (0 or 1) classification or it could have been a continuous range of numbers representing the probability of being a lens or it could have been a finite number of confidence levels.
The challenge was opened on November 25, 2016 and closed on February 5, 2017.
\section{the simulations}
\label{sec:simulation}
Creating the mock images started with a cosmological N-body simulation. In this case it was the Millennium simulation \citep{2009MNRAS.398.1150B}. A catalog of dark matter halos within a light-cone is constructed within the Millennium Observatory project \citep{2013MNRAS.428..778O}. The challenge sets were based on a 1.6 sq.deg. light cone extending out to redshift $z=6$. The halos are found with a friends-of-friends algorithm and characterised by a total mass, size and half mass radius. They include subhalos of larger halos. The halo catalog is read into the GLAMER lensing code \citep{2014MNRAS.445.1942M,2014MNRAS.445.1954P} to do all the ray-tracing. Within the code a Navarro, Frenk \& White (NFW) \citep{1996ApJ...462..563N} profile is fit to the three parameters given above. The halos are projected onto a series of lens planes, in this case 20, and the deflection angle at any point in each plane can be calculated by summing the effects of all the halos with a hybrid tree method. In this way the halos have the mass, concentration and clustering properties from the N-body simulation, but within each strong lens the mass resolution is not limited by the original simulation, but follows the analytic mass profile. An additional mass component that will be discussed later is added to each halo to represent the stellar mass.
With GLAMER we identify and map out all the caustics within the light-cone for 33 source planes -- z=1 to 3 in intervals of 0.1 and 3 to 6 in intervals of 0.25. We take every caustic that corresponded to a critical curve with an Einstein radius larger than 1.5 times the resolution of the final images. The Einstein radius is estimated here and in all that follows as $R_{\rm ein}=\sqrt{A_{\rm ein}/\pi}$ where $A_{\rm ein}$ is the angular area within the critical curve.
In the light cone there are many thousands of caustics for the higher source redshifts. These lenses could be used as is, but we wanted to produce a much larger number with more randomness. For each caustic we identify the lens plane with the highest convergence and identify all the halos within a three dimensional distance of 0.5 Mpc from the center of the critical curve and on this and its neighboring lens planes. This collection of halos is then used as the lens and rotated to produce more random lenses. It contains all the sub-halos and nearby companion halos, but not the large scale structure surrounding it.
To model the background sources that are lensed we use sources from the Hubble Ultra Deep Field (UDF) that have been decomposed into shapelet functions to remove noise. This is the same set of images as used in \citet{2008AandA...482..403M,2010AandA...514A..93M,2016arXiv160604548M}. There are 9,350 such sources with redshifts and separate shapelet coefficients in 4 bands.
To construct a mock lens, first a caustic on the highest redshift source plane is selected. This is done in order of Einstein area, but all the critical curves are used more than once. Since every lens with a caustic at a lower redshift will have a caustic at the highest redshift this is a selection from all of the caustics in the light-cone. The lens is extracted as explained above and rotated randomly in three dimensions. A source is selected at random from the shapelet catalog subject to a magnitude limit in a reference band. The redshift of the UDF source is used as the source redshift. If the source is at a lower redshift than the lens or within $\Delta z = 0.4$ another random source is selected.
The furthest point in the caustic is found from its own center and the source is placed randomly within 3 times this distance. This is a somewhat arbitrary length designed to be a compromise between producing only very clear strong lenses because all the sources are right in the center of the caustic and making the process inefficient because most of the sources are too far away from the caustic to produce clear lenses. If the source positions were taken completely at random the fraction of clear lenses would be very low.
The visible galaxies associated with the lens must also be simulated. There are too few bright galaxies in the UDF catalog to make enough mock lens galaxies for this purpose. Instead, for most of the lenses, we used an analytic model for the surface brightness of these galaxies. The Millennium Observatory provides parameters for the galaxies that inhabit the dark matter halos using the semi-analytic galaxy formation models of \citet{2011MNRAS.413..101G}. The parameters used here were the total magnitude, the bulge-to-disk ratio, the disk scale height and the bulge effective radius. The magnitude and bulge-to-disk ratio are a function of the pass band. Each galaxy is given a random orientation and inclination angle between 0 and $80^{\circ}$. The disk is exponential with no vertical height which is why the inclination is limited to $80^{\circ}$. The bulge is represented by a S\'{e}rsic profile with an axis ratio randomly sampled between 0.5 and 1. The S\'{e}rsic index, $n_s$, is given by
\begin{align}
\log( n_s )= 0.4 \log\left[ {\rm max}\left(\frac{B}{T},0.03\right)\right] + 0.1 x
\end{align}
where $\frac{B}{T}$ is the bulge to total flux ratio and $x$ is a uniform random number between -1 and 1. This very approximately reproduces the observed correlation between these quantities \citep{2001AJ...121..820G}.
In addition to the basic disk and bulge models we introduce some spiral arms. The surface brightness of
the disks are given by
\begin{align}
S(\theta,r) = e^{-r/R_h} \left( 1 + A \cos(N_a\theta + \phi_r \right) ~, \\
\phi_r = \alpha\log(2 r/ R_h) + \phi_d \nonumber
\end{align}
where $R_h$ is the scale height of the disk. The phase angle of the arms, $\phi_d$, is chosen at random. The parameters $A$, $\alpha$ and $N_a$ are chosen from distributions that are judged by eye to produce realistic
galaxies.
The bulge is also perturbed from a perfect S\'{e}rsic profile by multiplying the surface brightness by
\begin{align}
1+\sum_{n=1}^6 a_n \cos\left( n \theta + \phi_n \right)
\end{align}
where $\phi_n$ is a random phase. The coefficients are picked randomly from between -0.002 and 0.002.
These foreground galaxies are rotated in three dimensions with the halos of the lens each time a random lens is produced so that they remain in the same positions relative to the mass. All the random parameters are also reassigned with every realization of the lens.
These images of the of foreground galaxies are not intended to reproduce the true population of galaxies, but only to be sufficiently irregular to make them difficult to fit to a simple analytic model that might make them very easy to distinguish from a foreground plus a lensed image. As will be discussed later, more realistic models will be needed in the future and are a subject of current investigation.
To represent the mass of the galaxies we make a gridded map of the surface brightness at 3 times the resolution of the final image. The surface brightness map in converted into a mass map within GLAMER by assuming a uniform mass-to-light ratio of 1.5 solar in the reference band. These mass maps are added to the NFW dark matter halos discussed before to make the total lens mass distribution. The deflections caused by the mass maps are calculated by Fast Fourier Transform (FFT) and added to the halos' deflections for the ray tracing.
The code is able to produce any combination of foreground galaxies, lensed image and noise that is desired. For the training set we provided an image of the total lens with noise, an image of the foreground galaxies with noise and image of the lensed background source without noise.
For the test sets only the final images were provided to participants although all the information was stored for analysing the entries.
\subsection{Space-based}
\label{sec:sim-space-based}
The space-based datasets were meant to roughly mimic the data quality which is expected from observations by the Euclid telescope in the visible channel. To this end, the pixel size was set to 0.1 arcsec and a Gaussian PSF was applied with a FWHM of 0.18 arcsec. The Gaussian PSF is clearly a simplified model, but a realistic treatment of the PSF is outside the scope of this paper. The reference band for background and foreground galaxies was SDSS $i$, which is overlapping with the broader Euclid VIS band. The realisation of the mock images followed the same procedure described in \citet{2004PASP..116..750G} and \citet{2008AandA...482..403M}. As a result, the noise follows a Gaussian distribution with a realistic width and is uncorrelated between pixels. Characteristics of the instrument, filter and exposure times were taken from the Euclid Red Book \citep{2011arXiv1110.3193L}.
In the challenge set the limiting magnitude for background sources to 28 in $i$. 60\% of the cases had no background source and were thus labeled as non-lenses.
\subsection{Ground-based}
\label{sec:sim-ground-based}
For the ground-based images four bands (SDSS $u,g,r$, and $i$) where simulated. The reference band was $r$. For the challenge set 85\% of the images where made with purely simulated images as outlined above and the other 15\% used actual images taken from a preliminary sample of bright galaxies directly from the KiDS survey. Lensed source images where added to these real images at the same rate as for the mock images, in this case 50\%. No attempt was made to match the halo masses to the observed galaxies in these cases. These real images where added for more realism and so that, by comparing the results for real and mock images, we can evaluate how realistic our simulations are in this context. There were about 160,000 of these stamps from KiDS.
The KiDS survey provided a representative PSF map in each band that was applied to all mock images. The pixel size in this case was 0.2~arcsec.
Weight maps for the KiDS images were also provided. Some of these had masked regions from removed stars, cosmic rays, and bad pixels. For the mock images the noise was simulated by adding normally distributed numbers with the variance given by the weight maps. The weight maps were also randomly rotated and flipped. This resulted in many of the images having large masked regions in them.
By chance one of the original KiDS images appears to have been a lens. When an additional lensed source was added this made a double lens or "jackpot" lens \citep{2008ApJ...677.1046G}.
\section{The Entries}
\label{sec:entries}
\begin{table*}
\centering
\begin{tabular}{rlll}
\hline
& Name & type & authors \\
\hline
1 & AstrOmatic & Space-Based & Bertin \\
2 & GAHEC IRAP & Space-Based & Cabanac \\
3 & CAS Swinburne Melb$^\dagger$& Ground-Based & Jacobs \\
4 & ALL-star & Ground-Based & Avestruz, N. Li \& Lightman \\
5 & Manchester1 & Space-Based & Jackson \& Tagore \\
6 & CMU-DeepLens-Resnet-Voting & Space-Based & Ma, Lanusse \& C. Li \\
7 & Manchester SVM & Ground-Based & Hartley \& Flamary \\
8 & CMU-DeepLens-ResNet & Space-Based & Francois Lanusse, Ma, C. Li \& Ravanbakhsh \\
9 & CMU-DeepLens-Resnet-Voting & Ground-Based & Ma, Lanusse \& C. Li \\
10 & YattaLensLite & Space-Based & Sonnenfeld \\
11 & NeuralNet2 & Space-Based & Davies \\
12 & CAST & Ground-Based & Roque De Bom, Valent\'{\i}n \& Makler \\
13 & CMU-DeepLens-ResNet-ground3 & Ground-Based & Lanusse, Ma, Ravanbakhsh \& C. Li \\
14 & GAMOCLASS & Space-Based & Huertas-Company, Tuccillo, Velasco-Forero \& Decenci\`{e}re \\
15 & LASTRO EPFL (CNN) & Space-Based & Geiger, Sch\"{a}fer \& Kneib \\
16 & Manchester SVM & Space-Based & Hartley \& Flamary\\
17 & CMU-DeepLens-ResNet-aug & Space-Based & Ma, Lanusse, Ravanbakhsh \& C. Li \\
18 & LASTRO EPFL & Ground-Based & Geiger, Sch\"{a}fer \& Kneib \\
19 & CAST & Space-Based & Bom, Valent\'{\i}n \& Makler \\
20 & AstrOmatic & Ground-Based & Bertin \\
21 & All-now & Space-Based & Avestruz, N. Li \& Lightman \\
22 & Manchester-NA2 & Ground-Based & Jackson \& Tagore \\
23 & YattaLensLite & Ground-Based & Sonnenfeld \\
24 & Kapteyn Resnet & Space-Based & Petrillo, Tortora, Verdoes Kleijn, Koopmans \& Vernardos \\
\hline
& & &$\dagger$ No description was submitted for this paper.
\end{tabular}
\caption{Entries to the challenges.}
\label{table:entries}
\end{table*}
\section{lens finding methods}
\label{sec:methods}
There were 24 valid entries into the challenge. They are listed in Table~\ref{table:entries}. There were a variety of different methods used and participants come from a variety of different backgrounds, most were professional astronomers, but there were also entries from researchers outside of the field.
The following sections contain short descriptions of the lens finding methods that were used in
the challenge. Each subsection refers to a team which gave a separate entry. We have grouped the
methods into four categories according to the type of method used.
\subsection{Visual Inspection}
\subsubsection{Manchester/Manchester1 (Jackson,Tagore)}
All images (a total of 100000) were examined for each of the space- and
ground-based datasets. This was done by two observers; AT (Amit Tagore) examined 30000
images in each case and NJ (Neal Jackson) examined 70000. Observation was carried out
over a 48-hour period, at the rate of 5000/hr (NJ) and 2500/hr (AT).
The overall results, in terms of ROC curves, were very similar for
both observers. The space-based challenge produced areas of 0.800 and
0.812 for NJ and AT respectively, and the ground-based challenge yielded
0.891 and 0.884.
The Python scripts used for manual examination of multiple images are available
on GitHub\footnote{https://github.com/nealjackson/bigeye} and are described in more detail
in \citet{hartley2017support}. For one-colour data such as
the space-based training set, the images are individually colour-scaled using
square-root scaling. The bright limit of the colour-scale is determined
from the pixel values in a rectangle comprising the inner ninth of the
image area, with the limit being chosen as the $n$th centile of the pixel
values in this area. Values between $n=95$ and $n=98$ give optimum results,
judging by experiments on the training set. The number of images in each
grid was also optimised using the training set, with 16$\times$8 or
8$\times$4 giving good results on one-colour data. For three-colour data,
such as the ground-based challenge data, the individual bands for each
object are colour-scaled and then combined into an RGB image. In this case
8$\times$4 grids were used for examination, due to the generally lower
resolution of the images. The script also allows the user to adjust the
colour-scale in real time when examining and marking images, and records
the image name corresponding to the image within which the cursor resides
at the time any key is pressed, together with the key.
Images were classified by both observers into 5 categories, ranging from
0 (no evidence of any lensed objects in the image) to 4 (certain lenses).
For both observers, the rate of false positives in the ``certain'' lenses
was between 0.1\% and 0.3\%. The exception was the ground-based imaging
for one observer, where a 4.6\% rate resulted mainly from a
single decision to allow a false-positive ``double lens'' which occurred
repeatedly throughout the data at different orientations. The false-negative
rate among the class-0 identifications was similar for both observers, at
around 25\% for the space-based images and 20\% for the ground-based.
\subsection{Arc-Finders}
These methods seek to identify gravitationally lensed arcs and differentiate between them and
other objects such as spiral arms and edge on spirals using their width, color, curvature and other pre-selected criterion.
\subsubsection{ IRAP (Cabanac)}
\begin{figure}
\includegraphics[width=\columnwidth]{figures/arcmethod.pdf}
\caption{ (GAHEC IRAP) From top-left to bottom right, 1) a simulated arc extracted from SL challenge in which an tuned Arcfinder selects 3 candidates (green circles), 2) the smoothed image on which pixel wise elongation is computed, 3) the resulting elongated pixels after threshold, 4) the set of pixels selected for the computation of arc candidate properties. }
\label{fig:Cabanac}
\end{figure}
Arcfinder \citep{2006astro.ph..6757A,2007AandA...461..813C,2012ApJ...749...38M} illustrated in Figure \ref{fig:Cabanac}, is a fast linear method that computes a pixel wise elongation parameter (ratio of first-order moments in a n-pix window oriented in proper reference frame) for all pixels of mexican-hat-smoothed FITS images. Arcfinder then extracts contiguous pixels above a given background and computes the candidate arc's length, width, area, radius of curvature and peak surface brightness. A final thresholding is set to maximize purity over completeness on a few typical arcs of the dataset.
For the current SL challenge, arcfinder was tuned to detect long and narrow arcs, and was optimized on a subset of 1000 simulated images with a grid covering a range of elongation windows and arc areas. A python wrapper allows users to change parameters in a flexible way and run the arcfinder C code as a linux command line. Arcfinder took a couple of hours to run on the entire dataset with some overheads due to the dataset format. The code is publicly available at https://github.com/rcabanac/arcfinder.
\subsubsection{YattaLens Lite (Sonnenfeld)}
YattaLensLite is a simpler version of the algorithm YattaLens \citep{2017arXiv170401585S}, modified to meet the time constraints of the challenge.
YattaLensLite subtracts a model surface brightness profile describing the lens galaxy from the $g$-band image, then runs SExtractor to detect tangentially elongated or ring-shaped objects, which are interpreted as lensed images.
In the ground-based challenge, the model lens surface brightness profile is obtained by taking a rescaled version of the $i$-band image.
The difference in color between lens and source usually allows the lensed images to still be detectable after the lens subtraction process.
However, in order to avoid subtracting off the lensed images in systems with similar colors between lens source, we radially truncate the model lens surface brightness.
The model lens light is truncated at the smallest radius between the position where the surface brightness is comparable to the sky background level, or the position of a positive radial gradient in surface brightness, if detected.
In the space-based challenge, it is not possible to separate lens and source based on color, because only data in one band is provided. The lens light model then is produced by taking a centrally-inverted image and then using the same truncation prescription used with ground-based data. The central inversion step is taken to reduce the chances of subtracting flux from lensed images, which are in general not centrally symmetric as opposed to typical lens galaxies.
In the full version of YattaLens, a lens modeling step is performed to improve the purity of the sample. However, such a procedure is too time consuming and was not performed in this challenge.
\subsection{Machine Learning Methods that use pre-selected features}
These are methods that classify the objects by making linear or nonlinear boundaries in a feature space. The features are properties of the image and are typically chosen by the user with a combination of knowledge, intuition, and trial-and-error. Using the training set, the optimal boundaries are found according to a criterion that depends on the method. The machine learns how to use the features best for distinguishing between lenses and non-lenses.
\subsubsection{Gabor-SVM (Hartley, Flamary)}
A Support Vector Machine (SVM) is a supervised machine learning method which uses labeled training data to determine a classification model (see e.g., \citet{vapnik79estimation}, \citet{Cortes1995} and \citet{Burges1998}). A preprocessing stage first extracts a set of useful features from input samples, before projecting each sample as a vector into a high-, possibly infinite-dimensional space. The model then separates classes of data by maximising the margin between a defining hyperplane and a set of so-called support-vectors at the inner edge of each class. The process of classification is computationally inexpensive since the optimisation depends only on the dot products of the support vector subset. Feature extraction, however, requires both an extensive exploration of the feature space during the development of a model, and potentially intensive computer resources in order to transform the original samples. Our full method is described in detail in \citet{hartley2017support} and was developed using the Python scikit-learn and scikit-image packages \citep{scikit-learn,scikit-image}.
During our development of an SVM classifier for lens finding, feature extraction initially involved the decomposition of each image into a set of objects, using SExtractor \citep{1996AandAS..117..393B} and GALFIT \citep{2002AJ...124..266P} to recover and subtract objects iteratively. This method had previously been used in a static algorithm approach which assigned points according to the morphological properties of each image \citep[see][]{2014AandA...566A..63J}. Lensed-like objects displaying, for example, greater ellipticity and tangentiality were awarded more points. Since the SVM operates in a fixed dimensional space, properties of individual objects were collapsed into a fixed set describing the mean and variance of morphological properties of all the objects within an image. After training an SVM using these features we recorded a modest separation of lens and non-lens classes.
An alternative approach was to design a set of Gabor filters to be applied to each sample. The Gabor kernel is described by a sinusoidal function multiplied by a Gaussian envelope. We discard the imaginary part of the function to leave, in two-dimensional space:
\begin{equation}
G_c[i,j]=Be^{-\frac{(i^2+j^2)}{2\sigma^2}} \mathrm{cos}\left(\frac{2\pi}{\lambda} (i\, \mathrm{cos} \, \theta + j\, \mathrm{sin} \,\theta)\right),
\end{equation}
where harmonic wavelength $\lambda$, Gaussian spread $\sigma$ and orientation $\theta$ define the operation performed on each point $i,j$ in an image. Such a kernel is a popular image processing choice for edge detection and texture classification \citep[e.g.][]{Feichtinger98a,Springer-verlag97computationalmodels} and is thought to mimic some image processing functions of the mammalian brain \citep{Jones1233}.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{figures/polarfilter.pdf}
\caption{Example of our feature extraction procedure used to transform a ring. The image on the right shows the response of a set of Gabor filters after convolution with a polar transformed image of an Einstein ring. The strongest response is seen in the orientation perpendicular to the radial direction and at the frequency most closely matching that of the ring.}
\label{gaborring}
\end{figure}
Our final feature extraction procedure first applied a polar transform to each image in order to exploit the edge detection of the Gabor filter, picking out tangential components typical of galaxy-galaxy lensing. Each image was then convolved with several Gabor filters of varying frequency and rotation (see Fig.~\ref{gaborring}). Stability selection methods were used to investigate the classification performance using different combinations of filters. The responses for each Gabor filter when applied to each image were measured by calculating statistical moments for each filtered image. These moments formed our final input data on which the SVM could be trained and applied. We used brute-force optimisation methods to select a non-linear SVM containing radial basis function (RBF) kernel and tuned a small set of regularisation hyperparameters to achieve good generalisation performance. During training and testing, our final and best scores achieved when testing on the lens finding challenge training data were an AUC of 0.88 for the space set and 0.95 for the ground set. Classification was performed using a modest desktop PC.
\subsubsection{ALL (Avestruz, Li, Lightman)}
We detail the ALL team methodology in \citet{avestruz_etal17}. The
pipeline was originally developed to automatically classify strong
lenses in mock Hubble Space Telescope and Large Synoptic Sky Telescope
data generated with code described in \citet{li_etal16} and
\citet{collett_15}. We apply the same exact steps for the single-band
data for Euclid, but modify the feature extraction step for the
four-band KIDS data. We summarize the steps below.
We use tools from {\em Scikit-learn} \citep{pedregosa_etal12} and
perform minimal image preprocessing. First, we replace masked pixels
with the average of surrounding pixels, then enhance contrast in the
image by taking the normalized log of pixel values. The next step of
our pipeline consists of a feature extraction stage, where our feature
vector is a {\em histogram of oriented gradients} (HOG)
\citep{dalalandtriggs_05} that quantifies edges in the image. HOG has
three main parameters that determine the binning and resolution of
edges captured by the features. The result is a one dimensional
feature vector corresponding to the magnitude of oriented gradients
across the image. With the KIDS data, we extract a feature vector for
each of the four bands and concatenate the vectors to create a final
feature vector for each object that we use to train a model
classifier.
We use {\em Logistic Regression} (LR), a machine learning algorithm,
to train a classifier model. LR requires a parameter search over the
regression coefficient, $C_{LogReg}$. The parameters from both the
feature extrator, HOG, and the linear classifier, LR, contain
parameters that we optimize for peak model performance. We use {\em
GridSearchCV} from {\em Scikit-learn} to select cross-validated
parameters for HOG parameters and a subset of $C_{LogReg}$ values with
20\% of the test images provided. We then run a finer parameter
search over $C_{LogReg}$, splitting the test images into 80\% training
and and 20\% test to avoid overfitting the data. We use the best
parameters to then train the entire dataset for the final model
classifier that we used to evaluate the competition data.
\subsection{Convolutional Neural Networks}
Since Convolutional Neural Networks (CNNs) are central to many of the methods that will be described later, here we provide a brief general description of them.
A CNN \citep{Fukushima1980,Lecun1998} is a multi-layer feed-forward neural network model, which is particularly well-suited for processing natural images. With the very recent advances of the \textit{Deep Learning} framework \citep{Lecun2015}, models based on CNN architectures have reached or even surpassed human accuracy in image classification tasks \citep{He2015a}.
The fundamental building block of a CNN is the \textit{convolutional layer}. This element applies a set of convolution filters on an input image to produce a series of so-called \textit{feature maps}. The coefficients of these filters are free parameters that are learned by the model. The notation
$n \times n - n_c$ to denote a convolutional layer using filters of size $n \times n$ pixels and outputting $n_c$ feature maps will be used. In typical architectures, the size of these convolution filters is kept small (i.e. 3x3 or 5x5 pixels) to limit the complexity of the model.
Similarly to conventional fully-connected neural networks, convolution layers are typically followed by an element-wise activation function, which allows for the modeling of complex functional forms by introducing non-linearities in the model. Classical choices for activation functions include the sigmoid-shaped logistic function (or just sigmoid) $f(x) = 1/ (1 + \exp(-x))$ or the hyperbolic tangent function $f(x) = \tanh(x)$. However, much of the success of Deep Learning is due to the introduction of activation functions that do not saturate (become very close to one with very small derivatives), allowing for the efficient training of very deep architectures. The most common choice in modern deep learning models is the simple ReLU activation (for rectified linear unit) \citep{Nair2010} defined as $f(x) = \max(x, 0 )$. A closely related common alternative is the ELU activation (for Exponential Linear Unit) \citep{2015arXiv151107289C} defined as
\begin{equation}
f(x ) = \begin{cases}
x & \text{if } x\geq 0\\
e^{x } - 1, & \text{otherwise}
\end{cases}\;,
\end{equation}
which often leads to better results in practice.
Because the filters used in convolution layers are typically just a few pixels in size, to capture features on larger scales, CNNs rely on a multi-resolution approach by interleaving convolutional layers with \textit{pooling layers}, which apply a downsampling operation to the feature maps. The most common downsampling schemes are the max-pooling and average pooling strategies, which downsample an input image by taking respectively the maximum or average values within a given region (e.g. 2x2 patches for a downsampling of factor 2).
A CNN architecture is therefore a stack of convolution layers and pooling layers, converting the input image into an increasing number of feature maps of progressively coarser resolution. The final feature maps can capture information on large scales and can reach a high-level of abstraction. To perform the classification itself from these feature maps, the CNN is typically topped by a fully-connected neural network outputting the class probability of the input image.
For a binary classification problem such as the one involved in strong lens detection, the training is performed by optimizing the weights of the model as to minimize the \textit{binary cross-entropy} objective:
\begin{equation}
- \sum_{n=1}^{N} y[n] \log \hat{y}[n] + (1 - y[n])\log(1 - \hat{y}[n]) \;,
\end{equation}
where $N$ is the number of training instances, $y \in \{0,1\}$ is the true class of the image and $\hat{y} \in [0,1]$ is the class probability predicted by the model. This optimization is usually performed by a Stochastic Gradient Descent (SGD) algorithm or its variants (e.g. ADAM \citep{Kingma_2014}, Adagrad \citep{duchi12adaptive}, RMSprop \citep{Tieleman12RMSProp}, or accelerated gradients \citep{nesterov83method}). SGD updates the model iteratively by taking small gradient steps over randomly selected subsamples of the training set (so called \textit{mini-batches}). All the CNN-based methods presented in this work rely on the ADAM optimisation algorithm, which also uses past gradients from previous iterations to adaptively estimate lower-order moments. Empirically it has been found that in many problems ADAM converges faster than SGD \citep{Ruder16overview}.
Neural networks often suffer from overfitting to the training set. A common way to mitigate this is to use a regularisation scheme. For example, the Dropout regularisation technique \citep{2012arXiv1207.0580H,JMLR:v15:srivastava14a}, were a certain percentage of the neurones and their connections are randomly drop from the neural network. This regularisation techniques reduces overfitting by preventing complex co-adaptations of neurones on training data.
Training multi-layer neural networks with gradient descent based approaches can be very challenging. One of the main reasons behind this is the effect of \textit{vanishing gradients}: it has been empirically observed that in many multi-layer neural networks the gradients in higher-level layers often become too small to be effective in gradient descent based optimisation. Another difficulty is that the distribution of each layers inputs changes during training as the parameters of the previous layers change. These issues make it difficult to find the best learning rates.
Batch normalisation layers \citep{batch_norm} are one way of addressing these challenges. Let the activities of a given neurone in a mini-batch be denoted by $x_1,\ldots,x_m$. The batch normalisation layers i) calculate the empirical mean ($\mu = \frac{1}{m}\sum_{i=1}^m x_i $) and variance ($\sigma^2 = \frac{1}{m}\sum_{i=1}^m (x_i-\mu)^2 $) of the neural activities using the mini-batch data, ii) standardise the neurone activities to make them zero mean with variance one, that is $\hat x_i = (x_i-\mu)/\sigma$, iii) linearly transform these activities with adjustable parameters $\beta, \gamma \in \mathbb{R}$: $y_i=\gamma\hat x_i +\beta$. Here $y_i$ denotes the output after applying the batch normalisation layer on the neurone with activities $x_i$. It has been empirically demonstrated that batch normalisation can often accelerate the training procedure and help mitigate the above described challenges.
There are many variations on these techniques and concepts some of which are represented in the following descriptions of particular methods.
\subsubsection{AstrOmatic (Bertin)}
The lens detector is based on a CNN, trained with the provided training datasets. The CNN is implemented in Python, using the TensorFlow framework\footnote{http://www.tensorflow.org/}. Both ground multichannel and space monochannel image classifiers have the exact same CNN architecture.
The network itself consists of three convolutional layers (11x11-32, 5x5-64 and 3x3-64), followed by two fully-connected layers ($256$ and $64$ neurons) and an output softmax layer . The first five layers use the ELU activation function, which in our tests led to significantly faster convergence compared to ReLU and even SoftPlus activation. Dropout regularization is applied to both convolutional and fully connected layers, with ``keep'' probabilities $p=2/3$ and $p=1/2$, respectively.
Prior to entering the first convolutional layer, input image data are rescaled and dynamic-range compressed with function $f(x) =
\mathrm{arcsinh} (10^{11} x)$, and bad pixels are simply set to 0.
Data augmentation (increasing the amount of training data by modifying and reusing it) is performed in the form of random up-down and left-right image flipping, plus $k\pi/2$ rotations, where $k$ is a random integer in the $[0,3]$ range. Additionally, a small rotation with random angle $\theta$ is applied, involving bicubic image resampling. $\theta$ follows a Gaussian distribution with mean $\mu=0$ and standard deviation $\sigma_{\theta}=5^{\circ}$. No attempt was made to generate and randomize bad pixel masks in the data augmentation process.
The CNN weights are initialized to random values using a truncated Gaussian distribution with mean $\mu=0$ and standard deviation $\sigma=5.10^{-2}$. The network is trained on a Titan-X ``Pascal'' nVidia GPU using the \textit{Adam} gradient-based optimizer during 800 epochs, with an initial learning rate $\eta(t=0)=10^{-3}$ and a learning rate decay $\eta(t+1)/\eta(t)=0.99$, where $t$ is the epoch. Because of a lack of time, tests were limited to assessing the basic classification performance on a subset of the of 1,000 images/datacubes, using the 19,000 others for training.
\subsubsection{LASTRO EPFL (Geiger, Sch\"{a}fer)}
We used a CNN \citep{Fukushima1980,Lecun1998} with a simple architecture of 12 layers (inspired by \citep{symmetry}), see table \ref{tab:architecture}.
To avoid the problem of the data flow distribution getting out of the comfort zone of the activation functions ("Internal Covariate Shift"), we used a mix of normalization propagation \citep{norm_prop} (without the constraint on the weights but a proper initialization) and batch normalization \citep{batch_norm} (slowly disabled over the iterations).
As activation function, we used a scaled and shifted ReLU (Rectifier Linear Unit),
\begin{equation} \label{eq:relu}
\frac{1}{\sqrt{\pi-1}} (\sqrt{2 \pi} \max(0,x) - 1),
\end{equation}
to satisfy the properties required by the normalization propagation.
Our batch normalization implementation computes the mean of the activation function $\bar\mu_i$ using the following equation
\begin{equation} \label{eq:batchnorm}
\bar\mu_i \longleftarrow (1-\eta) \; \bar\mu_{i-1} + \eta \; \mu_i(\text{batch}).
\end{equation}
$\bar\mu_i$ is computed using the mean value $\mu_i$ over the batch in combination with the previous mean $\bar\mu_{i-1}$ using an inertia value $\eta$ set to $1$ at the beginning and decaying with the iterations.
For the training, the 20k provided images were split into two sets, 17k for training and 3k for validation.
Each iteration of the gradient descent (more precisely \textit{Adam} \citep{adam}) minimizes the cross entropy,
\begin{equation} \label{eq:xent}
\left\{
\begin{array}{ll}
- \log(p) & \text{if the image is a true lens} \\
- \log(1-p) & \text{if the image is a nonlens}
\end{array}
\right.,
\end{equation}
where $p$ is the output of the neural network, computed over a batch of 30 images, 15 lenses and 15 nonlenses, picked from the training set.
The small batches with only 30 images were easier to handle computationally but add more noise to the gradient which we considered negligible due to there being only two classes to classify.
To augment the training set, each image of the batch is transformed with a random transformation of the dihedral group (rotations of 90 degrees and mirrors), its pixel values multiplied by a factor picked between $0.8$ and $1.2$ and shifted by a random value between $-0.1$ and $0.1$.
To prevent the overfitting, we used some dropout \citep{dropout} (with a keeping probability decreasing with the iterations).
The masked region of the ground based images are handled by simply setting them to zero.
Each final prediction is made of the product of the predictions of the 8 transformations of the image by the dihedral group.
The architecture is implemented in Tensorflow\footnote{\url{http://tensorflow.org/}}.
Our code is accessible on github\footnote{\url{https://github.com/antigol/lensfinder-euclid}}. Additional details can be found in \citet{2017Schaefer}.
\begin{table*}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
Layer type & shape & activation & \# parameters \\ \hline \hline
\textbf{convolutional 4x4} & $101\!\times\! 101\!\times\!1/4 \to 98\!\times\!98\!\times\!16$ & rectifier & 256/1'024 + 16 \\ \hline
\textbf{convolutional 3x3} & $98\times98\times16 \to 96\times96\times16$ & rectifier & 2'304 + 16 \\ \hline
max pool /2 & $96\times96\times16 \to 48\times48\times16$ & - & - \\ \hline
batch normalization & $48\times48\times16$ & - & 16 + 16 \\ \hline
\textbf{convolutional 3x3} & $48\times48\times16 \to 46\times46\times32$ & rectifier & 4'608 + 32 \\ \hline
\textbf{convolutional 3x3} & $46\times46\times32 \to 44\times44\times32$ & rectifier & 9'216 + 32 \\ \hline
max pool /2 & $44\times44\times32 \to 22\times22\times32$ & - & - \\ \hline
batch normalization & $22\times22\times32$ & - & 32 + 32 \\ \hline
\textbf{convolutional 3x3} & $22\times22\times32 \to 20\times20\times64$ & rectifier & 18'432 + 64 \\ \hline
\textbf{convolutional 3x3} & $20\times20\times64 \to 18\times18\times64$ & rectifier & 36'864 + 64 \\ \hline
max pool /2 & $18\times18\times64 \to 9\times9\times64$ & - & - \\ \hline
batch normalization & $9\times9\times64$ & - & 64 + 64 \\ \hline
dropout & $9\times9\times64$ & - & - \\ \hline
\textbf{convolutional 3x3} & $9\times9\times64 \to 7\times7\times128$ & rectifier & 73'728 + 128 \\ \hline
dropout & $7\times7\times128$ & - & - \\ \hline
\textbf{convolutional 3x3} & $7\times7\times128 \to 5\times5\times128$ & rectifier & 147'456 + 128 \\ \hline
batch normalization & $5\times5\times128$ & - & 128 + 128 \\ \hline
dropout & $5\times5\times128$ & - & - \\ \hline
\textbf{fully-connected} & $5\times5\times128 \to 1024$ & rectifier & 3'276'800 + 1'024 \\ \hline
dropout & $1024$ & - & - \\ \hline
\textbf{fully-connected} & $1024 \to 1024$ & rectifier & 1'048'576 + 1'024 \\ \hline
dropout & $1024$ & - & - \\ \hline
\textbf{fully-connected} & $1024 \to 1024$ & rectifier & 1'048'576 + 1'024 \\ \hline
batch normalization & $1024$ & - & 1'024 + 1'024 \\ \hline
\textbf{fully-connected} & $1024 \to 1$ & sigmoid & 1'024 + 1 \\ \hline \hline
Total & - & - & $\approx$ 5'674'000 \\ \hline
\end{tabular}
\caption{LASTRO EPFL architecture}
\label{tab:architecture}
\end{table*}
\subsubsection{GAMOCLASS (Tuccillo, Huertas-Company, Velasco-Forero, Decenci\`ere)}
\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{figures/Fig1_gamoclass.pdf}
\caption{GAMOCLASS schematic}
\label{Fig1_gamoclass}
\end{figure*}
Our algorithm for classification of strong lensed galaxies was developed as a CNN used for binary classification task.
Our code was trained and tested on the space-based dataset (dataset 0) provided for the Ssrong lensing challenge, comprising 20,000 stamps of single band simulated galaxies. We used the full dataset in the proportion of 4/5 for training and 1/5 for validation. The training images were labelled with 1 if showing strong lensing and 0 otherwise. Our CNN gives as output a probability [0,1] of the input image being a strongly lensed system.
The final architecture of our model is illustrated in Fig. \ref{Fig1_gamoclass}. The input image (101x101 pixels) is first processed by a 2D convolution layer with a 3x3 filter size, then subsampled by a 3x3 max pooling layer. Another two identical units follow, with a growing dimensionality of the output space (depth, i.e.output number of filters) in the convolution, for a total of 3 convolutional layers and 3 max pooling layers. Each of these convolutional layers is followed by a rectified linear unit (ReLU) step. The output of these units is then processed through a single fully-connected layer follower by a dropout layer, and, finally, by a one-neuron fully connected layer with sigmoid activation function. For the classification problem we used the binary cross-entropy cost function and for weights updates we used an adaptive moment estimation (ADAM, \citealt{Kingma_2014}) optimization method. The use of the ADAM optimizer improves the learning rate, and previous tests with stochastic gradient descent (SGD) lead to worse results on the classification of the validation sample.
In order to increase the size of the training set and make the model invariant to specific transformations, we perform several data augmentation steps:
\begin{itemize}
\item Random rotation of the image in the range [0, 180$^{\circ}$], using a reflection fill mode to keep constant the size of the images.
\item The images were randomly shifted of 0.02 times the total width of the image.
\item Random horizontal and vertical flips of the images
\end{itemize}
During the training we initialize the weights of our model with random normal values and we warm up the training \citet{Huang_2016} of the CNN for 25 epochs, using an exponential decay rate ($10^{-6}$) and a staring learning rate of 0.001. Then the network is trained using an early stopping method, and for a maximum number of 300 epochs. The early stopping method is an effective method to prevent overfitting and consists in stopping the training if a monitored quantity does not improve for a fixed number (called \textit{patience}) of training epochs. The quantity that we monitored was the \textit{accuracy} of the classification of the validation sample. The best architecture was trained over 220 epochs with a parameter of patience equal to 20.
We implemented our code in the Keras framework \citep{Chollet_2015} on top of Theano \citep{Bastien_2012}.
Our architecture converges with a classification accuracy of 91\% on the validation sample. We further evaluated the performance of our classifier calculating the ROC curve of the classifier, i.e. the True Positive Rate (TPR) against the False Positive Rate. As shown in Fig. \ref{Fig2_gamoclass}, we reach a TPR higher than the 90\% with a FPR < 8\%.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{figures/Fig2_gamoclass.pdf}
\caption{GAMOCLASS ROC on training data.}
\label{Fig2_gamoclass}
\end{figure}
\subsubsection{CAST (Bom, Valent\'in, Makler)}
The CBPF Arc Search Team (CAST) tested several arcfinding schemes using as core a CNN. For both the space-based and ground-based samples we carried out a simple preprocessing phase to enhance the objects in the images in order to check if they improve the automated arc detection with CNN. We chose a contrast adjustment with $0.1\%$ pixel saturation and apply a low pass band Wiener filter \citep{wiener1964extrapolation} to reduce the effect of the noise.
We used a native CNN from Matlab\footnote{\texttt{https://www.mathworks.com/products/matlab.html}, \texttt{https://www.mathworks.com/help/nnet/convolutional-neural-networks.html}}, which has Convolutional 2D layers with 20 5x5 filters. This CNN can work either with one or three input images, representing greyscale and colour images.
Therefore, we employed different strategies for the two samples available for the challenge, which involve combinations of the available bands running in one or more CNN, using or not the preprocessing, and combining the output with the aid of other machine learning methods.
In each case we used the simulations made available for the challenge both to train and to validate the results and we used the area under the ROC to determine which combination of methods gives the best result.
We selected
$90\%$ of the images, chosen randomly, for the training and $10\%$ to validate. We repeated the process $10$ times to avoid bias due a specific choice of training/validation set and to define an uncertainty in our ROC.
For the space-based data set we tested only two configurations: i) using the CNN straightforwardly for classification and ii) with the preprocessing described above.
We found that the results, accounting for the uncertainties,
were clearly superior for the area under the ROC in the case
including the preprocessing. Therefore, this is the configuration we have used for the challenge.
As mentioned above, the CNN used can take 3 color images as input. Thus, if we want to use the information on the $4$ available bands, we will have to either combine $2$ of the $4$ bands to end up with $3$ bands for a single RGB CNN (configuration I below) or we use multiple CNNs to use the information available (configuration II to VI below). To combine the outputs of several CNNs we use a Support Vector Machine \citep[hereafter SVM; see e.g., ][]{rebentrost2014quantum} also implemented in Matlab. The SVM is used to combine the outputs $p_i$ of the several CNNs (configurations II, III, IV and VI). Instead of using only $p_i$ as inputs to the SVM we also tested providing the SVM with image features obtained by the CNN (feature maps, configuration V) as inputs. In all cases we tested with and without the preprocessing
A more detailed description of each configuration tested is presented below:
\begin{enumerate}
\item Combination of bands r and i with the average between bands u and g. Use one CNN for classification.
\item Creation of $1$ CNN for each band (total of $4$). The $4$ outputs are used as input to a SVM classifier which returns the final classification $p$.
\item Combination into 4 different combinations of bands: RGB $\rightarrow$ (\textit{u,g,r}), (\textit{u,g,i}), (\textit{u,r,i}) and (\textit{g,r,i}). One CNN for each combination of bands and then use of the output score as input to an SVM classifier.
\item Average of bands in different combinations RGB $\rightarrow$ (\textit{ug,r,i}), (\textit{u,gr,i}) and (\textit{u,g,ri}). The outputs of these 3 CNNs are inputs to a SVM classifier.
\item Use of CNN-activations (CNN feature maps) as inputs to a SVM classifier, using same combinations of bands of III. The output of each CNN is used as input to a SVM classifier.
\item Use of wiener filter and contrast adjustment on each band, then using the resulting images in the same architecture as in (iii).
\end{enumerate}
For the ground based cases three configurations with highest area under ROC were III,IV and VI. Although the areas are very similar between IV and VI the last one is superior in the low fake positives end. Thus, for the Strong Lensing Challenge in the ground base sample, we used configuration VI. This final scheme is illustrated in figure \ref{colorCASTCNN}.
The area under ROC, in both space based configurations were, in general, smaller than in the multi-band case, which suggests how the CNNs are sensitive to color information to find Strong Lensing. Particularly, ground base configuration II used one CNN per band thus not considering color information and has the similar area under ROC as our best single band configuration.
\begin{figure}
\includegraphics[width=\columnwidth]{figures/SLchallengeCAST.png}
\caption{CAST Lens finder pipeline for the ground based sample. Illustration of the chosen architecture for the CAST search in the case of ground-based simulated images.}
\label{colorCASTCNN}
\end{figure}
\subsubsection{CMU DeepLens (Lanusse, Ma, Li, Ravanbakhsh)}
\texttt{CMU DeepLens} is based on a residual network (or \textit{resnet}) architecture \citep{He2015a}, a modern variant of CNNs which can reach much greater depths (over 1000 layers) while still gaining accuracy. We provide a short overview of our model below but a full description of our architecture can be found in \citet{Lanusse2017}.
Much like conventional CNNs, resnets are based on convolutional and pooling layers. However, resnet differ from CNNs by the introduction of so-called shortcut connections bypassing blocks of several convolutional layers. As a result, instead of learning the full mapping from their input to their output these residual blocks only have to learn the difference to the identity. In practice, this difference allows residual networks to be trained even for very deep models. For a more thorough description of this architecture, we refer the interested reader to Section~2.3 of \citet{Lanusse2017}.
Our baseline model is composed of a first 7x7-32 convolutional layer which can accommodate either single-band or multi-band images. The rest of the model is composed of 5 successive blocks, each block being made of 3 resnet units (specifically, pre-activated bottleneck residual units \citep{He2016}). At each block, the signal is downsampled by a factor 2 and the number of feature maps is in turn multiplied by 2. The model is topped by an average-pooling layer followed by a single fully-connected sigmoid layer with a single output. Apart from the final layer, we use the ELU (Exponential Linear Unit) activation throughout. The weights of the model are initialized using random normal values, following the strategy advocated in \citet{He2015a}.
Training was performed using the ADAM optimizer with mini-batches of size 128 over 120 epochs, with an initial learning rate of $\alpha=0.001$, subsequently decreased by a factor 10 every 40 epochs. This multi-step training procedure is important to progressively refine the model parameters of the model and achieve our final accuracy.
We adopt a minimal pre-processing strategy for the input images, removing the mean image and normalising by the noise standard deviation $\sigma$ in each band, these statistic being evaluated over the whole training set. In addition, we clip extreme values above $250 \,\sigma$ to limit the dynamic range of the input. Bad-pixels are simply set to 0 after this pre-processing step.
Given the relatively small training set preventing overfitting is an important consideration. In our final model, we combine several data augmentation strategies: random rotations (in the range $[-90, \ 90^\circ]$), random mirroring along both axes, and random resizing (by a small factor in the range $[0.9, \ 1]$).
The architecture presented above is the one that lead to our best results in both branches of the challenge, i.e. \texttt{CMU-DeepLens-Resnet} for space-based and \texttt{CMU-DeepLens-Resnet-ground3} for ground-based. We also submitted results for two variants of this baseline model, named \texttt{-aug} and \texttt{-Voting}.
The first variant introduced several data-augmentation schemes, including the ones mentioned above and the addition of Gaussian noise to the input images. We found however that the introduction of noise was not necessary as the other methods were enough to prevent overfitting.
The second variant was used to explore a voting strategy between three different models. These models differed by the type of residual blocks (bottleneck vs wide) and by their handling of missing pixels (setting to 0 or to the median value of the image). The predictions of the best 2 out of 3 models were then averaged to produce the final classification probability.
Our model is implemented using the \texttt{Theano}\footnote{\url{http://deeplearning.net/software/theano/}} and \texttt{Lasagne}\footnote{\url{https://github.com/Lasagne/Lasagne}} libraries. On an Nvidia Titan Xp GPU, our full training procedure requires approximately 6 hours on the ground-based challenge, but classification of the whole testing set is performed in a couple of minutes. Finally, in the interest of reproducible research, our code is made publicly available on GitHub\footnote{\url{https://github.com/McWilliamsCenter} }.
This repository also contains a notebook detailing how to reproduce our challenge submission.
\subsubsection{Kapteyn Resnet (Petrillo, Tortora, Verdoes Kleijn, Koopmans) }
Our lens-finder is based on a CNN, following the strategy adopted recently in \citep{2017arXiv170207675P}.
We decide to treat the problem as a three-class classification problem where the classes are \textit{non-lenses}, \textit{clear lenses} and \textit{dubious lenses}. We define the \textit{dubious lenses} as the lenses with lensing features with less than 160 pixels and the \textit{clear lenses} those with more than 160 pixels belonging to the lensed source. This choice is motivated by the fact that specializing the network in recognizing different classes could lead to a more robust classification. In addition, in a hypothetical application of the method to real data from a survey, this could be a way to select the most blatant lenses.
Given the rules of the challenge, we had to submit a prediction for each source in the test-set as a single number: a number in the interval [0,1] or either a 0 or a 1. For satisfying this requirement we had to collapse one of the class into another: we provided a 0 when a source is classified as a \textit{non-lens} and a 1 when is classified as a \textit{clear lens} or as a \textit{dubious lens}. This choice did not allow to build a continuous ROC curve but only a binary one.
The final submission was produced averaging the values of the predictions from three CNNs with the same architecture.
{\bf Implementation}
The CNN is implemented in Python 2.7 using the open-source libraries \textsc{Lasagne}\footnote{\href{http://github.com/Lasagne/Lasagne/}{\tt http://github.com/Lasagne/Lasagne/}}
and \textsc{Theano}
\footnote{\href{http://deeplearning.net/software/theano/}{\tt http://deeplearning.net/software/theano/}} \citep{theano}.
The training of the CNN is executed on a GeForce GTX 760 in parallel with the data augmentation performed on the CPU using the \textsc{scikit-image}\footnote{\href{http://scikit-image.org/}{\tt http://scikit-image.org/}} package \citep{van2014scikit}.
{\bf Architecture}
We used the CNN architecture called ResNet described in \citep{he2015deep} with three stacks of residual blocks of 5 layers.
The output layer is composed by three units. Each units gives in output a number between 0 and 1 that represents, respectively, the probability of being a \textit{non-lens}, a \textit{dubious lens}, a \textit{certain lens}.
{\bf Preprocessing}
The training fits files are preprocessed with the software \textsc{STIFF}\footnote{\href{http://www.astromatic.net/software/stiff}{\tt http://www.astromatic.net/software/stiff}} which automatically converts the fits files to gray-scale TIFF images operating a non-linear intensity transformations to enhance the low-brightness features of the image.
Due to memory limitations we down-sampled the images to 84 by 84 pixels. Most likely, this leads to a worse classification accuracy than using the original image resolution.
We augmented the training images in the following way: i) random rotation of 90, 180 or 270 degrees; ii) random shift in both $x$ and $y$ direction between -2 and +2 pixels; iii) $50\%$ probability of horizontally flipping the image.
Finally, the image border is cropped in order to have 80 by 80 pixel input images.
{\bf Training}
The network is trained by minimizing the categorical cross-entropy loss function
\begin{equation}
L = -\sum_j t_{j}\log p_{j}
\label{EQloss}
\end{equation}
where the $t_j$ and $p_j$ are respectively the label and the prediction for the class $j$.
The minimization is done via mini-batch stochastic gradient descent with \textit{Adam} updates \citep{Kingma_2014}. We used a batch size of 56 and performed 46000 gradient updates. We started with a learning rate of $4 \times 10^{-4}$, decrease it to $4 \times 10^{-5}$ after 35000 updates and to $4 \times 10^{-6}$ after 43000 updates.
The initial values of the biases are set to zero and the weights of each filter are initialized from a random normal distribution with variance ${2/n}$ where $n$ is the number of inputs of the unit \citep{He2015a}.
We use L2-norm regularization with $\lambda= 9 \times 10^{-3}$.
\subsubsection{NeuralNet2 (Davies, Serjeant) }
Our lens finder included wavelet prefiltering. The image was convolved with the Mallat wavelet with a kernel size of 4 in both the horizontal and vertical directions, then combined and compared to the original image to make the input image; $input image = \sqrt[]{H^{2}+V^{2}}$. This prefiltering was performed to emphasise the edges in the images, improving the results from the Convolutional Neural Network (CNN). The CNN had 2 convolution layers each containing 32 $3\times3$ filters, incorporating dropout and max-pooling, and then 3 dense fully-connected layers to classify each image. The network was trained on 18000 of the 20000 training images; training took 15 epochs and was completed once the validation loss was minimised. The training was validated on the remaining 2000 images. Validation loss was calculated using binary cross entropy
\begin{equation}
\mathcal{L} = \sum_{i=1}^{n} \big[ y_{i}$ $log(p_{i}) + (1 $ - $ y_{i})$ $log(1 $ - $ p_{i}) \big]
\end{equation} where $\mathcal{L}$ is the loss function, $n$ is the number of inputs, $y_{i}$ is the true value of the $i^{th}$ input , and $p_{i}$ is the predicted value for the $i^{th}$ input from the network. A perfect loss of $0$ was generated once every predicted value matched the true value for every input. The network was made and trained in Python 2.7 using the open-source libraries \textsc{theano} and \textsc{keras}\footnote{\href{https://github.com/keras-team/keras}{\tt https://github.com/keras-team/keras}}. A more developed version of our lens finder will appear in Davies, Bromley and Serjeant (in preparation).
\section{results}
\label{sec:results}
In this section we summarize some of our analysis of the submissions. In section~\ref{sec:figure_of_merit} we discuss how to judge a classifier in this particular case and define some metrics of success.
\subsection{figures of merit}
\label{sec:figure_of_merit}
In deriving a good figure of merit for evaluating lens finding algorithms one needs to take into account the particular nature of this problem.
The traditional method for evaluating a classification algorithm is with the receiver operating characteristic curve, or {\bf ROC} curve. This is a plot of the true positive rate (TPR) and the false positive rate (FPR). In this case these are defined as
\begin{align}
{\rm TPR} &= \frac{\textrm{ number of true lenses classified as lenses}}{\textrm{ total number of true lenses}} \\
{\rm FPR} &= \frac{\textrm{number of non-lenses classified as lenses}}{\textrm{ total number of non-lenses}}
\end{align}
The classifier generally gives a probability of a case being a lens, $p$, in which case a threshold is set and everything with $p$ greater is classified as a lens and everything smaller is classified as not a lens. The TPR and FPR are then plotted as a curve parametrised by this threshold. At $p=1$ all of the cases are classified as non-lenses and so TPR=FPR=1 and at $p=0$ all of the cases are classified as lenses so TPR=FPR=0. These points are always added to the ROC curve. If the classifier made random guesses then the ratio of lenses to non-lenses would be the same as the ratio of the number of cases classified as lens to the number of cases classified as non-lenses and so TPR=FPR. The better a classifier is the smaller the FPR and the larger the TPR so the further away from this diagonal line it will be. When a classifier provides only a binary classification or a discrete ranking, the ROC connects the endpoints to the discrete points found by using each rank as a threshold.
A common figure of merit for a classifier is the area under the ROC ({\bf AUROC}). This evaluates the overall ability of a classifier to distinguish between cases. This was the criterion on which the challenge participants were told to optimise. However, in the case of gravitational lensing this is not the only thing, and not the most important thing, to consider. Gravitational lenses are rare events, but to improve the discrimination and training of the classifiers the fraction of lenses in test and training sets are boosted to something around half. In these circumstances it is important to consider the absolute number of cases that will be misclassified when the fraction of true cases is closer to what is expected in the data.
If the rates of false positives and false negatives remain the same in real data the contamination of the sample will be
\begin{align}
\frac{\rm FP}{\rm TP} \simeq \frac{\rm FPR}{\rm TPR} \left(\frac{\textrm{number of non-lenses in sample}}{\textrm{number of lenses in sample}} \right)
\end{align}
Since only about one in a thousand objects will be lenses (perhaps somewhat more depending on pre-selection) the contamination will be high unless the FPR is much less than the TPR. For this reason we consider some additional figures of merit.
The {\bf TPR$_0$} will be defined as the highest TPR reached, as a function of $p$ threshold, before a single false positive occurs in the test set of 100,000 cases. This is the point were the ROC meets the FPR = 0 axis. This quantity highly penalizes classifiers with discrete ranking which often get TPR$_0$ = 0 because their highest classification level is not conservative enough to eliminate all false positives. We also define {\bf TPR$_{10}$} which is the TPR at the point were less than ten false positives are made. If the TP rate is boosted from the FPR by a factor of 1,000 in a realistic data set this would correspond to about a 10\% contamination.
In addition to these considerations, the performance of a classifier is a function of many characteristics of the lens system. It might be that one classifier is good at finding systems with large Einstein radii and incomplete arcs, but not as good at finding small complete Einstein rings that are blended with the light of the lens galaxy. Also a lens may have a source that is too faint to be detected by any algorithm or is too far from the lens to be very distorted, but will be classified as a lens in the test dataset. We do not impose a definitive arc/ring magnification, brightness or surface brightness limit for a system to be considered a lens because we
want to include these objects because we want to test the limits of the classifiers. As we will see, if you restrict your objectives to detecting only lensed images with surface brightness above threshold, for example, the "best" algorithm might change and the TPR will change. For this reason we plot the AUROC, TPR$_0$ and TPR$_{10}$ as a function of several variables for all the entries. This is done by removing all the lenses that do not exceed the threshold and then recalculating these quantities, the number of non-lenses remains the same.
\subsection{Performance of methods}
\label{sec:performance}
\begin{table*}
\centering
\begin{tabular}{llrrrl}
\hline
Name & type & AUROC & TPR$_0$ & TPR$_{10}$ & short description \\
\hline
CMU-DeepLens-ResNet-ground3 & Ground-Based & 0.98 & 0.09 & 0.45 & CNN \\
CMU-DeepLens-Resnet-Voting & Ground-Based & 0.98 & 0.02 & 0.10 & CNN \\
LASTRO EPFL & Ground-Based & 0.97 & 0.07 & 0.11 & CNN \\
CAS Swinburne Melb & Ground-Based & 0.96 & 0.02 & 0.08 & CNN \\
AstrOmatic & Ground-Based & 0.96 & 0.00 & 0.01 & CNN \\
Manchester SVM & Ground-Based & 0.93 & 0.22 & 0.35 & SVM / Gabor \\
Manchester-NA2 & Ground-Based & 0.89 & 0.00 & 0.01 & Human Inspection \\
ALL-star & Ground-Based & 0.84 & 0.01 & 0.02 & edges/gradiants and Logistic Reg. \\
CAST & Ground-Based & 0.83 & 0.00 & 0.00 & CNN / SVM \\
YattaLensLite & Ground-Based & 0.82 & 0.00 & 0.00 & SExtractor \\
LASTRO EPFL & Space-Based & 0.93 & 0.00 & 0.08 & CNN \\
CMU-DeepLens-ResNet & Space-Based & 0.92 & 0.22 & 0.29 & CNN \\
GAMOCLASS & Space-Based & 0.92 & 0.07 & 0.36 & CNN \\
CMU-DeepLens-Resnet-Voting & Space-Based & 0.91 & 0.00 & 0.01 & CNN \\
AstrOmatic & Space-Based & 0.91 & 0.00 & 0.01 & CNN \\
CMU-DeepLens-ResNet-aug & Space-Based & 0.91 & 0.00 & 0.00 & CNN \\
Kapteyn Resnet & Space-Based & 0.82 & 0.00 & 0.00 & CNN \\
CAST & Space-Based & 0.81 & 0.07 & 0.12 & CNN \\
Manchester1 & Space-Based & 0.81 & 0.01 & 0.17 & Human Inspection \\
Manchester SVM & Space-Based & 0.81 & 0.03 & 0.08 & SVM / Gabor \\
NeuralNet2 & Space-Based & 0.76 & 0.00 & 0.00 & CNN / wavelets \\
YattaLensLite & Space-Based & 0.76 & 0.00 & 0.00 & Arcs / SExtractor \\
All-now & Space-Based & 0.73 & 0.05 & 0.07 & edges/gradiants and Logistic Reg. \\
GAHEC IRAP & Space-Based & 0.66 & 0.00 & 0.01 & arc finder \\
\hline
\end{tabular}
\caption{The AUROC, TPR$_0$ and TPR$_{10}$ for the entries in order of AUROC.}
\label{table:AUROC}
\end{table*}
Table~\ref{table:AUROC} shows the AUROC, TPR$_0$ and TPR$_{10}$ for the entries in order of AUROC and dataset type. It can be seen that CMU-DeepLens-ResNet-ground3 had the best AUROC for the ground-based set and LASTRO EPFL the best for the space-based set. The order is different if TPR$_0$ is used to rank the entries as seen in table~\ref{table:TPR0}. Here Manchester SVM and
CMU-DeepLens-ResNet get the best scores.
\begin{table*}
\centering
\begin{tabular}{llrrrl}
\hline
Name & type & AUROC & TPR$_0$ & TPR$_{10}$ & short description \\
\hline
Manchester SVM & Ground-Based & 0.93 & 0.22 & 0.35 & SVM / Gabor \\
CMU-DeepLens-ResNet-ground3 & Ground-Based & 0.98 & 0.09 & 0.45 & CNN \\
LASTRO EPFL & Ground-Based & 0.97 & 0.07 & 0.11 & CNN \\
CMU-DeepLens-Resnet-Voting & Ground-Based & 0.98 & 0.02 & 0.10 & CNN \\
CAS Swinburne Melb & Ground-Based & 0.96 & 0.02 & 0.08 & CNN \\
ALL-star & Ground-Based & 0.84 & 0.01 & 0.02 & edges/gradiants and Logistic Reg. \\
Manchester-NA2 & Ground-Based & 0.89 & 0.00 & 0.01 & Human Inspection \\
YattaLensLite & Ground-Based & 0.82 & 0.00 & 0.00 & SExtractor \\
CAST & Ground-Based & 0.83 & 0.00 & 0.00 & CNN / SVM \\
AstrOmatic & Ground-Based & 0.96 & 0.00 & 0.01 & CNN \\
CMU-DeepLens-ResNet & Space-Based & 0.92 & 0.22 & 0.29 & CNN \\
GAMOCLASS & Space-Based & 0.92 & 0.07 & 0.36 & CNN \\
CAST & Space-Based & 0.81 & 0.07 & 0.12 & CNN \\
All-now & Space-Based & 0.73 & 0.05 & 0.07 & edges/gradiants and Logistic Reg. \\
Manchester SVM & Space-Based & 0.80 & 0.03 & 0.07 & SVM / Gabor \\
Manchester1 & Space-Based & 0.81 & 0.01 & 0.17 & Human Inspection \\
LASTRO EPFL & Space-Based & 0.93 & 0.00 & 0.08 & CNN \\
GAHEC IRAP & Space-Based & 0.66 & 0.00 & 0.01 & arc finder \\
AstrOmatic & Space-Based & 0.91 & 0.00 & 0.01 & CNN \\
Kapteyn Resnet& Space-Based & 0.82 & 0.00 & 0.00 & CNN \\
CMU-DeepLens-ResNet-aug & Space-Based & 0.91 & 0.00 & 0.00 & CNN \\
CMU-DeepLens-Resnet-Voting & Space-Based & 0.91 & 0.00 & 0.01 & CNN \\
NeuralNet2 & Space-Based & 0.76 & 0.00 & 0.00 & CNN / wavelets \\
YattaLensLite & Space-Based & 0.76 & 0.00 & 0.00 & Arcs / SExtractor \\
\hline
\end{tabular}
\caption{The AUROC, TPR$_0$ and TPR$_{10}$ for the entries in order of TPR$_0$. }
\label{table:TPR0}
\end{table*}
\begin{figure*}
\includegraphics[width=2\columnwidth]{figures/roc_space.pdf}
\caption{The ROC curves for the space-based entries.}
\label{fig:roc_space}
\end{figure*}
\begin{figure*}
\includegraphics[width=2\columnwidth]{figures/roc_ground.pdf}
\caption{The ROC curves for the ground-based entries. Notice that these are generally better than in figure~\ref{fig:einstein_space} indicating that colour information is an important discriminant. }
\label{fig:roc_ground}
\end{figure*}
\begin{figure*}
\includegraphics[width=2\columnwidth]{figures/roc_kids.pdf}
\caption{The ROC curves for the ground-based entries including only the cases with authentic images taken from the KiDS survey. It can be seen that in all cases these are lower than in figure~\ref{fig:roc_ground}.}
\label{fig:roc_kids}
\end{figure*}
Figures~\ref{fig:roc_space} and \ref{fig:roc_ground} show the ROC curves for all the entries.
Note that ROC curves for the ground-based challenge (figure~\ref{fig:roc_ground}) are uniformly better than those for the space-based challenge (figure~\ref{fig:roc_space}). This is because of the importance of color information in discriminating lensed arcs from pieces of the foreground lens galaxy.
In addition, figure~\ref{fig:roc_kids} shows the ROC curves for only the ground-based images where an actual KiDS image was used (see section~\ref{sec:sim-ground-based}). It can be seen that the classifiers do uniformly less well on this subset. This indicates that the simulated galaxy images are different from the real ones and that the classifiers are able to distinguish fake foreground galaxies from lenses more easily than from real galaxies. Some methods are more affected by this than others, but none seem to be immune, not even the human classification. This is perhaps not unexpected, but does show that the simulated lenses need to be improved before the raw numbers can be directly used to evaluate the performance of a classifier on real data.
\begin{figure*}
\includegraphics[width=2\columnwidth]{figures/einstein_space.pdf}
\caption{The AUROC, TPR$_0$, TPR$_{10}$ and the fraction of lenses in the test sample after discarding the lenses with Einstein radii larger than the number indicated on the x-axis. }
\label{fig:einstein_space}
\end{figure*}
\begin{figure*}
\includegraphics[width=2\columnwidth]{figures/einstein_ground.pdf}
\caption{Same as figure~\ref{fig:einstein_space}, but for space-based entries.}
\label{fig:einstein_ground}
\end{figure*}
\begin{figure*}
\includegraphics[width=2\columnwidth]{figures/flux_space.pdf}
\caption{Same as figure~\ref{fig:einstein_space}, but here the x-axis is the flux within the pixels that are above 1 $\sigma$ in the lensed source only image.}
\label{fig:flux_space}
\end{figure*}
\begin{figure*}
\includegraphics[width=2\columnwidth]{figures/flux_ground.pdf}
\caption{Same as figure~\ref{fig:flux_space}, but for space-based entries.}
\label{fig:flux_ground}
\end{figure*}
\begin{figure*}
\includegraphics[width=2\columnwidth]{figures/npixel_space.pdf}
\caption{Same as figure~\ref{fig:einstein_space}, but here the x-axis is the number of pixels that are above 1 $\sigma$ in the lensed source only image. This is an indication of the lensed arcs' size.}
\label{fig:npixel_space}
\end{figure*}
\begin{figure*}
\includegraphics[width=2\columnwidth]{figures/npixel_ground.pdf}
\caption{Same as figure~\ref{fig:npixel_space}, but for space-based entries.}
\label{fig:npixel_ground}
\end{figure*}
\begin{figure*}
\includegraphics[width=2\columnwidth]{figures/flux_contrast_space.pdf}
\caption{Same as figure~\ref{fig:einstein_space}, but here the x-axis is the ratio of the flux coming from the lensed source to the total flux in the image in the index band.}
\label{fig:flux_contrast_space}
\end{figure*}
\begin{figure*}
\includegraphics[width=2\columnwidth]{figures/flux_contrast_ground.pdf}
\caption{Same as figure~\ref{fig:flux_contrast_space}, but for space-based entries.}
\label{fig:flux_contrast_ground}
\end{figure*}
Figures~\ref{fig:einstein_space} and \ref{fig:einstein_ground} shows the AUROC, TPR$_0$, TPR$_{10}$ and fraction of lenses as a function of a lower cutoff on the Einstein radius (area). There is also a vertical dotted line that indicates where no more than 100 lenses in the test sample had Einstein radii larger than this mark. Beyond this point one should be suspicious of small number statistics. When deriving the distribution of Einstein radii from data these curves would need to be used to correct for detection bias. It can be seen the CMU-DeepLens-ResNet, Manchester1, Manchester SVM and GAMOCLASS obtain significantly higher TPR$_0$ and TPR$_{10}$ for larger Einstein radii. Manchester1 is the human inspection method. In some cases the TPR$_0$'s are above 50\% of the lenses that meet this criterion. Remember that many of the so called lenses are very dim or there is no significant arc because of the source position is well outside the caustic. If an additional require was placed on the definition of a lens, such as the brightness of the arc being above a threshold, the TPRs would go up.
Figures~\ref{fig:flux_space} and \ref{fig:flux_ground} are the same except that the flux in the lensed image is used as the threshold. We count only the flux in pixels with flux over one $\sigma$ of the background. In some cases one can see an abrupt rise in the TPRs at some flux threshold. CMU-DeepLens-ResNet in particular reaches a TPR$_0$ above 75\% for the brightest $\sim$ 10\% of the lenses.
A lensed image can be bright without being visibly distorted as in the case of a unresolved images. Figures~\ref{fig:npixel_space} and \ref{fig:npixel_ground} use the number of pixels in the lensed image(s) that are over one $\sigma$ of the background. In this case also some classifiers show an abrupt improvement when the image is required to be larger than some threshold. Interestingly in some cases the TPRs go down with lensed image size after reaching a peak. This could be because they are not differentiating the arcs from companion galaxies as well in this regime. There were also cases where the arc intersects with the borders of the image that might cause them to be missed.
Figures~\ref{fig:flux_contrast_space} and \ref{fig:flux_contrast_ground} investigate how the flux contrast between the foreground objects and the lensed source affects the classifiers. Interestingly some method's TPRs go up with this quantity and some go down. We have not yet found any clear explanation for this variety of behaviors.
The two human inspector, NJ and AT, got significantly different scores on the ground based test set with individual AUROCs of 0.88 and 0.902 and TPR$_{10}$s of 0.01 and 0.06 respectively. They did not inspect the same images however so differences cannot be considered conclusive, but it does suggest that different inspectors will have different detection efficiencies and biases.
\section{Conclusions \& discussion}
\label{sec:conclusion}
A large variety of lens finding methods were tested on simulated images that were developed separately. We found that some methods could recover more than 50\% of the lenses above a lensed image brightness or size threshold without a single false positive out of 100,000 candidates. If the data closely resembled the simulations we would already have reasonably good methods whose efficiency and biases can be quantitatively characterized.
We have done a fairly good job of determining that lenses can be identified in a population of fairly "normal" galaxies. It is the rare "abnormal" objects that pose the greatest challenge. When real KiDS data was used in the simulations the classifiers were all less accurate and it was only human inspection that found the one jackpot lens (a double Einstein ring with two background sources) in the data. Things like ring galaxies, tidal tails in merging galaxies and irregular galaxies can be mistaken for lenses and were not well represented in the simulated data. Accurately reproducing these objects will be an objective of future work.
It was surprising to some of the authors how well CNN and SVM methods did relative to human inspection. These methods find differences in the classes of images that are not obvious to a human and can classify things as lenses with high confidence where a human would have doubt.
This ability comes with some danger of over fitting to the training set however. The distinguishing characteristics might only be a property of simulated data and not of real data. In principle, SVM methods might potentially mitigate this somewhat because with them one can choose which features to use based on knowledge of the properties of irregular galaxies or ring galaxies for example. This has yet to be shown however.
The confidence one will have in these machine learning methods is really based on the confidence one has in the realism of the simulations.
When initiating this project we had a concern that current methods would be too slow or require too much human intervention to handle large data sets. Happily this seems not to be a problem with most of the automatic methods. The CNN and SVM codes take some time to train, but once trained they are very fast in classifying objects. Billions of objects can be easily handled.
Another lesson is that colour information is very important. Even with lower noise levels, higher resolution, a simpler PSF and no masking, the lenses in the space-based set were harder to find than the lenses in the ground-based set (see figures~\ref{fig:roc_space} and \ref{fig:roc_ground}). Having multiple bands clearly makes a significant difference. Euclid will have several infrared bands with lower resolution than the visible images that were not included in the challenge. Even rather low resolution information from another instrument or telescope when combined with higher resolution data in one band might significantly improve the detection rates. Combining ground based data, such as LSST, with space based data, such as Euclid, would likely boost the detection rates by factors of several.
\section*{Acknowledgements}
AS was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
RBM's research was partly part of project GLENCO, funded under the European Seventh Framework Programme, Ideas, Grant Agreement n. 259349. AT acknowledges receipt of an STFC postdoctoral research
assistantship.
We thank the International Space Science Institute (ISSI) for hosting and funding our workshop.\footnote{http://www.issibern.ch/}
JPK and CS acknowledge support from the ERC advanced grant LIDA and the ESA-NPI grant 491-2016.
CA acknowledges support from the Enrico Fermi
Institute at the University of Chicago, and the University of Chicago
Provost's Office. NL would like to thank the funding support from
NSFC, grant no.11503064, and Shanghai Natural Science Foundation,
grant no. 15ZR1446700. This work was also supported in part by the
Kavli Institute for Cosmological Physics at the University of Chicago
through grant NSF PHY-1125897 and an endowment from the Kavli
Foundation and its founder Fred Kavli. LVEK, CEP, CT and GV are supported through an NWO-VICI grant (project number 639.043.308). GVK acknowledges
financial support from the Netherlands Research School for
Astronomy (NOVA) and Target. Target is supported by
Samenwerkingsverband Noord Nederland, European fund for regional
development, Dutch Ministry of economic affairs, Pieken in de
Delta, Provinces of Groningen and Drenthe.
\bibliographystyle{mnras}
|
1,108,101,566,754 | arxiv | \section{\label{sec:introduction}Introduction}
Electron cooling has become one of the most effective methods for increasing the phase space density of stored ion beams through their interaction with an electron beam copropagating at the same average velocity~\cite{Budker1967}. The first electron cooling experiment was successfully carried out at NAP-M (Novosibirsk) with nonrelativistic protons in 1974~\cite{Budker1976}. After decades of development, the electron cooling method has found a wide range of applications in several low- and medium-energy proton and ion storage rings. It has since become desirable to extend the method into the high-energy range of \SI{50}{\mega\electronvolt}, which will enable a high luminosity at future facilities such as the EIC~\cite{eic}.
However, most existing electron coolers are based on DC electron beams accelerated by electrostatic high voltage.
The highest-voltage cooler so far was successfully operated at FNAL at an energy of \SI{4.3}{\mega\electronvolt}~\cite{nagaitsev}.
Due to the technical limitations of high-voltage acceleration, providing cooling beams at much higher energies necessitates RF acceleration and thus the use of bunched electron beams.
A collaboration between Jefferson Lab (USA) and Institute of Modern Physics (IMP, China) was established in 2012 to conduct precursory bunched cooling experiments aimed at demonstrating the feasibility of such a scheme and investigating its beam-dynamical implications.
According to previous simulation results, the bunched electron beam cooling dynamics are different from those typically obtained with DC beams \cite{ZHAO2018219, PhysRevAccelBeams.21.023501}.
Because a dedicated facility for accelerating bunched electron beams for cooling purposes did not exist at the time, the availability and flexibility of the DC cooling setup installed in the ion ring CSRm at IMP led to the decision to add a pulsing option to the existing facility.
The first experiment, which took place in 2016, was the first demonstration of cooling with a pulsed beam \cite{Mao:COOL2017-TUP15, Zhang:IPAC2018-TUPAL069}, albeit with bunch lengths of several meters.
Four experiments were performed in total (in 2016, 2017, 2018, and 2019), the latter aiming at improving the data quality and addressing unresolved questions. Unlike the pioneer experiment, the most recent data set includes cooling of all stored bunches to improve statistics.
The beam current is also boosted by applying DC cooling during accumulation.
The first electron cooler based on RF acceleration of short ($\propto \si{\centi\meter}$) electron bunches was proposed in 2013 \cite{fedotov2013} and recently commissioned at RHIC with an electron energy of \SI{2}{\mega\electronvolt}~\cite{PhysRevLett.124.084801}.
We describe the pulsed-electron-beam cooling facility we set up at IMP and present the results of the most recent cooling experiment with ${}^{86}\mathrm{Kr}^{25+}$ ions at an energy of \SI{5}{\mega\electronvolt/nucleon}.
\section{\label{sec:setup}Experimental setup}
\subsection{Storage ring layout}
CSRm is a racetrack-shaped synchrotron with a circumference of \SI{161}{\meter}. An electron cooler with an effective length of \SI{3.4}{\meter} and an RF cavity are placed in the dispersion-free sections.
While the ring can be operated with a coasting beam, the cavity provides the option to create bunches of adjustable length.
The details of the facility are described in \cite{XIA200211}.
\subsection{Generation of the pulsed cooling beam}
The electron beam used for the cooling experiment is generated by the conventional magnetized electron cooler installed in the CSRm ring~\cite{BOCHAROV2004144}.
The existing electron source was modified to enable synchronization with the stored ion bunches in the following way:
The current emitted by the thermionic electron gun is a function of the voltage applied to the grid electrode. For normal operation, this voltage is set to a positive value that corresponds to a certain desired current, whereas it can be set to a negative value to shut off the electron beam completely.
By rapidly switching between these two voltages with a solid-state switch, we can generate rectangular electron pulses while leaving the other properties of the cooler essentially unmodified.
Using a reference signal generated by the RF system of the ring, these pulses are synchronized to the RF buckets, ensuring a stable temporal overlap between the electron bunches and the ion bunches.
The pulser setup is shown schematically in Fig.~\ref{fig:pulsed_gun} and the resulting voltage waveform in Fig.~\ref{fig:delay_setup}.
Because of the capacitance of the grid electrode and its cable, the currents flowing to charge and discharge the electrode with the desired high slew rate result in power dissipation in the switching element, which limits the available voltage and/or switching frequency.
Unlike a DC beam, which makes full use of energy recovery by default, a pulsed beam also loads the acceleration voltage supply of the cooler. The supply would therefore need to be bypassed at high frequencies to prevent excessive supply droop if a high peak current were to be extracted.
The beam parameters are a compromise to ensure that both the revolution frequency and the required voltages are well within the limits of the switching hardware \footnote{DEI PVX-4150}.
The choice of ${}^{86}\mathrm{Kr}^{25+}$ ions at an energy of $\SI{5}{\mega\electronvolt/nucleon}$, though far away from the properties of high-energy proton beams, is motivated by the use of this beam by existing users of the facility. Table~\ref{tab:all_params} lists the parameters used.
\begin{table}[!htbp
\caption{Beam and instrumentation parameters\label{tab:all_params}}
\begin{ruledtabular}
\begin{tabular}{lc}
\textbf{ion beam} & \\
\hline \\ [-1.5ex]
particle type & ${}^{86}\mathrm{Kr}^{25+}$ \\
beam current & $< \SI{100}{\micro\ampere}$ \\
rest mass & \SI{930.5}{MeV/nucleon} \\
kinetic energy & \SI{5.0}{MeV/nucleon} \\
$\beta$ & 0.103 \\
$\gamma$ & 1.005 \\
revolution frequency & \SI{191.5}{\kilo\hertz} \\
harmonic number & 2 \\
RF voltage & 0.6--\SI{2}{\kilo\volt} \\ [1ex]
\hline
\textbf{electron cooler} & \\
\hline \\ [-1.5ex]
acceleration voltage & \SI{2.7}{\kilo\volt} \\
positive grid voltage & \SI{50}{\volt} \\
negative grid voltage & \SI{-551}{\volt} \\
peak current & \SI{30}{\milli\ampere}
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}[htb]
\includegraphics{fig-pulsed-gun.pdf}
\caption{\label{fig:pulsed_gun}Schematic model of the electron pulse synchronization setup.}
\end{figure}
\begin{figure}[htb]
\includegraphics{fig-delay-setup.pdf}
\caption{\label{fig:delay_setup}Sketch of the fixed phase relation between the RF reference signal and the electron gun grid voltage. The delay $\tau_1$ and the bunch length $\tau_2$ are set independently.}
\end{figure}
\subsection{Measurement of beam properties}
Depending on the available beam instrumentation devices, there are multiple independent ways to gain information about the cooling process.
The usual way to determine the longitudinal momentum distribution is by measuring the revolution peaks and their respective synchrotron sidebands from a Schottky pickup with a spectrum analyzer~\cite{forck_schottky}.
However, the cooling time in this experiment is short ($\approx \SI{1}{\second}$), and spectra with sufficient resolution and signal-to-noise ratio cannot be readily obtained from the available hardware. We therefore only use the spectrum from the Schottky pickup to determine the synchrotron frequency, which is a more dependable measurement of the RF voltage than the setting in the RF hardware itself.
Instead, the longitudinal cooling process is observed by measuring the temporal bunch profile using a beam position monitor (BPM) that is mounted outside the cooling region and thus only detects the ion bunches. The electron signal is measured by a similar BPM near the electron collector.
Since the distance between the two BPMs---and, thus, the time-of-flight difference---is known, the longitudinal overlap of the bunches can be measured and adjusted.
To remove the transverse information from the BPM signal, the signals from two opposing plates are summed. The sum signal of both BPMs is then recorded by a digital oscilloscope that is triggered at \SI{12}{\hertz} and stores 20 frames in total. We use amplifiers with an input impedance of $R=\SI{50}{\ohm}$ close to the BPM devices to establish well-defined signal properties before summing. Neglecting parasitic properties such as resonances, the system of a BPM feeding a resistor can be modeled as a first-order highpass filter driven by a current source~\cite{forck_bpm}. The voltage at the output is:
\begin{equation}
U = Z I_\text{beam} \quad\text{with}\quad Z(\omega) \propto \frac{i \omega RC}{(1 + i \omega RC)} . \label{equation:ohmslaw}
\end{equation}
The pick-up capacitance $C$ is device-dependent, but it is cricital for signal reconstruction only if the maximum frequency of the signal of interest is on the order of $f_\text{cut}=(2\pi RC)^{-1}$ or higher.
With the amplifiers mounted directly on the BPM feedthroughs, the parasitic capacitance can be assumed to be negligible, but the transfer function was not measured in situ.
Our analysis conservatively assumes $f_\text{cut}=\SI{100}{\mega\hertz}$ for the ion BPM and $RC = \infty$ for the electron BPM, the latter being operated without amplifiers.
Since this transfer impedance acts like a differentiator at low frequencies, shortening the bunch length and, correspondingly, increasing the current slope increases the peak amplitude of the signal; the gain structure must be chosen such that the respective peak voltage is handled without clipping under all circumstances.
\begin{figure}[htb]
\includegraphics{fig-bpm-setup.pdf}
\caption{\label{fig:bpm_setup}Schematic model of the BPM data acquisition setup.}
\end{figure}
We observe the transverse cooling process by measuring the horizontal ion beam profile with an ionization profile monitor (IPM). The device is described in detail in \cite{IPM_Xie2020}.
\subsection{Top-level timing setup}
To prepare an ion beam for pulsed cooling, we first accumulate ions in the ring up to the desired beam current using the standard CSRm accumulation procedure, which takes about \SI{10}{\second} and is accompanied by DC operation of the electron cooler. The resulting beam is unbunched. After accumulation and an additional delay of \SI{2}{\second}, the cooling beam is switched off to let the ion beam heat up for \SI{3}{\second}. The RF system then starts ramping up the cavity voltage, eventually bunching the ion beam. An equilibrium is reached about 5 s after the start of heating. At this point, we start recording the BPM signal to determine the initial bunch profile and switch on the pulsed cooling beam after the first two recorded frames. The whole process after accumulation is recorded by a spectrum analyzer connected to the Schottky pickup to ensure proper timing.
Ideally, the acquisition of the transverse profile from the IPM would coincide with that of the BPM signals. However, for reasons specific to the top-level trigger hardware of the ring, the IPM is started at the same time as the spectrum analyzer.
The timing scheme for cooling a bunched ion beam is visualized in Fig.~\ref{fig:toplevel_timing}. The setup can also be used to cool a coasting ion beam with a pulsed electron beam, in which case the timing is the same except that the bunching cavity is not powered.
\begin{figure}[htb]
\includegraphics{fig-toplevel-timing.pdf}
\caption{\label{fig:toplevel_timing}Sketch of the top-level timing of beam instrumentation components. Components manipulating the beam are shown in red, components measuring its properties in green. The yellow band denotes the time window of interest in which the pulsed cooling process can be observed in all dimensions.}
\end{figure}
\section{\label{sec:results}Results}
\subsection{\label{sec:schottky_analysis}Spectrum of Schottky pickup signal}
We use the spectrum from the longitudinal Schottky pickup to determine the synchrotron frequency and, subsequently, the RF voltage.
An example of the evolution of this spectrum as a function of time is given in Fig.~\ref{fig:spectrogram}.
Given that its acquisition time extends from the end of accumulation to beyond the beam dump event, the spectrogram can also be used as a consistency check of the top-level timing.
Synchrotron motion creates sidebands around each harmonic of the revolution frequency with a spacing of the synchrotron frequency $f_\text{S}$~\cite{forck_schottky}.
By measuring this spacing, the RF voltage can be calculated using equation \ref{eq:rf_voltage}~\cite{2014arXiv1404.0927H}:
\begin{equation}
V_\text{RF,calc} = \left(\frac{f_\text{S}}{f_\text{rev}}\right)^2 \frac{2\pi \beta^2 E}{h \eta q}
\label{eq:rf_voltage}
\end{equation}
Here, $f_\text{rev}$ is the revolution frequency, $h=2$ the harmonic number, $\eta=0.952$ the phase slip factor, and $E/q=\SI{3.2}{\giga\volt}$ the total energy of the projectile (including its rest energy) divided by its total charge.
The resulting values of the RF voltage are given in Table~\ref{tab:real_voltage}.
It is evident that the calibration of the set values is inaccurate; in the following sections, the calculated values will be used instead.
\begin{figure}[htb]
\includegraphics{fig-spectrogram.pdf}
\caption{\label{fig:spectrogram}Example of a spectrogram of the 17\textsuperscript{th} revolution harmonic and its synchrotron sidebands measured by the Schottky pickup at $f_0 = \SI{3.256}{\mega\hertz}$. $V_\text{RF,set} = \SI{1}{\kilo\volt}$, electron bunch length \SI{500}{\nano\second}. The color corresponds to a logarithmic scale of spectral power density. $t=0$ denotes the beginning of BPM data acquisition. (1) DC cooler is switched off. (2) Beam starts being bunched. (3) Start of pulsed cooling. (4) RF is switched off. (5) End of pulsed cooling.}
\end{figure}
\begin{table}[!htbp
\caption{RF voltage derived from the measured synchrotron frequency.\label{tab:real_voltage}}
\begin{ruledtabular}
\begin{tabular}{rrr}
$V_\text{RF,set}$ (V) & $f_\text{S}$ (Hz) & $V_\text{RF,calc}$ (V) \\
\hline
600 & $287 \pm 3$ & $252 \pm 5$ \\
800 & $397 \pm 2$ & $483 \pm 6$ \\
1000 & $466 \pm 2$ & $664 \pm 6$ \\
1200 & $537 \pm 1$ & $882 \pm 4$ \\
1500 & $633 \pm 2$ & $1222 \pm 8$ \\
2000 & $753 \pm 2$ & $1730 \pm 9$
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{\label{sec:bpm_analysis}Analysis of BPM signals}
\begin{figure}[htb]
\includegraphics{fig-bpm-analysis.pdf}
\caption{\label{fig:bpm_analysis}Example of the procedure for processing BPM data frames (\SI{500}{\nano\second} electron bunches; $V_\text{RF,set} = \SI{1}{\kilo\volt}$; frame \#8). (a) Raw BPM sum signal as recorded by the oscilloscope during a single trigger event. (b) Frequency-domain representation. (c) Spectrum of filtered signal transformed to beam current. (d) Beam current in the time domain.}
\end{figure}
The output signal of the BPM, being insensitive to the transverse beam properties in this setup, allows for determination of the longitudinal beam profile as follows: First, the digitized BPM signal is windowed with a broad window that mitigates edge effects in the Fourier transform while only affecting the outer 10\,\% of the frame (Tukey; $\alpha=0.1$).
The signal is then transformed to the frequency domain using FFT, converted to beam current using eq.~\ref{equation:ohmslaw}, and transformed back to the time domain.
A 4\textsuperscript{th}-order zero-phase lowpass filter at \SI{200}{\mega\hertz} is used to remove broadband noise and resonances that are unrelated to the beam signal. An example of the transformation process is shown in Fig.~\ref{fig:bpm_analysis}.
The time-domain representation of the beam current shows significant unphysical background at frequencies below the bunch frequency.
As can be seen in the spectrum, this background cannot be filtered out completely in the digital domain because the limited acquisition time per frame causes a poor frequency resolution at the low end. However, the effect of the background on the measured bunch shape can be mitigated by performing a linear correction on each bunch.
Because the oscilloscope is triggered by a slow external signal that is not synchronized with the RF, there is no way to determine which physical bunch corresponds to which recorded bunch. Therefore, we treat the two stored bunches as one with double the revolution frequency so that each frame of data contains four or five bunches that can be averaged for further analysis.
The effect of beam cooling on the bunch shape is negligible during the time span of one frame, and the same is true for synchrotron and betatron motion.
Because the phase of the peaks is (pseudo-)random with respect to the acquisition trigger, the center of each peak is determined by the statistical mean of the signal windowed around the estimate of the respective peak center. Slight fluctuations of this value resulting from asymmetric changes of the bunch shape are unavoidable but inconsequential.
The electron beam current is reconstructed in a similar way, but the length and position of the electron bunches can be determined with greater certainty because their shape stays constant over time. As the rise and fall times are short compared to the total pulse duration, we define the bunch length as the time between the midpoints of the edges and, correspondingly, the bunch center as the point halfway between the edges, neglecting any asymmetries in the shape.
Figure~\ref{fig:single_bunch_analysis} shows an example of the bunch overlap after reconstruction.
This analysis allows a consistency check of the actual bunch overlap and length as a function of parameter settings. The delay between the bunches is shown in Fig.~\ref{fig:bunch_delay_analysis}.
The delay shows random deviations as a function of the bunch length setting, which is a result of manual adjustment between experimental runs. The systematic change in bunch delay as a function of RF voltage is assumed to be a property of the RF signal from which the pulse synchronization trigger is derived.
Figure \ref{fig:bunch_length_histogram} shows the distribution of measured electron bunch lengths; a statistical analysis is listed in table~\ref{tab:bunch_length_table}.
The bunch length is generally stable, having an RMS jitter level comparable to the temporal resolution of the BPM measurement.
However, the central value of the distribution differs from the nominal value by a varying amount, which is attributed to a deficiency of the signal transmission circuit driving the grid pulser.
The \SI{400}{\nano\second} case is peculiar in that this same issue results in two different bunch lengths being generated at random.
\begin{figure}[htb]
\includegraphics{fig-single-bunch-analysis.pdf}
\caption{\label{fig:single_bunch_analysis}Example of a frame reduced to a single bunch after corrections and averaging. The electron time axis is shifted to compensate for the time of flight between the BPMs. Parameters as in Fig.~\ref{fig:bpm_analysis}. Note that the visible irregularities in the ion bunch shape are not random and appear in all bunches within the frame.}
\end{figure}
\begin{figure}[htb]
\includegraphics{fig-bunch-delay-analysis.pdf}
\caption{\label{fig:bunch_delay_analysis}Delay between ion bunch center and electron bunch center as a function of RF voltage and electron bunch length; evaluated at the start of cooling.}
\end{figure}
\begin{figure}
\includegraphics{fig-bunch-length-histogram.pdf}
\caption{\label{fig:bunch_length_histogram}Histogram of observed electron bunch lengths (in ns), one plot per setting. Due to a deficiency of the circuit triggering the grid pulser, the central values of the distributions differ from the set point by varying amounts. In the \SI{400}{\nano\second} case, two different bunch lengths are created.}
\end{figure}
\begin{table}[htbp
\caption{Analysis of the observed electron bunch length distributions (see Fig.~\ref{fig:bunch_length_histogram})\label{tab:bunch_length_table}}
\begin{ruledtabular}
\begin{tabular}{ccc}
nominal (ns) & measured mean (ns) & measured $\sqrt{\sigma^2}$ (ns) \\
\hline
400 & 333 & 20.7 \\
500 & 441 & 1.4 \\
600 & 562 & 1.4 \\
700 & 622 & 1.6 \\
800 & 744 & 1.3 \\
900 & 864 & 1.3 \\
1000 & 925 & 2.2
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{\label{sec:ipm_analysis}Analysis of IPM signals}
The transverse beam profile projected to the horizontal axis is recorded as a function of time with an exposure time of $\SI{200}{\milli\second}$ per frame.
Because of the way the trigger initiating the acquisition is set up and the total number of frames is fixed, most of the data are taken during beam preparation prior to pulsed cooling. Only two frames are available that show the cooling process. An example of the data is shown in Fig.~\ref{fig:ipm_example}.
Because one transverse axis is averaged away during acquisition, the recorded profile has both high resolution and low noise, so its statistical properties can be analyzed without any preprocessing.
However, the observed total intensity systematically depends on the beam shape for unknown reasons, so the results must be interpreted with care.
The data are not corrupted by digital clipping or similar effects, but nonlinearities on the analog side cannot be ruled out.
\begin{figure}[htb]
\includegraphics{fig-ipm-example.pdf}
\caption{\label{fig:ipm_example}Example of the evolution of the horizontal profile during pulsed cooling. Parameters as in Fig.~\ref{fig:bpm_analysis}.}
\end{figure}
\subsection{\label{sec:coastingcooling}Cooling properties with a coasting ion beam}
\begin{figure}[htb]
\includegraphics{fig-coasting-lifetime.pdf}
\caption{\label{fig:coasting_lifetime}Dependence of the ion beam lifetime on the frequency mismatch between ion and electron pulses. The error bars result from averaging over multiple runs.}
\end{figure}
As a first check of the cooling dynamics, we investigated the process of a coasting ion beam being cooled by the pulsed electron beam with the RF cavity in the ring switched off. The frequency of the electron pulse repetition signal was varied around the ion revolution frequency. As shown in Fig.~\ref{fig:coasting_lifetime}, an anomalous reduction of the lifetime of the stored ion beam was observed at mismatched frequencies, while in the case of equal frequencies, the lifetime is equal to that observed with a DC electron beam.
As will be shown in section \ref{sec:spacecharge}, this particle loss is caused by space charge kicks perturbing the phase space in both the transverse and the longitudinal plane.
\begin{figure}[htb]
\includegraphics{fig-coasting-bunching.pdf}
\caption{\label{fig:coasting_bunching}Evolution of the longitudinal beam profile of the initially unbunched ion beam during the cooling process measured at different electron pulse lengths. (a) \SI{500}{\nano\second}, (b) \SI{700}{\nano\second}, (c) \SI{900}{\nano\second}. The color scale corresponds to an arbitrary unit of beam current and is normalized to an independent measurement of the respective DC current prior to cooling for the sake of comparability. The apparent increase in total intensity over time results from the inability of the BPM to detect the DC component of the beam. The non-uniform intensity distribution of the cooled ion bunch is identical among independent experiments and was also observed in our previous simulations, particularly for shorter electron pulses~\cite{ZHAO2018219}.}
\end{figure}
In the case of frequency-matched pulses, another interesting phenomenon is the bunching of the coasting beam as a result of the bucket created by the space charge field of the electron beam (called \enquote{grouping effect} in prior work \cite{ZHAO2018219, Mao:COOL2017-TUP15}). The evolution of the longitudinal ion beam profile in this case is shown in Fig.~\ref{fig:coasting_bunching}. It is evident that the ions are captured in a longitudinal space corresponding to the electron bunch length. Because the cooling force eventually reduces their momentum deviation below the bucket height, they cannot escape from the bucket.
\subsection{\label{sec:cooling}Cooling properties with a bunched ion beam}
In the main part of the experiment, the RF system was switched on, capturing the ions in corresponding buckets after accumulation.
This way, we were able to measure a bunched ion beam being cooled by a pulsed electron beam including synchrotron dynamics.
Using both the longitudinal and transverse beam profiles recorded during pulsed cooling with different RF voltages and electron bunch lengths, we can derive the respective cooling rates as a function of these parameters.
An example of the temporal evolution of the longitudinal bunch shape during cooling is shown in Fig.~\ref{fig:bunch_evolution}.
We observe an overall reduction of the bunch length as a function of time. Regardless of the initial shape, the resulting profile is non-Gaussian with irregularities both near the core and in the tails. While the overall shape of the profile is quantifiable by computing the sample moments, the details of these features vary so much across the data set that a dedicated systematic measurement will be warranted if they are to be fully understood.
\begin{figure}[htb]
\includegraphics{fig-bunch-evolution.pdf}
\caption{\label{fig:bunch_evolution}Example of the evolution of the longitudinal bunch profile. Parameters as in Fig.~\ref{fig:bpm_analysis}. The cooling beam is switched on at $t = \SI{0.1}{\second}$.}
\end{figure}
For lack of an analytic model of the bunch shape, we quantify its properties by extracting the following quantities directly from the intensity $I(t)$, with $t$ being an equidistantly spaced, discrete time axis:
\begin{align}
Q &= \sum I & \quad(\propto \text{bunch charge}) \\
\hat{t} &= \frac{\sum I(t) t}{\sum I(t)} & (\text{bunch center}) \\
\sigma^2 &= \frac{\sum I(t) (t-\hat{t})^2}{\sum I(t)} & (\text{variance}) \\
\kappa &= \frac{\sum I(t) (t-\hat{t})^4}{\sigma^4 \sum I(t)} & (\text{kurtosis})
\end{align}
$\sigma = \sqrt{\sigma^2}$ is a measure of the overall width of the distribution. The value of $\kappa$ is 3 for a Gaussian shape; a higher value indicates a shift of the probability density toward the tails of the distribution.
Figure \ref{fig:bpm_moments_1kv} shows the evolution of these quantities at a nominal RF voltage of \SI{1}{\kilo\volt}. The most salient aspect of this result is the particle loss taking place at an electron bunch length of \SI{400}{\nano\second}. This effect is visible at any RF voltage and can be attributed to single particles randomly being subject to both longitudinal and transverse heating as a result of the varying space-charge-induced focusing force brought about by the electron bunch length jitter, which is a unique property of the \SI{400}{\nano\second} setting (see Fig.~\ref{fig:bunch_length_histogram}).
A simulation showing this mechanism is described in section \ref{sec:spacecharge}.
Apart from particle loss, we observe a monotonic dependency between bunch length and cooling rate.
Since the longitudinal current density in the electron bunches is constant, i.e.~changing the bunch length changes only the longitudinal overlap but not the peak current, this result is to be expected as long as the length of the electron bunches does not significantly exceed that of the ion bunches.
Conversely, the evolution of the kurtosis hints at a tendency for longer electron bunches to be detrimental to preserving a Gaussian shape.
While our experiment was carried out at a constant phase between electron and ion bunches (albeit not perfectly, see Fig.~\ref{fig:bunch_delay_analysis}), this result warrants a dedicated measurement of how the evolution of the bunch tails is affected by the phase.
\begin{figure}[htb]
\includegraphics{fig-bpm-moments-1kv.pdf}
\caption{\label{fig:bpm_moments_1kv}Evolution of the statistical properties of the longitudinal ion bunch profile as a function of time, averaged over five identical experimental runs per setting. Each color corresponds to one setting of the electron bunch length. The cooling beam is switched on at $t = \SI{0.1}{\second}$. $V_\text{RF,set} = \SI{1}{\kilo\volt}$. The error bars represent the statistical error of the mean.}
\end{figure}
As shown in section \ref{sec:jspec}, the asymptotic behavior toward the end of the process is predominantly caused by intra-beam scattering (IBS) counteracting the cooling force.
Because this effect does not contribute significantly until the bunches become short, the longitudinal and transverse cooling rates are determined from the evolution of the bunch length or width, respectively, in a region of the curves where no asymptotic behavior is visible.
The relative slope of the $\sigma$ curve is summarized in Fig.~\ref{fig:bpm_moments_diff} for all parameter sets of our experiment.
The longitudinal cooling rate increases monotonically as a function of the electron bunch length, which is to be expected because longer bunches lead to a higher overlap between the bunches on average. It can, however, be observed that increasing the RF voltage leads to the opposite effect even though that, too, increases the overlap. We suspect this reduction in cooling rate with increasing RF voltage to be a result of the broadening of the longitudinal momentum distribution, which causes the average cooling force to decrease despite the higher bunch overlap.
This effect must be especially strong when the overlap is centered around the bunch center, where the momentum deviation of high-amplitude particles is highest by virtue of synchrotron motion.
The transverse cooling rate increases monotonically as a function of both the RF focusing strength and the electron bunch length because it only depends on the longitudinal overlap between the bunches. The transverse overlap is always the same, and stronger RF focusing does not broaden the transverse momentum distribution.
The change in focusing strength due to the bunch-length-dependent space charge potential of the electron beam should also have an effect on the bunch shape but cannot be determined in isolation in this experiment.
\begin{figure}[htb]
\includegraphics{fig-bpm-moments-diff.pdf}
\caption{\label{fig:bpm_moments_diff}Longitudinal and transverse cooling rates as a function of electron bunch length and RF voltage, calculated in the interval $\SI{0.2}{\second} \leq t \leq \SI{0.4}{\second}$ and averaged over five identical experimental runs per setting. The error bars represent the statistical error of the mean.}
\end{figure}
\section{\label{sec:jspec}Simulation of cooling and IBS rates}
\begin{table}[!htbp
\caption{Cooling simulation parameters\label{tab:jspec_parameters}}
\begin{ruledtabular}
\begin{tabular}{lc}
Transverse electron beam radius & \SI{15}{\milli\meter} \\
Effective cooler length & \SI{3.4}{\meter} \\
Magnetic field & \SI{0.1}{\tesla} \\
Average $\beta_x$ / $\beta_y$ in the cooler & \SI{10}{\meter} / \SI{17}{\meter} \\
Peak electron current (uniform bunch shape) & \SI{30}{\milli\ampere} \\
Transverse electron temperature & \SI{200}{\milli\electronvolt} \\
Longitudinal electron temperature & \SI{6}{\milli\electronvolt} \\
RMS normalized transverse emittance & \SI{0.6}{\milli\meter\milli\radian} \\
RMS long.~ion bunch size $\sigma_z$ & \SI{10.5}{\meter} \\
RMS long.~ion momentum deviation $\sigma_{p_z / p}$ & \num{7e-4}
\end{tabular}
\end{ruledtabular}
\end{table}
To gain confidence in the interpretation of the results, a set of simulations mimicking the experimental conditions was carried out using the code JSPEC~\cite{Zhang:IPAC2016-WEPMW014, jspec}.
In these simulations, the evolution of the bunch shape was computed on a turn-by-turn basis by applying theoretical models for both the cooling force and the heating effect caused by intra-beam scattering (IBS).
RF focusing is parametrized by the synchrotron frequency.
Table \ref{tab:jspec_parameters} shows the input parameters used. The longitudinal profile of the simulated electron bunches is rectangular; since the true bunch length in the experiment is different from the nominal setting as shown in Fig.~\ref{fig:bunch_length_histogram}, the averages of the respective measured lengths are used for the simulation.
While the code can consider an arbitrary temporal alignment between the bunches, a systematic study of the effect of the alignment was not part of the experiment, and the small offsets shown in Fig.~\ref{fig:bunch_delay_analysis} are considered negligible for the purpose of this study.
Because the cooling rate calculation is particularly sensitive to beam parameters that are hard to determine accurately, e.g.~the electron beam temperature, the resulting absolute numbers only serve as an order-of-magnitude estimate; however, the qualitative behavior and the relative dependence on the cooling bunch length can still be meaningfully compared to the experimental data.
Figure \ref{fig:jspec_rates} shows the evolution of the cooling rate, the IBS rate, and the resulting bunch length from the simulation at $f_\text{S} = \SI{466}{\hertz}$, corresponding to $V_\text{RF,set}=\SI{1}{\kilo\volt}$ as in Fig.~\ref{fig:bpm_moments_1kv}.
Here, we concentrate on the longitudinal behavior because the simulation is assumed to model the transverse cooling force inaccurately.
The bunch length evolution observed in the experiment is qualitatively reproduced.
While the beginning of the process is dominated by cooling, the contribution of IBS increases significantly as a result of the three-dimensional compression of the ion bunch, eventually canceling out the cooling effect and resulting in an equilibrium.
Table \ref{tab:jspec_comparison} shows a comparison of the measured and simulated bunch length reduction rates, the latter being determined from the slope of the bunch length curve in the same way as in Fig.~\ref{fig:bpm_moments_diff}.
While the simulation tends to overestimate the cooling rate toward lower electron bunch lengths, the overall behavior is in reasonable agreement considering the sensitivity to the models and beam parameters.
\begin{figure}[htb]
\includegraphics{fig-jspec-rates.pdf}
\caption{\label{fig:jspec_rates}Simulation of cooling rate, IBS rate, and ion bunch length with parameter sets similar to the experiment. Note that the definition of the rates used here refers to the emittance rather than the bunch length. The \SI{400}{\nano\second} case was omitted because this type of simulation cannot reproduce the effect of bunch length jitter. $f_\text{S}=\SI{466}{\hertz}$.}
\end{figure}
\begin{table}[!htbp
\caption{Comparison between simulated and measured bunch length reduction rates. $T_\text{set}$ and $T_\text{exp}$ are the nominal and measured electron bunch lengths, respectively. $R_\text{exp}$ is the longitudinal bunch length reduction rate as shown in Fig.~\ref{fig:bpm_moments_diff}; $R_\text{sim}$ is the corresponding simulation result.\label{tab:jspec_comparison}}
\begin{ruledtabular}
\begin{tabular}{ccccc}
$T_\text{set}$ (ns)& $T_\text{exp}$ (ns) & $R_\text{exp}$ (\si{\per\second})& $R_\text{sim}$ (\si{\per\second}) & $R_\text{sim} / R_\text{exp}$ \\
\hline
500 & 441 & 0.60 & 0.89 & 1.48 \\
600 & 562 & 0.92 & 1.17 & 1.28 \\
700 & 623 & 0.98 & 1.23 & 1.25 \\
800 & 744 & 1.27 & 1.45 & 1.14 \\
900 & 864 & 1.45 & 1.69 & 1.16 \\
1000 & 925 & 1.67 & 1.77 & 1.06
\end{tabular}
\end{ruledtabular}
\end{table}
\section{\label{sec:spacecharge}Simulation of space-charge effects}
The particle loss taking place in the runs with \SI{400}{\nano\second} electron bunch length was initially attributed to a transverse heating mechanism due to uneven space charge kicks caused by bunch length jitter. This mechanism had already been described both theoretically and with simulations for the case of LEReC, where, assuming a significant phase jitter, the heating is not negligible but still small enough not to cause considerable particle losses \cite{gangwang, Blaskiewicz:NAPAC2016-WEA3IO01}.
While the same basic principle applies to our experiment, the particles are nonrelativistic and space charge forces comparatively high as a result.
The possibility of the longitudinal dynamics also being affected should therefore not be ruled out.
In an effort to assess the magnitude of the effect, we carried out a tracking simulation including the space charge kicks from the electron beam in the cooler.
The simulation code tracks the 6-dimensional phase space coordinates of randomly generated single ions through the linear transport matrices of the ring and applies the longitudinal cavity kick at each revolution. The cooling section is discretized by a finite number of drifts, in which the electron charge distribution is placed according to the relative phase of the ion when it reaches each individual piece of drift. Changes in velocity and, thus, relative time of flight are accounted for. The charge distribution is then discretized on a cartesian, three-dimensional grid and the Coulomb force computed.
Collective effects of the ion beam are assumed to be negligible.
This model includes neither cooling nor IBS so that the space charge effect can be observed in isolation.
The assumptions made as input to the simulation are listed in table~\ref{tab:simulation_params}.
While the simplicity of the model causes some of them to be arbitrary, they do not affect the qualitative outcome significantly.
\begin{table}[!htbp
\caption{Parameters for tracking simulations and theoretical estimates\label{tab:simulation_params}}
\begin{ruledtabular}
\begin{tabular}{lc}
Peak electron current $I_\text{e}$ & \SI{30}{\milli\ampere} \\
Bunch length $T_\text{bunch}$ & \SI{320}{\nano\second} or \SI{370}{\nano\second} \\
Electron current rise/fall time & \SI{30}{\nano\second} \\
Transverse electron beam radius $a_\text{e}$ & \SI{15}{\milli\meter} \\
Cooler length $L_\text{cool}$ & \SI{3.4}{\meter} \\
Ion ring betatron tunes $Q_{x,y}$ & 3.62 / 2.61 \\
Ion ring RF voltage (see table \ref{tab:real_voltage}) & \SI{664}{\volt} \\
Transverse aperture limit $(\sqrt{x^2+y^2})_\text{max}$ & \SI{50}{\milli\meter} \\
Number of ions & 400 \\
RMS transverse ion bunch radius $\sigma_{x,y}$ & \SI{5}{\milli\meter} \\
RMS transverse ion momentum $\sigma_{p_{x,y}} / p_z$ & \num{1e-3} \\
RMS long.~ion bunch size $\sigma_z$ & \SI{7.5}{\meter} \\
RMS long.~ion momentum deviation $\sigma_{p_z / p}$ & \num{5e-4}
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}[!htb]
\includegraphics{fig-jitter-particle.pdf}
\caption{\label{fig:jitter_particle}Phase space trajectory of one example particle placed randomly in 6-d phase space and subjected to cooling bunches with bunch length jitter. The perturbation of the longitudinal motion is inconsequential, while fluctuations of the transverse action eventually lead to a collision with a limiting aperture. The plot only shows every 20\textsuperscript{th} revolution for clarity, whereas collisions with the aperture are checked after each machine element.}
\end{figure}
To simulate the bunch length jitter present in the \SI{400}{\nano\second} experiment, we assume the starting edge of any bunch to randomly arrive early by \SI{50}{\nano\second} with a probability of 1/3 to mimic the distribution in Fig.~\ref{fig:bunch_length_histogram}.
The phase space trajectory of an example particle subjected to these conditions is shown in Fig.~\ref{fig:jitter_particle}.
While the action in the longitudinal plane is visibly affected, there is no indication of a spontaneous blow-up or instability that would explain the loss of the particle by itself.
The transverse plane, however, shows a significant deformation of the trajectory over time.
Under the simplifying assumption that any particle with a transverse displacement of $\sqrt{x^2+y^2} > \SI{50}{\milli\meter}$ is lost, which is checked after each optical element, we obtain the number of lost particles as a function of time; the result is shown in Fig.~\ref{fig:jitter_lossrate}.
The observed loss of 18\,\% in \SI{0.5}{\second} is in reasonable agreement with the experiment considering the simplifications.
The same simulation without bunch length jitter gives no particle loss.
\begin{figure}[t]
\includegraphics{fig-jitter-lossrate.pdf}
\caption{\label{fig:jitter_lossrate}Relative particle loss with bunch length jitter as a function of time. The initial sample contains 400 particles randomly distributed according to table~\ref{tab:simulation_params}.}
\end{figure}
These results allow us to make a comparison with existing theoretical models: It is shown in \cite{gangwang} that a randomly distributed electron bunch arrival time of variance $\left<\delta_t^2\right>$ results in an exponential ion emittance growth, assuming longitudinally Gaussian-shaped electron bunches of length $\sigma_\text{e}$.
While the case at hand is slightly different in that our bunches are rectangular and their length varies, it is similar enough to make a comparison as a rough consistency check.
The growth time in either the $x$ or $y$ plane is given by \cite{gangwang}:
\begin{align}\label{tau_theo}
\tau_{\text{theo},x/y} = \frac{\exp(1)}{8 \pi^2 f_\text{rev}} \frac{\sigma_\text{e}^2}{\left<\delta_t^2\right>} \frac{1}{\Delta\nu_{\text{peak},x/y}^2} .
\end{align}
We choose the equivalent Gaussian bunch length $\sigma_\text{e}$ such that the FWHM is equal to the true rectangular bunch length; with the ion radius $r_0$, the number of electrons per bunch $N_\text{e}$, and the arrival time variance $\left<\delta_t^2\right>$ as given by
\begin{align}
r_0 &= \SI{4.5e-19}{\meter}\;\text{for}\;{}^{86}\mathrm{Kr}^{25+} ,\\
N_\text{e} &= \frac{I_\text{e}}{e_0} (T_\text{bunch} + T_\text{rise/fall}) = \num{7.1e10} , \\
\sigma_\text{e} &= \frac{T_\text{bunch}}{2\sqrt{2 \ln 2}} = \SI{161}{\nano\second} , \\
\left<\delta_t^2\right> &= (\SI{20.7}{\nano\second})^2 ,
\end{align}
we obtain \cite{gangwang}:
\begin{align}
\Delta \nu_{\text{peak},x/y} &= \frac{N_\text{e} r_0 L_\text{cool}}{(2\pi)^2 Q_{x/y} f_\text{rev} \sqrt{2\pi} \sigma_\text{e} \gamma^3 \beta^2 a_\text{e}^2} \\
&= \begin{pmatrix}\num{4.1e-3}\\\num{5.6e-3}\end{pmatrix} , \\
\tau_{\text{theo},x/y} &= \begin{pmatrix}\SI{0.66}{\second}\\\SI{0.34}{\second}\end{pmatrix} .
\end{align}
For comparison with this model, we can extract an emittance growth time from the full simulated ensemble, keeping in mind that the emittance ceases to be well-defined in the face of particle loss and that the heated ions spend considerable time outside of the cooling beam, which is in contrast to the assumptions made in \cite{gangwang}.
The RMS emittance in the $y$ plane (chosen here because of the non-dispersive optics) is calculated from the ensemble phase space via
\begin{equation}\label{eq:emittance}
\epsilon_y = \sqrt{\left<y^2\right> \left<y'^2\right> - \left<y y'\right>^2} ,
\end{equation}
where $y$ denotes the displacement and $y'$ the angle of each particle with respect to the reference orbit.
In this calculation, to avoid discontinuities in the data, lost particles are kept, but their emittance does not grow further.
Figure~\ref{fig:jitter_emittance_y} shows the evolution of the emittance.
While the data do not show a strictly exponential time dependence, applying an exponential fit allows us to extract a growth time of $\tau = \SI{0.22}{\second}$ for the sake of comparison with the model-based value $\tau_\text{theo}$.
The latter is higher by a factor of \num{1.5}.
\begin{figure}[htb]
\includegraphics{fig-jitter-emittance-y.pdf}
\caption{\label{fig:jitter_emittance_y}Growth of the RMS emittance as a result of bunch length jitter as obtained from the tracking simulation. The growth time $\tau$ is a free parameter, while the initial emittance $\epsilon_{y,0}$ is fixed.}
\end{figure}
Considering the number of assumptions going into both the simulation and the growth rate estimate, the discrepancy between the results is not surprising.
However, the qualitative agreement between the methods demonstrates their basic applicability and consistency.
While the results we obtained explain the particle loss observed in our experiment and imply that the bunch phase and length ought to be held as constant as possible in bunched-beam devices, the consequences of the effect are expected to be less severe for relativistic beams as space charge forces diminish at $\gamma \gg 1$.
The theory presented in \cite{gangwang} appears to be able to make a good prediction of the heating time in that case.
\section{\label{sec:conclusion}Conclusion}
We have demonstrated cooling of both coasting and bunched ion beams using a pulsed electron beam from a conventional cooler, allowing studies of bunched-beam cooling with relatively little technical effort and investment.
Even though the parameters of our experiment are far away from those of the most likely application of bunched-beam cooling, i.e.~high-energy protons \cite{eic}, the basic physics are similar.
The experimental result has been shown to be in reasonable albeit not perfect agreement with simulations.
Improving the input parameters of these simulations would require considerable beam diagnostics efforts.
While the level of agreement we achieved allows us to be confident in the overall scheme, it does not exclude the possibility of there being small heating effects in the set-up that have not been fully explored.
We have shown the observed particle loss to likely result from random transverse heating caused by uneven space charge kicks. This observation hints at the practical importance of maintaining the cooling bunch length and phase with high accuracy. An important implication is that sweeping the cooling bunch longitudinally on a short time scale is likely not an option for future machines. However, considering that the cooling time of conceivable high-energy coolers is between many minutes and an hour \cite{eic}, applying such a sweeping technique on a long time scale may still be feasible.
\begin{acknowledgments}
The authors would like to thank all staff of the CSR operation group at IMP.
We would also like to thank John Musson, Edith Nissen, Christiana Wilson, and Jianxun Yan of Jefferson Lab for supporting the experimental runs, and Robert McKeown and Mike Spata of Jefferson Lab for encouragement and support.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177.
This experiment is supported by the International Partnership Program of Chinese Academy of Sciences, Grant No.~113462KYSB20170051, and the National Natural Science Foundation of China, No.~11575264.
\end{acknowledgments}
|
1,108,101,566,755 | arxiv | \section{Introduction}
One of the basic problems of representation theory is the study of irreducible spherical unitary representations and their spherical functions. It is well known that spherical functions on classical symmetric spaces of compact type are expressed through multivariate orthogonal polynomials --- the Jack and Jacobi symmetric polynomials with certain special values of parameters.
The notion of spherical function also makes sense for infinite-dimensional symmetric spaces of the form
$$
G_\infty/K_\infty =\varinjlim G_N/K_N,
$$
where
$$
\cdots \to G_N/K_N \to G_{N+1}/K_{N+1}\to\cdots\,.
$$
is a chain of nested finite-dimensional symmetric spaces.
There are 10 infinite series $\{G_N/K_N\}$ of compact classical symmetric spaces of growing rank, with natural embeddings $G_N/K_N \to G_{N+1}/K_{N+1}$. For each such series, there are plenty of indecomposable spherical functions on $G_\infty/K_\infty$ indexed by countably many continuous parameters. It turns out that the description of spherical functions on $G_\infty/K_\infty$ is equivalent to finding the entrance boundary of a Markov chain obtained as a \emph{dualization} of the chain
$\{G_N/K_N\}$.
This reformulation is important for (at least) two reasons:
\smallskip
(1) the initial problem setting can be extrapolated to the case of Jack/Jacobi symmetric polynomials with \emph{general} parameters;
(2) the entrance boundary of the resulting Markov chain can be found by tools of algebraic combinatorics (see Okounkov and the author \cite{OO-IMRN}, \cite{OO-2006}).
\smallskip
The goal of the present paper is to move up the theory to the level of Macdonald polynomials. The main results are three theorems denoted as Theorem A, Theorem B, and Theorem C.
\medskip
$\bullet$ In Theorem A (see section \ref{results1}) we construct a certain Markov chain depending on $q$ and $t$; we call it the \emph{extended Macdonald chain}. The theorem is deduced from Okounkov's $q$-integral representation of Macdonald polynomials \cite{Ok-CM}.
\smallskip
$\bullet$ Theorem B (see section \ref{results2}) describes the entrance boundary of the extended Macdonald chain.
\smallskip
$\bullet$ Theorem C (see section \ref{results3}) is an approximation theorem. It shows that each probability measure on the boundary of the extended Macdonald chain can be obtained, in a canonical way, via a large-$N$ limit transition from random $N$-particle systems.
\medskip
These results are applied in \cite{Ols-MacdonaldTwo} to constructing a $(q,t)$-deformed combinatorial version of harmonic analysis on $U(\infty)$.
We proceed to a more detailed description of the contents of the paper.
\subsection{Formalism of projective chains}\label{sect1.1}
Recall that a (rectangular) matrix is said to be \emph{stochastic} if its entries are nonnegative and the row sums are equal to $1$. Stochastic matrices are a particular case of \emph{Markov kernels} (Meyer \cite{Meyer}). A Markov kernel is a map $L: S\to \P(S')$, where $S$ and $S'$ are two Borel (=measurable) spaces and $\P(S')$ denotes the space of probability measures on $S'$. When both $S$ and $S'$ are finite or countable sets, $L$ is given by a stochastic matrix of format $S\times S'$. That is, its rows are indexed by $S$ and the columns are indexed by $S'$.
Informally, one can treat $L$ as a `generalized map' from $S$ to $S'$. We denote such a surrogate of map by a dash arrow, $S\dasharrow S'$.
By a \emph{projective chain} we mean an infinite sequence $S_1, S_2, S_3, \dots$ of finite or countable sets linked by stochastic matrices $L^2_1, L^3_2, \dots$, where the matrix $L^N_{N-1}$ has format $S_N\times S_{N-1}$. This is symbolically represented by the diagram
\begin{equation}\label{eq1.A}
S_1\stackrel{L^2_1}{\dashleftarrow}S_2\stackrel{L^3_2}{\dashleftarrow}S_3\stackrel{L^4_3}{\dashleftarrow}\dots\,.
\end{equation}
For any projective chain one can define, in a canonical way, its \emph{boundary}. It is a Borel space $S_\infty$, which is linked to the sets $S_N$ via Markov kernels $L^\infty_N$ satisfying the relations $L^\infty_N L^N_{N-1}=\La^\infty_{N-1}$ (a composition of Markov kernels is read from left to right). The precise definition is given in section \ref{sect7.1}.
One can regard the diagram \eqref{eq1.A} as a non-stationary Markov chain with discrete time $N$ ranging in reverse direction (from $+\infty$ to $1$), time-dependent state spaces $S_N$, and transition kernels $L^N_{N-1}$; then $S_\infty$ is identified with what may be called the \emph{entrance boundary} of that chain, or the set of \emph{extreme entrance laws} in the terminology of Dynkin \cite[\S10.1]{Dynkin}. A comprehensive discussion is contained in Winkler's monograph \cite[ch. 4]{Winkler}.
The boundary $S_\infty$ can also be interpreted as the inverse limit of \eqref{eq1.A} in the category-theoretical sense. The corresponding category is formed by standard Borel spaces (as objects) and Markov kernels (as morphisms), see \cite[ch. 4]{Winkler}.
Here is an illustrative example.
\begin{example}[Boundary of Pascal graph]
Let $S_N:=\{0,1,\dots, N\}$ and $L^N_{N-1}$ be the two-diagonal matrix with the entries
$$
L^N_{N-1}(n,n-1)=\frac nN, \qquad L^N_{N-1}(n,n):=\frac{N-n}N,
$$
and all other entries being equal to $0$. One can show that the boundary $S_\infty$ of this chain is the closed interval $[0,1]$ with the Markov kernels from $S_\infty\dasharrow S_N$, $N=1,2,\dots,$ given by
$$
L^\infty_N(x,n)=\binom Nn x^n(1-x)^{N-n}, \qquad n=0,1,\dots,N.
$$
This result is in fact equivalent to classical de Finetti theorem (\cite[section 5]{BO-2017}). It is also related to other classical topics --- Bernstein polynomials and Hausdorff moment problem (Feller \cite[ch. VII]{Feller}).
\end{example}
Even in this simple example, finding the boundary requires some work. For more sophisticated projective chains, this task may require considerable efforts.
\subsection{Projective chains in representation theory}
In the body of the text there are no group representations, we work exclusively with symmetric polynomials and symmetric functions. However, because the extended Macdonald chain originated as a generalization of some representation-theoretic constructions, it makes sense to tell a little about these constructions --- otherwise the problem setup will not be sufficiently motivated.
Let $G\supset K$ be finite or separable compact groups forming a \emph{Gelfand pair} (Bump \cite[\S45]{Bump}); for instance, one may suppose that $G/K$ is a \emph{symmetric space of compact type} (Helgason \cite{H}). The \emph{spherical dual of $(G,K)$} is the set of indecomposable positive definite normalized functions $G\to\C$, constant on double $K$-cosets. This is a finite or countable set, which we denote by $\Om(G,K)$. It parameterizes the irreducible \emph{spherical} representations --- the irreducible unitary representations of $G$ possessing a $K$-invariant vector.
By a \emph{morphism $(G,K)\to(G',K')$ of Gelfand pairs} we mean a group homomorphism $\phi:G\to G'$ such that $\phi(K)\subseteq K'$. For noncommutative groups one cannot define a natural dual map $\phi^*$ from $\Om(G',K')$ to $\Om(G,K)$. However, there is a reasonable substitute --- a `generalized map' $\Om(G',K')\dasharrow \Om(G,K)$ given by a stochastic matrix of format $\Om(G',K')\times\Om(G,K)$.
This stochastic matrix, which we denote by $L^{G'}_G$, is defined in the following way. Given a function $\om'\in\Om(G',K')$, its composition with $\phi:G\to G'$ produces a positive definite normalized function $\om'\circ\phi$ on $G$; the latter is uniquely written as a convex combination of indecomposable spherical functions of $(G,K)$ with certain coefficients $L^{G'}_G(\om',\om)$,
\begin{equation}\label{eq1.D}
\om'\circ\phi=\sum_{\om\in\Om(G,K)}L^{G'}_G(\om',\om)\om,
\end{equation}
and these coefficients are just the entries of the matrix $L^{G'}_G$.
Now suppose we are given an infinite sequence
\begin{equation}\label{eq1.B}
(G_1,K_1)\to (G_2,K_2)\to (G_3,K_3)\to\dots
\end{equation}
of growing (finite or compact) Gelfand pairs. It gives rise to the dual projective chain
\begin{equation}\label{eq1.C}
\Om(G_1,K_1)\dashleftarrow \Om(G_2,K_2)\dashleftarrow \Om(G_3,K_3)\dashleftarrow\dots
\end{equation}
which has a certain boundary.
On the other hand, consider the inductive limit groups $G_\infty:=\varinjlim G_N$ and $K_\infty:=\varinjlim K_N$. In general, these are no longer compact groups, but the pair $(G_\infty,K_\infty)$ is still a Gelfand pair (in the sense explained in Olshanski \cite{Ols-1990}). Next, the corresponding spherical dual $\Om(G_\infty,K_\infty)$ is defined in exactly the same way as above, and the elements of $\Om(G_\infty,K_\infty)$ parametrize the irreducible spherical representations of $(G_\infty,K_\infty)$.
As pointed out above, there exist 10 series of classical symmetric spaces $G_N/K_N$ of compact type, and for each series, the spherical dual $\Om(G_\infty,K_\infty)$ is known (Olshanski \cite{Ols-1990}, Pickrell \cite{Pickrell}).
It is a direct consequence of definitions that $\Om(G_\infty,K_\infty)$ can be identified with the boundary of the projective chain \eqref{eq1.C}. Thus, this boundary has a representation-theoretic meaning.
\subsection{Projective chains related to Jack polynomials}
Introduce some notation:
\smallskip
$\bullet$ $\Sign(N)$ is the set of \emph{signatures of length $N$}; these are vectors $a=(a_1,\dots,a_N)\in\Z^N$ such that $a_1\ge\dots\ge a_N$ (the coordinates $a_i$ may be of arbitrary sign);
$\bullet$ $\tau$ is an arbitrary positive real number --- the parameter of Jack polynomials ($\tau$ is inverse to the parameter $\al$ used in Macdonald's book \cite{M});
$\bullet$ $P_a(u_1,\dots,u_N;\tau)$ is the Jack polynomial indexed by a signature $a\in\Sign(N)$ (in general, $P_a(\ccdot;\tau)$ is a Laurent polynomial; the definition given in \cite[ch. VI, \S10]{M} extends to the case of Laurent polynomials without difficulty);
$\bullet$ $u_1,\dots,u_N$ are variables;
$\bullet$ $(1^N)=(1,\dots,1)$ ($N$ times).
\smallskip
For three special values $\tau=\frac12, 1, 2$, the normalized Jack polynomials
$$
\frac{P_a(u_1,\dots,u_N;\tau)}{P_a((1^N); \tau)}
$$
give indecomposable spherical functions for the symmetric spaces
$$
U(N)/O(N), \qquad U(N)\times U(N)/ U(N), \qquad U(2N)/Sp(N),
$$
respectively. In terms of Jack polynomials, the expansion \eqref{eq1.D} takes the form
\begin{multline}\label{eq1.E}
\left.\frac{P_a(u_1,\dots,u_N;\tau)}{P_a((1^N); \tau)}\right|_{u_N=1}\\
=\sum_{b\in\Sign(N-1)} L^N_{N-1}(a,b;\tau)\frac{P_b(u_1,\dots,u_{N-1};\tau)}{P_b((1^{N-1});\tau)}, \qquad a\in\Sign(N),
\end{multline}
and the coefficients $L^N_{N-1}(a,b;\tau)$ defined by \eqref{eq1.E} form a matrix $L^N_{N-1}$ of format $\Sign(N)\times\Sign(N-1)$ depending on $\tau$.
A remarkable fact is that the coefficients $L^N_{N-1}(a,b;\tau)$ are nonnegative not only for special values $\tau=\frac12, 1,2$ corresponding to spherical functions, but also for any $\tau>0$. This implies that the matrices $L^N_{N-1}$ are stochastic matrices for any $\tau>0$ (the fact that the row sums equal $1$ is obvious).
In this way we obtain a projective chain with the states $S_N=\Sign(N)$, $N=1,2,\dots$, and parameter $\tau>0$; let us call it the \emph{Jack projective chain}. Its boundary was described in Okounkov--Olshanski \cite{OO-IMRN}.
For the remaining $10-3=7$ series of symmetric spaces, the spherical functions are expressed through Heckman--Opdam's multivariate Jacobi polynomials (see e.g. Heckman's lecture notes in \cite{Heckman}) with special values of parameters, and the definition \eqref{eq1.D} can again be extrapolated to general values of parameters. The corresponding boundary problem was solved in Okounkov--Olshanski \cite{OO-2006}.
\subsection{The Macdonald chain}\label{sect1.4}
There is a natural extension of \eqref{eq1.E}, where the Jack polynomials are replaced by the Macdonald polynomials and the specialization at $(1^N)$ is replaced by that at $(1,t^{-1},\dots,t^{1-N})$:
\begin{multline}\label{eq1.F}
\left.\frac{P_a(u_1,\dots,u_N;q,t)}{P_a(1,t^{-1},\dots,t^{1-N};q,t)}\right|_{u_N=t^{1-N}}\\
=\sum_{b\in\Sign(N-1)} L^N_{N-1}(a,b;q,t)\frac{P_b(u_1,\dots,u_{N-1};q,t)}{P_b(1,t^{-1},\dots,t^{2-N};q,t)}, \qquad a\in\Sign(N).
\end{multline}
An explicit expression for the coefficients $L^N_{N-1}(a,b;q,t)$ is provided by the \emph{branching rule for Macdonald polynomials} \cite[ch. VI, (7.14$'$)]{M}. In particular, the coefficients vanish unless the signatures $a$ and $b$ \emph{interlace}, meaning that
$$
a_i\ge b_i\ge a_{i+1}, \qquad 1\le i\le N-1.
$$
In this case we write $b\prec a$ or $a\succ b$.
(Macdonald deals with ordinary (not Laurent) polynomials, which are indexed by partitions, but extending the results that we need to the more general case of Laurent polynomials, which are indexed by signatures, presents no difficulty.)
Suppose $0<q<1$, $0<t<1$ (equally well one could take $q>1$, $t>1$); then the coefficients $L^N_{N-1}(a,b;q,t)$ are nonnegative. Next, from \eqref{eq1.F} it is evident that
$$
\sum_{b\in\Sign(N-1)}L^N_{N-1}(a,b;q,t)=1, \qquad \forall a\in\Sign(N).
$$
It follows that the matrices $L^N_{N-1}(q,t)$ with the entries $L^N_{N-1}(a,b;q,t)$ are stochastic matrices. In this way we obtain a projective chain,
\begin{equation}\label{eq1.L}
\Sign(1)\stackrel{L^2_1(q,t)}{\dashleftarrow}\Sign(2)\stackrel{L^3_2(q,t)}{\dashleftarrow}\Sign(3)\stackrel{L^4_3(q,t)}{\dashleftarrow}\cdots,
\end{equation}
which we call the \emph{Macdonald chain}; it depends on the two Macdonald parameters, $q$ and $t$, ranging over the open interval $(0,1)$.
Because of the limit relation
$$
P_a(u_1,\dots,u_N;\tau)=\lim_{q\to 1}P_a(u_1,\dots,u_N;q,q^\tau)
$$
(\cite[ch. VI, \S10]{M}) we have
$$
L^N_{N-1}(a,b;\tau)=\lim_{q\to1} L^N_{N-1}(a,b;q,q^\tau),
$$
so that the stochastic matrices defined by \eqref{eq1.E} are a limit case of the stochastic matrices defined by \eqref{eq1.F}.
The first work related to the Macdonald chain was that of Gorin \cite{G-AM}. He examined the special case of equal parameters, $q=t$, and obtained (among other things) the description of the boundary. In the case $q=t$ the Macdonald polynomials become the Schur polynomials, as do the Jack polynomials with $\tau=1$. However, as shown in \cite{G-AM}, the replacement of the specialization at $(1,\dots,1)$ by that at $(1,q^{-1},\dots,q^{1-N})$ drastically changes the structure of the boundary: it becomes a totally disconnected space.
Then Cuenca \cite{Cuenca} described the boundary of the Macdonald chain in a more general case, for $t=q^\tau$, where $\tau=1,2,3,\dots$\,.
In another direction, Sato \cite{Sato1}, \cite{Sato2} linked Gorin's results to characters of a quantum version of the group $U(\infty)$.
\subsection{Motivation for further generalization}\label{sect1.5}
The approach of the present work allows to describe the boundary of the Macdonald chain \eqref{eq1.L} for arbitrary values $q,t\in(0,1)$. However, this is only a side result, as our concern is to study an \emph{extension} of the chain \eqref{eq1.L}.
The need of such an extension was explained in the joint work by Gorin and the author \cite{GO-2016} about a $q$-version of the so called \emph{zw-measures}. In the initial version, the zw-measures arose from the problem of harmonic analysis on the infinite-dimensional unitary group \cite{Ols-2003}, \cite{BO-2005}; they form a four-parameter family of probability measures on the boundary of the Jack chain with $\tau=1$. The Jack deformation of the zw-measures was constructed in \cite{Ols-2003a}, so it was natural to ask if there exists a $q$-deformation, too.
Initially we tried to construct $q$-deformed zw-measures in the framework of Gorin's paper \cite{G-AM}, but that attempt failed. Then we understood the reason: the desired result can be achieved only after enlarging the sets $\Sign(N)$.
We proceed to necessary definitions.
\subsection{Double signatures and extended Gelfand Tsetlin graph}
\begin{definition}\label{def1.DSign}
By a \emph{double signature of length $N$} we mean an ordered pair of signatures $(a^+,a^-)$ such that $a^+\in\Sign(k)$, $a^-\in\Sign(l)$, and $k+l=N$. The set of all such pairs is denoted by $\DSign(N)$. We do not exclude the case when $k$ or $l$ equals $0$. Thus, $\DSign(N)$ is the disjoint union of the sets
$$
\Sign(N)\times\{\varnothing\}, \quad \Sign(N-1)\times\Sign(1),\; \dots,\; \Sign(1)\times\Sign(N-1), \quad \{\varnothing\}\times \Sign(N),
$$
where $\{\varnothing\}$ is interpreted as a singleton (the `empty signature').
\end{definition}
We identify $\Sign(N)$ with the subset $\Sign(N)\times\{\varnothing\}\subset \DSign(N)$. Thus, we may regard $\DSign(N)$ as an extension of $\Sign(N)$.
\begin{definition}\label{def1.interlace}
We say that the double signatures $(a^+,a^-)\in\DSign(N)$ and $(b^+,b^-)\in\DSign(N-1)$ \emph{interlace} if one of the following two conditions holds:
(i) $(a^+,a^-)\in\Sign(k)\times \Sign(l)$ with $k>0$, $(b^+,b^-)\in\Sign(k-1)\times \Sign(l)$, and
$$
a^+_1\ge b^+_1\ge\dots \ge a^+_{k-1}\ge b^+_{k-1}\ge a^+_{k}, \qquad a^-_1\ge b^-_1\ge\dots \ge a^-_{l}\ge b^-_{l};
$$
(ii) $(a^+,a^-)\in\Sign(k)\times \Sign(l)$ with $l>0$, $(b^+,b^-)\in\Sign(k)\times \Sign(l-1)$, and
$$
a^+_1\ge b^+_1\ge\dots \ge a^+_k\ge b^+_k, \qquad a^-_1\ge b^-_1\ge\dots \ge a^-_{l-1}\ge b^-_{l-1}\ge a^-_l.
$$
\end{definition}
Equivalently: $b^+\prec a^+$ and $b^-\prec a^-$ with the understanding that if $b^\pm$ has the same length as $a^\pm$, then $a^\pm$ should be replaced by $a^\pm\cup\{-\infty\}$.
We write the interlacement relation for double signatures as $(b^+,b^-)\prec (a^+,a^-)$ or $(a^+,a^-)\succ (b^+,b^-)$
Recall that the \emph{Gelfand--Tsetlin graph} is the graded graph whose vertex set is the disjoint union $\bigsqcup_{N=1}^\infty \Sign(N)$ and the edges are formed by the pairs $b\prec a$ of interlacing signatures.
\begin{definition}\label{def1.extGT}
The \emph{extended Gelfand--Tsetlin graph} is the graded graph whose vertex set is the disjoint union $\bigsqcup_{N=1}^\infty \DSign(N)$ and the edges are formed by the pairs $(b^+,b^-)\prec (a^+,a^-)$ of interlacing double signatures.
\end{definition}
This definition is equivalent to the one in \cite{GO-2016}.
The embedding $\Sign(N)\to \DSign(N)$ via the map $a\mapsto (a,\varnothing)$ induces an embedding of the conventional Gelfand--Tsetlin graph into the extended Gelfand--Tsetlin graph.
Note that if both components $a^+$ and $a^-$ are nonempty, then there are infinitely many vertices $(b^+,b^-)\prec(a^+,a^-)$. This is in sharp contrast with the conventional Gelfand--Tsetlin graph.
Note also that the extended Gelfand-Tsetlin graph \emph{is not} the product of two copies of the conventional Gelfand-Tsetlin graph.
\subsection{Point configurations attached to double signatures}
\begin{definition}\label{def1.C}
We fix two parameters $q,t\in(0,1)$ and two additional parameters $\zeta_+>0$ and $\zeta_-<0$. To an arbitrary double signature $(a^+,a^-)\in\DSign(N)$, where $a^+\in\Sign(k)$, $a^-\in\Sign(l)$, $k+l=N$, we assign an $N$-point configuration $X_N(a)=X_N(a^+,a^-)\subset\R^*$, as follows:
\begin{equation}\label{eq1.G}
X_N(a):=\{\zeta_+q^{-a^+_i}t^{i-1}: i=1,\dots,k\}\cup\{\zeta_-q^{-a^-_i}t^{i-1}: i=1,\dots,l\}.
\end{equation}
For each $N=1,2,3\dots$, we denote by $\Om_N$ the set of all configurations of the form \eqref{eq1.G} with $k+l=N$.
\end{definition}
For instance, let $N=5$, $k=3$, $l=2$, $a^+=(5,3,1)$, and $a^-=(4,2)$. Then the configuration \eqref{eq1.G} is the set
$$
\{\zeta_+ q^{-5}, \; \zeta_+q^{-3}t, \; \zeta_+q^{-1}t^2\}\cup\{\zeta_-q^{-4}, \; \zeta_- q^{-2}t\}.
$$
Or, listing the points in the ascending order,
$$
\{\zeta_-q^{-4}, \; \zeta_- q^{-2}t, \; \zeta_+q^{-1}t^2,\; \zeta_+q^{-3}t, \; \zeta_+ q^{-5}\}.
$$
Let us emphasize that the definition of $X_N(a)$ and $\Om_N$ depends on the quadruple of parameters $(q,t,\zeta_+,\zeta_-)$, but we suppress them from the notation for the sake of brevity.
Note that distinct double signatures produce distinct configurations. This will enable us to switch from double signatures to point configurations and back.
\subsection{The special case $t=q^\tau$, $\tau=1,2,3,\dots$}
In this case the description of $\Om_N$ simplifies.
Namely, consider the \emph{two-sided $q$-lattice}
\begin{equation}\label{eq1.H}
\L:=\{\zeta_+ q^n: n\in\Z\}\cup\{\zeta_- q^n: n\in\Z\}\subset\R^*.
\end{equation}
If $t=q$, then the configurations $X\in\Om_N$ are precisely the $N$-point subsets of $\L$.
Next, if $t=q^\tau$ with $\tau=2,3,\dots$, then the configurations $X\in\Om_N$ are the $N$-point subsets of $\L$ subject to the following constraint: between any two neighboring points of $X$ there are at least $\tau-1$ unoccupied nodes of the lattice.
\subsection{Construction of extended Macdonald chain (Theorem A)}\label{results1}
Let $\Y$ denote the set of all partitions; as in \cite{M}, we identify partitions with the corresponding Young diagrams. For $N=1,2,\dots$, let $\Y(N)\subset\Y$ denote the subset of partitions of length at most $N$. We have
$$
\Y(1)\subset \Y(2)\subset\Y(3)\subset\dots, \qquad \bigcup_{N=1}^\infty\Y(N)=\Y.
$$
Note that $\Y(N)$ may be viewed as a subset of $\Sign(N)$.
We denote by $P_{\nu\mid N}(x_1,\dots,x_N;q,t)$ the $N$-variate Macdonald polynomial with parameters $(q,t)$ and the index $\nu\in\Y(N)$. Every symmetric polynomial $f$ in $N$ variables may be viewed as a function $f(X)$ on $\Om_N$; in particular, the function corresponding to a Macdonald polynomial is written as $P_{\nu\mid N}(X;q,t)$, where $X\in\Om_N$. We denote by $[X]$ the smallest closed interval of $\R$ containing $X$.
For $z\in\C$ and a partition (=Young diagram) $\nu$ we set
\begin{equation}\label{eq1.J}
(z;q,t)_\nu:=\prod_{(i,j)\in\nu}(1-zq^{j-1}t^{1-i}),
\end{equation}
where $(i,j)$ denotes the box on the intersection of the $i$th row and $j$th column.
Our first main result is
\begin{theoremA}
Let $q,t\in(0,1)$ and $\zeta_+>0$, $\zeta_-<0$ be fixed.
For each $N\ge2$ there exists a unique stochastic matrix $\La^N_{N-1}$ of format $\DSign(N)\times \DSign(N-1)$, such that the following two conditions hold.
{\rm(i)} The entries $\La^N_{N-1}(a,b)$ of $\LaN$ are strictly positive if $b\prec a$ and are equal to\/ $0$ otherwise.
{\rm(ii)} For any $a\in\DSign(N)$ and any $\nu\in\Y(N-1)$,
\begin{equation}\label{eq1.I}
\sum_{b\in\DSign(N-1)}\LaN(a,b) \frac{P_{\nu\mid N-1}(X_{N-1}(b))}{(t^{N-1};q,t)_\nu}=\frac{P_{\nu\mid N}(X_N(a))}{(t^N;q,t)_\nu}.
\end{equation}
\end{theoremA}
\noindent\emph{Comments} 1. An explicit expression for the matrix entries $\LaN(a,b)$ is given in subsection \ref{sect5.10}. Like $X_N(a)$, it depends on $(q,t,\zeta_+,\zeta_-)$, but we suppress these parameters from the notation for the sake of brevity.
\smallskip
2. The matrix $\LaN$ extends the matrix $L^N_{N-1}(q,t)$ from \eqref{eq1.F}, in the sense that $L^N_{N-1}(q,t)$ can be identified with a submatrix (a diagonal block) of the matrix $\LaN$. This follows from the computation in Section \ref{sect2}. Note that the claim is not evident from the comparison of \eqref{eq1.I} with \eqref{eq1.F}: although both conditions are written in terms of Macdonald polynomials, they look quite different.
\smallskip
3. As pointed out in subsection \ref{sect1.4} above, the fact that the coefficients in the expansion \eqref{eq1.F} form a stochastic matrix immediately follows from the branching rule for the Macdonald polynomials. In the context of Theorem A, the definition of $\LaN$ relies on Okounkov's $q$-integral representation for Macdonald polynomials \cite{Ok-CM}, and the proof of the theorem that we can offer is rather long. It can be simplified when $t=q^\tau$ with $\tau=1,2,3,\dots$; this case is examined separately in Section \ref{sect4}, so that the reader may skip section \ref{sect5} if desired. In the special case $q=t$, considered in \cite{GO-2016}, the proof is direct and easy (\cite[Proposition 2.4]{GO-2016}), due to the fact that then the entries of $\LaN$ are given by a simple formula. However, the approach of \cite{GO-2016} does not apply in the two-parameter case.
\smallskip
4. We call \eqref{eq1.I} the \emph{coherency relation} for Macdonald polynomials.
\begin{definition}\label{def1.B}
Theorem A allows us to form the Markov chain
\begin{equation}\label{eq1.K}
\DSign(1)\stackrel{\La^2_1}{\dashleftarrow}\DSign(2)\stackrel{\La^3_2}{\dashleftarrow}\DSign(3)\stackrel{\La^4_3}{\dashleftarrow}\dots\,.
\end{equation}
We call it the \emph{extended Markov chain}.
\end{definition}
As is seen from claim (i) of the theorem, the transition probabilities of this Markov chain are attached to the edges of the extended Gelfand--Tsetlin graph.
\subsection{The boundary of the extended Macdonald chain (Theorem B)}\label{results2}
Our purpose is to find the boundary of the extended Macdonald chain \eqref{eq1.K} in the sense of Definition \ref{def7.A}. Informally, we call it the `$(q,t)$-boundary of the extended Gelfand--Tsetlin graph'.
\begin{definition}
(i) By an \emph{infinite signature} we mean an arbitrary infinite sequence of integers $a_1\ge a_2\ge\dots$\,. The \emph{infinite double signature} is a pair $(a^+,a^-)$ of signatures of which at least one is infinite. The set of infinite double signatures will be denoted by $\DSign(\infty)$.
(ii) To each infinite double signature $a=(a^+,a^-)$ one assigns an infinite point configuration $X_\infty(a)\subset\R^*$: it is defined as in \eqref{eq1.G}: the only change is that one of the indices $i, j$ (or both) will range over the whole set $\{1,2,3,\dots\}$.
\end{definition}
Note that $\DSign(\infty)$ has the power of the continuum. The following is our second main result.
\begin{theoremB}
The boundary of the extended Macdonald chain can be identified, in a natural way, with the set\/ $\DSign(\infty)$ of infinite double signatures. Under this identification, the Markov kernels $\La^\infty_N:\DSign(\infty)\dasharrow \DSign(N)$ can be characterized via the relations
\begin{equation}\label{eq1.I1}
\sum_{b\in\DSign(N)}\La^\infty_N(a,b) \frac{P_{\nu\mid N}(X_N(b);q,t)}{(t^N;q,t)_\nu}=P_\nu(X_\infty(a);q,t), \qquad \forall\nu\in\Y(N),
\end{equation}
where $P_\nu(\ccdot;q,t)$ is the Macdonald symmetric function indexed by $\nu$.
\end{theoremB}
\noindent\emph{Comments} 1. A bit more detailed form of this statement is given in Theorem \ref{thm8.B}.
\smallskip
2. The parametrization of the boundary does not depend on the parameters $(q,t, \zeta_+,\zeta_-)$ but the Markov kernels $\La^\infty_N$ do.
\smallskip
3. In the special case $t=q$ the result of Theorem B was proved in Gorin--Olshanski \cite[Theorem 3.12]{GO-2016},
4. As a corollary of Theorem B, one can obtain a description of the boundary for the (not extended) Macdonald chain \eqref{eq1.L} with arbitrary $q,t\in(0,1)$. To do this it suffices to use the computation of section \ref{sect2}, see comment 2 to Theorem A above. This extends earlier results of Gorin \cite{G-AM} (for the case $t=q$) and of Cuenca \cite{Cuenca} (for $t=q^\tau$, $\tau=1,2,3,\dots$). Our approach is different from those of these works.
\subsection{Coherent systems of measures and the approximation theorem (Theorem C)}\label{results3}
\begin{definition}
A sequence $\{M_N: N=1,2,3,\dots\}$ of probability measures on the sets $\DSign(N)$, $N=1,2,3,\dots$, is said to be a \emph{coherent system of measures} if for any $N\ge2$
\begin{equation}
\sum_{a\in\DSign(N)}M_N(a)\LaN(a,b)=M_{N-1}(b), \qquad \forall\, b\in\DSign(N-1).
\end{equation}
Or, in short form, $M_N\LaN=M_{N-1}$, where $M_N$ and $M_{N-1}$ are treated as row vectors.
\end{definition}
By the very definition of the boundary, every coherent system $\{M_N\}$ gives rise to a probability measure $M_\infty$ on $\DSign(\infty)$, uniquely determined by the relations
$$
M_\infty\La^\infty_N=M_N, \qquad N=1,2,\dots,
$$
where the left-hand side is the pushforward of $M_\infty$ by the Markov kernel $\La^\infty_N$ linking $\DSign(\infty)$ with $\DSign(N)$. We call $M_\infty$ the \emph{boundary measure} of the system $\{M_N\}$.
Consider the disjoint union
$$
\wt{\DSign}:=\DSign(\infty) \sqcup \bigsqcup_{N=0}^\infty \DSign(N),
$$
where $\DSign(0)$ is a singleton, interpreted as the pair of empty signatures. We equip $\wt{\DSign}$ with a topology in the following way.
\begin{definition}
Let us say that two signatures (finite or infinite, no matter) are \emph{$\varepsilon$-close} (where $\varepsilon>0$ is small), if they have the same set of coordinates exceeding $-\varepsilon^{-1}$. Likewise, we say that two double signatures, $(a^+,a^-)$ and $(b^+,b^-)$, are \emph{$\varepsilon$-close} if so are $a^\pm$ and $b^\pm$.
\end{definition}
The notion of $\varepsilon$-closeness just defined makes $\wt{\DSign}$ a uniform space, and hence a topological space. It is a non-discrete locally compact space in which both $\DSign(\infty)$ and $\bigsqcup_{N=0}^\infty \DSign(N)$ are dense subsets.
The definition of the boundary measure can be made more concrete due to the next theorem, which is our third main result. We call it the \emph{approximation theorem}.
\begin{theoremC}
For any coherent system $\{M_N\}$, the measures $M_N$ converge to the boundary measure $M_\infty$ in the weak topology of measures on the space $\wt{\DSign}$.
\end{theoremC}
\subsection{Hypergeometric point processes}
The main result of Gorin--Olshanski \cite{GO-2016} was the construction of the so called $q$-zw-measures --- a family of probability measures on the space $\DSign(\infty)$, depending on the parameter $q$ and providing a $q$-version of the spectral measures coming from harmonic analysis on $U(\infty)$ \cite{BO-2005}. The $q$-zw-measures are further studied in \cite{CGO}.
In the subsequent paper \cite{Ols-MacdonaldTwo} it is shown that the construction of the $q$-zw-measures can be extended by adding the second Macdonald parameter, $t$. This is achieved based on Theorems A, B, C and leads to a new family of random point processes.
\subsection{Organization of the paper}
Section \ref{sect2} establishes the link with \cite{GO-2016}. In section \ref{sect3} we state Okounkov's theorem about the $q$-integral representation of Macdonald polynomials in a form convenient for later use. In section \ref{sect4} we prove Theorem A for the case $t=q^\tau$ with $\tau\in\{1,2,3,\dots\}$. Section \ref{sect5} gives the proof of Theorem A for arbitrary $q,t\in(0,1)$. The essence of the argument is a delicate limit transition in Okounkov's formula. The section ends with remarks concerning a continuous analogue of Theorem A. Theorems B and C are proved in section \ref{sect8} after a preparation occupying sections \ref{sect6}-\ref{sect7}.
\section{Link between two families of stochastic matrices}\label{sect2}
The purpose of this short section is to justify the claim in comment 2 to Theorem A (subsection \ref{results1} above). Namely, in Proposition \ref{prop2.A} below we show how to transform the equations \eqref{eq1.F} to the same form as in \eqref{eq1.I}. This result establishes a link between our setup and that of Gorin \cite{G-AM} and Cuenca \cite{Cuenca}.
Note that the summation in \eqref{eq1.F} is actually taken over those signatures $b$ that interlace with $a$ (see \eqref{eq1.I}): indeed, this follows from the branching rule for Macdonald polynomials. Recall that the interlacement relation is denoted as $b\prec a$.
To simplify the notation we assume here that $\zeta_+=1$.
Given $a\in\Sign(N)$, we set
$$
X_N(a):=X_N(a,\varnothing)=\{q^{-a_i}t^{i-1}: i=1,\dots,N\}.
$$
Let $a\mapsto a^*$ denote the involutive map $\Sign(N)\to\Sign(N)$ defined by
$$
(a_1,\dots,a_N)\mapsto(-a_N,\dots,-a_1).
$$
\begin{proposition}\label{prop2.A}
Let the $L^N_{N-1}(a,b;q,t)$ be the coefficients from the expansion \eqref{eq1.F}. For any $N=2,3,\dots$, signature $a\in\Sign(N)$, and partition $\nu\in\Y(N-1)$ we have
\begin{equation}\label{eq2.B}
\sum_{b\prec a} L^N_{N-1}(a^*,b^*;q,t) \frac{P_{\nu\mid N-1}(X_{N-1}(b);q,t)}{(t^{N-1};q,t)_\nu}=\frac{P_{\nu\mid N}(X_N(a);q,t)}{(t^N;q,t)_\nu}.
\end{equation}
\end{proposition}
\begin{proof}
Using the fact that Macdonald polynomials are homogeneous we rewrite \eqref{eq1.F} as
\begin{multline}\label{eq2.C}
\frac{P_{a\mid N}(u_1t^{N-1},\dots,u_{N-1}t^{N-1},1;q,t)}{P_{a\mid N}(t^{N-1}, t^{N-2},\dots, 1;q,t)}\\
=\sum_{b\prec a} L^N_{N-1}(a,b;q,t)\frac{P_{b\mid N-1}(u_1t^{N-2},\dots,u_{N-1}t^{N-2};q,t)}{P_{b\mid N-1}(t^{N-2},t^{N-3},\dots,1;q,t)}.
\end{multline}
Take $\nu\in\Y(N-1)$ and substitute in \eqref{eq2.C}
$$
(u_1,\dots,u_{N-1})=(q^{\nu_1},\, q^{\nu_2}t^{-1}, \, \dots\, q^{\nu_{N-1}}t^{2-N}).
$$
Then we obtain
\begin{multline}\label{eq2.D}
\frac{P_{a\mid N}(q^{\nu_1}t^{N-1},\dots,q^{\nu_{N-1}}t, 1;q,t)}{P_{a\mid N}(t^{N-1}, t^{N-2},\dots, 1;q,t)}\\
=\sum_{b\prec a} L^N_{N-1}(a,b;qt)\frac{P_{b\mid N-1}(q^{\nu_1}t^{N-2},\dots,q^{\nu_{N-1}};q,t)}{P_{b\mid N-1}(t^{N-2},t^{N-3},\dots,1;q,t)}.
\end{multline}
Next, recall the \emph{label-argument symmetry relation} for Macdonald polynomials (\cite[ch. VI, (6.6)]{M}):
\begin{equation}\label{eq2.sym}
\frac{P_{\mu\mid N}(q^{\la_1}t^{N-1},\dots,q^{\la_{N-1}}t, q^{\la_N};q,t)}{P_{\mu\mid N}(t^{N-1}, t^{N-2},\dots, 1;q,t)}=\frac{P_{\la\mid N}(q^{\mu_1}t^{N-1},\dots,q^{\mu_{N-1}}t, q^{\mu_N};q,t)}{P_{\la\mid N}(t^{N-1}, t^{N-2},\dots, 1;q,t)}.
\end{equation}
In \cite{M}, this relation is established for ordinary (non-Laurent) Macdonald polynomials, so that $\la$, $\mu$ are assumed to be partitions of length at most $N$. But the result is immediately extended to the Laurent version of Macdonald polynomials labelled by signatures. Applying the symmetry relation to both sides of \eqref{eq2.D} we obtain
\begin{multline}\label{eq2.E}
\frac{P_{\nu\mid N}(q^{a_1}t^{N-1},\dots,q^{a_{N-1}}t, q^{a_N};q,t)}{P_{\nu\mid N}(t^{N-1}, t^{N-2},\dots, 1;q,t)}\\
=\sum_{b\prec a} L^N_{N-1}(a,b;q,t)\frac{P_{\nu\mid N-1}(q^{b_1}t^{N-2},\dots,q^{b_{N-1}};q,t)}{P_{\nu\mid N-1}(t^{N-2},t^{N-3},\dots,1;q,t)}.
\end{multline}
Then we replace $(a,b)$ with $(a^*,b^*)$, which gives
\begin{equation}\label{eq2.F}
\frac{P_{\nu\mid N}(X_N(a);q,t)}{P_{\nu\mid N}(t^{N-1}, t^{N-2},\dots, 1;q,t)}
=\sum_{b\prec a} L^N_{N-1}(a^*,b^*;q,t)\frac{P_{\nu\mid N-1}(X_{N-1}(b);q,t)}{P_{\nu\mid N-1}(t^{N-2},t^{N-3},\dots,1;q,t)}.
\end{equation}
Finally, from the principal specialization formula \cite[Ch VI, (6.11$'$)]{M} it follows that $P_{\nu\mid N}(1,t,\dots,t^{N-1};q,t)$ differs from $(t^N;q,t)_\nu$ by a factor that depends only on $\nu$. Therefore, \eqref{eq2.F} implies \eqref{eq2.B}.
\end{proof}
A similar computation is contained in Appendix B of Forrester--Rains \cite{FR}. See also the end of \S2 of \cite{GO-2016}.
\section{Okounkov's $q$-integral formula}\label{sect3}
In this section we formulate a result due to Okounkov \cite{Ok-CM}, which is substantially used in the sequel.
\subsection{Notation from $q$-calculus}
Throughout the paper we use the standard notation from $q$-calculus (see Gasper--Rahman \cite{GR}):
For $z\in\C$ and $m=0,1,2,\dots$
$$
(z;q)_\infty:=\prod_{n=0}^\infty(1-zq^n), \quad (z;q)_m:=\prod_{n=0}^{m-1}(1-zq^n)=\frac{(z;q)_\infty}{(zq^m;q)_\infty}.
$$
The $q$-integral of a function $f$ in the complex domain is defined by (below $z,z'\in\C$)
\begin{equation}\label{eq3.E}
\begin{gathered}
\int_{z'}^z f(w)d_q w:=\int_0^z f(w)d_q w-\int_0^{z'}f(w)d_q w, \\
\text{where} \quad \int_0^z f(w)d_q w:=(1-q)\sum_{n=0}^\infty f(zq^n) zq^n.
\end{gathered}
\end{equation}
These definitions make sense for any complex $q$ with $|q|<1$.
\subsection{Okounkov's formula}\label{sect3.2}
Introduce the following meromophic function in $2N-1$ variables:
\begin{multline}\label{eq3.A}
R(z_1,\dots,z_N;w_1,\dots,w_{N-1};q,t)\\
:=\prod_{r=1}^{N-1}\prod_{s=1}^N\frac{(w_rq/z_s;q)_\infty}{(w_rt/z_s;q)_\infty} \prod_{1\le i\ne j\le N}\frac{(z_it/z_j;q)_\infty}{(z_iq/z_j;q)_\infty}.
\end{multline}
Below we write $Z=(z_1,\dots,z_N)$, $W=(w_1,\dots,w_{N-1})$ and set
$$
V(Z):= \prod\limits_{1\le i<j\le N}(z_i-z_j), \quad V(W):=\prod\limits_{1\le r<s\le N-1}(w_r-w_s).
$$
A constant $C_N(q,t)$ is defined by
\begin{equation}\label{eq3.D}
C_N(q,t):=\frac{((t;q)_\infty)^N}{(1-q)^{N-1}(t^N;q)_\infty((q;q)_\infty)^{N-1}}.
\end{equation}
Recall that the symbol $(z;q,t)_\nu$ was defined in \eqref{eq1.J}.
Finally, recall that $P_{\nu\mid N}(\ccdot;q,t)$ is our notation for the $N$-variate Macdonald polynomial indexed by a partition $\nu\in\Y(N)$, with parameters $(q,t)$. Our normalization of these polynomials is the same as in \cite{M}; that is, the monomial $x_1^{\nu_1}\dots x_N^{\nu_N}$ enters $P_{\nu\mid N}(x_1,\dots,x_N;q,t)$ with coefficient $1$.
\begin{theorem}[Okounkov \cite{Ok-CM}, Theorem I]\label{thm3.A}
Let $q\in\C$, $|q|<1$. Next, let $N=2,3,\dots$ and $\nu\in\Y(N-1)$. Finally, let $Z=(z_1,\dots,z_N)\in\C^N$ and $t\in\C$ be in general position.
Then the following formula holds
\begin{multline}\label{eq3.B}
\frac{C_N(q,t)}{V(Z)}\,\int^{z_1}_{z_2}d_q w_1\int^{z_2}_{z_3}d_q w_2\dots \int^{z_{N-1}}_{z_N}d_q w_{N-1}\\
\times V(W)\, R(z_1,\dots,z_N;w_1,\dots,w_{N-1};q,t)\frac{P_{\nu\mid N-1}(w_1,\dots,w_{N-1};q,t)}{(t^{N-1};q,t)_\nu}\\
=\frac{P_{\nu\mid N}(z_1,\dots,z_N;q,t)}{(t^N;q,t)_\nu}.
\end{multline}
\end{theorem}
Note that the assumption about general position (not explicitly mentioned in \cite{Ok-CM}) is imposed in order to avoid possible singularities of the integrand on the $q$-contour of integration.
\subsection{A special case}
Suppose that $t=q^\tau$, where $\tau\in\{1,2,\dots\}$. Then the expression \eqref{eq3.A} can be simplified:
\begin{equation}\label{eq3.C}
R(z_1,\dots,z_N;w_1,\dots,w_{N-1};q,q^\tau)=\dfrac{\prod\limits_{r=1}^{N-1}\prod\limits_{s=1}^N(w_rq/z_s;q)_{\tau-1}}{\prod\limits_{1\le i\ne j\le N}(z_iq/z_j;q)_{\tau-1}}.
\end{equation}
In the case $t=q$ we simply have
$$
R(z_1,\dots,z_N;w_1,\dots,w_{N-1};q,q)\equiv1.
$$
\subsection{Another special case}
Another kind of simplification in Theorem \ref{thm3.A} occurs when $q=0$ (the case of Hall--Littlewood polynomials). Then \eqref{eq3.A} turns into
$$
R(z_1,\dots,z_N;w_1,\dots,w_{N-1};0,t)\\
:=\prod_{j=1}^N\dfrac{\prod_{i:\, i\ne j}(z_j-z_it)}{\prod_{i=1}^{N-1}(z_j-w_it)}
$$
and the $q$-integral \eqref{eq3.E} reduces to
$$
\left.\int_{z'}^z f(w)d_qw \right|_{q=0}=z f(z)-z'f(z'),
$$
so that the multiple $q$-integral \eqref{eq3.B} reduces to a finite sum (because of the factor $V(W)$, the sum actually comprises not $2^{N-1}$ but only $N$ summands).
\subsection{Remark on alternate derivation of Okounkov's formula}
Okounkov first shows (\cite[Proposition 3.4]{Ok-CM}) that the left-hand side of \eqref{eq3.B} is a symmetric polynomial in variables $(z_1,\dots,z_N)$, of the same degree (in this claim the polynomial $P_{\nu\mid N-1}(w_1,\dots,w_{N-1};q,t)$ can be replaced by an arbitrary symmetric polynomial). This part of his proof is relatively easy and short, while the remaining part is longer and more intricate. But the latter part can be replaced by the following argument. The aforementioned proposition allows to reduce the desired idenity \eqref{eq3.B} to the special case when, in the notation of section \ref{sect2},
$$
(z_1,\dots,z_N)=X_N(a), \qquad a\in\Sign(N).
$$
And then the computation of section \ref{sect2} allows to further reduce the identity to the branching rule for the Macdonald polynomials.
After I realized this, I found a similar remark in Forrester--Rains \cite[comment after (3.17)]{FR}.
\section{Proof of Theorem A: special case $t=q^\tau$, $\tau=1,2,3,\dots$} \label{sect4}
\subsection{Preliminaries}
Throughout this section we assume that $t=q^\tau$, where $0<q<1$ and $\tau$ is a fixed positive integer.
Recall the definition \eqref{eq1.G}: the \emph{two-sided $q$-lattice} $\L\subset\R\setminus\{0\}$ is the subset
\begin{equation}\label{eq4.A}
\L=\L_+\sqcup\L_-, \qquad \L_\pm:=\{\zeta_\pm q^n: n\in\Z\}\subset\R^*,
\end{equation}
where $\zeta_-<0$ and $\zeta_+>0$ are two fixed extra parameters. We need these two parameters only to define the lattice $\L$. Note that $\L$ does not change if $\zeta_+$ or $\zeta_-$ is multiplied by an integral power of parameter $q$. Points of $\L$ are called \emph{nodes}.
In this section, by a \emph{configuration} we always mean a subset $X\subset\L$.
We say that a configuration $X$ is \emph{$\tau$-sparse} if any two distinct points of $X$ are separated by at least $\tau-1$ empty nodes (of course, this is a real constraint only for $\tau\ge2$).
Equivalently, $X$ is $\tau$-sparse if for any two distinct points $x, x'$ of $X$ the following condition holds: $0<x'<x$ implies $x'\le x t$, and $x<x'<0$ implies $x'\ge xt$. One more equivalent formulation: for any two distinct points $x, \,x'\in X$ of the same sign, one has $|\log_q(x/x')|\ge \tau$.
\begin{definition}\label{def4.B}
Given two nodes $x'<x$ of $\L$, which are separated by at least $\tau-1$ other nodes, we introduce a special notion of \emph{$q$-interval} $\Int_\tau(x',x)$. This is a set of nodes whose definition depends on the position of the pair $(x',x)$ with respect to $0$:
\smallskip
1. If $0<x'<x$, then $\Int_\tau(x',x):=\{y\in\L_+: x' q^{-\tau}\le y\le x\}$.
2. If $x'<x<0$, then $\Int_\tau(x',x):=\{y\in\L_-: x'\le y\le xq^{-\tau}\}$.
3. If $x'<0<x$, then $\Int_\tau(x',x):=\{y\in\L: x'\le y\le x\}$.
\end{definition}
Note that in the third case $\Int_\tau(x',x)$ contains infinitely many nodes.
\begin{definition}\label{def4.C}
(i) For each $N=1,2,\dots$ we denote by $\Om_N$ the set of $\tau$-sparse $N$-point configurations on $\L$.
(ii) We say that two configurations $X\in\Om_N$ and $Y\in\Om_{N-1}$ \emph{interlace} if the following condition holds. Write $X=(x_1>\dots>x_N)$ and $Y=(y_1>\dots>y_{N-1})$; then we require that $y_i\in\Int_\tau(x_{i+1},x_i)$ for every $i=1,\dots,N-1$.
\end{definition}
This definition of $\Om_N$ in item (i) agrees with Definition \ref{def1.C}: the $N$-point $\tau$-sparse configurations are precisely the configurations $X_N(a^+,a^-)$ coming from double signatures $(a^+,a^-)\in\DSign(N)$. The definition of interlacement in item (ii) agrees with Definition \ref{def1.interlace}. Below we write the interlacement relation as $X\succ Y$ or $Y\prec X$.
\subsection{The matrices $\LaN$}\label{sect4.B}
Let us agree to enumerate the points of a given configuration $X\in\Om_N$ in the descending order: $X=(x_1>\dots>x_N)$. Keeping this in mind, we set
$$
V(X)=\prod_{1\le i<j\le N}(x_i-x_j).
$$
Thus, $V(X)>0$. In the case $N=1$ we agree that $V(X)=1$.
\begin{definition}\label{def4.A}
For each $N=2,3,\dots$ we define the matrix $\LaN$ of format $\Om_N\times\Om_{N-1}$ with the following entries $\LaN(X,Y)$:
$\bullet$ $\LaN(X,Y)=0$ unless $X\succ Y$.
$\bullet$ If $X\succ Y$, then
\begin{equation}\label{eq4.B}
\La^N_{N-1}(X,Y)=\frac{((t;q)_\infty)^N}{(t^N;q)_\infty((q;q)_\infty)^{N-1}}\cdot\frac{V(Y)}{V(X)}\cdot \prod_{y\in Y}|y|\cdot
\frac{\prod\limits_{y\in Y}\prod\limits_{x\in X}(yq/x;q)_{\tau-1}}{\prod\limits_{x,x'\in X: \, x\ne x'}(xq/x';q)_{\tau-1}}.
\end{equation}
\end{definition}
In the simplest case $\tau=1$ the expression on the right-hand side simplifies and reduces to
\begin{equation}\label{eq4.H}
(q;q)_{N-1} \prod\limits_{y\in Y}|y|\cdot\dfrac{V(Y)}{V(X)},
\end{equation}
which agrees with the definition given in \cite{GO-2016} and \cite{Ols-2016}.
\begin{lemma}\label{lemma4.A}
All the entries $\La^N_{N-1}(X,Y)$ are nonnegative.
\end{lemma}
\begin{proof}
We will show that $X\succ Y$ entails $\La^N_{N-1}(X,Y)>0$. In the case $\tau=1$ this follows immediately from \eqref{eq4.H}, so we will assume that $\tau\ge2$.
Suppose first that $X\subset\L_+$, that is, all points of $X$ are on the right of $0$. Then, according to Definition \ref{def4.C}, the following inequalities hold
\begin{gather}
x_1>\dots>x_N>0, \quad x_i\ge x_{i+1}q^{-\tau}, \qquad i=1,\dots,N-1; \label{eq4.C1}\\
y_1>\dots>y_{N-1}, \quad x_i\ge y_i\ge x_{i+1}q^{-\tau}\qquad i=1,\dots,N-1.\label{eq4.C2}
\end{gather}
We have to prove that the quantity
$$
\frac{\prod\limits_{y\in Y}\prod\limits_{x\in X}(yq/x;q)_{\tau-1}}{\prod\limits_{x,x'\in X: \, x\ne x'}(xq/x';q)_{\tau-1}}
$$
is strictly positive. Let us split it in two parts:
$$
\frac{\prod\limits_{N-1\ge i\ge j\ge1}(y_iq/x_j;q)_{\tau-1}}{\prod\limits_{N\ge i>j\ge1}(x_iq/x_j;q)_{\tau-1}}\cdot \frac{\prod\limits_{1\le i< j\le N}(y_iq/x_j;q)_{\tau-1}}{\prod\limits_{1\le i<j\le N}(x_iq/x_j;q)_{\tau-1}}.
$$
In the first part, all the factors in the numerator and denominator are strictly positive because of \eqref{eq4.C1} and \eqref{eq4.C2}: indeed, $i\ge j$ implies $0<y_iq/x_j<1$, while $i>j$ implies $0<x_iq/x_j<1$ (here we use the assumption $\tau\ge2$).
Next, the second part of our expression can be rewritten in the form
$$
\prod_{i=1}^{N-1}\prod_{j=i+1}^N\frac{(y_iq/x_j;q)_{\tau-1}}{(x_iq/x_j;q)_{\tau-1}}=\prod_{i=1}^{N-1}\prod_{j=i+1}^N\prod_{r=1}^{\tau-1}\frac{1-y_iq^r/x_j}{1-x_iq^r/x_j}.
$$
Now the inequalities \eqref{eq4.C1} and \eqref{eq4.C2} show that all the factors in the denominator and the numerator are strictly negative. Since they contain equally many factors, we conclude that the whole expression is strictly positive.
The same reasoning is applicable in the case $X\subset\L_-$ (observe that the definition of $\tau$-sparse configurations is symmetric with respect to reflection about zero, and so is the interlacement relation).
Now we turn to the case when $0$ sits somewhere inside $X$, that is, $x_{k+1}<0<x_k$ for some $k<N$. Let us split $X$ into two parts, $X_-\sqcup X_+$, where $X_-:=X\cap\L_-$ and $X_+:=X\cap\L_+$. Next, observe that $(yq/x;q)_{\tau-1}>0$ whenever $y$ and $x$ have opposite signs. Likewise,
$(xq/x';q)_{\tau-1}>0$ whenever $x$ and $x'$ have opposite signs. Let us discard the corresponding factors.
Note that $y_k\in[x_{k+1},x_k]$ and all factors $(y_kq/x;q)_{\tau-1}$, where $x$ has the same sign as $y_k$, are strictly positive, because $y_kq/x<1$. Thus, these factors may again be discarded. After that the problem is reduced to the case when $X=X_\pm$, examined above.
\end{proof}
The next remark will be used in the proof of Theorem \ref{thm4.A}
\begin{remark}\label{rem4.A}
Let $X=(x_1>\dots>x_N)\in\Om_N$ and let $Y=(y_1>\dots>y_{N-1})$ be an $(N-1)$-point configuration such that $y_i\in\Int_1(x_{i+1},x_i)$ for $i=1,\dots,N-1$. Then the right-hand side of \eqref{eq4.B} vanishes unless the stronger condition $y_i\in\Int_\tau(x_{i+1},x_i)$ holds for all $i$, meaning that $Y$ must interlace with $X$ in the sense of Definition \ref{def4.C}.
Indeed, we have to show that if $y_i\in\Int_1(x_{i+1}, x_i)\setminus\Int_\tau(x_{i+1}, x_i)$ for some $i$, then the right-hand side of \eqref{eq4.B} vanishes. Let us examine the possible cases.
(1) $x_{i+1}<0<x_i$. Then $\Int_1(x_{i+1},x_i)=\Int_\tau(x_{i+1},x_i)$, so the claim is trivial.
(2) $0<x_{i+1}=x_i q^\ell <x_i$ with $\ell\in\{\tau, \tau+1,\tau+2,\dots\}$. Then
\begin{gather*}
\Int_1(x_{i+1}, x_i)=\{y\in\L_+: x_{i+1}q^{-1}\le y\le x_i\}, \\
\Int_\tau(x_{i+1},x_i)=\{y\in\L_+: x_{i+1}q^{-\tau}\le y\le x_i\}.
\end{gather*}
If $y_i\in\Int_1(x_{i+1}, x_i)\setminus\Int_\tau(x_{i+1}, x_i)$, then $(y_iq/x_{i+1};q)_{\tau-1}=0$, so that the right-hand side of \eqref{eq4.B} vanishes.
(3) $x_{i+1}<x_i=x_{i+1}q^\ell<0$ with $\ell\in\{\tau, \tau+1,\tau+2,\dots\}$. Then
\begin{gather*}
\Int_1(x_{i+1}, x_i)=\{y\in\L_-: x_{i+1}\le y\le x_i q^{-1}\},\\
\Int_\tau(x_{i+1},x_i)=\{y\in\L_-: x_{i+1}\le y\le x_iq^{-\tau}\}.
\end{gather*}
If $y_i\in\Int_1(x_{i+1}, x_i)\setminus\Int_\tau(x_{i+1}, x_i)$, then $(y_iq/x_{i};q)_{\tau-1}=0$, so that the right-hand side of \eqref{eq4.B} vanishes.
\end{remark}
\subsection{The coherency relation}
Recall that $\Y$ denotes the set of partitions, which we identify with the corresponding Young diagrams. Next, the length of a partition $\nu\in\Y$ is denoted by $\ell(\nu)$, and $\Y(N):=\{\nu\in\Y: \ell(\nu)\le N\}$.
Let $\Sym(N)$ denote the algebra of symmetric polynomials with $N$ variables (as the base field one can take $\R$ or $\C$). Every polynomial $f\in\Sym(N)$ gives rise to a function on $\Om_N$: if $X=(x_1,\dots,x_N)\in\Om_N$, then we write $f(X):=f(x_1,\dots,x_N)$ (since $f$ is symmetric, the result does not depend on the enumeration of the points of $X$). In particular, this is applicable to Macdonald polynomials, and we write their values at configurations $X\in\Om_N$ as $P_{\nu\mid N}(X;q,t)$.
\begin{theorem}\label{thm4.A}
Let $N=2,3,\dots$ and $\LaN$ be the matrix of format\/ $\Om_N\times\Om_{N-1}$ introduced in Definition \ref{def4.A}. Recall that $t=q^\tau$, where $0<q<1$ and $\tau$ is a positive integer. For any $\nu\in\Y(N-1)$ and any $X\in\Om_N$ the following `coherency relation' holds
\begin{equation}\label{eq4.D}
\sum_{Y\in\Om_{N-1}}\LaN(X,Y) \frac{P_{\nu\mid N-1}(Y;q,t)}{(t^{N-1};q,t)_\nu}=\frac{P_{\nu\mid N}(X;q,t)}{(t^N;q,t)_\nu}.
\end{equation}
\end{theorem}
Recall that the symbol $(z;q,t)_\nu$ was defined in \eqref{eq1.J}. Note that $(z;q,t)_\nu\ne0$ if $z$ is real and less than $1$, hence $(t^N;q,t)_\nu\ne0$ and $(t^{N-1};q,t)_\nu\ne0$ for any $N=2,3,\dots$\,.
\begin{corollary}\label{cor4.A}
The $\LaN$, $N=2,3,\dots$, are stochastic matrices.
\end{corollary}
\begin{proof}[Proof of the corollary]
By Lemma \ref{lemma4.A}, the matrix entries are nonnegative. Next, take in \eqref{eq4.D} as $\nu$ the zero partition (= empty Young diagram). In this case the corresponding Macdonald polynomials are identically equal to $1$ and the generalized Pochhammer symbols in the denominators also equal $1$. Then \eqref{eq4.D} means that the row sums of $\LaN$ equal $1$. We conclude that $\LaN$ is a stochastic matrix.
\end{proof}
\begin{proof}[Proof of the theorem]
We enumerate the points of $X$ and $Y$ in descending order: $X=(x_1>\dots>x_N)$, $Y=(y_1>\dots>y_{N-1})$. Then $X\succ Y$ means exactly that $y_i\in\Int_\tau(x_{i+1},x_i)$ for all $i=1,\dots,N-1$.
First of all, observe that the series on the left-hand side of \eqref{eq4.D} converges. Indeed, if $X$ is entirely contained in $\L_+$ or $\L_-$, then there are only finitely many $Y$'s interlacing with $X$, so the sum is finite. If $X$ has points both in $\L_+$ and $\L_-$, then there exists a unique index $k$ such that $x_{k+1}<0<x_k$. It follows that the $q$-interval $\Int_\tau(x_{k+1},x_k)$ comprises infinitely many nodes, while all other $q$-intervals $\Int_\tau(x_{i+1},x_i)$ contain finitely many nodes. Then the series on the left-hand side is infinite, but its convergence is assured by the factor $|y_k|$ entering the right-hand side of \eqref{eq4.B}.
In the simplest case $\tau=1$, the Macdonald polynomials turn into the Schur functions, and then \eqref{eq4.D} admits a simple direct proof, see Gorin-Olshanski \cite[Proposition 2.7]{GO-2016} and Kim-Stanton \cite{KS}. In the case $\tau=2,3,\dots$ we derive \eqref{eq4.D} from Okounkov's formula \eqref{eq3.B}.
A simple but important observation is that if $x'<x$ is a pair of points of $\L$, then
\begin{equation}\label{eq4.G}
\int_{x'}^x f(y)d_qy=(1-q)\sum_{y\in\Int_1(x',x)}|y|f(y).
\end{equation}
Indeed, if $x'<0<x$, this follows directly from the definition \eqref{eq3.E} of the $q$-integral. If $x$ and $x'$ are of the same sign, then the terms corresponding to points $y$ lying outside the set $\Int_1(x',x)\subset[x',x]$ cancel out. Let us emphasize that the assumption $x',x\in\L$ is crucial here; if it is dropped, then it may well happen that the $q$-integral depends on the values of $f$ at some points $y$ outside $[x',x]$.
Let us return to Okounkov's formula \eqref{eq3.B} and show that it reduces to \eqref{eq4.D} when $x_1,\dots,x_N$ are the points of a configuration $X\in\Om_N$ enumerated in the descending order.
Indeed, due to \eqref{eq4.G}, the multiple $q$-integral in \eqref{eq3.B} becomes a sum over the set
$\Int_1(x_N,x_{N-1})\times \dots\times \Int_1(x_2,x_1)$. Let us compare this sum with the sum on the left-hand side of \eqref{eq4.D}. From comparison of \eqref{eq4.B} with \eqref{eq3.C} it is seen that both expressions have the same form, and the only apparent difference is that in \eqref{eq4.D}, the summation is taken over the smaller set $\Int_\tau(x_N,x_{N-1})\times\dots\times\Int_\tau(x_2,x_1)$.
However, this does not matter because of Remark \ref{rem4.A}, which shows that all the extra summands actually vanish.
This completes the proof of the theorem.
\end{proof}
Theorem \ref{thm4.A} and Corollary \ref{cor4.A} provide a proof of Theorem A in the case of $t=q^\tau$, $\tau\in\{1,2,,\dots\}$, except the uniqueness claim. The proof of the latter claim (in the general case) in given below in section \ref{sect5.3}.
\section{Proof of Theorem A: general case}\label{sect5}
Throughout this section $\zeta_+>0$ and $\zeta_-<0$ are fixed parameters; $q$ and $t$ are supposed to lie in the open interval $(0,1)$.
The goal of this section is to prove Theorem A (section \ref{results1}) and exhibit an explicit expression for the matrices $\LaN$ (see subsection \ref{sect5.10}).
In the course of the proof, we sometimes need to introduce the assumption that $t$ is in general position with respect to $q$, but each time this constraint is ultimately removed.
\subsection{The sets $\Om_N$}\label{sect5.1}
As in Section \ref{sect4}, it will be convenient for us to switch from double signatures $a\in\DSign(N)$ to the corresponding point configurations $X_N(a)\in\Om_N$. We start with a direct description of the sets $\Om_N$ and then translate the notion of interlacement $b\prec a$ into the language of configurations. This looks a bit more complicated as in the special case examined in Section \ref{sect4}.
Below we use the notation
$$
q^\Z:=\{q^m: m\in\Z\}, \quad \Z_{\ge0}:=\{0,1,2,\dots\}, \quad q^{\Z_{\ge0}}:=\{q^m: m\in\Z_{\ge0}\}.
$$
Given an $N$-tuple $X=(x_1>\dots>x_n)$ of nonzero real numbers, we set
$$
k=k(X):=\#\{i=1,\dots,N: x_i>0\}.
$$
\begin{lemma}\label{lemma5.Omega}
Let $N=1,2,\dots$\,. The set\/ $\Om_N$ introduced in Definition \ref{def1.C} consists of $N$-tuples $(x_1>\dots>x_N)$ of nonzero real numbers satisfying the following constraints {\rm(1) -- (4):}
\smallskip
{\rm(1)} If $k\ge1$, then $x_1\in \zeta_+ q^{\Z}$.
{\rm(2)} If $k\ge 2$, then $x_{i+1}\in x_iq^{\Z_{\ge0}}t$ for each $i=1,\dots,k-1$.
{\rm(3)} If $N-k\ge1$, then $x_N\in\zeta_- q^{\Z}$.
{\rm(4)} If $N-k\ge2$, then $x_{i-1}\in x_i q^{\Z_{\ge0}} t$ for each $i=N,\dots,k+2$.
\end{lemma}
\begin{proof}
Evident from Definition \ref{def1.C}.
\end{proof}
Let $X=(x_1>\dots>x_N)\in\Om_N$ and $Y=(y_1>\dots>y_{N-1})\in\Om_{N-1}$, where $N=2,3,\dots$\,. We say that $X$ and $Y$ \emph{interlace} (and then write $X\succ Y$ or $Y\prec X$) if the corresponding double signatures interlace in the sense of Definition \ref{def1.interlace}.
\begin{lemma}\label{lemma5.interlace}
Let $X\in\Om_N$ and $Y\in\Om_{N-1}$. Set $k=k(X)$. The configurations $X$ and $Y$ interlace if and only if for each $r=1,\dots,N-1$, the coordinate $y_r$ satisfies the following condition varying depending on the relative position of $r$ and $k${\rm:}
{\rm(1)} Suppose $1\le r<k$ and observe that this implies that $0<x_{r+1}<x_r$ and $x_{r+1}=x_rq^{l_r} t$ with some $l_r\in\Z_{\ge0}$. Then $y_r=x_r q^{m_r}$ with $0\le m_r\le l_r$.
{\rm(2)} Suppose $k<r\le N-1$ and observe that this implies that $x_{r+1}<x_r<0$ and $x_r=x_{r+1}q^{l_r}t$ with some $l_r\in\Z_{\ge0}$. Then $y_r=x_{r+1} q^{m_r} $ with $0\le m_r\le l_r$.
{\rm(3)} Suppose $0<k=r<N$ and observe that this implies $x_{k+1}<0<x_k$. Then $y_k=x_k q^{m_k}$ or $y_k=x_{k+1}q^{m_k}$, where $m_k\in\Z_{\ge0}$ may be arbitrary.
\end{lemma}
\begin{proof}
Evident from Definition \ref{def1.interlace}.
\end{proof}
Note that in all cases we have $x_{r+1}\le y_r\le x_r$.
\subsection{Scheme of proof}\label{sect5.2}
The coherency relation \eqref{eq1.I} can now be rewritten in the form
\begin{equation}\label{eq5.F}
\sum_{Y\in\Om_{N-1}}\LaN(X,Y)\frac{P_{\nu\mid N-1}(Y;q,t)}{(t^{N-1};q,t)_\nu}=\frac{P_{\nu\mid N}(X;q,t)}{(t^N;q,t)_\nu},
\end{equation}
where $X\in\Om_N$ and $\nu\in\Y(N-1)$ are arbitrary.
We are going to prove that \eqref{eq5.F} holds true for a certain matrix $\LaN$ of format $\Om_N\times\Om_{N-1}$ whose entries $\LaN(X,Y)$ are strictly positive for $X\succ Y$ and equal $0$ otherwise. Once this is done, the remaining claims of Theorem A will follow quickly. Indeed, the fact that the row sums of $\LaN$ equal $1$ (meaning that $\LaN$ is stochastic) is equivalent to the simplest particular case of \eqref{eq5.F} corresponding to $\nu=\varnothing$, and the uniqueness of the matrix is proved easily (Lemma \ref{lemma5.uni} below).
As in the context of section \ref{sect4}, we derive the desired coherency relation \eqref{eq5.F} from Okounkov's formula \eqref{eq3.B}. However, we can no longer simply substitute $Z=X$ into \eqref{eq3.B}. The reason is that in doing so, for general $(q,t)$ and $N\ge3$, we can stumble upon singularities. This is shown in Example \ref{example5.A}.
Our strategy is the following. First, we represent the multiple $q$-integral on the left-hand side of \eqref{eq3.B} as a multiple sum (Lemma \ref{lemma5.A} below). Next, instead of directly substitute $Z=X$ into this sum, we specialize $z_i\to x_i$ step by step, by sorting out the coordinates in a special order depending on $k=k(X)$, as indicated in Definition \ref{def5.C} below. (A somewhat similar trick works in a different context, related to the $R$-matrix formalism, see e. g. Nazarov--Tarasov \cite[\S2]{NT}.)
We show that, under assumption that $t$ is in general position with respect to $q$, in the course of this procedure, each term of our multiple sum remains well defined and finally has a limit. Many terms actually vanish in the limit, but those that survive give a weighted sum over the configurations $Y\prec X$, just as on the left-hand side of \eqref{eq5.F}.
After that we check that in the final sum, the constraint on $t$ becomes inessential and can be dropped. Moreover, the weights $\LaN(X,Y)$ turn out to be strictly positive.
\subsection{Uniqueness claim in Theorem A}\label{sect5.3}
This claim follows from the next lemma, which is similar to \cite[Lemma 4.1]{Ols-2016}. We fix $K\in\{1,2,\dots\}$ and define
$\wt\Om_K$ as the disjoint union of the sets $\Om_k$ with $0\le k\le K$. Next, given $A>0$, we denote by $\wt\Om_K[-A,A]$ the subset of those configurations in $\wt\Om_K$ that are entirely contained in the closed interval $[-A,A]$. Recall that $\Sym(K)$ is our notation for the algebra of symmetric polynomials in $K$ variables. Any polynomial $P\in\Sym(K)$ can be viewed as a function on $\wt\Om_K[-A,A]$, with zeroes added as arguments if needed. Given a finite measure $M$ on $\wt\Om_K[-A,A]$, we can form its pairing $\langle M, P\rangle$ with any $P\in\Sym(K)$:
$$
\langle M, P\rangle:=\sum_{X\in\wt\Om_K[-A,A]} P(X)M(X).
$$
This makes sense because $\wt\Om_K[-A,A]$ (as well as $\wt\Om_K$) is a countable set; here we tacitly assume that the sigma-algebra on $\wt\Om_K$ is generated by the singletons.
\begin{lemma}\label{lemma5.uni}
In this notation, any finite measure on $\wt\Om_K[-A,A]$ is uniquely determined by its pairings with the Macdonald polynomials indexed by partitions $\nu\in\Y(K)$.
\end{lemma}
\begin{proof}
In section \ref{sect6.1} we define a nontrivial Hausdorff topology on $\wt\Om_K$. In this topology, $\wt\Om_K[-A,A]$ is a compact subset. Although the topology is not discrete, this does not affect the Borel structure of the countable space $\wt\Om_K$: the Borel sets are simply arbitrary subsets, as for the discrete topology. Therefore, any finite measure on $\wt\Om_K[-A,A]$ is uniquely determined by its pairings with continuous functions.
Further, polynomials $P\in\Sym(K)$ are continuous functions on $\wt\Om_K[-A,A]$. Moreover, they separate points. Therefore, by virtue of the Stone--Weierstrass theorem, they form a dense subalgebra in the algebra of real-valued continuous functions on $\wt\Om_K[-A,A]$. Finally, the Macdonald polynomials $P_{\nu\mid K}(\ccdot;q,t)$ with $\nu\in\Y(K)$ form a basis in $\Sym(K)$. This proves the lemma.
\end{proof}
In the context of Theorem A, we apply the lemma to the measure $\LaN(X, \ccdot)$ by taking $K=N-1$ and choosing the interval $[-A,A]$ so large that it contains all points of $X$. This will guarantee that the measure $\La^N_{N-1}(X,\ccdot)$ is supported on
$\wt\Om_{N-1}[-A,A]$ because of the interlacement condition.
\subsection{Reorganization of Okounkov's formula}
To write the $q$-integral \eqref{eq3.B} as a multiple sum we need to introduce a suitable notation.
Let
$$
\varepsilon:=(\varepsilon(1),\dots,\varepsilon(N-1))\in\{0,1\}^{N-1}
$$
denote an arbitrary binary word of length $N-1$, and let
$$
m:=(m_1,\dots,m_{N-1})\in\Z_{\ge0}^{N-1}
$$
be an arbitrary $(N-1)$-tuple of nonnegative integers. Given $\varepsilon$ and $m$ as above, we assign to every ordered $N$-tuple $Z=(z_1,\dots,z_N)\in\C^N$ an $(N-1)$-tuple $\wt Z\in\C^{N-1}$:
$$
\wt Z:=(\wt z_1,\dots, \wt z_{N-1}):=(z_{1+\varepsilon(1)}q^{m_1},\dots, z_{N-1+\varepsilon(N-1)}q^{m_{N-1}}).
$$
Next, we set
\begin{equation}\label{eq5.R_epsm}
R_{\varepsilon,m}(Z;q,t):=\prod_{r=1}^{N-1}\prod_{s=1}^N\frac{(\wt z_rq/z_s;q)_\infty}{(\wt z_rt/z_s;q)_\infty} \prod_{1\le i\ne j\le N}\frac{(z_it/z_j;q)_\infty}{(z_iq/z_j;q)_\infty}.
\end{equation}
Or, in more detailed notation,
\begin{equation}\label{eq5.E}
R_{\varepsilon,m}(Z;q,t):=\prod_{r=1}^{N-1}\prod_{s=1}^N\frac{(z_{r+\varepsilon(r)}q^{m_r+1}/z_s;q)_\infty}{(z_{r+\varepsilon(r)} q^{m_r} t/z_s;q)_\infty} \prod_{1\le i\ne j\le N}\frac{(z_it/z_j;q)_\infty}{(z_iq/z_j;q)_\infty}.
\end{equation}
Note that $(a;q)_\infty$ is an entire function in $a\in\C$, with simple zeroes at the points $1,q^{-1}, q^{-2},\dots$\,. If follows that $ R_{\varepsilon,m}(Z;q,t)$ is a meromorphic function in $N$ variables $z_1,\dots,z_N$. Because this function actually depends only on the ratios of the variables, it can be regarded as a meromorphic function on the projective space $\C\mathbb P^{N-1}$.
Finally, we set
\begin{equation}\label{eq5.K}
\wt C_N(q,t):=(1-q)^{N-1} C_N(q,t)=\frac{((t;q)_\infty)^N}{(t^N;q)_\infty((q;q)_\infty)^{N-1}}
\end{equation}
\begin{lemma}\label{lemma5.A}
The left-hand side of Okounkov's formula \eqref{eq3.B} can be written as the following multiple series
\begin{equation*}
\frac{\wt C_N(q,t)}{V(Z)}\sum_{\varepsilon\in\{0,1\}^{N-1}}\sum_{m\in\Z_{\ge0}^{N-1}}V(\wt Z) R_{\varepsilon,m}(Z;q,t) \prod\limits_{r=1}^{N-1} (-1)^{\varepsilon(r)}\wt z_r\ F(\wt Z) ,
\end{equation*}
where
$$
F(\wt Z):=\frac{P_{\nu\mid N-1}(\wt z_1,\dots,\wt z_{N-1};q,t)}{(t^{N-1};q,t)_\nu}.
$$
\end{lemma}
\begin{proof}
This follows from the very definition of the $q$-integral.
Indeed, we represent each of the one-variate $q$-integrals in \eqref{eq3.B} as the difference of two $q$-integrals,
\begin{equation}\label{eq5.D}
\int^{z_r}_{z_{r+1}} (\ccdot)d_q w_r=\int^{z_r}_0 (\ccdot)d_q w_r-\int^{z_{r+1}}_0(\ccdot)d_q w_r,
\end{equation}
and then write the $q$-integrals on the right as infinite series over $m_r\in\Z_{\ge0}$. Namely, in the first $q$-integral we set $w_r=z_r q^{m_r}$, and in the second $q$-integral we set $w_r=z_{r+1} q^{m_r}$. Then the whole $(N-1)$-fold $q$-integral in \eqref{eq3.B} turns into the sum of $2^{N-1}$ summands each of which is a series over $m\in\Z_{\ge0}^{N-1}$. The summands are indexed by the binary words $\varepsilon$: namely, $\varepsilon(r)=0$ encodes the choice of the first summand on the right-hand side of \eqref{eq5.D}, while $\varepsilon(r)=1$ encodes the choice of the second one.
\end{proof}
\subsection{Singularities of the function $ R_{\varepsilon,m}(Z;q,t)$}
Let us fix arbitrary $\varepsilon\in\{0,1\}^{N-1}$, $m\in\Z_{\ge0}^{N-1}$, and $X=(x_1>\dots>x_N)\in\Om_N$. Next, suppose that $t$ is in general position with respect to $q$: in fact it suffices to require that
\begin{equation}\label{eq5.G}
t,t^2,\dots,t^{N-1}\notin q^\Z:=\{q^n: n\in\Z\}.
\end{equation}
\begin{lemma}\label{lemma5.B}
Under these assumptions, the only factors in the denominators of \eqref{eq5.E} that may vanish at the point $Z=X$ are those of the form $(z_{r+\varepsilon(r)}q^{m_r}t/z_s;q)_\infty$, where one the following two conditions holds
{\rm(1)} $x_{r+\varepsilon(r)}>0$, $x_s>0$, and $s=r+\varepsilon(r)+1$,
{\rm(2)} $x_{r+\varepsilon(r)}<0$, $x_s<0$, and $s=r+\varepsilon(r)-1$.
\end{lemma}
\begin{proof}
Let us begin with the second product in \eqref{eq5.E}. The factors in the denominator have the form $(z_i q/z_j;q)_\infty$ with $i\ne j$. We claim that they do not vanish at $Z=X$. Indeed, if $x_i$ and $x_j$ are of opposite sign, then $x_i q/x_j$ is negative and hence $(x_i q/x_j;q)_\infty>0$. If $x_i$ and $x_j$ are of same sign, then $x_i q/x_j$ lies in $t^{i-j}q^\Z$. Since $i\ne j$, we see from \eqref{eq5.G} that $x_i q/x_j$ is not in $q^\Z$, which entails $(x_i q/x_j;q)_\infty\ne0$.
Let us turn to the first product in \eqref{eq5.E}. The factors in the denominator, at $Z=X$, have the form
$$
(x_{r+\varepsilon(r)}q^{m_r} t/x_s;q)_\infty, \qquad r=1,\dots,N-1, \quad s=1,\dots,N, \quad m_r\in\Z_{\ge0}.
$$
Again, if $x_{r+\varepsilon(r)}$ and $x_s$ are of opposite sign, then vanishing is impossible. Assume they are of the same sign. Then vanishing may happen only if $x_{r+\varepsilon(r)} t/x_s\in q^\Z$, which exactly means that either (1) or (2) holds.
\end{proof}
Whether a singularity really occurs depends also on the factors in the numerator. At first glance, the whole picture looks complicated, but our limit procedure makes it possible to avoid singularities.
\begin{definition}\label{def5.A}
Given $X\in\Om_N$, let $k=k(X)$ be the number of positive coordinates in $X$, so that
$$
x_1>\dots>x_k>0>x_{k+1}>\dots>x_N.
$$
We say that a given binary word $\varepsilon\in\{0,1\}^{N-1}$ is \emph{$k$-adapted} if
$$
\varepsilon(r)=0 \quad \text{for} \quad 1\le r\le k-1 \quad \text{and} \quad \varepsilon(r)=1 \quad \text{for} \quad k+1\le r\le N-1.
$$
Note that if $k=N$ (that is, all $x_i$'s are positive) or $k=0$ (that is, all $x_i$'s are negative), then there is a unique choice for $\varepsilon$. In the remaining cases, when $1\le k\le N-1$, there are two $k$-adapted $\varepsilon$'s, because $\varepsilon(k)$ may take both values, $0$ and $1$.
\end{definition}
\begin{corollary}\label{cor5.A}
Suppose, as above, that $t$ satisfies the constraint \eqref{eq5.G}. Let $X\in\Om_N$ and assume that $\varepsilon$ is $k(X)$-adapted.
Then the meromorphic function $ R_{\varepsilon,m}(Z;q,t)$ is nonsingular at $Z=X$ for any $m\in\Z_{\ge0}^{N-1}$.
\end{corollary}
\begin{proof}
Set $k=k(X)$. We split $R_{\varepsilon,m}(Z;q,t)$ into the product of two expressions,
$$
R_{\varepsilon,m}(Z;q,t)= R^{(1)}_{\varepsilon,m}(Z;q,t) R^{(2)}_{\varepsilon,m}(Z;q,t),
$$
where
\begin{equation*}
R^{(1)}_{\varepsilon,m}(Z;q,t):=\prod_{r=1}^{k-1}\frac{(z_r t/z_{r+1};q)_\infty}{(z_r q^{m_r}t/z_{r+1};q)_\infty} \cdot \prod_{r=k+1}^{N-1}\frac{(z_{r+1} t/z_r;q)_\infty}{(z_{r+1} q^{m_r}t/z_r;q)_\infty}
\end{equation*}
and $ R^{(2)}_{\varepsilon,m}(Z;q,t)$ consists of the remaining factors. By virtue of Lemma \ref{lemma5.B}, all factors in the denominator of $R_{\varepsilon,m}(Z;q,t)$ that may vanish at $Z=X$ are assembled in the denominator of $R^{(1)}_{\varepsilon,m}(Z;q,t)$, so that $ R^{(2)}_{\varepsilon,m}(Z;q,t)$ is nonsingular at $X$. On the other hand, $ R^{(1)}_{\varepsilon,m}(Z;q,t)$ can be written as
\begin{equation}
R^{(1)}_{\varepsilon,m}(Z;q,t)=\prod_{r=1}^{k-1}(z_r t/z_{r+1};q)_{m_r} \cdot \prod_{r=k+1}^{N-1}(z_{r+1} t/z_r;q)_{m_r}
\end{equation}
and hence is nonsingular, too.
\end{proof}
\subsection{Example of singularity}
One can show that for $N=2$, the function $ R_{\varepsilon,m}(Z;q,t)= R_{\varepsilon,m}(z_1,z_2;q,t)$ is nonsingular at each point $X\in\Om_2$, for any $(\varepsilon,m)$. The next example shows that this is not always true for $N\ge3$.
\begin{example}\label{example5.A}
Below we use the standard shorthand notation
$$
(a_1,\dots,a_n;q)_\infty:=\prod_{i=1}^n (a_i;q)_\infty.
$$
Let $N=3$, $\varepsilon=(1,0)$, and $m=(0,0)$. The corresponding function $ R_{\varepsilon,m}(z_1,z_2,z_3;q,t)$ can be represented in the form
\begin{equation*}
\frac{(z_1t/z_3, z_3t/z_1, z_3t/z_2, z_2q/z_3, z_2q/z_1,z_1tq/z_2, q,q;q)_\infty\, z_3}
{(z_1q/z_2, z_1q/z_3, z_3q/z_1, z_3q/z_2,z_2t/z_1, z_2tq/z_3, t,t;q)_\infty\,z_2}
\cdot \frac{z_2-z_1t}{z_3-z_2t}.
\end{equation*}
Suppose that $t,t^2\notin q^\Z$. At the point $X=(1,t,t^2)$ (which is contained in $\Om_3$), the first fraction is nonsingular and nonvanishing, while the second fraction has a singularity. Hence the whole expression is also singular at $X$.
\end{example}
\subsection{Admissible pairs $(\varepsilon,m)$}\label{sect5.adm}
Let $X\in\Om_N$ and $k:=k(X)$ (Definition \ref{def5.A}). We use the description of $\Om_N$ given in Lemma \ref{lemma5.Omega}. As pointed out in Lemma \ref{lemma5.interlace}, there exist nonnegative integers $l_1,\dots,l_{N-1}$ such that
\begin{equation*}
x_{i+1}=x_i t q^{l_i}, \quad i=1,\dots,k-1; \qquad x_i=x_{i+1} t q^{l_i}, \quad i=k+1,\dots,N-1.
\end{equation*}
\begin{definition}\label{def5.B}
Let $\varepsilon\in\{0,1\}^{N-1}$ and $m\in\Z_{\ge0}^{N-1}$. Let us say that a pair $(\varepsilon,m)$ is \emph{$X$-admissible} if $\varepsilon$ is $k$-adapted in the sense of Definition \ref{def5.A} and, moreover, $m_i\le l_i$ for all $i=1,\dots,N-1$, except $i=k$.
\end{definition}
In the next lemma we restate the interlacement relation $Y\prec X$.
\begin{lemma}\label{lemma5.F}
There is a one-to-one correspondence between the $X$-admissible pairs $(\varepsilon,m)$ and the configurations $Y=(y_1>\dots>y_{N-1})\prec X$, defined by
$$
y_i=x_{i+\varepsilon(i)}q^{m_i}, \quad i=1,\dots,N-1.
$$
That is,
$$
y_i=x_i q^{m_i}, \quad i=1,\dots,k-1; \qquad y_i=x_{i+1}q^{m_i}, \quad i=k+1,\dots,N-1,
$$
and $y_k=x_k q^{m_k}$ or $y_k= x_{k+1}q^{m_k}$ depending on whether $\varepsilon(k)$ equals $0$ or $1$.
\end{lemma}
\begin{proof}
This follows at once from Lemma \ref{lemma5.interlace}.
\end{proof}
\subsection{Limit transition in $R_{\varepsilon;m}(Z;q,t)$ as $Z\to X$ ($t$ generic)}
Fix an arbitrary $X\in\Om_N$, where $N\ge 3$ and denote $k:=k(X)$. We would like to pass to a limit in \eqref{eq5.E} as $Z\to X$. Example \ref{example5.A} shows that one cannot do this directly, because, for some $(\varepsilon,m)$, the meromorphic function $ R_{\varepsilon,m}(Z;q,t)$ may be singular at $Z=X$. To circumvent this obstacle, we let $Z$ approach $X$ along a special path, as described in the following definition.
\begin{definition}[Step-by-step limit transition]\label{def5.C}
(1) If $k=N$, then we first set $z_1=x_1$ keeping $z_2,\dots,z_N$ in general position, next we let $z_2\to x_2$, then we let $z_3\to x_3$, and so on up to the step $z_N\to x_N$.
(2) If $k=0$, then we perform the same procedure in the inverse direction: $z_N=x_N$, $z_{N-1}\to x_{N-1}$, \dots, $z_1\to x_1$.
(3) If $1\le k\le N-1$, then we combine the two procedures, that is, we begin with $z_1=x_1$ and go up to the step $z_k\to x_k$; then we set $z_N=x_N$ and go up to the step $z_{k+1}\to x_{k+1}$. Equally well we may stop at the step $z_{k-1}\to x_{k-1}$, then pass to $z_N=x_N$ and proceed up to the step $z_{k+2}\to x_{k+2}$, and finally set $z_k=x_k$, $z_{k+1}=x_{k+1}$.
\end{definition}
\begin{lemma}\label{lemma5.D}
Assume $t$ satisfies the constraint \eqref{eq5.G}. Fix an arbitrary $X\in\Om_N$ and let $Z$ approach $X$ as described in Definition \ref{def5.C}. Then $ R_{\varepsilon,m}(X;q,t)$ has a limit for every $(\varepsilon,m)$, and the limit value equals\/ $0$ unless $(\varepsilon,m)$ is $X$-admissible in the sense of Definition \ref{def5.B}.
\end{lemma}
\begin{proof}
Suppose that $k\ge2$. As was pointed out above, the function $Z\mapsto R_{\varepsilon,m}(Z;q,t)$ depends only on the ratios $z_i/z_j$ of the variables. It follows that the first step, the specialization $z_1=x_1$, presents no difficulty, because all the ratios $z_i/z_j$ remain in general position. Let us justify the second step, the limit transition $z_2\to x_2$.
Setting
\begin{gather*}
X':=(x_2,\dots,x_N), \quad Z':=(z_2,\dots,z_N), \\
\varepsilon':=(\varepsilon(2),\dots,\varepsilon(N-1)), \quad m':=(m_2,\dots,m_{N-1}),
\end{gather*}
we may write
$$
R_{\varepsilon,m}(Z;q,t)= R_{\varepsilon',m'}(Z';q,t) \wt R_{\varepsilon(1),m_1}(Z;q,t),
$$
where $\wt R_{\varepsilon(1),m_1}(Z;q,t)$ collects all factors from $ R_{\varepsilon,m}(Z;q,t)$ that are not contained in $ R_{\varepsilon',m'}(Z';q,t)$. The exact form of $\wt R_{\varepsilon(1),m_1}(Z;q,t)$ depends on whether $\varepsilon(1)$ equals $0$ or $1$; let us examine these two possible variants separately.
$\bullet$ \emph{Variant} 1: $\varepsilon(1)=0$. Then we have
\begin{multline}\label{eq5.H}
\wt R_{0,m_1}(Z;q,t)\big|_{z_1=x_1}=\prod_{s=1}^N\frac{(z_1 q^{m_1+1}/z_s;q)_\infty}{(z_1 q^{m_1}t/z_s;q)_\infty}\cdot \prod_{j=2}^N\frac{(z_1 t/z_j, \, z_j t/x_1; q)_\infty}{(x_1 q/z_j,\, z_j q/z_1; q)_\infty}\bigg|_{z_1=x_1}\\
=\frac{(q^{m_1+1};q)_\infty}{(q^{m_1}t;q)_\infty}\cdot \prod_{s=2}^N \frac{(z_1 t/z_s;q)_{m_1}}{(x_1 q/z_s; q)_{m_1}}\cdot \prod_{j=2}^N \frac{(z_j t/x_1; q)_\infty}{(z_j q/x_1; q)_\infty}.
\end{multline}
Look at the final expression in \eqref{eq5.H}. Here the first fraction is a strictly positive constant. The product over $j$ causes no problem: it can be directly specialized at $Z=X$, because then all its factors will be of the form $(a;q)_\infty$ with $a<1$ and hence are strictly positive. Let us turn now to the product over $s$ and write it as
\begin{equation}\label{eq5.I}
\frac{(x_1 t/z_2)_{m_1}}{(x_1 q/z_2)_{m_1}}\cdot\prod_{s=3}^N \frac{(x_1 t/z_s)_{m_1}}{(x_1 q/z_s)_{m_1}}
\end{equation}
The product over $s$ does not involve $z_2$, so that it is not sensitive to the specialization $z_2\to x_2$. Finally, examine the fraction in front of the product in \ref{eq5.I}. At the point $z_2=x_2=x_1q^{l_1}t$, the denominator does not vanish, because $t\notin q^\Z$. As for the numerator, at the same point, it reduces to $(q^{-l_1};q)_{m_1}$, and this quantity vanishes unless $m_1\le l_1$.
$\bullet$ \emph{Variant} 2: $\varepsilon(1)=1$. Then we have
\begin{equation}\label{eq5.J}
\wt R_{1,m_1}(Z;q,t)=\prod_{s=1}^N\frac{(z_2 q^{m_1+1}/z_s;q)_\infty}{(z_2 q^{m_1}t/z_s;q)_\infty}\cdot \prod_{j=2}^N\frac{(z_1 t/z_j, \, z_j t/z_1; q)_\infty}{(z_1 q/z_j,\, z_j q/z_1; q)_\infty}.
\end{equation}
This expression makes sense under the substitution $(z_1,z_2)=(x_1,x_2)$: indeed, we use the fact that $(z_3,\dots,z_N)$ is in general position, $t$ is also generic, and $x_2/x_1<1$. But at the point $(z_1,z_2)=(x_1,x_2)$, the factor $(z_1 t/z_2;q)_\infty$ turns into $(q^{-l_1};q)_\infty$ and hence vanishes. So the whole expression in fact disappears.
We conclude that the limit as $z_2\to x_2$ does exist, and only the terms with $\varepsilon(1)=0$ and $m_1\le l_1$ survive.
Then we apply the same procedure to the function $ R_{\varepsilon',m'}(Z';q,t)$ and so on. If $k=N$, then we may go up to the end and obtain the desired result. Otherwise we stop as $k$ becomes equal $1$.
After that we begin to move in the opposite direction, starting from $z_N=x_N$. (Or we do that from the very beginning if $k=0$.) Here the argument is similar.
Finally, the only remaining case is the one with $N=2$, $k=1$. That is, $x_2<0<x_1$. Here the argument is trivial: in the limit, all $q$-Pochhammer factors are of the form $(a;q)_\infty$, where either $0<a<1$ or $a<0$; hence, they are strictly positive (see formulas \eqref{eq5.N=2(1)} and \eqref{eq5.N=2(2)} below). In this case, all the summands survive in the limit and lead to the desired (infinite) sum over $\{y\}\prec X$. \end{proof}
\subsection{Computation of $R_{\varepsilon;m}(X;q,t)$ for arbitrary $t$ and admissible $(\varepsilon;m)$}\label{sect5.9}
Let, as above, $X\in\Om_N$. The next lemma is a refinement of Corollary \ref{cor5.A}.
\begin{lemma}\label{lemma5.E}
Let us remove the constraints \eqref{eq5.G}, so that $t\in(0,1)$ may be arbitrary.
If a pair $(\varepsilon,m)$ is $X$-admissible in the sense of Definition \ref{def5.B}, then the function $ R_{\varepsilon,m}(Z;q,t)$ is nonsingular at the point $Z=X$ and $ R_{\varepsilon,m}(X;q,t)>0$.
\end{lemma}
\begin{proof}
Suppose $k\ge2$, so that $0<x_2<x_1$. Then from the proof of Lemma \ref{lemma5.D} (formula \eqref{eq5.H} and the argument after it) we obtain the recurrence relation
\begin{multline}\label{eq5.recurr1}
R_{\varepsilon_1,\dots,\varepsilon_{N-1}; m_1,\dots,m_N}(x_1,\dots,x_N;q,t)\\
=\frac{(q^{m_1+1};q)_\infty}{(q^{m_1}t;q)_\infty}\cdot \prod_{s=2}^N \frac{(x_1 t/x_s;q)_{m_1}}{(x_1 q/x_s; q)_{m_1}}\cdot \prod_{j=2}^N \frac{(x_j t/x_1; q)_\infty}{(x_j q/x_1; q)_\infty}\\
\times R_{\varepsilon_2,\dots,\varepsilon_{N-1}; m_2,\dots,m_N}(x_2,\dots,x_N;q,t),
\end{multline}
with the understanding that the expression on the last line equals $1$ if $N=2$.
We claim that the expression on the middle line is strictly positive, for any $t\in(0,1)$. Indeed, in the product over $j$, all factors are strictly positive. Next, write the product over $s$ in a more detailed way:
$$
\prod_{s=2}^N \frac{(x_1 t/x_s;q)_{m_1}}{(x_1 q/x_s; q)_{m_1}}=\prod_{s=2}^N \frac{(1-x_1 t/x_s)\dots(1-x_1tq^{m_1-1}/x_s)}{(1-x_1 q/x_s)\dots(1-x_1q^{m_1}/x_s)}.
$$
Since $x_s\le x_1tq^{l_1}$ and $m_1\le l_1$, we see that all factors are strictly negative, so that the whole expression is strictly positive.
Next, suppose $N-k\ge2$, so that $x_N<x_{N-1}<0$. Then we obtain a similar recurrence relation,
\begin{multline}\label{eq5.recurr2}
R_{\varepsilon_1,\dots,\varepsilon_{N-1}; m_1,\dots,m_N}(x_1,\dots,x_N;q,t)\\
=\frac{(q^{m_N+1};q)_\infty}{(q^{m_N}t;q)_\infty}\cdot \prod_{s=1}^{N-1} \frac{(x_N t/x_s;q)_{m_N}}{(x_N q/x_s; q)_{m_N}}\cdot \prod_{j=1}^{N-1} \frac{(x_j t/x_N; q)_\infty}{(x_j q/x_N; q)_\infty}\\
\times R_{\varepsilon_1,\dots,\varepsilon_{N-2}; m_1,\dots,m_{N-1}}(x_1,\dots,x_{N-1};q,t).
\end{multline}
The same argument shows that the expression on the middle line is strictly positive.
Using these two recurrence relations we reduce the problem to the case when $N=2$ and $k=1$, meaning $x_2<0<x_1$. Then we have
\begin{equation}\label{eq5.N=2(1)}
R_{\varepsilon_1; m_1}(x_1,x_2;q,t)=\dfrac{(q^{m_1+1};q)_\infty}{(q^{m_1}t;q)_\infty}\dfrac{(x_1t/x_2;q)_{m_1}(x_2t/x_1;q)_\infty}{(x_1q/x_2;q)_{m_1}(x_2q/x_1;q)_\infty}, \qquad \varepsilon(1)=0,
\end{equation}
and
\begin{equation}\label{eq5.N=2(2)}
R_{\varepsilon_1; m_1}(x_1,x_2;q,t)=\dfrac{(q^{m_1+1};q)_\infty}{(q^{m_1}t;q)_\infty}\dfrac{(x_2t/x_1;q)_{m_1}(x_1t/x_2;q)_\infty}{(x_2q/x_1;q)_{m_1}(x_1q/x_2;q)_\infty}, \qquad \varepsilon(1)=1.
\end{equation}
In both variants, all the factors are strictly positive for any $t\in(0,1)$, because $x_2<0<x_1$.
\end{proof}
\subsection{The matrices $\LaN$}\label{sect5.10}
\begin{definition}\label{def5.D}
For each $N=2,3,\dots$ we define a matrix $\LaN$ of format $\Om_N\times\Om_{N-1}$ as follows.
$\bullet$ If $Y$ and $X$ do not interlace, then $\LaN(X,Y):=0$.
$\bullet$. If $Y\prec X$, then we take the $X$-admissible pair $(\varepsilon,m)$ corresponding to $Y$ (see Lemma \ref{lemma5.F}) and set
\begin{equation}\label{eq5.L}
\LaN(X,Y):=\frac{((t;q)_\infty)^N}{(t^N;q)_\infty((q;q)_\infty)^{N-1}}\cdot\frac{V(Y)}{V(X)}\prod_{i=1}^{N-1} |y_i|\cdot R_{\varepsilon,m}(X;q,t).
\end{equation}
\end{definition}
\smallskip
The definition makes sense, because we know from Lemma \ref{lemma5.E} that the meromorphic function $R_{\varepsilon,m}(Z;q,t)$ is nonsingular at $Z=X$. Note also that the first fraction on the right-hand side is the constant $\wt C_N(q,t)$ given by \eqref{eq5.K}.
To make the definition \eqref{eq5.L} explicit we have to exhibit an explicit expression for $R_{\varepsilon,m}(X;q,t)$. Here are a few ways to do that.
\smallskip
(1) From the proof of Lemma \ref{lemma5.E} one can deduce the formula
\begin{multline}\label{eq5.R}
R_{\varepsilon;m}(X;q,t)=\prod_{r=1}^{N-1}\frac{(q^{m_r+1};q)_\infty}{(q^{m_r}t;q)_\infty}
\cdot\prod_{r=1}^{N-1} \prod_{\substack{s=1,\dots,N\\ s\ne r+\varepsilon(r)}}\frac{(x_{r+\varepsilon(r)}t/x_s;q)_{m_r}}{(x_{r+\varepsilon(r)}q/x_s;q)_{m_r}}\\
\times \prod_{\substack{j=1,\dots,N\\ j\ne k+1-\varepsilon(k)}}\frac{(x_{k+1-\varepsilon(k)}t/x_j;q)_\infty}{(x_{k+1-\varepsilon(k))}q/x_j;q)_\infty}.
\end{multline}
Recall that $\varepsilon(r)=0$ for $1<r<k$, $\varepsilon(r)=1$ for $k<r<N$, while for $r=k$ with $0< k<N$, both values $0$ and $1$ are admitted. Note also that the last product in \eqref{eq5.R} should be removed when $k=0,N$.
This formula can be checked directly as follows. It suffices to show that it agrees with the recurrence relations \eqref{eq5.recurr1}, \eqref{eq5.recurr2}, and with formulas \eqref{eq5.N=2(1)}, \eqref{eq5.N=2(2)}.
When we split off $x_1$, the remaining variables are renamed and $k$ is replaced by $k-1$. From this it is seen that \eqref{eq5.R} agrees with \eqref{eq5.recurr1}.
When we split off $x_N$, the enumeration does not change and $k$ remains intact. This agrees with \eqref{eq5.recurr2}.
Examine now the case when $N=2$ and $k=1$, meaning that $x_2<0<x_1$. There are two variants, $\varepsilon(1)=0$ and $\varepsilon(1)=1$. In both variants, agreement with \eqref{eq5.N=2(1)} and \eqref{eq5.N=2(2)} is seen directly.
(2) An alternative formula is obtained by specializing directly $Z:=X$, $\wt Z:=Y$ into \eqref{eq5.R_epsm}:
\begin{equation}\label{eq5.LaN}
R_{\varepsilon;m}(X;q,t)= \prod_{r=1}^{N-1}\prod_{s=1}^N\frac{(y_rq/x_s;q)_\infty}{(y_r t/x_s;q)_\infty} \prod_{1\le i\ne j\le N}\frac{(x_it/x_j;q)_\infty}{(x_iq/x_j;q)_\infty}
\end{equation}
However, in this formula we need to impose the constraint \eqref{eq5.G} to guarantee that the factors in the denominators do not vanish.
\smallskip
(3) Next, one can get rid of the constraint \eqref{eq5.G} in \eqref{eq5.LaN} by the following transformation of the right-hand side. The problem consists in possible vanishing of the factors $(y_r t/x_s;q)_\infty$ (where $y_r$ and $x_s$ are of the same sign and such that $|y_rt|\ge|x_s|$), as well as of the factors $(x_iq/x_j;q)_\infty$ (where $x_i$ and $x_j$ are of the same sign and such that $|x_iq|\ge|x_j|$). In \eqref{eq5.R}, that problem was resolved due to cancellations. Here is the idea of another solution.
Let, for definiteness, $1\le r<s<k$, so that $x_r\ge y_r>x_s>0$. Using the fact that $y_r\in x_r q^\Z$, one can show that
\begin{equation}\label{eq5.psi}
\frac{(y_rq/x_s;q)_\infty(x_rt/x_s;q)_\infty}{(y_rt/x_s;q)_\infty(x_rq/x_s;q)_\infty}
=\left(\frac{y_r}{x_r}\right)^{\tau-1}
\dfrac{(x_sq/(y_rt);q)_\infty (x_s/x_r;q)_\infty}{(x_s/y_r;q)_\infty(x_sq/(x_rt);q)_\infty}.
\end{equation}
Observe that on the right-hand side of \eqref{eq5.psi}, the factors in the denominator already do not vanish.
This trick makes it possible to transform \eqref{eq5.LaN} to the form which does not require the constraint \eqref{eq5.G}.
\subsection{Completion of proof of Theorem A}
As explained in section \ref{sect5.2}, for the proof of Theorem A it suffices to prove the identity \eqref{eq5.F} linking the matrices $\LaN$ with Macdonald polynomials.
Suppose first that $t$ satisfies the constraint \eqref{eq5.G}
By virtue of Lemma \ref{lemma5.A}, Okounkov's formula \eqref{eq3.B} can be written in the form
\begin{multline*}
\frac{\wt C_N(q,t)}{V(Z)}\sum_{\varepsilon\in\{0,1\}^{N-1}}\sum_{m\in\Z_{\ge0}^{N-1}}V(\wt Z) R_{\varepsilon,m}(Z;q,t) \prod\limits_{i=1}^{N-1} (-1)^{\varepsilon(i)}\wt z_i\\
\times \frac{P_{\nu\mid N-1}(\wt z_1,\dots,\wt z_{N-1})}{(t^{N-1};q,t)_\nu}
=\frac{P_{\nu\mid N}(z_1,\dots,z_{N-1})}{(t^N;q,t)_\nu}
\end{multline*}
Now we fix an arbitrary $X\in\Om_N$ and let $Z$ approach $X$ in the way described in Definition \ref{def5.C}. By Lemma \ref{lemma5.D}, in this limit regime, $ R_{\varepsilon,m}(Z;q,t)$ has a limit for each $(\varepsilon,m)$, but the result vanishes unless $(\varepsilon,m)$ is $X$-admissible. Therefore, in the limit, we obtain on the left a sum over the $X$-admissible pairs $(\varepsilon,m)$, which can be interpreted as a sum over the configurations $Y\prec X$. Note that, given an $X$-admissible pair $(\varepsilon, m)$, we have $\wt Z\to Y$ and $(-1)^{\varepsilon(i)}$ is the sign of $y_i$, so that $(-1)^{\varepsilon(i)}\wt z_i\to |y_i|$. Taking into account Definition \ref{def5.D} we see that in the limit, Okounkov's formula turns into \eqref{eq5.F}, as desired.
After that we may remove the constraint on $t$ by continuity, because the resulting formula makes sense for any $t\in(0,1)$ due to the results of section \ref{sect5.9}.
This completes the proof of Theorem A.
\subsection{Remarks}
\subsubsection{The Dixon--Anderson kernel}
Let $\Conf_N(\R)$ denote the set of $N$-point configurations on $\R$: an element of $\Conf_N(\R)$ is an $N$-tuple $\XX=(x_1>\dots>x_N)$ of real numbers. If $\XX\in\Conf_N(\R)$ and $\YY\in\Conf_{N-1}(\R)$, then we write $\XX\succ \YY$ or $\YY\prec\XX$ if $x_i>y_i>x_{i+1}$ for $i=1,\dots,N-1$.
Given $\XX\in\Conf_N(\R)$ with $N\ge2$, the following formula defines a probability measure $\LL^N_{N-1}(\XX, d\YY)$ on the domain $\{\YY: \YY\prec\XX\}\subset\Conf_{N-1}(\R)$:
\begin{equation}\label{eq5.DA}
\LL^N_{N-1}(\XX, d\YY)=\frac{\Ga(N\tau)}{(\Ga(\tau))^N} \, \frac{V(\YY)}{(V(\XX))^{2\tau-1}} \, (V(\XX;\YY))^{\tau-1} d\YY,
\end{equation}
where $\tau>0$ is a parameter and
$$
V(\XX):=\prod_{1\le i<j\le N}(x_i-x_j), \quad V(\YY):=\prod_{1\le r<s\le N-1}(y_r-y_s), \quad V(\XX;\YY):=\prod_{i=1}^N \prod_{r=1}^{N-1}|x_i-y_r|.
$$
The fact that $\LL^N_{N-1}(X,\ccdot)$ is indeed a probability measure is equivalent to the evaluation of what is called a Dixon--Anderson integral (Forrester--Warnaar \cite[sect. 2.1]{FW-2008}). Thus, $\LL^N_{N-1}$ is a Markov kernel from $\Conf_N(\R)$ to $\Conf_{N-1}(\R)$; let us call it the \emph{Dixon--Anderson kernel}.
For more about it, see Assiotis--Najnudel \cite{AN}.
\subsubsection{Degeneration $\LaN\to\LL^N_{N-1}$}
Fix $\tau>0$ and suppose $t=q^\tau$, as usual. One can show that, as $q\to1$, the stochastic matrices $\LaN$ (Definition \ref{def5.D}) converge to the Dixon--Anderson kernels $\LL^N_{N-1}$.
In the case $\tau\in\Z_{\ge1}$ the proof is easy: one can use the simple formula \eqref{eq4.B}. For arbitrary $\tau>0$ one has to deal with the more sophisticated definition \eqref{eq5.L} and the expression \eqref{eq5.LaN} (or rather its transformation described in item (3) of subsection \ref{sect5.10}). Then the proof relies on the asymptotic formula
$$
\lim_{q\to1}\frac{(u q^A;q)_\infty}{(u q^B;q)_\infty}=(1-u)^{B-A},
$$
which is valid on the domain $\C\setminus[1,+\infty)$ (Andrews--Askey--Roy \cite[Theorem 10.2.4]{AAR}, Gasper--Rahman \cite[ch. 1, (3.19)]{GR}).
\subsubsection{Continuous analogue of Theorem A}
The following identity is a continuous analogue of the coherency relation \eqref{eq5.F}:
\begin{equation}\label{eq5.F1}
\int_{\YY\prec\XX}\LL^N_{N-1}(\XX,d\YY) \frac{P_{\nu\mid N-1}(\YY;\tau)}{(N-1;\tau)_\nu}=\frac{P_{\nu\mid N}(\XX;\tau)}{(N;\tau)_\nu},
\end{equation}
where $X\in\Conf_N(\R)$, $\nu\in\Y(N-1)$,
$$
(u;\tau)_\nu:=\prod_{(i,j)\in\nu}((u+1-i)\tau+j-i),
$$
and the polynomials on the left and on the right are the Jack polynomials with parameter $\tau$, in $N-1$ and $N$ variables, respectively.
This formula appeared in \cite[sect. 6]{OO-1997}. Note that it is different from \eqref{eq1.E}.
\section{Extended stochastic matrices $\wt\La^N_{N-1}$}\label{sect6}
This section serves as a preparation to the proof of Theorem B. In that proof we use a compactness argument, for which we need to deal with some larger sets of configurations $\wt\Om_N\supset\Om_N$ equipped with a non-discrete topology. We introduce certain matrices $\wt\La^N_{N-1}$ of format $\wt\Om_N\times\wt\Om_{N-1}$ that extend the matrices $\LaN$, and we establish a technical result --- a property of $\wt\La^N_{N-1}$ stated as Theorem \ref{thm6.A}. It is used in the sequel, in sections \ref{sect7.3} and \ref{sect8.2}.
\subsection{Preliminaries}\label{sect6.1}
Theorem B was formulated in terms of double signatures, but it is more convenient to deal directly with point configurations, as in section \ref{sect5}. The definitions formulated below are a direct extension of those given in the author's paper \cite[\S6]{Ols-2016}. The reader is referred to that paper for more details.
\begin{definition}\label{def6.A}
Recall that an \emph{infinite signature} is an infinite sequence of non increasing integers $a=(a_1\ge a_2\ge\dots)$.
(i) By definition, the set $\Om_\infty$ consists of the configurations in $\R^*$ of the form
$$
X(a^+,a^-):=\{\zeta_+ q^{-a^+_i}t^{i-1}\}\cup\{\zeta_- q^{-a^-_j}t^{j-1}\},
$$
where $a^+$ and $a^-$ are two signatures, and at least one of them is infinite.
(ii) Next, $\wt\Om$ is defined as the union of the sets $\Om_0:=\{\varnothing\}$, $\Om_1$, $\Om_2$, \dots, and $\Om_\infty$.
\end{definition}
Thus, elements of the space $\wt\Om$ are certain configurations in $\R^*$ which may be finite or infinite (in the case $t=q$ the space $\wt\Om$ coincides with the set $\wt{\mathbb G}_\infty$ from \cite{Ols-2016}). We equip $\wt\Om$ with a structure of uniform space by proclaiming two configurations $X,X'\in\wt\Om$ to be \emph{$\varepsilon$-close} (where $\varepsilon>0$ is small) if they coincide outside the interval $(-\varepsilon,\varepsilon)$. In particular, this makes $\wt\Om$ a topological space. As such, it is locally compact and metrizable.
Both $\Om_\infty$ and $\bigcup_{N=0}^\infty\Om_N$ are dense subsets of $\wt\Om$.
For each $N=1,2,\dots$, we denote by $\wt\Om_N$ the closure of $\Om_N$ in $\wt\Om$; it is the union of the sets $\Om_0,\dots,\Om_N$.
\subsection{Construction of matrices $\wt\La^N_{N-1}$}
Below the symbol $\P(\ccdot)$ denotes the space of probability Borel measures on a given topological space.
\begin{proposition}\label{prop6.A}
Fix $N=2,3,\dots$ and let us interpret the matrix $\LaN$ as a map $\Om_N\to\P(\Om_{N-1})$. As such, it can be uniquely extended to a map $\wt{\La}^N_{N-1}: \wt\Om_N\to \P(\wt\Om_{N-1})$, which is continuous with respect to the topology on $\wt\Om_N$ inherited from $\wt\Om$ and the weak topology on $\P(\wt\Om_{N-1})$.
\end{proposition}
\begin{proof}
The argument is exactly the same as in the proof of \cite[Proposition 4.2]{Ols-2016}. The key ingredient is the coherency relation \eqref{eq5.F} for Macdonald polynomials, which is a generalization of a similar relation for the Schur polynomials from \cite[Proposition 2.4]{Ols-2016}.
\end{proof}
In the theorem below we use the fact that for each $N$, the $N$-variate Macdonald polynomials can be extended from $\Om_N$ to the ambient space $\wt\Om_N$ by continuity. This is equivalent to saying that the values on the subset $\Om_n\subset \wt\Om_N$ with $n<N$ are obtained by adding extra $N-n$ zeroes as variables.
\begin{theorem}\label{thm6.B}
Let $N=2,3,\dots$, $\nu\in\Y(N-1)$, and $X\in\wt\Om_N$. Then
\begin{equation}\label{eq6.A}
\sum_{Y\in\wt\Om_{N-1}}\wt{\La}^N_{N-1}(X,Y)\frac{P_{\nu\mid N-1}(Y;q,t)}{(t^{N-1};q,t)_\nu}=\frac{P_{\nu\mid N}(X;q,t)}{(t^N;q,t)_\nu}.
\end{equation}
\end{theorem}
\begin{proof}
This follows from \eqref{eq5.F}) and the definition of the extended matrices.
\end{proof}
\subsection{The support of $\wt\La^N_{N-1}(X^*, \ccdot)$: statement of the result}
Let $n<N$. We are going to define a modified interlacement relation, denoted as $Y^*\prec\!\!\prec X^*$, between configurations $X^*\in\Om_n\subset\wt\Om_N$ and $Y^*\in\Om_n\subset\wt\Om_{N-1}$. Introduce a notation:
\begin{equation}\label{eq6.B}
k(X^*):=\#\{x^*\in X^*: x^*>0\}, \quad l(X^*):=\#\{x^*\in X^*: x^*<0\}.
\end{equation}
We will often abbreviate and write $k=k(X^*)$, $l=l(X^*)$. Evidently, $k+l=n$.
\begin{definition}\label{def6.B}
We write $Y^*\prec\!\!\prec X^*$ if $Y^*\prec (X^*\cup\{x^0\})$, where $x^0$ is an arbitrary point in $\zeta_+ q^\Z t^k\sqcup \zeta_- q^{\Z}t^l$ sufficiently close to zero. Note that $X^*\cup\{x^0\}\in \wt\Om_N$ and the choice of $x^0$ does not matter (provided it is close to $0$).
\end{definition}
Equivalently, writing $X^*=(x^*_1>\dots>x^*_n)$ and $Y^*=(y^*_1>\dots>y^*_n)$, the relation $Y^*\prec\!\!\prec X^*$ means that
\begin{equation*}
\begin{gathered}
x^*_{i+1}t^{-1}\le y^*_i\le x^*_i \quad \text{for} \quad 1\le i\le k-1,\\
x^*_j\le y^*_j\le x^*_{j-1}t^{-1}\quad \text{for} \quad k+2\le j\le n,\\
0<y^*_k\le x^*_k, \qquad x^*_{k+1}\le y^*_{k+1}<0,
\end{gathered}
\end{equation*}
with the understanding that if $k=0$ or $l=0$, then some of the above inequalities disappear.
For instance, if $n=3$ and $X^*=(x^*_1,x^*_2,x^*_3)$, where $x^*_1>x^*_2>0>x^*_3$, then $k=2$, $l=1$, and $Y^*\prec\!\!\prec X^*$ means that $Y^*=(y^*_1,y^*_2,y^*_3)$, where
\begin{gather*}
y^*_1\in\zeta_+ q^{\Z}, \quad x^*_1\ge y^*_1\ge x^*_2 t^{-1},\\
y^*_2\in \zeta_+ q^{\Z}t, \quad x^*_2\ge y^*_2>0,\\
y^*_3\in \zeta_-q^{\Z}, \quad 0>y^*_3\ge x^*_3.
\end{gather*}
\begin{theorem}\label{thm6.A}
Let $X^*\in\Om_n\subset\wt\Om_N$, where $n<N$. Then the measure $\wt\La^N_{N-1}(X^*, \ccdot)$ is concentrated on the set $\{Y^*\in\Om_n: Y^*\prec\!\!\prec X^*\}\subset\wt\Om_{N-1}$.
\end{theorem}
The remaining part of the section is devoted to the proof of this theorem.
\subsection{Preparation to proof}
Let $X^*\in\Om_n\subset \wt\Om_N$ be fixed, $k:=k(X^*)$, $l:=l(X^*)$ (see \eqref{eq6.B}), and $d:=N-n$. We assume $d>0$. As usual, we enumerate the points of $X^*$ in the descending order: $X^*=(x^*_1>\dots>x^*_n)$. Suppose that both $k$ and $l$ are strictly positive (otherwise the argument is simplified, see section \ref{sect6.endproof}). Then $x^*_k>0>x^*_{k+1}$. We will also use the alternative notation
$$
x^+:=x^*_k, \qquad x^-:=x^*_{k+1}.
$$
Let $A$ be a large positive integer. We insert between $0$ and $x^*_k$ the $d$-point configuration
$$
X^0_A:=(x^+ q^A t, \, x^+ q^A t^2 ,\dots, x^+q^A t^d),
$$
and we set $X_A:=X^*\cup X^0_A$. Thus,
\begin{multline*}
X_A=(x^*_1,\dots,x^*_{k-1}, x^+;\; x^+ q^A t, \, x^+q^A t^2,\dots, x^+ q^A t^d;\; x^-, x^*_{k+2},\dots,x^*_n)\\ =:(x_1,\dots, x_N).
\end{multline*}
Obviously, $X_A\in\Om_N$ and $X_A\to X^*$ as $A\to\infty$. Therefore, by the definition of the extended matrix $\wt\La^N_{N-1}$, the measure $\wt\La^N_{N-1}(X^*,\ccdot)$ is the weak limit of the measures $\LaN(X_A,\ccdot)$.
Let $S_A$ denote the support of the pre-limit measure $\LaN(X_A,\ccdot)$: it consists of the configurations $Y\prec X_A$. Each $Y\in S_A$ contains the configuration
$$
Y^0_A:=(x^+ q^A t, \,x^+q^A t^2,\dots, x^+q^A t^{d-1}),
$$
and we set $Y^*:=Y\setminus Y^0_A$.
Thus, we may write
\begin{gather*}
Y=(y^*_1,\dots,y^*_k;\; x^+ q^A t, \, x^+q^A t^2,\dots, x^+ q^A t^{d-1};\; y^*_{k+1},\dots,y^*_n),\\
Y^*=(y^*_1,\dots,y^*_k; y^*_{k+1},\dots,y^*_n).
\end{gather*}
Note that the correspondence $Y\mapsto Y^*$ is one-to-one; we denote by $S^*_A$ the image of the set $S_A$ under this correspondence.
Next, observe that $y^*_{k+1}$ is the only point of the configuration $Y$ that can be both to the left and to the right of zero. It is important for us to distinguish these two possibilities, so we write
$$
S_A=S^-_A\sqcup S^+_A, \qquad S^-_A:=\{Y\in S_A: y^*_{k+1}<0\}, \quad S^+_A:=\{Y\in S_A: y^*_{k+1}>0\}
$$
and likewise
$$
S^*_A=S^{* -}_A\sqcup S^{* +}_A, \qquad S^{* -}_A:=\{Y^*\in S^*_A: y^*_{k+1}<0\}, \quad S^{* +}_A:=\{Y^*\in S^*_A: y^*_{k+1}>0\}.
$$
The bijection $S_A \leftrightarrow S^*_A$ gives rise to the bijections $S^\pm_A \leftrightarrow S^{* \pm}_A$.
Finally, observe that if $Y\in S^-_A$, then $Y^*\in \Om_n$ and $Y^*\pprec X^*$, while for $Y\in S^+_A$ this is wrong.
\subsection{Reduction of the problem}
The configurations $Y^*=(y^*_1,\dots,y^*_n)\in S^*_A$ satisfy the following constraints:
$\bullet$ Each of the $n-2$ points $y^*_1,\dots,y^*_{k-1}, y^*_{k+2},\dots,y^*_n$ may range only over a fixed finite set which does not depend on $A$.
$\bullet$ The point $y^*_k$ may range over the set $\{x^+, x^+q,\dots,x^+q^A\}$, which is a finite geometric progression of the growing length $A+1$.
$\bullet$ The range of $y^*_{k+1}$ is the disjoint union of two infinite geometric progressions:
\begin{equation}\label{eq6.C1}
\{x^-, x^-q,x^-q^2,\dots\}\cup \{x^+t^d q^A,\, x^+t^d q^{A+1},\, x^+t^d q^{A+2},\dots\}.
\end{equation}
Let $\de>0$ be small and $S^-_A(\de)$ denote the subset of configurations $Y\in\ S^-_A$ satisfying at least one of the conditions $y^*_k\le\de$, $|y^*_{k+1}|\le\de$.
Our first task is to reduce Theorem \ref{thm6.A} to the following two claims.
\smallskip
\emph{Claim} 1. We have
\begin{equation*}
\lim_{\de\to 0}\sum_{Y\in S^-_A(\de)} \LaN(X_A, Y)=0 \quad \text{uniformly on $A$}.
\end{equation*}
\emph{Claim} 2. We have
\begin{equation*}
\lim_{A\to\infty}\sum_{Y\in S^+_A} \LaN(X_A, Y)=0.
\end{equation*}
\smallskip
\begin{proposition} These claims imply Theorem \ref{thm6.A}.
\end{proposition}
\begin{proof}
Let us abbreviate
$$
M_A:=\LaN(X_A,\ccdot), \qquad M:=\wt\La^N_{N-1}(X^*,\ccdot).
$$
We can write $M_A=M^-_A+M^+_A$, where $M^\pm_A$ stands for the restriction of $M_A$ to the subset $S^\pm_A$ (more accurately, for this decomposition one should define $M^\pm_A$ as the result of multiplication of $M_A$ by the characteristic function of $S^\pm_A$).
We know that $M$ is the weak limit of the measures $M_A$ as $A\to+\infty$. On the other hand, Claim 2 tells us that the total mass of $M^+_A$ tends to $0$ as $A\to+\infty$. Therefore, $M$ is also the weak limit of the measures $M^-_A$.
Recall that a configuration $Y\in S^-_A$ differs from the corresponding configuration $Y^*\in S^{* -}_A$ only by the $(d-1)$-point configuration $Y^0_A$. The latter shrinks to $0$ as $A\to+\infty$. It follows that $Y$ and $Y^*$ get closer to each other as $A\to+\infty$, with respect to the uniform structure. Therefore, denoting by $M^{* -}_A$ the pushforward of $M^-_A$ under the bijection $S^-_A\leftrightarrow S^{*-}_A$, we conclude that the measures $M^-_A$ and $M^{* -}_A$ (which we regard as subprobability measures on the compact space $\wt\Om_N$) have a common weak limit. Thus, $M$ is the weak limit of the measures $M^{* -}_A$.
We forget now about the compact space $\wt\Om_N$ and regard the measures $M^{* -}_A$ as subprobability measures on the countable discrete space
$$
\Om^*_n:=\{Y^*\in\Om_n: Y^*\pprec X^*\}.
$$
We know that the total mass of $M^{* -}_A$ tends to $1$ as $A\to+\infty$. We also know that $M^{* -}_A(Y^*)\to M(Y^*)$ for any fixed $Y^*\in \Om^*_n$ (this is a consequence of the weak convergence $M^{* -}_A\to M$).
To finish the proof it remains to show that the family $\{M^{* -}_A\}$ is tight on $\Om^*_n$ (although our measures are not probability measures, only subprobability ones, this claim makes sense, because their masses tends to $1$).
Observe that, on the discrete space $\Om^*_n$, a configuration $Y^*\in\Om^*_n$ can escape to infinity only if $y^*_k\to0$ or $y^*_{k+1}\to0$, or both. From this it is seen that the desired tightness property is guaranteed by Claim 1.
\end{proof}
We proceed to the proof of Claims 1 and 2. It is based on formula \eqref{eq5.R}, which we apply to $X=X_A=X^*\cup X^0_A$ and $Y=Y^*\cup Y^0_A$. In our current notation, the configuration $X$ has $k+d$ points on the right of $0$, so that the parameter $k=k(X)$ in \eqref{eq5.L} should be replaced with $k+d$. Now \eqref{eq5.R} takes the form
\begin{multline}\label{eq6.E}
\LaN(X_A,Y)=\wt C_N(q,t)\frac{V(Y)}{V(X_A)}\cdot \prod_{r=1}^{N-1}|y_r|\cdot \prod_{r=1}^{N-1}\frac{(q^{m_r+1};q)_\infty}{(q^{m_r}t;q)_\infty} \\
\times \prod_{r=1}^{N-1} \prod_{\substack{s=1,\dots,N\\ s\ne r+\varepsilon(r)}}\frac{(x_{r+\varepsilon(r)}t/x_s;q)_{m_r}}{(x_{r+\varepsilon(r)}q/x_s;q)_{m_r}}\cdot \prod_{\substack{j=1,\dots,N\\ j\ne k+d+1-\varepsilon(k+d)}}\frac{(x_{k+d+1-\varepsilon(k+d)}t/x_j;q)_\infty}{(x_{k+d+1-\varepsilon(k+d))}q/x_j;q)_\infty},
\end{multline}
where
$$
X_A=(x_1,\dots,x_N)=(x^*_1,\dots,x^*_{k-1}, x^+;\; x^+ q^A t, \, x^+ q^A t^2,\dots, x^+ q^At^d;\; x^-, x^*_{k+2},\dots,x^*_n),
$$
$$
Y=(y_1,\dots,y_{N-1})=(y^*_1,\dots,y^*_k;\; x^+ q^A t, \, x^+ q^A t^2,\dots, x^+ q^A t^{d-1};\; y^*_{k+1},\dots,y^*_n),
$$
$$
\varepsilon(1)=\dots=\varepsilon(k+d-1)=0, \quad \varepsilon(k+d+1)=\dots=\varepsilon(N-1)=1,
$$
and
$$
\varepsilon(k+d)=\begin{cases} 1, & Y\in S^-_A, \\ 0, & Y\in S^+_A, \end{cases}
$$
so that
\begin{equation}\label{eq6.F}
k+d+1-\varepsilon(k+d)=\begin{cases} k+d, & Y^*\in S^-_A, \\ k+d+1, & Y^*\in S^+_A. \end{cases}
\end{equation}
Next, the parameters $m_1,\dots,m_{N-1}$ are as follows:
$$
m_{k+1}=\dots=m_{k+d-1}=0;
$$
$m_1,\dots,m_{k-1}$ and $m_{k+d+1},\dots,m_{N-1}$ are bounded from above by certain constants that depend only on $X^*$ but not on $A$; finally, $0\le m_k\le A$ and $m_{k+d}\in\Z_{\ge0}$.
Further, we set $m':=m_{k}$, $m'':=m_{k+d}$. Then
$$
y_k=y^*_k=x^+q^{m'}, \qquad m'=0,\dots,A,
$$
and
$$
y_{k+d}=y^*_{k+1}=\begin{cases} x^-q^{m''}, & Y^*\in S^-_A,\\ x^+ t^d q^{m''+A}, & Y^*\in S^+_A, \end{cases}\quad
\text{where} \quad m''\in\Z_{\ge0}.
$$
Let $f$ and $g$ be two expressions, possibly depending on $A$; then we write $f\lesssim g$ if $|f|\le \const |g|$ with some constant factor that does not depend on $A$. Let also agree that the symbol $\asymp$ will denote an equality up to a factor whose absolute value is bounded away from zero and infinity, uniformly on $A\to\infty$.
We will establish the following bounds on the quantities $\LaN(X_A,Y)$:
\begin{proposition}\label{prop6.C}
Assume $Y\in S^-_A$. Then
$$
\LaN(X_A,Y)\le\const\, t^{dm'} (\max(q,t))^{m''}
$$
with some constant factor which does not depend on $A$.
\end{proposition}
\begin{proposition}\label{prop6.D}
Assume $Y\in S^+_A$. Then
$$
\LaN(X_A, Y)\asymp t^{dm'+dA} q^{m'+m''}.
$$
\end{proposition}
Let us show that Propositions \ref{prop6.C} and \ref{prop6.D} imply Claims 1 and 2, respectively.
Indeed, in both claims, we have to estimate a double sum taken over two indices $(m',m'')$.
In Claim 1 we have to suppose that $-\de\le y_{k+d}=y^*_{k+1}<0$ with $\de\to0$, which amounts to saying that $m''\ge B$ with $B\to+\infty$. Summation over $m'$ produces an expression which is bounded by a constant, so we are left with the sum
$$
\sum_{m''\ge B} (\max(q,t))^{m''},
$$
which goes to $0$ as $B\to\infty$.
In Claim 2 there is no similar constraint, but we have instead the factor $t^{dA}$; due to it the double sum goes to $0$ as $A\to+\infty$.
Therefore, our problem is reduced to the proof of these two propositions.
\subsection{Proof of Proposition \ref{prop6.C}}
We examine formula \eqref{eq6.E} taking into account the assumption $Y\in S^-_A$.
According to \eqref{eq6.F} we have $k+d+1-\varepsilon(k+d)=k+d$. Thus, in the product over $j$, we have to substitute
$x_{k+d+1-\varepsilon(k+d)}=x_{k+d}$. As $A$ grows, the point $x_{k+d}$, which is in $X^0_A$, goes to $0$. From this it follows that the product over $j$ remains bounded.
Next, examine the double product over $(r,s)$. Observe that if $m$ remains bounded, then any fraction of the form $(z;q)_m/(zt^{-1}q;q)_m$ also remains uniformly bounded, even if $z$ grows together with $A$. It follows that all fractions with $r$ distinct from $k$ and $k+d$ remain bounded. Hence we are left with
\begin{equation}\label{eq6.G}
\prod_{r=k, k+d} \prod_{\substack{s=1,\dots,N\\ s\ne r+\varepsilon(r)}}\frac{(x_{r+\varepsilon(r)}t/x_s;q)_{m_r}}{(x_{r+\varepsilon(r)}q/x_s;q)_{m_r}}
\end{equation}
Since $\varepsilon(k)=0$ and $\varepsilon(k+d)=1$, we have
$$
k+\varepsilon(k)=k, \qquad k+d+\varepsilon(k+d)=k+d+1,
$$
so that
$$
x_{k+\varepsilon(k)}=x^+, \qquad x_{k+d+\varepsilon(k+d)}=x^-.
$$
Therefore, \eqref{eq6.G} is equal to
$$
\prod_{x\in X\setminus\{x^+\}}\frac{(x^+t/x;q)_{m'}}{(x^+q/x;q)_{m'}}\cdot \prod_{x\in X\setminus\{x^-\}}\frac{(x^-t/x;q)_{m''}}{(x^-q/x;q)_{m''}}
$$
In this expression, all the fractions with $x\notin X^0_A$ remain bounded, hence the only relevant part is
\begin{multline}\label{eq6.H}
\prod_{x\in X^0_A}\frac{(x^+t/x;q)_{m'}}{(x^+q/x;q)_{m'}}\cdot \prod_{x\in X^0_A}\frac{(x^-t/x;q)_{m''}}{(x^-q/x;q)_{m''}}\\
=\prod_{i=1}^d \frac{(t^{1-i}q^{-A};q)_{m'}}{(t^{-i}q^{1-A};q)_{m'}}\cdot \prod_{i=1}^d\frac{((x^-/x^+)t^{1-i}q^{-A};q)_{m''}}{((x^-/x^+)t^{-i}q^{1-A};q)_{m''}}.
\end{multline}
To handle the resulting two products in \eqref{eq6.H} we need two lemmas.
\begin{lemma}\label{lemma6.A}
Let $u>\max(q,t)$ be fixed. For $m\le A$,
$$
\frac{(u q^{-A};q)_m}{(u t^{-1} q^{1-A};q)_m}\asymp \left(\frac tq\right)^m
$$
\end{lemma}
\begin{proof}
We have
\begin{gather*}
\frac{(u q^{-A};q)_m}{(u t^{-1} q^{1-A};q)_m}=\prod_{i=1}^m\frac{1-uq^{i-A-1}}{1-ut^{-1}q^{i-A}}=\left(\frac tq\right)^m \frac{(u^{-1}q^{A-m+1};q)_m}{(u^{-1}tq^{A-m};q)_m}\asymp \left(\frac tq\right)^m,
\end{gather*}
where the last step is justified by the fact that, due to the assumption on $u$,
$$
0<u^{-1}q^{A-m+1}\le u^{-1}q<1 \quad \text{and} \quad 0<u^{-1}tq^{A-m}\le u^{-1}t<1.
$$
\end{proof}
\begin{lemma}\label{lemma6.B}
Let $w>0$ be fixed. Then
$$
\frac{(-wq^{-A};q)_m}{(-wt^{-1}q^{1-A};q)_m}\asymp \left(\frac tq\right)^{\min(m,A)}.
$$
\end{lemma}
\begin{proof}
Suppose $m\le A$. We have
\begin{gather*}
\frac{(-wq^{-A};q)_m}{(-wt^{-1}q^{1-A};q)_m}=\prod_{i=1}^m\frac{1+w q^{-A+i-1}}{1+wt^{-1}q^{-A+i}}=\left(\frac tq\right)^m\prod_{i=1}^m\frac{1+w^{-1} q^{A-i+1}}{1+w^{-1}q^{A-i}}\\
=\left(\frac tq\right)^m\frac{(-w^{-1}q^{A-m+1};q)_m}{(-w^{-1}tq^{A-m};q)_m}\asymp \left(\frac tq\right)^m,
\end{gather*}
where last step is justified by the fact that the quantities $w^{-1}q^{A-m+1}$ and $w^{-1}tq^{A-m}$ are bounded from above by a constant which does not depend on $A$ (here we use the hypothesis $m\le A$).
Suppose now $m>A$. Then we have
$$
\frac{(-wq^{-A};q)_m}{(-wt^{-1}q^{1-A};q)_m}
=\frac{(-wq^{-A};q)_A}{(-wt^{-1}q^{1-A};q)_A}
\cdot \frac{(-wq;q)_{m-A}}{(-wt^{-1}q^1;q)_{m-A}}\asymp \left(\frac tq\right)^A.
$$
This completes the proof of the lemma.
\end{proof}
\begin{corollary}\label{cor6.A}
The expression \eqref{eq6.H} is \quad $\asymp \left(\dfrac tq\right)^{dm'+d\min(m'',A)}. $
\end{corollary}
\begin{proof}
The first and the second product on the right-hand side of \eqref{eq6.H} are estimated by applying Lemma \ref{lemma6.A} and Lemma \ref{lemma6.B}, respectively. In Lemma \ref{lemma6.A} we set $u=t^{1-i}$, and in Lemma \ref{lemma6.B} we set $-w=(x^-/x^+)t^{1-i}$. In both cases, $i=1,\dots,d$.
\end{proof}
\begin{lemma}\label{lemma6.C}
Under the assumption that $Y\in S^-_A$, we have
$$
\frac{V(Y)}{V(X_A)}\prod_{r=1}^{N-1}|y_r|\lesssim q^{dm'+m''+(d-1)\min(m'',A)}.
$$
\end{lemma}
\begin{proof}
Indeed,
$$
\prod_{i=1}^{N-1}|y_r|\asymp q^{m'+m''+A(d-1)},
$$
$$
V(X_A)\asymp V(X^0_A)\asymp q^{Ad(d-1)/2},
$$
\begin{multline*}
V(Y)\asymp V(Y^0_A)\cdot(y_k-y_{k+d})\cdot \prod_{y\in\Y^0_A}(y-y_{k+d})\cdot\prod_{y\in Y^0_A}(y_k-y)\\
\asymp q^{A(d-1)(d-2)/2}\cdot(x^+q^{m'}+|x^-|q^{m''})\cdot \prod_{i=1}^{d-1}(x^+t^i q^A+|x^-|q^{m''})\cdot q^{m'(d-1)}\\ \lesssim q^{A(d-1)(d-2)/2}\cdot q^{(d-1)\min(m'',A)}\cdot q^{m'(d-1)}.
\end{multline*}
This implies the desired bound.
\end{proof}
From Corollary \ref{cor6.A} and Lemma \ref{lemma6.C} we obtain
$$
\LaN(X_A,Y)\le\const\, t^{dm'+d\min(m'',A)} q^{m''-\min(m'',A)}\le \const\, t^{dm'+\min(m'',A)} q^{m''-\min(m'',A)} ,
$$
because $d\ge1$.
It remains to check the inequality
$$
t^{\min(m'',A)} q^{m''-\min(m'',A)}\le (\max(q,t))^{m''}.
$$
If $m''\le A$, then it turns into
$$
t^{m''}\le (\max(q,t))^{m''},
$$
which is obvious.
Finally, suppose $m''>A$. Then the inequality takes the form
$$
t^A q^{m''-A}\le (\max(q,t))^{m''}.
$$
If $t\le q$, then this means $t^A q^{m''-A}\le q^{m''}$ or else $t^A q^{-A}\le1$, which holds true (because $t\le q$).
If $t>q$, then this means $t^A q^{m''-A}\le t^{m''}$ or else $q^{m''-A}\le t^{m''-A}$, which also holds true (because $m''-A>0$ and $t>q$).
This completes the proof of Proposition \ref{prop6.C}.
\subsection{Proof of Proposition \ref{prop6.D}}
Now we reexamine formula \eqref{eq6.E} assuming $Y\in S^+_A$.
According to \eqref{eq6.F}, we have $k+d+1-\varepsilon(k+d)=k+d+1$. Because $x_{k+d+1}=x^-$, the product over $j$ in \eqref{eq6.E} takes the form
\begin{multline}\label{eq6.I}
\prod_{j\ne k+d+1}\frac{(x^-t/x_j;q)_\infty}{(x^-q/x_j;q)_\infty}\asymp \prod_{x\in X^0_A}\frac{(x^-t/x;q)_\infty}{(x^-q/x;q)_\infty}\\
=\prod_{i=1}^d \frac{((x^-/x^+)t^{1-i}q^{-A};q)_\infty}{((x^-/x^+)t^{-i}q^{1-A};q)_\infty}\asymp\left(\frac tq\right)^{dA},
\end{multline}
where the last step is justified by the following lemma, which we apply for $w:=-(x^-/x^+)t^{1-i}$,
\begin{lemma}\label{lemma6.D}
Let $w>0$ be fixed. As $A\to+\infty$,
$$
\frac{(-wq^{-A};q)_\infty}{(-wt^{-1}q^{1-A};q)_\infty}\asymp \left(\frac tq\right)^A.
$$
\end{lemma}
\begin{proof}
We have
$$
\frac{(-wq^{-A};q)_\infty}{(-wt^{-1}q^{1-A};q)_\infty}=\frac{(-wq^{-A};q)_A}{(-wt^{-1}q^{1-A};q)_A} \, \frac{(-w;q)_\infty}{(-wt^{-1}q;q)_\infty}=\const\,\frac{(-wq^{-A};q)_A}{(-wt^{-1}q^{1-A};q)_A} \asymp \left(\frac tq\right)^A,
$$
where the last step follows from Lemma \ref{lemma6.B} in which we take $m=A$.
\end{proof}
Now we turn to the double product over $(r,s)$ in \eqref{eq6.E}. Again, its relevant part is \eqref{eq6.G}. We still have $\varepsilon(k)=0$, but now $\varepsilon(k+d)$ equals $0$ (not $1$, as above). The result is that
$$
k+\varepsilon(k)=k, \qquad k+d+\varepsilon(k+d)=k+d,
$$
and hence
$$
x_{k+\varepsilon(k)}=x_k=x^+, \qquad x_{k+d+\varepsilon(k+d)}=x_{k+d}\in X^0_A.
$$
It follows that the part of \eqref{eq6.G} related to $r=k+d$ is uniformly bounded, hence we are left with
\begin{equation}\label{eq6.J}
\prod_{\substack{s=1,\dots,N\\ s\ne k}}\frac{(x_k t/x_s;q)_{m_k}}{(x_k q/x_s;q)_{m_k}}\asymp \prod_{x\in X^0_A}\frac{(x^+ t/x;q)_{m'}}{(x^+q/x;q)_{m'}}=\prod_{i=1}^d\frac{(t^{1-i}q^{-A};q)_{m'}}{(t^{-i}q^{1-A};q)_{m'}}\asymp \left(\frac tq\right)^{dm'},
\end{equation}
where the last step is justified with the help of Lemma \ref{lemma6.A}.
Combining \eqref{eq6.I} and \eqref{eq6.J} we obtain:
\begin{corollary}\label{cor6.B}
The expression in the second line of \eqref{eq6.E} is \; $
\asymp \left(\dfrac tq\right)^{dm' +dA}.
$
\end{corollary}
The next lemma is an analogue of Lemma \ref{lemma6.C}:
\begin{lemma}\label{lemma6.E}
Under the assumption that $Y\in S^+_A$, we have
$$
\frac{V(Y)}{V(X_A)}\prod_{r=1}^{N-1}|y_r|\asymp q^{(d+1)m'+m''+dA} .
$$
\end{lemma}
\begin{proof}
Indeed,
$$
\prod_{i=1}^{N-1}|y_r|\asymp q^{m'+(m''+A)+A(d-1)},
$$
$$
V(X_A)\asymp V(X^0_A)\asymp q^{Ad(d-1)/2},
$$
\begin{multline*}
V(Y)\asymp V(Y^0_A)(y_k-y_{k+d})\cdot \prod_{y\in\Y^0_A}(y-y_{k+d})\cdot\prod_{y\in Y^0_A}(y_k-y)\\
\asymp q^{A(d-1)(d-2)/2}\cdot q^{m'}\cdot q^{A(d-1)}\cdot q^{m'(d-1)}
\end{multline*}
This implies the desired bound.
\end{proof}
Corollary \ref{cor6.B} and Lemma \ref{lemma6.E} together imply Proposition \ref{prop6.D}.
\subsection{End of proof of Theorem \ref{thm6.A}}\label{sect6.endproof}
We have finished the proof of the theorem in the case when $0<k<n$. It remains to examine the two extreme cases $k=n$ and $k=0$. By symmetry, they are equivalent, so we examine only the case $k=n$. Then all points of $X_A$ and $Y$ are on the right of $0$. We have $N=k+d$,
$$
X_A=(x_1,\dots,x_N)=(x^*_1,\dots,x^*_k;\; x^*_k q^A t, \, x^*_k q^A t^2,\dots, x^*_k q^At^d),
$$
$$
Y=(y_1,\dots,y_{N-1})=(y^*_1,\dots,y^*_k;\; x^*_k q^A t, \, x^*_k q^A t^2,\dots, x^*_k q^A t^{d-1}),
$$
and
$$
\varepsilon(1)=\dots=\varepsilon(k+d-1)=0.
$$
In the configuration $Y$, each of the points except $y_k=y^*_k$ is either fixed or ranges over a fixed finite set. As for $y_k$, it may take the values of the form $x^*_kt q^m$, where $0\le m\le A-1$.
In the present situation the theorem reduces to the following claim:
$$
\lim_{B\to+\infty}\sum_{Y:\, m\ge B}\LaN(X_A,Y)=0 \quad \text{uniformly on $A$}.
$$
But this follows from the next proposition, which is a simplified version of Propositions \ref{prop6.C} and \ref{prop6.D}:
\begin{proposition}\label{prop6.E}
In the case $k=n$ we have
$$
\LaN(X_A,Y)\asymp t^{dm}.
$$
\end{proposition}
\begin{proof}
We turn again to formula \eqref{eq6.E}. The product over $j$ now disappears and the only relevant part has the form
\begin{equation*}
\frac{V(Y)}{V(X_A)}\cdot \prod_{r=1}^{N-1}|y_r|\cdot \prod_{r=1}^{N-1} \prod_{\substack{s=1,\dots,N\\ s\ne r}}\frac{(x_rt/x_s;q)_{m_r}}{(x_r q/x_s;q)_{m_r}},
\end{equation*}
Arguing as in Lemma \ref{lemma6.E} we obtain
$$
\frac{V(Y)}{V(X_A)}\cdot \prod_{r=1}^{N-1}|y_r|\asymp q^{dm}.
$$
Next, the product over $(r,s)$ is handled with the aid of Lemma \ref{lemma6.A}, and the result is \; $\asymp\left(\dfrac tq\right)^{dm}$.
These two estimates yield the desired result.
\end{proof}
This completes the proof of Theorem \ref{thm6.A}.
\section{Boundaries of projective chains: general facts}\label{sect7}
\subsection{The boundary}\label{sect7.1}
Recall (section \ref{sect1.1}) that a \emph{projective chain} $\{S_N, L^N_{N-1}\}$ consists of an infinite sequence $S_1,S_2,\dots$ of countable sets linked by stochastic matrices $L^N_{N-1}$ of format $S_N\times S_{N-1}$, where $N=2,3,\dots$\,.
Recall that the symbol $\P(\ccdot)$ denotes the set of probability measures on a given measurable space. The matrix $L^N_{N-1}$ determines a map $\P(S_N)\to\P(S_{N-1})$, which we write as $M\mapsto M L^N_{N-1}$ (here any measure $M\in\mathscr P(S_N)$ is interpreted as a row-vector whose coordinates are indexed by $S_N$).
The maps $\P(S_N)\to\P(S_{N-1})$ allow us to form the projective limit space $\varprojlim \P(S_N)$. By the very definition, an element of this space is an infinite sequence $\{M_N\in \P(S_N): N=1,2,\dots\}$ with the property that $M_N L^N_{N-1}=M_{N-1}$ for all $N\ge2$; such sequences are called \emph{coherent systems}.
In what follows we assume that the space $\varprojlim\P(S_N)$ is nonempty. It possesses a natural structure of a convex set, which gives sense to the following definition.
\begin{definition}\label{def7.A}
By the \emph{boundary} of a projective chain $\{S_N, L^N_{N-1}: N=2,3,\dots\}$ we mean the set $S_\infty:=\operatorname{Ex}(\varprojlim\P(S_N))$ of extreme points of $\varprojlim\P(S_N)$.
\end{definition}
Given $X\in S_\infty$, we denote by $M^{(X)}=\{M^{(X)}_N\}$ the coherent system represented by $X$.
In the next theorem we use the natural Borel structure on $\varprojlim\P(S_N)$ generated by the cylinder sets.
\begin{theorem}\label{thm7.A}
The set\/ $S_\infty$ is a Borel subset of the space\/ $\varprojlim\P(S_N)$, so that we may form the space $\P(S_\infty)$ of probability Borel measures on\/ $S_\infty$.
For every coherent system $M=\{M_N\}$ there exists a unique measure $\sigma\in\P(S_\infty)$ such that $M=\int_{S_\infty} M^{(X)}\sigma(dX)$ in the sense that
$$
M_N(Y)=\int_{S_\infty} M^{(X)}_N(Y)\sigma(dX) \qquad \text{\rm for every $N=1,2,\dots$ and every $Y\in\Om_N$.}
$$
Conversely, every measure $\sigma\in\P(S_\infty)$ generates in this way a coherent system, so that we obtain a bijection $\P(S_\infty)\leftrightarrow\varprojlim\P(S_N)$.
\end{theorem}
\begin{proof} See Olshanski \cite[Theorem 9.2]{Ols-2016}.
\end{proof}
For later use it is convenient to slightly reformulate this result. Let us introduce the alternative notation
$$
L^\infty_N(X,Y):=M^{(X)}_N(Y), \qquad Y\in S_N.
$$
We may regard $L^\infty_N$ as a Markov kernel: this simply means that $L^\infty_N(X,\ccdot)$ is a probability measure on $S_N$ for any fixed $X\in S_\infty$, and the function $X\mapsto L^\infty_N(X,Y)$ is a Borel measurable function on $S_\infty$ for any fixed $Y\in S_N$. Next, we rename $\si$ by $M_\infty$. In this notation, Theorem \ref{thm7.A} claims that there is a one-to-one correspondence $\{M_N\}\leftrightarrow M_\infty$ between coherent systems and probability measures on $S_\infty$ given by
$$
M_N=M_\infty L^\infty_N, \qquad N=1,2,3,\dots,
$$
or, in more detail,
\begin{equation}\label{eq7.C}
M_N(Y)=\int_{S_\infty} M_\infty(dX) L^\infty_N(X,Y), \qquad N=1,2,3,\dots, \quad Y\in S_N.
\end{equation}
We call $M_\infty$ the \emph{boundary measure} of a given coherent system $\{M_N\}$. It is tempting to say that $M_\infty$ is the limit of the measures $M_N$ as $N\to\infty$. One cannot do it in the abstract setting, because $M_\infty$ and the $M_N$'s live on distinct spaces. However, in a number of concrete models one can use their specific properties and deduce from Theorem \ref{thm7.A} that the $M_N$'s do converge to $M_\infty$ in some natural sense. In particular, this can be done in our case, see Theorem \ref{thm8.C} below.
\subsection{The path space}
Given two elements $X\in S_N$ and $Y\in S_{N-1}$, we write $X\triangleright Y$ or equivalently $Y\triangleleft X$ if the matrix entry $L^N_{N-1}(X,Y)$ is nonzero (hence strictly positive).
A \emph{finite path} of length $N$ is a sequence $(X(1),\dots,X(N)$, where
$X(i)\in S_i$ for all $i=1,\dots,N$ and $X(1)\triangleleft X(2)\triangleleft\dots\triangleleft X(N)$. The set of all such paths will be denoted by $\Pi_N$.
Likewise, an \emph{infinite path} is an infinite sequence $(X(1)\triangleleft X(2)\triangleleft\dots)$, where $X(i)\in S_i$ for all $i=1,\dots$\,. The set of all such paths is denoted by $\Pi$ and called the \emph{path space}.
For each $N\ge2$ there is a natural projection $\Pi_N\to\Pi_{N-1}$:
$$
(X(1),\dots,X(N-1),X(N))\;\mapsto\; (X(1),\dots,X(N-1)).
$$
Evidently, $\Pi$ is the projective limit of the sets $\Pi_N$ with respect to these projections. We equip $\Pi$ with the corresponding Borel structure.
An \emph{elementary cylinder set of depth $N$} in $\Pi$ is the set of all infinite paths with a prescribed beginning $(X(1),\dots,X(N))$; let us denote such a set by $C(X(1),\dots,X(N))$.
\begin{definition}
We say that a probability Borel measure $\mathscr M$ on $\Pi$ is a \emph{Gibbs measure} if the following condition holds. Let $N$ and $X\in S_N$ be arbitrary, and consider all elementary cylinder sets $C(X(1),\dots,X(N))$ with $X(N)=X$. Then we require that
\begin{multline}\label{eq7.A}
\mathscr M(C(X(1),\dots,X(N))=m(X)L^N_{N-1}(X(N),\,X(N-1))\\
\times L^{N-1}_{N-2}(X(N-1),\,X(N-2)) \dots L^2_1(X(2),\,X(1)),
\end{multline}
where $m(X)\ge0$ is a quantity that depends on $X$ only. (Cf. \cite[sect. 7.4]{BO-2017}.)
\end{definition}
\begin{proposition}\label{prop7.A}
There is a natural bijective correspondence $\mathscr M\leftrightarrow \{M_N\}$ between Gibbs measures and coherent systems.
\end{proposition}
\begin{proof}
Let $\mathscr M$ be a Gibbs measure. For each $N$, we define a measure $M_N\in\P(S_N)$ by setting $M_N(X)=m(X)$ for $X\in S_N$, where $m(X)$ is taken from \eqref{eq7.A}. It is immediately checked the $M_N$ is a probability measure and the sequence $M_1,M_2,\dots$ is a coherent system.
Conversely, let $\{M_N\}$ be a coherent system. For each $N$, we define a probability measure $\mathscr M_N$ on $\Pi_N$ by setting
\begin{multline*}
\mathscr M_N(C(X(1),\dots,X(N))=M_N(X(N))L^N_{N-1}(X(N),\, X(N-1))\\
\times L^{N-1}_{N-2}(X(N-1), \,X(N-2))\dots L^2_1(X(2),\,X(1)),
\end{multline*}
where $(X(1),\dots,X(N))\in\Pi_N$. The measures obtained in this way are consistent with the projections $\Pi_N\to\Pi_{N-1}$. Hence, by Bochner's theorem (see Bochner \cite[Theorem 5.1.1]{Bochner} or Parthasarathy \cite[Ch. V]{Parth}), they give rise to a probability measure $\mathscr M$ on $\Pi$. By the very construction, it is a Gibbs measure.
\end{proof}
Let $\pi=(X(N))$ and $\pi'=(X'(N))$ be two infinite paths; let us say that they are \emph{equivalent} (and then write $\pi\sim\pi'$) if they have the same \emph{tail}, that is, $X(N)=X'(N)$ for all $N$ large enough. Let $G$ be the group of all bijections $g:\Pi\to \Pi$ such that $g\pi\sim\pi$ for every path $\pi$ and $g\pi\ne\pi$ for finitely many paths $\pi$ only. This is a countable group of transformations of $\Pi$. Associated with the action of $G$ on $\Pi$ is a $1$-cocycle $c(g,\pi)$: if $\pi=(X(N))$ and $g\pi=(X'(N))$, then
$$
c(g,\pi):=\prod_{N=2}^\infty\frac{L^N_{N-1}(X'(N),X'(N-1))}{L^N_{N-1}(X(N),X(N-1))}.
$$
The product on the right is actually finite and hence is well defined.
The notion of Gibbs measures can be reformulated as follows: these are precisely those probability measures $\mathscr M\in\P(\Pi)$ that are $G$-quasiinvariant and consistent with the cocycle $c(g,\pi)$, that is, for a test function $f$ on $\Pi$,
\begin{equation}\label{eq7.B}
\int_{\Pi}f(g^{-1}\pi)\mathscr M(d\pi)=\int_{\Pi}f(\pi)c(g,\pi)\mathscr M(d\pi) \qquad \forall g\in G.
\end{equation}
This fact is used in the next proposition. Before to state it, observe that the notion of Gibbs measures given above can be extended, in a natural way, to finite measures (not necessarily probability ones). Next, if $\mathscr M$ is a finite measure on $\Pi$ and $A\subset \Pi$ is a Borel subset, then we denote by $\mathscr M\big|_A$ the restriction of $\mathscr M$ to $A$, which we regard again as a measure on $\Pi$.
\begin{proposition}\label{prop7.B}
Let $A\subset\Pi$ be a Borel subset, which is saturated with respect to the tail equivalence relation {\rm(}that is, $A$ consists of whole equivalence classes{\rm)}. If $\mathscr M$ is a Gibbs measure, then so is $\mathscr M\big|_A$.
\end{proposition}
\begin{proof}
We have $\mathscr M\big|_A=\chi_A\mathscr M$, where $\chi_A$ denotes the characteristic function of $A$. Our assumption on $A$ means that $\chi_A$ is $G$-invariant. It follows that if $\mathscr M$ satisfies \eqref{eq7.B}, then so is $\chi_A\mathscr M$. This concludes the proof.
\end{proof}
\subsection{Decomposition on singular and nonsingular components}\label{sect7.3}
Now we apply the general formalism described above to two concrete projective chains, $\{\Om_N, \LaN\}$ and $\{\wt\Om_N, \wt\La^N_{N-1}\}$; the results are used below in the proof of Theorem \ref{thm8.B}.
Let $\Pi$ and $\wt\Pi$ denote the spaces of infinite paths for $\{\Om_N, \LaN\}$ and $\{\wt\Om_N, \wt\La^N_{N-1}\}$, respectively. Evidently, $\Pi$ is a subset of $\wt\Pi$. Let us say that a path $\pi\in\wt\Pi$ is \emph{nonsingular} if it is contained in $\Pi\subset\wt\Pi$; otherwise it is called \emph{singular}.
Write a path $\pi\in\wt\Pi$ as a sequence $\{X(N)\in\wt\Om_N: N=1,2,\dots\}$. In this notation, $\pi$ is nonsingular if and only if $X(N)\in\Om_N$ for each $N$; that is, $X(N)$ must contain exactly $N$ points. Therefore, $\pi$ is singular if this condition fails, that is, there exists and index $K$ such that $X(K)$ contains less than $K$ points, say, $k<K$ points. Then, as is seen from Theorem \ref{thm6.A}, for all $N>K$ the number of points in $X(N)$ is also equal to $k$.
Thus, the nonsingular paths have the form
$$
X(1)\prec X(2)\prec \dots, \qquad X(N)\in\Om_N, \quad N=1,2,\dots,
$$
while the singular paths have the form
$$
X(1)\prec\dots \prec X(k)\prec\!\!\prec X(k+1)\prec\!\!\prec X(k+2)\prec\!\!\prec\dots, \qquad
$$
where $X(N)\in\Om_k\subset\wt\Om_N$ for all $N\ge k$, with a certain $k$. Recall that the meaning of symbol $\prec\!\!\prec$ is explained in Definition \ref{def6.B}.
From this description we obtain a \emph{stratification} of the space $\wt\Pi$:
$$
\wt\Pi=\Pi\sqcup \bigsqcup_{k=0}^\infty \Pi_k,
$$
where $\Pi_k$ is formed by those paths $\{X(N): N=1,2,\dots\}$ for which $X(N)\in\Om_k$ for all $N\ge k$.
\begin{lemma}\label{lemma7.A}
Each of the strata $\Pi, \Pi_0,\Pi_1,\dots$ is a saturated Borel subset.
\end{lemma}
\begin{proof}
The fact that the strata are saturated follows directly from their definition. Next, for each $k=0,1,2,\dots$, the stratum $\Pi_k$ consists precisely of the paths $\pi=(X(N))$ with the property $X(k+1)\in\Om_k$. It follows that $\Pi_k$ is the union of countably many elementary cylinder sets and hence a Borel set. This in turn implies that $\Pi$ is a Borel set, too.
\end{proof}
\begin{corollary}\label{cor7.C}
Any extreme Gibbs measure on the space $\wt\Pi$ is concentrated on one of its strata $\Pi,\Pi_0,\Pi_1,\dots$\,.
\end{corollary}
\begin{proof}
Let $\mathscr M$ be an arbitrary finite Gibbs measure on the path space $\wt\Pi$. Lemma \ref{lemma7.A} makes it possible to restrict $\mathscr M$ to any of the strata. Moreover, the resulting measure (denote it by $\mathscr M_\infty$ or $\mathscr M_k$) will be a Gibbs measure by virtue of Proposition \ref{prop7.B}. We obtain the decomposition
$$
\mathscr M=\mathscr M_\infty+\mathscr M_0+\mathscr M_1+\dots
$$
in which each all the components are Gibbs measures. In the case when $\mathscr M$ is an extreme probability Gibbs measure it must coincide with one of its components, all other components being equal to zero. This completes the proof.
\end{proof}
\section{Proof of Theorems B and C}\label{sect8}
\subsection{The boundary of the chain $\{\wt\Om_N,\wt\La^N_{N-1}\}$}
Let $\Sym$ denote the algebra of symmetric functions. Observe that for any configuration $X\in\wt\Om$ and any positive integer $k$, the sum $\sum_{x\in X}|x|^k$ is finite. It follows that for any $f\in\Sym$ and any $X\in\wt\Om$, the value of $f$ at $X$ makes sense: namely, we enumerate the points $x\in X$ in an arbitrary way and set
$$
f(X):=f(x_1,x_2,\dots), \quad \text{where} \quad X=(x_1,x_2,\dots)
$$
(we add infinitely many $0$'s if $X$ if finite). An important remark is that $f$ is a continuous function in the topology of the space $\wt\Om$.
We denote by $P_\nu=P_\nu(x_1,x_2,\dots;q,t)$ the \emph{Macdonald symmetric function} with index $\nu\in\Y$ and parameters $q$ and $t$ (Macdonald \cite[Ch. VI, sect. 4]{M}). Its value at $X\in\wt\Om$ is denoted by $P_\nu(X;q,t)$.
Given $X\in\wt\Om$ (see Definition \ref{def6.A} (ii)), we denote by $[X]$ the smallest closed interval of $\R$ containing all points of $X$.
\begin{theorem}\label{thm8.A}
The elements of the boundary of the chain\/ $\{\wt\Om_N,\wt\La^N_{N-1}\}$ can be para\-met\-rized by the configurations $X\in\wt\Om$.
More precisely, to every $X\in\wt\Om$ there corresponds a coherent system $M^{(X)}=\{M^{(X)}_K: K=1,2,\dots\}$; here the $K$th measure $M^{(X)}_K\in\P(\wt\Om_K)$ is concentrated on the compact set $\{Y\in\wt\Om_K: Y\subset [X]\}$ and is uniquely determined by the relations
\begin{equation}\label{eq8.B}
\sum_{Y\in\wt\Om_K}M^{(X)}_K(Y)\frac{P_{\nu\mid K}(Y;q,t)}{(t^K; q,t)_\nu}=P_\nu(X;q,t),
\end{equation}
where $\nu$ is an arbitrary partition with $\ell(\nu)\le K$. The coherent families $M^{(X)}$ are pairwise distinct and are precisely the extreme ones.
Furthermore, the Borel structure on the boundary coincides with the Borel structure of the space $\wt\Om$ determined by its topology.
\end{theorem}
In the particular case $t=q$ this result was proved in \cite[Theorem 6.2]{Ols-2016}, and the same argument works in the general case. So we only sketch the proof and refer to \cite{Ols-2016} for more details.
\begin{proof}[Sketch of proof]
\emph{Step} 1. For $N>K$ we set
$$
\wt\La^N_K:=\wt\La^N_{N-1}\wt\La^{N-1}_{N-2}\dots\wt\La^{K+1}_K;
$$
this is a stochastic matrix of format $\wt\Om_N\times\wt\Om_K$. Below we use the following direct generalization of \eqref{eq6.A}: if $\nu\in\Y(K)$, then
\begin{equation}\label{eq8.D}
\sum_{Y\in\wt\Om_K}\wt{\La}^N_K(X,Y)\frac{P_{\nu\mid K}(Y;q,t)}{(t^K;q,t)_\nu}=\frac{P_{\nu\mid N}(X;q,t)}{(t^N;q,t)_\nu},\qquad \forall X\in\wt\Om_N.
\end{equation}
Note that $\wt\La^N_K(X,Y)$ vanishes unless $Y\subset[X]$.
Let $X\in\wt\Om$ be arbitrary. Take a sequence $\{X(N)\in\wt\Om_N: N=1,2,\dots\}$ such that $X(N)\subset[X]$ and $X(N)\to X$ (such a sequence always exists). We claim that for any fixed $K=1,2,\dots$ there exists a weak limit
\begin{equation}\label{eq8.A}
M^{(X)}_K:=\lim_{N\to\infty}\wt\La^N_K(X(N),\ccdot)\in\P(\wt\Om_K).
\end{equation}
Indeed, substitute $X=X(N)$ into \eqref{eq8.D} and rewrite the resulting equality in the form
\begin{equation}\label{eq8.D1}
\left\langle \wt{\La}^N_K(X(N),\ccdot), \; \frac{P_{\nu\mid K}(\ccdot;q,t)}{(t^K;q,t)_\nu}\right\rangle=\frac{P_{\nu\mid N}(X(N);q,t)}{(t^N;q,t)_\nu},
\end{equation}
where the angular brackets denote the canonical pairing between measures and functions.
Fix $\nu$ and let $N\to\infty$. Then $(t^N;q,t)_\nu\to1$ and the right-hand side of \eqref{eq8.D1} tends to $P_\nu(X;q,t)$. Thus, the left-hand side also has a limit for each $\nu$. Since the measures $\wt{\La}^N_K(X(N),\ccdot)$ are concentrated on a compact set, they have a weak limit, as it is seen from the argument of Lemma \ref{lemma5.uni}.
By the very construction, the limit measure, denoted by $M^{(X)}_K$, is concentrated on the compact set $\{Y\in\wt\Om_K: Y\subset [X]\}$ and is uniquely determined by the relations \eqref{eq8.B}. In particular, it does not depend on the choice of the approximation $X(N)\to X$. Furthermore, the sequence $\{M^{(X)}_K\}$ is a coherent system, and different configurations $X\in\wt\Om$ lead to different coherent systems. All these claims are proved exactly as in \cite{Ols-2016}.
(Note that a phrase in \cite{Ols-2016} has to be corrected: there, in the proof of Theorem 6.2, the beginning of step 1, it is written that any sequence $\{X(N)\}$ converging to $X$ is `regular', meaning that the measures $\wt\La^N_K(X(N),\ccdot)$ converge in a stronger sense, which is not true in general. However, we do not need this; for our purpose it suffices that these measures converge weakly.)
\emph{Step} 2. Let $\{M_K:K=1,2,\dots\}$ be an extreme coherent system. By a general theorem (see \cite[Theorem 6.1]{OO-IMRN}), there exists a sequence $\{X(N)\in\wt\Om_N\}$ such that, as $N$ goes to infinity, $\wt\La^N_K(X(N),Y)\to M_K(Y)$ for every $K$ and every $Y\in\wt\Om_K$. A fortiori, for every $K$, the measures $\wt\La^N_K(X(N), \ccdot)$ converge to $M_K$ weakly. In particular, this holds for $K=1$ which in turn implies that the measures $\wt\La^N_1(X(N), \ccdot)$ form a tight family of probability measures on $\wt\Om_1$. Now we apply Proposition \ref{prop8.A} (see below); it tells us that there exists a positive number $a$ such that $X(N)\subset [-a,a]$ for each $N$. Because the subset $\{X\in\wt\Om: X\subset [-a,a]\}\subset\wt\Om$ is compact, the sequence $\{X(N)\}$ has a limit point in $\wt\Om$. Therefore, one may choose a subsequence of indices $N$ such that, along this subsequence, $X(N)$ converges to some element $X\in\wt\Om$. Applying the result of step 1 we see that $M_K=M^{(X)}_K$ for every $K$. We conclude that the extreme coherent systems are contained among the systems of the form $\{M^{(X)}_K\}$.
\emph{Step} 3. Here we prove the converse claim: any coherent system of the form $\{M^{(X)}_K\}$ is extreme. The argument is the same as in \cite{Ols-2016}, with Schur symmetric functions being replaced by Macdonald symmetric functions.
\emph{Step} 4. Here we apply a general fact about Borel maps to prove the final claim of the theorem. This claim is also necessary to justify an argument in step 3.
\end{proof}
\subsection{A condition of tightness}\label{sect8.2}
Our task here is to prove the following proposition, which was used in the argument above, on step 2.
\begin{proposition}\label{prop8.A}
Let\/ $\{X(N)\in\wt\Om_N: N=1,2,\dots\}$ be a sequence of configurations such that the corresponding sequence $\{\wt\La^N_1(X(N),\ccdot)\}$ of probability measures on\/ $\wt\Om_1$ is tight. Then there exists $a>0$ such that $X(N)\subset[-a,a]$ for all $N$.
\end{proposition}
First we state a lemma.
\begin{lemma}\label{lemma8.A}
Let $X$ be a nonempty configuration from $\wt\Om_N$, where $N\ge2$, and let $x_0$ denote the point of $X$ with maximal absolute value, so that $x_0$ is either the leftmost or the rightmost point {\rm(}in the case these endpoints of $X$ have the same absolute value we take as $x_0$ any of them{\rm)}.
The number $\wt\La^N_1(X,x_0)$ is bounded from below by a universal positive constant{\rm:}
\begin{equation}\label{eq8.B1}
\wt\La^N_1(X,x_0)\ge c:=\frac{\prod\limits_{m=1}^\infty(t^m;q)_\infty}{(-1;q)_\infty\prod\limits_{m=1}^\infty(-t^m;q)_\infty}>0.
\end{equation}
\end{lemma}
Observe that the proposition immediately follows from the lemma. Indeed, if the configurations $X(N)$ are not uniformly bounded, then one can choose a subsequence of numbers $N_1<N_2<\dots$ such that for the corresponding configurations $X(N_i)$, at least one endpoint goes to infinity. By virtue of the lemma, this means that the measure $\wt\La^N_1(X_i,\ccdot)$ has an atom of size $\ge c>0$ that escapes to infinity as $i\to\infty$. But this contradicts the tightness assumption. Thus, it remains to prove the lemma.
\begin{proof}[Proof of the lemma]
First of all note that the two infinite products on the right-hand side of \eqref{eq8.B1} converge. Indeed, to see this, write each of them as a double product
$$
\prod_{m=1}^\infty (\pm t^m;q)_\infty=\prod_{m=1}^\infty\prod_{n=0}^\infty(1\mp t^mq^n)
$$
and observe that
$$
\sum_{m=1}^\infty\sum_{n=0}^\infty t^mq^n<\infty.
$$
We follow the proof of Lemma 4.3 in \cite{Ols-2016} which in turn relies on computations in \S3 of that paper.
\emph{Step} 1. Recall that $\wt\Om_1=\Om_0\cup\Om_1$, where $\Om_0$ consists of the empty configuration and $\Om_1=\L$. Since $X$ is assumed to be nonempty, the measure $\wt\La^N_1(X,\ccdot)$ is concentrated on $\Om_1$: here we use Theorem \ref{thm6.A}. Thus, we may regard $\wt\La^N_1(X,\ccdot)$ as a measure on $\L$.
\emph{Step} 2. Let us show that
\begin{equation}\label{eq8.C}
\sum_{y\in\L}\wt\La^N_1(X,y)\frac{(yz^{-1}t^N;q)_\infty}{(yz^{-1};q)_\infty}=\prod_{x\in X}\frac{(xz^{-1}t;q)_\infty}{(xz^{-1};q)_\infty}, \quad z\in\C\setminus\R,
\end{equation}
cf. \cite[Proposition 3.1]{Ols-2016}.
Indeed, setting $K=1$ and $\nu=(n)$ in \eqref{eq8.D} we get
\begin{equation}\label{eq8.E}
\sum_{y\in\L}\wt{\La}^N_1(X,y)\frac{(t^N;q)_n}{(t;q)_n}y^n=P_{(n)\mid N}(X;q,t),\qquad \forall X\in\Om_N, \quad n\in\Z_{\ge0}.
\end{equation}
Assume first that $|z|$ is large, multiply the both sides of \eqref{eq8.E} by $Q_{(n)\mid1}(z^{-1};q,t)$ (the univariate Macdonald $Q$-polynomial) and sum over all $n\in\Z_{\ge0}$. On the right-hand side we obtain the right-hand side of \eqref{eq8.C}, by virtue of the fundamental Cauchy identity for the Macdonald symmetric functions \cite[ch. VI, (4.13)]{M}.
Let us turn to the left-hand side. Here we may interchange the order of summation, which gives us
$$
\sum_{y\in\L}\wt\La^N_1(X,y)\left\{\sum_{n=0}^\infty\frac{(t^N;q)_n}{(t;q)_n}Q_{(n)\mid1}(z^{-1};q,t)y^n\right\}.
$$
We can compute the interior sum. From the definition of the Macdonald $Q$-functions (see \cite[ch. VI, (4,12), (4.11), and 6.19)]{M}) it follows that
$$
Q_{(n)\mid1}(z^{-1};q,t)=\frac{(t;q)_n}{(q,q)_n}z^{-n},
$$
Therefore, the interior sum is
$$
\sum_{n=0}^\infty\frac{(t^N;q)_n}{(q;q)_n}(yz^{-1})^n=\frac{(yz^{-1}t^N;q)_\infty}{(yz^{-1};q)_\infty},
$$
where the last equality follows from the $q$-binomial formula \cite{GR}. This gives the desired equality \eqref{eq8.C}. Finally, we get rid of the assumption that $|z|$ is large by using analytic continuation.
\emph{Step} 3. Let us derive from \eqref{eq8.C} the equality
\begin{equation}\label{eq8.F}
\wt\La^N_1(X,x_0)=\frac{(t;q)_\infty}{(t^N;q)_\infty}\,\frac{\prod\limits_{x\in X\setminus\{x_0\}}(xx_0^{-1}t;q)_\infty}{\prod\limits_{x\in X\setminus\{x_0\}}(xx_0^{-1};q)_\infty}.
\end{equation}
Indeed, the right-hand side of \eqref{eq8.C} is a meromorphic function in $z\in\C\setminus\{0\}$. It has a pole at $z=x_0$ with the residue
\begin{equation}\label{eq8.C1}
\Res_{z=x_0}\left\{\prod_{x\in X}\frac{(xz^{-1}t;q)_\infty}{(xz^{-1};q)_\infty}\right\}=x_0\frac{(t;q)_\infty}{(q;q)_\infty}\,\frac{\prod\limits_{x\in X\setminus\{x_0\}}(xx_0^{-1}t;q)_\infty}{\prod\limits_{x\in X\setminus\{x_0\}}(xx_0^{-1};q)_\infty}.
\end{equation}
On the hand, let us compute the same residue by looking at the left-hand side of \eqref{eq8.C}. Recall that the support of the measure $\wt\La^N_1(X,\ccdot)$ is contained in $[X]$, the smallest closed interval containing $X$. From this and by the very definition of $x_0$ we conclude that only the summand with $y=x_0$ contributes. Therefore, the residue in question is equal to
\begin{equation}\label{eq8.C2}
\wt\La^N_1(X,x_0)\Res_{z=x_0}\left\{\frac{(x_0z^{-1}t^N;q)_\infty}{(x_0z^{-1};q)_\infty}\right\} =\wt\La^N_1(X,x_0) x_0\frac{(t^N;q)_\infty}{(q;q)_\infty}.
\end{equation}
Equating \eqref{eq8.C1} to \eqref{eq8.C2} we obtain \eqref{eq8.F}.
\emph{Step} 4. It remains to find a lower bound for the right-hand side of \eqref{eq8.F}. By the definition of $x_0$, the numerator can be estimated as follows
$$
(t;q)_\infty \prod\limits_{x\in X\setminus\{x_0\}}(xx_0^{-1}t;q)_\infty\ge (t;q)_\infty (t^2;q)_\infty\dots (t^N;q)_\infty\ge\prod_{m=1}^\infty(t^m;q)_\infty.
$$
Likewise, the denominator can be estimated as follows
$$
(t^N;q)_\infty \prod\limits_{x\in X\setminus\{x_0\}}(xx_0^{-1};q)_\infty\le (-1;q)_\infty(-t;q)_\infty \dots (-t^{N-2};q)_\infty\le(-1;q)_\infty\prod_{m=1}^\infty(-t^m;q)_\infty.
$$
This completes the proof of \eqref{eq8.B}.
\end{proof}
\subsection{Proof of Theorem B}
The next theorem is similar to that of Theorem \ref{thm8.A}. Recall that the space $\Om_\infty$ was introduced in Definition \ref{def6.A}.
\begin{theorem}\label{thm8.B}
The elements of the boundary of the chain\/ $\{\Om_N,\La^N_{N-1}\}$ can be parametrized by the configurations $X\in\Om_\infty$.
More precisely, to every $X\in\Om_\infty$ there corresponds a coherent system $M^{(X)}=\{M^{(X)}_K: K=1,2,\dots\}$; here the $K$th measure $M^{(X)}_K\in\P(\Om_K)$ is concentrated on the compact set $\{Y\in\Om_K: Y\subset [X]\}$ and is uniquely determined by the relations
\begin{equation}\label{eq8.B3}
\sum_{Y\in\Om_K}M^{(X)}_K(Y)\frac{P_{\nu\mid K}(Y;q,t)}{(t^K; q)_\nu}=P_\nu(X;q,t),
\end{equation}
where $\nu$ is an arbitrary partition with $\ell(\nu)\le K$. The coherent families $M^{(X)}$ are pairwise distinct and are precisely the extreme ones.
Furthermore, the Borel structure on the boundary coincides with the Borel structure of the ambient space $\wt\Om$ determined by its topology.
\end{theorem}
This result is a reformulation (with a slight refinement) of Theorem B (see section \ref{results2}).
\begin{proof}
The results of Section \ref{sect7} show that the boundary of the chain $\{\Om_N,\La^N_{N-1}\}$ is contained in the boundary of the chain $\{\wt\Om_N,\wt\La^N_{N-1}\}$. We know (Theorem \ref{thm8.A}) that the latter boundary is the space $\wt\Om$, and we are going to prove that the former boundary is its subset $\Om_\infty$. After that the remaining claims will follow from the corresponding claims of Theorem \ref{thm8.A}.
Proposition \ref{prop7.A} allows us to switch to the language of Gibbs measures. Let $X\in\wt\Om$, $\{M^{(X)}_K:K=1,2,\dots\}$ be the corresponding coherent system, and $\mathscr M^{(X)}$ denote the corresponding Gibbs measure on the path space $\wt\Pi$. We know that $\mathscr M^{(X)}$ is extreme. Therefore, by virtue of Corollary \ref{cor7.C}, $\mathscr M^{(X)}$ is concentrated on one of the strata of $\wt\Pi$, and the boundary under question is the set of those configurations $X\in\wt\Om$ for which the corresponding stratum is $\Pi$, not $\Pi_k$.
Thus, it suffices to show that $\mathscr M^{(X)}$ is concentrated on some $\Pi_k$ if and only if $X\in\wt\Om\setminus\Om_\infty$. We proceed to the proof of this claim.
Suppose that there exists $k$ such that $\mathscr M^{(X)}$ is concentrated on the stratum $\Pi_k$. This implies that for any $K>k$, the measure $M^{(X)}_K$ is concentrated on $\Om_k\subset \wt\Om_K$. Now let $\nu$ be an arbitrary partition with $\ell(\nu)>k$. Take an arbitrary $K\ge\ell(\nu)$ and observe that the polynomial $P_{\nu\mid K}(\ccdot;q,t)$ vanishes on the subset $\Om_k\subset\wt\Om_K$. Then \eqref{eq8.B} shows that $P_\nu(X;q,t)=0$. In particular, all elementary symmetric functions $e_n$ with $n>k$ vanish at $X$ (here we use the fact that $e_n$ coincides with $P_{(1^n)}(\ccdot;q,t)$). Therefore, the generating function
$$
1+\sum_{n=1}^\infty e_n(X)z^n=\prod_{x\in X}(1+x z)
$$
is a polynomial in $z$ of degree at most $k$.
On the other hand, this function vanishes at each point of the form $z=-x^{-1}$ with $x\in X$. This implies that $X$ has at most $k$ points, so that $X\in\wt\Om\setminus\Om_\infty$.
Conversely, suppose that $X\in\wt\Om\setminus\Om_\infty$, so that $X\in\Om_k\subset\wt\Om$ for some $k$. For each $N>k$ let $X(N)$ denote the same configuration $X$ regarded as an element of $\Om_k\subset \wt\Om_N$. Then for each $K$ the limit relation \eqref{eq8.A} holds. It shows that the measure $M^{(X)}_K$ is concentrated on the subset of configurations with at most $k$ points. This in turn implies that $\mathscr M^{(X)}$ cannot be concentrated on $\Pi$. .
\end{proof}
\subsection{Proof of Theorem C}
The next theorem contains Theorem C (section \ref{results3}).
\begin{theorem}\label{thm8.C}
Let $\{M_N\}$ be a coherent system of probability distributions for the chain $\{\Om_N, \LaN\}$ or $\{\wt\Om_N, \wt\La^N_{N-1}\}$ and let $M_\infty$ be the corresponding boundary measure on $\Om_\infty$ or\/ $\wt\Om$, respectively. Then $M_N\to M_\infty$ in the weak topology of the space $\P(\wt\Om)$.
\end{theorem}
\begin{proof}
Recall that the space $\wt\Om$ is locally compact and suppose first that $M_\infty$ is compactly supported. Then there exists an interval $[a,b]\subset\R$ such that $M_\infty$ is concentrated on the compact subset $\wt\Om[a,b]:=\{X\in\wt\Om: X\subset[a,b]\}$. It follows that the same also holds for all measures $M_N$. The symmetric functions form a dense subset of the Banach space $C(\wt\Om[a,b])$, hence it suffices to prove that, as $N\to\infty$,
\begin{equation}\label{eq8.B2}
\langle M_N, P_\nu(\ccdot;q,t)\rangle \to \langle M_\infty, P_\nu(\ccdot;q,t)\rangle
\end{equation}
for any partition $\nu$,
where, as before, the angular brackets denote the canonical pairing between measures and functions, and $M_N$ is regarded as a measure on $\wt\Om$.
On the other hand, for large enough $N$ we may write
$$
\langle M_N, P_\nu(\ccdot;q,t)\rangle=\langle M_N, P_{\nu\mid N}(\ccdot;q,t)\rangle
$$
and then it follows from \eqref{eq8.B} and \eqref{eq8.B3} that, as $N\to\infty$,
$$
\langle M_N, P_{\nu\mid N}(\ccdot;q,t)\rangle=(t^N;q,t)_\nu\langle M_\infty, P_{\nu\mid N}(\ccdot;q,t)\rangle.
$$
Since $(t^N;q,t)_\nu\to1$, this implies \eqref{eq8.B2}.
In the general case we may write $M_\infty$ as a convex combination of two probability measures,
$$
M_\infty=(1-\varepsilon) M'_\infty+\varepsilon M''_\infty,
$$
where $M'_\infty$ is compactly supported and $\varepsilon>0$ is a small parameter. Denote by $\{M'_N\}$ and $\{M''_\infty\}$ the coherent systems corresponding to $M'_\infty$ and $M''_\infty$, respectively. For an arbitrary fixed bounded continuos function $F$ on $\wt\Om$ we have
\begin{multline*}
|\langle M_\infty,F\rangle - \langle M_N,F\rangle|\le
(1-\varepsilon)|\langle M'_\infty,F\rangle - \langle M'_N,F\rangle| + \varepsilon|\langle M''_\infty,F\rangle - \langle M''_N,F\rangle|\\
\le (1-\varepsilon)|\langle M'_\infty,F\rangle - \langle M'_N,F\rangle| +2\varepsilon \Vert F\Vert.
\end{multline*}
By virtue of the above argument, as $N$ gets large, $|\langle M'_\infty,F\rangle - \langle M'_N,F\rangle|$ goes to $0$. It follows that
$$
\lim_{N\to\infty}|\langle M_\infty,F\rangle - \langle M_N,F\rangle|=0,
$$
which completes the proof.
\end{proof}
\bigskip
\noindent \textbf{Funding}
\medskip
\noindent This work was supported by the Russian Science Foundation [project 20-41-09009].
\bigskip
\noindent\textbf{Acknowledgments}
\medskip
\noindent I am grateful to Cesar Cuenca and an anonymous referee for valuable comments.
|
1,108,101,566,756 | arxiv | \section{Conclusions}\vspace{-1mm}
We prove the first sublinear regret bounds for GP optimization with commonly
used kernels (see \figref{regrettable}), both for $f$ sampled from a known GP
and $f$ of low RKHS norm. We analyze \ucb, an intuitive, Bayesian upper confidence bound based sampling rule. Our regret bounds crucially depend on the information gain due to sampling, establishing a novel connection between bandit optimization and experimental design. We bound the information gain in terms
of the kernel spectrum, providing a general methodology for obtaining regret
bounds with kernels of interest.
Our experiments on real sensor network data indicate that \ucb performs at
least on par with competing criteria for GP optimization, for which no regret
bounds are known at present. Our results provide an interesting step towards
understanding exploration--exploitation tradeoffs with complex utility
functions.
|
1,108,101,566,757 | arxiv | \section{Introduction}
The motion picture industry is a multi-billion dollar market.
Therefore many attempts have been made at predicting how much money films
will bring in during their respective opening weekends in theaters \cite{Huberman:twitter, mestyan:wikipedia, sharda:predicting}. There is also a body of research on the effects of various factors on box office revenue which assists those attempting to predict it \cite{basuroy:critical, liu:word, terry:determinants, zhang:improving}.
Huberman et al. found that they could use chatter on Twitter\footnote{\url{https://twitter.com/}} about movies prior to their release to accurately predict how well they would do in theaters \cite{Huberman:twitter}. More recently, Mesty{\'a}n et al. successfully used data gathered from Wikipedia\footnote{\url{http://www.wikipedia.org/}}, including page views, total edits, and number of editors, to predict the amount of money a film (released in the United States) would bring in during its opening weekend \cite{mestyan:wikipedia}.
It is a rather unexpected result that the number of views a film's Wikipedia page has is so highly correlated with the money earned by that film during its opening weekend.
A possible explanation for this phenomenon is as follows. In modern society potential moviegoers who are unsure which movie to see often turn to the internet to aid them in making their decision.
It was recently suggested by Panaligan et al. that opening weekend box office sales for a film could be well approximated by the search volume for terms related to the movie. They were able to account for as much as 94\% of the variation in opening weekend box office performance using a combination of related search term volume, franchise status, and seasonality \cite{panaligan:google}. If we take this result into account along with the observation that upon searching for a film (using \textit{Google} or \textit{Bing}), that film's corresponding Wikipedia page is among the top search results, it is reasonable to hypothesize that the Wikipedia page acts as a proxy for searches relating to the film.
Our goals in this article are to recreate the results of Mesty{\'a}n et al. using a more recent set of films released in the United States and then to test whether or not the technique achieves comparable results when applied to films released in the United Kingdom, Australia, Japan, and Germany. First we discuss the methods we used to complete our study and then we present the results accompanied by a brief analysis.
\section{Methods}
\label{methods}
In order to carry out this study, we used a set of $325$ movies released in the United States in 2013, $141$ movies released in the United Kingdom in 2013 and 2014, $118$ movies released in Austrialia in 2013 and 2014, $95$ movies released in Japan in 2013, and $105$ movies released in Germany in 2013. We retrieved the box office data for the films released in each country from Box Office Mojo \footnote{\url{http://www.boxofficemojo.com/}}. From the same site we were also able to gather data on the number of screens each film was played on while in theaters (this data is often available prior to a film's release date).
Determining the web address of each film's Wikipedia page proved to be much more difficult in comparison. Our solution was to develop an automated procedure to accomplish this task (see the \textit{Finding Wikipedia URLs} Section).
Once we had the URLs of the Wikipedia pages corresponding to the films we were interested in, we used a website which records Wikipedia article traffic statistics\footnote{\url{http://stats.grok.se/}} to download the daily view counts of each page (see the \textit{Regression Model} Section). We then used a simple multivariate linear regression to attempt to predict each film's opening weekend box office revenue based on its Wikipedia page views and screen count. Finally, we analyzed the accuracy of the regression model using leave-one-out cross-validation (in the \textit{Cross-Validation} Section).
\subsection{Finding Wikipedia URLs}
\label{wiki urls}
Wikipedia URLs unambiguously identify films. Other sensors of public sentiment used for box office prediction, such as mentions of titles in Twitter text \cite{Huberman:twitter} or Google search queries \cite{panaligan:google}, are difficult to work with for titles based on common words (e.g. a tweet mentioning "Frozen" may not be referencing the film). Disambiguation becomes substantially more difficult when long lists of titles in multiple foreign languages are involved.
Wikipedia articles seldom contain information about opening weekend box office. In order to conduct our research we therefore need to construct a mapping between the opening weekend box office data available from Box Office Mojo and the associated language-specific Wikipedia URLs. Since Wikipedia URLs are not necessarily exact title names (e.g. \url{de.wikipedia.org/wiki/42_(Film)} ), this mapping must be constructed by hand (which is prohibitively time consuming) or through an automated fashion.
We check lists built from DBpedia\footnote{\url{http://dbpedia.org/}} against results obtained via the Google Custom Search API\footnote{\url{www.google.com/cse}} to resolve spelling differences between titles on Box Office Mojo and Wikipedia URLs. For each of the nations considered, we restrict the search API to to return links only from the national language's Wikipedia (i.e. \url{en.wikipedia.org}, \url{ja.wikipedia.org}, \url{de.wikipedia.org}) and query the search API for Box Office Mojo title names followed by the appropriate translation of the word ``film'' (i.e. ``film'' (en), ``film'' (de), or ``\begin{CJK}{UTF8}{min}映画\end{CJK}'' (ja) ).
Despite the addition of ``film'' to our searches, the highest ranked search results are not necessarily the Wikipedia pages for the queried films.
To ensure that the links retrieved from our searches indeed refer to the Wikipedia URLs for each film, we use language-specific variants of DBpedia \cite{auer2007dbpedia}
to build lists of URLs which are categorized as films and discard search results which do not appear in the lists.
For example, when entered into \url{http://ja.dbpedia.org/sparql}, the query below provides us with a list of 769 Wikipedia URLs which describe films (Category:\begin{CJK}{UTF8}{min}2012年の映画\end{CJK} ) or animated films (Category:\begin{CJK}{UTF8}{min}2013年のアニメ映画\end{CJK} ) which were released in Japan during 2012 or 2013:
\begin{CJK}{UTF8}{min}
\begin{verbatim}
PREFIX c: <http://ja.dbpedia.org/resource/
Category:>
PREFIX dcterms: <http://purl.org/dc/terms/>
SELECT ?film WHERE {
{?film dcterms:subject c:2012年の映画 .}
UNION
{?film dcterms:subject c:2013年の映画 .}
UNION
{?film dcterms:subject c:2013年のアニメ映
画 .}
UNION
{?film dcterms:subject c:2012年のアニメ映
画 .}
}
\end{verbatim}
\end{CJK}
These queries provide us with lists of Wikipedia URLs for: 769 Japanese films, 1026 German films, and 3079 English films.
Opening weekend box office data on Box Office Mojo is available for: 104 Japanese films, 166 German films, 219 Australian films and 225 UK films.
The technique provided here allows us to align Wikipedia URLs with box office data for: 73 Japanese films, 132 German films, 118 Australian films, and 141 UK films. Some alignments were also performed manually.
\subsection{Regression Model}
\label{regression}
As previously mentioned, we used a multivariate linear regression model\footnote{We used the linear regression (ordinary least squares) implementation provided by the \textit{Scikit-learn} Python package (\url{http://scikit-learn.org/stable/index.html})\cite{scikit-learn}.} of the following form to predict the revenue generated during each film during its opening weekend:
\begin{displaymath}
y_i = \alpha _1 x_{i,1} + \alpha _2 x_{i,2} (t) + \varepsilon _i ,
\end{displaymath}
where $y_i$, $x_{i,1}$, $x_{i,2}$, and $\varepsilon _i$ are the opening weekend box office revenue, screen count, number of Wikipedia page views\footnote{We chose to use the cumulative page views, i.e. the sum of the number of times the page was visited starting at some fixed number of days before release, up until day $t$.}, and the error in the prediction, for film $i$, respectively. $\alpha _1$ and $\alpha _2$ are the regression coefficients. Notice that $x_{i,2}$ is a function of time in the above equation. Since we have a range of dates for which we know how many views each film's Wikipedia page received, we have the ability to form a large number of regression models, one for each day leading up to the film's release. For example, we may wish to predict a movie's box office success using only data available one month before it premiers. Collecting the Wikipedia page hits over a long period of time allows us to discern how the accuracy of the regression model changes over time, as more information is introduced. One quantity we will use to measure accuracy is the R$^2$ Coefficient of Determination. We also found it prudent to use a second tool to evaluate our method's performance.
\subsection{Cross-Validation}
\label{validation}
Due to the large amount of potential training data available on movie ticket sales and Wikipedia page views, we decided that leave-one-out cross-validation was an appropriate technique for assessing the performance of our model\footnote{We used the leave-one-out cross-validation implementation provided by the \textit{Scikit-learn} Python library (\url{http://scikit-learn.org/stable/index.html}).}. Given a set of films along with their associated box office revenues, Wikipedia page views, and screen counts, we successively remove one film from consideration, form a linear regression model using the remaining films, and then use the resulting model to predict the box office revenue for the film we removed\footnote{Note that leave-one-out cross-validation is an appropriate method to apply here as it is relatively easy to obtain a large amount of data to train the regression model.}. Repeating this process for each film $i$ allows us to compute its associated \textit{relative error}, $e_i$. A film's relative error is given by
\begin{displaymath}
e_i = \frac{|y_i - p_i|}{y_i},
\end{displaymath}
where $y_i$ is the actual amount film $i$ earned during its opening weekend, and $p_i$ is the prediction generated by the multivariate regression\footnote{We used the number of Wikipedia page views for each film seven days before its premier to create the relative error plots in the \textit{Results and Discussion} Section.}. A significant relative error indicates a discrepancy between a movie's predicted box office revenue and its actual revenue which is large \textit{relative} to the movie's actual box office revenue.
\section{Results and Discussion}
\label{rnd}
\subsection{United States Box Office}
As was noted in the \textit{Methods} Section, once we had each US film's associated Wikipedia page view data along with its screen count, we formed a separate linear regression model on each day leading up to its premier.
Thus we were able to visualize the accuracy of our predictions as a function of time as in Figure \ref{us correlation}.
As one would expect, the accuracy of the model improves as the movie premiers draw near and movie awareness swells. This is the time period when potential moviegoers are presumably researching which movie to see during the coming weekend. Using this data set we obtain accuracy comparable to that reached by the model based only on Wikipedia page views used by Mesty{\'a}n et al.\footnote{Note that the coefficient of correlation plotted in \cite{mestyan:wikipedia} is the Pearson correlation, R, while we use R$^2$.}\cite{mestyan:wikipedia}. This is the level of accuracy we expected to obtain since the only major difference between the two models is that they were used on different lists of films.
In Figure \ref{us error} we have the relative errors (see the \textit{Cross-Validation} Section) for the first 50 films from the US data set. In this case the films have been sorted in descending order based on their box office revenues. The relative errors for these films carry more weight than those of the movies that were left out. This is because, in computing the relative error, we divide the difference between the actual box office revenue and the prediction by the actual box office revenue. For example, over predicting by one million dollars the box office revenue for a film which brought in ten million dollars during its opening weekend would result in a relative error of $0.1$ while over predicting by the same amount the box office revenue for a film which only brought in one hundred thousand dollars would yield a much greater relative error of $9.0$. A large relative error for a film that did well during its opening weekend indicates a considerable gap between our prediction and its actual box office revenue. We see in Figure \ref{us error} that the relative errors are reasonably small, as we would expect considering the R$^2$ value achieved predicting United States box office results.
\begin{figure}
\centering
\includegraphics[scale=0.45]{US_Correlation.png}
\caption{Evolution of the R$^2$ coefficient of determination in time (US films)}
\label{us correlation}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.2]{loocv_US.png}
\caption{Relative Error for 50 US films}
\label{us error}
\end{figure}
\subsection{Foreign Box Office}
After recreating one of the results of Mesty{\'a}n et al. we applied the same techniques to sets of films released in the United Kingdom, Australia, Germany, and Japan.
\subsubsection{English-Speaking Foreign Box Offices}
One possible problem with employing the same approach used to predict box office profits in the United States to anticipate those in the United Kingdom and Australia is that all three countries speak a common language. As a result internet users in all three nations use the same English branch of Wikipedia. This makes it impossible to discern which of the three countries a page view on a movie's Wikipedia page came from. Combined with the different release dates for films across the three markets, a large amount of noise is likely being introduced. For instance, if film ``A'' premiers in the United States before Australia, the United States premier could create an influx of Wikipedia page views earlier than normal, relative to Australia's premier. This, in turn, could cause the model to overestimate the film's expected box office revenue in the Australian market. Indeed there are over 90 films in both the lists from Australia and the United Kingdom which have different release dates from their counterparts in the United States
Regressing on the number of Wikipedia page views and screen counts for each of the $141$ UK films at each day leading up to the films' premiers, we produce a plot of the accuracy of our model as a function of time (Figure \ref{uk correlation}). The maximum value the R$^2$ coefficient of determination attains here is about $0.34$; much less than that for the United States predictions. The shape of the graph is also strikingly different from Figure \ref{us correlation}. The R$^2$ coefficient is at its maximum $50$ days before the film premiers and decreases as time progresses, eventually increasing marginally in the few days before the premiers. This strange behavior may be the result of differing movie release dates, as discussed above.
The relative errors for the films from the United Kingdom are given in Figure \ref{uk error}. There are a few outliers with very large relative errors, but the errors are larger on average, than for the US films. This is intuitive given that the coefficient of determination of the linear regression is much lower for the films from the United Kingdom.
\begin{figure}
\centering
\includegraphics[scale=0.45]{UK_Correlation.png}
\caption{Evolution of the R$^2$ coefficient of determination in time (UK films)}
\label{uk correlation}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.2]{loocv_UK.png}
\caption{Relative Error for 50 UK films}
\label{uk error}
\end{figure}
Repeating the steps taken for the films from the UK with the $118$ Australian movies, we obtain Figure \ref{au correlation}. Here the coefficient of determination evolves with time in a similar manner to that in Figure \ref{us correlation}; it increases as time does. The model attains a relatively high R$^2$ value of $0.57$ a day before the movie release dates. Figure \ref{au error} shows the relative error for 50 Australian films. There are fewer films with huge relative errors than we saw for the UK data set and the model produces lower relative errors, on average, for the Australian data set than the UK one.
\begin{figure}
\centering
\includegraphics[scale=0.45]{AU_Correlation.png}
\caption{Evolution of the R$^2$ coefficient of determination in time (Australian films)}
\label{au correlation}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.2]{loocv_AU.png}
\caption{Relative Error for 50 Australian films}
\label{au error}
\end{figure}
\subsubsection{Japanese and German Speaking Box Offices}
Since Wikipedia has separate versions written entirely in Japanese and German, it is a reasonable assumption that each is accessed by primarily Japanese and German citizens, respectively. This resolves the stumbling block we faced with the English-speaking nations, namely determining which market each viewer belonged to. Hence one might expect that predictions for Japanese and German opening weekend box office revenue using Wikipedia page views would outperform those for the UK and Australia.
Carrying out the same procedure as before for the $105$ German films yields Figure \ref{de correlation}. While the correlation changes in a similar manner to that in Figure \ref{us correlation} (i.e. it increases as the release date approaches), the overall accuracy achieved is considerably lower. The maximum R$^2$ value attained is $0.45$, compared to the almost $0.61$ R$^2$ coefficient associated with the United States films. Figure \ref{de error} shows the relative error for the German films.
\begin{figure}
\centering
\includegraphics[scale=0.45]{DE_Correlation.png}
\caption{Evolution of the R$^2$ coefficient of determination in time (German films)}
\label{de correlation}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.2]{loocv_DE.png}
\caption{Relative Error for 50 German films}
\label{de error}
\end{figure}
The results are even worse when the model is used to predict opening weekend box office revenues for $95$ films released in Japan. Figure \ref{ja correlation} shows the evolution of the coefficient of determination for this multivariate regression. The same pattern present in the United States', Australian, and German R$^2$ evolution plots does not appear here. The overall accuracy of the regression model is also markedly lower when it is applied to the Japanese market than the German market (maximum R$^2 \approx 0.30$). In Figure \ref{ja error} we have the relative error for the Japanese films. Using the regression model often predicts double the amount films actually earn during their opening weekends. This is unsurprising, given the low coefficient of determination attained with this set of films.\\
Overall, total Wikipedia page views do not appear to be strong predictors of how films will perform during their opening weekends in Germany, the UK, or Japan. It is likely that when researching which movie to see, citizens of Japan, Germany, and the UK simply do not use Wikipedia as a resource as often as people in the United States and Australia. Or, perhaps they use different means than the internet to aid their decisions altogether, or do not perform any research at all before going to the theaters. Another possible explanation for the disparity in accuracy between the predictions for the films released in the United States and those released in the UK, Germany, and Japan is the number of movies considered. Since there were two to three times as many films in the United States data set as the other sets, it included numerous low-budget, lesser-known movies. It is possible that the box office success of lesser-known movies is easier to predict. Indeed, if we remove the top 100 highest grossing films from the list of US films, our method obtains a maximum R$^2$ of almost $0.72$. However, this does not explain why we are able to predict box office revenue in Australia with more certainty than in other foreign countries. In any case, it is clear that a technique to predict opening weekend box office profits in the United Kingdom, Germany, or Japan must rely on more than Wikipedia page views alone.
\begin{figure}
\centering
\includegraphics[scale=0.45]{JA_Correlation.png}
\caption{Evolution of the R$^2$ coefficient of determination in time (Japanese films)}
\label{ja correlation}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.2]{loocv_JA.png}
\caption{Relative Error for 50 Japanese films}
\label{ja error}
\end{figure}
\section{Conclusion}
In this article we have shown that although the method proposed by Mesty{\'a}n et al. predicts films' opening weekend box office revenues in the United States and Australia with reasonable accuracy, its performance drops significantly when applied to various foreign markets. In particular, we constructed a multivariate linear regression using the number of views the Wikipedia pages of various films released in the United States received to predict their successes during their opening weekends in theaters. We automated the process of determining a film's corresponding Wikipedia page so that we could apply the same technique to foreign movies. However, when we used the model to predict the opening weekend box office revenues generated by films in British, Japanese, and German theaters, we found its accuracy to be far from satisfactory. Finally we gave brief discussions of possible causes of the discrepancies.
While Wikipedia page views may be a strong predictor of box office performance for films in the United States, the same cannot necessarily be said for films released in other nations. Before a model similar to that presented here is used to predict box office sales in a foreign market, it should be tested on backdata gathered from that market.
\section{Acknowledgments}
The authors would like to thank Dean Shaw for his help interpreting Japanese film titles and URLs.
\balance
\bibliographystyle{acm-sigchi}
|
1,108,101,566,758 | arxiv | \section{Introduction}
It is well known that for finite matrices image partition regularity behaves well with respect to central subsets of the underlying semigroup (Central sets were introduced by Furstenberg \cite{refF} and enjoy very strong combinatorial properties \cite[Proposition 8.21]{refF}). But the situation becomes totally different for infinite image partition regular matrices. It was shown in \cite{refHLSc} that some of very interesting properties for finite image partition regularity could not be generalized for infinite image partition regular matrices. To handle these situations the notion of centrally image partition regular matrices were introduced \cite{refHLSc}, while both these notions becomes identical for finite matrices. The same problem occurs in the setup of image partition regularity near zero over dense subsemigroup of $((0,\infty),+)$ which is stronger notion than image partition regularity. Again image partition regularity and image partition regularity near zero over dense subsemigroup of $((0,\infty),+)$ becomes identical for finite matrices. Also finite image partition regular matrices have images in any central sets as well as central set near zero for some nice dense subsemigroups of $((0,\infty),+)$. This situation motivates us to introduce the notion of {\em centrally image partition regular near zero over a dense subsemigroup of $((0,\infty),+)$ \/} which involve the notion of central sets near zero. Central sets near zero were introduced by Hindman and Leader \cite{refHL} and these sets also enjoy rich combinatorial structure like central sets.\\
We shall present the notion central sets and central sets near zero after giving a brief description of algebraic structure of $\beta S_d$ for a discrete semigroup $(S, +)$. We take the points of $\beta S$ to be the ultrafilters on $S$, identifying the principal ultrafilters with the points of S and thus pretending that $S\subseteq\beta S$. Given $A\subseteq S$,
$$c\ell A = \overline{A}= \{p\in\beta S : A\in p\}$$ is a basis for a topology on $\beta S$.
The operation $+$ on $S$ can be extended to the Stone-\v{C}ech compactification $\beta S$ of $S$ so that $(\beta S,+)$ is a compact right topological semigroup (meaning that for any $p \in \beta S$, the function $\rho_p : \beta S \rightarrow \beta S$ defined by $\rho_p(q) = q + p$ is continuous) with S contained in its topological center (meaning that for any $x \in S$, the function
$\lambda_x : \beta S \rightarrow \beta S$ defined by $\lambda_x(q) = x + q$ is continuous). Given $p,q\in \beta S$ and $A\subseteq S$, $A\in p + q$ if and only if $\{x\in S:-x+A\in q\}\in p$, where $-x+A=\{y\in S:x+y\in A\}$. \\
A nonempty subset $I$ of a semigroup $(T,+)$ is called a \emph{left ideal of $S$} if $T+I\subset I$, a \emph{right ideal} if $I+T\subset I$,
and a \emph{two sided ideal} (or simply an \emph{ideal}\/) if it is both a left and right ideal.
A \emph{minimal left ideal} is the left ideal that does not contain any proper left ideal.
Similarly, we can define \emph{minimal right ideal} and \emph{smallest ideal}.
Any compact Hausdorff right topological semigroup $(T,+)$
has a smallest two sided ideal
$$\begin{array}{ccc}
K(T) & = & \bigcup\{L:L \text{ is a minimal left ideal of } T\} \\
& = & \,\,\,\,\,\bigcup\{R:R \text{ is a minimal right ideal of } T\}\\
\end{array}$$
Given a minimal left ideal $L$ and a minimal right ideal
$R$, $L\cap R$ is a group, and in particular contains
an idempotent. An idempotent in $K(T)$ is called
a {\it minimal\/} idempotent. If $p$ and $q$ are idempotents in $T$
we write $p\leq q$ if and only if $p+q=q+p=p$. An idempotent
is minimal with respect to this relation if and only if it is a member of the smallest ideal.
See \cite{refHS} for an elementary introduction to the algebra of $\beta S$ and for any unfamiliar details.
\begin{defn}
Let $(S,+)$ be an infinite discrete semigroup. A set $C\subseteq S$ is central if and only if there is some minimal idempotent $p$ in $(\beta S, +)$ such that $C\in p$.
\end{defn}
We have been considering semigroups which are dense in $((0,\infty),+)$. Here ``dense" means with respect to the usual topology on $((0,\infty),+)$. When passing to the Stone-\v{C}ech compactification of such a semigroup $S$, we deal with $S_{d}$ which is the set $S$ with the discrete topology.
\begin{defn}
If $S$ is a dense subsemigroup of $((0,\infty),+)$, then $0^+(S)=\{ p\in\beta S_d: (\forall\epsilon>0)((0,\epsilon)\cap S \in p)\}$.
\end{defn}
It is proved in \cite[Lemma 2.5]{refHL}, that $0^+(S)$ is a compact right topological subsemigroup of $(\beta S_d,+)$. It was also noted that $0^+(S)$ is disjoint from $K(\beta S_d)$ and hence gives some new information which are not available from $K(\beta S_d)$. Being compact right topological semigroup $0^+(S)$ contains minimal idempotents. In \cite{refDH}, the authors applied the algebraic structure of $0^+(S)$ on their investigation of image partition regularity near zero of finite and infinite matrices. In \cite{refDPr} has been used algebraic structure of $0^+(\mathbb{R})$ to investigate image partition regularity of matrices with real entries from $\mathbb{R}$.\\
\begin{defn}
Let $S$ be a dense subsemigroup of $((0,\infty),+)$,
A set $C$ is central near $0$ if and only if there is some minimal idempotent $p$ in $0^+(S)$ such that $C\in p$.
\end{defn}
Next we present some well known characterizations of image partition regularity of matrices.
\begin{thm}\label{iprfinitech}
Let $u,v\in \mathbb{N}$ and let $M$ $u\times v$ matrix with entries from $\mathbb{Q}$. The following statements are equivalent.
\begin{itemize}
\item[(a)] M is image partition regular.
\item[(b)] For every central subset $C$ of $\mathbb{N}$, there exists $\vec x\in \mathbb{N}^{v}$ such that $M\vec x\in C^{u}$.
\item[(c)] For every central subset $C$ of $\mathbb{N}$, $\{\vec x\in \mathbb{N}^{v}$ : such that $M\vec x\in C^{u}\}$ is central in $\mathbb{N}^{v}$.
\item[(d)] For each $\vec r\in \mathbb{Q}^{v}\setminus \{\vec 0\}$ there exists $b\in \mathbb{Q}\setminus 0$ such that \\
$$\left(\begin{array}{c}b\vec r\\
M
\end{array}\right)\,
$$
is image partition regular.
\item[(e)] For every central subset $C$ of $\mathbb{N}$, there exists $\vec x\in \mathbb{N}^{v}$ such that $\vec y=M\vec x\in C^{u}$, all entries of $\vec x$ are distinct, and for all $i,j\in \{1,2,\ldots,u\}$, if rows $i$ and $j$ of $M$ are unequal, then $y_{i}\neq y_{j}$.
\end{itemize}
\end{thm}
\begin{proof}
~\cite[Theorem 2.10]{refHLSb}
\end{proof}
In paper ~\cite {refHLSc}, the authors presented some contrasts between finite and infinite partition regular matrices and so showed that some of very interesting properties for finite image partition regularity could not be generalized for infinite image partition regular matrices. \\
It is interesting to observe that an
important property is an immediate
consequence of Theorem \ref{iprfinitech}(b), namely that if
$M$ and $N$ are finite
image partition regular matrices, then the matrix
$$\left(\begin{array}{cc}M&\hbox{\bf O}\\ \hbox{\bf O}&N\\ \end{array}\right)$$
is also image partition regular. But this property does not hold good for infinite matrices as was shown in ~\cite[Theorem 2.2]{refHLSc}.
\begin{thm}\label{mksepn}
Let $\vec b$ be a compressed sequence with entries from $\mathbb{N}$ such that $\vec b\neq (1)$. Let $M$ be a matrix whose rows are all rows $\vec a\in \mathbb{Q}^{\omega}$ with only finitely many nonzero entries such that $c(\vec a)=\vec b$. Let $N$ be the finite sums matrix.
\begin{itemize}
\item[(a)] The matrices $M$ and $N$ are image partition regular.
\item[(b)] There is a subset $C$ of $\mathbb{N}$ which is a member of every idempotent in $\beta \mathbb{N}$ (and is thus, in particular, central) such that for no $\vec x\in \mathbb{N}^{\omega}$ does one have $M\vec x \in C^{\omega}$.
\item[(c)] The matrix $$\left(\begin{array}{cc}M&\hbox{\bf O}\\
\hbox{\bf O}&N
\end{array}\right)\,
$$
is not image partition regular.
\end{itemize}
\end{thm}
\begin{proof}
~\cite[Theorem 2.2]{refHLSc}
\end{proof}
To overcome the above situation the following notion was introduced in ~\cite[Definition 2.7]{refHLSc}.
\begin{defn}\label{centralipr}
Let $M$ be an $\omega\times \omega$ matrix with entries from $\mathbb{Q}$. Then $M$ is {\em centrally image partition regular \/} if and only if whenever $C$ is a central set in $\mathbb{N}$, there exists $\vec x\in \mathbb{N}^{\omega}$ such that $M\vec{x}\in C^{\omega}$.
\end{defn}
Note that the above definition ~\ref{centralipr} has a natural generalization for arbitrary subsemigroup $S$ of $((0,\infty),+)$, and hence forth we will abbreviate this by CIPR/$S$. Motivation behind the introduction this new notion was that the principal good properties of finite image partition regular matrices could not be extended with respect to infinite image partition regular matrices.\\
It is easy to see that whenever $M$ and $N$ are {\em centrally image partition regular \/} matrices over any subsemigroup $S$ of $((0,\infty),+)$, then so is
$$\left(\begin{array}{cc}M&\hbox{\bf O}\\
\hbox{\bf O}&N
\end{array}\right)\,.
$$\\
The above observation tells us that centrally image partition regular matrices are more natural candidate to generalize finite image partition regularity in case of infinite matrices.\\
In this course we introduce another natural candidate to generalize the properties of finite image partition regularity near zero in case of infinite matrices.
\begin{defn}
Let $M$ be an $\omega\times \omega$ matrix with entries from $\mathbb{Q}$ and let $S$ be a dense subsemigroup of $((0,\infty),+)$. Then $M$ is {\em centrally image partition regular near zero \/} if and only if whenever $C$ is a central set near zero in $S$, there exists $\vec x\in S^{\omega}$ such that $M\vec{x}\in C^{\omega}$.
\end{defn}
Hence forth for arbitrary subsemigroup $S$ of $((0,\infty),+)$, we will abbreviate {\em centrally image partition regular near zero \/} over $S$ by CIPR/$S_0$.
This is the simple fact that if $M$ and $N$ be two centrally image partition regular near zero matrices over a dense subsemigroup $S$ of $((0,\infty),+)$, then the diagonal sum $$\left(\begin{array}{cc}M&\hbox{\bf O}\\
\hbox{\bf O}&N
\end{array}\right)\,
$$
is also centrally image partition regular near zero matrix over a dense subsemigroup $S$ of $((0,\infty),+)$.
The following Examples show that there exists infinite matrices which are centrally image partition regular over $\mathbb{Q}^{+}$ but not centrally image partition regular near zero over $\mathbb{Q}^{+}$ and vice versa.
\begin{ex}\label{NnotRzp}
Let
$$M=\left(\begin{array}{ccccc}1&0&0&0&\ldots\\
2&1&0&0&\ldots\\
4&0&1&0&\ldots\\
8&0&0&1&\ldots\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right)\,.
$$
Then $M$ is CIPR/$\mathbb{Q}^{+}$ matrix but is not CIPR/$\mathbb{Q}^{+}_{0}$.
\end{ex}
\begin{proof}
To see that $M$ is centrally image partition regular matrix, let $C$ be any central set in $\mathbb{Q}^{+}$
and pick a monochromatic sequence $\langle y_n\rangle_{n=0}^\infty$ in $C$
such that for each $n\in \mathbb{N}$, $y_n>2^ny_0$. Let $x_0=y_0$ and for
each $n\in \mathbb{N}$, let $x_n=y_n-2^ny_0$. Then $M\vec x=\vec y$.
Now $(0,1)\cap\mathbb{Q^{+}}$ is a central set near zero in $\mathbb{Q^{+}}$ and suppose one has $\vec x\in (\mathbb{Q}^{+})^\omega$ such that $\vec y=M\vec x\in
((0,1)\cap \mathbb{Q^{+}})^\omega$.
Then $x_0=y_0> 0$. Pick $k\in \mathbb{N}$ such that $2^kx_0>1$. Then
$y_k=2^kx_0+x_k> 1$, a contradiction.
\end{proof}
\begin{ex}\label{QzspnotD} Let
$$M=\left(\begin{array}{cccccc}1&-1&0&0&0&\ldots\\
1/3&0&-1&0&0&\ldots\\
1/5&0&0&-1&0&\ldots\\
1/7&0&0&0&-1&\ldots\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right)\,.$$
Then $M$ is CIPR/$\mathbb{Q}^{+}_0$ but is not CIPR/$\mathbb{Q}^+$.
\end{ex}
\begin{proof} To see that $M$ is not CIPR/$\mathbb{Q}^{+}$, let $C$ be a central set in $\mathbb{Q}^{+}$ and we show that there
is no $\vec x\in (\mathbb{Q}^{+})^\omega$ such that $\vec y=M\vec x\in C^\omega$.
Indeed, suppose one has such $\vec x$ and pick $n\in\mathbb{N}$ such that
$x_0/(2n+1)\leqslant x_{0}$. Then $y_n=x_0/(2n+1)-x_{n+1}$ is also bounded by $x_{0}$ in $\mathbb{Q}^+$.
To see that $M$ is CIPR/$\mathbb{Q}^{+}_0$ near zero
let $C$ be a central set near zero in $\mathbb{Q}^+$ such that $0\in c\ell C$, and
pick a sequence $\langle y_n\rangle_{n=0}^\infty$ in $C$ which converges to $0$.
We may also assume that for each $n$, $y_n<1/(2n+1)$. Let $x_0=1$
and for $n\in\mathbb{N}$, let $x_n=1/(2n-1)-y_{n-1}$. Then $M\vec x=\vec y\in C^\omega$.
\end{proof}
In ~\cite{refHLSc}, we have seen that finite image partition regularity matrices hold some interesting properties but not infinite image partition regular matrices. In this paper we show this behaviour is also true for the notion of image partition regularity near zero. This is why we introduce the notion centrally image partition regularity near zero.
Now in section $2$ of this paper, we first prove that for two infinite image partition regular matrices near zero, i.e. $M$ and $N$, over $\mathbb{D}^{+}$ the diagonal sum $$\left(\begin{array}{cc}M&\hbox{\bf O}\\
\hbox{\bf O}&N
\end{array}\right)\,
$$
is not image partition regular near zero over $\mathbb{D}^{+}$. But we show that infinite image partition regular near zero matrices can be extended by finite ones. Also we show in proposition that how new types of centrally infinite image partition regular matrices near zero are constructed from old one.\\
In section $3$, we prove that a special type of infinite image partition regular matrices (i.e. segmented image partition regular matrices) are also centrally image partition regular near zero.
\section{Centrally image partition regularity of matrices near zero}
In Theorem ~\ref{mksepn} we have found two infinite image partition regular matrices $M$ and $N$ over $\mathbb{N}$ while the diagonal sum $$\left(\begin{array}{cc}M&\hbox{\bf O}\\
\hbox{\bf O}&N
\end{array}\right)\,
$$
is not image partition regular matrix over $\mathbb{N}$. But the central tool to prove the above Theorem is Milliken-Taylor separating theorem
~\cite[Theorem 3.2]{refDHLL}. Recently in ~\cite{refWd}, Milliken-Taylor separating theorem has been proved for dyadic rational numbers which employ to prove the following generalization of ~\ref{mksepn}. First we recall some Definitions from ~\cite{refWd}.
\begin{defn}
The set of dyadic rational numbers is given by\\ $\mathbb{D}=\{\frac{m}{2^{t}}$ : $m\in \mathbb{Z}$ and $t\in \omega\}$.
\end{defn}
We will be considering $\mathbb{D}^{+}$, the set of positive numbers contained in $\mathbb{D}$.
\begin{defn}
Let $x\in \mathbb{D}^{+}$. The {\em support\/} of $x$, denoted supp$(x)$, is the unique finite nonempty subset of $\mathbb{Z}$ such that $x=\sum_{t\in supp(x)}2^{t}$.
\end{defn}
\begin{defn}
Given a binary number, an \emph{even} $0$-\emph{block} is the occurrence of a positive even total of consecutive zeros between two consecutive ones.
\end{defn}
For $x\in \mathbb{D}^{+}$, define the \emph{start} of $x$ as the position of the first $1$ appearing in $x$ moving from left to right and the \emph{end} as the position of the last $1$. The formal definition is the following.
\begin{defn}
Let $x\in\mathbb{D}^+$. Then $x=\sum_{t\in\text{supp}(x)}2^t$ where $\text{supp}(x)\in\mathcal{P}_f(\mathbb{Z})$. Define the \emph{start} of $x$ as the max supp$(x)$ and the \emph{end} as the min supp$(x)$.
\end{defn}
Now we present the following Proposition from ~\cite[Proposition 2.12]{refWd} that play the key role to prove the following Theorem \ref{NotCentrally}.
\begin{prop}\label{separtmt}
Let $\varphi(z)$ be the number of even $0$-blocks between the start and end of $z$ for any $z\in \mathbb{D}\cap (0,2)$. For $i\in \{0,1,2\}$, let $C_{i}=\{c\in \mathbb{D}\cap (0,2)$ : $\varphi(c)\equiv i$ mod $3\}$. Then $\{C_{0}, C_{1}, C_{2}\}$ is a partition of $\mathbb{D}\cap (0,2)$ such that no $C_{i}$ contains $MT(\langle 1 \rangle,\langle x_i\rangle_{i=1}^\infty)\cup
MT(\langle 1,2 \rangle,\langle y_i\rangle_{i=1}^\infty)$ for any sequences
$\langle x_i\rangle_{i=1}^\infty$ and $\langle
y_i\rangle_{i=1}^\infty$ in $\mathbb{D}\cap (0,2)$.
\end{prop}
\begin{proof}
~\cite[Proposition 2.12]{refWd}
\end{proof}
\begin{thm}\label{NotCentrally}
Let $M$ be finite sum matrix and $N$ be the Milliken-Taylor matrix determined by compressed sequence $\langle 1,2\rangle$. Then
\begin{itemize}
\item[(a)] The matrices $M$ and $N$ are image partition regular near zero over $\mathbb{D}^{+}$.
\item[(b)] The matrix $$\left(\begin{array}{cc}M&\hbox{\bf O}\\
\hbox{\bf O}&N
\end{array}\right)\,
$$
is not image partition regular near zero over $\mathbb{D}^{+}$.
\item[(c)] The matrix $N$ is not centrally image partition regular near zero over $\mathbb{D}^{+}$.
\end{itemize}
\end{thm}
\begin{proof}
Statement [a] follows from ~\cite[Theorem 5.7]{refDH}.
From ~\ref{separtmt} the matrix is $$\left(\begin{array}{cc}M&\hbox{\bf O}\\
\hbox{\bf O}&N
\end{array}\right)\,
$$ not image partition regular near zero over $\mathbb{D}^{+}$.
Again, since the matrix $$\left(\begin{array}{cc}M&\hbox{\bf O}\\
\hbox{\bf O}&N
\end{array}\right)\,
$$
is not image partition regular near zero over $\mathbb{D}^{+}$. Therefore $N$ is not CIPR/$\mathbb{D}^{+}_0$ as $M$ has its image in every central set near zero.
Let $N$ is centrally image partition regular near zero. Again $M$ is centrally image partition regular near zero follows from ~\cite[Theorem 3.1]{refHL}. Then the matrix $$\left(\begin{array}{cc}M&\hbox{\bf O}\\
\hbox{\bf O}&N
\end{array}\right)\,
$$
is centrally image partition regular near 0 and hence also image partition regular near 0. But this is a contradiction. Therefore $N$ is not centrally image partition regular near zero over $\mathbb{D}^{+}$.
\end{proof}
Now we show that infinite image partition regular near zero matrices can be extended by finite ones.
\begin{thm}\label{extensionipr0}
Let $M$ be a finite image partition regular matrix over $\mathbb{N}$ and $N$ be an infinite image partition regular
near zero matrix over any dense subsemigroup $S$ of $((0,\infty),+)$. Then $$\left(\begin{array}{cc}M&\hbox{\bf O}\\
\hbox{\bf O}&N
\end{array}\right)\,
$$
is image partition regular near zero.
\end{thm}
\begin{proof}
Let $S$ be $r$-colored by $\varphi$ as $S=\bigcup_{i=1}^r C_i$ and $\epsilon>0$. By a standard compactness argument (see \cite[Section 5.5]{refHS} )
there exists $k\in\mathbb{N}$ such that whenever $\{1,2,\cdots,k\}=\bigcup_{i=1}^r D_i$ there exists
$\vec{x}\in\{1,2,\cdots,k\}^v$ and $i\in\{1,2,\cdots,r\}$ such that $M\vec{x}\in (D_i)^u$. Pick $z\in S \cap (0,\epsilon/k)$.
Now color $S$ with $r^k$ colors via $\psi$ as $S=\bigcup_{i=1}^{r^k} F_i$, where $\psi(x)=\psi(y)$ if and only if for all $t\in\{1,2,\cdots,k\}$, $\varphi(tx)=\varphi(ty)$. Choose $\vec{y}\in S^\omega$ such that the entries of $N\vec{y}$ are in $F_i\cap (0,z)$ for some $i\in \{1,2,\cdots,r^{k}\}$. Pick an entry $a$ of $N\vec{y}$ and for each $i\in\{1,2,\cdots,r\}$ let us set $D_i=\{t\in\{1,2,\cdots,k\}:ta\in C_i\}$. Then $\{1,2,\cdots,k\}=\bigcup_{i=1}^r D_i$. Note that since $a\in (0,z)$, $ta\in (0,\epsilon)$ for all $t\in\{1,2,\cdots,k\}$. If we express this coloring as $\gamma:\{1,2,\cdots,k\}\rightarrow \{1,2,\cdots,r\}$ then in fact $\gamma(p)=\varphi(ap)$.
So there exists $\vec{u}\in\{1,2,\cdots,k\}^v$ and $i\in\{1,2,\cdots,r\}$ such that $M\vec{u}\in (D_i)^u$ so that $a(M\vec{u})\in (C_i)^u$. Now $a(M\vec{u})=M(a\vec{u})$. Put $\vec{x}=a\vec{u}$. Then $M\vec{x}\in (C_i\cap (0,\epsilon))^u$. Choose an entry $i$ of $M\vec{u}$ and let $j=\gamma(i)$.
Let $\vec{z}=\left(\begin{array}{cc}a\vec{u}\\i\vec{y}\end{array}\right)$. We claim that for any row $\vec{w}$ of $\left(\begin{array}{cc}M & O\\O & N\end{array}\right)$, $\varphi(\vec{w}\cdot\vec{z})=j$. To observe this first assume that $\vec{w}$ is a row of $\left(\begin{array}{cc}M & O\end{array}\right)$, so that $\vec{w}={\vec{s}}^\frown\vec{0}$, where $\vec{s}$ is a row of $M$. Then $\vec{w}\cdot\vec{z}=\vec{s}\cdot (a\vec{u})=a(\vec{s}\cdot\vec{u})$. Therefore $\varphi(\vec{w}\cdot\vec{z})=\varphi(a(\vec{s}\cdot\vec{u}))=\gamma(\vec{s}\cdot\vec{u})=j$.
Next assume that $\vec{w}$ is a row of $\left(\begin{array}{cc}O & N\end{array}\right)$, so that $\vec{w}= \vec{0}^\frown\vec{s}$ where $\vec{s}$ is a row of $N$. Then $\vec{w}\cdot\vec{z}=i(\vec{s}\cdot\vec{y})$. Now $\psi(\vec{s}\cdot\vec{y})=\psi(a)$. So $\varphi(i(\vec{s}\cdot\vec{y}))=\varphi(ia)=\gamma(i)=j$.
\end{proof}
Now we shall show how new type of infinite centrally image partition regular matrix can be constructed from old one (that is extended up to infinite order i.e. here up to $\omega$).\\
Henceforth unless otherwise stated $S$ will be considered as dense subsemigroup of $((0,\infty),+)$ for which $cS$ is $central^{*}$ near zero for every $c\in \mathbb{N}$.\\
We now present the following theorem and corollary to prove the following proposition \ref{infinitecentralnear0}.
\begin{thm}\label{cardidempotent}
Let $S$ be a subsemigroup of $((0,\infty),+)$. Let $p\in
K(0^{+}(S))$, let $C\in p$, and let $R$ be the minimal right
ideal of $0^{+}(S)$ to which $p$ belongs. Then there are at
least countably infinitely many idempotents in $K(0^{+}(S))\cap
R\cap \overline{C}$.
\end{thm}
\begin{proof}
~\cite[Theorem 2.3]{refDHS09}
\end{proof}
\begin{cor}\label{infinitecentral}
Let $S$ be a dense subsemigroup of $((0,\infty),+)$ and let $C$ be a central set near zero. Then there exists a sequence $\langle C_{n} \rangle_{n=1}^{\infty}$ of pairwise disjoint central sets near zero in $S$ with $\bigcup _{n=1}^{\infty}C_{n}\subseteq C$.
\end{cor}
\begin{proof}
By the above Theorem ~\ref{cardidempotent}, there are at
least countably infinitely many idempotents in $\overline{C}$. hence contains an infinite strongly discrete subset. (Alternatively, there are two minimal idempotents in $\overline{C}$ so that $C$ can be split into two central sets near zero, $C_{1}$ and $D_{1}$. Then $D_{1}$ can be split into two central sets near zero, $C_{2}$ and $D_{2}$, and so on.)
\end{proof}
\begin{prop}\label{infinitecentralnear0}
For each $n\in \mathbb{N}$, let $M_{n}$ be a {\em centrally image partition regular near zero \/} matrix. Then the matrix
$$M=\left(\begin{array}{cccc}M_{1}&0&0&\ldots\\
0&M_{2}&0&\ldots\\
0&0&M_{3}&\ldots\\
\vdots&\vdots&\vdots&\ddots
\end{array}\right)\,.
$$
is also {\em centrally image partition regular near zero \/}.
\end{prop}
\begin{proof}
Let $C$ be a central sets near zero and choose by the above Corollary ~\ref{infinitecentral} a sequence $\langle C_{n} \rangle_{n=1}^{\infty}$ of pairwise disjoint central sets near zero in $S$ with $\bigcup _{n=1}^{\infty}C_{n}\subseteq C$. For each $n\in \mathbb{N}$ choose $\vec x^{(n)}\in S^{\omega}$ such that $\vec y^{(n)}=M_{n}\vec x^{(n)}\in C_{n}^{\omega}$. Let $$\vec z =\left(\begin{array}{c}\vec x^{(1)}\\
\vec x^{(2)}\\
\vdots
\end{array}\right)\,.
$$
Then all entries of $M\vec z$ are in $C$.
\end{proof}
\section{Some infinite Centrally image partition regularity of matrices near zero}
We now present a class of image partition regular matrices which are called segmented image partition regular matrices introduced in \cite{refHLi}. And we show that these class of matrices are also infinite centrally image partition regular matrices.
\begin{defn}
Let $M$ be an $\omega\times\omega$ matrix with entries from $\mathbb{Q}$. Then $M$ is a segmented image partition regular matrix if and only if
\begin{enumerate}
\item no row of $M$ is row is $\vec0$;
\item for each $i\in \omega$, $\{j\in \omega : a_{i,j}\neq\emptyset\}$ is finite; and
\item there is an increasing sequence $\langle\alpha_{n}\rangle_{n=0}^{\infty}$ in $\omega$ such that $\alpha_{0}=0$ and for each $n\in \omega$,\\
$\{\langle a_{i,\alpha_{n}},a_{i,\alpha_{n}+1},a_{i,\alpha_{n}+2},\ldots,a_{i,\alpha_{n+1}-1}\rangle : i\in \omega\}\setminus \{\vec 0\}$\\
is empty or is the set of rows of a finite image partition regular matrix.
\end{enumerate}
\end{defn}
If each of these finite image partition regular matrices is a first entries matrix, then $M$ is a segmented first entries matrix. If also the first nonzero entry of each $\langle a_{i,\alpha_{n}},a_{i,\alpha_{n}+1},a_{i,\alpha_{n}+2},\ldots,a_{i,\alpha_{n+1}-1}\rangle$, if any, is 1, then $M$ is a monic segmented first entries matrix.
\begin{thm}
Let $S$ be a dense subsemigroup of $((0,\infty),+)$ and let $M$ be a segmented image partition regular matrix with $\omega$. Then $M$ is \emph{centrally image partition regular near zero}.
\end{thm}
\begin{proof}
Let $\vec c_{0}, \vec c_{1}, \vec c_{2},\ldots$ denote the columns of $M$. Let $\langle\alpha_{n}\rangle_{n=0}^{\infty}$ be as in the definition of a segmented image partition regular matrix. For each $n\in \omega$, let $M_{n}$ be the matrix whose columns are $\vec c_{\alpha_{n}},\vec c_{\alpha_{n}+1},\ldots,\vec c_{\alpha_{n+1}-1}$. Then the set of non-zero rows of $M_{n}$ is finite and, if nonempty, is the set of rows of a finite image partition regular matrix. Let $B_{n} = (M_{0}$ $M_{1}\ldots M_{n})$.
Now by ~\cite[Lemma 2.5]{refHL}
$0^{+}(S)$ is a compact right topological semigroup so that we can choose an minimal idempotent $p\in 0^{+}(S)$. Let $C\subseteq S$
such that $C\in p$. Let $C^{*}=\{x\in C : -x+C\in p\}$. Then $C^{*}\in p$ and, for every $x\in C^{*}$, $-x+C^{*}\in p$ by ~\cite[Lemma 4.14]{refHS}.\\
Now the set of non-zero rows of $M_{n}$ is finite and, if nonempty, is the set of rows of a finite image partition regular matrix over $\mathbb{N}$ and hence by ~\cite[Theorem 2.3]{refDH} $IPR/S_{0}$. Then by ~\cite[Theorem 4.10]{refDH} , we can choose $\vec x^{(0)}\in S^{\alpha_{1}-\alpha_{0}}$ such that, if $\vec y=M_{0}\vec x^{(0)}$, then $y_{i}\in C^{*}$ for every $i\in \omega$ for which the $i^{th}$ row of $M_{0}$ is non-zero.\\
We now make the inductive assumption that, for some $m\in \omega$, we have chosen $\vec x^{(0)},\vec x^{(1)},\ldots,\vec x^{(1)}$ such that $\vec x^{(i)}\in S^{\alpha_{i+1}-\alpha_{i}}$ for every $i\in \{0,1,2,\ldots,m\}$, and, if
$$\vec y=B_{m}\left(\begin{array}{c}\vec x^{(0)}\\
\vec x^{(1)}\\ . \\.\\.\\\vec x^{(m)}\end{array}\right),$$
then $y_{j}\in C^{*}$ for every $j\in \omega$ for which the $j^{th}$ row of $B_{m}$ is non-zero.\\
Let $D=\{j\in \omega$ : row $j$ of $B_{m+1}$ is not $\vec 0\}$ and note that for each $j\in \omega, -y_{j}+C^{*}\in p$. (Either $y_{j}=0$ or $y_{j}\in C^{*}$) By ~\cite[Theorem 4.10]{refDH} we can choose $\vec x^{(m+1)}\in S^{\alpha_{m+2}-\alpha_{m+1}}$ such that, if $\vec z=M_{m+1}\vec x^{(m+1)}$, then $z_{j}\in \bigcap _{t\in D}(-y_{t}+C^{*})$ for every $j\in D$.\\
Thus we can choose an infinite sequence $\langle \vec x^{(i)} \rangle_{i\in \omega}$ such that, for every $i\in \omega$, $\vec x^{(i)}\in S^{\alpha_{i+1}-\alpha_{i}}$, and, if
$$\vec y=B_{i}\left(\begin{array}{c}\vec x^{(0)}\\
\vec x^{(1)}\\ . \\.\\.\\\vec x^{(i)}\end{array}\right),$$
then $y_{j}\in C^{*}$ for every $j\in \omega$ for which the $j^{th}$ row of $B_{i}$ is non-zero.\\
Let $$\vec x=\left(\begin{array}{c}\vec x^{(0)}\\
\vec x^{(1)}\\ \vec x^{(2)}\\\vdots \end{array}\right)$$
and let $\vec y=M\vec x$. We note that, for every $j\in \omega$, there exists $m\in \omega$ such that $y_{j}$ is the $j^{th}$ entry of
$$B_{i}\left(\begin{array}{c}\vec x^{(0)}\\
\vec x^{(1)}\\ . \\.\\.\\\vec x^{(i)}\end{array}\right)$$
whenever $i>m$. Thus all the entries of $\vec y$ are in $C^{*}$.
\end{proof}
\bibliographystyle{amsplain}
|
1,108,101,566,759 | arxiv | \section{Introduction}
In this paper we investigate some properties of the associative algebras
which were shown in \cite{1,2,3} to underly the rational Calogero model
\cite{4} and were denoted as $SH_N (\nu )\ $ in \cite{5}. Algebra $SH_N (\nu )$ is the
associative algebra of polynomials constructed from arbitrary elements
$\sigma$ of the symmetric group $S_N$ and the generating elements
$a^\alpha_i\,$ obeying the following relations \begin{equation} \label{begin}
\sigma\,a^\alpha_i=a^\alpha_{\sigma(i)}\,\sigma \,, \end{equation} \begin{equation} \label{gcom}
\left [ a^\alpha_i\,,\,a^\beta_j \right ]= \epsilon^{\alpha\,\beta} A_{ij}\,,
\end{equation} where $i,j=1,\,...,\,N$, $\ \alpha ,\beta=0,1$, $\
\epsilon^{\alpha\,\beta}$ =$- \epsilon^{\beta \,\alpha},$ $\
\epsilon^{0\,1}=1$ and \begin{equation} \label{A} A_{ij}=\delta_{ij} +\nu\tilde{A}_{ij}
\,,\qquad \tilde{A}_{ij}= \delta_{ij}\sum_{l=1}^N K_{il} - K_{ij}\,. \end{equation} Here
$K_{ij}\in S_N$ with $i,\,j\,=\,1,\,...\,,\,N\,,$ $i \neq j$, are the
elementary permutations $i \leftrightarrow j$ satisfying the relations $$
K_{ij}=K_{ji},\ K_{ij}\,K_{ij}=1,\ K_{ij}K_{jl}= K_{jl}K_{li}= K_{li}K_{ij}
$$ for $i\neq j \neq l\neq i$ and $$ K_{ij}\,K_{kl}=K_{kl}\,K_{ij} $$ if
$i,\,j,\,k,\,l$ are pairwise different. Note that in this paper repeated
Latin indices $i,j,k,\ldots$ do not imply summation.
The defining relations (\ref{begin})-(\ref{A}) are consistent. In particular,
the Jacobi identities \begin{equation} \label{jac} [a^\alpha_i\,,[a^\beta_j \,,a^\gamma_k
]]+ [a^\beta_j\,[a^\gamma_k\,, a^\alpha_i\,]]
+[a^\gamma_k\,,[a^\alpha_i\,,a^\beta_j ]]=0 \end{equation} are satisfied.
An important property of $SH_N (\nu )$ which allows one to solve the Calogero
model \cite{4} is that this algebra possesses inner $sl_2$ automorphisms with
the generators \begin{equation} \label {sl2} T^{\alpha\beta}= \frac 1 2 \sum _{i=1}^N
(a^{\alpha}_i a^{\beta}_i + a^{\beta}_i a^{\alpha}_i )\,, \end{equation} \begin{equation} \label
{csl2} [T^{\alpha\beta}, T^{\gamma\delta}]= \epsilon^{\alpha\gamma}
T^{\beta\delta} +\epsilon^{\alpha\delta} T^{\beta\gamma} +
\epsilon^{\beta\gamma} T^{\alpha\delta} +\epsilon^{\beta\delta}
T^{\alpha\gamma}\,, \end{equation} which act on the generating elements $a^\alpha_i$ as
on $sl_2$ vectors \begin{equation} \label {sl2vec} \left [ T^{\alpha\beta},\,a^\gamma_i
\right ]= \epsilon^{\alpha\gamma} a^\beta_i +\epsilon^{\beta\gamma}
a^\alpha_i \,. \end{equation} Calogero Hamiltonian is identified with the Cartan element
$T^{01}$ which according to (\ref {sl2vec}) induces $Z$ - gradation of $SH_N
(\nu )$. The latter property allows one \cite{2} to construct wave functions
via the standard Fock procedure with the Fock vacuum $ |0\rangle$ such that
$a_i^0 |0\rangle$=0. Thus, the elements $a_i^\alpha$ serve as generalized
oscillators underlying the Calogero problem. The concrete realization of
these oscillators in terms of Dunkl differential-difference operators
\cite{6} was presented in \cite{1,2}.
These properties characterize the algebra $SH_N (\nu)$ as a natural
generalization of the Heisenberg-Weyl algebra, the associative algebra of
harmonic oscillators. Since the Lie algebra of quantum operators in the
harmonic oscillator problem can be identified with the $W_{1+\infty}$ algebra
\cite{7}, the Lie (super)algebras constructed from $SH_N (\nu )$ with the aid
of supercommutators give rise to a class of the $W_{1+\infty}$ - type
algebras which have been denoted as $W_{N,\infty} (\nu )$ in \cite{8} where
it was shown that all these algebras contain the Virasoro algebra as a
subalgebra. The latter observation indicates that the algebras $SH_N (\nu )$
and $W_{N,\infty} (\nu )$ can be related to conformal models as well as to
other classes of models in the range from quantum Hall effect \cite{9} till
higher-spin gauge theories \cite{10} and KP hierarchy \cite{11} where
$W_{\infty}$ - type algebras prove to be important. An additional argument in
favor of the relationship of $SH_N (\nu )\ $ to the quantum Hall effect is due to the
known fact that the Calogero model can be interpreted as a one-dimensional
reduction of the full anyonic problem \cite{12, 3}.
For lower values of $N$, a nature of $SH_N (\nu )$ is rather well understood.
$SH_1 (\nu )$ is the ordinary Heisenberg-Weyl algebra (since $\nu$ -
dependence is artificial in this case we will use the notation $SH_1$).
Properties of this algebra are very well studied (see {\it e.g.} \cite{13}).
Note that since the center of mass coordinates $1/N \sum_{i=1}^{N}
a_i^\alpha$ decouple from everything else in the defining relations
(\ref{begin})-(\ref{A}), the associative algebra $SH_N (\nu )$ has the
structure $SH_N (\nu )$= $SH_1 \otimes SH^\prime_N (\nu )$ where, by
definition, $SH^\prime_N (\nu )$ is the algebra of elements depending only on
the relative coordinates $a_i^\alpha - a_j^\alpha$.
The properties of $SH^\prime_2 (\nu )$ are well studied too \cite{14}. The
algebra $SH^\prime_2 (\nu )$ is defined by the relations \begin{equation} \label{N2}
[a^\alpha ,a^\beta ]=\epsilon^{\alpha\beta}(1+2\nu K)\,, \n \end{equation} where $K$ is
the only nontrivial element of $S_2$ while $a^\alpha$ are the relative motion
oscillators. For the particular case of $\nu=0$ one recovers the algebra
$SH_1$ in the sector of the $K$ independent elements.
In \cite{14} it was shown that $SH^\prime_2 (\nu )$ admits a unique
supertrace operation defined by the simple formula \begin{equation} \label{str2}
str(1)=1\,,\qquad str(K)=-2\nu\,, \qquad str (W)=str (WK)=0 \n \end{equation} for any
polynomial $W\in SH^\prime_2$ of the form \begin{equation} \label{W} W=\sum_{n=1}^{\infty}
W_{\alpha_1 \ldots \alpha_n } a^{\alpha_1} \ldots a^{\alpha_n} \n \end{equation} with
arbitrary totally symmetric multispinors $W_{\alpha_1 \ldots \alpha_n }$.
For the particular case of $\nu=0$ one recovers the supertrace on $SH_1$.
Furthermore it was shown in \cite{14} by explicit evaluation of the invariant
bilinear form $B(x,\,y )\stackrel {def}{=} str(xy)$ that for $\nu = l+
\frac{1}{2}$ ($l$ is any integer) $SH^\prime_2 (\nu )$ reduces to
finite-dimensional matrix algebras up to some infinite-dimensional ideals
${\cal I}$ which decouple from everything under the supertrace operation
(\ref{str2}), $i.e.\ $ $str(xy)=0$, $\forall x \in {\cal I}$.
In \cite{15} it was then observed that $SH^\prime_2 (\nu )$ is isomorphic to
the factor of the enveloping algebra $U(osp(1;2))$ of $osp(1;2)$ over its
ideal generated by the quadratic Casimir operator $C_2$ by factoring out all
elements of the form $(C_2-c_2) U(osp(1;2))$ where $c_2=\frac 1 {16}
(4\nu^2-1)$ is an arbitrary constant. In its turn this observation clarified
the origin of the ideals of $SH^\prime_2 (\nu )$ at $\nu=l+\frac{1}{2}$ as
corresponding to the finite-dimensional representations of $osp(1;2)$.
Although the algebra $SH_N (\nu )$ is getting interesting applications for
any $N$, till now understanding of its algebraic properties for $N>2$ is far
from being satisfactory. In particular there is no interpretation of $SH_N
(\nu )$ in terms of enveloping algebras of finite-dimensional superalgebras
and nothing is known about ideals of $SH_N (\nu )$ which information is very
important in applications.
In this paper we analyze the existence of the invariant supertrace operation
on $SH_N (\nu)$ $i.e.\ $ such a complex valued linear function $str(f)$ on $SH_N
(\nu )$ that \begin{equation} \label{scom} str\left (\left [f\,,g\right \}\right
)=0\,,\qquad\forall f,g\in SH_N (\nu) \end{equation} with the convention that \begin{equation}
\label{com} \left [f\,,g\right \}=fg-(-1)^{\pi(f) \pi (g)}gf\,, \end{equation} where the
parity $\pi$ in $SH_N(\nu )$ is defined as follows: \begin{equation} \label {m0} \pi
(a^\alpha_i)=1\,,\ \ \pi (K_{ij})=0\,. \end{equation} Let us note that an attempt to
define differently graded traces like, $e.g.$, an ordinary trace ($\pi\equiv
0$) unlikely leads to interesting results.
Knowledge of the supertrace operations on $SH_N (\nu )\ $ is useful in various respects.
One of the most important applications of the supertrace is that it gives
rise to $n$-linear invariant forms \begin{equation} \label{form0} str(a_1 a_2 ...a_n ) \n
\end{equation} that allows one to work with the algebra essentially in the same way as
with the ordinary finite-dimensional matrix algebras and, for example,
construct Lagrangians when working with dynamical theories based on
$SH_N(\nu)$. Another useful property is that since null vectors of any
invariant bilinear form span a both-side ideal of the algebra, this gives a
powerful device for investigating ideals which decouple from everything under
the supertrace operation as it happens in $SH_2 (\nu )$ for half-integer
$\nu$. It is also worth mentioning that having an explicit form of the
trilinear form in one or another basis is practically equivalent to defining
a star-product law in the algebra.
An important motivation for the analysis of the supertraces of $SH_N (\nu )$
is due to its deep relationship with the analysis of the representations of
this algebra, which in its turn gets applications to the analysis of the wave
functions of the Calogero model. For example, given representation of $SH_N
(\nu )$, one can speculate that it induces some supertrace on this algebra as
(appropriately regularized) supertrace of (infinite) representation matrices.
When the corresponding bilinear form degenerates this would imply that the
representation becomes reducible.
As we show, the situation for $SH_N (\nu )$ is very interesting since
starting from $N=3$ it admits more than one independent supertrace in
contrast to the cases of $N=1$ and $N=2$. This fact is in agreement with the
results of \cite{5} where it was shown that there exist many inequivalent
lowest-weight type representations of $SH_N (\nu )$ for higher $N$ (these
representations are classified according to the representations of $S_N$.)
Another important consequence of this phenomenon is that the Lie
superalgebras $W_{N,\infty}(\nu)$ are not simple while appropriate their
simple subalgebras possess non-trivial outer automorphisms.
The paper is organized as follows. In Section \ref{sec2} we analyze
consequences of $S_N$ and $sl_2$ automorphisms of $SH_N(\nu)$. In Section
\ref{sec3} we discuss general properties of the supertraces and consequences
of the existence of several independent supertraces. In Section \ref{sec4} we
study the restrictions on supertraces of the group algebra of $S_N$
considered as a subalgebra of $SH_N (\nu )$, which follow from the defining
relations of $SH_N (\nu )$. These restrictions are called ground level
conditions ({\it GLC}). They play a fundamental role in the problem since as
we show in Section \ref{sec5} every solution of {\it GLC} admits a unique
extension to some supertrace on $SH_N(\nu)$. In Appendix \ref{appa} it is
shown that the number of independent supertraces on $SH_N (\nu )$ equals to
the number of partitions of $N$ into a sum of odd positive integers. Some
technical details of the proof of Section \ref{sec5} are collected in
Appendices \ref{appb} and \ref{appc}.
\section{Finite-Dimensional Groups of Automorphisms}\label{sec2}
The group algebra of $S_N$ is the finite-dimensional subalgebra of $SH_N (\nu
)$. The elements $\sigma \in S_N$ induce inner automorphisms of $SH_N(\nu)$.
It is well known, that any $\sigma \in S_N$ can be expanded into a product of
pairwise commuting cycles \begin{equation} \label{dec} \sigma=c_1 c_2 c_3\, ...\,c_t\,,
\end{equation} where $c_{\fam\frfam w}$, $ {\fam\frfam w}=1, \dots, t$, are cyclic permutations acting
on distinct subsets of values of indices $i$. For example, a cycle which acts
on the first $s$ indices as $1 \rightarrow 2 \rightarrow \,...\, \rightarrow
s \rightarrow 1$ has the form \begin{equation} \label{ex} c=K_{12}K_{23}\,...\,
K_{(s-1)\,s}\,. \end{equation} We use the notation $|c|$ for the length of the cycle
$c$. For the cycle (\ref{ex}), $|c|=s$. We take a convention that the cycles
of unit length are associated with all values of $i$ such that $\sigma
(i)=i$, so that the relation $\sum_{\fam\frfam w} |c_{\fam\frfam w}| = N$ is true.
Given permutation $\sigma \in S_N $, we introduce a new set of basis elements
${\fam\frfam B}_\sigma$=$\{b^I\}$ instead of $\{a^\alpha_i\}$ in the following way.
For every cycle $c_{\fam\frfam w}$ in the decomposition (\ref{dec}) ($ {\fam\frfam
w}=1,\,...\,,\,t $), let us fix some index $l_{\fam\frfam w}$, which belongs to the
subset associated with the cycle $c_{\fam\frfam w}$. The basis elements
$b^\alpha_{{\fam\frfam w}j}$, $j=1,\,...\,,|c_{\fam\frfam w}|$, which realize 1-dimensional
representations of the commutative cyclic group generated by $c_{\fam\frfam w}$,
have the form \begin{equation} \label {a} b^\alpha_{{\fam\frfam w}j} =\frac 1 {\sqrt{|c_{\fam\frfam
w}|}} \sum _{k=1}^{|c_{\fam\frfam w} |} (\lambda_{\fam\frfam w})^{jk} a^{\alpha}_{l({\fam\frfam
w},k)}\,, \end{equation} where $l({\fam\frfam w},k)=c_{\fam\frfam w}^{-k} (l_{\fam\frfam w})$ and
\begin{eqnarray}\label{lambda} \lambda_{\fam\frfam w}=exp(2\pi i /|c_{\fam\frfam w}|). \end{eqnarray}
{}From the definition (\ref{a}) it follows that \begin{equation}\label{eig} c_{\fam\frfam w}
b^\alpha_{{\fam\frfam w}j} = (\lambda_{\fam\frfam w})^j b^\alpha_{{\fam\frfam w}j} c_{\fam\frfam w}\,, \n
\end{equation} \begin{equation}\label{eignext} c_{\fam\frfam w} b^\alpha_{{\fam\frfam n}j} = b^\alpha_{{\fam\frfam n}j}
c_{\fam\frfam w}\mbox{ , for } {\fam\frfam n}\neq {\fam\frfam w} \n \end{equation} and therefore
\begin{equation}\label{eigs} \sigma b^\alpha_{{\fam\frfam w}j} = (\lambda_{\fam\frfam w}
)^jb^\alpha_{{\fam\frfam w}j} \sigma\,. \end{equation}
In what follows, instead of writing $b^\alpha_{{\fam\frfam w}j}$ we use the notation
$b^I$ with the label $I$ accounting for the full information about the index
$\alpha$, the index ${\fam\frfam w}$ enumerating cycles in (\ref{dec}), and the
index $j$ which enumerates various elements $b^\alpha_{{\fam\frfam w}j}$ related to
the cycle $c_{\fam\frfam w}$, $i.e.\ $ $I$ ($I = 1,\,...,\,2N$) enumerates all possible
triples $\{\alpha,{\fam\frfam w},j\}$. We denote the index $\alpha$, the cycle and
the eigenvalue in (\ref{eig}) corresponding to some fixed index $I$ as
$\alpha (I)$, $ c(I),\ $ and $\lambda_I=(\lambda_{\fam\frfam w})^j$, respectively.
The notation $\sigma(I)=\sigma_0$ implies that $b^I \in {\fam\frfam B}_{\sigma_0}$.
${\fam\frfam B}_{\bf 1}$ is the original basis of the generating elements
$a_i^\alpha$ (here ${\bf 1}$ is the unit permutation).
Let ${\fam\frfam M}(\sigma)$ be the matrix which maps ${\fam\frfam B}_{\bf
1}\longrightarrow {\fam\frfam B}_\sigma$ in accordance with (\ref{a}),
\begin{eqnarray}\label{frm} b^I=\sum_{i,\alpha} {\fam\frfam M}_{i\alpha}^I(\sigma)\,
a^\alpha_i\,. \end{eqnarray} Obviously this mapping is invertible. Using the matrix
notations one can rewrite (\ref{eigs}) as \begin{eqnarray}\label{eigmat} \sigma b^I
\sigma^{-1}=\sum_{J=1}^{2N} \Lambda^I_J(\sigma)\, b^J\,,\ \ \forall b^I \in
{\fam\frfam B}_\sigma\,, \end{eqnarray} where $\Lambda_I^J(\sigma)=\delta_I^J \lambda_I$.
Every polynomial in $SH_N(\nu)$ can be expanded into a sum of monomials of
the form \begin{equation} \label{[} b^{I_1} b^{I_2}...\,b^{I_s}\sigma\,, \end{equation} where all
$\sigma (I_k ) =\sigma$. Every monomial of this form realizes some
one-dimensional representation of the Abelian group generated by all cycles
$c_{\fam\frfam w}$ in the decomposition (\ref{dec}).
The commutation relations for the generating elements $b^I$ follow from
(\ref{gcom}) and (\ref{A}) \begin{equation}\label{f} \left [ b^I,\,b^J\right]= F^{IJ}=
{\cal C}^{IJ} + \nu f^{IJ}\,, \n \end{equation} where \begin{equation}\label{calC} {\cal
C}^{IJ}=\epsilon^{\alpha (I) \alpha (J)} \delta_{c(I) c(J)} \delta_{\lambda_I
\lambda_J^{-1}} \end{equation} and \begin{equation} \label{struc} f^{IJ}=\sum_{i,j,\alpha ,\beta}{\fam\frfam
M}^I_{i\alpha} (\sigma ) {\fam\frfam M}^J_{j\beta} (\sigma )
\epsilon^{\alpha\beta}\tilde{A}_{ij}. \end{equation}
The indices $I,J$ are raised and lowered with the aid of the symplectic form
$ {\cal C}^{IJ}$ \begin{equation}\label{rise} \mu^I=\sum_J{\cal C}^{IJ}\mu_J\,,\qquad
\mu_I=\sum_J\mu^J {\cal C}_{JI}\,; \qquad \sum_M{\cal C}_{IM}{\cal
C}^{MJ}=-\delta_I^J\,. \end{equation} Note that the elements $b^I$ are normalized in
(\ref{a}) in such a way that the $\nu$-independent part in (\ref{f}) has the
form (\ref{calC}).
Another important finite-dimensional algebra of inner automorphisms of $SH_N
(\nu )$ is the $sl_2$ algebra which acts on the indices $\alpha$. It is
spanned by the $S_N$-invariant second-order polynomials (\ref {sl2}).
Evidently, $SH_N(\nu)$ decomposes into the infinite direct sum of only
finite-dimensional irreducible representations of this $sl_2$ spanned by
various homogeneous polynomials (\ref{[}).
{}From the defining relations (\ref{begin})-(\ref{A}) it follows that $SH_N
(\nu )$ is $Z_2$ - graded with respect to the automorphism \begin{eqnarray}\label{auto}
f(a_j^\alpha)=-a_j^\alpha\,, \qquad f(K_{ij} )=K_{ij} \end{eqnarray} which gives rise
to the parity $\pi$ (\ref{m0}). In applications to higher-spin models, this
automorphism distinguishes between bosons and fermions.
The algebra $SH_N(\nu )$ admits the antiautomorphism $\rho$, \begin{equation} \label{ant}
\rho (a^\alpha_k )= ia^\alpha_k\,,\qquad \rho(K_{ij}) = K_{ij}\,, \n \end{equation}
which leaves invariant the basic relations (\ref{begin})-(\ref{A}) provided
that an order of operators is reversed according to the defining property of
antiautomorphisms: $\rho (AB)$= $\rho (B)\rho (A)$. {}From (\ref{dec}),
(\ref{ex}) and (\ref{eigs}) it follows that \begin{equation} \label{Ant} \rho (\sigma )=
\sigma^{-1}\,,\qquad \rho (b^I)= ib^J\,, \n \end{equation} where $J$ is related to $I$
in such a way that $\alpha(J)=\alpha(I)$, $\sigma (J) = (\sigma (I))^{-1}$,
$c(J)=(c(I))^{-1}$ and $\lambda_J = \lambda^{-1}_I$. Note that in higher-spin
theories the counterpart of $\rho$ distinguishes between odd and even spins
\cite{16}.
\section{General Properties of Supertrace}\label{sec3}
In this section we summarize some general properties to be respected by any
supertrace in $SH_N (\nu )$.
Let $A$ be an arbitrary associative $Z_2$ graded algebra with the parity
function $\pi(x) =0$ or $1$. Suppose that $A$ admits some supertrace
operations $str_p$ where the label $p$ enumerates different nontrivial
supertraces. We call a supertrace $str $ even (odd) if $str(x)=0 \ \forall x
\in A$ such that $\pi(x)=1(0)$. Let $T_A$ be a linear space of supertraces on
$A$. We say that $dim\,T_A$ is the number of supertraces on $A$.
Given parity-preserving (anti)automorphism $\tau$ and supertrace operation
$str$ on $A$, $str (\tau (x))$ is some supertrace as well. For inner
automorphisms $\tau$ ($\tau (x)= pxp^{-1},\ \pi (p)=0$) it follows from the
defining property of the supertrace that $str (\tau (x))$=$str (x)$. Thus,
$T_A$ forms a representation of the factor-group of the parity preserving
automorphisms and antiautomorphisms of $A$ over the normal subgroup of the
inner automorphisms of $A$. Applying this fact to the original parity
automorphism $(-1)^\pi$ one concludes that $T_A$ can always be decomposed
into a direct sum of subspaces of even and odd supertraces, $T_A =T_A^0
\oplus T_A^1$ and that $T_A^1 =0$ if the parity automorphism is inner.
In the sequel we only consider the case where $dim\,T_A \ <\infty$ and there
are no nontrivial odd supertraces. Let $A=A_1 \otimes A_2$ with the
associative algebras $A_1$ and $A_2$ endowed with some even supertrace
operations $t_1$ and $t_2$, respectively. The supertrace on $A$ can be
defined by setting $str(a_1 \otimes a_2 )$= $t_1(a_1)$ $t_2(a_2 )$, $\forall
a_1 \in A_1$, $\forall a_2 \in A_2$. As a result, one concludes that $\ T_A$=
$T_{A_1}\otimes T_{A_2}$. In the case of $SH_N (\nu )$ one thus can always
separate out a contribution of the center of mass coordinates as an overall
factor ($SH_1$ admits the unique supertrace).
If $A$ is finite-dimensional then the existence of two different supertraces
indicates that $A$ admits non-trivial both-side ideals. Actually,
con\-si\-der the bilinear form $B(f,g)$ = $\alpha_1 str_1 (fg) + \alpha_2
str_2 (fg)$ with arbitrary parameters $\alpha_1,\ \alpha_2\in {\fam\openfam C} $ and
elements $f,\ g \in A$. The determinant of this bilinear form is some
polynomial of $\alpha_{1}$ and $\alpha_2$. Therefore it vanishes for certain
ratios $\alpha_{1}/\alpha_{2}$ or $\alpha_{2}/\alpha_{1}$ according to the
central theorem of algebra. Thus, for these values of the parameters the
bilinear form $B$ degenerates and admits non-trivial null vectors $x$,
$B(x,g)$=0, $\forall g\in A$. It is easy to see that the linear space ${\cal
I}$ of all null vectors $x$ is some both-side ideal of $A$. For
infinite-dimensional algebras the existence of several supertraces does not
necessarily imply the existence of ideals. As mentioned in introduction the
existence of several supertrace operations may be related to the existence of
inequivalent representations. Also it is worth mentioning that for the case
of infinite-dimensional algebras and representations under investigation it
can be difficult to use the standard ($i.e.\ $ matrixwise) definition of the
supertrace. In this situation the formal definition of the supertraces on the
algebra we implement in this paper is the only rigorous one.
Let $l_A$ be the Lie superalgebra which is isomorphic to $A$ as a linear
space and is endowed with the product law (\ref{com}). It contains the
subalgebra $sl_A \in l_A $ spanned by elements $g$ such that $str_p (g)=0$
for all $p$. Evidently $sl_A$ forms the ideal of $l_A$. The factor algebra
$t_A$=$l_A$/$sl_A$ is a commutative Lie algebra isomorphic to $T_A^*$ as a
linear space. Elements of $t_A$ different from the unit element of $A$ (which
exist if $dim\,T_A >1$) can induce outer automorphisms of $sl_A$. Let us note
that it is this $sl_A$ Lie superalgebra which usually has physical
applications. For the case of $SH_N (\nu )$ under consideration the algebra
$l_{SH_N(\nu)}$ is identified with the algebra $W_{N,\infty }(\nu )$
introduced in \cite{8}. We therefore conclude that these algebras are not
simple for $N>2$ because it is shown below that $SH_N (\nu )$ admits several
supertraces for $N>2$. Instead one can consider the algebras $sW_{N,\infty}
(\nu )$.
Let $l_A$ contain some subalgebra ${\cal L} $ such that $A$ decomposes into a
direct sum of irreducible representations of ${\cal L}$ with respect to the
adjoint action of ${\cal L}$ on $A$ via supercommutators. Then, only trivial
representations of ${\cal L}$ can contribute to any supertrace on $A$.
Actually, consider some non-trivial irreducible representation $R$ of ${\cal
L}$. Any $r\in R$ can be represented as \begin{equation} \label{ir} r=\sum_j [l_j ,r_j
\}\,,\qquad l_j \in {\cal L},\quad r_j \in R\, \end{equation} since elements of the form
(\ref{ir}) span the invariant subspace in $R$. {}From (\ref{scom}) it follows
then that $str(r)=0\,,\forall r\in R$.
{}From the definition of the supertrace it follows that \begin{equation} str(a_1 a_2
)+str(a_2 a_1 )=0 \n \end{equation} for arbitrary odd elements $a_1$ and $a_2 $ of $A$.
A simple consequence of this relation is that \begin{equation} \label{odd} str(a_1
a_2\ldots a_n +a_2 \ldots a_n a_1 +\ldots + a_n a_1 \ldots a_{n-1} ) =0 \n
\end{equation} is true for an arbitrary even $n$ if all $a_i$ are some odd elements of
$A$. Since we assume that the supertrace is even (\ref{odd}) is true for any
$n$. This simple property turns out to be practically useful because, when
odd generating elements are subject to some commutation relations with the
right hand sides expressed via even generating elements like in (\ref{gcom}),
it often allows one to reduce evaluation of the supertrace of a degree-$n$
polynomial of $a_i$ to supertraces of lower degree polynomials.
Another useful property is that in order to show that the characteristic
property of the supertrace (\ref{scom}) is true for any $x,g\,\in A$, it
suffices to show this for a particular case where $x$ is arbitrary while $g$
is an arbitrary generating element of some fixed system of generating
elements. Then (\ref{scom}) for general $x$ and $g$ will follow from the
properties that $A$ is associative and $str$ is linear. For the particular
case of $SH_N (\nu )$ this means that it is enough to set either
$g=a_i^\alpha$ or $g=K_{ij}$.
Let us now turn to some specific properties of $SH_N (\nu)$ as a particular
realization of $A$.
By identifying ${\cal L}$ with $sl_2$ (\ref{sl2}) and taking into account
that $SH_N(\nu )$ decomposes into a direct sum of irreducible
finite-dimensional representations of $sl_2$, one arrives at the following
\noindent {\it {\bf Lemma 1}: $str(x)$ can be different from zero only when
$x$ is $sl_2$-singlet, $i.e.\ $ $[T^{\alpha\beta}, x]=0$.}
\noindent {\it Corollary}: Any supertrace on $SH_N (\nu )$ is even.
Analogously one deduces consequences of the $S_N$ symmetry. In particular,
one proves
\noindent {\it {\bf Lemma 2}: Given $c\in S_N$ such that $cF=\mu Fc$ for some
element $F$ and any constant $\mu\neq 1$, $str(F)=0\,$. Given monomial
$F=b^{I_1}b^{I_2}\,...\,b^{I_s} \sigma$ with $b^{I_k}\in {\fam\frfam B}_\sigma $ and
a cycle $c_0$ in the decomposition (\ref{dec}) of $\sigma$ one concludes that
$str(F)=0\,$ if $\,\prod_{k:\,c(I_k)=c_0} \lambda_{I_k} \neq 1 $ where
$\lambda_{I_k}$ are the eigenvalues (\ref{eigs}) of $b^{I_k}$.}
\section{Ground Level Conditions}\label{sec4} Let us analyze restrictions on
a form of $str(a)$, $a\in S_N$, which follow from the defining relations of
$SH_N (\nu )$.
Firstly, we describe supertraces on the group algebra of $S_N$. Let some
permutation $\sigma$ decomposes into $n_1$ cycles of length $1$, $n_2$ cycles
of length $2$, ... and $n_N$ cycles of length $N$. The non-negative integers
$n_k$ satisfy the relation \begin{equation}\label{m} \sum_{k=1}^N kn_k =N \end{equation} and fix
$\sigma$ up to some conjugation $\sigma \rightarrow \tau \sigma \tau^{-1}$,
$\tau \in S_N$. Thus \begin{equation}\label{varphi} str(\sigma)=\varphi
(n_1,n_2,\,...\,,n_N)\,, \end{equation} where $\varphi (n_1,n_2,\,...\,,n_N) $ is an
arbitrary function. Obviously the linear space of invariant functions on
$S_N$ ($i.e.\ $ such that $f(\tau\sigma\tau^{-1})=f(\sigma)$) coincides with the
linear space of supertraces on the group algebra of $S_N$. Therefore, the
dimension of the linear space of supertraces is equal to the number $p(N)$ of
independent solutions of (\ref{m}), the number of conjugacy classes of $S_N$.
One can introduce the generating function for $p(N)$ as
$P(q)=\sum_{n=0}^\infty p(n) q^n$=$\prod_{k=1}^\infty \frac 1 {(1-q^k)}$.
The properties of this generating function and of the quantities $p(N)$ are
discussed in details $e.g.\ $ in \cite{17}.
According to the general argument of the previous section, the existence of
several independent traces implies that the group algebra of $S_N$ must have
some ideals. Indeed it can be shown to decompose into a direct sum of matrix
algebras $Mat_n$.
Since the group algebra of $S_N$ is embedded into $SH_N (\nu)\,$ some
additional restrictions on the functions $\varphi (n_1,n_2,\,...,\,n_N) $
follow from (\ref{scom}) and the defining relations (\ref{gcom})-(\ref{A}) of
$SH_N (\nu )$. Actually, consider some elements $b^I$ such that
$\lambda_I=-1$. Then, one finds from (\ref{scom}) and (\ref{eigs}) that $str
\left ( b^I b^J \sigma \right )$= $ - str \left ( b^J \sigma b^I\right )$= $
str \left ( b^J b^I \sigma \right )$ and therefore \begin{equation}\label{ccc} str \left (
[ b^I, b^{J}] \sigma \right ) =0\,. \end{equation} Since these conditions restrict
supertraces of degree-0 polynomials of $a^\alpha_i$ we call them ground level
conditions ({\it GLC}). Thus for every permutation $\sigma$ and any even
integer $2k$ such that there exists some cycle $c$ of length $|c|=2k$ in the
decomposition (\ref{dec}) we have {\it GLC} (\ref{ccc}) with $b^I$ such that
$c(I)=c$. Note however that if $\lambda_J\neq -1$ or $c(J) \neq c(I)$ then
the relation $str ([ b^I, b^J] \sigma ) =0$ is trivially satisfied as a
consequence of {\it Lemma 2}.
It is convenient to rewrite {\it GLC} in the following form \begin{eqnarray} \label{eqss}
str(c_0\sigma_0)= -str\left(\Big ([b^0_{0k},b^1_{0k}]-1\Big )c_0\sigma_0 \right
), \end{eqnarray} where $c_0$ is any cycle of even length $2k$ in the decomposition of
the permutation $\sigma=c_0\sigma_0$ and $b_{0k}^\alpha$ is the corresponding
variable (\ref{a}) with $(\lambda_0)^k=-1$, $i.e.\ $ $c_0 b_{0k}^\alpha=
-b_{0k}^\alpha c_0\,,\ $ $\sigma_0 b_{0k}^\alpha= b_{0k}^\alpha\sigma_0$ and
$\lambda_0=exp(2\pi i/|c_0|)$.
To work out the explicit form of the restrictions on the functions $\varphi
(n_1,n_2,\,...\,n_N) $ which follow from {\it GLC} one has to use the
following simple facts from the theory of the symmetric group:
\noindent {\it {\bf Lemma 3}: Let $c_1$ and $c_2$ be two distinct cycles in
the decomposition (\ref{dec}). Let indices $i_1$ and $i_2$ belong to the
subsets of indices associated with the cycles $c_1$ and $c_2$, respectively.
Then the permutation $c= c_1 c_2 K_{i_1\, i_2}$ is a cycle of length $|c|=
|c_1| + |c_2| $.}
\noindent {\it {\bf Lemma 4}: Given cyclic permutation $c \in S_N$, let
$i\neq j$ be two indices such that $c^k (i) = j$, where $k$ is some positive
integer, $k<|c|$. Then $c K_{ij} = c_1 c_2 $ where $c_{1,2}$ are some
non-coinciding mutually commuting cycles such that $|c_1|=k$ and $|c_2|=
|c|-k$.}
Using the definition (\ref{a}), the commutation relations
(\ref{begin})-(\ref{A}) and {\it Lemmas 3} and {\it 4} one reduces {\it GLC}
to the following system of equations: \begin{eqnarray}\label {mm} &{}& n_{2k}\varphi
(n_1,\,...\,,n_{2k},\,...\,,n_N) \nn &{}&= -\nu n_{2k} \bigg ( 2 \sum_{s\neq
k,\,s=1}^{2k-1} O_s \varphi (n_1,\,...\,,n_s+1,\,...\,,
n_{2k-s}+1,\,...\,,n_{2k}-1,\,...\,,n_N) \nn &{}& + 2 O_k \varphi
(n_1,\,...\,,n_k+2,\,...\,,n_{2k}-1,\,...\,,n_N) \nn &{}&+ \sum_{s\neq
2k;\,s=1}^N sn_s \varphi (n_1,\,...\,,n_s-1,\,...\,,n_{2k}-1,\,...
\,,n_{2k+s}+1,\,...\,,n_N) \nn &{}&+ 2k(n_{2k}-1)\varphi
(n_1,\,...\,,n_{2k}-2,\,...\,,n_{4k}+1,\,...\,,n_N) \bigg ) \end{eqnarray} where
$O_k=0$ for $k$ even and $O_k=1$ for $k$ odd.
Let us note that by virtue of the substitution \begin{equation} \label{scal} \varphi
(n_1,\,\ldots \,,n_N)= \nu^{E(\sigma)} {\tilde \varphi} (n_1,\,\ldots
\,,n_N)\,, \end{equation} where $E(\sigma)$ is the number of cycles of even length in
the decomposition of $\sigma$ (\ref{dec})\,, $i.e.\ $ \begin{equation} E(\sigma)=n_2+n_4+\,...
\end{equation} one can get rid of the explicit dependence of $\nu$ from {\it GLC}
(\ref{mm}). As a result, there are two distinguishing cases, $\nu=0$ and $\nu
\neq 0$.
For lower $N$ the conditions (\ref {mm}) take the form \begin{equation}
\varphi(0,1)+2\nu\varphi(2,0)=0 \n \end{equation} for $N=2$ ({\it cf.} (\ref{str2})),
\begin{equation} \varphi(1,1,0)+2\nu\varphi(3,0,0)+\nu\varphi(0,0,1)=0 \n \end{equation} for $N=3$
and \begin{eqnarray}
\varphi(2,1,0,0)+2\nu\varphi(4,0,0,0)+2\nu\varphi(1,0,1,0)&=&0 \nn
\varphi(0,2,0,0)+2\nu\varphi(2,1,0,0)+2\nu\varphi(0,0,0,1)&=&0 \nn
\varphi(0,0,0,1)+4\nu\varphi(1,0,1,0)&=&0 \nonumber \end{eqnarray} for $N=4$.
As a result one finds 1-parametric families of solutions for $N=1$ and $N=2$
and 2-parametric families of solutions for $N=3$ and $N=4$.
Let $G_N$ be the number of independent solutions of (\ref{mm}). As we show in
the next section $G_N$=$dimT_{SH_N (\nu )}$ for all $\nu$. In other words all
other conditions on the supertrace do not impose any restrictions on the
functions $\varphi (n_1, \ldots ,n_N )$ but merely express supertraces of
higher order polynomials of $a_i^\alpha$ in terms of $\varphi (n_1, \ldots
,n_N )$.
In the Appendix \ref{appa} we prove the following
\noindent {\it {\bf Theorem 1}: $G_N=q(N)$ where $q(N)$ is a number of
partitions of $N$ into a sum of odd positive integers, $i.e.\ $ the number of the
solutions of the equation $\sum_{k=0}^\infty (2k+1) n_{k}=N$ for non-negative
integers $n_i$.}
One can guess this result from the particular case of $\nu=0$ where {\it GLC}
tell us that $\varphi(n_1,\,...\,,\,n_N)$ can be nonvanishing (and arbitrary)
only when all $n_{2k}=0$. Interestingly enough, $G_N$ remains the same for
$\nu\neq 0$.
\section{Supertrace for General Elements}\label{sec5}
In this section we prove
\noindent {\it {\bf Theorem 2}: $dimT_{SH_N (\nu )}$=$G_N$ where $G_N$ is the
number of independent solutions of the ground level conditions (\ref{mm}).}
The proof of the {\it Theorem 2} will be given in a constructive way by
virtue of the following double induction procedure:
\noindent {\bf (i)}. Assuming that {\it GLC} are true and $str\{b^I , P_p (a)
\sigma \}=0$ $\forall P_p (a),$ $\sigma$ and $I$ provided that $b^I \in {\fam\frfam
B}_\sigma$ and $$ \begin{array}{l} \mbox{$\lambda (I) \neq -1$; $p\leq k\,$
or}\\ \mbox{$\lambda (I) =-1$, $E(\sigma )\leq l$, $p\leq k\,$ or}\\
\mbox{$\lambda (I) =-1$; $p\leq k-2$}\,, \end{array} $$ where $P_p (a)$ is an
arbitrary degree $p$ polynomial of $a_i^{\alpha} $ ($p$ is odd) and $E(\sigma
)$ is the number of cycles of even length in the decomposition (\ref{dec}) of
$\sigma$, one proves that there exists such a unique extension of the
supertrace that the same is true for $l\rightarrow l+1$.
\noindent {\bf (ii)}. Assuming that $str\{b^I , P_p (a) \sigma \}=0$ $\forall
P_p (a)$, $\sigma$ and $b^I$ such that $\sigma (I)=\sigma$, $p\leq k$ one
proves that there exists such a unique extension of the supertrace that the
assumption {\bf (i)} is true for $k\rightarrow k+2$ and $l=0$.
As a result this inductive procedure extends uniquely any solution of {\it
GLC} to some supertrace on the whole $SH_N (\nu )$. (Let us remind ourselves
that the supertrace of any odd element of $SH_N (\nu )$ is trivially zero by
$sl_2$ invariance).
The inductive proof of the {\it Theorem 2} is based on the $S_N$ covariance
of the whole setting and the following important
\noindent {\it {\bf Lemma 5}: Given permutation $\sigma$ which has $E(\sigma
)$ cycles of even length in the decomposition (\ref{dec}), the quantity
$f^{IJ}\sigma $ for $\sigma (I)=\sigma (J)=\sigma $ and $\lambda_I=\lambda_J
=-1 $ can be uniquely expanded as $ f^{IJ}\sigma =\sum_q {\alpha}_q \sigma_q
$ where ${\alpha}_q$ are some coefficients and $E(\sigma_q )=E(\sigma )-1$
$\forall q$.}
{\it Lemma 5} is a simple consequence of the particular form of the structure
coefficients $f^{IJ}$ (\ref{struc}) and {\it Lemmas 3} and {\it 4}. The proof
is straightforward. Let us stress that it is {\it Lemma 5} which accounts for
the specific properties of the algebra $SH_N (\nu )$ in the analysis of this
section.
In practice it is convenient to work with the exponential generating
functions \begin{equation} \label{gf} \Psi_\sigma (\mu )= str\left ( e^S \sigma \right
)\,,\qquad S= \sum_{L=1}^{2N} (\mu_{L } b^{L} )\,, \n \end{equation} where $\sigma$ is
some fixed element of $S_N$, $b^L \in {\fam\frfam B}_\sigma $ and $\mu_{L } \in
{\fam\openfam C}$ are independent parameters. By differentiating over $\mu_{L }$ one
can obtain an arbitrary polynomial of $b^L$ in front of $\sigma$. The
exponential form of the generating functions implies that these polynomials
are Weyl ordered. In these terms the induction on a degree of polynomials is
equivalent to the induction on a degree of homogeneity in $\mu$ of the power
series expansions of $\Psi_\sigma (\mu )$.
As a consequence of the general properties discussed in the preceding
sections the generating function $\Psi_\sigma (\mu )$ must be invariant under
the $S_N$ similarity transformations \begin{eqnarray}\label{S_N}
\Psi_{\tau\sigma\tau^{-1}}(\mu)=\Psi_\sigma (\tilde{\mu})\,, \end{eqnarray} where the
$S_N$ transformed parameters are of the form \begin{eqnarray} \label{base}
\tilde{\mu}_I=\sum_J \left({\fam\frfam M}(\tau\sigma\tau^{-1}) {\fam\frfam
M}^{-1}(\tau)\Lambda^{-1}(\tau){\fam\frfam M}(\tau) {\fam\frfam M}^{-1}(\sigma)\right)_I^J
{\mu}_J \end{eqnarray} and matrices ${\fam\frfam M}(\sigma)$ and $\Lambda(\sigma)$ are defined
in (\ref{frm}) and (\ref{eigmat}).
In accordance with the general argument of Section \ref{sec3} the necessary
and sufficient conditions for the existence of even supertrace are the
$S_N$-covariance conditions (\ref{S_N}) and the condition that
\begin{equation}\label{start} str\left \{b^L , (exp S ) \sigma \right \}=0\qquad \mbox{for
any $\sigma$ and $L$} \,. \end{equation} To transform (\ref{start}) to an appropriate
form, let us use the following two general relations which are true for
arbitrary operators $X$ and $Y$ and the parameter $\mu \in {\fam\openfam C}$:
\begin{equation}\label{r1} Xexp(Y+\mu X)=\frac{\partial}{\partial\mu}exp (Y+\mu X )+ \int \,t_2 \,exp(t_1
(Y+\mu X))[X,Y] exp(t_2 (Y+\mu X))D^1t , \end{equation} \begin{equation}\label{r2} exp(Y+\mu
X)X=\frac{\partial}{\partial\mu}exp (Y+\mu X )- \int \,t_1 \,exp(t_1 (Y+\mu X))[X,Y] exp(t_2
(Y+\mu X))D^1t \end{equation} with the convention that \begin{equation}\label{t} D^{n-1}t=\delta (t_1
+\ldots +t_n -1)\theta (t_1 )\ldots \theta (t_n ) dt_1 \ldots dt_n \,. \end{equation}
The relations (\ref{r1}) and (\ref{r2}) can be derived with the aid of the
partial integration ($e.g.\ $ over $t_1$) and the following formula \begin{equation}\label{d}
\frac{\partial}{\partial\mu}exp (Y+\mu X ) = \int \, exp(t_1 (Y+\mu X)) X exp(t_2 (Y+\mu
X))D^1 t\,, \end{equation} which can be proven by expanding in power series. The
well-known formula \begin{equation} \label{r3} [X,exp(Y)]= \int \, exp(t_1 Y)[X,Y] exp(t_2
Y)D^1 t \end{equation} is a consequence of (\ref{r1}) and (\ref{r2}).
With the aid of (\ref{r1}), (\ref{r2}) and (\ref{eigs}) one rewrites
(\ref{start}) as \begin{equation}\label{nm1} (1+\lambda_L )\frac{\partial}{\partial\mu_L }\Psi_\sigma
(\mu )= \int \,(\lambda_L t_1 -t_2 ) str\Big ( exp (t_1 S)[b^L ,S]\,exp (t_2
S)\sigma\Big )\, D^1 t\,. \end{equation} This condition should be true for any $\sigma$
and $L$ and plays the central role in the analysis of this section.
There are two essentially distinguishing cases, $\lambda_L \neq -1$ and
$\lambda_L =-1$. In the latter case, the equation (\ref{nm1}) takes the form
\begin{equation}\label{m1} 0=\int \, str\Big ( exp (t_1 S)[b^L ,S]\,exp (t_2 S) \sigma \Big
)D^1 t\,,\qquad \lambda_L =-1\,. \end{equation}
In Appendix \ref{appb} we show by induction that the equations (\ref{nm1})
and (\ref{m1}) are consistent in the following sense \begin{eqnarray}\label{c1}
(1+\lambda_K )\frac{\partial}{\partial\mu_K }\int\,(\lambda_L t_1 -t_2 ) str\Big ( exp (t_1
S)[b^L ,S]\,exp (t_2 S) \sigma \Big )D^1 t -(L \leftrightarrow K )=0, & & \\
\lambda_L \neq -1, \ \lambda_K \neq -1 & & \nonumber \end{eqnarray} and \begin{equation} \label{c2}
(1+\lambda_K )\frac{\partial}{\partial\mu_K }\int \, str\Big ( exp (t_1 S)[b^L ,S]\,exp (t_2
S)\sigma\Big )D^1 t\,=0 ,\qquad \lambda_L=-1. \end{equation} Note that this part of the
proof is quite general and does not depend on a concrete form of the
commutation relations of $a_i^{\alpha}$ in (\ref{gcom}).
By expanding the exponential $e^S$ in (\ref{gf}) into power series in $\mu
_K$ (equivalently $b^K$) one concludes that the equation (\ref{nm1}) uniquely
reconstructs the supertrace of monomials containing $b^K$ with $\lambda_K\neq
-1$ (from now on called regular polynomials) via supertraces of some lower
order polynomials. The consistency conditions (\ref{c1}) and (\ref{c2}) then
guarantee that (\ref{nm1}) does not impose any additional conditions on the
supertraces of lower degree polynomials and allow one to represent the
generating function in the form \begin{eqnarray}\label{ex1} \Psi_\sigma &=&
\Phi_\sigma(\mu)\\ &+& \sum_{L:\,\lambda_L \neq -1} \int_0^1 \frac {\mu_L
d\tau} {1+\lambda_L}\int D^1 t\,(\lambda_L t_1 -t_2 ) str\Big ( e^{t_1 (\tau
S^{\prime\prime}+S^\prime)}[b^L ,(\tau S^{\prime\prime}+S^\prime)] \,e^{ t_2
(\tau S^{\prime\prime}+S^\prime)}\sigma\Big )\, , \nonumber \end{eqnarray} where we have
introduced the generating functions $\Phi_\sigma$ for the supertrace of
special polynomials, $i.e.\ $ the polynomials depending only on $b^L$ with
$\lambda_L=-1$, \begin{equation}\label{gff} \Phi_\sigma (\mu )\stackrel {def}{=} str\left
( e^{ S^\prime} \sigma \right ) = \Psi_\sigma (\mu)\Big |_ {(\mu_I=0\ \forall
I:\ \lambda_I \neq -1)} \n \end{equation} and \begin{equation} \label{spr} S^\prime = \sum_{L:\,b^L
\in {\fam\frfam B}_\sigma,\,\lambda_L=-1} (\mu_{L } b^L); \qquad
S^{\prime\prime}=S-S^\prime\,. \end{equation} The relation (\ref{ex1}) successively
expresses the supertrace of higher order regular polynomials via the
supertraces of lower order polynomials.
One can see that the arguments above prove effectively the inductive
hypotheses {\bf (i)} and {\bf (ii)} for the particular case where either the
polynomials $P_p (a)$ are regular and/or $\lambda_I \neq -1$. Note that for
this case the induction on the number of cycles of even length {\bf (i)} is
trivial: one simply proves that a power of polynomial can be increased by
two.
Let us now turn to the less trivial case of the special polynomials:
\begin{equation}\label{startprime} str\left \{b^I , (exp S^\prime ) \sigma \right
\}=0\,,\qquad \lambda_I =-1. \end{equation} Consider the part of $str\left \{b^I , (exp
S^\prime ) \sigma \right \}$ which is of order $k$ in $\mu$ and suppose that
$E(\sigma)=l+1$. According to (\ref{m1}) the conditions (\ref{startprime})
give \begin{equation} \label{m1prime} 0= \int \, str\left( exp (t_1 S^\prime)[b^I
,S^\prime]\, exp (t_2 S^\prime) \sigma \right) D^1 t\,. \n \end{equation}
Substituting $[b^I ,S^\prime]=\mu^I + \nu \sum_M f^{IM}\mu_M$, where the
quantities $f^{IJ}$ and $\mu^I$ are defined in (\ref{f})-(\ref{rise}), one
can rewrite the equation (\ref{m1prime}) in the form \begin{equation}\label{formprime}
\mu^I \Phi_\sigma(\mu ) = -\nu \int str\bigg ( exp (t_1 S^\prime)\sum_M
f^{IM}\mu_M\, exp (t_2 S^\prime) \sigma \bigg) D^1 t\,. \end{equation}
Now we use the inductive hypothesis {\bf (i)}. The right hand side of
(\ref{formprime}) is a supertrace of at most a degree $k-1$ polynomial of
$a^{\alpha}_i$ in the sector of degree $k$ polynomials in $\mu$. Therefore
one can use the inductive hypothesis {\bf (i)} to obtain $$ \int str\Big (
exp(t_1 S^\prime)\sum_M f^{IM}\mu_M\, exp(t_2 S^\prime) \sigma \Big )D^1t =
\int \, str\Big ( exp(t_2 S^\prime)exp(t_1 S^\prime) \sum_M f^{IM}\mu_M \sigma
\Big )D^1 t, $$ where we made use of the simple fact that $str(S^\prime F
\sigma)$ $=$ $ -str(F \sigma S^\prime) $= $str(F S^\prime \sigma)$ due to the
definition of $S^\prime$.
As a result, the inductive hypothesis allows one to transform
(\ref{startprime}) to the following form \begin{equation} \label{p9} X^I \equiv \mu^I
\Phi_\sigma(\mu ) +\nu str\bigg( exp (S^\prime ) \sum_Mf^{IM}\mu_M\sigma
\bigg)=0 \,. \end{equation}
By differentiating this equation with respect to $\mu_J$ one obtains after
symmetrization \begin{equation} \label{p10} \frac{\partial}{\partial\mu_J} \left( \mu^I \Phi_\sigma (\mu
)\right) +(I\leftrightarrow J )=-\nu \int str\Big (e^{t_1 S^\prime } b^Je^{t_2
S^\prime } \sum_Mf^{IM}\mu_M \sigma \Big )D^1 t +(I\leftrightarrow J ). \end{equation}
An important point is that the system of equations (\ref{p10}) is equivalent
to the original equations (\ref{p9}) except for the ground level part
$\Phi_\sigma (0)$. This can be easily seen from the simple fact that the
general solution of the system of equations $\frac{\partial}{\partial\mu_J} X^I(\mu) +
\frac{\partial}{\partial\mu_I} X^{J}(\mu) =0 $ for entire functions $X^I(\mu)$ is of the
form $X^I(\mu)=X^I(0)+\sum_{J}c^{IJ}\mu_J$ where $X^I(0)$ and
$c^{JI}$=$-c^{IJ}$ are some constants. The part of (\ref{p9}) linear in $\mu$
is however equivalent to the ground level conditions analyzed in the previous
section. Thus (\ref{p10}) contains all information additional to (\ref{mm}).
For this reason we will from now on analyze the equation (\ref{p10}).
Using again the inductive hypothesis we move $b^I$ to the left and to the
right with equal weights to get \begin{eqnarray}\label{p11} &{}& \frac{\partial}{\partial\mu_J}\mu^I
\Phi_\sigma (\mu )+(I\leftrightarrow J )= -\frac{\nu}{2} \sum_{M}str\Big (
exp(S^\prime )\{b^J ,f^{IM}\}\mu_M \sigma\Big )\nn &{}& -\frac{\nu}{2}\int
\,\sum_{L,M}(t_1 -t_2 ) str\Big (exp (t_1 S^\prime ) F^{JL}\mu_L exp (t_2
S^\prime ) f^{IM}\mu_M \sigma \Big )D^1 t + (I\leftrightarrow J ) \,. \end{eqnarray} The
last term on the right hand side of this expression can be shown to vanish
under the supertrace operation due to the factor of $(t_1 -t_2 )$, so that
one is left with the equation \begin{equation}\label{dm1} L^{IJ}\Phi_\sigma (\mu )= -\frac
{\nu}{2} R^{IJ} (\mu )\,, \end{equation} where \begin{eqnarray}\label{RIJ} R^{IJ} (\mu )=\sum_{M}
str\Big ( exp(S^\prime )\{b^J ,f^{IM}\}\mu_M \sigma \Big ) +(I\leftrightarrow J
) \end{eqnarray} and \begin{eqnarray}\label{LIJ} L^{IJ}=\frac{\partial}{\partial\mu_J}\mu^I +
\frac{\partial}{\partial\mu_I}\mu^J\,. \end{eqnarray}
The differential operators $L^{IJ}$ satisfy the standard $sp(2E(\sigma))$
commutation relations \begin{equation}\label{lcom} [L^{IJ},L^{KL}]= - \left( {\cal
C}^{IK}L^{JL}+ {\cal C}^{IL}L^{JK}+ {\cal C}^{JK}L^{IL}+ {\cal C}^{JL}L^{IK}
\right) \,. \end{equation} We show by induction in Appendix \ref{appc} that this algebra
is consistent with the right-hand side of the basic relation (\ref{dm1}) $i.e.\ $
that \begin{equation}\label{c3} [L^{IJ},\,R^{KL}]- [L^{KL},\,R^{IJ}]= -\left( {\cal
C}^{IK}R^{JL}+ {\cal C}^{JL}R^{IK}+ {\cal C}^{JK}R^{IL}+ {\cal C}^{IL}R^{JK}
\right) \,. \end{equation}
Generally, these consistency conditions guarantee that the equations
(\ref{dm1}) express $\Phi_\sigma (\mu ) $ in terms of $R^{IJ}$ in the
following way \begin{eqnarray} \label{ex2} \Phi_\sigma(\mu)&=& \Phi_\sigma(0)+\frac
{\nu}{8E(\sigma)}\sum_{I,J=1}^{2E(\sigma)} \int_0^1 \frac{dt}{t}
(1-t^{2E(\sigma )}) (L_{IJ} R^{IJ})(t\mu ) \,, \end{eqnarray} provided that \begin{equation}
\label{0} R^{IJ}(0)=0\,. \end{equation} The latter condition must hold for the
consistency of (\ref{dm1}) since its left hand side vanishes at $\mu_I =0$.
In the formula (\ref{ex2}) it guarantees that the integral on $t$ converges.
In the case under consideration the property (\ref{0}) is indeed true as a
consequence of the definition (\ref{RIJ}).
Taking into account {\it Lemma 5} and the explicit form of $R^{IJ}$
(\ref{RIJ}) one concludes that the equation (\ref{ex2}) expresses uniquely
the supertrace of special polynomials via the supertraces of polynomials of
lower degrees or via the supertraces of special polynomials of the same
degree with a lower number of cycles of even length provided that the $\mu$
independent term $\Phi_\sigma(0)$ is an arbitrary solution of {\it GLC}.
This completes the proof of {\it Theorem 2}.
\noindent {\it {\bf Comment 1:} The formulae (\ref{ex1}) and (\ref{ex2}) can
be effectively used in practical calculations of supertraces of particular
elements of $SH_N (\nu)$.}
\noindent {\it {\bf Comment 2:} Any supertrace on $SH_N (\nu )$ is determined
unambiguously in terms of its values on the group algebra of $S_N$.}
\noindent {\it Corollary:} Any supertrace on $SH_N (\nu )\ $ is $\rho$-invariant,
$str(\rho(x))=str(x)$ $\forall x \in SH_N(\nu)$, for the antiautomorphism
$\rho$ (\ref{ant}).
\noindent This is true due to the {\it Comment 2} because $\sigma$ and
$\sigma^{-1}=\rho (\sigma )$ belong to the same conjugacy class of $S_N$ so
that $str(\rho (\sigma ))=str(\sigma)$.
\section{Conclusions.}\label{sec6}
In this paper we have shown that the algebras $SH_N (\nu )$ can be endowed
with $q(N)$ independent supertrace operations where $q(N)$ is the number of
partitions of $N$ into a sum of odd positive integers. We hope to apply the
supertraces constructed in this paper to the analysis of the invariant forms
of $SH_N(\nu)$. Although a definition of the supertraces on $SH_N (\nu )\ $ behaves
regularly with the parameter $\nu$ (in particular, the number of supertraces
$q(N)$ is $\nu$ independent) one can expect that this is not the case for the
related bilinear forms which can degenerate for some special values of $\nu$
thus giving rise to ideals of $SH_N (\nu )\ $ as it happens \cite{14} for the simplest
case of $N=2$. The analysis of the structure of these ideals is a challenging
problem important for various application of $SH_N(\nu)$, including analysis
of its representations. We are going to study this problem for some lower
values of $N>2$ in the future publication.
In conclusion let us note that the method of the analysis of supertraces
presented in this paper is rather general. Practically, the only information
of the specific structure of $SH_N(\nu)$ is that {\it Lemma 5} is true.
Hopefully one can use the analogous methods for the analysis of supertraces
of other associative algebras.
\vskip 5 mm \noindent {\bf Acknowledgements} \vskip 3 mm \noindent Authors
are very grateful to M.~Soloviev for useful discussions. The research
described in this publication was made possible in part
by Grant $\mbox{N}^{\mbox{\underline o}}$
MQM300 from the International Science
Foundation and Government of Russian Federation.
This work was supported in part by
the Russian Basic Research Foundation, grant 93-02-15541, and INTAS grant
93-0633.
|
1,108,101,566,760 | arxiv | \section{Introduction}\label{Intro}
Conversion of kinetic collision energy into high multiplicity of newly made hadronic
particles is one of the most notable features of reactions observed at the Relativistic
Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory (BNL)~\cite{Back:2004je}.
In this process, aside of the light $u$ and $d$ quark pairs, present in all matter surrounding us,
the strange flavor quark pairs $s,\bar s$ are produced copiously (in general, in what follows,
the particle symbol will refer to the corresponding particle yield, either total or per unit of
rapidity, as appropriate). The final $s$-yield depends on the initial reactions, and
on the history of the fireball, and thus, also on the nature and properties
of the phase of matter formed.
On the time scale of hadronic interactions, strangeness flavor is conserved, and
prior to any weak interaction decays, we have $\bar s=s$ --- when we refer to strangeness
yield, production, etc, we always address yield, production, etc, of strange quark pairs.
This study is addressing strangeness under the physical conditions
achieved, at RHIC, at the highest attainable reaction
energy today, $\sqrt{s_{\rm NN}}=200$ GeV, and in future at LHC.
We are particularly interested in the sensitivity of strangeness production
to the nature and properties of the matter formed in the heavy ion reactions.
Theoretical studies have shown that strangeness is produced rapidly
in collisions (fusion)
of thermalized gluons~\cite{Rafelski:1982pu,Letessier:1996ad},
within the deconfined state,
the quark--gluon plasma (QGP) formed in the central collisions of
heavy nuclei.
On the scale of RHIC reaction time $\tau<10$ fm, the
hadron based reactions were found
to be too slow to allow copious
strangeness production after thermalization
of matter and are even more ineffective
to produce multi-strange hadrons~\cite{Koch:1984tz}.
On the other hand, strangeness can be produced fast in the QGP phase, as we shall see
achieving near chemical equilibrium in QGP phase formed at RHIC and even overshooting
the chemical equilibrium at hadronization at
the LHC. Thus, the situation is quite different when the deconfined QGP state breaks up
in a fast hadronization process: the enhancement of strange hadrons and most specifically
strange antibaryons, growing
with valance strange quark content of hadrons produced is the
predicted characteristic property of the
deconfined QGP phase~\cite{Koch:1986ud}. This happens since
in the breakup of the strangeness rich deconfined state, {\it i.e.}, hadronization, several
strange quarks formed in prior, and independent, reactions can combine into a multistrange hadron.
Our main objective in this work is to quantify the
mechanisms of kinetic strangeness production
occurring in thermal gluon collision (fusion) processes
during the expansion phase of the quark--gluon fireball.
We explore the centrality dependence
in a wide range between peripheral, and most central reactions,
in which up to 90\% of projectile and target nucleons participate. In this regard,
this paper is a theoretical companion to the more phenomenological analysis of
experimental RHIC data~\cite{Rafelski:2004dp,Letessier:2005kc}, and uses the insights gained
in this analysis, in particular, regarding the dynamics of the QGP expansion. This is then
applied to extrapolate our approach to the LHC energy domain, where our prior particle
yield study was based on a parametric consideration of final state strangeness
yield~\cite{Rafelski:2005jc}.
We also study in depth the centrality dependence of strangeness
production in the RHIC-LHC energy range. As the centrality of the nuclear reaction
and the number of participants $A$ decreases, the number of thermal
collisions gradually diminishes, and with it the strangeness enhancement
effect also diminishes gradually, similarly to the AGS-SPS energy range~\cite{Letessier:1996ad,ActaB}.
This decrease drives in turn a gradual decrease in the centrality dependent
production rate of multistrange hadrons.
This behavior we discuss in detail here
is an important and characteristic phenomenological feature
of the kinetic particle collision mechanism
of strangeness production and enhancement.
Interestingly, in this regard the
kinetic mechanism of strangeness production differs from
models, such as `canonical enhancement model', which are deriving the strange hadron
enhancement as a result of an always prevailing hadronic phase chemical
equilibrium~\cite{Koch:1982ij,Braun-Munzinger:2003zd}. The volume
dependence of the canonical phase space yields~\cite{Rafelski:1980gk}, and the
smallness of the N--N reference systems produce the centrality
enhancement effect~\cite{Hamieh:2000tk}. However, `canonical enhancement' is rising very
rapidly considering rather small collision systems, and with decreasing energy~\cite{Redlich:2001kb}.
The understanding of strangeness production
during the expansion phase of the quark--gluon fireball allows us to
study in depth the reaction mechanisms which are determining
the final state yield of strangeness. In this way, we learn how
`deep' into the history of QGP expansion this observable allows us
to look. Collective matter flow features observed
at RHIC suggest that thermalization of parton matter occurred
very fast, {\it i.e.}, the entropy $S$ has been produced in a not yet fully understood
fast process of parton thermalization, prior to the production
of strangeness pair yield $s$. We formulate the
kinetic equations allowing to address, in some detail, the growth in specific strangeness
per (fixed) entropy $s/S$ in the thermal QGP processes in both a longitudinally and
transversely expanding QGP fireball.
The time evolution of
$s/S$ has been considered for the first time early on in the
development of the QGP physics~\cite{Kapusta:1986cb}. However, the model of
dense matter evolution, and the range of statistical parameters considered at the time
is not appropriate for the RHIC and LHC physics environments. Moreover, we recompute here
the rate of strangeness production knowing the best current values of QCD parameters,
the coupling constant $\alpha_s$, and the strange quark mass $m_s$. These parameters
alter decisively the values of $s/S$ in chemical equilibrium, and thus the dynamical time
scale of the approach to chemical equilibrium. In order to
be able to compute the evolution in time of $s/S$,
we must also evaluate how near to chemical equilibrium is
strangeness in QGP: this nearness is characterized by a parameter $\gamma_s$, roughly
the ratio of prevailing strangeness density to chemical equilibrium density at
prevailing temperature.
Setting up the production of strangeness, we assume here that it follows in time the
chemical equilibration of the light $q=u,\,d$ quarks, and $g$ gluons~\cite{Alam:1994sc},
which are believed to occur at 1 fm scale. These effectively massless particles
can be produced by entirely soft processes which are intrinsically
non-perturbative~\cite{Geiger:1992si}. Their chemical equilibration can
be further driven by multi-particle
collisions~\cite{Xiong:1992cu,Xu:2004mz}.
Considering these studies, we assume here relatively short relaxation times for
$q,\,g$, we cannot compute these using the same perturbative method as will be developed
here to evaluate strangeness yield equilibration. It is the finite strangeness mass combined
with the measured strength of the running QCD coupling constant $\alpha_s$ which allows
us the use of perturbative formalism in study of strangeness
production~\cite{Rafelski:2001kc} with some minimal confidence.
The chemical relaxation times for the
strangeness approach to chemical equilibrium, in an expanding QGP,
has been considered several times
before~\cite{Koch:1986hf,Matsui:1985eu,Biro:1993qt,Letessier:1996ad,Rafelski:1999gq,Pal:2001fz,He:2004df}.
The study of QGP strangeness chemical equilibration must not be confused with the
phenomenological investigation of chemical equilibrium in the final state hadron
abundance. Because hadron phase space is generally smaller, chemical equilibrium and indeed
excess over equilibrium is much more easy to attain, and there is some continuing
discussion of this question~\cite{Letessier:1998ca,Becattini:2003wp,Torrieri:2005va}.
We offer in our work a
comprehensive exploration how the impact parameter dependence, and
consideration of energy dependence, influences chemical equilibration in QGP.
We evolve in time not the strangeness itself, but the specific strangeness
per entropy $s/S$. In this way, we can identify
more clearly the production processes of strangeness,
since entropy is produced earlier on, and is (nearly) conserved
during the time period of thermal strangeness production.
Importantly, $s/S$ is an experimental observable, practically
preserved in the fast hadronization process.
Thus, we can connect the final state of the
QGP evolution directly to experimental soft hadron yield experimental
results. We find that much of the variability about
the initial conditions, such as dependence on initial temperature, cancels in the
final result. This specific observable will be shown to yield nearly model independent
insights about thermal strangeness production in QGP.
In the following section \ref{over}, we begin with a brief discussion of
general features relevant in all considerations presented.
In subsection \ref{thermal}, we formulate the kinetic equations describing the
growth in specific strangeness per entropy $s/S$ and show
that, at RHIC, the observed specific per entropy
strangeness yield suggests that the
`direct' and `thermal' processes are of comparable strength in
most central reactions. We discuss the magnitude and importance
of QCD parameters in subsection \ref{QCDinput}.
In order to integrate as function of reaction time the strangeness yield equations,
we develop a simple collective expansion model of the plasma phase
in subsection \ref{expand}.
In section \ref{ress}, we study the thermal strangeness production processes.
We then obtain reference yields for two
different expansion geometries at RHIC in subsection \ref{bench}. We
extrapolate this to the LHC environment in subsection \ref{LHC}.
We explore how `deep' into the early history of the
hot and dense fireball the strangeness signature of thermal QGP
is allowing us to look, {\it i.e.}, the dependence on initial
conditions, in subsection \ref{initial} -- importantly we find that the selection of the initial
value of $s/S$ related to direct production of strangeness, has only minor impact on the
final results regarding strangeness yield. We then explore the influence of fundamental
uncertainties, such as the present day
limited knowledge about the strange quark mass and
the QCD thermal effects on the freezing of the strange quark degrees of freedom,
in subsection \ref{Fund}.
In section \ref{eval}, we connect the results we obtained to the experimental particle
yields. In subsection \ref{sh}, we consider the relationship between hadron multiplicity
and entropy yield, and obtain the strange hadron yield as function of $s/S$. This
allows an assessment how strangeness yield, at RHIC and LHC, influence physical observables.
In particular, we discuss how K$^+/\pi^+$ changes between these two experimental
environments. Then, we
discuss, in subsection \ref{PY}, for the two most often used statistical hadronization models
(sudden hadronization and chemical equilibrium hadronic gas (HG) hadronization) the production
of strange hadrons as function of participant number $A$, keeping the hadronization
condition independent of $A$. In subsection \ref{thc},
we apply the insights gained in study of thermal strangeness production to evaluate
thermal charm production at the RHIC and the LHC environments.
\section{Remarks about Strangeness Production and Density }\label{over}
\subsection{Parton equilibration and strangeness production}
The total final state hadron multiplicity is a measure of the entropy $S$ produced
prior to thermal production of strangeness $s$ in QGP:
once a quasi-thermal exponential energy distribution of partons
has been formed, the entropy production has been completed.
Further evolution of the dense deconfined fireball is nearly
entropy conserving, even though it is strangeness flavor producing:
fusion of gluons, or light quark pair annihilation into strangeness, is
a nearly entropy conserving process~\cite{Elze:2001ss}. We note that,
in reactions between two thermal particles into a strangeness pair, the energy
content of each initial state parton is transferred to the two reaction products,
so thermal partons produce thermal shape of strangeness spectrum.
The entropy produced, in RHIC reactions, has been evaluated in recent studies of
hadron multiplicities. In Au--Au reactions, at $\sqrt{s_{\rm NN}}=200$ GeV,
one sees $S\simeq 35,000$. Furthermore, at central rapidity, the yield
of entropy is $dS/dy\simeq 5000$. In the benchmark results we present
for LHC, we will assume that the the central
rapidity entropy yield is about 4 times greater than at RHIC.
The temporal evolution of the QGP fireball ends
when the temperature has decreased to the
QGP hadronization value. In the breakup of the QGP, the
yields of hadrons are established, and it is rather difficult in
the ensuing rather short lived evolution lasting not more than
1.5 fm/$c$ to alter these yields appreciably. Thus,
the hadronization volume, with the normalizing factor $dV/dy$,
provides the normalization of hadron particle
yields per unit of rapidity. The final value of $dV(\tau_f)/dy$ is result of
analysis of hadron particle yields and our model of the time
dependence of $dV(\tau)/dy$ will be constrained by the
magnitude of $dV_f/dy$ obtained in Ref.~\cite{Rafelski:2004dp}.
There are two separate stages of
strangeness (charm) production, corresponding to the two practically distinct periods
of the fireball evolution:\\
a) `direct' production
creates a `background' yield corresponding to what might be obtained in
a superposition model of independent nucleon--nucleon (N--N) reactions. \\
b) strangeness production in collisions between thermally equilibrated QGP
constituents.
For b) to be relevant prior formation of a thermal,
deconfined QGP phase is required. Without process b),
the yield of strangeness should not be enhanced in A--A collisions
as compared to scaled N--N reactions. We will not discuss in detail mechanisms a)
of direct particle production here, our interest is restricted to
(approximate) initial yields, which are the baseline for the thermal
mechanisms acting in the QGP, and define the strength of any
enhancement.
We note for the record, that
strangeness initial production, like other relatively soft parton
production processes, are believed to be due to
color string breaking mechanism~\cite{Sjostrand:2003wg} and the
Pythia 6 model of soft hadron production presumes that the relative
strength of $u:d:s$ production is $1:1:0.3$. The
(initial) charm production is due to
high energy parton collisions~\cite{Bedjidian:2003gd}.
Our study of the thermal strange particle production processes
is based on kinetic theory of particle collisions.
There is considerable uncertainty about the initial momentum
distributions of soft partons present in the initial state. However,
two recent theoretical studies argue that there is rapid thermalization.
The nonlinear gluon production
processes leads to the gluon momentum
distribution equilibration~\cite{Xu:2004mz}.
Axial asymmetry of the initial state causes collective
instabilities which further accelerate thermalization
of partons~\cite{Mrowczynski:2005ki}.
In this context, it is important to advance one result of our study, namely that
there is little sensitivity to the initial
thermal condition: a wide range of `reasonable' initial temperatures
leading to very similar strangeness production
results, as long as the entropy content is
preserved. In order to understand this, consider a
decrease in initial temperature. This requires, at fixed entropy, an increase
in initial volume, and this, then, is associated with increased lifespan
of the fireball in the high temperature strangeness producing domain. These
two effects combine to compensate the reduced strangeness production
rate per unit of time and volume that is associated with reduced
ambient temperature.
We believe that this mechanism
also implies that the precise form of the momentum
spectrum of the initial state partons is of minor practical
relevance for the purpose of evaluation of strangeness production,
and we do not study this.
Therefore, without loss of generality, we can assume that the parton
distributions we use in the kinetic strangeness formation process have
thermal shape, and the ambient temperature is determined considering
(lattice fitted) equations of state relation of initial temperature
and entropy density~\cite{Letessier:2003uj}, for a given geometric initial volume.
We already remarked above that the production of strangeness,
in a cascade of N--N reactions (without deconfinement), is
not able to add significantly to the initial strangeness yield considering the
short lifespan of the fireball. Thus, this alternative will not
be further considered in this work. Similarly, any additional strangeness produced in
the rapid hadronization of QGP into hadrons must be negligible compared to the
thermal production process which occurs at higher particle density (temperature), and
during a considerably longer lifespan.
\subsection{Approach to chemical yield equilibrium}\label{chemapp}
It has been shown, considering the entropy maximization principle,
that the approach of particle densities to chemical
equilibrium density $\rho_i^\infty$ can be characterized by the statistical parameter
$\gamma_i$~\cite{Letessier:1993qa}, which varies with the local proper time $\tau$
during the collision. For example, for gluons,
\begin{equation}\label{gamma}
\rho_{\rm g}(\tau)\equiv
\int d^3p \frac{\gamma_{\rm g}(\tau) e^{-E/T}}{1-\gamma_{\rm g}(\tau) e^{-E/T}},
\quad E=\sqrt{m^2+p^2}.
\end{equation}
Generally, the Lagrangian mass of gluons is zero. However, one may
be tempted to think that thermal mass $m(T)$ could change decisively
results, suppressing the collisional strangeness production. However,
one finds that instead the process of gluon decay becomes relevant
and if at all, there is a net rate increase of strangeness production~\cite{Biro:1990vj}.
One could argue that the scheme to study kinetic process of strangeness chemical
equilibration using
thermal mass amounts to a different resummation of reaction processes.
In this work, we will consider the evolution of $\gamma_{s}^{\rm QGP}(\tau)$
based on Lagrangian masses, allowing for $\gamma_{\rm g}^{\rm QGP}(\tau) $
and $\gamma_{q}^{\rm QGP}(\tau) $.
For strange quarks, we will keep only the Boltzmann term
(ignoring the denominator in Eq.\,(\ref{gamma}) above) and thus:
\begin{equation}\label{sdens}
\rho_s(\tau)\equiv \gamma_s^{\rm QGP}(\tau)\rho_s^\infty
=\gamma_s^{\rm QGP}\,{g_s\over 2\pi^2}z^2K_2(z),\ z=\frac{m_s}{ T}.
\end{equation}
Here, $g_s$ is the strange quark degeneracy, and $K_2$ is a Bessel function.
Since we will employ strangeness occupancy in hadron phase, $\gamma_s^{\rm h}$,
we have included the subscript {\small QGP}.
We will henceforth drop this subscript, and occupancy parameters without
superscript will, in general, refer to the QGP phase, while the hadronic gas
phase variable, when these are expected to differ from QGP, will have a superscript $h$.
Our target variable is the final QGP state specific yield of strangeness per entropy,
$s/S$, and the related phase space occupancy $\gamma_s$. Both these
variables have an important
physical relevance: $s/S$ determines the final yield of strange hadrons
compared to all hadrons, and its value implies some particular yield of
reference yields, such as, {\it e.g.}, K$^+/\pi^+$. $\gamma_s$ characterizes the
approach to chemical equilibrium, it measures
strangeness yield in terms of the chemical equilibrium yield.
The strangeness phase space of QGP and HG phases are different.
Strangeness in QGP is much denser than in the HG phase,
considering the range of strange quark masses, $0.080<m_s(\mu=2\, {\rm GeV})<0.125$ GeV.
Therefore, (near) chemical strangeness
equilibrium in the QGP phase $\gamma_s\simeq 1$, implies
a significantly oversaturated hadron phase space abundance
$\gamma_s^{\rm h}>1$ after hadronization. $\gamma_s^{\rm h}$ is directly
controlling the relative yields of hadrons with different $s+\bar s$
valance quark content and is thus observable.
\subsection{Role of initial conditions}\label{Rini}
We do not understand well the conditions in the QGP phase at time as early as
$\tau_0=0.25$ fm for RHIC, and $\tau_0=0.1$ fm
for LHC, when we presume that the thermal momentum
distribution is practically established. Thus, we must make
a number of assumptions and check if these impact our results.
The relevant parameters that could govern the strangeness production are:\\
$\gamma_{\rm g}(\tau)$ and, in particular, the initial value at $\tau_0$;\\
$\tau_0$, the time at which we assume thermal momentum of partons is reached;\\
$s/S|_{\tau_0}$ is the initial strangeness yield originating in direct parton collisions; \\
$R_\bot$ is the transverse radius dimension at initial time, related to the collision
geometry;\\
$v_\bot(\tau)$ is the transverse expansion velocity, and
in particular its maximum value at hadronization; \\
$\tau_{\rm g}$ and $\tau_q$, the relaxation time constant of gluon and quark fugacities,
considering that quarks are less relevant compared to gluons with regard
to strangeness production, we will assume $\tau_q$=1.5 $\tau_{\rm g}$ throughout this work.
We explore, here, a characteristic gluon thermalization
time $0.1< \tau_0 <1.5$ fm/$c$, with the longest
value applicable to most peripheral RHIC Au--Au reactions at $\sqrt{s_{\rm NN}}=200$ GeV,
and the shortest period assumed for the future LHC
central collisions. Knowing the exact dynamics of thermalization and how
long it takes will be, as we shall see, rather unimportant for the final insights we
obtain.
All initial state parameters are constrained in their value, either by
collision geometry, by final state particle yields observed at RHIC, or/and
by particle correlations. For example, the final yield of strangeness $ds/dy$
and of entropy $dS/dy$, and thus $s/S$ are known from an analysis of
particle production: as function of
centrality at RHIC~\cite{Rafelski:2004dp}, and as function of reaction energy
from top AGS, SPS to top RHIC energy~\cite{Letessier:2005qe}.
We will use these results in two ways. We compute, following the temporal
evolution, the final state $s/S$ ratio which we expect to converge at RHIC to
$s/S\simeq 0.033$. We need to specify the initial value at time $\tau_0$ for
this variable, and this value is chosen to be compatible with the peripheral
reactions. The entropy yield $dS/dy$, which we assume is
conserved during the evolution of QGP, determines, for a
known initial volume $dV(\tau=\tau_0)/dy$, the entropy density $\sigma=(dS/dy)/(dV/dy)$.
We then can obtain, from standard properties of QGP fitted to the lattice
results, the initial temperature $T_0/T_c\simeq 3$--4. This temperature decreases
as volume expands with $\tau$ given that the entropy is preserved.
We will show
that the two physical observables, $s/S$ and $\gamma_s$, we address
are largely independent of the model dependent details
of the initial conditions. Said differently, our important finding is
that the two global strangeness observables $s/S$ and $\gamma_s$ appear
to penetrate back only to about 2 fm/$c$ after the reaction has begun, and do not probe
earlier conditions in the QGP phase. The physical reason for this is,
of course, that once chemical
equilibrium is approached, one looses much of the event memory with regard to
intensive physical observables. We will further see that, in cases we studied,
that did not quite reach chemical equilibrium in the
QGP phase, this is also true, {\it i.e.}, there is little sensitivity
to what exactly happened to light quarks and gluons
in the first 2 fm/s.
The reason for this is that there is a strong correlation
between volume, temperature and degree of QGP (gluon and light quark)
chemical equilibration as we already discussed above. Repeating the argument
differently, we can say that when fewer gluons at fixed
entropy are in given volume, temperature has to be larger. Thus, any
decrease in the production of strangeness in gluon fusion due to absence of
gluons is compensated by the greater specific rate per colliding pair due
to greater ambient temperature. Hence, also when we do not quite reach chemical
equilibrium in QGP, be it due to large impact parameter or low reaction
energy (chemical nonequilibrium at lower energies is
not explored in this paper), there is little if any dependence
of thermal yield on initial conditions, and the results we arrive at regarding
near chemical equilibration are extraordinarily robust.
However, the thermal strangeness production does depend
on the degree of initial state strangeness equilibration,
simply because, if the initial yields were chemically equilibrated to start with,
there would be as much production as annihilation of strangeness and any
temporal evolution is driven by the time dependence of the evolution dynamics.
We will take as a measure of the pre-QGP thermal
phase strangeness production the specific
per hadron multiplicity yield of strangeness observed in
most peripheral RHIC reactions.
This is typically $s/S\simeq 0.016$ at $\tau=\tau_0$. This choice
allows to reproduce
the observed value $s/S=0.019$ attained in most peripheral nuclear
reactions at RHIC~\cite{Rafelski:2004dp},
with participant number about $\langle A\rangle =6.3$.
Clearly, with $s/S\to 0.033$ in most central collisions, the implication
of this choice is that the thermal process enhances total specific
yield by factor $1.9\pm 0.3$. As centrality and/or reaction energy
decreases, there is a gradual decrease of this enhancement, and
as the energy is increased (LHC) this enhancement rises somewhat.
\subsection{Strangeness production in thermal collisions}\label{thermal}
We follow the established
methods of evaluating thermal strangeness production~\cite{Letessier:2002gp}, expanding
our earlier more schematic model~\cite{Rafelski:1999gq}.
However, considerable simplification arises since we
focus attention on the specific yield of strangeness per entropy.
In the local (comoving) frame of reference, the rate
of change of strangeness is due to production
and annihilation reactions only:
\begin{eqnarray}
\frac{1}{V} {{d s}\over {d \tau}}=\frac{1}{V}{{d {\bar s}}\over {d \tau} }
&=&
\frac12 \rho_{\rm g}^2(t)\,\langle\sigma v \rangle_T^{gg\to s\bar s}
+
\rho_{q}(t)\rho_{\bar q}(t)
\langle\sigma \rangle_T^{q\bar q\to s\bar s}\nonumber \\ \label{qprod}
&&-
\rho_{s}(t)\,\rho_{\bar{\rm s}}(t)\,
\langle\sigma v\rangle_T^{s\bar s\to gg,q\bar q}.
\end{eqnarray}
The thermally average cross sections
are:
\begin{equation}\label{Tsig}
\langle\sigma v_{\rm rel}\rangle_T\equiv
\frac{\int d^3p_1\int d^3p_2 \sigma_{12} v_{12}f(\vec p_1,T)f(\vec p_2,T)}
{\int d^3p_1\int d^3p_2 f(\vec p_1,T)f(\vec p_2,T)}\,.
\end{equation}
$f(\vec p_i,T)$ are the relativistic Boltzmann/J\"uttner
distributions of two colliding particles $i=1,2$ of momentum $p_i$,
characterized by local statistical parameters.
A convenient way to address the dilution phenomena acting on the
density of strangeness $\rho_s\equiv s/V$ due to rapid expansion of
the QGP phase, Eq.\,(\ref{qprod}), is to consider the proper time evolution
of the specific strangeness per entropy yield:
\begin{equation}\label{qprod3}
{d\over d\tau} {s\over S}=\frac VS \ \frac1V\ {d {s}\over d \tau}.
\end{equation}
The entropy $S$
in a volume element is unchanged, as volume grows and temperature drops:
\begin{equation}\label{S1}
S=V{4\pi^2\over 90} g(T)T^3={\rm Const.},
\end{equation}
where we consider the quark and gluon degrees of freedom along with their
QCD corrections:
\begin{eqnarray}\label{ggq}
g&=&2_s8_c\left(1-\frac{ 15\alpha_s(T)}{4\pi}+\ldots\right)\nonumber\\
&&+\frac74 2_s3_cn_{\rm f} \left(1-\frac{ 50\alpha_s(T)}{21\pi}+\ldots\right).
\end{eqnarray}
We use as the number of quark flavors
$n_{\rm f}\simeq 2+ \gamma_s0.5 z^2\,K_2(z)$, where $z=m_s/T$.
The terms proportional to chemical potentials
are not shown in the expression for entropy, since $\mu/\pi T\ll 1$ at RHIC and LHC.
We have used, here, the lowest order QCD corrections to the effective degeneracies,
since these describe well the properties of QGP
phase obtained on the lattice~\cite{Letessier:2003uj}, when
the value of $\alpha_s(T)$ used is as described below, see Eq.\,(\ref{alfaseq}).
The agreement one sees for thermodynamic variables, such as $E,\,P,\,S$,
with the lattice results is very remarkable, including the temperature range near
to the phase boundary. Thus, the use of constraint Eq.\,(\ref{S1})
to evaluate the time dependence of temperature, considering also that the
third root of entropy is considered, should yield precise enough results.
We note, in passing, that we used as specified in Ref.~\cite{Letessier:2003uj}
the additional terms ${\cal A}$ in entropy, arising from differentiation
of the implicit temperature dependence of $g$, Eq.\,(\ref{ggq}) entering the
partition function.
In order to use the detailed balance
which relates production and annihilation reactions,
it is convenient to introduce the invariant rate per unit time and volume,
$A^{12\to 34}$, by incorporating the equilibrium densities into the thermally
averaged cross sections:
\begin{equation}
A^{12\to 34}\equiv\frac1{1+\delta_{1,2}}
\gamma_1\gamma_2 \rho_1^\infty\rho_2^\infty
\langle \sigma_{s} v_{12} \rangle_T^{12\to 34}.
\end{equation}
$\delta_{1,2}=1$ for the reacting particles being identical bosons,
and otherwise, $\delta_{1,2}=0$. Note also that
the evolution for $s$ and $\bar s$ in proper time of the
comoving volume element is identical as both change in pairs.
We find that the temporal evolution of $s/S$, in an expanding plasma,
is governed by:
\begin{eqnarray}\label{qprod3a}
{d\over { d\tau}} {s\over S}
&=&
{A^{gg\to s\bar s}\over (S/V) }
\left[\gamma_{\rm g}^2(\tau)-\gamma_{s}^2(\tau)\right] \nonumber\\
&&+
{A^{q\bar q\to s\bar s}\over (S/V) }
\left[\gamma_q^2(\tau)-\gamma_{ s}^2(\tau)\right]\,.
\end{eqnarray
When all $\gamma_i\to 1$, the Boltzmann collision term vanishes, and
equilibrium has been reached. The value arrived at for the observable $s/S$ depends
on the history of how the system evolves and, eventually, reaches equilibrium.
In order to be able to solve Eq.\,(\ref{qprod3a}), we need a relation
between $s/S$ and $ \gamma_{ s}$. This is obtained combining strangeness
density Eq.\,(\ref{sdens}) and entropy Eq.\,(\ref{S1}):
\begin{equation}\label{sS1}
\frac sS=\gamma_s {g_s \over g} \frac{90}{8\pi^4}z^2K_2(z),\quad z=m_s/T.
\end{equation}
In the initial period, gluons and quarks have not reached chemical equilibrium,
thus the actual
numerical integrals of Bose and Fermi distributions of the type Eq.\,(\ref{gamma}),
dependent on the values $\gamma_{q,{\rm g}}$ are employed instead, which modifies
the result seen in Eq.\,(\ref{sS1}).
The degeneracies we have considered in Eq.\,(\ref{S1})
for the entropy did include the effect of interactions, and
thus, we have to allow for the interaction effect in the
strange quark degeneracy as well:
\begin{equation}\label{gs}
g_s=2_s3_c\left(1-\frac{ k\alpha_s(T)}{\pi}+\ldots\right).
\end{equation}
The value of $k=2$ applies to massless strange quarks. At $T=0$
(or said differently, for $m\gg T$) the early study of quark matter
self-energy suggests that $k\to 0$~\cite{Chin:1979yb}. We will
present, in figure \ref{alfdep} below, results for varying the
value of $k$, and the reference value we use in our other studies
is $k=1$. We believe that this approach allows us to explore the
general behavior of the interactions effect on the strangeness
density, a more detailed study is not possible today.
\subsection{QCD parameters} \label{QCDinput}
We evaluate $A^{gg\to s\bar s}$ and $A^{q\bar q\to s\bar s}$ employing the available
strength of the QCD coupling, and range of accepted strange
quark masses. In our evaluation of strangeness production, in order
to account for higher order effects in quark and gluon fusion
reactions, we introduce a multiplicative $K=1.7$-factor.
The known properties of QCD strongly constrain our
results~\cite{Letessier:1996ad}. However, it turns out that the range
of strange quark masses remains sufficiently wide to impact the results
and we discuss this further in subsection~\ref{Fund}, see figure \ref{massdep} below.
We employ when not otherwise stated the central value from a recent PDG evaluation,
$m_s(\mu=2\,{\rm GeV})=0.10$ GeV which remains uncertain at the level of 25\%
at least. In fact, since our results were obtained, a more recent Particle Data Group study
recommends a 10\% smaller central value of $m_s$~\cite{Yao:2006px}.
We compute rate of reactions employing a running strange quark mass working
in two loops, and using as the energy scale the CM-reaction energy
$\mu\simeq \sqrt{s}$. Since the running of mass involves
a multiplicative factor, the uncertainty in the mass value discussed above
is the same for all values of $\mu$. Some simplification is further
achieved by taking at temperature $T$ the value $\mu\simeq 2\pi T$ which
is the preferred value of the thermal field theory, and agrees with
the value of the reaction energy. This means that we use $m_s(T)=m_s(\mu=2\pi T)$
with $m_s(T=318\,{\rm MeV})=0.1\,$GeV. The actual temperature $T(t)$ and thus time
dependent values of the strange (and charm) quark mas will be always shown in the
top panel of figures describing the evolution of the properties of the system
considered.
The strength of the QCD couping constant
is today much better understood. We use as reference value
$\alpha_s(\mu=m_{Z^0})=0.118$,
and evolve the value to applicable energy domain $\mu$ by using two loops.
Our ability to use perturbative methods of QCD to describe strangeness production,
a relatively soft process,
derives from two circumstances:\\
a) the reaction processes which change yield of strangeness
can compete with the fast $v_\bot>0.5c$ expansion of QGP only for
$T>220$ MeV, for lower temperatures the strange quark yields effectively
do not change (strange quark chemical freeze-out temperature in QGP).
Using the relation $\mu=2\pi T$, this implies that all
strangeness yield evolution occurs for $ \mu > 1.4 $\,GeV.\\
b) Because of the magnitude $\alpha_s(\mu=m_{Z^0})=0.118$, one can
quite well run $\alpha_s$ to the scale of interest, $\mu>1.2GeV$.\\
As we see in figure \ref{alfas}, this means that the strength of the interaction
remains $\alpha_s<0.5$. We also note that had the strength of $\alpha_s(\mu=m_{Z^0})$
been 15\% greater, strangeness production could not be studied in perturbative
approach.
\begin{figure}[t]
\hspace*{.2cm}
\psfig{width=8.5cm, figure=ALSMU4BB6JR.ps}
\caption{\label{alfas}
The running QCD couping constant $\alpha_s(\mu)$
fixed to $\alpha_s(\mu=m_{Z^0})=0.118$
(solid line) and several alternative strength scenarios excluded today
by the experimental measurement (dashed lines).
}
\end{figure}
We next express
$\alpha_s(\mu)$ (solid line in figure \ref{alfas}) as function of
temperature by the conditions $\alpha_s(T)=\alpha_s(\mu=2\pi T)$.
This leads to the expression (see also section 14 in~\cite{Letessier:2002gp}):
\begin{equation}\label{alfaseq}
\alpha_s(T)\simeq {\alpha_s(T_c)\over 1+C\ln (T/T_c)},\quad T<6 T_c,
\end{equation}
with $C=0.760\pm0.002$, $\alpha_s(T_c)=0.50\pm0.04$ at $T_c=0.16 $ GeV.
We stress that Eq.\,(\ref{alfaseq}) is a parametrization
corresponding to the result shown in figure \ref{alfas}. Only one logarithm
needs to be used to describe the two loop
running with sufficient precision, since the range we consider
is rather limited, $0.9T_c<T<6T_c$.
\subsection{Expansion and cooling of QGP} \label{expand}
We separate, in our work, the issue of strangeness production from
the even more complex and less understood questions about the
time evolution of the QGP. We assume that there is some active
volume at average temperature $T$, in which the strangeness is `cooked'.
We derive the time dependence of local temperature from the
hypothesis of a conserved entropy content and a reasonable model
describing the volume evolution in time. This is arrived at using
a hydrodynamically inspired model.
The volume at hadronization is
an implicit observable. All particle yields at hadronization are normalized
with a volume factor. Thus, our expansion model must be realistic
enough so that the hadronization conditions are in agreement
with data, and that the impact parameter dependence is
reproduced. In the geometry inspired model we consider, in
the central rapidity domain:
\begin{equation}\label{volmod}
{dV\over dy}= A_\bot(\tau) \left.{dz\over dy}\right\vert_{\tau={\rm Const.}}.
\end{equation}
$dV/dy$ is the normalization factor for the particle yields we are measuring
in an interval around central rapidity. The transverse expansion is
described by the transverse size $A_\bot(\tau)$. We further need to associate with
the domain of rapidity $dy$ a geometric region at the
source $dz$, from which particles emerge.
To accomplish this, we recall the space-time rapidity of
the scaling Bj\o rken hydrodynamical solution:
\begin{equation}\label{ybj}
y=\frac12 \ln {t+z\over t-z} .
\end{equation}
We see that $y$=0 corresponds to $z=0$.
In particular, if the transverse extend of the fireball is large, the Bj\o rken
space-time rapidity relation prevails.
We need this relation not at
a fixed laboratory time $t$ but at some fixed proper time $\tau$:
\begin{equation}\label{tau}
\tau=\sqrt{t^2- z^2 }.
\end{equation}
We eliminate in Eq.\,(\ref{ybj}) $t$ using Eq.\,(\ref{tau}):
\begin{equation}\label{zy}
z=\tau \sinh y,\quad {dz\over dy} = \tau \cosh y .
\end{equation}
$A_\bot$ is the transverse to scattering axes size of the evolving
QGP. For nearly homogeneous expanding bulk matter one can assume:
\begin{equation}
A_\bot =\pi R_\bot^2(\tau).
\end{equation}
However, if the matter is predominantly concentrated near
a narrow domain of width $d$, we
consider:
\begin{eqnarray}
A_\bot
&=&\pi \left[R_\bot^2(\tau)-(R_\bot^2(\tau)-d)^2\right],\nonumber\\
&=&2\pi d \left[R_\bot (\tau)-\frac d2\right] .\label{Adonut}
\end{eqnarray}
At central rapidity, we consider quantitatively the two
evolution scenarios, denoted here-forth as models V1 and V2. V1 will be
the most simple bulk homogeneous expansion while V2 simulates a
transverse donut, it corresponds to expansion with a cold
hole of matter in fireball (axial) center:
\begin{eqnarray}\label{V1}
&&{\bf\rm V1:}\quad {dV\over dy}= \pi R^2_\bot(\tau) \tau,\\
\label{V2}
&&{\bf\rm V2:}\quad {dV\over dy}= 2\pi d \left[R_\bot (\tau)-\frac d2\right] \tau,
\end{eqnarray}
with
\begin{equation}
R_\bot(\tau)=R_0+\int v(\tau) d\tau .
\end{equation}
Any model of transverse matter expansion dynamics $v(\tau)$
is constrained by the transverse mass shape of produced particle spectra,
too large transverse velocities would produce too hard spectra.
Accordingly, a hydro-inspired shape is assumed:
\begin{equation}\label{vexp}
v(\tau)= v_{\rm max} {2\over \pi} \arctan [4 (\tau-\tau_0)/\tau_v].
\end{equation}
Values of $v_{\rm max}$ we consider are in the range of 0.5--0.8$c$, the relaxation time
$\tau_c\simeq 0.5$ fm, and the onset of transverse expansion $\tau_0$
was tried in range 0.1--1 fm. None of these parameters matters for what
follows as long as one does not employ aberrant values.
The initial size $R_\bot$ is assumed, in what follows,
to be $R_\bot=5$\,fm for 5\% most central
collisions. When we study centrality dependence,
we will show results
for a series of centralities decreasing the transverse dimensions
$R_\bot$ and $d$ by factor $f_R=1.5$ in each step. We further scale entropy value
with $f_S=f_R^{2.2}$. This assures that
the dependence of entropy on the participant number $dS(A)/dy$ in
the final state follows the relationship,
\begin{equation}\label{entsca}
\frac{dS}{dy}\simeq 8 (A^{1.1}-1),
\end{equation}
obtained from the impact parameter dependent fit
to the RHIC impact parameter results~\cite{Rafelski:2004dp}.
\begin{figure}[t]
\vspace*{-0.3cm}
\psfig{width=8cm,height=8.5cm,figure=PLETRTAURHICLHCVOL.ps}
\vspace*{-0.6cm}
\caption{\label{Volume}
(Color online) QGP Volume related to central rapidity, $dV/dy$
as function of proper time $\tau$.
Top panel is for RHIC
with reference entropy content $dS/dy=5,000$ (central lines),
while bottom panel is for LHC with
4 times greater entropy content $dS/dy=20,000$ (central lines).
Three centralities are considered, with the middle
thicker lines corresponding to $R_\bot=5$ fm and
the upper/lower lines corresponding to $R_\bot=7 $, and,
respectively, $R_\bot=3 $ fm/$c$.
Solid lines are for
V1 model with transverse homogeneity, dashed lines for
V2 model of a transverse shell with widths (top to bottom)
$d=2.1,\ 3.5$ and 4.9 fm.
The volume expansion is shown in
the figure up to $T=140$ MeV. See text for more details.
}
\end{figure}
An overview of the resulting volume dynamic
behavior is given in figure \ref{Volume}, the top
panel applies to RHIC with $\sqrt{s_{\rm NN}}=200$ GeV,
the bottom panel presents a parallel study for LHC with the
{\it assumed} four times greater entropy content. The
solid lines are for transverse homogeneous volume (V1 model)
expansion, and dashed lines correspond
to a transverse region of thickness $d=3.5$\,fm (V2 model).
Three different centralities
were considered with $R_\bot=3$, 5 and 7 fm. For the second model of transverse
expansion, the transverse size $d$ is scaled with
$R_\bot/5$ fm. Thus, $d=2.1$\,fm for $R_\bot=3$\,fm and
$d=4.9$\,fm for $R_\bot=7$\,fm. Similarly,
entropy content, assumed to be $dS/dy= 5000$ at RHIC and $dS/dy=20,000$ at LHC
for $R_\bot=5$ fm, is scaled to values $dS(R_\bot=3 \,{\rm fm})/dy=1300$ and
$dS(R_\bot=7 \,{\rm fm})/dy=10,500 $, and correspondingly,
4 times greater values for LHC.
The temporal expansion of the volume is followed till
$T=140$ MeV is reached. In general, the maximum volume at LHC is
thus 4 times greater compared to RHIC. The expansion time is correspondingly longer,
with RHIC taking 6.5fm to freeze-out for $R_\bot=5$fm, the LHC lifespan is 10fm.
The QGP lifespan increase by as much as 60\% at LHC, when comparing to RHIC, if
the assumed initial entropy production is indeed increased by factor~4.
From the perspective of strangeness production, this
is one of the more interesting changes comparing RHIC to LHC.
Given the volume as function of $\tau$ and the associated
conserved entropy content, we can evaluate the prevailing temperature
$T$ for any given quark and gluon chemical yield condition
$\gamma_{q,s,{\rm g}}$. The solid lines in top panels
of the following figures \ref{TwoVol} to \ref{massdep} show this result, in the figures
\ref{TwoVol} and \ref{LHCVolVol2}, on
left for the V1 model and on right for the V2 (donut) model.
The assumed $\gamma_{\rm g}$ is presented as dashed line.
In some of the top panels, we also show by the dotted line
the time dependence of the applied transverse
velocity, $v_\bot$, see Eq.\,(\ref{vexp}).
\section{Results on strangeness production}\label{ress}
\subsection{The benchmark results for RHIC}\label{bench}
We present our results for strangeness production in QGP
in Figs. \ref{TwoVol}--\ref{massdep}.
In Figs. \ref{TwoVol} (RHIC) and \ref{LHCVolVol2} (LHC) we show the
centrality dependence, with the two volume models,
see Eqs.\,(\ref{V1} and \ref{V2}), corresponding to columns, with V1 on left
and V2 (donut model) on right. In the following
Figs. \ref{Gluedep}-- \ref{Gluedep} we explore
the dependence on the assumptions made. Here
we show RHIC results on left and LHC results on right.
In all Figs. \ref{TwoVol}--\ref{massdep} we show three panels above each other. As noted already,
we show in the top panel, by solid line(s), the model time-temperature profiles.
The experimental observables are shown as solid line(s)
in the middle panel ($\gamma_s$) and in the bottom panel ($s/S$).
The other lines illustrate as appropriate the key
inputs used to obtain these results. When several lines of the same type
are present, we are presenting the impact parameter dependence, scaling
the size and entropy content as discussed above.
In general, the temperature is followed down to a freeze-out at $T_f=0.14$ GeV.
\begin{figure*}[t]
\vskip -0.5cm
\psfig{width=7.9cm,figure= PLETRTAURHICVOL.ps }
\psfig{width=7.9cm,figure= PLETRTAURHICVOL2.ps }
\vspace*{-0.6cm}
\caption{\label{TwoVol}
(Color online) RHIC results.
Top panel: solid lines: temperature $T$; dashed lines: running mass $m_s^r(T)$;
dotted line: the assumed profile of transverse expansion velocity
$v_\bot(\tau)$. Different lines correspond
to different centralities.
Middle panel: Solid line(s) $\gamma_s$, which nearly coincide
for different centralities; dashed, the assumed $\gamma_{\rm g}(\tau)$,
dotted the profile of the assumed volume,
$[dV(\tau)/dy]/[dV(\tau_f)/dy]$ normalized by the freeze-out value.
$R_\bot(\tau_0)$ stepped down for each line by factor 1.5.
The end points, at maximum
$\tau$, allow to identify corresponding $s/S$ for different centrality in the
bottom panel.
Right and left: Comparison of the two transverse expansion models,
see Eqs.\,(\ref{V1},\ref{V2}): left bulk
expansion (model V1), right donut expansion (model V2).
}
\end{figure*}
In figure \ref{TwoVol}, we show what we believe is
the best $\gamma_s(\tau)$ (solid lines, middle panel)
and $s/S(\tau)$ (solid lines, bottom panel)
for RHIC at 100+100 GeV at varying reaction centrality. The solid lines in the
top panels show $T(\tau)$ for these 6 different centralities, with the lowest
temperatures seen for the least central collisions, all temperatures
continue to $T=0.14$\,GeV.
The slight increase in the initial temperature with increasing
centrality is result of the
scaling of initial entropy, which accommodates the observed change in
$dS/dy\vert_{\rm f}$ beyond participant scaling, see Eq.\,(\ref{entsca}).
For all centralities (and below also for RHIC) we assume the same initial
$s/S(\tau_0)=0.016$.
All lines shown begin at $\tau_0=1/4$ fm, where the
initial temperatures range $T_0\in (0.55 , 0.6)$ GeV.
For the V1 model, the range of $\tau$
spans the interval $\tau_f=2.2$ fm (most peripheral) to
$\tau_f=6.5$ fm (most central). In the donut
expansion model V2, this range is from $\tau_f=3$ to 8 fm.
\begin{figure*}[t]
\vskip -0.5cm
\psfig{width=7.9cm,figure= PLETRTAULHCVOL.ps }
\psfig{width=7.9cm,figure= PLETRTAULHCVOL2.ps }
\vspace*{-0.6cm}
\caption{\label{LHCVolVol2}
(color online) Case of LHC, see legend in figure \ref{TwoVol}.
See text for discussion of differences with RHIC.
}
\end{figure*}
In the top panel, we also show the growth with $\tau$ of the transverse
expansion velocity (dotted line), and the strangeness
pair energy threshold $2m_s^r$ using running strange
quark mass (dashed lines). We note that the temperature
drops below this threshold for the most peripheral reactions
considered already at $\tau=1$ fm/$c$, and this occurs for the most
central reactions at $\tau=2 $fm/$c$, for model V1, and respectively,
1.5 and 2.8 fm/$c$, for model V2. Thus, high strength thermal
strangeness production life span
varies by as much as factor 3, depending on centrality, and
the expansion model.
In the middle panel, we show dashed the rise of the gluon occupancy $\gamma_{\rm g}$
which we employed. The quark occupancy $\gamma_q$ is following the same
functional temporal
evolution starting with 2/3 smaller initial value and evolving
1.5 times slower. Because gluons dominate strangeness
production in QGP, we do not show $\gamma_q$ explicitly.
The dotted lines show
how the volume evolves toward its maximum value at freeze-out. Each line is
normalized to unity at freeze-out. The actual value of the volume can be
read of the figure \ref{Volume}, given the value of $\tau$.
To summarize the key results: we see a gradual increase of strangeness
yield with centrality, reaching near strangeness QGP equilibrium for the
most central collisions at RHIC. We have checked stability of this result
against variation of model assumptions. More detailed discussion
will be presented further below, see e.g. Fig.~\ref{alfdep}.
\subsection{Strangeness production predictions for LHC}\label{LHC}
We performed a similar evaluation of strangeness production at LHC,
see figure~\ref{LHCVolVol2} which follows the same pattern as
figure~\ref{TwoVol}. There are three modifications which
were introduced when we consider LHC:\\
\indent (i) To account
for the greater reaction energy, as already discussed, we
increase the entropy $dS/dy$ by factor 4, which implies an assumed
increase in rapidity density of hadrons by a similar factor. We assume that
in elementary parton interactions the relative strength of strangeness
and non-strange hadron production is unchanged and thus, we keep the
initial relative yield $s/S=0.016$ constant. Given the entropy yield
increase, we implicitly assumed
an increase in initial strangeness yield by a
factor 4 at LHC compared to RHIC.\\
\indent (ii)
In order to accommodate the greater transverse expansion pressure,
we increased the maximum transverse
flow velocity which can now attain $v_\bot=0.80$c (dotted line,
top panel figure \ref{LHCVolVol2}).\\
\indent (iii)
We further assume that thermalization time has dropped
from $\tau_0=1/4$ fm at RHIC to $\tau_0=1/10$ fm at LHC. However,
inspecting the slowly changing initial state evolution, in figure \ref{LHCVolVol2},
there would be little change in our results, were $\tau_0$ to remain
unchanged between RHIC and LHC. At this early time, $\tau_0=1/10$,
the value of $\gamma_s(\tau_0)$ at LHC is similar
to the situation at RHIC, compare the beginning of the solid line in middle panel
of figures \ref{TwoVol} and \ref{LHCVolVol2}.
This is so, since the magnitude of the phase space scales with $T^3$ and the
initial temperature $T(\tau_0)$ is considerably greater at LHC:
in the top panel, in figure \ref{LHCVolVol2}, we see that it
reaches up to $T=1.25$ GeV. For this reason, we have to show in the top panel
(dashed lines) $5 m_s^r$ rather than $2 m_s^r$, in order to fit it visibly into
the top panel of the figure, and this is the only difference in
the display of LHC results, in figure \ref{LHCVolVol2}, as
compared to the RHIC results, figure \ref{TwoVol}.
\begin{figure*}[t]
\vskip -0.5cm
\psfig{width=7.9cm,figure= PLETRTAURHICGAGVOL.ps }
\psfig{width=7.9cm,figure= PLETRTAULHCGAGVOL.ps }
\vspace*{-0.6cm}
\caption{\label{Gluedep}
(color online) Model V1 (volume expansion) at RHIC (left) and LHC (right)
for 5\% most central collisions.
$s/S$ (bottom panel) and $\gamma_s$ (solid lines middle panel)
as function of $\tau$, for widely
varying initial gluon conditions ($T$ in top panel,
$\gamma_{\rm g}$ dashed middle panel), constrained to same entropy content.
}
\end{figure*}
We note that despite a much greater expansion velocity, the evolution
time at LHC is significantly longer, with the most central collisions taking up
to 30\% longer to reach the freeze-out temperature, $T_f=0.14$ GeV.
The reader who prefers earlier freeze-out, at, {\it e.g.}, $T_f=0.17$ GeV can evaluate
the changes required by consulting the temperature profiles shown in the
top panel.
The different centralities, at LHC, are considered with the same scaling
of the transverse size and entropy content as we did for the case of RHIC.
Comparing the left to right set of results ({\it i.e.}, volume V1 to donut V2
expansion models)
for LHC, we see a more developed chemical equilibrium of strangeness with
clear evidence of (over)saturation of yields for a few centralities, see the
bottom panel on right in figure \ref{LHCVolVol2}. There is greater
final specific strangeness content at LHC than at RHIC, with visibly greater
thermal production leading to strangeness (over)saturation.
At RHIC, the thermal production raises the
value of $s/S$ from 0.016 to 0.028 for most central collisions
(V1 model of bulk volume expansion),
while at the LHC the thermal production raises $s/S$ from
0.016 to 0.032. We will discuss
below what this increase means for the $K/\pi$ and other particle ratios.
The same relative increase
in $s/S$ is seen in the model V2 of the expansion comparing RHIC and LHC. However, if
the homogeneous bulk expansion applies at RHIC, but a donut type expansion
arises at LHC,
the increases in strangeness yield and lifespan would be more spectacular. Depending
on its expansion dynamics, LHC clearly harbors the potential to surprise us.
\subsection{Study of the dependence on initial thermalization condition}\label{initial}
An important question is how the value of the
unknown initial conditions impacts
the results we presented above. We have studied this question in depth
in many different model approaches. The answer `practically no dependence'
is best illustrated in the figure \ref{Gluedep}, where
we show the more conservative volume expansion model V1 results on left
for RHIC, and and on right for LHC. We explore
a wide range of initial gluon (and quark) occupancy $\gamma_{\rm g}$,
which for consistency with other figures is shown in the middle panel
by dashed lines, the initial values we consider for glue occupancy
vary as $0.1<\gamma_{\rm g}(\tau_0)<2.1 $ in step of 0.5. The second
of these lines, from the bottom, is the reference behavior we use in
figures \ref{TwoVol} and \ref{LHCVolVol2}.
We recall that, with
$\gamma_{\rm g}(\tau_0)$, we also vary $\gamma_q(\tau_0)$,
which following the same functional temporal
evolution starting with 2/3 smaller initial value and evolving
1.5 times slower. Note also that the scale,
in top panel, varies between RHIC (left) and LHC (right) cases
the dashed lines denote $2m_s$ on left, and $5m_s$ on right.
Since the initial value of
$s/S=0.016$ and $dS/dy= 5,000$ on left for RHIC and, respectively
$s/S=0.016$ and $dS/dy=20,000$ on right for LHC is set, there is a
corresponding variation in $T_0$ (top panel, left end of solid lines)
and $\gamma_s$ (left end of solid lines in middle panel). The final
results for $\gamma_s(\tau_f)$ (right end of solid lines in middle panel)
and $s(\tau_f)/S$ (bottom panel) are impressively insensitive
to this rather exorbitant diversity of initial conditions at fixed entropy content.
The spread in $s/S(\tau)$ we see in the bottom panel could be seen as a wide
line width.
\begin{figure*}[t]
\vskip -0.5cm
\psfig{width=7.9cm,figure= PLETRTAURHICALFVOL.ps }
\psfig{width=7.9cm,figure= PLETRTAULHCALFVOL.ps }
\vspace*{-0.6cm}
\caption{\label{alfdep}
(color online) Study of of $s/S$ evolution with $\tau$,
on left for RHIC, on right for LHC, V1 volume expansion model.
Figure structure same as figure \ref{Gluedep}.
Middle panel, solid lines: computed evolution of $\gamma_s$ in the deconfined phase
for values of $k=2,\,1.5,\,1,\,0.5$ and 0, see Eq.\,(\ref{gs});
Bottom panel: corresponding evolution of $s/S$. The lowest $\gamma_s$ line
corresponds to $k=0$ and the largest to $k=2$, the opposite applies
for $s/S$ lines.
}
\end{figure*}
We conclude that strangeness cannot probe the very initial QGP conditions near
$\tau_0$ --- the memory of the initial history of the reaction is lost, the system
is opaque for $\tau<2$--3fm/$c$ to the strangeness signature. On the other hand,
and most importantly, for the study here undertaken,
this also means that
experimental observables are characteristic of the properties of nearly
chemically equilibrated QGP.
However, there remains dependence on the history of the QGP
fireball in models in which gluon (and light quark) chemical equilibrium
is not attained even in central reactions at
RHIC~\cite{Pal:2001fz,He:2004df}. Thus only if
gluons in QGP did not approach the chemical equilibrium
at $\tau_f$, a signature of this condition would be seen in the strangeness
yield as is seen considering the right hand
side of Eq.\,(\ref{qprod3a}).
Correspondingly smaller value of $s/S$ are then
expected.
Yet, our here presented results agree well with current RHIC strange hadron production
results regarding the total strangeness yield. This constitutes indirect
evidence for the achievement of light quark and gluon chemical equilibrium
in QGP formed in the most central, highest $A$ and highest energy RHIC reactions.
On the other hand, the relatively
small size and small lifespan of a QGP potentially also
formed in peripheral collisions could hinder the achievement
of QGP chemical equilibrium at hadronization.
\begin{figure*}[t]
\vskip -0.5cm
\psfig{width=7.9cm,figure= PLETRTAURHICMASVOL.ps }
\psfig{width=7.9cm,figure= PLETRTAULHCMASVOL.ps }
\vspace*{-0.6cm}
\caption{\label{massdep}
(color online) Study of $s/S$ evolution with $\tau$ for different $m_s$,
on left for RHIC, on right for LHC, V1 volume expansion model.
Figure structure same as figure \ref{Gluedep} and \ref{alfdep}.
Top panel: 5 (running) strange quark mass $m_s^r$. The middle of 5 lines
being our standard reference value, see text for more detail.
}
\end{figure*}
\subsection{Fundamental uncertainties in strangeness production}\label{Fund}
There are two QCD related uncertainties in our strangeness production
study which we explore in turn in figures \ref{alfdep} and \ref{massdep}:\\
\indent 1. the effect of interaction
on the number of strange quark degrees of freedom, Eq.\,(\ref{gs}), \\
\indent 2. the value of strange quark mass. \\
We will now show that these uncertainties lead to observable effects,
in particular regarding the final value of $s/S$, and to a lesser
extend also regarding the final
value of $\gamma_s$ (chemical yield equilibration).
We first recall that
$\gamma_s$ is introduced in Eq.\,(\ref{sdens})
in order to relate the prevailing strangeness
density to the chemically equilibrated density at
temperature $T$. Given a strangeness yield, the value of $\gamma_s$ depends on
the properties of QCD by the way of what the chemically equilibrated density is.
The actual value of $\gamma_s$ enters decisively
into the kinetic equation of strangeness production. Eq.\,(\ref{qprod3a}) shows
that the smaller the value of $\gamma_s$,
the bigger is the obtained change in the value of $s/S$.
Since, in our approach, we used Boltzmann
statistics for the strangeness degree of freedom, the effect
of Pauli blocking on strangeness production is not considered and in
fact is a minor effect.
The direct dependence of physical observables on $\gamma_s$ arises
from the process of strangeness reannihilation into gluons. We note that
Eq.\,(\ref{qprod3a}), describing the change in strange quark yield, can also
be written in the form:
\begin{equation}\label{qprod3aex}
\frac d{d\tau} \frac s S = a_{\rm g}
(\rho_{\rm g}^2\rho_s^{\infty\,2}\,-\rho_s^2\rho_{\rm g}^{\infty\,2} )
+a_q
(\rho_q^2\rho_s^{\infty\,2}\,-\rho_s^2\rho_q^{\infty\,2} ),
\end{equation}
where instead of $\gamma_i$ the actual densities of particles appear. We see that
for each particle both the equilibrium, and transient, density
must enter in order for the system to attain chemical equilibrium as function
of time. Consequently, the value of $\gamma_s$ and hence the QCD correction
in Eq.\,(\ref{sdens}) matters.
The figure \ref{alfdep}, which follows the pattern of
figure \ref{Gluedep}, illustrates this effect when the effective
degeneracy, see Eq.\,(\ref{gs}), varies.
We consider for values of $k=2,\,1.5,\,1.,\,0.5$ and 0. $k=2$ is
the perturbative effect seen for massless quarks, $k=0$ corresponds
to no effect of interaction, when $m_s\gg T$.
When $k=0$, the strange quark degeneracy is largest, thus
for a given strangeness yield $s$ the value of $\gamma_s$ is smallest.
and the production of strangeness biggest. Consequently, this
value corresponds to the smallest $\gamma_s$ in the middle panels of
figure \ref{alfdep} and greatest value of $s/S$ in the bottom panels.
Other lines follow, and the middle solid lines (red on line ) correspond to
$k=1$, which we used to obtain the reference results presented earlier.
A similar effect arises considering variation of strange quark mass as
shown by dashed lines in top panels of figure \ref{massdep}.
We vary by factor 2 the strange quark mass
as is seen in the top panel of the figure \ref{massdep}, on left for
RHIC and on right for LHC.
When the strange quark
mass is increased, the equilibrium strangeness density is decreased,
and thus, for a given strangeness yield $s/S$, the value of $\gamma_s$ is
increased, which in turn reduces strangeness
production strength. Smallest mass considered
(dotted line in top panels of figure \ref{massdep}) corresponds to the
largest final value of $s/S$ shown by dotted line in the bottom panels.
What is more surprising is that the effect of mass variation {\it cancels}
in the actual computed $\gamma_s$ shown in the middle panels.
The reason for this accidental cancellation
is that for a larger mass the smaller value of final $s/S$ solves for the same
value of $\gamma_s$, see Eq.\,(\ref{sS1}).
Since both $m_s$, and the interaction effect on the strange quark effective
degeneracy, see Eq.\,(\ref{gs}), are today not understood at sufficiently precise level,
the appearance of a possible range
of values at freeze-out, for both $\gamma_s$ and $s/S$,
in figures \ref{alfdep} and \ref{massdep}, signals (correlated) uncertainty
in the understanding of the results at RHIC and predictive power for LHC.
\section{Consequences for hadron yields and their evolution from RHIC to LHC}\label{eval}
\subsection{Strangeness and entropy}\label{sh}
The final value of $s/S$ is the key result which practically alone
determines relative strange
particle yields. It depends on the value of $dS/dy$ which we
start with. The value we take, $dS/dy=20,000$ at LHC (4 times RHIC), is a
guess arising from extrapolation of the energy dependence of
particle production. However, it fixes in effect the initial
conditions, and leads to the range of values for $s/S$ we obtained.
We had found that there is continued growth of strangeness yield beyond
the RHIC energy range where $dS/dy=5,000$, which suggests a further strong increase
in the strange hadron yield we will address below.
Variation of $dS/dy$ amounts to variation of the
produced hadron yield. We expect that $dS/dy\propto dh/dy$.
To quantify this, we present, in figure \ref{hsS}, how the yield of
charged hadrons relates to the entropy yield. These results were
obtained both for RHIC (left region, red on line) and LHC (right region)
by the methods described in Ref.~\cite{Rafelski:2005jc},
employing SHARE suite of programs~\cite{Torrieri:2004zz}.
In this evaluation
we have available for each $s/S$ (at RHIC and LHC) the corresponding
strangeness occupancy $\gamma_s$. We further can fix the
ratio of charge to baryon number as in the incoming nuclei, $q/b=0.39$.
\begin{figure}[t]
\psfig{width=7.9cm,figure= PLDHDSDY2.ps }
\vspace*{-0.6cm}
\caption{\label{hsS}
(color online) The yield of charged hadrons $d(h^-+h^+)/dy$
for different values of $dS/dy$, left domain for RHIC and
right upper domain for LHC.
Solid lines: before weak decays, dashed lines: after all weak decays.
}
\end{figure}
A further difference
between RHIC and LHC is that we take the thermal energy per baryon at RHIC
to be $E/b=39.3$ GeV and following Ref.~\cite{Rafelski:2005jc}
$E/b=412$ GeV at LHC. The consistent
ranges for $s/S$ at RHIC are $0.018< s/S< 0.03$ and, at LHC, we can address
$0.018<s/S<0.037$, we cannot otherwise find a smooth match of QGP to HG-phase space
within the physical range of phase space occupancies $\gamma_s,\,\gamma_q$.
We note that this LHC choice, $E/b=412$ GeV,
limits the range of possible entropy yield to just below the range
we explore in our present work, $dS/dy=20,000$, four times the RHIC range.
We next explore the resulting final $s/S$ and
$\gamma_s$. We show these quantities, on left in figure \ref{sSdvdydhdy},
as function of the $dS/dy$ input, and
as function of the charged hadron multiplicity $d(h^-+h^+)/dy$,
on right. The two expansion models we consider are as before
V1 homogeneous expansion (thin solid, lower line) and
V2 donut expansion model (thick solid, upper line).
We see a gradual rise of both $s/S$ and $\gamma_s$
as function of $dS/dy$ which begins to saturate for
$dS/dy>20,000$\,, but at a rather high values beyond
chemical equilibrium: the expansion is so fast that there
is no time to reannihilate the very abundant
strangeness before freeze-out.
\begin{figure*}[t]
\vskip -0.5cm
\psfig{width=7.9cm,figure= PLSOVERSGAMSS.ps }
\psfig{width=7.9cm,figure= PLSOVERSGAMSDHDY.ps }
\vspace*{-0.6cm}
\caption{\label{sSdvdydhdy}
(color online) $s/S$ (top panel) and $\gamma_s$ (bottom panel)
for the most central 5\% collisions
as function of $dS/dy$ (on left) and as function of $d(h^++h^-)/dy$
on right.
The bulk transverse expansion model V1 is the thin solid line (blue),
the thick line (red) is
the donut expansion model V2 with $d=3.5$ fm.
On right, solid lines: before weak decays, dashed lines: after weak decays
(excluding K$^\pm$ see text).
We recognize the RHIC domain results by the smaller $dh/dy$
range of results presented.
}
\end{figure*}
In Fig~\ref{sSdvdydhdy} on the right hand side, we see
the RHIC (smaller $dh/dy$ range) and LHC domains.
Dashed lines are the charged hadron multiplicity after weak
decays, in particular of neutral strange hadrons such
as K$_{\rm S}$. The negligible impact of charged
Kaon weak decays to hadron multiplicity is not accounted
for as this effect is small and experiment dependent considering
the quasi-stability of K$^\pm$.
In the RHIC domain, we have a nearly linear rise of $s/S$ (top
panel) and $\gamma_s$ (bottom panel) with $ d(h^++h^-)/dy$.
In the LHC domain chemical (over) saturation
of these quantities is clearly visible.
The left, and right hand side of figure \ref{sSdvdydhdy}
have at large $dS/dy$ and $ d(h^++h^-)/dy$, respectively,
a similar appearance indicating chemical (over)saturation
with the increasing hadron yield. We recall that while the
functional form of the figure is correct, the normalization of
the abscissa is somewhat arbitrary for the case of LHC, which
is based on an extrapolation of SPS and RHIC results.
We next study the growth of strange hadron
yield with $s/S$. In order to not confuse this
with the issue of final hadronization volume of QGP, we
will study particle ratios. The method of evaluating
particle yields is, as just discussed in study of the
charged hadron yield: SHARE program is asked to match micro-canonical
conditions in the QGP phase to those of HG phase space.
Given the statistical parameters, we can compute all particle
yields within the statistical hadronization approach.
We first consider, in the two top panels of
figure \ref{LKpFLXOsS}, the ratios $\Lambda/pi^+$
and K$^+/\pi^+$ as function of $s/S$, the
attained specific strangeness per entropy at QGP freeze-out.
Solid lines are before weak decays, and dashed
lines present the corresponding results
after weak decays. These enhance the pion
yield, but as we see, even more the $\Lambda$ yield. The K$^+$ yield is
practically not changed: considering the long life span of the charged
kaon it is common to present experimental results corrected for
any decay. Thus, K$^\pm$ are considered, in our study,
as if they were stable particles. The RHIC results are the thicker lines
and the LHC results are the thinner lines.
\begin{figure}[t]
\vskip -0.3cm
\psfig{width=7.7cm,figure= PLSSKALAPInoax.ps }\\
\vspace*{-1.7cm}
\psfig{width=7.7cm,figure= PLSSPHILAXIOKA.ps }
\vspace*{-0.6cm}
\caption{\label{LKpFLXOsS}
(color online) Relative particle yields as function of $s/S$:
from top to bottom $\Lambda/\pi^+$, K$^+/\pi^+$, and
$\phi/\mathrm{K}^+,\ \Lambda/\mathrm{K}^+,\
\Xi^-/\mathrm{K}^+,\ \Omega^-/\mathrm{K}^+$.
Solid lines (blue) are the primary relative yields,
dashed lines (red) give the yields after weak decays (K$^+$ is not decayed).
Thick line with $s/S<0.03$ are for RHIC, and thin lines with $s/S<0.037$
are for LHC physics environment.
Dotted lines guide the eye for the RHIC ${\rm K}^+/\pi^+$ ratio.
}
\end{figure}
The RHIC and LHC results
for K$^+/\pi^+$, $\phi/\mathrm{K}^+$ and $\Omega^-/\mathrm{K}^+$
are practically overlapping, since the influence of the difference in
baryochemical potential is not material for this ratio. However, the
greater final $s/S$ yield expected in most central LHC collisions
implies an increased yield at LHC compared to RHIC. The dotted lines
in the second from top panel, in figure \ref{LKpFLXOsS}, guide the eyes
both in the RHIC domain, and at the higher $s/S$, LHC domain.
We find, in this
perhaps easiest to measure K$^+/\pi^+$ ratio,
a noticeable and measurable relative
yield increase. This prediction is important, since
after dropping with energy at SPS, this ratio remained of the same constant
magnitude (within error) at the much higher RHIC energy domain.
There is less change expected
between RHIC and LHC for the $\Lambda/pi^+$ ratio, since the decreasing
baryochemical potential is compensating, to a large extend, the effect of
the increase in strangeness yield. We note that there will be the
opposite effect in the $\overline\Lambda/\pi^+$ yield.
We see that the ratios $\phi/\mathrm{K}^+$ and $\Omega/\mathrm{K}^+$
are most sensitive to the change in $s/S$ occurring between RHIC and LHC.
The change in baryochemical potential diminishes this sensitivity considerably in the
$\Xi/\mathrm{K}^+$ ratio. Except for $\Lambda/\mathrm{K}^+$
the weak decays have a negligible impact on all ratios with $\mathrm{K}^+$
considered here.
\subsection{Strange particle yields as function of centrality}\label{PY}
We have obtained the growth of strangeness yield in QGP with centrality
and this allows us to explore participant dependence of strange
hadron yields. Since we do not know well how $E/b$ and $E/TS$ changes
as centrality changes, we choose to follow a different approach in
order to present our results. We assume that the hadronization
occurs at a fixed condition, and first we consider $T=140$ MeV and
$\gamma_q=1.6$, corresponding to supercooled sudden hadronization.
In a second step, we compare these results to those obtained with
$T=160$ MeV and $\gamma_q=1$, which is the equilibrium hadron phase space.
The fixed value of statistical hadronization parameters as function of
centrality follows the pattern seen in the analysis of
RHIC results~\cite{Rafelski:2004dp}.
To obtain the results, we follow a similar computational
scheme as developed in the study of $s/S$ dependence, except that we
now fix $T$ MeV and $\gamma_q$, and not, {\it e.g.}, $E/TS$. These
conditions lead to similar hadronization conditions, but $E/TS$ had some variability
for small participant number, see figure 4 in Ref.~\cite{Rafelski:2004dp}.
In the figure \ref{LKpiA}, we show the results,
on left for RHIC, and on right LHC. We follow a similar display
scheme as in figure \ref{LKpFLXOsS}, and
show $\Lambda/pi^+$, K$^+/\pi^+$, followed by
$\phi/\mathrm{K}^+,\ \Lambda/\mathrm{K}^+,\
\Xi^-/\mathrm{K}^+,\ \Omega^-/\mathrm{K}^+$,
as function of participant number. The thinner (blue) lines are
for the V1 model of volume expansion, and the thicker (red) lines for the V2 model
of donut expansion. Dashed lines include the weak decays, which increase the
$\Lambda/pi^+$ ratio and decrease the K$^+/\pi^+$ ratio.
\begin{figure*}[t]
\vskip -0.5cm
\psfig{width=7.9cm,figure= PLVKALAPIRHICLOGnoax.ps }
\psfig{width=7.9cm,figure= PLVKALAPILHCLOGnoax2.ps }\\
\vspace*{-1.3cm}
\psfig{width=7.9cm,figure= PLVPHILAXIOKARHICLOG.ps }
\psfig{width=7.9cm,figure= PLVPHILAXIOKALHCLOG2.ps }
\vspace*{-0.8cm}
\caption{\label{LKpiA}
(color online) Relative strange particle yields as function of
participant number $A$, left RHIC and right LHC.
From top to bottom, $\Lambda/\pi^+$, K$^+/\pi^+$, and
$\phi/\mathrm{K}^+,\ \Lambda/\mathrm{K}^+,\
\Xi^-/\mathrm{K}^+,\ \Omega^-/\mathrm{K}^+$.
Solid lines primary relative yields, dashed lines (relative) yields
after all weak decays (not shown when difference is within line width).
Thin lines (blue), model V1 (volume
expansion) and thick lines (red), model V2 (donut expansion).
Results are for supercooled sudden hadronization.
}
\end{figure*}
We see, in figure \ref{LKpiA}, a steady, but rather slow, increase with centrality
of the relative strange hadron, to non strange pion yield.
The decay pions tend to flatten the K$^+/\pi^+$ ratio at RHIC. We obtain
predict more rise of $\phi/\mathrm{K}^+$ than is observed at RHIC~\cite{Adams:2004ux}, but
our variation is within the error bar of the experimental, nearly constant result,
$\phi/\mathrm{K}^+=0.15\pm0.03$.
As discussed earlier, the overall increase in $s/S$
expected at LHC (right side) compared to RHIC (left side) explains
the noticeably greater relative K$^+/\pi^+$ ratios at LHC.
The most noticeable rise, with centrality, is expected when the ratio
has the largest disparity in strangeness content. There is no
centrality dependence expected when there is no difference in
strangeness content, such as in $\Lambda/\mathrm{K}^+$, where
the weak decay produces very small centrality dependence.
One of the interesting questions is how sensitive is the study of these particle
ratios to the hadronization conditions. In figure \ref{LKpiART}, we compare the
sudden hadronization at $T=140$ MeV and $\gamma_q=1.6$ with the hadron
phase space equilibrium model at $T=160$ MeV and $\gamma_q=1$ forming
the ratio of particle yield ratios, that is the results obtained for
sudden hadronization are divided by those obtained
for the equilibrium HG phase space. The panels follow the same particle
ratios as in figure \ref{LKpiA}, on left we show RHIC and on right LHC
results.
The value of $\gamma_s^{\rm H}(T=160,\gamma_q^{\rm H}=1)$ varies
as function of centrality $1.26<\gamma_s^{\rm H}<0.88$. In general, the
higher hadronization temperature assumed in this chemical
equilibrium case favors the yield of the
more massive hadron. Thus, ratios of heavy to lighter particle
evaluated at the same value of $s/S$ as shown in figure \ref{LKpiA} are
bigger for the $\gamma_q^{\rm H}=1$ since we took a higher $T=160$ MeV
chemical equilibrium freeze-out value.
We note that the nonequilibrium model
is much better explaining available multistrange hadron data.
The equilibrium model needs to be
amended to explain the enhanced $\Omega$ yield. The way out
from this dilemma if one insists on HG equilibrium could be
a multi-freeze-out temperature interpretation. The freeze-out
temperature of the $\Omega^-$ in the equilibrium freeze-out
model would need to be noticeably higher than that of the
bulk of strange particles. We note that such multi-freeze-out models
could experience systematic difficulties as function of centrality.
Moreover, considering figure \ref{LKpiART}, in the chemical equilibrium
interpretation of hadron production a separate freeze-out for both $\Omega^-$ and
$\Xi^-$ will be required at LHC. Aside of being unpalatable,
such a multi-freeze-out HG equilibrium model
will not describe fluctuations well~\cite{Torrieri:2005va}.
\begin{figure*}[t]
\vskip -1.5cm
\psfig{width=7.9cm,figure= PLVKALAPIRAPRHICLOGnoax.ps }
\psfig{width=7.9cm,figure= PLVKALAPIRAPLHCLOGnoax2.ps }\\
\vspace*{-1.7cm}
\psfig{width=7.9cm,figure= PLVPHILAXIOKARAPRHICLOG.ps }
\psfig{width=7.9cm,figure= PLVPHILAXIOKARAPLHCLOG2.ps }
\vspace*{-0.6cm}
\caption{\label{LKpiART}
(color online) Ratio of relative yields obtained for sudden and equilibrium
hadronization (see text) as function of centrality. Left RHIC, and right LHC.
From top to bottom, $\Lambda/\pi^+$, K$^+/\pi^+$ and
$\phi/\mathrm{K}^+,\ \Lambda/\mathrm{K}^+,\
\Xi^-/\mathrm{K}^+,\ \Omega^-/\mathrm{K}^+$.
Solid lines primary relative yields, dashed lines (relative) yields
after all weak decays (when absent, no difference within line width with
solid lines). Thin lines (blue), model V1 (volume
expansion) and thick lines (red), model V2 (donut expansion).
}
\end{figure*}
\subsection{Thermal charm at RHIC and LHC}\label{thc}
\begin{figure*}
\vskip -0.5cm
\psfig{width=7.7cm,figure=PLETRTAURHICCHARMVOL.ps }
\psfig{width=7.7cm,figure=PLETRTAULHCCHARMVOL.ps }
\vspace*{-0.5cm}
\caption{\label{Charm2Vol}
(color online) Left RHIC and right LHC for charm production.
Figure structure same as figures \ref{TwoVol}--\ref{massdep}.
Top panel: solid lines $T$, dashed lines, running $m_c^r$, scaled with 10 for
RHIC on left, and with 2 on right for LHC.
Middle panel: dotted line $\gamma_{\rm g}$, solid lines the computed total charm
$\gamma_c$, dashed lines $\gamma_c$ corresponding to thermal charm production.
Bottom panel: specific charm yield per entropy, solid lines for
all charm, and dashed lines for thermally produced charm.
}
\end{figure*}
The direct initial high energy parton
collisions dominate production of the massive flavor, charm and bottom,
and a thermal process seems to be of no interest. However,
there are two questions we can investigate:\\
a) considering that $\gamma_c$ in the deconfined phase can be as large
as $\gamma_c\simeq 100$ is there any significant thermal annihilation of charm in
the QGP evolution?\\
b) how large is the thermally produced charm yield and can it lead to chemical
equilibrium of charmed quarks? \\
The interest in question a) needs no further discussion. Question b) is of interest since
the directly produced charmed quarks are in principle not easily thermalized,
while the thermally produced charmed quarks emerge naturally in a
momentum distribution,
imaging their `parent' particles thermal momentum distribution.
Consequently, the thermally produced charmed quarks provide
a solid thermal lower limit for the yield of charm, with the
directly produced charm contributing to thermal distribution
after charm has been subject to
collisions required for thermalization.
We see the results of this study
in figure \ref{Charm2Vol}, on left for RHIC, on right for LHC.
The top panels, as usual, presents the temperature and,
charmed quark mass, scaled with factor 1/10 at RHIC and 1/2 at LHC (on right).
The middle panel presents $\gamma_{\rm g}$, the charm
phase space occupancy, $\gamma_c$ and $\gamma_c$ obtained solely
by {\it thermal processes}. Similarly, in the bottom panels, the dashed lines
are the thermal yield at RHIC and LHC, while the horizontal lines are
the (little) evolving $c/S$ yields including the directly produced charm.
The direct charm production, at RHIC, is expected to be
600 times greater than the thermal process. At LHC, the higher initial
temperature, but unchanged specific direct yield $c/S$, in parton collisions,
suggests that thermal production is just factor 90 times smaller. However,
this factor depends on good understanding of both processes and the initial
conditions and surely cannot be fully trusted. Moreover, there is
the possibility that the directly produced charm at LHC may not well thermalized,
in which case an appreciable fraction of thermal charm yield could indeed
originate in thermal reactions.
The answers to the opening questions thus are\\
i) There is no visible charm reannihilation,
see the nearly horizontal lines at $c/S=3\cdot 10^{-3}$ (left, RHIC)
and $c/S=6\cdot 10^{-3}$ (right, LHC), in the bottom panel;
note that the direct charm yield, we assumed implicitly, is
obtained by multiplying the $c/S$ yields
with $dS/dy=5,000$ on left for RHIC, yielding $dc/dy|_{\rm RHIC}=15$,
and with $dS/dy=20,000$ on right for LHC, yielding $dc/dy|_{\rm LHC}=120$.\\
ii) In the middle panel, we
see that thermal production alone (dashed lines) oversaturates the charm phase space,
the thermal produced charm phase space occupancy $\gamma_c^{\rm th}$
(dashed lines middle panels) cross the gluon dotted
$\gamma_{\rm g}$ line at around 4 fm/$c$ for RHIC
(corresponding to $T\simeq 0.175$ GeV) and at
5 fm/$c$ (corresponding to $T\simeq 0.20$ GeV) at LHC.
\section{Summary and conclusions}
We have studied the thermal QGP based
strangeness production at RHIC and LHC,
and have interpreted the observed final $s$-yield in terms of
our theoretical knowledge about the properties of the QGP phase. Our aim
has been to understand how the overall final state strange quark flavor
has been produced, and to study in detail the mechanisms behind
strangeness enhancement. As a further objective we have
explored the impact of high strangeness on the strange hadron yields.
Our results suggest that strangeness enhancement
could be studied considering:
\begin{equation}\label{RsCP}
R^s_{\rm CP}\equiv {s/S|_{\rm central} \over s/S|_{\rm peripheral} }
= {s/S(\tau_f) \over s/S(\tau_0) }.
\end{equation}
The central strangeness yields is just the final value we find
at freeze-out, combining the initial yield with the additional
thermal production. The peripheral yield is the initial value
before thermal strangeness production begins. Our study shows that
$R^s_{\rm CP}\in [1.6, 2.2]$, with the precise result depending
on details such as strange quark mass, see figure~\ref{massdep},
reaction energy and dynamics of expansion, see figure~\ref{Volume}.
More generally, to separately consider $s$ and $S$
we can use $d(h^++h^-)/dy$ as a measure of entropy $dS/dy$
content, see figure~\ref{hsS}. Instead of the total strangeness,
on may consider enhancement of individual (multi)strange particles,
which we discussed in depth both as function of the achieved
$s/S$, in subsection \ref{sh},
and as function of centrality $A$ at fixed hadronization condition
for all centralities, in subsection \ref{PY}.
We note that the overall growth of the enhancement of the
strangeness yield with centrality, at the level
a factor 1.6--2.2 is accompanied by a further enhancement
of multistrange hadron yields, as is seen comparing the yields of
multistrange hadrons with the yield of kaons. One can show that
the $\phi/\mathrm{K}^+$ ratio is mainly dependent on value of
$s/S$ and not on hadronization temperature
(see appendix B2i ~\cite{Kuznetsova:2006bh})
when the strangeness conservation constraint is
implemented. This effect is unique to $\phi$ and
arises since $m_\phi\simeq 2m_{\rm K}$.
Aside of centrality dependence, we have explored, within the
framework of our model, the extrapolation
from RHIC to LHC physics environments. More generally,
we have presented, in figure \ref{sSdvdydhdy}, the reaction energy dependence
by considering the central rapidity $ s/S$
yields as function of $dS/dy$.
One of the interesting results obtained is the approach to chemical
strangeness equilibrium in the deconfined QGP phase formed in most central
and high energetic RHIC reactions. The evidence for this
is implicit in the experimentally reported yields of strange hadrons, which
lead to values of specific strangeness per entropy at the
level of $s/S\simeq 0.028$~\cite{Rafelski:2004dp}. Our study of QGP
based kinetic strangeness production
provide an explanation of this result, both, the value of $s/S$
and as function of centrality.
Our present study further shows
that the proximity of chemical strangeness
yield equilibration in QGP formed at RHIC and LHC, and
the effective opacity of QGP to this signature, is the reason that considerably less
sophisticated models of QGP evolution which we, and others, have considered
are equally successful in the study the strangeness
production, as long as these models yield
conditions near to chemically equilibrated QGP.
Given the near chemical equilibration at RHIC, and within the models considered,
we obtain some strangeness over saturation at LHC. Moreover, for the most central
5\% reactions there is no relevant dependence of strangeness
production on initial conditions prevailing in the reaction. We have
demonstrated this in a picture-book fashion, see figure \ref{Gluedep},
where, for a wide range of initial conditions, the same
final strangeness yield and equilibrium condition arises after $\Delta\tau=$2--3 fm/c.
On the other hand, the more peripheral reactions do
not saturate the phase space, in that both $\gamma_s^{\rm QGP}<1$,
and $s/S<0.03$. Thus, the peripheral yields, being sensitive to
the initial conditions, allow exploration of physical conditions in
the QGP prior to the onset of chemical reactions. Therefore, our
results for most peripheral reactions are also
somewhat dependent on model assumptions about initial state and evolution
dynamics.
We have shows, in figures \ref{TwoVol} and \ref{LHCVolVol2}, the impact parameter
dependence that arises in two volume expansion models. Since the analysis results
presented in Ref.~\cite{Rafelski:2004dp} have been used to fine tune
the dynamical evolution model at RHIC, there is good qualitative
agreement with experimental results. Our objective has
been here to learn how to extrapolate the dynamics of strangeness
production to the LHC environment. The gradual rise of
strangeness yield $s/S$ with $dS/dy$, seen in figure \ref{sSdvdydhdy}, is
reminiscent of the rise of $s/S$ with reaction energy obtained in a analysis of particle
yields obtained at SPS and RHIC at different reaction energies~\cite{Letessier:2005qe}.
We have made a (conservative) prediction regarding the increase in the K$^+/\pi^+$
ratio at LHC compared to RHIC, see figure \ref{LKpFLXOsS}. Though an important
result is that we expect an increase at LHC, we note that even a greater increase is
possible, signaling even greater values of $s/S$, depending on both:\\
1) the dynamics of the volume expansion --- this can
enhance the strangeness oversaturation of the final QGP state,
see figures \ref{TwoVol} and \ref{LHCVolVol2};\\
2) QCD details, such as strange quark density including
QCD interactions, and (still not well understood) strange quark mass,
see figures \ref{alfdep} and \ref{massdep}.\\
In any case, we believe that this simple observable will show again
strangeness production growing faster than entropy production,
its increase is directly coupled to an increase in $s/S$. We note again that in our
study the increase of $s/S$ with centrality implies that $\phi/{\rm K}^+$ also increases
with centrality.
A natural result is the finding of the chemical yield equilibration of strangeness in
the QGP formed in the 5\% most
central top RHIC energy reactions. This leads to
a better understanding of the resulting oversaturation of the hadron
phase space by strangeness. The magnitude of this effect,
dependent on the temperature of hadronization, can be considerable.
This can be easily seen considering the magnitude of $s/S$ in both
QGP and HG phases. The final state hadrons
formed far-off chemical equilibrium
cannot significantly adjust chemical composition, considering the
rapid breakup of the fireball, during the period
of about 1--2 fm/$c$ prior to onset of the free flow. Thus, our finding is
that strangeness rich QGP enhances decisively the yields of
multistrange hadrons. This phenomenon is more accentuated considering
charmed hadrons containing strangeness,
a topic under current investigation.
Furthermore, using the methods developed here,
we have considered thermal charm production. At LHC, we find
a nearly physically relevant thermal charm production, but not at RHIC. However,
the thermal process we consider is able to produce enough charm
to oversaturate the final state at both RHIC and LHC,
see figure \ref{Charm2Vol}. This also shows that the direct parton
collision based production at RHIC leads to extraordinarily large values
of $\gamma_c$. The chemical nonequilibrium of charm is thus more pronounced
than that of strangeness.
{\bf In conclusion:} The totality of our results shows
that, as function of entropy yield $dS/dy$
(equivalently, the reaction energy of $A_1$--$A_2$ collision) and
geometric reaction size (impact parameter dependence,
participant number $A$), the phenomenon
of strangeness enhancement is well described by the mechanism of QGP based
thermal gluon fusion strangeness
production. We find both, as function of centrality, and energy,
a gradual increase in specific strangeness yield, which agrees with all
available experimental results. We find that, as function of energy, this
continues from RHIC to LHC increasing our hopes for a more clear strangeness
signature of deconfinement.
\vspace*{.2cm}
\subsubsection*{Acknowledgments}
Work supported by a grant from: the U.S. Department of Energy DE-FG02-04ER4131.
LPTHE, Univ.\,Paris 6 et 7 is: Unit\'e mixte de Recherche du CNRS, UMR7589.
\vspace*{-0.3cm}
|
1,108,101,566,761 | arxiv | \section{Introduction}
A Bayesian network (BN) is a probabilistic graphical model consisting of a labeled directed acyclic graph (DAG) $\mathcal{G} = (V, E)$, in which the vertex set $V = \{V_1, \dots, V_m\}$ corresponds to $m$ random variables, and the edge set $E$ prescribes a decomposition of the joint probability distribution of the random variables based on their parents in $\mathcal{G}$. The edge set $E$ encodes Markov relations on the nodes in the sense that each node is conditionally independent of its non-descendents given its parents. BNs have been used in knowledge discovery \citep{spirtes2000causation, chen2018causal}, classification \citep{aliferis2010local}, feature selection \citep{gao2015structured}, latent variable discovery \citep{lazic2013structural} and genetics \citep{ott2003finding}. They also play a vital part in causal inference \citep{pearl2009causal}.
In this paper, we propose mixed-integer quadratic programming (MIQP) formulations for learning the optimal DAG structure of BNs given $n$ continuous observations from a system of linear structural equation models (SEMs). While there exist exact integer-programming (IP) formulations for learning DAG structure with \emph{discrete} data \citep{cussens2010maximum, cussens2012bayesian, hemmecke2012characteristic, studeny2013polyhedral, bartlett2013advances, JMLR:v17:14-479,oates2016exact, bartlett2017integer, cussens2017polyhedral,cussens2017bayesian}, the development of {tailored} computational tools for learning the optimal DAG structure from \emph{continuous} data has received less attention. In principle, exact methods developed for discrete data can be applied to {continuous} data. However, such methods result in exponentially sized formulations in terms of the number of binary variables. A common practice to circumvent the exponential number of binary variables is to limit the in-degree of each node \citep{cussens2012bayesian,cussens2017bayesian, bartlett2017integer}. But, this may result in sub-optimal solutions. On the contrary, MIQP formulations for learning DAGs corresponding to linear SEMs require a \textit{polynomial} number of binary variables. This is because for BNs with linear SEMs, the score function --- i.e., the penalized negative log-likelihood (PNL) --- can be explicitly written as a function of the coefficients of linear SEMs \citep{shojaie2010penalized, van2013ell, park2017bayesian, manzour2019integer}.
Continuous BNs with linear SEMs have witnessed a growing interest in the statistics and computer science communities \citep{van2013ell, raskutti2013learning, loh2014high, ghoshal2016information, solus2017consistency}.
In particular, it has been shown that the solution obtained from solving the PNL augmented by $\ell_0$ regularization achieves desirable statistical properties \citep{peters2013identifiability, van2013ell, loh2014high}.
Moreover, if the model is \emph{identifiable} \citep{peters2013identifiability, loh2014high}, such a solution is guaranteed to uncover the true causal DAG when the sample size $n$ is large enough.
However, given the difficulty of obtaining exact solutions, existing approaches for learning DAGs from linear SEMs have primarily relied on \emph{heuristics}, using techniques such as coordinate descent \citep{fu2013learning, aragam2015concave, han2016estimation} and non-convex continuous optimization \citep{zheng2018dags}.
Unfortunately, these heuristics are not guaranteed to achieve the desirable properties of the global optimal solution. Moreover, it is difficult to evaluate the statistical properties of a sub-optimal solution with no optimality guarantees \citep{koivisto2012advances}. To bridge this gap, in this paper we develop mathematical formulations for learning optimal BNs from linear SEMs using a PNL objective with $\ell_0$ regularization. By connecting the optimality gap of the mixed-integer program to the statistical properties of the solution, we also establish an \emph{early stopping criterion} under which we can terminate the branch-and-bound procedure and attain a solution which asymptotically recovers the true parameters with high probability.
Our work is related to recent efforts to develop exact tailored methods for DAG learning from continuous data.\ \cite{xiang2013lasso} show that $A^{\ast}$-lasso algorithm tailored for DAG structure learning from continuous data with $\ell_1$-regularization is more effective than the previous approaches based on dynamic programming \citep[e.g.,][]{silander2006simple} that are suitable for both discrete and continuous data.\ \cite{park2017bayesian} develop a mathematical program for DAG structure learning with $\ell_1$ regularization. \cite{manzour2019integer} improve and extend the formulation by \cite{park2017bayesian} for DAG learning from continuous data with both $\ell_0$ and $\ell_1$ regularizations. The numerical experiments by \cite{manzour2019integer} demonstrate that as the number of nodes grows, their MIQP formulation outperforms $A^{\ast}$-lasso and the existing IP methods; this improvement is both in terms of reducing the IP optimality gap, when the algorithm is stopped due to a time limit, and in terms of computational time, when the instances can be solved to optimality. In light of these recent efforts, the current paper makes important contributions to this problem at the intersection of statistics and optimization.
\begin{itemize}
\item The statistical properties of \textit{optimal} PNL with $\ell_0$ regularization have been studied extensively \citep{loh2014high,van2013ell}. However, it is often difficult to obtain an optimal solution and no results have been established on the statistical properties of approximate solutions. In this paper, we give an early stopping criterion for the branch-and-bound process under which the approximate solution gives consistent estimates of the true coefficients of the linear SEM. Our result leverages the statistical consistency of the PNL estimate with $\ell_0$ regularization \citep{van2013ell, peters2013identifiability} along with the properties of the branch-and-bound method wherein both lower and upper bound values on the objective function are available at each iteration. By connecting these two properties, we obtain a concrete early stopping criterion, as well as a simple proof of consistency of the approximate solution. To the best of our knowledge, this result is the first of its kind for DAG learning.
\item In spite of recent progress, a key challenge in learning DAGs from linear SEMs is enforcing bounds on arc weights. This is commonly modeled using the standard ``big-$M$ constraint" approach \citep{park2017bayesian, manzour2019integer}. As shown by \cite{manzour2019integer}, this strategy leads to poor continuous relaxations for the problem, which in turn results in slow lower bound improvement in the branch-and-bound tree. In particular, \cite{manzour2019integer} establish that all existing big-$M$ formulations achieve the same continuous relaxation objective function under a mild condition (see Proposition~\ref{Prop2}). To circumvent this issue, we present a mixed-integer second-order cone program (MISOCP), which gives a tighter continuous relaxation than existing big-$M$ formulations. This formulation can be solved by powerful state-of-the-art optimization packages. Our numerical results show the superior performance of MISOCP compared to the existing big-$M$ formulations in terms of improving the lower bound and reducing the optimality gap.
\end{itemize}
The rest of the paper is organized as follows. In Section~\ref{Sec: SEMs}, we define the DAG structure learning problem corresponding to linear SEMs, and give a general framework for the problem. In Section~\ref{Cons}, we present our early stopping criterion and establish the asymptotic properties of the solution obtained under this stopping rule.
We review existing mathematical formulations in Section~\ref{Sec: Previous work}, and present our proposed mathematical formulations in Section~\ref{Sec: Math models}.
Results of comprehensive numerical studies are presented in Section~\ref{Sec: Computational}. We end the paper with a summary in Section~\ref{Sec: Conclusion}.
\raggedbottom
\section{Problem setup: Penalized DAG estimation with linear SEMs} \label{Sec: SEMs}
Let $\mathcal{M} = (V, E)$ be an undirected and possibly cyclic super-structure graph with node set $V=\{1,2,\dots,m\}$ and edge set $E \subseteq V \times V$; let $\overrightarrow{\mathcal{M}} = (V, \overrightarrow{E})$ be the corresponding bi-directional graph with $\overrightarrow{E} =\{(j,k), (k,j) | (j,k) \in E\}$. We refer to undirected edges as \emph{edges} and directed edges as \emph{arcs}.
We assume that causal effects of continuous random variables in a DAG $\mathcal{G}_0$ are represented by $m$ linear regressions of the form
\begin{equation} \label{LSLM}
X_k = \sum_{j \in pa^{\mathcal{G}_0}_k} \beta_{jk} X_j + \epsilon_k, \quad k=1,\dots, m,
\end{equation}
\noindent where $X_k$ is the random variable associated with node $k$, $pa^{\mathcal{G}_0}_k$ represents the parents of node $k$ in $\mathcal{G}_0$, i.e., the set of nodes with arcs pointing to $k$; the latent random variable $\epsilon_k$ denotes the unexplained variation in node $k$; and BN parameter $\beta_{jk}$ specifies the effect of node $j$ on $k$ for $j \in pa^{\mathcal{G}_0}_k$. The above model is known as a linear SEM \citep{pearl2009causal}.
Let $\mathcal{X}=(\mathcal{X}_1, \dots , \mathcal{X}_m)$ be the $n \times m$ data matrix with $n$ rows representing i.i.d.\ samples from each random variable, and $m$ columns representing random variables $X_1, \ldots, X_m$.
The linear SEM \eqref{LSLM} can be compactly written in matrix form as $\mathcal{X} = \mathcal{X}{B} + \mathcal{E}$, where ${B} = [\beta] \in \mathbb{R}^{m \times m}$ is a matrix with $\beta_{kk}=0$ for $k=1,\dots,m$, $\beta_{jk}=0$ for all $(j,k) \notin E$, and $\mathcal{E}$ is the $n\times m$ `noise' matrix. Then, $\mathcal{G}(B)$ denotes the directed graph on $m$ nodes such that arc $(j,k)$ appears in $\mathcal{G}(B)$ if and only if $\beta_{jk} \neq 0$. Throughout the paper, we will use $B$ and $\beta$ to denote the matrix of coefficients and its vectorized version.
A key challenge when estimating DAGs by minimizing the loss function \eqref{eqn:lklhd} is that the true DAG is generally not identifiable from observational data. However, for certain SEM distributions, the true DAG is in fact identifiable from observational data. Two important examples are linear SEMs with possibly non-Gaussian homoscedastic noise variables \citep{peters2013identifiability}, as well as linear SEMs with unequal noise variances that are known up to a constant \citep{loh2014high}. In these special cases, the true DAG can be identified from observational data, without requiring the (strong) `faithfulness' assumption, which is known to be restrictive in high dimensions \citep{uhler2013geometry, sondhi2019reduced}. Given these important implications, in this paper we focus on learning Bayesian networks corresponding to the above \emph{identifiable} linear SEMs.
The negative log likelihood for an identifiable linear SEM \eqref{LSLM} with equal noise variances is proportional to
\begin{equation}\label{eqn:lklhd}
l(\beta; \mathcal{X}) =n\,\text{tr}\left\{(I-{B})(I-{B})^\top \widehat{\Sigma}\right\};
\end{equation}
here $\widehat{\Sigma} =n^{-1} \mathcal{X}^\top \mathcal{X}$ is the empirical covariance matrix, and $I$ is the identity matrix \citep{shojaie2010penalized, van2013ell}.
To learn \textit{sparse} DAGs, \citet{van2013ell} propose to augment the negative log likelihood with an $\ell_0$ regularization term. Given a super-structure $\mathcal{M}$, the optimization problem corresponding to this penalized negative log-likelihood (PNL$\mathcal{M}$) is given by \begin{subequations} \label{eq:PNLMform}
\begin{align}
\textbf{PNL$\mathcal{M}$} \quad & \underset{B \in {\mathbb R}^{m \times m}}{\min} \quad \Score(\beta):= l(\beta; \mathcal{X}) + \lambda_n \|\beta\|_0 \label{Eq: Opt} \\
\text{s.t.} \, \, & \mathcal{G}(B) \, \, \text{induces a DAG from} \, {\overrightarrow{\mathcal{M},}} \label{Eq: DAG const}
\end{align}
\end{subequations}
where the tuning parameter $\lambda_n$ controls the degree of the $\ell_0$ regularization, and the constraint \eqref{Eq: DAG const} stipulates that the resulting directed subgraph is a DAG induced from $\overrightarrow{\mathcal{M}}$. When $\mathcal{M}$ corresponds to a complete graph, PNL$\mathcal{M}$ reduces to the original PNL of \citet{van2013ell}.
The choice of $\ell_0$ regularization in \eqref{eq:PNLMform} is deliberate. Although $\ell_1$ regularization has attractive computational and statistical properties in high-dimensional regression \citep{bulmann2011statistics}, many of these advantages disappear in the context of DAG structure learning \citep{fu2013learning, aragam2015concave}. By considering $\ell_0$ regularization, \cite{van2013ell} establish the consistency of PNL under appropriate assumptions. More specifically, for a Gaussian SEM, they show that the estimated DAG has (asymptotically) the same number of edges as the DAG with minimal number of edges (minimal-edge I-MAP), and establish the consistency of PNL for learning sparse DAGs. These results are formally stated in Proposition~\ref{prop:van} in the next section.
\begin{remark}\label{rem:L2}
A Tikhonov ($\ell_2$) regularization term, $\mu \|\beta\|_2^2$, for a given $\mu > 0$, can also be added to the objective \eqref{Eq: Opt} to obtain more stable solutions \citep{bertsimas2016best}.
\end{remark}
In our earlier work \citep{manzour2019integer}, we observe that existing mathematical formulations are slow to converge to a provably optimal solution, $\beta^\star$, of \eqref{eq:PNLMform} using the state-of-the-art optimization solvers. Therefore, the solution process needs to be terminated early to yield a feasible solution, $\hat \beta$ with a positive optimality gap, i.e., a positive difference between the upper bound on $\Score(\beta^\star)$ provided by $\Score(\hat \beta)$ and a lower bound on $\Score(\beta^\star)$ provided by the best continuous relaxation obtained by the branch-and-bound algorithm upon termination.
However, statistical properties of such a sub-optimal solution are not well-understood. Therefore, there exists a gap between theory and computation: while the optimal solution has nice statistical properties, the properties of the solutions obtained from approximate computational algorithms are not known. Moreover, due to the non-convex and complex nature of the problem, characterizing the properties of the solutions provided by heuristics is especially challenging. In the next section, we bridge this gap by developing a concrete early stopping criterion and establishing the consistency of the solution obtained using this criterion.
\section{Early stopping criterion for DAG learning} \label{Cons}
In this section, we establish a sufficient condition for the approximate solution of PNL$\mathcal{M}$, $\hat{\beta}$ to be consistent for the true coefficients, $\beta^{0}$; that is $\|\beta^{0} - \hat{\beta}\|_2^2 = \mathcal{O}\left(s^0\log(m) / n \right)$, where $s^0$ is the number of arcs in the true DAG, and $x$ $\asymp$ $y$ means that $x$ converges to $y$ asymptotically.
This result is obtained by leveraging an important property of the branch-and-bound process for integer programming that provides both lower and upper bounds on the objective function $ \Score(\beta^\star)$ upon early stopping, as well as the consistency results of the PNL estimate with $\ell_0$ regularization. Using the insight from this new result, we then propose a concrete stopping criterion for terminating the branch-and-bound process that results in consistent parameter estimates.
Let $LB$ and $UB$ respectively denote the lower and upper bounds on the optimal objective function value \eqref{Eq: Opt} obtained from solving \eqref{eq:PNLMform} under an early stopping criterion (i.e., when the obtained solution is not necessarily optimal). We define the difference between the upper and lower bounds as the \emph{absolute} optimality gap: $GAP = UB -LB$. Let $\hat{\mathcal{G}}$ and $ \hat{\beta}$ denote the structure of the DAG and coefficients of the arcs from optimization model \eqref{eq:PNLMform} under the early stopping condition with sample size $n$ and regularization parameter $\lambda_n$.
Let ${\mathcal{G}^{\star}}$ and $\beta^{\star}$ denote the DAG structure and coefficients of arcs obtained from the optimal solution of \eqref{eq:PNLMform}, and $\mathcal{G}^{0}$ and $\beta^{0}$ denote the true DAG structure and the coefficient of arcs, respectively.
We denote the number of arcs in $\hat{\mathcal{G}}$, $\mathcal{G}^{0}$, and ${\mathcal{G}^{\star}}$ by $\hat{s}$, $s^0$, and $s^{\star}$, respectively. The score value in \eqref{Eq: Opt} of each solution is denoted by $\Score(\phi)$ where $\phi \in \{\beta^{\star}, \hat{\beta}, \beta^0\}$.
Next, we present our main result. Our result extends \citeauthor{van2013ell}'s result on consistency of PNL$\mathcal{M}$ for the optimal, but computationally unattainable, estimator, $\beta^{\star}$ to an approximate estimator, $\hat\beta$, obtained from early stopping. In the following (including the statement of our main result, Proposition~\ref{EarlyProp}), we assume that the super-structure $\mathcal{M}$ is known \emph{a priori}. The setting where $\mathcal{M}$ is estimated from data is discussed at the end of the section. We begin by stating the key result from \cite{van2013ell} and the required assumptions. Throughout, we consider a Gaussian linear SEM of the form \eqref{LSLM}. We denote the variance of error terms, $\epsilon_j$, by $\sigma_{jj}^2$ and the true covariance matrix of the set of random variables, $(X_1,\ldots, X_m)$ by the $m \times m$ matrix $\Sigma$.
\begin{cond}\label{cond:1}
For some constant $\sigma_0^2$, it holds that $\max_{j=1,\ldots,m}\sigma_{jj}^2 \leq \sigma_0^2$. Moreover, the smallest eigenvalue of $\Sigma$, $\kappa_{\min}(\Sigma)$, is nonzero.
\end{cond}
\begin{cond}\label{cond:2}
Let, as in \cite{van2013ell}, $\widetilde\Omega(\pi)$ be the precision matrix of the vector of noise variables for an SEM given permutation $\pi$ of nodes. Denoting the diagonal entires of this matrix by $\tilde \omega_{jj}$, there exists a constant $\omega_0 > 0$ such that if $\widetilde\Omega(\pi)$ is not a multiple of the identity matrix, then
\[
m^{-1} \sum_{j=1}^m\left( (\tilde\omega_{jj})^2 -1 \right)^2 > 1/ \omega_0.
\]
\end{cond}
\begin{prop} (Theorem 5.1 in \cite{van2013ell}) \label{prop:van}
Suppose Assumptions~\ref{cond:1} and \ref{cond:2} hold. Let $\alpha_0:= \min\{\frac{4}{m}, 0.05\}$. Then for an $\ell_0$ regularization parameter $\lambda \asymp \log(m)/n$, it holds with probability at least $1-\alpha_0$ that
\[
\|\beta^{\star}-\beta^{0}\|_2^2 + \lambda s^{\star} = \mathcal{O}\left(\lambda s^0\right).
\]
\end{prop}
Here, $\lambda=\lambda_n/n$, because the loss function \eqref{eqn:lklhd} is that of \cite{van2013ell} scaled by the sample size $n$.
Before presenting our main result, we state one more condition on the covariance matrix of the random variables generated by linear SEM. For a given subset $S \subset \{1,\ldots,m\}$, let $S^c$ denote its complement, i.e., $S^c:= \{1,\ldots,m\}\setminus S.$
\textbf{Definition \citep{raskutti2010restricted}.} Define the set $\mathcal{C}(S; \eta) := \{v \in \mathbb{R}^{m}\, \, | \, \, \| v_{S^c}\|_1 \leq \eta \|v_S\|_1\}$ for a given subset $S \subset \{1,\ldots,m\}$ and constant $\eta \geq 1$.
The $m \times m$ sample covariance matrix $\widehat{\Sigma} = n^{-1}\mathcal{X}^\top \mathcal{X}$ satisfies the \textit{restricted eigenvalue (RE) condition over $S$} with parameters $(\eta,\gamma) \in [1,\infty) \times [0,\infty)$ if
\[
\frac{1}{n} v^\top \mathcal{X}^{\top} \mathcal{X} v = \frac{1}{n} \|\mathcal{X}v\|_2^2 \geq \gamma^2 \|v\|_2^2, \quad \forall v \in \mathcal{C}(S;\eta).
\]
If this condition holds for all subsets $S$ with cardinality $s$, we say that $\widehat{\Sigma}$ satisfies a \textit{restricted eigenvalue (RE) condition of order $s$} with parameters $(\eta,\gamma)$. The $m \times m$ population covariance matrix $\Sigma$ is said to satisfy the RE condition if
\[
\| \Sigma^{1/2} v \|_2 \ge \gamma \| v \|_2 \quad \forall v \in \mathcal{C}(S;\eta).
\]
\citet{raskutti2010restricted} show that if $\Sigma$ satisfies the RE condition, then there exists constants $c$ and $c'$ such that with probability at least $1 - c' e^{-c n}$, $\widehat{\Sigma}$ also satisfies the RE condition with parameters $(\eta,\gamma/8)$. More specifically, their proof of Corollary~1 shows that for any $v \in \mathcal{C}(S; \eta)$,
\begin{equation}\label{eqn:REdef}
\| v \|_2^2 \le c_1 \| \mathcal{X} v \|_2^2,
\end{equation}
where $c_1 = n^{-1} \left\{\frac{\gamma}{4} - 9(1+\alpha) \sigma_0 \sqrt{\frac{s^0 \log(m)}{n}}\right\}^{-2}$ for $\sigma_0$ defined in Assumption~\ref{cond:1}. In fact, in the low-dimensional setting implied by condition \eqref{eqn:ncond}, the inequality \eqref{eqn:REdef} holds with probability one because, when $m \ll n$, for any $v \in \mathbb{R}^m$ we have
$
\|\mathcal{X}v\|_2^2 \geq \kappa_{\min}(\mathcal{X}) |\|v\|_2^2.
$
Thus, \eqref{eqn:REdef} holds with $c_1 = 1 / \kappa_{\min}(\mathcal{X})$.
\begin{prop} \label{EarlyProp}
Suppose $\Sigma$ satisfies the RE condition of order $s^0$ with parameters $(\eta,\gamma)$ and that for constants $c_2, c_3 > 0$,
\begin{equation}\label{eqn:ncond}
n > \max\left\{ c_2 \frac{\sigma_0^2(1+\eta)^2}{\gamma^2}s^0 \log(m), \, c_3 m\log(n) \right\}.
\end{equation}
Suppose also that Assumptions~\ref{cond:1} and \ref{cond:2} hold. Let $\alpha_{0} = \min \{\frac{4}{m}, 0.05\}$ and $\lambda \asymp \log(m) / n$.
Then, the estimator $\hat\beta$ obtained from early stopping of the branch-and-bound process such that GAP $\asymp \mathcal{O}\left(\frac{\log(m)}{n} s^0\right)$ satisfies $ \|\hat\beta - \beta^{0}\|_2^2 \asymp \mathcal{O}\left(\frac{\log(m)}{n}s^0\right)$ with probability $\min\{1- \alpha_0, 1- c' e^{-cn}\}$ for constants $c$ and $c'$ used for the RE condition.
\end{prop}
\begin{proof}
First, by the triangle inequality and the fact that $2ab \leq a^2 + b^2, \forall a,b \in \mathbb{R}$,
\begin{equation}\label{eqn:newbnd1}
\| \hat\beta - \beta^0 \|_2^2 \leq
2\| \hat\beta - \beta^\star \|_2^2 + 2\| \beta^\star - \beta^ 0 \|_2^2.
\end{equation}
Further, by the sparsity of $\beta^\star$ from Proposition~\ref{prop:van}, $\hat\beta - \beta^\star$ belongs to the set $\mathcal{C}(S^0; \eta)$, where $S^0 = \{j: \beta^0_j \ne 0 \}$ and $|S^0| = s^0$. Thus,
\begin{equation}\label{eqn:lowerbnd}
\| \hat\beta - \beta^\star \|_2^2 \leq c_1 \| \mathcal{X}(\hat\beta - \beta^\star) \|_2^2.
\end{equation}
Now, noting that $\ell(\beta;\mathcal{X}) = \| \mathcal{X} - \mathcal{X}\beta \|_2^2$ (see, e.g., the expanded version in Eq.~\eqref{CP-obj}), we can write a Taylor series expansion of $\ell(\hat\beta;\mathcal{X})$ around $\ell(\beta^\star;\mathcal{X})$ to get
\begin{align*}
\| \mathcal{X}(\hat\beta - \beta^\star) \|_2^2
= \ell(\hat\beta;\mathcal{X}) - \ell(\beta^\star;\mathcal{X}) - 2(\hat\beta - \beta^\star)^\top \mathcal{X}^\top \mathcal{X} (\beta^\star - \beta^0) +
2(\hat\beta - \beta^\star)^\top \mathcal{X}^\top \mathcal{E}.
\end{align*}
Here, we also use the fact that $\mathcal{X} = \mathcal{X}B^0 + \mathcal{E}$.
Thus, using triangle inequality again, we get
\begin{align*}
\| \hat\beta & - \beta^\star \|_2^2 \leq \\
& c_1\left| \ell(\hat\beta;\mathcal{X}) - \ell(\beta^\star;\mathcal{X}) \right| + 2c_1 \kappa_{\max}(\mathcal{X}^\top \mathcal{X}) \| \hat\beta - \beta^\star \|_2 \| \beta^\star - \beta^0 \|_2 +
2c_1 \| \hat\beta - \beta^\star \|_2 \| \mathcal{X}^\top\mathcal{E} \|_2,
\end{align*}
where $\kappa_{\max}$ denotes the maximum eigenvalue of the matrix.
Let $Z = \| \hat\beta - \beta^\star \|_2$, and denote
$
\Pi = 2c_1 \left[ \kappa_{\max}(\mathcal{X}^\top \mathcal{X}) \| \beta^\star - \beta^0 \|_2 + \| \mathcal{X}^\top\mathcal{E} \|_2 \right],
$
and
$
\Gamma = c_1\left| \ell(\hat\beta;\mathcal{X}) - \ell(\beta^\star;\mathcal{X}) \right|.
$
Then, the above inequality can be written as $Z^2 \leq \Pi Z + \Gamma$, which implies that $Z \leq \left(\Pi + \sqrt{\Pi^2 + 4\Gamma} \,\right) / 2$.
Let $\mathcal{T}$ be the event under which $\Pi = o(1)$. Then, on the set $\mathcal{T}$,
\begin{equation}\label{eqn:newbnd2}
\| \hat\beta - \beta^\star \|_2^2 \leq c_1\left| \ell(\hat\beta;\mathcal{X}) - \ell(\beta^\star;\mathcal{X}) \right| + o(1).
\end{equation}
Plugging in \eqref{eqn:newbnd2} into \eqref{eqn:newbnd1}, on the set $\mathcal{T}$ we get
\begin{align}\label{eqn:newbnd3}
\| \hat\beta - \beta^0 \|_2^2 & \leq
2 c_1\left| \ell(\hat\beta;\mathcal{X}) - \ell(\beta^\star;\mathcal{X}) \right| + 2\| \beta^\star - \beta^ 0 \|_2^2 + o(1) \nonumber\\
& \leq 2 c_1
\left| \ell(\hat\beta; \mathcal{X}) - \ell(\beta^{\star}; \mathcal{X}) + \lambda \hat{s} - \lambda s^\star \right| +
2\| \beta^{\star} - \beta^{0} \|_2^2 + 2 c_1 |\lambda s^{\star} - \lambda \hat{s}| + o(1) \nonumber \\
& \leq 2 c_1
\underset{\Score(\hat{\beta}) - \Score(\beta^{\star})}{\underbrace{ \left| \ell(\hat\beta; \mathcal{X}) - \ell(\beta^{\star}; \mathcal{X}) + \lambda \hat{s} - \lambda s^\star \right| } } +
2\| \beta^{\star} - \beta^{0} \|_2^2 + 2 c_1 \lambda s^{\star} + o(1) \nonumber \\
& \leq 2 c_1 GAP + 2\left(\| \beta^{\star} - \beta^{0} \|_2^2 + c_1 \lambda s^{\star}\right) + o(1),
\end{align}
where the last inequality follows from the fact that, by definition, $\left\vert \Score(\hat{\beta}) - \Score(\beta^{\star}) \right\vert \leq GAP$.
Now, by Proposition~\ref{prop:van}, we know that with probability at least $1 - \alpha_0$, $\| \beta^{\star} - \beta^{0} \|_2^2 = \mathcal{O}\left(s^0\log(m)/n\right)$, and $ c_1 \lambda s^{\star} = \mathcal{O}\left(c_1 s^0\log(m)/n\right)$. Moreover, by the RE condition, with probability at least $1- c' e^{-cn}$, $c_1 = \mathcal{O}(1)$. Finally, using concentration inequalities for the Gaussian SEM noise $\mathcal{E}$ \citep[e.g.][]{bulmann2011statistics}, the probability of the set $\mathcal{T}$ is lower bounded by the probability that $\| \beta^{\star} - \beta^{0} \|_2^2 = \mathcal{O}\left(s^0\log(m)/n\right)$, which is $1- \alpha_0$.
Thus, stopping the branch-and-bound algorithm when $GAP = \mathcal{O}(\lambda s^{0})$ guarantees that, with probability at least $\min\{1- \alpha_0, 1- c' e^{-cn}\}$, $\|\hat{\beta}-\beta^0\|_2^2 = \mathcal{O}\left( s^0 \log(m) / n \right)$.
\end{proof}
Proposition~\ref{EarlyProp} suggests that the algorithm can be stopped by setting a threshold $c^{\star} \lambda s^{0}$ on the value of $GAP = | UB - LB |$ for a constant $c^{\star} > 0$, say $c^{\star}=1$. Such a solution will then achieve the same desirable statistical properties as the optimal solution $\beta^{\star}$. However, while $\lambda$ can be chosen data-adaptively (as discussed in Section~\ref{Sec: Computational}), the value for $s^0$ is not known. However, one can find an upper bound for $s^0$ based on the number of edges in the super-structure $\mathcal{M}$. In particular, if $\mathcal{M}$ is the moral graph \citep{pearl2009causal} with $s_m$ edges, then $s^0 \leq s_m$.
Thus, in this case, a consistent parameter estimate can be obtained if the branch-and-bound process is stopped when $GAP \le s_m \lambda$.
The above results, including the specific choice of early stopping criterion, are also valid if the super-structure $\mathcal{M}$ corresponding to the moral graph is not known \emph{a priori}. That is because the moral graph can be consistently estimated from data using recent developments in graphical modeling; see \citet{drton2017structure} for a review of the literature. While some of the existing algorithms based on $\ell_1$-penalty require an additional \emph{irrepresentability} condition \citep{meinshausen2006high, saegusa2016joint}, this assumption can be relaxed by using instead an adaptive lasso penalty or by thresholding the initial lasso estimates \citep{bulmann2011statistics}.
In light of Proposition \ref{EarlyProp}, it is of great interest to develop algorithms that converge to a solution with a small optimality gap expeditiously. To achieve this, one approach is to obtain better lower bounds using the branch-and-bound process from strong mathematical formulations for \eqref{eq:PNLMform}. To this end, we next review existing formulations of \eqref{eq:PNLMform}.
\section{Existing Formulations of DAG Learning with Linear SEMs} \label{Sec: Previous work}
In this section, we review known mathematical formulations for DAG learning with linear SEMs. We first outline the necessary notation below. \\ \\
\noindent \textbf{Index Sets}\\
$V = \{1,2,\dots,m\}$: index set of random variables;\\
$\mathcal{D}= \{1,2,\dots,n\}$: index set of samples. \vspace{0.1in}\\
\noindent \textbf{Input} \\
$\mathcal{M}=(V,E)$: an undirected super-structure graph (e.g., the moral graph);\\
$\overrightarrow{\mathcal{M}}=(V,\overrightarrow{E})$: the bi-directional graph corresponding to the undirected graph $\mathcal{M}$; \\
$\mathcal{X} = (\mathcal{X}_1, \dots, \mathcal{X}_m)$, where $\mathcal{X}_v = (x_{1v}, x_{2v}, \dots, x_{nv})^{\top}$ and $x_{dv}$ denotes $d$th sample ($d \in \mathcal{D}$) of random variable $X_v$; note $\mathcal{X} \in \mathbb{R}^{n \times m}$; \\
$\lambda_n:$ tuning parameter (penalty coefficient for $\ell_0$ regularization).\\
\noindent \textbf{Continuous optimization variables} \\
$\beta_{jk}$: weight of arc $(j, k)$ representing the regression coefficients $\forall (j,k) \in \overrightarrow{E}$.\\
\noindent\textbf{Binary optimization variables} \\
$z_{jk}=1 \, \, \text{if arc} \, \, (j, k) \, \text{exists in a DAG}; \text{otherwise} \, 0, \, \forall (j,k) \in \overrightarrow{E}$, \\
$g_{jk}=1 \, \, \text{if} \, \, \beta_{jk} \neq 0; \, \text{otherwise} \, \, 0, \, \forall (j,k) \in \overrightarrow{E}$.
Let $F(\beta, g)= \sum_{k\in V}\sum_{d\in \mathcal{D}} \Big(x_{dk}-\sum_{(j,k) \in \overrightarrow{E}} \beta_{jk}x_{dj}\Big)^2 + \lambda_n\sum_{(j,k) \in \overrightarrow{E}} g_{jk}$.
The PNL$\mathcal{M}$ can be cast as the following optimization problem:
\begin{subequations}
\begin{alignat}{3}
\label{CP-obj} \quad \quad \underset{}{\min}\quad & \, F(\beta, g), \\
&\label{CP-con2}\mathcal{G}(B) \, \, \text{induces a DAG from} \, {\overrightarrow{\mathcal{M}}}, \\
& \label{CP-con1} \beta_{jk}(1-g_{jk})=0, \quad && \forall (j,k) \in \overrightarrow{E,}\\
&\label{CP-con3} g_{jk} \in \{0,1\},\quad && \forall (j,k) \in \overrightarrow{E}.
\end{alignat} \label{MIQP1}
\end{subequations}
\noindent The objective function \eqref{CP-obj} is an expanded version of $\mathcal L(\beta)$ in PNL$\mathcal{M}$, where we use the indicator variable $g_{jk}$ to encode the $\ell_0$ regularization. The constraints in \eqref{CP-con2} rule out cycles. The constraints in \eqref{CP-con1} are non-linear and stipulate that $\beta_{jk} \neq 0$ only if $g_{jk}=1$.
There are two sources of difficulty in solving \eqref{CP-obj}-\eqref{CP-con3}: (i) the acyclic nature of DAG imposed by the combinatorial constraints in \eqref{CP-con2}; (ii) the set of \textit{nonlinear} constraints in \eqref{CP-con1}, which stipulates that $\beta_{jk} \neq 0$ only if there exists an arc $(j,k)$ in $\mathcal{G}(B)$.\ In Section \ref{lit1}, we discuss related studies to address the former, whereas in Section \ref{lit2} we present relevant literature for the latter.
\subsection{Linear encodings of the acyclicity constraints \eqref{CP-con2}} \label{lit1}
There are several ways to ensure that the estimated graph does not contain any cycles. The first approach is to add a constraint for each cycle in the graph, so that at least one arc in this cycle must not exist in $\mathcal G(B)$.\ A \textit{cutting plane} (CP) method is used to solve such a formulation which may require generating an exponential number of constraints. Another way to rule out cycles is by imposing constraints such that the nodes follow a topological order \citep{park2017bayesian}. A topological ordering is a unique ordering of the nodes of a graph from 1 to $m$ such that the graph contains an arc $(j,k)$ if node $j$ appears before node $k$ in the order. We refer to this formulation as \textit{topological ordering} (TO). The \textit{layered network} (LN) formulation proposed by \cite{manzour2019integer} improves the TO formulation by reducing the number of binary variables. \cite{manzour2019integer} discuss these formulations in detail.
Let $\mathcal{C}$ be the set of all possible directed cycles and $\mathcal{C}_A \in \mathcal{C}$ be the set of arcs defining a cycle. The CP formulation removes cycles by imposing the following constraints for \eqref{CP-con2}
\begin{equation} \label{CE}
\textbf{CP} \quad \sum_{(j,k ) \in \, \mathcal{C}_A} g_{jk} \leq |\mathcal{C}_A|-1, \quad \forall \mathcal{C}_A \in \mathcal{C}.
\end{equation}
Define decision variables $z_{jk} \in \{0,1\}$ for all $(j,k) \in \overrightarrow{E}$ and $o_{rs} \in \{0,1\}$ for all $r, s \in \{1, \dots, m\}$. The variable $z_{jk}$ takes value 1 if there is an arc $(j,k)$ in the network, and $o_{rs}$ takes value 1 if the topological order of node $r$ equals $s$. The TO formulation rules out cycles in the graph by the following constraints
\begin{subequations}\label{TO}
\begin{alignat}{3}
\textbf{TO} \quad & \label{TO-con3} g_{jk} \leq z_{jk}, \quad && \forall (j,k) \in \overrightarrow{E}, \\
\label{TO-con4} & z_{jk} - m z_{kj} \leq \sum_{s \in V} s \, (o_{ks} - o_{js}), \quad&& \forall (j,k) \in \overrightarrow{E},\\
\label{TO-con5} & \sum_{s \in V} o_{rs} =1 \quad && \forall r \in V, \\
\label{TO-con6} & \sum_{r \in V} o_{rs} =1 \quad &&\forall s \in V.
\end{alignat}
\end{subequations}
The third way to remove cycles is by imposing the condition that the resulting graph is a layered network. This can be achived by the following set of constraints in the LN formulation:
\begin{subequations}\label{LN}
\begin{alignat}{3}
\label{LN-con3} \textbf{LN} \quad & g_{jk} \leq z_{jk} \quad&& \forall (j,k) \in \overrightarrow{E}, \\
\label{LN-con4} & z_{jk} - (m-1) z_{kj} \leq \psi_k - \psi_j \quad &&\forall (j,k) \in \overrightarrow{E}.
\end{alignat} \label{eq:LN}
\end{subequations}
\noindent Let $\psi_k$ be the \textit{layer value} for node $k$. The set of constraints in \eqref{LN-con4} ensures that if the layer of node $j$ appears before that of node $k$ (i.e., there is a direct path from node $j$ to node $k$), then $\psi_k \geq \psi_j + 1$. This rules out any cycles.
The set of constraints in \eqref{LN-con4} imposes that if $z_{ij} = 1$ and $z_{jk} = 1$, then $z_{ik} = 1$. Thus, additional binary vector $g$ along with the set of constraints in \eqref{LN-con3} is needed to correctly encode the $\ell_0$ regularization. Similar reasoning applies for the TO formulation; see \cite{manzour2019integer}.
\subsection{Linear encodings of the non-convex constraints \eqref{CP-con1}} \label{lit2}
The nonconvexity of the set of constraints in \eqref{CP-con1} causes challenges in obtaining provably optimal solutions with existing optimization software. Therefore, we consider convex representations of this set of constraints.
First, we consider a linear representation of the constraints in \eqref{CP-con1}. Although the existing formulations discussed in Section \ref{lit1} differ in their approach to ruling out cycles, one major commonality among them is that they replace the non-linear constraint \eqref{CP-con1} by so called \emph{big-$M$ constraints} given by
\begin{equation}\label{eq:bigM}
-M g_{jk} \leq \beta_{jk} \leq M g_{jk}, \forall (j,k) \in \overrightarrow{E},
\end{equation}
for a large enough $M$. Unfortunately, these big-$M$ constraints \eqref{eq:bigM} are poor approximations of \eqref{CP-con1}, especially in this problem, because no natural and tight value for $M$ exist. Although a few techniques have been proposed for obtaining the big-$M$ parameter for sparse regression problem \citep{bertsimas2017sparse,gomez2018mixed}, the resulting parameters are often too large in practice. Further, finding a tight big-$M$ parameter itself is a difficult problem to solve for DAG structure learning.
Consider \eqref{CP-obj}-\eqref{CP-con3} by substituting \eqref{CP-con1} by the linear big-$M$ constraints \eqref{eq:bigM} and writing the objective function in a matrix form. We denote the resulting formulation, which has a convex quadratic objective and linear constraints, by the following {MIQP}.
\begin{subequations}\label{eq:LNform}
\begin{alignat}{3}
\quad \label{L-obj} \textbf{MIQP}\quad \min & \quad \text{tr}\left[(I- B)(I-B)^{\top}\mathcal{X}^{\top}\mathcal{X}\right] + \lambda_n \sum_{(j,k) \in \overrightarrow{E}} g_{jk}\\
& \eqref{CP-con2}, \eqref{eq:bigM} \label{LN-con1} \\
& \label{LN-con5} g_{jk} \in \{0,1\} \quad \forall (j,k) \in \overrightarrow{E}.
\end{alignat}
\end{subequations}
Depending on which types of constraints are used in lieu of \eqref{CP-con2}, as explained in Section \ref{lit1}, {MIQP} \eqref{eq:LNform} results in three different formulations: {MIQP+CP}, which uses \eqref{CE}, {MIQP+TO}, which uses \eqref{TO}, and {MIQP+LN}, which uses \eqref{LN}, respectively.
To discuss the challenges of the big-$M$ approach, we give a definition followed by two propositions.
\begin{definition}\label{def:2}
A formulation $A$ is said to be \emph{stronger} than formulation $B$ if $\mathcal{R}(A) \subset \mathcal{R} (B)$ where $\mathcal{R}(A)$ and $\mathcal{R}(B)$ correspond to the feasible regions of continuous relaxations of $A$ and $B$, respectively.
\end{definition}
\begin{prop}{(Proposition 3 in \cite{manzour2019integer})}
{\it The {MIQP+TO} and {MIQP+CP} formulations are stronger than the {MIQP+LN} formulation.} \label{Prop:strong}
\end{prop}
\begin{prop}{(Proposition 5 in \cite{manzour2019integer}) \label{Prop2}}
\it{Let $\beta^{\star}_{jk}$ denote the optimal coefficient associated with an arc $(j,k) \in \overrightarrow{E}$ from problem \eqref{eq:PNLMform}.\ For the same variable branching in the branch-and-bound process, the continuous relaxations of the {MIQP+LN} formulation for $\ell_0$ regularizations attain the same optimal objective function value as {MIQP+TO} and {MIQP+CP}, if $M \geq 2 \underset{(j,k) \in \overrightarrow{E}}{\max} \, |\beta^{\star}_{jk}|$.} \label{Prop5: BB}
\end{prop}
Proposition \ref{Prop:strong} implies that the {MIQP+TO} and {MIQP+CP} formulations are stronger than the {MIQP+LN} formulation. Nonetheless, Proposition \ref{Prop5: BB} establishes that for sufficiently large values of $M$, stronger formulations attain the same continuous relaxation objective function value as the weaker formulation throughout the branch-and-bound tree. The optimal solution to the continuous relaxation of MIQP formulations of DAG structure learning may not be at an extreme point of the convex hull of feasible points. Hence, stronger formulations do not necessarily ensure better lower bounds. This is in contrast to a mixed-integer program (MIP) with linear objective, whose continuous relaxation is a linear program (LP). In that case, there exists an optimal solution that is an extreme point of the corresponding feasible set.\ As a result, a better lower bound can be obtained from a stronger formulation that better approximates the convex hull of a mixed-integer linear program; this generally leads to faster convergence. A prime example is the traveling salesman problem (TSP), for which stronger formulations attain better computational performance \citep{oncan2009comparative}. In contrast, the numerical results by \cite{manzour2019integer} show that {MIQP+LN} has better computational performance because it is a compact formulation with the fewest constraints and the same continuous relaxation bounds.
Our next result, which is adapted from \cite{dong2015regularization} to the DAG structure learning problem, shows that the continuous relaxation of {MIQP} is equivalent to the optimal solution to the problem where the $\ell_0$-regularization term is replaced with an $\ell_1$-regularization term (i.e., $\|\beta\|_1=\sum_{(j,k) \in \overrightarrow{E}}|\beta_{jk}|$) with a particular choice of the $\ell_1$ penalty. This motivates us to consider tighter continuous relaxation for MIQP.
Let $(\beta^R, g^R)$ be an optimal solution to the continuous relaxation of {MIQP}.
\begin{prop}
For $M \geq 2 \underset{(j,k) \in \overrightarrow{E}}{\max} \, |\beta^R_{jk}|$, a continuous relaxation of {MIQP} \eqref{eq:LNform}, where the binary variables are relaxed, is equivalent to the problem where the $\ell_0$ regularization term is replaced with an $\ell_1$-regularization term with penalty parameter $\tilde{\lambda}=\frac{\lambda_n}{M}$. \label{L1}
\end{prop}
\begin{proof}
For $M \geq 2 \underset{(j,k) \in \overrightarrow{E}}{\max} \, |\beta^R_{jk}|$, the value $g^R_{jk}$ is $\frac{\beta^R_{jk}}{M}$ in an optimal solution to the continuous relaxation of {MIQP} \eqref{eq:LNform}. Otherwise, we can reduce the value of the decision variable $g^R$ without violating any constraints while reducing the objective function. Note that since $M \geq 2 \underset{(j,k) \in \overrightarrow{E}}{\max} \, |\beta^R_{jk}|$, we have $\frac{\beta_{jk}^R}{M} \leq 1, \, \forall (j,k) \in \overrightarrow{E}$. To show that the set of constraints in \eqref{CP-con2} is satisfied, we consider the set of CP constraints. In this case, the set of constraints \eqref{CP-con2} holds, i.e., $\sum_{(j,k ) \in \, \mathcal{C}_A} \frac{\beta^{R}_{jk}}{M} \leq |\mathcal{C}_A|-1, \quad \forall \mathcal{C}_A \in \mathcal{C}$, because $M \geq 2 \underset{(j,k) \in \overrightarrow{E}}{\max} \, |\beta^R_{jk}|$.
This implies that $g_{jk}^R=\frac{\beta_{jk}^R}{M}$ is the optimal solution. Thus, the objective function reduces to $\ell_1$ regularization with the coefficient $\frac{\lambda_n}{M}$.
Finally, Proposition \ref{Prop2} establishes that for $M \geq 2 \underset{(j,k) \in \overrightarrow{E}}{\max} \, |\beta^\star_{jk}|$, the objective function value of the continuous relaxations of {MIQP+CP}, {MIQP+LN} and {MIQP+TO} are equivalent. This implies that the continuous relaxations of all formulations are equivalent, which completes the proof.
\end{proof}
Despite the promising performance of {MIQP+LN}, its continuous relaxation objective function value provides a weak lower bound due to the big-$M$ constraints. To circumvent this issue, a natural strategy is to improve the big-$M$ value. Nonetheless, existing methods which ensure a valid big-$M$ value or heuristic techniques \citep{park2017bayesian,gomez2018mixed} do not lead to tight big-$M$ values. For instance, the heuristic technique by \cite{park2017bayesian} to obtain big-$M$ values always satisfies the condition in Proposition \ref{Prop:strong} and exact techniques are expected to produce even larger big-$M$ values. Therefore, we next directly develop tighter approximations for \eqref{CP-con1}.
\section{New Perspectives for Mathematical Formulations of DAG Learning} \label{Sec: Math models}
In this section, we discuss improved mathematical formulations for learning DAG structure of a BN based on convex (instead of linear) encodings of the constraints in \eqref{CP-con1}.
Problem \eqref{MIQP1} is an MIQP with non-convex complementarity constraints \eqref{CP-con1}, a class of problems which has received a fair amount of attention from the operations research community over the last decade \citep{frangioni2006perspective, frangioni2007sdp, frangioni2009computational, frangioni2011projected, gomez2018mixed}. There has also been recent interest in leveraging these developments to solve sparse regression problems with $\ell_0$ regularization \citep{pilanci2015sparse, dong2015regularization, xie2018ccp, atamturk2019rank,wei2019convexification}.
Next, we review applications of MIQPs with complementarity constraints of the form \eqref{CP-con1} for solving sparse regression with $\ell_0$ regularization. \cite{frangioni2011projected} develop a so-called projected perspective relaxation method, to solve the perspective relaxation of mixed-integer nonlinear programming problems with a convex objective function and complementarity constraints. This reformulation requires that the corresponding binary variables are not involved in other constraints. Therefore, it is suitable for $\ell_0$ sparse regression, but cannot be applied for DAG structure learning. \cite{pilanci2015sparse} show how a broad class of $\ell_0$-regularized problems, including sparse regression as a special case, can be formulated exactly as optimization problems. The authors use the Tikhonov regularization term $\mu\|\beta\|_2^2$ and convex analysis to construct an improved convex relaxation using the reverse Huber penalty. In a similar vein, \cite{bertsimas2017sparse} exploit the Tikhonov regularization and develop an efficient algorithm by reformulating the sparse regression mathematical formulation as a saddle-point optimization problem with an outer linear integer optimization problem and an inner dual quadratic optimization problem which is capable of solving high-dimensional sparse regressions. \cite{xie2018ccp} apply the perspective formulation of sparse regression optimization problem with both $\ell_0$ and the Tikhonov regularization.
The authors establish that the continuous relaxation of the perspective formulation is equivalent to the continuous relaxation of the formulation given by \cite{bertsimas2017sparse}.
\cite{dong2015regularization} propose perspective relaxation for $\ell_0$ sparse regression optimization formulation and establish that the popular sparsity-inducing concave penalty function known as the minimax concave penalty \citep{zhang2010nearly} and the reverse Huber penalty \citep{pilanci2015sparse} can be obtained as special cases of the perspective relaxation -- thus the relaxations of formulations by \cite{zhang2010nearly,pilanci2015sparse, bertsimas2017sparse, xie2018ccp} are equivalent. The authors obtain an optimal perspective relaxation that is no weaker than any perspective relaxation. Among the related approaches, the optimal perspective relaxation by \cite{dong2015regularization} is the only one that does not explicitly require the use of Tikhonov regularization.
The perspective formulation, which in essence is a fractional non-linear program, can be cast either as a mixed-integer second-order cone program (MISOCP) or a semi-infinite mixed-integer linear program (SIMILP).
Both formulations can be solved directly by state-of-the-art optimization packages.
\cite{dong2015regularization} and \cite{atamturk2019rank} solve the continuous relaxations and then use a heuristic approach (e.g., rounding techniques) to obtain a feasible solution (an upper bound).
In this paper, we directly solve the MISOCP and SIMILP formulations for learning sparse DAG structures.
Next, we present how perspective formulation can be suitably applied for DAG structure learning with $\ell_0$ regularization. We further cast the problem as MISOCP and SIMILP.
To this end, we express the objective function \eqref{L-obj} in the following way:
\begin{subequations}\label{eq:LNform}
\begin{alignat}{3}
\label{PR-obj} \quad & \text{tr}[(I- B)(I-B)^{\top}\mathcal{X}^{\top}\mathcal{X}] + \lambda_n \sum_{(j,k) \in \overrightarrow{E}} g_{jk}\\
\label{PR-obj4} \quad & = \text{tr}[(I- B-B^\top)\mathcal{X}^{\top}\mathcal{X} + 2BB^\top \mathcal{X}^{\top}\mathcal{X} + \lambda_n \sum_{(j,k) \in \overrightarrow{E}} g_{jk}.
\end{alignat}
\end{subequations}
\noindent Let $\delta \in \mathbb{R}_{+}^{m}$ be a vector such that $\mathcal{X}^\top\mathcal{X} -{D}_\delta \succeq 0$, where $D_\delta=\text{diag}(\delta_1, \dots, \delta_m)$ and $A \succeq 0$ means that matrix $A$ is positive semi-definite. By splitting the quadratic term $\mathcal{X}^{\top}\mathcal{X}= (\mathcal{X}^{\top}\mathcal{X} -D_\delta)+D_\delta$ in \eqref{PR-obj4}, the objective function can be expressed as
\begin{equation}\label{eq:LNform1}
\text{tr}\left[(I- B-B^\top)\mathcal{X}^{\top}\mathcal{X} + BB^\top(\mathcal{X}^{\top}\mathcal{X} - D_\delta)\right] + \text{tr}\left(B B^\top D_\delta\right) +\lambda_n \sum_{(j,k) \in \overrightarrow{E}} g_{jk}.
\end{equation}
Let $Q= \mathcal{X}^{\top}\mathcal{X} - D_{\delta}$. (In the presence of Tikhonov regularization with tuning parameter $\mu> 0$, we let $Q= \mathcal{X}^{\top}\mathcal{X} + \mu I- D_{\delta}$ as described in Remark~\ref{rem:L2}.) Then, Cholesky decomposition can be applied to decompose $Q$ as $q^{\top}q$ (note $Q \succeq 0$). As a result, $\text{tr}\left(B B^\top Q\right) = \text{tr}\left(B B^\top {q^\top} {q}\right) = \sum_{i=1}^{m} \sum_{j=1}^{m} \left(\sum_{(\ell,j) \in \overrightarrow{E}} \beta_{\ell j}q_{i\ell}\right)^2$. The separable component can also be expressed as $\text{tr}\left(B B^\top D_\delta\right) = \sum_{j=1}^{m} \sum_{(j,k) \in \overrightarrow{E}} \delta_j\beta_{jk}^2$. Using this notation, the objective \eqref{eq:LNform1} can be written as
\begin{equation}
\nonumber \label{Obj} \text{tr}\left[(I- B-B^\top)\mathcal{X}^{\top}\mathcal{X} + BB^\top Q\right] +\sum_{j=1}^{m} \sum_{(j,k) \in \overrightarrow{E}} \delta_j\beta_{jk}^2 + \lambda_n\sum_{(j,k) \in \overrightarrow{E}} g_{jk}.
\end{equation}
\noindent The Perspective Reformulation (PRef) of {MIQP} is then given by
\begin{subequations}\label{eq:PR}
\begin{alignat}{3}
\textbf{PRef} \label{PR-Obj} \quad \min \quad &\text{tr}\big[(I- B-B^\top)\mathcal{X}^{\top}\mathcal{X} + BB^\top Q\big] + \\ & \nonumber \sum_{j=1}^{m} \sum_{(j,k) \in \overrightarrow{E}} \delta_j \frac{\beta_{jk}^2}{g_{jk}} + \lambda_n\sum_{(j,k) \in \overrightarrow{E}} g_{jk},\\
& \eqref{LN-con1}-\eqref{LN-con5}.
\end{alignat}
\end{subequations}
The objective function \eqref{PR-Obj} is formally undefined when some $g_{jk}$ = 0. More precisely, we use the convention that $\frac{\beta^2_{jk}}{g_{jk}}=0$ when $\beta_{jk} = g_{jk} = 0$ and $\frac{\beta^2_{jk}}{g_{jk}}=+\infty$ when $\beta_{jk} \neq 0$ and $g_{jk}=0$ \citep{frangioni2009computational}. The continuous relaxation of PRef, referred to as the perspective relaxation, is much stronger than the continuous relaxation of MIQP \citep{pilanci2015sparse}. However, an issue with PRef is that the objective function is nonlinear due to the fractional term.
There are two ways to reformulate PRef. One as a mixed-integer second-order conic program (MISOCP) (see, Section \ref{SOCP}) and the other as a semi-infinite mixed-integer linear program (SIMILP) (see, Section \ref{SIP}).
\subsection{Mixed-integer second-order conic program} \label{SOCP}
Let $s_{jk}$ be additional variables representing $\beta_{jk}^2$. Then, the MISOCP formulation is given by
\begin{subequations} \label{eq:misocp}
\begin{alignat}{3}
\textbf{MISOCP}\quad \min \quad &\text{tr}\left[(I- B-B^\top)\mathcal{X}^{\top}\mathcal{X} + BB^\top Q\right] + \\ & \nonumber \sum_{j=1}^{m} \sum_{(j,k) \in \overrightarrow{E}} \delta_j s_{jk} + \lambda_n \sum_{(j,k) \in \overrightarrow{E}} g_{jk},\\
& \label{SOCP-C1} s_{jk}g_{jk} \geq \beta_{jk}^2 \quad (j,k) \in \overrightarrow{E},\\
& \label{SOCP-C2} 0\le s_{jk} \leq M^2 g_{jk} \quad (j,k) \in \overrightarrow{E},\\
& \eqref{LN-con1}-\eqref{LN-con5}.
\end{alignat}
\end{subequations}
Here, the constraints in \eqref{SOCP-C1} imply that $\beta_{jk} \neq 0$ only when $z_{jk} = 1$. The constraints in \eqref{SOCP-C1} are second-order conic representable because they can be written in the form of $\sqrt{4\beta_{jk}^2+ (s_{jk}-g_{jk})^2}\leq s_{jk}+g_{jk}$. The set of constraints in \eqref{SOCP-C2} is valid since $\beta_{jk} \leq Mg_{jk}$ implies $\beta_{jk}^2 \leq M^2g^2_{jk}= M^2g^2_{jk}$ and $g_{jk}^2=g_{jk}$ for $g_{jk} \in \{0,1\}$. The set of constraints in \eqref{SOCP-C2} is not required, yet they improve the computational efficiency especially when we restrict the big-$M$ value. \cite{xie2018ccp} report similar behavior for sparse regression.
When we relax $g_{jk}\in \{0,1\}$ and let $g_{jk}\in[0,1]$, we obtain the continuous relaxation of {MISOCP} \eqref{eq:misocp}. Let us denote the feasible region of continuous relaxation of {MISOCP} \eqref{eq:misocp} and {MIQP} \eqref{eq:LNform} by $\mathcal{R}$MISOCP and $\mathcal{R}$MIQP, and the objective function values by OFV($\mathcal{R}$MISOCP) and OFV($\mathcal{R}$MIQP), respectively. For a more general problem than ours, \cite{cui2013convex} give a detailed proof establishing that the feasible region of the former is contained in the feasible region of latter i.e., $\mathcal{R}$MISOCP $\subset \mathcal{R}MIQP$. This implies that OFV($\mathcal{R}$MISOCP) $ \not > $ OFV($\mathcal{R}$MIQP). Therefore, we are able to obtain stronger lower bounds using MISOCP than MIQP.
\subsection{Mixed-integer semi-infinite integer linear program} \label{SIP}
An alternative approach to reformulate PRef is via \textit{perspective cuts} developed by \cite{frangioni2006perspective,frangioni2007sdp}. To apply perspective cuts, we use the reformulation idea first proposed in \cite{frangioni2006perspective} by introducing dummy decision matrix $D$ to distinguish the separable and non-separable part of the objective function; we also add the additional constraint $d = \beta$ where $d_{jk}$ is $(j,k)$ element of matrix $D$ and $\beta$ is the decision variable in the optimization problem. Following this approach, {MIQP} can be reformulated as an SIMILP:
\begin{subequations}
\begin{alignat}{3}
\textbf{SIMILP}\quad \min \quad &\text{tr}\left[(I- B-B^\top)\mathcal{X}^{\top}\mathcal{X} + DD^\top Q\right] + \\ & \nonumber \sum_{j=1}^{m} \sum_{(j,k) \in \overrightarrow{E}} \delta_j v_{jk} + \lambda_n \sum_{(j,k) \in \overrightarrow{E}} g_{jk}, \\
& \label{SIP-C0} d_{jk} = {\beta}_{jk} \quad (j,k) \in \overrightarrow{E}, \\
& \label{SIP-C1} v_{jk} \geq 2 \bar{\beta}_{jk}\beta_{jk} - \bar{\beta}_{jk}^2g_{jk} \quad \forall \bar{\beta}_{jk} \in [-M, M] \quad \forall (j,k) \in \overrightarrow{E}, \\
& \label{SIP-C2} \eqref{LN-con1}-\eqref{LN-con5}, \\
& v_{jk} \geq 0, \quad (j,k) \in \overrightarrow{E}.
\end{alignat}
\end{subequations}
The set of constraints in \eqref{SIP-C1} are known as perspective cuts. Note that there are infinitely many such constraints. Although this problem cannot be solved directly, it lends itself to a delayed cut generation approach whereby a (small) finite subset of constraints in \eqref{SIP-C1} is kept, the current solution $(\beta^{\star}, g^{\star}, v^{\star})$ of the relaxation is obtained, and all the violated inequalities for the relaxation solution are added for $\bar{\beta}_{jk} = \frac{\beta^{\star}_{jk}}{g^{\star}_{jk}}$ (assuming $\frac{0}{0} = 0$). This process is repeated until termination criteria are met. This procedure can be implemented using the cut callback function available by off-the-shelf solvers such as Gurobi or CPLEX.
\subsection{Selecting $\delta$} \label{deltavalue}
In the MISOCP and SIMILP formulations, one important question is how to identify a valid $\delta$. A natural choice is diag$(\delta) = (\lambda_{\min} - \varepsilon)e$ where $\lambda_{\min}$ is the minimum eigenvalue of $\mathcal{X}^\top\mathcal{X}$, $\varepsilon > 0$ is a sufficiently small number to avoid numerical instability of estimating eigenvalues, and $e$ is a column vector of ones. The issue with this approach is that if $\lambda_{\min} =0$, then $\text{diag}({\delta})$ becomes a trivial 0 matrix. If $\text{diag}(\delta)$ turns out to be a zero matrix, then MISOCP formulation reduces to the big-$M$ formulation. \cite{frangioni2007sdp} present an effective approach for obtaining a valid $\delta$ by solving the following semidefinite program (SDP)
\begin{subequations}
\begin{alignat}{3}
\label{delta} \max \left\{\sum_{i \in V} \delta_i \,|\, \mathcal{X}^\top \mathcal{X} - \diag(\delta) \succeq 0, \delta_i \geq 0\right\}.
\end{alignat}
\end{subequations}
This formulation can attain a non-zero $D_{\delta}$ even if $\lambda_{\min}=0$. Numerical results by \cite{frangioni2007sdp} show that this method compares favorably with the minimum eigenvalue approach. \cite{zheng2014improving} propose an SDP approach, which obtains $D_{\delta}$ such that the continuous relaxation of {MISOCP} \eqref{eq:misocp} is as tight as possible.
Similar to \cite{dong2015regularization}, our formulation does not require adding a Tikhonov regularization. In this case, PRef is effective when $\mathcal{X}^\top\mathcal{X}$ is sufficiently diagonally dominant.
When $n \geq m$ and each row of $\mathcal{X}$ is independent, then $\mathcal{X}^\top\mathcal{X}$ is guaranteed to be a positive semi-definite matrix \citep{dong2015regularization}. On the other hand, when $n < m$, $\mathcal{X}^\top\mathcal{X}$ is not full-rank. Therefore, a Tikhonov regularization term should be added with sufficiently large $\mu$ to make $\mathcal{X}^\top\mathcal{X} + \mu I \succeq 0 $ \citep{dong2015regularization} in order to benefit from the strengthening provided by PRef.
\section{Experiments} \label{Sec: Computational}
In this section, we report the results of our numerical experiments that compare different formulations and evaluate the effect of different tuning parameters and estimation strategies. Our experiments are performed on a cluster operating on UNIX with Intel Xeon E5-2640v4 2.4GHz.\ All formulations are implemented in the Python programming language. Gurobi 8.1 is used as the solver. Unless otherwise stated, a time limit of $50m$ (in seconds), where $m$ denotes the number of nodes, and an MIQP relative optimality gap of $0.01$ are imposed across all experiments after which runs are aborted. The \emph{relative} optimality gap is calculated by RGAP$:=\frac{UB(X)-LB(X)}{UB(X)}$ where UB(X) denotes the objective value associated with the best feasible integer solution (incumbent) and LB(X) represents the best obtained lower bound during the branch-and-bound process for the formulation $X \in \{{MIQP}, {SIMILP}, {MISOCP}\}$.
Unless otherwise stated, we assume $\lambda_n=\ln(n)$ which corresponds to the Bayesian information criterion (BIC) score. To select the big-$M$ parameter, $M$, in all formulations we use the proposal of \citet{park2017bayesian}. Specifically, given $\lambda_n$, we solve each problem without cycle prevention constraints and obtain $\beta^R$. We then use the upper bound $M = 2 \underset{(j,k) \in \overrightarrow{E}}{\max} \, |\beta^R_{jk}|$.
Although this value does not guarantee an upper bound for $M$, the results provided in \cite{park2017bayesian} and \cite{manzour2019integer} computationally confirm that this approach gives a large enough value of $M$.
The goals of our computational study are twofold. First, we compare the various mathematical formulations to determine which gives us the best performance in Subsection~\ref{sec:synth-data}, compare the sensitivity to the model parameters in Subsection~\ref{lambda}, and the choice of the regularization term in Subsection~\ref{sec:compl2}.
Second, in Subsection~\ref{sec:compearly} we use the best-performing formulation to investigate the implications of the early stopping condition on the quality of the solution with respect to the true graph. To be able to perform such a study, we use synthetic data so that the true graph is available.
We use the package \texttt{pcalg} in \texttt{R} to generate random graphs. First, we create a DAG by \texttt{randomDAG} function and assign random arc weights (i.e., $\beta$) from a uniform distribution, $\mathcal{U}[0.1, 1]$.
Next, the resulting DAG and random coefficients are fed into the \texttt{rmvDAG} function to generate multivariate data based on linear SEMs (columns of matrix $\mathcal X$) with the standard normal error distribution. We consider $m\in\{10,20,30,40\}$ nodes and $n=100$ samples. The average outgoing degree of each node, denoted by $d$, is set to two. We generate 10 random Erd\H{o}s-R\'enyi graphs for each setting ($m$, $n$, $d$).
We observe that in our instances, the minimum eigenvalue of $\mathcal{X}^\top \mathcal{X}$ across all instances is 3.26 and the maximum eigenvalue is 14.21.
Two types of problem instances are considered: (i) a set of instances with known moral graph corresponding to the true DAG; (ii) a set of instances with a complete undirected graph, i.e., assuming no prior knowledge. We refer to the first class of instances as \textit{moral} instances and to the second class as \textit{complete} instances. The observational data, $\mathcal{X}$, for both classes of instances are the same. The function \texttt{moralize(graph)} in the \texttt{pcalg} \texttt{R}-package is used to generated the moral graph from the true DAG. Although the moral graph can be consistently estimated from data using penalized estimation procedures with polynomial complexity
\citep[e.g.,][]{loh2014high}, the quality of moral graph affects all optimization models. Therefore, we use the true moral graph in our experiments.
\subsection{Comparison of Mathematical Formulations} \label{sec:synth-data}
We use the following MIQP-based metrics to measure the quality of a solution: relative optimality gap (RGAP), computation time in seconds (Time), Upper Bound (UB), Lower Bound (LB), objective function value (OFV) of the initial continuous relaxation, and the number of explored nodes in the branch-and-bound tree ($\#$ BB).
An in-depth analysis comparing the existing mathematical formulations that rely on linear encodings of the constraints in \eqref{CP-con1} for MIQP formulations is conducted by \cite{manzour2019integer}. The authors conclude that {MIQP+LN} formulation outperforms the other MIQP formulations, and the promising performance of MIQP+LN can be attributed to its size: (1) {MIQP+LN} has fewer binary variables and constraints than {MIQP+TO}, (2) {MIQP+LN} is a compact (polynomial-sized) formulation in contrast to {MIQP+CP} which has an exponential number of constraints. Therefore, in this paper,
we analyze the formulations based on the convex encodings of the constraints in \eqref{CP-con1}.
\subsubsection{Comparison of MISOCP formulations} \label{sec:MISOCP-experiments}
We next experiment with MISOCP formulations. For the set of constraints in \eqref{CP-con2}, we use LN, TO, and CP constraints discussed in Section \ref{lit1} resulting in three formulations denoted as {MISOCP+LN}, {MISOCP+TO}, {MISOCP+CP}, respectively. The {MISOCP+TO} formulation fails to find a feasible solution for instances with 30 and 40 nodes, see Table \ref{Details}. For moral instances, the optimality gap for {MISOCP+TO} are 0.000 and 0.021 for instances with 10 and 20 nodes, respectively; for complete instances, the optimality gap for {MISOCP+TO} formulation are 0.009 and 0.272 for instances with 10 and 20 nodes, respectively. Moreover, Table \ref{Details} illustrates that {MISOCP+LN} performs better than {MISOCP+TO} for even small instances (i.e., 10 and 20 nodes).
\begin{table}[t]
\caption{Optimality gaps for {MISOCP+TO} and {MISOCP+LN} formulations} \label{Details}
\centering
\footnotesize{
\begin{tabular}{l|l|ll|l|l}
\hline
& \multicolumn{2}{c}{Moral} & & \multicolumn{2}{c}{Complete} \\ \hline
$m$ & {MISOCP+TO} & {MISOCP+LN} & & {MISOCP+TO} & {MISOCP+LN} \\ \hline
10 & 0.000 & 0.000 & & 0.009 & 0.008 \\
20 & 0.021 & 0.006 & & 0.272 & 0.195 \\
30 & - & 0.010 & & - & 0.195 \\
40 & - & 0.042 & & - & 0.436 \\ \hline
\end{tabular}
\\ ``-" denotes that no feasible solution, i.e., UB, is obtained within the time limit, so optimality gap cannot be computed.
}
\end{table}
For {MISOCP+CP}, instead of incorporating all constraints given by \eqref{CE}, we begin with no constraint of type \eqref{CE}. Given an integer solution with cycles, we detect a cycle and impose a new cycle prevention constraint to remove the detected cycle. Depth First Search (DFS) can detect a cycle in a directed graph with complexity $O(|V|+|E|)$. Gurobi \texttt{LazyCallback} function is used, which allows adding cycle prevention constraints in the branch-and-bound algorithm, whenever an integer solution with cycles is found. The same approach is used by \cite{park2017bayesian} to solve the corresponding MIQP+CP. Note that Gurobi solver follows a branch-and-cut implementation and adds many general-purpose and special-purpose cutting planes.
Figures \ref{Figurea: MISOCP} and \ref{Figureb: MISOCP} show that {MISOCP+LN} outperforms {MISOCP+CP} in terms of relative optimality gap and computational time. In addition, {MISOCP+LN} attains better upper and lower bounds than {MISOCP+CP} (see, Figures \ref{Figurec: MISOCP} and \ref{Figured: MISOCP}). {MISOCP+CP} requires the solution of a second-order cone program (SOCP) after each cut, which reduces its computational efficiency and results in higher optimality gaps than {MISOCP+LN}. {MISOCP+TO} requires many binary variables which makes the problem very inefficient when the network becomes denser and larger as shown in Table \ref{Details}. Therefore, we do not illustrate the {MISOCP+TO} results in Figure \ref{Figure: MIQP}.
\begin{figure*}[t!]
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{MISOCPOptimalityGAP_MIPGAP_}
\caption{RGAPs}
\label{Figurea: MISOCP}
\end{subfigure}%
~
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{MISOCPTime_seconds_}
\caption{Time (in seconds)}
\label{Figureb: MISOCP}
\end{subfigure}
~
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{MISOCPUpperBound}
\caption{Best upper bounds}
\label{Figurec: MISOCP}
\end{subfigure}
~
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{MISOCPLowerBound}
\caption{Best lower bounds}
\label{Figured: MISOCP}
\end{subfigure}
~
\caption{Optimization-based measures for MISOCP+LN (green, left bar) and MISOCP+CP (yellow, right bar) formulations for $n=100$ and $\lambda_n=\ln(n)$.}
\label{Figure: MIQP}
\end{figure*}
\begin{figure*}[t!]
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{SIMILPOptimalityGAP_MIPGAP_}
\caption{RGAPs}
\label{Figurea: SIMILP}
\end{subfigure
~
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{SIMILPTime_seconds_}
\caption{Time (in seconds)}
\label{Figureb: SIMILP}
\end{subfigure}
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{SIMILPUpperBound}
\caption{Best upper bounds}
\label{Figurec: SIMILP}
\end{subfigure}
~
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{SIMILPLowerBound}
\caption{Best lower bounds}
\label{Figured: SIMILP}
\end{subfigure}
\caption{Optimization-based measures for \textbf{MISOCP+LN}, \textbf{MIQP+LN}, and \textbf{MISILP+LN} formulations for $n=100$ and $\lambda_n=\ln(n)$.}
\label{Figure: SIMLP}
\end{figure*}
\begin{figure*}[t!]
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{MIQPandMISOCPOptimalityGAP_MIPGAP_}
\caption{RGAPs}
\label{Figurea: Best}
\end{subfigure
~
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{MIQPandMISOCPTime_seconds_}
\caption{Time (in seconds)}
\label{Figureb: Best}
\end{subfigure}
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{MIQPandMISOCPUpperBound}
\caption{Best upper bounds}
\label{Figurec: Best}
\end{subfigure}
~
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{MIQPandMISOCPLowerBound}
\caption{Best lower bounds}
\label{Figured: Best}
\end{subfigure}
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{BB}
\caption{Number of Branch and Bound nodes}
\label{Figuree: Best}
\end{subfigure}
~
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[scale=0.22]{OFV}
\caption{Continuous relaxation objective function}
\label{Figuref: Best}
\end{subfigure}
\label{Figure: Best}
\caption{Optimization-based measures for \textbf{MISOCP+LN}, \textbf{MIQP+LN} formulations for $n=100$ and $\lambda_n=\ln(n)$.}
\label{Figure: MISOCP-MIQP}
\end{figure*}
\subsubsection{Comparison of MISOCP versus SIMILP} \label{sec:MISOCP ver MIMILP-experiments}
Our computational experiments show that the SIMILP formulation generally performs poorly when compared to {MISOCP+LN} and {MIQP+LN} in terms of optimality gap, upper bound, and computational time. We report the results for {SIMILP+LN}, {MISOCP+LN}, and {MIQP+LN} formulations in Figure \ref{Figure: SIMLP}. We only consider the LN formulation because that is the best performing model among the alternatives both for MISOCP and MIQP formulations.
Figures \ref{Figurea: SIMILP} and \ref{Figureb: SIMILP} show the relative optimality gaps and computational times for these three formulations. Figures \ref{Figurec: SIMILP} and \ref{Figured: SIMILP} demonstrate that {SIMILP+LN} attains lower bounds that are comparable with other two formulations. In particular, for complete instances with large number of nodes, {SIMILP+LN} attains better lower bounds than {MIQP+LN}. Nonetheless, {SIMILP+LN} fails to obtain good upper bounds. Therefore, the relative optimality gap is considerably larger for {SIMILP+LN}.
The poor performance of {SIMILP+LN} might be because state-of-the-art optimization packages (e.g., Gurobi, CPLEX) use many heuristics to obtain a good feasible solution (i.e., upper bound) for a compact formulation. In contrast, SIMILP is not a compact formulation, and we build the SIMILP gradually by adding violated constraints iteratively. Therefore, a feasible solution to the original formulation is not available while solving the relaxations with a subset of the constraints \eqref{SIP-C1}. Moreover, the optimization solvers capable of solving MISOCP formulations have witnessed noticeable improvement due to theoretical developments in this field. In particular, Gurobi reports 20\% and 38\% improvement in solution time for versions 8 and 8.1, respectively. In addition, Gurobi v8.1 reports over four times faster solution times than CPLEX for solving MISOCP on their benchmark instances.
\subsubsection{Comparison of MISOCP versus MIQP formulations} \label{sec:MISOCP ver MIQP experiments}
In this section, we demonstrate the benefit of using the second-order conic formulation {MISOCP+LN} instead of the linear big-$M$ formulation {MIQP+LN}. As before, we only consider the LN formulation for this purpose. Figures \ref{Figurea: Best} and \ref{Figureb: Best} show that {MISOCP+LN} performs better than MIQP+LN in terms of the average relative optimality gap across all number of nodes $m \in \{10,20,30,40\}$. The only exception is $m=40$ for moral instances, for which {MIQP+LN} performs better than {MISOCP+LN}. Nonetheless, we observe that {MISOCP+LN} clearly outperforms {MIQP+LN} for complete instances which are more difficult to solve.
Figures \ref{Figurec: Best} and \ref{Figured: Best} show the performance of both formulations in terms of the resulting upper and lower bounds on the objective function. We observe that {MISOCP+LN} attains better lower bounds especially for complete instances. However, {MISOCP+LN} cannot always obtain a better upper bound. In other words, {MISOCP+LN} is more effective in improving the lower bound instead of the upper bound as expected.
Figures \ref{Figuree: Best} and \ref{Figuref: Best} show that {MISOCP+LN} uses fewer branch-and-bound nodes and achieves better continuous relaxation values than {MIQP+LN}.
\subsection{Analyzing the Choices of $\lambda_n$ and $M$} \label{lambda}
We now experiment on different values for $\lambda_n$ and $M$ to assess the effects of these parameters on the performance of {MISOCP+LN} and {MIQP+LN}. First, we change $\lambda_n \in \{\ln{(n)}, 2\ln(n), 4\ln(n)\}$ while keeping the value of $M$ the same (i.e., $M=2\underset{(j,k) \in \overrightarrow{E}}{\max} \, |\beta^\star_{jk}|$). Table \ref{Table: lambda} shows that as $\lambda_n$ increases, {MISOCP+LN} consistently performs better than \linebreak MIQP+LN in terms of the relative optimality gap, computational time, the number of branch-and-bound nodes, and continuous relaxation objective function value. Indeed, the difference becomes even more pronounced for more difficult cases (i.e., complete instances). For instance, for $\lambda_n = 4 \ln(n)=18.4$, the optimality gap reduces from 0.465 to 0.374, an over 24\% improvement.
\begin{table}[t!]
\fontsize{60}{30}\selectfont
\caption{Computational results for different values of $\lambda_n = t \ln(n)$ for $t \in \{1,2,4\}$} \label{Table: IP}
\resizebox{1\textwidth}{!}{
\begin{adjustbox}{}{}
\begin{tabular}{llllllllllllllll|lllllllllllllll} \\ \Xhline{2\arrayrulewidth} \Xhline{2\arrayrulewidth}
&&& \multicolumn{11}{c}{Moral} &&& \multicolumn{11}{c}{Complete} \\
\Xhline{2\arrayrulewidth} \Xhline{2\arrayrulewidth}
& \multicolumn{2}{c}{Instances} & & \multicolumn{2}{c}{RGAP} & & \multicolumn{2}{c}{Time} & & \multicolumn{2}{c}{$\#$ nodes} & & \multicolumn{2}{c}{Relaxation OFV} & & \multicolumn{2}{c}{RGAP} & & \multicolumn{2}{c}{Time} & & \multicolumn{2}{c}{$\#$ nodes} & & \multicolumn{2}{c}{Relaxation OFV} \\
& $m$ &$\lambda_n$ & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP \\
\Xhline{2\arrayrulewidth}
&10 & 4.6 & & * & * & & 3 & 2 & & 1306 & 3715 & & 738.7 & 664.9 & & * & * & & 65 & 74 & & 38850 & 114433 & & 724.4 & 629.3 \\
&10 & 9.2 & & * & * & & 4 & 3 & & 1116 & 2936 & & 784.6 & 693.5 & & * & * & & 31 & 39 & & 15736 & 55543 & & 772.5 & 662.2 \\
&10 & 18.4 & & * & * & & 3 & 2 & & 1269 & 2457 & & 857.0 & 747.5 & & * & * & & 26 & 29 & & 18223 & 41197 & & 844.5 & 720.2 \\
&20 & 4.6 & & * & * & & 69 & 51 & & 46513 & 76261 & & 1474.2 & 1325.8 & &\textbf{0.195} & 0.275 & & 1000 & 1000 & & 101509 & 238765 & & 1404.9 & 1144.5 \\
&20 & 9.2 & & * & * & & 26 & 27 & & 10695 & 31458 & & 1589.6 & 1406.8 & & \textbf{0.152} & 0.250 & & 1000 & 1000 & & 152206 & 274514 & & 1526.9 & 1238.6 \\
&20 & 18.4 & & * & * & & 24 & 36 & & 9574 & 33788 & & 1763.7 & 1552.7 & &\textbf{0.113 }& 0.208 & & 944 & 1000 & & 159789 & 277687 & & 1697.1 & 1395.0 \\
&30 & 4.6 & & \textbf{0.010} & 0.011 & & 378 & 527 & & 121358 & 514979 & & 2230.1 & 2037.7 & & \textbf{0.298} & 0.441 & & 1500 & 1500 & & 38474 & 64240 & & 2024.0 & 1569.7 \\
&30 & 9.2 & & * & * & & 104 & 291 & & 33371 & 248190 & & 2392.4 & 2168.5 & & \textbf{0.239} & 0.395 & & 1500 & 1500 & & 59034 & 71475 & & 2217.5 & 1741.5 \\
&30 & 18.4 & & * & * & & 48 & 74 & & 15649 & 57909 & & 2608.3 & 2383.8 & & \textbf{0.215} & 0.318 & & 1500 & 1500 & & 74952 & 96586 & & 2449.2 & 2006.9 \\
&40 & 4.6 & & 0.042 & \textbf{0.037 }& & 1551 & 1615 & & 664496 & 2565247 & & {2979.3} & 2748.6 & & \textbf{0.436} & 0.545 & & 2000 & 2000 & & 23083 & 49050 & & 2582.0 & 1946.3 \\
&40 & 9.2 & & \textbf{0.024} & 0.036 & & 1125 & 1336 & & 353256 & 1347702 & & 3200.7 & 2923.5 & & \textbf{0.397} & 0.473 & & 2000 & 2000 & & 29279 & 73917 & & 2869.9 & 2216.9 \\
&40 & 18.4 & & \textbf{0.024} & 0.035 & & 1099 & 1375 & & 434648 & 1137666 & & 3521.8 & 3225.4 & & \textbf{0.374} & 0.465 & & 2000 & 2000 & & 31298 & 60697 & & 3240.1 & 2633.1 \\
\Xhline{2\arrayrulewidth}
\multicolumn{14}{l}{\huge{* indicates that the problem is solved to the optimality tolerance.}} && \multicolumn{11}{c}{} \\
\multicolumn{14}{l}{\huge{Better RGAPs are in bold.}} && \multicolumn{11}{c}{}
\end{tabular}
\end{adjustbox}}
\vskip 2ex
\label{Table: lambda}
\end{table}
\begin{table}[t!]
\fontsize{60}{30}\selectfont
\caption{Computational results for different values of $\gamma$} \label{Table: IP}
\resizebox{1\textwidth}{!}{
\begin{adjustbox}{}{}
\begin{tabular}{llllllllllllllll|lllllllllllllll} \\ \Xhline{2\arrayrulewidth} \Xhline{2\arrayrulewidth}
&&& \multicolumn{11}{c}{Moral} &&& \multicolumn{11}{c}{Complete} \\
\Xhline{2\arrayrulewidth} \Xhline{2\arrayrulewidth}
& \multicolumn{2}{c}{Instances} & & \multicolumn{2}{c}{RGAP} & & \multicolumn{2}{c}{Time} & & \multicolumn{2}{c}{$\#$ nodes} & & \multicolumn{2}{c}{Relaxation OFV} & & \multicolumn{2}{c}{RGAP} & & \multicolumn{2}{c}{Time} & & \multicolumn{2}{c}{$\#$ nodes} & & \multicolumn{2}{c}{Relaxation OFV} \\
& $m$ &$\gamma$ & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP \\
\Xhline{2\arrayrulewidth}
&10 & 2 & & * & * & & 3 & 2 & & 1306 & 3715 & & 738.7 & 664.9 & & * & * & & 65 & 74 & & 38850 & 114433 & & 724.4 & 629.3 \\
&10 & 5 & & * & * & & 5 & 2 & & 1433 & 3026 & & 717.9 & 647.1 & & * & * & & 81 & 82 & & 42675 & 130112 & & 705.1 & 607.8 \\
&10 & 10 & & * & * & & 5 & 2 & & 1523 & 2564 & & 712.5 & 641.1 & & * & * & & 74 & 100 & & 35576 & 174085 & & 699.8 & 600.3 \\
&20 & 2 & & * & * & & 69 & 51 & & 46513 & 76261 & & 1474.2 & 1325.8 & & \textbf{0.195} & 0.275 & & 1000 & 1000 & & 101509 & 238765 & & 1404.9 & 1144.5 \\
&20 & 5 & & * & * & & 103 & 156 & & 65951 & 209595 & & 1438.2 & 1274.2 & & \textbf{0.211} & 0.308 & & 1000 & 1000 & & 97940 & 225050 & & 1375.3 & 1080.9 \\
&20 & 10 & & * & * & & 215 & 207 & & 150250 & 349335 & & 1427.7 & 1256.6 & & \textbf{0.230} & 0.310 & & 1000 & 1000 & & 90864 & 257998 & & 1366.3 & 1058.2 \\
&30 & 2 & & \textbf{0.010} & 0.011 & & 378 & 527 & & 121358 & 514979 & & 2230.1 & 2037.7 & & \textbf{0.298} & 0.441 & & 1500 & 1500 & & 38474 & 64240 & & 2024.0 & 1569.7 \\
&30 & 5 & & \textbf{0.011} & 0.014 & & 571 & 620 & & 164852 & 527847 & & 2173.9 & 1950.3 & & \textbf{0.336} & 0.474 & & 1501 & 1500 & & 33120 & 64339 & & 1969.4 & 1448.4 \\
&30 & 10 & & 0.024 & \textbf{0.014} & & 630 & 638 & & 202635 & 585234 & & 2156.5 & 1919.6 & & \textbf{0.349} & 0.480 & & 1500 & 1500 & & 30579 & 77100 & & 1951.2 & 1404.0 \\
&40 & 2 & & 0.042 & \textbf{0.037} & & 1551 & 1615 & & 664496 & 2565247 & & 2979.3 & 2748.6 & & \textbf{0.436} & 0.545 & & 2000 & 2000 & & 23083 & 49050 & & 2582.0 & 1946.3 \\
&40 & 5 & & \textbf{0.045} & 0.047 & & 1643 & 1634 & & 638323 & 1347868 & & 2895.6 & 2635.0 & & \textbf{0.579} & 0.580 & & 2000 & 2000 & & 12076 & 30858 & & 2488.0 & 1751.7 \\
&40 & 10 & & \textbf{0.056} & 0.057 & & 1639 & 1632 & & 599281 & 1584187 & & 2869.2 & 2595.6 & & \textbf{0.585} & 0.594 & & 2000 & 2000 & & 11847 & 30222 & & 2456.1 & 1679.6 \\
\Xhline{2\arrayrulewidth}
\multicolumn{14}{l}{\huge{* indicates that the problem is solved to the optimality tolerance.}} && \multicolumn{11}{c}{} \\
\multicolumn{14}{l}{\huge{Better RGAPs are in bold.}} && \multicolumn{11}{c}{}
\end{tabular}
\end{adjustbox}}
\vskip 2ex
\label{Table: M}
\end{table}
Finally, we study the influence of the big-$M$ parameter. Instead of a coefficient $\gamma=2$ in \cite{park2017bayesian}, we experiment with $M = \gamma \underset{(j,k) \in \overrightarrow{E}}{\max} \, |\beta^{R}_{jk}|$ for $\gamma \in \{2, 5, 10\}$ in Table \ref{Table: M}, where $|\beta^{R}_{jk}|$ denotes the optimal solution of each optimization problem without the constraints to remove cycles. The larger the big-$M$ parameter, the worse the effectiveness of both models.\ However, {MISOCP+LN} tightens the formulation using the conic constraints whereas {MIQP+LN} does not have any means to tighten the formulation instead of big-$M$ constraints which have poor relaxation. For $M > 2 \underset{(j,k) \in \overrightarrow{E}}{\max} \, |\beta^{R}_{jk}|$, {MISOCP+LN} outperforms {MIQP+LN} in all measures, in most cases.
\subsection{The Effect of Tikhonov Regularization} \label{sec:compl2}
In this subsection, we consider the effect of adding a Tikhonov regularization term to the objective (see Remark \ref{rem:L2}) by considering $\mu \in \{0, \ln(n), 2\ln(n)\}$ while keeping the values of $\lambda_n = \ln(n)$ and $M$ the same as before. Table \ref{Table: mu} demonstrates that for all instances with $\mu>0$, {MISOCP+LN} outperforms {MIQP+LN}. For instance, for $m=40$ and $\mu=18.4$, {MISOCP+LN} improves the optimality gap from 0.445 to 0.366, an improvement over 21\%. The reason for this improvement is that $\mu>0$ makes the matrix more diagonally dominant; therefore, it makes the conic constraints more effective in tightening the formulation and obtaining a better optimality gap.
\begin{table}[t!]
\fontsize{60}{30}\selectfont
\caption{Computational results for different values of $\mu$} \label{Table: IP}
\resizebox{1\textwidth}{!}{
\begin{adjustbox}{}{}
\begin{tabular}{llllllllllllllll|lllllllllllllll} \\ \Xhline{2\arrayrulewidth} \Xhline{2\arrayrulewidth}
&&& \multicolumn{11}{c}{Moral} &&& \multicolumn{11}{c}{Complete} \\
\Xhline{2\arrayrulewidth} \Xhline{2\arrayrulewidth}
& \multicolumn{2}{c}{Instances} & & \multicolumn{2}{c}{RGAP} & & \multicolumn{2}{c}{Time} & & \multicolumn{2}{c}{$\#$ nodes} & & \multicolumn{2}{c}{Relaxation OFV} & & \multicolumn{2}{c}{RGAP} & & \multicolumn{2}{c}{Time} & & \multicolumn{2}{c}{$\#$ nodes} & & \multicolumn{2}{c}{Relaxation OFV} \\
& $m$ &$\mu$ & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP & & MISOCP & MIQP \\
\Xhline{2\arrayrulewidth}
&10 & 0 & & * & * & & 3 & 2 & & 1306 & 3715 & & 738.7 & 664.9 & & * & * & & 65 & 74 & & 38850 & 114433 & & 724.4 & 629.3 \\
&10 & 4.6 & & * & * & & 4 & 2 & & 1043 & 2758 & & 802.0 & 708.5 & & * & * & & 69 & 72 & & 38778 & 119825 & & 789.3 & 675.7 \\
&10 & 9.2 & & * & * & & 4 & 2 & & 1067 & 2231 & & 858.0 & 748.1 & & * & * & & 72 & 74 & & 36326 & 114383 & & 843.2 & 712.3 \\
&20 & 0 & & * & * & & 69 & 51 & & 46513 & 76261 & & 1474.2 & 1325.8 & & \textbf{0.195} & 0.2752 & & 1000 & 1000 & & 101509 & 238765 & & 1404.9 & 1144.5 \\
&20 & 4.6 & & * & * & & 45 & 45 & & 15111 & 55302 & & 1604.1 & 1426.5 & & \textbf{0.1666} & 0.2416 & & 1000 & 1000 & & 102467 & 249490 & & 1551.7 & 1267.1 \\
&20 & 9.2 & & * & * & & 43 & 55 & & 15384 & 62297 & & 1716.8 & 1515.7 & &\textbf{0.1422} & 0.2228 & & 1000 & 1000 & & 94360 & 258194 & & 1668.3 & 1355.1 \\
&30 & 0 & & \textbf{0.010} & 0.011 & & 378 & 527 & & 121358 & 514979 & & 2230.1 & 2037.7 & & \textbf{0.298} & 0.4408 & & 1500 & 1500 & & 38474 & 64240 & & 2024.0 & 1569.7 \\
&30 & 4.6 & & \textbf{0.008} & 0.011 & & 310 & 392 & & 76668 & 358544 & & 2432.5 & 2187.7 & & \textbf{0.2368} & 0.387 & & 1500 & 1500 & & 45473 & 69258 & & 2286.4 & 1788.5 \\
&30 & 9.2 & & \textbf{0.009} & 0.010 & & 67 & 377 & & 12410 & 320632 & & 2612.6 & 2311.4 & & \textbf{0.2092} & 0.3666 & & 1500 & 1500 & & 41241 & 68661 & & 2484.3 & 1915.7 \\
&40 & 0 & & 0.042 & \textbf{0.037} & & 1551 & 1615 & & 664496 & 2565247 & & 2979.3 & 2748.6 & & \textbf{0.4364} & 0.5452 & & 2000 & 2000 & & 23083 & 49050 & & 2582.0 & 1946.3 \\
&40 & 4.6 & & \textbf{0.027} & 0.029 & & 1331 & 1620 & & 422654 & 1303301 & & 3281.6 & 2972.8 & & \textbf{0.3538} & 0.4708 & & 2000 & 2000 & & 13209 & 30995 & & 2985.4 & 2261.3 \\
&40 & 9.2 & & \textbf{0.020} & 0.028 & & 870 & 1507 & & 239214 & 1762210 & & 3575.4 & 3165.3 & & \textbf{0.3668} & 0.4454 & & 2000 & 2000 & & 13884 & 54638 & & 3321.7 & 2468.7 \\
\Xhline{2\arrayrulewidth}
\multicolumn{14}{l}{\huge{* indicates that the problem is solved to the optimality tolerance.}} && \multicolumn{11}{c}{} \\
\multicolumn{14}{l}{\huge{Better RGAPs are in bold.}} && \multicolumn{11}{c}{}
\end{tabular}
\end{adjustbox}}
\vskip 2ex
\label{Table: mu}
\end{table}
\subsection{Practical Implications of Early Stopping}\label{sec:compearly}
In this subsection, we evaluate the quality of the estimated DAGs obtained from {MISOCP+LN} by comparing them with the ground truth DAG. To this end, we use the average structural Hamming distance $(\mathrm{SHD})$ which counts the number of arc differences (additions, deletions, or reversals) required to transform the estimated DAG to the true DAG. Since Gurobi sets a minimum relative gap RGAP$=1e^{-4}$, the solution obtained within this relative gap is considered optimal. Finally, because the convergence of the branch-and-bound process may be slow in some cases, we set a time limit of 100$m$.
To test the quality of the solution obtained with an early stopping criterion, we set the absolute optimality gap parameter as $GAP=\frac{\log(m)}{n}s_m$ and the $\ell_0$ regularization parameter as $\lambda_n=\log m$ as suggested by Proposition \ref{EarlyProp} for achieving a consistent estimate.
We compare the resulting suboptimal solution to the solution obtained by setting $GAP= UB -LB=0$ to obtain the truly optimal solution.
Table \ref{Early} shows the numerical results for the average solution time (in seconds) for instances that solve within the time limit, the number of instances that were not solved within the time limit, the actual absolute optimality gap at termination, the average SHD of the resulting DAGs, and in parenthesis, the standard deviation of the SHD scores, across 10 runs for moral instances. Table \ref{Early} indicates that the average SHD for $GAP=\frac{\log(m)}{n}s_m$ is close to that of the truly optimal solution. Note that a lower GAP does not necessarily lead to a better SHD score; see, e.g., $m=20$. From a computational standpoint, we observe that by using the early stopping criterion, we are able to obtain consistent solutions before reaching the time limit for more instances. In particular, four instances reach the time limit for GAP=0 before finding the optimal solution as opposed to only two for early stopping, Note that we only report the average solution time if the algorithm terminates before hitting the time limit, which explains why the average time appears smaller for optimal than early stopping. Taking into account the time to obtain the best integer solution, the average time for $m=40$ is 1678.485 seconds for GAP=0, whereas it is 954.79 seconds for the early stopping setting. Furthermore, stopping early does not sacrifice from the quality of the resulting DAG as can be seen from the SHD scores.
\begin{table}[t!]
\centering{
\caption{Structural Hamming distances (SHD) for early stopping with $n= 100, \lambda_n = \log m$, GAP $\leq \tau$ for moral instances. The superscripts $^{i}$ indicate that out of ten runs, $i$ instances reach the time limit of $100m$.}
\footnotesize{
\begin{tabular}{ll|ccc|cccc}
\hline
& & \multicolumn{3}{c|}{$\tau= 0$} & \multicolumn{3}{c}{$\tau=\frac{\log (m)}{n}s_m$} & \\ \hline
$m$ & $s_m$ & Time & GAP & SHD (std) & Time & GAP & SHD (std) \\ \hline
10 & 19 & 0.71 & 0.002 & 0 (0) & 0.64 & 0.002 & 0 (0) \\
20 & 58 & 31.99 & 0.062 & 0.80 (1.23) & 16.84 & 0.165 & 0.55 (1.01) \\
30 & 109 & $51.41^{2}$ & 0.210 & 1.25 (0.89) & $28.27^{2}$ & 0.557 & 1.29 (0.95) \\
40 & 138 & $784.85^{4}$ & 0.370 & 0.67 (0.52) & $1547.90^{2}$ & 1.411 & 0.71 (0.49) \\ \hline
\end{tabular}
\label{Early}}
}
\end{table}
\section{Conclusion} \label{Sec: Conclusion}
In this paper, we study the problem of learning an optimal directed acyclic graph (DAG) from continuous observational data, where the causal effect among the random variables is linear. The central problem is a quadratic optimization problem with regularization. We present a mixed-integer second order conic program ({MISOCP}) which entails a tighter relaxation than existing formulations with linear constraints. Our results show that {MISOCP} can successfully improve the lower bound and results in better optimality gap when compared with other formulations based on big-$M$ constraints, especially for dense and large instances. Moreover, we establish an early stopping criterion under which we can terminate branch-and-bound and achieve a solution which is asymptotically optimal.
\section*{Acknowledgments}
Simge K\"u\c{c}\"ukyavuz and Linchuan Wei were supported, in part, by ONR
grant N00014-19-1-2321.
Ali Shojaie was supported by NSF grant DMS-1561814 and NIH grant R01GM114029.
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\begin{document}
\title{Pathway Tools version 19.0: Integrated Software for Pathway/Genome Informatics and Systems Biology}
\author{
{\bf Peter D. Karp$^{*}$, Mario Latendresse, Suzanne M. Paley, } \\
{\bf Markus Krummenacker Quang Ong, Richard Billington, Anamika Kothari, } \\
{\bf Daniel Weaver, Tom Lee, Pallavi Subhraveti, Aaron Spaulding, } \\
{\bf Carol Fulcher, Ingrid M. Keseler, and Ron Caspi} \\
Bioinformatics Research Group, SRI International\\
333 Ravenswood Ave, Menlo Park, CA 94025 \\
[email protected] \\
\ \\
$^{*}$To whom correspondence should be addressed.\\
}
\date{}
\maketitle
\newcommand{{\sf BioVelo}}{{\sf BioVelo}}
\newcommand{{\sf MetaCyc}}{{\sf MetaCyc}}
\newpage
\section*{Abstract}
Pathway Tools is a bioinformatics software environment with a broad
set of capabilities. The software provides genome-informatics tools
such as a genome browser, sequence alignments, a genome-variant
analyzer, and comparative-genomics operations. It offers
metabolic-informatics tools, such as metabolic reconstruction,
quantitative metabolic modeling, prediction of reaction atom mappings,
and metabolic route search. Pathway Tools also provides
regulatory-informatics tools, such as the ability to represent and
visualize a wide range of regulatory interactions. The software
creates and manages a type of organism-specific database called a
Pathway/Genome Database (PGDB), which the software enables database
curators to interactively edit. It supports web publishing of PGDBs
and provides a large number of query, visualization, and omics-data
analysis tools. Scientists around the world have created more than
9,800 PGDBs by using Pathway Tools, many of which are curated databases
for important model organisms. Those PGDBs can be exchanged using a
peer-to-peer database-sharing system called the PGDB Registry.
\vspace{1 in}
\section*{Biographical Note}
Dr. Peter Karp is the Director of the Bioinformatics Research Group at
SRI International. He received the PhD degree in Computer Science
from Stanford University.
\newpage
\section{Introduction}
Pathway Tools \cite{PTools10,PToolsOverview06,PTools02,EcoCycJCB96} is a
software environment for management, analysis, simulation, and visualization of
integrated collections of genome, pathway, and regulatory
data. This article provides an overview of all current capabilities of
Pathway Tools. A shorter version of this article describing only those
capabilities of Pathway Tools developed since publication of a 2010
article on the software \cite{PTools10} was recently
published in {\em Briefings in Bioinformatics} \cite{PTools15BiB}.
New software capabilities developed since 2010 are typeset
in blue in this article.\footnote{This article incorporates significant text from
\cite{PTools10} by permission of the publisher.
}
Pathway Tools handles many types of information beyond pathways
and offers extensive capabilities. The software has been under continuous
development within the Bioinformatics Research Group (BRG) within SRI
International since the early 1990s. Pathway Tools serves several
different use cases in bioinformatics and systems biology:
\begin{itemize}
\item It supports development of organism-specific databases
(also called model-organism databases) that integrate many
bioinformatics data types.
\item It supports scientific visualization, web publishing, and dissemination of those
organism-specific databases.
\item It performs computational inferences from sequenced genomes including prediction of an
organism's metabolic network, prediction of metabolic pathway hole fillers,
and prediction of operons.
\item It enables creation of steady-state quantitative metabolic flux
models for individual organisms and for organism communities.
\item It provides tools for graph-based analysis of biological networks, such as
for identification of metabolic choke points, dead-end metabolites,
and blocked reactions.
\item It provides tools for analysis of gene expression, metabolomics,
proteomics, and multi-omics datasets.
\item It provides comparative analyses of organism-specific databases.
\item It supports metabolic engineering.
\end{itemize}
Pathway Tools is focused around a type of model-organism database
called a Pathway/Genome Database (PGDB). A PGDB integrates
information about an organism's genes, proteins, metabolic network, and
regulatory network.
Pathway Tools has several components. The {\bf PathoLogic} component
enables users to create a new PGDB from the annotated genome of an
organism, containing the genes,
proteins, biochemical reactions, and predicted metabolic pathways and
operons of the organism.
The {\bf Pathway/Genome Editors} let PGDB developers interactively
refine the contents of a PGDB, such as editing a metabolic pathway or
an operon, or defining the function of a newly characterized gene.
The {\bf Pathway/Genome Navigator} supports querying, visualization,
and analysis of PGDBs. Whereas all other Pathway Tools components run
as desktop applications only, the Navigator can run as both a desktop
application and as a web server. The Navigator enables scientists to
quickly find information, to display that information in familiar
graphical forms, and to publish a PGDB to the scientific community via
the web. The Navigator provides a platform for systems-level analysis
of high-throughput data by providing tools for painting
combinations of gene expression, protein expression, and metabolomics
data onto a full metabolic map of the cell, onto the full genome, and
onto a diagram of the regulatory network of the cell.
\blue{The {\bf MetaFlux} component enables construction and execution of
steady-state metabolic flux models from PGDBs. MetaFlux has modes to
accelerate development of metabolic models, and to use metabolic
models to simulate both gene and reaction knockouts. Pathway Tools
provides a unique environment for metabolic flux modeling: by
combining a tool for reconstructing metabolic networks from genome
annotations with metabolic-model debugging tools such as a reaction
gap-filler, the software enables rapid development of metabolic models
from sequenced genomes. And by tightly coupling the metabolic model
with other enriching information such as the sequenced genome,
chemical structures, and regulatory information, Pathway Tools-based
metabolic models are easier to understand, validate, reuse,
extend, and learn from.}
Pathway Tools includes a sophisticated ontology and database
application programming interface (API) that enables programs to
perform complex queries, symbolic computations, and data mining on the
contents of a PGDB. For example, the software has been used for
global studies of the {\em E. coli}\ metabolic network
\cite{KarpVcell00} and genetic network \cite{KarpSymTheory01}.
Pathway Tools is seeing widespread use across the bioinformatics
community to create Pathway/Genome Databases in all domains of
life. More than
\nPToolsLicensees\ groups to date have licensed the software. As well as supporting the
development of the EcoCyc \cite{EcoCycNAR13} and MetaCyc
\cite{MetaCycNAR14} databases (DBs) at SRI, and SRI's BioCyc
collection of 5,500 PGDBs \cite{MetaCycNAR14}, the software is used
by genome centers, experimental biologists, and groups that are
creating curated DBs for a number of different
organisms (see Section~\ref{sec:pgdbs} for a more detailed listing of
available PGDBs).
This article provides a comprehensive description of Pathway Tools.
Where possible, it references earlier publications that provide more algorithmic
details. However, in some cases, those earlier publications are
outdated by new developments in the software that are described here.
This article also emphasizes new aspects of the software that have not
been reported in earlier publications.
The organization of this article is as follows.
Section~\ref{sec:use-cases} articulates in more detail the use cases for which Pathway Tools was designed.
Section~\ref{sec:creating} relates how a new PGDB is created, and describes the
computational inference procedures within Pathway Tools. It
summarizes Pathway Tools' interactive editing capabilities and
the associated author-crediting system. It also describes tools for automatic
upgrading of a PGDB schema, and for bulk updating of the genome annotation
within a PGDB.
Section~\ref{sec:schema} describes the schema of a PGDB and the
ontologies used by Pathway Tools.
Section~\ref{sec:nav} relates the querying and visualization facilities
of Pathway Tools.
Section~\ref{sec:apis} summarizes the mechanisms for importing and
exporting data from Pathway Tools, and for accessing and updating
PGDB data via APIs.
Section~\ref{sec:analyses} describes multiple Pathway Tools modules
for performing network analyses of PGDBs, such as a tool for identifying dead-end metabolites.
Section~\ref{sec:comparative} describes comparative analysis tools
within the software.
Section~\ref{sec:arch} describes the software architecture of Pathway Tools.
Section~\ref{sec:pgdbs} lists the large family of PGDBs that have been
created by Pathway Tools users outside SRI International, and describes
a peer-to-peer data-sharing facility within Pathway Tools that enables
users to easily exchange their PGDBs.
Section~\ref{sec:metaflux} describes the metabolic modeling
capabilities of the software.
Section~\ref{sec:related} compares Pathway Tools to related efforts.
\section{Pathway Tools Use Cases}
\label{sec:use-cases}
This section articulates the objectives for which Pathway Tools was
designed. Please note that when we assert that Pathway Tools supports
a given type of use case, it does not mean that Pathway Tools provides
every type of computational tool needed in that area. For example,
omics data analysis is a huge field, and although Pathway Tools
contributes novel and useful omics data analysis capabilities, it does
not provide every omics data analysis method: in fact, it is
intended to be used in conjunction with other omics analysis tools
(such as for data normalization). Section~\ref{sec:limitations}
summarizes the limitations of Pathway Tools.
\Ssection{Development of Organism-Specific Databases}
Organism-specific DBs (also known as model-organism DBs) describe the
genome and other information about an organism
\cite{EcoCycNAR13,SGD14,ApiDB07,WormBase14,MGD15}.
We posit that every organism with a completely sequenced genome and an
experimental community of significant size requires an
organism-specific DB to fully exploit the genome sequence. Such DBs
should provide a central information resource that integrates
information dispersed through the scientific literature about the
genome, molecular parts, and cellular networks of the organism. Such
DBs both direct and accelerate further scientific investigations.
Pathway Tools facilitates rapid initial computational construction of
organism-specific DBs, followed by manual refinement of the PGDB, to
produce an extremely rich and accurate DB in minimal time. Our
approach differentiates experimental versus computationally inferred
information whenever possible. Rapid construction of PGDBs is
achieved by importing an annotated genome into a PGDB in the form of a
GenBank file, and by applying several computational inference tools to
infer new information within the PGDB, such as metabolic pathways.
Scientists can then employ the Pathway/Genome Editors to correct and
supplement computational inferences when necessary, and to perform
ongoing manual curation of the PGDB if desired.
The Pathway Tools DB schema (see
Section~\ref{sec:schema}) is significant in both its breadth and its
depth: it models an unusually broad set of bioinformatics data types
ranging from genomes to pathways to regulatory networks, and it
provides high-fidelity representations of those data types that allow
PGDBs to accurately capture complex biology.
\Ssection{Web Visualization and Querying of Organism-Specific Databases}
To speed user comprehension of the complex information within PGDBs,
the Pathway/Genome Navigator provides many scientific visualization
services, including a genome browser, visualization of single metabolic
pathways and entire metabolic maps, visualization of single operons
and of entire regulatory networks, and visualization of chemical
compounds and reactions (see Section~\ref{sec:nav} for more
details). These visualization tools operate within a web server,
permitting developers of PGDBs to publish their PGDBs to the
scientific community through a website. This form of PGDB publishing
supports interactive querying and browsing
using a three-tiered series of web query interfaces (see Section~\ref{sec:query}),
including a quick search, a set of object-specific query tools, and
a tool for interactively constructing queries whose power is
comparable to that of SQL.
We have developed other publishing paradigms to support computational
analysis and dissemination of PGDBs. Pathway Tools APIs exist in
four languages \cite{PTools05} and as web services. PGDBs can be exported in several formats and
imported into the BioWarehouse DB integration system
\cite{Biowarehouse06}. Finally, users can easily share and exchange
PGDBs using a peer-to-peer DB-sharing system that we have
developed.
\Ssection{Extend Genome Annotations with Additional Computational Inferences}
Pathway Tools extends the paradigm of genome analysis. After
traditional analyses such as gene calling and gene function prediction
are performed by external software packages, Pathway Tools provides additional computational genome
analyses that layer additional information above the traditional
genome annotation. Pathway Tools predicts the operons of the
organism. It predicts the metabolic pathways of the organism. It
also predicts which genes in the organism code for missing enzymes in
the predicted metabolic pathways, thus using pathway information to
predict additional gene functions. See Section~\ref{sec:creating} for
more details.
\Ssection{Analysis of Omics Data}
Pathway Tools provides three genome-scale viewers for animated visualization of omics
datasets in the context of the full metabolic network
\cite{PToolsOverview06}, full transcriptional regulatory network, and
full genome (see Sections~\ref{sec:cellov}--\ref{sec:genov} for more
details). It also provides enrichment analysis and SmartTable-based
analysis of omics data.
\Ssection{Quantitative Metabolic Flux Modeling}
The MetaFlux module of Pathway Tools supports development and execution
of steady-state metabolic flux models for individual organisms and
organism communities from PGDBs. MetaFlux supports a
{\em literate modeling} approach that makes metabolic flux models
highly accessible to and understandable by scientists.
\Ssection{Analysis of Biological Networks}
Pathway Tools includes programs for symbolic analysis of biological
networks (see Section~\ref{sec:analyses} for more details) that rely
on the detailed biological network ontology underlying Pathway Tools.
The software identifies dead-end metabolites
and blocked reactions, both of which usually reflect errors or
incompleteness of our knowledge of a metabolic network.
Pathway Tools indirectly supports a two-phased, pathway-based paradigm for drug
discovery. Phase I is the search for essential in vivo metabolic
pathways: pathways whose function is essential for microbial growth in
the host. Phase II is the search for targets within essential in vivo
pathways. Both phases are supported by a Pathway Tools module that
predicts choke-point reactions within the metabolic network as
likely drug targets \cite{Yeh04}.
\Ssection{Comparative Analyses of Organism-Specific Databases}
Pathway Tools provides a suite of comparative analysis operations that
can be applied to multiple user-selected PGDBs (see
Section~\ref{sec:comparative} for more details). Pathway Tools
emphasizes comparisons at the functional level, rather than the
sequence level. Example comparisons include (1) highlighting on the
Cellular Overview of one organism the reactions that it shares (or
does not share) with one or more other organisms; (2) a tabular
comparison of the reaction complements of several organisms,
organized by substrate type (e.g., small molecules, RNAs, proteins) or
by the number of isozymes per reaction; (3) a comparison of the pathway
complements of several organisms, where the tabular pathway comparison
is organized by a pathway ontology; (4) a table showing which genes have
orthologs in which PGDBs; and (5) a comparison of the genome organization of
orthologs using the genome browser.
\Ssection{Metabolic Engineering}
Metabolic engineering is a discipline that seeks to modify the
metabolic network of an organism in a desired fashion, such as to
achieve overproduction of desired end products, or degradation of
specified compounds \cite{Stephan98}. Pathway Tools is designed to
assist metabolic engineers in several respects. Its
metabolic-reconstruction capabilities aid in rapid characterization of
a host organism for metabolic engineering. Its editing tools permit
refinement of that pathway database. Its omics analysis capabilities
aid metabolic engineers in understanding the activity levels of
different portions of the metabolic network under different growth
conditions. Its RouteSearch tool supports design of novel reaction
pathways from a feedstock compound to a desired product compound, and its
metabolic modeling capabilities enable computational exploration of
modified flux routes.
\section{Creating and Curating a PGDB}
\label{sec:creating}
The life cycle of a PGDB typically includes the following three
procedures.
{\bf 1. Initial creation of the PGDB} starts with
one or more input files describing the functionally annotated genome of an
organism. The PathoLogic component of Pathway Tools transforms the
genome into an Ocelot \cite{Karp-JIIS-97a} DB structured according to the Pathway Tools
schema. Next, the user applies one or more computational inference
tools within PathoLogic to the genome to infer new information such as
metabolic pathways. For several of the PathoLogic inference tools, we
have created graphical user interfaces (GUIs) that enable
the user to review the inferences made by these tools, and to accept,
reject, or modify those inferences.
{\bf 2. PGDB curation.} Manual refinement and updating of a PGDB is
performed using the Pathway/Genome Editors. This phase can last for
years, or for decades, as in the case of EcoCyc \cite{EcoCycNAR13}.
Curation can be based on information found about the organism in the
experimental literature, information from in-house experiments, or
information inferred by the curator, perhaps with help from
computational tools. PGDB curation is multidimensional, involving addition and/or deletion of genes or
metabolic pathways to/from the PGDB; changing gene functions; altering
the structure of metabolic pathways; authoring of summary comments for
genes or pathways; attachment of Gene Ontology terms to
genes and gene products; entry of chemical structures for small
molecules; defining regulatory relationships; and entry of data
into many different PGDB fields including protein molecular weights,
pIs, and cellular locations.
{\bf 3. Bulk updating of a PGDB.} A PGDB developer might run an
external program that predicts cellular locations for hundreds of
genes, and want to load those predictions into the
PGDB. Some groups store the
authoritative genome annotation for an organism in another genome-data management
system, and want to periodically import the latest genome annotation
into Pathway Tools; the software provides a tool for this operation.
In addition, some of the individual
components within PathoLogic that were used to initially create a PGDB
can be run again at a later date to take advantage of updated
information.
The following subsections describe these procedures in more detail.
\Ssection{PathoLogic PGDB Creation}
PathoLogic performs a series of computational inferences that are
summarized in Figure~\ref{fig:pathologic}. These inferences can be
performed in an interactive mode, in which the user guides the system
through each step, and can review and modify the inferences made by
the system using interactive tools. PathoLogic can also execute in
a batch mode, in which all processing is automated, to process hundreds
or thousands of genomes.
The input to PathoLogic is
the annotated genome of an organism. PathoLogic does not perform
genome annotation; its input must supply the genome sequence, the
locations of genes, and functions of gene products.
The annotation is supplied as a set of files in GenBank format or PathoLogic format,
each of which describes the annotation of one replicon (chromosome or plasmid),
or of one contig for genomes that are not fully assembled.
When the annotation is provided in PathoLogic format, the sequence is
provided as one or more separate FASTA files.
The annotation specified in a GenBank or PathoLogic file can include
the start and stop positions of the coding region for each gene, and
intron positions. It can also include a description of the function
of the gene product as a text string, one or more EC numbers, and one
or more Gene Ontology terms. The annotation can also include a gene
name, synonyms for the gene name and the product name, links to other
bioinformatics databases, and comments.
\begin{figure}
\begin{center}
\includegraphics[width=6in]{pathologic-flow.jpg}
\caption{Inputs and outputs of the computational inference modules within PathoLogic.
The initial input to PathoLogic is either a GenBank-format file or a PathoLogic-format file.
The pink boxes all indicate that a PGDB is an input to or an output from some processing
step; the notations at the bottom of the pink boxes indicate what types of data have
been added by the previous processing step (for example, the Transport Inference Parser
adds transport reactions to a PGDB).
}
\label{fig:pathologic}
\end{center}
\end{figure}
PathoLogic initializes the schema of the new PGDB by copying from
MetaCyc into the new PGDB the definitions of the ontology
classes and the 350 slots (DB attributes) that define the schema of a
PGDB.
PathoLogic next creates a PGDB object for every replicon and contig
defined by the input files, and for every gene and gene product
defined in the input files. It populates these new objects with data
from the input files, such as gene names and their sequence
coordinates, and gene product names. As a result of these operations,
the new PGDB now mirrors the information in the input files.
\Ssection{PathoLogic Inference of Metabolic Pathways}
\label{sec:pwy-inference}
Pathway Tools predicts the metabolic pathway complement of an organism
by assessing what known pathways from the MetaCyc PGDB \cite{MetaCycNAR14} are present in
the annotated genome of that organism's PGDB. This inference is
performed in two steps that are described and evaluated in
\cite{KarpHpy02,KarpPathPred10,KarpPathPred11}.
{\bf Step 1:} The enzymes in the PGDB are assigned to their corresponding
reactions in MetaCyc, thus defining the reactome of the organism.
PathoLogic performs this assignment by matching to MetaCyc reactions
the gene-product names (enzyme names), the EC numbers, and the Gene
Ontology terms assigned to genes in the genome. The program can use
whatever combination of these three information types is available in
a given genome. For example, the {\em fabD} gene in {\em Bacillus
anthracis} was annotated with the function ``malonyl CoA-acyl carrier
protein transacylase.'' That name was recognized by PathoLogic as
corresponding to the MetaCyc reaction whose EC number is 2.3.1.39.
PathoLogic therefore imported that reaction and its substrates into the
{\em B. anthracis} PGDB, and created an enzymatic-reaction object
linking that reaction to that {\em B. anthracis} protein.
{\bf Step 2:}
Once the reactome of the organism has been established in the
preceding manner, PathoLogic imports into the new PGDB all MetaCyc
pathways that contain at least one reaction in the organism's
reactome. Once imported, PathoLogic then attempts to prune out
those pathways that are likely to be false-positive predictions. That
pruning process considers both the fraction of reaction steps in the
pathway that have assigned enzymes and how many of the reactions with
assigned enzymes are unique to that pathway (as opposed to being used
in additional metabolic pathways in that organism). The remaining
pathways are those predicted to occur in the organism under analysis.
MetaCyc records curated
information about the expected taxonomic groups in which a pathway is
expected to occur (e.g., in plants only) based on experimental observations of that pathway
to date.
The software prunes a predicted
pathway from organism $X$ if organism $X$ is outside the expected
taxonomic distribution of that pathway.
\SSsection{Calculation of Pathway Abundance for Metagenomics Analysis}
\blue{
PathoLogic computes abundances of metabolic pathways based on gene
abundances, which is useful for comparing the metabolic profiles of
different microbial communities. Gene abundances are specified in the
annotated genome file (PathoLogic format only).
}
\blue{
No pre-processing of the gene abundances (such as outlier removal) is done by PathoLogic.
The abundance of a pathway is computed based on the gene abundances
involved in the pathway. More precisely, assume that $R$ is the set of
reactions in pathway $P$ for which gene abundances are specified, $|R|$ is
the size of $R$, and $g_a$ is the given abundance of gene $g$. The
abundance of a pathway $P$ is
}
\blue{
$$\sum_{r \in P} r_a/|R|\;{\rm where\ } r_a = \sum_{g {\rm catalyzes}\; r} g_a$$
}
\blue{
That is, the abundance of a pathway is the sum of the abundances of
the genes catalyzing the reactions of the pathway, divided by the
number of reactions of the pathway for which gene abundances are
given. Notice that this formula does take into account all the known
isozymes catalyzing a reaction and the spontaneous reactions do not
take part in the computation. The abundances are provided, among other
results, in the file \texttt{pathways-report.txt}.
}
\Ssection{PathoLogic Inference of Operons}
The Pathway Tools operon predictor identifies operon boundaries by
examining pairs of adjacent genes $A$ and $B$ and using information
such as intergenic distance, and whether it can identify a functional
relationship between $A$ and $B,$ such as membership in the same
pathway \cite{KarpOperon04}, membership in the same multimeric protein
complex, or whether $A$ is a transporter for a substrate within a
metabolic pathway in which $B$ is an enzyme.
\Ssection{PathoLogic Inference of Pathway Holes}
A pathway hole is a reaction in a metabolic pathway for
which no enzyme has been identified in the genome that catalyzes that
reaction. Typical microbial genomes contain 200--300 pathway holes.
Although some pathway holes are probably genuine, we believe that the
majority are likely to result from the failure of the genome-annotation
process to identify the genes corresponding to those
pathway holes. For example, genome-annotation systems systematically
under-annotate genes with multiple functions, and we believe that the
enzyme functions for many pathway holes are unidentified second
functions for genes that have one assigned function.
Erroneous pathway holes can result from the prediction of pathways
that are not actually present in the organism.
The pathway hole-filling program PHFiller \cite{Green04} (a component
of PathoLogic) generates hypotheses as to which genes code for these
missing enzymes by using the following method. Given a reaction that
is a pathway hole, the program first queries the UniProt database to
find all known sequences for enzymes that catalyze that same reaction
in other organisms. The program then uses the BLAST tool to compare
that set of sequences against the full proteome of the organism in
which we are seeking hole fillers. It scores the resulting BLAST hits
using a Bayesian classifier that considers information such as genome
localization (that is, is a potential hole filler in the same operon
as another gene in the same metabolic pathway?). At a stringent
probability-score cutoff, our method finds potential hole fillers for
approximately 45\% of the pathway holes in a microbial genome
\cite{Green04}.
PHFiller includes a graphical interface that optionally presents each
inferred hole filler to the user along with information that helps the
user evaluate the hole fillers, and enables the user to
accept or reject the hole fillers that it has proposed.
\Ssection{PathoLogic Inference of Transport Reactions}
Membrane transport proteins typically make up 5--15\% of the gene
content of organisms sequenced to date. Transporters import nutrients
into the cell, thus determining the environments in which cell growth
is possible. The development of the PathoLogic Transport Inference
Parser (TIP) \cite{KarpTIP08} was motivated by the need to perform
symbolic inferences on cellular transport systems, and by the need to
include transporters on the Cellular Overview diagram. The motivating
symbolic inferences include the problems of computing answers to the
following queries: What chemicals can the organism import or export?
For which cellular metabolites that are consumed by metabolic
reactions but never produced by any reaction does no known
transporter exist (meaning that the origin of such metabolites is a mystery,
and indicates missing knowledge about transporters or reactions that
produce the compound)?
To answer such queries, Pathway Tools uses an ontology-based representation of
transporter function in which transport events are
represented as reactions in which the transported compound(s) are
substrates. Each substrate is labeled with the cellular compartment
in which it resides, and each substrate is a controlled-vocabulary
term from the extensive set of chemical compounds in MetaCyc
\cite{MetaCycNAR14}. The TIP program converts the free-text
descriptions of transporter functions found in genome annotations
(examples: ``phosphate ABC transporter'' and ``sodium/proline
symporter'') into computable transport reactions.
\Ssection{Atom Mappings}
\blue{The atom mapping of a reaction specifies for each reactant
non-hydrogen atom its corresponding atom in a product compound.
Pathway Tools contains an algorithm for computing atom mappings,
described in~\cite{latendresse2012}. Essentially, this approach computes atom
mappings that minimize the overall cost of bonds broken and made in
the reaction, given assigned propensities for bond creation and
breakage. This algorithm has been applied to compute atom mappings
for almost all of the reactions in
the MetaCyc database.
}
\blue{Atom mappings are used in two other parts of Pathway Tools. Atom
mappings are used in the rendering of Pathway Tools
reaction pages, to depict the conserved chemical moieties in a reaction.
Conserved moieties are depicted by using the same color on the reactant
and product sides. The bonds made or broken by a reaction are identified
from the atom mapping for the reaction, and are colored black.
Atom mappings are also used in the RouteSearch module of Pathway Tools
described in Section~\ref{sec:routesearch}.
}
\blue{Atom mappings are typically stored in the MetaCyc PGDB only, except
for the reactions unique to other PGDBs.
}
\Ssection{Computation of Metabolite Gibbs Free Energies}
\blue{The MetaCyc database provides the standard Gibbs free energy
of formation for its compounds, and the change in Gibbs free energy for
its reactions.
These data were calculated by an algorithm within Pathway Tools. The
algorithm first calculates the free energy of formation at pH 0 and
ionic strength 0 ($\Delta_f G^0$) by using a technique based on the
decomposition of the compounds into chemical groups with known
free-energy contributions to the overall energy, based on the method
of~\cite{Jankowski08}. Then, the standard Gibbs free energy at pH 7.3
and ionic strength 0.25 ($\Delta_f G^{'0}$) is computed based on a
technique developed by Robert A. Alberty~\cite{Alberty03}. In his
technique, Alberty proposes to use several protonation states for some
compounds, but we simplified the technique by always using only one
protonation state, the state stored in MetaCyc.
We use pH 7.3 because this is a common cellular pH, and computation of
the protonation state of all compounds in MetaCyc were performed at
that pH.
}
\blue{The change in standard Gibbs free energy of reactions, $\Delta_r G^{'0}$,
is computed based on the $\Delta_f G^{'0}$ values of the
compounds involved in the reaction.
The $\Delta_f G^{'0}$ could not be computed for some of the compounds
in MetaCyc due to the impossibility of decomposing them into the
groups provided by the technique
of~\cite{Jankowski08}. Consequently, the $\Delta_r G^{'0}$ is not
computed for any reaction which has a substrate for which its
$\Delta_f G^{'0}$ is not stored in MetaCyc.
}
\Ssection{Pathway/Genome Editors}
The Editors support PGDB curation through interactive modification and
updating of all the major data types supported by Pathway Tools.
The editing tools included in Pathway Tools are as follows:
\begin{itemize}
\item Gene Editor: Supports editing of gene name, synonyms, database
links, and start and stop position within the sequence.
\item Protein Editor: Supports editing of protein attributes as well
as of protein subunit structure and protein complexes (see Supplementary Figure 12). Enables users
to assign terms from the Gene Ontology controlled
vocabulary. Pathway Tools can store, edit,
and display features of interest on a protein; see
Section~\ref{sec:prot-schema} for more details. When editing a
protein feature the user selects a feature type (e.g., phosphorylation
site), defines the location
of the feature on the sequence, a bound or attached moiety where
appropriate, a textual label, an optional comment, citations, and
sequence motif.
\item Reaction Editor: Supports editing of metabolic reactions,
transport reactions, and signaling reactions. This editor checks
reactions for elemental balance and charge balance.
\item Pathway Editor: Enables users to interactively construct and edit
a metabolic pathway from its component reactions. (See
Supplementary Figure 11.)
\item \blue{Signaling Pathway Editor: Enables users to interactively
construct and edit a signaling-pathway diagram by using a toolkit of
icons and operations inspired by CellDesigner \cite{Funahashi03}
(See Supplementary Figure 14). Updates to the visual representation
are automatically translated back to changes to component reactions
and proteins.}
\item Regulation Editor: Enables definition of regulatory interactions
including regulation of gene expression by control of transcription
initiation, attenuation, and control of translation by proteins
and small RNAs (see Supplementary Figure 13). This editor also allows
creation of operons and definition of
their member genes, as well as specifying the positions of promoters
and transcription-factor binding sites.
\item Compound Editor: Supports editing of compound names, citations,
and database links. Pathway Tools has been interfaced to an
external chemical structure editor, Marvin \cite{MarvinURL}, which
is a JAVA applet. A chemical compound duplicate checker runs
whenever chemical structures are entered or modified, to inform the
user if the resulting structure duplicates another compound in that
user's PGDB or in MetaCyc. \blue{Additionally, we have added support for
displaying glycan structures in an icon-based style that follows the
conventions of CFG (Consortium for Functional Glycomics). To edit
these structures, we interfaced to another JAVA applet,
GlycanBuilder \cite{Ceroni07,Damerell12}, to which we have added
some extra functionality for better integration with Pathway Tools.}
\item Publication Editor: Supports entry of bibliographic references.
\item Organism Editor: Supports editing information about the organism
described by a PGDB, including species name, strain name, and synonyms;
taxonomic rank within the NCBI Taxonomy; \blue{sample collection
data (e.g., date, geographic location, host, body site); and phenotypic
information, such as pathogenicity and relationship to oxygen}.
\item \blue{Cellular Architecture Editor: Enables users to specify exactly
which set of cellular components are present in an organism or cell
type, with appropriate defaults derived from the organism's
taxonomy.}
\item \blue{Sequence Editor: Supports interactive, visual editing of
the nucleotide sequence for a replicon, allowing insertion,
deletion, and replacement of arbitrary sections of sequence.
Coordinates of all objects affected by the edits are updated
automatically.}
\end{itemize}
\SSsection{Author-Crediting System}
Often, multiple curators collaborate on development of a given PGDB,
and it is desirable to attribute their contributions accordingly, both
to identify who to ask if questions about particular entries arise,
and to provide an incentive for high-quality contributions, because
contributors will be able to clearly demonstrate their
accomplishments.
Most Pathway Tools editors thus create {\em credits} of several kinds.
When an object such as a pathway is first created, by default, a
``created'' credit is attached to the object, along with a timestamp.
A credit for an object can refer to curators, to organizations, or to
both. Pathway Tools provides a generated web page for every curator
and organization that lists all the objects for which they are
credited.
Other kinds of credits are ``revised'' when a curator substantially
edits an object that was created some time ago, and a ``last curated''
flag can be set to indicate when a curator has last researched the
literature available for a given object. The last-curated flag is useful
for those objects about which almost nothing is known, to distinguish
between the case where no curator ever looked at the object versus where
an extensive search was performed but still nothing new was found.
The ``reviewed'' credit is used to attribute reviews of DB objects by
external domain experts.
\Ssection{Incorporation of Genome-Annotation Revisions}
\label{sec:bulk-update}
Some groups choose to store the authoritative version of the
organism's genome annotation in a database external to the PGDB. Such
users need the ability to incorporate revisions to the genome
annotation into their PGDB without overwriting or otherwise losing any
manual curation they added to the PGDB. Pathway Tools
provides an interface for incorporating annotation revisions that takes as input one or
more update files, either in GenBank format or as a PathoLogic Format file.
The files can contain either a complete revised annotation for the
organism, or they can contain just the information that has changed.
The software will parse the update files and determine all differences
between the new data and the old. Types of changes that are detected
include new genes, as well as updated gene positions, names, synonyms,
comments, links to external databases, and updated functional
assignments. A graphical interface summarizes different classes of
changes to the user, and gives the user the option of either accepting all updates (e.g., creating
database objects for all the new genes) or of accepting/rejecting individual
updates. Once this phase is complete and any changes to functional
assignments have been made, the software will re-run the pathway-inference
procedure described in Section~\ref{sec:pwy-inference},
identify any new pathways that are inferred to be present and any
existing pathways that no longer have sufficient evidence, and allow
the curator to review those changes.
\Ssection{Consistency Checker and Aggregate Statistics}
\label{sec:ccheck}
Pathway Tools contains an extensive set of programs for performing
consistency checking of a PGDB to detect structural defects that
sometimes arise within PGDBs. Also included in this component are
tools for computing and caching aggregate statistics for a PGDB, such
as computing the molecular weights of all proteins from their amino
acid sequences. The statistics are cached so that they can be
displayed quickly.
Roughly half of the programs automatically repair PGDB problems that
they find. Such problems could be caused by either user data-entry errors
or errors in Pathway Tools itself.
Example checks include: ensuring that inverse relationship
links are set properly (e.g., that a gene is linked to its gene
product, and that the product links back to the gene); making sure
pathways do not contain duplicate reactions; validating and updating GO
term assignments with respect to the latest version of GO; performing
formatting checks in comment text; searching gene reading frames for internal stop codons;
and removing redundant bonds from
chemical structures.
\Ssection{Schema Upgrading and Propagation of MetaCyc Updates}
Most new releases of Pathway Tools include additions or modifications to
the Pathway Tools schema that are made to model the
underlying biology more accurately (such as adding support for introns
and exons), and to extend the data types within Pathway Tools (such as
adding support for features on protein sequences).
Because each new version of the software
depends on finding data within the fields defined by the associated
version of the schema, existing user PGDBs created by older versions
of the software will be incompatible with these new software versions.
Therefore, every release of Pathway Tools contains a program to
upgrade PGDBs whose schema corresponds to the previous version of the
software, to the new schema version. For users who have not upgraded the
software for several releases, several upgrade operations will be
performed consecutively. Example upgrade operations include adding
new classes to the PGDB from the MetaCyc PGDB; adding new slots to
PGDB classes; deleting PGDB classes; moving data values from one slot
to another; and moving objects from one class to another. The schema
upgrade leaves the user's curated data intact.
Every new release of Pathway Tools includes a new
version of the MetaCyc database, which, in addition to providing new
data content, typically contains updates and corrections to existing
pathways, reactions, and compounds. Pathway Tools includes an option
to propagate such updates and corrections to an existing organism
PGDB. However, because we do not want to override any manual edits
made to a PGDB, this tool does not run automatically. Much like the
tool for incorporating a revised genome annotation, described in
Section~\ref{sec:bulk-update}, this tool organizes the changes into
logical groups (such as all compounds with newly added structures, or
all reactions with changed reaction equations), and allows the user to
either accept an entire group of changes, or to examine and confirm
individual members of a group.
\section{The Pathway Tools Schema}
\label{sec:schema}
The Pathway Tools schema defines structured representations of a broad
range of biological data types to enable computational analyses and
integration of many types of data.
The schema consists of a set of classes and a set of slots.
Classes describe types of biological entities, such as genes and
pathways, and are arranged in a class--subclass hierarchy. Slots
define attributes of PGDB objects and
relationships between PGDB objects. Figure~\ref{fig:relationships}
provides an overview of the relationships among PGDB classes. For
example, user queries can follow the relationship from a gene to the
protein that it codes for, from a protein to a reaction that it
catalyzes, and from a reaction to a metabolic pathway in which it is a
component, to answer questions such as ``find all metabolic pathways in
which the products of a given gene play a role''.
Every PGDB object has a stable unique identifier (ID)---a
symbol that uniquely identifies that object within the PGDB. Example
unique IDs include TRP (an identifier for a metabolite), RXN0-2382 (an
identifier for a reaction), and PWY0-1280 (an identifier for a
pathway). Relationships within a PGDB are implemented by storing
object IDs within slots. For example, to state that the TRP
(L-tryptophan) object is a reactant in the reaction RXN0-2382, a slot
of RXN0-2382 called LEFT (meaning reactants) contains the value TRP.
Many PGDB relationships exist in both forward and backward directions
(for example, the TRP object contains a field called
APPEARS-IN-LEFT-SIDE-OF that lists all reactions in which TRP is a
reactant). The slots LEFT and APPEARS-IN-LEFT-SIDE-OF are called
inverses.
\begin{figure}
\includegraphics[width=6in]{PTools-Schema-13-0.png}
\caption{Major relationships among the major classes of the Pathway
Tools schema. Colors indicate biological areas: blue for reaction and
pathway information, green for genome and protein information, orange
for regulation.}
\label{fig:relationships}
\end{figure}
\Ssection{Metabolites, Reactions, and Pathways}
\label{sec:schema-metab}
There are two alternative ways in which one might choose to represent
the metabolic network in a computer: as a simple listing of all
metabolic reactions that occur in the cell, or by partitioning the
reaction list into a carefully delineated set of metabolic pathways
that describe small, functionally linked subsets of reactions. Which
approach is preferred? Both approaches have value,
and they are not mutually exclusive; therefore, Pathway Tools supports
both views of metabolism in a PGDB.
Pathway Tools conceptualizes the metabolic network in three layers.
The first layer consists of the small-molecule substrates upon which
metabolism operates. The second layer consists of the reactions
that interconvert the small-molecule metabolites. The third layer consists of
the metabolic pathways whose components are the metabolic reactions of
the second layer. Note that not all reactions in the second layer are
included in pathways in the third layer, because some metabolic
reactions have not been assigned to any metabolic pathway by biologists.
Scientists who choose to view the metabolic network within a PGDB solely as a
reaction list can operate on the second layer directly without
interference from the third layer.
The compounds, reactions, and pathways in levels 1--3 are each
represented as distinct database objects within a PGDB. The
relationships among the metabolic data types in a PGDB are depicted by
the blue region of Figure~\ref{fig:relationships}.
\blue{The representation of reactions can capture atom-mapping information
that records, for each atom in a reactant compound, its terminus atom
in a product compound. The representation of metabolites can capture
stereochemical structural information as well as glycan structures.}
The pathways in PGDBs are modules of the metabolic network of a single
organism. Pathway boundaries are defined by considering the following
factors. Pathways are often regulated as a unit (based on
substrate-level regulation of key enzymes, on regulation of gene
expression, and on other types of regulation). Pathway boundaries are
often defined at high-connectivity, stable metabolites
\cite{GreenK06}. Pathway conservation across multiple species is also
considered, as are pathway definitions from the
experimental literature \cite{MetaCycNAR14}.
\Ssection{The Proteome and the Genome}
PGDBs define the proteome and the genome of an organism in
the following manner, as depicted by the green region of
Figure~\ref{fig:relationships}.
The proteome of the organism is described as a set of PGDB objects,
one for each gene product in the organism, and one for each complex
formed from two or more (identical or nonidentical) polypeptides.
Furthermore, every chemically modified form of a monomer or of a
multimer is encoded by a distinct PGDB object. For example, we might
create one object representing an unmodified protein and another
representing the phosphorylated form. Each protein object is in turn
linked, through a slot in the object, to the metabolic reactions that
it catalyzes. Proteins can also be substrates of reactions.
Additional PGDB objects define features on proteins, as described in
Section~\ref{sec:prot-schema}.
Each protein product resulting from alternatively spliced forms of a
gene is also represented by a distinct protein object. Each protein
object records the exons of the gene that encodes it.
Protein objects are linked to gene objects that define the gene
encoding each protein. Each gene in the genome is defined by a
distinct PGDB object, as is every replicon (chromosome or plasmid) in
the genome. Genes are linked to the replicon on which they reside.
In addition, other features on the genome, such as operons, promoters,
and transcription-factor binding sites, are described by PGDB objects.
The associations between enzymes and the reactions they catalyze are
implemented by using an intermediary object called an enzymatic reaction,
as shown in Figure~\ref{fig:relationships}. This arrangement enables us to
capture the many-to-many relationship that exists between enzymes and
reactions---one reaction can be catalyzed by multiple enzymes, and
multifunctional enzymes catalyze multiple reactions. The purpose of
the enzymatic reaction is to encode information that is specific to
the pairing of the enzyme with the reaction, such as cofactors,
activators, and inhibitors. Consider a bifunctional enzyme with two
active sites, where one of the active sites is inhibited by pyruvate,
and the second active site is inhibited by lactate. We would
represent this situation with two enzymatic reactions linking the
enzyme to the two reactions it catalyzes, and each enzymatic reaction
would specify a different inhibitor.
\Ssection{Pathway Tools Regulation Ontology}
\label{sec:schema-reg}
The Pathway Tools schema can represent a wide range of regulatory
interaction types. A regulation object within a PGDB captures information about
each type of regulatory interaction. The available regulation types are as
follows:
(a) Substrate-level
regulation of enzyme catalytic activity, such as the allosteric
activation or competitive inhibition of an enzyme by a small molecule.
Slots of this class identify the regulator molecule, the
regulated enzymatic reaction object, encode the polarity of
regulation (activation or inhibition), and the mechanism of regulation
(allosteric, competitive, or noncompetitive).
(b) Regulation of a bacterial promoter by a transcription-factor protein.
The slots of this regulation class describe the
transcription factor, the promoter that is regulated, and the binding
site to which the regulator binds.
(c) Regulation via premature termination of transcription
(attenuation). This class of regulation is divided
into six subclasses, each describing a different attenuation mechanism
(e.g., ribosome-mediated, protein-mediated, or RNA-mediated). The slots
of these classes identify the regulated terminator region, the
regulator (a protein, RNA or small-molecule, depending on the type of
attenuation), and the regulator binding site if one exists.
(d) Regulation of the
translation of an mRNA transcript to the corresponding protein. This
regulation class is divided into two subclasses to distinguish between
regulation by a protein and regulation by a small RNA. The slots of
these classes identify the regulated transcription-unit (which
corresponds to a single transcript), the regulator protein or RNA,
and the mRNA binding site where the regulator binds. An additional
slot indicates whether regulation is by direct interference with the
translation machinery, by processing of the mRNA transcript to promote or
inhibit its degradation before translation, or both.
(e) Regulation of protein activity by chemical modification, such as by
phosphorylation, is represented by a reaction that converts the
unmodified form of the protein to the modified form.
\Ssection{Conditions of Cellular Growth}
\label{sec:growth-conditions}
\blue{A recent extension to the schema supports representation of
conditions of cellular growth that include the chemical composition
of the growth medium, pH, temperature, and aerobicity. This
representation enables us to capture low-throughput information about
conditions of cellular growth, and high-throughput information such
as Phenotype Microarray \cite{Bochner-etal01,Bochner09} datasets. }
\Ssection{Gene Essentiality}
\blue{
A recent extension to the schema supports representation of
gene-essentiality experiments. Our representation links growth
phenotype (no growth, limited growth, or growth) under a given gene
knockout with the conditions of cellular growth expressed as per Section~\ref{sec:growth-conditions}.
Multiple phenotypic observations can be recorded for a given gene
knockout and growth condition to express conflicting experimental outcomes.
}
\Ssection{Organism Phenotype Data and Genome Metadata}
\blue{
A recent extension to the schema supports representation of
microbial phenotypic data to enable users to query among the many
genomes stored within a Pathway Tools website to find organisms
pertinent to their research. Our representation adapts the MIGS
\cite{MIGS} standard to incorporate metadata about the
sample from which the organism was derived (e.g., geographic location,
depth, health-or-disease state of host, human microbiome site), plus phenotypic
information about the organism itself (e.g., relationship to oxygen,
temperature range, and pathogenicity).
}
\Ssection{Pathway Tools Evidence Ontology}
Database users want to know the type(s) of evidence that support assertions
within a DB, and they want to know the strength of that evidence.
We have developed an evidence ontology \cite{KarpEv04} that can encode
information about {\em why} we believe certain assertions in a PGDB,
the {\em sources} of those assertions, and the {\em degree of
confidence} scientists hold in those assertions (although in practice
the latter field is rarely populated). An example assertion is the
existence of a gene in a PGDB. Was the
gene predicted by using computational gene finding? Is it
supported by wet-lab experiments? The Pathway Tools evidence ontology builds upon
and substantially extends the Gene Ontology evidence ontology, which
applies only to gene products.
Evidence about object existence in PGDBs is recorded as a structured
{\em evidence tuple}. An evidence tuple enables us to associate
several types of information within one piece of evidence. Each {\em
evidence tuple} is of the form
\begin{verbatim}
Evidence-code : Citation : Curator : Timestamp : Probability
\end{verbatim}
\ \\
where
{\tt Evidence-code} is a unique ID for the type of evidence,
within a hierarchy of 54 evidence types described in \cite{PToolsEvOntologyURL,KarpEv04}.
{\tt Citation} is an optional citation identifier such as a PubMed ID that
indicates the source of the evidence. For computational evidence, the
citation refers to an article describing
the algorithm used.
{\tt Curator} identifies the curator who created this
evidence tuple. {\tt Timestamp} encodes when this evidence tuple was created.
{\tt Probability} is an optional real number indicating the
probability that the assertion supported by this evidence is correct,
such as a probability provided by an algorithm.
The Pathway Tools editors enable users to manually enter evidence
codes, and the PathoLogic pathway and operon predictors annotate
objects that they create with appropriate computational evidence
codes. The Navigator supports display and querying of evidence codes.
\Ssection{Pathway Tools Cell Component Ontology}
The Cell Component Ontology (CCO) is a controlled vocabulary of terms
describing cellular components and compartments, and relationships
between these terms \cite{PToolsCCOURL}. It was developed to provide
a controlled vocabulary of terms for annotating the subcellular
locations of enzymes, and the compartments involved in transport
reactions, in PGDBs. CCO spans all domains of life, and includes
terms such as cytoplasm, cell wall, and chloroplast. The ontology
currently contains 170 terms. CCO includes many terms and their
definitions from the Gene Ontology \cite{GO2008}, but substantially
extends Gene Ontology. \blue{A recent extension to CCO enables
any metabolic reaction to be annotated to one or more CCO compartments,
to allow metabolic models to span multiple compartments.}
\Ssection{Pathway Tools Protein Feature Ontology}
\label{sec:prot-schema}
We have developed an ontology of protein features to identify
and represent post-translational modifications, binding sites, active
sites, conserved regions, and other regions of interest on a protein.
Starting from the list of feature types described in the UniProt User
Manual \cite{SprotManual09}, with some suggested additions from the
SRI EcoCyc and MetaCyc database
curators, we created an ontology of 40 feature classes.
Features fall into two major classes. For amino acid site features, the
feature location is a list of one or more amino acid residue numbers
(or residue types, if the feature is associated with a generic protein
whose precise sequence is unspecified). For protein segment features,
the feature location is a range defined by its starting and
ending residue numbers.
Feature types that are classified as binding features (either covalent
or noncovalent) permit specification of an attached group. The
attached group could be a compound or compound fragment, as in the
case of a protein that binds a small molecule. The attached group can
also be another protein feature, as in the case of a disulfide bond or
other cross-link between two features on different proteins, or any
other type of molecule or binding site (such as a DNA binding site).
A different protein object is created in a PGDB for each biologically
relevant modified form of a protein, and a single feature may be
linked to multiple forms of the same protein. Some feature types are
capable of existing in multiple states. For example, an amino acid
modification feature can be in either the modified or the unmodified
state (as in the case of a phosphorylation feature, which will be in
the modified state when associated with the phosphorylated protein and
the unmodified state when associated with the unphosphorylated
protein), and a binding feature can be in either the bound or unbound
state (as in the case of a metal-binding feature whose state indicates
whether or not the metal ion is bound to the protein). We consider
the state to be not an attribute of the feature, but rather an
attribute of the pairing between a particular form of a protein and
the feature.
\section{Visualization and Querying of PGDBs}
\label{sec:nav}
The Pathway/Genome Navigator component of Pathway Tools provides
mechanisms for interrogating PGDBs and for visualizing the results of
those queries. We begin by describing the query tools. We then
describe visualization tools for individual biological entities
(such as genes and pathways). We next describe the new SmartTables
system, which enables the user to manipulate groups of PGDB objects,
such as gene sets and metabolite sets. Finally, we describe system-level visualization tools that
graphically display the entire metabolic network, entire regulatory
network, and entire genome map of an organism.
The Navigator runs as both a desktop application and a web server,
whereas the Pathway Tools components described in some other sections run as
desktop applications only (e.g., PathoLogic does not run as a web server).
The desktop mode is faster, and has more overall functionality (see
\cite{BioCycDesktopVsWebURL} for details), but the web mode
has some functionality that is not present in the desktop mode.
\Ssection{Query Tools}
\label{sec:query}
Pathway Tools provides a
three-tiered query paradigm, meaning that three different types of
query tools are available, each of which represents a different
trade-off between ease of use and query power. For example, the quick
search is designed to provide a fast and simple way for new or casual
users to find general information in the site.
The ``Quick Search'' box that appears at the top of most web pages
generated by a Pathway Tools server is extremely easy to use. The
user enters a search term and selects the organism whose PGDB the user
wants to query. Pathway Tools searches that PGDB for objects whose
primary name or synonyms contain the search term as a substring, and
presents the list of results, organized by object type (e.g., gene,
metabolite, pathway). The user can
click on an object name to navigate to the display page for that
object.
A set of intermediate-level query tools provides the ability to
construct more powerful and precise searches against objects of a
single class. One such query page exists for genes, proteins, and
RNAs (see Supplementary Figure~1); additional query pages exist
for pathways, reactions, chemical compounds, and growth media.
Finally, the Structured
Advanced Query Page (SAQP) described in the next section enables advanced users to construct
extremely powerful searches (that are approximately as powerful as those
provided by the SQL language). The graphical interactive nature of
this web form makes these searches much easier to construct than those
using the SQL language.
\SSsection{Structured Advanced Query Page}
The SAQP enables a biologist to construct precise structured
searches. A query
can be as simple as looking up a gene given a name, or as complex as
searching several databases and several object types interconnected by
several relations. The SAQP enables biologists to formulate queries
whose power and expressiveness closely approach SQL, but without
having to learn SQL. The SAQP translates a formulated query into
BioVelo, an OQL-like language \cite{88909}, before sending it to the web server.
The following explanation presents the elements of
this web user interface using the
example shown in Figure~\ref{fig:polypeptides2SAQP}, involving
a query against the class of protein monomers (Polypeptides) in the
EcoCyc DB.
\begin{figure*}
\begin{center}
\includegraphics[scale=0.4]{polypeptides2SAQP.png}
\end{center}
\caption{\label{fig:polypeptides2SAQP}
A query for the {\em E. coli}\ polypeptides whose experimental molecular
weight lies between 50 and 100 kilodaltons, whose pI is smaller than 7,
and whose gene is located after
the first 500 kb of the genome. An
output column is used to include the gene (or sometimes
genes) producing each polypeptide using the second
variable~\texttt{Z2}.
}
\end{figure*}
{\bf \noindent Step 1: Select Database and Class.}
The first step in building a query is to specify at least one database
(DB) and the class of objects to search.
{\bf \noindent Step 2: Specify Conditions.}
Most queries include one or more conditions on the desired objects
within the class. By clicking the button labeled \texttt{add a
condition} in the initial blank SAQP, a {\it where} clause is
added---visually boxed---in the search component. This operation adds a
selector for an {\it attribute} (e.g., \verb+name+) of the objects and
a selector for a relational operator (e.g., {\tt contains the
substring}). It also adds a {\it free text box} to enter a number or
string. Several other relational operators are provided, such as {\tt
is equal to}, and {\tt is a substring of}.
Regular expression matching is also available.
This new field forms an {\it atomic condition}. Additional atomic
conditions can be added to the query by using the button labeled ``add
a condition''.
When selecting a relational operator, the
list of relational operators provided is compatible with the type of
the selected attribute. In the case of the attribute {\tt name}, the
selectable operators are for strings, because the {\it type} of the
attribute {\tt name} is string. The query in
Figure~\ref{fig:polypeptides2SAQP} has three atomic conditions
to filter the selected polypeptides.
Quantifiers on relations within the SAQP enable a join-like
capability. For example, imagine that we want to extend the query
with an additional restriction that depends on the {\it gene encoding
the polypeptide,} not on the polypeptide itself.
To do so, the user would add an \verb+and+ condition and then select
the \verb+gene+ attribute, which represents the gene encoding the
polypeptide. We then select
the quantifier operator {\tt for some object...}, meaning that we want
to define a condition that applies to some of the genes in the
\verb+gene+ attribute of this polypeptide (although in the majority of
cases only one gene will be present).
At this point, the SAQP adds a new indented query clause, to enable a
condition to be defined on the gene. We have specified a constraint
that its nucleotide coordinate must lie after the first 500 kb of the
genome. Because several attributes and logical connectors can be
specified in this new clause, forming a complex condition by itself,
the web interface draws a box around this condition and introduces it
with the {\tt we have} keyword. A new unique variable, named
\verb+Z2+, is also introduced. This variable represents every value of
the \verb+gene+ attribute.
{\bf \noindent Step 3: Define Query Results.}
The section titled \texttt{Select attributes to include in the query
output} enables the user to describe the contents of the query results
by selecting the attributes to display for each result object. The
result of a query is a table containing zero or more rows, one for each
query result. Each column in the table is
a user-selected attribute.
\SSsection{Cross-Organism Search}
\blue{We recently added a tool for searching across all organisms within a
Pathway Tools web server, such as for the 5,500 organisms at
BioCyc.org. The cross-organism search tool
\cite{BioCycCrossOrgSearchURL} searches for user-specified
combinations of words in the Common-Name/Synonyms attributes, and/or
the Summary attribute. It can search all types of objects in a given
PGDB, or in user-specified object types, such as genes and/or pathways.
It can search all organism databases present in the Pathway Tools web
server, or it can search user-specified sets of organisms, such as all
organisms within a selected taxonomic group. Indexing and searching
is implemented using SOLR \cite{SOLRURL}.}
\Ssection{Visualization Tools for Individual Biological Entities}
\blue{ {\bf Genes/Proteins/RNAs:} Since our last overview, we merged
the individual display pages for a gene and its product (protein or
RNA) into a single page that combines all information in one central
place. These pages are quite extensive, listing information such as
the map position of the gene on the chromosome, a graphical
depiction of the chromosomal region containing the gene, and
available gene-essentiality information. A new diagram, the
regulation-summary diagram, integrates all known regulatory
influences on the gene and gene product into a single figure.
Common to all protein types is the ability to graphically display
information about protein regions (such as phosphorylation sites and
active sites) using a protein-feature ontology that we developed.
{\bf Enzymes:} The software displays the reaction catalyzed by the
enzyme and the name of the pathway that contains that reaction, if
any (see Supplementary Figure~3); the activators, inhibitors, and
cofactors required by the enzyme; and comments and citations for the
enzyme. {\bf Transporters:} The software displays the transport
reaction catalyzed by the transporter (see Supplementary Figure~4).
{\bf Transcription factors:} The software displays diagrams for all
operons controlled by the transcription factor (the regulon for the
transcription factor) (see Supplementary Figure~5). }
{\bf Reactions:} Reaction display applies to metabolic,
transport, and protein signaling reactions (see Supplementary Figure~2). The reaction display
shows the one or more enzymes that catalyze the reaction, the gene(s)
that code for the enzymes, and the pathway(s) that contain the
reaction. The display shows the EC number for the reaction and the
reaction equation. Conserved chemical moieties in the reaction
substrates are drawn in different colors based on atom-mapping information.
\blue{
{\bf Pathways:} All metabolic pathway visualizations are computed automatically
using pathway-layout algorithms. Pathway Tools can draw metabolic pathways at
multiple levels of detail, ranging from a skeletal view of a pathway
that depicts the compounds only at the periphery of the pathway and at
internal branch points, to a detailed view that shows full structures
for every compound, and EC numbers, enzyme names, and gene names at
every reaction step (see Supplementary Figure~6). The user can customize a pathway drawing to
include desired elements only, and to include superimposed omics data
using ``omics pop-ups'' (see Figure~\ref{fig:pathomics}). Support for display of signaling
pathways (see Figure~\ref{fig:sigpath}) was added recently; signaling pathway layout is performed by
the user.
}
\begin{figure}
\begin{center}
\includegraphics[width=7in]{pathomics.png}
\caption{The EcoCyc L-ornithine biosynthesis pathway shown with
omics pop-ups containing time-series data from a gene-expression experiment.}
\label{fig:pathomics}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=3in]{mapkinase-pwy.png}
\caption{The HumanCyc MAP kinase cascade, a signaling pathway diagram.}
\label{fig:sigpath}
\end{center}
\end{figure}
\blue{
{\bf Electron Transfer Reactions and Pathways:} A crucial role in
cellular metabolism is played by electron transfer reactions (ETRs),
which are of key importance in the energy household of a cell. In a
series of redox steps, the high-energy electrons from some
compounds drive the pumping of protons across a cell membrane, to
maintain the proton motive force needed for ATP synthesis. Because
electrons do not freely exist on their own, these redox reactions seem
less intuitive than other small molecule reactions. We designed and
implemented drawing code for a special ETR diagram, which shows the
enzyme complex embedded in a membrane, and which schematically depicts
the flow of electrons from one redox half reaction to another. Inside the
membrane, the quinone/quinol cofactor is shown together with an
indication of the cell compartments that are sources or sinks of the
protons. An additional vectorial proton transport reaction can be
added to the diagram. This results in displaying the flow of all
substrates and products relative to the cellular compartments, in a
similar way to what is customary in the biomedical literature.
Pathways consisting of several ETRs joined together can also be
depicted (see Figure~\ref{fig:etr-pwy}).
}
\begin{figure}
\begin{center}
\includegraphics[width=4in]{supp_etr_display.png}
\caption{An electron transfer pathway diagram.}
\label{fig:etr-pwy}
\end{center}
\end{figure}
{\bf Chemical compounds:} The metabolite display shows the chemical
structure for the compound (see Supplementary Figure~7). It lists all reactions in which the
compound appears, and it lists enzymes whose activity is regulated by
the compound.
{\bf Transcription units:} The display window for transcription units
diagrams the transcription unit and its regulatory sites
including promoters, transcription-factor binding sites, attenuators,
and binding sites for proteins and RNAs that regulate its translation.
The display contains sections describing each site within the
transcription unit. The promoter section describes which sigma factor
recognizes it. Sections for a transcription factor binding site
describe its sequence, which transcription factor it binds, ligands that influence
the activity of the transcription factor, and whether the effect of
binding is to activate or inhibit transcription initiation. Sections
for attenuators describe the signal that the attenuator senses, and
show the sequence regions that form the attenuator.
\Ssection{SmartTables: Large-Scale Manipulation of PGDB Object Groups}
\blue{SmartTables is a recent addition to Pathway Tools that enables
users to construct and manipulate groups of PGDB objects through a
spreadsheet-like user interface \cite{KarpGroups13} (SmartTables
were previously called Web Groups). SmartTables provide many
powerful operations to biologist end users that previously would
have required assistance from a programmer, and our user surveys
indicated that SmartTables are reasonably easy for biologists to use
\cite{KarpGroups13}. Altogether, 2700 users of BioCyc.org have
created more than 31,000 SmartTables.}
\blue{A typical SmartTables use case is for a user to define a SmartTable by
importing a list of PGDB objects from a file. For example, a user
could define a metabolite SmartTable by importing a list of
metabolites from a metabolomics experiment, where the metabolites are
specified by metabolite name, BioCyc identifier, PubChem identifier,
or KEGG identifier. (The set of objects in a SmartTable can also be
defined from a query result, from any column of an existing
SmartTable, or from the set of, say, all genes in a
PGDB.)}
\blue{The user can browse the set of objects in a SmartTable by paging
through the table, and can modify the information displayed about
each object by specifying which table columns to include (see
Figure~\ref{fig:smt}). SmartTable columns are derived from the PGDB
attributes available for each object, and can include information such
as chemical structures, molecular weights, links to other databases,
and nucleotide and protein sequence. A variety of filters and set
manipulations are provided for SmartTables, such as removing or
retaining all rows that match a user query; and computing the union,
intersection, and set difference of two SmartTables. SmartTables
are stored in the user's online web account, and a desktop version
of SmartTables is also provided. SmartTables are private by
default, but the user can make them public, share SmartTables with
selected other users, or archive them in a frozen form in
conjunction with a publication.}
\begin{figure}
\begin{center}
\includegraphics[width=7in]{smt.png}
\caption{A gene SmartTable. Column 1 shows the gene name, column 2 shows the {\em E. coli}\ genome
``b-number'' accession number for the gene (a property); column 3 shows the gene
product name (a property). Column 4 shows the result of a
transformation in which the regulator(s) of each gene were computed. }
\label{fig:smt}
\end{center}
\end{figure}
\blue{Several more advanced SmartTable operations are provided. {\em
Transformations} compute new columns from relationships in a PGDB.
For example, column~4 in Figure~\ref{fig:smt} is a transformation
column that shows one or more regulators for each gene in column~1
that have been computed from PGDB relationships. Other gene
transformations available include computing the metabolic pathways
in which a gene's product occurs and computing the amino-acid changes
caused by sequence variants.
Different transformations are available for different datatypes.
For example, the transformations available for a metabolite SmartTable
include computing the reactions in which a metabolite occurs, the
pathways in which a metabolite occurs, the proteins for which the
metabolite is a ligand, and mapping the compounds to their equivalents
in another PGDB.}
\blue{A user can perform a statistical enrichment analysis on a gene or
metabolite SmartTable to detect over-represented metabolic pathways or
GO terms, or over-represented metabolic pathways, respectively.
In addition, a SmartTable of genes or metabolites can be visualized
on the cellular overview.}
\Ssection{System-Level Visualization of Metabolic Networks}
\label{sec:cellov}
Pathway Tools can automatically generate organism-specific metabolic
charts that we call Cellular Overview diagrams
\cite{PToolsOverview06}. The diagram can be interrogated interactively and used to
analyze omics datasets. Recently, the diagram was re-engineered for the web
mode of Pathway Tools.
The diagram can be generated as a PDF file for
printing as a large-format poster. Example posters can be downloaded
from \cite{BioCycPostersURL}.
Figure~\ref{fig:cellov} depicts the Web Cellular Overview at low resolution
painted with gene-expression data. It contains all known metabolic
pathways and transporters of an organism (online example: \cite{OverviewURL};
example with animated display of omics data:
\cite{OverviewAnimationURL}). Each node in the diagram represents a
single metabolite, and each line represents a single bioreaction.
Omics data (e.g., gene-expression or metabolomics measurements) for a
given organism can be painted onto the cellular overview to place
this data in a pathway context and to enable the user to discern the
coordinated expression of entire pathways (such as the TCA cycle) or
of important steps within a pathway. \blue{Omics pop-ups show all
time points for
particular reactions or metabolites of interest.} Omics data may be loaded from a
data file\blue{, GEO or PortEco dataset, or SmartTable,} and
superimposed on the Overview diagram for that organism. \blue{In web
mode, the user has a choice of several color schemes---in desktop
mode the color scheme is fully customizable.}
Cellular Overview diagrams are generated automatically using an
advanced layout algorithm \cite{PToolsOverview06}. Automated layout
is essential to enable the diagram to accurately depict the underlying
database content as that content evolves, without requiring
time-consuming manual updates by curators that are bound to overlook
some updates. In addition, automated layout enables generation of
organism-specific cellular overviews that reflect the exact pathway
content of each organism-specific PGDB in large PGDB collections such
as BioCyc.
The Cellular Overview has
many capabilities (described in more detail in
\cite{PToolsOverview06}), including semantic zooming
of the diagram (where the highest magnification corresponds to the
detail shown in the poster version); highlighting of user-requested
elements of the diagram (such as metabolites or pathways);
highlighting large, biologically relevant subnetworks (such as all
reactions regulated by a given transcription factor); and
highlighting comparative analysis results, such as comparison of the
metabolic networks of two or more PGDBs. \blue{Because a metabolite
can appear in several different places in the diagram, in
desktop mode, the user can better
visualize the flow of material through the metabolic network by
selectively showing connections between a metabolite of interest, or
all the metabolites in a pathway of interest, and
everywhere else those metabolites appear in the network, as shown in Figure~\ref{fig:connections}}.
\begin{figure}
\begin{center}
\includegraphics[width=7in]{cellov.png}
\caption{The Pathway Tools Cellular Overview diagram for EcoCyc,
painted with gene-expression data. Omics pop-ups are shown for two genes.}
\label{fig:cellov}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7in]{connections.png}
\caption{The Pathway Tools Cellular Overview diagram for EcoCyc,
showing connections between metabolites of the serine biosynthesis
pathway and selected other pathways.}
\label{fig:connections}
\end{center}
\end{figure}
\Ssection{System-Level Visualization of Regulatory Networks}
\label{sec:regov}
The Pathway Tools Regulatory Overview depicts the full transcriptional
regulatory network stored in a PGDB in one screen, and enables the user
to interrogate and explore relationships within the network.
Figure~\ref{fig:regov} shows the Regulatory Overview for EcoCyc, after
the user has asked the system to highlight all genes annotated under
Gene Ontology term GO:0001539 (ciliary or flagellar motility). We can
see that a few transcription factors control all {\em E. coli}\
motility genes.
\begin{figure}
\begin{center}
\includegraphics[width=7in]{regov.png}
\caption{The Pathway Tools Regulatory Overview diagram for EcoCyc.
The diagram depicts a full regulatory network as three concentric
rings: the inner ring contains master regulator genes; the middle
ring contains other regulators; the outer ring contains genes that are
not regulators. An arrow (edge) from gene A to gene B indicates that
gene A regulates gene B. Initially, no arrows are shown; the user can
interactively add arrows, such as by clicking on a gene and requesting
that arrows are added to genes that it regulates, or from the genes
that regulate it.}
\label{fig:regov}
\end{center}
\end{figure}
\Ssection{System-Level Visualization of Genome Maps}
\label{sec:genov}
The Pathway Tools genome browser displays a selected replicon
(chromosome or plasmid), and enables
the user to zoom into a region of the replicon by gene name or by
coordinates. The browser supports semantic zooming: as the user moves
deeper into the genome, additional features are displayed, such as
promoters and terminators. \blue{At high magnification, the genome
sequence and the amino-acid sequence of coding regions become visible.
Intron and exon boundaries are shown.}
The genome browser can be used in a comparative mode that
displays replicon regions centered on orthologous genes across a
user-specified set of genomes to show the genomic context of those
genes (example: \cite{CompGenomeBrowserURL}). In comparative mode,
the user retains the ability to navigate left or right in the genome, and to
zoom in and out. The genome browser can also generate large-format genome posters
in PDF format; see examples at \cite{BioCycPostersURL}.
The genome browser also supports display of tracks, meaning the
ability to view positional data from external files along the genome,
such as viewing predicted transcription-factor binding sites.
A user who zooms out far enough is presented with a
depiction of all the genes on the replicon called the Genome Overview,
shown in Figure~\ref{fig:genov}. This diagram can be painted with
omics data to provide a global genome view of large-scale datasets.
\begin{figure}
\begin{center}
\includegraphics[width=7in]{genov.png}
\caption{Pathway Tools Genome Overview diagram for EcoCyc.
In this diagram genes are not shown to scale.
Adjacent genes drawn in the same color are in the same operon.
Left/right gene direction indicates transcription direction; up/down
gene direction indicates genes coding for proteins versus RNAs.
Horizontal lines under genes indicate transcript extents based on
promoter and terminator information in the PGDB.}
\label{fig:genov}
\end{center}
\end{figure}
\Ssection{Sequence-Based Query and Visualization Tools}
\label{sec:sequencetools}
\blue{The following tools support query and visualization of sequence data
from a Pathway Tools web server:}
\begin{itemize}
\item \blue{Nucleotide Sequence Viewer: Gene pages include links to view or
download the nucleotide or RNA/protein sequence for the gene. When
viewing the nucleotide sequence, an option is provided to
include an additional upstream and/or downstream flanking region of
any desired length. This option makes it easy to, for example, view
the sequence of a regulatory region surrounding a gene of interest.
Alternatively, the user can enter specific start and end coordinates
and the desired strand to view the sequence of any arbitrary portion
of the chromosome.}
\item \blue{BLAST Search: Users can use BLAST to search for all
sequences within a single specified genome that match a query
sequence.}
\item \blue{Sequence Pattern Search (PatMatch): The PatMatch facility
enables searching within a single genome for all occurrences of a
specified short nucleotide or peptide sequence (less than about 20
residues), with the ability to specify degenerate positions. The
user can specify the kind and number of allowable mismatches, and
whether to search coding regions only, intergenic regions, or the
entire genome. Examples of situations in which this facility might
be useful include searching for all occurrences of a particular
regulatory motif upstream of any gene, or all occurrences of a known
cofactor binding motif within proteins.}
\item \blue{Multiple Sequence Alignment: From a gene page, users can
request a multiple sequence alignment between the nucleotide or
amino acid sequences of that gene and its orthologs in a
user-specified set of organisms. Alignments are displayed using
MUSCLE \cite{MUSCLE04}.}
\end{itemize}
\section{Computational Access to PGDB Data}
\label{sec:apis}
In addition to the user-friendly graphical interfaces to PGDBs
provided through the web and desktop versions of Pathway Tools, the
software supports multiple methods for importing and exporting
data from files and via programmatic interactions, which are
summarized in Figure~\ref{fig:inputoutput}.
\begin{figure}
\begin{center}
\includegraphics{input-output.pdf}
\caption{Pathway Tools supported formats and APIs for data import and export.}
\label{fig:inputoutput}
\end{center}
\end{figure}
{\noindent \bf Programmatic Access through Application Programming
Interfaces (APIs).} Programmers can access and update PGDB data
directly \cite{PTools05,PToolsQueriesURL} by writing programs in the
Python, R, Java, Perl, and Common Lisp languages. R, Java, and Perl
queries are executed using systems called RCyc \cite{RCycURL}, JavaCyc
\cite{JavaCycURL}, and PerlCyc \cite{PerlCycURL}.
\blue{PythonCyc is a Python package that enables programmatic access to
Pathway Tools. The package provides the basic functions to access and
modify the data of any PGDB. It also exposes more than 150
functions of Pathway Tools, among them, the MetaFlux module. The
PythonCyc package is hosted on GitHub and is a separate installation
from Pathway Tools. A full API documentation and a tutorial is
available online. Please consult the URL~\cite{pythoncycURL} for
access to the package, the API documentation and the tutorial.}
\blue{{\noindent \bf Data Import Formats.} Pathway Tools can import data
from many sources into a PGDB. First of all, PathoLogic can create or
update a PGDB based on a PathoLogic Format (PF) \cite{PToolsGuide190}
or GenBank file \cite{GenbankFormatURL}. A PGDB can also be populated
from an SBML \cite{Hucka04,SBMLURL} or BioPAX
\cite{BioPAXURL,BioPAX10} file. Bulk updating of various data items
can be accomplished by a simple tab-delimited table format.}
\blue{Several specialized data types are also supported. The following
sources can be imported for proteins. UniProt sequence annotation
features \cite{UniProtProteinFeatureURL} can be fetched from a
Biowarehouse \cite{Biowarehouse06} server that was loaded with
SwissProt and TrEMBL. GO (Gene Ontology) annotations can be loaded
from a GAF \cite{GO2000,GOFormatsURL} file. PSORTdb \cite{Yu11}
cellular localization data can be imported from tab-delimited files.}
\blue{Gene regulatory data can be imported from RegTransBase
\cite{Cipriano13}, which is a SQL database. Growth conditions versus
growth media can be imported from Phenotype Microarray (PM)
\cite{Bochner-etal01,Bochner09} files. High-throughput expression
data can be obtained from NCBI GEO \cite{GEO13} and PortEco
\cite{PortEco14} via web services.}
\blue{{\noindent \bf File Export Formats.} Pathway Tools can export PGDBs
into several file formats that we have developed, which include
tab-delimited tables and an attribute-value format (see
\cite{BioCycFlatfilesURL}). Pathway Tools can also export subsets of
PGDB data to other common formats including SBML
\cite{Hucka04,SBMLURL}, BioPAX \cite{BioPAXURL,BioPAX10}, GO (in the
GAF format) \cite{GO2000,GOFormatsURL}, GenBank
\cite{GenbankFormatURL}, and FASTA \cite{FASTAFormatURL}. The
bioCycPlugin for Cytoscape \cite{bioCycPluginURL} makes use of web
services in conjunction with BioPAX, to select and export pathways into
the Cytoscape environment.}
{\noindent \bf Export for Relational Database Access via
BioWarehouse.} For scientists who want to query PGDB data through a
relational database system, the attribute-value files exported by
Pathway Tools can be loaded into SRI's BioWarehouse system
\cite{Biowarehouse06}. BioWarehouse is an Oracle or MySQL-based
system for integration of multiple public bioinformatics databases.
PGDB data can be queried through BioWarehouse alone or in combination
with other bioinformatics DBs such as UniProt, GenBank, NCBI Taxonomy,
ENZYME, and KEGG.
{\noindent \bf Queries Using the BioVelo Query Language,
and Web Services.}
Pathway Tools provides a powerful database query language for PGDBs,
called BioVelo \cite{BioVeloLangURL,BioVeloSAQPURL}. BioVelo queries
can be issued through an interactive web form, and through APIs.
\blue{Pathway Tools web services \cite{PToolsWebServicesURL} enable
programmatic retrieval of numerous data types, based on submitted HTTP
GET or POST commands, and have expanded substantially in recent years.
Users can access BioVelo queries, a Metabolite Translation Service,
and can also invoke omics visualization services and SmartTable
manipulations via web services. Services for SmartTables include
creation, retrieval, copying, and deletion; applying many
transformations; and changes like adding and deleting rows, columns
and cells.}
\section{Graph-Based Metabolic Network Analyses}
\label{sec:analyses}
This section describes Pathway Tools modules for performing
graph-based analyses of metabolic networks.
\Ssection{Metabolic Route Searching Using The RouteSearch Tool}
\label{sec:routesearch}
\blue{RouteSearch~\cite{Latendresse14} is a Pathway Tools component that
enables the exploration of the reaction network of a PGDB, and
engineering of
new metabolic pathways. RouteSearch computes optimal metabolic routes (that is,
an optimal series of biochemical reactions that connect start and
goal compounds), given various cost parameters to control the
optimality of the routes found. RouteSearch can display several of
the best routes it finds using an interactive graphical web page (see
Figure~\ref{fig:routesearch} for an example).
}
\blue{When RouteSearch is used in metabolic engineering, it uses the
MetaCyc database as its external reaction database for new reactions
to include in an organism. The cost for adding one reaction is selected
by the user. Typically this cost, an integer, is larger than the cost of
using one reaction from the organism. In such a case, a new reaction
would be added for a route if it conserve more atoms from the start
compound to the target compound. The cost of losing one atom from the
start compound is also selected by the user, and it is typically
larger than the cost of one reaction from the organism or the external
library of new reactions.
}
\begin{figure}
\includegraphics[width=6in]{routeSearch-2.png}
\caption{\blue{The RouteSearch web interface is shown with the result of one short
pathway found. The arrows represent reactions and are tagged with
the protein names catalyzing them. All atoms are conserved from the
start compound (succinate) to the target compound ((S)-Malate) in this
simplified example.}}
\label{fig:routesearch}
\end{figure}
\blue{In computing optimality, RouteSearch takes into account the
conservation of non-hydrogen atoms from the start compound to the goal
compound. The more atoms that are conserved, the more efficient the
transformation from start to goal. To compute the number of conserved
atoms, RouteSearch uses pre-computed atom mappings of reactions that
are available in MetaCyc. An atom mapping of a reaction gives a
one-to-one correspondence of each non-hydrogen atom from reactants to
products. RouteSearch is available only in web mode in Pathway Tools.
}
\Ssection{Dead-End Metabolite Analysis}
\label{sec:reachability}
Dead-end metabolites are metabolites that are only produced by the metabolic
network or only consumed by the network.
More precisely, the tool for identifying dead-end metabolites will
report a small-molecule metabolite $M$ as a dead-end metabolite in the cellular compartment
$C$ if and only if
one of the following conditions is true:
\begin{enumerate}
\item $M$ or parent classes of $M$ are only {\bf consumed} by small-molecule
reactions occurring in $C$, and $M$ or parent classes of $M$ are not transported into $C$.
\item $M$ or parent classes of $M$ are only {\bf produced} by small-molecule
reactions occurring in $C$, and $M$ or parent classes of $M$ are not
transported out of $C$, and no enzyme in the PGDB uses $M$ as a cofactor
\end{enumerate}
\Ssection{Computation of Blocked Reactions}
\label{sec:blockedreactions}
The MetaFlux component of Pathway Tools computes the blocked reactions
in a reaction network (see~\ref{sec:logfile})---reactions that can
never carry flux because of blockages in the network.
\Ssection{Prediction of Network Choke Points}
One application of a metabolic network model is to find network bottlenecks,
which if blocked could kill the cell. Such bottlenecks could
constitute antimicrobial drug targets. We have developed a
tool for predicting these so-called choke points.
The Pathway Tools choke-point detection algorithm examines the reactions attached to a given
metabolite, and processes one metabolite at a time. The first step is
to assemble the list of metabolites to examine. This is done by
collecting (1) all reactions that are in pathways, plus (2) reactions
that stand alone, but which use only small molecule metabolites. The
reactions that came from pathways may use some macromolecular
substrates, such as proteins that are modified by the reaction. From
this list of reactions, the algorithm collects all of their substrate
metabolites (meaning their reactant or product metabolites).
Definition \cite{Yeh04}: A ``choke point reaction'' is a reaction that
either uniquely consumes a specific substrate or uniquely produces a
specific product in a metabolic network, and is also balanced by at
least one reaction that respectively produces or consumes that
substrate. Specifically, the algorithm searches for two types of
choke point reactions: (a) Reactions $R_1$ such that only a single
reaction $R_1$ produces metabolite $M,$ and at least one reaction
consumes $M.$ (b) Reactions $R_2$ such that only a single reaction
$R_2$ consumes metabolite $M,$ and at least one reaction produces $M.$
These definitions imply that to find a choke point, all reactions
involving $M$ must be unidirectional. These choke point reactions are
collected and returned as the result. Note that the definition
excludes reactions directly connected to dead-end metabolites.
The resulting candidate choke point reactions can be painted onto the
cellular overview to facilitate further analysis.
\section{Comparative Tools}
\label{sec:comparative}
Pathway Tools contains a rich set of operations for comparing the
information in two or more PGDBs. These operations range from
comparison of genome-related information to comparison of pathway
information. These comparisons are of several types.
The comparative genome browser discussed in Section~\ref{sec:genov}
displays replicon regions centered on orthologous genes across a
set of genomes (see Supplementary Figure~9).
The user can generate a comparative table for a given metabolic
pathway across a specified set of organisms. For each organism the
table shows the presence of pathway enzymes and operon structures of
genes within the pathway.
A global comparison of the metabolic networks of multiple PGDBs can
be performed by highlighting on the Cellular Overview diagram (see
Section~\ref{sec:cellov}). This tool enables the user to highlight
in the Cellular Overview reactions that are shared, or not shared,
among a specified set of organisms.
Finally, a general comparative analysis facility enables the user to
generate comparative report tables for many aspects of a PGDB. As
well as being used for comparative analyses, these tools can be used
to generate statistics regarding the content of a single PGDB. These
tools are general in that they present their results in a standard
format, and they enable the user to drill down to specific results in a
consistent fashion. The initial report page shows summary statistics,
but the user can drill down to compare all instances of a category
by clicking on elements of a report table.
For example, consider the transporter report page
in Supplementary Figure~10. Table~2 within that report
summarizes the number of uptake transporters found in two organisms.
A user who wants to see the actual transported substrates
clicks on the text ``Compounds transported into the cell''
to generate a new report page containing a table listing the union of all substrates
imported by both organisms, along with an indication of which
organisms transport each substrate, and which transporter is utilized.
If the user clicks on a data cell within Table~2, such as the
number of imported substrates in {\em E. coli}\ K-12 (156), a page is
generated that lists those substrates only. Similar functionality
applies to most tables in these reports.
The following report types are provided. An example comparative
report is available at URL \cite{BioCycCompGenomicsExample1URL}.
\begin{itemize}
\item Reaction report includes the following statistics for each
selected organism:
\begin{itemize}
\item Number of reactions containing substrates of different types
(e.g., reactions for which all substrates are small molecules, and for
which some substrate is a protein or a tRNA)
\item Number of reactions in each Enzyme Commission (EC) category
\item Number of reactions containing different numbers of isozymes
\end{itemize}
\item Pathway report includes these statistics:
\begin{itemize}
\item Number of pathways in each category within the MetaCyc pathway
ontology
\item Number of pathways with different numbers of pathway holes
\end{itemize}
\item Compound report includes these statistics:
\begin{itemize}
\item Frequency with which different compounds appear in different
metabolic roles (substrate, cofactor, inhibitor, activator)
\end{itemize}
\item Protein report includes these statistics:
\begin{itemize}
\item General statistics on number of monomers versus multimeric
complexes, breakdown of multimers into heteromultimers and
homomultimers
\item Statistics on multifunctional enzymes
\end{itemize}
\item Transporter report includes these statistics:
\begin{itemize}
\item Number of efflux versus influx transporters
\item Number of genes whose products are transporters
\item Number of unique transported substrates, both overall and broken
down by efflux versus influx
\item Number of transported substrates that are substrates in
metabolic pathways or are enzyme cofactors
\item Transporters with multiple substrates, and substrates with
multiple transporters
\item Operon organization of transporters
\end{itemize}
\item Ortholog report includes these statistics:
\begin{itemize}
\item List of all orthologous proteins across the selected organisms
\item Proteins that are shared in all selected organisms, or unique to
one organism
\end{itemize}
\item Transcription Unit report includes these statistics:
\begin{itemize}
\item Distribution of number of genes per transcription unit
\item Distribution of number of operons into which metabolic pathway genes
are distributed
\end{itemize}
\end{itemize}
\section{Software and Database Architecture}
\label{sec:arch}
Pathway Tools is mostly implemented in the Common Lisp programming
language,\footnote{We use the Allegro Common Lisp implementation from
Franz Inc., Oakland, CA.} with some components implemented in JavaScript.
We chose Common Lisp because it is a
high-productivity programming environment. Because Lisp is a very
high-level language, one line of Lisp code is equivalent to several
lines of code in a language such as Java or C++. Therefore, the same
program can be written more quickly in Lisp, with fewer bugs. A study
by Gat found that compiled Lisp programs generally run faster than
Java programs, and that a given program can be developed 2--7 times
faster in Lisp than in Java \cite{Gat00}. Common Lisp also has a very
powerful interactive debugging environment.
Lisp has powerful dynamic capabilities that are illustrated
by a Pathway Tools feature called auto-patch. Imagine that a Pathway
Tools user site has reported a bug in the software. Once our group
has found a fix for the bug, we put a patch file that redefines the
offending Lisp function(s) on the SRI website. The next time Pathway
Tools is started at remote sites, it automatically downloads the patch
(in compiled form) from the SRI website, puts the patch in an
appropriate directory, and dynamically loads the patch file into the
running Pathway Tools to redefine the altered function(s).
Pathway Tools consists of \nLinesCodePTools\ lines of Common Lisp code, organized
into 20 subsystems. In addition, 24,000 lines of JavaScript code are
used within the Pathway Tools web interface. Pathway Tools runs on
the Macintosh, Linux, and Microsoft Windows platforms. Pathway Tools was ported
to 64-bit architectures in recent years.
The main bioinformatics modules of Pathway
Tools are the Navigator, Editors, and PathoLogic, plus a
chemoinformatics subsystem that includes tools such as SMILES
\cite{Smiles1} generation and parsing, a chemical substructure
matcher, plus a large set of shared utilities that we call the Pathway
Tools core. Pathway Tools uses an object-oriented database system
called Ocelot. The Pathway Tools user interface relies on a graph layout
and display package called Grasper \cite{GRASPER-CL}, and web and
desktop graphics packages called CWEST and CLIM
(the Common Lisp Interface Manager).
Other software used by and included with Pathway Tools are
(bioinformatics) textpresso; MUSCLE \cite{MUSCLE04};
PatMatch \cite{PatMatch05}; BLAST; libSBML \cite{LibSBML08};
(chemoinformatics) Marvin \cite{MarvinURL}; GlycanBuilder
\cite{Damerell12}; InChI \cite{InChI03};
(lisp) ARNESI; 5am; cl-json; cl-store;
(other) SCIP \cite{SCIPURL}; ghostscript; SKIPPY \cite{SKIPPYURL};
Yahoo User Interface library (YUI) \cite{YUIURL};
SOLR \cite{SOLRURL}; and MySQL.
Ocelot is an object/relational database management system (DBMS)
developed by our group at SRI \cite{EcoCycJCB96,Karp-JIIS-97a}.
Ocelot combines the expressive power of frame knowledge representation
systems \cite{KarpFreview} developed within the AI community (whose
object data model is far superior to the relational data model for
representing biological data\footnote{Superior aspects of the object data model
include the following. The object data model is better at managing very complex
schemas. That is, if the same domain is represented within the object data and within
the relational model, the object schema is usually much more compact and easier to
comprehend. One reason is that inheritance enables the object data model to define
subclasses by extending existing classes (e.g., the class Polypeptides is a subclass
of the class Proteins), whereas the relational model would force attributes shared between
the two tables to be duplicated in each, which both obscures the fact that the two
tables are related, and complicates schema evolution. Relational normalization also
increases the size of the schema by forcing the creation of new tables for every
multi-valued attribute, which is not required in the object data model. The object data model
used by Ocelot is particularly flexible in supporting any type of schema evolution without
forcing the entire database to be reloaded (unlike relational DBMSs), which is important in bioinformatics because
the complexity of biological data forces never-ending enhancements to the schema (note
that not every object DBMS provides such flexibility).}) with the scalability of relational
database management systems (RDBMSs). Ocelot DBs are persistently
stored within an Oracle or MySQL RDBMS. Ocelot objects are faulted on
demand from the RDBMS, and in addition are faulted by a background
process during idle time. Objects that were modified during a user
session are tracked and saved to the RDBMS during a save operation.
Ocelot uses optimistic concurrency control
\cite{Karp-JIIS-97a}---during
a save operation it checks for conflicts between the updates
made by the user and updates saved by other users since the saving
user began their session or last made a save operation. This approach
avoids the overhead of locking that becomes problematic in object
databases because modifications to one object often cascade to related
objects and could require a large number of lock operations. The
optimistic concurrency control works well in practice because curators
tend to focus in different biological areas and therefore rarely
update the same objects at the same time.
Ocelot DBs can also be saved to disk files, in which case the RDBMS is
not needed (see Figure~\ref{fig:arch-storage}). The file persistence
configuration is simpler to use since it does not require purchase or
installation of an RDBMS. It provides an easy and low-cost way to
begin a PGDB project; a project can switch to an RDBMS configuration
as its complexity grows. The advantage of an RDBMS configuration is
that it provides Ocelot with multi-user update capabilities, and it
permits incremental (and therefore faster) saving of DB updates. The
RDBMS configuration also enables Ocelot to maintain a history of all
DB transactions --- DB curators can examine the history of all updates
to a given object to determine when a given change was made, and by
whom. This functionality is very useful when diagnosing mistakes
within a PGDB.
Figure~\ref{fig:arch-graphics} shows the graphics architecture of Pathway Tools.
The Grasper graph toolkit is used in pathway layouts, and in the
cellular overview and regulatory overview. Grasper graphics, and
all other graphics generated by Pathway Tools, are rendered using the CLIM
Common Lisp graphics system, which is implemented using the X window system on Linux and Mac, and the native Windows API on Windows.
When Pathway Tools runs as a desktop application, CLIM graphics
directly update the user's screen.
Pathway Tools can also run as a web server, which is how it powers
websites such as BioCyc.org. Pathway Tools does not run with an
associated HTTP server such as Apache. Instead, Pathway Tools provides a
fully functional web server that includes services such as compression
and connection keep alive.
HTTP servers typically start a new operating-system process for each incoming web
request that terminates after the request has been serviced. In
contrast, Pathway Tools starts one long-lived web server process that
can service many thousands of web requests; that process forks an internal thread
to service each incoming request.
Pathway Tools processes an incoming web request in the following
manner. The top-level directory name within an incoming URL indicates
whether the operation is requesting a static file or dynamically
generated page.
\begin{itemize}
\item {\bf Static files:}
A small number of web pages, such as the home page and informational
pages for BioCyc.org, are implemented as disk files. Pathway Tools
can serve file-based web pages like a traditional web server.
\item {\bf Dynamically generated pages:} Most Pathway Tools pages are
generated dynamically by
processing that typically includes
querying PGDBs within the Pathway Tools virtual memory and generating
query outputs and visualizations, often using the same code as for
desktop mode. The CLIM graphics generated by software such as the
pathway layout code are dynamically converted to HTML and GIF images using CWEST \cite{CWEST2}, which uses Skippy
to generate GIF images. The resulting HTML and GIF images are
returned to the user's web browser, and the Pathway Tools web server
awaits the next query. The GIF images include generated
specifications of mouse-sensitive regions and of what operations
should be invoked when the user clicks on such a region.
\end{itemize}
\begin{figure}
\begin{center}
\includegraphics[height=3in]{arch-storage.pdf}
\caption{Database architecture and APIs for Pathway Tools.}
\label{fig:arch-storage}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[height=3in]{arch-graphics.pdf}
\caption{Graphics architecture of Pathway Tools.}
\label{fig:arch-graphics}
\end{center}
\end{figure}
\section{Metabolic Modeling with MetaFlux}
\label{sec:metaflux}
\blue{The MetaFlux component of Pathway Tools is used to develop and execute
quantitative metabolic flux models for individual organisms and for
organism communities. MetaFlux uses the steady-state modeling
technique of flux-balance analysis, which can be used to predict the
phenotypes of an organism, or a community of organisms, based on
a specification of available nutrients in the growth environment.}
\blue{MetaFlux offers several modes of operation:}
\blue{\begin{enumerate}
\item Solving mode: execute a metabolic model for a single organism or
for a community of organisms
\item Development mode: generate hypotheses on how to fill gaps in a
developing metabolic model
\item Knockout mode: run metabolic models under gene-knockout scenarios
\end{enumerate}}
\blue{MetaFlux can be controlled via a graphical user interface and via a Python API.}
\blue{The next section provides an overview of metabolic model development
using MetaFlux.}
\subsection{The MetaFlux Model Development Process}
\label{sec:metaflux-dev}
\blue{Pathway Tools provides a unique environment for the development of
metabolic flux models for several reasons. First, it includes a range
of tools that support fast and accurate development of metabolic
models from annotated genomes. Second, metabolic models developed
with MetaFlux are highly accessible to the user, and are coupled with
extensive enriching information, resulting in models that are easier
to understand and reuse.}
\blue{The high-level steps for developing a metabolic model from an annotated genome
using Pathway Tools are as follows. For more information on the
genome-scale metabolic reconstruction process, we suggest the
comprehensive COBRA-based \cite{COBRA2011} overview published by
Thiele and Palsson \cite{Thiele10}.}
\begin{enumerate}
\item \blue{The PathoLogic tool computes a metabolic
reconstruction by inferring the reactome of an organism from its
annotated genome. By combining the enzyme-name matching tool with the
extensive reaction information in the MetaCyc database, we obtain
a very complete mapping of annotated enzymes to metabolic reactions. }
\item \blue{Subsequent inference of metabolic pathways by PathoLogic
fills a significant number of missing (gap) reactions, because as well
as inferring the presence of pathways,
pathway inference also infers the presence of pathway
reactions that were not initially identified. }
\item \blue{The pathway hole filler identifies enzymes that catalyze those pathway-hole
reactions. (Note this step is optional and is informative in nature,
because it does not modify the set of reactions in the model.)}
\item \blue{The user can request that MetaFlux compute an initial
set of biomass metabolites for the organism. MetaFlux will do so if
the organism falls within the twelve taxonomic groups for which
MetaFlux has defined biomass compositions, obtained from the
experimental literature.}
\item \blue{The user supplies an objective function and constraints
on metabolite uptake and secretion. These can be based on experimental
observations of the organism under study, or can be set to arbitrary values
to explore the theoretical behavior of metabolism.}
\item \blue{The MetaFlux gap filler identifies missing reactions, nutrients,
and secretions that enable a model to be solved. It can be run on one
compartment at a time to simplify the gap-filling process for
eukaryotic organisms. MetaFlux tools for computing blocked reactions and dead-end metabolites
identify potential errors and omissions in the metabolic network definition.}
\item \blue{The reaction network, objective function, and constraints
of the metabolic model are adjusted by the user until its predictions
match experimental results.}
\end{enumerate}
\blue{When developing a metabolic model with MetaFlux, the reactions and
metabolites within the model are derived from (and stored in) the PGDB. MetaFlux
automatically generates the system of linear equations for the model
from the PGDB. Thus, to modify the reactions within a model, the user
edits the PGDB; to inspect the reactions and metabolites within a
model, the user can query the PGDB using the plethora of Pathway Tools query and
visualization operations. The entire PGDB/model can be published on a
website using Pathway Tools, where, all reactions and pathways
that utilize a given metabolite are listed on the Pathway Tools
metabolite page for that compound; all model metabolites within a given molecular weight
range can be found using metabolite searches; all reactions within a
given cellular location or using a given set of reactants and products
can be found using reaction searches.}
\blue{Furthermore, compared to other modeling environments, a metabolic
model stored within a PGDB contains extensive additional enriching
information. Chemical structures within a PGDB enable reaction mass
and charge balancing. Chemical structures and reaction atom mappings
aid users in understanding the chemistry of reaction transformations.
Pathways arrange reactions into biologically meaningful groupings.
Couplings between reactions, enzymes, and genes enable reasoning about
the roles of multi-subunit complexes, isozymes, and gene knockouts.
Regulatory information supports inferences about metabolic regulation.
Model testing and validation are facilitated by PGDB storage of growth
media and growth experiment results, and of gene-knockout experiment
results.}
\blue{Taken together, the Pathway Tools modeling environment---with its
extensive tools for inspecting metabolic-model content and its enriching information---renders MetaFlux models
significantly easier to understand, learn from, validate through
inspection, reuse, and extend than models produced with other
metabolic-modeling software environments.}
\subsection{Description of an FBA Model}
\blue{The description of a flux-balance analysis (FBA) model is provided to
MetaFlux partly via a text file called an FBA input file and partly
via a PGDB. Typically, the FBA file specifies many parameters, but we
will describe only the most important ones. Some parameters are only
relevant for specific modes of MetaFlux, so that we will present these
parameters when describing that mode. All parameter names end with a
colon, whereas keyword options for parameters start with a colon. In
the following explanation, we omit the colon character to reduce
clutter.}
\subsection{Solving Mode}
\blue{Solving mode computes flux values for the reactions in the metabolic
model given four inputs: a set of nutrient compounds, a set of
secreted compounds, a set of biomass metabolites that are synthesized
by the cell, and a set of metabolic reactions. The first three inputs
are supplied by the FBA file. The set of metabolic reactions are
provided by the PGDB, but may be altered by the FBA file.
For example, the following file describes a (very simple) FBA model for
{\em E. coli}: \dsw{Shouldn't MAL, FUM, and WATER have compartment specifications?}}
\begin{verbatim}
pgdb: ecoli
reactions:
metab-all # Include all metabolic reactions of PGDB ecoli.
FUMHYDR-RXN # Example of including a reaction according to its
# PGDB unique identifier.
mal -> fum + water # Example of including a reaction by reaction equation.
biomass:
CYS[CCO-CYTOSOL] 0.0054
GLN[CCO-CYTOSOL] 0.2987
GLT[CCO-CYTOSOL] 0.2987
GLY[CCO-CYTOSOL] 0.3431
nutrients:
GLC[CCO-PERI-BAC] :upper-bound 10.0
OXYGEN-MOLECULE[CCO-PERI-BAC] :upper-bound 1.0
secretions:
CARBON-DIOXIDE[CCO-PERI-BAC]
WATER[CCO-PERI-BAC]
\end{verbatim}
\blue{Note that this example is meant to show the syntax of the file
describing a model, it is not meant to show a working
model. The~\texttt{pgdb} parameter specifies the PGDB to be used in this
model. The set of model reactions is specified by using the
\texttt{reactions} parameter. The keyword \texttt{metab-all}
specifies all metabolic reactions from the PGDB, whereas
\texttt{FUMHYDR-RXN} specifies one reaction using a unique
identifier. A reaction equation can be provided to describe a
reaction that is to be present in the model, but is not present in the PGDB. }
\blue{The set of biomass metabolites are specified by parameter
\texttt{biomass}; each metabolite is specified either by metabolite
name or unique identifier, plus a compartment identifier in square
brackets, and an optional coefficient. Nutrients and secretions are
provided in the same manner, but no coefficients are allowed. Upper
and lower bounds can be provided to constrain nutrient uptake rates or
secretion production rates.}
\blue{Specifying the metabolites of the biomass
reaction using groups is also possible. The use of groups present a better
structure of the biomass reaction. For example,}
\begin{verbatim}
biomass:
:group val
Charged-VAL-tRNAs[CCO-CYTOSOL] 0.423162
VAL-tRNAs[CCO-CYTOSOL] -0.423162
:end-group
:group thr
Charged-THR-tRNAs[CCO-CYTOSOL] 0.253687
THR-tRNAs[CCO-CYTOSOL] -0.253687
:end-group
\end{verbatim}
\blue{These two groups, one named {\tt val}, the other named {\tt thr},
specify a relationship between a charged and an uncharged tRNA, for
valine and threonine. Essentially, the groups enable gathering the
compounds where one or more compounds, with negative coefficients, are
needed to produce the other(s) with positive coefficients. In other
words, these groups describe partial sub-reactions of the biomass
reaction because the metabolites with negative coefficients can be
considered as reactants, and the metabolites with positive coefficients
can be considered as products of a reaction, which is not necessarily
mass balanced. The negative and positive values do not need to be the
same absolute values because these sub-reactions are allowed not to be
mass balanced. Metabolites can be repeated in different groups, but
repetition of a metabolite is not allowed outside groups. Groups have
another use with the \texttt{try-biomass} parameter in development
mode (see subsection \ref{sec:developmentMode}). }
\subsubsection{Modeling a Community of Organisms}
\blue{Almost all organisms live in a larger ecosystem. MetaFlux can
solve a model describing an organism community by combining
individual FBA input files for each organism in the community, plus
one COM input file describing the organism
interactions. A typical COM description lists the FBA models to be combined, optionally
with the abundance of each organism in the community. It lists the
compartments shared by the organisms --- the compartments in which the
secretions of some organisms can be used as nutrients by other
organisms. For example, the following COM file
specifies two models, the exchange of metabolites in the periplasmic
space, and three nutrients that are provided to the community.}
\begin{verbatim}
community-name: ecoli-ile
fba-files:
ecoli-strain-A.fba :abundance 2
ecoli-strain-B.fba :abundance 3
exchange-compartments: [CCO-PERI-BAC]
override-fba-nutrients: no
community-nutrients:
GLC[CCO-PERI-BAC] :upper-bound 20
OXYGEN-MOLECULE[CCO-PERI-BAC] :upper-bound 10.0
AMMONIUM[CCO-CYTOSOL]
\end{verbatim}
\blue{The objective function defined by MetaFlux for the FBA model of a
community of organisms is the sum of the growth of the organisms
multiplied by their abundances. A solution file will be produced
that describes the growth of the community, and the growth rate for each
organism. We are currently developing an alternative community
modeling approach inspired by COMETS \cite{Harcombe14}, in which each organism
has an independent objective function, and dynamic FBA is used to create a
temporal simulation.}
\subsection{Development Mode}
\label{sec:developmentMode}
\blue{The development mode can be used to create an FBA model or to discover
what is wrong with a model that does not grow when growth is expected.
The main parameters used in development mode are the four parameters
\texttt{try-biomass}, \texttt{try-reactions}, \texttt{try-secretions}
and \texttt{try-nutrients}. }
\blue{When MetaFlux is provided with a list of metabolites for the \texttt{try-biomass} in development mode,
the software tries adding these metabolites to the biomass reaction.
That is, in development mode, MetaFlux computes
the maximum number of \texttt{try-biomass} metabolites
that can be produced by the model. In other words,
MetaFlux will output as a solution the largest subset of
metabolites it can produce as biomass.
In the early phases of developing a model, typically the entire biomass reaction is specified by
\texttt{try-biomass} and no metabolites are specified for the
\texttt{biomass} parameter. }
\blue{The \texttt{try-biomass} parameter cannot specify any metabolite with
a negative coefficient not included in a group. This is required
because any such metabolite could be used as a ``free'' nutrient
for the organism. On the other hand, inside a group, negative
coefficients are allowed because MetaFlux tries to produce the entire
group of metabolites, not any one of them independently. A group of metabolites
is supposed to form a cohesive unit where any metabolite specified
with a negative coefficient is used to produce some other metabolites
of that group. Therefore, declaring all metabolites that have negative
coefficients in some groups enables the entire biomass reaction to be used as
a try-biomass set, indicating for all metabolites which ones can or
cannot be produced.}
\blue{The \texttt{try-reactions} parameter can be used to try to add
candidate reactions to the model to increase the number of
\texttt{try-biomass} metabolites produced by the model. Therefore, the
\texttt{try-reactions} parameters is typically used when at least one
biomass metabolite is not produced. The candidate reactions are selected from MetaCyc.
A single keyword, \texttt{metacyc-metab-all}, instructs MetaFlux to try all
the metabolic reactions of MetaCyc. Alternatively, a list of candidate reactions
can be specified by their unique identifiers.
MetaFlux tries to produce as many
biomass metabolites as possible from the \texttt{try-biomass} section, by
adding as few reactions as possible from the \texttt{try-reactions} set.
This computation is performed using optimization as a
Mixed-Integer Linear Program (MILP), which can be computationally
expensive to solve. The amount of time to compute an optimal solution
varies depending on the number of candidate reactions. It can take a
few seconds to several hours, or even days.}
\blue{MetaFlux has also a fast development mode that can be used with the
\texttt{try-reactions} parameter only. It may run much faster than the
general development mode and may provide different solutions. It uses
an heuristic that does not necessarily provide an optimal solution. But
optimality is not always the best possible biological solution. Please
consult~\cite{LatendresseGapFill14} for a more complete
description of the advantages and disadvantages of fast
development mode.}
\subsection{Knockout Mode}
\blue{Knockout mode is used to computationally evaluate the impact of
removing genes or reactions from an FBA model. This mode is used to
predict essential genes of an organism for a given growth
environment, and can also be used to evaluate the accuracy of a model if
experimental gene-knockout data are available. MetaFlux can compute
single, double, or higher numbers of simultaneous knockouts.}
\blue{When run, MetaFlux solves the model without any knockouts, and
then solves for each gene to knockout by deactivating the reactions associated
with that gene. Note that a given gene may deactivate one, none, or several
reactions, because some genes may have isozymes, or catalyze several
reactions. Requesting a summary only of the results of
the knockout or requesting that, beside the summary, a complete solution
file be produced for each gene knockout is also possible. See the subsection~\ref{sec:sol}
for more details on the solution files produced.}
\subsection{Outputs Generated by MetaFlux}
\blue{Whichever mode is used to execute MetaFlux, the following output files are
generated: the solution file that contains a description of the active
reactions and metabolites used and produced, the log file to
describe issues that may exist in the model, and a data file
for the fluxes that can be displayed with the Cellular Overview map of
Pathway Tools.}
\blue{In solving mode, the main output is the computed optimal assignments
of reaction fluxes.
For an organism-community model, solution, log, and data files are produced
for each organism.}
\blue{In development mode, the outputs are the set of biomass metabolites
that can be produced; the try nutrients utilized (if any); the try
secretions produced (if any); the reactions that actively carry flux;
and a minimal list of suggested reactions to be added to the model,
and reactions whose directions are reversed (if any) to produce
otherwise unproducible biomass metabolites. Development mode also
identifies which biomass metabolites could not be produced by the
model (if any), despite the additions from the try sets.}
\subsubsection{Reports Generated by MetaFlux}
\label{sec:logfile}
\blue{The set of reactions specified by the PGDB and the FBA input file may contain
reactions that cannot, or that should not, be used in a model. MetaFlux
checks each reaction to ensure that its inclusion in the model
will not invalidate the model. In
particular, all reactions are verified to be mass and
charge balanced. Reactions that are unbalanced, or cannot be shown to be balanced, are not included in
the model, but are listed in the log file.}
\blue{Another step in model execution is to instantiate the
generic reactions of a model according to the compounds available in the PGDB. A
generic reaction has some compound classes as products or reactants (e.g.,
``sugars''). Each computed reaction instantiation
is added to the model. If a generic reaction instantiation
was not possible (for example, when which compound
instances should be used for classes on each side of a reaction it is ambiguous)
then such generic reactions are listed in the log file.}
\blue{{\noindent \bf Blocked reactions.} MetaFlux analyzes the network of
reactions specified by the PGDB plus the FBA input file to detect if
some reactions are blocked. A blocked reaction is a reaction that can
never have a positive flux, given the nutrients, secretions, biomass
metabolites, and reactions specified for a particular model
execution. That is, blocked reactions are a function not only of the
network, but of the cellular growth conditions. A blocked reaction
$R$ has at least one reactant $M$ that is not produced by any reaction
in the model, and that is not provided as a nutrient; or $R$ has a
product $M$ that is not used as a reactant by any other reaction and
that is not secreted, and is not specified as a biomass metabolite
with a positive coefficient. These reactions are the {\it basic
blocked reactions}, and the metabolites that caused basic blocked
reactions to be blocked are called {\it basic blocking metabolites}.}
\blue{Blocked reactions can never carry flux because in steady-state
modeling, the producing and consuming fluxes for every metabolite must
be balanced. But the preceding metabolites $M$ could not have
balanced fluxes if a blocked reaction $R$ carried flux, because
according to the preceding definition, $M$ must have either zero
reactions that produce it or zero reactions that consume it.}
\blue{Additional blocked reactions can be identified by eliminating basic
blocked reactions from the model, causing more reactions to become
basic blocked reactions. This process of removing basic blocked
reactions from the model is repeated until no more reactions become
blocked. The detection of blocked reactions is done before the linear
solver is called (that is, this detection does not depend on the
fluxes of reactions, but is a static evaluation of the model).}
\blue{The set of blocked reactions is listed in the log file, grouped by
basic blocking metabolite (the reactions are also grouped by
pathways). The basic blocking metabolites are, in a sense, the root
causes of blocked reactions, thus their identification is quite
valuable for model debugging.}
\subsubsection{The MetaFlux Solution File}
\label{sec:sol}
\blue{The content of the solution file generated by MetaFlux
depends on the mode of MetaFlux. In solving mode, the solution file
contains the uptake rates of the nutrients, the production rates of
the biomass metabolites and secretions, and the fluxes of the model
reactions. If no growth is obtained (that is, the flux of the biomass reaction is
zero, or very near zero), this condition is stated in the solution file. }
\blue{When solving a community model, a community solution file is generated
that contains a summary of the result, and a complete solution file is
produced for each organism in the community. The summary
lists many values: the growth rate of each model, the compartments
that were involved in the exchange of metabolites between the
organisms; the secretions used as nutrients by some organisms with
their flux; the secretions that were not used by any organisms; and the
secretions that were not produced by any organism of the community. In
essence, the summary gives the overall view of the exchange of
metabolites between the organisms and the rate of growth of each
organism.}
\blue{In development mode, the solution file contains essentially what is
given in solving mode, but the flux values are only meaningful as to
whether they are zero or non-zero. For example, if a
\texttt{try-biomass} parameter was specified with some metabolites,
the metabolites that could or could not be produced are
identified. This is different from the solving mode, where only one
biomass metabolite that cannot be produced will result in no growth,
but with no indication of which metabolites could not be produced.}
\blue{In knockout mode, a solution file with a summary of the results is
produced. The summary includes the growth of the model with no knockouts,
followed by a list of every knockout that was requested, based
on genes and/or reactions to knockout, and the resulting flux for the biomass
reaction. Requesting a complete solution file
for each knockout performed is also possible.}
\subsubsection{Painting MetaFlux Fluxes on the Cellular Overview}
\blue{MetaFlux also generates a data file that can be used with the Cellular
Overview of Pathway Tools (see Section~\ref{sec:cellov}). The graphical user interface of MetaFlux
enables the user to click one button to invoke the Cellular Overview
with the data from that file. The Cellular Overview displays all the
reactions, grouped by pathways, of the organisms, and highlights, with
colors indicative of flux values, the reactions that have positive fluxes. This overview
enables the user to visually assess the metabolic activities of a
model solution.}
\blue{For a community of organisms, a data file is generated for each
individual organism, and the graphical user interface gives direct
access to each individual Cellular Overview of the organisms involved in
the community.}
\section{Survey of Pathway Tools Compatible Databases}
\label{sec:pgdbs}
According to user surveys, Pathway Tools users have created 9,800
PGDBs for organisms from all domains of life, in addition to the
\NbiocycPGDBs\ PGDBs available from BioCyc.org. This section summarizes those
user-created PGDBs that are publicly available.
With highly curated PGDBs available for many important
organisms, it is not clear why users would consider using the
uncurated (and therefore lower quality) pathway DBs available for
these same organisms from other pathway DB providers. For example,
consider the highly curated AraCyc pathway database for {\em Arabidopsis thaliana}
\cite{Mueller03,AraCycURL}. AraCyc contains mini-review summaries for enzymes
and metabolic pathways; thousands of literature references; evidence
codes for enzyme functions and metabolic pathways (indicating which pathways are supported
by experimental evidence); and information on
enzyme subunit structure, activators, inhibitors, and cofactors. In
contrast, KEGG data on {\em Arabidopsis}
contains none of the preceding information. In addition, AraCyc
curators have carefully refined the metabolic reactions and pathways
present in AraCyc, such as to remove false-positive computational
predictions, and to add {\em Arabidopsis} reactions and pathways from
the biomedical literature to AraCyc. Although KEGG updates its
reference pathway map diagrams periodically to contain new pathways
and reactions from different organisms, the KEGG approach of
computationally coloring reactions within pathway maps based on the
presence of enzymes for those reactions within a genome results in
significant ambiguity. If AraCyc curators are reasonably certain that
a reaction or pathway is absent from {\em Arabidopsis}, they remove it
from the database. The KEGG model does not allow such removal, so whether an uncolored reaction is truly
absent from an organism or whether the gene for its enzyme has not
yet been identified in the genome
is never clear within KEGG. This situation results in a real
conundrum for a scientist who wishes to assemble the list of reactions
likely to be present in {\em Arabidopsis} from KEGG, because is no
way exists to distinguish the many uncolored reactions that are likely
present but for which no gene has been identified, from the many
uncolored reactions that are clearly known to be absent from {\em
Arabidopsis} (which curators have deleted from AraCyc).
Available PGDBs include those in Tables
\ref{tab:animal-pgdbs}--\ref{tab:microbe-pgdbs}, with curated PGDBs in
bold.
{\small
\begin{table}[!h]
\centerline{
\begin{tabular}{|l|l|l|} \hline
{\bf Hosting Site} & {\bf Organism} & {\bf Citations, URLs} \\ \hline \hline
BioCyc & \bf {\em Bos taurus} (CattleCyc) & \cite{CattleCyc09,BioCycURL} \\ \hline
FlyCyc & \bf {\em Drosophila melanogaster} (FlyCyc) & \cite{FlyCycURL} \\ \hline
BioCyc & \bf {\em Homo sapiens} (HumanCyc) & \cite{KarpHumanCyc04,HumanCycURL} \\ \hline
MouseCyc & \bf {\em Mus musculus} (MouseCyc) & \cite{MouseCycURL} \\ \hline
\end{tabular}
}
\caption{\label{tab:animal-pgdbs}
{\bf PGDBs for animals.}
}
\end{table}
}
{\small
\begin{table}[!h]
\centerline{
\begin{tabular}{|l|l|l|} \hline
{\bf Hosting Site} & {\bf Organism} & {\bf Citations, URLs} \\ \hline \hline
CoffeaCyc & {\em Coffea canephora} & \cite{CoffeaCycURL} \\ \hline
MediCyc & \bf {\em Medicago truncatula} & \cite{MediCyc07,MediCycURL} \\ \hline
Gramene & {\em Arabidopsis thaliana} & \cite{AraCycGrameneURL} \\ \hline
Gramene & {\em Brachypodium distachyon} & \cite{BrachyCycURL} \\ \hline
Gramene & {\em Coffea canephora} & \cite{CoffeaCycGrameneURL} \\ \hline
Gramene & {\em Medicago truncatula} & \cite{MediCycGrameneURL} \\ \hline
Gramene & {\em Oryza sativa} & \cite{RiceCycURL} \\ \hline
Gramene & {\em Populus trichocarpa} & \cite{PoplarCycGrameneURL} \\ \hline
Gramene & {\em Sorghum bicolor} & \cite{SorghumCycURL} \\ \hline
Gramene & {\em Solanum lycopersicum} & \cite{LycoCycGrameneURL} \\ \hline
Gramene & {\em Solanum tuberosum} & \cite{PotatoCycGrameneURL} \\ \hline
Gramene & {\em Zea mays} & \cite{MaizeCycURL} \\ \hline
sol genomics network \cite{SOLCYCURL} & {\em Capsicum annuum} & \cite{CapCycURL} \\ \hline
sol genomics network & {\em Nicotiana benthamiana} & \cite{BenthmianaCycURL} \\ \hline
sol genomics network & {\em Nicotiana tabacum} & \cite{TobaccoCycURL} \\ \hline
sol genomics network & {\em Petunia hybrida} & \cite{PetCycURL} \\ \hline
sol genomics network & {\em Solanum lycopersicum} & \cite{LycoCycURL} \\ \hline
sol genomics network & {\em Solanum tuberosum} & \cite{PotatoCycURL} \\ \hline
Plant Metabolic Network \cite{PlantCycURL} & \bf PlantCyc & \cite{PlantCycURL} \\ \hline
Plant Metabolic Network & \bf {\em Arabidopsis thaliana} (AraCyc) & \cite{Mueller03,AraCycURL} \\ \hline
Plant Metabolic Network & {\em Brachypodium distachyon} & \cite{BrachypodiumCycURL} \\ \hline
Plant Metabolic Network & {\em Brassica rapa} & \cite{ChineseCabbageCycURL} \\ \hline
Plant Metabolic Network & {\em Carica papaya} & \cite{PapayaCycURL} \\ \hline
Plant Metabolic Network & {\em Chlamydomonas reinhardtii}& \cite{ChlamyCycURL} \\ \hline
Plant Metabolic Network & {\em Manihot esculenta} & \cite{CassavaCycURL} \\ \hline
Plant Metabolic Network & {\em Hordeum vulgare} & \cite{BarleyCycURL} \\ \hline
Plant Metabolic Network & {\em Glycine max} & \cite{SoyCycURL} \\ \hline
Plant Metabolic Network & {\em Panicum virgatum} & \cite{SwitchgrassCycURL} \\ \hline
Plant Metabolic Network & {\em Physcomitrella patens} & \cite{MossCycURL} \\ \hline
Plant Metabolic Network & {\em Populus trichocarpa} & \cite{PoplarCycURL} \\ \hline
Plant Metabolic Network & {\em Oryza sativa} & \cite{OryzaCycURL} \\ \hline
Plant Metabolic Network & {\em Selaginella moellendorffii} & \cite{SelaginellaCycURL} \\ \hline
Plant Metabolic Network & {\em Setaria italica} & \cite{SetariaCycURL} \\ \hline
Plant Metabolic Network & {\em Sorghum bicolor} & \cite{SorghumBicolorURL} \\ \hline
Plant Metabolic Network & {\em Vitis vinifera} & \cite{GrapeCycURL} \\ \hline
Plant Metabolic Network & {\em Zea mays} & \cite{CornCycURL} \\ \hline
\end{tabular}
}
\caption{{\bf PGDBs for plants.}}
\label{tab:plant-pgdbs}
\end{table}
}
{\small
\begin{table}[!h]
\centerline{
\begin{tabular}{|l|l|l|} \hline
{\bf Hosting Site} & {\bf Organism} & {\bf Citations, URLs} \\ \hline \hline
Ano2Cyc & \bf {\em Anopheles gambiae} & \cite{Ano2CycURL} \\ \hline
CandidaCyc & \bf {\em Candida albicans} & \cite{CandidaCycURL} \\ \hline
DictyCyc & \bf {\em Dictyostelium discoideum} & \cite{DictyCycURL} \\ \hline
LeishCyc & \bf {\em Leishmania major} & \cite{LeishCycURL} \\ \hline
PchrCyc & \bf {\em Penicillium chrysogenum} & \cite{PchrCycURL} \\ \hline
YeastCyc & \bf {\em Saccharomyces cerevisiae} (YeastCyc)& \cite{YeastCycURL} \\ \hline
ThapsCyc & \bf {\em Thalassiosira pseudonana} & \cite{ThapsCycURL} \\ \hline
TrypanoCyc & \bf {\em Trypanosoma brucei} & \cite{TrypanoCycURL} \\ \hline
BioCyc & \bf 8 PGDBs for {\em Plasmodium, Cryptosporidium,} and {\em Toxoplasma} & \cite{BioCycURL} \\ \hline
\end{tabular}
}
\caption{\label{tab:other-eukaryote-pgdbs}
{\bf PGDBs for other eukaryotes.}
}
\end{table}
}
{\small
\begin{table}[!h]
\centerline{
\begin{tabular}{|l|l|l|} \hline
{\bf Hosting Site} & {\bf Organism} & {\bf Citations, URLs} \\ \hline \hline
BioCyc & \bf {\em Escherichia coli} (EcoCyc) & \cite{EcoCycNAR13,EcoCycURL} \\ \hline
BioCyc & \bf {\em Bacillus subtilis} & \cite{BsubCycURL} \\ \hline
BioCyc & \bf {\em Candidatus cardinium} & \cite{Cbtq1CycURL} \\ \hline
BioCyc & \bf {\em Candidatus portiera aleyrodidarum} & \cite{PabtqvlcCycURL} \\ \hline
BioCyc & \bf {\em Candidatus Evansia muelleri} & \cite{EvaCycURL} \\ \hline
BioCyc & \bf {\em Clostridium saccharoperbutylacetonicum} & \cite{ClossacCycURL} \\ \hline
BioCyc & \bf {\em Listeria monocytogenes} & \cite{10403s_rastCycURL} \\ \hline
BioCyc & \bf {\em Methylosinus trichosporium} & \cite{Mob3bCycURL} \\ \hline
BioCyc & \bf {\em Peptoclostridium difficile} & \cite{Pdif272563CycURL} \\ \hline
BioCyc & \bf {\em Streptomyces coelicolor} & \cite{ScoCycURL} \\ \hline
PseudoCyc & \bf {\em Pseudomonas aeruginosa} (PseudoCyc) & \cite{PseudoCycURL} \\ \hline
BioCyc & 5500 PGDBs & \cite{MetaCycNAR14,BioCycURL} \\ \hline
MicroCyc & 1702 PGDBs & \cite{MicroCycURL} \\ \hline
Taxonomically Broad EST Database & 49 PGDBs & \cite{TBestDBURL,OBrien2007} \\ \hline
BeoCyc & 23 PGDBs & \cite{BeoCycURL} \\ \hline
\end{tabular}
}
\caption{\label{tab:microbe-pgdbs}
{\bf PGDBs for microbes.}
}
\end{table}
}
To facilitate sharing of PGDBs among multiple users, we have created a
PGDB Registry that enables peer-to-peer sharing. PGDB sharing is
desirable because a user whose own computer has a copy of a PGDB can
use Pathway Tools functionality that would not be available through a
remote Pathway Tools web server, such as functionality that exists in
desktop mode only, and such as comparative operations. Comparative
analysis of two or more PGDBs is possible only when they are loaded
into the same instance of Pathway Tools.
The PGDB Registry uses a server maintained by SRI that tracks the
locations of available PGDBs that PGDB authors have registered for
downloading. The author of a PGDB can register that PGDB by
using a command within Pathway Tools that creates an entry for the
PGDB in the Registry server, and places the PGDB on an FTP or HTTP
server of the author's choosing. Users who want to download a PGDB
from the Registry can view available PGDBs by using a web browser (see
URL \cite{PToolsRegistryURL}) or using Pathway Tools itself. With a
few mouse clicks, a user can download a PGDB from the registry using
Pathway Tools.
\section{Comparison of Pathway Tools with Related Software Environments}
\label{sec:related}
Pathway Tools stands out with respect to related software tools in the
breadth of the functionality and the high level of integration that it
provides. It addresses a very large number of use cases. And it
provides schema, visualization, and editing support for an unusually
large number of data types in addition to metabolic and signaling pathways, including
chromosomes, genes, enzymes, transporters, regulatory networks, and compounds.
The following comparison is organized roughly according to the use
cases presented in Section~\ref{sec:use-cases}.
\Ssection{Development, Visualization, and Web Publishing of Organism Specific Databases}
\SSsection{Metabolic Pathway Information}
Other software systems for managing metabolic pathway information are
KEGG \cite{KEGG06,KEGG08}; The SEED \cite{Overbeek14};
GenMAPP \cite{GENMAPP07,PathVisio08};
PathCase \cite{PathCase03,PathCase06}; VisANT \cite{VisANT07}; and Reactome
\cite{Reactome14}. KEGG, Reactome, and GenMAPP employ static, pre-drawn pathway diagrams, a model that does not
scale to produce custom pathway diagrams for thousands of
{\em different} pathways in different organisms. Nor can the static approach produce
multiple views of a given pathway at different levels of
detail, as can the Customize Pathway option in Pathway Tools that enables
the user to choose exactly which graphical elements (e.g., gene names, EC
numbers, metabolite structures, activators and inhibitors)
appear in the pathway diagram, and to color a pathway diagram with
omics data.
PathCase and VisANT do have pathway layout capabilities,
but the resulting diagrams bear little resemblance to those found in
the biomedical literature, nor are they particularly compelling
visually. \blue{Their diagrams can be customized in various ways
(e.g., choice of layout algorithm, data overlays, and selective
compaction of nodes), but they do
not offer the same types of customization or multiple-detail views offered by
Pathway Tools.}
Cytoscape \cite{Cytoscape03} is a general tool for displaying
biological networks that embodies the philosophy that general graph
layout techniques can satisfactorily depict any biological network.
Although the Cytoscape layout algorithms are a terrific fit for
display of protein interaction maps, we assert that the general layout
algorithms do not
produce useful results for metabolic pathways. We believe that
superior visualization results are obtained when layout algorithms
are specifically tailored to metabolic pathways. For example, Pathway
Tools provides separate layout algorithms for circular, linear, and
tree-structured pathways to make the structure of those pathways stand
out prominently to the biologist. Biologists developed their
pre-computer depictions of metabolic pathways for important reasons,
namely to clearly present the topology of a pathway.
\blue{SEED models lack a unifying pathway visualization framework---they
use KEGG pathway diagrams, often supplemented with simple
text-based diagrams and/or images imported from other sources, with
limited interactivity beyond hyperlinks.}
GenMAPP, Reactome, VisANT, KGML-ED \cite{Klukas07}, and PathCase have
pathway-editing capabilities. KEGG and SEED lack pathway editing,
meaning that users cannot introduce new organism-specific pathways,
nor modify a reference pathway to customize it to a specific organism,
thus eliminating the possibility of removing erroneous reaction steps
from a pathway, or of adding missing reactions to a pathway.
KEGG Atlas \cite{KEGGatlas08} has a comparable
analog of our Cellular Overview diagram (which we introduced in 1999
\cite{Cyclone99,OverviewURL}). The KEGG Atlas diagram was constructed
by hand and provides a single set of overview metabolic maps for all
organisms in KEGG, as opposed to the organism-specific overviews that
Pathway Tools generates through advanced layout algorithms. The KEGG
diagram is not as flexibly queryable or interactive as the Pathway
Tools diagram, and it also lacks semantic zooming for adding increasing
details.
\blue{A notable improvement occurred with Pathway Projector
\cite{Tomita09}. It relies directly on KEGG for the overall
overview diagram and other data, but graphically, it is enhanced,
adding EC numbers and genes to reaction edges.}
\blue{Reactome includes a different type of high-level overview diagram
called Fireworks, which shows a hierarchical organization of
nodes representing
metabolic and signaling pathways, mostly the latter. Users can navigate to pathways of interest with this
diagram, and highlight pathways that contain objects such as compounds
or genes that are supplied in an analysis interface. Fireworks can
zoom in until pathway names are shown. From there, one can click to a
specific pathway display for the details. However, Fireworks does not
depict individual reactions or metabolites; therefore, it is not a
metabolic map diagram.}
\blue{Tools such as VisANT and
Cytoscape can be used to display very large networks, including the
entire metabolic network for an organism, and these diagrams can be
used to visualize omics data and interacted with in other ways, but
given that such diagrams are laid out using general-purpose network
layout algorithms, individual pathways are not likely to be
recognizable.}
\blue{No other general tool can show reactions with atom-mapping
information.}
\blue{Computationally generated diagrams for electron transfer reactions and
pathways, as well as for transport reactions, with their depiction of
compartments at a cellular membrane, are unique features of Pathway
Tools.}
\SSsection{Genome and Proteome Information}
\label{sec:gen}
\blue{Most model-organism DB tools include genome browsers and gene pages. A
representative sample of such systems includes GBrowse
\cite{GBrowse02,GBrowse07}; JBrowse \cite{Skinner09}; IMG
\cite{Markowitz14}; Entrez Genome \cite{EntrezGenomeURL}; the UC Santa
Cruz (UCSC) Genome Browser \cite{UCSC15}; Ensembl \cite{Ensembl15};
and PATRIC \cite{Wattam14}.}
\blue{The Pathway Tools genome browser uses a different approach than the others.
First, all the other genome browsers depict the chromosome as a single horizontal
line, with tracks information below it. Pathway Tools uses this approach
when the user explicitly enables tracks, but when tracks are not
enabled, a multi-line wrapped display is used to present the most
possible information in the available display window. Second, all the
other genome browsers use tracks to depict coding regions, promoters,
and other genome features. Although Pathway Tools does provide
a tracks capability, its assumption is that consensus curated
information on coding region extents, promoters, terminators, and
transcription-factor binding sites should be depicted ``inline'' on
the multi-line chromosome. This approach results in a much more
information-dense display.}
\blue{GBrowse is highly customizable through the addition of data tracks.
Wrapped multiline displays that will make full use of screen real
estate are not available. In GBrowse, the semantics of semantic
zooming has to be configured but provides flexibility. GBrowse can
provide comparative genomics with additional tools like SynView
\cite{Wang06} or GBrowse\textunderscore syn \cite{McKay10}.}
\blue{JBrowse is a JavaScript-based genome browser that could become a
replacement for GBrowse. Its main innovation is to rely on the web
browser to perform most of the processing and drawing, making it fast
and smooth. PATRIC uses JBrowse.}
\blue{The IMG genome browser (or ``chromosome viewer'', in their parlance)
is very basic, and does not support zooming. Some views are shown as
wrapped multiline displays, and several gene coloring schemes can be
selected, but no data tracks are offered. For comparative genomics,
three different, unrelated synteny viewers are available.}
\blue{The Entrez Genome sequence viewer has no real semantic viewing other
than showing the DNA sequence at maximum zoom level. It offers support
for adding extra tracks and an alignment view for comparative genomics.}
\blue{The UCSC Genome Browser is mainly geared towards eukaryotes and has an
extensive support for addition of data tracks. But no real semantic
viewing is available other than showing the DNA sequence at maximum zoom level.}
\blue{Ensembl is focused on eukaryotes, similarly to UCSC, and also has
extensive support for extra tracks. It offers no real semantic
viewing, other than showing the DNA sequence at maximum zoom level.}
\blue{Most sequenced genomes are circular, but Pathway Tools seems to have
the only genome browser that can seamlessly depict the junction between
the first and last base-pair to show a contiguous view. All other
genome browsers seem to artificially linearize all chromosomes. Also,
Pathway Tools seems to be unique in providing a compact overview of
the entire genome (the ``genome overview''), which enables coloring each gene with omics data.}
\SSsection{Regulatory Networks}
A number of bioinformatics databases include regulatory network
information; however, the majority of these databases and their
associated software environments can represent information on
transcription-factor-based regulation only, such as RegTransBase
\cite{RegTransBase07}; TRANSFAC \cite{Transfac06}; CoryneRegNet \cite{CoryneRegNet07}; ProdoNet
\cite{ProdoNet08}; and DBTBS \cite{DBTBS08}. The exception is
RegulonDB \cite{RegulonDB08}, which can also capture RNA-based
regulation, including riboswitches, attenuators, and small-RNA
regulators.
We are not aware of tools comparable to the Regulatory Overview in
being able to display and interrogate large, complete cellular
regulatory networks, although CoryneRegNet and ProdoNet
display smaller regulatory networks. CoryneRegNet
also displays omics data onto its regulatory network diagrams.
\SSsection{Query Tools}
Other organism DBs provide a subset of the three tiers of
queries provided by Pathway Tools (quick search, object-specific
searches, and Structured Advanced Query Page). Virtually all provide
a quick search. Sites providing particularly extensive
object-specific searches are FlyBase \cite{Flybase05}; Mouse Genome
Informatics \cite{MGD08}; EuPathDB \cite{PlasmoDB09}; and BioMart
\cite{Biomart09}. BioMart is used by bioinformatics DBs, including
WormBase, Rat Genome Database, UniProt, Reactome, and Galaxy. Its
underlying query language is Perl, using the BioMart libraries.
However, none of the preceding systems provides the query power of the
Pathway Tools SAQP. For example, BioMart does not allow the user to
construct arbitrary queries that perform joins (queries that combine
multiple data types); it provides only the ``and'' logical operator
(the ``or'' operator is not available); and it includes only a limited
form of ``not.'' Biozon \cite{Biozon06} (\verb+biozon.org+) provides
a web interface that provides fairly complex querying, including join
operations, over several biological DBs.
\Ssection{Extend Genome Annotations with Additional Computational Inferences}
KEGG, Model SEED, and Reactome are the only other tools that can
predict pathways from genome data. The pathway hole filler and
transport inference parser tools are unique to Pathway Tools. Many
genome-annotation pipelines include operon predictors.
\Ssection{Analysis of Omics Data}
\blue{Kono \etal\ introduced a tool based on the Google Maps API for
painting omics data onto an enhanced KEGG Atlas map, called Pathway
Projector \cite{Tomita09}. This tool does not produce animations as
our omics viewers do. However, it can depict time-series expression
data as small histograms that are reminiscent of Pathway Tools' omics pop-ups.
It also provides more powerful highlighting options than KEGG Atlas.
Sequence-based search and highlighting is available via KEGG BLAST,
although it can take minutes. Users can manually annotate the map
with custom markers and lines, which can be exported via XML and
shared with others.}
\blue{KEGG Atlas \cite{KEGGatlas08} enables some limited highlighting, but
mostly by using KEGG identifiers. No animation or omics pop-ups seem
to be available.}
\blue{Reactome \cite{Reactome14} can paint omics data onto the
Fireworks hierarchical pathway overview. Zooming to high
detail reveals the pathway names, but zooming does not go
to the reaction level. Showing time-series
data as animations is possible. No omics pop-ups seem to be available.}
GenMapp \cite{GENMAPP02}; VitaPad \cite{Holford04}; VisANT; and ArrayXPath
\cite{Chung04} paint omics data onto single pathways, rather than onto
a full metabolic overview.
\Ssection{Metabolic Modeling}
\blue{Other well-known tools providing constraint-based metabolic modeling
include COBRA \cite{COBRA2011} and Model SEED \cite{Henry10}.
Many other FBA tools exist, but a detailed comparison with all
of them is beyond the scope of this paper. Lakshmanan \etal\ provide a
comparison among 19 FBA tools, including MetaFlux (see their Tables 2,
3, and 4) \cite{Lakshmanan14}. This paper was first published in 2012
and therefore covers an old version of MetaFlux. Additions to MetaFlux
since that time include expansion to the Windows platform, import of
SBML files, solving of models for organism communities, and support
for dynamic FBA. Other tools have probably developed new capabilities as
well. We now examine some of the issues raised by Lakshmanan \etal\
in more detail.}
\blue{Our design of MetaFlux has emphasized ease of use and acceleration of
the very slow model development times of 12--18 months stated in some
FBA publications. At SRI we have reliably been able to create FBA
models for bacteria and fungi in 3--4 weeks (albeit to a lower level
of validation than some published models).
We note that in another recent survey of
metabolic-modeling software \cite{Hamilton14}, Pathway Tools
was the only software package recommended for users new to modeling.}
\blue{Elements of the MetaFlux approach that speed model
development are as follows.
(a) Use of MetaFlux does not require
programming ability as some other tools do.
(b) Lakshmanan mentions the collection of metabolic reactions for an
organism at the start of all metabolic modeling projects in passing,
without noting the strong link between model accuracy
and the completeness of the initial reaction set. The
PathoLogic enzyme name matcher (see Section~\ref{sec:pwy-inference})
combined with the extensive reaction
database of MetaCyc \cite{MetaCycNAR14} provide a powerful resource for
reactome prediction. Furthermore, when metabolic pathway prediction
follows reactome prediction, it produces a more complete metabolic
network, because it imports all reactions in a
pathway, even those that lack enzyme assignments, thus simplifying the
later gap-filling step. Furthermore, the fact that Model SEED contains
approximately $1/3$ as many metabolic pathways as does MetaCyc raises
a major question in the use of Model SEED for pathway
reconstruction.\footnote{This number was obtained by manually
reviewing the 1,320 subsystems present in the February 2015 version
of Model SEED, then removing all subsystems not related to metabolism,
then removing subsystems for individual metabolic enzymes. This
process yielded 730
subsystems that correspond to metabolic pathways, compared to the 2,300
pathways in the February 2015 version of MetaCyc. }
(c) MetaFlux has a more
powerful gap filler than any other tool: it can gap fill not
only reactions but also nutrients and secretions, and it can
identify which biomass metabolites cannot be synthesized by
the model, a critical step in helping the user discover which aspect of the
network is incomplete. This is a feature not found in other tools to our knowledge.
(d) MetaFlux computes blocked reactions and metabolites, reporting to the user the basic blocking
metabolites that are the root causes of model blockages.
(e) MetaFlux computes the reaction balance from the reaction equations plus chemical
structure data to ensure that unbalanced reactions are prohibited from
inclusion in a model.}
\blue{Many other tools use the GLPK solver (some not exclusively). MetaFlux
uses the SCIP solver \cite{SCIPURL}, which has a much faster
Mixed-Integer Linear Programming (MILP) solver, needed for gap filling,
and is free for research use by academics. The SCIP MILP solver is fast,
although not as fast as commercial solvers.}
\blue{The comment by Lakshmanan \etal\ on p8 that ``none of
the [network visualization] tools can handle large-scale models''
clearly does not apply to MetaFlux, as its cellular overview diagram
does handle genome-scale models.}
\blue{Lakshmanan \etal\ suggest (p10) that more tools should consider linking with
a biological model storage DB. Pathway Tools has taken this approach
to an extreme, as described in Section~\ref{sec:metaflux-dev}.}
\section{Limitations and Future Work}
\label{sec:limitations}
Here, we summarize limitations of Pathway Tools, organized by use
case. Some of these limitations are being addressed in current
research; many of the others will be addressed in future work.
{\noindent \bf Development of Organism Specific Databases.}
Pathway Tools has an emphasis on prokaryotic biology, although over
time we have added, and plan to add, more support for eukaryotic biology.
One remaining limitation is that although the software can capture many types of prokaryotic
regulation, we have not attempted comprehensive coverage of eukaryotic regulation.
Another limitation is that the editing tools within Pathway Tools are not web based, but require
installation of Pathway Tools on every computer that will be used for editing.
{\noindent \bf Visualization and Web Publishing of Organism Specific Databases.}
In recent years we have made significant progress in making the
capabilities of the web and desktop modes of the Pathway Tools
Navigator as similar as possible. For example, many cellular overview
capabilities previously present only in desktop mode are now also available
in web mode. However,
not all capabilities of Pathway Tools are available in
both the web and desktop modes. For example, many comparative tools
function in web mode only, whereas all aspects of PathoLogic and the
editing tools are available in desktop mode only.
{\noindent \bf Analysis of Omics Data.}
Pathway Tools is not a general-purpose environment for analysis of
omics data. Our assumption is that scientists will use one of the many
other software packages for the early stages of omics data analysis
(such as normalization), and will provide the output of those analyses to
Pathway Tools for display with the omics viewers.
{\noindent \bf Analysis of Biological Networks.}
We would like to see many additional network analysis tools present
within Pathway Tools, such for computing the scaling properties of
metabolic networks \cite{Jeong00}, and functional modules within metabolic
networks \cite{Ma04}.
{\noindent \bf Comparative Analysis of Organism-Specific Databases and
Metabolic Engineering.}
We are not aware of striking limitations in these areas.
\section{Summary}
Pathway Tools treats a genome as far more than a sequence and a set of
annotations. Instead, it links the molecular parts list of the cell
both to the genome and to a carefully constructed web of functional
interactions. The Pathway Tools ontology defines an extensive set of
object attributes and object relations that enables representing a rich
conceptualization of biology within a PGDB, along with enabling
querying and manipulation by the user. Furthermore, a PGDB can be
transformed into a quantitative metabolic model for the organism.
Pathway Tools provides a broad range of functionality. It can
manipulate genome data, metabolic networks, and regulatory networks.
For each datatype, it provides query, visualization, editing, and
analysis functions. It provides model-organism database development
capabilities, including computational inferences that support fast
generation of comprehensive databases, editors that enable
refinement of a PGDB, web publishing, and comparative analysis. A
family of curated PGDBs has been developed using these tools for
important model organisms.
The software also provides visual tools for analysis of omics
datasets, and tools for the analysis of biological networks.
\section{Software Availability}
Pathway Tools runs on Macintosh, Windows, and Linux. It is freely
available to academic and government researchers; a license fee
applies to commercial use. See \url{http://BioCyc.org/download.shtml}.
\section*{Funding}
This work was supported by grants GM75742, GM080746,
and GM077678 from the National Institutes of Health. The
contents of this article are solely the responsibility of the authors
and do not necessarily represent the official views of the National
Institutes of Health.
\section*{Acknowledgments}
Pathway Tools has benefited from advice, input, and contributions from
many scientists during its lifetime. We particularly wish to
recognize contributions from Ian Paulsen, Robert Gunsalus, Monica
Riley, John Ingraham, Jean-Francois Tomb, and Peifen Zhang.
Lukas Mueller developed PerlCyc and
has provided many helpful suggestions. Thomas Yan developed JavaCyc.
Tomer Altman developed RCyc. Jeremy Zucker developed the SBML
generation module, and contributed many other ideas. Christos
Ouzounis was a co-developer of the original metabolic pathway
prediction algorithm, contributed an early version of the
import/export system, and has been a source of much sound advice.
\section*{Key Points}
\begin{itemize}
\item The Pathway Tools software is a comprehensive
environment for creating model organism DBs that span
genome information, metabolic pathways, and regulatory networks.
\item Pathway Tools inference capabilities include
prediction of metabolic pathways, prediction of metabolic pathway hole
fillers, inference of transport reactions from transporter functions,
and prediction of operons.
\item Its metabolic modeling capabilities include flux-balance
analysis modeling for individual organisms and organism
communities, with model gap filling and the ability to model
gene knockouts.
\item Pathway Tools provides interactive editing tools for use by
database curators.
\item Omics data analysis tools paint genome-scale datasets onto a
complete genome diagram, complete metabolic network diagram, and
complete regulatory network diagram.
\item Other tools include comparative analysis operations,
dead-end metabolite and blocked-reaction analysis of metabolic networks,
and metabolic route searching.
\end{itemize}
|
1,108,101,566,763 | arxiv | \section{Introduction}
It is well known that exactly solvable systems play very
important role in quantum theory. Unfortunately number of
such systems is quite limited. This considerably narrows
their applications. Such a situation stimulates interest to
quasi-exactly solvable systems \cite{qes, shifman, ushv1,
ushv2, kamran91, kamran93, turb&post}. In contrast to
exactly solvable models in quasi-exactly solvable systems
the spectral problem can be solved partially. Nevertheless
such systems are very interesting. Besides modeling physical
systems \cite{appl, mbody, th0105223} they can be used as an
initial point of the perturbation theory or to investigate
various nonperturbative effects~\cite{Aoyama}. Furthermore,
recently in the series of papers~\cite{nsusy} (see also
Refs.~\cite{ andrian2, klish04}) it was revealed a
connection between quasi-exactly solvable models and
supersymmetric systems with nonlinear polynomial
superalgebras~\cite{andrian}.
There exist various approaches to constructing quasi-exactly
solvable systems \cite{qes, ushv2, shifman, Zaslavskii}.
Nevertheless all of them are not universal in the sense that
they do not cover all possible quasi-exactly solvable
systems. For example, the famous Lie-algebraic approach
\cite{qes, shifman} is used to construct quasi-exactly
solvable differential operators, but does not allow, for
example, to reproduce quasi-exactly solvable systems based
on hidden dynamical symmetries with nonlinear algebras
\cite{turb&post, kamran04, klish04}. In Ref. \cite{annih}
authors presented a general construction for quasi-exactly
solvable differential operators, linear and nonlinear. But
it cannot be directly applied to quasi-exactly solvable
noncommutative systems.
Considered in this paper approach is universal because it is
formulated in terms of algebraic relations and does not
depend on any space of representation. Besides we show that
this scheme reflects a general structure of quasi-exactly
solvable systems and and their connection with constrained
systems. Therefore particularly it can be applied to
construct both usual and noncommutative quasi-exactly
solvable systems.
The paper is organized as follows. In section \ref{alg} we
formulate a universal algebraic approach to quasi-exactly
solvable systems and demonstrate its connection with
constrained systems. In section \ref{app} the algebraic
approach is used to reproduce a family of two-dimensional
quasi-exactly solvable Hamiltonians, which in the
Lie-algebraic approach are derived from a finite-dimensional
representation of the Lie algebra $\mathfrak{su}(3)$. Brief discussion
of results is presented in section \ref{conc}.
\section{Formulation of the algebraic approach}
\label{alg}
Let us consider a set of linear operators, $A_k$ with $k=1,
\dots,n$ and $n\in\mathbb N$, on a Hilbert space $\mathcal H$ and suppose
that they have the following commutation relations:
\begin{align}\label{algA}
\left[A_k,\,A_l\right]&=\sum_{m=1}^nF_{klm}A_m,
\end{align}
where $F_{klm}$ are, in general, some linear
operators on $\mathcal H$. The subspace
$\mathcal F_A=\bigcap_{k=0}^n\mop{ker}A_k$ is the annihilator
\cite{annih} for the set of annihilating operators $A_k$.
Here we imply that the set $A_k$ is complete, i.e.
any linear operator $B$ on $\mathcal H$ with
$\mop{ker}B\subset\mathcal F_A$ can be represented as
$B=\sum_{k=1}^nC_kA_k$, where $C_k$ are some linear
operators on $\mathcal H$.
Thus from \eqref{algA} it follows that the annihilating
operators $A_k$ generally form nonlinear algebra. To come to
quasi-exactly solvable systems we have to require
\begin{align}\label{fdim}
\mop{dim}\mathcal F_A&<\infty,
\end{align}
i.e. $\mathcal F_A$ is a finite dimensional subspace of $\mathcal H$.
If a Hermitian linear operator $H$ on $\mathcal H$ has commutation
relations
\begin{align}\label{AH0}
\left[A_k,\,H\right]&=\sum_{l=1}^nM_{kl}A_l,
\end{align}
where $M_{kl}$ are, in general, some linear operators on
$\mathcal H$, then $H$ is a quasi-exactly solvable operator.
Indeed, the relations \eqref{AH0} imply that $H$ is an
invariant operator on finite dimensional space $\mathcal F_A$ and
can be diagonalized on this space by a finite procedure.
It is worth noting that the annihilating operators $A_k$ can
be treated as operator constraints while $\mathcal F_A$ can be
interpreted as a subspace of ``physical'' states. In this
context the commutation relations \eqref{AH0} can be
represented as
\begin{align}\label{AH1}
\left[A_k,\,H\right]&\thickapprox 0
\end{align}
or even as $A_kH\thickapprox 0$.\footnote{Here a linear
operator $B\thickapprox 0$ if $By=0$ $\forall y\in \mathcal F_A$.}
Thus, {\it a quasi-exactly solvable system with the
Hamiltonian $H$ obeying the commutation relations
\eqref{AH1} can be interpreted as a constrained Hamiltonian
system with the finite dimensional subspace of ``physical''
states $\mathcal F_A$.}
For $n=1$ the commutation relations \eqref{algA} and
\eqref{AH0} lead to the simplest superalgebra. Indeed, the
relation $\left[A,\,H\right]\thickapprox 0$ can be
represented as
\begin{align*}
\left[A,\,H\right]&=LA,
\end{align*}
where $L$ is a linear operator.
This relation can be rewritten in the matrix form
\begin{align*}
\left[Q,\,{\bf H}\right]=0,
\end{align*}
where $Q$ and $\bf H$ are matrix supercharge and
superhamiltonian:
\begin{align*}
Q&=\begin{pmatrix}
0&0\\mathfrak{A}&0
\end{pmatrix},&
{\bf H}&=\begin{pmatrix}
H&0\\0&L+H
\end{pmatrix}.
\end{align*}
In one-dimensional case, $\mathcal H={\cal C}^\omega(\mathbb R^1)$, we
have
\begin{align*}
\left\{Q,\,Q^\dag\right\}&=P(\bf H),
\end{align*}
where the order of the polynomial $P(.)$ is equal to the
order of the annihilating operator $A$ \cite{nsusy}. Thus we
came to the supersymmetry with a nonlinear polynomial
superalgebra. Various systems with such a nonlinear
supersymmetry and their relation to quasi-exactly solvable
systems were extensively studied \cite{nsusy}.
In multi-dimensional case, $\mathcal H={\cal C}^\omega(\mathbb R^d)$ with
$d>1$, the polynomial $P(.)$ has more involved structure and
can depend on some other operators. For example, the
two-dimensional systems with the nonlinear polynomial
supersymmetry were considered in Ref. \cite{d=2}.
The algebraic approach \eqref{algA}-\eqref{AH1} can be used
for constructing quasi-exactly solvable Hamiltonians.
Indeed, if we know the set of annihilating operators with
finite-dimensional annihilator, the corresponding
quasi-exactly solvable Hamiltonian can be constructed by
solving the commutation relations \eqref{AH0}.
Let $\mathcal F_A$ is a finite-dimensional functional space with a
basis of linearly independent analytical functions. Such a
basis can be used to construct the corresponding set of
annihilating operators \cite{annih}. Moreover, in
Ref.~\cite{annih} the general form of a quasi-exactly
solvable operator was derived. However, in practice such a
calculation of quasi-exactly solvable Hamiltonians can be
more complicated than that based on the proposed algebraic
approach. Besides, the algebraic approach is more universal
because its formulation is not restricted by specific
realization of the Hilbert space. For example, it can be
applied to noncommutative spaces.
It is worth noting that any quasi-exactly solvable system
admits formulation in the algebraic form
\eqref{algA}-\eqref{AH1}. Indeed, by definition in any
quasi-exactly solvable system there exits a
finite-dimensional subspace, say $\mathcal F$, which is invariant
for Hamiltonian of the system. For the finite-dimensional
subspace $\mathcal F$ it is always possible to construct a complete
set operators $A_k$ annihilating this subspace, i.e.
$A_ky=0$ $\forall y\in\mathcal F$ or $A_k\thickapprox 0$. Since the
Hamiltonian is invariant operator on $\mathcal F$ we conclude that
$A_kH\thickapprox 0$ because $Hy\in\mathcal F$ $\forall y\in\mathcal F$.
This is equivalent to \eqref{AH1}.
Thus the general scheme of building a quasi-exactly solvable
model is the following: (1)~We choose a finite set of
independent functions, an annihilator. (2)~The
corresponding complete set of annihilating operators has to
be constructed. (3)~We calculate the
quasi-exactly solvable operator of second order using the
relations \eqref{algA}-\eqref{AH1}. Existence of such an
operator depends on the set of independent functions.
Besides, for this operator to be a Hamiltonian it has to
obey to well-known conditions~\cite{shifman, kamran93,
klish07}.
\section{Application to 2D quasi-exactly solvable systems}
\label{app}
In Ref. \cite{klish04} the algebraic approach, discussed in
the last section, was applied to the annihilator
$\mathcal F_n=\left\{x^ky^l\,:\,0\le k\le n,\ 0\le l\le n\right\}$,
where $x,y\in\mathbb R$ and $k,l\in\mathbb Z_+$, while $n\in\mathbb N$.
The resulting Hamiltonian is equivalent to that derived from
the Lie-algebraic approach with Lie algebra
$\mathfrak{sl}(2,\,\mathbb R)\otimes\mathfrak{sl}(2,\,\mathbb R)$. In this section we
construct the quasi-exactly solvable Hamiltonian for the
following annihilator:
\begin{equation}\label{su3}
\mathcal F_n=\left\{x^ky^l\,:\,0\le k+l\le n\right\}.
\end{equation}
where $x,y\in\mathbb R$ and $k,l\in\mathbb Z_+$, while $n\in\mathbb N$.
For this annihilator the corresponding annihilating
operators can be taken in the form
\begin{equation}
A_k=\partial_x^k\partial_y^{n-k+1}
\end{equation}
with $k=0,\,1,\dots,n+1$.
Let us first construct general quasi-exactly solvable
operator of first order,
\begin{align}
L = L_1(x,y)\partial_x + L_2(x,y)\partial_y +L_0(x,y),
\end{align}
where $L_i(x,y)$ are real-valued analytical functions.
The commutation relations
\begin{align*}
\left[A_k,\,L\right]&\thickapprox 0,\qquad
\end{align*}
where $k=0,\,1,\dots,n+1$, lead to the following set of
differential equations:
\begin{multline}
(l+1)(m+1)L_0^{(m,l)}(x,y)
+ (m+1)(n-k-l+1)L_2^{(m,l+1)}(x,y)\\
{}+(l+1)(k-m)L_1^{(m+1,l)}(x,y)=0,
\end{multline}
where $-1\leq l\leq n-k+1$, $-1\leq m\leq k$, $l+m\geq 1$
and $L^{(i,j)}(x,y)=\partial_x^i\partial_y^jL(x,y)$.
This overdetermined system of differential equations can be
reduced to the following form:
\begin{align*}
L_1^{(0,2)}(x,y)&=0,&\ \ \quad
L_1^{(3,0)}(x,y)&=0,\ \ \quad&
L_1^{(2,1)}(x,y)&=0,
\end{align*}
\vspace{-8mm}
\begin{align*}
L_2{}^{(0,2)}(x,y) - 2L_1{}^{(1,1)}(x,y)&=0,&
L_0{}^{(0,1)}(x,y) + nL_1{}^{(1,1)}(x,y)&=0
\end{align*}
plus the equations obtained by exchange
$x\leftrightarrow y$. This system of differential equations
has 9-parametrical solution:
\begin{align*}
L_1(x,y)&= a_1 x + a_2 y + a_3 x^2 + a_4 xy + a_0,\\
L_2(x,y)&= b_1x + b_2 y + a_3 xy + a_4 y^2 + b_0,\\
L_0(x,y)&= c_0-n \left(a_3x + a_4y\right).
\end{align*}
It corresponds to the set of quasi-exactly solvable
differential operators
\begin{align}\label{J}
J=\left\{x\partial_x,\, y\partial_x,\, \partial_x,\, \partial_y,\, x\partial_y,\,
y\partial_y,\, x\left(x\partial_x+y\partial_y-n\right),\,
y\left(x\partial_x+y\partial_y-n\right),\, \hbox{{1\hskip -5.8pt 1}\hskip -3.35pt I}\right\},
\end{align}
which form a representation of the Lie algebra $\mathfrak{u}(3)$.
Now for the annihilator \eqref{su3} we construct a general
quasi-exactly solvable differential operator of second
order,
\begin{align*}
H&=H_{11}(x,y)\partial_x^2+H_{12}(x,y)\partial^2_{xy}+H_{22}(x,y)\partial_y^2
+ H_1(x,y)\partial_x + H_2(x,y)\partial_y+ H_0(x,y),
\end{align*}
where $H_{ij}(x,y)$ and $H_i(x,y)$ are real-valued
analytical functions. The commutation relations
\begin{align*}
\left[A_k,\,H\right]&\thickapprox 0,\qquad
\end{align*}
where $k=0,\,1,\dots,n+1$, lead to the following set of
differential equations:
\begin{multline*}
C_l^{n-k+1}C_m^kH_0^{(m,l)}
+ C_{l+1}^{n-k+1}C_m^kH_2^{(m,l+1)}\\
+ C_{l+2}^{n-k+1}C_m^kH_{22}^{(m,l+2)}
+ C_l^{n-k+1}C_{m+1}^kH_1^{(m+1,l)}\\
+ C_{l+1}^{n-k+1}C_{m+1}^kH_{12}^{(m+1,l+1)}
+ C_l^{n-k+1}C_{m+2}^kH_{11}^{(m+2,l)}=0,
\end{multline*}
where $C^k_m=\frac{k!}{(k-m)!m!}$.
This overdetermined system of differential equations can be
reduced to the following equations:
\begin{align*}
H_{11}^{(0,3)}&=0,&
H_{11}^{(4,1)}&=0,&
H_{11}^{(5,0)}&=0,&
6H_{11}^{(2,2)}-H_{22}^{(0,4)}&=0,
\end{align*}
\vskip -9mm
\begin{align*}
3H_{11}^{(1,2)}-H_{12}^{(0,3)}=0,\qquad
H_1^{(0,2)}+(n-1)H_{11}^{(1,2)}&=0, \\
3H_1^{(2,0)}-6H_2^{(1,1)} - (n-1)
\left(
3H_{22}^{(1,2)}-H_{11}^{(3,0)}
\right)&=0, \\
2H_0^{(0,1)} + n
\left(
2 H_1^{(1,1)}+(n-1)H_{11}^{(2,1)}
\right)&=0, \\
3H_{11}^{(2,1)} - 3H_{12}^{(1,2)}
+ H_{22}^{(0,3)}&=0
\end{align*}
plus the equations obtained by exchange
$x\leftrightarrow y$. This system of equations has the
following $36$-parametrical solution:
\begin{align*}
H_{11}&=\sum_{l=0}^2\sum_{k=0}^{4-l}a_{lk}x^ky^l ,\\
H_{22}&=\sum_{l=0}^2\sum_{k=0}^{3-l}b_{lk}x^ly^k
+ \left(a_{22}y^2 + a_{13}xy + a_{04}x^2\right)y^2,\\
H_{12}&=\sum_{k=0}^2\left(2a_{k\,4-k}xy + a_{k\,3-k}y
+ b_{2-k\,k+1}x\right)x^{2-k}y^k
+ \sum_{k=0}^2\sum_{l=0}^{2-k}c_{kl}x^ky^l,\\
H_1&= 2 (1-n)a_{04} x^3
+\left((1-n)
\left(a_{03}-b_{12}\right)+f_{11}\right)x^2
+ d_{01}x
\\&
+ (1-n)\left(a_{21}+2 x a_{22}\right)y^2
+ \left(-2(n-1)a_{13}x^2+d_{11} x+d_{10}\right)y
+ d_{00},\\
H_2&= 2 (1-n) a_{22}
y^3+\left((1-n) \left(b_{03}-a_{12}\right)+d_{11}\right)
y^2+f_{01}y
\\&
+ (1-n)x^2\left(2a_{04}y+b_{21}\right)
+ x\left(2 (1-n)a_{13}y^2+f_{11}y+f_{10}\right)
+ f_{00},\\
H_0&= n(n-1)a_{04} x^2
- n\left((n-1) b_{12}+f_{11}\right)x
+ n(n-1)a_{13}xy
\\&
+ n(n-1)a_{22}y^2
- n\left((n-1) a_{12}+d_{11}\right)y
+ h_{00},
\end{align*}
where all the coefficients $a_{lk}$, $b_{lk}$, $c_{kl}$,
$d_{kl}$, $f_{kl}$, $h_{00}$ are real.
By direct calculation it can be shown that the resulting
Hamiltonian is equivalent to the operator
$$
H=\sum_{\alpha,\beta=1}^9c_{\alpha\beta}J_\alpha J_\beta,
$$
where $c_{\alpha\beta}\in\mathbb R$,
which corresponds to the Lie-algebraic Hamiltonian for the
Lie algebra\footnote{In the case of the algebra
$\mathfrak{su}(3)$ number of independent components of the matrix
$c_{\alpha\beta}$ is equal to 36.} $\mathfrak{su}(3)$~\cite{shifman}.
For the operator $H$ to be a Hamiltonian its
coefficient functions have to obey to additional
constraints. The detail discussion of such constraints can
be found in Refs. \cite{shifman,klish04}.
\section{Conclusion}
\label{conc}
In this paper the we demonstrated that the algebraic
approach to constructing quasi-exactly solvable systems,
formulated in Ref.~\cite{klish04} can be reformulated in
terms of constrained Hamiltonian systems. This underlines
nontrivial relationship between such systems. Besides we
have shown that in the framework of this algebraic approach
one can reproduce well-known two-dimensional quasi-exactly
solvable Hamiltonian corresponding to the Lie algebra
$\mathfrak{su}(3)$ in the Lie-algebraic approach.
In contrast to the construction of quasi-exactly solvable
differential operators, proposed in Refs.~\cite{annih,
shifman, ushv1, Zaslavskii}, considered in this paper
approach is pure algebraic and not related to specific
realization of Hilbert space, where operators live.
Therefore it is universal. For example, it can be applied to
constructing quasi-exactly solvable integral operators or
Hamiltonians on noncommutative spaces where the other
approaches do not work. We will consider construction of
such quasi-exactly solvable systems elsewhere.
Also we hope that the noted connection with constrained
Hamiltonian systems will be helpful for further development
of the theory of quasi-exactly solvable systems.
|
1,108,101,566,764 | arxiv | \section{Introduction}
The concept of {\em heaps of pieces} was introduced by G. Viennot in 1986 \cite{Viennot86} (see
also \cite{Krattenthaler} for a review). Informally, a heap of pieces is a collection of elements
which are piled together. If two elements intersect in their horizontal projections, then the
resulting heap depends on the order in which the two are placed. In this case, the element which is
placed second is said to be \emph{above} the element placed first. On the other hand, the resulting
heap does not depend on the order in which two elements are placed if their horizontal projections
do not intersect. A special case of heaps are heaps of \emph{dimers}. The dimers can be drawn as
unit squares (boxes) which are not allowed to touch each other with their vertical edges. This
means that if we place a box in the $k$th column and one in the $(k\pm1)$th column afterwards,
then the resulting heap is different to the one obtained from placing the boxes in the inverse
order. Figure \ref{an:f01}c shows an example of a heap of dimers.
Heaps of dimers are particularly interesting due to their relation to the model of {\em directed
animals} (DA). The term ``lattice animals'' is used as a collective name for several related models
describing the growth of aggregates, for example, molecular layers on substrates. Two-dimensional
directed animals are structures of occupied and unoccupied nodes on a lattice strip of width $n$
and infinite height. In this paper we only consider triangular lattices. The occupied sites on the
lowest row are called roots (or source points) and the DA has to satisfy the condition that each
occupied site can be reached from at least one root along a directed path containing only occupied
sites \emph{via diagonal or vertical edges} (for the triangular lattice). Figure \ref{an:f01}a shows an
example of a directed animal on such a triangular lattice and figures \ref{an:f01}b and
\ref{an:f01}c illustrate the bijection between directed animals and heaps of dimers, which works as
follows. Given a directed animal, we draw boxes around the occupied sites. This way, the DA shown
in Figure \ref{an:f01}a is redrawn as shown in Figure \ref{an:f01}b. If a box is not supported from
below, then we shift it downwards so that it is now supported. This way, we obtain Figure
\ref{an:f01}c. It can be easily seen that this mapping is in fact invertible. Namely, for a given
heap like in Figure \ref{an:f01}c, we shift upwards each box which sits on top of a box in the same
column and with it all the boxes which are above it. Then we redraw the boxes as black circles,
place white circles everywhere else and connect everything by a triangular lattice to obtain our
DA. The arrows in \fig{an:f01}c illustrate the so-called ``Mikado'' enumeration of the nodes which
will be explained below. Since we are only considering heaps of dimers in this paper, we will
shortly refer to them as heaps from now on. Also, due to the described bijection, we use the term
directed animals and heaps synonymously.
\begin{figure}[ht]
\epsfig{file=an_f01.eps, width=8cm}
\caption{(a): example of a directed animal (DA); (b)-(c): the corresponding heap of dimers.
The arrows in (c) illustrate the ``Mikado'' enumeration (see text).}
\label{an:f01}
\end{figure}
The typical problem for $N$-site DA concerns the computation of the number $\Omega(N,n|\{C\})$ of
all \emph{distinct} DA configurations in the bounding box of $n$ columns for a given configuration
of roots (base) $\{C\}$ (for example, the base of the DA in \fig{an:f01} is $\{C\}=\{3,5\}$). This
function has been computed exactly for the first time by Hakim and Nadal \cite{Hakim83} by using
algebraic methods dealing with the transfer matrix diagonalization for some spin system.
In this work, we review a different algebraic approach to directed animals which consists in
representing each DA on a lattice strip of width $n$ by an ordered word, spelled by the generators
of a locally free semi-group of $n$ generators, and show that there is a deep connection between
this group-theoretical approach and the \emph{asymmetric simple exclusion process} (ASEP) on an
open line.
The ASEP is a stochastic process on a chain of $N$~sites which can be either occupied by a
particle or empty. A particle hops to its right with rate 1 if the right neighbouring site is
empty. For a detailed introduction and review of important results for this process, we refer the
reader to \cite{Derrida98}. The ASEP can be considered both with periodic and open boundary
conditions. In this paper, we consider open boundaries, where particles enter the chain from the
left with rate $\alpha$ and exit the chain on the right with rate $\beta$ (see \fig{an:f02}). For
this case, the probability distribution of the stationary state has been derived in
\cite{Derrida93} by a matrix ansatz. Combinatorial interpretations of the steady state weights of
the configurations have already been given in terms of pairs of paths (for $\alpha=\beta=1$)
\cite{Shapiro82}, in terms of weighted permutation tableaux \cite{Corteel07}, and in terms of
weighted binary trees \cite{Viennot07}.
\begin{figure}[ht]
\epsfig{file=an_f02.eps, width=8cm}
\caption{the asymmetric simple exclusion process on a chain of $N=11$ sites with entering rate
$\alpha$ and escape rate $\beta$.}
\label{an:f02}
\end{figure}
In what follows we will give a new combinatorial interpretation of the stationary weights of the
ASEP on an open line in terms of directed animals. More precisely, we will demonstrate that the
partition function of the ASEP steady state on an $N$-site segment with entrance and exit rates
equal to one coincides with the partition function of $(N+1)$-site directed animals on a triangular
semi-infinite lattice strip with the topmost particle (the ``roof'') being located at the left
boundary. This correspondence can be extended towards arbitrary entrance and exit rates by defining
an appropriate weighting of the position of the leftmost root and a ``sticky" left boundary. It is
then possible to relate the features of the steady state distribution of the ASEP in the different
regimes of the phase diagram to the geometric features of the associated generalized directed
animals.
\section{Algebraic approach to directed animals}
The algebraic approach to directed animals consists in assigning to each DA--configuration an
equivalence class of words in some semi--group with special local commutation relations
corresponding to local particle configurations as shown in \fig{an:f02}. To be specific, define the
\emph{locally free semi-group}, $F_n^+$ with $n$ generators $g_1,...,g_n$, determined by the
relations
\begin{equation}
g_k g_m = g_m g_k, \qquad |k-m| \ge 2.
\label{eq:01}
\end{equation}
Each pair of neighbouring generators, $(g_k, g_{k\pm 1})$ produces a free sub-semigroup of $F_n^+$.
\begin{figure}[ht]
\epsfig{file=an_f03.eps,width=5cm}
\caption{Commutation relations in the group $F_n^+$ and local configurations of dimers.}
\label{an:f03}
\end{figure}
The statistical properties of locally free groups and semi-groups were investigated in detail in
\cite{Nechaev00}, where it has been shown that the partition function of an $N$--site heap in a
bounding box of size $n$ coincides with the partition function of an $N$-step \emph{Markov chain}
on $F_n^+$, or, equivalently, by the total number of equivalence classes of $N$-letter words in
$F_n^+$. Namely, to any configuration of DA one can bijectively associate an equivalence class of
words in $F_n^+$. Now each equivalence class contains exactly one word which is in \emph{normal
form}, which means that in this word, the generators with smaller indices are pushed as left as
possible in accordance with the commutation relations \eq{eq:01}. Consequently, the word
\begin{equation}
W=g_{s_1} g_{s_2}\ldots g_{s_N},
\label{eq:02}
\end{equation}
is in ordered form if and only if the indices
$s_1,...,s_N$ satisfy the following conditions, graphically represented in \fig{an:f04}.
\begin{itemize}
\item[(i)] If $s_i=1$ then $s_{i+1}\in\{1, 2,..., n\}$;
\item[(ii)] If $s_i=x$ ($2\le x\le n-1$)\\
then $s_{i+1}\in\{x-1, x, x+1,...,n\}$;
\item[(iii)] If $s_i=n$ then $s_{i+1}\in\{n-1, n\}$.
\end{itemize}
\begin{figure}[ht]
\epsfig{file=an_f04.eps, width=6cm}
\caption{The set of possible values which can be taken by the index $s_{i+1}$ if the index $s_i$ is
equal to: 1 (i), $2,...,n-1$ (ii), $n$ (iii).}
\label{an:f04}
\end{figure}
Thus, any $N$-site heap in a bounding box of $n$ columns can be uniquely represented by
an $N$-letter ordered word, ``spelled'' by the generators of $F_n^+$. For example, the normally
ordered word
\begin{equation}
W=g_3\; g_2\; g_1\; g_1\; g_2\; g_5\; g_4\; g_5\; g_4\; g_3\; g_6\; g_6
\label{eq:03}
\end{equation}
uniquely represents the 12-site directed animal shown in \fig{an:f01}a.
For a given heap, the corresponding normally ordered word can be obtained by an algorithm which
sets a constructive geometrical way of normal ordering. We call this enumeration procedure the
``Mikado ordering" since it resembles the famous Mikado game, the goal of which consists in the
sequential removal of the boxes from a random pile, one-by-one, without disturbing the other
elements. To proceed, define in a heap a set of top sites, each of which can be removed from the
heap without disturbing the rest of the pile. We call these elements the ``roof", ${\cal T}$, of
the heap. Remove the rightmost element of ${\cal T}$. In the updated roof, ${\cal T}'$, remove
again the rightmost element to get ${\cal T}''$, and so on, until the heap is empty. The sequence
of one-by-one removed elements is normally ordered and uniquely enumerates the heap (i.e. the
directed lattice animal). This fact is established in Lemma 3 of \cite{Nechaev00}. For the heap
shown in \fig{an:f01}c, the Mikado ordering is depicted by the sequence of arrows and coincides
with \eq{eq:03} (note that the topmost element in the 4th column does not belong to the roof as it
cannot be removed without disturbing the topmost element in the 3rd column, which is above it).
\section{Matrix ansatz for generalized DA and ASEP}
We now introduce the partition function $\Omega_{i,j}(N+1,n)$, which enumerates all the
$(N+1)$-particle heaps in the bounding box of $n$ columns whose Mikado ordering has its first
element in the $i$-th and its last element in the $j$-th column. This function can be expressed in
terms of a local $(n\times n)$ transfer matrix, $M$, with transitions described by the rules
(i)-(iii) (see also \fig{an:f04}), namely
\begin{equation}
\Omega_{i,j}(N+1,n) = \left< v_{i} | M^N | v_{j} \right>,
\label{eq:04}
\end{equation}
where $\left< v_{k} \right| =(\overbrace{0,...,0,1,0...0}^{\rm n})$ with a one in the $k$-th position,
and, as usual, $\left| v_{k} \right>=\left< v_{k} \right|^{\top}$. For reasons which will become clear in
the following, we are mostly interested in the values of $\Omega_{i,j=1}(N+1,n)$. It is instructive
to introduce the generating function
\begin{equation}
Z_{N+1}(n,\alpha)=\sum_{i=1}^n \Omega_{i,1}(N+1,n) \alpha^{1-i} = \left< v_{in} | M^N | v_{1} \right>,
\end{equation}
where $\left< v_{in} \right| =(\overbrace{1,\alpha^{-1},\alpha^{-2},\alpha^{-3}...}^{n})$. Now the
transfer matrix $M$ allows a natural decomposition in ``forward'' ($D$) and ``backward'' ($E$)
parts, associated with arbitrarily far jumps to the right and one-step jumps to the left (see
\fig{an:f04}). Namely, we can write $M=D+E$, where
\begin{equation}
D= \left(\begin{array}{ccccc}
1 & 1 & 1 & \ldots & 1 \\
0 & 1 & 1 & \ldots & 1 \\
0 & 0 & 1 & \ldots & 1 \\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & \cdots & 0 & 1
\end{array}\right); \quad
E= \left(\begin{array}{ccccc}
0 & 0 & 0 & \ldots & 0 \\
1 & 0 & 0 & \ldots & 0 \\
0 & 1 & 0 & \ldots & 0 \\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & \cdots & 1 & 0
\end{array}\right).
\label{eq:05}
\end{equation}
There is a striking similarity between this result and the celebrated exact solution of the
asymmetric simple exclusion process (ASEP) on a line \cite{Derrida93}. Let us briefly recall this
well-known result. The steady state of this process can, according to \cite{Derrida93}, be
calculated via the following procedure know as the ``matrix ansatz". Introduce two formal operators
$\widetilde{D}$ and $\widetilde{E}$ which satisfy
\begin{equation}
\widetilde{D} + \widetilde{E} = \widetilde{D}\widetilde{E}
\label{eq:ans1}
\end{equation}
and two vectors $\left< \widetilde{v}_{\rm in}\right|$ and $\left|\widetilde{v}_{\rm out}\right>$, such that
\begin{equation}
\left< \widetilde{v}_{\rm in} \right|\widetilde{E} = \alpha^{-1} \left< \widetilde{v}_{\rm in}\right|~;~
\widetilde{D}\left|\widetilde{v}_{\rm out}\right> =
\beta^{-1} \left|\widetilde{v}_{\rm out}\right>.
\label{eq:ans2}
\end{equation}
Now one can show that the probability of observing any given ASEP configuration in the steady state
is proportional to a matrix element of the type $\left< \widetilde{v}_{\rm in}|...|\widetilde{v}_{\rm
out}\right>$, where for the dots one should insert a sequence of the operators $\widetilde{D}$ and
$\widetilde{E}$, with $\widetilde{D}$ and $\widetilde{E}$ corresponding to occupied and empty
sites, respectively. For example, the probability of the configuration shown in \fig{an:f02} is
proportional to $\left< \widetilde{v}_{\rm
in}\right|\widetilde{E}\widetilde{D}\widetilde{E}\widetilde{D}
\widetilde{D}\widetilde{D}\widetilde{E}
\widetilde{D}\widetilde{E}\widetilde{E}\widetilde{D}\left|\widetilde{v}_{\rm out}\right>$. The sum
$\widetilde{Z}_N(\alpha,\beta)$ of matrix elements over all possible configurations, which plays a
role very similar to the partition function of the steady-state ASEP, can be written as
\begin{multline}
\widetilde{Z}_N(\alpha,\beta) = \left< {\widetilde{v}}_{\rm
in}|(\widetilde{E}+\widetilde{D})^N|{\widetilde{v}}_{\rm out}\right> = \left< {\widetilde{v}}_{\rm
in}|(\widetilde{M})^N|{\widetilde{v}}_{\rm out}\right>.
\label{eq:07}
\end{multline}
For arbitrary $\alpha$ and $\beta$, the algebra defined by \eq{eq:ans1} and \eq{eq:ans2} has no
finite-dimensional representations. However, there exist many infinite-dimensional representations,
among which the most interesting for us is constructed as follows. Set $\left< \widetilde{v}_{\rm
in}\right|=(1,\alpha^{-1},\alpha^{-2}, \alpha^{-3},\dots)$, $\left< \widetilde{v}_{\rm out}
\right|=(1,0,0,\dots)$ and choose the matrices $\widetilde{D}$ and $\widetilde{E}$ as
\begin{equation}
\widetilde{D}= \left(\begin{array}{cccc}
\frac{1}{\beta} & \frac{1}{\beta} & \frac{1}{\beta} & ... \\
0 & 1 & 1 & \ldots \\
0 & 0 & 1 & \ldots \\
\vdots & \vdots & & \ddots
\end{array}\right); \quad
\widetilde{E}= \left(\begin{array}{cccc}
0 & 0 & 0 & \ldots \\
1 & 0 & 0 & \ldots \\
0 & 1 & 0 & \ldots \\
\vdots & \vdots & & \ddots
\end{array}\right).
\label{eq:08}
\end{equation}
It is easy to check that conditions \eq{eq:ans1} and \eq{eq:ans2} are satisfied. Furthermore, the
similarity between \eq{eq:05} and \eq{eq:08} is strikingly clear. Indeed, we immediately get
\begin{equation}
\widetilde{Z}_N(\alpha,\beta=1)= \lim_{n\to \infty} Z_{N+1}(n,\alpha).
\label{eq:08a}
\end{equation}
This explicitly demonstrates that in the limit $n\to\infty$, the generating function of
$N+1$-particle heaps with a topmost particle in the first column and activity $\alpha$ associated
to the position of the first particle (which is the leftmost particle in the lowest row), coincides
with the ``partition function'' of the stationary ASEP chain of $N$ sites in the case $\beta=1$.
The correspondence between DA and the ASEP for arbitrary values of $\alpha$ \emph{and} $\beta$ is
established as follows. Each ASEP configuration corresponds to a {\emph set} of \emph{pyramids}
\cite{Viennot86}, i.e. heaps with a roof consisting of a single element in column 1. Now we can
associate the sequence of ``backward'' and ``forward'' jumps with the ASEP configuration such that
backwards jumps correspond to a hole in the ASEP sequence, and forward jumps to a particle -- see
\fig{an:f04} (here ``no jump" is considered as a ``forward jump" with length zero). Since there can
be forward jumps of different lengths, there are, generally speaking, many different DAs
corresponding to a single ASEP configuration. This fact is depicted in \fig{an:f05}, where we show
two heaps of dimers corresponding to the same ASEP configuration.
Now the weight of a given ASEP configuration in the steady state on a line is proportional to the
sum of the weights of \emph{all} corresponding heaps, where the weight of a given heap equals
$\alpha^{1-x} \beta^{1-y}$ with $x$ being the coordinate of the column in which the leftmost root
is located, and $y$ being the number of elements of the heap in column 1. Using the Mikado
enumeration, one can rephrase this statement as follows: (a) the first letter in the normally
ordered word associated to a specific DA has weight $\alpha^{1-x}$, (b) each generator $g_1$
(except for the last letter) carries the weight $\beta^{-1}$, while all the other generators have
weight $1$, (c) the last letter in the normally ordered word is always $g_1$, (d) to get a weight
of an ASEP configuration one has to sum over all corresponding DAs in which any pair $g_i g_k$ with
$k\ge i$ ($(k,i)\in \{1,\dots,n\}^2$) corresponds to a particle, while a pair $g_i g_{i-1}$ ($i\in
\{2,\dots,n\}$) corresponds to a hole. In Table \ref{tab:01}, we summarize the correspondence
between the stationary ASEP and DA. We note several important facts about this correspondence.
\begin{figure}[ht]
\epsfig{file=an_f05.eps, width=7cm}
\caption{Two generalized directed animals corresponding to the same ASEP configuration (below) and
the Markov chains representing the associated ordered words. The horizontal coordinate stands for
the index of a letter in the word. All jumps starting from position $x=1$ carry the weight
$\beta^{-1}$ and a first letter $g_x$ contributes the weight $\alpha^{-(x-1)}$. The last letter is
always $g_1$.}
\label{an:f05}
\end{figure}
\begin{table}[ht]
\epsfig{file=an_f06.eps, width=8cm}
\caption{Correspondence between the asymmetric simple exclusion process and directed animals on
strip of width $n$ in the limit $n\to\infty$.}
\label{tab:01}
\end{table}
First, there is no straightforward analog of time in the heap picture, and there is no time
evolution imposed on the DAs. From the mapping revealed above one immediately gets the\,\emph{
steady state} probabilities of the ASEP configurations, but not the underlying time evolution.
Second, formally, the mapping is exact only for $n\to\infty$ (i.e., for directed animals on a
quarter-plane with no boundary on the right. However, the condition that the last particle should
be put in column 1 dictates that all $N$-step trajectories stay in columns $x\leq N$, so $n
\geq N$ is enough to make the mapping exact.
Third, the original ASEP problem has a well-known particle-hole symmetry, i.e. if one replaces the
particles with holes and vice versa, reverses the direction of the flow and interchanges $\alpha
\leftrightarrow \beta$, one returns to the original problem. This symmetry is evident in the formal
algebraic matrix ansatz, but broken down by the representation \eq{eq:08}, which makes it a
bit artificial in the original ASEP model. The interpretation in terms of Mikado-ordered DAs gives,
however, a natural, intuitive meaning to the representation \eq{eq:08}. The connection between the
DA and the ASEP problems also shows that there is actually a hidden symmetry in the DA model,
namely a symmetry between the \emph{position} of the leftmost root and the \emph{number of visits}
of the column $x=1$. In other words, the partition function of $N$-particle DAs with a single roof
particle in the first column, $s$ particles with weight $\beta^{-1}$ in the first column and
leftmost root in column $k$ with weight $\alpha^{1-k}$, coincides with the partition function of
$N$-particle DA with a single roof particle in the first column, $k$ particles in the first column
with weight $\alpha$ in total and leftmost root in column $s$ with weight $\beta^{1-s}$.
Now we can better understand the statistics of standard, non-weighted directed animals
($\alpha=\beta=1$). The point $(\alpha,\beta)=(1,1)$ lies deeply in the maximum-current phase of
the ASEP (\cite{Derrida98}, see also the next section). This means that the corresponding ASEP
steady state is dominated by configurations where the particle density is equal to $1/2$. In terms
of heaps, this means that for $N$ large enough, there are typically equal numbers of left and right
jumps in the Mikado ordering (compare with the numeric results shown in the next section). This
might seem counter-intuitive, as rightward jumps can have an arbitrarily big length while leftward
jumps always have length $1$. Note, though, that ``rightward" jumps can also have length $0$, so
the average length of a rightward jump can well turn out to be $1$.
Small changes in $\alpha$ or $\beta$ do not move the ASEP out of the maximum-current phase, and
thus the concentrations of forward and backward jumps stay equal also for $\alpha$ and $\beta$
slightly differing from $1$. One has to go as far as $\alpha=1/2$ or $\beta=1/2$ to see a big
change in the behavior of the heaps. We suggest the reader to compare this result to the
adsorption-desorption transition of a random walk (polymer chain) on a half-line with a potential
well at $x=0$, where the change in typical trajectories also occurs for the potential well depth
$\beta^{-1}=2$ \cite{Naidenov01}.
\section{Simulation of generalized heaps}
For different values of $\alpha$ and $\beta$, we have numerically generated corresponding
generalized heaps of $N=150$ particles. The simulation was carried out by generating random
$N=150$-step generalized Lukasiewicz paths with fixed endpoint at $x=1$ \cite{Lehner03} and weighting
of the steps and initial position according to \fig{an:f05}. The algorithm we used is described in
\cite{Kamenetskii81}. \fig{an:f07} shows the resulting pictures.
In the high density phase $\beta < \alpha < \frac 12$, the typical heaps are roughly vertical piles in
the first column with only a few boxes sticking out into the second column. The number of boxes
which are supported from below right, corresponding to a hole in the associated ASEP configuration,
is very small, the particle density of the corresponding ASEP configuration is close to 1 in this
case -- see \fig{an:f07}a.
In the low density phase $\alpha < \beta < \frac 12$, the typical heap roughly follows a diagonal
line, going from below right to the top left. This corresponds to a very low particle density of
the corresponding ASEP configuration -- see \fig{an:f07}b.\\
For both $\alpha$ and $\beta$ greater than $\frac 12$, one obtains less regular pictures with, on
average, as many boxes which are supported from below left or sit on top of another box as boxes
supported from below right. This means that the corresponding ASEP configuration has a particle
density close to $\frac 12$. The maximal current case $\alpha=\beta=1$ is depicted in \fig{an:f07}c.\\
At the first-order transition line $\alpha = \beta < \frac 12$, one observes heaps which roughly
consist of a diagonal line below, followed by a straight vertical pile in the first columnn. This
means that the corresponding ASEP configuration is divided into a region with very low density on
the left and a region with very high density on the right. However, the size of the two regimes
varies. The point in the ASEP chain at which the two density regimes meet each other is identified
as a \emph{shock} -- see \fig{an:f07}d.
\begin{figure}[ht]
\epsfig{file=an_f07.eps,width=8.5cm}
\caption{Numerically generated random generalized heaps of $N$=150 dimers for different values of
$\alpha$ and $\beta$. (a): $\alpha = 0.3$, $\beta = 0.1$, (b): $\alpha = 0.1$, $\beta = 0.3$, (c):
$\alpha = \beta = 1$, (d): $\alpha = \beta = 0.1$. The inserts in each picture show the position in
the phase diagram of the ASEP, the black lines mark the phase boundaries.}
\label{an:f07}
\end{figure}
\section{Stationary ASEP as polymer wetting}
Let us sketch the derivation of the stationary ASEP partition function (\ref{eq:07}). Although the
answer is well known since the pioneering works \cite{Derrida93} and has been derived with
different nuances in some subsequent works (see, for example, \cite{Blythe07,Depken04}), we would like
to emphasize the deep analogy of the ASEP generating function with the generating function of the
\emph{wetting problem} on a one-dimensional adsorbing substrate \cite{Naidenov01,Gangardt07}. In a
general setting, wetting implies the interface pinning by an impenetrable solid. Problems of
interface statistics in the presence of a hard wall were addressed in many publications (see, for
example, \cite{Abraham80,Abraham86} and references therein). The most interesting question concerns the
nature of the wetting or pinning-depinning transition of the interface controlled by parameters of
its interactions with the substrate. To the best of our knowledge, the similarity of the analytic
structures of the generating functions for asymmetric exclusion and wetting has been briefly touched
only in the review \cite{Blythe07}. The connection between the ASEP and pinned interface statistics allows
us, as we have seen in the previous section, to get a simple and transparent view on the nature
of shocks. Conversely, this connection raises open questions whether the fluctuations of the
interface density in vicinity of the pinning-depinning transition could exhibit the KPZ scaling
seen near the ASEP shock profiles as pointed out in \cite{Janowsky92}.
Define $Z_N(x,\alpha,\beta)$, the partition function of the $N$-step trajectories on the
semi-infinite discrete line ($n\to\infty$) with allowed steps and weighting as shown in
\fig{an:f04}, and with final position in $x$. For shortness we write $Z_N(x,\alpha,\beta) \equiv Z_N(x)$. This function can be expressed in terms of a matrix product as
\begin{equation}
Z_N(x) = \left< v_{\rm in} \right| \widetilde{M}^{N-1} \left| v_x \right>.
\label{matrixproduct}
\end{equation}
The quantity of our interest is $Z_N(x=1)$. From \eqref{matrixproduct}, we obtain the
following recursion relation, valid for any $N\ge 0$ (compare to
\cite{Derrida93}):
\begin{equation}
\left\{\begin{array}{rlll}
Z_{N+1}(x) & = & \disp \beta^{-1}Z_N(1)+\sum_{y=2}^{x+1} Z_N(y) & x = 1,2,..., \\
Z_{N=0}(x) & = & \disp \alpha^{x-1} & x = 1,2,..., \medskip \\
Z_N(x) & = & 0 & x=0.
\end{array}
\label{eq:09}
\right.
\end{equation}
Introduce the generating function
$$
W(s,x) = \sum_{N=0}^\infty Z_N(x) s^N;\; Z_N(x)=\frac{1}{2\pi i}\oint\limits_{C}
\frac{W(s,x)}{s^{N+1}} ds
$$
with a suitably chosen closed contour $C$ around the origin. In what follows we denote $W(s,x)
\equiv W(x)$ for shortness.
Defining now $Q(x)=s^{x/2}W(x)$ and using the Kronecker $\delta$--symbol, where $\delta_{x,1}=1$
for $x=1$, and 0 otherwise, we can rewrite \eq{eq:09} as a single equation in a symmetrized form,
which has straightforward interpretation in terms of the wetting generating function \cite{Gangardt07}
on a semi-infinite line $x\ge 0$. We get
\begin{multline}
\Big\{Q(x)-\sqrt{s}\big(Q(x-1)+Q(x+1)\big) \medskip \\
-s^{x/2}\alpha^{1-x}(1-\alpha)\Big\}(1-\delta_{x,1}) \medskip \\
+\left\{Q(x)\frac{\beta-s}{\beta}-\sqrt{s}Q(x+1)-s^{x/2}\right\}\delta_{x,1} = 0.
\label{eq:13}
\end{multline}
Applying the Fourier transform
$$
\mathcal{Q}(q) = \sum_{x=0}^\infty Q(x) \sin q x; \quad Q(x) =
\frac{2}{\pi}\int_{0}^\pi \mathcal{Q}(q) \sin q x\, dq
$$
to equation \eq{eq:13}, we obtain
\begin{multline}
(1-2\sqrt{s}\cos q)\mathcal{Q}(q)-\frac{s}{\beta}\sin q\, Q(1)- \medskip \\
-(\alpha-\alpha^2) \sum_{x=2}^\infty\left(\frac{\sqrt{s}}{\alpha}\right)^x \sin q x\,-\sqrt{s}\sin q=0.
\label{eq:18}
\end{multline}
The solution for $\mathcal{Q}(q)$ reads
\begin{equation}
\mathcal{Q}(q) = \frac{\frac{s}{\beta}\sin q\, Q(1)+f(q)}{1-2\sqrt{s}\cos q}.
\label{eq:19}
\end{equation}
where we have defined $f(q)$ as
\begin{equation}
\begin{array}{lcl}
f(q)&=&\disp(\alpha-\alpha^2)\sum_{x=2}^\infty \left(\frac{\sqrt{s}}{\alpha}\right)^x
\sin q x+\sqrt{s}\,\sin q,\medskip\\
\disp &=&\disp\frac{\alpha^2(1-2\sqrt{s}\cos q)+s\alpha} {\alpha^2-2\sqrt{s}\alpha\cos
q\,+s}\sqrt{s}\sin q,\medskip
\end{array}
\label{eq:14a}
\end{equation}
Remembering that $W(1)=s^{-1/2}Q(1)$, inserting the expression \eq{eq:14a} for $f(q)$ into
\eq{eq:19} and applying the inverse Fourier transform, we end up with
\begin{equation}
\begin{array}{lcl}
W(s,1) & = & \frac{\disp \frac{\disp 2}{\pi\sqrt{s}} \int_{0}^\pi \frac{f(q) \sin q}{1-2\sqrt{s}\cos
q}dq}{\disp 1-\frac{2s}{\pi\beta}\int_{0}^\pi \frac{\sin^2 q}{1-2\sqrt{s}\cos q}dq}\medskip\\
& = & \disp \frac{4\alpha\beta}{\left(2\alpha-1 + \sqrt{1-4s}\right)\left(2\beta-1 +
\sqrt{1-4s}\right)}
\end{array}
\label{eq:20}
\end{equation}
as an explicit expression for the generating function of the stationary ASEP partition function,
$Z_{N}(x=1,\alpha,\beta)$. Note that the roots in the denominator are positive for $s < \frac 14$. For
$\alpha$ and $\beta<\frac 12$, the generating function \eq{eq:20} has two pole singularities at
\begin{equation}
s_1=\alpha(1-\alpha),\quad s_2=\beta(1-\beta),
\label{eq:21}
\end{equation}
which are both smaller or equal to $\frac 14$. For $\alpha$ and $\beta > \frac 12$, these poles leave
the real axis and the branching point $s_3 = \frac 14$ becomes the dominant singularity. Depending on
which singularity is dominant, one recovers the known phase diagram of the ASEP, with the three
borders $\alpha = \beta < \frac 12$, $\alpha = \frac 12$ and $\beta = \frac 12$ \cite{Derrida98}.
As one sees, both generating functions, of the ASEP and of the wetting problem, have similar
analytic structures; they diverge at branching points, which signals the existence of a phase
transition. However, the behavior of the function $W(s,1)$ is far more rich: it has two possible
singularities $s_1$ and $s_2$ controlled by \textit{two} independent parameters, $\alpha$ and
$\beta$. Thus, in the thermodynamic limit $N\to\infty$, the ASEP ``free energy'', $f(\alpha,\beta)$
strongly depends on the parameters $\alpha$ and $\beta$ and is determined by the singularity which
is closest to zero :
\begin{equation}
f(\alpha,\beta)=-\ln \min\{s_1(\alpha), s_2(\beta),s_3\}.
\label{eq:23}
\end{equation}
\section{Summary}
In this letter we have established the connection between generalized directed animals on a
semi-infinite strip with adsorbing boundary and special initial particle distribution with the
stationary state configurations of the asymmetric simple exclusion process. Given the relation
between directed animals and the ASEP, we analysed how the features of one model translate into
features of the other one. We simulated generalized directed animals (heaps respectively) in the
different regimes of the ASEP phase diagram and discussed the shape of the typical pictures
obtained. In particular, we were able to observe shock configurations at the first order transition
line between the low and the high density phase of the ASEP. We also noted a hidden symmetry of the
directed animals model by making use of the known particle-hole symmetry in the ASEP.\\
\indent The random walk picture of directed animals which resulted from the normal order
representation of directed animal configurations (associated with the locally free group), allowed
us to regard the stationary ASEP as a sort of wetting model on a one-dimensional adsorbing
substrate. Using the evolution equation for this random walk, we provided a simple derivation of
the ASEP generating function on a one-dimensional line.
The authors are grateful to A. Vershik for numerous discussion of the problem. N.H. and M.T. would
like to thank the LPTMS for the warm hospitality. This work was partially supported by the grants
ANR-2011-BS04-013-01 WALKMAT, FP7-PEOPLE-2010-IRSES 269139 DCP-PhysBio, as well as by a MIT-France
Seed fund and the Higher School of Economics program for Basic Research.
|
1,108,101,566,765 | arxiv | \section{Introduction}
\def*}\footnotetext{These authors contributed equally to this work.{*}\footnotetext{These authors contributed equally to this work.}
Cooperative multi-agent reinforcement learning (MARL) \citep{busoniu2008comprehensive} has recently led to promising results in many real-world applications, such as robot control \citep{gupta2017cooperative} and autonomous driving \citep{shalev2016safe}. For example, in the path-finding task, MARL achieves similar performance as classic operation research algorithms but with much lower computational complexity \citep{sartoretti2019primal}. In the domain of games, well-trained agents have reached the master-level performance \citep{smac} and even won the game against professional players.
In these applications, centralized training decentralized execution (CTDE) is a widely adopted paradigm due to its scalability potential and ability to deal with non-stationarity.
Classic CTDE architectures usually employ a centralized value network that leverages global information to guide agents' local policy training \citep{MADDPG}. During execution, each agent utilizes its local information to make decisions without centralized coordination. In practice, however, the partial observation and stochastic nature of MARL environments make it difficult for agents to accurately predict others' actions in such communication-free CTDE schemes, and thus miscoordination often happens.
\begin{figure}[t]
\centering
\includegraphics[scale=0.33]{Fig1_Xinran_0811.pdf}
\caption{A didactic example in traffic junction environment. Consider agent $1$'s decision-making. In case 1, with only short-range communication, agent 1 only receives messages from agent 2, then it will enter the intersection and may collide with agent 3. In case 2, with both short-range and long-range communication, agent 1 is able to know agent $3$'s information in advance, therefore, it can make a better decision, i.e., waiting before entering the intersection.}
\label{Fig_example}
\end{figure}
To address this limitation, many recent studies on MARL enable communication to exchange information among agents in CTDE \citep{MADDPG, corr_commu}. These methods achieve superior performance compared with communication-free CTDE systems. Nevertheless, they often assume simplified communication models with a fully connected topology \citep{gcs}, i.e., each agent is able to receive a message from any other agent via point-to-point communication.
In real systems, an agent's communication range is limited, e.g., when communicating over a wireless channel, and thus the communication topology among agents should be partially connected. Moreover, this topology will dynamically change as agents move.
On the other hand, long-range information is desirable in MARL systems. As an example, consider a traffic junction task illustrated in Figure \ref{Fig_example}, where three cars need to pass through a traffic junction following predefined routes. In this case, obtaining long-range information helps agents plan ahead of time and avoid myopic decisions. Considering the realistic communication range constraint and the value of obtaining long-range information, an effective multi-hop communication mechanism is needed, which motivates our work.
In this paper, we consider a cooperative MARL system, where agents are assumed to have limited communication ranges. To facilitate effective communication, we propose a novel communication protocol called \textit{Adaptively Controlled Two-Hop Communication} (AC2C). Enabled by an attention-based communication module and a multi-layer perceptron (MLP)-based controller, agents learn to adaptively engage in two-hop communication to balance the cooperative task performance and the communication overhead.
The main contributions of this paper are summarized as follows:
\begin{itemize}
\item We consider a realistic MARL system with communication range constraints and propose a novel two-hop communication protocol, i.e., AC2C, to enable long-range information exchange.
\item Inspired by the gating mechanism \citep{mao2019learning}, we introduce an adaptive controller. This local controller adaptively determines whether the ego agent should ask for a subsequent communication round to obtain two-hop messages. In this way, expensive two-hop communication is only established whenever necessary.
\item We conduct experiments on three benchmark tasks, namely traffic junction, cooperative navigation, and predator prey. The superior performance compared with baselines demonstrates the effectiveness of our proposed method.
We further analyze the communication costs in our experiments and show that AC2C achieves a good trade-off between communication cost and cooperative task performance.
\end{itemize}
\section{Related Work}
Recent studies \citep{MADDPG, QMIX, COMA} have made remarkable progress in MARL under the CTDE paradigm. Compared with its counterparts, i.e., independent learning \citep{IQL} and centralized training centralized execution (CTCE) \citep{CTCE}, CTDE has demonstrated significant performance and scalability potential. Nevertheless, communication-free CTDE exacerbates the partially observable issue and hinders effective cooperation, leading to sub-optimal decisions.
Earlier works such as CommNet \citep{CommNet}, DIAL \citep{DIAL}, and BiCNet \citep{BiCNet} attempt to support communication in CTDE with a predefined topology. However, the performance of those methods often falls short in complex settings since relationships among agents constantly change in a dynamic multi-agent system, and a rigid communication graph cannot respond to such dynamics.
Subsequent works exploit state-dependent communication graphs to address the above shortcomings. In particular, ATOC \citep{atoc}, IC3Net \citep{ic3} and I2C \citep{i2c} introduce individual gating mechanisms to control the communication links among agents. The gating mechanisms are implemented with a classifier that determines whether to transmit messages based on local histories. Besides, VBC \citep{VBC} proposes a communication control unit depending on the local action confidence. The dynamically pruned communication graphs produced by these methods result in low communication costs. Nevertheless, the aforementioned works only consider single-round communication. In practical settings with limited communication ranges, agents cannot access information outside this range through single-round communication, which can severely limit the performance.
There have been some recent studies adopting multi-round communication to obtain more information from other agents. For instance, TarMAC \citep{Tarmac} utilizes multi-layer attention blocks to implement multi-round communication. But it requires a fully-connected communication topology, therefore causing prohibitive communication overhead. Furthermore, graph neural networks (GNNs) have recently been incorporated with MARL, owing to their power to enforce structural communication among agents. In particular,
DICG \citep{dicg} and DGN \citep{DGN} utilize graph convolutional networks (GCNs) to enable message passing among agents, while MAGIC \citep{magic} uses graph attention networks (GATs) to aggregate messages.
None of the above works dynamically prune the communication links (i.e. the edges in communication graphs), therefore they can be infeasible in realistic systems since densely connected communication graphs induce heavy communication overhead.
In this paper, we explicitly consider the effects of limited communication ranges on MARL systems. Our proposed method leverages multi-hop communication to enlarge the agents' reception fields.
To reduce the communication cost, a decentralized controller is designed to determine whether an agent should request a subsequent communication round based on the obtained single-hop messages. In this manner, agents acquire the ability to obtain information outside their communication ranges and dynamically prune two-hop communication, thus achieving a good performance-communication trade-off.
\section{System Model}
\subsection{Problem Formation}
We formalize the problem as a decentralized partially observable Markov decision process (Dec-POMDP) \citep{pomdp}. It is modeled by a tuple $\mathcal{M} = \langle \mathcal{S}, A, P, R, \Omega, O, N, \gamma \rangle$, where $N$ is the number of agents, and $\gamma \in [0, 1)$ is the discount factor. At each timestep, the
environment state is $s \in \mathcal{S}$. Each agent $i$ receives a local observation $ o_i \in \Omega$ drawn from the observation function $O(s, i)$. Then, it selects an action $a_i \in A$, forming a joint action $\boldsymbol{a} \in A^N$, which leads to a next state $s'$ according to the transition function $P(s'| s, a)$. The agents collaboratively gain a global reward according to the reward function $r = R(s, \boldsymbol{a})$. Each agent keeps a local action-observation history at the current timestep $\tau_i \in (\Omega \times A)$. The primary notations and descriptions are listed in Table \ref{table:notations}.
\subsection{Communication Protocol}
We consider a multi-agent system where agents are with limited communication ranges. For ease of illustration, we mainly consider distance-based communication constraints, i.e., an agent can only establish direct communication links with the ones within a range $L$.
Formally, we call the neighboring agents that are located within distance $L$ to agent $i$ its one-hop neighbors, denoted by $\mathcal{N}_i^{(1)}$. And agents that are within the distance $L$ to any agent in the set $\mathcal{N}_i^{(1)}$ are denoted as $\mathcal{\widetilde{N}}_i^{(2)}$. We then define the set of agents belonging to $\mathcal{\widetilde{N}}_i^{(2)}$, excluding agent $i$'s one-hop neighbors and itself, as agent $i$'s two-hop neighbors, denoted as $\mathcal{N}_i^{(2)}= (\mathcal{\widetilde{N}}_i^{(2)} \backslash \mathcal{N}_i^{(1)}) \backslash \{i\}$.
For the example in Figure \ref{Fig_comm}, agents $2$ and $3$ are one-hop neighbors of agent $1$, while agents $4$ and $5$ are its two-hop neighbors. In the following, we first describe the GNN-based communication protocol adopted by existing studies and then introduce the proposed AC2C communication protocol.
\subsubsection{GNN-based Communication Protocol}
GNN-based methods have recently been widely used for multi-agent communication \citep{dicg, DGN}. Before communication, agent $i$ holds a local feature embedding $c_i^{(0)}$. In the first communication round, it receives messages from its one-hop neighbors in $\mathcal{N}_i^{(1)}$ and updates it local embedding as $c_i^{(1)}$. In the second communication round, it communicates with nodes in $\mathcal{N}_i^{(1)}$ again, receiving their updated embeddings after the first communication round. In this way, each agent can obtain partial information from its two-hop neighbors, but in an indirect and inefficient sense. For example, in Figure \ref{Fig_comm}, agent 1 can obtain information from agents $4$ and $5$ after two communication rounds, but information on agents $6$ and $7$ is still unavailable.
\subsubsection{AC2C Communication Protocol}
The proposed AC2C communication protocol adopts a two-hop communication mechanism to obtain information from two-hop neighbors $\mathcal{N}_i^{(2)}$, which facilitates effective communication among agents. During the first communication round, the communication process is the same as the GNN-based protocol, where each agent exchanges its local feature embedding $c^{(0)}_i$ with its one-hop neighbors.
In the second communication round, however, agents exchange messages with their two-hop neighbors, while their one-hop neighbors only act as relay nodes. As illustrated in Figure \ref{Fig_comm}, the AC2C protocol can help agent 1 exploit information of agents $6$ and $7$ since agent $5$'s embedding (which contains partial information of agents 6 and 7) is transmitted to agent $1$ directly in the second communication round. Thus, AC2C can effectively enlarge agents' receptive fields compared to GNN-based communication protocol, leading to better performance. This performance gain comes with higher communication overhead, as it involves two-hop communication. Therefore, it is critical to adaptively control and reduce the frequency of evoking the expensive two-hop communication.
\begin{figure}[t]
\centering
\includegraphics[width=6cm]{communication_system_v6_Jiawei.pdf}
\caption{The two-hop communication model, where the blue and red nodes correspond to the one-hop and two-hop neighbors of node 1, respectively.
}
\label{Fig_comm}
\end{figure}
\section{Proposed Method}
In this work, we follow the conventional CTDE paradigm and augment it with the proposed AC2C communication protocol. In the proposed framework, each agent's local network consists of a GRU-based feature encoder, an AC2C communication module, and an MLP-based action policy network, as is shown in Figure \ref{AC2C_franework}.
At each timestep, the feature encoder first takes the local observation $o_i$ as input to update its historical representation $h_i$ and outputs an initial local embedding $c_i^{(0)}$. Then, the agent leverages the local embedding and conducts a two-round communication process with the AC2C modules to exchange information with other agents, obtaining the updated local embedding $c_i^{(1)}$ and $c_i^{(2)}$ after each communication round. After that, the agents feeds $c_i^{(0)}$, $c_i^{(1)}$ and $c_i^{(2)}$ to the action policy network to generate the local action $a_i$.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.48]{Fig3_Xinran_0219.drawio.pdf}
\end{center}
\caption{
The architecture of the proposed AC2C network. At each timestep, agent $i$ receives a local observation $o_i$ and utilizes its historical information $h_i$ to generate an initial embedding $c_i^{(0)}$ and update its local historical information as ${h_i’}$. In the first communication round, agent $i$ receives a message from its one-hop neighbor $j$ and outputs the updated embedding $c_i^{(1)}$.
Additionally, the local controller takes $c_i^{(0)}$ and $c_i^{(1)}$ as inputs and generates a binary signal $z_i$. If $z_i=0$, agent $i$ will not request the second communication round; if $z_i=1$, agent $i$ will request the second communication round. Upon receiving information from its two-hop neighbors $k$ and $g$ in the second communication round, agent $i$ aggregates messages again and produces $c_i^{(2)}$. After two communication rounds, agent $i$ generates an action based on $c_i^{(0)}$, $c_i^{(1)}$ and $c_i^{(2)}$ (if applicable).}
\label{AC2C_franework}
\end{figure}
\begin{table}[t]
\centering
\caption{Primary notations and descriptions.}
{
\begin{tabular}{c|c}
\hline
Notations & Description \\
\hline
$o_i$ & Agent $i$'s local observation \\
\hline
$h_i$ & Agent $i$'s historical representation \\
\hline
${c}_{i}^{(0)}$ & Agent $i$'s initial local embedding \\
\hline
${c}_{i}^{(1)}$ & \makecell[c]{Agent $i$'s local embedding \\after the first communication round}\\
\hline
${c}_{i}^{(2)}$ & \makecell[c]{Agent $i$'s local embedding \\after the second communication round} \\
\hline
$a_i$ & Agent $i$'s local action \\
\hline
$\mathcal{N}_i^{(1)}$ & The set of agent $i$'s one-hop neighbors\\
\hline
$\mathcal{\widetilde{N}}_i^{(2)}$ & The set of agent $i$'s two-hop neighbors\\
\hline
$\mathcal{N}_i^{(2)}$ & \makecell[c]{The set of agent $i$'s two-hop neighbors, \\ excluding its one-hop neighbors and itself}\\
\hline
$T$ & The controller threshold\\
\hline
${z}_{i}$ & Signal produced by agent $i$'s controller \\
\hline
\end{tabular}
}
\label{table:notations}
\end{table}
\subsection{Two-Hop Communication}
\subsubsection{Communication Protocol}
To enable effective communication and facilitate coordination among agents, AC2C implements a two-round communication mechanism.
In the first communication round, agent $i$ broadcasts its local embedding $c_i^{(0)}$ to its one-hop neighbors $\mathcal{N}_i^{(1)}$, and it also receives the embeddings from them. Upon receiving messages from $\mathcal{N}_i^{(1)}$ in the first communication round, agent $i$ aggregates the messages as well as its local embedding with an attention-based aggregation module and obtains the updated embedding $c_i^{(1)}$:
\begin{equation}
c_i^{(1)} = f^{(1)} \left(c_i^{(0)}, c_j^{(0)} \left|\right. j \in \mathcal{N}_i^{(1)} \right),
\end{equation}
where $f^{(1)}(\cdot,\cdot)$ denotes the first round aggregation function to be introduced in Section \ref{msg_agg}.
Then, agent $i$'s controller leverages $c_i^{(0)}$ and $c_i^{(1)}$ to locally determine whether a second communication round is needed for agent $i$ by outputting a binary signal $z_i$. We will defer the implementation details of the controller to Section \ref{controller}.
Upon deciding that the second communication round is needed, agent $i$ will inform its one-hop neighbors to initiate the second communication round. In this round, agent $i$ receives messages from its two-hop neighbors $\mathcal{N}_i^{(2)}$, with its one-hop neighbors only acting as relaying nodes. After receiving messages, agent $i$ again aggregates the messages and its local embedding $c_i^{(1)}$ by the aggregation module and obtains the updated embedding $c_i^{(2)}$:
\begin{equation}
c_i^{(2)} = f^{(2)} \left(c_i^{(1)}, c_j^{(1)} \left|\right. j \in \mathcal{N}_i^{(2)} \right),
\end{equation}
where $f^{(2)}(\cdot,\cdot)$ denotes the second round aggregation function to be described in Section \ref{msg_agg}.
After two communication rounds, each agent possesses embeddings $c_i^{(0)}$, $c_i^{(1)}$ and $c_i^{(2)}$ (if the second round communication was executed), which will all be concatenated together and fed into the action policy for decision making.
\subsubsection{Message Aggregation Strategy} \label{msg_agg}
For the message aggregation strategy, we implement an attention module for each communication round.
In the $n$-th communication round, we first calculate the key $k_i^{(n)}\in\mathbb{R}^{d}$, the query $q_i^{(n)}\in\mathbb{R}^{d}$ and the value $v_i^{(n)}\in\mathbb{R}^{d}$ from $c_i^{(n-1)}$ \citep{Attention,GAT} as:
\begin{equation}
k_i^{(n)}= W_k^{(n)} c_i^{(n-1)},
\end{equation}
\begin{equation}
q_i^{(n)}= W_q^{(n)} c_i^{(n-1)},
\end{equation}
\begin{equation}
v_i^{(n)}= W_v^{(n)} c_i^{(n-1)},
\end{equation}
where $W_k^{(n)}$, $W_q^{(n)}$, $W_v^{(n)}$ are model parameters.
Then, attention weights $\alpha_{ij}^{(n)}$ are obtained with a softmax function:
\begin{equation}
\alpha_{i j}^{(n)}=\operatorname{softmax}\left(e_{i j}^{(n)}\right)=\frac{\exp \left(e_{i j}^{(n)}\right)}{\sum_{k \in \mathcal{N}_{i}^{(n)}} \exp \left(e_{i k}^{(n)}\right)},
\end{equation}
where $e_{ik}^{(n)}=\text{LeakyRelu}(\frac{{q_i^{(n)}}^T{k_k^{(n)}}}{\sqrt{d}})$.
Finally the updated embedding $c_i^{(n)}$ is calculated as:
\begin{align}
c_i^{(n)}
&= f^{(n)} \left(c_i^{(n-1)}, c_j^{(n-1)} \left|\right. j \in \mathcal{N}_i^{(n)} \right) \\
&=\text{tanh}\left[\sum_{j\in{\mathcal{N}_i^{(n)}}}\alpha_{ij}^{(n)}v_{j}^{(n)}\right].
\end{align}
where $v_{j}^{(1)}$ is the value generated during the first-round communication.
We summarize the communication protocol from the receiver's side in Algorithm \ref{algo_comm_protocol}.
\begin{algorithm*}
\caption{AC2C Communication Protocol at agent $i$}\label{protocol_algorithm}
\begin{algorithmic}[1]
\State \textbf{Inputs:} Initial embedding $c_i^{(0)}$, $c_j^{(0)}$
\State Receive messages from agent $i$'s one-hop neighbors $\mathcal{N}_i^{(1)}$
\Comment{The first communication round}
\State Compute the local embedding
$c_i^{(1)} = f^{(1)} \left(c_i^{(0)}, c_j^{(0)} \left|\right. j \in \mathcal{N}_i^{(1)} \right)$ in Equation (1)
\State Compute the binary signal
$z_i = g(c_{i}^{(0)},c_{i}^{(1)};\theta_c, T)$ in Equation (9)
\If{$z_i == 1$}
\State Broadcast a request to initiate the second communication round
\State Receive messages from agent $i$'s two-hop neighbors $\mathcal{N}_i^{(2)}$ with one-hop neighbors acting as relaying nodes \\
\Comment{The second communication round}
\State Compute the local embedding
$c_i^{(2)} = f^{(2)} \left(c_i^{(1)}, c_j^{(1)} \left|\right. j \in \mathcal{N}_i^{(2)} \right)$ in Equations (7) and (8)
\Else
\State Assign $c_i^{(2)} = \bf{0}$
\EndIf
\State \textbf{Outputs:} Embedding $c_i^{(0)}$, $c_i^{(1)}$ and $c_i^{(2)}$
\end{algorithmic}
\label{algo_comm_protocol}
\end{algorithm*}
Note that while our method can easily be generalized to its multi-round variations by stacking the communication modules for more complicated applications, we confine it to the two-round case in this paper since we do not observe further performance gain when stacking more than two rounds in our experiments.
\subsection{Two-Hop Controller} \label{controller}
Since the second communication round may induce high communication costs, we propose a local two-hop controller to adaptively prune the unnecessary two-hop communication links to reduce the communication cost.
During execution, the agent $i$'s controller takes $c_{i}^{(0)}$ and $c_i^{(1)}$ as inputs and generates a signal $z_i$, determining whether to broadcast a request to initiate the second communication round:
\begin{equation} \label{controller_equation}
z_i=\mathds{1}\left[h \left(c_{i}^{(0)},c_{i}^{(1)};\theta_c\right) >T \right]
\end{equation}
where $\theta_c$ is controller's parameters , $z_i \in \{0, 1 \}$ is the binary signal, $T \in (0,1) $ is the threshold, and $\mathds{1} [ \cdot]$ is the indicator function.
We train the controller as a binary classifier in a self-supervised fashion. The training process for this controller is given in Algorithm \ref{training_algorithm}. The loss function for this auxiliary task is formulated as:
\begin{equation}\label{loss_c}
\begin{aligned}
\mathcal{L}(\theta_{c})=-\mathbb{E}_{\boldsymbol{o}, \boldsymbol{h}} &\left[y_i \log h \left(c_{i}^{(0)},c_{i}^{(1)};\theta_c\right) \right. \\
& \left.+ (1-y_i) \log \left(1- h \left(c_{i}^{(0)},c_{i}^{(1)};\theta_c\right)\right)\right],
\end{aligned}
\end{equation}
with
\begin{equation}
a^{\Rmnum{1}}_{i}=\pi\left(c_i^{(0)},c_i^{(1)}, \textbf{0} ; \theta_\pi \right),
\end{equation}
\begin{equation}
a^{\uppercase\expandafter{\romannumeral2}}_{i}=\pi \left(c_i^{(0)}, c_i^{(1)}, c_i^{(2)}; \theta_\pi \right),
\end{equation}
\begin{equation}\label{cls_target}
y_i =\mathds{1}\left[ \lVert a^\Rmnum{1}_{i}- a^{\uppercase\expandafter{\romannumeral2}}_{i} \rVert >T \right],
\end{equation}
where $\pi_{\theta}(\cdot)$ denotes the action policy, $a^{\Rmnum{1}}_{i}$ and $a^{\uppercase\expandafter{\romannumeral2}}_{i}$ denote the logit of agent $i$'s action decisions after receiving messages in the first and the second communication round, respectively.
The underlying idea of Equation \eqref{cls_target} is that once the second-round messages do not contribute much to agent $i$'s action decision, they would be eliminated as redundant information.
Built on this intuition and the training objective given by Equation \eqref{loss_c}, our controller only exploits the second round communication when necessary. In this way, the controller is able to cut off redundant information while maintaining satisfactory performance.
\begin{figure*}[t]
\centering
\subfigure[Traffic Junction]{
\includegraphics[width = 0.22\textwidth]{Env_tj_medium_v2.pdf}
\label{fig:tj}
}
\subfigure[Cooperative Navigation]{
\includegraphics[width = 0.22\textwidth]{Env_CN_v1_jiawei.pdf}
\label{fig:cn}
}
\subfigure[Predator Prey]{
\includegraphics[width = 0.22\textwidth]{Env_PP_v1_jiawei.pdf}
\label{fig:pp}
}
\caption{Three environments in the experiments: (a) traffic junction, (b) cooperative navigation, and (c) predator prey.
}
\label{fig:environments}
\end{figure*}
\begin{algorithm}[t]
\caption{Training Procedure for the Two-Hop Controller}\label{training_algorithm}
\begin{algorithmic}[1]
\State \textbf{Inputs:} Replay buffer $\mathcal{D}$, the controller threshold $T$
\State \textbf{Initializes:} Controller network parameters $\theta_c$
\State Sample a batch $\mathcal{B}$ with $| \mathcal{B} |$ transitions $(\boldsymbol{o}, \boldsymbol{h}, \boldsymbol{a}, r, \boldsymbol{o}')$ from replay buffer $\mathcal{D}$
\For{$i = 1 \cdots N$}
\State Compute the embeddings $c_i^{(0)}$, $c_i^{(1)}$, $c_i^{(2)}$ in Equations (1), (7) and (8)
\State Compute local action values $a^{\Rmnum{1}}_{i}=\pi_\theta(c_i^{(0)},c_i^{(1)}, \textbf{0})$ and $a^{\uppercase\expandafter{\romannumeral2}}_{i}=\pi_\theta(c_i^{(0)}, c_i^{(1)}, c_i^{(2)})$ in Equations (11) and (12)
\State Compute $y_i =\mathds{1}\left[ \lVert a^\Rmnum{1}_{i}- a^{\uppercase\expandafter{\romannumeral2}}_{i} \rVert >T \right]$
in Equation (13)
\State Update $\theta_{c}$ to minimize $\mathcal{L}(\theta_{c})$ in Equation \eqref{loss_c}
\EndFor
\State \textbf{Outputs:} $\theta_c$
\end{algorithmic}
\end{algorithm}
\subsection{Centralized Critic}
We adopt the actor-critic structure that has been wildly used for many single-agent and multi-agent algorithms. Following previous works \citep{COMA, MADDPG}, we leverage a centralized critic network to guide the policy optimization. The critic network shares a similar structure with the actor, but it takes the historical information $\boldsymbol{h}$, observations $\boldsymbol{o}$, and additional predicted actions $\boldsymbol{a}$ from all agents as inputs. A centralized critic network leverages all agents' information to update each agent's gradient. It can greatly alleviate the non-stationary problem. In order to make the implementation scalable, the centralized critic is not needed during execution.
\subsection{Training}
We implement the DDPG and REINFORCE algorithms for different experiments.
In the DDPG algorithm, we adopt a shared critic with a similar structure to the actor to guide each agent to update its policy under the CTDE paradigm. The centralized critic is updated by the standard TD loss:
\begin{equation}
\mathcal{L}(\theta_Q) = \mathbb{E}_{\boldsymbol{\tau}, \boldsymbol{h}, \boldsymbol{a}, r, \boldsymbol{\tau}'} \left[\left(y - Q\left(\boldsymbol{\tau}, \boldsymbol{a}; \theta_{Q}\right) \right)^2 \right],
\end{equation}
\begin{equation}
y = r + \gamma Q^{\prime}\left(\boldsymbol{\tau}', \pi^{\prime}\left(\boldsymbol{\tau}'; \theta_{\pi^{\prime}}\right) ; \theta_{Q^{\prime}}\right),
\end{equation}
where $Q^{\prime}$ is the target $Q$ network, $\pi^{\prime}$ is the target actor network, and $\theta_Q$ contains the parameters of the centralized critic network.
Besides, we update the actor network parameters $\theta_\pi$ by the sampled policy gradient:
\begin{equation}
\nabla_{\theta_\pi}J(\theta_\pi) = \mathbb{E}_{\boldsymbol{\tau}, \boldsymbol{h}, \boldsymbol{a}, r, \boldsymbol{\tau}'} \left[\nabla_{\theta_\pi}\pi (\tau_i ; \theta_{\pi})\nabla_{a}Q(\tau_i,a_i ; \theta_{Q})|_{a_{i}=\pi(\tau_i)} \right]
.
\end{equation}
In the traffic junction experiments, we adopt the REINFORCE algorithm with baseline \citep{sutton1999policy} to learn the actor policy. We update the policy network parameters $\theta_\mu$ by the following equation:
\begin{equation}
\nabla_{\theta_\pi}J(\theta_\pi) = \mathbb{E}_{\boldsymbol{\tau}, \boldsymbol{h}, \boldsymbol{a}, r, \boldsymbol{\tau}'} \left[ \left( G-b(\boldsymbol{\tau})\right) \nabla_{\theta_\pi} \log \pi\left(\tau_i; \theta_\pi \right) \right],
\end{equation}
where $G$ is the episodic return and $b(\cdot)$ is the counterfactual baseline.
In order to accelerate the training, we share the feature encoder and action policy parameters across agents.
\section{Experiments}
\begin{figure}[t]
\centering
\subfigure[Cooperative navigation]{
\includegraphics[width=0.32\textwidth]{cc_3_font1.pdf}
}
\vfill
\subfigure[Predator prey]{
\includegraphics[width=0.32\textwidth]{pp_3_font1.pdf}
}
\caption{Training curves of cooperative navigation and predator prey.}
\label{fig:PP_CN_training}
\end{figure}
We evaluate the proposed AC2C in three environments, namely, traffic junction, cooperative navigation and predator prey, as illustrated in Fig \ref{fig:environments}.
Following \citep{i2c,ic3}, our method as well as the baseline methods are implemented on top of the REINFORCE algorithm \citep{sutton1999policy} in the traffic junction environment, and on top of the DDPG \citep{DDPG} in the predator prey and cooperative navigation environments.
All presented results are average performance over five random seeds. The shaded area in each figure is the standard deviation.
\subsection{Baselines}
In this work, we compare our method with baselines including TarMAC \citep{Tarmac}, SARNet \citep{sarnet}, DICG \citep{dicg} and DGN \citep{DGN}. TarMAC achieves multi-round communication with a back-and-forth method \citep{Tarmac}. SARNet leverages a memory-based mechanism to solve cooperative multi-agent tasks. DICG and DGN are both typical GNN-based methods. DICG utilizes GCN layers and attention mechanisms to accomplish message aggregation, while DGN considers dynamic communication graphs and adopts GCN layers to conduct message aggregation.
In our AC2C communication protocol, each agent exchanges local information with its one-hop neighbors in the first communication round and exchanges messages with its two-hop neighbors in the second communication round. We set a GNN-based protocol for all baselines. Specifically, each agent communicates with its one-hop neighbors in the first communication round. Once all the agents have aggregated the information of one-hop neighbors, they will communicate again with one-hop neighbors in the next round to obtain information from further-away agents.
To quantify the communication cost, we calculate the number of active communication links, where $w$ bits messages are transmitted through each link. The cost of the first communication round $\text{Cost}^{(1)}$ for the GNN-based method and AC2C is calculated as:
\begin{equation}
\text{Cost}^{(1)}=\sum_{i} | \mathcal{N}_i^{(1)} | \cdot w
\end{equation}
where the $| \mathcal{N}_i^{(1)} |$ is the number of agent $i$'s one-hop neighbors.
The cost of the second communication round $\text{Cost}^{(2)}$ is computed as:
\begin{equation}
\text{Cost}^{(2)}=
\begin{cases}
\sum_{i}{| \mathcal{N}_i^{(2)} |}\cdot 2w,& \text{AC2C}\\
\sum_i{| \mathcal{N}_i^{(1)} |}\cdot w&, \text{GNN-based methods}
\end{cases}
\end{equation}
As the AC2C protocol transmits two-hop messages through a relaying node, the communication cost in the second round should be doubled.
\subsection{Environments}
\subsubsection{Traffic Junction}
The simulated traffic junction environment introduced by \cite{CommNet}, consists of 20 cars moving along predefined routes with one or more road junctions, as shown in Figure \ref{fig:tj}.
The goal for each car is to arrive at the destination while avoiding collisions with other cars.
Following \cite{i2c,ic3}, each car's field of view is set to 0, but it can communicate with other cars within its communication range.
Each car has only two actions: brake or gas (move forward).
The reward for each car includes a linear time penalty $-0.01\tau$, where $\tau$ is the number of timesteps after a car becomes active, and a penalty of -20 induced by collisions.
In the experiments, we consider two modes of traffic junction: a medium mode and a hard mode.
Particularly, the dimension of the map is set to $6\times6$, and the number of intersections is set as 1 in the medium mode. In the hard mode, the dimension of the map is $9\times9$, and the number of intersections is set as 4. In order to effectively compare the performance, we evaluate the success rate under 20000 testing episodes. We regard an episode as successful if no collision happens during this episode.
\subsubsection{Cooperative Navigation.}
The goal of cooperative navigation \cite{MPE1,MPE2} is for several agents to cover landmarks respectively, as shown in Figure \ref{fig:cn}.
In our experiments, 10 agents try to occupy 10 fixed landmarks, where each agent obtains partial observation of the environment. Specifically, an agent only knows its own position as well as velocity, and the positions of all the landmarks.
In this environment, each agent will get a bonus when it approaches the landmark but will receive a penalty when it collides with other agents. All agent are initialized at random positions in every episode. The episode length is set as 50 timesteps. We use the average reward per timestep of each agent as the evaluation metric.
\subsubsection{Predator Prey}
Following previous work \citep{i2c, sarnet}, the predator prey environment is a cooperative multi-agent task as is shown in Figure \ref{fig:pp}.
The goal is for the predators to capture as many preys as possible during a given time period.
The observations of each predator include its position, velocity, the two closest preys' positions, and two closest predators' positions.
As the preys move slightly faster than the predators, the predators need to learn how to capture the preys cooperatively.
We generate an environment with 10 agents (predators) and 10 preys, where the actions of the preys are controlled by the bots in I2C\citep{i2c}.
Each predator gets a bonus when it captures a prey while receiving a penalty when a collision among predators happens. The evaluation metric is set the same as cooperative navigation.
\begin{table}[t]
\centering
\caption{Success rates and communication overhead per timestep of traffic junction.}
\resizebox{0.465\textwidth}{!}{
\begin{tabular}{c|c|c|c|c}
\hline
& \multicolumn{4}{c}{Traffic junction} \\ \cline{2-5}
\quad& \multicolumn{2}{c}{Medium mode}& \multicolumn{2}{c}{Hard mode}\\
\cline{2-5}
\quad& Success rate& \begin{tabular}[c]{@{}c@{}} Communication\\ overhead $(10^5 \, \text{bits})$\end{tabular} & Success rate& \begin{tabular}[c]{@{}c@{}} Communication\\ overhead $(10^5 \, \text{bits})$ \end{tabular} \\
\hline
AC2C (ours) & \bf{95.33$\pm$0.21} & \bf{2.972$\pm$0.636} & \bf{71.85$\pm$1.45}&\bf{5.030$\pm$0.701}\\
TarMAC&93.81$\pm$0.17 & 3.790$\pm$0.245 & 49.23$\pm$1.06&6.979$\pm$0.705\\
SarNet&92.16$\pm$0.97 & 3.623$\pm$0.807 & 46.73$\pm$1.31&7.063$\pm$1.029\\
DICG&95.21$\pm$1.12 & 4.003$\pm$0.794 & 53.71$\pm$2.77&7.613$\pm$1.217\\
DGN&86.37$\pm$1.27 & 3.918$\pm$0.930 & 17.43$\pm$3.91&7.077$\pm$0.590\\
\hline
\end{tabular}
}
\label{table: TJ_tradeoff}
\end{table}
\begin{table}[t]
\centering
\caption{Performance and communication cost per timestep of cooperative navigation and predator prey.}
\resizebox{0.475\textwidth}{!}{
\begin{tabular}{c|c|c|c|c}
\hline
\quad& \multicolumn{2}{c|}{Cooperative navigation}& \multicolumn{2}{c}{Predator prey}\\
\hline
\quad& Reward & \begin{tabular}[c]{@{}c@{}} Communication\\ overhead $(10^5 \, \text{bits})$\end{tabular} & Reward & \begin{tabular}[c]{@{}c@{}} Communication\\ overhead $(10^5 \, \text{bits})$\end{tabular} \\
\hline
AC2C (ours) & \bf{-1.573$\pm$0.221} & \bf{5.636$\pm$0.614} & \bf{-2.502$\pm$0.402} & \bf{4.878$\pm$0.520}\\
TarMAC & -2.043$\pm$0.216 & 8.526$\pm$0.934 & -3.654$\pm$0.679 & 7.780$\pm$1.354\\
SarNet & -3.055$\pm$0.276 & 9.362$\pm$1.106 & -4.042$\pm$0.248 & 6.344$\pm$1.630\\
DICG & -5.538$\pm$0.378 & 7.770$\pm$1.484 & -5.478$\pm$0.438 & 7.046$\pm$2.108\\
DGN& -6.696$\pm$0.817 &7.688$\pm$1.496 & -9.250$\pm$0.956 &6.664$\pm$2.318\\
\hline
\end{tabular}
}
\label{table: CN_PP_tradeoff}
\end{table}
\subsection{Results}
\iffalse
\begin{figure}[t]
\centering
\subfigure[Cooperative Navigation]{
\includegraphics[width=3.5cm]{ss_comm.pdf}
\label{fig:CN_tradeoff}
}
\subfigure[Predator Prey]{
\includegraphics[width=3.5cm]{pp_comm_2.pdf}
\label{fig:PP_tradeoff}
}
\caption{Trade-off between communication cost and performance.}
\label{fig:PP_CN_tradeoff}
\end{figure}
\fi
We first investigate the trade-off between the performance and the communication overhead. As illustrated in Table \ref{table: TJ_tradeoff}, Table \ref{table: CN_PP_tradeoff} and Figure \ref{fig:PP_CN_training}, for all three environments, our proposed AC2C significantly outperforms all baselines while maintaining the lowest communication overhead.
Detailed observations are elaborated below for each environment.
\begin{table*}[!htbp]
\centering
\caption{Average test rewards and communication cost per timestep of each agent under different communication range in cooperative navigation.}
\resizebox{1\textwidth}{!}{
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}
\hline
\quad & \multicolumn{2}{c|}{AC2C (ours)} &\multicolumn{2}{c|}{TarMAC} &\multicolumn{2}{c|}{SARNet} &\multicolumn{2}{c|}{DICG} &\multicolumn{2}{c}{DGN} \\
\hline
\quad & Reward & \begin{tabular}[c]{@{}c@{}}Communication\\ overhead $(10^5 \, \text{bits})$\end{tabular} & Reward & \begin{tabular}[c]{@{}c@{}}Communication\\ overhead $(10^5 \, \text{bits})$\end{tabular} & Reward & \begin{tabular}[c]{@{}c@{}}Communication\\ overhead $(10^5 \, \text{bits})$\end{tabular} & Reward & \begin{tabular}[c]{@{}c@{}}Communication\\ $(10^5 \, \text{bits})$\end{tabular} & Reward & \begin{tabular}[c]{@{}c@{}}Communication\\ overhead $(10^5 \, \text{bits})$\end{tabular}\\
\hline
0.3 & \bf{-1.841}$\pm$0.221 & \bf{0.152$\pm$0.030}& -3.588$\pm$0.353 & 0.198$\pm$0.022&
-5.731$\pm$0.884 & 0.166$\pm$0.030 & -6.259$\pm$0.575 &0.194$\pm$0.028 &-10.04$\pm$1.025& 0.198$\pm$0.024\\
0.5& \bf{-1.623$\pm$0.357} & \bf{1.084$\pm$0.256} & -3.933$\pm$0.423 & 1.262$\pm$0.310 & -5.063$\pm$0.371 & 1.396$\pm$0.364 & -6.629$\pm$0.593 & 1.504$\pm$0.444& -8.166$\pm$0.732 & 1.424$\pm$0.334 \\
1.0& \bf{-1.573$\pm$0.221} &\bf{5.636$\pm$0.614}& -2.043$\pm$0.216 & 8.526$\pm$0.934& -3.055$\pm$0.276 &9.362$\pm$1.106& -5.538$\pm$0.378 & 7.770$\pm$1.484 &-6.696$\pm$0.817 & 7.688$\pm$1.496\\
1.5& \bf{-1.267$\pm$0.158} &\bf{7.132$\pm$0.652}& -1.418$\pm$0.034 &9.700$\pm$1.814& -1.500$\pm$0.054 &11.47$\pm$1.198& 1.700$\pm$0.344 &8.492$\pm$1.036& -4.603$\pm$0.551&9.634$\pm$0.924\\
\hline
\end{tabular}
}
\label{table: CN_comm_range}
\end{table*}
\begin{table*}[h]
\centering
\caption{Average test rewards and communication cost per timestep of each agent under different communication range in predator prey.}
\resizebox{1\textwidth}{!}{
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}
\hline
\quad & \multicolumn{2}{c|}{AC2C (ours)} &\multicolumn{2}{c|}{TarMAC} &\multicolumn{2}{c|}{SARNet} &\multicolumn{2}{c|}{DICG} &\multicolumn{2}{c}{DGN} \\
\hline
\quad & Reward & \begin{tabular}[c]{@{}c@{}}Communication\\ overhead $(10^5 \, \text{bits})$\end{tabular} & Reward & \begin{tabular}[c]{@{}c@{}}Communication\\ overhead $(10^5 \, \text{bits})$\end{tabular} & Reward & \begin{tabular}[c]{@{}c@{}}Communication\\ overhead $(10^5 \, \text{bits})$\end{tabular} & Reward & \begin{tabular}[c]{@{}c@{}}Communication\\ $(10^5 \, \text{bits})$\end{tabular} & Reward & \begin{tabular}[c]{@{}c@{}}Communication\\ overhead $(10^5 \, \text{bits})$\end{tabular}\\
\hline
0.3 & \bf{-5.443}$\pm$0.447 & \bf{0.166$\pm$0.014}& -7.504$\pm$1.080 & 0.188$\pm$0.016 &
-7.082$\pm$492.3 & 0.174$\pm$0.026 & -7.130$\pm$0.952 & 0.19$\pm$0.022 &-10.44$\pm$0.849& 0.178$\pm$0.018\\
0.5& \bf{-3.673$\pm$0.390} & \bf{0.784$\pm$0.062} & -5.714$\pm$0.651 & 1.388$\pm$0.192 & -5.343$\pm$0.404 & 1.346$\pm$0.382 & -6.026$\pm$0.624 & 1.428$\pm$0.634& -9.566$\pm$1.162 & 1.522$\pm$0.410 \\
1.0& \bf{-2.502$\pm$0.402} &\bf{4.878$\pm$0.520}& -3.654$\pm$0.679 & 7.780$\pm$1.354& -4.042$\pm$0.248 &6.344$\pm$0.630& -5.478$\pm$0.438 & 7.046$\pm$2.108 &-9.250$\pm$0.956&6.664$\pm$2.318\\
1.5& \bf{-2.034$\pm$0.211} &\bf{6.944$\pm$0.638}& -3.254$\pm$0.679 &8.586$\pm$1.024& -2.876$\pm$0.341 &7.370$\pm$0.486& 4.100$\pm$0.294 &8.194$\pm$0.944& -5.487$\pm$0.951&9.246$\pm$0.722\\
\hline
\end{tabular}
}
\label{table: PP_comm_range}
\end{table*}
In the traffic junction environment, we see from Table \ref{table: TJ_tradeoff} that in the medium mode, AC2C performs slightly better than other baselines with a lower communication overhead. We observe that, very few communication links are constructed since the cars are scattered sparsely across the map. Although agents sometimes obtain information of more distant agents, most of the time, they can only utilize the information of their one-hop neighbors. In the hard mode, AC2C demonstrates a substantial performance gain compared with all the baselines, in both the reward and communication overhead. It illustrates the importance of the communication mechanism design in such difficult environments.
For cooperative navigation and predator prey tasks, AC2C consistently outperforms the baselines. This is attributed to the long-range information exchange enabled by our protocol. We discover that information belonging to farther agents can effectively improve the current agent's action decisions, e.g., this information can point out where the landmarks are located. In addition, the controller effectively prunes irrelevant messages to make AC2C agents maintain the lowest communication cost.
We also examine the influence of the communication range $L$ on the performance.
In particular, we set the communication range as $L = 0.3, 0.5, 1.0, 1.5$ in the cooperative navigation and predator prey tasks. As illustrated in Tables \ref{table: CN_comm_range} and \ref{table: PP_comm_range}, AC2C consistently outperforms all the baselines and achieves low communication overhead.
For all the methods, as the communication range shrinks, both the rewards and communication overhead are decreased, which is due to the smaller number of one-hop and two-hop neighbors.
The results demonstrate that the controller in our method effectively identifies and prunes the irrelevant communication links without incurring performance degradation.
\iffalse
\begin{figure}[t]
\centering
\subfigure[Cutted Links]{
\includegraphics[width=3.5cm]{cutted_links.pdf}
}
\subfigure[Predator Prey]{
\includegraphics[width=3.5cm]{Figure_2.pdf}
}
\caption{Training curves of Cooperative Navigation and Predator prey.}
\label{fig:PP_CN_training}
\end{figure}
\fi
\begin{table}[ht]
\tiny
\centering
\caption{The impact of different threshold $T$ in cooperative navigation.}
\resizebox{0.45\textwidth}{!}{
\begin{tabular}{c|c|c}
\hline
\quad & Reward & Communication Overhead ($10^5$ bits) \\
\hline
$T=0$ & -1.579$\pm$0.195 & 9.546$\pm$0.878 \\
\hline
$T=0.1$ & -1.546$\pm$0.100 & 8.650$\pm$0.524 \\
\hline
$T=0.2$ & -1.539$\pm$0.195 & 8.072$\pm$0.622 \\
\hline
$T=0.3$ & -1.565$\pm$0.207& 6.672$\pm$0.724 \\
\hline
$T=0.4$ & -1.553$\pm$0.199 & 6.362$\pm$0.842 \\
\hline
$T=0.5$ & -1.573$\pm$0.221 & 5.636$\pm$0.614 \\
\hline
$T=0.6$ & -3.855$\pm$0.648 & 4.462 $\pm$0.686\\
\hline
\end{tabular}
}
\label{table: different_threshold}
\end{table}
\textbf{Ablation Study}. In order to demonstrate the effectiveness of the AC2C protocol, we conduct the ablation study on AC2C under the cooperative navigation environment. AC2C w/o controller refers to AC2C without the controller, where the second-round communication always happens; AC2C-GNN adopts the GNN-based protocol to communicate rather than ours; and AC2C one round indicates that the second-round communication never happens. As shown in Figure \ref{difference_AC2C}, AC2C achieves significant performance gains compared to AC2C-GNN and AC2C-one round. Moreover, it shows that the controller can effectively reduce the communication overhead without performance degradation. We next test the impact of the threshold $T$ on the controller in Table \ref{table: different_threshold}. When the value of $T$ varies from 0 to 0.5, the communication cost reduces, but the performance does not drop significantly. It illustrates that the controller helps to prune the irrelevant information. However, the performance dramatically deteriorates when the threshold reaches 0.6, where the controller cannot retain enough valuable information. In this case, the overall performance is similar to that of AC2C-one round. In other reported results, we did a grid search to find the optimal $T$.
\begin{figure}[t]
\centering
\includegraphics[width=5.8cm]{ablation_3_font1.pdf}
\caption{The performance between different AC2C versions.
}
\label{difference_AC2C}
\end{figure}
\iffalse
\subsection{Traffic Junction}
We used the REINFORCE algorithm in this experiment. When the difficulty is medium, the episode length is 60 timesteps, while the episode length is 80 timesteps for the hard mode. We adopted the \citep{i2c} method to evaluate agents: if no agent collides within an episode, then this episode can be called a success, otherwise, it would be a failure. The table below success rate tested by 20,000 episodes.
\textbf{\textit{Result Analysis}} As the results shown in Table 1, We compare that AC2C with TarMAC, SARNet, DICG and DGN in the medium and hard mode of traffic junction. In the medium mode, all baselines all perform well due to this mode is relatively easier for agents to learn and our AC2C method shows slightly higher performance than others. In the hard mode, the traffic junctions become more, and the agent needs more effective communication to guide the agent to move. Our method is the only one that exceeds 70 percent of success rates in this setting, while TarMAC and SarNet show a success rate of nearly 50\% and DICG performs 52.13\% success rate. This experiment proves that our model is effective and powerful. Our model setting is slightly different from other experiments conducted in this paper, We set different attention modules as shared-weights.
\subsection{Cooperative Navigation}
\textbf{\textit{Result Analysis}} We compare our method AC2C with TarMAC, SARNet, DICG and DGN under different communication range. Besides, we also introduce the AC2C w/o controller, which is the version AC2C removes the controller. Each agent only communicates with observed agents, and they can communicate with other agents who lied out of communication range in the next round. The Figure 6 shows the training curves of all these methods under the $L=1$, we test them on 5 different seeds. Our method AC2C and AC2C w/o controller outperforms other baselines. These 2 algorithms achieve over -800 reward per episode, but their performance are rather close. Additionally, TarMAC and SARNet perform slightly worse. Then follows DICG and DGN. DGN fails to develop cooperative strategy in this task. One of the reasons for poor performance of DGN is the setting is too difficult for it to handle; while for our AC2C algorithm, at the beginning of training, the controller can help AC2C to exclude some nodes, so that the whole training pays more attention to the communication of the surrounding neighbors. When the controller can accurately predict the nodes and is useful, it can help the agent to obtain more node information, thus Improve overall performance.
While the Figure 7 is the scatter graph of the performance versus communication cost. The communication cost is set the similar with \citep{VBC}, we count the total amount of the multi-round communication links between different agents set as $N_comm$. Then the x axis is the $\frac{N_comm}{N_full}$, where $N_full$ is the number of links under fully-connected scenario. In Figure 7, due to following the same communication protocol for all of the baselines, the communication cost of them are very close. our method cuts off a lot of communication links while still maintains satisfactory performance. Compared with other baselines without controller, AC2C cuts off nearly 20\% of the communication links. With the increase of the threshold, it shows the communication links slightly reduced, but the performance will also slightly exacerbate.
The Table 2 shows the performance of AC2C and other baselines under different communication range. We test them on $L=0.3, 0.5, 1.0, 1.5$ for 5 different seeds. When $L=0.3$ we can tell that all the methods perform undesirable due to the too short communication range, agents cannot get enough information from other agents to generate satisfactory action. Besides, the training process of the controller becomes extremely arduous, because the controller always receives too little information from neighbors, which causes the controller to always judge that the next round of communication is unnecessary. Therefore, the controller has became an obstacle to effective communication. As the communication range increases, the performance of AC2C surpasses the AC2C w/o controller. Especially in the case of communication range longer, the number of neighbors increases a lot, then selective communication is able to improve the overall performance.
\subsection{Predator Prey}
\textbf{\textit{Result Analysis}} We compare AC2C method with TarMAC, SARNet, DICG and DGN. As shown in Figure 8, AC2C clearly outperforms other baselines. Our ablation study AC2C w/o Controller performs slightly worse. But both these two methods converge to the similar value in the last. Besides, TarMAC, SARNet, DICG and DGN's performance following AC2C. DGN still performs miserably since this task is still too difficult for this model to cope with.
For Figure 9, we demonstrate the communication overhead of each algorithm. Our AC2C cuts off nearly 10\% communication links and maintains high performance. It validates our communication model is effective. It shows that the performance is the best when threshold=0.1, but the links removed by the controller are also the least at this time. Similar to the performance shown in the Cooperative Navigation, as the threshold gradually increases, the communication overhead gradually decreases, and the performance also decreases lightly.
Besides, we compare the AC2C with the non-controller AC2C version in the ablation study. Our AC2C performs slightly better than non-controller one; because the controller helps to remove the irrelevant nodes, the agent receives less noisy information so that training process can converge faster.
\fi
\section{Conclusion}
In this paper, we introduce an effective communication protocol for cooperative multi-agent reinforcement learning systems, which helps agents to obtain valuable messages from agents outside their communication range. A communication controller is introduced to reduce the communication overhead while maintaining performance. Extensive experiments show that the proposed method outperforms all the baselines regarding both the reward and communication overhead in the three considered environments. This study illustrates the importance of developing adaptive multi-hop communication protocols for multi-agent reinforcement learning systems.
\section{Acknowledgment}
This work was supported by the NSFC/RGC Collaborative Research Scheme (Project No. CRS\_HKUST603/22).
\bibliographystyle{ACM-Reference-Format}
\balance
|
1,108,101,566,766 | arxiv | \section{Introduction}
Legal reasoning problems can be addressed from different
perspectives.
From a lawyer's perspective, a trial may be best modeled as a
strategic game.
In a criminal trial, for example, the prosecutor may try
to convince the judge or jury of the defendant's guilt
while the defense attorney tries the opposite.
The problem is then to interpret the law and the evidence
in a way that maximizes the agent's utility.
From this perspective, a legal reasoning problem is best modeled
using tools from decision and game theory \cite{hanson2014game,prakken1996dialectical,riveret2007success}.
Our focus here is not on strategic considerations,
but on the decision process that leads
to the final verdict in a legal process like a trial.
Given different pieces of evidence and beliefs about
their authenticity and relevance, how can we merge them
to make a
plausible and transparent decision?
Different automated reasoning tools have been applied in
order to answer similar questions, for example,
case-based reasoning \cite{bench2003model,mccarty1995implementation},
argumentation frameworks \cite{dung2010towards,prakken2013formalization}
or Bayesian networks \cite{fenton2013general}.
Since lawyers and judges often struggle with the interpretation
of Bayesian networks, recent work also tries to explain Bayesian networks by argumentation tools \cite{vlek2016method}.
Here, we investigate the applicability of the probabilistic epistemic argumentation framework developed in \cite{hunter2013probabilistic,HunterPT2018Arxiv,HunterT16,thimm2012probabilistic}.
As opposed to classical argumentation approaches, this framework allows
expressing uncertainty by means of probability theory.
In particular, we can compute reasoning results in polynomial time
when we restrict the language \cite{potyka2019fragment}.
As it turns out, the resulting fragment is sufficiently expressive for our purpose,
so that our framework is computationally more efficient than
many other probabilistic reasoning approaches that suffer from
exponential runtime in the worst-case. At the same time, the graphical structure
is easily interpretable and allows to automatically generate
explanations for the final degrees of belief (probabilities) as we will explain later.
While we can incorporate objective probabilities in our framework,
our probabilistic reasoning is best described as subjective in the sense that we basically merge beliefs about pieces of evidence
and hypotheses (probabilities that can be either objective or subjective). In order to define the beliefs about pieces of
evidence from objective evidence and statistical information,
another approach like Bayesian networks or more general tools
from probability theory may be better suited. Our framework
can then be applied on top of these tools.
In this sense, our framework can be seen as a complement rather
than a replacement of alternative approaches.
The remainder of this paper is structured as follows:
Section 2 explains the necessary basics. We will introduce
a basic legal argumentation framework in Section 3 and
discuss more sophisticated building blocks in Section 4.
We will discuss and illustrate the explainability capabilities of our approach
as we proceed, but explain some more general ideas in Section 5.
Finally, we add some discussion about related work, the
pros and cons of our framework and future work in Sections 6
and 7.
\section{Probabilistic Epistemic Argumentation Basics}
Our legal reasoning approach builds up on the probabilistic epistemic argumentation approach
developed in \cite{thimm2012probabilistic,hunter2013probabilistic,HunterT16,HunterPT2018Arxiv}.
In this approach, we assign degrees of belief in the form of probabilities to arguments using probability functions
over possible worlds. A possible world basically interprets every argument as either accepted
or rejected. In order to restrict to probability functions that respect prior beliefs and
the structure of the argumentation graph, different constraints can be defined.
Afterwards, we can assign a probability interval to every argument based on these constraints.
We will restrict to a fragment of the constraint language here that allows polynomial-time
computations \cite{potyka2019fragment}.
Formally, we represent arguments and their relationships in a directed edge-weighted graph
$(\ensuremath{\mathcal{A}}, \ensuremath{\mathcal{E}}, \ensuremath{\operatorname{w}})$. $\ensuremath{\mathcal{A}}$ is a finite set of arguments,
$\ensuremath{\mathcal{E}} \subseteq \ensuremath{\mathcal{A}} \times \ensuremath{\mathcal{A}}$ is a finite set of directed edges between the arguments and
$\ensuremath{\operatorname{w}}: \ensuremath{\mathcal{E}} \rightarrow \mathbb{Q}$ assigns a rational number to
every edge.
If there is an edge $(A,B) \in \ensuremath{\mathcal{E}}$, we say that
\emph{$A$ attacks $B$} if $w((A,B)) < 0$ and \emph{$A$ supports $B$} if $w((A,B)) > 0$.
We let
$\ensuremath{\mathrm{Att}}(A) = \{B \in \ensuremath{\mathcal{A}} \mid (B,A) \in \ensuremath{\mathcal{E}}, w((A,B)) < 0\}$ be the set of attackers of an argument A
and
$\ensuremath{\mathrm{Sup}}(A) = \{B \in \ensuremath{\mathcal{A}} \mid (B,A) \in \ensuremath{\mathcal{E}}, w((A,B)) > 0\}$ be the set of supporters.
A \emph{possible world} is a subset of arguments $\omega \subseteq \ensuremath{\mathcal{A}}$. Intuitively,
$\omega$ contains the arguments that are accepted in a particular state of the world.
Beliefs about the true state of the world are modeled by rational-valued probability
functions $P: 2^\ensuremath{\mathcal{A}} \rightarrow [0,1]\cap \mathbb{Q}$ such that $\sum_{\omega \in 2^\ensuremath{\mathcal{A}}} P(\omega) = 1$.
The restriction to probabilities from the rational numbers is for computational reasons only. In practice, it does
not really mean any loss of generality because
implementations usually use finite precision arithmetic.
We denote the set of all probability functions over $\ensuremath{\mathcal{A}}$ by $\ensuremath{\mathcal{P}_\args}$.
The probability of an argument $A \in \ensuremath{\mathcal{A}}$ under $P$
is defined by adding the probabilities of all worlds in which $A$ is accepted, that is,
$P(A) = \sum_{\omega \in 2^\ensuremath{\mathcal{A}}, A \in \omega} P(\omega)$. $P(A)$ can be understood as a degree of belief,
where $P(A) = 1$ means complete acceptance and $P(A)=0$ means complete rejection.
The meaning of attack and support relationships can be defined by means of
constraints in probabilistic epistemic argumentation.
For example, the \emph{Coherence} postulate in \cite{HunterT16} intuitively demands that
the belief in an argument is bounded from above by the belief of its attackers.
Formally, a probability function $P$ respects \emph{Coherence}
iff $P(A) \leq 1 - P(B)$ for all $B \in \ensuremath{\mathrm{Att}}(A)$.
A more general constraint language has recently been introduced in \cite{HunterPT2018Arxiv}.
Here, we will restrict to a fragment of this language that allows solving our reasoning
problems in polynomial time \cite{potyka2019fragment}.
A \emph{linear atomic constraint} is an expression of the form
$$c_0 + \sum_{i=1}^n c_i \cdot \pi(A_i) \leq d_0 + \sum_{i=1}^m d_i \cdot \pi(B_i),$$
where $A_i, B_i \in \ensuremath{\mathcal{A}}$, $c_i, d_i \in \mathbb{Q}$, $n,m \geq 0$ (the sums can be empty)
and $\pi$ is a syntactic symbol that can be read as
'the probability of'.
For example, the \emph{Coherence} condition above can be expressed by a linear atomic constraint
with $m=n=1$, $c_0=0$, $c_1=1$, $A_1 = A$, $d_0 = 1$, $d_1 = -1$ and $B_1=B$. However, we can also define more complex
constraints that take the beliefs of more than just two arguments into account.
Usually, the arguments that occur in a constraint are neighbors in the graph
and the coefficients $c_i, d_i$ will often be based on the weight of the edges between the arguments.
We will see many examples later.
A probability function $P$ \emph{satisfies} a linear atomic constraint iff
$c_0 + \sum_{i=1}^n c_i \cdot P(A_i) \leq d_0 + \sum_{i=1}^m d_i \cdot P(B_i)$.
$P$ satisfies a set of linear atomic constraints $\ensuremath{\mathcal{C}}$, denoted as $P \models \ensuremath{\mathcal{C}}$, iff it satisfies all constraints $c \in C$.
If this is the case, we call $\ensuremath{\mathcal{C}}$ \emph{satisfiable}.
We are interested in two reasoning problems here that have been introduced in
\cite{HunterT16}.
First, the \emph{satisfiability problem} is, given a graph $(\ensuremath{\mathcal{A}}, \ensuremath{\mathcal{E}}, \ensuremath{\operatorname{w}})$
and a set of constraints $\ensuremath{\mathcal{C}}$ over this graph, to decide if the constraints
are satisfiable. This basically allows us to check that our modelling assumptions are
consistent.
Second, the \emph{entailment problem} is, given a graph $(\ensuremath{\mathcal{A}}, \ensuremath{\mathcal{E}}, \ensuremath{\operatorname{w}})$,
a set of satisfiable constraints $\ensuremath{\mathcal{C}}$ and an argument $A$, to compute lower and upper bounds
on the probability of $A$ based on the probability functions that satisfy the constraints.
For example, suppose we have $\ensuremath{\mathcal{A}} = \{A, B, C\}$, $\ensuremath{\mathcal{E}} = \{(A,B), (B,C)\}$, $\ensuremath{\operatorname{w}}((A,B)) = 1$,
$\ensuremath{\operatorname{w}}((B,C)) = -1$. We encode the meaning of the support relationship $(A,B)$ by
$w((A,B)) \cdot \pi(A) \leq \pi(B)$ (a supporter bounds the belief in the argument from below) and
the meaning of the attack relationship $(B,C)$ by $\pi(C) \leq 1 + w((B,C)) \cdot P(B)$ (an attacker bounds the belief in the argument from above). Say, we also tend to accept $C$ and model this
by the constraint $0.5 \leq \pi(C)$.
Then our constraints are satisfiable and the entailment
results are $P(A) \in [0, 0.5]$, $P(B) \in [0, 0.5]$, $P(C) \in [0.5, 1]$.
To understand the reasoning, let us consider the upper bound for $A$.
If we had $P(A) > 0.5$, we would also have $P(B) > 0.5$ because of the support
constraint. But then, we would have $P(C) < 0.5$ because of the attack constraint.
However, this would violate our constraint for $C$. Hence, we must have $P(A) \leq 0.5$.
In particular, if we would add the constraint $1 \leq \pi(A)$ (accept $A$), our
constraints would become unsatisfiable.
Both the satisfiability and the entailment problem can be automatically solved by
linear programming techniques.
In general, the linear programs can become exponentially large.
However, both problems
can be solved in polynomial time when we restrict to linear atomic constraints \cite{potyka2019fragment}.
\section{Basic Legal Argumentation Framework}
Legal reasoning problems can occur in many forms and an attempt to capture all of them
at once would most probably result in a framework that is hardly more
concrete than a general abstract argumentation framework.
We will therefore focus on a particular scenario, where the innocence of a defendant
has to be decided.
Modeling a single case may not be sufficient to illustrate the general applicability
of probabilistic epistemic argumentation.
We will therefore try to define a reasoning framework that can be instantiated for different cases, while still being easily comprehensible.
As with every formal model, there are some simplifying assumptions
about the nature of a trial.
However, we think that our framework
is sufficient to illustrate how real cases can be modeled and
structured by means of probabilistic epistemic argumentation.
We will make some additional comments about this as we proceed.
Following \cite{fenton2013general}, we regard a legal case roughly as
a collection of \emph{hypotheses} and \emph{pieces of evidence}
that support the hypotheses.
We model both as abstract arguments,
that is, as something that can be accepted or rejected to a certain degree
by a legal
decision maker like a judge, the jury or a lawyer.
To begin with, we introduce
three meta hypotheses that we model by
three arguments
$\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ (the defendant should be declared guilty because of the inculpatory evidence),
$\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ (the defendant should be declared innocent because of the exculpatory evidence) and
$\ensuremath{\mathrm{Innocence}}$ (the defendant is innocent).
We regard $\ensuremath{\mathrm{Innocence}}$ as the ultimate hypothesis
that is to be decided within the trial.
In general, it may be necessary to consider several
ultimate hypotheses that may correspond to different qualitative
degrees of legal liability (e.g. intent vs. accident vs. innocent).
If necessary, these can be incorporated by adding additional ultimate
hypotheses in an analogous way.
$\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ and $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ are supposed to merge
hypotheses and pieces of evidence that speak
against ($\ensuremath{\mathrm{E_{\mathrm{inc}}}}$) or for ($\ensuremath{\mathrm{E_{\mathrm{ex}}}}$)
the defendant's innocence
as illustrated in Figure \ref{fig:metagraph}.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{metagraph.png}
\caption{Meta-Graph for our Legal Reasoning Framework.}
\label{fig:metagraph}
\end{figure}
Support relationships are indicated by a plus and attack relationships by a minus sign.
There can also be attack and support relationships between
pieces of evidence and additional hypotheses.
Intuitively, as our belief in $\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ increases, our belief in
$\ensuremath{\mathrm{Innocence}}$ should decrease.
As our belief in $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ increases,
our belief in $\ensuremath{\mathrm{Innocence}}$ should increase.
From a classical perspective, accepting $\ensuremath{\mathrm{E_{\mathrm{inc}}}}$,
should result in rejecting $\ensuremath{\mathrm{Innocence}}$ and
accepting $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$,
should result in accepting $\ensuremath{\mathrm{Innocence}}$.
In particular, we should not accept
$\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ and $\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ at the same time.
Of course, in general, both the inculpatory
evidence and the exculpatory evidence can be convincing to a certain degree.
Probabilities are one natural way to capture this uncertainty.
Intuitively, our basic framework is based on the
following assumptions that
we will make precise in the subsequent definition.
\begin{description}
\item[Inculpatory Evidence (IE):] The belief in $\ensuremath{\mathrm{Innocence}}$ is bounded
from above by the belief in $\ensuremath{\mathrm{E_{\mathrm{inc}}}}$.
\item[Exculpatory Evidence (EE):] The belief in $\ensuremath{\mathrm{Innocence}}$ is bounded
from below by the belief in $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$.
\item[Supporting Evidence (SE):] The belief in $\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ and $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ is bounded from
below by the belief in their supporting pieces of
evidence.
\item[Presumption of Innocence (PI):] The belief in $\ensuremath{\mathrm{Innocence}}$
is the maximum belief that is consistent with all assumptions.
\end{description}
The following definition gives a more formal description
of our framework. Our four main assumptions
are formalized in items 4 and 5.
\begin{definition}[Basic Legal Argumentation Framework (BLAF)]
\label{def_blaf}
A BLAF is a quadruple $(\ensuremath{\mathcal{A}}, \ensuremath{\mathcal{E}}, \ensuremath{\operatorname{w}}, \ensuremath{\mathcal{C}})$,
where $\ensuremath{\mathcal{A}}$ is a finite set of arguments,
$\ensuremath{\mathcal{E}}$ is a finite set of directed edges between the arguments,
$\ensuremath{\operatorname{w}}: \ensuremath{\mathcal{E}} \rightarrow \mathbb{Q}$ is a weighting function
and $\ensuremath{\mathcal{C}}$ is a set of linear atomic constraints over $\ensuremath{\mathcal{A}}$
such that:
\begin{enumerate}
\item $\ensuremath{\mathcal{A}} = \ensuremath{\args_M} \uplus \ensuremath{\args_S} \uplus \ensuremath{\args_E}$
is partitioned into a set of \emph{meta-hypotheses}
$\ensuremath{\args_M} = \{\ensuremath{\mathrm{Innocence}}, \ensuremath{\mathrm{E_{\mathrm{inc}}}}, \ensuremath{\mathrm{E_{\mathrm{ex}}}}\},$
a set of \emph{sub-hypotheses} $\ensuremath{\args_S}$
and a set of \emph{pieces of evidence} $\ensuremath{\args_E}$.
\item $\ensuremath{\mathcal{E}} = \ensuremath{\edges_M} \uplus \ensuremath{\edges_S} \uplus \ensuremath{\edges_E}$
is partitioned into a set of \emph{meta edges}
$\ensuremath{\edges_M} = \{
(\ensuremath{\mathrm{E_{\mathrm{inc}}}}, \ensuremath{\mathrm{Innocence}}),
(\ensuremath{\mathrm{E_{\mathrm{ex}}}}, \ensuremath{\mathrm{Innocence}})\}$, a set of \emph{support edges}
$\ensuremath{\edges_S} \subseteq (\ensuremath{\args_S} \cup \ensuremath{\args_E}) \times \{\ensuremath{\mathrm{E_{\mathrm{inc}}}}, \ensuremath{\mathrm{E_{\mathrm{ex}}}}\}$
and a set of \emph{evidential edges}
$\ensuremath{\edges_E} \subseteq (\ensuremath{\args_S} \cup \ensuremath{\args_E}) \times (\ensuremath{\args_S} \cup \ensuremath{\args_E})$.
\item $\ensuremath{\operatorname{w}}((\ensuremath{\mathrm{E_{\mathrm{inc}}}}, \ensuremath{\mathrm{Innocence}}))=-1$ and
$\ensuremath{\operatorname{w}}((\ensuremath{\mathrm{E_{\mathrm{ex}}}}, \ensuremath{\mathrm{Innocence}}))=1$.
Furthermore, $0 \leq \ensuremath{\operatorname{w}}(e) \leq 1$ for all $e \in \ensuremath{\edges_S}$
\item $\ensuremath{\mathcal{C}}$ contains at least the following constraints:
\begin{description}
\item[IE:] $\pi(\ensuremath{\mathrm{Innocence}}) \leq 1 + \ensuremath{\operatorname{w}}((\ensuremath{\mathrm{E_{\mathrm{inc}}}}, \ensuremath{\mathrm{Innocence}})) \cdot \pi(\ensuremath{\mathrm{E_{\mathrm{inc}}}})$,
\item[EE:] $ \ensuremath{\operatorname{w}}((\ensuremath{\mathrm{E_{\mathrm{ex}}}}, \ensuremath{\mathrm{Innocence}})) \cdot \pi(\ensuremath{\mathrm{E_{\mathrm{ex}}}}) \leq \pi(\ensuremath{\mathrm{Innocence}})$,
\item[SE:] $\ensuremath{\operatorname{w}}((E,H)) \cdot \pi(E) \leq \pi(H)$ for all $(E,H) \in \ensuremath{\edges_S}$.
\end{description}
\item For all $A \in \ensuremath{\mathcal{A}}$, we call $\ensuremath{\underline{\belief}}(A) = \min_{P \models \ensuremath{\mathcal{C}}} P(A)$
the \emph{lower belief in $A$} and $\ensuremath{\overline{\belief}}(A) = \max_{P \models \ensuremath{\mathcal{C}}} P(A)$
the \emph{upper belief in $A$}. The \emph{belief in $\ensuremath{\mathrm{Innocence}}$} in is defined as
\begin{equation*}
PI: \ensuremath{\mathcal{B}}(\ensuremath{\mathrm{Innocence}}) = \ensuremath{\overline{\belief}}(A).
\end{equation*}
and the \emph{belief} in the remaining $A \in \ensuremath{\mathcal{A}} \setminus \{\ensuremath{\mathrm{Innocence}}\}$ is the interval
$\ensuremath{\mathcal{B}}(A) = [\ensuremath{\underline{\belief}}(A), \ensuremath{\overline{\belief}}(A)]$.
\end{enumerate}
\end{definition}
Items 1-3 basically give a more precise description of the graph illustrated in
Figure \ref{fig:metagraph}. Item 4 encodes our first three main assumptions as linear
atomic constraints.
The general form of our basic constraints is
$\pi(B) \leq 1 + w((A,B)) \cdot P(A)$ for attack relations $(A,B)$
(note that for $w((A,B)) = -1$, this is just the coherence constraint from \cite{HunterT16})
and $w((A,B)) \cdot \pi(A) \leq \pi(B)$ for support relations.
Intuitively, attacker bound beliefs from above and supporter bound beliefs
from below.
Item 5 defines lower and upper beliefs in arguments as the minimal
and maximal probabilities that are consistent with our constraints.
Following our fourth assumption (presumption of innocence), the belief in $\ensuremath{\mathrm{Innocence}}$ is defined
by the upper bound. The beliefs in the remaining arguments is the interval defined by the lower
and upper bound.
The following proposition summarizes some
consequences of our basic assumptions.
\begin{proposition}
\label{prop_basic_beliefs}
For every BLAF $(\ensuremath{\mathcal{A}}, \ensuremath{\mathcal{E}}, \ensuremath{\operatorname{w}}, \ensuremath{\mathcal{C}})$, we have
\begin{enumerate}
\item $\ensuremath{\overline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{inc}}}}) \leq 1 - \ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{ex}}}})$ and $\ensuremath{\overline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{ex}}}}) \leq 1 - \ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{inc}}}})$.
\item For all support edges $(a, E) \in \ensuremath{\edges_S}$, we have
\begin{itemize}
\item $\ensuremath{\overline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{ex}}}}) \leq 1 - \ensuremath{\operatorname{w}}((a,\ensuremath{\mathrm{E_{\mathrm{inc}}}})) \cdot \ensuremath{\underline{\belief}}(a)$ if $E = \ensuremath{\mathrm{E_{\mathrm{inc}}}}$,
\item $\ensuremath{\overline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{inc}}}}) \leq 1 - \ensuremath{\operatorname{w}}((a,\ensuremath{\mathrm{E_{\mathrm{ex}}}})) \cdot \ensuremath{\underline{\belief}}(a)$ if $E = \ensuremath{\mathrm{E_{\mathrm{ex}}}}$.
\end{itemize}
\end{enumerate}
\end{proposition}
\begin{proof}
1. We prove only the first statement, the second one follows analogously.
Consider an arbitrary $P \in \ensuremath{\mathcal{P}_\args}$ that satisfies $\ensuremath{\mathcal{C}}$.
Then $P(\ensuremath{\mathrm{E_{\mathrm{inc}}}}) \leq P(\ensuremath{\mathrm{Innocence}})
\leq 1 - P(\ensuremath{\mathrm{E_{\mathrm{ex}}}}) \leq 1 - \ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{ex}}}})$. The first inequality
follows from EE and the second from IE (Def. \ref{def_blaf}, item 4) along with the
conditions on $\ensuremath{\operatorname{w}}$ (Def. \ref{def_blaf}, item 3). The third inequality follows
because $\ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{ex}}}}) \leq P(\ensuremath{\mathrm{E_{\mathrm{ex}}}})$ by definition of $\ensuremath{\underline{\belief}}$.
2. Again, we prove only the first statement.
Note that SE (Def. \ref{def_blaf}, item 4) implies
$P(\ensuremath{\mathrm{E_{\mathrm{inc}}}}) \geq \ensuremath{\operatorname{w}}((a,\ensuremath{\mathrm{E_{\mathrm{inc}}}})) \cdot P(a)$
for all $P \in \ensuremath{\mathcal{P}_\args}$ that satisfy $\ensuremath{\mathcal{C}}$.
Therefore, $P(\ensuremath{\mathrm{E_{\mathrm{ex}}}}) \leq 1 - P(\ensuremath{\mathrm{E_{\mathrm{inc}}}})
\leq 1 - \ensuremath{\operatorname{w}}((a,\ensuremath{\mathrm{E_{\mathrm{inc}}}})) \cdot P(a) \leq 1 - \ensuremath{\operatorname{w}}((a,\ensuremath{\mathrm{E_{\mathrm{inc}}}})) \cdot \ensuremath{\underline{\belief}}(a) $, where the first and third inequalities can be derived like in 1.
\end{proof}
Intuitively, item 1 says that our upper belief that the defendant should be declared guilty
because of the inculpatory evidence is bounded from above by our lower belief that the defendant
should be declared innocent because of the exculpatory evidence and vice versa.
By rearranging the equations, we can see that the lower belief in $\ensuremath{\mathrm{E_{\mathrm{inc}}}}$
is also bounded from above by the upper belief in $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ and vice versa.
Item 2 explains that every argument $a$ that directly contributes to inculpatory (exculpatory) evidence $E$
gives an upper bound for the belief in $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ ($\ensuremath{\mathrm{E_{\mathrm{inc}}}}$) that is based on our lower
belief $\ensuremath{\underline{\belief}}(a)$ and the relevance $\ensuremath{\operatorname{w}}((a,E))$ of this argument.
In a similar way, we could bound the beliefs in contributors to $\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ by the belief in
contributors to $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ by taking their respective weights into account.
However, the general description becomes more and more difficult
to comprehend. Therefore, we just illustrate the interactions by means of a simple example.
\begin{example}
\label{example_blaf}
Let us consider a simple case of hit-and-run driving. The defendant is accused of having struck a car while parking at a shopping center. The plaintiff witnessed the accident from afar and denoted the registration number from the licence plate when the car left ($T_1$). The defendant denies the crime and testified that he was at home with his girlfriend at the time of the offence ($T_2$). His girlfriend confirmed his alibi ($T_3$). However, a security camera at the parking space recorded a person that bears strong resemblance to the defendant at the time of the crime ($E_1$). We consider a simple formalization shown in Figure \ref{fig:example_blaf}.
\begin{figure}[tb]
\centering
\includegraphics[width=0.42\textwidth]{example_blaf.png}
\caption{BLAF for Example \ref{example_blaf}.}
\label{fig:example_blaf}
\end{figure}
We designed the graph in a way that allows illustrating the interactions in our framework.
One may also want to regard $T_3$ as a supporter of exculpatory evidence and consider attack relationships between $E_1$ and
$T_1$ and $T_3$. We do not introduce such edges because we want to illustrate the indirect interactions
between arguments. In this example, we may weigh all edges with $1$ and control the uncertainty only about
the degrees of belief. However, we assign a weight of $0.9$ to the edge from $T_1$ in order to illustrate the effect
of the weight. This may capture the uncertainty that the plaintiff may have written down the wrong
registration number, for example.
The probability for $T_1$, $T_2$ and $T_3$ is our degree of belief that the corresponding testimonies are true. The probability of $E_1$ is our degree of belief that the camera does indeed show the
defendant and not just another person.
Without additional assumptions, we can only derive that our degree of belief in $\ensuremath{\mathrm{Innocence}}$ is
$1$ (presumption of innocence) as shown in the second column ($\ensuremath{\mathcal{B}}_1$) of Table \ref{tab:example_blaf}.
\def1.2{1.2}
\begin{table}
\centering
\begin{tabular}{l>{\raggedleft}p{0.9cm}>{\raggedleft}p{1.2cm}>{\raggedleft \arraybackslash}p{1.2cm}}
$\ensuremath{\mathcal{A}}$ & $\ensuremath{\mathcal{B}}_1$ & $\ensuremath{\mathcal{B}}_2$ & $\ensuremath{\mathcal{B}}_3$\\
\hline
$\ensuremath{\mathrm{Innocence}}$ & 1 & 1 & 0.1 \\
$\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ & [0, 1] & [0, 0.3] & [0.9, 1] \\
$\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ & [0, 1] & [0.7, 1] & [0, 0.1] \\
$T_1$ & [0, 1] & [0, 0.33] & [0, 1] \\
$T_2$ & [0, 1] & [0.7, 1] & [0, 0.1] \\
$T_3$ & [0, 1] & [\textbf{0.7}, 1] & [0, 0.1] \\
$E_1$ & [0, 1] & [0, 0.3] & [\textbf{0.9}, 1]
\end{tabular}
\caption{Beliefs under additional assumptions for Example \ref{example_blaf} (rounded to two digits). Directly constrained beliefs are highlighted in bold.}
\label{tab:example_blaf}
\end{table}
We could now start adding assumptions and looking at the consequences. For example, let us assume
that the statement of the defendant's girlfriend was very convincing. We could incorporate this by
adding the constraint $\pi(T_3) \geq 0.7$. The consequences are shown in the third column ($\ensuremath{\mathcal{B}}_2$) of Table \ref{tab:example_blaf}. However, if the person on the camera bears strong resemblance to the
defendant, we may find that the upper belief in $E_1$ is too low. This means that our assumption
is too strong and needs to be revised. Let us just delete the constraint $\pi(T_3) \geq 0.7$ and instead impose a
constraint on $E_1$. Let us assume that there is hardly any doubt that the camera shows the
defendant. We could incorporate this by adding the constraint $\pi(E_1) \geq 0.9$.
The consequences are shown in the fourth column ($\ensuremath{\mathcal{B}}_3$) of Table \ref{tab:example_blaf}.
\end{example}
The choice of probabilities (degrees of belief), weights (relevance) and additional attack or support relations is, of course, subjective. However, arguably, every court decision is subjective in that
the decision maker(s) have to weigh the plausibility and the relevance of the evidence in one way
or another. By making these assumptions explicit in a formal framework, the decision process can become more transparent. Furthermore, by computing probabilities while adding assumptions, possible inconsistencies can be detected and resolved early. Since we restrict to
linear atomic constraints, computing probabilities can be done within a second even when there are thousands of arguments.
Let us note that our framework also allows defining some simple rules that allow deriving
explanations for the verdict automatically.
For example, the belief in $\ensuremath{\mathrm{Innocence}}$ can be explained directly from the beliefs in
$\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ and $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$.
If both $\ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{inc}}}}) \leq 0.5$ and $\ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{ex}}}}) \leq 0.5$. our system may report
that the defendant is found innocent because of lack of evidence.
If $\ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{ex}}}}) > 0.5$, it could report that the defendant is found innocent because
the exculpatory evidence is more plausible than the inculpatory evidence (recall from Proposition
\ref{prop_basic_beliefs} that $\ensuremath{\overline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{inc}}}}) \leq 1 - \ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{ex}}}})$).
Finally, if $\ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{inc}}}})$ is sufficiently large, it could report that the defendant
is found guilty because of the inculpatory evidence.
The belief in $\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ and $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ can then be further explained
based on the belief in supporting hypotheses and pieces of evidence.
The influence of supporting arguments can be measured by their lower belief bounds and their weight.
To illustrate this, consider again Table \ref{tab:example_blaf}.
For $\ensuremath{\mathcal{B}}_1$, the system could report that the defendant is innocent because of lack of
convincing evidence,
while, for $\ensuremath{\mathcal{B}}_2$, it can explain that there is convincing exculpatory evidence. If desired, it
can then further report $T_2$ as the direct explanation and, going backwards, $T_3$ as an additional
explanation.
Similarly, for $\ensuremath{\mathcal{B}}_3$, the system could report that the defendant is probably not innocent because
of the inculpatory evidence. Again, the system could give further explanations by going backwards in
the graph. We will discuss the idea in more general form in Section 5.
\section{Adding Additional Structure to BLAFs}
BLAFs can capture a wide variety of cases. However, it is often desirable to add additional structure that captures recurring patterns in legal reasoning.
From a usability perspective, this makes the graph more easily comprehensible and allows modeling different cases in a consistent and standardized way.
From an automated reasoning perspective, it allows adding additional general rules that can automatically derive explanations for decisions.
Two natural subsets of inculpatory evidence are direct (\ensuremath{\mathrm{E_{\mathrm{d}}}}) and circumstantial (\ensuremath{\mathrm{E_{\mathrm{c}}}}) inculpatory evidence. While direct evidence provides direct inculpatory evidence, circumstantial evidence involves indirect evidence that requires multiple inferential steps \cite{fenton2013general}.
For example, a camera that recorded the defendant while committing the crime can be seen as direct
evidence, while a camera that recorded the defendant close to the crime scene like in Example \ref{example_blaf} can be seen as a piece of circumstantial evidence.
Two prominent categories of circumstantial evidence are \emph{motive}
(the defendant had a reason to commit the crime) and \emph{opportunity}
(the defendant had the opportunity to commit the crime).
Figure \ref{fig:blaf_advanced} shows a refined BLAF.
As indicated by the join of their support edges, the beliefs
in pieces of circumstantial evidence are merged and not considered independently.
Only if both a motive and the opportunity
(and perhaps some additional conditions)
were present, the defendant should be found guilty.
In contrast, pieces of direct evidence are standalone
arguments for the defendant's guilt.
Two recurring patterns of exculpatory evidence are \emph{alibi} and \emph{ability}. While an alibi indicates that the
defendant has not been at the crime scene at the time of the crime, \emph{ability} can contain
pieces of evidence that indicate that the defendant could not have committed the crime, for example,
due to lack of physical strength. Figure \ref{fig:blaf_advanced} shows an extended BLAF with six additional meta-hypotheses.
\begin{figure}[tb]
\centering
\includegraphics[width=0.43\textwidth]{metagraph_advanced.png}
\caption{Refined BLAF with additional meta-hypotheses.}
\label{fig:blaf_advanced}
\end{figure}
As before, we allow edges between all pieces of evidence and subhypotheses, but do not draw all possible direct connections in order to keep the graph comprehensible.
The meaning of the support edges pointing to inculpatory and exculpatory evidence is already defined
by SE in Definition \ref{def_blaf}, item 4. That is the corresponding support relations $(A,B)$
are associated with the constraint $w((A,B)) \cdot \pi(A) \leq \pi(B)$.
This constraint could also be naturally used for the evidential edges that point to direct evidence, alibi and ability.
However, the circumstantial evidence patterns motive and opportunity should not act independently, but complement each other. Neither a motive, nor the opportunity alone, are a good reason to
find the defendant guilty. However, if both a good motive and the opportunity are present, this
may be a good reason. We say that both items together provide \emph{collective support} for the guilt of the defendant. To formalize \emph{collective support}, we can consider a constraint
$\ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Motive}}, \ensuremath{\mathrm{E_{\mathrm{c}}}})) \cdot \pi(\ensuremath{\mathrm{Motive}}) +
\ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Opportunity}}, \ensuremath{\mathrm{E_{\mathrm{c}}}})) \cdot \pi(\ensuremath{\mathrm{Opportunity}})
\leq \pi(\ensuremath{\mathrm{E_{\mathrm{c}}}})$
such that $\ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Motive}}, \ensuremath{\mathrm{E_{\mathrm{c}}}})) + \ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Opportunity}}, \ensuremath{\mathrm{E_{\mathrm{c}}}})) \leq 1$.
For example, we could set $\ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Motive}}, \ensuremath{\mathrm{E_{\mathrm{c}}}})) = \ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Opportunity}}, \ensuremath{\mathrm{E_{\mathrm{c}}}})) = 0.4$.
Then the presence of a strong motive or the opportunity alone cannot decrease the belief in the defendant's
innocent by more than $0.4$ and both together cannot decrease the belief by more than $0.8$.
Opportunity is indeed considered a necessary requirement
for the defendant's guilt in the legal reasoning literature
and motive is, at least, widely accepted as such \cite{fenton2013general}.
Collective support is an interesting pattern in general, so that
we give a more general definition here.
Given arguments $A_1, \dots, A_n$ (pieces of evidence or sub-hypotheses) that support another argument $B$
such that $\sum_{i=1}^n \ensuremath{\operatorname{w}}((A_i,B)) \leq 1$, the \emph{collective support constraint} is defined as
\begin{description}
\item[CS:] $\sum_{i=1}^n \ensuremath{\operatorname{w}}((A_i,B)) \cdot \pi(A_i) \leq \pi(B)$.
\end{description}
The following example illustrates how the additional structure
can be applied.
\begin{example}
\label{example_blaf_extended}
Let us consider a simple robbery case.
The defendant $D$ is accused of having robbed
the victim $V$. The extended BLAF is shown in Figure \ref{fig:example_blaf_extended}.
\begin{figure}[tb]
\centering
\includegraphics[width=0.45\textwidth]{example_blaf_extended.png}
\caption{Extended BLAF for Example \ref{example_blaf_extended}.}
\label{fig:example_blaf_extended}
\end{figure}
Before the crime, $D$ and $V$ met in a bar and had a fight
about money that $V$ owed $D$.
$V$ testified that $D$ threatened to get the money one way or another ($V_1$).
$D$ acknowledged the fight, but denied the threat ($D_1$).
While D's testimony still contains a motive for the crime,
it is now significantly weaker. This can be reflected in the weights.
We could consider a more fine-grained view distinguishing the fight
and the threat and add an attack between the contradicting statements,
but in order to keep things simple, we refrain from doing so.
$V$ testified that he got robbed at 23:30 by a masked person
and that he recognized the defendant based on his voice and
stature ($V_2$). This can be seen as direct evidence for the crime, but since the accused is of average stature, it should have only a small weight.
A waiter working at the bar testified that the defendant left the
bar at about 23:00 ($W_1$). This may have allowed the defendant
hypothetically to commit the crime, but he could have went anywhere, so the weight
should be again low.
The defendant testified that he went to the movie theater
and watched a movie that started at 23:15 ($D_2$).
If true, this is a strong alibi and should therefore have a large weight.
An employee at the movie theater testified that the defendant
is a frequent guest and that he recalled him buying a drink ($E_1$).
However, he did not recall the exact time. So the alibi is somewhat weak
and should not have too much weight.
We weigh $\ensuremath{\mathrm{Motive}}$ and $\ensuremath{\mathrm{Opportunity}}$ equally with
$\ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Motive}}, \ensuremath{\mathrm{Innocence}})) = \ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Opportunity}}, \ensuremath{\mathrm{Innocence}})) = 0.3$.
The influence of the belief in motive and opportunity
on circumstantial evidence is defined by the collective support constraint that we described above.
All evidential edges $(E,A)$ that originate from a piece of evidence $E$ are associated with the constraint
$w((E,A)) \cdot \pi(E) \leq \pi(A)$.
Figure \ref{fig:example_blaf_extended} shows
the final graph structure and edge weights.
\def1.2{1.2}
\begin{table}
\centering
\resizebox{\linewidth}{!}{
\begin{tabular} {l>{\raggedleft}p{0.09\linewidth}>{\raggedleft}p{0.175\linewidth}>{\raggedleft}p{0.175\linewidth}>{\raggedleft\arraybackslash}p{0.175\linewidth}}
$\ensuremath{\mathcal{A}}$ & Basic & $W1$, $E1$ & $W1$, $E1$, $D1$ & $W1$, $E1$, $D1$, $V2$ \\
\hline
$\ensuremath{\mathrm{Innocence}}$ & [0, 1] & 0.94 & 0.91 & 0.8 \\
$\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ & [0, 1] & [0.06, 0.7] & [0.09, 0.7] & [0.2, 7] \\
$\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ & [0, 1] & [0.3, 0.94] & [0.3, 0.91] & [0.3, 0.8] \\
$\ensuremath{\mathrm{E_{\mathrm{c}}}}$ & [0, 1] & [0.06, 0.7] & [0.09, 0.7] & [0.09, 0.7]\\
$\ensuremath{\mathrm{E_{\mathrm{d}}}}$ & [0, 1] & [0, 0.7] & [0, 0.7] & [0.2, 0.7]\\
$\ensuremath{\mathrm{Alibi}}$ & [0, 1] & [0.3, 0.94] & [0.3, 0.91] &[0.3, 0.8]\\
$\ensuremath{\mathrm{Ability}}$ & [0, 1] & [0, 0.94] & [0, 0.91] & [0, 0.8]\\
$\ensuremath{\mathrm{Motive}}$ & [0, 1] & [0, 1] & [0.1, 1] & [0.1, 1]\\
$\ensuremath{\mathrm{Opportunity}}$ & [0, 1] & [0.2, 1] & [0.2, 1] & [0.2, 1]\\
$V1$ & [0, 1] & [0, 1] & [0, 1] & [0, 1]\\
$V2$ & [0, 1] & [0, 1] & [0, 1] & \textbf{1}\\
$D1$ & [0, 1] & [0, 1] & \textbf{1} & \textbf{1} \\
$D2$ & [0, 1] & [0.3, 1] & [0.3, 1] & [0.3, 0.89] \\
$W1$ & [0, 1] & \textbf{1} & \textbf{1}& \textbf{1}\\
$E1$ & [0, 1] & \textbf{1}& \textbf{1} &\textbf{1}
\end{tabular}}
\caption{Belief in $\ensuremath{\mathrm{Innocence}}$ and entailment results
under additional assumptions for Example \ref{example_blaf_extended} (rounded to two digits). Directly constrained beliefs are highlighted in bold.}
\label{tab:example_blaf_extended}
\end{table}
Having defined the structure of the graph and the meaning of the edges, we can start to assign beliefs to pieces of evidence.
Again, without making any assumptions about the beliefs, we can only infer that the degree of belief in $\ensuremath{\mathrm{Innocence}}$ is 1. This is shown in the second column of Table \ref{tab:example_blaf_extended}.
To begin with, we assume that the testimonies given by the cinema employee and the waiter of the bar are true ($\pi(E1) = 1, \pi(W1) = 1$). The third column of Table \ref{tab:example_blaf_extended} shows the consequences of
these assumptions.
We can see, for example, that the alibi $E1$ provides a lower bound for the belief in the exculpatory evidence and thus an upper bound for the beliefs in the inculpatory evidence and the related hypotheses.
It seems also safe to assume that the defendant did not lie about his participation in the fight, so we the constraint $\pi(D1) = 1$ next. The fourth column in Table \ref{tab:example_blaf_extended} shows the resulting belief intervals. The new support for motive adds to the support of the circumstantial evidence and the lower bound on the belief in the inculpatory evidence is raised. This lowers the belief in the innocence of the accused slightly. Note again that it also decreases the upper bound on the belief in exculpatory evidence indirectly.
Finally, let us assume that the defendant does not lie about having
recognized the defendant ($\pi(V2) = 1$) (recall that the uncertainty about
the recognition reliability is incorporated in the edge weight).
The fifth column in Table \ref{tab:example_blaf_extended} shows the new
beliefs. We can see that the belief in the defendant's innocence decreases
significantly. If we notice that a larger or smaller change is more plausible,
we could take account of this by adapting the edge weight.
In this way, legal cases can be analyzed in a systematic way and the
plausibility of assumptions can be checked on the fly by looking at their
ramifications.
\end{example}
In addition to the previously introduced additional categories of meta-hypotheses, another recurring pattern in legal cases are mutually
dependent pieces of evidence. One way to model this in our framework,
is to define a meta-argument that is influenced by the dependent pieces of evidence. The collective support constraint CS is well suited to capture this relationship accurately. We illustrate this with an example from \cite[pp.82-84]{fenton2013general}.
\begin{example}
Let us assume that a person was recorded by two video cameras from
different perspectives at a crime scene. If the person is the defendant,
the defendant should resemble the person on both images.
In the BLAF, we can incorporate the two camera observations as pieces of evidence $\ensuremath{\mathrm{Camera1}}, \ensuremath{\mathrm{Camera2}}$ supporting a meta-hypothesis $\ensuremath{\mathrm{Camera}}$
that says that the defendant was at the crime scene because of camera evidence. Note that if we use the SE constraint for the evidential edges
from $\ensuremath{\mathrm{Camera1}}, \ensuremath{\mathrm{Camera2}}$, each of the two cameras would independently determine a lower bound for $\ensuremath{\mathrm{Camera}}$ which seems to strong in this example.
Instead, we can use the CS constraint that we already used to capture the relationship between opportunity and motive.
In this example, the CS constraint becomes $\ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Camera1}}, \ensuremath{\mathrm{Camera}})) \cdot \pi(\ensuremath{\mathrm{Camera1}}) + \ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Camera2}}, \ensuremath{\mathrm{Camera}})) \cdot \pi(\ensuremath{\mathrm{Camera2}})\leq \pi(\ensuremath{\mathrm{Camera}})$,
where $\ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Camera1}}, \ensuremath{\mathrm{Camera}})) + \ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Camera2}}, \ensuremath{\mathrm{Camera}})) \leq 1$.
For example both camera weights could be set to $\ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Camera1}}, \ensuremath{\mathrm{Camera}})) = \ensuremath{\operatorname{w}}((\ensuremath{\mathrm{Camera2}}, \ensuremath{\mathrm{Camera}})) = 0.5$ to give equal relevance to both.
Then, if the person resembles the defendant only from one perspective, say we have $\pi(\ensuremath{\mathrm{Camera1}})=1$ and $\pi(\ensuremath{\mathrm{Camera2}})=0$, the induced lower bound on the belief in $\ensuremath{\mathrm{Camera}}$ will be only $0.5$. Only if the belief in
both cameras is larger than $0.5$, the lower bound can be larger than $0.5$. For example, if we have $\pi(\ensuremath{\mathrm{Camera1}})=0.7$ and $\pi(\ensuremath{\mathrm{Camera2}})=0.9$, the induced lower
bound is $0.8$.
\end{example}
\section{Automated Explanation Generation}
\label{sec:explanation}
As we already illustrated at the end of Section 3,
the structure of our framework allows generating
explanations for decisions automatically.
In general, explaining the meta-hypotheses $\ensuremath{\mathrm{Innocence}}, \ensuremath{\mathrm{E_{\mathrm{inc}}}}$ and $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ is
easier than explaining the beliefs in other arguments because of their restricted
form.
Note first that the only direct neighbors of $\ensuremath{\mathrm{Innocence}}$ are
$\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ and $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ and we know that
$\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ is an attacker and $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ is a supporter.
Therefore, we can basically distinguish three cases
that we already described at the end of Section 3.
\begin{enumerate}
\item $\ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{inc}}}}) \leq T$ and $\ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{ex}}}}) \leq 0.5$: The defendant is found
innocent due to lack of evidence.
\item $\ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{ex}}}}) > 0.5$: the defendant is found innocent because
the exculpatory evidence is more plausible than the inculpatory evidence.
\item $\ensuremath{\underline{\belief}}(\ensuremath{\mathrm{E_{\mathrm{inc}}}}) > T$: the defendant
is found guilty because of the inculpatory evidence.
\end{enumerate}
Here, $T$ is a threshold that should usually be chosen from the open interval $(0.5, 1)$. $0.5$ is sometimes
regarded as the acceptance threshold, but in a legal setting,
it may be more appropriate to choose a larger threshold
like $T=0.75$.
After having received a high-level explanation
of the verdict, the user may be interested in more
details and ask for reasons that explain the
plausibility of inculpatory or exculpatory
evidence.
Explaining $\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ and $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ is more complicated already because
we have an unknown number of neighbors in the graph
now. However, the only neighbors can be supporters
(parents) and $\ensuremath{\mathrm{Innocence}}$ (child).
By Definition \ref{def_blaf}, item 4,
their meaning is encoded by the \emph{SE}-constraint.
Assuming that the user did not add additional
constraints about the relationships between
$\ensuremath{\mathrm{E_{\mathrm{inc}}}}$, $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ and $\ensuremath{\mathrm{Innocence}}$, we
can again define some simple rules.
If additional constraints on $\ensuremath{\mathrm{E_{\mathrm{inc}}}}$ and $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ are desirable, these rules may need
to be refined, of course.
Otherwise, we can distinguish two cases.
If the user asks for an explanation for the lower belief, we can reason as follows:
a non-trivial lower bound ($>0$) can only result from a supporter with non-trivial lower bound.
So in this case, we can go through the supporters,
collect those supporters that induce the maximum
lower bound and report them as an explanation.
The user may also ask for an explanation for the upper belief. A non-trivial upper bound ($<1$) can only result
from a non-trivial bound on the belief in $\ensuremath{\mathrm{Innocence}}$.
Let us assume that we want to explain a non-trivial
upper bound on $\ensuremath{\mathrm{E_{\mathrm{inc}}}}$. From
the \emph{IE}-constraint in Definition \ref{def_blaf}, item 4, we can see that this must be caused by
a non-trivial lower bound on $\ensuremath{\mathrm{Innocence}}$.
This lower bound, in turn, must be caused by a
non-trivial lower bound on $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ by our
assumptions.
We could now report the lower bound on $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$
as an explanation. A more meaningful explanation
would be obtained by also explaining the lower bound on
$\ensuremath{\mathrm{E_{\mathrm{ex}}}}$. This can be done as explained before
by looking at the supporters of $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$.
A non-trivial upper bound on $\ensuremath{\mathrm{E_{\mathrm{ex}}}}$ can be
explained in a symmetrical manner.
Generating automatic explanations for the remaining
sub-hypotheses and pieces of evidence is most
challenging, but can be done as long as we can make
assumptions about the constraints that are involved.
For example, often the \emph{SE}-constraint gives
a natural meaning to support edges and the weighted \emph{Coherence} constraint gives a natural meaning
to attack edges. Intuitively, they cause a lower/upper
bound on the belief in an argument based on their own
lower belief.
If these are the only constraints that are employed,
explanations for lower bounds can again be generated by
collecting the supporters that induce the largest lower bound.
For explaining the upper bound, we now have to consider
two factors. The first factor are attackers with a non-trivial lower bound. The second factor are
other arguments that are supported and have a non-trivial upper bound
(then a too large belief in the supporting argument would cause an inconsistency).
Therefore, we do not only collect the attacking
arguments that induce the largest lower bound, but
we also collect supported arguments. We can order the
supported arguments by their upper belief multiplied
by the weight of the support edge.
If the smallest upper bound from the supported arguments
is $U$
and the largest lower bound from the attacking arguments is $L$,
we report the collected supported arguments as an explanation if $1-U > L$, the collected attacking
arguments as an explanation if $1-U < L$ or
both if it happens that $1-U = L$.
For additional constraints, we may have to refine
these rules again. One important constraint that
we discussed is the \emph{CS}-constraint.
In this case, we have to to treat the supporters involved in this constraint differently since
they all contribute to the induced lower bound.
When collecting supporters for explaining lower bounds
(the supporters are parents),
supporting edges that belong to one \emph{CS}-constraint
have to be considered jointly and not independently.
If they induce a lower bound that is larger than all
lower bounds caused by an \emph{SE}-constraint,
they can be reported collectively as an explanation.
When collecting supporters for explaining upper bounds
(the supporters are children), the reasoning becomes
more complicated because there can be various interactions
between the beliefs in the involved arguments.
We leave an analysis of this case and more general cases for future work.
\section{Related Work}
Our legal reasoning framework allows explicit formalization of uncertainty in legal decision making.
Other knowledge representation and reasoning formalisms have been applied for
this purpose.
Studies of different game-theoretical tools can be found in \cite{prakken1996dialectical,riveret2007success,roth2007strategic}.
\cite{dung2010towards} proposed a probabilistic argumentation framework where the beliefs
of different jurors are represented by individual probability spaces.
Intuitively, the jurors weigh the evidence and decisions can be made based on criteria like
majority voting or belief thresholds.
One particularly popular approach for probabilistic legal reasoning
are Bayesian networks. \cite{fenton2013general} provide a set of idioms used for the construction of Bayesian networks based on legal argument patterns and apply and discuss their framework for a specific case in \cite{fenton2019analyzing}. \cite{timmer2017two} developed an algorithm to extract argumentative information from a Bayesian network with an intermediate structure, a support graph and analyze their approach in a legal case study.
\cite{vlek2016method} propose a method to model different scenarios about crimes with Bayesian networks using scenario scheme idioms and to extract information about the scenario and the quality of the scenario.
Determining the weights and beliefs for the edges and items of evidence poses a problem for our framework as well as for other symbolic approaches. For some items of evidence the weights as well as the probabilities can be elicited based on statistical analysis and forensic evidence \cite{kwan2011sensitivity,fenton2012risk,zhang2016expert}.
To test the robustness of Bayesian networks with respect to minor changes in subjective beliefs, \cite{fenton2013general} propose to apply sensitivity analysis on the nodes in question. In our framework, the impact of subjective beliefs can be analysed in a similar manner, by altering the beliefs which are associated with the evidence or the weights associated with the edges. The automated explanation generation outlined in Section \ref{sec:explanation} can then provide information about the influence that differing beliefs have on hypotheses and sub-hypotheses in the framework. With this the perspective of different agents can be modeled, for example the defense and prosecution perspectives.
The clear structure of argumentation frameworks is well suited for generating explanations
automatically and related explanation ideas have been considered recently in \cite{cocarascu2019extracting,vcyras2019argumentation,zeng2018context}, for example.
In Bayesian networks, inconsistency is usually not an issue
because of the way how they are defined. In contrast,
in our framework, inconsistencies can easily occur.
For example, if a forensic expert judges both the accuracy
of an alibi and the relevance of a direct piece of evidence with $1$,
our constraints become inconsistent. While this may be inconvenient,
this inconsistency is arguably desirable. This is because the modeling
assumptions are inconsistent and this should be recognized and reported
by the system.
If automated merging of the inconsistent beliefs is desirable,
this can be achieved by different tools. One possibility is to
apply inconsistency measures for probabilistic logics in order to evaluate the
severity of conflicts \cite{de2015measuring,potyka2014linear,thimm2013inconsistency}.
In order to determine the sources of the inconsistency and their impact,
Shapley values can be applied \cite{hunter2010measure}.
Alternatively, we could replace our exact probabilistic reasoning algorithms
with inconsistency-tolerant reasoning approaches that resolve inconsistencies
by minimizing conflicts \cite{adamcik2014collective,muino2011measuring,potyka2015probabilistic} or based on priorities \cite{potyka2015reasoning}. This would be more convenient for
the knowledge engineer, but the resulting meaning of the probabilities
becomes less clear.
\section{Conclusions and Future Work}
We proposed a probabilistic abstract argumentation framework for automated reasoning in law
based on probabilistic epistemic argumentation \cite{HunterT16,HunterPT2018Arxiv}.
Our framework is best suited for merging beliefs in pieces of evidence and sub-hypotheses.
Computing an initial degree of belief for particular pieces of evidence based on
forensic evidence can often be better accomplished by applying Bayesian networks
or a conventional statistical analysis. Our framework can then be applied on top
in order to merge the different beliefs in pieces of evidence and subhypotheses
in a transparent and explainable way.
In particular, point probabilities are not required, but imprecise probabilities
in the form of belief intervals are supported as well.
It is also interesting
to note that the worst-case runtime of our framework is polynomial \cite{potyka2019fragment}.
Bayesian networks also have polynomial runtime guarantees in some special cases,
for example, when the Bayesian network structure is a polytree (i.e., it does not contain cycles when ignoring the direction of the edges).
The polynomial runtime in probabilistic epistemic argumentation is guaranteed by
restricting to a fragment of the full language. This fragment is sufficient for
many cases and is all that we used in this work. However, sometimes it may be
necessary to extend the language. For example, instead of talking only about the
probabilities of single pieces of evidence and subhypotheses, we may want to talk
about the probabilities of logical combinations. Similarly, one may want to merge
beliefs not only in a linear, but in a non-linear way. Both extensions are difficult
to deal with, in general. However, it seems worthwhile to study such cases in more
detail in order to identify some other tractable special cases.
Another interesting aspect for future work is extending the automated support tools for
designing and querying our legal argumentation frameworks. As explained in Section 5,
the basic framework can be explained well automatically. However, when beliefs are
merged in more complicated ways like by the collective support constraint, a deeper
analysis is required. We will study explanation generation for collective support
and other interesting merging patterns in more detail in future work.
For the design of the framework, it may also be helpful to generate explanations
for the sources of inconsistency. As explained in the related work section,
a combination of inconsistency measures for probabilistic logics and Shapley values
seems like a promising approach that we will study. It is also interesting
to apply different approaches for inconsistency-tolerant reasoning in order to
avoid inconsistencies altogether. However, while these approaches usually can
give some meaningful analytical guarantees, it is important to study empirically
if these guarantees
are sufficient in order to guarantee meaningful results in legal or other applications.
\bibliographystyle{kr}
|
1,108,101,566,767 | arxiv | \section{Introduction}
A \textit{singular modulus} is the $j$-invariant of an elliptic curve with complex multiplication. Given a singular modulus~$x$ we denote by $\Delta_x$ the discriminant of the associated imaginary quadratic order.
We denote by $h(\Delta)$ the class number of the imaginary quadratic order of discriminant~$\Delta$.
Recall that two singular moduli~$x$ and~$y$ are conjugate over~${\mathbb Q}$ if and only if ${\Delta_x=\Delta_y}$, and that there are $h(\Delta)$ singular moduli of a given discriminant~$\Delta$. In particular, ${[{\mathbb Q}(x):{\mathbb Q}]=h(\Delta_x)}$. For all details see, for instance, \cite[\S~7 and \S~11]{Co13}.
There has been much work on diophantine properties of singular moduli in recent years. In particular, studying algebraic equations where the unknowns are singular moduli \cite{ABP15,BLM17,BLP16} is interesting by virtue of its connection with the André--Oort property for affine space \cite{BMZ13,Ku12,Ku13}.
Pila and Tsimerman~\cite{PT17} proved that for every~$k$ there exists at most finitely many $k$-tuples $(x_1,\ldots, x_k)$ of distinct non-zero singular moduli with the property
``$x_1, \ldots,x_k$ are multiplicatively dependent, but any proper subset of them is multiplicatively independent''. Their argument is fundamentally non-effective.
Riffaut \cite[Th.~1.7]{Ri19} gave an effective (in fact, totally explicit) version of the theorem of Pila and Tsimerman in the case ${k=2}$. In fact, he did more: he classified all cases when ${x^my^n\in {\mathbb Q}^\times}$, where $x,y$ are singular moduli and $m,n$ non-zero integers.
In the present article we obtain an effective result for ${k=3}$. Moreover, like Riffaut did, we prove a stronger statement: we bound explicitly discriminants of singular moduli $x,y,z$ such that ${x^my^nz^r\in {\mathbb Q}^\times}$ for some non-zero integers $m,n,r$. Our bound is as follows.
\begin{theorem}
\label{thpizthree}
Let $x,y,z$ be distinct non-zero singular moduli and ${m,n,r}$ non-zero integers. Assume that ${x^my^nz^r\in {\mathbb Q}^\times}$. Then
$$
\max\{|\Delta_x|,|\Delta_y|,|\Delta_z|\}\le 10^{10}.
$$
\end{theorem}
The special case ${m=n=r}$ has been recently settled by Fowler~\cite{Fo20,Fo21}.
Note that there do exist triples of distinct singular moduli $x,y,z$ such that ${x^my^nz^r\in {\mathbb Q}^\times}$ for some non-zero ${m,n,r\in {\mathbb Z}}$. There are three types of examples.
\begin{description}
\item[``Rational type'']
Take distinct $x,y,z$ such that
$$
h(\Delta_x)=h(\Delta_y)=h(\Delta_z)=1,\qquad \Delta_x,\Delta_y,\Delta_z\ne -3.
$$
In this case ${x,y,z\in {\mathbb Q}^\times}$ and ${x^my^nz^r\in {\mathbb Q}^\times}$ for any choice of $m,n,r$. Pila and Tsimerman \cite[Example 6.2]{PT17} even found an example of ${x^my^nz^r=1}$:
$$
(2^63^3)^{10}(-2^{15})^6(-2^{15}3^3)^{-10}=1,
$$
the corresponding discriminants being $-4$, $-11$ and $-19$.
\item[``Quadratic type'']
Take distinct $x,y,z$ such that
$$
h(\Delta_x)=1,\qquad \Delta_x\ne -3, \qquad \Delta_y=\Delta_z, \qquad h(\Delta_y)=h(\Delta_z)=2.
$$
In this case ${x\in {\mathbb Q}^\times}$ and ${y,z}$ are of degree~$2$, conjugate over~${\mathbb Q}$. Hence ${x^my^nz^n\in {\mathbb Q}^\times}$ for any choice of $m,n$.
\item[``Cubic type'']
Take distinct $x,y,z$ such that
$$
\Delta_x=\Delta_y=\Delta_z,\qquad h(\Delta_x)=h(\Delta_y)=h(\Delta_z)=3.
$$
In this case ${x,y,z}$ are of degree~$3$, forming a full Galois orbit over~${\mathbb Q}$. Hence ${xyz\in {\mathbb Q}^\times}$.
\end{description}
We believe that, up to permuting $x,y,z$, there are no other examples, but to justify it, one needs to improve on the numerical bound $10^{10}$ in Theorem~\ref{thpizthree}.
The proof of Theorem~\ref{thpizthree} relies on the following result, which is a
partial common generalization (for big discriminants) of \cite[Theorem~1.7]{Ri19} and \cite[Theorem~1.3]{FR18}.
\begin{theorem}
\label{thprimel}
Let $x,y$ be distinct non-zero singular moduli and $m,n$ non-zero integers. Assume that
\begin{equation}\label{condthm}
\max\{|\Delta_x|,|\Delta_y|\}\ge 10^8.
\end{equation}
Then ${[{\mathbb Q}(x,y):{\mathbb Q}(x^my^n)]\le 2}$. More precisely, we have either
\begin{equation}
\label{esamefield}
{\mathbb Q}(x^my^n)={\mathbb Q}(x,y)
\end{equation}
or
\begin{equation}
\label{esubfield}
\Delta_x=\Delta_y, \qquad m=n, \qquad [{\mathbb Q}(x,y):{\mathbb Q}(x^my^m)]= 2.
\end{equation}
Moreover, in the latter case~$x$ and~$y$ are conjugate over the field ${{\mathbb Q}(x^my^m)}$.
If $\{ \Delta_x,\Delta_y \}$ is not of the form $\{\Delta,4\Delta\}$ for some $\Delta \equiv 1 \bmod 8$, then condition \eqref{condthm} can be relaxed to \begin{equation}
\max\{|\Delta_x|,|\Delta_y|\}\ge 10^6.
\end{equation}
\end{theorem}
\paragraph{Plan of the article}
In Section~\ref{sprelimi} we collect basic fact about singular moduli to be used throughout the article. In Section~\ref{slinear} we establish our principal tool: a linear relation between the exponents ${m_1, \ldots, m_k}$ stemming from the multiplicative relation ${x_1^{m_1}\cdots x_k^{m_k}=1}$. Theorems~\ref{thprimel} and~\ref{thpizthree} are proved in Sections~\ref{sprimel} and~\ref{spizthree}, respectively.
\paragraph{Acknowledgments} We thank Guy Fowler, who read an early version of this article and made a number of very helpful comments. We also thank Francesco Amoroso, Margaret Bilu, and Anatoly Vorobey for useful suggestions. Finally, we thank Bill Allombert and Amalia Pizarro, who allowed us to borrow Proposition~\ref{pbilly} from~\cite{ABP14}.
All calculations were performed using \textsf{PARI}~\cite{pari}. We thank Bill Allombert and Karim Belabas for the \textsf{PARI} tutorial. The reader may consult
\begin{center}
\url{https://github.com/yuribilu/multiplicative}
\end{center}
to view the \textsf{PARI} scripts used for this article.
\subsection{Notation and conventions}
\label{ssnota}
We denote by~$\H$ the Poincaré half-plane, and by~$\mathcal{F}$ the standard fundamental domain for the action of the modular group; that is, the open hyperbolic triangle with vertices
$$
\zeta_6=\frac{1+\sqrt{-3}}{2}, \quad \zeta_3=\frac{-1+\sqrt{-3}}{2}, \quad i\infty,
$$
together with the geodesics $[i,\zeta_6]$ and ${[\zeta_6,i\infty]}$.
We denote by $\log$ the principal branch of the complex logarithm:
$$
-\pi < \arg\log z\le \pi \qquad (z\in {\mathbb C}^\times).
$$
We use ${O_1(\cdot)}$ as a quantitative version of the ${O(\cdot)}$ notation: ${A=O_1(B)}$ means that ${|A|\le B}$.
We write the Galois action exponentially: ${x\mapsto x^\sigma}$. In particular, it is a right action: ${z^{(\sigma_1\sigma_2)}=(z^{\sigma_1})^{\sigma_2}}$. Most of the Galois groups occuring in this article are abelian, so this is not relevant, but in the few cases where the group is not abelian one must be vigilant.
Let~$R$ be a commutative ring, and ${a\in R}$. When this does not lead to confusion, we write $R/a$ instead of $R/aR$.
One point on references: item~Y of Proposition~X is quoted as Proposition~X:Y.
The following trivial lemma will be systematically used throughout the article.
\begin{lemma}
\label{llogs}
Let~$u$ be a complex numbers satisfying ${|u|<1}$. Then
\begin{equation*}
|\log(1+u)|\le \frac{|u|}{1-|u|}.
\end{equation*}
More generally, for ${n=1,2,\ldots}$ we have
\begin{equation*}
\log(1+u)= \sum_{k=1}^n\frac{(-1)^{k-1}u^k}{k} +O_1\left(\frac{1}{n+1}\frac{u^{n+1}}{1-|u|}\right)
\end{equation*}
\end{lemma}
\section{Class numbers, denominators, isogenies}
\label{sprelimi}
Unless the contrary is stated explicitly, the letter~$\Delta$ stands for an \textit{imaginary quadratic discriminant}, that is, ${\Delta<0}$ and ${\Delta\equiv 0,1\bmod 4}$.
We denote by $\mathcal{O}_\Delta$ the imaginary quadratic order of discriminant~$\Delta$, that is, ${\mathcal{O}_\Delta ={\mathbb Z}[(\Delta+\sqrt\Delta)/2]}$. Then ${\Delta=Df^2}$, where~$D$ is the discriminant of the number field ${K={\mathbb Q}(\sqrt\Delta)}$ (called the \textit{fundamental discriminant} of~$\Delta$) and ${f=[\mathcal{O}_D:\mathcal{O}_\Delta]}$ is called the \textit{conductor} of~$\Delta$.
We denote by $h(\Delta)$ the class number of~$\mathcal{O}_\Delta$.
Given a singular modulus~$x$, we denote $\Delta_x$ the discriminant of the associated CM order, and we write ${\Delta_x=D_xf_x^2}$ with~$D_x$ the fundamental discriminant and~$f_x$ the conductor.
Furthermore, we denote by~$K_x$ the associated imaginary quadratic field
$$
K_x=
{\mathbb Q}(\sqrt{D_x})={\mathbb Q}(\sqrt{\Delta_x}).
$$
We will call~$K_x$ the \textit{CM-field} of the singular modulus~$x$.
It is known (see, for instance, §11 in~\cite{Co13}) that a singular modulus~$x$ is an algebraic integer of degree $h(\Delta_x)$, and that there are exactly $h(\Delta)$ singular moduli of given discriminant~$\Delta$, which form a full Galois orbit over~${\mathbb Q}$.
\subsection{Class Numbers and Class Groups}
For a discriminant~$\Delta$ and a positive integer~$\ell$ set
\begin{equation}
\label{epsild} \Psi(\ell,\Delta)=\ell\prod_{p\mid \ell}\left(1-\frac{(\Delta/p)}{p}\right).
\end{equation}
It is useful to note that
\begin{equation}
\label{elowereuler}
\Psi(\ell,\Delta)\ge \ph(\ell),
\end{equation}
where ${\ph(\cdot)}$ is Euler's totient function. Note also the ``multiplicativity relation''
\begin{equation}
\label{emult}
\Psi(\ell_1\ell_2, \Delta)= \Psi(\ell_2, \Delta\ell_1^2) \Psi(\ell_1,\Delta).
\end{equation}
Recall the ``Class Number Formula'':
\begin{equation}
\label{eclnfr}
h(\Delta\ell^2) = \frac{1}{[\mathcal{O}_\Delta^\times:\mathcal{O}_{\Delta\ell^2}^\times]}h(\Delta)\Psi(\ell,\Delta) ,
\end{equation}
see \cite[Theorem~7.24]{Co13}. Note that in~\cite{Co13} it is proved only in the case when ${\Delta=D}$ is a fundamental discriminant. However, the general case easily follows from the case ${\Delta=D}$ using the ``multiplicativity relation''~\eqref{emult}. Note also that
$$
[\mathcal{O}_\Delta^\times:\mathcal{O}_{\Delta\ell^2}^\times]=
\begin{cases}
3,& \Delta=-3,\ell>1,\\
2,& \Delta=-4,\ell>1,\\
1,& \Delta\ne -3,-4.
\end{cases}
$$
\subsubsection{Discriminants with small class number}
Watkins~\cite{Wa04} classified fundamental discriminants~$D$ with ${h(D)\le 100}$. In particular, he proved that such discriminants do not exceed $2383747$ in absolute value. It turns out that the same upper bound holds true for all discriminants, not only for fundamental ones.
\begin{proposition}
\label{pwatki}
Let~$\Delta$ be a negative discriminant with ${h(\Delta) \le 100}$. Then we have ${|\Delta|\le 2383747}$. If $h(\Delta) \leq 64$, then $|\Delta| \leq 991027$.
\end{proposition}
\begin{remark}
As Guy Fowler informed us, the same result, with a similar proof, was also obtained in the 2012 Master Thesis of Janis Klaise~\cite{KL12}; but, apparently, it had never been published.
\end{remark}
\begin{proof}
Given a positive integer~$n$, denote
$$
D_{\max}(n):=\max\{|D|: D \text{ fundamental}\mathbin{,} h(D)\le n\}
$$
the biggest absolute value of a \textit{fundamental} discriminant~$D$ with ${h(D)\le n}$; the values of $D_{\max}$ for arguments up to $100$ can be found in Watkins \cite[Table~4]{Wa04}. For the reader's convenience, we give in Table~\ref{tadmax} the $D_{\max}$ of the arguments occurring in equation~\eqref{edeltale} below.
\begin{table}
\caption{Values of $D_{\max}(n)$ for~$n$ of the form $\lfloor 100/\varphi(f)\rfloor$}
\label{tadmax}
{\scriptsize
$$
\begin{array}{rr|ccccccccc}
&n&1&2&3&4&5&6&7&8&10\\
&\max f &420&210&120&90 & 66 & 60 & 42 & 42 & 30\\
&D_{\max}(n)&163&427&907&1555&2683&3763&5923&6307&13843\\
{ \ }\\
&n&12&16&25&50&100\\
&\max f&30 & 18 & 12 &6& 2 \\
&D_{\max}(n)& 17803 & 34483& 111763 & 462883 & 2383747
\end{array}
$$
\textbf{Explanation.\quad}
The first row contains all positive integers~$n$ of the form $\lfloor 100/\varphi(f)\rfloor$ for some positive integer~$f$. In the second row, for each~$n$ we give the biggest~$f$ with the property ${ 100/\varphi(f)\ge n}$. Finally, in the third row we display $D_{\max}(n)$.
}
\end{table}
Now let ${\Delta=Df^2}$ be such that ${h(\Delta) \le 100}$. Using the Class Number Formula~\eqref{eclnfr} (applied with~$D$ as~$\Delta$ and with~$f$ as~$\ell$), and the bound~\eqref{elowereuler} we get
\begin{equation}
\label{eintermed}
h(D) \varphi(f) \leq 100 [\mathcal{O}_D^\times:\mathcal{O}_\Delta^\times].
\end{equation}
If $D=-3$ or $-4$ then this implies $\varphi(f) \leq 300$: the largest such $f$ is $f=1260$, so that in this case ${|\Delta|\leq 6350400}$.
If $D \neq -3,-4$ then we find from \eqref{eintermed} that $h(D) \leq 100/\varphi(f)$, and hence
\begin{equation}
\label{edeltale}
|\Delta|=f^2|D| \leq f^2 D_{\max}(\left\lfloor 100/\varphi(f)\right\rfloor).
\end{equation}
Plugging in the values from Table~\ref{tadmax}, we find that the maximum of the right-hand side is attained for $f=420$ and is equal to $28753200$. This proves that ${|\Delta|\le 28753200}$.
To complete the proof, we run a \textsf{PARI} script computing the class numbers of all~$\Delta$ with ${|\Delta|\le 28753200}$. It confirms that the biggest~$\Delta$ with ${h(\Delta)\le 100}$ is ${-2383747}$, and the biggest~$\Delta$ with ${h(\Delta)\le 64}$ is ${-991027}$. The total running time was 50 minutes on a personal computer.
\end{proof}
\subsubsection{The $2$-rank}
\label{ssstwor}
Given a finite abelian group~$G$ and a prime number~$p$, we call the \textit{$p$-rank} of~$G$, denoted $\rho_p(G)$, the dimension of the ${\mathbb F}_p$-vector space ${G/G^p}$. If~$\Delta$ is a discriminant then we denote $\rho_p(\Delta)$ the $p$-rank of its class group.
The $2$-rank of a discriminant was determined by Gauss, see \cite[Theorem~3.15]{Co13}.
We do not need the full strength of the Gauss Theorem, but only the following consequence.
\begin{proposition}
\label{pgauss}
Let~$\Delta$ be a discriminant. Then
$$
\rho_2(\Delta) \in \{\omega(\Delta), \omega(\Delta)-1, \omega(\Delta)-2\},
$$
where, as usual, $\omega(n)$ denotes the number of distinct prime divisors of a non-zero integer~$n$. Moreover, if ${\Delta\equiv 4\bmod 16}$ then ${\rho_2(\Delta)=\omega(\Delta)-2}$, and if~$D$ is a fundamental discriminant then
${\rho_2(D) \in \{\omega(D)-1, \omega(D)-2\}}$.
\end{proposition}
\subsection{Ring Class Fields}
If~$x$ is a singular modulus with discriminant ${\Delta=Df^2}$ and ${K={\mathbb Q}(\sqrt D)}$ is its CM-field, then $K(x)$ is an abelian extension of~$K$ such that $\Gal(K(x)/K)$ is isomorphic to to the class group of~$\Delta$; in particular, ${[K(x):K]=h(\Delta)}$, and the singular moduli of discriminant~$\Delta$ form a full Galois orbit over~$K$ as well.
This leads to the useful notion of \textit{Ring Class Field}. Given an imaginary quadratic field~$K$ of discriminant~$D$ and a positive integer~$f$, the \textit{Ring Class Field} of~$K$ of conductor~$f$, denoted $K[f]$, is, by definition, $K(x)$, where~$x$ is some singular modulus of discriminant $Df^2$. It does not depend on the particular choice of~$x$ and is an abelian extension of~$K$.
Proofs of the statements above can be found, for instance, in §9--11 of~\cite{Co13}.
\bigskip
The following properties will be systematically used.
\begin{proposition}
\label{pdihedral}
Let~$K$ be an imaginary quadratic field and~$L$ a Ring Class Field of~$K$. Denote
\begin{equation}
\label{egh}
G=\Gal(L/{\mathbb Q}), \qquad H=\Gal(L/K).
\end{equation}
(As we have just seen,~$H$ is an abelian group.)
Then we have the following.
\begin{itemize}
\item
every element of ${G\smallsetminus H}$ is of order~$2$;
\item
if ${\gamma\in G\smallsetminus H}$ and ${\eta\in H}$ then ${\gamma\eta\gamma=\eta^{-1}}$.
\end{itemize}
In particular, any element of ${G\smallsetminus H}$ does not commute with any element of~$G$ of order bigger than~$2$.
\end{proposition}
For the proof see, for instance, \cite[Lemma~9.3]{Co13}.
\begin{proposition}
\label{pcompo}
Let~$K$ be an imaginary quadratic field of discriminant~$D$, and $\ell,m$ positive integers.
\begin{enumerate}
\item
Assume that either ${D\ne -3,-4}$ or ${\gcd(\ell,m)>1}$. Then the compositum $K[\ell]K[m]$ is equal to $K[\lcm(\ell,m)]$.
\item
Assume that ${D=-3}$ and ${\gcd(\ell, m)=1}$. Then $K[\ell]K[m]$ is either equal to $K[\lcm(\ell,m)]$ or is a subfield of $K[\lcm(\ell,m)]$ of degree~$3$.
\item
Assume that ${D=-4}$ and ${\gcd(\ell, m)=1}$. Then $K[\ell]K[m]$ is either equal to $K[\lcm(\ell,m)]$ or is a subfield of $K[\lcm(\ell,m)]$ of degree~$2$.
\end{enumerate}
\end{proposition}
For the proof see, for instance, \cite[Proposition~3.1]{ABP15}
\subsubsection{Two-elementary subfields of Ring Class Fields}
We call a group \textit{$2$-elementary} if all its elements are of order dividing~$2$. A finite $2$-elementary group is a product of cyclic groups of order~$2$. We call a number field $2$-elementary if it is Galois over~${\mathbb Q}$, with $2$-elementary Galois group.
The following is well-known, but we include the proof for the reader's convenience.
\begin{proposition}
\label{pabelian}
Let~$F$ be a number field abelian over~${\mathbb Q}$ and contained in some Ring Class Field.
Then~$F$ is $2$-elementary.
\end{proposition}
\begin{proof}
This is an easy consequence of Proposition~\ref{pdihedral}.
Let~$K$ be an imaginary quadratic field such that its ring class field, denoted~$L$, contains~$F$.
We use the notation of~\eqref{egh}.
For ${\gamma \in G}$ let ${\tilde{\gamma} \in \Gal(F/{\mathbb Q})}$ denote the restriction of~$\gamma$ to~$F$. Each element of $\Gal(F/{\mathbb Q})$ is a restriction of either some ${\gamma\in G\smallsetminus H}$ or some ${\eta\in H}$. In the former case ${\tilde{\gamma}^2=1}$ because ${\gamma^2=1}$. Now consider~$\tilde{\eta}$ for some ${\eta\in H}$. Pick ${\gamma \in G\smallsetminus H}$. Then ${\tilde{\gamma}\tilde{\eta}\tilde{\gamma} =\tilde{\eta}^{-1}}$. But ${\Gal(F/{\mathbb Q})}$ is abelian, which implies that ${\tilde{\gamma}\tilde{\eta}\tilde{\gamma} =\tilde{\gamma}^2\tilde{\eta} =\tilde{\eta}}$. Hence ${\tilde{\eta}^2=1}$ as well. Thus, every element of ${\Gal(F/{\mathbb Q})}$ is of order dividing~$2$, as wanted.
\end{proof}
The only positive integers~$m$ with the property ``the multiplicative group ${({\mathbb Z}/m{\mathbb Z})^\times}$ is $2$-elementary'' are the divisors of~$24$. Hence we have the following corollary.
\begin{corollary}
\label{crofoneinrcf}
The group of roots of unity in a Ring Class Field is of order dividing~$24$.
\end{corollary}
Another famous case of $2$-elementary fields is the intersection ${{\mathbb Q}(x)\cap{\mathbb Q}(y)}$, where~$x$ and~$y$ singular moduli with distinct fundamental discriminants. This is known since ages (see, for instances, the articles of André~\cite{An98} or Edixhoven~\cite{Ed98}), but we again include a proof for the reader's convenience.
\begin{proposition}
\label{pintersect}
Let~$x$ and~$y$ be singular moduli with distinct fundamental discriminants: ${D_x\ne D_y}$. Then the field ${{\mathbb Q}(x)\cap {\mathbb Q}(y)}$ is $2$-elementary. In particular, if ${{\mathbb Q}(x)\subset {\mathbb Q}(y)}$ then ${\mathbb Q}(x)$ is $2$-elementary.
\end{proposition}
\begin{proof}
It suffices to prove that the field ${{\mathbb Q}(x)\cap {\mathbb Q}(y)}$ is abelian: Proposition~\ref{pabelian} will then complete the job.
Recall that we denote ${K_x={\mathbb Q}(\sqrt{D_x})}$ the CM field for~$x$. We will denote $K_{xy}$ the compositum of~$K_x$ and~$K_y$, that is, the field ${\mathbb Q}(\sqrt{D_x},\sqrt{D_y})$. Furthermore, we denote~$M$ be the compositum and~$L$ be the intersection of the fields $K_{xy}(x)$ and $K_{xy}(y)$:
$$
M=K_{xy}(x,y), \qquad L= K_{xy}(x)\cap K_{xy}(y).
$$
It suffices to prove that~$L$ is abelian, because ${L\supset {\mathbb Q}(x)\cap {\mathbb Q}(y)}$. To start with, let us prove that~$L$ is $2$-elementary over the field $K_{xy}$.
Since ${K_x\ne K_y}$, there exists ${\iota \in \Gal(M/{\mathbb Q})}$ such that
$$
\iota\vert_{K_x}=\id, \qquad \iota\vert_{K_y}\ne \id.
$$
Proposition~\ref{pdihedral} implies that for ${\eta \in \Gal(M/K_{xy})}$ we have
$$
\iota^{-1}\eta \iota\vert_{K_x(x)}= \eta\vert_{K_x(x)}, \qquad \iota^{-1}\eta \iota\vert_{K_y(y)}= \eta^{-1}\vert_{K_y(y)}.
$$
We also have ${\eta\vert_{K_{xy}}=\id}$ by the choice of~$\eta$. Hence ${\eta\vert_L=\eta^{-1}\vert_L}$. Since every element of $\Gal(L/K_{xy})$ is a restriction to~$L$ of some ${\eta \in \Gal(M/K_{xy})}$, this proves that the Galois group $\Gal(L/K_{xy})$ is $2$-elementary, as wanted.
To complete the proof, we must show that~$L$ is abelian over~${\mathbb Q}$. Clearly,~$L$ is Galois over~${\mathbb Q}$, being the intersection of two Galois extension. We have to show that ${\Gal(K_{xy}/{\mathbb Q})}$ acts trivially on ${\Gal(L/K_{xy})}$. This means proving the following: for every ${\eta,\gamma \in \Gal(M/{\mathbb Q})}$ such that ${\eta\vert_{K_{xy}}=\id}$ we have
${\gamma^{-1}\eta\gamma\vert_L=\eta}$.
We denote ${\eta^\gamma=\gamma^{-1}\eta\gamma}$. Proposition~\ref{pdihedral} implies that
$$
\eta^\gamma\vert_{K_x(x)}\in \bigl\{\eta\vert_{K_x(x)}, \eta^{-1}\vert_{K_x(x)}\bigr\}.
$$
We also have
${\eta^\gamma\vert_{K_y}= \id\vert_{K_y}=\eta\vert_{K_y}=\eta^{-1}\vert_{K_y}}$.
It follows that
$$
\eta^\gamma\vert_{K_{xy}(x)}\in \bigl\{\eta\vert_{K_{xy}(x)}, \eta^{-1}\vert_{K_{xy}(x)}\bigr\}.
$$
In particular,
${\eta^\gamma\vert_L\in \bigl\{\eta\vert_L, \eta^{-1}\vert_L\bigr\}}$.
Since ${\eta\vert_{K_{xy}}=\id}$ and ${L/K_{xy}}$ is $2$-elementary, we have ${\eta\vert_L= \eta^{-1}\vert_L}$. Hence ${\eta^\gamma\vert_L=\eta\vert_L}$. The proposition is proved.
\end{proof}
\subsubsection{(Almost) $2$-elementary discriminants}
We will need a slight generalization of the notion of a $2$-elementary group. A finite abelian group~$G$ will be called \textit{almost $2$-elementary} if it has a $2$-elementary subgroup of index~$2$. This means that either~$G$ is $2$-elementary, or it is ${{\mathbb Z}/4{\mathbb Z}}$ times a $2$-elementary group.
We call a discriminant (almost) $2$-elementary if so is its class group. (Almost) $2$-elementary discriminants can be conveniently characterized in terms of the $2$-rank, see Subsection~\ref{ssstwor}:
\begin{align}
\label{etwoel}
\text{$\Delta$ is $2$-elementary} &\Longleftrightarrow h(\Delta)=2^{\rho_2(\Delta)};\\
\label{ealmtwoel}
\text{$\Delta$ is almost $2$-elementary} &\Longleftrightarrow h(\Delta)\mid 2^{\rho_2(\Delta)+1}.
\end{align}
\begin{proposition}\label{pconductors}
Let ${\Delta=Df^2}$ be a $2$-elementary discriminant. Then ${f\mid 24}$ when ${D\ne -3,-4}$, and ${f\le 8}$ when ${D\in \{-3,-4\}}$.
If ${\Delta=Df^2}$ is almost $2$-elementary, then ${f\mid 240}$ when ${D\ne -3,-4}$, and ${f\le 20}$ when ${D\in \{-3,-4\}}$.
\end{proposition}
\begin{proof}
Denote by ${\mathcal{O}=\mathcal{O}_K}$ the ring of integers of ${K={\mathbb Q}(\sqrt D)}$ and by $(\mathcal{O}^\times)_f$ the image of $\mathcal{O}^\times$ in $(\mathcal{O}/f)^\times$. The class group of~$\Delta$ has a subgroup isomorphic to ${(\mathcal{O}/f)^\times \big / ({\mathbb Z}/f)^\times(\mathcal{O}^\times)_f}$, see \cite[§~7.D and Exercise 7.30]{Co13} or \cite[Section~2.1]{ABP15}.
If ${D\ne -3,-4}$ then ${({\mathbb Z}/f)^\times(\mathcal{O}^\times)_f=({\mathbb Z}/f)^\times}$, which means that the group ${(\mathcal{O}/f)^\times \big / ({\mathbb Z}/f)^\times}$ must be (almost) $2$-elementary. By the Chinese Remainder Theorem for groups, this group is the product of the groups ${(\mathcal{O}/p^k)^\times \big / ({\mathbb Z}/p^k)^\times}$ for all prime powers ${p^k\,\|\,f}$; hence these groups must be (almost) $2$-elementary as well.
For ${p>2}$ we have
$$
(\mathcal{O}/p^k)^\times \big / ({\mathbb Z}/p^k)^\times \cong {\mathbb Z}/\bigl(p^{k-1}(p-(D/p))\bigr),
$$
The latter group is $2$-elementary only if\footnote{Note that here and below it is exactly``only if'', a necessary condition. We do not claim that the converse is true (and, obviously, it is not).} ${p^k=3}$, and almost $2$-elementary only if ${p^k \in \{3,5\}}$. For ${p=2}$ we have
$$
(\mathcal{O}/2^k)^\times \big / ({\mathbb Z}/2^k)^\times \cong
\begin{cases}
{\mathbb Z}/\bigl(2-(D/2)\bigr), & k=1, \\
{\mathbb Z}/2\times {\mathbb Z}/\bigl(2^{k-2}(2-(D/2))\bigr), & k\ge 2.
\end{cases}
$$
This group is $2$-elementary only if ${2^k\in \{2,4,8\}}$, and almost $2$-elementary only if ${2^k\in \{1,2,4,8,16\}}$. Thus, in the $2$-elementary case~$f$ divides ${8\cdot 3=24}$ and in the almost $2$-elementary case it divides ${16\cdot3\cdot5=240}$. This proves the statement for ${D\ne -3,-4}$.
If ${D=-3}$ or $-4$ then the extra factor $(\mathcal{O}^\times)_f$ must be taken into account. For instance, for ${D=-3}$ a similar argument shows that, in the $2$-elementary case, we must have ${p^k\in \{2,4,8,3,5,7\}}$, and in the almost $2$-elementary case one should add ${16,11,13}$ to the list. This gives finitely many possible~$f$, which can be verified by inspection. The discriminant ${D=-4}$ is treated similarly. The details are routine and we omit them.
\end{proof}
{\sloppy
\begin{proposition}
\label{pbilly}
There exists a fundamental discriminant~$D^\ast$ such that ${h(D^\ast)\ge 128}$ and the following holds. Let ${\Delta=Df^2}$ be either $2$-elementary or almost $2$-elementary. Then either ${D=D^\ast}$ or
$$
h(\Delta) \le
\begin{cases}
16, & \text{if $\Delta$ is $2$-elementary},\\
64, & \text{if $\Delta$ is almost $2$-elementary}.
\end{cases}
$$
\end{proposition}
}
This is proved in~\cite{ABP14}, an early version of~\cite{ABP15}, see Corollary~2.5 and Remark~2.6 therein. This result was not included in the published version of~\cite{ABP15}, so we reproduce here the proof (adding some details missing in~\cite{ABP14}). The proof broadly follows the strategy in Weinberger \cite{We73}; in particular, it rests on a classical bound of Tatuzawa \cite[Theorem~2]{Ta51}.
\begin{lemma}[Tatuzawa]
\label{ltatuzawa}
Let $0<\varepsilon<1/2$. There exists a fundamental discriminant~$D^\ast$ such that the following holds. Let $D\neq D^\ast$ be a fundamental discriminant,~$\chi$ the associated quadratic character, defined by ${\chi(n)=(D/n)}$, the Kronecker symbol, and $L(s,\chi)$ the attached $L$-function. Then we either have ${|D| \leq \max\{e^{1/\varepsilon},73130 \}}$ or ${ L(1,\chi) \geq 0.655 \varepsilon |D|^{-\varepsilon}}$.
\end{lemma}
\begin{proof}[Proof of Proposition \ref{pbilly}]
If ${D=-3}$ or~$-4$ then the result easily follows from Proposition~\ref{pconductors} and a quick calculation with \textsf{PARI}. Hence we may assume that ${D\ne -3,-4}$. In this case the analytic class number formula states that ${h(D)=\pi^{-1}|D|^{1/2} L(1,\chi) }$. If $\Delta$ is almost $2$-elementary then so is $D$. By~\eqref{ealmtwoel} and Proposition \ref{pgauss} we have ${h(D) \le 2^{\rho_2(D)+1}\le 2^{\omega(D)}}$.
We pick $\varepsilon=0.048$ throughout and we use the corresponding~$D^\ast$ from Lemma~\ref{ltatuzawa}.
Assuming that ${D\ne D^\ast}$, Lemma \ref{ltatuzawa} implies that
$$
2^{\omega(D)}\geq h(D)=\pi^{-1}|D|^{1/2}L(1,\chi) \geq 0.655 \pi^{-1}\varepsilon |D|^{1/2-\varepsilon}
$$
as long as $|D| \geq 1116353418$. This implies that
$$ |D| \leq \left((0.655\varepsilon)^{-1} \pi 2^{\omega(D)} \right)^{1/(1/2-\varepsilon)}\leq 26549 \cdot 4.635^{\omega(D)};$$ we conclude that
\begin{equation}\label{econd1}
|D| \leq \max\{ 26549 \cdot 4.635^{\omega(D)}, 1116353418\}.
\end{equation}
Note moreover that, since $D$ is fundamental and
$$
1116353418<4\cdot(3 \cdot 5 \cdot 7\cdot 11 \cdots 37 )
$$
($4$ times the product of the first $11$ odd primes), we must have ${\omega(D) \leq 11}$ whenever ${|D| \leq 1116353418}$. More generally, $|D|$ is not smaller than $4$ times the product of the first $\omega(D)-1$ odd primes. Hence, when ${\omega (D) \geq 12}$, we have
$$
|D|\ge 4\cdot(3 \cdot 5 \cdot 7\cdot 11 \cdots 37 ) \cdot 41^{\omega(D)-12}.
$$
Combining this observation with the upper bound~\eqref{econd1}, we conclude that, when ${\omega(D) \geq 12}$, we must have
$$
4\cdot(3 \cdot 5 \cdot 7\cdot 11 \cdots 37 ) \cdot 41^{\omega(D)-12} \leq |D| \leq 26549 \cdot 4.635^{\omega(D)}.
$$
This is easily seen to be a contradiction for ${\omega(D) \geq 12}$. We conclude that
\begin{equation}
\label{econd2}
\omega(D) \leq 11
\end{equation}
for any almost $2$-elementary fundamental ${D\ne D^\ast}$.
Thus, we are now left with the task of examining discriminants ${\Delta=Df^2}$ such that the corresponding fundamental~$D$ satisfies conditions~\eqref{econd1} and~\eqref{econd2}. We want to show that
\begin{itemize}
\item
if such~$\Delta$ is $2$-elementary then ${h(\Delta)\le 16}$;
\item
if such~$\Delta$ is almost $2$-elementary then ${h(\Delta)\le 64}$.
\end{itemize}
Proving this is a numerical check using \textsf{PARI}. We distinguish two cases: ${\omega(D)\le 6}$ and ${7\le\omega(D)\le 11}$.
When ${\omega(D) \le 6}$ and~$D$ is almost $2$-elementary then ${h(D)\mid 64}$. Table~4 of Watkins~\cite{Wa04} implies that in this case ${|D|\le 693067}$. Now we run a \textsf{PARI} script, which, for every fundamental discriminant~$D$ satisfying ${|D|\le 693067}$, decides whether it is (almost) $2$-elementary. If it is,
the script counts, using Proposition~\ref{pconductors}, all (almost) $2$-elementary discriminants ${\Delta=Df^2}$.
Our script found 101 discriminants that are $2$-elementary, the biggest being ${-7392=-1848\cdot2^2}$. The class numbers of all these discriminants do not exceed $16$. Similarly, the script found 425 almost $2$-elementary discriminants, ${87360=-5460\cdot4^2}$ being the biggest, and their class numbers do not exceed $64$. The total running time on a personal computer was less than 2 minutes. This completes the proof in the case ${\omega(D) \le 6}$.
When ${7\le\omega(D)\le 11}$, we can no longer use~\cite{Wa04}, and the only upper bound that we have is~\eqref{econd1}. Checking all fundamental discriminants below this bound is too costly, so we proceed differently. For every ${n=7,\ldots,11}$ we determine all fundamental discriminants~$D$ satisfying
\begin{equation}
\label{enappears}
\omega(D)=n, \qquad |D|\le 26549 \cdot 4.635^n
\end{equation}
(note that ${26549 \cdot 4.635^n>1116353418}$ for ${n\ge 7}$), and for each of them we check whether it is almost $2$-elementary. Our script found no almost $2$-elementary discriminants satisfying~\eqref{enappears} with ${7\le n\le 11}$. The total running time was about 6 minutes for ${n=7}$, about 2 minutes for each of ${n=8,9}$, and negligible for ${n=10,11}$. This completes the proof of Proposition~\ref{pbilly}.
\end{proof}
\begin{corollary}
\label{cxy}
Let~$x$ and~$y$ be singular moduli of distinct fundamental discriminants: ${D_x\ne D_y}$.
\begin{enumerate}
\item
\label{isame}
Assume that ${{\mathbb Q}(x)= {\mathbb Q}(y)}$. Then ${h(\Delta_x)= h(\Delta_y) \le 16}$.
\item
\label{itwo}
Assume that ${{\mathbb Q}(x)\subset {\mathbb Q}(y)}$ and ${[{\mathbb Q}(y):{\mathbb Q}(x)]=2}$. Then ${h(\Delta_x) \le 16}$ and ${h(\Delta_y) \le 32}$.
\end{enumerate}
\end{corollary}
\begin{proof}
If ${{\mathbb Q}(x)= {\mathbb Q}(y)}$ then both $\Delta_x$ and~$\Delta_y$ are $2$-elementary by Proposition~\ref{pintersect}. Since ${D_x\ne D_y}$, one of the two is distinct from $D^\ast$; say, ${D_x\ne D^\ast}$. Then ${h(\Delta_x)\le 16}$. Hence ${h(\Delta_y) =h(\Delta_x) \le 16}$ as well. This proves item~\ref{isame}.
Now assume that we are in the situation of item~\ref{itwo}. Then $\Gal({\mathbb Q}(x)/{\mathbb Q})$ is $2$-elementary by Proposition~\ref{pintersect}. Hence so is $\Gal(K_y(x)/K_y)$. Since
$$
[K_y(y):K_y(x)]\le [{\mathbb Q}(y):{\mathbb Q}(x)]=2,
$$
the group $\Gal(K_y(y)/K_y)$ is almost $2$-elementary. If ${D_x\ne D^\ast}$ then ${h(\Delta_x)\le 16}$ and ${h(\Delta_y) \le 32}$, so we are done. If ${D_y\ne D^\ast}$ then ${h(\Delta_y) \le 64}$. It follows that ${h(\Delta_x)\le 32}$ and we must have ${D_x\ne D^\ast }$, so we are done again.
\end{proof}
\subsection{Gauss Reduction Theory, denominators}
\label{ssgauss}
Denote~$T_\Delta$ the set of triples ${(a,b,c)\in {\mathbb Z}^3}$ with ${\Delta=b^2-4ac}$ satisfying
\begin{align}
&
\text{$\gcd(a,b,c)=1$},\nonumber\\
\label{ekuh}
&
\text{either\quad $-a < b \le a < c$\quad or\quad $0 \le b \le a = c$}.
\end{align}
Note that condition~\eqref{ekuh} is equivalent to
$$
\frac{b+\sqrt\Delta}{2a}\in \mathcal{F}.
$$
For every singular modulus~$x$ of discriminant~$\Delta$ there exists a unique triple ${(a_x,b_x,c_x)\in T_\Delta}$ such that, denoting
$$
\tau_x= \frac{b_x+\sqrt\Delta}{2a_x},
$$
we have ${x=j(\tau_x)}$. This is, essentially, due to Gauss; see \cite[Section~2.2]{BLP16} for more details.
We will call~$a_x$ the \textit{denominator} of the singular modulus~$x$.
Note that, alternatively,~$\tau_x$ can be defined as the unique ${\tau\in \mathcal{F}}$ such that ${j(\tau)=x}$.
We will say that a positive integer~$a$ is a denominator for~$\Delta$ if it is a denominator of some singular modulus of discriminant~$\Delta$; equivalently, there exist ${b,c\in {\mathbb Z}}$ such that ${(a,b,c)\in T_\Delta}$.
It will often be more convenient to use the notation $a(x)$, $b(x)$, $\tau(x)$ etc. instead of~$a_x$,~$b_x$,~$\tau_x$, etc.
\begin{remark}
\label{rparity}
It is useful to note that~$b_x$ and~$\Delta_x$ are of the same parity: ${b_x\equiv \Delta_x\bmod 2}$. This is because ${\Delta_x=b_x^2-4a_xc_x\equiv b_x^2\bmod 4}$.
\end{remark}
For every~$\Delta$ there exists exactly one singular modulus of discriminant~$\Delta$ and of denominator~$1$, which will be called the \textit{dominant} singular modulus of discriminant~$\Delta$. Singular moduli with denominator~$2$ will be called \textit{subdominant}.
\begin{proposition}
\label{pcountden}
Let~$\Delta$ be a discriminant. Then for every ${a\in\{2,3,4,5\}}$ there exist at most~$2$ singular moduli~$x$ with ${\Delta_x=\Delta}$ and ${a_x=a}$. For every ${A\in \{13,18,30\}}$ there exists at most $S(A)$ singular moduli~$x$ with ${\Delta_x=\Delta}$ and ${a_x<A}$, where $S(A)$ is given in the following table:
$$
\begin{array}{r|ccc}
A&13&18&30\\
\hline
S(A)&32&48&99
\end{array}.
$$
\end{proposition}
\begin{proof}
Let~$a$ be a positive integer. For a residue class ${r\bmod 4a}$ denote $B(r)$ the number of ${b\in {\mathbb Z}}$ satisfying
${-a< b\le a}$ and ${b^2\equiv r\bmod 4a}$. Denote $s(a)$ the biggest of all $B(r)$:
$$
s(a)=\max\{B(r): r\bmod4a \}.
$$
The number of triples ${(a,b,c)\in T_\Delta}$ with given~$a$ does not exceed $B(\Delta)$; hence it does not exceed $s(a)$ either.
A quick calculation shows that ${s(a)=2}$ for ${a\in \{2,3,4,5\}}$, and
$$
\sum_{a<A}s(a)=S(A)
$$
for ${A\in \{13,18,30\}}$. The proposition is proved.
\end{proof}
We will also need miscellaneous facts about (non)-existence of singular moduli of some specific shape.
The following proposition will be used in this article only for ${p=3}$. We, however, state it for general~$p$, for the sake of further applications.
\begin{proposition}
\label{poddpden}
Let~$\Delta$ be a discriminant and~$p$ an odd prime number.
\begin{enumerate}
\item
\label{ionemodthree}
Assume that ${(\Delta/p)=1}$. If ${|\Delta|\ge 4p^2-1}$ then~$\Delta$ admits exactly~$2$ singular moduli with denominator~$p$. More generally, if ${|\Delta|\ge 4p^k-1}$ then~$\Delta$ admits exactly~$2$ singular moduli with denominator~$p^k$.
\item
\label{idivbyp}
Assume that ${p^2\mid \Delta}$, and let~$a$ be a denominator for~$\Delta$. Then either ${p\nmid a}$ or ${p^2\mid a}$. In particular,~$p$ is not a denominator for~$\Delta$.
\end{enumerate}
\end{proposition}
\begin{proof}
By Hensel's lemma, the assumption ${(\Delta/p)=1}$ implies that the congruence ${b^2\equiv \Delta\bmod p^k}$ has exactly two solutions satisfying ${0<b<p^k}$, and exactly one of these solutions satisfies ${b^2\equiv \Delta\bmod 4p^k}$. If~$b$ is this solution and ${|\Delta|\ge 4p^{k}-1}$ then the two triples ${(p^k,\pm b, (b^2-\Delta)/4p^k)}$ belong to $T_\Delta$. This proves item~\ref{ionemodthree}.
If ${p^2\mid \Delta}$ and ${p\mid a}$ then ${p\mid b}$ and ${p\nmid c}$. Hence ${p^2\mid 4ac=b^2-\Delta}$, which implies that ${p^2\mid a}$. This proves item~\ref{idivbyp}.
\end{proof}
Here is an analogue of Proposition~\ref{poddpden} for the ``oddest'' prime~$2$.
\begin{proposition}
Let~$\Delta$ be a discriminant.
\label{pevenden}
\begin{enumerate}
\item
\label{ionemodeight}
Assume that ${\Delta\equiv 1\bmod 8}$. If ${|\Delta|>15}$, then~$\Delta$ admits exactly~$2$ subdominant singular moduli, which are ${j\bigl((\pm1+\sqrt\Delta)/4\bigr)}$. More generally, if ${|\Delta|\ge 4^{k+1}-1}$ then~$\Delta$ admits exactly~$2$ singular moduli with denominator~$2^k$.
\item
\label{inonemodeight}
If ${\Delta\ne 1\bmod 8}$ then it admits at most one subdominant singular modulus.
\item
\label{i432hensel}
Let~$\Delta$ satisfy ${\Delta\equiv 4\bmod 32}$ and ${|\Delta|\ge 252}$. Then it admits exactly~$2$ singular moduli of denominator~$8$. These are
\begin{equation}
\label{ebprime}
j\left(\frac{\pm b'+\sqrt{\Delta/4}}8\right),
\qquad
b'=
\begin{cases}
1, & \Delta\equiv 36\bmod 64,\\
3, & \Delta\equiv 4\bmod 64.
\end{cases}
\end{equation}
More generally, if ${k\ge 3}$ and ${|\Delta|\ge 4^{k+1}-4}$ then~$\Delta$ admits exactly~$2$ singular moduli with denominator~$2^k$.
\item
\label{i432none}
Let~$\Delta$ satisfy ${\Delta\equiv 4\bmod 32}$ and let~$a$ be a denominator for~$\Delta$. Then either~$a$ is odd, or ${8\mid a}$. In particular,~$2$,~$4$ and~$6$ are not denominators for~$\Delta$.
\item
\label{i16none}
Let~$\Delta$ be divisible by~$16$ and let~$a$ be a denominator for~$\Delta$. Then either~$a$ is odd, or ${4\mid a}$. In particular,~$2$ is not a denominator for~$\Delta$.
Furthermore,~$\Delta$ admits at most one singular modulus with denominator~$4$.
\item
\label{ieveryn}
Assume that~$\Delta$ is even, but ${\Delta\ne 4\mod32}$, and that ${|\Delta|>48}$. Then~$2$ or~$4$ is a denominator for~$\Delta$.
\end{enumerate}
\end{proposition}
\begin{proof}
Items~\ref{ionemodeight} and~\ref{i432hensel} are proved using Hensel's Lemma exactly like item~\ref{ionemodthree} of Proposition~\ref{poddpden}; we omit the details.
Item~\ref{inonemodeight} follows from \cite[Proposition~2.6]{BLP16}, and item~\ref{ieveryn} is \cite[Proposition~3.1.4]{BLP20}.
Note that in~\cite{BLP20} ``denominators'' are called ``suitable integers''.
We are left with items~\ref{i432none} and~\ref{i16none}. If ${\Delta\equiv 4\bmod 32}$ and ${(a,b,c)\in T_{\Delta}}$ with ${2\mid a}$ then ${2\|b}$ and~$c$ is odd. Hence
${b^2\equiv 4\bmod 32}$, which implies that ${4ac\equiv 0\bmod 32}$. This shows that ${8\mid a}$, which proves item~\ref{i432none}.
Finally, if ${16\mid \Delta}$ and ${2\|a}$ then ${2\mid b}$ and ${2\nmid c}$, which implies that
$$
b^2=\Delta+4ac\equiv 8 \bmod 16,
$$
a contradiction. This proves the first statement in item~\ref{i16none}. Similarly, if ${a=4}$ then ${4\mid b}$ and ${2\nmid c}$; in particular, ${4ac\equiv 16 \bmod 32}$. Hence
$$
b=
\begin{cases}
0,& \text{if $\Delta\equiv 16 \bmod 32$},\\
4,& \text{if $\Delta\equiv 0 \bmod 32$}.
\end{cases}
$$
Thus, in any case there is only one choice for~$b$, which proves the second statement in item~\ref{i16none}.
The proposition is proved.
\end{proof}
It is useful to note that the dominant singular modulus is real; in particular, there exists at least one real singular modulus of every discriminant. This has the following consequence.
\begin{proposition}
\label{preal}
Let~$x$ be a singular modulus, and let~$K$ be a subfield of ${\mathbb Q}(x)$. Assume that~$K$ is Galois over~${\mathbb Q}$. Then~$K$ is a real field.
\end{proposition}
Since ${\mathbb Q}(x)$ is Galois over~${\mathbb Q}$ when $\Delta_x$ is $2$-elementary, this implies that singular moduli of $2$-elementary discriminants are all real.
\subsection{Isogenies}
Let~$\Lambda$ and~$\mathrm{M}$ be lattices in~${\mathbb C}$. We say that~$\Lambda$ and $\mathrm{M}$ are isogenous if~$\Lambda$ is isomorphic to a sublattice of~$\mathrm{M}$.
More specifically, given a positive integer~$n$,
we say that~$\Lambda$ and~$\mathrm{M}$ are $n$-isogenous if~$\mathrm{M}$ has a sublattice~$\Lambda'$, isomorphic to~$\Lambda$, such that the quotient group ${\mathrm{M}/\Lambda'}$ is cyclic of order~$n$. This relation is symmetric.
Recall that we denote~$\H$ the Poincaré plane and~$\mathcal{F}$ the standard fundamental domain. It is well-known that, for ${z,w\in \H}$, the lattices ${\langle z,1\rangle}$ and ${\langle w,1\rangle}$ are $n$-isogenous if and only if there exists ${\gamma \in \mathrm{M}_2({\mathbb Z})}$ with coprime entries and determinant~$n$ such that ${w=\gamma(z)}$. Imposing upon~$z$ and~$w$ certain reasonable conditions, one may show that matrix~$\gamma$ is upper triangular.
\begin{proposition}
\label{pisog}
Let ${z, w \in \H}$ and let~$n$ be a positive integer. Assume that
\begin{equation}
\label{eimzgen}
w\in \mathcal{F}, \qquad \Im\, z \ge n.
\end{equation}
Then the following two conditions are equivalent.
\begin{enumerate}
\item
\label{iisog}
The lattices ${\langle z,1\rangle}$ and ${\langle w,1\rangle}$ are $n$-isogenous.
\item
\label{iabd}
We have
$$
w=\frac{pz+q}{s},
$$
where ${p,q,s\in {\mathbb Z}}$ satisfy
$$
p,s>0, \qquad ps=n, \qquad \gcd(p,q,s)=1.
$$
\end{enumerate}
\end{proposition}
\begin{proof}
Implication \ref{iabd}$\Rightarrow$\ref{iisog} is trivial and does not require~\eqref{eimzgen}.
Now assume that condition~\ref{iisog} holds, and let ${\gamma\in M_2({\mathbb Z})}$ be a matrix with coprime entries and determinant~$n$ such that ${w=\gamma(z)}$. There exists ${\delta \in \mathrm{SL}_2({\mathbb Z})}$ such that
${\delta \gamma}$ is an upper triangular matrix. Replacing~$\delta$ by ${\bigl(\begin{smallmatrix}1&\nu\\0&1\end{smallmatrix}\bigr)\delta}$ with a suitable ${\nu\in {\mathbb Z}}$, we may assume that
${w':=\delta\gamma(z)}$ satisfies
\begin{equation}
\label{erew}
-\frac 12<\Re \, w'\le \frac12.
\end{equation}
Write ${\delta\gamma= \bigl(\begin{smallmatrix}p&q\\0&s\end{smallmatrix}\bigr)}$. Replacing, if necessary,~$\delta$ by $-\delta$, we may assume that ${p,s>0}$.
Since ${ps=n}$ and
$$
w'=\frac{pz+q}{s},
$$
we only have to prove that ${w'=w}$.
Note that
$$
\Im\, w' = \frac1s\Im\, z\ge 1
$$
by~\eqref{eimzgen}. Together with~\eqref{erew} this implies that ${w'\in \mathcal{F}}$.
But~$w$ belongs to~$\mathcal{F}$ as well, by our assumption~\eqref{eimzgen}. Since ${w'=\delta w }$ and each $\mathrm{SL}_2({\mathbb Z})$-orbit has exactly one point in~$\mathcal{F}$, we must have ${w'=w}$.
\end{proof}
We say that two singular moduli are $n$-isogenous if, writing ${x=j(\tau)}$ and ${y=j(\upsilon)}$, the lattices ${\langle\tau,1\rangle}$ and ${\langle\upsilon,1\rangle}$ are $n$-isogenous.
Singular moduli~$x$ and~$y$ are $n$-isogenous if and only if ${\Phi_n(x,y)=0}$, where ${\Phi_n(X,Y)}$ denotes the modular polynomial of level~$n$. Since ${\Phi_n(X,Y)\in {\mathbb Q}[X,Y]}$, being $n$-isogenous is preserved by Galois conjugation: for any ${\sigma\in \Gal(\bar{\mathbb Q}/{\mathbb Q})}$ the singular moduli~$x^\sigma$ and~$y^\sigma$ are $n$-isogenous as long as~$x$ and~$y$ are.
For a positive integer~$n$ define
$$
\mathcal{Q}(n)=\left\{\frac rs: r,s\in {\mathbb Z}, rs=n\right\}.
$$
For example,
$$
\mathcal{Q}(12)=\left\{\frac1{12},\frac13,\frac34,\frac43,3,12\right\}.
$$
The following property is an immediate consequence of Proposition~\ref{pisog}.
\begin{corollary}
\label{cisog}
Let~$x$ and~$y$ be $n$-isogenous singular moduli. Assume that ${|\Delta|_x^{1/2}\ge 2na_x}$. Then ${(a_y/f_y)/(a_x/f_x)\in \mathcal{Q}(n)}$. In particular,
$$
\frac1n\le\frac{a_y/f_y}{a_x/f_x}\le n.
$$
When ${n=p}$ is a prime number, we have
${(a_y/f_y)/(a_x/f_x)\in \left\{p,1/p\right\}}$.
\end{corollary}
The following simple facts will be repeatedly used, often without special reference.
\begin{proposition}
\label{pvarisog}
Let~$x$ and~$y$ be singular moduli.
\begin{enumerate}
\item
\label{idenoms}
Assume that ${\Delta_x=\Delta_y}$ and
${\gcd(a_x,a_y)=1}$.
Then~$x$ and~$y$ are ${a_xa_y}$-isogenous.
\item
\label{ibothdom}
Assume that
${a_x=a_y=1}$ and ${\Delta_x/e_x^2=\Delta_y/e_y^2}$,
where~$e_x$ and~$e_y$ are coprime positive integers. Then~$x$ and~$y$ are ${e_xe_y}$-isogenous.
\item
\label{isubdomisog}
Two subdominant singular moduli of the same discriminant are either equal or $4$-isogenous.
\end{enumerate}
\end{proposition}
\begin{proof}
To prove item~\ref{idenoms}, note that
$$
\tau_y=\frac{a_x\tau_x+(b_y-b_x)/2}{a_y}.
$$
Since ${b_x\equiv b_y\bmod 2}$ (see Remark~\ref{rparity}), this proves that~$x$ and~$y$ are $a_xa_y$-isogenous.
For item~\eqref{ibothdom} we have
$$
\tau_x= \frac{b_x+e_x\sqrt\Delta}{2}, \qquad \tau_y= \frac{b_y+e_y\sqrt\Delta}{2},
$$
where ${\Delta=\Delta_x/e_x^2=\Delta_y/e_y^2}$.
Hence
$$
\tau_y=\frac{e_y\tau_x +(b_ye_x-b_xe_y)/2}{e_x}
$$
Remark~\ref{rparity} implies now that ${b_xe_y\equiv b_ye_x\bmod 2}$, and we conclude that~$x$ and~$y$ are $e_xe_y$-isogenous.
To prove item~\ref{isubdomisog}, note that distinct subdominant singular moduli of the same discriminant~$\Delta$ must be of the form ${j(\tau)}$ and ${j(\tau')}$, where
$$
\tau=\frac{-1+\sqrt\Delta}{4}, \qquad \tau'=\frac{1+\sqrt\Delta}{4},
$$
see Proposition~\ref{pevenden}:\ref{ionemodeight}.
We have
${\tau'= (2\tau+1)/2}$,
which implies that ${j(\tau)}$ and ${j(\tau')}$ are $4$-isogenous.
\end{proof}
\subsection{Galois-theoretic lemmas}
In this subsection we collect some lemmas with Galois-theoretic flavor that will be repeatedly used in the proofs of Theorems~\ref{thpizthree} and~\ref{thprimel}.
\begin{lemma}
\label{lnorou}
Let~$m$ be a positive integer and~$x$ a singular modulus. Then ${{\mathbb Q}(x)={\mathbb Q}(x^m)}$. In other words: if~$x$ and~$y$ are distinct singular moduli of the same discriminant then ${x^m\ne y^m}$.
\end{lemma}
\begin{proof}
See \cite[Lemma~2.6]{Ri19}.
\end{proof}
\begin{lemma}
\label{lperm}
Let~$x$ and~$y$ be distinct singular moduli of the same discriminant, ${K=K_x=K_y}$ their common CM field, ${L=K(x)=K(y)}$ the Ring Class Field and ${\sigma \in \Gal(L/{\mathbb Q})}$ a Galois morphism. Assume that~$\sigma$ permutes~$x$ and~$y$:
$$
x^\sigma=y, \qquad y^\sigma=x.
$$
Then~$\sigma$ is of order~$2$.
\end{lemma}
\begin{proof}
If ${\sigma \notin \Gal(L/K)}$ then it is of order~$2$ because every element of ${\Gal(L/{\mathbb Q})}$ not belonging to ${\Gal(L/K)}$ is of order~$2$. And if ${\sigma \in \Gal(L/K)}$ then ${\sigma^2=1}$ because ${x^{\sigma^2}=x}$ and ${L=K(x)}$.
\end{proof}
\begin{lemma}
\label{lmainimproved}
Let~$x$,~$y$ be distinct singular moduli of the same discriminant and let~$K$ and~$L$ be as in Lemma~\ref{lperm}.
Let~$F$ be a proper subfield of ${\mathbb Q}(x,y)$; we denote ${G=\Gal(L/F)}$. Then one of the following options takes place.
\begin{enumerate}
\item
\label{ifixed}
There exists ${\sigma\in G}$ such that, up to switching~$x,y$, we have ${x^\sigma=x}$ but ${y^\sigma\ne y}$.
\item
\label{icyclictwo}
We have ${[{\mathbb Q}(x,y):F]=2}$ and the non-trivial automorphism of ${\mathbb Q}(x,y)/F$ permutes~$x$ and~$y$.
\item
\label{icyclicthree}
We have ${L={\mathbb Q}(x,y)}$ and ${[L:F]=3}$. Moreover, there exists a singular modulus~$z$ and ${\sigma \in G}$ such that
$$
x^\sigma=y, \quad y^\sigma=z, \quad z^\sigma=x.
$$
\item
\label{isigmaimproved}
There exists ${\sigma \in \Gal(L/F)}$ such that
\begin{equation}
\label{esigmaimproved}
x^\sigma\ne x,y, \qquad y^\sigma \ne x,y.
\end{equation}
\end{enumerate}
\end{lemma}
(Versions of this lemma were used, albeit implicitly, in~\cite{FR18} and elsewhere, but it does not seem to have appeared in the literature in this form.)
\begin{proof}
We may assume that every element in~$G$ which fixes~$x$ or~$y$ fixes both of them; otherwise we have option~\ref{ifixed}. We may also assume that~$x$ and~$y$ are conjugate over~$F$; otherwise, any ${\sigma \in G}$ not belonging to ${\Gal(L/{\mathbb Q}(x,y))}$ satisfies~\eqref{esigmaimproved}.
Assume first that ${L={\mathbb Q}(x,y)}$. Then the only element of~$G$ that fixes~$x$ or~$y$ is identity. Since~$x$ and~$y$ are conjugate over~$F$, there is exactly one ${\sigma \in G}$ with the property ${x^\sigma= y}$ and exactly one with the property ${y^\sigma=x}$. Hence we must have option~\ref{isigmaimproved} if ${[L:F]\ge 4}$. And if ${[L:F]\le 3}$ then we have one of the options~\ref{icyclictwo} or~\ref{icyclicthree}. This completes the proof in the case ${L={\mathbb Q}(x,y)}$.
Now assume that ${L\ne{\mathbb Q}(x,y)}$. Then ${{\mathbb Q}(x)={\mathbb Q}(y)}$ is a subfield of~$L$ of degree~$2$. Option~\ref{isigmaimproved} holds if ${[{\mathbb Q}(x):F]\ge 4}$, and option~\ref{icyclictwo} holds if ${[{\mathbb Q}(x):F]=2}$.
We are left with the case ${[{\mathbb Q}(x):F]=3}$. In this case the Galois orbit of~$x$ over~$F$ consists of~$3$ elements: $x,y$ and a certain~$z$.
The group~$G$ must be either cyclic of order~$6$ or symmetric~$S_3$. In the latter case~$G$ acts by permutations on the set ${\{x,y,z\}}$. But in this case~$G$ has an element fixing~$x$ and permuting $y,z$, a contradiction.
Thus,~$G$ is cyclic. Let~$\gamma$ be the non-trivial element of ${\Gal(L/{\mathbb Q}(x))}$. Then ${\gamma \notin \Gal(L/K)}$; otherwise, from ${L=K(x)}$ and ${x^\gamma=x}$ we would obtain that~$\gamma$ is the identity. It follows (see Proposition~\ref{pdihedral}) that~$\gamma$ does not commute with the elements of~$G$ of order~$3$, again a contradiction. The lemma is proved.
\end{proof}
\begin{lemma}
\label{ldeq}
Let~$x$ and~$y$ be singular moduli with the same fundamental discriminant~$D$, and let ${K={\mathbb Q}(\sqrt D)}$ be their common CM field. Assume that ${K(x)=K(y)}$. Then either ${h(\Delta_x)=h(\Delta_y)=1}$ or ${\Delta_x/\Delta_y\in \{4,1,1/4\}}$. Moreover, if, say, ${\Delta_x=4\Delta_y}$ then ${\Delta_y\equiv 1\bmod 8}$.
\end{lemma}
\begin{proof}
See \cite[Proposition~4.3]{ABP15} and \cite[Subsection~3.2.2]{BLP16} (where the congruence ${\Delta_y\equiv 1\bmod 8}$ is proved). Note that in~\cite{ABP15} a formally stronger hypothesis ${{\mathbb Q}(x)={\mathbb Q}(y)}$ is imposed, but in the proof it is only used that ${K(x)=K(y)}$.
\end{proof}
\begin{lemma}
\label{lndeq}
Let $x,x',y,y'$ be singular moduli. Assume that
$$
\Delta_x=\Delta_{x'}, \quad \Delta_y=\Delta_{y'}.
$$
Furthermore, assume that ${{\mathbb Q}(x,x')={\mathbb Q}(y,y')}$. Then we have the following.
\begin{enumerate}
\item
\label{inefudi}
If ${D_x\ne D_y}$ then ${{\mathbb Q}(x)={\mathbb Q}(y)}$.
\item
\label{iefudi}
If ${D_x=D_y}$ then ${K(x)=K(y)}$, where ${K=K_x=K_y}$ is the common CM-field for~$x$ and~$y$.
\end{enumerate}
\end{lemma}
\begin{proof}
See \cite[Lemma~7.1]{BFZ20}.
\end{proof}
\section{The linear relation}
\label{slinear}
Let ${x_1, \ldots, x_k}$ are non-zero singular moduli of the same fundamental discriminant~$D$ and ${m_1, \ldots, m_k\in {\mathbb Z}}$. We want to show that, under some reasonable assumption, the multiplicative relation
\begin{equation}
\label{emultdep}
x_1^{m_1}\cdots x_k^{m_k}=1
\end{equation}
implies the linear relation
\begin{equation}
\label{elinrelf}
\sum_{i=1}^k\frac{f(x_i)}{a(x_i)}m_i=0.
\end{equation}
Recall that we denote~$f_x$ or $f(x)$ the conductor, and~$a_x$ or $a(x)$ the denominator of the singular modulus~$x$, as in Subsection~\ref{ssgauss}.
Let us denote
\begin{align}
\label{edefxy}
X=\max\{|\Delta(x_i)|: 1\le i\le k\},\qquad
Y=\min\{|\Delta(x_i)|: 1\le i\le k\}.
\end{align}
\begin{proposition}
\label{plinearel}
Let~$A$ be a positive number such that
\begin{equation}
a(x_i)\le A \qquad (1\le i\le k).
\end{equation}
Assume that
\begin{equation}
\label{erootofy}
Y^{1/2} > \frac13Ak(\log X+\log A+\log k+20).
\end{equation}
Then~\eqref{emultdep} implies~\eqref{elinrelf}.
\end{proposition}
It often happens that we control only a part of the denominators of ${x_1,\ldots, x_k}$. In this case we cannot expect an identity like~\eqref{elinrelf}, but we may have good bounds for the part of the sum corresponding to the terms with small denominators.
We need some extra notation. Set ${f=\gcd(f_{x_1}, \ldots, f_{x_k})}$ and ${\Delta=Df^2}$. We also define
\begin{align*}
e_{x_i}=e(x_i)=f(x_i)/f, \qquad
m_i'=e(x_i)m_i \qquad (i=1, \ldots, k).
\end{align*}
Then we have
${\Delta(x_i)=e(x_i)^2\Delta}$,
and~\eqref{elinrelf} can be rewritten as
\begin{equation}
\label{elinrele}
\sum_{i=1}^k\frac{m_i'}{a(x_i)}=0.
\end{equation}
As indicated above, we want to obtain a less precise result, in the form of an inequality, which however holds true without the assumption that all the denominators are small. It will be practical to estimate separately the sums with positive and with negative exponents~$m_i$.
\begin{proposition}
\label{pineq}
Let $A,\eps$ be real numbers satisfying ${A\ge 1}$ and ${0<\eps \le 0.5}$. Assume that
\begin{equation}
\label{eassumpdelta}
|\Delta|^{1/2}\ge \max\left\{k\eps^{-1}\log X, \frac13A\bigl(\log(k\eps^{-1})+4\bigr)\right\}.
\end{equation}
Then
\begin{align}
\label{eupperforpos}
\sum_{\overset{a(x_i)<A}{m_i>0}}\frac{m_i'}{a(x_i)}&\le \sum_{m_i<0} \frac{|m_i'|}{\min\{a(x_i),A\}} +\eps\|\mathbf{m}'\|, \\
\label{eupperforneg}
\sum_{\overset{a(x_i)<A}{m_i<0}}\frac{|m_i'|}{a(x_i)}&\le \sum_{m_i>0} \frac{m_i'}{\min\{a(x_i),A\}} +\eps\|\mathbf{m}'\|.
\end{align}
Here we denote by $\|\mathbf{m}'\|$ the sup-norm:
${\|\mathbf{m}'\|=\max\{|m_1'|, \ldots, |m_k'|\}}$.
\end{proposition}
Propositions~\ref{plinearel} and~\ref{pineq} will be our principal tools in the proofs of Theorems~\ref{thpizthree} and~\ref{thprimel}. They will be
proved in Subsection~\ref{sslinearproofs}, after some preparatory work in Subsections~\ref{ssest} and~\ref{ssmasser}.
\subsection{Estimates for singular moduli}
\label{ssest}
For ${\tau\in \H}$ denote ${q=q_\tau:=e^{2\pi i\tau}}$.
Recall that the $j$-invariant function has the Fourier expansion
$$
j(\tau)=\sum_{k=-1}^\infty c_kq^k,
$$
where
$$
c_{-1}=1, \quad c_0=744, \quad c_1= 196884, \ldots
$$
are positive integers.
\begin{proposition}
Let ${\tau\in \H}$ satisfy ${\Im\tau=v \ge 5}$. Then
\begin{align}
j(\tau)
\label{etwoterms}
&=q^{-1}+744+ O_1(2\cdot10^5|q|), \\
\label{ethreeterms}
j(\tau)&=q^{-1}+744+196884q+ O_1(3\cdot10^7|q|^2),\\
\label{elogsimple}
\log |j(\tau)|&=2\pi v + O_1(800|q|)\\
\label{elogoneterm}
\log(qj(\tau))&=744q + O_1(5\cdot10^5|q|^2), \\
\label{elogtwoterms}
\log(qj(\tau))&=744q -79884q^2+O_1(2\cdot10^8|q|^3)
\end{align}
Finally, if
\begin{equation}
\label{esmalltau}
\tau \in \mathcal{F}, \qquad \Im\tau\le V, \qquad V\ge 5,
\end{equation}
then
\begin{equation}
\label{ecapv}
\log |j(\tau)|\le 2\pi V + 3000e^{-2\pi V}.
\end{equation}
\end{proposition}
\begin{proof}
Write ${\tau=u+vi}$. Then ${q=e^{2\pi ui}e^{-2\pi v}}$ and
${|q|= e^{-2\pi v}\le e^{-10\pi}}$.
For ${n\ge 0}$ denote
$$
j_n(\tau) = \sum_{k=n+1}^\infty c_kq^k.
$$
In particular,
$$
j_0(\tau) =j(\tau)-q^{-1}-744, \qquad j_1(\tau) =j(\tau)-q^{-1}-744-196884q.
$$
Positivity of the coefficients~$c_k$ implies that
$$
|j_n(\tau)q^{-n-1}| \le\sum_{k=n+1}^\infty c_k|q|^{k-n-1}
\le \sum_{k=n+1}^\infty c_ke^{-10\pi (k-n-1)}= e^{10\pi(n+1)}j_n(5i).
$$
In particular,
$$
|j_0(\tau)q^{-1}| \le e^{10\pi}j_0(5i)<2\cdot10^5,\qquad |j_1(\tau)q^{-2}| \le e^{20\pi}j_1(5i)<3\cdot10^7
$$
which proves expansions~\eqref{etwoterms},~\eqref{ethreeterms}. Using Lemma~\ref{llogs}, we deduce from them expansions~\eqref{elogoneterm},~\eqref{elogtwoterms}. Furthermore,~\eqref{elogsimple} easily follows from~\eqref{elogoneterm}.
Finally, if~\eqref{esmalltau} is satisfied, then ${v=\Im \tau}$ satisfies ${v\ge \sqrt3/2}$. Hence
$$
|j(\tau)|\le |q|^{-1}+ 744+j_0(\sqrt3/2) \le e^{2\pi v}+ 2079\le e^{2\pi V}(1+2079e^{-2\pi V}).
$$
Using Lemma~\ref{llogs}, we obtain~\eqref{ecapv}.
\end{proof}
\begin{corollary}
\label{clogx}
Let~$x$ be a singular modulus of discriminant~$\Delta$ and denominator~$a$.
Then we have the following.
\begin{enumerate}
\item
\label{iequ}
Assume that ${a\le 0.1|\Delta|^{1/2}}$. Then
$$
\log|x|= \pi \frac{|\Delta|^{1/2}}{a}+O_1(e^{-3|\Delta|^{1/2}/a}).
$$
\item
\label{inequ}
Let ${A\ge1}$ be such that ${a\ge A}$ and ${A\le 0.1|\Delta|^{1/2}}$. Then
$$
\log|x|\le \pi \frac{|\Delta|^{1/2}}{A}+e^{-3|\Delta|^{1/2}/A}.
$$
In particular, if ${|\Delta|\ge 10^4}$ then
$$
\log|x|\le \pi |\Delta|^{1/2}+e^{-3|\Delta|^{1/2}} \le 4|\Delta|^{1/2}.
$$
\item
\label{icomplog}
Define~${\tau_x}$
as in Subsection~\ref{ssgauss}.
Assume that ${a\le 0.1|\Delta|^{1/2}}$. Then
\begin{align}
\label{efirst}
\log(xq)& =744q + O_1(5\cdot10^5|q|^2), \\
\label{esecond}
\log(xq)&=744q -79884q^2+O_1(2\cdot10^8|q|^3).
\end{align}
where ${q= e^{2\pi i\tau_x} }$.
\end{enumerate}
\end{corollary}
We also need a lower bound. The following is (a weaker version of) \cite[Theorem~6.1]{BFZ20}, applied with ${y=0}$.
\begin{proposition}
\label{pbifazhu}
Let~$x$ be a singular modulus with discriminant ${\Delta_x\ne -3}$. Then ${|x|\ge|\Delta_x|^{-3}}$.
\end{proposition}
\subsection{Bounding the exponents}
\label{ssmasser}
Let ${\alpha_1, \ldots, \alpha_k}$ be non-zero algebraic numbers. The set of
${\mathbf{m}=(m_1, \ldots, m_k) \in {\mathbb Z}^k}$ such that
$$
\alpha_1^{m_1}\cdots \alpha_k^{m_k}=1
$$
is a subgroup in ${\mathbb Z}^k$, denoted here ${\Gamma(\alpha_1, \ldots,\alpha_k)}$ (or simply~$\Gamma$ if this does not cause confusion).
Masser~\cite{Ma88} showed that~$\Gamma$ admits a ``small'' ${\mathbb Z}$-basis. To state his result, let us introduce some notation. Let~$L$ be a number field.
We denote by ${\omega=\omega(L)}$ the order of the group of roots of unity belonging to~$L$, and by ${\eta=\eta(L)}$ the smallest positive height of the elements of~$L$:
$$
\eta=\min\{\height(\alpha): \alpha \in L,\ \height(\alpha)>0\}.
$$
Here $\height(\cdot)$ is the usual absolute logarithmic height.\footnote{There is no risk of confusing the height $\height(\cdot)$ and the class number $h(\cdot)$, not only because the former is roman and the latter is italic, but, mainly, because class numbers do not occur in this section, and heights do not occur outside this section.}
Finally, we define the norm of a vector ${\mathbf{m}=(m_1,\ldots,m_k)\in {\mathbb Z}^k}$ as
${\|\mathbf{m}\|=\max\{|m_1|, \ldots, |m_k|\}}$.
\begin{proposition}[Masser]
\label{pmasser}
Let ${\alpha_1, \ldots\alpha_k}$ be elements in ${L^\times}$.
Denote
$$
\height=\max\{\height(\alpha_1), \ldots, \height(\alpha_k),\eta\}.
$$
Then ${\Gamma(\alpha_1, \ldots,\alpha_k)}$
has a ${\mathbb Z}$-basis consisting of vectors with norm bounded by
${\omega(k\height/\eta)^{k-1}}$.
\end{proposition}
We want to adapt this result to the case when our algebraic numbers are singular moduli.
\begin{proposition}
\label{pbasrel}
Let ${x_1, \ldots, x_k}$ be non-zero singular moduli. Set
$$
X=\max\{|\Delta_{x_1}|,\ldots, |\Delta_{x_k}|\},
$$
and assume that, among ${D_{x_1},\ldots, D_{x_k}}$, there are~$\ell$ distinct fundamental discriminants. Then the group $\Gamma(x_1, \ldots, x_k)$ has a ${\mathbb Z}$-basis consisting of vectors with norm bounded by
${ 24 (c(\ell)kX^{1/2})^{k-1}}$,
where ${c(\ell)=3^{4^\ell+2^{\ell+1}+8}}$. In particular, if ${\ell=1}$ (that is, ${x_1, \ldots, x_k}$ all have the same fundamental discriminant) then ${c(\ell)=3^{16}}$.
\end{proposition}
The proof uses the following result due to Amoroso and Zannier~\cite[Theorem~1.2]{AZ10}.
\begin{lemma}
\label{laz}
Let~$K$ be a number field of degree~$d$, and let~$\alpha$ be an algebraic number such $K(\alpha)$ is an abelian extension of~$K$. Then either ${\height(\alpha)=0}$ or ${\height(\alpha)\ge 3^{-d^2-2d-6}}$.
\end{lemma}
\begin{proof}[Proof of Proposition~\ref{pbasrel}]
Denote
$$
K={\mathbb Q}(\sqrt{D_{x_1}}, \ldots, \sqrt{D_{x_k}}), \qquad L=K(x_1, \ldots, x_k).
$$
To apply Proposition~\ref{pmasser},
we have to estimate the quantities~$\height$,~$\eta$ and~$\omega$.
Since~$x_i$ is an algebraic integer, and every one of its conjugates~$x_i^\sigma$ satisfies ${\log|x_i^\sigma|\le 4|\Delta|^{1/2}}$ (see Corollary~\ref{clogx}:\ref{inequ}),
we have
${\height(x_i) \le 4X^{1/2}}$.
It follows that
${\height\le 4X^{1/2}}$.
Since ${[K:{\mathbb Q}]\le 2^\ell}$ and~$L$ is an abelian extension of~$K$, Lemma~\ref{laz} implies that
${\eta \ge 3^{-4^\ell-2^{\ell+1}-6}}$. Finally, we have ${\omega\le 24}$ from Corollary~\ref{crofoneinrcf}.
Putting all of these together, the result follows.
\end{proof}
In this article we will often work with relations of the form
$$
x_1^{m_1}\cdots x_k^{m_k}= (x_1')^{m_1}\cdots (x_k')^{m_k}.
$$
It is useful to have a bounded basis for the group of these relations as well.
\begin{proposition}
\label{pbasrelsigma}
Let ${x_1, \ldots, x_k, x_1', \ldots, x_k'}$ be singular moduli of discriminants not exceeding~$X$, and let~$\ell$ be the numbers of distinct fundamental discriminants among ${D_{x_1}, \ldots, D_{x_k'}}$.
Then the group $\Gamma(x_1/x_1', \ldots, x_k/x_k')$ has a ${\mathbb Z}$-basis consisting of vectors with norm bounded by
${ 24 (c(\ell)kX^{1/2})^{k-1}}$,
where ${c(\ell)=3^{4^\ell+2^{\ell+1}+8}}$. In particular, ${c(1)=3^{16}}$.
\end{proposition}
\begin{proof}
Same as for Proposition~\ref{pbasrel}, only with ${\height\le 8X^{1/2}}$.
\end{proof}
\subsection{Proofs of Propositions~\ref{plinearel} and~\ref{pineq}}
\label{sslinearproofs}
\begin{proof}[Proof of Proposition~\ref{plinearel}]
By Proposition~\ref{pbasrel} we may assume that
\begin{equation}
\label{embounded}
\|\mathbf{m}\|\le 24 (3^{16}kX^{1/2})^{k-1}.
\end{equation}
Using Corollary~\ref{clogx}, we
obtain
\begin{equation}
\label{ezero=}
0=\sum_{i=1}^km_i\log|x_i|= \pi|D|^{1/2}L+O_1(k\|\mathbf{m}\|e^{-3Y^{1/2}/A}),
\end{equation}
where~$L$ is the left-hand side of~\eqref{elinrelf}. (Recall that~$D$ denotes the common fundamental discriminant of ${x_1, \ldots, x_k}$.) Using~\eqref{erootofy} and~\eqref{embounded}, we deduce from this the estimate
${|L|\le 0.5A^{-k}}$. Since~$L$ is a rational number with denominator not exceeding~$A^k$, we must have ${L=0}$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{pineq}]
We will prove only~\eqref{eupperforpos}, because~\eqref{eupperforneg} is totally analogous.
Using Corollary~\ref{clogx} and Proposition~\ref{pbifazhu}, we obtain
\begin{align*}
0&=\frac{1}{\pi|\Delta|^{1/2}}\sum_{i=1}^km_i\log |x_i| \\
&\ge \sum_{\overset{1\le i\le k}{a(x_i)<A}}\frac{m_i'}{a(x_i)} - \sum_{m_i<0} \frac{|m_i'|}{\min\{a(x_i),A\}}+O_1\left(k\|\mathbf{m}'\| \frac{3\log X+e^{-3|\Delta|^{1/2}/A}}{\pi|\Delta|^{1/2}}\right),
\end{align*}
Using our hypothesis~\eqref{eassumpdelta}, we obtain
\begin{align*}
\frac{3k\log X}{\pi|\Delta|^{1/2}}&\le \frac3\pi\eps,\qquad \frac{ke^{-3|\Delta|^{1/2}/A}}{\pi|\Delta|^{1/2}} \le 0.01\eps,
\end{align*}
and the result follows.
\end{proof}
\section{Proof of Theorem~\ref{thprimel}}
\label{sprimel}
In this section we prove Theorem~\ref{thprimel}.
Thus, throughout this section, unless the contrary is stated explicitly,
$x$ and~$y$ are distinct singular moduli satisfying
\begin{equation}
\label{elowerprimel}
\max\{|\Delta_x|,|\Delta_y|\}\ge 10^8,
\end{equation}
and ${m,n}$ are non-zero integers such that ${{\mathbb Q}(x^my^n)\ne {\mathbb Q}(x,y)}$. We want to show that
\begin{equation}
\label{ewewant}
\begin{aligned}
&\Delta_x=\Delta_y, \qquad m=n, \qquad [{\mathbb Q}(x,y):{\mathbb Q}(x^my^m)]= 2, \\
&\text{$x$ and~$y$ are conjugate over ${\mathbb Q}(x^my^m)$}.
\end{aligned}
\end{equation}
Moreover, we want to show that, unless ${\{\Delta_x,\Delta_y\}}$ is of the form ${\{\Delta,4\Delta\}}$ for some ${\Delta\equiv 1\bmod 8}$, assumption~\eqref{elowerprimel} may be relaxed to
\begin{equation}
\label{elowerprimelsix}
\max\{|\Delta_x|,|\Delta_y|\}\ge 10^6.
\end{equation}
We assume that $x,y$ have the same fundamental discriminant; if this is not the case, then the argument is much simpler, see Subsection~\ref{ssdisfudis}. We denote by~$K$ the common CM field of $x,y$, and we set
${L=K(x,y)}$. We also denote
$$
\alpha=x^my^n, \qquad F={\mathbb Q}(\alpha), \qquad G= \Gal (L/F).
$$
Since~$F$ is a proper subfield of ${\mathbb Q}(x,y)$, there exists ${\sigma \in G}$ such that ${x^\sigma\ne x}$ or ${y^\sigma \ne y}$. We claim that
\begin{equation}
\label{ebothxy}
x^\sigma \ne x \quad \text{and}\quad y^\sigma \ne y.
\end{equation}
Indeed, if, say, ${y^\sigma=y}$ then ${(x^\sigma)^m=x^m}$, which implies ${x^\sigma=x}$ by Lemma~\ref{lnorou}.
The argument reduces to the study of the multiplicative relations
\begin{equation}
\label{emainmultrelation}
x^my^n(x^\sigma)^{-m}(y^\sigma)^{-n}=1,
\end{equation}
with various choices of ${\sigma \in G}$ satisfying~\eqref{ebothxy},
using Propositions~\ref{plinearel} and~\ref{pineq}.
In our usage of Propositions~\ref{plinearel} and~\ref{pineq} the parameters therein will satisfy the following restrictions:
\begin{equation}
\label{eparameters}
\begin{split}
k\le 4, \quad X=\max\{|\Delta_x|,|\Delta_y|\}\ge 10^8\ \textrm{resp.}\ 10^6, \\ Y=|\Delta|\ge \frac14X, \quad A\le 9, \quad \eps=0.016\ \textrm{resp.}\ 0.16.
\end{split}
\end{equation}
It is easy to verify that for any choice of parameters satisfying~\eqref{eparameters}, conditions~\eqref{erootofy} and~\eqref{eassumpdelta} are met, so using the propositions is justified.
\begin{remark}
\label{repses}
Everywhere throughout the proof until Subsection~\ref{sssfourdems} we assume that ${\max\{|\Delta_x|,|\Delta_y|\}\ge 10^6}$ and we use Proposition~\ref{pineq} with ${\eps=0.16}$. Starting from Subsection~\ref{sssfourdems} we have ${\{\Delta_x,\Delta_y\}=\{\Delta,4\Delta\}}$ with ${\Delta\equiv 1\bmod 8}$, which will allow us to assume ${\max\{|\Delta_x|,|\Delta_y|\}\ge 10^8}$ and use Proposition~\ref{pineq} with ${\eps=0.016}$.
\end{remark}
\subsection{A special case}
\label{ssspecialcase}
In this subsection we study the special case
\begin{equation}
\label{especial}
m=-n, \qquad \Delta_x=\Delta_y.
\end{equation}
We will need the result on this case to treat the general case. It will also be a good illustration of how our method works in a simple set-up.
Let ${\sigma \in G}$ be such that~\eqref{ebothxy} holds. We will apply Proposition~\ref{pineq} to the multiplicative relations
\begin{align}
\label{erelwithsigma}
x^my^{-m}(x^\sigma)^{-m}(y^\sigma)^m&=1,\\
\label{erelwithsigmaminone}
x^my^{-m}(x^{\sigma^{-1}})^{-m}(y^{\sigma^{-1}})^m&=1.
\end{align}
We may assume, up to Galois conjugation, that $x$ is dominant. Then neither of $y,x^\sigma, x^{\sigma^{-1}}$ is:
$$
a(x)=1, \qquad a(y),\ a(x^\sigma),\ a(x^{\sigma^{-1}})\ge 2.
$$
If one of ${a(y), a(x^\sigma)}$ is $\ge 3$, then, applying Proposition~\ref{pineq} to~\eqref{erelwithsigma} with ${A=3}$ and ${\eps=0.16}$, we obtain
$$
m \le \left(\frac{1}{\min\{3, a(y)\}}+ \frac{1}{\min\{3, a(x^\sigma)\}} + 0.16\right)m \le \left(\frac12+\frac13+0.16\right)m,
$$
a contradiction.
Thus, we must have ${a(y)=a(x^\sigma)=2}$. This means that~$y$ and~$x^\sigma$ are either equal or $4$-isogenous, see Proposition~\ref{pvarisog}:\ref{isubdomisog}. Hence so are $y^{\sigma^{-1}}$ and~$x$. Corollary~\ref{cisog} now implies that
${a(y^{\sigma^{-1}})\le 4}$.
Applying Proposition~\ref{pineq} to~\eqref{erelwithsigmaminone} with ${A=5}$ and ${\eps=0.16}$, we obtain
\begin{align*}
m+\frac{m}{a(y^{\sigma^{-1}})} &\le m\left(\frac{1}{\min\{5,a(y)\}}+ \frac{1}{\min\{5,a(x^{\sigma^{-1}})\}}+0.16\right)\\
& \le m\left(\frac12+\frac12+0.16\right).
\end{align*}
Since ${a(y^{\sigma^{-1}})\le 4}$, it is a contradiction.
We have proved that~\eqref{especial} is impossible.
\subsection{The general case: preparations}
\label{ssprepage}
Now we are ready to treat the general case.
Pick ${\sigma \in G}$ satisfying~\eqref{ebothxy}.
We have ${(x/x^\sigma)^m=(y^\sigma/y)^n}$, and, in particular,
$
{{\mathbb Q}((x/x^\sigma)^m)={\mathbb Q}((y/y^\sigma)^n)}
$.
The result of Subsection~\ref{ssspecialcase} implies that
$$
{\mathbb Q}((x/x^\sigma)^m)={\mathbb Q}(x,x^\sigma), \qquad {\mathbb Q}((y/y^\sigma)^n)={\mathbb Q}(y,y^\sigma).
$$
Lemma~\ref{lndeq}:2 now implies that ${K(x)=K(y)=L}$, and Lemma~\ref{ldeq} implies that there exists a discriminant~$\Delta$ such that
$$
\Delta_x=e_x^2\Delta, \quad \Delta_y=e_y^2\Delta, \qquad (e_x,e_y)\in \{(1,1), (2,1),(1,2)\},
$$
and, moreover,
\begin{equation}
\label{emoreover}
\text{if $(e_x,e_y) \ne (1,1)$ then $\Delta\equiv 1\bmod 8$}.
\end{equation}
We may and will assume in the sequel that
\begin{equation}
\label{ebasicassum}
m>0, \qquad e_xm\ge e_y|n|, \qquad a_x=1.
\end{equation}
If ${(e_x,e_y) \ne (1,1)}$ then~$x$ and~$y$ are not conjugate over~${\mathbb Q}$, and~\eqref{ebothxy} becomes
\begin{equation}
\label{emainoption}
x^\sigma \ne x,y, \qquad y^\sigma \ne x,y.
\end{equation}
When ${(e_x,e_y)=(1,1)}$, Lemma~\ref{lmainimproved} implies that, besides~\eqref{emainoption}, two more options are possible:
\begin{align}
\label{etwocyc}
&x^\sigma=y, \quad y^\sigma =x, \quad \hphantom{z^\sigma=x,} \qquad [{\mathbb Q}(x,y):F]=2;\\
\label{ethreecyc}
&x^\sigma=y, \quad y^\sigma=z,\quad z^\sigma=x, \qquad L={\mathbb Q}(x,y), \quad [L:F]=3.
\end{align}
Option~\eqref{etwocyc} is easy: relation~\eqref{emainmultrelation} becomes ${x^{m-n}=y^{m-n}}$, and Lemma~\ref{lnorou} implies that ${m=n}$, which means that we have~\eqref{ewewant}.
We have to show that the other two options are impossible. For~\eqref{ethreecyc} this is done in Subsection~\ref{sscycl}. Option~\eqref{emainoption} is much harder to dispose of, we deal with it in Subsection~\ref{ssmainopone}.
\subsection{Option~\eqref{ethreecyc}}
\label{sscycl}
We have
$$
x^my^{n-m}z^{-n}=1.
$$
Recall that ${m\ge |n|}$ and ${a_x =1}$, see~\eqref{ebasicassum}. In particular,
we must have ${a_y,a_z\ge 2}$.
Assume first that ${n>0}$. Then
$$
\max\{m,|n-m|, |-n|\}=m.
$$
Using Proposition~\ref{pineq} with ${\eps=0.16}$ and ${A=5}$, we obtain
$$
m \le
\frac{m-n}{\min\{5,a_y\}}+\frac{m}{\min\{5,a_z\}}n+0.16m \le \frac12(m-n)+\frac12n+0.16m,
$$
a contradiction.
{\sloppy
Now assume that ${n<0}$. Then
$$
\max\{m,|n-m|, |-n|\}\le 2m.
$$
If ${a_y\ge 3}$ then Proposition~\ref{pineq} with ${\eps=0.16}$ and ${A=3}$ implies that
$$
m\le \frac{1}{3}\cdot2m+0.16\cdot 2m,
$$
a contradiction. If ${a_y=2}$ then~$x$ and~$y$ are $2$-isogenous (see Proposition~\ref{pvarisog}:\ref{idenoms}), and so are ${y=x^\sigma}$ and ${z=y^\sigma}$. This implies that ${a_z\in \{1,4\}}$, see Corollary~\ref{cisog}. But ${a_z\ge 2}$, and so ${a_z=4}$. Proposition~\ref{plinearel} now implies that
$$
m+ \frac{n-m}{2}-\frac{n}{4}=0,
$$
yielding ${n=-2m}$, again a contradiction. Thus,~\eqref{ethreecyc} is impossible.
}
\subsection{Option~\eqref{emainoption}}
\label{ssmainopone}
We will use notation
$$
m'=e_xm, \qquad n'=e_yn.
$$
Recall that ${m'\ge |n'|}$ and ${a(x)=1}$, see~\eqref{ebasicassum}. This implies, in particular, that ${a(x^\sigma)\ge 2}$.
\subsubsection{One of~$y$,~$y^\sigma$ is dominant}
We start by showing that either~$y$ or~$y^\sigma$ is dominant.
\begin{proposition}
\label{pinside}
If ${n>0}$ then ${a(y^\sigma)=1}$ and ${\sigma^2=1}$. If ${n<0}$ then ${a(y)=1}$. In both cases we have ${(e_x,e_y)\ne (1,1)}$ and ${\Delta\equiv 1\bmod 8}$.
\end{proposition}
\begin{proof}
We treat separately the cases ${n>0}$ and ${n<0}$.
\bigskip
Assume first that ${n>0}$, but ${a(y^\sigma) \ge 2}$.
We know already that ${a(x^\sigma) \ge 2}$. If one of $a(x^\sigma),a(y^\sigma)$ is $\ge 3$ then, applying Proposition~\ref{pineq} with ${A=3}$ and ${\eps=0.16}$ to the relation
${x^my^n(x^\sigma)^{-m}(y^\sigma)^{-n}=1}$,
we obtain
$$
m'\le \frac{m}{\min\{3,a(x^\sigma)\}}+ \frac{n}{\min\{3,a(y^\sigma)\}}+0.16m\le \left(\frac12+\frac13+0.16\right)m,
$$
a contradiction.
Thus, ${a(x^\sigma)=a(y^\sigma)=2}$. This implies that ${e_x=e_y=1}$; in the opposite case ${\Delta=1\bmod 8}$ by~\eqref{emoreover}, and one of $\Delta_x,\Delta_y$, being ${4\bmod 32}$, cannot admit singular moduli with denominator~$2$, see Proposition~\ref{pevenden}:\ref{i432none}.
Since ${a(x^\sigma)=a(y^\sigma)=2}$ but ${x^\sigma\ne y^\sigma}$, the singular moduli~$x^\sigma$ and~$y^\sigma$ must be $4$-isogenous, see Proposition~\ref{pvarisog}:\ref{isubdomisog}. Hence so are~$x$ and~$y$. Corollary~\ref{cisog} now implies that ${a_y=4}$, and Proposition~\ref{plinearel} yields
$$
m'+\frac{n'}{4}-\frac{m'}{2}-\frac{n'}{2}=0.
$$
Hence ${n'=2m'}$, again a contradiction.
Thus, ${n>0}$ implies that ${a(y^\sigma)=1}$.
Note that if~\eqref{emainoption} holds for some ${\sigma\in G}$ then it also holds with~$\sigma$ replaced by $\sigma^{-1}$. Hence ${n>0}$ implies that ${a(y^{\sigma^{-1}})=1}$ as well. Since there can be only one dominant singular modulus of given discriminant, we must have ${y^{\sigma}=y^{\sigma^{-1}}}$. Hence ${\sigma^2=1}$ by Lemma~\ref{lperm}.
\bigskip
Now assume that ${n<0}$ but ${a(y) \ge 2}$.
The same argument as above shows that
${a(x^\sigma)=a(y)=2}$ and ${e_x=e_y=1}$.
The singular moduli $x^\sigma$ and~$y$ must be $4$-isogenous. Hence so are~$x$ and~$y^{\sigma^{-1}}$, which implies that ${a(y^{\sigma^{-1}})=4}$. Applying Proposition~\ref{pineq} with ${A=5}$ and ${\eps=0.16}$ to
$$
x^my^{-|n|}(x^{\sigma^{-1}})^{-m}(y^{\sigma^{-1}})^{|n|}=1,
$$
we obtain
$$
m'+\frac{|n'|}{4}\le
\frac{|n'|}{2}+ \frac{m'}{\min\{5,a(x^{\sigma^{-1}})\}}+0.16m'.
$$
Since ${|n'|\le m'}$ and ${a(x^{\sigma^{-1}})\ge 2}$, this is impossible.
Thus, we proved that ${n<0}$ implies that ${a(y)=1}$.
\bigskip
Finally, ${(e_x,e_y)\ne (1,1)}$, because there cannot be two distinct dominant singular moduli of the same discriminant. Hence ${\Delta\equiv 1\bmod8}$ by~\eqref{emoreover}.
The proposition is proved.
\end{proof}
\subsubsection{Controlling the four denominators}
\label{sssfourdems}
Thus, we know that two of the singular moduli ${x,y,x^\sigma,y^\sigma}$ are dominant. Unfortunately, we have no control over the denominators of the other two.
We will show now that, with a suitably chosen Galois morphism~$\theta$, we can control the denominators of all four of
${x^\theta,y^\theta,x^{\sigma\theta},y^{\sigma\theta}}$.
Note that, so far, we assumed that ${\max\{|\Delta_x|, |\Delta_y|\}\ge 10^6}$ and used Proposition~\ref{pineq} with ${\eps=0.16}$. However, now we know that
$$
\{\Delta_x,\Delta_y\}=\{\Delta, 4\Delta\} \quad \text{with}\quad \Delta\equiv 1\bmod 8,
$$
which allows us (See Remark~\ref{repses}) to assume that ${\max\{|\Delta_x|, |\Delta_y|\}\ge 10^8}$ and to use Proposition~\ref{pineq} with ${\eps=0.016}$.
\begin{proposition}
\label{ptwoeight}
There exists ${\theta \in \Gal(L/K)}$ such that, when ${(e_x,e_y)=(1,2)}$, we have
\begin{equation}
\label{etwoeight}
a(x^\theta)=a(x^{\sigma\theta})=2, \qquad a(y^\theta)=a(y^{\sigma\theta})=8,
\end{equation}
and when ${(e_x,e_y)=(2,1)}$, we have~\eqref{etwoeight} with~$x$ and~$y$ switched.
\end{proposition}
To prove this proposition, we need to bound ${|n'|}$ from below.
\begin{lemma}
Assume that ${(e_x,e_y)=(2,1)}$. Then ${|n'|\ge 0.85m'}$.
\end{lemma}
A similar estimate can be proved when ${(e_x,e_y)=(1,2)}$, but we do not need this.
\begin{proof}
Assume first that ${n<0}$. Then ${a(x)=a(y)=1}$. In particular,~$x$ and~$y$ are $2$-isogenous, and so are $x^\sigma,y^\sigma$.
Write
$$
x^my^{-|n|}(x^\sigma)^{-m}y^{|n|}=1.
$$
When ${a(x^\sigma)\ge 8}$ we use Proposition~\ref{pineq} with ${A=8}$ and ${\eps=0.016}$ to obtain
$$
m'\le |n'|+\frac{m'}{8}+0.016m',
$$
which implies that ${|n'|\ge 0.85m'}$.
When ${a(x^\sigma)\le 7}$, we must have ${a(x^\sigma)\in\{3,5,7\}}$ by Proposition~\ref{pevenden}:\ref{i432none}, because ${\Delta_x=4\Delta\equiv 4\bmod 32}$. Since~$x^\sigma$ and~$y^\sigma$ are $2$-isogenous, Corollary~\ref{cisog} implies that ${a(y^\sigma)\in \{a(x^\sigma),a(x^\sigma)/4\}}$, and we must have ${a(y^\sigma)= a(x^\sigma)}$. Using Proposition~\ref{plinearel} with ${A=7}$ we obtain
$$
m'-|n'|-\frac {m'}{a(x^\sigma)}+\frac{|n'|}{a(x^\sigma)}=0,
$$
which shows that ${|n'|=m'}$. This proves the lemma in the case ${n<0}$.
Now assume that ${n>0}$. Then ${a(x)=a(y^\sigma)=1}$ and ${\sigma^2=1}$. In particular,~$x$ and~$y^\sigma$ are $2$-isogenous, and so are ${x^{\sigma^{-1}}=x^\sigma}$ and~$y$. Arguing as above, we obtain that either ${a(x^\sigma)\ge 8}$ and ${n'\ge 0.85m'}$, or ${a(x^\sigma)\in\{3,5,7\}}$ and ${n'=m'}$.
The lemma is proved.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{ptwoeight}]
Let us assume first that ${n<0}$ and ${(e_x,e_y)=(1,2)}$.
We have
\begin{equation}
\label{edeltaxdeltay}
\Delta_x=\Delta\equiv1\bmod8, \qquad \Delta_y=4\Delta\equiv 4\bmod 32.
\end{equation}
By Proposition~\ref{pevenden}:\ref{ionemodeight}, there exist two distinct morphisms ${\theta \in \Gal(L/K)}$ such that ${a(x^\theta)=2}$. Of the two, there can be at most one with the property ${a(x^{\sigma\theta})=1}$. Hence we may find~$\theta$ satisfying
$$
a(x^\theta)=2, \qquad a(x^{\sigma\theta})\ge 2.
$$
Since ${n<0}$, we have ${a(y)=1}$ by Proposition~\ref{pinside}. Hence~$x$ and~$y$ are $2$-isogenous, and so are~$x^\theta$ and~$y^\theta$. It follows that ${a(y^\theta)\in \{2,8\}}$. But
${a(y^\theta)\ne 2}$ by Proposition~\ref{pevenden}:\ref{i432none}. Hence
${a(y^\theta)=8}$.
Proposition~\ref{pineq}, applied to
$$
(x^\theta)^m(y^\theta)^{-|n|}(x^{\sigma\theta})^{-m}(y^{\sigma\theta})^{|n|}=1
$$
with ${A=9}$ and ${\eps=0.016}$, implies that
$$
\frac{m'}2\le
\frac{|n'|}8+ \frac{m'}{\min\{9,a(x^{\sigma\theta})\}}+0.016m'.
$$
If ${a(x^{\sigma\theta})\ge 3}$ then this implies that
$$
m'\left(\frac12-\frac13\right) \le \frac{|n'|}8+0.016m',
$$
which is impossible because ${m'\ge |n'|}$. Hence
${a(x^{\sigma\theta})= 2}$,
and, as above, this implies that
${a(y^{\sigma\theta})= 8}$.
\bigskip
Now assume that ${n<0}$ and ${(e_x,e_y)=(2,1)}$. We again have~\eqref{edeltaxdeltay}, but with~$x$ and~$y$ switched.
Arguing as before, we find ${\theta \in \Gal(L/K)}$ such that
$$
a(y^\theta)=2, \qquad a(y^{\sigma\theta})\ge 2, \qquad a(x^\theta)=8.
$$
As before, in the case ${a(y^{\sigma\theta})\ge 3}$ we apply Proposition~\ref{pineq} to
$$
(x^\theta)^{-m}(y^\theta)^{|n|}(x^{\sigma\theta})^{m}(y^{\sigma\theta})^{-|n|}=1
$$
and obtain
$$
|n'|\left(\frac12-\frac13\right) \le \frac{m'}8+0.016m',
$$
which is impossible because ${|n'|\ge 0.85m'}$. Hence
${a(y^{\sigma\theta})= 2}$,
and, as above, this implies that
${a(x^{\sigma\theta})= 8}$.
\bigskip
Finally, let us assume that ${n>0}$. Then ${a(y^\sigma)=1}$ and ${\sigma^2=1}$. In particular,~$x$ and~$y^\sigma$ are $2$-isogenous, and so are~$x^\sigma$ and ${y^{\sigma^2}=y}$. Now writing
$$
x^m(y^\sigma)^{-n}(x^\sigma)^{-m}y^n=1,
$$
we repeat the previous argument with $y,y^\sigma$ switched, and with~$n$ replaced by~$-n$. The proposition is proved.
\end{proof}
\subsubsection{Completing the proof}
\label{ssscomplete}
Now we are ready to prove impossibility of option~\eqref{emainoption}. Let us summarize what we have. After renaming, we have distinct singular moduli $x_1,x_2$ of discriminant~$\Delta$ and $y_1,y_2$ of discriminant $4\Delta$ such that
$$
a(x_1)=a(x_2)=2, \qquad a(y_1)=a(y_2)=8,
$$
and
\begin{equation}
\label{eonetwo}
x_1^{m_1}y_1^{n_1}x_2^{-m_1}y_2^{-n_1}=1,
\end{equation}
where ${m_1,n_1}$ is a permutation of $m,n$. We want to show that this impossible.
Proposition~\ref{pbasrelsigma} implies that we may assume
\begin{equation}
\label{emaxmn}
\max\{|m_1|,|n_1|\}\le 10^{10}|\Delta|^{1/2}.
\end{equation}
Note also that
\begin{equation}
\label{elowerdeltabefore}
|\Delta|\ge 10^{7}
\end{equation}
by the assumption \eqref{elowerprimel}.
Proposition~\ref{pevenden} implies that, after possible renumbering, we have
\begin{align*}
\tau(x_1)& =\frac{1+\sqrt\Delta}{4}, & \tau(x_2)&=\frac{-1+\sqrt\Delta}{4},\\
\tau(y_1)&=\frac{b+\sqrt\Delta}{8},& \tau(y_2)&=\frac{-b+\sqrt\Delta}{8},
\end{align*}
where ${b\in \{\pm1,\pm3\}}$.
Denote ${t=e^{-\pi|\Delta|^{1/2}/4}}$ and ${\xi=e^{b\pi i/4}}$. Then
$$
e^{2\pi i\tau(x_1)}= it^2, \quad e^{2\pi i\tau(x_2)}= -it^2, \quad e^{2\pi i\tau(y_1)}= \xi t, \quad
e^{2\pi i\tau(y_2)}= \bar\xi t.
$$
We deduce from~\eqref{eonetwo} that
\begin{equation}
\label{elogofrel}
m_1\log(it^2x_1)-m_1\log(-it^2x_2)+n_1\log(\xi ty_1)-n_1\log(\bar\xi ty_2) \in \frac{1}{4}\pi i {\mathbb Z}.
\end{equation}
Transforming the left-hand side of~\eqref{elogofrel} using Corollary~\ref{clogx}:\ref{icomplog}, we obtain
$$
744(\xi-\bar\xi)tn_1+O_1\bigl(10^7t^2\max|m_1|,|n_1|\}\bigr) =\frac{1}{4}\pi i k
$$
for some ${k\in {\mathbb Z}}$.
An easy estimate using~\eqref{emaxmn} and~\eqref{elowerdeltabefore} shows that the left-hand side does not exceed~$10^{-1000}$ in absolute value. Hence ${k=0}$, and we obtain, again using~\eqref{emaxmn} and~\eqref{elowerdeltabefore}, that
\begin{equation*}
744|\xi-\bar\xi||n_1| \le 10^{17}|\Delta|^{1/2}e^{-\pi|\Delta|^{1/2}/4}< 10^{-900}.
\end{equation*}
Hence ${n_1=0}$, a contradiction.
This proves impossiblity of~\eqref{emainoption}, completing thereby the proof of Theorem~\ref{thprimel} in the case of equal fundamental discriminants.
\subsection{Distinct fundamental discriminants}
\label{ssdisfudis}
In this subsection ${D_x\ne D_y}$. Arguing as in the beginning of Subsection~\ref{ssprepage}, we find a Galois morphism~$\sigma$ such that ${{\mathbb Q}(x,x^\sigma)={\mathbb Q}(y,y^\sigma)}$. Lemma~\ref{lndeq}:\ref{inefudi} now implies that ${{\mathbb Q}(x)={\mathbb Q}(y)}$. Corollary~\ref{cxy} now implies that ${h(\Delta_x) =h(\Delta_y) \le 16}$, and our hypothesis
${\max\{|\Delta_x|, \Delta_y|\}\ge 10^6}$ contradicts Proposition~\ref{pwatki}.
Theorem~\ref{thprimel} is proved.
\section{Proof of Theorem~\ref{thpizthree}}
\label{spizthree}
In this section we prove Theorem~\ref{thpizthree}.
Throughout this section, unless the contrary is stated explicitly,
$x,y,z$ are distinct singular moduli satisfying
\begin{equation}
\label{elower}
\max\{|\Delta_x|,|\Delta_y|,|\Delta_z|\}\ge 10^{10},
\end{equation}
such that there exist non-zero integers ${m,n,r}$ with the property ${x^my^nz^r\in {\mathbb Q}^\times}$.
We assume that $x,y,z$ have the same fundamental discriminant~$D$; if this is not the case, then the argument is much simpler, see Subsection~\ref{ssdis}.
We denote by ${K={\mathbb Q}(\sqrt D)}$ the common CM field of $x,y,z$, and we denote
$$
L=K(x,y,z), \qquad G=\Gal(L/K).
$$
We set
$$
f=\gcd(f_x,f_y,f_z), \quad e_x=\frac{f_x}f, \quad e_y=\frac{f_y}f, \quad e_z=\frac{f_z}f, \quad \Delta=Df^2.
$$
Then ${\gcd(e_x,e_y,e_z)=1}$ and
\begin{align}
\label{eexeyez}
\Delta_x=e_x^2\Delta, \qquad \Delta_y=e_y^2\Delta, \qquad \Delta_z=e_z^2\Delta.
\end{align}
\subsection{The discriminants}
The following property, showing that the discriminants of $x,y,z$ have ``very much in common'', is the basis for everything.
\begin{proposition}
\label{pexeyez}
\begin{enumerate}
\item
\label{ikxykxzkyz}
We have
\begin{equation}
\label{ekxykxzkyz}
L=K(x,y)=K(x,z)=K(y,z).
\end{equation}
\item
\label{ikxkykz}
Each of the fields $K(x),K(y),K(z)$ is a subfield of~$L$ of degree at most~$2$.
\item
\label{iexeyez}
Up to permuting $x,y,z$ (and permuting correspondingly $m,n,r$) we have one of the options from Table~\ref{taexeyez}.
\end{enumerate}
\end{proposition}
\begin{table}
\caption{Data for Proposition~\ref{pexeyez}}
\label{taexeyez}
{\scriptsize
$$
\begin{array}{cccccclc}
e_x&[L:K(x)]&e_y&[L:K(y)]&e_z&[L:K(z)]&\Delta\equiv&\text{remark}\\
\hline
1&
1&
1&
1&
1&
1&\\
1&
1&
1&
1&
2&
1&
1\bmod8\\
1&
1&
2&
1&
2&
1&
1\bmod8\\
1&
2&
2&
1&
2&
1&
0\bmod4
&n=r\\
1&
2&
3&
1&
3&
1&
1\bmod3
&n=r\\
2&
2&
3&
1&
3&
1&
1\bmod24
&n=r\\
1&
2&
4&
1&
4&
1&
1\bmod8&
n=r\\
1&
2&
6&
1&
6&
1&
1\bmod24
&n=r\\
\end{array}
$$}
\end{table}
\begin{proof}
By the assumption, ${K(x^m)= K(y^nz^r)\subset K(y,z)}$. Lemma~\ref{lnorou} implies that ${K(x)= K(y^nz^r)}$, and, in particular, ${x\in K(y,z)}$. Hence ${L=K(y,z)}$. By symmetry, we obtain~\eqref{ekxykxzkyz}. This proves item~\ref{ikxykxzkyz}.
From~\eqref{elower} we may assume that, for instance, ${|\Delta_z|\ge 10^{10}}$.
Theorem~\ref{thprimel} implies that the field ${K(x)= K(y^nz^r)}$ is a subfield of~$L$ of degree at most~$2$, and the same holds true for the fields $K(y)$.
Unfortunately, we cannot make the same conclusion for $K(z)$, because, \textit{a priori}, we cannot guarantee that ${\max\{|\Delta_x|,|\Delta_y|\}\ge 10^8}$, which is needed to apply Theorem~\ref{thprimel} in this case. So a rather lengthy extra argument is required to prove that ${[L:K(z)]\le 2}$. We split it into two cases.
Assume first that
\begin{equation}
\label{egoodcases}
\Delta\ne -3,-4 \quad \text{or} \quad \gcd(e_x,e_z) >1.
\end{equation}
In this case, setting ${\ell=\lcm(e_x,e_y,e_z)}$, Proposition~\ref{pcompo} implies that ${L=K[\ell f]}$, the Ring Class Field of~$K$ of conductor $\ell f$. The Class Number Formula~\eqref{eclnfr} implies that
\begin{equation}
\label{ecnf}
2\ge [L:K(x)] = \Psi(\ell/e_x, \Delta_x),
\end{equation}
which results in one of the following six options:
\begin{equation}
\label{esixoptions}
\begin{aligned}
&\ell=e_x, && L=K(x);\\
&\ell=2e_x, &&L=K(x), && \Delta_x\equiv 1\bmod 8;\\
&\ell=2e_x, &&[L:K(x)]=2, && \Delta_x\equiv 0\bmod 4;\\
&\ell=4e_x, &&[L:K(x)]=2, && \Delta_x\equiv 1\bmod 8;\\
&\ell=3e_x, &&[L:K(x)]=2, && \Delta_x\equiv 1\bmod 3;\\
&\ell=6e_x, &&[L:K(x)]=2, && \Delta_x\equiv 1\bmod 24.
\end{aligned}
\end{equation}
In particular,
${e_x\ge \ell/6\ge e_z/6}$, which implies that ${|\Delta_x|\ge |\Delta_z|/36\ge 10^8}$. Hence we may again apply Theorem~\ref{thprimel} to conclude that ${[L:K(z)]\le 2}$ in case~\eqref{egoodcases}.
We are left with the case
\begin{equation}
\label{eimpossi}
\Delta \in \{-3,-4\}, \qquad \gcd(e_x,e_z)=1.
\end{equation}
We want to show that it is impossible. We claim that in this case ${\ph(e_z) \le 6}$.
Indeed, if ${x\in K}$ then
$$
2\ge [K(x,z):K(x)]=[K(z):K]= \frac{\Psi(e_z,\Delta)}{[\mathcal{O}_K^\times:\mathcal{O}_{K(z)}^\times]} \ge \frac{\Psi(e_z,\Delta)}{3},
$$
which proves that ${\ph(e_z)\le \Psi(e_z, \Delta)\le 6}$. And if ${x\notin K}$ then
$$
\Psi(e_z, e_x^2\Delta) = \bigl [K[e_ze_x]:K[e_x]\bigr]= \bigl [K[e_ze_x]:K(x,z)\bigr]\cdot [K(x,z) :K(x)].
$$
We have ${\bigl [K[e_ze_x]:K(x,z)\bigr]\le 3}$ by Proposition~\ref{pcompo}, and
$$
[K(x,z) :K(x)]=[L:K(x)]\le 2,
$$
as we have seen above. It follows that ${\ph(e_z) \le \Psi(e_z, e_x^2\Delta)\le 6}$.
From
${\ph(e_z) \le 6}$
we deduce ${e_z\le 18}$. Hence ${|\Delta_z|\le 4\cdot18^2<10^{10}}$, a contradiction. This shows impossibility of~\eqref{eimpossi}, completing thereby the proof of item~\ref{ikxkykz}.
We are left with item~\ref{iexeyez}.
As we have just seen, we have one of the six options~\eqref{esixoptions}, and similarly with~$x$ replaced by~$y$ or by~$z$.
It follows that ${\ell\mid 12}$.
Moreover,
\begin{align}
\label{etwo}
\text{if ${2\mid \ell}$\ } &\text{then ${\Delta\equiv 1\bmod 8}$ or ${\Delta\equiv 0\bmod 4}$}; \\
\label{efour}
\text{if ${4\mid \ell}$\ } &\text{then ${\Delta\equiv 1\bmod 8}$}; \\
\label{ethree}
\text{if ${3\mid \ell}$\ } &\text{then ${\Delta\equiv 1\bmod 3}$}.
\end{align}
Indeed, assume that ${2\mid \ell}$ but ${\Delta\equiv 5\bmod 8}$. Since ${e_x,e_y,e_z}$ are coprime, one of them, say,~$e_x$, is not divisible by~$2$. Then ${(\Delta_x/2)=-1}$, and ${\Psi(\ell/e_x, \Delta_x)}$ must be divisible by~$3$, which contradicts~\eqref{ecnf}. This proves~\eqref{etwo}.
In a similar fashion, one shows that
${\Psi(\ell/e_x, \Delta_x)}$ is divisible by~$4$ in each of the cases
\begin{align*}
&4\mid \ell, \qquad 2\nmid e_x, \qquad \Delta \equiv 0\bmod 4,\\
&3\mid \ell, \qquad 3\nmid e_x, \qquad \Delta \equiv 2\bmod 3,
\end{align*}
and it is divisible by~$3$ in the case
$$
3\mid \ell, \qquad 3\nmid e_x, \qquad \Delta \equiv 0\bmod 3.
$$
This proves~\eqref{efour} and~\eqref{ethree}.
Finally, it also follows from Theorem~\ref{thprimel} that, when, say, ${K(x)\ne L}$, we must have
${e_y=e_z}$ and ${n=r}$.
A little \textsf{PARI} script (or verification by hand) shows that, up to permuting $x,y,z$, all possible cases are listed in Table~\ref{taexeyez}.
\end{proof}
Recall that we denote ${f=\gcd(f_x,f_y,f_z)}$. Denote ${L_0=K[f]}$. Recall also that ${G=\Gal(L/K)}$. Then we have the following consequence.
\begin{corollary}
\label{cocases}
\begin{enumerate}
\item
\label{ioneortwo}
We have either
$$
L=L_0=K(x)=K(y)=K(z),
$$
or ${[L:L_0]=2}$, in which case exactly one of the fields $K(x),K(y),K(z)$ is~$L_0$ and the other two are~$L$.
\item
\label{ihundredone}
We have ${[L_0:K]\ge 101}$.
\item
\label{ibigdensthree}
There exists ${\sigma \in G}$ such that
${a(x^\sigma), a(y^\sigma),a(z^\sigma)\ge 13}$,
\item
\label{ibigdenstwo}
There exists ${\sigma \in G}$ such that
${a(x^\sigma), a(y^\sigma)\ge 18}$,
and the same statement holds true for $x,z$ and for $y,z$.
\item
\label{ibigdenone}
There exists ${\sigma \in G}$ such that
${a(x^\sigma)\ge 30}$,
and the same statement holds true for~$y$ and for~$z$.
\end{enumerate}
\end{corollary}
\begin{proof}
Item~\ref{ioneortwo} is proved just by exploring Table~\ref{taexeyez}. To prove item~\ref{ihundredone}, note that, since ${\max\{e_x,e_y,e_z\}\le 6}$, we have
\begin{equation}
\label{elowerdelta}
|\Delta|\ge \max\{|\Delta_x|,\Delta_y|,\Delta_z|\}/36\ge 10^8
\end{equation}
by~\eqref{elower}. Hence
${[L_0:K] =h(\Delta) \ge 101}$
by Proposition~\ref{pwatki}.
In proving item~\ref{ibigdensthree} we must distinguish the case ${L=L_0}$ and ${[L:L_0]=2}$. In the former case
${L=K(x)=K(y)=K(z)}$ and ${|G|=[L:K]\ge 101}$. Proposition~\ref{pcountden} implies that there exist at most~$32$ elements ${\sigma\in G}$ such that ${a(x^\sigma)< 13}$, and the same for~$y$ and~$z$. Since ${|G|\ge 101>96}$, we can find ${\sigma \in G}$ as wanted.
If ${[L:L_0]=2}$ then, say,
$$
K(x)=L_0, \qquad K(y)= K(z)=L,
$$
and
${|G|=[L:K]\ge 202}$. Again using Proposition~\ref{pcountden}, there exist at most~$32$ elements ${\sigma\in G}$ such that ${a(y^\sigma)<13}$, the same for~$z$, and at most~$64$ elements ${\sigma\in G}$ such that ${a(x^\sigma)<13}$. Since ${|G|\ge 202>128}$, we again can find a~$\sigma$ as wanted. This proves item~\ref{ibigdensthree}.
Item~\ref{ibigdenstwo} is proved similarly. In the case ${L=K(x)=K(y)}$ there exist at most~$48$ elements ${\sigma\in G}$ such that ${a(x^\sigma)< 18}$, and the same for~$y$. Since ${|G|\ge 101>96}$, we are done. In the case when one of $K(x),K(y)$ is~$L$, the other is~$L_0$, and
${[L:L_0]=2}$, we have ${48+96=144}$ unsuitable ${\sigma\in G}$; since ${|G|\ge 202}$, we are done again.
Item~\ref{ibigdenone} is similar as well: there exist at most~$99$ unsuitable~$\sigma$ when ${L=K(x)}$, and at most $198$ when ${[L:K(x)]=2}$; in both cases we conclude as before.
\end{proof}
In the sequel we denote
$$
m'=me_x, \qquad n'=ne_y, \qquad r'=re_z.
$$
We may and will assume that ${m>0}$ and that
\begin{equation}
\label{empgenprp}
m'\ge \max\{|n'|,|r'|\}, \qquad a_x=1.
\end{equation}
In the course of the argument we will study multiplicative relations
\begin{equation}
\label{emainmultrelationthree}
x^my^nz^r(x^\sigma)^{-m}(y^\sigma)^{-n}(z^\sigma)^{-r}=1,
\end{equation}
with various choices of ${\sigma \in G}$,
using Propositions~\ref{plinearel} and~\ref{pineq}.
In our usage of Propositions~\ref{plinearel} and~\ref{pineq} the parameters therein will satisfy the following restrictions:
\begin{equation}
\label{eparametersthree}
\begin{aligned}
&k= 6, \quad X=\max\{|\Delta_x|,\Delta_y|,|\Delta_z|\}\ge 10^{10}, \quad Y=|\Delta|\ge \frac1{36}X, \\
& A \le 162 \quad \text{for Proposition~\ref{plinearel}}, \\
&A\le 30, \quad \eps=0.01 \quad \text{for Proposition~\ref{pineq}}.
\end{aligned}
\end{equation}
It is easy to verify that for any choice of parameters satisfying~\eqref{eparametersthree}, conditions~\eqref{erootofy} and~\eqref{eassumpdelta} are met, so using the propositions is justified.
\subsection{The denominators}
We already know that~$x$ is dominant. Our principal observation is that either one of $y,z$ is dominant as well, or they both are subdominant. More precisely, we have the following.
\begin{proposition}
\label{pdomorsubd}
With~$y$ and~$z$ possibly switched, we have one of the following options:
\begin{align}
\label{edomopt}
&a_y=1,\hphantom{=a_z=2} \quad n<0;\\
\label{esubdopt}
&a_y=a_z=2,\hphantom{=1} \quad n,r<0.
\end{align}
\end{proposition}
\begin{proof}
With~$y$ and~$z$ possibly switched, we may assume that we are in one of the following cases:
\begin{align}
\label{eplusplus}
&n,r>0;\\
\label{eplusminus}
&n<0, \quad r>0;\\
\label{eminusminus}
&n,r<0.
\end{align}
We consider them separately.
\bigskip
Assume~\eqref{eplusplus}. Let~$\sigma$ be like in Corollary~\ref{cocases}:\ref{ibigdensthree}.
Applying Proposition~\ref{pineq} to
$$
x^my^nz^r(x^\sigma)^{-m}(y^\sigma)^{-n}(z^\sigma)^{-r}=1
$$
with ${A=13}$ and ${\eps=0.01}$,
and using that ${\max\{|m'|,|n'|,|r'|\}=m'}$ by~\eqref{empgenprp}, we obtain
\begin{equation*}
m' \le \frac{m'}{13}+\frac{n'}{13}+\frac{r'}{13}+0.01m'\le m'\left(\frac3{13}+0.01\right),
\end{equation*}
a contradiction. This shows that~\eqref{eplusplus} is impossible.
\bigskip
Now assume~\eqref{eplusminus}. We want to show that ${a_y=1}$ in this case. Thus, assume that ${a_y\ge 2}$. Using Corollary~\ref{cocases}:\ref{ibigdenstwo}, we find ${\sigma \in G}$ such that ${a(x^\sigma),a(z^\sigma)\ge 18}$.
Applying Proposition~\ref{pineq} to
$$
x^my^{-|n|}z^r(x^\sigma)^{-m}(y^\sigma)^{|n|}(z^\sigma)^{-r}=1
$$
with ${A=18}$ and ${\eps=0.01}$, we obtain
\begin{equation*}
m' \le \frac{|n'|}{\min\{18,a(y)\}}+\frac{m'}{18}+\frac{r'}{18}+0.01m'\le m'\left(\frac12+\frac2{18}+0.01\right),
\end{equation*}
a contradiction. This shows that in the case~\eqref{eplusminus} we must have~\eqref{edomopt}.
\bigskip
Finally, let us assume~\eqref{eminusminus}. If one of $a_y,a_z$ is~$1$ then we have~\eqref{edomopt}, possibly after switching. If ${a_y=a_z=2}$ then we have~\eqref{esubdopt}. Now let us assume that none of these two option happens; that is, both $a_y,a_z$ are $\ge2$ and one of them is $\ge3$.
Again using Corollary~\ref{cocases}, we may find ${\sigma \in G}$ such that ${a(x^\sigma)\ge 30}$. Applying Proposition~\ref{pineq}, we obtain, in the same fashion as in the previous cases, the inequality
$$
m'\le m'\left(\frac12+\frac13+\frac1{30}+0.01\right),
$$
a contradiction.
The proposition is proved.
\end{proof}
We study options~\eqref{edomopt} and~\eqref{esubdopt} in Subsections~\ref{ssdomopt} and~\ref{sssubdopt}, respectively.
\subsection{The ``dominant option''}
\label{ssdomopt}
In this subsection we assume~\eqref{edomopt}. Thus, we have the following:
$$
m>0, \quad n<0, \quad m'\ge\max\{ |n'|,|r'|\}, \qquad a_x=a_y=1.
$$
Since both~$x$ and~$y$ are dominant, we must have ${e_x\ne e_y}$. Exploring Table~\ref{taexeyez}, we find ourselves in one of the following cases:
\begin{align}
\label{ecaseonetwoeight}
&\{e_x,e_y\}=\{1,2\},&& e_z\in\{1,2\}, && \Delta\equiv 1\bmod 8,\\
\label{ecaseonetwotwofour}
&\{e_x,e_y\}=\{1,2\}, && e_z=2, && \Delta\equiv 0\bmod 4,\\
\label{ecaseonethreethreethree}
&\{e_x,e_y\}=\{1,3\},&& e_z=3, && \Delta\equiv 1\bmod 3,\\
\label{ecasetwothreethreetwentyfour}
&\{e_x,e_y\}=\{2,3\},&& e_z=3, && \Delta\equiv 1\bmod 24,\\
\label{ecaseonefourfoureight}
&\{e_x,e_y\}=\{1,4\},&& e_z=4, && \Delta\equiv 1\bmod 8,\\
\label{ecaseonesixsixtwentyfour}
&\{e_x,e_y\}=\{1,6\},&& e_z=6, && \Delta\equiv 1\bmod 24.
\end{align}
\begin{remark}
It is absolutely crucial that, in each of the cases above, a non-trivial congruence condition is imposed on~$\Delta$. This would allow us to use Propositions~\ref{poddpden} and~\ref{pevenden} to find Galois morphisms~$\sigma$ with well-controlled denominators of $x^\sigma,y^\sigma,z^\sigma$, which is needed for the strategy described in Subsection~\ref{sssstra} to work.
\end{remark}
Here are some more specific observations.
\begin{enumerate}
\item
We have either ${e_z=e_x}$ or ${e_z=e_y}$, which implies, in particular, that
\begin{equation}
\label{eazneone}
a_z\ne 1.
\end{equation}
\item
In case~\eqref{ecaseonetwoeight} we have ${K(x)=K(y)=K(z)=L}$.
\item
In cases \eqref{ecaseonetwotwofour}--\eqref{ecaseonesixsixtwentyfour} we have ${K(z)=L}$, and one of the fields $K(x)$ or~$K(y)$ is~$L$ as well, while the other is a degree~$2$ subfield of~$L$. More precisely:
\begin{itemize}
\item
if ${e_x<e_y=e_z}$ then ${K(y)=L}$ and ${[L:K(x)]=2}$;
\item
if ${e_y<e_x=e_z}$ then ${K(x)=L}$ and ${[L:K(y)]=2}$.
\end{itemize}
\item
Theorem~\ref{thprimel} implies that in
cases \eqref{ecaseonetwotwofour}--\eqref{ecaseonesixsixtwentyfour} we have
${n=r}$ when ${e_x<e_y}$, and ${m=r}$ when ${e_x>e_y}$.
\end{enumerate}
\subsubsection{The strategy}
\label{sssstra}
In each of cases \eqref{ecaseonetwoeight}--\eqref{ecaseonesixsixtwentyfour} we apply the following strategy.
\begin{itemize}
\item
Find possible values for~$a_z$.
\item
Using Proposition~\ref{poddpden} or~\ref{pevenden}, find several ${\sigma\in G}$ such that we can control the denominators
\begin{equation}
\label{ethreedens}
a(x^\sigma), \qquad a(y^\sigma), \qquad a(z^\sigma).
\end{equation}
\item
For every such~$\sigma$, and every possible choice of~$a_z$ and of denominators~\eqref{ethreedens}, Proposition~\ref{plinearel} implies the linear equation
$$
m'+n' + \frac{r'}{a_z} = \frac{m'}{a(x^{\sigma})}+ \frac{n'}{a(y^{\sigma})}+ \frac{r'}{a(z^{\sigma})}.
$$
With sufficiently many choices of~$\sigma$, we may hope to have sufficiently many equations to conclude that ${m'=n'=r'=0}$, a contradiction.
\end{itemize}
Practical implementation of this strategy differs from case to case. For instance, in cases~\eqref{ecaseonetwotwofour}--\eqref{ecaseonesixsixtwentyfour} we have ${m'=r'}$ or ${n'=r'}$, so wee need only two independent equations to succeed, while in case~\eqref{ecaseonetwoeight} three independent equations are needed.
Case~\eqref{ecaseonetwotwofour} is somewhat special, because we get only one equation. To complete the proof in that case, we need to use an argument similar to that of Subsection~\ref{ssscomplete}.
Below details for all the cases follow.
\subsubsection{Cases~\eqref{ecaseonethreethreethree}--\eqref{ecaseonesixsixtwentyfour}}
\label{sssfourcases}
In these cases ${K(z)=L}$, and one of the fields~$K(x)$, $K(y)$ is also~$L$ while the other is a degree~$2$ subfield of~$L$ . In this subsection we make no use of the assumption ${m'\ge |n'|}$. Hence we may assume that ${e_x<e_y=e_z}$, in which case we have
\begin{equation}
\label{eyzxdegs}
K(y)=K(z)=L, \qquad [L:K(x)]=2.
\end{equation}
Moreover, Theorem~\ref{thprimel} implies that in this case ${n=r}$, and that $y,z$ are conjugate over $K(x)$:
\begin{equation}
\label{exthetay}
y^\theta=z, \qquad z^\theta=y,
\end{equation}
where~$\theta$ is the non-trivial element of $\Gal(L/K(x))$.
Let us specify the general strategy described in Subsection~\ref{sssstra} for the cases~\eqref{ecaseonethreethreethree}--\eqref{ecaseonesixsixtwentyfour}.
\begin{enumerate}
\item
\label{iaz}
Proposition~\ref{pvarisog} implies that~$x$ and~$y$ are $\ell$-isogenous, where ${\ell=e_xe_y}$.
Hence ${x=x^\theta}$ and ${z=y^\theta}$ are $\ell$-isogenous as well. Using Corollary~\ref{cisog}, we may now shortlist possible values of the denominator $a_z$. Precisely,
$$
a_z \in \left(\frac{e_z}{e_x}\mathcal{Q}(\ell)\right) \cap{\mathbb Z}_{\ge 2},
$$
where we use notation ${\lambda S=\{\lambda s: s\in S\}}$.
For instance, in case~\eqref{ecasetwothreethreetwentyfour} we have ${\ell=6}$, and
$$
a_z\in \left(\frac32\left\{\frac16,\frac23,\frac32,6\right\}\right)\cap{\mathbb Z}_{\ge 2} = \{9\}.
$$
\item
Propositions~\ref{poddpden} and~\ref{pevenden} imply existence of morphisms~$\sigma_1$ and~$\sigma_2$ such that the three denominators
$a(x)$ (which is~$1$), $a(x^{\sigma_1})$ and $a(x^{\sigma_2})$
are distinct. Precisely,
\begin{itemize}
\item
if ${\Delta_x\equiv1\bmod 3}$ then~$3$ and~$9$ are denominators for~$\Delta_x$;
\item
if ${\Delta_x\equiv1\bmod 8}$ then~$2$ and~$4$ are denominators for~$\Delta_x$.
\end{itemize}
For instance, in case~\eqref{ecasetwothreethreetwentyfour} we may find~$\sigma_1$ and~$\sigma_2$ to have
$$
a(x^{\sigma_1})=3, \qquad a(x^{\sigma_2})=9.
$$
\item
Using again Corollary~\ref{cisog}, we may now shortlist the denominators $a(y^{\sigma_i})$ and $a(z^{\sigma_i})$. Precisely,
$$
a(y^{\sigma_i}), a(z^{\sigma_i}) \in \left(a(x^{\sigma_i})\frac{e_z}{e_x}\mathcal{Q}(\ell)\right) \cap{\mathbb Z}_{\ge 1}.
$$
For instance, in case~\eqref{ecasetwothreethreetwentyfour} we have
\begin{align*}
a(y^{\sigma_1}), a(z^{\sigma_1}) &\in \left(3\cdot\frac32\left\{\frac16,\frac23,\frac32,6\right\}\right)\cap{\mathbb Z}_{\ge 1} = \{3,27\}, \\
a(y^{\sigma_2}), a(z^{\sigma_2}) &\in \left(9\cdot\frac32\left\{\frac16,\frac23,\frac32,6\right\}\right)\cap{\mathbb Z}_{\ge 1} = \{9,81\}.
\end{align*}
\item
Now, Proposition~\ref{plinearel} implies the system of linear equations
\begin{equation}
\label{eequations4}
m'+\left(1 + \frac{1}{a(z)}\right)n' = \frac{m'}{a(x^{\sigma_i})}+ \left(\frac{1}{a(y^{\sigma_i})}+ \frac{1}{a(z^{\sigma_i})}\right)n'\qquad (i=1,2).
\end{equation}
(recall that ${n'=r'}$).
Solving the system, we find that ${m'=n'=0}$ in every instance, a contradiction. This shows impossibility of cases~\eqref{ecaseonethreethreethree}--\eqref{ecaseonesixsixtwentyfour}.
For instance, in case~\eqref{ecasetwothreethreetwentyfour},
equations~\eqref{eequations4} become
\begin{align*}
m'+\left(1+\frac19\right)n'&= \frac{m'}{3}+ \lambda n', \qquad \lambda\in \left\{\frac23,\frac13+\frac{1}{27}, \frac{2}{27}\right\}\\
m'+\left(1+\frac19\right)n'&= \frac{m'}{9}+\mu n', \qquad \mu\in \left\{\frac29,\frac19+\frac{1}{81}, \frac{2}{81}\right\},
\end{align*}
so nine systems in total, each of them having ${m'=n'=0}$ as the only solution.
\end{enumerate}
The numerical data obtained following these steps can be found in Table~\ref{tafourcases}. Note that we have 390 linear systems to solve: nine systems in cases \eqref{ecaseonethreethreethree}, \eqref{ecasetwothreethreetwentyfour}, 72 systems in case~\eqref{ecaseonefourfoureight}, and 300 systems in case \eqref{ecaseonesixsixtwentyfour}. Doing this by hand is impractical, and we used a \textsf{PARI} script for composing Table~\ref{tafourcases} and solving the systems.
\begin{table}
\caption{Cases~\eqref{ecaseonethreethreethree}--\eqref{ecaseonesixsixtwentyfour} and case~\eqref{ecaseonetwotwofour} with ${\Delta\equiv 4\bmod 32}$}
\label{tafourcases}
{\tiny
$$
\begin{array}{l|lc@{\ }ccc@{}c@{}c@{}c@{}c@{}c}
\text{case}&\Delta\equiv&e_x&e_y,&\ell& a_z&a(x^{\sigma_1})&a(y^{\sigma_1}), &a(x^{\sigma_2})&a(y^{\sigma_2}),&\text{total} \\
&&&e_z&& &&a(z^{\sigma_1}) &&a(z^{\sigma_2})&\text{systems} \\
\hline
\text{\eqref{ecaseonethreethreethree}}&1\bmod 3&1&3&3&9&3&\in\{3,27\}&9&\in\{9,81\}&9\\
\text{\eqref{ecasetwothreethreetwentyfour}}&1\bmod 24&2&3&6&9&3&\in\{3,27\}&9&\in\{9,81\}&9\\
\text{\eqref{ecaseonefourfoureight}}&1\bmod 8&1&4&4&\in\{4,16\}&2&\in\{2,8,32\}&4&\in\{4,16,64\}&72\\
\text{\eqref{ecaseonesixsixtwentyfour}}&1\bmod 24&1&6&6&\in\{4,9,36\}&2&\in\{2,8,18,72\}&3&\in\{3,12,27,108\}&300\\
\text{\eqref{ecaseonetwotwofour}}&4\bmod 32&1&2&2&4&8&\in \{8,32\}&16&\in \{16,64\}&9
\end{array}
$$
}
\end{table}
\begin{remark}
Using Propositions~\ref{poddpden} and~\ref{pevenden}, we can further refine the lists of possible denominators for~$z$,~$y^{\sigma_i}$ and~$z^{\sigma_i}$. For instance,
if the discriminant ${\Delta_y=\Delta_z}$ is ${\equiv0\bmod 9}$ then it cannot have denominators divisible by~$3$ but not by~$9$. Thus, in case~\eqref{ecasetwothreethreetwentyfour}, number~$3$ cannot be the denominator of~$y^{\sigma_1}$ or of~$z^{\sigma_1}$, and so we must have ${a(y^{\sigma_1})= a(z^{\sigma_1})=27}$. Arguments of this kind, used systematically, allow one to decimate the number of systems to solve.
However, the computational time for solving our systems being insignificant, we prefer to disregard this observation.
\end{remark}
\subsubsection{Case~\eqref{ecaseonetwotwofour}}
\label{sssonetwotwofour}
This case is similar to cases~\eqref{ecaseonethreethreethree}--\eqref{ecaseonesixsixtwentyfour}, but somewhat special. Let us reproduce our data for the reader's convenience:
$$
\{e_x,e_y\}=\{1,2\}, \quad e_z=2, \quad \Delta\equiv 0\bmod 4, \quad a_x=a_y=1.
$$
We may again assume that ${e_x<e_y}$, which means now that ${e_x=1}$ and ${e_y=2}$, and we again have~\eqref{eyzxdegs},~\eqref{exthetay}. Furthermore, step~\ref{iaz} of the strategy described in Subsection~\ref{sssfourcases} works here as well: we prove that each of~$y,z$ is $2$-isogenous to~$x$, which, in particular, allows us to determine
${a_z=4}$. For the latter use, let us note that
\begin{equation}
\label{etauxyz}
\tau_x=\frac{\sqrt\Delta}{2}, \qquad \tau_y=\sqrt\Delta, \qquad \tau_z= \frac{b'+\sqrt\Delta}{4} , \quad
\end{equation}
where
\begin{equation}
\label{ebprimebis}
b'=
\begin{cases}
0, &\text{if $\Delta\equiv 4\bmod 8$}, \\
2, &\text{if $\Delta\equiv 0\bmod 8$}.
\end{cases}
\end{equation}
The rest of the argument splits into two subcases. If ${\Delta\equiv 4 \bmod 32}$ then we may act as in Subsection~\ref{sssfourcases}. Proposition~\ref{pevenden} implies that there exist ${\sigma_1,\sigma_2\in G}$ such that
${a(x^{\sigma_1})=8}$ and ${a(x^{\sigma_2})=16}$.
As before, we can now determine possible denominators of $y^{\sigma_i}, z^{\sigma_i}$ (see the bottom line of Table~\ref{tafourcases}) and solve the resulting systems~\eqref{eequations4}, concluding that ${m=n=0}$.
Now assume that ${\Delta\not\equiv 4 \bmod 32}$. In this case~$2$ or~$4$ is a denominator for~$\Delta$, see Proposition~\ref{pevenden}:\ref{ieveryn}. Since ${\Delta_x=\Delta}$, there exists ${\sigma \in G}$ such that ${a(x^\sigma) \in \{2,4\}}$. We claim that
\begin{equation}
\label{ealldiff}
y^\sigma \ne y,z, \qquad z^\sigma \ne y, z.
\end{equation}
Indeed, if, say, ${y^\sigma=y}$ then ${\sigma = \id}$ because ${L=K(y)}$; but ${x^\sigma\ne x}$, a contradiction. For the same reason, ${z^\sigma\ne z}$. Now assume that ${y^\sigma=z}$. Theorem~\ref{thprimel} implies that~$y$ and~$z$ are conjugate over $K(x)$. Hence there exists ${\theta \in G}$ such that
$$
x^\theta=x, \qquad y^\theta=z, \qquad z^\theta =y.
$$
Then ${z^{\theta\sigma}=z}$, and, as before, ${\theta\sigma=\id}$, which is again a contradiction because ${x^{\theta\sigma}=x^\sigma\ne x}$. Similarly one shows that ${z^\sigma\ne y}$. This proves~\eqref{ealldiff}.
The cases ${a(x^\sigma)=2}$ and ${a(x^\sigma)=4}$ are very similar, but each one has some peculiarities, so we consider them separately.
\bigskip
Assume that ${a(x^\sigma)=2}$. Then ${a(y^\sigma)=a(z^\sigma)=8}$. Proposition~\ref{plinearel} gives
$$
m'+n'\left(1+\frac{1}{4}\right)=\frac{m'}{2}+n'\left(\frac18+\frac18\right),
$$
which is just ${m'=-2n'}$. Hence ${m=-4n}$. It follows that
${(x/x^\sigma)^4(y^\sigma/y)(z^\sigma/z)}$ is a root of unity. Since the roots of unity in~$L$ are of order dividing~$24$ (see Corollary~\ref{crofoneinrcf}), we obtain
\begin{equation}
\label{e96}
\bigl(x^{4}(x^\sigma)^{-4}y^{-1}z^{-1}y^\sigma z^\sigma\bigr)^{24}=1.
\end{equation}
Now we are going to argue as in Subsection~\ref{ssscomplete}. This means:
\begin{itemize}
\item
We give explicit expressions for the $\tau$- and $q$-parameters of all the six singular moduli occurring in~\eqref{e96}. Note that the $\tau$-parameters for $x,y,z$ are already given in~\eqref{etauxyz}, so we need to determine them only for $x^\sigma,y^\sigma,z^\sigma$.
\item
Taking the logarithm of~\eqref{e96}, we deduce that a certain linear combination of logarithms is a multiple of $\pi i/12$.
\item
Using the $q$-expansion from Corollary~\ref{clogx}, we obtain a contradiction.
\end{itemize}
Note however that in Subsection~\ref{ssscomplete} the first order expansion~\eqref{efirst} was sufficient, while now we would need the second order expansion~\eqref{esecond}.
Since $y,z,x^\sigma$ are distinct and $2$-isogenous to~$x$, we must have, in addition to~\eqref{etauxyz},~\eqref{ebprimebis},
\begin{equation}
\label{ebsecond}
\tau(x^\sigma)= \frac{b_2+\sqrt\Delta}{4}, \quad \text{where} \quad
b_2=
\begin{cases}
0, &\text{if $b'=2$}, \\
2, &\text{if $b'=0$}.
\end{cases}
\end{equation}
Furthermore, since $x,y^\sigma, z^\sigma$ are distinct and $2$-isogenous to~$x^\sigma$, we must have
$$
\{\tau(y^\sigma), \tau(z^\sigma)\} =\left\{\frac{b_2+\sqrt\Delta}{8},\frac{b_2'+\sqrt\Delta}{8}\right\}, \qquad
b_2' \in \{b_2+4,b_2-4\}.
$$
Denote ${t=e^{-\pi|\Delta|^{1/2}/4}}$ and ${\xi=e^{\pi ib_2/4}}$. Note that ${\xi\in \{1,i\}}$, and that
$$
e^{\pi ib'/2}= - \xi^2, \qquad e^{\pi ib_2'/4}=-\xi.
$$
We obtain
\begin{align*}
&e^{2\pi i\tau_x}=t^4, \qquad e^{2\pi i \tau_y} =t^8, \qquad e^{2\pi i \tau_z} = -\xi^2t^2, \qquad
e^{2\pi i \tau(x^\sigma)} = \xi^2t^2,\\
& \bigl\{e^{2\pi i \tau(y^\sigma)}, e^{2\pi i \tau(z^\sigma)}\bigr\} =\{\xi t,-\xi t\}.
\end{align*}
Taking the logarithm of~\eqref{e96}, we obtain
$$
4 \log(t^4x)-4\log (\xi^2t^2x^\sigma) -\log(t^8y)-\log(-\xi^2t^2z)+\log(-\xi t \cdot \xi t \cdot y^\sigma\cdot z^\sigma) \in \frac{\pi i}{12}{\mathbb Z}.
$$
The $q$-expansion~\eqref{esecond} from Corollary~\ref{clogx} implies that for some ${k\in {\mathbb Z}}$ we have
$$
\frac{\pi i}{12}k=-162000\xi^2t^2+O_1(10^{10}t^3).
$$
This easily leads to contradiction, exactly like in Subsection~\ref{ssscomplete}.
\bigskip
Now assume that ${a(x^\sigma)=4}$. Since~$x^\sigma$ is $2$-isogenous to~$y^\sigma$ and to~$z^\sigma$, Corollary~\ref{cisog} and Proposition~\ref{pevenden} imply that ${a(y^\sigma),a(z^\sigma) \in \{4,16\}}$. Note however that
$$
\Delta_y=\Delta_z=4\Delta\equiv 0\bmod 16,
$$
and Proposition~\ref{pevenden}:\ref{i16none} implies that there may be at most one singular modulus of this discriminant with denominator~$4$. But we already have ${a_z=4}$, and so neither of $a(y^\sigma),a(z^\sigma)$ can equal~$4$ by~\eqref{ealldiff}.
Thus, ${a(y^\sigma)=a(z^\sigma)=16}$. Proposition~\ref{plinearel} gives
$$
m'+n'\left(1+\frac{1}{4}\right)=\frac{m'}{4}+n'\left(\frac1{16}+\frac1{16}\right),
$$
which is ${m=-3n}$. Arguing as before,
we obtain
\begin{equation}
\label{e72}
\bigl(x^{3}(x^\sigma)^{-3}y^{-1}z^{-1}y^\sigma z^\sigma\bigr)^{24}=1.
\end{equation}
We have
$$
\tau(x^\sigma) =\frac{b_4+\sqrt\Delta}{8},
$$
and we want to specify this $b_4$. Since ${(b_4)^2\equiv \Delta \bmod 16}$, we must have
$$
\Delta \equiv 0,4\bmod 16, \qquad
b_4=
\begin{cases}
0, &\text{if ${\Delta \equiv 16 \bmod 32}$}, \\
4, &\text{if ${\Delta \equiv 0 \bmod 32}$},\\
\pm2, & \text{if ${\Delta \equiv 4 \bmod 16}$}.
\end{cases}
$$
In particular,
$$
b'\in \{b_4+2,b_4-2\}.
$$
Finally, since both $y^\sigma,z^\sigma$ have denominators~$16$ and are $2$-isogenous to~$x^\sigma$, we have
$$
\{\tau(y^\sigma), \tau(z^\sigma)\} =\left\{\frac{b_4+\sqrt\Delta}{16},\frac{b_4'+\sqrt\Delta}{16}\right\}, \qquad
b_4'\in \{b_4+8,b_4-8\}.
$$
Denote ${t=e^{-\pi|\Delta|^{1/2}/8}}$ and ${\xi=e^{\pi ib_4/8}}$. Note that ${\xi\in \{1,i, e^{\pm\pi i/4}\}}$, and that
$$
e^{\pi ib'/4}= \pm i\xi^2, \qquad e^{\pi ib_4'/8}=-\xi.
$$
We obtain
\begin{align*}
&e^{2\pi i\tau_x}=t^8, \qquad e^{2\pi i \tau_y} =t^{16}, \qquad e^{2\pi i \tau_z} = \eps i \xi^2t^4, \qquad
e^{2\pi i \tau(x^\sigma)} = \xi^2t^2,\\
& \bigl\{e^{2\pi i \tau(y^\sigma)}, e^{2\pi i \tau(z^\sigma)}\bigr\} =\{\xi t,-\xi t\},
\end{align*}
where ${\eps\in \{1,-1\}}$.
Taking the logarithm of~\eqref{e72}, we obtain
$$
3 \log(t^8x)-3\log (\xi^2t^2x^\sigma) -\log(t^{16}y)-\log(-\eps i\xi^2t^4z)+\log(-\xi t \cdot \xi t \cdot y^\sigma\cdot z^\sigma) \in \frac{\pi i}{12}{\mathbb Z}.
$$
The $q$-expansion~\eqref{esecond} implies that for some ${k\in {\mathbb Z}}$ we have
$$
\frac{\pi i}{12}k=-162000\xi^2t^2+O_1(10^{10}t^3),
$$
which again leads to a contradiction.
\subsubsection{Case~\eqref{ecaseonetwoeight}}
We want to adapt the procedure described in Subsection~\ref{sssfourcases} to this case. We reproduce our data for the reader's convenience:
\begin{equation}
\label{esixtwenty}
\{e_x,e_y\}=\{1,2\},\quad e_z\in\{1,2\}, \quad \Delta\equiv 1\bmod 8,\quad a_x=a_y=1.
\end{equation}
The singular moduli~$x$ and~$y$ are $2$-isogenous by Proposition~\ref{pvarisog}. However, now we have
${K(x)=K(y)=K(z)=L}$, which means that there does not exist ${\theta \in G}$ with the properties ${x^\theta=x}$ and ${y^\theta =z}$. Hence, a priori we
have no control of the degree of isogeny between~$x$ and~$z$. To gain such control we need to locate the denominator~$a_z$.
\begin{proposition}
\label{pazletwo}
Assume~\eqref{esixtwenty}. Then ${(e_x,e_y)=(1,2)}$ and
we have one of the options
\begin{equation}
\label{e1423}
e_z=1, \quad a_z=4 \qquad \text{or}\qquad
e_z=2, \quad a_z= 3.
\end{equation}
\end{proposition}
The proof consists of several steps. To start with, we eliminate the subcase ${(e_x,e_y)=(2,1)}$.
\begin{proposition}
\label{pazleone}
In case~\eqref{esixtwenty} we must have ${(e_x,e_y)=(1,2)}$.
\end{proposition}
\begin{proof}
Note that ${a_z>1}$, see~\eqref{eazneone}.
We will assume that ${(e_x,e_y)=(2,1)}$ and get a contradiction.
Since ${\Delta_y=\Delta\equiv 1\bmod 8}$, Proposition~\ref{pevenden} implies that there are~$2$ elements ${\sigma\in G}$ with the property ${a(y^\sigma)=2}$. Since ${L=K(z)}$, at most one of them may satisfies ${a(z^\sigma)=1}$. Hence there exists ${\sigma \in G}$ with the properties
${a(y^\sigma)=2}$ and ${a(z^\sigma) \ge 2}$.
Since~$x$ and~$y$ are $2$-isogenous, we must have ${a(x^\sigma)\in \{2,8\}}$. But~$2$ is not a denominator for~${\Delta_x=4\Delta}$ by Proposition~\ref{pevenden},
which implies that ${a(x^\sigma)=8}$. Thus, we found~$\sigma$ such that
$$
a(x^\sigma)=8, \qquad a(y^\sigma)=2, \qquad a(z^\sigma)\ge 2.
$$
We now want to arrive to contradiction in each of the cases
\begin{align}
\label{eone2}
&\text{one of $a(z),a(z^\sigma)$ is~$2$},\\
\label{eboth3}
&\text{both $a(z),a(z^\sigma)\ge3$}.
\end{align}
Assume~\eqref{eone2}. Then ${e_z=1}$, again by the same reason:~$2$ is not a denominator
for $4\Delta$. Hence there exists ${\theta\in G}$ such that ${y^\theta=z}$. Since $y,y^\sigma$ are $2$-isogenous, so are ${z=y^\theta}$ and ${z^\sigma=y^{\theta\sigma}=y^{\sigma\theta}}$. It follows that if one of the denominators $a(z),a(z^\sigma)$ is~$2$, then the other must be~$4$. Proposition~\ref{plinearel} now implies that
$$
m'+n'+\frac{r'}{a'}= \frac{m'}{8}+\frac{n'}{2}+\frac{r'}{a''}, \qquad \{a',a''\}=\{2,4\}.
$$
Hence
$$
\frac78m'= \frac{|n'|}{2}+ r'\left (\frac{1}{a''}-\frac{1}{a'}\right) \le m'\left(\frac12+\frac14\right),
$$
a contradiction. This eliminates~\eqref{eone2}.
\bigskip
In case~\eqref{eboth3}
Proposition~\ref{pineq} used with ${A=9}$ implies that
$$
m' +\frac{|n'|}{2}\le \frac{m'}{8}+ |n'|+\frac{|r'|}{d}+0.01m', \qquad
d=\begin{cases}
\min\{9, a(z^\sigma)\}, & r>0, \\
\min\{9, a(z)\}, & r<0.
\end{cases}
$$
Since ${d\ge 3}$, we obtain
$$
\left(\frac78-0.01\right)m' \le \frac{|n'|}{2}+\frac{|r'|}{3} \le m'\left(\frac12+\frac13\right),
$$
a contradiction. This rules~\eqref{eboth3} out as well. The proposition is proved.
\end{proof}
Next, we show impossibility of ${a_z=2}$.
\begin{proposition}
\label{paznotwo}
In case~\eqref{esixtwenty} we must have ${a_z\ge 3}$.
\end{proposition}
\begin{proof}
We already know that ${a_z\ge2}$ and that ${(e_x,e_y)=(1,2)}$.
We also note the statement is immediate for ${e_z=2}$, because~$2$ is not a denominator for $4\Delta$, see Proposition~\ref{pevenden}. Thus, let us assume that
$$
(e_x,e_y,e_z)=(1,2,1), \qquad a_z=2,
$$
and show that this is impossible.
Arguing as in the proof of Proposition~\ref{pazleone} but with~$x$ and~$y$ exchanging roles, we find~$\sigma$ satisfying
\begin{equation*}
a(x^\sigma)=2, \qquad a(y^\sigma)=8, \qquad a(z^\sigma)\ge 2.
\end{equation*}
Since $x,z$ are $2$-isogenous, we have ${a(z^\sigma)\in \{1,4\}}$, and we must have ${a(z^\sigma)=4}$ because ${a(z^\sigma)\ge 2}$.
Next, let ${\theta\in G}$ be defined by ${z^\theta=x}$. Since $x,z$ are $2$-isogenous, we must have ${a(x^\theta)=2}$, which implies that ${a(y^\theta)=8}$.
Applying Proposition~\ref{plinearel} to the relation
$$
(x^\sigma)^m(y^\sigma)^n(z^\sigma)^r= (x^\theta)^m(y^\theta)^n(z^\theta)^r,
$$
we obtain
$$
\frac{m'}{2}+\frac{n'}{8}+\frac{r'}{4}= \frac{m'}{2}+\frac{n'}{8}+\frac{r'}{1},
$$
which implies ${r=0}$, a contradiction.
\end{proof}
We also need to know that $|n'|$ is not ``much smaller'' than~$m'$.
\begin{proposition}
\label{pnprimebelow}
When ${r>0}$ we have ${|n'|> 0.87 m'}$.
When ${r<0}$ and ${a_z\ge a}$ we have ${|n'|> \lambda(a) m'}$, where
$$
\lambda(a)=0.956-\frac{1}{\min\{30,a\}}.
$$
\end{proposition}
Here are lower bounds for ${\lambda(a)}$ for some values of~$a$ that will emerge below:
$$
\begin{array}{r|rrrrr}
a=&3&5&6&24&30\\
\hline
\lambda(a) >&0.62&0.75&0.78&0.91&0.92
\end{array}
$$
\begin{proof}
When ${r>0}$ we use Corollary~\ref{cocases} to find~$\sigma$ such that ${a(x^\sigma),a(z^\sigma)\ge 18}$. Now Proposition~\ref{pineq} gives
$$
m' \le |n'| + \frac{m'}{18}+\frac{r'}{18}+0.01 m' \le |n'|+m'\left(\frac2{18}+0.01\right),
$$
which implies ${|n'|> 0.87 m'}$.
When ${r<0}$ we use Corollary~\ref{cocases} to find~$\sigma$ such that ${a(x^\sigma)\ge 30}$. When ${a_z\ge a}$, we obtain
$$
m'\le |n'|+ \frac{m'}{30}+\frac{|r'|}{\min\{30,a\}} +0.01 m' \le |n'|+m'\left(\frac1{30}+\frac{1}{\min\{30,a\}}+0.01\right),
$$
which implies ${|n'|\ge \lambda(a) m'}$.
\end{proof}
Now we are ready prove Proposition~\ref{pazletwo}.
\begin{proof}[Proof of Proposition~\ref{pazletwo}]
The proof is similar to that of Proposition~\ref{pazleone}, but with the roles of~$x$ and~$y$ exchanged. This means that, instead of the clean inequality ${m'\ge |n'|}$, we have to use weaker inequalities from Proposition~\ref{pnprimebelow}. This is why we cannot rule out options~\eqref{e1423}.
We already know that ${a_z\ge3}$ and that ${(e_x,e_y)=(1,2)}$.
We also note that~$4$ is not a denominator for $4\Delta$, see Proposition~\ref{pevenden}. Hence it suffices to
show that each of the options
\begin{align}
\label{e13}
&e_z=1, \qquad a_z\ge3, \qquad a_z\ne 4, \\
\label{e25}
&e_z=2, \qquad a_z \ge 5
\end{align}
leads to a contradiction. As in the proof of Proposition~\ref{paznotwo}, we fix ${\sigma\in G}$ satisfying
$$
a(x^\sigma)=2, \qquad a(y^\sigma)=8, \qquad a(z^\sigma)\ge 2.
$$
\bigskip
Let us assume~\eqref{e13}. In the same fashion as in the proof of Proposition~\ref{pazleone}, we show that $z,z^\sigma$ are $2$-isogenous. Hence ${\{a(z),a(z^\sigma)\}=\{a',2a'\}}$, where ${a'\ge 3}$.
If ${a'\ge 6}$ then, using
Proposition~\ref{pineq}, we obtain
$$
\frac{m'}2 +|n'|\le m'+ \frac{|n'|}8+\frac{|r'|}{6}+0.01m',
$$
which implies that
$$
|n'|\le \frac87m'\left(\frac12+\frac16+0.01\right)<0.78 m',
$$
contradicting the lower bound ${|n'|> 0.78m'}$ from Proposition~\ref{pnprimebelow}.
If ${a'\in \{3,4,5\}}$ then Proposition~\ref{plinearel} gives
$$
m'+n'+\frac{r'}{a(z)}=\frac{m'}{2}+\frac{n'}{8}+\frac{r'}{a(z^\sigma)}.
$$
This can be rewritten as
\begin{equation}
\label{enprimeid}
|n'|= \frac87\left(\frac{m'}{2}+r'\left(\frac{1}{a(z)}-\frac{1}{a(z^\sigma)}\right)\right),
\end{equation}
which implies
\begin{equation}
\label{e77}
|n'| \le \frac87m'\left(\frac12+\frac16\right)<0.77m',
\end{equation}
When ${r>0}$ this contradicts the lower bound ${|n'|> 0.87m'}$ from Proposition~\ref{pnprimebelow}. When ${r<0}$ and ${a_z=2a'}$ this contradicts the lower bound ${|n'|> 0.78m'}$ from Proposition~\ref{pnprimebelow}. Finally,
when ${r<0}$ and ${a_z=a'}$, we deduce from~\eqref{enprimeid} the sharper upper bound ${|n'|\le (4/7)m'}$, contradicting
the lower bound ${|n'|\ge 0.62m'}$ from Proposition~\ref{pnprimebelow}. This shows impossibility of~\eqref{e13}.
\bigskip
Now let us assume~\eqref{e25}.
Proposition~\ref{pineq} implies that
$$
\frac{m'}2 +|n'|\le m'+ \frac{|n'|}8+\frac{|r'|}{d}+0.01m', \qquad
d=\begin{cases}
\min\{9, a(z)\}, & r>0, \\
\min\{9, a(z^\sigma)\}, & r<0.
\end{cases}
$$
If ${r>0}$ then ${d\ge 5}$, and we obtain
$$
|n'|\le \frac87m'\left(\frac12+\frac15+0.01\right)<0.82 m',
$$
contradicting the lower bound ${|n'|> 0.87m'}$ from Proposition~\ref{pnprimebelow}.
If ${r<0}$ and ${d\ge 8}$ then
$$
|n'|\le \frac87m'\left(\frac12+\frac18+0.01\right)<0.73 m',
$$
contradicting the lower bound ${|n'|> 0.75m'}$ from Proposition~\ref{pnprimebelow}.
Thus,
$$
r<0, \qquad 3\le a(z^\sigma)\le 7.
$$
Since ${e_z=2}$, we must have
${a(z^\sigma)=p\in \{3,5,7\}}$. Hence~$y^\sigma$ and~$z^\sigma$ are $8p$-isogenous, and so are~$y$ and~$z$. It follows that ${a_z=8p\ge 24}$, and Proposition~\ref{pnprimebelow} implies the lower bound ${|n'|> 0.91m'}$.
On the other hand, Proposition~\ref{plinearel} implies that
$$
m'+n'+\frac{r'}{8p}= \frac{m'}{2}+\frac{n'}{8}+\frac{r'}{p},
$$
which yields
$$
|n'|= \frac{4m'}{7}+\frac{|r'|}{p} <0.91m',
$$
a contradiction. This shows impossibility of~\eqref{e25}.
The proposition is proved.
\end{proof}
Now it is easy to dispose of case~\eqref{ecaseonetwoeight}, alias~\eqref{esixtwenty}. We define~$\sigma_1$ as~$\sigma$ from the proof of Proposition~\ref{pazletwo}; that is,
${a(x^{\sigma_1})=2}$ and ${a(z^{\sigma_1})\ne 1}$.
Next, we define~$\sigma_2$ from
$$
z^{\sigma_2}=
\begin{cases}
x,& e_z=1,\\
y,& e_z=2.
\end{cases}
$$
Finally, we set ${\sigma_3=\sigma_2\sigma_1}$.
Using Proposition~\ref{pevenden} and Corollary~\ref{cisog}, we calculate the possible denominators, see Table~\ref{tadensonetwotwoeight}. A verification shows that, in each case, the system of~$3$ linear equation
\begin{equation}
\label{esystemthree}
m'+n' + \frac{r'}{a_z} = \frac{m'}{a(x^{\sigma_i})}+ \frac{n'}{a(y^{\sigma_i})}+ \frac{r'}{a(z^{\sigma_i})}
\qquad(i=1,2,3)
\end{equation}
has only the zero solution.
\begin{table}
\caption{Denominators for case~\eqref{ecaseonetwoeight} }
\label{tadensonetwotwoeight}
{\scriptsize
$$
\begin{array}{ccc}
(e_x,e_y,e_z)=(1,2,1), \quad a_z=4&&
(e_x,e_y,e_z)=(1,2,2), \quad a_z=3\\
&&\\
\begin{array}{cccc}
i&a(x^{\sigma_i})&a(y^{\sigma_i})&a(z^{\sigma_i})\\
\hline
1&2&8&\in \{2,8\}\\
2&4&16&1\\
3&\in\{2,8\}&\in \{8,32\}&2
\end{array}&&
\begin{array}{cccc}
i&a(x^{\sigma_i})&a(y^{\sigma_i})&a(z^{\sigma_i})\\
\hline
1&2&8&24\\
2&3&3&1\\
3&\in \{6,24\}&24&8
\end{array}\\
&&\\
\text{8 systems in total}&&\text{2 systems in total}
\end{array}
$$}
\end{table}
\subsection{The ``subdominant option''}
\label{sssubdopt}
In this subsection we assume~\eqref{esubdopt}. Thus, we have the following:
$$
m>0, \quad n,r<0, \quad m'\ge\max\{ |n'|,|r'|\}, \qquad a_x=1, \quad a_y=a_z=2.
$$
To start with, note that
\begin{equation}
\label{eyez}
e_y=e_z, \qquad \Delta_y=\Delta_z\equiv 1\bmod 8.
\end{equation}
Indeed, among the three numbers $e_x,e_y,e_z$ there are only two distinct integers, see Table~\ref{taexeyez}. If ${e_y\ne e_z}$ then, switching, if necessary,~$x$ and~$y$, we may assume that ${e_x=e_y}$. Hence ${K(x)=K(y)}$, and we have one of the following two possibilities:
\begin{align}
\label{exyzl}
&K(x)=K(y)=K(z)=L, \\
&K(x)=K(y)=L, \qquad [L:K(z)]=2. \nonumber
\end{align}
In the latter case we must have ${m=n}$ by Theorem~\ref{thprimel}, which is impossible because ${m>0}$ and ${n<0}$.
Thus, we have~\eqref{exyzl}. Lemma~\ref{ldeq} now implies that ${e_z=2}$ and ${\Delta_x=\Delta_y=\Delta\equiv 1\bmod 8}$. But then ${\Delta_z\equiv 4 \bmod 32}$, and we cannot have ${a_z=2}$ by Proposition~\ref{pevenden}:\ref{i432none}.
Thus, ${e_x=e_y}$ is impossible, which proves that ${e_y=e_z}$. Now Proposition~\ref{pevenden}:\ref{inonemodeight} implies that ${\Delta_y=\Delta_z\equiv 1\bmod 8}$, which completes the proof of~\eqref{eyez}.
Exploring Table~\ref{taexeyez} and taking note of~\eqref{eyez}, we end up with one of the following cases:
\begin{align}
\label{ealll}
&e_x\in \{1,2\}, && e_y=e_z=1, && \Delta\equiv 1\bmod 8, && L=K(x)=K(y)=K(z);\\
\label{exylznl}
&e_x\in\{1,2\},&& e_y=e_z=3, && \Delta\equiv 1 \bmod 24, && [L:K(x)]=2, \quad n=r.
\end{align}
Each of these cases can be disposed of using the strategy described in Subsection~\ref{sssstra}; moreover, the very first step of that strategy can be skipped, because~$a_z$ is already known.
Case~\eqref{ealll} is analogous to case~\eqref{ecaseonetwoeight}, but is much simpler, because, as indicated above, we already know~$a_z$. We define $\sigma_1,\sigma_2,\sigma_3$
by
$$
a(y^{\sigma_1})=1, \qquad a(z^{\sigma_2})=1, \qquad a( y^{\sigma_3})=8.
$$
Note that there can be several candidates for~$\sigma_3$, we just pick one of them. The possible denominators, determined using Corollary~\ref{cisog} and Proposition~\ref{pevenden}:\ref{i432none}, are given in Table~\ref{tadensalll}. A verification with \textsf{PARI} shows that each of the 12 possible systems
$$
m'+\frac{n'}{2}+ \frac{r'}{2}= \frac{m'}{a(x^{\sigma_i})}+\frac{n'}{a(y^{\sigma_i})}+\frac{r'}{a(z^{\sigma_i})} \qquad(i=1,2,3)
$$
has only the trivial solution ${m'=n'=r'=0}$.
\begin{table}
\caption{Denominators for case~\eqref{ealll} }
\label{tadensalll}
{\scriptsize
$$
\begin{array}{ccc}
(e_x,e_y,e_z)=(1,1,1)&&
(e_x,e_y,e_z)=(2,1,1) \\
&&\\
\begin{array}{cccc}
i&a(x^{\sigma_i})&a(y^{\sigma_i})&a(z^{\sigma_i})\\
\hline
1&2&1&4\\
2&2&4&1\\
3&\in\{4,16\}&8&\in\{2,8,32\}
\end{array}&&
\begin{array}{cccc}
i&a(x^{\sigma_i})&a(y^{\sigma_i})&a(z^{\sigma_i})\\
\hline
1&8&1&4\\
2&8&4&1\\
3&\in \{16,64\}&8&\in\{2,8,32\}
\end{array}\\
&&\\
\text{6 systems in total}&&\text{6 systems in total}
\end{array}
$$}
\end{table}
\bigskip
In case~\eqref{exylznl} we have ${n=r}$ (and also ${n'=r'}$), and so we need only~$\sigma_1$ and~$\sigma_2$. We do as in Subsection~\ref{sssfourcases}. Since ${\Delta_x\equiv 1 \bmod 3}$, we can find ${\sigma_1,\sigma_2}$ satisfying
${a(x^{\sigma_1})=3}$ and ${a( y^{\sigma_2})=9}$. Defining
$$
\ell=
\begin{cases}
6, &\text{when ${e_x=1}$}, \\
12, &\text{when ${e_x=2}$},
\end{cases}
$$
a quick verification shows that singular moduli $x,y$ are $\ell$-isogenous, and so are $x,z$. Using Corollary~\ref{cisog} and Proposition~\ref{poddpden}:\ref{idivbyp}, we determine the possible denominators: in both cases ${e_x=1}$ and ${e_x=2}$ we find that
$$
a(y^{\sigma_1}), a(z^{\sigma_1})=54, \qquad a(y^{\sigma_2}), a(z^{\sigma_2}) \in\{18,162\}.
$$
It follows that ${m',n'}$ satisfy one of the three linear systems
\begin{align*}
&\begin{cases}
\displaystyle m'+ n'\left(\frac12+\frac12\right) = \frac{m'}3 + \left(\frac{1}{54}+\frac{1}{54}\right)n', &\\
&\\
\displaystyle m'+ n'\left(\frac12+\frac12\right) = \frac{m'}9 +\lambda n',
\end{cases}\\
&\text{where}\quad \lambda \in \left\{\frac{1}{18}+\frac{1}{18}, \frac{1}{18}+ \frac{1}{162}, \frac{1}{162}+\frac{1}{162}\right\}.
\end{align*}
A verification shows that each of these systems has only the trivial solution ${m'=n'=0}$. This completes the proof of Theorem~\ref{thpizthree} for equal fundamental discriminants.
\subsection{Distinct fundamental discriminants}
\label{ssdis}
We are left with the case when the fundamental discriminants $D_x,D_y,D_z$ are not all equal. We may assume that
${|\Delta_z|\ge |\Delta_y| \ge |\Delta_x|}$. In particular, ${|\Delta_z|\ge 10^{10}}$ and ${\Delta_z\ne \Delta_x}$.
Theorem~\ref{thprimel} and Lemma~\ref{lnorou} imply that
\begin{equation*}
{\mathbb Q}(y)={\mathbb Q}(y^n)= {\mathbb Q}(x^mz^r) ={\mathbb Q}(x,z).
\end{equation*}
In particular,
\begin{equation}
\label{exziny}
{\mathbb Q}(x), {\mathbb Q}(z)\subset {\mathbb Q}(y).
\end{equation}
Furthermore, Theorem~\ref{thprimel} and Lemma~\ref{lnorou} imply that ${{\mathbb Q}(x)={\mathbb Q}(x^m)={\mathbb Q}(y^nz^r)}$ is a subfield of ${\mathbb Q}(y,z)$ of degree at most~$2$. Since ${z\in {\mathbb Q}(y)}$, this implies that
$$
[{\mathbb Q}(y):{\mathbb Q}(x)]\le 2.
$$
Unfortunately, we cannot claim similarly that ${[{\mathbb Q}(y):{\mathbb Q}(z)]\le 2}$, because we do not know whether the singular moduli $x,y$ satisfy the hypothesis of Theorem~\ref{thprimel}.
The rest of the proof splits into the following subcases:
\begin{align*}
D_y&\ne D_x;\\
|\Delta_y|&\ge10^8, \qquad D_y\ne D_z;\\
10^6\le |\Delta_y|&\le 10^8, \qquad D_y\ne D_z;\\
|\Delta_y|&\le 10^6, \qquad D_y\ne D_z.
\end{align*}
\subsubsection{The subcase ${D_x\ne D_y}$}
Since ${[{\mathbb Q}(y):{\mathbb Q}(x)]\le 2}$, Corollary~\ref{cxy} implies that ${[{\mathbb Q}(y):{\mathbb Q}]\le 32}$. It follows that ${h(\Delta_z)=[{\mathbb Q}(z):{\mathbb Q}]\le 32}$ as well. This contradicts Proposition~\ref{pwatki} because
${|\Delta_z|\ge 10^{10}}$.
\bigskip
In the remaining subcases we have ${D_y\ne D_z}$. Since ${z\in {\mathbb Q}(y)}$, the field ${\mathbb Q}(z)$ must be $2$-elementary by Proposition~\ref{pintersect}. Hence the proof will be complete if we show one of the following:
\begin{align}
\label{erhotwoz}
\rho_2(\Delta_z) &\le 6;\\
\label{eqyqz}
[{\mathbb Q}(y):{\mathbb Q}(z)]&\le 2.
\end{align}
Recall that $\rho_2(\cdot)$ is the $2$-rank, see Subsection~\ref{ssstwor}.
Indeed, assume that~\eqref{erhotwoz} holds. Since ${\mathbb Q}(z)$ is $2$-elementary, we have ${h(\Delta_z) = 2^{\rho_2(\Delta_z)}\le 64}$, contradicting Proposition~\ref{pwatki}.
Similarly, if~\eqref{eqyqz} holds then ${h(\Delta_z)\le 16}$ by Corollary~\ref{cxy}, again contradicting Proposition~\ref{pwatki}.
\subsubsection{The subcase $|\Delta_y|\ge10^8$, ${D_y\ne D_z}$}
\label{sssteneight}
Since ${|\Delta_y|\ge10^8}$, Theorem~\ref{thprimel} applies to the singular moduli $x,y$. It follows that
${{\mathbb Q}(z)={\mathbb Q}(z^r)={\mathbb Q}(x^my^n)}$ is a subfield of ${{\mathbb Q}(x,y)}$ of degree at most~$2$. Since ${x\in {\mathbb Q}(y)}$, we obtain
${[{\mathbb Q}(y):{\mathbb Q}(z)]\le 2}$, which is~\eqref{erhotwoz}.
\subsubsection{The subcase ${10^6\le |\Delta_y|\le 10^8}$, ${D_y\ne D_z}$}
If ${\Delta_y\not\equiv 4\bmod 32}$ then Theorem~\ref{thprimel} applies to the singular moduli $x,y$, and we may argue as in Subsection~\ref{sssteneight}. Now assume that ${\Delta_y\equiv 4\bmod 32}$. Then ${\rho_2(\Delta_y)\le \omega(\Delta_y)-2}$, see Proposition~\ref{pgauss}. Since
$$
10^8<4\cdot 3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot 19\cdot 23=446185740,
$$
we have ${\omega(\Delta_y) \le 8}$. Hence ${\rho_2(\Delta_y)\le6}$.
Now denote ${K={\mathbb Q}(\sqrt{\Delta_y})}$ the CM-field of~$y$. Since ${\mathbb Q}(z)$ is $2$-elementary,~$K$ is not contained in ${\mathbb Q}(z)$ by Proposition~\ref{preal}. Hence the group $\Gal({\mathbb Q}(z)/{\mathbb Q})$ is isomorphic to $\Gal(K(z)/K)$, which is a quotient group of $\Gal(K(y)/K)$. In particular,
${\rho_2(\Gal({\mathbb Q}(z)/{\mathbb Q}))\le \rho_2(\Gal(K(y)/K))\le 6}$, which is~\eqref{erhotwoz}.
\subsubsection{The subcase ${|\Delta_y|\le 10^6}$, ${D_y\ne D_z}$}
We note that
${10^6< 4\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13 \cdot 17= 1021020}$.
Hence every~$\Delta_y$ with ${|\Delta_y|\le 10^6}$ satisfies
${\rho_2(\Delta_y)\le \omega(\Delta) \le 6}$, again by Proposition~\ref{pgauss}. As we have seen above, this implies~\eqref{erhotwoz}.
The theorem is proved.
{\footnotesize
\bibliographystyle{amsplain}
|
1,108,101,566,768 | arxiv | \section{Introduction}
Plasmonic nanofluids, which contain a suspension of plasmonic nanoparticles in a base fluid, have been proposed as effective working fluids to directly convert solar radiation to thermal energy \cite{lee2012radiative}. Owing to the resonance characteristics of the localized surface plasmon (LSP), the absorption efficiency of the nanoparticles can be greatly enhanced with the excitation of the LSP, offering great potential in solar thermal applications. For instance, a direct-absorption solar collector (DASC) combined with a plasmonic nanofluid has drawn much attention for solar thermal energy harvesting in recent decades \cite{lee2012radiative, jeon2014optical, duan2018photothermal, mallah2018blended, mehrali2018full}. Recently, Qin \textit{et al.} \cite{qin2018optimization} showed how the spectral absorption coefficient of a plasmonic nanoparticle should be tuned (i.e., either uniformly or following the solar spectrum) by engineering nanoparticle suspensions to exploit the solar radiation maximally with the given constraint of the total particle concentration. Therefore, the effective tuning of the spectral absorption coefficients of plasmonic nanofluids is crucial for improving the thermal performance capabilities of DASCs.
As suggested by Lee \textit{et al.} \cite{lee2012radiative}, broadband absorption spectra can be designed by blending multiple types of nanoparticles given that the resonance wavelength of the LSP depends on the material, size and shape of the nanoparticles. The simplest and the most common structure is a spherical nanoparticle \cite{encina2010plasmon, schaeublin2012does}. Nano-spheres made with noble metals are widely utilized in various disciplines, such as in medical \cite{huang2010gold} and biological \cite{murphy2008gold, schaeublin2012does} applications. However, because the resonance peak of the LSP associated with nano-spheres mainly depends on the material properties \cite{maier2007plasmonics}, nano-spheres themselves may not be suitable for thermal applications. Thus, additional types of nanoparticles, such as silica core-metallic shell \cite{khanadeev2017optical, ma2017controllable} and nano-rod \cite{yu1997gold, schaeublin2012does} nanoparticles, have also been considered given their potential for better controllability. For core-shell nanoparticles, the ratio between the core radius and the shell thickness serves as a factor when tuning the absorption response \cite{oldenburg1998nanoengineering}, while the aspect ratio of the nano-rod performs this function \cite{link1999spectral}. Thus, the critical question is \textit{``What would the optimal combination of various types of nanoparticles be for the effective tuning of the spectral absorption coefficient?''} Note that finding the optimal combination of plasmonic nanoparticles is not a straightforward task due to the diversity and complexity of nanoparticles with regards to their materials and shapes. For instance, Taylor \textit{et al.} \cite{taylor2012nanofluid} just employed the Monte-Carlo approach (i.e., random generation and selection) to find the optimal blending combination of core-shell particles for a nanofluid-based optical filter.
In general, an inverse problem is a problem that requires the determination of the design of a system from its output response. It is known that finding a proper solution to an inverse problem is often challenging because most inverse problems are ill-posed and nonlinear \cite{modest2013radiative}. Nevertheless, if a particular solution-finding technique of an inverse problem is available, it can be readily applied to diverse engineering fields, such as magnetic resonance imaging \cite{sumpf2011model}, combustion \cite{liu2019inverse}, and radiative heat transfer \cite{yadav2019inverse, radfar2019application}. Tailoring the absorption spectrum of a plasmonic nanofluid can also be treated as an inverse problem when designing a system (i.e., combination of plasmonic nanoparticles) at a given response (i.e., the desired spectral absorption coefficient). Because a blended combination of nanoparticles is represented with a broad range of variables, it is difficult to determine the optimal composition of a nanofluid to have the desired absorption spectrum. Therefore, a modern solution technique, such as a genetic algorithm \cite{yadav2019inverse, radfar2019application}, can be employed to solve our blending problem.
Genetic algorithms (GAs) have been widely employed to solve many design problems by customizing a description of an individual chromosome (i.e., member of a population) and a fitness function (i.e., score of the chromosome) properly \cite{goldberg1988genetic, harman2001search}. For example, the dimensions of a tandem-grating nanostructure for a solar thermal absorber \cite{choi2018robust}, the spectral distribution of absorption coefficients for DASC \cite{qin2018optimization}, and the dimensions of a multi-layer microcylinder for a plasmonic nanojet \cite{huang2019optimization} have been optimized based on carefully defined chromosomes and fitness functions. Here, we also apply a GA to find the optimal combination of plasmonic nanoparticles to achieve the desired spectral absorption coefficient of a nanofluid. To maximize the diversity of the plasmonic response of the nanoparticles, we consider two materials (gold and silver) and three types of nanoparticle shapes (nano-sphere, core-shell, and nano-rod). The target spectral absorption coefficient of the plasmonic nanofluid is first set to be either uniform or to follow the solar spectrum \cite{qin2018optimization}. In addition, a step-function-like absorption coefficient will be designed for a hybrid solar PV/T application \cite{taylor2012nanofluid, jia2019development}. Finally, how the inhomogeneous broadening effect caused by the fabrication uncertainty of the nanoparticles \cite{khanadeev2017optical, jeon2014optical} changes their optimal combination is also investigated.
\section{Modelling}
\subsection{Absorption coefficient of a blended plasmonic nanofluid}
It is well known that subwavelength-size metallic nanoparticles can support a localized surface plasmon (LSP), whose resonance condition depends strongly on the materials, sizes, and shapes of the nanoparticles \cite{bohren1983absorption, hutter2004exploitation, jain2006calculated}.
In the present study, we consider two materials and three shapes of nanoparticles (see Fig.\ \ref{fig:particleScheme}) to diversify a number of possible blending combinations. The ranges of each design variable are carefully constrained according to the literatures \ \cite{khanadeev2017optical, koktan2017magnetic, jeon2014optical}. These are listed in Table\ \ref{tbl:variableRange}.
\begin{figure}[!t]
\centering
\includegraphics[width=0.75\textwidth]{Fig1}
\caption{Schematic of the nano-sphere, core-shell, and nano-rod shapes. The design variables are the radius ($r$) for the nano-sphere, the core radius ($r_c$) and the shell thickness ($t_s$) for the core-shell, and the radius ($r_r$) and the length ($l_r$) for the nano-rod.}
\label{fig:particleScheme}
\end{figure}
\begin{table}[!b]
\centering
\caption{Ranges of each design variable when training the surrogate model}
\begin{tabular}{c|c|c}
\hline
Type & Design variable & Range (nm) \\ \hline
Nano-sphere & Radius ($r$) & 10 $\sim$ 100 \\ \hline
\multirow{2}{*}{Core-shell} & Core radius ($r_c$) & 5 $\sim$ 90\\
& Shell thickness ($t_s$) & 5 $\sim$ ($100 - r_c$)\\ \hline
\multirow{2}{*}{Nano-rod} & Radius ($r_r$) & 6 $\sim$ 30\\
& Length ($l_r$) & $\text{max}(16, 2r_r)$ $\sim$ 200\\ \hline
\end{tabular}
\label{tbl:variableRange}
\end{table}
For a given nanoparticle, its spectral absorption efficiency, $Q_{a}(\lambda)$, can be calculated by solving Maxwell's equations. For nano-sphere and core-shell particles, Mie-scattering theory \cite{bohren1983absorption} and a modified version of it \cite{lee2012radiative} were used to determine $Q_{a}(\lambda)$. For the nano-rod, a boundary element method (BEM) was applied to obtain the polarization- and direction-averaged absorption efficiency. In this study, the open-source BEM software MNPBEM\cite{waxenegger2015plasmonics} was used. In the calculations, the permittivities of gold, silver, silicon dioxide, and water (i.e., the base fluid) were used from tabulated data \cite{palik1998handbook}. As discussed by Lee \textit{et al.}\cite{lee2012radiative}, if the radius (or thickness) of a metal is smaller than its mean-free-path of conduction electrons, we also must consider that the size-dependent permittivity of the metal should differ from that of the bulk metal due to electron-boundary scattering. Furthermore, a broadening effect will arise due to the modification of the permittivity. In this work, the effect of electron-boundary scattering is neglected for simplicity, though this effect will be discussed later.
With the calculated absorption efficiency of the $i$-th particle in water, $Q_{a, i}(\lambda)$, the corresponding absorption coefficient of the nanofluid, $\alpha_{a, \lambda}$, can be expressed as\cite{bohren1983absorption}:
\begin{equation}
\alpha_{i, \lambda} = \frac{3 f_i}{2 D_i}Q_{a, i}(\lambda)\label{eq:eachAbsorptionCoefficient}
\end{equation}
where, $f_i$ is the volume concentration of the nanoparticle and $D_i = \sqrt[3]{6\times\text{(volume of particle)}/\pi}$ is the effective diameter of each particle \cite{jeon2014optical}. If $N$ types of nanoparticles are dispersed together, the resulting absorption coefficient of blended plasmonic nanofluid ($\alpha_\lambda$) is then given by:
\begin{equation}
\alpha_\lambda = \left ( 1 - \sum^N_{i=1}{f_i} \right ) \alpha_{w, \lambda} + \sum^N_{i=1}{\alpha_{i, \lambda}}\label{eq:absorptionCoeffcient}
\end{equation}
where, $\alpha_{w, \lambda}$ is the absorption coefficient of the water itself. Because solar irradiance begins at approximately $\lambda= 300$ nm and the absorption coefficient of water becomes dominant after $\lambda= 1,100$ nm, we calculate the absorption efficiency spectrum of each particle from 300 nm to 1,100 nm in 10 nm intervals (i.e., 81 spectral data points).
In principle, $Q_{a, i}(\lambda)$ must be known \textit{a priori} in each computation of the fitness function of a GA. Because the calculation of $Q_{a, i}(\lambda)$ takes about 3 min and the average number of fitness calculations in our GA numbers into the thousands, it is not feasible to compute it every time. To reduce the computational cost, we decided to build a surrogate model to estimate the absorption efficiency of each nanoparticle, i.e., [Input: geometry of $i$-th particle and $\lambda$ $\rightarrow$ Output: $Q_{a, i}(\lambda)$]. To ensure the accuracy of the surrogate model, an artificial neural network model\cite{seo2019design} was employed as part of a modelling technique. To train the neural network models, samples were composed with 2 nm intervals of the design variables and a 10 nm interval of the wavelength. Consequently, we constructed and applied accurate surrogate models with correlation values exceeding $R=0.999$ with a regulated amount of sample data. Note that neural networks were modelled with three fully connected hidden layers and 10 nodes in each layer, which was enough to construct a surrogate model for estimating the $Q_{a}$ spectrum. The accuracy of the model was estimated with the difference between the predicted and actual $Q_{a, i}(\lambda)$ values at the peak location, where the maximum $Q_{a, i}$ value was achieved. As a result, the accuracy of model used in this work was found to be between 0.3\% (for the nano-spheres) and 1.6\% (for the core-shells and the nano-rods) on average.
\subsection{Customized genetic algorithm}
The genetic algorithm (GA) is a powerful method for solution processes owing to its ranges for diverse applicability for a variety of problems \cite{goldberg1988genetic, harman2001search}. In the world of a GA, the population consists of individuals. Each individual has its own chromosome (i.e., set of genes) and evolves along descent generations. Based on simple and bio-mimicking procedures, the population of the GA will evolve to obtain the best individual, which has the best chromosome, through a process of selection, crossover and mutation. The GA can be utilized with proper modification of its gene description, fitness function, and any embedded algorithms or hyper-parameters. In this study, (1) descriptions of the chromosomes (or genes), (2) fitness function, (3) crossover, and (4) mutation algorithms are customized especially for solving our inverse problem, designing of the system (a combination of plasmonic nanoparticles) at the given response (the desired spectral absorption coefficient).
The most significant aspect of customization is to define the chromosomes of the GA. Because the chromosomes of individuals must be related to a combination of plasmonic nanoparticles, we defined the chromosome to possess a set of nanoparticles as a genes. Each gene on the chromosome has particle properties, such as the material (gold or silver), shape (nano-sphere, core-shell, nano-rod), design variables (geometric parameters), and volume concentration. A chromosome is implemented as a list of genes. When a chromosome is created, the properties of each gene are determined randomly within their ranges. Note that the volume concentration is intentionally set to be less than 0.005\% divided by the number of genes (i.e., the number of nanoparticle types in Eq.\ \eqref{eq:absorptionCoeffcient}) to match the scale of each particle's volume fraction to 0.0001\% \cite{jeon2014optical}.
A fitness function is usually set to be a distance or a loss function, as the optimization process evolves to minimize a score. In this work, the fitness function is defined as the sum of square error (SSE):
\begin{equation}
\text{SSE} = \sum_{j=0}^{80}{(\alpha_{\lambda_j} - \alpha_{\text{target}, \lambda_j})^2}
\end{equation}
where, $\lambda_i$ is the wavelength in interest with a 10 nm interval (i.e., $300 + 10j$ nm) and $\alpha_{\text{target}, \lambda}$ is the target absorption coefficient spectrum defined in Section\ \ref{sec:targetAbs}. The absorption coefficient of the blended nanofluid was calculated from Eqs.\ \eqref{eq:eachAbsorptionCoefficient} and \eqref{eq:absorptionCoeffcient}. To obtain $Q_{a, i}(\lambda)$, the type and design variables of the nanoparticles described in the chromosome are required. The GA scored each individual with this SSE value and caused the population to evolve to minimize the score.
After the scores of individuals are evaluated by the fitness function, the GA will prepare the population of the next generation. Initially, the highest scoring individual will remain based on elitism. By default, the top 5\% individuals in terms of their scores will move to the next generation. Next, the GA selects individuals as the parents of the rest (i.e., 95\% of the next generation) according to rule of natural selection. That is, individuals with better fitness values are more probable to be a parent, and a stochastic uniform selection rule \cite{baker1987reducing} is applied. For the crossover process, an offspring individual will have a gene list, which basically consists of the first parent's genes. In addition, an arbitrary gene fragment cut from the second parent is inserted into a randomly chosen location. Finally, for the mutation process, simple one-point mutation is used. A mutated child will have a gene list from a parent with one point of a gene replaced by newly created nanoparticle. In this work, a 20\% mutation rate is used by default. In other words, 80\% of the remaining children are generated by crossover while 20\% are generated by mutation. Hence, 162\% ($95\% \times [80\times2 + 20]\%$) of the total population is selected by the selection rule, and the offspring for the next generation is born from them by following the crossover and mutation process.
\begin{figure}[!t]
\centering
\includegraphics[width=.99\textwidth]{Fig2}
\caption{Target absorption coefficient of the blended plasmonic nanofluid (red line): (a) uniform distribution, (b) solar-spectrum-like distribution, and (c) step-function-like distribution. The spectral absorption coefficient of water is also illustrated by the blue line.}
\label{fig:targetAbs}
\end{figure}
\subsection{Target spectrum of the absorption coefficient} \label{sec:targetAbs}
To demonstrate how the customized GA can effectively tune the absorption coefficient of a blended plasmonic nanofluid, we consider three target absorption coefficient spectra, as illustrated in Fig.\ \ref{fig:targetAbs}. The first two target spectra are for solar thermal applications, especially for a direct-absorption solar collector, i.e., a uniform distribution (Fig.\ \ref{fig:targetAbs}a) and a solar-spectrum-like distribution (Fig.\ \ref{fig:targetAbs}b). As reported by Qin \textit{et al.} \cite{qin2018optimization}, a uniform absorption coefficient is more efficient for a highly concentrated nanofluid because the heat loss can be minimized. On the other hand, a solar-spectrum-like absorption coefficient is more suitable when the system only requires an insufficient particle concentration. For simplicity, the average of the absorption coefficient was set to 1 cm$^{-1}$ considering that the magnitude of the absorption coefficient is scalable according to the particle concentration [see Eq.\ \eqref{eq:absorptionCoeffcient}]. It should be noted from Fig.\ \ref{fig:targetAbs} that the absorption coefficient of water does not play much of a role in the absorption process.
Thus, the nanoparticles should be carefully designed to achieve broadband absorption, associated with their LSP resonances.
In addition, a step-function-like absorption coefficient (Fig.\ \ref{fig:targetAbs}c) is designed for hybrid solar photovoltaic/thermal (PV/T) applications \cite{jia2019development, taylor2012nanofluid}. At the zero-absorption regime of the nanofluid, incident solar irradiance will directly reach the PV cell and be converted into electricity. In the opaque regime of the nanofluid, the solar irradiance will be converted to heat by the nanofluid. Here, we select the step point of the spectrum to be the bandgap of the PV cell used in a hybrid PV/T system with a high bandgap with 1.84 eV (approximately 680 nm) \cite{abdulraheem2014optical}. Note also that a high-bandgap PV cell is widely applied for common solar cell systems \cite{todorov2017ultrathin} or for special purposes \cite{jenkins2013high}.
\section{Results and Discussion}
The main idea when tailoring the absorption coefficient of a blended plasmonic nanofluid is to distribute the absorption peaks associated with each type of nanoparticle along the target spectrum. It is thus expected that more types of nanoparticles makes the corresponding absorption coefficient a better fit to the target spectrum. Considering the productivity of a plasmonic nanofluid, the number of types of nanoparticles ($N$) should not be excessive. Given that plasmonic nanofluids with 3 to 5 types of nanoparticles have been experimentally demonstrated \cite{jeon2014optical, mallah2018blended}, $N$ is limited to 6 or less.
Although not shown here, a higher value of $N$ can achieve a smaller root-mean-square error (RMSE), defined as $\text{RMSE}=\sqrt{\text{SSE}/81}$. Henceforth, we discuss the optimal blending combination case of $N=6$, which can retain the smallest RMSE value (i.e., the closest absorption coefficient spectrum to the target). The detailed dimensions as well as the locations of the major absorption peaks of each type of nanoparticles are listed in Table\ \ref{tbl:blendingResult}.
\begin{table}[!t]
\centering
\caption{Optimal combination of plasmonic nanoparticles for the desired spectral absorption coefficient of the nanofluid. The design variables are $r$ for the nano-sphere, ($r_c$, $t_s$) for the core-shell, and ($r_r$, $l_r$) for the nano-rod.}
\begin{tabular}{c | c | c | c | c}
\hline
Particle type & \multirow{2}{*}{Property} & Uniform & Solar-spectrum-like & Step-function-like\\
Index & & distribution & distribution & distribution \\ \hline
\multirow{5}{*}{\#1} & Material & Silver & Gold & Silver \\
& Type & Core-shell & Core-shell & Core-shell \\
& Design & (73, 14) & (15, 11) & (86, 8) \\
& $f_i\times 10^{6}$ & 1.053 & 0.544 & 3.870 \\
& Peak [nm] & 580 & 550 & 720 \\ \hline
\multirow{5}{*}{\#2} & Material & Silver & Silver & Silver \\
& Type & Core-shell & Nano-rod & Core-shell \\
& Design & (80, 9) & (16, 120) & (87, 5) \\
& $f_i\times 10^{6}$ & 3.602 & 0.187 & 1.547 \\
& Peak [nm] & 670 & 380, 810 & 850 \\ \hline
\multirow{5}{*}{\#3} & Material & Gold & Silver & Gold \\
& Type & Nano-rod & Core-shell & Core-shell \\
& Design & (28, 181) & (49, 13) & (86, 6) \\
& $f_i\times 10^{6}$ & 0.605 & 4.224 & 0.036 \\
& Peak [nm] & 510, 890 & 500, 690 & 810 \\ \hline
\multirow{5}{*}{\#4} & Material & Gold & Silver & Silver \\
& Type & Core-shell & Core-shell & Core-shell \\
& Design & (52, 12) & (88, 10) & (89, 7) \\
& $f_i\times 10^{6}$ & 3.116 & 2.642 & 1.933 \\
& Peak [nm] & 610, 740 & 680 & 760 \\ \hline
\multirow{5}{*}{\#5} & Material & Gold & Silver & Gold \\
& Type & Core-shell & Core-shell & Core-shell \\
& Design & (83, 6) & (53, 7) & (46, 5) \\
& $f_i\times 10^{6}$ & 1.298 & 0.970 & 0.599 \\
& Peak [nm] & 800, 1080 & 610, 820 & 860 \\ \hline
\multirow{5}{*}{\#6} & Material & Gold & Gold & Silver \\
& Type & Core-shell & Core-shell & Core-shell \\
& Design & (88, 5) & (87, 5) & (15, 70) \\
& $f_i\times 10^{6}$ & 1.434 & 1.909 & 0.003 \\
& Peak [nm] & 860 & 860 & 400 \\ \hline
\end{tabular}
\label{tbl:blendingResult}
\end{table}
\begin{figure}[!t]
\centering
\includegraphics[width=.55\textwidth]{Fig3}
\caption{Absorption coefficient of a blended plasmonic nanofluid for the target spectrum: (a) uniform distribution, (b) solar-spectrum-like distribution, and (c) step-function-like distribution. The effect of water is illustrated by the blue dotted line.}
\label{fig:blendingResult}
\end{figure}
Figure \ref{fig:blendingResult}a shows the absorption coefficient of a blended plasmonic nanofluid for the target spectrum with a uniform distribution. The designed plasmonic nanofluid results in a RMSE value of 0.099 cm$^{-1}$, which is less than 10\% of the target absorption coefficient (i.e., 1 cm$^{-1}$). The absorption peaks of each type of nanoparticle are well distributed along the visible and near-infrared spectral regions for broadband absorption. Although the \#4 particle (i.e., the Au core-shell) mainly contributes to the visible absorption, the \#2 and \#3 particles successfully compensate for the absorption dips of the \#4 particle. Moreover, all six types of nanoparticles shows minor absorption at the wavelengths greater than 900 nm, where water (i.e., the base fluid) starts to play a role. It is interesting to note that no nano-spheres are used in the six types of nanoparticles. This is mainly due to the tunability of the LSP resonance condition in the core-shell and nano-rod structures. In other words, the nano-sphere is less tunable as the polarizability of its Clausius-Mossotti relation is predetermined by the material properties \cite{bohren1983absorption} and the resulting absorption peak is often confined to a narrow spectral range (i.e., Au: $500 \sim 540$ nm and Ag: $380 \sim 450$ nm).
The absorption coefficient of a blended plasmonic nanofluid for a solar-spectrum-like target spectrum is shown in Fig.\ \ref{fig:blendingResult}b. To follow the solar spectrum, the absorption peaks should be confined to the major spectral regions of solar radiation (i.e., from 400 to 700 nm and from 800 to 900 nm). It can be observed that the \#1, \#3, and \#4 particles mainly contribute to the absorption in the aforementioned spectral region. Although the RMSE value is 0.232 cm$^{-1}$ (about twice that the uniform case) for the solar-spectrum-like distribution, the designed absorption coefficient reasonably follows the solar spectrum except for wavelengths between 300 and 400 nm.
Finally, we also demonstrate the blended plasmonic nanofluid for the step-function-like distribution in Fig.\ \ref{fig:blendingResult}c.
As in the case of the solar-spectrum-like distribution, the designed absorption coefficient captures the features of the target spectrum. However, there exists non-negligible absorption of approximately 0.2 cm$^{-1}$ in the wavelengths between 300 and 680 nm, mainly due to intrinsic absorption by silver and gold. The resulting RMSE value is 0.209 cm$^{-1}$, which is slightly less than that in Fig.\ \ref{fig:blendingResult}b. Interestingly, Table \ref{tbl:blendingResult} reveals that the volume fractions of the \#4 and \#6 particles were greatly suppressed by the GA, meaning that the \#4 and \#6 particles were considered to be "useless" by the GA. Therefore, we can simply use four types of particles for the step-function-like distribution without seriously compromising the fitness. In Fig.\ \ref{fig:blendingResult}, the customized GA clearly demonstrates its excellent design capability for a blended plasmonic nanofluid with the desired spectral absorption coefficient.
Table\ \ref{tbl:blendingResult} also indicates that only core-shell nanoparticles are used for the step-function-like distribution. Although the nano-rod can induce the multiple LSP peaks \cite{yu1997gold}, its resonance condition is polarization-dependent due to its geometrical anisotropy. Hence, the polarization-averaged absorption efficiency becomes less significant as compared to the geometrically isotropic core-shell structure. The present optimization results clearly indicate that the core-shell nanoparticle is superior to the nano-sphere and the nano-rod structures in terms of the tunability of the LSP resonance condition as well as the enhanced absorption efficiency associated with the LSP.
\begin{figure}[!b]
\centering
\includegraphics[width=0.55\textwidth]{Fig4}
\caption{Absorption coefficient of a blended plasmonic nanofluid considering inhomogeneous broadening due to polydispersed nanoparticles in reality. The effect of water is also illustrated by the blue dotted line.}
\label{fig:broadeningResult}
\end{figure}
\begin{table}[!t]
\centering
\caption{Optimal combination of plasmonic nanoparticles considering inhomogeneous broadening due to polydispersed nanoparticles in reality. The notation of the design variables follows that in Table\ \ref{tbl:blendingResult}.}
\begin{tabular}{c | c c c c c}
\hline
Property & \#1 & \#2 & \#3 & \#4 & \#5 \\ \hline
Material & Gold & Gold & Silver & Silver & Silver \\
Type & Nano-rod & Core-shell & Core-shell & Core-shell & Core-shell\\
Design & (21, 139) & (76, 15) & (88, 5) & (89, 8) & (71, 15) \\
$f_i\times 10^{6}$ & 0.41 & 4.84 & 1.50 & 3.23 & 4.28 \\
Peak [nm] & 510, 830 & 670 & 840 & 740, 1050 & 560 \\ \hline
\end{tabular}
\label{tbl:broadeningResult}
\end{table}
Thus far, we have demonstrated how to achieve broadband absorption by blending nanoparticles made of noble metals (such as Au and Ag), which usually exhibits sharp resonance peaks. In reality, however, there could be many factors that give rise to a broadening effect, such as the electron-boundary scattering effect when the characteristic size of the metal is smaller than the mean-free-path of electrons or inhomogeneous broadening due to a non-uniform size distribution of the nanoparticles. Because the electron-boundary scattering effect occurs only for the core-shell structure with an extremely thin metallic shell \cite{lee2012radiative}, for instance, its effect may not be prominent as compared to the inhomogeneous broadening that occurs inevitably due to polydispersed nanoparticles \cite{khanadeev2017optical, jeon2014optical}. Here, we examine how inhomogeneous broadening occurring in reality can affect the optimal combination of plasmonic nanoparticles by applying a randomized distribution of design variables. To do this, a Gaussian distribution with the mean value of the design variable and a standard deviation of 10\% of the mean is assumed. For instance, the core radius of core-shell particle ($r_c$) is treated as a random variable following a normal distribution, $N(r_c, (0.1r_c)^2)$. In the calculation, 100 random particles following a Gaussian distribution were calculated using the surrogate neural network model and their absorption spectra were averaged to determine the broadened absorption spectrum of randomized nanoparticles.
Figure \ref{fig:broadeningResult} shows the absorption coefficient of a blended plasmonic nanofluid considering inhomogeneous broadening due to polydispersed nanoparticles in reality. It is remarkable that we can achieve an even lower RMSE value (i.e., 0.097 cm$^{-1}$) than the previous blending result ($\text{RMSE}=0.099$ cm$^{-1}$) with only five types of nanoparticles if inhomogeneous broadening is taken into account. The optimal combination of plasmonic nanoparticles considering inhomogeneous broadening is listed in Table\ \ref{tbl:broadeningResult}. As noted in Table\ \ref{tbl:broadeningResult}, the absorption peaks of each type of nanoparticle are well distributed, spanning the entire spectral region of interest, and the inhomogeneous broadening causes the absorption coefficient of the blended plasmonic nanofluid to be more uniform. Similarly, it is also expected that the electron-boundary scattering effect eventually makes the designed spectrum more uniform, possibly leading to a reduction in the required number of nanoparticle types. It should be noted that the optimal combination in Table \ref{tbl:broadeningResult} is wholly different from that in Table \ref{tbl:blendingResult}, suggesting that the customized GA is very effective at finding the solution under any given constraint.
\section{Summary}
We have employed a customized genetic algorithm to tailor the spectral absorption coefficient of a blended plasmonic nanofluid made of nano-sphere, core-shell, and/or nano-rod structures. The chromosome description, fitness function, crossover and mutation process in a conventional GA were customized to be suitable for the inverse problem of finding the optimal combination of plasmonic nanoparticles for the prescribed distribution of the absorption coefficient. In addition, neural network models estimating the absorption coefficient of a plasmonic nanoparticle were constructed and coupled with the customized GA to reduce the computational cost of the optimization process. In this work, three different target absorption coefficients, specifically a uniform distribution, solar-spectrum-like distribution and step-function-like distribution, were considered. The resulting absorption coefficient of a designed plasmonic nanofluid was in good agreement well with the prescribed spectral distribution within about 10\% to 20\% of error when six types of nanoparticles were used. Finally, we also considered inhomogeneous broadening mainly due to polydispersed nanoparticles during the optimization process. It was found that we can achieve an even lower RMSE value (i.e., 0.097 cm$^{-1}$) than in the previous blending result ($\text{RMSE}=0.099$ cm$^{-1}$) with fewer types of nanoparticles if inhomogeneous broadening is considered. The design methodology proposed here will facilitate the future development of a direct-absorption solar collector using a blended plasmonic nanofluid.
\section*{Data availability}
All data that support the findings of this study are available from the corresponding author upon request.
|
1,108,101,566,769 | arxiv | \subsection{Optimising Target Objects}\label{subsec:SpontTarget}
\begin{figure*}[!t]
\centering
\includegraphics[width=0.45\textwidth]{Scatter_Plot1.pdf}
\includegraphics[width=0.45\textwidth]{Scatter_Plot2.pdf} \\
\caption{In the {\em left} panel. Signal strength as given by $\Sigma/\Delta v \propto T_{RJ}$. We assume an \textit{identical} object and beam size $\Sigma = M_{\rm obj}/(\theta_{\rm obj} D_{\rm obj})^2$ taking values from table~\ref{ClusterCalculations}. Note we normalised $\Sigma$ by the background value $1.2\times 10^{-9}\,{\rm kg}\,{\rm m}^{-3}\,{\rm s}$ for the data in table appears relatively flat - the solid green line - and is compatible with the simple argument presented in the text, albeit with a somewhat higher value ($\approx 500$) relative to the background value. Possibly there is a trend with mass which we denoted with a line $\propto M^{-0.2}$ which could be due to the concentration parameter varying as a function of mass and the fact that the angular sizes are probably the scale radius for some fitted profile function rather than the virial radius. We note that much of this trend is driven by the outliers at low mass, ultra-faint dwarf spheroidal, and high mass, the galaxy clusters, VIRGO and COMA. As with the bottom panel we present results using both $\Delta v_{\rm obj}$ and $\Delta v_{\rm int}$ which are broadly consistent. In the {\em right} panel, we present the quantity in \eqref{eqn:CaputoQuantity} for the data in table~\ref{ClusterCalculations} which clearly increases like $\theta_{\rm obj}^2$ as denoted by the line in the plot. Note that the starred data points, which use observational measurements of the velocity width $\Delta v_{\rm obj}$, and the circular points, which correspond to the inferred width $\Delta v_{\rm inf} \approx \left(GM_{\rm obj}/R_{\rm obj}\right)^{1/2}$, show the same trend. }
\label{fig:ScatterPlots}
\end{figure*}
In the previous two sections we have explained that, if one targets a halo with surface mass density $\Sigma_{\rm beam}\approx 0.07\,{\rm kg}\,{\rm m}^{-2}$ and velocity width $\Delta v\approx 200\,{\rm Km}\,{\rm s}^{-1}$, the signal from spontaneous decay combined with stimulated emission from the CMB for $g_{\rm a\gamma\gamma}=10^{-10}\,{\rm GeV}^{-1}$ is too weak to be detected even for an array of receivers with $N\lesssim 10^{6}$. We came to this conclusion by estimating the integration time required to detect the signal focusing on the expression for the signal expressed in terms of the brightness temperature (\ref{ref:brighttemp_again}).
Examination of this equation makes it clear that the largest possible signal is obtained by maximising $\Sigma_{\rm beam}/\Delta v$. If the object is such that $\theta_{\rm FHWM}\approx \theta_{\rm vir}$, we estimate the quantity to be $\approx 3.5\times 10^{-7}\,{\rm kg}\,{\rm m}^{-3}\,{\rm s}^{-1}$ for the strawman object used in the previous section which is around 300 times larger than the background value. This value is based on what we think, at a level of better than a factor two, are realistic values, but precise knowledge of it is absolutely critical to any attempt to improve the CAST limits of $g_{\rm a\gamma\gamma}$ using this approach. In this section, we will discuss, using theoretical arguments and comparing to observations, the range of values for $\Sigma_{\rm beam}/\Delta v$ that might be available for us to be observed in the Universe.
One might wonder how $\Sigma_{\rm vir}/\Delta v$ depends on the size of the object. If we consider a halo with virial overdensity $\Delta_{\rm vir}\sim 100$, then $M_{\rm vir}=\tfrac{4\pi}{3}\Delta_{\rm vir}\rho_{\rm a} R_{\rm vir}^3$, where $\rho_{\rm a}=\Omega_{\rm a}\rho_{\rm crit}$ is the background density of axions and $\rho_{\rm crit}$ is the critical density. An estimate for the velocity width, up to order one factors, is $\Delta v=(GM_{\rm vir}/R_{\rm vir})^{1/2}$ and hence we find that
\begin{equation}\label{eqn:SigmaQuantity}
\frac{\Sigma_{\rm vir}}{\Delta v}\approx 0.7\left(\frac{\Delta_{\rm vir}\rho_{\rm a}}{G}\right)^{1/2}\approx 3.5\times 10^{-7}\,{\rm kg}\,{\rm m}^{-3}\,{\rm s}\,,
\end{equation}
which is independent of the size of the object - that is, there is no dependence on $M_{\rm vir}$ or $R_{\rm vir}$. If $\Delta_{\rm vir}$ is universal and independent of the size of the object, as it is supposed to be almost by definition, then the expected brightness temperature averaged over a virialised halo will be independent of the size and hence the optimal detection for a specific halo size and telescope configuration would be obtained by matching the size of the object approximately to the telescope beam width. This is the standard practice to optimise detection efficiency in all branches of astronomy.
This suggestion that there is no optimal size of object appears to be contrary to the conclusions of \cite{ref:Caputo,ref:Caputo1}, who claimed that the optimal detection would be for dwarf spheroidal galaxies, that is the very lowest mass halos. They came to this conclusion considering the quantity
\begin{equation}\label{eqn:CaputoQuantity}
\frac{1}{\Delta v}\int \mathrm{d}\Omega \mathrm{d}l\rho_{\rm a}\propto \frac{M_{\rm beam}}{d^2\Delta v}\propto S_{\rm beam}\,,
\end{equation}
where $d$ is the distance to the object and the angular integration over the angular size of the object - or, as they state it, for a telescope beam which has the same size as the object. This quantity is $\propto S_{\rm beam}$ defined in \eqref{eqn:FluxValue} which is equivalent to (\ref{ref:brighttemp}) if one is careful with the choice of $\Sigma_{\rm beam}$. But we have already explained that one can come to the wrong conclusion if one uses the wrong value of $M_{\rm beam}$ for a specific halo and that it is actually better to think in terms of the surface mass density $\Sigma_{\rm beam}$. If one were to integrate out to the size of the object $S_{\rm beam}=S_{\rm tot}$, one would expect this to be $\propto\theta_{\rm FWHM}^2$ as they find.
In fig.~\ref{fig:ScatterPlots}, we have plotted the quantities in \eqref{eqn:CaputoQuantity} and \eqref{eqn:SigmaQuantity} using the data in table~\ref{ClusterCalculations} which is similar to, but not exactly the same as, that used in \cite{ref:Caputo,ref:Caputo1}. In particular we have added some galaxies and galaxy clusters to the dwarf galaxies which they focus on that enables us to probe a wider lever arm in mass. The table contains values for the distance to and the mass of the object $D_{\rm obs}$ and $M_{\rm obj}$, respectively, the angular size $\theta_{\rm obj}$ and the velocity width $\Delta v_{\rm obj}$. These are inferred in a heterogeneous way, but should at least be indicative of some truth. We would not necessarily expect these values to be those for a virialised halo and therefore we denoted them with the suffix ``obj" to distinguish them as being observationally determined. From the observed information, we can infer the radius, $R_{\rm obj}=\theta_{\rm obj}/(2D_{\rm obj}$ and also check consistency with our analytic estimates above by inferring $\Delta v_{\inf}=(GM_{\rm obj}/R_{\rm obj})^{1/2}$, as well as calculating the surface mass density appropriate to an average over the object radius, $\Sigma_{\rm obj}=M_{\rm obj}/(\pi R_{\rm obj}^2)$.
Firstly, we find in the right panel of fig.~\ref{fig:ScatterPlots} that \eqref{eqn:CaputoQuantity} which was plotted in \cite{ref:Caputo1} is indeed $\propto\theta_{\rm obj}^2$ as claimed. But on the basis of the theoretical argument above, this is exactly what one would expect for the total flux density $S_{\rm tot}\propto \Sigma_{\rm ave}\theta^2/\Delta v$, where $\Sigma_{\rm ave}$ is some average surface mass density for the objects, and hence, while it provides some confidence that the modelling is correct, it does not yield any obvious information about which objects would be optimal.
In the left panel of fig.~\ref{fig:ScatterPlots} we have plotted $\Sigma_{\rm obj}/\Delta v$ for the data presented in table~\ref{ClusterCalculations}, using both $\Delta v_{\rm obj}$ and $\Delta v_{\inf}$ with consistent results. We find that the data are compatible with $\Sigma_{\rm beam}/\Delta v$ being a constant over eight orders of magnitude and for it to be $\approx 500$ times the background value - slightly higher than for our strawman object - within the kind of uncertainties that we might expect coming from a heterogeneous sample such as the one which we have used. Visually, there could be some evidence for a trend $\sim M^{-0.2}$ which we have also included to guide the eye, but the evidence for this is largely due to a few outliers at the low- and high-mass ends where perhaps the observational estimate are most uncertain. So it could be that there is some preference for lower mass halos over high mass halos, but the effect is not very dramatic. Note that on the $y-$axis, we plot $\left(\Sigma/\Delta v\right)^{\rm rel} \equiv \frac{\Sigma/\Delta v}{1.2\times 10^{-9}\,{\rm kg\,m^{-3}\,s}}$, where the denominator is the value associated to the background.
It could be that the possible trend seen in the left panel of fig.~\ref{fig:ScatterPlots} is related to the concentration parameter of the halo. It is likely that the observationally determined angular size, $\theta_{\rm obj}$, is not the virial radius but some scale radius from a fitting function used in conjunction with images. If this is the case, then we might expect a weak trend with mass.
The concentration parameter has been computed in numerical simulations and is usually assumed to be universal for halos of a given mass, $M$. A recently proposed expression is~\cite{ref:MultiDark}
\begin{equation}
{\hat c}(M,z)={\hat c}_0(z)\left(\frac{M}{ M_0}\right)^{-\gamma(z)}\left[1+\left(\frac{M}{M_1(z)}\right)^{0.4}\right]\,,
\end{equation}
where $M_0=10^{12}h_{100}^{-1}M_{\odot}$ and $\hat{c}_0(z)$, $\gamma(z)$ and $M_1(z)$ are fitted parameters which are redshift dependent. We will focus on low redshifts where $\hat{c}_0(z)\approx 7.4$, $\gamma(0)\approx 0.12$ and $M_1(0)=5.5\times 10^{17}h_{100}^{-1}M_{\odot}$.
From this we see that at $z=0$, $\hat{c}\propto M^{-0.12}$, that is, lower mass halos typically are more concentrated than higher mass halos, and therefore there will be more mass inside the scale radius, and for observations focussing on the region inside this scale radius $\Sigma_{\rm beam}$ might be larger.
\begin{table*}
\centering
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Object & $D_{\rm obj}$ & $M_{\rm obj}$ ($\rm M_{\odot}$) & $\theta_{\rm obj}$ & $\Delta v_{\rm obj}$ [$\rm km\,s^{-1}$] & Reference(s) \\
\hline
Leo 1 & $250\pm 30$ kpc & $2.2\times 10^{7}$ & $12.6\pm1.5$ arcmin & 8.8 & \cite{ref:Mateo} \\
NGC 6822 & $490\pm 40$ kpc & $1.6\times 10^9$ & $40\pm 10$ arcmin & 8 & \cite{ref:Mateo} \\
Draco & $82\pm 6$ kpc & $2.2\times10^{7}$ & $28.3\pm 2.4$ arcmin & 9.5 & \cite{ref:Mateo} \\
Wilman 1 & 45 kpc & $4\times 10^5$ & 9 arcmin & 4 & \cite{ref:DwarfGalMass} \\
Reticulum 2 & 30 kpc & $5.6 \pm 2.4\times 10^5$ & $3.64^{+0.21}_{-0.12}$ arcmin & 3.3 &\cite{ref:Beasts, ref:Reticulum2} \\
Sextans B & 1345 kpc& $3.95\times 10^8$ & 3.9 arcmin & 18 & \cite{ref:Mateo}\\
Pegasus & 955 kpc& $5.8 \times 10^{7}$ & 3.9 arcmin & 8.6 & \cite{ref:Mateo} \\
Antlia & 1235 kpc& $1.2\times 10^7$ & 5.2 arcmin & 6.3 & \cite{ref:Mateo} \\
NGC 205 & 815 kpc& $7.4\times 10^{8}$ & 6.2 arcmin& 16 & \cite{ref:Mateo}\\
\hline
NGC 5128 & 3.75 Mpc & $5.07\times 10^{11}$ & 34.67 arcmin & 477 & \cite{ref:NearGalCat} \\
NGC 5194 & 15.85 Mpc & $4.17\times 10^{10}$ & 8.4 arcmin & 175 & \cite{ref:NearGalCat} \\
Maffei2 & 2.8 Mpc & $4.17\times 10^{10}$ & 3.8 arcmin & 306 & \cite{ref:NearGalCat} \\
IC2574 & 4.02 Mpc & $4.57\times 10^9$ & 13.18 arcmin & 107 & \cite{ref:NearGalCat} \\
SexA & 1.32 Mpc & $2.51\times 10^8$ & 5.89 arcmin & 46 & \cite{ref:NearGalCat} \\
NGC 3556 & 9.9 Mpc & $3.31\times 10^{10}$ & 5.01 arcmin & 308 & \cite{ref:NearGalCat} \\
IC 0342 & 3.28 Mpc & $1.41\times 10^{11}$ & 21.38 arcmin & 181 & \cite{ref:NearGalCat} \\
NGC 6744 & 8.3 Mpc & $2.24\times 10^{11}$ & 21.38 arcmin & 323 &\cite{ref:NearGalCat}\\
ESO 300-014 & 9.8 Mpc & $10^{10}$ & 7.08 arcmin & 130 & \cite{ref:NearGalCat}\\
NGC 3184 & 11.12 Mpc & $6.3\times 10^{10}$ & 7.41 arcmin & 128 & \cite{ref:NearGalCat}\\
\hline
Virgo & $18 \pm 1.2$ Mpc & 2.91$\times 10^{15}$ & 7 degrees & 1100 & \cite{ref:VirgoMass1,ref:VirgoMass2} \\
Coma & 100 Mpc & 3$\times 10^{15}$ & 100 arcmin & 1100 & \cite{ref:ComaMass, ref:ComaDistance}\\
\hline
\end{tabular}
\caption{Table of masses ($M_{\rm obj}$), distances ($D_{\rm obj}$), angular sizes ($\theta_{\rm obj})$ and velocity widths ($\Delta v_{\rm obj})$ extracted from the literature and used in fig.~\ref{fig:ScatterPlots}. In each case we have specified the reference of the paper from which the numbers are calculated. From paper to paper the methods employed are very different and hence the overall sample is relatively heterogeneous. For each object we can infer a radius $R_{\rm obj}=\theta_{\rm obj}D_{\rm obj}/2$ and a velocity width $\Delta v_{\rm inf}=(GM_{\rm obj}/R_{\rm obj})^{1/2}$. We find that $\Delta v_{\rm obj}$ is strongly correlated with $\Delta v_{\rm inf}$ as we would expect and indeed that $M_{\rm obj}$ is also correlated with $R_{\rm obj}$.}
\label{ClusterCalculations}
\end{table*}
\begin{figure}[t]
\centering
\includegraphics[width = 0.45\textwidth,height=0.45\textwidth]{G_Func.pdf}
\caption{The function $G(\hat{c}, \tilde{R})$ as a function of its arguments. In the left panel, we plot $G$ as a function of $\hat{c}$ for different values constant $\tilde{R}$, and vice versa in the right panel. }
\label{fig:GFunction}
\end{figure}
This leads us on to an important caveat in this discussion: one does not have to choose to focus on trying to detect the entire signal from a halo and indeed it will be optimal, as well as practical, to not do this. Using (\ref{ref:mbeam}) we can deduce that
\begin{equation}
\Sigma_{\rm beam}=G\left(\hat{c},{\frac{R_{\rm beam}}{R_{\rm vir}}}\right)\Sigma_{\rm vir}\,,
\end{equation}
for an NFW profile where $G(x,y)=x^2\log(2y/x)/f(x)$ for $y/x\ll 1$. We anticipate that one could derive a similar expression for any halo profile. We plot the function $G(\hat{c}, \frac{R_{\rm beam}}{R_{\rm vir}})$ as a function of $\hat{c}$ and $\tilde{R} = R_{\rm beam}/R_{\rm vir}$, in fig.~(\ref{fig:GFunction}) which indicates that enhancements of up to 100 might easily be possible and that these are likely to be larger in lower mass objects than those of higher mass. Therefore, it is clear that, for a fixed experimental set up ($R_{\rm vir}/R_{\rm beam}$ fixed), one should search for an object with the largest concentration, a general result which we already anticipated in section~\ref{subsec:SpontSignal}.
In conclusion, we have argued that maximizing $\Sigma_{\rm beam}/\Delta v$ will give the largest possible brightness temperature signal. Theoretical arguments suggest that if the beam encloses the virial radius of a particular object, this will be independent of mass and a very rudimentary search of the literature for specific values suggests that this could be true. However, for fixed observational setup, and hence fixed resolution, one might find a significant enhancement of the signal due to the fact that the surface mass density will increase as one probes the more central regions of a halo. These are likely to be larger for smaller mass objects since they are on average more concentrated. This is the reason we have presented our sensitivity estimates as a function of $\Sigma_{\rm beam}$ and results for range of values $\Sigma_{\rm beam}=0.07-0.7\,{\rm kg}\,{\rm m}^{-2}$ in fig.~\ref{fig:CMB_Enhancement}.
\section{Introduction}
\label{sec:intro}
Understanding the exact nature of dark matter remains one of the major challenges in particle physics and cosmology. One particularly simple solution to the dark matter problem is offered by the QCD axion which results from the breaking of Peccei-Quinn (PQ) symmetry~\cite{ref:PQ}, proposed as a resolution to the strong CP problem of Quantum Chromodynamics (QCD). There are a number of specific ways to incorporate the axion into the Standard Model of particle physics; the most common being the KSVZ~\cite{ref:K, ref:SVZ} and the DFSZ~\cite{ref:DFSZ, ref:Zhit} models. Soon after the realisation that the axion was a natural consequence of PQ symmetry, it was pointed out that it could be produced by the non-thermal misalignment mechanism~\cite{ref:misalign1, ref:misalign2, ref:misalign3} and that its relic abundance and low momentum would allow it to be a Cold Dark Matter (CDM) candidate. The axion has since been subject of extensive theoretical work and has been proposed as a candidate for a number of other cosmological phenomena (see \cite{ref:Marsh} for a recent review). In what follows, we will make the assumption that axions are responsible for all the CDM in the Universe and discuss their detection in the radio/mm-waveband.
A recent detailed calculation~\cite{ref:WS} of the misalignment production of axions yielded
\begin{equation}
\Omega_{\rm a}h_{100}^2\approx 0.54g_{\star}^{-0.41}\theta_{\rm i}^2\left(\frac{f_{\rm a}}{10^{12}~\rm GeV}\right)^{1.19}\,,
\end{equation}
where $g_{\star}\approx 10$ is the number of relativistic degrees of freedom during the realignment process, $\theta_{\rm i}$ is the initial angle of misalignment, $h_{100}$ is defined by the Hubble constant $H_0=100h_{100}\,{\rm km}\,{\rm sec}^{-1}\,{\rm Mpc}^{-1}$ and $f_{\rm a}$ is the axion decay constant which is related to the axion mass, $m_{\rm a}$, by $m_{\rm a}c^2 = 6\,{\rm \mu eV} \left(f_{\rm a}/10^{12}{\rm GeV}\right)^{-1}$ (see also \cite{ref:BaeMisalign,Kawasaki2013,ref:Marsh,Enander2017} for other recent treatments of this issue). Recent measurements of the Cosmic Microwave Background (CMB) by the \textit{Planck} satellite~\cite{ref:Planck, ref:Planck2018} yield an estimate for the CDM density, $\Omega_{\rm c}h_{100}^2 \approx 0.12$. Assuming that this is the case, taking into account the uncertainty in the value of $g_{\star}$ and the standard assumption $\langle\theta_{\rm a}^2\rangle=\pi^2/3$, we can predict a mass range of $19\,\mu{\rm eV}\leq m_{\rm a}c^2\leq 23\,\mu{\rm eV}$.
This particular choice of $\theta_{\rm a}$ is based on a scenario where the value at each position in space is assigned randomly and eventually homogenised by expansion. We will use it in what follows as our baseline choice (as done by many authors) but we note that it is not really a firm prediction at all. In inflationary scenarios one would expect a random value anywhere in the range $0<\theta_{\rm a}\le\pi$. One might expect that, in order to avoid an anthropic solution to the strong CP problem, there is a lower limit for $\theta_{\rm a}$ and hence $10^{-2}<\theta_{\rm a}<\pi$. In this case, we come up with a wider prediction for the range of masses from misalignment, $6\times 10^{-3}\,\mu{\rm eV}<m_{\rm a}c^2< 6\times 10^2\,\mu{\rm eV}$.
We note that there is a lower limit to the detection approaches we are advocating due to the emission from neutral hydrogen, which would prevent detection of the axion signal for $m_{\rm a}c^2<12\,\mu{\rm eV}$. This happens because there will be a degeneracy between the spectral line associated to the axion and the HI emission line with $\lambda$ $\approx 21$ cm. At higher redshifts, this value will shift to smaller frequencies (larger wavelengths) and it will make it more difficult to disentangle the signal due to the axion decay. We also note that the spectral lines from organic molecules, for example, $\rm {CO, CS, HCO, HCN, H_2O}$ and ${\rm NH_3}$ can also be a source of degeneracy at frequencies greater than 10 GHz, although the impact of these lines is less clear.
PQ symmetry is a $U(1)$ symmetry and therefore one would expect cosmic strings to form via the Kibble Mechanism when the symmetry is broken. The expected relic abundance from this process is expected to dominate if the symmetry breaking transition takes place after inflation, and comprises two contributions from long strings and loops~\cite{ref:Battye1,ref:Battye}
\begin{equation}
\label{eqn:strings}
\Omega_{\rm a}h^2_{100}\approx \left [1+10J\left({\frac{\alpha}{\kappa}}\right)\right]\Delta\left(\frac{f_{\rm a}}{10^{12}\rm GeV}\right)^{1.18}\,,
\end{equation}
where $\alpha$ is the loop production size relative to the horizon, $\kappa$ quantifies the rate of decay of the string loops, $J(x)=x^{3/2}\left[1-(1+x)^{-3/2}\right]$ and $1/3<\Delta<3$ is the theoretical uncertainty associated with the QCD phase transition. This estimate was recently refined~\cite{ref:WS}, notably improving the estimate of $\Delta$ and making the assumption that $\alpha/\kappa=0.5\pm 0.2$ to deduce $100\,\mu{\rm eV}<m_{\rm a}c^2<400\,\mu{\rm eV}$ under the assumption that the axions are the cold dark matter. Note that this axion mass range cannot be probed by standard axion haloscope experiments.
The axion couples to ordinary matter very weakly, most notably to photons and this is quantified by the axion-photon coupling constant $g_{\rm a\gamma\gamma}$ for the axion decaying spontaneously into two photons with a lifetime given by~\cite{ref:KT}
\begin{align}\label{eqn:DecayTime}
\tau_{2\gamma} = &\, {\frac{64\pi\hbar}{g_{\rm a\gamma\gamma}^2m_{\rm a}^3c^6}}\,,\\
\approx &\, 8\times 10^{35}\,{\rm sec}\left(\frac{g_{\rm a\gamma\gamma}}{10^{-10}~{\rm GeV^{-1}}}\right)^{-2}\left(\frac{m_{\rm a}c^2}{\rm 250~\mu eV}\right)^{-3}\,,\nonumber
\end{align}
with a rest-frame emission frequency of $f_{\rm emit}=m_{\rm a}c^2/(2h)$ which is $\approx 2.4\,{\rm GHz}$ for $m_{\rm a}c^2=20\,\mu{\rm eV}$ and $\approx 30\,{\rm GHz}$ for $m_{\rm a}c^2=250\,\mu{\rm eV}$ which correspond to the misalignment (with $\langle\theta_{\rm a}^2\rangle\approx 3$) and string prediction ranges, respectively. In what follows, we will use $m_{\rm a}c^2=250\,\mu{\rm eV}$ and $m_{\rm a}c^2=20\,\mu{\rm eV}$ as particular fiducial values in order to calculate specific numbers, but it is worth pointing out that we have argued that it is possible for there to be an axion signal anywhere in the frequency range $\sim 70\,{\rm MHz}$ to $\sim 100\,{\rm GHz}$.
For specific models there is a relation between $g_{\rm a\gamma\gamma}$ and $m_{\rm a}$, which depends on the choice of $E/N$, which is the ratio of electromagnetic and colour anomalies \citep{diCortona2016}
\begin{equation}\label{eq:QCDg}
g_{\rm a\gamma\gamma} = 5.1 \times 10^{-14} \, \text{GeV}^{-1}\left(\frac{m_{\rm a}c^2}{250 \mu \text{eV}}\right)\left|\frac{E}{N} - 1.92 \right|\,.
\end{equation}
The KSVZ model has $E/N$ = 0, while DFSZ model has $E/N = 8/3$ making the latter more weakly coupled to photons. At present, the most sensitive experimental limits come from the ADMX haloscope collaboration which constrains $g_{\rm a \gamma\gamma} < 10^{-15}$\,GeV$^{-1}$ for $1.90\,\mu$eV $\leq m_{\rm a}c^2 \leq 3.69\,\mu$eV, under the assumption that the local galactic dark matter density $\rho_{\rm gal} c^2 \approx 0.45\,{\rm GeV}\,{\rm cm}^{-3}$~\cite{ref:ADMXl, ref:ADMXf}. This limit was further improved recently to rule out DFSZ axions in the narrow mass range $2.66\,\mu$eV $\leq m_{\rm a}c^2\leq 2.81\,\mu$eV~\cite{ref:ADMX2018} with a limit of $g_{\rm a\gamma\gamma}< 4\times 10^{-16}\,$ GeV$^{-1}$. A number of experiments have been proposed to speed up these searches so that much wider ranges of mass can be probed \citep{Majorovits2017, Brun2019, Droster2019, McAllister2017}. Typically these approaches find it more difficult, for practical reasons, to be sensitive to higher axion masses and therefore we believe that the strongest motivation for the ideas we present in this work is to search for axions in the multi GHz frequency range and hence we have centred the estimates presented in subsequent sections on $m_{\rm a}c^2=250\,\mu{\rm eV}$, although they apply more widely.
There is an upper limit $g_{\rm a\gamma\gamma}<0.66\times 10^{-10}\,{\rm GeV}^{-1}$ from the CAST solar axion experiment which applies for $m_{\rm a}c^2< 10^{-2}$\,eV~\cite{ref:CAST}. Given this limit, the predicted range of axion masses and the limits on the mass from terrestrial haloscopes, it seems sensible to search for astrophysical signals from dark matter axions in virialized halos (for example, galaxies and galaxy clusters) in the frequency range $f_{\rm obs}\approx 1-100\,{\rm GHz}$ which might be loosely described as the radio/mm-waveband and for decay times $\tau_{2\gamma}\sim 8\times 10^{35}\,{\rm sec}$ and higher with the aim of achieving a limit which is better than the limit from CAST~\footnote{There has been a previous attempt to obtain limits on dark matter axions using 6 days of integration on the dwarf galaxies Leo 1, LGS 2 and Pegasus using the Haystack 37\,m telescope \cite{ref:Blout}. A limit of $g_{\rm a\gamma\gamma} < 10^{-9}$\,GeV$^{-1}$ was published for axion masses $298\,\mu$eV $\leq m_{\rm a}c^2 \leq$ $363\,\mu$eV, but given the estimates we make for the strength of the signal in subsequent sections we believe that there must have been an error in the analysis. We will comment further on this at the end of section~\ref{subsec:SpontSensitivity}.}. There have been a number of recent studies~\cite{ref:KQ1, ref:KQ2, ref:Caputo, ref:Caputo1} of this subject in the context of future radio telescope, such as the Square Kilometre Array~\cite{ref:SKA} (see \cite{ref:redbook} for a recent summary of the current SKA science case in the context of cosmology), and one aim is to clarify and extend this work.
These studies have explored enhanced decay mechanisms such as the effects of astrophysical magnetic fields and stimulated emission due to the CMB. In \cite{ref:KQ1,ref:KQ2} it was suggested that magnetic fields of amplitude $\sim 10\mu{\rm G}$, already detected in galaxies and clusters, could lead to a strong and eminently detectable signal. However, \cite{ref:Sigl} pointed out that the decay lifetime into a single photon with $f_{\rm emit}=m_{\rm a}c^2/h$ expected for such a process is
\begin{equation}
\tau_{\rm B} = \frac{m_{\rm a}}{2\pi^2 \hbar^2 cg_{\rm a\gamma\gamma}^2}\frac{\mu_0 V}{k^3|\hat{B}(k_{\rm a})|^2}\,,
\end{equation}
where $\hat{B}(k_{\rm a})$ is the Fourier transform of the magnetic field evaluated at a wavenumber corresponding to the inverse Compton wavelength of the axion $k_{\rm a}=m_{\rm a}c/\hbar$, $\mu_0$ is the vacuum permeability and $V$ is the volume over which the conversion takes place. The coherence length of the magnetic fields in typical halos is expected to be of the order of the size of the halo, which is $\sim 100\,{\rm kpc}$ for a galaxy. For dark matter axions, which we have already argued will have Compton wavelengths in the cm/mm range, and some decaying spectrum of magnetic turbulence (for example, a Kolmogorov spectrum $k^3|{\hat B}(k)|^2\propto k^{-2/3}$), one finds that $\tau_{\rm B}\gg\tau_{2\gamma}$. In fact, \cite{ref:Sigl} explained that there is a maximum possible flux density that one might expect via this mechanism, and it is far too weak to be detected. For this reason we will ignore this in what follows.
The decay of axions into two photons can be enhanced in the presence of a photon background and, by contrast to the enhancement due to magnetic fields, this may be very significant. References \cite{ref:Caputo, ref:Caputo1} have shown that the effective decay lifetime can be reduced to $\tau=\tau_{2\gamma}/(1+{\cal F}^{\rm eff}_\gamma)$ where ${\cal F}^{\rm eff}_\gamma$ is the photon occupation number associated to the relevant sources considered. Sources of photons include the CMB, the radio background and galactic emission with ${\cal F}^{\rm eff}_\gamma={\cal F}_{\rm CMB}+{\cal F}_{\rm radio}+{\cal F}_{\rm gal}+\ldots$ For the CMB, this is given by
\begin{equation}
{\cal F}_{\rm CMB}=2\left[\exp\left(\frac{m_{\rm a}c^2}{2k_{\rm B}T_{\rm CMB}}\right)-1\right]^{-1}\,,
\end{equation}
where $T_{\rm CMB}=2.725\,{\rm K}=235\,\mu{\rm eV}/k_{\rm B}$ which can be approximated by ${\cal F}_{\rm CMB}\approx 4k_{\rm B}T_{\rm CMB}/(m_{\rm a}c^2)$ for $m_{\rm a}c^2\ll 470~\mu{\rm eV}$ which can provide a potentially very significant enhancement of the signal. The CMB and the radio background are both isotropic sources, and so the factor ${\cal F}$ is easily worked out to be proportional to the brightness temperature measured by experiments \cite{ref:ARCADE2, ref:RadioBack}.
The contribution from the radio background is very uncertain for a number of reasons. Firstly, making absolute measurement of the background temperature is inherently difficult. But perhaps more important is that this measurement is made from the point of view of telescopes on Earth and it may not be the same elsewhere in the Universe and also at higher redshifts. In principle, it would be necessary to model the sources contributing to the radio background and quantify the uncertainty in order set limits on $g_{\rm a\gamma\gamma}$.
A dedicated study of specific sources, which might be easier to model than the overall background, could result in significant effective enhancement in values of $\cal F$ for the axion masses between 1 and 20 $\mu$eV/$c^2$. \cite{ref:Caputo1} suggested that ${\cal F_{\rm source}} \approx I_{\rm source}/E_{\nu}^3$ where $E_{\nu} = hf_{\rm obs}$ is the energy of the photons. We will adopt this relation for our later order-of-magnitude estimates of the signal from the galactic centre including the enhancement due to diffuse radio emission (eg. synchrotron emission) as well as the radio background.
We note that there have been attempts to search for the axion signal in the infra-red waveband~\cite{ref:InfraredAxionSearch}. In particular axions with masses $m_{\rm a}c^2\approx 1-10\,{\rm eV}$ have been considered which could have been produced thermally - in the absence of strong non-thermal production mechanisms such as misalignment and string decay. Thermal production predicts
\begin{equation}
\Omega_{\rm a}h_{100}^2\approx \frac{m_{\rm a}c^2}{130~{\rm eV}}\left(\frac{10}{g_{\star}}\right)\,,
\end{equation}
and the published limit is $g_{\rm a\gamma\gamma}<10^{-12}\,{\rm GeV}^{-1}$ for axions in the mass range $4.5\,{\rm eV}<m_{\rm a}c^2<7.7\,{\rm eV}$\footnote{There is also a limit of $m_{\rm a}c^2< 0.529$ eV~\cite{ref:MelDi1} from Planck temperature and polarisation data. As these axions may be produced in the early universe also via thermal processes, they constitute a hot dark matter component with masses strongly degenerate with those of the active neutrinos, as their signature on observables is identical to neutrinos. Hence, when axions are relativistic, they contribute to the effective number of relativistic degrees of freedom $N_{\rm eff}$.}. In section~\ref{sect:decay} we will discuss applying exactly the same ideas in the radio/mm waveband. We note additionally that axions with large masses are also subject to constraints from astrophysics, specifically due to axion cooling competing with that from neutrinos in stars and supernovae; the most stringent limit being from the observations of the neutrino burst from SN 1987A, which appears to preclude axions in the mass range $10^{-3}\,{\rm eV}-2\,{\rm eV}$~\cite{ref:KT}. This is based on detailed modelling of the interaction of axions with stellar material and the detailed modelling of stars and hence could be considered to be less direct and more susceptible to uncertainties than other probes.
In the latter half of this paper, we discuss the resonant mixing of photons and axion dark matter in pulsar magnetospheres \cite{Pshirkov:2007st,ref:NS-Hook,ref:NS-Japan,ref:LaiHeyl}. The idea is a simple one: namely that in regions of the plasma where the photon plasma mass and axion mass become degenerate, there is enhanced conversion of dark matter axions to photons, just as in a regular haloscope whose density is tuned to a particular axion mass range. In addition, the ultra-strong magnetic fields of neutron stars also greatly enhance the overall magnitude of the effect. Our analysis falls into roughly two parts. The first focuses on theoretical fundamentals of axion electrodynamics in magnetised plasma, beginning with an examination of one-dimensional (1D) propagation in planar geometries (the standard approach to axion-photon mixing). We clarify two important aspects, firstly how to treat distinct and locally varying dispersion relations of the photon, which we do via a controlled gradient expansion, incorporating the mass-shell constraints systematically. Next we are able to unify two apparently disparate analytic results for the conversion amplitude. The first is the perturbative $\mathcal{O}(g_{\rm a \gamma \gamma}^2)$ formula for the conversion process of e.g., \cite{ref:NS-Hook}, while the second is non-perturbative and given by the well-known Landau-Zener formula~\cite{Brundobler,ref:LaiHeyl} derived by computing the S-matrix for conversion as dictated by the mixing equations. Our analysis unifies these two approaches and reveals the perturbative result to be a truncation of the full Landau-Zener formula in the non-adiabatic limit. For a given plasma background, this allows one to see precisely for what axion masses and momenta the evolution becomes non-adiabatic and therefore where a perturbative treatment is justified (see fig.~\ref{fig:GammaPlot}).
Next we question to what extent the 1D mixing equations (which dominate the literature on axion-photon conversion in stellar environments) are valid, and examine how three-dimensional (3D) effects excite a wider variety of plasma modes and polarisations. This component of our work is important in illustrating the need for a more systematic analysis of 3D effects in axion electrodynamics in magnetised plasmas, as we show qualitatively that if one is not in a specialised 1D geometric setup, then new polarisation modes of the photon are excited. We discuss the difficulties in analytically solving such a system, and leave any further investigation of what this might imply for the overall signal for future work.
We finish our study of conversion in neutron star magnetospheres with some observational considerations, reviewing telescope sensitivities and Doppler broadening of the signal from the motion of the star.
The structure of the paper is as follows. In section \ref{sect:decay} we discuss axion observations in virialised structures and outline the targets with the best prospects for axion decay detection. We devote section \ref{sect:mixing} to the analysis of the evolution of the axion field in neutron star magnetospheres. After a formulation of the problem from first principles, we first investigate a one-dimensional set-up which paves the way for the study of the mixing in two and three dimensions. In this way, we can highlight differences and similarities arising from the geometrical set-up of the problem. We then proceed to estimate the single dish and interferometer sensitivities to the axion-photon parameter space in the context of the resonant conversion in section \ref{sect:resdecay}. We compare previous approaches to this work and explore the simplest way to optimise and to determine the best candidate neutron stars to target in an experiment. We conclude in section~\ref{sect:conclusions}. Some technical details are left in the appendices: in appendix~\ref{App:MassinBeam} we discuss how to evaluate the mass contained in a beam and in appendix~\ref{Density}, we give a detailed derivation of the Wentzel–Kramers–Brillouin (WKB) expansion of axion-photon mixing, with a careful discussion of dispersion relations and a derivation of the Landau-Zener formula.
In sections~\ref{sect:decay} and \ref{sect:resdecay} we will include all factors due to fundamental physics and present quantities in SI units or other appropriately practical units, whereas in section~\ref{sect:mixing} we will present theoretical calculations using natural units $c=k_{\rm B}=\hbar=1$ with the Lorentz–Heaviside convention $\varepsilon_0 = \mu_0 =1$ for the vacuum permittivity and permeability.
\section{Detecting Dark Axions emitted by Virialised Halos}\label{sect:decay}
In this section we will derive estimates for the signal due to the spontaneous decay, present some estimates of what might be possible with current and planned facilities operating in the radio/mm-waveband, concluding that amounts of integration time required are too large to be feasible, and discuss how one might optimise the detection and improve current constraints on the axion-photon parameter space. In order to present estimates of the signal strength we will set up a strawman object which is a galaxy with a virial mass, $M_{\rm vir}=10^{12}\,M_{\odot}$, virial radius $R_{\rm vir}=100\,{\rm kpc}$ at a distance $d=5\,{\rm Mpc}$ and a velocity width of $200\,{\rm km}\,{\rm sec}^{-1}$ which corresponds to an object similar to the nearby galaxy Centaurus A \citep{ref:NearGalCat}. We have chosen these values to be broadly consistent with the model for the virial radius ($\propto M_{\rm vir}^{1/3}$) from a given mass that we will use later in the subsequent discussion. As part of that discussion, we focus on our suggestion that the basic signal strength will be relatively independent of the object mass. Such an object would be expected to have an average surface mass density $\Sigma_{\rm vir}\approx M_{\rm vir}/(\pi R_{\rm vir}^2)\approx 0.07\,{\rm kg}\,{\rm m}^{-2}$ over an angular diameter of $\theta_{\rm vir}=2R_{\rm vir}/d\approx 40\,{\rm arcmin}$. We will see that this value, which we will use in the subsequent signal estimates, is probably quite conservative and that values up to a thousand times larger than this might be accessible in some objects, albeit over smaller areas, typically in the centre of the object. The basic conclusion will be that it will be difficult to imagine a telescope with a single pixel receiver system achieving a limit on $g_{\rm a\gamma\gamma}$ better than that from CAST. In order to be competitive with the CAST limit, we find that it is easier to optimise future experiments if one quantifies the signal in terms of the brightness temperature, rather than the flux density. We show that the brightness temperature is proportional to the surface-mass-density $\Sigma_{\rm beam}$ associated with the telescope beam, which makes it clear that future experiments must target the centres of virialised objects where this quantity is the largest possible value. From our analysis, the main conclusion is that the larger surface-mass density at the galactic centre/Virgo cluster centre coupled with large amounts of radio emission at the relevant frequencies could enhance the signal enough to probe couplings below the CAST limit.
\begin{comment}
In order to aid understanding, below we provide a glossary of notation used in the following sections.
\begin{table}[h]
\centering
\begin{tabular}{|l|l|}
\hline
\textbf{Term} & \textbf{Description}\\
\hline
$S_{\rm tot}$ & total flux density (i.e. luminosity divided by bandwidth) from virialised halo\\
$I$ & Intensity - def.\\
$d_L(z)$ & luminosity distance of halo at redshift $z$\\
$R_{\rm beam}$ & width of telescope beam\\
$D, \, A_{\rm eff}$ & telescope diameter and area\\
$t_{\rm int}$ & integration time required for observation \\
$T_{\rm sys}$ & system temperature of telescope\\
$\Delta v$ & width of axion velocity distribution in halo \\
$f_{\rm obs.}, \, \Delta {f}_{\rm obs.}$ & observing frequency ($\sim m_{\rm a}$) and its associated bandwidth\\
$\theta_{\rm FWHM}$& (maximum?) observing angle for a given wavelength and distance to object: $\theta_{\rm FWHM} = \lambda_{obs}/D$ \\
$M, \, \Sigma$ & total halo mass, and its surface density,
respectively \\
$M_{\rm beam}, \, \Sigma_{\rm beam}$ & total mass of halo and surface mass contained in telescope beam - see fig.~\ref{fig:HaloBeam}.\\
$S_{\rm beam}$ & flux density from halo contained within telescope beam\\
$T_{\rm RJ}$ & Raleigh Jeans Temperature - which is?\\
GBT & \textit{Green Bank Telescope} - $100$m diameter collecting area in range $0.1–116$GHz $\simeq$ eV. \\
SKA& \textit{Square Kilometer Array} area + freq. \\
c& concentration parameter - def./which is?\\
\hline
\end{tabular}
\label{tab:glossary}
\end{table}
\end{comment}
\subsection{Estimates of the signal amplitude for axion decay from virialised halos}\label{subsec:SpontSignal}
Clearly the first and most important task in determining whether or not dark matter axions can be detected via spontaneous decays is to obtain a reliable estimate for the strength of the resulting signal. Let us consider a virialised halo of mass $M$ and at redshift $z$. We further assume that axions constitute its whole mass. The total bolometric flux from the object is
\begin{equation}\label{eqn:BolometricFlux}
\int \,S_{\rm tot} \,\mathrm{d}f_{\rm obs} = \frac{L_{\rm obs}}{4\pi [r(z)]^2} = \frac{N_{\rm a}E_{\rm obs}}{\tau_{\rm obs}}\frac{1}{4\pi r(z)^2}\,,
\end{equation}
where $r(z)$ is the comoving distance to redshift $z$, $S_{\rm tot}$ the total flux density, $E_{\rm obs}=2hf_{\rm emit}/(1+z)$ and $\tau_{\rm obs}=(1+z)\tau_{2\gamma}/(1+{\cal F}^{\rm eff}_\gamma)$ are the emitted photon energy and decay life-time in the observer's frame, respectively and $\mathcal{F}_{\gamma}^{\rm eff}$ is the photon distribution discussed in the previous section. The luminosity in the observer's frame is $L_{\rm obs}=N_{\rm a}E_{\rm obs}/\tau_{\rm obs}$ and $N_{\rm a}=M/m_{\rm a}$ is the number of axions in the halo. One can obtain an estimate of the observed flux density by assuming that all the flux is detected (the point source approximation) and that it is equally distributed across a bandwidth $\Delta f_{\rm obs}$, effectively assuming a top-hat line profile, in the observer's frame
\begin{equation}\label{Eqn:Flux}
S_{\rm tot} =\frac{Mc^2}{4\pi [d_{\rm L}(z)]^2\tau_{2\gamma}\Delta f_{\rm obs}}(1+{\cal F}^{\rm eff}_\gamma)\,,
\end{equation}
where $d_{\rm L}(z)=(1+z)r(z)$ is the luminosity distance to redshift $z$. We note that this formula is equivalent to that for the emission of neutral Hydrogen due to the spin-flip transition under the exchange of $M$ with the neutral Hydrogen mass, $M_{\rm HI}$, and $\tau_{2\gamma}$ with the effective lifetime of the spin state.
Neither of the assumptions will be true in reality. The assumption of a top-hat frequency profile should only lead to a small correction if $\Delta f_{\rm obs}$ is set by the velocity width of the halo $\Delta v/c=\Delta f_{\rm obs}/f_{\rm obs}$. From first principles, this is set by the halo mass as $\Delta v\propto M^{1/3}$. In what follows, it will be convenient to specify the measured value of $\Delta v$ for a specific object rather than calculate it self-consistently from the halo mass. For typical values, and a halo at redshift $z$, we find
\begin{align}\label{Eqn:Bandwidth}
\Delta f_{\rm obs} = &\,{\frac{f_{\rm emit}\Delta v}{c(1+z)}}\,,\\
\approx &\, \frac{20\,{\rm MHz}}{{1+z}}\left(\frac{\Delta v}{200\,\rm km\,sec^{-1}}\right)\left(\frac{m_{\rm a}c^2}{250\,\rm \mu eV}\right)\,.
\nonumber
\end{align}
Typical receiver systems can produce spectra with the resolution in eq.~(\ref{Eqn:Bandwidth}) in all but the most extreme situations. The question of whether one is sensitive to flux from the entire halo is more complicated. Unless the telescope beam is larger than the projected angular size of the cluster, the total flux-density can be less than that of eq.~\eqref{Eqn:Flux} as illustrated in fig.~\ref{fig:HaloBeam}. Let us now estimate the importance of finite angular resolution.
\begin{figure*}
\centering
\includegraphics[scale=1.2]{Halo.pdf}
\caption{Schematic illustration of the telescope beam of width $R_{\rm beam}$ given in eq.~\eqref{eqn:rbeam} and virialised halo with surface density $\Sigma$ and virial radius $R_{\rm vir}$.}
\label{fig:HaloBeam}
\end{figure*}
We define $R_{\rm beam}$ as the {\em radius} corresponding to the Full-Width Half-Maximum (FWHM) angular {\em diameter} $\theta_{\rm FWHM}\approx \lambda_{\rm obs}/D$, where $\lambda_{\rm obs}$ is the observed wavelength and $D$ is the effective diameter of the observing telescope. In the case of a single dish telescope this is the actual size, whereas for an interferometer it will be given by the longest baseline. The beam radius can be estimated by $R_{\rm beam} = d_{\rm A}(z)\sin{(\theta_{\rm FWHM}/2)}$, where $d_{\rm A}(z)$ is the angular diameter distance which can be expanded for small $\theta_{\rm FWHM}$ to give
\begin{align}\label{eqn:rbeam}
R_{\rm beam} = &\,{\frac{hr(z)}{Dm_{\rm a}c}}\,,\\
\approx &\, 0.5\,{\rm kpc}\left(\frac{r(z)}{5\,{\rm Mpc}}\right)\left(\frac{D}{100\,{\rm m}}\right)^{-1}\left(\frac{m_{\rm a}c^2}{250\,\mu{\rm eV}}\right)^{-1}\,,\nonumber
\end{align}
where we have adopted a fiducial diameter of $100\,{\rm m}$ such as for the Green Bank Telescope (GBT). If $M_{\rm beam}\le M_{\rm vir}$ is the mass enclosed in the projected cylinder, then the observed flux density will be
\begin{align}\label{eqn:FluxValue}
S_{\rm beam} \approx &\, 4\,\mu{\rm Jy}\left(1+{\cal F}^{\rm eff}_\gamma\right)\times
\left(\frac{\tau_{2\gamma}}{8\times 10^{35}~\rm s}\right)^{-1} \times\nonumber \\
& \left(\frac{\Delta f_{\rm obs}}{20\,\rm MHz}\right)^{-1}
\left(\frac{M_{\rm beam}}{10^{12}~M_{\rm \odot}}\right)\left(\frac{d_{\rm L}(z)}{5\,\rm Mpc}\right)^{-2} \,.
\end{align}
If we substitute (\ref{eqn:DecayTime}) and (\ref{Eqn:Bandwidth}) into (\ref{eqn:FluxValue}) we find that
\begin{equation}\label{eqn:FluxValue_again}
\begin{split}
S_{\rm beam} \approx &\, 4\,\mu{\rm Jy}\left(1+{\cal F}^{\rm eff}_\gamma\right)\left(\frac{g_{\rm a\gamma\gamma}}{10^{-10}\,{\rm GeV}^{-1}}\right)^2\times\\
& \left(\frac{m_{\rm a}c^2}{250\,\mu{\rm eV}}\right)^2
\left(\frac{M_{\rm beam}}{10^{12}~M_{\rm \odot}}\right)\times\\
& \left(\frac{\Delta v}{{200\,\rm km}{\rm sec}^{-1}}\right)^{-1}\left(\frac{d_{\rm L}(z)}{5\,\rm Mpc}\right)^{-2}\,.
\end{split}
\end{equation}
From this we see that, if ${\cal F}^{\rm eff}_\gamma=0$, the expected flux density is $\propto m_{\rm a}^2$ for a fixed value of $M_{\rm beam}$. This reflects the fact that the size of the object which is inside the beam is dependent on $m_{\rm a}$ via the fact that $f_{\rm obs}\propto\theta_{\rm FWHM}$. This is an undesirable feature of using the flux density to assess the detectability of the axion signal, although it is possible to take into account the dependence of $M_{\rm beam}$ on $\theta_{\rm FWHM}$. Note that there will be additional dependence on $m_{\rm a}$ from ${\cal F}^{\rm eff}_\gamma$; for example, there is a component from the CMB which is $\propto m_{\rm a}^{-1}$.
It is possible to express the expected signal in terms of the intensity $I$, or equivalently the Rayleigh-Jeans brightness temperature
\begin{equation}
I=\frac{2f_{\rm obs}^2k_{\rm B}T_{\rm RJ}}{c^2}\,,
\end{equation}
and we shall see that this is a much clearer way of quantifying the signal. For a source of axions at redshift $z$ with surface mass-density $\Sigma=\int\rho_{\rm a}\mathrm{d}l$, taking into account that the flux density is the integral of the intensity over the solid angle subtended by the source, the integrated line intensity is given by
\begin{equation}
\int I_{\rm beam}\,{\rm d}f_{\rm obs}={\frac{c^2 \Sigma_{\rm beam}}{4\pi\tau_{2\gamma}(1+z)^4}}(1+{\cal F}^{\rm eff}_\gamma)\,,
\end{equation}
where the appropriate surface mass density is that integrated over the beam profile of the telescope, $\Sigma_{\rm beam}$. To obtain this expression, we used eq.~(\ref{Eqn:Flux}) and Etherington's reciprocity theorem $d_{\rm L}(z)=(1+z)^2d_{\rm A}$, as the solid angle of the object is defined as $\Delta\Omega=R^2/d_{\rm A}^2$. For the surface mass-density $\Sigma_{\rm beam}=\Sigma_{\rm vir}\approx 0.07\,{\rm kg}\,{\rm m}^{-2}$ of our strawman object, we can deduce an intensity
\begin{equation}
\begin{split}
I_{\rm beam}\approx & {\frac{3\,{\rm mJy}\,{\rm sr}^{-1}}{(1+z)^4}}(1+{\cal F}^{\rm eff}_\gamma)\left(\frac{\tau_{2\gamma}}{8\times 10^{35}\,{\rm sec}}\right)^{-1}\times\\
& \left(\frac{\Delta f_{\rm obs}}{20\,{\rm MHz}}\right)^{-1}\left(\frac{\Sigma_{\rm beam}}{0.07\,{\rm kg}\,{\rm m}^{-2}}\right)\,,
\end{split}
\end{equation}
and a brightness temperature
\begin{align}\label{ref:brighttemp}
T^{\rm beam}_{\rm RJ} \approx & {\frac{100\,{\rm pK}}{(1+z)^2}}(1+{\cal F}^{\rm eff}_\gamma)\left(\frac{\tau_{2\gamma}}{8\times 10^{35}\,{\rm sec}}\right)^{-1}\times\\
& \left(\frac{\Delta f_{\rm obs}}{20\,{\rm MHz}}\right)^{-1}\left(\frac{m_{\rm a}c^2}{250\,\mu{\rm eV}}\right)^{-2}\left(\frac{\Sigma_{\rm beam}}{0.07\,{\rm kg}\,{\rm m}^{-2}}\right)\,.\nonumber
\end{align}
This can be simplified by substituting in eqs.~(\ref{eqn:DecayTime}) and (\ref{Eqn:Bandwidth}) to yield
\begin{equation}\label{ref:brighttemp_again}
\begin{split}
T^{\rm beam}_{\rm RJ} \approx & {\frac{100\,{\rm pK}}{1+z}}(1+{\cal F}^{\rm eff}_\gamma)\left(\frac{g_{\rm a\gamma\gamma}}{10^{-10}\,{\rm GeV}^{-1}}\right)^2\times\\
& \left(\frac{\Sigma_{\rm beam}}{0.07\,{\rm kg}\,{\rm m}^{-2}}\right)\left(\frac{\Delta v}{{200\,\rm km}{\rm sec}^{-1}}\right)^{-1}\,.
\end{split}
\end{equation}
This expression does not have any explicit dependence on $m_{\rm a}$ and tells us that the key parameters dictating the signal strength are $g_{\rm a\gamma\gamma}$, $\Sigma_{\rm beam}/\Delta v$ and ${\cal F}^{\rm eff}_\gamma$. The only dependence on $m_{\rm a}$ is via the observation frequency and consequently the size of the area over which $\Sigma_{\rm beam}$ is computed. The size of the signal could be larger than this for our strawman object which is relevant to an average over the virial radius - see subsequent discussions.
\begin{figure*}[t]
\centering
\includegraphics[width = 0.65\textwidth]{cmb_stimulated_flux1.pdf}
\caption{Estimates of the brightness temperature for a halo as a function of axion mass including spontaneous decay and the enhancement due to stimulated emission from the CMB (solid lines) and the pure spontaneous decay (dashed lines). We have fixed $g_{\rm a\gamma\gamma} = 10^{-10}\,{\rm GeV}^{-1}$ which is close to the CAST limit and is the goal signal level. We have also fixed $\Delta v=200\,{\rm km}\,{\rm s}^{-1}$ and used different values for $\Sigma_{\rm beam}=0.07$, $0.7, 7$ and $70\,{\rm kg}\,{\rm m}^{-2}$ which lead to brightness temperatures $\approx 100\,{\rm pK}$, $1$, $10$ and $100\,{\rm nK}$ respectively for $m_a\gg 470\,\mu{\rm eV}$ where spontaneous decay is dominant. For lower values of $m_{\rm a}$, we see the increase $\propto m_{\rm a}^{-1}$ due to stimulated emission from the CMB which could be added to other sources such as the radio background and galactic emission. We have also included some sample noise levels (dotted lines) due to 1 year of integration time with instantaneous sensitivities of $10\,{\rm mK}{\rm s}^{1/2}$, $100$ and $1\,\mu{\rm K}{\rm s}^{1/2}$ at $m_{\rm a}c^2=250\,\mu{\rm eV}$ with the scaling $m_a^{-1/2}$ necessary for a fixed velocity width. The two vertical lines represent $m_{\rm a}c^2 = 20\, \mu$eV and 250 $\mu$eV, respectively, which are illustrative values that we have used in the text.}
\label{fig:CMB_Enhancement}
\end{figure*}
As a prelude to more detailed discussions of specific telescopes in the next subsection, we comment that a typical flux density of $S_{\rm beam}=4\,\mu {\rm Jy}$ might seem to be a quite accessible number for future large radio telescopes - many papers report detection of radio signals in the $\mu{\rm Jy}$ range using presently available facilities. Conversely a brightness temperature of $T_{\rm RJ}^{\rm beam}=100\,{\rm pK}$ is very low and much weaker than any value usually discussed. These numbers can be reconciled in realising that the flux density is averaged over a region $\Omega\approx \pi (R_{\rm vir}/d)^2\approx 1.2\times 10^{-3}\,{\rm sr}$ and it is also worth noting that most published radio detections are for bandwidths much larger than $20\,{\rm MHz}$. In the subsequent discussion we will argue that it is easier to understand whether the signal is detectable by considering the intensity or brightness temperature and that this gives a clearer picture of the potential for detection.
We can also calculate the background intensity due to all axions in the Universe with comoving density $\rho_{\rm a}$
\begin{equation}
I_{\rm back}={\frac{c^2\rho_{\rm a}}{4\pi [r(z)]^2\tau_{2\gamma}f_{\rm emit}}}{\frac{\mathrm{d}V}{\mathrm{d}z\mathrm{d}\Omega}}\,,
\end{equation}
where $\tfrac{\mathrm{d}V}{\mathrm{d}z\mathrm{d}\Omega}=cr(z)^2/H(z)$ is the comoving volume element and $H(z)$ is the Hubble parameter at redshift $z$. Using this we can deduce a background brightness temperature
\begin{equation}
T_{\rm RJ}^{\rm back} = \frac{3h^3c^5}{8\pi^2k_{\rm B}G}\frac{H_0\Omega_{\rm a}}{\tau_{2\gamma}}\left(\frac{1}{m_{\rm a}c^2}\right)^3\frac{(1+z)^2}{E(z)}\,.
\end{equation}
Assuming that $\Omega_{\rm a}h_{100}^2 \approx 0.12$ and $h_{100} = 0.7$, we obtain
\begin{align}\label{ref:backbrighttemp}
T_{\rm RJ}^{\rm back} \approx &\, 0.3\,{\rm pK}{\frac{(1+z)^2}{E(z)}}\left(\frac{m_{\rm a}c^2}{250\,\rm \mu eV}\right)^{-3}\left(\frac{\tau_{2\gamma}}{8\times 10^{35}\,\rm s}\right)^{-1}\,, \nonumber\\
\approx & 0.3\,{\rm pK}{\frac{(1+z)^2}{E(z)}}\left(\frac{g_{\rm a\gamma\gamma}}{10^{-10}\,{\rm GeV}^{-1}}\right)^2\,.
\end{align}
In making this background estimate we have ignored possible stimulated emission which would, of course, contribute at lower frequencies as was the case for the signal from virialised halos. The fact that this value is significantly lower than that for a halo means that there will be enough contrast to detect the signal from a halo against the background.
One can recover eq.~(\ref{ref:backbrighttemp}) by substituting the background value for $\Sigma/\Delta v$ into (\ref{ref:brighttemp_again}). This background value is given by
\begin{equation}
\frac{\mathrm{d}\Sigma}{\mathrm{d}v}=\rho_{\rm a}(z)\frac{\mathrm{d}l}{\mathrm{d}v}=\frac{(1+z)^3}{ E(z)} \frac{\rho_{\rm a}(0)}{H_0}\,,
\end{equation}
so that at $z=0$ this is $\rho_{\rm a}(0)/H_0\approx 1.2\times 10^{-9}\,{\rm kg}\,{\rm m}^{-3}\,{\rm s}$ using $\Omega_{\rm a}h_{100}\approx 0.17$. Note that one can make a rough estimate for the surface mass density of the background by multiplying the density of axions by the size of the Universe given by the Hubble radius, that is, $\Sigma_{\rm back}\approx \rho_{\rm a}c/H_0\approx 0.36\,{\rm kg}\,{\rm m}^{-2}$. This value is a factor of a few larger than the fiducial value we used for the halo surface mass density. To explain why this is the case, it is useful to notice that $\Sigma^{\rm halo}\approx \rho_{\rm a}\Delta_{\rm vir}R$, where $\Delta_{\rm vir}$ represents the virial overdensity of the halo. This quantity can be evaluated, given a cosmological model, using the virial theorem (see the Appendix in \cite{Pace2017a} for details on the implementation and \cite{Pace2019b} for a recent discussion on the topic), but here we will consider it to be of the order of 200 (higher values are also often used). The ratio between the two expressions, $\Sigma^{\rm back}/\Sigma^{\rm halo}\approx \tfrac{c/H_0}{\Delta_{\rm vir}R}\gg 1$ for our strawman object, but it is of the order of a few for $\Delta_{\rm vir}$ (a few hundred) and $R$ (a few Mpc).
In fig.~\ref{fig:CMB_Enhancement}, we present estimates of the brightness temperature expected from a halo with a fixed velocity width $\Delta v=200\,{\rm km}{\rm sec}^{-1}$ and a range of values for $\Sigma_{\rm beam}$ computed using (\ref{ref:brighttemp}). We have fixed $g_{\rm a\gamma\gamma}=10^{-10}\,{\rm GeV}^{-1}$ which is close to the upper limit from the CAST experiment (and hence the target goal) and have included the effects of stimulated emission by the CMB which leads to an increase $\propto m_{\rm a}^{-1}$ for $m_{\rm a}\ll 470\,\mu{\rm eV}$. We have chosen $\Sigma_{\rm beam}=0.07\,{\rm kg}{\rm m}^{-2}$ which is $\Sigma_{\rm vir}$ for our strawman object, along with ten, hundred and a thousand times this value. In subsequent sections, we will discuss that such values might be attainable by observing more concentrated regions of the halo close to their centres.
In addition we have also added noise curves for a total integration time of 1 year with instantaneous sensitivities of $10\,{\rm mK}{\rm s}^{1/2}$, 100 and $1\mu{\rm K}{\rm s}^{1/2}$ at $m_{\rm a}c^2=250\,\mu{\rm eV}$ with the scaling $\propto (m_{\rm a}/250\,\mu{\rm eV})^{-1/2}$ so that the noise level remains that for a fixed velocity width as $m_{\rm a}$ varies. We see that a sensitivity of $\sim 10\,{\rm mK}\,{\rm s}^{1/2}$ - which we will argue in section \ref{subsec:SpontSensitivity} is typical of a single pixel receiver at the relevant frequencies and bandwidths - is not sufficient to get anywhere near detecting the signal for $g_{\rm a\gamma\gamma}=10^{-10}\,{\rm GeV}^{-1}$, never mind that expected for the KSVZ and DFSZ models for typical values of $\Sigma_{\rm beam}$ as large as $7\,{\rm kg}\,{\rm m}^{-2}$. One might imagine that this can be reduced by having $N$ receivers/telescopes in which case the instantaneous sensitivity will be $\approx 10\,{\rm mK}\,{\rm s}^{1/2}/\sqrt{N}$. Looking at fig.~\ref{fig:CMB_Enhancement}, it appears that $N\sim 10^{2}$ would be necessary to probe signals created by $\Sigma_{\rm beam}\approx 70\,{\rm kg}\,{\rm m}^{-2}$, $\sim 10^{4}$ to probe $7\,{\rm kg}\,{\rm m}^{-2}$, $\sim 10^{6}$ to probe $0.7\,{\rm kg}\,{\rm m}^{-2}$ and $\sim 10^8$ for our strawman value of $0.07\,{\rm kg}\,{\rm m}^{-2}$. Therefore, it is clear that one would need to target sufficiently concentrated parts of haloes to probe this decay, which might be possible in haloes with supermassive black holes at their centres. While this enhancement would not allow one to probe the benchmark QCD models for the axion, one could at least probe the parameter space below the well-established CAST limit [see fig.(\ref{subsec:SpontSensitivity}) for sensitivity estimates].
\subsection{Sensitivity estimates for current and planned telescopes}\label{subsec:SpontSensitivity}
\begin{table*}[th]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Telescope & $N$ & $A_{\rm eff}$ [$\rm m^2$] & $T_{\rm sys}$ [K] & Frequency [GHz] & $\theta_{\rm FWHM} $[arcmin] & $R_{\rm beam}$ [kpc] \\
\hline
GBT & 1 & 5500 & 30 & 30 & 0.3 & 0.5\\
FAST & 1 & 50000 & 20 & 2.4 & 1.4 & 2.1\\
SKA1:Band 5 & 200 & 120 & 20 & 4.6-13.6 & 5.1-14.9 & 7.3-21.7 \\
SKA2:Band 5 & 10000 & 120 & 20 & 4.6-13.6 & 5.1-14.9 & 7.3-21.7\\
\hline
\end{tabular}
\caption{Table of telescope parameters which we have used in section~\ref{subsec:SpontSensitivity} that are {\em indicative} of what might be possible using current and planned facilities. $N$ is the number of dishes, $A_{\rm eff}$ the effective collecting area, $T_{\rm sys}$ the overall system temperature (in Rayleigh-Jeans regime), $\theta_{\rm FWHM}$ the beam size and $R_{\rm beam}$ the radius corresponding to the beam size assuming a distance of 5 Mpc. GBT is the Green Bank Telescope and FAST is the Five hundred metre Aperture Spherical Telescope. They are currently operational and can cover a range of frequencies (up to $\approx 100\,{\rm GHz}$ in the case of GBT and up to $\approx 3\,{\rm GHz}$ for FAST). For the purposes of the discussion we have chosen to focus on one frequency for each and have chosen values of $T_{\rm sys}$ indicative of the noise levels that would be possible. We refer the reader to their webpages \url{https://greenbankobservatory.org} and \url{http://fast.bao.ac.cn/en/} for more detailed information about the capabilities. The Square Kilomtere Array (SKA) is currently being designed/built in two phases. Phase I is much more certain that phase II. Again we believe that our numbers are indicative of what might ultimately transpire.}
\label{tab:Telescopes}
\end{table*}
In this section we assess the possibility of detecting the decay of dark matter axions emitted from virialised halos using current and planned telescopes operating in the radio/mm waveband. We have tabulated the numbers we have used in the sensitivity calculations below in table~\ref{tab:Telescopes}. Typically, previous analyses have focused on comparing the flux density to the expected telescope noise. As we have already alluded to and indeed we will explain below that it is best to frame the discussion of sensitivity in terms of the intensity, or more commonly the brightness temperature.
\subsubsection*{Flux Density Signal}
Having discussed the signal strength associated to axion decays in the previous sub-section, we turn now to another key parameter in determining the feasibility of detection - the integration time. The integration time required to detect a flux density $S_{\sigma}$ in a bandwidth $\Delta f_{\rm obs}$ can be deduced from the radiometer equation
\begin{equation}\label{Eqn:Radiometer}
t_{\rm int} = \left(\frac{2k_{\rm B}T_{\rm sys}}{A_{\rm eff}S_{\sigma}}\right)^2\frac{1}{\Delta f_{\rm obs}}\,,
\end{equation}
where $T_{\rm sys}$ is the system temperature, $S_{\sigma}$ is the flux density noise level and $A_{\rm eff}$ is the effective area. For a signal-to-noise ratio of unity, $S_{\rm beam} = S_{\sigma}$. For a single dish telescope with aperture efficiency $\eta$ (typically $\approx 0.5-0.7$), this is given by $A_{\rm eff}=\eta\pi D^2/4$. Using this, we can deduce that for a $1\sigma$ detection of the flux described by eq.~(\ref{eqn:FluxValue}) for a fiducial $M_{\rm beam}=10^{12}M_{\odot}$, the integration time is given by
\begin{align}\label{eqn:IntegrationTime}
t_{\rm int} \approx & \frac{10\,{\rm days}}{(1+{\cal F}^{\rm eff}_\gamma)^2 (1+z)}\left(\frac{T_{\rm sys}}{30 \rm~K}\right)^2\left(\frac{A_{\rm eff}}{5500 ~\rm m^2}\right)^{-2}\times \nonumber\\
& \left(\frac{\Delta v}{200\,{\rm km\,s^{-1}}}\right)\left(\frac{g_{\rm a\gamma\gamma}}{10^{-10}\rm \,GeV^{-1}}\right)^{-4}\times \\
& \left(\frac{m_{\rm a}c^2}{250\,{\rm \mu eV}}\right)^{-5}
\left(\frac{M_{\rm beam}}{10^{12}\,M_{\rm \odot}}\right)^{-2} \left(\frac{d_{\rm L}(z)}{5\,\rm Mpc}\right)^{4}\,,\nonumber
\end{align}
where the specific choice for $T_{\rm sys}$ and $A_{\rm eff}$ have been chosen to be indicative of what might be possible for observations at $30\,{\rm GHz}$ with a $100\,{\rm m}$ telescope such as the GBT which would have a resolution $\approx 20\,{\rm arcsec}$ operating in a band around $30\,{\rm GHz}$ and an axion mass $m_{\rm a}c^2\approx 250\,\mu{\rm eV}$. Despite this particular choice, the expression for $t_{\rm int}$ should be applicable to the whole range of frequencies observed by the GBT, and indeed any single dish radio telescope, provided $M_{\rm beam}$ is chosen appropriately. We chose the GBT to illustrate this since it is the largest telescope in the world operating at these frequencies and possibly as high as $\lesssim100\,{\rm GHz}$. Setting a 95\% exclusion limit - which is the standard thing to do in constraining dark matter - would require approximately 40 days. Detection at the $5\sigma$ level would take 25 times longer, that is 250 days of on-source integration time. Achieving an exclusion limit for the flux expected for the KSVZ model in this mass range would require ruling out $\tau_{2\gamma}\approx 6\times 10^{40}\,{\rm s}$ which would take $5\times 10^9$ times longer, and the level expected for DFSZ will be even lower, neither of which are practical. We note that ${\cal F}_{\rm CMB}\approx 0.5$ for $m_{\rm a}c^2=250\mu{\rm eV}$ and $\approx 12$ for $m_{\rm a}c^2=20\,\mu{\rm eV}$ which will reduce the required integration times, but probably not enough to make much difference to the conclusions.
Despite this, one might think that integration times of a few tens of days might allow one to impose stronger limits than the CAST bounds. However, the numerical value in \eqref{eqn:IntegrationTime} is quite misleading since such a telescope would have a resolution of $\approx 20\,{\rm arcsec}$ at these frequencies and therefore we would expect $M_{\rm beam}\ll M_{\rm vir}$. From eq.~(\ref{eqn:rbeam}) we have that $R_{\rm beam}\approx 0.5\,{\rm kpc}$ when the galaxy would be expected to have a total radius of $R_{\rm vir}\approx 100\,{\rm kpc}$, which is a factor of $200$ larger.
We can obtain an estimate for the total halo mass contained within the beam by using the canonical halo dark matter distribution given by the Navarro-Frenk-White (NFW) profile \citep{ref:NFW} parameterized by the concentration parameter, $\hat{c}$, which is the ratio of the virial radius and the scale radius of the halo. It quantifies the amount of mass within the scale radius relative to that in the total halo, with large values of $\hat{c}$ having more mass {\em concentrated} in the centre than lower values. In appendix~\ref{App:MassinBeam} we have calculated for $\hat{c}R_{\rm beam}/R_{\rm vir} = R_{\rm beam}/r_{\rm s}\ll 1$, that is, a beam size much less than the characteristic scale of the NFW profile, the following estimate for the halo mass contained within the telescope beam:
\begin{figure}
\centering
\includegraphics[width =0.9 \linewidth]{conc_trend.pdf}
\caption{Projected mass within the beam as a function of $\bar{R} = cR_{\rm beam}/R_{\rm vir}$ assuming an NFW profile. From top to bottom, we consider three different concentration parameters, ranging from clusters to dwarf galaxies. The solid lines represent an analytic approximation for $\bar{R} \ll 1$, while the dotted lines are given by full numerical integration - see appendix \ref{App:MassinBeam} for details.}
\label{fig:VirgoMass}
\end{figure}
\begin{equation}
\frac{M_{\rm beam}}{M_{\rm vir}}= \frac{R_{\rm beam}^2}{R_{\rm vir}^2} \cdot \frac{\hat{c}^2}{2 f(\hat{c})}\log{\left(\frac{{2R_{\rm vir}}}{\hat{c}R_{\rm beam}}\right)} \,,
\label{ref:mbeam}
\end{equation}
where $f(x)=\log(1+x)-\frac{x}{1+x}$. The behaviour of the beam mass is plotted in fig.~\ref{fig:VirgoMass}. Using this expression we deduce that $M_{\rm beam}\approx 0.8\times 10^{9}\,{\rm M_{\odot}}\,, 1.9\times 10^{9}\,{\rm M_{\odot}}$ and $6.2\times 10^{9}\,{\rm M_{\odot}}$ for ${\hat c}=3,5$ and 10, respectively. As one would expect, there is a trend for $M_{\rm beam}$ to increase as $\hat{c}$ increases, but even for relatively large values we find that in this case $M_{\rm beam}\ll M_{\rm vir}$. Clearly, this reduction in $M_{\rm beam}$ has a deleterious effect on the ability of a single dish telescope to even post an upper limit on the spontaneous decay of dark matter axions since $t_{\rm int}\propto M_{\rm beam }^{-2}$ with $t_{\rm int}\approx 3\times 10^4\,{\rm years}$ for $M_{\rm beam}=10^{9}M_{\odot}$. Therefore, one needs to be very careful in using (\ref{eqn:IntegrationTime}).
It is possible to think in terms of the flux density, but as we have explained above one has to be very careful to use the mass inside the beam radius and not the total mass of the object since they will typically be very different. Our view is that it is much easier to think in terms of the brightness temperature (or equivalently the intensity, although telescope sensitivities are more commonly expressed in terms of a brightness temperature).
\subsubsection*{Brightness Temperature Signal}
The calculation of the noise temperature is simpler. The noise level in intensity is simply given by $I_{\sigma} = S_{\sigma}/\Omega_{\rm beam}$. Substituting for the intensity in terms of Rayleigh-Jean's law and setting $\Omega_{\rm beam} = \lambda^2/D_{\rm tel}^2$, we obtain the well-known Radiometer equation for brightness temperature
\begin{equation}
T_{\sigma} = \frac{T_{\rm sys}}{\eta\sqrt{\Delta f_{\rm obs} t_{\rm int}}} \,,
\end{equation}
for a single telescope with system temperature $T_{\rm sys}$ and aperture efficiency $\eta$ observing in a bandwidth of $\Delta f_{\rm obs}$. The instantaneous sensitivity is just given by $T_{\rm sys}/(\eta\sqrt{\Delta f_{\rm obs}})\approx 10\,{\rm mK}\,{\rm s}^{1/2}\left(T_{\rm sys}/30\,{\rm K}\right)(\Delta f_{\rm obs}/20\,{\rm MHz})^{-1/2}$ for $\eta=0.7$ and hence the integration time required to detect a surface mass density of $\Sigma_{\rm beam}$, which is that averaged over the beam radius, at $1\sigma$ is
\begin{equation}\label{eqn:tintegrate}
\begin{split}
t_{\rm int} \approx & 3\times 10^{8}\,{\rm years} \frac{(1+z)^3}{(1+{\cal F}^{\rm eff}_\gamma)^2}\left(\frac{T_{\rm sys}}{30\,{\rm K}}\right)^2\times\\
& \left(\frac{g_{\rm a\gamma\gamma}}{10^{-10}\,{\rm GeV}^{-1}}\right)^{-4}\left(\frac{\Sigma_{\rm beam}}{0.07\,{\rm kg}\,{\rm m}^{-2}}\right)^{-2}\times\\
&\left(\frac{\Delta v}{{200\,\rm km}{\rm sec}^{-1}}\right)\left(\frac{m_{\rm a}c^2}{250\mu{\rm eV}}\right)^{-1}\,.
\end{split}
\end{equation}
Note that this is independent of the telescope collecting area, as one would expect for an unresolved detection, and also there is no explicit dependence on the distance, although there is a dependence on the redshift. Many of the other dependencies, for example, on $T_{\rm sys}$, $\Delta v$ and $g_{\rm a\gamma\gamma}$ are the same. Moreover, this expression makes it very obvious that the discussion above based on (\ref{eqn:IntegrationTime}) can be very misleading since the number at the front of the expression (remembering that the surface mass density of $0.07\,{\rm kg}\,{\rm m}^{-2}$ was chosen to correspond to the average across an object of mass $10^{12}\,M_{\odot}$ and radius $100\,{\rm kpc}$) is very much larger than in (\ref{eqn:IntegrationTime}).
The fact that $t_{\rm int}$ is dependent on $\Sigma_{\rm beam}$ has two advantages. The first is that it is clear that in order to increase the size of the signal and hence reduce $t_{\rm int}$ to a practical length of time one has to increase $\Sigma_{\rm beam}$. From our earlier discussion, we calculated, assuming an NFW profile, $M_{\rm beam}\sim 10^{9}M_{\odot}$ for our fiducial galaxy and telescope configuration for which $R_{\rm beam}\approx 0.5\,{\rm kpc}$, assuming a sensible range of concentration parameters. In this case the appropriate surface mass density would be\footnote{We note that (\ref{eqn:tintegrate}) and (\ref{eqn:IntegrationTime}) would be identical if $\Sigma_{\rm beam}$, $M_{\rm beam}$ and $R_{\rm beam}$ were chosen to be consistent with each other.}
\begin{equation}\label{eqn:Sigma7}
\Sigma_{\rm beam}\approx 7\,{\rm kg}\,{\rm m}^{-2}\left(\frac{M_{\rm beam}}{2.3\times 10^{9}M_{\odot}}\right)\left(\frac{R_{\rm beam}}{0.5\,{\rm kpc}}\right)^{-2}\,.
\end{equation}
Of course this only gives one a factor of around $200$ improvement but it makes it clear in what direction one might have to go in optimising the signal strength. We will return to this issue in sect.~\ref{subsec:SpontTarget}.
The other advantage is that it makes clear what one would have to do to establish an upper bound on the signal: one would need an estimate of $\Sigma_{\rm beam}$ over the region which one was observing. Fortunately, the amplitude of any gravitational lensing signal that one might measure is directly related to the surface mass density. The measurement of the amplification and shear can be related to the surface mass density of the lenses. One of the largest surface mass densities measured from strong lensing on the scale of a few kiloparsecs (which corresponds to the typical beam sizes) is 50 $\rm kg\,{m^{-2}}$ \cite{ref:Winn2004SMD}. Such values are typically found towards the centre of virialised haloes. This motivates high resolution observations and detailed study of high-density sources with rich ambient radio emission for an accurate estimate of $\Sigma_{\rm beam}$ and ${\cal F}^{\rm eff}$.
The discussion so far has focused on the axion mass range $m_{\rm a}c^2\approx 250\,\mu{\rm eV}$, but we have also motivated searches at lower masses, for example, $m_{\rm a}c^2=20\,{\mu \rm eV}$ which corresponds to $f_{\rm obs}=2.4\,{\rm GHz}$. The Five hundred meter Aperture Spherical Telescope (FAST) might be a candidate large telescope for the detection of axions in this mass range. Despite its name, it can only illuminate beams with $D\approx 300\,{\rm m}$ corresponding to a resolution of $\approx 1.4\,{\rm arcmin}$ and $R_{\rm beam}\approx 2\,{\rm kpc}\ll R_{\rm vir}$. The bandwidth corresponding to $\Delta v=200\,{\rm km}\,{\rm sec}^{-1}$ at $z=0$ is $\Delta f_{\rm obs}=1.6\,{\rm MHz}$. The instantaneous sensitivity to such $T_{\rm sys}/(\eta\sqrt{\Delta f_{\rm obs}})\approx 20\,{\rm mK\,s}^{1/2}\left(T_{\rm sys}/20\,{\rm K}\right)(\Delta f_{\rm obs}/1.6\,{\rm MHz})^{-1/2}$ which is a little larger than for our estimate for the GBT at $30\,{\rm GHz}$ despite having a lower system temperature. The formula \eqref{eqn:tintegrate} should apply here as well with the values of $T_{\rm sys}$ and $\Sigma_{\rm beam}$ adjusted to take into account $R_{\rm beam}$ being a little larger. Ultimately, we come to the same conclusion.
If a focal plane array or phased array were fitted to the telescope, it might be possible to observe with $N$ beams and this would reduce the amount of integration time required by a factor of $1/N$. However, there are practical limitations on the size of array which one can deploy on telescope since the physical size of the region over which one can focus is limited; much more than $N\sim 100$ would be difficult to imagine. Moreover, the beams cannot point at the same region of the sky and just serve to increase the field-of-view. This does reduce the noise level, but over a wider area which would likely result in the decrease in the expected signal strength.
A number of recent works \citep{ref:KQ1,ref:KQ2,ref:Caputo,ref:Caputo1} have suggested that it might be possible to use the Square Kilometre Array (SKA) to search for axions. Naively the very large collecting area of the SKA in the formula (\ref{eqn:IntegrationTime}) would substantially reduce the necessary integration time. The proposed band 5 of the SKA, which has a frequency range of $4.6-13.7\,{\rm GHz}$, could potentially be of interest for the detection of axions in the mass range $40-110\,\mu{\rm eV}$. However, it is not valid to use the entire collecting area of the SKA in this way because the beam size, since it is an interferometer, is set by the longest baseline and this would be far too small. If one thinks in terms of brightness temperature, there is an extra factor, known as the filling factor, $\eta_{\rm FF}\ll 1$, which will increase the noise level $\propto \eta_{\rm FF}^{-1}$.
An interesting alternative approach would be to use each of the SKA telescopes as single telescopes in auto-correlation mode as it is envisaged for HI intensity mapping~\cite{ref:Bull}. The SKA dishes will have a diameter of $D=15\,{\rm m}$ and a sensitivity defined by $A/T_{\rm sys}\approx 6\,{\rm m}^2\,{\rm K}^{-1}$. Operating in band 5, this will have a resolution of $\theta_{\rm FWHM}\approx 15\,{\rm arcsec}$ at the lower end of the band and $\approx 6\,{\rm arcsec}$ at the higher end. In the first instance the SKA - SKA phase 1, sometimes called SKA1 - will have $\approx 200$ such dishes but may eventually - SKA2 - have $\approx 10000$. As before, the integration time for the telescopes decreases by a factor of $N$, the number of telescopes, but unlike a phased array on a single telescope they can co-point at the same region of sky which is advantageous. With 200 telescopes, we estimate an integration time of about 1.5$\times 10^6$ years, while for $10^4$ telescopes, we obtain $t_{\rm int} \approx 3\times 10^4$ years. This estimate will be smaller for lower masses (around 2 orders of magnitude at $m_{\rm a}c^2 = 20\,{\rm \mu eV}$) due to the enhancement from the stimulated decay. However, this will be mitigated to some extent by the factor $m_{\rm a}c^2$ in the denominator of \eqref{eqn:tintegrate}. The values used are for a strawman object, while if we use the surface mass density of (\ref{eqn:Sigma7}), we would estimate integration times $\approx 10^4$ times smaller, which might bring this in the realms of possibility. We note that our integration time estimate for dwarf galaxies is consistent with that of reference \cite{ref:Caputo1} up to a factor of a few, although it is difficult to make a precise comparison. We believe that any minor discrepancies might be due to the fact that observational measurements of the size of the individual dwarf galaxies might lead to slight overestimation of the signal from them. This point is borne out in fig.~(\ref{fig:tintPlots}), where we obtain slightly lower integration times for higher mass objects when we determine object size from the virial overdensity parameter, via the relationship between the virial mass and radius.
We have already mentioned that \cite{ref:Blout} published an upper limit for $g_{\rm a\gamma\gamma}$ based on 6 days of observations using the Haystack radio telescope for axions in the mass range around $m_{\rm a}c^2\approx 300\,\mu{\rm eV}$. In \cite{ref:Blout} they state that $T_{\rm sys}\approx 100\,{\rm K}$ and we estimate $A_{\rm eff}\approx 750\,{\rm m}^2$ (assuming $\eta\approx 0.6$) and hence flux density and brightness temperature sensitivities of $100\,{\rm mJy}\,{\rm s}^{1/2}$ and $40\,{\rm mK}\,{\rm s}^{1/2}$, respectively, in an observing bandwidth of $\Delta f_{\rm obs}\approx 4\,{\rm MHz}$. They assume a mass of $\approx 10^{7}M_{\odot}$ and a diameter of $\approx 10\,{\rm kpc}$ for the dwarf galaxies which they probe at distances in the range $d\approx 200\,{\rm kpc}$ with velocity width of $\Delta v\approx 30\,{\rm km}\,{\rm s}^{-1}$ equivalent to $\Delta f_{\rm obs}\approx 3.6$\,MHz. For $\tau_{2\gamma}=5\times 10^{33}\,{\rm s}$, which corresponds to their upper limit of $g_{\rm a\gamma\gamma}<10^{-9}\,{\rm GeV}^{-1}$, we predict a flux density of $S\approx 4\,{\rm mJy}$ which would take $3\times 10^{3}\,{\rm s}$ to obtain a 95\% exclusion limit. However, the typical angular diameter of these objects is $\approx 3\,{\rm deg}$, which is very much larger - by around more than a factor of 100 - than the beam size which would mean that $M_{\rm beam}\ll M_{\rm vir}$. For the reasons explained earlier, it is clear that they must have made some error in their calculations and this limit should be discounted.
\subsection{Optimising Target Objects}\label{subsec:SpontTarget}
\begin{figure*}[!t]
\centering
\includegraphics[width=0.45\textwidth]{Scatter_Plot1.pdf}
\includegraphics[width=0.45\textwidth]{Scatter_Plot2.pdf} \\
\caption{In the {\em left} panel. Signal strength as given by $\Sigma/\Delta v \propto T_{\rm RJ}$. We assume an \textit{identical} object and beam size $\Sigma = M_{\rm obj}/(\theta_{\rm obj} D_{\rm obj})^2$ taking values from table~\ref{ClusterCalculations}. Note we normalised $\Sigma$ by the background value $1.2\times 10^{-9}\,{\rm kg}\,{\rm m}^{-3}\,{\rm s}$. The trend appears relatively flat for the data in the table - the solid green line - and is compatible with the simple argument presented in the text, albeit with a somewhat higher value ($\approx 500$) relative to the background value. Possibly there is a trend with mass which we denoted with a line $\propto M^{-0.2}$ which could be due to the concentration parameter varying as a function of mass and the fact that the angular sizes are probably the scale radius for some fitted profile function rather than the virial radius. We note that much of this trend is driven by the outliers at low mass, ultra-faint dwarf spheroidal, and high mass, the galaxy clusters, Virgo and Coma. In the {\em right} panel, we present the quantity in \eqref{eqn:CaputoQuantity} for the data in table~\ref{ClusterCalculations} which clearly increases like $\theta_{\rm obj}^2$ as denoted by the line in the plot. Note that the starred data points, which use observational measurements of the velocity width $\Delta v_{\rm obj}$, and the circular points, which correspond to the inferred width $\Delta v_{\rm inf} \approx \left(GM_{\rm obj}/R_{\rm obj}\right)^{1/2}$, show the same trend.}
\label{fig:ScatterPlots}
\end{figure*}
In the previous two sections we have explained that, if one targets a halo with surface mass density $\Sigma_{\rm beam}\approx 0.07\,{\rm kg}\,{\rm m}^{-2}$ and velocity width $\Delta v\approx 200\,{\rm km}\,{\rm s}^{-1}$, the signal from spontaneous decay combined with stimulated emission from the CMB for $g_{\rm a\gamma\gamma}=10^{-10}\,{\rm GeV}^{-1}$ is too weak to be detected even for an array of receivers with $N\lesssim 10^{6}$. We came to this conclusion by estimating the integration time required to detect the signal focusing on the expression for the signal expressed in terms of the brightness temperature (\ref{ref:brighttemp_again}).
\subsubsection{Maximising brightness temperature}
Examination of this equation makes it clear that the largest possible signal is obtained by maximising $\Sigma_{\rm beam}/\Delta v$. If the object is such that $\theta_{\rm FWHM}\approx \theta_{\rm vir}$, we estimate the quantity to be $\approx 3.5\times 10^{-7}\,{\rm kg}\,{\rm m}^{-3}\,{\rm s}^{-1}$ for the strawman object used in the previous section which is around 300 times larger than the background value. This value is based on what we think, at a level of better than a factor two, are realistic values, but precise knowledge of it is absolutely critical to any attempt to improve the CAST limits of $g_{\rm a\gamma\gamma}$ using this approach. In this section, we will discuss, using theoretical arguments and comparing to observations, the range of values for $\Sigma_{\rm beam}/\Delta v$ that might be available for us to be observed in the Universe.
Consider now the possibility that the effective beam size is sufficiently large to capture the full object flux so that
$S_{\rm beam}=S_{\rm tot}$. From the beam geometry, one expects that $S_{\rm tot} \propto M_{\rm vir}$ - the scenario considered by \cite{ref:Caputo}. Indeed this setup can be realised by considering the resolution of the SKA dishes at 2.4 GHz ($m_{\rm a}c^2 = 20\,{\rm \mu eV}$) for which most of our candidate objects (table \ref{ClusterCalculations}) are within the beam of the telescope. Put simply, this means that we are in the regime where the surface mass density within the beam is that of the whole object, i.e., $\Sigma_{\rm beam}=\Sigma_{\rm vir}$. Similarly, $M_{\rm beam} = M_{\rm vir}$. Throughout the subsequent discussion we therefore identify $\Sigma_{\rm beam} = \Sigma_{\rm vir}$ and phrase our analysis purely in terms of $\Sigma_{\rm vir}$.
One might wonder how $\Sigma_{\rm vir}/\Delta v$ depends on the size of the object. If we consider a halo with virial overdensity $\Delta_{\rm vir}\sim 100$, then $M_{\rm vir}=\tfrac{4\pi}{3}\Delta_{\rm vir}\rho_{\rm a} R_{\rm vir}^3$, where $\rho_{\rm a}=\Omega_{\rm a}\rho_{\rm crit}$ is the background density of axions and $\rho_{\rm crit}$ is the critical density. An estimate for the velocity width, up to order one factors, is $\Delta v=(GM_{\rm vir}/R_{\rm vir})^{1/2}$ and hence we find that
\begin{equation}\label{eqn:SigmaQuantity}
\frac{\Sigma_{\rm vir}}{\Delta v}\approx 0.7\left(\frac{\Delta_{\rm vir}\rho_{\rm a}}{G}\right)^{1/2}\approx 3.5\times 10^{-7}\,{\rm kg}\,{\rm m}^{-3}\,{\rm s}\,,
\end{equation}
which is independent of the size of the object - that is, there is no dependence on $M_{\rm vir}$ or $R_{\rm vir}$. If $\Delta_{\rm vir}$ is universal and independent of the size of the object, as it is supposed to be almost by definition, then the expected brightness temperature averaged over a virialised halo will be independent of the size and hence the optimal detection for a specific halo size and telescope configuration would be obtained by matching the size of the object approximately to the telescope beam width. This is the standard practice to optimise detection efficiency in all branches of astronomy.
This suggestion, that there is no optimal size of object, appears to be contrary to the conclusions of \cite{ref:Caputo}, who claimed that the optimal detection would be for dwarf spheroidal galaxies, that is, the very lowest mass halos. They came to this conclusion considering the quantity
\begin{equation}\label{eqn:CaputoQuantity}
\frac{1}{\Delta v}\int \mathrm{d}\Omega \mathrm{d}l\rho_{\rm a}\propto \frac{M_{\rm beam}}{d^2\Delta v}\propto S_{\rm beam}\,,
\end{equation}
where $d$ is the distance to the object and the angular integration is over the angular size of the object - or, as they state it, for a telescope beam which has the same size as the object. This quantity is $\propto S_{\rm beam}$ defined in \eqref{eqn:FluxValue} which is equivalent to (\ref{ref:brighttemp}) if one is careful with the choice of $\Sigma_{\rm beam}$. But we have already explained that one can come to the wrong conclusion if one uses the wrong value of $M_{\rm beam}$ for a specific halo and that it is actually better to think in terms of the surface mass density $\Sigma_{\rm beam}$.
In fig.~\ref{fig:ScatterPlots}, we have plotted the quantities in \eqref{eqn:CaputoQuantity} and \eqref{eqn:SigmaQuantity} using the data in table~\ref{ClusterCalculations} which is similar to, but not exactly the same as, that used in \cite{ref:Caputo}. In particular, we have added some galaxies and galaxy clusters to the dwarf galaxies which they focus on that enables us to probe a wider lever arm in mass. The table contains values for the distance to and the mass of the object $D_{\rm obj}$ and $M_{\rm obj}$, respectively, the angular size $\theta_{\rm obj}$ and the velocity width $\Delta v_{\rm obj}$. These are inferred in a heterogeneous way, but should at least be indicative of some truth. We would not necessarily expect these values to be those for a virialised halo and therefore we denoted them with the suffix ``obj" to distinguish them as being observationally determined. From the observed information, we can infer the radius, $R_{\rm obj}=\theta_{\rm obj}/(2D_{\rm obj})$ and also check consistency with our analytic estimates above by inferring $\Delta v_{\inf}=(GM_{\rm obj}/R_{\rm obj})^{1/2}$, as well as calculating the surface mass density appropriate to an average over the object radius, $\Sigma_{\rm obj}=M_{\rm obj}/(\pi R_{\rm obj}^2)$.
Firstly, we find in the right panel of fig.~\ref{fig:ScatterPlots} that \eqref{eqn:CaputoQuantity} which was plotted in \cite{ref:Caputo} is indeed $\propto\theta_{\rm obj}^2$ as claimed. But on the basis of the theoretical argument above, this is exactly what one would expect for the total flux density $S_{\rm tot}\propto \Sigma_{\rm ave}\theta^2/\Delta v$, where $\Sigma_{\rm ave}$ is some average surface mass density for the objects, and hence, while it provides some confidence that the modelling is correct, it does not yield any obvious information about which objects would be optimal.
In the left panel of fig.~\ref{fig:ScatterPlots} we have plotted $\Sigma_{\rm obj}/\Delta v$ for the data presented in table~\ref{ClusterCalculations}, using both $\Delta v_{\rm obj}$ and $\Delta v_{\inf}$ with consistent results. We find that the data are compatible with $\Sigma_{\rm beam}/\Delta v$ being a constant over eight orders of magnitude and for it to be $\approx 500$ times the background value - slightly higher than for our strawman object - within the kind of uncertainties that we might expect coming from a heterogeneous sample such as the one which we have used. Visually, there could be some evidence for a trend $\sim M^{-0.2}$ which we have also included to guide the eye, but the evidence for this is largely due to a few outliers at the low- and high-mass ends where perhaps the observational estimates are most uncertain. So it could be that there is some preference for lower mass halos over high mass halos, but the effect is not very dramatic. Note that on the $y-$axis, we plot $\left(\Sigma/\Delta v\right)^{\rm rel} \equiv \frac{\Sigma/\Delta v}{1.2\times 10^{-9}\,{\rm kg\,m^{-3}\,s}}$, where the denominator is the value associated to the background.
It could be that the possible trend seen in the left panel of fig.~\ref{fig:ScatterPlots} is related to the concentration parameter of the halo. It is likely that the observationally determined angular size, $\theta_{\rm obj}$, is not the virial radius but some scale radius from a fitting function used in conjunction with images. If this is the case, then we might expect a weak trend with mass.
The concentration parameter has been computed in numerical simulations and is usually assumed to be universal for halos of a given mass, $M$. A recently proposed expression is~\cite{ref:MultiDark}
\begin{equation}
{\hat c}(M,z)={\hat c}_0(z)\left(\frac{M}{ M_0}\right)^{-\gamma(z)}\left[1+\left(\frac{M}{M_1(z)}\right)^{0.4}\right]\,,
\end{equation}
where $M_0=10^{12}h_{100}^{-1}M_{\odot}$ and $\hat{c}_0(z)$, $\gamma(z)$ and $M_1(z)$ are fitted parameters which are redshift dependent. We will focus on low redshifts where $\hat{c}_0(z)\approx 7.4$, $\gamma(0)\approx 0.12$ and $M_1(0)=5.5\times 10^{17}h_{100}^{-1}M_{\odot}$.
From this we see that at $z=0$, $\hat{c}\propto M^{-0.12}$, that is, lower mass halos typically are more concentrated than higher mass halos, and therefore there will be more mass inside the scale radius, and for observations focusing on the region inside this scale radius $\Sigma_{\rm beam}$ might be larger.
\begin{table*}
\centering
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Object & $D_{\rm obj}$ & $M_{\rm obj}$ ($\rm M_{\odot}$) & $\theta_{\rm obj}$ & $\Delta v_{\rm obj}$ [$\rm km\,s^{-1}$] & Reference(s) \\
\hline
Leo 1 & $250$ kpc & $2.2\times 10^{7}$ & $12.6$ arcmin & 8.8 & \cite{ref:Mateo} \\
NGC 6822 & $490$ kpc & $1.6\times 10^9$ & $40$ arcmin & 8 & \cite{ref:Mateo} \\
Draco & $82$ kpc & $2.2\times10^{7}$ & $28.3$ arcmin & 9.5 & \cite{ref:Mateo} \\
Wilman 1 & 45 kpc & $4\times 10^5$ & 9 arcmin & 4 & \cite{ref:DwarfGalMass} \\
Reticulum 2 & 30 kpc & $5.6 \times 10^5$ & $3.6$ arcmin & 3.3 &\cite{ref:Beasts, ref:Reticulum2} \\
Sextans B & 1345 kpc& $3.9\times 10^8$ & 3.9 arcmin & 18 & \cite{ref:Mateo}\\
Pegasus & 955 kpc& $5.8 \times 10^{7}$ & 3.9 arcmin & 8.6 & \cite{ref:Mateo} \\
Antlia & 1235 kpc& $1.2\times 10^7$ & 5.2 arcmin & 6.3 & \cite{ref:Mateo} \\
NGC 205 & 815 kpc& $7.4\times 10^{8}$ & 6.2 arcmin& 16 & \cite{ref:Mateo}\\
\hline
NGC 5128 & 3.8 Mpc & $5.1\times 10^{11}$ & 34.7 arcmin & 477 & \cite{ref:NearGalCat} \\
NGC 5194 & 15.8 Mpc & $4.2\times 10^{10}$ & 8.4 arcmin & 175 & \cite{ref:NearGalCat} \\
Maffei2 & 2.8 Mpc & $4.2\times 10^{10}$ & 3.8 arcmin & 306 & \cite{ref:NearGalCat} \\
IC2574 & 4.0 Mpc & $4.6\times 10^9$ & 13.2 arcmin & 107 & \cite{ref:NearGalCat} \\
SexA & 1.3 Mpc & $2.5\times 10^8$ & 5.9 arcmin & 46 & \cite{ref:NearGalCat} \\
NGC 3556 & 9.9 Mpc & $3.3\times 10^{10}$ & 5.0 arcmin & 308 & \cite{ref:NearGalCat} \\
IC 0342 & 3.3 Mpc & $1.4\times 10^{11}$ & 21.4 arcmin & 181 & \cite{ref:NearGalCat} \\
NGC 6744 & 8.3 Mpc & $2.2\times 10^{11}$ & 21.4 arcmin & 323 &\cite{ref:NearGalCat}\\
ESO 300-014 & 9.8 Mpc & $10^{10}$ & 7.1 arcmin & 130 & \cite{ref:NearGalCat}\\
NGC 3184 & 11.1 Mpc & $6.3\times 10^{10}$ & 7.4 arcmin & 128 & \cite{ref:NearGalCat}\\
\hline
Virgo & $18$ Mpc & 2.9$\times 10^{15}$ & 7 degrees & 1100 & \cite{ref:VirgoMass1,ref:VirgoMass2} \\
Coma & 100 Mpc & 3$\times 10^{15}$ & 100 arcmin & 1100 & \cite{ref:ComaMass, ref:ComaDistance}\\
\hline
\end{tabular}
\caption{Table of masses ($M_{\rm obj}$), distances ($D_{\rm obj}$), angular sizes ($\theta_{\rm obj})$ and velocity widths ($\Delta v_{\rm obj})$ extracted from the literature and used in fig.~\ref{fig:ScatterPlots}. In each case we have specified the reference of the paper from which the numbers are extracted/calculated. From paper to paper the methods employed are different and hence the overall sample is relatively heterogeneous. For each object we can infer a radius $R_{\rm obj}=\theta_{\rm obj}D_{\rm obj}/2$ and a velocity width $\Delta v_{\rm inf}=(GM_{\rm obj}/R_{\rm obj})^{1/2}$. We find that $\Delta v_{\rm obj}$ is strongly correlated with $\Delta v_{\rm inf}$ as we would expect and indeed that $M_{\rm obj}$ is also correlated with $R_{\rm obj}$.}
\label{ClusterCalculations}
\end{table*}
\begin{figure}[t]
\centering
\includegraphics[width = 0.45\textwidth,height=0.45\textwidth]{G_Func.pdf}
\caption{The function $G(\hat{c}, \tilde{R})$ as a function of its arguments. In the left panel, we plot $G$ as a function of $\hat{c}$ for different values constant $\tilde{R}$, and vice versa in the right panel.}
\label{fig:GFunction}
\end{figure}
This leads us on to an important caveat in this discussion: one does not have to choose to focus on trying to detect the entire signal from a halo and indeed it will be optimal, as well as practical, to not do this. Using (\ref{ref:mbeam}), we can eliminate $M_{\rm beam}$ and $M_{\rm vir}$ in terms of $\Sigma_{\rm beam}$ and $\Sigma_{\rm vir}$. To do this we first recall the definition of the beam surface-mass density (see appendix \ref{App:MassinBeam})
\begin{align}
\Sigma(R_{\rm beam}) &= \int_{R_{\rm beam}}^{\hat c} \frac{r\rho(r)}{\sqrt{r^2 - R_{\rm beam}^2}} dr\,, \\
M_{\rm beam} &= \, 2\pi\int_0^{R_{\rm beam}}R\Sigma(R)\mathrm{d}R\,,
\end{align}
where $r$ is the radial coordinate of the object in question and $R_{\rm beam}$ is the projected distance which we identify to be given by the beam size.
Explicitly for an NFW profile $\rho(r)=\rho_{\rm s}F(r/r_{\rm s})$ with $F(y)=y^{-1}(1+y)^{-2}$, where $r_{\rm s}$ is the scale radius, $R_{\rm vir}$ the virial radius and the ratio of the two ${\hat c}=R_{\rm vir}/r_{\rm s}$. Next we can expand these integrals in small beam radius limit $\bar{R}_{\rm beam}\ll1$ to find the relation
\begin{equation}\label{eq:SigmaBeamNFW}
\Sigma_{\rm beam}\simeq G\left(\hat{c},{\frac{R_{\rm beam}}{R_{\rm vir}}}\right)\Sigma_{\rm vir}\,,\qquad \bar{R}_{\rm beam} \ll 1\,,
\end{equation}
for an NFW profile $G(x,y)=x^2\log(2y/x)/f(x)$ for $y/x\ll 1$. We anticipate that one could derive a similar expression for any halo profile.
We plot the function $G(\hat{c}, \frac{R_{\rm beam}}{R_{\rm vir}})$ as a function of $\hat{c}$ and $\tilde{R} = R_{\rm beam}/R_{\rm vir}$, in fig.~\ref{fig:GFunction} which indicates that enhancements of up to 1000 might easily be possible and that these are likely to be larger in lower mass objects than those of higher mass. Therefore, at a first glance it would appear that, for a fixed experimental set up ($R_{\rm vir}/R_{\rm beam}$ fixed), one should search for an object with the largest concentration, a general result which we already anticipated in section~\ref{subsec:SpontSignal}. However, one should also note that for small $\tilde{R}$, which is fixed by the resolution of the telescope, the enhancement across the different concentration parameters is comparable. Furthermore, for a fixed resolution $\theta$, $R_{\rm beam}/R_{\rm vir}$ is significantly smaller for larger mass halos, since $R_{\rm vir}$ is much larger. As a result, $\Sigma_{\rm beam}$ is larger for larger mass halos.
In conclusion, we have argued that maximising $\Sigma_{\rm beam}/\Delta v$ will give the largest possible brightness temperature signal. Theoretical arguments suggest that if the beam encloses the virial radius of a particular object, this will be independent of mass and a very rudimentary search of the literature for specific values suggests that this could be true. However, for fixed observational setup, and hence fixed resolution, one might find a significant enhancement of the signal due to the fact that the surface mass density will increase as one probes the more central regions of a halo. These are likely to be larger for larger mass objects since the telescope beam probes denser regions of larger mass halos. This is the reason we have presented our sensitivity estimates as a function of $\Sigma_{\rm beam}$ and results for range of values $\Sigma_{\rm beam}=0.07-70\,{\rm kg}\,{\rm m}^{-2}$ in fig.~\ref{fig:CMB_Enhancement}.
\subsubsection{Minimising Integration Time}
\begin{figure*}[!t]
\centering
\includegraphics[width = 0.74\textwidth]{t_int1.pdf}
\caption{The integration time for the 1$\sigma$ detection of the brightness temperature signal for the objects in table \ref{ClusterCalculations}, assuming a single-pixel detector in a GBT-like telescope and the stimulated enhancement from both the CMB and the radio background. In this case, we have used \eqref{eq:SigmaBeamNFW} to evaluate $\Sigma_{\rm beam}$ assuming the resolution of the GBT, that is, the virial mass and the virial radii are related by the virial overdensity parameter, $M_{\rm vir} \propto R_{\rm vir}^3 \Delta_{\rm vir}$. Note that we assume the fiducial signal strength corresponding to $g_{\rm a\gamma\gamma} = 10^{-10}\,{\rm GeV^{-1}}$.}
\label{fig:tintPlots}
\end{figure*}
From \eqref{eqn:IntegrationTime} and \eqref{eqn:tintegrate} we see that the integration time can be expressed either in terms of $M_{\rm beam}$ or $\Sigma_{\rm beam}$. Here we shall use the latter measure. We have just seen how brightness temperature is proportional to $\Sigma_{\rm beam}/\Delta v$ and therefore largest when this ratio is maximal. However, whilst brightness temperature is a key observable, the ultimate arbiter of feasibility of detection is of course the integration time. From \eqref{eqn:IntegrationTime} we see the integration time has a slightly different dependence on the halo parameters $\Sigma_{\rm beam}$ and $\Delta v$ to that of the brightness temperature, scaling instead as $t_{\rm int} \propto \frac{1}{\Delta v} (\Sigma_{\rm beam}/\Delta v)^{-2}$, with the additional factor of $1/\Delta v$ arising from the bandwidth of the signal. In light of the different parametric dependence of the integration time and brightness temperature on the halo parameters $\Sigma_{\rm beam}$ and $\Delta v$, and from table \ref{ClusterCalculations} since $\Delta v$ varies significantly between objects, formally maximising $\Sigma_{\rm beam}/\Delta v$ (brightness temperature) is slightly different to minimising $\Delta v/\Sigma_{\rm beam}^2$ (integration time). Thus, it is natural to re-run the analysis of the previous discussion and check whether there is also no preferred object group for $t_{\rm int}$.
We can then estimate the beam surface mass density $\Sigma_{\rm beam}$ using the NFW profile as found in \eqref{eq:SigmaBeamNFW} and take values of $\Delta v$ from table \ref{ClusterCalculations} as before. Thus, we must know $R_{\rm vir}$, $\hat{c}$ and $\Delta v$. We can infer the virial radius from the mass of the object $M_{\rm vir}= M_{\rm obj} =\frac{4\pi}{3}\Delta_{\rm vir}\rho_{\rm a} R_{\rm vir}^3$, using the values in the table. The results for the integration time for different objects are plotted in fig.~\ref{fig:tintPlots}. We have assumed the resolution of the GBT, i.e., $\theta_{\rm FWHM} \approx 10^{-4}$ at 30 GHz.
At $m_{\rm a}c^2 = 250\, {\rm \mu eV}$, the stimulated enhancement factor is quite small. However, the decay time $\tau_{2\gamma}$ is significantly smaller than at $m_{\rm a}c^2 = 20\,{\rm \mu eV}$. The values of $\cal F^{\rm eff}_{\gamma}$ at lower mass aren't large enough to compensate for the increase of the decay time. Note that $\Sigma_{\rm beam}$ is roughly a factor of 2-3 smaller for lower mass, since the resolution is a factor $\approx 12$ larger. Therefore, the integration time is lower at larger masses. As mentioned before, we see that the larger mass halos give a slightly lower integration time, since we are probing smaller values of $\tilde{R}$, i.e., denser regions of the halo. The Virgo cluster at $m_{\rm a}c^2 = 250 \,{\rm \mu eV}$ has an integration time of around 350 years. Ideally, one would want to find objects where $1 + {\cal F}_{\gamma} \gg 1$ at $m_{\rm a}c^2 \geq 100\,{\rm \mu eV}$. Therefore, this motivates a more detailed study of the radio emission from the centre of the Virgo cluster.
In \cite{ref:Caputo1} it was suggested that the Galactic Centre could be a target since it would benefit from a large signal enhancement from the CMB, the measured radio background, but perhaps most importantly from the diffuse radio emission associated with the high density region and supermassive black hole located there. The size of the enhancement in this direction, ${\cal F}_{\gamma}^{\rm GC}$, due to the photon occupation number density, will depend on the resolution of the telescope used in the measurement since ${\cal F} \approx I_{\nu}/E^3$. Hence, we need to estimate the intensity of radio emission from the Galactic Centre.
A measurement of the flux density of Sagittarius A$^{\ast}$ at 30 GHz is presented in the Planck Point Source Catalogue~\cite{ref:PSC} and we will assume an intensity power law spectral index $\alpha = -2.8$ indicative of synchrotron emission and compatible with the spectrum of the Galactic Centre~\citep{ref:PlanckIV_2018}. For any observation for which this source is effectively point-like, the intensity can be estimated as $I=S/\Omega_{\rm beam}\times (f/30{\rm GHz})^{-2.8}$ where $S\approx 200 {\rm Jy}$ is the flux density from the catalogue, $f$ is the frequency of observation and $\Omega_{\rm beam}$ is the area of the beam, which scales with frequency like $f^{-2}$.
For a GBT-like instrument, this gives us an intensity estimate $\approx 5\times 10^{5} \,{\rm Jy}\,{\rm sr}^{-1}$ and hence the enhancement is
\begin{equation}
{\cal F_{\gamma}^{\rm GC}} \approx 50 \left(\frac{250\,{\rm \mu eV}}{m_{\rm a}c^2}\right)^{0.8} \,.
\end{equation}
Clearly, this suggests that the galactic centre might be a good candidate to target for future studies. Of course, we are assuming in this calculation that the synchrotron index is the dominant contributor to the frequency dependence of the signal, which might be an oversimplification. However, this estimate clearly demonstrates that one can achieve similar sensitivity to the galactic centre with just a 100 m single-dish telescope rather than an array of many dishes used in auto-correlation mode, as done in reference \cite{ref:Caputo1} (which indicates that our order-of-magnitude estimate approximately agrees with their analysis). To make an accurate estimate of the stimulated enhancement factor, a dedicated study of the synchrotron, free-free as well as anomalous microwave emission(s) needs to be carried out, ideally on a pixel-by-pixel basis, from high-resolution observations of the galactic centre.
\subsection{Observational conclusions}
In the previous sections we have argued that the brightness temperature is a more robust quantity to measure, since one does not have to optimise to a specific solid angle for a given resolution. As a result, we have concluded that the appropriate quantity to optimise is $\Sigma_{\rm beam}/\Delta v$. Higher resolution measurements of objects can benefit from an enhancement in the measured $\Sigma_{\rm beam}$. For a flux density measurement, such an arrangement would result in $M_{\rm beam}\ll M_{\rm vir}$, which, of course, implies a weaker signal. Therefore, for single dish observations, the clear way forward is to target smaller regions of the Universe where one may obtain an enhancement for the surface mass density. Clearly, for such observations, one will require higher resolution which is easy for instruments like the GBT.
\begin{figure*}
\centering
\includegraphics[width = 0.45\textwidth]{sensitivity_vir.pdf}
\includegraphics[width = 0.45\textwidth]{sensitivity_vir1.pdf}
\caption{The sensitivity to axion-photon coupling as a function of axion mass observing a source with surface mass density $\Sigma_{\rm beam}$ and a velocity dispersion of $200\,{\rm km\,s^{-1}}$. In the left panel, we assume $N = 10^4$ telescopes (SKA2:Band 5), used in single-dish mode for an integration time of 4 days and a system temperature of $30\,{\rm K}$. The frequency coverage is as given on table \ref{tab:Telescopes}. We include e the enhancement due to the CMB and the radio background in this case, but note that the enhancement from the radio background is very uncertain. In the right panel, we show the sensitivity from observations of the galactic centre between 1 and 100 GHz, assuming a 100 m single-dish telescope with a system temperature of $30\,{\rm K}$, such as the GBT, and an integration time of 4 days. We included estaimtes of the stimulated emission enhancement from the CMB, the radio background and the synchrotron emission from the supermassive black hole, Sagittarius A$^{\ast}$ discussed in the text. We note that in reality, the system temperature for most radio telescope receivers varies with frequency, which would need to be modelled in an experiment. The sky-blue shaded region is the parameter region excluded by the CERN Axion Solar Telescope (CAST) \cite{ref:CAST} \footnote{We thank Igor Irarstorza for sharing the CAST data.}. The green and magenta exclusions are from the ADMX \cite{ref:ADMX2018} and HAYSTAC \cite{ref:HAYSTAC} haloscope experiments. We also highlight the axion mass ranges predicted by the misalignment mechanism (red) and the string decay (cyan).}
\label{fig:SensitivityVir}
\end{figure*}
We have also discussed the stimulated decay enhancement of the signal and noted that this enhancement is substantial at lower mass. A future experiment would greatly benefit from a dedicated study of specific sources for which high intensity radio emission has been measured. In our previous section, we motivated the Virgo cluster and the galactic centre. Note that for our sensitivity estimates for the galactic centre, we have assumed a constant $\Sigma_{\rm beam}$ for all axion masses, since the presence of the black hole results in a density spike at the galactic centre out to a few parsecs from the position of Sagittarius $\rm A^{\ast}$.
For the radio background, we use the power law derived in \cite{ref:ARCADE2}, given by
\begin{equation}
T_{\rm ARCADE-2}\approx 1.2\,{\rm K}\left(\frac{1\,{\rm GHz}}{f_{\rm obs}}\right)^{2.62}\,.
\end{equation}
Substituting this expression back in, one obtains
\begin{equation}
{\cal F_{\gamma}^{\rm RB}} \approx 1.6\times 10^3 \left(\frac{1\,{\rm GHz}}{f_{\rm obs}}\right)^{3.62} \,.
\end{equation}
We note that this is probably an over-estimate of ${\cal F}^{\rm RB}$ since the ARCADE measurement would require an additional population of radio sources at the relevant frequencies. In principle, there is also a free-free component as well as anomalous microwave emission from the galactic plane, some of which will contribute to the photon occupation number associated to the galactic centre. We remark that while a complete study of the sensitivity to the galactic centre is outside the purpose of this work, our order of magnitude estimate motivates a more detailed future study.
In the near future, the SKA will go into operation. With $1\,{\rm km}^2$ of collecting area, the SKA brings the possibility of very high radio sensitivity. However, we note an sparse interferometer is, by construction, most suited to measuring flux densities with high resolution. One can use Rayleigh-Jeans law to convert the noise level on the flux density, which is set by the collecting area into a brightness temperature temperature sensitivity
\begin{equation}
T_{\sigma} = \frac{T_{\rm sys}}{\eta_{\rm FF}\sqrt{\Delta f_{\rm obs}t_{\rm obs}}}\,.
\end{equation}
The factor $\eta_{\rm FF} \equiv (N A_{\rm eff})/D_{\rm baseline}^2\ll 1$ is known as the filling factor and this increases the expected noise level for the brightness temperature. Here, $N$ is the number of telescopes in the interferometric setup and $A_{\rm eff}$ is the effective collecting area of each telescope. However, if the telescopes are all used in single dish mode, then the integration time for a measurement decreases by a factor $N$ since all the telescopes can point at the same region of the sky.
The high resolution associated with interferometers also means that their large collecting area is offset by the small beam size, again decreasing $M_{\rm beam}$ by several orders of magnitude. As mentioned before, the flux density sensitivity can be increased by using the telescope in single-dish mode, which results in a factor of $N$ decrease in integration time.
We conclude from our analysis that the brightness temperature is the appropriate quantity to optimise radio telescope searches for the spontaneous decay. In fig.~\ref{fig:SensitivityVir} we show our estimates of the radio sensitivity to the spontaneous decay. In both the panels, we have set the integration time, $t_{\rm int}$ to be 4 days. The left panel shows the SKA2:Band 5 sensitivity operating in the single dish mode for the Virgo cluster and the Reticulum 2 dwarf galaxy using the numbers explained in the caption. Note that in principle, the sensitivity to the Virgo cluster could be significantly better, as we assume there is no radio emission from the centre of Virgo at frequencies larger than 10 GHz. In the right panel, we show the sensitivity to the galactic centre, assuming $\Sigma_{\rm beam} \approx 7$ and $70\,{\rm kg\,m^{-2}}$ and a single pixel detector in a GBT-like telescope. It is clear the galactic centre is a promising target for future experiments, which motivates a more detailed study of the different sources of stimulated enhancement.
\section{Resonant mixing in neutron stars}\label{sect:mixing}
There has recently been renewed interest in the possibility of detecting radio signals from the resonant conversion of dark matter axions in neutron star magnetospheres \cite{ref:NS-Hook,ref:NS-Japan}, originally proposed in \cite{Pshirkov:2007st} together with a number of follow-up studies \cite{Camargo:2019wou,Safdi:2018oeu,Edwards:2019tzf}. The conversion happens in some small critical region within the magnetosphere where the plasma mass $\omega_{\rm pl}$ is approximately equal to the axion mass $m_{\rm a}$. This part of the magnetosphere -- whose width $\propto 1/\left|\nabla \omega_{\rm pl}\right|$ is determined by the gradients of the background plasma -- acts essentially as a stellar haloscope. The characteristic frequencies for non-relativistic axions are given by the axion mass. The emitted radiation then results in a radio line peaked at frequencies $\omega \simeq m_{\rm a}$.
The effect is similar to the Mikheyev–Smirnov–Wolfenstein (MSW) mechanism for neutrino inter-conversion \cite{Kuo:1989qe} where a finite density of background charge carriers can endow neutrinos with an effective mass so that when the mass-splitting becomes small, flavour mixing is enhanced. Relativistic axion-photon mixing in neutron stars has also been studied in \cite{ref:LaiHeyl} where, by contrast with the dark matter axion case, it was assumed that all particles are in the \textit{weak dispersion} regime $\omega \simeq \left| \textbf{k} \right|$, as in earlier references \cite{Raffelt:1987im}.
The principal aim of this section is to re-examine the canonical assumptions made in the study of axion-photon mixing in a medium and determine to what extent they can be justified in a neutron star setup. Our analysis focuses on the following points:
\begin{enumerate}
\item Unlike for simple haloscopes with constant magnetic fields and uniform plasma densities, magnetospheres are inhomogeneous with a non-trivial 3D structure. We, therefore, examine to what extent the axion-Maxwell equations can be reduced to a two-flavour mixing system in a 1D planar geometry, whose evolution depends on a single integration parameter along the line of sight.
\item We go beyond refs.~\cite{ref:NS-Hook,ref:LaiHeyl} and perform a controlled gradient expansion (appendix \ref{Density}) of the mixing equations similar to ref.~\cite{Prokopec:2003pj}. This allows us to obtain in a systematic way the leading order WKB behaviour of the mixing system and has the particular advantage of providing a careful treatment of dispersion relations which are in general distinct for the axion and photon away from the resonance region. Our treatment is also valid away from purely relativistic/non-relativistic regimes, with our final form of the first order mixing equations valid for arbitrary values of the momenta.
\item We establish in which regions of the axion phase-space $(k,m_{\rm a})$ the evolution can be considered non-adiabatic. This determines when the $a \rightarrow \gamma$ conversion can be treated perturbatively in the coupling $g_{\rm a \gamma \gamma}$ and where a non-perturbative Landau-Zener formula \cite{Brundobler,ref:LaiHeyl} for two-level mixing must be applied.
\item We examine the role of higher dimensional structure in producing a longitudinal mode $\nabla \cdot \textbf{E}\neq 0$ for the photon and to what extent geometry affects the decoupling of polarisations $\textbf{E}_\parallel$ and $\textbf{E}_{\perp}$, parallel and normal to the background magnetic field.
\end{enumerate}
\subsection{Axion Electrodynamics}
Our starting point is the standard Lagrangian for the axion and photon, with medium effects described by a current $j^{\mu}$:
\begin{equation}\label{eqn:ClassicalEulerHeisenberg}
\begin{split}
\mathcal{L} = & -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - A_{\mu}j^{\mu} \\
& + \frac{1}{2}\left(\partial_{\mu}a\partial^{\mu}a - m_{\rm a}^2a^2 \right) + \frac{1}{4}g_{\rm a\gamma\gamma}aF_{\mu\nu}\tilde{F}^{\mu\nu} \,,
\end{split}
\end{equation}
where $F^{\mu\nu}$ and $\tilde{F}^{\mu\nu}$ are the electromagnetic field tensor and its dual, respectively. The equations of motion for the electromagnetic (EM) fields are given by
\begin{align}
\nabla \cdot \textbf{E}& = \rho - g_{\rm a \gamma \gamma} \textbf{B} \cdot \nabla a\,, \label{Gauss}\\
\nabla \times \textbf{B} - \dot{\textbf{E}} &= \textbf{J} + g_{\rm a \gamma \gamma}\dot{a} \textbf{B} - g_{\rm a \gamma \gamma} \textbf{E}\times \nabla a\,,\label{curlB}\\
\nabla \cdot \textbf{B} &=0\,, \\
\dot{\textbf{B} } + \nabla \times \textbf{E}& =0 \label{Bianchi2}\,.
\end{align}
Next, we linearise the equations of motion about the background solutions satisfying the $g_{\rm a \gamma \gamma}=0$ equations of motion by setting $\textbf{E} \rightarrow \textbf{E}_0 + \textbf{E}$ and $\textbf{B}\rightarrow \textbf{B}_0 + \textbf{B}$, with a corresponding ansatz for $\rho$ and $\textbf{J}$. We also neglect the background electric field, setting $\textbf{E}_0=0$, since for neutron stars the magnetic component typically dominates in the magnetosphere, see, e.g., \cite{Melrose:2016kaf}. The electromagnetic fluctuations must be self-consistently accompanied by perturbations of charge carriers in the plasma via Lorentz forces. This can be modelled via an Ohm's law relation between the current and electric fluctuations $\textbf{E}$ and $\textbf{J}$,
\begin{equation}\label{Ohm}
\textbf{J} = \sigma \cdot \textbf{E}\,,
\end{equation}
where the three-by-three matrix $\sigma$ is the conductivity tensor. Note that together with current conservation $\dot{\rho} + \nabla \cdot \textbf{J}=0$, this closes the system of equations. To obtain a simple system of mixing equations, we specialise to a stationary background throughout the remainder of this section assuming $\textbf{B}_0$ and $\sigma$ to be time-independent, as would be the case for an aligned rotator neutron star model. One then obtains the following system of mixing equations for $\textbf{E}$ and $a$,
\begin{align}
\square \, a + m_{\rm a}^2 a & = g_{\rm a \gamma \gamma}\textbf{E} \cdot \textbf{B}_0\,,\label{axionEOM}\\
\square \, \textbf{E} + \nabla (\nabla \cdot \textbf{E}) + \sigma \cdot \dot{\textbf{E}} &= -g_{\rm a \gamma \gamma} \ddot{a}\textbf{B}_0\,,
\label{EOMPerts}
\end{align}
where \eqref{EOMPerts} was obtained by taking the curl of \eqref{Bianchi2} and combing with \eqref{curlB} and \eqref{Ohm}. We have thus completely parametrised the axion-photon fluctuations in terms of two physical fields, $\textbf{E}$ and $a$. Note that the magnetic component is determined immediately from integration of \eqref{Bianchi2}. We see from \eqref{EOMPerts} that, in general, different polarisations of $\textbf{E}$ will mix owing to the presence of a longitudinal mode $\nabla \cdot \textbf{E} \neq 0$, which can be sourced via the axion [see eq.~\eqref{Gauss}] or when $\sigma$ has off-diagonal components. Note, furthermore, that in a stationary background, the fields have simple harmonic time-dependence $\sim e^{-\imath\omega t}$. The conductivity in a magnetised plasma takes the form \cite{Gurevich2006}
\begin{equation}\label{sigma}
\sigma(\omega) = \frac{\imath\,e^2 n_e}{m_e} R_B(\theta)
\left(
\begin{array}{ccc}
\frac{\omega}{\omega^2 -\omega_{\rm B}^2 } & \, \frac{i\omega_{\rm B}}{\omega^2 -\omega_{\rm B}^2 } & \, 0 \\
-\frac{i\omega_{\rm B}}{\omega^2 -\omega_{\rm B}^2 } & \frac{\omega}{\omega^2 -\omega_{\rm B}^2 } &\, 0 \\
0 & \, 0 & \, \frac{1}{\omega}
\end{array}
\right)R_B(\theta)^{-1}\,,
\end{equation}
where $\theta=\theta(\textbf{x})$, $\omega_{\rm B}=eB_0/m_e$ is the gyrofrequency, $R_B(\theta)$ is the local rotation matrix which rotates $\textbf{B}_0$ into the $z$-direction and $B_0=|\mathbf{B}_0|$. We assume furthermore that $\omega \ll \omega_{\rm B}$, which is easily satisfied for neutron stars with $B \simeq 10^{9}$-$10^{14}\text{G}$ and frequencies $\omega \simeq m_{\rm a} \sim \mu\text{eV}$ associated to non-relativistic axions. In this case, one has $\sigma(\omega)\cdot \textbf{E} = \imath(\omega_{\rm pl}^2/\omega) \textbf{E}_{\parallel}$, where $\textbf{E}_{\parallel}$ is the component of $\textbf{E}$ along $\textbf{B}_0$.
\subsection{Resonant mixing in 1D}\label{sec:1D}
Here we spell out what are the precise physical assumptions needed to reduce the plasma \eqref{axionEOM}-\eqref{EOMPerts} to a simple 1D problem.
Consider first a planar geometry in which all background fields depend on a single parameter $z$, i.e., $\textbf{B}_0=\textbf{B}_0(z)$. Then, since $\textbf{B}_0$ is transverse ($\nabla \cdot \textbf{B}_0 = 0$), it follows immediately that $\textbf{B}_0$ has no polarisation in the $z$-direction. Consider also that the wavefronts propagate in the same direction, such that $a = a(z)$ and $\textbf{E} = \textbf{E}(z)$. Crucially, these geometric assumptions ensure
\begin{equation}
\textbf{B}_0(z) \cdot \nabla (\nabla \cdot \textbf{E}(z)) = 0\,,
\end{equation}
since by construction there are no gradients in the direction of $\textbf{B}_0$. Thus, by geometric considerations and assumptions, we are able to exclude the effects of a longitudinal component $\nabla \cdot \textbf{E}$ from the mixing equations. One can then project \eqref{EOMPerts} onto $\textbf{B}_0$ to arrive at the following set of mixing equations,
\begin{equation}
\left(
\begin{array}{cc}
\partial_z^2 - m_{\rm a}^2 + \omega^2 & \quad \omega g_{\rm a \gamma \gamma} B_0(z)\\
\omega g_{\rm a \gamma \gamma} B_0(z) & \partial_z^2 - \omega_{\rm pl}^2(z) +\omega^2
\end{array}
\right)
\left(
\begin{array}{cc}
a\\
\mathcal{E}
\end{array}
\right) = 0\,, \label{eq:2by2}
\end{equation}
where $\mathcal{E} = E_{\parallel}/\omega$, $E_{\parallel}=\textbf{E}\cdot\textbf{B}_0/|\textbf{B}_0|$ is the component of $\textbf{E}$ parallel to $\textbf{B}_0$ and $\omega_{\rm pl}^2 = e^2 n_{e}/ m_{\rm e}$ is the plasma frequency. The remaining component $E_{\perp}$ normal to $\textbf{B}_0$, from Gauss' law can be seen to satisfy $\partial_z E_{\perp} = 0$ and thus by boundary conditions must vanish. Thus, in such a geometry, the mixing simplifies to only two degrees of freedom. To fully solve these equations, one should ensure that solutions have the appropriate ingoing and outgoing waves at infinity,
\begin{align}
z \rightarrow -\infty: \quad
&\left(
\begin{array}{c}
a \\
\mathcal{E}
\end{array}
\right) = \left(
\begin{array}{c}
a_I \, e^{\imath k_{\rm a} z} \\
0
\end{array}
\right)
+
\left(
\begin{array}{c}
a_R \,e^{-\imath k_{\rm a} z} \\
\gamma_R \ e^{-\imath k_\gamma z}
\end{array}
\right)\,,\\
%
z \rightarrow \infty : \quad
&\left(
\begin{array}{c}
a \\
\mathcal{E}
\end{array}
\right) =
\left(
\begin{array}{c}
a_T \,e^{\imath k_{\rm a} z} \\
\gamma_T \ e^{\imath k_{\gamma} z}
\end{array}
\right)\,, \label{1DMixing}
\end{align}
where $a_I$ is the amplitude of the incident wave and $\gamma_R$ and $a_R$, $\gamma_T$ and $a_T$ are the amplitudes of the reflected and transmitted waves, respectively.
There are two principal analytic formulae which describe the resonant conversion, one of which, as we now show, is the truncation of the other. The first result \cite{ref:NS-Hook,Camargo:2019wou,Safdi:2018oeu,Edwards:2019tzf} is perturbative, whilst the second explicitly solves the mixing equations with appropriate boundary conditions, providing a non-perturbative conversion amplitude in $g_{\rm a \gamma \gamma}$ - this is the Landau-Zener formalism \cite{ref:LaiHeyl,Brundobler}.
The first step in deriving analytic results is to reduce the system to a first order equation. This involves two stages, firstly a gradient expansion with respect to background fields and secondly imparting information about local dispersion relations into the resulting equations. A somewhat heuristic derivation of a first order equation is given in the classic reference \cite{Raffelt:1987im} for relativistic particles $k \gg \omega_{\rm pl}, \, m_{\rm a}$ with trivial dispersion $\omega \simeq k$. This is the so-called ``weak dispersion" regime also examined in \cite{ref:LaiHeyl}. However, here we deal with non-relativistic dark matter axions which have $\omega \simeq m_{\rm a}$, and since we are interested also in a photon whose dispersion varies locally according to $\omega^2 = k^2 + \omega_{\rm pl}^2$, a more subtle analysis is required. We therefore derive explicitly in appendix~\ref{Density} the following first-order analogue of \eqref{eq:2by2},
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}z}
\left(
\begin{array}{c}
\psi_{\rm a}\\
\psi_\gamma
\end{array}
\right) = \frac{\imath}{2 \bar{k}(z)} \left(
\begin{array}{cc}
m_{\rm a}^2 & \omega g_{\rm a\gamma \gamma} B_0(z)\\
\omega g_{\rm a\gamma \gamma}B_0(z) & \omega_{\rm pl}^2(z)
\end{array}
\right)
\left(
\begin{array}{c}
\psi_{\rm a}\\
\psi_\gamma
\end{array}
\right)\,, \label{eq:Schrodinger}
\end{equation}
with $\bar{k} \equiv \sqrt{\omega^2 - \bar{M}^2}$ and where the key difference from refs.~\cite{ref:NS-Hook} or \cite{ref:LaiHeyl} is the realisation that the distinct axion and photon mass-shell conditions express themselves in a local \textit{average momentum} associated to the average $\bar{M}^2 = (M_1^2 + M_2^2)/2 = (\omega_{\rm pl}^2 + m_{\rm a}^2)/2$ of the two eigenmasses,
\begin{equation}
M_{1,2}^2 = \frac{1}{2}\left\{m_{\rm a}^2 + \omega_{\rm pl}^2 \pm \left[ (m_{\rm a}^2 - \omega_{\rm pl}^2)^2 + 4 B_0^2g_{\rm a\gamma\gamma}^2 \omega^2 \right]^{1/2} \right\}\,.
\end{equation}
In particular, it also varies throughout space. Note that in the relativistic limit $\bar{k} \rightarrow \omega$ reproduces the weak dispersion equations of \cite{ref:LaiHeyl} and at the critical point, one can set $\bar{k} \rightarrow k$ to the axion momentum $\omega^2 = k^2 + m_{\rm a}^2$, giving the localised version of ref.~\cite{ref:NS-Hook} about $z=z_{\rm c}$, where $z_{\rm c}$ is the location of the resonance at which $m_{\rm a} = \omega_{\rm pl}$. Here $\psi_{\rm a}$ and $\psi_\gamma$ appearing in eq.~\eqref{eq:Schrodinger} can be viewed as axion and photon states which have been put on-shell. For compactness of notation we also define
\begin{equation}
\Delta_{\rm a} = m_{\rm a}^2/2\bar{k}\,, \quad
\Delta_\gamma = \omega_{\rm pl}^2/2\bar{k}\,, \quad
\Delta_B = \omega g_{\rm a\gamma \gamma} B_0/2\bar{k}\,.
\end{equation}
\subsubsection{Perturbative calculation}
As was done in \cite{ref:NS-Hook} following the approach of \cite{Raffelt:1987im}, these equations can be solved perturbatively. Following the latter of these references, by going to the interaction picture, one can derive the following conversion probability
\begin{equation}\label{eq:Pintegral}
P_{\rm a \rightarrow \gamma} = \left| \int_{-\infty }^\infty dz^{\prime} \Delta_B (z^{\prime}) e^{\imath \int_0^{z^{\prime}} dz^{\prime\prime} \left[\Delta_\gamma(z^{\prime\prime}) - \Delta_{\rm a}(z^{\prime\prime})\right]} \right|^2\,.
\end{equation}
The exponent is stationary at the resonance, allowing one to perform the integral using the stationary phase approximation to get
\begin{equation}
P_{\rm a \rightarrow \gamma} = \frac{2 \pi \Delta_B^2(z_{\rm c})}{ |\Delta^{\prime}_{\gamma}(z_{\rm c})|} \equiv 2 \pi \gamma\,. \label{eq:gammaDef}
\end{equation}
where $z_{\rm c}$ is defined by $\omega_{\rm pl}(z_{\rm c}) = m_{\rm a}$ and the prime represents the derivative with respect to $z$. In order to make contact with the Landau-Zener formula for the conversion probability of ref.~\cite{ref:LaiHeyl}, we note that by using the definition of the mixing angle
\begin{equation}
\tan{2 \theta} = \frac{\omega B_0(z)g_{\rm a \gamma \gamma}}{m_{\rm a}^2 - \omega_{\rm pl}^2}\,, \label{eq:Mangle}
\end{equation}
we can write
\begin{equation}
\gamma = 2\pi \frac{\Delta M^2(z_{\rm c})/2\bar{k}_{\rm c}}{4 |\theta^{\prime}(z_{\rm c})|} + \mathcal{O}\left(\bar{k}^{\prime}(z_{\rm c}), B_0^{\prime}(z_{\rm c}) \right)\,,
\end{equation}
where $\Delta M^2 = M_1^2 - M_2^2$ is the mass-splitting in the mass-diagonal basis. Thus, up to gradients in the dispersion relation and the magnetic field, the result is precisely that of \cite{ref:LaiHeyl}. Note that by looking at the exponent in the stationary phase approximation, the width of the corresponding Gaussian gives the characteristic width $\Delta z_{\rm c}$ of the resonant region
\begin{equation}
\left(\Delta z_{\rm c}\right)^2 = \frac{\pi}{|\Delta^{\prime}_{\gamma}(z_{\rm c})|}\,.
\end{equation}
We mimic the $\sim 1/r^3$ behaviour of the near-field dipole of the neutron star by taking
\begin{equation}\label{eq:B1D}
B_0(z) = \frac{B_{\ast} R^3}{z^3}\,,
\end{equation}
and use the Goldreich-Julian density \cite{ref:GJ} for the plasma frequency, with $n_e = \Omega B_0(z)$ and $\Omega$ the rotation frequency of the neutron star, from which it follows that
\begin{equation}\label{eq:deltazc}
\Delta z_{\rm c} \simeq \sqrt{\frac{2\pi z_{\rm c} \bar{k}}{3 m_{\rm a}^2}}\,, \qquad
z_{\rm c} = R \left[\frac{B_{\ast} \Omega e^2}{m_e m_{\rm a}^2} \right]^{1/3}\,.
\end{equation}
This allows one to write the conversion probability explicitly as
\begin{equation}\label{eqn:Hook}
P_{\rm a\rightarrow\gamma} = \frac{1}{2}\,\frac{\omega^2}{\bar{k}^2(z_{\rm c})} g_{\rm a \gamma \gamma}^2\, B(z_{\rm c})^2 \Delta z_{\rm c}^2\,.
\end{equation}
There is a pleasing interpretation of this result in terms of a resonant forced oscillator solution - as can be seen from the form of \eqref{EOMPerts}. The photon field $\mathcal{E} = E_{\parallel}/\omega$ can be viewed as a harmonic oscillator with local ``frequency" $k_{\gamma} = \sqrt{\omega^2 - \omega_{\rm pl}^2}$ which becomes equal to that of the axion forcing $k_{\rm a} = \sqrt{\omega^2 - m_{\rm a}^2}$ when $\omega_{\rm pl} = m_{\rm a}$. Since the particular solution to the forced resonant oscillator grows linearly with $z$ behaving as $\sim z e^{\imath k_{\gamma} z}$ and since the overall magnitude of the forcing is set by $\omega g_{\rm a \gamma \gamma} B_0$, the total resonant growth in the photon amplitude is then given by multiplying the size of the region (linear $z$ behaviour) by the magnitude of the forcing - which gives precisely the amplitude-squared of \eqref{eqn:Hook}.
\subsubsection{Landau-Zener}
It is also interesting to quote the well-known Landau-Zener expression for the conversion probability in a two-state system \cite{Brundobler} which is obtained by linearising $\Delta_{\gamma}$ in \eqref{eq:Schrodinger} about $z=z_{\rm c}$ and neglecting gradients in the mixing $\Delta_B$, leading to (see appendix \ref{Density})
\begin{equation}\label{eq:LZ}
P_{\rm a\rightarrow \gamma} = 1 - e^{-2 \pi \gamma }\,, \qquad
\gamma = \frac{\Delta_B^2(z_{\rm c})}{\left|\Delta^{\prime}_\gamma(z_{\rm c})\right|}\,.
\end{equation}
The physical interpretation of this result is that $\gamma$ controls the \textit{adiabaticity} of the evolution - i.e., how rapidly the background is varying. Formally this corresponds to the size of background plasma gradients. We see immediately that the perturbative result \eqref{eq:gammaDef} (refs.~\cite{ref:NS-Hook,Raffelt:1987im}) is precisely the truncation of the Landau-Zener probability \eqref{eq:LZ} (\cite{ref:LaiHeyl,Kuo:1989qe}) in the non-adiabatic limit for small $\gamma$.
It is intriguing to note the link between these results. Of course mathematically speaking, the stationary phase approximation used to compute \eqref{eq:Pintegral} amounts to a linearisation of the plasma mass about the critical point and our use of the Landau-Zener result is formally valid in the limit for which the mass-splitting $m_{\rm a}^2 - \omega_{\rm pl}^2$ varies linearly with $z$ implying the same implicit assumption. However, given that the derivation of each of these results seems a priori to be quite different - it is striking to see that their agreement is exact in the $\gamma \ll 1$ limit.
The size of $\gamma$ -- and therefore the regime in which a perturbative treatment is appropriate -- is given in fig.~\ref{fig:GammaPlot} for the QCD axion with canonical neutron star parameters. Note that our systematic treatment of mass-shell constraints allows us to study $\gamma$ across the full range of relativistic and non-relativistic axion parameter space.
\begin{figure}
\centering
\includegraphics{GammaPlotQCD.pdf}
\caption{The adiabaticity parameter $\gamma$ of \eqref{eq:LZ} for the QCD axion with $g_{a \gamma \gamma}$ given by \eqref{eq:QCDg} with $E/N=8/3$. We considered a magnetic field \eqref{eq:B1D} with $B_{\ast}=10^{14}\text{G}$, a rotation period $P=0.1 {\rm s}$ with $R=10\,{\rm km}$. We also show the velocity at the critical point $v_{\rm c} =k_{\rm c}/m_{\rm a}$ for the value $10^{-1}$ which can be reached via gravitational acceleration.}
\label{fig:GammaPlot}
\end{figure}
Fig.~\ref{fig:massplot} summarises our results for conversion in 1D and compares the full numerical results of the second order equation \eqref{eq:2by2} against analytic approximations. The numerical conversion probability was computed by assuming an incident axion from $z \rightarrow - \infty$ with the magnetic field background $\eqref{eq:B1D}$ and solving the equations for the photon up to a finite depth inside the region of plasma overdensity defined by $\omega_{\rm pl}>\omega$ in which the photon amplitude becomes exponentially suppressed. This was implemented numerically as a Dirichlet and Neumann boundary condition by setting the electric field and its first derivative to zero at some finite depth inside the $\omega_{\rm pl} > \omega$ region.
Figs.~\ref{fig:GammaPlot} and \ref{fig:massplot} show that the conversion of dark matter axions in neutron star magnetospheres typically involves non-adiabtic evolution for which a perturbative treatment in $g_{\rm a \gamma \gamma}$ is valid. The fact one does not stray into the adiabatic regime arises from two considerations. Firstly, for asymptotic values of the axion velocity $v_{\rm a} \equiv k_{\rm a}/m_{\rm a}$ given by $10^{-3}$, gravitational acceleration can bring these up to around $10^{-1}$ shown by the purple line in fig.~\ref{fig:GammaPlot}. Secondly there is an upper limit on the axion mass beyond which the resonance region would be pushed inside the neutron star. These two facts together restrict one to the non-adiabatic region of dark matter axions.
Of course there are some caveats to the above assumptions. Firstly axions with very high or very low momenta can in principle be pushed into the adiabatic regime. However, the gravitational acceleration of the neutron star puts a lower bound $v_{\rm c} \geq GM/z_{\rm c}$, which is saturated by axions which are asymptotically at rest. Meanwhile for large $v$, the distribution is exponentially suppressed by the velocity dispersion $v_0$.
\begin{figure}[t]
\centering
\includegraphics[width = 0.45\textwidth]{nonrel.pdf}
\caption{The analytic Landau-Zener probability, adiabaticity parameter $\gamma$ and the numerical solution of the full second order equations of motion as a function of the axion mass. Here, we assume $g_{\rm a\gamma\gamma} = 10^{-12}$ GeV$^{-1}$, $B_0 = 10^{14}$ G and $k_{\rm a} = 0.1 m_{\rm a}$. For these parameters, we find from energy conservation that the average velocity of the axion at the resonant conversion region $v_{\rm c} \approx \frac{2GM}{z_{\rm c}}$ is roughly 10\% of the speed of light.}
\label{fig:massplot}
\end{figure}
\subsection{Mixing in higher dimensions}
Firstly we review some of the canonical assumptions made in reducing the system of equations \eqref{axionEOM}-\eqref{EOMPerts} to a simple 1D form and then explain why these assumptions may break down in more complicated geometries.
\subsubsection{Polarisation, geometry and the longitudinal mode}\label{sec:1DLimitations}
It is common to assume a transverse photon \cite{Raffelt:1987im,Kartavtsev:2016doq}, such that $\nabla \cdot \textbf{E} = 0$. In this instance, the purely transverse field $\textbf{E}$ can be projected onto the magnetic field in such a way that the polarisation normal to $\textbf{B}_0$ decouples
\begin{equation}
\nabla \cdot \textbf{E} = 0 \quad \Rightarrow \quad
\left.
\begin{array}{c}
\! \! \! \square \, a + m_{\rm a}^2 a = g_{\rm a \gamma \gamma} E_{\parallel} B_t,\\
\square \, E_{\parallel} + \omega_{\rm pl}^2 E_\parallel = -g_{\rm a \gamma \gamma} \ddot{a} B_t
\end{array}
\right.\,,
\end{equation}
where $B_{\rm t}$ is the projection of $\textbf{B}_0$ onto the (assumed to be transversely polarised) $\textbf{E}$ field. This form is valid either for isotropic conductivities or $\omega, \omega_{\rm pl} \ll \omega_{\rm B}$ such that $\sigma\cdot\textbf{E} = \imath \omega_{\rm pl}^2/\omega E_{\parallel}$. Under such an assumption the system will reduce to the mixing of two scalar degrees of freedom, $E_{\parallel}$ and $a$.
However, in the absence of special geometric considerations described in sec.~\ref{sec:1D} the presence of plasma and the axion itself will source a longitudinal component $\nabla \cdot \textbf{E} \neq 0$, as can be seen explicitly from Gauss' equation \eqref{Gauss}, which, by using current conservation in a stationary background together with Ohm's law reads
\begin{equation}
\nabla \cdot \textbf{E} = \nabla \cdot \left(\frac{\sigma}{\omega} \cdot \textbf{E}\right)- g_{\rm a \gamma \gamma} \textbf{B}_0 \cdot \nabla a\,.
\end{equation}
If one chooses a geometry such that the axion field has no gradients in the direction of $\textbf{B}_0$, then the longitudinal mode will not be excited by the axion. If, for instance, the axion gradients are negligible over the scale of the experiment in question, it will have no effect, and in a homogeneous background one would then have $(1-\omega_{\rm pl}^2/\omega^2)\nabla \cdot \textbf{E} = 0$, allowing to neglect the longitudinal mode. However, for the neutron star case, these simplifications need not apply so straightforwardly as we now demonstrate explicitly.
\subsubsection{2D example}\label{eq:2DExample}
Several authors have studied axion-electrodynamics in non-planar geometries \cite{Ouellet:2018nfr,Knirck:2019eug}. We also note in passing a more detailed examination of axion-plasma effects \cite{Mendonca:2019eke,Tercas:2018gxv}.
We solve \eqref{EOMPerts} in a stationary background, working perturbatively in $g_{\rm a \gamma \gamma}$ such that the back-reaction onto the axion beam can be neglected. We, therefore, consider the axion as a fixed source, to solve for $\textbf{E}$
\begin{align}\label{eq:EStationary}
-\nabla^2 \, \textbf{E} + \nabla (\nabla \cdot \textbf{E}) + \omega_{\rm p}^2 \textbf{E} - \omega^2 \textbf{E} & = \omega^2 g_{\rm a \gamma \gamma} a \textbf{B}_0\,,
\end{align}
with $a = a_0 e^{\imath \textbf{k}_{\rm a} \cdot \textbf{x}}$.
This form implicitly assumes that enough time has elapsed since the axions last scattering that an initially localised axion packet will have dispersed to scales much larger than the neutron star via quantum diffusion by the time it approaches the resonant region, justifying the infinite transverse extent of the axion wave-fronts in \eqref{eq:EStationary}. Such an approximation is easily justified by the low density of particles in the inter-stellar medium, and the weakness with which they couple to the axion \cite{ref:Marsh}. The wave-optics picture used above can be viewed as summing over all possible rays parallel to $\textbf{k}_{\rm a}$ which pass-through the neutron star, since by virtue of the uncertainty principle, only the axion's momentum $\textbf{k}_{\rm a}$, not its location, is known.
To solve \eqref{eq:EStationary}, we implement a finite element method solver by constructing a mesh over a given integration region. In order to resolve the wave front structures, the characteristic length of the mesh elements must be less than the wavelength $\lambda_{\rm a} = 1/k$ of the axion. Furthermore, the size of the resonance region can be written as $r_{\rm c} = v_{\rm a} \lambda_{\rm a} [r_{\rm c}/(3 \lambda_{\rm a})]^{1/2}$ where $v_{\rm a}$ is defined by $v_{\rm a} = k/m_{\rm a}$. Since the critical radius $r_{\rm c}\gg \lambda_{\rm a}$ is set by neutron star scales, we see that $\Delta r_{\rm c} \gg \lambda_{\rm a}$. In other words, the effective haloscope size is many orders of magnitude larger than the axion wavelength. This should be contrasted with the results of \cite{Knirck:2019eug}, where axion wavelengths are comparable to the size of the experiment. This hierarchy presents a numerical challenge in that one must integrate over many wavelengths along the conversion region, with a sufficiently high resolution over each wavelength, resulting in a large number of mesh cells. The situation is clearly exacerbated in higher dimensions, where even more cells will be required.
Therefore, we consider a 2D setup, which allows one to study the mixing process in non-planar geometries, whilst keeping the computational cost low. We consider the following ``2D dipole" magnetic field:
\begin{align}
& \textbf{B}_0 = \nabla \times \textbf{A}\,, \quad
\textbf{A} = f(r,\theta)\hat{\textbf{z}}\,, \label{eq:Bfield2D}\\
& f(r,\theta) = e^{\imath m\theta}B_{\ast}\left(\frac{R}{r}\right)^m\,, \qquad
R \leq r < \infty\,,\nonumber
\end{align}
which satisfies $\nabla \cdot \textbf{B}_0 = 0$ automatically and $\nabla \times \textbf{B}_0 = 0$ by virtue of $f(\theta,\varphi)$ being a solution to the cylindrical Laplace equation. We take $m=1$ to mimic a dipole-like configuration explicitly, then
\begin{equation}\label{eq:Bcar}
\textbf{B}_0(x,y) = B_{\ast} R \left \{\frac{-2xy}{(x^2+y^2)^{2}}, \frac{x^2-y^2}{(x^2+y^2)^{2}} \right\}\,.
\end{equation}
The conductivity is taken in the high-magnetisation limit $\omega_{\rm B} \gg \omega$, and mimics the Goldreich-Julian density of an aligned rotator by projecting on the direction of the magnetic dipole:
\begin{equation}\label{eq:plasma2D}
\sigma = \frac{\imath \omega_{\rm pl}^2}{\omega} \left(\hat{\textbf{B}}_0 \otimes \hat{\textbf{B}}_0\right)\,, \qquad
n_{\rm GJ}^{\rm 2D} = \Omega(\hat{\textbf{y}}\cdot \textbf{B}_0)\,.
\end{equation}
\begin{figure*}
\centering
\includegraphics[width = 0.75\textwidth]{BandGJ.pdf}
\caption{Magnetic field and 2D Goldreich-Julian density of eqs.~\eqref{eq:Bfield2D}-\eqref{eq:plasma2D} normalised to surface values.}
\label{fig:BandGJ}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width = 0.75\textwidth]{EPlots.pdf}
\caption{The electric fields components perpendicular and normal to $\textbf{B}_0$ from solving \eqref{eq:EStationary} (normalised to $B_{\ast} a_0g_{\rm a \gamma \gamma}$) with the profiles in \eqref{eq:plasma2D} and the boundary conditions
\eqref{eq:BCRadiation}-\eqref{eq:BCPerfectConductor}. We took the values $\textbf{k} = (0,3)$ and $m_{\rm a} = 1$ in units of $R^{-1}$.}
\label{fig:Efields}
\end{figure*}
The resulting background configurations are shown in fig.~\ref{fig:BandGJ}. We implement the following boundary conditions which are correspondingly of Robin and Dirichlet type
\begin{align}
& \textbf{n} \times \nabla \times \textbf{E} - \imath \omega \textbf{n} \times (\textbf{n} \times \textbf{E}) = 0\,, &
r \rightarrow \infty\,, \label{eq:BCRadiation} \\
& \textbf{E} = 0\,, & r = R\,, \label{eq:BCPerfectConductor}
\end{align}
where $\textbf{n}$ is the unit normal to the outer boundary of the integration region. The first of these implements purely outgoing waves so that the outer boundary is absorptive. It can be derived by considering asymptotic solutions of the vector Helmholtz equation \cite{Peterson,Knirck:2019eug}. The second condition assumes a perfect conductor at $r < R$ so that $\textbf{E}$ vanishes for $r \leq R$, with no surface charges at $r=R$ such that the electric field continuity conditions $\textbf{n} \cdot \left[\textbf{E}(r \rightarrow R^+) - \textbf{E} (r \rightarrow R^-) \right]= 0$ and $\textbf{n} \times \left[\textbf{E}(r \rightarrow R^+) - \textbf{E} (r \rightarrow R^-) \right] =0$ on the inner boundary at $r=R$. The results are shown in figs.~\ref{fig:Efields} and \ref{fig:DivE}.
\begin{figure}[t]
\centering
\includegraphics[scale=1]{DivEPlot.pdf}
\caption{The divergence of the electric field (arbitrary units) with values as in fig.~\ref{fig:Efields}.}
\label{fig:DivE}
\end{figure}
It is clear to see that in those regions where $\nabla a \cdot \textbf{B}_0 \sim \textbf{k}_{\rm a} \cdot \textbf{B}_0 \neq 0$ one has $\nabla \cdot \textbf{E} \neq 0$ whose profile tracks those axion wave-fronts parallel to $\textbf{B}_0$. Since the decoupling procedure of the different polarisations breaks down in a non-planar geometry, we also see in fig.~\ref{fig:Efields} that $\textbf{E}_{\perp}$ enters in the mixing equations and becomes dynamical.
In general then, we see that a simple decoupling of polarisations need not hold in non-planar geometries, suggesting that the 2-component mixing equations applied to a neutron star context in \cite{ref:NS-Hook} are at best an order of magnitude approximation. That said, since the sourcing of $\nabla \cdot \textbf{E}$ arises via axion gradients, it may be that for sufficiently non-relativistic axions these terms could be neglected in a controlled way, however such an analysis is beyond the scope of the present work. In addition, even if one can decouple polarisations, one still has to contend with multi-directional gradients, such that the 1D Landau-Zener formulae would need to be adapted to a 3D setting.
Ultimately, it may be that accurate results can only be obtained by full 3D simulations of the mixing equations as in \cite{Knirck:2019eug}. However, as discussed in previous paragraphs, resolving the wave-front structure across the resonance region requires a large number of mesh cells. One remedy could be a coarse-graining procedure in which one tracks the field amplitudes, but integrates out structures below wavelength scales. This is akin to a gradient expansion used to derive \eqref{eq:Schrodinger} in appendix~\ref{Density} and would entail performing the same expansion in 3D on \eqref{eq:EStationary} to derive a three dimensional set of transport equations similar to those of \cite{Prokopec:2003pj,Stirner:2018ojk} encountered in flavour mixing in leptogenesis or neutrino oscillations in supernovae.
\section{Estimating the Signal and Radio Sensitivity for the Resonant Decay}\label{sect:resdecay}
The radio sensitivity to resonant conversion in neutron star magnetospheres has been previously discussed in \cite{ref:NS-Hook, Safdi:2018oeu}. In particular, \cite{Safdi:2018oeu} discussed the radio sensitivity to neutron star populations. The conclusion of their study was that the radio lines from the individual ``brightest" neutron stars (where bright here means where the resonant conversion is the strongest) offer better sensitivity to the axion-photon coupling than a population. An important factor contributing to this is that the frequency width of the signal in the case of observing a population of stars is proportional to the inverse of the velocity dispersion, compared to the inverse square of the velocity dispersion in the case of single neutron stars, which increases integration time considerably. Therefore, one needs to increase the field of view of the observation considerably to observe a large enough population in order to get a larger signal compared to the isolated bright neutron star case. Unfortunately, one is then limited by the fact that telescopes with large collecting areas have higher resolution, which lead to smaller fields of view. We remark that it might be possible to design a bespoke instrument optimised to try and maximise this signal, but such an undertaking is beyond the scope of this current work.
In this section, we work out the flux density associated with the resonant mixing and explore the possibility of a detection with current and future telescopes. We discuss the impact of the velocity dispersion near a neutron star and Doppler broadening due to its overall motion in section \ref{subsec:ResSensVelocity}. We then discuss the single-dish sensitivity to the axion-photon decay in neutron stars and describe potential neutron star targets in section \ref{subsec:ResSensHook}, where we also compare and contrast our sensitivity calculations to that of \cite{ref:NS-Hook}. Our sensitivity estimates are for the resonant production in a single neutron star, where we assume that the signal is a spectral line broadened by the velocity dispersion of the axions. We also forecast sensitivities of single-dish telescopes (for which we assume the GBT or the Arecibo telescope to be typical examples) and interferometers, like the the SKA.
\subsection{Velocity Dispersion and Doppler Broadening}\label{subsec:ResSensVelocity}
Since the pulsar magnetosphere in general is not a stationary configuration, the energy of test particles moving in this background is not conserved.
While a somewhat rich structure is indicated by simulations~\cite{Philippov:2014mqa,Kalapotharakos:2017bpx}, we consider here the minimal model of an oblique rotating magnetic dipole field that also determines the electron density according to Goldreich and Julian~\cite{ref:GJ} and hence the critical surface. In order to arrive at an analytically transparent picture, we make some additional simplifying assumptions.
For an oblique rotator, the intersection of a plane perpendicular to the rotation axis with the critical surface takes the shape of an ellipse.
When the lengths of the semi-major and semi-minor axes are $a$ and $b$, respectively, the numerical eccentricity is $\varepsilon=\sqrt{1-b^2/a^2}$. This ellipse rotates at an angular velocity $\Omega$ about its middle point.
Consider a corotating point on the critical surface, where an axion may be converted into a photon.
We can further distinguish the cases of reflection and transmission. For reflection, an infalling axion reaches the critical surface from the outside and the photon is subsequently reflected when further climbing the potential barrier made up by the plasma. For transmission, the axion is coming from the inside region and the photon then continues to travel outbound. The instantaneous velocity of
the tangential plane of the critical surface in general is not parallel to the plane itself (unless the point considered is aligned with one of the axes of the ellipse or in the degenerate case of a circle). Physically, a particle that interacts with the critical surface transfers momentum to the magnetosphere, corresponding to the Doppler effect from the reflection by a moving mirror\footnote{We thank Georg Raffelt for bringing this issue to our attention.}.
We therefore calculate the reflection or transmission of a ray of a particle of mass $m_{\rm a}$ in the $xy$-plane that approaches the origin at an angle $\alpha$ (all angles refer here to the $x$-axis), where it falls on a plane whose normal vector points in the direction $\varphi$.
Upon reflection or transmission, the particle is converted into a massless state.
The plane moves at a constant velocity $v$ in the direction of the angle $\vartheta$, see fig.~\ref{fig:doppler}.
The calculation can be carried out by first boosting the four-momentum of the massive initial state from the rest frame of the observer to the rest frame
of the critical surface. In that frame, the zero-component of the four-momentum is conserved as well as
the spatial components of the momentum parallel to the surface. The component perpendicular to the
surface is then found by imposing the energy-momentum relation of a massless particle. The final answer
is obtained when boosting back to the frame of the observer.
To clarify this approach, we first quote the result for the situation where the surface moves toward
the incoming massive particle, $\alpha=\vartheta=\varphi=0$, such that we obtain
\begin{align}
k^{0\prime}=\frac{c \sqrt{m_{\rm a}^2+k^2}+k v}{c\mp v}\,,
\end{align}
where $k$ and $k^{\prime}$ are the moduli of the wave vectors of incoming and reflected wave, respectively.
Throughout this section, an upper sign refers to the case of reflection and a lower one to transmission.
Clearly, when setting $m_{a}=0$, we obtain the classic result for Doppler shift
for reflection as well as zero change in the frequency for transmission. We may therefore anticipate
that for non-relativistic axions, the Doppler shift for axions leaving the magnetosphere is not suppressed
compared to infalling axions.
To arrive at a conservative estimate of the Doppler broadening in the magnetosphere, we now assume that the shape is only mildly elliptical such that the misalignment angle $\varphi-\vartheta+\pi/2\ll 1$ between the tangential plane and its velocity corresponds to a small parameter that we can expand in.
The Doppler shift then takes the simple form
\begin{align}
\frac{k^{\prime 0}}{k^0} = &\, 1\mp \frac{v}{c}\frac{\sqrt{k^2+m_{\rm a}^2-k^2\cos^2(\alpha-\vartheta)}\pm k\sin(\alpha-\vartheta)}{\sqrt{k^2+m_{\rm a}^2}\sqrt{1-\frac{v^2}{c^2}}}\times \nonumber\\
&\, \left[2(\varphi-\vartheta+\pi/2) + {\cal O}\left[(\varphi-\vartheta+\pi/2)^2\right]\right]\,,
\end{align}
In the limit of a relativistic incident particle, $m_{\rm a}/k\to 0$, this reduces to
\begin{align}
\frac{k^{\prime 0}}{k^0} = &\, 1-\frac{v}{c}\frac{\sin(\alpha-\vartheta)}{\sqrt{1-\frac{v^2}{c^2}}}
2(\varphi-\vartheta+\pi/2) + \\
&\, {\cal O}\left[(\varphi-\vartheta+\pi/2)^2\right]\,,\nonumber
\end{align}
for reflections and $k^{\prime 0}/k^0\approx 1$ for transmissions.
In the opposite limit, $m_{\rm a}\gg k$, we find
\begin{align}
\frac{k^{\prime 0}}{k^0} = &\, 1\mp\frac{v}{c}\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
2(\varphi-\vartheta+\pi/2) + \\
&\, {\cal O}\left[(\varphi-\vartheta+\pi/2)^2\right]\,,\nonumber
\end{align}
which is the expression useful for the present context.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.35]{Doppler.eps}
\end{center}
\caption{
\label{fig:doppler}
Parametrization of the Doppler shift on a moving, misaligned mirror.}
\end{figure}
In order to estimate the average effect for the conversion in the magnetosphere, we note that, for a given eccentricity, the angle $\varphi-\vartheta+\pi/2$ can assume values between $\pm\varepsilon^2/4$ within one rotation . Furthermore, depending on the impact parameter, the angle $\alpha-\theta$ approximately takes values between $-\pi/2$ (for trajectories that come very close to the core of the pulsar) and $\pi$ (for trajectories that just about touch the critical surface on the far side of the pulsar). A full quantitative analysis involving the axion and photon trajectories should be straightforward, but it is probably not of obvious benefit since the oblique rotator model of the magnetosphere is likely to be oversimplified, and hence we just make an estimate of the size of the effect. Assuming further $v/c\ll 1$, we estimate that
\begin{equation}\label{doppler:estimate}
\left\langle\left|\frac{k^{\prime0}}{k^0}-1\right|\right\rangle\sim\frac{\Omega r_{\rm c} \varepsilon^2}{c}\approx 6\times 10^{-4} \left(\frac{\Omega}{1\,{\rm Hz}}\right) \left(\frac{r_{\rm c}}{200\,\rm km}\right)\varepsilon^2\,.
\end{equation}
This is to be compared with the width from the velocity dispersion of the axion dark matter
\begin{equation}
\frac{1}{2} v_0^2/c^2\approx 8\times 10^{-7} \left(\frac{v_0}{100\,{\rm km}{\rm s}^{-1}}\right)^2\,.
\end{equation}
We see that the impact of Doppler broadening depends very strongly on the axion velocity in the resonant conversion region.
When the axion is non-relativistic, the Doppler broadening dominates in the width of the spectral line over the velocity dispersion.
For axions that are relativistic at the point of conversion, there is the interesting possibility that the Doppler broadening
for transmissions is strongly suppressed, which may be of importance for line searches. For our subsequent estimates, we use the
the non-relativistic expression~(\ref{doppler:estimate}) for the Doppler broadening.
As stated above, the oblique rotator model with the electron density proposed
by Goldreich and Julian is chosen here because it is analytically tractable.
Eventually, it should be replaced with a more realistic model of the magnetosphere.
Even for the Goldreich--Julian model, we have made simplifying assumptions that we now comment on.
First, the conversion from
the axion to the photon takes place during some finite time during which the
location $z_{\rm c}$ of the critical surface, where the conversion takes place, changes its position due to acceleration.
The width (\ref{eq:deltazc}) of the surface in which
the conversion occurs can be estimated as $\Delta z_{\rm c}\sim (z_{\rm c} m_{\rm a})^{1/2} v_{\rm c}/c$. Assuming that the converting axion passes through this region at a speed $v_{\rm c}/c\sim0.1$ (created in the gravitational potential of the neutron star), it is clear that this takes a time much smaller than the rotation time, $2 \pi/\Omega$, for axions in the GHz mass range.
Second, more important are corrections that should arise from the fact that
the outgoing photon can only be considered relativistic when the Lorentz factor
$\gamma\approx m_{\rm a}/\omega_{\rm pl}$ is large, which occurs for large $(z/z_{\rm c})^{3/2}$. Integration of time along the photon trajectory implies that the point
$z$ is reached after the time $z/c+{\cal O}(z_{\rm c}/c)$. If during that time the
background plasma changes significantly because of the rotation of the pulsar, one
should anticipate order-one corrections to the Doppler effect.
Finally, due to the curvature of the contours of equal plasma mass and the finite distance to be traversed before it becomes relativistic, there should be corrections
due to the continuous refraction of the escaping photon. For axions traversing the
critical surface at a small angle, these can also be of order one.
In case there is additional structure in the magnetosphere beyond the Goldreich-Julian model, then the estimate~(\ref{doppler:estimate}) should be considered as conservative
when applied within its range of validity, which is $\varepsilon\ll 1$.
This is because for structures in the magnetosphere that are indicated by simulations,
the critical surfaces appear to move at large velocities $\sim \Omega r_{\rm c}$.
It would, therefore, be desirable to numerically compute the broadening for realistic magnetosphere models on a full statistical average of axion trajectories and, if possible, to devise of methods of correcting for the Doppler effect.
We stress that since the estimate in \eqref{doppler:estimate} is significantly larger than the background velocity dispersion, the amplitude of the radio signal will be weaker, as we will show in the subsequent sections of the paper.
\subsection{Single-Dish Sensitivity to Resonant Conversion}\label{subsec:ResSensHook}
From \eqref{eqn:BolometricFlux}, the flux density reads
\begin{equation}\label{eqn:FluxNS}
S = \frac{c^2}{4\pi\Delta f_{\rm obs}r(z)^2}\int \frac{\rho_{\rm a}}{\tau_{\rm obs}}\mathrm{d}V \,,
\end{equation}
where we have set the total energy from the decay to be equal to the volume integral of the axion density, i.e., $N_{\rm a}E_{\rm obs} = \int \rho_{\rm a}c^2\mathrm{d}V$. Since we are interested in the flux due to the resonant axion-photon decay at a distance $z_{\rm c}$ over a thin shell of width $\Delta z_{\rm c}$, we have that
\begin{equation}
\int \rho_{\rm a} \mathrm{d}V = \int \rho_{\rm c} z_{\rm c}^2\Delta z_{\rm c} \mathrm{d}\Omega \,,
\end{equation}
where $\rho_{\rm c}$ is the density of the axions in the resonant conversion region. If $v_{\rm c}$ is the velocity of the axions at $z_{\rm c}$, there is then a characteristic time scale over which the axions traverse the width of the shell, $T_{\rm c} = \Delta z_{\rm c}/v_{\rm c}$. Substituting these expressions into \eqref{eqn:FluxNS}, the flux density can be expressed as
\begin{equation}\label{eqn:STc}
S = \frac{c^2}{4\pi \Delta f_{\rm obs}r(z)^2}\int \rho_{\rm c}z_{\rm c}^2 v_{\rm c}\frac{T_{\rm c}}{\tau_{\rm obs}}\mathrm{d}\Omega \,,
\end{equation}
where we identify $T_{\rm c}/\tau_{\rm obs} \equiv P_{\rm a\rightarrow\gamma}$. Therefore, we have a pleasing interpretation of the probability of conversion as the ratio of the resonant crossing time $T_{\rm c}$ to the decay time $\tau_{\rm obs}$, which means that when the two timescales are equal, the probability of conversion becomes unity. This implies that the integral in eq.~(\ref{eqn:STc}) is equivalent to a specific intensity integrated over the area of the source associated with this decay, $\int I \mathrm{d}A_{\rm source} \approx 4\pi z_{\rm c}^2\,I$, which is consistent with previous work, where this quantity was viewed as the power radiated by the flux of photons sweeping across the resonance shell at a velocity $v_{\rm c}$ \cite{ref:NS-Hook}. The estimated decay time is, therefore,
\begin{equation}\label{eqn:decaytime}
\tau_{\rm obs} = T_{\rm c}/p_{\rm a\rightarrow \gamma} = \frac{\Delta z_{\rm c}/v_{\rm c}}{p_{\rm a\rightarrow\gamma}} \approx 190~{\rm s} \,,
\end{equation}
assuming that $p_{\rm a\rightarrow \gamma} \approx 10^{-8}$. For comparison, the axion decay time derived by \cite{ref:Sigl} is given by
\begin{align}
\tau_{\rm obs}^{\rm Sigl} \approx & 2\times 10^4~{\rm s} \left(\frac{B_0}{10^{14}~\rm G}\right)^{-2}\left(\frac{g_{\rm a\gamma\gamma}}{10^{-12}~\rm GeV^{-1}}\right)^{-2}\times \nonumber\\
& \left(\frac{r_0}{10~\rm km}\right)^{-3}\left(\frac{z_{\rm c}}{200~\rm km}\right)^{3}\,.
\end{align}
This expression has been derived directly from the rate of conversion of axions to photons in an astrophysical magnetic field using a non-resonant perturbative calculation and it is two orders of magnitude larger than the decay time in (\ref{eqn:decaytime}).
With current, and realistically possible, telescopes, it is impossible to resolve objects on the scales of $z_{\rm c}$. Therefore, we assume that the neutron star is a point source and hence, in contrast the resolved sources discussed in section~\ref{sect:decay} is better to talk in terms of the flux density rather than the brightness temperature. To determine whether it is possible to detect this conversion, we estimate the flux density
\begin{equation}
S \approx \frac{c^2}{\Delta f_{\rm obs}} \frac{\rho_{\rm c} z_{\rm c}^2 v_{\rm c}}{4\pi r(z)^2} \,,
\end{equation}
where the total flux is given by integrating the specific intensity over the solid angle subtended by the source, $S = \int I \mathrm{d}\Omega$, We note that in \cite{ref:NS-Hook}, it was assumed that $\Delta f \approx m_{\rm a}v_0^2/c^2$. We have shown that the broadening of the signal is dominated by the relative motion of the critical surface with respect to the observer (see sect.~\ref{subsec:ResSensVelocity}). Therefore, using \eqref{doppler:estimate} we can deduce that
\begin{equation}
\begin{split}
\Delta f_{\rm obs} = 7 \,{\rm MHz}&\left(\frac{\Omega}{1\,\rm Hz}\right)^{4/3}\left(\frac{m_{\rm a}c^2}{6.6\,{\rm \mu eV}}\right)^{1/3} \\
&\left(\frac{B_0}{10^{14}\,{\rm G}}\right)^{1/3}\epsilon^2 \,.
\end{split}
\end{equation}
In the subsequent projections we will use $\epsilon^2=1$ and 0.1 as spanning the likely range of values for this geometrical factor.
If we now define the dimensionless quantities $\tilde{\rho} = \rho_{\rm c}/\rho_0$ and $\tilde{v} = v_{\rm c}/v_0$, where $\rho_0$ and $v_0$ are the density and velocity of the axions in the neighbourhood of the neutron star, we can write $S = \tilde{S}\tilde{v}\tilde{\rho}$, where $\tilde{S}$ is a characteristic flux density given by
\begin{align}
\tilde{S} = &\, \frac{c^2}{\Delta f_{\rm obs}}\frac{\rho_0 v_0 z_{\rm c}^2}{4\pi r(z)^2}\frac{T_{\rm c}}{\tau_{\rm obs}} \,, \\
= &\, 1.6\,{\rm \mu Jy}\left(\frac{\rho_0}{\rm GeV~cm^{-3}}\right)
\left(\frac{r(z)}{300~ \rm pc}\right)^{-2}\times
\nonumber\\
&\, \left(\frac{P_{\rm a\rightarrow\gamma}}{10^{-8}}\right) \left(\frac{z_{\rm c}}{224~\rm km}\right)^{-3}\left(\frac{\Delta f_{\rm obs}}{7\,\rm MHz}\right)^{-1}\left(\frac{m_{\rm a}c^2}{6.6\,{\rm \mu eV}}\right)^{-1}\, \nonumber \\
=&\, 1.6\,{\rm \mu Jy}\left(\frac{\rho_0}{\rm GeV~cm^{-3}}\right) \left(\frac{\Omega}{1\rm Hz}\right)^{-7/3}\times\nonumber\\
&\, \left(\frac{m_{\rm a}c^2}{6.6\,{\rm \mu eV}}\right)^{2/3}
\left(\frac{B_0}{10^{14}\,{\rm G}}\right)^{2/3}\left(\frac{r(z)}{300~ \rm pc}\right)^{-2}\times\nonumber\\
&\, \left(\frac{g_{\rm a\gamma\gamma}}{10^{-12}\,{\rm GeV^{-1}}}\right)^2\,.\nonumber
\end{align}
The velocity of the axions near the neutron star, $v_{\rm c}$, can be estimated in terms of the dark matter virial velocity and the neutron star mass from energy conservation, i.e., the kinetic energy of the axion far away from the neutron star must be equal to its total energy near the neutron star. According to this argument, one finds that \cite{ref:NS-Hook}
\begin{equation}
v_{\rm c}^2 = v_0^2 + \frac{2GM}{z_{\rm c}} \approx \frac{2GM}{z_{\rm c}} \,,
\end{equation}
since the escape velocity from the neutron star is much larger than the background virial velocity, $v_0$. It is easy to see that $v_{\rm c}$ is roughly 10\% of the speed of light and therefore $v_{\rm c} \gg v_0$. This implies that the axion velocity in the conversion region is non-relativistic suggesting that the width of the spectral line is likely to be dominated by the Doppler broadening effect. A direct consequence of this is the signal being enhanced by 2 orders of magnitude, since $\tilde{v} \approx150$.
In \cite{ref:NS-Hook}, the authors estimate the dark matter density at $z_{\rm c}$ from Liouville's theorem for the distribution function in the phase-space. Assuming a time-independent Maxwell-Boltzmann distribution function $f(\textbf{v})$ for the dark matter velocity, one may obtain an expression for $\rho_{\rm c}$ integrating the distribution function by expanding in the small parameter $v_0^2/v_{\rm c}^2 = \tilde{v}^{-2} \ll 1$. The result is that the density at the resonant conversion region is enhanced by a factor $\tilde{v}$, which means a further 2 orders of magnitude increase in the flux. Under these assumptions, we obtain a flux of about $0.04\,\mu$Jy. We note that the integration time required to detect this flux using the a GBT-like instrument assuming a bandwidth of about 7~MHz is $\approx 640$ years. If one were to consider the Arecibo telescope instead, the collecting area being approximately a factor of 9 larger, the total integration times decreases by a factor of $\approx 80$. Clearly, this decrease cannot make this signal detectable.
However, we note that the flux increases with axion mass. If we assume that $z_{\rm c}$ scales as $m_{\rm a}^{-2/3}$ [see (\ref{eq:deltazc})], then the resonance occurs closer to the neutron star and the magnetic field at the resonant conversion region will be stronger. However, the resonant shell has a smaller radius, which means the density at $z_{\rm c}$ is integrated over a smaller volume for larger masses. The increase in the magnetic field dominates over the decrease in volume for a dipole magnetic field that scales as $1/z^3$. An upper mass limit exists due to the condition that $z_{\rm c} \geq R^{\ast}$. This hard limit obviously varies for different neutron stars. We note the subtlety that while low-period neutron stars are preferred since $S\propto \Omega^{-7/3}$, the larger the period of the neutron star, the smaller the range of masses one can probe in a radio observation. Therefore, we conclude that the period of the neutron star is perhaps not the best parameter to optimise an experiment for.
\subsubsection*{Potential Neutron Star Targets}
Our analysis until now has suggested that the decay due to neutron stars cannot be detected at the level of $g_{\rm a\gamma\gamma} \lesssim 10^{-12}\, {\rm GeV^{-1}}$, as claimed in \cite{ref:NS-Hook}. Using eq.~\eqref{Eqn:Radiometer}, we estimate the integration time for an axion mass of around 82.5 $\mu$eV/$c^2$, assuming the collecting area of the Arecibo telescope, to be
\begin{equation}
\begin{split}
t_{\rm int} = 50\,{\rm days}& \left(\frac{\tilde{S}}{8.5 \times 10^{-6}\,{\rm \mu Jy}}\right)^{-2}\left(\frac{\tilde{\rho}}{150}\right)^{-2}\left(\frac{\tilde{v}}{150}\right)^{-2}\times \\
&\left(\frac{\Delta f_{\rm obs}}{17 \,{\rm MHz}}\right)^{-1}\left(\frac{A_{\rm eff}}{50000\,{\rm m^2}}\right)^{-2}\left(\frac{T_{\rm sys}}{30\,{\rm K}}\right)^2 \,.
\end{split}
\end{equation}
Note that this value of the axion mass corresponds to about 10 GHz, which is the largest frequency the Arecibo telescope can presently operate at. With the larger frequency coverage of the GBT, one may probe axion masses in the range 1 - 825 $\mu$eV$c^2$. For the fiducial mass we used in sect.~\ref{sect:decay} of 250 $\mu$eV$c^2$, we obtain an integration time of around 3.5 years with a GBT-like instrument. Clearly, to probe larger axion masses with current telescopes, one would have to design an optimisation procedure that might alleviate the difficulties to some extent.
Assuming that $\tilde{v}$ is set by the mass of the neutron star via its gravitational potential and is therefore fixed, one would require a larger $\tilde{\rho}$ to enhance the signal enough for detection. In other words, a simple optimisation procedure would be to look for low-period neutron stars in regions where $\rho_{\rm DM}$ is several orders of magnitude larger than the background value. Another scheme of detection could be to target neutron stars of the largest magnetic fields, like magnetars which are associated to magnetic fields of up to $10^{15}\,{\rm G}$ \citep{Mori2013,Kennea2013,Shannon2013,Eatough2013}. As mentioned in \cite{ref:NS-Hook}, such a candidate magnetar exists near the density spike due to the black hole Sagittarius $\rm A^{\ast}$ at the galactic centre (see sect.~\ref{sect:decay}), for which $\tilde{\rho}$ could be $\approx 10^9$. Of course, with the advent of the SKA2 interferometer with a collecting area of $10^6\,{\rm m^2}$, one could think of probing down to model sensitivities for KSVZ and DFSZ axions.
In fig.~\ref{fig:NS_Sensitivity}, we plot the sensitivity of the Arecibo telescope (left panel) and the SKA2:Band 5 (right panel) assuming a system temperature of 30 K and pulsar mass and radius of $1 \,M_{\odot}$ and $10\,{\rm km}$, respectively. For our standard sensitivity estimate, we consider the pulsar RX J0806.4-4123 \cite{Kaplan2009}. For this pulsar, $\Omega \approx 0.5\, {\rm Hz}$, $B_0 \approx 2.5\times 10^{13}$ G and $r(z) \approx 250$ pc. We also consider the magnetar near the galactic centre SGR J1745-2900, for which $B_0 \approx 1.4 \times 10^{14}\,{\rm G}$ and $\Omega \approx 1.67\,{\rm Hz}$. Note that our sensitivity estimates are more than 2 orders of magnitude weaker than those of reference \cite{ref:NS-Hook}. This is because our estimate of the bandwidth is approximately 2 orders of magnitude larger. Furthermore, no radio telescope is 100\% efficient and therefore the system-equivalent-flux-density (SEFD) for the Arecibo telescope is actually a little larger than 2 Jy. We would like to stress that radio observations of the axion-photon decay are most useful when they are complementary to the haloscope searches, which cannot probe arbitrarily high axion masses\footnote{We note that the MADMAX axion haloscope \citep{Majorovits2017} is sensitive to axions masses predicted by the string decay mechanism ($100~\mu{\rm eV/c^2}\lesssim m_{\rm a}\lesssim 400\mu{\rm eV/c^2}$).}.
\begin{figure*}
\centering
\includegraphics[width = 0.49\textwidth]{sensitivity_NS.pdf}
\includegraphics[width = 0.49\textwidth]{sensitivity_NS_SKA.pdf}
\caption{The sensitivity of radio telescopes to the resonant axion-photon decay. In the left panel, we plot the sensitivity of a single dish telescope between 0.3 and 10 GHz, assuming a system temperature of $30\,{\rm K}$, a diameter $D_{\rm tel} = 300\,{\rm m}$ and efficiency $\eta = 0.5$. These numbers are representative of an Arecibo-like single dish system. In the right panel, we plot the sensitivity representative of SKA2:Band 5, assuming $\eta = 0.7$, $T_{\rm sys} = 30\,{\rm K}$ and $A_{\rm eff} = 10^6\,{\rm m^2}$. The yellow line is for the isolated neutron star RX J0806.4-4123, while the red line is for the neutron star observed to be near Sagittarius $\rm A^{\ast}$ with $\tilde{\rho} \approx 10^9$. We have use dashed lines for the case where the geometrical factor $\epsilon^2=1$ in (\ref{doppler:estimate}), that is, where the Doppler broadening is maximal and dot-dashed lines for when $\epsilon^2 = 0.1$. We remark that these estimates have been chosen to be representative of the typical sensitivity one might expect to achieve. In reality, one would need to take into account the variation in system temperature as a function of frequency.}
\label{fig:NS_Sensitivity}
\end{figure*}
\section{Summary and Discussion}\label{sect:conclusions}
In this work we have clarified and extended the analysis of spontaneous decays and resonant conversion of dark matter axions, with an extensive discussion of both theory and observations. Axion masses larger than 100 ${\rm \mu eV}/c^2$ cannot be probed by axion haloscopes and these masses have been motivated by studies of axion string decay \cite{ref:Battye, ref:WS} and the non-linear substructure formed as a result \cite{ref:KhlopovArchioles}. In the case of the detection of the spontaneous decay, previous work \citep{ref:Caputo,ref:Caputo1} has suggested that nearby dwarf spheroidal galaxies are ideal candidates to observe under the claim they maximise the flux density.
In our analysis, we argue that a procedure to maximise the flux density signal can be non-trivial. This is because the resolution of most single-dish radio telescopes is such that the beam size is smaller than the apparent size of these dwarf galaxies, resolving them. As a result, it becomes confusing to optimise an experiment where one is interested in maximising the flux-density signal since it is difficult to observationally determine the virial radius of dwarf galaxies \cite{ref:DwarfSpheroidalDiff}.
Our analysis highlights the fact that one need not carry out a matching procedure of sources to the resolution of the telescope. Indeed, the relevant quantity that determines the specific intensity is the ratio of the surface-mass density to the velocity dispersion, $\Sigma_{\rm beam}/\Delta v$. Our results show that, except a weak trend in the halo concentration parameter with respect to the mass, this ratio is independent of the halo mass. We infer that a high resolution is in fact desirable, since the surface-mass density along the line of sight is enhanced for a more concentrated beam. This result motivates a search for structures that are characterised by large values of $\Sigma_{\rm beam}$.
An important point that was first studied in \cite{ref:Caputo, ref:Caputo1} is the enhancement due to simulated emission, since photons and axions are both described by Bose-Einstein statistics and are therefore indistinguishable. Thus, the presence of an ambient radiation field at the same frequency as that of the axions results in an effective enhancement of the decay. This is quantified by the photon occupation number.
Our analysis of the sensitivity to virialised objects shows that it is virtually impossible to design a conventional interferometer that can constrain the axion-photon coupling below the CAST limit. This is due to the fact that the sensitivity of an interferometer to any brightness temperature signal is weakened by a filling factor that increases the integration time to unachievable values. On the other hand, one may use an interferometer as a ``light-bucket", where all dishes are used in single-dish auto-correlation mode. We show that even in this case, one would require 4 days of on-source integration time with band 5 of SKA2 observing the Virgo cluster to improve on the CAST constraints \cite{ref:CAST} on the axion-photon coupling.
Our previous results indicate that the ideal source for detecting the spontaneous decay is characterised by large values of $\Sigma_{\rm beam}$ and large amounts of ambient radiation at the same frequency - corresponding to ${\cal F}_{\rm eff}\gg 1$. Our order of magnitude estimate of the sensitivity to the galactic centre due to synchrotron emission (using the Planck Point Source catalogue \cite{ref:PSC}) from Sagittarius $\rm A^{\ast}$ shows that this may be an ideal target to improve the CAST limit. This motivates a further detailed study of the radiation field in the galactic centre, which in principle could include components from Anomalous Microwave Emission (AME) (dust), free-free emission as well as synchrotron radiation.
For the resonant conversion in neutron star magnetospheres, we began our analysis with a careful discussion axions in magnetised plasmas. Our results on mixing in 1D formalise many of the aspects of axion-photon conversion in magnetised plasmas, where we derived a controlled WKB expansion, mass-shell conditions in inhomogeneous backgrounds, adiabatic/non-adiabatic limits and their application to the neutron star case. We also hope that our arguments lay bare the precise geometrical assumptions needed to simplify the axion-Maxwell equations to a simple 1D form involving only the parallel electric component and the axion. It seems that these assumptions seem not always transparent in the literature.
Our results on 3D mixing, though preliminary, raise some interesting questions about the role geometry can play in exciting additional plasma modes and highlights the qualitative importance of both the perpendicular photon polarisations $\textbf{E}_\perp$ and the emergence of a longitudinal mode. Of course, one might be tempted to argue that it is a well-known and unsurprising fact that $\textbf{E}_\perp$ also becomes active in a magnetised plasma due to mixing between $\textbf{E}_\perp$ and $\textbf{E}_\parallel$. This is a typical feature of anisotropic media of which a magnetic background is but one example. Formally, this can be seen to happen in the off-diagonal components of the conductivity \eqref{sigma} (see also equation (29) of ref.~\cite{ref:LaiHeyl}). However, our argument is more subtle than this. For the neutron star case, one is typically in the high-magnetisation limit where the cyclotron frequency greatly exceeds that of the photon, $\omega_{\rm B} \gg \omega$, thus conductivity-induced mixing is, in fact, switched off, but nonetheless a mode $\textbf{E}_\perp$ is activated. For example, the Gauss equation shows the complications arising from longitudinal modes in the high-magnetisation limit
\begin{equation}
\left(1 - \frac{\omega_{\rm pl}^2}{\omega^2}\right)\nabla \cdot \textbf{E}_\parallel + \nabla \cdot \textbf{E}_\perp = - g_{\rm a \gamma \gamma} \nabla a \cdot \textbf{B}\,,
\end{equation}
which clearly couples $\textbf{E}_\perp$, $\textbf{E}_\parallel$ and $a$ in a non-trivial way with the axion fundamentally changing the form of Maxwell's equations. Given the arguments of subsec.~\ref{sec:1DLimitations}, it seems that the only way to prevent the additional modes becoming active and to completely decouple $\textbf{E}_\perp$ and the effects of $\nabla \cdot \textbf{E} \neq 0$ terms, is to choose a very peculiar geometry in which the momentum of the axion and photon are normal to the magnetic field and all background gradients are dominated by a particular preferred direction parallel to the momentum. For a neutron star magnetosphere, this is clearly too simplistic and the plasma will exert gradient forces on the photon causing it to deflect away from the initial axion trajectory.
It is also worth noting that in going from the planar 1D setup to 3D backgrounds, results are not only modified by the presence of additional polarisations, but also the nature of mixing itself may change. Specifically, any analytic formulae for conversion probabilities must be sensitive to the dimensionality of the underlying differential equations to be solved. For example, an analytic expression such as the Landau-Zener formula \eqref{eq:LZ} is a 1D result and is expected to be modified in higher dimensions regardless of which polarisations are most important.
The ultimate goal is clearly to determine \textit{quantitatively} what are the effects of 3D geometries since this is the most relevant aspect from an observational perspective. This will either entail more work to obtain analytic results in 3D and/or cross-checking these results against numerical simulations. Of course, the latter option presents some numerical challenges, as explained in sec.~\ref{eq:2DExample} owing principally to the fact wavelengths are much smaller than background scales over which the equations must be integrated, in contrast to \cite{Knirck:2019eug} where laboratory halosccopes contain only a few wavelengths. Clearly this limit strongly suggests the problem should be amenable to a WKB expansion, which would require performing a gradient expansion of the axion-Maxwell's equations in 3D as done in 1D in appendix \ref{Density}. The resulting 3D transport equations may allow one to circumvent the issue resolving the wavefront structure and track only field amplitudes/number densities relevant for computing the flux.
We showed that Doppler broadening of the signal due to the relative motion of the neutron star results in a bandwidth that is at least two orders of magnitude larger than the value estimated in reference \cite{ref:NS-Hook} if the axion velocity is non-relativistic in the resonant conversion region. As a result, our sensitivity estimates are significantly weaker in the mass range $1-10\,{\rm \mu eV}$. Therefore, we emphasise the need to identify candidate pulsars that are located in regions of dark matter density peaks large enough to make the flux density detectable. We find that, in possibly realistic scenarios, it is possible to significantly improve on the CAST constraints from 4 days of observations using the Arecibo telescope. We also find that the SKA2:Band 5 can possibly rule out DFSZ axions of $m_{\rm a}c^2 \geq 20\,{\rm \mu eV}$, assuming an enhancement of $10^9$ in the dark matter density at the location of the magnetar near the galactic centre.
\begin{center}
\textbf{Acknowledgements}
\end{center}
JIM would like to thank Carlos Tamarit and Francesca Chadha-Day for general discussions and Georg Raffelt for useful conversations on wave optics and mixing in higher dimensions as well as Juan Cruz for help with the cluster. This work is supported by an Alexander von Humboldt Fellowship and by the Collaborative Research Centre SFB 1258 of the Deutsche Forschungsgemeinschaft. RAB and FP acknowledge support from Science and Technology Facilities Council (STFC) grant ST/P000649/1. SS is supported by a George Rigg Scholarship from the University of Manchester. SS would like to thank Keith Grainge, Dominic Viatic, Joel Williams and Joshua Hayes for useful advice.
|
1,108,101,566,770 | arxiv | \section{Introduction}
\label{sec:intro}
For the last ten years, three international collaborations have been collecting precise timing observations of the most stable millisecond pulsars, with the aim of identifying the imprint of gravitational waves (GWs) in the nanohertz frequency band \citep{h13,kc13,ml13}. The best-motivated GW source for such \emph{pulsar-timing arrays} (PTAs) is the stochastic \emph{background} signal from the cosmological population of gravitationally bound supermassive black-hole--binaries (SMBHBs: \citealt{rm95,jb03,wl03}) that are expected to form at the centers of galaxies after these merge \citep{shm+04,svc08}. If the binary inspirals are driven purely by GW radiation reaction at orbital separations corresponding to the PTA frequency band, the resulting timing-residual signal has a power-law spectrum with a characteristic exponent,
\begin{equation}
S(f) = \frac{h_c(f)^2}{12\pi^2 f^3} = \frac{A_{\rm GW}^2}{12 \pi^2} \left(\frac{f}{\mathrm{yr}^{-1}} \right)^{-13/3} \, \mathrm{yr}^3 \, ,
\label{eq:powerlaw}
\end{equation}
where $h_c(f)$ is the characteristic strain spectrum of the GW background. Theoretical expectations for the GW amplitude $A_\mathrm{GW}$ center around a few $10^{-15}$ \citep{mop14,s13,rws+14}. Recent observational upper limits from the three international pulsar-timing consortia \citep{ltm+15,arz+15,s+15} imply $A_\mathrm{GW} \lesssim 10^{-15}$ at 95\% confidence. These limits appear in tension with theoretical expectations, so much so that it seems plausible that environmental effects in galactic centers either stall the formation of gravitationally bound SMBHBs, or accelerate their inspiral; in both cases, the effective $A_\mathrm{GW}$ is reduced at the frequencies where PTAs are most sensitive \citep{arz+15,s+15}.
Indeed, the PTA detection of the stochastic GW background (GWB) from SMBHBs, considered imminent only a few years ago \citep{sejr13,mop14}, now seems to be receding toward the future. Is this really the case? In this letter, we answer this question quantitatively. Namely, given the upper limit $A_\mathrm{ul} = 10^{-15}$ obtained by the Parkes PTA (PPTA; \citealt{s+15}), we ask \emph{when} we can expect to make a positive detection using different PTAs: the PPTA \citep{h13}, limited to the four low--timing-noise pulsars used for the upper limit; the North American Nanohertz Observatory for Gravitational-waves (NANOGrav; \citealt{ml13}); the European PTA (EPTA; \citealt{kc13}); the International PTA (IPTA; see, e.g., \citealt{haa10}); and the pulsar-timing project that will be supported by SKA1 \citep{2015aska.confE..37J}.
We adopt the frequentist formalism developed by \citet{rsg15}, and we characterize the PTAs simply by listing, for each pulsar, the duration of the observation and the levels of measurement and timing noise.
While GW searches and upper limits with PTAs have recently been given a Bayesian treatment \citep{vlml09,vl13,l+13}, the frequentist approach based on \emph{optimal statistics} is both convenient and appropriate for the purpose of this paper, because it dispenses with the simulation of actual datasets, and because the quantification of detection probability is intrinsically a frequentist statement (see, e.g., \citealt{2012PhRvD..86h2001V}). Furthermore, experience shows that frequentist upper limits and detection prospects are rather close to Bayesian results (see, e.g., \citealt{arz+15}).
We proceed as follows: for a specified PTA and true GW background amplitude ($A_\mathrm{true}$), we compute the detection probability (DP) as a function of observation time, using the detection statistic described below. Detection is a probabilistic endeavor since the actual realization of measurement and timing noise may obscure or expose the underlying GW signal.
Indeed, we do \emph{not} currently have access to $A_\mathrm{true}$, but only to the observed upper limit $A_\mathrm{ul}$.
Setting upper limits is also a probabilistic endeavor, because different realizations of noise lead to different $A_\mathrm{ul}$ given the same $A_\mathrm{true}$, so we use the upper-limit optimal statistic (also described below) to compute $p(A_\mathrm{ul}|A_\mathrm{true})$. By introducing an astrophysically motivated prior $p(A_\mathrm{true})$, we can then use Bayes' theorem to obtain $p(A_\mathrm{true}|A_\mathrm{ul}) \propto p(A_\mathrm{ul}|A_\mathrm{true}) p(A_\mathrm{true})$. Finally, we obtain the \emph{expected} detection probability $\mathrm{DP}(A_\mathrm{ul})$ as $\int \mathrm{DP}(A_\mathrm{true}) p(A_\mathrm{true}|A_\mathrm{ul}) \, \mathrm{d}A_\mathrm{true}$.\footnote{By assuming a prior, we are introducing a Bayesian element in a frequentist scheme, but this is necessary because we wish to make a statement about a \emph{range} of astrophysical possibilities, which are all compatible to varying degrees with the observed upper limit.}
We conclude that, while small arrays of a few low--timing-noise pulsars (represented here by the \citealt{s+15} configuration) provide the most constraining upper limits on the nanohertz GWB, they are suboptimal for detection in comparison to larger arrays such as NANOGrav, EPTA, full-PPTA, and the IPTA, which are expanded regularly with newly discovered pulsars.
Indeed, detection and upper limits have rather different demands, and the PPTA itself may employ a $\sim 20$-pulsar array for detection in the near future \citep{rhc+15}.
Encouragingly, we find that larger PTAs have $\sim 80\%$ probability of making a detection in the next ten years, even allowing for a reduced GWB signal due to significant binary stalling or environmental influences.
\section{Statistics} \label{sec:stats}
The details of the frequency-domain statistical framework used here can be found in \citet{rsg15}. In the frequentist context, a statistic $X$ is a single number that summarizes the (noisy) data with respect to the presence of a (GW) signal, so that, on average, a higher $A_\mathrm{true}$ yields a higher $X$, with fluctuations due to the range of possible noise realizations. Given a set of data, there is \emph{one} observed $\hat{X}$, and setting an upper limit $A_\mathrm{ul}^{95\%}$, amounts to stating that if $A_\mathrm{true}$ were equal to $A_\mathrm{ul}^{95\%}$, for 95\% of noise realizations the observed $X$ would have been higher than $\hat{X}$.
Detection schemes based on statistics, on the other hand, proceed as follows: one considers separately the case where a signal is present in the data at a certain level, and the case where it is not, and computes the probability distribution of the statistic (over the ensemble of noise realizations) in both cases. The zero-signal distribution is used to set a \emph{detection threshold} as a function of a false-alarm probability; the signal distribution is used to compute the probability that for a certain $A_\mathrm{true}$ the statistic will exceed the threshold---that is, the detection probability (or efficiency).
In the following we use cross-correlation statistics for both upper-limit and detection considerations, making use of the correlated influence of a GWB signal on pulsars which are widely separated on the sky.
{\it Upper limits.}-- For the discussion of upper limits in this work we will use the so-called $A$-statistic of \citet{rsg15} modified so that the standard deviation is measured in units of squared GWB amplitude and that the value of the statistic is equal to the injected GWB amplitude on average, as is done in \cite{ccs+15}. Assuming that the distribution of this statistic is Gaussian (a reasonable approximation at low signal-to-noise ratios \citep{ccs+15}) we obtain
\begin{equation}
p(A^{2}|A_{\rm true}) = \frac{1}{\sqrt{2\pi\sigma_{0}^{2}}}\exp\left(-\frac{(A^{2}-A_{\rm true}^{2})^{2}}{2\sigma_{0}^{2}}\right)
\end{equation}
with
\begin{equation}
\sigma_{0} = A_{\rm true}^{-2}\left( 2\sum_{k}\sum_{ij}\Gamma_{ij}^{2}\frac{S_{h0}^{2}}{P_{i}P_{j}} \right)^{1/2},
\label{eq:asig}
\end{equation}
where $\sum_{k}$ denotes a sum over frequencies $f_{k}$, $\sum_{ij}$ denotes a sum over pulsar pairs, $\Gamma_{ij}$ is the overlap reduction function (i.e. this is the \citet{hd83} curve for an isotropic GWB) for pulsars $i$ and $j$, $S_{h0}\equiv S_{h0}(f_{k})$ is the modeled cross-power spectral density, and $P_{i}\equiv P_{i}(f_{k})$ is the intrinsic noise power spectral density. From this Gaussian probability distribution, it can be shown that
\begin{equation}
A_{\rm ul}^{2} = \hat{A}^{2} + \sqrt{2}\sigma_{0}{\rm erfc}^{-1}(2(1-C)),
\end{equation}
where $\hat{A}$ is the measured value of the GWB amplitude and $C$ is our upper limit confidence (e.g., 95\%). Thus, the distribution $p(A_{\rm ul}^{2}|A_{\rm true})$ is a Gaussian distribution with standard deviation $\sigma_{0}$ and mean $A_{\rm true}^{2}+\sqrt{2}\sigma_{0}{\rm erfc}^{-1}(2(1-C))$. Note, to compute $p(A_{\rm ul}|A_{\rm true})$ we perform a coordinate transformation to obtain $p(A_{\rm ul}|A_{\rm true})$ = $2A_{\rm ul}p(A_{\rm ul}^{2}|A_{\rm true})$.
\begin{figure}
\includegraphics[angle=0, width=0.5\textwidth]{pAinj_plotfinal_v2.pdf}
\vspace{-18pt}
\caption{The normalized probability distribution for the SMBHB GW amplitude $A_\mathrm{true}$, given upper limits from the PPTA \citep{s+15} and a large PTA which regularly adds pulsars (specifically NANOGrav \citep{arz+15}). The darker blue curves assume an amplitude prior based on \citet{s13}; the faded red curves modify it by assuming 90\% binary stalling, reducing the amplitude $\sqrt{10}$-fold. The dashed curves reflect turnover spectra due to binary stellar hardening. The dash-dotted curves correspond to the astrophysically-motivated priors on the amplitude for no stalling (blue) and $90\%$ stalling (red).}
\label{fig:pAtruegivenAul}
\end{figure}
{\it Detection.}-- In the previous section we used the $A$-statistic for upper limits in order to compare our results with upper limits computed using the statistic presented in \cite{ccs+15}. For assessing our detection probability, however, we will make use of the $B$-statistic of \cite{rsg15}. In this case, we wish to determine the detection probability
\begin{equation}
{\rm DP}=\frac{1}{2}{\rm erfc}\left[ \frac{\sqrt{2}\sigma_{0}{\rm erfc}^{-1}(2\alpha_{0})-\mu_{1}}{\sqrt{2}\sigma_{1}} \right],
\label{eq:dp}
\end{equation}
where $\alpha_{0}$ is the false alarm probability (FAP); $\mu_{1}$, $\sigma_{1}$, and $\sigma_{0}$ (distinct from previous) are the mean in the presence of a signal, the standard deviation in the presence (subscript 1) and the standard deviation in the absence (subscript 0) of a signal (see Eqs.\@~A16--A18 of \citealt{rsg15} for more details). Both $\mu_1$ and $\sigma_1$ grow with increasing signal amplitude. Here we choose a FAP of $0.13\%$ to match the 3-sigma detection threshold of \citet{sejr13}.\footnote{This choice is clear if one notices that the 3-sigma range is a \emph{two}-sided confidence limit, thus 0.27\% of the probability density function is outside of this range. To convert to a standard FAP, we need the one-sided limit which is simply 0.27/2 $\sim$ 0.13.}
{\it Implementation.}-- In the work of \cite{rsg15}, it is argued that timing model subtraction (namely subtraction of the quadratic term of the timing model; due to pulsar spin down) can be emulated by not including the lowest frequency in the sum over $k$ frequencies. However, we have found that these analytic results agree much better with simulations including timing model subtraction if all frequencies are included in the sum. Furthermore, to emulate data sets with different time spans we only include frequencies in the sum that are \emph{greater} than $1/T_{\rm min}$, where $T_{\rm min}$ is the shortest time span for the given pulsar pair.
In this work we always include the GWB power spectral density in the \emph{intrinsic} pulsar noise $P_{i}$. This is meant to mimic a real optimal statistic analysis where the intrinsic noise would be based on single-pulsar-analyses in which there is no way to distinguish the intrinsic noise from GWB power. This will make our analysis conservative in the sense that we are overestimating the intrinsic noise.
\section{Results \& discussion}
In the following, we characterize detection prospects in terms of small and large PTAs, and specifically consider the following five configurations:
\begin{enumerate}
\item PPTA4, consisting of the four pulsars described in \citet{s+15} with associated measurement (white) and timing (red) noise \citep{s+15,rhc+15}.
\item NANOGrav$+$, consisting of the $37$ pulsars described in Table $3$ of \citet{arz+15b} with their associated noise properties, to be expanded in future observations by adding four new pulsars each year with $250$-ns TOA measurement noise, in line with current expectations.
\item EPTA$+$, consisting of the $42$ pulsars described in Table $3$ of \citet{cll+15} with their associated noise properties, and regularly adding pulsars as in $(2)$.
Some of these additional pulsars are monitored with the Large European Array for Pulsars (LEAP; \citealt{kc13,bassa15}), which synthesizes a 194-m steerable dish from the five EPTA telescopes.
\item A conservative IPTA+, consisting of the $49$ pulsars of the first IPTA data release, described in \citet{vlh+15} with measurement and timing noise\footnote{For IPTA pulsars timing noise is taken to consist only of red spin noise, and not of any other system- or band-specific red noise. The latter components may be isolated with multi-system and multi-frequency observations, while the former is completely conflated with a GWB signal in the absence of cross-correlation measurements.} properties as in \citet{lsc+15}, and expanding by six new pulsars each year, again with $250$-ns TOA measurement noise.
\item A theorist's PTA (TPTA), consisting of the toy configuration of \citet{rsg15} which may be supported by an advanced radio telescope such as SKA$1$: $50$ pulsars with $100$-ns TOA measurement noise and no intrinsic timing noise.
\end{enumerate}
These configurations are summarized in Table~\ref{tab:array_properties}.\footnote{We rescale PPTA, NANOGrav, EPTA, and IPTA measurement noises (but not the timing noise) by a common factor for each PTA, so that the corresponding $A$-statistic upper limits match the results of \citet{s+15}, \citet{arz+15}, \citet{ltm+15}, and \citet{vlh+15}; this rescaling has little impact on detection probability in the future, since white measurement noise becomes subdominant to red timing noise at low frequencies.}
\begin{table*}
\caption{PTAs considered, as characterized in our analysis.}
\label{tab:array_properties}
\centering
\scriptsize
\begin{tabular}{l@{\;\;}l@{\;\;}l}
\hline
PTA & configuration & number of pulsars \\
\hline
\hline
PPTA$4$ & PPTA \citep{s+15} & 4 \\[2ex]
NANOGrav$+$ & NANOGrav & $37$ $+$ $4$ new/year with $250$-ns\\
& \citep{arz+15b} & TOA measurement precision\\[2ex]
EPTA$+$ & EPTA \citep{cll+15} & $42$ $+$ $4$ new/year with $250$-ns\\
& & TOA measurement precision\\[2ex]
IPTA$+$ & IPTA Data Release $1$ & $49$ $+$ $6$ new/year with $250$-ns\\
& \citep{vlh+15} & TOA measurement precision\\
& \citep{lsc+15} & \\[2ex]
Theorist's PTA & $100$-ns TOA measurement precision & $50$\\
\hline\hline
\end{tabular}
\end{table*}
We consider four combined estimates for the amplitude and spectral shape of the stochastic GWB from SMBHBs, ranging from more optimistic (detection wise) to more conservative.
For the amplitude, we adopt the observationally motivated log-normal distribution of \citet{s13} with mean $\log_{10}A_\mathrm{true}=-15$ and standard deviation of $0.22$; however, we also consider the case where $90\%$ of binaries require longer than $\sim 9 - 13.7$ Gyr before reaching the PTA band (i.e., the binaries stall, \citealt{bs15,mop14}), which corresponds to a mean $\log_{10}A_\mathrm{true}=-15.5$, with the same standard deviation. For each of these amplitude priors, we examine the purely GW-driven power-law spectrum of Eq.\ \eqref{eq:powerlaw}, as well as the case where the GW strain has a turnover at $f = 1/(11\,\mathrm{yr})$ caused by SMBHB interactions with stars in galactic nuclei, which accelerate binary inspiral and remove low-frequency power with respect to the pure power law (see, e.g., \citealt{scm15} and \citealt{arz+15}). The $1/(11\,\mathrm{yr})$ turnover frequency was chosen to lie just beyond the low-frequency reach of the current PPTA dataset, since the \citet{s+15} analysis still found $\sim 9\%$ consistency with the purely GW-driven model of \citet{s13} for $f\geq 1/(11\,\mathrm{yr})$. Similarly, the recent NANOGrav \citep{arz+15} analysis finds 20\% consistency using a 9-year dataset \citep{arz+15b}. Furthermore, our choice of a turnover at $1/(11\,\mathrm{yr})$ corresponds to reasonable assumptions about the density of stellar populations interacting with SMBHBs in galactic nuclei (see Figs. 10 and 11 of \citealt{arz+15}).
At very low frequencies, the frequency scaling of the timing-residual spectrum becomes $S(f)\propto 1/f$ due to the environmental influences.
In the top panel of {Fig.}~\ref{fig:pAtruegivenAul} we show the probability distribution $p(A_\mathrm{true}|A_\mathrm{ul})$ obtained by setting the 95\% $A$-statistic upper limit to the \citet{s+15} value, $1\times 10^{-15}$, and by assuming no-stalling (solid blue line) and 90\%-stalling (faded solid red line) amplitude priors, with pure power-law true spectra. The dashed lines correspond to turnover spectra.\footnote{When analyzing these latter cases, we use the ``wrong'' spectrum -- a pure power law -- for the $P_i$ in Eq.\ \eqref{eq:asig}, for consistency with existing analyses that assume GW-driven inspirals in deriving limits. As discussed in \citet{rsg15}, doing so has negligible effects on the distribution of the $A$-statistic.} For comparison, in the bottom panel of {Fig.}~\ref{fig:pAtruegivenAul} we show also $p(A_\mathrm{true}|A_\mathrm{ul})$ as computed for NANOGrav's optimal-statistic upper limit $1.3\times 10^{-15}$ \citep{arz+15}. The underlying astrophysically-motivated priors for no-stalling and $90\%$-stalling are shown in each panel as blue and red dash-dot lines, respectively.
\begin{figure*}
\centering
\includegraphics[angle=0, width=\textwidth]{dp_total_plot_v4.png}
\caption{Detection probabilities (DPs) for all arrays considered in this work, as a function of the true GW background amplitude $A_\mathrm{true}$ and of observation time $T$ beyond the existing dataset. The left panels were derived for pure power-law GW background spectra, and the right panels for turnover spectra with a knee at $f=1/(11 \,\mathrm{yrs})$ due to stellar hardening.
}
\label{fig:otherpta_dp}
\end{figure*}
Next, we compute the $B$-statistic GW detection probabilities, Eq.\ \eqref{eq:dp}, for each PTA, as a function of $A_\mathrm{true}$ and of the observation time $T$ beyond current datasets. The resulting DPs are shown in {Figs.}\ \ref{fig:otherpta_dp}: the left and right columns correspond to the pure power-law and turnover spectra, respectively.\footnote{In computing the $B$-statistic, we do assume that the analysis accounts correctly for the true spectral shape. Again, this choice has minimal effects.}
As mentioned above, for NANOGrav$+$/EPTA$+$ and IPTA+ we add new pulsars at rates of four and six per year, respectively. This annual expansion has a positive impact on the DP, since it provides more independent pulsar pairs at differing angular separations, which help discriminate GWs with quadrupolar spatial correlations (the \citealt{hd83} curve, or the more general anisotropic signatures discussed by \citealt{msmv13} and \citealt{grtm14}) from non--spatially-correlated red noise processes \citep{sejr13}. The unique GW correlation signature is the smoking gun for detection, and we must coordinate future efforts to maximize its measurement.
We obtain the \emph{expected} detection probabilities $\langle \mathrm{DP} \rangle_{A_\mathrm{true}}$ as a function of time by integrating $\mathrm{DP}(A_\mathrm{true},T)$ against the $p(A_\mathrm{true}|A_\mathrm{ul})$ curves of {Fig.}\ \ref{fig:pAtruegivenAul} (top panel). The resulting $\langle \mathrm{DP} \rangle_{A_\mathrm{true}}$ are shown in {Figs.}\ \ref{fig:summary}. As in {Fig.}\ \ref{fig:pAtruegivenAul}, the darker blue and lighter red curves correspond to no-stalling and 90\%-stalling amplitude priors. For the PPTA$4$ array, expected detection probabilities remain below $10\%$ throughout the next $20$ years of observation, even in the most optimistic GW-signal scenario. This is unsurprising---a PTA consisting of a few exquisitely timed pulsars may provide very tight upper limits, but it will not usually yield convincing detection statistics, which require an array of many pulsars to map out the expected spatial correlation signature.
\begin{figure}
\includegraphics[angle=0, width=0.5\textwidth]{summary_plot_final_v4.pdf}
\caption{Summary of results for the growth of GWB detection probability (DP) with further observation time in each array. Blue and red lines are for zero and $90\%$ binary stalling, respectively. Dashed lines correspond to the case where the true background spectrum has a turnover at $f=1/(11 \,\mathrm{yrs})$ due to binary stellar hardening.}
\label{fig:summary}
\end{figure}
By contrast, large pulsar arrays such as NANOGrav$+$, EPTA$+$, and IPTA$+$ provide high detection probabilities even with strong binary environmental couplings, since they allow the quadrupolar spatial correlation signature of a GWB to be mapped by many different pulsar pairings. We expect these results to be qualitatively the same for a full PPTA that regularly adds pulsars to the array. In NANOGrav$+$, EPTA$+$, and IPTA+, detection probabilities begin to grow rapidly after only five years of observation beyond current datasets.
Binary stalling (the red curves) stunts this growth by three years at most. While the low-frequency turnover reduces detection probability, its effect is mitigated by the large number of pulsars and the annual catalog expansion.\footnote{\citet{s+15} advocate performing observations with higher cadence to improve sensitivity at $f\geq 0.2$ yr$^{-1}$, where the GWB would be less affected by environmental couplings. In fact, higher-cadence observations would improve sensitivity \emph{across all frequencies}, but not if they come at the cost of reducing the number of monitored pulsars because of limited observing-time allocations. Furthermore, GW searches are actually sensitive to the spectrum of timing residuals rather than of strain itself. The two are related by $S_h(f)\propto h_c(f)^2/f^3$, so even a turnover spectrum will leave a steep red-noise signature in the pulsar timing residuals.}
Finally, we see that the large number of well-timed pulsars in the TPTA builds highly convincing detection probabilities after only a few years of operation. By the time the pulsars have been observed long enough that the influence of the turnover at $f=1/(11\,\mathrm{yr})$ may be noticeable, the DP is already close to unity. The same is true were we to add timing noise at currently known levels to these TPTA pulsars--the DP will already be close to unity when low-frequency timing noise begins to have a significant influence.
We stress that there are several important caveats to our analysis. We have focused on detecting the stochastic background rather than deterministic signals. Even for these, using data from more pulsars is desirable, in order to build evidence for a coherent signal. We have assumed for each pulsar a single value of measurement noise that does not improve over time (as occurs in reality when receivers and backends are upgraded), nor do we consider interstellar medium effects such as dispersion (see, e.g., \citealt{stinebring2013}). The influence of a GWB spectral turnover depends on its frequency, which is a function of the typical environmental properties of galactic nuclei (either directly or in how these properties drive SMBHB orbital eccentricity)---our choice corresponds to reasonable assumptions about the stellar mass density of SMBHB environments. The timelines for the growth of detection in each PTA are approximate, intended to emphasize the differences between PTAs suited to upper-limits versus detection, and to demonstrate the influence of various binary stalling and environmental scenarios on detection probabilities.
We conclude by emphasizing the different demands of placing stringent upper limits on the stochastic background versus actually detecting it. To wit:
\begin{itemize}
\item Highly constraining, astrophysically significant upper limits are achievable with only a few exquisitely timed pulsars, but such a PTA is suboptimal for the detection of a stochastic GW background.
\item Timing many pulsars allows for the quadrupolar spatial correlation signature of the SMBHB background to be sampled at many different angular separations, enhancing prospects for detection.
\item Adding more pulsars regularly to PTAs will continually improve detection probability, in addition to the gains already made by timing existing pulsars for longer, and will help to mitigate the deleterious influences of binary stalling and environmental couplings.
\end{itemize}
\section{Acknowledgements}
It is our pleasure to thank Pablo Rosado, Alberto Sesana, Jonathan Gair, Lindley Lentati, Sarah Burke-Spolaor, Xavier Siemens, Maura McLaughlin, Joseph Romano, and Michael Kramer for very useful suggestions. We also thank the full NANOGrav collaboration for their comments and remarks. SRT was supported by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory, administered by Oak Ridge Associated Universities through a contract with NASA. MV acknowledges support from the JPL RTD program. JAE and RvH acknowledge support by NASA through Einstein Fellowship grants PF4-150120 and PF3-140116, respectively. CMFM was supported by a Marie Curie International Outgoing Fellowship within the European Union Seventh Framework Programme. This work was supported in part by National Science Foundation Physics Frontier Center award no.\ 1430284, and by grant PHYS-1066293 and the hospitality of the Aspen Center for Physics. This research was performed at the Jet Propulsion Laboratory, under contract with the National Aeronautics and Space Administration.
\bibliographystyle{apj}
|
1,108,101,566,771 | arxiv | \section{Introduction}
In QCD, charge symmetry (CS) is a symmetry of the Lagrangian under the
exchange of the up and down quarks~\cite{MNS}. This symmetry has many
consequences at the hadronic level, where it translates into, {\textrm{e.g.}}, the
invariance of the strong nuclear force under the exchange of protons
and neutrons. However, CS is broken by the different masses of the up
and down quarks and thus the strong interaction manifests charge
symmetry breaking (CSB). The different electromagnetic properties of
the up and down quarks also contribute to CSB. An important
consequence of the first CSB effect (strong CSB) is that the neutron is
heavier than the proton, since if CSB was only an electromagnetic effect
the proton would be heavier and prone to decay.
This would make our world very different, since big-bang nucleosynthesis
is dependent on the relative proton and neutron abundances.
While there are a number of pieces of experimental evidences for
CSB~\cite{MNS}---including recent results in $dd\to\alpha\pi^0$ at
IUCF~\cite{IUCFCSB} and $np\to d\pi^0$ at TRIUMF~\cite{Allena}---one of the
most fundamental to nuclear physics is the difference between the
neutron-neutron ($a_{nn}$) and proton-proton ($a_{pp}$) scattering lengths.
The scattering lengths parameterize the nucleon-nucleon interaction at low
relative energies through the effective range expansion.
In this low-energy region the ($^1S_0$) phase shift $\delta_0$ can be
expressed as
\begin{equation}
p \cot\delta_0 = -\frac{1}{a}+\frac12 r_0p^2,
\label{eq:ER}
\end{equation}
where $p$ is the center-of-momentum (c.m.) relative nucleon momentum and
$r_0$ the effective range.
This expansion is reliable for $p\alt150$~MeV/$c$.
Since there are no free neutron targets it is very difficult to make a
direct measurement of the $^1S_0$ neutron-neutron scattering length,
though proposals to do so have been made. The latest of these
suggests using the pulsed nuclear reactor YAGUAR in Snezhinsk,
Russia~\cite{yaguar}. However, the more common---and so far more
successful---approach is to rely on suitable reactions involving two free
neutrons and corresponding theoretical calculations to extract $a_{nn}$ from
indirect data. By choosing the
kinematics carefully one can detect the neutrons in a low-energy
relative $S$-wave which can be accurately described by the effective
range expansion (\ref{eq:ER}). The currently accepted value
\begin{equation}
a_{nn}=-18.5\pm 0.3\ {\rm fm}
\label{eq:annvalue}
\end{equation}
is deduced from the break-up reaction $nd\to nnp$ and the pion radiative
capture process $\pi^-d\to nn\gamma$.
The first reaction is, however, complicated by the possible presence of
three-body forces, but even after they are taken into account there
are significant disagreements between values extracted by the two
techniques. A recent $nd$ break-up experiment reports
$a_{nn}=-16.1\pm0.4$~fm~\cite{nd1}, {\textrm{i.e.}}, more than five standard
deviations from the standard value~(\ref{eq:annvalue}).
The result of Ref.~\cite{nd1} is also in disagreement with another $nd$
experiment that claims $-18.7\pm0.6$~fm~\cite{nd2}.
Earlier data had an even larger spread, see Ref.~\cite{Slaus} for a review.
Since the proton-proton scattering length is
$a_{pp}=-17.3\pm0.4$~fm (after corrections of electromagnetic
effects~\footnote{There is a small electromagnetic correction
($-0.3$~fm) to $a_{nn}$~\protect\cite{MNS}, which for the rest of this
paper will be ignored.}), there is even uncertainty about the sign
of the difference $a_{pp}-a_{nn}$. The more negative $a_{nn}$ is
favored by nuclear structure calculations, where the small (but
important) CSB piece of the AV18 potential is fitted to reproduce
$a_{pp} - a_{nn}$ with $a_{nn}=-18.5$~fm~\cite{Wi95}. The
binding-energy difference between $^3{\rm H}$ and $^3{\rm He}$, which
has a small contribution from CSB effects, is then very accurately
reproduced. This would not occur were $a_{pp} - a_{nn}$ to take the
opposite sign~\cite{AV18}.
Because of these issues the accepted value is weighted toward the
$a_{nn}=-18.50\pm0.05(\rm stat.)\pm0.44(syst.)\pm0.30(theory)$~fm
reported by the most recent $\pi^-d\to nn\gamma$ experiment~\cite{LAMPF}.
The extraction in this is case done by fitting the shape of the neutron
time-of-flight spectrum using the model of
Gibbs, Gibson, and Stephenson (GGS)~\cite{GGS}. This model was
developed in the mid-70s and explored many of the relevant mechanisms
and the dependence on various choices of wave functions.
Gibbs, Gibson, and Stephenson calculated the single-nucleon radiative
pion capture tree-level amplitude to order $p/M$, and consequently ignored
the pion loops that would enter at the next chiral order.
Two-body diagrams were not fully implemented in this model.
The theoretical error was dominated
by uncertainties in the scattering wave function.
Similar results were obtained in earlier $\pi^-d\to nn\gamma$ experiments
carried out at the Paul Scherrer Institut (PSI) (then Swiss Institute of
Nuclear Research)~\cite{Gabioudetal}.
In the PSI experiments, only the FSI peak was fitted, while at LAMPF the entire
spectrum was fitted.
The theoretical work for the PSI results compared the GGS model with work done
by de~T\'eramond and collaborators~\cite{deTeramond}.
The latter used a dispersion relation approach for the final state interaction,
with a theoretical error of the order 0.3~fm, {\textrm{i.e.}}, similar to GGS.
In this paper we recalculate the $\pi^-d\to nn\gamma$ reaction using
chiral perturbation theory ($\chi$PT). The one-body and two-body
mechanisms are thus consistent and the constraints of chiral
symmetry are respected, which is of crucial importance in this
threshold regime. At third order, $O(Q^3)$, all the amplitudes of the
previous calculation are included, as well as pion loops and three
pion-rescattering diagrams. Additional advantages of the chiral power
counting are that it gives a clearly defined procedure to estimate the
theoretical error and provides a systematic and consistent way to
improve the calculation if needed.
In this first paper we establish the machinery necessary for a precise
extraction of the $nn$ scattering length.
We isolate the sources of the largest remaining errors and suggest means for
their reduction, which should make it possible to reach the desired high
precision in future work.
The paper is organized as follows. In Sec.~\ref{sec:layout} we will
develop the main ingredients of our calculation: the Lagrangian, the
explicit forms of the one- and two-body amplitudes, and a
description of our wave functions. The numerical results are
presented in Sec.~\ref{sec:results} together with our estimate of the
theoretical error. We conclude in Sec.~\ref{sec:end}.
\section{Layout of calculation}
\label{sec:layout}
The LAMPF experiment~\cite{LAMPF} used stopped pions, captured into
atomic orbitals around the deuteron. The subsequent radiative decay
occurs for pionic $s$-wave orbitals only. Thus the c.m.\ and
laboratory frames coincide and the pion momentum is vanishingly small.
The neutron time-of-flight distribution of the c.m.\ $\pi^-d$ decay
width (with the photon and one neutron detected) can be expressed as
\begin{equation}
\frac{d^2\Gamma}{dt_1d\theta_3} = \frac{1}{2(2\pi)^3M_{\pi d}t_1}
\frac{p_1^3E_1 k\sin\theta_3}{M_{\pi d}-E_1-p_1\cos\theta_3}
\frac{1}{3}\sum_{\rm pols.}|\mathcal{M}|^2,
\end{equation}
where $t_1$, $p_1$, and $E_1$ are the time-of-flight, momentum, and
energy of the detected neutron, $\theta_3$ is the supplement of the
angle between this neutron and the photon, and $k$ is the photon
momentum. Here the sum is over deuteron and photon polarizations
and $M_{\pi d}$ is the mass of the deuteron-pion bound
system, which to a very good approximation is given by the sum of the
pion ($\mu$) and deuteron ($M_d$) masses.
The matrix element $\mathcal{M}$ is the sum of four interfering parts,
the quasi-free (QF) one-body, the one-body with final state
interaction (FSI), and the two-body contributions, with and without
FSI. These can be symbolized by the generic diagrams shown in
Fig.~\ref{fig:gendiag}. In this first calculation we restrict FSI to
$S$-waves only, and subtract the plane-wave from the scattering wave
function and include it in the QF contribution.
\begin{figure}[t]
\includegraphics{gendiag.eps}
\caption{Generic diagrams of the contributions to $\pi^-d\to nn\gamma$.
Diagram (a) and (b) are the one-body contributions, without and with FSI.
Diagram (c) and (d) are the corresponding for the two-body currents.
The amplitudes $\mathcal{A}_{\rm I}$ and $\mathcal{A}_{\rm II}$ are described
in the text.}
\label{fig:gendiag}
\end{figure}
We will derive the matrix elements for $\gamma nn\to\pi^-d$ rather
than $\pi^-d\to nn\gamma$ in order to reduce the possibility of
relative phase errors when using the $\gamma n\to\pi^-p$ amplitudes.
The $\pi^-d\to nn\gamma$ decay rate can of course then be obtained by
detailed balance. An explicit expression for the matrix
element in terms of $\pi d$ atomic ($\Phi_{\pi d}$) and deuteron
($\varphi_d$) wave functions and the pion-photon amplitude
$\mathcal{A}$ is given by
\begin{eqnarray}
\mathcal{M}(\gamma nn\to\pi^-d) & = & \int
\frac{d^3qd^3p'd^3p''}{(2\pi)^9}\frac{M\sqrt{2E_{\pi d}}}
{\sqrt{E_1E_22E_\pi}}
\Phi^\ast_{\pi d}(0;\qi{})\varphi^\ast_{d}(-\qi{};{\bf p}'')
\mathcal{A}
\Psi_{\bf -k}({\bf p}',{\bf p}),
\end{eqnarray}
where ${\bf p}$, ${\bf p}'$, and ${\bf p}''$ are the initial, intermediate, and final
relative momenta of the two nucleons (for $\gamma nn\to\pi^- d$), while
$\qi{}$ is the pion c.m.\ momentum and $E_x$ the energy of the indicated
particle.
The meaning of these kinematic variables can also be inferred from
Fig.~\ref{fig:kin}.
\begin{figure}[t]
\includegraphics{kinfig.eps}
\caption{Basic diagram for $\pi^-d\to nn\gamma$, defining kinematic variables.}
\label{fig:kin}
\end{figure}
Here $\Psi_{\bf -k}({\bf p}',{\bf p})$ is the $nn$ scattering wave function at total
momentum ${\bf -k}$, normalized such that
\begin{equation}
\int \frac{d^3q}{(2\pi)^3}
\Psi^{\ast}_{\bf -k}({\bf p}',\qi{})\Psi_{\bf -k}({\bf p},\qi{}) =
(2\pi)^3\delta^3({\bf p}'-{\bf p}).
\end{equation}
Its Fourier transform is given by
\begin{equation}
\Psi_{\bf -k}({\bf p}',{\bf p}) = \int\frac{d^3r}{(2\pi)^3}\expup{-i{\bf p}'\cdot\ri{}}
\Psi_{\bf -k}(\ri{},{\bf p}),
\end{equation}
where
$\Psi_{\bf -k}({\bf p},\ri{})=\sum_l(2l+1)i^l\frac{v_l(r)}{r}P_l(\bmath{\widehat{\bf p}}\cdot\bmath{\widehat{\bf r}})$,
which could be compared to the plane wave expansion
$\expup{i{\bf p}\cdot\ri{}}=\sum_l(2l+1)i^lj_l(pr)P_l(\bmath{\widehat{\bf p}}\cdot\bmath{\widehat{\bf r}})$.
The form of $v_l(r)$ will be discussed in Sec.~\ref{sec:wf}.
The pion-photon amplitude $\mathcal{A}$ can be separated into the one- and
two-body amplitudes $\mathcal{A}_{\rm I}$ and $\mathcal{A}_{\rm II}$;
\begin{eqnarray}
\mathcal{A} & = & (2\pi)^3\frac{E}{M}
\delta^{(3)}(\tilde{\bf p}\pm\frac{\ki{}-\qi{}}{2})
\mathcal{A}_{\rm I}(\ki{},\qi{})+
\mathcal{A}_{\rm II}(\pm\tilde{\bf p},\ki{},\qi{}),
\end{eqnarray}
where the upper(lower) sign is for interaction on nucleon 1(2) of the deuteron
and $\tilde{\bf p}={\bf p}'-{\bf p}''$.
The one- and two-body matrix elements are then
\begin{eqnarray}
\mathcal{M}_{\rm I}(\gamma nn\to\pi^-d) & = & \int \frac{d^3qd^3p'}{(2\pi)^6}
\frac{E\sqrt{2E_{\pi d}}}{\sqrt{E_1E_22E_\pi}}
\Phi^\ast_{\pi d}(0;\qi{})
\varphi^\ast_{d}(-\qi{};{\bf p}'\pm\frac{\bf k-q}{2})
\mathcal{A}_{\rm I}(\ki{},\qi{})
\Psi_{-\bf k}({\bf p}',{\bf p}), \nonumber \\
\mathcal{M}_{\rm II}(\gamma nn\to\pi^-d) & = &
\int \frac{d^3qd^3p'd^3p''}{(2\pi)^9}
\frac{M\sqrt{2E_{\pi d}}}{\sqrt{E_1E_22E_\pi}}
\Phi^\ast_{\pi d}(0;\qi{})
\varphi^\ast_{d}(-\qi{};{\bf p}'')
\mathcal{A}_{\rm II}(\pm\tilde{\bf p},\ki{},\qi{}) \nonumber \\
& \times & \Psi_{-\bf k}({\bf p}',{\bf p}).
\end{eqnarray}
In configuration space the matrix elements are given by
\begin{eqnarray}
\mathcal{M}_{\rm I}^{\rm FSI} & = & \sum\Phi_{\pi d}(0)
\sqrt{\frac{M_{\pi d}}{\mu}}
\int dr\frac{d\Omega_r}{\sqrt{4\pi}}
[S_0u(r)+S_2(\bmath{\widehat{\bf r}})w(r)]\expup{\pm\frac{i}{2}(\ki{}-\qi{})\cdot{\bf r}}
\mathcal{A}_{\rm I}(\ki{},\qi{}) \tilde{v}_0(r)\chi_0, \\
\mathcal{M}_{\rm I}^{\rm QF} & = & \sum\Phi_{\pi d}(0)
\sqrt{\frac{M_{\pi d}}{\mu}}
\int rdr\frac{d\Omega_r}{\sqrt{4\pi}}
[S_0u(r)+S_2(\bmath{\widehat{\bf r}})w(r)]\expup{\pm \frac{i}{2}(\ki{}-\qi{})\cdot{\bf r}}
\mathcal{A}_{\rm I}({\bf k},\qi{})\expup{i{\bf p}\cdot\ri{}}\chi_S, \\
\mathcal{M}_{\rm II}^{\rm FSI} & = & \sum\Phi_{\pi d}(0)
\sqrt{\frac{M_{\pi d}}{\mu}}
\int dr \frac{d\Omega_r}{\sqrt{4\pi}}
[S_0u(r)+S_2(\bmath{\widehat{\bf r}})w(r)]\mathcal{A}^\pm_{\rm II}({\bf r},{\bf k},\qi{})
\tilde{v}_0(r)\chi_{0}, \\
\mathcal{M}_{\rm II}^{\rm PW} & = & \sum\Phi_{\pi d}(0)
\sqrt{\frac{M_{\pi d}}{\mu}}
\int rdr \frac{d\Omega_r}{\sqrt{4\pi}}
[S_0u(r)+S_2(\bmath{\widehat{\bf r}})w(r)]\mathcal{A}^\pm_{\rm II}({\bf r},{\bf k},\qi{})
\expup{\pm i{\bf p}_i\cdot\ri{}}\chi_S,
\end{eqnarray}
where ${\bf p}_{1,2}=-\frac{1}{2}{\bf k}\pm{\bf p}$,
$\mathcal{A}^\pm_{\rm II}({\bf r},{\bf k},{\bf q})=\int
\frac{d^3\tilde{p}}{(2\pi)^3}\expup{-i\tilde{\bf p}\cdot{\bf r}}
\mathcal{A}_{\rm II}(\pm\tilde{\bf p},{\bf k},{\bf q})$,
and $\tilde{v}_0(r)$ is the subtracted scattering wave function as defined in
Sec.~\ref{sec:wf}.
The sums are over the two nucleons.
In these expressions $\Phi_{\pi d}(0)=(\mu_{\pi d}\alpha)^{3/2}/\sqrt{\pi}$ is
the pion-deuteron atomic $s$-orbital wave function evaluated at the origin
with $\mu_{\pi d}$ the reduced $\pi d$ mass, while
$S_0=-\frac{1}{\sqrt{2}}\bmath{\sigma}\cdot\bmath{\epsilon}_d^\dagger$ and
$S_2(\bmath{\widehat{\bf r}})=\frac{1}{2}(3\bmath{\sigma}\cdot\bmath{\widehat{\bf r}}\,\bmath{\widehat{\bf r}}\cdot\bmath{\epsilon}_d^\dagger-
\bmath{\sigma}\cdot\bmath{\epsilon}_d^\dagger)$
are the $S$- and $D$-wave spin structures of the deuteron, $\bmath{\epsilon}_d^\dagger$
being the deuteron polarization vector.
The $\chi_S$'s are the neutron-neutron spin wave functions for spin $S$.
In the following subsections we will derive explicit expressions for the
amplitudes and show how the coordinate-space wave functions are obtained.
\subsection{Power counting}
\label{sec:EFT}
We start from the relativistic Lagrangian
\begin{eqnarray}
\mathcal{L} & = & {\cal L}_{\pi N}^{(1)} + {\cal L}_{\pi N}^{(2)} +
{\cal L}_{\pi \pi}^{(2)}; \nonumber \\
{\cal L}_{\pi N}^{(1)} & = & \bar N\left[i\gamma\cdot D_N-M
-\frac{1}{4f_\pi^2}\epsilon^{abc}\tau^c\pi^a\gamma\cdot\partial\pi^b
-\frac{g_A}{2f_\pi}(\gamma\cdot D^{ab} \pi^a)\tau^b\gamma_5\right] N,
\nonumber\\
{\cal L}_{\pi N}^{(2)} & = & -\bar
N\left[\frac{e(\kappa_0+\kappa_1\tau^3)}{8M}\sigma^{\mu\nu}F_{\mu\nu}\right]N,
\nonumber \\
{\cal L}_{\pi \pi}^{(2)}&=& \frac12D_\mu^{ab}\pi^a D^{\mu
cb}\pi^c-\frac12\mu^2\bmath{\pi}^2,
\label{eq:Lagrange}
\end{eqnarray}
where $D^\mu_N = \partial^\mu-ieQ_NA^\mu$ and
$D^{\mu ab}_\pi=\delta^{ab}\partial^\mu-ieQ^{ab}_\pi A^\mu$ are the covariant
derivatives for the nucleon and pion ($e<0$), $Q_N=\frac12(1+\tau^3)$ and
$Q^{ab}_\pi=i\epsilon^{ab3}$ are the electric charge isospin operators
(with $a$/$b$ isospin indices of incoming/outgoing pion), and
$\kappa_{0,1}=\kappa_p\pm\kappa_n$ ($\kappa_p=1.793$ and $\kappa_n=-1.913$)
represent the nucleon anomalous magnetic moments.
The electromagnetic field tensor is given by
$F_{\mu\nu}=\partial_\nu A_\mu-\partial_\mu A_\nu$ and
$\sigma^{\mu\nu}=\frac{i}{2}[\gamma^\mu,\gamma^\nu]$ as usual.
The Lagrangian (\ref{eq:Lagrange}) is organized according to the
number of powers of small momenta $Q$ (here $e$ counts as one ``small''
momentum). The chiral order of any graph to any amplitude involving
nucleons, together with pions and photons with energies of order
$\mu$ can be assessed by multiplying the $Q$-scaling factors of the
individual units of the graph by one another. These factors are as
follows:
\begin{itemize}
\item Each vertex from ${\cal L}^{(n)}$ contributes $Q^n$.
\item Each nucleon propagator scales like $1/Q$ (provided that
the energy flowing through the nucleon line is $\sim\mu$);
\item Each pion propagator scales like $1/Q^2$;
\item Each pion loop contributes $Q^4$;
\item Graphs in which two nucleons participate in the reaction
acquire an extra factor of $Q^3$.
\end{itemize}
In practice tree-level relativistic graphs must be calculated,
and then expanded in powers of $p/M$ in order to establish
contact with the usual heavy-baryon formulation of chiral
perturbation theory. The expressions for loop graphs that we use
are also computed in heavy-baryon $\chi$PT, and so we do not
need to employ any special subtraction schemes to remove
pieces of loop integrals which scale with positive powers
of the nucleon mass~\cite{BL99,Fu03}.
Note that we will employ the Coulomb gauge in all calculations.
\subsection{One-body amplitudes}
\label{sec:oneB}
There are four basic one-body diagrams, shown in Fig.~\ref{fig:one}:
the Kroll-Ruderman (KR) term (a), the pion pole (b), and the $s$- and
$u$-channel nucleon pole terms (c) and (d).
\begin{figure}[t]
\includegraphics{onebodies.eps}
\caption{Single-nucleon pion-photon diagrams relevant for pion photoproduction.
In this and all other figures a solid line represent a nucleon,
a dashed line a pion and a wavy line a photon.}
\label{fig:one}
\end{figure}
They can be calculated directly from a non-relativistic reduction of
the relativistic Lagrangian or from the amplitudes of heavy-baryon
$\chi$PT (HB$\chi$PT)~\cite{ulfreview}. In addition, there are also
pion-loop corrections at $O(Q^3)$ as shown in Fig.~\ref{fig:piloops}.
\begin{figure}[t]
\includegraphics{pionloops.eps}
\caption{Pion loops at NNLO for $\gamma N\to \pi N$ in Coulomb gauge.
Note that not all time orderings are shown.}
\label{fig:piloops}
\end{figure}
The loop corrections, together with the corresponding counterterms from
${\cal L}_{\pi N}^{(3)}$~\cite{EM} have already been calculated in the
$\gamma N$ c.m.~frame to $O(Q^3)$ for radiative pion capture on a
nucleon~\cite{Fearing} using Coulomb gauge. The low-energy constants
(LECs) from the third-order chiral Lagrangian were fitted to
experiment, yielding an excellent description of the near-threshold
data.
The third-order piece of the one-body amplitudes of
Ref.~\cite{Fearing} can hence be taken as is, without introducing any
new unknown parameters, and combined with our evaluation to $O(Q^2)$
of the tree-level diagrams in Fig.~\ref{fig:one} using the
relativistic Lagrangian (\ref{eq:Lagrange})~\footnote{The amplitudes
of Ref.~\protect\cite{Fearing} are based on the third-order
heavy-baryon Lagrangian of Ecker and
Moj\v{z}i\v{s}~\protect\cite{EM}. There are some differences between
the results obtained in this way and those found when the tree-level
relativistic amplitude for $\gamma n \rightarrow \pi^- p$ is expanded
to relative order $p^2/M^2$. Thus the tree-level terms we find at $O(Q^3)$
are slightly different to those listed in Ref.~\cite{Fearing}, but
this can be accounted for by a redefinition of the LECs in ${\cal
L}_{\pi N}^{(3)}$. This redefinition only affects the $O(Q^3)$
(NNLO) photoproduction amplitude and thus---at the order we consider
here---it is not relevant to the boost or other issues associated with
embedding the $\gamma n \rightarrow \pi^- p$ amplitude in the $A=2$
system.}. In order to incorporate these amplitudes in the two-body
system they should be evaluated at the relevant subthreshold
kinematics, corrected for a boost to the overall rest frame, and
corrected for off-shell effects. These issues will all be discussed
below.
The full one-body amplitude is given (in Coulomb gauge) by
\begin{eqnarray}
\mathcal{A}_{\rm I}(\gamma N\to\pi N) & = &
F_1(E_\pi,x)i\bmath{\sigma}\cdot\bmath{\epsilon}_\gamma+
F_2(E_\pi,x)\bmath{\sigma}\cdot\bmath{\widehat{\bf q}}\,\bmath{\sigma}\cdot(\bmath{\widehat{\bf k}}\times\bmath{\epsilon}_\gamma)+
F_3(E_\pi,x)i\bmath{\sigma}\cdot\bmath{\widehat{\bf k}}\,\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_\gamma \nonumber \\
& + & F_4(E_\pi,x)i\bmath{\sigma}\cdot\bmath{\widehat{\bf q}}\,\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_\gamma,
\label{eq:Fis}
\end{eqnarray}
where the $F_i$ are the Chew-Goldberger-Low-Nambu (CGLN) amplitudes~\cite{CGLN}
and $\bmath{\epsilon}_\gamma$ is the photon polarization vector.
The isospin channels are separated as
\begin{equation}
F_i^a(E_\pi,x) = F_i^{(-)}(E_\pi,x)i\epsilon^{a3b}\tau^b+
F_i^{(0)}(E_\pi,x)\tau^a+F_i^{(+)}(E_\pi,x)\delta^{a3},
\label{eq:isospin}
\end{equation}
where $a$ is the pion isospin index. For $\gamma n\to\pi^-p$ this
implies that $F_i=\sqrt{2}[F_i^{(0)}-F_i^{(-)}]$. The $F_i$'s of
Ref.~\cite{Fearing} are evaluated with the pion energy $E_\pi$ and
photon-pion cosine $x=\bmath{\widehat{\bf k}}\cdot\bmath{\widehat{\bf q}}$ in the $\pi^-p\to\gamma n$ rest
frame. In our case $\qi{}=0$, $E_\pi=\mu$, and $x$ is undetermined.
Thus only $F_1$ survives, the other spin amplitudes being proportional
to the pion momentum. In charged pion photo-production $F_1$ is
dominated by the KR contribution [Fig.~\ref{fig:one}(a)].
\subsubsection{Subthreshold extrapolation}
\label{sec:subthreshold}
If the $\pi^-p\to\gamma n$ process was completely free, the CGLN
amplitudes should be evaluated at the pion threshold $E_\pi=\mu$.
However, since the proton is bound in the deuteron, the $\pi^-p$
energy is actually less than $\mu$, which means that we must extrapolate
to the sub-threshold regime. To do this we need
a prescription to calculate the invariant two-body energy
$s_{\pi^-p}$. The pion and photon energy in the $\pi^-p\to\gamma n$
rest frame can then be calculated using the well-known relations
\begin{eqnarray}
E_\pi^\ast & = & \frac{s_{\pi^-p}-m_p^2+\mu^2}{2\sqrt{s_{\pi^-p}}}, \\
\omega^\ast & = & \frac{s_{\pi^-p}-m_n^2}{2\sqrt{s_{\pi^-p}}}.
\end{eqnarray}
The energy available to the $\pi^- p$ subsystem, $s_{\pi^- p}$ would
seem to be different depending on whether FSI or QF kinematics are
considered. Furthermore, there are two QF situations, {\textrm{i.e.}}, the
detected neutron can originate from the one-body vertex or it can be
the spectator. In fact, the first case is overwhelmingly favored by
the kinematics of the LAMPF experiment and is also the one closest to
threshold. The second, spectator, scenario is suppressed by kinematics, so
even though it is further from threshold and so results in a larger
shift in $s_{\pi^- p}$, any correction resulting from this shift is
small compared to other, included, effects.
For the QF kinematics where the detected neutron originates from the
one-body vertex the rest frame coincides, by definition, with the
overall $\gamma nn$ c.m. But, in the FSI region one has to make a
choice. The invariant energy of the $\pi^-p\to\gamma n$ system can be
established from
\begin{equation}
s_{\pi^- p} = (M_d+\mu)^2+m_n^2-2(M_d+\mu)\epsilon_s,
\label{eq:spipspect}
\end{equation}
where $\epsilon_s=\sqrt{m_n^2+p_s^2}$ is the energy of
the spectator nucleon. We choose to assume that the spectator nucleon
is on-shell and that its typical momentum $p_s$ can be estimated
through calculating the expectation value $\langle p_s^2\rangle$
between initial and final state wave functions. Using the $S$-state of
the deuteron only, the average is given by
\begin{eqnarray}
\langle p_s^2\rangle & = & \frac{k^2}{4}-
\frac{\int dr(MB-p^2+2MV_{SS})u(r)j_0(\frac{kr}{2})\tilde{v}_0(r)}
{\int dr u(r)j_0(\frac{kr}{2})v(r)}.
\label{eq:pssq}
\end{eqnarray}
We then use free kinematics for the one-body amplitudes in the QF
region, and the formulas (\ref{eq:spipspect}) and (\ref{eq:pssq})
to calculate the energy at which the one-body amplitude should
be evaluated in the FSI peak.
The one-body amplitudes are then calculated using $E_\pi^\ast$ according
to the different kinematics of the QF and FSI configurations.
The theoretical uncertainty due to this procedure is assessed in
Sec.~\ref{subsec:extrap}.
\subsubsection{Boost corrections}
In general the $\gamma n\to\pi^-p$ rest frame does not coincide with
the overall c.m., so we have to adjust the $F_i$ for boost effects.
The boost corrections can be calculated by replacing the $\gamma n\to
\pi^-p$ rest frame kinematics by the overall $\gamma nn$ c.m.\
kinematics in the evaluation of the one-body amplitudes. This changes
the incoming and outgoing nucleon momenta, but not the photon and pion
momenta. The Coulomb gauge condition $\epsilon_\gamma^0=0$ is
retained. From the Lagrangian~(\ref{eq:Lagrange}) one can then
deduce the following boost corrections for the reduced amplitudes, up
to order $Q^2/M^2$ for the $\gamma N\to\pi N$ reactions;
\begin{eqnarray}
\Delta F_1^{(0)}(E_\pi) & = & \frac{eg_A}{2f_\pi}
\frac{-(E_\pi{\bf p}_n\cdot\bmath{\widehat{\bf k}}+E_\pi^2)}{2M^2}(\mu_p+\mu_n),
\label{eq:corrF10} \\
\Delta F_1^{(-)}(E_\pi) &=& \frac{eg_A}{2f_\pi}
\frac{E_\pi{\bf p}_n\cdot\bmath{\widehat{\bf k}}+E_\pi^2}{M^2},
\label{eq:corrF1m}
\end{eqnarray}
where ${\bf p}_n$ is the outgoing nucleon momentum ($=-\ki{}$ in the $\gamma n$
rest frame, which makes these amplitudes vanish).
As before we have assumed Coulomb gauge, $\qi{}=0$, and the same isospin
designations as in Eq.~(\ref{eq:isospin}).
These corrections should thus be added to the amplitudes of Eqs.~(\ref{eq:Fis})
and (\ref{eq:isospin}) as given in~\cite{Fearing}, except for an overall
factor of $M/4\pi\sqrt{s}$, which is included in the phase
space in our formalism.
There are also terms with new spin-momentum structures:
\begin{eqnarray}
G^{(0)}(E_\pi) & = & \frac{eg_A}{2f_\pi}
\frac{iE_\pi{\bf p}_n\cdot\bmath{\epsilon}_\gamma\bmath{\sigma}\cdot\bmath{\widehat{\bf k}}}{2M^2}
(\mu_p+\mu_n-1), \\
G^{(-)}(E_\pi) & = & \frac{eg_A}{2f_\pi}\left(
\frac{E_\pi{\bf p}_n\cdot(\bmath{\widehat{\bf k}}\times\bmath{\epsilon}_\gamma)}{2M^2}(\mu_p-\mu_n+\frac12)-
\frac{i{\bf p}_n\cdot\bmath{\epsilon}_\gamma\bmath{\sigma}\cdot(2{\bf p}_n+E_\pi\bmath{\widehat{\bf k}})}{M^2}
\right),
\end{eqnarray}
which will also vanish in the limit ${\bf p}_n\to-\ki{}$.
In the case of non-vanishing pion momentum, additional terms will show up.
One would expect that the first term of $G^{(-)}$ should give the largest
contribution since $\mu_p-\mu_n+\frac12=5.2$ is a big number.
However, because of the particular kinematics of the present problem,
${\bf p}_n\approx-\ki{}$ and the triple scalar product
${\bf p}_n\cdot(\ki{}\times\bmath{\epsilon}_\gamma)\approx E_\pi^2\sin\theta_3$
with $\theta_3=0.075$.
Thus, ultimately this piece of $G^{(-)}$ is very small because of the
kinematics.
Similarly ${\bf p}_n\cdot\bmath{\epsilon}_\gamma\approx E_\pi\sin\theta_3$.
(Additionally, only one of the photon polarizations can contribute.)
In fact it turns out that the new spin-momentum structures have a negligible
effect on the pion-photon amplitude and the only possible relevant boost
corrections come from the terms in Eqs.~(\ref{eq:corrF10}) and
(\ref{eq:corrF1m}).
In the actual calculations the subthreshold value $E_\pi^\ast$
and not $E_\pi=\mu$ was used in evaluating these.
\subsection{Two-body amplitudes}
\label{sec:twoB}
At third order there are three pion-exchange diagrams, displayed in
Fig.~\ref{fig:two}.
\begin{figure}[t]
\includegraphics{twobodies.eps}
\caption{The third order pion rescattering diagrams relevant for
$\pi^-d\to nn\gamma$.
They are shown in order of expected importance as explained in the text.}
\label{fig:two}
\end{figure}
Of these, the first one (a) is expected to give the largest contribution since
its propagator is coulombic, {\textrm{i.e.}}, behaves like $1/\qi{}^2$, where $\qi{}$ is
the momentum of the exchange pion~\cite{BLvK}.
This is because (for our kinematics) the pion energy is transferred completely
to the photon and vanishes against the pion mass in the propagator, thus
putting the pion effectively on-shell.
It has been argued in the literature that when the intermediate nucleon-nucleon
state for this diagram (as interpreted in time-ordered perturbation theory) is
Pauli-allowed, corrections due to nucleon recoil need
to be taken into account~\cite{Baru}.
However, in our case the intermediate nucleon pair is in a triplet-isotriplet
state implying a relative $P$-wave, which is Pauli-suppressed.
The recoil correction evaluated in Ref.~\cite{Baru} is thus small.
The second graph (b) has an extra pion propagator which is off-shell, reducing
the magnitude of this diagram.
The third two-body amplitude (c) has two off-shell pion propagators and is
hence suppressed compared to the other two.
More importantly, this diagram is further suppressed since in Coulomb gauge it
is proportional to the (vanishingly small) pion momentum.
The two-body amplitudes, corresponding to the diagrams of Fig.~\ref{fig:two}
(a-c), are
\begin{eqnarray}
\mathcal{A}_{{\rm II}a}(\tilde{\bf p},{\bf k},{\bf q}) & = & \frac{eg_A}{2f_\pi}
\frac{(-2iE_\pi)}{4f_\pi^2}
\left[\frac{\tau_1^a\tau_2^3\bmath{\sigma}_1\cdot\bmath{\epsilon}_\gamma}
{(\tilde{\bf p}+\frac{\ki{}+\qi{}}{2})^2}
+\frac{\tau_1^3\tau_2^a\bmath{\sigma}_2\cdot\bmath{\epsilon}_\gamma}
{(\tilde{\bf p}-\frac{\ki{}+\qi{}}{2})^2} \right], \\
\mathcal{A}_{{\rm II}b}(\tilde{\bf p},{\bf k},{\bf q}) & = & \frac{eg_A}{2f_\pi}
\frac{4iE_\pi}{4f_\pi^2}
\left[\frac{\tau_1^a\tau_2^3\bmath{\sigma}_1\cdot\left(\tilde{\bf p}-
\frac{\ki{}-\qi{}}{2}\right)
\bmath{\epsilon}_\gamma\cdot(\tilde{\bf p}+\qi{})}
{(\tilde{\bf p}+\frac{\ki{}+\qi{}}{2})^2[\mu^2+(\tilde{\bf p}-
\frac{\ki{}-\qi{}}{2})^2]}
\right.\nonumber \\ & + & \left.
\frac{\tau_1^3\tau_2^a\bmath{\sigma}_2\cdot\left(\tilde{\bf p}+
\frac{\ki{}-\qi{}}{2}\right)
\bmath{\epsilon}_\gamma\cdot(\tilde{\bf p}-\qi{})}
{(\tilde{\bf p}-\frac{\ki{}+\qi{}}{2})^2[\mu^2+(\tilde{\bf p}+
\frac{\ki{}-\qi{}}{2})^2]}
\right], \\
\mathcal{A}_{{\rm II}c}(\tilde{\bf p},{\bf k},{\bf q}) & = & \frac{eg_A}{2f_\pi}
\frac{2E_\pi-\omega}{4f_\pi^2}
\frac{(\tau_1^a\tau_2^3-\tau_1^3\tau_2^a)i\bmath{\epsilon}_\gamma\cdot\qi{}}
{\omega(E_\pi-qy)}
\left[ \frac{\bmath{\sigma}_1\cdot(\tilde{\bf p}-\frac{\ki{}-\qi{}}{2})}
{\mu^2+(\tilde{\bf p}-\frac{\ki{}-\qi{}}{2})^2}
+\frac{\bmath{\sigma}_2\cdot(\tilde{\bf p}+\frac{\ki{}-\qi{}}{2})}
{\mu^2+(\tilde{\bf p}+\frac{\ki{}-\qi{}}{2})^2}\right], \nonumber \\
\end{eqnarray}
where $y=\bmath{\widehat{\bf k}}\cdot\bmath{\widehat{\bf q}}$ is the pion-photon cosine in the overall c.m.
In configuration space (for $\qi{}=0$) the two-body amplitudes can be expressed
as
\begin{eqnarray}
\mathcal{A}_{{\rm II}a}({\bf r},{\bf k},{\bf q=0}) & = &
\frac{eg_A}{8f_\pi^3}\frac{-2iE_\pi}{4\pi r}
\left(\tau_1^a\tau_2^3\bmath{\sigma}_1\cdot\bmath{\epsilon}_\gamma
\expup{\frac{i}{2}{\bf k\cdot r}}
+\tau_1^3\tau_2^a\bmath{\sigma}_2\cdot\bmath{\epsilon}_\gamma
\expup{-\frac{i}{2}{\bf k\cdot r}} \right), \\
\mathcal{A}_{{\rm II}b}({\bf r},{\bf k},{\bf q=0}) & = &
-\frac{eg_A}{8f_\pi^3}\frac{2E_\pi}{4\pi}
\tau_1^a\tau_2^3 \int d\alpha \expup{-\tilde\mu r}
\left[\bmath{\epsilon}_\gamma\cdot\bmath{\widehat{\bf r}}
\left( \bmath{\sigma}_1\cdot[(1-\alpha){\bf k}+i(\tilde\mu+\frac{1}{r})\bmath{\widehat{\bf r}}]
\expup{i(\frac12-\alpha){\bf k\cdot r}} \right.\right.\nonumber \\
& + & \left.
\bmath{\sigma}_2\cdot[(1-\alpha){\bf k}-i(\tilde\mu+\frac{1}{r})\bmath{\widehat{\bf r}}]
\expup{-i(\frac12-\alpha){\bf k\cdot r}}\right) \nonumber \\
& - & \left. \frac{i}{r}
(\bmath{\sigma}_1\cdot\bmath{\epsilon}_\gamma\expup{i(\frac12-\alpha){\bf k\cdot r}}-
\bmath{\sigma}_2\cdot\bmath{\epsilon}_\gamma\expup{-i(\frac12-\alpha){\bf k\cdot r}})
\right], \\
\mathcal{A}_{{\rm II}c}({\bf r},{\bf k},{\bf q=0}) & = & 0,
\end{eqnarray}
where $\tilde\mu^2=\alpha(\mu^2+\omega^2)-\alpha^2\omega^2$.
These expressions agree with the ones derived in Ref.~\cite{BLvK} after the
sign correction of Ref.~\cite{Be97}.
\subsection{Matrix elements}
\label{sec:M}
The full matrix elements for the QF amplitudes, projected on spin-0 and spin-1
final states are, after taking the trace over nucleon spins and isospins
\begin{eqnarray}
\mathcal{M}_0 & = & Ci \left[\bmath{\epsilon}_d^\dagger\cdot\bmath{\epsilon}_\gamma(F_1-xF_2)+
\bmath{\widehat{\bf k}}\cdot\bmath{\epsilon}_d^\dagger\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_\gamma(F_2+F_3)+
\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_d^\dagger\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_\gamma F_4\right] f(p_2)
\nonumber \\
& + & \frac{3Ci}{\sqrt2} \bmath{\widehat{\bf p}}_2\cdot\bmath{\epsilon}_d^\dagger
\left[\bmath{\widehat{\bf p}}_2\cdot\bmath{\epsilon}_\gamma(F_1-xF_2)+\bmath{\widehat{\bf p}}_2\cdot\bmath{\widehat{\bf k}}\bmath{\widehat{\bf q}}\cdot
\bmath{\epsilon}_\gamma(F_2+F_3)+
\bmath{\widehat{\bf p}}_2\cdot\bmath{\widehat{\bf q}}\qhat\cdot\bmath{\epsilon}_\gamma F_4\right] g(p_2)+(2\rightarrow1),
\nonumber \\
\mathcal{M}_1 & = & C\left\{\bmath{\epsilon}_d^\dagger\cdot
(\bmath{\epsilon}_\gamma\times\bmath{\epsilon}_{nn})F_1-
\left[\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_d^\dagger\bmath{\epsilon}_{nn}-\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_{nn}\bmath{\epsilon}_d^\dagger
+\bmath{\epsilon}_d^\dagger\cdot\bmath{\epsilon}_{nn}\bmath{\widehat{\bf q}}\right]\cdot(\bmath{\widehat{\bf k}}\times\bmath{\epsilon}_\gamma)
F_2 \right.
\nonumber \\ & + & \left.
\bmath{\epsilon}_d^\dagger\cdot(\bmath{\widehat{\bf k}}\times\bmath{\epsilon}_{nn})\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_\gamma F_3+
\bmath{\epsilon}_d^\dagger\cdot(\bmath{\widehat{\bf q}}\times\bmath{\epsilon}_{nn})\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_\gamma F_4\right\}
f(p_2) \nonumber \\
& + & \frac{3C}{\sqrt2}\bmath{\widehat{\bf p}}_2\cdot\bmath{\epsilon}_d^\dagger\left[
\bmath{\widehat{\bf p}}_2\cdot(\bmath{\epsilon}_\gamma\times\bmath{\epsilon}_{nn})F_1
-\left(\bmath{\widehat{\bf p}}_2\cdot\bmath{\widehat{\bf q}}\bmath{\epsilon}_{nn}-\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_{nn}\bmath{\widehat{\bf p}}_2
+\bmath{\widehat{\bf p}}_2\cdot\bmath{\epsilon}_{nn}\bmath{\widehat{\bf q}}\right)\cdot(\bmath{\widehat{\bf k}}\times\bmath{\epsilon}_\gamma)F_2 \right.
\nonumber \\
& + & \left. \bmath{\widehat{\bf p}}_2\cdot(\bmath{\widehat{\bf k}}\times\bmath{\epsilon}_{nn})\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_\gamma F_3
+\bmath{\widehat{\bf p}}_2\cdot(\bmath{\widehat{\bf q}}\times\bmath{\epsilon}_{nn})\bmath{\widehat{\bf q}}\cdot\bmath{\epsilon}_\gamma F_4 \right]
g(p_2)-(2\rightarrow1) ,
\end{eqnarray}
where $\bmath{\epsilon}_{nn}$ is the polarization vector of a spin-1 neutron pair,
\begin{eqnarray}
C & = & \sqrt{4\pi}\Phi_{\pi d}(0)\sqrt{\frac{M_{\pi d}}{\mu}},\\
f(p) & = & \int \, r \, dr\left[u(r)j_0(pr)-{{1}\over{\sqrt 2}}w(r)j_2(pr)
\right],\\
g(p) & = & \int \,r \, dr w(r) j_2(pr).
\end{eqnarray}
The corresponding spin-0 FSI matrix element is easily obtained by the
replacement ${\bf p}_2\to\ki{}$ and letting
\begin{eqnarray}
f(k)&=&\int \, dr \, \left[u(r)j_0\left(\frac{kr}{2}\right)-\frac{1}{\sqrt 2}
w(r)j_2\left(\frac{kr}{2}\right)\right]\tilde{v}_0(r),\\
g(k)&=&\int \, dr \, w(r)j_2\left(\frac{kr}{2}\right)\tilde{v}_0(r).
\end{eqnarray}
The symmetrization ($2\rightarrow1$) is then equivalent to an overall
factor of two in the spin-0 FSI matrix element. Similar expressions
can be derived for the two-body amplitudes and for higher partial
waves.
\subsection{Wave functions}
\label{sec:wf}
It is possible to calculate quite accurate deuteron and
nucleon-nucleon scattering wave functions from the well-established
asymptotic states. By using data extracted from the Nijmegen
phase-shift analysis~\cite{NijmPWA} as well as a one-pion-exchange
potential we ensure that the behavior of the wave function at
$r\agt\frac{1}{m_\pi}$ is correct. This yields wave functions that are
consistent with those obtained from $\chi$PT potentials at leading
order~\cite{weinNN}. In order to be fully consistent with the
$O(Q^3)$, or NNLO, operators we have derived here one should of course
include $O(Q^2)$ corrections to the $NN$ potential, {\textrm{i.e.}}, incorporate
at least the ``leading'' chiral two-pion exchange
(TPE)~\cite{Or96,Ka97,Ep99,Re99}. This will be done in future work.
\subsubsection{The deuteron wave function}
The deuteron wave function at large distances is described by the
asymptotic $S$- and $D$-state wave functions:
\begin{eqnarray}
u^{(0)}(r) & = & A_S\expup{-\gamma r}, \\
w^{(0)}(r) & = & \eta A_S\left(1+\frac{3}{\gamma r}+
\frac{3}{(\gamma r)^2}\right) \expup{-\gamma r},
\end{eqnarray}
where $\gamma=\sqrt{MB}=45.70223(9)$~MeV/$c$ [$B=2.224575(9)$~MeV],
$A_S=0.8845(8)$~fm$^{-1/2}$ is the asymptotic normalization, and
$\eta=0.0253(2)$ the asymptotic $D/S$ ratio~\cite{Nijmd}.
The (un-regulated) deuteron wave functions $u(r)$ and $w(r)$ can be obtained
from the asymptotic ones and the radial Schr\"odinger equation by integrating
in from $r=\infty$~\cite{PC}
\begin{eqnarray}
u(r) & = & u^{(0)}(r)-M\int_r^\infty dr' G_0(r',r)[V_{SS}(r')u(r')+
V_{SD}(r')w(r')], \nonumber \\
w(r) & = & w^{(0)}(r)-M\int_r^\infty dr' G_2(r',r)[V_{DS}(r')u(r')+
V_{DD}(r')w(r')].
\label{eq:inteqd}
\end{eqnarray}
Here, $G_{0/2}(r',r)$ is the $S$- and $D$-wave Green function (propagator) and
$V_{L'L}$ the standard projections of the Yukawa OPE potential:
\begin{eqnarray}
V_{SS} & = & -f^2\frac{\expup{-\mu r}}{r}, \nonumber \\
V_{SD}=V_{DS} & = & -2\sqrt2 f^2
\frac{\expup{-\mu r}}{r} \left(1+\frac{3}{\mu r}+\frac{3}{(\mu r)^2}\right),
\nonumber \\
V_{DD} & = & -f^2\frac{\expup{-\mu r}}{r}
+2 f^2\frac{\expup{-\mu r}}{r}
\left(1+\frac{3}{\mu r}+\frac{3}{(\mu r)^2}\right),
\end{eqnarray}
where $f^2=0.0750(5)$ is the $\pi NN$ coupling constant
squared~\cite{NijmpiNN}.
The coupled integral equation (\ref{eq:inteqd}) is solved using standard
numerical techniques.
The integrated wave functions are divergent at small distances, reflecting that
short-range physics has been ignored.
Instead of trying to model this piece as done in many phenomenological $NN$
potentials, we choose to regulate it by matching with a spherical well solution
at $r=R_d$.
This procedure is motivated by the fundamental EFT hypothesis that results
should not be sensitive to the behavior at small $r$.
If it is, there are some short-distance physics that need to be included in
the calculation, {\textrm{i.e.}}, that the parametrization is incomplete.
This hypothesis can be tested by varying the cut-off $R_d$ over some sensible
range.
A thorough discussion of the boundary between long- and short-distance
physics along these lines can be found in the lectures by
Lepage~\cite{Lepage:1997cs}.
The $D$-wave ($w$) is matched at the boundary $R_d$ by the continuity of the
logarithmic derivative, which determines the depth of the well.
The $S$-wave is matched assuming continuity and that the deuteron wave function
is normalized to unity.
The matching condition is then
\begin{equation}
1-\int_{R_d}^\infty dr u^2(r)-\int_0^\infty dr w^2(r)=\int_0^{R_d} dr u^2(r),
\end{equation}
where the left-hand side is calculated numerically and the right-hand
side analytically.
In Fig.~\ref{fig:wfsd} these wave functions are compared to each other, to
the modern chiral NLO wave function of Epelbaum~{\textit{et al.}}~\cite{Ep99}, and to the
wave function of the Nijm93 potential~\cite{nijmPot}.
Choosing $R_d$ to be in the range 1.5--2~fm gives wave functions that are very
close to the high-precision wave functions on the market.
\begin{figure}[t]
\includegraphics[width=135mm]{wfdplotsBW.eps}
\caption{The $S$- (top panel) and $D$-state (bottom panel) parts of the
deuteron wave functions.
Our wave functions are labeled by the choice of matching radius $R_d$.
These are compared to the NLO chiral wave function of
Ref.~\protect\cite{Ep99} and Nijm93~\protect\cite{nijmPot}.}
\label{fig:wfsd}
\end{figure}
Note that the chiral wave function (thick dashed line in Fig.~\ref{fig:wfsd})
deviates considerably from the potential model wave function (solid line)
between 1.5 and 2.0~fm.
We take this as an indication that short-range $NN$ dynamics are at play even
at distances as large as 2.0~fm.
Thus we vary our matching point $R_d$ (and $R_{nn}$ below) between 1.5 and
2~fm, where the lower limit is set because $R_d$ cannot be reduced much further
without the interaction becoming non-Hermitian.
It is possible to use the asymptotic wave functions (dotted lines in
Fig.~\ref{fig:wfsd}) in the calculation if the matrix element has the
necessary factors of $r$ to cancel the divergences of the wave functions
as $r\to0$.
Such an approach is similar in spirit to the pionless effective field theory
[EFT($\not\!\pi$)] and gives analytic expressions for the matrix element.
\subsubsection{The $nn$ scattering wave function}
The scattering wave function can be calculated in a similar way.
However, the asymptotic state is now described by the phase shift according to
\begin{equation}
\Psi_{\bf -k}({\bf r},{\bf p}) \sim \frac{v^{(0)}_0(r)}{r} =
\frac{\expup{i\delta_0}\sin(pr+\delta_0)}{pr},
\end{equation}
where $\delta_0$ is calculated from Eq.~(\ref{eq:ER}) with given values of $p$,
$a_{nn}$, and $r_0$.
In the limit of vanishing momentum this wave function reduces to $1-a/r$ as it
should.
If higher partial waves can be neglected the only integral equation we need is
the one for the $^1S_0$ channel, which is
\begin{equation}
v_0(r) = v^{(0)}_0(r)-M\int_r^\infty dr'
\widetilde{G}_0(r',r)V_{SS}(r')v_0(r'),
\end{equation}
where $\widetilde{G}_0(r',r)$ is the free $S$-wave two-body propagator.
This wave function is regularized at $r=R_{nn}$ by matching the logarithmic
derivative of a spherical well solution.
The depth of the spherical well is hence energy dependent, a treatment related
to the energy-dependent potential used by Beane~{\textit{et al.}},~\cite{towards}.
As for the deuteron we vary $R_{nn}$ between 1.5 and 2~fm in order to
test our sensitivity to short-range dynamics.
\begin{figure}[t]
\includegraphics*[width=140mm]{nnwfGGSvsGPBW.eps}
\caption{Our $^1S_0$ $nn$ scattering wave functions (thin lines) at
$p=10$~MeV/c and for varying $R_{nn}$ as indicated.
Here $a_{nn}=-16$~fm and $r_0=2.8$~fm.
A comparison is made with the wave functions of the GGS model (thick lines).
The latter are explained in~\protect\cite{GGS}, from which reference the figure
was adapted.
Reprinted figure with permission from
W.~R.~Gibbs, B.~F.~Gibson, and G.~J.~Stephenson, Jr.,
Phys\ Rev.\ C {\bf 11}, 90 (1975).
Copyright (1975) by the American Physical Society.}
\label{fig:nnwf}
\end{figure}
In Fig.~\ref{fig:nnwf} our $nn$ wave functions are plotted together
with the wave functions used by GGS. The wave functions are quite
similar in that they tend to the same asymptotic limit (the curve
labeled Zero Range) for larger $r$. Thus they are very close to each
other for $r\agt1.5$~fm. There are, however, a few important
differences between the $nn$ wave functions in the two calculations:
Firstly, the GGS wave functions have been derived using the Reid
soft-core potential (RSC) (with the old larger value for the $\pi NN$
coupling constant) for the long-range part, while we used one-pion
exchange only.
This explains the slight difference in the size of $v(r)$ at $r=1.4$~fm.
Secondly, GGS match at a fixed value of $R_{nn}=1.4$~fm, while we vary
$R_{nn}$.
Thirdly, our wave function uses a spherical well solution ($\sin\kappa r$)
and GGS assume a polynomial of fifth order, where the magnitude and first
two derivatives vanish at $r=0$ and are matched to the RSC solution at
$r=R_{nn}$.
The assumption of vanishing derivatives is equivalent
to using a hard-core potential at short distances, while many chiral
potentials have a softer behavior for small $r$ (see, {\textrm{e.g.}},~Ref.~\cite{Ep99}).
[The different shapes of the GGS wave functions were obtained by adding an
extra term $\eta r^3(r-R_{nn})^3/pr$ to the short-range piece of their wave
function.]
The combined effect of all this is that our wave function
has most variation around $r=1$~fm, while the GGS wave functions
varies most around $\sim0.7$~fm. As we will see later, these
differences have a strong influence on the assessment of the size
of the theoretical error in the extracted $a_{nn}$.
An obvious improvement of our calculation would be to use wave
functions whose short-range behavior is constrained by other
observables, thus reducing the uncertainty. We could also compare to
wave functions of modern high-precision potentials,
{\textrm{e.g.}},~Refs.~\cite{Wi95,nijmPot}, or the recent N$^3$LO chiral
potentials~\cite{EM03,Ep04}. Another extension would be to include
higher partial waves.
In the actual calculations we subtract off the plane-wave $S$-wave contribution
$j_0(pr)$ from the scattering wave function [$\tilde{v}_0=v_0-rj_0(pr)$] and
then calculate the full plane-wave (QF or PW) contribution using
$\expup{i{\bf p}\cdot\ri{}}$ without partial-wave decomposition.
\section{Results}
\label{sec:results}
\subsection{Convergence}
The calculated differential decay width is shown in Fig.~\ref{fig:KR}
for the LO KR term only and with the NLO and NNLO one- and two-body
amplitudes added in succession.
\begin{figure}[t]
\includegraphics{KRorallBW.eps}
\caption{Time-of-flight distribution for the $\pi^-d\to nn\gamma$ decay rate.
This and all following spectra are calculated assuming $a_{nn}=-18$~fm,
$r_0=2.75$~fm, and $\theta_3=0.075$~rad, unless otherwise indicated.
The plot shows the contributions from the LO KR (dotted line), NLO
one-body (short-dashed line), NNLO one-body (long-dashed line), and NNLO one-
and two-body (solid line) amplitudes.
The two peaks are labeled QF and FSI from their dominant contributions.}
\label{fig:KR}
\end{figure}
The spectrum shows two separate peaks, labeled QF and FSI from the
dominant contributions that give rise to them.
It is clear that the LO curve is very similar to the full
calculation and that the corrections of higher orders mainly affect the
magnitude, but do have some impact on the shape.
The evolution is most easily assessed by forming the QF to FSI peak
ratio at the various orders.
At LO the ratio is 2.58, at NLO 2.59, at NNLO one-body 2.60, and at full NNLO
2.71. Thus the LO, NLO, and NNLO one-body results are very close to
each other, but do not contain the full dynamics that the NNLO
two-body amplitudes provide. For a good quantitative result it is
important to include the full NNLO amplitude---using only one-body
amplitudes would give a wrong answer at this order.
\subsection{Sensitivity to $a_{nn}$}
\label{sec:errorA}
In Fig.~\ref{fig:spectra} the decay rate is plotted for various
choices of $a_{nn}$. The curves have been rescaled to coincide at the
QF peak, to facilitate comparison of the relative height of the QF and
FSI peak. This is done since in the LAMPF experiment~\cite{LAMPF} the
scattering length is extracted by fitting the shape of the spectrum,
not the magnitude of the decay rate.
\begin{figure}[t]
\includegraphics{paper12rescaled.eps}
\caption{Time-of-flight distribution for the $\pi^-d\to nn\gamma$
decay rate. The spectrum is calculated for different choices of
$a_{nn}$ as indicated. The curves for $a_{nn}=-20$ and -16~fm have
been (slightly) rescaled to coincide with the QF peak of the
$a_{nn}=-18$~fm curve.}
\label{fig:spectra}
\end{figure}
Note that only the height of the FSI peak changes, the valley between
the two peaks is largely unaffected by the value of $a_{nn}$.
The theoretical error in the extraction of the scattering length has
several sources and they will be investigated and estimated in the
following paragraphs. The error in the extracted $a_{nn}$ can be
related to the error in the decay rate $\Gamma$ by
\begin{equation}
\frac{\Delta\Gamma}{\Gamma} = \frac{d\Gamma}{da_{nn}}\frac{a_{nn}}{\Gamma}
\frac{\Delta a_{nn}}{a_{nn}},
\end{equation}
where the actual calculations (Fig.~\ref{fig:spectra}) give that
$\frac{d\Gamma}{da_{nn}}\frac{a_{nn}}{\Gamma}=1.21$ at $a_{nn}=-18$~fm.
Thus
\begin{equation}
\frac{\Delta a_{nn}}{a_{nn}}=0.83\frac{\Delta\Gamma}{\Gamma},
\end{equation}
a result we shall use repeatedly in what follows.
\subsection{Theoretical error bar}
We estimate the theoretical error under the assumption that the entire
time-of-flight spectrum is fitted.
To the best of our knowledge, previous work have only considered fitting the
FSI peak, which limits the kinematics~\cite{GGS,deTeramond,Gabioudetal}.
Because of the large relative momentum in the QF region, this extended analysis
will have significant importance for the size of the error.
We will use a nominal value of $a_{nn}=-18$~fm in our estimate of the error.
\subsubsection{Neglected higher orders in the $\pi^- p \rightarrow
\gamma n$ amplitude}
\label{subsec:extrap}
The present calculation ignore pieces of the $\pi^-p\to\gamma n$ amplitude of
$O(Q^4)$ or higher, which is thus three orders down from the leading piece of
$O(Q)$.
One might think that the error would then be of the order
$(\omega/\Delta)^3$, since the first dynamical effects not explicitly
included in our Lagrangian are associated with $\Delta$-isobar
excitation and so the `high'-energy scale is $\Delta$, the
$\Delta$-nucleon mass difference, rather than $\Lambda_{\chi}$. This
is supported by the results of Ref.~\cite{Fearing} where the fitted
counter terms had unnaturally large coefficients when expressed in
units of ${\rm GeV}^{-2}$.
However, in Ref.~\cite{Fearing} the $O(Q^3)$ one-body (single-nucleon
pion photo-production) amplitude was fitted to actual data for
$\omega_0=142$~MeV and higher (roughly 10~MeV above threshold).
The error in our calculation is thus introduced only in our extrapolation
of the amplitude to a subthreshold energy, denoted by $\omega^\ast$.
Compared to the leading $O(Q)$ term, this gives a correction
$(\omega_0^3-{\omega^\ast}^3)/\Delta^3\sim4\%$.
This is a special, very beneficial feature of the pion absorption process:
since the pion momentum is vanishing there is no angular dependence and the
amplitudes depend only on the photon energy.
This error should include the errors due to uncertainties in the LECs fitted
in~\cite{Fearing}.
A simple calculation based on the LEC fit errors and the formulas for the CGLN
amplitudes gives corrections of the order 3\% or smaller, which is in line
with the above 4\%.
Since the extrapolation photon energy is roughly the same at the QF
and the FSI peak, this error should add roughly equally to both peaks,
which reduces the error on the neutron time-of-flight spectrum to less than the
$\sim4\%$ estimated above, since
now only the shape is fitted.
The actual calculations confirm this: the spectrum using extrapolated
amplitudes (see Sec.~\ref{sec:subthreshold}) differs by only 1.1\%
in the FSI peak from the spectrum with amplitudes evaluated at threshold.
The corresponding error in $a_{nn}$ is thus 0.95\% or 0.17~fm for
$a_{nn}=-18$~fm.
\subsubsection{Boost corrections}
The contribution of the boost corrections [Eqs.(\ref{eq:corrF10}) and
(\ref{eq:corrF1m})] is of the order $\mu^2/2M^2\sim1\%$, but the change occurs
in the same direction in both peaks.
Thus the relative change is much reduced ($0.14\%$ in
$\Gamma_{\rm FSI}/\Gamma_{\rm QF}$, {\textrm{i.e.}}, 0.11\% or 0.02~fm in $a_{nn}$) and can
be completely neglected for the present purposes.
After rescaling, the boosted curve cannot be distinguished from the original
one.
We use the calculated $O(Q^3)$ boost correction of 0.14\% as a conservative
estimate of the boost error introduced at higher orders.
The boost correction is included in all plots.
\subsubsection{Off-shellness}
In calculating the one-body amplitudes we tacitly assumed that both
nucleons were on-shell. This introduces an ``off-shellness'' error
that should be estimated. It is well known that field transformations
can be employed to trade dependence of the one-body amplitude on the
``off-shellness'' of the nucleon, $p^0 - \frac{{\bf p}^2}{2 M}$, for a
two-body amplitude~\cite{Haag,FH}. This is done as
follows: using field transformations such as those employed in
Refs.~\cite{FS,Fe} the dependence of the $\gamma n \rightarrow \pi^-p$
amplitude on the nucleon energy $p^0$ is replaced by dependence on
$\frac{{\bf p}^2}{2 M}$ plus terms in ${\cal L}^{(3)}_{\pi N}$ and
beyond. This means that the one-body amplitude for the photoproduction
process now has no ``off-shell ambiguity'', although we do acquire
additional pieces of the two-body amplitude for the charged-pion
photoproduction process. (This argument is shown graphically in the
sequence of diagrams in Fig.~\ref{fig:offshell}.) The new
contribution, depicted in Fig.~\ref{fig:offshell}(c), involves a
$\gamma \pi \pi$ vertex from ${\cal L}_{\pi N}^{(3)}$, and so is
$O(Q^5)$. This two-body effect is thus $p^2/M^2\sim\mu^2/M^2\sim2\%$
down from the NNLO two-body diagrams, which contribute $7.1\%$ to the
rescaled decay rate (according to Fig.~\ref{fig:KR}). Consequently the
error in $a_{nn}$ from any potential ``off-shell ambiguity'' is
approximately $0.02\times0.071\times0.83=0.12\%$ or 0.02~fm.
\begin{figure}
\includegraphics{offshell.eps}
\caption{The off-shell nucleon in (a) can be taken care of by extracting a
meson exchange from the deuteron wave function as in (b).
The off-shell part absorbs the closest propagator and becomes the two-body
diagram (c), splitting off the on-shell amplitude (d).
The cross indicates an on-shell nucleon.}
\label{fig:offshell}
\end{figure}
\subsubsection{$O(Q^4)$ two-body pieces of the amplitude}
A larger effect comes from $O(Q^4)$ two-body pieces of the $\gamma nn
\rightarrow \pi^- d$ amplitude, such as the one depicted in
Fig.~\ref{fig:oq4tb}.
\begin{figure}[t]
\includegraphics{twoOQ4.eps}
\caption{Typical two-body operator at $O(Q^4)$.
The sliced photo-nucleon vertex is from $\mathcal{L}_{\pi N}^{(2)}$.}
\label{fig:oq4tb}
\end{figure}
A naive estimate indicates they should be
$\sim p/\Lambda_\chi \sim \mu/\Lambda_\chi \sim 20$\% of the
$O(Q^3)$ two-body diagrams. This estimate is supported
by studies of pion photoproduction to $O(Q^4)$ in $\chi$PT~\cite{Be97}.
This suggests that $O(Q^4)$ two-body effects are roughly a
0.7\% effect in $a_{nn}$.
\subsubsection{Error from wave functions}
\label{sec:errorwf}
\begin{figure}[t]
\includegraphics[width=140mm]{errornnwfBW.eps}
\caption{Time-of-flight distribution for the $\pi^-d\to nn\gamma$ decay rate.
The spectra are calculated for $a_{nn}=-18$~fm and different
choices of $R_{nn}$ as indicated.
The case of $R_{nn}=2.0$~fm and $a_{nn}=-18.6$~fm is also plotted.}
\label{fig:spectrawf}
\end{figure}
By changing the matching points $R_d$ and $R_{nn}$ between 1.5 and 2.0
fm, we tested the error introduced by our ignorance of short-distance
physics in the $NN$ wave functions. This change was significant for
the $nn$ scattering wave functions, as shown in
Fig.~\ref{fig:spectrawf}. The resulting error in $a_{nn}$
turns out to be $-0.6$~fm (3.3\%) or smaller. A similar spread
was obtained with wave functions calculated from ``high-quality'' $NN$
potentials, {\textrm{e.g.}}~Nijm I and Nijm II~\cite{nijmPot}. We note that both
these potentials have $\chi^2/{\rm d.o.f.}=1.03$ with respect to the
1993 Nijmegen database and identical $nn$ scattering lengths. They
differ only in their treatment of the heavy mesons, indicating that
our calculation is sensitive to truly short-range parts of the $NN$
interaction. Our results with the NijmI and NijmII potentials also
suggest that this short-range sensitivity has a greater impact on the
extracted $a_{nn}$ than does our neglect of two-pion exchange. We are
confident that this uncertainty could be considerably reduced by
finding other observables that constrain the wave function, in
particular its short-range behavior. This will be pursued in future
work. Note that the change in $R_{nn}$ not only changes the height of
the FSI peak, but also the valley region. This feature could
potentially be used in a fit procedure to distinguish the $R_{nn}$
dependence from a change in $a_{nn}$.
Indeed, if one focuses only on the FSI peak then the variation in the
spectrum due to the use of different wave functions is significantly
smaller than the one discussed in the previous paragraph. If we adjust
both calculations to agree in the valley region we find that the FSI
peak height only differs by 0.6\%. (See Fig.~\ref{fig:FSIspectrawf}.)
This corresponds to an uncertainty in $a_{nn}$ of $\pm 0.1$ fm.
\begin{figure}[t]
\includegraphics[width=140mm]{errornnwfFSIBW.eps}
\caption{Time-of-flight distribution for the $\pi^-d\to nn\gamma$ decay rate
in the region of the FSI peak.
The spectra are calculated for $a_{nn}=-18$~fm and different
choices of $R_{nn}$ as indicated.}
\label{fig:FSIspectrawf}
\end{figure}
Meanwhile, effects due to the bound-state wave function chosen are
also small. Changing the deuteron wave function by varying $R_d$ from
2.0~fm to 1.5~fm would alter the extracted $a_{nn}$ by 0.55\% or
0.10 fm. Using the Bonn B deuteron wave function instead of the
EFT-motivated wave function yields a $\Delta a_{nn}$ of 0.56\% or
0.10~fm.
\subsubsection{Higher partial waves}
The error from neglecting higher partial waves in the rescattering wave
function can be estimated in the following way.
The higher partial waves are only substantial for large relative energies and
are thus negligible in the FSI peak region.
In the QF peak, the relative $nn$ momentum is roughly 80~MeV/$c$, which means
that the $S$-wave phase shift is $\delta_0\alt60^\circ$, while the $P$-wave
phase shifts are typically $\delta_1\alt5^\circ$.
The $P$- to $S$-wave amplitude ratio can then be estimated as
$A_1/A_0\sim \sin\delta_1/\sin\delta_0=0.10$.
\begin{figure}[t]
\includegraphics{spinsepBW.eps}
\caption{The decay rate separated into spin-0 and spin-1 contributions.
The subtracted FSI $^1S_0$ partial wave is also distinguished.}
\label{fig:hipw}
\end{figure}
From Fig.~\ref{fig:hipw} the $S$-wave FSI amplitude at the QF peak is
$A_0=\sqrt{0.030}=0.17$ and thus the $P$-wave FSI amplitude is
$A_1\sim0.10A_0=0.017$. However, since the $P$-waves are spin-1 and
the $S$-waves spin-0 and the two do not interfere, the influence of
the $P$-wave should be related to the QF spin-1 amplitude, which is
$B_1=\sqrt{0.295}=0.543$. The error in the calculated QF peak is then
$2|A_1||B_1||\cos\theta|$, where $\theta$ is the unknown phase angle
between $A_1$ and $B_1$. Using the maximal possible error (setting
$\cos\theta=1$) seems overly pessimistic, so we instead choose the
average $\langle|\cos\theta|\rangle=2/\pi$. The relative error at the
QF peak is then $\frac{4}{\pi}|A_1||B_1|/\Gamma_{\rm QF}\sim2.9\%$,
yielding an error in the extracted scattering length of 2.4\% or
0.43~fm. This should be regarded as a conservative estimate of the
error of neglecting $P$-waves for two reasons. Firstly, the $P$-waves
could interfere destructively with each other. And secondly, we
implicitly assume that the radial integral for $P$-waves is of the
same magnitude as for $S$-waves, whereas it is probably smaller.
Most importantly, it is possible to actually calculate and include the
$P$-waves. This error can thus easily be pushed to higher partial
waves and so made substantially smaller. This will be done in future
work.
Note that this $P$-wave error is much larger than the one estimated by
GGS~\cite{GGS}. The reason is that the $P$-waves only contribute at
large relative energy, {\textrm{i.e.}}, under the QF peak, where they can
interfere with the QF amplitudes, thus changing the QF to FSI peak
ratio and the extracted $a_{nn}$. The GGS error estimate assumes that
the $nn$ opening angle is smaller than $30^\circ$, which restricts the
kinematics to the FSI peak region only and thus does not apply to the
entire range of neutron energies used in the LAMPF extraction. Also
in the work of de~T\'eramond~{\textit{et al.}}~\cite{deTeramond} as used by the PSI
group~\cite{Gabioudetal}, only the FSI peak region is fitted, which
gives a small $P$-wave contribution. Thus, as far as we can
ascertain, our analysis is the first that estimates
the interference of the FSI $P$-waves with the spin-1 QF amplitude.
If this effect was not included in the analysis of the data in
Ref.~\cite{LAMPF} then the $\Delta a_{nn}$ of approximately 0.43 fm we
have found here should be included in the theoretical uncertainty
quoted in that work. However, correspondence with one of the authors
of Ref.~\cite{LAMPF} suggests that FSI in $NN$ $P$-waves was included in
the version of the GGS model used for the
$a_{nn}$ extraction there~\cite{Ben}. This source of uncertainty
would then not be present in Ref.~\cite{LAMPF}'s value for $a_{nn}$.
\subsubsection{Sensitivity to $r_0$}
An estimate of a change in $r_0$ due to CSB can be obtained by
assuming that the relative change in $r_0$ is similar to the relative
change in $a_{NN}$.
Thus $\frac{\Delta r_0}{r_0}\approx\frac{\Delta a}{a}$ so that
$\Delta r_0=\frac{r_0}{a}\Delta a=\frac{2.75}{18}1.5=0.23$~fm.
The sensitivity to the effective-range parameter $r_0$ was tested by varying
it away from its nominal value $2.75$~fm, using a conservative spread of
$\pm 0.25$~fm.
This changes the FSI peak by $1.4\%$ (after rescaling to the QF peak) and
thus indicates a change in the extracted $a_{nn}$ of 1.2\% or 0.21~fm.
On the other hand, the error suggested by analysis of different
experimental determinations of $r_0$ is $\pm0.11$~fm~\cite{Slaus}.
If $r_0$ is instead varied over this narrower range the resultant $\Delta
a_{nn}$ is only 0.5\% or 0.09~fm.
We will use the latter, smaller, error in our error budget.
\subsection{Error budget}
The errors are summarized in Table~\ref{tab:errors}.
\begin{table}[t]
\caption{Error budget for the extraction of $a_{nn}$ from the
$\pi^-d\to nn\gamma$ reaction as it was performed in Ref.~\cite{LAMPF}.
The calculation of the absolute errors assumes a scattering length of -18~fm.
The total error is summed in quadrature.}
\label{tab:errors}
\begin{ruledtabular}
\begin{tabular}{rdd}
Source & \multicolumn{1}{c}{Relative error (\%)} &
\multicolumn{1}{c}{Absolute error (fm)} \\
\hline
Off-shell & 0.07 & 0.01 \\
Boost & <0.11 & <0.02 \\
Subthreshold & 0.95 & 0.17 \\
$O(Q^4)$ 2B & 0.7 & 0.12 \\
$r_0$ & 0.5 & 0.09 \\
Dep.\ on $R_d$ & 0.55 & 0.10 \\
$p$-wave in FSI & <2.4 & <0.43 \\
Dep.\ on $R_{nn}$ & <3.3 & <0.60 \\
total & <4.3 & <0.78
\end{tabular}
\end{ruledtabular}
\label{table:errors}
\end{table}
The first four errors are due to uncertainties in the amplitudes,
while the last four are due to the wave functions~\footnote{We realize
that such a separation is, strictly speaking, not meaningful, since
unitary transformations can be employed to trade ``wave-function''
effects for ``operator'' effects. However, the separation makes sense
within the approach to the calculation we have adopted here.}.
We consider the total error of $<4.3\%$ to be a very conservative estimate.
Note that if $a_{nn}$ is extracted only from data in the FSI region
then the last two errors drop to 0.2\% and 0.5\% respectively, while a
number of the other errors listed in Table~\ref{table:errors} are
also reduced. We find that an extraction performed using only data from
this section of the neutron time-of-flight spectrum would have a
theoretical uncertainty of $\pm 0.2$ fm. This confirms the conclusion
of GGS from thirty years ago. The significantly reduced theoretical
uncertainty comes at a price though: one must sacrifice the large
number of counts acquired under the QF peak. We have argued above that
the last two errors quoted in Table~\ref{table:errors} can be
decreased by additional theoretical work on radiative pion-capture on
deuterium, and therefore we hold out hope that in future a $\chi$EFT
extraction of $a_{nn}$ which has an accuracy of $\pm 0.3$~fm (or better)
and uses the full neutron spectrum obtained in Ref.~\cite{LAMPF} can be
performed.
One reason for this optimism is the convergence of the chiral
expansion for this reaction, which can be made more explicit by computing the
QF to FSI peak ratio for the different orders. From Fig.~\ref{fig:KR}
we obtain
\begin{equation}
\frac{\Gamma_{\rm QF}}{\Gamma_{\rm FSI}}=(2.580+0.014+0.112\pm0.039)
(1\pm0.05),
\end{equation}
where the first parenthesis contains (in order) the contribution of
the LO, NLO, NNLO, and the error in the chiral expansion. The second
parenthesis shows the error due to effects in the wave functions.
Note that modifying the wave functions by including two-pion
exchanges, $P$-waves, or different short-distance dynamics would
already change the LO calculation, which is why we choose to write
this error as an overall factor.
The smallness of the NLO and NNLO one-body terms can perhaps be an effect of
the particular kinematics of the present problem, especially that the pion
momentum is vanishing.
On the other hand, the comparatively large NNLO two-body contribution is most
likely a result of a combination of two effects:
Firstly, the two-body currents allow for momentum sharing between the nucleons,
which would be of importance in the QF region.
Secondly, in the leading two-body diagram [Fig.~\ref{fig:two}(a)], the
coulombic propagator was power-counted as $1/\mu^2$.
However, because of the small deuteron binding energy, the typical momentum
is instead of the order $\gamma=\sqrt{MB}=45.7$~MeV~\cite{pid}.
Since $\gamma\ll\mu$ this further enhances this diagram.
\section{Conclusions}
\label{sec:end}
In this paper we have calculated the $\pi^-d\to nn\gamma$ reaction,
using $\chi$PT pion-photon amplitudes and EFT-inspired wave functions.
The errors in the extracted scattering length from the operators are
of the order 1\%. These errors include effects that were not
considered by Gibbs, Gibson, and Stephenson (GGS)~\cite{GGS}, {\textrm{e.g.}},
errors from extrapolating the single-nucleon amplitudes sub-threshold,
the boost of the $\gamma n\to\pi^-p$ amplitude from the $\gamma n$
rest frame to c.m., the effects of off-shell nucleons, and more
complicated two-body mechanisms. A key improvement is that we have
included the full two-body amplitude at third chiral order and have
found that on the scale of the other errors it has a substantial
influence on the extraction of the scattering length.
Nevertheless, if $a_{nn}$ is extracted from the FSI region alone our
analysis within $\chi$EFT confirms GGS's result for the theoretical
uncertainties, putting them at $\pm 0.2$ fm. On the other hand,
if---as was done in the most recent $a_{nn}$
extraction~\cite{LAMPF}---the entire shape of the neutron spectrum,
including both the QF and FSI peaks, is used for the extraction, then
the uncertainty in the scattering wave function at small distances and
the neglect of higher partial waves is a potentially large source of
errors, maybe as large as 4.3\%. This might seem like a large
uncertainty, since it is almost three times larger than the 1.5\%
estimated by GGS. But, as was argued in Sec.~\ref{sec:errorwf}, some
of the assumptions behind their error estimate do not seem to apply
for the entire kinematic range spanned by the data from the LAMPF
experiment. This tempts us to suggest that the error estimate given
in Ref.~\cite{LAMPF} is optimistic and should be increased.
We plan to improve our model in the near future by constraining the
short-distance part of the $nn$ wave function using other observables
and by incorporating higher partial waves. We will report these
results in a future publication. We also plan to fold our model with
the neutron detector acceptance and the experimental geometry in order
to extract the $nn$ scattering length from the data of
Ref.~\cite{LAMPF}.
Overall we conclude that the $\pi^-d\to nn\gamma$ reaction has some
very desirable features that makes it extremely suitable for the
extraction of the neutron-neutron scattering length. The vanishing
pion momentum obviously favors a $\chi$PT calculation and also reduces
the number of contributing terms dramatically, leading to the
dominance of the Kroll-Ruderman term. The fact that the extraction is
done by fitting the shape of time-of-flight spectra rather than an
absolute decay rate reduces many errors further still.
The reaction $\gamma d\to nn\pi^+$ could be used as an alternative and
complementary way to extract the neutron-neutron scattering length.
This reaction has been considered before, see the review \cite{Laget}
and later papers, {\textrm{e.g.}}, \cite{Ulla}.
A chiral calculation should be feasible for this reaction, and could
benefit from the work of the present paper.
With a threshold photon laboratory energy of $149$~MeV it should be
accessible at existing experimental facilities, {\textrm{e.g.}}, HI$\gamma$S@TUNL after
the planned upgrade and MAX-lab in Lund, Sweden.
After the submission of the manuscript, a calculation of $\gamma d\to nn\pi^+$
using chiral perturbation theory along lines similar to ours
has become available~\cite{Lensky:2005hb}.
\begin{acknowledgments}
We are grateful to T.~Hemmert for clarifications regarding the
single-nucleon radiative pion absorption amplitudes, and to
B.~F.~Gibson and C.~Howell for information on details of the
theoretical model used in Ref.~\cite{LAMPF}. A. G. thanks
C.~J.~Horowitz for discussions that led to a better understanding of
the scattering wave functions. This work was supported by the DOE
grants DE-FG02-93ER40756 and DE-FG02-02ER41218.
\end{acknowledgments}
\bibliographystyle{apsrev}
|
1,108,101,566,772 | arxiv | \section{Introduction}
Twisted bilayer graphene (tBG) with a “magic angle” ($\backsim 1.1^{\circ}$) has gained extensive attention since the discovery of gate-tunable unconventional superconductivity and strongly correlated insulating phases, which are due to the presence of ultraflat bands near the Fermi level\cite{cao2018unconventional,cao2018correlated}. Recently, experimental and theoretical investigations have been shown that twisted trilayer graphene (tTLG) has a better tunability of its superconducting phases than the twisted bilayer graphene, which makes it a good platform to study the correlated properties\cite{fischer2021unconventional,guerci2021higher,xie2021tstg,zhu2020twisted,lopez2020electrical,cao2021pauli,phong2021band,hao2021electric}. Plenty of degrees of freedom, for instance, the twist angle, stacking configurations, external electric field, and interlayer separation, are available to tune the electronic properties of the tTLG. For example, the interlayer coupling strength can be precisely controlled by the external vertical pressure.
Previous studies of the pressure effect are mainly focus on bilayer cases. It has been studied experimentally that the hydrostatic pressure can tune the interlayer coupling and hence the band structure of graphene moir\'{e} superlattices\cite{yankowitz2018dynamic,gao2020band}. Interestingly, for twisted bilayer graphene with a moderate twist angle that shows relatively weak correlation under ambient pressure, an appropriate hydrostatic pressure induces robust insulating phases and superconductivity with higher $T_c$ than that in zero-pressure magic-angle case\cite{feldman2019squeezing,yankowitz2019tuning}. Theoretically, vertical pressure can be used to achieve the ultraflat bands in twisted bilayer graphene with arbitrary twist angle\cite{carr2018pressure,padhi2019pressure,chittari2018pressure,lin2020pressure,ge2021emerging}. Consequently, when studying the strong correlation, we can reduce the impact of structural inhomogeneity by using a moir\'{e} pattern with a short wavelength. However, the vertical pressure effects on the electronic properties of twisted trilayer graphene have not been explored yet. Furthermore, it remains unclear if the flat band tuned by the vertical pressure has similarly peculiar properties as that of the tTLG with zero-pressure magic angle.
Recently, plasmons were detected by utilizing a scattering-type scanning near-field optical microscope (s-SNOM) at tBG with $1.35^\circ$\cite{plasexp2019cao}. Different from the monolayer graphene which displays damped plasmons, tBG with magic angle has collective modes that are damping free. The flat band plasmon modes are ultraflat over the whole wave vector, and with the energy determined by the band width of the flat bands. Such different electronic response to various band widths can be used to identify the magic angle in samples that far beyond the ability of the first-principles methods. Moreover, it has been theoretically predicted that unconventional superconductivity in tBG is mediated by the purely collective electronic modes\cite{sharma2020superconductivity,lewandowski2021pairing}. A deep understanding of the collective excitations in flat band materials, for instance, the tTLG with zero-pressure magic angle and with pressure-induced ``magic angle'', may shed light on the plasmonic superconductivity. Up to now, flat bands are detected in several graphene moir\'{e} superlattices, for example, tBG with magic angle\cite{cao2018unconventional,cao2018correlated}, tBG under moderate pressure\cite{yankowitz2018dynamic,gao2020band}, tTLG with magic angle\cite{fischer2021unconventional,guerci2021higher,xie2021tstg,zhu2020twisted,lopez2020electrical,cao2021pauli,phong2021band,hao2021electric}, trilayer graphene boron-nitride moir\'{e} superlattices\cite{chittari2019gate}, and so on. Natural questions are whether these flat band materials display similar plasmon excitations and the collective modes have similar mechanisms. All in all, a systematic investigation of the collective plasmon modes in graphene moir\'{e} superlattices with flat bands is demanding.
In this paper, we study the pressure effects on the electronic properties of tTLG by means of a full tight-binding (TB) model. We find that an experimental accessible vertical pressure (with the value almost half of that in the tBG case) can push a large twist angle system to reach the flat-band regime\cite{carr2018pressure,feldman2019squeezing,xia2021strong}. Then, the plasmonic properties of the flat band materials are investigated by utilizing the Lindhard function. We observe distinct collective plasmon modes in the tTLG with zero-pressure and pressure-induced magic angles. The outline of the paper is as follows: In Sec. \ref{method}, the TB model and the computational methods are introduced, then followed by the response of the band width and band gap of tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ to the vertical pressure. In sec. \ref{plasm}, we compare the plasmonic properties of the flat band twisted multilayer graphene, in particular, of tTLG-A$\mathrm{\mathrm {\tilde{A}}}$A with zero-pressure magic angle and pressure-induced magic angle. Finally, we give a summary and discussion of our work.
\section{Numerical methods}
\label{method}
The moir\'{e} supercell of twisted trilayer graphene is constructed by identifying a common periodicity between the three layers\cite{shi2020large}. Generally, the electronic properties of the tTLG vary with different stacking configurations\cite{wu2021lattice}. In this paper, we only focus on a mirror-symmetric structure, the so-called tTLG-A$\mathrm{\mathrm {\tilde{A}}}$A, which starts with a AAA stacking ($\theta = 0^\circ$) and with the middle layer twisted an angle $\theta$ with respect to both the top and bottom layers. The rotation origin is chosen at an atom site. As shown in Fig. \ref{band_dos}(a), the supercell is composed of various high-symmetry stacking patterns, that is, the AAA, ABA, and BAB stackings. Moreover, the lattice relaxation (both the out-of-plane and in-plane) is also considered by utilizing the classical simulation package LAMMPS in all calculations\cite{plimpton1995fast}. The intralayer and interlayer interactions in twisted trilayer graphene are simulated with the LCBOP\cite{los2003intrinsic} and Kolmogorov-Crespi potential\cite{kolmogorov2005registry}, respectively.
The electronic properties of the tTLG are obtained by using a full tight-binding model based on $p_{z}$ orbitals. The Hamiltonian of the system has the form\cite{wu2021lattice}:\\
\begin{equation}
H=\displaystyle\sum_{i}\epsilon_i|i\rangle\langle i|+\displaystyle\sum_{\langle i,j\rangle}t_{ij}|i\rangle \langle j|,
\label{ham}
\end{equation}
where $|i\rangle$ is the $p_z$ orbital located at $\mathbf r_i$, $\epsilon_i$ is the on-site energy of orbital $i$, and $\langle i,j\rangle$ is the sum over indices $i$ and $j$ with $i\neq j$.
The hopping integral $t_{ij}$, interaction between sites i and j, is:
\begin{equation}
t_{ij} = n^2 V_{pp\sigma}(r_{ij}) +(1-n^2)V_{pp\pi}(r_{ij}).
\end{equation}
Here $r_{ij}=|\mathbf r_{ij}|$ is the distance between two sites located at $\mathbf r_i$ and $\mathbf r_j$, n is the direction cosine of $\mathbf r_{ij}$ along the direction $\mathbf e_{z}$ that perpendicular to the graphene layer. The Slater and Koster parameters $V_{pp\sigma}$ and $V_{pp\pi}$ are expressed as distance-dependent functions\cite{shi2020large}:
\begin{eqnarray}
V_{pp\pi}(r_{ij}) = -\gamma_{0}e^{2.218(b_0-r_{ij})}F_{c}( r_{ij} ),\nonumber \\
V_{pp\sigma}(r_{ij}) = \gamma_{1}e^{2.218(h_0-r_{ij})}F_{c}(r_{ij}),
\label{KS}
\end{eqnarray}
where $b_0=1.42$ \AA\; and $h_0=0.335$ \AA\; represent the nearest carbon-carbon distance and interlayer distance in equilibrium, respectively. The intralayer and interlayer hopping parameters $\gamma_{0}$ = 3.2 eV and $\gamma_{1}$ = 0.48 eV are used in all calculations. $F_{c}(r)=(1 + e^{(r-0.265)/5})^{-1}$ is a smooth function. All the hoppings with $r_{ij}\leq 8.0$ \AA \; are considered in the calculations.
To calculate the electronic properties of the tTLG under a vertical pressure, we extend the hopping parameters in Eq. (\ref{KS}). It has been proven that the vertical compression has a negligible influence on the intralayer interactions, whereas significantly modify the interlayer hoppings\cite{carr2018pressure,lin2020pressure}. Therefore, we only modify the interlayer hopping term $V_{pp\sigma}$ as\cite{lin2020pressure}:
\begin{equation}
V_{pp\sigma}(r_{ij}) = \gamma_{1}e^{2.218(h-r_{ij})}e^{-(h-h_{0})/\lambda^{\prime}}F_{c}(r_{ij}),
\end{equation}
where $\lambda^{\prime} = 0.58$ \AA, h is the out-of-plane projection of $r_{ij}$. The evolution of the lattice constants with the vertical pressure in twisted multilayer graphene has been theoretically investigated, which has the expression as\cite{carr2018pressure,yu2020pressure,gao2020band}:
\begin{equation}
\mathrm{Pressure} = A\cdot(e^{B(1 - h/h_0)}-1),
\label{press}
\end{equation}
with $A = 5.73$ GPa and $B = 9.54$. Here, $\delta=1-h/h_0$ is the compression.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{figure1_bs_dos.pdf}
\caption{(a) The upper panel shows the top view of the atomic configuration of tTLG-A$\mathrm {\mathrm {\tilde{A}}}$A-$6.01^{\circ}$. High-symmetry stacking regions of AAA, ABA, BAB are marked by black, red and blue circles, respectively. The lower panel shows a schematic representation of these high-symmetry stacking patterns. The number 6.01 stands for the twist angle $\theta=6.01^\circ$. (b) Band structure and density of states of relaxed tTLG-A$\mathrm {\tilde{A}}$A-$1.35^{\circ}$ with no pressure. (c), (d) Band structure and density of states of relaxed tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ without pressure and with 4 GPa vertical pressure, respectively.}
\label{band_dos}
\end{figure}
By direct diagonalization of the Hamiltonian in Eq. (\ref{ham}), we calculate the band structure of tTLG with different twist angles. Moreover, we use the tight-binding propagation method in the frame of a full TB model to calculate the density of states (DOS) as\cite{yuan2010modeling}:
\begin{equation}\label{dos}
D(E)=\frac{1}{2\pi N}\displaystyle\sum_{p=1}^{N}\int_{-\infty}^{\infty}e^{iEt}\langle\varphi_p(0)|e^{-Ht}|\varphi_p(0)\rangle dt,
\end{equation}
where $|\varphi_p(0)\rangle$ is one initial state with the random superposition of basis states at all sites $N$.
To investigate the plasmonic properties of the twisted trilayer graphene, we obtain firstly the dynamical polarization by using the Lindhard function in a full TB model as\cite{kuang2021collective,yuan2011excitation}:
\begin{eqnarray}
\Pi (\mathbf q,\omega) &&=-\frac{g_{s}}{(2\pi)^{2} }\int_{BZ} d^{2}\mathbf k\sum_{l,l^{\prime}}\frac{n_{F}(E_{\mathbf kl})-n_{F}(E_{\mathbf k^{\prime}l^{\prime}})}{E_{\mathbf kl}-E_{\mathbf k^{\prime}l^{\prime}}+\hbar\omega+i\delta} \nonumber \\
&&\quad\times|\langle \mathbf k^{\prime}l^{\prime}\mid e^{i\mathbf{q}\cdot\mathbf{r} } \mid \mathbf kl \rangle|^{2}.
\end{eqnarray}
Here, $n_F(H)=\frac{1}{e^{\beta (H-\mu)}+1}$ is the Fermi-Dirac distribution operator, $\beta = \frac{1}{k_BT}$ being $T$ the temperature and $k_B$ the Boltzmann constant, and $\mu$ is the chemical potential; $|\mathbf{k}l \rangle$ and $E_{\mathbf{k}l}$ are the eigenstates and eigenvalues of the TB Hamiltonian in Eq. (\ref{ham}), respectively, with $\mathit{l}$ being the band index, $\mathbf{k^{'}}$=$\mathbf{k}$+$\mathbf{q}$, $\delta \rightarrow 0^+$, the integral is taken over the whole Brillouin zone (BZ). Then, based on the random phase approximation (RPA), the dielectric function is given by the formula\cite{slotman2018plasmon,jin2015screening,yuan2011excitation}:
\begin{equation}
\varepsilon(\mathbf q,\omega) = 1-V(q)\Pi(\mathbf q,\omega)
\end{equation}
where $V(q)= \frac{2\pi e^{2}}{k\mid q \mid}$ is the Fourier component of the two-dimensional Coulomb interaction, and $\kappa$ is the background dielectric constant. We set $\kappa = 3.03$ to simulate the hexagonal boron nitride substrate environment in our calculations. Finally, the energy loss function has the form:
\begin{equation}
S(\mathbf q,\omega) = -\mathrm{Im}(1/\varepsilon(\mathbf q,\omega))
\label{loss}
\end{equation}
In principle, undamped plasmons with frequency $\omega_p$ exist if both $\mathrm{Re}\;\varepsilon(\mathbf q,\omega_p)=0$ and the loss function $S(\mathbf q, \omega)$ is peaked around $\omega_p$ with width $\gamma \ll \omega_p$. The loss function can be directly measured by the s-SNOM in the experiment. As a consequence, we will mainly focus on the calculation of the loss function in the paper.
\section{Evolution of bands in twisted trilayer graphene by pressure}
\begin{figure}[t]
\centering
\includegraphics[width=0.46\textwidth]{figure_3.pdf}
\caption{(a) The band width and band gap of tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ versus the vertical pressure for the rigid and relaxed cases, respectively. The red and blue triangular symbols stand for the band width and band gap in relaxed case, respectively. The red and blue star symbols stand for the band width and band gap in the rigid case, respectively. (b) Pressure-induced magic angle as a function of the critical compression. The blue dashed line is for the tTLG case and the red dashed line is for the tBG case that is extracted from Ref. \cite{carr2018pressure}.}
\label{band_gap_width}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figure_2.pdf}
\caption{The electronic properties of relaxed tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ under ambient and high pressures. (a) and (c) The layer-projected weights of band eigenstates of tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ under ambient and 4 GPa vertical pressures, respectively. The thickness of the lines represent the weight of band eigenstates. (b), (d) The band structure and inverse participation ratio (IPR) of the tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ under ambient and 4 GPa external pressure, respectively. (e), (f) Calculated local density of states (LDOS) mappings of Van Hove singularities near the Fermi level of tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ without pressure and with 4 GPa pressure.}
\label{OW_IPR}
\end{figure*}
Fig. \ref{band_dos}(b) shows the band structure and density of states of relaxed tTLG-A$\mathrm {\tilde{A}}$A with twist angle $\theta=1.35^{\circ}$ under ambient pressure. The band gap (energy difference between the valence band edge and its higher energy band at the $\Gamma$ point of the Brillouin zone) is about 55.4 meV and the band width (the energy difference between the $\Gamma$ and K points of the valence band edge) is about 10 meV. Similar to the tBG case, the states of four nearly flat bands around the Fermi level show strong localization at the AAA stacking region. One significant difference between the tTLG and tBG is the coexistence of the flat bands with a Dirac cone close to one another only in the mirror-symmetric tTLG-A$\mathrm {\tilde{A}}$A\cite{carr2020ultraheavy}. The relative energy of the Dirac cone with respective to the flat bands is sensitive to the computational parameters of the TB model\cite{wu2021lattice,carr2020ultraheavy}. Here, the Dirac cone is below the flat bands about 18.8 meV. Theoretically, one definition of the ``magic angle'' is the angle where the Fermi velocity at the K and K' points of the BZ vanishes. Another definition is those which lead to the narrowest bands\cite{tarnopolsky2019origin}. In the relaxed tTLG-A$\mathrm {\tilde{A}}$A cases, the narrowest bands appear in sample with $\theta=1.35^{\circ}$--the so-called zero-pressure magic angle. When the twist angle increase to $\theta=1.89^{\circ}$, as illustrated in Fig. \ref{band_dos}(c), the band width is significantly enlarged and the band gap has a value of 48 meV due to the reduced interlayer interactions\cite{yan2012angle}. Moreover, two van Hove singularities flank the Dirac point. Two different sets of linear dispersion bands with different Fermi velocities located at the K point of the BZ. One preserved the monolayer band has Fermi velocity around $9.35*10^{5}$ m/s, and the other has reduced Fermi velocity around $1.53*10^{5}$ m/s due to the interlayer interaction. When applying a vertical pressure with the value of 4 GPa, four nearly flat bands appear near the Fermi level, which can be attributed to pressure-enhanced interlayer correlations. The distortion of the flat bands is different from that of the relaxed tTLG-A$\mathrm {\tilde{A}}$A with zero-pressure magic angle, and the band gap has an obvious decrease. Note that we simulate the pressure by changing the height between two monolayers according to the expression in Eq. (\ref{press}) and the atomic relaxation by allowing the atoms to fully relax in all cases.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figure_4_plasmon.pdf}
\caption{Plasmonic properties of relaxed tTLG-A$\mathrm {\tilde{A}}$A. The figure is organized in columns. In each column, the upper panel shows the loss function ($-\mathrm{Im}(1/\varepsilon)$) and the lower panel shows the imaginary part of the frequency-dependent dynamic polarization function $-\mathrm{Im}(\Pi(\mathbf q,\omega))$. (a) Result for tTLG-A$\mathrm {\tilde{A}}$A-$1.35^{\circ}$ under ambient pressure. (b) and (c) Results for tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ under vertical pressure of 4 GPa and under ambient pressure, respectively. The wave vectors is along $\Gamma$ to M in the first Brillouin zone. The temperature is 1 K and chemical potential is $\mu = 0$.}
\label{plasmon}
\end{figure*}
Obviously, similar to the method of precisely controlling the twist angle, pressure is an efficient way of tuning the tTLG-A$\mathrm {\tilde{A}}$A into the magic regime. Next, we investigate how the band structures evolve with the external vertical pressure. As shown in Fig. \ref{band_gap_width}(a), the band gap and band width vary nonmonotonically with the pressure, and such tendency is similar to that of the tBG case\cite{lin2020pressure}. In the rigid sample, the band gap is zero when the pressure is higher than 3 GPa. The band width decreases linearly with the pressure growing up to 2.5 GPa, and then increases linearly after the pressure go beyond the turning point 2.5 GPa, whereas the band gap remains unchanged with the pressure higher than 2.5 GPa. In the relaxed sample, the critical pressure is around 4 GPa, where both the band gap and band width reach their minimum values. That is, for the tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ under 4 GPa vertical pressure, flat-band regime is achieved. Such value of pressure can be achievable experimentally. Recent experimental progress that make use of a hydrostatic pressure allowed to continually tune the interlayer separation in van der Waals heterostructures with pressure up to 2.3 GPa\cite{yankowitz2018dynamic}. Higher pressure would be achievable with diamond anvil cells. By assuming that the interlayer coupling strength has quadratic dependence on compression and neglecting the momentum scattering that the twist angle introduces, we can write the critical value $\theta_c(\delta)$ of the magic angle as a function of compression $\delta$ as\cite{carr2018pressure}:
\begin{equation}
\theta_c(\delta)=\theta_0[(t_2/t_0)\delta^2-(t_1/t_0)\delta+1],
\end{equation}
Here $\theta_0=1.35^\circ$ is the magic angle under ambient pressure, and the numerical parameters are $t_{[0,1,2]}=[1.117, 2.466, 192.496]$. From the results in Fig. \ref{band_gap_width}(b), it is obvious that the pressure needed to induce the flat band in tTLG is lower than that in the tBG case. Furthermore, the pressure-induced magic angle $\theta_c=3^\circ$ could be achievable when apply a pressure around 10 GPa, where no significant reconstruction appears.
To understand the pressure effect, we compare the electronic properties of tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ with ambient and critical pressures. We calculate the layer-projection weights of band structure of tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ under ambient and moderate pressures in Fig. \ref{OW_IPR}(a) and (c), separately. Firstly, let us focus on the conduction and valence band edges. The middle layer has $50\%$ weight in tTLG-A$\mathrm{\tilde{A}}$A-$1.89^{\circ}$ under zero and critical pressures. The weights are always identical in top and bottom layers, which means that the mirror symmetry is still maintained under pressure. Next, we investigate the localization of the states in tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ with and without critical pressures. The inverse participation ratio (IPR), which is defined as $\sum_{i=1}^N|a_i|^2/(N\sum_{i=1}^N|a_i|^4)$, where $a_i$ is the state at site $i$ and $N$ is the total number of sites, are shown in Fig. \ref{OW_IPR}(b) and (d). After applying a pressure to the tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ sample, the IPR in the conduction and valence band edges change from 0.5 to 0.1, which means the states become more localized. Our LDOS mappings in Fig. \ref{OW_IPR} (e), (f) (the high-symmetry stacking regions are marked by AAA, ABA, BAB in Fig. \ref{OW_IPR}(e)) further clarify the charge concentration process. After applying a vertical pressure, the charges in the AAA of the top and bottom layer gathering to the AAA center, while in the protected middle layer, charges from the ABA and BAB concentrating to the center of AAA. The charge distribution for the tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ under critical pressure in Fig. \ref{OW_IPR} (f) is almost the same as magic angle tTLG-A$\mathrm {\tilde{A}}$A-$1.35^{\circ}$ under ambient pressure\cite{wu2021lattice}. All in all, the vertical pressure has similar effect to modify the electronic properties of tTLG as that of tuning the twist angle.
\section{Plasmonic properties of twisted trilayer graphene with magic angles}
\label{plasm}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figure_5_eps.pdf}
\caption{(a) The real part of the frequency-dependent dielectric function of the relaxed tTLG-A$\mathrm {\tilde{A}}$A-$1.35^{\circ}$ under ambient pressure. (b) and (c) The real part of the frequency-dependent dielectric function of relaxed tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ under 4 GPa and ambient vertical pressures, respectively. The red dashed line indicates the zero of the real part of the dielectric function.}
\label{epsilon}
\end{figure*}
In the previous part, we show that pressure can trigger the appearance of flat bands in tTLG with twist angles larger than the zero-pressure magic angle $\theta=1.35^\circ$. One question arises: will the pressure-induced flat bands show similarly peculiar properties as that of the zero-pressure flat bands? To answer the question, we investigate the plasmon mode in tTLG with magic angles. Generally, when a plasmon mode with frequency $\omega_p$ exists, the electron energy loss spectra possesses a sharp peak at frequency $\omega=\omega_p$. The loss function can be obtained theoretically by using the Eq. (\ref{loss}). For the relaxed tTLG-A$\mathrm {\tilde{A}}$A$-1.35^{\circ}$, as shown in Fig. \ref{plasmon}(a), several collective plasmon modes with energies between 0.01 eV and 0.15 eV appear. Similar to the tBG case\cite{kuang2021collective}, a plasmon mode with energy around 0.15 eV is attributed to the interband transitions from the valence band near the Fermi level to the conduction bands located at the energy around 0.15 eV. Two collective modes appear in the low energy range. One has an energy of 0.01 eV and stretches to large q. Such plasmon comes from both the interband and intraband transitions of flat bands, and has a weak dependence of the wave vectors, which is due to the interband transition between the flat bands and higher bands\cite{pnasplas2019intrinsically}. The interband transition between the flat bands is suppressed by the interband polarization of flat bands with higher energy bands. This effect is more significant in the second mode with the energy around 0.027 eV, which only appears in small q. The second plasmon is contributed only by the interband transition of flat bands. Such interband polarization is suppressed in the large q range (results not shown here). From the imaginary part of the dynamic polarization functions ($-\mathrm{Im}(\Pi(q,\omega))$) plotted in the bottom panel of Fig. \ref{plasmon}(a), we can see clearly that these plasmon modes are free from Landau damping. The excitons would not exchange energy with other collective excitations nor have single particle-hole transitions, which means the plasmon mode near 0.15 eV is a long-lived plasmon mode.
For the tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ under critical pressure, a collective plasmon mode locates at energy 0.15 eV, and from the dynamic polarization functions ($-\mathrm{Im}(\Pi(q,\omega))$) in the bottom panel, we can see this kind of plasmon behave the same as the magic angle tTLG-A$\mathrm {\tilde{A}}A-1.35^{\circ}$ under ambient pressure. However, only one plasmon mode appear in the low energy range, which may due to the different shape of the flat bands. Such flat-band plasmon stretches to large wave vector $\bm{q}$, whereas it disperses within particle-hole continuum, as shown in the bottom panel of Fig. \ref{plasmon}(b). This is due to the reduction of the band gap in the pressure-induced magic angle tTLG. For the tTLG-A$\mathrm {\tilde{A}}$A$-1.89^{\circ}$ under ambient pressure, a collective plasmon mode exists around 0.05 eV, and this plasmon mode is a long-lived plasmon mode. The tunable plasmons in Fig. \ref{plasmon} strengthen the finding that a sample with relative larger twist angle can be pushed to the flat-band regime by applying a vertical pressure, which could be justified in experiments by s-SNOM\cite{plasexp2019cao}, electron energy loss spectroscopy\cite{EELS2011electron}.
In principle, there are two different ways to identify the plasmon mode with frequency $\omega_p$. One is the energy where the peak of the loss functions located, which can be seen clearly from the loss functions ($-\mathrm{Im}(1/\varepsilon)$) in the top panel of Fig. \ref{plasmon}; Another way is via identifying the frequencies at which $\mathrm{Re}(\varepsilon(q,\omega))=0$\cite{yuan2011excitation,jin2015screening,slotman2018plasmon}. In Fig. \ref{epsilon}, we plot the frequency-dependent real part of the dielectric functions with varied wave vectors. For the tTLG-A$\mathrm {\tilde{A}}$A-$1.35^{\circ}$ under ambient pressure, the real part of the dielectric functions cross 0 at the energy around 0.15 eV. For the low energy one, there exist dips in the real part of the dielectric functions, and these dips with varied wave vectors approach but never cross the zero dashed line. That means the flat-band plasmon is not a genuine plasmon\cite{nano2016plas}. Such low energy plasmon mode can be tuned to a damping free one by external factors, for instance, an external electric field. Similarly, for the tTLG-A$\mathrm {\tilde{A}}$A$-1.89^\circ$ under critical pressure, the high-energy plasmon mode is free of damping and the low energy flat-band mode damps into particle-hole continuum. The vertical pressure shift the energy of the undamped plasmon in tTLG-A$\mathrm {\tilde{A}}$A$-1.89^\circ$ from 0.05 eV to 0.15 eV. Based on the dielectric properties of the tTLG-A$\mathrm {\tilde{A}}$A with and without pressure, we found that the vertical pressure can push a larger twist angle to reach the flat-band regime, and make their dielectric properties similar to the magic angle tTLG-A$\mathrm {\tilde{A}}$A$-1.35^{\circ}$ under ambient pressure. Furthermore, pressure-induced plasmon mode has a blue shift. That is, a collective of plasmon mode with different energies can be realized continuously by vertical pressure.
\section{CONCLUSION}
We have systematically investigated the evolution of the band widths and band gaps of the tTLG-A$\mathrm {\tilde{A}}$A with an external pressure. The electronic properties are obtained by employing a full tight-binding model, and the relaxation effects have been taken into account by using the LAMMPS package to fully relax the sample. When applying a vertical pressure with the value around 4 GPa, tTLG-A$\mathrm {\tilde{A}}$A-$1.89^{\circ}$ reach the flat-band regime, both the band gap and band width approach their minimum values. Based on the layer-projected band structure and the LDOS mapping, we found that the appearance of the pressure-induced flat bands is due to the charge concentration in each layer as a result of the enhanced interlayer correlations. The dielectric and plasmonic properties further strengthen our finding that a relatively larger twist angle can be pushed to reach the flat-band regime by vertical pressure. Two plasmonic modes are predicted in tTLG with zero-pressure and pressure-induced magic angles. For the high energy long-lived plasmon, pressure-induced high energy plasmon mode is almost the same as that with zero-pressure magic angle. However, the low energy plasmon mode has obvious divergence, which is probably due to the different shapes of the flat bands in these two kinds of magic angle samples. Recent theory predicts that unconventional superconductivity in TBG is mediated by the purely collective electronic modes\cite{sharma2020superconductivity,lewandowski2021pairing}. This may provide a platform to justify the prediction. Furthermore, we may observe a much higher superconducting $T_c$ in the tTLG with large pressure-induced magic angle\cite{carr2020ultraheavy}. Last but not least, zero-energy high-order van Hove singularity (VHS) has recently emerged as a fascinating playground to study correlated and exotic superconducting phases\cite{yuan2019magic,bi2019designing,guerci2021higher}. Such high-order VHS can be achieved by tuning the band structure with a single parameter in moir\'{e} superlattice, for instance, the twist angle, external pressure, heterostrain and external electric field. It will be worth to explore if a high-order VHS could be induced in tTLG by applying a vertical pressure, which will be our future work.
\section{ACKNOWLEDGEMENTS}
This work was supported by the National Natural Science Foundation of China (Grants No.11774269 and No.12047543), the National Key R\&D Program of China (Grant No. 2018FYA0305800), and the Natural Science Foundation of Hubei Province, China (2020CFA041). Numerical calculations presented in this paper were performed on the supercomputing system in the Supercomputing Center of Wuhan University.\\
|
1,108,101,566,773 | arxiv | \section{Introduction}
Random walks, and the associated continuous-time Brownian motion (BM),
are ubiquitous in nature. As such, they are not only
the cornerstones of statistical physics \cite{chandrasekhar, feller, hughes} but have also found many
applications in a variety of areas such as biology \cite{koshland},
computer science \cite{asmussen, satya_functionals}
and finance \cite{williams}. Continuous time Brownian motion is simply
defined by the equation of motion
\begin{eqnarray}\label{def_BM}
x(0) = 0 \;, \; \frac{\rmd x(t)}{\rmd t} = \eta(t) \;,
\end{eqnarray}
where $\eta(t)$ is a Gaussian white noise of zero mean $\langle
\eta(t) \rangle = 0$ and short range correlations $\langle \eta(t)
\eta(t')\rangle = 2 D \delta(t-t')$ where $D$ is the diffusion
constant (in the following we set $D = 1$). An interesting variant
of Brownian motion in a given time interval $[0,T]$ is the so called
Brownian bridge $x_B(t)$ which is a Brownian motion conditioned to start and end at zero, {\it i.e.} $x_B(T) = x_B(0) = 0$. Here we
focus on two interesting observables associated with this bridge, namely
\begin{itemize}
\item{the distribution of the area $A$ under the bridge (see Fig. \ref{fig_1} a))
\begin{eqnarray}\label{def_A}
A = \int_0^T x_B(t)\, \rmd t \;,
\end{eqnarray}
which is obviously a random variable, being the sum of (strongly
correlated) random variables. For the Brownian bridge, the distribution of $A$ can easily be computed
using the fact that $x_B(t)$ is a Gaussian random variable. This
can be seen from the well known identity in law \cite{feller}
\begin{eqnarray}\label{id_bb}
x_B(t) := x(t) - \frac{t}{T} x(T) \;,
\end{eqnarray}
where $x(t)$ is a standard Brownian motion (\ref{def_BM}).
For the Brownian bridge, $A$ is thus also a centered Gaussian random variable. A direct
computation of the second moment $\langle A^2 \rangle$ yields
straightforwardly
\begin{eqnarray}\label{dist_A_bb}
P_B(A,T) = \sqrt{\frac{3}{\pi T^3}} \exp{\left(-\frac{3 A^2}{T^3}
\right)} \;.
\end{eqnarray}
}
\item{the average shape of the bridge, $\langle \tilde x_B(t)\rangle$ for
a fixed area $A$ (see Fig. \ref{fig_1} b)). For the Brownian bridge, it takes a simple form \cite{rivasseau}
\begin{eqnarray}\label{shape_bb}
\langle \tilde x_B(t)\rangle = \frac{A}{T} f\left(\frac{t}{T} \right)
\;, \; f(x) = 6x(1-x) \;.
\end{eqnarray}
}
\end{itemize}
\begin{figure}[ht]
\begin{center}
\includegraphics [width = \linewidth]{Fig_Intro.pdf}
\end{center}
\caption{{\bf a)}: The blue line is the trajectory of a Brownian bridge $x_B(t)$ on $[0, T]$, $\it x_B(0)=x_B(T)=0$ and in red is the area $A$ under this Brownian bridge. {\bf b)}: The typical trajectory of a Brownian bridge $\tilde x_B(t)$ (in blue) with a fixed area $A$ (in red) on the interval $[0,T]$. The solid black line is the
average profile $\langle \tilde x_B(t) \rangle$ as given by Eq. (\ref{shape_bb}).}\label{fig_1}
\end{figure}
The distribution of the area under a Brownian bridge (\ref{dist_A_bb}) is a standard result and its extension to various constrained Brownian motions has recently
attracted much attention \cite{satya_airy,kearney,schehr_airy,welinder,janson_review,rajabpour,rambeau_airy}. For instance, the distribution of the area under a Brownian excursion ({\it i.e.} a Brownian motion conditioned
to start and end at $0$ and constrained to stay positive in-between), the so called Airy-distribution, describes the statistics of
the maximal relative height of one-dimensional elastic interfaces \cite{satya_airy, schehr_airy, rambeau_airy}. Another example is the area $A$
under a Brownian motion till its first-passage time $t_f$ \cite{kearney}, which has an interesting application to the description of the avalanches in the directed Abelian sandpile model proposed in Ref. \cite{dhar}, such that $t_f$ relates to the avalanche duration and $A$ to the size of the avalanche cluster. Related quantities were recently studied in the statistics of avalanches near the depinning transition of elastic manifolds in random media \cite{pld}. On the other hand, the average shape of random walk bridges with a fixed area $A$ (\ref{shape_bb}) has been studied some time ago in the context of wetting \cite{rivasseau} to prove the validity of the Wulff construction in $1+1$ dimensions and more recently in the context of mass transport
models \cite{waclaw}. In these models, where the transport rules depend on the environment of the departure site, the steady state has a pair-factorized form \cite{satya_condensation}, which generalizes the factorized steady states found in simpler system like the zero range process \cite{evans_review, godreche_review,satya_review_condensation}. As the mass density crosses some critical
value, the system exhibits a condensation transition which is governed by interactions, which in turn give rise to a spatially extended condensate. It was shown in Ref. \cite{waclaw} that the shape of this condensate can be described by the average shape of a random walk bridge with fixed area, and the results
of Ref.~\cite{rivasseau} were recovered.
While these quantities are well understood for a Brownian bridge, much less is known for the case of a L\'evy bridge. The aim of the present paper is to compute the distribution of the area and the average shape for a fixed area $A$ in that case. To
this purpose, it is convenient to consider a random walk $x(m)$, in discrete time (see Fig. \ref{fig_rw} a)),
starting at $x(0)=x_0$ at time $0$ and evolving according to
\begin{eqnarray}\label{rw}
x(m) = x(m-1) + \eta(m) \;, \
\end{eqnarray}
where $\eta(m)$ are independent and identically distributed (i.i.d.) random variables distributed according to a common distribution $\phi(\eta)$. Here we focus on the
case where $\phi(\eta) = {\cal S}_{\alpha}(\eta)$ where ${\cal S}_{\alpha}(\eta)$ is a symmetric $\alpha$-stable (L\'evy) distribution.
Its characteristic function is given by $\int_{-\infty}^\infty {\cal
S}_\alpha(\eta) e^{i k \eta} \rmd \eta = e^{- |k|^\alpha}$. In
particular, for large $\eta$, the distribution of $\eta$ has a power
law tail $\phi(\eta) \sim \eta^{-(1+\alpha)}$, with $0< \alpha < 2$. The
L\'evy bridge $x_B(m)$, on the interval $[0,n]$, is a L\'evy random
walk conditioned to start and end at $0$, {\it i.e.} $x_B(n)=x_B(0)=0$. In the following we will compute the distribution $P_B(A,n)$ of the area $A$ under the bridge of length $n$, {\it i.e.} $A = \sum_{m=0}^n x_B(m)$ and the average shape of a bridge $\langle \tilde x_B(m) \rangle$, with $0 \leq m \leq n$, for fixed $A$. Here we consider the natural scaling limit where $x_B \sim n^{1/\alpha}$, while $A \sim n^{1+1/\alpha}$, whereas the aforementioned previous works \cite{rivasseau, waclaw} focused on a different scaling limit which, for Brownian motion, corresponds to $x_B \sim \sqrt{n}$ and $A \sim n^2$. Note that, even in this natural scaling limit, the identity in law valid for the Brownian bridge (\ref{id_bb}) does
not hold for a L\'evy bridge \cite{knight, bertoin} and one thus expects the distribution of the area $A$ to be non trivial. Our results can be summarized as follows:
\begin{itemize}
\item {for a free L\'evy random walk starting at $x_0 = 0$, one finds that the distribution of the area $P(A,n)$ takes the form
\vspace*{0.3cm}
\hspace*{4cm}\fbox{
\begin{minipage}[c]{0.5\textwidth}
\begin{eqnarray}
\hspace*{-2cm} P(A,n) = \frac{1}{\gamma_n} {\cal S}_{\alpha}\left(\frac{A}{\gamma_n} \right)\;, \; \gamma_n = \left(\sum_{m=1}^n m^\alpha \right)^{1/\alpha} \;, \nonumber
\end{eqnarray}
\end{minipage}
}\hfill
\begin{minipage}[c]{0.1\textwidth}
\centering
\begin{equation}
\hspace*{-10cm}
\end{equation}
\end{minipage}
\vspace*{0.3cm}
where ${\cal S}_\alpha(x)$ is a symmetric $\alpha$-stable distribution (\ref{def_stable}). For a L\'evy bridge, one finds, in the scaling limit $A \to \infty$, $n \to \infty$, keeping $A/n^{1+1/\alpha}$ fixed, that the distribution of the area $P_B(A,n)$ takes the scaling form
\hspace*{4cm}\fbox{
\begin{minipage}[c]{0.4\textwidth}
\begin{eqnarray}
\hspace*{-2cm} P_B(A,n) \sim \frac{1}{n^{1+1/\alpha}} F_{\alpha}\left(\frac{A}{n^{1+1/\alpha}} \right)\;, \nonumber
\end{eqnarray}
\end{minipage}
}\hfill
\begin{minipage}[c]{0.1\textwidth}
\centering
\begin{equation}
\hspace*{-10cm}
\end{equation}
\end{minipage}
\vspace*{0.3cm}
where $F_\alpha(y)$ is a monotonically decreasing function, with asymptotic behaviors
\vspace*{0.3cm}
\hspace*{4cm}\fbox{
\begin{minipage}[c]{0.4\textwidth}
\begin{eqnarray}
\hspace*{-2cm} F_\alpha(Y) \sim
\cases{
F_\alpha(0) \;, \; Y \to 0 \;, \\
\frac{a_\alpha}{Y^{2(1+\alpha)}} \;, \; Y \to \infty \;,
}\nonumber
\end{eqnarray}
\end{minipage}
}\hfill
\begin{minipage}[c]{0.1\textwidth}
\centering
\begin{equation}
\hspace*{-10cm}
\end{equation}
\end{minipage}
\vspace*{0.3cm}
where $F_\alpha(0)$, see Eq. (\ref{expr_constant}), and $a_\alpha$, see Eq. (\ref{large_y}), are computable constants. For $\alpha=1$, we obtain an explicit expression for $F_1(Y)$ in terms of elementary functions (\ref{elem}).}
\item{on the other hand, in the aforementioned scaling limit, one
obtains the average profile $\langle \tilde x_B(m) \rangle$ for a L\'evy
bridge as well as the average profile $\langle \tilde x(m) \rangle$ for a
free L\'evy walk with a fixed area $A$. For $\langle \tilde x(m) \rangle$ one obtains a farely simple expression
\vspace*{0.3cm}
\hspace*{4cm}\fbox{
\begin{minipage}[c]{0.4\textwidth}
\begin{eqnarray}
\hspace*{-2cm} \langle \tilde x(m) \rangle = \frac{A}{n} \frac{\alpha+1}{\alpha} \left[1-\left(1-\frac{m}{n} \right)^\alpha \right] \;. \nonumber
\end{eqnarray}
\end{minipage}
}\hfill
\begin{minipage}[c]{0.1\textwidth}
\centering
\begin{equation}
\hspace*{-10cm}
\end{equation}
\end{minipage}
\vspace*{0.3cm}
For a L\'evy bridge, the expression is more involved. For generic $\alpha$ one finds the scaling form
\vspace*{0.3cm}
\hspace*{4cm}\fbox{
\begin{minipage}[c]{0.4\textwidth}
\begin{eqnarray}
\hspace*{-2cm} \langle \tilde x_B(m) \rangle = n^{1/\alpha} H_\alpha\left(\frac{m}{n},\frac{A}{n^{1+1/\alpha}} \right) \;, \nonumber
\end{eqnarray}
\end{minipage}
}\hfill
\begin{minipage}[c]{0.1\textwidth}
\centering
\begin{equation}
\hspace*{-10cm}
\end{equation}
\end{minipage}
\vspace*{0.3cm}
which, in general, has a non-trivial dependence on $A$. One recovers a linear dependence in $A$ (as for the Brownian bridge in Eq. (\ref{shape_bb})) only in the limits $A \to 0$ (\ref{small_y_shape}) and $A \to \infty$ (\ref{large_y_shape}).
}
\end{itemize}
The paper is organized as follows. In section 2, we compute the joint distribution of the position and the area under a L\'evy random walk. In section 3, we use these results to compute the distribution of the area $A$ under a L\'evy bridge of size $n$ while in section 4, we use them to compute the average profile of a L\'evy walk
with a fixed area. Finally, in section 5 we present a numerical method, based on a Monte-Carlo algorithm, to compute numerically $P_B(A,n)$ and $\langle \tilde x_B(m)\rangle$ before we conclude in section 6. Some technical (and useful) details have been left in Appendices A,B and C.
\section{Free L\'evy walk : joint distribution of the position and the area}
We start with the computation of the joint distribution $P(x,A,m|x_0,x_0,0)$ of the position and the area after $m$ steps given that $x(0)=x_0$ (see Fig. \ref{fig_rw} a)). If we denote by $A(m)$ the area under the random walk after $m$ time steps, this random variable evolves according to the equation
\begin{eqnarray}\label{area}
&&A(0) = x_0 \;, \\
&&A(m) = A(m-1) + x(m) \;.
\end{eqnarray}
Therefore $P(x,A,m|x_0,x_0,0)$ satisfies the following recursion relation:
\begin{eqnarray}\label{recurrence}
&&P(x,A,0|x_0,x_0,0) = \delta(x-x_0) \delta(A-x_0) \;, \nonumber \\
&&P(x,A,m|x_0,x_0,0) = \int_{-\infty}^\infty P(x-\eta, A-x,m-1|x_0,x_0,0) \phi(\eta) \rmd \eta \;.
\end{eqnarray}
Introducing $\hat \phi(k) = \int_{-\infty}^\infty \phi(\eta) e^{i k \eta} \rmd \eta$ the Fourier transform of $\phi(\eta)$, and thus
$\hat \phi(k) = e^{-|k|^\alpha}$ for a L\'evy random walk, and $\hat P(k_1, k_2,m|x_0,x_0,0)$ the double Fourier transform of $P(x,A,m|x_0,x_0,0)$
with respect to both $x$ and $A$, {\it i.e.} $\hat P(k_1, k_2,m|x_0,x_0,0) = \int_{-\infty}^\infty \rmd x \int_{-\infty}^\infty \rmd A P(x,A,m|x_0,x_0,0) e^{i k_1 x + i k_2 A}$ the
recursion relation (\ref{recurrence}) reads
\begin{eqnarray}\label{recurrence_fourier}
&& \hat P(k_1,k_2,0|x_0,x_0,0) = e^{i (k_1+k_2) x_0 } \;, \\
&& \hat P(k_1, k_2,m|x_0,x_0,0) = \hat \phi(k_1 + k_2) \hat P(k_1+k_2,k_2,m-1|x_0,x_0,0) \;,
\end{eqnarray}
which can be solved, yielding
\begin{eqnarray}\label{expr_Fourier_joint}
\hat P(k_1, k_2,n|x_0,x_0,0) = \prod_{m=1}^n \hat \phi(k_1 + m k_2) e^{i (k_1 + (n+1)k_2) x_0} \;.
\end{eqnarray}
Hence for a L\'evy walk of index $\alpha$ one has simply
\begin{eqnarray}\label{start_expr}
\fl P(x,A,n|x_0,x_0,0) = \int_{-\infty}^\infty \frac{\rmd k_1}{2 \pi} \int_{-\infty}^\infty \frac{\rmd k_2}{2 \pi} e^{- \sum_{m=1}^n |k_1 + m k_2|^\alpha}e^{- ik_1 (x-x_0)} e^{-ik_2 (A-(n+1)x_0)} \;.
\end{eqnarray}
Note that this expression (\ref{start_expr}) can also be obtained directly by noticing that $x(n) = x_0 + \sum_{m=1}^n \eta(i)$ and thus $A(n) - (n+1)x_0 = \sum_{m=1}^n x(m) = \sum_{m=1}^n \sum_{l=1}^m \eta(l) = \sum_{m=1}^n m \eta(n+1-m)$ such that
\begin{eqnarray}\label{direct_joint}
\fl && P(x,A,n|x_0,x_0,0) = \prod_{m=1}^n \int_{-\infty}^\infty \rmd
\eta(m) \prod_{m=1}^n \phi\left[\eta(m) \right]\delta\left(x-x_0-\sum_{m=1}^n
\eta(m)\right) \nonumber \\
&& \times \delta\left(A - (n+1)x_0 - \sum_{m=1}^n m \eta(n+1-m)\right) \;.
\end{eqnarray}
After a double Fourier transform with respect to $x$ and $A$, this
Eq. (\ref{direct_joint}) yields immediately the expression $\hat
P(k_1, k_2,n|x_0,x_0,0)$ in Eq. (\ref{expr_Fourier_joint}). Of course
the marginal distribution of the position $P(x,n|x_0,0)$ and of
the area $P(A,n|x_0,0)$ are also stable laws. Indeed one has
\begin{eqnarray}\label{marginals}
\fl && P(x,n|x_0,0) = \int_{-\infty}^\infty P(x,A,n|x_0,x_0,0) {\rm d} A = \frac{1}{n^{1/\alpha}} {\cal S}_\alpha \left( \frac{x-x_0}{n^{1/\alpha}} \right) \;, \\
\fl && P(A,n|x_0,0) = \int_{-\infty}^\infty P(x,A,n|x_0,x_0,0) {\rm d} x = \frac{1}{\gamma_n} {\cal S}_\alpha \left( \frac{A-(n+1)x_0}{\gamma_n} \right) \;, \nonumber \\
\fl && \hspace*{1.6cm}\gamma_n = \left( \sum_{m=1}^n m^\alpha \right)^{1/\alpha} \sim \frac{n^{1+1/\alpha}}{(\alpha+1)^{1/\alpha}} \;, \; n \gg 1 \;, \nonumber
\end{eqnarray}
where
\begin{eqnarray}\label{def_stable}
{\cal S}_\alpha(x) = \int_{-\infty}^\infty \, e^{-|k|^\alpha - i k x} \frac{\rmd k}{2 \pi} \;.
\end{eqnarray}
For example, ${\cal S}_1(x)$ is the Cauchy distribution while ${\cal S}_2(x)$ is a Gaussian distribution :
\begin{eqnarray}\label{stable_explicit}
{\cal S}_1(x) = \frac{1}{\pi} \frac{1}{1+x^2} \;, \; {\cal S}_2(x) = \frac{1}{2 \sqrt{\pi}} e^{-\frac{x^2}{4}} \;.
\end{eqnarray}
Note also the explicit expression
\begin{eqnarray}\label{salpha_zero}
{\cal S}_\alpha(0) = \int_{-\infty}^\infty \, e^{-|k|^\alpha} \frac{\rmd k}{2 \pi} = \frac{\Gamma(1+\alpha^{-1})}{\pi} \;,
\end{eqnarray}
which will be useful in the following.
\begin{figure}
\centering
\includegraphics[width = \linewidth]{figure_rw.pdf}
\caption{{\bf a)}: The area $A$ under a random walk in discrete time. {\bf b)}: Illustration of the method, see Eq. (\ref{start_free}), to compute the probability $\tilde P(x,m|A,n)$ that the position of the random walker, starting in $x(0)=0$ is $x$ after $m$ time steps, given that the area, after $n$ time steps, is fixed to $A$. The light area corresponds to $A_1$, the area under the walk over the interval $[0,m]$ while the shaded area corresponds to the area $A-A_1$, the area under the walk over the interval $[m,n]$.}\label{fig_rw}
\end{figure}
We now want to study $P(x,A,n|x_0,x_0,0)$ in the limit of large $n$. The marginal distributions in Eq. (\ref{marginals}) suggest the scaling $x \sim n^{1/\alpha}$ and $A \sim n^{1/\alpha + 1}$. From the expression in Eq. (\ref{start_expr}) one checks explicitly that in the limit $n \to \infty$, keeping $X = x/n^{1/\alpha}$ and $Y = A/n^{1/\alpha+1}$ fixed, the joint distribution takes the scaling form
\begin{eqnarray}\label{scaling_form}
P(x,A,n|x_0,x_0,0) = \frac{1}{n^{2/\alpha+1}} G\left(
\frac{x}{n^{1/\alpha}}, \frac{A}{n^{1/\alpha+1}} \bigg |
\frac{x_0}{n^{1/\alpha}} \right) \;,
\end{eqnarray}
where the function $G(X,Y|X_0)$ is given by
\begin{eqnarray}
\fl G(X,Y|X_0) &=& \int_{-\infty}^\infty \frac{\rmd k_1}{2 \pi} \int_{-\infty}^\infty \frac{\rmd k_2}{2 \pi} e^{-\int_0^1 |k_1 + k_2 z |^\alpha dz - i k_1 (X-X_0) - i k_2 (Y-X_0) } \;.
\end{eqnarray}
After the change of variable $k_2 = k$ and $k_1 = k r$, we obtain
\begin{eqnarray} \label{expr_scaling}
\fl G(X,Y|X_0) &=& \int_{-\infty}^\infty \frac{\rmd r}{2\pi} \int_{-\infty}^\infty\frac{\rmd k}{2\pi} |k| e^{-|k|^\alpha \gamma(r) - i k r (X-X_0 ) - i k (Y - X_0)}
\;, \; \gamma(r) = \int_0^1 |r+z|^\alpha \rmd z \;,
\end{eqnarray}
where the function $\gamma(r)$ is explicitly given by
\begin{eqnarray}\label{expr_gamma}
\gamma(r) =
\cases{
\frac{1}{\alpha+1} \left( (-r)^{\alpha+1} - (-1-r)^{\alpha+1} \right) \;, \; r < -1 \;, \\
\frac{1}{\alpha+1} \left((r+1)^{\alpha+1} + (-r)^{\alpha+1} \right) \;, \; - 1 \leq r \leq 0 \;, \\
\frac{1}{\alpha+1}\left((r+1)^{\alpha+1} - r^{\alpha+1} \right) \;, \; r > 0 \;.
}
\end{eqnarray}
For $\alpha=2$, one has $\gamma(r) = r^2+r+1/3$ and one finds
\begin{eqnarray}
G(X,Y|X_0) = \frac{\sqrt{3}}{2 \pi} \exp{\left[- 3(Y-X_0)(Y-X) - (X-X_0)^2 \right]} \;,
\end{eqnarray}
which yields back the propagator of the so-called random acceleration process \cite{rap}.
\section{L\'evy bridge}
In the absence of any constraint for the walker, the area under a L\'evy random walk is the sum of L\'evy random variables
and is thus again a L\'evy random variable. However, if one considers constrained L\'evy walks, this is not true anymore
and the area may become different from a simple L\'evy random variable. In the first subsection, we compute the distribution of the position for a L\'evy bridge while the second subsection is devoted to the distribution of the area under this bridge.
\subsection{Distribution of the position}
Here we study the L\'evy bridge $\{ x_B(m) \}_{0\leq m \leq n}$ which starts at $0$ at time $0$, $x_B(0) =0$, and is constrained to come back to $0$ after $n$ time steps, {\it i.e.} $x_B(n) =0$. In that case, one can compute the distribution of the position $P_B(x,m)$ after $m$ time steps for such a bridge as $P_B(x,m) = P(x,m|0,0) P(x,n-m|0,0)/P(0,n|0,0)$, such that in the scaling limit one has
\begin{eqnarray}\label{marginal_bridge1}
&& P_B(x,m) = \frac{1}{n^{1/\alpha}} G_B\left(\frac{x}{n^{1/\alpha}},\frac{m}{n} \right) \;, \; \nonumber \\
&& G_B(X,\tau) = \frac{\pi}{\Gamma(1+\alpha^{-1})}\frac{1}{(\tau(1-\tau))^{1/\alpha}} {\cal S}_\alpha \left(\frac{X}{\tau^{1/\alpha}} \right) {\cal S}_\alpha \left( \frac{X}{(1-\tau)^{1/\alpha}}\right) \;,
\end{eqnarray}
where we have used ${\cal S}_\alpha(0) = \Gamma(1+\alpha^{-1})/\pi$, see Eq. (\ref{salpha_zero}). For $\alpha=2$ it is easy to see from Eq. (\ref{marginal_bridge1}) that $x_B(m)$ is a Gaussian variable. However, for $\alpha < 2$, the L\'evy bridge is not any more a L\'evy random variable. For instance, for $\alpha = 1$ one obtains a non trivial distribution
\begin{eqnarray}\label{marginal_bridge2}
G_B(X,\tau) = \frac{1}{\pi} \frac{\tau(1-\tau)}{(\tau^2 + X^2)((1-\tau)^2+X^2)} \;.
\end{eqnarray}
It is also easy to see that, for any $\alpha < 2$ one has the asymptotic behavior
\begin{eqnarray}\label{asympt_bridge}
G_B(X,\tau) \sim c'_\alpha \tau(1-\tau) X^{-2(\alpha+1)} \;, \; X \gg 1 \;,
\end{eqnarray}
where $c'_\alpha$ is independent of $\tau$, which implies that $\langle (x_B(m))^2 \rangle$ is well defined for $\alpha > 1/2$ (of course $\langle x_B(m) \rangle = 0$ by symmetry for all $\alpha$). A straightforward calculation shows that
\begin{eqnarray}\label{variance_bridge}
\frac{\langle x^2_B(m) \rangle}{n^{2/\alpha}} = \tilde a_\alpha \frac{m}{n}\left(1-\frac{m}{n} \right) \;, \; \tilde a_\alpha = \frac{\alpha \Gamma(2-\alpha^{-1})}{\Gamma(1+\alpha^{-1})} \;.
\end{eqnarray}
It is interesting to notice that ${\langle x^2_B(m) \rangle}/{n^{2/\alpha}}$ depends on $\alpha$ only through the amplitude $\tilde a_\alpha$ but the parabolic shape in $(m/n)(1-m/n)$ holds for all values of $2 \geq \alpha > 1/2$. Besides $\tilde a_\alpha$ is diverging for $\alpha \to 1/2^+$ while one has ${\tilde a}(1) = 1$ and ${\tilde a}(2) = 2$. One finds, curiously, that it reaches a minimum for a non-trivial value $\alpha^* = 0.74122 \dots$ for which ${\tilde a}(\alpha^*) = 0.85264 \dots$. In view of these properties (\ref{marginal_bridge1}, \ref{marginal_bridge2}) one expects that, for $\alpha < 2$, the area under such a L\'evy bridge has a non trivial distribution, which we now focus on.
\subsection{Distribution of the area}
In this subsection, we consider a L\'evy bridge, {\it i.e.} a L\'evy walk which starts at the origin $x_0=0$ and is conditioned to come back to the origin
after $n$ steps and we ask : what is the distribution ${P}_B(A,n)$ of the area $A$ under this {\it L\'evy} bridge ? One
can obtain ${P}_B(A,n)$ from Eqs (\ref{start_expr}, \ref{marginals}) as
\begin{eqnarray}
{P}_B (A,n) = \frac{P(0,A,n|0,0,0)}{P(x=0,n|0,0)} \;.
\end{eqnarray}
Therefore, using $P(x=0,n|0,0) = n^{-1/\alpha} {\cal S}_\alpha(0) = n^{-1/\alpha} \Gamma(1+\alpha^{-1})/\pi$, see Eq. (\ref{salpha_zero}), together with the scaling form (\ref{scaling_form}, \ref{expr_scaling}) one obtains, in the limit $n \to \infty$, keeping $A/n^{1+1/\alpha}$ fixed:
\begin{eqnarray}\label{expr_scaling_genalpha}
{P}_B (A,n) = \frac{1}{n^{1+1/\alpha}}
{F}_\alpha\left(\frac{A}{n^{1+1/\alpha}} \right) \;, \nonumber \\
{F}_\alpha(Y) =\frac{1}{2\Gamma(1+\alpha^{-1})} \int_{-\infty}^\infty \, \rmd r \int_{-\infty}^\infty \frac{\rmd k}{2 \pi} |k| e^{-|k|^\alpha \gamma(r) - i Y k} \;.
\end{eqnarray}
Using the explicit expression of $\gamma(r)$ above (\ref{expr_gamma}), one computes the Fourier transform $\hat {F}_\alpha(k)$ of ${F}_\alpha(Y)$ as
\begin{eqnarray}\label{expr_fourier_gen_alpha}
&&\hat {F}_\alpha(k) = \int_{-\infty}^\infty {F}_\alpha(Y) e^{i k Y} \rmd Y \;, \\
&& = \frac{|k|}{\Gamma(1+\frac{1}{\alpha})} \left[ \int_0^\infty e^{-\frac{|k|^\alpha}{\alpha+1} \left[(r+1)^{\alpha+1}-r^{\alpha+1}\right]} \rmd r + \int_0^{1/2} e^{-\frac{|k|^\alpha}{\alpha+1} \left[(1/2+r)^{\alpha+1}+(1/2-r)^{\alpha+1}\right]} \rmd r
\right] \;. \nonumber
\end{eqnarray}
For generic $\alpha$, it seems quite difficult to perform explicitly the integrals over $r$ and $k$ in the expression for the distribution ${F}_\alpha(Y)$ in Eq. (\ref{expr_scaling_genalpha}). One can however extract from this expression the asymptotic behaviors both for $Y \to 0$ and $Y \to \infty$.
{\bf Asymptotic behavior for small argument.} For small argument, it is straightforward to see on the expression (\ref{expr_scaling_genalpha}) above that the leading behavior of $F_\alpha(Y)$ when $Y \to 0$ is given by
\begin{eqnarray}\label{expr_constant}
F_\alpha(Y) \sim F_\alpha(0) \;, \; Y \to 0 \;, {\rm with} \; \nonumber \\
\fl F_\alpha(0) = \frac{(\alpha+1)^{\frac{2}{\alpha}}}{2 \pi} \frac{\Gamma(1+\frac{2}{\alpha})}{\Gamma(1+\frac{1}{\alpha})} \left( \int_0^\infty \frac{\rmd r}{((r+1)^{\alpha+1} - r^{\alpha+1})^{\frac{2}{\alpha}}} + \int_0^{1/2} \frac{\rmd r}{((1/2+r)^{\alpha+1} + (1/2-r)^{\alpha+1} )^{\frac{2}{\alpha}}}\right) \nonumber \;.
\end{eqnarray}
A study of this function $F_\alpha(0)$ shows that it is a decreasing function of $\alpha$ on the interval $]0,2]$, which is diverging when $\alpha \to 0$. For $\alpha = 1$ and $\alpha =2 $, $F_\alpha(0)$ assumes simple values
\begin{eqnarray}
F_1(0) = 1 + \frac{4}{\pi} = 2.27324... \;, \; F_2(0) = \sqrt{\frac{3}{\pi}} = 0.977205...
\end{eqnarray}
{\bf Asymptotic behavior for large argument.} The analysis of the large argument behavior of $F_\alpha(Y)$ is more involved. A careful analysis, left in \ref{app_asympt}, shows that for large $Y$ one has:
\begin{eqnarray}\label{large_y}
&& F_\alpha(Y) \sim \frac{a_\alpha}{Y^{2(1+\alpha)}} \;, \; Y \gg 1 \;, \\
&& a_\alpha = \frac{2^{-2(2+\alpha)} \sqrt{\pi} \Gamma(2+2\alpha) \tan{(\alpha \pi/2)}}{\Gamma(2+\alpha^{-1}) \Gamma(1-\alpha) \Gamma(\frac{5}{2} + \alpha)} \;.
\end{eqnarray}
When $\alpha \to 2$, one has from (\ref{large_y}), $a_{\alpha} \sim \frac{\sqrt{\pi}}{21}(\alpha-2)^2$. From Eq. (\ref{large_y}) one obtains that the second moment of the distribution $\langle Y^2 \rangle$ is defined only for $\alpha > 1/2$ where it takes the value (see Eq. (\ref{expr_appendix}))
\begin{eqnarray}\label{area_variance}
\langle Y^2 \rangle = \frac{{\tilde a}_\alpha}{12} = \frac{\alpha \Gamma(2-\alpha^{-1})}{12 \Gamma(1+\alpha^{-1})} \;, \; \alpha > 1/2 \;,
\end{eqnarray}
where the amplitude ${\tilde a}_\alpha$ appears in the expression for $\langle x^2_B(m) \rangle$ computed above (\ref{variance_bridge}). This power law tail of the area distribution (\ref{large_y}) with an exponent $2(1+\alpha)$ is quite interesting. Indeed, the area itself is the sum of non-identical and strongly correlated variables $x_B(m)$ all having a similar power law tail also with exponent $2(1+\alpha)$ (\ref{asympt_bridge}). For $\alpha > 1/2$, their variance is finite and the non-Gaussianity
of $A$ can a priori be due both to the correlations between the $x_B(m)$'s and to the fact that $A$ is the sum of non-identical random variables. To test which of these features is responsible for the non-Gaussianity of $A$, we study the sum of $n$ random variables $X_m$ which are {\it independent} and such that $X_m$ has the same distribution as $x_B(m)$. Defining $S_n = \sum_{m=1}^n X_m$ and $\Sigma_n^2 = \sum_{m=1}^n \langle X_m^2 \rangle$, it is known that $S_n/\Sigma_n$ converges to a centered Gaussian variable of unit variance if the following condition (known as the Lindeberg's condition) is satisfied \cite{feller}
\begin{eqnarray}\label{lindeberg}
\lim_{n \to \infty} \frac{1}{\Sigma_n^2} \int_{|x| > \epsilon \Sigma_n} x^2 {\rm Proba}(X_m = x) \rmd x = 0 \;, \; \forall \epsilon > 0 \;.
\end{eqnarray}
Intuitively, this Lindeberg condition (\ref{lindeberg}) ensures that the probability that any term $X_m$ will be of the same order of magnitude as the sum $S_n$ must tend to zero (see \ref{appendix_lindeberg} for an example of non-identical independent random variables which do not satisfy the Lindeberg condition). In the present case of the L\'evy bridge, one can check (see \ref{appendix_lindeberg}) that if $X_m$ is distributed like $x_B(m)$ then the above Lindeberg condition (\ref{lindeberg}) is satisfied (see \ref{appendix_lindeberg}). Therefore the deviations from Gaussianity (\ref{large_y}) are purely due to the {\it strong correlations} between the positions of the walker $x_B(m)$'s.
{\bf The special case $\alpha = 1$}. In the Cauchy case, $\alpha=1$, the integral over $k$ can be done in Eq. (\ref{expr_scaling}) to obtain
\begin{eqnarray}
G(X,Y|0) = \int_{-\infty}^\infty \frac{\gamma^2(r) - (r X + Y)^2}{\left(\gamma^2(r) + (r X + Y)^2\right)^2} \frac{\rmd r}{2 \pi^2} \;,
\end{eqnarray}
where the function $\gamma(r)$ (\ref{expr_gamma}) takes here a rather simple form
\begin{eqnarray}
\gamma(r) =
\cases{ - r - \frac{1}{2} \;, \; r < -1 \\
r^2+r + \frac{1}{2} \;, \; - 1 \leq r \leq 0 \\
r + \frac{1}{2} \;, r > 0 \;.
}
\end{eqnarray}
The distribution of the area under a Cauchy bridge is thus given by
\begin{eqnarray}\label{cauchy_integral}
&& {P}_B (A,n) = \frac{1}{n^{2}} F_1 \left(\frac{A}{n^{2}} \right) \;, \nonumber \\
&& F_1 (Y) = \frac{1}{\pi}\frac{2}{1+4Y^2} + \frac{1}{\pi} \int_0^{1/2} \frac{(u^2+1/4)^2 - Y^2}{((u^2+1/4)^2+Y^2)^2} \rmd u \;.
\end{eqnarray}
Under this form (\ref{cauchy_integral}), one can easily obtain the asymptotic behaviors as
\begin{eqnarray}
F_1(Y) \sim
\cases{
1 + \frac{4}{\pi} \;, \; Y \to 0 \\
\frac{1}{20 \pi Y^4} \;, \; Y \to \infty
}
\end{eqnarray}
in agreement with the asymptotic behaviors obtained above (\ref{expr_constant}, \ref{large_y}). In fact the integral over $u$ in the expression
above (\ref{cauchy_integral}) can be done explicitly yielding the expression
\begin{eqnarray}\label{expr_arctan}
&& F_1(Y) = \frac{1}{\pi}\frac{2}{1+4Y^2} \\
&&+ \frac{1}{\pi} \left( \frac{2(1-8Y^2)}{(1+4Y^2)(1+16Y^2)} + \frac{4}{(1+16Y^2)^{\frac{3}{2}}} {\rm Re}\left[{(1- 4 i Y)^{\frac{3}{2}} \arctan{\left((1+4 i Y)^{-\frac{1}{2}}\right)}}\right] \right) \;, \nonumber
\end{eqnarray}
where ${\rm Re}{(z)}$ denotes the real part of the complex number $z$. In \ref{appendix_elementary} we show how this expression~(\ref{expr_arctan})
can be written explicitly in terms of elementary functions (\ref{def_ab}, \ref{def_lm}, \ref{elem}).
\section{Average profile for fixed area $A$.}
\subsection{The case of a free L\'evy walk}
We first consider the case of a free L\'evy walk of constrained
area. We compute the probability $\tilde P(x,m | A,n)$ that the
position of the random walker, starting at $x(0)=x_0=0$ at time $0$, is $x$
after $m$ time steps given that
the area, after $n$ time steps, is fixed to $A$. From this
probability, one obtains the average profile as $\langle \tilde x(m)
\rangle = \int_{-\infty}^\infty x \tilde P(x,m|A,n) \rmd x$. To compute this
probability $\tilde P(x,m|A,n)$, we divide the interval $[0,n]$ into two intervals $[0,m]$ and $[m,n]$. Over
$[0,m]$ the process starts at $x_0 = 0$ with area $A_0=0$ and reaches to $x$ with area $A_1$ (see the light area on Fig. \ref{fig_rw} b)). Over the interval
$[m,n]$, the process starts in $x$ and reaches to $x_F$ with area $A-A_1$ (see the shaded area on Fig. \ref{fig_rw} b)). Therefore this probability $\tilde P(x,m|A,n)$ can be simply expressed in terms of the
propagator $P(x,A,n|x_0,x_0,0)$ computed above (\ref{start_expr}) as (see Fig. \ref{fig_rw} b)):
\begin{eqnarray}\label{start_free}
\fl \tilde P(x,m | A,n) = \frac{1}{P(A,n)} \int_{-\infty}^\infty \rmd x_F \int_{-\infty}^\infty \rmd A_1 P(x,A_1,m|0,0,0) P(x_F,A-A_1,n-m|x,x,0) \;,
\end{eqnarray}
where we have used the Markov property of the L\'evy random walk. In the above expression (\ref{start_free}), $x_F$ is the end point of the walk (see Fig. \ref{fig_rw} b)), which is free here. Hence $\tilde P(x,m | A,n)$ is obtained by integration over this end point $x_F$. Notice that it is normalized according to
$\int_{-\infty}^\infty \tilde P(x,m | A,n) \rmd x = 1$ (and therefore we have divided by $P(A,n)$ in the expression above (\ref{start_free}) because the measure is restricted to random walks of fixed area $A$ after $n$ time steps). Using the explicit expressions computed above (\ref{start_expr}) one obtains after integration over $x_F$ and $A_1$:
\begin{eqnarray}
\fl \tilde P(x,m | A,n) = \frac{1}{P(A,n|0,0)} \int_{-\infty}^\infty \frac{\rmd k_1}{2\pi} \int_{-\infty}^\infty \frac{\rmd k_2}{2\pi} e^{-\sum_{\nu=0}^m |k_1+\nu k_2|^\alpha - |k_2|^\alpha \sum_{\nu=0}^{n-m} |\nu|^\alpha} e^{- i k_1 x -ik_2 (A - (n-m) x)} \;. \nonumber \\
\end{eqnarray}
In the large $n$ limit, keeping $X = x/n^{1/\alpha}$, $Y = A/n^{1+1/\alpha}$ and $\tau = m/n$ fixed one has
\begin{eqnarray}
\fl && \tilde P(x,m | A,n) = \frac{1}{n^{1/\alpha}} \tilde G(X, \tau | Y) \;, \nonumber \\
\fl && \tilde G(X, \tau | Y) = \frac{(1+\alpha)^{-1/\alpha}}{{\cal S}_\alpha((\alpha + 1)^{1/\alpha} Y)} \int_{-\infty}^\infty \frac{\rmd r}{2\pi} \int_{-\infty}^\infty \frac{\rmd k}{2\pi} |k| e^{-|k|^\alpha\left( \tilde \gamma(r,\tau) + \tilde \gamma(0,1-\tau) \right)} e^{- i k r X - i k (Y - (1-\tau) X)} \;,
\end{eqnarray}
where we have used the expression of $P(A,n|0,0)$ given in Eq. (\ref{marginals}) and we have introduced
\begin{eqnarray}
\tilde \gamma(r,\tau) = \int_0^\tau |r + z|^\alpha dz \;,
\end{eqnarray}
which is a generalization of the function $\gamma(r) \equiv \tilde \gamma(r,1)$ in (\ref{expr_scaling}). It reads
\begin{eqnarray}\label{expr_gen_gamma}
\tilde \gamma(r , \tau) =
\cases{
\frac{1}{\alpha+1} \left( (-r)^{\alpha+1} - (-\tau-r)^{\alpha+1} \right) \;, \; r \leq -\tau \;, \\
\frac{1}{\alpha+1} \left( (r+\tau)^{\alpha+1} + (-r)^{\alpha+1} \right) \;, \; -\tau \leq r \leq 0 \;, \\
\frac{1}{\alpha+1} \left( (r+\tau)^{\alpha+1} - r^{\alpha+1} \right) \;, \; r \geq 0 \;.
}
\end{eqnarray}
We can now compute $\langle \tilde x(m) \rangle$ which, in the large $n$ limit, takes the scaling form
\begin{eqnarray}\label{free_1}
\fl &&\frac{\langle \tilde x(m) \rangle}{n^{1/\alpha}} = h_\alpha\left(\frac{m}{n}, \frac{A}{n^{1+1/\alpha}}\right) \\
\fl &&h_\alpha(\tau, Y) = \frac{(1+\alpha)^{-1/\alpha}}{{\cal S}_\alpha((\alpha + 1)^{1/\alpha} Y)} \int_{-\infty}^\infty \rmd X X \int_{-\infty}^\infty \frac{\rmd r}{2\pi} \int_{-\infty}^\infty \frac{\rmd k}{2\pi} |k| e^{-|k|^\alpha g(r,\tau)} e^{- i k r X - i k (Y - (1-\tau) X)} \;. \nonumber
\end{eqnarray}
with $g(r,\tau) = \tilde \gamma(r,\tau) + \tilde \gamma(0,1-\tau)$. This function $h_\alpha(\tau, Y)$ can be written as
\begin{eqnarray}
\fl &&h_\alpha(\tau, Y) = \frac{(1+\alpha)^{-1/\alpha}}{{\cal S}_\alpha((\alpha + 1)^{1/\alpha} Y)} \int_{-\infty}^\infty \rmd X \int_{-\infty}^\infty \frac{\rmd r}{2\pi} \int_{-\infty}^\infty \frac{\rmd k}{2\pi} \frac{|k|}{(- i k)} e^{-|k|^\alpha g(r,\tau)} \frac{\partial}{\partial r} \left( e^{- i k r X - i k (Y - (1-\tau) X)}\right)
\end{eqnarray}
which suggests to perform an integration by part in the integral over $r$, yielding (one can check that the boundary terms vanish)
\begin{eqnarray}\label{intermediaire}
\fl h_\alpha(\tau, Y) = \frac{(1+\alpha)^{-1/\alpha}}{{\cal S}_\alpha((\alpha + 1)^{1/\alpha} Y)} \int_{-\infty}^\infty \rmd X \int_{-\infty}^\infty \frac{\rmd r}{2\pi} \int_{-\infty}^\infty \frac{\rmd k}{2\pi} \frac{|k|^{1+\alpha}}{(- i k)}\frac{\partial g(r,\tau)}{\partial r} e^{-|k|^\alpha g(r,\tau)} e^{- i k r X - i k (Y - (1-\tau) X)} \nonumber \\
\end{eqnarray}
On this expression (\ref{intermediaire}), the integral over $X$ can be done yielding simply a delta function of $r$, namely $2 \pi |k|^{-1}\delta(r - (1-\tau))$. This allows us to perform then the integral over $r$ to obtain
\begin{eqnarray}\label{free_2}
&& h_\alpha(\tau, Y) = \frac{(1+\alpha)^{-1/\alpha}}{{\cal S}_\alpha((\alpha + 1)^{1/\alpha} Y)} i \left(\frac{\partial g(r,\tau)}{\partial r} \right)_{r = 1-\tau}
\int_{-\infty}^\infty |k|^\alpha k^{-1} e^{-\frac{|k|^\alpha}{1+\alpha}} e^{-ikY} \frac{\rmd k}{2\pi} \;,
\end{eqnarray}
where we have used the relation $\tilde \gamma(1-\tau,\tau) + \tilde \gamma(0,1-\tau) = 1/(1+\alpha)$. It is then easy to check that
\begin{eqnarray}
i \int_{-\infty}^\infty |k|^\alpha k^{-1} e^{-\frac{|k|^\alpha}{1+\alpha}} e^{-ikY} \frac{\rmd k}{2\pi} = \frac{\alpha+1}{\alpha} Y (1+\alpha)^{1/\alpha} {\cal S}_\alpha\left[(1+\alpha)^{1/\alpha} Y\right] \;,
\end{eqnarray}
so that finally one obtains the simple result
\begin{eqnarray}\label{explicit_free}
h_\alpha(\tau, Y) = Y \frac{\alpha+1}{\alpha} (1- (1-\tau)^\alpha) \;,
\end{eqnarray}
where we have used $\left(\frac{\partial g(r,\tau)}{\partial r} \right)_{r = 1-\tau} = 1-(1-\tau)^\alpha$. In Fig. (\ref{fig_profiles}) a), we show a plot of $h_\alpha(\tau, Y)/Y$ as a function of $\tau$.
\begin{figure}[ht]
\begin{center}
\includegraphics [width = \linewidth]{figure_profile.pdf}
\end{center}
\caption{Average shape of a free L\'evy walk in {\bf a)} and of a L\'evy bridge in {\bf b)}. {\bf a)}: Plot of $h_{\alpha}(\tau,Y)/Y$ given in Eq. (\ref{explicit_free}) as a function of $\tau$ for $\alpha = 2$ (solid line), $\alpha=1$ (dotted line) and $\alpha=1/2$ (dashed line). {\bf b)}: Plot of $H_1(\tau,Y)/Y$ as a function of $\tau$ in the limit $A \to 0$ (solid line), as given by Eq. (\ref{small_y_shape}), and in the limit $A \to \infty$ (dotted line), as given in Eq. (\ref{large_y_shape}). For comparison, we have also plotted the parabola (dashed line) corresponding to the result for a Brownian bridge, {\it i.e.} $\alpha=2$, as given in Eq. (\ref{shape_bb}) }\label{fig_profiles}
\end{figure}
\subsection{The case of a L\'evy bridge}
Here we consider a L\'evy bridge, i.e. a L\'evy random walker starting in $0$ at initial time and constrained to come back to the origin
after $n$ time steps. We compute the probability $\tilde P_B(x,m |
A,n)$ that the position of the random walker is $x$ after $m$ time
steps given that the area, after $n$ time steps, is fixed to $A$. From
this probability, one obtains the average profile as $\langle \tilde x_B(m)
\rangle = \int_{-\infty}^\infty x \tilde P_B(x,m | A,n) \rmd
x$. This probability $\tilde P_B(x,m | A,n)$ can be expressed, as in Eq. (\ref{start_free}) in terms
of the propagator $P(x,A,n| x_0,x_0,0)$ computed above in Eq. (\ref{start_expr}) as:
\begin{eqnarray}\label{start_dist_profile}
\fl \tilde P_B(x,m | A,n) = \frac{1}{P(0,A,n|0,0,0)}\int_{-\infty}^\infty P(x,A_1,m|0,0,0) P(x,A-A_1,n-m|0,0,0) \rmd A_1 \;,
\end{eqnarray}
where we have used the Markov property of the L\'evy random walk. It
is normalized according to $\int_{-\infty}^\infty \tilde P_B(x,m | A,n) \rmd
x = 1$ (and therefore we have divided by $P(0,A,n|0,0,0)$ because the
measure is restricted to bridges of fixed area $A$). Using the explicit
expressions obtained above (\ref{start_expr}) one has
\begin{eqnarray}\label{dist_profile}
\tilde P_B(x,m | A,n) = &&\frac{1}{P(0,A,n|0,0,0)}\int_{-\infty}^\infty \frac{\rmd k_1}{2\pi} \int_{-\infty}^\infty \frac{\rmd k'_1}{2\pi} \int_{-\infty}^\infty \frac{\rmd k_2}{2\pi} e^{-i(k_1 + k'_1) x - i k A} \nonumber \\
&& \times
e^{- \sum_{\nu=1}^n |k_1 + \nu k_2|^\alpha - \sum_{\nu=1}^{n-m} |k'_1 + \nu k_2|^\alpha} \;.
\end{eqnarray}
In the large $n$ limit, keeping $X = x/n^{1/\alpha}$, $Y = A/n^{1+1/\alpha}$ and $\tau = m/n$ fixed one has
\begin{eqnarray}\label{start_expr_profile}
&& \tilde P_B(x,m | A,n) = \frac{1}{n^{1/\alpha}} \tilde G_B(X,\tau|Y) \;, \nonumber \\
&& \tilde G_B(X,\tau|A) = \frac{\pi}{\Gamma(1+\alpha^{-1}) F_\alpha(Y)}\int_{-\infty}^\infty \frac{\rmd r}{2\pi} \int_{-\infty}^\infty \frac{\rmd r'}{2\pi}\int_{-\infty}^\infty \frac{\rmd k}{2\pi} k^2 e^{-i k (r + r') X - i k A} \\
&& \times e^{ - |k|^\alpha [\tilde \gamma(r,\tau) + \tilde \gamma(r',1-\tau) ] }
\end{eqnarray}
We can now compute $\langle \tilde x_B(m) \rangle$, which in the large $n$ limit takes the scaling form
\begin{eqnarray}
\frac{\langle \tilde x_B(m) \rangle}{n^{1/\alpha}} = H_\alpha\left(\frac{m}{n}, \frac{A}{n^{1+1/\alpha}} \right) \;,
\end{eqnarray}
where
\begin{eqnarray}\label{expr_bridge_stable}
\!\!\!\!\!H_\alpha(\tau, Y) &=& i \frac{1}{2\Gamma(1+\alpha) F_\alpha(Y)} \int_{-\infty}^\infty \frac{\rmd k}{2 \pi} \int_{-\infty}^\infty {\rmd} r |k|^{\alpha-1} k \partial_r \tilde \gamma(r,\tau) e^{- |k|^\alpha \tilde \Gamma(r,\tau) - i k Y} \\
\!\!\!\!\!\! &=& - \frac{1}{2\Gamma(1+\alpha) F_\alpha(Y)} \frac{\partial}{\partial Y} \left( \int_{-\infty}^\infty \frac{\rmd k}{2 \pi} \int_{-\infty}^\infty {\rmd} r |k|^{\alpha-1}
\partial_r \tilde \gamma(r,\tau) e^{- |k|^\alpha \tilde \Gamma(r,\tau) - i k Y} \right)\;, \nonumber
\end{eqnarray}
where we have introduced the notation $\tilde \Gamma(r,\tau) = \tilde \gamma(r,\tau) + \tilde \gamma(-r,1-\tau)$ which we compute
straightforwardly from Eq. (\ref{expr_gen_gamma}) as:
\begin{eqnarray}\label{def_biggamma}
\fl \tilde \Gamma(r,\tau) = \tilde \gamma(r,\tau) + \tilde \gamma(-r,1-\tau) =
\cases{
\frac{1}{\alpha+1} \left((-r+1-\tau)^{\alpha+1} - (-r-\tau)^{\alpha+1} \right) \;, \; r \leq -\tau \\
\frac{1}{\alpha+1} \left( (r+\tau)^{\alpha+1} + (-r+1-\tau)^{\alpha+1} \right) \;, \; -\tau \leq r \leq 1-\tau \\
\frac{1}{\alpha+1} \left( (r+\tau)^{\alpha+1} - (r-1+\tau)^{\alpha+1} \right) \;, \; r \geq 1-\tau \;.
}
\end{eqnarray}
Note that this function $\tilde \Gamma(r,\tau)$ satisfies the identity
\begin{eqnarray}
\tilde \Gamma(1-\tau-r,\tau) = \int_0^{1} |z-r|^\alpha \rmd z = \gamma(-r) \;,
\end{eqnarray}
independently of $\tau$.
For generic $\alpha$, the expression above (\ref{expr_bridge_stable})
is quite difficult to handle. For $\alpha = 2$ (Brownian motion) and
$\alpha = 1$, further analytical progress is however possible. For
$\alpha = 2$, one has $\partial_r \tilde \gamma(r,\tau) =
\tau(2r+\tau)$ and $\tilde \Gamma(r,\tau) = r^2 + r (2\tau-1) -
\tau(1-\tau)+1/3$ and therefore one checks
\begin{eqnarray}\label{identity}
\int_{-\infty}^\infty \partial_r \tilde \gamma(r,\tau) e^{-k^2
\tilde \Gamma(r,\tau) } \rmd r = \frac{\sqrt{\pi}}{|k|} e^{-\frac{k^2}{12}} \tau(1-\tau) \;.
\end{eqnarray}
Using this identity (\ref{identity}) and integrating over $r$ in the expression above (\ref{expr_bridge_stable}), and using $F_2(Y) = \sqrt{3/\pi}\exp{(-3Y^3)}$ one obtains for $\alpha=2$:
\begin{eqnarray}
H_2(\tau, Y) = 6 Y \tau(1-\tau) \;.
\end{eqnarray}
Another interesting case where analytical progress is possible is $\alpha =1$. Given the expression of $\tilde \gamma(r,\tau)$ in Eq. (\ref{expr_gen_gamma}) and $\tilde \Gamma(r,\tau)$ in Eq. (\ref{def_biggamma}), one
observes that the integral over $r$ in Eq. (\ref{expr_bridge_stable}) gives rise to $4$ different terms, corresponding to $r \in ]-\infty, -\tau]$, $r \in [-\tau,0]$,
$r \in [0, 1-\tau]$ and finally $r \in [1-\tau,+\infty[$. For $\alpha = 1$ it turns out that the first and fourth terms, corresponding to $r \in ]-\infty, -\tau]$ and $r \in [1-\tau, +\infty[$ do cancel each other (which is the case only for $\alpha=1$) resulting in the following expression:
\begin{eqnarray}
\fl H_1(\tau, Y) = \frac{Y}{\pi \Gamma(1+\alpha) F_1(Y)} \left(
\int_{-\tau}^0 (2r+\tau) \frac{\tilde \Gamma(r,\tau)}{(\tilde \Gamma(r,\tau)^2 +
Y^2)^2} \rmd r + \tau \int_{0}^{1-\tau}
\frac{\tilde \Gamma(r,\tau)}{(\tilde \Gamma(r,\tau)^2+Y^2)^2} \rmd r
\right) \;,
\end{eqnarray}
with $\tilde \Gamma(r,\tau) = \frac{1}{2} ((r+\tau)^2+(r+\tau-1)^2)$. In the asymptotic limit $Y \to 0$ one obtains
\begin{eqnarray}\label{small_y_shape}
H_1(\tau, Y) \sim \frac{Y}{4 + \pi} \left( 3\pi-2 + \frac{2}{1+2\tau(\tau-1)} + 12 (2\tau-1) \arctan{(1-2\tau)} \right) \;.
\end{eqnarray}
In the opposite limit $Y \to \infty$ one obtains
\begin{eqnarray}\label{large_y_shape}
H_1(\tau, Y) \sim Y \frac{10}{3} \tau(1-\tau)(2 - \tau(1-\tau)) \;.
\end{eqnarray}
Note that although the two functions of $\tau$ entering these asymptotic expansions in Eq. (\ref{small_y_shape}) and Eq. (\ref{large_y_shape}) have very different analytical expressions, they are actually quite close to each other on the interval $[0,1]$ (see Fig. (\ref{fig_profiles}) b)).
\section{Numerical results}
We now come to numerical simulations of L\'evy bridges. As mentioned above, one can not use the relation above (\ref{id_bb}), which is only valid for $\alpha=2$ \cite{knight} to simulate a L\'evy bridge. Instead, we consider the joint probability distribution function (pdf) of the increments $\eta(m)$ for a L\'evy bridge of size $n$. Indeed, these increments are independent random variables, distributed
according to $\phi(\eta)$ with the global constraint that $x(n) = \sum_{m=1}^n \eta(n) = 0$. Therefore the joint pdf of the increments $P_B\left(\eta(1), \eta(2), \cdots, \eta(n)\right)$ is simply given by
\begin{eqnarray}\label{joint_increments}
P_{B}\left(\eta(1), \eta(2), \cdots, \eta(n) \right) &\propto& \prod_{m=1}^n \phi\left[\eta(m)\right] \delta\left(\sum_{m=1}^n \eta(m)\right) \\
&\propto& \exp{\left[ \sum_{m=1}^n \ln{\left [\phi\left[\eta(m)\right] \right]} \right ]} \delta \left(\sum_{m=1}^n \eta(m) \right) \;. \nonumber
\end{eqnarray}
This joint distribution can thus be considered as a Boltzmann weight with an effective
energy $E = - \sum_{m=1}^n \ln{\left [\phi\left[\eta(m)\right] \right] }$ and effective inverse temperature $\beta = 1$. This thus leads us to use a Monte-Carlo algorithm, with a global constraint, to generate "configurations" of the increments distributed according to the distribution above (\ref{joint_increments}). We implement it in the following way. We start with a random initial configuration of the $\eta(m)$'s which satisfies the global constraint $\sum_{m=1}^n \eta(m)=0$ (it can also be $\eta(m) = 0$, for all $m$). At each time step we choose randomly two sites $i$ and $j$ among $1, 2, \cdots, n$ and the simple following moves are proposed
\begin{eqnarray}\label{move}
&&\eta(i) \to \eta'(i) = \eta(i) + \Delta \eta \;, \nonumber \\
&&\eta(j) \to \eta'(j) = \eta(j) - \Delta \eta \;,
\end{eqnarray}
such that the global constraint of zero sum is automatically satisfied. This move is then accepted, in Metropolis
algorithm that we use here, with a probability $P_{ij}$ given by
\begin{eqnarray}
P_{ij} &=& \min{\left(1, \frac{\phi \left[ \eta'(i) \right] \phi\left[\eta'(j)\right]}{\phi \left[\eta(i) \right] \phi\left[\eta(j) \right]} \right)} \\
&=& \min(1,\exp{(-\Delta E)}) \;, \; \Delta E = \log{\left(\frac{\phi \left[\eta(i) \right] \phi\left[\eta(j) \right]}{\phi \left[ \eta'(i) \right] \phi\left[\eta'(j)\right]}\right)}
\end{eqnarray}
This Monte Carlo algorithm is thus very similar to the Kawasaki dynamics for ferromagnetic spin systems relaxing
towards equilibrium with a conserved global magnetization \cite{kawasaki}. Once the increments $\eta(k)$'s are generated according to this joint
probability (\ref{joint_increments}), we can generate the random walk bridge $x_B(m) = \sum_{k=1}^m \eta(k)$ and compute the distribution of the area $A = \sum_{m=1}^n x_B(m)$ under the L\'evy bridge. In Fig. \ref{fig_numerics} a), we show a plot of this distribution $P_B(A,n)$ for $\alpha=1$ and $n=100$. To compute it we have first run $10^7$ Monte Carlo steps to equilibrate the system and the distribution was then computed as an average over $10^7$ samples generated in the time interval $[10^7, 2 . 10^7]$. In Fig. \ref{fig_numerics}, we also show a plot of the exact explicit expression for $F_1(Y)$ given in Eq. (\ref{elem}), showing a very good agreement with our numerics. We have also computed numerically this distribution for other values of $\alpha \in ]0,2[$, showing a good agreement with the power law tail obtained in Eq. (\ref{large_y}). Note however that for small $\alpha$, it is actually quite difficult to equilibrate the system such that a precise estimate of the exponent characterizing the power law tail of $P_B(A,n)$ is quite difficult for $\alpha < 1$.
\begin{figure}[ht]
\begin{center}
\includegraphics [width = \linewidth]{numerics.pdf}
\end{center}
\caption{{\bf a)}: The squares represent the numerical data for $n^2 P_B(A,n)$ as a function of $Y \equiv A/n^2$ for a L\'evy bridge of index $\alpha=1$ of length $n=100$. The solid line is our exact expression for $F_1(Y)$ given in Eq. (\ref{elem}). {\bf b)}: The squares represent our numerical data for $n \langle \tilde x_B(m) \rangle/A$ as a function of $\tau =m/n$ for a L\'evy bridge of index $\alpha=1$ of length $n=100$ and $A/n^2 = 20$. The solid line corresponds to our asymptotic result
for large $A$ given in Eq. (\ref{large_y_shape}) while the dotted line represents the result for the Brownian bridge in Eq. (\ref{shape_bb}).}\label{fig_numerics}
\end{figure}
We can use a similar Monte Carlo approach to generate a random walk bridge with a fixed area $A = \sum_{m=1}^n (n+1-m) \eta(m) $. In that case, the joint
pdf of the increments $\tilde P_B\left(\eta(1), \eta(2), \cdots, \eta(n)\right)$ is simply given by
\begin{eqnarray}\label{joint_increments_area}
\fl P_{B}\left(\eta(1), \eta(2), \cdots, \eta(n) \right) &\propto& \prod_{m=1}^n \phi\left[\eta(m) \right ]\delta\left(\sum_{m=1}^n \eta(m)\right) \delta(\sum_{m=1}^n (n+1-m) \eta(m)-A) \;.
\end{eqnarray}
We start with an initial configuration of the $\eta(m)$'s which satisfies the both global constraints. In practice, we start with $\eta(m) = 6 A (N+1-2 m)/((N-1)N(N+1))$. Then, to satisfy both constraints (\ref{joint_increments_area}), at each time step we choose randomly three sites $i$, $j$ and $k$ among $1, 2, \cdots, n$ and the simple following moves are proposed
\begin{eqnarray}\label{move_area}
&&\eta(i) \to \eta'(i) = \eta(i) + \Delta \eta \;, \nonumber \\
&&\eta(j) \to \eta'(j) = \eta(j) + \frac{i-k}{k-j} \Delta \eta \;, \nonumber \\
&&\eta(k) \to \eta'(k) = \eta(k) + \frac{j-i}{k-j} \Delta \eta \;.
\end{eqnarray}
Note that to converge to the correct probability measure (\ref{joint_increments_area}) one has to choose $\eta$ either positive or negative with equal probability.
This move (\ref{move_area}) is then accepted with a probability $P_{ijk}$ given by
\begin{eqnarray}
P_{ijk} &=& \min{\left(1, \frac{\phi\left[\eta'(i)\right] \phi\left[\eta'(j)\right] \phi\left[\eta'(k)\right]}{\phi\left[ \eta(i)\right] \phi\left[\eta(j)\right] \phi\left[\eta(k) \right]} \right)} \\
&=& \min(1,\exp{(-\Delta E)}) \;, \; \Delta E = \log{\left(\frac{\phi \left[\eta(i) \right] \phi\left[\eta(j) \right] \phi\left[\eta(k) \right] }{\phi \left[ \eta'(i) \right] \phi\left[\eta'(j)\right] \phi\left[\eta'(k) \right] }\right)}
\end{eqnarray}
Once the increments $\eta(k)$'s are generated according to this joint
probability (\ref{joint_increments_area}), we can generate the random walk bridge $\tilde x_B(m) = \sum_{k=1}^m \eta(k)$ with fixed area $A$ and compute
the profile $\langle \tilde x_B(m) \rangle$. In Fig.~\ref{fig_numerics} b), we show a plot of this average profile for $\alpha=1$, $n=100$ and $A/n^2 \sim 20$. To compute it we have first run $10^7$ Monte Carlo steps to equilibrate the system and the average was then computed over $10^7$ samples generated in the time interval $[10^7, 2 . 10^7]$. In Fig. \ref{fig_numerics} b), we also plot, with a solid line, our asymptotic result in Eq. (\ref{large_y_shape}), showing a relatively good agreement with our numerics (note that here $A/n^2 = 20$). On the same plot, Fig. \ref{fig_numerics} b), we also show in dotted line, the result for the Brownian bridge (\ref{shape_bb}), which is independent of $A$. It is quite remarkable that these two profiles are very similar which show that the global constraints that we impose here have strong consequences on the statistics of the L\'evy random walk.
\section{Conclusion}
To conclude, we have studied two main properties of a L\'evy bridge $x_B(m)$ of length $n$ : (i) the distribution $P_B(A,n)$ of the area under a L\'evy bridge and (ii) the average profile $\langle \tilde x_B(m) \rangle$ of a L\'evy bridge with fixed area $A$.
\begin{itemize}
\item{
For $P_B(A,n)$ we have found the scaling form, valid for large $n$, $P_B(A,n) \sim n^{-1-1/\alpha} F_\alpha(Y)$ with an interesting power law behavior $F_\alpha(Y) \sim Y^{-2(1+\alpha)}$. For $\alpha=1$, we have obtained an explicit expression for $F_1(Y)$ in terms of elementary functions (\ref{elem}). We have also shown, using the Lindeberg condition that the non-Gaussianity of $P_B(A,n)$, for $\alpha > 1/2$, is due only to the correlations between the positions of the walkers $x_B(m)$'s.}
\item{For the average profile, $\langle \tilde x_B(m) \rangle$, we have found the scaling form $\langle \tilde x_B(m) \rangle \sim n^{1/\alpha} H_{\alpha}(m/n,A/n^{1+1/\alpha})$ where, at variance with Brownian motion, $H_\alpha(X,Y)$ is a non trivial function of the rescaled area $Y$. For $\alpha=1$, we have obtained simple analytical expressions for $H_1(X,Y)$ in both limits $Y \to 0$ and $Y \to \infty$. In particular, we have shown that the average profile of the L\'evy random walk with a fixed area is not very far from the profile of a Brownian bridge with fixed area.}
\item{We have finally compared our analytical results with Monte Carlo simulations of these L\'evy random walks with global constraints.}
\end{itemize}
In view of recent developments in the study of area distributions for variants of Brownian motions \cite{janson_review, satya_airy, schehr_airy, rambeau_airy}, it would be very interesting to extend the results presented here to other constrained L\'evy walk, including in particular L\'evy random walks conditioned to stay positive (L\'evy excursion), which is a challenging open problem.
\newpage
|
1,108,101,566,774 | arxiv | \section*{Introduction}
Let $S=k[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $k$. We order lexicographically the monomials of $S$ such that $x_1>x_2>\ldots> x_n$. Let $d\geq 2$ be an integer and $\mathcal{M}_d$ the set of monomials of degree $d$. For two monomials $u,v\in\mathcal{M}_d$, with $u\geq_{lex}v$, the set $$\mathcal L(u,v)=\{w\in\mathcal{M}_d\ |\ u\geq_{lex}w\geq_{lex}v\}$$ is called a lexsegment. A lexsegment ideal in $S$ is a monomial ideal of $S$ which is generated by a lexsegment. Lexsegment ideals have been introduced by Hulett and Martin \cite{HM}. Arbitrary lexsegment ideals have been studied by A. Aramova, E. De Negri, and J. Herzog in \cite{ADH} and \cite{DH}. They characterized the lexsegment ideals which have linear resolutions.
In this paper we show that any lexsegment ideal with linear resolution has linear quotients with respect to a suitable order of the generators.
Let $I\subset S$ be a monomial ideal and $G(I)$ its minimal monomial set of generators. $I$ has linear quotients if there exists an ordering $u_1,\ldots,u_m$ of the elements of $G(I)$ such that for all $2\leq j\leq m$, the colon ideals $(u_1,\ldots,u_{j-1}):u_j$ are generated by a subset of $\{x_1,\ldots,x_n\}$.
Lexsegment ideals which have linear quotients with respect to the lexicographical order of the generators have been characterized by the third author in \cite{S}.
In Section \ref{Section1} we show that any completely lexsegment ideal with linear resolution has linear quotients with respect to the following order of the generators. Given two monomials of degree $d$ in $S$, $w=x_1^{\alpha_1}\ldots x_n^{\alpha_n}$ and $w'=x_1^{\beta_1}\ldots x_n^{\beta_n},$ we set $w\prec w'$ if $\alpha_1<\beta_1$ or $\alpha_1=\beta_1$ and $w>_{lex}w'$.
Let $u,v\in\mathcal{M}_d$ which define the completely lexsegment ideal $I=(\mathcal{L}(u,v))$ with linear resolution. If $\mathcal{L}(u,v)=\{w_1,\ldots, w_r\}$, where $w_1\prec w_2\prec\ldots\prec w_r$, we show that $I$ has linear quotients with respect to this ordering of the generators. The non-completely lexsegment ideal will be separately studied in Section $2$.
For the completely lexsegment ideals with linear resolution it will turn out that their decomposition function with respect to the ordering $\prec$ is regular . Therefore, one may apply the procedure developed in \cite{HT} to get the explicit resolutions for this class of ideals.
In the last section of our paper we study the depth and the dimension of lexsegment ideals. Our results show that one may compute these invariants just looking at the ends of the lexsegment. As an application, we characterize the Cohen-Macaulay lexsegment ideals.\\
We acknowledge the support provided by the Computer Algebra Systems \textsc{CoCoA} \cite{Co} and \textsc{Singular} \cite{GPS} for the extensive experiments which helped us to obtain some of the results of this work.
\section{Completely lexsegment ideals with linear resolutions}\label{Section1}
In the theory of Hilbert functions or in extremal combinatorics usually one considers initial lexsegment ideals, that is ideals generated by an initial lexsegment $\mathcal{L}^i(v)=\{w\in\mathcal{M}_d\ |\ w\geq_{lex} v\}$. Initial lexsegment ideals are stable in the sense of Eliahou and Kervaire (\cite{EH}, \cite{AH}) and they have linear quotients with respect to lexicographical order \cite[Proposition 2.1]{S}.
One may also define the final lexsegment $\mathcal{L}^f(u)=\{w\in\mathcal{M}_d\ |\ u\geq_{lex}w\}$. Final lexsegment ideals are generated by final lexsegments. They are also stable in the sense of Eliahou and Kervaire with respect to $x_n>x_{n-1}>\ldots>x_1$. Therefore they have linear quotients.
Throughout this paper we use the following notations. If $m=x_1^{\alpha_1}\ldots x_n^{\alpha_n}$ is a monomial of $S$, we denote by $\nu_i(m)$ the exponent of the variable $x_i$ in $m$, that is $\nu_i(m)=\alpha_i$, $i=1,\ldots,n$. Also, we will denote $\max(m)=\max\{i\mid x_i|m\}.$
Hulett and Martin call a lexsegment $L$ \it{completely lexsegment} \rm if all the iterated shadows of $L$ are again lexsegments. We recall that the shadow of a set $T$ of monomials is the set $\shad(T)=\{vx_i\mid v\in T,\ 1\leq i\leq n\}$. The $i$-th shadow is recursively defined as $\shad^i(T)=\shad(\shad^{i-1}(T))$. The initial lexsegments have the property that their shadow is again an initial lexsegment, a fact which is not true for arbitrary lexsegments. An ideal spanned by a completely lexsegment is called a \it completely lexsegment ideal\rm. All the completely lexsegment ideals with linear resolution are determined in \cite{ADH}:
\begin{Theorem}\cite{ADH}\label{completelylex} Let $u=x_1^{a_1}\ldots x_n^{a_n}$, $v=x_1^{b_1}\ldots x_n^{b_n}$ be monomials of degree $d$ with $u\geq_{lex}v$, and let $I=(\mathcal{L}(u,v))$ be a completely lexsegment ideal. Then $I$ has linear resolution if and only if one of the following conditions holds:
\begin{itemize}
\item [(a)] $u=x_1^ax_2^{d-a},\ v=x_1^ax_n^{d-a}$ for some $a,\ 0< a\leq d;$
\item [(b)] $b_1< a_1-1;$
\item [(c)] $b_1 = a_1-1$ and for the largest $w <_{lex} v, w$ monomial of degree $d,$ one has $x_1 w/x_{\max(w)}\leq_{lex} u.$
\end{itemize}
\end{Theorem}
\begin{Theorem}\label{colex}
Let $u=x_1^{a_1}\ldots x_n^{a_n},$ with $a_1>0,$ and $ v=x_1^{b_1}\ldots x_n^{b_n}$ be monomials of degree $d$ with $u\geq_{lex} v,$ and let $I=({\mathcal L}(u,v))$ be a completely lexsegment ideal. Then $I$ has linear resolution if and only if $I$ has linear quotients.
\end{Theorem}
\begin{proof}
We have to prove that if $I$ has linear resolution then $I$ has linear quotients, since the other implication is known \cite{H}. By Theorem \ref{completelylex}, since $I$ has linear resolution, one of the conditions (a), (b), (c) holds.
We define on the set of the monomials of degree $d$ from $S$ the following total order: for $$w=x_1^{\alpha_1}\ldots x_n^{\alpha_n},\ w^{\prime}=x_1^{\beta_1}\ldots x_n^{\beta_n},$$ we set $$w\prec w^{\prime}\text{ if }\alpha_1 < \beta_1 \text{ or }\alpha_1=\beta_1 \text{ and } w>_{lex} w^{\prime}.$$ Let $$\mathcal{L}(u,v)=\{w_1,\ldots,w_r\}, \text{ where }w_1\prec w_2\prec\ldots\prec w_r.$$ We will prove that $I=(\mathcal{L}(u,v))$ has linear quotients with respect to this ordering of the generators.
Assume that $u,v$ satisfy the condition (a) and $a<d$ (the case $a=d$ is trivial). Then $I$ is isomorphic as $S$--module with the ideal generated by the final lexsegment ${\mathcal L}^f(x_2^{d-a}) \subset S$ and the ordering $\prec$ of its minimal generators coincides with the lexicographical ordering $>_{lex}.$ The ideal $(\mathcal{L}^f(x_2^{d-a}))\cap k[x_2,\ldots,x_n]$ is the initial ideal in $k[x_2,\ldots,x_n]$ defined by $x_n^{d-a},$ which has linear quotients with respect to $>_{lex}.$ Hence $I$ has linear quotients with respect to $\prec$ since it is obvious that the extension in the ring $k[x_1,\ldots, x_n]$ of a monomial ideal with linear quotients in $k[x_2,\ldots,x_n]$ has linear quotients too.
Next we assume that $u,v$ satisfy the condition (b) or (c).
By definition, $I$ has linear quotients with respect to the monomial generators $w_1,\ldots,w_r$ if the colon ideals $(w_1,\ldots,w_{i-1}):w_i$ are generated by variables for all $i\geq 2,$ that is for all $j < i$ there exists an integer $1\leq k <i$ and an integer $l\in [n]$ such that $w_k/\gcd(w_k,w_i)=x_l \text{ and }x_l\text{ divides } w_j/\gcd(w_j,w_i).$
In other words, for any $w_j\prec w_i, w_j,w_i\in \mathcal{L}(u,v),$ we have to find a monomial $w^{\prime}\in \mathcal{L}(u,v)$ such that
\begin{eqnarray}
w^{\prime}\prec w_i,\ \frac{w^{\prime}}{\gcd(w^{\prime},w_i)}=x_l, \text{ for some }l\in [n] \text{, and }x_l\text{ divides } \frac{w_j}{\gcd(w_j,w_i)}. \eqname{$*$}\label{*}
\end{eqnarray}
Let us fix $w_i=x_1^{\alpha_{1}}\ldots x_n^{\alpha_{n}}$ and $w_j=x_1^{\beta_{1}}\ldots x_n^{\beta_{n}}, \ w_i,w_j\in \mathcal{L}(u,v),$ such that $w_j\prec w_i.$ By the definition of the ordering $\prec,$ we must have $$\beta_{1} < \alpha_{1} \text { or }\beta_{1} = \alpha_{1} \text{ and } w_j>_{lex} w_i.$$
\underline{\textit{Case $1$}}: Let $\beta_1<\alpha_1$. One may find an integer $l$, $2\leq l\leq n$, such that $\alpha_s\geq \beta_s$ for all $s<l$ and $\alpha_l<\beta_l$ since, otherwise, $\deg(w_i)>\deg(w_j)=d$ which is impossible. We obviously have $\max(w_j)\geq l$. If $l\geq \max(w_i),$ one may take $\bar{w}=x_lw_i/x_1$ which satisfies the condition (\ref{*}) since the inequalities $\bar{w}\prec w_i,$ $\bar{w}\leq_{lex}w_i\leq_{lex} u$ hold, and we will show that $\bar{w}\geq_{lex}w_j$. This will imply that $\bar{w}\geq_{lex}v$, hence $\bar{w}\in\mathcal{L}(u,v)$.
The inequality $\bar{w}\geq_{lex}w_j$ is obviously fulfilled if $\alpha_1-1>\beta_1$ or if $\alpha_1-1=\beta_1$ and at least one of the inequalities $\alpha_s\geq\beta_s$ for $2\leq s<l$, is strict. If $\alpha_1-1=\beta_1$ and $\alpha_s=\beta_s$ for all $s<l$, comparing the degrees of $w_i$ and $w_j$ it results $d=\alpha_1+\ldots+\alpha_l=\beta_1+1+\beta_2+\ldots+\beta_{l-1}+\alpha_l<(\beta_1+1)+\beta_2+\ldots+\beta_l.$ It follows that $d\geq\beta_1+\beta_2+\ldots+\beta_l>d-1,$ that is $\beta_1+\beta_2+\ldots+\beta_l=d.$ This implies that $l=\max(w_j)$ and $\beta_l=\alpha_l+1$, that is $\bar{w}=x_lw_i/x_1=x_1^{\alpha_1-1}x_2^{\alpha_2}\ldots x_l^{\alpha_l+1}=x_1^{\beta_1}\ldots x_l^{\beta_l}=w_j.$
From now on, in the Case $1$, we may assume that $l<\max(w_i)$. We will show that at least one of the following monomials: $$w'=\frac{x_lw_i}{x_{\max(w_i)}},\ w''=\frac{x_lw_i}{x_1}$$ belongs to $\mathcal{L}(u,v)$. It is clear that both monomials are strictly less than $w_i$ with respect to the ordering $\prec.$ Therefore one of the monomials $w',\ w''$ will satisfy the condition (\ref{*}).
The following inequalities are fulfilled: $$w'>_{lex}w_i\geq_{lex}v,\ \text{and}$$ $$w''<_{lex}w_i\leq_{lex}u.$$
Let us assume, by contradiction, that $w'>_{lex}u$ and $w''<_{lex}v$. Comparing the exponents of the variable $x_1$, we obtain $a_1-1\leq\alpha_1-1\leq b_1$. Since the ideal generated by $\mathcal{L}(u,v)$ has linear resolution, we must have $b_1=a_1-1$. Let $z$ be the largest monomial of degree $d$ such that $z<_{lex}v$. Then, by our assumption on $w''$, we also have the inequality $w''\leq_{lex}z$.
Now we need the following
\begin{Lemma}\label{lema1} Let $m=x_1^{a_1}\ldots x_n^{a_n},\ m'=x_1^{b_1}\ldots x_n^{b_n}$ be two monomials of degree $d$. If $m\leq_{lex}m'$ then $m/x_{\max(m)}\leq_{lex}m'/x_{\max(m')}$.
\end{Lemma}
\begin{proof} Let $m <_{lex}m'$. Then there exists $s\geq1$ such that $a_1=b_1,\ldots, a_{s-1}=b_{s-1}$ and $a_s<b_s$. It is clear that $\max(m')\geq s$. Comparing the degrees of $m$ and $m'$ we get $\max(m)>s.$
If $\max(m)>s$ and $\max(m')>s$, the required inequality is obvious.
Let $\max(m)>s$ and $\max(m')=s$. Let us suppose, by contradiction, that $m/x_{\max(m)}>_{lex}m'/x_{\max(m')}=m'/x_s.$ This implies that $a_s\geq b_s-1$, and, since $a_s<b_s$, we get $a_s=b_s-1$.
Looking at the degree of $m'$ we obtain $d=b_1+b_2+\ldots+b_s=a_1+a_2+\ldots+ a_{s-1}+a_s+1,$ that is $a_1+\ldots+a_s=d-1$. It follows that $a_{\max(m)}=1$ and $m/x_{\max(m)}=x_1^{a_1}\ldots x_s^{a_s}=x_1^{b_1}\ldots x_{s-1}^{b_{s-1}}x_s^{b_s-1}=m'/x_{\max(m')}$, contradiction.
\end{proof}
Going back to the proof of our theorem,
we apply the above lemma for the monomials $w''$ and $z$ and we obtain $w''/x_{\max(w'')}\leq_{lex}z/x_{\max(z)},$ which implies that $x_1w''/x_{\max(w'')}\leq_{lex}x_1z/x_{\max(z)}.$
By using condition (c) in the Theorem \ref{completelylex} it follows that $x_1w''/x_{\max(w'')}\leq_{lex}u.$ On the other hand, $x_1w''/x_{\max(w'')}=x_1x_lw_i/(x_1x_{\max(w_i)})$ $=x_lw_i/x_{\max(w_i)}=w'.$ Therefore, it results $w'\leq_{lex}u$, which contradicts our assumption on $w'$.
Consequently, we have $w'\leq_{lex}u$ or $w''\geq_{lex}v$, which proves that at least one of the monomials $w',\ w''$ belongs to $\mathcal{L}(u,v)$.
\underline{\textit{Case $2$:}} Let $\beta_1=\alpha_1$ and $w_j>_{lex}w_i$. Then there exists $l$, $2\leq l\leq n$, such that $\alpha_s=\beta_s$, for all $s<l$ and $\alpha_l<\beta_l$. If $\max(w_i)\leq l$, then, looking at the degrees of $w_i$ and $w_j$, we get $d=\alpha_1+\alpha_2+\ldots+\alpha_l<\beta_1+\beta_2+\ldots+\beta_l,$ contradiction. Therefore, $l<\max(w_i)$. We proceed in a similar way as in the previous case. Namely, exactly as in the Case $1,$ it results that at least one of the following two monomials $w'=x_lw_i/x_{\max(w_i)},\ w''=x_lw_i/x_1$ belongs to $\mathcal{L}(u,v)$. It is clear that both monomials are strictly less than $w_i$ with respect to the order $\prec$.
\end{proof}
\begin{Example}\rm
Let $S=k[x_1,x_2,x_3]$. We consider the monomials: $u=x_1x_2x_3$ and $v=x_2x_3^2$, $u>_{lex} v,$ and let $I$ be the monomial ideal generated by $\mathcal{L}(u,v)$. The minimal system of generators of the ideal $I$ is
$$G(I)=\mathcal{L}(u,v)=\{x_1x_2x_3,\ x_1x_3^2,\ x_2^3,\ x_2^2x_3,\ x_2x_3^2\}.$$
Since $I$ verifies the condition (c) in Theorem \ref{completelylex}, it follows that $I$ is a completely lexsegment ideal with linear resolution. We denote the monomials from $G(I)$ as follows: $u_1=x_1x_2x_3,\ u_2=x_1x_3^2,\ u_3=x_2^3,\ u_4=x_2^2x_3,\ u_5=x_2x_3^2$, so $u_1>_{lex}u_2>_{lex}\ldots>_{lex}u_5.$ The colon ideal $(u_1,u_2):u_3=(x_1x_3)$ is not generated by a subset of $\{x_1,x_2,x_3\}$. This shows that $I$ is not with linear quotients with respect to lexicographical order.
We consider now the order $\prec$ and check by direct computation that $I$ has linear quotients. We label the monomials from $G(I)$ as follows: $u_1=x_2^3,\ u_2=x_2^2x_3,\ u_3=x_2x_3^2,\ u_4=x_1x_2x_3,\ u_5=x_1x_3^2$, so $u_1\prec u_2\prec\ldots\prec u_5$. Then $(u_1):u_2=(x_2),\ (u_1,u_2):u_3=(x_2),\ (u_1,u_2,u_3):u_4=(x_2,x_3),$ $(u_1,u_2,u_3,u_4):u_5=(x_2)$.
\end{Example}
We further study the decomposition function of a completely lexsegment ideal with linear resolution. The decomposition function of a monomial ideal was introduced by J. Herzog and Y. Takayama in \cite{HT}.
We recall the following notation. If $I\subset S$ is a monomial ideal with linear quotients with respect to the ordering $u_1,\ldots,u_m$ of its minimal generators, then we denote $$\set(u_j)=\{k\in[n]\ |\ x_k\in(u_1,\ldots,u_{j-1}):u_j\}$$
for $j=1,\ldots,m$.
\begin{Definition}\cite{HT} \rm Let $I\subset S$ be a monomial ideal with linear quotients with respect to the sequence of minimal monomial generators $u_1,\ldots, u_m$ and set $I_j=(u_1,\ldots, u_j)$, for $j=1,\ldots,m$. Let $\mathcal{M}(I)$ be the set of all monomials in $I$. The map $g:M(I)\rightarrow G(I)$ defined as: $g(u)=u_j$, where $j$ is the smallest number such that $u\in I_j$, is called \it the decomposition function \rm of $I$.
\end{Definition}
We say that the decomposition function $g:M(I)\rightarrow G(I)$ is \it regular \rm if $\set(g(x_su))$$\subseteq\set(u)$ for all $s\in\set(u)$ and $u\in G(I)$.
We show in the sequel that completely lexsegment ideals which have linear quotients with respect to $\prec$ have also regular decomposition functions.
In order to do this, we need some preparatory notations and results.
For an arbitrary lexsegment $\mathcal{L}(u,v)$ with the elements ordered by $\prec$, we denote by $I_{\prec w}$, the ideal generated by all the monomials $z\in\mathcal{L}(u,v)$ with $z\prec w$. $I_{\preceq}w$ will be the ideal generated by all the monomials $z\in\mathcal{L}(u,v)$ with $z\preceq w$.
\begin{Lemma}\label{1nset} Let $I=(\mathcal{L}(u,v))$ be a lexsegment ideal which has linear quotients with respect to the order $\prec$ of the generators. Then, for any $w\in\mathcal{L}(u,v)$, $1\notin\set(w)$.
\end{Lemma}
\begin{proof} Let us assume that $1\in\set(w)$, that is $x_1w\in I_{\prec w}$. It follows that there exists $w'\in\mathcal{L}(u,v)$, $w'\prec w$, and a variable $x_j$ such that $x_1w=x_jw'$. Obviously, we have $j\geq2$. But this equality shows that $\nu_1(w')>\nu_1(w)$, which is impossible since $w'\prec w$.
\end{proof}
\begin{Lemma}\label{desc} Let $I=(\mathcal{L}(u,v))$ be a completely lexsegment ideal which has linear quotients with respect to the ordering $\prec$ of the generators. If $w\in\mathcal{L}(u,v)$ and $s\in\set(w)$, then
\[g(x_sw)=
\left\{\begin {array}{cc}
x_s w/x_1, & \mbox{if}\ x_sw\geq_{lex} x_1v,\\
&\\
x_s w/x_{\max(w)}, & \mbox{if}\ x_sw<_{lex} x_1v.
\end{array}\right. \]
\end{Lemma}
\begin{proof} Let $u=x_1^{a_1}\ldots x_n^{a_n},\ v=x_1^{b_1}\ldots x_n^{b_n},\ a_1>0,$ and $w=x_1^{\alpha_1}\ldots x_n^{\alpha_n}$.
In the first place we consider $$x_sw\geq_{lex}x_1v.$$ Since, by Lemma \ref{1nset}, we have $s\geq 2$, the above inequality shows that $\nu_1(w)\geq1$. We have to show that $g(x_sw)=x_s w/x_1$, that is $x_s w/x_1=\min_{\prec}\{w'\in\mathcal{L}(u,v)\ |\ x_sw\in I_{\preceq w'}\}$. It is clear that $v\leq_{lex} x_sw/x_1<_{lex}w\leq_{lex}u$, hence $x_sw/x_1\in\mathcal{L}(u,v)$. Let $w'\in\mathcal{L}(u,v)$ such that $x_sw\in I_{\preceq w'}$. We have to show that $x_sw/x_1\preceq w'$. Let $w''\in\mathcal{L}(u,v),\ w''\preceq w'$ such that $x_sw=w''x_j$, for some variable $x_j$. Then
$w''=x_sw/x_j\succeq x_sw/x_1
$
by the definition of our ordering $\prec$. This implies that $w'\succeq x_sw/x_1$.
Now we have to consider the second inequality,
\begin{eqnarray}x_sw<_{lex}x_1v.\label{star}
\end{eqnarray}
Since $s\in\set(w)$, we have $x_sw\in I_{\prec w}$, that is there exists $w'\in\mathcal{L}(u,v)$, $w'\prec w$, and a variable $x_j,\ j\neq s$, such that
\begin{eqnarray}x_sw=x_jw'.\label{1}
\end{eqnarray}
If $j=1$, then $x_sw=x_1w'\geq_{lex}x_1v$, contradiction. Hence $j\geq2$. We also note that $x_j|w$ since $j\neq s$, thus $j\leq\max(w)$. The following inequalities hold:
\begin{eqnarray}
x_s w/x_{\max(w)}\geq_{lex} x_s w/x_j= w'\geq_{lex} v. \label{4}
\end{eqnarray}
If $\nu_1(w)<a_1$, we obviously get $x_sw/x_{\max(w)}\leq_{lex}u$. Let $\nu_1(w)=a_1$. From the inequality (\ref{star}) we obtain $a_1\leq b_1+1$.
If $a_1=b_1$ then $u=x_1^{a_1}x_2^{d-a_1}$ and $v=x_1^{a_1}x_n^{d-a_1}$ by Theorem \ref{completelylex}. Since $w\leq_{lex} u$, by using Lemma \ref{lema1}, we have
$x_sw/x_{\max(w)}\leq_{lex}x_su/x_{\max(u)}=x_su/x_2\leq_{lex} u,
$ the last inequality being true by Lemma \ref{1nset}. Therefore, $x_s w/x_{\max(w)}\in \mathcal{L}(u,v).$
If $a_1=b_1+1$ then the condition (c) in Theorem \ref{completelylex} holds. Let $z$ be the largest monomial with respect to the lexicographical order such that $z<_{lex}v$. Since $x_sw/x_1<_{lex}v$ by hypothesis, we also have $x_sw/x_1\leq_{lex}z$. By Lemma \ref{lema1} we obtain
$x_sw/(x_1x_{\max(x_sw/x_1)})\leq_{lex}z/x_{\max(z)}.
$
Next we apply the condition (c) from Theorem \ref{completelylex} and get the following inequalities:
\begin{eqnarray}
x_1\frac{x_sw}{x_1x_{\max\left(\frac{x_sw}{x_1}\right)}}\leq_{lex}x_1\frac{z}{x_{\max(z)}}\leq_{lex}u.\label{2}
\end{eqnarray}
From the equality (\ref{1}) we have $w'=x_sw/x_j$. As $j\neq1,\ \nu_1(w')=\nu_1(w),$ and the inequality $w'\prec w$ gives $w'>_{lex} w,$ that is $x_s w/x_j>_{lex}w,$ which implies that $x_s>_{lex} x_j$. This shows that $s<j\leq\max(w)$. Now looking at the inequalities (\ref{2}), we have
\begin{eqnarray}x_sw/x_{\max(w)}\leq_{lex}u.\label{3}
\end{eqnarray}
From (\ref{3}) and (\ref{4}) we obtain $x_sw/x_{\max(w)}\in\mathcal{L}(u,v)$.
It remains to show that $x_s w/x_{\max(w)}=\min_{\prec}\{w'\in\mathcal{L}(u,v)\mid x_sw\in I_{\preceq w'}\}.$ Let $\tilde{w}=\min_{\prec}\{w'\in\mathcal{L}(u,v)\mid x_sw\in I_{\preceq w'}\}.$ We obviously have $\tilde{w}\preceq x_s w/x_{\max(w)}\prec w.$ By the choice of $\tilde{w}$ we have $$x_sw=x_t\tilde{w}$$ for some variable $x_t.$
If $t=s$ we get $w=\tilde{w}$ which is impossible since $\tilde{w}\prec w.$ Therefore, $t\neq s.$
Then $x_t| w,$ so $t\leq\max(w)$. It follows that $\tilde{w}=x_sw/x_t\leq_{lex}x_sw/x_{\max(w)}$. If $t=1$ we have $x_1\tilde{w}=x_sw <_{lex}x_1 v,$ which implies that $\tilde{w}<_{lex}v,$ contradiction. Therefore $t\neq 1$ and, moreover, $\tilde{w}\succeq x_sw/x_{\max(w)}$, the inequality being true by the definition of the ordering $\prec$.
This yields $\tilde{w}=x_sw/x_{\max(w)}.$ Therefore we have proved that $x_sw/x_{\max(w)}=g(x_sw)$.
\end{proof}
After this preparation, we prove the following
\begin{Theorem} Let $u=x_1^{a_1}\ldots x_n^{a_n},\ v=x_1^{b_1}\ldots x_n^{b_n},\ u,v\in\mathcal{M}_d$, with $u\geq_{lex}v$, and $I=(\mathcal{L}(u,v))$ be a completely lexsegment ideal which has linear resolution. Then the decomposition function $g:M(I)\rightarrow G(I)$ associated to the ordering $\prec$ of the generators from $G(I)$ is regular.
\end{Theorem}
\begin{proof} Let $w\in\mathcal{L}(u,v)$ and $s\in\set(w)$. We have to show that $\set(g(x_sw))\subset\set(w)$.
Let $t\in\set(g(x_sw))$. In order to prove that $t\in\set(w)$, that is $x_tw\in I_{\prec w}$, we will consider the following two cases:
\underline{\textit{Case 1}}: Let $x_sw\geq_{lex}x_1v$. By Lemma \ref{desc}, $g(x_sw)=x_sw/x_1$. Since $t\in\set(g(x_sw))$, we have $$\frac{x_tx_sw}{x_1}\in I_{\prec\frac{x_sw}{x_1}},$$ so there exists $w'\prec x_sw/x_1$, $w'\in\mathcal{L}(u,v)$, and a variable $x_j$, such that
$x_t x_sw/x_1=x_jw',
$
that is
\begin{eqnarray}x_tx_sw=x_1x_jw'.\label{6}
\end{eqnarray}
By Lemma \ref{1nset}, $s,t\neq1$ and, since $w'\prec\ x_sw/x_1$, we have $j\neq t$. Note also that $w'\prec w$ since $\nu_1(w')<\nu_1(w)$. If $j=s$ then $x_tw=x_1w'\in I_{\prec w}$ and $t\in\set(w)$.
Now let $j\neq s$. If $j=1$, we have
$x_tx_sw=x_1^2w',
$
which implies that $\nu_1(w')=\nu_1(w)-2$. The following inequalities hold:
$v<_{lex}x_1w'/x_s<_{lex}w\leq_{lex}u,
$
the first one being true since $v\leq_{lex}w'$, so $\nu_1(v)\leq\nu_1(w')$. These inequalities show that $x_1w'/x_s\in\mathcal{L}(u,v)$. But we also have $x_1w'/x_s\prec w$, hence $x_1w'/x_s\in I_{\prec w}$.
To finish this case we only need to treat the case $j\neq1,\ j\neq s$. We are going to show that at least one of the monomials $x_1w'/x_s$ or $x_jw'/x_s$ belongs to $I_{\prec w}.$ In any case this will lead to the conclusion that $x_t w\in I_{\prec w}$ by using (\ref{6}).
From the equality (\ref{6}), we have $x_j|w$, hence $j\leq\max(w)$, and $\nu_1(w')=\nu_1(w)-1$. Since $w'\prec\ x_sw/x_1$ and $\nu_1(w')=\nu_1(w)-1=\nu_1(x_sw/x_1)$, we get
\begin{eqnarray}w'>_{lex}x_sw/x_1,\label{7}
\end{eqnarray} which gives
\[x_1w'/x_s>_{lex}v.
\]
If the inequality \begin{eqnarray}x_1w'/x_s\leq_{lex}u\label{8}\end{eqnarray}holds, then we get $x_1w'/x_s\in\mathcal{L}(u,v)$. We also note that $\nu_1(x_1w'/x_s)=\nu_1(w)$ and $x_1w'/x_s>_{lex}w$ (by (\ref{7})). Therefore $x_1w'/x_s\prec w$ and we may write
$x_tw=x_j(x_1w'/x_s)\in I_{\prec w}.
$
This implies that $t\in\set(w)$.
Now we look at the monomial $x_jw'/x_s$ for which we have $\nu_1(x_jw'/x_s)=\nu_1(w')<\nu_1(w)$, so
$x_jw'/x_s<_{lex}w\leq_{lex}u.
$
If the inequality \begin{eqnarray}x_jw'/x_s\geq_{lex}v\label{9}\end{eqnarray}holds, we obtain $x_jw'/x_s\in\mathcal{L}(u,v)$. Obviously we have $x_jw'/x_s\prec w$. By using (\ref{6}), we may write
$x_tw=x_1(x_jw'/x_s)\in I_{\prec w},
$
which shows that $t\in\set(w)$.
To finish the proof in the Case $1$ we need to consider the situation when both inequalities (\ref{8}) and (\ref{9}) fail. Hence, let
\[x_1w'/x_s>_{lex}u\ \mbox{and}\ x_jw'/x_s<_{lex}v.
\]
We will show that this inequalities cannot hold simultaneously. Comparing the exponents of $x_1$ in the monomials involved in the above inequalities, we obtain $\nu_1(w')=b_1\geq a_1-1$. Since, by hypothesis, $x_sw>_{lex} x_1v,$ we have $\nu_1(w)>b_1.$ On the other hand, $w\leq_{lex} u$ implies that $\nu_1(w)\leq a_1.$ So $b_1=a_1-1$ and $\mathcal{L}(u,v)$ satisfies the condition (c) in Theorem \ref{completelylex}. Let, as usually, $z$ be the largest monomial with respect to the lexicographical order such that $z<_{lex}v$.
Since $x_jw'/x_s<_{lex}v$, we have $x_jw'/x_s\leq_{lex}z$. By Lemma \ref{lema1} and using the condition $x_1z/x_{\max(z)}\leq_{lex}u$, we obtain:
$x_1x_jw'/(x_sx_{\max(x_jw'/x_s)})\leq_{lex}u.
$
But our assumption was that
$u<_{lex}x_1w'/x_s.
$
Therefore, combining the last two inequalities, after cancellation, one obtains that
$x_j<_{lex}x_{\max(x_jw'/x_s)}=x_{\max(x_tw/x_1)}=x_{\max(x_tw)}.
$
This leads to the inequality $j>\max(x_tw)$ and, since $j\leq\max(w)$, we get $\max(w)>\max(x_tw)$, which is impossible.
\underline{\textit{Case 2}}: Let $x_sw<_{lex}x_1v$. Then $g(x_sw)=x_sw/x_{\max(w)}$. In particular we have $x_sw/x_{\max(w)}\prec w$. Indeed, since $s\in \set(w),$ we have $x_sw\in I_{\prec w}$, that is there exists $w'\in \mathcal{L}(u,v), w'\prec w,$ such that $x_sw\in I_{\preceq w'}.$ By the definition of the decomposition function we have $g(x_sw)\preceq w'$ and next we get $g(x_sw)\prec w.$ Since $\nu_1(x_sw/x_{\max(w)})=\nu_1(w)$, the above inequality implies that $x_sw/x_{\max(w)}>_{lex}w$, that is $x_s>_{lex}x_{\max(w)}$ which means that $s<\max(w)$.
As $t\in\set(g(x_sw))$, there exists $w'\prec\ x_sw/x_{\max(w)}$, $w'\in\mathcal{L}(u,v)$, and a variable $x_j$, such that
\[x_tx_sw/x_{\max(w)}=x_jw',
\]
that is
\begin{eqnarray}x_tx_sw=x_jx_{\max(w)}w'.\label{10}
\end{eqnarray}
As in the previous case, we would like to show that one of the monomials $x_{\max(w)}w'/x_s$ or $x_jw'/x_s$ belongs to $\mathcal{L}(u,v)$ and it is strictly less than $w$ with respect to $\prec$. In this way we obtain $x_tw\in I_{\prec w}$ and $t\in\set(w)$.
We begin our proof noticing that $s,t\neq1$, by Lemma \ref{1nset}. The equality $j=t$ is impossible since $w'\neq\ x_sw/x_{\max(w)}$. If $j=s$, then $x_tw=w'x_{\max(w)}\in I_{\preceq w'}$. But $w'\prec\ x_sw/x_{\max(w)}\prec w$, hence $x_tw\in I_{\prec w}$.
Let $j\neq s,t$. From the equality (\ref{10}) we have $x_j|w$, so $j\leq\max(w)$. We firstly consider $j=1$. Then the equality (\ref{10}) becomes
\begin{eqnarray}x_tx_sw=x_1x_{\max(w)}w'.\label{11}
\end{eqnarray}
Since $s< \max(w)$, we have
$x_{\max(w)}w'/x_s<_{lex}w'\leq_{lex}u.
$
If the inequality $x_{\max(w)}w'/x_s$ $\geq_{lex}v$ holds too, then $x_{\max(w)}w'/x_s\in\mathcal{L}(u,v)$ and, as $\nu_1(w')<\nu_1(w)$, it follows that $x_{\max(w)}w'/x_s\prec w$. From (\ref{11}), we have $x_tw=x_1(x_{\max(w)}w'/x_s)\in I_{\prec w}$, hence $t\in\set(w)$.
From the inequality $x_sw<_{lex}x_1v$, we get
\[x_sw<_{lex}x_1w',
\]
so
\[x_1w'/x_s>_{lex}w.
\]
Let us assume that $x_1w'/x_s\leq_{lex}u$. Since $\nu_1(x_1w'/x_s)=\nu_1(w)$, by using the definition of the ordering $\prec$ we get $x_1w'/x_s\in I_{\prec w}$. Then we may write $x_tw=x_{\max(w)}(x_1w'/x_s)\in I_{\prec w}$.
It remains to consider that
$x_{\max(w)}w'/x_s<_{lex}v\ \mbox{and}\ x_1w'/x_s>_{lex}u.
$
Proceeding as in the case 1 we show that we reach a contradiction and this ends the proof for $j=1$.
We only need to notice that we have to consider $b_1\leq a_1-1.$ Indeed, we can not have $b_1=a_1$ since one may find in $\mathcal{L}(u,v)$ at least two monomials, namely $w$ and $w',$ with $\nu_1(w')<\nu_1(w).$
Finally, let $j\neq1$. Recall that in the equality (\ref{10}) we have $j\neq 1,t,s $ and $s<\max(w)$. From (\ref{10}) we obtain $\nu_1(w)=\nu_1(w')$. Since $w'\prec \ x_s w/x_{\max(w)}$, we have $w'>_{lex} x_sw/x_{\max(w)}$, that is \begin{eqnarray}w'x_{\max(w)}>_{lex}x_sw.\label{12}\end{eqnarray}
Replacing $w'x_{\max(w)}$ by $x_tx_sw/x_j$ in (\ref{12}), we get $x_t>_{lex}x_j$, which means $t<j$. It follows that:
$x_{\max(w)}w'/x_s=x_tw/x_j>_{lex} w\geq_{lex}v.
$
Since $s<\max(w)$, as in the proof for $j=1$, we have $x_{\max(w)}w'/x_s\leq_{lex}u$. Therefore $x_{\max(w)}w'/x_s\in\mathcal{L}(u,v)$. In addition, from (\ref{12}), $x_{\max(w)}w'/x_s>_{lex}w$ and $\nu_1(x_{\max(w)}w'/x_s)=\nu_1(w)$, so $x_{\max(w)}w'/x_s\prec w$. In other words, we have got that
$x_tw=x_j(x_{\max(w)}w'/x_s)\in I_{\prec w}
$
and $t\in\set(w)$.
\end{proof}
The general problem of determining the resolution of arbitrary lexsegment ideals is not completely solved. The resolutions of the lexsegment ideals with linear quotients are described in \cite{HT} using iterated mapping cones. We recall this construction from \cite{HT}. Suppose that the monomial ideal $I$ has linear quotients with respect to the ordering $u_1,\ldots,u_m$ of its minimal generators. Set $I_j=(u_1,\ldots,u_j)$ and $L_j=(u_1,\ldots,u_j):u_{j+1}.$ Since $I_{j+1}/I_j\simeq S/L_j,$ we get the exact sequences $$0\rightarrow S/L_j \rightarrow S/I_j\rightarrow S/I_{j+1}\rightarrow 0,$$ where the morphism $S/L_j \rightarrow S/I_j$ is the multiplication by $u_{j+1}.$ Let $F^{(j)}$ be a graded free resolution of $S/I_j$, $K^{(j)}$ the Koszul complex associated to the regular sequence $x_{k_1},\ldots,x_{k_l}$ with $k_i\in\set(u_{j+1}),$ and $\psi^{(j)}:K^{(j)}\rightarrow F^{(j)}$ a graded complex morphism lifting the map $S/L_j \rightarrow S/I_j$. Then the mapping cone $C(\psi^{(j)})$ of $\psi^{(j)}$ yields a free resolution of $S/I_{j+1}.$ By iterated mapping cones we obtain step by step a graded free resolution of $S/I.$
\begin{Lemma}\cite{HT} Suppose $\deg\ u_1 \leq \deg\ u_2 \leq \ldots\leq \deg\ u_m.$ Then the iterated mapping
cone $\mathbb{F}$, derived from the sequence $u_1,\ldots,u_m,$ is a minimal graded free resolution
of $S/I$, and for all $i > 0$ the symbols
\[f(\sigma; u)\ \mbox{with}\ u\in G(I),\ \sigma \subset \set(u),\ |\sigma| = i - 1
\]
form a homogeneous basis of the $S-$module $F_i$. Moreover $\deg(f(\sigma; u)) = |\sigma| +\deg(u)$.
\end{Lemma}
\begin{Theorem}\cite{HT} Let $I$ be a monomial ideal of $S$ with linear quotients, and $\mathbb{F}_{\bullet}$ the graded
minimal free resolution of $S/I$. Suppose that the decomposition function $g : M(I) \rightarrow G(I)$ is regular. Then the chain map $\partial$ of $\mathbb{F}_{\bullet}$ is given by
\[\partial(f(\sigma; u)) = -\sum_{s\in\sigma}(-1)^{\alpha(\sigma;s)}x_sf(\sigma\setminus s;u)+\sum_{s\in\sigma}(-1)^{\alpha(\sigma;s)}\frac{x_su}{g(x_su)}f(\sigma\setminus s;g(x_su)),\]
if $\sigma\neq\emptyset$, and
\[\partial(f(\emptyset; u)) = u\] otherwise.
Here $\alpha(\sigma;s)=|\{t\in\sigma\ |\ t<s\}|$.
\end{Theorem}
In our specific context we get the following
\begin{Corollary} Let $I=(\mathcal{L}(u,v))\subset S$ be a completely lexsegment ideal with linear quotients with respect to $\prec$ and $\mathbb{F}_{\bullet}$ the graded minimal free resolution of $S/I$. Then the chain map of $\mathbb{F}_{\bullet}$ is given by
\[\partial(f(\sigma; w)) = -\sum_{s\in\sigma}(-1)^{\alpha(\sigma;s)}x_sf(\sigma\setminus s;w)+\sum_{\stackrel{s\in\sigma:}{x_sw\geq_{lex}}x_1v}(-1)^{\alpha(\sigma;s)}x_1f\left(\sigma\setminus s;\frac{x_sw}{x_1}\right)+\]\[+\sum_{\stackrel{s\in\sigma:}{x_sw<_{lex}}x_1v}(-1)^{\alpha(\sigma;s)}x_{\max(w)}f\left(\sigma\setminus s;\frac{x_sw}{x_{\max(w)}}\right),\]
if $\sigma\neq\emptyset$, and
\[\partial(f(\emptyset; w)) = w\] otherwise. For convenience we set $f(\sigma;w)=0$ if $\sigma\nsubseteq\set{w}$.
\end{Corollary}
\begin{Example}\rm
Let $u=x_1^2x_2$ and $v=x_2^3$ be monomials in the polynomial ring $S=k[x_1,x_2,x_3]$. Then $$\mathcal{L}(u,v)=\{x_2^3,\ x_1x_2^2,\ x_1x_2x_3,\ x_1x_3^2,\ x_1^2x_2\}.$$ The ideal $I=(\mathcal{L}(u,v))$ is a completely lexsegment ideal with linear quotients with respect to this ordering of the generators. We denote $u_1=x_2^3,\ u_2=x_1x_2^2,\ u_3=x_1x_2x_3,\ u_4=x_1x_3^2,\ u_5=x_1^2x_2$. We have $\set(u_1)=\emptyset,\ \set(u_2)=\{2\},\ \set(u_3)=\{2\},\ \set(u_4)=\{2\},\ \set(u_5)=\{2,3\}$. Let $\mathbb{F}_{\bullet}$ be the minimal graded free resolution of $S/I$.
Since $\max\{|\set(w)|\mid w\in\mathcal{L}(u,v)\}=2$, we have $F_i=0$, for all $i\geq4$.
A basis for the $S-$module $F_1$ is $\{f(\emptyset;u_1),\ f(\emptyset;u_2),\ f(\emptyset;u_3),\ f(\emptyset;u_4),\ f(\emptyset;u_5)\}$.
A basis for the $S-$module $F_2$ is $$\{f(\{2\};u_2),\ f(\{2\};u_3),\ f(\{2\};u_4),\ f(\{2\};u_5),\ f(\{3\};u_5)\}.$$
A basis for the $S-$module $F_3$ is $\{f(\{2,3\};u_5)\}$.
We have the minimal graded free resolution $\mathbb{F}_{\bullet}$:
\[0\rightarrow S(-5)\stackrel{\partial_2}{\rightarrow} S(-4)^5\stackrel{\partial_1}{\rightarrow} S(-3)^5\stackrel{\partial_0}{\rightarrow} S\rightarrow S/I\rightarrow 0
\]
where the maps are
\[\partial_0(f(\emptyset;u_i))=u_i,\ \mbox{for}\ 1\leq i\leq 5,
\]
so \[\partial_0=
\left(\begin {array}{ccccc}
x_2^3& x_1x_2^2& x_1x_2x_3& x_1x_3^2& x_1^2x_2
\end{array}\right) .\]
\[\begin{array}{lll}
\partial_1(f(\{2\};u_2))&= &-x_2f(\emptyset;u_2)+x_1f(\emptyset;u_1),\\
\partial_1(f(\{2\};u_3))&= &-x_2f(\emptyset;u_3)+x_3f(\emptyset;u_2),\\
\partial_1(f(\{2\};u_4))&= &-x_2f(\emptyset;u_4)+x_3f(\emptyset;u_3),\\
\partial_1(f(\{2\};u_5))&= &-x_2f(\emptyset;u_5)+x_1f(\emptyset;u_2),\\
\partial_1(f(\{3\};u_5))&= &x_3f(\emptyset;u_5)-x_1f(\emptyset;u_3),\\
\end{array}\]
so
\[\partial_1=
\left(\begin {array}{ccccc}
x_1& 0& 0& 0& 0\\
-x_2& x_3& 0& x_1& 0\\
0& -x_2& x_3& 0& -x_1\\
0& 0& -x_2& 0& 0\\
0& 0& 0& -x_2& x_3
\end{array}\right) .\]
\[\partial_2(f(\{2,3\};u_5))=-x_2f(\{3\};u_5)+x_3f(\{2\};u_5)+x_1f(\{3\};u_2)-x_1f(\{2\};u_3)=\]\[=-x_2f(\{3\};u_5)+x_3f(\{2\};u_5)-x_1f(\{2\};u_3),
\]
since $\{3\}\nsubseteq\set(u_2)$, so\[\partial_2=
\left(\begin {array}{c}
0\\
-x_1\\
0\\
x_3\\
-x_2
\end{array}\right) .\]
\end{Example}
\section{Non-completely lexsegment ideals with linear resolutions}
\begin{Theorem}\label{noncompletely}
Let $u=x_1^{a_1}\ldots x_n^{a_n},\ v=x_2^{b_2}\ldots x_n^{b_n}$ be monomials of degree $d$ in $S,$ $a_1> 0.$ Suppose that the ideal $I=({\mathcal L}(u,v))$ is not completely lexsegment ideal. Then $I$ has linear resolution if and only if $I$ has linear quotients.
\end{Theorem}
\begin{proof}
We only have to prove that if $I$ has linear resolution then $I$ has linear quotients for a suitable ordering of its minimal monomial
generators. By \cite[Theorem 2.4]{ADH}, since $I$ has linear resolution, $u$ and $v$ have the form: $$u=x_1x_{l+1}^{a_{l+1}}\ldots
x_n^{a_n},\ v=x_lx_n^{d-1}, \text {for some } l\geq 2.$$ Then the ideal $I=({\mathcal L}(u,v))$ can be written as a sum of ideals $I=J+K,$ where $J$ is the ideal generated by all the monomials of ${\mathcal L}(u,v)$ which are not divisible by $x_1$ and $K$ is
generated by all the monomials of ${\mathcal L}(u,v)$ which are divisible by $x_1.$ More precise, we have $$J=(\{w\mid
x_2^d\geq_{lex} w\geq_{lex} v\})$$ and $$K=(\{w\mid u\geq_{lex} w\geq_{lex} x_1x_n^{d-1}\}).$$ One may see that $J$ is generated by
the initial lexsegment $\mathcal{L}^i(v)\subset k[x_2,\ldots,x_n],$ and hence it has linear quotients with respect to
lexicographical order $>_{lex}.$ Let $G(J)=\{g_1\prec \ldots\prec g_m\},$ where $g_i\prec g_j\ \text{if and only if}\
g_i>_{lex} g_j.$ The ideal $K$ is isomorphic with the ideal generated by the final lexsegment of degree $d-1$
$$\mathcal{L}^f(u/x_1)=\{w\mid u/x_1\geq_{lex} w\geq_{lex} x_n^{d-1},\ \deg(w)=d-1\}.$$ Since final lexsegments
are stable with respect to the order $x_n>\ldots >x_1$ of the variables, it follows that the ideal $K$ has linear quotients
with respect to $>_{\overline{lex}},$ where by $\overline{lex}$ we mean the lexicographical order corresponding to
$x_n>\ldots >x_1.$ Let $G(K)=\{h_1\prec \ldots\prec h_p\},$ where $h_i\prec h_j\ \text{if and only if}\
h_i>_{\overline{lex}} h_j.$ We consider the following ordering of the monomials of $G(I):$ $$G(I)=\{g_1\prec \ldots\prec
g_m\prec h_1\prec \ldots\prec h_p\}.$$ We claim that, for this ordering of its minimal monomial generators, $I$ has linear quotients. In order to check this, we firstly notice that $I_{\prec g}:g=J_{\prec g}:g$ for every $g\in G(J).$ Since $J$
has linear quotients with respect to $\prec$ it follows that $J_{\prec g}:g$ is generated by variables. Now it is enough
to show that, for any generator $h$ of $K,$ the colon ideal $I_{\prec h}:h$ is generated by variables.
We note that $$I_{\prec h}:h=J:h + K_{\prec h}:h.$$ Since $K$ is with linear quotients, we already know that
$K_{\prec h}:h$ is generated by variables. Therefore we only need to prove that $J:h$ is generated by variables. We
will show that $J:h=(x_2,\ldots,x_l)$ and this will end our proof. Let $m\in J:h$ be a monomial. It follows that $mh\in J.$ Since $h$ is a generator of $ K,$ $h$ is of the form $h=x_1x_{l+1}^{\alpha_{l+1}}\ldots x_n^{\alpha_n},$ that is $h\not\in (x_2,\ldots,x_l).$ But this implies that $m$ must be in the ideal $(x_2,\ldots,x_l).$ For the reverse inclusion, let $2\leq t\leq l.$ Then $x_t h=x_1 \gamma$ for some monomial $\gamma,$ of degree $d.$ Replacing $h$ in the equality we get $\gamma=x_tx_{l+1}^{\alpha_{l+1}}\ldots x_n^{\alpha_n}$ which shows that $\gamma$ is a generator of $J.$ Hence $x_th\in J.$
\end{proof}
\begin{Example}\rm
Let $I=(\mathcal{L}(u,v))\subset k[x_1,\ldots,x_6]$ be the lexsegment ideal of degree $4$ determined by the monomials $u=x_1x_3^2x_5$ and $v=x_2x_6^3.$ $I$ is not a completely lexsegment ideal as it follows applying \cite[Theorem 2.3]{DH}, but $I$ has linear resolution by \cite[Theorem 2.4]{ADH}. $I$ has linear quotients if we order its minimal monomial generators as indicated in the proof of the above theorem. On the other hand, if we order the generators of $I$ using the order relation defined in the proof of Theorem \ref{colex} we can easy see that $I$ does not have linear quotients. Indeed, following the definition of the order relation from Theorem \ref{colex} we should take $$G(I)=\{x_2^4\prec x_2^3x_3\prec\ldots\prec x_2x_6^3\prec x_1x_3^2x_5\prec x_1x_3^2x_6\prec x_1x_3x_4^2\prec\ldots\prec x_1x_6^3\}.$$ For $h=x_1x_3x_4^2$ one may easy check that $I_{\prec h}:h$ is not generated by variables.
\end{Example}
\begin{Example}\rm Let $u=x_1x_3^2$, $v=x_2x_4^2$ be monomials in $k[x_1,\ldots, x_4]$. Then $I=(L(u,v))\subset k[x_1,\ldots, x_4]$ is a non-completely ideal with linear resolution and, by the proof of Theorem \ref{noncompletely}, $I$ has linear quotients with respect to the following ordering of its minimal monomial generators:
$$x_2^3,\ x_2^2x_3,\ x_2^2x_4,\ x_2x_3^2,\ x_2x_3x_4,\ x_2x_4^2,\ x_1x_4^2,\ x_1x_3x_4,\ x_1x_3^2.$$
We note that $\set(x_1x_4^2)=\{2\}$ and $\set(g(x_1x_2x_4^2))=\set(x_2x_4^2)=\{2,3\}\nsubseteq\set(x_1x_4^2)$, so the decomposition function is not regular for this ordering of the generators.
\end{Example}
\section{Cohen-Macaulay lexsegment ideals} \label{Section3}
In this section we study the dimension and the depth of arbitrary lexsegment ideals. These results are applied to describe the lexsegments ideals which are Cohen-Macaulay. We begin with the study of the dimension. As in the previous sections, let $d\geq 2$ be an integer. We denote $\frak{m}=(x_1,\ldots,x_n).$ It is clear that if $I=(\mathcal{L}(u,v))\subset S$ is a lexsegment ideal of degree $d$ then $\dim(S/I)=0$ if and only if $I=\frak{m}^d.$
\begin{Proposition}\label{dim} Let $u=x_1^{a_1}\ldots x_n^{a_n},\ v=x_q^{b_q}\ldots x_n^{b_n}$, $1\leq q\leq n$, $a_1,b_q>0,$ be two monomials of degree $d$ such that $u\geq_{lex}v$ and let $I$ be the lexsegment ideal generated by $\mathcal{L}(u,v)$. We assume that $I\neq\frak{m}^d$. Then
\[\dim(S/I)=
\left\{\begin {array}{ll}
n-q, & \mbox{if}\ 1\leq q<n,\\
1, & \mbox{if}\ q=n.
\end{array}\right. \]
\end{Proposition}
\begin{proof} For $q=1$, we have $I\subset(x_1)$. Obviously $(x_1)$ is a minimal prime of $I$ and $\dim(S/I)=n-1$.
Let $q=n$, that is $v=x_n^d$ and $\mathcal{L}(u,v)=\mathcal{L}^f(u)$. We may write the ideal $I$ as a sum of two ideals,
$I=J+K,$ where
$J=(x_1\mathcal{L} (u/x_1,x_n^{d-1}))
$
and
$K=(\mathcal{L}(x_2^d,x_n^d)).
$
Let $p\supset I$ be a monomial prime ideal. If $x_1\in p$, then $J\subseteq p$. Since $p$ also contains $K$, we have $p\supset (x_2,\ldots,x_n)$. Hence $p=(x_1,x_2,\ldots,x_n)$. If $x_1\notin p$, we obtain $(x_2,\ldots,x_n)\subset p$. Hence, the only minimal prime ideal of $I$ is $(x_2,\ldots,x_n)$. Therefore, $\dim(S/I)=1$.
Now we consider $1<q<n$ and write $I$ as before,
$I=J+K,
$
where $J=(x_1\mathcal{L}(u/x_1,x_n^{d-1}))$ and $K=(\mathcal{L}(x_2^d,v))$.
Firstly we consider $u=x_1^d.$ Let $p\supset I$ be a monomial prime ideal. Then $p\ni x_1$ and, since $p\supset K,$ we also have $p\supset (x_2,\ldots,x_q).$ Hence $(x_1,\ldots,x_q)\subset p.$ Since $I\subset (x_1,\ldots,x_q),$ it follows that $(x_1,\ldots,x_q)$ is the only minimal prime ideal of $I.$ Therefore $\dim(S/I)=n-q.$
Secondly, let $a_1>1$ and $u\neq x_1^d.$ The lexsegment $\mathcal{L}(u/x_1,x_n^{d-1})$ contains the lexsegment $\mathcal{L}(x_2^{d-1},x_n^{d-1})$. Let $p$ be a monomial prime ideal which contains $I$ and such that $x_1\not\in p$. Then $p\supset\mathcal{L}(x_2^{d-1},x_n^{d-1})$ which implies that $(x_2,\ldots,x_n)\subset p$. Obviously we also have $I\subset(x_2,\ldots,x_n)$, hence $(x_2,\ldots,x_n)$ is a minimal prime ideal of $I$.
Let $p\supset I$ be a monomial prime ideal which contains $x_1$. Since $p\supset K$, we also have $(x_2,\ldots,x_q)\subset p$. This shows that $(x_1,\ldots, x_q)$ is a minimal prime ideal of $I$. In conclusion, for $a_1>1$, the minimal prime ideals of $I$ are $(x_1,\ldots,x_q)$ and $(x_2,\ldots, x_n)$. Since $q\leq n-1$, we get $\height(I)=q$ and $\dim(S/I)=n-q$.
Finally, let $a_1=1$, that is $u=x_1 x_l^{a_l}\ldots x_n^{a_n}$, for some $a_l>0$, $l\geq2$. As in the previous case, we obtain $(x_1,\ldots,x_q)$ a minimal prime ideal of $I$. Now we look for those minimal prime ideals of $I$ which do not contain $x_1$.
If $a_l=d-1$, the ideal $J=(x_1\mathcal{L}(u/x_1,x_n^{d-1}))$ becomes $J=(x_1\mathcal{L}(x_l^{d-1},x_n^{d-1}))$. If $p\supset I$ is a monomial prime ideal such that $x_1\notin p$, we get $(x_l,\ldots,x_n)\subset p$, and, since $p$ contains $K$, we obtain $(x_2,\ldots,x_q)\subset p$. This shows that if $q<l$ then $(x_2,\ldots,x_q,x_l,\ldots,x_n)$ is a minimal prime ideal of $I$ of height $q+n-l\geq q$, and if $q\geq l$, then $(x_2,\ldots,x_n)$ is a minimal prime ideal of height $n-1\geq q$. In both cases we may draw the conclusion that $\height(I)=q$ and, consequently, $\dim(S/I)=n-q$.
The last case we have to consider is $a_l<d-1$. Then $l<n$ and, with similar arguments as above, we obtain $\dim(S/I)=n-q$.
\end{proof}
In order to study the depth of arbitrary lexsegment ideals, we note that one can restrict to those lexsegments defined by monomials of the form $u=x_1^{a_1}\ldots x_n^{a_n},\ v=x_1^{b_1}\ldots x_n^{b_n}$ of degree $d$ with $a_1>0$ and $b_1=0$.
Indeed, if $a_1=b_1$, then $I=(\mathcal{L}(u,v))$ is isomorphic, as an $S-$module, with the ideal generated by the lexsegment $\mathcal{L}(u/x_1^{a_1},v/x_1^{b_1})$ of degree $d-a_1$. This lexsegment may be studied in the polynomial ring in a smaller number of variables.
If $a_1>b_1$, then $I=(\mathcal{L}(u,v))$ is isomorphic, as an $S-$module, with the ideal generated by the lexsegment $\mathcal{L}(u',v')$, where $u'=u/x_1^{b_1}$ has $\nu_1(u')=a_1-b_1>0$ and $v'=v/x_1^{b_1}$ has $\nu_1(v')=0$.
Taking into account these remarks, from now on, we consider lexsegment ideals of ends $u=x_1^{a_1}\ldots x_n^{a_n}$, $v=x_q^{b_q}\ldots x_n^{b_n}$, for some $q\geq2$, $a_1,b_q>0$.
The first step in the depth's study is the next
\begin{Proposition}\label{depthzero} Let $I=(\mathcal{L}(u,v))$, where $u=x_1^{a_1}\ldots x_n^{a_n}$, $v=x_q^{b_q}\ldots x_n^{b_n}$, $q\geq2$, $a_1,b_q>0$. Then $\depth(S/I)=0$ if and only if $x_nu/x_1\geq_{lex} v$.
\end{Proposition}
\begin{proof} Let $x_nu/x_1\geq_{lex} v$. We claim that
$(I\colon (u/x_1))=(x_1,\ldots,x_n).
$
Indeed, for $1\leq j\leq n$, the inequalities
$u\geq_{lex}x_ju/x_1\geq_{lex}x_nu/x_1\geq_{lex}v
$ hold.
They show that $x_ju/x_1\in I$ for $1\leq j\leq n$. Therefore $(x_1,\ldots,x_n)\subseteq(I\colon (u/x_1))$. The other inclusion is obvious. We conclude that $(x_1,\ldots,x_n)\in \Ass(S/I)$, hence $\depth(S/I)=0$.
For the converse, let us assume, by contradiction, that $x_nu/x_1<_{lex}v$. We will show that $x_1-x_n$ is regular on $S/I$. This will imply that $\depth(S/I)>0$, which contradicts our hypothesis. We firstly notice that, from the above inequality, we have $a_1-1=0$, that is $a_1=1$. Therefore, $u$ is of the form $u=x_1x_l^{a_l}\ldots x_n^{a_n}$, $l\geq2,\ a_l>0$. Moreover, we have $l\geq q.$
Let us suppose that $x_1-x_n$ is not regular on $S/I$, that is there exists at least a polynomial $f\notin I$ such that $f(x_1-x_n)\in I$. One may assume that all monomials of $\supp(f)$ do not belong to $I$. Let us choose such a polynomial $f=c_1w_1+\ldots+c_tw_t,\ c_i\in k,\ 1\leq i\leq t$, with $w_1>_{lex}w_2>_{lex}\ldots>_{lex}w_t,\ w_i\notin I,\ 1\leq i\leq t$.
Then $\ini_{lex}((x_1-x_n)f)=x_1w_1\in I$. It follows that there exists $\alpha\in G(I)$ such that
\begin{eqnarray}x_1w_1=\alpha\cdot\alpha\,'.\label{3.1}
\end{eqnarray}
for some monomial $\alpha\,'$. We have $x_1\nmid\alpha\,'$ since, otherwise, $w_1\in I$, which is false. Hence $\alpha$ is a minimal generator of $I$ which is divisible by $x_1$, that is $\alpha$ is of the form $\alpha=x_1\gamma$, for some monomial $\gamma$ such that $x_n^{d-1}\leq_{lex}\gamma\leq_{lex}u/x_1$. Looking at $(\ref{3.1})$, we get $w_1=\gamma \alpha\,'$. This equality shows that $x_1\nmid w_1$. We claim that the monomial $x_nw_1$ does not cancel in the expansion of $f(x_1-x_n)$. Indeed, it is clear that $x_nw_1$ cannot cancel by some monomial $x_nw_i$, $i\geq2$. But it also cannot cancel by some monomial of the form $x_1w_i$ since $x_nw_1$ is not divisible by $x_1$. Now we may draw the conclusion that there exists a monomial $w\notin I$ such that $w(x_1-x_n)\in I$, that is $wx_1,\ wx_n\in I$.
Let $w\notin I$ be a monomial such that $wx_1,\ wx_n\in I$, let $\alpha,\ \beta\in\mathcal{L}(u,v)$ and $\alpha\,',\ \beta\,'$ monomials such that
\begin{eqnarray}x_1w=\alpha\cdot\alpha\,'\label{3.2}
\end{eqnarray}
and
\begin{eqnarray}x_nw=\beta\cdot\beta\,'.\label{3.3}
\end{eqnarray}
As before, we get $x_1\nmid w$, hence $\beta$ must be a minimal generator of $I$ such that
$x_2^d\geq_{lex}\beta\geq_{lex}v.
$ By using (\ref{3.3}), we can see that $x_n$ does not divide $\beta\,',$ hence $x_n|\beta.$ It follows that $w$ is divisible by $\beta/ x_n$. $w$ is also divisible by $\alpha/x_1$. Therefore,
$\delta=\lcm(\alpha/x_1,\beta/x_n) | w.$ If $\deg \delta\geq d$ there exists a variable $x_j,$ with $j\geq 2,$ such that $(x_j\beta/x_n )| \delta,$ thus $(x_j\beta/x_n) | w.$ It is obvious that $x_2^d\geq_{lex} x_j\beta/x_n \geq_{lex} \beta\geq_{lex} v,$ hence $x_j\beta/x_n $ is a minimal generator of $I$ which divides $w,$ contradiction. This implies that $\delta$ has the degree $d-1.$ This yields $\alpha/x_1=\beta/x_n.$ Then $\beta=x_n\alpha /x_1\leq_{lex} x_n u/x_1<_{lex} v,$ contradiction.
\end{proof}
\begin{Corollary}
Let $I=(\mathcal{L}(u,v))$, where $u=x_1^{a_1}\ldots x_n^{a_n}$, $v=x_q^{b_q}\ldots x_n^{b_n}$, $q\geq2$, $a_1,b_q>0$. Then $\projdim(S/I)=n$ if and only if $x_nu/x_1\geq_{lex} v$.
\end{Corollary}
\begin{Corollary}Let $I=\left(\mathcal{L}^f(u)\right)$ be the ideal generated by the final lexsegment defined by $u=x_1^{a_1}\ldots x_n^{a_n},\ a_1>0$. Then $\depth(S/I)=0$.
\end{Corollary}
\begin{Corollary}
Let $I=\left(\mathcal{L}^i(v)\right)$ be the ideal generated by the initial lexsegment defined by the monomial $v$. Then $\depth(S/I)=0$ if and only if $v\leq_{lex}x_1^{d-1}x_n$.
\end{Corollary}
Next we are going to characterize the lexsegment ideals $I$ such that $\depth\ S/I>0$, that is $x_nu/x_1<_{lex}v$, which implies that $u$ has the form $u=x_1x_l^{a_l}\ldots x_n^{a_n}$, for some $l\geq2,\ a_l>0$ and $l>q,$ or $l=q$ and $a_q\leq b_q$. We denote $u\,'=u/x_1=x_l^{a_l}\ldots x_n^{a_n}$. Then we have $x_nu\,'<_{lex}v$. From the proof of Proposition \ref{depthzero} we know that $x_1-x_n$ is regular on $S/I$. Therefore
\[\depth(S/I)=\depth(S\,'/I\,')+1,
\]
where $S\,'=k[x_2,\ldots,x_n]$ and $I\,'$ is the ideal of $S\,'$ whose minimal monomial generating set is $G(I\,')=x_n\mathcal{L}(u\,',x_n^{d-1})\cup\mathcal{L}^i(v)$.
\begin{Lemma}\label{depthneq0} In the above notations and hypothesis on the lexsegment ideal $I$, the following statements hold:
\begin{itemize}
\item[(a)] If $v=x_2^d$ and $l\geq4$, then $\depth(S\,'/I\,')=l-3$.
\item[(b)] If $v=x_2^{d-1}x_j$ for some $3\leq j\leq n-2$ and $l\geq j+2$ then $\depth(S\,'/I\,')=l-j-1$.
\item[(c)] $\depth(S\,'/I\,')=0$ in all the other cases.
\end{itemize}
\end{Lemma}
\begin{proof} (a) Let $v=x_2^d$ and $l\geq4$. The ideal $I\,'\subset S '$ is minimally generated by all the monomials $x_n\gamma$, where $x_n^{d-1}\leq_{lex}\gamma\leq_{lex}u\,'$, $\deg(\gamma)=d-1$, and by the monomial $x_2^d$. Then it is clear that $\{x_3,\ldots,x_{l-1}\}$ is a regular sequence on $ S\,'/I\,'$, hence
$$\depth S\,'/I\,'=\depth\frac{S\,'/I\,'}{(x_3,\ldots,x_{l-1})S\,'/I\,'}+l-3.
$$We have
$$\frac{S\,'/I\,'}{(x_3,\ldots,x_{l-1})S\,'/I\,'}\cong\frac{k[x_2,x_l,\ldots,x_n]}{I\,'\cap k[x_2,x_l,\ldots,x_n]}.
$$In this way we may reduce the computation of $\depth(S\,'/I\,')$ to the case (c).
(b) Let $v=x_2^{d-1}x_j$, for some $3\leq j\leq n-2$ and $l\geq j+2$. Hence $I\,'$ is minimally generated by the following set of monomials
\[\{x_n\gamma\ |\ \gamma\ \mbox{monomial of degree}\ d-1\ \mbox{such that}\ x_n^{d-1}\leq_{lex}\gamma\leq_{lex}u\,' \}\cup\]\[\cup\{x_2^d,\ x_2^{d-1}x_3,\ldots,x_2^{d-1}x_j\}.
\]
Then $\{x_{j+1},\ldots,x_{l-1}\}$ is a regular sequence on $S\,'/I\,'$ and
$$\depth S\,'/I\,'=\depth\frac{S\,'/I\,'}{(x_{j+1},\ldots,x_{l-1})S\,'/I\,'}+(l-j-1).
$$
Since
$$\frac{S\,'/I\,'}{(x_{j+1},\ldots,x_{l-1})S\,'/I\,'}\cong\frac{k[x_2,\ldots,x_j,x_l,\ldots,x_n]}{I\,'\cap k[x_2,\ldots,x_j,x_l,\ldots,x_n]},
$$
we may reduce the computation of $\depth(S\,'/I\,')$ to the case (c).
(c) In each of the cases that it remains to treat, we will show that $(x_2,\ldots,x_n)\in \Ass(S '/I ')$, that is there exists a monomial $w\notin I\,'$ such that $I\,':w=(x_2,\ldots,x_n)$. This implies that $\depth(S\,'/I\,')=0$.
{\textit{Subcase $C_1$}:} $v=x_2^d,\ l=2$. Then $w=x_n^{d-1}\notin I\,'$ and
$x_n^{d-1}\leq_{lex}x_jw/x_n=x_jx_n^{d-2}\leq_{lex}x_2x_n^{d-2}\leq_{lex}x_l^{a_l}\ldots x_n^{a_n}=u',
$
for all $2\leq j\leq n$. Hence $\gamma=x_jw/x_n$ has the property that $x_n\gamma\in G(I\,')$. Therefore, $x_j\in I\,':w$ for all $2\leq j\leq n$. It follows that $I\,':w=(x_2,\ldots,x_n)$.
{\textit{Subcase $C_2$}:} $v=x_2^d,\ l=3$. Then $w=x_2^{d-1}x_n^{d-1}\notin I\,'$. Indeed, $x_2^d\nmid w$ and if we assume that there exists $x_n^{d-1}\leq_{lex}\gamma\leq_{lex}u\,'$, $\deg\gamma=d-1,$ such that $x_n\gamma|w$, we obtain $x_n\gamma|x_n^{d-1}$ which is impossible.
We show that $x_jw\in I\,'$ for all $2\leq j\leq n$. Indeed, $x_2w=x_2^dx_n^{d-1}\in I\,'$. Let $3\leq j\leq n$. Then $x_n^{d-1}\leq_{lex}x_jx_n^{d-2}\leq_{lex}x_3x_n^{d-2}\leq_{lex} u\,'$. It follows that $\gamma=x_jx_n^{d-2}$ has the property that $x_n\gamma=x_jx_n^{d-1}\in G(I\,')$. Since $x_n\gamma|x_jw$, we have $x_jw\in I\,'$. This arguments shows that $I\,':w=(x_2,\ldots,x_n)$.
{\textit{Subcase $C_3$}:} $v=x_2^{d-1}x_j$ for some $3\leq j\leq n-1$ and $2\leq l\leq j+1$. Let us consider again the monomial $w=x_2^{d-1}x_n^{d-1}$. It is clear that $x_tw\in I$ for all $2\leq t\leq j$. Let $t\geq j+1$. Then $x_tw$ is divisible by $x_tx_n^{d-1}$. Since $x_tx_n^{d-2}$ satisfies the inequalities
$x_n^{d-1}\leq_{lex}x_tx_n^{d-2}\leq_{lex}u\,',
$
we have $x_tx_n^{d-1}\in G(I\,')$. It follows that $x_t w\in I\,'$ for $t\geq j+1$. Assume that $w\in I\,'$. Since $x_2^{d-1}x_t\nmid w$ for $2\leq t\leq j$, we should have $x_n\gamma|w$ for some $\gamma$ of degree $d-1$ such that $x_n^{d-1}\leq_{lex}\gamma\leq_{lex}u\,'$. Since $\gamma|x_2^{d-1}x_n^{d-2}$ and $\gamma\leq_{lex}u\,'$, we get $l=2$ and $a_2=\nu_2(u\,')\geq\nu_2(\gamma)$. Let $\gamma=x_2^ax_n^{d-1-a}$, for some $a\geq1$. In this case we change the monomial $w.$ Namely, we consider the monomial $w'=x_2x_n^{d-2}$ which does not belong to $G(I\,')$ since it has degree $d-1$.
If $a_2\geq2$, for any $j$ such that $2\leq j\leq n$, we have
$x_n^{d-1}<_{lex}x_jw'/x_n=x_2x_jx_n^{d-3}<_{lex}x_l^{a_l}\ldots x_n^{a_n}=u\,'.
$
This shows that $x_jw'\in I\,'$ for $2\leq j\leq n$ and hence, $I\,':w=(x_2,\ldots,x_n)$.
If $a_2=1$, we take $w\,''=x_n^{d-1}\notin I\,'$. For all $j$ such that $2\leq j\leq n$, we have
$x_n^{d-1}\leq_{lex}x_j w\,''/x_n=x_jx_n^{d-2}\leq_{lex}x_2x_n^{d-2}\leq_{lex} u\,'.
$
Therefore $x_jw\,''\in I\,'$ for $2\leq j\leq n$, hence
$I\,':w\,''=(x_2,\ldots,x_n).
$
In conclusion we have proved that in every case one may find a monomial $w\notin I\,'$ such that $I':w=(x_2,\ldots, x_n)$.
{\textit{Subcase $C_4$}}: Finally, let $v\leq_{lex} x_2^{d-1}x_n$. In this case, the ideal $I\,':x_2^{d-1}$ obviously contains $(x_2,\ldots,x_n)$. Since the other inclusion is trivial, we get $I\,':x_2^{d-1}=(x_2,\ldots,x_n)$. It is clear that $x_2^{d-1}\notin I\,'$.
\end{proof}
By using Lemma \ref{depthneq0} we get:
\begin{Proposition}\label{depth} Let $I=(\mathcal{L}(u,v))$ be a lexsegment ideal defined by the monomials $u=x_1x_l^{a_l}\ldots x_n^{a_n},\ v=x_q^{b_q}\ldots x_n^{b_n}$ where $a_l,\ b_q>0$, $l,q\geq2$ and $x_n u/x_1<_{lex} v$. Then the following statements hold:
\begin{itemize}
\item[(a)] If $v=x_2^d$ and $l\geq4$ then $\depth(S/I)=l-2$;
\item[(b)] If $v=x_2^{d-1}x_j$ for some $3\leq j\leq n-2$ and $l\geq j+2$ then $\depth(S/I)=l-j$;
\item[(c)] $\depth(S/I)=1$ in all the other cases.
\end{itemize}
\end{Proposition}
\begin{proof} Since $x_1-x_n$ is regular on $S/I$ if $x_nu/x_1<_{lex}v$, we have $\depth(S/I)=\depth(S\,'/I\,')+1$. The conclusion follows applying Lemma \ref{depthneq0}.
\end{proof}
\begin{Corollary}
Let $I=(\mathcal{L}(u,v))$ be a lexsegment ideal defined by the monomials $u=x_1x_l^{a_l}\ldots x_n^{a_n},\ v=x_q^{b_q}\ldots x_n^{b_n}$ where $a_l,\ b_q>0$, $l,q\geq2$ and $x_nu/x_1<_{lex} v$. Then the following statements hold:
\begin{itemize}
\item[(a)] If $v=x_2^d$ and $l\geq4$ then $\projdim(S/I)=n-l+2$;
\item[(b)] If $v=x_2^{d-1}x_j$ for some $3\leq j\leq n-2$ and $l\geq j+2$ then $\projdim(S/I)=n-l+j$;
\item[(c)] $\projdim(S/I)=n-1$ in all the other cases.
\end{itemize}
\end{Corollary}
As a consequence of the results of this section we may characterize the Cohen--Macaulay lexsegment ideals.
In the first place, we note that the only Cohen--Macaulay lexsegment ideal such that $\dim(S/I)=0$ is $ I=\frak m^d$. Therefore it remains to consider Cohen--Macaulay ideals $I$ with $\dim(S/I)\geq1$.
\begin{Theorem} Let $n\geq3$ be an integer, let $u=x_1^{a_1}\ldots x_n^{a_n}$, $v=x_1^{b_1}\ldots x_n^{b_n}$, with $a_1>b_1\geq0,$ monomials of degree $d,$ and $I=(\mathcal{L}(u,v))\subset S$ the lexsegment ideal defined by $u$ and $v.$ We assume that $\dim(S/I)\geq 1$. Then $I$ is Cohen--Macaulay if and only if one of the following conditions is fulfilled:
\begin{itemize}
\item[(a)] $u=x_1x_n^{d-1}$ and $v=x_2^d$;
\item[(b)] $v=x_{n-1}^ax_n^{d-a}$ for some $a>0$ and $x_n\ u/x_1<_{lex}v$.
\end{itemize}
\end{Theorem}
\begin{proof} Let $u,v$ be as in (a). Then $\dim(S/I)=n-2$, by Proposition \ref{dim} and $\depth(S/I)=n-2$ by using (a) in Proposition \ref{depth} for $n\geq4$ and (c) for $n=3$.
Let $u,\ v$ as in (b). Then $\dim(S/I)=1$ by Proposition \ref{dim}. By using Proposition \ref{depth}(c), we obtain $\depth(S/I)=1$, hence $S/I$ is Cohen--Macaulay.
For the converse, in the first place, let us take $I$ to be Cohen--Macaulay of $\dim(S/I)=1$. By Proposition \ref{dim} we have $q=n$ or $q=n-1$. If $q=n$, then $v=x_n^d$ and $x_n u/x_1\geq_{lex}v$. By Proposition \ref{depthzero}, $\depth(S/I)=0$, so $I$ is not Cohen--Macaulay.
Let $q=n-1$, that is $v=x_{n-1}^ax_n^{d-a}$ for some $a>0$. By Proposition \ref{depthzero}, since $\depth(S/I)>0$, we must have $x_nu/x_1<_{lex}v$, thus we get (b).
Finally, let $\dim(S/I)\geq2$, that is $q\leq n-2$. By using Proposition \ref{depth}, we obtain $q=2$. Therefore $\dim(S/I)=\depth(S/I)=n-2$. Using again Proposition \ref{depth} (a),(b), it follows that $u=x_1x_n^{d-1}$ and $v=x_2^d$.
\end{proof}
|
1,108,101,566,775 | arxiv | \section*{Methods}
\subsection*{Numerical method}
System parameters are chosen to correspond to a fluid bath of density 0.95 g/cm$^3$, viscosity 16 cS and surface tension 20.9 dynes/cm vibrating vertically with amplitude $A_0$ and frequency $\omega_0 = 80$Hz. The resonant bouncing of the particle at the Faraday frequency triggers a quasi-monochromatic damped wave pattern with a corresponding
Faraday wavelength of $\lambda_F=4.75$~mm.
Each of the four cavities has a fixed length of 1.2 cm, corresponding to approximately 2.5$\lambda_F$. In all simulations, we set the coupling cavity depth to $d_c=6.3~\lambda_F $ which ensures strong inter-cavity coupling. We thus describe our bipartite tunneling system in terms of two coupled, two-level systems, as shown schematically in Figure~\ref{fig:Fig1}b.
\begin{figure*}
\includegraphics[width=\textwidth]{Fig4.png}
\caption{{\bf Strategy to optimize \bm{$S(a,a,a',a')=M(\alpha=a,\beta=a)-M(\alpha= a^\prime,\beta=a^\prime)+2M(\alpha=a,\beta=a')$} }. $S$ is optimized by searching a parameter regime $(a,a')$ near the maximum of $\delta M(a,a')=M(\alpha = a,\beta= a)-M(\alpha=a',\beta=a')$ and a range of $a'$ that maximizes $M(a,a')$ with $a$ fixed. {\bf a} Evolution of $M(\alpha=a, \beta =a)$ as function of $a$. The indicated values $a^*$ = 0.099 cm and $a^{\prime *}$ = 0.1033 cm are the S-maximizing values used in Figs. 2-3. Note $\delta a= a'-a$. The difference of the corresponding correlation functions $\delta M(a,a+\delta a)=\delta M(a,a')=M(\alpha=a,\beta=a)-M(\alpha=a',\beta=a')$ is marked in orange. {\bf b} Optimization of $\delta M(\alpha=a, \beta =a+\delta a)$ as a function of barrier depth $a$ and $\delta a $. The domain for which $(\max_{a,a'} (\delta M)-\delta M)/\max_{a,a'}(\delta M)>0.9$ is bound by the black dashed lines. {\bf c} 2$M(\alpha=a^*,\beta =a')$ as a function of depth $a’$ for fixed $a = a^*$. {\bf d}. Evolution of $S (a,a,a',a')$ in the correlation representation space $(M(a,a');M(a',a'))$ with $a=a^*$. The direction of increasing $a'$ is indicated by the blue arrow. The dots are colored with respect to their $S$ values. A grey dashed line indicates the limiting case $S=2$. }
\label{fig:Fig4}
\end{figure*}
Nachbin et al. \cite{Nachbin2017,Nachbin2018} formulated a theoretical model for the one-dimensional motion of walking droplets over a vibrating liquid bath with complex topography. Here we adjust this model in order to consider the cooperative tunneling of two identical particles in the geometry depicted in Figure \ref{fig:Fig1}b. The positions, $x_j$ ($j=1,2$), of the two identical particles of mass $m$ evolve according to Newton's Law:
\begin{equation}
m\ddot{x}_j+ c~F(t)\dot{x}_j = - F(t)\; \frac{\partial\eta}{\partial x}(x_j(t),t).
\label{Drop1ODE}
\end{equation}
The particle moves in response to gradients of the wave elevation $\eta(x,t)$, which thus plays the role of a time-dependent potential. The particle motion is resisted by a drag force proportional to its speed. The drag constant $c$ follows from the modeling presented in Molacek \& Bush \cite{Molacek2013b}. The time dependence of these propulsive and drag forces is prescribed by $F(t)$, as arises in the walker system owing to the droplet's bouncing~\cite{Bush2015a,Milewski2015,Nachbin2017}. In terms of their lateral motion, the particles are viewed as horizontal oscillators that can transition unpredictably between two neighboring cavities. The dichotomic property $X$ assessed for Bell’s inequality is assigned according to the particle location $x_j = x_j(t_m)\; (j = A,B)$ at the measurement time $t_m$. Specifically, $X = +1$ if the drop is in the outer, excited state, and $X = -1$ if it is in the inner, ground state.
The particles serve as moving wave sources that establish their own time-dependent wave potential that is computed as follows. The velocity potential of the liquid bath $\phi(x,z,t)$ is a harmonic function satisfying Laplace's equation. In the bulk of the fluid, the velocity field is given by $(u,v)=\nabla\phi$.
The wave model is formulated in the bath's reference frame, where the effective gravity is $g(t)= g+\gamma_0 \sin(\omega_0 t)$, where $g$ is the gravitation acceleration, and {$\gamma_0$} is the amplitude of the bath's vibrational acceleration. The wave field thus evolves according to \cite{Milewski2015,Nachbin2017}:
\begin{equation}
\frac{\partial \eta}{\partial t} = \frac{\partial \phi}{\partial z} + 2\nu
\frac{\partial^2\eta}{\partial x^2},
\label{Kin}
\end{equation}
\begin{equation}
\frac{\partial \phi}{\partial t} = - g(t) \eta + \frac{\sigma}{\rho} \frac{\partial^2\eta}{\partial x^2} + 2\nu
\frac{\partial^2\phi}{\partial x^2} -\!\! \sum_{j=1,2} \frac{P_d(x-x_j(t))}{\rho}.
\label{Bern}
\end{equation}
The particles ($j=A,B$) generate waves on the free surface by applying local pressure terms $P_d$. The wave forcing term $P_d(x-x_j(t))$ and the coefficient $F(t)$ are activated
only during a fraction of the Faraday period $T_F$, corresponding to the contact time $T_c$ in the walking-droplet system and approximated by $T_c=T_F/4$. The particle is assumed to be in resonance with the
most unstable (subharmonic) Faraday mode of the bath~\cite{Milewski2015}, a key feature of pilot-wave hydrodynamics \cite{Protiere2006,Bush2015a,BushOza}. The numerical approach to simulating Eqs. (2)-(3) is detailed in the Supplementary Information.
\subsection*{Measurement procedure and data collection}
To initialize the runs, the wave and velocity field of the bath are set to zero, and the particle positions are assigned random, uniformly distributed values. Then, the model runs for 2000 Faraday periods, a measurement is made, and all fields are reset back to zero to initialize the subsequent run.
This cycle is repeated for each set of parameter settings until the relative error in the running average of $M(\alpha = a,\beta = b)$ is reduced to an acceptably small value. We set this tolerance to be 3$\%$ for parameters that violate the inequality and 7$\%$ for those that do not. While extremely accurate, this `discrete' technique is computationally intensive; thus
we have used it only for the most critical points of the parameter space, in which the maximal Bell violations occurred. To explore the parameter space more efficiently, we adopt an alternative, relatively expedient, `continuous' approach, in which the final conditions of one run serve as the initial conditions of the next.
We demonstrated the statistical equivalence of the two approaches as follows. For specific selected data points, we performed approximately 30 different runs using the two techniques, and found the results of the ‘discrete’ and ‘continuous’ runs to be in agreement to within 3$\%$. We then executed continuous runs for 48,000 Faraday periods, during which measurements are performed frequently at uniformly distributed random times. After a sufficiently long run, the full range of initial conditions will have been effectively explored. The consistency of the results deduced with the discrete and continuous approaches demonstrates that the system is ergodic; specifically, the long-time emergent statistics are independent of the initial conditions.
Since the inequality involves four different correlation functions (three for the symmetric case considered here), finding the combinations of measurement settings that maximized $S$ was not entirely straightforward. Figure 4 summarises the strategy we followed in seeking violations. We first investigated the evolution of a single correlation function $M(\alpha =a, \beta =a)$ as a function of $a$. This gave us a good sense of parameters that maximize the difference $\delta M(a,a')=M(a,a)-M(a',a')$ (see Figure 4a). $\delta M(a,a')$ involves two of the correlation functions of Eq. 1, in the symmetric case of interest where $a$=$b$ and $a’$=$b’$. Figure 4b shows a 2D plot of the optimisation of $\delta M$ as function of $a$ and $a'$. The black dashed lines highlight the domain in which $(\max_{a,a'} (\delta M)-\delta M)/\max_{a,a'}(\delta M)>0.9$. The other term in the inequality, specifically $2M(\alpha=a,\beta=a')$, represents a combination of measurements from unequal barrier depths at the two measurement stations. Figure 4c represents the dependence of 2$M(\alpha =a^*, \beta =a’)$ on depth $a’$ for fixed $a=a^*$, the $S$-maximizing value considered in Figures 2 and 3. Finally, Figure 4d shows the evolution of the correlation functions $(M(a,a');M(a',a'))$ with increasing $a'$ and fixed $a=a^*$. \\
\vspace*{0.1in}
\section{Data availability}
The data that support the findings of our study are available upon request.
{\section{Code Availability}}
{The code that generated the data is available upon request.}
\bibliographystyle{unsrt}
|
1,108,101,566,776 | arxiv |
\section{INTRODUCTION}
\label{sec:intro}
Consider
motion planning for robots such as UAVs \cite{liu2017planning}, autonomous ships \cite{chiang2018colreg}, and spacecrafts \cite{richards2002spacecraft}. The planning solution needs to satisfy two criteria. First, the solution path must be feasible, meaning that the path must be collision-free and satisfy kinodynamic constraints, e.g. velocity and acceleration bounds even in the presence of sensor noise.
Second, the path needs to be efficient, i.e. near optimal with respect to objectives such as time to reach the goal. For example, a motion plan for a car-like robot should avoid obstacles, reach the goal promptly, not make impossibly sharp turns, and maintain enough clearance to compensate for sensor noise.
\begin{figure}[t]
\centering
\subfloat[\scriptsize RL-RRT and SST in Map 1 (46.1 x 49.5 m) ]{\includegraphics[width=0.35\textwidth,height=4.0cm,keepaspectratio=true]{figs/coverPic.png}\label{fig:coverPic}}
\subfloat[\scriptsize The Fetch robot]{\includegraphics[height=4cm,keepaspectratio=true]{figs/fetch_small.png}\label{fig:fetchPic}}
\subfloat[\scriptsize Trjectory execution of Fetch in Map 2 (46.1 x 49.5 m) ]{\includegraphics[width=0.33\textwidth,height=4cm,keepaspectratio=true]{figs/physicalTrajV2.png}\label{fig:physicalPath}}
\caption{
\revised{}{(a) Example trees constructed with RL-RRT (yellow) and SST \cite{li2016asymptotically} (blue) for a kinodynamic car navigating from start (S) to goal (G).
(b) The Fetch robot. (c) RL-RRT (green) and the real-world trajectory executed (cyan) from the start (green dot) towards the goal (blue dot) in Map 2. Map 2 is a SLAM map of an actual office building.}
}
\label{}
\end{figure}
\revised{E.2}{Current state of the art kinodynamic motion planners
search the robot's feasible state space by building a tree data structure of possible robot motions rooted at the robot's current state. The methods iteratively use a local planner to attempt to grow the tree until the goal is reached.
While some tree-based methods grow the tree by randomly propagating actions, others guide the tree growth to focus state space expansion thus requiring the local planner to be a steering function, a control policy that guides a robot to a specific goal in obstacle-free space, while satisfying the kinodynamic constraints. For example, consider a car-like robot needing to translate a small distance to the left, a motion resembling parallel parking. This motion plan is difficult, even in the absence of obstacles, and requires a steering function to steer the car to the goal. Computing the steering function requires solving an optimal control problem, and is generally NP-Hard \cite{wolfslag2018rrt}. To date, only very limited robot dynamics such as linear \cite{webb2013kinodynamic} and differential drive \cite{palmieri2015distance} systems have optimal steering functions.
}
\revised{E.2}{Besides the existence of steering functions, there are two additional difficulties facing efficient tree-based kinodynamic motion planning. First, tree-based methods that use steering functions require identifying the state in the tree from which to grow. This requires a function that compare the distance between states and return those that are expected to be easily solved by the steering function.
An effective distance function for kinodynamic planning is the Time To Reach (TTR) between states using an optimal steering function \cite{palmieri2015distance}.
TTR, however, is often expensive to compute as it involves numerically integrating the steering function \cite{palmieri2015distance}.
Second, neither the steering functions nor the related TTR are informed by sensors, and, as a result, do not account for potential obstacles. For example, if a goal is occluded by a wall, the steering function is not able to see the wall due to the lack of sensory input, and TTR would return a value as if an agent could go through the wall.}
Recently, deep Reinforcement Learning (RL) emerged as a promising near optimal steering function for kinodynamic systems \cite{layek2017deep}.
\revised{}{In addition, deep RL algorithms can learn policies that map noisy lidar or camera observations directly to robot actions, thus enabling obstacle avoidance while navigating between states for differential drive robots \cite{autorl,unstuck-dinesh}.}
\revised{}{With the recent development of AutoRL \cite{autorl}, which uses evolutionary algorithms to largely eliminate the need to hand-tune hyper-parameters, network structure and reward functions.} This combination offers the promise of deep RL being employed for local planning, i.e., providing both steering function and obstacle avoidance.
However, RL policies often lack long-term planning capabilities \cite{mnih2015human} and get trapped in environments with complex obstacles \cite{prm-rl}.
To address \revised{E.2}{the lack of available steering functions, good distance functions for aiding tree growth, and obstacle-awareness} facing kinodynamic motion planning, we propose RL-RRT, which combines RL and sampling-based planning.
It works in three steps.
First, we learn an obstacle-avoiding point-to-point (P2P) policy with AutoRL.
\revised{E.2}{This is a mapless, goal-conditioned policy that maps sensor readings to control. These P2P policies generalize to new environments without re-training \cite{autorl}.
Second, we train a supervised obstacle-aware reachability estimator that predicts the time it takes the P2P policy to guide the robot from a start to goal state in presence of obstacles, using local observations such as lidar. The key insight is that the AutoRL policy and the estimator implicitly learn the topology of the obstacles. This allows reasonably accurate estimates of time to reach in new environments.}
Lastly, \revised{E.2}{presented with a motion planning problem in a new envrionment,} in a RRT setting, we use the RL policy as a local planner and the reachability estimator as the distance function. The combination of these two learning solutions offers two primary advantages.
First, by using RL policies as an obstacle avoiding local planner, RL-RRT can be applied to a variety of kinodynamic systems without optimal steering functions.
Second, by using the obstacle-aware reachability estimator, RL-RRT can prune out randomly sampled states that are un-reachable from the tree, e.g., the policy is expected to be unsuccessful, and identify nodes with small TTR to the sampled state. \revised{E.2}{In the example of a car in front of a wall, the RL policy will go around the wall, and the estimator will predict that the time to reach will be longer because of the wall.}
We evaluate RL-RRT in two environments with three kinodynamic robots.
Results indicate that AutoRL policies are effective obstacle-avoiding local planners.
The obstacle-aware reachability estimators, one for each robot, are 74-80\% accurate in identifying if a goal state is reachable.
Compared to a state of the art steering function free method,
SST \cite{li2016asymptotically}, RL-RRT is up to 2.3 times more likely to identify a path within a fixed time budget and the identified path is up to 4.5 times shorter.
RL-RRT typically identifies dynamically-feasible paths in very few iterations -- 11 in this case -- thanks to intelligent node selection and the obstacle-avoiding local planner (Figure \ref{fig:coverPic}).
The enclosed video demonstrates RL-RRT tree construction and trajectory execution on a physical differential drive robot.
\section{RELATED WORK}
\label{sec:related-work}
Steering function-based kinodynamic planners, such as kinodynamic RRT* \cite{webb2013kinodynamic} and D-FMT \cite{schmerling2015optimal}
rely on a steering function to ``pull'' the tree to achieve rapid exploration \cite{phillips2004guided} and a proper distance function \cite{webb2013kinodynamic,palmieri2015distance,wolfslag2018rrt}.
\revised{}{RL-RRT uses AutoRL \cite{autorl} to learn steering functions, thus bypassing the challenging two-point boundary value problem.}
\revised{}{Steering function free-based approaches, such as EST \cite{phillips2004guided} and SST \cite{li2016asymptotically}, propagate random actions from a selected node.
These methods can be applied to a variety of robot dynamics, although they tend to ``wander'' \cite{allen2016real}, thus they can take a long time to identify a solution.}
Recent research has offered several solutions for P2P obstacle-avoidance policies on a differential drive robot from raw sensory input, including learning from demonstration \cite{Pfeiffer2017FromPT}, curriculum learning \cite{successor-features}, and reinforcement learning \cite{virtualtai2017,autorl}.
Other research offers hierarchical solutions to navigation, where the RL agent executes a path identified by another planner, e.g., from a grid \cite{unstuck-dinesh}, PRMs \cite{prm-rl, francis2019long}, or manually selected waypoints \cite{ddqn-topological}.
However, none of those methods are designed for kinodynamic robots, leading to failures at milestones due to dynamic constraints \cite{francis2019long}.
Designing an effective distance function for sampling-based kinodynamnic motion planning is challenging \cite{palmieri2015distance}.
The commonly used Euclidean and weighted Euclidean distance for configuration space planning is inefficient as kinodynamic robot states have limited reachability \cite{li2011learning}.
The minimum TTR between states is a highly effective distance function \cite{palmieri2015distance,wolfslag2018rrt} but is often too computationally-expensive to be used as a distance function \cite{palmieri2015distance}.
While learned TTR of a near-optimal differential drive steering function \cite{palmieri2015distance} can overcome the computational complexity, this approach still requires a near-optimal steering function.
Indirect optimal control has also been used to generate training samples composed of minimum TTR and optimal control actions along trajectories \cite{wolfslag2018rrt}.
However, this approach currently only works for low dimensional systems such as inverted pendulum and does not handle limited action bounds.
\revised{}{Our approach addresses} these challenges by bypassing the necessity of a near-optimal steering function via RL.
Unlike previous methods, we also take into account obstacle avoidance, which can significantly change the minimum TTR.
\section{METHODS}
\label{sec:methods}
\revised{3.7}{RL-RRT is a kinodynamic motion planner that learns local planner and distance function w.r.t the individual robot dynamics.
It has three main steps.}
First, we learn an obstacle-avoiding point to point policy with AutoRL \cite{autorl}.
Next, since the RL policy avoids obstacles, we can use the policy to generate obstacle-aware reachability training samples by repeatedly rolling out the learned policy.
An obstacle-aware reachability estimator is trained to predict the time to reach between two robot states in the presence of obstacles. \revised{E.2}{Policy and estimator training is done once per robot in training environments.}
Third, during planning, RL-RRT creates dynamically-feasible motion plans using the RL policy as the local planner and the reachablity estimator as the distance function. \revised{E.2}{Note, that the training and planning simulators require simulated depth measurements (e.g. lidar) around the robot, which can be thought of as analogous to motion planning with information about clearance.}
\subsection{AutoRL Local Planner}
\label{sec:methodAutoRl}
We train a RL agent to perform a P2P task that avoids obstacles.
The output of the training is a policy that is used as a local planner, an execution policy, and a data generation source for the obstacle-aware reachability estimator.
\revisedV{Using one RL policy for both local planning and path execution is inspired by \cite{francis2019long}. This allows the planner to account for potential noise during path execution.}
\revisedV{To train a policy robust against noise, we model the RL policy is a solution for a continuous state, continuous action, partially observable Markov decision process (POMDP) given as a tuple $(\Omega, S, A, D, R, \gamma, O)$ of observations, state, actions, dynamics, reward, scalar discount, $\gamma \in (0,1) $, and observation probability.}
The observations are the last three measurements of the noisy robot lidar \revisedV{ and potentially noisy relative goal position and robot velocity.}
\revisedV{We define states as the true robot configuration and its derivative.}
A black-box robot dynamics simulator, which maps states-action pairs to states, is an input to the RL training environment.
\revisedV{Another black-box simulator maps the robot state to noisy lidar observations w.r.t. obstacles.}
The goal is to train the agent to reach a goal state, $G$, within radius, $d_{G}.$
\revised{2.1}{Note that AutoRL identifies a policy that maps noisy sensor and state observations to action.
We explore simulated lidar measurement noise in this work and left state estimation and process noise to future work.}
\revised{}{AutoRL training is required only once for a given robot.}
AutoRL \cite{autorl} over DDPG \cite{ddpg}, used for learning the RL agent policy, takes as input: observations, actions, dynamics, goal definition, $(G, r)$, and a parametrized reward, $R:O \times \theta_r \rightarrow \R{},$.
\revised{}{The agent is trained to maximize the probability of reaching the goal without collision.}
This is achieved by using evolutionary algorithms over populations of agents to find a dense reward that maximizes successful goal reaching. Each generation of agents is trained with a new reward, selected based on the previous experience. At the end, the fittest agent that performs P2P tasks best, is selected as the P2P policy. In this work, all three agents use the same observations, goal definitions, and neural network architectures, but differ in the robot dynamics and reward features used.
As an example, we explain the training of the Asteroid robot here (details of the robot are in the Appendix). Details for the Differential Drive and Car robot can be found in \cite{autorl} and \cite{francis2019long}.
\revisedV{The observation is a vector of $3N_\text{beams}$ noisy lidar returns concatenated with the relative planar position of the goal, the robot velocity and orientation ($3N_\text{beams}+5$ dimensional vector).}
\revisedV{The state is the planar position, velocity and orientation of the robot.}
The action is the amount of forward thrust and turn rate.
The parameterized reward includes
\begin{equation}
R_{\myvec{\theta_{r_\text{DD}}}} =
\myvec{\theta}^T [
r_\text{goal}
r_\text{goalDist} \,
r_\text{collision} \,
r_\text{clearance} \,
r_\text{speed} \,
r_\text{step} \,
r_\text{disp} \,
],
\nonumber
\label{eq:asteroid_reward}
\end{equation}
where
$r_\text{goal}$ is 1 when the agent reaches the goal and 0 otherwise,
$r_\text{goalDist}$ is the negative Euclidean distance to the goal,
$r_\text{collision}$ is \revisedV{-1} when the agent collides with obstacles and 0 otherwise,
$r_\text{clearance}$ is the distance to the closest obstacle,
$r_\text{speed}$ is the agent speed when the clearane is below 0.25m,
$r_\text{step}$ is a constant penalty step with value 1, and
$r_\text{disp}$ is sum of displacement between the current and positions 3, 6 and 9 steps before.
\revisedV{$\myvec{\theta}$ is the weight vector tuned by AutoRL.}
\subsection{Obstacle-Aware Reachablity Estimator}
\label{sec:methodReachability}
We further improve upon work in \cite{palmieri2015distance} by learning the TTR of an obstacle-avoiding P2P RL policy learned in Section \ref{sec:methodAutoRl}.
Our obstacle-aware reachability estimator provides the following benefits:
1) It does not need an engineered near-optimal steering function for each robot dynamics. This allows TTR learning for robot systems without near-optimal steering functions.
2) Due to the presence of obstacles, the minimum TTR between states is a function of both robot dynamics and obstacles.
Since RL policies can also learn to avoid obstacles, the obstacle-aware reachability estimator can provide additional benefit over TTR estimators that consider only obstacle dynamics such as \cite{palmieri2015distance}.
\subsubsection{Training data collection}
\begin{algorithm}[tb]
\caption{Training data collection}
\label{alg:trainingData}
\begin{algorithmic}[1]
\footnotesize
\item[\algorithmicinput] $\myvec{\pi}(\myvec{o})$: Obstacle avoiding P2P RL policy, $N_\text{episode}$: Number of episodes, $\Delta t$: Time step size, $T_\text{horizon}$: Reachability horizon
\OUTPUT $\text{trainingData} = (\myvec{o}_1, y_1), (\myvec{o}_2, y_2), \cdots, (\myvec{o}_N, y_N)$.
\FOR {$i=1,\cdots N_\text{episode}$ }
\STATE $\myvec{s}, \myvec{g}$ = sampleStartAndGoal()
\STATE elapsedTime = 0
\WHILE{isDone is False}
\STATE elapsedTime += $\Delta t$
\STATE $\myvec{o}$ = makeObservation()
\STATE executePolicy($\myvec{\pi}(\myvec{o})$, $\Delta t$)
\STATE obsHistory.append($\myvec{o}$)
\STATE \revised{2.3}{$c$}, isDone = \revised{2.3}{getTTRCost}(elapsedTime, $T_\text{horizon}$)
\STATE \revised{2.3}{costHistory}.append(\revised{2.3}{$c$})
\ENDWHILE
\STATE \revised{2.3}{cfc} = \revised{2.3}{computeCumulativeFutureCost}(\revised{2.3}{costHistory})
\FOR {j=0, len(obsHistory)}
\STATE trainingData.append(($\myvec{o}=$ obsHistory[j], $y=$ \revised{2.3}{cfc}[j]))
\ENDFOR
\STATE obsHistory.clear(); \revised{2.3}{costHistory}.clear()
\ENDFOR
\RETURN trainingData
\end{algorithmic}
\end{algorithm}
Algorithm \ref{alg:trainingData} summarizes the training data collection.
First, for each episode, we initialize the robot with randomly chosen start and goal states (Alg. \ref{alg:trainingData} line 2).
Next, we execute the policy until the episode terminates (lines 4-11) \revised{3.1}{due to reaching the goal, collision, or reaching a time horizon $T_\text{horizon}$.}
During execution, we record the robot observation at each time step (line 8) and
compute and record the TTR \revised{2.3}{cost} (lines 9-10).
The TTR \revised{2.3}{cost} is set to $\Delta t$ at every time step.
\revised{3.1}{To classify whether the robot can reach the goal, we use a simple heuristic that penalizes trajectories that do not reach the goal.
If the robot is in collision or the time horizon is reached (elapsedTime equals to $T_\text{horizon}$), the TTR cost of that time step is set to $\Delta t + T_\text{horizon}$, and the episode is terminated immediately by setting isDone to true.
}
After an episode terminates, \revised{2.3}{we compute the cumulative future TTR cost for all states along the trajectory, i.e., remaining cost-to-go to the end of the trajectory (line 12)}.
The observation and cumulative future cost of each time step form a training sample and is recorded (line 14).
The process repeats for $N_\text{episode}=1000$ episodes.
\revised{3.1}{We designed the TTR cost heuristic such that if the robot reaches the goal, the cumulative future cost of each state along the trajectory is the TTR between that state and the goal.}
Conversely, if the robot failed to reached the goal due to collision or the episode reaches time horizon, all \revised{3.1}{cumulative future cost} along the trajectory will be larger than $T_\text{horizon}$.
\revised{3.1}{By employing a common machine learning technique that uses a regressor and a threshold value as a classifier \cite{goodfellow2016deep},
we can quickly classify whether a goal state can be reached during planning.}
\subsubsection{Reachability Estimator Network}
We train the obstacle-aware reachability estimator network with the training data collected above.
The network input is the robot observation $\myvec{o}$ and the output is the estimated TTR.
We use a simple three-layer fully-connected network with [500, 200, 100] hidden neurons with each a dropout probability of 0.5.
We use the L2 loss between estimated TTR and the V-value label from the training data.
\subsection{RL-RRT}
Alg. \ref{alg:rlrrt} describes RL-RRT.
While the standard RRT algorithm was utilized, modifications were made to efficiently utilize the obstacle-aware reachability estimator and the obstacle-avoiding RL local planner.
\begin{algorithm}[tb]
\begin{algorithmic}[1]
\footnotesize
\item[\algorithmicinput] $\myvec{\pi}(\myvec{o})$: Obstacle avoiding P2P RL policy, $\Delta t_\text{tree}$: Tree extension time step size, $\Delta t$: policy time step size, $T_\text{horizon}$: Reachability horizon, $P_\text{goalBias}$: Goal bias, $\myvec{x}_\text{root}$: Current robot state, $k_c$: Number of candidate nodes
\OUTPUT $\mathcal{P}$: Motion plan.
\STATE iteration = 0
\STATE $\mathcal{T}$.add(makeNode($\myvec{x}_\text{root}$, None))
\WHILE {termination condition not met}
\STATE iteration += 1
\STATE goodXrndFound = False
\WHILE {not goodXrndFound}
\STATE $\myvec{x}_\text{rnd}$ = sampleCollisionFreeStateSpace($P_\text{goalBias}$)
\STATE candidateNodes = findNearestNodesEu($\mathcal{T}$, $\myvec{x}_\text{rnd}$, $k_c$)
\STATE $n_\text{nearest}$ = findNearestNode(candidateNodes, $\myvec{x}_\text{rnd}$)
\STATE TTR = getAvgTTR($n_\text{nearest}$, $\myvec{x}_\text{rnd}$)
\IF {TTR $<$ TTR$_\text{threshold}$ or rnd $> P_\text{prune}$}
\STATE goodXrndFound = True
\ENDIF
\ENDWHILE
\STATE $\myvec{x}_\text{new} = n_\text{nearest}$.state; $t_\text{extend}$ = 0
\WHILE {not ($t_\text{extend} >$ $t_\text{maxExtend}$ or reach($\myvec{x}_\text{new}$, $\myvec{x}_\text{rnd}$) or $\myvec{x}_\text{new}$ is in collision)}
\STATE $t_\text{extend}$ += $\Delta t$
\STATE $\myvec{o}$ = makeObservation($\myvec{x}_\text{new}$, $\myvec{x}_\text{rnd}$)
\STATE $\myvec{x}_\text{new}$ = propagateDynamics($\myvec{\pi}(\myvec{o})$, $\myvec{x}_\text{new}$)
\IF {$\myvec{x}_\text{new}$ is not in collision and $t_\text{extend}$ \% $\Delta t_\text{tree} = 0$}
\STATE $\mathcal{T}$.add(makeNode($\myvec{x}_\text{new}$, $\myvec{x}_\text{rnd}$))
\ENDIF
\ENDWHILE
\ENDWHILE
\RETURN $\mathcal{P}$ = extractMotionPlan($\mathcal{T}$)
\end{algorithmic}
\caption{RL-RRT}
\label{alg:rlrrt}
\end{algorithm}
Within RL-RRT, the obstacle-aware reachability estimator can provide insight into the best samples to enhance tree growth. However, as we began to use the estimator, it became clear that the obstacle-aware reachability estimator can take longer than the standard Euclidean distance metric to compute (about 0.5 ms vs. 7 $\mu$s for Euclidean). Therefore, to enhance computation time in large trees, the estimator was integrated into a hierarchical nearest neighbor selector. Similar to \cite{chiang2018fast}, the method first identifies $k_c$ candidate nodes closest to $\myvec{x}_\text{rnd}$ using Euclidean distance (Alg. \ref{alg:rlrrt}, line 8), and subsequently these choices are filtered by the obstacle-aware TTR between each candidate node and $\myvec{x}_\text{rnd}$. To alleviate noise in the TTR estimator, we take the average of the TTR between the selected node and $N_\text{TTR sample}$=10 target states around $\myvec{x}_\text{rnd}$, i.e., within a hypercube of $d_{TTR sample}$=0.3 units (line 10).
The node with the lowest average TTR is selected for
RRT extension (line 9).
In addition,
the obstacle-aware reachability estimator can also be used to check whether the randomly sampled state $\myvec{x}_\text{rnd}$ is reachable from the nearest node $n_\text{nearest}$.
Recall that the TTR reward in Section \ref{sec:methodReachability} is setup such that any $\myvec{x}_\text{rnd}$ unreachable from $n_\text{nearest}$.state has an associated V-value larger than $T_\text{horizon}$.
As the result, the estimated TTR can be used to prune out $\myvec{x}_\text{rnd}$ that are un-reachable from the tree within $T_\text{horizon}$.
However, since the estimated TTR is not exact, we made the pruning probabilistic, i.e., if $\myvec{x}_\text{rnd}$ is deemed unreachable, it will be pruned with probability $P_\text{prune}$ (line 10).
If $\myvec{x}_\text{rnd}$ is pruned, it is rejected and a new $\myvec{x}_\text{rnd}$ is sampled (line 6).
After the nearest node is selected, RL-RRT uses the RL policy $\myvec{\pi}$ as the local planner (lines 15-24).
Specifically, an observation $\myvec{o}$ which includes \revised{3.8}{simulated lidar}, robot state, and goal information is made at every policy time step $\Delta t$ (line 17).
This observation is fed to the RL policy, which produces an action that can be used to forward propagate the dynamics to a new state $\myvec{x}_\text{new}$ (line 18).
This process repeats and a new node storing $\myvec{x}_\text{new}$ is created, and added to the tree every $\Delta_\text{tree}$ seconds (line 21), until $\myvec{x}_\text{new}$ is in collision, a maximum extension time is reached (line 20), or $\myvec{x}_\text{rnd}$ is reached (line 20).
RL-RRT terminates when either the tree reaches the goal or after a fixed amount of computation time is exhausted (line 3).
If the tree reaches the goal, a dynamically-feasible motion plan can be returned (line 25).
\section{Evaluation}
\label{sec:results}
\revised{3.7}{To demonstrate RL-RRT, we evaluate our method on three kinodynamic robots in two environments unseen during training}, and we experimentally verify the method on a physical differential drive Fetch robot from Fetch Robotics.
\subsection{Setup}
\label{sec:setup}
The three robots we evaluate are: Car, Asteroid, and Fetch.
Car is a kinematic car with inertia \cite{paden2016survey} with a maximum steering angle $30^\circ,$ and a 1.0 $m/s^2$ maximum acceleration and speed of 1.0 $m/s$.
Asteroid has similar dynamics to those found in the popular video game Asteroid, and we chose it since it is highly kinodynamic, unintuitive for a human to control, and has no known optimal steering function. The details are available in the supplemental materials.
The Fetch robot has a radius of \dist{0.3}, 1.0 m/s maximum speed and 2.0 rad/s turn rate.
\revised{}{The sensor noise is simulated by a zero mean Gaussian with a standard deviation of 0.1 m.}
We use the Fetch robot as a differential drive platform for on-robot experiments.
All point-to-point policies are trained in the environment depicted in Figure \ref{fig:envTrain}.
We evaluate RL policies and plan in two office building environments, Map 1 (Figure \ref{fig:coverPic}) and Map 2 (Figure \ref{fig:physicalPath}), which are roughly 15 and 81 times larger than the training environment, respectively.
Map 1 is is generated from a floor plan, while Map 2 is generated using a noisy SLAM of the Fetch physical testbed where we ran the experiments.
These environments include parts that are cluttered, as seen in Map 1, and very narrow corridors, such seen in Map 2.
We compare RL-RRT to SST \cite{li2016asymptotically}, a state of the art steering function free kinodynamic motion planner.
For Fetch robot, we also compare to RRT with Dynamic Window Approach (DWA) \cite{fox1997dynamic} as local planner (denoted RRT-DW).
Additionally, we test disabling the clearance term of DWA, essentially turning it into a MPC-based steering function (denoted RRT-S).
\revised{}{All experiment are repeated 50 times. Besides AutoRL training, all computation was done} on an Intel Xeon E5-1650 @ 3.6GHz using TensorFlow 1.x (Google release) and Python 2.7. AutoRL policies were implemented with Google Vizier \cite{vizier} and TFAgents \cite{tfagents}.
\subsection{AutoRL Policy Performance}
\label{sec:rlPerf}
We use pre-trained P2P policies for Fetch \cite{autorl} and Car \cite{francis2019long} robots. Their short description is available in the Appendix. The Asteroid P2P policy is original to this paper. All agents are trained with AutoRL over DDPG \cite{autorl}. The goals are randomly placed within \dist{10}.
\revised{}{We train 100 agents in parallel over 10 generations as in \cite{autorl}. The training took roughly 7 days.}
Figure \ref{fig:p2pSucc} shows the success rate of the P2P agents compared to goal distance. Notice that when the goal distance is \dist{10} or farther than the trained policy, the performance degrades. We also notice that the Car policy is best performing, while the Asteroid policy is the most challenging.
These results show that AutoRL produces, without hand-tuning, effective local planners, i.e., both a steering function and an obstacle avoidance policy for a variety of robot dynamics.
\begin{figure*}[h]
\begin{center}
\begin{tabular}{ccc}
\subfloat[\scriptsize Differential Drive]{\includegraphics[width=0.3\textwidth,height=2.1cm,keepaspectratio=false]{figs/p2pSuccFetchV2.png}\label{fig:p2pSuccAst}}&
\subfloat[\scriptsize Car]{\includegraphics[width=0.3\textwidth,height=2.1cm,keepaspectratio=false]{figs/p2pSuccCarV2.png}\label{fig:p2pSuccCar}}&
\subfloat[\scriptsize Asteroid]{\includegraphics[width=0.3\textwidth,height=2.1cm,keepaspectratio=false]{figs/p2pSuccAstV2.png}\label{fig:p2pSuccAsoh ot}}
\\
\end{tabular}
\caption{\footnotesize AutoRL P2P navigation success rate as a function of start and goal distance for (a) Fetch, (b) Car and (c) Asteroid robot. \revised{3.6}{The success rates are evaluated in Map 1 with randomly sampled start and goal states.}
\label{fig:p2pSucc}}
\end{center}
\end{figure*}
\subsection{Reachability Estimator Performance}
\label{sec:ttrEstimatorPerf}
The obstacle-aware reachability estimator is trained in the training environment with goals sampled within \dist{20} from the initial states, twice the distance used for P2P training.
The estimator network was trained on 1000 episodes with about 100,000 samples. \revised{2.5}{Data generation takes about 10 minutes.}
The reachability thresholds are 20 seconds for differential drive and Asteroid, and 40 seconds for Car.
\revised{2.5}{Each estimator was trained over 500 epochs and took about 30 minutes.}
\begin{table}[tb]
\centering
\begin{tabular}{l| l | r r | r | r | r }
\multirow{ 2}{*}{Robot} & \multicolumn{3}{|c|}{Confusion Matrix} & Prec. & Recall &Accur. \\ \cline{2-4}
& & \multicolumn{2}{|c|}{True (\%)} & (\%) & (\%) & (\%) \\ \hline
\multirow{ 2}{*}{Fetch} & \multirow{ 1}{*}{Predicted} & 42.7 & 21.6 & \multirow{ 2}{*}{66.4} & \multirow{ 2}{*}{92.2} & \multirow{ 2}{*}{74.8} \\
& (\%) & 3.6 & 32.1 & & & \\ \hline
\multirow{ 2}{*}{Car} & \multirow{ 1}{*}{Predicted} & 44.5 & 14.2 & \multirow{ 2}{*}{75.8} & \multirow{ 2}{*}{90.2} & \multirow{ 2}{*}{81.0} \\
& (\%) & 4.8 & 36.5 & & & \\ \hline
\multirow{ 2}{*}{Asteroid} & \multirow{ 1}{*}{Predicted} & 26.5 & 16.3 & \multirow{ 2}{*}{61.9} & \multirow{ 2}{*}{73.4} & \multirow{ 2}{*}{74.1} \\
& (\%) & 9.6 & 47.6 & & & \\
\hline
\end{tabular}
\caption{ \footnotesize
Reachability estimator confusion matrix, precision, recall, and accuracy in the training environment.
}
\label{tab:reachConfusion}
\end{table}
Accuracy of the models is between 70\% and 80\% (Table \ref{tab:reachConfusion}). Notice that a high recall means that the estimator misses fewer nodes, and suggests that the paths RL-RRT produces should be near-optimal. On the other hand, relatively low precision implies that RL-RRT will explore samples that end up not being useful. This means that we can speed-up RL-RRT further by learning a more precise predictor.
\begin{figure*}[h]
\centering
\begin{tabular}{ccc}
\subfloat[\scriptsize Differential Drive]{\includegraphics[width=0.3\textwidth,height=3.1cm,keepaspectratio=false]{figs/ttrScatterFetchV3.png}\label{fig:ttrScatterFetch}} &
\subfloat[\scriptsize Car ]{\includegraphics[width=0.3\textwidth,height=3.1cm,keepaspectratio=false]{figs/ttrScatterCarV3.png}\label{fig:ttrScatterCar}}&
\subfloat[\scriptsize Asteroid]{\includegraphics[width=0.3\textwidth,height=3.1cm,keepaspectratio=false]{figs/ttrScatterAstV3.png}\label{fig:ttrScatterAst}}
\end{tabular}
\caption{\footnotesize \revised{3.11}{Predicted cumulative future time to reach cost v.s. true value for various robots.}}
\label{fig:ttrScatter}
\end{figure*}
\revised{R3.2}{ The reachability estimator overestimates the TTR of reachable states across all robots (Fig. \ref{fig:ttrScatter}). However, overestimation disappears when trained and evaluated only on reachable states (see Fig. 1 in Appendix for more detail).
This suggests that the overestimation of TTR is likely due to the TTR cost heuristic uses a penalty for states unreachable within $T_\text{horizon}$.
We leave identifying better TTR cost heuristics and estimator network architectures for future work.
}
\begin{figure*}[tb]
\centering
\subfloat[\scriptsize Training environment (22.7 x 18.0 m)]{\includegraphics[width=0.2\textwidth,keepaspectratio=false,trim=40mm 0mm 40mm 0mm,clip]{figs/env1965.png}\label{fig:envTrain}}
\subfloat[\scriptsize Predicted]{\includegraphics[width=0.3\textwidth,keepaspectratio=false]{figs/valueContourNet.png}\label{fig:valueLandscapeNet}}
\subfloat[\scriptsize Ground truth ]{\includegraphics[width=0.3\textwidth,keepaspectratio=false]{figs/valueContourTruth.png}\label{fig:valueLandscapeTruth}}
\caption{\footnotesize \revised{}{(a) The training environment. Contour plot of (b) Predicted \revised{3.6}{future cumulative time to reach cost} v.s. (c) the true value for \revised{3.6}{Car} to reach the goal near the center marked by the blue dot. The white regions have time to reach value over the 40s horizon, i.e., un-reachable. All start states and the goal have 0 as linear speed and orientation.}}
\label{fig:valueLandscape}
\end{figure*}
In general, the estimator captures the regions of start states that cannot reach the goal (blue dot) (Fig. \ref{fig:valueLandscape}).
This is most visible at the bottom right region of the environment, which has a TTR larger than the 40s horizon which indicates that the policy failed to escape that region.
We also see that the estimated TTR captures the dynamics of Car robot, i.e., since the goal orientation is facing right, it takes less time to reach the goal from the left, top or bottom than from the right.
Note that the network is never trained on trajectories that start inside of obstacles and thus cannot accurately predict TTR starting from those states, an event which should not occur in sampling-based planning.
\subsection{Planning Results}
\label{sec:planningResults}
\begin{figure*}[h]
\centering
\addtocounter{subfigure}{-1}
\subfloat{\includegraphics[width=0.6\textwidth,height=0.5cm,keepaspectratio=false]{figs/legend.png}}
\subfloat[\scriptsize Differential Drive.]{\includegraphics[width=0.3\textwidth,height=3.0cm,keepaspectratio=false]{figs/perfSuccTwoEnvFetchV5.png}\label{fig:perfSuccFetch}}
\subfloat[\scriptsize Car ]{\includegraphics[width=0.3\textwidth,height=3.0cm,keepaspectratio=false]{figs/perfSuccTwoEnvCarV5.png}\label{fig:perfSuccCar}}
\subfloat[\scriptsize Asteroid]{\includegraphics[width=0.3\textwidth,height=3.0cm,keepaspectratio=false]{figs/perfSuccTwoEnvAstV5.png}\label{fig:perfSuccAstroid}}
\subfloat[\scriptsize Differential Drive.]{\includegraphics[width=0.3\textwidth,height=3.0cm,keepaspectratio=false]{figs/perfFinishTwoEnvFetchV4.png}\label{fig:perfFinishFetch}}
\subfloat[\scriptsize Car]{\includegraphics[width=0.3\textwidth,height=3.0cm,keepaspectratio=false]{figs/perfFinishTwoEnvCarV4.png}\label{fig:perfFinishCar}}
\subfloat[\scriptsize Asteroid]{\includegraphics[width=0.3\textwidth,height=3.0cm,keepaspectratio=false]{figs/perfFinishTwoEnvAstV4.png}\label{fig:perfFinishAstroid}}
\caption{\footnotesize Success rate (top) and Finish time (bottom) of RL-RRT (black) compared to, SST (blue), RRT-DW (red, RRT with DWA obstacle-avoiding steering function), RRT-S (yellow, RRT with DWA as the steering function) and RL-RRT-E (magenta, RL-RRT using Euclidean distance instead of the reachability estimator) in Map 1 (M1) and Map 2 (M2). }
\label{fig:perf}
\end{figure*}
RL-RRT finds a solution faster than SST for all three robots in both environments (Fig. \ref{fig:perfSuccFetch}, \ref{fig:perfSuccCar}, \ref{fig:perfSuccAstroid}). Note that Car shows the best improvement over the baseline (up to 2.3 times faster), which matches the high success rate of the P2P Car policy.
Conversely, the least improvement is for Asteroid, which is the most challenging for the RL agent.
Figure \ref{fig:perfSuccFetch} also shows that RL-RRT finds a solution faster than steering function-based methods, where DWA was used as the steering function (yellow, RRT-S) and obstacle-avoiding steering function (red, RRT-DW).
These results are expected as RL-RRT learns a obstacle-avoiding local planner that can often go through very narrow corridors and move around corners (Figure \ref{fig:coverPic}).
In comparison, DWA often gets stuck around corners.
To separate the impact of the RL local planner as compared to the reachability estimator, we tested RL-RRT without the estimator and use Euclidean distance to identify the nearest state in the tree instead.
Figures \ref{fig:perfSuccFetch}, \ref{fig:perfSuccCar} and \ref{fig:perfSuccAstroid} show that RL-RRT without the reachability estimator (magenta curves) performs worse than RL-RRT for all robots.
This is expected as the reachability estimator prunes potentially infeasible tree-growth, thereby biasing growth towards reachable regions.
Also, the reachabilty estimator encodes the TTR and is thus more informative than the Euclidean distance for kinodynamic robots such as Asteroid.
The finish time of trajectories identified by RL-RRT are significantly shorter (up to 6 times shorter) than SST for all robots (Fig. \ref{fig:perfFinishFetch}, \ref{fig:perfFinishCar}, \ref{fig:perfFinishAstroid}) and comparable to RRT-DWA and RRT-S on differential drive.
This is expected as SST does not use steering functions. Instead, it randomly propagates actions, resulting in a ``jittery'' behavior (visible in Figure \ref{fig:coverPic}) and long finish time.
The comparable finish time with steering function-based methods show that RL-RRT learns a near-optimal steering function.
\subsection{Physical Robot Experiments}
\label{sec:physical}
In order to verify that the RL-RRT produces motion plans that can be used on real robots, we executed the motion plans on the Fetch robot (Figure. \ref{fig:fetchPic}) in Map 2 environment.
We ran 10 different motion plans, repeated 3 times.
Figure \ref{fig:physicalPath} presents one such trajectory. The straight line distance between the start and goal is 20.8 m. In green are tree nodes for a path, and the blue line is the executed robot path with the P2P AutoRL policy. We notice two things. First, the path is similar to the one humans would take. The shortest path leads through cubicle space, which is cluttered. Because the P2P policy does not consistently navigate the cubicle space, the TTR estimates are high in that region and the tree progress slowly in that area. At the same time, in the uncluttered space near the start position (left and right) the tree grows quickly. The executed trajectory (in blue) stays close to the planned path.
Enclosed video contains the footage of the robot traversing the path.
\section{DISCUSSION}
\label{sec:discussion}
\begin{figure}[h]
\centering
\subfloat[\scriptsize Two Astroid trajectories.]{\includegraphics[width=0.24\textwidth,height=2.4cm,keepaspectratio=false]{figs/ddpgTraj.png}\label{fig:ddpgTraj}}
\subfloat[\scriptsize V-value and TTR.]{\includegraphics[width=0.25\textwidth,height=2.4cm,keepaspectratio=false]{figs/ddpgVal.png}\label{fig:ddpgVals}}
\caption{\footnotesize (a) Two trajectories (green and red) of the Asteroid robot from the yellow dots to blue dots. (b) The corresponding predicted TTR \revised{3.6}{(solid lines)} and the negative of V-value from DDPG's critic net \revised{3.6}{(dashed lines)}.}
\label{fig:xrndReach}
\end{figure}
\revised{3.9}{Deep actor-critic RL methods approximate the cumulative future reward, i.e., state-value function with the critic net. Intuitively, the state-value function captures the progress towards the goal and may be used as a distance function during planning.
Here we show that this is \textit{not} the case when proxy rewards are used. AutoRL uses proxy rewards (shown in Section \ref{sec:methodAutoRl}) since they significantly improve learning performance, especially for tasks with sparse learning sigals such as navigation \cite{autorl}.}
Fig \ref{fig:ddpgTraj} shows examples of two Asteroid trajectories and Fig. \ref{fig:ddpgVals} shows the corresponding the estimated TTR (solid lines) and negative of DDPG state-value function extracted form the critic net (dashed lines). The obstacle-aware reachability estimator correctly predicted the TTR while the DDPG's critic net has a significant local maximum, thus unsuitable as a distance function. This finding motivated the supervised reachability estimator.
\begin{figure}[h]
\centering
\subfloat[\scriptsize Predicted]{\includegraphics[width=0.18\textwidth]{figs/mazeVLandscapePred.png}\label{fig:valueLandscapeNetMaze}}
\subfloat[\scriptsize Ground truth ]{\includegraphics[width=0.18\textwidth]{figs/mazeVLandscapeTruth.png}\label{fig:valueLandscapeTruthMaze}}
\caption{\footnotesize \revised{1.1}{Contour plot of (a) Predicted future cumulative time to reach cost v.s. (b) the true value for Car to reach the goal near the center marked by the blue dot. The white regions have time to reach value over the 40s horizon, i.e., un-reachable. All start states and the goal have 0 as linear speed and orientation. The environment size is 50 m by 40 m.}}
\label{fig:xrndReachMaze}
\end{figure}
\revised{1.1}{One limitation of RL-RRT is that the obstacle-aware reachability estimator approximates reachability using only local information such as simulated lidar measurements around the robot.
However, the true reachability is often impacted significantly by large-scale obstacle structures.
Figure \ref{fig:xrndReachMaze} demonstrates this limitation.
The ground truth shows that the Car policy generally fails to reach the goal outside of the center box due to the complex maze-like obstacles (Figure \ref{fig:valueLandscapeTruthMaze}).
The reachability estimator failed to predict this as some regions outside of the center box are incorrectly predicted as reachable (Figure \ref{fig:valueLandscapeNetMaze}).
On the other hand, we also demonstrated that the estimator performs well when the training and planning environments are similar (Figure \ref{fig:valueLandscape}).
This suggests that the reachability estimator should to be trained in environments similar to the planning environment or perform online adaptation/learning during planning.
We leave the latter to future work.
}
\section{CONCLUSIONS}
\label{sec:conclusions}
This paper contributes RL-RRT, a kinodynamic planner which works in three steps: 1) learning obstacle-avoiding local planner; 2) training an obstacle-aware reachability estimator for the learned local planner; and 3) using the estimator as the distance function and to bias sampling in RRT.
\revised{3.7}{Unlike traditional kinodynamic motion planners, RL-RRT learns a suitable steering and distance function. The robot is trained once, and the policy and estimator transfer to the new envrionments.}
We evaluated the method on three kinodynmic robots in two simulated environments. Compared to the baselines, RRT plans faster and produces shorter paths. We also verified RL-RRT on a physical differential drive robot.
\revisedV{For future work, following PRM-RL, we plan to improve the noise robustness of RL-RRT by Monte Carlo roll-outs during tree extensions.} We also plan to identify better TTR cost heuristics, network architectures and online adaptation of the reachability estimator.
\section*{ACKNOWLEDGMENT}
\small
We thank Tsang-Wei Edward Lee for assisting with robot experiments, and Brian Ichter for the helpful feedback.
Tapia and Chiang are partially supported by the National Science Foundation under Grant Numbers IIS-1528047 and IIS-1553266 (Tapia, CAREER).
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
\bibliographystyle{abbrv}
|
1,108,101,566,777 | arxiv | \section{Partially Ordered Sets}
\subsection{Preliminaries}
Suppose that $(S, \preceq)$ is a discrete partially ordered set. Recall that $C \subseteq S$ is a {\em chain} if $C$ is totally ordered under $\preceq$. We make the following assumptions:
\begin{enumerate}
\item \label{a.m} There is a minimum element $e$.
\item \label{a.f} For every $x \in S$, every chain in $S$ from $e$ to $x$ is finite.
\end{enumerate}
Recall that $y$ {\em covers} $x$ if $y$ is a minimal element of $\{t \in S: t \succ x\}$. The {\em covering graph} (or Hasse graph) of $(S, \preceq)$ is the directed graph with vertex set $S$ and edge set $E = \{(x, y) \in S^2: y \text{ covers } x\}$. From the assumptions, it follows that for each $x \in S$, there is a (directed) path from $e$ to $x$ in the graph, and every such path is finite. For $x \in S$, let
\[ A_x = \{y \in S: y \text{ covers } x\}, \quad B_x = \{w \in S: x \text{ covers } w\} \]
That is, $A_x$ is the set of elements immediately {\em after} $x$ in the partial order, while $B_x$ is the set of elements immediately {\em before} $x$ in the partial order. Note that $A_x$ could be empty or infinite. On the other hand, $B_e = \emptyset$, but for $x \ne e$, $B_x \ne \emptyset$ since there is at least one path from $e$ to $x$.
An {\em upward run chain} on $(S, \preceq)$ is a Markov chain that, at each transition, moves to a state immediately above the current state or back to $e$, the minimum state. A {\em downward run chain} is a Markov chain that, at each transition, moves to a state immediately below the current state, unless the current state is $e$ in which case the chain can move anywhere in $S$. For particular posets, upward and downward runs can have applications in reliability theory, communications theory, queuing theory and other areas. Generally, posets are the natural mathematical home for these stochastic processes. In this article, we are interested in general issues of recurrence, invariant distributions, time reversal, and results for special types of posets. See \cite{Evstigneev} for another class of Markov chains on posets.
\subsection{Uniform posets} \label{ss.uniform0}
An interesting case is when the partially ordered set $(S, \preceq)$ is {\em uniform} in the sense that for each $x \in S$, all paths from $e$ to $x$ have the same length. It then follows that the all paths from $x$ to $y$ have the same length for any $x, \, y \in S$ with $x \preceq y$; we denote this length by $d(x, y)$. Let $S_n = \{x \in S: d(e, x) = n\}$ for $n \in \N$. Of course, $S_0 = \{e\}$ and $\{S_n: n \in \N\}$ partitions $S$.
\subsection{Rooted trees and path space} \label{ss.paths}
Another important special case is when the covering graph of $(S, \preceq)$ is a rooted tree with root $e$. In this case, $B_x$ has a single element, which we will denote by $x^-$, for each $x \ne e$. There is a unique path from $e$ to $x$ for each $x \in S$. Thus the poset $(S, \preceq)$ is uniform, so the definitions in Section \ref{ss.uniform0} apply.
In fact, rooted trees form an essential special case, because we will show that upward and downward runs on an arbitrary poset can be constructed from upward and downward runs on a certain rooted tree of paths. Specifically, suppose that $(S, \preceq)$ is a poset, and let $\hat S$ denote the set of finite, directed paths in $S$, starting at $e$. We define the partial order $\preceq$ on $\hat S$ by $a \preceq b$ if and only if $a$ is a prefix of $b$. The covering graph of $(\hat S, \preceq)$ is a tree rooted at $e$ (the degenerate path consisting only of $e$). For $a \in \hat S$, let $m(a)$ denote the endpoint of $a$. If $y$ covers $x$ in $S$ then for every $a \in \hat S$ with $m(a) = x$, $ay$ covers $a$ (where $ay$ denotes that path obtained by appending $y$ to the end of $a$). Let $\hat S(x) = \{a \in \hat S: m(a) = x\}$, the set of paths ending in $x$. Note that for $a \ne e$, $a^-$ is the path obtained by removing the endpoint of $a$. If $a \in \hat S$, the unique path from $e$ to $a$ in $\hat S$ simply consists of the successive prefixes of $a$. Thus in the notation of Section \ref{ss.uniform0}, $d(e, a)$ is the length of the path $a$ and hence $\hat S_n$ is the set of paths of length $n$.
\subsection{Positive semigroups} \label{ss.semigroups0}
Another important special case is when the partially ordered set $(S, \preceq)$ is associated with a {\em positive semigroup} $(S, \cdot)$. That is, $\cdot$ is an associative binary operation on $S$ with an identity element $e$, no non-trivial inverses, and satisfying the left-cancellation law. In this case, $x \preceq y$ if and only if there exists $t \in S$ with $xt = y$. Positive semigroups are essentially characterized by the fact that $xS = \{y \in S: y \succeq x\}$ is order-isomorphic to $S$ for each $x \in S$; the mapping $t \mapsto xt$ is an isomorphism. Assumption \ref{a.m} is always satisfied since $e$ is the minimum element. We will assume that $[e, x] = \{t \in S: t \preceq x\}$ is finite for each $x \in S$, so Assumption \ref{a.f} is satisfied as well. Probability distributions on positive semigroups are studied in \cite{Rowell, Siegrist, Siegrist2, Siegrist3, Siegrist4}
An element $i \in S$ is {\em irreducible} if $i$ cannot be factored, except for the trivial factoring $i = ie = ei$. If $I$ is the set of irreducible elements of $(S, \cdot)$ then $A_x = \{xi: i \in I\}$, so in particular, $\#(A_x)$ is the same for each $x$. The poset $(S, \preceq)$ will be uniform if and only if for every $x \in S$, all factorings of $x$ over $I$ have the same number of factors. In this case, $d(e, x)$ is the number of factors.
The path space associated with $(S, \cdot)$ is isomorphic to the {\em free semigroup} on the set of irreducible elements $I$. This is the set of finite ``words'' over the alphabet $I$, with concatenation as the semigroup operation. That is, a path from $e$ to $x$ in path space is uniquely associated with a factoring of $x$ over $I$: $x = i_1 i_2 \cdots i_n$; the string on the right is a word in the free semigroup.
\subsection{Probability Distributions}
Suppose that $(S, \preceq)$ is a poset and that $X$ is a random variable with support $S$. As usual, the probability density function (PDF) of $X$ is the function $f$ given by $f(x) = \Pr(X = x)$ for $x \in S$. The {\em upper probability function} (UPF) of $X$ is the function $F$ given by
\[ F(x) = \Pr(X \succeq x) = \sum_{y \succeq x} f(y), \quad x \in S \]
Finally, the {\em rate function} of $X$ is the function $r$ given by
\[ r(x) = \frac{f(x)}{F(x)} = \Pr(X = x | X \succeq x), \quad x \in S \]
In particular, $X$ has {\em constant rate} if $r$ is constant on $S$. For general posets, the distribution of $X$ is not uniquely determined by the UPF $F$ (and certainly not by the rate function $r$). These issues and the existence of constant rate distributions are explored in \cite{Siegrist5}. A special case of a general expected value result in \cite{Siegrist5} is
\begin{equation} \label{eq.expect}
\sum_{x \in S} F(x) = \E[\#(D[x])]
\end{equation}
where $D[x] = \{t \in x: t \preceq x\}$. If $(S, \preceq)$ is a rooted tree, then the UPF $F$ of $X$ does determine the distribution of $X$, since clearly the PDF $f$ of $X$ is given by
\[ f(x) = F(x) - \sum_{y \in A(x)} F(y) \]
Moreover, when $(S, \preceq)$ is a rooted tree, $\#(D[x]) = 1 + d(e, X)$
\section{The Upward Run Chain} \label{s.upward}
\subsection{Basic definitions and results}
A Markov chain $\bs{X} = (X_0, X_1, \ldots)$ on a poset $(S, \preceq)$ is an {\em upward run} chain if the transition function $P$ satisfies $P(x, y) > 0$ if and only if $y \in A_x$ or $y = e$. Of course, we must have
\[ P(x, e) = 1 - P(x, A_x), \quad x \in S\]
Thus, in state $x \in S$, the chain next moves to a state $y \in A_x$, or back down to $e$. The chain is irreducible since $e$ leads to every state, and every state leads back to $e$. The chain is aperiodic since $e$ leads back to $e$ in one step. If the state space is $(\N, \le)$, then an upward run chain is simply a {\em success-runs} chain; these are commonly studied in reliability theory.
For $x \in S$, let $T_x$ denote the first positive hitting time to $x$:
\[ T_x = \min \{n \in \N_+: X_n = x\} \]
Define $F: S \to (0, 1]$ by $F(x) = \Pr_e(T_x \le T_e)$. Of course, $\Pr_e(T_x = T_e) = 0$ if $x \ne e$. Thus, $F(x) = \Pr_e(T_x < T_e)$ if $x \ne e$ while $F(e) = 1$.
\begin{proposition}
Suppose that $\bs{X}$ is recurrent. Then $F$ is left-invariant for $P$.
\end{proposition}
\begin{proof}
Suppose first that $y \ne e$. Starting at $e$, the chain moves upward to $y$ without an intermediate return to $e$ if and only if the chain moves upward to some $x \in B_y$, without an intermediate return to $e$, and then moves in one step from $x$ to $y$. Therefore
\[ F(y) = \sum_{x \in B(y)} F(x) P(x, y) = (F P)(y) \]
On the other hand, to return to $e$, starting at $e$ the chain must go directly back to $e$ or move upward to some intermediate state $x \in S$ and then go back to $e$ in one step. Thus, since the chain is recurrent,
\[ (F P)(e) = \sum_{x \in S} F(x) P(x, e) = \Pr_e(T_e < \infty) = 1 = F(e) \]
\end{proof}
We will refer to $F$ as the {\em standard invariant function} for $\bs{X}$. In the recurrent case, the chain is positive recurrent if and only if
\begin{equation} \label{eq.precurrent}
\mu(e) := \sum_{x \in S} F(x)
\end{equation}
is finite. In this case, the invariant PDF $f$ is given by $f(x) = F(x) / \mu(e)$ for $x \in S$. Equivalently, $\mu(x) := \E_x(T_x) = \mu(e) / F(x)$ for $x \in S$.
\subsection{Upward runs on rooted trees} \label{ss.upTrees}
Consider now the special case where the covering graph of $(S, \preceq)$ is a rooted tree. If $x \in S$ and $e \, x_1 \, x_2 \cdots x_{n-1} \, x$ is the unique path in $S$ from $e$ to $x$, then clearly
\begin{equation} \label{eq.up1}
F(x) = P(e, x_1) P(x_1, x_2) \cdots P(x_{n-1},x)
\end{equation}
the product of the transition probabilities along the path from $e$ to $x$.
It follows that
\begin{equation} \label{eq.up2}
\Pr_e (T_e > n) = \sum_{x \in S_n} F(x), \quad n \in \N
\end{equation}
Thus the chain is recurrent if and only if the sum in (\ref{eq.up2}) has limit 0 as $n \to \infty$.
Suppose that the chain is recurrent. Then clearly $F$ is the UPF of $X_{T(e) - 1}$, the last state vistied before returning to $e$ (starting at $e$). That is
\[ F(x) = \Pr_e(T_x \le T_e) = \Pr_e(X_{T(e) - 1} \succeq x), \quad x \in S \]
Note from (\ref{eq.expect}) that $\mu(e) = \E_e(T_e)$ can be written as
\[ \mu(e) = 1 + \E_e \left[ d \left( e, X_{T(e) - 1} \right) \right] \]
In the positive recurrent case, the invariant PDF $f$ is the function obtained by normalizing $F$ with $\mu(e)$. Note that in general, $f$ is {\em not} the PDF of $X_{T(e) - 1)}$ given $X_0 = e$. In fact, the two PDFs are the same if and only if the invariant distribution has constant rate.
Conversely, given an UPF $F$ on $S$, we can construct a recurrent upward run chain with $F$ as the standard invariant function. Specifically, suppose that $X$ is random variable with support $S$ and with UPF $F$ and PDF $f$. Define $P$ by
\begin{align*}
P(x, y) &= \Pr(X \succeq y | X \succeq x) = \frac{F(y)}{F(x)}, \quad x \in S, \, y \in A_x \\
P(x, e) &= \Pr(X = x | X \succeq x) = \frac{f(x)}{F(x)}, \quad x \in S
\end{align*}
Then $P$ is the transition probability function for an upward run chain $\bs{X} = (X_0, X_1, \ldots)$. Moreover, $F$ and $P$ satisfy (\ref{eq.up1}), so the notation is consistent and $F$ is the standard invariant function. Note that $x \mapsto P(x, e)$ is the rate function of $X$. In particular, if $P(x, e) = \alpha$ for all $x \in S$, then $X$ has constant rate $\alpha$, the chain is positive recurrent, and $f$ is the invariant PDF.
\subsection{Upward runs on path space}
Now return to the general case where $(S, \preceq)$ is an arbitrary poset and $(\hat S, \preceq)$ is the corresponding path space discussed in Section \ref{ss.paths} (a rooted tree). Suppose that $P$ is the transition probability function of an upward run chain on $S$. We define $\hat P$ on $\hat S$ by
\[ \hat P(a, ay) = P(x, y), \; \hat P(a, e) = P(x, e), \quad a \in \hat S(x), \, x \in S \]
Clearly $\hat P$ is a valid transition probability function and corresponds to an upward run chain on $\hat S$. Moreover, starting at $e$, we can define the two chains on a common probability space. To do this, we need some notation. If $\bs{\hat X}$ is a process on $\hat S$, define the process $\bs{X}$ on $S$ by $X_n = m(\hat X_n)$.
\begin{proposition} \label{p.upCouple}
If $\bs{\hat X}$ is an upward run chain on $\hat S$ with transition probability function $\hat P$ and starting at $e$, then $\bs{X}$ is an upward run chain on $S$ with transition probability function $P$ and starting at $e$.
\end{proposition}
\begin{proof}
The proof follows easily from the definition of the path space $\hat S$ and the fact that
\[ \hat P(a, b) = P(m(a), m(b)), \quad a, \, b \in \hat S \]
\end{proof}
Of course the results of Section \ref{ss.upTrees} apply to $\bs{\hat X}$. Also, with the chains coupled as in Proposition \ref{p.upCouple}, note that $T_e = \hat T_e$. If $a = e \, x_1 \, x_2 \cdots x_n \in \hat S$ then from (\ref{eq.up2}) and the definition of $\hat P$,
\begin{align*}
\hat F(a) &= \hat P(e, e x_1) \hat P(e x_1, e x_1 x_2) \cdots \hat P(e x_1 \cdots x_{n-1}, e x_1 \cdots x_n)\\
&= P(e,x_1) P(x_1, x_2) \cdots P(x_{n-1}, x_n)
\end{align*}
Now, for $\bs{X}$ to go from $e$ to $x$ without an intermediate return to $e$, $\bs{X}$ must move along some path from $e$ to $x$ so
\[ F(x) = \sum_{a \in \hat S(x)} \hat F(a), \quad x \in S \]
It then follows that
\begin{equation} \label{eq.path2}
\Pr_e (T_e > n) = \sum_{a \in \hat S_n} \hat F(a)
\end{equation}
The chains $\bs{X}$ and $\bs{\hat X}$ have the same classification: recurrent if and only if the sum in (\ref{eq.path2}) has limit $0$ as $n \to \infty$, and of course positive recurrent if and only if the sum in (\ref{eq.precurrent}) is finite. In the positive recurrent case, the invariant PDFs $\hat f$ and $f$ are related by
\[ f(x) = \sum_{a \in \hat S(x)} \hat f(a), \quad x \in S \]
Since $(\hat S, \preceq)$ is a rooted tree, $\hat F$ is an UPF if the chains are recurrent. On the other hand $F$ may not be an UPF on $S$, and conversely, there may exist UPFs $F$ on $S$ that cannot be realized as the standard invariant function for a recurrent upward run chain on $S$.
\section{The Downward Run Chain} \label{s.downward}
\subsection{Basic definitions and results}
A Markov chain $\bs{Y} = (Y_0, Y_1, \cdots)$ on $(S \preceq)$, is a {\em downward run chain} if the transition probability function $Q$ satisfies $Q(x, y) > 0$ if and only if $x = e$ and $y \in S$, or if $x \in S - \{e\}$ and $y \in B_x$. Thus, the chain moves downward from a state to one of its predecessors, until it gets to $e$; then it can move anywhere in $S$. The chain is irreducible since every state leads to $e$ and $e$ leads to every state. The chain is aperiodic since $e$ leads to $e$ in one step. If the poset is $(\N, \le)$, then a downward run chain is simply a {\em remaining life} chain; these are commonly studied in reliability theory.
We denote the first positive hitting time to $x \in S$ by
\[ U_x = \min \{n \in \N_+: Y_n = x\} \]
A downward chain is always recurrent. Since all paths from $e$ to $x$ are finite, $\Pr_x(U_e < \infty) = 1$ for $x \in S - \{e\}$. Hence
\[ \Pr_e(U_e < \infty) = Q(e, e) + \sum_{x \in S - \{e\}} Q(e, x) \Pr_x (U_e < \infty) = 1 \]
Define $G: S \to (0, 1]$ by $G(x) = \Pr_e(U_x \le U_e)$. Of course, $\Pr_e(U_x = U_e) = 0$ if $x \ne e$. Thus, $G(x) = \Pr_e(U_x < U_e)$ if $x \ne e$ while $G(e) = 1$. Note that $G$ is defined for the downward run chain just as $F$ is defined for the upward run chain.
\begin{proposition}
The function $G$ is left-invariant for $Q$.
\end{proposition}
\begin{proof}
For $y \in S$,
\begin{align*}
(G Q)(y) &= \sum_{x \in S} G(x) Q(x, y)\\
&= G(e) Q(e, y) + \sum_{x \in A(y)} G(x) Q(x, y)\\
&= Q(e, y) + \sum_{x \in A(y)} G(x) Q(x, y) = G(y)
\end{align*}
For the last line, note that starting at $e$, the chain hits $y$ before returning to $e$ if and only if the chain jumps immediately to $y$ or hits some $x \in A_y$, before returning to $e$, and then moves from $x$ to $y$ in one step.
\end{proof}
As before, we will refer to $G$ as the {\em standard invariant function} for $\bs{Y}$. The chain is positive recurrent if and only if
\[\nu(e) := \sum_{x \in S} G(x)\]
is finite. In this case, the invariant PDF is given by $g(x) = G(x) / \nu(e)$ for $x \in S$. Equivalently, $\nu(x) := \E_x(U_x) = \nu(e) / G(x)$ for $x \in S$.
\subsection{Downward runs on rooted trees} \label{ss.downTrees}
Consider the special case where the covering graph of $(S, \preceq)$ is a rooted tree. Since a non-root vertex has a single parent, we have
\[ Q(x, x^-) = 1, \quad x \in S - \{e\} \]
Thus, the invariant function $G$ is simply the UPF of $Y_1$, given $Y_0 = e$:
\[ G(x) = \sum_{y \succeq x} Q(e, y) = \Pr_e(Y_1 \succeq x), \quad x \in S \]
From (\ref{eq.expect}), $\nu(e) = \E_e(U_e)$ can be written as
\[\nu(e) = 1 + \E_e [d(e, Y_1)] \]
Note again that in the positive recurrent case, the invariant PDF is the function $g$ obtained by normalizing $G$, and is not in general the PDF of $Y_1$ given $Y_0 = e$. The two PDFs are the same if and only if the distribution has constant rate.
Conversely, given an UPF $G$ on $S$, it's trivial to construct a downward run chain $\bs{Y}$ with $G$ as the standard invariant function. Specifically, suppose that $G$ is the UPF corresponding to the PDF $g$. We just need to define $Q(x, e) = g(x)$ for $x \in S$, and of course $Q(x, x^-) = 1$ for $x \in S - \{e\}$.
\subsection{Downward runs on path space}
Now return to the general case where $(S, \preceq)$ is an arbitrary poset and $(\hat S, \preceq)$ is the corresponding path space discussed in Section \ref{ss.paths} (a rooted tree). Suppose that $Q$ is the transition probability function of a downward run chain on $S$. We define $\hat Q$ on $\hat S$ as follows:
\begin{align*}
\hat Q(e, e \, x_1 \cdots x_n) &= Q(e, x_n) Q(x_n, x_{n-1}) \cdots Q(x_1, e), \quad e \, x_1 \cdots x_n \in \hat S \\
\hat Q(a, a^-) &= 1, \quad a \in \hat S - \{e\}
\end{align*}
It's easy to see $\hat Q$ is a valid transition probability function and corresponds to a downward run chain on $\hat S$. Moreover, starting at $e$, we can define the two chains on a common probability space. As before, if $\bs{\hat Y}$ is a process on $\hat S$, define the process $\bs{Y}$ on $S$ by $Y_n = m(\hat Y_n)$. The proof of the following proposition is straightforward.
\begin{proposition}
If $\bs{\hat Y}$ is a downward run chain on $\hat S$ with transition probability function $\hat Q$ and starting at $e$, then $\bs{Y}$ is a downward run chain on $S$ with transition probability function $Q$ and starting at $e$.
\end{proposition}
Of course the results of Section \ref{ss.downTrees} apply to $\bs{\hat Y}$. Also, with the chains coupled as in Proposition \ref{p.upCouple}, note that $T_e = \hat T_e$. The chains $\bs{Y}$ and $\bs{\hat Y}$ have the same classification.
Since $(\hat S, \preceq)$ is a rooted tree, $\hat G$ is an UPF. On the other hand $G$ may not be an UPF on $S$, and conversely, there may exist UPFs $G$ on $S$ that cannot be realized as the standard invariant function for a recurrent downward run chain on $S$.
\section{Time Reversal}
The class of recurrent upward run chains and the class of downward run chains are time reversals of each other.
\subsection{Reversing an upward run chain}
Suppose that $\bs{X}$ is a recurrent upward run chain with the transition probability function $P$ and standard invariant function $F$ (and other notation) as in Section \ref{s.upward}. Then the transition probability function $Q$ associated with the time reversed chain satisfies
\[ F(y)Q(y, x) = F(x) P(x, y), \quad x, \, y \in S \]
Thus, $Q(y, x) > 0$ if and only if $y = e$, or $y \ne e$ and $x \in B_y$. Hence the time reversed chain $\bs{Y}$ is a downward run chain and
\begin{equation}\label{eq.tr}
Q(y, x) = \frac{F(x)}{F(y)} P(x, y)
\end{equation}
Of course $F$ is also invariant for $Q$ and satisfies $F(e) = 1$ so $F = G$. In the positive recurrent case, $f = g$ and $\mu = \nu$.
We can interpret (\ref{eq.tr}) as
\[ Q(y, x) = \Pr_e[X_{T(y) - 1} = x | T_y \le T_e] \]
In particular, when $y = e$, we have
\[ Q(e, x) = \Pr_e[X_{T(e) - 1} = x] \]
Note that if $P(x, e)$ is constant in $x$, then the chains are positive recurrent, $f(x) = Q(e, x)$ is the invariant PDF. That is,
\[ Q(e, x) = \frac{F(x)}{\mu(e)}, \quad x \in S \]
\subsection{Reversing a downward run chain}
Suppose that $\bs{Y}$ is a downward run chain with transition probability function $Q$ and standard invariant function $G$ (and other notation) as in Section \ref{s.downward}. The transition probability function $P$ associated with the time reversal satisfies
\[ G(x) P(x, y) = G(y) Q(y, x), \quad x, \, y \in S \]
Thus $P(x, y) > 0$ if and only if $y \in A_x$ or $y = e$. Hence, the time reversed chain $\bs{X}$ is an upward run chain and
\begin{equation} \label{eq.tr2}
P(x, y) = \frac{G(y)}{G(x)} Q(y, x)
\end{equation}
As before, $F = G$, and in the positive recurrent case, $f = g$, $\mu = \nu$.
We can interpret (\ref{eq.tr2}) as
\[ P(x, y) = \Pr_e[Y_{U(x) - 1} = y | U_x \le U_e] \]
When $y = e$, (\ref{eq.tr2}) gives
\begin{equation} \label{eq.tr3}
P(x, e) = \frac{Q(e, x)}{G(x)}
\end{equation}
which we can interpret as
\[ P(x, e) = \Pr_e[U_x = 1 | U_x \le U_e] = \Pr_e(Y_1 = x | U_x \le U_e) \]
If $P(x, e)$ is constant in $x \in S$ then from (\ref{eq.tr3}), the chains are positive recurrent and $g(x) = Q(e, x)$ is the invariant PDF.
\section{Examples and Special Cases}
\subsection{Uniform Posets} \label{ss.uniform}
Suppose that the poset $(S, \preceq)$ is uniform, as defined in Section \ref{ss.uniform0}. The general results in Sections \ref{s.upward} and \ref{s.downward} simplify significantly.
For the upward run chain $\bs{X} = (X_0, X_1, \ldots)$, note that
\[ F(x) = P^{d(e, x)}(e, x), \quad x \in S \]
That is, $F(x)$ is the probability, starting at $e$, that the chain moves strictly upward in $S$, reaching state $x$ in the minimum time $d(e, x)$. Thus,
\begin{align}
&F(x) = P^n(e, x), \quad x \in S_n, \, n \in \N \notag \\
&\Pr_e(T_e > n) = P^n(e, S_n), \quad n \in \N \label{eq.uni2} \\
&\mu(e) = \E_e(T_e) = \sum_{n=0}^\infty P^n(e, S_n) \label{eq.uni3}
\end{align}
The chain is recurrent if and only if the expression in (\ref{eq.uni2}) has limit 0 as $n \to \infty$ and positive recurrent if and only if the sum in (\ref{eq.uni3}) is finite. In the positive recurrent case, the invariant PDF $f$ is given by
\[ f(x) = \frac{P^n(e, x)}{\mu(e)}, \quad x \in S_n, \, n \in \N \]
Consider the downward run chain $\bs{Y} = (Y_0, Y_1, \cdots)$. If $Y_0 = e$ then $U_e = n + 1$ if and only if $Y_1 \in S_n$. Hence
\[ \Pr_e (U_e = n + 1) = \Pr_e(Y_1 \in S_n) = Q(e, S_n) \]
and so
\begin{equation} \label{eq.nu}
\nu(e) := \E_e(U_e) = \sum_{n = 0}^\infty (n + 1) Q(x, S_n)
\end{equation}
Thus the chain is positive recurrent if and only if the sum in (\ref{eq.nu}) is finite. The standard invariant function $G$ also simplifies
\[ G(x) = \sum_{y \succeq x} Q(e, y) Q^{d(x, y)}(y, x), \quad x \in S \]
Now, for the upward run chain $\bs{X}$, consider the special case where
\[ P(x, A_x) = \alpha_n, \; P(x, e) = 1 - \alpha_n; \quad x \in S_n, \, n \in \N \]
where $\alpha_n \in (0, 1)$ for $n \in \N$. Thus, the chain moves from level $n$ to level $n + 1$ with probability $\alpha_n$, and resets to $e$ with probability $1 - \alpha_n$. Let $P_+(x, y) = P(x, y) / \alpha_n$ for $x \in S_n$ and let $N_n = d(e, X_n)$ for $n \in \N$. Then clearly $\bs{N} = (N_0, N_2, \ldots)$ is an ordinary success-runs chain on $(\N, +)$ while $P_+$ is the transition probability for a Markov chain that moves strictly upward in the poset. The behavior of the upward run chain $\bs{X}$ and the corresponding reversed downward run chain $\bs{Y}$ can be explained simply in terms of $\bs{N}$ and $P_+$.
Of course, the transition matrix $\hat P$ for $\bs{N}$ is given by
\begin{equation*}
\hat P(n, n+1) = \alpha_n, \; \hat P(n, 0) = 1 - \alpha_n, \quad n \in \N
\end{equation*}
Clearly $\hat T_0 = T_e$ where $\hat T_0$ is the hitting time to 0 for the chain $\bs{N}$. The standard invariant function $\hat F$ for $\bs{N}$ is
\[ \hat F(n) = \alpha_0 \cdots \alpha_{n-1}, \quad n \in \N \]
Both chains are recurrent if and only if $\prod_{k=0}^\infty \alpha_k = 0$ and both chains are positive recurrent if and only if $\mu(0) = \sum_{n=0}^\infty \alpha_0 \cdots \alpha_{n-1} < \infty$. In the positive recurrent case, the invariant PDF for $\bs{N}$ is
\[ \hat f(n) = \frac{\hat F (n)}{\mu(0)}, \quad n \in \N \]
Returning to the upward run chain $\bs{X}$, note that the standard invariant function $F$ satisfies
\[ F(x) = \hat F(n) P_+^n(e, x), \quad x \in S_n, \, n \in \N \]
In the positive recurrent case, the invariant PDF similarly satisfies
\[ f(x) = \hat f(n) P_+^n(e, x), \quad x \in S_n, \, n \in \N \]
The downward run chain obtained by reversing $\bs{N}$ has transition probabilities
\[ \hat Q(n+1, n) = 1, \; \hat Q(0, n) = \alpha_0 \cdots \alpha_{n-1} (1 - \alpha_n), \quad n \in \N \]
The downward run chain obtained by reversing $\bs{X}$ has transition probabilities
\begin{align*}
Q(y, x) &= \frac{P_+^n(e, x) P_+(x, y)}{P_+^{n+1}(e, y)}, \quad x \in S_n, y \in A_x, n \in \N \\
Q(e, x) &= \hat Q(0, n) P_+^n(e, x), \quad x \in S_n, n \in \N
\end{align*}
Note in particular that for $y \succ e$ and $x \in B_y$ the downward probability $Q(y, x)$ is independent of the parameters $(\alpha_0, \alpha_1, \ldots)$.
\subsection{Positive semigroups} \label{ss.ps}
Suppose now that the poset $(S, \preceq)$ is associated with a positive semigroup $(S, \cdot)$ with $I$ as the set of irreducible elements. It's natural to consider upward and downward runs that take advantage of the self-similarity noted in Section \ref{ss.semigroups0}.
An upward run chain $\bs{X}$ on $(S, \cdot)$ is {\em spatially homogeneous} if $P(x, xi) = r_i$, for all $x \in S$ and $i \in I$ where $r_i > 0$ and $r := \sum_{i \in I} r_i < 1$. It follows of course that $P(x, e) = 1 - r$ for $x \in S$. The standard invariant function for the corresponding chain on the free semigroup $(I^*, \cdot)$ is
\[ F^*(i_1 \, i_2 \cdots i_n) = \prod_{k=1}^n r_{i_k}, \quad (i_1, i_2, \ldots, i_n) \in I^n \]
or equivalently $\ F^*(a) = \prod_{i \in I} r_i^{n_i(a)}$ where $n_i(a)$ is the number of times that letter $i$ appears in word $a$. The chain is positive recurrent and moreover, the invariant distribution is ``exponential'' (see \cite{Siegrist}). The corresponding standard invariant function $F$ for $\bs{X}$ is given by
\[ F(x) = \sum_{i_1 i_2 \cdots i_n = x} F^*(i_1 i_2 \cdots i_n) \]
Of course, these results are a special case of the construction in Section \ref{s.upward}, since $(I^*, \cdot)$ is isomorphic to the path space $(\hat S, \preceq)$.
For the downward run chain $\bs{Y}$ obtained by reversing $\bs{X}$, the function $x \mapsto Q(e, x)$ is the invariant PDF $f$, since $P(x, e)$ is constant in $x$.
\subsection{Upward and downward runs on $(\N^k, +)$}
Consider the case of the uniform positive semigroup $(\N^k, +)$. For $i = 1, \ldots, k$, let $\bs{u}_i \in \N^k$ be the element with 1 in position $i$ and 0 in all other positions; these are the irreducible elements. For $\bs{x} \in \N^k$, let
\[ C(\bs{x}) = \frac{\left(\sum_{i=1}^k x_i\right)!}{\prod_{i=1}^k x_i !} \]
This is a multinomial coefficient and gives the number of factorings of $\bs{x}$ over $I$; in each factoring, $\bs{u}_i$ must occur $x_i$ times.
Consider the upward run chain with uniform probabilities, as in Sections \ref{ss.uniform} and \ref{ss.ps}. Specifically, let $r_i = P(\bs{x}, \bs{x} + \bs{u}_i)$, independent of $\bs{x} \in \N^k$, where $r := \sum_{i=1}^k p_i < 1$. For the upward run chain, the standard invariant function $F$ is given by
\[ F(\bs{x}) = C(\bs{x}) \prod_{i=1}^k r_i^{x_i}, \quad \bs{x} \in \N^k \]
The chain is positive recurrent and the invariant PDF is given by
\[ f(x) = (1 - r) F(\bs{x}) = (1 - r) C(\bs{x}) \prod_{i=1}^k r_i^{x_i}, \quad \bs{x} \in \N^k \]
If $\bs{Z} = (Z_1, \ldots, Z_k$) is a random vector with the invariant distribution then the (marginal) distribution of $Z_i$ is geometric on $\N$ with rate parameter
\[ \alpha_i := 1 - \frac{r_i}{\sum\{r_j : j \ne i\}} \]
For the downward run chain $\bs{Y}$ obtained by reversing $\bs{X}$, the downward probabilities are given by
\[ Q(\bs{x}, \bs{x} - \bs{u}_i) = \frac{x_i}{\sum_{j=1}^k x_j}, \quad \bs{x} \in \N - \{\bs{0}\} \]
Note that these probabilities are independent of $(r_1, r_2, \ldots, r_k)$.
|
1,108,101,566,778 | arxiv | \section{Introduction}
\noindent Paul Dirac, in 1963, published a paper on \textit{A Remarkable Representation of the $3+2$ de-Sitter Group} \cite{pamd}. He further showed that, among the infinitesimal generators of this group, there are four cyclic ones while the rest are hyperbolic. The representation is very simple with the wavefunctions depending on two variables. \\
\hspace*{2mm} The de-Sitter group is an important space-time symmetry in both particle physics and general relativity. It was initially introduced into physics for describing curved background \cite{hpr}, leading to significant cosmological consequences. It has several interesting subgroups and is also the starting point for building representations of the Poincare group for relativistic particles \cite{epw1}. The algebra presented in \cite{pamd} has found many applications \cite{kn}, especially in providing an explicit example of connecting the standard Schroedinger approach to the quantum mechanics with Wigner's phase space approach \cite{epw2}. Its mathematical construction has been used for building the two mode squeezed states in quantum optics \cite{hkn, hkny, ymk}. Thus, it has an important role in contemporary physics.\\
\hspace*{2mm} In this paper we give a simple derivation of Dirac's representation that is based on a generalisation of the well known Jordan-Schwinger map that yields a realisation of $SU(2)$ generators. This generalisation is motivated by the fact that the algebraic structure of Dirac's representation may be understood by infinitesimal symplectic transformations related to the group $Sp(4)$. It is possible to reproduce the complete structure of the representation \cite{pamd} using these transformations. Finally, some other features are discussed that provide new insights into this representation. \\
\hspace*{2mm} In section II, we briefly review the salient features of Dirac's remarkable representation. This is followed by showing, in section III, that this representation may be obtained from the $Sp(4)$ group using infinitesimal symplectic transformations. In section IV, we show that all results may be simply obtained by using a generalised Jordan-Schwinger type map. Section V discusses some physical aspects of Dirac's representation. Finally, Section VI contains our conclusion, including possible future directions.
\section{The Representation}
\noindent Dirac considered a pair of coupled harmonic oscillators. The ladder operators, denoted by $a_{i}$ and $a_{i}^{\dagger}$ satisfy the basic commutation relations, $$[a_{i}, a_{j}] = [a_{i}^{\dagger}, a_{j}^{\dagger}] = 0$$
\begin{equation}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[a_{i}, a_{j}^{\dagger}] = \delta_{ij} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (i,j = 1,2).
\label{eq1}
\end{equation}
\noindent Now sixteen quadratic operators can be constructed from these $a_{i}$ and $a_{i}^{\dagger}$. However, because of the relations (\ref{eq1}), only ten of these are independent. Appropriate linear combinations of these were used to define the following ten operators \cite{pamd}, $$ L_{1} = \dfrac{1}{2}(a_{1}^{\dagger}a_{2} + a_{2}^{\dagger}a_{1}) ~~ , ~~ L_{2} = \dfrac{i}{2}(a_{2}^{\dagger}a_{1} - a_{1}^{\dagger}a_{2}) $$ $$L_{3} = \dfrac{1}{2}(a_{1}^{\dagger}a_{1} - a_{2}^{\dagger}a_{2}) ~~ , ~~ H = \dfrac{1}{2}(a_{1}^{\dagger}a_{1} + a_{2}a_{2}^{\dagger}) $$ $$ K_{1} = -\dfrac{1}{4}(a_{1}^{\dagger}a_{1}^{\dagger} + a_{1}a_{1} - a_{2}^{\dagger}a_{2}^{\dagger} - a_{2}a_{2}) ~~~~~~~~~~~$$ $$ K_{2} = \dfrac{i}{4}(a_{1}^{\dagger}a_{1}^{\dagger} + a_{2}^{\dagger}a_{2}^{\dagger} - a_{1}a_{1} - a_{2}a_{2}) ~~~~~~~~~~~~~$$ $$ K_{3} = \dfrac{1}{2}(a_{1}^{\dagger}a_{2}^{\dagger} + a_{1}a_{2})~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$
$$ B_{1} = -\dfrac{i}{4}(a_{1}^{\dagger}a_{1}^{\dagger} - a_{1}a_{1} - a_{2}^{\dagger}a_{2}^{\dagger} + a_{2}a_{2}) ~~~~~~~~~~~$$ $$ B_{2} = -\dfrac{1}{4}(a_{1}^{\dagger}a_{1}^{\dagger} + a_{2}^{\dagger}a_{2}^{\dagger} + a_{1}a_{1} + a_{2}a_{2}) ~~~~~~~~~~~$$
\begin{equation}
B_{3} = \dfrac{i}{2}(a_{1}^{\dagger}a_{2}^{\dagger} - a_{1}a_{2}) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\label{eq2}
\end{equation}
\noindent The first four $(L_{1}, L_{2}, L_{3}, H)$ are cyclic rotations with $H$ separate from the other three, since it commutes with all of them. The remaining six correspond to hyperbolic rotations. These operators provide a representation of the $3+2$ de-Sitter group. This is the group of rotations of five real variables $q_{1}, q_{2}, q_{3}, q_{4}, q_{5}$ that keeps the quadratic form, $$q_{1}^{2}+q_{2}^{2}+q_{3}^{2}-q_{4}^{2}-q_{5}^{2}$$ invariant. Denoting the generators of this group by $J_{ij}$, we have,
\begin{equation}
[J_{ij}, J_{kl}] = i(J_{ik}\eta_{jl} - J_{il}\eta_{jk} + J_{jl}\eta_{ik} - J_{jk}\eta_{il}) ~~~~~ i,j = 1, 2, 3, 4, 5 ~~,~~ \eta_{ij} = (+ + + - -)
\label{eq3}
\end{equation}
\noindent The correspondance with the operators (\ref{eq2}) is given by, $$J_{ij} = \varepsilon_{ijk}L_{k} ~~~~ for ~~i, j, k = 1, 2, 3$$ $$J_{i4} = K_{i} ~~~~~~~~for ~~ i = 1,2,3~~~~~~~$$ $$J_{i5} = B_{i} ~~~~~~~~for ~~ i = 1,2,3~~~~~~~$$
\begin{equation}
J_{45} = H ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\label{eq4}
\end{equation}
\noindent There are three spacial rotation operators $(L_{1}, L_{2}, L_{3})$, two boosts $(K_{i}, B_{i})$ corresponding to two times and a hamiltonian $(H)$. It may be observed that this $H$ in (\ref{eq2}) is half the energy of two harmonic oscillators, as also mentioned in \cite{pamd}. In section V we shall give an explanation for this half factor. The individual algebra satisfied by these operators is given by, $$[L_{i}, L_{j}] = i\varepsilon_{ijk}L_{k} ~~,~~ [L_{i}, H] = 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$[L_{i}, K_{j}] = i\varepsilon_{ijk}K_{k} ~,~~ [K_{i}, K_{j}] = -i\varepsilon_{ijk}L_{k}~~~~~~~~~~~~~~~~~~~$$ $$[L_{i}, B_{j}] = i\varepsilon_{ijk}B_{k} ~~,~~ [B_{i}, B_{j}] = -i\varepsilon_{ijk}L_{k}~~~~~~~~~~~~~~~~~~~~$$
\begin{equation}
[K_{i}, H] = iB_{i} ~~,~~ [B_{i}, H] = -iK_{i} ~~,~~ [K_{i}, B_{j}] = i\delta_{ij}H ~~~
\label{eq5}
\end{equation}
\noindent The operators $L_{i}$ satisfy the $SU(2)$ algebra. Also, there are three sets of operators $(K_{1}, B_{1}, H)$, $(K_{2}, B_{2}, H)$ and $(K_{3}, B_{3}, H)$ that satisfy the hyperbolic $SU(1, 1)$ algebra. We will return to these issues later.
\section{Symplectic Transformations and Dirac's Representation}
\noindent Since the $3+2$ de-Sitter group is isomorphic to the four dimensional symplectic group $Sp(4)$, the algebra (\ref{eq5}) also serves to characterise the latter. This fact was utilised in \cite{kn, hkn} to illuminate the nature of the operators given in (\ref{eq2}). It was shown that the unitary transformations generated by these operators are translated into linear canonical transformations of the Wigner function for a pair of coupled oscillators. The corresponding group in this case is just $Sp(4)$. \\
\hspace*{2mm} In this section we provide an alternative approach that does not require the introduction of the Wigner function. The structure of the generators (\ref{eq2}) is deduced solely from properties of symplectic transformations. Starting from the matrix representation of the ten generators of $Sp(4)$, we construct the corresponding vector fields. From these fields, quadratic homogeneous polynomials involving four phase space variables are obtained. Finally, replacing the phase space variables by the oscillator variables, the ten generators in (\ref{eq2}) are reproduced. \\
\hspace*{2mm} A symplectic matrix $M$ is defined by the condition,
\begin{equation}
MJM^{T} = J
\label{eq6}
\end{equation}
\noindent where $M^{T}$ is the transpose of $M$ and $J$ has the canonical form,
\begin{equation}
J =
\begin{pmatrix}
0 ~&~ I \\
-I ~&~ 0
\end{pmatrix}
\label{eq7}
\end{equation}
\noindent with $I$ being the identity matrix. For two pairs of canonical variables relevant for $Sp(4)$, $J$ takes the antisymmetric structure,
\begin{equation}
J =
\begin{pmatrix}
0 ~&~ 0 ~&~ 1 ~&~ 0 \\
0 ~&~ 0 ~&~ 0 ~&~ 1 \\
-1 ~&~ 0 ~&~ 0 ~&~ 0 \\
0 ~& -1 ~&~ 0 ~&~ 0 \\
\end{pmatrix}
\label{eq8}
\end{equation}
\noindent Introducing the generators $G_{i}$ of $Sp(4)$ we have,
\begin{equation}
M = e^{-i\alpha_{i}G_{i}}
\label{eq9}
\end{equation}
\noindent where $G_{i}$ is a set of ten pure imaginary $4\times4$ matrices. The symplectic condition (\ref{eq6}) now implies that $G$ be either antisymmetric and commute with $J$ (first set) or, be symmetric and anticommute with $J$ (second set) \cite{kn}. \\
\hspace*{2mm} Using the Pauli spin matrices and the $2\times2$ identity matrix it is possible to construct the relevant generators. The first set is given by,
$$
L_{1} = \dfrac{i}{2}
\begin{pmatrix}
0 ~&~ \sigma_{1} \\
-\sigma_{1} ~&~ 0
\end{pmatrix}
~~,~~ L_{2} = \dfrac{1}{2}
\begin{pmatrix}
\sigma_{2} ~&~ 0\\
0 ~&~ \sigma_{2}
\end{pmatrix}
$$
\begin{equation}
L_{3} = \dfrac{i}{2}
\begin{pmatrix}
0 ~&~ \sigma_{3} \\
-\sigma_{3} ~&~ 0
\end{pmatrix}
~~,~~ H = \dfrac{i}{2}
\begin{pmatrix}
0 ~&~ I\\
-I ~&~ 0
\end{pmatrix}
\label{eq10}
\end{equation}
Likewise, the second set is given by,
$$
K_{1} = \dfrac{i}{2}
\begin{pmatrix}
0 ~&~ \sigma_{3} \\
\sigma_{3} ~&~ 0
\end{pmatrix}
~~,~~ K_{2} = \dfrac{i}{2}
\begin{pmatrix}
I ~&~ 0\\
0 ~&~ -I
\end{pmatrix}
~~,~~K_{3} = -\dfrac{i}{2}
\begin{pmatrix}
0 ~&~ \sigma_{1} \\
\sigma_{1} ~&~ 0
\end{pmatrix}
$$
\begin{equation}
B_{1} = -\dfrac{1}{2}
\begin{pmatrix}
\sigma_{3} ~&~ 0 \\
0 ~&~ -\sigma_{3}
\end{pmatrix}
~~,~~ B_{2} = \dfrac{i}{2}
\begin{pmatrix}
0 ~&~ I\\
I ~&~ 0
\end{pmatrix}
~~,~~B_{3} = \dfrac{i}{2}
\begin{pmatrix}
\sigma_{1} ~&~ 0 \\
0 ~&~ -\sigma_{1}
\end{pmatrix}
\label{eq11}
\end{equation}
\noindent Using the result,
\begin{equation}
[\sigma_{i}, \sigma_{j}] = 2i\varepsilon_{ijk}\sigma_{k}
\label{eq12}
\end{equation}
\noindent it is possible to show that the matrices (\ref{eq10}, \ref{eq11}) satisfy the algebra (\ref{eq5}). \\
\hspace*{2mm} We now show that the above matrices can be translated into oscillator variables that precisely reproduce the representation (\ref{eq2}). Let us consider the space $X = \rm I\!R^{2n} = \rm I\!R^{n} + \rm I\!R^{n}$ which has the symplectic form $w$,
\begin{equation}
w = \Sigma~dq_{i} \land dp_{i}
\label{eq13}
\end{equation}
\noindent Let $\xi$ be the vector field on $X$ and $i(\xi)$ denote the interior product by $\xi$. Then there is a smooth function $H$ determined up to a local constant such that \cite{gs},
\begin{equation}
i(\xi)w = dH
\label{eq14}
\end{equation}
\noindent Hence, for $w$ defined in (\ref{eq13}), the vector field is given by,
\begin{equation}
\xi_{H} = \Sigma ~ \bigg(\dfrac{\partial H}{\partial p_{i}}\dfrac{\partial}{\partial q_{i}} -\dfrac{\partial H}{\partial q_{i}}\dfrac{\partial}{\partial p_{i}}\bigg)
\label{eq15}
\end{equation}
\noindent The function $H$ is the hamiltonian and determines the evolution of the vector field (\ref{eq15}) from the differential equations,
\begin{equation}
\dfrac{dq_{i}}{dt} = \dfrac{\partial H}{\partial p_{i}} ~~,~~ \dfrac{dp_{i}}{dt} = -\dfrac{\partial H}{\partial q_{i}}
\label{eq16}
\end{equation}
\noindent The poisson bracket of two functions $f_{1}$ and $f_{2}$ is defined as,
\begin{equation}
\{f_{1}, f_{2}\} = -\xi_{f_{1}}f_{2}
\label{eq17}
\end{equation}
so that the familiar Hamilton's equations are obtained from (\ref{eq16}),
\begin{equation}
\dfrac{dq_{i}}{dt} = \{q_{i}, H\} ~~,~~ \dfrac{dp_{i}}{dt} = \{p_{i}, H\}
\label{eq18}
\end{equation}
Using the group property it can be shown that \cite{gs},
\begin{equation}
[\xi_{f_{1}}, \xi_{f_{2}}] = -\xi_{\{f_{1}, f_{2}\}}
\label{eq19}
\end{equation}
This is illustrated by taking a simple example. Consider the functions $f_{1}$ and $f_{2}$ to be the hamiltonians for the harmonic oscillator and free particle. respectively,
\begin{equation}
f_{1} = \dfrac{1}{2}(p^{2} + q^{2}) ~~,~~ f_{2} = \dfrac{p^{2}}{2}
\label{eq20}
\end{equation}
Then the corresponding vector fields are obtained from (\ref{eq15}),
\begin{equation}
\xi_{f_{1}} = p\dfrac{\partial}{\partial q} - q\dfrac{\partial}{\partial p} ~~,~~\xi_{f_{2}} = p\dfrac{\partial}{\partial q}
\label{eq21}
\end{equation}
The vector field corresponding to the Poisson bracket $\{f_{1}, f_{2}\}$ (defined in (\ref{eq17})) turns out to be,
\begin{equation}
\xi_{\{f_{1}, f_{2}\}} = \xi_{qp} = q\dfrac{\partial}{\partial q} - p\dfrac{\partial}{\partial p}
\label{eq22}
\end{equation}
It is now easy to prove (\ref{eq19}) using (\ref{eq21}) and (\ref{eq22}). This shows an isomorphism between the algebra of quadratic homogeneous polynomials (in two variables) and that of their corresponding vector fields. The isomorphism can be pushed further to include the symplectic algebra. To do this we first write the appropriate matrices. \\
\hspace*{2mm} For the function $f_{1}$ the equations of motion (\ref{eq16}) are given by,
\begin{equation}
\dfrac{dq}{dt} = \dfrac{\partial f_{1}}{\partial p} = p ~~,~~ \dfrac{dp}{dt} = -\dfrac{\partial f_{1}}{\partial q} = -q
\label{eq23}
\end{equation}
In matrix notation,
\begin{equation}
\dfrac{d}{dt}
\begin{pmatrix}
q \\
p
\end{pmatrix}
=
\begin{pmatrix}
0 ~&~ 1 \\
-1 ~&~ 0
\end{pmatrix}
\begin{pmatrix}
q \\
p
\end{pmatrix}
\label{eq24}
\end{equation}
Thus the function $f_{1}$ corresponds to the vector field $\xi_{f_{1}}$ and the matrix $\begin{pmatrix}
0 ~&~ 1 \\
-1 ~&~ 0
\end{pmatrix}
$ belonging to the Lie algebra of the symplectic group $Sp(2)$. Likewise we can find the matrices corresponding to $f_{2}$ and $\{f_{1}, f_{2}\}$. Collecting all results,
\begin{equation}
M_{f_{1}} =
\begin{pmatrix}
0 ~&~ 1 \\
-1 ~&~ 0
\end{pmatrix}
~~,~~
M_{f_{2}} =
\begin{pmatrix}
0 ~&~ 1 \\
0 ~&~ 0
\end{pmatrix}
~~,~~
M_{\{f_{1}, f_{2}\}} =
\begin{pmatrix}
1 ~&~ 0 \\
0 ~&~ -1
\end{pmatrix}
\label{eq25}
\end{equation}
It is easy to see that the algebra closes,
\begin{equation}
[M_{f_{1}}, M_{f_{2}}] = M_{\{f_{1}, f_{2}\}}
\label{eq26}
\end{equation}
Associating a self adjoint operator $f_{q}$ to each homogeneous quadratic polynomial $f$, it is possible to construct the map $f \longrightarrow -if_{q}$ such that Poisson brackets get replaced by commutators. We then have an isomorphim between the commutator algebra of the quadratic polynomials and the Lie algebra of the symplectic group. Finally, the quadratic polynomials may be equivalently represented in terms of the oscillator variables. Following this approach Dirac's representation (\ref{eq2}) is derived starting from the generators (\ref{eq10}, \ref{eq11}) of the symplectic group $Sp(4)$. \\
\hspace*{2mm} Let us explicitly work out one specific example. Take $L_{1}$ from the set (\ref{eq10}).
\begin{equation}
L_{1} = \dfrac{i}{2}
\begin{pmatrix}
0 ~&~ \sigma_{1} \\
-\sigma_{1} ~&~ 0
\end{pmatrix}
= \dfrac{i}{2}
\begin{pmatrix}
0 ~&~ 0 ~&~ 0 ~&~ 1 \\
0 ~&~ 0 ~&~ 1 ~&~ 0 \\
0 ~& -1 ~&~ 0 ~&~ 0 \\
-1 ~&~ 0 ~&~ 0 ~&~ 0
\end{pmatrix}
\label{eq27}
\end{equation}
The corresponding vector field is given by,
\begin{equation}
\xi_{L_{1}} = \dfrac{i}{2}\bigg(p_{2}\dfrac{\partial}{\partial q_{1}} + p_{1}\dfrac{\partial}{\partial q_{2}} - q_{2}\dfrac{\partial}{\partial p_{1}} - q_{1}\dfrac{\partial}{\partial p_{2}}\bigg)
\label{eq28}
\end{equation}
The homogeneous quadratic polynomial leading to (\ref{eq28}) is obtained from (\ref{eq15}),
\begin{equation}
L_{1} = \dfrac{i}{2}(p_{1}p_{2} + q_{1}q_{2})
\label{eq29}
\end{equation}
As a cross check we reproduce (\ref{eq27}) from (\ref{eq29}). The equations of motion (\ref{eq16}) are,
\begin{equation}
\dfrac{dq_{1}}{dt} = \dfrac{\partial L_{1}}{\partial p_{1}} = \dfrac{i}{2}p_{2} ~,~ \dfrac{dq_{2}}{dt} = \dfrac{i}{2}p_{1} ~,~ \dfrac{dp_{1}}{dt} = -\dfrac{i}{2}q_{2} ~,~ \dfrac{dp_{2}}{dt} = -\dfrac{i}{2}q_{1}
\label{eq30}
\end{equation}
so that,
\begin{equation}
\dfrac{d}{dt}
\begin{pmatrix}
q_{1} \\
q_{2} \\
p_{1} \\
p_{2}
\end{pmatrix}
= \dfrac{i}{2}
\begin{pmatrix}
0 ~&~ 0 ~&~ 0 ~&~ 1 \\
0 ~&~ 0 ~&~ 1 ~&~ 0 \\
0 ~& -1 ~&~ 0 ~&~ 0 \\
-1 ~&~ 0 ~&~ 0 ~&~ 0
\end{pmatrix}
\begin{pmatrix}
q_{1} \\
q_{2} \\
p_{1} \\
p_{2}
\end{pmatrix}
\label{eq31}
\end{equation}
thereby yielding the matrix (\ref{eq27}). Next, replacing $L_{1}$ by $-iL_{1}$ and substituting $q, p$ by corresponding operators, we obtain the operator version of (\ref{eq29}),
\begin{equation}
\hat{L_{1}} = \dfrac{1}{2}(\hat{p_{1}}\hat{p_{2}} + \hat{q_{1}}\hat{q_{2}})
\label{eq32}
\end{equation}
which is already hermitian since,
\begin{equation}
[\hat{q_{i}}, \hat{p_{j}}] = i\delta_{ij} ~~,~~ [\hat{q_{i}}, \hat{q_{j}}] = [\hat{p_{i}}, \hat{p_{j}}] = 0
\label{eq33}
\end{equation}
Introducing the ladder (creation and destruction) operators for oscillators,
\begin{equation}
\hat{q_{i}} = \dfrac{1}{\sqrt{2}}(a_{i} + a^{\dagger}_{i}) ~~,~~ \hat{p_{i}} = -\dfrac{i}{\sqrt{2}}(a_{i} - a^{\dagger}_{i})
\label{eq34}
\end{equation}
we finally obtain from (\ref{eq32}),
\begin{equation}
\hat{L_{1}} = \dfrac{1}{2}(a^{\dagger}_{1}a_{2} + a^{\dagger}_{2}a_{1})
\label{eq35}
\end{equation}
which reproduces the first relation in (\ref{eq2}). \\
\hspace*{2mm} In cases, unlike (\ref{eq32}), which are not hermitian, one has to just add the hermitian conjugate. As an example, consider $K_{2}$ from (\ref{eq11}),
\begin{equation}
K_{2} = \dfrac{i}{2}
\begin{pmatrix}
1 ~&~ 0 ~&~ 0 ~&~ 0 \\
0 ~&~ 1 ~&~ 0 ~&~ 0 \\
0 ~&~ 0 ~& -1 ~&~ 0 \\
0 ~&~ 0 ~&~ 0 ~& -1
\end{pmatrix}
\label{eq36}
\end{equation}
The vector field is,
\begin{equation}
\xi_{K_{2}} = \dfrac{i}{2}\bigg(q_{1}\dfrac{\partial}{\partial q_{1}} + q_{2}\dfrac{\partial}{\partial q_{2}} - p_{1}\dfrac{\partial}{\partial p_{1}} - p_{1}\dfrac{\partial}{\partial p_{2}}\bigg)
\label{eq37}
\end{equation}
and the corresponding polynomial follows from (\ref{eq15}),
\begin{equation}
K_{2} = \dfrac{i}{2}(q_{1}p_{1} + q_{2}p_{2})
\label{eq38}
\end{equation}
The operator version is given by,
\begin{equation}
\hat{K_{2}} = \dfrac{1}{4}(\hat{q_{1}}\hat{p_{1}} + \hat{p_{1}}\hat{q_{1}} + \hat{q_{2}}\hat{p_{2}} + \hat{p_{2}}\hat{q_{2}})
\label{eq39}
\end{equation}
where we have included the hermitian conjugate so that $\hat{K_{2}}$ is hermitian. Using the basic brackets (\ref{eq33}), this reduces to,
\begin{equation}
\hat{K_{2}} = \dfrac{1}{2}(\hat{q_{1}}\hat{p_{1}} + \hat{p_{2}}\hat{q_{2}})
\label{eq40}
\end{equation}
Translating into the oscillator variables (\ref{eq34}), this reproduces the corresponding result in (\ref{eq2}). \\
\hspace*{2mm} In this manner it is possible to derive the complete representation (\ref{eq2}) starting from (\ref{eq10}, \ref{eq11}).
\section{Generalised Jordan-type map}
\noindent It is useful to note that the matrices (\ref{eq10}, \ref{eq11}) which eventually led to the representation (\ref{eq2}) are all composed of the Pauli matrices and the identity matrix. This suggests the possibility of obtaining (\ref{eq2}) directly from the Pauli and identity matrices. Here we show how this can be done. \\
\hspace*{2mm} Before coming to the actual solution, let us first show that there are other possible representations of the de-Sitter group in terms of oscillator variables, which are different from (\ref{eq2}). This is based on Jordan's map,
\begin{equation}
\hat{M_{i}} = \dfrac{1}{2}a^{\dagger}M_{i}a
\label{neweq41}
\end{equation}
which states that the algebra of the operators $\hat{M_{i}}$ is isomorphic to that of the matrices $M_{i}$, where $a$ and $a^{\dagger}$ are the usual ladder operators. In other words if the $M_{i}$ satisfy,
\begin{equation}
[M_{i}, M_{j}] = \omega_{ijk}M_{k}
\label{neweq42}
\end{equation}
where $\omega_{ijk}$ are some structure constants, then,
\begin{equation}
[\hat{M_{i}}, \hat{M_{j}}] = \omega_{ijk}\hat{M_{k}}
\label{neweq43}
\end{equation}
so that $\hat{M_{i}}$ may be interpreted as an operator representation of the matrices $M_{i}$. Since we have earlier provided the ten generators of $Sp(4)$ in matrix form (\ref{eq10}, \ref{eq11}), it is straightforward to compute their corresponding operator versions from (\ref{neweq41}). Thus for $L_{1}$ (\ref{eq10}), we have,
\begin{equation}
\hat{L_{1}} = \dfrac{i}{4}(a_{1}^{\dagger}a_{4}+a_{2}^{\dagger}a_{3}-a_{3}^{\dagger}a_{2}-a_{4}^{\dagger}a_{1})
\label{neweq44}
\end{equation}
Likewise, it is possible to obtain the relevant operators for all the matrices (\ref{eq10}, \ref{eq11}). Clearly, these will satisfy the de-Sitter algebra (\ref{eq5}).\\
\hspace*{2mm} The structure found in this way is obviously different from the representation (\ref{eq2}). It now involves four coupled oscillators instead of two. Among other things, the representation (\ref{eq2}) is minimal in the sense that, to construct ten independent quadratic operators in the ladder variables, the minimum number of the oscillators needed is two. It is now clear that to reproduce (\ref{eq2}) in this fashion, we have to work with ($2\times2$) matrices instead of ($4\times4$)ones. As indicated, the obvious choice would be the Pauli matrices and the ($2\times2$) identity matrix.\\
\hspace*{2mm} There is a well known representation of the $SU(2)$ generators in terms of the oscillator (ladder) operators. This is called the Jordan-Schwinger map. It is a special version of (\ref{neweq41}) with $M_{i}$ replaced by the Pauli matrices and $\hat{M_{i}}$ by $L_{i}$, the angular momentum operator,
\begin{equation}
L_{i} = \dfrac{1}{2}~a^{\dagger}\sigma_{i}a
\label{eq41}
\end{equation}
from which follows,
\begin{equation}
[L_{i}, L_{j}] = i\varepsilon_{ijk}L_{k}
\label{eq42}
\end{equation}
which is the first relation in (\ref{eq5}). Indeed, using the explicit structure of the Pauli matrices, $L_{1}, L_{2}$ and $L_{3}$ exactly reproduce the corresponding relations in (\ref{eq2}). \\
\hspace*{2mm} The point is that (\ref{eq41}) is not the only possible way of expressing a hermitian operator in terms of a product of the Pauli matrices and quadratic functions of the oscillator variables. For instance, we can have the form,
\begin{equation}
W_{i} = \dfrac{1}{4}(a^{\dagger}\sigma_{i}a^{\dagger} + a\sigma_{i}a)
\label{eq43}
\end{equation}
which, like (\ref{eq41}), is also hermitian. Inserting the explicit form for the $\sigma$-matrices, we find,
\begin{equation}
W_{1} = K_{3} ~~,~~ W_{3} = -K_{1}
\label{eq44}
\end{equation}
Note that $W_{2} = 0$ due to reasons of symmetry. Another hermitian construction, similar to (\ref{eq43}), is given by,
\begin{equation}
Z_{i} = \dfrac{i}{4}(a^{\dagger}\sigma_{i}a^{\dagger} - a\sigma_{i}a)
\label{eq45}
\end{equation}
We find,
\begin{equation}
Z_{1} = B_{3} ~~,~~ Z_{3} = -B_{1}
\label{eq46}
\end{equation}
while $Z_{2} = 0$. This exhausts possible (quadratic) hermitian combinations involving the Pauli matrices. Now we consider the identity matrix. \\
\hspace*{2mm} The analogues of (\ref{eq43}) and (\ref{eq45}) involving the identity matrix are given by,
\begin{equation}
W = \dfrac{1}{4}(a^{\dagger}_{i}a^{\dagger}_{i} + a_{i}a_{i})
\label{eq47}
\end{equation}
and,
\begin{equation}
Z = \dfrac{i}{4}(a^{\dagger}_{i}a^{\dagger}_{i} - a_{i}a_{i})
\label{eq48}
\end{equation}
which yield,
\begin{equation}
W = -B_{2} ~~,~~ Z = K_{2}
\label{eq49}
\end{equation}
Finally, we have the structure,
\begin{equation}
H = \dfrac{1}{4}(a^{\dagger}_{i}a_{i} + a_{i}a^{\dagger}_{i})
\label{eq50}
\end{equation}
which reproduces the corresponding expression in (\ref{eq2}). Thus the complete representation (\ref{eq2}) may be deduced by using a generalisation of the Jordan map to include all possible quadratic combinations of the oscillator operators sandwitching the Pauli and identity matrices. These combinations have to be hermitian.
\section{Physical Interpretation}
In this section we discuss certain features of the representation (\ref{eq2}) that manifest a physical structure. In the cyclic sector comprising $L_{1}, L_{2}, L_{3}$ and $H$, while $L_{1}$ and $L_{2}$ have coupled expressions, $L_{3}$ and $H$ correspond to decoupled oscillators. Indeed it is possible to reproduce these expressions by considering a two dimensional oscillator as a pair of chiral oscillators, one rotating in the clockwise direction while the other in the anticlockwise direction. \\
\hspace*{2mm} The lagrangian for a one-dimensional harmonic oscillator is given by,
\begin{equation}
L = \dfrac{1}{2}(\dot{q}^{2} - q^{2})
\label{eq51}
\end{equation}
By introducing an additional variable it can be converted into its first order form \cite{bg},
\begin{equation}
L_{\pm} = \dfrac{1}{2}(\pm ~\varepsilon_{\alpha\beta}q_{\alpha}\dot{q_{\beta}} - q^{2}_{\alpha})
\label{eq52}
\end{equation}
where $\alpha = 1, 2$ is an internal index with $\epsilon_{12} = 1$. Eliminating either $q_{1}$ or $q_{2}$ in favour of the other yields (\ref{eq51}) with $q = q_{1}$ or $q_{2}$. The lagrangians $L_{\pm}$ are a pair of chiral oscillators with the sign of the first term determining their chirality. \\
\hspace*{2mm} Although the lagrangians (\ref{eq52}) each seem to have two degrees of freedom, only one of them is independent due to the presence of constraints which imply the symplectic structure,
\begin{equation}
\{q_{\alpha}, q_{\beta}\} = \mp~ \varepsilon_{\alpha\beta}
\label{eq53}
\end{equation}
for $L_{\pm}$. The corresponding hamiltonians are identical due to the first order nature of (\ref{eq52}),
\begin{equation}
H_{\pm} = \dfrac{1}{2}(q_{1}^{2} + q_{2}^{2}) = \tilde{H}
\label{eq54}
\end{equation}
The angular momentum is given by,
\begin{equation}
J_{\pm} = \varepsilon_{\alpha\beta}q_{\alpha}p_{\beta}
\label{eq55}
\end{equation}
where the canonical momenta $p_{\beta}$ is,
\begin{equation}
p_{\beta} = \dfrac{\partial L_{\pm}}{\partial \dot{q_{\beta}}} = \pm ~ \dfrac{1}{2} \varepsilon_{\rho\beta}q_{\rho}
\label{eq56}
\end{equation}
Thus,
\begin{equation}
J_{\pm} = \pm~ \dfrac{1}{2}q_{\alpha}^{2} = \pm~ \tilde{H}
\label{eq57}
\end{equation}
so that the angular momenta have the same magnitude, but differ in sign, a consequence of chirality. \\
\hspace*{2mm} Let us now show explicitly how $L_{+}$ and $L_{-}$ may be combined to yield a two dimensional oscillator. Consider the following combination of $L_{+}(q_{\alpha})$ and $L_{-}(r_{\alpha})$,
\begin{equation}
L = L_{+}(q_{\alpha}) + L_{-}(r_{\alpha}) = \dfrac{1}{2}\varepsilon_{\alpha\beta}q_{\alpha}\dot{q_{\beta}} - \dfrac{1}{2}q_{\alpha}^{2} - \dfrac{1}{2}\varepsilon_{\alpha\beta}r_{\alpha}\dot{r_{\beta}} - \dfrac{1}{2}r_{\alpha}^{2}
\label{eq58}
\end{equation}
Introducing new variables,
\begin{equation}
q_{\alpha} + r_{\alpha} = Q_{\alpha} ~~,~~ q_{\alpha} - r_{\alpha} = R_{\alpha}
\label{eq59}
\end{equation}
we get,
\begin{equation}
L = \dfrac{1}{2}\varepsilon_{\alpha\beta}R_{\alpha}\dot{Q_{\beta}} - \dfrac{1}{4}(Q_{\alpha}^{2} + R_{\alpha}^{2})
\label{eq60}
\end{equation}
where a total time derivative term has been dropped. Eliminating either $R$ or $Q$ in favour of the other yields,
\begin{equation}
L = \dfrac{1}{4}(\dot{Z_{\alpha}}^{2} - Z_{\alpha}^{2}) ~~;~~ Z_{\alpha} = R_{\alpha} ~~or~~ Q_{\alpha}
\label{eq61}
\end{equation}
which is the standard lagrangian for the two dimensional oscillator but with an averall normalisation of $\dfrac{1}{2}$. \\
\hspace*{2mm} From (\ref{eq53}, \ref{eq54}) we find that the hamiltonians for the chiral oscillators have identical expressions as normal oscillators.\hyperlink{footnote}{$^{1}$} Hence, accounting for the $\dfrac{1}{2}$ factor mentioned above, the final hamiltonian has the form \\
\begin{minipage}[b]{1\textwidth}
~~~~~~~~~~~~~~~~~~\\
\rule{15mm}{0.005pt}\\
\hypertarget{footnote}{$^{1}$}\footnotesize{Note that the canonical pairs ($q, p$) for $L_{+}$ and $L_{-}$ are ($q_{2}, q_{1}$) and ($q_{1}, q_{2}$), respectively.}
\end{minipage}
\begin{equation}
H = \dfrac{1}{2}(H_{+} + H_{-}) = \dfrac{1}{2}\bigg((a^{\dagger}_{1}a_{1} + \dfrac{1}{2}) + (a^{\dagger}_{2}a_{2} + \dfrac{1}{2})\bigg) = \dfrac{1}{2}(a^{\dagger}_{1}a_{1} + a^{\dagger}_{2}a_{2} + 1)
\label{eq62}
\end{equation}
while the angular momentum, which is the difference between $H_{+}$ and $H_{-}$, becomes,
\begin{equation}
J = \dfrac{1}{2}\bigg((a^{\dagger}_{1}a_{1} + \dfrac{1}{2}) - (a^{\dagger}_{2}a_{2} + \dfrac{1}{2})\bigg) = \dfrac{1}{2}(a^{\dagger}_{1}a_{1} - a^{\dagger}_{2}a_{2})
\label{eq63}
\end{equation}
These expressions match with the corresponding ones in (\ref{eq2}) (i.e. $H$ and $L_{z}$). Thus two chiral oscillators rotating in th $x-y$ plane in opposite (clockwise and anticlockwise) directions effectively yield the hamiltonian ($H$) and angular momentum ($L_{z}$ or $L_{3}$) given in (\ref{eq2}). From (\ref{eq62}) it is seen that all eigenvalues of the hamiltonian are positive with the minimum being $\dfrac{1}{2}$, corresponding to the zero point energy of the two chiral oscillators. If they had been usual oscillators then this minimum would have been 1. Likeiwse, the $z$-component of the angular momentum (\ref{eq63}) can have both integral and half integral values with the sign determined by the difference in the number of clockwise and anticlockwise rotating chiral oscillators. \\
\hspace*{2mm} A similar analysis may be done to understand the physical content of the remaining generators containing only uncoupled terms (like $K_{1}, K_{2}, B_{1}, B_{2}$). Incidentaly, ($K_{1}, B_{1}, H$) and ($K_{2}, B_{2}, H$) satisfy an algebra isomorphic to $SU(1,1)$, as easily checked from (\ref{eq5}). Thus the starting point is to consider the basic matrices defining this algebra.\\
\hspace*{2mm}Consider the following $2\times2$ matrices,
\begin{equation}
S_{1} = \dfrac{i}{2}
\begin{pmatrix}
1 ~&~ 0 \\
0 ~& -1
\end{pmatrix}
~~,~~
S_{2} = \dfrac{i}{2}
\begin{pmatrix}
0 ~&~ 1 \\
-1 ~&~ 0
\end{pmatrix}
~~,~~
S_{3} = \dfrac{i}{2}
\begin{pmatrix}
0 ~&~ 1 \\
1 ~&~ 0
\end{pmatrix}
\label{eq64}
\end{equation}
which satisfy the $SU(1,1)$ algebra,
\begin{equation}
[S_{1}, S_{2}] = iS_{3} ~~,~~ [S_{2}, S_{3}] = iS_{1} ~~,~~ [S_{3}, S_{1}] = -iS_{2}
\label{eq65}
\end{equation}
Following the technique discussed in sction III, the vector field corresponding to $S_{1}$ is given by,
\begin{equation}
\xi_{S_{1}} = \dfrac{i}{2}\bigg(q\dfrac{\partial}{\partial q} - p\dfrac{\partial}{\partial p}\bigg)
\label{eq66}
\end{equation}
Using (\ref{eq15}), the relevant polynomial generating this vector is found to be,
\begin{equation}
S_{1} = \dfrac{i}{2}(qp)
\label{eq67}
\end{equation}
and the corresponding hermitian operator is,
\begin{equation}
\hat{S_{1}} = -\dfrac{1}{4}(\hat{q}\hat{p} + \hat{p}\hat{q})
\label{eq68}
\end{equation}
Translating into oscillator variables,
\begin{equation}
\hat{S_{1}} = \dfrac{i}{4}(a^{\dagger2} - a^{2})
\label{eq69}
\end{equation}
Similarly $S_{2}$ and $S_{3}$ translate into,
\begin{equation}
\hat{S_{2}} = \dfrac{1}{4}(a^{\dagger}a + aa^{\dagger}) ~~,~~ \hat{S_{3}} = -\dfrac{1}{4}(a^{\dagger2} + a^{2})
\label{eq70}
\end{equation}
The operators $\hat{S_{1}},\hat{S_{2}}$ and $\hat{S_{3}}$ satisfy the same algebra as (\ref{eq65}). \\
\hspace*{2mm} We may now identify, with appropriate modifications, the set $\hat{S_{1}}, \hat{S_{2}}, \hat{S_{3}}$ with $K_{2}, H$ and $B_{2}$, respectively, as defined in (\ref{eq2}). One has to just consider two independent oscillators instead of a single one. Then the transition of $\hat{S_{1}}$ and $\hat{S_{3}}$ to $K_{2}$ and $B_{2}$ is straightforward. For $\hat{S_{2}}$, taking two independent oscillators yields,
$$ \hat{S_{2}}|_{1} + \hat{S_{2}}|_{2} = \dfrac{1}{4}(a^{\dagger}_{1}a_{1} + a_{1}a^{\dagger}_{1} + a^{\dagger}_{2}a_{2} + a_{2}a^{\dagger}_{2} )$$
\begin{equation}
= \dfrac{1}{2}(a^{\dagger}_{1}a_{1} + a_{2}a^{\dagger}_{2})~~~~
\label{eq71}
\end{equation}
obtained on using basic commutation relations. The final form is just $H$ in (\ref{eq2}). Incidently, the distinctive roles of $SU(2)$ and $SU(1)$ groups in interferometry has been discussed in \cite{ymk}. Here we see another example in the case of the de-Sitter representation (\ref{eq2}).
\section{Conclusion}
\noindent In this paper we have looked at the remarkable representation (\ref{eq2}) of the $3+2$ de-Sitter group found by Dirac \cite{pamd}. Although this work was done in 1963, its importance can be gauged from the fact that it is still relevant in contemporary physics. Among other applications \cite{kn}, this construction became the basic mathematical language for discussing two mode squeezed states in quantum optics \cite{hkn,hkny,ymk}. \\
\hspace*{2mm}Since the algebra of the $3+2$ de-Sitter group is isomorphic to the four dimensional symplectic group $Sp(4)$, it is usually customary to analyse the representation (\ref{eq2}) through this group. This is because the symplectic group, being the group of canonical transformations, provides a natural connection between physics and mathematics.\\
\hspace*{2mm} The matrices representing the generators of $Sp(4)$ were translated into homogeneous quadratic polynomials involving a pair of phase space variables ($q_{1}, p_{1}, q_{2}, p_{2}$). The step by step sequence of this construction was discussed. Once the polynomial was obtained, it was straightforward to express these in terms of the oscillator (ladder) operators using the standard expressions. In this way the complete representation (\ref{eq2}) was obtained.\\
\hspace*{2mm} It may be recalled that, in refs.\cite{kn,hkn,hkny}, the set (\ref{eq2}) was derived by using the ten matrices of $Sp(4)$ and performing canonical transformations on Wigner's phase space function involving a pair of canonical variables. Our derivation is more geometrical and does not need the Wigner function. While the connection of this function with density matrices is well known [~], the present analysis reveals its geometrical origin. Indeed the vector fields found here by symplectic considerations agree with those found by Wigner's phase space approach \cite{kn}.\\
\hspace*{2mm}Motivated by the use of Pauli and identity matrices in the building of the $Sp(4)$ representation, we have given a simple derivation of (\ref{eq2}) that generalises the well known Jordan-Schwinger map for $SU(2)$ matrices. By constructing the most general quadratic hermitian combinations of the ladder operators with the Pauli and identity matrices, this result was obtained. Effectively, the set (\ref{eq2}) may be interpreted as this generalised Jordan type map for the de-Sitter group. Since $SU(2)$ is a subgroup of the de-Sitter group, the usual Jordan map is just a subset of this map that is restricted to the three operators $L_{1}, L_{2}$ and $L_{3}$ in (\ref{eq2}).\\
\hspace*{2mm}A physical interpretation of the results has also been provided. It is easy to see from (\ref{eq2}) that there are some operators that involve a coupling of the pair of oscillators, but some are uncoupled. The uncoupled set $L_{3}$ and $H$ was shown to fit in the interpretation of regarding ($a_{1}, a_{1}^{\dagger}$) and ($a_{2}, a_{2}^{\dagger}$) as the ladder operators corresponding to chiral oscillators rotating in clockwise and anticlockwise directions. Furthermore, this interpretation explains the appearance of the half factor in the hamiltonian $H$ in (\ref{eq2}). Also, some other operators of (\ref{eq2}) were shown to be a simple addition of the corresponding operators for two independent oscillators whose ladder operators are used to define the $SU(1,1)$ representation. \\
\hspace*{2mm} The techniques developed here may be generalised in a systematic way to higher dimensions. For example, knowing the twenty one generators of $Sp(6)$ ($Sp(2n)$ has $2n^{2}+n$ generators) it would be possible to build a similar representation as (\ref{eq2}) comprising three coupled oscillators. Like (\ref{eq2}), this would also be the optimal or minimal representation since the number of independent quadratic functions of $a_{i}, a_{i}^{\dagger}~(i = 1, 2, 3)$ is exactly twentyone. Similarly, with more operators at hand, other generalised Jordan type maps analogous to (\ref{eq43}) and (\ref{eq45}) can be constructed. It is known that the Wigner phase space approach gets connected to other formalisms (like the Schoedinger approach) through density matrices. Our symplectic analysis provides another route, which is complimentary to the convensional group theoretic approach \cite{kn}. This aspect may be studied further by looking at specific problems in quantum mechanics.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\noindent {\bf{Acknowledgements}}: I thank Ankur Srivastav for typing the manuscript.
|
1,108,101,566,779 | arxiv | \section{Introduction}
\label{sec:intro}
Smart devices have gained significant market share in the mobile handset and personal computer markets, and this trend is expected to continue in the near future. Smartphone shipments were $43.7\%$ of all handset shipments in 2012, and smartphones' share in the handset market is expected to grow to $65.1\%$ in 2016~\cite{bib:canalysMobileMarket13}. Furthermore, while smartphones represented only $18\%$ of total global handsets in use in 2012, they were responsible for $92\%$ of total global handset traffic~\cite{bib:ciscoMobileForecast13}. Therefore, smart devices evidently have a central role in the current and future mobile landscape. The growing popularity of smart mobile devices, Android and iOS devices in particular~\cite{bib:idcMobileOS13}, has not gone unnoticed by cyber-criminals, who have started to address these smart mobile ecosystems in their activities. Smart mobiles are highly attractive targets since they combine personal data such as lists of contacts, social networks, and, increasingly, financial data and security credentials for online banking, mobile payments and enterprise intranet access, in a frequently used and always connected device. While most users are aware of security risks with using a traditional PC or laptop and therefore are more cautious, smart mobile devices have been found to provide a false sense of security~\cite{bib:ncsaMobileSurvey12}, which exacerbates the mobile risk.
Smart devices also provide access to mobile networks for cyber-criminals and attackers to cross service and network boundaries by exploiting the vulnerabilities of the multiple communication technologies that smart devices have. Evolution of the mobile networks also introduces additional vulnerabilities, for example via the adoption of new radio access technologies such as femtocells~\cite{bib:goldeFemtocell12}. Although the use of femtocells and other complementary access is recent and not yet widespread, their effect should not be underestimated. For example in 2012, $429$ petabytes per month of global mobile data was offloaded onto the fixed network through \mbox{Wi-Fi} or femtocell radio access, which accounts for $33\%$ of total mobile traffic~\cite{bib:ciscoMobileForecast13}. Mobile devices are also increasingly at the center of security systems for managing small or large emergencies in built environments, or during sports or entertainment events~\cite{bib:gelenbeICCCN12,bib:gelenbeWuSimEvac12}, and they are used increasingly for online search of difficult-to-get sensitive information~\cite{bib:gelenbeSearch10,bib:abdelrahmanSearch13}. Thus they will necessarily be targeted and breached in conjunction with other physical or cyber attacks, as a means of disrupting safety and confidentiality of individuals and emergency responders~\cite{bib:gorbilANT11,bib:gorbilISCIS11,bib:gorbilPernem13}.
The ability of smart devices to install and run applications from official stores and third-party application markets has significantly increased the mobile malware threat~\cite{bib:feltSurveyMobileMalware11,bib:zhouAndroidMalware12}. While the mobile malware threat is not new~\cite{bib:dagonMobileVirus04}, it is decidedly evolving and growing as attackers experiment with new business models by targeting smart mobile users~\cite{bib:lookoutMobileSecurity12,bib:kasperskyMalwareEvolution12}. For example, the number of detected malware was more than $35,000$\footnote{This number is for Android alone, which accounts for $99\%$ of all encountered malware in 2012~\cite{bib:kasperskyStatistics12}.} in 2012, which reflects a six-fold increase from 2011~\cite{bib:kasperskyMalwareEvolution12}. 2012 has also seen the emergence of the first mobile botnets~\cite{bib:kasperskyStatistics12}. A botnet is a collection of Internet-connected devices acting together to perform tasks, often under the control of a command and control server. Most malicious botnets are used to generate various forms of spam, phishing, and distributed denial-of-service (DDoS) attacks. In addition to giving cyber-criminals the advantages of control and adaptability, mobile botnets are also a significant threat to the mobile core network as they could be used to launch debilitating signaling-based DDoS attacks~\cite{bib:leeDetectionDoS3G09,bib:traynorCellularBotnet09}. In order to address the growing mobile threat, there is an urgent need to detect, analyze and understand the new vulnerabilities and threats in the smart mobile ecosystem. These new vulnerabilities and threats are a result of the evolution of mobile networks and smart devices, the changing way users interact with technology, the popularity of smart devices, and the heterogeneity of the wireless interfaces, supported platforms and offered services. We need to be proactive and work on predicting threats and vulnerabilities to build our defenses before threats materialize in order to advance in the fast moving field of cyber-security and to counter existing and potential mobile threats. In the next section, the approach adopted in the NEMESYS project for this purpose is described.
Thus the EU FP7 research project NEMESYS\footnote{http://www.nemesys-project.eu/nemesys/index.html} will develop a novel security framework for gathering and analyzing information about the nature of cyber-attacks targeting mobile devices and the mobile core network, as well as the identification and prediction of abnormal behaviours observed on smart mobile devices so that appropriate countermeasures can be taken to prevent them. We aim to understand the modus operandi of cyber-criminals, and to identify and reveal the possible shift in the way they launch attacks against mobile devices through root cause analysis and correlation of new findings with known patterns of attacks on wireline networks.
\section{The Data Collection Infrastructure}
\label{sec:dataCollection}
\begin{figure}[tbp]
\centering
\includegraphics[height=6.5cm,width=0.99\linewidth]{architecture2.eps}
\caption{The NEMESYS architecture}
\label{fig:architecture}
\end{figure}
Figure \ref{fig:architecture} shows the system architecture that will be developed within the NEMESYS project. The core of the NEMESYS architecture consists of a data collection infrastructure (DCI) that incorporates a high-interaction honeyclient and interfaces with virtualized mobile honeypots (VMHs) in order to gather data regarding mobile attacks. The honeyclient and VMHs collect mobile attack traces and provide these to the DCI, which are then enriched by analysis of the data and by accessing related data from the mobile core network and external sources. For example, TCP/IP stack fingerprinting in order to identify the remote machine's operating system, and clustering of the traces are passive methods of data enrichment. DNS reverse name lookup, route tracing, autonomous system identification, and geo-localization are methods to improve characterization of remote servers which may require access to external sources, possibly in real time. The enriched mobile attack data is made available to anomaly detection module, and the visualization and analysis module (VAM) running on the mobile network operator site.
As an initial step in the design of the DCI, we are identifying available external data sources relating to wireline network attacks which will enable correlation of attack-related data coming from multiple heterogeneous sources. Different sources of information that NEMESYS partners maintain and have access to will be used for this purpose, for example SGNET~\cite{bib:leitaSgnet08}, HARMUR~\cite{bib:leitaHarmur11}, and VirusTotal. A source aggregator will be designed and developed to harvest and consolidate data from these sources, and the honeyclient and VMHs, in a scalable database. Scalable design of the database is important in order to be able to efficiently store and handle large heterogeneous data sets. As a final step, the DCI will help in the definition of the appropriate inputs representing normal and malicious network activity, which will then be used as the fundamental representation of information in the VAM.
The \textit{honeyclient} being developed as part of the DCI is responsible for interacting with web servers to find websites with malicious content targeting mobile users, and for collecting related mobile threat data. The honeyclient consists of the crawler, client, and detector components. The crawler will generate a list of websites of interest for the client to visit. The client component will run an Android emulator which will generate, queue, and execute the requests corresponding to each discovered website, and record traces of changes in the system state that occur as a result. The malware detector component will be used to detect malicious content. Data relating to identified malicious websites will be provided to the DCI by the honeyclient, which is described in more detail in~\cite{bib:delosieresMalwareDetection13}.
\section{Virtualized Mobile Honeypots}
\label{sec:honeypot}
We adopt the high-interaction virtualized mobile client honeypot scheme in order to attract and collect mobile attack traces. Honeypots are networked computer system elements that are designed to be attacked and compromised so we can learn about the methods employed by the attackers~\cite{bib:provosHoneypot07}. Traditional honeypots are servers that passively wait to be attacked, whereas client honeypots are security devices that actively search for malware, compromised websites and other forms of attacks. High-interaction client honeypots are fully functional, realistic client systems which do not impose any limitations on the attacker other than those required for containing the attack within the compromised system. Despite their complexity and difficulty of maintenance, high-interaction client honeypots are effective at detecting unknown attacks, and are harder to detect by the attacker~\cite{bib:provosHoneypot07}. They also enable in-depth analysis of the attacks during and after the attack has taken place.
In NEMESYS, we are developing a high-interaction virtualized client honeypot for the Android mobile platform. We have chosen Android considering its popularity among mobile users and the extremely high ratio of malware targeting Android~\cite{bib:kasperskyStatistics12,bib:baltatuNemesys13}. We are developing a virtualization technology that addresses the problems we have identified in the system- and application-level security mechanisms of Android and enables secure support for new schemes of smart device use such as ``bring your own device''~\cite{bib:liebergeldAndroidSecurity13}. Our virtualization technology logically partitions the physical device into two virtual machines (VMs): the \textit{honeypot VM} and the \textit{infrastructure VM}. The honeypot VM will host the largely unmodified mobile device operating system, and it will not have direct access to the device's communication hardware. The infrastructure VM will mediate all access to the communication hardware, and employ sensors to wiretap any communication and detect suspicious behaviour. It will also provide the event monitoring, logging and filesystem snapshot facilities, as well as transmit threat information to the DCI. It will host a \textit{lightweight malware detection module} in order to identify malicious applications running on the honeypot VM. For this purpose, both signature-based and behaviour-based approaches will be considered. In order to improve the efficiency of malware detection, we will identify and prioritize the most important attributes in the system state space to monitor.
Our virtualization technology will ensure that an attack is confined within the compromised device so that it will not put other devices in the network at risk. It will also stop malware from using premium rate services and from subscribing the user to services without her knowledge. Thus, the user will be spared from any financial distress that may arise as a result of using the mobile honeypot. The virtualization solution also enables taking full snapshots of the honeypot VM filesystem for further forensic analysis of an attack, as well as improving honeypot maintenance since a compromised honeypot could be restored more quickly.
Current IP-based attacks encountered on mobile devices~\cite{bib:wahlischMobileHoneypot13} have been found to be largely similar to non-mobile devices~\cite{bib:gelenbeLoukasDoS07,bib:gelenbeSelfAware09}, but we are more interested in the traits of attacks that are tailored specifically for mobile devices. Our initial research has shown that the infection vector of most mobile malware is social engineering, where users are ``tricked'' into installing the malware themselves. Upcoming malware will also employ attack vectors that require interaction with the user; for example, we have already witnessed the first malicious QR codes, which need to be scanned by the user for their activation. These observations have led us to the conclusion that the user should not be ignored in the construction of an effective mobile honeypot. To this end, we introduce the \textit{nomadic honeypot} concept~\cite{bib:liebergeldHoneypot13}, which utilizes real smartphone hardware running the virtualization solution being developed by NEMESYS. We plan to deploy nomadic honeypots by handing them out to a chosen group of volunteers, who will use the honeypot as their primary mobile device. It will be up to these human users to get the honeypot infected by visiting malicious sites, installing dubious applications, etc. Traces from malware and other types of mobile attacks collected by the honeypots will be provided to the DCI.
\section{Anomaly Detection Using Control Plane and Billing Data}
\label{sec:anomalyDetection}
The purpose of the anomaly detection mechanisms is to identify and predict deviations from normal behaviour of mobile users and the core network. These mechanisms will utilize Charging Data Records (CDR) of the users and control-plane protocol data, together with enriched mobile attack traces provided by the DCI. In addition to attacks targeting mobile users, mobile networks are vulnerable to a novel DoS attack called the signaling attack~\cite{bib:leeDetectionDoS3G09}, which seeks to overload the control plane of the mobile network using low-rate, low-volume attack traffic by exploiting the structure and characteristics of mobile networks, for example by repeatedly triggering radio channel allocations and revocations. We will use control-plane protocol data such as traces of signaling events in order to identify such DoS attacks against the mobile network. Sanitized (anonymized) billing data will mostly be used to identify attacks targeting mobile users. For these purposes, we will use normal user behaviour statistics, as well as synthetic ``typical'' user playbacks, to create traces of signaling events and CDRs so as to characterize and extract their principal statistics such as frequencies, correlations, times between events, and possible temporal tendencies over short (milliseconds to seconds) and long (hours to days) intervals. We will employ Bayesian techniques such as maximum likelihood detection, neuronal techniques based on learning, and a combination of these two in order to design and develop robust and accurate change detection algorithms to detect the presence of an attack, and classification algorithms to identify with high confidence the type of attack when it is detected. Novel femtocell architectures provide a specific opportunity for user-end observation of network usage, while they also have specifics for attacks within the femtocells. To address femtocell-specific attacks, we will conduct a survey and evaluation of how users may be monitored and attacks detected within a femtocell, and how these are linked to overall mobile network events.
In these environments a number of novel ideas are being exploited. The structure of the signaling and billing network is being modeled as a queueing network~\cite{bib:gelenbeMuntzProb76} to capture the main events that involve hundreds of thousands of mobile calls and interactions among which only a few may be subject to an intrusion or attack at any given time. Detection of abnormalities is studied using learning techniques based on neural network models~\cite{bib:gelenbeRNN99,bib:gelenbeNatural12} that can provide the fast low-order polynomial or linear detection complexity required from the massive amount of real-time data, and the need to detect and respond to threats in real-time. Such techniques can also benefit from distributed task decomposition and distributed execution for greater efficiency~\cite{bib:aguilarTask97}. Our approach to anomaly detection is discussed in more detail in~\cite{bib:abdelrahmanAnomalyDetection13}.
\section{Root Cause Analysis, Correlation and Visualization}
\label{sec:analysisVisualization}
Enriched attack traces and mobile network data collected by the DCI, and the output of the anomaly detection modules are fed into the visualization and analysis module (VAM). The VAM's purpose is to aid the detection of existing and emerging threats in the mobile ecosystem through attack attribution, root cause identification, and correlation of observed mobile attacks with known attack patterns. The data provided to the VAM represents a large and heterogeneous data set that needs to be presented in a meaningful way to the security analyst without overwhelming her or restricting available views and actions on the data. In addition to mere representation of data, the VAM aims to provide visual analytics tools to the analyst. This task is compounded by different uses of visualization: (i) real-time monitoring of the status of mobile users and the mobile network, and (ii) exploratory data analysis. For real-time monitoring, the security status of a large set of mobile users, and more importantly the mobile network, need to be presented. This includes providing early alerts for abnormal behaviour, DoS attacks, malware spreading among the users of the mobile network, etc. The VAM must also provide visual analytics tools so the analyst can perform hypothesis formulation and testing, attack attribution, and correlation analysis, with the help of algorithms running in the background.
In order to effectively visualize and explore large sets of heterogeneous, dynamic, complex data, it is necessary to create multiple coordinated views of the data that allow a multi-faceted perception and the discovery of any hidden attributes. The analysis methods also need to be scalable for early network alerts and fast access to the underlying data. We will therefore focus on enabling a real-time analysis framework by means of incremental analysis and visualization methods, such as multi-level hierarchical screen visualizations that update smoothly rather than showing abrupt changes. Visualization of mobile network data is discussed in more detail in~\cite{bib:papaVisualNetwork13}.
\section{Conclusions}
\label{sec:conclusion}
In the NEMESYS Project, we will address and understand the new and potential vulnerabilities, threats, and operating methods of cyber-criminals, and provide new insight into next generation network security for the smart mobile ecosystem. We will contribute to the research novel security technologies for the identification and prediction of abnormal behavior observed on smart mobile devices, and to the gathering and analyzing of information about cyber-attacks that target mobile devices, so that countermeasures can be taken. We will develop virtualized honeypots for mobile devices, a data collection infrastructure, and novel attack attribution and visual analytics technologies for mining, presentation of large amounts of heterogeneous data regarding the smart mobile ecosystem.
\begin{acknowledgement}
The work presented in this paper was supported by the EU FP7 collaborative research project NEMESYS (Enhanced Network Security for Seamless Service Provisioning in the Smart Mobile Ecosystem), under grant agreement no. 317888 within the FP7-ICT-2011.1.4 Trustworthy ICT domain.
\end{acknowledgement}
\bibliographystyle{IEEEtranSortedNoDash}
|
1,108,101,566,780 | arxiv | \section{Introduction} \label{sec1}
It is generally believed that a neutron star (NS) is born as a result
of the gravitational collapse of the iron core of a massive evolved
progenitor star ($M\sim 8\,-\,25\,M_{\sun}$) in a Type-II supernova
\cite[e.g.][]{Bet90}. The iron core of such a star collapses when its
mass reaches the Chandrasekhar limit
\begin{equation}
\label{1.1}
M_\mathrm{Ch} \sim 5.8Y^{2}_\mathrm{e} M_{\sun}
\sim 1.2 \ldots 1.5 M_{\sun}\, ,
\end{equation}
where $Y_\mathrm{e}$ denotes the number of electrons per baryon which
depends on the mass of the progenitor star. Due to the Fermi pressure of
the nucleons, the collapse stops when nuclear matter density is reached
and the core bounces back. Shortly after core bounce (some
10 ms) a hot, lepton rich NS, called {\it protoneutron} star (PNS), is
formed. This PNS consists of a shocked envelope with an entropy per
baryon\footnote{$k_\mathrm{B} =
\hbar = c = 1$ throughout this paper.} $s$ $\sim$ 4 - 10 and an
unshocked core with $s \sim 1$ \cite[]{BHF95}.
The envelope and the core contain
nearly the same mass of about $0.6$ - $0.8 M_{\sun}$ \cite[slightly
depending on the mass of the progenitor star, see][]{BHF95, KJM96}.
During the so-called Kelvin-Helmholtz cooling phase
\cite[e.g.][]{Jan93} the lepton number decreases in the PNS due to the
loss of neutrinos and consequently the PNS evolves in about 10 to 30
seconds into a hot, lepton poor neutron star (HNS) with an entropy per
baryon, $s\sim
1\,-\,2$, depending on the model \cite[e.g.][]{BL86, KJ95, SST95,
PRPLM98}. After several minutes this HNS cools to a cold neutron star
(CNS) with
temperatures $T < 1$ MeV throughout the star
\cite[e.g.][]{KJ95}. Finally, it slowly cools via neutrino and photon
emission until its thermal radiation is too weak to be observable after
about $10^7$~yr \cite[e.g.][]{Tsuruta66,Schaab96}.
The PNS is in $\beta$-equilibrium during its lifetime, since the time
scale of the weak-interaction is much smaller than the
evolutionary time scale, i.e. the neutrino diffusion time scale or the
neutrino cooling time scale, respectively. Hence the evolution of a
PNS can be studied by considering quasi-stationary models at different
times.
The properties of PNS's were investigated by different authors. For
example, the case of non-rotating PNS's was studied by \cite{TNH94},
\cite{Bom95}, and \cite{Pra97}. The case of rotating PNS's was treated
by \cite{RDIMP92} and \cite{Tak95, Tak96} by means of an empirical
formula for the Kepler frequency which was developed for CNS's
\cite[e.g.][]{HSB95}. \cite{HOE94} and \cite{GHZ97} account for rapid
rotation by using an exact, general relativistic approach. Finally,
\cite{GHZ98} have performed the case of differential rotation of PNS's.
Most of these authors did not utilize an equation of state (EOS) of
hot matter throughout the whole star \cite[except][]{RDIMP92, GHZ97,
GHZ98}, but used an EOS of cold matter for the envelope of the star
instead. As we will show, this simplification leads to radii (Kepler
frequencies) which are too small (large).
The aim of this work is to study the properties of non-rotating and
rapidly, uniformly rotating PNS's. We use an exact, general
relativistic approach to rapid rotation \cite[]{Schaab98}. The hot
dense matter is described by a recently devolped EOS \cite[]{Str98},
which is based on a modern parametrisation of the Thomas-Fermi
approach for finite nuclei and cold nuclear matter performed by
\cite{MS90, MS91, MS96}. We generalize this approach to hot dense matter
by taking the thermal effects on both the kinetic and the interaction
energy into account. In this way, we
construct a set of EOS's with different profiles of the entropy per
baryon and different lepton numbers. We can follow the evolution of
the PNS into a CNS, by means of this set, at different
evolutionary stages. We also investigate the influence of the
location and the shape of the neutrino sphere as well as the influence
of the value of the temperature in the star's envelope.
The paper is organized as follows. Firstly, in Sect.~\ref{sec2} we
will briefly review the physics in the interior of PNS's and describe
the different evolutionary stages of PNS's and NS's. Furthermore, we
discuss the location and the shape of the neutrino sphere.
The EOS's of PNS and NS matter are described in Sect.~\ref{sec3}, where we
emphasize the influence of finite temperature and trapped lepton
number. The properties of rotating and non-rotating PNS's and NS's are
presented in Sect.~\ref{sec4}. Finally, discussion of our results
and conclusions are given in Sect.~\ref{sec5}.
\section{Inside a protoneutron star} \label{sec2}
A PNS differs in several respects from a CNS: At the beginning of its
lifetime the PNS contains a high lepton number, $Y_\mathrm{l}$, since
the core is opaque with respect to neutrinos. A further difference is
the high temperature which cannot be neglected with respect to the
Fermi temperature throughout the whole PNS. We define, as usual, the
lepton number, $Y_\mathrm{l}=Y_\mathrm{e}+Y_{\nu_\mathrm{e}}$, as the
sum of the net electron fraction,
$Y_\mathrm{e}=(n_\mathrm{e^{-}}-n_\mathrm{e^{+}})/n$, (where $n$,
$n_\mathrm{e^{-}}$, and $n_\mathrm{e^{+}}$ are the baryon number
density, the electron number density, and the positron number density,
respectively) and the net electron neutrino fraction,
$Y_{\nu_\mathrm{e}}=(n_{\nu_\mathrm{e}}-n_{\bar \nu_\mathrm{e}})/n$,
(where $n_{\nu_\mathrm{e}}$ and $n_{\bar \nu_\mathrm{e}}$ are the
electron neutrino number density and the electron anti-neutrino number
density, respectively). Since the muon number density is small in a
PNS, they are neglected here. The reason for the
small muon number is that the muon lepton family number is conserved,
$Y_{\mu} + Y_{\nu_\mu}=0$, while the neutrinos are
trapped\footnote{This statement is also true for massive neutrinos
recently detected by the Super-Kamiokande collaboration
\cite[see][]{Fuk98}. The detected mass is too small to allow
considerable flavour oscillations during the first seconds of a
supernova \cite[see discussion in Section 9.5.3 of][]{Raf96}}.
Relativistic calculations lead to the conclusion that NS's and PNS's
are composed not only of nucleons and leptons but also of hyperons
and, possibly, of nucleon isobars \cite[see, e.g.][]{Pan71,
SM96, BG97, HWWS98}.
Nevertheless, we shall not take these particle species into
account. In view of the rather large uncertainties of the hyperon
couplings, we shall, as a first approach to this problem, neglect
these additional degrees of freedom. In the following we select four
different stages in the evolution of PNS's and NS's, namely at times
$t_{1} \sim 50 - 100$ ms, $t_{2} \sim 0.5 - 1$ s, $t_{3} \sim 10-30$
s, and $t_{4} = $ some minutes after core bounce.
\subsection{Protoneutron stars about 50 - 100 ms after core bounce}
\label{sec21}
\begin{table*}
\caption[]{Entropies, temperatures, densities, and lepton numbers
used in this paper.
The entries are: entropy per baryon or temperature in the
envelope, $s_\mathrm{env}$, $T_\mathrm{env}$;
entropy per baryon or temperature in the
core, $s_\mathrm{core}$, $T_\mathrm{core}$;
maximum baryon number density of the envelope correlated
with the
entropy per baryon or temperature in the envelope,
$n(s_\mathrm{env}$, $T_\mathrm{env}$);
minimum baryon number density of the core correlated
with the
entropy per baryon or temperature in the core,
$n(s_\mathrm{core}$, $T_\mathrm{core}$);
baryon number density below which the neutrinos are not trapped,
$n(Y_\mathrm{l,env})$;
baryon number density above which the neutrinos are totally trapped,
$n(Y_\mathrm{l,core})$;
lepton fraction inside the core, $Y_\mathrm{l,core}$.}
\label{EOSs}
\begin{tabular}{ l c c c c c c c }
\hline
& & & & & & & \\
Label & $s_\mathrm{env}$, $T_\mathrm{env}$ &
$s_\mathrm{core}$, $T_\mathrm{core}$ &
$n(s_\mathrm{env}, T_\mathrm{env})$ &
$n(s_\mathrm{core}, T_\mathrm{core})$ &
$n(Y_\mathrm{l,env})$ & $n(Y_\mathrm{l,core})$ &
$Y_\mathrm{l,core}$ \\
& & & [fm$^{-3}$] &
[fm$^{-3}$] & [fm$^{-3}$] & [fm$^{-3}$] & \\
& & & & & & & \\
\hline
& & & & & & & \\
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ & 5.0 & 1.0 & 0.02 & 0.1 &
0.0006 & 0.0006 &
0.4 \\
EPNS$^\mathrm{s4s1}_\mathrm{YL04}$ & 4.0 & 1.0 & 0.02 & 0.1 &
0.0006 & 0.0006 &
0.4 \\
& & & & & & & \\
LPNS$^\mathrm{s2}_\mathrm{YL04}$ & 2.0 & 2.0 & - & - &0.0006 &
0.0006 & 0.4 \\
LPNS$^\mathrm{T06s2}_\mathrm{YL04}$ & 0.6 MeV & 2.0 & 0.0004 &
0.0004 &
0.0006 & 0.0006 &
0.4 \\
LPNS$^\mathrm{s2}_\mathrm{YL04(64-63)}$$^*$ & 2.0 & 2.0 & - & - &
0.0006 & 0.006 &
0.4 \\
LPNS$^\mathrm{s2}_\mathrm{YL04(64-22)}$& 2.0 & 2.0 & - & - &0.0006 &
0.02 & 0.4 \\
LPNS$^\mathrm{s2}_\mathrm{YL04(64-62)}$& 2.0 & 2.0 & - & - &0.0006 &
0.06 & 0.4 \\
LPNS$^\mathrm{s2}_\mathrm{YL03}$ & 2.0 & 2.0 & - & - &0.0006 &
0.0006 & 0.3 \\
LPNS$^\mathrm{s1}_\mathrm{YL04}$ & 1.0 & 1.0 & - & - & 0.0006 &
0.0006 & 0.4 \\
LPNS$^\mathrm{s1}_\mathrm{YL03}$ & 1.0 & 1.0 & - & - & 0.0006 &
0.0006 & 0.3 \\
LPNS$^\mathrm{s0}_\mathrm{YL04}$ & 0 & 0 & - & - &
0.0006 & 0.0006 & 0.4 \\
& & & & & & & \\
HNS$^\mathrm{s2}$ & 2.0 & 2.0 & - & - & - & - & - \\
HNS$^\mathrm{T06s2}$ & 0.6 MeV & 2.0 & 0.0004 &
0.0004 & - & - & - \\
HNS$^\mathrm{T0s2}$ & 0 MeV & 2.0 & $0.06$ &
$0.06$ & - & - & - \\
HNS$^\mathrm{s1}$ & 1.0 & 1.0 & - & - & - & - & - \\
HNS$^\mathrm{T03s1}$ & 0.3 MeV & 1.0 & 0.0004 &
0.0004 & - & - & - \\
HNS$^\mathrm{T0s1}$ & 0 MeV & 1.0 & $0.06$ &
$0.06$ & - & - & - \\
& & & & & & & \\
CNS & 0 MeV & 0 MeV & - & - & - & -
& - \\
& & & & & & & \\
\hline
\end{tabular} \\
* This notation means that the first number in parentheses
classifies the lower density
boundary of the neutrino sphere, i.e. $6 \times 10^{-4}$, the second
number gives the upper density boundary, i.e. $6 \times 10^{-3}$.
\end{table*}
This early type protoneutron star (EPNS) is characterized by a hot
shocked envelope with an entropy per baryon of $s \sim $ 4 - 5 for
densities $n<0.02$ fm$^{-3}$, an unshocked core with $s \sim 1$ for
densities $n>0.1$ fm$^{-3}$, and a transition region between these
densities \cite[]{BHF95}. The entropy per baryon in the very outer
layers of an EPNS is larger than $s \sim 10$. However, these layers
have only a small influence on the EPNS structure and are therefore
neglected. We investigate EPNS models with constant lepton number,
$Y_\mathrm{l} = 0.4$ for densities above $n = 6 \times 10^{-4}$
fm$^{-3}$ where the neutrinos are trapped \cite[]{BHF95}. Below this
density, the neutrinos can freely escape and the chemical potential of
the neutrinos vanishes, $\mu_{\nu_\mathrm{e}} = 0$
\cite[]{Coo88}. We refer to Table~\ref{EOSs} for the detailed
parameters of the EPNS models studied here.
\subsection{Protoneutron stars at $t \sim 0.5 - 1$ s after core bounce}
\label{sec22}
At this later stage, the entropy per baryon is approximately constant
throughout the star, $s \sim 2$, except in some outer regions
\cite[]{BL86, KJ95, Kei96}. The lepton number is approximately constant
since the neutrino diffusion time, $\tau_{\nu-\rm{diff}}\sim 10$~s, is
by an order of magnitude larger than the PNS's age. We model this
late type protoneutron star (LPNS) with a neutrino transparent
envelope with densities $n < n_\mathrm{env} = 6 \times 10^{-4}$
fm$^{-3}$ and a neutrino opaque core with densities $n >
n_\mathrm{core}$ and $Y_\mathrm{l} = 0.4$ (see Table~\ref{EOSs}). The
transition region between $n_\mathrm{env}$ and $n_\mathrm{core}$ is
called \emph{neutrino sphere} \cite[]{Jan93}. We choose four
different values for $n_\mathrm{core}$ to simulate the influence of
the shape of the neutrino sphere on the structure of LPNS's. We show
also a LPNS with $Y_\mathrm{l} = 0.3$ for densities $n > 6 \times
10^{-4}$ fm$^{-3}$, for the sake of comparison, since different
evolution calculations show different lepton numbers \cite[]{BL86,
KJ95, PRPLM98}.
These different choices are motivated by recent theoretical
calculations of the neutrino-nucleon cross-section that include
modifications due to the nucleon-nucleon interaction and spin-spin
correlations. The problem was treated, for instance, by \cite{Raf96},
\cite{RPL98}, \cite{BS98}, and \cite{PRPLM98} using static correlation
functions and by \cite{Raf96}, \cite{JKRS96}, \cite{RS97}, and
T. Strobel (work in preparation)
by using dynamical correlation functions. Because of the
high complexity of the problem, the behaviour of the cross section and
thus of the location and shape of the neutrino sphere is rather
uncertain.
Another reason for the different choices is the uncertainty due to
convection, that might considerably influence the cooling of PNS's
\cite[e.g.][]{BL88, KJM96, Mez98}.
For the sake of comparison, we investigate also a model with an
isothermal envelope with
temperature\footnote{We choose $T$ and not $T^*$ for simplicity,
because the values of $T$ and $T^*$ are comparable in the envelope;
for the definition of $T^*$, which includes metric corrections, see
\cite{GHZ97} and \cite{Gon97}.} $T = 0.6$ MeV
for densities below $n \sim 4 \times 10^{-4}$ fm$^{-3}$. This
temperature value is motivated by the fact that the temperature in the
central parts of the progenitor star approximately raises to this value
before the onset of the core collapse \cite[e.g.][]{ST83, Bet90}. It
is certainly an upper limit of the true temperature since it
corresponds to an increase of the entropy per baryon by three or four
orders of magnitude for densities $n \sim 6 \times 10^{-10}$
fm$^{-3}$ (outer most layer of the star). This high
entropy seems to be possible only in hot bubbles \cite[see][]{MWW93}.
For comparison we investigate also an unphysical, cold EOS,
LPNS$^\mathrm{s0}_\mathrm{YL04}$, with a trapped
lepton number of $Y_\mathrm{l} = 0.4$ for densities
$n > 6 \times 10^{-4}$ fm$^{-3}$.
\subsection{Deleptonized hot neutron star at $t \sim 10 - 30$ s
after core bounce} \label{sec23}
After $10 - 30$~s the neutrinos can freely escape and the HNS is
nearly deleptonized. This also means that the lepton family number is
not conserved anymore. The $\beta$-equilibrium is thus given by
$\mu_\mathrm{p} + \mu_\mathrm{e} = \mu_\mathrm{n}$, $\mu_\mathrm{e} =
\mu_{\mu}$, and $\mu_{\nu} = 0$ for all neutrino species. At this
stage muons have to be taken into account since the muon number
density is comparable to the electron number density above nuclear matter
density. The entropy per baryon is nearly constant, $s\sim 1-2$,
during the evolution from the LPNS to the HNS
\cite[e.g.][]{BL86, KJ95, SST95, PRPLM98}.
We again compare the models with isentropic envelopes with models with
isothermal envelopes, $T = 0.3$ MeV or $T = 0.6$ MeV for
densities below $n \sim 4 \times 10^{-4}$ fm$^{-3}$. The models for
the deleptonized HNS's are summarized in Table \ref{EOSs}.
\subsection{Cold neutron star some minutes after core bounce}
\label{sec24}
After some minutes the NS has a temperature of $T < 1$ MeV throughout
the star and the EOS for cold NS matter can be used to describe the
CNS, because the thermal effects are negligibly small
\cite[see][]{ST83}. We shall adopt the model derived by \cite{BPS71}
for densities below neutron drip density, $n<2.6\times
10^{-4}$~fm$^{-3}$, and \cite{NV73} for densities between neutron
drip density and the transition density, $2.6\times
10^{-4}<n<0.1$~fm$^{-3}$. Above this transition density, we use the
model for CNS matter in $\beta$-equilibrium without neutrinos derived
by \cite{Str97}.
\section{Equation of state for protoneutron stars}
\label{sec3}
The EOS of PNS matter is the basic input quantity whose knowledge over
a wide range of densities, ranging from the density of iron at the
star's surface up to about eight times the density of normal nuclear
matter reached in the cores of the most massive stars of a sequence, is
necessary to solve the structure equations. Due to the high lepton
number, the EOS of PNS's is different from the EOS's for cold and hot
NS's with low lepton numbers. The nuclear EOS used in this paper for
the description of a PNS is a Thomas-Fermi model of average nuclear
properties, with a momentum- and density-dependent, effective
nucleon-nucleon interaction developed by \cite{MS90, MS91}. The
parameters of the nuclear EOS were adjusted to reproduce a wide range
of properties of normal nuclear matter and nuclei \cite[]{MS95, MS96,
MS98, WCS97}. \cite{Str98} extended this
approach to the case of finite
temperature\footnote{The EOS's used in this paper are available at:
http:\hspace{0cm}//www.\hspace{0cm}physik.\hspace{0cm}uni-muenchen.\hspace{0cm}de\hspace{0cm}/sektion\hspace{0cm}/suessmann\hspace{0cm}/astro.},
where they use exact numerical solutions for the integration
over the Fermi-Dirac distribution functions.
Appendix A contains a brief description of this approach.
We extend the nuclear EOS to subnuclear
densities and to different compositions of PNS and HNS matter
(i.e. trapped neutrinos, constant entropy per baryon, $\ldots$). In
the subnuclear regime, the EOS was also obtained by means of the
homogeneous Thomas-Fermi model. The electron number is derived by
fitting the pressure to the subnuclear EOS's of \cite{BPS71} and
\cite{NV73} for densities below the density of the neutrino sphere in
the case of EPNS and LPNS models and below the nuclear density in the
case of HNS models. Our results for subnuclear densities are
comparable to the EOS's derived by
\cite{LS91} and used in the investigations of
\cite{GHZ97, GHZ98} and \cite{Gon97}.
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig1.ps}}}
\caption[]{Pressure versus baryon number density for densities
$n < 10^{-5}$ fm$^{-3}$ of hot dense matter.
The curve CNS corresponds to cold matter (see
Sect. \ref{sec24}). The
LPNS$^\mathrm{T06s2}_\mathrm{YL04}$ and
HNS$^\mathrm{T06s2}$ curve is the isothermal
envelope part of an isentropic core with entropy
per baryon $s = 2$ (see Sects. \ref{sec22} and
\ref{sec23} for explanation). Finally the curve
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ corresponds to
the envelope part of this EOS with $s=5$ and
no trapped lepton number in this density region.}
\label{pn1a}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig2.ps}}}
\caption[]{Pressure versus baryon number density in the density region
$10^{-5}$ fm$^{-3}< n < 10^{-1}$ fm$^{-3}$ for different EOS's
of hot dense matter. The abbreviations are
described in Table~\ref{EOSs}. }
\label{pn1b}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig3.ps}}}
\caption[]{Pressure versus baryon number density for densities
$n > 0.1$ fm$^{-3}$ for different EOS's of our model
of hot dense matter. The abbreviations are
described in Table~\ref{EOSs}.
The pressure of the HNS$^\mathrm{s2}$ EOS is nearly
identical to the LPNS$^\mathrm{s1}_\mathrm{YL04}$ case in
this density region and is not shown for that reason.}
\label{pn1c}
\end{figure}
Figures \ref{pn1a} - \ref{pn1c} show the pressure, $P(n)$, as function
of the baryon number density, $n$, for different physical scenarios.
The envelope region with densities, $n \leq 10^{-5}$~fm$^{-3}$ is
depicted in Fig. \ref{pn1a} for the isothermal part with $T =
0.6$ of the LPNS$^\mathrm{T06s2}_\mathrm{YL04}$ EOS and the
HNS$^\mathrm{T06s2}$ EOS, the isentropic
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ EOS, and the EOS for a CNS. The
neutrinos do not contribute to the pressure for densities $n < 6
\times 10^{-4}$~fm$^{-3}$, since they are not trapped in this
region. In the EOS with $T = 0.6$~MeV, the pressure is dominated
by the contribution of the photons, $p_\mathrm{Ph} = 2.817
\times 10^{-7}\,T^4$\,MeV$^{-3}$\,fm$^{-3}$ = $3.65
\times 10^{-8}$\,MeV\,fm$^{-3}$, for low densities.
Nevertheless, the
influence of this low density region on gross properties of LPNS's and
HNS's is only small (see Sect.\ \ref{sec4}). In contrast to the
isothermal EOS, the isentropic EOS is almost identical with the cold
EOS in this density region.
Figure \ref{pn1b} shows the general trend that the leptons dominate
the pressure in the density region around and above the neutrino
sphere, $10^{-4}$~fm$^{-3} < n < 10^{-1}$~fm$^{-3}$ for LPNS's. Both
thermal effects and the trapped lepton number contribute significantly
to the pressure and increase it by a factor of $\sim 3 - 4$ each at
$n=0.01$~fm$^{-3}$. The thermal effects are even larger for the high
entropy model labeled EPNS$^\mathrm{s5s1}_\mathrm{YL04}$. The impact
of the shape of the neutrino sphere can be inferred by comparison of
the curves labeled LPNS$^\mathrm{s2}_\mathrm{YL04}$ and
LPNS$^\mathrm{s2}_\mathrm{YL04(64-22)}$.
With increasing density the temperature dependence of the nuclear EOS
increases. At the highest densities possible in PNS's ($n \sim 1 -
1.2$~fm$^{-3}$) the pressure increase due to thermal effects and due
to high lepton numbers become comparable (see Fig.~\ref{pn1c}). This
feature is clearly expressed by the nearly identical pressure of the
HNS$^\mathrm{s2}$ and the LPNS$^\mathrm{s1}_\mathrm{YL04}$ EOS's.
The EOS of \cite{LS91} shows a smaller temperature dependence at high
densities \cite[see][]{GHZ97} in comparison with our EOS. The reason for
this is that the temperature dependence of the EOS of \cite{LS91} lies
entirely in the kinetic part of the enegy density, since they choose
$m^* = m$ in their approach\footnote{See Sect. 2.3 of \cite{LS91},
espacially Eqs. 2.8, 2.13, and 2.18. This will change if $m^*$ is chosen to be
smaller than $m$.}. At this point it should be mentioned, that the behaviour
of $m^*$ at high densities is highly uncertain. At $n\sim0.5$~fm$^{-3}$
we obtain, for instance, a pressure increase due to thermal effects of
$28\,\%$ (from the LPNS$^\mathrm{s0}_\mathrm{YL04}$ to the
LPNS$^\mathrm{s2}_\mathrm{YL04}$ case), whereas \cite{LS91} obtain an
increase of only $8\,\%$ \cite[see Fig. 1.b in][]{GHZ97}.
The impact on the structure of PNS's and
HNS's are discussed in Section \ref{sec4}, where we compare our
results with the results of \cite{GHZ97, GHZ98} and \cite{Gon97}
who used the EOS of \cite{LS91}.
The number density and the mean energy of the neutrinos $y = \nu_{e}$,
$\bar \nu_{e}$, $\nu_{\mu}$, $\bar \nu_{\mu}$, $\nu_{\tau}$, $\bar
\nu_{\tau}$ are given by\footnote{The recently detected mass
of neutrinos \cite[see][]{Fuk98} is much smaller than the mean energy
of the neutrinos. A finite neutrino mass therefore can be neglected.}:
\begin{equation}
\label{nany}
n_{y} = \frac{g}{2 \pi^2} \int_{0}^{\infty}
\frac{p^2}{ 1 + \exp (\frac{1}{T}(p - \mu_{y})) } \mathrm{d}p
\end{equation}
and
\begin{equation}
\label{uany}
u_{y} = \frac{g}{2 n_{y} \pi^2}
\int_{0}^{\infty}
\frac{p^3}{ 1 + \exp (\frac{1}{T}(p - \mu_{y})) } \mathrm{d}p,
\end{equation}
respectively. The trapped electron neutrinos and anti-neutrinos are in
chemical equilibrium, $\mu_{\bar
\nu_\mathrm{e}}=-\mu_{\nu_\mathrm{e}}$. The chemical potentials of all
other neutrino species vanish, $\mu_{x} = 0$ (with $x = \nu_{\mu}$,
$\bar \nu_{\mu}$, $\nu_{\tau}$, $\bar \nu_{\tau}$) due to the lepton
family number conservation (the muon number density is small compared to
the electron number density in PNS's and
is therefore neglected, for simplicity). The factor $g$ denotes the spin-degeneracy
factor and is related to the spin, $\vec s$, of the particles by $g =
2|\vec s| + 1$. Since only positive helicity neutrinos and negative
helicity anti-neutrinos exist\footnote{No negative helicity neutrinos
nor positive helicity anti-neutrinos were found in experiment until
now; CP violation of the weak interaction.}, the degeneracy factor is
equal 1 for the neutrinos. In the case of vanishing chemical
potential Eqs. (\ref{nany}) and (\ref{uany}) lead to a temperature
dependence of the number density:
\begin{equation}
\label{rhonyx}
n_{x} = 1.19 \times 10^{-8}\,T^3~\mathrm{MeV}^{-3}~\mathrm{fm}^{-3}
\end{equation}
and a linear temperature dependence of the mean neutrino energy:
\begin{equation}
\label{unyx}
u_{x} = 3.15\,T ,
\end{equation}
with $x = \nu_{\mu}$, $\bar \nu_{\mu}$, $\nu_{\tau}$, $\bar
\nu_{\tau}$. Due to the high chemical potential of the electron
neutrinos in the case of a high trapped lepton number
($\mu_{\nu_\mathrm{e}} \gg T$; see Fig. \ref{my} and Fig. \ref{T}),
the number density and the mean energy of the electron anti-neutrinos
can be approximated by:
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig4.ps}}}
\caption[]{Chemical potential versus baryon number density
for electrons and electron-neutrinos
of the LPNS$^\mathrm{s1}_\mathrm{YL04}$ and
LPNS$^\mathrm{s2}_\mathrm{YL04}$ cases.
The dotted lines correspond to the CNS,
the HNS$^\mathrm{s1}$ and the HNS$^\mathrm{s2}$ EOS's.}
\label{my}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig5.ps}}}
\caption[]{Temperature versus baryon number density
for different EOS's.}
\label{T}
\end{figure}
\begin{equation}
\label{3.1}
n_{\bar \nu_\mathrm{e}} = \frac{T^{3}}{\pi^{2}}
\exp({- \frac{\mu_{\nu_\mathrm{e}}}{T}})
\end{equation}
and
\begin{equation}
\label{3.2}
u_{\bar \nu_\mathrm{e}} = \frac{3T^{4}}{n_{\bar \nu_\mathrm{e}} \pi^{2}}
\exp({- \frac{\mu_{\nu_\mathrm{e}}}{T}}) .
\end{equation}
This leads to the simple, linear temperature dependence of the mean
energy of the electron anti-neutrinos:
\begin{equation}
\label{3.3}
u_{\bar \nu_\mathrm{e}} = 3\,T ,
\end{equation}
the energy of an ultra-relativistic Boltzmann gas.
The mean neutrino energies of all neutrino species are shown in
Fig. \ref{Eny} for the case LPNS$^\mathrm{s2}_\mathrm{YL04}$.
Figure \ref{my} shows the chemical potential of the electron neutrinos
and the electrons for different models of the EOS.
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig6.ps}}}
\caption[]{Mean neutrino energies versus baryon number density
for all neutrino types in the case
LPNS$^\mathrm{s2}_\mathrm{YL04}$ for densities larger than
$6 \times 10^{-4}$ fm$^{-3}$
($\nu_x = \nu_\mu, \bar\nu_\mu, \nu_\tau$ and $\bar\nu_\tau$).}
\label{Eny}
\end{figure}
In Fig. \ref{T} we show the density dependence of the temperature for
different EOS's. One can see the temperature drop in the
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ EOS at the interface between the
hot shocked envelope ($n<0.02$ fm$^{-3}$) and the unshocked core
($n>0.1$ fm$^{-3}$)\cite[see][]{BL86, BHF95}.
The temperature increases with increasing entropy per baryon
(see discussion in Sect.~\ref{ssec:evol.seq})
and decreases with increasing lepton number, since the neutron
fraction and the proton fraction become more equal
\cite[see][]{Pra97}. The maximum temperature in the most massive
PNS's and HNS's reaches values between 80 and 120 MeV (see
Fig.~\ref{T} and Table~\ref{max_bar_mass}). The temperature in PNS's
and HNS's with a typical baryonic mass of $1.5 M_{\sun}$ has values
between 20 and 40 MeV (see Table~\ref{neutrinosphere}).
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig7.ps}}}
\caption[]{Lepton fractions versus baryon number density
for different EOS's. The lepton fractions of the
unphysical LPNS$^\mathrm{s0}_\mathrm{YL04}$ EOS are
nearly identical
to that of the LPNS$^\mathrm{s1}_\mathrm{YL04}$ EOS
and is not shown for that reason.
The lower dotted line corresponds to
the CNS EOS, the middle dotted line to the
HNS$^\mathrm{s1}$ EOS and the upper dotted line to the
HNS$^\mathrm{s2}$ EOS.
The dotted lines show the sum of the electron and muon
fraction. The stars correspond to the positron fraction
of the HNS$^\mathrm{s2}$ case. The positron fraction of all
other EOS's lies below the resolution of this figure.}
\label{Y}
\end{figure}
The fractions of electrons and electron-neutrinos are shown in
Fig.~\ref{Y} for the LPNS$^\mathrm{s1}_\mathrm{YL04}$ and
LPNS$^\mathrm{s2}_\mathrm{YL04}$ cases with constant lepton number.
The lepton fractions of the unphysical LPNS$^\mathrm{s0}_\mathrm{YL04}$
EOS are nearly identical to that of the LPNS$^\mathrm{s1}_\mathrm{YL04}$
EOS and are not shown in Fig.~\ref{Y} for that reason.
The fractions are nearly constant ($29 - 32\%$ for electrons and $8 -
11\%$ for electron neutrinos) in a wide range of densities. The
electron fraction decreases for $n > 0.7$~fm$^{-3}$ since the symmetry
energy decreases for densities $n > 0.4$~fm$^{-3}$ \cite[]{Str97}.
Non-relativistic EOS's derived within variational approaches behave
similar in this respect, whereas the symmetry energy derived in
relativistic and non-relativistic Br\"uckner-Bethe calculations
monotonically increases with density \cite[see][]{Str97}. Higher
temperatures cause a small decrease (increase) of the electron
(electron neutrino) fraction \cite[see][]{TNH94}. Without trapped
neutrinos the sum of the electron and muon fraction increases, in
first approximation, quadratically with increasing temperature
\cite[see Fig. \ref{Y} and][]{KJ95}.
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig8.ps}}}
\caption[]{Effective mass of neutrons and protons versus baryon number
density. The solid line shows the effective mass of neutrons
and protons for symmetric nuclear matter (the difference
in the values for neutrons and protons are negligibly
small). The long dashed line shows the effective mass
of neutrons in pure neutron matter. The dot-dashed line shows
the effective mass of protons brought into pure neutron
matter.}
\label{effmass}
\end{figure}
The effective masses, $m^*_\tau / m_\tau$, of neutrons and protons for
symmetric and pure neutron matter are shown in Fig.~\ref{effmass}.
The effective mass for symmetric nuclear matter at saturation density is
found to be $m^*_\tau / m_\tau = 0.867$ for neutrons and protons in our
model. This is in good agreement with the generally accepted, experimental
value \cite[e.g.][]{BHHQ82}.
In the case of pure neutron matter, the effective neutron mass
increases up to $m^*_\mathrm{n} / m_\mathrm{n} = 0.935$, at nuclear matter
density, whereas the effective proton mass in pure neutron matter
decreases to $m^*_\mathrm{p} / m_\mathrm{p} = 0.808$.
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig9.ps}}}
\caption[]{Adiabatic index versus baryon number density
for different EOS's.}
\label{gamma}
\end{figure}
The adiabatic index $\gamma$:
\begin{equation}
\label{3.4}
\gamma = \left. \frac{\mathrm{d}\ln P}{\mathrm{d}\ln n} \right|_{s}
= \left. \frac{n}{P} \frac{\mathrm{d}P}{\mathrm{d}n} \right|_{s},
\end{equation}
is shown for different EOS's in Fig.~\ref{gamma}. The adiabatic index
decreases with increasing temperature and lepton numbers for densities
around and above nuclear density. In contrast, it decreases with
increasing lepton number for densities $n < 0.05$~fm$^{-3}$
\cite[see][]{Gon97}. The steep behaviour of the adiabatic index
$\gamma$ around nuclear matter density is the reason for the core
bounce of the collapsing iron core of the progenitor star
\cite[]{ST83}.
\begin{table*}
\caption[]{The speed of sound $v_\mathrm{s}$ in the density region
around and above nuclear matter density for different EOS's.
The abbreviations are described in Table~\ref{EOSs}.
The maximum value is reached in the CNS EOS:
$n^\mathrm{max}_\mathrm{c} (\mathrm{CNS}) = 1.246$ fm$^{-3}
\rightarrow v_\mathrm{s}=0.964$ (in units of $c$).}
\label{soundspeed}
\begin{tabular}{ l c c c c c c c c c c }
\hline
& & & & & & & & & & \\
& & & & & $n$ [fm$^{-3}$] & & & & & \\
& & & & & & & & & & \\
\cline{2-11}
& & & & & & & & & & \\
EOS & $0.02$ & $0.06$ & $0.1$ & $0.2$ & $0.3$ & $0.4$ &
$0.6$ & $0.8$ & $1.0$ & $1.2$ \\
& & & & & & & & & & \\
\hline
& & & & & & & & & & \\
LPNS$^\mathrm{s2}_\mathrm{YL04}$ & 0.1242 & 0.1627 & 0.2067 & 0.3270 & 0.4411 &
0.5434 & 0.7089 & 0.8276 & 0.9055 & 0.9594\\
LPNS$^\mathrm{s1}_\mathrm{YL04}$ & 0.1046 & 0.1320 & 0.1737 & 0.2968 & 0.4172 &
0.5240 & 0.7008 & 0.8257 & 0.9113 & 0.9617\\
HNS$^\mathrm{s2}$ & 0.1011 & 0.1473 & 0.1917 & 0.3128 & 0.4280 &
0.5305 & 0.6963 & 0.8176 & 0.9073 & 0.9750\\
HNS$^\mathrm{s1}$ & 0.0684 & 0.1095 & 0.1559 & 0.2735 & 0.3905 &
0.4962 & 0.6665 & 0.7921 & 0.8862 & 0.9606\\
CNS & 0.0507 & 0.0807 & 0.1255 & 0.2469 & 0.3699 &
0.4802 & 0.6569 & 0.7863 & 0.8812 & 0.9543\\
& & & & & & & & & & \\
\hline
\end{tabular}
\end{table*}
The speed of sound $v_\mathrm{s}$ in units of the speed of light $c$:
\begin{equation}
\label{3.5}
v_\mathrm{s} = \sqrt{\frac{\mathrm{d}P}{\mathrm{d}\varepsilon}}
= \sqrt{\gamma \frac{P}{\varepsilon + P}},
\end{equation}
is tabulated in Table~\ref{soundspeed} for different EOS's. The speed
of sound $v_\mathrm{s}$ increases with density up to nearly the speed
of light, in the most massive stars of a sequence. Nevertheless,
$v_\mathrm{s}$ is always smaller than the speed of light and all EOS's
used in this paper are causal. The speed of sound increases with
temperature and lepton number at fixed density in contrast to the
results of \cite{GHZ98}. This is probably caused by the
smaller temperature dependence of the EOS of
\cite{LS91}, which was discussed before in this section.
\section{Structure of rotating and non-rotating protoneutron stars} \label{sec4}
The structure of rotating PNS's and NS's is governed by the Einstein
equations in stationary, axisymmetric, and asymptotic flat
space-time. Under these special conditions to the space-time symmetry
the ten Einstein equations reduce to four non-trivial equations which
are elliptic in quasi-isotropic coordinates \cite[]{BGSM93}. The
four non trivial Einstein equations together with the energy-momentum
conservation are solved via a finite difference scheme
\cite[]{Schaab98}. We follow \cite{BGSM93} in compactifying
the outer space to a finite region by using the transformation $r
\rightarrow 1/r$. The boundary condition of approximate flatness can
then be exactly fulfilled. The neutron star model is uniquely
determined by fixing one of the parameters: central density,
gravitational mass, or baryon number, as well as one of the
parameters: angular velocity, angular momentum, or stability parameter
$\beta=E_\mathrm{kin}/|E_\mathrm{grav}|$. The models of maximum mass
and/or maximum rotation velocity can also be calculated.
\subsection{Protoneutron star and neutron star sequences} \label{ssec:pns.seq}
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig10.ps}}}
\caption[]{The gravitational mass versus central energy-density
of non-rotating NS's and PNS's.
The lower long dashed line corresponds to the
HNS$^\mathrm{s1}$ EOS and the upper long dashed
line to the HNS$^\mathrm{s2}$ EOS.
The lower solid line corresponds to the
LPNS$^\mathrm{s1}_\mathrm{YL04}$ EOS and the upper solid
line to the LPNS$^\mathrm{s2}_\mathrm{YL04}$ EOS.}
\label{mg_ec_nrot}
\end{figure}
Figure~\ref{mg_ec_nrot} shows the gravitational mass as function of the
central energy-density for the non-rotating PNS and NS models.
Only models whose gravitational masses increase with central
energy-density are stable against axisymmetric perturbations.
Whereas the maximum mass differs only by $\sim 5\,\%$ between the
various star models, the minimum mass of the EPNS models,
$M_{\rm min}\sim 0.9-1.2M_{\sun}$, is much larger than the minimum mass
of the CNS models, $M_{\rm min}\sim 0.1M_{\sun}$.
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig11.ps}}}
\caption[]{Kepler frequency versus gravitational mass of PNS's and NS's
for different EOS's. The upper long dashed line corresponds
to the HNS$^\mathrm{s1}$ EOS and the lower long dashed
line to the HNS$^\mathrm{s2}$ EOS.
The upper solid line corresponds to the
LPNS$^\mathrm{s1}_\mathrm{YL04}$ EOS and the lower solid
line to the LPNS$^\mathrm{s2}_\mathrm{YL04}$ EOS.}
\label{omegaK}
\end{figure}
Figure \ref{omegaK} shows the Kepler frequency, i.e. the frequency at
which mass shedding sets in, as function of the gravitational mass.
Also shown is the rotational frequency, $\Omega=4033$~s$^{-1}$, of the
fastest pulsar known, PSR 1937+214 \cite[]{Bac82}. The
gravitational mass of this pulsar is unfortunately unknown but is
typically assumed to be in the range 1.0--2.0\,$M_{\sun}$. Since the
Kepler frequency of the CNS models is larger than the rotational
frequency of PSR 1937+214 for this mass range, this model is
consistent with the observation.
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig12.ps}}}
\caption[]{The gravitational mass versus central energy-density
of NS's and PNS's rotating at their Kepler frequency.
The lower long dashed line corresponds to the
HNS$^\mathrm{s1}$ EOS and the upper long dashed
line to the HNS$^\mathrm{s2}$ EOS.
The lower solid line corresponds to the
LPNS$^\mathrm{s1}_\mathrm{YL04}$ EOS and the upper solid
line to the LPNS$^\mathrm{s2}_\mathrm{YL04}$ EOS.}
\label{mg_ec_rot}
\end{figure}
Figure~\ref{mg_ec_rot} shows the sequences of stars rotating with
Kepler frequency. As expected the masses are now larger at fixed
central energy density, but the maximum mass is reached for lower
central energy densities (see also Table~\ref{max_bar_mass}). The
mass increase due to rotation is smaller for PNS's than for CNS's,
since the radii of PNS's are larger and thus the Kepler frequencies are
smaller.
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig13.ps}}}
\caption[]{The gravitational mass versus stellar radius (as measured by
an observer located at infinity) of non-rotating NS's and PNS's.
The abbreviations for the different EOS's are
described in Table~\ref{EOSs}.
The lower long dashed line corresponds to the
HNS$^\mathrm{s1}$ EOS and the upper long dashed
line to the HNS$^\mathrm{s2}$ EOS.
The lower solid line corresponds to the
LPNS$^\mathrm{s1}_\mathrm{YL04}$ EOS and the upper solid
line to the LPNS$^\mathrm{s2}_\mathrm{YL04}$ EOS.}
\label{mg_r_nrot}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig14.ps}}}
\caption[]{The gravitational mass versus equatorial radius (as measured by
an observer located at infinity) of NS's and PNS's
rotating at their Kepler frequency.
The lower long dashed line corresponds to the
HNS$^\mathrm{s1}$ EOS and the upper long dashed
line to the HNS$^\mathrm{s2}$ EOS.
The lower solid line corresponds to the
LPNS$^\mathrm{s1}_\mathrm{YL04}$ EOS and the upper solid
line to the LPNS$^\mathrm{s2}_\mathrm{YL04}$ EOS.}
\label{mg_r_rot}
\end{figure}
We also show the mass-radius relation for PNS's in
Figs.~\ref{mg_r_nrot} and \ref{mg_r_rot}\footnote{The equatorial
radius $R_\mathrm{inf}$ is the circumferential radius as measured in
infinity \cite[see, e.g.,][]{Schaab98}}. In comparison with CNS's,
where the radius only slightly changes in the relevant mass region
around $M_\mathrm{G} \sim 1.4\,M_{\sun}$, one obtains for PNS's a
much stronger increase of the radius with decreasing mass, which is caused
by the stiffer EOS's for the PNS's.
\begin{figure}
\resizebox{\hsize}{!}{{\includegraphics{fig15.ps}}}
\caption[]{Iso-energy-density surfaces for a
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ model with a baryonic mass of
$1.5 M_{\sun}$ rotating at its Kepler frequency. The surfaces
correspond to energy-densities (beginning at the centre),
$\varepsilon=0.99,0.1,0.01,10^{-3},10^{-4},10^{-5},10^{-6}
\varepsilon_\mathrm{c}$,
where $\varepsilon_\mathrm{c}=8.07\times10^{14}$~g~cm$^{-3}$ denotes
the central energy-density.} \label{edenss5}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{{\includegraphics{fig16.ps}}}
\caption[]{Iso-energy-density surfaces for a
CNS model with a baryonic mass of
$1.5 M_{\sun}$ rotating at its Kepler frequency. The surfaces
correspond to energy-densities (beginning at the centre),
$\varepsilon=0.99,0.9,0.8,0.7,0.6,0.5,0.4,0.3,0.2,0.1
\varepsilon_\mathrm{c}$,
where $\varepsilon_\mathrm{c}=8.11\times10^{14}$~g~cm$^{-3}$ denotes
the central energy-density.} \label{edensT0}
\end{figure}
To demonstrate the differences of the internal structure of an EPNS
and a CNS, we show in Figs.~\ref{edenss5} and \ref{edensT0} the
iso-energy-density surfaces for models with $M_{\rm B}=1.5M_{\sun}$. It
turns out that EPNS's contain mainly matter of densities below nuclear
matter density (Fig.~\ref{edenss5}). Contrary to this, matter in a
CNS is dominated by matter of densities around and above nuclear
matter density (Fig.~\ref{edensT0}).
Another important point concerns the stability of the star models
against secular or dynamical instabilities. A certain configuration is
dynamically stable against axisymmetric perturbations if the
gravitational mass is minimum with respect to variations at fixed
baryon number and angular momentum. Along a star sequence with fixed
angular momentum, this is the case if the gravitational mass increases
with the central energy density \cite[see][p. 151]{ST83}. Please
note that the angular momentum is not fixed in the sequences of stars
rotating with Kepler frequency. The maximum mass configuration differs
therefore slightly from the marginally stable configuration. A fully
general relativistic analysis of dynamical and secular stability
against non-axisymmetric perturbations is extremely difficult and has
been performed only by means of approximations and/or special
assumptions in literature. It turned out however that the
configurations can be classified by a stability parameter defined by
the ratio of the kinetical energy and the absolute value of the
gravitational energy, $\beta = E_\mathrm{kin}/|E_\mathrm{grav}|$. For
values, $\beta\gtrsim 0.26$, the models are probably dynamically
unstable against the bar mode ($m=2$), whereas they become already
secularly unstable for $\beta\gtrsim 0.14$ \cite[see,
e.g.,][]{Dur75,Man85,IFD85,BFG96}. Higher modes,
$m\gtrsim 5$, of perturbations are suppressed by the finite viscosity
of neutron star matter. It is however uncertain whether modes with
$m=3$ or 4 are also suppressed. If this is not the case, secular
instability sets already in above a critical value, $\beta\gtrsim
0.08$ \cite[]{FIP86}.
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig17.ps}}}
\caption[]{Stability parameter $\beta = E_\mathrm{kin}/|E_\mathrm{grav}|$
versus gravitational mass
of PNS's and NS's rotating at their Kepler frequency
for different EOS's. The upper long dashed line corresponds
to the HNS$^\mathrm{s1}$ EOS and the lower long dashed
line to the HNS$^\mathrm{s2}$ EOS.
The upper solid line corresponds to the
LPNS$^\mathrm{s1}_\mathrm{YL04}$ EOS and the lower solid
line to the LPNS$^\mathrm{s2}_\mathrm{YL04}$ EOS.}
\label{e_rot_e_grav}
\end{figure}
Figure~\ref{e_rot_e_grav} shows the value of the stability parameter
for stars rotating with their Kepler frequency. Since the timescale
for the growing of the secular instabilities is larger than the
evolution timescale of PNS's and HNS's, only the dynamical instability
is important for PNS's and HNS's. The critical value for the dynamical
instability, $\beta\gtrsim 0.26$, is reached by none of the
models. Depending on the internal structure of the CNS's, the CNS's
models rotating with Kepler frequency may become secular unstable
against modes with $m=3$ or $m=4$ for star masses $M\gtrsim 0.8
M_{\sun}$.
\subsection{Evolution of a non-rotating protoneutron star} \label{ssec:evol.seq}
\begin{table*}
\caption[]{Properties of non-rotating and with Kepler frequency rotating
PNS's and NS's,
for a fixed baryonic mass $M_\mathrm{B} = 1.5\,M_{\sun}$.
The EOS's are
summarized in Table~\ref{EOSs}.
The entries are: gravitational mass,
$M_\mathrm{G}$;
baryonic mass, $M_\mathrm{B}$;
circumferential radius (as measured in infinity),
$R_\mathrm{inf}$;
central baryon number density, $n_\mathrm{c}$;
central temperature, $T_\mathrm{c}$;
Kepler frequency, $\Omega_\mathrm{K}$;
angular momentum, $J$
($M_{\sun}$~km $\hat =\,5.966\times10^{48}$ g~cm$^{2}$~s$^{-1}$).}
\label{neutrinosphere}
\begin{tabular}{ l c c c c @{~~~~~~~~~~} c c c c c }
\hline
& & & & & & & & & \\
& \multicolumn{4}{c}{$\Omega = 0$} &
\multicolumn{5}{c}{$\Omega = \Omega_\mathrm{K}$} \\
& & & & & & & & & \\
\cline{2-10}
& & & & & & & & & \\
EOS & $M_\mathrm{G}$ &
$R_\mathrm{inf}$ & $n_\mathrm{c}$ & $T_\mathrm{c}$ &
$M_\mathrm{G}$ &
$R_\mathrm{inf}$ & $n_\mathrm{c}$ & $\Omega_\mathrm{K}$
& $J$ \\
& [$M_{\sun}$] & [km] & [fm$^{-3}$] & [MeV] &
[$M_{\sun}$] & [km] & [fm$^{-3}$] & [s$^{-1}$] &
[$M_{\sun}$ km] \\
& & & & & & & & & \\
\hline
& & & & & & & & & \\
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ & 1.425 & 25.66 & 0.440 & 20.8 &
1.430 & 38.55 & 0.427 & 1879 & 0.634\\
EPNS$^\mathrm{s4s1}_\mathrm{YL04}$ & 1.425 & 18.66 & 0.456 & 21.3 &
1.433 & 27.46 & 0.436 & 3056 & 0.924\\
LPNS$^\mathrm{s2}_\mathrm{YL04}$ & 1.431 & 17.38 & 0.385 & 39.5 &
1.437 & 25.62 & 0.353 & 3410 & 1.234\\
LPNS$^\mathrm{T06s2}_\mathrm{YL04}$ & 1.432 & 17.47 & 0.386 & 39.6 &
1.437 & 25.77 & 0.355 & 3381 & 1.219\\
LPNS$^\mathrm{s2}_\mathrm{YL04(64-63)}$ & 1.430 & 16.72 & 0.386 & 39.6 &
1.438 & 24.68 & 0.348 & 3610 & 1.340\\
LPNS$^\mathrm{s2}_\mathrm{YL04(64-22)}$ & 1.430 & 16.34 & 0.386 & 39.6 &
1.438 & 24.13 & 0.344 & 3739 & 1.408\\
LPNS$^\mathrm{s2}_\mathrm{YL04(64-62)}$ & 1.430 & 15.85 & 0.388 & 39.7 &
1.439 & 23.36 & 0.341 & 3930 & 1.497\\
LPNS$^\mathrm{s2}_\mathrm{YL03}$ & 1.414 & 16.01 & 0.400 & 42.9 &
1.423 & 23.37 & 0.361 & 3897 & 1.362\\
LPNS$^\mathrm{s1}_\mathrm{YL04}$ & 1.411 & 14.70 & 0.459 & 21.4 &
1.423 & 21.26 & 0.417 & 4482 & 1.389\\
LPNS$^\mathrm{s1}_\mathrm{YL03}$ & 1.391 & 13.59 & 0.475 & 22.8 &
1.406 & 19.55 & 0.424 & 5076 & 1.552\\
LPNS$^\mathrm{s0}_\mathrm{YL04}$ & 1.410 & 12.39 & 0.519 & 0 &
1.429 & 17.73 & 0.450 & 5949 & 1.800\\
HNS$^\mathrm{s2}$ & 1.391 & 15.22 & 0.412 & 49.7 &
1.403 & 22.13 & 0.367 & 4204 & 1.431\\
HNS$^\mathrm{T06s2}$ & 1.392 & 15.29 & 0.412 & 49.7 &
1.404 & 22.24 & 0.369 & 4171 & 1.413\\
HNS$^\mathrm{T0s2}$ & 1.385 & 13.53 & 0.404 & 49.0 &
1.407 & 19.86 & 0.324 & 5075 & 2.072\\
HNS$^\mathrm{s1}$ & 1.362 & 12.78 & 0.501 & 26.8 &
1.382 & 18.36 & 0.437 & 5562 & 1.669\\
HNS$^\mathrm{T03s1}$ & 1.363 & 12.80 & 0.502 & 26.9 &
1.382 & 18.39 & 0.438 & 5550 & 1.662\\
HNS$^\mathrm{T0s1}$ & 1.362 & 12.18 & 0.504 & 27.0 &
1.387 & 17.59 & 0.418 & 6023 & 1.957\\
CNS & 1.346 & 11.45 & 0.551 & 0 &
1.374 & 16.36 & 0.459 & 6689 & 1.953\\
& & & & & & & & & \\
\hline
\end{tabular}
\end{table*}
The evolution of a PNS to a CNS can be followed by means of several
``snapshots'' taken at different times after core bounce (see
Sect.~\ref{sec2}). The evolution path is determined by fixing the
baryonic mass and the angular momentum if accretion of matter and loss of
angular momentum is neglected. First, we study the evolution of a
non-rotating PNS with $M_{\rm B}=1.5M_{\sun}$ (see Table
\ref{neutrinosphere} and Fig. \ref{mg_mb}).
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig18.ps}}}
\caption[]{Gravitational mass versus baryonic mass for the
non-rotating EPNS$^\mathrm{s4s1}_\mathrm{YL04}$ EOS and
the non-rotating CNS EOS.
The EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ EOS lies
exactly on the EPNS$^\mathrm{s4s1}_\mathrm{YL04}$
line and is not shown by that reason.
The arrows show the evolution (from left to right)
for the minimum mass
EPNS$^\mathrm{s4s1}_\mathrm{YL04}$, the minimum mass
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$, the
$M_\mathrm{B} = 1.5~M_{\sun}$, and the maximum mass
configurations.}
\label{mg_mb}
\end{figure}
The first snapshot corresponds to approximately 50 - 100 ms when the
envelope is characterized by a high entropy per baryon and high lepton
number (see curves labeled EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ and
EPNS$^\mathrm{s4s1}_\mathrm{YL04}$). After 0.5 - 1 seconds the PNS
reached our LPNS stage, which is characterized by an approximately
constant entropy per baryon, $s\sim 2$, throughout the star (model
LPNS$^\mathrm{s2}_\mathrm{YL04}$). Due to the higher entropy per
baryon in the core of the PNS, the central density and thus the
gravitational binding energy decreases. On the other hand the entropy
per baryon decreases in the envelope and therefore the radius
decreases, too. For a lower value of the entropy per baryon and/or a
lower value for the lepton number, the gravitational binding energy
increases compared to the EPNS stage (models
LPNS$^\mathrm{s2}_\mathrm{YL03}$, LPNS$^\mathrm{s1}_\mathrm{YL04}$,
LPNS$^\mathrm{s1}_\mathrm{YL03}$). After about 10 - 30 seconds the
neutrinos escape from the star and its EOS softens (HNS models). The
gravitational binding energy thus increases by roughly 3\,\%. Finally,
the CNS model is even more compressed.
Due to a smaller increase of the pressure with increasing temperature,
\cite{BL86} and \cite{PRPLM98} obtain a monotonous
increase of the central density during the evolution of the PNS to the CNS.
In contrast, our results, and also those of \cite{KJ95}, show a more
complex behaviour of the central density \cite[see discussion in the
paper of][]{PRPLM98}.
This difference has mainly two reasons:
Firstly, \cite{PRPLM98} use an entropy profile obtained in a supernova
collapse simulation
of a 1.08~$M_{\sun}$ NS, which leads to a higher central entropy per
baryon, $s \sim 1.5$, at the EPNS stage and an increase to $s \sim 2$
at the LPNS stage. We use the calculations of \cite{BHF95} who
simulated the supernova collapse of a NS with $M \sim 1.5~M_{\sun}$.
They obtained a smaller starting central entropy per baryon,
$s \sim 1$, and an increase to $s \sim 2$ at the LPNS stage.
The second reason is the fact, that they use an approximation for
the temperature influence on the pressure \cite[derived by][]{Pra97}
of the nucleons:
\begin{equation}
\label{4.1}
s = \frac{\pi^2}{2}\,T \sum_{\tau = n, p}
\frac{Y_\mathrm{\tau}}{T_\mathrm{F, \tau}} ,
\end{equation}
\begin{equation}
\label{4.1a}
\frac{P_\mathrm{th}}{P_0} = \frac{5 s}{3 \pi^2} \frac{\sum\nolimits_\tau \frac{Y_\tau}{T_\mathrm{F, \tau}} \left( 1 - \frac{3}{2} \frac{\mathrm{d~ln} m^*_\tau}{\mathrm{d~ln} n_\tau} \right)}{\left(\sum\nolimits_\tau \frac{Y_\tau}{T_{F, \tau}}\right)^2\left(\sum\nolimits_\tau Y_\tau T_{F, \tau}\right)}\left( 1 + \frac{P_\mathrm{pot}}{P_\mathrm{kin}}\right)^{-1} ,
\end{equation}
where $T_\mathrm{F, \tau}$ denotes the Fermi temperature of the quasi particles
($T_\mathrm{F, \tau} = p_\mathrm{F, \tau}^2/2m^*_\tau$), $P_\mathrm{th}$, the
thermal pressure and
$P_0 = P_\mathrm{kin} + P_\mathrm{pot}$, the pressure at zero temperature
as sum of kinetic and potential pressure.
This approximation holds under the assumption that in the dense parts
of the PNS the temperature is small in comparison with
the Fermi temperature, i.e. $T/T_\mathrm{F} \ll 1$. However in our case
these ratio reaches $T/T_\mathrm{F} \approx 0.45$
($T/T_\mathrm{F} \approx 0.6$) at $n = n_{0}$ and $T/T_\mathrm{F} \approx 0.29$
($T/T_\mathrm{F} \approx 0.44$) at $n = 1$ fm$^{-3}$ for neutrons (protons) in
the LPNS$^\mathrm{s2}_\mathrm{YL04}$ model (see also the Fermi-Dirac
distribution function in Fig.~\ref{fn} of Appendix A and
\cite{TNH94} for this purpose, in which it can be seen that more than
$10\,\%$ of the matter is non-degenerate).
Therefore this approximation underestimates the pressure increase due to
thermal effects, e.g. the thermal pressure $P_\mathrm{th}$ is exact
(approximative) 2.37~MeVfm$^{-3}$ (1.26~MeVfm$^{-3}$) at $n = n_0$ and
54.62~MeVfm$^{-3}$ (28.83~MeVfm$^{-3}$) at $n = 1$~fm$^{-3}$ for the
nucleons of the LPNS$^\mathrm{s2}_\mathrm{YL04}$ case of our model. Hence
in our opinion, higher order terms should be included in the treatment.
Since the temperature dependence of the GM3 model of \cite{PRPLM98} is
smaller than in our case, the deviations between the exact solution and
the approximation may be small.
\subsection{Maximum rotational frequency of a neutron star}
\begin{table*}
\caption[]{Evolution of 1.5 $M_{\sun}$ star at constant angular
momentum, $J=0.634$~$M_{\sun}$\,km and
$J=0.924$~$M_{\sun}$\,km for the sequences
starting with the model EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ and
EPNS$^\mathrm{s4s1}_\mathrm{YL04}$, respectively.}
\label{evolv15}
\begin{tabular}{ l c c c c c @{~~~~~~~~~~} c c c c c }
\hline
& & & & & & & & & & \\
& \multicolumn{5}{c}{EPNS$^\mathrm{s5s1}_\mathrm{YL04}$} &
\multicolumn{5}{c}{EPNS$^\mathrm{s4s1}_\mathrm{YL04}$} \\
& & & & & & & & & & \\
\cline{2-11}
& & & & & & & & & & \\
EOS & $M_\mathrm{G}$ & $R_\mathrm{inf}$ & $n_\mathrm{c}$ &
$T_\mathrm{c}$ & $\Omega$
& $M_\mathrm{G}$ & $R_\mathrm{inf}$ & $n_\mathrm{c}$ &
$T_\mathrm{c}$ & $\Omega$ \\
& [$M_{\sun}$] & [km] & [fm$^{-3}$] & [MeV] & [s$^{-1}$]
& [$M_{\sun}$] & [km] & [fm$^{-3}$] & [MeV] & [s$^{-1}$] \\
& & & & & & & & & & \\
\hline
& & & & & & & & & & \\
EPNS & 1.430 & 38.55 & 0.427 & 20.4 & 1879 &
1.433 & 27.46 & 0.436 & 20.7 & 3056\\
LPNS$^\mathrm{s2}_\mathrm{YL04}$ & 1.349 & 14.83 & 0.350 & 37.0 & 2091 &
1.341 & 15.40 & 0.336 & 36.0 & 2961 \\
LPNS$^\mathrm{s2}_\mathrm{YL03}$ & 1.363 & 14.27 & 0.374 & 40.9 & 2171 &
1.361 & 14.74 & 0.362 & 39.9 & 3073 \\
LPNS$^\mathrm{s1}_\mathrm{YL04}$ & 1.391 & 13.17 & 0.442 & 20.9 & 2377 &
1.391 & 13.54 & 0.430 & 20.5 & 3390 \\
LPNS$^\mathrm{s1}_\mathrm{YL03}$ & 1.385 & 12.76 & 0.462 & 22.4 & 2474 &
1.387 & 13.09 & 0.451 & 22.0 & 3525 \\
HNS$^\mathrm{s2}$ & 1.349 & 14.06 & 0.387 & 46.5 & 2246 &
1.348 & 14.53 & 0.375 & 46.3 & 3178 \\
HNS$^\mathrm{s1}$ & 1.362 & 12.49 & 0.489 & 26.3 & 2609 &
1.365 & 12.83 & 0.478 & 25.9 & 3709 \\
CNS & 1.348 & 11.83 & 0.535 & 0 & 2833 &
1.351 & 12.14 & 0.523 & 0 & 4031 \\
& & & & & & & & & & \\
\hline
\end{tabular}
\end{table*}
We follow now the evolution of an EPNS with $M_{\rm B}=1.5M_{\sun}$,
which rotates with Kepler frequency at its EPNS state. During its
evolution, the angular momentum is assumed to be conserved (see Table
\ref{evolv15}). As in the non-rotating case, the star becomes more and
more compressed during its evolution. With only a few exceptions, the
central baryon density increases, whereas the gravitational mass and
the circumferential radius decreases. It is obvious that this trend
has to be counterbalanced by an increasing angular velocity in order
to keep the angular momentum constant. Compared to the EPNS state, the
angular velocity in the CNS state is increased by 51\,\% or 32\,\% for
the sequences starting with the model
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ or
EPNS$^\mathrm{s4s1}_\mathrm{YL04}$, respectively. Nevertheless, the
angular velocity in the CNS state reaches at most 60\,\% of the Kepler
frequency.
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig19.ps}}}
\caption[]{Minimum rotational period $P$ versus baryonic mass of the
EPNS's and CNS's. The figure shows the spin up during the evolution
of the EPNS's to CNS's. The dot-dashed (short dashed) line shows
the EPNS$^\mathrm{s5s1}_\mathrm{YL04}$
(EPNS$^\mathrm{s4s1}_\mathrm{YL04}$) model rotating at its Kepler
frequency. The solid (long dashed) line shows the corresponding CNS
model obtained by cooling at constant angular momentum. The dotted
line shows CNS's rotating at their Kepler frequency for comparison.
The arrows represent the evolution from the EPNS stage to the CNS
stage.} \label{maxrot}
\end{figure}
In this respect, the minimum rotational period is determined by the
Kepler rotating EPNS model. Figure \ref{maxrot} shows the the general
evolution of the rotational period for the two EPNS models. For a CNS
with a typical baryonic mass of $1.5 M_{\sun}$ one obtains a minimum
rotational period, $P$, between $1.56$ and $2.22$~ms.
Recently, \cite{GHZ97, GHZ98} found similar results for the minimum
rotational period, which confirms our calculations.
The period of
the fastest known pulsar PSR 1937+214, $P=1.56$~ms \cite[]{Bac82} is
at the lower limit of this range. Pulsars that rotate even faster
cannot be born with such small periods but have to be accelerated
after their formation, as long as a typical baryonic mass, $M_{\rm
B}=1.5 M_{\sun}$ is assumed. \cite{And98} has recently shown
that not too cold NS's rotating with $\Omega\gtrsim 0.1\Omega_{\rm K}$
are unstable against r-modes. This means that
the minimum rotational period for a young NS is even higher,
$P_{\rm min}\sim 10$~ms.
Both the results by \cite{And98} and the results presented here
strengthen the view of millisecond pulsars as being recycled by
accretion \cite[]{Lor96}.
\begin{figure}
\resizebox{\hsize}{!}{\rotatebox{-90}{\includegraphics{fig20.ps}}}
\caption[]{Variation of the stability parameter
$\beta=E_\mathrm{kin}/|E_\mathrm{grav}|$ during the evolution from
EPNS's to CNS's at constant angular momentum. The arrows represent
the evolution from the EPNS stage to the CNS stage.} \label{beta_evolv}
\end{figure}
Figure~\ref{beta_evolv} shows the behaviour of $\beta$ during the
evolution from the EPNS to the CNS state. The stability parameter
increases during the evolution. Nevertheless, its maximum value,
$\beta\sim 0.082$, reached for the most massive star of the
EPNS$^\mathrm{s4s1}_\mathrm{YL04}$ EOS, is only a little larger than
the critical value $\beta\sim 0.08$ for the onset of the secular
instabilities with $m=3,4$ (see Sect.\ \ref{ssec:pns.seq}).
\subsection{Maximum and minimum mass of a neutron star}
\begin{table*}
\caption[]{Properties of the maximum gravitational mass configurations
of non-rotating and with Kepler frequency rotating PNS's
and NS's.}
\label{max_bar_mass}
\begin{tabular}{ l c c c c c @{~~~~~~~~~~} c c c c c c }
\hline
& & & & & & & & & & & \\
& \multicolumn{5}{c}{$\Omega = 0$} &
\multicolumn{6}{c}{$\Omega = \Omega_\mathrm{K}$} \\
& & & & & & & & & & & \\
\cline{2-12}
& & & & & & & & & & & \\
EOS & $M_\mathrm{G}$ & $M_\mathrm{B}$ &
$R_\mathrm{inf}$ & $n_\mathrm{c}$ & $ T_\mathrm{c}$ &
$M_\mathrm{G}$ & $M_\mathrm{B}$ &
$R_\mathrm{inf}$ & $n_\mathrm{c}$ &
$\Omega_\mathrm{K}$ & $J$\\
& [$M_{\sun}$] & [$M_{\sun}$] & [km] & [fm$^{-3}$] & [MeV] &
[$M_{\sun}$] & [$M_{\sun}$] & [km] & [fm$^{-3}$] & [s$^{-1}$]
& [$M_{\sun}$ km] \\
& & & & & & & & & & & \\
\hline
& & & & & & & & & & & \\
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ & 2.05 & 2.33 & 12.59 & 1.09 &
40.4 & 2.18 &
2.45 & 16.93 & 1.08 & 7609 & 3.06\\
EPNS$^\mathrm{s4s1}_\mathrm{YL04}$ & 2.05 & 2.33 & 11.83 & 1.10 &
40.7 & 2.22 &
2.50 & 15.44 & 1.09 & 8803 & 3.69\\
LPNS$^\mathrm{s2}_\mathrm{YL04}$ & 2.09 & 2.33 & 11.91 & 1.03 &
81.6 & 2.30 &
2.54 & 15.90 & 0.97 & 8602 & 4.29\\
LPNS$^\mathrm{s2}_\mathrm{YL03}$ & 2.08 & 2.37 & 11.29 & 1.09 &
92.0 & 2.33 &
2.62 & 15.43 & 0.95 & 9067 & 4.61\\
LPNS$^\mathrm{s1}_\mathrm{YL04}$ & 2.05 & 2.33 & 10.82 & 1.14 &
41.8 & 2.28 &
2.56 & 15.38 & 0.93 & 9384 & 4.49\\
LPNS$^\mathrm{s1}_\mathrm{YL03}$ & 2.04 & 2.38 & 10.70 & 1.18 &
45.6 & 2.31 &
2.65 & 14.43 & 0.93 & 9701 & 4.94\\
LPNS$^\mathrm{s0}_\mathrm{YL04}$ & 2.05 & 2.27 & 10.29 & 1.12 &
0 & 2.28 &
2.54 & 14.31 & 0.88 & 9956 & 4.98\\
HNS$^\mathrm{s2}$ & 2.08 & 2.41 & 11.24 & 1.12 &
120.6 & 2.34 &
2.68 & 15.47 & 0.92 & 9073 & 4.80\\
HNS$^\mathrm{s1}$ & 2.01 & 2.38 & 10.48 & 1.20 &
62.1 & 2.28 &
2.66 & 14.14 & 0.92 & 9864 & 5.03\\
CNS & 1.99 & 2.41 & 9.83 & 1.25 &
0 & 2.30 &
2.70 & 13.74 & 0.95 & 10631 & 5.64\\
& & & & & & & & & & & \\
\hline
\end{tabular}
\end{table*}
Table \ref{max_bar_mass} shows, that the maximum baryonic mass of a
non-rotating (rotating) EPNS is $2.33 M_{\sun}$ ($2.45 - 2.5
M_{\sun}$), whereas the maximum baryonic mass of a CNS is 2.41
$M_{\sun}$ (2.7 $M_{\sun}$) (see also Fig.\ \ref{mg_mb}). Since the
maximum baryonic mass of a LPNS and HNS is larger than the maximum
baryonic mass of a EPNS, too, it is obvious that a PNS based on our
nuclear EOS cannot collapse to a black hole during its
Kelvin-Helmholtz cooling phase if any further accretion is neglected
\cite[see][]{BJKST96}. The situation changes if hyperons, meson
condensation, or a quark-hadron phase transition are included
\cite[see, e.g.,][]{BB94, PPT95, PCL95, Gle95}.
Another important point concerning the maximum mass of a CNS is
due to the different maximum masses of EPNS's and CNS's rotating at
their Kepler frequencies. The EPNS can support a baryonic mass of
$2.45 - 2.5 M_{\sun}$, which is only slightly higher than the maximum
baryonic mass of the non-rotating CNS ($2.41 M_{\sun}$), but
$0.2 - 0.25 M_{\sun}$ less than the maximum baryonic mass of the CNS
rotating at its Kepler frequency. Such supramassive CNS's enter unstable
regions during their spin down evolution due to unstability against
axisymmetric perturbations and may finally collapse to a black hole.
For a discussion of these supramassive CNS's, see \cite{CST94}
and \cite{SBGH94}.
In first approximation, the gravitational mass of non-rotating PNS's
and HNS's increases quadratically with the entropy per baryon, which is
taken constant throughout the star \cite[see][]{Pra97}:
\begin{equation}
\label{massincreas}
M_\mathrm{G}(s) = M_\mathrm{G}(T=0) \left( 1 + \lambda s^2 + \dots \right).
\end{equation}
We obtain for $\lambda$ approximate values $\sim 0.010 - 0.011$
($\sim 0.005$)
for the cases without (with) trapped lepton number (see
Table~\ref{max_bar_mass}). These values are in agreement with the
values derived by \cite{Pra97}. In the investigation of \cite{GHZ97}
the gravitational mass shows the opposite behaviour: The gravitational
mass decreases slightly with increasing entropy per baryon.
This is probably caused by the smaller temperature dependence of the
EOS of \cite{LS91}, as it was pointed out in
Sect.~\ref{sec3}. In the case of rotating stars the behaviour seems
to be more complex since the value of $\lambda$ is negative for
HNS$^\mathrm{s1}$ and LPNS$^\mathrm{s1}_\mathrm{YL04}$ and positive
for HNS$^\mathrm{s2}$ and LPNS$^\mathrm{s2}_\mathrm{YL04}$. Another
interesting point is, that the maximum gravitational and baryonic mass
of the non-rotating PNS's are determined by the EOS used in the core
and do not depend on the EOS used in the envelope (compare the models
EPNS$^\mathrm{s4s1}_\mathrm{YL04}$,
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$, and
LPNS$^\mathrm{s1}_\mathrm{YL04}$ in Table \ref{max_bar_mass} and
Figs.~\ref{mg_ec_nrot} and \ref{mg_r_nrot}). In the case of rotating
PNS's, the maximum gravitational mass of the EPNS is however smaller
compared to the corresponding LPNS model (see Table \ref{max_bar_mass}
and Figs.~\ref{mg_ec_rot} and \ref{mg_r_rot}). This behaviour is caused
by the higher Kepler frequency which can be supported by the LPNS
models (see Sect.\ \ref{ssec:pns.seq}).
\begin{table*}
\caption[]{Evolution of minimum mass star for the non-rotating
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ EOS
($M_\mathrm{B} = 1.238 M_{\sun}$)
and the EPNS$^\mathrm{s4s1}_\mathrm{YL04}$ EOS
($M_\mathrm{B} = 0.950 M_{\sun}$).}
\label{evolvmin}
\begin{tabular}{ l c c c c @{~~~~~~~~~~} c c c c }
\hline
& & & & & & & \\
& \multicolumn{4}{c}{EPNS$^\mathrm{s5s1}_\mathrm{YL04}$} &
\multicolumn{4}{c}{EPNS$^\mathrm{s4s1}_\mathrm{YL04}$} \\
& & & & & & & \\
\cline{2-9}
& & & & & & & \\
EOS & $M_\mathrm{G}$ & $R_\mathrm{inf}$ & $n_\mathrm{c}$ &
$T_\mathrm{c}$
& $M_\mathrm{G}$ & $R_\mathrm{inf}$ & $n_\mathrm{c}$ &
$T_\mathrm{c}$ \\
& [$M_{\sun}$] & [km] & [fm$^{-3}$] & [MeV]
& [$M_{\sun}$] & [km] & [fm$^{-3}$] & [MeV] \\
& & & & & & & \\
\hline
& & & & & & & \\
EPNS & 1.207 & 45.61 & 0.340 & 17.3 &
0.942 & 44.49 & 0.281 & 15.1 \\
LPNS$^\mathrm{s2}_\mathrm{YL04}$ & 1.199 & 19.77 & 0.308 & 33.8 &
0.926 & 25.34 & 0.219 & 26.7 \\
LPNS$^\mathrm{s2}_\mathrm{YL03}$ & 1.186 & 17.49 & 0.327 & 37.1 &
0.922 & 20.38 & 0.250 & 30.7 \\
LPNS$^\mathrm{s1}_\mathrm{YL04}$ & 1.185 & 15.83 & 0.384 & 18.9 &
0.926 & 18.05 & 0.307 & 16.1 \\
LPNS$^\mathrm{s1}_\mathrm{YL03}$ & 1.168 & 14.24 & 0.404 & 20.3 &
0.915 & 15.31 & 0.331 & 17.6 \\
HNS$^\mathrm{s2}$ & 1.168 & 16.28 & 0.341 & 43.2 &
0.911 & 18.26 & 0.267 & 36.5 \\
HNS$^\mathrm{s1}$ & 1.146 & 13.18 & 0.429 & 23.7 &
0.898 & 13.75 & 0.354 & 20.5 \\
CNS & 1.134 & 11.53 & 0.481 & 0 &
0.890 & 11.58 & 0.410 & 0 \\
& & & & & & & \\
\hline
\end{tabular}
\end{table*}
The minimum baryonic mass of our non-rotating EPNS sequences is in the
range $0.95 - 1.24 M_{\sun}$. The minimum mass is increased in the
case of Kepler rotating stars to $0.97 - 1.29 M_{\sun})$. If accretion
is neglected the baryonic mass is conserved during the evolution of
the EPNS to the CNS. We follow the evolution of the minimum mass
EPNS$^\mathrm{s5s1}_\mathrm{YL04}$ and
EPNS$^\mathrm{s4s1}_\mathrm{YL04}$ model to a CNS via the LPNS and the
HNS stage (see Table \ref{evolvmin}). The minimum mass of a CNS born
in a supernova is therefore determined by the minimum mass model of
the EPNS sequence. The above ranges of baryonic mass correspond to a
lower limit of the gravitational CNS's mass in the range $0.89 - 1.13
M_{\sun}$. This is by a factor of ten larger than the minimum mass of
the CNS sequence.
This property of EPNS's was also recently found by \cite{GHZ98}.
Variations of the location of the transition region between the envelope
(high entropy per baryon) and the core (low entropy per baryon) of the
EPNS's do not change the minimum mass considerably.
Starting the transition region at lower densities will lead to smaller
minimum masses of the EPNS's, but the lower the initial mass of the
EPNS is, the higher is the entropy per baryon in the envelope and in
the core \cite[see e.g.][]{KJM96, PRPLM98}. This effect drives the
minimum mass back to higher values, so that our results of the
minimum mass range is a good approximation \cite[see also][]{GHZ98}.
\cite{TWW96} examine the most likely masses of NS's using the
numerical data of \cite{WW95} who simulated Type-II supernovae with
progenitor stars in the mass range between 11 and 40\,$M_{\sun}$.
They obtain a lower limit of the NS mass, which depends on the mass
and the composition of the progenitor, of $1.15-1.27$~$M_{\sun}$. This
lower limit is comparable to our results.
\subsection{Sensitivity of the results}
In Table \ref{neutrinosphere}, we compare the properties of our PNS
and NS models for a fixed baryonic mass, $M_{\rm B}=1.5\,M_{\sun}$. This
canonical value corresponds to the measured gravitational masses,
$1.35 \pm 0.27\,M_{\sun}$, of neutron stars in binary systems
\cite[see][]{TAMT93,VVZ95}.
As can be inferred by comparing the LPNS models
LPNS$^\mathrm{s2}_\mathrm{YL04(64-62)}$,
LPNS$^\mathrm{s2}_\mathrm{YL04(64-22)}$, and
LPNS$^\mathrm{s2}_\mathrm{YL04(64-63)}$ with the model
LPNS$^\mathrm{s2}_\mathrm{YL04}$ (see Sect.~\ref{sec22}), the
location of the neutrino sphere has nearly no effect on the
gravitational mass and the central density. However, the
circumferential radius and the Kepler frequency vary by up to $10\,\%$
and $15\,\%$, respectively.
The use of an isothermal, instead of an isentropic, EOS in the
envelope of the HNS models HNS$^\mathrm{T03s1}$ and
HNS$^\mathrm{T06s2}$ has only small effects on the properties of the
HNS's (see Table \ref{neutrinosphere}). If thermal effects in the
envelope are however neglected (models HNS$^\mathrm{T0s1}$ and
HNS$^\mathrm{T0s2}$), the circumferential radius is reduced by $\sim
10\,\%$. This yields to an increase of the Kepler frequency by $\sim
20\,\%$. Though the assumption of zero temperature in the envelope
does not change the resulting mass and central densities, the error
made in the circumferential radius and in the Kepler frequency might
be rather large.
\section{Discussion and conclusion} \label{sec5}
The aim of this paper was the investigation of the properties of
rapidly rotating PNS's consisting of nuclear matter (n, p, e$^{-}$,
$\mu^{-}$) under the influence of trapped neutrinos. We used a
recently developed nuclear EOS for hot, dense matter in the nuclear
Thomas-Fermi approach \cite[]{Str98}. The nuclear EOS was
extended in this paper to subnuclear densities and to different
compositions of PNS and HNS matter (i.e. trapped neutrinos, constant
entropy per baryon, $\ldots$). Our results for subnuclear densities
are comparable to the results derived by \cite{LS91} used in the
investigations of \cite{GHZ97, GHZ98} and \cite{Gon97}.
However, considerable deviations occur at densities around and
above nuclear matter density,
since the EOS of \cite{LS91} shows a smaller temperature dependence.
In general, this difference to our results has only a small
impact on the properties of PNS's, but impacts on HNS's with canonical
mass. The impact becomes considerable for more massive stars.
An investigation of this kind should investigate the properties of
PNS's and HNS's for the whole range of possible masses. So far lepton
concentration profiles and entropy per baryon profiles
were derived only for one fixed mass in simulations of PNS evolution
\cite[e.g.][]{BL86, KJ95, BHF95}. In our calculation, the extension
to the whole range of masses was done by assuming that the lepton
concentration and the entropy per baryon does not depend on the mass
of the PNS.
As a result we find that the minimum gravitational mass of a NS is
determined at the earliest stage of a PNS, so that the mass of a NS
formed in a Type-II supernova is larger than 0.89 - 1.13\,$M_{\sun}$,
which confirms similar results of a recent investigation by
\cite{GHZ98}.
The exact lower limit of the NS mass depends on the used entropy per
baryon in the EPNS model. The quoted mass range was obtained by using
$s = 4$ and $s = 5$ as lower and upper limit of the entropy
per baryon in the envelope of the EPNS \cite[]{BHF95}. The minimum
mass of EPNS's is approximately by a factor of ten larger than the
minimum stable mass of CNS's.
The maximum possible baryonic mass of a CNS is always larger than the
maximum possible baryonic mass of the PNS's, this means that once a
PNS was formed, it cannot collapse into a black hole, if further
accretion is neglected \cite[see also][]{Tak95, Bom96}.
One exception is the case of the most massive stars
rotating near or at its Kepler frequency, which possibly
collapse to a black hole during their time evolution,
for a discussion, see \cite{CST94} and \cite{SBGH94}. This
statement holds for stars with a pure nucleonic/leptonic composition. If
one includes hyperons and/or quarks, the maximum baryonic mass of a
CNS decreases \cite[e.g.][]{HWWS98, BLC99} and may be smaller than the
maximum baryonic mass of a PNS. Then, a sufficiently massive PNS may
collapse to a black hole during deloptonization \cite[]{Pra97}. The
maximum gravitational mass increases slightly with increasing entropy
per baryon in contrast to the maximum baryonic mass \cite[see
also][]{Pra97}. As it was pointed out, \cite{GHZ97} got a different
result due to the use of the EOS derived by \cite{LS91}.
It turned out that the influence of rotation has several impacts on
properties of PNS's and NS's. Whereas the minimum mass changes only
slightly due to rotation, the effect on the maximum mass is rather
large, particularly for CNS's. The central baryon density is nearly
unaffected by rotation at the early stages of the evolution, whereas
the impact on the the later stages is rather large. The effect on the
circumferential radius decreases with increasing mass and the impact
on EPNS's and LPNS's is slightly larger than the impact on CNS's.
We investigated the influence of different shapes of the neutrino
sphere on the structure of LPNS's. As expected, we obtained considerable
differences (up to 10\,\%) in the circumferential radii and, due to
this, in the Kepler frequency. Other properties, as the central
density, are barely changed by the location of the neutrino sphere.
We have also considered different temperatures in the envelope of
LPNS's and HNS's. If a very high temperature of 0.6 MeV is used in
the envelope instead of a constant entropy per baryon, the gross
properties of the LPNS change by less than 1\,\%. Larger deviations
were obtained if the envelope is assumed to be cold. The finite
temperature effects should therefore not be neglected in the
envelope. Furthermore, we obtain that the thermal effects are
comparable to the effects due to trapped neutrinos. This is in
contrast to the results of \cite{Pra97}, who found the thermal effects
to be much smaller than the effect of high lepton numbers.
In our model we assumed that accretion onto the emerging NS stopped
after the formation of the EPNS \cite[see][]{BHF95}. Furthermore, we
kept the angular momentum constant during the deleptonization and the
thermal cooling period. Under these presumptions, we obtain a lower
limit on the periods of young NS's with a typical baryonic mass of 1.5
$M_{\sun}$ between 1.56 and 2.22\,ms, which is in accordance with
similar values obtained by \cite{GHZ97, GHZ98}. These results
support strongly the hypothesis that millisecond pulsars were
accelerated due to accretion. With the
same reasoning, one obtains an upper limit of the stability parameter
$\beta\lesssim 0.082$, which is smaller, for almost all NS's, than the
critical value for the onset of dynamical and secular instabilities.
Kepler rotating massive CNS's possess stability parameter values up
to $\beta \sim 0.13$ and might therefore be secular unstable against
$m=2$ and $m=3$ non-axisymmetric perturbations.
Though we have assumed uniform rotation, it seems to be very
reasonable that PNS's and young NS's rotate differentially
\cite[e.g.][]{JM89}. As it was found by \cite{Schaab98} and \cite{GHZ98},
differential rotation may considerably effect the structure of PNS's
and NS's, and we will pursue our investigations to differential
rotation in the future. Another issue that will be addressed in a
future work is the effect of additional degrees of freedom
(e.g. hyperons) on the evolution of PNS's.
\begin{acknowledgements}
We want to thank Wolfgang Keil, Georg Raffelt, Thomas Strobel and
Fridolin Weber for many helpful discussions. Two of us, K.~S. and
Ch.~S., gratefully acknowledge the Bavarian State for financial
support.
\end{acknowledgements}
|
1,108,101,566,781 | arxiv | \section{Introduction}
Traditionally, the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi
(DGLAP) evolution equations \cite{DGLAP1,DGLAP2,DGLAP3,DGLAP4}
approach is used to obtain the quark, anti-quark and gluon
densities, i.e., the parton distribution functions (PDF),
$a(x,\mu^2)$. These functions depend on the Bjorken variable x
and the hard scale ${\mu ^2}$, and can be easily used in the
collinear QCD factorization formalisms. In these DGLAP evolution approaches, the
transverse momentum ($k_t$) components of partons are integrated
over and there is not any degree of freedom for the initial gluon
radiations and the transverse momentum ($k_t$) of the partons in the
PDF. But, the outcome of hadron-hadron colliders at high energies
indicates that the explicit inclusion of the intrinsic transverse
momentum of initial hadron constituents is important to get
accurate results and predictions. Therefore, the important inputs are the
transverse momentum dependent parton distribution functions or the
so-called "unintegrated" PDF (UPDF).
Theoretically, various methods are proposed to generate these
fundamental quantities, i.e., UPDF, and among them the BFKL
\cite{BFKL1,BFKL2,BFKL3,BFKL4,BFKL5} (which is valid for the small x
and the scale $k_t^2$) and CCFM
\cite{CCFM1,CCFM2,CCFM3,CCFM4,CCFM5} (which is applicable at both
the small and large x, the scales $k_t^2$ and $\mu^2$) evolution
equations are considered extensively. Nevertheless, the CCFM
approach is both mathematically and numerically more complicated and
time consuming.
Recently, Kimber et al \cite{KMR} and Martin et al \cite{MRW}
proposed the KMR and MRW formalisms, respectively, in the leading
order
(LO) and next-to-leading order (NLO) levels. These formalisms were extensively used to extract
the UPDF, $f(x,k^2_t,\mu^2)$, from the ordinary PDF based on the ${k_{t}}$-factorization
approach of pQCD as well as probing the partonic structures of hadrons
\cite{10,11,12,13,14}. The main difference between these two
approaches is turned back into the various types of imposing the
angular ordering constraints (AOC). These formalisms were analyzed
by us to calculate the proton structure functions and the different
hadron-hadron differential cross sections in the references
\cite{15,16,17,18,19}.
The analysis of Drell-Yan lepton pair production (DY) in the
hadron-hadron collisions
at high energies is the subject of intense studies \cite{20,21,22,23,24,25,26,27,phi3}, since it provides an ideal
ground for testing the QCD predictions \cite{book1,book2,Klasen,28}. Many experimental groups like the CDF,
CMS, ATLAS, D0 and LHCb collaborations \cite{20,21,22,23,24,25,26,phi3} in the available energies of the Tevatron and LHC colliders, try to
compare the experimental measurements of these DY events to the corresponding theoretical predictions from
the pQCD and the parton level Monte Carlo programs, such as
ResBos \cite{ResBos1}, DYNNLO \cite{DYNNLO1} and POWHEG+PYTHIA \cite{pythia1} event generators.
The ResBos method simulates the vector-boson
production and its decay,
using a resumed treatment of the soft-gluon emissions at the NLO-logarithm
(NNLL) and the $\gamma ^*$
and Z/$\gamma ^*$ contributions are simulated at the NLO accuracies.
The DYNNLO approach is a parton level Monte Carlo program that computes the
cross sections for the vector boson production in the P-P(\={P}) collisions up to the NNLO in the pQCD theory.
The PYTHIA program generates the LO QCD interactions via its parton shower algorithms.
In the recent investigation \cite{26} the distribution of dilepton
transverse momentum and angular variable $\phi _\eta ^*$
were calculated perturbatively at $\sqrt s = 8$ TeV using the ResBos Monte Carlo
generator at the NNLO accuracy
and compared to the ATLAS data. Although, the results at low values of
$p_T$ and $\phi _\eta ^*$ show a good agreement
with data, but this is not the case at high values of $p_T$ and $\phi _\eta ^*$.
In the present report, it is intended to calculate the DY differential cross sections
based on the KMR and MRW ${k_{t}}$-factorization approaches
using the corresponding off-shell transition amplitudes. We consider the three sub-processes
namely, (1)
$q^* + \bar q^* \to {{{\gamma ^*}} \mathord{\left/
{\vphantom {{{\gamma ^*}} {Z \to {l^ + } + {l^ - }}}} \right.
\kern-\nulldelimiterspace} {Z \to {l^ + } + {l^ - }}}$ and (2) $q^* + {g^*} \to {{{\gamma ^*}} \mathord{\left/
{\vphantom {{{\gamma ^*}} {Z \to {l^ + } + {l^ - } + q}}} \right.
\kern-\nulldelimiterspace} {Z \to {l^ + } + {l^ - } + q}}$ and (3) $ q^* + \bar q^* \to {{{\gamma ^*}} \mathord{\left/
{\vphantom {{{\gamma ^*}} {Z + g \to {l^ + } + {l^ - }}}} \right.
\kern-\nulldelimiterspace} {Z + g \to {l^ + } + {l^ - }}} + g$
at the LO and NLO levels, respectively.
The dependence of the DY differential cross sections on the dilepton transverse momentum, the invariant mass
and the rapidity distributions as well as the angular correlation between the produced
leptons are calculated in the above frameworks and compared with the experimental data developed by the CDF,
CMS, ATLAS and LHCb collaborations in both the Tevatron and LHC
energies. The Harland-Lang et al. (MMHT2014) PDF libraries \cite{mmht2014} in the LO
for both the KMR \cite{KMR} and MRW \cite{MRW} formalisms and the NLO for NLO-MRW approaches are considered. These calculations
are performed for the off-shell incoming partons.
We should also
point out here that, recently, the consideration of various
angular ordering, as well as the generation of different UPDF becomes
the subject of several reports \cite{re1,re2}. In the reference \cite{re1}, it is pointed out that the KMR UPDF, which are generated using the differential and integral approaches, give different results in the region where $k_t > \mu$. Therefore it is concluded that, the integral form of the KMR UPDF gives the correct result, while for the application of differential form, one should use the cutoff dependent PDF. On the other hand, in the reference \cite{Guiot:2019vsm}, the above idea is rejected, and a new term is added to the Sudakov form factor via a Heaviside step function, to set the Sudakov form factor equal to $1$ in the region where $k_t > \mu$. Further, it is claim that the above two forms of KMR approach give the same result, i.e., there is no need to introduce cutoff dependent PDF. Finally, in the reference \cite{Guiot:2019vsm}, by referring to this report, i.e., \cite{Guiot:2018kfy} ( in which the predictions of the KMR approach with the AOC overestimates the data in case of the heavy quark production), it is suggested that the above problem is due to the freedom of parton to have transverse momentum larger than $\mu$, and concluded that it is much suitable to use the KMR UPDF with the strong ordering constraint (SOC), which harshly cuts the transverse momentum in $k_t > \mu$ region. Because of the above statements, and all of the problems appear in $k_t > \mu$, we compute the DY differential cross sections with respect to $M^{ll}$ and $p_T^{ll}$ using the SOC KMR UPDF to check the sensitivity of our results in this region (see the figures 1 (panel f) and 2 (panel d)). One should also note that, as it is discussed
in the reference \cite{Watt:2003mx}, the result of $k_t$-factorization should not be as good as collinear factorization approaches, in covering the experimental data, on the other hand, it is more simplistic, considering computer time consuming.
It should be also noted that the Sudakov form factor of the KMR approach does not obey the multiplication law according to the reference \cite{Hautmann:2019biw}, but despite of this fact, it is interesting to point out that the normalization condition (see the equation (\ref{NOR})) is approximately satisfied in the KMR formalism, which is a critical issue in constructing any new UPDF.
The $k_t$-factorization calculations were also performed by
considering one or two of the above three sub-processes with the
MSTW2008 PDF
\cite{lipatov2011,reggeiz1,reggeiz2}. In these works, although the authors declare that they use the KMR
formalism, but they do not take into account the factor $1/{k_t^2}$ in the cross section nor in the normalization
formulas:
\begin{equation}
xa(x,\mu^2) \simeq \int^{\mu^2} {{dk_t^2}\over{k_t^2}}
f(x,k^2_t,\mu^2).\label{NOR}
\end{equation}
Beside these, they use different angular ordering conditions with
respect to the KMR prescriptions (we refer to them as semi-KMR).
However, their results are surprisingly close to the experimental
data. In the above references \cite{reggeiz1,reggeiz2}, it is
claimed that the second and the third of the above
sub-processes can be omitted by effectively using only the
reggeized (off-shell) quark approach in the first
sub-process. A brief discussion about the result of these reports
and the comparison with our predictions are presented in the section III. In the reference \cite{reggeiz1},
although the off-shell initial quarks are used, it is shown that utilizing
the reggeized model and the effective vertexes guaranteed the gauge invariance of the transition matrix elements (TME). However,
in our previous work \cite{diphoton}, we showed that using the off-shell initial quarks in the
$k_t$-factorization dynamics and in the small x regions leads to the gauge invariance of the TME, too.
To check the validity of our calculated cross sections the KaTie parton-level event generator \cite{KaTie} is used, in which the off-shell partonic cross
sections are taken care of, and gives the hadronic cross section with desirable accuracy. However,
we are not intended to solely show the result of cross section as it can be simply done with the KaTie parton-level event generator \cite{KaTie} (see the figures 1 (panel f) and 2 (panel h)). Indeed, our intention from one side is to check the effects of different impositions of the cutoff $\Delta$, which is additionally imposed on the quark radiation term in the KMR approach. On the other side, we also want to check the other forms of the DGLAP based UPDF, i.e. NLO-MRW, in which Martin et. al. used the virtuality $k^2 = k_t^2/(1-z)$. The UPDF of this form are rarely investigated in the phenomenological applications of the $k_t\textrm{-factorization}$. We include the sub-process $q^*+\bar{q}^*\rightarrow \gamma^*/Z\rightarrow l^++l^-+g$
which usually is neglected, e.g. \cite{lipatov2011}. In the others works, including those that are cited in our report \cite{lipatov2011}, incorrectly, the combinations of KMR and MRW formalisms are used and they forget about the importance of the normalization constraint (the equation (1)) on the UPDF. This point is discussed in details in the reference \cite{diphoton}. One should note that the KMR prescription is a semi-NLO. Another important item is the fragmentation effect, which does not present in the processes that are discussed in our report. Because of that, as it is explained in the paper, the KMR and NLO-MRW procedures demonstrate better agreement to the experimental data.
The outline of our paper is as following. In
the section II, we briefly present the basic cross section formulas
of ${k_{t}}$-factorization (\ref{CS}) approach and the derivation of
input UPDF (\ref{U}). In the section III we present numerical
calculations (\ref{1}), results presentations (\ref{2}) and
discussions (\ref{2}). The section IV expresses our conclusions.
\section{The Theoretical framework of DY}
\subsection{The $k_t$-factorization cross section formulas}
\label{CS}
Our DY differential cross section calculations are based
on the $k_t$-factorization in the KMR and MRW UPDF \cite{KMR,MRW}
approaches. Therefore in this section, we describe the theoretical framework
of these approaches as well as the corresponding matrix elements
(also see the appendix \ref{a}). As we pointed out in the
introduction, we include all the sub-processes contributions up to
the $\alpha\alpha_s$ levels, namely: $q^* + \bar q^* \to {{{\gamma ^*}}
\mathord{\left/
{\vphantom {{{\gamma ^*}} {Z \to {l^ + } + {l^ - }}}} \right.
\kern-\nulldelimiterspace} {Z \to {l^ + } + {l^ - }}}$,
$q^* + {g^*} \to {{{\gamma ^*}} \mathord{\left/
{\vphantom {{{\gamma ^*}} {Z \to {l^ + } + {l^ - } + q}}} \right.
\kern-\nulldelimiterspace} {Z \to {l^ + } + {l^ - } + q}}$ and
$q^* + \bar q^* \to {{{\gamma ^*}} \mathord{\left/
{\vphantom {{{\gamma ^*}} {Z + g \to {l^ + } + {l^ - }}}} \right.
\kern-\nulldelimiterspace} {Z + g \to {l^ + } + {l^ - }}} + g$.
From the kinematical point of view, if we show the four-momenta of
the incoming protons (partons) by ${P^{(1)}}$ (${k^{(1)}}$) and
${P^{(2)}}$ (${k^{(2)}}$) and neglect their masses, then in the
proton center of mass framework we have:
\begin{equation}
{P^{(1)}} = \frac{{\sqrt s }}{2}(1,0,0,1), \ \ \ {P^{(2)}} =
\frac{{\sqrt s }}{2}(1,0,0, - 1),
\end{equation}
where ${\sqrt s }$ is the total center of mass energy. In the high
energy and the leading-log-approximation kinematics, the
corresponding partons four-momenta can be written in terms of their
transverse momenta ${k_{1t}}$ and ${k_{2t}}$ and the fraction
($x_i$) of the incoming protons momentum as:
\begin{equation}
{k_1} = {x_1}{P^{(1)}} + {k_{1t}}, \ \ \ {k_2} = {x_2}{P^{(2)}} +
{k_{2t}}.
\end{equation}
There are some relations for the above three sub-processes due to
the energy-momentum conservation law as following:
\begin{equation}
{k_{1t}} + {k_{2t}} = {p_{1t}} + {p_{2t}},
\end{equation}
\begin{equation}
{x_1} = \frac{1}{{\sqrt s }}({m_{1t}}{e^{{y_1}}} +
{m_{2t}}{e^{{y_2}}}),
\end{equation}
\begin{equation}
{x_2} = \frac{1}{{\sqrt s }}({m_{1t}}{e^{ - {y_1}}} + {m_{2t}}{e^{ -
{y_2}}}).
\end{equation}
for the first sub-process and
\begin{equation}
{k_{1t}} + {k_{2t}} = {p_{1t}} + {p_{2t}} + {p_{3t}},
\end{equation}
\begin{equation}
{x_1} = \frac{1}{{\sqrt s }}({m_{1t}}{e^{{y_1}}} + {m_{2t}}{e^{{y_2}}} + {m_{3t}}{e^{{y_3}}}),
\end{equation}
\begin{equation}
{x_2} = \frac{1}{{\sqrt s }}({m_{1t}}{e^{ - {y_1}}} + {m_{2t}}{e^{ -
{y_2}}} + {m_{3t}}{e^{ - {y_3}}}) \label{9}.
\end{equation}
for the second and third sub-processes, where ${p_{it}}$,
${y_i}$ and ${m_{it}}$ are the transverse momenta,
the rapidities and the transverse masses ($m_{it}^2
= m_i^2 + p_{it}^2$) of the produced particles, ($i$=1 and 2 for
leptons and $i$=3 for (anti-)quark or gluon), respectively.
To calculate the matrix elements squared in the $k_t$-factorization
framework \cite{43}, the summation over the incoming off-shell gluon
polarizations is carried out as:
\begin{equation}
\sum {{\varepsilon ^\mu }{\varepsilon ^\nu } = k_{2t}^\mu }
k_{2t}^\nu /k_{2t}^2,
\end{equation}
where ${k_{t}}$ is the gluon transverse momentum. For the
off-shell quarks spinors with momentum $k$ (after imposing the Sudakov decomposition in the high
energy and the leading-log-approximation kinematics \cite{n1}), we have $\sum {u(k)\bar u(k)} \simeq {x\hat P \over k_t^2}$,
where $x$ represents the fractional longitudinal momentum of
proton, see the references \cite{diphoton,n2,n3}. Also, the effective vertices are used to calculate the Feynman amplitudes to test and ensure the gauge invariance of the different matrix elements \cite{LLIP1,LLIP2p,LLIP2}. It is worth to point out that a similar technique is also developed, using the Slavnov-Taylor identities by the means of the helicity amplitude \cite{Kutak0,Kutak1,Kutak2} and being checked against those obtained by usage of Lipatov’s effective action \cite{LLIP1,LLIP2p,LLIP2}, as we pointed out.
To calculate the differential cross sections of DY, according to
the ${k_{t}}$-factorization theorem, for $2 \to 2$ sub-process we
have:
\[\sigma_1 = \sum\limits_q {\int {\frac{1}{{16\pi {{({x_1}{x_2}s)}^2}}}} } {\left| {\cal M}_1^{\gamma^*}+ {\cal M}_1^{Z} \right|^2} \times\]
\begin{equation}
{f_q}({x_1},k_{1t}^2,{\mu ^2}){f_{\bar{q}}}({x_2},k_{2t}^2,{\mu ^2})\frac{{d{k _{1t}^2}}}{{k_{1t}^2}}\frac{{d{k _{2t}^2}}}{{k_{2t}^2}}dp_{1t}^2dp_{2t}^2
d{y_1}d{y_2}\frac{{d{\phi _1}}}{{2\pi }}\frac{{d{\phi _2}}}{{2\pi
}},
\end{equation}
and for $2 \to 3$ sub-process one finds:
\[\sigma_{2(3)} = \sum\limits_q {\int {\frac{1}{{256{\pi ^3}{{({x_1}{x_2}s)}^2}}}} } {\left| {\cal M}_{2(3)}^{\gamma^*}+ {\cal M}_{2(3)}^{Z} \right|^2} \times \]
\begin{equation}
{f_q}({x_1},k_{1t}^2,{\mu ^2}){f_{g(\bar{q})}}({x_2},k_{2t}^2,{\mu
^2})\frac{{d{k _{1t}^2}}}{{k_{1t}^2}}\frac{{d{k
_{2t}^2}}}{{k_{2t}^2}}dp_{1t}^2dp_{2t}^2d{y_1}d{y_2}d{y_3}\frac{{d{\phi
_1}}}{{2\pi }}\frac{{d{\phi _2}}}{{2\pi }}\frac{{d{\psi _1}}}{{2\pi
}}\frac{{d{\psi _2}}}{{2\pi }},
\end{equation}
where ${{f_q}({x_i},k_{it}^2,{\mu ^2})}$ are the UPDF, which depend
on the two hard scales, ${k_{t}^2}$ and ${\mu ^2}$, and they can be
written in terms of the usual PDF. As we pointed out in the
introduction, in the present calculations, the MMHT2014 PDF
\cite{mmht2014} is used for calculating the UPDF. In the above
formula, ${\cal M}_i^j$ are the off-shell matrix elements which are
presented for the three different sub-processes in the appendix A.
Note that when we squared the matrix element of each three
sub-processes we get the interference effect between $\gamma^*$ and
$Z$ production, which will be discussed in the section III. The
azimuthal angles of the initial partons and the produced leptons are
presented by ${\phi _1}$ and ${\phi _2}$, and ${\psi _1}$ and ${\psi _2}$, respectively.
Then the total cross section can be written as:
\begin{equation}
\sigma_{Total}=\sigma_1+\sigma_2+\sigma_3.
\end{equation}
To calculate the
UPDF of (anti-)quarks and gluons in a proton, we apply the LO KMR,
LO MRW and NLO-MRW approaches \cite{KMR,MRW}. In the following each
of them will be described.
\subsection{The KMR and MRW UPDF}
\label{U}
In the KMR method the UPDF of each parton, which means the
probability to find a parton with transverse momentum $k_t$ and
fractional momentum $x$ at hard scale $\mu^2$ are given by:
\begin{equation}
f_a(x,k_t^2,\mu^2) = T_a(k_t^2,\mu^2)\sum_{b=q,g} \left[
{\alpha_S(k_t^2) \over 2\pi} \int^{1-\Delta}_{x} dz P_{ab}^{(0)}(z)
b\left( {x \over z}, k_t^2 \right) \right] , \label{eq56}
\end{equation}
where ${T_a(k_t^2,\mu^2)}$ is
\begin{equation}
T_a(k_t^2,\mu^2) = exp \left( - \int_{k_t^2}^{\mu^2} {\alpha_S(k^2)
\over 2\pi} {dk^{2} \over k^2} \sum_{b=q,g} \int^{1-\Delta}_{0} dz'
P_{ab}^{(0)}(z') \right). \label{eq5}
\end{equation}
which is the familiar Sudakov survival form factor
and limits the emissions of partons between
${k_t^{2}}$ and $\mu^2 $ scales \cite{KMR,MRW}. ${P_{ab}^{(0)}}(z)$
are the usual LO splitting functions. In this formula the
angular-ordering constraint (AOC)
\cite{CCFM1,CCFM2,CCFM3,CCFM4,CCFM5,44,45}, $\Delta$, is applied in
the upper limit of the integration, which is an infrared cutoff to
prevent the soft gluon singularities arise from the splitting
functions and defined as:
\begin{equation}
\Delta= {k_t \over \mu + k_t}.
\end{equation}
Note that this constraint is imposed on both quark and gluon
radiations. ${b\left( {x \over z}, k_t^2 \right)}$ are the LO PDF,
and in this work they are taken from the MMHT2014 libraries
\cite{mmht2014}.
To determine the UPDF we also apply the MRW prescription which is
similar to the KMR formalism, but the AOC only acts on the terms
which include the on shell gluon emissions. For the quarks and the
gluons they take the following forms:
$$
f_q^{LO}(x,k_t^2,\mu^2)= T_q(k_t^2,\mu^2) {\alpha_S(k_t^2) \over
2\pi} \int_x^1 dz \left[ P_{qq}^{(0)}(z) {x \over z} q \left( {x
\over z} , k_t^2 \right) \Theta \left( {\mu \over \mu + k_t}-z
\right) \right.
$$
\begin{equation}
\left. + P_{qg}^{(0)}(z) {x \over z} g \left( {x \over z} , k_t^2
\right) \right],
\end{equation}
with
\begin{equation}
T_q(k_t^2,\mu^2) = exp \left( - \int_{k_t^2}^{\mu^2} {\alpha_S(k^2)
\over 2\pi} {dk^{2} \over k^2} \int^{z_{max}}_{0} dz'
P_{qq}^{(0)}(z') \right), \end{equation}
and
$$
f_g^{LO}(x,k_t^2,\mu^2)= T_g(k_t^2,\mu^2) {\alpha_S(k_t^2) \over
2\pi} \int_x^1 dz \left[ P_{gq}^{(0)}(z) \sum_q {x \over z} q \left(
{x \over z} , k_t^2 \right)
\right.$$
\begin{equation}
\left. + P_{gg}^{(0)}(z) {x \over z} g \left( {x \over z} , k_t^2
\right) \Theta \left( {\mu \over \mu + k_t}-z \right)
\right],
\end{equation}
with
\begin{equation}
T_g(k_t^2,\mu^2) = exp \left( - \int_{k_t^2}^{\mu^2} {\alpha_S(k^2)
\over 2\pi} {dk^{2} \over k^2}
\left[ \int^{z_{max}}_{z_{min}} dz' z' P_{gg}^{(LO)}(z')
+ n_f \int^1_0 dz' P_{qg}^{(0)}(z') \right] \right),
\end{equation}
respectively. In the above equations, ${{z_{max}=1-{z_{min}}}= {\mu
\over \mu + k_t}}$ \cite{watt}.
By expanding MRW to the NLO level, we have:
$$
f_a^{NLO}(x,k_t^2,\mu^2)= \int_x^1 dz T_a \left( k^2={k_t^2 \over
(1-z)}, \mu^2 \right) {\alpha_S(k^2) \over 2\pi}
\sum_{b=q,g} \tilde{P}_{ab}^{(0+1)}(z)
$$
\begin{equation}
\times b^{NLO} \left( {x \over z} , k^2 \right) \Theta \left(
1-z-{k_t^2 \over \mu^2} \right).
\label{eq11}
\end{equation}
In this formalism the Sudakov form factor is defined as:
\begin{equation}
T_q(k^2,\mu^2) = exp \left( - \int_{k^2}^{\mu^2} {\alpha_S(q^2)
\over 2\pi} {dq^{2} \over q^2} \int^1_0 dz' z' \left[
\tilde{P}_{qq}^{(0+1)}(z') + \tilde{P}_{gq}^{(0+1)}(z') \right]
\right),
\end{equation}
\begin{equation}
T_g(k^2,\mu^2) = exp \left( - \int_{k^2}^{\mu^2} {\alpha_S(q^2)
\over 2\pi} {dq^{2} \over q^2} \int^1_0 dz' z' \left[
\tilde{P}_{gg}^{(0+1)}(z') + 2n_f\tilde{P}_{qg}^{(0+1)}(z') \right]
\right) .
\end{equation}
The higher order splitting functions
are presented in the appendix B.
\section{Numerical results and discussions}
\subsection{Numerical calculations}
\label{1} In this section, we present the kinematics and theoretical
inputs of our calculations. First, we calculate the UPDF based on the
different ${k_{t}}$-factorization schemes by using two different
methods, i.e., KMR and MRW. Through our calculations, we set the
renormalization and factorization scales to be equal to $\mu _R =
\mu _F = \zeta M$, in which,
$M=\sqrt{2p_{1t}p_{2t}[\cosh(y_1-y_2)-\cos(\phi_1-\phi_2)]}$ is the
invariant mass of produced dilepton and as usual we consider the
default value $\zeta = 1$ \cite{diphoton}. We let this parameter to
vary from 1/2 to 2, to estimate the scale uncertainties of our
calculations. We also set ${m_Z} = 91.187$ $GeV$ and ${\Lambda
_{QCD}} = 200 MeV$ with ${n_f} = 4$ active quark flavors. Using the
LO coupling constant, we get ${\alpha _s}(M_Z^2) = 0.123$ ($g_W= 0.66$). Second,
with the massless quarks approximation, the calculation of
transition matrix elements squared is carried out, using the small x
approximation presented in the appendix \ref{a}, by the means of
FeynCalc \cite{feyncalc}, i.e., the mathematica package for symbolic
semi-automatic evaluation of Feynman diagrams. In the present
report, the non-logarithmic loop corrections to the $q$-$\bar q$
annihilation cross section are taken into account by applying the
effective K-factor with a particular scale choice of ${\mu ^2} =
p_T^{4/3}{M^{2/3}}$ as it was done, for example, in the references
\cite{19,lipatov2011,field}, i.e.,
\[K = \exp [{C_F}\frac{{{\alpha _s}({\mu ^2})}}{{2\pi }}{\pi ^2}]\]
where ${p_T}$, ($p_T=\mid \vec{p_{1t}}+\vec{p_{2t}} \mid$, see
above the equation (\ref{9})), is the transverse momentum of
produced dilepton and ${C_F}$ is the color factor. To calculate the
multidimensional integration, the VEGAS routine \cite{vegas} is
used. The differential cross sections at several center of mass
energies, i.e., 1.96, 7 and 8 TeV as a function of the dilepton
invariant mass ($M$), rapidity ($y$), transverse momentum ($p_T$)
and the variable $\phi _\eta ^*$ \cite{phi1,phi2,phi3,phi4}, i.e.,
\[\phi _\eta ^* = \tan (\frac{{{\phi _{acop}}}}{2}){[cos(\frac{{\Delta \eta }}{2})]^{ - 1}},\]
are calculated, with ${\phi _{acop}} = \pi - \left|
{\Delta \phi } \right|$, where ${\Delta \eta }$ and ${\Delta \phi }$ are the
pseudorapidity and azimuthal angles differences between the produced
leptons, as well, respectively. The variable $\phi _\eta ^*$ is correlated
to the quantity ${{\left| {{p_T}} \right|} \mathord{\left/
{\vphantom {{\left| {{p_T}} \right|} M}} \right.
\kern-\nulldelimiterspace} M}$ and both of them probes the same physics as the
dilepton transverse momentum, but it gives a better experimental
resolution \cite{56,57,58}.
\subsection{Results presentations}
\label{2} The results of above numerical calculations are compared
with the experimental data of DY at the Tevatron and LHC
laboratories with the total center of mass energy $\sqrt s = 1.8$
TeV and $\sqrt s =7$ and $8$ TeV, respectively. We use the data
from different groups such as the CDF, CMS, ATLAS and LHCb
collaborations. The available pQCD predictions are also presented in
each figure.
The above comparisons are demonstrated in the figures 1 to 10 as
follows:
\\
(1): In all of the figures, the numerical results related to the KMR
UPDF are shown in the left panels in which the dash, dotted-dash
and dotted histograms correspond to the contribution of individual
sub-processes, i.e., $q^* + \bar q^* \to {{{\gamma ^*}} \mathord{\left/
{\vphantom {{{\gamma ^*}} {Z \to {l^ + } + {l^ - }}}} \right.
\kern-\nulldelimiterspace} {Z \to {l^ + } + {l^ - }}}$, $q^* + {g^*} \to {{{\gamma ^*}} \mathord{\left/
{\vphantom {{{\gamma ^*}} {Z \to {l^ + } + {l^ - } + q}}} \right.
\kern-\nulldelimiterspace} {Z \to {l^ + } + {l^ - } + q}}$ and $q^* + \bar q^* \to {{{\gamma ^*}} \mathord{\left/
{\vphantom {{{\gamma ^*}} {Z + g \to {l^ + } + {l^ - }}}} \right.
\kern-\nulldelimiterspace} {Z + g \to {l^ + } + {l^ - }}} + g$,
respectively. The shaded bands indicate the
corresponding uncertainty (${1\over 2}\leq \zeta\leq 2$) due to the
hard scale variation with KMR UPDF in cross sections evaluation.
Unlike the present report, most of the previous phenomenological
works with the semi-KMR UPDF for the DY differential cross sections
did not present the contribution of each sub-process in their final
results and also, did not take into account the contribution of the
third sub-process, assuming the possible double counting
\cite{lipatov2011,reggeiz1} between the first and the third
sub-processes. However, according to our previous reports
\cite{18,diphoton}, we do not believe that there is any
double-counting among the first and the third sub-processes. This
point will be discussed in section \ref{3}.
\\
(2): In the right panels of each figure, the results of different UPDF schemes applications, namely KMR, LO-MRW and NLO-MRW, in the
differential cross sections are shown by the solid, dash and
dotted-dash histograms, respectively, for the possible comparisons.
\\
(3): The figures 1, 2-4, 5-8 and 9-10, demonstrate the DY
differential cross sections versus the diplepton invariant mass
($M$), the transverse momentum ($p_T$), the variable $\phi _\eta ^*$
and the rapidity ($y$), respectively (see the caption of each figure
for more details).
\subsection{Discussions}
\label{3} First, we generally start by analyzing the calculated
cross sections related to the medium and high center of mass
energies for $\sqrt s = 1.8$ and $7$ TeV. Although the results
show that the KMR UPDF describe reasonably the wide range of data of
Tevatron and LHC, but for the two sets of differential cross section
data which are in terms of $p_T$ and $\phi _\eta ^*$ parameters,
this is not the case. Indeed, in these two cases, the input NLO-MRW
UPDF describe the data better than other schemes. To be sure about
this conclusion, we try to include the newer data from the ATLAS
collaboration at $\sqrt s = 8$ $TeV$. The results of this double
check are presented in the different panels of figures 2-8. The
final comparisons, as we will be discussed below, indicates that
among the three different schemes, i.e., KMR, LO-MRW and NLO-MRW
UPDF, on average the NLO-MRW one is more suitable for describing
the experimental data and it confirms other groups report of NNLO
pQCD calculations \cite{60}. We should point out here that in our
previous works, e.g., the reference \cite{diphoton}, because of the
possible fragmentation effects, the KMR and LO-MRW had a better
agreement to the data.
The results of double and single differential cross sections
${{d\sigma}\over {dMdy}}$ and ${{d\sigma}\over {dM}}$ versus the
invariant mass of the dilepton are compared to the experimental
data at $\sqrt s = 1.8$ and $7$ TeV are shown in the figure 1, the panels
(a) and (b) and (c) to (f), respectively. In the panels (a) to (d) of
this figure, as it is expected, the Z boson mass peak is observed
around the $M=91$ GeV. In these panels, it is clear that in the
small M region $(M<M_z)$ which corresponds to the medium and large
$p_T$, the contribution of LO $q$-$\bar q$ sub-process to the cross
section is less than the other ones. However, by increasing the
invariant mass of dilepton, its effect become larger than the other
sub-processes. In the panels (b), (d) and (f), the comparison
between
all three approaches, i.e., KMR, LO-MRW and NLO-MRW, are presented and a similar behavior spatially at the
Z boson mass region is observed. In the panel (d), it is clear that the results of three schemes are more
or less the same, but in the panel (f) the KMR one shows more agreement with the experimental
data. Our results in the panels (c) to (f) are also close to
those of PYTHIA \cite{pythia1} and SHERPA \cite{d/dmATLAS}, especially in the small $M$ regions,
but the reggeized \cite{reggeiz1,reggeiz2} model is below our
predictions. According to these panels,
although our results show a overestimate and underestimate in the low and high dilepton invariant
mass region, the uncertainty bands of our calculations cover the experimental data. In addition, the results of the SOC and the KaTie parton-level event generator are presented in panel (f) of this figure as well. It is clear that there is not any significant difference between these results.
In the figures 2-4, the normalized differential cross-sections of DY
as a function of $p_T$, at $\sqrt s = 7$ and $8$
TeV are compared to the CMS and ATLAS collaborations data. As it is
expected, in all of the three figures, the first subprocess has
the main contribution while as we go to the higher center of mass
energy, i.e., $8$ TeV, the second subprocess also becomes important,
especially with the increase of dilepton mass (see the panels
(e) and (g) of the figure 2). In this figure, the results of applying the SOC and the KaTie parton-level event generator are compared in the panels (d) and (h), respectively. It is observed that
in most of the regions, there isnot significant difference between the two schemes. Also in the figure 3 and 4 in which the
rapidity is increased, only the second subprocess is sensitive to
the rapidity only in small $p_T$. However in the large $p_T$ region
($p_T>10$) the contribution of second subprocess, i.e., $q$-$g$,
becomes enhanced and in the middle of $p_T$ region only the first
and the second sub-processes are in the same order. Now by
considering the above three UPDF schemes, one can find that for
$p_T<40 $, they behave very similar, while they are separated from
each other such that the LO-MRW and NLO-MRW
are the upper and lower band of KMR, respectively. The experimental data also pass through the AOC band, see
the reference \cite{diphoton}. A comparison between our results and
the parton level Monte Carlo programs
such as PYTHIA and SHERPA are also presented. According to these panels,
although our results show a overestimate and underestimate in the low and high dilepton invariant
mass region, the uncertainty band of our calculations covers the experimental data.
In the figures 5-8, the normalized differential cross-sections of DY
at LHC as a function of the variable $\phi _\eta ^*$ and different
experimental conditions on the dilepton rapidity and invariant mass
are presented. According to these figures (beside the figure 5), it is
clear that in the small $\phi _\eta ^*$ region, which corresponds to
the back-to-back
leptons, the contribution of the first and the second sub-processes are dominated and approximately in the same
order. But in the figure 5, which is demonstrated for the small mass
interval, all of the three sub-processes are in the same order for the small $\phi _\eta ^*$
region. On the other hand for $\phi _\eta ^*> 0.01$ the three
sub-processes are separated and as it is expected contribution
of the first sub-process becomes enhanced and ${{d\sigma_1}\over {\sigma d\phi _\eta ^*}}>
{{d\sigma_2}\over {\sigma d\phi _\eta ^*}}\gg{{d\sigma_3}\over {\sigma d\phi _\eta
^*}}$.
In the right panels of these figures, the same conclusion as above
can be made about the effect of different UPDF schemes in which up
to $\phi _\eta ^*< 0.1$ they behave the same, and for larger $\phi _\eta
$ as we discussed before, the NLO-MRW UPDF cross section
calculations predict closer results to the corresponding data. The
AOC and uncertainty bands approximately cover the ATLAS
collaboration data.
The differential cross sections of DY with respect to the rapidity of
dilepton versus y are plotted in the various panels of the figures 9
and 10. It is observed that on average the second sub-process ($qg$)
is dominant, especially in the mid-y region, compared to the
other two sub-processes. On the other hand by comparing the right
panels of these figures, one can conclude that again the NLO-MRW and
KMR schemes give closer results to experimental data with respect to
LO-MRW procedure.
In addition, our results are slightly different from the reference
\cite{reggeiz1} as we use different PDF, UPDF and method. Indeed, we
use the original method introduced by Kimber et al and consider the
correct form of the normalization equation (\ref{NOR}).
It is notable that the redefined form of normalization equation as
$xa(x,\mu^2) \simeq \int^{\mu^2} {{dk_t^2}} f(x,k^2_t,\mu^2)$ without
the factor $1/{k_t^2}$ does not lead to the collinear form of cross section
after integrating over $k_t^2$. In the reference \cite{reggeiz1} the unpolarized
DY in the pp collisions is investigated at the LHC energies by CCFM
and semi-KMR within the reggeized quark formalism
\cite{reggeiz1,reggeiz2} to be sure about the gauge invariance of
matrix elements. However, as we discussed in our previous work
\cite{diphoton}, the gauge invariance is guaranteed because of
applying the small-x-approximation in our calculations. As we
pointed out before, in the figures 1 (panel f) and 2 (panel b), our results are
compared with those of references \cite{reggeiz1,reggeiz2}. On the
other hand, our results are compared with those of PYTHIA
\cite{CDFM} (figure 1 (panels c-f)), SHERPA \cite{d/dmATLAS} (figures
1 ( panels e-f) and 3 ( panels a,c,e)),
FEWZ \cite{LHCb7000} (figure 7 (panels g-h)) and RESBOS \cite{ResBos2} (figures 4 (panel e) and 7 (panel e)). It is observed that
in the regions in which the higher order calculations are not
important our results are similar to those of pQCD. However, in
some parts, spatially for high $p_T$ and $\phi _\eta ^*$, the
NLO-MRW UPDF scheme shows slightly different behavior with respect to the data and the
pQCD methods.
In several papers such as the reference \cite{60}, the authors
denote that the
description of two observables, including the $p_T$ and $\phi _\eta ^*$ distributions,
are
improved, if the higher order perturbative contributions are taken into account, which is in agreement with the cross check we performed.
On the other hand as the scale of energy increases, for the LHC
energies, the $q$-$g$
sub-process has the largest contribution to the differential cross section in the most intervals
of $p_T$ and $\phi _\eta ^*$, as it is expected.
We also checked the interference effect between the $\gamma^*$ and
$Z$ in the cross sections and find out that the interference is ignorable in
all regions.
Finally, we would like to point out that there is a new CMS measurement on the differential cross sections of the Z boson production in the P-P collisions \cite{CMS2019}. In the figures 7 (in terms of $p_T^Z$) and 8 (in terms of $\phi^*_\eta$ of dilepton) of this report, the CMS data are compared to the theoretical works presented in the reference \cite{R1,R2,R3,R4}, in which the UPDF (the so called transverse momentum dependent distribution functions (TMD)) are calculated, using the Parton Branching (PB) model. In this PB TMD model, the resummation to NLL accuracy, the fixed-order results at NLO, and the nonperturbative contributions are taken into account. The PB TMD results can predict the data well at low $p_T^Z$, but deviates from the measurements at
high $p_T^Z$, because of missing contributions from Z+jets matrix element calculations. Furthermore, in the present work, we do not use the LHAPDF \cite{LHAPDF} or TMDlib \cite{TMDlib} repositories, but we hope in our future reports, we can analyze the difference between the applications of present PDF and UPDF with those can be generated through LHAPDF and TMDlib repositories.
\section{Conclusions}
We investigated the lepton pair production in the $p$-$p$ and
$p$-$\bar p$ collisions within the framework of
${k_t^{}}$-factorization approach. We used the transverse momentum
dependent parton distribution functions of three different
prescriptions, i.e., KMR, LO-MRW and NLO-MRW. We calculated the
matrix element square for the three different sub-processes among
which the matrix element square for the $q$-$\bar q$ in the NLO
level is rarely taken into account. We calculated several
differential cross sections in terms of the dilepton invariant
mass, transverse momentum and rapidity, as well as the angular
correlation between produced leptons of the Drell-Yan process. In
addition, we obtained the uncertainty band for the cross section
distribution in the case of KMR by changing the scale factor as
described in the section III. We considered the contribution of each
sub-processes separately based on the off-shell and massless quarks.
We found that although some of the results show that using the KMR
framework, rather than LO-MRW and NLO-MRW schemes, represents more
agreement with the experimental data, in the case of $p_T$ and $\phi
_\eta ^*$ probing the NLO-MRW gave better predictions. It is shown that the AOC and SOC constraints give similar results and our direct calculations of the off shell matrix elements and the method of integration for evaluation of the cross section give the same prediction as those of KaTie parton-level event generator.
Finally, in this work we consider the
renormalization and factorization scales to be equal, i.e., $\mu _R =
\mu _F = \zeta M$, in which,
$M$ is the
invariant mass of produced dilepton and $\zeta$ can
vary from 1/2 to 2, to estimate the scale uncertainties of our
calculations. However as stated in the reference \cite{CMS2019}, one can vary each scale independently. Beside this it is possible to find the uncertainty, which come through the implementation of PDF through UPDF. But it should not be as large as the uncertainty effect due to the variation of renormalization and factorization scales. We hope to verify these effects in our future reports.
\begin{acknowledgements}
M. Modarres and R. Taghavi would like to acknowledge the research
support of University of Tehran and the Iran National Science
Foundation (INSF) for their grants.
\end{acknowledgements}
|
1,108,101,566,782 | arxiv | \section{Introduction}
\subsection{Rohrlich's formula}
The classical Jensen's formula is a well-known theorem of complex analysis which characterizes,
for a meromorphic function $f$ on the unit disc, the value of the integral of
$\operatorname{log}|f(z)|$ on the unit circle in terms of the zeros and poles of $f$ inside the unit disc.
An important theorem of Rohrlich \cite{ROH} establishes a version of Jensen's formula for modular
functions $f$ with respect to the full modular group $\mathrm{PSL}_2(\mathbb{Z})$ and expresses
the integral of $\operatorname{log}|f(z)|$ over a fundamental domain in terms of special values of Dedekind's eta
function.
To be more precise, let $\mathbb H^2=\{ \tau=x+iy\,|\, x,y\in \mathbb R, y>0\}$, $\Gamma=\mathrm{PSL}_2(\mathbb Z)$, and
$X=\mathrm{PSL}_2(\mathbb Z)\backslash \mathbb H^2$. Let $\Gamma_{\tau}$ denote the
stabilizer subgroup of $\tau$ in $\Gamma$ and let $\nu(\tau)$ denote its order.
The hyperbolic measure on $X$ is given by $d\mu(\tau)=dxdy/y^2$ and the hyperbolic Laplacian on $X$
is given by
\begin{align*}
\Delta=-y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right).
\end{align*}
The quotient space $X$ has the structure of a hyperbolic Riemann surface of finite hyperbolic
volume $\operatorname{vol}(X)=\pi/3$, admitting one cusp which we denote by $\infty$.
The field of modular functions on $X$ is given by $\mathbb{C}(j(\tau))$, with $j(\tau)$ denoting Klein's $j$-invariant \cite{Serre} satisfying
\begin{align*}
j(\tau)=\frac{1}{q_{\tau}}+744+O(q_{\tau}),
\end{align*}
as $\tau\to\infty$, where $q_{\tau}=e^{2\pi i \tau}$.
Consider now the class $\mathcal{M}$ of functions $F:\mathbb H^2\to \mathbb R\cup\{\infty\}$ satisfying the following properties:
\begin{enumerate}
\item[($\mathcal{M}1$)] The function $F(\tau)$ is $\Gamma$-invariant and can therefore be considered as a function on $X$.
\item[($\mathcal{M}2$)] There exist distinct points $\tau_1,\ldots ,\tau_m\in X$ together with constants $n_1,\ldots ,n_m\in \mathbb Z$ satisfying $\sum_{\ell=1}^{m}n_{\ell}=0$ such that, for $\ell\in\{1,\ldots ,m\}$,
the bound
\begin{equation*}
F(\tau)=n_{\ell}\, \nu(\tau_{\ell}) \operatorname{log}|\tau-\tau_{\ell}|^{-1}+O(1),
\end{equation*}
as $\tau \to \tau_{\ell}$, holds
and such that $F(\tau)$ is smooth
at any point $\tau\in X$ with $\tau\not= \tau_{\ell}$ for $\ell\in\{1,\ldots ,m\}$.
\item[($\mathcal{M}3$)] For $\tau\in X$ with $\tau\not=\tau_{\ell}$ for $\ell\in\{1,\ldots ,m\}$,
we have $\Delta F(\tau)=0$.
\item[($\mathcal{M}4$)] The function $F(\tau)$ is square-integrable on $X$.
\end{enumerate}
If $F:\mathbb H^2\to \mathbb{R}\cup\{\infty\}$ satisfies the properties $(\mathcal{M}1)$--$(\mathcal{M}4)$,
then the limit $F(\infty):=\lim_{\tau \to \infty}F(\tau)$ exists
and we have the equality
\begin{align}\label{expression}
F(\tau)=\operatorname{log}|f(\tau)|,\quad \mbox{ with } f(\tau)=e^{F(\infty)}\prod_{\ell=1}^m \left(j(\tau)-j(\tau_{\ell})\right)^{-n_{\ell}}.
\end{align}
Now, Rohrlich's Theorem can be rephrased as follows
\begin{theorem}[Rohrlich \cite{ROH}]\label{thm-rohrlich}
Let $F:\mathbb H^2\to \mathbb R\cup\{\infty\}$ be in $\mathcal{M}$, the class of functions satisfying
the properties $(\mathcal{M}1)$--$(\mathcal{M}4)$. Then, we have the equality
\begin{align*}
\frac{3}{\pi}\int_{X}F(\tau) d\mu(\tau)=F(\infty)+6\sum_{\ell=1}^{m}
n_{\ell}\operatorname{log} \big(|\eta(\tau_{\ell})|^4\, \mathrm{Im}(\tau_{\ell})\big),
\end{align*}
where
$\eta(\tau)=q_{\tau}^{1/24}\prod_{n=1}^{\infty}\left(1 - q_{\tau}^{n}\right)$
is the classical Dedekind's eta function.
\end{theorem}
Observe that the function on the right hand side of the equality in Theorem \ref{thm-rohrlich}
is given by the constant term in the Laurent expansion of the non-holomorphic Eisenstein series
$E_{\infty}(\tau,s)$ at $s=1$. For $\tau\in\mathbb{H}^2$ and $s\in\mathbb{C}$ with $\mathrm{Re}(s)>1$, this series is defined by
\begin{align*}
E_{\infty}(\tau,s)=\sum_{\gamma\in \Gamma_{\infty}\backslash \Gamma}\mathrm{Im}(\gamma \tau)^{s}.
\end{align*}
The Eisenstein series is $\Gamma$-invariant with respect to $\tau$ and holomorphic in $s$, and it admits a meromorphic continuation to the whole complex $s$-plane
with a simple pole at $s=1$ with residue
\begin{align*}
\operatorname{res}_{s=1}E_{\infty}(P,s)=\frac{1}{\operatorname{vol}(X)}=\frac{3}{\pi}.
\end{align*}
In this context,
the well-known Kronecker's limit formula for $\textrm{PSL}_2(\mathbb{Z})$
(see, e.g., \cite{Siegel80}) states
\begin{align*}
\lim_{s\to1}\left(E_{\infty}(z,s)-
\frac{3}{\pi(s-1)}\right)=
-\frac{3}{\pi}\operatorname{log}\bigl(|\eta(z)|^{4}\operatorname{Im}(z)\bigr)+C,
\end{align*}
where $C=6(1-12\,\zeta'(-1)-\operatorname{log}(4\pi))/\pi$ and $\zeta(s)$ denotes the Riemann zeta function.
Note that the constant $C$ does not appear in Theorem \ref{thm-rohrlich}, since $\sum_{\ell=1}^{m}n_{\ell}=0$.
The proof of Rohrlich's formula is an application of this Kronecker's limit formula.
The formula admits several generalizations and has many applications in number theory,
see, e.g. \cite{funke}, \cite{kudla}.
There is also an extension of Rohrlich's formula which has applications
to the computation of arithmetic intersection numbers in Arakelov theory, see, e.g., \cite{kuehn}.
\subsection{Purpose of the article}
The goal of this paper is to give an analogue of Rohrlich's formula in $\mathbb H^3$, the hyperbolic 3-space.
We write $\mathbb H^3=\{P=z+rj\mid z\in \mathbb C,r\in\mathbb{R}_{>0}\}$, which is a subset of the usual quaternions
$\mathbb R[i,j,k]$, and we will view $z$ and $r$ as coordinate functions on $\mathbb H^3$. The quaternionic norm on $\mathbb R[i,j,k]$ induces a norm on $\mathbb H^3$ given
explicitly by $\|P\|=\sqrt{|z|^2+r^2}$.
We let $K$ be an imaginary quadratic field, $\mathcal{O}_K$ its ring of integers, $h_K$ its class number,
and $d_K$ its discriminant. From now on, we let
$\Gamma=\mathrm{PSL}_2(\mathcal{O}_K)\subset \mathrm{PSL}_2(\mathbb{C})$, which is
a discrete and cofinite subgroup, and we let $X=\Gamma\backslash \mathbb H^3$.
By $\Gamma_{P}$ we denote the
stabilizer subgroup of $P$ in $\Gamma$ and by $\nu(P)$ its order.
By $d\mu(P)$ we denote the hyperbolic measure on $X$
and by $\Delta$ the hyperbolic Laplacian on $X$ (see \ref{hyp}).
The quotient space $X$ has finite hyperbolic volume, which is explicitly given by
\begin{equation}\label{volX}
\operatorname{vol}(X)=\frac{|d_K|^{3/2}}{4\pi^2}\,\zeta_{K}(2)
\end{equation}
with $\zeta_K(s)$ denoting the Dedekind zeta function, and it admits $h_K$ cusps (see Section \ref{section2}).
For $P\in \mathbb H^3$ and $s\in \mathbb C$ with $\operatorname{Re}(s)>1$, the Eisenstein series
associated to the cusp $\infty$ is defined by
\begin{align*}
E_{\infty}(P,s)=
\sum_{\gamma\in \Gamma_{\infty}^{\prime}
\backslash \Gamma}r(\gamma P)^{s+1},
\end{align*}
where $\Gamma_{\infty}^{\prime}$ is the maximal unipotent subgroup of the
stabilizer group $\Gamma_{\infty}$ of $\infty$ in $\Gamma$.
The Eisenstein series is $\Gamma$-invariant with respect to $P$ and holomorphic in $s$,
and it admits a meromorphic continuation to the whole complex $s$-plane
with a simple pole at $s=1$ with residue
\begin{align}\label{ResEisInfinity}
\operatorname{res}_{s=1}E_{\infty}(P,s) =\frac{\operatorname{covol}(\mathcal{O}_{K})}{\operatorname{vol}(X)}
=\frac{2\pi^2}{|d_K|\zeta_K(2)}.
\end{align}
Here, $\operatorname{covol}(\mathcal{O}_{K})$ denotes the euclidean covolume of the
lattice $\mathcal{O}_{K}$ in $\mathbb C$. In this case, Kronecker's limit formula states
\begin{align}\label{KLF}
\lim_{s\to 1}\left(E_{\infty}(P,s)-
\frac{2\pi^2}{|d_K|\zeta_K(2)(s-1)}\right)=-\frac{2\pi^2}{|d_K|\zeta_K(2)}
\operatorname{log}\left(\eta_{\infty}(P)\, r(P)\right)
+ C_K,
\end{align}
where $C_K$ is an explicit constant depending only on $K$. Here,
the function $\eta_{\infty}:\mathbb H^3\to \mathbb R$ satisfies
$
\eta_{\infty}(\gamma P)=\|cP+d\|^{2}\eta_{\infty}(P)
$
for any $\gamma=\begin{psmallmatrix}a&b\\c&d\end{psmallmatrix}\in\Gamma$
and can be considered as the analogue of the weight 2 real-analytic
modular form $|\eta(z)|^4$. The function $\eta_{\infty}$ is essentially the function
defined by Asai in \cite{ASA}. More precisely, we have
\begin{align*}
-\frac{2\pi^2\operatorname{log}\left(\eta_{\infty}(P)\right)}{|d_K|\zeta_K(2)}=
\frac{|\mathcal{O}_K^{\times}|}{2} r^{2}
+
4 \pi
\sum_{\substack{\mu \in \mathcal{D}^{-1}\\ \mu\neq 0}}
|\mu| \varphi_{\infty,\infty}(\mu;1)\, r K_{1}(4\pi |\mu|r)e^{2\pi i \operatorname{tr}(\mu z)}.
\end{align*}
Here, we employed the notation of Section \ref{section2}.
The value $\varphi_{\infty,\infty}(\mu;1)$ can be explicitly given in
terms of special values of certain generalized divisors sums. For these results,
we refer the reader to \cite{EGM}, Chapter 8, Sections 1--3.
Consider now the class $\mathcal{A}$ of functions $F:\mathbb H^3\to \mathbb R\cup\{\infty\}$ satisfying the
following properties:
\begin{enumerate}
\item[($\mathcal{A}1$)] The function $F(P)$ is $\Gamma$-invariant and can therefore
be considered
as a function on $X$.
\item[($\mathcal{A}2$)] There exist distinct points $Q_1,\ldots,Q_m\in X$ together
with constants $c_1,\ldots,c_m\in \mathbb R$ satisfying $\sum_{\ell=1}^{m}c_{\ell}=0$ such
that, for $\ell\in\{1,\ldots,m\}$, the bound
\begin{equation*}
F(P)=c_{\ell}\,\nu(Q_{\ell}) \frac{r_{\ell}}{ \|P-Q_{\ell}\|}+O(1),
\end{equation*}
as $P \to Q_{\ell}=z_{\ell}+r_{\ell} j$, holds and $F(P)$ is smooth at any point $P\in X$
with $P\not=Q_{\ell}$ for $\ell\in\{1,\ldots ,m\}$.
\item[($\mathcal{A}3$)] For $P\in X$ with $P\not=Q_{\ell}$ for $\ell\in\{1,\ldots ,m\}$,
we have $\Delta F(P)=0$.
\item[($\mathcal{A}4$)] The function $F(P)$ is square-integrable on $X$.
\end{enumerate}
We note that the bounds in $(\mathcal{M}2)$ and $(\mathcal{A}2)$ are the natural bounds that arise from the type of singularities of the corresponding Green's functions.
In Proposition \ref{thm_block} of Section \ref{section_proofofmainthm}, we
will show that, if $F:\mathbb H^3\to \mathbb R\cup\{\infty\}$ satisfies the properties $(\mathcal{A}1)$--$(\mathcal{A}4)$,
then the limit $F(\infty):=\lim_{r \to \infty}F(P)$ exists, and we will
prove the analogue of \eqref{expression} in this case. Our main theorem is
\begin{theorem}\label{mainth}
Let $F:\mathbb H^3\to \mathbb R\cup\{\infty\}$ be in $\mathcal{A}$, the class of functions
satisfying the properties $(\mathcal{A}1)$--$(\mathcal{A}4)$.
Then, we have the equality
\begin{align*}
\frac{1}{\operatorname{vol}(X)}\int_{X}F(P) d\mu(P)= F(\infty)+\frac{2\pi}{\operatorname{vol}(X)}
\sum_{\ell=1}^{m} c_{\ell} \operatorname{log}\big(\eta_{\infty}(Q_{\ell})\, r_{\ell}\big).
\end{align*}
\end{theorem}
Note that, in analogy with Rohrlich's Theorem, the constant $C_K$
arising in \eqref{KLF} does not appear in Theorem \ref{mainth}, since $\sum_{\ell=1}^{m}c_{\ell}=0$.
It is known to the experts that Rohrlich's formula can be proven using
the theory of the resolvent kernel of the hyperbolic Laplacian and our proof of
Theorem \ref{mainth} is a generalization of this method to the hyperbolic 3-space.
The advantage of this method is that it can be
generalized to other settings such as the case of the
hyperbolic $n$-space.
This method also naturally leads to an
analogue of the function $\operatorname{log}|j(\tau_1)-j(\tau_2)|$
(see the function defined in \eqref{def_analogj}). The properties of this function play a central role in our proof of Theorem \ref{mainth}, and the proof of these follow from properties of the resolvent kernel and of Niebur type Poincar\'e series.
\subsection{Outline of the article}
The paper is organized as follows.
In Section \ref{section2}, we begin by collecting background information. In Section \ref{section-Fourier-expansions}, we compute the Fourier expansion of the resolvent kernel associated to the hyperbolic Laplacian on $X$. In addition, we give the Fourier expansion of the Niebur type Poincar\'e series which appear as coefficients in the Fourier expansion of the resolvent kernel.
To the best of the authors' knowledge these expansions have not been explicitly stated
elsewhere in the literature and are of independent interest.
In Section \ref{section-analytic-continuation},
we study some of the analytic properties of the Niebur
type Poincar\'e series and we prove the meromorphic continuation of the resolvent
kernel via its Fourier expansion.
In Section \ref{section_proofofmainthm}, we construct the above mentioned analogue of $\operatorname{log}|j(\tau_1)-j(\tau_2)|$, prove its main properties, and give our proof of Theorem \ref{mainth} using these properties.
Identities involving special functions that are needed in the paper as well as some technical
lemmas are given in the Appendix and in Section \ref{section-technical-lemma}, respectively.
\subsection{Acknowledgements}
The authors would like to thank the anonymous referee for helpful comments on an earlier version of this paper.
Herrero, von Pippich, and T\'oth thank the Institute for Mathematical Research FIM at ETH Z\"urich for providing a stimulating and comfortable atmosphere during their visits to Z\"urich. Herrero, Imamo{\={g}}lu, and von Pippich
thank J\"urg Kramer and the Department of Mathematics at Humboldt-Universit\"at zu Berlin for their kind hospitality during the preparation of this work. T\'oth thanks the support of the MTA R\'enyi Int\'ezet Lend\"ulet Automorphic Research Group and the NKFIH (National Research, Development and Innovation Office) grant ERC$\underline{\ }$HU$\underline{\ }$15 118946.
\section{Background material}\label{section2}
\subsection{The hyperbolic 3-space and the group $\operatorname{PSL}_2(\mathcal{O}_K)$}
Let
$
\mathbb H^3:=\{ P=z+rj\,|\, z\in\mathbb C, r\in\mathbb R_{>0}\}
$
denote the upper half-space model of the three-dimensional hyperbolic space,
where $\{1,i,j,k\}$ is the standard basis for the quaternions $\mathbb R[i,j,k]$.
The quaternionic norm on $\mathbb R[i,j,k]$
induces a norm on $\mathbb H^3$ given explicitly by $\|P\|=\|z+rj\|=\sqrt{|z|^2+r^2}.$
For $z\in\mathbb C$, we set $\operatorname{tr}(z):=z+\overline{z}$. The hyperbolic volume element, resp.~the
hyperbolic Laplacian are given as
\begin{align}\label{hyp}
d\mu (P):=\frac{dx\,dy\, dr}{r^{3}},\quad\textrm{resp.}
\quad\Delta:=-r^{2}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial
r^{2}}\right)+r \frac{\partial}{\partial r}\,.
\end{align}
Let $d(P,Q)$ denote the hyperbolic distance between the points $P$ and $Q$.
An explicit formula is given by
\begin{align}\label{formula_cosh}
\cosh\left(d(P,Q)\right)=
\frac{|z_1-z_2|^2+r_1^2+r_2^2}{2r_1r_2},
\end{align}
where $P=z_1+r_1j$ and $Q=z_2+r_2j$.
An element $\gamma=\bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\bigr)\in \mathrm{PSL}_2(\mathbb C)$
acts on $\mathbb H^3$ by
\begin{align*}
\gamma P=
\frac{(az+b)\overline{(cz+d)}+a\overline{ c} r^2}{|cz+d|^2+|c|^2r^2}+ \frac{r}{|cz+d|^2+|c|^2r^2}\,j,
\end{align*}
where $P=z+rj$. By abuse of notation, we represent an element of $\mathrm{PSL}_2(\mathbb C)$ by a matrix.
As mentioned in the Introduction, we let $K$ be an imaginary quadratic field, $\mathcal{O}_K$
its ring of integers, $h_K$ its class number, and $d_K$ its discriminant.
We let $\Gamma=\mathrm{PSL}_2(\mathcal{O}_K)\subset \mathrm{PSL}_2(\mathbb{C})$
and we let $X:=\Gamma\backslash \mathbb H^3$. By $\Gamma_{P}$ we denote the
stabilizer subgroup of $P$ in $\Gamma$ and by $\nu(P)$ its order.
In a slight abuse of notation, we will at times identify $X$ with a
fundamental domain in $\mathbb H^3$ and identify points on $X$ with their preimages in
such a fundamental domain.
The hyperbolic volume $\operatorname{vol}(X)$ of $X$ is given by formula \eqref{volX} in terms
of a special value of Dedekind's zeta function, which is defined by
\begin{align*}
\zeta_{K}(s)=\sum_{\substack{I\subseteq \mathcal{O}_K \text{ideal}\\ I \not=(0)}}\mathcal{N}(I)^{-s},
\end{align*}
where $s\in\mathbb{C}$ with $\operatorname{Re}(s)>1$ and $\mathcal{N}(I)$ denotes the norm of $I$.
A cusp of $X$ is the $\Gamma$-orbit of a parabolic fixed point of $\Gamma$, and
$X$ has $h_K$ cusps. From now on we fix a complete set of representatives
$C_{\Gamma}\subseteq \mathbb{P}^1(K)$ for the cusps of $X$.
We write elements of $C_{\Gamma}$ as $[a:b]$
for $a,b\in\mathcal{O}_K$, not both equal to $0$, and we write $\infty:=[1:0]$ and
assume that $\infty\in C_{\Gamma}$.
Furthermore, for any cusp $\kappa=[a:b]\in C_{\Gamma}$, we fix a scaling matrix
$\sigma_{\kappa}=\begin{psmallmatrix}a &\ast \\b& \ast \end{psmallmatrix}\in\mathrm{PSL}_2(K)$
such that $\sigma_{\kappa}\infty=\kappa$ and
\begin{align}\label{first-scaling-sigma}
\sigma_{\kappa}^{-1}\Gamma_{\kappa}\sigma_{\kappa}=\left\{ \begin{pmatrix}u& \lambda
\\0&u^{-1}\end{pmatrix}\bigg|\,
u\in\mathcal{O}_K^{\times}, \lambda \in \Lambda_{\kappa}\right\}
\end{align}
with the full lattice $\Lambda_{\kappa}=(a\mathcal{O}_K+b\mathcal{O}_K)^{-2}\subseteq\mathbb{C}$
(see, e.g., \cite{mazanti}).
For the cusp $\infty$,
we choose $\sigma_{\infty}$ to be the identity.
Furthermore, for the
maximal unipotent subgroup $\Gamma_{\kappa}^{\prime}$, which consists
of all the parabolic elements of $\Gamma_{\kappa}$ together with the identity,
we have
\begin{align}\label{scaling-sigma}
\sigma_{\kappa}^{-1}\Gamma^{\prime}_{\kappa}\sigma_{\kappa}=
\left\{ \begin{pmatrix}1&\lambda\\0&1\end{pmatrix}\bigg|\,
\lambda\in \Lambda_{\kappa} \right\}.
\end{align}
We let $\Lambda_{\kappa}^{*}=\{\nu\in \mathbb C: \operatorname{tr}(\nu \lambda)\in \mathbb Z \text{ for any }\lambda \in \Lambda_{\kappa}\}$ denote its dual lattice.
In particular, we have $\Lambda_{\infty}=\mathcal{O}_{K}$
and $\Lambda_{\infty}^{*}=\mathcal{D}^{-1}$ with
$\mathcal{D}^{-1}=\{\nu\in K\mid\operatorname{tr}(\nu\lambda)\in \mathbb Z \text{ for any } \lambda\in \mathcal{O}_K\}$
denoting the inverse different.
\subsection{Fourier expansion of automorphic functions}\label{subsec_auto_functions}
A function $f:\mathbb H^3\to \mathbb C$ is called automorphic with respect to $\Gamma$ if it is
$\Gamma$-invariant, that is, $f(\gamma P)=f(P)$ for any $\gamma\in\Gamma$. An important
tool to study the behavior of an automorphic function $f$
at a cusp $\xi\in C_{\Gamma}$, with scaling matrix $\sigma_{\xi}$, is its Fourier expansion.
More precisely, since the function $P\mapsto f(\sigma_{\xi}P)$ is
$\sigma_{\xi}^{-1}\Gamma^{\prime}_{\xi}\sigma_{\xi}$-invariant, employing \eqref{scaling-sigma}, we have
$
f(\sigma_{\xi}(P+\lambda))=f(\sigma_{\xi}P)
$
for any $\lambda\in\Lambda_{\xi}$.
If $f$ is smooth, the Fourier expansion of
$f$ with respect to the cusp $\xi$ is therefore of the form
\begin{align}\label{fourier_expansion_general}
f(\sigma_{\xi}P)=\sum_{\mu \in \Lambda_{\xi}^{*}}
a_{\mu}(r)e^{2\pi i \operatorname{tr}(\mu z)},
\end{align}
where $P=z+rj$ and with Fourier coefficients given by
$$a_{\mu}(r)=\frac{1}{\mathrm{covol}(\Lambda_{\xi})}
\int_{\mathbb C/\Lambda_{\xi}}f(\sigma_{\xi}P)e^{-2\pi i \operatorname{tr}(\mu z)}dz.$$
If we assume that $f$ is an eigenfunction of the hyperbolic Laplacian, satisfying
$\Delta f=(1-s^2)f$ for some $s\in\mathbb{C}$ with $s\not=0$,
and that $f$ is of polynomial growth
as $r\to \infty$, that is $f(z+rj)=O(r^C)$ as $r\to \infty$ for some constant $C$,
then the expansion \eqref{fourier_expansion_general} has the form
(see, e.g., \cite{EGM}, Theorem 3.1, p.~105)
\begin{align}\label{fourier_expansion_special}
f(\sigma_{\xi}P)=a_0r^{1+s}+b_0 r^{1-s}
+\sum_{\substack{\mu \in \Lambda_{\xi}^{*}\\ \mu\neq 0}}a_{\mu}\,r K_{s}(4\pi |\mu|r)e^{2\pi i \operatorname{tr}(\mu z)}
\end{align}
with $a_0, b_0, a_{\mu}\in\mathbb{C}$ and with $K_{s}(\cdot)$ denoting the modified
Bessel function of the second kind.
\subsection{Poincar\'e series}\label{subsection_poincare}
For later purposes, we define two families of eigenfunctions of the hyperbolic Laplacian, namely the Eisenstein series and the Niebur type Poincar\'e series, which from now one will be called Niebur--Poincar\'e series for simplicity. For this, let $\kappa\in C_{\Gamma}$ be a cusp with scaling matrix $\sigma_{\kappa}$.
For $P\in \mathbb H^3$ and $s\in \mathbb C$ with $\operatorname{Re}(s)>1$, the Eisenstein series
associated to the cusp $\kappa$ is given by
\begin{align*}
E_{\kappa}(P,s)=
\sum_{\gamma\in \Gamma_{\kappa}^{\prime}\backslash \Gamma}
r(\sigma_{\kappa}^{-1}\gamma P)^{s+1}.
\end{align*}
The Eisenstein series is an automorphic function for $\Gamma$ and it is holomorphic
in $s$ in the region $\mathrm{Re}(s)>1$. Moreover, it satisfies the differential
equation
\begin{align*}
\left(\Delta-(1-s^2)\right)E_{\kappa}(P,s)=0,
\end{align*}
i.e.~it is an eigenfunction of $\Delta$. For $s\in\mathbb{C}$ with $\mathrm{Re}(s)>1$,
the Eisenstein series admits a Fourier expansion of the form \eqref{fourier_expansion_special}
given by (see, e.g., \cite{EGM}, Theorem 4.1, p.~111)
\begin{align}\label{equ_fourier_expansion_series_eis_gen}
E_{\kappa}(\sigma_{\xi}P,s)&=
\delta_{\kappa,\xi} [\Gamma_{\kappa}:\Gamma_{\kappa}^{\prime}]\,
r^{1+s}
+
\varphi_{\kappa,\xi}(0;s)\,r^{1-s}
+
\frac{2^{1+s}\pi^{s}}{\Gamma(s)}
\sum_{\substack{\mu \in \Lambda_{\xi}^{*}\\ \mu\neq 0}}|\mu|^{s}\varphi_{\kappa,\xi}(\mu;s)\,r K_{s}(4\pi |\mu|r)e^{2\pi i \operatorname{tr}(\mu z)},
\end{align}
where $\delta_{\kappa,\xi}$ is Kronecker's delta symbol and, for $s\in\mathbb{C}$ with $\mathrm{Re}(s)>1$, we have set
\begin{align}\label{def_varphiscattering}
\varphi_{\kappa,\xi}(\mu;s):=\frac{\pi}{\operatorname{covol}(\Lambda_{\xi})s}
\sum_{\begin{psmallmatrix}*&*\\c&d \end{psmallmatrix}\in \sigma_{\kappa}^{-1}\mathcal{R}_{\kappa,\xi}\sigma_{\xi}}
\frac{e^{2\pi i \operatorname{tr}(\mu \frac{d}{c})}}{|c|^{2s+2}}
\end{align}
with
\begin{align}\label{def_R}
\mathcal{R}_{\kappa,\xi}:=\Gamma_{\kappa}^{\prime}\backslash \{\gamma\in \Gamma: \gamma \xi\neq \kappa\}/\Gamma_{\xi}^{\prime}.
\end{align}
Note that $\{\gamma\in \Gamma: \gamma \xi\neq \kappa\}=\Gamma$ if
$\xi\not=\kappa$. It is known (see, e.g., \cite{EGM}, \cite{SAR}) that the function $\varphi_{\kappa,\xi}(\mu;s)$
admits a meromorphic continuation to all $s\in\mathbb{C}$, which is holomorphic at $s=1$ if $\mu \neq 0$.
It is also well-known that one can use the above Fourier expansion in order to prove that
$E_{\kappa}(P,s)$ admits a meromorphic continuation to the whole complex $s$-plane.
There is always a simple pole at $s=1$ with residue given by
\begin{equation}\label{resEis}
\operatorname{res}_{s=1}E_{\kappa}(P,s)
=\frac{\operatorname{covol}(\Lambda_{\kappa})}{\operatorname{vol}(X)}
=\operatorname{res}_{s=1}\varphi_{\kappa,\kappa}(0;s).
\end{equation}
In case that $\kappa=\infty$, the residue is explicitly given by \eqref{ResEisInfinity} and
we have
\begin{equation*}
\varphi_{\infty,\infty}(0;s)=\frac{\pi |\mathcal{O}_K^{\times}|}{h_K|d_K|^{1/2}s}\sum_{\chi}\frac{L(s,\chi)}{L(s+1,\chi)},
\end{equation*}
where the sum runs over all characters $\chi$ of the class group of $K$ and
$L(s,\chi)$ denotes the associated $L$-function (see, e.g., \cite{EGM}, Chapter 8, Theorems 1.5 and 2.11). From these, a straight-forward computation yields
the Kronecker's limit formula \eqref{KLF} stated in the Introduction.
Finally, we recall the definition of the Niebur--Poincar\'e series.
For $P\in \mathbb H^3$ and $s\in \mathbb C$ with $\operatorname{Re}(s)>1$, the Niebur--Poincar\'e series
associated to the cusp $\kappa$ and to $\nu\in \Lambda_{\kappa}^{*}$,
$\nu \neq 0$, is given by
\begin{align}\label{def-niebur-series}
F_{\kappa,\nu}(P,s)=
\sum_{\gamma \in \Gamma_{\kappa}'\backslash \Gamma}
r(\sigma_{\kappa}^{-1}\gamma P)\,
I_{s}\left(4\pi |\nu | r(\sigma_{\kappa}^{-1} \gamma P)\right)
e^{ 2\pi i \operatorname{tr}\left(\nu z(\sigma_{\kappa}^{-1}\gamma P)\right)},
\end{align}
where $I_s(\cdot)$ denotes the modified Bessel function of the first kind.
We recall that
the Niebur--Poincar\'e series converges absolutely and defines an automorphic function,
which is holomorphic for $s\in\mathbb{C}$
with $\mathrm{Re}(s)>1$ (see, e.g., \cite{matthes}).
Moreover, it satisfies the differential equation
\begin{align*}
\left(\Delta-(1-s^2)\right)F_{\kappa,\nu}(\cdot,s)=0,
\end{align*}
i.e.~it is an eigenfunction of $\Delta$.
\subsection{The resolvent kernel}\label{Green-subsection}
The resolvent kernel for the hyperbolic Laplacian is given by the automorphic Green's function.
For $P,Q\in \mathbb{H}^3$ with $P\not=\gamma Q$ for any $\gamma\in\Gamma$,
and $s\in \mathbb C$ with $\operatorname{Re}(s)>1$, it is defined by
\begin{align*}
G_s(P,Q)=\frac{1}{2\pi}\sum_{\gamma\in \Gamma}
\varphi_s\left(\cosh (d(P,\gamma Q))\right),
\end{align*}
where $\varphi_s(t)=(t+\sqrt{t^2-1})^{-s}(t^2-1)^{-1/2}$. The series defining $G_s(P,Q)$ converges uniformly on compact subsets of $\{(P,Q)\in \mathbb H^3\times \mathbb H^3:P\neq \gamma Q \text{ for any }\gamma \in \Gamma)\}\times \{s\in \mathbb C: \mathrm{Re}(s)>1\}$.
We recall the following well-known properties of $G_s(P,Q)$ (see, e.g., \cite{EGM}):
\begin{enumerate}
\item[(G1)]
The function $G_s(P,Q)$ is $\Gamma$-invariant in each variable
and can therefore be considered as a function on $X\times X$, away from the diagonal.
Moreover, we have $G_s(P,Q)=G_s(Q,P)$.
\item[(G2)]
For fixed $Q\in X$,
we have a singularity of the form
\begin{equation*}
G_s(P,Q)= \frac{\nu(Q)}{2\pi} \frac{1}{ d(P,Q)}+O_Q(1),
\end{equation*}
as $P \to Q$.
\item[(G3)] For $P,Q\in X$ with $P\neq Q$, we have $(\Delta_P-(1-s^2)) G_s(P,Q)=0.$
\end{enumerate}
The Green's function is holomorphic for $s\in \mathbb C$ with $\operatorname{Re}(s)>1$ and it
admits a meromorphic continuation to the whole complex $s$-plane
with a simple pole at $s=1$ with residue
\begin{equation}\label{Greenpole}
\operatorname{res}_{s=1}\,G_s(P,Q)=\frac{1}{\operatorname{vol}(X)}.
\end{equation}
Moreover, using the spectral expansion of $G_s(P,Q)$ given in \cite{EGM} (Proposition 4.6, p.~285),
it is easy to see that the function
\begin{align*}\label{prop_greens}
P \mapsto \lim_{s\to 1}\left(G_s(P,Q)-\frac{2}{\operatorname{vol}(X)(s^2-1)}\right)
\end{align*}
is square-integrable on $X$, for fixed $Q\in X$, and orthogonal to the constant functions, i.e.
\begin{equation}\label{G-orthogonal}
\int_X\lim_{s\to 1}\left(G_s(P,Q)-\frac{2}{\operatorname{vol}(X)(s^2-1)}\right)d\mu(P)=0.
\end{equation}
\section{Fourier expansions}\label{section-Fourier-expansions}
In this section, we compute the Fourier expansion of the Green's function and that of
the Niebur--Poincar\'e series. Part of the computations involve explicit evaluations of
certain integrals in terms of special functions. The proof of these technical identities
is postponed to Section \ref{section-technical-lemma} in order to keep the exposition simple.
\begin{proposition}\label{prop-fourier-expansion-greens}
Let $P=z+rj\in \mathbb H^3$ with $r>r(\sigma^{-1}_{\xi}\gamma Q)$ for any $\gamma\in\Gamma$,
and $s\in\mathbb{C}$ with $\operatorname{Re}(s)>1$. Then, we have the following Fourier expansion
\begin{equation*}
G_s(\sigma_{\xi}P,Q)=\frac{1}{\operatorname{covol}(\Lambda_{\xi})}\biggl(
\frac{r^{1-s}}{s}E_{\xi}(Q,s)+2\sum_{\substack{\mu \in \Lambda_{\xi}^{*}\\ \mu\neq 0}} F_{\xi,-\mu}(Q,s) \, r K_{s}(4\pi |\mu| r) e^{2\pi i \operatorname{tr} (\mu z )}\biggr).
\end{equation*}
\end{proposition}
\begin{proof}
The Fourier coefficient $a_{\mu}(r)=a_{\mu,s}(r,Q)$ in the Fourier expansion
\eqref{fourier_expansion_general} of the function $P\mapsto G_s(P,Q)$ with respect to the cusp $\xi$ is given
by
$$a_{\mu,s}(r,Q)=\frac{1}{\operatorname{covol}(\Lambda_{\xi})}\int_{\mathbb C/\Lambda_{\xi}}
G_s(\sigma_{\xi}P,Q)e^{-2\pi i \operatorname{tr} ( \mu z )}dz.
$$
To compute this integral, we start by writing
\begin{align*}
G_s(\sigma_{\xi}P,Q)
&=
\frac{1}{2\pi}\sum_{\gamma\in \Gamma_{\xi}'\backslash \Gamma}
\sum_{\eta\in \Gamma_{\xi}'}
\varphi_s\left(\cosh (d(\eta^{-1}\sigma_{\xi}P,\gamma Q))\right)\\
&
=\frac{1}{2\pi}\sum_{\gamma\in \Gamma_{\xi}'\backslash \Gamma}
\sum_{\lambda\in \Lambda_{\xi}}
\varphi_s\left(\cosh (d(P+\lambda,\sigma_{\xi}^{-1}\gamma Q))\right),
\end{align*}
where for the last equality we employed \eqref{scaling-sigma},
namely the identity
$
\sigma_{\xi}^{-1}\Gamma^{\prime}_{\xi}\sigma_{\xi}=
\left\{\begin{psmallmatrix}1&\lambda\\0&1\end{psmallmatrix}|\,
\lambda\in \Lambda_{\xi} \right\}
$.
Hence, we get
\begin{align*}
a_{\mu,s}(r,Q)=\frac{1}{2\pi\operatorname{covol}(\Lambda_{\xi})}
\sum_{\gamma\in \Gamma_{\xi}'\backslash \Gamma}\,\int_{\mathbb C}
\varphi_s\left(\cosh (d(P,\sigma_{\xi}^{-1}\gamma Q))\right)e^{-2\pi i \operatorname{tr} ( \mu z)}dz.
\end{align*}
Now, we set $\tilde{z}:=z(\sigma_{\xi}^{-1}\gamma Q)$ and $\tilde{r}:=
r(\sigma_{\xi}^{-1}\gamma Q)$.
Using formula \eqref{formula_cosh}, namely
$$
\cosh (d(P,\sigma_{\xi}^{-1}\gamma Q))=
\frac{|z-\tilde{z}|^2+r^2+\tilde{r}^2}{2r\tilde{r}},
$$
we obtain by a change of variables
($z\mapsto z+\tilde{z}$),
$$
a_{\mu,s}(r,Q)
=\frac{1}{2\pi\operatorname{covol}(\Lambda_{\xi})}
\sum_{\gamma\in \Gamma_{\xi}'\backslash \Gamma}
e^{-2\pi i \operatorname{tr} (\mu \tilde{z} )}
I_{\mu,s}(r,\tilde{r}),
$$
where we have set
\begin{align*}
I_{\mu,s}(r,\tilde{r}):=\int_{\mathbb C}\varphi_s\left(\frac{|z|^2+r^2+\tilde{r}^2}{2r\tilde{r}}\right)e^{-2\pi i \operatorname{tr} (\mu z )}dz.
\end{align*}
By Lemma \ref{Iintegral}, we have
\begin{align*}
I_{\mu,s}(r,\tilde{r})=
\begin{cases}
2\pi s^{-1}r^{1-s}\tilde{r}^{s+1}, &\text{ if }\mu= 0,\\
4\pi r\tilde{r} K_{s}(4\pi |\nu|r) I_{s}(4\pi |\nu|\tilde{r}), &\text{ if }\mu \neq 0.
\end{cases}
\end{align*}
Summing up and recalling that $\tilde{r}=r(\sigma_{\xi}^{-1}\gamma Q)$, we conclude
\begin{align*}
a_{0,s}(r,Q)
=\frac{1}{\operatorname{covol}(\Lambda_{\xi})} \frac{r^{1-s}}{s}
\sum_{\gamma\in \Gamma_{\xi}'\backslash \Gamma}
r(\sigma_{\xi}^{-1}\gamma Q)^{s+1}=\frac{1}{\operatorname{covol}(\Lambda_{\xi})} \frac{r^{1-s}}{s}E_{\xi}(Q,s)
\end{align*}
and, for $\mu\neq 0$, we derive
\begin{align*}
a_{\mu,s}(r,Q)
&=\frac{2}{\operatorname{covol}(\Lambda_{\xi})} r K_{s}(4\pi |\mu| r)
\sum_{\gamma\in \Gamma_{\xi}'\backslash \Gamma}
\tilde{r}I_{s}(4\pi |\mu| \tilde{r}) e^{-2\pi i \operatorname{tr} (\mu \tilde{z} )}\\
&=\frac{2}{\operatorname{covol}(\Lambda_{\xi})} r K_{s}(4\pi |\mu| r)
F_{\xi,-\mu}(Q,s),
\end{align*}
as asserted. This completes the proof.
\end{proof}
We proceed by computing the Fourier expansion of the Niebur--Poincar\'e series
$F_{\kappa,\nu}(P,s)$, where $\kappa\in C_{\Gamma}$ and $\nu\in \Lambda_{\kappa}^{*}$, $\nu \neq 0$.
For this, we define the function $\mathcal J_s:\mathbb C^{\times}\to \mathbb C$ by
\begin{align}\label{def_Lniebur}
\mathcal J_s(z):=
\begin{cases}
J_s(4\pi \sqrt{z})J_s(4\pi \sqrt{\overline{z}}), &\text{ if } \operatorname{Re}(z)\geq 0,\\
I_s(4\pi \sqrt{-z})I_s(4\pi \sqrt{-\overline{z}}), & \text{ if } \operatorname{Re}(z)\leq 0.
\end{cases}
\end{align}
Using the identity $I_s(z)=e^{\mp s \pi /2}J_s(ze^{\pm \pi i /2})$ for $z\in \mathbb C$ with $\operatorname{Re}(z)>0$,
it is easy to verify that $\mathcal J_s(z)$ is well-defined for $z\in \mathbb C$, $z\neq 0$, with $\operatorname{Re}(z)=0$.
With this, we have
\begin{proposition}\label{FENgeneral}
Let $P=z+rj\in \mathbb H^3$ and $s\in\mathbb{C}$ with $\operatorname{Re}(s)>1$. Then,
we have the following Fourier expansion
\begin{align*}
F_{\kappa,\nu}(\sigma_{\xi} P,s)
&= \delta_{\kappa,\xi}\, rI_{s}(4\pi |\nu| r)
\sum_{ u\in \mathcal{O}_{K}^{\times}/\{\pm1\}}
e^{2\pi i \operatorname{tr}( \nu u^2z )} +\frac{\mathrm{covol}(\Lambda_{\kappa})}{\mathrm{covol}(\Lambda_{\xi})}
\frac{(2\pi|\nu|)^s}{s\Gamma(s)}\varphi_{\xi,\kappa}(-\nu;s)\,r^{1-s}\\
&\mathrel{\phantom{=}} + \sum_{\substack{\mu \in \Lambda_{\xi}^{\ast}\\ \mu\neq 0}}
\mathcal{B}_{\kappa,\xi}(\nu,\mu;s)\,rK_s(4\pi |\mu|r) e^{2\pi i \operatorname{tr}(\mu z)}.
\end{align*}
Here,
\begin{align*}
\mathcal{B}_{\kappa,\xi}(\nu,\mu;s)
:=\frac{2\pi}{\operatorname{covol}(\Lambda_{\xi})}\,
\sum_{\begin{psmallmatrix}a&*\\c&d \end{psmallmatrix} \in \sigma_{\kappa}^{-1}\mathcal{R}_{\kappa,\xi}\sigma_{\xi}}\frac{e^{2\pi i \operatorname{tr}\left( (\nu a+\mu d)/c\right)}}{|c|^{2}}
\,\mathcal J_s\left(\frac{\nu \mu}{c^2}\right),
\end{align*}
and $\varphi_{\xi,\kappa}(-\nu;s)$, $\mathcal{R}_{\kappa,\xi}$, and $\mathcal J_s(\cdot)$ are given by \eqref{def_varphiscattering}, \eqref{def_R}, \eqref{def_Lniebur}, respectively.
\end{proposition}
\begin{proof}
To simplify the notation, we set $f(P):=r(P)I_{s}\left(4\pi |\nu | r(P)\right)e^{ 2\pi i \operatorname{tr}\left(\nu z(P)\right)}$
and we define
\begin{align*}
\widehat{F}_{\kappa,\nu}(P,s):=
\sum_{\gamma \in \mathcal{R}_{\kappa,\xi}}\sum_{\eta\in\Gamma'_{\xi} }
f(\sigma_{\kappa}^{-1} \gamma \eta P).
\end{align*}
Recalling the definition \eqref{def-niebur-series} of the Niebur-Poincar\'e series, we
then deduce
\begin{align}
F_{\kappa,\nu}(\sigma_{\xi} P,s)
&=\sum_{\gamma \in \Gamma_{\kappa}'\backslash \Gamma}
f\left(\sigma_{\kappa}^{-1}\gamma \sigma_{\xi}P\right)\notag\\
&=\delta_{\kappa,\xi}\sum_{\gamma \in \Gamma_{\kappa}'\backslash \Gamma_{\kappa}}
f\left(\sigma_{\kappa}^{-1}\gamma \sigma_{\xi}P\right)
+ \widehat{F}_{\kappa,\nu}(\sigma_{\xi}P,s).\label{splitting}
\end{align}
To treat the first term in \eqref{splitting}, we assume that
$\delta_{\kappa,\xi}=1$, that is $\kappa=\xi$ and $\sigma_{\kappa}= \sigma_{\xi}$.
Then
\begin{align*}
\delta_{\kappa,\xi}\sum_{\gamma \in \Gamma_{\kappa}'\backslash \Gamma_{\kappa}}
f\left(\sigma_{\kappa}^{-1}\gamma \sigma_{\xi}P\right)
=\sum_{\gamma \in \sigma_{\kappa}^{-1}(\Gamma_{\kappa}'\backslash \Gamma_{\kappa})\sigma_{\kappa}}
f\left(\gamma P\right)
= rI_{s}(4\pi |\nu| r)
\sum_{ u\in \mathcal{O}_{K}^{\times}/\{\pm1\}}
e^{2\pi i \operatorname{tr}( \nu u^2z )},
\end{align*}
where for the second equality we note that
$\sigma_{\kappa}^{-1}(\Gamma_{\kappa}'\backslash \Gamma_{\kappa})\sigma_{\kappa}\cong
\left\{ \begin{psmallmatrix}u&0
\\0&u^{-1}\end{psmallmatrix}\mid
u\in\mathcal{O}_K^{\times}\right\}/\{\pm1\}$, which is an immediate consequence of
\eqref{first-scaling-sigma} and \eqref{scaling-sigma}, and we
used
the identity $\begin{psmallmatrix}u&0\\0&u^{-1} \end{psmallmatrix} P=
u^2z+rj$ for $u\in \mathcal{O}_{K}^{\times}$.
To treat the second term in \eqref{splitting}, we note that
the function $\widehat{F}_{\kappa,\nu}(\sigma_{\xi} \cdot ,s)$ is $\sigma_{\xi}^{-1}\Gamma'_{\xi} \sigma_{\xi}$-invariant.
The Fourier coefficient $b_{\mu}(r)=b_{\mu,\kappa,\nu}(r,s)$ in the Fourier expansion
\eqref{fourier_expansion_general} of the function $\widehat{F}_{\kappa,\nu}(P,s)$ with respect to the cusp $\xi$ is given
by
$$b_{\mu,\kappa,\nu}(r,s)
=\frac{1}{\operatorname{covol}(\Lambda_{\xi})}\int_{\mathbb C/\Lambda_{\xi}}
\widehat{F}_{\kappa,\nu}(\sigma_{\xi} P,s)e^{-2\pi i \operatorname{tr} ( \mu z )}dz.
$$
To compute this integral, we start by writing
\begin{align*}
\widehat{F}_{\kappa,\nu}(\sigma_{\xi} P,s)
&=
\sum_{\gamma \in \mathcal{R}_{\kappa,\xi}}\sum_{\eta\in\Gamma'_{\xi} }
f\left(\sigma_{\kappa}^{-1} \gamma \eta \sigma_{\xi} P\right)\\
&=\sum_{\gamma\in \mathcal{R}_{\kappa,\xi}}
\sum_{\lambda\in \Lambda_{\xi}}
f\left(\sigma_{\kappa}^{-1} \gamma \sigma_{\xi}(P+\lambda)\right),
\end{align*}
where for the last equality we employed \eqref{scaling-sigma},
namely the identity
$
\sigma_{\xi}^{-1}\Gamma^{\prime}_{\xi}\sigma_{\xi}=
\left\{\begin{psmallmatrix}1&\lambda\\0&1\end{psmallmatrix}|\,
\lambda\in \Lambda_{\xi} \right\}
$.
Hence, we get
\begin{align*}
b_{\mu,\kappa,\nu}(r,s)&=\frac{1}{\operatorname{covol}(\Lambda_{\xi})}
\sum_{\gamma \in\sigma_{\kappa}^{-1} \mathcal{R}_{\kappa,\xi}\sigma_{\xi}}\,\int_{\mathbb C}
f\left(\gamma P\right)e^{-2\pi i \operatorname{tr} ( \mu z)}dz.
\end{align*}
Now, writing
$
z(\gamma P)=\tfrac{a}{c}-\tfrac{1}{c} \tfrac{\overline{cz+d}}{|cz+d|^2+|c|^2r^2}
$ with $\gamma=\bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\bigr)$ and using
\begin{align*}
f\left(\gamma P\right)
&=\frac{r}{|cz+d|^2+|c|^2r^2}\,I_s\left(\frac{4\pi |\nu|r}{|cz+d|^2+|c|^2r^2}\right)
e^{2\pi i \operatorname{tr}( \nu\frac{a}{c})}
e^{-2\pi i \operatorname{tr} \left(\tfrac{\nu}{c} \tfrac{\overline{cz+d}}{|cz+d|^2+|c|^2r^2}\right)},
\end{align*}
we obtain
by a change of variables ($z\mapsto z-\tfrac{d}{c}$)
\begin{align*}
b_{\mu,\kappa,\nu}(r,s)&=\frac{1}{\mathrm{covol}(\Lambda_{\xi})}\sum_{\begin{psmallmatrix}a&*\\c&d \end{psmallmatrix} \in \sigma_{\kappa}^{-1}\mathcal{R}_{\kappa,\xi}\sigma_{\xi}}
e^{2\pi i \operatorname{tr} \left(\nu\frac{a}{c}+ \mu \frac{d}{c}\right)}
\mathcal I(r,\nu,\mu,c),
\end{align*}
where we have set
\begin{align*}
\mathcal I(r,\nu,\mu,c):=\int_{\mathbb C}\frac{r}{|c|^2(|z|^2+r^2)}
I_s\left(\frac{4\pi |\nu|r}{|c|^2(|z|^2+r^2)}\right)
e^{-2\pi i \operatorname{tr}\left(\nu\tfrac{\overline{z}}{c^2(|z|^2+r^2)}+\mu z\right)}dz.
\end{align*}
By Lemma \ref{Gintegral},
we have
\begin{align*}
\mathcal I(r,\nu,\mu,c)=
\begin{cases}
\dfrac{2^s\pi^{s+1}|\nu|^s}{|c|^{2(s+1)}s^2\Gamma(s)}r^{1-s}, & \text{ if }\mu =0,\\[0.5cm]
\dfrac{2\pi}{|c|^2}\mathcal J_s\left(\frac{\nu \mu}{c^2}\right) rK_s(4\pi |\mu |r), & \text{ if }\mu \neq 0.
\end{cases}
\end{align*}
Summing up, we conclude
\begin{align*}
b_{0,\kappa,\nu}(r,s)&=\frac{1}{\mathrm{covol}(\Lambda_{\xi})}
\frac{2^s\pi^{s+1}|\nu|^s}{s^2\Gamma(s)}r^{1-s}
\sum_{\begin{psmallmatrix}a&*\\c&* \end{psmallmatrix} \in \sigma_{\kappa}^{-1}\mathcal{R}_{\kappa,\xi}\sigma_{\xi}}\frac{e^{2\pi i \operatorname{tr}( \nu \frac{a}{c})}}{|c|^{2s+2}}\\
&=\frac{1}{\mathrm{covol}(\Lambda_{\xi})}
\frac{2^s\pi^{s+1}|\nu|^s}{s^2\Gamma(s)}r^{1-s}
\sum_{\begin{psmallmatrix}*&*\\c&d \end{psmallmatrix} \in \sigma_{\xi}^{-1}\mathcal{R}_{\xi,\kappa}\sigma_{\kappa}}\frac{e^{2\pi i \operatorname{tr}( -\nu \frac{d}{c})}}{|c|^{2s+2}}.
\end{align*}
Recalling definition \eqref{def_varphiscattering}, we get that
\begin{align*}
b_{0,\kappa,\nu}(r,s)
&=\frac{\mathrm{covol}(\Lambda_{\kappa})}{\mathrm{covol}(\Lambda_{\xi})}
\frac{2^s\pi^{s}|\nu|^s}{s\Gamma(s)}r^{1-s}
\varphi_{\xi,\kappa}(-\nu;s),
\end{align*}
as asserted. Furthermore, for $\mu\not=0$, we conclude
\begin{align*}
b_{\mu,\kappa,\nu}(r,s)&=\frac{2\pi}{\mathrm{covol}(\Lambda_{\xi})}
rK_s(4\pi |\mu |r)
\sum_{\begin{psmallmatrix}a&*\\c&d \end{psmallmatrix} \in \sigma_{\kappa}^{-1}\mathcal{R}_{\kappa,\xi}\sigma_{\xi}}
\frac{e^{2\pi i \operatorname{tr}(\nu\frac{a}{c}+ \mu \frac{d}{c})}}{|c|^2}\mathcal J_s\left(\frac{\nu \mu}{c^2}\right) \\
&= \mathcal{B}_{\kappa,\xi}(\nu,\mu;s)rK_s(4\pi |\mu |r).
\end{align*}
This completes the proof.
\end{proof}
\section{Analytic continuation}\label{section-analytic-continuation}
The main goal of this section is to prove the meromorphic continuation of the
Green's function via its Fourier expansion. We remark here that the existence
of this meromorphic continuation is well-known and follows from the spectral
expansion of the Green's function (see, e.g., \cite{EGM}, Proposition 4.6, p.~285).
Here we choose a different approach as we also need precise information about the
growth at the cusp $\infty$ of this meromorphic continuation. In order to do this,
we first analytically continue the Niebur--Poincar\'e series $F_{\infty,\nu}(P,s)$
by using the explicit Fourier expansion given in Proposition \ref{FENgeneral}
with $\xi=\infty$. Before doing so, we need the following result.
\begin{lemma}\label{Lbound}
We have the bounds
\begin{align*}
|\mathcal J_s(z)|=
\begin{cases}
O\left(| z|^{\operatorname{Re}(s)}\right), & \text{ for }0<|z|\leq 1,\\[0.1cm]
O\left(e^{8\pi \sqrt{|z|}}|z|^{\operatorname{Re}(s)}\right), & \text{ for } |z|> 1,
\end{cases}
\end{align*}
holding uniformly for $s$ in any compact set contained in $\mathrm{Re}(s)>-1/2$.
\end{lemma}
\begin{proof}
Using the asymptotic formula \eqref{IJasympt}, we conclude
$$\mathcal J_s(z)\sim \frac{|4\pi^2 z|^s}{\Gamma(s+1)^2},$$
for $z\to 0$.
This implies the first bound. In order to obtain the second bound, we use \eqref{Iinfinity}
and get
$$
I_s(4\pi \sqrt{-z})I_s(4\pi \sqrt{-\overline{z}})=O\left(|z|^{\operatorname{Re}(s)}
e^{8\pi \operatorname{Re}(\sqrt{-z})
}
\right),
$$
for $z\to \infty,\operatorname{Re}(z)\leq 0$. On the other hand, formula \eqref{Jinfinity}
gives
$$
J_s(4\pi \sqrt{z})J_s(4\pi \sqrt{\overline{z}})=O\left(|z|^{\operatorname{Re}(s)}
e^{8\pi |\operatorname{Im}(\sqrt{z})|}
\right),
$$
for $z\to \infty, \operatorname{Re}(z)\geq 0$. The second bound follows easily from these estimates. Since the used asymptotic formulas and bounds are uniform for $s$ in any compact set contained in $\operatorname{Re}(s)>-1/2$, we conclude that these estimates are also uniform. This completes the proof of the Lemma.
\end{proof}
Given $\nu,\mu\in \mathcal{D}^{-1}$ both non zero and $s\in \mathbb C$ define
$$
\mathcal{Z}(\nu,\mu;s)
:=\sum_{\substack{c\in \mathcal{O}_K/\{\pm 1\}\\ c\neq 0}}
\frac{|\mathcal{S}(\nu,\mu,c)|}{|c|^{2+2s}},
$$
where
$$\mathcal{S}(\nu,\mu,c):=\sum_{\substack{u,u^{\ast}\in \mathcal{O}_K/c\mathcal{O}_K\\ uu^{\ast}=1}}e^{2\pi i \operatorname{tr}((u\nu +u^{\ast}\mu)/c)}.$$
By using the trivial bound for $|\mathcal{S}(\nu,\mu,c)|$, namely $|\mathcal{S}(\nu,\mu,c)|\leq \mathcal{N}(c)=|c|^2$, one sees that the series $\mathcal{Z}(\nu,\mu;s)$ converges absolutely for $\mathrm{Re}(s)>1$.
\begin{lemma}\label{contLZseries}
The series $\mathcal{Z}(\nu,\mu;s)$ converges absolutely for $\operatorname{Re}(s)>1/2$. Moreover, there exists $\alpha>0$ such that the bound
$$|\mathcal{Z}(\nu,\mu;s)|=O\left(\mathcal{N}(\nu \mu \mathcal{D}^2)^{\alpha}\right)$$
holds uniformly for $s$ in any compact set contained in $\operatorname{Re}(s)> 1/2$.
\end{lemma}
\begin{proof}
This result is essentially due to Sarnak. Indeed, from the proof of Proposition 3.4 in \cite{SAR}
we have
$$\sum_{\substack{c\in \mathcal{O}_K/\{\pm 1\}\\ c\neq 0}} \frac{|\mathcal{S}(\nu,\mu,c)|}{|c|^{2+2\sigma}}
\leq
\frac{|\mathcal{O}_K^{\times}|}{2}
\prod_{\substack{P\subset \mathcal{O}_K\\ \nu\mu\mathcal{D}^2\subseteq P}}\left(1-\mathcal{N}(P)^{-\sigma}\right)^{-1} \prod_{\substack{P\subset \mathcal{O}_K\\ P \neq (0)}}\left(1+2\mathcal{N}(P)^{-\frac{1}{2}-\sigma}+\frac{\mathcal{N}(P)^{-2\sigma}}{1-\mathcal{N}(P)^{-\sigma}}\right),
$$
where $\sigma=\operatorname{Re}(s)$ and the products run over prime ideals $P\subset \mathcal{O}_K$. The infinite product
$$\prod_{\substack{P\subset \mathcal{O}_K\\ P \neq (0)}}\left(1+2\mathcal{N}(P)^{-\frac{1}{2}-\sigma}+\frac{\mathcal{N}(P)^{-2\sigma}}{1-\mathcal{N}(P)^{-\sigma}}\right) $$
converges for $\sigma>1/2$, proving the absolute convergence of $\mathcal{Z}(\nu,\mu,s)$ for $\operatorname{Re}(s)>1/2$. On the other hand, since the function $x\mapsto (1-x^{-\sigma})^{-1}$ is decreasing for $x>1$ and $\mathcal{N}(P)\geq 2$ for any prime ideal $P$, we have
$$\prod_{\substack{P\subset \mathcal{O}_K\\ \nu\mu\mathcal{D}^2\subseteq P}}\left(1-\mathcal{N}(P)^{-\sigma}\right)^{-1}\leq (1-2^{-\sigma})^{-\ell},$$
where $\ell$ is the number of prime ideals dividing $\nu\mu \mathcal{D}^2$. But $\ell\leq 2\omega^{\#}(\mathcal{N}(\nu \mu \mathcal{D}^2))$, where $\omega^{\#}(n)$ is the number of prime divisors of $n\in \mathbb{N}$. It is known that $\omega^{\#}(n)=O(\operatorname{log}(n))$, which gives $(1-2^{-\sigma})^{-\ell} \leq \mathcal{N}(\nu \mu \mathcal{D}^2)^{\alpha}$ for some $\alpha>0$ depending on $\sigma$. Moreover, one can choose $\alpha>0$ such that this bound holds uniformly for $\sigma$ in any fixed compact set contained in $\sigma>1/2$. This implies the desired bound for $|\mathcal{Z}(\nu,\mu,s)|$.
\end{proof}
\begin{lemma}\label{Bbound}
Let $\nu,\mu\in \mathcal{D}^{-1}$ both non zero. Then, the series $\mathcal{B}_{\infty,\infty}(\nu,\mu;s)$
converges absolutely for $\operatorname{Re}(s)>1/2$ and the bound
\begin{align*}
|\mathcal{B}_{\infty,\infty}(\nu,\mu;s)|=
O\left( e^{8\pi \sqrt{|\nu \mu |}}|\nu \mu|^{\operatorname{Re}(s)+1} \right)
\end{align*}
holds uniformly for $s$ in any compact set contained in $\operatorname{Re}(s)>1/2$.
\end{lemma}
\begin{proof}
We start by writing
\begin{align*}
\mathcal{B}_{\infty,\infty}(\nu,\mu;s)=\frac{2\pi}{\operatorname{covol}(\mathcal{O}_K)}\sum_{\substack{c\in \mathcal{O}_K/\{\pm 1\}\\ c\neq 0}}\frac{\mathcal{S}(\nu,\mu,c)}{|c|^2}\,\mathcal J_s\left(\frac{\nu \mu}{c^2}\right).
\end{align*}
For fixed $s\in \mathbb C$ with $\mathrm{Re}(s)>1/2$, Lemma \ref{Lbound} gives
\begin{align*}
\sum_{\substack{c\in \mathcal{O}_K/\{\pm 1\}\\ c\neq 0}}\left|\frac{\mathcal{S}(\nu,\mu,c)}{c^2}\,\mathcal J_s\left(\frac{\nu \mu}{c^2}\right)\right|&=
O(|\nu \mu|^{\operatorname{Re}(s)} \sum_{|\frac{\nu \mu}{c^2}|\leq 1 }\frac{|\mathcal{S}(\nu,\mu,c)|}{|c|^{2+2\operatorname{Re}(s)}} \\
&\mathrel{\phantom{=}} +
|\nu \mu|^{\operatorname{Re}(s)} \sum_{\left|\frac{\nu \mu}{c^2}\right|>
1 }\frac{|\mathcal{S}(\nu,\mu,c)|}{|c|^{2+2\operatorname{Re}(s)}}
e^{8\pi \sqrt{|\nu \mu/c^2 |}}\bigg).
\end{align*}
Since
\begin{align*}
\sum_{\left|\frac{\nu \mu}{c^2}\right|>
1 }\frac{|\mathcal{S}(\nu,\mu,c)|}{|c|^{2+2\operatorname{Re}(s)}} e^{8\pi \sqrt{|\nu \mu/c^2 |}}
\leq e^{8\pi \sqrt{|\nu \mu |}} \cdot \#\left\{c\in \mathcal{O}_K:|c|^2<|\nu \mu|\right\}
=O\left(e^{8\pi \sqrt{|\nu \mu |}}|\nu \mu|\right),
\end{align*}
we have
\begin{align*}
\sum_{\substack{c\in \mathcal{O}_K/\{\pm 1\}\\ c\neq 0}}\left|\frac{\mathcal{S}(\nu,\mu,c)}{c^2}\mathcal J_s\left(\frac{\nu \mu}{c^2}\right)\right|
= O\left(|\nu \mu|^{\operatorname{Re}(s)} \mathcal{Z}(\nu,\mu,\operatorname{Re}(s))+ e^{8\pi \sqrt{|\nu \mu |}}|\nu \mu|^{\operatorname{Re}(s)+1}\right).
\end{align*}
This together with Lemma \ref{contLZseries} implies the absolute convergence of $\mathcal{B}_{\infty,\infty}(\nu,\mu;s)$ and the desired bound for $|\mathcal{B}_{\infty,\infty}(\nu,\mu;s)|$. This completes the proof of this Lemma.
\end{proof}
We now give the analytic continuation of the Niebur--Poincar\'e series.
\begin{proposition}\label{ACNieburPointacare}
The Niebur--Poincar\'e series $F_{\infty,\nu}(P,s)$ has an analytic continuation to $\mathrm{Re}(s)>1/2$. Moreover, for fixed $P\in \mathbb H^3$ and $\delta>1$, the bound
$$F_{\infty,\nu}(P,s)=O_{P,\delta}\left(\max\left\{|\nu|^{\operatorname{Re}(s)}e^{4\pi |\nu|r},|\nu|^{\operatorname{Re}(s)+1}e^{\frac{4\delta \pi |\nu|}{r}}\right\}\right)$$
holds uniformly for $s$ in any compact set contained in $\operatorname{Re}(s)>1/2$.
\end{proposition}
\begin{proof}
By Proposition \ref{FENgeneral}, we have
\begin{align}\label{FENinfinity}
F_{\infty,\nu}( P,s) &= rI_{s}(4\pi |\nu| r) \sum_{u\in \mathcal{O}_K^{\times}/\{\pm 1\} }
e^{2\pi i \operatorname{tr}( \nu u^2z )} +\frac{(2\pi|\nu|)^s}{s\Gamma(s)}r^{1-s}
\varphi_{\infty,\infty}(-\nu;s)\\
& \mathrel{\phantom{=}} + \sum_{\substack{\mu \in \Lambda_{\xi}^{\ast}\\ \mu\neq 0}}
\mathcal{B}_{\infty,\infty}(\nu,\mu;s) rK_s(4\pi |\mu|r) e^{2\pi i \operatorname{tr}(\mu z)}. \nonumber
\end{align}
For $s$ in a fixed compact set in $\mathrm{Re}(s)>1/2$ we have, by Lemma \ref{Bbound} and \eqref{Kasymptotic}, the bound
\begin{eqnarray*}
\sum_{\substack{\mu \in \Lambda_{\xi}^{\ast}\\ \mu\neq 0}}
|\mathcal{B}_{\infty,\infty}(\nu,\mu;s) rK_s(4\pi |\mu|r)| = O\bigg(\sqrt{r}|\nu|^{\sigma+1}\sum_{\substack{\mu\in \mathcal{D}^{-1}\\ \mu\neq 0}}
|\mu|^{\sigma+1/2}e^{8\pi \sqrt{|\nu||\mu|}}e^{-4\pi |\mu|r}\bigg),
\end{eqnarray*}
where $\sigma=\operatorname{Re}(s)$. The inequality
$$
e^{8\pi \sqrt{|\nu| |\mu|}}e^{-4\pi |\mu| r}\leq
e^{\frac{4\delta |\nu| \pi }{r}}e^{-\pi (1-\delta^{-1}) |\mu| r},
$$
which holds for $ \delta>1$, gives
\begin{align}\label{boundNP2}
\sum_{\substack{\mu \in \Lambda_{\xi}^{\ast}\\ \mu\neq 0}}
\left|\mathcal{B}_{\infty,\infty}(\nu,\mu;s) rK_s(4\pi |\mu|r)\right|=O_s\bigg(\sqrt{r}|\nu|^{\sigma+1}e^{\frac{4\delta |\nu| \pi }{r}}
\sum_{\substack{\mu\in \mathcal{D}^{-1}\\ \mu\neq 0}}|\mu|^{\sigma+1/2}e^{-\pi (1-\delta^{-1}) |\mu| r}\bigg).
\end{align}
In particular, the series on the left hand side converges. This, together with the Fourier expansion \eqref{FENinfinity} and the analytic continuation of $\varphi_{\infty,\infty}(-\nu;s)$,
give the analytic continuation of $F_{\infty,\nu}(P,s)$. Now, by the asymptotic bound \eqref{Iinfinity}, we have
\begin{equation}\label{boundconstanttermNP}
rI_{s}(4\pi |\nu| r) =O\left(r^{\operatorname{Re}(s)+1}|\nu|^{\operatorname{Re}(s)}e^{4\pi |\nu|r}\right).
\end{equation}
On the other hand, as mentioned in the Introduction, the function $\varphi_{\infty,\infty}(-\nu;s)$
can be expressed in terms of certain generalized divisors sums and it therefore
has at most polynomial growth with respect to $|\nu|$,
uniformly for $s$ in any fixed compact set contained in $\operatorname{Re}(s)>0$.
This, together with \eqref{boundNP2} and \eqref{boundconstanttermNP}, gives the result on the growth
of $|F_{\infty,\nu}(P,s)|$. This completes the proof of the Proposition.
\end{proof}
We can now state the existence of the meromorphic continuation of $G_s(P,Q)$ together with precise information about its growth at the cusp $\infty$.
\begin{theorem}\label{GreenAC}
For fixed $P,Q\in \mathbb H^3$ with $r=r(P)>\max \{r(Q),r(Q)^{-1}\}$,
the automorphic Green's function $G_s(P,Q)$ has an analytic continuation to
$\mathrm{Re}(s)>1/2$, $s\neq 1$, with a simple pole at $s=1$. Moreover,
we have
\begin{equation}\label{Greenasymptotic}
\lim_{s\to 1}\bigg(G_s(P,Q)-\frac{1}{\operatorname{covol}(\mathcal{O}_K)}E_{\infty}(Q,s)\bigg)=-\frac{1+\operatorname{log}(r)}{\mathrm{vol}(X)}+o\big(1\big),
\end{equation}
as $r\to\infty$.
\end{theorem}
Note that the analyticity of $G_s(P,Q)$ for $\operatorname{Re}(s)>\frac{1}{2}$, $s\neq 1$, is equivalent to Sarnak's lower bound for the first ``exceptional'' discrete eigenvalue of the Laplacian on $X$ (\cite{SAR}, Theorem 3.1).
\begin{proof}
Let $P,Q\in \mathbb H^3$ and assume that $r=r(P)>\max\{r(Q),r(Q)^{-1}\}$. By Theorem \ref{prop-fourier-expansion-greens}, we have
\begin{equation*}
G_s(P,Q)=\frac{1}{\operatorname{covol}(\mathcal{O}_K)}\bigg(
\frac{r^{1-s}}{s}E_{\infty}(Q,s)+2\sum_{\substack{\mu \in \mathcal{D}^{-1}\\ \mu\neq 0}} F_{\infty,-\mu}(Q,s) \, r K_{s}(4\pi |\mu| r) e^{2\pi i \operatorname{tr} (\mu z )}\bigg).
\end{equation*}
By Proposition \ref{ACNieburPointacare}, together with the asymptotic bound \eqref{Kasymptotic}, we have
$$\sum_{\substack{\mu \in \mathcal{D}^{-1}\\ \mu\neq 0}} |F_{\infty,-\mu}(Q,s) \, r K_{s}(4\pi |\mu| r)|=O_{Q,\delta}\bigg(\sqrt{r} \sum_{\substack{ \mu \in \mathcal{D}^{-1}\\ \mu\neq 0}}|\mu|^{\operatorname{Re}(s)-\frac{1}{2}} e^{-4\pi |\mu|r} \max\left\{e^{4\pi |\mu|r(Q)},|\mu|e^{\frac{4\delta \pi |\mu|}{r(Q)}}\right\}\bigg)$$
for any $\delta>1$, uniformly for $s$ in any compact set contained in $\operatorname{Re}(s)>1/2$. Choosing $\delta$ such that $r>\max\{r(Q),\delta r(Q)^{-1}\}$ we conclude that the series on the left hand side is convergent. This proves that $G_s(P,Q)$ has a meromorphic continuation to $\operatorname{Re}(s)>1/2$ having poles only where $E_{\infty}(P,s)$ has poles, in which case the multiplicities also agree. Since $E_{\infty}(P,s)$ admits an analytic continuation to $\operatorname{Re}(s)>0$, $s\neq 1$, with a simple pole at $s=1$, we conclude the same for $G_s(P,Q)$. Now, we note that the above computations also give
\begin{equation}\label{Greenasymp1}
\lim_{s\to 1}\bigg(G_s(P,Q)-\frac{r^{1-s}}{\operatorname{covol}(\mathcal{O}_K)s}E_{\infty}(Q,s)\bigg)=o\big(1\big),
\end{equation}
as $r\to\infty$.
A straight-forward computation using \eqref{ResEisInfinity} gives
\begin{equation}\label{Eisdifference}
\lim_{s\to 1}\bigg(\frac{r^{1-s}}{\operatorname{covol}(\mathcal{O}_K)s}E_{\infty}(Q,s)-\frac{1}{\operatorname{covol}(\mathcal{O}_K)}E_{\infty}(Q,s)\bigg)=-\frac{1+\operatorname{log}(r)}{\operatorname{vol}(X)}.
\end{equation}
Formula \eqref{Greenasymptotic} follows by combining \eqref{Greenasymp1} with \eqref{Eisdifference}. This completes the proof of the Theorem.
\end{proof}
\section{Proof of the main theorem}\label{section_proofofmainthm}
To prove our main theorem, we first introduce a building block for the class of functions
in $\mathcal{A}$. More precisely, for $P,Q\in \mathbb{H}^3$ with $P\not=\gamma Q$ for any $\gamma\in\Gamma$,
we define
\begin{align}\label{def_analogj}
\mathcal L(P,Q):=\lim_{s\to 1}\left(
G_s(P,Q)-\frac{1}{\operatorname{covol}(\mathcal{O}_K)}
\Big(E_{\infty}(Q,s)
+
E_{\infty}(P,s)
-
\varphi_{\infty,\infty}(0;s) \Big)\right).
\end{align}
Recalling \eqref{Greenpole} and \eqref{resEis},
the above limit exists. This function can be seen as the analogue of $\operatorname{log}|j(\tau_1)-j(\tau_2)|$ (see Proposition 5.1 in \cite{GZ}).
The next lemma summarizes the properties of the function $\mathcal L(P,Q)$.
\begin{lemma}\label{lemm_properties_J}
The function $\mathcal L(P,Q)$ satisfies the following properties:
\begin{enumerate}
\item[$(\L1)$] The function $\mathcal L(P,Q)$ is $\Gamma$-invariant in each variable
and can therefore be considered as a function on $X\times X$. Moreover, we have $\mathcal L(P,Q)=\mathcal L(Q,P)$.
\item[$(\L2)$] For fixed $Q\in X$,
we have a singularity of the form
\begin{equation*}
\mathcal L(P,Q)= \frac{\nu(Q)}{2\pi} \frac{ r(Q) }{ \|P-Q\|}+O_Q(1),
\end{equation*}
as $P \to Q$, and the function $P\mapsto \mathcal L(P,Q)$ is smooth at any point $P\in X$ with $P\neq Q$.
\item[$(\L3)$] For $P,Q\in X$ with $P\neq Q$, we have $\Delta_{P}\mathcal L(P,Q)=0$.
\item[$(\L4)$] For fixed $Q\in X$, we have
$$\mathcal L(P,Q)=-\frac{1}{\mathrm{vol}(X)}-\frac{|\mathcal{O}_K^{\times}|}{2\operatorname{covol}(\mathcal{O}_K)}r^2+o(1),$$
as $r=r(P)\to \infty$.
\end{enumerate}
\end{lemma}
\begin{proof}
Properties $(\L1)$, $(\L2)$, and $(\L3)$ follow from properties $(G1)$, $(G2)$, and $(G3)$ of the Green's function $G_s(P,Q)$ together with the equality
$$\frac{1}{d(P,Q)}=\frac{r(Q)}{\|P-Q\|}+O_Q(1),$$
as $P\to Q$.
In order to prove property $(\L4)$, we consider the Fourier expansion
\eqref{equ_fourier_expansion_series_eis_gen}
of $E_{\infty}(P,s)$, namely the equality
\begin{align*}
E_{\infty}(P,s)&=
\frac{|\mathcal{O}_K^{\times}|}{2}r^{1+s}
+
\varphi_{\infty,\infty}(0;s)\,r^{1-s}
+
\frac{2^{1+s}\pi^{s}}{\Gamma(s)}
\sum_{\substack{\mu \in \mathcal{D}^{-1}\\ \mu\neq 0}}|\mu|^{s}
\varphi_{\infty,\infty}(\mu;s)\,r K_{s}(4\pi |\mu|r)e^{2\pi i \operatorname{tr}(\mu z)},
\end{align*}
where we employed the identity $[\Gamma_{\infty}:\Gamma_{\infty}^{\prime}]=|\mathcal{O}_K^{\times}|/2$.
A straight-forward computation using \eqref{resEis} gives
\begin{align*}
\lim_{s\to 1}\varphi_{\infty,\infty}(0;s)(r^{1-s}-1)=-\frac{\operatorname{covol}(\mathcal{O}_K)}{\operatorname{vol}(X)}\operatorname{log}(r).
\end{align*}
Since $|\varphi_{\infty,\infty}(\mu;1)|$ is of at most polynomial growth in $|\mu|$ and $K_1(r)$ has exponential
decay as $r\to \infty$, we therefore get
\begin{align*}
\lim_{s\to 1}\bigg(E_{\infty}(P,s)-\varphi_{\infty,\infty}(0;s)\bigg)
=\frac{|\mathcal{O}_K^{\times}|}{2}r^2-\frac{\operatorname{covol}(\mathcal{O}_K)}{\operatorname{vol}(X)}\operatorname{log}(r)+o(1),
\end{align*}
as $r\to \infty$. Property $(\L4)$ now follows from this together with \eqref{Greenasymptotic}.
This completes the proof of the Lemma.
\end{proof}
The function $\mathcal L(P,Q)$ is a building block for functions in $\mathcal A$. More precisely, we have the following proposition which can be seen as an analogue of \eqref{expression}.
\begin{proposition}\label{thm_block}
Let $F:\mathbb H^3\to \mathbb R\cup\{\infty\}$ be in $\mathcal{A}$, the class of functions
satisfying $(\mathcal{A}1)$--$(\mathcal{A}4)$.
Then, the limit $F(\infty):=\lim_{r\to \infty}F(P)$ exists and we have the equality
\begin{align*}
F(P)=F(\infty)+2\pi \sum_{\ell=1}^{m}
c_{\ell}\,\mathcal L(P,Q_{\ell}),
\end{align*}
for any $P\in X$ with $P\neq Q_{\ell}$, for $\ell=1,\ldots ,m$.
\end{proposition}
\begin{proof}
Let us define $\widetilde{F}(P)$, for $P\in X$ with $P\neq Q_{\ell}$, for $\ell=1,\ldots ,m$, by
$$\widetilde{F}(P)=F(P)-2\pi \sum_{\ell=1}^m c_{\ell}\,\mathcal L(P,Q_{\ell}).$$
By properties $(\mathcal{A}3)$ and $(\L3)$, we have that $\Delta \widetilde{F}(P)=0$ for $P\neq Q_{\ell}$, $\ell=1,\ldots ,m$. On the other hand, properties $(\mathcal{A}2)$ and $(\L2)$ imply that $\widetilde{F}(P)$ is locally bounded around any point in $X$. This implies that $\widetilde{F}(P)$ extends to a smooth function $\widetilde{F}:X\to \mathbb R$ satisfying $\Delta \widetilde{F}(P)=0$ everywhere. Indeed, by taking geodesic normal coordinates around any point, one can reduce the problem to the case where $\widetilde{F}(P)$ is a harmonic function with respect to the euclidean Laplacian, at least locally. The existence of the harmonic extension of $\widetilde{F}(P)$ then follows from Theorem 2.3 in \cite{ABW}.
Using $ \sum_{\ell=1}^m c_{\ell}=0$, we note that
\begin{align*}
\sum_{\ell=1}^{m}
c_{\ell}\,\mathcal L(P,Q_{\ell}) &= \sum_{\ell=1}^m c_{\ell}\lim_{s\to 1}\left(
G_s(P,Q_{\ell})-\frac{1}{\operatorname{covol}(\mathcal{O}_K)}
E_{\infty}(Q_{\ell},s) \right)\\
&= \sum_{\ell=1}^m c_{\ell}\lim_{s\to 1}\left(G_s(P,Q_{\ell})-\frac{2}{\operatorname{vol}(X)(s^2-1)}\right)\\
&\mathrel{\phantom{=}}+\sum_{\ell=1}^m c_{\ell}\lim_{s\to 1}\left(\frac{2}{\operatorname{vol}(X)(s^2-1)}-\frac{1}{\operatorname{covol}(\mathcal{O}_K)} E_{\infty}(Q_{\ell},s)\right)\\
&= \sum_{\ell=1}^m c_{\ell}\lim_{s\to 1}\left(G_s(P,Q_{\ell})-\frac{2}{\operatorname{vol}(X)(s^2-1)}\right)
+\frac{1}{\operatorname{vol}(X)}\sum_{\ell=1}^m c_{\ell}\operatorname{log}(\eta_{\infty}(Q_{\ell}) r_{\ell}).
\end{align*}
As mentioned in Section \ref{Green-subsection}, the function
$$P\mapsto \lim_{s\to 1}\left(G_s(P,Q_{\ell})-\frac{2}{\operatorname{vol}(X)(s^2-1)}\right)$$
is square-integrable on $X$, for fixed $Q_{\ell}$. This implies that the function
$$P\mapsto \sum_{\ell=1}^{m}c_{\ell}\,\mathcal L(P,Q_{\ell})$$
is also square-integrable. By property $(\mathcal{A}4)$, we conclude
that $\widetilde{F}(P)$ is square-integrable over $X$. By Theorem 4.1.8 in \cite{EGM}, p.~140, we know that any smooth, harmonic, square-integrable function on $X$ is constant. We conclude that $\widetilde{F}(P)$ is constant. Finally, using $(\L4)$ together with $\sum_{\ell=1}^m c_{\ell}=0$, we have
\begin{align*}
\sum_{\ell=1}^mc_{\ell}\,\mathcal L(P,Q_{\ell})=o(1),
\end{align*}
as $r\to \infty$.
We conclude that $\widetilde{F}(P)=F(\infty)$. This proves the result.
\end{proof}
We now prove our main theorem.
\begin{proof}[Proof of Theorem \ref{mainth}]
Let $F:\mathbb H^3\to \mathbb R\cup\{\infty\}$ be a function in the class $\mathcal{A}$ satisfying the properties
$(\mathcal{A}1)$--$(\mathcal{A}4)$. By Proposition \ref{thm_block} we have
\begin{align*}
\frac{1}{\operatorname{vol}(X)}\int_XF(P)d\mu(P)=
F(\infty)+\frac{2\pi}{\operatorname{vol}(X)} \int_X\sum_{\ell=1}^{m}
c_{\ell}\,\mathcal L(P,Q_{\ell})d\mu(P).
\end{align*}
Since $\sum_{\ell}c_{\ell}=0$, we have
\begin{align*}
\int_X \sum_{\ell=1}^{m}
c_{\ell}\,\mathcal L(P,Q_{\ell})d\mu(P)&=\int_X \lim_{s\to 1}\sum_{\ell=1}^m c_{\ell}\bigg(G_s(P,Q_{\ell})-\frac{1}{\operatorname{covol}(\mathcal{O}_K)}E_{\infty}(Q_{\ell},s)\bigg)d\mu(P)\\
&=\int_X\sum_{\ell=1}^m c_{\ell}\lim_{s\to 1}\left(G_s(P,Q_{\ell})-\frac{2}{\operatorname{vol}(X)(s^2-1)}\right)d\mu(P)\\
& \mathrel{\phantom{=}} +\int_X\sum_{\ell=1}^m c_{\ell}\lim_{s\to 1}\left(\frac{2}{\operatorname{vol}(X)(s^2-1)}-\frac{1}{\operatorname{covol}(\mathcal{O}_K)} E_{\infty}(Q_{\ell},s)\right)d\mu(P).
\end{align*}
Using \eqref{G-orthogonal} and \eqref{KLF}, we obtain
\begin{align*}
\int_X \sum_{\ell=1}^{m}
c_{\ell}\,\mathcal L(P,Q_{\ell})d\mu(P)&=
\lim_{s\to 1}\sum_{\ell=1}^m c_{\ell}\bigg(\frac{2}{s^2-1}-\frac{\operatorname{vol}(X)}{\operatorname{covol}(\mathcal{O}_K)}E_{\infty}(Q_{\ell},s)\bigg)\\
&= \sum_{\ell=1}^{m}c_{\ell} \operatorname{log}(\eta_{\infty}(Q_{\ell})\, r_{\ell}).
\end{align*}
This completes the proof of Theorem \ref{mainth}.
\end{proof}
\section{Technical lemmas}\label{section-technical-lemma}
In this section we prove two lemmas that were used
in Section \ref{section-Fourier-expansions} for the computation
of the Fourier coefficients of the Green's function and of the
Niebur--Poincar\'e series.
\begin{lemma}\label{Iintegral}
For $\mu,s\in \mathbb C$ with $\operatorname{Re}(s)>0$ and $r>\tilde{r}>0$, let
\begin{align*}
I_{\mu,s}(r,\tilde{r}):=\int\limits_{\mathbb C}\varphi_s\left(\frac{|z|^2+r^2+\tilde{r}^2}{2r\tilde{r}}\right)e^{-2\pi i \operatorname{tr} (\mu z )}dz
\end{align*}
with $\varphi_s(t)=(t+\sqrt{t^2-1})^{-s}(t^2-1)^{-1/2}$. Then, we have
\begin{align*}
I_{\mu,s}(r,\tilde{r})=
\begin{cases}
2\pi s^{-1}r^{1-s}\tilde{r}^{s+1}, & \text{ if }\mu= 0,\\
4\pi r\tilde{r} K_{s}(4\pi |\nu|r) I_{s}(4\pi |\nu|\tilde{r}), & \text{ if }\mu \neq 0.
\end{cases}
\end{align*}
\end{lemma}
\begin{proof}
Using polar coordinates $z=\rho e^{i\theta}$, we get
\begin{align*}
I_{\mu,s}(r,\tilde{r})
&=\int_{0}^{2\pi}\int_0^{\infty}
\varphi_s\left(\frac{\rho^2+r^2+\tilde{r}^2}{2r\tilde{r}}\right)
e^{-2\pi i \rho \operatorname{tr}( \mu e^{i\theta} )}\rho\, d\rho d\theta.
\end{align*}
Letting $t=\rho^2$ and $f(t):=(t+r^2+\tilde{r}^2)/2r\tilde{r}$, we have
\begin{align*}
I_{0,s}(r,\tilde{r})
=\pi \int_0^{\infty}\varphi_s\left(f(t)\right)dt
=-\frac{ 2\pi r\tilde{r}}{s} \left[ \left (f(t)+\sqrt{f(t)^2-1}\right)^{-s}
\right]_{t=0}^{t=\infty}
=\frac{ 2 \pi }{s}r^{1-s}\tilde{r}^{s+1},
\end{align*}
where we have used that $r>\tilde{r}$. This proves the first formula. For $\mu\not=0$, we write $\mu=|\mu|e^{i\alpha}$ and get
\begin{align*}
\int_{0}^{2\pi}e^{-2\pi i \rho \operatorname{tr}( \mu e^{i\theta})} d\theta
=\int_{0}^{2\pi}e^{-4\pi i \rho |\mu|\sin(\theta) } d\theta
= 2\pi J_0(4\pi |\mu|\rho ),
\end{align*}
by using formula \eqref{A1}. Replacing this in the above formula for $I_{\mu,s}(r,\tilde{r})$ and
making the change of variables $t=\rho^2/r^2$,
we get
\begin{align*}
I_{\mu,s}(r,\tilde{r})
&=\pi r^2\int_0^{\infty}
\varphi_s\left(\frac{r}{2\tilde{r}} \left(t+1+\frac{\tilde{r}^2}{r^2}\right)\right)
J_0(4\pi |\mu| r \sqrt{t})dt.
\end{align*}
Using formula \eqref{A2}, we have
\begin{align*}
\varphi_s(b/a)=a\,
\int_0^{\infty}I_s(a u)e^{-b u}du,
\end{align*}
for $b>a>0$.
Using this identity with $a=2\tilde{r}/r$ and $b=t+1+\frac{\tilde{r}^2}{r^2}$,
we get
\begin{align*}
I_{\mu,s}(r,\tilde{r})
&=2\pi r\tilde{r} \int_0^{\infty}I_s\left(\frac{2\tilde{r}u}{r}\right)e^{-\left(1+\frac{\tilde{r}^2}{r^2}\right) u}
\int_0^{\infty}J_0(4\pi |\mu| r \sqrt{t})e^{-t u}dtdu.
\end{align*}
Formula \eqref{A3} with $a=u$ and $b=4\pi|\mu|r$ yields
\begin{align*}
I_{\mu,s}(r,\tilde{r})
&=2\pi r\tilde{r} \int_0^{\infty}I_s\left(\frac{2\tilde{r}u}{r}\right)e^{-\left(1+\frac{\tilde{r}^2}{r^2}\right) u}
e^{-4\pi^2 |\mu|^2 r^2/u}
\frac{du}{u}.
\end{align*}
Next, we make the change of variables $t=8\pi^2 |\mu|^2 r^2/u$
and we get
\begin{align*}
I_{\mu,s}(r,\tilde{r})
&=2\pi r\tilde{r} \int_0^{\infty}
I_s\left(\frac{a b}{t}\right) e^{-\frac{a^2+b^2}{2 t} }e^{-t/2}
\frac{dt}{t},
\end{align*}
with $a=4\pi |\mu| r$ and $b=4\pi |\mu| \tilde{r}$.
Observing that $a>b>0$ and using \eqref{A4} we conclude
\begin{align*}
I_{\mu,s}(r,\tilde{r})=4\pi r\tilde{r} K_{s}(a) I_{s}(b).
\end{align*}
This completes the proof of the Lemma.
\end{proof}
\begin{lemma}\label{Gintegral}
For $\nu,\mu,c\in \mathbb C$ with $\nu,c$ both non zero, $r>0$, and $s\in \mathbb C$ with $\operatorname{Re}(s)>0$, put
\begin{align*}
\mathcal I(r,\nu,\mu,c)=\int_{\mathbb C}\frac{r}{|c|^2(|z|^2+r^2)}I_s\left(\frac{4\pi |\nu|r}{|c|^2(|z|^2+r^2)}\right)e^{-2\pi i \operatorname{tr}\left(\nu\frac{\overline{z}}{c^2(|z|^2+r^2)}+\mu z\right)}dz.
\end{align*}
Then, we have
\begin{align*}
\mathcal I(r,\nu,\mu,c)=
\begin{cases}
\dfrac{\pi^{1+s}2^s|\nu|^s}{|c|^{2(1+s)}s\Gamma(1+s)}\,r^{1-s}, & \text{ if }\mu =0,\\[0.5cm]
\dfrac{2\pi}{|c|^2}\,rK_s(4\pi |\mu |r)\,\mathcal J_s\left(\frac{\nu \mu}{c^2}\right), & \text{ if }\mu \neq 0,
\end{cases}
\end{align*}
where $\mathcal J_s(z)$ is given in \eqref{def_Lniebur}.
\end{lemma}
\begin{proof}
We start with the case $\mu=0$. Using polar coordinates $z=\rho e^{i\theta}$, we get
\begin{align*}
\mathcal I(r,\nu,0,c)=\frac{r}{|c|^2}\int_0^{\infty}\int_0^{2\pi}\frac{\rho}{\rho^2+r^2}I_s\left(\frac{4\pi |\nu| r}{|c|^2(\rho^2+r^2)}\right)e^{-\frac{4\pi i |\nu| \rho}{|c|^2(\rho^2+r^2)}\cos(\theta)}d\theta d\rho.
\end{align*}
Using formula \eqref{A1} and making the change of variables $\xi=\rho/r$,
we get
$$\mathcal I(r,\nu,0,c)=\frac{2\pi r}{|c|^2}\int_0^{\infty}\frac{\xi}{\xi^2+1}I_s\left(\frac{4\pi |\nu| }{|c|^2r(\xi^2+1)}\right)J_0\left(\frac{4\pi |\nu| \xi}{|c|^2r(\xi^2+1)}\right)d\xi. $$
Using Lemma \ref{IsJ0Integral} with $a=\frac{4\pi |\nu| }{|c|^2r}$, we get
$$\mathcal I(r,\nu,0,c)=\frac{\pi^{1+s}2^s|\nu|^s}{|c|^{2(1+s)}s\Gamma(1+s)}\,r^{1-s}.$$
This proves the first formula. For $\mu\neq 0$, we start by writing
$$|\nu|\mathcal I\left(\frac{r}{|\mu|},\nu,\mu,c\right)=\int_{\mathbb C}\frac{|\beta|r}{|\mu z|^2+r^2}I_s\left(\frac{4\pi |\beta|r}{|\mu z|^2+r^2}\right)e^{-2\pi i \operatorname{tr}\left(\beta \frac{\overline{z\mu }}{(|z\mu |^2+r^2)}\right)-2\pi i \operatorname{tr}(\mu z)}dz$$
with $\beta=\frac{\nu \mu}{c^2}$.
Making the change of variables $\xi =\mu z$ and using polar coordinates $\xi=\rho e^{i\theta}$,
we obtain
$$|\nu|\mathcal I\left(\frac{r}{|\mu|},\nu,\mu,c\right)=\frac{|\beta| r}{|\mu|^2}\int_{0}^{\infty}\frac{\rho}{\rho^2+r^2}
I_s\left(\frac{4\pi |\beta|r}{\rho^2+r^2}\right)\int_0^{2\pi}e^{-2\pi i \left( \frac{\beta \rho e^{-i\theta}+\overline{\beta}\rho e^{i\theta} }{\rho^2+r^2}+2\rho \cos(\theta)\right)}d\theta d\rho.$$
Now, we compute
\begin{align*}
\int_0^{2\pi}e^{-2\pi i \left( \frac{\beta \rho e^{-i\theta}+\overline{\beta}\rho e^{i\theta} }{\rho^2+r^2}+2\rho \cos(\theta)\right)}d\theta
&= \int_0^{2\pi}e^{-4\pi i \frac{\rho}{\rho^2+r^2}|\beta+\rho^2+r^2|\sin(\theta)}d\theta \\
&= 2\pi J_0\left(\frac{4\pi \rho}{\rho^2+r^2}|\beta+\rho^2+r^2|\right),
\end{align*}
by formula \eqref{A1}. We conclude
\begin{align*}
|\nu|\mathcal I\left(\frac{r}{|\mu|},\nu,\mu,c\right)=\frac{2\pi |\beta| r}{|\mu|^2}\int_{0}^{\infty}\frac{\rho}{\rho^2+r^2}
I_s\left(\frac{4\pi |\beta|r}{\rho^2+r^2}\right)
J_0\left(\frac{4\pi \rho}{\rho^2+r^2}|\beta+\rho^2+r^2|\right)d\rho.
\end{align*}
Making the change of variables $t=\rho/r$, we get
\begin{align*}
|\nu|\mathcal I\left(\frac{r}{|\mu|},\nu,\mu,c\right)
&=\frac{2\pi |\beta| r}{|\mu|^2}\int_{0}^{\infty}\frac{t}{t^2+1}I_s\left(\frac{4\pi |\beta|}{r(t^2+1)}\right)
J_0\left(\frac{4\pi r t}{t^2+1}\left|\frac{\beta}{r^2}+t^2+1\right|\right)dt.
\end{align*}
Now, by formula \eqref{JmultiplicationTh} with $\lambda=\bigl|\frac{\beta}{r^2(t^2+1)}+1\bigr|$,
$z=4\pi rt$, and $s=0$,
we get
\begin{align*}
J_0\left(\frac{4\pi rt}{t^2+1}\left|\frac{\beta}{r^2}+t^2+1\right|\right)
= \sum_{k=0}^{\infty}\frac{(-1)^k}{k!}\frac{1}{(t^2+1)^{2k}}\left(\frac{|\beta|^2}{r^4}
+\frac{\operatorname{tr}(\beta)}{r^2} (t^2+1)\right)^k (2\pi r t)^k J_{k}(4\pi r t).
\end{align*}
Therefore, using the binomial theorem, we have
\begin{align*}
&|\nu|\mathcal I\left(\frac{r}{|\mu |},\nu,\mu,c\right)=\\
&\frac{2\pi |\beta| r}{|\mu|^2}\sum_{k=0}^{\infty}\frac{(-1)^k(2\pi)^k}{k!}
\sum\limits_{j=0}^{k} {k\choose j }
\frac{|\beta|^{2j} \operatorname{tr}(\beta)^{k-j}}{r^{k+2j}}
\int_{0}^{\infty}\frac{t^{k+1} }{(t^2+1)^{k+j+1}}
I_s\left(\frac{4\pi |\beta|}{r(t^2+1)}\right)
J_{k}(4\pi r t)
dt.
\end{align*}
Using the power expansion for $I_s(z)$ given in formula \eqref{Iseries}, we have
\begin{align*}
&\int_{0}^{\infty}\frac{t^{k+1} }{(t^2+1)^{k+j+1}}
I_s\left(\frac{4\pi |\beta|}{r(t^2+1)}\right)
J_{k}(4\pi r t)
dt
\\
&=\left(\frac{2\pi |\beta|}{r}\right)^s
\sum_{\ell=0}^{\infty}\frac{(2\pi|\beta|)^{2\ell}r^{-2\ell}}{\ell!\,\Gamma(s+\ell+1)}\int_{0}^{\infty}\frac{t^{k+1} }{(t^2+1)^{s+k+j+2\ell+1}}
J_{k}(4\pi r t)dt
\\
&=\left(\frac{2\pi |\beta|}{r}\right)^s
\sum_{\ell=0}^{\infty}\frac{(2\pi|\beta|)^{2\ell}r^{-2\ell}}{\ell!\,\Gamma(s+\ell+1)}
\frac{(2\pi r)^{s+k+j+2\ell}K_{s+j+2\ell}(4\pi r)}{\Gamma(s+k+j+2\ell+1)},
\end{align*}
by formula \eqref{A5} with $s=k$, $\mu=s+k+j+2\ell$, and $a=4\pi r$. This gives
\begin{align*}
\frac{|\nu| |\mu|^2}{2\pi |\beta| r}\mathcal I\left(\frac{r}{|\mu|},\nu,\mu,c\right)
&=\sum_{k=0}^{\infty}
\sum\limits_{j=0}^{k}\sum_{\ell=0}^{\infty}
\frac{(-1)^k(2\pi)^{2s+2k+j+4\ell}|\beta|^{s+2j+2\ell} \operatorname{tr}(\beta)^{k-j} K_{s+j+2\ell}(4\pi r)}
{(k-j)!j!\ell!\,\Gamma(s+\ell+1)\Gamma(s+k+j+2\ell+1)\,r^{j}}\\
&=\sum_{k=0}^{\infty}
\sum\limits_{j=0}^{\infty}\sum_{\ell=0}^{\infty}
\frac{(-1)^{k+j}(2\pi)^{2s+2k+3j+4\ell}|\beta|^{s+2j+2\ell} \operatorname{tr}(\beta)^{k} K_{s+j+2\ell}(4\pi r)}{k!j!\ell!\,\Gamma(s+\ell+1)\Gamma(s+k+2j+2\ell+1)\,r^{j}},
\end{align*}
using a well-known identity for the double sum over $k$ and $j$.
Applying this identity again for the double sum over $j$ and $\ell$
and using Lemma \ref{Krecursion} with $z=4\pi r$, we get
\begin{align*}
\frac{|\nu| |\mu|^2}{2\pi |\beta| r}\mathcal I\left(\frac{r}{|\mu|},\nu,\mu,c\right)
&=\sum_{k=0}^{\infty}
\sum\limits_{j=0}^{\infty}
\frac{(-1)^{k+j}(2\pi)^{2s+2k+3j}|\beta|^{s+2j} \operatorname{tr}(\beta)^{k}}
{k!j!\,\Gamma(s+k+2j+1)r^{j}}
\sum_{\ell=0}^{j}
{j\choose \ell }
\frac{(-2\pi r)^{\ell} K_{s+j+\ell}(4\pi r)}
{\Gamma(s+\ell+1)}\\
&= K_s(4\pi r)\sum_{j=0}^{\infty}
\frac{(2\pi)^{2s+4j}|\beta|^{s+2j} }
{j!\,\Gamma(s+j+1)}
\sum_{k=0}^{\infty}
\frac{(-1)^{k}(2\pi)^{2k}\operatorname{tr}(\beta)^{k}}{k!\,\Gamma(s+k+2j+1)}.
\end{align*}
Assuming that $\operatorname{tr}(\beta)>0$, using formula \eqref{Jseries} and
recalling that $\beta=\frac{\nu \mu}{c^2}$, we therefore obtain
\begin{align*}
\mathcal I\left(r,\nu,\mu,c\right)
&=
\frac{2\pi }{|c|^2}r K_s(4\pi |\mu|r)\sum_{j=0}^{\infty}
\frac{\bigl((2\pi)|\beta|/\sqrt{\operatorname{tr}(\beta)}\bigr)^{s+2j} }
{j!\,\Gamma(s+j+1)}
J_{s+2j}\left(4\pi \sqrt{\operatorname{tr}(\beta)}\right).
\end{align*}
Applying Lemma \eqref{Jproductlemma} with $x=4\pi \sqrt{\operatorname{tr}(\beta)}$ and $A=\sqrt{\beta/\operatorname{tr}(\beta)}$,
we get
\begin{align*}
\mathcal I\left(r,\nu,\mu,c\right)
&=
\frac{2\pi }{|c|^2}
r K_s(4\pi |\mu|r)J_s\left(4\pi\sqrt{\beta}\right)J_s\left(4\pi\sqrt{\overline{\beta}}\right)\\
&=\frac{2\pi }{|c|^2}r
K_s(4\pi |\mu|r)\,\mathcal J_s\left(\frac{\nu \mu}{c^2}\right).
\end{align*}
This implies the second formula in the case $\mu\neq 0, \operatorname{Re}\left(\frac{\nu \mu}{c^2}\right)>0$.
The case $\operatorname{Re}\left(\frac{\nu \mu}{c^2}\right)<0$ is completely analogous, so we omit the details. Finally, the case $\operatorname{Re}\left(\frac{\nu \mu}{c^2}\right)=0$ follows from any of the two other cases by taking the limit $\beta\to it, t\in \mathbb R,t\neq 0$. This completes the proof of the Lemma.
\end{proof}
|
1,108,101,566,783 | arxiv | \chapter{COLLECTIVE DYNAMICS OF LOHE TYPE AGGREGATION MODELS}
\markboth{Seung-Yeal Ha and Dohyun Kim}{Collective dynamics of Lohe type aggregation models}
\author{Seung-Yeal Ha}
\address{Department of Mathematical Sciences and Research Institute of Mathematics\\
Seoul National University, Seoul 08826, and \\
Korea Institute for Advanced Study, \\
Hoegiro 85, Seoul 02455, Republic of Korea\\
[email protected]}
\author{Dohyun Kim}
\address{School of Mathematics, Statistics and Data Science\\
Sungshin Women's University\\
Seoul 02844, Republic of Korea \\
[email protected]}
\begin{abstract}
In this paper, we review state-of-the-art results on the collective behaviors for Lohe type first-order aggregation models. Collective behaviors of classical and quantum many-body systems have received lots of attention from diverse scientific disciplines such as applied mathematics, control theory in engineering, nonlinear dynamics of statistical physics, etc. To model such collective dynamics, several phenomenological models were proposed in literature and their emergent dynamics were extensively studied in recent years. Among them, we present two Lohe type models: {\it the Lohe tensor(LT) model} and {\it the Schr\"odinger-Lohe(SL) model}, and present several sufficient conditions in unified frameworks via the Lyapunov functional approach for state diameters and dynamical systems theory approach for two-point correlation functions. We also present several numerical simulation results for the SL model.
\end{abstract}
\section{Introduction}
\setcounter{equation}{0}
The purpose of this paper is to continue a recent review \cite{H-K-P-Z} on the collective behaviors of classical and quantum synchronization (or aggregation) models. From the beginning of this century, collective behaviors of many-body systems have received a lot of attention from various scientific disciplines, e.g., synchronization and swarming behaviors in biology \cite{B-B,Pe,T-B-L,T-B,Wi1,Wi2}, decentralized control in engineering \cite{L-P-L-S,P-L-S,Re}, non-convex stochastic optimization algorithms in machine learning community \cite{C-C-T-T,C-J-L-Z,F-H-P-S1,F-H-P-S2,K-T,K-E,P-T-T-M}, etc. Mathematical modeling of such collective behaviors was begun by several pioneers, Winfree \cite{Wi2}, Kuramoto \cite{Ku2} and Vicsek \cite{Vi} whose works have established milestones of collective dynamics as major research subjects in applied mathematics, control theory, statistical physics, etc. Since then, several phenomenological models have been proposed, to name a few, the Cucker-Smale model \cite{C-S}, the swarm sphere model \cite{C-C-H,C-H1,C-H2,C-H4,H-K-L-N,O1}, matrix aggregation models \cite{B-C-S,D-F-M-T,D,G-H,H-K2,Lo2,Lo3}, etc. We also refer the reader to survey articles \cite{A-B,A-B-F,D-B1,H-K-P-Z,M-T,P-R-K,St,VZ,Wi2} for a brief introduction to collective dynamics.
In what follows, we are mainly interested in the Lohe type aggregation models (the LT model and the SL model). The LT model is a finite-dimensional aggregation model on the space of tensors with the same rank and size, and it encompasses previously introduced first-order aggregation models, whereas the SL model is an infinite-dimensional toy model describing synchronous behaviors in a quantum regime. In fact, Lohe first focused on the similarity between classical and quantum synchronizations and proposed a merely phenomenological model to capture common properties between two emergent behaviors in different regimes. However, when we focus on the dissimilarity between classical and quantum systems, one encounters several limitations of the SL model due to its quantum nature, namely \textit{entanglement} which is a genuine quantum feature and is not observed in the classical world. We here omit detailed descriptions on the relation between quantum synchronization and entanglement which is beyond our scope.
In this paper, we briefly review state-of-the-art results on the collective dynamics of the aforementioned two Lohe type aggregation models from a universal platform for collective behaviors, Lyapunov functional approach and dynamical systems theory approach.
The rest of the paper is organized as follows. In Section \ref{sec:2}, we introduce two Lohe type aggregation models, namely the LT model and the SL model. In Section \ref{sec:3}, we review state-of-the-art results on the emergent dynamics of the LT model for homogeneous and heterogeneous ensembles by providing several sufficient frameworks for the complete state aggregation and practical aggregation. In Section \ref{sec:4}, we review parallel results for the SL model compared to the LT model in Section \ref{sec:3}. Finally, Section \ref{sec:5} is devoted to a brief summary and discussion on some remaining interesting issues for a future direction.
\vspace{1cm}
\noindent {\bf Notation}: Throughout the paper, we will see several models. As long as there are no confusion, we use handy acronyms for such models:
\begin{align*}
\begin{aligned}
& \mbox{LT:~ Lohe tensor}, \quad \mbox{SL:~Schr\"{o}dinger-Lohe}, \quad \mbox{LM:~ Lohe matrix,} \\
& \mbox{LHS:~ Lohe hermitian sphere}, \quad \mbox{SDS:~swarm double sphere}, \\
& \mbox{SDM:~swarm double matrix}.
\end{aligned}
\end{align*}
Moreover, we use $| \cdot |$ to denote $\ell^2$-norm of vectors in $\mathbb R^d$ or $\mathbb C^d$, where $\| \cdot \|$ represents $L^2$-norm.
\section{Preliminaries} \label{sec:2}
\setcounter{equation}{0}
In this section, we briefly introduce two first-order aggregation models whose emergent dynamics will be discussed in the following two sections, namely {\it ``the Lohe tensor model''} and {\it ``the Schr\"odinger-Lohe model''}, separately.
\subsection{The Lohe tensor model} \label{sec:2.1}
To set up the stage, we begin with basic terminologies on tensors. A complex rank-$m$ tensor can be visualized as a multi-dimensional array of complex numbers with multi-indices. The {\it ``rank}'' (or {\it ``order}'') of a tensor is the number of indices, say a rank-$m$ tensor with size $d_1 \times \cdots \times d_m$ is an element of ${\mathbb C}^{d_1 \times \cdots \times d_m}$. For example, scalars, vectors and matrices correspond to rank-0, 1 and 2 tensors, respectively.
Let $T$ be a rank-$m$ tensor with a size $d_1 \times \cdots \times d_m$. Then, we denote $(\alpha_1, \cdots, \alpha_m)$-th component of $T$ by $[T]_{\alpha_1 \cdots \alpha_m}$, and we also set $\overline{T}$ by the rank-$m$ tensor whose components are simply the complex conjugate of the corresponding elements in $T$:
\[ [\overline{T}]_{\alpha_1 \cdots \alpha_m} :=\overline{[T]_{\alpha_1 \cdots \alpha_m}}, \quad 1 \leq \alpha_i \leq d_i,~~1 \leq i \leq m. \]
In other words, each component of $\overline T$ is defined as the complex conjugate of the corresponding element of $T$. Let ${\mathcal T}_m(\mathbb C; d_1 \times\cdots\times d_m)$ be the collection of all rank-$m$ tensors with size $d_1 \times\cdots\times d_m$. For notational simplicity, we set
\[ {\mathbb C}^{d_1 \times \cdots \times d_m}:= {\mathcal T}_m(\mathbb C; d_1 \times\cdots\times d_m). \]
Then, it is a complex vector space. Several well-known first-order aggregation models, for instance, the Kuramoto model \cite{Ku2}, the swarm sphere model \cite{O1} and the Lohe matrix model \cite{Lo3} can be regarded as aggregation models on the subsets of $\mathbb R, \mathbb C^d$ and $\mathbb C^{d \times d}$, respectively. Furthermore, we also introduce the following handy notation:~for $T \in \mathbb C^{d_1 \times \cdots \times d_m}$ and $A \in \mathbb C^{d_1 \times \cdots \times d_m \times d_1 \times \cdots \times d_m}$, we set
\begin{align*}
\begin{aligned}
& [T]_{\alpha_{*}}:=[T]_{\alpha_{1}\alpha_{2}\cdots\alpha_{m}}, \quad [T]_{\alpha_{*0}}:=[T]_{\alpha_{10}\alpha_{20}\cdots\alpha_{m0}}, \quad [T]_{\alpha_{*1}}:=[T]_{\alpha_{11}\alpha_{21}\cdots\alpha_{m1}}, \\
& [T]_{\alpha_{*i_*}}:=[T]_{\alpha_{1i_1}\alpha_{2i_2}\cdots\alpha_{mi_m}}, \quad [T]_{\alpha_{*(1-i_*)}}:=[T]_{\alpha_{1(1-i_1)}\alpha_{2(1-i_2)}\cdots\alpha_{m(1-i_m)}}, \\
& [A]_{\alpha_*\beta_*}:=[A]_{\alpha_{1}\alpha_{2}\cdots\alpha_{m}\beta_1\beta_2\cdots\beta_{m}}.
\end{aligned}
\end{align*}
Moreover, we can associate inner product $\langle \cdot, \cdot \rangle_\textup{F}$, namely {\it ``Frobenius inner product''} and its induced norm $\| \cdot \|_\textup{F}$ on $ \mathbb C^{d_1 \times \cdots \times d_m}$:~for $T, S \in \mathbb C^{d_1 \times \cdots \times d_m}$,
\[ \langle T, S \rangle_\textup{F} := \sum_{\alpha_* \in \prod_{i=1}^{m} \{1, \cdots, d_i\} } [{\bar T}]_{\alpha_*} [S]_{\alpha_*}, \quad \|T \|_\textup{F}^2 := \langle T, T \rangle_\textup{F}. \]
Let $A_j$ be a {\it block skew-hermitian} rank-$2m$ tensor with size $(d_1 \times\cdots\times d_m) \times (d_1 \times \cdots\times d_m)$ such that
\begin{equation*}
[\bar A_j]_{\alpha_{*0} \alpha_{*1}} = -[A_j]_{\alpha_{*1} \alpha_{*0}}.
\end{equation*}
In other words, if two blocks with the first $m$-indices are interchanged with the rest $m$-indices, then its sign is changed.
Now, we are ready to introduce the LT model on the finite ensemble $\{T_j \}_{j=1}^{N} \subset \mathbb C^{d_1 \times \cdots \times d_m}$:
\begin{align} \label{LT}
\begin{aligned}
&\dot{[T_j]}_{\alpha_{*0}} = [A_j]_{\alpha_{*0}\alpha_{*1}}[T_j]_{\alpha_{*1}} \\
&\hspace{0.5cm}+ \sum_{i_* \in \{0, 1\}^m}\kappa_{i_*} \Big([T_c]_{\alpha_{*i_*}}\bar{[T_i]}_{\alpha_{*1}}[T_i]_{\alpha_{*(1-i_*)}}-[T_i]_{\alpha_{*i_*}}\bar{[T_c]}_{\alpha_{*1}}[T_i]_{\alpha_{*(1-i_*)}} \Big),
\end{aligned}
\end{align}
where $\kappa_{i_*}$'s are coupling strengths, $T_c := \frac{1}{N} \sum_{k=1}^{N} T_k$ is the f average of $\{T_j\}_{j1}^N$, and we used the Einstein summation convention for repeated indices in the R.H.S. of \eqref{LT}. Although the LT model \eqref{LT} looks so complicated, interaction terms inside the parenthesis of \eqref{LT} are designed to include interactions for the Kuramoto model, the swarm sphere model and the Lohe matrix model, and they certainly have ``gain term$-$loss term'' structure and are cubic in nature. These careful designs of interactions lead to the existence of a constant of motion.
\begin{proposition} \label{P2.1}
\emph{\cite{H-P7}}
Let $\{ T_j \}$ be a solution to \eqref{LT}. Then, for each $j = 1, \cdots, N$, $\|T_j \|_\textup{F}$ is a first integral for \eqref{LT}:
\[ \|T_j(t) \|_\textup{F} = \|T_j(0) \|_\textup{F}, \quad t \geq 0.\]
\end{proposition}
\subsection{The Schr\"odinger-Lohe model} \label{sec:2.2}
Consider a network consisting of $N$ nodes denoted by $1, \cdots, N,$ and we assume that the network topology is registered by an adjacent weighted matrix $(a_{ij})$.
For each $j = 1, \cdots, N$, let $\psi_j=\psi_j(x,t)$ be the wave-function of the quantum subsystem lying on the $j$-th node. Then, we assume that the spatial-temporal dynamics of $\psi_j$ is governed by the Cauchy problem to the SL model: for $(x,t)\in \mathbb R^d\times \mathbb R_+$,
\begin{equation} \label{S-L}
\begin{cases}
\displaystyle {\mathrm i} \partial_t \psi_j = - \frac{1}{2} \Delta \psi_j + V_j\psi_j + \frac{{\mathrm i}\kappa}{2N}\sum_{k=1}^N a_{jk} \left( \psi_k - \frac{ \langle \psi_j,\psi_k\rangle}{\langle \psi_j,\psi_j\rangle} \psi_j \right), \\
\displaystyle \psi_j(x,0) = \psi_j^0(x), \quad \|\psi_j^0 \| = 1, \quad j=1,\cdots,N,
\end{cases}
\end{equation}
where $\kappa$ is a coupling strength, and we have taken mass and normalized Planck's constant to be unity for the simplicity of presentation. Like the classical Schr\"odinger equation, system \eqref{S-L} satisfies $L^2$-conservation.
\begin{proposition}
Let $\{ \psi_j \}$ be a solution to \eqref{S-L}. Then, $\|\psi_j(t)\|$ is a first integral:
\[ \|\psi_j(t)\| = \| \psi_j^0 \|, \quad t \geq 0, \quad 1 \leq j \leq N. \]
\end{proposition}
Then, $L^2$-conservation and standard energy estimates yield a global existence of unique weak solutions to \eqref{S-L}.
\begin{theorem} \label{T2.1}
\emph{\cite{A-M,H-H}}
Suppose initial data satisfy $\psi_j^0 \in L^2(\mathbb R^d)$ for each $j= 1, \cdots, N$. Then, the Cauchy problem \eqref{S-L} admits a unique global-in-time weak solution satisfying
\[
\psi_j \in C([0,\infty); L^2(\mathbb R^d)), \quad j = 1, \cdots, N.
\]
Moreover, if we assume $\psi_j^0 \in H^1(\mathbb R^d)$, then the corresponding global weak solution satisfies $\psi_j \in C([0,\infty);H^1(\mathbb R^d))$.
\end{theorem}
\vspace{0.5cm}
Before we close this section, we show that the SL model \eqref{S-L} reduces to the Kuramoto model as a special case. For this, we assume
\[ V_{j}(x) = :\nu_j: \mbox{constant}, \qquad \psi_j(x,t) =: e^{-{\mathrm i} \theta_j(t)}, \quad (x, t) \in \mathbb R^d \times \mathbb R_+. \]
We substitute the above ansatz into the SL model \eqref{S-L} to get
\[ {\dot \theta}_j \psi_j = \nu_j \psi_j + \frac{{\mathrm i} \kappa}{N} \sum_{k=1}^{N} a_{jk} \Big(\psi_k - e^{-{\mathrm i}(\theta_j - \theta_k)} \psi_j \Big). \]
Then, we multiply $\overline{\psi}_j$ to the above relation, use $|\psi_j(t)|^2 =1$, and compare the real part of the resulting relation to obtain the Kuramoto model \cite{Ku2}:
\[ {\dot \theta}_j = \nu_j + \frac{{\bar \kappa}}{N} \sum_{k=1}^{N} a_{jk} \sin (\theta_k - \theta_j), \quad {\bar \kappa} := 2 \kappa. \]
\vspace{0.5cm}
In the following two sections, we review emergent dynamics of the LT model and the SL model, separately. Most presented results in the following section will be provided without detailed proofs, but we may discuss brief ideas or key ingredients to give a feeling to see how proofs go.
\section{The Lohe tensor model} \label{sec:3}
\setcounter{equation}{0}
In this section, we review the emergent dynamics of the LT model and two explicit low-rank LT models which can be related to the swarm sphere model and the Lohe matrix model.
\subsection{Emergent dynamics} \label{sec:3.1}
In this subsection, we review emergent dynamics of the Lohe tensor model \eqref{LT}. First, we recall two concepts of aggregations (or synchronizations) as follows.
\begin{definition} \label{D3.1}
\emph{\cite{H-P1,H-P5}}
Let $\{T_j \}$ be a finite ensemble of rank-m tensors whose dynamics is governed by \eqref{LT}.
\begin{enumerate}
\item The ensemble exhibits complete state aggregation (synchronization) if relative states tend to zero asymptotically:
\[ \lim_{t\rightarrow \infty}\max_{1\leq i,j\leq N } \| T_i(t)-T_j(t) \|_\textup{F} = 0. \]
\item The ensemble exhibits practical aggregation (synchronization) if magnitudes of relative states can be controlled by the principal coupling strength $\kappa_{0\cdots0}$ as follows:
\[ \lim_{\kappa_{i_*} \to \infty} \limsup_{t\rightarrow\infty} \max_{1\leq i,j\leq N } \| T_i(t)-T_j(t) \|_\textup{F} =0, \]
for some $i_* \in \{0, 1 \}^m$.
\end{enumerate}
\end{definition}
\begin{remark} The jargon ``{\it synchronization''} is often used in control theory and physics communities instead of aggregation. In fact, synchronization represents an adjustment of rhythms in oscillatory systems. In contrast, our systems under consideration might not be oscillatory. Thus, the authors feel more comfortable to use aggregation instead of synchronization.
\end{remark}
For a given state ensemble $\{T_j \}$ and free flow ensemble $\{A_j \}$, we define diameters for both ensembles:
\[
\mathcal D(T) := \max_{1\leq i,j\leq N} \|T_i - T_j \|_\textup{F}, \quad \mathcal D(A) := \max_{1\leq i,j\leq N} \|A_i - A_j \|_\textup{F}.
\]
Note that $\mathcal D(T)$ is time-varying and Lipschitz continuous. Thus, it is differentiable a.e. and $\mathcal D(A)$ is constant, since $A_j$ is a constant tensor. \newline
Let $\{T_j \}$ be a solution to \eqref{LT} with $ \|T_j \|_\textup{F} = 1.$ Then, after tedious and delicate analysis, one can derive a differential inequality for $\mathcal D(T)$ (see Proposition 4.2 in \cite{H-P7}): for a.e. $t>0$,
\begin{equation} \label{C-1}
\left|{\d\over{\textup{d}t}} {\mathcal D}(T)+\kappa_0 {\mathcal D}(T) \right|
\leq 2\kappa_0 {\mathcal D}(T)^2+ 2 \hat{\kappa}_0 \|T_c^{0}\|_\textup{F} {\mathcal D}(T)+ {\mathcal D}(A),
\end{equation}
where $T_c^{0}, \kappa_0$ and ${\hat \kappa}_0$ are defined as follows:
\[ T_c^{0} := \frac{1}{N} \sum_{j=1}^{N} T_j^{0}, \qquad \kappa_0 := \kappa_{0\cdots 0}, \qquad \hat{\kappa}_0 := \sum_{i_*\neq (0,\cdots, 0)}\kappa_{i_*} . \]
Depending on whether the ensemble $\{T_j \}$ has the same free flows or heterogeneous free flows, we have the following two cases:
\[ {\mathcal D}(A) = 0 : \mbox{homogeneous ensemble}, \quad {\mathcal D}(A) > 0 : \mbox{heterogeneous ensemble}. \]
Then, the emergent dynamics of \eqref{LT} can be summarized as follows.
\begin{theorem} \label{T3.1} \cite{H-P7}
The following assertions hold.
\begin{enumerate}
\item
(Emergence of complete state aggregation):~Suppose system parameters and initial data $\{T_j^0 \}$ satisfy
\begin{align}
\begin{aligned} \label{C-2}
& {\mathcal D}(A) = 0, \quad \|T_j^{0}\|_\textup{F} = 1, \quad j = 1, \cdots, N, \quad \kappa_0 > 0, \\
& {\hat \kappa}_{0} < \frac{\kappa_{0}}{2 \|T_c^{0}\|_\textup{F}},\quad 0< {\mathcal D}(T^{0})<\frac{\kappa_{0}- 2{\hat \kappa}_0 \|T_c^{0}\|_\textup{F}}{2\kappa_0},
\end{aligned}
\end{align}
and let $\{T_j \}$ be a global solution to \eqref{LT}. Then, there exist positive constants $C_0$ and $C_1$ depending on $\kappa_{i_*}$ and $\{ T_j ^{0} \}$ such that
\[
C_0 e^{-\left(\kappa_{0}+ 2 \hat{\kappa}_0 \|T_c^{0}\|_\textup{F} \right)t} \leq {{\mathcal D}(T(t))} \leq C_1 e^{-\left(\kappa_{0}- 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F} \right)t}, \quad t \geq 0.
\]
\item
(Emergence of practical aggregation):~Suppose system parameters and initial data satisfy
\begin{align}
\begin{aligned} \label{C-2-1}
& \|T_j^{0}\|_\textup{F} = 1, \quad j = 1, \cdots, N, \quad \kappa_0 > 0, \\
& 0\leq {\mathcal D}(T^{0})\leq\eta_2,\quad 0 < {\mathcal D}(A)< \frac{|\kappa_0- 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F} |^2}{8 \kappa_0},
\end{aligned}
\end{align}
where $\eta_2$ appearing in \eqref{C-2-1} is the largest positive root of the following quadratic equation:
\[
-2\kappa_0 x^2+(\kappa_{0}- 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F} )x = {\mathcal D}(A).
\]
Let $\{T_j\}$ be a global solution to system \eqref{LT}. Then, practical aggregation emerges asymptotically:
\begin{equation} \label{C-2-2}
\lim_{ \kappa_0 \to \infty} \limsup_{t\rightarrow\infty} {\mathcal D}(T(t))=0.
\end{equation}
\end{enumerate}
\end{theorem}
\begin{proof}
For a detailed proof, we refer the reader to \cite{H-P1}. Below, we instead provide a brief sketch on the key ingredient of proofs. \newline
\noindent (i)~Suppose that ${\mathcal D}(A) = 0$. \newline
\noindent $\bullet$~Case A (Upper bound estimate):~It follows from \eqref{C-1} that
\begin{equation} \label{C-4}
{\d\over{\textup{d}t}} {\mathcal D}(T)\leq {\mathcal D}(T) \Big[ 2\kappa_0 {\mathcal D}(T)-(\kappa_{0}- 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F}) \Big ], \quad \mbox{a.e.~$t > 0$}.
\end{equation}
Under the assumptions \eqref{C-2}, coefficients appearing in the R.H.S. of \eqref{C-4} satisfy
\[ 2\kappa_0 >0, \quad \kappa_{0}- 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F} > 0. \]
Now, we directly solve the differential inequality \eqref{C-4} to derive desired upper bound estimates. \newline
\noindent $\bullet$~Case B (Lower bound estimate): Again, it follows from \eqref{C-1} that
\begin{equation} \label{C-5}
\frac{\d}{\textup{d}t} {\mathcal D}(T) \geq {\mathcal D}(T) \Big[ 2\kappa_0 {\mathcal D}(T) - (\kappa_0 + 2 {\hat \kappa}_0 \| T^{0}_c \|_\textup{F}) \Big], \quad \mbox{a.e.}~t > 0.
\end{equation}
Similar to Case A, we integrate \eqref{C-5} to find the desired lower bound estimate. \newline
\noindent (ii)~~Suppose that ${\mathcal D}(A) > 0$. Then, it follows from \eqref{C-1} that
\begin{equation} \label{F-2}
{\d\over{\textup{d}t}} {\mathcal D}(T)\leq 2\kappa_0 {\mathcal D}(T)^2-(\kappa_{0}- 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F}) {\mathcal D}(T)+ {\mathcal D}(A), \quad \mbox{a.e.~$t > 0$}.
\end{equation}
In order to use a comparison principle, we introduce a quadratic function $f$ defined by
\begin{equation} \label{F-3}
f(x) :=-2\kappa_0 x^2+(\kappa_{0}- 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F} )x.
\end{equation}
Since we are interested in the regime $\kappa_0 \to \infty$, the term $\kappa_0-2\hat{\kappa}_0 \|T_c^{0}\|$ will be positive. Thus, it follows from \eqref{F-2} and \eqref{F-3} that
\begin{equation*} \label{F-4}
\frac{\d}{\textup{d}t} {\mathcal D}(T)\leq{}{\mathcal D}(A)-f({\mathcal D}(T)), \quad \mbox{a.e.}~t > 0.
\end{equation*}
Considering the geometry of the graph of $f$, one can see that the quadratic equation $f(x)=D(A)$ has two positive roots $\eta_1$ and $\eta_2$ satisfying
\[ 0<\eta_1< \frac{\kappa_0- 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F} }{4\kappa_0} <\eta_2< \frac{\kappa_0- 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F} }{2\kappa_0}. \]
Moreover, one can claim: (see Lemma 5.3 in \cite{H-P7})
\begin{enumerate}
\item
${\d\over{\textup{d}t}} {\mathcal D}(T(t))\leq{0}$ almost every $t>0$ when ${\mathcal D}(T(t))\in[\eta_1, \eta_2]$.
\vspace{0.1cm}
\item
$ \mathcal{S}(\eta_2):=\{ {\mathcal D}(T(t))< \eta_2 \}$ is a positively invariant set for the LT flow generated by \eqref{LT}.
\vspace{0.1cm}
\item
There exist $t_e\geq0$ such that
\[ {\mathcal D}(T(t)) < \eta_1, \quad t \geq t_e. \]
\end{enumerate}
In fact, the smaller positive root $\eta_1$ can be calculated explicitly:
\[ \eta_1 = \frac{\mathcal D(A)}{\kappa_0- 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F} } \left( \frac{2}{1+\sqrt{1- \frac{8\kappa_0 \mathcal D(A)}{\kappa_0- 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F}}}} \right). \]
Then, it follows from the claim above that
\[ \limsup_{t\rightarrow\infty}\mathcal D(T(t) )\leq\eta_1={\mathcal D(A)\over{\kappa_0 - 2\hat{\kappa}_0 \|T_c^{0}\|_\textup{F} }}{2\over{1+\sqrt{1-8 \kappa_0 \mathcal D(A)\over{\kappa_0-2 \hat{\kappa}_0 \|T_c^{0}\|_\textup{F} }}}}.
\]
By letting $\kappa_0 \to \infty$, one has the desired estimate \eqref{C-2-2}.
\end{proof}
\subsection{Low-rank LT models} \label{sec:3.2}
In the previous subsection, we have reviewed the emergent dynamics of the LT model in a general setting. In this subsection, we study two low-rank LT models which can be derived from the LT model on $\mathbb C^{d_1 \times d_2}$ and $\mathbb C^d$, respectively.
\subsubsection{The generalized Lohe matrix model} \label{sec:3.2.1}
In this part, we first derive a matrix aggregation model on $\mathbb C^{d_1 \times d_2}$ that can be reduced from the LT model. In the case of a square matrix $d_1 = d_2$, the Lohe matrix model serves as a first-order aggregation model on the subset of $\mathbb C^{d_1 \times d_2}$ which is a unitary group. Thus, the LT model can provide an aggregation model on the space of nonsquare matrices. More precisely, consider the Lohe tensor model \eqref{LT} with $m=2$:
\begin{align}
\begin{aligned} \label{C-6}
\dot{T}_j &= A_j T_j + \kappa_{00}(\mathrm{tr}(T_j^\dagger T_j)T_c-\mathrm{tr}(T_c^\dagger T_j)T_j) + + \kappa_{11}\mathrm{tr}(T_j^\dagger T_c-T_c^\dagger T_j)T_j \\
&\hspace{0.2cm} + \kappa_{10}(T_j T_j^\dagger T_c-T_j T_c^\dagger T_j) + \kappa_{01}(T_cT_j^\dagger T_j -T_jT_c^\dagger T_j),
\end{aligned}
\end{align}
where $T_j^{\dagger}$ is a hermitian conjugate of $T_j$ and the free flow term $A_jT_j$ is defined as a rank-2 tensor via tensor contraction between a rank-4 tensor and a rank-2 tensor:
\[
[T_j]^{\dagger}_{\alpha \beta} = [{\overline T_j}]_{\beta \alpha}, \quad [A_jT_j]_{\alpha\beta}=[A_j]_{\alpha\beta\gamma\delta}[T_j]_{\gamma\delta}.
\]
For simplicity, we set
\[ \kappa_{00}=\kappa_{11}=0, \quad \kappa_1 := \kappa_{01}, \quad \kappa_2 := \kappa_{10}. \]
Then, system \eqref{C-6} reduces to the following simplified model, namely {\it ``the generalized Lohe matrix model''} \cite{H-P3}:
\begin{equation}
\begin{cases} \label{C-7}
\vspace{0.3cm} \displaystyle {\dot T}_j =A_j T_j +\kappa_{1}(T_cT_j^\dagger T_j -T_j T_c^\dagger T_j)+\kappa_{2}(T_j T_j^\dagger T_c-T_j T_c^\dagger T_j), \quad t >0, \\
\displaystyle T_j(0) =T_j^0 \in \mathbb C^{d_1 \times d_2}, \quad j = 1, \cdots, N,
\end{cases}
\end{equation}
where $\kappa_{1}$ and $\kappa_{2}$ are nonnegative coupling strengths, and we have used Einstein summation convention.
Next, we define a functional measuring deviations from the centroid of configuration for \eqref{C-7}:
\[ {\mathcal V}[T(t)] :=\frac{1}{N}\sum_{k=1}^N\|T_k(t)-T_c(t)\|_\textup{F}^2 = 1 - \|T_c(t)\|_\textup{F}^2, \quad t \geq 0. \]
Then, we show that $\mathcal V[T]$ converges to a nonnegative constant $\mathcal V_\infty$.
\begin{theorem}\label{T3.2}
\emph{\cite{H-P3}}
Let $\{ T_j \}$ be a global solution to \eqref{C-7} with $\|T_j^0 \|_\textup{F} = 1$. Then, the following assertions hold.
\begin{enumerate}
\item
There exists a nonnegative constant ${\mathcal V}_\infty$ such that
\[ \lim_{t \to \infty } {\mathcal V}[T] = {\mathcal V}_\infty. \]
\item
The orbital derivative of ${\mathcal V}(T)$ tends to zero asymptotically:
\begin{align*}
&\lim_{t\rightarrow\infty} \frac{\d}{\textup{d}t} {\mathcal V}[T(t)] = 0 \quad \textup{and} \\ &\lim_{t\rightarrow\infty} \Big( \|T_j T_c^\dagger -T_cT_j^\dagger \|_\textup{F} + \|T_j^\dagger T_c-T_c^\dagger T_j \|_\textup{F} \Big) = 0.
\end{align*}
\end{enumerate}
\end{theorem}
\vspace{0.2cm}
In what follows, we consider {\it ``the reduced Lohe matrix model}'' which corresponds to the generalized Lohe matrix model \eqref{C-7} with $\kappa_2=0$: \begin{align}\label{C-8}
\begin{cases}
{\dot T}_j =A_j T_j + \kappa_{1}(T_cT_j^\dagger T_j -T_j T_c^\dagger T_j), \quad t > 0, \\
T_j(0)=T_j^0, \quad j = 1, \cdots, N,
\end{cases}
\end{align}
subject to the initial conditions:
\begin{equation*} \label{C-9}
T_i^{0,\dagger} T_i^0 = T_j^{0,\dagger} T_j^0, \quad 1 \leq i, j \leq N.
\end{equation*}
Let $\{T_j\}$ be a solution to \eqref{C-8} with a specific natural frequency tensor $A_j$:
\begin{equation*}
[A_j]_{\alpha\beta\gamma\delta} :=[B_j]_{\alpha\gamma}\delta_{\beta\delta},
\end{equation*}
where $B_j$ is a rank-2 tensor. Then, by singular value decomposition of $T_j(t)$, one has
\[
T_j(t)=U_j(t)\Sigma_j V_j^\dagger .
\]
Here, $\Sigma_j$ and $V_j$ are time-independent constant matrices, whereas $U_j=U_j(t)$ is a time-dependent unitary matrix satisfying
\begin{align}\label{C-11}
\begin{cases}
\dot{U}_j= B_j U_j + \kappa_{1}(U_cD-U_j D^\dagger U_c^\dagger U_j), \quad t > 0, \\
U_j(0)=U_j^0, \quad j = 1, \cdots, N,
\end{cases}
\end{align}
where $D$ is a diagonal matrix and $B_jU_j$ is a usual matrix multiplication between $B_j$ and $U_j$. Moreover, if complete state aggregation occurs for \eqref{C-8}, then it also occurs for \eqref{C-11}, and vice versa. By algebraic manipulation, one can find a differential inequality for the diameter of $\{U_j\}$:
\[
\mathcal D(U) := \max_{1\leq i,j \leq N} \|U_i - U_j\|_\textup{F},\quad \mathcal D(B) := \max_{1\leq i,j\leq N} \|B_i - B_j\|_\textup{F}.
\]
For notational simplicity, we denote
\begin{align*}
\begin{aligned}
& D :=\mathrm{diag}(\lambda_1^2, \cdots, \lambda_{d_1}^2), \quad
\langle \lambda^2 \rangle :=\frac{1}{d_1}(\lambda_1^2+\lambda_2^2+\cdots+\lambda_{d_1}^2), \\
& \Delta(\lambda^2):=\max_{1\leq k\leq d_1}|\lambda_k^2- \langle \lambda^2 \rangle |, \quad \mathcal{A} :=\langle \lambda^2 \rangle +\Delta(\lambda^2),\quad \mathcal{B} :=\langle \lambda^2 \rangle -\Delta(\lambda^2).
\end{aligned}
\end{align*}
Now, we are ready to provide the emergent dynamics of \eqref{C-11} as follows.
\begin{theorem}\label{T3.3}
\emph{\cite{H-P3}}
The following assertions hold.
\begin{enumerate}
\item
(Complete state aggregation):~Suppose system parameters and initial data satisfy
\[ {\mathcal D}(B) = 0, \quad {\mathcal A} > 0, \quad {\mathcal B} > 0, \quad \kappa_1 > 0, \quad {\mathcal D}(U^0)\leq \sqrt{\frac{2\mathcal{B}}{\mathcal{A}}}. \]
Then for any solution $\{U_j\}$ to \eqref{C-11}, we have
\[
\lim_{t\to\infty} \mathcal D(U) =0.
\]
Moreover, the convergence rate is exponential.
\vspace{0.1cm}
\item
(Practical aggregation):~Suppose system parameters and initial data satisfy
\[ {\mathcal A} > 0, \quad {\mathcal B} > 0, \quad \kappa_1 > {\mathcal D}(B)\cdot\sqrt{\frac{27\mathcal{A}}{32\mathcal{B}^3}}>0,\quad {\mathcal D}(U^0)<\alpha_2, \]
where $\alpha_2$ is a largest positive root of $g(x)= \mathcal{A}x^3-2\mathcal{B}x+\frac{{\mathcal D}(B)}{\kappa_1} = 0$, and let $\{U_j\}$ be a solution to \eqref{F-4}. Then, one has practical aggregation:
\[
\lim_{\kappa_1\rightarrow\infty}\limsup_{t\rightarrow\infty} {\mathcal D}(U)=0.
\]
\end{enumerate}
\end{theorem}
\begin{proof} The first statement for the complete state aggregation is based on the following differential inequalities:
\[ -2\kappa_1 \mathcal{A} {\mathcal D}(U)+\kappa_1 \mathcal{A} {\mathcal D}(U)^3 \leq\frac{\d}{\textup{d}t} {\mathcal D}(U) \leq-2\kappa_1 \mathcal{B} {\mathcal D}(U)+\kappa_1 \mathcal{A} {\mathcal D}(U)^3,
\]
where $\mathcal A$ and $\mathcal B$ are constants determined by the diagonal matrix $D$. This implies the desired upper and lower bound estimates for ${\mathcal D}(U)$. In contrast, the proof of the second statement will be done by similar arguments as in Theorem \ref{T3.1}. For details, we refer the reader to \cite{H-P3}.
\end{proof}
\subsubsection{The Lohe hermitian sphere model} \label{sec:3.2.2}
In this part, we consider a reduction of the LT model on the hermitian sphere $\{z\in \mathbb C^d: |z|=1\}$. In this case, the LT model reduces to
\begin{equation} \label{C-13}
\dot{z}_j= \Omega_j z_j +\kappa_{0} (z_c\langle z_j, z_j \rangle-z_j \langle z_c. z_j \rangle )+\kappa_1(\langle{z_j, z_c}\rangle- \langle z_c z_j\rangle) z_j,
\end{equation}
where $\langle z, w \rangle$ and $\Omega_j$ are standard inner product in $\mathbb C^d$ and a skew-hermitian $d \times d$ matrix satisfying
\[ \langle z, w \rangle = [{\bar z}]_{\alpha} [w]_{\alpha}, \quad \Omega_j^\dagger = -\Omega_j. \]
Here we used the Einstein summation convention. We refer the reader to \cite{B-H-H-P,B-H-P,H-H-P} for the emergent dynamics of \eqref{C-13}.
Note that for a real-valued rank-1 tensor $z_j \in \mathbb R^d$, the coupling terms involving $\kappa_1$ become identically zero thanks to the symmetry of inner product in ${\mathbb R}^d$. Hence, we can recover the swarm sphere model on $\mathbb S^{d-1}$:
\begin{equation}\label{C-14}
\dot{x}_j=\Omega_j x_j+\kappa_0 \Big (\langle{x_j, x_j}\rangle x_c-\langle{x_c, x_j}\rangle x_j \Big ).
\end{equation}
The emergent dynamics of \eqref{C-14} has been extensively studied in a series of papers
\cite{C-C-H,C-H1,C-H2,C-H3,C-H4,J-C,Lo2,Lo3,M-T-G,T-M,Zhu}.
In what follows, to investigate the nonlinear effect on the collective behaviors of \eqref{C-13} due to two coupling terms, we consider for a while the following two special cases:
\[ \textup{(i)} ~~\Omega_j = \Omega, \quad \kappa_1 = 0; \qquad \textup{(ii)}~~ \Omega_j = \Omega, \quad \kappa_0 = 0,\quad j=1,\cdots,N. \]
Then, the corresponding models for each case read as follows:
\begin{align} \label{C-15}
\begin{aligned}
&\textup{(i)}~~ \dot{z}_j = \Omega z_j + \kappa_0(\langle{z_j, z_j}\rangle z_c-\langle{z_c, z_j}\rangle z_j). \\
&\textup{(ii)}~~\dot{z}_j = \Omega z_j + \kappa_1(\langle{z_j, z_c}\rangle-\langle{z_c, z_j}\rangle)z_j.
\end{aligned}
\end{align}
From the models in \eqref{C-15}, we define a functional for $\{z_j\}$:
\[
\mathcal D(Z):= \max_{1\leq i,j\leq N} |1-\langle z_i,z_j\rangle|
\]
In the following proposition, we summarize results on the emergent dynamics of the models in \eqref{C-15} without proofs.
\begin{proposition} \label{P3.1}
\emph{\cite{H-P6}}
The following assertions hold.
\begin{enumerate}
\item
Suppose system parameters and initial data satisfy
\[
\kappa_0 > 0, \quad |z_j^{0} |=1,\quad \max_{i\neq j}|1-\langle{z_i^{0}, z_j^{0}}\rangle|<\frac12,
\]
and let $ \{ z_j \}$ be a global solution to $\eqref{C-15}_1$ with the initial data $\{z_j^0\}$. Then, there exists a positive constant $\Lambda$ depending on the initial data such that
\[ {\mathcal D}(Z(t)) \leq {\mathcal D}(Z^{0}) e^{- \kappa_0 \Lambda t}, \quad t \geq 0. \]
\item
Suppose system parameters and initial data satisfy
\[
\kappa_1 > 0, \quad |z_j^{0} |=1,
\]
and let $\{ z_j \}$ be a global solution to $\eqref{C-15}_2$ with the initial data $\{z_j^{0} \}$. Then, there exist a time-dependent phase function $\theta_j=\theta_j(t)$ such that
\[ z_j(t)= e^{{\mathrm i} \theta_j(t)} z^{0}_j, \quad j = 1, \cdots, N, \]
and $\theta_j$ is a solution to the Kuramoto-type model with frustrations:
\begin{equation} \label{C-16}
\begin{cases}
\displaystyle \dot{\theta}_j=\frac{2 \kappa_1}{N}\sum_{k=1}^N R_{jk}^{0} \sin(\theta_k-\theta_j+\alpha_{jk}^{0}), \quad t > 0, \\
\displaystyle \theta_j(0)=0,\quad j=1,\cdots,N,
\end{cases}
\end{equation}
where $R_{jk}^{0}$ and $\alpha_{jk}^{0}$ are determined by initial data:
\[ \langle{z_j^{0}, z_k^{0}} \rangle=R^{0}_{jk}e^{\mathrm{i} \alpha^0_{jk}}. \]
\end{enumerate}
\end{proposition}
\begin{remark}
\textup{(i)}~~In Proposition \ref{P3.1}(2), $R_{jk}^{0}$ and $\alpha^0_{jk}$ satisfy symmetry and anti-symmetry properties, respectively:
\[ R_{jk}^{0} = R_{kj}^{0}, \qquad \alpha^0_{jk} = -\alpha^0_{kj}, \qquad j,k = 1, \cdots, N. \]
\textup{(ii)}~~System \eqref{C-16} can be rewritten as a gradient flow with the following potential $V$:
\[ \dot{\Theta}=-\nabla_{\Theta} V[\Theta], \quad t > 0, \quad V[\Theta] := \frac{\kappa_1}{N}\sum_{i, j = 1}^{N} R^{0}_{ij} \Big(1- \cos(\theta_i-\theta_j+\alpha^0_{ji}) \Big). \]
\end{remark}
We now return to the full model \eqref{C-13} with the same free flows. Note that the term involving $\kappa_0$ corresponds to the swarm sphere model and the term with $\kappa_1$ describes the complex nature of underlying phase space. For an ensemble $\{z_j \}$, we define the norm of the centroid:
\[ \rho (t):= \left | \frac{1}{N} \sum_{j=1}^{N} z_j(t) \right |. \]
\begin{theorem} \label{T3.4}
\emph{\cite{H-P6}} Suppose system parameters and initial data satisfy
\begin{equation} \label{Z-10}
\Omega_j = \Omega, \quad j = 1, \cdots, N, \quad 0< \kappa_1 < \frac{1}{4} \kappa_0, \quad \rho^{0} > \frac{N-2}{N},
\end{equation}
and let $\{z_j \}$ be a solution to \eqref{C-13}. Then for each $i, j = 1, \cdots, N$, two-point correlation function $\langle z_i, z_j \rangle$ converges to 1 exponentially fast, i.e., complete state aggregation emerges asymptotically.
\end{theorem}
\begin{proof} We give a brief sketch for a proof. Details can be found in Theorem 4.1 \cite{H-P6}. First, note that
\[ |z_i - z_j |^2 = |z_i |^2 + |z_j |^2 - 2 \mbox{Re} \langle z_i, z_j \rangle = 2 (1 - \mbox{Re} \langle z_i, z_j \rangle) \leq 2 |1 - \langle z_i, z_j \rangle|. \]
Thus, once we can show that $ |1 - \langle z_i, z_j \rangle|$ tends to zero exponentially fast, then it directly follows that ${\mathcal D}(Z)$ tends to zero exponentially fast. Thus, we introduce a functional
\[ \mathcal{L}(Z) :=\max_{1\leq i, j \leq N}|1-\langle{z_i, z_j}\rangle|^2. \]
By detailed and straightforward calculation, one can derive differential inequality for ${\mathcal L}(Z)$:
\[
\frac{\d}{\textup{d}t}\mathcal{L}(Z)\leq-\kappa_0\mathcal{L}(Z)\left(\mathrm{Re}(\langle{z_{i_0}+ z_{j_0}, z_c}\rangle)-\frac{4\kappa_1}{\kappa_0}\right),
\]
where $i_0$ and $j_0$ are extremal indices such that
\[ \mathcal{L}(Z) =:|1-\langle{z_{i_0}, z_{j_0}}\rangle|^2. \]
On the other hand, one can show that under the assumption \eqref{Z-10} on $\rho^0$, the quantity $\langle z_i, z_c \rangle$ tends to 1 asymptotically. Again, by the assumption on the coupling strengths, there exist positive constants $T$ and $\varepsilon$ such that
\[
\mathrm{Re}(\langle{z(t)_{i_0}+z(t)_{j_0}, z_c}\rangle)-\frac{4\kappa_1}{\kappa_0}>\varepsilon, \quad \mbox{$t > T$}.
\]
This yields
\[ \frac{\d}{\textup{d}t}\mathcal{L}(Z)\leq-\kappa_0 \varepsilon \mathcal{L}(Z), \quad t > T. \]
Hence, one gets the exponential decay of ${\mathcal L}(Z)$.
\end{proof}
Before we close this subsection, we discuss the swarm double sphere (SDS) model on $\mathbb S^{d_1-1} \times \mathbb S^{d_2-1}$ which was recently introduced by Lohe \cite{Lo1}:
\begin{align}\label{SDS}
\begin{cases}
\dot{u}_i=\Omega_iu_i+\displaystyle\frac{\kappa}{N}\sum_{j=1}^N\langle v_i, v_j\rangle (u_j-\langle u_i, u_j\rangle u_i) ,\quad t>0, \\
\dot{v}_i=\Lambda_i v_i+\displaystyle\frac{\kappa}{N}\sum_{j=1}^N \langle u_i, u_j\rangle(v_j-\langle v_i, v_j\rangle v_i),\\
(u_i, v_i)(0)=(u_i^0, v_i^0) \in\mathbb S^{d_1-1} \times \mathbb S^{d_2-1},\quad 1\leq i \leq N,
\end{cases}
\end{align}
where $\Omega_i \in {\mathbb R}^{d_1 \times d_1}$ and $\Lambda_i \in {\mathbb R}^{d_2 \times d_2}$ are real skew-symmetric matrices, respectively, and $\kappa$ denotes the (uniform) nonnegative coupling strength. For homogeneous zero free flows
\[ \Omega_i = O_d, \quad \Lambda_i = O_d, \quad i = 1, \cdots, N, \]
system \eqref{SDS} can be represented as a gradient flow. More precisely, we set an analytical potential ${\mathcal E}_s$ as
\begin{equation*} \label{C-16-1}
\mathcal{E}_s(U, V) := 1-\frac{1}{N^2}\sum_{i, j=1}^N\langle u_i,u_j\rangle \langle v_i, v_j\rangle.
\end{equation*}
Then, system \eqref{SDS} can recast as a gradient system on the compact state space $(\mathbb S^{d_1-1} \times \mathbb S^{d_2-1})^N$:
\begin{equation}
\begin{cases} \label{C-16-2}
\vspace{0.3cm} \displaystyle {\dot u}_i =-\frac{N\kappa}{2} {\mathbb P}_{T_{u_i}\mathbb S^{d_1-1}} \Big( \nabla_{u_i}\mathcal{E}_s(U, V) \Big), \\
\displaystyle {\dot v}_i =-\frac{N\kappa}{2} {\mathbb P}_{T_{v_i}\mathbb S^{d_2-1}} \Big( \nabla_{v_i}\mathcal{E}_s(U, V)\Big),
\end{cases}
\end{equation}
where projection operators onto the tangent spaces of $\mathbb S^{d_1-1}$ and $\mathbb S^{d_2-1}$ at $u_i$ and $v_i$, respectively, are defined by the following explicit formula: for $w_1 \in \mathbb R^{d_1}$ and $w_2 \in \mathbb R^{d_2}$,
\begin{equation*} \label{C-16-3}
{\mathbb P}_{T_{u_i}\mathbb S^{d_1-1}} (w_1) := w_1 - \langle w_1, u_i \rangle u_i, \quad {\mathbb P}_{T_{v_i}\mathbb S^{d_2-1}} (w_2) := w_2 - \langle w_2, v_i \rangle v_i.
\end{equation*}
By the standard convergence result on a gradient system with analytical potential on a compact space, one can derive the following convergence result for all initial data.
\begin{proposition}\label{P3.2}
The following assertions hold.
\begin{enumerate}
\item
Let $\{(u_i, v_i)\}$ be a solution to \eqref{C-16-2}. Then, there exists a constant asymptotic state $(U^{\infty}, V^{\infty}) \in (\mathbb S^{d_1 - 1})^{N} \times (\mathbb S^{d_2 - 1})^{N}$ such that
\[
\lim_{t\to\infty} (U(t), V(t)) = (U^\infty, V^\infty).
\]
\item
Suppose initial data satisfy
\begin{equation} \label{C-16-3}
\min_{1\leq i, j\leq N }\langle u_i^0, u_j^0\rangle>0,\quad \min_{1\leq i, j\leq N}\langle v_i^0, v_j^0\rangle>0,
\end{equation}
and let $\{(u_i, v_i)\}$ be a solution to system \eqref{C-16-2}. Then, one has complete state aggregation:
\[
\lim_{t\to\infty} \max_{1\leq i,j\leq N} |u_i(t)-u_j(t) |=0,\quad \lim_{t\to\infty} \max_{1\leq i,j\leq N} |v_i(t)-v_j(t) |=0.
\]
\end{enumerate}
\end{proposition}
\begin{remark}
1. We give some comments on the results in Proposition \ref{P3.2}. In the first statement, the asymptotic state $(U^\infty, V^{\infty})$ may depend on initial data. Thus, the result does not tell us whether the complete state aggregation occurs or not as it is.\newline
\noindent 2. The second result says that relative states $u_i - u_j$ and $v_i - v_j$ tend to zero asymptotically, but if we combine both results (1) and (2), we can show that the initial data satisfying \eqref{C-16-3} lead to complete state aggregation.
\end{remark}
\vspace{0.5cm}
So far, we have considered sufficient conditions for the emergence of complete state aggregation and practical aggregation. However, this emergent dynamics does not tell us on the solution structure of the LT model. In the following subsection, we consider a special set of solutions, namely tensor product states which can be expressed as tensor products of lower rank tensors.
\subsection{Tensor product states} \label{sec:3.3}
In this subsection, we review tensor product states for the LT model which can be written as a tensor product of rank-1 tensors or rank-2 tensors. In the following definition, we provide concepts of two special tensor product states.
\begin{definition} \label{D3.2}
\emph{\cite{H-K-P1,H-K-P2}}
\begin{enumerate}
\item Let $\{T_i\}$ be a ``completely separable state" if it is a solution to \eqref{LT}, and it is the tensor product of only rank-1 tensors with unit modulus: for $1\leq i \leq N$ and $1\leq k \leq m$,
\begin{equation*} \label{C-17}
T_i = u^1_i \otimes u_i^2 \otimes \cdots \otimes u_i^m, \quad u^k_i \in \mathbb C^{d_k}, \quad |u^k_i | = 1,
\end{equation*}
where $ |\cdot |$ is the standard $\ell^2$-norm in $\mathbb C^d$.
\vspace{0.2cm}
\item Let $\{T_i \}$ be a ``quadratically separable state" if it is a solution to \eqref{LT}. and it is the tensor product of only rank-2 tensors (or matrices) with unit Frobenius norm: for $1\leq i \leq N$ and $1\leq k \leq m$,
\begin{equation*}
T_i = U_i^1 \otimes U_i^2 \otimes \cdots \otimes U_i^m, \quad U_i^k \in \mathbb C^{d_1^k \times d_2^k},\quad \|U_i^k\|_\textup{F} = 1,
\end{equation*}
where $\|\cdot\|_\textup{F}$ is the Frobenius norm induced by Frobenius inner product.
\end{enumerate}
\end{definition}
\subsubsection{Completely separable state} \label{sec:3.3.1}
To motivate our discussion, we begin with rank-2 tensors that can be decomposed into two rank-1 tensors for all time. Then, since its extension to the case of a rank-$m$ tensor will be straightforward, it suffices to focus on rank-2 tensors, and we refer the reader to \cite{H-K-P1,H-K-P2} for details. In next proposition, we show that \eqref{C-7} and \eqref{SDS} are equivalent in the following sense.
\begin{proposition} \label{P3.3}
\emph{\cite{H-K-P2}} The following assertions hold.
\begin{enumerate}
\item
(Construction of a completely separable state):~ Suppose $\{(u_i,v_i) \}$ is a global solution to \eqref{SDS}. Then, a real rank-2 tensor $T_i$ defined by $T_i :=u_i \otimes v_i$ is a completely separable state to \eqref{C-7} with a well-prepared free flow tensor $A_i$ and coupling strengths:
\begin{equation*} \label{C-18}
A_i T_i:= \Omega_i T_i + T_i \Lambda_i^\top, \quad \kappa_1 = \kappa_2 =:\kappa.
\end{equation*}
\item
(Propagation of complete separability):~Suppose $T_i$ is a solution to \eqref{C-7} with completely separable initial data:
\[
T_i^0 =: u_i^0 \otimes v_i^0, \quad 1 \leq i, j \leq N,
\]
for real rank-1 tensors $u_i^0 \in \mathbb S^{d_1-1}$ and $v_i^0 \in \mathbb S^{d_2-1}$. Then, there exist two unit vectors $u_i=u_i(t)$ and $v=v_i(t)$ such that
\[ T_i(t) = u_i(t) \otimes v_i(t), \quad t>0, \]
where $(u_i,v_i)$ is a solution to \eqref{SDS} with $(u_i,v_i)(0) = (u_i^0,v_i^0)$.
\end{enumerate}
\end{proposition}
Thus, in order the investigate the emergent dynamics of some classes of solutions to \eqref{C-7}, it suffices to study the dynamics of \eqref{SDS}.
\begin{theorem} \label{T3.5}
\emph{\cite{H-K-P2}}
Let $\{T_i = u_i \otimes v_i \}$ be a completely separable state to \eqref{C-7} with the initial data $\{ u_i^0 \otimes v_i^0 \}$ satisfying
\[
\min_{1 \leq i,j\leq N } \langle u_i^0,u_j^0\rangle>0 \quad\textup{and}\quad \min_{1 \leq i,j\leq N } \langle v_i^0,v_j^0\rangle>0.
\]
Then, we have complete state aggregation:
\[ \lim_{t \to \infty} \| T_i(t) - T_j(t) \|_\textup{F} = 0, \quad 1 \leq i, j \leq N. \]
\end{theorem}
\begin{proof}
Note that
\[ T_i - T_j = u_i \otimes v_i - u_j \otimes v_j = u_i v_i^{\top} - u_j v_j^{\top} = (u_i - u_j) v_i^{\top} + u_j (v_i^{\top} - v_j^{\top}). \]
This yields
\begin{align*}
\begin{aligned}
\| T_i - T_j \|_\textup{F} &\leq \|(u_i - u_j) v_i^{\top} \|_\textup{F} + \| u_j (v_i^{\top} - v_j^{\top}) \|_\textup{F} \\
&\leq \| u_i - u_j \|_\textup{F} \cdot \| v_i^{\top} \|_\textup{F} + \| u_j \|_\textup{F} \cdot \| v_i^{\top} - v_j^{\top} \|_\textup{F} \\
&= |u_i - u_j | \cdot |v_i| + |u_j | \cdot |v_i - v_j | = |u_i - u_j | + |v_i - v_j |.
\end{aligned}
\end{align*}
By the result of Proposition \ref{P3.3}, one has complete state aggregation of \eqref{C-7} under suitable conditions on initial data and coupling strengths, and the desired estimates follow.
\end{proof}
\begin{remark}
Extension to rank-m tensors of the results in Theorem \ref{T3.5} can be found in Section 6 and Section 7 of \cite{H-K-P2}.
\end{remark}
\subsubsection{Quadratically separable state} \label{sec:3.3.2}
Similar to the previous part, we consider only a finite ensemble of rank-4 tensors that can be decomposed into a tensor product of two rank-2 tensors, and extension to rank-$2m$ tensors will be straightforward.
Now, we introduce the {\it swarm double matrix (SDM) model} induced from the LT model whose elements have rank-4 with a specific condition on natural frequencies $B_j$ and $C_j$ in the same spirit of the SDS model \eqref{SDS}:
\begin{align} \label{C-19}
\begin{cases}
\dot{U}_j=B_jU_j+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^N\left(
\langle V_j, V_k\rangle_\textup{F}~U_kU_j^\dagger U_j
-\langle V_k, V_j\rangle_\textup{F}~U_jU_k^\dagger U_j\right)\\
\hspace{2cm}+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^N\left(
\langle V_j, V_k\rangle_\textup{F}~U_jU_j^\dagger U_k
-\langle V_k, V_j\rangle_\textup{F}~U_jU_k^\dagger U_j\right),\\
\dot{V}_j=C_jV_j+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^N\left(
\langle U_j, U_k\rangle_\textup{F}~V_kV_j^\dagger V_j
-\langle U_k, U_j\rangle_\textup{F}~V_jV_k^\dagger V_j\right)\\
\hspace{2cm}+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^N\left(
\langle U_j, U_k\rangle_\textup{F}~V_jV_j^\dagger V_k
-\langle U_k, U_j\rangle_\textup{F}~V_jV_k^\dagger V_j\right),\\
\end{cases}
\end{align}
where $B_j\in\mathbb C^{d_1\times d_2\times d_1\times d_2}$ and $C_j\in \mathbb C^{d_3\times d_4\times d_3\times d_4}$ are block skew-hermitian rank-4 tensors, respectively. Similar to the SDS model in Section \ref{sec:3.2.2}, the SDM model \eqref{C-19} with homogeneous zero free flows can be rewritten as a gradient system for the following analytical potential:
\begin{equation*} \label{C-19-1}
\mathcal{E}_m(U, V) :=1-\frac{1}{N^2}\sum_{i, j=1}^N \langle U_i, U_j\rangle_\textup{F} \langle V_i, V_j\rangle_\textup{F}.
\end{equation*}
We refer the reader to \cite{H-K-P1} for details. \newline
Next, we consider a special case for the SDM model:
\[ d_1 = d_2 = n, \quad d_3 = d_4 = m, \quad U_j^0 \in {\mathbf U}(n), \quad V_j^0\in\mathbf{U}(m), \]
where $\mathbf{U}(n)$ and $\mathbf{U}(m)$ denote $n\times n$ and $m \times m$ unitary groups, respectively. In this case, one can easily show that unitary properties of $U_j$ and $V_j$ are propagated along \eqref{C-19}:
\begin{equation} \label{C-20}
U_j(t) \in {\mathbf U}(n), \quad V_j(t) \in\mathbf{U}(m), \quad j = 1, \cdots, N,\quad t > 0.
\end{equation}
Note that natural frequency tensors $B_j$ and $C_j$ are rank-4 tensors satisfying block skew-hermitian properties. In order to give a meaning of Hamiltonian, we associate two hermitian matrices, namely, $H_j \in \mathbb C^{n\times n}$ and $G_j \in \mathbb C^{m \times m}$:
\begin{equation} \label{C-21}
[B_j]_{\alpha_1\beta_1\alpha_2\beta_2}=:[-\mathrm{i} H_j]_{\alpha_1\alpha_2}\delta_{\beta_1\beta_2},\quad [C_j]_{\gamma_1\delta_1\gamma_2\delta_2}=:[-\mathrm{i}G_j]_{\gamma_1\gamma_2}\delta_{\delta_1\delta_2}.
\end{equation}
Under the setting \eqref{C-20} and \eqref{C-21}, system \eqref{C-19} reduces to the model on $\Un\times \Um$:
\begin{align} \label{C-22}
\begin{cases}
\dot{U}_j=-\mathrm{i} H_j U_j+\displaystyle\frac{\kappa}{N}\sum_{k=1}^N\left(
\langle V_j, V_k\rangle_\textup{F}~U_k
-\langle V_k, V_j\rangle_\textup{F}~U_jU_k^\dagger U_j\right),\\
\dot{V}_j=-\mathrm{i}G_j V_j+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^N\left(
\langle U_j, U_k\rangle_\textup{F}~V_k
-\langle U_k, U_j\rangle_\textup{F}~V_jV_k^\dagger V_j\right),
\end{cases}
\end{align}
where $H_j U_j$ and $G_j V_j$ are now usual matrix products. We recall two concepts of definitions for emergent behaviors.
\begin{definition} \label{D3.3}
\emph{\cite{H-R}}
Let $(\mathcal U,\mathcal V):=\{U_j,V_j\}_{j=1}^N$ be a solution to \eqref{C-22}.
\begin{enumerate}
\item
System \eqref{C-22} exhibits complete state aggregation if the following estimate holds:
\begin{equation*} \label{Z-0-3}
\lim_{t\to\infty} \max_{1\leq i,j\leq N} \Big( \|U_i(t) - U_j(t) \|_\textup{F} + \|V_i(t) - V_j(t)\|_\textup{F} \Big) =0.
\end{equation*}
\item
System \eqref{C-22} exhibits state-locking if the following relations hold:
\begin{equation*}
\exists~ \lim_{t\to\infty} U_i(t) U_j(t)^\dagger \quad \textup{and}\quad \exists~ \lim_{t\to\infty} V_i(t) V_j(t)^\dagger.
\end{equation*}
\end{enumerate}
\end{definition}
\vspace{0.2cm}
For the emergent dynamics of \eqref{C-22}, we introduce several functionals measuring the degree of aggregation:
\begin{align}
\begin{aligned} \label{C-22-1}
& \mathcal L(t):= \mathcal D(\mathcal U(t)) +\mathcal D(\mathcal V(t)) + \mathcal S(\mathcal U(t) ) +\mathcal S(\mathcal V(t)) , \\
&\mathcal D(\mathcal U(t)) := \max_{1\leq i,j\leq N} \|U_i(t) - U_j(t)\|_\textup{F},~~ \mathcal S(\mathcal U(t)):= \max_{1\leq i,j\leq N} |n- \langle U_i,U_j\rangle_\textup{F} (t)|, \\
&\mathcal D(\mathcal V(t)) := \max_{1\leq i,j\leq N} \|V_i(t) - V_j(t)\|_\textup{F}, ~~ \mathcal S(\mathcal V(t)):= \max_{1\leq i,j\leq N} |m- \langle V_i,V_j\rangle_\textup{F} (t)|.
\end{aligned}
\end{align}
By using the unitarity of $U_i$ and $V_i$, we see
\[
\|U_i - U_j\|_\textup{F}^2 = 2\textup{Re}(n-\langle U_i,U_j\rangle_\textup{F}),\quad \|V_i -V_j\|_\textup{F}^2 = 2\textup{Re}(m-\langle V_i,V_j\rangle_\textup{F}).
\]
Thus, one has
\[
\mathcal D(\mathcal U)^2 \leq 2 \mathcal S(\mathcal U),\quad \mathcal D(\mathcal V)^2 \leq 2 \mathcal S(\mathcal V).
\]
From the relation above, we observe
\[ \lim_{t\to\infty} \mathcal L(t) = 0 \quad \Longleftrightarrow \quad \mbox{complete state aggregation}, \]
and for given positive integers $n$ and $m$, we set
\[ \alpha_{n,m} := \frac{-(12n+27) + \sqrt{ (12n+27)^2 + 48(m-4\sqrt n)(3n+4) } }{4(3n+4)}. \]
Then, the following emergent estimate can be verified by deriving a suitable dissipative differential inequality for ${\mathcal L}$ defined in \eqref{C-22-1}.
\begin{theorem} \label{T3.6}
\emph{\cite{H-K-P1}}
Suppose system parameters and initial data satisfy
\[ H_j = O_n, \quad G_j = O_m, \quad j = 1, \cdots, N, \quad n\geq m >4\sqrt n, \quad \mathcal L^0 <\alpha_{n,m}, \]
and let $\{(U_j,V_j)\}$ be a global solution to \eqref{C-22}. Then, complete state aggregation emerges asymptotically.
\end{theorem}
\begin{proof} By tedious calculation, one can derive for a.e. $t>0$,
\begin{equation} \label{C-23}
\dot {\mathcal L} \leq -2\kappa (m-4\sqrt n)\mathcal L + \kappa (4n+9)\mathcal L^2 +\kappa \left( 2n + \frac 83\right)\mathcal L^3 =:\kappa \mathcal L f(\mathcal L),
\end{equation}
where $f$ is a quadratic polynomial defined by
\[
f(s):= \left( 2n + \frac83\right) s^2 + (4n+9)s - 2(m-4\sqrt n).
\]
By assumption on $n$ and $m$, the coefficient $- 2(m-4\sqrt n) < 0$ and $f=0$ admit a unique positive root $\alpha_{n,m}$. Moreover, one can show that the set $\{ {\mathcal L}(t) < \alpha_{n,m} \}$ is positively invariant under the flow \eqref{C-22}. Thus, by the phase line analysis for \eqref{C-23}, one can see that
\[ \lim_{t \to \infty} {\mathcal L}(t) =0. \]
We refer the reader to \cite{H-K-P1} for details.
\end{proof}
\begin{remark}
For heterogeneous free flows, we can also find a sufficient framework leading to state-locking (see Definition \ref{D3.3}) in terms of system parameters with a large coupling strength and initial data in \cite{H-K-P1}.
\end{remark}
Finally, we consider a special ansatz for a solution $T_i$ with rank-4 to \eqref{LT} as follows:
\begin{equation*}
T_i(t)\equiv U_i(t)\otimes V_i(t),\quad t>0,
\end{equation*}
where $\{(U_j,V_j)\}$ is a solution to \eqref{C-19}. Parallel to Proposition \ref{P3.3}, we show the propagation of the tensor product structure along \eqref{C-19}.
\begin{proposition} \label{P3.4}
\emph{\cite{H-K-P1}}
The following assertions hold.
\begin{enumerate}
\item
(Construction of a quadratically separable state):~Suppose $\{(U_i,V_i) \}$ is a solution to \eqref{C-19}. Then, a rank-4 tensor $T_i$ defined by $T_i :=U_i \otimes V_i$ is a quadratically separable state to \eqref{LT} with a well-prepared free flow tensor $A_i$ satisfying
\begin{align}
\begin{aligned} \label{C-26}
[A_j]_{\alpha_1\beta_1\gamma_1\delta_1\alpha_2\beta_2\gamma_2\delta_2} &=[B_j]_{\alpha_1\beta_1\alpha_2\beta_2}\delta_{\gamma_1\gamma_2}\delta_{\delta_1\delta_2} \\
&\hspace{0.5cm} + [C_j]_{\gamma_1\delta_1\gamma_2\delta_2}\delta_{\alpha_1\alpha_2}\delta_{\beta_1\beta_2}.
\end{aligned}
\end{align}
\vspace{0.2cm}
\item
(Propagation of quadratic separability):~Suppose a rank-4 tensor $T_i$ is a solution to \eqref{LT} with \eqref{C-26} and quadratically separable initial data:
\begin{equation*} \label{C-3}
T_i^0 =: U_i^0 \otimes V_i^0, \quad 1 \leq i \leq N,
\end{equation*}
for rank-2 tensors $U_i^0 \in \mathbb C^{d_1\times d_2}$ and $V_i^0 \in \mathbb C^{d_3 \times d_4}$ with unit Frobenius norms. Then, there exist two matrices with unit Frobenius norms $U_i=U_i(t)$ and $V=V_i(t)$ such that
\[ T_i(t) = U_i(t) \otimes V_i(t), \quad t>0, \]
where $(U_i,V_i)$ is a solution to \eqref{C-19} with $(U_i,V_i)(0) = (U_i^0,V_i^0)$.
\end{enumerate}
\end{proposition}
As we have seen in Theorem \ref{T3.5}, complete state aggregation of a quadratically separable state to the LT model will be completely determined by Proposition \ref{P3.4}.\newline
\section{The Schr\"odinger-Lohe model} \label{sec:4}
\setcounter{equation}{0}
In this section, we review emergent dynamics, standing wave solutions, and numerical simulations for the SL model on a network.
\subsection{Emergent dynamics} \label{sec:4.1} In this subsection, we study sufficient frameworks leading to the emergent dynamics of the SL model over a network in terms of system parameters and initial data, as we have seen in previous section. First, we recall definitions of aggregation for the SL model.
\begin{definition} \cite{A-M,C-H3,H-K1,H-H,H-H-K}
Let $\Psi = \{\psi_j\}$ be a global solution to \eqref{S-L}.
\begin{enumerate}
\item $\Psi$ exhibits complete state aggregation if the following estimate holds:
\[
\lim_{t\to\infty} \max_{1\leq i,j\leq N } \| \psi_i (t) - \psi_j(t) \| =0.
\]
\item $\Psi$ exhibits practical aggregation if the following estimate holds:
\[
\lim_{\kappa\to\infty} \limsup_{t\to\infty} \max_{1\leq i,j\leq N } \| \psi_i (t) - \psi_j(t) \| =0.
\]
\item $\Psi$ exhibits state-locking if the following relation holds:
\[
\exists \lim_{t\to\infty} \langle \psi_i(t),\psi_j(t) \rangle.
\]
\end{enumerate}
\end{definition}
As we have seen from Section 3, the emergent dynamics of the SL model will be different whether the corresponding linear free flows are homogeneous or heterogeneous. Precisely, we define diameter for external potentials:
\[ {\mathcal D}(V) := \max_{1\leq i,j \leq N} \| V_i - V_j \|_{L^{\infty}}.\]
Then, we have the following two cases:
\[ \mathcal D(V)=0:\mbox{homogeneous ensemble}, \quad {\mathcal D}(V) > 0 : \mbox{heterogeneous ensemble}.
\]
\subsubsection{All-to-all network} \label{sec:4.1.1}
In this part, we list up sufficient frameworks for the emergent dynamics of \eqref{S-L} without detailed proofs. First, we consider identical one-body potentials over all-to-all network:
\[
V_i = V_j,\quad i,j=1,\cdots,N,\quad \textup{i.e.,}\quad \mathcal D(V) =0 \quad \textup{and} \quad a_{ik} \equiv 1.
\]
For a given ensemble $ \Psi = \{ \psi_j \}$, we set several Lyapunov functionals measuring the degree of aggregation:
\begin{equation*} \label{D-1}
{\mathcal D}(\Psi) := \max_{1 \leq i,j \leq N} \| \psi_i - \psi_j \|, \quad h_{ij} := \langle \psi_i, \psi_j \rangle, \quad \rho := \left \| \frac{1}{N} \sum_{k=1}^N \psi_k \right \|,
\end{equation*}
where $\langle \cdot, \cdot \rangle$ and $\| \cdot \|$ are $L^2$-inner product and its associated $L^2$-norm, respectively.
\begin{theorem} \label{T4.1}
\emph{\cite{A-M,C-C-H,C-H3,H-H,H-H-K2}}
(Homogeneous ensemble)~Suppose system parameters and initial data satisfy
\[ \kappa >0, \quad a_{jk} \equiv 1, \quad \mathcal D(V) = 0, \quad \| \psi_j^0 \| = 1, \quad 1 \leq j,k \leq N, \]
and let $\Psi= \{ \psi_j \}$ be a global solution to \eqref{S-L} with the initial data $\Psi^0$. Then, the following assertions hold:
\begin{enumerate}
\item
(Emergence of dichotomy): one of the following holds: either complete state aggregation:
\begin{equation*}
\lim_{t\to\infty} \langle \psi_i,\psi_j\rangle =1 \quad \textup{for all $i,j=1,\cdots,N$},
\end{equation*}
or bi-polar aggregation:~ there exists a single index $\ell_0 \in \{1,\cdots,N\}$ such that
\begin{equation*}
\lim_{t\to\infty} \langle \psi_i ,\psi_j \rangle = 1 ~~ \textup{for $i,j\neq \ell_0$} ~~ \textup{and} ~~ \lim_{t\to\infty} \langle \psi_{\ell_0},\psi_i \rangle = -1 ~~ \textup{for $i\neq \ell_0$.}
\end{equation*}
\item
If initial data satisfy ${\mathcal D}(\Psi^0) <\frac{1}{2}$, then complete state aggregation occurs exponentially fast:
\[ {\mathcal D}(\Psi(t)) \lesssim e^{-\kappa t}, \quad \textup{as $t \geq 0$}. \]
\item
If initial data satisfy $\textup{Re} \langle \psi_i^0,\psi_j^0\rangle >0$ for all $i,j$, then complete state aggregation occurs in $H^1$-framework as well:
\begin{equation*}
\lim_{t\to\infty} \|\psi_i-\psi_j\|_{H^1(\mathbb R^d)} =0, \quad \textup{for all $i,j=1,\cdots,N$.}
\end{equation*}
\end{enumerate}
\end{theorem}
\begin{proof} Below, we provide some ingredients without technical details. \newline
\noindent (i)~By direct calculation, we can see that $\rho$ satisfies
\begin{align} \label{D-1-1}
\begin{aligned}
& \frac{d\rho}{dt}= \kappa\left( \rho^2 - \frac1N \sum_{k=1}^N \textup{Re}( \langle \zeta,\psi_k\rangle^2 ) \right), \\
& 1-\rho(t)^2 = \frac{1}{2N^2}\sum_{j,k=1}^N \|\psi_j(t)-\psi_k(t)\|^2.
\end{aligned}
\end{align}
Note that complete state aggregation emerges if and only if $\rho(t)$ converges to 1 as $t \to \infty$. By $\eqref{D-1-1}_1$ and boundedness of $\rho$, the order parameter $\rho$ is non-decreasing in time, and hence it converges to a definite value $\rho_\infty$. After careful analysis of $\rho_\infty$, one can see that the disired dichotomy holds (see \cite{H-H-K}).
\vspace{0.2cm}
\noindent (ii)~By direct calculation, one can derive Gronwall's inequality for ${\mathcal D}(\Psi)$ (see \cite{C-H3}):
\[ {\dot {\mathcal D}}(\Psi) \leq \kappa \Big(-{\mathcal D}(\Psi) + 2 {\mathcal D}(\Psi)^2 \Big), \quad t > 0.
\]
This yields
\[ {\mathcal D}(\Psi(t)) \leq \frac{{\mathcal D}(\Psi^0)}{{\mathcal D}(\Psi^0) + (1-2 {\mathcal D}(\Psi^0)) e^{\kappa t}}. \]
Note that the condition ${\mathcal D}(\Psi^0) <\frac{1}{2}$ is needed to exclude the finite-time blowup of ${\mathcal D}(\Psi(t))$.
\vspace{0.2cm}
\noindent (iii)~In \cite{A-M,H-H}, the authors provided ``{\it finite dimensional approach''} based on the two-point correlation function $h_{ij}=\langle \psi_i,\psi_j \rangle$ measuring the degree of aggregation. Then, the correlation function $h_{ij}$ with $a_{ik} \equiv 1$ satisfies
\begin{equation*}
\frac{\d}{\textup{d}t} h_{ij} = \frac{\kappa}{2N} \sum_{k=1}^N (h_{ik} + h_{kj})(1-h_{ij}), \quad 1 \leq i, j \leq N.
\end{equation*}
By applying dynamical systems theory, we can obtain the desired result.
\end{proof}
\vspace{0.5cm}
Next, we consider a heterogenous ensemble with distinct one-body potentials. In the aforementioned work \cite{H-H}, the authors considered the case of a two-oscillator system whose external potentials are assumed to be the vertical translation of a common potential:
\[
V_j(x) =V(x) + \nu_j,\quad x\in \mathbb R^d,\quad \nu_j \in \mathbb R,\quad j=1,2.
\]
Then for the system, they exactly find the critical coupling strength for the bifurcation where the system undergoes a transition from the emergence of periodic motion to the existence of equilibrium. More precisely,
\begin{align*}
&\textup{(i)}~\kappa < \nu := |\nu_1 - \nu_2| : \textup{$h(t)$ is a periodic function with the period $\frac{\pi}{\sqrt{\nu^2-\kappa^2}}$.} \\
&\textup{(ii)}~\kappa = \nu: \lim_{t\to\infty} h(t) = -{\mathrm i}. \\
&\textup{(iii)}~\kappa>\nu: \lim_{t\to\infty} h(t) = -{\mathrm i}\frac{\nu}{\kappa} + \sqrt{1-\frac{\nu^2}{\kappa^2}.}
\end{align*}
The emergent dynamics of the SL model with $N\geq3$ can be studied similarly as in previous section for some restricted class of initial data and system parameters.
\begin{proposition}
\emph{\cite{Cho-C-H,H-H-K21}}
(Heterogeneous ensemble). The following assertions hold.
\begin{enumerate}
\item
Suppose system parameters and initial data satisfy
\[
\kappa > 54 {\mathcal D}(V) > 0, \quad {\mathcal D}(\Psi^0) < \alpha_2,
\]
where $\alpha_2$ is a larger positive root of $f(x) := 2x^3-x^2+ \frac{2{\mathcal D}(V)}{\kappa}=0$, and let $\{\psi_j\}$ be a solution to \eqref{S-L}. Then, practical aggregation emerges:
\[ \lim_{\kappa \to \infty} \limsup_{t \to \infty} {\mathcal D}(\Psi(t)) = 0. \]
\vspace{0.1cm}
\item
Suppose system parameters and initial data satisfy
\begin{equation} \label{Z-1}
\kappa> 4 {\mathcal D}(\nu) =: 4 \max_{1 \leq i, j \leq N} |\nu_i - \nu_j |, \quad \mathcal D(\Psi^0)^2 < \frac{ \kappa + \sqrt{ \kappa^2- 4\kappa {\mathcal D}(\nu)}}{\kappa},
\end{equation}
and let $\{\psi_j\}$ be a global solution to \eqref{S-L}. Then for each $i,j$, there exist a complex number $\alpha_{ij}^\infty \in \mathbb C$ such that
\[
\lim_{t\to\infty} \langle \psi_i,\psi_j\rangle(t) = \alpha_{ij}^\infty, \quad |\alpha^{\infty}_{ij}|\leq1.
\]
In other words, state-locking occurs. Moreover, the convergence rate is exponential.
\end{enumerate}
\end{proposition}
\begin{proof}
\noindent (i)~ Consider the equation:
\[ f(x) := 2x^3-x^2+ \frac{2{\mathcal D}(V)}{\kappa}=0, \quad x \in [0, \infty), \qquad \kappa > 54 {\mathcal D}(V). \]
Then, the cubic equation $f = 0$ has a positive local maximum $ \frac{2{\mathcal D}(V)}{K}$ and a negative local minimum
$\frac{2D({\mathcal V})}{K} -\frac{1}{27}$ at $x = 0$ and $\frac{1}{3}$, respectively. Moreover, it has two positive real roots $\alpha_1 < \alpha_2 $:
\[ 0 < \alpha_1 < \frac{1}{3} < \alpha_2 < \frac{1}{2}. \]
Clearly, the roots depend continuously on $\kappa$ and ${\mathcal D}(V)$, and
\[ \label{Fact-1}
\lim_{\kappa \to \infty} \alpha_1 = 0, \quad \lim_{\kappa \to \infty} \alpha_2 = \frac{1}{2}.
\]
The flow issued from the initial data satisfying ${\mathcal D}(\Psi^0) < \alpha_2$ tends to the set $\{ \Psi~:~{\mathcal D}(\Psi) < \alpha_1 \}$ in finite-time. Moreover, this set is positively invariant, i.e.,
\[ {\mathcal D}(\Psi(t)) < \alpha_1, \quad t \gg1. \]
On the other hand one has $ \lim_{\kappa \to \infty} \alpha_1(\kappa) = 0$. This yields the desired estimate. Detailed argument can be found in \cite{Cho-C-H}.
\vspace{0.1cm}
\noindent (ii)~In order to show that the limit of $h_{ij} =\langle \psi_i,\psi_j\rangle$ exists, we consider any two solutions to \eqref{S-L} denoted by $\{\psi_j\}$ and $\{\tilde{\psi}_j\}$, and we write
\[
h_{ij} = \langle \psi_i,\psi_j\rangle, \quad \tilde h_{ij} := \langle \tilde\psi_i,\tilde\psi_j\rangle,
\]
and define the diameter measuring the dissimilarity of two correlation functions:
\[
d(\mathcal H, \tilde {\mathcal H})(t) := \max_{1\leq i,j\leq N } |h_{ij}(t) - \tilde h_{ij}(t)|.
\]
Then, we find a differential inequality for $d(\mathcal H,\tilde{\mathcal H})$:
\[
\frac\d\textup{d}t d(\mathcal H,\tilde {\mathcal H}) \leq -\kappa (1- \mathcal D(\Psi)^2) d(\mathcal H,\tilde {\mathcal H}) ,\quad t>0.
\]
Under the assumption \eqref{Z-1}, we show that $d(\mathcal H,\tilde{\mathcal H})$ tends to zero. Once we establish the zero convergence of $d(\mathcal H,\tilde{\mathcal H})$, since our system is autonomous, we can choose $\tilde h_{ij}$ as $\tilde h_{ij}(t) = h_{ij}(t+T)$ for any $T>0$. By discretizing the time $t\in \mathbb R_+$ as $n\in \mathbb Z_+$, one can deduce that $\{h_{ij}(n)\}_{n\in \mathbb Z_+}$ becomes a Cauchy sequence in the complete space $\{z\in \mathbb C:|z|\leq1\}$. Thus, we find the desired complex number $\alpha_{ij}^\infty$.
\end{proof}
\subsubsection{Network structure} \label{sec:4.1.2}
For the interplay between emergent behaviors and network structures, the authors in \cite{H-H-K} considered the following three types of network structures: cooperative, competitive and cooperative-competitive networks depending on their signs:
\begin{align*}
&\textup{(i)}~\textup{Cooperative: $a_{ik}>0$ for all $i,k=1,\cdots,N$.} \\
&\textup{(ii)}~\textup{Competitive: $a_{ik}<0$ for all $i,k=1,\cdots,N$.} \\
&\textup{(iii)}~\textup{Cooperative-competitive: $a_{ik}=(-1)^{i+k}$ for $i<k$ and $a_{ki} =-a_{ik}$. }
\end{align*}
For the cooperative network $a_{ik}$, all values of $a_{ik}$ have positive values so that we can expect complete state aggregation. Here, we associate statistical quantities for the cooperative network $\{a_{ik}\}$:
\begin{align*}
d(\mathcal A) := \max_{1\leq i,j,k\leq N } |a_{ik} - a_{jk}|, ~~ a_m^c := \min_{1\leq i \leq N } \frac1N \sum_{k=1}^N a_{ik},~~ a_M := \max_{1\leq i,j\leq N } a_{ij}
\end{align*}
On the other hand, for the competitive network, all values of $a_{ik}$ are assumed to be negative and they considered the simplest case $a_{ik}\equiv -1$. In this case, each oscillator would exhibit repulsive behaviors. Lastly for the cooperative-competitive network, some of $a_{ik}$ are positive and some are negative. Hence, we expect interesting dynamical patterns other than aggregation, such as periodic orbit or bi-polar aggregation. The arguments above are summarized in the following theorem.
\begin{theorem}
\emph{\cite{H-H-K}}
Let $\{\psi_j\}$ be a global solution to \eqref{S-L}.
\begin{enumerate}
\item Suppose that $a_{ik}$ and initial data satisfy
\begin{equation*}
a_{ik}>0,\quad d(\mathcal A) < a_m^c,\quad \mathcal D(\Psi^0)^2 <\frac{2(a_m^c - d(\mathcal A))}{a_M}.
\end{equation*}
Then, system \eqref{S-L} exhibits complete state aggregation.
\vspace{0.2cm}
\item
Suppose that $a_{ik}$ and initial data satisfy
\begin{equation*}
a_{ik} \equiv -1,\quad \rho^0 >0.
\end{equation*}
Then, a solution to \eqref{S-L} tends to the splay state.
\vspace{0.2cm}
\item Suppose for $N=4$ that $a_{ik}$ and initial data satisfy
\begin{equation*}
a_{ik} = (-1)^{i+k},\quad 2 + h_{12}^0 + h_{14}^0 + h_{23}^0 + h_{34}^0 < h_{13}^0 + h_{24}^0.
\end{equation*}
Then, we have
\begin{equation*}
\lim_{t\to\infty} h_{ij}(t) = (-1)^{i+j}.
\end{equation*}
\end{enumerate}
\end{theorem}
\subsection{Standing wave solution} \label{sec:4.2}
In this subsection, we consider a specific type of a solution whose shape is invariant under the flow, namely, a standing wave solution. We begin with the ansatz for $\psi_i$:
\begin{equation} \label{E-1-1}
\psi_j(x,t) = u(x) e^{-{\mathrm i} E t} \quad \text{with} \quad \|u\|=1, \quad j = 1, \cdots, N,
\end{equation}
where $E$ is a real number.
We substitute the ansatz \eqref{E-1-1} into \eqref{S-L} with an identical harmonic potential $V_i(x) = |x|^2$ to derive
\begin{equation} \label{E-1-2}
-\Delta u + |x|^2 u = Eu, \quad x\in \mathbb R^d.
\end{equation}
In what follows, we consider the one-dimensional $d=1$ case and generalization to the multi-dimensional case can be constructed from the tensor product of a one-dimensional solution. For the one-dimensional case, equation \eqref{E-1-2} becomes
\begin{equation} \label{E-2}
-u_{xx} + x^2u = Eu, \quad x \in \mathbb R.
\end{equation}
Then, it is well known that the equation \eqref{E-2} has eigenvalues $E_k$ and orthonormal eigenfunctions (or constant multiple of the Hermite functions) $u_k$:
\begin{equation} \label{E-2-1}
E_k=2k+1, \qquad u_k(x)= \left( \frac{1}{\sqrt{\pi} 2^k k!} \right)^{1/2} e^{-x^2/2} H_k(x), \quad k = 0,1, \cdots,
\end{equation}
where $H_k$ is the $k$-th Hermite polynomial defined by
\begin{equation*}
H_k(x)=(-1)^k e^{x^2} \frac{d^k}{dx^k}e^{-x^2}, \quad k = 0, 1, \cdots.
\end{equation*}
We now consider the following Cauchy problem:
\begin{equation} \label{E-3}
\begin{cases}
\displaystyle {\mathrm i} u_t = -u_{xx} + x^2 u, \quad x \in \mathbb R,~~t > 0, \\
\displaystyle u(x,0)=u_0(x) = \sum_{k=0}^\infty a_ku_k(x),
\end{cases}
\end{equation}
where $\{u_k\}$ is defined in \eqref{E-2-1} and $\{ a_n\}$ is a $\ell^2$-sequence of complex numbers. Then, it can be easily seen that the solution $u(x,t)$ of the Cauchy problem \eqref{E-3} is given as follows:
\begin{equation*}
u(x,t) = \sum_{k=0}^\infty a_ku_k(x) e^{-{\mathrm i} (2k+1)t} .
\end{equation*}
Next, we study the stability issue for the two types of the standing wave solutions whose existence is guaranteed by the previous argument:
\begin{align*}
\textup{(I)}~~ &\psi_j(x,t) =u_k(x) e^{-\mathrm{i} (2k+1)t} \,\,\textup{ for } j=1, \,2, \cdots, N. \\
\textup{(II)}~~&\psi_1(x,t)=- u_k(x) e^{-\mathrm{i} (2k+1)t} \quad \textup{and}\quad \psi_j = u_k(x) e^{-\mathrm{i} (2k+1)t} ~~ \textup{ for } j\neq1,
\end{align*}
where $u_k$ is given by the formula \eqref{E-2-1}. \newline
Note that the family (I) corresponds to the completely aggregated state, whereas the family (II) corresponds to bi-polar state. As in Theorem \ref{T4.1}, bi-polar state is unstable. Below, under the initial condition for which complete state aggregation occurs, the SL model becomes stable.
\begin{theorem} \label{T4.3}
\emph{\cite{H-H-K2}}
The family $(I)$ is stable in the following sense: for all $\varepsilon>0$, there exists $\delta>0$ such that
\[
\|\psi^0_j - u_k\|<\delta \quad \Longrightarrow \quad \lim_{t\to\infty} \|\psi_j(t) - u_k e^{-{\mathrm i}(2k+1)t} \| < \varepsilon.
\]
\end{theorem}
\begin{remark}
Note that we cannot expect an asymptotic stability of $u_k(x) e^{-{\mathrm i} (2k+1)t}$. In fact, we consider the following form of solution:
\begin{equation*}
\psi_j(x,t) = (1-a) u_k(x) e^{-{\mathrm i} (2k+1)t}+ b u_{k+1}(x) e^{-{\mathrm i} (2k+3)t} \quad \textup{for } j=1, \,2, \cdots, N,
\end{equation*}
where $ |1-a|^2 + |b|^2=1$. Then, it is easy to check that the above $\psi_j$ is a solution to \eqref{S-L} and
\begin{align*}
\| \psi_j(x,t)-u_k(x) e^{-{\mathrm i} (2k+1)t}\|^2= |a|^2+ |b|^2 = 2 \textup{Re} (a).
\end{align*}
\end{remark}
\subsection{Numeric scheme}
In \cite{B-H-K-T}, the authors consider nonlinearly coupled Schr\"odinger-Lohe type system by employing cubic nonlinearity so that the model would be reduced to the Gross-Pitaevskii equation when the coupling is turned off:
\begin{equation} \label{G-P-L}
\begin{cases}
\displaystyle {\mathrm i} \partial_t \psi_j = - \frac{1}{2} \Delta \psi_j + V_j\psi_j + \sum_{k=1}^N \beta_{jk} |\psi_k|^2 \psi_j \\
\displaystyle \hspace{3.7cm}+ \frac{{\mathrm i}\kappa}{2N}\sum_{k=1}^N a_{jk} \left( \psi_k - \frac{ \langle \psi_j,\psi_k\rangle}{\langle \psi_j,\psi_j\rangle} \psi_j \right),\\
\displaystyle \psi_j(x,0) = \psi_j^0(x),\quad (x,t)\in \mathbb R^d\times \mathbb R_+, \quad j=1,\cdots,N.
\end{cases}
\end{equation}
Next, we discuss an efficient and accurate numerical method for discretizing \eqref{G-P-L}. Several numerical examples will be carried out and compared with corresponding analytical results shown in previous section. Due to the external trapping potential $V_j(x)$ ($j=1,\cdots,N$), the wave functions $\psi_j$ ($j=1,\cdots,N$) decay exponentially fast as $|x|\rightarrow\infty$. Therefore, it suffices to truncate the problem \eqref{G-P-L} into a sufficiently large bounded domain $\mathcal{D}\subset\mathbb{R}^d$ with periodic boundary condition (BC). The bounded domain $\mathcal{D}$ is chosen as a box $[a,b]\times[c,d]\times[e,f]$ in 3D, a rectangle $[a,b]\times[c,d]$ in 2D, and an interval $[a,b]$ in 1D.
\subsubsection{A time splitting Crank-Nicolson spectral method}
First, we begin with the description of \eqref{G-P-L} combining a time splitting spectral method and the Crank-Nicolson method. Choose $\Delta t>0$ as the time step size and denote time steps $t_n:=n\Delta t$ for $n\ge0$. From time $t=t_n$ to $t=t_{n+1}$, the GPL is solved in three splitting steps. One solves first
\begin{equation}
\label{eq:TS-L}
{\mathrm i} \partial_t \psi_j = - \frac{1}{2} \Delta \psi_j , \quad x\in\mathcal{D}, \quad j=1,\cdots, N,
\end{equation}
with periodic BC on the boundary $\partial\mathcal{D}$ for the time step of length $\Delta t$, then solves
\begin{equation}
\label{eq:TS-NonL01}
{\mathrm i} \partial_t \psi_j =V_j\psi_j + \sum_{k=1}^N \beta_{jk} |\psi_k|^2 \psi_j, \quad j=1,\cdots, N,
\end{equation}
for the same time step, and finally solves
\begin{equation}
\label{eq:TS-NonL}
{\mathrm i} \partial_t \psi_j = \frac{{\mathrm i}\kappa}{2N}\sum_{k=1}^N a_{jk} \left( \psi_k - \frac{ \langle \psi_j,\psi_k\rangle}{\langle \psi_j,\psi_j\rangle} \psi_j \right),\quad j=1,\cdots, N,
\end{equation}
for the same time-step. The linear subproblem \eqref{eq:TS-L} is discretized in space by the Fourier pseudospectral method and integrated in time analytically in the phase space \cite{B,B-C3,B-C4,B-J-M}. For the nonlinear subproblem \eqref{eq:TS-NonL01}, it conserves $|\psi_k|^2$ pointwise in time, i.e. $|\psi_k(x,t)|^2\equiv |\psi_k(x,t_n)|^2$ for $t_n\le t\le t_{n+1}$ and $k=1,\ldots,N$ \cite{B,B-C3,B-C4,B-J-M}.
Thus it collapses
to a linear subproblem and can be integrated in time analytically \cite{B,B-C3,B-C4,B-J-M}.
For the nonlinear subproblem \eqref{eq:TS-NonL}, due to the presence of the Lohe term involving $\kappa$, it cannot be integrated analytically (or explicitly) in the way for the standard GPE \cite{B,B-C3}. Therefore, we will apply a Crank-Nicolson scheme \cite{B-C40} to further discretize the temporal derivate of \eqref{eq:TS-NonL}.
To simplify the presentation, we will only provide the scheme for 1D. Generalization to $d>1$ is straightforward for tensor grids. To this end, we choose the spatial mesh size as $\Delta x=\frac{b-a}{M}$ with $M$ a even positive integer, and let the grid points be
\begin{equation*}
x_\ell=a+ \ell \Delta x, \qquad \ell=0,\cdots,M.
\end{equation*}
For $1\le j\le N$ denote $\psi_{j,\ell}^n$ as the approximation of $\psi_j(x_\ell, t_n)$ ($0\le \ell\le M$) and $\bm{\psi}_j^n$ as the solution vector with component $\psi_{j,\ell}^n$. Combining the time splitting \eqref{eq:TS-L}--\eqref{eq:TS-NonL} via the Strang splitting and the Crank-Nicolson scheme for \eqref{eq:TS-NonL}, a second order \textit{Time Splitting Crank-Nicolson Fourier Pseudospectral} (TSCN-FP) method to solve GPL on $\mathcal{D}$ reads as:
\begin{eqnarray}
\nonumber
\psi^{(1)}_{j,\ell}
&=&\sum_{p=-M/2}^{M/2-1}e^{-i\Delta t\,\mu_p^2/4}\,\widehat{(\bm{\psi}_j^n)}_p\,e^{i\mu_p(x_\ell-a)},\\[0.5em]
\label{eq:TSCNFP12}
\psi^{(2)}_{j,\ell}
\nonumber
&=&e^{-i\Delta t\left(V_{j}(x_\ell)+\sum_{k=1}^N \beta_{jk}|\psi^{(1)}_{k,\ell}|^2 \right)/2}\;\psi^{(1)}_{j,\ell},\\[0.5em]
\label{eq:TSCNFP2}
i\frac{\psi^{(3)}_{j,\ell}-\psi^{(2)}_{j,\ell}}{\Delta t}
&=&\textcolor{black}{\frac{{\mathrm i}\kappa}{2N}}\sum_{k=1}^{N}a_{jk}\bigg[\psi_{k,\ell}^{(\frac{5}{2})} -\frac{\big\langle \bm{\psi}_{j}^{(\frac{5}{2})},\bm{\psi}_{k}^{(\frac{5}{2})}\big\rangle_{\Delta x} }{\big\langle \bm{\psi}_j^{(\frac{5}{2})}, \bm{\psi}_j^{(\frac{5}{2})}\big\rangle_{\Delta x}}\;\psi_{j,\ell}^{(\frac{5}{2})} \bigg], \\[0.5em]
\nonumber
\psi^{(4)}_{j,\ell}
&=&e^{-i\Delta t\left(V_{j}(x_\ell)+\sum_{k=1}^N \beta_{jk}|\psi^{(3)}_{k,\ell}|^2\right)/2}\;\psi^{(3)}_{j,\ell},\quad
0\le \ell\le M,\\[0.5em]
\nonumber
\psi^{n+1}_{j,\ell}&=&\sum_{p=-M/2}^{M/2-1}e^{-i\Delta t\,\mu_p^2/4}\,\widehat{(\bm{\psi}_j^{(4)})}_p\,e^{i\mu_p(x_\ell-a)},\quad j=1,\cdots,N.
\end{eqnarray}
Here, $\mu_p=\frac{p\pi}{b-a}$, $\widehat{(\bm{\psi}_j^{n})}_p$ and $\widehat{(\bm{\psi}_j^{(4)})}_p$ ($p=-\frac{M}{2},\cdots,\frac{M}{2}$) are the discrete Fourier transform coefficients of the vectors $\bm{\psi}_j^{n}$ and $\bm{\psi}_j^{(4)}$ ($j=1,\cdots, N$), respectively. Moreover,
\begin{equation*}
\psi^{(\frac{5}{2})}_{j,\ell}=:\frac{1}{2}\Big( \psi^{(3)}_{j,\ell}+\psi^{(2)}_{j,\ell}\Big),\qquad
\big\langle \bm{\psi}_{j}^{(\frac{5}{2})},\bm{\psi}_{k}^{(\frac{5}{2})}\big\rangle_{\Delta x}
=:\Delta x \sum_{\ell=0}^{M-1}\psi^{(\frac{5}{2})}_{j,\ell}\,
\bar{\psi}^{(\frac{5}{2})}_{k,\ell}.
\end{equation*}
\textcolor{black}{Although the Crank-Nicolson step \eqref{eq:TSCNFP2} is fully implicit, it can be either solved efficiently by Krylov subspace iteration method with proper preconditioner \cite{AD1} or the fixed-point iteration method with a stabilization parameter \cite{B-C-L}. In addition, TSCN-FP is of spectral accuracy in space and second-order accuracy in time. By following the standard procedure, it is straightforward to show that the TSCN-FP conserve mass of each component in discrete level, i.e.,
$\|\bm{\psi}_j^{n}\|_{l^2}^2:=\big\langle \bm{\psi}_{j}^{n},\bm{\psi}_{j}^{n}\big\rangle_{\Delta x}\equiv \|\bm{\psi}_j^{0}\|_{l^2}^2$
for $n\ge0$ and $j=1,2,\ldots,N$. We omit the details here for brevity.}
\subsubsection{Numerical Results}
In this subsection, we apply the TSCN-FP schemes proposed in the previous subsection to simulate some interesting dynamics.
For our simulation, we choose
\begin{equation*}
\beta=1,\quad \Delta t=\textcolor{black}{2\times10^{-4}},\quad \mathcal{D}=[-12, 12]^d, \quad d=1,2.
\end{equation*}
The potentials and initial data are chosen as follows:
\begin{equation*}
V_j(x)=\pi^2\alpha_j^2\ |x|^2,\quad
\psi_j^0=\sqrt{a_j/\pi}\ e^{-a_j |x-x^{j}_0|^2}.
\end{equation*}
Here, $\alpha_j$ and $x^{j}_0$ are real constants to be given later. In fact, complete state aggregation and practical aggregation estimates do not depend on the form of the initial data and the relative $L^2$-distances of the initial data play a crucial role. However, when we deal with the center-of-mass $x_c$, we used the Gaussian initial data so that they have the symmetric form (see Remark 4.2 in \cite{B-H-K-T}). For the numerical experiment, we introduce the following quantities (see \cite{B-H-K-T} for details):
\begin{align*}
& R(t) := \textup{Re} \langle \psi_1,\psi_2\rangle(t),\quad \mathcal R_{ijk\ell} := \frac{(1-h_{ij})(1-h_{k\ell})}{(1-h_{i\ell})(1-h_{kj})}, \\
& \mathcal B := (\beta_{ij})_{1\leq i,j\leq N},\quad x_c^j(t) := \int_{\mathbb R} x|\psi_j(x,t)|^2 \d x, \\
& \mathcal E[\Psi] := \sum_{j=1}^N \int_{\mathbb R^d}\left[ \frac12 |\nabla\psi_j|^2 + V_j |\psi_j|^2 + \frac12\sum_{k=1}^N \beta_{jk} |\psi_k|^2 |\psi_j|^2 \right] \d x.
\end{align*}
\begin{example}
\label{eg:1d-case}
{\em
Here, we consider the two-component system in 1D, i.e., we take $N=2$ and $d=1$ in \eqref{S-L}. To this end, we
take $(x_1^0, x_2^0) = (2.5, -5)$ and consider the following two cases: for $j=1,2,$
\begin{itemize}
\item[]{\bf Case 1.} fix $\alpha_j=\beta_{j\ell}=1$ ($\ell=1,2$) and vary $\kappa=\textcolor{black}{0,2,20}$.
\item[]{\bf Case 2.} fix $\alpha_j=j$, $\beta_{12}=\beta_{21}=1$, $\beta_{11}=4\beta_{22}=2$ and vary $\kappa=\textcolor{black}{0,2,10, 20}$.
\end{itemize}
Figure \ref{fig:quant_case1} and Figure \ref{fig:quant_case2} depict the time evolution of the quantity $1-R(t)$ (where $R(t)$ is the real part of the correlation function $h
_{12}(t)$), the center of mass $x_c^j(t)$, the component mass $\|\psi_j\|^2$ and the total energy
$\mathcal{E}(t)$ for {\bf Case 1} and {\bf Case 2}, respectively. From these figures and
other numerical experiments not shown here for brevity, we can see the following observations: \newline
\noindent (i). For all cases, we observe that the mass is conserved along time.
\vspace{0.2cm}
\noindent (ii). If the Lohe coupling is off, i.e., $\kappa=0$, both the mass and energy are conserved well, and the center of mass ($x_c^1(t),x_c^2(t))$ are periodic in time with the same period. In addition, for the identical case, i.e., $\mathcal{B}=J_2$ and $V_1(x)=V_2(x)$, $R(t)$ is conserved for identical case.
\vspace{0.2cm}
\noindent (iii). If the Lohe coupling is on, i.e., $\kappa>0$, the phenomena become complicated. The energy is no longer conserved, indeed it decays to some value for large $\kappa$ while it oscillates for small $\kappa$.
\vspace{0.2cm}
\noindent (iv). Moreover, for the identical case, $R(t)$ converges exponentially to 1, which coincides with the theoretical results. Thus, complete state aggregation occurs in this case. After complete state aggregation,
$\|\psi_1(x,t)-\psi_2(x,t)\|_\infty$ will converge to zero and the center of mass $x_c^1(t)$ and $x_c^2(t)$ will become the same and swing periodically along the line
connecting $-\bar{x}_c^0$ and $\bar{x}_c^0$ (here, $\bar{x}_c^0:=(x_c^1(0)+x_c^2(0))/2$).
\vspace{0.2cm}
\noindent (v). Furthermore, for the non-identical case, i.e., $\mathcal{B}\ne J_2$ and $V_1(x)\ne V_2(x)$, $R(t)$ does not
converge to 1, i.e., complete state aggregation cannot occur. However, for large $\kappa$,
$R(t)$ indeed converges to some definite constant $R_\infty<1$. The larger $\kappa$, the smaller value $1-R_\infty$. Meanwhile, $|x_c^1(t)-x_c^2(t)|$ also converges to zero, which could be also justified in a similar process as shown in Corollary 4.1 of \cite{B-H-K-T}.
}
\begin{figure}[h]
\centering
\emph{(a)
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_K0_Rt.eps}}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_K0_Center_of_Mass.eps}}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_K0_Mass_Energy.eps}}
}
\centering{\emph{(b)}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_K1_Rt.eps}}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_K1_Center_of_Mass.eps}}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_K1_Mass_Energy.eps}} }
\centering{\emph{(c)}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_K10_Rt.eps}}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_K10_Center_of_Mass.eps}}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_K10_Mass_Energy.eps}} }
\vspace{-0.3cm}
\caption{Time evolution of the quantity $1-R(t)$ (left), the center of mass $x_c^j(t)$ (middle),
and the component mass $\|\psi_j\|^2$ and the total energy $\mathcal{E}(t)$ (right) for {\bf Case 1} in Example \ref{eg:1d-case}
for \textcolor{black}{$\kappa=0, 2, 20$} (top to bottom).}
\label{fig:quant_case1}
\end{figure}
\begin{figure}[h]
\centering{
\emph{(a)}
\subfigure{\;\,\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K0_Rt.eps}}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K0_Center_of_Mass.eps}}
\subfigure{\,\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K0_Mass_Energy.eps}}
\centering{
\emph{(b)}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K1_Rt.eps}}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K1_Center_of_Mass.eps}}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K1_Mass_Energy.eps}}
}
\centering{
\emph{(c)}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K5_Rt.eps}}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K5_Center_of_Mass.eps}}
\subfigure{\,\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K5_Mass_Energy.eps}}
}
\centering{
\emph{(d)}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K10_Rt.eps}}
\subfigure{\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K10_Center_of_Mass.eps}}
\subfigure{\,\includegraphics[width=0.3\textwidth]{Ex1_1D_NonID_K10_Mass_Energy.eps}}
}
\vspace{-0.3cm}
\caption{Time evolution of the quantity $1-R(t)$ (left), the center of mass $x_c^j(t)$ (middle),
and the component mass $\|\psi_j\|^2$ and the total energy $\mathcal{E}(t)$ (right) for {\bf Case 2} in Example \ref{eg:1d-case}
for \textcolor{black}{$\kappa=0, 2, 10, 20$} (top to bottom). }
\label{fig:quant_case2}
\end{figure}
\end{example}
\begin{example}
\label{eg:2d-case}{\em
Here, we consider the six-component system in 2D, i.e., we take $N=6$ and $d=2$ in \eqref{S-L}. To this end, we here only consider the
identical case, i.e., we choose $\alpha_j=1=\beta_{j\ell}=1$ ($j,\ell=1,\cdots,6$ ). Let \textcolor{black}{$\kappa=20$}, we
consider four cases of initial setups:
\begin{itemize}
\item[]{\bf Case 3.} $x_0^j=\big(6\, \cos((j-1)\pi/3), 6\, \sin((j-1)\pi/3)\big)$, \quad $j=1,\cdots,6$.
\item[]{\bf Case 4.} $x_0^j=\big(2+4\, \cos(j\pi/3-\pi/12), 2+4\, \sin(j\pi/3-\pi/12)\big)$, \quad $j=1,\cdots,6$.
\item[]{\bf Case 5.} $x_0^j=\big(6\, \cos((j-1)\pi/5), 6\, \sin((j-1)\pi/5) \big)$, \quad $j=1,\cdots,6$.
\item[]{\bf Case 6.} Random location:
\begin{eqnarray}\nonumber
&&x_0^1=(3.4707, 2.7526 ), \quad x_0^2=(-0.8931, 1.9951), \\
\nonumber
&& x_0^3=(0.1809, -1.1538), \quad x_0^4=(0.0937, -5.8995 ), \\
\nonumber
&& x_0^5=(-2.9235, -2.4171),\quad x_0^6=(-3.6423, 4.3714).
\end{eqnarray}
\end{itemize}
\textcolor{black}{For {\bf Case 3-Case 6}, Figure \ref{fig:quant_case3to6} illustrates the trajectory and time evolution of the center of mass $x_c^j(t)=:(x_{c1}^j(t), \,x_{c2}^j(t))$, Figure \ref{fig:dyn_R_ijkl} depicts the time evolution of $|\mathcal{R}_{1256}(t)-\mathcal{R}_{1256}(0)|$, $|\mathcal{R}_{2456}(t)-\mathcal{R}_{2456}(0)|$ and $|\mathcal{R}_{3456}(t)-\mathcal{R}_{3456}(0)|$, }and
Figure \ref{fig:density_case3to6} shows the contour plots of $|\psi_1(x,t)|^2$ at different times.
From these figures and other numerical experiments not shown here for brevity, we can see the following observations. \newline
\noindent (i). Complete state aggregation occurs for all cases.
\vspace{0.2cm}
\noindent (ii). All the center of mass $x_c^j(t)$ ($j=1,\cdots,6$) will converge to the same periodic function $\bar{x}_c(t)$, which swings exactly along the line connecting the points ($-\bar{x}_{c1}^0, -\bar{x}_{c2}^0$) and ($\bar{x}_{c1}^0, \bar{x}_{c2}^0$ which are defined as the average of the initial center of mass of the six oscillators:
$$
\left(\bar{x}_{c1}^0, \bar{x}_{c2}^0\right):=\left(\frac16\sum_{j=1}^{6} x^j_{c1}(0), \frac16\sum_{j=1}^{6} x^j_{c2}(0)\right).
$$
Thus, when $\bar{x}_{c1}^0=\bar{x}_{c2}^0=0$, the center of mass will stay steady at
the origin (cf. Figure \ref{fig:quant_case3to6} (a)), which also agrees with the conclusion in Remark 4.2 of \cite{B-H-K-T}. \newline
\noindent \textcolor{black}{(iii)
Before complete state aggregation, all density profiles $|\psi_j(x,t)|^2$ ($j=1,\cdots,6$) will evolve similarly, i.e., the same dynamical pattern as those shown in Figure \ref{fig:density_case3to6} for $|\psi_1|^2$ (only differ from the `color', i.e., the more blurred humps imply the centers of the other five component, while the lighter one shows the one of the current component). While after complete state aggregation (around $t=0.4$, which corresponds to the moment the center of mass $x_c^j$ ($j=1,\cdots,6$) meet together in Figure \ref{fig:quant_case3to6}), all $\psi_j(x,t)$ (hence also for all density profiles) will converge to the same function, whose density changes periodically in time (as shown in columns 4--6 in Figure \ref{fig:density_case3to6}, which also indicate the periodic dynamics for the center of mass that illustrated in Figure \ref{fig:quant_case3to6}).
In addition, before complete state aggregation, although numerical schemes cannot conserve the cross-ratio like quantities $\mathcal{R}_{ijkl}(t) (1\le i, j, k, l \le 6)$ in discretized level, the difference of those quantities from their initial ones are still small (cf. Figure \ref{fig:dyn_R_ijkl}).}
}
\end{example}
\begin{figure}[h]
\centering{
\mbox{(a)
\subfigure{\includegraphics[width=0.23\textwidth]{Ex2_Case1_Traj_T0_to_T1_5.eps}}
\subfigure{\includegraphics[width=0.23\textwidth]{Ex2_Case1_Traj_T1_5_T10.eps}}
\subfigure{\includegraphics[width=0.45\textwidth]{Ex2_Case1_TimeEvoL_Xc_Yc.eps}}
}
\mbox{(b)
\subfigure{\includegraphics[width=0.23\textwidth]{Ex2_Case3_Traj_T0_to_T05.eps}}
\subfigure{\includegraphics[width=0.23\textwidth]{Ex2_Case3_Traj_T05_to_T10.eps}}
\subfigure{\includegraphics[width=0.45\textwidth]{Ex2_Case3_TimeEvoL_Xc_Yc.eps}}
}
\mbox{(c)
\subfigure{\includegraphics[width=0.23\textwidth]{Ex2_Case2_Traj_T0_to_T05.eps}}
\subfigure{\includegraphics[width=0.23\textwidth]{Ex2_Case2_Traj_T05_to_T10.eps}}
\subfigure{\includegraphics[width=0.45\textwidth]{Ex2_Case2_TimeEvoL_Xc_Yc.eps}}
}
\mbox{(d)
\subfigure{\includegraphics[width=0.23\textwidth]{Ex2_Case4_Traj_T0_to_T05.eps}}
\subfigure{\includegraphics[width=0.23\textwidth]{Ex2_Case4_Traj_T05_to_T10.eps}}
\subfigure{\includegraphics[width=0.45\textwidth]{Ex2_Case4_TimeEvoL_Xc_Yc.eps}}
}
}
\vspace{-0.3cm}
\caption{First two columns: trajectory of center of mass $x_c^j(t)$ in $t\in[0, t_c]$ and $t\in[t_c, 10]$ ($t_c=1.5$ for first row while 0.5 for the others).
The third column: time evolution of $x_{c1}^j(t)$ and $x_{c2}^j(t))$ (right). $\circ$ denotes location of $x_c^j(0)$,
while $\Diamond$ denotes the one of $x_c^j(t_c)$.}
\label{fig:quant_case3to6}
\end{figure}
\begin{figure}[h]
\centering{
\subfigure{\includegraphics[width=0.23\textwidth]{Case1_2D_R_IJKL.eps}}
\subfigure{\includegraphics[width=0.23\textwidth]{Case2_2D_R_IJKL.eps}}
\subfigure{\includegraphics[width=0.23\textwidth]{Case3_2D_R_IJKL.eps}}
\subfigure{\includegraphics[width=0.23\textwidth]{Case4_2D_R_IJKL.eps}}
}
\vspace{-0.3cm}
\caption{Time evolution of $|\mathcal{R}_{1256}(t)-\mathcal{R}_{1256}(0)|$, $|\mathcal{R}_{2456}(t)-\mathcal{R}_{2456}(0)|$ and $|\mathcal{R}_{3456}(t)-\mathcal{R}_{3456}(0)|$ for {\bf Case} 3-6 (Left to right). }
\label{fig:dyn_R_ijkl}
\end{figure}
\begin{figure}[h]
\centering{
\mbox{(a)
\subfigure{\includegraphics[width=0.15\textwidth]{Case1_Density_Psi_1_t_0.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case1_Density_Psi_1_t_02.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case1_Density_Psi_1_t_04.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case1_Density_Psi_1_t_05.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case1_Density_Psi_1_t_06.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case1_Density_Psi_1_t_2.eps}}
}
\mbox{(b)
\subfigure{\includegraphics[width=0.15\textwidth]{Case3_Density_Psi_1_t_0.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case3_Density_Psi_1_t_02.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case3_Density_Psi_1_t_04.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case3_Density_Psi_1_t_05.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case3_Density_Psi_1_t_06.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case3_Density_Psi_1_t_1.eps}}
}
\mbox{(c)
\subfigure{\includegraphics[width=0.15\textwidth]{Case2_Density_Psi_1_t_0.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case2_Density_Psi_1_t_02.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case2_Density_Psi_1_t_04.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case2_Density_Psi_1_t_05.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case2_Density_Psi_1_t_06.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case2_Density_Psi_1_t_1.eps}}
}
\mbox{(d)
\subfigure{\includegraphics[width=0.15\textwidth]{Case4_Density_Psi_1_t_0.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case4_Density_Psi_1_t_02.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case4_Density_Psi_1_t_04.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case4_Density_Psi_1_t_05.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case4_Density_Psi_1_t_06.eps}}
\subfigure{\includegraphics[width=0.15\textwidth]{Case4_Density_Psi_1_t_1.eps}}
}
\mbox{
\subfigure{\includegraphics[width=0.5\textwidth,height=1.3cm]{Colorbar_2.eps}}
\subfigure{\includegraphics[width=0.5\textwidth,height=1.3cm]{Colorbar_1.eps}}
}}
\vspace{-0.8cm}
\caption{Contour plots of \textcolor{black}{$|\psi_1(x,t)|^2$} at different time $t$ for {\bf Cases 3-5} in Example \ref{eg:2d-case} (the top 4 rows)
and color bars of the contour plots at $t=0.5$ (bottom left) and other time $t$ (bottom right).
} \label{fig:density_case3to6}
\end{figure}
\section{Conclusion} \label{sec:5}
\setcounter{equation}{0}
In this survey paper, we have reviewed state-of-the-art results on the collective behaviors for two Lohe type aggregation models (the Lohe tensor model and the Schr\"{o}dinger-Lohe model). The former deals with the aggregation dynamics of tensors with the same rank and size, and it turns out to be a generalized model for previously known first-order aggregation models such as the Kuramoto model, the swarm sphere model and the Lohe matrix model. Of course, the Lohe tensor model can be reduced to aggregation models on the hermitian sphere and nonsquare matrix group which are not known in previous literature. For the collective dynamics, we adopt two concepts of aggregation (complete state aggregation and practical aggregation). When all state aggregates to the same state, we call it as complete state aggregation. This phenomenon occurs for a homogeneous ensemble in which all particles follow the same free flow. In contrast, when states are governed by different free flows, complete state aggregation cannot occur. Of course, the rate of state change (we call it velocity) can aggregate to the same value. At present, this strong estimate is not available for the aforementioned models yet.
The latter describes the collective aggregation of the coupled Schr\"{o}dinger equations for wave functions. For a special case, it can be reduced to the swarm sphere and Kuramoto models. For this model, we can also adopt the same universal approaches (Lyapunov functional approach and dynamical systems theory approach for two-point correlation functions). Similar to the Lohe tensor model, this model exhibits the same aggregation phenomena for homogeneous and heterogeneous ensembles. As we have already mentioned, we do not have a complete theory for dealing with a heterogeneous ensemble except for a weak aggregation estimate (practical aggregation). Thus, we would say that our reviewed results for a heterogeneous ensemble are still far from completeness. This will be an interesting research direction for those who are interested in collective dynamics.
\section*{Acknowledgments}
The content of this paper is based on a lecture given by the first author at the Institute of Mathematical Sciences of
National University of Singapore in December 2019. The authors acknowledge the Institute of Mathematical Sciences for generous support and especially thank Prof. Weizhu Bao for the invitation and his warm hospitality. The work of S.-Y. Ha is supported by the NRF grant (2020R1A2C3A01003881) and the work of D. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.2021R1F1A1055929).
|
1,108,101,566,784 | arxiv | \section{Introduction}
\label{sec:intro}
Encoder-decoder based methods have been widely used for generating information preserving embeddings for different data modalities. In the past decades, several encoder-decoder based methods have been proposed for 2D images \cite{segnet, ronneberger2015unet, deeplab} and have shown impressive results for image compression and filtering tasks \cite{ffdnet, deepcnn}. However, it is challenging to extend these methods for 3D point cloud data due to its irregular structure as compared to that of images and 3D voxels. In this work, we focus on designing a deep-learning based encoder-decoder framework for processing point cloud data.
Several methods have been proposed for encoding-decoding of point cloud data \cite{qi2017pointnet, pointcnn, voxelnet, pointrcnn}. PointNet \cite{qi2017pointnet} is a pioneer deep-learning-based method for encoding the point cloud data to lower-dimensional embeddings. These embeddings carry rich information about the point clouds and can be used for downstream tasks like segmentation and classification of point clouds. In a recent work \cite{shu20193d}, the authors propose a tree-structured decoder which uses the idea of graph convolution \cite{gcnn} to generate a point cloud using a noise vector $z \in R^{96}$ sampled from a normal distribution $\mathcal{N}(0, I)$. They propose aggregating the information from parent nodes at each layer instead of spatially adjacent nodes to leverage the tree-structured decoder architecture when applying graph convolution. The unique definition of graph convolution proposed in \cite{gcnn} highlights the effect of using information from parent nodes at multiple levels during the aggregation stage.
Inspired by the tree-structured decoder architecture in \cite{shu20193d}, we propose a tree-structured encoder and a graph convolution mechanism for down-sampling. \comment{which aggregates information from parent nodes instead of spatially neighbouring points similar to \cite{qi2017pointnet, yang2018foldingnet}}We combine our proposed encoder with the decoder proposed in \cite{shu20193d} to create a complete tree-based encoder-decoder framework called TreeGCN-ED \footnote{Code is available at: https://github.com/prajwalsingh/TreeGCN-ED} for processing point clouds. To show the effectiveness of proposed framework, we compare its results with FoldingNet \cite{yang2018foldingnet} architecture. The results show that TreeGCN-ED performs better than FoldingNet on two different evaluation metrics - Chamfer Distance (CD) \cite{fan2016point} and Fr\'echet point cloud distance (FPD) \cite{fan2016point}. We also observe that TreeGCN-ED learns inherent semantic information of the point cloud and hence, performs semantic segmentation without any explicit training. This highlights that our encoder generates more information-preserving embeddings. We also perform ablation studies to determine the effect of feature embedding dimension and data augmentation on the proposed TreeGCN-ED network. We further use the learned embeddings in a transfer learning setup and compare the results for point cloud classification on ModelNet10 and ModelNet40 datasets \cite{wu20153d}. Finally, we demonstrate the applicability of proposed framework for 3D point cloud completion and single image based 3D reconstruction.
\textbf{Contributions.} The following are the major contributions of this work.
\begin{itemize}
\item A tree-structured encoder to generate robust embeddings for point cloud processing using graph convolution.
\item An autoencoder based framework formed by the proposed encoder with the tree-based decoder \cite{shu20193d} for better point cloud reconstruction.
\end{itemize}
\begin{figure*}[!htbp]
\centering
\includegraphics[width=1.0\textwidth]{images/TreeGCN-ED.pdf}
\caption{\textbf{TreeGCN-ED Architecture}. The encoder part (on the left) consist of downsampling and graph convolution modules for encoding the input 3D point cloud into a feature embedding $ \psi \in \mathbb{R}^{n}$. The decoder architecture (on right) is taking the embedding as input from the encoder and reconstructing the 3D point cloud.}
\label{fig:treegcn_ed_model}
\vspace{-3mm}
\end{figure*}
\begin{figure*}[!htbp]
\centering
\includegraphics[width=0.55\textwidth]{images/3dv_TreeGCN-ED_sub_model.pdf}
\caption{\textbf{TreeGCN-ED Down-Branching and Up-Branching Architecture}. The down-branching module (on the left) consists of a fully connected layer followed by max-pooling for feature extraction. Output of the fully connected layer is divided into $\mathbb{C}$ equal components that are passed to the max-pooling layer. The up-branching architecture is responsible for collecting information from the feature embedding of the ancestors and upsampling. The upsampled feature is passed to graph convolution layer for further processing.}
\label{fig:treegcn_ed_submodel}
\vspace{-3mm}
\end{figure*}
\vspace{-2mm}
\section{RELATED WORK}
\label{sec:RelatedWork}
Point cloud is an important data structure that can be used to store information about the geometry of any 3D shape. There are various applications related to point cloud processing. Some of the applications and methods are point cloud classification and segmentation \cite{qi2017pointnet, UnstructuredPC, objectpart, Dohan2015LearningHS, Hackel2016FASTSS, Huang2016PointCL, dynamicgraphcnn}, point cloud completion \cite{yuan2019pcn, chang2015shapenet, Tchapmi_2019_CVPR, yang2018foldingnet}, point cloud auto-encoder \cite{yang2018foldingnet, qi2017pointnet, Kazhdan:2003:RIS, lfd, Girdhar16b, vconvdae, gan3d, lgan}, and Generative Adverserial Networks (GANs) for point cloud \cite{shu20193d, Li2019PointCG}.
In \cite{qi2017pointnet}, the authors propose the first end-to-end deep auto-encoder to directly process point cloud data. The encoder uses 1D CNN and global max-pooling to extract features of the input point cloud. This makes the model permutation invariant. The decoder reconstructs the point cloud using a three-layer fully connected network. FoldingNet \cite{yang2018foldingnet} model is build up on the idea of PointNet \cite{qi2017pointnet}, by proposing an auto-encoder network that uses graph-based method to learn encoding of a point cloud. Edge based convolution method is proposed in \cite{dynamicgraphcnn} to learn the local neighborhood as well as global properties of the 3D shape. In \cite{shu20193d}, the authors have proposed a deep generative model for 3D point cloud generation. This method is unique because the authors have used graph convolution on point cloud data which inherently does not contain any edge connections.
\section{METHOD}
\label{sec:method}
\subsection{Tree-GAN}
Tree-GAN \cite{shu20193d} proposes a deep generative model for 3D point cloud generation. It uses a branching method to gather information from neighbouring points. The accumulated information is then distributed to other points using graph convolution. The point cloud thus generated through this method is implicitly segmented.
In Tree-GAN \cite{shu20193d}, a noise vector $z \in \mathbb{R}^{96}$ is sampled from $\mathcal{N}(0, I)$ and is given as input to the generator network. Each layer of generator consists of a branching network and a graph convolution layer. The branching network accumulates the feature vectors from the previous layers which is then upsampled by the graph convolution layer to generate a new feature vector for that layer. This is repeated until the point cloud of the desired dimension $\mathbb{R}^{3 \times n}$ is obtained at the output. Note that the feature vector for the first layer is the noise vector $z$ itself. The generator and discriminator are trained under WGAN \cite{arjovsky2017wasserstein}.
\subsection{TreeGCN Based Point Cloud Encoder-Decoder}
\begin{figure*}[!htbp]
\centering
\includegraphics[width=1.0\textwidth]{images/3dv_interpolation2.pdf}
\caption{\textbf{Interpolation Result}. Illustration of intra-class (on left) and inter-class (on right) point cloud interpolation.}
\label{fig:interpolation_result}
\vspace{-3mm}
\end{figure*}
In this section, we discuss the proposed method for 3D point cloud processing. The key idea of this approach is inspired from the tree-GAN \cite{shu20193d}, a deep learning based model for generating a 3D point cloud from a noise vector. We use the idea of tree-based graph convolution from \cite{shu20193d} to develop an encoder that extracts rich embeddings to performs well on the unseen point cloud data. Our model takes a 3D point cloud of size $\mathbb{R}^{n \times 3}$ as input, then passes it through sequences of graph-based operation to generate encoding for the point cloud. The generated encoding is then passed through the decoder network where a sequence of graph-based operations upsample the encoding to obtain a $\mathbb{R}^{n \times 3}$ point cloud as the output. The complete network is trained end-to-end by minimizing chamfer loss \cite{fan2016point} and is called TreeGCN-ED.
Fig. \ref{fig:treegcn_ed_model} represents the proposed architecture of TreeGCN-ED. A $\mathbb{R}^{n \times 3}$ point cloud is given as input to the model, which then passes through a down branching network for gathering features from the ancestors of each node. Fig. \ref{fig:treegcn_ed_submodel} shows the down branching network and here, we are first passing each ancestor to a sequence of fully connected. Then, max pooling is applied on it to extract dominant features and we pass this feature to the next network, i.e. graph convolution network. The point cloud continuously passes through the down branching and graph convolution network sequence until the desired encoding $\psi$ is obtained. The generated encoding is given as input to the decoder network. In the decoder, the embedding is first passed through up branching network. The internal working of up branching network is shown in Fig. \ref{fig:treegcn_ed_submodel}, where the encoding is first passed through the fully connected layer. Then the feature vector is concatenated with the ancestor information, which can be useful for reconstructing point clouds. Then the constructed feature is passed on to the graph convolution network. This process is repeated till the point cloud of size $\mathbb{R}^{n \times 3}$ is reconstructed. The decoder architecture is similar to tree-GAN \cite{shu20193d}. The overall model is trained in an end-to-end manner using the chamfer loss function \cite{fan2016point}.
The branching network is an essential part of the TreeGCN-ED network. It helps in accumulating information from ancestors for each node. Every ancestor feature is passed through a fully connected layer at each stage to help the network learn the relation between a node and its neighbour. This is also useful because the point cloud does not have edge connections between points. We use max pooling for selecting important features from the encoded point cloud. Max pooling has been proved to be a permutation invariant function \cite{qi2017pointnet}. We experimented with other pooling functions, such as averaging and adding feature vectors, but max-pooling works better than other methods. Tree graph convolution network learns the semantic segmentation of point cloud implicitly \cite{shu20193d}.
\subsection{Loss Function}
To train the TreeGCN-ED, we have used the chamfer loss \cite{fan2016point} function as it shows promising results for point cloud based reconstruction \cite{shu20193d}.
\begin{equation}
\resizebox{0.5\textwidth}{!}{
\label{eqn:chamferloss}
$
\begin{aligned}
\mathcal{L}_{chamfer}(S_{1}, S_{2}) &= \sum_{x \in S_{1}} min_{y \in S_{2}} || x - y ||_{2}^{2} + \sum_{y \in S_{2}} min_{x \in S_{1}} || x - y ||_{2}^{2}
\end{aligned}$
}
\end{equation}
In Equation \ref{eqn:chamferloss}, $S_{1} \in \mathbb{R}^{n \times 3}$ and $S_{2} \in \mathbb{R}^{n \times 3}$ represents two different point clouds. There are two specific reasons for using this loss function. First, it is permutation invariant \cite{fan2016point}. Second, it penalises the loss function if a point from one set is not matched with its corresponding nearest neighbour in another set and vice-versa. This forces the model to learn information preserving embedding for the point cloud.
\subsection{Data Preprocessing}
To train our model, we have used ShapeNetBenchmarkV0 dataset \cite{chang2015shapenet} consisting of 16 object classes. The dataset is split into training, validation, and testing as per the standard ratio proposed in \cite{chang2015shapenet}. We uniformly sample $2048$ points from the meshes of the ShapeNet dataset \cite{chang2015shapenet}. To ensure uniform sampling of points, we make use of barycentric coordinates for the surface sampling.
\section{EXPERIMENTS AND RESULTS}
\label{sec:exp_eval}
\subsection{Training and Comparison of Encoder-Decoder Model}
We have used ShapeNetBenchmarkV0 dataset \cite{chang2015shapenet}, which consists of 16 object classes, with train-test split officially available along with the dataset. The dataset is first uniformly sampled for 2048 points and then passed on to the network. We have trained the complete network using the chamfer distance \cite{fan2016point} as the loss function till convergence. We compare the performance of TreeGCN-ED with the FoldingNet architecture \cite{yang2018foldingnet} on the test set of \cite{chang2015shapenet}. We use two different metrics for the evaluation task: Chamfer Distance (CD) \cite{fan2016point} and Fr\'echet Point Cloud Distance (FPD) \cite{shu20193d}. The results are shown in Table \ref{table:pointclouded} on ShapeNetBenchmarkV0 dataset \cite{chang2015shapenet}. For fair evaluation, in case of FoldingNet \cite{yang2018foldingnet}, we have considered the first $2025$ minimum point cloud distances for calculating CD, owing to the difference in input and output point cloud size. The quantitative results shown in the table highlight that our proposed method performs better than FoldingNet \cite{yang2018foldingnet} for chamfer distance (CD) as well as Fr\'echet point cloud distance (FPD).
\begin{table*}[!htbp]
\resizebox{\textwidth}{!}{
\begin{tabular}{clllllllllllllllll|l}
\hline
\multirow{2}{*}{Models} & \multirow{2}{*}{Metrics} & \multicolumn{16}{c|}{Object Class} & \\
& & \multicolumn{1}{c}{Airplane} & \multicolumn{1}{c}{Bag} & \multicolumn{1}{c}{Cap} & \multicolumn{1}{c}{Car} & \multicolumn{1}{c}{Chair} & \multicolumn{1}{c}{Earphone} & \multicolumn{1}{c}{Guitar} & \multicolumn{1}{c}{Knife} & \multicolumn{1}{c}{Lamp} & \multicolumn{1}{c}{Laptop} & \multicolumn{1}{c}{Motorbike} & \multicolumn{1}{c}{Mug} & \multicolumn{1}{c}{Pistol} & \multicolumn{1}{c}{Rocket} & \multicolumn{1}{c}{Skateboard} & \multicolumn{1}{c|}{Table} & Average \\ \hline
FoldingNet \cite{yang2018foldingnet} & \multicolumn{1}{c}{\begin{tabular}[c]{@{}c@{}}CD\\ FPD\end{tabular}} & \begin{tabular}[c]{@{}l@{}}0.67\\ 11.10\end{tabular} & \begin{tabular}[c]{@{}l@{}}3.12\\ 87.45\end{tabular} & \begin{tabular}[c]{@{}l@{}}2.82\\ 117.36\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.76\\ 28.47\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.47\\ 12.00\end{tabular} & \begin{tabular}[c]{@{}l@{}}3.34\\ 152.04\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.44\\ 19.55\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.55\\ 19.56\end{tabular} & \begin{tabular}[c]{@{}l@{}}2.60\\ 45.19\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.01\\ 11.19\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.48\\ 33.91\end{tabular} & \begin{tabular}[c]{@{}l@{}}2.28\\ 40.17\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.16\\ 30.14\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.88\\ 32.53\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.35\\ 47.17\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.70\\ 24.62\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.48\\ 44.52\end{tabular} \\ \hline
TreeGCN-ED & \multicolumn{1}{c}{\begin{tabular}[c]{@{}c@{}}CD\\ FPD\end{tabular}} & \begin{tabular}[c]{@{}l@{}}0.50\\ 5.79\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.88\\ 21.02\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.62\\ 16.14\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.45\\ 9.47\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.32\\ 7.85\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.91\\ 51.79\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.40\\ 13.90\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.41\\ 14.80\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.97\\ 21.82\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.88\\ 2.56\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.14\\ 14.67\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.72\\ 12.70\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.79\\ 9.62\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.61\\ 23.91\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.78\\ 13.90\end{tabular} & \begin{tabular}[c]{@{}l@{}}1.41\\ 13.90\end{tabular} & \textbf{\begin{tabular}[c]{@{}l@{}}1.21\\ 11.54\end{tabular}} \\ \hline
\end{tabular}
}
\caption{Comparison of the efficiency for 3D point cloud encoding-decoding between our proposed architecture and the FoldingNet \cite{yang2018foldingnet} model on ShapeNetBenchmarkV0 dataset \cite{chang2015shapenet}.}
\label{table:pointclouded}
\vspace{-2mm}
\end{table*}
\subsection{Point Cloud Interpolation}
To show that our proposed encoder architecture is learning information rich embedding, we perform inter-class and intra-class interpolation experiments between the source and the target point clouds. The interpolation results are shown in Fig. \ref{fig:interpolation_result}.
The intra-class interpolation results illustrate the ability of our model to synthesize novel shapes between two given shapes. We observe that the generated shapes faithfully represent the object class at each interpolation stage and the interpolation is observed to be very smooth. Similarly, in the case of inter-class interpolation, we observe a smooth transition of characteristic class features from one object class to another.
\subsection{t-SNE Visualization}
We use t-SNE \cite{tsne} plot to show how well our encoder model can generate feature embedding for each class. We set the perplexity value to 40. Based on the results of t-SNE plot shown in Fig. \ref{fig:embedding_space}, the inter-class separation is higher. This signifies the discriminative capacity of our proposed encoder model.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.5\textwidth]{images/3dv_1.png}
\caption{The visualization of t-SNE \cite{tsne} clustering of the feature embeddings obtained from TreeGCN-ED model.}
\label{fig:embedding_space}
\vspace{-5mm}
\end{figure}
\subsection{Ablation Studies}
We perform ablation studies to determine how feature embedding $\psi$ dimension and data augmentation affect the ability of TreeGCN-ED to learn a meaningful feature representation. Four different training regimes are compared on the ShapeNetCore.v2 test-set \cite{largeshapenet} which consist of 55 classes. In \textit{Regime 1 and 2}, the dimension of feature embedding is fixed to 256 and 512, respectively, without augmentation. Similarly, in \textit{Regime 3 and 4}, the dimension of feature embedding is fixed to 256 and 512, respectively, but with augmentation. We use ShapeNetCore.v2 dataset \cite{largeshapenet} to train TreeGCN-ED for all the four regimes. Table \ref{table:exp_cd} shows the variation in chamfer distance for the above mentioned regimes. Based on the results, Regime 4 gives the best model performance.
\begin{table}[!htbp]
\centering
\footnotesize
\begin{tabular}{lllll}
\hline
\multicolumn{1}{c}{\multirow{2}{*}{Dataset}} & \multicolumn{2}{c}{No Augmentation} & \multicolumn{2}{c}{Rotation Augmentation} \\
\multicolumn{1}{c}{} & $\psi = 256$ & $\psi = 512$ & $\psi = 256$ & $\psi = 512$ \\ \hline
ShapeNetCore.v2 & 10.90 & 10.07 & 8.82 & 7.88 \\ \hline
\end{tabular}
\caption{Quantitative results for all four regimes for the task of 3D point cloud encoding-decoding. Chamfer distance \cite{fan2016point} is used as the metric for comparison.}
\label{table:exp_cd}
\vspace{-5mm}
\end{table}
Furthermore, we also evaluate the efficiency of feature representation learning of TreeGCN-ED on ModelNet10 and ModelNet40 datasets \cite{wu20153d} for all the four regimes. We follow the same procedure as mentioned in \cite{yang2018foldingnet} to train a linear SVM classifier on features extracted from trained TreeGCN-ED for the ModelNet datasets \cite{wu20153d}. Table \ref{table:exp_acc} shows the variation in classification accuracy for all the four regimes on the test set of ModelNet datasets \cite{wu20153d}. Based on the results, Regime 4 gives the best model performance.
\begin{table}[!htbp]
\centering
\footnotesize
\begin{tabular}{lllll}
\hline
\multicolumn{1}{c}{\multirow{2}{*}{Dataset}} & \multicolumn{2}{c}{No Augmentation} & \multicolumn{2}{c}{Rotation Augmentation} \\
\multicolumn{1}{c}{} & $\psi = 256$ & $\psi = 512$ & $\psi = 256$ & $\psi = 512$ \\ \hline
ModelNet10 & 0.83 & 0.83 & 0.85 & 0.85 \\
ModelNet40 & 0.71 & 0.72 & 0.73 & 0.73 \\ \hline
\end{tabular}
\caption{Quantitative results for all four regimes for the task of 3D point cloud classification using transfer learning on ModelNet10 and ModelNet40 dataset \cite{wu20153d}.}
\label{table:exp_acc}
\vspace{-2mm}
\end{table}
It can be easily argued that the tree-GAN \cite{shu20193d} decoder itself is enough for point cloud processing at hand. However, to establish the need and examine the strength of the proposed encoder, we perform an additional experiment by replacing it with the PointNet \cite{qi2017pointnet} encoder to train the complete network on ShapeNetBenchmarkV0 dataset \cite{chang2015shapenet}. We observed that the average CD is $8.65$ with the PointNet encoder and $1.21$ with the proposed encoder. This clearly establishes the efficacy of the proposed encoder.
\subsection{Applications}
We showcase two potential applications of our proposed method: 3D point cloud completion and single image based 3D reconstruction.
\subsubsection{3D Point Cloud Completion}
The task of 3D point cloud completion is to reconstruct the incomplete input point cloud. To achieve this, we train TreeGCN-ED on the Completion 3D benchmark dataset \cite{yuan2019pcn, chang2015shapenet, Tchapmi_2019_CVPR} and perform a qualitative evaluation on the officially available test set. The qualitative results are shown in Fig. \ref{fig:shapecompletion}. Since the ground truth for the test set is not available, we only perform qualitative analysis.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.5\textwidth]{images/shapecompletion.pdf}
\caption{Qualitative results on the test set of Completion 3D benchmark dataset \cite{yuan2019pcn, chang2015shapenet, Tchapmi_2019_CVPR}.}
\label{fig:shapecompletion}
\vspace{-5mm}
\end{figure}
\subsubsection{Single Image Based 3D Reconstruction}
3D reconstruction from a single image is an ill-posed problem. This problem arises due to the ambiguity involved in the occluded part of the object, which is not visible in the image. We attempt to solve this problem using TreeGCN-ED architecture. We train the TreeGCN-ED model on 16 different classes of the ShapeNetBenchmarkV0 dataset \cite{chang2015shapenet} till convergence. Later, we replace the encoder of TreeGCN-ED with a CNN based architecture to extract image features. We freeze the trained weights of the decoder and train the image encoder network end-to-end for 3D reconstruction. We use Chamfer Distance (CD) \cite{fan2016point} as the loss function. We use the synthesized images available in ShapeNetBenchmarkV0 dataset \cite{chang2015shapenet} to train the single image to 3D shape reconstruction model. The qualitative results are shown in Fig. \ref{fig:image23d}.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.5\textwidth]{images/imageto3d.pdf}
\caption{Qualitative results for single image to 3D reconstruction on the test set of ShapeNetBenchmarkV0 dataset \cite{chang2015shapenet}.}
\label{fig:image23d}
\vspace{-5mm}
\end{figure}
\section{CONCLUSION}
\label{sec:conclusion}
In this work, we propose a tree-structured graph convolution-based encoder architecture and combine it with the decoder of tree-GAN to create a complete tree-structured encoder-decoder model for processing 3D point cloud data. The experimental results of our proposed architecture highlight the effectiveness of the encoder model in learning information-rich features. We also showcase that TreeGCN-ED can be used for the task of point cloud completion and single image based 3D reconstruction.
\section{Acknowledgments}
This research is supported by SERB MATRICS and SERB IMPRINT-2 grants. Also, we would like to thank Ashish Tiwari and Dhananjay Singh for their constructive and valuable feedback.
\bibliographystyle{IEEEbib}
\small{
|
1,108,101,566,785 | arxiv | \section*{Some macros}
\begin{verbatim}
\newcommand{\mathrm{Gr}}{\mathrm{Gr}}
\newcommand{\mathcal R}{\mathcal R}
\newcommand{\mathcal F}{\mathcal F}
\newcommand{\mathcal T}{\mathcal T}
\newcommand{\mathcal K}{\mathcal K}
\newcommand{\mathcal O}{\mathcal O}
\newcommand{\mathscr P}{\mathscr P}
\newcommand{\mathrm{St}}{\mathrm{St}}
\newcommand{\Uh}[2][G]{{\mathcal U}_{#1}^{#2}}
\newcommand{\Bun}[2][G]{\operatorname{Bun}_{#1}^{#2}}
\newcommand{\mathbb D}{\mathbb D}
\newcommand{\boldsymbol\omega}{\boldsymbol\omega}
\newcommand{\operatorname{Stab}}{\operatorname{Stab}}
\end{verbatim}
\verb+\cAh+ yields $\cAh$.
\verb+\cAh[G,\mathbf N]+ yields $\cAh[G,\mathbf N]$.
\verb+\cAh[G]+ yields $\cAh[G]$.
\verb+\cAh[]+ yields $\cAh[]$.
\end{NB}
\begin{CJK}{UTF8}{ipamp}
\section{複素シンプレクティック多様体と変形量子化}
$G$を複素簡約群とし、$\mathbf M$をそのシンプレクティックな表現とする。すなわち、$\mathbf M$は${\mathbb C}$上のシンプレクティック形式を持つベクトル空間であり、$G$はシンプレクティック形式を保って線形に作用している。
{\bf Coulomb枝}$\mathcal M_C\equiv\mathcal M_C(G,\mathbf M)$は、理論物理における場の量子論の研究に動機付けられて発見された、$(G,\mathbf M)$からアファイン複素シンプレクティック多様体\footnote{一般には特異点を持つ。高々Beauvillの意味でシンプレクティックな特異点しか持たないと期待されているが、証明は与えられていない。}を作るレシピである。
\begin{equation*}
(G,\mathbf M) \leadsto \mathcal M_C(G,\mathbf M)
\end{equation*}
作り方は、これまで知られている代数多様体の与え方、多項式の零点、商空間、等々とはかなり毛色が異なる。まず座標環${\mathbb C}[\mathcal M_C]$を、幾何学的表現論でよく使われるホモロジー群とその上の合成積を考える方法で作る。そしてその可換環のスペクトラムとして$\mathcal M_C$を定め、その幾何学的な性質を調べる、という手法を取る。
あとで説明するように、$\mathcal M_C$は $T^* T^\vee/W$と双有理同型である。
\begin{equation*}
\mathcal M_C \approx T^*T^\vee/W = \mathfrak t\times T^\vee/W
\end{equation*}
ここで、$T^\vee$は$G$の極大トーラス$T$の双対トーラスであり、$W$はワイル群である。$T^*T^\vee$は\linebreak[3]$T^\vee$のco\-tangent bundleで、$\mathfrak t$は$T$のリー環である。
特に、$\mathcal M_C$の双有理類は表現$\mathbf M$には依存しない。
上で言及したように、ホモロジー群とその上の合成積を用いて環を作るレシピは、幾何学的表現論ではよく使われてきた。表現論の研究が目的であるから、そこでは、非可換環を構成するのが普通である。
実際、Coulomb枝においてもその構成法から、$\mathcal M_C$の変形量子化$\cAh$が同時に作られる。ここで、変形量子化とは、${\mathbb C}[\hbar]$上で定義された非可換環 $\cAh$ であって、$\cAh/\hbar\cAh$が
$\mathcal M_C$の座標環 ${\mathbb C}[\mathcal M_C]$ に等しく、Poisson 括弧
\begin{equation*}
\{ f, g\} = \left.
\frac{\tilde f\tilde g - \tilde g\tilde f}{\hbar}\right|_{\hbar=0}, \qquad
\tilde f|_{\hbar=0} = f, \quad \tilde g|_{\hbar=0} = g
\end{equation*}
が、シンプレクティック構造から来るものに一致しているものをいう。これを{\bf 量子化されたCoulomb枝}とよぶ。
振り返って考えれば、表現論で研究されてきた非可換環は、可換環の変形として得られているものが多い。しかし、合成積を用いて可換環を新しく系統的に構成しようという発想は、今回の研究で初めて現れたものだと認識している。
最初の論文\cite{2015arXiv150303676N}では、一般の$\mathbf M$を考えていたが${\mathbb C}[\mathcal M_C]$のベクトル空間としての構成にとどまり、積の定義はあとの\cite{main}で与えられた。その際に、$\mathbf M = \mathbf N\oplus\mathbf N^*$という形であると仮定した。この仮定は技術的なものなのか、もしくはより本質的なものなのかはまだ分からないが、物理ではこの仮定を満たさないCoulomb枝も考察されており、どんな条件が満たされていれば定義ができるのか、検討の余地が残されている。なお$\mathbf M=\mathbf N\oplus\mathbf N^*$を仮定する次々節以降では、$\mathcal M_C(G,\mathbf N)$という記号を使うが、混乱のおそれはないと思われる。
$(G,\mathbf M)$ に対して、アファイン複素シンプレクティック多様体を与える、よく知られたレシピがある。それは、シンプレクティック商
\begin{equation*}
\mathbf M/\!\!/\!\!/ G = \mu^{-1}(0)/\!\!/ G
\end{equation*}
であり、物理では{\bf Higgs枝}とよばれる。Coulomb枝に対応して$\mathcal M_H \equiv \mathcal M_H(G,\mathbf M)$であらわす。
上の式で、$\mu\colon\mathbf M\to\operatorname{Lie}G^*$は運動量写像であり、$\mu^{-1}(0)$を$G$で(幾何学的不変式論の意味で)割ってできる商空間が$\mu^{-1}(0)/\!\!/ G$である。
$\mathbf M=\mathbf N\oplus\mathbf N^*$となっているときには、$\mathbf N$の上の多項式係数の微分作用素の全体のなす非可換環$\mathcal D(\mathbf N)$が、$\mathbf M$の変形量子化になる。($\hbar$を入れるには、次数によるフィルターに関してRees代数を作る。) シンプレクティック商と同様に、$\mathcal D(\mathbf N)$の$G$作用に関する`商'を作る構成法が、量子シンプレクティック簡約として知られており、それが $\mathcal M_H$の変形量子化を与える。
Higgs枝として、箙多様体やトーリック超ケーラー多様体を例として、表現論的に興味深いシンプレクティック多様体や、その変形量子化が現れることを経験している。
一方、Coulomb枝の研究は始まったばかりであるが、Higgs枝としては得られないシンプレクティック多様体(正確には、有限次元のシンプレクティック・ベクトル空間のシンプレクティック商としての記述が知られていない空間)も現れるので、今後重要性が高まるレシピであると期待している。
また、同じ $(G,\mathbf M)$ からできるHiggs枝とCoulomb枝は、Braden-Licata-Proudfoot-Webster \cite{2014arXiv1407.0964B}の意味で、シンプレクティック双対であることが期待されている。シンプレクティック双対は、複素シンプレクティック多様体のペアの間に不可思議な関係があることを期待するもので、全体像はまだまだ研究途中で見えていないが、少なくともHiggs枝とCoulomb枝の両方を同時に研究することに意味があり、重要であることを示唆している。
\cite{2014arXiv1407.0964B}においては、そのような複素シンプレクティック多様体のペアの例が例示されていたにとどまっていたが、Coulomb枝による系統的な構成が与えられたことになる。ただし、\cite{2014arXiv1407.0964B}で期待されている不可思議な関係のチェックは、今後の課題である。特に、\cite{2014arXiv1407.0964B}は定式化において二つの複素シンプレクティック多様体は、ともにシンプレクティックな特異点解消を持つことが仮定されていたが、多くのHiggs枝、Coulomb枝においてこの仮定は成立しないので、何を期待するのか、ということまで含めて検討する必要がある。
\section{物理的な背景}
前節の説明で、Coulomb枝の数学的な研究に意味があることが伝えられたと期待するが、今節では物理的な背景について、筆者の理解できる範囲内で説明を試みる。ここに書いてあることを理解する必要はないし、筆者自身もよく理解したとは思っていないが、次節で説明する定義がどこから発見されたのかを理解するためと、今後新たな研究成果を上げるためには、背景にある物理のある程度の理解が必要であろうと思っている。
先を急ぐ読者は、この節を飛ばして読んでも構わないが、より深い理解を求める方は、今節を読み、また物理の文献に挑戦していただきたい。物理の文献は\cite{2015arXiv150303676N}にあげたので、これを参照すること。
また、この論説は\cite{main}と同様に、この節以外は物理を知らなくても読めるように書かれており、物理の文献の引用をしない。これは、あくまで読みやすさのためのもので、原典は\cite{2015arXiv150303676N}の文献表から見つけてあたってもらいたい。
物理では、微分幾何と同様に複素簡約群 $G$ の代わりに、その極大コンパクト部分群$G_c$ を取り扱う。同様に、$\mathbf M$には$G_c$で保たれる内積が入っているものとする。
組$(G_c,\mathbf M)$に対して、物理学者は$3$次元の$\mathcal N=4$超対称ゲージ理論を定める。
これは、場の全体のなす無限次元の空間の上の汎関数(ラグランジアン)を与え、量子化して得られる場の量子論の例である。
場のうちで主要なものは、${\mathbb R}^3$上の$G_c$主束$P$上の接続と、$\mathbf M$に値を持つような$P$の切断である。さらにいくつかのベクトル束の切断を場に加える。足される場は物理的には必要であるが、ここでは雑な理解しか与えないので、説明は省く。いずれにせよ、物理学者は、接続の曲率や切断の微分を含んだラグランジアンを書き、場の量子論を考える。
ラグランジアンが極小値を取るような接続や切断(と説明を省略した場)のconfigurationは、量子力学でいうところの古典解に対応しており、基本的な対象である。
今の状況では、極小値をとる場は、ただ一つではなく、有限自由度を持った空間になっている。これは、物理では{\bf 真空のモジュライ空間}とよばれる。
上に述べたように、ラグランジアンは接続の曲率や切断の微分などの和として与えられる。極小値を与える場は、和のうちのいくつかの項が消えているものであり、どの項が消えているかで分けて、真空の枝という考え方をする。その中の典型的なものがHiggs枝$\mathcal M_H$と\linebreak[3]Coulomb枝$\mathcal M_C$である。
Higgs枝$\mathcal M_H$は前節に述べたシンプレクティック商であり、微分幾何学的には超ケーラー商である。接続は自明接続で、定数な切断だけが生き残るので、$\mathbf M$の情報だけが残って超ケーラー商になる。ここでは、超ケーラー商の定義は復習しないので、例えば \cite{MR1193019}を参照してほしい。
例えば筆者が長年に渡り研究している箙多様体や、トーリック超ケーラー多様体は、Higgs枝の例になっている。
一方、Coulomb枝は $(T^\vee_c\times({\mathbb R}^3\otimes\mathfrak t_c))/W$ となる。$T^\vee_c$は、$G_c$の極大トーラス $T_c$ の双対であり、$\mathfrak t_c$は$T_c$のリー環で、$W$はワイル群である。前節にでてきた$T^*T^\vee/W$と同じものである。
Coulomb枝では、切断は$0$であり、$({\mathbb R}^3\otimes\mathfrak t_c)$は、ここで省略した場の成分から来るものである。$T^\vee_c$成分は接続から来るのだが、物理で`双対'とよんでいる、無限次元の接続の空間でのFourier変換を取るために、双対トーラス$T_c^\vee$に値をとり、かつスカラーになっている。この議論は、そのまま数学にのせるのは難しいと思うが、\subsecref{subsec:torus}と定理\ref{thm:classical}で見るように数学的に厳密な定義から出発して、$T^*T^\vee/W$ を再現することができ、なぜ双対トーラスになるのかも説明される。
$\mathcal M_C$と$\mathcal M_H$、より一般に真空のモジュライ空間は、超対称ゲージ理論の重要な情報を含んでおり、物理的にはゲージ理論を解析する上で、これを理解することは大切なステップである。特に、最初に与えた超対称性ゲージ理論が、真空のモジュライ空間をtargetにするような写像にいろいろな場を足して定められる超対称性場の理論と、低エネルギーにおいて等価になる。(ここで出てくる超対称性場の理論は、トポロジカル捻りをするとRozansky-Witten不変量を与えるものである。)
しかし、古典解に対応するような、ラグランジアンの最小値だけを見ていて、量子的な効果を含んだ場の理論の等価性を導くのは、過度な期待である。
物理学者は、そこでCoulomb枝は量子補正を受ける、と主張する。すなわち、Coulomb枝が$(T_c^\vee\times({\mathbb R}^3\otimes\mathfrak t_c))/W$であるのは古典的な記述であって、量子的な効果を受けたあとのCoulomb枝は、変更される、と主張する。ただし、超対称性から超ケーラー多様体であることは量子効果のあとも保たれる。
この変更が、超ケーラー構造の存在以外にどの程度$(T_c^\vee\times({\mathbb R}^3\otimes\mathfrak t_c))/W$を変更するのか、筆者には想像ができないが、数学的な定義のもとでは、$\mathcal M_C$は$(T_c^\vee\times({\mathbb R}^3\otimes\mathfrak t_c))/W$と双有理同値であり、確かに変更していると取れなくもない。
というわけで、物理学者による$\mathcal M_C$の定義は、$\mathcal M_H$とは違って数学的には厳密とはいえず、そのままでは数学的に取り扱うことができない。筆者は、1996年11月にケンブリッジのニュートン研究所に滞在中に、Wittenの連続講演で初めてCoulomb枝の説明を聞いたが、研究対象として扱うことは長らくできなかった。出てくる超ケーラー多様体はよく知っているものであったので、頭の隅にずっと置いていたが、解決するのは難しいと考えていた。
新しい着想を得たのは、2014年秋にウォーリックでHananyの講演を聞いたときである。\linebreak[3]Hananyは、$\mathcal M_C$の座標環${\mathbb C}[\mathcal M_C]$の${\mathbb C}^\times$作用に関する指標を与える一般的な公式(モノポール公式)が成立すると説明した。この公式は、$G$のコウェイトに関する和で与えられ、足される項はコウェイトで定める具体的な式である。そして、知られているCoulomb枝の多くの例で、モノポール公式が確かに成立していることが、確かめられていた。
そこで、モノポール公式を再現するような空間を実現するためにはどうしたらいいかを逆に考えて発見したのが、\cite{2015arXiv150303676N}であり、その修正版の\cite{main}である。私が、どのように試行錯誤したかは\cite{2015arXiv150303676N}に説明したので、興味ある読者は参照されるとよいだろう。特に、$3$次元の位相的場の理論があると仮想して試行をしているところは、今後の発展の手がかりになるはずである、と期待しているところである。
\section{数学的な定義}\label{sec:def}
この節以降は、$G$は複素簡約群、$\mathbf N$はその有限次元表現とする。$\mathbf N$は既約でなくてもよく、$0$であってもよい。最初の節で述べた$\mathbf M$は$\mathbf N\oplus\mathbf N^*$として与えられるが、ここから先は$\mathbf M$は少なくとも表面上は出てこない。
$D = \operatorname{Spec}\nolimits{\mathbb C}[[z]]$をformal disk、$D^\times = \operatorname{Spec}\nolimits{\mathbb C}((z))$をformal punctured diskとする。$\mathbf N((z))$, $\mathbf N[[z]]$をそれぞれ $\mathbf N_\mathcal K$, $\mathbf N_\mathcal O$ で表す。同様に$G_\mathcal K = G((z))$, $G_\mathcal O = G[[z]]$とする。
アファイン・グラスマン$\mathrm{Gr}_G$は、モジュライ空間
\begin{equation*}
\left.\left\{ (\mathscr P,\varphi) \middle|
\begin{aligned}[m]
& \text{$\mathscr P$は$D$上の(代数的な)$G$-主束}\\
& \text{$\varphi\colon\mathscr P|_{D^\times}\to G\times D^\times$は、$\mathscr P$の$D^\times$上での自明化}
\end{aligned}
\right\}\middle/\text{isom.}\right.
\end{equation*}
として定義される。射影多様体の直極限としてのind-schemeの構造を持つことが知られている。集合論的には$\mathrm{Gr}_G = G_\mathcal K/G_\mathcal O$ と表される。すなわち、$\mathscr P$の$D$上での自明化をとって、$\varphi$を$G_\mathcal K$の元で表し、最初の自明化のambiguityの分の$G_\mathcal O$で割って、$G_\mathcal K/G_\mathcal O$となる。
さらに、これに表現$\mathbf N$に付随したベクトル束$\mathscr P\times_G\mathbf N$の切断$s$を付け加えた三つ組$(\mathscr P,\varphi,s)$のモジュライ空間を$\mathcal T$で表す。集合論的には$G_\mathcal K\times_{G_\mathcal O}\mathbf N_\mathcal O$である。$s$の展開を途中で止めることによって、$\mathcal T$は射影多様体上のベクトル束の逆極限の直極限になる。
以下では、$\mathcal T$や、その閉部分多様体のホモロジー群を取り扱うが、厳密には有限次元の空間のホモロジー群の極限として取り扱われる。
$\mathcal T$の閉部分多様体$\mathcal R$として、$\varphi(s)$が$D$まで伸びるという条件を課して、定められる空間と定義する。
\begin{equation*}
\mathcal R = \{ (\mathscr P,\varphi,s) \mid \varphi(s)\in\mathbf N_\mathcal O \}/\text{isom.}
\end{equation*}
$\varphi$は$D^\times$上の自明化でしかないから、$\varphi(s)$は一般には原点に極を持つ有理型切断であって、その特異部分が$0$であるという条件を課したものが$\mathcal R$である。集合論的には、
\(
\mathcal R = \{ [g,s]\in G_\mathcal K\times_{G_\mathcal O}\mathbf N \mid gs\in\mathbf N_\mathcal O \}
\)
と記述できる。
この空間$\mathcal R$が主要登場人物である。その意味は、より大きな空間
\begin{equation*}
\{ (\mathscr P_1, \varphi_1, s_1, \mathscr P_2, \varphi_2, s_2)
\in \mathcal T\times\mathcal T \mid \varphi_1(s_1) = \varphi_2(s_2) \}/\text{isom.}
\end{equation*}
を考えると、分かりやすいだろう。これは、$D$上の$G$主束と$D^\times$上の自明化および$\mathbf N$に付随したベクトル束の切断の組が二つあって、切断が$D^\times$上で自明化を通じて等しい、というファイバー積$\mathcal T\times_{\mathbf N_\mathcal K}\mathcal T$に他ならない。$(\mathscr P_2,\varphi_2)$が$\mathrm{Gr}_G$の原点、すなわち$\varphi_2$が$D$上の自明化に伸びているもの、になっているものが$\mathcal R$に他ならない。
逆に、$\mathcal R$への$G_\mathcal O$の作用を用いて$G_\mathcal K\times_{G_\mathcal O}\mathcal R$を考えると、これが上で出てきた空間$\mathcal T\times_{\mathbf N_\mathcal K}\mathcal T$に他ならない。
ゲージ理論的な視点では、$\mathcal T\times_{\mathbf N_\mathcal K}\mathcal T$は、二次元空間の上にある接続と切断の組が、原点のまわりでひねられている様子をあらわす空間である。$(\mathscr P_1,\varphi_1,s_1)$がひねられる前で、$(\mathscr P_2,\varphi_2,s_2)$がひねられる後であり、原点でひねられるだけなので、原点の外では一致している。本来は$3$次元のゲージ理論であるが、時間方向の動きは見ずに、前後の二つの瞬間だけを切り取って比べているので、$2$次元の記述になっている。
空間を準備し、次に$\mathcal R$の$G_\mathcal O$-同変Borel-Mooreホモロジー群$H^{G_\mathcal O}_*(\mathcal R)$を考える。厳密には、$\mathcal T$の原点におけるファイバーの基本類が次数$0$になるように、次数をうまく定義する必要があるが、この点の詳細は略す。また、奇数次のホモロジーが消えていること、$H^*_G(\mathrm{pt})$上自由な加群になっていることなどは、アファイン・グラスマン多様体のSchubert胞体分割を考えると、ただちに従う。
$H^{G_\mathcal O}_*(\mathcal R)$には合成積
\begin{equation*}
\ast\colon H^{G_\mathcal O}_*(\mathcal R)\otimes H^{G_\mathcal O}_*(\mathcal R)
\to H^{G_\mathcal O}_*(\mathcal R)
\end{equation*}
が定義される。詳しい定義は、技術的なのでここでは略す。同変ホモロジー群のinduction
$H^{G_\mathcal K}_*(\mathcal T\times_{\mathbf N_\mathcal K}\mathcal T)\cong H^{G_\mathcal O}_*(\mathcal R)$が有限次元の空間のときと同様に成り立っていると仮想的に考えて、さらに$\mathcal T$は非特異であるとすると、通常の合成積の定義が、$(i,j)$成分への射影
\begin{equation*}
\mathcal T\times_{\mathbf N_\mathcal K}\mathcal T\times_{\mathbf N_\mathcal K}\mathcal T \xrightarrow{p_{ij}}
\mathcal T\times_{\mathbf N_\mathcal K}\mathcal T \qquad (i,j) = (1,2), (2,3), (1,3)
\end{equation*}
を用いて
\begin{equation*}
c\ast c' = p_{13*}(p_{12}^*c \cap p_{23}^*c')
\end{equation*}
と定義される。$H^{G_\mathcal K}_*(\mathcal T\times_{\mathbf N_\mathcal K}\mathcal T)$ が定義されるかどうかは不明であり、$\mathcal T$は非特異でないので、このままの定義がうまくできているのかどうかは分からないが、実際には$H^{G_\mathcal O}_*(\mathcal R)$の上に合成積$\ast$が定義される。
このとき次が成立する。
\begin{Theorem}\rm
$(H^{G_\mathcal O}_*(\mathcal R),\ast)$ は可換環である。
\end{Theorem}
合成積で環を構成する手法は、幾何学的表現論で広く使われており、ワイル群の群環がSteinberg多様体から作られること、Kac-Moody Lie環の普遍展開環が箙多様体におけるSteinberg多様体の類似物から作られることなどが知られている。これらの例では得られるものは、非可換環であり、合成積の一般論からは$\ast$が可換になる理由はなく、上の定理は今の状況の特殊性を表している。
ただし、幾何学的佐武対応を思い起こせば、可換性は不思議ではない。幾何学的佐武対応では、アファイン・グラスマン$\mathrm{Gr}_G$上の$G_\mathcal O$-同変な偏屈層のなすアーベル圏を考え、その上に合成積によってテンソル圏の構造を導入し、これが$G$のLanglands双対の有限次元表現の全体のなすテンソル圏と同値であることを主張する。後者のテンソル圏は可換、すなわち $V\otimes W\cong W\otimes V$ であるので、前者もそうである。この同型を幾何学的に説明するのがBeilinson-Drinfeldによるアファイン・グラスマンの1-パラメータ変形であり、これを使って上の定理が証明される。(論文では、計算による直接証明も与えている。)
さて、$(H^{G_\mathcal O}_*(\mathcal R),\ast)$は可換環になったので、そのスペクトラムとしてアファイン多様体を導入することができる。これが、Coulomb枝の数学的な定義である。
\begin{equation*}
\mathcal M_C = \operatorname{Spec}\nolimits (H^{G_\mathcal O}_*(\mathcal R),\ast)
\end{equation*}
さらに、$(H^{G_\mathcal O}_*(\mathcal R),\ast)$が有限生成であることや、整であることを証明できるので、$\mathcal M_C$は既約なアファイン多様体である。また正規であることも示されている。
変形量子化は、次のようにして与えられる。formal disk $D$に、${\mathbb C}^\times$がloop rotation$z\mapsto tz$により作用する。この作用は、今まで使ってきた様々な空間への作用を引き起こす。特に、$G_\mathcal O$に作用して、半直積$G_\mathcal O\rtimes{\mathbb C}^\times$を考えることができ、$\mathcal R$に$G_\mathcal O\rtimes{\mathbb C}^\times$が作用する。そこで、同変Borel-Mooreホモロジー群$H^{G_\mathcal O\rtimes{\mathbb C}^\times}(\mathcal R)$を考え、合成積を同じように導入する。こうして{\bf 量子化されたCoulomb枝}を
\begin{equation*}
\cAh = (H^{G_\mathcal O\rtimes{\mathbb C}^\times}(\mathcal R),\ast)
\end{equation*}
と定義する。
なお、アファイン・グラスマン多様体や、その類似の合成積を考えるのは、以前から\cite{MR3013034,MR2135527,MR2422266}で考えられており、合成積の定義を厳密に書き下す際には、それを参考にした。\cite{MR3013034}では、アファイン・グラスマンの代わりにアファイン旗多様体、同変ホモロジー群の代わりに同変K群が用いられているが、$\mathbf N = \mathfrak g$の場合を扱っていると思ってよい。出てくる代数は、Cherednikの二重アファイン・ヘッケ代数(DAHA)である。Coulomb枝のようにアファイン・グラスマン多様体にすれば、そのspherical partになり、同変ホモロジー群になれば楕円版の代わりに三角関数版のDAHAになる。対応するCoulomb枝は$\mathfrak t\times T^\vee/W$であり、量子補正がない、ということになる。
\cite{MR2135527,MR2422266}では、$\mathbf N=0$の場合を取り扱っている。出てくるものは、$G$のLanglands双対の戸田格子であるが、詳細は略す。
\section{例}
前節の構成は、無限次元空間のホモロジーを使うもので、ずいぶんと回りくどい構成に見えるかもしれないので、簡単な例をあげよう。
\subsection{}\label{subsec:torus}
$G = {\mathbb C}^\times$とし、$\mathbf N = 0$ とする。これは、一番自明な例である。$\mathbf N=0$なので、$\mathcal R$はアファイン・グラスマン$\mathrm{Gr}_G$に他ならず、また$G={\mathbb C}^\times$なので、$\mathrm{Gr}_G$は$D$上の直線束とその$D^\times$上での自明化のモジュライ空間に他ならない。reduced schemeを取ると、$\mathrm{Gr}_G$は整数${\mathbb Z}$でパラメトライズされた離散的な空間になる。実際、$\varphi(z) = z^n$ ($n\in{\mathbb Z}$)が対応する点をあらわす。よって
\begin{equation*}
H^{G_\mathcal O}_*(\mathcal R) = \bigoplus_n H^{{\mathbb C}^\times}_*(\mathrm{pt})
\end{equation*}
となる。$H^{{\mathbb C}^\times}_*(\mathrm{pt})$は、一変数の多項式環${\mathbb C}[w]$である。これが各整数$n$の上に乗っているので、$m$の上の多項式と$n$の上の多項式を掛けるとどうなるかを、合成積の定義に戻って計算する。合成積の定義を説明しなかったので、チェックすることはできないが、$G={\mathbb C}^\times$の場合には、テンソル積を取る写像
\begin{equation*}
\mathrm{Gr}_{{\mathbb C}^\times}\times\mathrm{Gr}_{{\mathbb C}^\times}\xrightarrow{\otimes}\mathrm{Gr}_{{\mathbb C}^\times}
\end{equation*}
があり、これがホモロジー群に引き起こすpushforward凖同型が$\ast$に他ならない。すると$m$の上の$f(w)$と$n$の上の$g(w)$を掛けたものは、$m+n$の上の$f(w)g(w)$になる。すなわち、$n=1$ の上の$1$(基本類に対応する)を$x$であらわすと、
\begin{equation*}
H^{G_\mathcal O}_*(\mathcal R) \cong {\mathbb C}[w,x^\pm] = {\mathbb C}[{\mathbb C}\times{\mathbb C}^\times]
\end{equation*}
となる。したがって、今のばあいのCoulomb枝は${\mathbb C}\times{\mathbb C}^\times$である。これは${\mathbb R}^3\times S^1$であるから、この場合のCoulomb枝は量子補正を受けないことを意味しており、ゲージ理論が自明なことの反映である。
もう一歩、精密に見るために$G$はトーラス$T$で、表現はやはり$0$であるとする。$\mathrm{Gr}_T$は離散的な空間で、$\operatorname{Hom}({\mathbb C}^\times,T)$でパラメトライズされている。従って、$H^{T_\mathcal O}_*(\mathcal R) = \bigoplus_{\lambda\in\operatorname{Hom}({\mathbb C}^\times,T)} H^*_T(\mathrm{pt})$である。$H^*_T(\mathrm{pt})$は、$T$のLie環$\mathfrak t$上の多項式環${\mathbb C}[\mathfrak t]$である。一方、$\lambda$に対応する元を$e^\lambda$と書くと、上と同様に$e^\lambda\ast e^\mu = e^{\lambda+\mu}$ となる。これは、$T$の双対 $T^\vee$ の指標 ($\operatorname{Hom}(T^\vee,{\mathbb C}^\times) = \operatorname{Hom}({\mathbb C}^\times, T)$) と見なすことができるから、Coulomb枝は$\mathfrak t\times T^\vee = T^* T^\vee$である。
\subsection{}\label{subsec:C2}
次に$G$は${\mathbb C}^\times$のままで, 表現を$\mathbf N = {\mathbb C}$ と標準表現に取ろう。
$\mathrm{Gr}_{{\mathbb C}^\times}$は上で説明したように${\mathbb Z}$でパラメトライズされる離散的な空間であり、$\mathcal R$は各整数$n$の上にベクトル空間が乗っているものである。条件は$\varphi(z) = z^n$によって原点に特異点が生じないというものであるから、
\begin{equation*}
\mathcal R = \bigsqcup_{n\in{\mathbb Z}} z^{n}{\mathbb C}[z]\cap {\mathbb C}[z] =
\bigsqcup_{n\in{\mathbb Z}} z^{\max(0,n)}{\mathbb C}[z]
\end{equation*}
である。各整数の上に乗っているものは、ベクトル空間でありThom同型により$H^{G_\mathcal O}(\mathcal R) \cong \bigoplus_n H^{{\mathbb C}^\times}_*(\mathrm{pt})$となる。すなわちベクトル空間としては、上の例と同じである。しかし合成積は、上の例とは$n>0$の上のホモロジー類と$n<0$の上のホモロジー類の積が変わってくる。定義を省略したので、最後のポイントだけいうと、$n=1$の基本類と$n=-1$の基本類を掛けたものが
\begin{equation*}
z{\mathbb C}[z] \to {\mathbb C}[z]
\end{equation*}
の押し出し写像による、基本類の像になる。これは余次元$1$の部分空間であるから、同変ホモロジー群の元としては、$w$を基本類に掛けたものになる。したがって$n=1$の基本類を$x$, $n=-1$の基本類を$y$とすると、$xy = w$が成り立つ。この計算から
\begin{equation*}
H^{G_\mathcal O}_*(\mathcal R) \cong {\mathbb C}[w,x,y]/(w=xy) \cong {\mathbb C}[x,y] = {\mathbb C}[{\mathbb C}^2]
\end{equation*}
が従う。よって今の場合のCoulomb枝は${\mathbb C}^2$である。
表現をウェイトが$N$の一次元表現に取り替えると、最後の部分の計算が$z^{|N|}{\mathbb C}[z]\to {\mathbb C}[z]$のpushforwardに置き換わり、座標環は${\mathbb C}[w,x,y]/(w^{|N|}=xy)$となる。これは、$A_{|N|-1}$型の単純特異点に他ならない。
\section{いくつかの構造}
この節では、Coulomb枝$\mathcal M_C$が持つ構造について解説する。いずれも物理的には発見されていたが、これが\secref{sec:def}の定義のもとで数学的に厳密に実現されることがポイントである。
\subsection{}
$H^{G_\mathcal O}_*(\mathcal R)$は、ホモロジーの次数の半分により次数付けられた環になる。つまり、${\mathbb C}[\mathcal M_C] = \bigoplus_d {\mathbb C}[\mathcal M_C]_d$ と分解し、${\mathbb C}[\mathcal M_C]_d\cdot{\mathbb C}[\mathcal M_C]_{d'}\subset{\mathbb C}[\mathcal M_C]_{d+d'}$となる。これは、$\mathcal M_C$に${\mathbb C}^\times$の作用が与えられていることを意味する。実際、${\mathbb C}[\mathcal M_C]_d$は、${\mathbb C}^\times$がウェイト$d$で作用するウェイト空間である。
上の例では、${\mathbb C}\times{\mathbb C}^\times$と${\mathbb C}^2 = {\mathbb C}\times{\mathbb C}$のそれぞれ第一成分への標準的な作用になっている。(正確には、後者は$x$がウェイト$1$で、$y$がウェイト$0$である。)
なお、次数の定義を省いたので説明が不足しているが、次数は非負とは限らず、一般にはすべての整数の値を取りうる。従って、$\mathcal M_C$は一般的には錘であるとは限らない。ここで、$\mathcal M_C$が錘であるとは、${\mathbb C}[\mathcal M_C]_d = 0$ ($d < 0$), ${\mathbb C}[\mathcal M_C]_0 = {\mathbb C}$が成り立つときをいう。
物理的には、この${\mathbb C}^\times$作用は、$\operatorname{\rm SU}(2)$作用の$S^1$への制限から来るものに、ある修正のあと一致すると期待されている。修正については説明しないが、次に述べる$\mathcal M_C$へのハミルトニアンなトーラス作用を適当に組み合わせるものである。特に、$G$が半単純のときには、修正は必要ない。$\operatorname{\rm SU}(2)$作用は、超ケーラー構造$I$, $J$, $K$がなす二次元球面 $S^2 = \{ a I + bJ + cK \mid a^2 + b^2 + c^2 = 1\}$ に$\operatorname{\rm SU}(2)\to\operatorname{\rm SO}(3)$を通じて標準的に作用するもので、一つの複素構造$I$を固定すると、$I$を保つ$S^1$の作用しか見えない。現在のところ、超ケーラー構造の定義は与えられていないので、$\operatorname{\rm SU}(2)$の作用を数学的に与えることはできていないが、制限の$S^1$だけが見えている。
上の例では、${\mathbb C}\times{\mathbb C}^\times = {\mathbb R}^3\times S^1$として${\mathbb R}^3$を$\operatorname{\mathfrak{su}}(2)$と見れば、確かに$\operatorname{\rm SU}(2)$の作用がある。ウェイトは半分になっている。${\mathbb C}^2$の場合は、$x$, $y$がそれぞれウェイト$-1/2$, $1/2$のハミルトニアンな$S^1$作用と合わせて、ウェイトが共に$1/2$の作用に直せば、やはりウェイトが半分になっていることを除き、${\mathbb C}^2$を四元数体$\mathbb H$とみた$\operatorname{\rm SU}(2) = \operatorname{\rm Sp}(1)$の作用に一致する。(複素線形でないので、$\operatorname{\rm SU}(2)$の${\mathbb C}^2$への標準的な表現とは異なり、四元数の右掛け算と左掛け算の違いがある。)
\subsection{}
$H^{G_\mathcal O}_*(\mathcal R)$は同変ホモロジー群であるから、$H^*_{G_\mathcal O}(\mathrm{pt})\cong H^*_G(\mathrm{pt})$からの準同型を持つ。(ただし、$H^*_G(\mathrm{pt})$上の代数ではなく、合成積 $c\ast c'$は第二成分$c'$に関しては、自然には$H^*_G(\mathrm{pt})$-線形にはならず、変形量子化したものについては、確かに線形でない。)
これのスペクトラムを取ると、
\begin{equation*}
\varpi\colon\mathcal M_C\to \operatorname{Spec}\nolimits H^*_G(\mathrm{pt})
\end{equation*}
を得る。よく知られているように、
\begin{equation*}
H^*_G(\mathrm{pt}) = {\mathbb C}[\operatorname{Lie}G]^G
= {\mathbb C}[\mathfrak t]^W
\end{equation*}
であるから、$\operatorname{Spec}\nolimits H^*_G(\mathrm{pt}) = \mathfrak t/W$であり、これはアファイン空間である。ここで、$\mathfrak t = \operatorname{Lie}T$である。
この構成は、変形量子化したあとも残り、
\begin{equation*}
H^{G\times{\mathbb C}^\times}_*(\mathrm{pt})\to
\cAh = H^{G_\mathcal O\rtimes{\mathbb C}^\times}(\mathcal R)
\end{equation*}
という単射な環準同型がある。これは、変形量子化が大きな可換環を含んでいることを意味しており、これの $\hbar = 0$を考えることにより、$\varpi$はポアソン可換であることを導く。すなわち、$\operatorname{Lie}T/W$上の関数 $f$, $g$ を $\varpi$で引き戻したものは、Poisson可換である：$\{ \varpi^*f, \varpi^*g\} = 0$.
さらに次が成立する。
\begin{Theorem}\label{thm:classical}\rm
$\varpi$のgenericなファイバーは、$T^\vee$である。より強く、次の可換図式がある。上の横矢印は双有理写像である。
\begin{equation*}
\xymatrix{
\mathcal M_C \ar@{.>}[rr] \ar[dr]_\varpi &&
T^* T^\vee/W = \mathfrak t\times T^\vee/W \ar[dl]^{\text{第一射影}}
\\
& \mathfrak t/W &}
\end{equation*}
\end{Theorem}
これは、同変ホモロジー群の局所化定理の帰結である。局所化定理は、$H^*_T(\mathrm{pt})$の商体を$\mathbb F$とするとき、
\begin{equation*}
H^{T_\mathcal O}_*(\mathcal R)\otimes_{H^*_T(\mathrm{pt})}\mathbb F
\cong H^{T_\mathcal O}_*(\mathcal R^T)\otimes_{H^*_T(\mathrm{pt})}\mathbb F
\end{equation*}
が成り立つという主張である。ここで、$\mathcal R^T$は$\mathcal R$の$T$-固定点の集合であり、同型写像は、包含写像$\mathcal R^T\hookrightarrow\mathcal R$のpushforward準同型である。これと、$H^{G_\mathcal O}_*(\mathcal R)$は、$H^{T_\mathcal O}_*(\mathcal R)$の$W$-不変部分であるという事実を組み合わせると、$\mathcal R^T$の同変ホモロジー群を決定すれば良いことになるが、$\mathcal R^T$が$\mathrm{Gr}_T\times\mathbf N^T$であることと、\subsecref{subsec:torus}の計算から、$\mathfrak t\times T^\vee$であることが分かる。
操作$\otimes_{H^*_T(\mathrm{pt})}\mathbb F$は、$\mathfrak t/W$のgeneric pointに制限することであり、同変ホモロジー群を考えることが$\mathfrak t/W$上の族を考えるという幾何学的な描像に対応しているという、よく知られた哲学の有効性をあらわす典型的な議論である。
以上で、$\varpi$はポアソン交換していて、ファイバーが代数的トーラスであることから、$\varpi$はLiovilleの意味で可積分系である。変形量子化$\cAh$はその量子化である。
\subsection{}
アファイン・グラスマン $\mathrm{Gr}_G$は位相的には基点付きループ群 $\Omega G$であることが知られており、特にその連結成分は$G$の基本群 $\pi_1(G)$に一致する。ホモロジー群は連結成分に応じて分解するが、これは合成積とcompatibleである。すなわち$\gamma\in\pi_1(G)$に対応する$\mathcal R$の連結成分を$\mathcal R_\gamma$と書くとき、$H^{G_\mathcal O}_*(\mathcal R_\gamma)\ast H^{G_\mathcal O}_*(\mathcal R_{\gamma'})\subset
H^{G_\mathcal O}_*(\mathcal R_{\gamma+\gamma'})$となる。($\pi_1(G)$が可換であることはよく知られている。) 従って $H^{G_\mathcal O}_*(\mathcal R)$は、$\pi_1(G)$で次数付けられた環である。
これを $\mathcal M_C = \operatorname{Spec}\nolimits H^{G_\mathcal O}_*(\mathcal R)$側で考えると、$\pi_1(G)$のポントリャーギン双対$\operatorname{Hom}(\pi_1(G),\linebreak[3]{\mathbb C}^\times)$が$\mathcal M_C$に作用することになる。たとえば、上の例では$\pi_1(G) = \pi_1({\mathbb C}^\times) = {\mathbb Z}$であり、ポントリャーギン双対は${\mathbb C}^\times$である。${\mathbb C}^\times$作用は、最初の例では${\mathbb C}\times{\mathbb C}^\times$の第二成分への自然な作用であり、二番目の例の場合は$x$がウェイト$1$で、$y$がウェイト$-1$である。
この作用は、変形量子化 $H^{G_\mathcal O\rtimes{\mathbb C}^\times}_*(\mathcal R)$にも自然に伸びていることから、シンプレクティック形式を保っていることも従う。
$G$が半単純のときには、$\pi_1(G)$は有限群で、そのポントリャーギン双対も有限群になってしまうが、トーラスが現れるのは$\operatorname{Hom}(G,{\mathbb C}^\times)$が自明でない場合である。このとき、運動量写像は$\varpi$に$\operatorname{Lie}G\to\operatorname{Lie}{\mathbb C}^\times$を合成したもので与えられ、特に作用はハミルトニアンである。
\subsection{}
$\mathbf N$が、$G$を正規部分群として含む大きな群$\widetilde G$の表現の制限として現れている場合を考える。物理では、商群 $\widetilde G/G$はフレーバー対称性の群とよばれる。これを$G_F$で表わす。
$\widetilde G_\mathcal O$は$\mathcal R$に作用するので、大きな群の同変ホモロジー$H^{\widetilde G_\mathcal O}_*(\mathcal R)$を考えることができる。合成積により、$H^*_{G_F}(\mathrm{pt})$上の代数になり、対応するスペクトラムは、$\operatorname{Spec}\nolimits H^*_{G_F}(\mathrm{pt}) = {\mathbb C}[\operatorname{Lie}G_F]^{G_F}$上の多様体の族になり、原点のファイバーが元の$\mathcal M_C$である。すなわち、$\mathcal M_C$は$\operatorname{Lie}G_F/\!\!/ G_F$でパラメトライズされた変形を持つ。
また、説明は省略するが、変形に対応するような$\mathcal M_C$の部分特異点解消(の候補)を構成することもできる。
この節と前節では、$\operatorname{Hom}(G,{\mathbb C}^\times)$と$G_F$が$\mathcal M_C$に導く構造を調べたが、Higgs枝$\mathcal M_H$に導く性質を考えることは有用である。まず、$G_F$であるが、$\mathcal M_H = \mathbf M/\!\!/\!\!/ G$であるから、$G_F$は$\mathcal M_F$に作用する。
一方で、$\operatorname{Hom}(G,{\mathbb C}^\times)$があると、対応する$\zeta\in\operatorname{Hom}(\operatorname{Lie}G,\operatorname{Lie}{\mathbb C}^\times)$を考えて、運動量写像のレベル集合を$\mu=0$から$\mu=\zeta$に変形することができる。
すなわち、$\operatorname{Hom}(G,{\mathbb C}^\times)$と$G_F$が$\mathcal M_C$と\linebreak[3]$\mathcal M_H$に誘導する構造は、それぞれ群作用と変形であるが、両者は$\mathcal M_C$と$\mathcal M_H$で入れ替わっている。
\subsection{}
前節と、前々節の構造の例として、トーリック超ケーラー多様体を考える。こ
れには、トーラスの完全列
\begin{equation*}
1 \to T = ({\mathbb C}^\times)^{d-n} \to \widetilde T = ({\mathbb C}^\times)^d
\to T_F = ({\mathbb C}^\times)^n \to 1
\end{equation*}
が与えられたとする。$\widetilde T$の標準的な表現$\mathbf N = {\mathbb C}^d$を取り、その$T$への制限も$\mathbf N$で表わす。さて、$\mathcal M_C(\widetilde T,\mathbf N)$は
\subsecref{subsec:C2}の計算より${\mathbb C}^{2d}$となる。前々節の構成により$\pi_1(\widetilde T)$のポントリャーギン双対が${\mathbb C}^{2d}$に作用するが、これは$\widetilde T$の双対トーラス$\widetilde T^\vee$に他ならない。$T_F$の双対トーラスは$T_F^\vee$はその部分トーラスであり、前々節の構成をもう少し進めると $T$に関するCoulomb枝 $\mathcal M_C(T,\mathbf N)$は、${\mathbb C}^{2d}$の$T_F^\vee$に関するシンプレクティック商
${\mathbb C}^{2d}/\!\!/\!\!/ T_F^\vee$に他ならない。これは、双対トーラスの完全列
\begin{equation*}
1\to T_F^\vee \to \widetilde T^\vee \to T^\vee \to 1
\end{equation*}
を考えて、$\widetilde T^\vee$の表現$\mathbf M = {\mathbb C}^d\oplus({\mathbb C}^d)^*$に関するHiggs枝ということもできる。すなわち、$T$と$T_F^\vee$を入れ替えることによって、Higgs枝とCoulomb枝が入れ替わっている。
\section{箙ゲージ理論}
現在のところ、Higgs枝が箙多様体になるような$(G,\mathbf N)$に対応するCoulomb枝についてが一番よく調べられている。$Q$ を箙とし、$Q_0$をその頂点の集合、$Q_1$を向きの付けられた辺の集合とする。$h\in Q_1$に対し、その始点と終点を$\vout{h}$, $\vin{h}$で表わす。二つの$Q_0$で次数付けられたベクトル空間 $V = \bigoplus V_i$, $W = \bigoplus W_i$が与えられたとき、
\begin{equation*}
\begin{split}
& G = \prod_{i\in Q_0}\operatorname{GL}(V_i),
\\
& \mathbf N = \bigoplus_{h\in Q_1}\operatorname{Hom}(V_{\vout{h}},V_{\vin{h}})
\oplus\bigoplus_{i\in Q_0} \operatorname{Hom}(W_i, V_i)
\end{split}
\end{equation*}
が、箙ゲージ理論である。ただし、$G$の$\mathbf N$への作用は自然なものである。
$Q$が$ADE$型の場合には、$\mathcal M_C$は原点に特異点を持った${\mathbb R}^3$の上のモノポールのモジュライ空間になると物理的には洞察されていたが、この空間の代数幾何的な対応物が、先の数学的な定義のもとで示されている。(\cite{2016arXiv160403625B}) ここで、モノポールの構造群は、$Q$に対応する(adjoint型の)複素単純リー群$G_Q$であり、$V$の次元は、モノポールの次数に対応し、$W$の次元は特異点の情報を与える。
代数幾何的な対応物は、一般の場合は記述は簡単ではないが、$\mu = \sum \mathop{\text{\rm dim}}\nolimits W_i \varpi_i - \mathop{\text{\rm dim}}\nolimits V_i \alpha_i$が支配的なときには、$G_Q$のアファイン・グラスマンを考え、$\lambda = \sum \mathop{\text{\rm dim}}\nolimits W_i \varpi_i$と$\mu$に対応するSchubert多様体$\overline{\mathrm{Gr}}_{G_Q}^{\lambda}$, $\overline{\mathrm{Gr}}_{G_Q}^{\mu}$を取って、$\overline{\mathrm{Gr}}_{G_Q}^\mu$の横断切片と$\overline{\mathrm{Gr}}_{G_Q}^{\lambda}$の交わりが$\mathcal M_C$である。
幾何学的佐竹対応によってアファイン・グラスマンは $G_Q$のラングランズ双対の表現論と結びついていたが、一方で箙多様体のホモロジー群には、$G_Q$の表現の構造が入っていた。始めに述べたシンプレクティック双対性は、この二つの構成が`双対'によって結びついていることを主張するように定式化される(べきである)。
この結果の証明のためには、$\mathcal M_C$を決定する次のような処方箋を用いる。
\begin{enumerate}
\item
まず、$\mathcal M_C$の候補になる空間を作る。これは、多くの場合は、物理学者の答えを採用する。
\item
次に、その候補の空間に、$\varpi$に対応すると期待される可積分系を作る。
\item
その可積分系が平坦な族であること、$\mathcal M_C$が正規であることをチェックする。
\item
$\mathcal M_C$とその候補の間の $T^* T^\vee/W$を通じた双有理写像が$\mathfrak t/W$の余次元$2$の集合を除いて拡張することをチェックする。
\end{enumerate}
最後の余次元$2$の集合を除けば十分であるところは、正規性の帰結である。
同変ホモロジーの局所化定理の応用で、genericには$T^*T^\vee$であることを説明したが、余次元$1$のところも同様の議論で、階数$1$の群のCoulomb枝を決定する問題に帰着できる。階数$1$の場合は、Coulomb枝は${\mathbb C}^3$の超曲面として実現できることが示されており、決定されている。したがって、(4)は、易しいステップである。現状では、(3)を示す部分が、ケースバイケースで行われていてキーポイントになっている。
アファイン$ADE$型の場合は、有限型のモノポールの代わりにインスタントンを考えればよい。ただし、${\mathbb R}^4$上のインスタントンではなく、Taub-NUT空間上のインスタントンにするのが正確なので、微妙な問題があり、特に上でいうところの$\mu$が支配的な場合は${\mathbb R}^4$上でもTaub-NUT空間上でも、複素シンプレクティック多様体としては変わらないと期待されている。
インスタントンのモジュライ空間については、(3)の性質が証明されていないので、現在のところCoulomb枝の決定までは至っていない。
(3)は微妙な性質である。例えばべき零軌道は、$A$型のときは常に正規であるが、一般にはそうでない。一方、Coulomb枝は常に正規である。古典型のべき零軌道やそのSlodowy横断切片は、Higgs枝として現れることが知られているので、対応するCoulomb枝も、そうなっていると安直には考えられるが、正規性の問題から、そうそう単純ではなさそうである。Hananyらは、正規化を取ればよいと考えているようであるが、まだまだ十分な根拠があるとはいえないのではないだろうか？
アファイン$A$型のときには、インスタントンのモジュライ空間を直接取り扱う代わりに、Cherkisの弓箭多様体(bow varietyの和訳)を用いる。弓箭多様体は、Nahm方程式とよばれる非線形常微分方程式の解を用いてあらわされているので、そのままでは取扱いにくいが、\cite{2016arXiv160602002N}により、箙多様体の変種として書き直し、(3)の性質を証明した。したがって、アファイン$A$型の箙ゲージ理論のCoulomb枝は決定された。
\section{量子化されたCoulomb枝}
量子化されたCoulomb枝$\cAh$については、多様体の決定に比べると分かっている例は少ない。
前に、随伴表現$\mathbf N=\mathfrak g$のときにspherical DAHAが出てくることを言及したが、$G=\operatorname{GL}(k)$のときは、ジョルダン箙に対応する箙ゲージ理論で、$V={\mathbb C}^k$, $W=0$の場合であると思うことができる。これを一般化して$V={\mathbb C}^k$, $W={\mathbb C}^r$と変えると、$\cAh$はwreath積${\mathbb Z}/r{\mathbb Z}\wr S_k = ({\mathbb Z}/r{\mathbb Z})^k\rtimes S_k$の有理Cherednik代数のspherical partになる。\cite{2016arXiv160800875K} 対応するCoulomb枝は$\operatorname{Sym}^k{\mathbb C}^2/({\mathbb Z}/r{\mathbb Z})$である。
有限$ADE$型の箙ゲージ理論の場合は、\cite{2016arXiv160403625B}のAppendixにおいて$\cAh$がshited Yangianとして同型であることが示された。ただし、前節で言及した$\mu$が支配的という条件を仮定した下で証明されており、一般の場合は未解決である。
\bibliographystyle{myamsalpha}
|
1,108,101,566,786 | arxiv | \section{Introduction}\label{introduction}
Given an ideal $I$ in a Noetherian ring $R$, one can associate an algebra to $I$ known as the Rees algebra $\hbox{\ensuremath{{\mathcal R}(I)}}$ of $I$. This algebra $\hbox{\ensuremath{{\mathcal R}(I)}}=\bigoplus_{i\ge 0}I^i t^i$ is a subalgebra of $R[t]$, where $t$ is an indeterminate. It was introduced by Rees in 1956 in order to prove what is now known as the Artin-Rees Lemma \cite{Rees}.
Geometrically, the Rees algebra corresponds to the blowup of ${\rm Spec}(R)$ along $V(I)$. If $R$ is local with maximal ideal ${\mathfrak m}$ or graded with homogeneous maximal ideal ${\mathfrak m}$, the special fiber ring of $I$ is the algebra $\hbox{\ensuremath{{\mathcal F}(I)}}=\hbox{\ensuremath{{\mathcal R}(I)}}\otimes R/{\mathfrak m}$. This algebra corresponds to the special fiber of the blowup of ${\rm Spec}(R)$ along $V(I)$. Besides its connections to resolution of singularities, the study of Rees algebras plays an important role in many other active areas of research including multiplicity theory, equisingularity theory, asymptotic properties of ideals, and integral dependence.
Although blowing up is a fundamental operation in the study of birational varieties,
an explicit understanding of this process remains an open problem. In particular, a key objective in this area is to express the Rees algebra and the special fiber ring as quotients of a polynomial ring, henceforth to determine their defining ideals.
This question is wide open even for the simplest classes of ideals, including ideals generated by forms of the same degree in a polynomial ring. These are precisely the ideals generated by forms parametrizing a variety in projective space. The implicit equations of these varieties can be obtained from the defining ideal of the Rees ring. Indeed, the bihomogenous coordinate ring of the graph of the morphism defined by the forms is the Rees algebra of the ideal $I$. The homogeneous coordinate ring of the variety parametrized by the forms is the special fiber ring.
As the graph of a map carries more information than its image, even a partial understanding of the Rees ring such as the bigraded degrees of its defining equations, the Betti numbers, or the regularity of the defining ideal can be instrumental to the study of the variety. Determining the defining equations of the Rees algebra is a difficult problem in elimination theory, studied by commutative algebraists, algebraic geometers, and applied mathematicians in geometric modeling, see e.g. \cite{Buse, BJ,CWL, Cox, SCG}. Answers to these questions also have applications to the study of chemical reaction networks~\cite{CLS}.
The goal of this paper is to determine the defining equations of the Rees algebra and of the special fiber ring of the ideals generated by the maximal minors of sparse $2 \times n$ matrices. Sparse matrices are matrices whose entries are either zeroes or distinct variables. Their degeneracy loci were first studied by Giusti and Merle in the 80's. In~\cite{GM} they compute the codimension of their defining ideals and characterize when these ideals are prime or Cohen-Macaulay. Boocher in 2012 proved in~\cite{Boocher} that a minimal free resolution of the ideals of maximal minors of sparse matrices can be obtained from the Eagon-Northcott complex via a pruning method. In the same paper, he shows that the natural generators form a universal Gr\"obner basis.
In the case of a generic matrix, that is a matrix whose entries are distinct variables, the special fiber ring of the ideal of maximal minors is the coordinate ring of a Grassmannian variety. The fact that the Pl\"{u}cker relations define the Grassmannian variety is a classical theorem, see e.g.~\cite{BV}. The Rees algebra and the special fiber ring of ideals of maximal minors of generic matrices are Algebras with Straightening Laws (ASLs) in the sense of~\cite{DEP}, see~\cite{E, EHu}. Since the straightening relations come from the Pl\"{u}cker relations
and the defining equations of the symmetric algebra, it follows that $\hbox{\ensuremath{{\mathcal R}(I)}}$ is of fiber type, see \cite{DEP, E}, \cite[Lemma~2.2.1]{BST}, \cite[Proposition~4.2]{BV}.
In addition, as the posets defining $\hbox{\ensuremath{{\mathcal R}(I)}}$ and $\hbox{\ensuremath{{\mathcal F}(I)}}$ are wonderful in the sense of~\cite{EHu}, it follows that both algebras are Cohen-Macaulay, see \cite[Proposition~2.6]{EHu}. The normality of $\hbox{\ensuremath{{\mathcal F}(I)}}$ follows immediately from the Cohen-Macaulay property since the Grassmannian variety is smooth. Trung in \cite{Trung} proved that the powers and symbolic powers of $I$ coincide and therefore $\hbox{\ensuremath{{\mathcal R}(I)}}$ is normal. From the deformation theorem developed in \cite{CHV}, see for instance \cite[Proposition~3.6]{BCV}, one can see that the Rees algebra of the ideal of maximal minors is defined by a Gr\"{o}bner basis of quadrics. The same statement holds for $\hbox{\ensuremath{{\mathcal F}(I)}}$.
Therefore, in the generic case both $\hbox{\ensuremath{{\mathcal R}(I)}}$ and $\hbox{\ensuremath{{\mathcal F}(I)}}$ are Koszul algebras and according to \cite{Bl} the ideal $I$ and all its powers have a linear resolution.
Our main result shows that the above properties of the blowup algebras of ideals of maximal minors of a generic matrix still hold in the case of sparse $2\times n$ matrices.
Now let $I$ be the ideal of maximal minors of a $2\times n$ sparse matrix. Inspired by the pioneering work of Conca, Herzog, and Valla~\cite{CHV}, we study the initial algebra of the Rees algebra of $I$. Our main technique is SAGBI bases~\cite{KM, RS}, an analogue for algebras of Gr\"obner bases for ideals.
This approach was successfully used to study the Rees algebras of other families of ideals~\cite{ALL,CHV,LS2}.
First we prove that the initial algebra of the Rees algebra of $I$ is the Rees algebra of the initial ideal of $I$ with respect to a suitable order (see Theorem~\ref{initial rees thm}). Using deformation theory, we transfer properties from the Rees algebra of the initial ideal to the Rees algebra of the ideal itself. One advantage of this approach is that it allows us to reduce to the study of the Rees algebra of the initial ideal, which is not just a monomial algebra, but also the Rees algebra of the edge ideal of a graph and a ladder determinantal ring. These objects have been studied extensively and one can draw a plethora of information that allows us to describe these algebras in full detail, see among others~\cite{Conca, CN, SVV, V, Wang, Wang1}.
A key step in our proof that $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ is the initial algebra of $\hbox{\ensuremath{{\mathcal R}(I)}}$ is Lemma~\ref{I^2=}, where we prove that taking the initial ideal commutes with powers. The main idea behind the proof is a comparison of the Hilbert functions of $I^2$ and $(\ini(I))^2$, an approach which was first used in~\cite{GMN}. We then use a lifting technique to obtain the defining equations of the Rees algebra and of the special fiber ring. Interestingly, they turn out to be the specialization of the defining equations of the Rees algebra and of the special fiber ring in the generic case.
The general question of understanding the Rees algebra and the special fiber ring of the ideal $I$ of maximal minors of a sparse $m\times n$ matrix is still open. In Remark~\ref{ASL} we propose a different approach, which applies to sparse matrices whose zero region has a special shape. These sparse matrices are exactly those that have the property that a maximal minor is non-zero if and only if the product of the elements on its diagonal is non-zero. This yields a nice combinatorial description of the initial ideal of $I$. Our arguments allow us to compute the equations of the special fiber ring and the Rees algebra and to establish algebraic properties such as normality, Cohen-Macaulayness, and Koszulness. We conjecture that these properties hold in general.
Our main results are summarized in the following.
\begin{theorem}\label{mainresults}
Let $X$ be a sparse $2 \times n$ matrix and $I = I_2(X)$.
\begin{enumerate}[$($a$)$]
\item The defining ideal of $\hbox{\ensuremath{{\mathcal F}(I)}}$ is generated by the Pl\"ucker relations on the $2\times 2$-minors of $X$ and these form a Gr\"obner basis of the ideal they generate.
\item $\hbox{\ensuremath{{\mathcal R}(I)}}$ is of fiber type, that is, its defining ideal is generated by the relations of the symmetric algebra of $I$ and by the Pl\"ucker relations on the $2\times 2$-minors of $X$. Moreover, these equations form a Gr\"obner basis of the ideal they generate.
\item $\hbox{\ensuremath{{\mathcal R}(I)}}$ and $\hbox{\ensuremath{{\mathcal F}(I)}}$ have rational singularities in characteristic zero and they are $F$-rational in positive characteristic. In particular, they are Cohen-Macaulay normal domains.
\item $\hbox{\ensuremath{{\mathcal R}(I)}}$ and $\hbox{\ensuremath{{\mathcal F}(I)}}$ are Koszul algebras. In particular, the powers of $I$ have a linear resolution.
\item The natural algebra generators of $\hbox{\ensuremath{{\mathcal R}(I)}}$ and $\hbox{\ensuremath{{\mathcal F}(I)}}$ are SAGBI bases of the algebras they generate.
\end{enumerate}
\end{theorem}
\section{Notation}\label{sectsetting}
This section is devoted to the setup and notation we will use throughout the paper.
Let $K$ be a field, $n\ge 3$ a positive integer,
and $X=(x_{ij})$ a $2 \times n$ sparse matrix of indeterminates over $K$, i.e., a matrix whose entries $x_{ij}$ are either distinct indeterminates or zero. We use the notation $x_{i,j}$ when it is not clear what the two subscripts are. Let $R=K[X]$ denote the polynomial ring over $K$ in the variables appearing in $X$ and let $I$ be the ideal generated by the $2 \times 2$ minors of $X$.
Up to permuting the rows and the columns of $X$, we may assume that there exist $r$ and $s$ with $0\le n-r-s\le r < n$ such that
$$
X= \begin{pmatrix}
x_{1,1} & \cdots & x_{1,r} & x_{1,r+1} & \cdots & x_{1,r+s} & 0 & \cdots & 0 \\
0 & \cdots & 0 & x_{2,r+1} & \cdots & x_{2,r+s} & x_{2,r+s+1} & \cdots & x_{2,n}
\\
\end{pmatrix}.
$$
If $r = 0$, then $n=s$ and $X$ is a generic matrix. Since the results obtained in this paper are known in this case,
we may assume without loss of generality that $r\ge 1$.
Let $f_{ij}$ denote the $2 \times 2$ minor of $X$ that involves columns $i$ and $j$.
Then $$I=(f_{ij}\mid 1\le i<j\le n)\subseteq R.$$
Notice that $f_{ij} = - f_{ji}$ and that $f_{ij}$ is a monomial or zero, unless $r+1\le i, j\le r+s$.
By the form of $X$, if a minor is non-zero, then the product of the entries on its diagonal is also non-zero. This means that we can choose a diagonal term order on $K[X]$, that is, a term order with the property that the leading term of each minor of $X$ is the product of the elements on its diagonal. An example of a diagonal term order is the lexicographic order with $x_{1,1}>\ldots>x_{1,r+s}>x_{2,r+1}>\ldots>x_{2,n}$. The maximal minors of $X$ form a diagonal Gr\"obner basis for $I$ by \cite[Proposition~5.4]{Boocher}.
Throughout the paper, we fix a diagonal term order $\tau$. The minimal generating set for $\ini(I)$ with respect to $\tau$ is described next.
\begin{proposition}[\cite{Boocher}, Proposition~5.4]\label{gb of ini}
Let $X$ be a sparse $2\times n$ matrix, $I$ the ideal of $2\times 2$ minors of $X$, and $\tau$ a diagonal term order.
The initial ideal of $I$ is $$\ini(I)=(x_{1i}x_{2j}\mid 1\le i\le r+s, \max\{r,i\}<j\le n).$$
\end{proposition}
It turns out that $\ini(I)$ corresponds to a {\it Ferrers diagram} as in Figure~\ref{fig:1}.
\begin{figure}[h]
\centering
\begin{tikzpicture}[ every node/.style={scale=0.9}]
\node [label = above: {\small$x_{2,n}$}] at (5.3,5) {};
\node [label = above: {\small $\cdots$}] at (6.5,5) {};
\node [label = above: {\small$x_{2,r+s+1}$}] at (7.42,5) {};
\node [label = above: {\small$x_{2,r+s}$}] at (8.5,5) {};
\node [label = above: {\small$\cdots$}] at (9.5,5) {};
\node [label = above: {\small$x_{2,r+1}$}] at (11,5) {};
\node [label = left: { \small$x_{1,1}$}] at (5,4.8){};
\node [label = left: {\small$\vdots$}] at (4.7,4){};
\node [label = left: {\small$x_{1,r}$}] at (5,3.2){};
\node [label = left: {\small$x_{1,r+1}$}] at (5,2.8){};
\node [label = left: {\small$\vdots$}] at (4.7,1){};
\node [label = left: {\small$x_{1,r+s}$}] at (5,-0.4){};
\draw [line width=1pt, color=black] (5,5)--(11.5,5);
\draw[line width=1pt] (5,5)--(5,-.5);
\draw[line width=1pt](5,3)--(11.5,3);
\draw[line width=1pt](8,5)--(8,-.5);
\draw[line width=1pt](9,1)--(9,0.5);
\draw[line width=1pt](8.5,0.5)--(8.5,0);
\draw[line width=1pt](10,2)--(10,1.5);
\draw[line width=1pt](9.5,1.5)--(9.5,1);
\draw[line width=1pt](11,3)--(11,2.5);
\draw[line width=1pt](10.5,2.5)--(10.5,2);
\draw[line width=1pt](11.5,5)--(11.5,3);
\draw[line width=1pt](10.5,2.5)--(11,2.5);
\draw[line width=1pt](10,2)--(10.5,2);
\draw[line width=1pt](9.5,1.5)--(10,1.5);
\draw[line width=1pt](9,1)--(9.5,1);
\draw[line width=1pt](8,0)--(8.5,0);
\draw[line width=1pt](8.5,0.5)--(9,0.5);
\draw[line width=1pt](5,-0.5)--(8,-.5);
\node [label = left: { \Large $\bf{A}$}] at (7,4){};
\node [label = left: { \Large $\bf{B}$}] at (9.5,4){};
\node [label = left: { \Large $\bf{C}$}] at (7,2){};
\node [label = left: { \Large $\bf{D}$}] at (9.5,2){};
\end{tikzpicture}
\caption{ Ferrers diagram }
\label{fig:1}
\end{figure}
\begin{corollary} \label{cor init Ferrers}
The ideal $\ini(I)$ is the edge ideal of a Ferrers bipartite graph
whose vertex sets are
$$V=\{x_{1,1}, \ldots, x_{1,r+s}\}, \;\; W=\{x_{2,n}, \ldots ,x_{2,r+1}\},$$
and whose partition is $$\lambda=(\underbrace{n-r, \ldots, n-r}_{r \ {\rm{ times }}}, n-r-1, n-r-2, \ldots, n-r-s).$$
In other words, the first $r$ elements of $V$ have edges connecting them to all $n-r$ elements of $W$ Moreover, for $i>r$, the $i$th element of~$V$ is connected by an edge to the first $n-i$ elements of $W$.
\end{corollary}
We say that $\ini(I)$ is the {\it Ferrers ideal} $I_{\lambda}$.
See \cite{CN} for more on Ferrers graphs, diagrams, and ideals.
\begin{example} \label{running ex}
For the matrix
$$
X= \begin{pmatrix}
x_{11} & x_{12}& x_{13} & x_{14} & x_{15} & x_{16} &x_{17}&0&0\\
0 & 0 & 0 &x_{24} &x_{25}&x_{26}&x_{27} &x_{28} &x_{29}\\
\end{pmatrix}
$$
the ideal $\ini(I)$ is the Ferrers ideal $I_{\lambda}$
for the partition $\lambda=(6,6,6,5,4,3,2)$. Its Ferrers diagram is depicted in Figure~\ref{fig:2}.
\end{example}
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.8, every node/.style={scale=0.6}]
\node [label = above: {\Large $x_{29}$}] at (5.5,5) {};
\node [label = above: {\Large $x_{28}$}] at (6.5,5) {};
\node [label = above: {\Large $x_{27}$}] at (7.5,5) {};
\node [label = above: {\Large $x_{26}$}] at (8.5,5) {};
\node [label = above: {\Large $x_{25}$}] at (9.5,5) {};
\node [label = above: {\Large $x_{24}$}] at (10.5,5) {};
\node [label = left: {\Large $x_{11}$}] at (5,4.5){};
\node [label = left: {\Large $x_{12}$}] at (5,3.5){};
\node [label = left: {\Large $x_{13}$}] at (5,2.5){};
\node [label = left: {\Large $x_{14}$}] at (5,1.5){};
\node [label = left: {\Large $x_{15}$}] at (5,0.5){};
\node [label = left: {\Large $x_{16}$}] at (5,-0.5){};
\node [label = left: {\Large $x_{17}$}] at (5,-1.5){};
\draw [line width=1pt, color=black] (5,5)--(11,5);
\draw [line width=1pt] (5,5)--(5,-2);
\draw [line width=1pt] (5,4)--(11,4);
\draw [line width=1pt](6,5)--(6,-2);
\draw [line width=1pt](5,3)--(11,3);
\draw [line width=1pt](7,5)--(7,-2);
\draw [line width=1pt](8,5)--(8,-1);
\draw [line width=1pt](9,5)--(9,0);
\draw [line width=1pt](10,5)--(10,1);
\draw [line width=1pt](11,5)--(11,2);
\draw [line width=1pt](7,2)--(7,-1);
\draw [line width=1pt](5,2)--(11,2);
\draw [line width=1pt](5,1)--(10,1);
\draw [line width=1pt](5,0)--(9,0);
\draw [line width=1pt](5,-1)--(8,-1);
\draw [line width=1pt](5,-2)--(7,-2);
\end{tikzpicture}
\caption{Ferrers diagram for $\lambda=(6,6,6,5,4,3,2)$} \label{fig:2}
\end{figure}
\begin{definition}[\cite{CN}]\label{L lambda}
The {\it one-sided ladder associated to} a Ferrers ideal $I_{\lambda}$
is the ladder $L_{\lambda}$ with the same shape as the Ferrers diagram for $\lambda$ and for which the entry in row $x_{1i}$ and column $x_{2j}$ is $y_{ij}$.
The {\it ladder associated to} $\ini(I)=I_{\lambda}$
is the two-sided ladder $L_{\lambda}'$
obtained from~$L_{\lambda}$ by adding a row of $n-r$ boxes at the top and a column of $r+s$ boxes on the left. The new boxes are filled with the variables from $W$ and $V$, respectively,
in their listed order.
\end{definition}
Illustrations of $L_{\lambda}$ and $L'_{\lambda}$
for $\lambda = (6,6,6,5,4,3,2)$ are in Figure~\ref{fig:3}.
\begin{figure}[h]
\begin{tikzpicture}[scale=0.8, every node/.style={scale=0.6}]
\node [label = above: {\Large $x_{29}$}] at (5.5,5) {};
\node [label = above: {\Large $x_{28}$}] at (6.5,5) {};
\node [label = above: {\Large $x_{27}$}] at (7.5,5) {};
\node [label = above: {\Large $x_{26}$}] at (8.5,5) {};
\node [label = above: {\Large $x_{25}$}] at (9.5,5) {};
\node [label = above: {\Large $x_{24}$}] at (10.5,5) {};
\node [label = left: {\Large $x_{11}$}] at (5,4.4){};
\node [label = left: {\Large $x_{12}$}] at (5,3.4){};
\node [label = left: {\Large $x_{13}$}] at (5,2.4){};
\node [label = left: {\Large $x_{14}$}] at (5,1.4){};
\node [label = left: {\Large $x_{15}$}] at (5,0.4){};
\node [label = left: {\Large $x_{16}$}] at (5,-0.6){};
\node [label = left: {\Large $x_{17}$}] at (5,-1.6){};
\draw [line width=1pt, color=black] (5,5)--(11,5);
\draw [line width=1pt] (5,5)--(5,-2);
\draw [line width=1pt] (5,4)--(11,4);
\draw [line width=1pt](6,5)--(6,-2);
\draw [line width=1pt](5,3)--(11,3);
\draw [line width=1pt](7,5)--(7,-2);
\draw [line width=1pt](8,5)--(8,-1);
\draw [line width=1pt](9,5)--(9,0);
\draw [line width=1pt](10,5)--(10,1);
\draw [line width=1pt](11,5)--(11,2);
\draw [line width=1pt](7,2)--(7,-1);
\draw [line width=1pt](5,2)--(11,2);
\draw [line width=1pt](5,1)--(10,1);
\draw [line width=1pt](5,0)--(9,0);
\draw [line width=1pt](5,-1)--(8,-1);
\draw [line width=1pt](5,-2)--(7,-2);
\node [label = above: {\Large $y_{19}$}] at (5.5,4) {};
\node [label = above: {\Large $y_{18}$}] at (6.5,4) {};
\node [label = above: {\Large $y_{17}$}] at (7.5,4) {};
\node [label = above: {\Large $y_{16}$}] at (8.5,4) {};
\node [label = above: {\Large $y_{15}$}] at (9.5,4) {};
\node [label = above: {\Large $y_{14}$}] at (10.5,4) {};
\node [label = above: {\Large $y_{29}$}] at (5.5,3) {};
\node [label = above: {\Large $y_{28}$}] at (6.5,3) {};
\node [label = above: {\Large $y_{27}$}] at (7.5,3) {};
\node [label = above: {\Large $y_{26}$}] at (8.5,3) {};
\node [label = above: {\Large $y_{25}$}] at (9.5,3) {};
\node [label = above: {\Large $y_{24}$}] at (10.5,3) {};
\node [label = above: {\Large $y_{39}$}] at (5.5,2) {};
\node [label = above: {\Large $y_{38}$}] at (6.5,2) {};
\node [label = above: {\Large $y_{37}$}] at (7.5,2) {};
\node [label = above: {\Large $y_{36}$}] at (8.5,2) {};
\node [label = above: {\Large $y_{35}$}] at (9.5,2) {};
\node [label = above: {\Large $y_{34}$}] at (10.5,2) {};
\node [label = above: {\Large $y_{49}$}] at (5.5,1) {};
\node [label = above: {\Large $y_{48}$}] at (6.5,1) {};
\node [label = above: {\Large $y_{47}$}] at (7.5,1) {};
\node [label = above: {\Large $y_{46}$}] at (8.5,1) {};
\node [label = above: {\Large $y_{45}$}] at (9.5,1) {};
\node [label = above: {\Large $y_{59}$}] at (5.5,0) {};
\node [label = above: {\Large $y_{58}$}] at (6.5,0) {};
\node [label = above: {\Large $y_{57}$}] at (7.5,0) {};
\node [label = above: {\Large $y_{56}$}] at (8.5,0) {};
\node [label = above: {\Large $y_{69}$}] at (5.5,-1) {};
\node [label = above: {\Large $y_{68}$}] at (6.5,-1) {};
\node [label = above: {\Large $y_{67}$}] at (7.5,-1) {};
\node [label = above: {\Large $y_{79}$}] at (5.5,-2) {};
\node [label = above: {\Large $y_{78}$}] at (6.5,-2) {};
\end{tikzpicture}
\hskip1cm
\begin{tikzpicture}[scale=0.8, every node/.style={scale=0.6}]
\node [label = above: {\Large $x_{29}$}] at (5.5,5) {};
\node [label = above: {\Large $x_{28}$}] at (6.5,5) {};
\node [label = above: {\Large $x_{27}$}] at (7.5,5) {};
\node [label = above: {\Large $x_{26}$}] at (8.5,5) {};
\node [label = above: {\Large $x_{25}$}] at (9.5,5) {};
\node [label = above: {\Large $x_{24}$}] at (10.5,5) {};
\node [label = left: {\Large $x_{11}$}] at (5,4.4){};
\node [label = left: {\Large $x_{12}$}] at (5,3.4){};
\node [label = left: {\Large $x_{13}$}] at (5,2.4){};
\node [label = left: {\Large $x_{14}$}] at (5,1.4){};
\node [label = left: {\Large $x_{15}$}] at (5,0.4){};
\node [label = left: {\Large $x_{16}$}] at (5,-0.6){};
\node [label = left: {\Large $x_{17}$}] at (5,-1.6){};
\draw [line width=1pt, color=black] (4,5)--(11,5);
\draw [line width=1pt] (4,5)--(4,-2);
\draw [line width=1pt] (5,6)--(5,-2);
\draw [line width=1pt] (4,4)--(11,4);
\draw [line width=1pt] (5,6)--(11,6);
\draw [line width=1pt](6,6)--(6,-2);
\draw [line width=1pt](4,3)--(11,3);
\draw [line width=1pt](7,6)--(7,-2);
\draw [line width=1pt](8,6)--(8,-1);
\draw [line width=1pt](9,6)--(9,0);
\draw [line width=1pt](10,6)--(10,1);
\draw [line width=1pt](11,6)--(11,2);
\draw [line width=1pt](7,2)--(7,-1);
\draw [line width=1pt](4,2)--(11,2);
\draw [line width=1pt](4,1)--(10,1);
\draw [line width=1pt](4,0)--(9,0);
\draw [line width=1pt](4,-1)--(8,-1);
\draw [line width=1pt](4,-2)--(7,-2);
\node [label = above: {\Large $y_{19}$}] at (5.5,4) {};
\node [label = above: {\Large $y_{18}$}] at (6.5,4) {};
\node [label = above: {\Large $y_{17}$}] at (7.5,4) {};
\node [label = above: {\Large $y_{16}$}] at (8.5,4) {};
\node [label = above: {\Large $y_{15}$}] at (9.5,4) {};
\node [label = above: {\Large $y_{14}$}] at (10.5,4) {};
\node [label = above: {\Large $y_{29}$}] at (5.5,3) {};
\node [label = above: {\Large $y_{28}$}] at (6.5,3) {};
\node [label = above: {\Large $y_{27}$}] at (7.5,3) {};
\node [label = above: {\Large $y_{26}$}] at (8.5,3) {};
\node [label = above: {\Large $y_{25}$}] at (9.5,3) {};
\node [label = above: {\Large $y_{24}$}] at (10.5,3) {};
\node [label = above: {\Large $y_{39}$}] at (5.5,2) {};
\node [label = above: {\Large $y_{38}$}] at (6.5,2) {};
\node [label = above: {\Large $y_{37}$}] at (7.5,2) {};
\node [label = above: {\Large $y_{36}$}] at (8.5,2) {};
\node [label = above: {\Large $y_{35}$}] at (9.5,2) {};
\node [label = above: {\Large $y_{34}$}] at (10.5,2) {};
\node [label = above: {\Large $y_{49}$}] at (5.5,1) {};
\node [label = above: {\Large $y_{48}$}] at (6.5,1) {};
\node [label = above: {\Large $y_{47}$}] at (7.5,1) {};
\node [label = above: {\Large $y_{46}$}] at (8.5,1) {};
\node [label = above: {\Large $y_{45}$}] at (9.5,1) {};
\node [label = above: {\Large $y_{59}$}] at (5.5,0) {};
\node [label = above: {\Large $y_{58}$}] at (6.5,0) {};
\node [label = above: {\Large $y_{57}$}] at (7.5,0) {};
\node [label = above: {\Large $y_{56}$}] at (8.5,0) {};
\node [label = above: {\Large $y_{69}$}] at (5.5,-1) {};
\node [label = above: {\Large $y_{68}$}] at (6.5,-1) {};
\node [label = above: {\Large $y_{67}$}] at (7.5,-1) {};
\node [label = above: {\Large $y_{79}$}] at (5.5,-2) {};
\node [label = above: {\Large $y_{78}$}] at (6.5,-2) {};
\end{tikzpicture}
\caption{Ladders $L_\lambda$ and $L_\lambda'$ for $\lambda=(6,6,6,5,4,3,2)$}
\label{fig:3}
\end{figure}
\FloatBarrier
Notice that the $x_{ij}$ are only row and column markers in $L_\lambda$,
whereas in $L'_\lambda$ they are entries of the ladder.
The entries in $L_\lambda$ and in $L'_\lambda$
are distinct variables. Therefore, results for ladder determinantal ideals apply, see for instance \cite{Conca, Nar}.
\section{Rees algebras of $I$ and of its initial ideal}\label{sect Rees alg}
The Rees algebra of an ideal can be realized as a quotient of a polynomial ring.
When $I$ is an ideal generated by $n$ elements, say $f_1, \ldots, f_n$, we let $y_1, \ldots, y_n$ be variables over $R$ and consider the map from $R[y_1, \ldots, y_n]$ to $\hbox{\ensuremath{{\mathcal R}(I)}}$ that maps $y_i$ to $f_it$. Hence $\hbox{\ensuremath{{\mathcal R}(I)}}\cong R[y_1, \ldots, y_n]/J$, where $J$ is the {\it defining ideal} of the algebra, and the {\it defining equations} are a system of generators of $J$.
The defining equations of the Rees algebra
are in general difficult to compute or determine theoretically. The largest $y$-degree of a minimal generator of the defining ideal $J$ is the {\it relation type} of the ideal and plays an important role in the study of blowup algebras. Finally, we say that the Rees algebra of $I$ is of {\it fiber type} if the defining ideal of the Rees algebra is generated by the defining equations of the symmetric algebra and the defining equations of the special fiber ring.
Given a term order $\tau$ on a polynomial ring $R$ over a field $K$, one can extend it to a term order $\tau'$ on $R[t]$ as follows. Let $a,b\in R$ be monomials and let $i,j$ be non-negative integers. Define
\begin{equation}\label{tau'}
at^i<_{\tau'} bt^j \ \ \Leftrightarrow \ \ i<j \ \ \mbox{ or } \ \ i=j \mbox{ and } a<_{\tau}b.
\end{equation}
For a $K$-subalgebra $A$ of $R[t]$, one defines ${\rm{in}_{\tau'}}(A)$ as the $K$-algebra generated by all initial monomials of elements in $A$.
When $A$ is homogeneous in the variable $t$, then ${\rm{in}_{\tau'}}(A)=\oplus_{i \ge 0} {\rm{in}_{\tau'}}(A_i)$, where $A_i$ is the set of elements of $A$ that are homogeneous of degree $i$ in $t$. In particular, in our setting $${\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\oplus_{i \ge 0} {\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}}_i)=\oplus_{i\ge 0} \ini(I^i)t^i. $$ Since $\hbox{\ensuremath{{\mathcal R}(\lt I)}}=\oplus_{i \ge 1} (\ini(I))^it^i$, in order to prove that ${\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\hbox{\ensuremath{{\mathcal R}(\lt I)}}$, it suffices to show that $(\ini(I))^i=\ini(I^i)$ for all $i \ge 1$. A result by Conca, Herzog, and Valla states that it suffices to check the equality up to the relation type of $\ini(I)$.
\begin{theorem}\cite[Corollary~2.8]{CHV} \label{reltype thm}
Let $R$ be a polynomial ring, $I$ a homogeneous ideal in $R$, and $\tau$ a term order on $R$. Suppose that $(\ini(I))^i=\ini(I^i)$ for $1\le i \le {\rm{reltype}}(\ini(I))$. Then
$$(\ini(I))^i=\ini(I^i) \mbox{ for all } i \ge 1 \ \ \mbox{ and } \ \ {\rm{reltype}}(I) \le {\rm{reltype}}(\ini(I)).$$
\end{theorem}
The equality ${\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ was established by Conca in~\cite[Theorem~2.1]{ConcaGB} for the ideal $I$ of maximal minors of a generic $m\times n$ matrix. The following lemma is the key to establishing ${\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ in our case.
\begin{lemma}\label{I^2=}
Let $X$ be a sparse $2\times n$ matrix, $I$ the ideal of $2\times 2$ minors of $X$, and $\tau$ a diagonal term order on $R=K[X]$. Then $\ini(I^2)=(\ini(I))^2$.
\end{lemma}
\begin{proof}
We proceed by induction on $n\ge 2$. For $n=2$, the ideal $I$ is principal and the result holds automatically. To proceed we choose a decomposition of $I$ as follows.
Let $I_1$ be the ideal generated by the variables $x_{2j}$ for $r+1 \le j \le n$ and let $I_2$ be the ideal generated by the $2 \times 2$ minors of the submatrix of $X$ obtained by deleting the first column of $X$. Since $r\ge 1$, then $I = x_{11} I_1 + I_2$. By Proposition~\ref{gb of ini}, $\ini(I) = x_{11} I_1 + \ini(I_2)$. Notice here that for $I_2$ we may have that its corresponding $r$ is $0$. In that case $\ini(I_2^2)=(\ini(I_2))^2$ by \cite[Theorem~2.1]{ConcaGB}. Otherwise, the equality holds by induction.
Certainly $(\ini(I))^2 \subseteq \ini(I^2)$. To prove equality it is enough to show that
the Hilbert functions of $(\ini(I))^2$ and $I^2$ are the same, since the Hilbert function of $I^2$ is the same as the Hilbert function of $\ini(I^2)$.
By induction on $n$, the Hilbert functions of $I_2^2$ and $(\ini(I_2))^2$ are the same. We have $$I^2=(x_{11}^2I_1^2+x_{11}I_1I_2)+I_2^2 \quad \mbox{and }\quad (\ini(I))^2=(x_{11}^2I_1^2+x_{11}I_{1}\ini(I_2))+(\ini(I_2))^2.$$
Both of these ideals are of the form
$J^2 = (x_{11}^2 I_1^2 + x_{11}I_1J_2) + J_2^2$,
with $J_2 = I_2$ and $J = I$ in the former case
and with $J_2 = \ini(I_2)$ and $J = \ini(I)$ in the latter case.
Each case gives rise to the short exact sequence:
\begin{eqnarray} \label{ses}
0 \longrightarrow (x_{11}^2I_1^2+x_{11}I_1J_2)\cap J_2^2\longrightarrow
(x_{11}^2I_1^2+x_{11}I_1J_2)\oplus J_2^2\longrightarrow J^2 \longrightarrow
0.
\end{eqnarray}
Since the variable $x_{11}$ is a non-zerodivisor on $J_2^2$,
and since $J_2 \subseteq I_1$,
the intersection on the left-hand side of the sequence is:
\begin{align*}
(x_{11}^2 I_1^2 + x_{11} I_1 J_2) \cap J_2^2
&= (x_{11}^2 I_1^2 + x_{11} I_1 J_2) \cap (x_{11}) \cap J_2^2 \\
&= (x_{11}^2 I_1^2 + x_{11} I_1 J_2) \cap x_{11} J_2^2 \\
&= x_{11} J_2^2.
\end{align*}
This means that the short exact sequence simplifies to
\begin{eqnarray}
0 \longrightarrow x_{11}J_2^2\longrightarrow
(x_{11}^2I_1^2+x_{11}I_1J_2)\oplus J_2^2\longrightarrow J^2 \longrightarrow
0.
\end{eqnarray}
By the induction hypothesis,
the two incarnations of $J_2^2$ and hence of $x_{11}J_2^2$
have the same Hilbert function,
so that to prove that the two incarnations $I^2$ and $(\ini(I))^2$ of $J^2$
have the same Hilbert function
it suffices to prove that the two incarnations
of $x_{11}^2I_1^2+x_{11}I_1J_2$ have the same Hilbert function.
As before we get a short exact sequence:
\begin{eqnarray} \label{ses3}
0 \longrightarrow x_{11}I_1^2\cap I_1J_2 \longrightarrow x_{11}I_1^2\oplus I_1J_2\longrightarrow x_{11}I_1^2+I_1J_2 \longrightarrow 0,
\end{eqnarray}
and since $x_{11}$ is a non-zerodivisor on $I_1J_2$
and $J_2 \subseteq I_1$,
the short exact sequence simplifies to
\begin{eqnarray} \label{ses2}
0 \longrightarrow x_{11} I_1 J_2 \longrightarrow x_{11}I_1^2\oplus I_1J_2\longrightarrow x_{11}I_1^2+I_1J_2 \longrightarrow 0.
\end{eqnarray}
The conclusion
will follow once we show that in the two incarnations of $J_2$ the Hilbert function of $I_1 J_2$ is the same, that is $\ini(I_1I_2)=I_1\ini(I_2)$.
Clearly, $I_1\ini(I_2) \subseteq \ini(I_1I_2)$.
Define a grading on $R$ by setting $\deg(x_{1i})= 0$ and $\deg(x_{2j}) = 1$ for all $i, j$.
With this grading, $I_1$ is generated by elements of degree $1$,
$I_2$ is generated by elements of degree $1$,
and $I_1I_2$ is generated by elements of degree $2$.
It suffices to show that for every homogeneous $f \in I_1I_2$,
$\ini(f) \in I_1 \ini(I_2)$.
Clearly $\ini(f)$ has degree at least $2$ and is in $\ini(I_2)$.
By Proposition~\ref{gb of ini},
$\ini(I_2)$ is generated by elements of degree $1$.
Thus by degree reasons,
to write $\ini(f)$ as an element of $\ini(I_2)$,
the coefficient must have degree at least $1$,
i.e.,
the coefficient must be in
$(x_{2i} \mid i = r+1, \ldots, n) = I_1$.
It follows that $\ini(f) \in I_1 \ini(I_2)$.
\end{proof}
We are now ready to prove the main theorem of this section. Let $A$ be a subalgebra of a polynomial ring over a field $K$.
We recall that a subset $C$ of $A$ is a \emph{SAGBI basis} for $A$ with respect to a term order $\tau$ if $\ini(A)$ is generated as a $K$-algebra by the initial monomials of the elements in $C$ with respect to $\tau$. In general, SAGBI bases are difficult to compute and may not even be finite. When $I$ is the ideal of maximal minors of a generic $m \times n$ matrix $X$, Conca showed that ${\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ with respect to a diagonal term order $\tau$~\cite[Theorem~2.1]{ConcaGB}.
Moreover, Narasimhan \cite[Corollary~3.4]{Nar} showed that the maximal minors of $X$ form a Gr\"{o}bner basis for $I$. It then follows that the natural algebra generators of $\hbox{\ensuremath{{\mathcal R}(I)}}$ are a SAGBI basis of it. The following theorem extends these results to the case of the ideal maximal minors of a sparse $2\times n$ matrix.
\begin{theorem}\label{initial rees thm}
Let $X$ be a sparse $2\times n$ matrix and let $I$ be the ideal of $2\times 2$ minors of $X$. Let $\tau$ be a diagonal term order on $R=K[X]$ and let $\tau'$ be its extension to $R[t]$ as in~(\ref{tau'}). Then $${\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\hbox{\ensuremath{{\mathcal R}(\lt I)}} \ \ \mbox{ and } \ \ {\rm{reltype}}(I)\le 2.$$
Moreover, the set $\{x_{1i}, x_{2j} \mid 1\le i \le r+s, r+1 \le j \le n\} \cup\{f_{ij}t\mid 1\le i<j \le n\}$ is a SAGBI basis of $\hbox{\ensuremath{{\mathcal R}(I)}}$ with respect to $\tau'$.
\end{theorem}
\begin{proof}
According to \cite[Proposition~5.1]{CN}, the defining ideal of the special fiber ring of~$I_{\lambda} = \ini(I)$ is the ideal generated by the $2 \times 2$ minors of $L_{\lambda}$. By~\cite[Theorem~3.1]{V} we know that $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ is of fiber type and hence the relation type of $\ini(I)$ is at most $2$.
By Lemma~\ref{I^2=} we have that $\ini(I^2)=(\ini(I))^2$, and thus by Theorem~\ref{reltype thm}, the relation type of $I$ is at most~$2$ and $\ini(I^i)=(\ini(I))^{i}$ for all $i$.
Therefore,
$${\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\oplus_{i\ge 0} \ini(I^i)=\oplus_{i\ge 0} (\ini(I))^i=\hbox{\ensuremath{{\mathcal R}(\lt I)}}.$$
The last part of the claim now follows, since the $2\times 2$ minors of $X$ are a Gr\"{o}bner basis of $I$ by \cite[Proposition~5.4]{Boocher}. Therefore the set $\{x_{1i}, x_{2j} \mid 1\le i \le r+s, r+1 \le j \le n\} \cup\{f_{ij}t\mid 1\le i<j \le n\}$ is a SAGBI basis of $\hbox{\ensuremath{{\mathcal R}(I)}}$, since the leading terms of these elements generate $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ as $K$-algebra and ${\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\hbox{\ensuremath{{\mathcal R}(\lt I)}}$.
\end{proof}
An immediate consequence of the equality $\ini(I^i)=(\ini(I))^i$ is the following corollary.
\begin{corollary}
Let $X$ be a sparse $2\times n$ matrix and let $I$ the ideal of $2\times 2$ minors of $X$. Let $\tau$ be a diagonal term order on $R=K[X]$. The $i$-fold products of maximal minors of $X$ are a $\tau$-Gr\"{o}bner basis of $I^i$ for every $i$.
\end{corollary}
Using the theory of SAGBI bases, one can now deduce properties of the Rees algebra of $I$ from those of its initial algebra.
\begin{corollary}\label{rees sing}
Let $X$ be a sparse $2\times n$ matrix and let $I$ the ideal of $2\times 2$ minors of $X$. Then $\hbox{\ensuremath{{\mathcal R}(I)}}$ has rational singularities if the field $K$ has characteristic $0$ and it is $F$-rational if $K$ has positive characteristic. In particular, $\hbox{\ensuremath{{\mathcal R}(I)}}$ is a Cohen-Macaulay normal domain.
\end{corollary}
\begin{proof}
Let $\tau$ be a diagonal term order on $R=K[X]$ and let $\tau'$ be its extension to $R[t]$ as in~(\ref{tau'}).
By \cite[Corollary 5.3 and Theorem 5.9]{SVV}, $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ is a Cohen-Macaulay normal domain. Thus ${\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})$ is a Cohen-Macaulay normal domain by Theorem~\ref{initial rees thm}.
The conclusion then follows from \cite[Corollary~2.3]{CHV}.
\end{proof}
\section{The Defining Equations of the Rees Algebra}
In this section we obtain the defining equations of the Rees algebra of the ideal of $2\times 2$ minors of a sparse $2\times n$ matrix $X$. In addition to the setup and notation from Section~\ref{sectsetting}, we will also adopt the following.
\begin{setting}\label{setting 2}
Let $S=K[L_{\lambda}']$ and $T=K[L_{\lambda}]$ be the polynomial rings over $K$ in the variables that appear in $L_{\lambda}'$ and $L_{\lambda}$, respectively.
To simplify our notation, we make the identifications:\begin{itemize}
\item $x_{ui}=0$ if $u=1$ and $i>r+s$ or if $u=2$ and $i\le r$,
\item $y_{ij}=0$ if $i,j\in \{1,\ldots,r\}$ or $i,j \in \{r+s+1,\ldots, n\}$,
\end{itemize}
to give meaning to all other $x_{ui}, y_{ij}$ with $u \in \{1,2\}$, $1 \le i< j \le n$.
Define the following standard presentations
of the symmetric algebra $\hbox{\ensuremath{{\mathcal S}(I)}}$,
the Rees algebra $\hbox{\ensuremath{{\mathcal R}(I)}}$ of $I$,
and the special fiber ring $\hbox{\ensuremath{{\mathcal F}(I)}}$:
\begin{eqnarray*}
\sigma: S \longrightarrow \hbox{\ensuremath{{\mathcal S}(I)}},\\
\rho:S\longrightarrow \hbox{\ensuremath{{\mathcal R}(I)}},\\
\varphi:T\longrightarrow \hbox{\ensuremath{{\mathcal F}(I)}},
\end{eqnarray*}
where for all $i,j$,
$\sigma(x_{ij})=\rho(x_{ij})=x_{ij}$, $\sigma(y_{ij})=\varphi(y_{ij})=f_{ij}$, and $\rho(y_{ij})=f_{ij}t$.
We let $\mathcal{L}=\ker(\sigma)$, $\mathcal{J}=\ker(\rho)$, and $\mathcal{K}=\ker (\varphi)$. The ideals $\mathcal{L}, \mathcal{J}, \mathcal{K}$ are called the defining ideals of the symmetric algebra, the Rees algebra, and the special fiber ring of $I$, respectively.
We similarly define the presentations of the symmetric algebra,
of the Rees algebra, and of the special fiber ring of $\ini(I)$:
\begin{eqnarray*}
\sigma': S &\longrightarrow& \hbox{\ensuremath{{\mathcal S}(\ini(I))}},\\
\rho': S &\longrightarrow&\hbox{\ensuremath{{\mathcal R}(\lt I)}}= {\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}}), \\
\varphi':T&\longrightarrow& \hbox{\ensuremath{{\mathcal F}(\lt I)}},\\
\end{eqnarray*}
where for all $i,j$,
$\sigma(x_{ij})=\rho(x_{ij})=x_{ij}$, $\sigma(y_{ij})=\varphi(y_{ij})=\ini(f_{ij})$, and $\rho(y_{ij})=\ini(f_{ij})t$.
Moreover, let $\mathcal{L}'=\ker(\sigma')$, $\mathcal{J}'=\ker(\rho')$, and $\mathcal{K'}=\ker (\varphi')$ denote the defining ideals of the symmetric algebra, the Rees algebra, and the special fiber of $\ini(I)$, respectively.
\end{setting}
\begin{remark} \label{definition ell, plucker}
Let $\ell_{uijk}=x_{ui}y_{jk}-x_{uj}y_{ik}+x_{uk}y_{ij}$ and $p_{ijkl}=y_{ij}y_{kl}-y_{ik}y_{jl}+y_{il}y_{jk}$
for $u \in\{1, 2\}$ and $1\le i<j<k<l \le n$.
With the identifications from the Setting and Notation~\ref{setting 2}, $\ell_{uijk}, p_{ijkl}$ are in $S$ for all $u\in\{1,2\}$, $1\le i<j<k<l\le n$. We call the $\ell_{uijk}$'s the linear relations of $I$ and the $p_{ijkl}$'s the Pl\"{u}cker relations of $I$. Clearly $\ell_{uijk} \in \mathcal{L}$ and $\ell_{uijk}, p_{ijkl} \in \mathcal{J}$ for all $u \in\{1,2\}$ and $1\le i<j<k<l \le n$.
\end{remark}
In the next proposition we describe the defining equations of the symmetric algebra, the special fiber, and the Rees algebra of $\ini(I)$. In addition, we show that the $\ell_{uijk}$'s are indeed the defining equations of the symmetric algebra of $I$.
\begin{proposition} \label{L L' K' J'}
Adopt Setting and Notation~\ref{setting 2}. Then
\begin{enumerate}[$($a$)$]
\item $\mathcal{L} =(\ell_{uijk}\mid u = 1, 2; 1\le i <j <k\le n)$.
\item $\mathcal{L} '$ is generated by the $2 \times 2$ minors of $L_{\lambda}'$ that involve either the first column or the first row.
\item $\mathcal{K}' =I_2(L_{\lambda})$.
\item $\mathcal{J}' =I_2(L_{\lambda}')=\mathcal{L}' +I_2(L_{\lambda})S$.
\end{enumerate}
Moreover, the natural generators of $I_2(L_{\lambda})$ and $I_2(L_{\lambda}')$ are Gr\"{o}bner bases of
$\mathcal{K}'$ and $\mathcal{J}'$, respectively, with respect to a diagonal term order.
\end{proposition}
\begin{proof}
(a) By \cite[Theorem~4.1 and its proof]{Boocher} the presentation matrix of $I$ is obtained from the Eagon-Northcott resolution by ``pruning''.
In particular, the relations on the generators~$f_{ij}$ of~$I$ arise from taking the $3 \times 3$ minors of
the $3 \times n$ matrix obtained from $X$ by doubling one of the two rows.
These yield precisely the relations on the symmetric algebra of the given form.
Item (b) follows from \cite[Theorem~5.1]{HHV} since Ferrers ideals satisfy the $\ell$-exchange property, see e.g.~\cite[Lemma~6.3]{LS}.
(c) This follows from \cite[Proposition~5.1]{CN}.
(d) Since $\ini(I)$ is a Ferrers ideal, it is the edge ideal of a bipartite graph. Therefore the Rees algebra of $\ini(I)$ is of fiber type by \cite[Theorem~3.1]{V}.
The last part follows from the fact that $I_2(L_{\lambda})$ and $I_2(L_{\lambda}')$ are ladder determinantal ideals. Therefore, by~\cite[Corollary~3.4]{Nar}, the $2\times 2$ minors of $L_{\lambda}$ and $L_{\lambda}'$ are Gr\"{o}bner bases of $I_2(L_{\lambda})$ and $I_2(L_{\lambda}')$, respectively, with respect to a diagonal term order.
\end{proof}
The description of the defining equations of the Rees algebra of $\ini(I)$, in combination with our result from the previous section that shows that ${\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\hbox{\ensuremath{{\mathcal R}(\lt I)}}$, allows us to deduce that the Rees algebra of $I$ is a Koszul algebra.
\begin{corollary}\label{Rees koszul}
Let $X$ be a sparse $2\times n$ matrix, $I$ the ideal of $2\times 2$ minors of $X$. Then $\hbox{\ensuremath{{\mathcal R}(I)}}$ is a Koszul algebra and $I$ has linear powers. In particular, ${\rm{reg}} (I^k)=2k$ for all $k \in \mathbb{N}$.
\end{corollary}
\begin{proof}
The defining ideal of $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ has a Gr\"{o}bner basis of quadrics by Proposition~\ref{L L' K' J'}, so $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ is a Koszul algebra, see~\cite{Fro} and~\cite[Theorem~6.7]{EH}. By Theorem~\ref{initial rees thm} we have $\ini(\hbox{\ensuremath{{\mathcal R}(I)}})=\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ and since $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ is a Koszul algebra, then $\hbox{\ensuremath{{\mathcal R}(I)}}$ is a Koszul algebra, by \cite[Corollary~2.6]{CHV}.
According to Blum~\cite[Corollary~3.6]{Bl}, if the Rees algebra of an ideal is Koszul, then the ideal has linear powers, i.e., the powers of the ideal have a linear resolution.
\end{proof}
We now are ready to prove the main result of this article.
\begin{theorem}\label{gens of rees}
Let $X$ be a sparse $2\times n$ matrix, $I$ the ideal of $2\times 2$ minors of $X$. Adopt Setting and Notation~\ref{setting 2}.
\begin{enumerate}[$($a$)$]
\item The defining ideal $\mathcal{J}$ of $\hbox{\ensuremath{{\mathcal R}(I)}}$
is generated by the linear relations $\ell_{uijk}$ for $u \in\{1, 2\}$ and $1\le i<j<k\le n$, and the Pl\"{u}cker relations $p_{ijkl}$ for $1\le i<j<k<l \le n$. Moreover, these generators form a Gr\"obner basis of $\mathcal{J}$ with respect to a suitable weight.
\item The Rees algebra $\hbox{\ensuremath{{\mathcal R}(I)}}$ is of fiber type, i.e., $$\mathcal{J}=\mathcal{L}+\mathcal{K}S.$$
\item The Pl\"ucker relations $p_{ijkl}$ for $1\le i<j<k<l\le n$ are the defining equations of $\hbox{\ensuremath{{\mathcal F}(I)}}$.
\end{enumerate}
\end{theorem}
\begin{proof}
(a) We let $G$ be the set of all $\ell_{uijk}$ and $p_{ijkl}$ as in Remark~\ref{definition ell, plucker}.
We claim that $\mathcal{J}=(G)$. Define a weight $\omega$ on $R[t]$ and a weight $\pi$ on $S$ as follows:
$\omega(x_{1j})= 1,$ $\omega(x_{2j})= j,$ $\omega(t)=1,$
$\pi(x_{1j})= 1,$ $\pi(x_{2j})= j,$ $\pi(y_{ij})=\omega(\ini(f_{ij})t)=j+2.$
By Proposition~\ref{L L' K' J'}, $I_2(L_{\lambda}')$ is the defining ideal of $\hbox{\ensuremath{{\mathcal R}(\lt I)}}= {\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})$.
The weight $\omega$ represents the term order $\tau$, that is, $\inid_{\omega}(f_{ij})=\ini(f_{ij})$. Therefore, by~\cite[Proposition in Lecture 3.1]{Stur} (see also \cite[Theorem~11.4]{Stur2}) we conclude that $I_2(L_\lambda')=\inid_\pi(\mathcal{J})$.
Since $G \subseteq \mathcal{J}$, to prove that $G$ is a Gr\"obner basis of $\mathcal{J}$ with respect to $\pi$, it suffices to prove that each $2\times 2$ minor of $L_\lambda'$ is the leading form with respect to $\pi$ of some element in $G$. This also implies that $\mathcal{J}$ is generated by $G$.
We first analyze the $2 \times 2$ minors of $L'_\lambda$ that involve the first column.
These are of the form $E_{1ijk} = x_{1i} y_{jk} - x_{1j} y_{ik}$ with $i<j<k$ and they are homogeneous of weight $k+3$. Since $x_{1k}y_{ij}$ has weight $j+3$, $E_{1ijk}$ is the leading form of $\ell_{1ijk} \in G$.
We next analyze the $2 \times 2$ minors of $L'_\lambda$ that involve the first row.
These are of the form $E_{2ijk} = x_{2k} y_{ij} - x_{2j} y_{ik}$ with $i<j<k$ and they are homogeneous of weight $j+k+2$. Since $x_{2i}y_{jk}$ has weight $i+k+2$, $E_{2ijk}$ is the leading form of $\ell_{2jik} \in G$.
It remains to prove that each $2 \times 2$ minor of $L_\lambda$ is a unit multiple of a leading form of an element in $G$. Such a minor is of the form $F_{ijkl} = y_{il} y_{jk} - y_{ik} y_{jl}$ for some $i<j<k<l$ and it is homogeneous of weight $k+l+4$. Since $y_{ij}y_{kl}$ has weight $j+l+4$, $F_{ijkl}$ is the leading form of $p_{ijkl} \in G$.
Items (b) and (c) follow immediately from (a).
\end{proof}
The following corollary is a direct consequence of Theorem~\ref{gens of rees}.
\begin{corollary} \label{ini F}
Let $X$ be a sparse $2\times n$ matrix, $I$ the ideal of $2\times 2$ minors of $X$, and $\tau$ a diagonal term order on $R=K[X]$. Then $\ini(\hbox{\ensuremath{{\mathcal F}(I)}})=\hbox{\ensuremath{{\mathcal F}(\lt I)}}$. Moreover, the $2\times 2$ minors of $X$ are a SAGBI basis of the $K$-algebra $\hbox{\ensuremath{{\mathcal F}(I)}}$, the Pl\"ucker relations form a Gr\"obner basis for $\mathcal{K}$, and $\hbox{\ensuremath{{\mathcal F}(I)}}$ is a Koszul algebra.
\end{corollary}
\begin{proof}
By Proposition~\ref{L L' K' J'}, the defining equations of $\hbox{\ensuremath{{\mathcal F}(\lt I)}}$ are the $2 \times 2$ minors of $L_\lambda$, i.e., $\hbox{\ensuremath{{\mathcal F}(\lt I)}}=K[\ini(f_{ij})]\cong T/I_2(L_\lambda)$. For any $i<j<k<l$, let $F_{ijkl}=y_{il} y_{jk} - y_{ik} y_{jl}$ be a generator of $I_2(L_\lambda)$.
In the proof of Theorem~\ref{gens of rees} we showed that $p_{ijkl}=F_{ijkl}+y_{ij}y_{kl}$. Hence $\varphi(F_{ijkl})=-\varphi (y_{ij}y_{kl})$ and $\ini \varphi(F_{ijkl})= \ini \varphi (y_{ij}y_{kl}) $. Then by \cite[Proposition~1.1]{CHV} the generators of $I$ are a SAGBI basis of $\hbox{\ensuremath{{\mathcal F}(I)}}$, in particular $\ini(\hbox{\ensuremath{{\mathcal F}(I)}})=\hbox{\ensuremath{{\mathcal F}(\lt I)}}$. Moreover, the Pl\"ucker relations form a Gr\"obner basis of $\mathcal{K}$ by~\cite[Corollary~11.6]{Stur2}, hence $\hbox{\ensuremath{{\mathcal F}(I)}}$ is a Koszul algebra.
\end{proof}
Finally, we obtain the analogous result for $\hbox{\ensuremath{{\mathcal F}(I)}}$ as in Corollary~\ref{rees sing}.
\begin{corollary}\label{FI CM normal}
Let $X$ be a sparse $2\times n$ matrix and $I$ the ideal of $2\times 2$ minors of $X$. Then $\hbox{\ensuremath{{\mathcal F}(I)}}$ has rational singularities if the field $K$ has characteristic $0$ and is $F$-rational if $K$ has positive characteristic. In particular, $\hbox{\ensuremath{{\mathcal F}(I)}}$ is a Cohen-Macaulay normal domain.
\end{corollary}
\begin{proof}
By \cite[Theorem 7.1]{SVV} and Corollary~\ref{rees sing},
$\hbox{\ensuremath{{\mathcal F}(\lt I)}}$ is normal.
Thus, by \cite[Theorem 1]{Ho}
$\hbox{\ensuremath{{\mathcal F}(\lt I)}}$ is Cohen-Macaulay as well.
By Corollary~\ref{ini F}, we have $\ini(\hbox{\ensuremath{{\mathcal F}(I)}})=\hbox{\ensuremath{{\mathcal F}(\lt I)}}$.
The result now follows from \cite[Corollary~2.3]{CHV}.
\end{proof}
\begin{corollary}\label{invFi}
Let $X$ be a sparse $2\times n$ matrix and $I$ the ideal of $2\times 2$ minors of $X$. Then
\begin{enumerate}[$($a$)$]
\item $\hbox{\ensuremath{{\mathcal F}(I)}}$ has dimension $\min\{n+s-1,2n-r-2\}$.
\item $\hbox{\ensuremath{{\mathcal F}(I)}}$ is Gorenstein if $r\le 2$.
\item ${\rm{reg}}(\hbox{\ensuremath{{\mathcal F}(I)}})=\min\{n-3,n-r-1,r+s-1\}$.
\item The $a$-invariant of $\hbox{\ensuremath{{\mathcal F}(I)}}$ is $a(\hbox{\ensuremath{{\mathcal F}(I)}})=\min\{-r-s,r-n,-s-2\}$, unless $n=r+s$ in which case $a(\hbox{\ensuremath{{\mathcal F}(I)}})=-n$ if $r=1$ or $a(\hbox{\ensuremath{{\mathcal F}(I)}})=-n+1$ otherwise.
\end{enumerate}
\end{corollary}
\begin{proof}
(a) The formula follows from \cite[Corollary~2.6]{CHV} in combination with \cite{SVV} or \cite[Section~4]{Conca}.
(b) By \cite[Proposition~2.5]{Conca} $\hbox{\ensuremath{{\mathcal F}(\lt I)}}$ is Gorenstein if and only if $r\le 2$. The statement now follows from \cite[Corollary~2.3]{CHV}.
(c) From \cite[Proposition~5.7]{CN} we obtain the regularity of $\hbox{\ensuremath{{\mathcal F}(\lt I)}}$,
which simplifies to ${\rm{reg}}(\hbox{\ensuremath{{\mathcal F}(\lt I)}})=\min\{n-3,n-r-1,r+s-1\}$. By \cite[Corollary~2.5]{CHV} we have $a(\hbox{\ensuremath{{\mathcal F}(\lt I)}})=a(\hbox{\ensuremath{{\mathcal F}(I)}})$ and since $\hbox{\ensuremath{{\mathcal F}(\lt I)}}$ is Cohen-Macaulay, then ${\rm{reg}}(\hbox{\ensuremath{{\mathcal F}(I)}})={\rm{reg}}(\hbox{\ensuremath{{\mathcal F}(\lt I)}})$.
(d) The result follows from the fact that $a(\hbox{\ensuremath{{\mathcal F}(I)}})=-\dim \hbox{\ensuremath{{\mathcal F}(I)}} +{\rm{reg}}(\hbox{\ensuremath{{\mathcal F}(I)}})$. Alternatively, the formula is a special case of \cite[Corollary~9]{GK}. In the special case $n=r+s$, we have $\dim \hbox{\ensuremath{{\mathcal F}(I)}}=2n-r-2$ and ${\rm{reg}}(\hbox{\ensuremath{{\mathcal F}(I)}})=\min\{n-3, n-r-1\}$.
\end{proof}
\begin{corollary}\label{eFI}
Let $X$ be a sparse $2\times n$ matrix and $I$ the ideal of $2\times 2$ minors of $X$. Let $\lambda$ be the partition for the Ferrers ideal $\ini(I_{\lambda})$.
Then the normalized Hilbert series of $\hbox{\ensuremath{{\mathcal F}(I)}}$ is $p_{\lambda}(z)=1+h_1(\lambda)z+\ldots +h_{r+s-1}(\lambda)z^{r+s-1}$, where $h_k(\lambda)=\binom{r+s-1}{k}\binom{n-r-1}{k}-\binom{s+1}{k+1}\binom{n-3}{k-1}$ with the convention that $\binom{j}{i}=0$ if $j<i$.
Moreover, the multiplicity of $\hbox{\ensuremath{{\mathcal F}(I)}}$ is $e(\hbox{\ensuremath{{\mathcal F}(I)}})=\binom{n+s-2}{n-r-1}-\binom{n+s-2}{n-1}$.
\end{corollary}
\begin{proof} First notice that the $h$-vectors for $\hbox{\ensuremath{{\mathcal F}(I)}}$ and $\hbox{\ensuremath{{\mathcal F}(\lt I)}}$ coincide and in particular, $e(\hbox{\ensuremath{{\mathcal F}(I)}})=e(\hbox{\ensuremath{{\mathcal F}(\lt I)}})$ by \cite[Corollary~2.5]{CHV}. Since $\hbox{\ensuremath{{\mathcal F}(\lt I)}}$ is a ladder determinantal ring the formula for the Hilbert series is obtained in \cite[Theorem~4.7]{Wang}. One can then deduce the formula for the multiplicity immediately. The formula for the multiplicity is also worked out explicitly in \cite[Corollary~4.2]{Wang1}.
\end{proof}
\begin{corollary}
Let $X$ be a sparse $2\times n$ matrix and $I$ the ideal of $2\times 2$ minors of $X$. Then
\begin{enumerate}[$($a$)$]
\item The regularity of $\hbox{\ensuremath{{\mathcal R}(I)}}$ is ${\rm{reg}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\min\{n-1,n-r,r+s\}$.
\item The $a$-invariant of $\hbox{\ensuremath{{\mathcal R}(I)}}$ is $a(\hbox{\ensuremath{{\mathcal R}(I)}})=\min\{-s-2,-s-r-1,r-n-1\}$.
\item The normalized Hilbert series of $\hbox{\ensuremath{{\mathcal R}(I)}}$ is $p_{\lambda}(z)=1+h_1(\lambda)z+\ldots +h_{r+s}(\lambda)z^{r+s}$, where $h_1(\lambda)=(r+s)(n-r)-\binom{s+1}{2}-1$ and $h_k(\lambda)=\binom{r+s}{k}\binom{n-r}{k}-\binom{s+1}{k+1}\binom{n-1}{k-1}$ for $k\neq 1$, with the convention that $\binom{j}{i}=0$ if $j<i$.
\item The multiplicity of $\hbox{\ensuremath{{\mathcal R}(I)}}$ is $e(\hbox{\ensuremath{{\mathcal R}(I)}})=\binom{n+s}{n-r}-\binom{n+s}{n+1}-1$.
\end{enumerate}
\end{corollary}
\proof
Let \begin{align*}
\lambda&=(\underbrace{n-r, \ldots, n-r}_{r \ {\rm{ times }}}, n-r-1, n-r-2, \ldots, n-r-s), \\
\mu&=(\underbrace{n-r+1, \ldots, n-r+1}_{r+1 \ {\rm{ times }}}, n-r, n-r-1, \ldots, n-r-s+1),
\end{align*}
and let $I_2(L'_\lambda)$ be the defining ideal of $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$. By~\cite[Theorem~2.1]{Gorla}, the ideal $I_2(L_\mu)$ is obtained from the ideal $I_1(L_\lambda)$ generated by the entries of $L_\lambda$ by an ascending G-biliaison of height 1 on $I_2(L'_\lambda)$. Specifically, one has
\begin{equation}\label{biliaison}
yI_2(L_\mu)+I_2(L'_\lambda)=fI_1(L_\lambda)+I_2(L'_\lambda),
\end{equation}
where $y$ is the variable appearing in position $(2,2)$ of $L_\mu$ and $f$ is the $2\times 2$-minor of the first two rows and columns of $L_\mu$.
(a) By \cite[Corollary~2.5]{CHV} we have $a(\hbox{\ensuremath{{\mathcal R}(\lt I)}})=a(\hbox{\ensuremath{{\mathcal R}(I)}})$ and since $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ and $\hbox{\ensuremath{{\mathcal R}(I)}}$ are Cohen-Macaulay, then ${\rm{reg}}(\hbox{\ensuremath{{\mathcal R}(I)}})={\rm{reg}}(\hbox{\ensuremath{{\mathcal R}(\lt I)}})$. Since ${\rm{reg}}(K[L_\mu]/I_1(L_\lambda))=0$, then $${\rm{reg}}(\hbox{\ensuremath{{\mathcal R}(I)}})={\rm{reg}}(K[L_\mu]/I_2(L_\mu))=\min\{n-1,n-r,r+s\},$$ where the first equality follows from~\cite[Theorem~3.1]{DG} and the second from Corollary~\ref{invFi}.
(b) The formula for the $a$-invariant of $\hbox{\ensuremath{{\mathcal R}(I)}}$ now follows from the fact that $a(\hbox{\ensuremath{{\mathcal R}(I)}})=-\dim \hbox{\ensuremath{{\mathcal R}(I)}} +{\rm{reg}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\min\{-s-2,-s-r-1,r-n-1\}$.
We now prove (c) and (d) together. The normalized Hilbert series of $\hbox{\ensuremath{{\mathcal R}(I)}}$ and $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ coincide and in particular $e(\hbox{\ensuremath{{\mathcal R}(I)}})=e(\hbox{\ensuremath{{\mathcal R}(\lt I)}})$ by \cite[Corollary~2.5]{CHV}. Using the short exact sequences
$$0\longrightarrow I_2(L'_\lambda)(-1) \longrightarrow I_2(L_\mu)(-1) \oplus I_2(L'_\lambda) \longrightarrow yI_2(L_\mu)+I_2(L'_\lambda)\longrightarrow 0$$
and
$$0\longrightarrow I_2(L'_\lambda)(-2) \longrightarrow I_1(L_\lambda)(-2) \oplus I_2(L'_\lambda) \longrightarrow fI_1(L_\lambda)+I_2(L'_\lambda)\longrightarrow 0$$ together with (\ref{biliaison}),
one obtains $$HS_{K[L_\mu]/I_2(L_\mu)}(z)=zHS_{K[L_\mu]/I_1(L_\lambda)}(z)+(1-z)HS_{K[L_\mu]/I_2(L'_\lambda)}(z),$$
where $HS_A(z)$ denotes the Hilbert series of the algebra $A$.
Therefore, by Corollary~\ref{eFI} $$h_1(\lambda)=h_1(\mu)-1=(r+s)(n-r)-\binom{s+1}{2}-1$$
and
$$h_k(\lambda)=h_k(\mu)=\binom{r+s}{k}\binom{n-r}{k}-\binom{s+1}{k+1}\binom{n-1}{k-1}
\ \mbox{ for }\ k\neq 1,$$ where $1+h_1(\mu)z+\ldots+h_{r+s}(\mu)z^{r+s}$ is the normalized Hilbert series of $K[L_\mu]/I_2(L_\mu)$.
In particular, the multiplicity of $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ is $$e(\hbox{\ensuremath{{\mathcal R}(\lt I)}})=e(K[L_\mu]/I_2(L_\mu))-e(K[L_\mu]/J)=\binom{n+s}{n-r}-\binom{n+s}{n+1}-1. \eqno\hbox{\quad\qedbox}$$
We close with the following remark that gives an alternative path for a proof of our results for special types of $m\times n$ sparse matrices.
\begin{remark}\label{ASL}
Let $X$ be an $m\times n$ sparse matrix, $m\leq n$. We assume that, after row and column permutations, the variables that appear in $X$ form a two-sided ladder as in Figure~\ref{fig:stair}.
\begin{figure}[h]
\centering
\begin{tikzpicture}[ every node/.style={scale=0.9}]
\draw [line width=1pt, color=black] (0,5)--(10,5);
\draw [line width=1pt, color=black] (0,-1)--(10,-1);
\draw [line width=1pt, color=black] (0,-1)--(0,5);
\draw [line width=1pt, color=black] (10,-1)--(10,5);
\draw [line width=1pt, color=black] (0,4)--(2,4);
\draw [line width=1pt, color=black] (4,1)--(6,1);
\draw [line width=1pt, color=black] (4,3)--(4,5);
\draw [line width=1pt, color=black] (2,2)--(3,2);
\draw [line width=1pt, color=black] (2,4)--(2,2);
\draw [line width=1pt, color=black] (7,1)--(7,3);
\draw [line width=1pt, color=black] (3,1)--(5,1);
\draw [line width=1pt, color=black] (3,1)--(3,2);
\draw [line width=1pt, color=black] (9,1)--(9,0);
\draw [line width=1pt, color=black] (4,3)--(7,3);
\draw [line width=1pt, color=black] (7,1)--(9,1);
\draw [line width=1pt, color=black] (6,1)--(6,-1);
\draw [line width=1pt, color=black] (9,0)--(10,0);
\filldraw [gray!60](0,4)--(0,5)--(4,5)--(4,3)--(7,3)--(7,1)--(9,1)--(9,0)--(10,0)--(10,-1)--(6,-1)--(6,1)--(3,1)--(3,2)--(2,2)--(2,3)--(2,4);
\end{tikzpicture}
\caption{A matrix with a shaded two-sided ladder.}
\label{fig:stair}
\end{figure}
The case of a sparse $2\times n$ matrix $X$ is a special case of the above type of matrix with $m=2$, where the ladder has one lower inside corner in position $(1,r+1)$ and one upper inside corner in position $(2,r+s)$. Let $I=I_m(X)$ and let $\tau$ be a diagonal term order. Notice that an $m\times m$-minor of $X$ is non-zero if and only if all the entries on its diagonal are non-zero. Moreover, the $m\times m$-minors of $X$ form a universal Gr\"obner basis of $I_m(X)$ by~\cite[Proposition~5.4]{Boocher}. It follows that $\ini(I)$ is generated by the products of the elements on the diagonals of the $m\times m$ non-zero minors of $X$.
One can show that $\ini(I)$ is a sortable ideal with respect to the lexicographic order induced by the following order of the variables: $x_{i,j}>x_{k,l}$ if either $i<k$ or $i=k$ and $j<l$. Therefore, the defining ideal of $\hbox{\ensuremath{{\mathcal F}(\lt I)}}$ is the toric ideal generated by the binomial relations obtained by the sorting \cite[Theorem~6.16]{EH}. Moreover, these generators are a quadratic Gr\"{o}bner basis of the defining ideal of $\hbox{\ensuremath{{\mathcal F}(\lt I)}}$. In particular, the sorting we use allows us to conclude that $\ini(I)$ is a generalized Hibi ideal and $\hbox{\ensuremath{{\mathcal F}(\lt I)}}$ is a generalized Hibi ring, see \cite{HH05} and \cite[Section 6.3]{EH}.
Therefore, $\hbox{\ensuremath{{\mathcal F}(\lt I)}}$ is a normal, Cohen-Macaulay, Koszul algebra.
Furthermore, $\ini(I)$ is a weakly polymatroidal ideal with respect to the same order on the variables as above. For the definition of a weakly polymatroidal ideal, see \cite{KH} or \cite[Definition~6.25]{EH}. Hence, by \cite[Proposition~6.26]{EH} $\ini(I)$ satisfies the $\ell$-exchange property with respect to the sorting order. Therefore, $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ is of fiber type and the defining equations of $\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ are precisely the defining equations of the special fiber ring $\hbox{\ensuremath{{\mathcal F}(\lt I)}}$ and the linear relations by \cite[Theorem~6.24]{EH}.
Because of the special shape of our matrix, one can use the criterion in \cite{RS} (see also~\cite[Proposition~1.1]{CHV}) and proceed in a similar manner as in \cite[Theorem~6.46]{EH} to show that the Pl\"{u}cker relations are the defining equations of $\hbox{\ensuremath{{\mathcal F}(I)}}$ and the maximal minors of $X$ are a SAGBI basis of $\hbox{\ensuremath{{\mathcal F}(I)}}$.
One can also show that Pl\"{u}cker relations along with the linear relations of $\mathcal{S}(I)$ are the defining equations of $\hbox{\ensuremath{{\mathcal R}(I)}}$ and the maximal minors of $X$ along with the variables of $R$ are a SAGBI basis of $\hbox{\ensuremath{{\mathcal R}(I)}}$. In particular ${\rm{in}_{\tau'}}(\hbox{\ensuremath{{\mathcal R}(I)}})=\hbox{\ensuremath{{\mathcal R}(\lt I)}}$ and $\ini(\hbox{\ensuremath{{\mathcal F}(I)}})=\hbox{\ensuremath{{\mathcal F}(\lt I)}}$, where $\tau^\prime$ is a suitable term order. One then obtains similar results for $\hbox{\ensuremath{{\mathcal R}(I)}}$ and $\hbox{\ensuremath{{\mathcal F}(I)}}$ as in Corollaries~\ref{rees sing},~\ref{Rees koszul},~\ref{ini F}, and ~\ref{FI CM normal}.
\end{remark}
\acknowledgement{We are grateful to the Women in Commutative Algebra (WICA) group for organizing the first ``Women in Commutative Algebra" workshop at BIRS, Banff, Canada, where this project began. We also thank the staff at BIRS for their great hospitality during the workshop. The WICA workshop was funded by the National Science Foundation grant DMS 1934391 and by the Association for Women in Mathematics grant NSF-HRD 1500481.}
|
1,108,101,566,787 | arxiv | \section{Introduction}
\label{sec:intro}
Ever since it has been demonstrated that convolutional neural networks (CNNs) generalize better than the alternatives for image classification~\cite{726791,10.1145/3065386}, further improvements have relied on simple but effective architectural changes that allow training of deeper CNNs~\cite{https://doi.org/10.48550/arxiv.1409.1556,https://doi.org/10.48550/arxiv.1409.4842,https://doi.org/10.48550/arxiv.1512.03385,https://doi.org/10.48550/arxiv.1704.04861,https://doi.org/10.48550/arxiv.1709.01507,https://doi.org/10.48550/arxiv.1608.06993}. On the other hand, adapting transformers that were developed for natural language processing (NLP) for vision tasks was a departure from using convolutional processing, which showed that further improvements in generalization in image classification are possible by using architectures that scale with larger labeled datasets and computational resources~\cite{https://doi.org/10.48550/arxiv.2004.13621,dosovitskiy2021image}. However, forsaking inductive priors suitable for images, such as 2D convolutional weight sharing, and adopting global self-attention with quadratic complexity meant that the training requirements increased to a prohibitive extent for most applications~\cite{Khan_2022}. Linear approximations to quadratic attention reduce the computational requirements only to some extent, and they do not resolve the problem of not having an appropriate inductive bias for images~\cite{Jeevan_2022_WACV}.
More recently, there have been attempts to combine the inductive prior of convolutional design with the scalability of transformers in hybrid architectures for image analysis to reduce the training requirements of neural networks (pre-training dataset size, parameters, floating point operations (FLOPs))~\cite{graham2021levit,hassani2021escaping, dosovitskiy2021image,wu2021cvt,Jeevan_2022_WACV,https://doi.org/10.48550/arxiv.2201.10271}. The expensive training requirements of pure transformers also led to research into alternative token-mixing architectures that can replace self-attention~\cite{tolstikhin2021mlpmixer,leethorp2021fnet,https://doi.org/10.48550/arxiv.2111.13587,trockman2022patches}. These works indicate that it is worth exploring priors other than shift-invariance (via convolutions) for images to reduce the training requirements with little to no sacrifice of generalization. However, most of these architectures cannot be easily adapted for multiple image sizes and other vision tasks, such as segmentation and object detection~\cite{trockman2022patches}.
An important prior in images that has been much less explored in deep neural networks is the multi-resolution self-similarity (across spatial scales) and sparseness of edges (finite spatial extents of objects), with some notable exceptions~\cite{https://doi.org/10.48550/arxiv.2004.13824}. Two-dimensional discrete wavelet transforms (2D-DWT) capture this prior well and have been widely used in the pre-deep neural network era for various imaging applications, especially for compression and denoising~\cite{136601,5598411}.
In this work, we explore the use of a 2D-DWT as a spatial token mixer directly into a largely convolutional architecture, which we call \emph{WaveMix-Lite}. WaveMix-Lite has the following benefits:
\begin{enumerate}
\item \textbf{Incorporate multi-resolution inductive bias without increasing training costs:} A pre-defined 2D-DWT allows us to mix features spatially in a multi-resolution manner without introducing extra parameters. Additionally, the 2D-DWT also scales the image dimension by positive integral powers of $\frac{1}{2}\times\frac{1}{2}$, which reduces the GPU RAM and FLOPs required per training image per forward or backward pass for the subsequent trainable layers. The resultant efficiency in terms of parameters, FLOPs, and GPU RAM allowed us to conduct experiments using a \emph{single} GPU on Google Colab Pro+\textsuperscript{\textregistered}.
\item \textbf{Impart versatility in architectural design for image recognition as well as segmentation:} Unlike other CNN and multi-head attention (MHA) models which need complete redesigning of the architecture to provide good performance in different vision tasks, the WaveMix-Lite architecture is versatile and can perform multiple vision tasks, such as image classification and semantic segmentation, without the need for special architectural modifications. We find this design simplicity and reusable block structure to be attractive in its own right.
\item \textbf{Obtain high accuracy with reduced training requirements:} For image classification and segmentation on multiple datasets WaveMix-Lite was able to match the test accuracy of some widely used convolutional and transformer architectures while using 5 to 10 times fewer parameters and 2 to 50 times lesser GPU RAM for a fixed batch size. Consequently, its training and testing throughputs were 1.5 to 6 times higher than the models compared. WaveMix-Lite also achieved state-of-the-art accuracy on 5 EMNIST datasets (Byclass, Bymerge, Letters, Digits and Balanced). Additionally, it gives compelling results on TinyImageNet, CIFAR 10 and 100, STL-10, Places-365, Caltech-256 and ImageNet-1K for classification; and Cityscapes for semantic segmentation.
\end{enumerate}
After describing how WaveMix-Lite is related to previous works in Section~\ref{sec:related}, we describe its insights and details in Section~\ref{sec:architecture}. We compare WaveMix-Lite with other models on image classification and semantic segmentation and provide empirical evidence of its scalability and examine the importance of its components in Section~\ref{sec:experiments}. We conclude and list potential future directions in Section~\ref{sec:conclusions}.
\section{Related Works}
\label{sec:related}
\textbf{Statistical properties of natural images}, which have been studied for the last several decades, include shift-invariance (stationarity), scale-invariance (especially in 2D projections of a 3D world viewed from various distances), high spatial auto-correlation (monochromatic objects or regions), spatial sparseness of edges (finite spatial extent of objects), and certain chromatic contrasts (preponderance of certain colors)~\cite{field1993scale,ruderman1994statistics,lee1996image,parraga2002spatiochromatic}. Of these, only the shift-invariance has been widely exploited in neural architectures for image analysis in the form of convolutional filters~\cite{lecun-gradientbased-learning-applied-1998} and other architectural elements. There have been some exceptions that incorporate rotational-invariance for remote sensing images~\cite{cheng2016learning}, and multi-resolution analysis for segmentation of histopathology images~\cite{https://doi.org/10.48550/arxiv.2205.01777,VANRIJTHOVEN2021101890}, but these methods have not been tested on general computer vision benchmarks.
\textbf{Advances in CNN performance} have mainly come from architectural changes with the goals of easing gradient flow to deeper layers~\cite{he2015deep,https://doi.org/10.48550/arxiv.1409.4842}, or reducing parameters per layer by restricting convolutional kernel size~\cite{https://doi.org/10.48550/arxiv.1409.1556} or their scope to only one dimension~\cite{https://doi.org/10.48550/arxiv.1610.02357}. Attention mechanisms for space or channel~\cite{Chen_2017_CVPR} also seem to improve performance of CNNs, although it has not been explored why a stack of additional convolutional layers cannot model the same function as that of spatial or channel attention.
\textbf{Vision transformers and hybrid architectures}, inspired by their success on NLP tasks, have pushed the image classification accuracy beyond those of the largest CNNs, albeit at the cost of several times more data and network parameters~\cite{dosovitskiy2021image}. Training such data hungry models with hundreds of millions of parameters requires access to large GPU clusters, which is impractical for resource-constrained applications. Reduction in the computational requirements of vision transformers have been made possible by architectural changes that provide image specific inductive biases creating hybrid models with elements including distillation~\cite{https://doi.org/10.48550/arxiv.2012.12877}, convolutional embeddings~\cite{Jeevan_2022_WACV,hassani2021escaping}, convolutional tokens~\cite{wu2021cvt}, and encoding overlapping patches~\cite{https://doi.org/10.48550/arxiv.2101.11986}. The quadratic complexity with respect to the sequence length (number of pixels) for vanilla transformers has also led to the search for other linear approximations of self-attention to efficiently mix tokens~\cite{Jeevan_2022_WACV}.
\textbf{Token mixers} that replace the self-attention in transformers with fixed token mixing mechanisms, such as the Fourier transform (FNet), achieves comparable generalization with lower computational requirements~\cite{leethorp2021fnet}. Other token-mixing architectures have also been proposed that use standard neural components, such as convolutional layers and multi-layer perceptrons (MLPs) for mixing visual tokens. MLP-mixer~\cite{tolstikhin2021mlpmixer} uses two MLP layers (cascade of $1\times1$ convolutions) applied first to image patch sequence and then to the channel dimension to mix tokens. ConvMixer~\cite{trockman2022patches} uses standard convolutions along image dimensions and depth-wise convolutions across channels to mix token information. These token mixing models perform well with lower computational costs compared to transformers without compromising generalization.
\textbf{Wavelets for images:} Extensive prior research has uncovered and exploited various multi-resolution analysis properties of wavelet transforms on image processing applications, including denoising~\cite{5598411}, super-resolution~\cite{8014882}, recognition~\cite{8536280}, and compression~\cite{136601}. Features extracted using wavelet transforms have also been used extensively with machine learning models~\cite{1028240}, such as support vector machines and neural networks~\cite{7754470}, especially for image classification~\cite{Nayak2016BrainMI}. Representative instances of integration with neural architectures include the following. ScatNet architecture cascades wavelet transform layers with nonlinear modulus and average pooling to extract translation invariant features that are robust to deformations and preserves high-frequency information for image classification~\cite{6522407}. WaveCNets replaces max-pooling, strided-convolution, and average-pooling of CNNs with 2D-DWT for noise-robust image classification~\cite{li2020wavelet}. Multi-level wavelet CNN (MWCNN) has been used for image restoration as well with U-Net architectures for better trade-off between receptive field size and computational efficiency~\cite{liu2018multilevel}. Wavelet transform has also been combined with a fully convolutional neural network for image super resolution ~\cite{kumar2017convolutional}.
We propose using the two-dimensional discrete wavelet transform (2D-DWT) for token mixing. Among the different types of mother wavelets available, we used the Haar wavelet (a special case of the Daubechies wavelet~\cite{57199} , also known as Db1), which is frequently used due to its simplicity and faster computation. Haar wavelet is both orthogonal and symmetric in nature, and has been used to extract basic structural information from images~\cite{Porwik2004TheHT}. For even-sized images, it reduces the dimensions exactly by a factor of $2$, which simplifies the designing of subsequent layers.
\section{WaveMix-Lite Architecture}
Image pixels have several interesting co-dependencies. The localized and stationary nature of certain image features (e.g., edges) have been exploited using linear space-invariant filters (convolutional kernels) of limited size. Scale-invariance of natural images has been exploited to some extent by pooling~\cite{lecun-gradientbased-learning-applied-1998}. However, we think that scale-invariance can be better modeled by wavelet decomposition due to its natural multi-resolution analysis properties. Additionally, the finer scale of a multi-level wavelet decomposition also incorporates the idea of linear space-invariant feature extraction using convolutional filters of small support; albeit using predefined weights. The basic idea, therefore, behind our proposed architecture is to alternate between learnable spatially repeated feature extraction using convolutional layers (includes, $3\times3$ conv, MLP, as well as upconv layers), and fixed token mixing using 2D-DWT for a few layer blocks. Injecting fixed (unlearnable) spatial token-mixing that also reduces the image dimensions by a factor of $\frac{1}{2}\times\frac{1}{2}$, similar to a pooling layer (which we do not use), reduces the number of computations in some of the subsequent learnable layers and increases the effective receptive field to capture distant spatial relationships more efficiently with the number of layers. This combination requires far fewer layers and parameters than using only convolutional layers with pooling. On the other hand, while transformers and other token mixers have very large effective receptive fields right from the first few layers, they do not utilize inductive priors that are suitable for images. This is where the wavelet transform plays its role in both increasing the effective receptive field at an exponential rate per layer (unlike the linear rate of convolutional layers), while still retaining the essence of convolutional design to keep the architecture flexible. Additionally, compared to pooling, wavelet is a lossless transform.
\label{sec:architecture}
\begin{figure}
\centering
\includegraphics[scale=0.8]{wavemixlite.drawio.pdf}
\caption{WaveMix-Lite architecture: Overall architecture for (a) classification and (b) semantic segmentation, along with (c) details of the WaveMix-Lite block}
\label{fig:wavemixlite}
\end{figure}
\subsection{Overall architecture}
As shown in Figure \ref{fig:wavemixlite}, the input image is first passed through a convolutional layer that creates feature maps of the image. The use of trainable convolutions \emph{before} the wavelet transform is a key aspect of our architecture, as it allows the extraction of only those feature maps that are suitable for the chosen wavelet family. This is followed by a series of WaveMix-Lite blocks\footnote{Our code is available at \url{https://github.com/pranavphoenix/WaveMix}}. A task specific output layer is then attached to the end. For image classification, we add an MLP head, a global average pooling layer, and a softmax layer for generating the class probabilities. For semantic segmentation, we use deconvolution layers to expand the output from WaveMix-Lite block back to the input resolution. A pixel-wise softmax layer is then added to generate the class probabilities for the required number of semantic classes. For both the tasks, the core architecture remains the same and we only replace the classification head with the segmentation head. WaveMix-Lite processes image as a 2D graph and not as sequence of pixels/patches. That is, at no point in the model do we unroll the image into a sequence of pixels/patches as done in transformer models. This key feature allows transfer learning from a source to a target dataset even when the two have different image sizes or tasks.
\subsection{WaveMix-Lite block}
In a WaveMix-Lite block (Figure \ref{fig:wavemixlite}(c)), the input is first passed through a convolutional layer which decreases the embedding dimension by a factor of four, so that the concatenated output after 2D-DWT has the same dimension as input.\footnote{Base code: \url{https://pytorch-wavelets.readthedocs.io/en/latest/readme.html}} We only use one level 2D-DWT in WaveMix-Lite to reduce the parameters and computations. The 2D-DWT produces four output channels (one approximation and three details~\cite{57199}) for each input channel. The four outputs are concatenated together (depth or channel-wise) and this output has the same number of channels as the input to the WaveMix-Lite block (embedding dimension). The output resolution (height $\times$ width) after 2D-DWT will be half that of the input; i.e., if the input is 64 $\times$ 64, the output will be 32 $\times$ 32.
The concatenated output from 2D-DWT is passed to an MLP layer (two $1\times1$ convolutional layers separated by a GELU non-linearity) having a multiplication factor more than one where channel mixing is performed by the MLP. Since wavelet transform reduces the image resolution by half, the GPU consumption and computations needed by the MLP significantly reduces in each layer. The image size reconciliation is performed using transposed convolutions (up-convolutions) which resizes the image back to the original input resolution. The kernel size and stride of deconvolutional layers were chosen such that the output has the same size as the input to WaveMix-Lite block. We chose deconvolutional layer rather than an inverse 2D-DWT because the former is much faster and consumes less GPU than the latter. The outputs from the deconvolutional layers are then passed through batch normalization. A residual connection~\cite{he2015deep} is provided within each WaveMix-Lite block so that the model can be made deeper with a larger number of blocks, if necessary.
\section{Experiments and Results}
\label{sec:experiments}
\subsection{Datasets}
To demonstrate the applicability of WaveMix-Lite for image classification, we used multiple types of publicly available (under MIT Licenses) datasets based on the number of images and image size. Small datasets of smaller image sizes included CIFAR-10, CIFAR-100~\cite{Krizhevsky09learningmultiple}, EMNIST~\cite{7966217}, Fashion MNIST~\cite{DBLP:journals/corr/abs-1708-07747}, and SVHN~\cite{Netzer2011}. Small datasets of larger image sizes included STL-10~\cite{pmlr-v15-coates11a}, Caltech-256~\cite{griffin2007caltech} and Tiny ImageNet~\cite{Le2015TinyIV}. For benchmarking, we could not load the whole ImageNet-1K in $224\times224$ resolution due to the resource constraints of Google Colab Pro+\textsuperscript{\textregistered}. Therefore, we used ImageNet-1K of lower image resolution $64\times64$. We also used larger datasets with larger images sizes such as, Places-365~\cite{zhou2017places} and iNaturalist2021-10k (iNAT mini)~\cite{horn2021benchmarking}. We used Cityscapes~\cite{https://doi.org/10.48550/arxiv.1604.01685} dataset for semantic segmentation experiments and evaluated performance in the Cityscapes validation dataset.
\subsection{Models compared}
WaveMix-Lite was compared with various other CNNs, transformers, and token-mixing models. These include ResNets~\cite{he2015deep}, MobileNetV2~\cite{https://doi.org/10.48550/arxiv.1801.04381}, UNets and DeepLabV2 as CNNs; ViT (vision transformer)~\cite{dosovitskiy2021image}, hybrid ViN (vision Nystromformer)~\cite{Jeevan_2022_WACV}, CPV (convolutional performer for vision)~\cite{https://doi.org/10.48550/arxiv.2201.10271}, CCT (compact convolutional transformer)~\cite{hassani2021escaping}, CvT (convolutional vision transformer)~\cite{wu2021cvt}, and SegFormer~\cite{https://doi.org/10.48550/arxiv.2105.15203} as transformers; and FNet~\cite{leethorp2021fnet}, ConvMixer~\cite{trockman2022patches} and MLP-Mixer~\cite{tolstikhin2021mlpmixer} as token-mixers. Results of the other models that were directly taken from their original papers are cited in results tables.
For model notation we use the format \emph{Model$\_$Name-Embedding Dimension/Layers$\times$Heads} for transformers and exclude the \emph{heads} for the other architectures. For example, CCT with embedding dimension of 128 having 4 layers and 4 heads is labelled as CCT-$128/4\times4$.
\subsection{Implementation details}
We trained models using AdamW optimizer ($\alpha = 0.001, \beta_{1} = 0.9, \beta_{2}=0.999, \epsilon = 10^{-8}$) with a weight decay coefficient of 0.01 during initial epochs and then used SGD (stochastic gradient descent) with learning rate of $0.001$ and momentum $= 0.9$ during the final 20 epochs~\cite{keskar2017improving}. We used automatic mixed precision in PyTorch during training to optimize speed and memory consumption. Almost all experiments were done with a 16 GB Tesla V100-SXM2 GPU available in Google Colab Pro+\textsuperscript{\textregistered}. \emph{No image augmentations were used while training the models}. Maximum number of epochs in all experiments was set to 150. GPU usage for a batch size of 64 was reported for image classification along with top-1\% accuracy from best of three runs with random initialization based on prevailing protocols~\cite{hassani2021escaping}. We report the semantic segmentation performance using mean intersection over union (mIoU) metric. Cross-entropy loss was used for image classification and pixel-wise focal loss was used for semantic segmentation. Few segmentation experiments were run on A100-SXM4 GPU in Google Colab Pro+\textsuperscript{\textregistered}. Due to resource constraints, for segmentation experiments in 16 GB V100 GPU, the original $1024\times2048$ image was resized to $256\times512$ and for 40 GB A100 GPU, it was resized to $512\times 1024$. We also adjusted the stride of the initial convolutional layers in all WaveMix-Lite models that handled high-resolution images to ensure that smaller side of input was always 64 before it reached WaveMix-Lite blocks. The classification head of ConvMixer was replaced with a segmentation head similar to WaveMix-Lite for segmentation. \emph{No pre-training was performed on any of the WaveMix-Lite models}.
\subsection{Image classification}
\begin{table}[]
\centering
\caption{Image classification on ImageNet-1K dataset (64x64) shows improved accuracy as well as throughput due to decreased parameter count and GPU RAM consumption by WaveMix-Lite (arrows show desired directions)}
\label{tab:imagenet}
\small\addtolength{\tabcolsep}{-2.5pt}
\begin{tabular}{lrrrrrr}
\toprule
Architecture & {\begin{tabular}[c]{@{}c@{}}Top-1 Accu. \\ (\%) ↑\end{tabular}} & {\# Param. ↓} & {\begin{tabular}[c]{@{}c@{}}GPU RAM for \\ batch size 64 ↓\\\end{tabular}} & {\begin{tabular}[c]{@{}c@{}}Max batch size\\ in 16 GB GPU ↑\\\end{tabular}} & \multicolumn{2}{c}{\begin{tabular}[c]{@{}c@{}}Throughput (im/s)\\ \hspace{.3cm}Train ↑ \hspace{.3cm} Test ↑\end{tabular}} \\
\toprule
ResNet-18 & 50.67 & 11.2 M & 2.6 GB & 384 & \hspace{.7cm}764 & 1,786 \\
ResNet-34 & 55.04 & 21.3 M & 3.5 GB & 288 & 357 & 943 \\
ResNet-50 & 55.66 & 23.6 M & 11.3 GB & 96 & 266 & 920 \\
ResNet-101 & 56.05 & 44.6 M & 15.1 GB & 64 & 163 & 568 \\
CPV-128/5x4 & 50.92 & 1.6 M & 7.8 GB & 128 & 375 & 1,190 \\
ViT-128/4x4 & 30.91 & 0.7 M & 14.7 GB & 64 & 340 & 1,020 \\
CCT-128/4x4 & 48.82 & 1.0 M & 20.4 GB & 48 & 149 & 514 \\ \midrule
WaveMix-Lite-128/7 & 47.27 & 2.5 M & 4.7 GB & 216 & 1,056 & 3,125 \\
WaveMix-Lite-128/14 & 53.06 & 4.8 M & 7.8 GB & 131 & 555 & 1,471 \\
WaveMix-Lite-256/7 & 56.66 & 9.0 M & 9.0 GB & 112 & 159 & 420 \\
WaveMix-Lite-256/12 & \textbf{60.56} & 16.5 M & 14.1 GB & 72 & 118 & 344 \\
\bottomrule
\end{tabular}
\end{table}
Table~\ref{tab:imagenet} shows the performance of WaveMix-Lite compared to the other architectures on image classification using supervised learning on ImageNet-1K scaled down to 64 $\times$ 64 to manage training on a single GPU within a reasonable time. Similarly, for the transformer models, in order to train on a single GPU, fewer layers gave better results, for which we used smaller patch sizes. We see that WaveMix-Lite models outperform ResNets, transformers, hybrid xformers, and other token-mixing models while requiring lesser GPU RAM and number of parameters. WaveMix-Lite does not need a large number of parameters to give performance comparable to large ResNets and they require very less GPU RAM compared to transformer models for achieving similar results.
Even though convolution has been widely regarded as a GPU-efficient operation, the need for deeper architectures have necessitated the use of networks having over tens to hundreds of layers for achieving high generalization. Even though a single convolutional operation is comparatively cheaper than a 2D-DWT, we can achieve generalization comparable to deep convolutional networks with far fewer layers of the wavelet transforms. This ability of the wavelet transform to provide competitive performance without needing large number of layers helps in improving the efficiency of the network by consuming much lesser GPU RAM than deep convolutional models like ResNets. We also observe that deeper WaveMix-Lite models perform better and this suggests that even further scale-up of WaveMix-Lite in multi-GPU setting could be possible.
\begin{table}[]
\centering
\caption{Results for image classification on small datasets (32x32, 64x64) show improved accuracy as well as decreased parameter count and GPU RAM consumption by WaveMix-Lite}
\label{tab:cifar}
\small\addtolength{\tabcolsep}{-2.5pt}
\begin{tabular}{lrrrrr}
\toprule
{{Model}} & {\#Param. ↓} & {\begin{tabular}[c]{@{}c@{}}GPU RAM for\\batch size 64 ↓\end{tabular}} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Accuracy (\%) ↑\\\hspace{.1cm}CIFAR-10 \hspace{.1cm} CIFAR-100\hspace{.1cm} TinyImNet\end{tabular}}\\%{Accuracy (\%)} \\ \cmidrule(l){4-6} \\ CIFAR-10 CIFAR-100 TinyImageNet \\
\midrule
ResNet-18 \cite{hassani2021escaping} & 11.20 M & 1.2 GB& \hspace{.7cm}90.27 & \hspace{.8cm}63.41 & 48.11 \\
ResNet-34 \cite{hassani2021escaping} & 21.30 M & 1.4 GB& 90.51 & 64.52 & 45.60 \\
ResNet-50 \cite{hassani2021escaping} & 25.20 M & 3.3 GB& 90.60 & 61.68 & 48.77 \\
MobileNetV2 \cite{hassani2021escaping} & 8.72 M & - & 91.02 & 67.44 & - \\
ViT-128/4×4 & 0.53 M & 13.8 GB& 56.81 & 30.25 & 26.43 \\
ViT-384/12x6 \cite{hassani2021escaping} & 85.60 M & - & 76.42 & 46.61 & - \\
ViT-Lite-256/6x4 \cite{hassani2021escaping} & 3.19 M & - & 90.94 & 69.20 & - \\
HybridViN-128/4×4 & 0.62 M & 4.8 GB& 75.26 & 51.44 & 34.05 \\
CCT-128/4×4 & 0.91 M & 15.8 GB& 82.23 & 57.09 & 39.05 \\
CvT-128/4×4 & 1.12 M & 15.4 GB& 79.93 & 48.29 & 40.69 \\
MLP-Mixer-512/8 & 2.41 M & 1.0 GB& 72.22 & 44.23 & 26.83 \\ \midrule
WaveMix-Lite-16/7 & 0.04 M & 0.1 GB& 64.98 & 23.03 & 19.15 \\
WaveMix-Lite-32/7 & 0.15 M & 0.3 GB& 84.67 & 46.89 & 34.34 \\
WaveMix-Lite-64/7 & 0.60 M & 0.6 GB& 87.81 & 62.72 & 46.31 \\
WaveMix-Lite-128/7 & 2.42 M & 1.1 GB& 91.08 & 68.40 & 52.03 \\
WaveMix-Lite-144/7 & 3.01 M & 1.2 GB& \textbf{92.97} & 68.86 & \textbf{52.38} \\
WaveMix-Lite-256/7 & 9.62 M & 2.3 GB& 90.72 & \hspace{.35cm}\textbf{70.20} & 51.37 \\ \bottomrule
\end{tabular}
\end{table}
In Table~\ref{tab:cifar} we see that on CIFAR and TinyImageNet datasets, WaveMix-Lite performs much better than the other models, giving accuracy higher than ResNets and MobileNets with 4 to 10 times fewer parameters and less GPU consumption. GPU consumption of WaveMix-Lite is sometimes 50 times lower for similar performance when compared to transformer models. The results of WaveMix-Lite on several large resolution image datasets are provided in the Appendix.
\begin{table}[]
\centering
\caption{WaveMix-Lite outperforms ResNets \cite{https://doi.org/10.48550/arxiv.2203.15331} for image classification on various EMNIST, MNIST, and Fashion MNIST datasets (28$\times$28) and achieves SOTA on 5 datasets}
\label{tab:emnist}
\begin{tabular}{@{}lrrrrrrr@{}}
\toprule
Models & Byclass & Bymerge & Letters & Digits & Balanced & MNIST & Fashion MNIST \\ \midrule
ResNet-18 &87.98 &91.09 &94.76 &99.67 &89.00 &99.64 &93.97 \\
ResNet-34 &88.10 &91.13 &95.04 &99.68 &89.17 &99.60 &93.91 \\
ResNet-50 &88.18 &91.29 &94.64 &99.62 &89.76 &99.56 &93.81 \\
WaveMix-Lite &\textbf{88.43} &\textbf{91.80} &\textbf{95.96} &\textbf{99.80}
&\textbf{91.06} &\textbf{99.75} &\textbf{94.32} \\ \bottomrule
\end{tabular}
\end{table}
We can see from Table~\ref{tab:emnist} that WaveMix-Lite models outperform ResNets on all datasets ($28 \times 28$) tested. It also establishes a new state-of-the-art by outperforming the previous best results~\cite{Kabir2020SpinalNetDN,Pad_2020_CVPR} by 0.01, 0.08, 0.01, 0.31 and 0.01 percentage points, respectively for Balanced, Letters, Digits, Byclass and Bymerge subsets within EMNIST~\cite{7966217}.
\subsection{Semantic segmentation}
WaveMix-Lite can be directly used for semantic segmentation by replacing the classifier head with deconvolution layers and a linear layer to generate the segmentation maps. On the other hand, architectural changes -- such as encoder-decoder and UNet structures~\cite{https://doi.org/10.48550/arxiv.1505.04597} -- are required for base CNNs and transformers, including SegFormer~\cite{https://doi.org/10.48550/arxiv.2105.15203}. The WaveMix-Lite performs on par with the other models on half-resolution Cityscapes validation set. We can see from Table~\ref{tab:segment} that WaveMix-Lite performs better than deep architectures like DeepLabV2~\cite{https://doi.org/10.48550/arxiv.1606.00915} and SegFormer model which uses an encoder pre-trained on ImageNet-1K dataset. The low mIoU obtained by replacing the classification head of ConvMixer~\cite{trockman2022patches} with segmentation head similar to WaveMix-Lite shows that token-mixing architectures which work well for classification cannot translate that performance in segmentation tasks without significant architectural modifications. This shows the versatility of our model which can provide high performance efficiently in multiple tasks. See Appendix for detailed results.
\begin{table}[]
\centering
\caption{Results for semantic segmentation on Cityscapes validation dataset show improved mIoU by WaveMix-Lite without compromising throughput}
\label{tab:segment}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lrrrrrrl}
\toprule
{{Architecture}} & \multicolumn{1}{r}{{mIoU ↑}} & {{\# Param. ↓}} & \multicolumn{1}{r}{{\begin{tabular}[r]{@{}r@{}}GPU RAM for \\ batch size 64 ↓\end{tabular}}} & \multicolumn{1}{r}{{\begin{tabular}[r]{@{}r@{}}Max batch size\\ for 16 GB ↑\end{tabular}}} & \multicolumn{2}{r}{\begin{tabular}[r]{@{}r@{}}Throughput (im/s)\\ \hspace{.3cm}Train ↑ \hspace{.3cm} Test ↑\end{tabular}} & {{Notes}} \\
\toprule
\begin{tabular}[c]{@{}l@{}}UNet \\ \cite{https://doi.org/10.48550/arxiv.1803.02758}\end{tabular} & 57.90 & 28.9 M & -& - & \hspace{1cm}1 &- & \begin{tabular}[c]{@{}l@{}}ResNet-18\\ Encoder\end{tabular} \\
\begin{tabular}[c]{@{}l@{}}UNet \\ \cite{https://doi.org/10.48550/arxiv.1803.02758}\end{tabular} & 61.00 & 9.0 M & - & - & - & - & \begin{tabular}[c]{@{}l@{}}MobileNetV2\\ Encoder\end{tabular} \\
\begin{tabular}[c]{@{}l@{}}DeepLabV2-CRF \\ \cite{https://doi.org/10.48550/arxiv.1606.00915}\end{tabular} & 71.40 & 20.5 M & - & - & 5 &- & \begin{tabular}[c]{@{}l@{}}ResNet-101,\\ Augmentations\end{tabular} \\
\begin{tabular}[c]{@{}l@{}}SegFormer (MiT-B0) \\ \cite{https://doi.org/10.48550/arxiv.2105.15203}\end{tabular} & 71.90 & 3.4 M & -& - & 47 & - & \begin{tabular}[c]{@{}l@{}}Augmentations,\\ ImageNet Pretrained\end{tabular} \\
ConvMixer-512/16 & 53.40 & 7.8 M & 42 GB & 24 & 10 & 11 & $256\times512$, 16 GB \\
SegFormer (MiT-B0) & 62.56 & 7.7 M & 232 GB & 4 & 16 & 16 & $512\times1024$, 40 GB \\ \midrule
WaveMix-Lite 128/8 & 63.33 & 2.9 M & 19 GB & 55 & 18 & 18 & $256\times512$, 16 GB \\
WaveMix-Lite 256/12 & 67.46 & 16.9 M & 38 GB & 25 & 11 & 12 &$256\times512$, 16 GB\\
WaveMix-Lite 256/16 & \textbf{71.75} & 44.1 M & 49 GB & 21 & 9 & 11 &$256\times512$, 16 GB \\
WaveMix-Lite 256/16 & \textbf{75.32} & 22.2 M & 189 GB & 6 & 14 & 16 & $512\times1024$, 40 GB \\ \bottomrule
\end{tabular}%
}
\end{table}
\subsection{Parameter efficiency}
Since WaveMix-Lite heavily uses unlearnable token-mixers, it uses much fewer parameters compared to the commonly used architectures. Table~\ref{tab:paraeff} shows that WaveMix-Lite can achieve all the tasks mentioned with far fewer parameters compared to previous models~\cite{https://doi.org/10.48550/arxiv.2101.02268,https://doi.org/10.48550/arxiv.1809.02209}. We can further reduce the parameter count of WaveMix-Lite by replacing the deconvolution layers with Upsampling layers using unlearnable interpolation techniques (e.g., IDWT, bilinear or bicubic).
\begin{table}[]
\centering
\caption{WaveMix-Lite needs very few parameters to achieve the benchmark results on the tasks mentioned below compared to other architectures}
\label{tab:paraeff}
\begin{tabular}{@{}llrl@{}}
\toprule
Task & Model & Parameters & Expansion\\ \midrule
99\% accuracy on MNIST & WaveMix-Lite-8/10 & 3,566 & Upsampling \\
90\% accuracy on Fashion MNIST & WaveMix-Lite-8/5 & 7,156 & Deconvolution\\
80\% accuracy on CIFAR-10 & WaveMix-Lite-32/7 & 37,058 & Upsampling \\
90\% accuracy on CIFAR-10 & WaveMix-Lite-64/6 & 520,106 & Deconvolution \\ \bottomrule
\end{tabular}
\end{table}
\subsection{Ablation studies}
We performed multiple ablations on WaveMix-Lite using the CIFAR-10 dataset to understand the effect of each type of layer on performance.
When we removed the 2D-DWT layers from WaveMix-Lite, the GPU RAM requirement of the model increased by 61.8 \% and accuracy fell by 5\%. This is due to the MLP receiving the full resolution instead of the half-resolution feature map from 2D-DWT.
Replacing the 2D-DWT with the real part of a 2D-discrete Fourier transform showed 12\% decrease in accuracy along with 73\% increase in GPU consumption as the Fourier transform also does not downscale an image. Additionally, the Fourier transform has global and spatially smoothly varying kernels, which do not model objects in images in a sparse manner. Objects have finite and abruptly-ending spatial extents, which are better modeled by wavelet functions that have the local kernels (finite support set) with sharp transitions.
GPU RAM consumption increased by 7.8\% and accuracy decreased by 5\% when we replaced the 2D-DWT with 2D-MaxPooling, indicating that the loss of information by the latter hurts generalization.
Additional ablation studies on the number of layers, embedding dimension, MLP dimension, and multi-level 2D-DWT are included in the Appendix.
\section{Conclusions, Future Directions, and Impact}
\label{sec:conclusions}
We proposed a novel and versatile neural architecture -- WaveMix-Lite -- that can generalize at par with both self-attention networks (transformers), CNNs, and their hybrids for image classification and segmentation, while needing fewer parameters, GPU RAM, and FLOPs. In addition to convolutions, WaveMix-Lite uses a 2D wavelet transform for token mixing in images, which exploits additional image priors, such as scale-invariance and finite spatial extents of objects. It is also better tailored for computer vision applications than transformers as it handles the data in a 2D format without unrolling it as a sequence. It is easy to adapt WaveMix-Lite for different image sizes and tasks, such as image classification and semantic segmentation, without changing its core architecture, as shown by our experiments on multiple datasets and tasks.
\textbf{Limitations:} Although our own lack of access to a large GPU cluster inspired the objective of designing a neural architecture that can generalize well with limited resources, scalability of WaveMix-Lite to larger datasets (in terms of image size and the number of images) needs to be tested with larger compute resources. Additionally, WaveMix-Lite should also be tested on other vision tasks, such as object detection and image enhancement (e.g., super-resolution, deblurring, denoising, and inpainting) using task-specific architectural variations.
\textbf{Impact}: The high accuracy of image classification by transformers and CNNs comes with high costs in terms of training data, computations, GPU RAM, hardware costs, form factors, and energy consumption~\cite{li2020train}. Our research shows that architectural innovations can still reduce these computational requirements by exploiting priors of natural images, such as scale-invariance and finite spatial extents (in addition to shift-invariance that is already exploited by convolutional design elements), without sacrificing accuracy. Such architectures would also be more environment-friendly due to lower energy consumption and accessible for modification to more researchers who may not have access to large computational resources. We hope that our research has shed light into the less traversed area of resource-efficient models that exploit more priors of natural images.
This work can be extended in several directions, some of which are mentioned in its limitations above. Exploitation of additional properties of images and videos can also be explored. Instead of using a fixed function (e.g., Haar), we can also make the learning of the wavelet function itself as a part of the training process.
|
1,108,101,566,788 | arxiv | \section{Introduction}
Integrable models are ideal testing grounds of various methods and
ideas in quantum field theories. The simplest interacting model of
this type is the sinh-Gordon theory, which has a single particle type
and the full finite volume energy spectrum can be calculated from
the scattering phase of these particles. There is a hope that similar
exact results can be obtained also for finite volume matrix elements.
The finite volume matrix elements of local operators are essentially
the building blocks of finite volume correlations functions, which
are relevant in statistical and solid state systems \cite{Mussardo:2010mgq,Samaj:2013yva}.
Their non-local counterparts can be used in the AdS/CFT correspondence
to describe three-point functions in the gauge theory and the string
field theory vertex in string theory \cite{Beisert:2010jr,Basso:2015zoa,Bajnok:2015hla,Basso:2017muf,Bajnok:2017mdf}.
Diagonal matrix elements play a special role there, as they describe
the HHL type correlation functions \cite{Bajnok:2014sza,Bajnok:2016xxu,Hollo:2015cda,Jiang:2015bvm,Jiang:2016dsr}.
There were two alternative approaches for the calculation of finite
volume matrix elements. For generic operators and theories one can
try to use the infinite volume form factors \cite{Smirnov:1992vz,Babujian:2003sc}
and the scattering matrix \cite{Zamolodchikov:1978xm} to develop
a systematic large volume expansion. Polynomial volume corrections
originate from momentum quantization \cite{Luscher:1986pf}, while
exponentially small finite size corrections from the presence of virtual
particles \cite{Luscher:1985dn}. The LeClair-Mussardo formula \cite{Leclair:1999ys}
provides an infinite series for the exact finite volume one-point
function, where each term contains the contribution of a given number
of virtual particles in terms of their infinite volume connected form
factors and a weight function, which is related to the Thermodynamic
Bethe ansatz (TBA) densities of these particles \cite{Zamolodchikov:1989cf}.
This formula was then generalized by analytical continuation for diagonal
matrix elements, which replaces the ground-state TBA densities with
the excited state ones and contains additional factors, which can
be interpreted as partial density of states \cite{Pozsgay:2013jua,Pozsgay:2014gza}.
Alternatively, there is an other approach which focuses on specific
theories and exploits their hidden (Grassmann) structure to provide
compact expressions for finite volume matrix elements \cite{HGSV}.
These specific continuum models arise as limits of integrable lattice
models and the most studied examples are the sinh-Gordon and sine-Gordon
models. There have been active work and relevant progress in deriving
finite volume one-point functions for the exponential operators and
their descendants in these theories \cite{Negro:2013rxa,Negro:2013wga}.
These results were then extended for diagonal matrix elements in the
sine-Gordon theory \cite{Hegedus:2017muz,Hegedus:2017zkz,Hegedus:2019zks}
and the aim of our paper is to provide similar expressions in the
sinh-Gordon theory.
The paper is organized as follows: Section 2 reviews the description
of the finite size energy spectrum of the sinh-Gordon theory. A multi-particle
state for large volumes can be labelled by momentum quantum numbers,
which we relate at small volume to the spectrum of the
Liouville conformal field theory by matching the eigenvalues of the
conserved charges. In Section 3 we formulate our main conjecture for
the finite volume exceptions values in the fermionic basis. The novelty
compared to the vacuum expectation values is the discrete part of
the convolutions, which carries information on the particles' rapidities.
We check this conjecture for large volumes in Section 4. The discrete
part of the convolution contains the polynomial, while the continuous
part the exponentially small corrections in the volume. In Section
5 we compare our conjecture with Liouville three-point functions for
low lying states including non-degenerate and degenerate $L_{0}$
subspaces. All the checks performed confirm our conjecture, thus we
close the paper with conclusions in Section 6.
\section{Energy spectrum}
In this section we summarize the exact description of the finite volume
energy spectrum together with its large and small volume formulations \cite{Teschner:2007ng}.
The sinh-Gordon theory is defined by the Lagrangian:
\begin{equation}
\mathcal{L}=\frac{1}{4\pi}(\partial\phi)^{2}+\frac{2\mu^{2}}{\sin\pi b^{2}}\cosh(b\varphi)\,.
\nn\end{equation}
In the literature there is an abundance of notations for the parameters of this model.
We decided to follow the paper \cite{Zamolodchikov:2000kt} by introducing
the background charge of the related Liouville model and the renormalized coupling
constant as
$$Q=b+b^{-1}\,,\qquad p=\frac{b^2}{1+b^2}\,.$$
The sinh-Gordon model is the simplest integrable interacting two dimensional quantum field
theory. It has one single particle of mass $m$ with the corresponding
two particle scattering matrix
\begin{equation}
S(\theta)=\frac{\sinh\theta-i\sin(\pi p)}{\sinh\theta+i\sin(\pi p)}\,\,.
\nn\end{equation}
The finite size energy spectrum, in a volume $R$, can be formulated
in terms of the $\Q$ function, which satisfies the following functional relations:
\begin{equation}
\Q(\theta+{\textstyle \frac{i\pi}{2}})\Q(\theta-{\textstyle \frac{i\pi}{2}})=1+\Q(\theta+{\textstyle \frac{i\pi}{2}}(1-2p))\Q(\theta-{\textstyle \frac{i\pi}{2}}(1-2p))\equiv1+e^{-\epsilon(\theta)}\,,\nn
\end{equation}
where we introduced the TBA pseudo-energy $\epsilon.$ Excited states
can be labeled by the zeros of $\Q$ as: $\{\theta_{1},\dots,\theta_{N}\}$.
With the prescribed large $\theta$ asymptotics, $\log \Q(\theta)\simeq-\frac{r}{2}\frac{\cosh\theta}{\sin\pi p}$
there is a unique solution
\begin{equation}
\Q(\theta)=\prod\limits _{k=1}^{N}\tanh\Bigl(\frac{\theta-\theta_{k}}{2}\Bigr)\exp\Bigl(-\frac{r\cosh(\theta)}{2\sin(\pi p)}+\frac{1}{2\pi}\int\limits _{-\infty}^{\infty}\frac{1}{\cosh(\theta-\theta')}\log\ensuremath{(1+e^{-\epsilon(\theta')})}d\theta'\Bigr)\,.
\nn\end{equation}
Here $r=mR$ is the dimensionless volume. Thanks to the functional
relation $\epsilon(\theta)$ can be fixed from the following TBA equation
\begin{align}
\epsilon(\theta)=r\cosh\theta+\sum\limits _{k=1}^{N}\log S(\theta-\theta_{k}-\pp
)-\int\limits _{-\infty}^{\infty}K(\theta-\theta')\log(\ensuremath{1+e^{-\epsilon(\theta')})}d\theta'\,.\label{TBA}
\end{align}
where the kernel is related to the scattering matrix as
\begin{equation}
K(\theta)=\frac{1}{2\pi i}\frac{d}{d\theta}\log S(\theta)=\frac{1}{2\pi i}\Bigl(\frac{1}{\sinh(\theta-\pi ip)}-\frac{1}{\sinh(\theta+\pi i p)}\Bigr)\,.
\nn\end{equation}
The finite size spectrum can be characterized by a set of integers
$\{N_{k}\}$, denoted by $\mathcal{N}$, via the zeros of the $\Q$
function, written equivalently as
\begin{align}
f(\theta_{k})=\pi N_{k}\,,\label{BY}
\end{align}
where
\begin{equation}
f(\theta)=r\sinh\theta+\sum\limits _{k=1}^{N}\arg\ensuremath{(-S(\theta-\theta_{k}))}-\int\limits _{-\infty}^{\infty}K(\theta-\theta'+
\pp
)\log(\ensuremath{1+e^{-\epsilon(\theta')})}d\theta'\,.
\nn\end{equation}
coincides at the positions $\theta_{k}$ with the analytical continuation
of $-i\epsilon(\theta+i\pi/2)$ (with certain choice of the branches
of logarithms).
Here we use $-S(\theta-\theta_{k})$ under $\arg$ for computational convenience (notice that $-S(0)=1$).
These equations are called the Bethe Ansatz (BA) equations
and can be interpreted as the momentum quantization equations of the
particles with rapidity $\theta_{k}$. The energy of the multi-particle
state with rapidities $\{\theta_{1},\dots,\theta_{N}\}$ can be written
as
\begin{equation}
E_{\mathcal{N}}=\sum_{i=1}^{N}m\cosh\theta_{i}-m\int_{-\infty}^{\infty}\cosh\theta\log(1+e^{-\epsilon(\theta)})\frac{d\theta}{2\pi}\,.\label{eq:TBAenergy}
\end{equation}
\subsection{Large volume expansion}
Since in the large volume limit the TBA pseudo-energy behaves as $\epsilon=r\cosh\theta+O(1)$
the integral terms are of order $O(e^{-r})$
and can be neglected. This results in the large volume limit of the
BA equations
\begin{equation}
r\sinh\theta_{j}+\sum\limits _{k=1}^{N}\arg\ensuremath{(-S(\theta_{j}-\theta_{k}))}=\pi N_{j}\,.\label{eq:BAlargeR}
\end{equation}
Let us assume that rapidities are labeled such that $\{\theta_{1}>\theta_{2}>\dots>\theta_{m}\}$.
We recall that in \cite{Teschner:2007ng} it was proven that for any
given set of integers $\{n_{1},\dots,n_{m}\}$ the equations
\begin{equation}
r\sinh\theta_{j}-\sum_{k=1}^{j-1}\arg(S(\theta_{k}-\theta_{j}))+\sum_{k=j+1}^{N}\arg(S(\theta_{j}-\theta_{k}))=2\pi n_{j}\,,\label{eq:BAargS}
\end{equation}
have a unique solution\footnote{Note that, although here the quantum numbers $\{n_{j}\}$ can be equal,
the solutions for the rapidities $\{\theta_{j}\}$ can not, so the
system is nevertheless fermionic and not bosonic type.}. The idea of the proof was to introduce $P(\theta)=\int_{0}^{\theta}\arg(S(v))dv$
and to show that the rapidities $\{\theta_{j}\}$ minimize the positive
definite Yang-Yang functional
\begin{equation}
\sum_{j}(r\cosh\theta_{j}-2\pi n_{j}\theta_{j})+\sum_{j<k}P(\theta_{j}-\theta_{k})\,.
\nn\end{equation}
In order to compare eq. (\ref{eq:BAlargeR}) to eq. (\ref{eq:BAargS})
we recall that for positive arguments
\begin{equation}
\arg(S(\theta))=\arg(-S(\theta))-i\pi\quad;\qquad\theta>0\,.
\nn\end{equation}
This leads to the following relation between the quantum numbers $\{N_{j}\}$
and $\{n_{j}\}$:
\begin{equation}
N_{j}=2n_{j}-N-1+2j\,.
\nn\end{equation}
In particular, the state labelled by $\{0,\dots,0\}$ in eq. (\ref{eq:BAargS})
will be mapped to $\{-M+1,-M+3,\dots,M-3,M-1\}$, with all quantum
numbers being distinct. This also shows that states with even number
of particles are labelled by odd quantum numbers, while states with
odd number of particles with even quantum numbers in $\mathcal{N}$.
Once the equations (\ref{eq:BAlargeR}) are solved for the rapidities
the large volume energy is
\begin{equation}
E_{\mathcal{N}}=\sum_{i=1}^{N}m\cosh\theta_{i}\,.
\nn\end{equation}
In the following we analyze the small volume limit of the energies.
\subsection{Small volume limit}
In the small volume limit we compare the energy eigenvalues with the
spectrum of the Liouville theory \cite{Zamolodchikov:1995aa}. In
this description the sinh-Gordon theory is understood as the perturbation
of the Liouville theory with the operator $e^{-b\varphi}$:
\begin{equation}
\mathcal{L}=\mathcal{L}_{CFT}+\frac{\mu^{2}}{\sin\pi b^{2}}e^{-b\varphi}\,.
\nn\end{equation}
There are infinitely many conserved charges and the energy is related
to the first two as $E=-\frac{\pi}{12R}(I_{1}+\bar{I}_{1})$, where
\begin{equation}
I_{1}=-\frac{6r}{\pi}\biggl(\sum\limits _{k=1}^{N}e^{\theta_{k}}-\int\limits _{-\infty}^{\infty}e^{\theta}\log(\ensuremath{1+e^{-\epsilon(\theta)}})\frac{d\theta}{2\pi}\biggr)\,,\quad\bar{I}_{1}=-\frac{6r}{\pi}\biggl(\sum\limits _{k=1}^{N}e^{-\theta_{k}}-\int\limits _{-\infty}^{\infty}e^{-\theta}\log(\ensuremath{1+e^{-\epsilon(\theta)}})\frac{d\theta}{2\pi}\biggr)\,.\nn
\end{equation}
The Liouville theory is a conformal field theory with a continuous
spectrum. Its Hilbert space is built up from the non-compact zero
mode and the oscillators. The zero mode determines the dimension of
the primary fields, while the oscillators create descendants. Once
the perturbation is introduced the spectrum of primary fields can
be approximated by the quantization of the zero mode \cite{Zamolodchikov:1995aa}:
\begin{align}
& 4P_{L}(r)Q\log\Bigl(Z(p)rb^{\frac{b^{2}-1}{b^{2}+1}}\Bigr)=-\pi L+\frac{1}{i}\log\ensuremath{\frac{\Gamma(1+2iP_{L}(r)b)\Gamma(1+2iP_{L}(r)/b)}{\Gamma(1-2iP_{L}(r)b)\Gamma(1-2iP_{L}(r)/b)}}\,.\label{primary}
\end{align}
with $L=1,2,\dots$. Here and later we use the mass scale
\begin{equation}
Z(p)=\frac{1}{16Q\pi^{3/2}}\Gamma\Bigl(\frac{p}{2}\Bigr)\Gamma\Bigl(\frac{1-p}{2}\Bigr).
\nn\end{equation}
The eigenvalues of the conserved charges in the CFT can be written
as
\begin{equation}
I_{1}^{\mathrm{CFT}}=P_{L}(r)^{2}-\frac{1}{24}+M\,,\quad\bar{I}_{1}^{\mathrm{CFT}}=P_{L}(r)^{2}-\frac{1}{24}+\overline{M}\,,
\nn\end{equation}
where $M,\overline{M}$ are levels of descendants for the two chiralities.
By comparing the energies in the TBA and the perturbed Liouville descriptions
we can relate the quantum numbers ${\cal N}$ to $\{L,M,\bar{M}\}$.
To give an example, we claim that $\mathcal{N}=\{-2,0,2\}$ corresponds
to the primary field with $L=4$. Indeed, by numerically solving the
TBA and BA equations (\ref{TBA},\ref{BY}) on the one hand and the
zero mode quantization (\ref{primary}) on the other we found for
$r=.001$ the ratios:
\begin{equation}
\frac{I_{1}}{I_{1}^{\mathrm{CFT}}}=1.00003\,,\quad\frac{\bar{I}_{1}}{\bar{I}_{1}^{\mathrm{CFT}}}=0.999989\,.
\nn\end{equation}
In this way we obtain the following correspondence, which we present
both at the language of $\mathcal{N}$ and that of $\{n_{j}\}$ with
$n_{j}=(N_{j}+M+1)/2-j$:
\begin{align}
\begin{tabular}{|c|c|c|c|c|}
\hline \ensuremath{\mathcal{N}} & \{\ensuremath{n_{i}}\} & \ensuremath{L} & \ensuremath{M} & \ensuremath{\bar{M}}\\
\hline\hline \{\ \} & \{\} & 1 & 0 & 0 \\
\hline \{0\} & \{0\} & 2 & 0 & 0 \\
\hline \{-1,1\} & \{0,0\} & 3 & 0 & 0 \\
\hline \{-2,0,2\} & \{0,0,0\} & 4 & 0 & 0 \\
\hline \{2\} & \{1\} & 1 & 1 & 0 \\
\hline \{-1,3\} & \{0,1\} & 2 & 1 & 0 \\
\hline \{-2,0,4\} & \{0,0,1\} & 3 & 1 & 0 \\
\hline \{-3,3\} & \{-1,1\} & 1 & 1 & 1 \\
\hline \{4\} & \{2\} & 1 & 2 & 0 \\
\hline \{1,3\} & \{1,1\} & 1 & 2 & 0 \\
\hline \{-1,5\} & \{0,2\} & 2 & 2 & 0 \\
\hline \{-2,2,4\} & \{0,1,1\} & 2 & 2 & 0 \\
\hline \{-3,5\} & \{-1,2\} & 1 & 2 & 1 \\
\hline \{-5,5\} & \{-2,2\} & 1 & 2 & 2 \\
\hline \{1,5\} & \{1,2\} & 1 & 3 & 0 \\
\hline \{0,2,4\} & \{1,1,1\} & 1 & 3 & 0 \\
\hline \{-3,-1,1,5\} & \{0,0,0,1\} & 4 & 1 & 0 \\
\hline
\end{tabular}\label{tab}
\end{align}
Clearly in the parametrization $\{n_{j}\}$ the number of zeros is
$L-1$, while the sum of positive/negative numbers is $M/\bar{M}$,
in agreement with \cite{Teschner:2007ng}.
Starting from $M=2$ the
spectrum of $L_0$ is degenerate.
The
degeneracy can be lifted using the second integral of motion as we demonstrate in Section \ref{degenerate}.
We will use all these states
later to compare our form factor conjecture to the Liouville three-point
functions in the small volume limit.
\section{Finite volume expectation values}
In the following we provide formulas for the expectation values
\begin{equation}
\langle\theta_{1
},\dots,\theta_{m}\vert\mathcal{O}\vert\theta_{m},\dots,\theta_{1}\rangle_{R}\,,
\nn\end{equation}
where $\vert\theta_{m},\dots,\theta_{1}\rangle_{R}$ is a normalized
finite volume energy eigenstate (\ref{eq:TBAenergy}) and $\mathcal{O}$
is a local operator. Expectation values of local operators obtained
by commuting with a conserved charge, $[I_{n},\mathcal{O}]$, vanish,
thus we consider only the quotient space, where these operators are
factored out.
Local operators in massive perturbed conformal field theories are
in one-to-one correspondence with the states of the conformal Hilbert
space of the unperturbed model. The sinh-Gordon theory can be considered
either as the perturbation of the free massless boson with $\cosh(b\varphi)$
or as the perturbation of the Liouville theory with the operator $e^{-b\varphi}$.
Local operators are the exponentials $\Phi_{\alpha}=e^{\frac{Q\alpha}{2}\varphi}(0)$
together with their descendants $\mathcal{O}_{\alpha}$, which can
be generated in two different ways \cite{Fateev:1998xb,Negro:2013rxa,Negro:2013wga}.
If the modes of the free massless boson are used the operators are
called \emph{Heisenberg descendants} and the expectation values of
the corresponding operators in the quotient space have the $\sigma_{1}:\alpha\to-\alpha$
symmetry. In the perturbed Liouville scheme \emph{Virasoro descendants}
are generated by the Virasoro modes and the expectation values have
the symmetry $\sigma_{2}:\alpha\to2-\alpha$.
Relating these two descriptions should provide a basis of the CFT adapted to
the integrable perturbation. Direct attempt to find such a basis failed for level higher than $2$
because it requires solving a rather complicated Riemann-Hilbert problem. The solution came from a
rather distant study of lattice integrable models which lead to the discovery of the fermionic basis.
The latter provides in the scaling limit the fermionic basis for the sine-Gordon model
\cite{HGSIV,HGSV}. As has been shown
in \cite{Negro:2013rxa} this fermionic basis
brings the Riemann-Hilbert problem in question to the diagonal form.
\subsection{Fermionic basis}
The definition of the fermionic basis in the CFT case can be considered as a purely algebraic one,
that is why it is equally suitable for the sinh-Gordon case. Analytical advantage of using the fermionic
basis in the sine-Gordon model is due to the fact that the expectation values of the elements of the fermionic
basis are expressed as determinants. We do not know how to prove similar fact for the sinh-Gordon case,
so, like in \cite{Negro:2013wga} we shall formulate it as a conjecture and then perform numerous checks.
The fermionic basis is
created by the anti-commutative operators $\beta^{*},\gamma^{*}$
(and $\bar{\beta}^{*},\bar{\gamma}^{*}$ for the other chirality). They
can be used to generate the quotient space as
\begin{equation}
\beta_{M}^{*}\gamma_{N}^{*}\bar{\beta}_{\bar{M}}^{*}\bar{\gamma}_{\bar{N}}^{*}\Phi_{\alpha}=\beta_{m_{1}}^{*}\dots\beta_{m_{k}}^{*}\gamma_{n_{1}}^{*}\dots\gamma_{n_{k}}^{*}\bar{\beta}_{\bar{m}_{1}}^{*}\dots\bar{\beta}_{\bar{m}_{\bar{k}}}^{*}\bar{\gamma}_{\bar{n}_{1}}^{*}\dots\bar{\gamma}_{\bar{n}_{\bar{k}}}^{*}\Phi_{\alpha}\,,\nn
\end{equation}
with all modes being odd and positive.
Later we shall have these operators for negative
indices. By definition they are related to annihilation operators $\beta_{-j}^{*}=\gamma_{j}$
, $\gamma_{-j}^{*}=\beta_{j}$ (together with similar relations for
the other chirality) such that their anti-commutator is
\begin{equation}
\{\beta_{m},\beta_{n}^{*}\}=\{\bar{\gamma}_{m},\bar{\gamma}_{n}^{*}\}=-t_{m}(\alpha)\delta_{m,n}\quad;\qquad t_{n}(\alpha)=\frac{1}{2\sin{\pi}(\al-np)}\,.\nn
\end{equation}
The relation between the fermionic basis and the Heisenberg or Virasoro
basis is a very complicated problem and requires a case by case study.
Later we shall have examples.
What is particularly nice about the fermionic basis is that the finite
volume expectation values take a very simple determinant form. Indeed,
the main result of our paper is a conjecture of the form
\begin{equation}
\frac{\langle\theta_{1},\dots,\theta_{m}\vert\beta_{M}^{*}\gamma_{N}^{*}\bar{\beta}_{\bar{M}}^{*}\bar{\gamma}_{\bar{N}}^{*}\Phi_{\alpha}\vert\theta_{m},\dots,\theta_{1}\rangle_{R}}{\langle\theta_{1},\dots,\theta_{m}\vert\Phi_{\alpha}\vert\theta_{m},\dots,\theta_{1}\rangle_{R}}=\mathcal{D}(\{M\cup(-\bar{M})\}\vert\{N\cup(-\bar{N})\}\vert\alpha)\nn\,.
\end{equation}
where for the index sets $A=\{a_{1},\dots,a_{n}\}$ and $B=\{b_{1},\dots,b_{n}\}$
the determinant is
\begin{equation}
\mathcal{D}(A\vert B\vert\alpha)=\prod_{j=1}\frac{\text{sgn}(a_{j})\text{sgn}(b_{j})}{\pi}\text{Det}_{j,k}\left(\Omega_{a_{j},b_{k}}\right)\quad;\quad\Omega_{n,m}=\omega_{n,m}-\pi\text{sgn}(n)\delta_{n,-m}t_{n}(\alpha)\,.\nn
\end{equation}
The construction of the matrix $||\omega_{m,n}||$ is explained in the next section.
\subsection{The matrix $\omega_{m,n}$}
The matrix $\omega_{m,n}$ is built via a deformation of a linear operator
involved in the linearization of the TBA equations. We start by explaining this linearization.
Consider the variation of the TBA equations \eqref{TBA},\eqref{BY} with respect to $r$.
The functions $\epsilon(\theta)$ and $f(\theta)$ depend actually on $\theta$ and $r$, while the points of the discrete spectrum
$\theta_k$ depends on $r$. We have
\begin{align}
&\partial_r\epsilon(\theta)=\cosh\theta-2\pi i\sum\limits _{k=1}^{N}K(\theta-\theta_{k}+
\p
)\frac{d \theta_k}{dr}+\int\limits _{-\infty}^{\infty}K(\theta-\theta')\partial_r\epsilon(\theta')\frac 1 {1+e^{\epsilon(\theta')}}d\theta'\label{variation}\\
&=\cosh\theta+2\pi i\sum\limits _{k=1}^{N}K(\theta-\theta_{k}+
\p
)
\frac{1}{\partial_{\theta}f(\theta_k)}\partial_{r}f(\theta_k)+\int\limits _{-\infty}^{\infty}K(\theta-\theta')\partial_r\epsilon(\theta')\frac 1 {1+e^{\epsilon(\theta')}}d\theta'\,,\nn
\end{align}
where we used that $$\partial_{r}f(\theta_k)+\partial_{\theta}f(\theta_k)\frac{d \theta_k}{dr}=0\,,$$
following from \eqref{BY}.
Consider functions on discrete and continuous spectra $G=\{g_1,\cdots, g_k,g(\theta)\}$.
Motivated by \eqref{variation} we introduce paring for two such functions
\begin{equation}
G* H=2\pi i \sum\frac{1}{\partial_{\theta}f(\theta_k)}
g_k h_k+\int\limits_{-\infty}^{\infty}g(\theta)h(\theta)\frac{d\theta}{1+e^{\epsilon(\theta)}}\,.
\label{eq:convolution}
\end{equation}
By using this convolution the matrix element $\omega_{n,m}$ entering in our conjecture can be written as
\begin{align}
\omega_{n,m} & =e_{n}*(1+\mathcal{K}_{\alpha}+\mathcal{K}_{\alpha}*\mathcal{K}_{\alpha}+\dots)*e_{m}\equiv e_{n}*(1+\mathcal{R}_{\mathrm{dress},\alpha})*e_{m}\nn\,,
\end{align}
where $e_{n}=\{e^{n(\theta_{1}+
\p
)},\dots,e^{n(\theta_{m}+\p
)},e^{n\theta}\}$
and $\mathcal{K}_{\alpha}$ has a matrix structure
\begin{equation}
\mathcal{K}_{\alpha}=\begin{pmatrix}K_{\alpha}(\theta_{k}-\theta_{l}) & K_{\alpha}(\theta_{k}-\theta+
\p
)\\
K_{\alpha}(\theta-\theta_{l}-
\p
) & K_{\alpha}(\theta-\theta')
\end{pmatrix}\nn\,.
\end{equation}
reflecting the fact that the convolution has a discrete and the continuous
part. Here $K_{\alpha}$ is the deformation of the TBA kernel
\begin{equation}
K_{\alpha}(\theta)=\frac 1 {2\pi i}\Bigl(\frac{e^{-i\pi\alpha}}{\sinh(\theta-\pi ip)}-\frac{e^{i\pi\alpha}}{\sinh(\theta+\pi i p)}\Bigr)\,,\nn
\end{equation}
which satisfy $K_{\alpha+2}(\theta)=K_{\alpha}(\theta)$.
Similar determinant expression to ours was proposed and tested for vacuum expectation values
in \cite{Negro:2013wga}. Our formulae are the extensions of VEVs
for excited states and the novel complication is the discrete part
of the convolutions. In the next section we explain how to work with
these expressions.
There was a nice observation in \cite{HGSV} that one might
relax the condition that the number of $\beta^{*}$ and $\gamma^{*}$
are the same, but in the same time maintain the determinant form.
By this way operators with different sectors can be connected as
\begin{equation}
\beta_{M}^{*}\gamma_{N}^{*}\bar{\beta}_{\bar{M}}^{*}\bar{\gamma}_{\bar{N}}^{*}\Phi_{\alpha+2mp}=\frac{C_{m}(\alpha)}{\prod_{j=1}^{m}t_{2j-1}(\alpha)}\beta_{M+2m}^{*}\gamma_{N-2m}^{*}\bar{\beta}_{\bar{M}-2m}^{*}\bar{\gamma}_{\bar{N}+2m}^{*}\beta_{\{m\}}^{*}\bar{\gamma}_{\{m\}}^{*}\Phi_{\alpha}\nn\,.
\end{equation}
where $\{m\}=1,3,\dots,2m-1$ and $C_{m}(\alpha)$ is the ratio of
the infinite volume vacuum expectation values \cite{Fateev:1998xb}:
\begin{equation}
C_{m}(\alpha)=\frac{\langle\Phi_{\alpha-2mp}\rangle_{\infty}}{\langle\Phi_{\alpha}\rangle_{\infty}}\,.\nn
\end{equation}
The simplest of these relations is
\begin{equation}
\frac{\Phi_{\alpha-2p}}{\langle\Phi_{\alpha-2p}\rangle_{\infty}}=\frac{1}{t_{1}(\alpha)}\beta_{1}^{*}\bar{\gamma}_{1}^{*}\frac{\Phi_{\alpha}}{\langle\Phi_{\alpha}\rangle_{\infty}}\,.\nn
\end{equation}
This relation is understood in the weak sense, i.e. for matrix elements.
In the next section we take diagonal matrix elements of this relation
and compare its large volume expansion with Pozsgay's result \cite{Pozsgay:2013jua}.
\section{Large volume checks}
In this section we make some IR checks of our formulae for the diagonal
finite volume matrix elements, which we normalize as
\begin{equation}
F(\theta_{1},\dots,\theta_{m}\vert\alpha)=\frac{\langle\theta_{1},\dots,\theta_{m}\vert\Phi_{\alpha}\vert\theta_{m},\dots,\theta_{1}\rangle_{R}}{\langle\Phi_{\alpha}\rangle_{\infty}}\,.\nn
\end{equation}
Reflection properties with $\sigma_{1}$ and $\sigma_{2}$ ensure the invariance under the $\alpha\to\alpha+2$
shift. The finite volume state $\vert\theta_{m},\dots,\theta_{1}\rangle_{R}$
is symmetric in the rapidity variables, which satisfy the BA
equations $f(\theta_{k})=\pi N_{k}$. These states can be labelled either
by the discrete quantum numbers $N_{k}$ or by the rapidities
$\theta_{k}$ and are naturally normalized to Kronecker delta
functions.
In the following we investigate the simplest non-trivial example
\begin{equation}
\frac{F(\theta_{1},\dots,\theta_{m}\vert\al-2p)}{F(\theta_{1},\dots,\theta_{m}\vert\alpha)}=1+\frac{2}{\pi}\sin\pi(p-\al)(e_1*e_{-1}+e_1*\mathcal{R}_{\mathrm{dress},\alpha}*e_{-1})\,.\label{eq:e1Rem1}
\end{equation}
where $e_n$ is related to $e^{n\theta}$. For each function $g$ we have a discrete and a continues part: $(g_{1},\dots,g_{m},g(\theta))$
with $g_{j}=g(\theta_{j}+i\frac{\pi}{2})$ and the convolution is
understood as in \eqref{eq:convolution}. We would like to compare these
formulae with the available results in literature which we recall
now.
\subsection{Form factor expansion of the diagonal finite volume matrix elements}
A \emph{finite} volume diagonal form factor can be expressed in terms
of the \emph{infinite }volume connected form factors, which are defined
to be the finite ($\epsilon$-independent) part in the crossed expression
\begin{equation}
F_{2n}^{\mathcal{O}}(\theta_{1}+i\pi+\epsilon_{1},\dots,\theta_{m}+i\pi+\epsilon_{m},\theta_{m},\dots,\theta_{1})=\frac{O(\epsilon^{m})}{\epsilon_{1}\dots\epsilon_{m}}+F_{2n,c}^{\mathcal{O}}(\theta_{1},\dots,\theta_{m})+O(\epsilon)
\,.\nn
\end{equation}
These connected form factors have interesting properties \cite{Pozsgay:2014gza}.
For the exponential operators, normalized by the VEV's, $\Phi_{\alpha}/\langle\Phi_{\alpha}\rangle_{\infty}$,
the first two connected form factors read as \cite{Leclair:1999ys,Negro:2013wga}:
\begin{align}
&F_{2,c}^{\alpha}=4\sin(\pi p)[k_{\alpha}]^{2}\,,\nn\\
&F_{4,c}^{\alpha}(\theta_1,\theta_2)=4\pi F_{2,c}^{\alpha}K(\theta_1-\theta_2)\Bigl(\cosh(\theta_1-\theta_2)[k_{\alpha}]^{2}-\frac{[k_{\alpha}-1][k_{\alpha}+1]}{\cosh(\theta_1-\theta_2)}\Bigr)\,,\nn
\end{align}
where
\[
[k]=\frac{\sin({\pi} p k)}{\sin({\pi}p)}\quad;\qquad k_{\alpha}=\frac{\alpha}{2p}\,.
\]
The six particle connected form factor
would fill a half page and there is no closed form available for the
general case. In principle such expressions could be obtained from
the determinant representation of form factors \cite{Koubek:1993ke}
using the limiting behavior of the symmetric polynomials \cite{Pozsgay:2014gza}.
But this procedure is quite cumbersome and the results do not seem to have any nice structure, thus we restrict our investigations for these first two form factors only.
The exact formula for the finite volume diagonal matrix elements was
conjectured in \cite{Pozsgay:2013jua} based on carefully evaluating
the contour deformation trick in the LM formula for 1 and 2 particle
states. For $m$ particles the conjecture takes the form
\begin{equation}
F(\theta_{1},\dots,\theta_{m}\vert\alpha)=\frac{\sum_{I\subseteq M}\mathcal{F}_{m-\vert I\vert}^{\alpha}(M\setminus I)\rho_{\vert I\vert}(I)}{\rho_{m}(M)}\,,\label{eq:FVexact}
\end{equation}
where the full index set is denoted by $M=\{1,2,\dots,m\}$, and an
index set $I=\{i_{1},\dots,i_{k}\}$ in the argument abbreviates the
set
of rapidities $\theta_{i_{1}},\dots,\theta_{i_{k}}$. The appearing
densities $\rho_{\vert I\vert}(I)$ are defined to be the determinant
\begin{equation}
\rho_{k}(\theta_{i_{1}},\dots,\theta_{i_{k}})=\det_{j,l}\left|\partial_{\theta_{i_{j}}}f(\theta_{i_{l}})\right|\,.\nn
\end{equation}
For $I=M$ this is simply the density of the finite volume $m$-particle
states. The quantity $\mathcal{F}_{k}^{\alpha}$ is the generalization of
the LM expansion for the connected form factor $F_{2k,c}^{\alpha}$:
\begin{equation}
\mathcal{F}_{k}^{\alpha}(\theta_{1},\dots,\theta_{k})=\sum_{n=0}^{\infty}\frac{1}{n!}\prod_{j=1}^{n}\int\frac{dm(v_{j})}{2\pi}F_{2(k+n),c}^{\alpha}(\theta_{1}+
\p
,\dots,\theta_{k}+
\p
,v_{1},\dots,v_{n})\,,\nn
\end{equation}
where $dm(v)=\frac{dv}{1+e^{\epsilon(v)}}$.
\subsection{Checks at polynomial order}
The finite volume diagonal form factor at any polynomial order in
the inverse of the volume can be obtained from (\ref{eq:FVexact})
by neglecting the integral terms both in $f$ and also in $\mathcal{F}$.
At this order
\begin{equation}
\mathcal{F}_{k}^{\alpha}(\theta_{1},\dots,\theta_{k})=F_{2k,c}^{\alpha}(\theta_{1},\dots,\theta_{k})+O(e^{-r})\,,\nn
\end{equation}
and
\begin{equation}
\rho_{k}(\theta_{i_{1}},\dots,\theta_{i_{k}})=\det_{j,l}\left|\delta_{j,l}(r\cosh\theta_{j}+2\pi\sum_{n=1}^{m}K(\theta_{i_{j}}-\theta_{n}))-2\pi K(\theta_{i_{j}}-\theta_{i_{l}})\right|+O(e^{-r})\,,\nn
\end{equation}
This asymptotic expression was conjectured in \cite{Pozsgay:2007gx} based on form
factor perturbation theory and later proved in \cite{Bajnok:2017bfg}, moreover, it leads
to the proof of the LM series, whose analytic continuation provided the exact conjecture (\ref{eq:FVexact}).
In the following we recover these asymptotic results from our fermionic expression (\ref{eq:e1Rem1}).
We proceed in the particle number.
\subsubsection{1-particle case}
We need to check the relation
\begin{align}
\frac{F(\theta_{1}\vert\alpha -2p)}{F(\theta_{1}\vert\alpha)} & =\frac{F_{2,c}^{\alpha -2p}+r\cosh\theta_{1}}{F_{2,c}^{\alpha}+r\cosh\theta_{1}}+O(e^{-r})=\label{eq:1pff}\\
& =1+\frac{2}{\pi}\sin\pi(p-\al)(e_1*e_{-1}+e_1*\mathcal{R}_{\mathrm{dress},\alpha}*e_{-1})\,,\nonumber
\end{align}
where $e_1$ represents one discrete particle with $\theta_{1}$
and the function $e^{v_{1}}$ as $(e^{\theta_{1}+\frac{i\pi}{2}},e^{v_{1}})$.
Thus the first convolution in \eqref{eq:convolution} explicitly reads
as
\begin{equation}
e_1*e_{-1}=\frac{2\pi}{f'(\theta_{1})}+\int dm(v_{1})=\frac{2\pi}{r\cosh\theta_{1}+2\pi K(0)}+O(e^{-r})\,,\label{eq:1pLO}
\end{equation}
where we evaluated it neglecting exponentially small terms. Clearly,
each convolution in the discrete part introduces a polynomial suppression
factor $r^{-1}$ while in the continuous part an exponential one $e^{-r}$.
In order to obtain all polynomial volume corrections we need to sum
up the iterated series in the discrete part
\begin{align}
e_1*e_{-1}+e_1*\mathcal{R}_{\mathrm{dress},\alpha}*e_{-1} & =e_1*(1+K_{\alpha}(0)+K_{\alpha}(0)*K_{\alpha}(0)+\dots)*e_{-1}\label{eq:1pferm}\\
& =\frac{2\pi}{f'(\theta_{1})}\left(1+\sum_{n=1}^{\infty}\frac{(2\pi K_{\alpha}(0))^{n}}{f'(\theta_{1})^{n}}\right)=\frac{2\pi}{f'(\theta_{1})-2\pi K_{\alpha}(0)}\nonumber \\
& =\frac{2\pi}{r\cosh\theta_{1}+2\pi(K(0)-K_{\alpha}(0))}+O(e^{-r})\,.\nonumber
\end{align}
The relation $2\pi(K(0)-K_{\alpha}(0))=F_{2,c}^{\alpha}$ together with $F_{2,c}^{\al-2p}-F_{2,c}^{\alpha}=4\sin(\pi(p-\al))$
imply that the two forms (\ref{eq:1pff}) and (\ref{eq:1pferm})
are indeed equivalent.
\subsubsection{2-particle case}
For two particles the form factor expression, neglecting exponential
corrections, gives
\begin{equation}
\frac{F(\theta_{1},\theta_{2}\vert\al-2p)}{F(\theta_{1},\theta_{2}\vert\alpha)}=\frac{F_{4,c}^{\al-2p}(\theta_{1},\theta_{2})+F_{2,c}^{\al-2p}(\rho_{1}(\theta_{1})+\rho_{1}(\theta_{2}))+\rho_{2}(\theta_{1},\theta_{2})}{F_{4,c}^{\alpha}(\theta_{1},\theta_{2})+F_{2,c}^{\alpha}(\rho_{1}(\theta_{1})+\rho_{1}(\theta_{2}))+\rho_{2}(\theta_{1},\theta_{2})}\,.\label{eq:2pff}
\end{equation}
In the fermionic formulation we consider only the discrete part, thus
in the convolution $e_1$ represents $i(e^{\theta_{1}},e^{\theta_{2}})$,
while $e_{-1}$ is nothing but $-i(e^{-\theta_{1}},e^{-\theta_{2}})$.
The kernel is a $2\times 2$ matrix: $(\hat{K}_{\alpha})_{ij}\equiv K_{\alpha}(\theta_{i}-\theta_{j})$.
The convolution in the discrete part can be traded for ordinary matrix
multiplication by introducing an extra matrix factor\footnote{Note that $f'(\theta_{j})-\partial_{\theta_{j}}f(\theta_{j})=2\pi K(0)+O(e^{-r})$.} $(\hat{f})_{ij}=\delta_{ij}f'(\theta_{j})$.
As a result we obtain
\begin{equation}
\frac{F(\theta_{1},\theta_{2}\vert\al-2p)}{F(\theta_{1},\theta_{2}\vert\alpha)}=1+4\sin\pi(p-\al)(e^{\theta_{1}},e^{\theta_{2}})(\hat{f}-2\pi\hat{K}_{\alpha})^{-1}\text{\ensuremath{\left(\begin{array}{c}
e^{-\theta_{1}}\\
e^{-\theta_{2}}
\end{array}\right)}}\,,\label{eq:2pferm}
\end{equation}
where exponential corrections are neglected, but all polynomial corrections
are summed up. We have checked explicitly that this result (\ref{eq:2pferm})
agrees with the form factor description (\ref{eq:2pff}).
\subsubsection{m-particle case}
Similar calculation can be repeated for the generic $m$-particle
case. Now $\hat{K}_{\alpha}$ and $\hat{f}$ are $m\times m$ matrices
with entries
\begin{equation}
(\hat{K}_{\alpha})_{ij}=K_{\alpha}(\theta_{i}-\theta_{j})\quad;\qquad(\hat{f})_{ij}=\delta_{ij}(r\cosh\theta_{j}+2\pi\sum_{k=1}^{m}K(\theta_{j}-\theta_{k}))\,,\nn
\end{equation}
leading to the analogous formula
\begin{align}
\frac{F(\theta_{1},\dots,\theta_{m}\vert\al-2p)}{F(\theta_{1},\dots,\theta_{m}\vert\alpha)} & =1+4\sin\pi(p-\al)(e^{\theta_{1}},\dots,e^{\theta_{m}})(\hat{f}-2\pi\hat{K}_{\alpha})^{-1}\text{\ensuremath{\left(\begin{array}{c}
e^{-\theta_{1}}\\
\vdots\\
e^{-\theta_{m}}
\end{array}\right)}}\,.\nn
\end{align}
Since higher than two-particle connected form factors are very complicated
we did not check explicitly this result, although we have no doubts
about its correctness. However, we would like to point out that substituting
$r=0$ in the formula provides a very compact and simple expression
for the ratios of diagonal matrix elements. These are actually nothing
but the symmetric evaluations of the form factors \cite{Pozsgay:2007gx}.
We believe that this observation could be used to find some nice parametrization
of these form factors in the generic case.
\subsection{Checks at the leading exponential order }
We now check the leading exponential correction for the simplest 1-particle form factor
\begin{equation}
\frac{F(\theta_{1}\vert\al-2p)}{F(\theta_{1}\vert\alpha)}=\frac{F_{2,c}^{\al-2p}+\rho_{1}(\theta_{1})+\int\frac{dm(v_{1})}{2\pi}(F_{4,c}^{\al-2p}(\theta_{1}+\frac{i\pi}{2},v_{1})+\rho_{1}(\theta_{1})F_{2,c}^{\al-2p})}{F_{2,c}^{\alpha}+\rho_{1}(\theta_{1})+\int\frac{dm(v_{1})}{2\pi}(F_{4,c}^{\alpha}(\theta_{1}+\frac{i\pi}{2},v_{1})+\rho_{1}(\theta_{1})F_{2,c}^{\alpha})}+O(e^{-2r})\,,\label{eq:1pNLO}
\end{equation}
where we also need to expand $\rho_{1}(\theta_{1})$. In doing so
we recall that
\begin{equation}
\rho_{1}(\theta_{1})=\partial_{\theta_{1}}f(\theta_{1})=r\cosh\theta_{1}-i\int dm(\theta)K(\theta_{1}+i\frac{\pi}{2}-\theta)\biggl(\frac{\partial\epsilon(\theta)}{\partial\theta}+\frac{\partial\epsilon(\theta)}{\partial\theta_{1}}\biggr)\,.\nn
\end{equation}
By differentiating the TBA equation wrt. to both $\theta_{1}$ and
$\theta$ we obtain linear integral equations with solutions
\begin{align}
&\frac{\partial\epsilon(\theta)}{\partial\theta}=r\sinh\theta+2\pi iK(\theta-\frac{i\pi}{2}-\theta_{1})+\int dm(v)R_{\mathrm{dress}}(\theta-v)(r\sinh v+2\pi iK(v-\frac{i\pi}{2}-\theta_{1}))\,,\label{eq:depstheta}
\\&
\frac{\partial\epsilon(\theta)}{\partial\theta_{1}}=-2\pi iK(\theta-\frac{i\pi}{2}-\theta_{1})-2\pi i\int dm(v)R_{\mathrm{dress}}(\theta-v)K(v-\frac{i\pi}{2}-\theta_{1}))\,,\label{eq:depstheta1}
\end{align}
where the resolvent $R_{\mathrm{dress}}$ satisfies the equation
\begin{equation}
R_{\mathrm{dress}}(\theta)-\int dm(v)R_{\mathrm{dress}}(\theta-v)K(v)=K(\theta)\,.\nn
\end{equation}
Thus at the leading exponential order
\begin{equation}
\rho_{1}(\theta_{1})=r\Bigl(\cosh\theta_{1}-i\int dm(\theta)K(\theta_{1}+i\frac{\pi}{2}-\theta)\sinh\theta\Bigr)+O(e^{-2r})\,.\nn
\end{equation}
This allows us to expand the denominator and keep only the leading
exponential piece in order to compare with the formula coming from
the fermionic description (\ref{eq:e1Rem1}).
Evaluating the leading piece in the fermionic formula provides (\ref{eq:1pLO}).
To get the remaining terms we sum up the iterative terms. Keeping in mind
that $e_{1}$ represents the discrete and the continuous parts $(ie^{\theta_{1}},e^{v_{1}})$
the $k^{th}$ convolution gives
\begin{align}
e_{1}*\mathcal{K}_{\alpha}*\dots*\mathcal{K}_{\alpha}e_{-1} & =\frac{2\pi}{f'(\theta_{1})}\frac{(2\pi K_{\alpha}(0))^{k-2}}{f'(\theta_{1})^{k-2}}\Biggl(\frac{(2\pi K_{\alpha}(0))^{2}}{f'(\theta_{1})^{2}}+\frac{2\pi K_{\alpha}(0))}{f'(\theta_{1})}\times\\
& \qquad i\int dm(v_{1})(K_{\alpha}(\theta_{1}-v_{1}+\frac{i\pi}{2})e^{\theta_{1}-v_{1}}-K_{\alpha}(v_{1}-\theta_{1}-\frac{i\pi}{2})e^{v_{1}-\theta_{1}})\nonumber \\
& \qquad+(k-1)\frac{(2\pi)}{f'(\theta_{1})}\int dm(v_{1})K_{\alpha}(\theta_{1}-v_{1}+\frac{i\pi}{2})K_{\alpha}(v_{1}-\theta_{1}-\frac{i\pi}{2})\Biggr)\,,\nonumber
\end{align}
where we kept only terms with at most one continuous convolution.
We need to sum the first line from $k=0$, the second from $k=1$
, while the last from $k=2$ to infinity. Also there is one more convolution
from (\ref{eq:1pLO}). Let us recall that
\begin{equation}
f'(\theta_{1})=r\cosh\theta_{1}+2\pi K(0)-i\int dm(\theta)K(\theta_{1}+\frac{i\pi}{2}-\theta)\frac{\partial\epsilon(\theta)}{\partial\theta}
\,,\nn
\end{equation}
with the solution given by (\ref{eq:depstheta}). Expanding this formula
up to the leading exponential order and plugging back to the expressions
summed up agrees with (\ref{eq:1pNLO}). Let us emphasize that to obtain
the leading exponential contribution we need to sum up infinitely many terms in the discrete parts. Thus the agreement
found is a highly non-trivial test of our approach.
\section{Small volume checks}
For small volume we compare the ratios of the expectation values to
the ratios of three-point functions in the Liouville conformal field
theory in a cylindrical geometry shown on Figure \ref{cylinder}.
\begin{figure}
\begin{centering}
\includegraphics[width=3cm]{CFTcylinder.eps}
\par\end{centering}
\caption{Cylindrical geometry for the conformal three-point functions.}
\label{cylinder}
\end{figure}
The general three-point function in the CFT takes the form
\begin{equation}
\langle\Delta_{+}\vert{\cal O}_{\alpha}(0)\vert\Delta_{-}\rangle=\langle\Delta\vert L_{n_{1}}\dots L_{n_{i}}(\mathbf{l}_{-m_{1}}\dots\mathbf{l}_{-m_{j}}\Phi_{\al})L_{-p_{1}}\dots L_{-p_{r}}\vert\Delta\rangle\,,\label{eq:cft3pt}
\end{equation}
where two different Virasoro modes are introduced. Both are related
to the same energy momentum tensor, but expanded around different
points.
Let us introduce a complex coordinate on the cylinder as $z=x+iy$
with $y\equiv y+2\pi$. By expanding $T(z)$ around the origin we
can act and change the operator, which is inserted. This action is
called the \emph{local} action:
\begin{align}
T(z)=\sum_{n=-\infty}^{\infty}\mathbf{l}_{n}z^{-n-2} & \quad;\qquad\mathbf{l}_{n}\Phi_{\alpha}=\oint\frac{dz}{2\pi i}z^{n+1}T(z)\Phi_{\alpha}\,.\nn
\end{align}
For diagonal matrix elements, i.e. for expectation values, only even
mode numbers are used to generate the quotient space, where the action
of the conserved charges is factored out.
By expanding $T(z)$ at $z\to\pm\infty$ we obtain the \emph{global
}action of the Virasoro algebra which can alter the initial and final
states:
\begin{align}
T(z)=\sum_{n=-\infty}^{\infty}L_{n}e^{nz}-\frac{c}{24}\,.\nn
\end{align}
In order to relate the three-point function of the descendants (\ref{eq:cft3pt})
to that of the primary $\langle\Delta\vert \Phi_{\al}\vert\Delta\rangle\equiv\langle \Phi_{\al}\rangle_{\Delta}$
we use the cylinder conformal Ward identities:
\begin{align}
\langle T(z_{k})\cdots T(z_{1})\Phi_{\al}\rangle_{\Delta} & =-\frac{c}{12}\sum_{j=2}^{k}\chi'''(z_{1}-z_{j})\langle T(z_{k})\cdots\overset{j}{\widehat{\phantom{T}}}\cdots T(z_{2})\Phi_{\al}\rangle_{\Delta}\\
& \quad+\Bigl\{\sum_{j=2}^{k}\bigl(-2\chi'(z_{1}-z_{j})+(\chi(z_{1}-z_{j})-\chi(z_{1}))\frac{\partial}{\partial z_{j}}\bigr)\nonumber \\
& \quad\qquad\,\,\,\,\,\,\,\,\,-\Delta_{\al}\chi'(z_{1})+\Delta-\frac{c}{24}\Bigr\}\langle T(z_{k})\cdots T(z_{2})\Phi_{\al}\rangle_{\Delta}\nonumber
\end{align}
where $\chi(z)=\frac{1}{2}\coth\left(\frac{z}{2}\right)\,$.
In calculating (\ref{eq:cft3pt}) we follow the prescription of \cite{Boos:2010ww}: we first take $k=i+j+r$ and send
$z_{1},\dots,z_{i}$ to $-\infty$, $z_{i+1},\dots,z_{i+j}$ to $0$,
while $z_{i+j+1},\dots,z_{k}$ to $\infty$. By picking up the coefficient
of the appropriate power of $e^{\pm z}$ at $\mp\infty$ and $z$
around $0$ the three-point function (\ref{eq:cft3pt}) can be calculated.
In the following we first analyze non-degenerate $L_{0}$ subspaces,
i.e. highest weight states and their first descendants, and then level
$2$ states.
\subsection{Non-degenerate $L_{0}$ eigenspaces}
We perform this analysis for the low lying operators and states with
a non-degenerate $L_{0}$. This includes the state $\vert\Delta\rangle$
and $\vert\Delta+1\rangle\equiv L_{-1}\vert\Delta\rangle$, thus from
the table (\ref{tab}) we take all rows with $L=1,2,3,4$ and $M=0,1,\bar{M}=0$.
The computation using Ward identities provides
\begin{align}
\frac{\langle\mathbf{l}_{-2}\Phi_{\al}\rangle_{\Delta}}{\langle \Phi_{\al}\rangle_{\Delta}} & =\Delta-\frac{c}{24}-\frac{\Delta_{\al}}{12}\,,\label{different}\\
\frac{\langle\mathbf{l}_{-4}\Phi_{\al}\rangle_{\Delta}}{\langle \Phi_{\al}\rangle_{\Delta}} & =\frac{\Delta_{\al}}{240}\,,\nonumber \\
\frac{\langle\mathbf{l}_{-2}^{2}\Phi_{\al}\rangle_{\Delta}}{\langle \Phi_{\al}\rangle_{\Delta}} & =\Delta^{2}-\Delta\frac{2\Delta_{\al}+c+2}{12}+\frac{20\Delta_{\al}^{2}+56\Delta_{\al}+20c\Delta_{\al}+5c^{2}+22c}{2880}\,.\nonumber
\end{align}
\begin{align}
\frac{\langle \Phi_{\al}\rangle_{\Delta+1}}{\langle \Phi_{\al}\rangle_{\Delta}}\quad & =2\Delta+\Delta_{\al}^{2}-\Delta_{\al}\,,\label{eq:different2}\\
\frac{\langle\mathbf{l}_{-2}\Phi_{\al}\rangle_{\Delta+1}}{\langle \Phi_{\al}\rangle_{\Delta}} & =2\Delta^{2}+\Delta\frac{12\Delta_{\al}^{2}+34\Delta_{\al}+24-c}{12}-\frac{(\Delta_{\al}-1)\Delta_{\al}(2\Delta_{\al}-24+c)}{24}\,,\nonumber \\
\frac{\langle\mathbf{l}_{-4}\Phi_{\al}\rangle_{\Delta+1}}{\langle \Phi_{\al}\rangle_{\Delta}} & =\Delta\frac{241\Delta_{\al}}{120}-\frac{\Delta_{\al}^{2}}{240}+\frac{\Delta_{\al}^{3}}{240}\,,\nonumber \\
\frac{\langle\mathbf{l}_{-2}^{2}\Phi_{\al}\rangle_{\Delta+1}}{\langle \Phi_{\al}\rangle_{\Delta}} & =2\Delta^{3}+\Delta^{2}\frac{70-c+40\Delta_{\al}+6\Delta_{\al}^{2}}{6}\nonumber \\
& \,\,\,+\Delta\frac{2400-218c+5c^{2}+4616\Delta_{\al}-340c\Delta_{\al}+1940\Delta_{\al}^{2}-120c\Delta_{\al}^{2}-240\Delta_{\al}^{3}}{1440}\nonumber \\
& \,\,\,+\frac{(\Delta_{\al}-1)\Delta_{\al}(2400-218c+5c^{2}-424\Delta_{\al}+20c\Delta_{\al}+20\Delta_{\al}^{2})}{2880}\,\,.\nonumber
\end{align}
where the central charge and the scaling dimensions of the operator and of the asymptotical state
are
\begin{equation}
c=1+6Q^{2},\quad\Delta_{\al}=\frac{Q^2} 4\al(2-\al),\quad \Delta=\frac{P^2} 2+\frac{Q^2} 4\,.
\end{equation}
We also need the ratio of the three-point functions for the primary
fields:
\begin{align}
\frac{\langle \Phi_{\al-2p}\rangle_{\Delta}}{\langle \Phi_{\al}\rangle_{\Delta}}=\frac{\gamma^{2}(ab-b^{2})}{\gamma(2ab-2b^{2})\gamma(2ab-b^{2})}\gamma(ab-b^{2}-2ibP)\gamma(ab-b^{2}+2ibP)\,;\quad a=\frac{\al Q} 2\,.
\end{align}
where $\gamma(x)=\Gamma(x)/\Gamma(1-x)$, and we used notations close to \cite{Zamolodchikov:1995aa} in order to simplify comparison.
Using the results of \cite{HGSIV,HGSV} we can relate the fermionic
basis to the low lying Virasoro descendants as
\begin{align}
& \Omega_{1,1}\simeq r^{-2}D_{1}(\al,p)D_{1}(2-\al,p)\frac{\langle\mathbf{l}_{-2}\Phi_{\al}\rangle}{\langle \Phi_{\al}\rangle}\,,\label{ch11}\\
& \Omega_{3,1}\simeq r^{-4}\frac{1}{2}D_{3}(\al,p)D_{1}(2-\al,p)\Bigl\{\frac{\langle\mathbf{l}_{-2}^{2}\Phi_{\al}\rangle}{\langle \Phi_{\al}\rangle}+\Bigl(\frac{2c-32}{9}+\frac{2}{3}d(\al,p)\Bigr)\frac{\langle\mathbf{l}_{-4}\Phi_{\al}\rangle}{\langle \Phi_{\al}\rangle}\Bigr\}\,,\label{ch13}\\
& \Omega_{1,3}\simeq r^{-4}\frac{1}{2}D_{1}(\al,p)D_{3}(2-\al,p)\Bigl\{\frac{\langle\mathbf{l}_{-2}^{2}\Phi_{\al}\rangle}{\langle \Phi_{\al}\rangle}+\Bigl(\frac{2c-32}{9}-\frac{2}{3}d(\al,p)\Bigr)\frac{\langle\mathbf{l}_{-4}\Phi_{\al}\rangle}{\langle \Phi_{\al}\rangle}\Bigr\}\,,\label{ch31}\\
& \Omega_{1,-1}\simeq r^{2(\Delta_{\al}-\Delta_{\al-b})}t_{1}(\al)F(\al,p)\frac{\langle \Phi_{\al-2p}\rangle}{\langle \Phi_{\al}\rangle}\,,\label{chprim}
\end{align}
where
$$d(\al,p)=\frac{2p-1}{p(p-1)}(\al-1)\,,$$
and the expectation values are taken
in the finite volume eigenstate of the conserved charges. The appearing
coefficients for descendants originate from the normalization of the
fermionic operators
\begin{align}
D_{m}(\al,p)
=\frac{1}{2i\sqrt{\pi}}Z(p)^{-m}\Gamma\Bigl(\frac {\al+mp} 2\Bigr)\Gamma\Bigl(\frac {\al+m(1-p)} 2\Bigr)\,,
\end{align}
while for primaries they are essentially the ratio of two Lukyanov-Zamolodchikov
one-point functions
\begin{align}
F(\al,p)=Z(p)^{2(\Delta_{\al}-\Delta_{\al-2p})}\frac 2 {1-p}\ \gamma\Bigl(\frac{\al+1-p}{2}\Bigr)\gamma\Bigl(\frac{2-\al+p}{2}\Bigr)\gamma\Bigl(\frac{\al-p}{1-p}\Bigr)\,,
\end{align}
For asymptotical states we consider either primary fields parametrized
by the quantum number $L$ or their first descendants. For the primary
fields the formulae above are taken literally. For the descendants
we have to use the formulae (\ref{different}) carefully as, for instance,
\begin{align}
\frac{\langle\mathbf{l}_{-2}\Phi_{\al}\rangle_{\Delta+1}}{\langle \Phi_{\al}\rangle_{\Delta+1}} & =\frac{\langle\mathbf{l}_{-2}\Phi_{\al}\rangle_{\Delta+1}}{\langle \Phi_{\al}\rangle_{\Delta}}\frac{\langle \Phi_{\al}\rangle_{\Delta}}{\langle \Phi_{\al}\rangle_{\Delta+1}}\nonumber \\
& =\frac{48\Delta^{2}+2\Delta(12\Delta_{\al}^{2}+34\Delta_{\al}+24-c)-{(\Delta_{\al}-1)\Delta_{\al}(2\Delta_{\al}-24+c)}}{{24}(2\Delta+\Delta_{\al}^{2}-\Delta_{\al})}\,.
\end{align}
In the Table 1 we compare the numerical values of $\Omega_{i,j}$ obtained
for
\begin{align}
r=.001,\ \ a=\frac{87}{80},\ \ b=\frac{2}{5}\,,\label{don}
\end{align}
to their CFT limits.
\begin{table}[h]
\begin{centering}
\begin{tabular}{|c|c|c|c|c|}
\hline
state & \multicolumn{1}{c}{$M=0$} & $L=1$ & \multicolumn{1}{c}{$M=1$} & $L=1$\tabularnewline
\hline
& numerical & CFT & \multicolumn{1}{c}{numerical} & CFT\tabularnewline
\hline
\hline
$\Omega_{1,1}$ & $\ensuremath{3.85677\cdot10^{6}}$ & $3.85677\cdot10^{6}$ & $-6.60202\cdot10^{7}$ & $\ensuremath{-6.60203\cdot10^{7}}$\tabularnewline
\hline
$\Omega_{3,1}$ & $\ensuremath{1.00405\cdot10^{14}}$ & $1.00405\cdot10^{14}$ & $1.07476\cdot10^{16}$ & $1.07475\cdot10^{16}$\tabularnewline
\hline
$\Omega_{1,3}$ & $1.04361\cdot10^{14}$ & $1.04361\cdot10^{14}$ & $1.05988\cdot10^{16}$ & $1.05987\cdot10^{16}$\tabularnewline
\hline
$\Omega_{1,-1}$ & $-0.0028363$ & $-0.0028363$ & $-0.00231607$ & $-0.00231668$\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\vspace{.5cm}
\begin{centering}
\begin{tabular}{|c|c|c|c|c|}
\hline
state & \multicolumn{1}{c}{$M=0$} & $L=2$ & \multicolumn{1}{c}{$M=1$} & $L=2$\tabularnewline
\hline
& numerical & CFT & \multicolumn{1}{c}{numerical} & CFT\tabularnewline
\hline
\hline
$\Omega_{1,1}$ & $\ensuremath{3.79053\cdot10^{6}}$ & $3.79053\cdot10^{6}$ & $-6.61159\cdot10^{7}$ & $\ensuremath{-6.61132\cdot10^{7}}$\tabularnewline
\hline
$\Omega_{3,1}$ & $\ensuremath{9.93725\cdot10^{13}}$ & $9.93725\cdot10^{13}$ & $1.0771\cdot10^{16}$ & $1.07703\cdot10^{16}$\tabularnewline
\hline
$\Omega_{1,3}$ & $1.03188\cdot10^{14}$ & $1.03188\cdot10^{14}$ & $1.06245\cdot10^{16}$ & $1.06237\cdot10^{16}$\tabularnewline
\hline
$\Omega_{1,-1}$ & $-0.00276414$ & $-0.00276451$ & $-0.00225609$ & $-0.00225862$\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\vspace{.5cm}
\begin{centering}
\begin{tabular}{|c|c|c|c|c|}
\hline
state & \multicolumn{1}{c}{$M=0$} & $L=3$ & \multicolumn{1}{c}{$M=1$} & $L=3$\tabularnewline
\hline
& numerical & CFT & \multicolumn{1}{c}{numerical} & CFT\tabularnewline
\hline
\hline
$\Omega_{1,1}$ & $3.68203\cdot10^{6}$ & $3.68197\cdot10^{6}$ & $-6.62725\cdot10^{7}$ & $\ensuremath{-6.62653\cdot10^{7}}$\tabularnewline
\hline
$\Omega_{3,1}$ & $\ensuremath{9.77084\cdot10^{13}}$ & $9.77074\cdot10^{13}$ & $1.08094\cdot10^{16}$ & $1.08076\cdot10^{16}$\tabularnewline
\hline
$\Omega_{1,3}$ & $1.01297\cdot10^{14}$ & $1.01296\cdot10^{14}$ & $1.06668\cdot10^{16}$ & $1.06648\cdot10^{16}$\tabularnewline
\hline
$\Omega_{1,-1}$ & $-0.00265524$ & $-0.00265529$ & $-0.00216529$ & $-0.0021703$\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\vspace{.5cm}
\begin{centering}
\begin{tabular}{|c|c|c|c|c|}
\hline
state & \multicolumn{1}{c}{$M=0$} & $L=4$ & \multicolumn{1}{c}{$M=1$} & $L=4$\tabularnewline
\hline
& numerical & CFT & \multicolumn{1}{c}{numerical} & CFT\tabularnewline
\hline
\hline
$\Omega_{1,1}$ & $3.53306\cdot10^{6}$ & $3.53306\cdot10^{6}$ & $-6.64869\cdot10^{7}$ & $-6.64737\cdot10^{7}$\tabularnewline
\hline
$\Omega_{3,1}$ & $9.54791\cdot10^{13}$ & $9.5479\cdot10^{13}$ & $1.08622\cdot10^{16}$ & $1.08589\cdot10^{16}$\tabularnewline
\hline
$\Omega_{1,3}$ & $9.87646\cdot10^{13}$ & $9.87645\cdot10^{14}$ & $1.07248\cdot10^{16}$ & $1.07212\cdot10^{16}$\tabularnewline
\hline
$\Omega_{1,-1}$ & $-0.00251988$ & $-0.00252043$ & $-0.00205411$ & $-0.00206125$\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{We calculate numerically $\Omega_{i,j}$ for various states and compare
them to their exact conformal counterparts. }
\end{table}
\subsection{Checks with degenerate $L_{0}$ spaces}\label{degenerate}
Here we consider the simplest case of degeneracy: level 2. We work
in the basis:
\begin{equation}
L_{-2}|\Delta\rangle,\quad L_{-1}^{2}|\Delta\rangle\,.\label{eq:level2basis}
\end{equation}
There are two eigenvectors of the local integrals of motion. Since
$I_{1}=L_{0}-c/24$ does not distinguish between them we consider
the next conserved charge:
\begin{equation}
I_{3}=2\sum\limits _{n=1}^{\infty}L_{-n}L_{n}+L_{0}^{2}-\frac{c+2}{12}L_{0}+\frac{c(5c+22)}{2880}\,.
\end{equation}
This integral of motion is a $2\times2$ matrix in the basis above,
with eigenvalues
\begin{align}
\lambda_{\pm}(\Delta)=\frac{17}{3}+\frac{c(5c+982)}{2880}-\frac{c-142}{12}\Delta+\Delta^{2}\pm\frac{1}{2}\sqrt{288\Delta+(c-4)^{2}}\,,
\end{align}
and eigenvectors
\begin{equation}
\psi_{\pm}=\begin{pmatrix}\frac{1}{12}\ensuremath{c-4\pm\sqrt{288\Delta+(c-4)^{2}}}\\
1
\end{pmatrix}
\end{equation}
For simplicity we consider $L=1$. In table (\ref{tab}) we present
two cases with $L=1,M=2,\bar{M}=0$: $\{1,3\}$, $\{4\}$. We first
identify which one corresponds to $\lambda_{+}$ and which one to
$\lambda_{-}$. In doing so we recall the general eigenvalue of the
local integral of motion:
\begin{align}
I_{n}(r)=\frac{1}{C_{n}(p)}\Bigl(-\frac{1}{n}\sum_{j=1}^{m}e^{n\theta_{k}}+(-1)^{\frac{n-1}{2}}\frac{1}{2\pi}\int\limits _{-\infty}^{\infty}e^{n\theta}\log\Bigl(1+e^{-\epsilon(\theta)}\Bigr)d\theta\Bigr)\,,
\end{align}
where
\begin{equation}
C_{n}(p)=-\frac{Z(p)^{-n}}{4\sqrt{\pi}Q\ensuremath{\frac{n+1}{2}}!}\Gamma(np)\Gamma(n(1-p))\,.
\end{equation}
The normalized eigenvalue $\tilde{I}_{n}(r)=r^{n}I_{n}(R)$ should
approach the CFT limit. For $r=10^{-3}$ we obtained the following
numerical results:
\begin{align}
& \mathcal{N}_{-}=\{1,3\}\,,\quad\tilde{I}_{3}(r)=21.3773\,,\quad\lambda_{-}(\Delta(r))=21.3767\,,\\
& \mathcal{N}_{+}=\{4\}\,,\ \ \ \quad\tilde{I}_{3}(r)=74.8405\,,\quad\lambda_{+}(\Delta(r))=74.8399\,.\nonumber
\end{align}
which establishes the required correspondence.
For any local operator \emph{$\mathcal{O}$} we introduce a $2\times2$
matrix, which contains its matrix elements in the basis (\ref{eq:level2basis}).
We denote this matrix by $\frac{\langle\mathbf{l}_{-2}O\rangle_{\Delta+2}}{\langle \Phi_{\al}\rangle_{\Delta}}$.
We need the following two cases
\begin{align}
\frac{\langle \Phi_{\al}\rangle_{\Delta+2}}{\langle \Phi_{\al}\rangle_{\Delta}} & =\begin{pmatrix}4\Delta-4\Delta_{\al}+4\Delta_{\al}^{2}+\frac{c}{2} & 2(3\Delta-\Delta_{\al}+\Delta_{\al}^{3})\\
2(3\Delta-\Delta_{\al}+\Delta_{\al}^{3}) & 8\Delta^{2}+\Delta(4-8\Delta_{\al}+8\Delta_{\al}^{2})-2\Delta_{\al}+3\Delta_{\al}^{2}-2\Delta_{\al}^{3}+\Delta_{\al}^{4}
\end{pmatrix}\,.
\end{align}
and
\begin{equation}
\frac{\langle\mathbf{l}_{-2}\Phi_{\al}\rangle_{\Delta+2}}{\langle \Phi_{\al}\rangle_{\Delta}}=\begin{pmatrix}M_{1,1} & M_{1,2}\\
M_{1,2} & M_{2,2}
\end{pmatrix}\nn\,,
\end{equation}
with entries
\begin{align}
M_{1,1} & =\frac{1}{48}\bigl(48c-c^{2}-672\Delta_{\al}+102c\Delta_{\al}+976\Delta_{\al}^{2}-8c\Delta_{\al}^{2}-16\Delta_{\al}^{3}\nonumber \\
& \quad\quad+\Delta(384+16c+560\Delta_{\al}+192\Delta_{\al}^{2})+192\Delta^{2}\bigr)\,,\nonumber \\
M_{1,2} & =\frac{1}{12}\bigl(-72\Delta_{\al}+7c\Delta_{\al}+14\Delta_{\al}^{2}+6c\Delta_{\al}^{2}+84\Delta_{\al}^{3}-c\Delta_{\al}^{3}-2\Delta_{\al}^{4}\nonumber \\
& \quad\quad+\Delta(144-3c+258\Delta_{\al}+144\Delta_{\al}^{2}+24\Delta_{\al}^{3})+72\Delta^{2}\bigr)\,,\nonumber \\
M_{2,2} & =\frac{1}{24}\bigl(-96\Delta_{\al}+2c\Delta_{\al}+52\Delta_{\al}^{2}-3c\Delta_{\al}^{2}-6\Delta_{\al}^{3}+2c\Delta_{\al}^{3}+52\Delta_{\al}^{4}-c\Delta_{\al}^{4}-2\Delta_{\al}^{5}\nonumber \\
& \quad\quad+\Delta(192-4c+232\Delta_{\al}+8c\Delta_{\al}+568\Delta_{\al}^{2}-8c\Delta_{\al}^{2}+128\Delta_{\al}^{3}+24\Delta_{\al}^{4})\nonumber \\
& \quad\quad+\Delta^{2}(480-8c+560\Delta_{\al}+192\Delta_{\al}^{2})+192\Delta^{3}\bigr)\,.\nn
\end{align}
We now rewrite the general formulae (\ref{ch11}), (\ref{chprim})
for the present case
\begin{align}
& \Omega_{1,1}^{\pm}\simeq r^{-2}D_{1}(\al,p)D_{1}(2-\al,p)\frac{\psi_{\pm}^{t}\cdot\langle\mathbf{l}_{-2}\Phi_{\al}\rangle_{\Delta+2}\cdot\psi_{\pm}}{\psi_{\pm}^{t}\cdot\langle \Phi_{\al}\rangle_{\Delta+2}\cdot\psi_{\pm}}\,,\nonumber \\
& \Omega_{1,-1}^{\pm}\simeq r^{2(\Delta_{\al}-\Delta_{\al-2p})}t_{1}(a,b)F(\al,p)\frac{\psi_{\pm}^{t}\cdot\langle \Phi_{\al-2p}\rangle_{\Delta+2}\cdot\psi_{\pm}}{\psi_{\pm}^{t}\cdot\langle \Phi_{\al}\rangle_{\Delta+2}\cdot\psi_{\pm}}\,.\nn
\end{align}
We compute these quantities at the numerical values (\ref{don}).
The results are summarized in the table
\begin{center}
\begin{tabular}{|c||c|c||c|c|}
\hline
eigenvalue & $\Omega^{-}$ & CFT & $\Omega^{+}$ & CFT\tabularnewline
\hline
\hline
$\Omega_{1,1}$ & $-1.08278\cdot10^{8}$ & $-1.08276\cdot10^{8}$ & $\ensuremath{-1.88722\cdot10^{8}}$ & $-1.88716\cdot10^{8}$\tabularnewline
\hline
$\Omega_{1,-1}$ & $\ensuremath{-0.00210992}$ & $-0.00211103$ & $-0.00245252$ & $-0.00245289$\tabularnewline
\hline
\end{tabular}
\par\end{center}
Thus we see that our procedure works well in the case with degenerate
$L_{0}$, too. This completes the small volume check of our conjecture.
\section{Conclusions}
We conjectured compact expressions for the finite volume diagonal
matrix elements of exponential operators and their descendants in
the sinh-Gordon theory. By using the fermionic basis to create the
descendant operators we could relate their finite volume expectation
values to that of the primaries in terms of a determinant with entries,
which satisfies a linear integral equation. Careful choice of the
fermionic creation operators can relate the matrix elements of two
different exponential operators allowing, in principle, they complete
determination. The linear integral equation contains a measure, which
is built up from the pseudo-energy of the excited state TBA equations
and a kernel, which is a deformation of the TBA kernel. Excited states
are characterized by the discrete rapidities of the particles and
the continuous pseudo-energy and the two parts are connected by the
TBA and BA equations. They both appear in the linear integral equations,
which can be solved by iterations. The discrete part is responsible
for the polynomial finite size corrections, while the continuous part
for the exponentially small ones. We checked for low number of particles
that summing up all the polynomial corrections the asymptotic diagonal
finite volume form factors can be recovered. We also checked the leading
exponential correction against Pozsgay's formula and found complete
agreement. The integral equation can also be solved numerically. The
small volume limit of the solution allows us to map multi-particle
states to the spectrum of the Liouville conformal field theory and
compare our conjecture to the CFT three-point functions providing
ample evidence for its correctness.
In calculating the asymptotic expressions for the finite volume form
factors we used a deformation of the TBA kernel, which is the logarithmic
derivative of the scattering matrix. We believe that this alternative
form for the connected and symmetric form factors can be used to find
a compact and closed expression for them and we initiate a study into
this direction.
It would be very nice to extend our exact finite volume results for
non-diagonal form factors. These results then could be tested for large
volumes against the leading exponential correction of form factors
\cite{Bajnok2019prep}.
Finally, the knowledge of all form factors could give rise
to the determination of finite volume correlation functions relevant both in statistical
and solid state physics.
It is an interesting question whether the very nice structure we obtained
for the sinh-Gordon model extends to other integrable models such
as O(N) models or the AdS/CFT correspondence.
\subsection*{Acknowledgments}
This research was supported by the NKFIH research Grant K116505, by the
Lend\"ulet Program of the Hungarian Academy of Sciences and by a Hungarian-French
bilateral exchange project. FS is grateful to Wigner Research Centre
for Physics where this work was started for kind hospitality.
|
1,108,101,566,789 | arxiv | \section{Introduction}
The O(3) nonlinear sigma model has long been the subject of intense research due to its theoretical and phenomenological basis. This theory describes classical (anti) ferromagnetic spin systems at their critical points in Euclidean space, while in
the Minkowski one it delineates the long wavelength limit of quantum antiferromagnets. The model exhibits solitons,
Hopf instantons and novel spin and statistics in 2+1 space-time dimensions with inclusion of the Chern-Simons term.
The
soliton solutions of the model exhibit scale invariance which poses difficulty in the particle interpretation on quantization. A popular means of breaking this scale invariance is to gauge a U(1) subgroup of the O(3) symmetry of the model by coupling the
sigma model fields with a gauge field through the corresponding U(1)
current. {\footnote{This is different from the minimal coupling via the topological
current discussed previously \cite{wil}.}} This class of gauged O(3) sigma models in three dimensions have
been studied over a long time\cite{{sch},{gho},{lee},{muk1},{muk2},{mend}, {land}}. Initially
the gauge field dynamics was assumed to be dictated by the Maxwell term
\cite{sch}. Later the extension of the model with
the Chern - Simons coupling was investigated \cite{gho}. A particular form of
self - interaction was required to be included in these models in order to
saturate
the Bogomol'nyi bounds \cite{bog}. The form of the assumed self - interaction
potential is of crucial importance. The minima of the
potential determine the vaccum structure of the theory. The solutions change remarkably
when the vaccum structure exhibits spontaneous breaking of the symmetry
of the gauge group. Thus it was demonstrated that the observed degeneracy
of the solutions of \cite{sch,gho} is lifted when potentials with
symmetry breaking minima were incorporated \cite{muk1,muk2}. The studies of the gauged O(3) sigma model is important due to their intrinsic interest and also due to the fact that the soliton solutions of the gauged
O(3) Chern-Simons model may be relevant in planar condensed matter systems \cite{pani,pani1,han}. Recently gauged
nonlinear sigma model was considered in order to obtain self-dual cosmic string solutions \cite{verb, ham}. This explains the continuing interest in such models in the literature \cite{{sch},{gho},{lee},{muk1},{muk2},{mend}, {land}}.
A particular aspect of the gauged O(3) sigma models where the gauge field dynamics is governed by the Maxwell term can be identified by comparing the results of \cite{sch} and \cite{muk2}. In \cite{sch} the vaccum is symmetric and the $n = 1$ soliton solution does not exist,
$n$ being the topological charge. Here, solutions exists for $n = 2$ onwards. Moreover, these soliton solutions have arbitrary magnetic flux. When we achieve symmetrybreaking vaccum by chosing the potential appropriately \cite{muk2} soliton solutions are obtained for $n = 1$. {\footnote{The disappearence of the $n = 1$ soliton has been shown to follow from general analytical method in \cite{sch1}.}}These solutions have quntized magnetic flux and qualify as magnetic vortices. It will be interesting to follow the solutions from the symmetrybreaking to the symmetric phase. This is the motivation of the present paper.
We will consider a generalisation of the models of \cite{sch} and \cite{muk2}
with an adjustable real parameter $v$ in the
expression of the self - interaction potential which interpolates between the symmetric and the symmetrybreaking vaccua. This will in particular
allow us to investigate the soliton solutions in the entire regime of the symmetrybreaking vacuum structures and also to follow the collapse of the $n = 1$ soliton as we move from the assymmetric
to the symmetric phase. The solitons of the model are obtained as the solutions of the self -- dual equation obtained by saturating the Bogomoln'yi bounds. Unfortunately, these equations fall outside the Liouville class even after assuming a rotationally symmetric ansatz. Thus
exact analytical solutions are not obtainable and numerical methods are to be invoked.
The organisation of the paper is as follows. In the following section we present a brief review of the O(3) nonlinear sigma model. This will be helpful in presenting our work in the proper context. In section 3 our model is introduced. General topological classifications of the soliton solutions of the model has been discussed here. In section 4 the saturation of the self-dual limits has been examined and the Bogomol'nyi equations have been written down. Also the analytical form of the Bogomol'nyi equations has been worked out assuming a rotationally symmetyric ansatz. These equuations, even in the rotationally symmetric scenario, are not exactly integrable. A numerical solution has been performed to understand the details of the solution. A fourth order Runge -- Kutta algorithm is adopted with provision of tuning the potential appropriately.
In section 5 the numerical method and some results are presented. We conclude in section 6.
\section { O(3) nonlinear sigma models }
It will be useful to start with a brief review of the
nonlinear O(3) sigma model \cite{bel}. The lagrangian of the model
is given by,
\begin {equation}
{\cal L} =\frac {1} {2}\partial_\mu {\mbox{\boldmath$\phi$}}\cdot
\partial^\mu
{\mbox{\boldmath$\phi$}}\label{LO3}
\end {equation}
Here ${\mbox{\boldmath$\phi$}}$ is a triplet of scalar fields
constituting a vector in the internal space with unit norm
\begin {eqnarray}
\phi_a = {\bf n_a}\cdot {\mbox{\boldmath$\phi$}}, ~(a=1,2,3)\\
{\mbox{\boldmath$\phi$}}\cdot{\mbox{\boldmath$\phi$}} =\phi_a\phi_a= 1
\label{CONST}
\end {eqnarray}
The vectors ${\bf n}_a$
constitute a basis of unit orthogonal vectors in the internal space.
We work in the Minkowskian space - time with the metric tensor
diagonal, $g_{\mu\nu} = (1,-1,-1)$.
The finite energy solutions of the model (\ref{LO3})
satisfies the boundary condition
\begin{equation}
{\rm lim} \phi^a\hspace{.2cm}=\hspace{.2cm}\phi^a_{(0)}\label{BN}
\end{equation}
at physiacal infinity.
The condition (\ref{BN}) corresponds to one point compactification of the
physical infinity.
The physical space $R_2$ becomes topologically equivalent to $S_2$
due to this compactification. The static finite energy solutions of the
model are then maps from this sphere to the internal sphere. Such
solutions are
classified by the homotopy \cite{dho}
\begin{equation}
\Pi_2(S_2) = Z\label{HOMO}
\end{equation}
We can construct a current
\begin{equation}
K_\mu = {\frac{1}{8\pi}}\epsilon_{\mu\nu\lambda}\bf{\phi}\cdot(
\partial^\nu\bf{\phi}\times\partial^\lambda\bf{\phi})\label{TCUR}
\end{equation}
which is conserved irrespective of the equation of motion. The corresponding
charge
\begin{eqnarray}
T &=& \int d^2{\bf x} K_0\nonumber\\
&=& {\frac{1}{8\pi}}\int d^2{\bf x} \epsilon_{ij}
\bf{\phi}\cdot (\partial^i\bf{\phi}\times\partial^j\bf{\phi})\label{TCHAR}
\end{eqnarray}
gives the winding number of the mapping (\ref{HOMO}) \cite{raj}.
\section{Our model - topological classification of the soliton solutions}
In the class of gauged models of our interest here
a U(1) subgroup of the rotation symmetry
of the model (\ref{LO3}) is gauged. We chose this to be
the SO(2) [U(1)] subgroup of rotations
about the 3 - axis in the internal space. The Lagrangian of our model is given
by
\begin {equation}
{\cal L} ={\frac{1}{2}}D_\mu {\mbox{\boldmath$\phi$}}\cdot D^\mu
{\mbox{\boldmath$\phi$}}
-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+ U({\mbox{\boldmath$\phi$}})\label{LGO3}
\end {equation}
$D_\mu {\mbox{\boldmath$\phi$}}$ is the covariant derivative given by
\begin {equation}
D_\mu {\mbox{\boldmath$\phi$}}=\partial_\mu {\mbox{\boldmath$\phi$}}
+ A_\mu {\bf n}_3\times {\mbox{\boldmath$\phi$}}
\end {equation}
The SO(2)
(U(1)) subgroup is gauged by the vector potential $A_\mu$ whose dynamics is
dictated
by the Maxwell term. Here $F_{\mu\nu}$ are the electromagnetic field tensor,
\begin {equation}
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu
\end {equation}
$ U(\mbox{\boldmath$\phi$})$ is the self - interaction potential
required for saturating the self - dual limits. We chose
\begin {equation}
U({\mbox{\boldmath$\phi$}})=-{\frac{1}{2}}(v - \phi_3)^2\label{POT}
\end {equation}
where $v$ is a real parameter. Substituting $v = 0$ we get back the model
of \cite{muk2} whereas $v = 1$ gives the model of \cite{sch}.
We observe that
the minima
of the potential arise when ,
\begin {equation}
\phi_3 = v\label{N3B}
\end {equation}
which is equivalent to the condition
\begin {equation}
\phi_1^2+\phi_2^2 =1-v^2\label{NB}
\end {equation}
on account of the constraint (\ref{CONST}).
The values of v must be restricted to
\begin{equation}
{\left |v\right |}\le 1\label{VB}
\end{equation}
The condition (\ref{NB}) denotes a latitudinal circle
(i.e. circle with fixed latitude) on the unit sphere
in the internal space.
By varying v from -1 to +1 we span the sphere from the south pole to
the north pole. It is clear that the finite energy solutions of the model
must satisfy (\ref{NB}) at physical infinity. For $v\ne 1$ this boundary
condition corresponds to the spontaneous breaking of the symmetry of the
gauge group and in the limit
\begin {equation}
{\left |v\right |} \to 1\label{VLIM}
\end{equation}
the asymmetric phase changes to the symmetric phase.
We call the potential (\ref{POT}) interpolating in this sense.
In the asymmetric phase the soliton solutions are classified according
to the homotopy
\begin{equation}
\Pi_1(S_1) = Z \label{HOM1}
\end {equation}
instead of (\ref{HOMO}). In the symmetric phase, however, this new topology
disappears and the solitons are classified according to (\ref{HOMO})
as in the usual sigma model (\ref{LO3}).
A remarkable fallout of this change of topology is the disappearence of the soliton with unit charge. The fundamental solitonic
mode $n = 1$ ($n$ being the vorticity) ceases to exist in the symmetric phase.
The modes corresponding to n = 2 onwards still persist but the magnetic flux
associated with them ceases to remain quantised.
In the asymmetric phase the vorticity is the
winding number i.e. the number of times by which
the infinite physical circle winds over the latitudinal circle (\ref{NB}).
Associated with this is a uniqe mapping of the internal sphere where the
degree of mapping is usually fractional. By inspection we
construct a current
\begin {equation}
K_\mu = {\frac{1} {8\pi}}\epsilon_{\mu\nu\lambda}[
{\mbox{\boldmath$\phi$}}\cdot D^\nu {\mbox{\boldmath$\phi$}}
\times D^\lambda {\mbox{\boldmath$\phi$}} - F^{\nu\lambda}(v - \phi_3)]\label{TCUR1}
\end {equation}
generalising the topological current (\ref{TCUR}). The current (\ref{TCUR1})
is manifestly gauge invariant and differs from (\ref{TCUR}) by the
curl of a vector field. The conservation principle
\begin {equation}
\partial_\mu K^\mu = 0
\end {equation}
thus automatically follows from the conservation of (\ref{TCUR}).
The corresponding conserved charge is
\begin {equation}
T = \int d^2x K_0\label{TCH}
\end {equation}
Using (\ref{TCUR1}) and (\ref{TCH}) we can write
\begin {eqnarray}
T &=&\int d^2x[{\frac{1}{8\pi}}\epsilon_{ij}{\mbox{\boldmath$\phi$}}
\cdot(\partial^i{\mbox{\boldmath$\phi$}}
\times \partial^j {\mbox{\boldmath$\phi$}})]\nonumber\\
&-&{ \frac{1}{4\pi}}\int_{boundary}(v -\phi_3) A_\theta r d\theta\label{tnw}
\end {eqnarray}
where r,$\theta$ are polar coordinates in the physical space and $A_\theta
= {\bf e}_\theta \cdot {\bf A}$.
Using the boundary condition (\ref{N3B})
we find that T is equal to the degree of the
mapping
of the internal sphere. Note that this situation is different from \cite{lee}
where the topological charge usually differs from the degree of the mapping.
In this context
it is interesting to observe that
the current (\ref{TCUR1}) is not unique because
we can always add an arbitrary multiple of
\begin{equation}
{\frac{1}{8\pi}}\epsilon_{\mu\nu\lambda}F^{\nu\lambda}\nonumber
\end{equation}
with it without affecting its conservation. We chose (\ref{TCUR1})
because it generates proper topological charge.
\section{Self dual equations in the rotationally symmetric ansatz}
In the previous section we have discussed the general topological classification of the solutions of the equations of motion following from
(\ref{LGO3}). In the present section we will discuss the solution of the equations of motion.
The Euler - Lagrange equations of the system (\ref{LGO3}) is derived subject to the
constraint (\ref{CONST}) by the Lagrange multiplier technique
\begin {eqnarray}
D_\nu (D^\nu {\mbox{\boldmath$\phi$}})& =& [D_\nu (D^\nu
{\mbox{\boldmath$\phi$}})
\cdot {\mbox{\boldmath$\phi$}}] {\mbox{\boldmath$\phi$}} +{\bf n}_3(v - \phi_3)\nonumber\\
&+& (v - \phi_3)\phi_3{\mbox{\boldmath$\phi$}}\label{elphi}\\
\partial_\nu F^{\nu\mu} =j^\mu\label{ela}
\end {eqnarray}
where
\begin {equation}
j^\mu = -{\bf n}_3\cdot{\bf J}^\mu\hspace{.2cm} and\hspace{.2cm}
{\bf J}^\mu ={\mbox{\boldmath$\phi$}}\times D^\mu {\mbox{\boldmath$\phi$}}\label{jmu}
\end {equation}
Using (\ref{elphi}) we get
\begin {equation}
D_\mu {\bf J}^\mu = -(v - \phi_3)
({\bf n}_3 \times {\mbox{\boldmath$\phi$}})\phi_3
\end {equation}
From (\ref{ela}) we find,for static configurations
\begin {equation}
\nabla^2 A^0 = -A^0(1 - \phi_3^2)
\end {equation}
From the last equation it is evident that we can chose
\begin {equation}
A^0 = 0
\end {equation}
As a consequence we find that the excitations of the model are electrically
neutral.
The equations (\ref{elphi}) and (\ref{ela}) are second order differential equations in time. As is well known, first order equations which are the solutions of the equations of motion can be derived by minimizing the energy functional in the static limit. Keeping this goal in mind
we now construct the
energy functional from the symmetric energy - momentum tensor following from (\ref{LGO3}).
The energy
\begin{equation}
E = {\frac{1}{2}}\int d^2{\bf x}\left [ D_0\mbox{\boldmath$\phi$}\cdot D_0\mbox{\boldmath$\phi$}
- D_i\mbox{\boldmath$\phi$}\cdot D^i\mbox{\boldmath$\phi$} + (v - \phi_3)^2
- 2(F_0^\sigma F_{0\sigma} - {\frac{1}{4}}F_{\rho\sigma}F^{\rho\sigma})
\right ].\label{E}
\end{equation}
For static configuration and the choice $A^0$ = 0, $E$ becomes
\begin {equation}
E = {\frac{1}{2}}\int d^2x[(D_i{\mbox{\boldmath$\phi$}})
\cdot(D_i{\mbox{\boldmath$\phi$}})+ F_{12}^2+(v -\phi_3)^2]\label{EST}
\end {equation}
Several observations about the finite energy solutions can be made at
this stage from (\ref{EST}). By defining
\begin{equation}
\psi = \phi_1 + i\phi_2
\end{equation}
we get
\begin {equation}
D_i {\mbox{\boldmath$\phi$}}\cdot D_i{\mbox{\boldmath$\phi$}}
= |(\partial_i + iA_i)\psi|^2 +(\partial_i\phi_3)^2
\end {equation}
The boundary condition (\ref{NB}) dictates that
\begin {equation}
\psi \approx (1 - v^2)^{{\frac{1}{2}}}e^{in\theta}
\end {equation}
at infinity.
From (\ref{EST}) we observe that for finite energy configurations we
require
\begin {equation}
{\bf{A}}={\bf{e_\theta}}{\frac{n}{r}}\label{AB}
\end {equation}
on the boundary. This scenario is exactly identical with the
observations of \cite{muk2} and leads to the quantisation of
the magnetic flux
\begin {equation}
\Phi = \int B d^2x = \int_{boundary}A_\theta r d\theta = 2\pi n
\end {equation}
The basic mechanism leading to this quantisation remains operative
so far as $v$ is less than 1. At $v = 1$, however, the gauge field ${\bf A}$
becomes arbitrary on the boundary except for the requirement that
the magnetic field B should vanish on the boundary. Remember
that not all the vortices present in the broaken phase survives this demand.
Specifically, the $n = 1$ vortex becomes inadmissible.
Now the search for the self - dual conditions proceed in the usual way.
We rearrange the energy functional as
\begin {equation}
E = {\frac{1}{2}} \int d^{2}x[{\frac{1}{2}}(D_i{\mbox{\boldmath$\phi$}} \pm
\epsilon_{ij}{\mbox{\boldmath$\phi$}}\times D_{j}{\mbox{\boldmath$\phi$}})^2
+ (F^{12} \mp (v -\phi_3))^2] \pm 4\pi T\label{EBOG}
\end {equation}
Equation (\ref{EBOG}) gives the Bogomol'nyi conditions
\begin {eqnarray}
D_i{\mbox{\boldmath$\phi$}}\pm \epsilon_{ij} {\bf\phi}
\times D_j{\mbox{\boldmath$\phi$}} = 0\label{SDPHI}\\
F_{12}\mp (v -\phi_3) =0\label{SDA}
\end {eqnarray}
which minimize the energy functional in a particular topological sector,
the upper sign corresponds to +ve and the lower sign corresponds to -ve
value of the topological charge.
We will now
turn towards the analysis of the self - dual equations using the
rotationally symmetric ansatz \cite{wu}
\begin {eqnarray}
\phi_1(r,\theta) = \sin g(r) \cos n\theta\nonumber\\
\phi_2(r,\theta) = \sin g(r) \sin n\theta\nonumber\\
\phi_3(r,\theta) = \cos g(r)\nonumber\\
{\bf A}(r,\theta)= -{\bf e}_\theta {\frac{na(r)}{r}}\label{ANS}
\end {eqnarray}
From (\ref{N3B}) we observe that we require the boundary condition
\begin {equation}
g(r) \to \cos^{-1}v \hspace{.2cm}
{\rm as}\hspace{.2cm} r \to \infty\label{GB}
\end {equation}
and equation (\ref{AB}) dictates that
\begin {equation}
a(r) \to -1 \hspace{.2cm}{\rm as}\hspace{.2cm} r \to \infty\label{aB}
\end {equation}
Remember that equation (\ref{AB}) was obtained so as the solutions have finite
energy.
Again for the fields to be well defined at the origin we require
\begin {equation}
g(r) \to 0 \hspace{.2cm}{\rm or}\hspace{.2cm}
\pi \hspace{.2cm}{\rm and} \hspace{.2cm}
a(r) \to 0\hspace{.2cm} {\rm as} \hspace{.2cm}r \to 0\label{ag0}
\end {equation}
Substituting the Ansatz(\ref{ANS}) into (\ref{SDPHI}) and (\ref{SDA})
we find that
\begin {eqnarray}
g^\prime (r) = \pm {\frac{n(a+1)}{r}} \sin g,\label{eqg}\\
a^\prime (r) = \pm {\frac{r}{n}}(v- \cos g)\label{eqa}
\end {eqnarray}
where the upper sign holds for +ve T and the lower sign corresponds to
-ve T.Equations (\ref{eqg}) and (\ref{eqa}) are not exactly integrable.
In the following section we will discuss the numerical solution
of the boundary value problem defined by (\ref{eqg})
and (\ref{eqa}) with (\ref{GB}) to (\ref{ag0}).
Using the Ansatz (\ref{ANS}) we can explicitly compute the topological charge T
by performing the integration in (\ref{TCH}).The result is
\begin {equation}
T = -{\frac{n}{2}}[\cos g(\infty)-\cos g(0)]-{\frac{1}{2}}[v - \cos g(\infty)]\label{t}
\end {equation}
The second term of (\ref{t}) vanishes due to the boundary condition (\ref{GB}).
Also, when g(0) = 0,
\begin{equation}
T = {\frac{n}{2}}(1 - v)\label{A}
\end{equation}
and, when g(0) = $\pi$,
\begin{equation}
T = -{\frac{n}{2}}(1 + v)\label{B}
\end{equation}
It is evident that T is in general fractional. Due to (\ref{tnw}) it is
equal to the degree of mapping of the internal sphere. This can also be checked
explicitly.
From the above analysis we find that g(0)
= 0
corresponds to +ve T
and g(0) = $\pi$ corresponds to -ve T.
We shall restrict our attention on negetive T
which will be useful for comparision of results with those available in the literature. The boundary value problem of interest is then
\begin {eqnarray}
g^\prime (r) = - {\frac{n(a+1)}{r}} \sin g\label{eqg1}\\
a^\prime (r) = - {\frac{r}{n}}(v- \cos g)\label{eqa1}
\end {eqnarray}
with
\begin{eqnarray}
g(0) = \pi,a(0) = 0\nonumber\\
g(\infty) = \cos^{-1}v, a(\infty) = 0\label{BOUN}
\end{eqnarray}
In addition we require $a^\prime (r)$ $\to$ 0 as $r\to \infty$. This condition follows from
(\ref{eqg1}), (\ref{eqa1}) and (\ref{BOUN}) and should be considered as a consistency condition to be satisfied by their soloutions.
\section{Numerical solution}
The simultaneous equations (\ref{eqg1}) and (\ref{eqa1}) subject to the boundary conditions (\ref{BOUN}) are not amenable to exact solution.
They can however be integrated numerically. We have already mentioned the quenching of the $n = 1$ solution in the limit $v\to 1$.
This is connected with the transition from the symmetry breaking to the symmetric phase. The numerical solution is thus interesting because it will enable us to see how the solutions change as we follow them from the deep assymetric phase $( v = 0)$ to the symmetric phase $(v = 1)$. In the following we provide the results of numerical solution to highlight these issues.
Let us note some details of the numerical method. A fourth order Runge -- Kutta method was employed. The point $r=0$ is a regular singular point of the equation. So it was not possible to start the code from $r=0$. Instead, we start it from a small value of $r$. The behaviour of the functions near r = 0 can be easily derived from (\ref{eqg1}) and(\ref{eqa1})
\begin{equation}
g(r)\approx \pi + Ar^n
\label{bc1}
\end{equation}
\begin{equation}
a(r) \approx -{\frac{r^2}{2n}}(1+v)
\label{bc2}
\end{equation}
Here $A$ is an arbitrary constant which fixes the values of g and a at
infinity. In the symmetrybreaking phase the numerical solution depends sensitively on the value of $A$. {\footnote{This should be contrasted with the symmetric vacuum solution where $A$ may have arbitrary values.}}There is a critical value of $A$, $A = A^{crit}$ for which the boundary
conditions are satisfied.
If the value of $A$ is larger than $A^{crit}$
the conditions at infinity are overshooted,
whereas, if the value is smaller than the critical value
g(r) vanishes asymptotically
after reaching a maximum. The situation is comparable with similar
findings elsewhere \cite{jac}.
The values of $g$ and $a$ were calulated at a small value of $r$ using (\ref{bc1}) and (\ref{bc2}). The parameter $A$ was tuned to match boundary conditions at the other end. Interestingly, this matching is not obtainable when $n=1$ and $v =1$. This is consistent with the quenching of the $n = 1$ mode in the symmetric vacuum situation.
After the brief discussion of the numerical technique we will present a summary of the results. As may be recalled, the purpose of the paper is to study the solutions throughout the asymmetric phase with an eye to the disappearence of the $n=1$ mode. Accordingly
profiles of $g$ and $a$ will be given for $n = 1$, for different values of $v$. In figures 1 and 2 these profiles are shown for $v = 0,.2,.4,.6,.8$. The corresponding magnetic field distributions are given in figure 3. Another interesting issue is the change of the matter and the gauge profiles with the topological charge. In figures 4 and 5 this is demonstrated for different $n$ values for a constant $v$.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=5.5cm,angle=270]{fig1.eps}
\caption{\label{fg01} {\it The matter profile.} The function $g(r)$ is plotted against $r$ for $n = 1$ and different $v$ values, indicated on the right hand top corner.
}
\end{center}
\end{figure}
\begin{figure}[ht1!]
\begin{center}
\includegraphics[width=5.5cm,angle=270]{fig2.eps}
\caption{\label{fg02} {\it The gauge field profile.} The function $a(r)$ is plotted against $r$ for $n = 1$ and different $v$ values.
}
\end{center}
\end{figure}
\begin{figure}[ht2!]
\begin{center}
\includegraphics[width=5.5cm,angle=270]{fig3.eps}
\caption{\label{fg03} {\it The magnetic field profiles.} Plot of the function $B(r)$ against $r$.
}
\end{center}
\end{figure}
\begin{figure}[ht4!]
\begin{center}
\includegraphics[width=5.5cm,angle=270]{fig5.eps}
\caption{\label{fg00} {\it The matter profiles for different n (v = .4).}
}
\end{center}
\end{figure}
\begin{figure}[ht5!]
\begin{center}
\includegraphics[width=5.5cm,angle=270]{fig6.eps}
\caption{\label{fg00} {\it The gauge field profiles for different n (v =.4).}
}
\end{center}
\end{figure}
\newpage
\section{Conclusion} The O(3) sigma model in (2+1) dimensional space -- time with its U(1) subgroup gauged was mooted \cite{sch} as a possible mechanism to break the scale -- invariance of the soutions of the original 3 - dimensional O(3) sigma model. The model finds possible applications in such diverse areas such as planar condensed matter physics \cite{pani,pani1,han}, gravitating cosmic strings \cite{verb, han} and as such is being continuously explored in the literature \cite{sch,gho,sch1,lee,muk1,muk2,mend,land}. An interesting aspect of the gauged O(3) sigma models is the qualitative change of the soliton modes in the symmetric and symmetrybreaking vacuum scenario, as can be appreciated by a comparision of solutions given in \cite{sch, muk2}. In this paper
we have considered a gauged O(3) sigma model with the gauge field
dynamics determined by the Maxwell term as in \cite{sch, muk2}. An interpolating potential was included to invesigate the solutions in the entire symmetrybreaking regime
. This potential depends on a free parameter, the
variation of which effects transition from the asymmetric to symmetric phase. We have discussed the transition of the associated topology of the
soliton solutions. The Bogomol'nyi bound was saturated to give the self -- dual solutions of the equation of motion.
The self - dual equations
are, however, not exactly solvable. They were studied
numerically to trace out the solutions in the entire asymmetric phase with particular emphasis on the $n = 1$ mode.
Our analysis may be interesting from the point of view of
applications, particularly in condensed matter physics.
\section{Acknowledgement}
The author likes to thank Muktish Acharyya for his assistance in the mumerical solution.
|
1,108,101,566,790 | arxiv | \section{Acknowledgements}
We thank the ARC Laser team and NIF Experimental Operations team for implementing these experiments.
We gratefully acknowledge the NIF Discovery Science Program for the experimental allocation.
This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and funded by the LLNL LDRD program under tracking code 17-ERD-010. Target fabrication was performed with General Atomics under contract DE-NA0001808.
|
1,108,101,566,791 | arxiv | \section{Introduction}
\label{sec-intro}
Power-law components in the X-ray spectra of accreting black holes
are attributed to hot, tenuous, quasi-spherical plasmas called {\it
accretion disk coronae} (hereafter ADCe) by analogy with the solar
corona, though comptonization rather than atomic lines or bremsstrahlung
is thought to be the main emission mechanism in accreting systems
\citep{Bisnovatyi-Kogan_Blinnikov76, Liang_Price77}.
The vertical extent of ADCe is open to doubt since they are spatially
unresolved and since the electron temperature inferred from high-energy
spectral cutoffs, $T_e\sim 100\,\mbox{keV}$, is typically small compared
to the virial temperature of the ions, $T_i\sim 100\,\mbox{MeV}$.
Direct simulations of magnetorotational (hereafter MRI) turbulence in
radiation-pressure-dominated disks show very large density contrasts
and bulk velocities \citep{Turner_Stone_Sano02}, so perhaps the power-laws
are made within the disk itself \citep{Socrates_Davis_Blaes04}.
If a comptonizing corona indeed exists, then it is difficult to
avoid the conclusion that it should be magnetically dominated.
The electrons themselves cannot store much energy
\citep{Merloni_Fabian01}: their Compton cooling time is at most
comparable to the local dynamical time at luminosities $\gtrsim
10^{-2}L_{\rm Edd}$ and radii $\lesssim 20 GM_{\rm bh}/c^2$.
Ions at virial temperatures would store much more energy but
could not transfer it efficiently to the electrons by Coulomb
collisions \citep{Rees_Begelman_Blandford_Phinney82}. Also,
as in the solar case, magnetic fields are probably needed to
convert mechanical energy of the optically thick regions into
coronal heat. And since accretion is believed to be driven by
MRI turbulence within the disk proper, it is expected that fields
float up into the corona by Parker and interchange
instabilities~\citep{Galeev_Rosner_Vaiana79,Tout_Pringle92}.
Despite widespread recognition of these points, modelers of ADC
emission rarely concern themselves with the dynamics of coronal
magnetic fields. This is perceived to be too hard; certainly
the long and arduous struggle to understand the heating of solar
corona---based on much more abundant data---tends to discourage
hopes of solving the corresponding problem for accreting systems
any time soon \citep[\emph{e.g.}][]{Walsh_Ireland03,Klimchuk06}.
Attempts to model coronae through direct three-dimensional (3D)
magnetohydrodynamic shearing-box \citep{Miller_Stone00,Hirose_Krolik_Stone06}
and global \citep{Machida_etal00} simulations have been made, however.
To date, such simulations suggest that while a magnetically dominated
region does form, its vertical extent, when defined by distributions
of shear stress or dissipation rate, exceeds that of the optically
thick regions only modestly.
There are nevertheless good reasons to expect dynamically dominant
coronae and to question the contrary evidence from simulations. On
the one hand, a strongly magnetized corona may be required to extract
power from black hole spin \citep{Blandford_Znajek77} or from plunging
gas inside the marginally stable orbit \citep{Gammie99,Krolik99}, or
to drive a wind from the disk \citep{Blandford_Payne82}. A less
widespread motivation, which however we feel strongly, is to reduce
self-gravity in the outer parts of accretion disks in active galactic
nuclei (at $r\gtrsim 10^3 GM_{\rm bh}/c^2$): a magnetized corona or
wind might transport angular momentum more quickly than stresses
limited to the optically thick layer, and thereby reduce the mass
density within the disk for a given accretion rate \citep{Goodman03}.
On the other hand, codes designed for pressure-dominated plasmas
may not be reliable when applied to magnetically dominated coronae.
Shearing-box calculations cannot be trusted to represent magnetic
structures much larger than a disk scale height ($H$) unless
\emph{all three} dimensions of the box are $\gg H$, which has not
yet been achieved. Global simulations are unable to resolve thin
disks and extended coronae unless the grid is made coarser in the
corona than near the midplane, which increases numerical diffusion
in the corona. This is not serious when $H\sim r$, as for the innermost
parts of near-Eddington or radiatively inefficient accretion disks,
where the X-ray evidence for coronae is strongest; but it would be
a severe limitation for simulations of the marginally self-gravitating
parts of AGN disks, where $H/r\lesssim 10^{-2}$. These problems of
spatial dynamic range will eventually be overcome with sufficient
computer power.
A more fundamental difficulty for simulations has to do with magnetic
reconnection. We will argue that the efficiency of
reconnection\footnote{It would be natural to write ``rate of reconnection''
here, but that this phrase is often used to mean the speed at which
field lines of opposing polarities approach one another in a localized
reconnection event, whereas we are concerned here with the global
efficiency of reconnection in reducing magnetic energy.} is crucial
to the storage and dissipation of magnetic energy in ADCe, as it
appears to be in the solar corona \citep{Parker83,Parker88,Klimchuk06}.
Reconnection in solar flares is observed to be ``fast,'' meaning that
annihilating field lines come together at a significant fraction of the
local Alfv\'en speed. This is not well understood; because of the very
high conductivity of the corona, MHD predicts reconnection rates slower
by many orders of magnitude \citep{Sweet58,Parker57}. Collisionless
plasma effects outside conventional resistive MHD may be necessary
\citep[and references therein]{Uzdensky07a,Uzdensky07b}. Another
mystery is what triggers fast reconnection, which does not always
occur immediately but seems to require special conditions that have
not been fully identified \citep{Klimchuk06}. The overall rate of
magnetic dissipation depends both on the rate at which reconnection
events are triggered and on the speed of reconnection during such
events.
These complexities are elided by the astrophysical MHD codes used
to study magnetorotational turbulence. Often in these codes,
reconnection is purely numerical, that is, produced by truncation
errors due to limited grid resolution. Explicit diffusivities are
sometimes used, but if these are large enough make truncation errors
unimportant, then they are necessarily orders of magnitude larger
than astrophysical reality. Until recently, the general view among
MRI simulators seems to have been that the microphysics of reconnection
is unimportant. This view may be inspired by analogies with the
dissipation of kinetic energy in hydrodynamic turbulence, both
compressible and incompressible. Supersonic turbulence involves
shocks, and as is well known, shock dissipation is independent of
transport coefficients in the limit that these are small. On the
other hand, three-dimensional incompressible turbulence involves
inertial cascades such that dissipation, though occurring on small
(viscous) scales, is entirely controlled by the dynamics on large
scales and proceeds at rates that are again asymptotically independent
of transport coefficients.
Whether or when turbulent \emph{magnetic} dissipation is similarly
independent of small---and therefore numerically unresolved---scales
is an open question. Recent work indicates that magnetorotational
turbulence is sensitive to the magnetic Prandtl number
$P_m\equiv\nu/\eta$
\citep{Lesur_Longaretti07,Fromang_Papaloizou_etal07}, at least in
the range of fluid and magnetic Reynolds numbers ($Re\equiv VL/\nu$,
$Re_m\equiv P_m Re$) accessible to direct simulations. It may be that
$Re_m$ and $Re$ become unimportant when both are sufficiently large,
but this has not yet been established even for kinematic dynamos,
where the field is dynamically unimportant on all scales by
construction \citep[and references
therein]{Boldyrev_Cattaneo04,Schekochihin_etal07}, much less for MRI
turbulence. Even if this is true of systems in which fluid motions
dominate the energy density, as they are presumed to do near the
midplane of an accretion disk, the answer could be different for
magnetically dominated systems such as ADCe. Perhaps relevant here
is the case of turbulence in the presence of a dominant mean field,
which has become somewhat better understood since the seminal paper
of \citet[hereafter GS95]{Goldreich_Sridhar95}. Cascades do exist
in such turbulence, with different scalings along and perpendicular
to the mean field, so that the large scales are insulated from details
at the resistive and viscous scales. However, there are at least two
important differences between such systems and the nearly force-free
ADCe contemplated in this paper. In the former systems, the
Alfv\'enic propagation time along the field is assumed to be longer
than the timescale of the cascade, so that the turbulent dynamics are
local; by contrast, communication along the entire length of a line-tied
coronal loop is effectively instantaneous compared to the timescale
on which energy is injected into the loop by footpoint motions.
The magnetic dynamics are therefore nonlocal, and not appropriately
characterized as turbulent; a better description is a progression
of force-free equilibria driven by changes at the boundary.
[See, however, \cite{Rappazzo_etal07} for a contrary view.]
Secondly, in GS95's Alfv\'enic cascades, the deviations from the
mean field are small, and the cascade does not affect the energy
in the large-scale mean field.
In this paper, by contrast, we are concerned with coronal flux loops
that reconnect with one another at large angles between their field
lines. A significant fraction of the loops' magnetic energy may be
liberated in such reconnection events, or in the relaxation to a new
force-free equilibrium following topological changes brought about by
reconnection.
It might be hoped that the MRI simulations could predict the total
dissipation rate of the corona, if not its vertical distribution,
because the rate of injection of energy to the corona is determined
at its base, where thermal and kinetic energies dominate. This is a
false hope, however. The rate of work done on the corona by the disk
is proportional to the magnetic stress tensor at its base, specifically
the $rz$ and $\phi z$ components of the stress. Insofar as the corona
is approximately force-free, its total energy is expressible in terms
of a boundary integral involving the same stress components. Thus if
the coronal energy and magnetic configuration are incorrectly calculated,
then the coronal dissipation rate is probably also incorrect.
More concretely, because most of the kinetic energy available from
the disk is the large-scale differential rotation rather than local
turbulence, coronal field loops with large radial separations between
their footpoints may be particularly important for the energy input;
such loops are not possible in shearing-box simulations whose radial
dimensions are no larger than the disk scale height, and even if the
dimensions were increased, spurious reconnection might suppress the
large loops. In short, for geometrical reasons and because they do
not model reconnection correctly, present-day MRI simulations may
underestimate the energies and dissipation rates of disk coronae.
The dynamics of ADCe may not be fully understood without great progress
on all of the fronts described above: more powerful computations, better
understanding of fast reconnection, and of course more incisive observations.
Since all of this may take years or decades to accomplish, our purpose here
is to try to imagine, in a disciplined way, some aspects of that ultimate
understanding.
Our approach is clearly indebted to \citet[henceforth TP96]{Tout_Pringle96}
but is richer in physical elements. We model the ADC as an assembly
of closed magnetic loops with footpoints on the disk; open field is
probably important but is deferred to a later paper because we do not
wish to address winds here. We do not resolve the dynamics within the
optically thick disk at all but treat the base of the corona as a
dynamic boundary. It is assumed that the disk thickness is much less
than its radius in the parts of the ADC that we model and that the
loop lengths ($L$) lie at intermediate scales ($H\ll L\ll r$).
Thus the lower boundary is conceived as an infinite plane.
Small new loops are injected from this boundary, and existing loops are
energized by the Keplerian shear if their footpoints lie at different
radii. Loops reconnect in pairs according to prescribed rules with a
frequency scaled relative to the shear by a dimensionless parameter.
Each reconnection results in a new pair involving the same four footpoints
differently connected, rather than a single loop as in~TP96. The loop
population is described by a distribution function over the length and
orientation of the displacement from the negative to the positive-polarity
footpoint (rather than the length alone as in TP96), and the processes
of injection, stretching, and reconnection are incorporated in an
integro-differential kinetic equation for the evolution of the loop
distribution function.
Our model is not appropriate for all forms of ADCe. Following
\citet{Liang_Price77}, we presuppose a ``sandwich'' geometry in which
the optically thick but geometrically thin disk coexists with its
corona at the same radius. Such a geometry seems most appropriate to
high/soft and very high states of galactic X-ray sources where the
X-ray continuum shows both thermal and power-law features, and also to
cases, both galactic and extragalactic, where a relativistically broad
iron K$\alpha$ line indicates that an optically thick disk, fluorescing
under illumination by hard X-rays from the corona, extends down to the
marginally stable orbit or beyond
\citep[\emph{e.g.}][]{Wilms_Reynolds_etal01,Miller_Fabian_etal02}.
In low/hard states where only a power-law is seen, it may be that
the inner parts of the thin disk have evaporated so that those regions
are all ``corona'' \citep{Esin_McClintock_Narayan97}; alternatively,
the thin disk may persist down to the marginally stable orbit, but
the corona may take the form of a mildly relativistic outflowing
wind or jet, whose emission is directed away from the disk
\citep{Beloborodov99,Miller_Homan_etal06}. The coronal model presented here
would not apply to either of these cases without substantial modification.
In this paper, we explore what controls the magnetic energy,
integrated stress, and dissipation rate of the corona; specifically,
we explore the roles of shear and reconnection in this balance. Our
overarching motivation is to determine under what conditions the
corona contributes importantly to outward angular momentum transport:
that is, to the torque that drives accretion through the disk.
Secondary goals are to examine the distribution of energy, stress, and
dissipation with height. We recognize that because of a number of
questionable assumptions and simplifications, our model will hardly be
the last word on this subject. We hope , however, at least to set up
a target for future simulations to aim at, and to draw attention to
certain quantities that could be extracted from existing simulations,
such as the rate of emergence of flux dipoles (small loops) from the
disk.
The outline of our paper is as follows. \S~\ref{sec-statistical-model}
introduces our conception of magnetic loops and the loop distribution
function. \S~\ref{sec-self-consistent} explores some properties of
loops in equilibrium with the mean-field pressure exerted by neighboring
loops: shape, maximum height, and energy. \S~\ref{sec-kinetic-eqn}
constructs the kinetic equation, including the rules for reconnection
and other processes important to the evolution of the loop distribution.
The numerical set-up used to solve the loop kinetic equation, including
the boundary conditions at both small and large scales, is described
in~\S~\ref{sec-numerical}. In the same section we also present numerical
solutions and discuss their implications for ADCe.
In~\S~\ref{sec-discussion} we discuss limitations of our model in the light
of these results, and we indicate ways in which the model might be made
more realistic. We attempt to relate what we have done to the present
theoretical understanding of ADCe, and we discuss how simulations in
the near future might be used to calibrate some features of our model,
for example the rate of emergence of small loops from the disk.
Finally, \S~\ref{sec-conclusions} summarizes our main conclusions.
\section{Statistical Description of the ADC Magnetic Field}
\label{sec-statistical-model}
As noted in \S1, there are fundamental physical similarities between
the formation processes of the solar and accretion disk coronae.
At the same time, one has to keep in mind the important differences
between them. These include: (1) differences in the underlying
sub-photospheric turbulence (thermal convection vs.\ MRI);
(2) a strong large-scale differential rotation and shear in
disks, whereas small scales dominate the shear in the Sun;
(3) the possibility of strong magneto-centrifugally- and radiation-driven
winds from an accretion disk, as compared to the relatively weak thermally
driven solar wind; (4) Compton cooling of the disk corona in black hole
systems; (5) a greater separation of spatial scales in thin disks
between the disk thickness and its radius, as opposed to the solar case,
where the convection zone spans a significant fraction of the Sun's radius,
which results in the generation and emergence of very large magnetic
structures associated with sunspots in active regions. Finally, we have
only one Sun, whereas there is a large variety of astrophysical
accretion-disk systems, some of which have coronae.
In addition to the above, there is an important basic difference in
our observational capabilities: whereas the Sun is so close that we
can spatially resolve {\it individual} events and structures in the
solar corona (such as flares, loops, etc), such resolution is not available
for ADCe. Therefore, we can study only the spatially-integrated spectral
and timing properties of the disk corona. This fact provides a strong
motivation for focusing on a {\it statistical description} for the
magnetic field in the~ADC.
Here is, briefly, the basic physical picture of the magnetized corona
above a turbulent accretion disk. The corona is a dynamic, self-organized
system that can be represented by a statistical ensemble of flux loops
\citep{Tout_Pringle96,Hughes_Paczuski_etal03}.
The loops continuously emerge out of (and submerge into) the disk as a result
of magnetic buoyancy. Once above the surface, they constantly evolve due to
a number of physical processes.
They are twisted and stretched by the differential keplerian rotation and
by the random motions of their footpoints on the disk's surface, which causes
the individual loops to inflate. As a result, the magnetic field in the
corona becomes non-potential and highly stressed; an appreciable amount
of free magnetic energy can thus be stored in the corona. However, in
the process of twisting and expansion the loops may undergo internal
disruptions due to MHD instabilities and also may reconnect with other
loops. Such relaxation events manifest themselves as flares; they bring
the field closer to the potential state and thus enable the inflation
process to resume. At the same time, reconnection between loops sometimes
produces more spatially-extended magnetic structures in the corona (the
coronal ``inverse cascade''). Finally, all these complicated processes
occur repeatedly over and over, simultaneously on various spatial scales.
Thus, the corona can be viewed as a {\it boiling magnetic foam}, in which
magnetic loops repeatedly swell and grow because of the magnetic energy
pumped into them by the footpoint motions and then snap and contract back
due to reconnective disruptions and sometimes merge to form bigger structures.
What is the appropriate statistical language for describing a chaotic,
highly intermittent magnetized corona above a turbulent accretion disk?
At a most general level, we describe the corona by an ensemble of some
{\it elementary magnetic structures}. Each of these fundamental individual
constituents of the corona is characterized by a small set of primary physical
parameters. The individual elements evolve with time according to certain
rules that reflect the relevant physical processes that we believe are most
important in shaping the corona. These processes generally include various
forms of interaction between elements. Mathematically, the evolution rules
are represented by stochastic (Langevin) equations of motion of magnetic
elements in the primary parameter space. Since we are interested in a
statistical description, we introduce a distribution function of our magnetic
elements in the primary-parameter phase-space. Correspondingly, one of our
main goals is to derive the equation for the evolution of this distribution
function, using the equations of motion of individual elements. This is done
by analogy with the way the Boltzmann kinetic equation for the particle
distribution function in a gas is derived in Statistical Mechanics, but
obviously is more {\it ad hoc} in our case. Finally, there are several
important integral quantities in our theoretical framework, which are
related to moments of the distribution function. These self-consistent
quantities represent the mean-field interaction between the magnetic
elements and they affect the evolution of the distribution function.
\begin{figure}[t]
\plotone{discorona-bw.eps}
\figcaption{Schematic view of ADC as an ensemble of many magnetic loops.
\label{fig-corona}}
\end{figure}
\subsection{The Loop Distribution Function}
\label{subsec-F}
Based on the above discussion, we shall now build a calculable model
of the corona. Our first task is to select the most appropriate and
most fundamental elementary magnetic constituents of the corona.
We shall then need to select the most natural set of parameters
describing these elements.
Guided by the analogy with the solar corona, we shall use simple
(anchored in the disk at both ends, see Fig.~\ref{fig-loop-1})
magnetic loops, or flux tubes, as our fundamental magnetic elements
--- the main structural constituents of the corona. This choice is
influenced by the existing theoretical work in both solar physics
\citep[\emph{e.g.}][]{Hughes_Paczuski_etal03}, and also in ADC
\citep[]{Tout_Pringle96}.
Such loops represent the closed magnetic field corresponding
to zero net vertical flux through the disk. This is a natural
assumption for the case when the magnetic field in the corona
comes from the flux emergence of the field generated by the dynamo
in the disk itself. In this paper we shall assume that this is indeed
the case. In principle, however, one may also wish to consider a more
general situation where, in addition to the closed coronal loops, there
is also an external large-scale open magnetic field through the disk,
such as may be coming from the central star or the interstellar medium
(see \S~\ref{subsec-open}).
For simplicity, we shall characterize each loop by only two primary parameters:
{\it (i)} the radial footpoint separation $\Delta r = r_{-}-r_{+}$,
and
{\it (ii)} the azimuthal footpoint separation
$r\Delta\phi = \Delta y =y_{-}-y_{+}$ (see Fig.~\ref{fig-loop-1}).
Thus, we shall measure $\Delta r$ and~$\Delta y$ from the ``+'' magnetic
footpoint to the ``-'' magnetic footpoint. This means that $\Delta r$
and~$\Delta y$ can be positive or negative depending on the orientation of
the loop. Alternatively, sometimes we will use an equivalent representation
in terms of the loop's projected length (the distance between the loop's
footpoints): $L\equiv (\Delta r^2 + \Delta y^2)^{1/2}$; and the orientation
angle, $\theta$, measured clockwise with respect to the toroidal direction:
$\theta \equiv \arctan(\Delta r/\Delta y)$.
\begin{figure}
\plotone{loop-1-bw.eps}
\figcaption{Closed magnetic loop as the main structural element
of a magnetized accretion disk corona.
\label{fig-loop-1}}
\end{figure}
In a more general description, one may enlarge the parameter
space to include additional parameters, such as the magnetic
flux~$\Delta\Psi$ contained within a loop, or the loop's twist
(see~\S~\ref{subsec-twisted}).
However, since our goal here is to build the simplest version of this
already very complicated theory, we shall assume that all the loops
have the same magnetic flux~$\Delta\Psi$ and, furthermore, that they
are not twisted. The latter assumption means that the magnetic field
within each loop is purely potential, {\it i.e.}, that the bulk of the
corona is nearly current-free and all the coronal currents flow along
inter-loop boundaries.
In addition to the above two primary parameters, we shall also need
some secondary parameters describing a given loop, such as the loop's
overall shape, its maximum height~$Z_{\rm top}$ above the disk; and
its thickness at a given height, $d(z)$.
These quantities will be useful for estimating the loop expansion rate and
for analyzing binary interaction (reconnection) of loops with each other.
In our model, these secondary parameters are uniquely determined by the
primary ones in combination with the self-consistent mean field~$\bar{B}(z)$
(see \S~\ref{subsec-Bbar}).
Note that a loop carrying a finite flux $\Delta\Psi$ has a certain finite
thickness (in the radial and azimuthal directions) at the disk surface.
For a typical loop, this thickness is generally of order~$H$. Therefore,
instead of a pair of footpoints that
a field {\it line} would have, a finite-thickness {\it loop}
has a pair of footspots. Thus, we need to be a little bit more
precise in our definitions of~$\Delta r$ and~$\Delta y$. We shall
define them as the radial and azimuthal separations between the
centers of the two footspots, that is between the two footpoints
of the central field line of the loop (magnetic axis for a twisted loop).
Following \cite{Tout_Pringle96}, we introduce the {\it loop distribution
function}, $F(\Delta r,\Delta y)$, defined so that $F(\Delta r,\Delta y)
d\Delta r d\Delta y$ is the number of loops with the values of primary
parameters in the range $([\Delta r,\Delta r + d\Delta r],
[\Delta y,\Delta y + d\Delta y])$, per unit disk area. Alternatively,
we may write the distribution function in terms of loop length
$L=(\Delta r^2 + \Delta y^2)^{1/2}$ and orientation
$\theta=\arctan(\Delta r/\Delta y)$, {\it i.e.}, $F(L,\theta)$.
The overall normalization of the distribution function is determined
by the requirement that loops cover the entire disk surface; it will
be discussed in more detail in~\S~\ref{subsec-Bbar}.
\subsection{Role of Magnetic Reconnection in ADCe}
\label{subsec-recn-role}
At the most basic level, the corona (either solar or ADC) is perfectly
conducting almost everywhere. However, as it evolves driven by the complex
turbulent motions of the magnetic footpoints on the surface, the corona
may develop numerous current sheets on a variety of scales \citep{Parker72,
Parker83}.
These current sheets are possible sites of dissipation of magnetic energy
{\it via} reconnection. In fact, reconnection is one of the most essential
nontrivial physical processes that govern the complex dynamical behavior
of the corona. In particular, it controls the vertical extent of the corona
({\it e.g.}, the coronal magnetic scale-height~$H_B$).
Indeed, if reconnection were too efficient, then the coronal field would
be nearly potential and~$H_B\sim H$; then, the free magnetic energy stored
in the corona would be small, as would the magnetic dissipation rate.
On the other hand, if no reconnection were allowed at all, then,
magnetic loops would, over time, grow bigger and bigger in height
because of the differential Keplerian rotation. Unable to dissipate,
magnetic energy would continuously accumulate in the corona as the
characteristic magnetic scale-height~$H_B$ increases, essentially
linearly in time. This phase would continue until~$H_B$ becomes
comparable to the disk radius, $H_B\sim r\gg H$. After that, radial
gradients become important and the subsequent evolution would enter
a qualitatively different regime characterized by accelerated expansion
of the magnetic loops, which effectively would become open, perhaps even
in a finite time \citep{vanBallegooijen94,Lynden-Bell_Boily94,Aly95,
Sturrock_Antiochos_Roumeliotis95,Uzdensky02b}.
The corona would then consist of a dense forest of open flux tubes of
alternating polarity separated by a multitude of current sheets.
Unless there are significant mass-loaded winds (violating the force-free
assumption), the power pumped into the corona would then go down, and the
accumulated free magnetic energy would saturate at a value corresponding
to a fully open (split-monopole) magnetic field~\citep{Aly91,Sturrock91}.
Although this asymptotic energy would be very large, of order~$r/H$ larger
than that of the fully closed potential field, it would still remain finite;
this is because the toroidal magnetic field at the disk surface and hence
the work done on the coronal magnetic field by the Keplerian disk shear
would both go to zero. Similarly, the angular momentum exchange between
different parts of the disk due to coronal loops would also go down. As
new flux tubes emerged from the disk, the magnetic forest would become ever
more dense. The corona would thus look very different from what we expect.
We thus see that reconnection is necessary for maintaining a {\it meaningful}
statistical steady state. It enables open field lines to close back and thus
restores the magnetic connection between different parts of the disk.
This, in turn, facilitates angular-momentum transport {\it via} the
coronal magnetic field [coronal~MRI, \citep{Goodman03}; see also \citep
{Heyvaerts_Priest89,Pavlidou_etal01}]. Strong magnetic dissipation and
large torque thus require some intermediate reconnection efficiency,
neither so rapid as to keep the field nearly potential, nor so slow as
to allow it to become fully open; in both limits, the torque and energy
dissipation rate vanish.
Another reason why reconnection is important is that a growing magnetic
loop may reconnect with another one connected to a very different place
on the disk. This process may lead to an {\it ``inverse cascade''} of
magnetic loops \citep{Tout_Pringle96}. It is an important avenue towards
building up a population of loops with large radial footpoint separation.
Indeed, whereas Keplerian differential rotation increases the azimuthal
footpoint separation of a loop, it does not affect its radial footpoint
separation. Therefore, without reconnection, the radial footpoint separation
of a coronal flux loop would change only relatively slowly by the random
walk of its footpoints due to the underlying disk turbulence. In principle,
the footpoints will drift radially apart in direct response to the angular
momentum transfer by the coronal loop itself (we call this process
``coronal~MRI''). The characteristic velocity of this drift is on
the order of $B^2/4\pi\Sigma \Omega$, where $\Sigma$ is the surface
density of the disk. The resulting relative increase in~$\Delta r$
on the rotation-period timescale is of the order of $\delta\Delta r/\Delta r
\sim B^2/4\pi\Sigma \Delta r \Omega^2$. Using $\Sigma\simeq H\rho$ and
$H\sim c_s/\Omega$, where $c_s$ and~$\rho$ are the sound speed and the
gas density within the disk, we can estimate that $\delta\Delta r/\Delta r
\sim (V_A/c_s)^2 \, H/\Delta r \equiv \beta^{-1}\, H/\Delta r$, where
$V_A^2 \equiv B^2/4\pi\rho$ is the Alfv\'en speed within the disk.
Thus, since we are mostly interested in large loops, $\Delta r\gg H$,
we see that $\Delta r$ cannot grow appreciably without reconnection.
{\it Thus reconnection is necessary for the coronal ``inverse cascade''.}
In addition, magnetic reconnection in the corona regulates the fraction
of the magnetic flux that is open at any given time and also the
effective radial transport of a large-scale vertical magnetic field
\cite{Spruit_Uzdensky05,Fisk05}. Both of these processes are important
for establishing large-scale disk outflows.
Finally, as in the solar corona, reconnection is believed to be the main
mechanism of releasing the accumulated magnetic energy, leading to coronal
heating and observed high-energy coronal emission.
\section{The Self-Consistent Corona}
\label{sec-self-consistent}
In principle, the loop distribution function~$F(L,\theta)$ contains
enough information to fully describe the statistical magnetic structure
of the corona. This means that, once ~$F(L,\theta)$ is known, one should
be able to answer most of the questions posed in~\S\ref{sec-intro}.
In particular, one should be able to derive the actual shapes and heights,
$Z_{\rm top}(L)$, of coronal loops, the distribution of magnetic energy
with height, $\bar{B}^2(z)/8\pi$, the energy~$\mathcal{E}(L)$ associated
with a loop of a given size, the torque transmitted by the coronal magnetic
field, etc. In this section we demonstrate how to do all this.
\subsection{Equilibrium Shape of a Loop in a Stratified Atmosphere}
\label{subsec-shape}
First, we shall work out the correct shape of an isolated slender
(with a cross-sectional diameter $d\ll L$) untwisted loop~$\mathcal{A}$
carrying magnetic flux~$\Delta\Psi$, immersed in a medium with some isotropic
but, in general, nonuniform pressure~$P(z)$ (see Fig.~\ref{fig-loop-2}).
This pressure represents the confining magnetic pressure of all other loops;
thus, for actual calculations, it will be convenient to write~$P(z)$ as
$P(z)\equiv\bar{B}^2(z)/8\pi$.
The shape of the loop is then determined by the requirement that the loop
be in magnetostatic equilibrium with this external pressure.
\begin{figure}[h]
\plotone{loop-2.eps}
\figcaption{Untwisted isolated loop confined by external isotropic
pressure~$P(z)$.
\label{fig-loop-2}}
\end{figure}
First, the local pressure balance across the loop gives
us the magnetic field strength inside it as a function of
height:
\begin{equation}
B(z) = \sqrt{8\pi P(z)} \equiv \bar{B}(z)\, .
\label{eq-pressure-balance}
\end{equation}
Then, since the magnetic flux is constant along the loop,
we can write the local cross-sectional area~$a(z)$ of the
loop in terms of~$\bar{B}(z)$:
\begin{equation}
a(z) = {\Delta\Psi\over{\bar{B}(z)}} \, .
\end{equation}
Now let us discuss equilibrium shape~$x(z)$ of a slender loop as
a whole, represented by the shape of the loop's central field line.
First, we would like to note that, for a curved but untwisted loop
confined by a {\it uniform} external pressure~$P_0$, it is impossible
to find an equilibrium shape.
Indeed, since the field inside the loop has no twist and is
purely axial ({\it i.e.}, runs along the loop), it is a potential
field produced by perpendicular (to the direction of the loop)
currents flowing on the loop's surface. Therefore, at any given
location along the loop, the magnetic field strength is slightly
nonuniform in the cross-loop direction: it drops off as~$1/R$,
where~$R$ is the local curvature radius (``major radius'' in
tokamak terminology). In other words, the magnetic force balance
between the magnetic pressure and the magnetic tension inside a
curved loop means that there must be a magnetic pressure gradient
to balance the tension force due to the curvature. Therefore,
the magnetic field on the underside (the ``inboard'', in tokamak
terminology) of the loop is larger than that on the upside (the
``outboard''). On the other hand, however, the magnetic field at
each point on the surface of the loop has to be in pressure balance
with the external pressure~$P(z)$. If~$P(z)=P_0={\rm const}$, this
pressure balance means that the magnetic field has to be uniform
along the loop's boundary. Thus, we have a clear incompatibility
between the assumption that the loop is curved but untwisted and
the condition of local force balance with an external uniform
pressure.
This phenomenon can also be understood using the notion of Pfirsch--Schl\"uter
currents, a well-known concept in tokamak physics. Since the magnetic field
inside an untwisted flux tube is strictly in the axial direction (along the
tube), the currents that produce it flow on the skin of the tube in the
perpendicular direction. But since the currents have to close, and since
the tube is curved, the surface current density ({\it i.e.}, the current
per unit length along the tube) is bigger on the underside than on the upside.
Then, according to Ampere's law, the magnetic field is also stronger
on the underside, and thus cannot be in pressure equilibrium with a
uniform external pressure on both sides simultaneously. The only way
a curved loop can be in equilibrium with uniform external pressure, is
when the surface current density is also uniform. Since the current has
to be conserved, this requires that some surface current should also
flow along the loop (the so-called Pfirsch--Schl\"uter current), and
hence the magnetic field must be twisted.
These considerations show that an untwisted flux tube confined by a uniform
external pressure has to be straight. If, however, the pressure is not
uniform, then an equilibrium shape for a curved tube can be found, as we
now show.
Let us consider a case with pressure~$P(z)$ decreasing monotonically
with height. Consider a slender loop in the $(x,z)$ plane, symmetric
with respect to $x=0$, with its two footpoints at $x=\pm\,L/2$
(see Fig.~\ref{fig-loop-2}).
The shape of the loop as a whole, described by a symmetric function~$z(x)$,
is determined by the perpendicular (to the magnetic field) force-balance
between tension and the pressure-gradient forces. The force-balance is
established at every point along the loop. To analyze it at a given
location on the loop, let us consider a small loop segment. We shall denote
the arc-length along the loop, measured from its left footpoint, by~$l$.
For clarity of discussion, let us represent the loop segment locally
by a slightly curved cylinder of length~$\delta l\ll L$, cross-sectional
area~$a(l)$, and curvature radius~$R(l)$ (see Fig.~\ref{fig-force-balance}).
Let us also introduce the angle~$\alpha(l)$ between the magnetic field and
the vertical direction:
\begin{equation}
\cos\alpha = {dz\over{dl}} \, .
\label{eq-def-alpha}
\end{equation}
For definiteness, we consider the ascending leg of the loop, where $B_z>0$,
so that $0\leq \alpha \leq \pi/2$. The local curvature radius~$R(l)$ of the
loop is related to~$\alpha(l)$ {\it via}
\begin{equation}
{1\over R} = {d\alpha\over{dl}} \, .
\label{eq-curvature-radius}
\end{equation}
Here we treat $\bar{B}(z)$ as a known function and our goal is to determine
the geometrical shape of the loop described by~$\alpha(z)$.
\begin{figure}
\plotone{force-balance.eps}
\figcaption{Infinitesimal loop segment.
\label{fig-force-balance}}
\end{figure}
The magnetic tension force on the loop segment acts in
the direction of its curvature radius~$\hat{R}$ and is
equal to
\begin{equation}
\delta f_{\rm tension} = {{B^2}\over{4\pi R}}\, a(l)\, \delta l \, ,
\end{equation}
The projection of the external-pressure force on the loop segment
onto~$\hat{R}$ can be written as
\begin{equation}
\delta f_{\nabla P} =
-\, {dP\over{dz}}\, \sin\alpha\ a(l)\, \delta l \, .
\end{equation}
Then, the force-balance condition can be written simply as
\begin{equation}
{B^2\over{4\pi}}\, {1\over R} = -\, \sin\alpha \, {dP\over{dz}} \, ,
\end{equation}
or, making use of equation~(\ref{eq-pressure-balance}),
\begin{equation}
{{\bar{B}(z)}\over{R}} = -\sin\alpha \, {{d\bar{B}}\over{dz}} \, .
\label{eq-loopshape-1}
\end{equation}
Combining this equation with the geometrical relation~(\ref
{eq-curvature-radius}), we immediately obtain:
\begin{equation}
{d\alpha\over{dl}}={1\over R} = -\,{{d\log\bar{B}}\over{dz}}\, \sin\alpha\, .
\end{equation}
[Note that, as one can immediately see, a magnetic loop can be in equilibrium
with a uniform [$\bar{B}(z)={\rm const}$] external pressure only if it is
straight, $\alpha={\rm const}$.]
Using~(\ref{eq-def-alpha}), we get
\begin{equation}
\cot\alpha \, {d\alpha\over{dz}} = -\,{{d\log\bar{B}}\over{dz}} \qquad
\Rightarrow \qquad \sin\alpha(z) = {C\over{\bar{B}(z)}} \, .
\label{eq-alpha-Bbar}
\end{equation}
It is interesting to note that the same result can be obtained
in a simple and elegant way by using a variational principle,
namely, by by minimizing the loop's magnetic energy~$E_{\rm magn}=
(\Delta\Psi/8\pi)\,\int B(l)\,dl$ (see \S~\ref{subsec-energy})
viewed as a functional of~$z(x)$. Using the relationship
$dl = dx\, \sin^{-1}\alpha$, we have
\begin{eqnarray}
\delta E_{\rm magn}[z(x)] &=&
{\Delta\Psi\over{8\pi}}\,\delta \int B[z(x)]\sqrt{1+z'^2(x)}\,dx \nonumber \\
&=&
{\Delta\Psi\over{8\pi}}\,\int\biggl[B'(z)-B(z)\,{{z''}\over{1+z'^2}}\biggr]\,
{{\delta z(x) dx}\over{\sqrt{1+z'^2}}} \, .
\end{eqnarray}
From the condition $\delta E_{\rm magn} = 0$, we thus immediately get
\begin{equation}
B = C \sqrt{1+z'^2(x)} = C {dl\over{dx}} = {C\over{\sin\alpha}} \, ,
\end{equation}
which, taking into account that $B=\bar{B}(z)$, is the same as
the above result~(\ref{eq-alpha-Bbar}).
The result~(\ref{eq-alpha-Bbar}) means that the horizontal ($x$) component
of the magnetic field is constant along the loop:
\begin{equation}
B_x(l) = B \sin\alpha = \bar{B}(z) \sin\alpha = C = {\rm const} \, .
\label{eq-B_x=B_top}
\end{equation}
The integration constant $C$ is just equal to the magnetic field~$B_{\rm top}$
at the top of the loop (where $\alpha=\pi/2$). Thus, the shape of the loop
is given by the equation
\begin{equation}
\sin\alpha(z) = {B_{\rm top}\over{\bar{B}(z)}} \, .
\label{eq-loopshape-2}
\end{equation}
We can now work out an explicit expression for the field line shape~$z(x)$
in terms of the function~$b(z)\equiv\bar{B}(z)/\bar{B}(z=0)$. Let us
denote the magnetic field strength at the base by $B_0\equiv \bar{B}(z=0)$
and the angle between the loop and the vertical direction at the base by
$\alpha_0(L)=\alpha(z=0;L)$. We then have
\begin{equation}
b_{\rm top}(L) = {{\bar{B}_{\rm top}(L)}\over{B_0}} = \sin\alpha_0(L) \, .
\label{eq-b_top=sin-alpha_0}
\end{equation}
Then, the shape of the ascending leg of a loop of length $L$ is given by
\begin{equation}
x(z) = -\, b_{\rm top}\, \int\limits_z^{z_{\rm top}} \,
{{dz'}\over{\sqrt{b^2(z') - b^2_{\rm top}}}} =
-\, {L\over 2} + b_{\rm top}\, \int\limits_0^z \,
{{dz'}\over{\sqrt{b^2(z') - b^2_{\rm top}}}} \, .
\label{eq-loopshape-3}
\end{equation}
The height of the loop, $z_{\rm top}$, is determined implicitly by
the condition $x(z_{\rm top})=0$:
\begin{equation}
{L\over 2} = b_{\rm top}\ \int\limits_0^{z_{\rm top}} \
{{dz'}\over{\sqrt{b^2(z') - b^2_{\rm top}}}} \, .
\label{eq-z_top-1}
\end{equation}
Analogous expressions have been obtained by \cite{Parker75}
and \cite{Browning_Priest84}.
Here are a few analytical examples of the use of this relationship.
1) Example I:
\begin{equation}
b(z) = {1\over{1+\zeta}} \, ,
\label{eq-b(z)-example1}
\end{equation}
where $\zeta = z/z_0$; $z_0$ represents the characteristic magnetic
scale height of the corona.
Then we get $b_{\rm top}(L)= 2z_0\, (4 z_0^2+L^2)^{-1/2}$,
and $z_{\rm top}(L)=(z_0^2+L^2/4)^{1/2}-z_0$.
2) Example II: exponential atmosphere,
\begin{equation}
b(z) = e^{-z/H}
\label{eq-b(z)-example2}
\end{equation}
Performing the integration~(\ref{eq-z_top-1}), we obtain
$x(z)= -\, H\, \arctan\, [e^{2(z_{\rm top}-z)/H} - 1]^{1/2}$,
and correspondingly,
\begin{equation}
L(Z_{\rm top}) = 2 H\, \arccos\,[{b(Z_{\rm top})}] =
2 H\, \biggl[{\pi\over 2} - \alpha_0(Z_{\rm top})\biggr] \, .
\end{equation}
Notice that for tall loops with $Z_{\rm top}\gg H$ and $b_{\rm top}\ll 1$,
the dependence $L(Z_{\rm top})$ saturates: $L(Z_{\rm top}) \rightarrow
L_{\rm max} = \pi H$. This example illustrates an important point: if
the external pressure drops off sufficiently steeply, then there is a
maximum projected length, $L_{\rm max}$, that a loop in equilibrium can
have. This implies that if one tries to insert a slender loop with a
footpoint separation $L>L_{\rm max}$, then such a loop will not be able
to attain an equilibrium and will instead grow in height without bound,
{\it i.e.}, will tend to open up.
This fact points to an important feedback: for a given external pressure
profile, large loops extend to larger heights, but this has an effect
of increasing the contribution of these loops to the pressure at these
large heights (see below), and hence may make the pressure profile less
steep.
\subsection{Magnetic Energy Density~$\bar{B}^2(z)/8\pi$
as a Self-Consistent Mean Field}
\label{subsec-Bbar}
Our next step is to determine $\bar{B}$ as a function of~$z$.
However, the best way to do this is first to express $\bar{B}$
directly in terms of the length of the smallest loop $L$ that
reaches the given height~$z$, {\it i.e.}, to find $\bar{B}(L)$.
This can be done directly in terms of the orientation-integrated
distribution function $\bar{F}(L)\equiv \int F(L,\theta) d\theta$,
since, at any given height~$z$, contributions to the magnetic
pressure~$\bar{B}^2/8\pi$ come only from those loops that extend
to this height or higher. Since the dependence~$Z_{\rm top}(L)$
is presumed to be monotonic, then $\bar{B}$ at a given height~$z$
will be proportional, roughly speaking, to the integral over all
loops with lengths $L>L(z)$, where $L(z)$ is the function inverse
to~$Z_{\rm top}(L)$. Thus, naively, we anticipate a result that
looks something like this:
\begin{equation}
\bar{B}(L) \sim \Delta\Psi \int\limits_L^\infty \bar{F}(L)\, dL \, .
\end{equation}
In Appendix A we perform a rigorous analysis and derive an exact
(within our model) result:
\begin{equation}
db = -\, {{\pi\,\Delta\Psi}\over{B_0}}\ \bar{F}(L)\, dL \quad\Rightarrow\quad
b(L)={{\pi\,\Delta\Psi}\over{B_0}}\ \int\limits_L^\infty \bar{F}(L')\, dL' \,,
\end{equation}
that is,
\begin{equation}
B_{\rm top}(L) = \pi\,\Delta\Psi\ \int\limits_L^\infty \bar{F}(L')\, dL' \,.
\label{eq-Btop(L)}
\end{equation}
The condition $b(z=0)=1$, in conjunction with $Z_{\rm top}(L=0)=0$,
gives us the normalization condition for the function~$\bar{F}(L)$:
\begin{equation}
\int\limits_0^\infty \bar{F}(L') dL' = {B_0\over{\pi\,\Delta\Psi}} \, .
\label{eq-normalization}
\end{equation}
If all the loops were perpendicular to the disk surface,
the normalization coefficient would be~1/2 (since each loop
has two footpoints). The extra factor~$(2/\pi)$ in the above
expression reflects the fact that small low-lying loops are
not perpendicular to the disk surface, and hence occupy larger
horizontal projected area on this surface.
For a given loop distribution function~$\bar{F}(L)$, one can
thus compute, in principle, the function~$b(L)$ that we shall
need in the next subsection.
\subsection{Self-Consistent Loop Height $Z_{\rm top}(L)$}
\label{subsec-Ztop}
Once $L(b_{\rm top})$ is thus determined, we can substitute
it into equation~(\ref{eq-z_top-1}) and thus reduce the whole
problem to the following integral equation for the function~$U(b)
\equiv dz/db$:
\begin{equation}
{{L(b)}\over{2b}} =
-\, \int\limits_b^1 {{U(b')\, db'}\over{\sqrt{b'^2 - b^2}}} \, .
\label{eq-Volterra-I}
\end{equation}
Once this equation is solved, we can integrate $U(b)$ to find~$z(b)$,
thus completing the solution.
Mathematically, equation (\ref{eq-Volterra-I}) is a linear Volterra
integral equation of the first kind for the function~$U(b)$ in terms
of a known function~$L(b)$. It can be solved exactly. In particular,
by a simple transformation of variables:
$t \equiv 1-b'^2$,
$s \equiv 1-b^2$,
$G(s) \equiv L(b)/2b$,
and $V(t) \equiv -\, U(b')/2b'$,
it can be transformed into the Abel equation:
\begin{equation}
\int\limits_0^s {{V(t)\,dt}\over\sqrt{s-t}} = G(s) \, .
\label{eq-Abel}
\end{equation}
whose solution is
\begin{equation}
V(s) =
{1\over\pi}\, {d\over{ds}}\, \int\limits_0^s {{G(t)\,dt}\over\sqrt{s-t}} =
{1\over\pi}\, \int\limits_0^s {{G'(t)\,dt}\over\sqrt{s-t}} \, ,
\label{eq-Abel-soln}
\end{equation}
where we have used $G(s=0)=L(z=0)/2=0$.
Actually, the most useful form of the solution is
the first equality in equation~(\ref{eq-Abel-soln}).
By substituting the definitions of~$V(s)$, $G(s)$ and~$U(b)$
into this equation, multiplying by $(-2b)$ and integrating
with respect to~$b$, we get the following elegant final
expression for the function~$z(b)$:
\begin{equation}
z(b) = {1\over\pi} \int\limits_b^1 {{L(b')\,db'}\over\sqrt{b'^2-b^2}} \, .
\label{eq-z-of-b}
\end{equation}
Using equation~(\ref{eq-Btop(L)}), we can rewrite this
in terms of the functions~$\bar{F}(L)$ and~$b(L)$ as
\begin{equation}
Z_{\rm top}(L) = {{\Delta\Psi}\over{B_0}}
\int\limits_0^L {{L'\,\bar{F}(L')\,dL'}\over\sqrt{b^2(L')-b^2(L)}} \, .
\label{eq-Ztop-of-L}
\end{equation}
\subsection{Loop Energy}
\label{subsec-energy}
One of the main goals of this section is to address the {\it energetics}
of the magnetized corona. Relevant issues include the energy distribution
of flares as well as the torque on the disk due to the coronal magnetic
fields.
In order to be able to address this, we must first determine the energy,
$\mathcal{E}(\mathcal{A})$, associated with a loop of type~$\mathcal{A}$.
The loop energy is given by the work done by the footpoints against
magnetic forces as the loop's footpoint separation ({\it i.e.}, the
projected length of the loop) is increased from zero to its present
value~$L$. Since we regard $P(z)$ as isotropic, the energy depends
only on the length but not on the orientation of the loop:
\begin{equation}
\mathcal{E}(\mathcal{A}) = \mathcal{E}(L) =
\int\limits_0^L\, f_{\rm fp}(L')\, dL' \, .
\label{eq-energy-1}
\end{equation}
Here, $f_{\rm fp}(L)$ is the magnetic force on each of the two footpoints;
it is proportional to the horizontal magnetic field~$B_{\rm hor}(z=0)$ at
the disk surface:
\begin{equation}
f_{\rm fp}(L)={{B_z B_{\rm hor}}\over{4\pi}}\biggl|_{z=0}\,a_{\rm hor}(z=0)=
{\Delta\Psi\over{4\pi}}\, B_{\rm hor}(z=0;L) \, ,
\label{eq-f_footpt}
\end{equation}
where $\Delta\Psi$ is the magnetic flux carried by the loop and
$a_{\rm hor}(z=0)=\Delta\Psi/B_z(z=0)$ is the area of the loop's
footspot on the disk surface.%
\footnote
{Note that here, instead of fixing $\Delta\Psi$ and~$B_z(z=0)$,
we fix $\Delta\Psi$ and the total magnetic field~$B(z=0)$, which
includes the horizontal component.}
Thus, we have
\begin{equation}
\mathcal{E}(L) =
{\Delta\Psi\over{4\pi}}\ \int\limits_0^L\,B_{\rm hor}(z=0;L')\, dL' \, .
\label{eq-energy-2}
\end{equation}
Another quantity of interest is the magnetic energy~$E_{\rm magn}$
contained within the loop:
\begin{equation}
E_{\rm magn} = \int a(l)\, {{B^2(l)}\over{8\pi}}\, dl \, ,
\label{eq-E_magn-1}
\end{equation}
where the integral is taken along the loop from one footspot to the other.
Using flux conservation, $\Delta\Psi(l) = a(l) B(l) = {\rm const}$, this
energy can be written simply as
\begin{equation}
E_{\rm magn} = {\Delta\Psi\over{8\pi}}\, \int B(l)\, dl \, .
\label{eq-E_magn-2}
\end{equation}
This expression actually has a very simple physical meaning.
The integral $\int B(l)\, dl$ is the circulation of the magnetic
field along the loop; according to the Ampere's law, this is
just the total surface current flowing around the loop in the
perpendicular direction. Thus, the above expression for $E_{\rm magn}$
is just a manifestation of the well-known result that the magnetic energy
of a current circuit is proportional to the product of the magnetic flux
enclosed by the circuit and its total current.
We would like to remark that $\mathcal{E}(L)$ can be viewed as
a magnetic enthalpy, $H_{\rm magn}$. It includes both the magnetic
energy~$E_{\rm magn}$ stored within the loop and the work~$W$ done
by the loop on the surrounding gas with a fixed (but not necessarily
uniform) pressure profile:
\begin{equation}
d\mathcal{E} = dH_{\rm magn} = dE_{\rm magn} + dW \, .
\end{equation}
This is analogous to calculating the amount of heat~$Q$ required
to inflate a hot-air balloon at constant atmospheric pressure~$P_0$.
Indeed, when the balloon air is heated, the energy is expended both
to increase the internal energy~$U$ of the hot air inside the balloon
and to perform work against atmospheric pressure (neglecting heat losses
from the balloon through its skin). Thus, the amount of heat that needs
to be supplied is equal to the change in balloon's enthalpy~$H$:
\begin{equation}
dQ = dH = dU + P_0 dV = {1\over{\gamma-1}}\, d(P_0V) + P_0 dV =
{\gamma\over{\gamma-1}}\, P_0 dV \, ,
\label{eq-balloon}
\end{equation}
where $\gamma$ is the adiabatic index of air. In our case of a magnetic loop
confined by external pressure, the adiabatic index is~$\gamma=2$. Therefore,
we expect that the total energy that needs to be supplied by the footpoint
motions to inflate the loop is twice the internal magnetic field energy within
the loop:
\begin{equation}
\mathcal{E} = E_{\rm magn} + W = 2\, E_{\rm magn} \, .
\end{equation}
In Appendix B we demonstrate that this is indeed so.
Note that, in addition to the work $W$ done against the external
gas pressure as the loop expands and increases its cross-section,
there is also the work done against the magnetic tension force as
the length of the loop is increased and the work done on the loop
by the external pressure-gradient force. However, as long as the loop
expands quasi-statically, always maintaining its equilibrium shape,
the last two forces precisely balance each other (see \S~\ref{subsec-shape}),
and so their corresponding works cancel out.
Using the formalism developed in the previous subsections,
we can now easily calculate the energy associated with a
given loop. Substituting equation~(\ref{eq-B_x=B_top}) into
equation~(\ref{eq-energy-2}) and using expression~(\ref{eq-Btop(L)})
for $B_{\rm top}(L')$, we get
\begin{equation}
\mathcal{E}(L)={\Delta\Psi\over{4\pi}} \int\limits_0^L\,B_{\rm top}(L')\,dL'=
{{\Delta\Psi^2}\over{4}}\
\int\limits_0^L \int\limits_{L'}^\infty \bar{F}(L'')\, dL'' \, dL' \, .
\label{eq-energy-3}
\end{equation}
The total magnetic energy in the corona is then
\begin{eqnarray}
E_{\rm tot} &=&
{1\over 2}\ \int\limits_0^\infty\ \bar{F}(L)\,\mathcal{E}(L) \, dL =
{{\Delta\Psi^2}\over{8}} \
\int\limits_0^\infty dL \int\limits_0^L dL' \int\limits_{L'}^\infty dL''
\ \bar{F}(L)\, \bar{F}(L'') \nonumber \\
&=& {{\Delta\Psi^2}\over{4}} \
\int\limits_0^\infty dL \int\limits_0^L dL'\ L'\, \bar{F}(L)\, \bar{F}(L') \, .
\label{eq-E_tot}
\end{eqnarray}
It is instructive to consider a case in which $\bar{F}(L)$ has a power law
tail, $\bar{F}\sim L^{-\alpha}$, truncated at some large $L_{\rm max}\gg H$.
Then, as can be seen from equation~(\ref{eq-energy-3}), the energy of
the largest loop, $\mathcal{E}(L_{\rm max})$, is almost independent of
$L_{\rm max}$ for $\alpha>2$, but starts to grow as~$L_{\rm max}^{2-\alpha}$
for $\alpha<2$.
In a similar manner, from equation~(\ref{eq-E_tot}) it follows that
the total coronal energy becomes dominated by the large-$L$ tail if
$\bar{F}(L)$ drops off sufficiently slowly, {\it i.e.}, $\alpha<3/2$.
In this case the total energy scales with~$L_{\rm max}$ as $E_{\rm tot}
\propto L_{\rm max}^{3-2\alpha}$ and may become much larger than the
potential magnetic field (whose characteristic magnetic scale height
is of order~$H$). Physically, we expect $L_{\rm max}$ to be at most
about the local disk radius, $r$, so that $E_{\rm tot}$ is bounded.
Finally, let us consider the angular momentum transfer by the coronal
magnetic field. The torque due to a single loop is given by
\begin{equation}
\Delta G(L,\theta) =
-\, {{\Delta\Psi}\over{4\pi}}\ |B_{\rm hor}|\ {{\Delta y\,\Delta r}\over L}=
-\, {{\Delta\Psi B_0}\over{8\pi}}\ b_{\rm top}(L)\ L\, \sin{2\theta} \, ,
\label{eq-loop-torque}
\end{equation}
and hence the total torque per unit disk area is
\begin{equation}
G = -\, {{\Delta\Psi B_0}\over{8\pi}}\ \int\int dL\, d\theta\
\bar{F}(L,\theta)\, b_{\rm top}(L)\ L\, \sin{2\theta} \, .
\label{eq-total-torque}
\end{equation}
Let is again consider the truncated power-law example, $\bar{F}\sim
L^{-\alpha(\theta)}$. For a fixed degree of anisotropy, {\it e.g.},
a fixed the characteristic angular scale~$|\sin{\theta_{\rm min}}|$
at which the function $\alpha(\theta)$ has a minimum~$\alpha_{\rm min}$,
the torque becomes dominated by large loops if $\alpha_{\rm min}<3/2$,
similar to~$E_{\rm tot}$ [where we used equation~(\ref{eq-Btop(L)}].
It then scales as~$L_{\rm max}^{3-2\alpha}$ and thus may become by
a factor $(L_{\rm max}/H)^{3-2\alpha}\gg 1$ larger than the usual
torque exerted directly by MRI turbulence within the disk, assuming
that the magnetic field at the base of the corona is comparable to
that in the disk. In reality, however, a decrease in $\alpha_{\rm min}$
may come hand-in-hand with an increase in the degree of anisotropy of
the distribution function~$\bar{F}(L,\theta)$ (see \S~\ref{subsec-results}),
manifested as a simultaneous decrease in $|\sin{\theta_{\rm min}}|$.
If this is the case, the torque amplification will be not as strong.
\section{The Loop Kinetic Equation}
\label{sec-kinetic-eqn}
In this section we discuss how to calculate the loop distribution function.
In particular, we construct the loop kinetic equation that governs the
evolution of this function.
\subsection{Physical Assumptions of our Model}
\label{subsec-assumptions}
In order to build a quantitative model of the magnetic field
in the corona, we need to make some specific assumptions about
the most important physical processes that govern the life of
individual coronal loops, including their interactions with each
other. These assumptions are the main building blocks of our model;
we shall discuss them in this section.
1) The Alfv\'en velocity in the corona is much faster than both the disk's
rotational velocity at the given radius and the thermal velocity of the
coronal gas; therefore, the corona is considered to be in a slowly-evolving
force-free magnetostatic equilibrium at all times and almost everywhere
(except for rapid rearrangements due to reconnection events, see below).
2) The disk is geometrically {\it thin}, with the gas scale-height
much smaller than the distance from the central object, $H\ll r$.
This gives us an important small parameter that can be used in the
analysis. For example, this assumption gives us an ``inertial range''
of spatial scales much smaller than~$r$ but much larger than~$H$.
This enables us to perform an analysis that is local in~$r$.
In particular, this means that we can neglect geometrical effects
resulting from cylindrical geometry when considering the flux-loop
expansion process. Note, however, that the validity of the thin-disk
assumption is questionable close to a black hole accreting near its
Eddington limit.
3) The disk is {\it differentially-rotating} ({\it e.g.}, keplerian).
As a result, coronal loops with radially-separated footpoints
are subject to continuous stretching in the toroidal direction.
This generates toroidal magnetic field whose pressure inflates
the loops and ultimately leads to the creation of a vertically-extended
corona (see below).
4) At any given time, the shape and the overall height of each loop
are determined by the magnetostatic equilibrium of the loop as if it
were confined by a stratified atmosphere with certain external isotropic
pressure~$P(z)$. In turn, this pressure represents the effective magnetic
pressure of all the neighboring loops, and we shall denote it as
$P=\bar{B}^2(z)/8\pi$. This equilibrium shape is maintained at all
times, since it adjusts on the Alfv\'en time scale, which is assumed to
be much faster than the disk rotation.
5) The disk is {\it turbulent} due to the usual internal MRI
(as opposed to coronal that may act simultaneously).
The characteristic spatial scale of the turbulence is~$H$,
and the characteristic time scale is~$\Omega_K$.
The important effects of the disk turbulence on the corona are:
5a) {\it Flux Emergence} plays a very important role
in the solar corona and, by analogy, is also believed
to be important in the case of the disk \cite{Galeev_Rosner_Vaiana79}.
We generally expect the emerging magnetic loops to be relatively small
in size (of order the disk thickness~$H$) and to have typical magnetic
fields of order~$\alpha_{\rm ss}^{1/2} B_{\rm eq}$, where the dimensionless
parameter $\alpha_{\rm ss}\sim 0.01-0.1$ is the \cite{Shakura_Sunyaev73}
viscosity coefficient and $B_{\rm eq}$ is the field strength that
corresponds to equipartition with the gas pressure inside the disk.
In addition, numerical studies of MRI turbulence
show that the toroidal field in the disk tends to be larger than
the radial field by a factor of 5-10. Thus, flux emergence is expected
to be anisotropic, with newly emergent loops elongated in the toroidal
direction by a factor of a few.
5b) Again, similar to the Sun, the disk turbulence leads to a
two-dimensional random walk of the coronal loops' footpoints
on the disk surface. We expect this random walk to be characterized
by spatial and temporal scales of the order of~$H$ and~$\Omega^{-1}$,
respectively.
However, similar to the process of flux emergence discussed above,
the random walk, in general, may be anisotropic, with characteristic
steps in the azimuthal direction being somewhat larger than in the
radial direction.
6) {\it Reconnection:}
In our model, two loops may {\it reconnect} with each other, forming
two new loops (see Fig.~\ref{fig-recn}). Thus, reconnection represents
a binary interaction between individual magnetic structures, analogous
to binary collisions between particles in a gas.
We shall assume that, once triggered, a reconnection event (a flare)
happens very quickly, essentially instantaneously on the orbital time
scale. This assumption can be justified by noting that the corona is
assumed to be a very low-density, and hence collisionless, environment.
Therefore, reconnection there proceeds in the Petschek-like fast
collisionless regime, enabled by anomalous resistivity or by the
two-fluid (Hall-MHD) effects.
The characteristic reconnection time-scale is then only by a factor
of 10-100 slower than the Alfv\'en crossing time~$\tau_A$.
Thus, since we assume that $V_A \gg V_K$, it is reasonable to expect
that the typical duration of coronal reconnection events may still be
fairly short compared with the orbital time-scale~$\Omega^{-1}$.
Then, to the extent that~$\Omega^{-1}$ is the main dynamical time-scale
in our problem, characterizing differential rotation, flux emergence,
and turbulent random walk, we can, for the purposes of our study,
regard reconnection between loops, once triggered, as being essentially
instantaneous.
Thus, we arrive at a picture in which magnetic loops evolve slowly
({\it i.e.}, on the orbital time scale), but from time to time they
suddenly and instantaneously reconnect. This picture is similar
to the observed behavior of solar coronal loops, where the characteristic
reconnection (or flare) time is typically much shorter than the typical
loop lifetimes. Thus, from the standpoint of viewing the corona as an
ensemble of many loops, reconnection events can be regarded as relatively
infrequent binary collisions between loops, analogous to the binary
collisions between particles in Boltzmann's gas.
An important corollary from this is that the footpoints of the loops
do not have time to move significantly during the reconnection event.
As we shall see in~\S~\ref{subsec-LKE}, this will give us the rules
that determine the footpoint separations of newly-formed loops.
We shall also assume that these newly-formed loops quickly assume
their equilibrium shapes (see above).
\begin{figure}[t]
\plotone{recn.eps}
\figcaption{Reconnection between two loops as a binary process.
\label{fig-recn}}
\end{figure}
\subsection{The Loop Kinetic Equation}
\label{subsec-LKE}
Based on the above assumptions, we shall now work out an
evolution equation for the loop distribution function~$F$.
We shall call this the {\it Loop Kinetic Equation} (LKE).
This equation should have the following terms, reflecting
the relevant physical processes:
1) Flux emergence/submergence acts as a source/sink of new coronal loops.
It can be modeled by a source term, $S(\mathcal{A})$, that describes the
rate at which the loops emerge into the corona, their characteristic sizes,
magnetic field strengths, etc., (or, in a more elaborate model, by specifying
the distributions of these quantities). Specifically, one can add loops at
some characteristic ``injection scale'', somewhat larger than the disk
thickness~$H$, and remove very small loops, say, of size~$H$ or less,
as it was done in the model by \cite{Hughes_Paczuski_etal03}.
In addition, in the ADC case, we expect flux emergence to be anisotropic,
with the emerging loops being by a factor of a few longer in the toroidal
direction than in the radial direction, as indicated by numerical
simulations \cite[\emph{e.g.}][]{Hirose_Krolik_Stone06}.
An alternative way to take flux emergence into account is {\it via}
the boundary conditions for $F$ at small scales (of order~$H$). This
is the view adopted in our model. This choice is justified by arguing
that the population of smallest loops comprising the ``magnetic carpet''
is predominantly determined by a detailed balance that is quickly
established with the magnetic fields in the disk itself. This process
turns over (operates) very quickly and hence the distribution of the
very small loops is basically independent of what happens in the
larger-scale corona.
2) Random footpoint motions due to the disk turbulence. Since we expect
the characteristic steps of this random walk to be relatively small
(of order~$H$, see \S~\ref{subsec-assumptions}) compared with the sizes
of most loops under consideration, we can employ a Fokker--Planck-like
approach to this process. This results in a diffusion operator
with the diffusion coefficient of the order of the \cite{Shakura_Sunyaev73}
$\alpha$-viscosity coefficient: $D\simeq \alpha_{\rm ss} c_s H \sim
\alpha_{\rm ss} \Omega H^2$.
In general, however, this diffusion may be anisotropic, with~$D$
being a tensor ({\it e.g.}, a diagonal tensor with $D_{yy}> D_{rr}$).
We expect the effect of the random walk to be relatively unimportant
for large loops, $L\gg H$.
3) Keplerian differential rotation leads to a secular evolution of~$\Delta y$,
\begin{equation}
{d\Delta y\over{dt}} = -\, {3\over 2}\, \Omega\, \Delta r \, .
\label{eq-shear}
\end{equation}
Here, $\Omega = \Omega_K(r)$ can be regarded as constant because
we consider spatial scales that are small compared with the disk
radius, $\Delta r \ll r$.
In the Loop Kinetic Equation this process is described by an advection
term, $(3/2)\, \Omega\, \Delta r \, (\partial F/\partial\Delta y)$.
4) Coronal MRI backreaction term~$\dot{F}_{\rm backreaction}$:
in a geostrophic approximation, this is obtained by balancing
the magnetic force on the footpoints (per unit area) with the
Coriolis force (also per unit area) due to the rotation of the
loop it induces:
\begin{equation}
2\, {{B_z {\bf B}_{\rm hor}}\over{4\pi}} =
2\, \Sigma\, [{\bf \Omega} \times \delta {\bf v}_{\rm backreaction}] \, ,
\end{equation}
in which ${\bf v}_{\rm backreaction}$ represents the departure
from the keplerian rotation velocity.
5) Interaction of two loops by reconnecting with each other,
yielding two new loops. This process is described by a binary
collision term~$\dot{F}_{\rm rec}$.
This is the most non-trivial term and we devote the entire
next subsection (\S~\ref{subsec-recn-term}) to a detailed
discussion of it.
At the end, we arrive at the following general form of the kinetic equation
for the loop distribution function:
\begin{eqnarray}
{\partial F\over{\partial t}}(\Delta r,\Delta y,t) &=&
S(\Delta r,\Delta y) +
\biggl( D_{rr}\,{\partial^2\over{\partial \Delta r^2}} +
D_{yy}\,{\partial^2\over{\partial \Delta y^2}} \biggr) F \nonumber \\
&+& {3\over 2}\,\Omega \Delta r\,{{\partial F}\over{\partial\Delta y}} +
\dot{F}_{\rm backreaction} + \dot{F}_{\rm rec}
\label{eq-LKE}
\end{eqnarray}
The simplest meaningful case of this equation is when one neglects
the source, diffusion, and feedback terms and looks for a {\it steady
state} that is produced by the balance between Keplerian shear and
reconnection:
\begin{equation}
{3\over 2}\, \Omega \Delta r\,
{{\partial F}\over{\partial \Delta y}} = \dot{F}_{\rm rec} \, .
\label{eq-steady-state}
\end{equation}
Our approach will thus be analogous to, and can be regarded as an
extension of, the previous work by \cite{Tout_Pringle96}, whose
main goal was to study the formation of large magnetic structures
{\it via} the reconnective ``inverse cascade'' in the corona.
Following them, we also represent the coronal magnetic field by
an ensemble of flux loops described by a distribution function.
However, our model is more general and uses more realistic physics.
We take into account a number of effects ignored in \cite{Tout_Pringle96},
such as inflation of the loops as they are stretched by the Keplerian
differential rotation. Also, in their model reconnection was taking place
only at the disk surface and thus one of two newly reconnected loops was
vanishingly small and was assumed to just disappear; as a result,
the reconnection process did not conserve the number of loops.
We, on the other hand assume that reconnection occurs higher in
the corona, and hence two new loops form and the loop number is
conserved (similar to the model by \cite{Hughes_Paczuski_etal03}).
\subsection{Reconnection Described as a Collision Integral}
\label{subsec-recn-term}
Two loops $\mathcal{A}$ and~$\mathcal{B}$ may interact by reconnecting
with each other and forming two new loops~$\mathcal{C}$ and~$\mathcal{D}$
as a result (see Fig.~\ref{fig-recn}). Following \cite{Tout_Pringle96},
we shall describe this process by a nonlinear binary-collision integral,
similar to the collision integral in the Boltzmann kinetic equation.
In reality, of course, interaction between loops is more complicated
and so such a description is oversimplified. Moreover, magnetic loops
fill up the entire coronal space, and so they resemble more a non-ideal
liquid rather than an almost ideal rarefied gas with infrequent binary
encounters. Nevertheless, we believe that this binary-collision representation
of reconnection can lead to some valuable physical insight into the
complicated dynamics of the coronal magnetic field.
The Boltzmann collision integral can be split into two: the source term and
the sink term. The sink term $\dot{F}_{\rm coll,-}(\mathcal{A})$ describes
the rate of reduction in the number of loops of a given type~$\mathcal{A}$
due to reconnection between these loops and all other loops.
The source term $\dot{F}_{\rm coll,+}(\mathcal{A})$ describes
the rate of increase in the number of loops of type $\mathcal{A}$
when they are a product of reconnection of other-type loops.
By ``type'' we here mean a set of loops with the same values
of their primary parameters $(L,\theta)$ or $(\Delta r,\Delta y)$.
Thus, each of these terms is a quadratic integral operator, with
a kernel that depends both on the types of the two loops and also
on their relative position (see below). Thus, we can write the
reconnection term schematically as
\begin{equation}
\dot{F}_{\rm rec}(\mathcal{A}) =
\dot{F}_{\rm rec,-}(\mathcal{A}) + \dot{F}_{\rm rec,+}(\mathcal{A}) \, ,
\label{eq-Fdot_rec}
\end{equation}
where
\begin{eqnarray}
\dot{F}_{\rm rec,-}(\mathcal{A}) &=& - \int d\mathcal{B}\
Q_{\mathcal{A}\mathcal{B}} \, F(\mathcal{A}) \, F(\mathcal{B})\, ,
\label{eq-Fdot-} \\
\dot{F}_{\rm rec,+}(\mathcal{A}) &=&
{1\over 2}\, \int\int d\mathcal{C}\, d\mathcal{D}\
Q_{\mathcal{C}\mathcal{D}\rightarrow\mathcal{A}} \
F(\mathcal{C})\,F(\mathcal{D})\, .
\label{eq-Fdot+}
\end{eqnarray}
Here, $d\mathcal{B}\equiv d\Delta r_B\, d\Delta y_B$, etc.,
The kernels $Q_{\mathcal{A}\mathcal{B}}$ in the sink term and
$Q_{\mathcal{C}\mathcal{D}\rightarrow\mathcal{A}}$ in the source
term are related {\it via}
\begin{equation}
Q_{\mathcal{A}\mathcal{B}} = {1\over 2}\, \int d\mathcal{C}\
Q_{\mathcal{A}\mathcal{B}\rightarrow\mathcal{C}} \, .
\end{equation}
Using this relationship, the two terms can be combined as
\begin{equation}
\dot{F}_{\rm rec}(\mathcal{A}) =
{1\over 2}\, \int \int d\mathcal{C}\, d\mathcal{D}\ F(\mathcal{D})\,
\biggl[\, Q_{\mathcal{C}\mathcal{D}\rightarrow\mathcal{A}}\, F(\mathcal{C})-
Q_{\mathcal{A}\mathcal{D}\rightarrow\mathcal{C}}\, F(\mathcal{A})\, \biggr]\, .
\label{eq-Fdot_rec-2}
\end{equation}
In order to go from this general expression to a specific
operational procedure, we need to formulate the rules that
govern the reconnection process. Indeed, the two new loops
$\mathcal{C}$ and~$\mathcal{D}$ formed as products of reconnection
between two loops~$\mathcal{A}$ and~$\mathcal{B}$ cannot be arbitrary
and certain selection rules must be applied. More specifically, from
the assumption that reconnection is instantaneous on the orbital time
scale, it follows that the footpoints of the reconnecting loops do
not move significantly during the reconnection event. Only the way
they are connected to each other changes. Therefore, the primary
parameters ({\it i.e.}, footpoint separations) of loops~$\mathcal{C}$
and~$\mathcal{D}$ are uniquely determined by the footpoint positions
of the two incoming loops~$\mathcal{A}$ and~$\mathcal{B}$:
\begin{eqnarray}
\Delta r_{C} &\equiv & r_{C-} - r_{C+} = r_{A-} - r_{B+} \, ,
\label{eq-rec-rule-C} \\
\Delta r_{D} &\equiv & r_{D-} - r_{D+} = r_{B-} - r_{A+} \, ,
\label{eq-rec-rule-D}
\end{eqnarray}
and similarly for $\Delta y_{C}$, $\Delta y_{D}$.
Here, $A_+$, $A_-$ are the positive and negative polarity footpoints
of the loop~$\mathcal{A}$, {\it etc}. Mathematically, these rules play
a role similar to the momentum and energy conservation conditions
for particle collisions in kinetic theory of gases; they enter as
$\delta$-functions in the interaction integral in our loop kinetic
equation. Thus, one can easily see that the parameters of the new
loops depend not only on the parameters of the old loops, but also
on the positions of~$\mathcal{A}$ and~$\mathcal{B}$ relative to each
other (see below).
The kernel $Q_{\mathcal{A}\mathcal{B}}$ in equation~(\ref{eq-Fdot-}) is
the probability rate ({\it i.e.}, probability per unit time) that two loops
of types~$\mathcal{A}$ and~$\mathcal{B}$ will come together and reconnect.
Thus, $Q_{\mathcal{A}\mathcal{B}}$ describes the rate of reconnection events
(the number of such events per unit time). This should not be confused with
the concept of ``reconnection rate'', a widely-used term in reconnection
research with a completely different meaning.
Now let us discuss on which physical parameters~$Q_{\mathcal{A}\mathcal{B}}$
should generally depend. Whereas Tout \& Pringle (1996) just took
$Q_{\mathcal{A}\mathcal{B}}={\rm const}$, we want to develop a more
realistic and more sophisticated model, taking into account several
important factors.
First, notice that~$Q_{\mathcal{A}\mathcal{B}}$ has dimensions of
$[{\rm cm}^2/{\rm sec}]$. Based on dimensional arguments, it should
then be proportional to the characteristic rate at which the coronal
magnetic field is reconfigured. In addition, it should also reflect
the fact that larger loops have larger ``interaction cross-section''
(see below), and thus should be roughly proportional to the squares
of loop sizes.
Let us first address the characteristic reconfiguration time-scale.
The only fundamental dynamical time-scale in the corona, {\it i.e.},
the time-scale on which the corona, seen as an ensemble of elementary
coronal structures, rearranges itself, is the orbital time, $\Omega^{-1}$
(or the inverse of the shear rate, $3/2\Omega$, which is not independent).
This means that, if we represent the evolution by a sequence of discrete
steps, each step representing a noticeable change in the relative position
and/or orientation of the coronal elements, then the most appropriate choice
for duration of these steps is of order~$\Omega^{-1}$. Therefore, in general,
$Q$ should scale with~$\Omega$. Next, if we follow a given magnetic element,
at each new step there will be a certain probability $\kappa<1$ that the
resulting new magnetic configuration around this element becomes favorable
for reconnection of this element with another. We shall treat~$\kappa$ as
a constant number, independent of the loops primary parameters. Thus, the
overall rate at which the loops are disrupted through reconnection with
other elements should be proportional to~$\kappa\Omega$:
\begin{equation}
Q_{\mathcal{A}\mathcal{B}} = \kappa\, \Omega\, \sigma_{AB} \, ,
\end{equation}
where we introduced the ``reconnection cross-section''~$\sigma_{AB}$.
The cross-section~$\sigma_{AB}$ should in some way scale with the loop sizes.
It involves contributions from all possible relative positionings of
the two interacting loops for which the two loops ``effectively intersect''.
We shall describe this relative positioning by two impact parameters,
$b_\parallel$ and~$b_\perp$, defined as the offsets between the centers
of the two loops in the direction parallel and perpendicular to
loop~$\mathcal{A}$, respectively. We shall assume that once the impact
parameters are in a range such that the two loops ``effectively intersect'',
the probability that these loops will reconnect is constant, independent
of their positions or their parameters. Furthermore, for simplicity we
shall assume~$b_\parallel$ and~$b_\perp$ to be uniformly-distributed
independent random variables. Thus, $\sigma_{AB}$ is just equal to
the ``interaction area'' in the $(b_\parallel,b_\perp)$ space that
corresponds to an ``effective intersection'' of the given two loops:
\begin{equation}
\sigma_{AB} = \int d\sigma_{AB} = \int\int db_\parallel \, db_\perp \, .
\end{equation}
But what do we mean by ``effectively intersecting''?
If the two loops are approximated by their central lines
(one-dimensional objects), then the set of values $(b_\parallel,b_\perp)$
for which they intersect is also one-dimensional (a line segment),
and thus has measure zero in the two-dimensional $(b_\parallel,b_\perp)$
space. In other words, the probability that two randomly-drawn lines
intersect is zero in the three-dimensional space. Therefore,
to get a meaningful result, we need to take into account finite
thicknesses of the loops. In particular, we shall say that two
loops ``effectively intersect'' when the closest distance between
their central lines is less than a certain fraction of the combined
loop thicknesses at the intersection height.
Operationally, for a given value of~$b_\parallel$, say, we can introduce
the $y$-cross-section $\sigma_{AB,\perp}(b_\parallel)$ as the spread in
the values of~$b_\perp$ which result in an effective intersection of two
given loops~$\mathcal{A}$ and~$\mathcal{B}$. We can then write
\begin{equation}
\sigma_{AB} = \int \sigma_{AB,\perp}(b_\parallel)\, db_\parallel \, .
\label{eq-sigma_perp}
\end{equation}
where the integral is taken over the range of impact parameters~$b_\parallel$
for which an intersection between loops~$\mathcal{A}$ and~$\mathcal{B}$
is at all possible.
Correspondingly, the sink-term part of the reconnection integral
can be written as
\begin{equation}
\dot{F}_{\rm rec,-}(\mathcal{A}) =
-\, \kappa\Omega\ \int\int d\mathcal{B}\ db_\parallel\,
\sigma_{\mathcal{A}\mathcal{B},\perp}(b_\parallel)\,
F(\mathcal{A})\, F(\mathcal{B})\, .
\label{eq-Fdot-2}
\end{equation}
Now let us consider the source term~(\ref{eq-Fdot+}).
Employing the arguments given earlier in this subsection, we
introduce the cross-section for two given loops~$\mathcal{C}$
and~$\mathcal{D}$ to reconnect giving a loop $\mathcal{A}$ as
a result: $Q_{\mathcal{C}\mathcal{D}\rightarrow\mathcal{A}}=
\kappa\,\Omega\,\sigma_{CD\rightarrow A}$. We can then write
\begin{equation}
\dot{F}_{\rm coll,+}(\mathcal{A}) =
{{\kappa\Omega}\over 2}\ \int\int d\mathcal{C}\, d\mathcal{D}\,
\sigma_{\mathcal{C}\mathcal{D}\rightarrow\mathcal{A}}\
F(\mathcal{C})\, F(\mathcal{D}) \, .
\label{eq-Fdot+2}
\end{equation}
The $\mathcal{D}$ integral in this equation is taken over all the loops
that can yield a loop of type~$\mathcal{A}$ as a result of reconnection
with a loop of a given type~$\mathcal{C}$. Note that, whereas one did not
need to know reconnection product loops to compute the sink term~(\ref
{eq-Fdot-2}), in order to calculate the source term, this knowledge is
in fact necessary. It is contained in our ``reconnection rules'', such
as those given by equations~(\ref{eq-rec-rule-C})--(\ref{eq-rec-rule-D}).
For definiteness, let us consider the case when the resulting
loop~$\mathcal{A}$ starts from the ``+'' footpoint of loop~$\mathcal{C}$
and ends at the ``-'' footpoint of loop~$\mathcal{D}$.
Then, for given~$\mathcal{C}$ and~$\mathcal{A}$, there is a well-defined
range of impact parameters~$b_\parallel$ for which one can find one
(or sometimes two) loop~$\mathcal{D}=\mathcal{D}_{\mathcal{A}-
\mathcal{C}}(b_\parallel)$ that intersects loop~$\mathcal{C}$
and ends at the end footpoint of loop~$\mathcal{A}$. Thus, in
principle, for a given~$b_\parallel$ one can formulate the rules
that relate the primary parameters~$\Delta r_D$ and~$\Delta y_D$
to those of loops~$\mathcal{C}$ and~$\mathcal{A}$. We shall denote these
relationships by $\Delta r_D^{A-C}(b_r)$ and $\Delta y_D^{A-C}(b_\parallel)$.
In general, there may be one or two such solutions.
Recalling now that flux tubes have a finite thickness,
we have, by analogy with equation~(\ref{eq-sigma_perp}),
\begin{equation}
\sigma_{CD\rightarrow A} =
2\, \int db_\parallel\, \sigma_{CD,\perp}(b_\parallel)\
\delta[\mathcal{D}-\mathcal{D}_{\mathcal{A}-\mathcal{C}}(b_\parallel)] \, ,
\end{equation}
where
$\delta[\mathcal{D}-\mathcal{D}_{\mathcal{A}-\mathcal{C}}(b_\parallel)] \equiv
\delta[\Delta r_D - \Delta r_D^{A-C}(b_\parallel)]\, \times
\delta[\Delta y_D - \Delta y_D^{A-C}(b_\parallel)]$.
The factor~2 accounts for the fact that in the preceding paragraph
we considered only one half of all possible configurations, requiring
the starting footpoint of loop~$\mathcal{A}$ to be the starting footpoint
of loop~$\mathcal{C}$. For each such configuration there will also
be an identical contribution from interchanging loops~$\mathcal{C}$
and~$\mathcal{D}$.
Substituting this cross-section into our expression~(\ref{eq-Fdot+2})
for~$\dot{F}_{\rm coll,+}(\mathcal{A})$, and using the $\delta$-function
to integrate over~$d\mathcal{D}=d\Delta r_D d\Delta y_D$, we get
\begin{equation}
\dot{F}_{\rm coll,+}(\mathcal{A}) =
\kappa\, \Omega \int \int d\mathcal{C}\, db_\parallel\
\sigma_{CD_{A-C},\perp}(b_\parallel)\,
F(\mathcal{C})\, F[\mathcal{D}_{\mathcal{A}-\mathcal{C}}(b_\parallel)]\, .
\label{eq-Fdot+3}
\end{equation}
Combining equations~(\ref{eq-Fdot-2}) and~(\ref{eq-Fdot+3}), we can write:
\begin{eqnarray}
\dot{F}_{\rm rec}(\mathcal{A}) &=&
\kappa\, \Omega\ \int\int d\mathcal{C}\ db_\parallel\, F(\mathcal{C})\,
\nonumber\\
&\times& \biggl[F[\mathcal{D}_{\mathcal{A}-\mathcal{C}}(b_\parallel)]
\sigma_{CD_{A-C},\perp}(b_\parallel) - F(\mathcal{A})
\sigma_{AC,\perp}(b_\parallel)\biggr] .
\label{eq-Fdot_rec-3}
\end{eqnarray}
\subsubsection{Effect of Finite Loop Thickness}
For a given value of~$b_\parallel$, one first finds the coordinates
$[r(b_\parallel),y(b_\parallel),z(b_\parallel)]$ of the point where
the central lines of the two loops would intersect. After that, one
calculates the $\perp$-extent of each of the two loops at this point,
which hence gives one~$\sigma_{AB,\perp}(b_\parallel)$.
When doing this, one should take into account the following important effect.
If the loops have no internal twist, as we assume here, the longitudinal
magnetic field is approximately constant
across a loop and is roughly equal to the characteristic magnetic
field $\bar{B}(z)$ at a given height~$z$.
Then, by flux conservation, the cross-section of the loop (normal
to its central line) varies along its length, and, in particular,
may increase greatly at large heights if $\bar{B}(z)$ drops off rapidly.
More precisely, approximating a loop's cross-section at a given height~$z$
as a circle of some radius~$d(z)$, we can estimate this radius as
$d(z)\sim [\Delta\Psi/\pi\bar{B}(z)]^{1/2}= d_0 [\bar{B}(z)/B_0]^{-1/2}$.
As long as the loop is slender, $d\ll L$, this should be a good approximation.
Thus, if the intersection point lies high above the disk, so that
$\bar{B}(z)\ll B_0$, the characteristic loop thickness is much larger
than near the disk surface. Consequently, the reconnection cross-section
is increased, which has important implications for the ``inverse
cascade'' of magnetic loops in the corona. It is also interesting to
note that this process is controlled by the self-consistent field~$\bar{B}(z)$.
[Note that, in the case of twisted loops this effect is not as profound
and a more accurate approximation is probably given by
$d(z)\simeq d_0={\rm const}$ (see~\S~\ref{subsec-twisted}).]
Thus, typically we expect $\sigma_{AB,\perp}$ to scale as
\begin{equation}
\sigma_{AB,\perp}(b_\parallel) \sim 2\, d[z(b_\parallel)]
\sim 2\, \sqrt{{\Delta\Psi}\over{\pi\bar{B}[z(b_\parallel)]}}
\sim 2\, d_0 \sqrt{B_0\over{\bar{B}[z(b_\parallel)]}} \, .
\label{eq-sigma_ABperp-2}
\end{equation}
For example, if $\mathcal{B}$ is the smaller of the two loops,
we expect the typical interaction height to be of the order of
this loop's height~$Z_B$.
Correspondingly, we expect $\sigma_{AB,\perp} \sim d_0
b^{-1/2}(Z_B)$, where $b(z)\equiv \bar{B}(z)/B_0$.
This estimate for $\sigma_{AB,\perp}(b_\parallel)$ will be roughly
valid for almost the entire allowed range of~$b_\parallel$,
which is of order $4\,\Delta r_B$.
To sum up, larger loops have larger cross-sections for reconnection,
for two complementary reasons. First, the cross-section is enhanced
because larger loops have larger range of impact parameters
(in the radial direction, say) for which intersection of
their central lines is possible. Second, larger loops
extend to, and may interact at, larger heights, where
the mean magnetic field is weaker and hence where the
loops become fatter. They thus have a greater chance
of overlapping with each other. As a result, the reconnection
cross-section scales with the size~$L_B$ of the {\it smaller}
of the two loops as
\begin{equation}
\sigma_{AB} \sim L_B\, d_0\, b^{-1/2}(Z_B)\, ,
\end{equation}
[for $L_B\gg d(Z_B)$]. We thus see that the function~$\bar{B}(z)$
affects the evolution of the loop distribution function. Since,
according to~\S~\ref{subsec-Bbar}, $\bar{B}(z)$ is itself determined
by the distribution function, this means that determining these two
functions together in a self-consistent way requires an iterative
procedure.
What is important here, is that larger loops have a tendency to
reconnect with each other quickly, probably leading to a rapid
``inverse cascade'' to even larger loops. In addition, the fact
that reconnection with large loops cannot be neglected suggests
that a Fokker-Planck-like approximation to the reconnection term
will not work. This is because reconnection events that lead to
large changes in loop parameters are important and so our collision
integral cannot be described by a differential diffusion-like
operator.
\section{Numerical Solution}
\label{sec-numerical}
\subsection{Numerical Setup}
\label{subsec-setup}
We solve the LKE numerically and obtain a steady state solution.
For simplicity we leave out the turbulent diffusion term, the coronal
backreaction term, and the source term (flux emergence is then treated
{\it via} the boundary condition at small scale, see \S~\ref{subsec-bc}).
Thus, we include only the two processes which we believe
are the most important: Keplerian shear and reconnection
between loops. Correspondingly, we aim here at investigating
the effect of the relative importance of these two processes
on the steady-state loop distribution function.
In the numerical implementation we work in the $(L,\theta)$ parameter
space where $L=(\Delta r^2+\Delta y^2)^{1/2}$ is the distance between
the footpoints and $\theta$ is the angle the vector ($\Delta r,\Delta y$)
makes with the toroidal direction, measured clockwise:
$\tan\theta = \Delta r/\Delta y$.
We use a grid that is uniform in $\theta$ (between 0 and $2\pi$)
but logarithmic in~$L$. The ${L}$~grid spans from $L_{\rm min}=1$
to some $L_{\rm max}\gg 1$ (usually we take $L_{\rm max}$ to be~10
or~20) in length units such that $H\simeq 1$.
The advection term resulting from Keplerian shear is very easy to implement,
we just use one-sided derivatives.
The reconnection collision integral is obviously more complicated and we
devote the rest of this subsection to our numerical implementation of it.
At each time step we go over all possible pairs of loops~$\mathcal{A}$
and~$\mathcal{B}$. Furthermore, for each given pair we go over all
possible reconnecting configurations, distinguished by the impact
parameter~$b_\parallel$, defined as the displacements between the
centers of the two loops in the direction along loop~$\mathcal{A}$
(see \S~\ref{subsec-recn-term}). Thus, at each timestep, we are
performing a five-dimensional (5D) integration, which makes increasing
resolution extremely numerically costly.
We found, however, that numerical convergence is very good and
relatively modest resolution suffices. To study numerical convergence,
we performed calculations with $N_L=20$ and~40 points in~$\log{L}$,
$N_\theta=20$, 40, 60, and~80 points in~$\theta$, and 30, 50, or~100
points in~$b_\parallel$. We found the resulting~$F(L,\theta)$ to
be essentially unchanged, although small values of $\kappa$ required
a higher $\theta$-resolution for convergence (see below).
For simplicity, in our treatment of reconnection, we assume the loops
to be {\it semi-circular} in shape, instead of using equilibrium shapes
discussed in~\S~\ref{subsec-shape}. This simplification has two benefits.
First, since the shapes are described by relatively simple analytical
expressions, we can derive explicit analytical relationships expressing
the parameters of the two product loops in terms of the parameters of
the interacting loops and the impact parameter (we call these
relationships the ``reconnection rules''). Having such expressions
in an explicit form greatly simplifies the numerical procedures.
The second advantage of the semicircular approximation stems from
the observation that, because all the loops have the same shape, the
reconnection rules essentially depend only on the ratio of loop sizes,
whereas their dependence on the absolute loop size is a trivial rescaling.
Likewise, the reconnection process is, by itself, isotropic, {\it i.e.},
does not depend on the absolute orientation of the loops, only on the
angle between them.
This enables us to to reduce the analysis of reconnection between
two given loops $\mathcal{A}$ and~$\mathcal{B}$ to considering
a template that corresponds to the given angle between the loops
and the given length ratio~$L_B/L_A$. That is, we can analyze
reconnection between any two loops in a rotated and rescaled
system of coordinates, $(x',y')$, in which one of the loops
(loop $\mathcal{A}$ for definiteness) is in the positive $x'$-direction,
{\it i.e.}, has $\theta_A'=\pi/2$ and has unit length, $L_A'=1$.
In practice, we first (even before we start the evolution of~LKE),
create a lookup table describing the reconnection rules for interaction
of this loop~$\mathcal{A}'$ with all other loops~$\mathcal{B}'$.
The lookup table is~3D, two of the coordinates being~$\theta_B'$
and~$L_B'$, and the third coordinate being the rescaled impact
parameter~$b_\parallel'$; the latter lies within the range from
$-(1+\Delta x_B')/2$ to $(1+\Delta x_B')/2$, where $\Delta x_B'=
L_B'\sin\theta_B'$. For each given~$\theta_B'$, $L_B'$ and~$b_\parallel'$,
it is just a matter of simple algebra to figure out the perpendicular
displacement $b_\perp'$ corresponding to the intersection between the
semi-circular loops.
Simultaneously one finds the 3D position of the intersection point between
loops~$\mathcal{A'}$ and~$\mathcal{B'}$ for given~$b_\parallel'$ (including
the height of the reconnection point). One can then readily deduce the
parameters of the two product loops~$\mathcal{C'}$ and~$\mathcal{D'}$.
It is easy to see that the problem reduces to quadratic equations and hence
for each~$\theta_B'$, $L_B'$, and~$b_\parallel'$ there may be zero, one, or
two solutions.
During the time evolution we use this table as follows. For each given
pair of loops~$\mathcal{A}$ and~$\mathcal{B}$ we find their corresponding
template pair by rotating them by $\theta_A-\pi/2$ and rescaling the loop
sizes by~$L_A$. We also rescale the impact parameter by~$L_A$, {\it i.e.},
$b_\parallel'=b_\parallel/L_A$.
We then use the template table to find the two template product loops
and we transform them back by multiplying by~$L_A$ and rotating by
$\pi/2-\theta_A$ to find the actual product loops~$\mathcal{C}$
and~$\mathcal{D}$. Having a grid uniform in~$\theta$ and~$\log{L}$
makes this procedure especially convenient and straightforward.
Once the product loops are found, one also needs to figure out
the reconnection cross-sections corresponding to given $\mathcal{A}$,
$\mathcal{B}$, and~$b_\parallel'$. In accordance with the above
discussion, $d\sigma_{AB}(b_\parallel')$ is proportional to
$d b_\parallel=L_A\,(1+\Delta x_B')/N_b)$ and to the loop combined
deprojected thickness~$db_\perp'$ at the intersection point. Apart
from simple geometrical projection factors, $db_\perp'$ is proportional
to $d(z) \sim d_0 [B_0/\bar{B}(z)]^{1/2}$, where~$z$ is the height of
the reconnection point above the disk. This factor enhances the
reconnection probability for large loops intersecting at large
heights. Since $\bar{B}(z)/B_0$ itself depends on~$F(L,\theta)$,
this procedure requires iteration.
Once the reconnection cross-section $\sigma_{AB}(b_\parallel')$ is
found, one can proceed to evolve the number of incoming~($\mathcal{A}$
and~$\mathcal{B}$) and product ($\mathcal{C}$ and~$\mathcal{D}$) loops.
Namely, $F(\mathcal{A})d\mathcal{A}$ and~$F(\mathcal{B})d\mathcal{B}$
are reduced by
\begin{equation}
F(\mathcal{A})\, F(\mathcal{B})\, d\mathcal{A}\, d\mathcal{B}\
L_A\, db_\parallel'\ \kappa\, \Omega\,\sigma_{AB}(b_\parallel')\, \Delta t \,,
\end{equation}
and $F(\mathcal{C})\,d\mathcal{C}$ and $F(\mathcal{D})\,d\mathcal{D}$
are increased by the same amount at each timestep~$\Delta t$.
Finally, we varied the initial conditions and found that our resulting
steady state solutions are insensitive to them.
\subsection{Boundary Conditions}
\label{subsec-bc}
For various physical reasons, we expect our model to break down
at both small scales and large scales. We thus need to discuss how
to prescribe the boundary conditions for the~LKE at both of these
scales.
At small scales, $L\sim H$, the model is expected to become invalid
because the magnetic field is no longer force-free, as plasma pressure
starts to become dynamically important at small heights. In addition,
the characteristic thickness of magnetic structures near the disk
surface is expected to be of order~$H$ and hence small loops with $L\sim H$
are not going to remain slender as our model assumes. Thus we need to
understand what is a plausible way to describe these small loops.
At small scales the dynamics of magnetic loops is strongly affected
by rapid flux-exchange ({\it e.g.} flux-emergence) processes with the
turbulent disk. One can therefore argue that the overall distribution
of the smallest loops is largely determined by the detailed balance
equilibrium that is rapidly established between smallest loops
and the turbulent disk. Characterizing this intense interaction
with the disk turbulence and calculating the small-scale distribution
function is completely beyond our present model. It would require thorough
understanding of 3D MHD turbulence in stratified disks with simultaneous
actions of the MRI and Parker instability \citep{Tout_Pringle92};
most likely, this task will be accomplished using sophisticated numerical
simulations. However, what matters for our model here is that we expect
that the number of small loops be essentially insensitive to what happens
to larger coronal loops.
This means that we can mimic the disk-corona interaction by setting-up
a Dirichlet-type boundary condition at small scales, {\it i.e.}, by
prescribing the distribution function at some small-cutoff scale
$L_{\rm min}\sim H$:
\begin{equation}
F(L=L_{\rm min},\theta) = F_1(\theta) \, .
\end{equation}
(In our model we set $L_{\rm min}=1$.)
Notice that $F_1(\theta)$ is in general expected not to be isotropic
because emerging loops will be preferentially azimuthally elongated
($|\sin\theta|\ll 1$). In the present simulations, however, we take
it to be isotropic, $F_1(\theta)={\rm const}$.
Our model also breaks down at large scales. In particular, when a loop's
size becomes comparable to the disk radius~$r$, the local cartesian geometry
adopted here is no longer applicable, and toroidal effects become important.
This leads to a much faster expansion of stretched loops, resulting in a
complete opening of the field, as discussed in~\S~\ref{subsec-recn-role}.
In the future, we plan to implement a physically realistic way of
treating the opening of large loops by incorporating open field lines
into our model (see~\S~\ref{subsec-recn-collisionless}).
In the present model, however, we just introduce some large cut-off
scale, $L_{\rm max}$, and we set boundary conditions at this scale.
We experimented with two types of boundary conditions:
in one case, we set $F(L_{\rm max})=0$, {\it i.e.}, we simply remove all
loops that reach the cut-off scale. In the other case, we just limit the
growth of loops beyond $L_{\rm max}$; {\it i.e.}, as soon as a loop's
length exceeds~$L_{\rm max}$, we reset it back to $L_{\rm max}$.
This leads to a gradual pile-up of loops near~$L_{\rm max}$, but does
not affect the loop distribution on smaller scales; in particular, it
does not change the power law tail for sizes just slightly smaller
than~$L_{\rm max}$.
\subsection{Results}
\label{subsec-results}
\subsubsection{Distribution Function}
We performed a series of calculations with several different values of~$\kappa$
($\kappa=\infty$, 1, 0.3, 0.1, 0.03, 0.02, 0.01, 0.005, and~0.003). Some
of the resulting steady state loop distribution functions for our fiducial
resolution $N_L=40$, $N_{\theta}=80$ and for the large-scale boundary
condition $F(L_{\rm max}, \theta)=0$, $L_{\rm max}=10$, are presented
in Figures~\ref{fig-radial-F}--\ref{fig-alpha-theta}.
In particular, Figures~\ref{fig-radial-F} and~\ref{fig-azimutal-F}
show (in log-log coordinates) $F$ as a function of~$L$ for purely
radial loops ($\theta=\pi/2$) and purely toroidal loops ($\theta=0$),
respectively.
In the case $\kappa=\infty$ (keplerian shear turned off), the distribution
function is isotropic. This is of course expected, since loop-loop
reconnection --- the only process determining the distribution function
in this case --- by itself is independent of the absolute orientation of
the reconnecting loops. As the frequency of reconnection events relative
to shear, quantified by~$\kappa$, is decreased, a given loop (especially
if it has a large~$|\Delta r|$) experiences, on average, larger stretching
in the toroidal direction by the shear before it undergoes reconnection
with another loop. As a result, the loops become predominantly azimuthal
and the loop distribution function becomes more and more anisotropic:
$F(\theta=\pi/2,L)$ steepens, whereas $F(\theta=0,L)$ becomes shallower
with decreasing~$\kappa$.
\begin{figure}[h]
\plotone{radial-F-lke.eps}
\figcaption{Distribution function for purely radial loops ($\theta=\pi/2$),
for dimensionless reconnection parameter $\kappa=1.0$ (pluses), 0.3 (crosses),
and~0.1 (asterisks).
\label{fig-radial-F}}
\end{figure}
\begin{figure}[h]
\plotone{azimutal-F-lke.eps}
\figcaption{Distribution function for purely toroidal loops ($\theta=0$),
for $\kappa=1.0$ (pluses), 0.1 (crosses), and~0.01 (asterisks).
\label{fig-azimutal-F}}
\end{figure}
Because our problem lacks a preferred length-scale between~$L_{\rm min}$
and~$L_{\rm max}$, we find that, along each ray $\theta={\rm const}$,
$F(L)$ is well described by a power law with the orientation-angle-dependent
exponent:
\begin{equation}
F(L,\theta) \sim L^{-\alpha_\kappa(\theta)} \, .
\end{equation}
Figure~\ref{fig-alpha-theta} presents the function~$\alpha_\kappa(\theta)$
for $\kappa=1$, 0.3, 0.1, and~0.03.
\begin{figure}[h]
\plotone{power-law-of-theta.eps}
\figcaption{Power-law exponent $\alpha_\kappa(\theta)$ for
$\kappa=1.0$, 0.3, 0.1, 0.03.
\label{fig-alpha-theta}}
\end{figure}
Finally, in order to estimate the total magnetic energy in the corona
and the total magnetic torque (see~\S~\ref{subsec-energy}), one needs
to know the $\theta$-integrated distribution function,
\begin{equation}
\bar{F}(L) \equiv \int\limits_0^{2\pi} F(L,\theta)\, d\theta \, .
\end{equation}
In Figure~\ref{fig-averaged-F-q} we plot $\bar{F}(L)$ for $\kappa=1.0$,
0.3, 0.1, 0.03, 0.01, and~0.005 in log-log coordinates. In general, of
course, one cannot expect a $\theta$-integral of exponents
$L^{-\alpha(\theta)}$ to be itself a power law of~$L$.
However, we find that the integral is strongly dominated by the range
of values~$\theta$ corresponding to the ridge in~$F(L,\theta)$, and so
a power law $\bar{F}(L)\sim L^{-\bar{\alpha}(\kappa)}$ is actually a
reasonably good approximation, especially for relatively large values
of~$\kappa$. Because the power-law fit is not ideal, there is some
degree of uncertainty in determining the value of~$\bar{\alpha}$;
we estimate the characteristic error to be~$\lesssim 0.1$.
The dependence of the power-law exponent~$\bar{\alpha}$ on the reconnection
parameter~$\kappa$ is plotted in Figure~\ref{fig-alpha_bar-q-log}.
We find that when reconnection is strong compared with keplerian shear
($\kappa \gtrsim 0.2$), the power-law exponent stays close to~3.1,
independent of~$\kappa$. This is because, as long as the keplerian
shear can be neglected, reconnection is the only term on the right-hand
side of the~LKE. In this case, ~$\kappa$ cannot affect the steady state
solution, it can only regulate how fast this solution is achieved.
As~$\kappa$ is decreased, however, the keplerian shear term becomes
important and $\bar{\alpha}(\kappa)$ starts to decrease.
We find that its overall behavior can be approximated by
$\bar{\alpha}(\kappa)\simeq 3.1 = {\rm const}$ for~$\kappa\gtrsim 0.2$
and $\bar{\alpha}(\kappa)\simeq 3.55+(1/3)\, \ln{\kappa}$ for~$\kappa
\lesssim 0.2$. Correspondingly, we expect $\bar{\alpha}(\kappa)$ to
crosses the critical values~2 and~3/2 (see~\S~\ref{subsec-energy})
at $\kappa_2\simeq 0.01$ and~$\kappa_{3/2} \simeq 0.002$, respectively.
As a reminder, $\bar{\alpha}<3/2$ means that the total magnetic energy
of the corona is dominated by the largest loops.
\begin{figure}[h]
\plotone{averaged-F-of-q.eps}
\figcaption{Orientation-averaged loop distribution function~$\bar{F}(L)$
for $\kappa=1.0$, 0.3, 0.1, 0.03, 0.01, and~0.005.
\label{fig-averaged-F-q}}
\end{figure}
\begin{figure}[h]
\plotone{alpha_bar-of-q-log.eps}
\figcaption{Power-law exponent $\bar{\alpha}$ for the orientation-averaged
loop distribution function~$\bar{F}(L)\sim L^{-\bar{\alpha}}$, as a function
of~$\kappa$.
The dotted horizontal lines at $\bar{\alpha}=2$ and $\bar{\alpha}=3/2$
correspond, respectively, to the critical values at which the magnetic
energy of the largest loop, $\mathcal{E}(L_{\rm max})$, and the total
magnetic energy in the corona, $E_{\rm tot}$, start to become dominated
by the large-scale cut-off~$L_{\rm max}$ (see~\S~\ref{subsec-energy}).
\label{fig-alpha_bar-q-log}}
\end{figure}
Finally, although most of our runs were done with $L_{\rm max}=10$,
we also conducted some runs with~$L_{\rm max}=20$. We found the
distribution function to be essentially the same; the power-law
just extended further in the $L_{\rm max}=20$ case but the slope
and the normalization were unchanged.
\subsubsection{Energetics and Torque}
Using the computed loop distribution functions, we calculate
the magnetic energy density as a function of height above the disk,
$\bar{B}^2(z)/8\pi$, and the energy of a loop as a function of its
length, $\mathcal{E}(L)$, for several different values of~$\kappa$
(see Figs.~\ref{fig-Bbar-squared-z} and~\ref{fig-Energy-L}).
We see that these functions remain essentially independent of~$\kappa$
as long as it is large enough, $\kappa \gtrsim 0.1$), but start changing
for smaller values of~$\kappa$. In particular, we find that for a given
value of~$\kappa$, $\mathcal{E}(L)$ saturates to a finite value at large~$L$.
This is because, although a large loop occupies a relatively large volume
at large~$z$, the magnetic energy density very high above the disk is very
low, and so the contribution of the large-$z$ part of a loop to its total
energy is relatively small.
However, we find that the asymptotic value of~$\mathcal{E}(L)$ at large~$L$
begins to increase rapidly as~$\kappa$ is lowered below roughly~$\kappa_2
\simeq 0.01$, in agreement with the expectations of~\S~\ref{subsec-energy}.
We also calculate the total torque~$G$, transmitted by the coronal
magnetic field and the total magnetic energy in the corona, $E_{\rm tot}$.
We show these quantities as functions of~$\kappa$ in Figures~\ref
{fig-torque-of-q-log} and~\ref{fig-Etot-of-q-log}.
Since our code is only first-order accurate with respect to
the number~$N_\theta$ of grid points in the $\theta$~direction,
the finite $\theta$-resolution becomes an issue at very small
values of~$\kappa$, where the distribution function is strongly
anisotropic. Therefore, to get more accurate values for the torque
and the magnetic energy corresponding to~$N_\theta=\infty$, we use
linear extrapolation based on the calculations with $N_\theta=40$,
60, and~80, separately for each value of~$\kappa$ (see Figs.~\ref
{fig-torque-of-q-log} and~\ref{fig-Etot-of-q-log}).
We find that, as $\kappa$ is decreased, the coronal angular
momentum transfer~$G(\kappa)$ increases steadily as~$\kappa^{-1}$
for $\kappa\gtrsim 0.1$, as is expected due to the increased degree
of anisotropy of the loop distribution function.
As~$\kappa$ becomes $\lesssim 0.1$, further growth of~$G(\kappa)$ slows
down and becomes $G\sim \kappa^{-1/3}$. The fact that $G(\kappa)$ is
a power law in this range is understandable in light of the discussion
at the end of~\S~\ref{subsec-energy}, combined with the logarithmic
dependence of~$\bar{\alpha}(\kappa)$ reported above.
We also find that the total magnetic energy in the corona,
$E_{\rm tot}(\kappa)$, is nearly flat for large values of~$\kappa$,
but starts to increase roughly as~$\kappa^{-1/2}$ as~$\kappa$ is
lowered below $\kappa\lesssim 0.05$ (see Fig.~\ref{fig-Etot-of-q-log}).
\begin{figure}[h]
\plotone{Bbar-squared-of-Z.eps}
\figcaption{Magnetic energy density as a function of height,
$\bar{B}^2(z)/8\pi$, for several values of the dimensionless
reconnection parameter~$\kappa$.
\label{fig-Bbar-squared-z}}
\end{figure}
\begin{figure}[h]
\plotone{Energy-of-L.eps}
\figcaption{Energy associated with a loop of length~$L$, $\mathcal{E}(L)$,
for several values of the dimensionless reconnection parameter~$\kappa$.
\label{fig-Energy-L}}
\end{figure}
\begin{figure}[h]
\plotone{torque-of-q-log.eps}
\figcaption{Coronal magnetic torque $G$ (normalized to $\Delta\Psi^2/4\pi$)
as a function of~$\kappa$ in logarithmic coordinates. The small insert shows
an expanded view of convergence with respect to the resolution in the~$\theta$
direction for $\kappa=0.005$ ($\log_{10}{\kappa}=-2.3$). The plus signs mark
the values obtained with $N_\theta=40$, 60, and 80 (top to bottom) and the
asterisks correspond to the extrapolation to~$N_\theta=\infty$.
\label{fig-torque-of-q-log}}
\end{figure}
\begin{figure}[h]
\plotone{Etot-of-q-log.eps}
\figcaption{Total coronal magnetic energy (normalized to~$\Delta\Psi^2/4$)
as a function of~$\kappa$ in logarithmic coordinates. As in Figure~\ref
{fig-torque-of-q-log}, the plus signs correspond to $N_\theta=40$, 60,
and 80 (top to bottom) and the asterisks corresponds to the extrapolations
to $N_\theta=\infty$, for each~$\kappa$.
\label{fig-Etot-of-q-log}}
\end{figure}
Furthermore, in agreement with the discussion in~\S~\ref{subsec-energy},
we find that both the magnetic torque and the total magnetic energy are
substantially larger in the $L_{\rm max}=20$ case than in the $L_{\rm max}=10$
case for small values to~$\kappa$,
whereas they are essentially the same for both values of~$L_{\rm max}$
for larger~$\kappa$.
\section{Discussion}
\label{sec-discussion}
There are several extensions of our model that we plan to develop in the near
future. They include: (1) characterizing the coronal field backreaction on
the disk motions; (2) including open magnetic flux; (3) taking into account
magnetic twist inside the loops; (4) incorporating a more realistic
prescription for reconnection; (5) investigating the mass exchange
between the disk and the corona and its effect on regulating coronal
energy release; (6) studying the interaction between the coronal magnetic
field and a super-imposed external large-scale field;
(7) assessing the implications of our theory for observations,
{\it e.g.}, in terms of time delay between hard and soft X-ray emission.
We discuss some of these issues in more detail in this section.
\subsection{Open Flux Tubes, Outflows, and Net Vertical Flux}
\label{subsec-open}
In this paper we have assumed that closed loops are the only magnetic
structures in the corona. In principle, however, one should also
consider a population of open flux tubes. If considered to be
force-free along their entire (infinite) length, open loops exert
no torques on the disk (in the non-relativistic limit) but contribute
to the averaged magnetic pressure~$\bar{B}^2(z)$ everywhere and thus
prevent it from becoming too small at large heights. Open tubes are
also essential for investigating the role of the corona in launching
large-scale outflows; relaxing the assumption of force-free fields
allows the inertia of the outflow to exert a torque on the disk
along open field lines \citep{Blandford_Payne82} by analogy with
the angular-momentum loss of the Sun to the solar wind \citep{Weber_Davis67}.
In our model, open loops could be introduced through the large-scale
cut-off for the closed loops. That is, whenever a closed loop grows to
exceeds a certain maximal footpoint separation~$L_{\rm max}$, it could
be replaced by a pair of open flux tubes of opposite polarities.
The large-scale cut-off may be physically associated with the disk
radius,~$r$. It should determine the fraction of magnetic flux that
is open at any given time. The rules that govern the interaction of
open tubes with closed ones and with themselves, are straightforward:
two open tubes of the same sign do not interact; two open tubes of
opposite sign can annihilate by reconnection, forming one single
closed loop. An open tube can also reconnect with a closed loop,
forming again an open tube of the same sign and a new closed loop.
In addition to such pairs of positive and negative open flux tubes,
there may also be a net vertical magnetic flux imposed on the disk
by the central object or the ambient inter-stellar medium.
The radial transport of such externally-imposed field across a turbulent
accretion disk is an important problem with significant consequences for
understanding the different spectral states of accreting black holes
\citep{Spruit_Uzdensky05}, for production and collimation of disk-driven
winds and jets, and for star--disk magnetic interaction \citep[\emph{e.g.}]
[]{Uzdensky_Koenigl_etal02a,Uzdensky_Koenigl_etal02b}, which is believed
to regulate the spin evolution of accreting neutron stars in X-ray pulsars
\citep{Ghosh_Lamb78}, as well as young stars \citep{Koenigl91,Matt_Pudritz05}.
This problem, however, is highly non-trivial \citep[\emph{e.g.}][]
{Lubow_Spruit95,Heyvaerts_Priest_Bardou96,Livio_Ogilvie_Pringle99,
Spruit_Uzdensky05}, in part because the effective transport of the
large-scale open flux may be greatly affected by reconnection with
the small- and intermediate-scale closed coronal magnetic loops
\citep{Spruit_Uzdensky05,Fisk05}. Incorporating a net large-scale
flux into our statistical model is very straightforward and its overall
transport should come out automatically. We therefore believe that our
theoretical model can be a very useful tool for addressing this problem.
\subsection{Twisted Loops}
\label{subsec-twisted}
In this paper we have assumed, for simplicity, that coronal loops have no
longitudinal current, and hence have no internal twist. In a more general
situation, however, there may be force-free field-aligned currents along
the loops, generated in response to certain disk footpoint motions:
specifically, rotation of field-line footpoints around each other.
Internal twist would have two major consequences for our coronal
model. First, the pinch force of the associated longitudinal current
(parallel to the magnetic field because of the force-free assumption)
will tend to reduce the width of loop and its cross section for
reconnection with other loops. One might consider as a limiting case
that the effective thickness of the loop is constant along its length.
There is indeed observational evidence that in the solar corona bright
loops usually have a nearly constant thickness along their length
\cite{Klimchuk00}.
Second, if the loop twist becomes too large, then the entire loop may
become kink-unstable and, as a result, the loop makes a transition to
a different equilibrium, where the internal twist is partially
transformed into the global writhe of the loop.
That is, the loop no longer lies in one plane but rather has a twisted,
helical shape. Such $S$-like loops (so called sigmoidal loops) are
routinely observed in the solar corona \citep{Rust_Kumar96}.
\subsection{Incorporating Realistic Reconnection Physics
(Collisionless Reconnection Condition)}
\label{subsec-recn-collisionless}
The physics of reconnection is notoriously complex.
However, significant progress has been achieved in recent years and
the picture that emerges can be summarized as follows~\citep{Uzdensky07b}.
There are two regimes of reconnection: a slow (Sweet--Parker) regime
in collisional plasmas and a fast regime in collisionless plasmas.
The slow reconnection regime is just as important as the fast one,
since, without it, it would be difficult for the system to accumulate
significant free magnetic energy before releasing it suddenly {\it via}
fast flare-like events. For practical purposes, the actual rate of fast
reconnection is not very critical in our problem, as long as it is faster
than the main dynamical time scale ({\it i.e.}, the orbital period).
More important is the fast-reconnection {\it onset}, or {\it trigger},
problem, {\it i.e.}, the question of the transition from the slow to
fast reconnection regime occur.
The physics of fast collisionless reconnection is very complicated;
it involves either two-fluid effects, such as the Hall effect, and/or
anomalous resistivity due to current-driven plasma micro-instabilities.
However, despite this complexity, one can formulate a rough criterion
for the transition from the slow collisional to the fast collisionless
regime \citep{Cassak_Shay_Drake05,Yamada_etal06,Uzdensky07a,Uzdensky07b}.
We thus plan to utilize this condition to formulate a physically-motivated
prescription for handling reconnection. This prescription can then be used
directly in our statistical theory or as a sub-grid model in actual MHD
simulations of the corona. In our present model in this paper, the
reconnection parameter~$\kappa$, which for simplicity we take to be
constant, effectively subsumes all this complexity.
The condition for transition to fast collisionless reconnection
involves several physical parameters of the system, including the
ambient plasma density. The dependence of the reconnection regime
on the density is critical, since it establishes an important feedback
that the dynamically subdominant coronal gas exerts on the coronal
magnetic field \citep{Uzdensky07a,Uzdensky07b}.
In turn, the plasma density in the corona is determined by the disk--corona
mass exchange processes, such as evaporation in response to coronal heating,
precipitation due to gradual radiative cooling, and magneto-centrifugally
and radiatively driven winds. Coupled together, disk--corona mass exchange
and the transition to fast collisionless reconnection ensure that the corona
is maintained near the state of marginal collisionality and regulate the
overall level of coronal activity and its intermittency, as well as the
vertical distribution of magnetic energy density and of magnetic dissipation.
These ideas, especially the concept of marginal collisionality, have recently
been successfully applied to the solar corona to explain the self-regulating
nature of the coronal heating process \citep{Uzdensky07a,Uzdensky07b} and
to the coronae of other main-sequence stars \citep{Cassak_Mullan_Shay07}.
They also have proved very useful for providing a natural explanation for
the observed optical depth in the coronae of accreting black holes (Goodman
\& Uzdensky 2008, in preparation).
Finally, especially for studies of the ADC's interaction with
a large-scale disk wind \citep[\emph{e.g.}][]{Brandenburg_vonRekowski07},
it is important to understand the transition from the force-free
regime to the wind regime in which the plasma inertia becomes
dynamically important \citep{Uzdensky_Koenigl_etal02b}.
This transition happens near the Alfv\'en critical surface of
the outflow and has a strong effect on magnetic reconnection.
In particular, it is expected that, beyond the Alfv\'en surface,
open magnetic field lines will not be able to close back {\it via}
reconnection \cite[\emph{e.g.}][]{Uzdensky04}.
\subsection{Flux Emergence}
\label{subsec-flux-emergence}
Our model, by construction, is not complete---it needs to be connected
to what happens in the disk. In particular, it needs as input a
statistical description of the population of small loops, or of their
rate of emergence from the disk. This information determines the overall
normalization of the loop distribution function and hence the net rates of
angular momentum transport and dissipation in the corona. In the models
computed here, we have simply assumed a fixed, isotropic distribution
of small loops.
The ultimate source of coronal activity is the MHD turbulence in the disk
\citep[\emph{e.g.}][]{Galeev_Rosner_Vaiana79,Tout_Pringle92,Miller_Stone00}.
Therefore, in order to estimate the rate and form of magnetic flux
emergence into the corona, one first needs to understand the properties
of MHD turbulence in a stratified disk, with a particular emphasis on
the production, evolution, and buoyant rise of magnetic flux tubes
\citep[\emph{e.g.}][]{Schramkowski_Achterberg93}. The best prospect
for developing this understanding is through appropriate statistical
analyses of MRI turbulence in stratified shearing boxes.
However, the notion of a flux tube in most theoretical discussions
(including the present paper) is not sufficiently precise to be applied
directly to simulations.
What well-defined, measurable, and statistically meaningful quantities
correspond to flux tubes or to their rates of emergence? Candidates
exist, but it is not clear which is best. Field lines can
be found as integral curves of the field and their motions determined,
but by what prescription should they be grouped into tubes?
Alternatively one might work with Fourier decompositions of the
vertical Poynting flux on horizontal planes, or with more general
$n$-point correlation functions of the field.
\section{Summary and Conclusions}
\label{sec-conclusions}
In this paper we construct a general theoretical framework for understanding
the structure of a strongly-magnetized corona above a turbulent accretion
disk. This study is motivated by the need to provide a more solid physical
foundation for ADC spectral modeling efforts. It should also act as a
connecting bridge between numerical MHD simulations of MRI-turbulent
disks and semi-empirical coronal models, and stimulate further theoretical
studies of accretion disks coupled to their coronae. We also hope that some
of the theoretical tools and ideas developed in this paper will prove useful
in solar physics.
One of the major goals of our study is to develop a statistical
language appropriate for describing the chaotic coronal magnetic
field. Here, we are interested in spatial scales larger than the
disk thickness but smaller than its radius and in temporal scales
longer than the orbital period but shorter than the overall accretion
time. Our approach builds upon the previous work by Tout \& Pringle (1996),
but uses much more realistic physics in several key aspects and also
goes further in analyzing and interpreting the results.
To construct the statistical theory, we represent the corona by an ensemble
of elementary magnetic structures, namely, loops connecting two spots on
the disk surface (Fig.~\ref{fig-corona}). Each loop is characterized by
several primary parameters ({\it e.g.}, the distance between the footpoints
and the orientation). The main object in this study is the {\it distribution
function}~$F$ of loops in this parameter space. One of our main goals is to
formulate, and then solve, the {\it loop kinetic equation} (LKE) for this
distribution function, similar to the Boltzmann kinetic equation in the
statistical theory of gases.
To do this, we first analyze the key physical processes that govern
the evolution of individual coronal loops. First, there are several
processes that affect the loops individually, such as:
(1) emergence of small loops into corona;
(2) random footpoint motions due to the disk turbulence;
(3) Keplerian shear, stretching loops azimuthally and thereby
also making them grow in height.
On average, these processes pump energy from the disk into the corona,
creating a stressed non-potential force-free field. In addition, loops
may interact with each other {\it via} episodic reconnection between two
individual loops \citep[\emph{c.f.}][]{Tout_Pringle96}, forming two new loops
(see Fig.~\ref{fig-recn}). In our theory reconnection is represented
as a binary collision, analogous to binary collisions between atoms
in a gas. Correspondingly, we describe loop-loop reconnection by a
nonlinear integral operator, similar to Boltzmann's collision operator.
In contrast to processes (1)--(3), magnetic reconnection relaxes
the accumulated magnetic stresses and dissipates the accumulated
free magnetic energy. Overall, a magnetically-active ADC can be
described as a Boiling Magnetic Foam.
Based on these processes we are able to construct the loop kinetic
equation. In this equation we characterize the overall rate of
reconnection events relative to the Keplerian shear rate by a
dimensionless parameter~$\kappa$. In order to investigate the
role of magnetic reconnection in the corona, we solve the loop
kinetic equation numerically for several different values of~$\kappa$.
We obtain a {\it statistical steady state} for each value of~$\kappa$
and find that the steady-state loop distribution function is generally
well represented by a orientation-dependent power-law, $F(L,\theta)
\sim L^{-\alpha_\kappa(\theta)}$. When Keplerian shear is absent,
the distribution function is isotropic, $\alpha_\infty(\theta)=
{\rm const}$. As the rate of shear relative to reconnection increases
({\it i.e.}, $\kappa$ decreases), the distribution becomes more and more
anisotropic, with a predominance of toroidal orientation. At the same
time, a typical loop grows to a larger size by a stronger shear before
its growth is disrupted by reconnection. Thus, the orientation-averaged
distribution function becomes shallower as~$\kappa$ is decreased.
Once the distribution function is known, we use it to calculate several
important integral quantities related to the energetics of the corona.
First, we use a self-consistent {\it mean-field} approach to compute
the magnetic energy density as a function of height, $\bar{B}^2(z)/8\pi$.
This quantity represents the collective magnetic pressure of all the
neighboring loops that confine any given loop and thus represents another
(in addition to reconnection) way in which loops interact with each other
in our theory. Although it doesn't enter explicitly into the loop kinetic
equation, $\bar{B}(z)$ is very important in our model. In particular, it
controls the thickness of loops as a function of height, which affects in
turn the cross-section for reconnection. In addition, by requiring that
the vertical gradient of the magnetic pressure $\bar{B}^2(z)/8\pi$ be
balanced by the magnetic tension within each loop, we self-consistently
calculate the equilibrium shape and vertical extent, $Z_{\rm top}(L)$,
of the loops. This, in turn, enables us to calculate some important
quantities such as the energy associated with a given loop, the force
exerted on its footpoints, etc. We then use these quantities to assess
various issues of coronal energetics, including the overall magnetic
energy stored in the corona, statistical distribution of coronal energy
release events (flares), and the overall angular momentum transferred
by the coronal magnetic field. As a result of our parametric study with
respect to the reconnection parameter~$\kappa$, we find that if~$\kappa$
is decreased ({\it i.e.}, reconnection in the corona is inhibited) beyond
a certain value, the slope of the loop distribution function becomes so
shallow (namely, shallower than~$L^{-3/2}$) that the contribution of large
loops to both the magnetic energy and torque starts to dominate, leading
to a significant enhancement in these quantities. In our specific model,
the critical value of~$\kappa$ is found to be $\kappa_{3/2}\simeq 0.002$.
These results demonstrate that the energetic dominance of coronae
is inextricably linked to reconnection processes.
They thus motivate further efforts to develop more realistic physical
description of reconnection.
To reiterate an important point made in the Introduction, the tenuous
corona above an accretion disk is likely to be marginally collisionless
(Goodman \& Uzdensky 2008, in preparation), unlike the dense plasma inside
the disk itself. This means that the corona cannot be described by
traditional MHD simulations with constant resistivity because of their
inability to control or resolve magnetic reconnection, which, as we have
shown in this paper, may influence the coronal magnetic energy and angular
momentum transfer. Therefore, some kind of a physically-motivated subgrid
prescription for reconnection is needed.
\begin{acknowledgments}
We would like to acknowledge fruitful discussions
with V.~Titov, Z.~Mikic, A.~Pankin, and D.~Schnack.
We are grateful to R.~Blandford, S.~Cowley, and V.~Titov
for drawing our attention to several useful references.
This work is supported by National Science Foundation Grant
No.\, PHY-0215581 (PFC: Center for Magnetic Self-Organization
in Laboratory and Astrophysical Plasmas).
\end{acknowledgments}
\bibliographystyle{apj}
|
1,108,101,566,792 | arxiv | \section{Introduction}
\label{sec:intro}
In the past decade, \textit{Hubble Space Telescope}
(\textit{HST}) observations helped reveal the eMSTO
phenomenon, now considered a
ubiquitous feature of young star clusters in the
Magellanic Clouds (e.g., \citealt{ADMBNP2007,
ADM2008, KG2008, APM2009, PG2009}), i.e., those
with ages less than 2 Gyr. However, the origin of the
eMSTO has remained a mystery. Recently
revealed to be common in clusters of the
Milky Way by \textit{Gaia} (e.g.,
\citealt{AFM2018a, GC2018, BN2018}) as well,
suggesting that the eMSTO is a natural property of
young star clusters, rather than a peculiarity of
the Magellanic Clouds.
The eMSTO appears in a
star cluster's CMD as a
broadened MSTO, as if the cluster hosts multiple
single age MSTOs. At face value, this suggests that
such clusters contain stars born over some period of
eSF in the past (e.g., \citealt{RuSt2010,
RuSt2011, PG2011b, PG2011a, CCDS2011, SK2011,
ADM2013}), perhaps lasting a few hundred Myr.
Thus, first instincts were to view the eMSTO as
a consequence of eSF, which was surprising under
conventional beliefs that regard young clusters as
simple stellar populations (SSPs) with coeval stars.
Alternate theories have arisen since then, but a
definitive consensus on the cause of the eMSTO has
not been met. Ultimately, uncovering the physical
process(es) behind this phenomenon could open new
chapters in stellar evolution and cluster formation.
Much effort to reveal the true origin of the eMSTO
has focused on searching for observational evidence
of eSF. Such evidence could elucidate the
contemporaneous puzzle of ``multiple populations''
(MPs) observed in globular clusters (GCs) older
than 2 Gyr, as reviewed by \cite{NBCL2018}.
Theoretically, eSF can reproduce observed eMSTO
morphologies (e.g., \citealt{CL2017}), but it is
challenged by a lack of empirical support, despite
numerous observational campaigns. For example,
\cite{ICZ2016} found no evidence for eSF events
when modeling the spectral properties of the
young massive cluster W3 in NGC 7252, despite this
cluster's young age and exceptionally large escape
velocity (possibly conducive to entrapment of
stellar ejecta and subsequent eSF). Relatedly,
\cite{APNB2016} and \cite{GC2018} found that many
young LMC and Galactic clusters possess eMSTOs as
well, in spite of their relatively low masses (e.g.,
around 2400 $M_{\odot}$ for NGC 2818, \citealt{BN2018});
presenting additional evidence that eMSTOs exist
independent of the cluster's potential for gas
retention. Observations have not revealed how
these young clusters might retain enough gas to
experience a (up to) several hundred Myr episode
of star formation required to explain the largest
eMSTOs. This is especially so, given that much more massive
clusters like W3 (roughly 600 Myr old and
$1.13\times10^{8}M_{\odot}$, \citealt{ICZ2016}, and
see also \citealt{CZ2014}) appear to lack
evidence of eSF.
An alternative explanation has
developed alongside the eSF theory. \cite{BdM2009}
made a case for stellar rotation as being the
cause of the eMSTO. Rotation can grant stars a
greater core fuel supply, thereby extending main
sequence (MS) lifetimes (see e.g.,
\citealt{MeMa2000, MaMe2010, LG2011, SE2012,
JC2016}). Rotation also causes gravity
darkening, i.e., structural deformations that alter
the apparent magnitudes and colors of rotating
stars in a viewing angle dependent manner, see e.g.,
\cite{HvZ1924, LBL1967, ELR2011}. Combined,
these effects allow coeval rotating stars to
take on a range of colors and magnitudes that
can mimic an age spread (e.g., \citealt{BH2015c,
SG2018}) and create a broad MSTO. Additional
observational evidence has accumulated for
stellar rotation within the eMSTO (e.g.
\citealt{BdM2009, ZL2012, WY2013, NF2015,
BH2015c, BN2016, NF2016, BN2018, FDAn2018}, and
\citealt{CG2019}). Furthermore,
\cite{JoC2019} have recently shown, using non-rotating
stellar models calibrated on \textit{Kepler} astroseismic
data, that convective core overshooting can produce an eMSTO
similar to observations. Thus, rotation and convective
mixing are observed stellar processes at play in these
clusters that may contribute to the eMSTO; we limit our
study to the exploration of stellar rotation and age
spreads as a matter of simplicity, but acknowledge that
variable convective mixing efficiencies may also play a
role.
A central finding in eMSTO studies has been that as
clusters age, the eMSTO width increases (i.e., the
$\Delta$(Age)-age trend highlighted by
\citealt{NF2015}). This behavior may
suggest that eMSTO width is related to stellar
evolution, possibly the evolution of rotation. This
interpretation says that the eMSTO is not due to a
range of ages, but rather coeval stars with a
distribution of rotation rates. If the
$\Delta$(Age)-age trend is due to eSF, it would
require that SF took longer in the past. This
explanation is not impossible, but it is difficult
to find a natural reason as to why the length of SF
has steadily decreased over time. Furthermore,
\cite{GC2018} have found a similar $\Delta$(Age)-age
trend for young Milky Way clusters, suggesting that
the trend is not specific to the environments of
the LMC/SMC. Many lines of evidence now point
towards stellar rotation at least playing some
role in causing the eMSTO. This is often
acknowledged, but it is still argued that some
degree of eSF may be present in these clusters.
Hiding underneath all of this could
be additional effects due to stellar
binary interactions. For instance, binary
mergers could replenish stars nearing
terminal age MS (TAMS), or otherwise alter
their evolutionary trajectory. Relatively
fewer studies have incorporated models
that account for binary evolution, but
work has been done by e.g., \cite{WY2011}
and \cite{ZL2012, ZL2016}. As high mass
stars can still exist in young clusters,
binary effects may be especially important
there (e.g., \citealt{WY2018} and
suggested by \citealt{ERB2019}). Binary
fractions are still expected to be
significant ($\sim30$\% in many studies) in
intermediate age clusters (1-2 Gyr),
so some degree of binary interaction could
be expected, but may be minimal. We
acknowledge that binary interaction has a
potentially strong effects, but do not model
it in this work.
Along these lines, \cite{FDAn2017} have suggested
that braking due to binary interaction may
contribute to the rotation rate evolution of
cluster stars and result in populations of fast
and slow rotators. Therefore, this mechanism could
be part of the physical explanation not only for
eMSTO spreads, but also for the observed split
MS in younger star clusters. The split MS shows
up in a CMD as a blue/redward bifurcation of the
main sequence. It is now widely thought to be
due to a bimodal distribution of stellar
rotation rates (see e.g., \cite{APM2018} for
examples). \cite{AFM2018b} found first spectroscopic
evidence for the split MS hosting slow and fast
rotators in the redward and blueward portions of
the split MS, respectively, for the young
globular cluster NGC 1818. This feature does not
appear to be due to metallicity variations,
nor photometric errors, and has a numerically dominant
population in the redward MS (around 60-70\% of the
split MS stars), which are the proposed fast rotators
(\citealt{APM2016,APM2017}). Star clusters younger
than about 600 Myr (\citealt{APM2018}) appear to
host a split MS, in addition to their eMSTO. In our
cluster sample, NGC 1866 is one such cluster. Our
analysis of its split MS is highlighted in later
sections. Whether or not the binary braking
scenario of \cite{FDAn2017} causes this is
uncertain at this point, rotation distributions
in the split MS may reflect the initial distribution
that stars are born with instead.
Until now, a quantitative assessment of the
CMD morphology with flexible stellar models
has been missing, and we aim to provide this
here. Previous studies have variously used
stellar models that are non-rotating, e.g.,
PARSEC (\citealt{AB2012}), with rotation often
modeled according to either GENEC
(\citealt{SE2012}), or MESA (\citealt{BP2013}).
PARSEC has traditionally been a non-rotating model set
(though see \citealt{GCosta2019}), while GENEC
provides a dense grid of stellar rotation
rates, but is limited in stellar mass to
$1.7\ M_{\odot}$ and above, and a relatively coarse
metallicity sampling. GENEC has been the
workhorse model set in these studies for
including the effects of stellar rotation,
and has done much to form our current
understanding. We have extended MIST to now
include a dense grid of rotation rates.
Meanwhile, in comparison to GENEC, our new
MIST model set also includes masses ranging
down to $0.1\ M_{\odot}$, and finer metallicity
sampling. These new rotating stellar models
can now investigate the effects of stellar
rotation in older clusters than previous
studies. We form synthetic stellar populations
that can span a range of ages, metallicities,
and stellar rotation rates. In our analysis, we
opt for a non-parametric rotation rate
distribution whose predictions are presented
in our results.
We explore the two main scenarios proposed to
explain the eMSTO: eSF and stellar rotation.
To this end, we build synthetic stellar
populations according to three scenarios: 1)
populations that experienced eSF, or 2)
experienced no eSF but could have stars at
a range of rotational velocities, and 3) a
cluster that experienced both eSF and has
stars at a range of rotational velocities.
We fit these synthetic populations to
observations of the clusters NGC 1866, 1831,
2818, 2249, and 2203 and determine the
best fit model populations, thereby deriving
cluster properties like age, age spread, and
the presence of rotation distributions. In
Section \ref{sec:data}, we present the
sources for our data and give brief
descriptions of each cluster. Section
\ref{sec:methods} provides details of our
models, lays out our fitting procedure, and
presents mock tests of the methods. Section
\ref{sec:results} gives our results,
sequentially for each scenario along with
brief commentary. Finally, our conclusions
are summarized in Section \ref{sec:conclusions},
along with some discussion of caveats and suggestions
for future work. The photometric zero point is Vega
for all magnitudes shown.
\begin{deluxetable}{cccccc}[!t]
\tablecaption{Adopted Cluster Parameters}
\tablehead{ \colhead{Cluster} & \colhead{$\mu$\tablenotemark{1}}
& \colhead{Age [Gyr]} & \colhead{[Fe/H]\tablenotemark{2}}
& \colhead{$A_{\rm{V}}$} & \colhead{Binary \%}}
\startdata
NGC 2203 & 18.37 & 1.55 & -0.30 & 0.16 & 18 \\
NGC 2249 & 18.20 & 1.0 & -0.46 & 0.07 & 30 \\
NGC 2818 & 12.76 & 0.7 & 0.0 & 0.90 & 29 \\
NGC 1831 & 18.35 & 0.7 & -0.25 & 0.14 & 20 \\
NGC 1866 & 18.31 & 0.2 & -0.36 & 0.34 & 25 \\
\enddata
\vspace{0.1cm}
\tablecomments{All parameters listed are fixed in our fits,
except the age. Age initialized at the listed
value but is allowed to vary. See the text for
the literature sources to these values.}
\tablenotetext{1}{Distance modulus}
\tablenotetext{2}{Based on \cite{MA2009} protosolar abundances}
\label{t:cparams}
\end{deluxetable}
\section{Data}
\label{sec:data}
Our data comes from \textit{HST}
observations and the recent \textit{Gaia}
DR2. Our target clusters are NGC 2203, 2249,
1831, and 1866, located in the LMC, and
NGC 2818, located in the Milky Way. We chose
this set of clusters because they cover a
good portion of the age range where eMSTOs
are observed, i.e., younger than about 1.5 Gyr.
As discussed in Section \ref{sec:methods}, our
models are limited to $<5\ M_{\odot}$, and so we
cannot model clusters younger than about 200 Myr
currently. For each cluster, we list the values of
mean log age, distance modulus, binary fraction,
$A_{\rm{V}}$, and metallicity [Fe/H]; the
mean log age listed here serves to inform
the initial position of our parameter
search in age during our fits, while we
keep all other parameters fixed. The adopted
parameters are collected in Table \ref{t:cparams}.
CMDs produced by the data are shown in Figure
\ref{fig:photometry}, where black, dashed
boxes show the regions we fit models to
(focusing on the MSTO, ignoring the red
clump, or RC). It is known that some
models can have trouble simultaneously
fitting the RC and MSTO (e.g.,
see \citealt{BN2018}) and this is the reason
we exclude it in our fits. Red isochrones
are also plotted, showing the SSP expected
by the adopted cluster parameters, which
are listed in the following subsections.
\begin{figure*}[!ht]
\center
\includegraphics[width=\textwidth]{cluster_photometry.pdf}
\caption{CMDs of our five chosen star clusters,
in order of descending age: NGC 1866, 1831, 2818, 2249, 2203.
Black rectangles show the CMD area used for fitting
models to data. Red (non-rotating, MIST) isochrones
represent SSPs according to our adopted cluster parameters,
listed in the text. The red clump is ignored in our fits, for
reasons explained in the text.}
\label{fig:photometry}
\end{figure*}
\subsection{NGC 2203}
\label{ssec:ngc2203_data}
The intermediate age cluster NGC 2203,
located in the LMC is the oldest of our
target clusters according to the literature,
at about 1.55 Gyr (\citealt{PG2014,
PR2017}). Photometry for this cluster
is the same as that used in
\cite{PR2017}, where the data
reduction process is also described.
In brief, this data is part of a
larger set that was re-reduced from
two \textit{HST} programs: GO-9891
(PI: Gilmore) and GO-12257 (PI:
Girardi). So, this photometry comes
from archival ACS and WFC3 data,
re-reduced with the University of
Washington data reduction
pipeline, designed to reduce the
\textit{HST} treasury programs ANGST
(\citealt{JJD2009}) and PHAT
(\citealt{JJD2012}); see
\cite{BFW2014} for further details.
\texttt{ASteCA} (\citealt{PVP2015}) was
used to determine the
cluster center, found via the maximum
spatial density using a 2D Gaussian kernel
density estimator. The cluster radius
corresponds to where the radial density
profile becomes indistinguishable
from the background stellar density.
\texttt{ASteCA} was also used to determine
contamination, utilizing a non-parametric
Bayesian decontamination algorithm based
on the method of \cite{CCA1990}. The final
membership was limited to stars within the
cluster radius with $>70\%$ membership
probability.
We adopt the cluster parameters
cited in \cite{PG2014}, i.e., their
age, [Fe/H], $A_{V}$, and distance
estimated via best-fit isochrones
from the Padova group (\citealt{PM2008}).
Respectively, these are values of
1.55 Gyr, -0.30, 0.16 dex, and 18.37.
We also adopt the binary fraction of
0.18 used by \cite{PG2014}. Taking into
account that slight differences in fits
to CMD features can arise due to different
input physics between our MIST and those
Padova models. Although, see that in
Figure \ref{fig:photometry} that these
parameters still provide a reasonable fit
with our models.
\subsection{NGC 2249}
\label{ssec:ngc2249_data}
On the boundary of the intermediate
age regime, NGC 2249 is a star
cluster also located in the LMC.
Membership for this cluster was
determined in an identical way to
what was described for NGC 2203.
Isochrone analysis in \cite{MC2014}
has estimated the cluster to be
about 1 Gyr old. Data reduction for
this cluster is the same as
described for NGC 2203, with further
details in \cite{PR2017}.
In this work, we adopt the
same mean age, [Fe/H], $A_{\rm{V}}$,
and distance modulus for NGC 2249 as
cited by \cite{MC2014}; namely, 1
Gyr, -0.46, 0.07 dex, and 18.2 mag,
respectively. We adopt the binary
fraction of 0.30 cited by
\cite{MC2014}.
\subsection{NGC 2818}
\label{ssec:ngc2818_data}
NGC 2818 enters the regime of
young cluster ages for our target
clusters, at about 700 Myr, and it
is located in the Milky Way. Our
NGC 2818 data is taken from
the publicly available data
gathered by \cite{BN2018},
originating from \textit{Gaia}
DR2 (\citealt{GAIA2016,
GAIA2018}). Details of the member
selection for this cluster are
given in \cite{BN2018}, but briefly,
members were selected via proper
motion and parallax cuts. NGC 2818
is one example of a number of young
Galactic clusters with eMSTOs that
have been revealed with \textit{Gaia}
DR2 data in the last year or so,
thanks to \textit{Gaia}'s enhanced
photometric precision. The adopted
cluster parameters are based on
values used in \cite{BN2018}: with an
age of 700 Myr, a solar [Fe/H] of 0.0,
$A_{\rm{V}} = 0.90$, distance modulus
of 12.76. We adopt a binary
fraction of 0.29 from \cite{GC2018}.
\subsection{NGC 1831}
\label{ssec:ngc1831_data}
NGC 1831 is also an approximately
700 Myr old cluster as well,
located in the LMC. For this
cluster, we carried out PSF
photometry on the flat-field
corrected, and bias-subtracted
\textit{HST} `flc' images (Program ID:
GO-14688) using the WFC3 module of
\texttt{DOLPHOT}, a modified
version of \texttt{HSTphot}
(\citealt{AD2000}) and following the
procedure described in \cite{EB2009}.
For our analysis, and for the CMD
shown in Figure \ref{fig:photometry},
we chose stars within a half-light
radius (from \citealt{McLvdM2005}),
$r_h=33.85''$ of the cluster's center.
We have adopted the parameters based
on those used by \cite{PG2018} for
NGC 1831, determined from PARSEC
isochrones. Thus, an age of 700 Myr,
[Fe/H] of -0.25, $A_{\rm{V}}$ of 0.14,
distance modulus of 18.35, and a
binary fraction of 0.20.
\subsection{NGC 1866}
\label{ssec:ngc1866_data}
NGC 1866 is the youngest cluster
in this study, at about 200 Myr old,
and it is located in the LMC.
This photometry was obtained in the
same manner as described for NGC
1831 (except from \textit{HST} Program
ID: GO-14204). Here we have taken stars
within the half-light radius
$r_h=42.9''$ of the cluster's
center, based on \cite{McLvdM2005}.
Our adopted cluster parameters
for NGC 1866 are adopted from
\cite{APM2017} for the metallicity
and binary fraction. These
parameters are an age of 200 Myr,
[Fe/H]=-0.36, $A_{\rm{V}}=0.34$, a
distance modulus of 18.31, and a
binary fraction of 0.25.
\section{Methodology}
\label{sec:methods}
Major aspects of our models are identical
to those described in \cite{SG2018}, but
topical details are recapitulated, and
additions are noted here. The greatest
difference in our current models and
those described in the aforementioned
paper is the inclusion of higher
rotation rates. Our mass range has
changed and is from $0.1-5\ M_{\odot}$, with
metallicities from [Fe/H] = $-0.60$ to
$0.45$ in $0.15$ dex steps. This mass
range extends lower than what is
available in SYCLIST (\citealt{CG2014},
a stellar population synthesis tool, and
stellar isochrone and track database). The
SYCLIST isochrones and tracks are (similar
to how MIST models are based on MESA) based
on the GENEC stellar evolution code whose
lowest stellar mass is $1.7\ M_{\odot}$.
SYCLIST has been the primary model set
used to study stellar rotation in
these clusters so far. As our new models
extend to 0.1 $M_{\odot}$, we can model
older clusters than SYCLIST has traditionally
been able to, however, with
the caveat that magnetic braking is crudely
accounted for in these lower mass stars (more
on this in Section \ref{ssec:mesamods}). We
do not evolve binary systems, or investigate
the effects of mass transfer, nor tidal braking
in our model set. Our models also allow for
a finer metallcity range than GENEC does
publicly, which offers Z = 0.014, 0.006,
and 0.002 (roughly [Fe/H] = 0.0, -0.35, and
-0.85).
\begin{figure*}[!th]
\center
\includegraphics[width=0.95\linewidth]{fakephot.pdf}
\caption{\textbf{Top row}: synthetic clusters, with
each point color coded by the stellar model's average
surface velocity. These points were generated at a single
age of 0.7, 1, and 1.5 Gyr, from left to right. Binaries
are shown as cyan points. Solid red lines mark the
youngest and oldest age non-rotating MIST isochrones that
span the eMSTO width; rotation effects can mimic this
span. The red dashed lines show the mean age. Turn off
masses (left to right): 1.5 - 1.8 $M_{\odot}$, 1.60 - 2.15
$M_{\odot}$, 2.1 - 2.85 $M_{\odot}$, respectively, according to
our models. The black dashed line approximately marks
the magnitude below which our velocities are ramped down
as a proxy for magnetic braking. \textbf{Bottom row}: data
for NGC 2818, 2249, and 2203 as black dots, with the same
non-rotating isochrones from above overlaid for comparison.
Blue crosses are data excluded from the fits.}
\label{fig:fakephot}
\end{figure*}
\subsection{Stellar Population Models}
\label{ssec:mesamods}
The starting point for our models is
the \texttt{MESA} stellar
evolution code (\citealt{BP2011, BP2013,
BP2015, BP2018, BP2019}), version
\texttt{r7503}, which is a modular and
open source 1D stellar evolution code.
We closely followed the physics used
for the MIST database (\citealt{JC2016}),
adopting the protosolar abundances of
\cite{MA2009} and using boundary
conditions from ATLAS12, while SYNTHE is
used for bolometric corrections
(\citealt{RLK1970, RLK1993}). Our models
are evolved to the end of core helium
burning. Hereafter, we will refer to our
models as ``MIST models'' or
``MIST-based''.
The MIST models are set rotating at
the zero age MS (ZAMS) with a given
velocity denoted by the ratio of
equatorial angular velocity
$\Omega_{\rm{ZAMS}}$ at ZAMS, over the
critical $\Omega_{\rm{c}}$. The
critical velocity $\Omega_{\rm{c}}$ is
a property intrinsic to the star that
depends on its mass (see e.g.,
\citealt{AM2009}); it represents the
limit where centrifugal force overcomes
the star's gravity.
The ratio $\Omega/\Omega_{\rm{crit,ZAMS}}$ is equivalent to the
linear velocity form $v/v_{\rm{c,ZAMS}}$
in the \texttt{MESA} formalism.
Previously, our models were limited to
$\Omega/\Omega_{\rm{crit,ZAMS}} = 0.6$ in \cite{SG2018}; we
now include models ranging up to
$\Omega/\Omega_{\rm{crit,ZAMS}} = 0.9$, in steps of $0.1$. The
initial MIST models released
by \cite{JC2016} only included
non-rotating and $\Omega/\Omega_{\rm{crit,ZAMS}}=0.4$ models.
MIST models rotation under the shellular
approximation developed by \cite{KT1970},
with chemical and angular momentum
transport described by the equations of
diffusion equations of \cite{ES1978}. This
diffusive formalism is also adopted in the
stellar evolution codes STERN
(\citealt{IB2011}) and the recent version of
PARSEC (\citealt{GCosta2019}). The
shellular approximation is standard in 1D
stellar evolution codes. The treatment
of angular momentum and chemical transport varies
between codes. For instance, GENEC uses a
diffusive-advective formalism, described in
\cite{JPZ1992, MaZh1998, MaMe2000a}. The
differences in these two formalisms have
significant effects on the models, e.g.,
leading to different MS lifetime extensions
(by up to 20\% or so) and color-magnitude
variations (see \citealt{JC2016, SG2018} for
examples). Our models possess a stronger convective
mixing with weaker rotation mixing, whereas
GENEC features the opposite. Consequently,
our rotating models are primarily affected
by the structural changes of gravity
darkening when they rotate, they don't see
a dramatic MS lifetime extension or
luminosity enhancement from rotational
mixing, as is seen in the GENEC models.
Gravity darkening is handled by the
equations of \cite{ELR2011} (recently
adopted by \cite{BP2019} in MESA as well)
in determining the (surface averaged)
luminosity and temperature of a given
stellar model at viewing angle $i$. The
viewing angle corresponds to $i=90^{\circ}$
when viewing is equator-on, versus
$0^{\circ}$ when viewing pole-on. Gravity
darkening is the effect of centrifugal
force reducing the surface gravity of a
rotating star. This effect is stronger at
the equator than at the poles, due to the
greater centrifugal force at the equator.
Thus, gravity darkening causes the
equator of a rotating star to become
cooler and dimmer than the poles,
introducing a viewing angle dependence on
the apparent magnitude and color of a
rotating star. The effects can be
substantial; examples
for our models exist in \cite{SG2018}.
The formalism that we adopt from
\cite{ELR2011} is similar to the gravity
darkening formalism used for the
GENEC-based SYCLIST stellar population
models.
Another important aspect of rotation, at
least for masses $\lesssim1.8\ M_{\odot}$
is magnetic braking. Modeling this process
is an active area of research (e.g.,
\citealt{CeGa2016, SA2017, CeGa2018,
JF2019}). \cite{NF2015} predicted the
$\Delta$(Age)-age trend should stop after
magnetic braking becomes effective. This
limit is expected to be reached by TO stars in
clusters with ages older than about 1.5
Gyr, depending on the metallicity. In
their recent work, \cite{CG2019} used
models developed with the STAREVOL code
(\citealt{LA2016}), for masses between 1
and 2 $M_{\odot}$, including a prescription
for magnetic braking according to
\cite{SPM2015}. This mass range is roughly
where surface convection zones develop,
leading to surface magnetic fields that can
act on extended stellar material,
braking the star. We take a crude approach to
simulate this, in absence of a proper
model of the effects of magnetic braking.
Below $M = 1.3\ M_{\odot}$, models are forced to
be non-rotating; from $M = 1.3-1.8\ M_{\odot}$
models have $\Omega/\Omega_{\rm{crit,ZAMS}}$ scaled up to the full
value. The SYCLIST models do not
model magnetic braking, but also exclude
stellar masses below $M = 1.7\ M_{\odot}$
(\citealt{CG2014}). The MSTOs of NGC 2203
and 2249 have TO masses that are low enough
for magnetic braking to become important, so our
results for these clusters in particular
will be affected by uncertainties due to
magnetic braking.
From our MIST-based stellar models, we compute
synthetic stellar populations, as in
\cite{SG2018}, using the code MATCH
\citep{AD2001}. Specifically, we use MATCH to
compute Hess diagrams of CMDs, including
unresolved binaries, at fixed values of Z,
age, and $\Omega/\Omega_{\rm{crit,ZAMS}}$. Photometric errors are
simulated with MATCH via artificial star
tests. The populations are created at
distinct ages, covering log Age = 8.0 to
9.5 (in 0.02 dex steps), each of which is
also created at $\Omega/\Omega_{\rm{crit,ZAMS}}=0.0,\ 0.1,...$, up to
0.9. Our synthetic populations include the
effect of gravity darkening via randomly
drawn viewing angles for constituent stars.
Stellar models are drawn according to a
\cite{PK2001} initial mass function (IMF).
We combine these synthetic populations
(weighting them as described below
in Section \ref{ssec:fitmethod}) to form a
composite stellar population that may possess
stars from a range of ages and rotation rates.
Colored points in the top row of Figure
\ref{fig:fakephot} shows several examples
of these models. In this figure, models have
fixed age pertaining to the representative
cluster, from left to right: NGC 2818, 2249,
and 2203. The colors map the full range
of rotation rates (i.e., $\Omega/\Omega_{\rm{crit,ZAMS}}=0.0$
to $0.9$), according to a flat distribution of
$\Omega/\Omega_{\rm{crit,ZAMS}}$, to clearly show the full effects of
stellar rotation on the MSTO in these clusters.
Red lines show non-rotating MIST-based isochrones,
dashed at the mean age, and solid at $\pm\Delta$(age).
This $\Delta$(Age) was chosen so that the isochrones
roughly covered the full extent of the eMSTO. The
bottom row of panels shows the same non-rotating
isochrones overlaid on the data of NGC 2818, 2249,
and 2203. Broadly, both stellar rotation and a range
of ages can cover the extent of the eMSTO, but in
different ways, hence the contention between the
theories.
Figure \ref{fig:fakephot} shows how increasing age
reddens and decreases the luminosity of TO stars in a
similar manner to the effects of gravity darkening.
The effects of stellar rotation manifest in a confined
region of CMD space at a given age, in comparison to
an age spread which can modify the luminosity and
temperatures of stars over as wide a space as is
useful for covering the eMSTO. In this sense, the
effects of an age spread have relatively more freedom
than the effects of a rotation rate distribution.
MATCH was used to generate the models displayed in the
top row, but the code currently lacks the ability to
perform a straightforward fit for rotation rate
distributions. We opted for a separate fitting method
outside of what is provided by MATCH (outlined below in
Section \ref{ssec:fitmethod}) to handle this.
\begin{figure*}[!htb]
\center
\includegraphics[width=0.95\linewidth]{mock_50thpercentile_summary.pdf}
\caption{Recovered rotation rate distributions from
mock tests described in the text. Black hatched bins represent
the input ``true'' values, blue bins are best-fit distributions, with
error bars (84th, 16th percentiles) shown as red vertical lines. The
top row corresponds to cases where the $\sigma_t\Omega$-model was
tested against mock data created with the scenario written in the
annotations and described in the text. The bottom row shows the same,
but for the $\Omega$-model tests.}
\label{fig:mock_weights}
\end{figure*}
\begin{figure}[!htb]
\center
\includegraphics[width=0.95\linewidth]{small_mock_sigsummary.pdf}
\caption{Selected residuals from the mock tests. The top row shows
results from cases where the $\Omega$-model was fit to either
of the other two models ($\sigma_t\Omega$ or $\sigma_t$). The
bottom row shows the same, but for the $\sigma_t$-model (fit
to either $\sigma_t\Omega$ or $\Omega$-mock data). Cases for the
$\sigma_t\Omega$-model, and all 1:1 fits (e.g., $\Omega$-model to
$\Omega$-mock data) were essentially zero throughout, and are
not displayed.}
\label{fig:mock_res}
\end{figure}
\subsection{Fitting to Data}
\label{ssec:fitmethod}
We build composite stellar populations
according to three scenarios (below ``weights'' measure
the total number of stars in a Hess diagram whose stars
may be observations or simulations):
\begin{enumerate}
\item
\textbf{Gaussian log(Age) spread ($\sigma_t$-model)}:
This has a Gaussian SFH. More complex SFH are
imaginable, but would expand parameter space
further, and so are omitted for now. Here
$\Omega/\Omega_{\rm{crit,ZAMS}}$ is restricted to 0.0 (non-rotating).
This model has three parameters
describing the overall weight, or number of stars
in the composite $\Omega/\Omega_{\rm{crit,ZAMS}}=0.0$ population (i.e.,
amplitude), plus the Gaussian log(Age)
distribution's mean and standard deviation.
\item
\textbf{Non-parametric rotation rate distribution
($\Omega$-model)}: This model considers 10 free
weights ranging from 0 to the total weight
of observed stars. Each free weight corresponds to
one of the 10 possible $\Omega/\Omega_{\rm{crit,ZAMS}}$ populations. All
stars are assumed to have a single age, in this
model, which is also fit as an 11th free
parameter.
\item
\textbf{Age spread with rotation
($\sigma_t\Omega$-model)}: This model combines
the $\sigma_t$- and $\Omega$-models. A Gaussian
age distribution is allowed (mean age and
standard deviation are free parameters), as
are the 10 free weights for a non-parametric
rotation distribution, giving a total of 12
parameters. Like the $\sigma_t$-model, the age
distribution here is in terms of log(Age).
\end{enumerate}
We measure the probability of models matching the
data using Hess diagrams and a Poisson likelihood as
described in \cite{AD2001}. Our fitting
considers up to 10 independent weights
for the density of stars at our 10
values of $\Omega/\Omega_{\rm{crit,ZAMS}}$, plus up to two more
parameters describing the Gaussian age
distribution's mean and standard
deviation. We take this standard
deviation of the Gaussian age distribution
to represent the ``age spread''. So, given a
derived mean log age $\mu_{\tau}$, we take
the age spread to be
$10^{(\mu_{\tau}+\sigma_{\tau})} -
10^{(\mu_{\tau}-\sigma_{\tau})}$, with
$\sigma_{\tau}$ being the standard deviation.
We use Markov chain Monte Carlo (MCMC) to
sample the probability distributions,
determining the most likely parameter values for
rotation rate weights, mean ages, and age
spreads. We employ \texttt{emcee}'s affine-
invariant ensemble sampling algorithm
(\citealt{DFM2013}). To initialize our
ensemble of walkers, we chose randomized
positions from a uniform distribution within
$\pm 0.2$ dex of the chosen mean log age for
a cluster, and within $\pm 0.05$ dex of the
arbitrarily chosen initial age spread of
$\Delta$ log age = 0.05, if applicable. In
initializing the walker positions for the
various $\Omega/\Omega_{\rm{crit,ZAMS}}$ weights was done
using a Dirichlet distribution. The
reasoning behind this choice being that we
desired these random initial positions to
lie within the solution plane, such that all
rotation rate weights add up to the total
combined weight of all bins for the data. In
other words, we set these positions
in a way that preserves a cluster's total
number of stars, rather than initializing in an invalid
portion of parameter space where the total
counts is far off from that of the data. Thus,
our model for rotation distributions is a
non-parametric model consisting of 10 free
parameters; our eSF model is a Gaussian model
described by its mean and standard deviation.
\begin{figure*}[!ht]
\center
\includegraphics[width=0.95\linewidth]{cluster_age_summary.pdf}
\caption{Derived ages according to the $\sigma_t\Omega$- and
$\sigma_t$-models (red and blue respectively). As
the $\sigma_t\Omega$-model allows non-zero rotation
rates, it typically finds a smaller age spread and
younger age than the $\sigma_t$-model, as discussed
in the text. Vertical black dashed lines indicate
ages cited from literature in Section \ref{sec:data};
transparent cyan solid lines indicate the best-fit
age found by the $\Omega$-model for the respective
cluster. The black solid curves are pseudo-age distributions
from \cite{PG2014} and \cite{GGC2017} for NGC 2203
and 2249 (respectively) for comparison.}
\label{fig:clusterage}
\end{figure*}
\subsection{Mock Tests}
\label{ssec:mocks}
Here we present the results of mock tests
examining accuracy in parameter recovery.
These tests were carried out for our three
scenarios of eMSTO presence under
consideration: population age spread ($\sigma_t$-
model), stellar rotation distribution
($\Omega$-model), or both
($\sigma_t\Omega$-model). We generated mock
data according to each of these scenarios
using MATCH and applied our models to check
that the input age and relative weights of
populations at various $\Omega/\Omega_{\rm{crit,ZAMS}}$
were recovered. The mock data is generated
with a mean log age 9.0 in all cases,
metallicity of [Fe/H] = -0.40,
$A_{\rm{V}} = 0.07$, akin to NGC 2249,
as determined in \cite{MC2014}. We generate
mock data according to each scenario, and the
log(age [yr]) spread is $0.05$, while the input
rotation distribution is a Gaussian centered
at $\Omega/\Omega_{\rm{crit,ZAMS}}=0.4$ with a standard deviation of
$0.2$ dex in applicable cases.
The recovered weights shown in Figure
\ref{fig:mock_weights} find values near the
inputs (black, hatched), and the truth is
contained within errors in most cases, except
the two shown in the bottom-left and -right
panels (d) and (f). These two cases correspond
to fitting the $\Omega$-model to mock data
created with either the $\sigma_t$- in panel (d),
or $\sigma_t\Omega$-model in panel (f).
In both of these cases, the model forms
roughly a bimodal distribution of rotation
rates; both of these cases are the
$\Omega$-model fit to mock data containing an
age spread. Thus, we expect that our
$\Omega$-model sees an age spread as a
bimodal distribution of rotation rates. The
reason for this is that fast rotators can
enhance the eMSTO spread with more dramatic
gravity darkening effects, as shown in e.g.,
\cite{BdM2009}, \cite{BH2015c}, and
\cite{SG2018}. Additionally, our MIST-based
rotating models mostly become redder as
$\Omega/\Omega_{\rm{crit,ZAMS}}$ rises, populating a reddened MSTO,
allowing a greater TO spread but leaving
behind a de-populated blue MSTO. Low rotation
rates refill the bluer side of the MS when
added to the full ensemble of stars. The
bimodal nature of rotation seems to arise
from fast rotators being used in an effort to
fit the eMSTO spread, leaving a de-populated
blue MSTO, alongside slow rotators
counteracting this offset and populating the
red MSTO. So, the properties of fast and
slow rotators appear to drive our models to
favor the presence of both in explaining an
age spread, or otherwise broad MSTO. Unless
the underlying rotation distribution in the
data is something specific, like a Gaussian,
we expect to see a bimodal distribution arise
in order to model the width of the eMSTO. This
is shown in panel (e) for $\Omega$-model fit
to itself when the input rotation distribution
is a Gaussian. The full model (i.e., the
$\sigma_t\Omega$-model) is capable of recovering
the input rotation rate distribution in all
cases, as may be seen in panels (a), (b), (c).
The additional degrees of freedom allowed by
the Gaussian age spread in the $\sigma_t\Omega$
-model removes the necessity for a bimodal
rotation rate distribution.
Figure \ref{fig:mock_res} shows residuals for
our mock tests, again omitting 1:1 comparisons
and only showing cases where the comparisons
produced interesting residuals. In panel (b)
of Figure \ref{fig:mock_res}, one may see that
the $\Omega$-model does not match the smooth
variation of stellar densities created by a
Gaussian age spread (as in the $\sigma_t$-model).
It does better matching the $\sigma_t\Omega$-mock,
seen in panel (a) of Figure \ref{fig:mock_res},
(i.e. it acquires a higher likelihood), but
similarly misses the continuous morphology of
a Gaussian age spread; the placement of rotating
models on the CMD imposes a combination of
relatively specific morphologies, discretized
according to a corresponding rotation rate. Thus,
there is a low chance that each of these
morphologies matches the more ambiguous morphology
of a Gaussian age spread, creating distinct
features in the residuals and relatively poor fits
when the two try to match each other.
Finally, the bottom row of Figure
\ref{fig:mock_res} shows the $\sigma_t$-model
fit to the $\sigma_t\Omega$-model in panel (c) and
to the $\Omega$-model in panel (d). The $\sigma_t$
-model is generally capable of achieving higher
likelihoods than the $\Omega$-model shown in the
top row, but it still has difficulty reproducing
the densities of stars at non-zero rotation rates,
leading to the features shown in the residuals.
This indicates that the $\sigma_t$-model possesses
enough ambiguity to smooth out inconsistencies and
produce a higher fit statistic with disregard to
the presence of a distribution of rotation rates. In
comparing panels (a) and (c), it is noticeable that
the sub-giant branch (SGB) of the $\sigma_t\Omega$-
mock data is better matched by the $\Omega$-model.
\section{Results}
\label{sec:results}
We provide results in this section. Section
\ref{ssec:resparam} provides a description of
the derived rotation rate and
ages distributions. Section \ref{ssec:res}
presents the residuals between our best-fit
models and data. Section \ref{ssec:dage} discusses
our resulting $\Delta$(Age)-age trend with
comparison to previous studies.
\begin{figure*}[!ht]
\center
\includegraphics[width=0.95\linewidth]{50thpercentile_summary.pdf}
\caption{Rotation rate distributions resulting from our fits to (in columns
going left to right) NGC 1866, 1831, 2818, 2249, and 2203. The
top row shows results for the full $\sigma_t\Omega$-model, and
results for the $\Omega$-model are in the bottom row. The
$\sigma_t\Omega$-model tends to favor a single population of
moderate-fast rotators, while the $\Omega$-model tends to favor a
bimodal distribution of fast and slow rotators, with few
intermediate rates, similar to some recent observations of eMSTO
stars.}
\label{fig:rotdists}
\end{figure*}
\subsection{The Age and $\Omega/\Omega_{\rm{crit,ZAMS}}$ Distributions}
\label{ssec:resparam}
Derived Gaussian age distributions for the
$\sigma_t$- (blue shaded region) and
$\sigma_t\Omega$-models are shown in Figure
\ref{fig:clusterage}, with the best-fit age
of the $\Omega$-model as a solid cyan line.
Black dashed lines show the literature ages
described in Section \ref{sec:data}. For NGC
2203 and 2249, black solid curves indicate
the ``pseudo-age'' distributions determined
by \cite{PG2014} and \cite{GGC2017},
respectively. It is also mentioned in those
works that the pseudo-age distributions are
broader than what photometric errors allow,
and so are not spurious in that manner.
Briefly, the pseudo-age distribution is
one method of determining the age spread on
the eMSTO. It is created with a
parallelogram that encloses the width of
the eMSTO. The colors and magnitudes of
stars within this parallelogram are
translated into an age distribution. This
is done by taking the ages predicted by
stellar models of these stars. Like our
Gaussian age distributions shown in Figure
\ref{fig:clusterage}, the pseudo-age
is not reflective of the SF history of the
cluster on its own (\citealt{PG2014}), but
rather emulates the distribution of ages
that may be present in the eMSTO at the
time of observation.
In comparison to our derived age distributions,
the pseudo-age distributions have multiple
peaks, with the strongest amplitude at younger
ages than we find. This may be due to the
different models used (Padova in
\citealt{PG2014} and SYCLIST in
\cite{GGC2017}, while ours are MIST-based).
Additionally, it could be due to our age
spread model only allowing a single peak,
causing it to compromise for a peak in
between the multiple peaks found by
\cite{PG2014, GGC2017}. In spite of these
offsets in mean age, the widths of the
pseudo-age and Gaussian age distributions
are comparable, suggesting that both methods
predict similar age spreads.
However, a common trend seen with all
clusters is that the $\sigma_t\Omega$-model
predicts a smaller age spread. This is
expected, as stellar rotation also
contributes to the eMSTO morphology with
this model. Gravity darkening can introduce
substantial color variations and eMSTO width,
as demonstrated in Figure \ref{fig:fakephot}.
Furthermore, it may be seen that the inclusion
of stellar rotation tends to reduce the
predicted mean age.
Due to the reddening effect of gravity darkening,
it is also expected that stellar rotation would
reduce the mean age. Age is primarily determined
by the CMD position of the MSTO, which is fixed
by the data. As stellar rotation tends to redden
stellar models, selecting a younger age
counteracts this effect by making the rotating
stars blue again. Hence, stellar rotation is seen
to derive a younger age than non-rotating models
(the $\sigma_t$-model) in Figure \ref{fig:clusterage}.
The ages found by the $\Omega$- and $\sigma_t\Omega$-
models are either similar or coincide.
Although the $\sigma_t\Omega$- and $\Omega$-model
agree on cluster ages, the derived $\Omega/\Omega_{\rm{crit,ZAMS}}$
distributions shown in Figure \ref{fig:rotdists}
highlight where these scenarios disagree. The
top row shows the distributions found by the
$\sigma_t\Omega$-model, while results for the
$\Omega$-model are on the bottom row. In all
cases, the $\Omega$-model finds a more
distinct population of fast (e.g., $\Omega/\Omega_{\rm{crit,ZAMS}}
\geq 0.5$) and slow rotators ($\Omega/\Omega_{\rm{crit,ZAMS}} < 0.5$).
In contrast, the $\sigma_t\Omega$-model
generally finds a smaller presence, or lack
of slow rotators.
\begin{figure*}[!t]
\center
\includegraphics[height=0.9\textheight]{sigsummary.pdf}
\caption{Residuals for the fits to real data. Each row
pertains to a star cluster (top to bottom):
NGC 1866, 1831, 2818, 2249, 2203. Each column
contains results for a particular model (left to
right): $\sigma_t\Omega$-, $\Omega$-, $\sigma_t$-model.
Blue pixels correspond to where the model
overestimates the data, and red pixels to where it
underestimates. Fit statistics are annotated in the
upper right corners of each panel. The black dashed
lines are non-rotating isochrones at ages chosen to
roughly span the observed eMSTO width of each
cluster, for reference.}
\label{fig:residuals}
\end{figure*}
The $\Omega$-model's rotation rate
distributions agree with observations more
than the $\sigma_t\Omega$-model in
this context. They qualitatively reproduce
the observed fast and slow rotators found by
e.g., \cite{AKD2017, BN2018}. In the
observations, slow rotators reside blueward,
while fast rotators lie redward. The stellar
models capture this behavior as well, as
seen in Figure \ref{fig:fakephot}; gravity
darkening tends to redden fast rotators,
causing the two populations to occupy
distinct color spaces in the CMD. The two
populations are necessary in the stellar
rotation scenario so that the full
blue to redward extent of the eMSTO is
reproduced.
The two populations are not required in
the $\sigma_t\Omega$-model. The Gaussian
age spread can compensate for a lack
of slow rotators. This does lead to a
clear lack of slow rotators in
NGC 2203 and 2249 (the oldest clusters).
For these clusters, the predicted lack of
slow rotators is not in line with
recent findings for younger clusters.
\subsection{Residuals}
\label{ssec:res}
Figure \ref{fig:residuals} shows the residuals
of the data compared to the best-fit
models for the $\sigma_t\Omega$- (left
column), $\Omega$-
\noindent
\clearpage
\noindent
(middle), and
$\sigma_t$-model (right). Dashed black lines
show non-rotating MIST isochrones placed
to trace the eMSTO width (similar to the
isochrones in Figure \ref{fig:fakephot}).
Each row corresponds to a cluster, and the
global likelihoods (-2lnP) are kept in
the upper right corner of each panel. While
the $\sigma_t\Omega$-model may not reproduce
the rotation rate distributions one might
expect, it is the best-fit model overall,
considering the global likelihoods.
In NGC 2249, 2818, and 1831, the
likelihoods of all three scenarios are
comparable, and the best matched clusters
on the basis of the residuals. Meanwhile,
NGC 2203 and 1866 (the oldest and youngest
clusters) are the worst matched. In the
latter two cases, performance between the
three scenarios is more disparate. The
$\sigma_t\Omega$-model formally achieves he
best-fit with all clusters (perhaps
unsurprisingly given more free parameters),
but clearly so with these two
clusters in particular. None of the
residuals are clean, showing that all
scenarios produce imperfect
fits, although in different ways.
Inclusion of an age spread does tend to
deliver a higher likelihood. At the same
time, the residuals show that the $\sigma_t$-
and $\sigma_t\Omega$-models often create eMSTOs
that are broader than the data. This is seen
both by pixels over fit by these models tend
to lie outside of the bounding isochrones
that roughly trace the eMSTO. The $\Omega$
-model tends to over predict within the
isochrones, without the extended behavior of
the Gaussian age spread models.
The $\Omega$-model can find comparable
likelihoods to the age spread models, but
also finds mismatches. In fact, it is formally
the worst fit in all cases. Yet, it does appear
to match the eMSTO extent well qualitatively,
suggesting that it fails moreso in getting the
precise stellar densities correct. In other words,
the $\Omega$-model appears to reproduce the data
morphologically, though it does not show a strong
statistical advantage over the other two models.
\begin{figure}[!t]
\center
\includegraphics[width=0.95\linewidth]{mist_n15.pdf}
\caption{Blue and red points show derived age and age spreads
according to the $\sigma_t$-model and $\sigma_t\Omega$
-models, respectively. Also shown lines comparing
the same trend found via the SYCLIST models
in \cite{NF2015} (magenta solid), with our MIST-
based model prediction (black solid).}
\label{fig:nh15}
\end{figure}
\subsection{The $\Delta$(Age)-Age Trend}
\label{ssec:dage}
The $\Delta$(Age)-age trend, such as
the one derived here and shown in
Figure \ref{fig:nh15}, has been a
central point in eMSTO studies such as
\cite{NF2015}. The points in Figure
\ref{fig:nh15} mark age spreads and
ages found by the $\sigma_t$- (blue) and
$\sigma_t\Omega$-models (the latter in
red). For a series of ages,
\cite{NF2015} determined the effective
$\Delta$(Age) that a non-rotating
GENEC-based, SYCLIST isochrone would need
in order to match a $\Omega/\Omega_{\rm{crit,ZAMS}}=0.5$ isochrone.
They selected distinct points on each
isochrone, and determined how much the
non-rotating isochrone's age needed to
shift from its original value in order
to match the CMD position of the
rotating isohrone. Remarkably, they found a
$\Delta$(Age)-age trend that agreed
with measured eMSTO widths from a
range of studies. The lines in Figure
\ref{fig:nh15} show such model predicted
trends. These $\Delta$(Age)-age trends
imply that the effect of stellar
rotation can mimic an age spread.
The magenta line in Figure \ref{fig:nh15}
is the trend formed by their second
isochrone point, ``$M_V$ at MSTO''. We
performed the same analysis to see what the
MIST models predict, and this trend is
shown as the solid black line. The MIST
models produce a shallower trend. In
\cite{SG2018}, we found that MIST models
predict a smaller apparent age spread as a
result of rotation when compared to SYCLIST,
due to our weaker rotational mixing. This
causes the luminosities of our rotating and
non-rotating isochrones to separate less
than SYCLIST models, leading to a smaller
effective ``age spread''. We also found
that gravity darkening appeared to be the
strongest effect of stellar rotation in the
MIST models.
The black and magenta lines in Figure
\ref{fig:nh15} do not show the inclination
angle dependence of rotationally induced
eMSTO width. These lines were measured with
the luminosity and temperature of the stellar
models averaged over all inclination angles.
Thus, this trend does vary slightly depending
on the chosen inclination angle, but it remains
that on average the MIST models predict a
shallower trend than SYCLIST when measuring the
eMSTO spread with this isochrone-based
method. In a synthetic population (as opposed to
single isochrones), the combined effects of a
distribution of rotation rates and inclination
angles allow the rotating MIST models to achieve
comparable eMSTO widths to the observed
$\Delta$(Age)-age trends.
\begin{figure*}[!t]
\center
\includegraphics[width=0.95\linewidth]{ngc1866_milone_alt.pdf}
\caption{Schematic demonstrations of the age spread/rotation
rate distribution contention in NGC 1866. \textbf{Left:} Non-rotating
and rotating ($\Omega/\Omega_{\rm{crit,ZAMS}}=0.5, 0.7$) MIST isochrones show a reproduction of the
split MS (highlighted in the inset), although they do not reproduce the
CMD location of the bluest and reddest TO stars. The age here is the age
found by the best-fit $\Omega$-model. Here, $\Omega/\Omega_{\rm{crit,ZAMS}}=0.5$ is the largest population
of fast rotators found by the best fit model, while $\Omega/\Omega_{\rm{crit,ZAMS}}=0.7$ is included to
show how it aids in reproducing the eMSTO. \textbf{Right:} Non-rotating
isochrones varying in age show that eSF could in theory match the eMSTO
width, covering the bluest and reddest regions of the cluster TO. The
age range is chosen so that the isochrones roughly span the eMSTO width.}
\label{fig:n1866isos}
\end{figure*}
\begin{figure}[!t]
\center
\includegraphics[width=0.95\linewidth]{n1866_ptblend.pdf}
\caption{Colored points are the
best-fit $\Omega$-model for NGC 1866, with its derived
age in the lower left. The inset focuses on the
split MS, showing that this model qualitatively
replicates the split MS with a distribution of
rotation rates, mainly slow rotators combined with
fast rotators at $\Omega/\Omega_{\rm{crit,ZAMS}}=0.5$. Grey points are the data.
The yellow points are stars rotating near critical,
which in our modeling does not match the data; this
population appears to be spurious. The reddest TO
stars are also missed; these may be fast rotators
reddened by decretion disks (Be stars).}
\label{fig:n1866best}
\end{figure}
In the derived ages and age spreads of the
$\sigma_t$- and $\sigma_t\Omega$-models, we see
that the inclusion of stellar rotation does
not reduce the age spread to zero. The age
spreads derived by the $\sigma_t\Omega$-model
are smaller than those found by the $\sigma_t$
-model. This is expected, as stellar rotation
and age spread compete to explain the eMSTO
morphology in the $\sigma_t\Omega$-model. The
fact that the $\sigma_t\Omega$-model's age
spreads still correlate with cluster
age suggests that they trace a stellar
evolution effect, rather than SF. The
fact that the $\sigma_t\Omega$-model finds a
reduced, but non-zero age spread may be
indicative of missing physics in the rotating
models, compensated for by an age spread.
\subsection{The Split MS of NGC 1866}
\label{ssec:splitms}
NGC 1866 has a split MS, in
addition to its eMSTO, that warrants further
analysis. This additional complexity may contribute
to the relatively poor fit achieved for NGC 1866 (Figure
\ref{fig:residuals}). The split MS has been argued
to imply that star clusters host both eSF and a
distribution of rotation rates. The fact that the
$\sigma\Omega$-model achieves the best-fit here,
and the $\Omega$-model the worst, serves to
demonstrate how eSF may be compensating
for an incomplete modeling of stellar evolutionary
effects (e.g., binary mergers and decretion disks).
In spite of the lower statistical likelihood of the
$\Omega$-model (Figure \ref{fig:residuals}), we
find that a coeval distribution of rotation rates
is capable of reproducing the split MS and the
eMSTO simultaneously, albeit with caveats.
Schematically highlighted in Figure \ref{fig:n1866isos} is
the debate between whether the split MS
is due to a rotation rate distribution,
or whether eSF may be present. Figure \ref{fig:n1866isos}
displays content similar to Figure 10
from \cite{APM2017}. A range of ages clearly
does not reproduce the split MS (right panel).
As also noted by \cite{APM2017, APM2018},
modeling the split MS appears to require
fast and slow rotators in the cluster. This is
shown for the simple case of a non-rotating and
$\Omega/\Omega_{\rm{crit,ZAMS}}=0.5$ isochrone ($\Omega/\Omega_{\rm{crit,ZAMS}}=0.7$ is included as well to
show how they aid in extending the MSTO). The split MS
may be driven by bimodal rotation rate distributions,
but $V\sin i$ confirmations still need to be obtained.
As plotted here, and as seen in Figure
\ref{fig:rotdists}, the fast rotators are mainly
stars rotating at $\Omega/\Omega_{\rm{crit,ZAMS}}=0.5$, in contrast
to the predictions of Geneva shown in
\cite{APM2017, APM2018} where the red MS appears to
be comprised of stars rotating near critical velocity.
The right panel of Figure \ref{fig:n1866isos} shows that a range of
stellar ages can aid in spanning the bluemost and
redmost regions of the eMSTO. This was also shown in
the corresponding Figures 10 and 11 of \cite{APM2017},
where isochrones at several rotation rates and multiple
ages are plotted together. This is one example of how
eSF in models (e.g., $\sigma\Omega$) may optimize a
fit to the data, but it is uncertain whether the invoked
age spread is physical. As mentioned, there is sparse
empirical evidence for eSF, and alternatives are known
(although, see recent work by \citealt{GCosta2019a},
finding possible evidence for eSF in NGC 1866 via
modeling of its Cepheid stars).
Our best fit $\Omega$-model (which obtains the worst
likelihood) for NGC 1866 is shown in Figure
\ref{fig:n1866best}, with an inset focusing on the
split MS. The recovered rotation rate distribution
for NGC 1866 is indeed bimodal (for both the
$\sigma\Omega$- and $\Omega$-model), and the model
qualitatively matches the split MS. The eMSTO is
matched in some respects, e.g., that it predicts a
mixed population of slow and fast rotators, as
observed spectroscopically by \cite{AKD2017}. In
other areas, a rotation rate distribution is
inconsistent with the data as modeled
here. Particularly, it misses stars blueward (i.e.,
about F336W - F814W $<$ -0.65 mag) and redward (about
F336W - F814W $>$ -0.35) in the eMSTO, but these areas
of the CMD are known to be affected by complex
stellar evolution effects, e.g., interacting binaries and
Be stars, respectively.
For instance, our models do not include properties such
as decretion disks, nor binary merger products.
The population of fast rotators extending blueward may
not reproduce the findings of \cite{AKD2017},
and could be a spurious result. Increasing the binary
fraction to 50\% and removing the possible ``blue
stragglers'' does not influence the presence of
these near critical rotators in the best fit; so, their
presence appears to aid in matching stellar densities
along the MS, rather than the eMSTO, with these near
critical rotators extending blueward of the TO as a
possible side effect. On the matter of reddened, fast
rotators, \cite{BN2017} found
evidence of a high number of H$\alpha$ emitters
(suspected Be stars) throughout the eMSTOs of these
young clusters. A number of which lie in this redward
region. Though, \cite{MC2017} examined whether H$\alpha$
emitters account for the redward extension of the
MSTO in the young cluster NGC 1850; they found that
H$\alpha$ emitters could not in NGC 1850, in comparing
their data with SYCLIST stellar models. See the modeling done by
\cite{GA2018} for more on Be stars and further examples
in young clusters. As for the blueward extent, \cite{CL2019}
show some examples of the blue straggler populations that
might exist in Magellanic Cloud clusters, but \cite{DE2019}
warn that such stars may also be unaccounted for field
contaminants. Presently for our NGC 1866 data, these stars
do not reside in the field, and so appear to be cluster
members.
Rather than eSF, it is possible that the
redmost and bluemost populations of eMSTO stars (where
the $\Omega$-model does not reproduce the data well, thus
calling to question if this is evidence of eSF) are
affected by decretion disks and binary interaction.
These features are also missed by the $\Omega$-model
in NGC 1831, 2818, and 2249. The
$\sigma\Omega$-model may compensate for these missing
factors by invoking an age spread. We will need to
improve our stellar modeling, or obtain further
data on what sort of stars these are before we can say
for sure.
\section{Discussion \& Conclusions}
\label{sec:conclusions}
In this paper, we have compared the
statistical ability of three proposed
scenarios to explain the eMSTO
morphology: 1) eSF, 2) a rotation rate
distribution, and 3) both combined. In
analyzing the statistics, we also
considered whether the three scenarios
could reproduce observed properties of
eMSTOs, such as their rotation rate
distributions. eSF remains a possibility
based on this analysis,
but it seems unlikely in our interpretation.
The results highlight that a
distribution of rotation rates is
capable of solely accounting for
eMSTO morphology and observed
populations of fast and slow
rotators in the eMSTO. Additionally, a
distribution of rotation rates may
simultaneously account for the split
MS, and the eMSTO of the roughly 200
Myr cluster NGC 1866, given current
uncertainties in stellar modeling. Yet, the
$\sigma_t$- and $\sigma_t\Omega$-
models formally achieve the highest
likelihoods. Whether or not
these age spreads are physical is put
into question by our residuals. All
three scenarios over/under-predict in
different areas. The youngest and
oldest clusters (NGC 1866 and 2203)
are the worst fit clusters, but these
are also the only cases where an age
spread appears to significantly
outperform the $\Omega$-model.
Incidentally, these two cases are
both affected by quite uncertain and
unaccounted for aspects of stellar
rotation: magnetic braking in
NGC 2203, and Be stars in NGC 1866. In
each case, the $\sigma_t\Omega$-
and $\sigma_t$-model show strong
over predictions blue- and redward of
the observed eMSTO. This suggests that
the Gaussian age spread may optimize
the fit in these cases, but create
broader eMSTOs than the bulk of the data
to do so. In contrast, the $\Omega$
-model tends to find mismatches within
the observed eMSTO region, while not
producing such broad artifacts. It is known
that our models are incomplete (e.g.,
in not modeling certain effects like the
decretion disks of very fast rotators and
in approximating rotational and
convective mixing with a 1D framework).
Thus, it seems plausible that a Gaussian
age spread is compensating for missing
aspects. In this section we discuss
caveats, uncertainties, and suggestions
for future work.
\begin{figure*}[!ht]
\center
\includegraphics[width=0.95\linewidth]{ptblend.pdf}
\caption{Each panel shows the best-fit $\Omega$-model (colored
points) for each cluster, overlaid on top of the data
(gray crosses). Similar to observations, our models
find a blueward population of slow rotators, while
fast rotators tend to lie redward. The color scale
corresponds to the average surface velocity of our
models. The best-fit age found by the $\Omega$-model
is shown in the panels.}
\label{fig:rotblend}
\end{figure*}
Figure \ref{fig:rotblend} shows
the best-fit $\Omega$-models for each
cluster. Grey crosses show the data,
colored points show the best-fit
$\Omega$-models, with colors mapped to
the average surface velocity of the
stellar models. Visually, the
$\Omega$-model provides a good match,
but is not perfect. For instance, models
miss the redward MSTO of NGC 1866 which
could be due to not including decretion
disks that can further redden rotating
stars (see Section \ref{ssec:splitms}).
It is worth highlighting that NGC 2818
contains a relatively low number of stars,
so the model shown in Figure
\ref{fig:rotblend} is subject to
stochasticity when stars are drawn from the
IMF. On multiple draws, the eMSTO of NGC 2818
is visually reproduced more or less well as
a result; it can become narrower or broader
on subsequent draws. We have chosen a draw
that matches the data fairly well. In NGC
2203 and NGC 2249, the redward shelf-like
structure roughly at magnitudes of 20 and
19.3 for each cluster, respectively, is only
matched by fast rotators (Figures
\ref{fig:fakephot} and
\ref{fig:rotblend} show this), but is
over predicted in each case (see Figure
\ref{fig:residuals}) with the
$\Omega$-model. It is conceivable that
missing physics related to stellar
rotation, such as magnetic braking,
could lead to model mismatches here.
Additionally, mismatches could come
from associated effects, like stellar
binary evolution. However, the effect
of binary interaction may be minimal
(\citealt{WY2011, ZL2016}) in
intermediate age clusters, in
comparison to the effects of
rotation or a possible age spread.
Though we have included unresolved
binaries in our models, we have not
incorporated the effects of binary
interactions, which could affect
rotation rates via tidal forces, or
directly impact evolution through
mass transfer, possibly producing objects like
blue stragglers. Observations of
eMSTO stars focused
on determining their binary status
could help shed light on the
importance of these effects.
The origin and termination of
rotation rate distributions is
currently undetermined. Braking of
stellar rotation via various mechanisms
may be a source of influence. At the older
end of eMSTO hosting star clusters,
magnetic braking affects stars with
convective envelopes. Such stars have
masses $<1.8M_{\odot}$ or so, and start to
arrive on the MSTO near 1 Gyr.
Disappearance of the eMSTO appears to
occur for clusters older than about 2 Gyr,
where stars now possess larger convective
envelopes and much stronger magnetic
braking, consequently driving all TO stars
in these older clusters towards slow
rotation rates. E.g., \cite{SM2018a} found
NGC 1978 to host an MSTO that is consistent
with no spread within the
observational errors. Although, the
age of eMSTO disappearance may be
metallicity dependent, as early work
by \cite{CG2019} appears to suggest. Going
towards younger (than about 600 Myr)
clusters, such as NGC 1866, the
split MS becomes a prominent feature in
addition to the eMSTO. The origin of the
split MS could be linked to a braking
mechanism (as \citealt{FDAn2017, FDAn2018} suggest
with tidal braking) that makes fast rotators
transition to slow rotators, producing the
observed bimodal MS in these young clusters. Just
as well, at this point it is unclear if this bimodal
distribution is an imprint of the initial velocities
that stars may be born with.
Neither magnetic braking, nor binary evolution
are modeled here, but both appear to be integral
in understanding how rotation rate distributions
evolve with time in these clusters.
Furthermore, several inconsistencies
between the age spread scenario and CMD
structure are relevant to whether
significant age spreads are physical
phenomena (\citealt{ICZ2018} gives an
overview). Figure \ref{fig:fakephot}
shows a broad SGB is expected in intermediate
age clusters if eSF is present, while a
distribution of rotation rates predicts a
narrow SGB (also see \citealt{BdM2009}).
However, models showing a narrow SGB are those
of \cite{BdM2009}, which lacked interior
rotational fuel mixing, and ours, which
possess relatively weak rotational mixing.
This is in comparison to either an age
spread, or models with stronger rotational
mixing, as those used in \cite{NF2015},
which may produce a broad SGB at these ages.
A broad SGB does not show up clearly in
the observations of NGC 2203 or 2249 used
here. \cite{LdeGD2014} found age spreads
were inconsistent with the SGB of NGC 1631,
while \cite{PG2015} found that the SGB
structure could be consistent with an age
spread in some cases. The absence or
presence of a broad SGB could be a useful
determinant in constraining the physics at
hand.
Another important diagnostic could be the RC.
The RC (excluded from our fits) should be
broadened in intermediate age clusters,
generally according to some effect (e.g.,
rotation rate distributions or an age spread)
that can create a spread in stellar mass within
a cluster. \cite{LG1999} laid out the physics
of the extended RC; it requires that a
range of stellar masses exists in the cluster,
such that some RC stars developed degenerate
He cores, while others were massive enough to
bypass this. A range of ages is capable of
creating this spread in stellar mass (see e.g.,
\citealt{PG2014, PG2015}), although rotating
models have not been widely tested as an
alternative. We aim to study the eRC according
to the stellar rotation scenario in upcoming work.
Alongside the eMSTO, studying the SGB and RC
structure will inform a more complete picture
of the role of stellar rotation, and could reveal
missing physical ingredients within the models.
The physical basis
of eSF is questioned further by the
$\Delta$(Age)-age trend (Figure
\ref{fig:nh15}). \cite{GGC2017} and
\cite{APM2018} found that age spreads
combined with a distribution of
rotation rates provides a better
reproduction of data than the latter
acting solely. However, as
\cite{BN2018} alluded to in their
Figure 4, age spreads still appear to
correlate with cluster age. This
suggests that the age spreads
determined in those cases, and here
with the $\sigma_t\Omega$-model emanate
from a stellar evolution effect, rather
than true eSF. In general, the ability
of stellar rotation to largely account
for eMSTO structure on its own (as
shown in Figure \ref{fig:rotblend})
may suggest that residual
mismatches are signs of imperfect
stellar modeling, rather than a true
age spread.
In our interpretation,
a distribution of rotation
rates appears to be the overall most
physically motivated explanation for the
eMSTO phenomenon. We
can not rule out eSF here, as it does
aid in creating a better fit to the data.
However, the age spread may also be
compensating for known
missing ingredients in the rotating models,
such as braking (tidal or magnetic), Be
star disks, effects of binary interaction,
and uncertainties in 1D convective mixing.
This contention may resolve as stellar models
improve. Direct comparison of observed eMSTO
$V\sin i$ with stellar models (such as those
in Figure \ref{fig:rotblend}) should provide
strong constraints on the physics of
stellar rotation. There is known
uncertainty between the formalisms that
are used to simulate stellar
rotation. A detailed assessment of
MIST- with GENEC-based models should
illuminate the consequent range of model
behaviors. The SGB and RC, in
combination with the eMSTO, may provide
an opportunity to study, constrain, and
reconcile many complex, and as of now
uncertain stellar processes within young
clusters.
\section*{Acknowledgments}
SG acknowledges the National Science
Foundation Graduate Research Fellowship
under grant No. DGE1745303. CC
acknowledges support from the Packard
Foundation. This paper is based upon
work supported by the National
Aeronautics and Space Administration
(NASA) under Contract No. NNG16PJ26C
issued through the WFIRST
Science Investigation Teams Program.
Some of this material is based upon work
supported by the National Science
Foundation under Award No. 1501205. ICZ
acknowledges support from NASA through
Hubble Fellowship grant
HST-HF2-51387.001-A, awarded by the Space
Telescope Science Institute (STScI), which is
operated by the Association of Universities
for Research in Astronomy, Incorporated, for
NASA, under contract NAS5-26555. BFW acknowledges
support from NASA through grant AR-13901
from STScI under NASA contract NAS5-26555.
NB gratefully acknowledges financial
support from the European Research
Council (ERC-CoG-646928, Multi- Pop).
and the Royal Society (University
Research Fellowship). We would also
like to thank Bill Paxton and the
MESA community for making this work
possible. The authors also acknowledge
Benjamin D. Johnson for helpful insights
during the development of our methods.
|
1,108,101,566,793 | arxiv | \section{Introduction}
\label{sec:Intro}
Currently, there is a strong interest in in-memory computing concept. In
particular, there are expectations that in-memory computing architectures
may help to overcome the Von Neumann bottleneck problem~\cite{Backus78a} of
conventional computers and thus provide us with better computing machines.
Memristive~\cite{chua76a} (memory resistive) and memcapacitive~\cit
{diventra09a} (memory capacitive) elements that combine information
processing and storage functionalities in simple device structures of
nanoscale dimensions have received a great deal of attention in the context
of in-memory computing (memcomputing~\cite{diventra13a})
paradigm. In fact, the material implication gate was demonstrated
experimentally with non-volatile memristive devices several years ago~\cit
{Borghetti10}. This idea has been further developed and reviewed in a number
of papers \cite{yang2013memristive,Kvatinsky14a,Kvatinsky14b,traversa14a,Linn15a,Linn15b,pershin15a}.
While there is a wide variety of physical systems with memory~\cit
{Pershin11}, it is generally agreed that the non-volatile memory devices are
the most suitable candidates for in-memory computing, and for good reason.
In this paper, however, we explore a different route to in-memory computing
based on \textit{volatile} memristive devices. It is shown that, in
principle, simple circuits of volatile memristors can provide some useful
logic functions. Here, we do not aim to develop a practical
in-memory computing architecture, but rather present a proof of concept
application of volatile memristors. Eventually, it may find its own
application niche.
To make our description physically based, in this paper we consider the
hysteretic behavior of carbon-based field emitters~\cite{Eletskii10,
Fedoseeva15, Li15, Kleshch15, Gorodetskiy16}. For concreteness, we
have chosen a hysteretic graphene-based field emission structure~\cit
{Kleshch15} as a prototype of volatile memristor. The memory effect in such
a structure is attributed to a field-induced detachment of a portion of graphene
sheet from substrate~\cite{Kleshch15}. As in this system the minimum voltage required to induce
an OFF to ON transition $V_{\mathrm{ON}}$ is larger than that needed for the
transition from ON to OFF, $V_{\mathrm{OFF}}$, there is a voltage interval
where the structure remembers its state (defined by the history of voltage
applied).
Thus, there are two main results reported in this paper: (i) the memristive
model of graphene field emitters, and (ii) realization of in-memory
computing gates based on such devices. Accordingly, this paper is organized
as follows. We develop a memristive model of hysteretic graphene-based field
emitters in Sec.~\ref{sec:model}. In particular, in the first part of this
Section we formulate general equations of the model, while in the second
part (that may be skipped by those
readers who are not interested in model details), we formulate the model
parameters based on our understanding of
physical processes associated with graphene detachment
from substrate. In Sec.~\ref{sec:logic}, an implementation
of logic gates based on volatile memristors is explored. We conclude in
Sec.~\ref{sec:conclusion} with a summary of our study.
\section{Memristive model of graphene field emitters}
\label{sec:model}
In this Section we develop a memristive model of graphene field emitters
\cite{Kleshch15} showing that such devices can be classified as first-order
voltage-controlled memristive systems. Our model is well suited for the
description of experimental results, as it captures both the switching
dynamics and physics of field emission. We emphasize that the suggested
memristive model can be adopted for the description of other nanomechanical
systems with memory including those~\cite{Nieminen02,Sun16} that do not
require high voltages for their operation.
\begin{figure}[b]
\centering{\includegraphics[width=8cm]{fig1}}
\caption{(a) and (b): Schematic representation of low- and high-current
states of the graphene field emitter: (a) the low-current state (the edge is
attached to substrate, $x=0$) and (b) the high-current state (the edge is
detached/standing, $x=1$). Both states are stable at $V_{\mathrm{OFF}}<V<V_{\mathrm{ON}}$. (c)
Memristive circuit model of circuits in (a) and (b).}
\label{fig1}
\end{figure}
\subsection{Memristive model}
In a recent experiment~\cite{Kleshch15} a strong hysteresis in
current-voltage characteristics of field emission from the edge of graphene
on SiO$_{2}$ was observed. This behavior was explained by a field-assisted
local detachment of the graphene edge from the substrate (for a schematic
illustration see Fig.~\ref{fig1}). In particular, it was demonstrated that
when the system is subjected to an increasing voltage $V$ from 0 to a
maximum value, there is a rapid increase in the current at a certain
V_{switch}$ (in what follows, denoted by $V_{\mathrm{ON}}$). On the way back, a
current drop is observed at $V_{\mathrm{OFF}}<V_{\mathrm{ON}}$ such that $V_{\mathrm{ON}}/V_{\mathrm{OFF}
\approx 7$. Importantly, in the hysteretic region (ranging from $V_{\mathrm{OFF}}$ to
$V_{\mathrm{ON}}$) the current is stable in the sense that the system can stay
arbitrary long in one of two (in some samples, many) possible current states. Thus, the
memory of such field emitters can be classified as long-term and volatile
(the memory is lost at small $V$ including $V=0$). A similar memory effect
in field emission from graphene was also observed in our own in-house
experiments \cite{in-house}.
In order to describe the hysteretic field emission from graphene, we use the
formalism of memristive devices developed by Chua and Kang~\cite{chua76a}.
According to the definition, an $N$-order voltage-controlled memristive
system is given by
\begin{eqnarray}
I(t) &=&R_{M}^{-1}(\mathbf{x},V,t)V(t), \label{eq:I(t)} \\
\dot{\mathbf{x}} &=&f(\mathbf{x},V,t), \label{eq:x_dot}
\end{eqnarray
where $R_{M}$ is the memristance (memory resistance), which depends on the
input voltage $V$ and vector $\mathbf{x}$ of $N$ internal state variables. The
function $f$ in Eq.~(\ref{eq:x_dot}) defines the dynamics of internal state.
Nowadays, Eqs.~(\ref{eq:I(t)})-(\ref{eq:x_dot}) are widely used to model a
broad range of emergent non-volatile memory devices~\cite{Pershin11}.
Moreover, the present authors applied Eqs.~(\ref{eq:I(t)})-(\ref{eq:x_dot})
to field emission from carbon nanotubes \cite{Gorodetskiy16}.
It is natural to select the internal state variable $x$ as $x=L_{p}/L_{tot}
, where $L_{p}$ is the length of detached (standing) portion of the edge,
and $L_{tot}$ is the edge length. Two limit cases (completely attached, $x=0
, and detached, $x=1$, edges) are schematically depicted in Fig.~\ref{fig1}.
Generally, $x$ can take any intermediate value between 0 and 1. To formulate
the memristive model of graphene field emitters, we assume that the current
in $x=0$ and $x=1$ states can be described by the Fowler-Nordheim law~\cit
{FN28a}. Note that this assumption is in agreement with experimental
observations~\cite{Kleshch15}.
The total current can be written as a sum of currents through the attached
and detached regions of the edge:
\begin{equation}
I=\left( 1-x\right) I_{\mathrm{OFF}}+xI_{\mathrm{ON}}, \label{I}
\end{equation
where $I_{\mathrm{OFF}}$ and $I_{\mathrm{ON}}$ are the total emission currents at $x=0$ and
x=1$, respectively. $I_{\mathrm{OFF}}$ and $I_{\mathrm{ON}}$ are represented using the
Fowler-Nordheim law as
\begin{equation}
I_{\mathrm{OFF(ONN)}}=A_{\mathrm{OFF(ON)}}V^{2}e^{-\frac{B_{\mathrm{OFF(ON)}
}{V}}. \label{eq:FN}
\end{equation
Here, $A_{\mathrm{OFF(ON)}}$ and $B_{\mathrm{OFF(ON)}}$ are constants discussed in Subsec. \re
{sec2b}.
\begin{figure}[t]
\centering{\includegraphics[width=8cm]{fig2}}
\caption{Hysteretic curves for the internal state variable $x$ of (a)
volatile (graphene field emitter) and (b) hypothetical nonvolatile
memristor. Insets demonstrate respective non-pinched and pinched hysteretic
I-V$ curves.}
\label{fig2}
\end{figure}
In order to reproduce experimental results~\cite{Kleshch15}, it is
sufficient to select the function $f$ in the Eq.~(\ref{eq:x_dot}) as
\begin{equation}
f(V)=\left\{
\begin{array}{cc}
\gamma & \;\textnormal{if}\;V\geq V_{\mathrm{ON}} \\
-\gamma & \;\textnormal{if}\;V\leq V_{\mathrm{OFF}} \\
0 & \textnormal{otherwise
\end{array
,\right. \label{eq:f(V,M)}
\end{equation
where $\gamma >0$ is the rate of change of $x$. In fact, the function $f$ defined by Eq.~
\ref{eq:f(V,M)}) can describe both types of memristors: non-volatile and
volatile. Assuming a positive $V_{\mathrm{ON}}$, the memristor type is defined by inequalities
\begin{eqnarray}
V_{\mathrm{ON}} &>&V_{\mathrm{OFF}}\geq0\text{ : volatile,} \\
V_{\mathrm{ON}} &>&0>V_{\mathrm{OFF}}\text{ : non-volatile.} \notag
\end{eqnarray
Figure~\ref{fig2} schematically shows examples of the dynamics of $x$ in a
volatile memristor (such as the graphene field emitter), Fig. \ref{fig2}(a), and in a hypothetical
non-volatile memristor, Fig. \ref{fig2}(b), subjected to a periodic quasistatic waveform voltage.
\begin{figure}[t]
\centering{\includegraphics[width=8cm]{fig3}}
\caption{$I-V$ curve of the graphene field emitter found using Eqs.~(\protect
\ref{eq:I(t)})-(\protect\ref{eq:f(V,M)}) with the following set of parameter
values: $V_{\mathrm{OFF}}=50$ V, $V_{\mathrm{ON}}=350$ V, $A_{\mathrm{ON}}=2.32\cdot 10^{-9}$ A/V$^{2}
, $B_{\mathrm{ON}}=662.2$ V, $A_{\mathrm{OFF}}=1.99\cdot 10^{-14}$ A/V$^{2}$, $B_{\mathrm{OFF}}=160.6$
V, $\protect\gamma T=100$, where $T$ is the voltage period. Inset: the same
curve shown in the linear scale.}
\label{fig3}
\end{figure}
A calculated $I-V$ curve of graphene field emitter subjected to a triangular waveform
voltage is shown in Fig.~\ref{fig3}. We emphasize that our volatile
memristor exhibits a non-pinched hysteresis.
\subsection{Physical basis of the model}
\label{sec2b}
Here, we briefly discuss the expressions for the model parameters $A_{\mathrm{OFF(ON
}}$ and $B_{\mathrm{OFF(ON)}}$.
Consider the field emission from a graphene-based cathode, as presented in
Fig.~\ref{fig1}. The potential difference $V(t)$ between the cathode and
anode results in the electric field $E=\beta V/D$, where $D$ is the
distance between the electrodes and $\beta$ is the form factor. Then the
current is described by the Fowler-Nordheim formula \cite{Eletskii10,
Sheshin01, in-house
\begin{eqnarray}
I(V) &=&AV^{2}\exp \left( -B/V\right) , \label{FN} \\
A &=&\frac{e^{3}}{16\pi ^{2}\hbar }\frac{1}{\varphi }\left( \frac{\beta }{D
\right) ^{2}S,\text{ \ \ }B=\frac{4\sqrt{2m}}{3e\hbar }\varphi ^{3/2}\left(
\frac{\beta }{D}\right) ^{-1}, \notag
\end{eqnarray
where $e$ and $m$ are the electron charge and mass, $\hbar $ is the Planck constant,
$S$ is the effective emitting surface, and $\varphi =4.8$ eV
is the work function.
In Fig.~\ref{fig1}, the situation (a) corresponds to the
graphene sheet entirely attached to the substrate, while in the case (b), the
edge of the sheet is detached. Following the arguments put forward in Refs.~\cite{Kleshch15, in-house},
we believe that the main effect is likely associated with the change in
the form factor $\beta$ and effective emitting surface $S$.
Introducing $\beta _{\mathrm{OFF(ON)}}$ and $S_{\mathrm{OFF(ON)}}$ for the low- and high-current
states, the model parameters are defined as $A_{\mathrm{OFF(ON)}}\equiv A(S_{\mathrm{OFF(ON)}},\beta _{\mathrm{OFF(ON)}})$ and $B_{\mathrm{OFF(ON)}}\equiv B(S_{\mathrm{OFF(ON)}},\beta _{\mathrm{OFF(ON)}})$.
An intermediate situation is described by the superposition state,
Eq.~(\ref{I}).
\section{Logic gates}
\label{sec:logic}
\begin{figure}[t]
\centering{\includegraphics[width=7cm]{fig4}}
\caption{In-memory computing circuit considered in this work. The circuit
combines two memristors M$_i$, resistor R and two voltage sources.}
\label{fig4}
\end{figure}
\subsection{Circuit and calculation of the operation code}
The possibility of in-memory computing with volatile memristors is
investigated employing Fig.~\ref{fig4} circuit, which is similar to the
circuit used in the demonstration of the material implication with
non-volatile memristors~\cite{Borghetti10}. In what follows this circuit
is simulated based on the Kirchhoff's circuit laws equation for $V_R(t)$
\begin{equation}
\frac{ V_{1}-V_R(t)}{R_{M,1}} +\frac{ V_{2}-V_R(t)}{R_{M,2}} =\frac
V_{R}(t)}{R}, \label{Kirchhoff}
\end{equation}
which is supplemented by Eqs. (\ref{eq:I(t)}), (\ref{eq:x_dot}) for the dynamics of
memristances $R_{M,1}$ and $R_{M,2}$. In Eq. (\ref{Kirchhoff}), $V_R(t)$ is the
voltage across R.
\begin{table}[bh]
\caption{Codes~\protect\cite{pershin15a} of logic operations calculated
according to Eq.~(\protect\ref{code}). These codes are defined with respect
to different pairs of initial states of M$_{1}$ and M$_{2}$ and can describe
the final state of any device of interest (in our case, M$_{1}$ or M$_{2}$).
For more information, see the text and Ref.~\cite{pershin15a}.
\renewcommand{\arraystretch}{1.3} \centering
\begin{tabular}{|c|c||c|c||c|c|}
\hline
set to 0 & 0 & XOR & 6 & copy M$_1$ & 12 \\ \hline
NOR & 1 & NAND & 7 & IMP$_2$ & 13 \\ \hline
NOT(IMP$_2$) & 2 & AND & 8 & OR & 14 \\ \hline
NOT M$_1$ & 3 & NOT(XOR) & 9 & set to 1 & 15 \\ \hline
NOT(IMP$_1$) & 4 & copy M$_2$ & 10 & & \\ \hline
NOT M$_2$ & 5 & IMP$_1$ & 11 & & \\ \hline
\end{tabular
\label{codes}
\end{table}
\begin{figure*}[t]
\centering{(a)\includegraphics[width=6cm]{fig5a} \hspace{1cm}
(b) \includegraphics[width=6cm]{fig5b}}
\caption{Operation type as a function of applied voltages calculated using
Fig.~\protect\ref{fig4} circuit with $R=10^6$ $\Omega$. The final states of
$_1$ and M$_2$ hold the logic function outputs presented in (a) and (b),
respectively. These plots were obtained with the same parameters of M$_1$ and M
_2$ as in Fig.~\protect\ref{fig3}.}
\label{fig5}
\end{figure*}
\begin{figure*}[tb]
\centering{(a)\includegraphics[width=6cm]{fig6a} \hspace{1cm}
(b) \includegraphics[width=6cm]{fig6b}}
\caption{Operation type as a function of applied voltages calculated using
Fig.~\protect\ref{fig4} circuit with $R=10^{8}$ $\Omega $ and $10^{13}$
\Omega $. These plots were obtained with the same parameters of M$_{1}$ and
$_{2}$ as in Fig.~\protect\ref{fig3}.}
\label{fig6}
\end{figure*}
Following Ref.~\cite{pershin15a}, we analyze the simulations results
calculating a numerical code that can be associated with a specific logic
operation. Taking $w_{i}=1,2,4,8$ as weights for the input
combinations (0,0), (0,1), (1,0) and (1,1), the numerical code is calculated
as a weighted sum of the final state of a selected memristor,
\begin{equation}
\mathnormal{code}=\sum\limits_{i=1}^{4}w_{i}b_{ij}^{f}, \label{code}
\end{equation
where $b_{ij}^{f}$ is the final state ($0$ or $1$) of the device of interest
$j$ (in our case, M$_{1}$ or M$_{2}$) for $i$-th input combination $(0,0)$,
(0,1)$, $(1,0)$ or $(1,1)$ that correspond to $i=1,2,3,4$. Table~\ref{codes}
summarizes logic functions for all possible code values. In this Table the standard
notations are used for the logic functions, e.g., NOT is the logical
negation, IMP is the material implication (in particular, IMP$_{1}$ is M
_{1}\rightarrow $M$_{2}$), etc. In our numerical simulations of Fig.~\re
{fig4} circuit, we have encountered the following operation codes: $0$, $2$,
$4$, $10$-$13$, $15$.
\subsection{Diagrams of logic operations}
Figs.~\ref{fig5} and \ref{fig6} show some selected results of our
simulations. In order to obtain each point of these plots, we simulated the
dynamics of Fig.~\ref{fig4} circuit for all possible pairs of initial states
of M$_{1}$ and M$_{2}$ subjected to $V_{1}$ and $V_{2}$. The operation code
was found with Eq.~(\ref{code}) and interpreted based on the Table~\re
{codes}.
\begin{figure*}[tb]
\centering{(a)\includegraphics[width=6cm]{Appfig1a} \hspace{1cm}
(b) \includegraphics[width=6cm]{Appfig1b}} \centering{(c
\includegraphics[width=6cm]{Appfig1c} \hspace{1cm} (d)
\includegraphics[width=6cm]{Appfig1d}} .
\caption{Effect of variability of memristor parameters. To obtain these
plots we used $R=10^6$ $\Omega$, and higher $V_{\mathrm{ON(OFF)}}$ for M$_{2}$: $V_{\mathrm{OFF}}=60$ V
and $V_{\mathrm{ON}}=420$ V in (a) and (b), and $V_{\mathrm{OFF}}=70$ V and $V_{\mathrm{ON}}=490$ V in
(c) and (d). All other model parameters were as in Fig.~\protect\ref{fig3}. Compare with Fig.~\protect\ref{fig5}.}
\label{figApp1}
\end{figure*}
According to Fig.~\ref{fig5}, the logic operations are symmetric for M$_{1}$
and M$_{2}$ with respect to $V_{1}=V_{2}$ line. As expected, at low voltages
applied to M$_{i}$, $x_{i}$ changes to 0, at high voltages -- to 1, and
there is also a stability region (copy to M$_{i}$). At $R=0$, the common
stability region is a square defined by the lines $V_{i}=V_{\mathrm{ON(OFF)}}$. This
square is deformed at $R>0$ (this can be seen by placing Fig.~\ref{fig5}(b)
over Fig.~\ref{fig5}(a) or vice versa). The most important voltages regions,
however, are those providing the material implication (IMP) and negation of
implication (NOT(IMP)) gates. The importance of the material implication
stems from the fact that it is a fundamental logic gate~\cit
{whitehead1912principia}, which, together with 'set to 0' (FALSE) operation
form a computationally complete logic basis.
Fig.~\ref{fig6} shows the effect of the resistance of R on logic operations
regions. One can notice that, generally, an increase in $R$ scales the
operation regions in Fig.~\ref{fig5} (a) to higher voltages. In particular,
one can notice the disappearance of 'set to 1' regions (these regions are now beyond
the scales presented) and, in fact, an increase of the region of NOT(IMP).
This observation, actually, is of value as the proper choice of R simplifies
the experimental realization of logic gates and improves reliability.
In order to demonstrate the proposed logic gates experimentally,
one can implement, for example, the following operation
protocol. First of all, the memristors can be independently initialized by
grounding the common point of their connection with R and applying suitable voltage sequences
$V_1(t)$ and $V_2(t)$.
Next, the grounding of the connection point is released while
$V_{1}$ and $V_{2}$ are kept in the stability region of memristors
(operation codes $10$ and $12$). Third, $V_{1}$
and $V_{2}$ can be simultaneously placed into the desired operation point
and switched back into the stability region. The calculation results will be
stored in the final states of memristors.
\subsection{Parameter variability effects}
In this subsection we investigate the effect of variability of memristor
parameters on the logic functions realized with Fig.~\ref{fig4} circuit.
Specifically, we consider the operation of Fig.~\ref{fig4} circuit employing
memristors with different threshold voltages.
For this purpose, the simulations are performed using higher values of threshold voltages
of M$_{2}$ keeping all other simulation parameters as in Fig.~\ref{fig5}
simulations. Fig.~\ref{figApp1} presents two examples of such calculations
showing the diagrams found at about $20\%$ and $40\%$ higher threshold
voltages of M$_{2}$ compared to M$_{1}$.
In Fig.~\ref{figApp1}, one can notice that the diagrams for M$_{1}$ and M
_{2} $ are no more symmetric. At the same time, the general topologies of
diagrams are the same as these in Fig.~\ref{fig5}. Importantly, the areas of
useful logic functions for M$_{1}$ (the implication and negation of
implication) increase with an increase in $V_{\textnormal{OFF}}$ and $V_{\textnormal{ON}}$ of M$_{2}$. This
observation can be used, e.g., to achieve more stable operation of such
memristive logic gates.
\section{Conclusion}
\label{sec:conclusion}
We considered the possibility of in-memory computing (in the form of boolean
logic) based on volatile memristive devices. As a prototype of such
structures, a hysteretic graphene field emitter was adopted. A memristive
model of field emission from the graphene cathode was developed. This model
is practical for the description of real experiments.
Moreover, it was shown that simple circuits of volatile memristors can serve
as a polymorphic logic gate. Specifically, we have demonstrated that in
addition to the trivial operation set (FALSE, TRUE and hold the state) the
same circuit can implement the material implication and the negation
of implication. We expect that volatile memristors could find their own
applications, e.g., in low-level information processing circuits.
\section{Acknowledgment}
This work has been supported by the Russian Scientific Foundation grant No.
15-13-20021. The authors gratefully acknowledge fruitful discussions with A.~V.~Okotrub and D.~V.~Gorodetskiy.
\section*{References}
|
1,108,101,566,794 | arxiv | \section{Introduction}
The study of phase transitions and critical phenomena has attracted much attention in recent decades
\cite{71Stanley,14cjpHu,18cjpIsing,19preDimer,19cjpIsing}. The key concepts in such studies include
critical point, critical exponent, universality, scaling, and finite-size scaling function
\cite{84prbPF,94jpaHu,95prlHu,96prlHu,98preLinHu,99preOkabe}.
In this paper, we will address the problem of critical behavior of network percolation.
Network percolation has been playing an important
role as a simplified model to understand spreading processes of
message, disease, matter and dynamic processes in complex
systems~\cite{percolation1,percolation2,percolation3,percolation4,PRR2019,Newman,Bianconi,Radicchi2015,Allard}.
It has been attracting more and more
attention from physics and other research communities. With the
paradigm of complex networks, nodes representing individuals and
links interactions among them, percolation in networks serves as a
bridge connecting classical model of statistical physics and
practical problems in various fields~\cite{Achlioptas}. However,
further application of the theory is somewhat limited since links
in networks are often in the sense of topology, i.e., connecting
relations without taking into account the geometric distance. By contrast, it is
necessary to have a geometric controllability in network
percolation, i.e., to facilitate or inhibit network
percolation in link-adding processes based on the geometric distance,
which motivates us to free ourselves from the constraint of purely
topological connection between nodes in previous models. As the consequence,
intervening strategies for this kind of correlated percolation \cite{Weinrib,PhysRept2017,Cherag,rigid}
lead to new scaling relations and finite-size scaling.
In some systems, the connecting probability (and thus the percolation
process) between two sites depends on the geometric distance between them. Mobile ad
hoc network (MANET)~\cite{MANET}, as an example, is a new wireless
communication system for temporal assembly of moving members. The
flooding mechanism \cite{MANET} of its message pervading can be viewed as a
percolation process. A MANET should assign proper transmission
range~\cite{MANET,global connectivity,epidemics2} for all nodes to prevent interference among
themselves,
and to save energy for longer lifetime of the network since they
could not be recharged during motion. Therefore, direct
communications can happen only inside speaking nodes' transmission
circles~\cite{Wangli2,epidemics2}, outside which nodes are linked in a manner of multi-hop
(indirect wireless connections through successive relays). Here,
global connectivity~\cite{global connectivity} relies on a suitable
design of transmission range adapting to the occupation density of
nodes on a two-dimensional (2D) plane. Besides, the traffic flux and
bilateral trade volumes between two cities or countries are found to
be proportional to the gross economic quantity of each side, and
inversely proportional to the distance between them. Therefore,
gravitation models ~\cite{gravitation1,gravitation2} are often used to understand
empirical data in various situations. The spread of the ground traffic
congestion could be viewed as another kind of distance-related
percolation in which Manhattan distance (the summation over
projected lengths of geometric distance along two perpendicular
directions) plays a key role. Therefore, Li, et al.~\cite{Li} pointed
out that a power-law distance-decaying link-adding probability in a
2D lattice could optimize ground traffic under certain constraints
on total cost. Moreover, a disaster gravity
mobility model~\cite{gravitation ad hoc} for MANET defines a maximum
distance at which an event affects objects in a gravitational style. That is
why pervasive disasters or rush-hour congestion can cause
percolation-like phenomena between objects \cite{pervasive and rushhour}.
In short, to properly
understand percolation in some real networks, we should not ignore
linking effect related to the geometric distance.
In practice, people often need to combine percolation process with
strategies to achieve better results of coevolutionary processes. In
the situation of massive disaster, base stations of mobile
communication often suffer from black-out, yielding a large scale of
disconnected population. In order to deliver messages, energy, and
matter supplies in disaster relief efforts to all panicked people as
soon as possible, one needs to facilitate percolation in link-adding
networks of vehicles equipped with MANET nodes or other systems.
While in other situations, such as the spread of ground traffic
congestion and epidemics \cite{epidemics1,epidemics2} which depend on the geometric
distance, one
should design effective measures to inhibit percolation. One possible
algorithm for such inhibition is the product rule (PR)
proposed by Achlioptas, et al.~\cite{Achlioptas} and other models
suggested afterwards~\cite{2,3,4,16,5,6,7,8,9,10,11,12,13,14,15,17,18,19,Cho2013,Ziff2012}.
Original PR starts from a network with isolated nodes
as the initial condition. During the evolution process, a
node $i$ is labeled by its mass (or called size) $m_i$ which is the
number of connected nodes in the cluster that includes node $i$.
Two topological links are randomly put into the set of the nodes at
every time step, and only the one connecting two nodes $i$ and $j$
with smaller product of masses ($m_i m_j$) is retained. This rule
postpones the development of the giant component, and a sharp change
of the fraction of the nodes in the largest cluster is observed, which has been called
``explosive percolation". Instead of investigating
the nature of such an unusual continuous ~\cite{Dorogov, 2011science, BJKim,Grassberg}
or discontinuous \cite{Cho2013,Ziff2012} phase transition, we are concerned with
how to facilitate or inhibit percolation in a kind of extended scheme in network growth process.
In this paper, we propose several percolation schemes on a 2D plane with
link-adding rules depending on the geometric distance, which takes the
form similar to Newton's gravity. Simply by adopting the strategy of
either maximum or minimum gravity in successive linking steps for different cases,
one can facilitate or inhibit network percolation in a systematic
way. The observed size of the largest component (cluster), and average connection
lengths of various link types, are revealed to follow
scaling relations which were not recognized in purely
topological percolation models. The present scheme gives a generic
picture for percolation processes in real systems which are often
inevitably geometrically constrained.
\section{Models}
Suppose $N$ isolated nodes are uniformly scattered on a two-dimensional (2 D)
plane with the edge length $L$, hence $N = L^2$. For the convenience to
calculate distance, the plane is discretized with a triangular lattice $G$. Each vertex of the triangles
is occupied by a node.
As in product rule (PR) of Achlioptas process~\cite{Achlioptas} , we pick randomly two pairs [$(i,j)$ and
$(k,l)$] of nodes in the plane at every time step. For the pair
$(i,j)$ (and for $(k,l)$ likewise), we compute the generalized
gravity defined by $g_{ij} \equiv m_i m_j/r_{ij}^\emph{d}$, where
$m_i$ and $m_j$ are the number of sites of the clusters which include site $i$ and site $j$, respectively,
$r_{ij}$ is the geometric distance between $i$ and $j$, and $\emph{d}$ is
an adjustable decaying exponent. Once we have $g_{ij}$ and
$g_{kl}$, we have two choices in selecting which link should be retained.
For the case of the maximum gravity strategy (we call it $S_{\rm
max}$) we connect the pair with the larger value of the gravity,
e.g., the link $(i,j)$ is made if $g_{ij} > g_{kl}$, and the link
$(k,l)$ otherwise. We also use the minimum gravity strategy
($S_{\rm min}$) in which we favor the pair of nodes with smaller gravity to make
connection. The two strategies, $S_{\rm max}$ and $S_{\rm min}$,
lead the link-adding networks to evolve along the opposite
percolation processes. Generally speaking, $S_{\rm max}$
facilitates the percolation process, whereas $S_{\rm min}$
inhibits it similar to explosive percolation
\cite{Achlioptas,Cho2013,Dorogov, 2011science,BJKim}. All such generalized gravity values are
calculated inside the circular transmission range with the radius
$R$ centered at one of nodes $i$ and $j$ as the speaking
node~\cite{Wangli2,epidemics2} in a MANET, for example.
For the different limits of parameters $R$ and $\emph{d}$, we have
two cases. Case 1: With the transmission range
$R\rightarrow \infty$, we have a generalized gravitation rule
which is an extension~\cite{gravitation ad hoc} of widely used
gravitation model $(\emph{d} = 1)$~\cite{gravitation1,gravitation2}
with the decaying exponent $\emph{d}$ tunable.
Case 2: With both adjustable values of radius $R$ and
exponent $\emph{d}$, we have the gravity rule \cite{gravitation1,gravitation2} inside the transmission
range. It can describe the communication or traffics with
constrained power or resources.
\section{Simulation results}
All simulations are carried out on the $L\times L$ triangular lattice of the size
$N = L\times L$ with $L = 32, 64, 128$ and $256$, respectively. We simulate either of strategy $S_{\rm max}$ or
$S_{\rm min}$. The total number of links
equating to that of time-steps is divided by $N$, which is defined
as $T$. The mass of the largest component divided by $N$ makes up
the observable $C_{1}$, the node fraction of the largest component.
The algorithm in the present model is similar to that of
Ref.\ \cite{3} including the rule of intra-cluster priority, except
distance-decaying exponent $\emph{d}$ and transmission radius $R$ used.
And similar time-dependent variation of fractions of different
types (I: both inter-clusters; II: one inter-cluster and the other
intra-cluster; and III: both intra-clusters, see Fig. 2 in \cite{3}) of links
retained \cite{3, 19} are also observed near the threshold of
percolation. All results presented in this work are obtained from
the average over 100 different realizations of network
configurations.
For strategy $S_{\rm max}$ in Case 1, the percolation threshold decreases
from the limit $T_c = 0.5 $ for the Erd\H os-R\'enyi (ER) random graph
[see Fig.~1(a)]. As the exponent $\emph{d}$ decreases, $T_c$ shifts
downward (e.g.,$T_c = 0.37$ for $\emph{d} = 0.2$ and $T_c = 0.36$ for $\emph{d} = 0.01$).
Following the standard manipulation ~\cite{Christensen}, we
obtain a group of decaying - exponent
$\emph{d}$ - dependent percolation thresholds $T_c$, and corresponding critical exponents
$\beta_1$ in the probability for a node to be in the percolating cluster:
\begin{equation}
C_{1} \sim (T - T_c)^{\beta_1} \ \ {\rm for} \ \ T \to T_c +.
\end{equation}
Numerical results for $S_{\rm max}$ in Case 1 are listed in Table I. Obviously, $T_c$ and $\beta_1$ increase with $d$.
Figure 1(b) shows that for $S_{\rm min}$, $T_c$ and the exponent $\beta_1$ also depends on $d$.
In addition, another special point $T_0$ attracts our attention.
Curves $C_{1}(T)$ of Fig. 1(a) for different $\emph{d}$ cross approximately at a point
$T_0~( >T_c )$. Let $t = (T - T_0)/T_0$, then, $C_{1}(T)$ can be roughly re-scaled
as
\begin{equation}
C_{1} \sim \emph{d}^{-\omega} f( t \emph{d}^{\epsilon})
\end{equation}
for different exponents $0.2 < \emph{d} \le 2$
(inset of Fig.~1(a), except the situation $\emph{d} = 0.2$ with a dashed green line),
where $T_0 = 0.78$, , $\omega = 0.01$, $\epsilon = 0.20$,
and $f(x)$ is a universal scaling function, which is
similar to the super-scaling behavior studied by Watanabe and Hu~\cite{08preWH}.
With the strategy $S_{\rm min}$ in Case 1, PR ~\cite{Achlioptas} can be resumed by
letting $\emph{d} \rightarrow 0$, with the threshold $T_c$ approaching
$0.888$ which is the transition point of Achlioptas-type percolation~\cite{Achlioptas}.
On the other hand, $\emph{d} ~\rightarrow \infty$, the gravity values for both candidate links
become indistinguishable and thus any one of the two is selected
arbitrarily, which resumes the case of percolation in growing ER
random graph. Fig.~1 (b) illustrates these two limiting cases and
intermediary ones between them with $L = 128$.
According to the priority rules distinguishing candidate links into three types as shown in Figure 2 of \cite{3},
we calculated the average lengths $\emph{l}_{I},\emph{l}_{II}$ and $\emph{l}_{III}$ of
type-I, type-II and type-III links, respectively, as the summations of specific link-lengths over
corresponding numbers of such types of links.
The finally saturated average lengths of both type-II and type-III links are $\emph{l}_{0} = 131.9$ for $L = 128$ (see Fig.2).
Such saturated value is reached for $T \ge T_s=1.0$.
To find the average length $\emph{l}$ till time step $T$ with
strategy $S_{\rm max}$ in case 1, we do ensemble average on geometric lengths of
retained links under different exponents ($\emph{d} = 0.2, 0.5, 1.0, 2.0, 3.0$ and $5.0$ )
for the lattice with the edge length $L = 128$. Simulation results for three types of links \cite{3, 19} are shown in
Fig. 2. Temporal variations of normalized average lengths of type-III links \cite{3, 19} are re-scaled to collapse very well
into a single curve as shown in Fig. 3. Therefore, we get the following scaling behavior:
\begin{center}
\begin{equation}
\emph{l}/\emph{l}_{0} \sim \emph{d}^{-\lambda} F(\emph{d}^{\tau}T)
\end{equation}
\end{center}
where $\emph{l}_{0} = 131.90$ (see Fig. 2), $\lambda =-0.001$, $\tau = 0.005$ and $F(x)$ is
a universal scaling function.
As seen in Fig.2, averaged lengthes $\emph{l}$ of both type-II and type-III links grow monotonically
until they get saturated. Actually,
they approach the saturated average length $\emph{l}_{0} =131.9$ (see Fig. 2) in $\emph{d}$-dependent paces.
Average length $\emph{l}$ in any growth step (T)
for a smaller $\emph{d}$ is longer than those with larger $\emph{d}$, because strategy $S_{\rm max}$ favors the former links,
and the links with a larger $\emph{d}$ starts
to be realized later on average than those with smaller $\emph{d}$ due to the same reason. Interestingly, $\emph{d}$-dependent average
lengths for each type of links have their own universal scaling functions, which are illustrated in Fig. 4 and Fig. 5,
respectively. The scaling behavior for type-II links in case 1 to arrive at saturated average length $\emph{l}_{0}$ reads:
\begin{center}
\begin{equation}
p_{2} \sim g((T - 1.0)^{\alpha_{2}} \emph{d}^{\gamma_{2}})
\end{equation}
\end{center}
where $\alpha_{2} = - 0.35$ and $\gamma_{2} = -0.03$, respectively and $g(x)$ is a universal scaling
function valid for $0.2 \le \emph{d} \le 5.0$.
Meanwhile, the scaling behavior for type-III links in case 1 to arrive at saturated average
length $\emph{l}_{0}$ reads:
\begin{center}
\begin{equation}
p_{3} \sim S((T - 1.0)^{\alpha_{3}}\emph{d}^{\gamma_3})
\end{equation}
\end{center}
where $\alpha_{3} = - 1.0$ and $\gamma_{3} = -0.08$, respectively and $S(x)$ is a universal scaling function valid for $0.2 \le \emph{d} \le 5.0$.
In addition, the difference of average lengths between type-II and type-III
links $(\emph{l}_{II} - \emph{l}_{III})$ is exactly coherent with the difference of average fractions
between these two types of links $(F_{II} - F_{III})$ at the same $T$, which is
shown in Fig. 6. Therefore, a universal function exists for
$(\emph{l}_{II} - \emph{l}_{III})$ vs. $(F_{II} - F_{III})$ in the simulated range of \emph{d} $(0.2 \le \emph{d} \le 5.0)$.
While pure Achlioptas process~\cite{Achlioptas} does not share the same property (shown in blue line).
Obviously, Fig.3 - Fig.6 and corresponding scaling relations (formulas (3), (4), and (5)) can not be accounted as
trivial ones, since they only happen to the present schemes based on the classification in ref. \cite{3,19}.
Simulations for Case 2 reveal combined effect of transmission
range and gravitation. Following the standard manipulation in ~Ref. \cite{Christensen},
we obtain a group of decaying - exponent
$\emph{d}$ - dependent percolation thresholds $T_c$, and corresponding critical exponents
$\beta_2$ in the probability for a node to be in the percolating cluster, with the same
form as formula (1) but different exponents $\beta_2$.
\begin{equation}
C_{2} \sim (T - T_c)^{\beta_2} for T ~\rightarrow T_c +
\end{equation}
Numerical results for $S_{\rm min}$ in Case 2 are listed in Table II which shows that $T_c$ and $\beta_2$ depend on $d$.
In addition, another special point $T_0$ attracts our attention.
Rough scaling relations with strategy $S_{\rm min}$ are obtained for a range of $R$ $(3 < R \le 8)$ and distance-decaying exponent
$\emph{d}$ $(0.2 < \emph{d} < 2.0)$:
\begin{center}
\begin{equation}
C_{2} \sim (\emph{d}/\emph{d}_0)^{-\theta} h[t(\emph{d}/\emph{d}_0)^{\phi}]
\end{equation}
\end{center}
for different \emph{d}, where $T_{0} = 1.0$, $\theta = 0.005$,
$\phi = -0.50$, $\emph{d}_0 = 0.5$, and $h(x)$ is an approximate universal scaling function. For
this scaling relation, the validation range of decaying exponent $\emph{d}$
and transmission range $R$ need to adapt to each other, since the effect of
a weak decay with a small exponent $\emph{d}$ would be diminished by a small enough $R$ (e.g., we must
have $\emph{d} > 0.5$ for $R = 4$), and strong enough decay (large $\emph{d}$) would
ruin the effect of a large $R$ (e.g., we must have $\emph{d} < 5.0$ for
$R = 8$).
A modest example for $R = 5$ is shown in Fig.~7(a)
(The scaling is roughly valid for $0.2 < \emph{d} < 2.0 $ ).
Besides, a rough scaling behavior with strategy $S_{\rm max}$ for different $R$ and
$\emph{d}$ reads:
\begin{center}
\begin{equation}
C_{2} \sim R^{-\delta} H(t\rho^{\eta})
\end{equation}
\end{center}
for $R > 3$, where $\rho = (R - R_0)/R_0$, $R_{0} = 2$, $\eta = -0.10$,
$\delta = -0.005$, $T_0 = 1.0$, and $H(x)$ is an approximate universal scaling function, which
is shown in Fig. 7(b) (scaling is only valid for a small range $(4 < R \le 8 )$. Here, $T_0$ is
another special point where
average lengths of type-II and type-III links arrive at the same
level, and fractions of type I and III links \cite{3, 19} get a
balance, meanwhile the fraction of type II links arrives at its
summit \cite{to be published}. Also, suitable match between parameters $\emph{d}$ and $R$
is required. Otherwise, this scaling behavior is invalid, just as the case $R = 4$
in the inset of Fig. 7(b).
Inhibitory strategy $S_{\rm min}$ in Case 2 produces the largest threshold
on a 2D plane to the best of our knowledge. Through finite-size
transformation we check the critical point $T_c$. The scaling
behaviors of node fraction $C_{2}$ and susceptibility $\chi$ defined
as $\chi \equiv [{\langle C_{2}^2 \rangle - \langle
C_{2}\rangle^2}]/N$~~\cite{4,84prbHu} are
\begin{center}
\begin{equation}
C_2 \sim N^{-\beta / \nu} Q\left(\left(T - T_{c}\right) N^{1/\nu}\right),
\end{equation}
\end{center}
\begin{center}
\begin{equation}
\chi \sim N^{\gamma / \nu} Z\left(\left(T - T_{c}\right) N^{1/\nu}\right),
\end{equation}
\end{center}
where $1/\nu$=0.2, $\beta$/$\nu$ =0.005, $\gamma$/$\nu$ =0.995,
and $Q(x)$ and $Z(x)$ are universal scaling functions. Therefore, a scaling law
of continuous phase transition
\begin{center}
\begin{equation}
\beta/\nu + \gamma/\nu = 1.
\end{equation}
\end{center}
remains valid for two scaling relations for different parameter sets $(R, \emph{d})$,
which is verified well although scaling relations (7) is limited within a small range for
$S_{\rm min}$ in Case 2. Similar scaling law has been obtained
by Radicchi {\it et al.}~\cite{4} for scale-free networks but with
different sets of exponents. Therefore, the present one in Fig. 8 should be concluded into a different universality class.
Numerical evidence of $S_{\rm min}$ in Case 2 for $R = 2$, $\emph{d} = 2.0$ with $L = 32, 64, 128$ and $256$ are shown as an example
in Fig. 8(a) and Fig. 8(b), with the percolation threshold as large as
$T_c\simeq 1.5$, as an example of $S_{\rm min}$ in Case 2. Insets of them illustrate the re-scaled results of $C_{2}$
and $\chi$ (see formulas (9) and (10)), respectively.
\section{Discussion and Conclusions}
It should be noted that scaling relations illustrated in Fig. 1, Fig. 3, Fig. 4, Fig. 5 and Fig. 7
(formulas (2), (3), (4), (5), (7) and (8)) are not
referring to critical points $T_c$ of pertinent percolation in specific gravitational distance
- decaying schemes. Instead, they are referring to kind of special points $T_0$ which are $\emph{d}$
governing or $(\emph{d}, R)$
coordinately controlled, and worthy of further investigation. Among
them formula (1) around $T_0$ for $S_{\rm max}$ in case 1 is approximately valid. And formula
(5) and (6)
are valid only for properly matched sets of $R$ and $\emph{d}$. By contrast, Fig. 8 and
corresponding scaling relations, $\emph{i.e.}$
formulas (7), (8) and (9) referring to order parameter $C_{2}$ and $\chi$ around $T_c$ are
quite solid.
The gravitationally correlated lattice percolation models (GCLPMs) introduced in this paper
are new models of long-range correlated percolation
\cite{Weinrib,PhysRept2017,Cherag,rigid}, and they are in different
universality class from the existing correlation percolation model, e.g. the scaling law mentioned in \cite{Weinrib}
is violated and the PR is merged into the
bond-occupation schemes. From this viewpoint we can
understand a different saturation effect of $S_{\rm max}$ for decaying
exponent $d\geq 3.0$ in Fig. 1(b) and limited validation ranges of $\emph{d}$
for all scaling relations relevant to correlations in Case 2 with strategy $S_{\rm min}$.
Intervening schemes in the present gravitational correlated percolation have predicted rich
scaling relations.
With the link-adding network schemes depending on gravitational distance-decaying strategies
$S_{\rm max}$ or $S_{\rm min}$,
we designed different ways to facilitate or inhibit network
percolation on the 2D plane from a generic view of continuous phase
transition. The adjustable
transition threshold covers the range from $0.36$ to $1.5$ with the
present simulations, which provides an approach to tuning critical point $T_c$ precisely according
to requirement of different systems.
Moreover, the approaches to re-scale time (the number
of edges $T$) of a growing network with distance information would
reveal more critical spatiotemporal properties of co-evolutionary
processes. They could get broader applications than previous network percolation
models constrained in topological sense when parameters $\emph{d}$ and $R$
are properly selected for practical problems.
The GCLPMs introduced in this paper can inspire many interesting
problems for further studies. In the present paper, we only simulate the GCLPM on the plane triangular (pt) lattice and obtain
the finite-size scaling function only for the pt lattice.
It has been found that bond and site percolation models on the square (sq), plane triangular (pt), and honeycomb (hc)
lattices can have universal finite-size scaling functions when the aspect ratios of the sq, pt, and hc lattices
are chosen to have the relative sizes 1: $\sqrt{3}/2$: $\sqrt{3}$ \cite{95prlHu,95PhysicaAHu,96prlHu}. An argument about
why to choose such aspect ratios can be found in the Appendix C of \cite{14cjpHu}. We can simulate the GCLPM
on the sq, pt and hc lattices whose aspect ratios have the relative sizes 1: $\sqrt{3}/2$: $\sqrt{3}$ to obtain
the universal finite-size scaling functions of the GCLPM on the sq, pt and hc lattices.
The Ising model and the Potts model are important lattice models \cite{71Stanley,14cjpHu,98preIzma,82rmpPotts-WuFY,96prlChenCN,96prlPotts2}.
It has been found that the Ising model on the sq, pt and hc lattices whose aspect ratios have the relative sizes 1: $\sqrt{3}/2$: $\sqrt{3}$
can have universal finite-size scaling functions \cite{99preOkabe,97preWangHu,03preWuMC}.
It has been shown that the Ising model and the Potts model are corresponding to the 2-state and the $q$-state bond-correlated
percolation models (qBCPM) \cite{14cjpHu,84prbHu,84prbHuPotts}, respectively. The 2-state bond correlated percolation
model (2BCPM) is a special case of the qBCPM when $q=2$. The random bond percolation model
is a special case of the qBCPM when $q=1$ \cite{82rmpPotts-WuFY}.
To simulate the qBCPM, Swendsen-Wang has proposed a Swendsen-Wang algorithm \cite{87prlSW}, which can overcome the critical slowing
down. Hu and Mak had used this algorithm to simulate the qBCPM on the sq and the simple cubic lattices \cite{89prbQPM}.
Chen, Hu and Mak had developed a FORTRAN program to simulate the qBCPM on D-dimensional hypercubic lattices \cite{91cpcQPM} based on
the Swendsen-Wang algorithm \cite{87prlSW}. In this paper, we modify the bond random percolation model to introduce the GCLPM.
In the future, we can modify the qBCPM to include the concepts from the GCLPM.
Such a model can be denoted as qBCPM-GCLPM. We can simulate the qBCPM-GCLPM
on the sq, pt and hc lattices whose aspect ratios have the relative sizes 1: $\sqrt{3}/2$: $\sqrt{3}$ to find the
universal finite-size scalings for the qBCPM-GCLPM. We can also study whether and how the qBCPM-GCLPM can show
a first-order phase transition as parameters of the model, e.g. $q$ and $d$, are changed.
In summary, the GCLPM introduced in this paper can inspire many interesting problems for further studies.
\begin{acknowledgments}
C.P.Z. thanks H. Park, P. Holm, X.-S. Chen, M.-X. Liu and Z.-M. Gu for useful
discussion. C. K. H. is indebted to R. M. Ziff for a critical reading of the manuscript.
C.P.Z., L.T.J. and L.L.S. acknowledge financial support from National Natural
Science Foundation of China (NNSFC) under Grants No. 11175086,
10775071 and 11775111. B.J.K. acknowledges the support from the National
Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIT)
Grant No. 2017R1A2B2005957. C.K.H. is supported by Grant MOST 108-2112-M-259 -008.
\end{acknowledgments}
|
1,108,101,566,795 | arxiv | \section{Introduction}
Constellation shaping is now a well-established technique in the optical communications research community for improving the system throughput and operate close to the theoretically achievable information rate and at high spectral efficiency.
Both probabilistic shaping \cite{b01, b02, b03} and geometric shaping \cite{b06, b07, b08, b10, b12}, each having its benefits and drawbacks, are of significant interest to both system vendors and researchers.
Probabilistic shaping refers to optimizing the probabilities of the constellation points, typically applied to rectangular quadrature amplitude modulation (QAM), whereas geometric shaping refers to optimizing the positions of the constellation points.
Probabilistic shaping provides slightly higher shaping gains than geometric shaping in terms of the generalized mutual information (GMI), since non-rectangular constellations lack an obvious Gray-like code. This problem can be avoided, in trade-off for increased complexity, by employing Geometric shaping with iterative demapping and decoding~\cite{b07}, or with non-binary forward error correction (FEC)~\cite{b06}.
Also probabilistic shaping includes similar trade-off, as it requires a shaping encoder, such as the constant composition distribution matcher~\cite{b03,b04,b05}. Such encoders may be problematic due to their error-propagation and inherent serialized processing, making it challenging for parallelized implementation.
Hence, a geometric constellation shape with Gray-like code combines low implementation complexity and shaping gain. Such constellations have been proposed in~\cite{b10}, yet the applied method switches in between two independent optimization methods and is not scalable in a straight forward manner.
In this paper, an autoencoder is used to optimize a geometric shaped constellation in terms of GMI by jointly optimizing the bit mapping and the position of the constellation points.
The proposed method makes no assumption on the channel model
and is easily scalable to constellations of higher order and higher dimension. The optimization builds upon the autoencoder-based geometric shaping, optimized for mutual information (MI) only \cite{b12}. The following myths about geometric shaping are debunked with the proposed method:
\begin{enumerate}
\item \textit{Conventional bit-interleaved coded modulation (BICM) is penalized with geometric shaping due to the non-Gray labeling.} We show that the proposed autoencoder arrives at a Gray-like code, which does not exhibit this problem.
\item \textit{The implementation penalty is higher for geometric shaping than rectangular QAM.} We show that in the operating regions of interest and with the application of modulation-format independent digital signal processing (DSP) chain, the penalty is the same.
\item \textit{Iterative demapping or non-binary FEC are required for geometric shaping schemes.} We show that the proposed labelings do not have this requirement because they are GMI optimized.
\end{enumerate}
\begin{figure}[t]
\centering\input{autoencoder.tikz}
\caption{Schematic autoencoder model applied directly on bits.}
\label{fig0}
\end{figure}
\section{Methology}
\subsection{Autoencoder}
Similar to the setup in~\cite{b12}, an autoencoder with embedded channel model is trained for a geometric constellation shape of order $M$:
\begin{equation}
\begin{split}
\vec{x} &= f(\vec{s}),\\
\vec{y} &= c(\vec{x},P^3,\kappa,\kappa_3),\\
\vec{r} &= g(\vec{y})
\end{split}
\end{equation}
where $f(\cdot)$ is the encoder neural network, $c(\cdot)$ is a fiber channel model and $g(\cdot)$ is the decoder neural network. \begin{figure}[t]
\centering\includegraphics[width=0.60\linewidth]{const256-eps-converted-to.pdf}
\centering\includegraphics[width=0.60\linewidth]{const256Zoom-eps-converted-to.pdf}
\caption{Geometric constellation shape of order $M$=256~\textbf{(top)} and a zoomed-in version~\textbf{(bottom)} including bit mappings.}
\label{fig1}
\end{figure}
The used fiber channel model~\cite{b11} is dependent on the launch power $P$, and on the fourth and sixth order moment of the transmitted constellation, $\kappa$ and $\kappa_3$. The channel input $\vec{x}$ represents samples from a constellation and the channel output $\vec{y}$ the respective observations.
In contrast to the setup in~\cite{b12}, where the autoencoder is trained with a dataset of one-hot encoded vectors and softmax output layer, here the input space is $\vec{s} \in \mathcal{S}=\{0,1\}^m$, the set of all possible bit sequences of length $m=\log_2{M}$. Accordingly, the target space is given by
$\vec{r} \in \mathcal{R} = \{x~\in~\mathbb{R} | 0<x<1\}^m$, which refers to an output layer with sigmoid activation function, and represents the posterior probabilities of the bits being 1 or 0, $p(\vec{s}|\vec{y})$.
A training batch of size~$K$,
$\{ \vec{s}^{(1)}, \vec{s}^{(2)}, ..., \vec{s}^{(K)}\}$,
is uniformly sampled from $\mathcal{S}$ and propagated through the autoencoder.
The weights of the autoencoder are updated with stochastic gradient descent according to a loss function:
\begin{equation}
\begin{split}
& L(\{ \vec{s}^{(1)}, ..., \vec{s}^{(K)}\}) = \frac{1}{K} \sum_{k=1}^{K} l(\vec{s}^{(k)},\vec{r}^{(k)}),\\
& l(\vec{s},\vec{r}) =-\frac{1}{m} \sum_{i=1}^{m} s_i \log(r_i) + (1 - s_i) \log(1 - r_i).
\end{split}
\end{equation}
Since the input and output vectors represent bit sequences, the encoder neural network is learning both a geometric constellation shape and a bit mapping rule.
Following the mismatched decoding principle (described in e.g. \cite{b01}), the log-likelihood loss in Eq. (2) is an upper bound to the conditional entropy $L \cdot m \ge {\mathcal{H}}(\mathbf{S}|\mathbf{Y})$ and minimizing it thus leads to the maximization of the $\text{GMI}={\mathcal{H}}(\mathbf{S})-{\mathcal{H}}(\mathbf{S}|\mathbf{Y})$, where ${\mathcal{H}}(\mathbf{S})=m$ is the entropy of the input.
In Fig.~\ref{fig1}~(top), an example constellation of order $M$=256 is shown, with a zoomed-in version~(bottom), illustrating that a Gray-like bit mapping is achieved (only a selected subset of labels shown for clarity). An implementation of the presented autoencoder is available online as Python/TensorFlow programs~\cite{b13}.
\begin{figure}[b]
\centering\includegraphics[width=0.93\linewidth]{DSP_cropped.pdf}
\caption{Setup of pilot based system including transceiver impairments.}
\label{fig2}
\end{figure}
\subsection{System simulation and DSP chain}
A QPSK-hybrid quasi-pilot-based DSP chain is considered which also takes advantage of the distribution of the modulation format. The system is adopted from~\cite{b09}, and is briefly summarized below and in Fig.~\ref{fig2} for completeness. The FEC-encoded modulation symbols are interleaved with 10\% QPSK symbols, which carry data at rate 2 bits/symbol. A Zaddoff-Chu pre-amble sequence is inserted before the 1st transmitted block for frame synchronization. Square-root raised cosine pulse shape is applied at the transmitter with a roll-off factor of 0.01. The entire waveform is sent on a standard single mode fiber ($\alpha = 0.2 \frac{dB}{km}, D = 17 \frac{ps}{nm \cdot km}, \gamma = 1.3 \frac{1}{W \cdot km}$), simulated with the split-step Fourier method. At the receiver, the pre-amble is used to detect the start of transmission. Then the QPSK symbols are detected with the Viterbi\&Viterbi method. The high spectral efficiency operating point easily results in error-free QPSK symbols, which after detection are used as pilots for two sample per symbol CMA equalization and one sample per symbol carrier phase estimation (CPE). After phase tracking, MI is used to estimate the achievable information rate, the symbols are demapped and the soft bits are sent for FEC decoding. The autoencoder decoder part $g(\vec{y})$ is only used for the optimization and is replaced by classical Gaussian auxiliary channel for the actual MI and GMI estimation and FEC decoding as in~\cite{b01, b09}. Iterative demapping and decoding is not performed. The FEC is the LTE standard turbo code \cite{b14}. The data rate is controlled easily by puncturing the FEC. For example, a data rate of 6~bits/QAM symbol is achieved with 256QAM by puncturing the rate R=$\frac{1}{3}$ mother code to a new code rate of R=$\frac{3}{4}$, corresponding to 33\% overhead. For simplicity, a data rate step of 0.5~bits/symbol is considered in this paper, although a smaller step is straight forward to achieve.
\section{Results}
The proposed designs are evaluated with a wavelength division multiplexed (WDM) system of 5 channels, 20~GBd each, with 22~GHz spacing. An Erbium doped fiber amplifier-based link is considered with amplifier spacing of 80~km and noise figure of 5~dB. At each distance, the launch power is swept and the performance at the optimal launch power is reported in each case for the central WDM channel. For simpler comparisons, the data rate achieved by the QAM symbols is reported only, and the constant (for all formats) addition of the 2~bit/symbol 10\% QPSK symbols is not shown.
\begin{figure*}[t
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=0.92\linewidth]{MI_vs_distance_imp-eps-converted-to.pdf}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.92\linewidth]{GMI_vs_distance-eps-converted-to.pdf}
\end{minipage}
\caption{\textbf{(left)} MI vs. distance for the studied modulation formats. Geometric shaping achieves similar gain to probabilistic shaping, both with and without impairments. \textbf{(right)} GMI and achieved error-free data rates as a function of the transmission distance with the impairments on. The shaping gain is 1 span, and comes for free.}
\label{fig3}
\end{figure*}
In order to evaluate the geometric shaping method in a realistic environment, the following impairments are introduced at the receiver and transmitter: 1) laser linewidth of 10~kHz, modelled with a Wiener process; 2) frequency offset between transmitter laser and local oscillator of 50~MHz; 3) ADC sampling frequency of 80~GSa/s; 4) ADC resolution of 6~bits, modelled with a uniform quantization step. A family of three geometric shapes optimized as in Section 2 are evaluated. The shapes are optimized for transmission at 2, 5 and 10 spans. The optimal of the three is reported at each distance. For reference, a standard, Gray-coded 256QAM and a probabilistic shaped Gray-coded 256QAM achieved with the Maxwell-Boltzmann probability mass function \cite{b02} are studied. The latter is optimized at each distance and each launch power. The MI at the optimal launch power of all formats is given in Fig.~\ref{fig3}~(left) as a function of the transmission distance both with (solid lines) and without (dashed lines) impairments.
Geometric shaping achieves between $\approx$~50\% and $\approx$~90\% of the probabilistic shaping gain, both with and without impairments, demonstrating that geometric shapes are not penalized more than rectangular constellations by the transceiver impairments, provided that a proper DSP is employed. Up to 0.2 bits/symbol of implementation penalty can be noticed for all formats, slightly more pronounced at the short distances/high rates. It should be noted that an efficient transmission system should switch to a larger constellation for those operating points, for which the quantization loss is no longer the dominant effect, but rather the linear and nonlinear transmission noise.
Finally, the effects of the mapping on the total achievable throughput is studied in Fig.~\ref{fig3}~(right) with the impairments on. The GMI at the optimal launch power is reported with solid lines, and \textit{the maximum distance, at which error-free transmission was achieved at the optimal launch power for a given input data rate} is given in dashed lines. As seen, the GMI gain is slightly lower (0.2~bits/symbol) than the MI gain (up to 0.3~bits/symbol) and is translated to one span of achieved error-free distance. Since there is no change to the BICM structure, the demapper and decoder, but only to the mapping function, this gain comes for free.
\section{Conclusion}
Autoencoder based GMI optimization of geometric shapes and mapping functions was proposed for high spectral efficiency WDM systems. Neither changes to the BICM coding structure nor iterative demapping and/or non-binary FEC are required in order to achieve shaping gain. The shaping gain of 0.2~bits/symbol or alternatively 1~span of transmission therefore comes for free, including in the presence of typical transceiver impairments, such as ADC quantization and limited sampling frequency, as well as laser phase noise.
\begin{footnotesize}
\textbf{Acknowledgements}
This work was financially supported by the European Research Council through the ERC-CoG FRECOM project (grant agreementno. 771878).
\end{footnotesize}
\clearpage
\section*{5 References}
|
1,108,101,566,796 | arxiv | \section{Introduction}
Depending on how we count, there are roughly 7000 languages spoken around the world today. The field of linguistic typology is concerned with the study and categorization of the world's languages based on their linguistic structural properties \cite{comrie1988linguistic, crofttypology}. While two languages may share structural properties across some typological dimensions, they may vary across others. For example, two languages could have identical speech sounds in their phonetic inventory, yet be perceived as dissimilar because each has its own unique set of phonological rules governing possible sound combinations. This leads to tremendous variation and diversity in speech patterns across the world languages \cite{tucker2020introduction}, the effects of which are understudied across many downstream applications due in part to lack of available resources. Building robust speech technologies which are applicable to any language is crucial to equal access as well as the preservation, documentation, and categorization of the world's languages, especially for endangered languages with a declining speaker community.
Unfortunately, robust (spoken) language technologies are only available for a small number of languages, mainly for speaker communities with strong economic power. The main hurdle for the development of speech technologies for under-represented languages is the lack of high-quality transcribed speech resources (see \citet{joshi2020state} for a detailed discussion on linguistic diversity in language technology research). The largest multilingual speech resource in terms of language coverage is the CMU Wilderness dataset \cite{black2019wilderness}, which consists of read speech segments from the Bible in $\sim$700 languages. Although this wide-coverage resource provides an opportunity to study many endangered and under-represented languages, it has a narrow domain and lacks speaker diversity as the vast majority of segments are recorded by low-pitch male speakers. It remains unclear whether such resources can be exploited to build generalizable speech technologies for under-resourced languages.
Spoken language identification (SLID) is an enabling technology for multilingual speech communication with a wide range of applications. Earlier SLID systems addressed the problem using the phonotactic approach whereby generative models are trained on sequences of phones transduced from the speech signal using an acoustic model \cite{lamel1994language, li2005phonotactic}. Most current state-of-the-art SLID systems are based on deep neural networks which are trained end-to-end from a spectral representation of the acoustic signal (e.g., MFCC feature vectors) without any intermediate symbolic representations \cite{lopez2014automatic, gonzalez2014automatic}. In addition to their ability to effectively learn to discriminate between closely related language varieties \cite{gelly2016language, shon2018convolutional}, it has been shown that neural networks can capture the degree of relatedness and similarity between languages in their emergent representations \cite{abdullah-etal-2020-rediscovering}.
Several SLID evaluation campaigns have been organized in the past, including the NIST Language Recognition Evaluation \cite{lee20162015, sadjadi20182017}, focusing on different aspects of this task including closely related languages, and typically used conversational telephone speech. However, the languages were not sampled according to typologically-aware criteria but rather were geographic or resource-driven choices. Therefore, while the NIST task languages may represent a diverse subset of the world's languages, there are many languages and language families which have not been observed in past tasks. In this shared task, we aim to address this limitation by broadening the language coverage to a set of typologically diverse languages across seven languages families. We also aim to assess the degree to which single-speaker speech resources from a narrow domain can be utilized to build robust speech language technologies.
\section{Task Description}
While language identification is a fundamental speech and language processing task, it remains a challenging task, especially when going beyond the small set of languages past evaluation has focused on. Further, for many low-resource and endangered languages, only single-speaker recordings may be available, demanding a need for domain and speaker-invariant language identification systems.
We selected 16 typologically diverse languages, some of which share phonological features, and others where these have been lost or gained due to language contact, to perform what we call robust language identification: systems were to be trained on largely single-speaker speech from one domain, but evaluated on data in other domains recorded from speakers under different recording circumstances, mimicking more realistic low-resource scenarios.
\begin{table*}[th]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{llllllll}
\toprule
\textbf{ISO} & \textbf{Wilderness ID} & \textbf{Language name} & \textbf{Family} & \textbf{Genus} & \textbf{Macroarea} & \textbf{Train} & \textbf{Eval} \\
\midrule
kab & KABCEB & Kabyle & Afro-Asiatic & Berber & Africa & Wilderness & CV \\
iba & IBATIV & Iban & Austronesian & Malayo-Sumbawan & Papunesia & Wilderness & SLR24 \\
ind & INZTSI & Indonesian & Austronesian & Malayo-Sumbawan & Papunesia & Wilderness & CV \\
sun & SUNIBS & Sundanese & Austronesian & Malayo-Sumbawan & Papunesia & Wilderness & SLR36 \\
jav & JAVNRF & Javanese & Austronesian & Javanese & Papunesia & Wilderness & SLR35 \\
eus & EUSEAB & Euskara & Basque & Basque & Eurasia & Wilderness & CV \\
tam & TCVWTC & Tamil & Dravidian & Southern Dravidian & Eurasia & Wilderness & SLR65 \\
kan & ERVWTC & Kannada & Dravidian & Southern Dravidian & Eurasia & Wilderness & SLR79 \\
tel & TCWWTC & Telugu & Dravidian & South-Central Dravidian & Eurasia & Wilderness & SLR66 \\
hin & HNDSKV & Hindi & Indo-European & Indic & Eurasia & Wilderness & SS \\
por & PORARA & Portuguese & Indo-European & Romance & Eurasia & Wilderness & CV \\
rus & RUSS76 & Russian & Indo-European & Slavic & Eurasia & Wilderness & CV \\
eng & EN1NIV & English & Indo-European & Germanic & Eurasia & Wilderness & CV \\
mar & MARWTC & Marathi & Indo-European & Indic & Eurasia & Wilderness & SLR64 \\
cnh & CNHBSM & Chin, Hakha & Niger-Congo & Gur & Africa & Wilderness & CV \\
tha & THATSV & Thai & Tai-Kadai & Kam-Tai & Eurasia & Wilderness & CV \\
\bottomrule
\end{tabular}
}%
\caption{Provided data with language family and macroarea information. \textbf{ISO} shows ISO 639-3 codes. Training data (\textbf{Train}) for all languages is taken from CMU Wilderness dataset; validation and evaluation data (\textbf{Eval}) is derived from multiple data sources. }
\label{tab:provided-data}
\end{table*}
\subsection{Provided Data}
To train models, we provided participants with speech data from the CMU Wilderness dataset \cite{black2019wilderness}, which contains utterance-aligned read speech from the Bible in 699 languages,\footnote{Data source: \url{bible.is}} but predominantly recorded from a single speaker per language, typically male.
Evaluation was conducted on data from other sources---in particular, multi-speaker datasets recorded in a variety of conditions, testing systems' capacity to generalize to new domains, new speakers, and new recording settings.
Languages were chosen from the CMU Wilderness dataset given availability of additional data in a different setting, and include several language families as well as more closely-related challenge pairs such as Javanese and Sundanese.
These included data from the Common Voice project \cite[CV;][]{ardila-etal-2020-common-voice} which is read speech typically recorded using built-in laptop microphones;
radio news data \cite[SLR24;][]{SLR24-1,SLR24-2};
crowd-sourced recordings using portable electronics \cite[SLR35, SLR36;][]{SLR35-36};
cleanly recorded microphone data \cite[SLR64, SLR65, SLR66, SLR79;][]{SLR64-65-66-79};
and a collection of recordings from varied sources \cite[SS;][]{SS}.
\autoref{tab:provided-data} shows the task languages and their data sources for evaluation splits for the robust language identification task.
We strove to provide balanced data to ensure signal comes from salient information about the language rather than spurious correlations about e.g. utterance length.
We selected and/or trimmed utterances from the CMU Wilderness dataset to between 3 to 7 seconds in length.
Training data for all languages comprised 4,000 samples each.
We selected evaluation sources for validation and blind test sets to ensure no possible overlap with CMU Wilderness speakers.
We held out speakers between validation and test splits, and balanced speaker gender within splits to the degree possible where annotations were available.
We note that the Marathi dataset is female-only.
Validation and blind test sets each comprised 500 samples per language.
We release the data as derivative MFCC features.
\section{Evaluation}
The robust language identification shared task allowed two kinds of submissions: first, \textit{constrained} submissions, for which only the provided training data was used; and second, \textit{unconstrained} submissions, in which the training data may be extended with any external source of information (e.g. pre-trained models, additional data, etc.).
\subsection{Evaluation Metrics}
We evaluate task performance using precision, recall, and F$_1$.
For each metric we report both micro-averages, meaning that the metric average is computed equally-weighted across all samples for all languages, and macro-averages, meaning that we first computed the metric for each language and then averaged these aggregates to see whether submissions behave differently on different languages.
Participant submissions were ranked according to macro-averaged F$_1$.
\subsection{Baseline}
For our baseline SLID system, we use a deep convolutional neural network (CNN) as sequence classification model. The model can be viewed as two components trained end-to-end: a segment-level feature extractor ($f$) and a language classifier ($g$). Given as input a speech segment parametrized as sequence of MFCC frames $\mathbf{x}_{1:T} = (\mathbf{x}_1, \dots, \mathbf{x}_T) \in \mathbb{R}^{k \times T}$, where $T$ is the number of frames and $k$ is the number of the spectral coefficients, the segment-level feature extractor first transforms $\mathbf{x}_{1:T}$ into a segment-level representation as $\mathbf{u} = f(\mathbf{x}_{1:T}; \boldsymbol{\theta}_f) \in \mathbb{R}^d$. Then, the language classifier transforms $\mathbf{u}$ into a logit vector $\mathbf{\hat{y} \in \mathbb{R}^{|\mathcal{Y}|}}$, where $\mathcal{Y}$ is the set of languages, through a series of non-linear transformations as $\mathbf{\hat{y}} = g(\mathbf{u}; \boldsymbol{\theta}_g)$. The logit vector $\mathbf{\hat{y}}$ is then fed to a softmax function to get a probability distribution over the languages.
The segment-level feature extractor consists of three 1-dimensional, temporal convolution layers with 64, 128, 256 filters of widths 16, 32, 48 for each layer and a fixed stride of 1 step. Following each convolutional operation, we apply batch normalization, ReLU non-linearity, and unit dropout with probability which was tuned over $\{0.0, 0.4, 0.6\}$. We apply average pooling to downsample the representation only at the end of the convolution block, which yields a segment representation $\mathbf{u} \in \mathbb{R}^{256}$. The language classifier consists of 3 fully-connected layers (256 $\rightarrow$ 256 $\rightarrow$ 256 $\rightarrow$ 16), with a unit dropout with probability $0.4$ between the layers, before the softmax layer. The model is trained with the ADAM optimizer with a batch size of 256 for 50 epochs. We report the results of the best epoch on the validation set as our baseline results.
\subsection{Submissions}
We received three constrained submissions from three teams, as described below.
\textbf{Anlirika} \cite[composite]{anlirika2021sigtyp}
The submitted system (constrained) consists of several recurrent, convolutional, and dense layers.
The neural architecture starts with a dense layer that is designed to remove sound harmonics from a raw spectral pattern. This is followed by a 1D convolutional layer that extracts audio frequency patterns (features). Then the features are fed into a stack of LSTMs that focuses on `local' temporal constructs. The output of the stack of LSTMs is then additionally concatenated with the CNN features and is fed into one more LSTM module. Using the resulting representation, the final (dense) layer evaluates a categorical loss across 16 classes. The network was trained with Adam optimizer, the learning rate was set to be $5\times10^{-4}$. In addition, similar to Lipsia, the team implemented a data augmentation strategy: samples from validation set have been added to the training data.
\textbf{Lipsia} \cite[Universität Leipzig]{lipsia2021sigtyp} submitted a constrained system based on the ResNet-50 \citep{he2016deep}, a deep (50 layers) CNN-based neural architecture. The choice of the model is supported by a comparative analysis with more shallow architectures such as ResNet-34 and a 3-layer CNNs that all were shown to overfit to the training data. In addition, the authors proposed transforming MFCC features into corresponding 640x480 spectrograms since this data format is more suitable for CNNs. The output layer of the network is dense and evaluates the probabilities of 16 language classes.%
\footnote{The submitted system actually predicts one out of 18 classes as two other languages that weren't part of the eventual test set were included. The system predicted these two languages for 27 of 8000 test examples, i.e., $\approx~0.34\%$.}
Finally, the authors augmented the training data with 60\% of the samples from the validation set because the training set did not present enough variety in terms of domains and speakers while the validation data included significantly more. Use of the validation data in this way seems to have greatly improved generalization ability of the model.
The model performed relatively well with no fine-tuning or transfer-learning applied after augmentation.\footnote{The authors trained ResNet-50 from scratch.}
\textbf{NTR} \cite[NTR Labs composite]{ntr2021sigtyp}, submitted an essentially constrained\footnote{Although technically external noise data was used when augmenting the dataset, no language-specific resources were.} system which uses a CNN with a self-attentive pooling layer.
The architecture of the network was QuartzNet ASR following \citet{kriman2020quartznet}, with the decoder mechanism replaced with a linear classification mechanism.
The authors also used a similar approach in another challenge on low-resource ASR, Dialog-2021 ASR\footnote{\url{http://www.dialog-21.ru/en/evaluation/}}.
They applied several augmentation techniques, namely shifting samples in range (-5ms; +5ms),
MFCC perturbations \cite[SpecAugment;][]{park2019specaugment},
and adding background noise.
\begin{table*}[tbh]
\newcommand{\cellcolor{gray!10}\color{black!50}}{\cellcolor{gray!10}\color{black!50}}
\centering
\begin{adjustbox}{width=\linewidth}
\begin{tabular}{ccccccccccccc}
\toprule
\textbf{ISO} & \multicolumn{2}{c}{\textbf{Anlirika}} && \multicolumn{2}{c}{\textbf{Baseline}} && \multicolumn{2}{c}{\textbf{Lipsia}} && \multicolumn{2}{c}{\textbf{NTR}} \\
\cmidrule{2-3}\cmidrule{5-6}\cmidrule{8-9}\cmidrule{11-12}
& Valid. & Test && Valid. & Test && Valid. & Test && Valid. & Test \\
\midrule
\cellcolor{gray!10}\color{black!50}\textit{Family: Afro-Asiatic} & \cellcolor{gray!10}\color{black!50}\textit{.329} & \cellcolor{gray!10}\color{black!50}.214 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.181} & \cellcolor{gray!10}\color{black!50}.235 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.670} & \cellcolor{gray!10}\color{black!50}\textbf{.453} &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.102} & \cellcolor{gray!10}\color{black!50}.082 \\
kab & \textit{.329} & .214 && \textit{.181} & .235 && \textit{.670} & \textbf{.453} && \textit{.102} & .082 \\
\midrule
\cellcolor{gray!10}\color{black!50}\textit{Family: Austronesian} & \cellcolor{gray!10}\color{black!50}\textit{.429} & \cellcolor{gray!10}\color{black!50}.368 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.082} & \cellcolor{gray!10}\color{black!50}.094 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.578} & \cellcolor{gray!10}\color{black!50}\textbf{.498} &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.065} & \cellcolor{gray!10}\color{black!50}.060 \\
iba & \textit{.692} & .696 && \textit{.029} & .018 && \textit{.980} & \textbf{.968} && \textit{.020} & .031 \\
ind & \textit{.350} & .108 && \textit{.033} & .105 && \textit{.700} & \textbf{.338} && \textit{.096} & .074 \\
sun & \textit{.406} & \textbf{.369} && \textit{.160} & .149 && \textit{.090} & .140 && \textit{.086} & .082 \\
jav & \textit{.267} & .300 && \textit{.106} & .106 && \textit{.540} & \textbf{.547} && \textit{.059} & .053 \\
\midrule
\cellcolor{gray!10}\color{black!50}\textit{Family: Basque} & \cellcolor{gray!10}\color{black!50}\textit{.565} & \cellcolor{gray!10}\color{black!50}.405 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.100} & \cellcolor{gray!10}\color{black!50}.090 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.850} & \cellcolor{gray!10}\color{black!50}\textbf{.792} &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.077} & \cellcolor{gray!10}\color{black!50}.016 \\
eus & \textit{.565} & .405 && \textit{.100} & .090 && \textit{.850} & \textbf{.792} && \textit{.077} & .016 \\
\midrule
\cellcolor{gray!10}\color{black!50}\textit{Family: Dravidian} & \cellcolor{gray!10}\color{black!50}\textit{.351} & \cellcolor{gray!10}\color{black!50}.246 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.202} & \cellcolor{gray!10}\color{black!50}.138 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.807} & \cellcolor{gray!10}\color{black!50}\textbf{.572} &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.074} & \cellcolor{gray!10}\color{black!50}.053 \\
tam & \textit{.342} & .272 && \textit{.348} & .204 && \textit{.800} & \textbf{.609} && \textit{.172} & .046 \\
kan & \textit{.188} & .168 && \textit{.000} & .042 && \textit{.820} & \textbf{.557} && \textit{.004} & .015 \\
tel & \textit{.523} & .298 && \textit{.259} & .168 && \textit{.800} & \textbf{.550} && \textit{.046} & .097 \\
\midrule
\cellcolor{gray!10}\color{black!50}\textit{Family: Indo-European} & \cellcolor{gray!10}\color{black!50}\textit{.439} & \cellcolor{gray!10}\color{black!50}.225 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.130} & \cellcolor{gray!10}\color{black!50}.144 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.722} & \cellcolor{gray!10}\color{black!50}\textbf{.402} &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.114} & \cellcolor{gray!10}\color{black!50}.047 \\
hin & \textit{.458} & .378 && \textit{.091} & .099 && \textit{.780} & \textbf{.635} && \textit{.021} & .011 \\
por & \textit{.211} & .143 && \textit{.157} & .166 && \textit{.550} & \textbf{.358} && \textit{.102} & .068 \\
rus & \textit{.630} & \textbf{.034} && \textit{.014} & .014 && \textit{.900} & \textbf{.065} && \textit{.050} & \textbf{.049} \\
eng & \textit{.194} & .148 && \textit{.161} & .179 && \textit{.460} & \textbf{.414} && \textit{.270} & .099 \\
mar & \textit{.701} & .423 && \textit{.229} & .263 && \textit{.920} & \textbf{.539} && \textit{.126} & .010 \\
\midrule
\cellcolor{gray!10}\color{black!50}\textit{Family: Niger-Congo} & \cellcolor{gray!10}\color{black!50}\textit{.516} & \cellcolor{gray!10}\color{black!50}.403 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.138} & \cellcolor{gray!10}\color{black!50}.063 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.860} & \cellcolor{gray!10}\color{black!50}\textbf{.763} &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.122} & \cellcolor{gray!10}\color{black!50}.038 \\
cnh & \textit{.516} & .403 && \textit{.138} & .063 && \textit{.860} & \textbf{.763} && \textit{.122} & .038 \\
\midrule
\cellcolor{gray!10}\color{black!50}\textit{Family: Tai-Kadai} & \cellcolor{gray!10}\color{black!50}\textit{.362} & \cellcolor{gray!10}\color{black!50}.156 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.086} & \cellcolor{gray!10}\color{black!50}.052 &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.780} & \cellcolor{gray!10}\color{black!50}\textbf{.401} &\cellcolor{gray!10}\color{black!50}& \cellcolor{gray!10}\color{black!50}\textit{.025} & \cellcolor{gray!10}\color{black!50}.015 \\
tha & \textit{.362} & .156 && \textit{.086} & .052 && \textit{.780} & \textbf{.401} && \textit{.025} & .015 \\
\midrule\midrule
F1, Macro Avg. & \textit{.421} & .282 && \textit{.131} & .122 && \textit{.719} & \textbf{.508} && \textit{.086} & .049 \\
F1, Micro Avg. & \textit{.436} & .298 && \textit{.145} & .137 && \textit{ } & \textbf{.532} && \textit{ } & .063 \\
\midrule
Accuracy & & 29.9\% && & 13.7\% && & \textbf{53.1\%} && & 6.3\% \\
\bottomrule
\end{tabular}
\end{adjustbox}
\caption{F$_1$ scores, their macro-averages per family, and overall accuracies of submitted predictions on validation and test data (validation results are self-reported by participants). The Lipsia system performed best across nearly all languages and consistently achieves the highest averages.}
\label{tab:results-subm}
\end{table*}
\begin{figure*}[tbh]
\centering
\hspace*{-1em}
\includegraphics[width=.95\linewidth]{figs/bigscatter.pdf}
\caption{Correlating submitted systems' F$_1$ scores for our 16 languages on the test set. The lines are linear regressions as described in \autoref{sec:results}.}
\label{fig:scatter-systems-against-baseline}
\end{figure*}
\section{Results and Analysis}\label{sec:results}
The main results in \autoref{tab:results-subm} show all systems greatly varying in performance, with the Lipsia system clearly coming out on top, boasting best accuracy and average F$_1$ score, and reaching the best F$_1$ score for nearly each language individually.%
\footnote{
Each of the ``wins'' indicated by boldface in \autoref{tab:results-subm} is statistically significant under a paired-permutation significance test (note that as we are not in a multiple-hypothesis testing setting, we do not apply Bonferroni or similar corrections). There are no significant differences between the baseline and the Anlirika system for kab, ind, por, rus, and eng; between the baseline and the Lipsia system for sun; between the baseline and the NTR system for ind, iba, and cnh; between Anlirika and Lipsia on rus; between Lipsia and NTR on rus; between Anlirika and NTR on ind and rus.
}
All four systems' performance varies greatly on average, but nevertheless some interesting overall trends emerge.
\autoref{fig:scatter-systems-against-baseline} shows that while the Anlirika and Lipsia systems' performance on the different languages do not correlate with the baseline system (linear fit with Pearson's $R^2=0.00$ and $p>0.8$ and $R^2=0.02$ and $p>0.5$, respectively), the NTR system's struggle correlates at least somewhat with the same languages that the baseline system struggles with: a linear fit has $R^2=0.15$ with $p>0.1$.
More interestingly, in correlations amongst themselves, the Anlirika and Lipsia systems do clearly correlate ($R^2=0.57$ and $p<0.001$), and the NTR system correlates again at least somewhat with the Anlirika system ($R^2=0.11$ and $p>0.2$) and the Lipsia system ($R^2=0.19$ and $p>0.05$).
Note that most systems submitted are powerful enough to fit the training data: our baseline achieves a macro-averaged F$_1$ score of $.98$ ($\pm .01$) on the training data, the Lipsia system similarly achieves $.97$ ($\pm .03$), the NTR system reaches a score of $.99$ ($\pm .02$). An outlier, the Anlirika system reaches only $.75$ ($\pm .09$).
On held-out data from CMU Wilderness which matches the training data domain, the baseline achieves $.96$ F1.
This suggests an inability to generalize across domains and/or speakers without additional data for adaptation.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{figs/confusions.pdf}
\caption{Visualization of Precision (P), Recall (R), and confusion matrices (scores are counts normalized by number of gold entries) for the Anlirika, baseline, Lipsia, and NTR system, grouped by language families.}
\label{fig:confusion}
\end{figure*}
Diving deeper into performance on different languages and families, \autoref{fig:confusion} shows confusion matrices for precision and recall, grouped by language family. We can see the superiority of the Lipsia system and to a lesser degree the Anlirika system over the generally more noisy and unreliable baseline system and the NTR system which was likely overtrained: it classifies 23\% of examples as tel, 20\% as kab, and 16\% as eng, with the remaining 41\% spread across the remaining 13 languages (so $\approx$ 3.2\% per language).
Interestingly, the other three systems all struggle to tell apart sun and jav, the Anlirika and baseline systems classifying both mostly as sun and the Lipsia system classifying both mostly as jav. Note that the baseline system tends to label many languages' examples as sun (most notably mar, the test data for which contains only female speakers), eus (most notably also rus), and eng (most notably also iba), despite balanced training data. In a similar pattern, the Anlirika predicts tam for many languages, in particular ind, the other two Dravidian languages kan and tel, por, rus, eng, cnh, and tha.
Looking more closely at the clearly best-performing system, the Lipsia system, and its performance and confusions, we furthermore find that the biggest divergence from the diagonal after the sun/jav confusion is a tendency to label rus as por, and the second biggest divergence is that mar examples are also sometimes labeled as kan and tel; while the first one is within the same family, in the second case, these are neighbouring languages in contact and mar shares some typological properties with kan (and kan and tel belong to the same language family).
\section{Conclusion}
This paper describes the SIGTYP shared task on robust spoken language identification (SLID).
This task investigated the ability of current SLID models to generalize across speakers and domains.
The best system achieved a macro-averaged accuracy of 53\% by training on validation data, indicating that even then the task is far from solved.
Further exploration of few-shot domain and speaker adaptation is necessary for SLID systems to be applied outside typical well-matched data scenarios.
|
1,108,101,566,797 | arxiv | \section{Introduction}
Entanglement describes nonlocal quantum correlations, and is one of the
characteristic peculiar features of quantum mechanics. Motivated by recent
studies showing intimate connections between entanglement and quantum phase
transitions \cite{fazio1,nielsen}, the understanding of the degree of
entanglement in quantum many-body systems has prompted an enormous effort at
the interface between condensed matter physics, quantum information theory and
quantum field theory \cite{fazio}.
A fundamental question in this research field is concerned with the scaling of
the entropy quantifying the degree of entanglement between a spatially confined
region and its complement in a quantum many-body system. Suppose a system,
combined by two subsystems $\mathcal{A}$ and $\mathcal{B}$, is in a pure quantum
state $|\psi\rangle$, with density matrix $\rho=| \psi\rangle\langle\psi |$.
The entanglement entropy is just the von Neumann entropy of either subsystem
given by
\begin{equation}
S_{\cal A}=-\Tr (\rho_{\cal A} \log_2 \rho_{\cal A}) =
-\Tr (\rho_{\cal B} \log_2 \rho_{\cal B}) =
S_{\cal B},
\end{equation}
where the reduced density matrix for $\cal A$ is constructed by tracing over
the degrees of freedom in $\cal B$, given by $\rho_{\cal A}=\Tr_{\cal B}\rho$.
Analogously, $\rho_{\cal B}=\Tr_{\cal A}\rho$. In order to explore the behavior
of quantum entanglement at different length scales, one is particularly
interested in how the entanglement entropy depends on the linear size $\ell$ of
the subsystem considered. An early conjectured scaling law relates the
entanglement entropy to the surface area $\ell^{d-1}$, not the volume, of the
region in a $d$-dimensional system \cite{AREA}. This area law of entropy
scaling has been established for gapped quantum many-body systems where the
correlation length is finite. In one-dimensional (1D) systems, the
entanglement behavior changes drastically at a quantum phase transition where
the absence of gaps leads to long range correlations and results in a
logarithmically diverging entanglement entropy as the system size $L$ goes to
infinity, i.e. $S_{\mathcal{A}}\sim \log_2 \ell$ for $L\to \infty$
\cite{vidal,conformal,calabrese_cardy,refael}. This connection between
entanglement entropy and quantum phase transitions is however lost in quantum
systems in higher dimensions \cite{FREE-F,BOSON-NUM,BOSON-ANA,lir07,yu}.
The scaling behavior of quantum entanglement for $1+1$-dimensional conformally
invariant systems has been derived by several authors
\cite{conformal,calabrese_cardy}. Here we summarize some known results. For a
critical chain of length $L$ with {\it periodic} boundary conditions, the
entanglement entropy of a subsystem of length $\ell$ embedded in the chain
scales as
\begin{equation}
S^{(p)}_L(\ell)=\frac{c}{3} \log_2\left[\frac{L}{\pi}
\sin\left(\frac{\ell \pi}{L}\right)\right] + c_1,
\label{S_L_p}
\end{equation}
where the prefactor $c$ is universal and given by the central charge of the
associated conformal field theory, whereas the constant $c_1$ is non-universal.
For the leftmost segment of length $\ell$ in a finite {\it open} chain of
length $L$ at criticality, the entanglement entropy reads
\begin{equation}
S^{(o)}_L(\ell)= \frac{c}{6} \log_2\left[\frac{2L}{\pi} \sin\left(\frac{\ell \pi}{L}\right)\right] +
\log_2 g +\frac{c_1}{2},
\label{S_L_o}
\end{equation}
where $\log_2 g$ is the boundary entropy\cite{boundary_entropy} and the
constant $c_1$ is the same to the one in Eq.~(\ref{S_L_p}). For an infinite
system $L\to\infty$, the critical entanglement entropy becomes
\begin{equation}
S_\infty(\ell)=\frac{c}{3}\log_2 \ell + c_1.
\label{S_l}
\end{equation}
Away from the critical point, where the correlation length $\xi\ll \ell$, we have
\begin{equation}
S_\infty \simeq b\frac{c}{6} \log_2 \xi.
\label{S_xi}
\end{equation}
where $b$ is the number of boundary points between the subsystem and the rest
of the chain. Some of the results given above have been verified by analytic
and numerical calculations on integrable 1D quantum spin chains, in particular
on the antiferromagnetic $XX$-chain and on the quantum Ising chain
\cite{vidal,peschel05,jin_korepin,boundary}. Notice that an exact relationship
between the entanglement entropy of these two models has been recently
established \cite{IJ07}.
Remarkably, the logarithmic scaling law of entanglement entropy in the
thermodynamic limit is valid even for critical quantum chains that are not
conformally invariant. In those cases the central charge determining the
prefactor of the logarithmic scaling law is replaced by an effective one. For
disordered quantum Ising and $XX$ chains at infinite-randomness fixed points,
the effective central charge was determined as $c_{\rm eff}=c\ln 2$ for the
disorder-average entropy \cite{refael} by using strong disorder
renormalization group method \cite{mdh,fisher,review}. Also the average entropy
of other types of random quantum spin chains with infinite-randomness fixed
points has been studied by similar methods\cite{Santachiara,Bonesteel,s=1}. In
aperiodic quantum Ising chains, where the couplings follow some quasi-periodic
or aperiodic sequence, the coefficient in Eq.~(\ref{S_l}) is shown to depend
on the ratio of the couplings\cite{ijz07}, provided the perturbation caused by
the aperiodicity is marginal or relevant.
In this paper we consider the quantum Ising chain with three different types of
couplings: homogeneous, periodically modulated and random. We calculate the
entanglement entropy for large finite systems up to $L=4096$ by free fermionic
techniques. For the homogeneous chain, conformal predictions about the entropy
at the critical point for finite chains with different boundary conditions are
checked, and subleading corrections are investigated. We also study the
finite-size scaling behavior of $S_L(L/2)$ around its maximum and use the
position of the maximum to identify the finite-size critical transverse field.
The model with periodically modulated couplings belongs to the same critical
universality class as the homogeneous model. In this case we study the entropy
for finite chains and check whether the logarithmic scaling law is valid.
Finally, for random chains we calculate the average entropy, check the validity
of the strong disorder renormalization group prediction, and compare the
average entropy with the corresponding conformal result in Eq.(\ref{S_L_p}).
The structure of the paper is the following. In Sec. \ref{sec:fermionic} we
present the model, its free-fermion solution and the way of calculating the
entanglement entropy. Results of the numerical calculations at the critical
point are shown in Sec. \ref{sec:crit} for homogeneous, periodically modulated
and random chains. For homogeneous chains finite-size scaling of the maximum of
the entropy close to the critical point is analyzed in Sec. \ref{sec:fss}. Our
results are discussed in Sec. \ref{sec:disc}. In \ref{sec:hom}, the correlation
matrix, which is relevant to the calculation of entanglement entropy, is
determined for the homogeneous chain at its critical point. In \ref{sec:shift}
the shift exponent of homogeneous closed chains is calculated.
\section{The quantum Ising chain and its entropy in the fermionic representation}
\label{sec:fermionic}
\subsection{The model and its free-fermion representation}
The model we consider is an Ising chain with nearest neighbor couplings $J_i$ in
a transverse field of strength $h_i$, defined by the Hamiltonian:
\begin{equation}
{\cal H} =
-\frac{1}{2}\sum_{i=1}^L J_{i}\sigma_i^x \sigma_{i+1}^x-
\frac{1}{2}\sum_{i=1}^L h_i \sigma_i^z
\label{eq:H}
\end{equation}
in terms of the Pauli-matrices $\sigma_i^{x,z}$ at site $i$. Here we consider
three types of couplings: (i) homogeneous case with $J_i=1$ and $h_i=h\,(>0)$;
(ii) staggered case with $J_{2i-1}=\lambda\,(>0)$, $J_{2i}=1/\lambda$, and
$h_i=h$; (iii) random case with $\{J_i\}$ and $\{h_i\}$ being independent and
identically distributed random variables.
The essential technique in the solution of ${\cal H}$ is the mapping to
spinless free fermions \cite{lsm,pfeuty}. First we express the spin operators
$\sigma_i^{x,y,z}$ in terms of fermion creation (annihilation) operators
$c_i^\dagger$ ($c_i$) by using the Jordan-Wigner
transformation: $c^\dagger_i=a_i^+\exp\left[\pi i \sum_{j}^{i-1}a_j^+a_j^-\right]$
and $c_i=\exp\left[\pi i
\sum_{j}^{i-1}a_j^+a_j^-\right]a_i^-$, where $a_j^{\pm}=(\sigma_j^x \pm
i\sigma_j^y)/2$. Doing this, ${\cal H}$ can be rewritten in a quadratic form
in fermion operators:
\begin{eqnarray}
{\cal H}&=&
-\sum_{i=1}^{L}h_i\left( c^\dagger_i c_i-\frac{1}{2} \right) -
\frac{1}{2}\sum_{i=1}^{L-1} J_i(c^\dagger_i-c_i)(c^\dagger_{i+1}+c_{i+1})\cr
&+&\frac{1}{2}w J_L(c^\dagger_L-c_L)(c^\dagger_{1}+c_{1}).
\label{ferm_I}
\end{eqnarray}
Here the parameter $w=\exp(i\pi {\cal N}_c)$ depends on the number of fermions
${\cal N}_c=\sum_{i=1}^L c_i^\dagger c_i =1/2 \sum_{i=1}^L(1+\sigma_i^z)$, therefore one
should consider two separated sectors depending on the parity of ${\cal N}_c$.
The ground state corresponds to the fermionic vacuum, thus $w=1$.
In the second step, the Hamiltonian is diagonalized by a Bogoliubov transformation:
\begin{equation}
\eta_k=\sum_{i=1}^L\left[ \frac{1}{2}\left(\Phi_k(i)+\Psi_k(i)\right)c_i+
\frac{1}{2}\left(\Phi_k(i)-\Psi_k(i)\right)c_i^\dagger\right]
\end{equation}
where the $\Phi_k(i)$ and $\Psi_k(i)$ are real and normalized: $\sum_i^L
\Phi_k^2(i)=\sum_i^L \Psi^2_k(i)=1$, so that we have
\begin{equation}
\mathcal{H}=\sum_{k=1}^L \Lambda_k(\eta_k^\dagger \eta_k-1/2).
\label{free_fermion}
\end{equation}
The fermionic excitation energies, $\Lambda_k$, and the components of the
vectors, ${\mathbf \Phi}_k$ and ${\mathbf \Psi}_k$, are obtained from the
solution of the following eigenvalue problem\cite{it}: ${\mathbf T}{\mathbf
V}_k=\Lambda_k {\mathbf V}_k$. Here ${\mathbf T}$ is a symmetric $2L \times 2L$
matrix:
\begin{equation}
\mathbf{T}=\left(
\begin{array}{cccccc}
0 & h_1 & & & & -w J_L\\
h_1 & 0 & J_1 & & & \\
& J_1 & 0 & h_2 & & \\
& & \ddots & \ddots &\ddots & \\
& & & J_{L-1} & 0 & h_L \\
-w J_L & & & & h_L & 0
\end{array}
\right)
\label{T}
\end{equation}
and the eigenvectors have the components: ${\mathbf
V}_k=(-\Phi_k(1),\Psi_k(1),-\Phi_k(2),\Psi_k(2),\dots,\\-\Phi_k(L),\Psi_k(L))$.
Transforming $\Phi_k(i)$ into $- \Phi_k(i)$, $\Lambda_k$ is changed to
$-\Lambda_k$. Thus we only restrict ourselves to the sector corresponding to
$\Lambda_k \ge 0$, $k=1,2,\dots,L$. To obtain the quantum critical point of the
system we make use of the condition that the energy of the first fermionic
excitation vanishes in the thermodynamic limit. From Eq.(\ref{T}) with $w=-1$
we obtain \cite{pfeuty,ilrm07}:
\begin{equation}
\lim_{L \to \infty} \frac{1}{L}\sum_{i=1}^L \ln J_i=\lim_{L \to \infty} \frac{1}{L} \sum_{i=1}^L \ln h_i
\label{crit_point}
\end{equation}
Consequently, the critical point of the homogeneous chain as well as the
staggered chain is located at $h_c=1$. For the random chain the criticality
condition is given by $\overline{\ln J}=\overline{\ln h}$, where the overbar
denotes an average over quenched disorder.
\subsection{Calculation of the entanglement entropy}
Now we turn to the procedure for calculating the entanglement entropy of
the system in its ground state $|0\rangle$. We consider a subsystem of length
$\ell$, consisting of spins $i=1,2,\dots,\ell$. The reduced density matrix
${\mathbf \rho}_{\ell}=\Tr_{L-\ell} |0\rangle \langle 0 |$ can be
calculated from the restricted correlation matrix ${\mathbf G}$
\cite{peschel,vidal}, the elements of which are given by
\begin{eqnarray}
G_{m,n}=\langle 0 |(c^\dagger_n-c_n)(c^\dagger_m+c_m)|0 \rangle\cr
=-\sum_{k=1}^L \Psi_k(m) \Phi_k(n),\quad m,n=1,2,\dots,\ell
\label{G}
\end{eqnarray}
For the homogeneous critical chain, the eigenvalue problem of $\mathbf{T}$
in Eq.~(\ref{T}) and thus the matrix elements of ${\mathbf G}$ can be solved
analytically. The results, both for periodic and for open finite chains
with an even $L$, as well as for $L\to\infty$, are given in \ref{sec:hom}.
The von Neumann entropy of the considered subsystem,
$S_L(\ell)=-\Tr(\rho_\ell\log_2 \rho_\ell)$, is fully determined by the
spectrum of the reduced density matrix ${\mathbf \rho}_{\ell}$. To diagonalize
${\mathbf \rho}_{\ell}$, we transform the $\ell$ fermionic modes into
non-correlated fermions with operators:
\begin{equation}
\mu_q=\sum_{i=1}^\ell\left[ \frac{1}{2}\left(v_q(i)+u_q(i)\right)c_i+
\frac{1}{2}\left(v_q(i)-u_q(i)\right)c_i^\dagger\right]\;,
\label{mu_q}
\end{equation}
where the $v_q(i)$ and $u_q(i)$ are real and normalized: $\sum_i^{\ell} v_q^2(i)=\sum_i^{\ell} u_q^2(i)=1$.
In the transformed basis, we have
\begin{equation}
\langle 0 |\mu_q \mu_p|0 \rangle=0,\quad \langle 0 |\mu_q^\dagger \mu_p|0 \rangle=\delta_{qp}\frac{1+\nu_q}{2}\;,
\label{uncorr}
\end{equation}
for $p,q=1,2,\dots \ell$, which means that the fermionic modes are uncorrelated.
Thus the reduced density matrix is the direct
product $\rho_\ell=\bigotimes_{q=1}^{\ell} \rho_q$, where $\rho_q$ has eigenvalues
$(1 \pm \nu_q)/2$. The entanglement
entropy is then given by the sum of binary entropies:
\begin{equation}
S_L(\ell)=-\sum_{q=1}^{\ell} \left(\frac{1+\nu_q}{2} \log_2 \frac{1+\nu_q}{2}
+\frac{1-\nu_q}{2} \log_2 \frac{1-\nu_q}{2}\right).
\label{binary}
\end{equation}
The $\nu_q$-s in Eq.~(\ref{binary}) are the solutions of the equations
\begin{equation}
{\mathbf G}{\mathbf u}_q=\nu_q {\mathbf v}_q,\quad
{\mathbf G}^T{\mathbf v}_q=\nu_q {\mathbf u}_q\;,
\end{equation}
or, equivalently, are related to the eigenvalue problem:
\begin{equation}
{\mathbf G}{\mathbf G}^T{\mathbf v}_q=\nu_q^2 {\mathbf v}_q,\quad
{\mathbf G}^T{\mathbf G}{\mathbf u}_q=\nu_q^2 {\mathbf u}_q\;.
\label{G_GT}
\end{equation}
In numerical calculations, many eigenvalues $\nu_q^2$ are found to be very
close to zero and these small eigenvalues are often out of the computer
precision, resulting in instability in the calculations. To circumvent the
problem, we can introduce a symmetric $2\ell \times 2\ell$ matrix ${\mathbf U}$
with elements:
\begin{equation}
U_{i,j}=
\left[\begin{array}{cc}
0 & G_{i,j} \cr
G_{j,i} & 0
\end{array}\right]\;,
\end{equation}
whose eigenvalue problem corresponds to ${\mathbf U}{\mathbf W}_q=\nu_q {\mathbf W}_q$.
Here the eigenvector ${\mathbf W}_q$ is given by
${\mathbf W}_q=\left(-v_q(1),u_q(1),-v_q(2),u_q(2),\dots,-v_q(\ell),u_q(\ell) \right)$.
Only non-negative eigenvalues $\nu_q \ge 0$ are taken into account.
\section{Scaling at the critical point}
\label{sec:crit}
Here we calculate the entanglement entropy of the quantum Ising chain for different types of
interactions (homogeneous, staggered and random) at the quantum critical point, defined
in Eq.(\ref{crit_point}). The numerical results obtained for finite periodic and open
chains are compared with the conformal results in Eqs.(\ref{S_L_p}) and (\ref{S_L_o}),
respectively.
\subsection{Homogeneous chain}
\label{sec:crit_hom}
For a critical Ising chain of finite length $L$ with periodic boundary
conditions, the expression for the entanglement entropy of a subsystem of size
$\ell$ is given by Eq.~(\ref{S_L_p}) with the central charge $c=1/2$, while
with open boundary conditions it corresponds to Eq.~(\ref{S_L_o}) with the
theoretical value $g=1$ \cite{cardy_g} for the boundary entropy $\log_2 g$. To
calculate the non-universal constant $c_1$ we use the exact relationship:
$S_{2L}^{XX}(2\ell)=2S_L(\ell)$ \cite{IJ07}, between the entropy of the
$XX$-chain, $S_L^{XX}(\ell)$, and the entropy of the quantum Ising chain,
yielding $c_1=\frac{1}{2}c_1^{XX}+\frac{1}{3} c$, where $c_1^{XX}$ is the
constant for the $XX$-chain and is given in Ref.~\cite{jin_korepin} in terms of
a definite integral. In this way we obtain $c_1=0.6904132738\cdots$, which
agrees with the value evaluated in a recent paper using a different method
\cite{cardy}. In Table \ref{table:check}, the constant $c_1$ is calculated by
\begin{equation}
c_1(L)=S_L^{(p)}(\ell)-\frac{c}{3} \log_2\left[\frac{L}{\pi} \sin\left(\frac{\ell \pi}{L}\right)\right],
\label{c_1}
\end{equation}
with $\ell=L/2$. It, indeed, converges to the asymptotic value of $c_1$ as the system
size $L$ is increasing.
\begin{table}
\caption{Finite-size dependence of the ratio $r(L)$ defined in Eq.~(\ref{ratio}) for
chains with periodic boundary conditions, the constant $c_1(L)$ in Eq.(\ref{c_1}) with
$\ell/L=1/2$, and $g$ for the boundary entropy $\log_2 g$ in Eq.(\ref{S_L_o}).
The finite-size correction coefficients are:
$r_2=0.3334(2)$ and $g_1=0.3095(2)$\label{table:check}
}
\begin{indented}
\item[]\begin{tabular}{|c|c|c|c|} \hline
L & $r$ & $c_1$ & $g$ \\ \hline
128 & 0.5000203756 & 0.6904174985 & 0.9975945594 \\
256 & 0.5000050926 & 0.6904143299 & 0.9987944047 \\
512 & 0.5000012731 & 0.6904135378 & 0.9993964859 \\
1024 & 0.5000003182 & 0.6904133397 & 0.9996980631 \\
2048 & 0.5000000795 & 0.6904132903 & 0.9998489873 \\
4096 & & 0.6904132780 & 0.9999244833 \\ \hline
$\infty$ & $0.5+r_2/L^{2}$ & 0.6904132738 & $1-g_1/L$ \\ \hline
\end{tabular}
\end{indented}
\end{table}
A comparison between the conformal expressions (in Eq.~(\ref{S_L_p}) and
Eq.~(\ref{S_L_o})) and the entropy calculated by exact diagonalization for
$L=2048$ is shown in Fig.~\ref{fig:profile}, both with periodic and open
boundary conditions; an excellent agreement is achieved. The
accuracy of the functional form can be checked by the ratio
\begin{equation}
r(L)=\frac{S_L(L/2)-S_L(L/4)}{S_{2L}(L)-S_{L}(L/2)}\quad \underset{L\to\infty}{\rightarrow}\quad \frac{1}{2}\;.
\label{ratio}
\end{equation}
The numerical results for $r(L)$ for different system sizes $L$ up to $L=2048$ are given
in Table \ref{table:check}, and the first correction term is found to be $O(L^{-2})$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=3.2in,angle=0]{fig1.eps}
\end{center}
\caption{
\label{fig:profile}
Entropy of finite chains of length $L=2048$ vs subsystem size $\ell$: the full
curves for $\ell \le L/2$ are calculated numerically; the dashed curves for
$\ell \ge L/2$ are the corresponding conformal results, described by
Eq.~(\ref{S_L_p}) with $c=1/2$ and $c_1=0.690413$ for homogeneous chains with
periodic boundary conditions, and by Eq.~(\ref{S_L_o}) with $g=1$ for open
homogeneous chains. For chains with staggered interactions, there are four
branches, depending on the type of the couplings on the boundary of the
subchain. Data presented here are for staggered chains with $\lambda=0.5$ and
with periodic boundary conditions. The fits for $\ell \ge L/2$ using
Eq.~(\ref{S_L_p}) are fulfilled with $c=1/2$, $c_1^{(++)}=1.02009$ (two strong
couplings on the boundary), $c_1^{(+-)}=c_1^{(-+)}=0.78446$ (one strong and one
weak coupling) and $c_1^{(--)}=0.54883$ (two weak couplings), respectively.}
\end{figure}
Furthermore, we are interested in the finite-size correction terms for the
coefficient $c$ and the boundary entropy $\log_2 g$ given in Eq.~(\ref{S_L_o}).
To evaluate $c$ for different system size, we first calculate the entropy
difference $\Delta S(L)=S_L(L/2)-S_{L/2}(L/4)$. For a chain with periodic
boundary conditions, we have $\Delta S(L)=c(L)/3$ and obtain an $L^{-2}$ -
correction for the coefficient: $c(L)=0.5-0.623(1)/L^2+O(L^{-3})$; for an open
chain, we obtain a $L^{-1}$ - correction, $c(L)=0.5+1.339(1)/L+O(L^{-2})$, via
$\Delta S(L)=c(L)/6$. To compute the boundary entropy $g(L)$, we make use the
relation between $S_L^{(p)}$ and $S_L^{(o)}$ for periodic and open boundary
conditions, respectively, via $2S_L^{(o)}(\ell)-S_L^{(p)}(\ell)=\log_2
g(L)+c/3$. In Table \ref{table:check}, the values of $g(L)$ using $\ell=L/2$
and $c=1/2$ are given for system sizes up to $L=4096$, and it shows a
correction of $O(1/L)$.
In conclusion, our numerical results for the entanglement entropy of
the homogeneous quantum Ising chain agree with all the known conformal predictions.
\subsection{Chains with staggered interactions}
\label{sec:stagg}
Now we consider the quantum Ising chain with periodically varying interactions
of period 2, corresponding to a chain with staggered interactions:
$J_{2i-1}=\lambda$ and $J_{2i}=1/\lambda$. According to
Eq.~(\ref{crit_point}), the critical point of the system is located at $h_c=1$.
This quantum Ising chain with staggered interactions has been solved in
Ref.~\cite{iz88}, and its critical singularities were found to be the same as for
the homogeneous chain. Here we study the entanglement entropy of the staggered
chain and check its relationship with the entropy of the homogeneous chain.
First, we calculate the entanglement entropy, $S_L(\ell)$, as a function of the
subsystem size $\ell$, for a finite chain. As shown in
Fig.~\ref{fig:profile} for a chain of length $L=2048$ with $\lambda=0.5$,
there are four branches with a twofold degeneracy, depending on the type of the
couplings ($\lambda$ or $\lambda^{-1}$) at the boundaries of the subsystem. For
each $\ell$, the largest and smallest value of $S_L(\ell)$ correspond to the
case in which both boundary couplings are strong (denoted by $(++)$) and weak
($(--)$), respectively, and the twofold degeneracy lying in between occurs when
one boundary coupling is strong and one weak ($(+-)$ and $(-+)$). All branches
are well fitted by the conformal form in Eq.~(\ref{S_L_p}) with coefficient
$c=1/2$ corresponding to the central charge of the homogeneous case, but with
different additive constants $c_1$. The additive constants for the above mentioned
four branches satisfy the relation: $c_1^{(++)}+c_1^{(--)}=2c_1^{(+-)}$.
This means that the boundary effect is strictly additive:
$c_1^{(++)}=2c_1^{(+)}$, $c_1^{(+-)}=c_1^{(+)}+c_1^{(-)}$ and
$c_1^{(--)}=2c_1^{(-)}$, here the subscript $+$ ($-$) corresponds to one strong
(weak) boundary coupling. For $\lambda=0.5$ we have $c_1^{(+)}=0.510045$ and
$c_1^{(-)}=0.274415$, whereas for $\lambda=0.25$ these are $c_1^{(+)}=0.663435$
and $c_1^{(-)}=0.262845$. Furthermore, $c_1^{(+)}$ ($c_1^{(-)}$) is found to
be a monotonously increasing (decreasing) function of $1/\lambda \ge 1$, and the
average, $c_1^{(+-)}/2$, is minimal for the homogeneous chain $\lambda=1$.
Consequently, for irrelevant perturbations represented by the staggered
interaction the average critical entanglement entropy is increasing, compared
with the fixed point value of the homogeneous chain.
To see how the coefficient $c(L)$ for a finite chain of length $L$ approaches
the conformal value $c=1/2$, we follow the procedure described in
Sec.~\ref{sec:crit_hom} for the homogeneous chain. Like the homogeneous chain,
the leading term of the finite-size correction to $c(L)$ is found to be
$O(L^{-2})$ for periodic boundary conditions, and $O(L^{-1})$ for
open boundary conditions.
\subsection{Random chains}
The entanglement entropy of the quantum Ising model with random couplings
and/or transverse fields can be conveniently studied by the strong disorder
renormalization group (RG) method\cite{refael,review}. In this RG
representation, the ground state of the quantum Ising model consists of a
collection of independent ferromagnetic clusters of various sizes; each cluster
of $n$ spins is in a $n$-site entangled state
$\frac{1}{\sqrt{2}}(\left|\uparrow\right\rangle^{\otimes
n}+\left|\downarrow\right\rangle^{\otimes n})$. The entanglement entropy of a subsystem
is just given by the number of the clusters that cross the boundary of the
subsystem. In 1D the asymptotic number of such clusters that contribute
to the entropy of a subsystem of length $\ell$ has been analytically calculated
by Refael and Moore \cite{refael} and the disorder average entropy in the long chain limit
is found to scale as:
\begin{equation}
\overline{S}(\ell)=\frac{c_{\rm eff}}{3} \log_2 \ell + c_1'
\label{S_l_r}
\end{equation}
where the effective central charge, $c_{\rm eff}=\ln 2/2$, is expected to be universal,
i.e. does not depend on the form of disorder, whereas the additive constant, $c_1'$,
is disorder dependent.
For a finite chain of length $L$ with periodic boundary conditions, the entropy of
a subsystem of length $\ell$ is expected to behave as:
\begin{equation}
\overline{S}_L(\ell)=\frac{c_{\rm eff}}{3} \log_2[ L f(\ell/L)]+ c_1'
\label{S_c_eff}
\end{equation}
where the scaling function $f(v)$ is reflection symmetric, $f(v)=f(1-v)$, and
$\lim_{v \to 0} f(v) \simeq v$. Consequently $f(v)$ can be expanded as a
Fourier series: $f(v)=\sum_{k=1}^{\infty} A_k \sin (2k-1) \pi v$, with
$\sum_{k=1}^{\infty} A_k (2k-1) \pi=1$. We note that for
conformally invariant models only the first term of this expansion exists
(cf. Eq.~(\ref{S_L_p})).
In our numerical calculations we used a power-law distribution:
\begin{equation}
P_D(x)=\frac{1}{D} x^{-1+1/D}\;,
\end{equation}
both for the couplings and the transverse fields, which ensures that the random
model is at the critical point. Here $D^2=\overline{\ln^2x}-\overline{\ln x}^2$
measures the strength of disorder. For the random chains we have treated
finite chains up to a length $L=1024$, and considered at least $10^4$
independent realizations for each length $L$, plus different positions of
a subsystem in the chain for a given $\ell$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=3.2in,angle=0]{fig2.eps}
\end{center}
\caption{
\label{fig:random}
Average entropy of the random chain with uniform disorder (with $D=1$) as a
function of $\ln \ell$ for different system sizes $L$. The slope of the broken
straight line is given by $1/6$, corresponding to the RG prediction $c_{\rm
eff}=\ln 2/2$. Inset: scaling plot of the entropy vs. $\ell$ for different sizes
$L$ using the scaling prediction in Eq.~(\ref{S_c_eff}). The solid line corresponds
to the conjecture of conformal invariance, given in Eq.~(\ref{S_L_p}),
with $c$ replaced by $c_{\rm eff}=\ln(2)/2$ and an additive constant as the fit parameter.
}
\end{figure}
In Fig.~\ref{fig:random} we plot the average entropy $\overline{S}_L(\ell)$ vs.
$\ln \ell$. The curves tend to approach an asymptotic linear behavior with a
slope which is, in the large $\ell$ regime, well described by the
renormalization group prediction $c_{\rm{eff}}=\ln(2)/2$. To estimate
$c_{\rm eff}(L)$ quantitatively for different chain sizes $L$, we average over
the entropy in the large $\ell$ region and make use of the relation between
$\overline{S}_{2L}$ and $\overline{S}_{L}$, given by
\begin{equation}
\frac{1}{2n+1} \sum_{\ell=L/2-n}^{L/2+n}[\overline{S}_{2L}(2l)-\overline{S}_L(l)]=
c_{\rm eff}(2L)/3.
\label{c_L_n}
\end{equation}
The estimated values of $c_{\rm eff}(L)/\ln 2$ are presented in Table
\ref{table:random} for uniform disorder. The results for the two largest finite
systems are compatible with the estimate: $c_{\rm eff}/\ln 2=0.501(3)$, which is
in excellent agreement with the RG prediction.
\begin{table}
\caption{Finite-size estimates of $c_{\rm eff}/\ln 2$ for the random model
using the relation in Eq.(\ref{c_L_n}). \label{table:random}\\}
\begin{indented}
\item[]\begin{tabular}{|c|c|c|c|} \hline
n & $L=128$ & $L=256$ & $L=512$ \\ \hline
2 & 0.531 & 0.503 & 0.502 \\
4 & 0.532 & 0.502 & 0.506 \\
8 & 0.533 & 0.502 & 0.501 \\
16 & 0.534 & 0.504 & 0.499 \\ \hline
\end{tabular}
\end{indented}
\end{table}
Finally we turn to a study of the form of the average entropy
$\overline{S}_L(\ell)$ as a function of $\ell$, in particular we are interested
in how well it can be approximated by the conjecture of conformal invariance
given in Eq.~(\ref {S_L_p}) with an effective central charge
$c_{\rm{eff}}$. As shown in the inset of Fig.~\ref{fig:random}, the
approximation is seemingly good. To have a quantitative comparison, we have
calculated the ratio $\overline{r}(L)$, similar to Eq.~(\ref {ratio}), defined
as
\begin{equation}
\overline{r}(L)=\frac{\overline{S}_L(L/2)-
\overline{S}_L(L/4)}{\overline{S}_{2L}(L)-\overline{S}_{L}(L/2)},
\label{ratio_r}
\end{equation}
whose asymptotic value is given, in terms of the
Fourier coefficients, by:
\begin{eqnarray}
\overline{r}&=&\frac{1}{2} + \log_2 \left[ \sum_{k=1}^{\infty} (-1)^{k+1}A_k \right]\cr
&-&\log_2 \left[ \sum_{k=1}^{\infty} (-1)^{k+1}(A_{2k-1}+A_{2k}) \right] \;.
\end{eqnarray}
For conformal invariant cases, we have $\overline{r}=1/2$. The numerically calculated
values of $\overline{r}(L)$, presented in Table~\ref{table:rand} for disorder
strength $D=0.5$ and $D=1$, deviate significantly from $\overline{r}=1/2$ for large $L$.
This means that the higher order terms in the Fourier expansion are not
negligible. The scaling function $f(v)$ is presumably universal, i.e.
independent of the form of the disorder.
\begin{table}
\caption{The ratio defined in Eq.~(\ref {ratio_r}) for the average entropy $\overline{S}_L(\ell)$
for disorder strength $D=0.5$ and $D=1$.\label{table:rand}\\}
\begin{indented}
\item[]
\begin{tabular}{|c|c|c|} \hline
L & $D=0.5$ & $D=1.0$ \\ \hline
32 & 0.548 & 0.587 \\
64 & 0.588 & 0.605 \\
128 & 0.573 & 0.584 \\
256 & 0.619 & 0.608 \\
512 & 0.615 & 0.606 \\ \hline
\end{tabular}
\end{indented}
\end{table}
\section{Scaling close to the critical point}
\label{sec:fss}
So far we have studied the entanglement entropy at the critical point. In this
section, we consider the entanglement between two halves of a finite chain with
homogeneous interactions, and study its behavior approaching to the critical point
$h_c=1$.
\begin{figure}
\begin{center}
\includegraphics[width=3.2in,angle=0]{fig3.eps}
\end{center}
\caption{
\label{fig:S_h}
The entanglement entropy
of a half of a chain with $J=1$ as a function of the strength of the
transverse field $h$, for periodic (a)
and open (b) boundary conditions. On increasing the system size
$L$, the maximum gets more pronounced, and the position of the
maximum tends towards the critical point $h_c=1$.
}
\end{figure}
According to Eq.~(\ref{S_l}) and Eq.~(\ref{S_xi}), a divergence of the maximal
entanglement entropy occurs at the quantum critical point, which can be traced
back to the divergence of the correlation length with $\xi\sim |h-h_c|^{-\nu}$.
In a finite system of length $L$, the finite size effects induce a rounding and
a shift of the maximum of the entropy, as shown in Fig.~\ref{fig:S_h} for
$S_L(L/2)$ vs. $h$. In the following we denote the entropy of a half of a
finite chain of length $L$ as a function of the transverse field $h$ by
$S_L(L/2,h) \equiv \hat{\cal S}(L,h)$. The position of the maximum of
$\hat{\cal S}(L,h)$, denoted by $h_m(L)$, can be used to define a finite-size
effective critical point and its shift from the true critical point is expected
to scale as: $h_c-h_m(L) \sim L^{-\lambda}$, where $\lambda$ is the shift
exponent. The numerically calculated finite-size transition points are listed
in Table~\ref{table:fss} both for closed and open chains. The maximum of the
entropy $\hat{\cal S}(L,h_m(L))$, like $\hat{\cal S}(L,h_c)$, depends
logarithmically on the system sizes $L$ [insets in Fig.~\ref{fig:S_sc_o} and
Fig.~\ref{fig:S_sc_p}].
As a matter of fact the difference $\Delta S(L)=\hat{\cal S}(L,h_m(L))- \hat{\cal
S}(L,h_c)$ approaches a well defined limiting value for $L \to \infty$. Note,
however that for open chains $\Delta S(L)$ tends to a finite value, whereas for
closed chains the entropy difference goes to zero.
We first study the rounding of the maximum of the entropy. Making
use of the finite-size scaling ansatz\cite{barber}:
\begin{equation}
\hat{\cal S}(L,h)-\hat{\cal S}(L,h_m(L))= \widetilde{F}[L^{1/\nu}(h-h_m(L))]\;.
\label{diff_S1}
\end{equation}
with $\nu=1$, we can make all data for different system sizes perfectly
collapse onto a single curve, as shown in Fig.~\ref{fig:S_sc_o} for open chains
and Fig.~\ref{fig:S_sc_p} for closed chains. In both cases the scaling function
is $\widetilde{F}[\tilde{x}] \sim \tilde{x}^2$ for small $\tilde{x}$.
In order to obtain the shift of the finite-size critical points, we take the
derivative of both sides of Eq.(\ref{diff_S1}) at $h=h_c$:
\begin{equation}
\left.\frac{\partial \hat{\cal S}(L,h)}{\partial h}\right|_{h_c} \sim L^{2/\nu}(h_c-h_m(L))\;.
\label{der_S}
\end{equation}
For open chains the derivative at the l.h.s. is proportional to $L^{1/\nu}$,
leading to conventional finite-size scaling relation: $h_c-h_m(L) \sim
L^{-1/\nu}$. For closed chains this derivative has a much weaker
$L$-dependence, which can be identified as logarithmic in $L$. From
Eq.~(\ref{der_S}), we then expect the relation: $h_c-h_m(L) \sim \log_2 L/L^2$.
This prediction can be checked by calculating the shift exponent $\lambda$
through the finite-size estimates:
$\lambda(L)=\log_2[h_c-h_m(L/2)]-\log_2[h_c-h_m(L)]$ [Table \ref{table:fss}].
For open chains the exponent approaches $\lambda=1/\nu=1$ for large $L$, in
accordance with our previous discussion. For closed chains the effective shift
exponent is around $1.87$ for the largest system size, which, however, cannot
rule out a true value $\lambda=2$ with a logarithmic finite-size-correction.
In \ref{sec:shift} we present an argument in favor of the
$\log_2 L/L^2$ behavior of the shift for closed chains.
As a numerical check of this scenario we have calculated
the scaling combination: $sc =(1-h_m(L))\times L^2/(\log_2 L+a)$,
which are shown in Table \ref{table:fss}. Indeed the value of $sc$
seems to approach a finite limiting value.
\begin{figure}
\begin{center}
\includegraphics[width=3.2in,angle=0]{fig4.eps}
\end{center}
\caption{
\label{fig:S_sc_o}
A scaling plot of the entanglement entropy
for chains with open boundary conditions. Data collapse is obtained for
$\nu=1$, consistent with the universality hypothesis. In the inset is shown
the divergence of the value at the maximum as the system size increases. The slope
is given by 0.08, consistent with the exact value $1/12$ (cf. Eq.~(\ref{S_xi})).}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=3.2in,angle=0]{fig5.eps}
\end{center}
\caption{
\label{fig:S_sc_p}
The finite size scaling is performed for
chains with periodic boundary conditions, using the scaling form in
Eq.~(\ref{diff_S1}) and $\nu=1$. The inset shows the logarithmic dependence
of the value at the maximum on the system size $L$. The slope is consistent with
the theoretical value $1/6$ for a half subchain taken from a closed chain.}
\end{figure}
Having clarified the finite-size scaling behavior of the rounding and the shift
of the maximum of the entropy, let us consider the scaling form of the entropy
in the critical region. In the conventional finite-size scaling theory we have the
ansatz:
\begin{equation}
\hat{\cal S}(L,h)-\hat{\cal S}(L,h_c)= F[L^{1/\nu}(h-h_c)]\;.
\label{diff_S2}
\end{equation}
The l.h.s. of Eq.(\ref{diff_S2}) can be rewritten as:
$\hat{\cal S}(L,h)-\hat{\cal S}(L,h_m(L))+\Delta S(L)=\widetilde{F}[\tilde{x}]+\Delta S(L)$,
where
the argument of $\widetilde{F}(\tilde{x})$ is given by: $\tilde{x}=x-x_c$ with
$x=L^{1/\nu}(h-h_c)$ and $x_c=L^{1/\nu}(h_c-h_m(L))$. Now using the fact
that $\widetilde{F}[\tilde{x}]$ is quadratic for small $\tilde{x}$ we obtain
for the scaling function in Eq.(\ref{diff_S2}):
\begin{equation}
F(x)\approx A(x-x_c)^2+\Delta S, \quad x \approx x_c\;.
\end{equation}
For open chains in the large $L$ limit, we have $x_c>0$ and $\Delta S>0$,
so that shift exponent is $\lambda=1/\nu$. On the other hand, for closed chains
both limiting values vanish: $x_c=0$ and $\Delta S=0$. Therefore
conventional finite-size scaling is not valid and the shift
exponent is $\lambda > 1/\nu$.
We close this section by two remarks. First we note that in Ref~\cite{fazio1}
the derivatives of the nearest-neighbor concurrence, $\partial_J C(1)$, with
respect to the control parameter $J$ is studied. The position of the minimum of
$\partial_J C(1)$, which defines the effective quantum critical point of a
finite closed chain of length $L$, is shifted from the true critical point by
$L^{-1.87}$ (see Fig. 1, in Ref.~\cite{fazio1}), with a shift exponent that is
very close to the effective exponent given in Table \ref{table:fss}. One might
think that the shift of the minimum of $\partial_J C(1)$ has the same scaling
behavior as discussed here for the position of the maximum of the entropy.
Our second remark concerns random chains. The position of the maximum of the
average entanglement entropy of a half chain can be used to define sample
dependent pseudo\-critical point. Its scaling has been studied in detail in
Ref.~\cite{ilrm07}.
\begin{table}
\caption{Finite-size critical transverse fields of the homogeneous quantum Ising
chain of $L$ sites with periodic [p] and open [o] boundary conditions calculated
from the location of the maxima of the entropy for $\ell=L/2$. The effective shift
exponents, $\lambda(L)$, are calculated by two point fits. The scaling
combinations are: $sc=(1-h_m(L))\times L^2/(\log_2 L + a)$ with $a=2-1/\ln 2$
(s. Appendix~\ref{sec:shift}), for periodic chains and $x_c=(1-h_m(L)) \times L$ for
open chains.\label{table:fss}}
\begin{indented}
\item[]
\begin{tabular}{|c||c|c|c||c|c|c|} \hline
L & $h_m(L)[p]$ & $\lambda(L)$&$sc$& $h_m(L)[o]$ &$\lambda(L)$& $x_c$ \\ \hline
128 & 0.9983031 & 1.813 & 3.679 & 0.9636656 & 1.077 & 4.651 \\
256 & 0.9995225 & 1.829 & 3.656 & 0.9822266 & 1.045 & 4.550 \\
512 & 0.9998671 & 1.845 & 3.644 & 0.9912353 & 1.027 & 4.487 \\
1024 & 0.9999633 & 1.858 & 3.641 & 0.9956543 & 1.015 & 4.450 \\
2048 & 0.9999900 & 1.876 & 3.629 & 0.9978379 & 1.007 & 4.424 \\ \hline
\end{tabular}
\end{indented}
\end{table}
\section{Discussion}
\label{sec:disc}
In this paper we have studied finite size effects of entanglement entropy of
the quantum Ising chain at/near its order-disorder quantum phase transition.
The model considered can be expressed in terms of free fermions, which enables
us to perform large scale numerical investigations. Three types of couplings
were considered: homogeneous, periodically modulated and random couplings.
For the homogeneous system at the critical point we have verified the finite
size form predicted by the conformal field theory, both for periodic and open
boundary conditions. We have also calculated the additive constant to the
entropy and subleading corrections. In the off-critical region, we have studied
the finite-size scaling behavior of the entropy, $S_L(L/2)$, in the vicinity of
its maximum, and confirmed the intimate connection between entanglement and
universality. The position of the maximum, $h_m(L)$, can be regarded as an
indicator of the effective critical point in the finite sample. For an open
chain the shift of $h_m(L)$ from the true critical point is shown to be
$O(L^{-1})$, whereas for a chain with periodic boundary conditions it is
$O(L^{-2} \ln L)$. We have provided analytical results for the off-critical entropy
in infinite chains to explain these findings. We expect that the shift of $h_m(L)$
for other critical quantum spin chain has the same type differences for open
and periodic boundary conditions.
The quantum Ising chain with periodically modulated couplings belongs to the
same critical universality class as the homogeneous model. In the case of
staggered couplings, we have found that the critical entropy is split into four
branches, each of which has the same prefactor (the central charge) of the
logarithm but has different additive constants. This is expected to be generic
to critical quantum spin chains with all kinds of periodically modulated
couplings.
For random quantum Ising chains, we have numerically verified the prefactor of
the logarithm predicted by the analysis of the strong disorder renormalization
group. The functional form of the average entropy versus subsystem size, which
is presumably universal for any strength of disorder, has been found to
deviate from the results for conformally invariant models.
The results obtained in this paper, though only based on quantum Ising chains,
are expected to be valid in some other cases of quantum spin chains. For
example, the XY-chain is related with the Ising chain via an exact mapping, so
that the results obtained for the Ising chain can be directly transferred to
those for the XY-chain through the mapping. This mapping is also applicable for
random cases. Moreover, for random cases the criticality of many quantum spin
chains belongs to the same universality class (cf. random XX-chains and random
Heisenberg chains), known from the strong-disorder renormalization group,
and the universality of the associated effective central charge was numerically
confirmed \cite{num}. Therefore, our results for the random cases,
e.g. the functional form of the average entropy vs. $\ell$, should be universal
for a wide range of models.
\ack{We thank R. Juh\'asz, C. Monthus, H. Rieger and Z. Zimbor\'as for useful
discussions. This work has been supported by the National Office of Research
and Technology under Grant No. ASEP1111, by German-Hungarian exchange programs
(DAAD-M\"OB and DFG-MTA) and by the Hungarian National Research Fund under
grant No OTKA TO48721, K62588, MO45596 and M36803.}
|
1,108,101,566,798 | arxiv | \section{Introduction}
This paper describes our approach to the Natural Language Inference (NLI) subtask of the MEDIQA 2019 shared task \cite{MEDIQA2019}. As it is not yet clear the extent to which knowledge-based embeddings may provide task-specific improvement over recent advances in contextual embeddings, we provide an analysis of the differences in performance between these two methods. Additionally, it is not yet clear from the literature the extent to which information stored in contextual embeddings overlaps with that in knowledge-based embeddings for which we provide a preliminary analysis of the attention weights of models that use these two representation methods as input. We compare BERT fine-tuned to MIMIC-III \cite{Johnson2016} and PubMed to Embeddings of Semantic Predications (ESP) trained on SemMedDB and a baseline that uses Cui2Vec embeddings trained on clinical and biomedical text.
Two recent advances in the unsupervised modeling of natural language, Embeddings of Language Models (ELMo) \cite{Peters} and Bidirectional Encoder Representations from Transformers (BERT) \cite{Devlin2018}, have led to drastic improvements across a variety of shared tasks. Both of these methods use transfer learning, a method whereby a multi-layered language model is first trained on a large unlabeled corpus. The weights of the model are then frozen and used as input to a task specific model \cite{Peters, Devlin2018, Liu2019}. This method is particularly well-suited for work in the medical domain where datasets tend to be relatively small due to the high cost of expert annotation.
However, whereas clinical free-text is difficult to access and share in bulk due to privacy concerns, the biomedical domain is characterized by a significant amount of manually-curated structured knowledge bases. The BioPortal repository currently hosts 773 different biomedical ontologies comprised of over 9.4 million classes. SemMedDB is a triple store that consists of over 94 million predications extracted from PubMed by SemRep, a semantic parser for biomedical text \cite{SemRep, Kilicoglu2012}. These available resources make a strong case for the evaluation of knowledge-based methods for the Medical Natural Language Inference (MedNLI) task \cite{romanov2018}.
\section{Related Work}
In this section, we provide a brief overview of methods for distributional and frame-based semantic representation of natural language. For a more detailed synthesis, we refer the reader to the review of Vector Space Models (VSMs) by Turney and Pantel \shortcite{Turney2010}.
\subsection{Distributional Semantics}
The distributed representation of words has a long history in computational linguistics, beginning with latent semantic indexing (LSI) \cite{Deerwester1990, Hofmann1999, Kanerva2000RandomIO}, maximum entropy methods \cite{Berger1996}, and latent Dirichlet allocation (LDA) \cite{Blei2003}. More recently, neural network methods have been applied to model natural language \cite{Bengio2000ANP, Weston2008, Ratinov2010}. These methods have been broadly applied as a method of improving supervised model performance by learning word-level features from large unlabeled datasets with more recent work using either Word2Vec \citep{Mikolov, Pavlopoulos2014} or GloVe \cite{Pennington2014} embeddings. Recent work has learned a continuous representation of Unified Medical Language System (UMLS) \cite{Aronson2006} concepts by applying the Word2Vec method to a large corpus of insurance claims, clinical notes, and biomedical text where UMLS concepts were replaced with their Concept Unique Identifiers (CUIs) \cite{Beam2018}.
Models that incorporate sub-word information are particularly useful in the medical domain for representing medical terminology and out-of-vocabulary terms common in clinical notes and consumer health questions \cite{romanov2018}. Most approaches use a temporal convolution over a sliding window of characters and have been shown to improve performance on a variety of tasks \cite{Kim2015, Zhang2015, Minjoon2016, Bojanowski2017EnrichingWV}.
Embeddings from Language Models (ELMo) computes word representations using a bidirectional language model that consist of a character-level embedding layer followed by a deep bidirectional long short-term memory (LSTM) network \cite{Peters}.
Bidirectional Encoder Representations from Transformers (BERT) replaces the each forward and backward LSTMs with a single Transformer that simultaneously computes attention in both the forward and backward directions and is regarded as the current state-of-the-art method for language representation \cite{Vaswani2017, Devlin2018}. This method additionally substitutes two new unsupervised training objectives in place of the classical language models, i.e., masked language modeling (MLM) and next sentence prediction (NSP). In the case of MLM, a percentage of the words in the corpus are replaced by a [MASK] token. The task is then for the system to predict the masked token. For NSP, the task is given two sentences, $s1$ and $s2$, from a document to determine whether $s2$ is the next sentence following $s1$.
While ELMo has been shown to outperform GloVe and Word2Vec on consumer health question answering \cite{Kearns2018ResourceAR}, BERT has outperformed ELMo on various clinical tasks \cite{si_enhancing_2019} and has been fine-tuned and applied to the biomedical literature and clinical notes \cite{alsentzer_publicly_2019, huang_clinicalbert:_2019, si_enhancing_2019, lee_biobert:_2019}. BERT supports the transfer of a pretrained general purpose language model to a task-specific application through fine-tuning. The next sentence prediction objective in the pre-training process suggests this method would be inherently suitable for NLI. In addition, BERT utilizes character-based and WordPiece tokenization \cite{wu_googles_2016} to learn the morphological patterns among inflections. The subword segmentation such as \textit{\#\#nea} in the word \textit{dyspnea} makes it capable to understand the context of an out-of-vocabulary word making it a particularly suitable representation for clinical text.
\subsection{Frame-based Semantics}
FrameNet is a database of sentence-level frame-based semantics that proposes human understanding of natural language is the result of frames in which certain roles are expected to be filled \cite{Baker1997}. For example, the predicate \textit{``replace"} has at least two such roles, the thing being replaced and the new object.
A sentence such as \textit{``The table was replaced."} raises the question \textit{``With what was the table replaced?"}. Frame-based semantics is a popular approach for semantic role labeling (SRL) \cite{Swayamdipta2018}, question answering (QA) \cite{Shen2007, Roberts2012, He2015, Michael}, and dialog systems \cite{Larsson1998, Gupta2018}.
Vector symbolic architectures (VSA) are an approach that seeks to represent semantic predications by applying binding operators that define a directional transformation between entities \cite{Levy2008}. Early approaches included binary spatter code (BSC) for encoding structured knowledge \cite{Kanerva1996BinarySO, Kanerva1997358FD} and Holographic Embeddings that used circular convolution as a binding operator to improve the scalability of this approach to large knowledge graphs \cite{Plate1995HolographicRR}. The resurgence of neural network methods has focused attention on extending these methods as there is a growing interest in leveraging continuous representations of structured knowledge to improve performance on downstream applications.
Knowledge graph embeddings (KGE) are one approach that represents entities and their relationships as continuous vectors that are learned using TransE/R \cite{Bordes2009}, RESCAL \cite{Nickel2011ATM}, or Holographic Embeddings \cite{Plate1995HolographicRR, Nickel2015}. Stanovsky et. al \shortcite{Stanovsky2017} showed that RESCAL embeddings pretrained on DbPedia improved performance on the task of adverse drug reaction labeling over a clinical Word2Vec model. RESCAL uses tensor products whose application to representation learning dates back to Smolensky \shortcite{Smolensky1986, Smolensky1990TensorPV} that used the inner product and has recently been applied to the bAbI dataset \cite{Smolensky2016BasicRW, Weston2016TowardsAQ}. Embeddings of Semantic Predications (ESP) are a neural-probabilistic representational approach that uses VSA binding operations to encode structured relationships \cite{Cohen2017}. The Embeddings Augmented by Random Permutations (EARP) used in this paper are a modified ESP approach that applies random permutations to the entity vectors during training and were shown to improve performance on the Bigger Analogy Test Set by up to 8\% against a fastText baseline \cite{CohenCoNLL2018}.
\section{Methods}
In this section, we provide details on the three representation methods used in this study, i.e. BERT, Cui2Vec, and ESP. We continue with a description of the inference model used in each experiment to predict the label for a given hypothesis/premise pair.
\begin{figure*}[t!]
\includegraphics[width=\textwidth, height=11cm]{29.png}
\caption{An example of a correct BERT prediction demonstrating its general domain coverage and contextual embedding. Premise: \textit{``He will be spending time with family and friends who are coming in from around the country to see him."} Hypothesis: \textit{``his family and friends do not yet have plans to visit."}}
\label{fig:HeatmapTopBert}
\end{figure*}
\begin{figure*}[t!]
\includegraphics[width=\textwidth,
height=11cm]{119.png}
\caption{An example of a correct ESP prediction demonstrating its ability to associate Advil as a subclass of NSAIDs. Premise: \textit{``She is on a daily ASA, and denies other NSAID use."} Hypothesis: \textit{``She takes Advil regularly."}}
\label{fig:HeatmapTopESP}
\end{figure*}
\subsection{Representation Layer}
There are many publicly available biomedical BERT embeddings which were initialized from the original BERT Base models. BioBERT was trained on PubMed Abstracts and PubMed Central Full-text articles \cite{lee_biobert:_2019}. In this study, we applied ClinicalBERT that was initialized from BioBERT and subsequently trained on all MIMIC-III notes \cite{alsentzer_publicly_2019}.
For Cui2Vec, we used the publicly available implementation from Beam et al. \shortcite{Beam2018} that was trained on a corpus consisting of 20 million clinical notes from a research hospital, 1.7 million full-text articles from PubMed, and an insurance claims database with 60 million members.
For ESP, we used a 500-dimensional model trained over SemMedDB using the recent Embeddings Augmented by Random Permutations (EARP) approach with a $10^{-7}$ sampling threshold for predications and a $10^{-5}$ sampling threshold for concepts excluding concepts that had a frequency greater than $10^{6}$ \cite{CohenCoNLL2018}.
To apply Cui2Vec and ESP, we first processed the MedNLI dataset \cite{romanov2018} with MetaMap to normalize entities to their concept unique identifier (CUI) in the UMLS \cite{Aronson2006}. MetaMap takes text as input and applies biomedical and clinical entity recognition (ER), followed by word sense disambiguation (WSD) that links entities to their normalized concept unique identifiers (CUIs). Entities that mapped to a UMLS CUI were assigned a representation in Cui2Vec and ESP. Other tokens were assigned vector representations using fastText embeddings trained on MIMIC-III data \cite{Bojanowski2017EnrichingWV, romanov2018}.
\subsection{Inference Model}
For all experiments, we used the AllenNLP implementation \cite{Gardner2017} of the Enhanced Sequential Inference Model (ESIM) architecture \cite{ESIM}. This model encodes the premise and hypothesis using a Bidirectional LSTM (BiLSTM) where at each time step the hidden state of the LSTMs are concatenated to represent its context. Local inference between the two sentences is then achieved by aligning the relevant information between words in the premise and hypothesis. This alignment based on soft attention is implemented by the inner product between the encoded premise and encoded hypothesis to produce an attention matrix (Figure \ref{fig:HeatmapTopBert} and \ref{fig:HeatmapTopESP}). These attention values are used to create a weighted representation of both sentences. An enhanced representation of the premise is created by concatenating the encoded premise, the weighted hypothesis, the encoded premise minus the weighted hypothesis, and the element-wise multiplication of the encoded premise and the weighted hypothesis. The enhanced representation of the hypothesis is created similarly. This operation is expected to enhance the local inference information between elements in each sentence. This representation is then projected into the original dimension and fed into a second BiLSTM inference layer in order to capture inference composition sequentially. The resulting vector is then summarized by max and average pooling. These two pooled representations are concatenated and passed through a multi-layered perceptron followed by a sigmoid function to predict probabilities for each of the sentence labels, i.e. \textit{entailment}, \textit{contradiction}, and \textit{neutral}.
\section{Results}
The ESIM model achieved an accuracy of 81.2\%, 65.2\%, and 77.8\% for the MedNLI task using BERT, Cui2Vec, and ESP, respectively. Table \ref{tab:predictionAccuracyData} shows the number of correct predictions by each embedding type. The BERT model has the highest accuracy on predicting \textit{entailment} and \textit{contradiction} labels, while the ESP model has the highest accuracy on predicting \textit{neutral} labels. However, the difference is only significant in the case of \textit{entailment}.
To evaluate the ability to set a predictive threshold for use in clinical applications, we sought to measure the certainty with which the model made its predictions. To achieve this goal, we used the predicted probabilities of each embedding type on their respective subset of correct predictions such that. We found the predicted probability of ESP to be much higher than the others as depicted in Figure \ref{fig:boxplot}. ESP's minimum predicted probability as well as the variance of its distribution is the lowest among all embedding types.
\begin{table*}[t!]
\begin{tabular}{@{}llll@{}}
& \multicolumn{3}{c}{\textbf{Embedding Type}} \\
\textbf{Label} & \textbf{BERT} & \textbf{Cui2Vec} & \textbf{ESP} \\
\hline Entailment & \textbf{82.22\% (n=111)} & 60.00\% (n=81) & 71.85\% (n=97) \\
Contraction & \textbf{88.15\% (n=119)} & 74.81\% (n=101) & 87.41\% (n=118) \\
Neutral & 73.33\% (n=99) & 60.74\% (n=82) & \textbf{74.07\% (n=100)} \\
\end{tabular}
\centering
\caption{Model accuracy for each label by embedding type.}
\label{tab:predictionAccuracyData}
\end{table*}
\begin{figure*}[t!]
\includegraphics[width=\textwidth]{boxplot}
\caption{Distribution of predicted probability of the gold label from the subset of correct predictions for each representation method.}
\label{fig:boxplot}
\end{figure*}
\subsection{Error Analysis}
To examine the relationship between embedding prediction performance and hypothesis focus, we first annotated the test set for:
\begin{itemize}
\item hypothesis focus (e.g. \textit{medications, procedures, symptoms}, etc.)
\item hypothesis tense (e.g. \textit{past, current, future})
\end{itemize}
\subsubsection{Focus}
A total of eleven, non-mutually exclusive hypothesis focus classes were arrived at by consensus of the three authors after an initial blinded round of annotation by two annotators. The remaining data was annotated by one of these annotators. We provide definitions of the classes and their overall counts in Table \ref{table:Foci}. The classes are: \textit{State}, \textit{Anatomy}, \textit{Disease}, \textit{Process}, \textit{Temporal}, \textit{Medication}, \textit{Clinical Finding}, \textit{Location}, \textit{Lab/Imaging}, \textit{Procedure}, and \textit{Examination}.
We then performed Pearson's chi-squared test with Yates' continuity correction on 2x2 contingency tables for each embedding sentence pair prediction (correct or incorrect) with each hypothesis focus (presence or absence) using the \textit{chisq.test} function in R software and results reported in Table \ref{table:chiSquared}.
The only significant relationships between hypothesis focus and embedding accuracy were found between BERT and \textit{Disease} (p-value = 0.01) and Cui2Vec and \textit{Disease} (p-value = 0.01) through Pearson's Chi-squared test with Yates' continuity correction.
Both embeddings achieved higher accuracy on sentence pairs with a hypothesis focus labeled \textit{Disease} (BERT=90.4\%; Cui2Vec=76.6\%) than without (BERT=78.5\%; Cui2Vec=61.7\%).
\begin{table*}[t!]
\begin{tabular}{lll}
\hline \textbf{Hypothesis Focus} & \textbf{Definition} & \textbf{Count(\%)} \\ \hline
State & Patient state or symptoms (e.g. \textit{``...has high blood pressure..."}) & 251 (62.0) \\
Anatomy & Specific body part referenced (e.g. \textit{``... has back pain"}) & 115 (28.4) \\
Disease & Similar to state, but a defined disease (e.g. \textit{``...has Diabetes"}) & 95 (23.5) \\
\multirow{2}{*}{Process} & Events like transfers, family visiting, scheduling, or vague & 52 (12.8) \\
& references to interventions (e.g. \textit{``...received medical attention"}) &\\
\multirow{2}{*}{Temporal} & Reference to time (e.g. \textit{``...initial blood pressure was low"}) & 51 (12.6)\\
& besides tense or history & \\
\multirow{2}{*}{Medication} & Any reference to medication (e.g. \textit{``antibiotics", ``fluids"}, & 32 (7.9) \\
& \textit{``oxygen", ``IV"}) including administration and patient habits & \\
Clinical Finding & Results of an exam, lab/image, procedure, or a diagnosis & 28 (6.9)\\
Location & Specific physical location specified (e.g.\textit{``...discharged home"}) & 28 (6.9) \\
Lab/Imaging & Laboratory tests or imaging (e.g. \textit{histology, CBC, CT scan}) & 24 (5.9) \\
\multirow{2}{*}{Procedure} & Physical procedure besides Lab/Image or exam & 14 (3.5) \\
& (e.g.\textit{ ``intubation", ``surgery", ``biopsies"}) & \\
Examination & Physical examination or explicit use of the word exam(ination) & 3 (0.7)
\end{tabular}
\caption{\label{font-table} Hypothesis foci definitions, examples, and count for all 405 hypotheses in the test set.}
\label{table:Foci}
\centering
\end{table*}
\begin{table*}[t!]
\begin{center}
\begin{tabular}{@{}llllllllll@{}}
& \multicolumn{9}{c}{\textbf{Embedding Type}} \\
& \multicolumn{3}{c}{\textbf{BERT}} & \multicolumn{3}{c}{\textbf{Cui2Vec}} & \multicolumn{3}{c}{\textbf{ESP}} \\
\cmidrule(lr){2-4}\cmidrule(lr){5-7}\cmidrule(lr){8-10}
\textbf{Focus} & \textbf{(+)} & \textbf{(-)} & \textbf{p-value} & \textbf{(+)} & \textbf{(-)} & \textbf{p-value} & \textbf{(+)} & \textbf{(-)} & \textbf{p-Value} \\
\hline Anatomy & 93 & 22 & 1 & 73 & 42 & 0.74 & 90 & 25 & 0.99 \\
Clinical Finding & 24 & 4 & 0.71 & 16 & 12 & 0.47 & 24 & 4 & 0.42 \\
Disease & 85 & 9 & \textbf{0.01} & 72 & 22 & \textbf{0.01} & 78 & 16 & 0.21 \\
Examination & 3 & 0 & 0.93 & 2 & 1 & 0.58 & 3 & 0 & 0.82 \\
Lab/Imaging & 30 & 7 & 1 & 22 & 15 & 0.55 & 31 & 6 & 0.48 \\
Location & 21 & 7 & 0.53 & 14 & 14 & 0.12 & 19 & 9 & 0.28 \\
Medication & 27 & 5 & 0.81 & 24 & 8 & 0.30 & 28 & 4 & 0.25 \\
Procedure & 12 & 2 & 0.93 & 7 & 7 & 0.35 & 11 & 3 & 1 \\
Process & 41 & 11 & 0.78 & 35 & 17 & 0.85 & 40 & 12 & 1 \\
State & 198 & 53 & 0.16 & 158 & 93 & 0.27 & 191 & 60 & 0.36 \\
Temporal & 38 & 12 & 0.41 & 37 & 13 & 0.22 & 41 & 9 & 0.56
\end{tabular}
\caption{Results from chi-squared (with Yates' continuity correction) test of correct(+) and incorrect(-) predictions by embedding and hypothesis focus type.}
\label{table:chiSquared}
\centering
\end{center}
\end{table*}
\subsubsection{Tense}
Each hypothesis was annotated for tense into one of three mutually exclusive classes: \textit{Past, Current,} and \textit{Future}.
Test set hypotheses were predominantly \textit{Current} (n=273; 67.4\%) or \textit{Past} (n=131; 32.3\%) tense.
Only one hypothesis (0.2\%) was \textit{Future} tense.
A subset (n=22; 7.9\%) of the \textit{Current} tense hypotheses explicitly described patient history (e.g. \textit{``The patient has a history of PE"}).
\section{Discussion}
Our preliminary analysis, identified several patterns from the attention heatmaps that differentiated the three representation methods. We describe two here and provide the entire set of attention matrices along with supplemental analysis on Github \footnote{\url{https://kearnsw.github.io/MEDIQA-2019/}}.
The coverage of entities and their associations was characteristic of BERT predictions (Figure \ref{fig:HeatmapTopBert}). BERT associated \textit{``spending time"} with \textit{``plans"} in addition to the lexical overlap of the word \textit{``family"} which is attended by each experimental condition in this example. All three embeddings identified the contradictory significance of the word \textit{``not"} in the hypothesis. However, BERT associated it with both spans \textit{``will be"} and \textit{``are coming"} in the premise, which led to the correct prediction. Cui2Vec over-attended the lexical match of the words \textit{``and"}, \textit{``to"} and \textit{``C0079382"}, which led to the wrong prediction.
The ESP model recognized hierarchical relationships between entities, e.g. \textit{``Advil"} and \textit{``NSAIDs"} (Figure \ref{fig:HeatmapTopESP}). In this example, the ESP approach attends to the daily use of \textit{``ASA"} (acetyl-salicylic acid), i.e. aspirin, and the patient denying the use of \textit{``other NSAIDs"}. This pattern was recognized multiple times in our analysis and provides a strong example of how continuous representations of biomedical ontologies may be used to augment contextual representations.
\section{Limitations}
The results presented in this paper compare a single model for each representation method fine-tuned to the development set. However, it is well known that the weights of the same model may vary slightly between training runs. Therefore, a more comprehensive approach would be to present the average attention weights across multiple training runs and to examine the weights at each attention layer of the models which we leave for future work.
\section{Conclusion}
We have presented our analysis of representation methods on the MedNLI task as evaluated during the MEDIQA 2019 shared task. We found that BERT embeddings fine-tuned using PubMed and MIMIC-III outperformed both Cui2Vec and ESP methods. However, we found that ESP had the lowest variance and highest predictive certainty, which may be useful in determining a minimum threshold for clinical decision support systems. \textit{Disease} was the only hypothesis focus to show a significant positive relationship with embedding prediction accuracy. This association was present for BERT and Cui2Vec embeddings - but not ESP. Overall, contradiction was the easiest label to predict for all three embeddings, which may be the result of an annotation artifact where contradiction pairs had higher lexical overlap often differentiated by explicit negation. However, overfitting on the negation can lead to lower accuracy on other entailment labels. Further, our preliminary results indicate that recognition of hierarchical relationships is characteristic of ESP suggesting that they can be used to augment contextual embeddings which, in turn, would contribute lexical coverage including sub-word information. We propose combining these methods in future work.
\section*{Acknowledgments}
We would like to acknowledge Trevor Cohen for sharing the Embeddings of Semantic Predications used in this study. Author Jason A. Thomas' work was supported, in part, by the National Library of Medicine (NLM) Training Grant T15LM007442. This work was facilitated, in part, through the use of the advanced computational, storage, and networking infrastructure managed by the Research Computing Club at the University of Washington and funded by an STF award. \\
|
1,108,101,566,799 | arxiv | \section{Introduction}\label{SECintro}
In the braneworld models, the observable universe is a 3-brane (domain wall) to which the matter fields are confined, while the gravity field enters the extra spatial dimensions of size that could strongly exceed the Planck length scale \cite{Ark-Dim-Dva:1998:}. Gravity can be localized near the brane at low energies even with an infinite size extra dimension with warped spacetime satisfying the 5D Einstein equations containing negative cosmological constant \cite{Ran-Sun:1999:} and an arbitrary energy-momentum tensor allowed on the brane \cite{Shi-Mae-Sas:1999:}. The Randall-Sundrum model gives 4D Einstein gravity in low energies, while significant deviations from the Einstein gravity occur at very high energies, in the early universe and in vicinity of compact objects as black holes and neutron stars \cite{Maar:2004:}.
Although no exact solution of the full 5D Einstein equations is know recently, there is a variety of astrophysically plausible special solutions of the 4D effective Einstein equations constrained to the brane. Such solutions describe black holes with spherical symmetry \cite{Dad-Maar-Pap-Rez:2000:PHYSR4:} and axial symmetry \cite{Ali-Gum:2005:}, or compact objects that could represent neutron (quark) stars \cite{Ger-Maar:2001:}, but there is no guarantee that these solutions have a regular bulk spacetime. The black hole spacetimes are determined by geometry of the spherical Reissner-Nordstr\"{o}m (RN) and axial Kerr-Newman (KN) type where the electric charge squared is substituted by the braneworld ``tidal charge'' parameter representing the tidal (Weyl tensor) effects of the bulk space onto the 4D black hole structure. Astrophysically relevant properties of the braneworld black holes were studied in a series of papers devoted to both motion of matter in their vicinity \cite{Stu-Kot:2009:,Ali-Tal:2009:,Abdu-Ahme:2010:PHYSR4,Mam-Hak-Toj:2010:MPLA:,Mor-Ahme-Abdu-Mam:2010:ASS:} and optical phenomena \cite{Sche-Stu:2009:a,Sche-Stu:2009:b,Bin-Nun:2010:PHYSR4:a,Bin-Nun:2010:PHYSR4:b}.
In the Randall-Sundrum model, full 5D bulk spacetime solutions for relativistic compact stars and black holes have to be inevitably solved by numerical methods by properly imposing regularity conditions in the bulk. For uniform energy density stars such a solution was obtained and discussed by Wiseman, who demonstrated possible existence of very compact stars~\cite{Wis:2002:}. Using the numerical methods developed in~\cite{Wis:2002:,Wis:2003:}, some 5D black hole solutions were constructed for limited radii~\cite{Kud-Tan-Nak:2003:,Kud:2004:}. Very recently, a black hole 5D solution localized to a brane has been numerically constructed and existence of dynamically stable black holes was demonstrated for all radii~\cite{Fig-Wis:2011:}. Here, we restrict our attention to the solution of the effective Einstein equations that can be given in a simple, analytic form.
In the simple model of spherically symmetric stars with uniform energy density profile a variety of special solutions having asymptotically Schwarzschildian character and satisfying the braneworld boundary conditions were found for the effective Einstein equations~\cite{Ger-Maar:2001:}. The most popular is the one with external spacetime described by the Reissner-Nordstr\"{o}m geometry with the braneworld tidal charge parameter $b$ reflecting the tidal effects of the bulk and related to the energy density and brane tension. Usually, the braneworld tension is assumed positive while the related tidal charge has to be negative \cite{Ger-Maar:2001:}, but the negative tension and related positive tidal charge are not excluded \cite{Dad-Maar-Pap-Rez:2000:PHYSR4:}, so we consider here both positive and negative tidal charge of the compact object. Notice that in such a case, the exterior of a neutron star with positive charge can be described by both black-hole and naked-singularity types of the external RN spacetime (for details see \cite{Kot-Stu-Tor:2008:CLASQG:}). In the later case, it should be stressed that there is no naked singularity in the solution as this part of the solution is replaced by a star. Properties of the compact stars with RN external geometry were extensively studied both in the weak field limit \cite{Boh-Har-Lob:2008:CLAQG:,Boh-Ris-Har-Lob:2010:CLASQG} and strong field limit when some restrictions on the brane tension were implied from the data of kHz quasiperiodic oscillations (QPOs) observed in low mass X-ray binary systems containing a neutron (quark) star \cite{Kot-Stu-Tor:2008:CLASQG:} or for trapping of neutrinos in extremely compact stars allowing for existence of trapped null geodesics \cite{Stu-Hla-Urb:2011:}.
Here we focus to the tidal charge influence on the redshift of photons radiated from the compact star surface and neutrinos radiated from the whole compact star interior. For simplicity, we assume the photon and neutrino motion along radial null geodesics that are usually considered in estimating the parameters of neutron (quark) stars~\cite{Lat-Pra:2007:PhysRep:}. The neutron star is considered in the framework of the simple model with external spacetime described by the RN geometry --- then our results can be compared to a wide variety of recent studies related to this kind of the braneworld solutions. We also consider Extremely Compact Stars (ECS) containing trapped null geodesics \cite{Abr-Mil-Stu:1993:PHYSR4:,Stu:2000:ACTPS2:,Nil-Cla:2000:GRRelStarsPolyEOS:,Stu-Tor-Hle-Urb:2009:}. For negative tidal charges, their external spacetime is of the black-hole type with one unstable photon circular orbit and the ECS surface at radius $R$ has to be located under the photon circular orbit. For positive tidal charges, their external spacetime can be of both black-hole (with an unstable photon circular orbit) and naked-singularity type when two photon circular orbits (stable and unstable) can exist, or none such orbit exists. In the case of positive tidal charges, the ECS can exist with surface radius $R$ located both under and above the unstable photon circular orbit (which is located above the stable photon circular orbit in the naked-singularity type external spacetimes) and can exist even with external naked-singularity spacetimes allowing none photon circular orbit \cite{Stu-Hla-Urb:2011:}
Our paper is organized as follows. In Section~\ref{SECbrwns}, we summarize properties and matching conditions of the internal uniform energy density spacetime and the Reissner-Nordstr\"{o}m external spacetime. Null geodesics of both the internal and external spacetime are described in terms of properly given effective potential and the ECS are classified according to the properties of the trapping region of the null geodesics. In Section~\ref{SECredsh}, the surface redshift and the redshift profile through the internal spacetime are determined under assumption of purely radial motion, and their properties are discussed, especially with respect to the ECS and photon circular orbits of the external spacetime. In Section~\ref{SECconcl}, observational characteristic of the redshift profiles are introduced and possibilities to determine the surface radius and tidal charge are discussed. Finally, concluding remarks are presented. Throughout the paper, we shall use the high-energy units with $\hbar = c = k_\mathrm{B} = 1$, if not stated otherwise.
\section{Braneworld neutron stars}\label{SECbrwns}
We consider the braneworld model of neutron stars with spherical symmetry and uniform distribution of the energy density in their interior. The external spacetime is given by the Reissner-Nordstr\"{o}m geometry with the braneworld tidal charge representing the influence of the tidal effects of the bulk space \cite{Dad-Maar-Pap-Rez:2000:PHYSR4:}.
\subsection{Internal and external spacetime and matching conditions}
In the standard Schwarzschild coordinates and the high-energy units, the line element of the spherically symmetric spacetimes reads
\begin{equation}
\mathrm{d}s^2 = -A^{2}(r)\mathrm{d}t^2 + B^{2}(r)\mathrm{d}r^2 + r^{2}\mathrm{d}\Omega.
\end{equation}
The internal solution, matched to the external geometry at the surface of the star $r = R$, is characterized by the uniform energy density distribution --- $\varrho = \mathrm{const}$, and by the tension of the brane --- $\lambda$. We assume $\varrho > 0$ and both possibilities $\lambda > 0$, $\lambda < 0$ for the brane tension, although the positive value is more realistic \cite{Dad-Maar-Pap-Rez:2000:PHYSR4:}. The line element of the internal geometry is given by the metric coefficients $A^{-}(r)$, $B^{-}(r)$ that are determined by \cite{Ger-Maar:2001:}
\begin{equation}
A^{-}(r) = \frac{\Delta(R)}{\left(1 + p(r)/\varrho\right)}
\end{equation}
with the pressure radial profile given by
\begin{equation}
\frac{p(r)}{\varrho} = \frac{\left[\Delta(r) - \Delta(R)\right]\left(1 + \frac{\varrho}{\lambda}\right)}{3\Delta(R) - \Delta(r) + \left[3\Delta(R) - 2\Delta(r)\right]\left(\frac{\varrho}{\lambda}\right)}.
\end{equation}
and
\begin{equation}
\left(B^{-}(r)\right)^2 = \frac{1}{\Delta^{2}(r)} = \left[1 - \frac{2G M}{r}\left(\frac{r}{R}\right)^3 \left(1 + \frac{\varrho}{2\lambda}\right)\right]^{-1},
\end{equation}
where $M = \frac{4}{3}\pi\varrho R^3$. The maximum of the pressure profile is at $r = 0$, while there is $p(r)/\varrho = 0$ at $r = R$.
The reality condition on the metric coefficient $B^{-}(r)$ (taken at $r = R$) implies a relation between $\lambda$, $\varrho$ and $R$ that can be expressed in the form
\begin{equation}\label{EQRm2GM}
\frac{G M}{R - 2 G M} \geq \frac{\varrho}{\lambda}.
\end{equation}
Considering the restriction $R > 2 G M$ ($R < 2 G M$), we can see that the reality condition~(\ref{EQRm2GM}) is satisfied for all $\lambda < 0$ (forbidden for all $\lambda > 0$), while for positive tension $\lambda > 0$ (negative tension $\lambda < 0$), we obtain a limit on the positive (negative) tension given by~\cite{Ger-Maar:2001:}
\begin{equation}
\lambda \geq \left(\frac{R - 2 G M}{G M}\right)\varrho .
\end{equation}
The line element of the external geometry is given by the metric coefficients $A^{+}(r)$, $B^{+}(r)$ that are determined by
\begin{equation}
\left( A^{+}(r)\right)^{2} = \left( B^{+}(r)\right)^{-2} = 1 - \frac{2G\mathcal{M}}{r} + \frac{q}{r^2},
\end{equation}
where, due to the matching conditions on the neutron star surface, i.e., $A^{-}(R) = A^{+}(R)$, $B^{-}(R) = B^{+}(R)$, the external mass parameter $\mathcal{M}$ and external tidal charge parameter $q$ are related to the internal geometry parameters by $\varrho$, $\lambda$ (and $M$, $R$) by the relations \cite{Ger-Maar:2001:}
\begin{equation}
q = -3 G M R\frac{\varrho}{\lambda}
\end{equation}
\begin{equation}
\mathcal{M} = M \left(1 - \frac{\varrho}{\lambda}\right).
\end{equation}
For $\lambda > 0$, the tidal charge $q < 0$ and $\mathcal{M} < M$, while for $\lambda < 0$, there is $q > 0$ and $\mathcal{M} > M$. Notice that for $\lambda > 0$, the condition $\varrho < \lambda$ has to be satisfied in order to have $\mathcal{M} > 0$.
For our purposes, it is convenient to express the internal spacetime using the parameters of the external spacetime that can be directly determined from observations of accretion and optical phenomena in vicinity of the neutron (quark) stars. Since the matching conditions imply the relations
\begin{equation}
q = - \frac{3 G M R \varrho}{\lambda} = \frac{3 G \mathcal{M} R}{\left(1 - \frac{\lambda}{\varrho}\right)},
\end{equation}
\begin{equation}
\frac{\lambda}{\varrho} = 1 - \frac{3 G \mathcal{M} R}{q},
\end{equation}
the internal metric can be expressed in terms of the gravitational radius $r_\mathrm{g}$ and dimensionless braneworld tidal charge $b$
\begin{equation}
r_\mathrm{g} \equiv G \mathcal{M},\qquad b \equiv \frac{q}{r_\mathrm{g}^{2}},
\end{equation}
in the form
\begin{equation}
A^{-} (r) = \frac{3\Delta(R)\left[\Delta(R) \left(2b - 3 R/r_\mathrm{g}\right) + \Delta(r)\left(R/r_\mathrm{g} - b\right)\right]}{2\Delta(R)\left(2b - 3R/r_\mathrm{g}\right)-\Delta(r)b},
\end{equation}
\begin{equation}
\left(B^{-}(r)\right)^2 = \Delta^{-2}(r) = \left[1 - \frac{r_\mathrm{g}}{r}\left(\frac{r}{R}\right)^3 \left(2 - b\frac{r_\mathrm{g}}{R}\right)\right]^{-1},
\end{equation}
\begin{equation}
\Delta^2 (R) = 1 - 2 \frac{r_\mathrm{g}}{R} + b \left(\frac{r_\mathrm{g}}{R}\right)^{2}.
\end{equation}
In the following, we shall use the dimensionless units ($r_\mathrm{g} = 1$). Then pressure function reads
\begin{equation}
\frac{p}{\varrho}\left(r, R, b\right) = \frac{\left(\Delta(r) - \Delta(R)\right) \left(2b - 3R\right)}{3 \Delta(R) \left(2b - 3R\right) - 3\Delta(r) \left(b - R\right)}.
\end{equation}
The pressure increases monotonously with radius decreasing and its central value is given by
\begin{equation}
\frac{p}{\varrho}\left(r = 0, R, b\right) = \frac{3R - 2b}{3R \left(\sqrt{b + (R - 2)R} + R - 3\right) + 6b}.
\end{equation}
\subsection{Limits on existence of uniform density stars}
The limit on the existence of the uniform density spherical configuration is related to compactness of the star and is given by the condition of pressure finiteness in the center. The limit on compactness of the star expressed in terms of the gravitational mass related to the internal spacetime reads~\cite{Ger-Maar:2001:}
\begin{equation}\label{EQGMR}
\frac{G M}{R}\leq \frac{4}{9}\left[\frac{1 + \frac{5}{4}\frac{\varrho}{\lambda}}{\left(1 + \frac{\varrho}{\lambda}\right)^2}\right].
\end{equation}
In terms of the external gravitational mass parameter $\mathcal{M}$, the compactness limit~(\ref{EQGMR}) is transformed to the form
\begin{equation}
\frac{G \mathcal{M}}{R}\leq \frac{4}{9}\left[\frac{\left(1 + \frac{5}{4}\frac{\varrho}{\lambda}\right)\left(1 - \frac{\varrho}{\lambda}\right)}{\left(1 + \frac{\varrho}{\lambda}\right)^2}\right].
\end{equation}
\begin{figure}[t]
\centering\includegraphics[width=0.8\hsize,keepaspectratio=true]{figure1.pdf}
\caption{Classification of the compact stars. For the extremely compact stars (ECS), the parameter space $(b-R)$ is separated into four Zones~I--IV; their properties are given in the text.}\label{figure1}
\end{figure}
The reality condition of the central pressure reads $b + (R - 2) R\geq 0$ being equivalent to the condition $\Delta(R)\in \mathbb{R}$.
The pressure in the center of the star must be finite and positive implying the limits $R_\mathrm{min}(b)$ on the existence of uniform density stars. The central pressure $p(r = 0, R, b)$ diverges when the surface radius $R$ satisfies the condition
\begin{equation}
3R \left(\sqrt{b + (R - 2) R} + R - 3\right) + 6b = 0
\end{equation}
that leads to a cubic equation relative to $R$ giving one solution that is relevant for $b\leq 1$ and reads
\begin{eqnarray}
R & = & \frac{1}{4} \left(\frac{(b - 9) (b - 1)}{\sqrt[3]{8 \sqrt{(b - 1)^2 b^3}+b \left[b (b + 17) - 45\right] + 27}}\right. + \qquad \qquad \qquad \nonumber \\
& &\qquad \qquad \qquad \left.+ \sqrt[3]{8 \sqrt{(b - 1)^2 b^3} + b \left[b (b + 17) - 45\right] + 27} + b + 3\right).
\end{eqnarray}
The solution is depicted on Figure~\ref{figure1} as the part of $R_\mathrm{min}(b)$ denoted by $p \rightarrow \infty$. For $b = 0$, we arrive at the standard condition $R > \frac{9}{4}$ (see~\cite{Stu-Tor-Hle-Urb:2009:}). For $b > 1$ the relevant limit is given by the condition of positiveness of the central pressure. It reads
\begin{equation}
R_\mathrm{min}(b) = \frac{2}{3}b
\end{equation}
and is denoted by $p \rightarrow 0$. For smaller $R$ the central pressure is not positive.
Note that for $3/4 < b < 1$ there is a region of surface radii given by the condition $2b/3 < R < 1 - \sqrt{1 - b}$ where $p(r = 0)$ is positive. However, the star surface is located under the inner horizon of the external spacetime that belongs to the black-hole type RN spacetimes. Such a configuration has to be hidden under the inner horizon of the external RN spacetime and is irrelevant for our considerations.
\subsection{Extremely compact stars}\label{SECecs}
The extremely compact stars (neutron, quark, or hybrid) are defined as compact stars allowing for existence of trapped null geodesics in their interior \cite{Stu-Tor-Hle-Urb:2009:,Stu-Hla-Urb:2011:}. In the case of internal uniform density Schwarzschild spacetimes ($b = 0$), such objects appear just when the surface of the compact star is located under the photon circular geodesic of the external vacuum Schwarzschild spacetime located at $r_\mathrm{ph} = 3r_\mathrm{g}$ \cite{Stu-Tor-Hle-Urb:2009:}. In the braneworld uniform density compact stars this simple rule does not hold and the situation is more complex due to different character of the external spacetime and its relation to the internal spacetime caused by the bulk space tidal effect on the matching of the spacetimes reflected by a non-standard way of the relations of the effective potential of the null geodesics in the internal and external spacetimes of extreme compact stars \cite{Stu-Hla-Urb:2011:}.
The motion along null-geodesics is independent of energy (frequency) and can conveniently be described in terms of the impact parameter
\begin{equation}
\ell = \frac{L}{E},
\end{equation}
where $E$ and $L$ are the constants of motion related to the Killing vector fields of the internal and external spacetimes.
The relevant equation governing the radial motion along null geodesics takes the form~\cite{Stu-Tor-Hle-Urb:2009:}
\begin{equation}
(p^{r})^{2} = A^{-2}(r)B^{-2}(r)E^{2} \left(1 - A^{2}(r)\frac{\ell^{2}}{r^{2}}\right).
\end{equation}
The energy $E$ is irrelevant and can be used for rescalling of the affine parameter $\lambda$. The radial motion is restricted by an effective potential related to the impact parameter $\ell$, defined by the relations
\begin{equation}
\ell^{2} \leq V{}_{\mathrm{eff}} =\left\{
\begin{array}{lll}
V{}_{\mathrm{eff}}^{\mathrm{int}} = \displaystyle\frac{r^2}{(A^{-}(r))^2} & \quad\mbox{for} & r\leq R,\\
V{}_{\mathrm{eff}}^{\mathrm{ext}} = \displaystyle\frac{r^2}{(A^{+}(r))^2} & \quad\mbox{for} & r > R.
\end{array}\right.
\end{equation}
$V{}_{\mathrm{eff}}^{\mathrm{int}}$ is the effective potential of the null-geodetical motion in the internal braneworld spacetime and $V{}_{\mathrm{eff}}^{\mathrm{ext}}$ is the effective potential of the null-geodetical motion in the external, vacuum RN spacetime. Due to the bulk space tidal effects on the matching of the internal and external spacetimes (see \cite{Ger-Maar:2001:} for details) we obtain a non-standard variety of relations of the effective potentials in the ECS interior and exterior.
Using the dimensionless radial coordinate expressed in terms of the gravitational radius ($r/G\mathcal{M} \rightarrow r$), and dimensionless tidal charge $b$, we obtain the relations
\begin{equation}
V{}_{\mathrm{eff}}^{\mathrm{int}} = \frac{r^2 R^2 \left[b Y - 2R (2b - 3R) Z\right]^2}{9 \left[b + (R - 2) R\right] \left[(R - b) + R (2b -3R) Z\right]^2},
\end{equation}
\begin{equation}
V{}_{\mathrm{eff}}^{\mathrm{ext}} = \frac{r^4}{b + (r - 2) r},
\end{equation}
where
\begin{eqnarray}
Y &\equiv & R^2 \Delta(r) = \sqrt{r^2 (b - 2R) + R^4},\\
Z &\equiv & R \Delta(R) = \sqrt{b + (R - 2) R}.
\end{eqnarray}
Circular null geodesics, located at $r_\mathrm{c(i)}$ and $r_\mathrm{c(e)}$, respectively, are given by the local extrema of the effective potential ($\partial V_\mathrm{eff}/\partial r = 0$). In the internal spacetime we have to solve a nontrivial equation~\cite{Stu-Hla-Urb:2011:}:
\begin{equation}
\begin{array}{l}
2 r R^2 \left[b (Y - 4 R Z) + 6 R^2 Z\right] \left\{-b^3 \left[r^2 (Y - 4 R Z) + 8 R^2 Y\right] + b^2 R \left[r^2 (3 Y - 14 R Z) + R^2 \times \right.\right. \\
\left.\times \left(6 R^2 Z - 9 R Y + 40 Y\right)\right] + b R^2 \left[R^2 \left(-13 R^2 Z + 25 R Y - 66 Y\right) - 2 r^2 (Y - 6 R Z) \right] + 6 R^5 \times \\
\left.\left. \times \left(R^2 Z - 3 R Y + 6 Y\right)\right\}\right]
\left[9 \left\{b + (R - 2) R\right\} Y \left\{R (Y - 3 R Z) - b \left(Y - 2 R Z\right)\right\}^3\right]^{-1} = 0.
\end{array}
\end{equation}
This equation can be solved by using numerical methods and determines the loci $r_\mathrm{c(i)}(R, b)$ of the internal circular null geodesics that are stable and correspond to a local maximum of the internal effective potential (see~\cite{Stu-Hla-Urb:2011:} for details). The existence of the internal stable circular null geodesics is allowed only for the surface radius $R$ limited by values explicitly given by
\begin{equation}
R_\mathrm{I\pm} = \frac{3}{2} \pm \sqrt{\frac{9}{4} - \frac{5}{3}b}
\end{equation}
that has to satisfy naturally also the condition $R > R_\mathrm{min}(b)$.
In the external spacetime, the photon circular geodesics and the related local extrema of the effective potential of the Reissner-Nordstr\"{o}m (RN) type, with the tidal charge substituting the electric charge squared appearing in the standard RN spacetimes, are given in the dimensionless units by the condition
\begin{equation}
r^2 - 3r + 2b = 0.
\end{equation}
For negative tidal charges ($b < 0$) external spacetimes of the black-hole type are allowed only~\cite{Kot-Stu-Tor:2008:CLASQG:}.
The radii of the photon circular orbits are given by~\cite{Bal-Bic-Stu:1989:,Stu-Hle:2002:}
\begin{equation}
r_{\mathrm{ph}\pm} = \frac{3}{2}\left(1 \pm \sqrt{1 - \frac{8}{9}b}\right). \label{EQph}
\end{equation}
However, only the outer photon circular geodesic at $r_\mathrm{ph+}$ is physically relevant and we see immediately that for $b < 0$ there is $r_{\mathrm{ph}}>3$.
\begin{figure}[t]%
\begin{minipage}[b]{.499\hsize}
\centering\includegraphics[width=\hsize,keepaspectratio=true]{figure2a.pdf}
\centering\includegraphics[width=\hsize,keepaspectratio=true]{figure2c.pdf}
\end{minipage}\hfill%
\begin{minipage}[b]{.499\hsize}
\centering\includegraphics[width=\hsize,keepaspectratio=true]{figure2b.pdf}
\centering\includegraphics[width=\hsize,keepaspectratio=true]{figure2d.pdf}
\end{minipage}
\caption{The effective potential $V_{\mathrm{eff}}$ of both internal (dashed lines) and external spacetimes (full lines) given for the black-hole type ($b < 1$) and naked-singularity ($b > 1$) type of the external spacetime. All characteristic cases of its behaviour are presented for both ECS and non-extreme compact stars.}\label{figure2}
\end{figure}
For braneworld neutron stars with a positive tidal charge ($b > 0$) both black-hole and naked-singularity RN spacetimes are relevant \cite{Kot-Stu-Tor:2008:CLASQG:}. In the black-hole type spacetimes ($b \leq 1$), the photon circular orbits are given by the relation~\eqref{EQph}, but only the outer solution corresponding to an unstable orbit can be astrophysically relevant since the surface of the compact star has to be located above radius of the RN outer horizon. In the naked-singularity spacetimes ($b > 1$) two photon circular orbits can be astrophysically relevant when $b < 9/8$, the inner one being stable, the outer one --- unstable. In the spacetimes with $b > 9/8$, no photon circular orbits can exist. Therefore, in the braneworld spacetimes, the existence of ECS is governed by the interplay of the behaviour of the internal and external effective potential of the motion and will be determined and classified in the following way.
When we take into account the (non-)existence of the local extrema of the internal/external part of the effective potential $V_\mathrm{eff}$, we obtain the whole region of existence of ECS.
The function $R{}_\mathrm{max}^\mathrm{ECS}(b)$ is determined by the relation
\begin{equation}
R{}_\mathrm{max}^{\mathrm{ECS}} =\left\{
\begin{array}{lll}
r_\mathrm{ph} \equiv \frac{1}{2} \left(3 + \sqrt{9 - 8 b}\right) & \quad\mbox{for} & b \leq 0,\\
R_\mathrm{I+} \equiv \frac{3}{2} + \sqrt{\frac{9}{4} - \frac{5}{3}b} & \quad\mbox{for} & 0 < b < 27/20,
\end{array}
\right.
\end{equation}
while for $R{}_\mathrm{min}^\mathrm{ECS}(b)$ we have
\begin{equation}
R{}_\mathrm{min}^{\mathrm{ECS}} = \left\{
\begin{array}{lll}
R_\mathrm{min} & \quad\mbox{for} & b \leq 1,\\
R_\mathrm{I-} \equiv \frac{3}{2} - \sqrt{\frac{9}{4} - \frac{5}{3}b} & \quad\mbox{for}& 1 < b < 27/20.
\end{array}
\right.
\end{equation}
The region of ECS in the parameter space is represented in Figure~\ref{figure1}.
Typical behaviour of the effective potential of the null-geodetical motion $V{}_{\mathrm{eff}}$ is demonstrated in Figure~\ref{figure2} for appropriately chosen values of the tidal charge $b$. Here we express the radii in units of the gravitational radius $r_\mathrm{g} = G \mathcal{M}$. The selection of the tidal charge values used in Figure~\ref{figure2} demonstrates the full classification of the behaviour of the effective potential in both internal and external spacetimes. When the effective potential in the internal or external spacetime (or in both of them) has a local extreme corresponding to a photon circular geodesic, trapped null geodesics can appear if the surface radius $R$ is properly chosen giving an ECS. On the other hand, we find an ordinary compact star if the surface is chosen in such a way that no local extrema of the effective potentials exist (see Figure~\ref{figure2}). Separation of the zones of compact stars of different character, both extremely compact and ordinary compact, in the parameter space $(b-R)$, is determined by the functions $R_\mathrm{min}(b)$, $R_\mathrm{max}(b)$ and $r_\mathrm{ph}(b)$ where the last function governs radius of the photon circular geodesics in the external spacetime (both of them for $1 < b < 9/8$). The subdivision of the Zone IV in the tidal charge range $1 < b < 9/8$ is given by the relation of the magnitude of the effective potential at the surface of the ECS and the magnitude of the external effective potential at its local minimum, and is determined numerically.
The parameter space $R-b$ is divided into regions corresponding to the existence of four different zones of ECS. The classification is based on the existence of local maxima/minima of $V{}^{\mathrm{int/ext}}_{\mathrm{eff}}$ in the following way (see Figures~\ref{figure1} and \ref{figure2}):
\begin{itemize}
\item[$\bullet$]{\emph{Zone I} there exist maximum of $V{}^{\mathrm{int}}_{\mathrm{eff}}$ and minimum of $V{}^{\mathrm{ext}}_{\mathrm{eff}}$, both located at $r \neq R$, $\mathrm{min} V{}^{\mathrm{ext}}_{\mathrm{eff}} < V{}^{\mathrm{ext}}_{\mathrm{eff}}(r = R)$, and there is no local maximum of $V{}^{\mathrm{ext}}_{\mathrm{eff}}$ at $r > R$;}
\item[$\bullet$]{\emph{Zone II} there exist maximum of $V{}^{\mathrm{int}}_{\mathrm{eff}}$ at $r = R$ and minimum of $V{}^{\mathrm{ext}}_{\mathrm{eff}}$ at $r \neq R$, $\mathrm{min} V{}^{\mathrm{ext}}_{\mathrm{eff}} < \mathrm{max} V{}^{\mathrm{int}}_{\mathrm{eff}}$, and there is no local maximum of $V{}^{\mathrm{ext}}_{\mathrm{eff}}$ at $r > R$;}
\item[$\bullet$]{\emph{Zone III} there exist maximum of $V{}^{\mathrm{int}}_{\mathrm{eff}}$ at $r < R$ and minimum of $V{}^{\mathrm{ext}}_{\mathrm{eff}}$ at $r = R$; there is no local maximum of $V{}^{\mathrm{ext}}_{\mathrm{eff}}$ at $r > R$;}
\item[$\bullet$]{\emph{Zone IV} there exist both the local maximum and local minimum of $V{}^{\mathrm{ext}}_{\mathrm{eff}}$ at $r > R$. The last zone can be divided into two parts according to the criterion~\cite{Stu-Hla-Urb:2011:}
\begin{itemize}
\item[a)]{minimum of $V{}^{\mathrm{ext}}_{\mathrm{eff}} < V{}^{\mathrm{int}}_{\mathrm{eff}}\left(r = R\right)$,}
\item[b)]{ minimum of $V{}^{\mathrm{ext}}_{\mathrm{eff}} > V{}^{\mathrm{int}}_{\mathrm{eff}}\left(r = R\right)$.}
\end{itemize}}
\end{itemize}
\section{Redshift of radiation from the braneworld RN compact stars}\label{SECredsh}
We can find interesting restrictions on the parameters of the braneworld compact stars considering redshift of radiation; electromagnetic radiation emitted from the surface of the compact stars, and neutrino radiation emitted from the interior of cooling compact stars, assuming compact stars cooled down enough to allow the approximation of the geodesic motion of neutrinos. Of course, we have to expect development of sophisticated observational techniques enabling measurement of the energy distribution in the case of neutrinos radiated by newly born neutron stars that cooled down sufficiently to allow for the neutrino free motion in the whole star interior.
The geodetical motion of photons in the external vacuum spacetime is fully justified, contrary to the neutrino motion in the neutron star interior. The approximation of free motion of neutrinos in the internal spacetime is correct when the mean free path of neutrinos $\lambda > R$. Neutrinos have inelastic scatter on electrons (muons) and elastic scatter on neutrons. The scatter cross section on electrons (neutrons) $\sigma_\mathrm e$ ($\sigma_\mathrm n$) gives the mean free path in the form $\lambda = (\sigma_i n_i)^{-1}$ where $n_i$ $(i\in \{\mathrm{e, n}\})$ denotes the number density of electrons (neutrons). It was shown \cite{Sha-Teu:1983:BHWDNS:} that
\begin{equation}
\lambda_\mathrm{e}\sim 9\times 10^7\left(\frac{\rho_\mathrm{nucl}}{\rho} \right)^{4/3}\left(\frac{100\mathrm{keV}}{E_\nu}\right)^{3}~\mathrm{km},
\end{equation}
while
\begin{equation}
\lambda_\mathrm{n}\sim 300\frac{\rho_\mathrm{nucl}}{\rho} \left(\frac{100\mathrm{keV}}{E_\nu}\right)^{2}~\mathrm{km}.
\end{equation}
There is $\lambda_\mathrm e \gtrsim 10$~km for $E_\nu \lesssim 20$~MeV and $\lambda_\mathrm n \gtrsim 10$~km for $E_\nu \lesssim 500$~keV. Therefore, in a few hours old neutron star, see \cite{Lat-Pra:2007:PhysRep:,Sha-Teu:1983:BHWDNS:,Gle:2000:CompactStars:,Web:1999:Pul:}, at temperatures $T \lesssim 10^9$~K ($E_\nu \sim 100$~keV), the neutrino motion could be considered geodetical through whole the internal spacetime. Of course, our results can be applied at any time of the neutrino radiation as they give the upper limit on the observed redshift of neutrinos.
It is usual to make estimates on the redshift of radiation of the surface and interior of the compact stars using the simplest possibility of the purely radial motion of photons and neutrinos. The redshift factor related to the neutron star interior or surface is given (under the assumption of a static source and radially emitted photons and neutrinos) by the standard formula $\mathrm{d}\tau^2 = -g_{tt} \mathrm{d}t^2$ relating proper time of the source ($\tau$) and the observer ($t$) that implies
\begin{equation}
\frac{\nu_\mathrm{e}}{\nu_\mathrm{o}} = \frac{1}{\sqrt{-g_{tt}}} = 1 + z.
\end{equation}
In Figure~\ref{figure3} we give an illustration of the redshift profile in dependence on the tidal charge $b$ and the surface radius $R$ in the interval corresponding to the lower limit of the radius of extremely compact stars $R_\mathrm{min}$ when the redshift diverges at the centre and some relatively large and astrophysically acceptable radius $R = 4.2$ representing relatively flat profile related to realistic compact stars.
We first study in detail the redshift of photons radiated from the surface of the braneworld compact stars.
\begin{figure}[t
\centering\includegraphics[width=0.6\hsize]{figure3.pdf}
\caption{The dependence of $1 + z$ on the brane parameter $b \in \{-1, -0.5, 0, 0.5, 1)$ and $r$ is demonstrated for several values of $R$. Curves for $b = -1$ are plotted thick. For $R \rightarrow R_\mathrm{min}$ the central redshifts diverge; this effect is amplified with $b$ getting closer to $1$, even so much, that there is no curve on the plot for case $b \rightarrow 1$, $R \rightarrow R_\mathrm{min}(b \rightarrow 1)$.}\label{figure3}
\end{figure}
\subsection{Surface redshift}
The redshift factor related to the neutron star surface implies
\begin{equation}
\frac{\nu_e}{\nu_o} = (1 + z)(r = R) = \frac{1}{\sqrt{-g_{tt}(r = R)}} .
\end{equation}
\begin{figure}[t]%
\begin{center}
\includegraphics[width=.499\hsize]{figure4a.pdf}\hfill\includegraphics[width=.499\hsize]{figure4b.pdf}
\end{center}
\centering\includegraphics[width=.499\hsize]{figure4c.pdf}
\caption{Surface redshifts of the compact stars given for typical values of the tidal charge --- for negative ($b < 0$) and positive ($b > 0$) tidal charge spacetimes of the black-hole type and the $b > 1$ case of naked-singularity type external spacetime.}\label{figure4}
\end{figure}
The external geometry at the surface radius
\begin{equation}
-g_{tt}(r = R) = \left[1 - 2\frac{r_\mathrm{g}}{R} + b\left(\frac{r_\mathrm{g}}{R}\right)^2\right].
\end{equation}
Using the dimensionless units $r_\mathrm{g} = 1$ here and in the following, the surface redshift factor reads
\begin{equation}
1 + z = \left[1 - \frac{2}{R} + \frac{b}{R^2}\right]^{-1/2},
\end{equation}
where the condition $R > R_\mathrm{min}(b)$ must be satisfied. Behaviour of the surface redshift is illustrated in Figure~\ref{figure3}, where the loci $R = r_\mathrm{ph}$ are depicted by circles. The surface redshift is given in Figure~\ref{figure4} for negative and positive tidal charges with surface radius going down to $R = R_\mathrm{min}$. For the surface located at the unstable photon circular orbit ($R = r_\mathrm{ph+}$) the redshift slightly decreases (increases) with descending $b < 0$ (increasing $b > 0$). For $R \rightarrow R_\mathrm{min}(b)$, the surface redshift slightly decreases with decreasing $b < 0$, but it substantially increases with increasing $b > 0$ and diverges for $b \rightarrow 1$.
For positive tidal charges $b > 1$ corresponding to naked-singularity spacetimes the redshift radial profile reaches a maximum at $R = b$ where
\begin{equation}
(1 + z)_\mathrm{max}(r = R = b) = \sqrt{\frac{b}{b - 1}};
\end{equation}
its position is depicted in Figure~\ref{figure4} by black points. For $b\leq 9/8$ we depict also the radius corresponding to photon circular orbits.
It is important to determine the redshift of the braneworld neutron stars with surface located at the radii corresponding to the photon circular geodesics of the external spacetime --- in the case of the standard internal Schwarzschild spacetimes with $b = 0$ the case of limiting (maximally extended) extremely compact objects coincides with their surface being located at $R = r_{\mathrm{ph}} = 3$. Then we find the well known value of the surface redshift
\begin{equation}
(1 + z)(r = R = r_{\mathrm{ph+}}, b=0) = \left(1 - \frac{2}{3}\right)^{-1/2} = \sqrt{3}.
\end{equation}
\begin{figure}[t]%
\centering\includegraphics[width=0.7\hsize]{figure5.pdf}
\caption{The dependence of surface redshift and central redshift on the brane parameter $b\in(-1.2,27/20)$ for ECS.}\label{figure5}
\end{figure}
On the other hand, in the braneworld case with $b \neq 0$ we have to distinguish the black-hole type external spacetimes with one photon circular orbit, and the naked-singularity type spacetimes when two photon circular orbits exist --- the internal one being stable, the external one being unstable. It is important to distinguish these cases because of the qualitatively different character of optical phenomena at the external field with strong influence on the accretion discs \cite{Stu-Schee:2010:CLAQG:,Stu-Hle:2000:} that could be astrophysically very important and could provide an independent evidence on the tidal charge presence. The surface redshift at $R = r_\mathrm{ph}$ in the black-hole type spacetimes is given by the relation
\begin{equation}
(1 + z)(r = R = r_\mathrm{ph}, b) = 2 \sqrt{\frac{2b}{4b - 3 + \sqrt{9 - 8b}}}.
\end{equation}
This dependence is illustrated in Figure~\ref{figure4}.
In the naked-singularity-type spacetimes admitting existence of the photon circular orbits, i.e., for $1 < b < 9/8$, the surface redshift is determined by
\begin{equation}
(1 + z)(r = R = r_\mathrm{ph\pm}, b) = 2\, \sqrt{\frac{2b}{4 b\pm\sqrt{9 - 8 b} - 3}},
\end{equation}
and for both circular orbits is given in Figure~\ref{figure4}. For $b > 9/8$ no photon circular orbits exist.
\begin{figure}[t]%
\centering\includegraphics[width=0.7\hsize]{figure6.pdf}
\caption{The figure illustrates the span of $1 + z$ for the interior of the star with $R = r_\mathrm{ph}$. }\label{figure6}
\end{figure}
For the limiting, minimally and maximally extended extremely compact braneworld spacetimes the value of surface redshift is varying in the range starting from $(1+z){}_\mathrm{min}^{\mathrm{ECS}}$ and finishing at $(1+z){}_\mathrm{max}^{\mathrm{ECS}}$, determined by relations
\begin{equation}
(1 + z){}_\mathrm{min}^{\mathrm{ECS}} =\left\{
\begin{array}{lll}
(1+z)(r = R = r_\mathrm{ph+}, b) & \quad\mbox{for} & b \leq 0,\\
(1+z)(r = R = R_\mathrm{I+}, b) & \quad\mbox{for} & 0 < b < 27/20,
\end{array}
\right.
\end{equation}
\begin{equation}
(1+z){}_\mathrm{max}^{\mathrm{ECS}} =\left\{
\begin{array}{lll}
(1+z)(r = R = R_\mathrm{min}, b) = \Delta(R_\mathrm{min})^{-1} & \quad\mbox{for} & b \leq 1,\\
(1+z)(r = R = R_\mathrm{I-}, b) & \quad\mbox{for} & 1 < b < 27/20,
\end{array}
\right.
\end{equation}
where
\begin{equation}
(1+z)(r = R = R_\mathrm{I\pm}, b) = 5\, \sqrt{\frac{2b}{20 b\pm \sqrt{81 - 60 b} - 9}}.
\end{equation}
We give the dependencies of the surface redshift limiting values on the braneworld tidal charge parameter $b$ in Figures~\ref{figure4} and \ref{figure5}. For special cases, when the compact star surface is located at the radii corresponding to photon circular orbits of the external spacetime, the dependence of surface redshift on tidal charge is illustrated in Figure~\ref{figure6}.
\subsection{Redshift factor of neutrinos radiated at the interior of the star}
In principle, distribution of the internal redshift factor enables to estimate the internal structure of the neutron star if the energy (and frequency shift) of outcoming neutrinos can be measured precisely enough.
The redshift factor gives frequency shift of the neutrinos radiated by static sources inside the star. We assume radially emitted neutrinos --- this implies some restrictions on the observations but we shall not consider here the effect of bending of null geodesics since it is not too relevant for the static sources in the compact star interior. Of course, in the case of the source at the center of the star, only radially emitted neutrinos are, in fact, allowed, putting thus a natural restriction on the observed range of the redshift.
The general redshift formula for static sources and purely radial emission reads
\begin{equation}
1 + z = \frac{\nu_e}{\nu_o} = \left[-g_{tt}(r)\right]^{-1/2} = \left[A^{-}(r)\right]^{-1} = \frac{1 + p(r)/\varrho}{\Delta(R)}.
\end{equation}
We can see immediately that $(1+z) \rightarrow \infty$ if $p(r) \rightarrow \infty$. This occurs for all $R \rightarrow R_\mathrm{min}(b < 1)$. On the other hand, for fixed parameters $R$ and $b$, the behaviour of $1+z$ is simply given by the behaviour of the pressure function $p(r)$ that can be studied from its partial derivative
\begin{equation}
\frac{\partial p(r)/\varrho}{\partial r}=\frac{r R Z (2 b - 3 R) (b - 2 R)^2}{3 Y \left[R (Y - 3 R Z) - b (Y - 2 R Z)\right]^2}.
\end{equation}
Because $r > 0$ and $R > 2b/3$ for all considered braneworld compact stars (see Figure~\ref{figure1}), there is $\partial p(r)/\partial r < 0$, and we can conclude that $(1+z)(r,R,b)$ is (for fixed $R$ and $b$) monotonously decreasing function of increasing $r$; therefore, the tidal effects does not change this standard and intuitively expected behaviour of the pressure (and redshift) profile.
Using the internal metric coefficients expressed in terms of the external metric parameters, we arrive at the redshift expressed in the form
\begin{equation}
(1 + z)(r, R, b) = \frac{2\Delta(R)\left(2b-3R\right)-\Delta(r)b}{3\Delta(R)\left[\Delta(R) \left(2b - 3 R\right)+\Delta(r)\left(R-b\right)\right]}.
\end{equation}
For a fixed $b > 1$ the function $(1 + z)(r = 0, R, b)$ has a maximum occurring for a specific value of surface radius $R$. Its value can be found numerically by solving equation
\begin{equation}
\begin{array}{l}
8 b^4 + b^3 R \left[9 R - 4 \left(\sqrt{b + (R - 2) R} + 12\right)\right] + b^2 R^2 \left(9 \sqrt{b + (R - 2) R} - 33 R + 106\right) + b R^3 \times\\
\times \left[R \left(-\sqrt{b + (R - 2) R} + R + 41\right) - 2 \left(2 \sqrt{b + (R - 2) R} + 51\right)\right] - 18 (R - 2) R^4 = 0.\qquad
\end{array}
\end{equation}
Solution is given in Figure~\ref{figure1} as the curve $R{}_\mathrm{max}^{\mathrm{RS}|r=0}$.
\begin{figure}[t]%
\begin{center}
\includegraphics[width=.499\hsize]{figure8a.pdf}\hfill\includegraphics[width=.499\hsize]{figure8b.pdf}
\end{center}
\centering\includegraphics[width=.499\hsize]{figure8c.pdf}
\caption{Plot of the redshifts $1 + z$ from the center of the star for several values of $b$.}\label{figure8}
\end{figure}
For the surface redshift at $r = R$, where $p(R) = 0$, we obtain the first limiting value of the redshift to be
\begin{equation}
(1 + z)(r = R, R, b) = \Delta(R)^{-1} = \left[1 - 2\frac{1}{R} + b\left(\frac{1}{R}\right)^2\right]^{-1/2}.
\end{equation}
For the fixed $R$ and $b$, the second limiting value of the redshift is given by $(1 + z)(r = 0, R, b)$ i.e., by its value at the center of the star. Since $\Delta(r = 0) = 1$, we find
\begin{equation}
(1 + z)(r = 0, R, b) = \frac{2\Delta(R)(3R - 2b) + b}{3\Delta(R)\left[\Delta(R)(3R - 2b) + (b - R)\right]}.
\end{equation}
In the standard internal Schwarzschild spacetime ($b = 0$), the central redshift reads
\begin{equation}
(1 + z)(r = 0, R, b = 0) = \frac{2}{3 \sqrt{1 - 2/R} - 1}
\end{equation}
and for the maximally extended ECS there is
\begin{equation}
(1 + z)(r = 0, R = 3, b = 0) = 1 + \sqrt{3}.
\end{equation}
Because for given $b$, $R$ the pressure function $p(r)$ is monotonously decreasing function of increasing $r$, the values of redshifts are varying between those two limiting values. Behaviour of the central pressure is demonstrated in Figure~\ref{figure8}. In the case of compact stars with external spacetime of the black-hole type ($b < 1$), the central redshift diverges for $R \rightarrow R_\mathrm{min}(b)$. On the other hand, in the case of external spacetime of the naked-singularity type ($b > 1$), the central pressure remains finite for $R \rightarrow R_\mathrm{min}(b)$; in fact it reaches a maximum value for $R > R_\mathrm{min}(b)$. This special behaviour is also reflected by redshift profile given for fixed representative values of $b$ in Figure~\ref{figure7}.
\begin{figure}[t]%
\begin{minipage}[b]{.445\hsize}
\centering\includegraphics[width=\hsize,keepaspectratio=true]{figure7a.pdf}
\centering\includegraphics[width=\hsize,keepaspectratio=true]{figure7c.pdf}
\end{minipage}\hfill%
\begin{minipage}[b]{.445\hsize}
\centering\includegraphics[width=\hsize,keepaspectratio=true]{figure7b.pdf}
\centering\includegraphics[width=\hsize,keepaspectratio=true]{figure7d.pdf}
\end{minipage}
\centering\includegraphics[width=.445\hsize,keepaspectratio=true]{figure7e.pdf}
\caption{Contour plots of the redshift profiles $(1 + z)(r/R, R, b)$ for selected representative values of $b$. The area where surface extend beyond the plotting range is colored white. Notice the qualitative different character of the redshift profiles for compact stars with $b>1$.}\label{figure7}
\end{figure}
\afterpage{\clearpage}
It is convenient to find the redshift formula for compact stars with some special values of the surface radius related to the existence of extremely compact stars (when special phenomenon of neutrino trapping inside the star is important~\cite{Stu-Hla-Urb:2011:} and the photon circular orbits that can be relevant for accretion and optical phenomena outside the compact star).
The central redshifts when the surface radius $R$ is located at the photon circular orbits of the naked-singularity spacetime (with $b > 1$) takes the form
\begin{equation}
(1 + z)(r = 0, R = r_\mathrm{ph+}, b) = \frac{\left(-4 b + \sqrt{9 - 8 b} + 3\right) \left(3 \sqrt{2} \sqrt{\frac{4 b + \sqrt{9 - 8 b} - 3}{b}} + 4\right)}{24 (1 - b)},
\end{equation}
\begin{equation}
(1 + z)(r = 0, R = r_\mathrm{ph-}, b) = \frac{4 \sqrt{2} \left(\sqrt{2} \sqrt{\frac{4 b - \sqrt{9 - 8 b} - 3}{b}} b - 8 b - 4 \sqrt{9 - 8 b} + 12\right)}{3 \left(\sqrt{2} \sqrt{\frac{4 b - \sqrt{9 - 8 b} - 3}{b}} - 4\right) \sqrt{\frac{4 b - \sqrt{9 - 8 b} - 3}{b}} \left(2 b + \sqrt{9 - 8 b} - 3\right)}.
\end{equation}
For the limiting, maximally (minimally) extended extremely compact braneworld spacetimes the value of the central and minimal (maximal) redshift is given by $(1 + z)_\mathrm{min}^{\mathrm{ECS}}$ ($(1 + z)_\mathrm{max}^{\mathrm{ECS}}$) that are determined by
\begin{equation}
(1 + z){}_\mathrm{min}^{\mathrm{ECS}} =\left\{
\begin{array}{lll}
(1 + z)(r =0, R = r_\mathrm{ph+}, b) & \quad\mbox{for} & b \leq 0,\\
(1 + z)(r =0, R = R_\mathrm{I+}, b) & \quad\mbox{for} & 0 < b < 27/20,
\end{array}
\right.
\end{equation}
where
\begin{equation}
(1 + z)(r = 0, R = R_\mathrm{I+}, b) = \frac{-\sqrt{6} \left(-4 b + \sqrt{81 - 60 b} + 9\right) - \frac{\left(\sqrt{81 - 60 b} + 9\right) b}{\sqrt{-4 b + \sqrt{81 - 60 b} + 9}}}{\sqrt{\frac{3}{2}} \left(-6 b + \sqrt{81 - 60 b} + 9\right) - \frac{9 \left(-4 b + \sqrt{81 - 60 b} + 9\right)^{3/2}}{\sqrt{81 - 60
b} + 9}},
\end{equation}
and by
\begin{equation}
(1 + z){}_\mathrm{max}^{\mathrm{ECS}} =\left\{
\begin{array}{lll}
(1 + z)(r =0, R = R_\mathrm{min}, b) \rightarrow \infty & \quad\mbox{for} & b \leq 0,\\
(1 + z)(r =0, R = R_\mathrm{I-}, b) & \quad\mbox{for} & 0 < b < 27/20,
\end{array}
\right.
\end{equation}
where
\begin{equation}
(1 + z)(r = 0, R = R_\mathrm{I-}, b) = \frac{\sqrt{6} \left(4 b + \sqrt{81 - 60 b} - 9\right) - \sqrt{\frac{\left(\sqrt{81 - 60 b} - 9\right)^2}{-4 b - \sqrt{81 - 60 b} + 9}} b}{\sqrt{\frac{3}{2}} \left(-6 b - \sqrt{81 - 60 b} + 9\right) - 9 \sqrt{\frac{\left(-4 b - \sqrt{81 - 60 b} + 9\right)^3}{\left(\sqrt{81 - 60 b} - 9\right)^2}}}.
\end{equation}
We give the central redshift limiting values in dependence on the braneworld parameter $b$ in Figure~\ref{figure5}. The central redshift of compact stars with surface located at the photon circular orbits of the external spacetime is illustrated in dependence on the tidal charge in Figure~\ref{figure6}. Notice that for compact stars with $b \in (1, 1.0749)$ the surface redshift from the compact star with $R = r_\mathrm{ph-}$ overcomes the central redshift of the compact star with $R = r_\mathrm{ph+}$.
\section{Discussion}\label{SECconcl}
\begin{figure}[t]%
\centering\includegraphics[width=0.499\hsize]{figure9a.pdf}\includegraphics[width=0.499\hsize]{figure9b.pdf}
\caption{The parameter $\alpha_\mathrm{z}$ relating the central and surface redshift given as a function of tidal charge $b$ and surface of radius $R$. On the plot on the right the values of parameter $b$ are given near the correspondig curves.}\label{figure9}
\end{figure}
\subsection{The role of the tidal charge}
In order to obtain a theoretical tool for estimating possible presence of the tidal charge influence using observational data on the surface redshift of photons and internal redshift of neutrinos, it could be useful to compare the surface redshift $(1 + z)(r = R, R, b)$ to the central redshift $(1 + z)(r = 0, R, b)$ and the internal redshift interval
\begin{equation}
\Delta_z (R, b) \equiv (1 + z)(r = 0, R, b) - (1 + z)(r = R, R, b) = \frac{p(r = 0, R, b)}{\varrho\Delta(R, b)}.
\end{equation}
We can introduce a characteristic quantity
\begin{equation}
\alpha_\mathrm{z}(R, b) \equiv \frac{(1 + z)(r = 0, R, b)}{(1 + z)(r = R, R, b)}= 1 + \frac{p(r = 0, R, b)}{\varrho}.
\end{equation}
The plot of $\alpha_\mathrm{z}(R, b)$ is given in Figure~\ref{figure9}. We see that in ECS with $R \rightarrow R_\mathrm{min}(b\leq1)$ the parameter $\alpha$ diverges while in compact stars with the naked-singularity type external spacetime the redshift remains finite for whole range of allowed surface radii. The plot of $\alpha_z$ is given in Figure~\ref{figure9} also explicitly for $R = r_\mathrm{ph}$ and $R = R{}_\mathrm{I}$.
\begin{figure}[t]%
\centering\includegraphics[width=.6\hsize]{figure10a.pdf}\\
\centering\includegraphics[width=.499\hsize]{figure10b.pdf}\includegraphics[width=.499\hsize]{figure10c.pdf}
\caption{Plot of the surface redshifts $1 + z$ and the difference of central and surface redshift $\Delta_\mathrm{z}$ for $b = -1.2, 0, 1.08, 1.2$.}\label{figure10}
\end{figure}
It is useful to relate the surface redshift and the redshift difference $\Delta_\mathrm{z}$. In the standard Schwarzschild spacetimes ($b = 0$), the $(1 + z)(r = R, R, b = 0)$ and $\Delta_\mathrm{z}(R, b = 0)$ are illustrated in Figure~\ref{figure10}. We can see that for the maximal limit of ECS ($R = r_\mathrm{ph}$), there is $1 + z = \sqrt{3}$ and $\Delta_\mathrm{z} = 1$. For $R > r_\mathrm{ph}$ increasing $(1 + z)(r = R, R, b = 0)$ decreases slowly, but $\Delta_\mathrm{z}$ decreases fast to zero. On the other hand, for $R \rightarrow R_\mathrm{min} = 9/4$, the surface redshift slowly increases to $(1 + z)(r = R\rightarrow R_\mathrm{min}) = 3$, but $\Delta_\mathrm{z}$ fast diverges. There is $(1+z)(r = R, R, b)=\Delta_\mathrm{z}(R)$ for $R = 8/3$. For negative (positive) tidal charges both the surface redshift (1 + z)(r = R) and the difference $\Delta_\mathrm{z}(R, b)$ decrease (increases) with decreasing (increasing) $b$. Notice that for $b \in (1, 1.10254)$ there are two common points of $(1 + z)(r = R)$ and $\Delta_\mathrm{z}$, while for $b > 1.10254$ there are no common points.
\subsection{Determination of the compact star parameters}
\begin{figure}[t]%
\centering\includegraphics[width=.7\hsize]{figure11.pdf}
\caption{Plot for graphical determination of parameters $R$ and $b$ from values $\Delta_z$ and $(1 + z)(r = R, R, b)$. Area of ECS is denoted by dark gray colour.}\label{figure11}
\end{figure}
From the knowledge of values $\Delta_z$ and $(1 + z)(r = R, R, b)$ (both of them are in principle observable measurable) we can determine the radius of the star $R$ and braneworld parameter $b$ via relations ($(1 + z)_R \equiv (1 + z)(r = R, R, b)$)
\begin{equation}
R = \frac{(1 + z)_{R}^{2} \left[3 \Delta_z + (1 + z)_R\right]}{\left[(1 + z)_R + 1\right] \left\{3 \Delta_z \left[(1 + z)_R - 2\right] + 2 \left[(1 + z)_R - 1\right](1 + z)_R\right\}},
\end{equation}
and
\begin{equation}
b = \frac{3 (1 + z)_R^2 \left[3 \Delta_z + (1 + z)_R\right] \left\{\Delta_z \left[(1 + z)_R - 3\right] + \left[(1 + z)_R - 1\right] (1 + z)_R\right\}}{\left[(1 + z)_R + 1\right]\left\{3 \Delta_z \left[(1 + z)_R - 2\right] + 2 \left[(1 + z)_R - 1\right] (1 + z)_R\right\}^2}.
\end{equation}
The results obtained by this procedure can be confronted with relations derived above for limiting extremely compact stars and the compact star with surface coinciding with the photon circular orbits of the external spacetime. Determination of the parameters $R$ and $b$ from the observationally determined quantities $\Delta_z$ and $(1 + z)(r = R, R, b)$ can be represented by nomograms. We illustrate the nomogram in Figure~\ref{figure11} where the region of ECS is explicitly depicted and we can immediately find if the observed compact star could belong to the ECS, or could allow for an unusual character of accretion phenomena in vicinity of compact stars having their surface located under the photon circular orbits of the external spacetime.
\section{Conclusions}
In our study we have focused our attention on the simple model of the braneworld neutron (quark) stars and we have studied the surface (photon) redshift and the internal (neutrino) redshift under assumption of zero neutrino rest energy and the motion along null geodesics. We postpone to future investigations the realistic case of non-zero neutrino rest energy and the role of neutrino flavour mixing along the trajectory to the observers \cite{alt:2007:,nuno:2007:,thomas:2008:}. (In any case it is reasonable to neglect the effect of neutrino flavour missing on the short distances inside the compact stars.) We believe that our results will keep relevance even in such more complex realistic situations. Of special interest could be a study of the redshift phenomena related to the full 5D uniform energy density stars numerically constructed in~\cite{Wis:2002:}.
We can conclude that our results indicate at least a principal possibility to find the compact star parameters (both surface radius and the tidal charge) from measurements of photon surface redshift and the range of internal redshift of neutrinos. From the observational data we are able to decide if the observed braneworld compact star is an extremely compact star allowing for gravitational trapping of some part of radiated neutrinos. Of course, the results giving the compact stars parameters from internal phenomena have to be further confronted to the observational phenomena related to accretion discs orbiting the compact stars where the tidal charge can yield some other specific signatures related to optical effects of accretion discs.
\clearpage
\acknowledgments{The present work was supported by the Czech grants MSM~4781305903, LC~06014, GA\v{C}R 205/09/H033 and the internal grant SGS/2/2010. One of the authors (Z.\,S.) would like to express his gratitude to the Czech Committee for collaboration with CERN and the Theory Division of CERN for perfect hospitality.}
|
1,108,101,566,800 | arxiv | \section{Introduction}
Hamiltonians of the form
\begin{eqnarray}\label{HN}
H=\frac{1}{2} p^2-g (ix)^N,
\end{eqnarray}
which are are $PT$-symmetric, but not Hermitian,
have been shown\cite{BB, Dorey} to have a real, positive
spectrum for $N\ge2$. For $N<4$ the energy eigenvalue problem can
be posed on the real $x$-axis, but for $N\ge4$ it must instead be imposed
in the complex $x$-plane, along a contour which asymptotically remains
within the Stokes wedges of the Hamiltonian. Such is indeed the case
for $N=4$, when the potential on the real axis is an ``upside down"
quartic. To that end we rewrite Eq.~(\ref{HN}) for $N=4$ in terms of
the complex variable $z$:
\begin{eqnarray}\label{H4}
H=\frac{1}{2} p_z^2 -gz^4.
\end{eqnarray}
A general framework encompassing $PT$-symmetric Hamiltonians was given by
Mostafazadeh\cite{AM}, who showed that such Hamiltonians were related
by a similarity transformation to an equivalent Hermitian Hamiltonian
$h$ possessing the same spectrum. In only a few cases is it possible to
construct $h$ explicitly, but this was done recently for Eq.~(\ref{H4}) in Ref.~\cite{JM},
rediscovering an earlier result of Buslaev and Grecchi\cite{BG}. The calculation
was performed using the particular real parametrization
\begin{eqnarray}\label{param}
z=-2i\sqrt{1+ix}
\end{eqnarray}
of the contour in the
complex plane on which Eq.~(\ref{H4}) was defined, and using operator techniques
to perform the similarity transformation to $h$, whose form is ($\alpha\equiv16g$)
\begin{eqnarray}\label{h}
h=\frac{1}{2} p_y^2+\frac{1}{4} \alpha y^4 -\sqrt{\frac{\alpha}{8}}y.
\end{eqnarray}
We should emphasize that this latter form is obtained only after a Fourier transform,
which means that $y$ is really a momentum rather than a position, as will become
apparent in the path-integral treatment. When a proper dimensional analysis is made\cite{An1},
the linear term turns out to be proportional to $\hbar$, which means that it is
in the nature of an anomaly.
In fact we will present various path-integral treatments, of various levels of
sophistication. In the next section we give the simplest of these, a continuum version, which fails
to reproduce the anomaly. Realizing that for a general change of variables in a functional
integral there is an additional term, $\Delta V$, to be included over and above the functional determinant,
we show, in Section~\ref{sec:Lee}, how $\Delta V$ is traded for the anomaly in an intermediate functional integration.
In Section 4 we give the discretized version of this calculation, in order to provide a benchmark
for comparison with the results obtained in Section 5 by two variants of the discretized calculation
of Ref.~\cite{An1}. In effect this latter section gives an alternative derivation of $\Delta V$, as a consequence
of the mismatch between the functional determinant and the coefficient of the kinetic term when our
particular change of variable is made.
\section{Na\"{\i}ve Continuum Treatment}
The original path integral is the single Euclidean functional integral
\begin{eqnarray}
Z=\int_C [D\psi] \exp \left\{-\int dt\left[\frac{1}{2}\dot{\psi}^2-g\psi^4\right]\right\},
\end{eqnarray}
where the subscript $C$ is to remind us that the integrals are to be performed on an
appropriate curve in the complex $\psi$ plane. Making the change of variable
\begin{eqnarray}\label{cv}
\psi=-2i\surd(1+i\,\varphi),
\end{eqnarray}
in analogy with Eq.~(\ref{param}),
we obtain
\begin{eqnarray}\label{Znaive}
Z=\int \frac{[D\varphi]}{{\rm Det}\surd(1+i\,\varphi)} \exp \left\{-\int dt\left[\frac{1}{2}\frac{\dot{\varphi}^2}{1+i\,\varphi}
-\alpha(1+i\,\varphi)^2\right]\right\},
\end{eqnarray}
where the $\varphi$ integrals are now along the real axis.
We now rewrite this as a double functional
integral in $\varphi$ and $\pi$ by means of the identity
\begin{eqnarray}\label{Pi1}
\frac{1}{{\rm Det}\surd(1+i\,\varphi)}=\frac{1}{N}\int [D\pi] \exp \left\{ -\int dt\,\frac{1}{2}
(1+i\,\varphi)\left(\pi-i\frac{\dot{\varphi}}{1+i\,\varphi}\right)^2\right\},
\end{eqnarray}
where $N$ is an appropriate normalization constant.
This leads to the expression
\begin{eqnarray}
Z=\frac{1}{N}\int [D\varphi][D\pi] \exp \left\{-\int dt\left[\frac{1}{2}(1+i\,\varphi)\pi^2-i\pi\dot{\varphi}-\alpha(1+i\,\varphi)^2
\right]\right\}.
\end{eqnarray}
Replacing $\pi\dot{\varphi}$ by -$\dot{\pi}\varphi$ under the $t$ integration, we have an exponent that is
quadratic in $\varphi$, so the $\varphi$ functional integration can be performed.
The result, after applying the compensating rescalings
$\varphi\to \varphi/\sqrt{2\alpha}$ and $\pi\to \pi\sqrt{2\alpha}$, is
\begin{eqnarray}
Z=\int[D\pi] \exp \left\{-\int dt\left[\frac{1}{2}\dot{\pi}^2-\sqrt{2\alpha}\dot{\pi}\left(1-\frac{1}{2}\pi^2\right)+
\frac{\alpha}{4}\pi^4
\right]\right\}.
\end{eqnarray}
The middle term in the integrand is a perfect derivative, and so may be discarded, leaving
the equivalent Lagrangian $\ell$
in the exponent, written in terms of $\pi$:
\begin{eqnarray}\label{eq:naive}
Z=\int[D\pi] \exp \left\{-\int dt\left[\frac{1}{2}\dot{\pi}^2+
\frac{\alpha}{4}\pi^4
\right]\right\},
\end{eqnarray}
\emph{except} that the linear term is missing. The simple classical calculation we have just performed is unable
to obtain this term. Note that the Fourier transform of the Schr\"{o}dinger treatment occurs here naturally,
since $Z$ is expressed in terms of the momentum variable $\pi$.
\section{Correct Continuum Treatment}\label{sec:Lee}
It is shown in various standard books on functional integration, for example \cite{Lee,CD}, that when a general change
of variables, such as that of Eq.~(\ref{cv}), is made, an additional potential term $\Delta V$ must
be included in the Lagrangian. Ultimately this term, which is actually of order $\hbar^2$, is derived from the
discretized form of the functional integral when the particular form of discretization
\begin{eqnarray}
\dot{\varphi_n}&\equiv& \frac{1}{a}(\varphi_{n+1}-\varphi_n)\\
\bar{\varphi_n}&\equiv&\frac{1}{2} (\varphi_{n+1}+\varphi_n)
\end{eqnarray}
is adopted, where $a$ is the lattice spacing, corresponding to Weyl ordering in the operator treatment.
The importance of this prescription was indeed emphasized in Ref.~\cite{An1}.
The general form of $\Delta V$ for a change of variable from $\psi$ to $\varphi$ is\cite{Lee}
\begin{eqnarray}\label{Vcgen}
\Delta V=\frac{1}{8}\left[\frac{d}{d\varphi}\left(\frac{d\varphi}{d\psi}\right)\right]^2,
\end{eqnarray}
which for the particular transformation of Eq.~(\ref{cv}) turns out to be
\begin{eqnarray}\label{Vc}
\Delta V=-\frac{1}{32}\frac{1}{1+i\varphi}.
\end{eqnarray}
The correct version of Eq.~(\ref{Znaive}) is thus
\begin{eqnarray}\label{ZLee}
Z&=&\int \frac{[D\varphi]}{{\rm Det}\surd(1+i\,\varphi)} \exp \left\{-\int dt\left[\frac{1}{2}\frac{\dot{\varphi}^2}{1+i\,\varphi}
-\frac{1}{32}\frac{1}{1+i\varphi}-\alpha(1+i\,\varphi)^2\right]\right\}.\nonumber\\
&&
\end{eqnarray}
It is now possible to write down a variant of the Gaussian identity (\ref{Pi1}),
\begin{eqnarray}\label{Pi2}
\frac{1}{{\rm Det}\surd(1+i\,\varphi)}=\frac{1}{N}\int [D\pi] \exp \left\{ -\int dt\,\frac{1}{2}
(1+i\,\varphi)\left(\pi-\frac{i\dot{\varphi}+\frac{1}{4}}{1+i\,\varphi}\right)^2\right\},
\end{eqnarray}
which serves to cancel the $\Delta V$ term as well as the kinetic term. In turn it introduces two additional
terms in the exponent: (i) a term in $i\dot{\varphi}/(1+i\varphi)$, which is a perfect derivative and so can be
discarded under the $t$ integration, and (ii) the anomaly $\int dt\ \pi/4$.
Finally, after rescaling as before, the corrected version of Eq.~(\ref{eq:naive}) is
\begin{eqnarray}
Z=\int[D\pi] \exp \left\{-\int dt\left[\frac{1}{2}\dot{\pi}^2-\sqrt{\frac{\alpha}{8}}\pi+
\frac{\alpha}{4}\pi^4
\right]\right\},
\end{eqnarray}
\section{Discretized Version}
In this section we will go through the discretized version of the previous calculation, in
order to provide a standard discretized formula with which we can compare the results of the (corrected)
calculation of Ref.~\cite{An1} and another calculation whereby the kinetic term is expanded in powers of
the lattice spacing $a$.
In place of Eq.~(\ref{ZLee}) we have
\begin{eqnarray}
Z=\prod_n\int \frac{d\varphi_n}{\sqrt{1+i\bar{\varphi}_n}} \exp\left\{-a\left[\frac{1}{2}\frac{\dot{\varphi}_n^2}{1+i\bar{\varphi}_n}
-\frac{1}{32}\frac{1}{{1+i\bar{\varphi}_n}}-\alpha(1+i\bar{\varphi}_n)^2 \right]\right\}.
\end{eqnarray}
The Gaussian identity we will use is
\begin{eqnarray}
\frac{1}{\sqrt{1+i\bar{\varphi}_n}}=\frac{1}{N}\int d\bar{\pi}_n e^{-\frac{1}{2}\lambda(\bar{\pi}_n-B)^2},
\end{eqnarray}
where $\lambda=a(1+i\bar{\varphi}_n)$ and $B=(i\dot{\varphi}_n+\frac{1}{4})/(1+i\bar{\varphi}_n)$.
Written out in full this is
\begin{eqnarray*}
\frac{1}{\sqrt{1+i\bar{\varphi}_n}}&=&\frac{1}{N}\int d\bar{\pi}_n\exp\left\{-a\left[\frac{1}{2} \bar{\pi}_n^2(1+i\bar{\varphi}_n)
-i\bar{\pi}_n\dot{\varphi}_n- \frac{1}{4}\bar{\pi}_n\right.\right.\\
&&\left.\left.\hspace{2.7cm}+\frac{1}{4}\frac{i\dot{\varphi}_n}{1+i\bar{\varphi}_n}-\frac{1}{2}\frac{\dot{\varphi}_n^2}{1+i\bar{\varphi}_n}+\frac{1}{32}\frac{1}{1+i\bar{\varphi}_n}
\right]\right\}.
\end{eqnarray*}
Neglecting the term in $i\dot{\varphi}_n/(1+i\bar{\varphi}_n)$ because the identity
\begin{equation}
\log\left(\frac{1+i\varphi_{n+1}}{1+i\varphi_n}\right)=\frac{ia\dot{\varphi}_n}{1+i\bar{\varphi}_n}
+O(a^3)
\end{equation}
shows it to be a perfect difference up to a correction of order $a^3$, we obtain
\begin{equation}\label{bench}
Z=\frac{1}{N}\prod_n\int d\bar{\pi}_n d\varphi_n \exp\left\{-a\left[ \frac{1}{2}(1+i\bar{\varphi}_n)\bar{\pi}_n^2-i\bar{\pi}_n\dot{\varphi}_n-
\frac{1}{4}\bar{\pi}_n-\alpha(1+i\bar{\varphi}_n)^2) \right]\right\}
\end{equation}
Thus the $\Delta V$ term has been cancelled, and we are left with the anomaly $\frac{1}{4}a\bar{\pi}_n$
in the exponent.
Now $\bar{\pi}_n\dot{\varphi}_n+\bar{\varphi}_n\dot{\pi}_n$ is a perfect difference:
\begin{equation}
a(\bar{\pi}_n\dot{\varphi}_n+\bar{\varphi}_n\dot{\pi}_n)=
\pi_{n+1}\varphi_{n+1}-\pi_n\varphi_n.
\end{equation}
So now we can ``integrate by parts", in the form $\bar{\pi}_n\dot{\varphi}_n\to-\bar{\varphi}_n\dot{\pi}_n$.\\
Changing the integration measure
from $\int d\bar{\pi}_n d\varphi_n$ to
$\int d\pi_n d\bar{\varphi}_n$, which does not introduce any additional factors, Eq.~(\ref{bench}) becomes
\begin{eqnarray*}
Z&=&\frac{1}{N}\prod_n\int d\pi_n d\bar{\varphi}_n \exp\left\{-a\left[
\alpha\left((\bar{\varphi}_n-i)+\frac{i}{4\alpha}(\bar{\pi}_n^2+2\dot{\pi}_n)\right)^2\right.\right.\\
&&\hspace{4cm}\left.\left.+\frac{1}{16\alpha}\left( \bar{\pi}_n^2+2\dot{\pi}_n\right)^2 -\frac{1}{4}\bar{\pi}_n\right]\right\},
\end{eqnarray*}
having dropped a perfect difference proportional to $\dot{\pi}_n$.
Now we rescale: $\bar{\varphi}_n\to \bar{\varphi}_n/\sqrt{2\alpha}$ and $\pi_n\to \pi_n\sqrt{2\alpha} $ and perform
the $\bar{\varphi}_n$ integration, with the result
\begin{eqnarray*}
Z=\frac{1}{N}\prod_n\int d\pi_n \exp\left\{-a\left[
\frac{\alpha}{4}\bar{\pi}_n^4+\sqrt{\frac{\alpha}{2}}\bar{\pi}_n^2\dot{\pi}_n+\frac{1}{2}\dot{\pi}_n^2
-\sqrt{\frac{\alpha}{8}}\bar{\pi}_n \right]\right\}
\end{eqnarray*}
This is the desired result, provided that we can neglect the term $\bar{\pi}_n^2\dot{\pi}_n$.
The identity
\begin{eqnarray}
3a\bar{\pi}_n^2\dot{\pi}_n=\pi_{n+1}^3-\pi_n^3-(\pi_{n+1}-\pi_n)^3/4
\end{eqnarray}
shows that is a perfect difference up to a correction of order $a^3$, so that it can indeed be neglected.
The resulting expression for Z is
\begin{eqnarray}
Z=\prod_n\int d\pi_n \exp\left\{-a\left[
\frac{1}{2}\dot{\pi}_n^2
-\sqrt{\frac{\alpha}{8}}\bar{\pi}_n +\frac{\alpha}{4}\bar{\pi}_n^4\right]\right\},
\end{eqnarray}
the discrete version of Eq.~(\ref{ZLee}).
Equation (\ref{bench}), from which this was derived, will be a reference point for the calculations of the following two sections, which
arise out of the treatment of Ref.~\cite{An1}.
The basis of that treatment was an exact, discretized, treatment of the kinetic term.
In the original functional integral written in terms of $\psi$, the time derivative is
defined as
$\dot\psi_n=(\psi_{n+1}-\psi_n)/a.$
In terms of $\psi_n=-2i\surd(1+i\varphi_n)$, this becomes
$\dot\psi_n=\dot{\varphi_n}/A_n$,
where
\begin{eqnarray}
A_n=\frac{1}{2}(\sqrt{1+i\varphi_{n+1}}+\sqrt{1+i\varphi_n})
\end{eqnarray}
The relation between $A_n$ and $\sqrt{1+i\varphi_n}$ can be written exactly as
\begin{eqnarray}\label{determinant}
\frac{1}{\sqrt{1+i\varphi_n}}=\frac{1}{A_n}\left(1+\frac{ia\dot{\varphi_n}}{4A_n\sqrt{1+i\varphi_n}}\right)
\end{eqnarray}
In this formulation the anomaly arises because the denominator $A_n$ in the expression for $\dot{\varphi}_n$ is not
quite the same as that in the determinant of Eq.~(\ref{determinant}). That is, we have the expression
$\exp[-a\dot{\varphi_n}^2/(2A_n^2)]/\surd(1+i\varphi_n)$,
and we need to expand one of these in
terms of the other.
\section{Expanding the Kinetic Term}\label{KE}
In this case we take $\lambda=a(1+i\varphi_n)$, $x=\bar{\pi}_n$ and $B=i\dot{\varphi_n}/(1+i\varphi_n)$ in the
Gaussian identity
\begin{eqnarray}\label{GI}
\frac{1}{\sqrt{1+i\varphi_n}}&=&\frac{1}{N}\int dx \exp\left\{-\frac{1}{2}\lambda(x-B)^2\right\}\\
&=&\frac{1}{N}\int d\bar{\pi}_n \exp\left\{-\frac{1}{2} a(1+i\varphi_n)\bar{\pi}_n^2
+ia\dot{\varphi}_n\bar{\pi}_n+\frac{1}{2}\frac{a\dot{\varphi}_n^2}{1+i\varphi_n}\right\}\nonumber
\end{eqnarray}
Now we need to expand the kinetic term $-\frac{1}{2} a\dot{\varphi_n}^2/A_n^2$ in terms of $\frac{1}{2}\lambda B^2$, namely
$\frac{1}{2}\lambda B^2=-\frac{1}{2} a\dot{\varphi_n}^2/(1+i\varphi_n)$.
First write Eq.~(\ref{determinant}) in terms of $B$:
\begin{eqnarray*}
\frac{1}{\sqrt{1+i\varphi_n}}=\frac{1}{A_n}\left(1+\frac{aB}{4A_n}\sqrt{1+i\varphi_n}\right)
\end{eqnarray*}
In terms of $R\equiv{\sqrt{1+i\varphi_n}}/{A_n}$, this reads
\begin{eqnarray}
\frac{1}{R}=1+\frac{1}{4}aBR,
\end{eqnarray}
which is a quadratic equation for $R$, with solution
\begin{eqnarray*}
R=\frac{2}{1+\sqrt{1+aB}}.
\end{eqnarray*}
So the kinetic term is $-\frac{1}{2} a\dot{\varphi_n}^2/A_n^2=\frac{1}{2}\lambda B^2 R^2=
\frac{1}{2} \lambda B^2+\frac{1}{2}\lambda B^2 (R^2-1).$\\
Hence
\begin{equation}\label{keep}
e^{-\frac{1}{2} a\dot{\varphi_n}^2/A_n^2}
=e^{\frac{1}{2}\lambda B^2}\left[1-\frac{1}{4} a\lambda B^3 +\frac{a^2}{32}(5\lambda B^4
+\lambda^2 B^6) +\dots \right]
\end{equation}
Now we need the Gaussian identities, under ${\large\int} dt \exp\{-\frac{1}{2}\lambda(x-B)^2\}$:
\begin{eqnarray}
B^3&\equiv& x^3-\frac{3x}{\lambda}, \nonumber\\
B^4&\equiv& x^4-\frac{6x^2}{\lambda}+\frac{3}{\lambda^2},\\
B^6&\equiv& x^6-\frac{15x^4}{\lambda}+\frac{45x^2}{\lambda^2}-\frac{15}{\lambda^3},\nonumber
\end{eqnarray}
to get
\begin{eqnarray}\label{wrongK}
e^{-\frac{1}{2} a\dot{\varphi_n}^2/A_n^2}=e^{\frac{1}{2}\lambda B^2}\left(1+\frac{3}{4}a\bar{\pi}_n + O(a^2)\right).
\end{eqnarray}
Note that it is crucial to keep terms in Eq.~(\ref{keep}) which are nominally of order $a^2$ (and
possibly higher). The point is that the Gaussian identities bring in terms of order $1/\lambda$, $1/\lambda^2$
etc., which means that such terms may actually be of order $a$.
The net result so far is that we appear to have produced the anomaly, but with the wrong coefficient, 3/4 versus 1/4. However,
it is important to realize that after these transformations we are
left with $-\frac{1}{2} a(1+i\varphi_n)\bar{\pi}_n^2$ in the exponent, as opposed to
the $-\frac{1}{2} a(1+i\bar{\varphi}_n)\bar{\pi}_n^2$ of Eq.~(\ref{bench}).
The difference between them is
\begin{eqnarray*}
-\frac{1}{2} a(1+i\varphi_n)\bar{\pi}_n^2&=&-\frac{1}{2} a(1+i\bar{\varphi}_n)\bar{\pi}_n^2+\frac{1}{4} i a^2\dot{\varphi}_n \bar{\pi}_n^2\\
&=&-\frac{1}{2} a(1+i\bar{\varphi}_n)\bar{\pi}_n^2+\frac{1}{4}\lambda aB \bar{\pi}_n^2
\end{eqnarray*}
In order to implement Gaussian identities with the new kinetic term we need to
write the correction in terms of a new $\lambda$ and $B$, namely
$\bar\lambda= a(1+i\bar{\varphi}_n)$ and $\bar{B}= i\dot{\varphi}_n/(1+i\bar{\varphi}_n)$,
with the same property that $\bar{\lambda}\bar{B}=\lambda B=ia\dot{\varphi}_n$.
The exponential of the additional term $\frac{1}{4}\bar{\lambda} a\bar{B} x^2$ expands to
\begin{eqnarray*}
\exp(\frac{1}{4}\bar{\lambda} a\bar{B} x^2)=1+\frac{1}{4}\bar{\lambda} a\bar{B}x^2+\frac{1}{32}\bar{\lambda}^2\bar{B}^2x^4+\dots
\equiv 1-\frac{1}{2} ax+O(a^2),
\end{eqnarray*}
using the further Gaussian equivalences
\begin{eqnarray}
\bar{B}x^2&\equiv&x^3-\frac{2x}{\bar{\lambda}},\nonumber\\
\bar{B}^2x^4&\equiv&x^6-\frac{9x^4}{\bar{\lambda}}+\frac{12x^2}{\bar{\lambda}^2}.
\end{eqnarray}
Thus, the additional term precisely corrects
the coefficient of the anomaly from 3/4 in Eq.~(\ref{wrongK}) to 1/4.
A necessary ingredient for this to work is the lack of a term in $1/\bar{\lambda}^3$ in the
Gaussian equivalence for $\bar{B}^2x^4$.
\section{Expanding the Determinant}\label{detsec}
In this case we take $x=\bar{\pi}_n$, $\lambda=aA_n^2$ and $B=i\dot{\varphi_n}/A_n^2$ in the
Gaussian identity
\begin{eqnarray}\label{Gauss}
\frac{1}{A_n}&=&\frac{1}{N}\int dx \exp \left\{
-\frac{1}{2}\lambda(x-B)^2\right\}\\
&=&\frac{1}{N}\int dx \exp \left\{-\frac{1}{2} aA_n^2\bar{\pi}_n^2+ia\dot{\varphi}_n\bar{\pi}_n+\frac{a\dot{\varphi}_n^2}{2A_n^2}\right\}
\end{eqnarray}
In fact we can solve Eq.~(\ref{determinant}) for $\sqrt{1+i\varphi_n}$ in terms of $A_n$:
\begin{eqnarray}
\frac{1}{A_n}=\frac{1}{\sqrt{1+i\varphi_n}}\left(1-\frac{ia\dot{\varphi_n}}{4A_n^2}\right)
=\frac{1}{\sqrt{1+i\varphi_n}}\left(1-\frac{1}{4}aB\right)
\end{eqnarray}
Thus, in terms of the new $B$,
$R\equiv\sqrt{1+i\varphi_n}/A_n=1-aB/4$.
So the determinant can be written as
\begin{equation}
\frac{1}{\sqrt{1+i\varphi_n}}=\frac{1}{A_n}\cdot \frac{1}{R}
=\frac{1}{A_n}\left(1+\frac{1}{4}aB+\frac{1}{16}a^2B^2 +\dots\right)
\end{equation}
Now we use two further identities under the Gaussian integration of Eq.~(\ref{Gauss}):
\begin{eqnarray}
B&\equiv& x \nonumber\\
B^2&\equiv& x^2-\frac{1}{\lambda}
\end{eqnarray}
to obtain
\begin{eqnarray}\label{first}
\frac{1}{\sqrt{1+i\varphi_n}}
\equiv \frac{1}{A_n}\left(1+\frac{1}{4}ax-\frac{a}{16A_n^2} +O(a^2)\right)
\end{eqnarray}
The second term in $\big(\hspace{1cm}\big)$ correctly gives the anomaly when elevated to the
exponent, but the third term, which was missed in Ref.~\cite{An1}, appears to be an unwanted addition.
The final result is
\begin{equation}\label{eqdet}
Z=\frac{1}{N} \prod_n\int d\varphi_n d\bar{\pi}_n \exp\left\{-a\left[\frac{1}{2} A_n^2 \bar{\pi}_n^2-i\bar{\pi}_n\dot{\varphi}_n
-\frac{1}{4}\bar{\pi}_n +\frac{1}{16A_n^2}-\alpha(1+i\varphi_n)^2\right]\right\}\nonumber
\end{equation}
This differs from Eq.~(\ref{bench})
in two ways: we have a term resembling $\Delta V$, and the coefficient of $\bar{\pi}_n^2$
is $\frac{1}{2} A_n^2$ rather than $\frac{1}{2}(1+i\bar{\varphi}_n)$. The difference between $\varphi_n$ and $\bar{\varphi}_n$ in the interaction
term is not important.
Now
\begin{eqnarray*}
aA_n^2\bar{\pi}_n^2=a(1+i\varphi_n)\bar{\pi}_n^2+\lambda\bar{\pi}_n^2\left(1-R^2\right).
\end{eqnarray*}
So, since $ia\varphi_n=ia(\bar{\varphi}_n-\frac{1}{2} a \dot{\varphi}_n)=ia\bar{\varphi}_n-\frac{1}{2}\lambda aB$,
\begin{eqnarray*}
-\frac{1}{2} aA_n^2\bar{\pi}_n^2=-\frac{1}{2} a(1+i\bar{\varphi}_n)\bar{\pi}_n^2+\frac{1}{4}\lambda (aB) \bar{\pi}_n^2
-\frac{1}{2} \lambda\bar{\pi}_n^2\left(1-R^2\right).
\end{eqnarray*}
The additional terms expand to
\begin{eqnarray*}
\frac{1}{4} \lambda\bar{\pi}_n^2 (aB)-\frac{1}{2}\lambda\bar{\pi}_n^2\left[\frac{1}{2}(aB)-\frac{1}{16}(aB)^2\right]
=\frac{1}{32}\lambda\bar{\pi}_n^2(aB)^2
\end{eqnarray*}
and their exponential to $1+(aB)^2\lambda\bar{\pi}_n^2/32+\dots$.
Again, because we have changed the coefficient of the kinetic term to $\frac{1}{2}(1+i\bar{\varphi}_n)$ we
strictly need to rewrite $\lambda B^2$ in terms of $\bar{\lambda}$ and $\bar{B}$
according to $\lambda B^2 = (\bar{\lambda}\bar{B}^2)\times (\bar{\lambda}/\lambda)$,
where the correction factor is
\begin{eqnarray*}
\frac{\bar{\lambda}}{\lambda}=\frac{1+i\bar{\varphi}_n}{A_n^2}=1+O(a\bar{B})^2
\end{eqnarray*}
The additional term does not in fact contribute to order $a$.
Thus we need the final equivalence
\begin{eqnarray}\label{xsqBsq}
\bar{B}^2x^2&\equiv&x^4-\frac{5x^2}{\bar{\lambda}}+\frac{2}{\bar{\lambda}^2}
\end{eqnarray}
for $\bar{B}^2x^2$ under Gaussian integration, which gives
$1+a^2/(16\bar{\lambda})+O(a^3)$.
When exponentiated this term precisely cancels the $\Delta V$-like term $a/(16A_n^2)$ in
Eq.~(\ref{eqdet}) up to $O(a^2)$.
\section{Conclusions}
The na\"ive functional integral formulation of the operator calculation can be regarded as the
classical result, in that it takes no account of operator ordering, or equivalently of any particular
discretization, and so fails to produce the linear, anomalous term, which is of order $\hbar$.
We have shown an elegant method of producing this term in the continuous formalism, provided we take
account of the additional $\Delta V$ term that has been shown (by careful discretization) to occur whenever
a general change of variables is made.
The calculation of Ref.~\cite{An1} indeed attempted a careful discretization, but missed terms that were
nominally of higher order in the lattice spacing $a$, but were in fact of the same order as the terms kept
because of the particular nature of the Gaussian equivalences, which bring in factors of $O(1/\lambda)=O(1/a)$.
Starting from the recognition that the coefficient of the kinetic term does not exactly match the argument of
the functional determinant we proceeded in two ways, expanding either the kinetic term or the functional determinant.
In both cases we used the discretized version of the corrected continuum calculation as a canonical form with
which to compare our results. The expansion of the kinetic term turned out to be the easier, needing only a
change from the point variable $\varphi_n$ to the averaged variable $\bar{\varphi}_n$ to obtain agreement. The method
of expanding the determinant used in Ref.~\cite{An1} eventually produced the same result after more lengthy
manipulations, which incidentally showed that the simple passage from Eq.~(57) to Eq.~(58) in Ref.~\cite{An1}
was not correct.
One can ask whether the same phenomenon of trading a $\Delta V$ term for an anomaly by means of
some analogue of Eq.~(\ref{Pi2}) is possible for other changes of variable. The answer seems to
be in the negative: the parametrization of Eq.~(\ref{cv}) seems to be very special, as indeed
it is in other respects.
In general, the coefficient of the kinetic term in $\varphi$ starting from a standard kinetic term in
$\psi$ is $\frac{1}{2} M$, where
\begin{equation}
M=\left(\frac{d\psi}{d\varphi}\right)^2,
\end{equation}
to be compared with the expression for $\Delta V$ in Eq.~(\ref{Vcgen}). For the two to be proportional
we require
\begin{equation}
\frac{d\psi}{d\varphi}\propto\frac{d}{d\varphi}\left(\frac{d\varphi}{d\psi}\right),
\end{equation}
or equivalently $d^2\varphi/d\psi^2=\mbox{const}$. This leads back to a relation of the general form of
Eq.~(\ref{cv}) up to a shift in $\psi$.
The main motivation for reformulating the quantum mechanical problem in path-integral terms in
Ref.~\cite{An1} was in order to attempt a generalization to higher dimensions, where there are
indications from the non-Hermitian formulation that the theory is asymptotically free and naturally
possesses a non-vanishing vacuum expectation value, making it an attractive alternative to the standard
Higgs model. It is therefore
natural to ask whether additional potentials of the nature of $\Delta V$, and the consequent
production of an anomaly, occur in dimensions greater than 1. The answer appears to be negative,
at least in the context of dimensional regularization. This is because algebraic field transformations
lead to $O(\hbar^2)$ terms proportional to $(\delta^{(n)}(0))^2$ in $n$ spatial dimensions,
which vanish in such a regularization scheme. This is consistent with a diagrammatic expansion,
for which no additional terms are necessary \cite{HV}. The reason for their presence in quantum
mechanics versus their absence in field theory has been analyzed in some detail by Salomonson\cite{PS}.
Even though it appears that this particular problem does not occur in field theory, there remain
severe difficulties in carrying out the programme. The most promising approach (3) put forward
in Ref.~~\cite{An1} still contains uncancelled Jacobian factors, which can only be represented
in the Lagrangian by the introduction of several auxiliary fields.\\
\noindent{\large\bf Acknowledgements}\\
This work has been partially supported by SpanishMEC (ProjectMTM2005-
09183) and Spanish JCyL (Excellence Project VA013C05). We are grateful
to C.~M. Bender and J.~M. Cerver\'o for helpful discussions.
|
1,108,101,566,801 | arxiv | \section{Introduction}\label{sec:intro}
Given a Riemannian manifold $(M, g)$, one can consider the Laplace operator $\Delta_g$ acting on the space of smooth functions on $M$; the spectrum of $\Delta_g$ is the set of eigenvalues of $\Delta_g$ on $\mathcal{C}^\infty(M)$. From a spectral-theoretic point of view, one is interested in how much about the geometry of $(M,g)$ is determined by the spectrum of $\Delta_g$. There are examples of Riemannian manifolds with the same spectrum which are not isometric (e.g., \cite{Gor_survey}, \cite{Mil_tori}, \cite{Sunada_method}), and there are also positive results showing that manifolds within a certain class are spectrally determined (e.g., \cite{Tanno}). In the setting of symplectic toric geometry, Miguel Abreu \cite{m2} asked
\begin{question}\label{q:Abreu}
Let $M$ be a toric manifold equipped with a toric K\"ahler metric $g$. Does the spectrum of the Laplacian $\Delta_g$ determine the moment polytope of $M$?
\end{question}
In \cite{dgs} the authors considered a modified version of this question, replacing the spectrum of the
Laplacian by the \emph{equivariant spectrum} of the Laplacian. This is simply the spectrum of the Laplacian together with, for each eigenvalue, the weights of the representation of $\mathbb{T}^n$ on the eigenspace corresponding to the given eigenvalue. Question \ref{q:Abreu} then becomes
\begin{question}\label{q:ours}
Let $M$ be a toric manifold equipped with a toric K\"{a}hler metric $g$. Does the equivariant spectrum of $\Delta_g$ on $\mathcal{C}^\infty(M)$ determine the moment polytope of $M$?
\end{question}
Thomas Delzant \cite{Delzant} proved that the moment polytope of a toric symplectic manifold $M$ determines $M$ up to symplectomorphim. Thus, if the answer to Question \ref{q:ours} is ``yes,'' the spectrum of the Laplacian of a symplectic toric manifold determines its symplectomorphism type. We showed that the answer is positive for many generic toric $4$-manifolds, up to translation and a small number of choices; however, we could not resolve the question when the moment polytope of $M$ has ``many'' parallel sides, a case which occurs with positive probability.
Toric orbifolds are a natural generalization of toric manifolds. They admit toric K\"{a}hler metrics, i.e., metrics that are determined by a symplectic form and a compatible, integrable almost complex structure and that are invariant under the torus action. Thus one may again define the Laplacian and its associated equivariant spectrum. Toric orbifolds also have moment polytopes associated to them, so it is natural to ask Question \ref{q:ours} in the context of toric orbifolds. The same issues as in \cite{dgs} arise when the moment polytope has parallel facets, but unlike for manifolds such facets occur for orbifolds with zero probability. Hence we are able to prove the following theorem.
\begin{thm}\label{main_theorem}
Let $\mathcal{O}$ be a generic toric orbifold with a fixed torus action and a toric K\"ahler metric. Then the equivariant spectrum of $\mathcal{O}$ determines the moment polytope $P$ of $\mathcal{O}$, and hence the equivariant symplectomorphism type of $\mathcal{O}$, up to two choices and up to translation.
\end{thm}
\noindent Note that the two choices determined by the equivariant spectrum have symplectomorphic underlying manifolds (see \S \ref{sec:mainthm}).
The main tool in \cite{dgs} is a result of Harold Donnelly \cite{Don1} which gives an asymptotic expansion for the heat kernel in the presence of an isometry on a manifold. A key step in the proof of Theorem \ref{main_theorem} is to generalize this tool to the setting of orbifolds (see \S \ref{sec:eq_heat_oflds}). Our approach is similar to the generalization of the asymptotic expansion of the heat kernel to orbifolds that was done in \cite{dggw}; the resulting expansion should be of independent interest.
\begin{thm}\label{main_asymptotic}
Let $\mathcal{O}$ be a closed Riemannian orbifold, let $K(t,x,y)$ be the heat kernel of $\mathcal{O}$, and let $f$ be a nontrivial liftable isometry of $\mathcal{O}$. Then $\int_\mathcal{O}\,K(t,x,f(x))\text{dvol}_{\mathcal{O}}(x)$ is asymptotic as $t\to 0^+$ to
\[
\sum_{S \in \mathcal{S}(\mathcal{O})} \frac{1}{|\operatorname{Iso}(S)|} (4\pi t)^{-\frac{\text{dim}(\text{Fix} f \cap S)}{2}} \int_{\text{Fix} f \cap S} \sum_{k=0}^{\infty} b_k(f,S) t^k \text{dvol}_{\text{Fix} f \cap S}(x),
\]
where $S(\mathcal{O})$ is a stratification of $\mathcal{O}$ and $|\operatorname{Iso}(S)|$ denotes the order of the isotropy of any point $p \in S$.
\end{thm}
\noindent In a subsequent paper the authors will extend this result to more general operators than the Laplacian using semi-classical analysis techniques.
Theorem \ref{main_theorem} holds for any toric K\"ahler metric on $\mathcal{O}$, and it is well known (see \cite{g1}, \cite{m1}) that toric orbifolds admit many toric K\"ahler metrics. Thus one is led to ask what the equivariant spectrum tells us about the toric metric itself.
\begin{question}
Does the equivariant spectrum corresponding to a toric K\"ahler metric on a toric orbifold determine the toric K\"ahler metric?
\end{question}
A positive answer to this question is unlikely, but one could hope that the equivariant spectrum might determine some properties of the metric. Finding ``special'' K\"ahler metrics on K\"ahler manifolds or orbifolds is currently an active research topic, with especial attention to K\"ahler Einstein metrics and extremal metrics in the sense of Calabi. As a particular instance of these, one often looks for constant scalar curvature metrics. It is known that such metrics do not always exist, but it is conjectured that their existence is equivalent to a stability condition on the underlying manifold. For $4$-dimensional toric manifolds this conjecture was proved recently by Simon Donaldson \cite{Doncsc}. We will use the asymptotic expansion in Theorem \ref{main_asymptotic} to show that one can equivariantly hear constant scalar curvature toric K\"ahler metrics.
\begin{thm}\label{thm:csc}
Let $\mathcal{O}$ be a generic toric orbifold endowed with a toric K\"ahler metric $g$. Then the equivariant spectrum of $\mathcal{O}$ determines if $g$ has constant scalar curvature.
\end{thm}
The paper is organized as follows. In \S \ref{sec:ofld_back} we give the necessary background on orbifolds, with particular emphasis on orbifold strata and isotropy groups. This allows us to prove Theorem \ref{main_asymptotic} in \S \ref{sec:eq_heat_oflds}. We then specialize to the setting of toric orbifolds, giving relevant background in \S \ref{sec:toric_back}. The proof of Theorem \ref{main_theorem} is given in \S \ref{sec:mainthm}, followed by the proof of Theorem \ref{thm:csc} in \S \ref{sec:csc}.
\vspace{.5cm}
\noindent \textbf{Acknowledgments:} The first and third authors appreciate the hospitality shown to them by the Mathematics Department at MIT during their visits there. The first author's visit during Summer 2010 was partially supported by an NSF-AWM Mentoring Travel Grant, and the second author was partially supported by NSF grant DMS-1005696. We thank Yael Karshon and Isabella Novik for making us aware of the work by Daniel Klain on the Minkowski problem.
\section{Background on orbifolds}\label{sec:ofld_back}
We begin by reviewing some of the basic definitions related to orbifolds that are relevant to our work. Our presentation and notation will follow that used in \S 2 of \cite{dggw}, which the reader may consult for more details.
A $k$-dimensional \emph{orbifold} $\mathcal{O}$ is a second-countable Hausdorff topological space $X$ that is equipped with a maximal orbifold atlas. Each chart in this atlas consists of a connected open subset $\tilde{U} \subset \mathbb{R}^k$, a finite group $G_U$ acting on $\tilde{U}$ by diffeomorphisms, and a mapping $\pi_U: \tilde{U} \rightarrow U$, where $U$ is an open subset of $X$ and $\pi_U$ induces a homeomorphism from $G_U \backslash \tilde{U}$ onto $U$. We will assume that the action of $G_U$ on $\tilde{U}$ is effective.
Points in $\mathcal{O}$ are either \emph{singular} or \emph{regular}. A point $x \in \mathcal{O}$ is singular if for some (hence every) orbifold chart $(\tilde{U}, G_U, \pi_U)$ about $x$, the points in the inverse image of $x$ in $\tilde{U}$ have nontrivial isotropy in $G_U$. The isomorphism class of this isotropy group is called the \emph{abstract isotropy type} of $x$ and is independent of the choice of point in the inverse image of $\pi_U$ and of the choice of orbifold chart about $x$.
An orbifold $\mathcal{O}$ can be endowed with a \emph{Riemannian structure} by assigning to each orbifold chart $(\tilde{U}, G_U, \pi_U)$ a $G_U$-invariant Riemannian metric on $\tilde{U}$ satisfying an appropriate compatibility condition among charts. Every Riemannian orbifold has an associated orthonormal frame bundle, as we now briefly describe. For an orbifold $\mathcal{O}$ of the form $G \setminus M$, where $M$ is a Riemannian manifold and $G$ is a discrete subgroup of the isometry group of $M$, we begin by considering the orthonormal frame bundle of $M$, $F(M) \rightarrow M$. Since each element of $G$ induces a diffeomorphism of $F(M)$ that takes fibers to fibers, we have an action of $G$ on $F(M)$ that covers the action of $G$ on $M$. We then define the orthonormal frame bundle of $O$, $F(\mathcal{O})$, to be $G \setminus F(M) \rightarrow \mathcal{O}$, with the fiber over a point $x \in \mathcal{O}$ defined as the preimage of $x$ in $G \setminus F(M)$. Note that $F(M)$ admits a right action of the orthogonal group $O(k)$, and this action commutes with the left action of $G$; hence we have a right $O(k)$-action on $F(\mathcal{O})$. For an arbitrary orbifold $\mathcal{O}$, one may define an orthonormal frame bundle using the above construction on the local charts on $\mathcal{O}$ with an appropriate compatibility condition among charts; this bundle is an \emph{orbibundle} whose total space is again a smooth manifold that admits a right action of the orthogonal group. The orbifold $\mathcal{O}$ can then be viewed as the orbit space $F(\mathcal{O}) / O(k)$.
Let $x \in \mathcal{O}$ be a singular point and view it as an element of $F(\mathcal{O})/ O(k)$. As $\tilde{x} \in F(\mathcal{O})$ ranges over the fibers in the preimage of $x$ in $F(\mathcal{O})$, the stabilizer $\operatorname{Iso} (\tilde{x})$ ranges over a conjugacy class of subgroups of $O(k)$. It turns out that this conjugacy class is independent of the choice of Riemannian metric used to construct $F(\mathcal{O})$, so it makes sense to call it the \emph{isotropy type} of $x \in \mathcal{O}$, denoted $\operatorname{Iso}(x)$. Moreover, the subgroups in a given conjugacy class can be shown to lie in the isomorphism class defined by the abstract isotropy type of $x$. The cardinality $|\operatorname{Iso}(x)|$ is the \emph{order} of the isotropy at $x$, and the equivalence classes of points with the same isotropy type are called the \emph{isotropy equivalence classes}.
\begin{defn}
A smooth \emph{stratification} of an orbifold $\mathcal{O}$ is a locally finite partition of $\mathcal{O}$ into submanifolds. Each submanifold, called a \emph{stratum}, is locally closed and its closure is the union of the stratum with a collection of lower-dimensional strata.
\end{defn}
Given a stratification of an orbifold $\mathcal{O}$, the strata of maximal dimension are open in $\mathcal{O}$ and their union has full measure in $\mathcal{O}$. From general results about smooth actions of Lie groups on manifolds, one can prove
\begin{prop}\cite[Cor. 2.11]{dggw}
Let $\mathcal{O}$ be an orbifold. Then the action of $O(k)$ on the frame bundle $F(\mathcal{O})$ gives rise to a (Whitney) stratification of $\mathcal{O}$. The strata are connected components of the isotropy equivalence classes in $\mathcal{O}$. The set of regular points of $\mathcal{O}$ intersects each connected component $\mathcal{O}_0$ of $\mathcal{O}$ in a single stratum that constitutes an open dense submanifold of $\mathcal{O}_0$.
\end{prop}
The strata of $\mathcal{O}$ will be called \emph{$\mathcal{O}$-strata}; given an orbifold chart $(\tilde{U}, G_U, \pi_U)$ on $\mathcal{O}$, the stratifications of sets $U$ and $\tilde{U}$ induced by $G_U$ will be referred to as \emph{$U$-strata} and \emph{$\tilde{U}$-strata}, respectively. The following result was proved in \cite{dggw}.
\begin{prop}\cite[Prop. 2.13]{dggw}
Let $\mathcal{O}$ be a Riemannian orbifold and
$(\tilde{U},G_U,\pi_U)$ be an orbifold chart. Then:
\begin{enumerate}
\item The $U$-strata are precisely the connected components of the intersections of
the $\mathcal{O}$-strata with $U$.
\item Any two elements of the same $\tilde{U}$-stratum have the same stabilizers
in $G_U$ (not just conjugate stabilizers).
\item If $H$ is a subgroup of $G_U$, then each connected component $W$ of the
fixed point set $\operatorname{Fix}(H)$ of $H$ in $\tilde{U}$ is a closed submanifold of $\tilde{U}$. Any
$\tilde{U}$-stratum that intersects $W$ nontrivially lies entirely in $W$. Thus the
stratification of $\tilde{U}$ restricts to a stratification of $W$.
\end{enumerate}
\end{prop}
It follows from this proposition that if $\tilde{N}$ is any $\tilde{U}$-stratum in $\tilde{U}$, then all the points in $\tilde{N}$ have the same isotropy group in $G_U$; we denote this isotropy group of $\tilde{N}$ by $\operatorname{Iso}(\tilde{N})$. The set of $\gamma \in \operatorname{Iso}(\tilde{N})$ such that $\tilde{N}$ is open in $\operatorname{Fix}(\gamma)$ will be denoted $\operatorname{Iso}^{\operatorname{max}} (\tilde{N})$. Note that the union of the $\tilde{U}$-strata $\tilde{N}$ for which $\gamma$ is an element of $\operatorname{Iso}^{\operatorname{max}} (\tilde{N})$ has full measure in $\operatorname{Fix}(\gamma)$.
\begin{exa}\label{exa1}
Let $M \subset \mathbb{R}^3$ be a sphere of radius $1$ centered at the origin. The quotient of $M$ by a rotation of order $p$ about the $z$-axis is a closed orbifold called a $(p,p)$-football. Each of $(1,0,0)$ and $(0,0,-1)$ is a singular stratum with isotropy of order $p$.
We may also consider a $(p,q)$-football with $p \neq q$; this does not arise as a global quotient of a manifold, but is an orbifold with two antipodal singular points whose underlying topological space is a sphere. Taking the quotient of this orbifold with respect to reflection in the plane containing the origin and the singular points, we get an orbifold whose underlying space is a disk. The points on the boundary of the disk are singular points (not boundary points) of the orbifold and comprise four strata: the image of each antipodal point forms a single stratum with isotropy of order $2p$ or $2q$, respectively, and each open edge forms a stratum with isotropy of order $2$. The intersection $U$ of the disk with a smaller disk centered at one of the ``poles'' is the image of an orbifold chart $(\tilde{U}, G, \pi_U)$, where $\tilde{U}$ is a disk in $\mathbb{R}^2$ centered at the origin and $G$ is the dihedral group of appropriate order. The $\tilde{U}$-strata with respect to $G$ are the origin and the intersection of $\tilde{U}$ with, say, the positive and negative $x$-axis. Let $\tilde{N}$ be the intersection of $\tilde{U}$ with one of the half-axes; then $\operatorname{Iso}(\tilde{N})$ contains the reflection and the identity, but $\operatorname{Iso}^{\operatorname{max}}(\tilde{N})$ is just the reflection. If $\tilde{N} = \{0\}$, then $\operatorname{Iso}(\tilde{N})=G$ but $\operatorname{Iso}^{\operatorname{max}}(\tilde{N})$ is just the rotation through angle $\frac{2 \pi}{p}$ (respectively, $\frac{2\pi}{q}$) about the origin.
\end{exa}
\noindent We will return to variations on this example in \S \ref{sec:toric_back}.
\section{Equivariant heat kernel asymptotics for orbifolds}\label{sec:eq_heat_oflds}
We now develop the asymptotic expansion of the heat kernel on an orbifold $\mathcal{O}$ in the presence of a liftable isometry. To show that such an expansion exists and to find it, one approach is to use the local structure of orbifolds. In particular, one may take a local covering of $\mathcal{O}$ by convex geodesic balls and piece together a parametrix for the equivariant heat operator on $\mathcal{O}$ using locally defined parametrices: working in a convex geodesic ball $U \subset \mathcal{O}$ with orbifold chart $(\tilde{U}, G_U, \pi_U)$, define a local parametrix $\tilde{H} (t,\tilde{x},\tilde{y})$ on $(0, \infty) \times \tilde{U} \times \tilde{U}$. Suppose that $f: \mathcal{O} \rightarrow \mathcal{O}$ is an isometry of $\mathcal{O}$ that lifts to an isometry $\tilde{f}: \tilde{U} \rightarrow \tilde{U}$ with $\tilde{f} \circ \gamma = \gamma \circ \tilde{f}$ for all $\gamma \in G_U$. Then the function
\[
(t, \tilde{x}, \tilde{y}) \longmapsto \sum_{\gamma \in G_U} \tilde{H} (t, \tilde{x}, \tilde{f} \circ \gamma (\tilde{y}))
\]
descends to a well-defined function on $(0, \infty) \times U \times U$. The argument to patch together these locally defined parametrices to get a globally defined parametrix and thus an equivariant heat kernel on $\mathcal{O}$ follows as in \S 3 of \cite{dggw}.
To find the asymptotic expansion of the equivariant heat kernel, we generalize the following theorem of Donnelly.
\begin{thm}\label{prop:Don}\cite{Don1}. Let $M$ be a closed Riemannian manifold,
let $K(t,x,y)$ be the heat kernel of $M$, and let $\gamma$ be a nontrivial isometry
of $M$.
Then
$\int_M\,K(t,x,\gamma(x))\text{dvol}_M(x)$ is asymptotic as $t\to 0^+$ to
$$\sum_{W\subset\operatorname{Fix}(\gamma)}\,(4\pi
t)^{-\frac{\dim(W)}{2}}\sum_{k=0}^\infty\,t^k\int_W\,b_k(\gamma,a)\text{dvol}_W(a)$$
where $W$ ranges over connected components of the fixed point set of $\gamma$, $b_k(\gamma,a)$ is a real-valued function on the fixed point set of $\gamma$
and $\text{dvol}_W$ is the volume form on $W$ defined by the Riemannian metric
induced from $M$.
\end{thm}
The function $b_k(\gamma, x)$ has several key properties (see \cite[\S 4]{dggw}). First, its restriction to any $W \subset \operatorname{Fix}(\gamma)$ is smooth. Second, it is local in that it only depends on the germs at $x$ of the Riemannian metric and of the isometry $\gamma$. Finally, it is universal in that it behaves as one would hope with respect to isometries; namely, if $M$ and $M'$ are Riemannian manifolds admitting the isometries $\gamma$ and $\gamma'$, respectively, and $\sigma: M \rightarrow M'$ is an isometry satisfying $\sigma \circ \gamma = \gamma' \circ \sigma$, then $b_k(\gamma,x) = b_k (\gamma', \sigma(x))$ for all $x \in \operatorname{Fix} (\gamma)$. Donnelly gave explicit formulas for $b_0$ and $b_1$, with the general definition of $b_k$ as follows. Let $x \in W$. Note that the orthogonal complement of $T_xW$ in $T_xM$ is invariant under $\gamma_*$; let $A_\gamma(x)$ be the nonsingular matrix transformation defined by $\gamma_*$, and set $B_\gamma(x) = (I - A_\gamma(x))^{-1}$. Then we can define
\[
b_k(\gamma, x) = |\det (B_\gamma(x))| b'_k(\gamma, x),
\]
where $b'_k (\gamma, x)$ is a universal invariant polynomial in the components of $B_\gamma$ and in the curvature tensor $R$ of $M$ and its covariant derivatives at $x$.
With this definition in mind, we prove a special case of Theorem \ref{main_asymptotic}.
\begin{lemma}\label{lemma:global}
Let $\mathcal{O}=G\backslash M$, where $M$ is a Riemannian manifold and $G$ is a finite group acting effectively on $M$. Let $f: \mathcal{O} \rightarrow \mathcal{O}$ be an isometry that lifts to an isometry $\tilde{f}: M \rightarrow M$ with $\tilde{f} \circ \gamma = \gamma \circ \tilde{f}$ for all $\gamma \in G$. Then
\[
\sum_{i=1}^{\infty}\operatorname{Tr} (f_{\lambda_i})^* e^{-t\lambda_i} \sim
\sum_{S \in \mathcal{S}(\mathcal{O})} \frac{1}{|\operatorname{Iso}(S)|} (4\pi t)^{-\frac{\dim(\operatorname{Fix} f \cap S)}{2}} \sum_{k=0}^{\infty} t^k \int_{\operatorname{Fix} f \cap S} b_k(f,x) \text{dvol}_{\operatorname{Fix} f \cap S}(x)
\]
as $t \to 0^+$, where $|\operatorname{Iso}(S)|$ is the order of the isotropy at every point in $S$ as defined in \S \ref{sec:ofld_back}.
\end{lemma}
\begin{proof}
Note that $(M,G,\pi)$ is a global orbifold chart where
$\pi:M\to\mathcal{O}$ is the projection. If $K$ denotes the heat kernel of $M$, then the heat kernel $K^\mathcal{O}$ of $\mathcal{O}$ is given by
$$K^\mathcal{O}(t,x,y)=\sum_{\gamma\in G}\,K(t,\widetilde{x},\gamma(\widetilde{y}))$$
where $\widetilde{x}$, respectively $\widetilde{y}$, are any elements of $\pi^{-1}(x)$, respectively
$\pi^{-1}(y)$. Thus
\[
\int_\mathcal{O} K^\mathcal{O}(t,x,x) \text{dvol}_\mathcal{O}(x)=\frac{1}{|G|}\sum_{\gamma\in
G}\,\int_M\,K(t,\widetilde{x},\gamma(\widetilde{x}))\text{dvol}_M(\widetilde{x}).
\]
Let us examine what happens to this heat kernel in the presence of an isometry $f: \mathcal{O} \rightarrow \mathcal{O}$. Letting $K_f^{\mathcal{O}}$ denote the $f$-equivariant heat kernel of $\mathcal{O}$, we have that
\[
\int_\mathcal{O} K_f^\mathcal{O}(t,x,x) \text{dvol}_\mathcal{O}(x)=\frac{1}{|G|}\sum_{\gamma\in
G}\,\int_M\,K(t,\widetilde{x},\tilde{f} (\gamma(\widetilde{x})))\text{dvol}_M(\widetilde{x}).
\]
Applying Theorem \ref{prop:Don} to this expression gives
\begin{equation}\label{eqn:Dondirect}
\int_\mathcal{O} K_f^\mathcal{O}(t,x,x) \text{dvol}_\mathcal{O}(x) \sim \frac{1}{|G|}\sum_{\gamma\in G} \sum_{W \in \operatorname{Fix}(\tilde{f} \circ \gamma)} (4\pi
t)^{-\frac{\dim(W)}{2}}\sum_{k=0}^\infty\,t^k\int_W\,b_k(\tilde{f} \circ \gamma,\widetilde{x})\text{dvol}_W(\widetilde{x})
\end{equation}
as $t\to 0^+$.
In order to express the right side of \eqref{eqn:Dondirect} in terms of intrinsic orbifold data, we need to analyze the fixed point sets arising from $\tilde{f} \circ \gamma$ for $\gamma \in G$. We begin by relating these fixed point sets to $M$-strata.
Let $W$ be a connected component in $\operatorname{Fix}(\tilde{f} \circ \gamma)$ and let $\widetilde{N}$ be an $M$-stratum contained in $W$. Then either $\widetilde{N}$ has measure zero in $W$ (in which case
$\gamma\notin\operatorname{Iso}^{\operatorname{max}}(\widetilde{N})$) or $\widetilde{N}$ is open in $W$ and $\gamma \in \operatorname{Iso}^{\operatorname{max}}(\widetilde{N})$.
Suppose $\widetilde{x} \in \widetilde{N}$ and $\gamma \in \operatorname{Iso}^{\operatorname{max}}(\widetilde{N})$. Since $\widetilde{N} \subset W \in \operatorname{Fix}(\tilde{f} \circ \gamma)$, we have $\tilde{f}(\gamma(\widetilde{x})) = \widetilde{x}$. But $\gamma \in \operatorname{Iso}^{\operatorname{max}}(\widetilde{N})$, so $\gamma(\widetilde{x})=\widetilde{x}$. Thus $\tilde{f}(\widetilde{x}) = \widetilde{x}$, or $\widetilde{x} \in \operatorname{Fix}(\tilde{f} \circ I)$. This means that we only need to consider the contribution from the identity element in $G$, and we may replace the integral over $W$ with integrals over the $M$-strata that are open in $W$. Thus the right side of \eqref{eqn:Dondirect} becomes
\begin{equation}\label{eqn:trace_up}
\frac{1}{|G|} \sum_{\widetilde{N} \in \mathcal{S}_f(M)} (4 \pi t)^{-\frac{\dim(\widetilde{N})}{2}} \sum_{k=0}^{\infty} t^k \int_{\widetilde{N}} b_k(\tilde{f} \circ \gamma,\widetilde{x})\text{dvol}_{\widetilde{N}}(\widetilde{x}),
\end{equation}
where $\mathcal{S}_f(M)$ denotes the strata in $\mathcal{S}(M)$ that are open in $\operatorname{Fix}(\tilde{f} \circ \gamma)$.
Our next task is to relate the data on the manifold $M$ to data on $\mathcal{O}$. Let $N$ be an $\mathcal{O}$-stratum that is open in a component of $\operatorname{Fix}(f)$. Then $\pi^{-1}(N)$ is a union of finitely many mutually isometric strata in $\mathcal{S}_f(M)$ and $\pi: \pi^{-1}(N) \to N$ is a covering map of degree $\frac{|G|}{|\operatorname{Iso}(N)|}$. Moreover, the total contributions to \eqref{eqn:trace_up} from the elements of $\pi^{-1}(N)$ are equal to
\[
\frac{|G|}{|\operatorname{Iso}(N)|} (4 \pi t)^{-\frac{\dim(N)}{2}} \sum_{k=0}^{\infty} t^k \int_{N} b_k(f, x)\text{dvol}_{N}(x).
\]
Thus \eqref{eqn:trace_up} becomes
\begin{equation}\label{eqn:tracedown}
\sum_{N \in \mathcal{S}_f(\mathcal{O})}\frac{1}{|\operatorname{Iso}(N)|} (4 \pi t)^{-\frac{\dim(N)}{2}} \sum_{k=0}^{\infty} t^k \int_{N} b_k(f, x)\text{dvol}_{N}(x)
\end{equation}
where $\mathcal{S}_f(\mathcal{O})$ denotes the strata in $\mathcal{S}(\mathcal{O})$ that are open in $\operatorname{Fix}(f)$. This proves the lemma.
\end{proof}
The argument in the proof of Lemma \ref{lemma:global} can be applied to orbifold charts, and one may piece together the resulting computations via a partition of unity to prove Theorem \ref{main_asymptotic}. The ideas are exactly the same as those used in \cite{dggw} to pass from the asymptotic expansion of the heat trace for an orbifold of the form $\mathcal{O}=G\backslash M$ to the expansion for a general orbifold. We refer the interested reader to \S 4 of \cite{dggw} for details.
Using computations from \cite{Don1} and \cite{dggw} one can find explicit expressions for the first few terms in the asymptotic expansion in Theorem \ref{main_asymptotic}. We will denote the scalar curvature by $s$, the Ricci tensor by $\rho$ and the full curvature tensor by $R$. Let $S$ be a connected component of $\mathcal{S}(\mathcal{O})$ and $x\in S$. Let $\gamma$ be an element of $\operatorname{Iso}^{\operatorname{max}}(S)$ and $W_f$ be a local lift of $S\cap \operatorname{Fix}(f)$ via an orbifold chart. Let $A_{f,\gamma}$ be the isometry
\[
df\circ d\gamma :{T_{\widetilde{x}}W_f^\perp}\rightarrow T_{\widetilde{x}}W_f^\perp,
\]
where $T_{\widetilde{x}}W_f^\perp$ denotes the normal space to $W_f$, and set $B_{f,\gamma}(\widetilde{x})=(I-A_{f,\gamma}(\widetilde{x}))^{-1}$. Then we have
\[
b_0(f,x)=\sum_{\gamma \in \operatorname{Iso}^{\operatorname{max}}(S)}|\det (B_{f,
\gamma}(\widetilde{x}))|,
\]
and
\[
b_1(f,x)=\sum_{\gamma \in \operatorname{Iso}^{\operatorname{max}}(S)}|\det (B_{f,\gamma}(\widetilde{x}))|\tau_\gamma(\widetilde{x})
\]
where
\[
\tau_\gamma=\frac{s}{6}+\frac{1}{6}\rho_{kk}+\frac{1}{3}R_{iksh}
B_{ki}B_{hs}+\frac{1}{3}R_{ikth}
B_{kt}B_{hi}-R_{kaha}
B_{ks}B_{hs}.
\]
Here the indices $k,i,s,h,t$ correspond to normal directions in $T_{\widetilde{x}}W_f^\perp$ and we sum over repeated indices.
When $f$ is the identity and we are in a neighborhood of a regular point $x \in \mathcal{O}$, the function $b_k$ equals the usual heat invariant $a_k$ for manifolds. In particular, we note for use in \S \ref{sec:csc} that
\begin{equation}\label{eqn:b2}
b_2(x)=\frac{1}{360} \left(2|R|^2-2|\rho|^2+5s^2 \right).
\end{equation}
\section{Background on toric orbifolds} \label{sec:toric_back}
We now specialize to the setting of toric orbifolds, providing the definitions and background that are needed to understand the proof of Theorem \ref{main_theorem}.
The notion of symplectic manifold generalizes easily to the orbifold setting: an orbifold is said to be symplectic if it admits a $2$-form $\omega$ which is nondegenerate and closed. One can then specify what it means for an orbifold to be \emph{toric}.
\begin{defn}
Let $(\mathcal{O},\omega)$ be a symplectic orbifold of real dimension $2n$. Then $(\mathcal{O},\omega)$ is said to be toric if admits an effective Hamiltonian $\mathbb{T}^n$-action, where $\mathbb{T}^n$ is the real torus of dimension $n$.
\end{defn}
An action of a Lie group $G$ on a symplectic orbifold $(\mathcal{O},\omega)$ is said to be \emph{Hamiltonian} if it admits a \emph{moment map}. This is a map $\phi:\mathcal{O} \rightarrow \mathfrak{G}^*$, where $\mathfrak{G}^*$ denotes the dual of the Lie algebra of $G$, satisfying
\[
d\phi(x)(v)=v^\sharp \lrcorner \omega,
\]
where $v$ is an element of the Lie algebra of $G$ and $v^\sharp$ is the vector field determined by $v$ on $\mathcal{O}$. That is, if $g(t)$ is a smooth path on $G$ with $g(0)=\text{id}$ and
\[
v=\frac{d}{dt}_{|t=0}g(t),
\]
then
\[
v^\sharp(x)=\frac{d}{dt}_{|t=0}g(t)\cdot x.
\]
Note that this is well-defined up to a constant.
For toric manifolds it is known that the image of the moment map determines the manifold up to symplectomorphism. For toric orbifolds the image of the moment map is insufficient to give this result, but Eugene Lerman and Sue Tolman \cite{LT} have indicated what additional data one needs to determine the orbifold. To state their result, we begin by describing the image of the moment map.
Identifying the dual of the Lie algebra of $\mathbb{T}^n$ with $\mathbb{R}^n$, Lerman and Tolman showed that the image of the moment map of a toric orbifold is a special type of convex polytope in $\mathbb{R}^n$.
\begin{defn}\label{defn:rat_simple}
A convex polytope $P$ in $\mathbb{R}^n$ is \emph{rational simple} if
\begin{enumerate}
\item there are $n$ facets meeting at each vertex;
\item for every facet of $P$, a primitive outward normal can be chosen in $\mathbb{Z}^n$;
\item for every vertex of $P$, the outward normals corresponding to the facets meeting at that vertex form a basis for $\mathbb{Q}^n$.
\end{enumerate}
Note that a facet is a face in $P$ of codimension $1$.
\end{defn}
\noindent The triangle with vertices $(0,0), (0,1), (2,0)$ is an example of a rational simple polytope. Moreover, this polytope cannot be the moment map image of a toric manifold; in the manifold setting, we replace $\mathbb{Q}^n$ in (3) of Definition \ref{defn:rat_simple} by $\mathbb{Z}^n$.
It is also important to understand what types of singularities can occur in toric orbifolds. Lerman and Tolman show that all the points over the interior of the moment polytope of the toric orbifold are regular and that the points over the facets of the polytope have cyclic isotropy type.
\begin{thm}\cite[Theorem 6.4]{LT}\label{thm:isotropy}
Let $(\mathcal{O},\omega)$ be a toric orbifold with moment map $\phi$. Let $F$ be an open facet of $\phi(\mathcal{O})$. Then there exists an integer $m_F$ such that all points in $\phi^{-1}(F)$ have isotropy group $\mathbb{Z}_{m_F}$.
\end{thm}
The integer $m_F$ is called the label of the open facet $F$, and we call the moment polytope together with the facet labels a \emph{labeled polytope}. Thus to each toric orbifold of dimension $2n$, we can associate a labeled rational simple polytope in $\mathbb{R}^n$. Lerman and Tolman proved that these labeled polytopes essentially determine the associated symplectic orbifolds.
\begin{prop}\cite[Proposition 6.5]{LT}
If two toric orbifolds have the same labeled moment polytopes up to $SL(n,\mathbb{Z})$-transformations and translations, then the orbifolds are equivariantly symplectomorphic.
\end{prop}
\begin{exa}
We return to the $(p,q)$-footballs of Example \ref{exa1}. These are symplectic orbifolds and they admit a Hamiltonian $S^1$-action given by rotation about the north-south axis. The labeled polytope associated to this toric orbifold is the interval $[-1,1]$ with labels $p$ and $q$ at the upper and lower endpoints, respectively.
\end{exa}
In the proof of Theorem \ref{main_theorem}, we will need two further results relating the moment polytope to its associated orbifold. These results are stated and proved in the setting of toric manifolds in \cite{dgs}, but the proofs are exactly the same in the orbifold case. First we examine the fixed point set of an isometry of an orbifold.
\begin{lemma}\cite[Lemma 2.9]{dgs}\label{fixed_point_set}
Let $\theta\in \mathbb{R}^n$. The fixed point set of $\psi(e^{i\theta})$, denoted $F_\theta$, is the union of the pre-images via the moment map of all faces to which $\theta$ is normal in a face of lower codimension.
\end{lemma}
Finally, we give the relationship between the volume of a face in the polytope and the volume of its pre-image under the moment map.
\begin{lemma}\cite[Lemma 2.10]{dgs}\label{vol}
Consider a face $F$ of dimension $q$ in the labeled polytope $P$ of a symplectic toric orbifold $(\mathcal{O}, \omega)$. Let $\phi$ be the moment map of the torus action with respect to the form $\omega$. Then
\[
\operatorname{Vol}_{\omega} (\phi^{-1}(F)) = (2 \pi)^q \operatorname{Vol}(F).
\]
\end{lemma}
Since we want to study the spectrum of the Laplacian on toric orbifolds, we need to understand metrics on such orbifolds. We will restrict to metrics that are compatible with the symplectic structure and that are K\"ahler. That is, we consider metrics that come from an integrable almost complex structure $J$ on $(\mathcal{O},\omega)$ which is compatible with $\omega$; more precisely,
\[
g(\cdot,\cdot)=\omega(\cdot,J\cdot)
\]
defines a positive definite metric on $\mathcal{O}$.
It was shown in \cite{g1} that all toric manifolds admit a special K\"ahler structure which is invariant under the torus action, called the reduced K\"ahler structure. We will refer to K\"ahler structures which are invariant under the torus action as toric K\"ahler structures. In \cite{m2} Abreu showed how to construct all other toric K\"ahler structures from the reduced K\"ahler structure using functions on the moment polytope of the toric manifold. One may generalize the results in \cite{g1} and \cite{m2} to orbifolds; for a discussion of this generalization, see \cite{m3}.
\begin{thm}\cite{g1},\cite{m2}
Any toric orbifold admits many toric K\"ahler structures.
\end{thm}
As mentioned in \S \ref{sec:intro} the problem of finding the ``best" such K\"ahler structures has been the source of much work in differential geometry, and the K\"ahler structures which correspond to metrics whose scalar curvature is constant are of particular interest. We return to these metrics in \S \ref{sec:csc}.
\section{Hearing a generic toric orbifold}\label{sec:mainthm}
In this section we prove Theorem \ref{main_theorem}. First we give a precise definition of equivariant spectrum; it is entirely analogous to the corresponding definition for manifolds.
\begin{defn}
Let $\mathcal{O}^{2n}$ be a toric orbifold with a fixed torus action. Denote by $\psi:\mathbb{T}^n\rightarrow Sympl(\mathcal{O})$ the corresponding group homomorphism, and let $g$ be a toric metric on $\mathcal{O}$. The \emph{equivariant spectrum} is the list of all the eigenvalues of the Laplacian on $(\mathcal{O},g)$ together with the weights of the action induced by $\psi(e^{i\theta})$ on the corresponding eigenspaces, for all $\theta \in \mathbb{R}^n$. The eigenvalues and weights are listed with multiplicities.
\end{defn}
By studying the asymptotic expansion in Theorem \ref{main_asymptotic}, we see that the equivariant spectrum provides significant information about the moment polytope of a toric orbifold. We begin with the case in which the moment polytope has no parallel facets.
\begin{prop}\label{spectral_data}
The equivariant spectrum associated to a toric orbifold $\mathcal{O}$ whose moment polytope has no parallel facets determines
\begin{enumerate}
\item the (unsigned) normal directions to the facets;
\item the volumes of the corresponding facets;
\item the labels of the facets.
\end{enumerate}
\end{prop}
\begin{proof}
Let $\phi$ be the moment map of the torus action on $\mathcal{O}$ and let $P$ be its image, with $P$ given by
\begin{displaymath}
P=\{x\in \mathbb{R}^n: x\cdot u_i\geq c_i,\,\, i=1,\ldots,d\}
\end{displaymath}
for some collection of $u_i$ in $\mathbb{R}^n$ and $c_i \in \mathbb{R}$.
For each $u$ in $\mathbb{R}^n$ let $\psi(u)$ denote the isometry of $\mathcal{O}$ given by the $e^{u}$-action on $\mathcal{O}$. For each $\lambda$ in the spectrum of $\mathcal{O}$, the map $\psi(u)$ induces a linear action on the $\lambda$-eigenspace of $\mathcal{O}$ which we denote by $\psi^\sharp_\lambda(u)$. The asymptotic expansion from Theorem \ref{main_asymptotic} gives
\begin{equation}\label{eqn:asymp}
\sum_\lambda \text{tr} ( \psi^\sharp_\lambda(u))e^{-t\lambda} \simeq
\sum_{V_i} \frac{1}{|\operatorname{Iso}(V_i)|}(4\pi t)^{-\frac{\dim (V_i)}{2}}\text{Vol}(V_i)D_i +O(t) \ ,
\end{equation}
where the $V_i$ are the connected components of the fixed point set of $\psi(u)$ and $D_i$ is calculated as follows. Let $x$ be in $V_i$ and choose an orbifold chart $(\tilde{U}, G_U, \pi_U)$ about $x$, where $G_U = \operatorname{Iso}(x)$. We can locally lift $\psi(u)$ to an isometry $\tilde\psi(\tilde{u})$ on $\tilde{U}$ which commutes with $G_U$. Then
\[
D_i(x)=\sum_{\gamma\in \operatorname{Iso}^{\operatorname{max}}(V_i)} \left| \det((I-A_{\psi, \gamma}(\tilde{x}))^{-1}) \right|
\]
where $A_{\psi, \gamma}(\tilde{x})$ is as in \S \ref{sec:eq_heat_oflds}. Note that $D_i$ is $G_U$-invariant and locally constant, so it is indeed a constant function on $V_i$.
We will now untangle the polytope data contained in the right side of \eqref{eqn:asymp}.
Set $u=ru_i$ for some $r$ and some $u_i$. If $r\ne 0$, Lemma \ref{fixed_point_set} implies that the connected components of the fixed point set of $\psi(u)$ are the pre-image via the moment map of all the faces whose normal is parallel to $u_i$. The highest-dimensional connected components have dimension $2(n-1)$ and we can determine the $u_i$'s up to sign: they are the vectors for which the right side of \eqref{eqn:asymp} has power $-(n-1)$ in $t$. When our polytope does not have parallel facets there is a single highest-dimensional connected component of $\text{Fix}(\psi(u))$, namely the pre-image of a facet $F$; we can determine its volume and its label by considering how $D_r$ varies with $r$. More precisely, given $x \in \phi^{-1}(F)$ we can find an orbifold chart $(\tilde{U}, G_U, \pi_U)$ about $x$ and we know by Theorem \ref{thm:isotropy} that $G_U$ is a cyclic group, say of order $\Omega$. Let $\tilde{\psi}_r$ be the local lift of $\psi(ru_i)$. We have
\[
D_r(x)=\sum_{l=1}^{\Omega-1} \left|\det((I-A_{\tilde{\psi_r}, \gamma^l}(\tilde{x}))^{-1})\right|
\]
where $\gamma$ generates $G_U$. Now $T_{\tilde{x}}\operatorname{Fix}(\tilde{\psi_r})^{\perp}$ is a two-dimensional vector space and $A_{\tilde{\psi_r}, \gamma^l}$ are isometries of that space, so they must be rotations. Note that $d\gamma^{\Omega}$ is the identity and therefore the rotation angle of $d\gamma$ is a multiple of $2\pi/\Omega$; by reordering we may assume that the rotation angle equals $2\pi/\Omega$.
Since $\tilde\psi_r$ is a group homomorphism from $S^1$ to the group of isometries of $\mathcal{O}$, we see that $d\tilde\psi_r$ is a rotation whose angle is $r\theta$ for some fixed $\theta$. Therefore $A_{\tilde{\psi_r}, \gamma^l}$ is a rotation of angle $r\theta+\frac{2l\pi}{\Omega}$, and the matrix representation of $I-A_{\tilde{\psi_r}, \gamma^l}$ is
\[
\begin{pmatrix}
1-\cos(r\theta+\frac{2l\pi}{\Omega})&-\sin(r\theta+\frac{2l\pi}{\Omega})\\
\sin(r\theta+\frac{2l\pi}{\Omega})&1-\cos(r\theta+\frac{2l\pi}{\Omega})
\end{pmatrix}.
\]
Thus
\[
D_r=\sum_{l=1}^{\Omega-1} \frac{1}{2-2\cos(r\theta+\frac{2l\pi}{\Omega})},
\]
and the coefficient corresponding to the lowest-order term in the right side of \eqref{eqn:asymp} is
\[
\frac{(4 \pi)^{-(n-1)}}{\Omega}\text{Vol}(F)\sum_{l=1}^{\Omega-1} \frac{1}{2-2\cos(r\theta+\frac{2l\pi}{\Omega})}.
\]
Hence this coefficient is spectrally determined for all $r\ne 0$, which implies that $\text{Vol}(F)$ and $\Omega$ are spectrally determined. Note that we have used that the volume of $\phi^{-1}(F)$ is proportional to the volume of $F$, as indicated in Lemma \ref{vol}.
\end{proof}
We next address the fact that the spectrum only determines the normals of the facets up to sign. Note that a convex polytope with associated facet normals and volumes $\{(u_i,\nu_i),i=1,\dots,d\}$ always satisfies
\begin{displaymath}
\sum_{i=1}^d \nu_i u_i=0.
\end{displaymath}
\begin{defn}
Let $P$ be a convex polytope in $\mathbb{R}^n$ with associated facet normals and volumes $\{(u_i,\nu_i),i=1,\ldots,d\}$. We say that $P$ has no subpolytopes if
\begin{displaymath}
\sum_{i\in I}\nu_i u_i=0 \ \ \text{implies} \ \ I=\{1,\ldots,d \} \,\,\text{or}\,\, I=\emptyset.
\end{displaymath}
\end{defn}
For convex polytopes with no subpolytopes the set of normals up to sign determines the actual normals up to a finite number of sign choices.
\begin{lemma}\label{two_choices}
Let $P$ be a convex polytope in $\mathbb{R}^n$ with no subpolytopes and facet volumes $\nu_1,\dots,\nu_d$. Assume that the facet normals to $P$ are $u_1, \dots, u_d$ up to sign. Then, up to translation, there are only $2$ choices for the set of signed normals.
\end{lemma}
\begin{proof}
For each choice $\xi_i=\pm u_i, i=1,\dots , d$, corresponding to a convex polytope we have
\begin{displaymath}
\sum_{i=1}^d \nu_i \xi_i=0.
\end{displaymath}
Adding the sums corresponding to two different choices, we get a relation
\begin{displaymath}
\sum_I \nu_i \xi_i=0.
\end{displaymath}
Our assumption that $P$ has no subpolytopes ensures that $I$ is actually empty and the two choices must be $(\xi_1, \dots, \xi_d)$ and $(-\xi_1, \dots, -\xi_d)$, which clearly give rise to convex polytopes.
\end{proof}
Thus we see that the equivariant spectrum of a toric orbifold with no parallel facets and no subpolytopes determines two collections of facet normals and corresponding facet volumes. The question we are asking becomes a purely combinatorial one.
\begin{question}
Do the normal directions to the facets of a rational simple polytope and the corresponding facet volumes determine the rational simple polytope uniquely?
\end{question}
For convex polytopes, this question and its answer are known as the Minkowski problem. Daniel Klain \cite{klain} recently gave an elegant solution to this problem.
\begin{thm}\cite[Thm. 2]{klain} \label{klain}
Given a list
$\{(u_i,\nu_i), u_i \in \mathbb{R}^n, v_i\in \mathbb{R}^+,\,\,\,i=1,\dots, d\}$
where the $u_i$ are unit vectors that span $\mathbb{R}^n$, there exists a convex polytope $P$ with facet normals $u_1, \dots, u_d$ and corresponding facet volumes $\nu_1, \dots, \nu_d$ if and only if
\[
\sum_{i=1}^d \nu_i u_i = 0.
\]
Moreover, this polytope is unique up to translation.
\end{thm}
Klain proves uniqueness of the solution to the Minkowski problem using a clever inductive argument involving the Minkowski and Brunn-Minkowski inequalities. These inequalities are generalized isoperimetric inequalities for so-called \emph{mixed volumes}, which encode the relationship between a compact convex set and its orthogonal projections onto subspaces. For the base case in his induction, Klain gives the solution to the Minkowski problem in dimension $2$ and derives the inequalities as consequences. He then assumes that the Minkowski and Brunn-Minkowski inequalities hold in dimension $n-1$ and uses them to prove uniqueness in dimension $n$. The Minkowski inequality in dimension $n$ follows, and this in turn implies the Brunn-Minkowski inequality in dimension $n$. See \cite{klain} for more details.
Together with Proposition \ref{spectral_data} and Lemma \ref{two_choices}, Theorem \ref{klain} implies the following preliminary result.
\begin{cor}
Let $(\mathcal{O},\omega)$ be a toric orbifold with a toric K\"ahler metric such that the moment polytope of $\mathcal{O}$ has no parallel facets and no subpolytopes. Then the equivariant spectrum of $\mathcal{O}$ determines the moment polytope of $\mathcal{O}$ up to translation and $2$ choices, and hence determines $\mathcal{O}$ up to symplectomorphism.
\end{cor}
In order to prove Theorem \ref{main_theorem}, we only need to show that rational simple polytopes with no parallel facets and no subpolytopes are generic among all rational simple polytopes.
\begin{lemma}
Close to any rational simple polytope in $\mathbb{R}^n$, there is a rational simple polytope that has no parallel facets and has no subpolytopes.
\end{lemma}
\begin{proof}
Let $P=\{x \in \mathbb{R}^n: x\cdot u_i\geq c_i, i=1,\dots, d\}$ be a rational simple polytope and let $\nu_1,\dots, \nu_d$ be the corresponding facet volumes.
It is easy to perturb our polytope to a rational simple polytope without parallel facets, and this is precisely the advantage of toric orbifolds over toric manifolds. Suppose that $P$ has two parallel facets, say those with labels $1$ and $i$. Choose $\tilde u_1 \in \mathbb{Q}^n$ very close to $u_1$. Let
\[
\tilde{P}=\{x \in \mathbb{R}^n: x\cdot u_i\geq c_i, i=2,\dots d, \,\,\,x\cdot \tilde{u}_1\geq c_1\}.
\]
It is easy to see that $\tilde{P}$ is simple and rational: one can multiply $\tilde{u}_1$ by an appropriate integer multiple so that $\tilde{u}_1 \in \mathbb{Z}^n$, and if $\tilde{u}_1$ is sufficiently close to $u_1$, then $\{\tilde{u}_1,u_2,\dots,u_n\}$ will remain a basis of $\mathbb{Q}^n$.
Now we show that one can also perturb our polytope to a rational simple polytope without subpolytopes. If $P$ has a subpolytope then there is a proper subset $I$ of $\{1,\dots,d\}$ such that
\begin{displaymath}
\sum_I \nu_i u_i=0.
\end{displaymath}
Choose $j \in \{1, \dots, d\} \setminus I$ and perturb $P$ by moving the facet $F_j$ perpendicular to $u_j$ along $u_j$ by distance $\epsilon$, thus changing $c_i$. For $\epsilon$ sufficiently small, we will not introduce new subpolytopes. The only $\nu_i$ 's that are changed by this perturbation are the ones corresponding to facets which intersect $F_j$, and one obtains a one parameter family of polytopes. If the sum
\begin{displaymath}
\sum_I \nu_i(t) u_i
\end{displaymath}
is nonzero for some small value of the parameter $t$, then we have the desired perturbation. If not, then taking derivatives with respect to $t$ yields
\begin{displaymath}
\sum_{I'} \nu_i'(t) u_i=0,
\end{displaymath}
where $I'$ is the subset of $i\in I$ such that $F_i\cap F_j \neq \emptyset$. It is not hard to see that
\begin{displaymath}
\nu_i'(0)=\text{Vol}(F_i\cap F_j),
\end{displaymath}
so that
\begin{displaymath}
\sum_{I'} \text{Vol}(F_i\cap F_j)u_i=0.
\end{displaymath}
Choose $k \in I'$ such that $k \neq j$. By slightly perturbing $u_k$ in $\mathbb{Q}^n$ without changing the volume $F_k\cap F_j$ (i.e., such that $u_j^\perp\cap u_k^\perp$ is unchanged), the preceding sum equals a nonzero vector and one obtains the desired rational simple polytope without subpolytopes.
\end{proof}
The two possibilities in Theorem \ref{main_theorem} corresponding to the polytopes $P$ and $-P$ may not be equivariantly symplectomorphic, and thus the equivariant spectrum does not determine a single pair (toric orbifold, torus action). However, the underlying toric manifolds \emph{are} symplectomorphic (but not equivariantly so).
\begin{prop}\label{unique_symplectic}
Let $(\mathcal{O},\omega)$ be a generic toric orbifold endowed with a toric K\"ahler metric. Then the equivariant spectrum of $\mathcal{O}$ determines the symplectomorphism type of $\mathcal{O}$.
\end{prop}
\begin{proof}
Since we have proved Theorem \ref{main_theorem}, we only need to show that the orbifolds $\mathcal{O}_P$ and $\mathcal{O}_{-P}$ are symplectomorphic. Let $P$ be given by
\begin{equation} \label{eqn:P}
P=\{x\in \mathbb{R}^n: x\cdot m_i u_i\geq c_i,\,\, i=1,\ldots,d\}
\end{equation}
where the $u_i$ are the primitive inward-pointing normals to the facets of $P$ with corresponding weights $m_i$, and $c_i \in \mathbb{R}$.
Then $-P$ is given by \eqref{eqn:P} with $-u_i$ replacing $u_i$. The explicit construction of the toric symplectic orbifolds associated to such polytopes (e.g., \cite[pp. 8-9]{m3}) shows that these two orbifolds are in fact the same orbifold, with different torus actions. For the sake of completeness we briefly describe the construction. For the orbifold associated with the polytope $P$ consider the exact sequences
\begin{eqnarray*}\label{exactseq}
0 \rightarrow N \rightarrow \mathbb{T}^d \xrightarrow{\beta'} \mathbb{T}^n \rightarrow 0 \\
0 \rightarrow \mathfrak{n} \xrightarrow{\iota} \mathbb{R}^d \xrightarrow{\beta} \mathbb{R}^n \rightarrow 0
\end{eqnarray*}
where $\beta$ takes an element $e_i$ in the canonical basis for $\mathbb{R}^d$ to $m_iu_i$, and $\mathfrak{n}$ is the Lie algebra of $N$. The group $N$ acts symplectically on $\mathbb{C}^d$ with moment map
\begin{displaymath}
\phi(z)=\sum|z_i|^2\iota^*e_i.
\end{displaymath}
The toric orbifold associated to $P$ is $\mathcal{O}=\phi^{-1}(c)/N$,
where $c \in \mathfrak{n}$ is determined by the $c_i$ as $c=-\sum c_i\iota^*e_i$. The symplectic structure on $\mathcal{O}$ comes from the canonical symplectic structure on $\mathbb{C}^d$ via symplectic quotient in the usual way. We see that this construction will yield the same orbifold $\mathcal{O}$ with the same symplectic form when we replace $\beta$ by the map $-\beta$ defined by $-\beta(e_i) = -m_i u_i$. Hence $\mathcal{O}_P$ and $\mathcal{O}_{-P}$ are symplectomorphic.
\end{proof}
Note that the $\mathbb{T}^n$-actions on $\mathcal{O}_P$ and $\mathcal{O}_{-P}$ differ. For example, for $\mathcal{O}_P$ we have
\begin{displaymath}
e^{itu_l}\cdot [z_1,\ldots, z_d]=[z_1,\ldots,e^{it}z_l,\ldots,z_d],
\end{displaymath}
whereas we have
\begin{displaymath}
e^{itu_l}\cdot [z_1,\ldots, z_d]=[z_1,\ldots,e^{-it}z_l,\ldots,z_d]
\end{displaymath}
for $\mathcal{O}_{-P}$.
\section{Constant scalar curvature metrics are audible}\label{sec:csc}
The goal of this section is to prove Theorem \ref{thm:csc}.
In this theorem, ``generic'' has the same meaning as in Theorem \ref{main_theorem}.
In particular, the equivariant spectrum of $(\mathcal{O},g)$ must determine the moment polytope of $\mathcal{O}$ (up to translation and two choices) as well as its labels. Theorem \ref{thm:csc} also holds in the setting of smooth toric $2n$-manifolds, provided the corresponding Delzant polytope is audible (again, up to translation and two choices). From \cite{dgs} we know that this occurs for $n=2$ if the manifold is generic and has at most $3$ pairs of parallel sides.
Throughout this section we will make use of Chern classes for orbifolds. For the relevant background, see \cite[\S2]{gRR}. As for manifolds, Chern classes for orbifolds are diffeomorphism invariants.
We start with a few preliminary results.
\begin{lemma}\label{lem:charclasses}
The quantities $\int_\mathcal{O} c_2\wedge \omega^{n-2}$ and $\int_\mathcal{O} c_1^2\wedge \omega^{n-2}$ are determined by the equivariant spectrum of $\mathcal{O}$, as is the cohomology class of $\omega$.
\end{lemma}
Henceforth we will write $\int_\mathcal{O} c_2$ and $\int_\mathcal{O} c_1^2$ for $\int_\mathcal{O} c_2\wedge \omega^{n-2}$ and $\int_\mathcal{O} c_1^2\wedge \omega^{n-2}$, respectively.
\begin{proof}
We know from \S \ref{sec:mainthm} that the equivariant spectrum determines a set of two rational simple polytopes $P$ and $-P$ that can arise as moment map images of $\mathcal{O}$. It also determines a unique set of labels $L$ associated to the faces of $P$ (or $-P$). The data $(P,L)$ determine a symplectic toric orbifold $\mathcal{O}_P$ and the data $(-P, L)$ determine another symplectic toric orbifold $\mathcal{O}_{-P}$. We know that $\mathcal{O}$ is equivariantly symplectomorphic to either $\mathcal{O}_P$ or $\mathcal{O}_{-P}$. Proposition \ref{unique_symplectic} ensures that $\mathcal{O}_P$ and $\mathcal{O}_{-P}$ are symplectomorphic even though they are not equivariantly symplectomorphic. This then implies that the characteristic classes of $\mathcal{O}$ are determined from its equivariant spectrum, and so is the cohomology class of $\omega$.
\end{proof}
Since the labels of the moment polytope of $\mathcal{O}$ are uniquely determined, the isotropy groups of points in $\mathcal{O}$ are uniquely determined.
\begin{lemma}\label{hear_iso}
The isotropy groups associated to all points in $\mathcal{O}$ are determined by the equivariant spectrum.
\end{lemma}
\begin{proof}
A point $p\in \mathcal{O}$ is associated to certain facets of its moment polytope $P$; let $\mathcal{F}(p)$ denote the set of facets of $P$ containing $\phi(p)$. For each facet $F_i$ in $\mathcal{F}(p)$, let $u_i$ denote its primitive outward normal and $m_i$ its label.
Define
\[
\Lambda_p=\text{Span}_{\mathbb{Z}}\{u_i: F_i\in \mathcal{F}(p)\},\,\,\,\hat{\Lambda}_p=\text{Span}_{\mathbb{Z}}\{m_iu_i: F_i\in \mathcal{F}(p)\}.
\]
In \cite{LT}, Lerman and Tolman show that $\text{Iso}(p)$ is $\Lambda_p/\hat{\Lambda}_p$. Thus $P$ and $L$ determine the isotropy groups of all points in $\mathcal{O}$, and one can check that the above construction gives the same isotropy groups for $-P$.
\end{proof}
\begin{rem}
Note that the above characterization of isotropy groups shows that all elements in the pre-image of an open facet in the moment polytope have the same isotropy group. In fact we see that the orbifold stratification is as follows. The highest-dimensional stratum $\mathcal{S}_0$ is the pre-image of the interior of the polytope. The strata of codimension $1$, $\mathcal{S}_{1}$, are the pre-image of the union of the interior of codimension $1$ facets. In general, the codimension $i$ strata, $\mathcal{S}_{i}$, are the pre-image of the union of the relative interior of the faces of codimension $i$.
\end{rem}
We now prove Theorem \ref{thm:csc}.
\begin{proof}
Let us first apply Theorem \ref{main_asymptotic} to $\mathcal{O}$ with $f$ equal to the identity. This is the case that is treated in \cite{dggw}. The heat trace is asymptotic to
\[
\sum_{S \in \mathcal{S}(\mathcal{O})} \frac{1}{|\text{Iso}(S)|} (4\pi t)^{-\frac{\text{dim}(S)}{2}} \int_{ S} \sum_{k=0}^{\infty} b_k(I, S) t^k \text{dvol}_{S}(x).
\]
In this expansion the term in $t^{-n+2}$ is spectrally determined. It is given by
\begin{equation}\label{eqn:id_exp}
(4\pi)^{-n}b_2(\mathcal{O})+(4\pi)^{-n+1}\sum_{i=1}^d \frac{\int_{F_i} b_1(F_i)}{m_i}+(4\pi)^{-n+2}\sum_{a\in I} \frac{\int_{F_a} b_0(F_a)}{m_a},
\end{equation}
where $F_i$ denotes the pre-image via the moment map of a face and $m_i$ is the corresponding label; $I$ denotes some set that indexes codimension $2$ faces; $F_a$ denotes the pre-image via the moment map of a codimension $2$ face; and $m_a=|\operatorname{Iso}(F_a)|$.
Note that we use $F_k$ to denote both the $k$th face and its pre-image under the moment map. Although this is an abuse of notation, it should not cause confusion. We show that each of the terms in \eqref{eqn:id_exp} is determined by the equivariant spectrum.
We begin with the last term. We have that $\int_{F_a}b_0(F_a)=\text{Vol}(F_a)$ is determined by $P$ (equivalently, by $-P$), hence it is determined by the equivariant spectrum. By Lemma \ref{hear_iso} we also hear $m_a$, implying that the last term in \eqref{eqn:id_exp} is determined by the equivariant spectrum.
Next we consider the middle term in \eqref{eqn:id_exp}.
Let $F$ be a face and $u$ the corresponding normal. Consider the isometry $\psi_u$ defined by
\[
\psi_u(p)=e^{iu}.p \ ;
\]
it follows from Theorem \ref{main_asymptotic} that the asymptotic behavior of the $\psi_u$-invariant heat trace is given by
\[
\sum_{S \in \mathcal{S}(\mathcal{O})} \frac{1}{|\text{Iso}(S)|} (4\pi t)^{-\frac{\text{dim}(\text{Fix} \psi_u \cap S)}{2}} \int_{\text{Fix} \psi_u \cap S} \sum_{k=0}^{\infty} b_k(f,S) t^k \text{dvol}_{\text{Fix} \psi _u \cap S}(x).
\]
Now $\operatorname{Fix} \psi_u=F$, which has dimension $2(n-1)$ and thus does not intersect $\mathcal{S}_0$. The coefficient of $t^{-n+2}$ in this expansion is given by
\[
(4\pi)^{-n+1}\frac{\int_F b_1(F)}{m}+(4\pi)^{-n+2}\sum_{a\in I} \frac{\int_{F_a} b_0(F_a)}{m_a},
\]
where $m$ is the label corresponding to $F$; $I$ denotes some set that indexes the codimension $2$ faces that intersect $F$; $F_a$ denotes the pre-image via the moment map of a codimension $2$ face; and $m_a=|\operatorname{Iso}(F_a)|$.
The same argument as above shows that each summand in the second term of this expansion is determined by the equivariant spectrum. The index set $I$ is also determined by $P$, so that the sum over $I$ is spectrally determined. Thus we hear $\frac{\int_F b_1(F)}{m}$ for each face, and hence the middle term in \eqref{eqn:id_exp} is determined by the equivariant spectrum.
Since the middle and last terms in \eqref{eqn:id_exp} are spectrally determined, so is $b_2(\mathcal{O})$.
It follows from \eqref{eqn:b2} that
\begin{equation}\label{b_2}
360 b_2(\mathcal{O})= \int _\mathcal{O} b_2(x) \text{dvol}_{\mathcal{O}}(x) =\int_\mathcal{O} (2|R|^2-2|\rho|^2+5s^2) \text{dvol}_{\mathcal{O}}(x)
\end{equation}
where $R$ denotes the full curvature tensor, $\rho$ denotes the Ricci tensor, and $s$ denotes the scalar curvature. Our next goal is to show how the right side of \eqref{b_2} can be expressed as a linear combination of $\int_\mathcal{O} s^2$, $\int_\mathcal{O} c_1^2$, and $\int_\mathcal{O} c_2$, where the coefficient in $\int_\mathcal{O} s^2$ is nonzero.
We first show that $\int_\mathcal{O} |\rho|^2$ can be written as a linear combination of $\int_\mathcal{O} s^2$ and $\int_\mathcal{O} c_1^2$. Recall that the complex dimension of $\mathcal{O}$ is $n$. Let $\rho_0$ be the primitive part of the Ricci curvature so that
\[
\rho=\frac{\text{Tr}(\rho)}{n} \omega+\rho_0.
\]
By definition, $\text{Tr}(\rho)=s$ and the decomposition above is orthogonal, implying that
\[
|\rho|^2=\frac{s^2}{n^2}|\omega|^2+|\rho_0|^2.
\]
Since $|\omega|^2=n$ this becomes
\[
|\rho|^2=\frac{s^2}{n}+|\rho_0|^2.
\]
Using the Apte formula (see \cite[p. 80]{besse}) we have
\begin{equation}\label{eqn:c1squared}
\frac{4\pi^2}{(n-2)!}\int_\mathcal{O} c_1^2=\frac{n-1}{4n}\int_\mathcal{O} s^2 -\int _\mathcal{O} |\rho_ 0|^2,
\end{equation}
and thus
\begin{align}\nonumber
\int_\mathcal{O} |\rho|^2=&\frac{1}{n}\int_\mathcal{O} s^2+\frac{n-1}{4n}\int_\mathcal{O} s^2-\frac{4\pi^2}{(n-2)!}\int_\mathcal{O} c_1^2\\ \nonumber
=& \frac{n+3}{4n} \int_\mathcal{O} s^2-\frac{4\pi^2}{(n-2)!}\int_\mathcal{O} c_1^2\ . \label{ricci}
\end{align}
Next we write $\int_\mathcal{O} |R|^2$ as a linear combination of $\int_\mathcal{O} s^2$, $\int_\mathcal{O} c_1^2$, and $\int_\mathcal{O} c_2$. One can view $R$ as an endomorphism of $\Omega^2\mathcal{O}$ and decompose it as
\[
R=U+Z+W,
\]
where the tensor $U$ is determined by the scalar curvature, $Z$ is related to the trace-free part of the Ricci tensor, and $W$ is the usual Weyl tensor.
This decomposition is also orthogonal so that $|R|^2=|U|^2+|Z|^2+|W|^2$. We have the relations
\[
|W|^2=|B_0|^2+\frac{3(n-1)}{n+1}|U|^2+\frac{n-2}{n}|Z|^2, \, |U|^2=\frac{s^2}{4n(2n-1)},\, |Z|^2=\frac{|\rho_0|^2}{n-1},
\]
where $B_0$ is the trace-free part of a tensor that arises in a different decomposition of $R$ (see \cite[p. 77]{besse}). Thus
\begin{eqnarray}
|R|^2& = & |B_0|^2+\frac{2(2n-1)}{n+1}|U|^2+\frac{2(n-1)}{n}|Z|^2 \nonumber \\
& = & |B_0|^2+\frac{1}{2n(n+1)}s^2+\frac{2}{n}|\rho_0|^2. \label{eqn:Rsquared}
\end{eqnarray}
Using the Apte formula again gives
\[
\frac{8\pi^2}{(n-2)!}\int_\mathcal{O} c_2=\frac{n-1}{4(n+1)}\int_\mathcal{O} s^2 -\frac{2(n-1)}{n}\int _\mathcal{O} |\rho_ 0|^2+\int_\mathcal{O} |B_0|^2\ ;
\]
we substitute this expression and that from \eqref{eqn:c1squared} into \eqref{eqn:Rsquared} to get
\begin{align}\nonumber
\int_\mathcal{O} |R|^2 = & \frac{8\pi^2}{(n-2)!}\int_\mathcal{O} c_2-\frac{n-1}{4(n+1)}\int_\mathcal{O} s^2 +\frac{2(n-1)}{n}\int _\mathcal{O} |\rho_ 0|^2 +\frac{1}{2n(n+1)}s^2+\frac{2}{n} \int _\mathcal{O} |\rho_ 0|^2\\ \nonumber
=& \frac{8\pi^2}{(n-2)!}\int_\mathcal{O} c_2+\frac{2+n-n^2}{4n(n+1)}\int_\mathcal{O} s^2 +2\int _\mathcal{O} |\rho_ 0|^2\\ \nonumber
=& \frac{8\pi^2}{(n-2)!}\int_\mathcal{O} c_2+\frac{2+n-n^2}{4n(n+1)}\int_\mathcal{O} s^2 -\frac{8\pi^2}{(n-2)!}\int_\mathcal{O} c_1^2+\frac{n-1}{2n}\int_\mathcal{O} s^2 \\ \nonumber
=&\frac{8\pi^2}{(n-2)!}\int_\mathcal{O} (c_2-c_1^2)+\frac{1}{4}\int_\mathcal{O} s^2 \ .
\end{align}
We can replace $\int_\mathcal{O} |\rho|^2$ and $\int_\mathcal{O} |R|^2$ in (\ref{b_2}) by the equivalent expressions we have found to get
\[
360 b_2(\mathcal{O})=\frac{16\pi^2}{(n-2)!}\int_\mathcal{O} (c_2-c_1^2)+\frac{1}{2}\int_\mathcal{O} s^2-\frac{n+3}{2n} \int_\mathcal{O} s^2+\frac{8\pi^2}{(n-2)!}\int_\mathcal{O} c_1^2+5\int_\mathcal{O} s^2,
\]
which simplifies to
\[
360 b_2(\mathcal{O})=\frac{8\pi^2}{(n-2)!}\int_\mathcal{O} (2c_2-c_1^2)+\frac{10n-3}{2n}\int_\mathcal{O} s^2 \ .
\]
We saw above that $b_2(\mathcal{O})$ is determined by the equivariant spectrum, and so are $\int_\mathcal{O} c_1^2$ and $\int_\mathcal{O} c_2$ by Lemma \ref{lem:charclasses}; hence $\int_\mathcal{O} s^2$ is spectrally determined.
We conclude our proof of Theorem \ref{thm:csc} by giving a characterization of constant scalar curvature that is amenable to our spectral setting. Consider the integral
\[
\mathcal{C}(g)=\int_\mathcal{O} (s-\bar{s})^2\frac{\omega^n}{n!},
\]
where $\bar{s}$ denotes the average of the scalar curvature, i.e., $\bar{s}=\frac{\int_\mathcal{O} s}{\text{Vol}(\mathcal{O})}$.
It is known that $\bar{s}$ is determined by the symplectic topology of $\mathcal{O}$ (see \cite{Don_Calabi}): we have
\[
\frac{s\omega^n}{n!}=\frac{2\pi c_1\wedge \omega^{n-1}}{(n-1)!}
\]
so that we may write $\bar{s}$ as
\[
\bar{s}=\frac{2\pi c_1\wedge [\omega]^{n-1}}{\text{Vol}(\mathcal{O})(n-1)!}.
\]
The metric $g$ has constant scalar curvature if and only if $\mathcal{C}(g)$ is zero. One can write $\mathcal{C}(g)$ as
\begin{eqnarray*}
\mathcal{C}(g)&=&\int_\mathcal{O} (s^2-2\bar{s}s+\bar{s}^2)\frac{\omega^n}{n!} \\
&=& \int_\mathcal{O} s^2\frac{\omega^n}{n!}-2\bar{s}\int_\mathcal{O} s\frac{\omega^n}{n!}+\bar{s}^2 \text{Vol}(\mathcal{O})\\
&=&\int_\mathcal{O} s^2\frac{\omega^n}{n!}-2\bar{s}^2\text{Vol}(\mathcal{O})+\bar{s}^2 \text{Vol}(\mathcal{O})\\
&=& \int_\mathcal{O} s^2\frac{\omega^n}{n!}-\bar{s}^2\text{Vol}(\mathcal{O})
\end{eqnarray*}
so that $g$ has constant scalar curvature exactly when
\[
\int_\mathcal{O} s^2\frac{\omega^n}{n!}=\frac{1}{\text{Vol}(\mathcal{O})}\left(\frac{2\pi c_1\wedge [\omega]^{n-1}}{(n-1)!}\right)^2\ .
\]
By Lemma \ref{lem:charclasses}, the expression on the right is determined by the equivariant spectrum.
Hence to determine if $g$ has constant scalar curvature we ``hear'' $\int_\mathcal{O} s^2$, we ``hear'' $\frac{1}{\text{Vol}(\mathcal{O})}\left(\frac{2\pi c_1\wedge [\omega]^{n-1}}{(n-1)!}\right)^2$, and we compare the two quantities.
\end{proof}
\bibliographystyle{plain}
|
1,108,101,566,802 | arxiv | \section{Introduction}
\IEEEPARstart{F}{UTURE} wireless networks are envisioned to offer an unprecedented opportunity to connect the global world via a massive number of low-power heterogeneous smart devices, enabled by the internet of Things (IoTs) \cite{Akpakwu}. A major bottleneck for the application of such untethered nodes is their finite battery capacity, requiring the need to be recharged/replaced rather frequently. In this context, simultaneous wireless information and power transfer (SWIPT) has emerged as a promising technology to address the conflicting design goals of perpetual lifetime and uninterrupted network performance. In a SWIPT-enabled system, a wireless node is powered up by a received Radio Frequency (RF) signal and, simultaneously, information processing is carried out using the same signal \cite{Varshney2008}.
\par SWIPT-based relaying was proposed as a promising technique to provide advantages in two fold. First, the network itself can benefit from the relays in throughput improvement, communication reliability enhancement, and coverage range extension. Second, the harvested energy can be used to charge the relay nodes, and therefore, the overall power consumption of the network may be considerably reduced \cite{Nasir2013, Mohjazi3}. From this perspective, the theoretical and implementation aspects of SWIPT relay networks have been areas of active research interest (see \cite{Rabie1,Al-habob,Fang,Ojo} and the references therein).
\par Although there has been a growing literature on SWIPT, particularly in the context of relay networks (see e.g., \cite{Rabie1,Al-habob,Fang,Ojo} and the references therein), all research studies were based upon the classical assumption of additive white Gaussian noise (AWGN). However, many communication channels are additionally impaired by impulsive man-made electromagnetic interference or atmospheric noise encountered in various metropolitan and indoor wireless applications, such as, automotive ignition, electronic devices, household appliances, medical equipment, and industrial equipment. \cite{Blankenship, Blackard,Sanchez}. A practical foreseen scenario of such a situation is future IoTs, for instance, where nodes can be implanted in environments that are susceptible to impulsive noise such as in industrial locations or in fields close to power lines. Although these nodes are envisioned to be powered by RF energy through SWIPT to achieve advantages, such as, dual use of RF signals for information and power transfer, extended network lifetime, etc., their performance in terms of error rate is not yet studied when impulsive noise is considered. Nonetheless, it is considered as a prevalent source of performance degradation. It has been demonstrated in \mbox{\cite {Spaulding}} that communication systems designed under the AWGN assumption typically suffer from severe performance degradations when exposed to impulsive noise. This elevates the need for studying the performance of SWIPT systems, which are not only disturbed by multipath fading, but also by impulsive (non-Gaussian) noise, in order to provide pragmatic information for the system designer.
\par Several statistical models have been proposed to approximate the behaviour of impulsive noise, such as Bernoulli-Gauss \cite{Ghosh}, the symmetric alpha stable distribution \cite{Ilow}, and the Middleton's models \cite{Middleton1, Middleton2}. However, Middleton's models have been widely accepted to model the effects of impulse noise in communication systems due to its accuracy in approximating the behaviour of this noise over many communication channels and since its validity was confirmed by many measurement campaigns. Among the three distinct noise categories of Middleton's models, the most popular is the so-called Middleton Class-A (MCA) noise model \cite{Middleton2}. Additionally, this model presents the advantage to be a generic model which only depends on three physical parameters, namely, the noise power, the impulsive index that describes the average number of impulses during some interference time, and the Gaussian factor which resembles the ratio of the variances of the background Gaussian noise to the impulsive noise. Furthermore, the MCA noise model is characterized by a simple probability density function (PDF) expression which enables designing an optimum receiver with low complexity.
\par Several research studies in the open literature have investigated the effect of the MCA impulsive noise on conventional non-energy harvesting (EH) communication systems \cite{Alhussein1,Ping,Al-Dharrab,Gao,Schober1} and the references therein. However, these studies focus on examining the impact of impulsive noise on the process of information delivery only. Nonetheless, SWIPT systems are characterized both by information and power delivery simultaneously. Therefore, a thorough analysis of the effect of impulsive noise is an inevitable prerequisite for the appropriate design of impulsive noise combating mechanisms and robust receivers for such systems.
\par While most of the current literature on SWIPT systems is based upon the assumption of the classical AWGN noise assumption, there have been recent results \cite{Rabie2,Rabie3} which study the performance of a point-to-point SWIPT system under the assumption of impulsive noise following the Bernoulli-Gauss model. To the best of our knowledge, the impact of impulsive noise on the performance of SWIPT is not comprehensively understood yet, since it has not been addressed in the related open literature, which demands for a thorough investigation. We note that such an investigation is imperative for the actual realization of SWIPT and for determining the actual performance limits in terms of error rate performance.
\par Aiming to fulfil this research gap, we propose an accurate mathematical framework to analyse the pairwise error probability (PEP) performance of SWIPT relaying systems over Rayleigh fading channels subject to MCA. PEP constitutes the stepping stone for the derivation of union bounds to the error probability. It is widely used in the literature to analyse the achievable diversity order, where closed-form error probability expressions are unavailable. In particular, we assume that SWIPT relaying is enabled by a power splitting (PS) receiver architecture \cite{Nasir2013} and adopt the amplify-and-forward (AF) relaying protocol with two schemes depending on the availability of channel state information (CSI) at the relay node, namely, a CSI-assisted relaying scheme and a blind relaying scheme. Additionally, we adopt two EH techniques: EH based on average CSI (AEH) \cite{Liu} and EH based on instantaneous CSI (IEH) \cite{Mohjazi4}. Specifically, the main contributions and results of this paper are summarized as follows:
\begin{itemize}
\item We derive novel exact closed-form PEP expressions for a two-relay dual-hop SWIPT relaying system with blind and CSI-assisted relaying schemes employing AEH and IEH.
\item The derived analytical PEP expressions are used to numerically evaluate the diversity order of the considered schemes. Specifically, we demonstrate that CSI-assisted relaying with AEH is superior to the other three relaying techniques achieving the highest diversity order of three. We further demonstrate that the lowest diversity order of two is obtained by the blind relaying scheme employing IEH suffering from cascaded fading resulting from IEH.
\item We demonstrate that under severe noise impulsiveness, the convergence to full spatial diversity becomes slower and that the associated performance loss increases with the diversity order.
\item We demonstrate through our numerical results that for all considered relaying techniques, the best performance is achieved when the two relays are located closer to the source node than the destination node and conclude that the optimal location of the relays is independent from the noise type, i.e., MCA or AWGN.
\item Finally, a comprehensive computer-based Monte Carlo simulation study is presented to verify the accuracy of the analytical results and to further investigate several design choices within the considered relay-assisted transmission scenarios.
\end{itemize}
\par The remainder of the paper is organized as follows. In Section II, we describe the noise model and the two-relay SWIPT transmission model in conjunction with blind and CSI-assisted relaying. In Section III, we present the analytical derivations of the PEP expressions for each of the relaying techniques under consideration. Section IV provides extensive Monte-Carlo simulation results to corroborate the analytical results and to provide detailed performance comparisons among the competing schemes for various scenarios. Concluding remarks are given in Section V. The appendices include mathematical details of the PEP derivations.
\par \underline{\textit{Notation}:} Bold lower case letters denote vectors. $(.)^T, (.)^*$, $\mathbb{E}[z]$, and $|z|$ stand for the transpose, conjugate, expectation of the random variable $z$, and magnitude of a complex variable $z$, respectively.
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{systemmodel.jpg}
\caption{Schematic representation of relay-assisted transmission.}
\label{blockdiagram}
\end{figure}
\section{System Model}
\label{sec:model}
We consider a dual-hop AF SWIPT relaying system as shown in Fig.~\ref{blockdiagram}, where a source node, $S$, communicates with a destination node, $D$, via two intermediate relay nodes, $R_1$ and $R_2$. The source and the destination nodes are assumed to be energy unconstrained nodes powered by either a battery or a power grid. On the other hand, the relay nodes have no dedicated power supply and harvest energy from the received signal which is then used over the second hop. In our work, we assume that a direct link exists between the source node and the destination node. We consider the PS protocol for wireless EH, and assume that all nodes are equipped with a single antenna. We also assume that all nodes operate in the presence of impulsive noise. In what follows, we introduce the adopted noise and transmission models.
\subsection{Noise Model}
We assume that each noise sample in the $t$-th time slot at any node is given by
\begin{equation}\label{noisesample}
n(t)=n_G(t)+n_I(t),
\end{equation}where $n_G(t)$ and $n_I(t)$ denote the background zero-mean complex Gaussian noise with variance $\sigma^2_G$ and the impulsive noise with variance $\sigma^2_I$, respectively. Adopting the MCA noise model and assuming that the active interfering sources emit independently, the PDF of the complex-valued noise sample, given in \eqref{noisesample}, at any of the nodes can be expressed as \cite{Middleton}
\begin{equation}\label{MCApdf}
f(n(t)) =\sum_{m(t)=0}^\infty \frac{\alpha_{m(t)}}{\pi\sigma^2_{m(t)}}\text{exp}\left(-\frac{|n(t)|^2}{\sigma^2_{m(t)}}\right),
\end {equation}where\footnote{Hereafter, we drop the time index in $m(t)$ and use $m$ instead.}
\begin{equation}\label{alpha}
\alpha_{m}=\frac{e^{-A}A^{m}}{m!},
\end{equation}with $A$ denoting the impulsive noise index that describes the average number of impulses during the interference time \cite{Middleton}. When it takes small values, i.e., $A\to 0$, it results in a highly structured and more impulsive noise, whereas it results in a near-Gaussian noise when it is large, i.e., $A\to\infty$. Furthermore, in \eqref{MCApdf}, $\sigma^2_m$ is the conditional variance given that $m$ impulses are affecting the receiver and is calculated as $\sigma_m^2=\sigma_n^2\beta_m$, where $\sigma_n^2$ denotes the mean variance of impulsive noise $n(t)$ and is equal to $N_0$ and $\beta_m$ is given by
\begin{equation}\label{beta}
\beta_m=\left(\frac{mA^{-1}+\delta}{1+\delta}\right),
\end{equation}where $\delta=\sigma^2_G/\sigma^2_I$ is called the Gaussian noise factor \cite{Middleton}, which is equal to the ratio of the variance of the background Gaussian noise component to the impulsive noise component. It is worth noting that the noise PDF in \eqref{MCApdf} reduces to the Gaussian distribution when $\delta\to\infty$ while it tends to be more impulsive when $\delta\to 0$.
Throughout this work, we assume that $\delta>0$ which implies that the Gaussian noise component is always present.
\par As clearly seen from \eqref{MCApdf}, the noise sample $n(t)$ in \eqref{noisesample} is not Gaussian, however, it can be viewed as conditionally Gaussian, such that, when conditioned on the state $m$, $n(t)$ is Gaussian with zero-mean and variance $\sigma_m^2$. The sequence of states $m(t)$ is an independent and identically distributed (i.i.d) random process, and a particular state $m(t)=m$ occurs with probability $C_0=\alpha_m$, $0\leq m<\infty$, where $m(t)$ follows a Poisson distribution with parameter $A$. Therefore, it is interpreted from that the integer random variable $C_0$ is the state of the noise indicating that there is no impulse $(C_0=0)$, or impulses are present $(C_0>0)$.
\par Although the distribution of MCA includes an infinite summation, it is completely characterized by two parameters, $A$ and $\delta$. In this work, we assume that $A$, $\delta$, and $\sigma_n^2$ are perfectly known at the receiver. In practice, these parameters can be estimated using the expectation maximization (EM) method proposed in \cite{Zabin}. We can see that the noise state probability $\beta_m$ in \eqref{beta} tends to zero as $m$ approaches infinity. Therefore, in the subsequent analysis, we truncate the sum in \eqref{MCApdf} to $M$ terms to reduce the computational complexity without compromising the performance accuracy \cite{Al-Dharrab}.
\par In this paper, we assume that the impulsive noise samples are temporally dependent during a transmission frame, following the widely used assumption in literature \cite{Delaney}. Furthermore, from the perspective of spatial dimension, we consider two models, namely, dependent and independent impulsive noise models. In Model I, which assumes spatially dependent noise samples, the same set of interfering sources affects the destination and relay nodes together. This scenario is applicable when the destination and relay nodes are at relatively the same distance to the interfering sources \cite{Gao, Al-Dharrab}. On the contrary, in Model II, it is assumed that each of the destination and relay nodes are affected by different sets of interfering sources and therefore, their respective noise samples are spatially independent.
\begin{figure}[!t]
\centering
\includegraphics[width=4in]{transmissionmodelch3.pdf}
\caption{Transmission allocation of the source node, $S$, and the two relay nodes, $R_1$ and $R_2$ over the two-Phase transmission scheme with each phase consisting of time slots.}
\label{transmissionmodelch3}
\end{figure}
\subsection{Transmission Model}
We consider a wireless communication scenario where the source node $S$ transmits information to the destination node $D$ with the assistance of two EH relay nodes $R_1$ and $R_2$. We adopt the so-called Protocol II of \cite{Laneman, Nabar} as the relaying protocol, which is completed over two signalling intervals, namely, \textit{Phase-1} and \textit{Phase-2} (see Fig.~\ref{transmissionmodelch3}). We further assume that the source node communicates with the two relays and the destination nodes using the Alamouti's code \cite{Alamouti}. Specifically, the transmission of two Alamouti coded symbols is performed over four time slots ${t}=1,...,4$. During \textit{Phase-1}, spanning two time slots ${t}=1$ and ${t}=2$, the source node communicates with the relays and destination nodes. In \textit{Phase-2}, spanning two time slots ${t}=3$ and ${t}=4$, the source node remains silent, whereas the two relays employ the AF relaying technique to retransmit a scaled version of their received signals to the destination node using Alamouti coding\footnote{This protocol realizes a maximum degree of broadcasting and exhibits no receive collision \cite{Nabar}.}. Also, it is assumed that the system is perfectly synchronized at the symbol level, i.e., relays transmit at the same time \mbox{\cite{Nabar}}. Protocol II is logical in a scenario where the source node engages in data reception from another node in the network over the second time slot, thereby rendering it unable to transmit \cite{Nabar}. The implementation of the Alamouti coding scheme has been considered in the literature of SWIPT networks in \cite{Zhai,Liu2016}. We assume that the relays harvest energy from the received source signals during \textit{Phase-1}, which is then used to forward the information to the destination in \textit{Phase-2}.
\par We further categorize the adopted AF relaying schemes based on the applied amplifying coefficient at the relay nodes, referred to as blind relaying \cite{Laneman2,Hasna} and CSI-assisted \cite{Laneman1} relaying. In the former scheme, the relays have no access to instantaneous CSI of their respective $S\to R$ links and hence, employ a fixed amplifying coefficient, which ensures that an average output power is maintained \cite{Laneman2}. While in the latter scheme, the relays use the receive CSI of their respective $S\to R$ link to ensure that the output power is limited to the power available at the relay, and therefore, a constant power is maintained for each realization \cite{Laneman1}.
\par Let $h_{sd}$, $h_{sr,n}$ and $h_{rd,n}$, respectively denote the complex small-scale fading coefficients over the $S\to D$ link, $S\to R_n$ link from the source to the $n$-th relay, $n \in \lbrace{1,2\rbrace}$, and $R_n\to D$ link from the $n$-th relay to the destination. These channel coefficients are modelled as i.i.d zero-mean complex Gaussian random variables (RVs) with variance 0.5 per dimension leading to the well-known Rayleigh fading channel model. It is also assumed that all channel coefficients remain constant over the block duration and vary independently and identically from one block to another. In addition to the small-scale fading, we further assume that all links are subject to large-scale path-loss that reflects the effect of the relative relays' locations on the performance of the system. Under this assumption, the received power is inversely proportional to $d_{ij}^{\lambda}$, where $d_{ij}$ is the propagation distance between transmitter $i$ and receiver $j$ and $\lambda> 2$ denotes the path-loss exponent. We set the reference distance equal to the distance from the source to the destination and assume that it is equal to unity, and hence, $d_{sr,n}= 1-d_{rd,n}, n\in\lbrace{1,2\rbrace}$. Consequently, the relative gains of $S\to R_n$ and $R_n\to D$ links are defined as $L_{sr,n}=(d_{sr,n}/d_{sd})^{\lambda}$ and $L_{rd,n}=(d_{rd,n}/d_{sd})^{\lambda}$, where $n \in \lbrace{1,2\rbrace}$.
\par Let the two consecutive signals transmitted by the source in \textit{Phase-1} be denoted as $s_1(t)$ and $s_2(t)$. We assume a binary phase shift keying (BPSK) signal constellation with normalized energy for the signals i.e., $\mathbb{E}[|s_p(t)|^2]=1$, $p\in \lbrace{1,2\rbrace}$. More specifically, during the first phase, the received signals at the destination in time slots $t=1$ and $t=2$ are given by
\begin{equation}\label{yd1}
y_{d}(t)=\sqrt{P_s}h_{sd}s_p(t)+n_{d}(t), \quad t=1,2,
\end{equation}where $P_s$ is the source transmit power and $s_p(t)$, $p\in \lbrace{1,2\rbrace}$, is the symbol sent from the source in the $t$-th time interval. Also, $n_{d}(t)$ represents the overall background and impulsive noise at the destination node with conditional variance $\sigma_{m,d}^2=\beta_{m,d} N_{0_d}$, associated with the $t$-th symbol. It is recalled that the parameter $\beta_{m,d}$ depends on the occurrence of a particular random impulsive state $m$ with probability $\alpha_{m,d}$, which follows a Poisson distribution.
\par During \textit{Phase-1}, the $n$-th relay node assigns a portion $\theta_n$ (called the PS ratio) of the received signal power in the $t$-th symbol interval for EH, and the remaining power $(1-\theta_n)$ is assigned for information processing at the information receiver. Accordingly, the received signal at its information receiver is given by
\begin{equation}
\label{YIR}
y_{r,n}(t)=\frac{\sqrt{\kappa_n P_s}}{\sqrt{L_{sr,n}}}h_{sr,n}s_p(t)+ n_{r,n}(t),
\end{equation} where $\kappa_n=(1-\theta_n)$. In this paper, we assume that $0<\theta_n<1$, corresponding to a general SWIPT system featuring both wireless information transfer and wireless EH. Furthermore, $n_{r,n}(t)$ is the overall background and impulsive noise at the $n$-th relay node associated with the $t$-th symbol, which is given by $n_{r,n}(t)=\sqrt{\kappa_n}n_{ra,n}(t)+n_{rc,n}(t)$, such that $n_{ra,n}(t)$ and $n_{rc,n}(t)$ are the receive antenna noise and the noise due to the RF-baseband signal conversion at the $n$-th relay, respectively, with mean variances of $N_{0_{ra,n}}$ and $N_{0_{rc,n}}$, respectively. Therefore, the conditional variance of $n_{r,n}(t)$ is $\sigma^2_{m,r,n}=\beta_{m,r,n} (\kappa_n N_{0_{ra,n}}+ N_{0_{rc,n}})$. For simplicity of the ensuing analysis, we assume that $N_{0_{ra,n}}=N_{0_{rc,n}}=N_0$.
\par The remaining portion of the received signal at $R_n$ in the $t$-th time slot is forwarded to the energy harvester, hence, the power available at $R_n$ at the end of each of the two symbol intervals of the first phase can be expressed as
\begin{equation}\label{Pr}
P_{r,n}=\frac{\eta_n\theta_n P_s|h_{sr,n}|^2}{L_{sr,n}},
\end{equation} with $0<\eta_n<1$ denoting the energy conversion efficiency factor at $R_n$. It should be noted that the EH process at $R_n$ is independent of the power scaling process and it is assumed that EH is performed instantaneously. The harvested instantaneous energy is simply used as a transmit power in the second phase of transmission. Note that the assumption of instantaneous EH was adopted in \cite{Nasir2013}.
\newcounter{tempequationcounter}
\begin{figure*}[t]
\vspace*{-0.3cm}
\normalsize
\setcounter{equation}{7}
\begin{equation}\label{yd3}
y_d(3)=\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}}h_{sr,1}h_{rd,1} s_1(3) +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}h_{rd,2} s_2(3)+\hat{n}_d(3)
\end{equation}
\hrulefill
\vspace*{5pt}
\end{figure*}
\newcounter{tempequationcounter1}
\begin{figure*}[t]
\vspace*{-0.3cm}
\normalsize
\begin{equation}\label{yd4}
y_d(4)=-\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}}h_{sr,1}^*h_{rd,1} s_2(4)^* +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}^*h_{rd,2} s_1(4)^*+\hat{n}_d(4),
\end{equation}
\vspace*{4pt}
\hrulefill
\vspace*{-0.3cm}
\end{figure*}
\newcounter{tempequationcounter2}
\begin{figure*}[t]
\normalsize
\setcounter{equation}{12}
\begin{equation}\label{tildey3}
\tilde{y}_d(3)=\frac{y_d(3)}{\Omega}=\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{\Omega G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}}h_{sr,1}h_{rd,1} s_1(3) +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{\Omega G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}h_{rd,2} s_2(3)+\frac{\hat{n}_d(3)}{\Omega},
\end{equation}
\hrulefill
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\newcounter{tempequationcounter3}
\begin{figure*}[t]
\normalsize
\setcounter{equation}{13}
\begin{equation}\label{tildey4}
\tilde{y}_d(4)=\frac{y_d(4)}{\Omega}=-\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{\Omega G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}}h_{sr,1}^*h_{rd,1} s_2(4)^* +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{\Omega G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}^*h_{rd,2} s_1(4)^*+\frac{\hat{n}_d(4)}{\Omega},
\end{equation}
\hrulefill
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\newcounter{tempequationcounter4}
\begin{figure*}[t]
\normalsize
\setcounter{equation}{14}
\begin{equation}\label{matrixmod}
\mathbf{y}_d=\begin{bmatrix}
y_d(1)\\y_d(2)\\y_d(3)\\y_d(4)\end{bmatrix}=\begin{bmatrix}
\sqrt{P_s}h_{sd}s_1(1)+n_{d}(1) \\
\sqrt{P_s}h_{sd}s_2(2)+n_{d}(2) \\
\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{\Omega G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}} h_{sr,1}h_{rd,1} s_1(3) +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{\Omega G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}h_{rd,2} s_2(3)+\tilde{n}_d(3) \\
-\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{\Omega G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}}h_{sr,1}^*h_{rd,1} s_2(4)^* +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{\Omega G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}^*h_{rd,2} s_1(4)^*+\tilde{n}_d(4)\end{bmatrix}.
\end{equation}
\hrulefill
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\begin{figure*}[t]
\normalsize
\setcounter{equation}{18}
\begin{equation}\label{matrixmod2}
\bold{y}_d=\begin{bmatrix}
y_d(1)\\y_d(2)\\y_d(3)\\y_d(4)\end{bmatrix}=
\begin{bmatrix}
\sqrt{P_s}h_{sd}s_1(1)+n_{d}(1) \\
\sqrt{P_s}h_{sd}s_2(2)+n_{d}(2) \\
\Phi_1|h_{sr,1}| h_{sr,1}h_{rd,1} s_1(3) +\Phi_2|h_{sr,2}| h_{sr,2}h_{rd,2} s_2(3)+\tilde{n}_d(3) \\
-\Phi_1|h_{sr,1}|h_{sr,1}^*h_{rd,1} s_2(4)^* +\Phi_2|h_{sr,2}| h_{sr,2}^*h_{rd,2} s_1(4)^*+\tilde{n}_d(4)\end{bmatrix}.
\end{equation}
\hrulefill
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\par During \textit{Phase-2} spanning two symbol intervals, the received signals are processed at the relay nodes using the Alamouti scheme in a distributed manner. The resulting signals are then forwarded to the destination nodes using the energy harvested in \textit{Phase-1}. Specifically, the signals received at the destination through the $R_n\to D$ links over time slots ${t}=3$ and ${t}=4$ are given by \eqref{yd3} and \eqref{yd4}, respectively, at the top of this page. In \eqref{yd3} and \eqref{yd4}, $G_{r,n}, n\in\lbrace{1,2\rbrace}$ is the scaling term at the $n$-th relay which depends on the type of amplifying coefficient deployed at $R_n$ (i.e. blind relaying or CSI-assisted relaying), which will be discussed in details in the subsequent section. This normalization does not alter the signal-to-noise ratio SNR but simplifies the ensuing presentation \cite{Nabar}. Furthermore, $\hat{n}_d(3)$ and $\hat{n}_d(4)$ are the effective noise terms associated with the third and fourth symbols, respectively, defined as
\setcounter{equation}{9}
\begin{equation}
\hat{n}_d(3)=\frac{\sqrt{P_{r,1}}}{G_{r,1}\sqrt{L_{rd,1}}}h_{rd,1}n_{r,1}+\frac{\sqrt{P_{r,2}}}{G_{r,2}\sqrt{L_{rd,2}}}h_{rd,2}n_{r,2}+n_d(3)
\end{equation} and
\setcounter{equation}{10}
\begin{equation}
\hat{n}_d(4)=\frac{-\sqrt{P_{r,1}}}{G_{r,1}\sqrt{L_{rd,1}}}h_{rd,1}n_{r,1}^{*}+\frac{\sqrt{P_{r,2}}}{G_{r,2}\sqrt{L_{rd,2}}}h_{rd,2}n_{r,2}^{*}+n_d(4).
\end{equation}\\
Assuming the so-called average power scaling (APS) \cite{Al-Dharrab}, the destination node normalizes the received signals given by \eqref{yd3} and \eqref{yd4} with
\begin{equation}
\Omega=\left(\frac{\eta_1\theta_1 P_s (\kappa_1+1) }{L_{sr,1}L_{rd,1}\mathbb{E}[|G_{r,1}|^2]}+\frac{\eta_2\theta_2 P_s (\kappa_2+1) }{L_{sr,2}L_{rd,2}\mathbb{E}[|G_{r,2}|^2]}+1\right)^{1/2},
\end{equation}resulting in \eqref{tildey3} and \eqref{tildey4}, respectively, at the top of this page. With the aforementioned signal models in mind, by letting $\tilde{n}_d(q)=\hat{n}_d(q)/\Omega, q\in\lbrace{3,4\rbrace}$, the received signal vector over four time slots is expressed as \eqref{matrixmod} at the top of the next page.
Introducing $\bold{h}=[\sqrt{P_s}h_{sd}, \sqrt{P_s}h_{sd}, D_1 h_{sr,1}^*h_{rd,1}, D_2 h_{sr,2}^*h_{rd,2}] $, where $ h_{sr,n}^*$ is chosen as $h_{sr,n}$ or $h_{sr,n}^*$ based on the code matrix $\bold{S}$ given by
\setcounter{equation}{15}
\begin{equation}\label{codemat}
\mathbf{S}=\begin{bmatrix}
s_1(1) & 0 & 0 & 0\\
0&s_2(2)&0&0\\
0&0&s_1(3)&-s_2(4)^*\\
0&0&s_2(3)&s_1(4)^*
\end{bmatrix},
\end{equation}and $\bold{n}=[n_{d}(1), n_{d}(2), n_{d}(3), n_{d}(4)]$, the received signal vector over the whole observation period can be obtained as
\begin{equation}\label{shortsys}
\bold{y}_d=\bold{hS}+\bold{n}.
\end{equation}
\par After setting up the relay-assisted transmission model given by \eqref{matrixmod} and \eqref{shortsys}, we will now introduce the details of the signal models for blind and CSI-assisted relaying techniques.
\subsection{Blind Relaying}
Under this relaying technique, it is assumed that the $n$-th relay node does not have knowledge of its relative $S\to R_n$ link fading coefficient, therefore, it scales the received signal $y_{r,n}(t)$ by a factor of
\begin{equation}
G_{r,n}=\sqrt{\mathbb{E}[|y_{r,n}|^2]}=\sqrt{(\kappa_n P_s/L_{sr,n})+N_{0_{r,n}})}
\end{equation} to normalize the average energy to unity \cite{Laneman2}\footnote{This power constraint is called fixed gain relaying in \cite{Hasna}}. Replacing the scaling term $G_{r,n}$, $P_{r,1}$, and $P_{r,2}$ in \eqref{matrixmod}, we can rewrite the vector form of the received signal model $\bold{y}_d$ as \eqref{matrixmod2} at the top of the next page, where $\Phi_1^2$ and $\Phi_2^2$ are given as
\setcounter{equation}{19}
\begin{equation}\label{phi1APS}
{\Phi_{1}^2}=\frac{\eta_1\theta_1\kappa_1 P_s (P_s/N_{0_{r,1}})}{\Omega^2 L_{sr,1}^2L_{rd,1}[(\kappa_1/L_{sr,1})(P_s/N_{0_{r,1}})+1]}
\end{equation} and
\begin{equation}\label{phi2APS}
{\Phi_2^2}=\frac{\eta_2\theta_2\kappa_2 P_s (P_s/N_{0_{r,2}})}{\Omega^2 L_{sr,2}^2L_{rd,2}[(\kappa_2/L_{sr,2})(P_s/N_{0_{r,2}})+1]},
\end{equation} respectively.
\subsection{CSI-assisted Relaying}
Under this relaying technique, it is assumed that the relays $R_1$ and $R_2$ have knowledge about the CSI of their relative $S\to R_n$ links and accordingly, the scaling factor of the $n$-th relay becomes
\begin{equation}
G_{r,n}=\sqrt{(\kappa_n P_s/L_{sr,n})|h_{sr,n}|^2+\beta_{I,r,n}^mN_{0_{r,n}}}.
\end{equation} Replacing this scaling term $G_{r,n}$, $P_{r,1}$, and $P_{r,2}$ in \eqref{matrixmod}, we can rewrite the vector form of the received signal at the destination as \eqref{matrixmod2}, where $\Phi_1^2$ and $\Phi_2^2$ are now written as \\
\begin{equation}\label{phi1IPS}
{\Phi_{1}^2}=\frac{\eta_1\theta_1\kappa_1 P_s (P_s/(\beta_{m,r,1} N_{0_{r,1}}))}{\Omega^2 L_{sr,1}^2L_{rd,1}[(\kappa_1/L_{sr,1})(P_s/(\beta_{m,r,1} N_{0_{r,1}}))|h_{sr,1}|^2+1]}
\end{equation} and\\
\begin{equation}\label{phi2IPS}
{\Phi_2^2}=\frac{\eta_2\theta_2\kappa_2 P_s (P_s/(\beta_{m,r,2} N_{0_{r,1}}))}{\Omega^2 L_{sr,2}^2L_{rd,2}[(\kappa_2/L_{sr,2})(P_s/(\beta_{m,r,2}N_{0_{r,2}}))|h_{sr,2}|^2+1]},
\end{equation} \\respectively. To simplify the ensuing analysis, we assume that $N_{0}\triangleq N_{0_{sr,1}}= N_{0_{sr,2}}=N_{0_{d}}$.
\section{Pairwise Error Probability Analysis}
Based on the described noise and transmission models in the preceding section, we proceed to investigate the performance of the SWIPT relay system for each of the considered relaying techniques by deriving the PEP expressions for noise Models I and II.
\subsection{Performance Under Noise Model I}
We start by considering the spatially dependent impulsive noise model and investigate its relative effect on the underlying SWIPT relaying system. Specifically, under Model I, the number of impulses affecting $R_1$, $R_2$, and $D$ are statistically dependent and follow the same Poisson random variable $C_0$, i.e., $\alpha_{m,d}=\alpha_{m,r,1}=\alpha_{m,r,2}=\alpha_m$.
\par We will assume minimum distance decoding with perfect knowledge of the individual CSIs of the $S\to R_n$, $R_n\to D$, and $S\to D$ links at the receiver which is considered to be optimal when the noise is Gaussian, but is suboptimal over the impulsive noise channel \cite{Gao}. However, since the minimum distance receiver (MDR) is practical with a low detection complexity technique, we are motivated to derive its PEP performance which is mathematically tractable.
\par Let $\bold{s}$ and $\hat{\bold{s}}$ denote the originally transmitted codeword, $\bold{s}=[s_1, s_2]$, and the erroneously-decoded codeword, $\hat{\bold{s}}=[\hat{s}_1, \hat{s}_2]$, vectors at the destination, respectively. Recalling that for the spatially dependent case, $\beta_{m,d}=\beta_{m,r,1}=\beta_{m,r,2}=\beta_{m}$, after normalising \eqref{yd3} and \eqref{yd4} by $\Omega$, then conditioned on the conditional noise variance $\beta_m$, $\tilde{n}_d(q), q\in\lbrace{3,4\rbrace},$ turns out to be a zero-mean complex Gaussian random variable with variance $\beta_m N_0$. Accordingly, the exact conditional PEP is obtained following the derivation of the conditional PEP in the Gaussian noise case as
\begin{equation}\label{exactPEPMDR}
P(\bold{s}\to\hat{\bold{s}}|\bold{h})=\sum_{m=0}^{M-1}\alpha_m Q\left(\sqrt{\frac{d^2(\bold{s},\hat{\bold{s}})}{2\beta_m N_0}}\right),
\end{equation} where all possible realizations of the Poisson random variable $C_0$ are considered. Also, $Q(.)$ is the Gaussian-$Q$ function \cite{Rizhik} and $d^2(\bold{s},\hat{\bold{s}})$ is the Euclidean distance between $\bold{s}$, and $\hat{\bold{s}}$ written as
\begin{equation}
d^2(\bold{s},\hat{\bold{s}})=d^2_{S\to D}(\bold{s},\hat{\bold{s}})+d^2_{S\to R_1\to D}(\bold{s},\hat{\bold{s}})+d^2_{S\to R_2\to D}(\bold{s},\hat{\bold{s}})
\end{equation}
Applying the standard Chernoff bound on the $Q(.)$ function in \eqref{exactPEPMDR}, the conditional PEP can be upper bounded by \cite{Tarokh}
\begin{equation}\label{chernoff}
P(\bold{s}\to\hat{\bold{s}}|\bold{h})\leq\sum_{m=0}^{M-1}\alpha_m\text{exp}\left(\frac{-d^2(\bold{s},\hat{\bold{s}})}{4\beta_m N_0}\right).
\end{equation}
\subsubsection{PEP for Blind Relaying}
The Euclidean distance for the blind relaying scheme can be written as
\begin{align}\label{distAPS}
d^2(\bold{s},\hat{\bold{s}})=\bold{h}(\bold{S}-\hat{\bold{S}})(\bold{S}-\hat{\bold{S}})^H\bold{h}^H \quad \quad\quad \quad \quad \quad\quad\quad \quad \quad \quad\nonumber \\
\quad=\Delta P_s |h_{sd}|^2+\epsilon_1 \Phi_1^2|h_{sr,1}|^4|h_{rd,1}|^2+\epsilon_2\Phi_2^2 |h_{sr,2}|^4|h_{rd,2}|^2.
\end{align} where $\Phi_1$ and $\Phi_2$ are defined in \eqref{phi1APS} and \eqref{phi2APS}, respectively, $\Delta=|s_1-\hat{s}_1|^2+|s_2-\hat{s}_2|^2$ and $\epsilon_n$ denote the eigenvalues of the codeword difference matrix $(\bold{S}-\hat{\bold{S}})(\bold{S}-\hat{\bold{S}})^H$, $n\in\lbrace{1,2\rbrace}$.
It is worth noting that the term $|h_{sr_n}|^4$, $n\in\lbrace{1,2\rbrace}$, appears due to the process of instantaneous EH taking place at the $n$-th relay. Henceforth, we call this relaying schemes as blind IEH-relaying. Substituting \eqref{distAPS} in \eqref{chernoff}, the PEP expression is obtained in the following proposition.
\newcounter{tempequationcounter6}
\begin{figure*}[t]
\normalsize
\setcounter{equation}{30}
\begin{equation}\label{matrixmod3}
\bold{y}_d=\begin{bmatrix}
y_d(1)\\y_d(2)\\y_d(3)\\y_d(4)\end{bmatrix}=
\begin{bmatrix}
\sqrt{P_s}h_{sd}s_1(1)+n_{d}(1) \\
\sqrt{P_s}h_{sd}s_2(2)+n_{d}(2) \\
\Phi_1h_{sr,1}h_{rd,1} s_1(3) +\Phi_2 h_{sr,2}h_{rd,2} s_2(3)+\tilde{n}_d(3) \\
-\Phi_1h_{sr,1}^*h_{rd,1} s_2(4)^* +\Phi_2 h_{sr,2}^*h_{rd,2} s_1(4)^*+\tilde{n}_d(4)\end{bmatrix}.
\end{equation}
\hrulefill
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\setcounter{equation}{28}
\begin{prop}
The unconditional PEP performance of the considered SWIPT blind IEH-relaying system in the presence of impulsive noise can be expressed in a closed-form as\\
\begin{align}\label{finalPEPmodIAPS}
P(\bold{s}\to\hat{\bold{s}})&\leq\sum_{m=0}^{M-1}\alpha_m \left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1} \nonumber \\
&\times \prod_{n=1}^2\frac{1}{\sqrt{\pi}} G^{1,3}_{3,1}\left[\frac{\epsilon_n\Phi_n^2}{\beta_m N_0} \ \Big\vert \ {0.5, 0,0 \atop 0}\right],
\end{align}\end{prop}
\noindent where $ G^{m,n}_{p,q}[.\vert .]$ is the Meijer G-function defined in \cite[Eq. (8.2.1.1)]{Prudnikov}. Furthermore, $\alpha_m$ and $\beta_m$ can be calculated using \eqref{alpha} and \eqref{beta}, respectively. Note that the Meijer G-function in \eqref{finalPEPmodIAPS} can be easily and accurately computed by standard mathematical software packages such as Mathematica$^{\copyright}$, Matlab$^{\copyright}$, and Maple$^{\text{TM}}$.
\begin{IEEEproof}
See Appendix \ref{Appendix A}.
\end{IEEEproof}
\vspace*{0.1cm}
\par \textbf{Special Case (Blind AEH-relaying):} We assume that $R_1$ and $R_2$ perform AEH which corresponds to a practical scenario where the relay nodes are equipped with a battery. Under this assumption, \eqref{Pr} which represents the power available at the $n$-th relay at the end of \textit{Phase-1} is written as
\begin{equation}\label{PrAEH}
P_{r,n}=\frac{\eta_n\theta_n P_s}{L_{sr,n}}.
\end{equation}Replacing \eqref{PrAEH} in \eqref{matrixmod}, the vector form of the received signal model $\bold{y}_d$ is now given as \eqref{matrixmod3} at the top of this page. Under this scenario, $d^2(\bold{s},\hat{\bold{s}})$ is given by
\setcounter{equation}{31}
\begin{align}\label{convdit}
d^2(\bold{s},\hat{\bold{s}})&=\Delta P_s |h_{sd}|^2 +\epsilon_1 \Phi_1^2|h_{sr,1}|^2|h_{rd,1}|^2\nonumber \\
&+\epsilon_2\Phi_2^2 |h_{sr,2}|^2|h_{rd,2}|^2.
\end{align}It can be easily verified that \eqref{convdit} has a similar form to that in \cite[Eq. (31)]{Muhaidat1} and \cite[Eq. (26)]{Al-Dharrab} for the conventional non-EH case. Therefore, the unconditional PEP is found as
\begin{align}\label{PEPcon}
P(\bold{s}\to\hat{\bold{s}})&\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1}\nonumber\\
&\times\prod_{n=1}^2\left(\frac{\epsilon_n\Phi_n^2}{4\beta_m N_0}\right)^{-1} \text{exp}\left(\frac{4 \beta_m N_0}{\epsilon_n \Phi_n^2}\right)\Gamma\left(0,\frac{4 \beta_m N_0}{\epsilon_n \Phi_n^2}\right),
\end{align} where $\Gamma(a,b)=\int_b^\infty x^{a-1}\text{exp}(-x)dx$ \cite{Rizhik} denotes the upper incomplete gamma function.
\newcounter{tempequationcounter7}
\begin{figure*}[b]
\normalsize
\hrulefill
\setcounter{equation}{40}
\begin{align}\label{PEPmodII}
P(\bold{s}\to\hat{\bold{s}}|\bold{h})&=\sum_{m,r,1=0}^{M-1}\sum_{m,r,2=0}^{M-1}\sum_{m,d=0}^{M-1}\left(\prod_{k=1}^3\alpha_{m,k}\right) \nonumber \\
&\times Q\left(\frac{d^2(\bold{s},\hat{\bold{s}})}{\sqrt{2\left[\Delta P_s|h_{sd}|^2\beta_{m,d}+\epsilon_1\Phi_1|h_{sr,1}|^4|h_{rd,1}|^2\beta_{m,r,1}+\epsilon_2\Phi_2|h_{sr,2}|^4|h_{rd,2}|^2\beta_{m,r,2}\right]N_0}}\right).
\end{align}
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\setcounter{equation}{33}
\subsubsection{PEP for CSI-assisted Relaying}
The Euclidean distance for the CSI-assisted relaying scheme can be written as \eqref{distAPS} where $\Phi_1^2$ and $\Phi_2^2$ are now given by \eqref{phi1IPS} and \eqref{phi2IPS}, respectively. Note that, unlike \eqref{phi1APS} and \eqref{phi2APS} for the blind relaying case, \eqref{phi1IPS} and \eqref{phi2IPS} are functions of $|h_{sr,1}|^2$ and $|h_{sr,1}|^2$, respectively. To this effect, substituting \eqref{phi1IPS} and \eqref{phi2IPS} in \eqref{distAPS}, we can write $d^2(\bold{s},\hat{\bold{s}})$ as
\begin{align}\label{distIPS}
d^2(\bold{s},\hat{\bold{s}})&= \Delta P_s |h_{sd}|^2 +\epsilon_1 \zeta_1 \frac{|h_{sr,1}|^4|h_{rd,1}|^2}{\xi_1 |h_{sr,1}|^2+1}\nonumber \\
&+\epsilon_2\zeta_2 \frac{|h_{sr,2}|^4|h_{rd,2}|^2}{\xi_2 |h_{sr,2}|^2+1} \nonumber \\
& =\Delta P_s |h_{sd}|^2 +\epsilon_1 \zeta_1 \frac{X_1^2Y_1}{\xi_1 X_1+1}+\epsilon_2\zeta_2 \frac{X_2^2 Y_2}{\xi_2 X_2+1}.
\end{align} where $\xi_n=[(\kappa_n/L_{sr,n})(P_s/(\beta_I^m N_0))]$, $\zeta_n$ is given as \\
\begin{equation}\label{zeta}
\zeta_n =\frac{\eta_n\theta_n\kappa_n P_s (P_s/(\beta_m N_0))}{\Omega^2 L_{sr,n}^2L_{rd,n}}, \quad n\in \lbrace{1,2\rbrace}.
\end{equation}and $X_n\triangleq|h_{sr,n}|^2$, $Y_n\triangleq|h_{rd,n}|^2$. To obtain an expression for the PEP for the CSI-assisted IEH-relaying, let $Z_n=U_n/V_n$, where $U_n=X_n^2Y_n$ and $V_n=\xi_n X_n+1, n\in \lbrace{1,2\rbrace}$. Then, one could obtain the unconditional PEP by taking the expectation of \eqref{chernoff} with respect to the RVs $|h_{sd}|^2, Z_1$ and $Z_2$. In the following proposition, we derive the unconditional PEP expression. \\
\begin{prop}
The unconditional PEP performance of SWIPT CSI-assisted IEH-relaying system in the presence of impulsive noise can be expressed as
\vspace*{-0.5cm}
\end{prop}
\begin{align}\label{pepIEHIPS}
P(\bold{s}\to\hat{\bold{s}})&\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1}\nonumber \\
&\prod_{n=1}^2\frac{1}{2B_n\psi_n}\left[\text{exp}(\Lambda_n)\text{Ei}(\Lambda_n)\text{D}_1+\text{exp}(\Psi_n)\text{Ei}(\Psi_n)\text{D}_2\right],
\end{align}where $\text{Ei}(.)$ is the exponential integral function \cite{Prudnikov}, $\psi=\sqrt{\xi_n^2-4B_n}$ where $\xi_n$ is defined before \eqref{zeta}, $\text{D}_1=-\xi_n^2-\xi_n\psi_n+2 B_n$, $\text{D}_2=\xi_n^2-\xi_n\psi_m-2 B_n$, and $\Lambda_n$ and $\psi_n$ are given by
\begin{equation}
\Lambda_n=\frac{\xi_n+\psi_n}{2B_n},
\end{equation}and
\begin{equation}
\Psi_n=\frac{\xi_n-\psi_n}{2B_n},
\end{equation}respectively.
\begin{IEEEproof}
See Appendix \ref{Appendix B}.
\end{IEEEproof}\vspace*{0.1cm}
\par \noindent \textbf{Special Case (Asymptotic PEP in high SNR):} To give more insight into the PEP performance, we consider the high SNR assumption, i.e., $\xi_n\to \infty$. Under this assumption, the second factor in the denominators of \eqref{distIPS} can be negligible. Consequently, $d^2(\bold{s},\hat{\bold{s}})$ in \eqref{distIPS} is reduced to \eqref{convdit}, yielding the PEP expression to be given as \eqref{PEPcon}. \\
\par \noindent \textbf{Special Case (CSI-assisted AEH-relaying):} Similar to the blind-relaying scenario, we assume here that $R_1$ and $R_2$ perform average EH. Under this assumption, the power available at the $n$-th relay at the end of \textit{Phase-1} is given by \eqref{PrAEH}. Hence, we get
\begin{align}\label{distIPS2}
d^2(\bold{s},\hat{\bold{s}})&= \Delta P_s |h_{sd}|^2 +\epsilon_1 \zeta_1 \frac{|h_{sr,1}|^2|h_{rd,1}|^2}{\xi_1 |h_{sr,1}|^2+1}\nonumber \\
&+\epsilon_2\zeta_2 \frac{|h_{sr,2}|^2|h_{rd,2}|^2}{\xi_2 |h_{sr,2}|^2+1},
\end{align}where $\xi_n$ and $\zeta_n$ are given below \eqref{distIPS}. Substituting \eqref{distIPS2} in \eqref{chernoff}, followed by taking the expectation with respect to $|h_{sd}|^2, |h_{sr,1}|^2, |h_{sr,2}|^2, |h_{rd,1}|^2$ and $|h_{rd,2}|^2$, the unconditional PEP is given in the following proposition.
\begin{prop}
The unconditional PEP performance of SWIPT CSI-assisted AEH-relaying system can be expressed as
\end{prop}
\vspace*{-0.5cm}
\begin{align}\label{PEPAEHIPS}
P(\bold{s}\to\hat{\bold{s}})\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\nonumber \\
\quad\quad\times \prod_{n=1}^2\left(\gamma_n^{-1}G^{1,2}_{2,1}\left[\gamma_n \ \big\vert \ {1, 1 \atop 1}\right]+\xi_n \gamma_n^{-2}G^{1,2}_{2,1}\left[\gamma_n \ \big\vert \ {1, 2 \atop 2}\right]\right). \quad\quad\quad\quad
\end{align}
\begin{IEEEproof}
See Appendix \ref{AppendixC}.
\end{IEEEproof}
\vspace*{0.3cm}
\noindent It is noted from each of \eqref{finalPEPmodIAPS}, \eqref{PEPcon}, \eqref{pepIEHIPS}, and \eqref{PEPAEHIPS} that these expressions include the conventional AWGN assumption as a special case. It is recalled from \eqref{beta} that as $\delta\to\infty$, $\beta_m$ converges to 1. Therefore, the summation in \eqref{finalPEPmodIAPS}, \eqref{PEPcon}, \eqref{pepIEHIPS}, and \eqref{PEPAEHIPS} will be equal to 1, reducing these expressions to the PEP expressions for the conventional AWGN case. It is worth mentioning that due to the presence of the summation term, in the above mentioned expressions, the convergence to asymptotic diversity order under impulsive noise is slower compared to the AWGN case.
\par Based on the previously derived PEP expressions the diversity order \mbox{$D$} can be computed as \cite{Simon}
\begin{equation}\label{dorder}
D= -\lim_{\text{SNR} \rightarrow \infty}\frac{\text{log}\left(P(\bold{s}\to\hat{\bold{s}})\right)}{\text{log}\left(\text{SNR}\right)}.
\end{equation}Since the only source of power is, \mbox{$P_s$}, the performance of the entire system is parametrized by SNR \mbox{$\triangleq P_s/N_0$}. Using \mbox{\eqref{dorder}}, the diversity order of blind IEH-relaying, blind AEH-relaying, CSI-assisted IEH-relaying, and CSI-assisted AEH-relaying are numerically evaluated by substituting \mbox{\eqref{finalPEPmodIAPS}, \eqref{PEPcon}, \eqref{pepIEHIPS}, and \eqref{PEPAEHIPS} in \eqref{dorder}}, respectively.
\subsection{Performance Under Noise Model II}
In the following, we will study the performance of the considered SWIPT relay system under the assumption of spatially independent impulsive noise model, where $R_1$, $R_2$, and $D$ nodes are affected by statistically independent number of impulses, respectively, following Poisson random variables $C_{r,1}$, $C_{r,2}$, and $C_d$, i.e., $\alpha_{m,d}, \alpha_{m,r,1}$, and $\alpha_{m,r,2}$ may not necessarily be equal. In particular, the conditional variances $\beta_{m,d}$, $\beta_{m,r,1}$, and $\beta_{m,r,2}$ are not necessarily equal. To address the independence in the number of impulses occurring at $R_1$, $R_2$, and $D$, the PEP expression has to be averaged over all possible realizations of each of $C_{r,1}$, $C_{r,2}$, and $C_d$, and thus, the conditional PEP is given by \eqref{PEPmodII} at the bottom of this page, where $\alpha_{m,1}=\alpha_{m,d}$, $\alpha_{m,2}=\alpha_{m,r,1}$, and $\alpha_{m,3}=\alpha_{m,r,2}$.
\par To evaluate the unconditional exact PEP for each of the relaying schemes described in Section II, the expression in \eqref{PEPmodII} has to be averaged over fading coefficients $\bold{h}$, which is mathematically intractable. However, we can obtain an approximate expression for the conditional PEP in \eqref{PEPmodII} by setting $\beta_{m,r,1}=\beta_{m,r,2}=\beta_{m,d}=\bar{\varphi}$, which denotes the average number of impulses affecting $R_1$, $R_2$, $D$ nodes during a transmission frame and is given by \cite{Al-Dharrab}
\setcounter{equation}{41}
\begin{equation}
\bar{\varphi}=\frac{2(\beta_{m,r,1}+\beta_{m,r,2})+4\beta_{m,d}}{8}.
\end{equation}Then, by using the Chernoff upper bound, taking the expectation over the fading coefficients $\bold{h}$, and following the same line of analysis performed in the derivation of the PEP expressions of the blind and CSI-assisted relaying schemes under noise Model I, the PEP performance under noise Model II can be evaluated.
\section{Numerical and Simulation Results}\label{sec:results}
In this section, we provide a variety of numerical and Monte Carlo simulation results to validate the accuracy of the proposed analytical framework and to compare the performance of the considered blind and CSI-assisted relaying techniques employed for a SWIPT relaying system under the MCA noise Models I and II. The term Monte Carlo simulations refers to the use of actual fading channel variates with a number of repetitions of $10^6$ trials. We further assume that the two relays are located on the straight line between the source and the destination nodes. Unless otherwise specified, in order to study various degrees of noise impulsiveness, we use three sets of values for the impulsive noise parameters $A$ and $\delta$: $(A,\delta)=(1,0.1)$, $(A,\delta)=(0.1,0.1)$, and $(A,\delta)=(0.001,0.1)$ to represent \textit{near-Gaussian} (NG), \textit{moderately impulsive} (MI), and \textit{highly impulsive} (HI) noise channels, respectively, which fit well within the practical ranges of $A$ and $\delta$ \cite{Delaney}.
\par Unless otherwise stated, we set the EH efficiency factor $\eta_1=\eta_2=0.3$ as a worst case, capturing the effects of low-cost hardware, the PS factors $\theta_1=\theta_2=0.5$, the normalized distances of both relays for their respective $S\to R$ links are set to $d_{sr,1}=d_{sr,2}=0.5$, the source transmission power $P_s=1$ Watt and the path-loss exponent $\lambda=2.7$ \cite{Nasir2013}. The simulation parameters are summarized in Table \ref{t2}.
\begin{table}[!t]
\centering
\caption{Simulation Parameters}
\label{t2}
\begin{tabular}{|l|c|c|}
\hline
\multicolumn{1}{|c|}{\textbf{Name}} & \textbf{Symbol} & \textbf{Value} \\ \hline
Impulsive noise index & \textit{A} & 1 (NG), 0.1 (MI), 0.001 (HI) \\ \hline
Gaussian noise factor & $\delta$ & 0.1 \\ \hline
Number of interfering sources & \textit{M} & 5 \\ \hline
PS ratio for $R_1$ & $\theta_1$ & 0.5 \\ \hline
PS ratio for $R_2$ & $\theta_2$ & 0.5 \\ \hline
Normalized $S\to R_1$ distance & $d_{sr,1}$ & 0.5 \\ \hline
Normalized $S\to R_2$ distance & $d_{sr,2}$ & 0.5 \\ \hline
Path-loss exponent & $\lambda$ & 2.7 \\ \hline
EH efficiency of $R_1$ & $\eta_1$ & 0.3 \\ \hline
EH efficiency of $R_2$ & $\eta_2$ & 0.3 \\ \hline
Source transmit power & $P_s$ & 1 Watt \\ \hline
\end{tabular}
\end{table}
\begin{figure}[!t]
\centering
\includegraphics[width=3.7in]{PEP_allschemes_SNR.pdf}
\caption{PEP performance with respect to SNR for blind and CSI-assisted relaying techniques over Rayleigh fading channels in the presence of HI, MI, and NG MCA noise for Model I.}
\label{f3}
\vspace*{-0.5cm}
\end{figure}
\par In Fig. \ref{f3}, we compare the PEP performance of the blind and CSI-assisted relaying techniques when IEH or AEH are considered under three MCA noise environments, namely, HI, MI, and NG, for noise Model I. Furthermore, to evaluate the accuracy of our mathematical models presented in \eqref{finalPEPmodIAPS}, \eqref{PEPcon}, \eqref{pepIEHIPS}, and \eqref{PEPAEHIPS}, we present in Fig. \ref{f3} the corresponding Monte Carlo simulation results. It is observed that the analytical PEP curves are in full agreement with the simulation results over the whole SNR operating range. This finding directly reflects the accuracy of our proposed mathematical framework and its effectiveness in quantifying the performance of the considered relaying techniques under MCA noise. It is illustrated in Fig. \ref{f3} that for all the studied relaying techniques, namely, blind relaying with IEH or AEH and CSI-assisted relaying with IEH or AEH, the PEP curves undergo a flattening when the SNR is between 5 - 20 dB under the HI noise environment, which dramatically differs from those of the NG noise environment. This behaviour is also reported for non-EH systems cooperative systems \cite{Al-Dharrab, Alhussein1} and is due to the fact that the tails of the PDF of the MCA noise becomes wider as the impulsive index $A$ decreases. However, as $A$ increases, the tails of the MCA density asymptotically approach those of a Gaussian density, resulting in the behaviour observed for the PEP performance. Moreover, for the three noise scenarios, it is shown that the performance exhibited by the CSI-assisted AEH-relaying is superior to that of the other three relaying techniques. Although, CSI-assisted relaying schemes are intuitively expected to outperform their blind relaying counterparts, our results show that the CSI-assisted IEH-relaying and blind AEH-relaying technique experience identical PEP performance. This indicates that the extra power consumption, resulting from CSI estimation, can be avoided without causing performance loss. However, this comes at the expense of requiring a battery to perform AEH.
\par In an attempt to gain more insights about the performance of the considered relaying techniques, we investigate the achievable diversity order. Specifically, in Fig. \ref{f4}, we utilize the expressions obtained in \eqref{finalPEPmodIAPS}, \eqref{PEPcon}, \eqref{pepIEHIPS}, and \eqref{PEPAEHIPS} to calculate the diversity order, defined as the negative of the asymptotic slope of the PEP on a log-log scale \cite{Tarokh}. The achievable diversity order in the presence of the well-known AWGN case is included as a benchmark.
\par Fig. \ref{f4} demonstrates that the CSI-assisted AEH-relaying scheme enables the system to achieve the highest diversity order $(d=3$, at NG), whereas the lowest $(d=2$, at NG) is obtained by the blind IEH-relaying scheme, where the performance is severely degraded. This is due to the effect of cascaded fading resulting from IEH. Meanwhile, the attainable diversity order for both the blind AEH-relaying and CSI-assisted IEH-relaying is identical $(d=2.85$, at NG). In Table \ref{t1}, we present the achievable diversity order levels observed by the investigated four relaying techniques under the three MCA noise environments, along with the corresponding AWGN case. It is noted that for all the studied relaying techniques: as the impulsive noise index $A$ becomes smaller, (i.e., the noise becomes highly impulsive), the convergence to full spatial diversity, represented by the AWGN case, becomes slower. This can be attributed to the performance loss introduced by the impulsive nature of the noise incurred by the MDR. Additionally, the full diversity order of all relaying techniques in the MCA noise environments are not realized due to the noise impulsiveness severity. Interestingly, as the noise impulsiveness level increases from NG to HI, the associated performance loss increases with the diversity order. This result is consistent with the conclusion reported in \cite{Gao} for a non-cooperative non-EH wireless communication system.
\begin{figure}[!t]
\centering
\includegraphics[width=3.2in]{Diversity_Vs_SNR3.pdf}
\caption{Diversity order of blind and CSI-assisted relaying schemes in the presence of HI, MI, and NG noise for Model I.}
\label{f4}
\end{figure}
\begin{table}[h]
\centering
\caption{Achievable diversity order under MCA noise and AWGN}
\label{t1}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{Relaying Technique}} & \multicolumn{3}{c|}{MCA Noise} & \multicolumn{1}{c|}{\multirow{2}{*}{AWGN}} \\ \cline{2-4}
\multicolumn{1}{|c|}{} & HI & MI & NG & \multicolumn{1}{c|}{} \\ \hline
Blind IEH & 1.86 & 1.96 &1.99 &\multicolumn{1}{c|}{2}
\\ \hline
Blind AEH &2.59 &2.78 &2.85 & \multicolumn{1}{c|}{2.87} \\ \hline
CSI-assisted IEH &2.59 &2.78 &2.85 &\multicolumn{1}{c|}{2.87} \\ \hline
CSI-assisted AEH &2.38 &2.86 &3 & \multicolumn{1}{c|}{3} \\ \hline
\end{tabular}
\end{table}
\par To explore the effect of the relays' locations on the PEP performance of the considered blind and CSI-assisted relaying techniques with IEH, we illustrate in Fig. \ref{f5} the performance of the Alamouti-based scheme, under the assumption of HI MCA noise. This study is conducted for six distinct scenarios of the geometrical layout of the two relays:
\begin{itemize}
\item Scenario 1: $d_{sr,1}$=0.8 and $d_{sr,2}$=0.8,
\item Scenario 2: $d_{sr,1}$=0.5 and $d_{sr,2}$=0.8,
\item Scenario 3: $d_{sr,1}$=0.2 and $d_{sr,2}$=0.8,
\item Scenario 4: $d_{sr,1}$=0.5 and $d_{sr,2}$=0.5,
\item Scenario 5: $d_{sr,1}$=0.5 and $d_{sr,2}$=0.2,
\item Scenario 6: $d_{sr,1}$=0.2 and $d_{sr,2}$=0.2.
\end{itemize}
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{PEP_multiplerelayLoc_SNR_IEHAPS2.pdf}
\caption{PEP performance with respect to SNR for various relay locations over HI noise under Model I.}
\label{f5}
\vspace*{-0.6cm}
\end{figure}
It is shown in Fig. \ref{f5} that the best performance for both blind and CSI-assisted IEH-relaying schemes is exhibited by Scenario 6, where both relays are close to the source, while locating the two relays close to the destination represented by Scenario 1 leads to the worst performance. This is expected, since the power available at the relay nodes resulting from EH during \textit{Phase-1}, as defined in \eqref{Pr}, is inversely proportional to the distance between the source and the relay node. Specifically, as $d_{sr,n}, n\in\lbrace{1,2\rbrace}$ increases, both the harvested energy and the received signal strength at the relay node decrease due to the increased path-loss, and consequently, deteriorating the performance. A similar result is noted for both of the relaying techniques when AEH is employed, however their performance is not plotted to avoid repetition.
\par This observation suggests the support for the conclusion in \cite{Nasir2013} for a SWIPT wireless cooperative systems under the general AWGN noise assumption. On the contrary, this finding is different from the conventional case where EH is not considered at the relays \cite{Al-Dharrab}, wherein the best performance is attained by Scenario 4, where both relays are equidistant from the source and destination nodes and the worst performance is observed in Scenario 3 where one of the relays is placed closer to the source node and the other is placed closer to the destination node. The aforementioned result along with the ones reported in \cite{Nasir2013} and \cite{Al-Dharrab} lead us to conclude that the optimal position of the relays in a SWIPT relaying system may be independent from the channel noise type.
\par Remarkably, for both blind and CSI-assisted relaying techniques, as the two relays become closer to the source the flat region observed in the case of HI noise is significantly diminished, thereby, considerably outperforming the non-EH case presented in \cite{Al-Dharrab} from this perspective. Therefore, the results obtained in this examination are two fold. First, it is noted that EH relaying systems are more robust towards impulsive noise. Second, the location of the relays plays a crucial role in the underlying system performance. Further examinations of the impact of the relays' location on the system performance are carried out in Fig. \ref{f6}.
\par Fig. \ref{f6} depicts the PEP performance of blind and CSI-assisted relaying for both IEH and AEH as a function of the normalized $S\to R$ link distances of $R_1$ and $R_2$. The study is carried out for the NG and HI noise environments, considering Model I, under the assumption of both low (15~dB) and high (40~dB) SNR regimes. As it can be readily observed for all four relaying schemes, in general, the PEP increases as $d_{sr,1}$ and $d_{sr,2}$ increase, i.e., the distance between the source and the two relays increases. As explained earlier, this is because the farther away the two relay nodes are from the source node, the larger the experienced path-loss is, leading to less signal power to be received at $R_n$. Accordingly, the received signal power at the destinations node is poor, yielding inferior PEP performance. This result is in accordance with the majority of the research work in the literature of SWIPT relaying networks \cite{Nasir2013,Rabie1,Ojo,Rabie,Liu} and the references therein, where it is demonstrated that the best performance of the network was achieved when the relay nodes are located closer to the source node than the destination node. In our work, we demonstrate that this finding also holds when the network is operating under the impulsive noise. Moreover, we notice that in the case of low SNR regime (SNR=15dB), which is included in the flat region of the PEP performance under the HI noise, the PEP performance does not notably change with the change in the distance and that the performance is irrespective of the adopted relaying schemes. However, a rather more noticeable change is observed in the high SNR regime. This is in contrast to the NG noise environment case, where more rapid improvements can be seen at both low and high SNR regimes as the relays move closer to the source. Therefore, it turns out that moving the relays closer to the source is more rewarding in the NG noise environment. It can be further deduced from Fig. \ref{f6} that the performance gap between the four analyzed relaying schemes is more pronounced in the NG noise environment in the high SNR scenario. Finally, one can observe that the PEP performance does not notably change by increasing $d_{sr,1}$ and $d_{sr,2}$ beyond a certain value ($d_{sr,n}>0.8$), since as the relays get closer to the destination, smaller values of harvested energy are required to support the reliable communication through the $R_n\to D$ link. A similar conclusion can be drawn for all the presented relaying techniques for EH relays which are solely powered by the source. This suggests that the harvested energy at the relay nodes is the dominant performance limiting factor, rendering the $R_n\to D$ link to be the bottleneck of the system performance.
\begin{figure}[!t]
\centering
\includegraphics[width=3.7in]{PEP_Vs_dist.pdf}
\caption{PEP performance with respect to the normalized distances $d_{sr,n}, n\in \lbrace{1,2\rbrace}$ over NG and HI noise under Model I.}
\label{f6}
\vspace*{-0.3cm}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3.7in]{PEP_vs_theta.pdf}
\caption{PEP performance with respect to the PS factor $\theta_1$ at relay $R_1$ over NG and HI noise under Model I, where $\theta_2=\theta_1$.}
\label{f7a}
\vspace*{-0.3cm}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3.7in]{PEP_Vs_theta2.pdf}
\caption{PEP performance with respect to the PS factor $\theta_1$ at relay $R_1$ over NG and HI noise under Model I, where $\theta_2=1-\theta_1$.}
\label{f7b}
\vspace*{-0.6cm}
\end{figure}
\par In Figs. \ref{f7a} and \ref{f7b}, we investigate the impact of the PS factor $\theta_n$ at the relays on the associated PEP performance of the competing relaying techniques for NG and HI noise environments under Model I. The examination is carried out for low and high SNR regimes. Furthermore, in our work, we consider two scenarios for the PS factor of the two relays. The first scenario is depicted in Fig. \ref{f7a}, where we plot the PEP performance as a function of the PS factor of relay $R_1$, $\theta_1$, and we set the PS factor at the second relay $R_2$ to be $\theta_2=\theta_1$. In the second scenario, illustrated in Fig. \ref{f7b}, we set $\theta_2=1-\theta_1$. This is done to provide a deeper understanding on the behaviour of the system when equal or different power settings are imposed on the two relays. Interestingly, one can arrive at the same observation on the PEP performance from Fig. \ref{f7a} and Fig. \ref{f7b}. Specifically, it is noted that the PEP performance is insensitive to the change in the value of the PS factors at the two relays in the HI noise environment under the low SNR assumption due to the detrimental effects of the impulsive noise. On the contrary, it is demonstrated that for the other three scenarios (low SNR with NG noise and low and high SNR with NG and HI noise), there exists an optimal value for the PS factor that minimizes the PEP for the scenario in Fig. \ref{f7a}. This stems from the fact that when the value of $\theta_n, n\in\lbrace{1,2\rbrace}$ is smaller than the optimal, there is less power available for EH. Consequently, less transmission power is available at the two relay nodes causing the performance to deteriorate gradually. On the other hand, as the value of $\theta_n$ increases beyond the optimal value, more power is spent on EH at the expense of the power available for data transmission which considerably degrades the PEP performance. This phenomenon is expected, since the performance of dual-hop systems is constrained by the quality of the weakest hop \cite{Muhaidat2}. Comparing the two setups, we observe from Fig. \ref{f7a} that the minimum PEP performance is attained when $\theta_1=\theta_2=0.22$. However, when the PS factors are different, we observe from Fig. \ref{f7b} that the minimum PEP is achieved for $\theta_1=\theta_2=0.5$. This finding suggests that allocating equal PS factors displays a performance gain gap over the non-equal PS factors at the two relays. A final observation for both Fig. \ref{f7a} and \ref{f7b} is that when blind IEH-relaying is adopted, varying $\theta_n$ only makes a rather small change to the PEP performance. This trend is similar for all the examined noise and SNR scenarios. The aforementioned two scenarios imply that the PS factor for EH must be optimized for best performance.
\begin{figure}[!t]
\centering
\includegraphics[width=3.7in]{PEP_indp_dep_SNR.pdf}
\caption{PEP performance over NG and HI noise under Model I and Model II.}
\label{f8}
\end{figure}
\par To address the effect of the spatial independence, we plot in Fig. \ref{f8} the PEP performance for Models I and II under both NG and HI noise environments against the AWGN benchmark case. It is recalled that Model I refers to the case when the same set of interfering sources affects the relay and destination nodes together, while Model II refers to the case when different sets of interfering sources affect the relay and destination nodes. Fig. \ref{f8} illustrates that when the noise is HI, Model I outperforms Model II in the sufficiently low SNR regime (SNR $<22$dB). This behaviour is reversed in the higher SNR region and the performance over Model II becomes superior to that exhibited by Model I. On the other hand, both models exhibit a similar performance in the NG noise over the whole inspected SNR region. These results are in accordance with the ones reported in \cite{Al-Dharrab}.
\section{Conclusions}\label{sec:conc}
In this paper, we have investigated the performance of distributed Alamouti codes for SWIPT AF relaying systems in the presence of MCA noise. Assuming the PS receiver architecture, we have derived novel closed-form PEP expressions which are then exploited to provide detailed performance comparisons among the four relaying techniques under consideration. Besides the fact that our results are accurate and mathematically tractable, they provide efficient means for the design and evaluation of SWIPT relaying networks in practical scenarios where impulsive noise is present. In particular, the proposed analytical model is exploited to study the diversity gains of blind AF and CSI-assisted AF schemes considering AEH and IEH. In addition, we have illustrated that the performance of CSI-assisted AEH-relaying is superior to that exhibited by the other three relaying techniques, achieving the highest diversity order of 3. Furthermore, we have demonstrated that the performance loss incurred by the severity of noise impulsiveness increases with the diversity order and that the performance of the system in the low and medium SNR regions depends on the impulsive nature of the noise, resulting in different diversity orders to dominate the performance. Significant performance gains have been observed by locating the relays close to the source, offering a potential solution to mitigate the deleterious effect of MCA noise. Our results highlight the importance of accurately characterising the performance of the system for the successful implementation of SWIPT relay networks in the presence of impulsive noise.
\vspace*{-0.5cm}
\appendices
\newcounter{tempequationcounter8}
\begin{figure*}[b]
\normalsize
\hrulefill
\setcounter{equation}{48}
\begin{align}\label{I0}
I_0=-\frac{\xi_n}{B_n}\int_0^\infty\frac{\text{exp}\left(\frac{-t}{\xi_n}\right) dt}{\left(t+\left(\frac{\xi_n^2+\xi_n\sqrt{\xi_n^2-4B_n}}{2B_n}\right)\right)\left(t+\left(\frac{\xi_n^2-\xi_n\sqrt{\xi_n^2-4B_n}}{2B_n}\right)\right)} \nonumber \\
-\frac{\xi_n}{B_n}\int_0^\infty \frac{t\quad\text{exp}\left(\frac{-t}{\xi_n}\right) dt}{\left(t+\left(\frac{\xi_n^2+\xi_n\sqrt{\xi_n^2-4B_n}}{2B_n}\right)\right)\left(t+\left(\frac{\xi_n^2-\xi_n\sqrt{\xi_n^2-4B_n}}{2B_n}\right)\right)}
\end{align}
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\newcounter{tempequationcounter9}
\begin{figure*}[b]
\normalsize
\hrulefill
\setcounter{equation}{49}
\begin{align}\label{pepprop3}
P(\bold{s}\to\hat{\bold{s}}||h_{sr_1}|^2,|h_{sr_2}|^2)\leq\sum_{m=0}^{M}\alpha_m\left(\frac{\Delta_s}{4\beta_m N_0}+1\right)^{-1}\quad \quad\quad \quad\quad \quad\quad \quad\quad \quad\quad \quad\quad \quad\nonumber \\
\times \left(\frac{\epsilon_1\zeta_1/(4\beta_m N_0))|h_{sr_1}|^2}{\xi_1|h_{sr_1}|^2+1}+1\right)^{-1}\left(\frac{(\epsilon_2\zeta_2/(4\beta_m N_0))|h_{sr_2}|^2}{\xi_1|h_{sr_2}|^2+1}+1\right)^{-1}.
\end{align}
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\setcounter{equation}{42}
\section{Proof of Proposition 1}
\label{Appendix A}
Starting from the conditional PEP expression in \eqref{chernoff}, we take the expectation with respect to fading coefficients $|h_{sd}|^2$, $|h_{r_1d}|^2$, and $|h_{r_2d}|^2$, which follow an exponential distribution, resulting in
\begin{align}
P(\bold{S}\to\hat{\bold{S}}||h_{sr_1}|^4,|h_{sr_2}|^4)&\leq\sum_{m=0}^{M}\alpha_m\left(\frac{\Delta_s}{4\beta_m N_0}+1\right)^{-1}\nonumber \\
&\times\prod_{n=1}^2 \left(\frac{\epsilon_n\Phi_n^2 }{4\beta_m N_0}|h_{sr_n}|^4+1\right)^{-1}.
\end{align}
Performing an expectation with respect to the random variables $|h_{sr_1}|^4$, $|h_{sr_2}|^4$, which also follow an exponential distribution, yields the unconditional PEP, which is written as\\
\begin{align}\label{uncondPEP1}
P(\bold{s}\to\hat{\bold{s}})&\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1} \nonumber \\
&\times\prod_{n=1}^2\int_0^\infty\left(\frac{\epsilon_n \Phi_n^2 }{4\beta_m N_0}t^2+1\right)^{-1}\text{exp}(-t) dt,
\end{align}where $t$ is the integration variable and $n\in\lbrace{1,2\rbrace}$. Using the equality in \cite[Eq. (8.4.2.5)]{Prudnikov} to express the first integrand of \eqref{uncondPEP1} as \\
\begin{equation}
\left(\frac{\epsilon_n \Phi_n^2}{4\beta_m N_0}t^2+1\right)^{-1}=G_{1, 1}^{1, 1}\left[\frac{\epsilon_n \Phi_n^2}{4\beta_m N_0}t^2 \Big \vert \ {0 \atop 0} \right], \quad n\in\lbrace{1,2\rbrace},
\end{equation}\\then making use of the equality $e^{-t}=G_{0, 1}^{1, 0}\left[t\ \vert \ {- \atop 0} \right]$\cite[Eq. (8.4.3.1)]{Prudnikov} to rewrite the second integrand in \eqref{uncondPEP1}, the unconditional PEP can be derived in a closed-form as in \eqref{finalPEPmodIAPS} by exploiting the integral identity \cite[Eq. (2.24.1.2)]{Prudnikov}.\\
\section{Proof of Proposition 2}
\label{Appendix B}
In order to derive the PEP expression, we first obtain the exact PDF of the RV $Z_n$. It is recalled that RVs $X$ and $Y$\footnote{We drop in the proof the index $n$ for the convenience of analysis.} are independent RVs drawn from the exponential distribution. Therefore, their joint PDF is $f_{X,Y}=e^{-x-y}$ \cite{papoulis}. Expressing $X$ and $Y$ in terms of $U$ and $V$ as $X=(V-1)/\xi$ and $Y=U\xi^2/(V-1)^2$, then with the help of the Jacobian transformation method \cite{papoulis}, $(X,Y)$ are transformed to $(U,V)$. Consequently, the PDF of $(U,V)$ is obtained as \\
\begin{equation}\label{pdfuv}
f_{U,V}=J_d f_{X,Y}\left(\frac{(v-1)}{\xi},\frac{ub^2}{(v-1)^2} \right),
\end{equation}where $J_d=-\xi/(V-1)^2$ is the Jacobian of the transformation. Then using \cite[Eq. (6.60)]{papoulis} and \eqref{pdfuv}, and after some algebraic manipulations, the exact PDF of $Z$ is derived as\\ \\
\begin{align}\label{pdfz}
f(z)=\int_1^\infty v f_{U,V}(vz,v) dv \quad\quad\quad\quad\quad\quad\quad\quad \quad \quad \quad \quad \nonumber \\
=-\int_0^\infty \frac{\xi(t+1)}{t^2} f_{X,Y}\left(\frac{t}{\xi},\frac{(t+1)z\xi^2}{t^2}\right) dt\quad \quad \quad \quad \nonumber \\
=-\int_0^\infty \frac{\xi (t+1)}{t^{2}}\text{exp}\left(-\frac{t}{\xi}-\frac{\xi^2 (t+1) z}{t^2}\right)dt, \quad \quad\quad
\end{align}\\where the second equality in \eqref{pdfz} stems from the fact that $v>1$, as shown in \eqref{pdfuv}. To the best of the authors' knowledge, the integral in \eqref{pdfz} does not lend itself to a closed-form. However, we can obtain the exact PEP expression in a closed-form by substituting \eqref{distIPS} in the conditional PEP expression given in \eqref{chernoff}. Then, the desired unconditional PEP expression is deduced in \eqref{pepIEHIPS} by taking the expectation with respect to the RVs $|h_{sd}|^2, Z_1$ and $Z_2$, where we used the fact that the PDF of $|h_{sd}|^2$ follows the exponential distribution and that the PDF of each of $Z_1$ and $Z_2$ is computed using \eqref{pdfz}, yielding
\begin{align}\label{peppdfz}
P(\bold{s}\to\hat{\bold{s}})\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1}\quad\quad\quad\quad\quad\quad\quad\quad\nonumber\\
\times\prod_{n=1}^2\underbrace{\int_0^\infty -\xi_n (t+1) t^{-2}\left(\frac{\xi_n^2 (t+1)}{t^2}+B_n\right)^{-1}\text{exp}\left(\frac{-t}{\xi_n}\right) dt}_{I_0},
\end{align}where $B_n=(\epsilon_n\zeta_n)/(4\beta_m N_0)$. Finally, by rewriting $I_0$ as \eqref{I0} at the bottom of this page, followed by some algebraic manipulations, and invoking \cite[Eq. (3.354.3)]{Rizhik} and \cite[Eq. (3.354.4)]{Rizhik}, the integral in \eqref{peppdfz} is obtained in a closed-form as in \eqref{pepIEHIPS}.
\vspace*{-0.3cm}
\section{Proof of Proposition 3}
\label{AppendixC}
Substituting \eqref{distIPS2} in the conditional PEP expression in \eqref{chernoff}, then taking the expectation with respect to fading coefficients $|h_{sd}|^2$, $|h_{r_1d}|^2$, and $|h_{r_2d}|^2$, which follow an exponential distribution, to yield \eqref{pepprop3} at the bottom of this page. Performing an expectation with respect to the random variables $|h_{sr_1}|^2$, $|h_{sr_2}|^2$, which also follow an exponential distribution, yields the unconditional PEP which is written as
\setcounter{equation}{50}
\begin{align}\label{uncondPEP}
P(\bold{s}\to\hat{\bold{s}})\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1}\quad \quad\quad\quad \quad\quad\quad \quad\quad\nonumber \\
\times\prod_{n=1}^2\underbrace{\int_0^\infty\left(\frac{(\epsilon_1\zeta_1/(4\beta_m N_0))|h_{sr_1}|^2}{\xi_1|h_{sr_1}|^2+1}+1\right)^{-1}\text{exp}(-t) dt}_{\Upsilon},
\end{align}where $t$ is the integration variable and $n\in\lbrace{1,2\rbrace}$. To solve the integral $\Upsilon$, we perform simple algebraic manipulations to get
\begin{align}
\Upsilon&=\int_0^\infty\left(\xi_n t+1\right)\left(\left(\frac{\epsilon_n\zeta_n}{4\beta_m N_0}+\xi_n\right) t+1\right)^{-1}\text{exp}(-t) dt \nonumber \\
&=\underbrace{\int_0^\infty \left(\left(\frac{\epsilon_n\zeta_n}{4\beta_m N_0}+\xi_n\right) t+1\right)^{-1}\text{exp}(-t) dt}_{I_1}\nonumber \\
&+\xi_n\underbrace{\int_0^\infty t\left(\left(\frac{\epsilon_n\zeta_n}{4\beta_m N_0}+\xi_n\right) t+1\right)^{-1}\text{exp}(-t) dt}_{I_2} \label{47a}.
\end{align}
Then, with the aid of the equality in \cite[Eq. (8.4.2.5)]{Prudnikov}, followed by applying the transformation \cite[Eq. (8.2.2.14)]{Prudnikov}, the first and second integrands of $I_1$ and $I_2$, respectively, are expressed in terms of their Meijer G-function representations as\\
\begin{equation}
\left(\gamma_n t+1\right)^{-1}=G_{1, 1}^{1, 1}\left[\frac{1}{\gamma_n t } \ \Big \vert \ {1 \atop 1} \right].
\end{equation}where $\gamma_n=\left(\frac{\epsilon_n\zeta_n}{4\beta_m N_0}+\xi_n\right)$. Similarly, the second and third integrands of $I_1$ and $I_2$, respectively, are rewritten by making use of the equality $e^{-t}=G_{0, 1}^{1, 0}\left[t \ \vert \ {- \atop 0} \right]$\cite[Eq. (8.4.3.1)]{Prudnikov}, yielding
\begin{align}
\Upsilon&=\int_0^\infty G^{1,1}_{1,1}\left[\gamma_n t \ \Big\vert \ {0 \atop 0}\right]G^{1,0}_{0,1}\left[t \ \Big\vert \ {- \atop 0}\right]dt \nonumber \\
&+\xi_n\int_0^\infty t G^{1,1}_{1,1}\left[\gamma_n t \ \Big\vert \ {0 \atop 0}\right]G^{1,0}_{0,1}\left[t \ \Big\vert \ {- \atop 0}\right]dt. \label{b47}
\end{align}
Then, by exploiting the integral identity \cite[Eq. (3.356.4)]{Prudnikov}, followed by performing some algebraic manipulations, $\Upsilon$ can be derived in a closed-form as
\begin{align}
\Upsilon&=\gamma_n^{-1}G^{1,2}_{2,1}\left[\gamma_n \ \Big\vert \ {1, 1 \atop 1}\right] +\xi_n \gamma_n^{-2}G^{1,2}_{2,1}\left[\gamma_n \ \Big\vert \ {1, 2 \atop 2}\right] \label{c47}
\end{align}
Finally, after substituting \eqref{c47} in \eqref{uncondPEP}, the desired result in \eqref{PEPAEHIPS} is derived.
\balance
\bibliographystyle{IEEEtran}
\bstctlcite{BSTcontrol}
\section{Introduction}
\IEEEPARstart{F}{UTURE} wireless networks are envisioned to offer an unprecedented opportunity to connect the global world via a massive number of low-power heterogeneous smart devices, enabled by the internet of Things (IoTs) \cite{Akpakwu}. A major bottleneck for the application of such untethered nodes is their finite battery capacity, requiring the need to be recharged/replaced rather frequently. In this context, simultaneous wireless information and power transfer (SWIPT) has emerged as a promising technology to address the conflicting design goals of perpetual lifetime and uninterrupted network performance. In a SWIPT-enabled system, a wireless node is powered up by a received Radio Frequency (RF) signal and, simultaneously, information processing is carried out using the same signal \cite{Varshney2008}.
\par SWIPT-based relaying was proposed as a promising technique to provide advantages in two fold. First, the network itself can benefit from the relays in throughput improvement, communication reliability enhancement, and coverage range extension. Second, the harvested energy can be used to charge the relay nodes, and therefore, the overall power consumption of the network may be considerably reduced \cite{Nasir2013, Mohjazi3}. From this perspective, the theoretical and implementation aspects of SWIPT relay networks have been areas of active research interest (see \cite{Rabie1,Al-habob,Fang,Ojo} and the references therein).
\par Although there has been a growing literature on SWIPT, particularly in the context of relay networks (see e.g., \cite{Rabie1,Al-habob,Fang,Ojo} and the references therein), all research studies were based upon the classical assumption of additive white Gaussian noise (AWGN). However, many communication channels are additionally impaired by impulsive man-made electromagnetic interference or atmospheric noise encountered in various metropolitan and indoor wireless applications, such as, automotive ignition, electronic devices, household appliances, medical equipment, and industrial equipment. \cite{Blankenship, Blackard,Sanchez}. A practical foreseen scenario of such a situation is future IoTs, for instance, where nodes can be implanted in environments that are susceptible to impulsive noise such as in industrial locations or in fields close to power lines. Although these nodes are envisioned to be powered by RF energy through SWIPT to achieve advantages, such as, dual use of RF signals for information and power transfer, extended network lifetime, etc., their performance in terms of error rate is not yet studied when impulsive noise is considered. Nonetheless, it is considered as a prevalent source of performance degradation. It has been demonstrated in \mbox{\cite {Spaulding}} that communication systems designed under the AWGN assumption typically suffer from severe performance degradations when exposed to impulsive noise. This elevates the need for studying the performance of SWIPT systems, which are not only disturbed by multipath fading, but also by impulsive (non-Gaussian) noise, in order to provide pragmatic information for the system designer.
\par Several statistical models have been proposed to approximate the behaviour of impulsive noise, such as Bernoulli-Gauss \cite{Ghosh}, the symmetric alpha stable distribution \cite{Ilow}, and the Middleton's models \cite{Middleton1, Middleton2}. However, Middleton's models have been widely accepted to model the effects of impulse noise in communication systems due to its accuracy in approximating the behaviour of this noise over many communication channels and since its validity was confirmed by many measurement campaigns. Among the three distinct noise categories of Middleton's models, the most popular is the so-called Middleton Class-A (MCA) noise model \cite{Middleton2}. Additionally, this model presents the advantage to be a generic model which only depends on three physical parameters, namely, the noise power, the impulsive index that describes the average number of impulses during some interference time, and the Gaussian factor which resembles the ratio of the variances of the background Gaussian noise to the impulsive noise. Furthermore, the MCA noise model is characterized by a simple probability density function (PDF) expression which enables designing an optimum receiver with low complexity.
\par Several research studies in the open literature have investigated the effect of the MCA impulsive noise on conventional non-energy harvesting (EH) communication systems \cite{Alhussein1,Ping,Al-Dharrab,Gao,Schober1} and the references therein. However, these studies focus on examining the impact of impulsive noise on the process of information delivery only. Nonetheless, SWIPT systems are characterized both by information and power delivery simultaneously. Therefore, a thorough analysis of the effect of impulsive noise is an inevitable prerequisite for the appropriate design of impulsive noise combating mechanisms and robust receivers for such systems.
\par While most of the current literature on SWIPT systems is based upon the assumption of the classical AWGN noise assumption, there have been recent results \cite{Rabie2,Rabie3} which study the performance of a point-to-point SWIPT system under the assumption of impulsive noise following the Bernoulli-Gauss model. To the best of our knowledge, the impact of impulsive noise on the performance of SWIPT is not comprehensively understood yet, since it has not been addressed in the related open literature, which demands for a thorough investigation. We note that such an investigation is imperative for the actual realization of SWIPT and for determining the actual performance limits in terms of error rate performance.
\par Aiming to fulfil this research gap, we propose an accurate mathematical framework to analyse the pairwise error probability (PEP) performance of SWIPT relaying systems over Rayleigh fading channels subject to MCA. PEP constitutes the stepping stone for the derivation of union bounds to the error probability. It is widely used in the literature to analyse the achievable diversity order, where closed-form error probability expressions are unavailable. In particular, we assume that SWIPT relaying is enabled by a power splitting (PS) receiver architecture \cite{Nasir2013} and adopt the amplify-and-forward (AF) relaying protocol with two schemes depending on the availability of channel state information (CSI) at the relay node, namely, a CSI-assisted relaying scheme and a blind relaying scheme. Additionally, we adopt two EH techniques: EH based on average CSI (AEH) \cite{Liu} and EH based on instantaneous CSI (IEH) \cite{Mohjazi4}. Specifically, the main contributions and results of this paper are summarized as follows:
\begin{itemize}
\item We derive novel exact closed-form PEP expressions for a two-relay dual-hop SWIPT relaying system with blind and CSI-assisted relaying schemes employing AEH and IEH.
\item The derived analytical PEP expressions are used to numerically evaluate the diversity order of the considered schemes. Specifically, we demonstrate that CSI-assisted relaying with AEH is superior to the other three relaying techniques achieving the highest diversity order of three. We further demonstrate that the lowest diversity order of two is obtained by the blind relaying scheme employing IEH suffering from cascaded fading resulting from IEH.
\item We demonstrate that under severe noise impulsiveness, the convergence to full spatial diversity becomes slower and that the associated performance loss increases with the diversity order.
\item We demonstrate through our numerical results that for all considered relaying techniques, the best performance is achieved when the two relays are located closer to the source node than the destination node and conclude that the optimal location of the relays is independent from the noise type, i.e., MCA or AWGN.
\item Finally, a comprehensive computer-based Monte Carlo simulation study is presented to verify the accuracy of the analytical results and to further investigate several design choices within the considered relay-assisted transmission scenarios.
\end{itemize}
\par The remainder of the paper is organized as follows. In Section II, we describe the noise model and the two-relay SWIPT transmission model in conjunction with blind and CSI-assisted relaying. In Section III, we present the analytical derivations of the PEP expressions for each of the relaying techniques under consideration. Section IV provides extensive Monte-Carlo simulation results to corroborate the analytical results and to provide detailed performance comparisons among the competing schemes for various scenarios. Concluding remarks are given in Section V. The appendices include mathematical details of the PEP derivations.
\par \underline{\textit{Notation}:} Bold lower case letters denote vectors. $(.)^T, (.)^*$, $\mathbb{E}[z]$, and $|z|$ stand for the transpose, conjugate, expectation of the random variable $z$, and magnitude of a complex variable $z$, respectively.
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{systemmodel.jpg}
\caption{Schematic representation of relay-assisted transmission.}
\label{blockdiagram}
\end{figure}
\section{System Model}
\label{sec:model}
We consider a dual-hop AF SWIPT relaying system as shown in Fig.~\ref{blockdiagram}, where a source node, $S$, communicates with a destination node, $D$, via two intermediate relay nodes, $R_1$ and $R_2$. The source and the destination nodes are assumed to be energy unconstrained nodes powered by either a battery or a power grid. On the other hand, the relay nodes have no dedicated power supply and harvest energy from the received signal which is then used over the second hop. In our work, we assume that a direct link exists between the source node and the destination node. We consider the PS protocol for wireless EH, and assume that all nodes are equipped with a single antenna. We also assume that all nodes operate in the presence of impulsive noise. In what follows, we introduce the adopted noise and transmission models.
\subsection{Noise Model}
We assume that each noise sample in the $t$-th time slot at any node is given by
\begin{equation}\label{noisesample}
n(t)=n_G(t)+n_I(t),
\end{equation}where $n_G(t)$ and $n_I(t)$ denote the background zero-mean complex Gaussian noise with variance $\sigma^2_G$ and the impulsive noise with variance $\sigma^2_I$, respectively. Adopting the MCA noise model and assuming that the active interfering sources emit independently, the PDF of the complex-valued noise sample, given in \eqref{noisesample}, at any of the nodes can be expressed as \cite{Middleton}
\begin{equation}\label{MCApdf}
f(n(t)) =\sum_{m(t)=0}^\infty \frac{\alpha_{m(t)}}{\pi\sigma^2_{m(t)}}\text{exp}\left(-\frac{|n(t)|^2}{\sigma^2_{m(t)}}\right),
\end {equation}where\footnote{Hereafter, we drop the time index in $m(t)$ and use $m$ instead.}
\begin{equation}\label{alpha}
\alpha_{m}=\frac{e^{-A}A^{m}}{m!},
\end{equation}with $A$ denoting the impulsive noise index that describes the average number of impulses during the interference time \cite{Middleton}. When it takes small values, i.e., $A\to 0$, it results in a highly structured and more impulsive noise, whereas it results in a near-Gaussian noise when it is large, i.e., $A\to\infty$. Furthermore, in \eqref{MCApdf}, $\sigma^2_m$ is the conditional variance given that $m$ impulses are affecting the receiver and is calculated as $\sigma_m^2=\sigma_n^2\beta_m$, where $\sigma_n^2$ denotes the mean variance of impulsive noise $n(t)$ and is equal to $N_0$ and $\beta_m$ is given by
\begin{equation}\label{beta}
\beta_m=\left(\frac{mA^{-1}+\delta}{1+\delta}\right),
\end{equation}where $\delta=\sigma^2_G/\sigma^2_I$ is called the Gaussian noise factor \cite{Middleton}, which is equal to the ratio of the variance of the background Gaussian noise component to the impulsive noise component. It is worth noting that the noise PDF in \eqref{MCApdf} reduces to the Gaussian distribution when $\delta\to\infty$ while it tends to be more impulsive when $\delta\to 0$.
Throughout this work, we assume that $\delta>0$ which implies that the Gaussian noise component is always present.
\par As clearly seen from \eqref{MCApdf}, the noise sample $n(t)$ in \eqref{noisesample} is not Gaussian, however, it can be viewed as conditionally Gaussian, such that, when conditioned on the state $m$, $n(t)$ is Gaussian with zero-mean and variance $\sigma_m^2$. The sequence of states $m(t)$ is an independent and identically distributed (i.i.d) random process, and a particular state $m(t)=m$ occurs with probability $C_0=\alpha_m$, $0\leq m<\infty$, where $m(t)$ follows a Poisson distribution with parameter $A$. Therefore, it is interpreted from that the integer random variable $C_0$ is the state of the noise indicating that there is no impulse $(C_0=0)$, or impulses are present $(C_0>0)$.
\par Although the distribution of MCA includes an infinite summation, it is completely characterized by two parameters, $A$ and $\delta$. In this work, we assume that $A$, $\delta$, and $\sigma_n^2$ are perfectly known at the receiver. In practice, these parameters can be estimated using the expectation maximization (EM) method proposed in \cite{Zabin}. We can see that the noise state probability $\beta_m$ in \eqref{beta} tends to zero as $m$ approaches infinity. Therefore, in the subsequent analysis, we truncate the sum in \eqref{MCApdf} to $M$ terms to reduce the computational complexity without compromising the performance accuracy \cite{Al-Dharrab}.
\par In this paper, we assume that the impulsive noise samples are temporally dependent during a transmission frame, following the widely used assumption in literature \cite{Delaney}. Furthermore, from the perspective of spatial dimension, we consider two models, namely, dependent and independent impulsive noise models. In Model I, which assumes spatially dependent noise samples, the same set of interfering sources affects the destination and relay nodes together. This scenario is applicable when the destination and relay nodes are at relatively the same distance to the interfering sources \cite{Gao, Al-Dharrab}. On the contrary, in Model II, it is assumed that each of the destination and relay nodes are affected by different sets of interfering sources and therefore, their respective noise samples are spatially independent.
\begin{figure}[!t]
\centering
\includegraphics[width=4in]{transmissionmodelch3.pdf}
\caption{Transmission allocation of the source node, $S$, and the two relay nodes, $R_1$ and $R_2$ over the two-Phase transmission scheme with each phase consisting of time slots.}
\label{transmissionmodelch3}
\end{figure}
\subsection{Transmission Model}
We consider a wireless communication scenario where the source node $S$ transmits information to the destination node $D$ with the assistance of two EH relay nodes $R_1$ and $R_2$. We adopt the so-called Protocol II of \cite{Laneman, Nabar} as the relaying protocol, which is completed over two signalling intervals, namely, \textit{Phase-1} and \textit{Phase-2} (see Fig.~\ref{transmissionmodelch3}). We further assume that the source node communicates with the two relays and the destination nodes using the Alamouti's code \cite{Alamouti}. Specifically, the transmission of two Alamouti coded symbols is performed over four time slots ${t}=1,...,4$. During \textit{Phase-1}, spanning two time slots ${t}=1$ and ${t}=2$, the source node communicates with the relays and destination nodes. In \textit{Phase-2}, spanning two time slots ${t}=3$ and ${t}=4$, the source node remains silent, whereas the two relays employ the AF relaying technique to retransmit a scaled version of their received signals to the destination node using Alamouti coding\footnote{This protocol realizes a maximum degree of broadcasting and exhibits no receive collision \cite{Nabar}.}. Also, it is assumed that the system is perfectly synchronized at the symbol level, i.e., relays transmit at the same time \mbox{\cite{Nabar}}. Protocol II is logical in a scenario where the source node engages in data reception from another node in the network over the second time slot, thereby rendering it unable to transmit \cite{Nabar}. The implementation of the Alamouti coding scheme has been considered in the literature of SWIPT networks in \cite{Zhai,Liu2016}. We assume that the relays harvest energy from the received source signals during \textit{Phase-1}, which is then used to forward the information to the destination in \textit{Phase-2}.
\par We further categorize the adopted AF relaying schemes based on the applied amplifying coefficient at the relay nodes, referred to as blind relaying \cite{Laneman2,Hasna} and CSI-assisted \cite{Laneman1} relaying. In the former scheme, the relays have no access to instantaneous CSI of their respective $S\to R$ links and hence, employ a fixed amplifying coefficient, which ensures that an average output power is maintained \cite{Laneman2}. While in the latter scheme, the relays use the receive CSI of their respective $S\to R$ link to ensure that the output power is limited to the power available at the relay, and therefore, a constant power is maintained for each realization \cite{Laneman1}.
\par Let $h_{sd}$, $h_{sr,n}$ and $h_{rd,n}$, respectively denote the complex small-scale fading coefficients over the $S\to D$ link, $S\to R_n$ link from the source to the $n$-th relay, $n \in \lbrace{1,2\rbrace}$, and $R_n\to D$ link from the $n$-th relay to the destination. These channel coefficients are modelled as i.i.d zero-mean complex Gaussian random variables (RVs) with variance 0.5 per dimension leading to the well-known Rayleigh fading channel model. It is also assumed that all channel coefficients remain constant over the block duration and vary independently and identically from one block to another. In addition to the small-scale fading, we further assume that all links are subject to large-scale path-loss that reflects the effect of the relative relays' locations on the performance of the system. Under this assumption, the received power is inversely proportional to $d_{ij}^{\lambda}$, where $d_{ij}$ is the propagation distance between transmitter $i$ and receiver $j$ and $\lambda> 2$ denotes the path-loss exponent. We set the reference distance equal to the distance from the source to the destination and assume that it is equal to unity, and hence, $d_{sr,n}= 1-d_{rd,n}, n\in\lbrace{1,2\rbrace}$. Consequently, the relative gains of $S\to R_n$ and $R_n\to D$ links are defined as $L_{sr,n}=(d_{sr,n}/d_{sd})^{\lambda}$ and $L_{rd,n}=(d_{rd,n}/d_{sd})^{\lambda}$, where $n \in \lbrace{1,2\rbrace}$.
\par Let the two consecutive signals transmitted by the source in \textit{Phase-1} be denoted as $s_1(t)$ and $s_2(t)$. We assume a binary phase shift keying (BPSK) signal constellation with normalized energy for the signals i.e., $\mathbb{E}[|s_p(t)|^2]=1$, $p\in \lbrace{1,2\rbrace}$. More specifically, during the first phase, the received signals at the destination in time slots $t=1$ and $t=2$ are given by
\begin{equation}\label{yd1}
y_{d}(t)=\sqrt{P_s}h_{sd}s_p(t)+n_{d}(t), \quad t=1,2,
\end{equation}where $P_s$ is the source transmit power and $s_p(t)$, $p\in \lbrace{1,2\rbrace}$, is the symbol sent from the source in the $t$-th time interval. Also, $n_{d}(t)$ represents the overall background and impulsive noise at the destination node with conditional variance $\sigma_{m,d}^2=\beta_{m,d} N_{0_d}$, associated with the $t$-th symbol. It is recalled that the parameter $\beta_{m,d}$ depends on the occurrence of a particular random impulsive state $m$ with probability $\alpha_{m,d}$, which follows a Poisson distribution.
\par During \textit{Phase-1}, the $n$-th relay node assigns a portion $\theta_n$ (called the PS ratio) of the received signal power in the $t$-th symbol interval for EH, and the remaining power $(1-\theta_n)$ is assigned for information processing at the information receiver. Accordingly, the received signal at its information receiver is given by
\begin{equation}
\label{YIR}
y_{r,n}(t)=\frac{\sqrt{\kappa_n P_s}}{\sqrt{L_{sr,n}}}h_{sr,n}s_p(t)+ n_{r,n}(t),
\end{equation} where $\kappa_n=(1-\theta_n)$. In this paper, we assume that $0<\theta_n<1$, corresponding to a general SWIPT system featuring both wireless information transfer and wireless EH. Furthermore, $n_{r,n}(t)$ is the overall background and impulsive noise at the $n$-th relay node associated with the $t$-th symbol, which is given by $n_{r,n}(t)=\sqrt{\kappa_n}n_{ra,n}(t)+n_{rc,n}(t)$, such that $n_{ra,n}(t)$ and $n_{rc,n}(t)$ are the receive antenna noise and the noise due to the RF-baseband signal conversion at the $n$-th relay, respectively, with mean variances of $N_{0_{ra,n}}$ and $N_{0_{rc,n}}$, respectively. Therefore, the conditional variance of $n_{r,n}(t)$ is $\sigma^2_{m,r,n}=\beta_{m,r,n} (\kappa_n N_{0_{ra,n}}+ N_{0_{rc,n}})$. For simplicity of the ensuing analysis, we assume that $N_{0_{ra,n}}=N_{0_{rc,n}}=N_0$.
\par The remaining portion of the received signal at $R_n$ in the $t$-th time slot is forwarded to the energy harvester, hence, the power available at $R_n$ at the end of each of the two symbol intervals of the first phase can be expressed as
\begin{equation}\label{Pr}
P_{r,n}=\frac{\eta_n\theta_n P_s|h_{sr,n}|^2}{L_{sr,n}},
\end{equation} with $0<\eta_n<1$ denoting the energy conversion efficiency factor at $R_n$. It should be noted that the EH process at $R_n$ is independent of the power scaling process and it is assumed that EH is performed instantaneously. The harvested instantaneous energy is simply used as a transmit power in the second phase of transmission. Note that the assumption of instantaneous EH was adopted in \cite{Nasir2013}.
\newcounter{tempequationcounter}
\begin{figure*}[t]
\vspace*{-0.3cm}
\normalsize
\setcounter{equation}{7}
\begin{equation}\label{yd3}
y_d(3)=\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}}h_{sr,1}h_{rd,1} s_1(3) +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}h_{rd,2} s_2(3)+\hat{n}_d(3)
\end{equation}
\hrulefill
\vspace*{5pt}
\end{figure*}
\newcounter{tempequationcounter1}
\begin{figure*}[t]
\vspace*{-0.3cm}
\normalsize
\begin{equation}\label{yd4}
y_d(4)=-\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}}h_{sr,1}^*h_{rd,1} s_2(4)^* +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}^*h_{rd,2} s_1(4)^*+\hat{n}_d(4),
\end{equation}
\vspace*{4pt}
\hrulefill
\vspace*{-0.3cm}
\end{figure*}
\newcounter{tempequationcounter2}
\begin{figure*}[t]
\normalsize
\setcounter{equation}{12}
\begin{equation}\label{tildey3}
\tilde{y}_d(3)=\frac{y_d(3)}{\Omega}=\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{\Omega G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}}h_{sr,1}h_{rd,1} s_1(3) +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{\Omega G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}h_{rd,2} s_2(3)+\frac{\hat{n}_d(3)}{\Omega},
\end{equation}
\hrulefill
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\newcounter{tempequationcounter3}
\begin{figure*}[t]
\normalsize
\setcounter{equation}{13}
\begin{equation}\label{tildey4}
\tilde{y}_d(4)=\frac{y_d(4)}{\Omega}=-\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{\Omega G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}}h_{sr,1}^*h_{rd,1} s_2(4)^* +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{\Omega G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}^*h_{rd,2} s_1(4)^*+\frac{\hat{n}_d(4)}{\Omega},
\end{equation}
\hrulefill
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\newcounter{tempequationcounter4}
\begin{figure*}[t]
\normalsize
\setcounter{equation}{14}
\begin{equation}\label{matrixmod}
\mathbf{y}_d=\begin{bmatrix}
y_d(1)\\y_d(2)\\y_d(3)\\y_d(4)\end{bmatrix}=\begin{bmatrix}
\sqrt{P_s}h_{sd}s_1(1)+n_{d}(1) \\
\sqrt{P_s}h_{sd}s_2(2)+n_{d}(2) \\
\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{\Omega G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}} h_{sr,1}h_{rd,1} s_1(3) +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{\Omega G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}h_{rd,2} s_2(3)+\tilde{n}_d(3) \\
-\frac{\sqrt{\kappa_1 P_{r,1}P_s}}{\Omega G_{r,1}\sqrt{L_{rd,1}L_{sr,1}}}h_{sr,1}^*h_{rd,1} s_2(4)^* +\frac{\sqrt{\kappa_2 P_{r,2}P_s}}{\Omega G_{r,2}\sqrt{L_{rd,2}L_{sr,2}}} h_{sr,2}^*h_{rd,2} s_1(4)^*+\tilde{n}_d(4)\end{bmatrix}.
\end{equation}
\hrulefill
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\begin{figure*}[t]
\normalsize
\setcounter{equation}{18}
\begin{equation}\label{matrixmod2}
\bold{y}_d=\begin{bmatrix}
y_d(1)\\y_d(2)\\y_d(3)\\y_d(4)\end{bmatrix}=
\begin{bmatrix}
\sqrt{P_s}h_{sd}s_1(1)+n_{d}(1) \\
\sqrt{P_s}h_{sd}s_2(2)+n_{d}(2) \\
\Phi_1|h_{sr,1}| h_{sr,1}h_{rd,1} s_1(3) +\Phi_2|h_{sr,2}| h_{sr,2}h_{rd,2} s_2(3)+\tilde{n}_d(3) \\
-\Phi_1|h_{sr,1}|h_{sr,1}^*h_{rd,1} s_2(4)^* +\Phi_2|h_{sr,2}| h_{sr,2}^*h_{rd,2} s_1(4)^*+\tilde{n}_d(4)\end{bmatrix}.
\end{equation}
\hrulefill
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\par During \textit{Phase-2} spanning two symbol intervals, the received signals are processed at the relay nodes using the Alamouti scheme in a distributed manner. The resulting signals are then forwarded to the destination nodes using the energy harvested in \textit{Phase-1}. Specifically, the signals received at the destination through the $R_n\to D$ links over time slots ${t}=3$ and ${t}=4$ are given by \eqref{yd3} and \eqref{yd4}, respectively, at the top of this page. In \eqref{yd3} and \eqref{yd4}, $G_{r,n}, n\in\lbrace{1,2\rbrace}$ is the scaling term at the $n$-th relay which depends on the type of amplifying coefficient deployed at $R_n$ (i.e. blind relaying or CSI-assisted relaying), which will be discussed in details in the subsequent section. This normalization does not alter the signal-to-noise ratio SNR but simplifies the ensuing presentation \cite{Nabar}. Furthermore, $\hat{n}_d(3)$ and $\hat{n}_d(4)$ are the effective noise terms associated with the third and fourth symbols, respectively, defined as
\setcounter{equation}{9}
\begin{equation}
\hat{n}_d(3)=\frac{\sqrt{P_{r,1}}}{G_{r,1}\sqrt{L_{rd,1}}}h_{rd,1}n_{r,1}+\frac{\sqrt{P_{r,2}}}{G_{r,2}\sqrt{L_{rd,2}}}h_{rd,2}n_{r,2}+n_d(3)
\end{equation} and
\setcounter{equation}{10}
\begin{equation}
\hat{n}_d(4)=\frac{-\sqrt{P_{r,1}}}{G_{r,1}\sqrt{L_{rd,1}}}h_{rd,1}n_{r,1}^{*}+\frac{\sqrt{P_{r,2}}}{G_{r,2}\sqrt{L_{rd,2}}}h_{rd,2}n_{r,2}^{*}+n_d(4).
\end{equation}\\
Assuming the so-called average power scaling (APS) \cite{Al-Dharrab}, the destination node normalizes the received signals given by \eqref{yd3} and \eqref{yd4} with
\begin{equation}
\Omega=\left(\frac{\eta_1\theta_1 P_s (\kappa_1+1) }{L_{sr,1}L_{rd,1}\mathbb{E}[|G_{r,1}|^2]}+\frac{\eta_2\theta_2 P_s (\kappa_2+1) }{L_{sr,2}L_{rd,2}\mathbb{E}[|G_{r,2}|^2]}+1\right)^{1/2},
\end{equation}resulting in \eqref{tildey3} and \eqref{tildey4}, respectively, at the top of this page. With the aforementioned signal models in mind, by letting $\tilde{n}_d(q)=\hat{n}_d(q)/\Omega, q\in\lbrace{3,4\rbrace}$, the received signal vector over four time slots is expressed as \eqref{matrixmod} at the top of the next page.
Introducing $\bold{h}=[\sqrt{P_s}h_{sd}, \sqrt{P_s}h_{sd}, D_1 h_{sr,1}^*h_{rd,1}, D_2 h_{sr,2}^*h_{rd,2}] $, where $ h_{sr,n}^*$ is chosen as $h_{sr,n}$ or $h_{sr,n}^*$ based on the code matrix $\bold{S}$ given by
\setcounter{equation}{15}
\begin{equation}\label{codemat}
\mathbf{S}=\begin{bmatrix}
s_1(1) & 0 & 0 & 0\\
0&s_2(2)&0&0\\
0&0&s_1(3)&-s_2(4)^*\\
0&0&s_2(3)&s_1(4)^*
\end{bmatrix},
\end{equation}and $\bold{n}=[n_{d}(1), n_{d}(2), n_{d}(3), n_{d}(4)]$, the received signal vector over the whole observation period can be obtained as
\begin{equation}\label{shortsys}
\bold{y}_d=\bold{hS}+\bold{n}.
\end{equation}
\par After setting up the relay-assisted transmission model given by \eqref{matrixmod} and \eqref{shortsys}, we will now introduce the details of the signal models for blind and CSI-assisted relaying techniques.
\subsection{Blind Relaying}
Under this relaying technique, it is assumed that the $n$-th relay node does not have knowledge of its relative $S\to R_n$ link fading coefficient, therefore, it scales the received signal $y_{r,n}(t)$ by a factor of
\begin{equation}
G_{r,n}=\sqrt{\mathbb{E}[|y_{r,n}|^2]}=\sqrt{(\kappa_n P_s/L_{sr,n})+N_{0_{r,n}})}
\end{equation} to normalize the average energy to unity \cite{Laneman2}\footnote{This power constraint is called fixed gain relaying in \cite{Hasna}}. Replacing the scaling term $G_{r,n}$, $P_{r,1}$, and $P_{r,2}$ in \eqref{matrixmod}, we can rewrite the vector form of the received signal model $\bold{y}_d$ as \eqref{matrixmod2} at the top of the next page, where $\Phi_1^2$ and $\Phi_2^2$ are given as
\setcounter{equation}{19}
\begin{equation}\label{phi1APS}
{\Phi_{1}^2}=\frac{\eta_1\theta_1\kappa_1 P_s (P_s/N_{0_{r,1}})}{\Omega^2 L_{sr,1}^2L_{rd,1}[(\kappa_1/L_{sr,1})(P_s/N_{0_{r,1}})+1]}
\end{equation} and
\begin{equation}\label{phi2APS}
{\Phi_2^2}=\frac{\eta_2\theta_2\kappa_2 P_s (P_s/N_{0_{r,2}})}{\Omega^2 L_{sr,2}^2L_{rd,2}[(\kappa_2/L_{sr,2})(P_s/N_{0_{r,2}})+1]},
\end{equation} respectively.
\subsection{CSI-assisted Relaying}
Under this relaying technique, it is assumed that the relays $R_1$ and $R_2$ have knowledge about the CSI of their relative $S\to R_n$ links and accordingly, the scaling factor of the $n$-th relay becomes
\begin{equation}
G_{r,n}=\sqrt{(\kappa_n P_s/L_{sr,n})|h_{sr,n}|^2+\beta_{I,r,n}^mN_{0_{r,n}}}.
\end{equation} Replacing this scaling term $G_{r,n}$, $P_{r,1}$, and $P_{r,2}$ in \eqref{matrixmod}, we can rewrite the vector form of the received signal at the destination as \eqref{matrixmod2}, where $\Phi_1^2$ and $\Phi_2^2$ are now written as \\
\begin{equation}\label{phi1IPS}
{\Phi_{1}^2}=\frac{\eta_1\theta_1\kappa_1 P_s (P_s/(\beta_{m,r,1} N_{0_{r,1}}))}{\Omega^2 L_{sr,1}^2L_{rd,1}[(\kappa_1/L_{sr,1})(P_s/(\beta_{m,r,1} N_{0_{r,1}}))|h_{sr,1}|^2+1]}
\end{equation} and\\
\begin{equation}\label{phi2IPS}
{\Phi_2^2}=\frac{\eta_2\theta_2\kappa_2 P_s (P_s/(\beta_{m,r,2} N_{0_{r,1}}))}{\Omega^2 L_{sr,2}^2L_{rd,2}[(\kappa_2/L_{sr,2})(P_s/(\beta_{m,r,2}N_{0_{r,2}}))|h_{sr,2}|^2+1]},
\end{equation} \\respectively. To simplify the ensuing analysis, we assume that $N_{0}\triangleq N_{0_{sr,1}}= N_{0_{sr,2}}=N_{0_{d}}$.
\section{Pairwise Error Probability Analysis}
Based on the described noise and transmission models in the preceding section, we proceed to investigate the performance of the SWIPT relay system for each of the considered relaying techniques by deriving the PEP expressions for noise Models I and II.
\subsection{Performance Under Noise Model I}
We start by considering the spatially dependent impulsive noise model and investigate its relative effect on the underlying SWIPT relaying system. Specifically, under Model I, the number of impulses affecting $R_1$, $R_2$, and $D$ are statistically dependent and follow the same Poisson random variable $C_0$, i.e., $\alpha_{m,d}=\alpha_{m,r,1}=\alpha_{m,r,2}=\alpha_m$.
\par We will assume minimum distance decoding with perfect knowledge of the individual CSIs of the $S\to R_n$, $R_n\to D$, and $S\to D$ links at the receiver which is considered to be optimal when the noise is Gaussian, but is suboptimal over the impulsive noise channel \cite{Gao}. However, since the minimum distance receiver (MDR) is practical with a low detection complexity technique, we are motivated to derive its PEP performance which is mathematically tractable.
\par Let $\bold{s}$ and $\hat{\bold{s}}$ denote the originally transmitted codeword, $\bold{s}=[s_1, s_2]$, and the erroneously-decoded codeword, $\hat{\bold{s}}=[\hat{s}_1, \hat{s}_2]$, vectors at the destination, respectively. Recalling that for the spatially dependent case, $\beta_{m,d}=\beta_{m,r,1}=\beta_{m,r,2}=\beta_{m}$, after normalising \eqref{yd3} and \eqref{yd4} by $\Omega$, then conditioned on the conditional noise variance $\beta_m$, $\tilde{n}_d(q), q\in\lbrace{3,4\rbrace},$ turns out to be a zero-mean complex Gaussian random variable with variance $\beta_m N_0$. Accordingly, the exact conditional PEP is obtained following the derivation of the conditional PEP in the Gaussian noise case as
\begin{equation}\label{exactPEPMDR}
P(\bold{s}\to\hat{\bold{s}}|\bold{h})=\sum_{m=0}^{M-1}\alpha_m Q\left(\sqrt{\frac{d^2(\bold{s},\hat{\bold{s}})}{2\beta_m N_0}}\right),
\end{equation} where all possible realizations of the Poisson random variable $C_0$ are considered. Also, $Q(.)$ is the Gaussian-$Q$ function \cite{Rizhik} and $d^2(\bold{s},\hat{\bold{s}})$ is the Euclidean distance between $\bold{s}$, and $\hat{\bold{s}}$ written as
\begin{equation}
d^2(\bold{s},\hat{\bold{s}})=d^2_{S\to D}(\bold{s},\hat{\bold{s}})+d^2_{S\to R_1\to D}(\bold{s},\hat{\bold{s}})+d^2_{S\to R_2\to D}(\bold{s},\hat{\bold{s}})
\end{equation}
Applying the standard Chernoff bound on the $Q(.)$ function in \eqref{exactPEPMDR}, the conditional PEP can be upper bounded by \cite{Tarokh}
\begin{equation}\label{chernoff}
P(\bold{s}\to\hat{\bold{s}}|\bold{h})\leq\sum_{m=0}^{M-1}\alpha_m\text{exp}\left(\frac{-d^2(\bold{s},\hat{\bold{s}})}{4\beta_m N_0}\right).
\end{equation}
\subsubsection{PEP for Blind Relaying}
The Euclidean distance for the blind relaying scheme can be written as
\begin{align}\label{distAPS}
d^2(\bold{s},\hat{\bold{s}})=\bold{h}(\bold{S}-\hat{\bold{S}})(\bold{S}-\hat{\bold{S}})^H\bold{h}^H \quad \quad\quad \quad \quad \quad\quad\quad \quad \quad \quad\nonumber \\
\quad=\Delta P_s |h_{sd}|^2+\epsilon_1 \Phi_1^2|h_{sr,1}|^4|h_{rd,1}|^2+\epsilon_2\Phi_2^2 |h_{sr,2}|^4|h_{rd,2}|^2.
\end{align} where $\Phi_1$ and $\Phi_2$ are defined in \eqref{phi1APS} and \eqref{phi2APS}, respectively, $\Delta=|s_1-\hat{s}_1|^2+|s_2-\hat{s}_2|^2$ and $\epsilon_n$ denote the eigenvalues of the codeword difference matrix $(\bold{S}-\hat{\bold{S}})(\bold{S}-\hat{\bold{S}})^H$, $n\in\lbrace{1,2\rbrace}$.
It is worth noting that the term $|h_{sr_n}|^4$, $n\in\lbrace{1,2\rbrace}$, appears due to the process of instantaneous EH taking place at the $n$-th relay. Henceforth, we call this relaying schemes as blind IEH-relaying. Substituting \eqref{distAPS} in \eqref{chernoff}, the PEP expression is obtained in the following proposition.
\newcounter{tempequationcounter6}
\begin{figure*}[t]
\normalsize
\setcounter{equation}{30}
\begin{equation}\label{matrixmod3}
\bold{y}_d=\begin{bmatrix}
y_d(1)\\y_d(2)\\y_d(3)\\y_d(4)\end{bmatrix}=
\begin{bmatrix}
\sqrt{P_s}h_{sd}s_1(1)+n_{d}(1) \\
\sqrt{P_s}h_{sd}s_2(2)+n_{d}(2) \\
\Phi_1h_{sr,1}h_{rd,1} s_1(3) +\Phi_2 h_{sr,2}h_{rd,2} s_2(3)+\tilde{n}_d(3) \\
-\Phi_1h_{sr,1}^*h_{rd,1} s_2(4)^* +\Phi_2 h_{sr,2}^*h_{rd,2} s_1(4)^*+\tilde{n}_d(4)\end{bmatrix}.
\end{equation}
\hrulefill
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\setcounter{equation}{28}
\begin{prop}
The unconditional PEP performance of the considered SWIPT blind IEH-relaying system in the presence of impulsive noise can be expressed in a closed-form as\\
\begin{align}\label{finalPEPmodIAPS}
P(\bold{s}\to\hat{\bold{s}})&\leq\sum_{m=0}^{M-1}\alpha_m \left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1} \nonumber \\
&\times \prod_{n=1}^2\frac{1}{\sqrt{\pi}} G^{1,3}_{3,1}\left[\frac{\epsilon_n\Phi_n^2}{\beta_m N_0} \ \Big\vert \ {0.5, 0,0 \atop 0}\right],
\end{align}\end{prop}
\noindent where $ G^{m,n}_{p,q}[.\vert .]$ is the Meijer G-function defined in \cite[Eq. (8.2.1.1)]{Prudnikov}. Furthermore, $\alpha_m$ and $\beta_m$ can be calculated using \eqref{alpha} and \eqref{beta}, respectively. Note that the Meijer G-function in \eqref{finalPEPmodIAPS} can be easily and accurately computed by standard mathematical software packages such as Mathematica$^{\copyright}$, Matlab$^{\copyright}$, and Maple$^{\text{TM}}$.
\begin{IEEEproof}
See Appendix \ref{Appendix A}.
\end{IEEEproof}
\vspace*{0.1cm}
\par \textbf{Special Case (Blind AEH-relaying):} We assume that $R_1$ and $R_2$ perform AEH which corresponds to a practical scenario where the relay nodes are equipped with a battery. Under this assumption, \eqref{Pr} which represents the power available at the $n$-th relay at the end of \textit{Phase-1} is written as
\begin{equation}\label{PrAEH}
P_{r,n}=\frac{\eta_n\theta_n P_s}{L_{sr,n}}.
\end{equation}Replacing \eqref{PrAEH} in \eqref{matrixmod}, the vector form of the received signal model $\bold{y}_d$ is now given as \eqref{matrixmod3} at the top of this page. Under this scenario, $d^2(\bold{s},\hat{\bold{s}})$ is given by
\setcounter{equation}{31}
\begin{align}\label{convdit}
d^2(\bold{s},\hat{\bold{s}})&=\Delta P_s |h_{sd}|^2 +\epsilon_1 \Phi_1^2|h_{sr,1}|^2|h_{rd,1}|^2\nonumber \\
&+\epsilon_2\Phi_2^2 |h_{sr,2}|^2|h_{rd,2}|^2.
\end{align}It can be easily verified that \eqref{convdit} has a similar form to that in \cite[Eq. (31)]{Muhaidat1} and \cite[Eq. (26)]{Al-Dharrab} for the conventional non-EH case. Therefore, the unconditional PEP is found as
\begin{align}\label{PEPcon}
P(\bold{s}\to\hat{\bold{s}})&\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1}\nonumber\\
&\times\prod_{n=1}^2\left(\frac{\epsilon_n\Phi_n^2}{4\beta_m N_0}\right)^{-1} \text{exp}\left(\frac{4 \beta_m N_0}{\epsilon_n \Phi_n^2}\right)\Gamma\left(0,\frac{4 \beta_m N_0}{\epsilon_n \Phi_n^2}\right),
\end{align} where $\Gamma(a,b)=\int_b^\infty x^{a-1}\text{exp}(-x)dx$ \cite{Rizhik} denotes the upper incomplete gamma function.
\newcounter{tempequationcounter7}
\begin{figure*}[b]
\normalsize
\hrulefill
\setcounter{equation}{40}
\begin{align}\label{PEPmodII}
P(\bold{s}\to\hat{\bold{s}}|\bold{h})&=\sum_{m,r,1=0}^{M-1}\sum_{m,r,2=0}^{M-1}\sum_{m,d=0}^{M-1}\left(\prod_{k=1}^3\alpha_{m,k}\right) \nonumber \\
&\times Q\left(\frac{d^2(\bold{s},\hat{\bold{s}})}{\sqrt{2\left[\Delta P_s|h_{sd}|^2\beta_{m,d}+\epsilon_1\Phi_1|h_{sr,1}|^4|h_{rd,1}|^2\beta_{m,r,1}+\epsilon_2\Phi_2|h_{sr,2}|^4|h_{rd,2}|^2\beta_{m,r,2}\right]N_0}}\right).
\end{align}
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\setcounter{equation}{33}
\subsubsection{PEP for CSI-assisted Relaying}
The Euclidean distance for the CSI-assisted relaying scheme can be written as \eqref{distAPS} where $\Phi_1^2$ and $\Phi_2^2$ are now given by \eqref{phi1IPS} and \eqref{phi2IPS}, respectively. Note that, unlike \eqref{phi1APS} and \eqref{phi2APS} for the blind relaying case, \eqref{phi1IPS} and \eqref{phi2IPS} are functions of $|h_{sr,1}|^2$ and $|h_{sr,1}|^2$, respectively. To this effect, substituting \eqref{phi1IPS} and \eqref{phi2IPS} in \eqref{distAPS}, we can write $d^2(\bold{s},\hat{\bold{s}})$ as
\begin{align}\label{distIPS}
d^2(\bold{s},\hat{\bold{s}})&= \Delta P_s |h_{sd}|^2 +\epsilon_1 \zeta_1 \frac{|h_{sr,1}|^4|h_{rd,1}|^2}{\xi_1 |h_{sr,1}|^2+1}\nonumber \\
&+\epsilon_2\zeta_2 \frac{|h_{sr,2}|^4|h_{rd,2}|^2}{\xi_2 |h_{sr,2}|^2+1} \nonumber \\
& =\Delta P_s |h_{sd}|^2 +\epsilon_1 \zeta_1 \frac{X_1^2Y_1}{\xi_1 X_1+1}+\epsilon_2\zeta_2 \frac{X_2^2 Y_2}{\xi_2 X_2+1}.
\end{align} where $\xi_n=[(\kappa_n/L_{sr,n})(P_s/(\beta_I^m N_0))]$, $\zeta_n$ is given as \\
\begin{equation}\label{zeta}
\zeta_n =\frac{\eta_n\theta_n\kappa_n P_s (P_s/(\beta_m N_0))}{\Omega^2 L_{sr,n}^2L_{rd,n}}, \quad n\in \lbrace{1,2\rbrace}.
\end{equation}and $X_n\triangleq|h_{sr,n}|^2$, $Y_n\triangleq|h_{rd,n}|^2$. To obtain an expression for the PEP for the CSI-assisted IEH-relaying, let $Z_n=U_n/V_n$, where $U_n=X_n^2Y_n$ and $V_n=\xi_n X_n+1, n\in \lbrace{1,2\rbrace}$. Then, one could obtain the unconditional PEP by taking the expectation of \eqref{chernoff} with respect to the RVs $|h_{sd}|^2, Z_1$ and $Z_2$. In the following proposition, we derive the unconditional PEP expression. \\
\begin{prop}
The unconditional PEP performance of SWIPT CSI-assisted IEH-relaying system in the presence of impulsive noise can be expressed as
\vspace*{-0.5cm}
\end{prop}
\begin{align}\label{pepIEHIPS}
P(\bold{s}\to\hat{\bold{s}})&\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1}\nonumber \\
&\prod_{n=1}^2\frac{1}{2B_n\psi_n}\left[\text{exp}(\Lambda_n)\text{Ei}(\Lambda_n)\text{D}_1+\text{exp}(\Psi_n)\text{Ei}(\Psi_n)\text{D}_2\right],
\end{align}where $\text{Ei}(.)$ is the exponential integral function \cite{Prudnikov}, $\psi=\sqrt{\xi_n^2-4B_n}$ where $\xi_n$ is defined before \eqref{zeta}, $\text{D}_1=-\xi_n^2-\xi_n\psi_n+2 B_n$, $\text{D}_2=\xi_n^2-\xi_n\psi_m-2 B_n$, and $\Lambda_n$ and $\psi_n$ are given by
\begin{equation}
\Lambda_n=\frac{\xi_n+\psi_n}{2B_n},
\end{equation}and
\begin{equation}
\Psi_n=\frac{\xi_n-\psi_n}{2B_n},
\end{equation}respectively.
\begin{IEEEproof}
See Appendix \ref{Appendix B}.
\end{IEEEproof}\vspace*{0.1cm}
\par \noindent \textbf{Special Case (Asymptotic PEP in high SNR):} To give more insight into the PEP performance, we consider the high SNR assumption, i.e., $\xi_n\to \infty$. Under this assumption, the second factor in the denominators of \eqref{distIPS} can be negligible. Consequently, $d^2(\bold{s},\hat{\bold{s}})$ in \eqref{distIPS} is reduced to \eqref{convdit}, yielding the PEP expression to be given as \eqref{PEPcon}. \\
\par \noindent \textbf{Special Case (CSI-assisted AEH-relaying):} Similar to the blind-relaying scenario, we assume here that $R_1$ and $R_2$ perform average EH. Under this assumption, the power available at the $n$-th relay at the end of \textit{Phase-1} is given by \eqref{PrAEH}. Hence, we get
\begin{align}\label{distIPS2}
d^2(\bold{s},\hat{\bold{s}})&= \Delta P_s |h_{sd}|^2 +\epsilon_1 \zeta_1 \frac{|h_{sr,1}|^2|h_{rd,1}|^2}{\xi_1 |h_{sr,1}|^2+1}\nonumber \\
&+\epsilon_2\zeta_2 \frac{|h_{sr,2}|^2|h_{rd,2}|^2}{\xi_2 |h_{sr,2}|^2+1},
\end{align}where $\xi_n$ and $\zeta_n$ are given below \eqref{distIPS}. Substituting \eqref{distIPS2} in \eqref{chernoff}, followed by taking the expectation with respect to $|h_{sd}|^2, |h_{sr,1}|^2, |h_{sr,2}|^2, |h_{rd,1}|^2$ and $|h_{rd,2}|^2$, the unconditional PEP is given in the following proposition.
\begin{prop}
The unconditional PEP performance of SWIPT CSI-assisted AEH-relaying system can be expressed as
\end{prop}
\vspace*{-0.5cm}
\begin{align}\label{PEPAEHIPS}
P(\bold{s}\to\hat{\bold{s}})\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\nonumber \\
\quad\quad\times \prod_{n=1}^2\left(\gamma_n^{-1}G^{1,2}_{2,1}\left[\gamma_n \ \big\vert \ {1, 1 \atop 1}\right]+\xi_n \gamma_n^{-2}G^{1,2}_{2,1}\left[\gamma_n \ \big\vert \ {1, 2 \atop 2}\right]\right). \quad\quad\quad\quad
\end{align}
\begin{IEEEproof}
See Appendix \ref{AppendixC}.
\end{IEEEproof}
\vspace*{0.3cm}
\noindent It is noted from each of \eqref{finalPEPmodIAPS}, \eqref{PEPcon}, \eqref{pepIEHIPS}, and \eqref{PEPAEHIPS} that these expressions include the conventional AWGN assumption as a special case. It is recalled from \eqref{beta} that as $\delta\to\infty$, $\beta_m$ converges to 1. Therefore, the summation in \eqref{finalPEPmodIAPS}, \eqref{PEPcon}, \eqref{pepIEHIPS}, and \eqref{PEPAEHIPS} will be equal to 1, reducing these expressions to the PEP expressions for the conventional AWGN case. It is worth mentioning that due to the presence of the summation term, in the above mentioned expressions, the convergence to asymptotic diversity order under impulsive noise is slower compared to the AWGN case.
\par Based on the previously derived PEP expressions the diversity order \mbox{$D$} can be computed as \cite{Simon}
\begin{equation}\label{dorder}
D= -\lim_{\text{SNR} \rightarrow \infty}\frac{\text{log}\left(P(\bold{s}\to\hat{\bold{s}})\right)}{\text{log}\left(\text{SNR}\right)}.
\end{equation}Since the only source of power is, \mbox{$P_s$}, the performance of the entire system is parametrized by SNR \mbox{$\triangleq P_s/N_0$}. Using \mbox{\eqref{dorder}}, the diversity order of blind IEH-relaying, blind AEH-relaying, CSI-assisted IEH-relaying, and CSI-assisted AEH-relaying are numerically evaluated by substituting \mbox{\eqref{finalPEPmodIAPS}, \eqref{PEPcon}, \eqref{pepIEHIPS}, and \eqref{PEPAEHIPS} in \eqref{dorder}}, respectively.
\subsection{Performance Under Noise Model II}
In the following, we will study the performance of the considered SWIPT relay system under the assumption of spatially independent impulsive noise model, where $R_1$, $R_2$, and $D$ nodes are affected by statistically independent number of impulses, respectively, following Poisson random variables $C_{r,1}$, $C_{r,2}$, and $C_d$, i.e., $\alpha_{m,d}, \alpha_{m,r,1}$, and $\alpha_{m,r,2}$ may not necessarily be equal. In particular, the conditional variances $\beta_{m,d}$, $\beta_{m,r,1}$, and $\beta_{m,r,2}$ are not necessarily equal. To address the independence in the number of impulses occurring at $R_1$, $R_2$, and $D$, the PEP expression has to be averaged over all possible realizations of each of $C_{r,1}$, $C_{r,2}$, and $C_d$, and thus, the conditional PEP is given by \eqref{PEPmodII} at the bottom of this page, where $\alpha_{m,1}=\alpha_{m,d}$, $\alpha_{m,2}=\alpha_{m,r,1}$, and $\alpha_{m,3}=\alpha_{m,r,2}$.
\par To evaluate the unconditional exact PEP for each of the relaying schemes described in Section II, the expression in \eqref{PEPmodII} has to be averaged over fading coefficients $\bold{h}$, which is mathematically intractable. However, we can obtain an approximate expression for the conditional PEP in \eqref{PEPmodII} by setting $\beta_{m,r,1}=\beta_{m,r,2}=\beta_{m,d}=\bar{\varphi}$, which denotes the average number of impulses affecting $R_1$, $R_2$, $D$ nodes during a transmission frame and is given by \cite{Al-Dharrab}
\setcounter{equation}{41}
\begin{equation}
\bar{\varphi}=\frac{2(\beta_{m,r,1}+\beta_{m,r,2})+4\beta_{m,d}}{8}.
\end{equation}Then, by using the Chernoff upper bound, taking the expectation over the fading coefficients $\bold{h}$, and following the same line of analysis performed in the derivation of the PEP expressions of the blind and CSI-assisted relaying schemes under noise Model I, the PEP performance under noise Model II can be evaluated.
\section{Numerical and Simulation Results}\label{sec:results}
In this section, we provide a variety of numerical and Monte Carlo simulation results to validate the accuracy of the proposed analytical framework and to compare the performance of the considered blind and CSI-assisted relaying techniques employed for a SWIPT relaying system under the MCA noise Models I and II. The term Monte Carlo simulations refers to the use of actual fading channel variates with a number of repetitions of $10^6$ trials. We further assume that the two relays are located on the straight line between the source and the destination nodes. Unless otherwise specified, in order to study various degrees of noise impulsiveness, we use three sets of values for the impulsive noise parameters $A$ and $\delta$: $(A,\delta)=(1,0.1)$, $(A,\delta)=(0.1,0.1)$, and $(A,\delta)=(0.001,0.1)$ to represent \textit{near-Gaussian} (NG), \textit{moderately impulsive} (MI), and \textit{highly impulsive} (HI) noise channels, respectively, which fit well within the practical ranges of $A$ and $\delta$ \cite{Delaney}.
\par Unless otherwise stated, we set the EH efficiency factor $\eta_1=\eta_2=0.3$ as a worst case, capturing the effects of low-cost hardware, the PS factors $\theta_1=\theta_2=0.5$, the normalized distances of both relays for their respective $S\to R$ links are set to $d_{sr,1}=d_{sr,2}=0.5$, the source transmission power $P_s=1$ Watt and the path-loss exponent $\lambda=2.7$ \cite{Nasir2013}. The simulation parameters are summarized in Table \ref{t2}.
\begin{table}[!t]
\centering
\caption{Simulation Parameters}
\label{t2}
\begin{tabular}{|l|c|c|}
\hline
\multicolumn{1}{|c|}{\textbf{Name}} & \textbf{Symbol} & \textbf{Value} \\ \hline
Impulsive noise index & \textit{A} & 1 (NG), 0.1 (MI), 0.001 (HI) \\ \hline
Gaussian noise factor & $\delta$ & 0.1 \\ \hline
Number of interfering sources & \textit{M} & 5 \\ \hline
PS ratio for $R_1$ & $\theta_1$ & 0.5 \\ \hline
PS ratio for $R_2$ & $\theta_2$ & 0.5 \\ \hline
Normalized $S\to R_1$ distance & $d_{sr,1}$ & 0.5 \\ \hline
Normalized $S\to R_2$ distance & $d_{sr,2}$ & 0.5 \\ \hline
Path-loss exponent & $\lambda$ & 2.7 \\ \hline
EH efficiency of $R_1$ & $\eta_1$ & 0.3 \\ \hline
EH efficiency of $R_2$ & $\eta_2$ & 0.3 \\ \hline
Source transmit power & $P_s$ & 1 Watt \\ \hline
\end{tabular}
\end{table}
\begin{figure}[!t]
\centering
\includegraphics[width=3.7in]{PEP_allschemes_SNR.pdf}
\caption{PEP performance with respect to SNR for blind and CSI-assisted relaying techniques over Rayleigh fading channels in the presence of HI, MI, and NG MCA noise for Model I.}
\label{f3}
\vspace*{-0.5cm}
\end{figure}
\par In Fig. \ref{f3}, we compare the PEP performance of the blind and CSI-assisted relaying techniques when IEH or AEH are considered under three MCA noise environments, namely, HI, MI, and NG, for noise Model I. Furthermore, to evaluate the accuracy of our mathematical models presented in \eqref{finalPEPmodIAPS}, \eqref{PEPcon}, \eqref{pepIEHIPS}, and \eqref{PEPAEHIPS}, we present in Fig. \ref{f3} the corresponding Monte Carlo simulation results. It is observed that the analytical PEP curves are in full agreement with the simulation results over the whole SNR operating range. This finding directly reflects the accuracy of our proposed mathematical framework and its effectiveness in quantifying the performance of the considered relaying techniques under MCA noise. It is illustrated in Fig. \ref{f3} that for all the studied relaying techniques, namely, blind relaying with IEH or AEH and CSI-assisted relaying with IEH or AEH, the PEP curves undergo a flattening when the SNR is between 5 - 20 dB under the HI noise environment, which dramatically differs from those of the NG noise environment. This behaviour is also reported for non-EH systems cooperative systems \cite{Al-Dharrab, Alhussein1} and is due to the fact that the tails of the PDF of the MCA noise becomes wider as the impulsive index $A$ decreases. However, as $A$ increases, the tails of the MCA density asymptotically approach those of a Gaussian density, resulting in the behaviour observed for the PEP performance. Moreover, for the three noise scenarios, it is shown that the performance exhibited by the CSI-assisted AEH-relaying is superior to that of the other three relaying techniques. Although, CSI-assisted relaying schemes are intuitively expected to outperform their blind relaying counterparts, our results show that the CSI-assisted IEH-relaying and blind AEH-relaying technique experience identical PEP performance. This indicates that the extra power consumption, resulting from CSI estimation, can be avoided without causing performance loss. However, this comes at the expense of requiring a battery to perform AEH.
\par In an attempt to gain more insights about the performance of the considered relaying techniques, we investigate the achievable diversity order. Specifically, in Fig. \ref{f4}, we utilize the expressions obtained in \eqref{finalPEPmodIAPS}, \eqref{PEPcon}, \eqref{pepIEHIPS}, and \eqref{PEPAEHIPS} to calculate the diversity order, defined as the negative of the asymptotic slope of the PEP on a log-log scale \cite{Tarokh}. The achievable diversity order in the presence of the well-known AWGN case is included as a benchmark.
\par Fig. \ref{f4} demonstrates that the CSI-assisted AEH-relaying scheme enables the system to achieve the highest diversity order $(d=3$, at NG), whereas the lowest $(d=2$, at NG) is obtained by the blind IEH-relaying scheme, where the performance is severely degraded. This is due to the effect of cascaded fading resulting from IEH. Meanwhile, the attainable diversity order for both the blind AEH-relaying and CSI-assisted IEH-relaying is identical $(d=2.85$, at NG). In Table \ref{t1}, we present the achievable diversity order levels observed by the investigated four relaying techniques under the three MCA noise environments, along with the corresponding AWGN case. It is noted that for all the studied relaying techniques: as the impulsive noise index $A$ becomes smaller, (i.e., the noise becomes highly impulsive), the convergence to full spatial diversity, represented by the AWGN case, becomes slower. This can be attributed to the performance loss introduced by the impulsive nature of the noise incurred by the MDR. Additionally, the full diversity order of all relaying techniques in the MCA noise environments are not realized due to the noise impulsiveness severity. Interestingly, as the noise impulsiveness level increases from NG to HI, the associated performance loss increases with the diversity order. This result is consistent with the conclusion reported in \cite{Gao} for a non-cooperative non-EH wireless communication system.
\begin{figure}[!t]
\centering
\includegraphics[width=3.2in]{Diversity_Vs_SNR3.pdf}
\caption{Diversity order of blind and CSI-assisted relaying schemes in the presence of HI, MI, and NG noise for Model I.}
\label{f4}
\end{figure}
\begin{table}[h]
\centering
\caption{Achievable diversity order under MCA noise and AWGN}
\label{t1}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{Relaying Technique}} & \multicolumn{3}{c|}{MCA Noise} & \multicolumn{1}{c|}{\multirow{2}{*}{AWGN}} \\ \cline{2-4}
\multicolumn{1}{|c|}{} & HI & MI & NG & \multicolumn{1}{c|}{} \\ \hline
Blind IEH & 1.86 & 1.96 &1.99 &\multicolumn{1}{c|}{2}
\\ \hline
Blind AEH &2.59 &2.78 &2.85 & \multicolumn{1}{c|}{2.87} \\ \hline
CSI-assisted IEH &2.59 &2.78 &2.85 &\multicolumn{1}{c|}{2.87} \\ \hline
CSI-assisted AEH &2.38 &2.86 &3 & \multicolumn{1}{c|}{3} \\ \hline
\end{tabular}
\end{table}
\par To explore the effect of the relays' locations on the PEP performance of the considered blind and CSI-assisted relaying techniques with IEH, we illustrate in Fig. \ref{f5} the performance of the Alamouti-based scheme, under the assumption of HI MCA noise. This study is conducted for six distinct scenarios of the geometrical layout of the two relays:
\begin{itemize}
\item Scenario 1: $d_{sr,1}$=0.8 and $d_{sr,2}$=0.8,
\item Scenario 2: $d_{sr,1}$=0.5 and $d_{sr,2}$=0.8,
\item Scenario 3: $d_{sr,1}$=0.2 and $d_{sr,2}$=0.8,
\item Scenario 4: $d_{sr,1}$=0.5 and $d_{sr,2}$=0.5,
\item Scenario 5: $d_{sr,1}$=0.5 and $d_{sr,2}$=0.2,
\item Scenario 6: $d_{sr,1}$=0.2 and $d_{sr,2}$=0.2.
\end{itemize}
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{PEP_multiplerelayLoc_SNR_IEHAPS2.pdf}
\caption{PEP performance with respect to SNR for various relay locations over HI noise under Model I.}
\label{f5}
\vspace*{-0.6cm}
\end{figure}
It is shown in Fig. \ref{f5} that the best performance for both blind and CSI-assisted IEH-relaying schemes is exhibited by Scenario 6, where both relays are close to the source, while locating the two relays close to the destination represented by Scenario 1 leads to the worst performance. This is expected, since the power available at the relay nodes resulting from EH during \textit{Phase-1}, as defined in \eqref{Pr}, is inversely proportional to the distance between the source and the relay node. Specifically, as $d_{sr,n}, n\in\lbrace{1,2\rbrace}$ increases, both the harvested energy and the received signal strength at the relay node decrease due to the increased path-loss, and consequently, deteriorating the performance. A similar result is noted for both of the relaying techniques when AEH is employed, however their performance is not plotted to avoid repetition.
\par This observation suggests the support for the conclusion in \cite{Nasir2013} for a SWIPT wireless cooperative systems under the general AWGN noise assumption. On the contrary, this finding is different from the conventional case where EH is not considered at the relays \cite{Al-Dharrab}, wherein the best performance is attained by Scenario 4, where both relays are equidistant from the source and destination nodes and the worst performance is observed in Scenario 3 where one of the relays is placed closer to the source node and the other is placed closer to the destination node. The aforementioned result along with the ones reported in \cite{Nasir2013} and \cite{Al-Dharrab} lead us to conclude that the optimal position of the relays in a SWIPT relaying system may be independent from the channel noise type.
\par Remarkably, for both blind and CSI-assisted relaying techniques, as the two relays become closer to the source the flat region observed in the case of HI noise is significantly diminished, thereby, considerably outperforming the non-EH case presented in \cite{Al-Dharrab} from this perspective. Therefore, the results obtained in this examination are two fold. First, it is noted that EH relaying systems are more robust towards impulsive noise. Second, the location of the relays plays a crucial role in the underlying system performance. Further examinations of the impact of the relays' location on the system performance are carried out in Fig. \ref{f6}.
\par Fig. \ref{f6} depicts the PEP performance of blind and CSI-assisted relaying for both IEH and AEH as a function of the normalized $S\to R$ link distances of $R_1$ and $R_2$. The study is carried out for the NG and HI noise environments, considering Model I, under the assumption of both low (15~dB) and high (40~dB) SNR regimes. As it can be readily observed for all four relaying schemes, in general, the PEP increases as $d_{sr,1}$ and $d_{sr,2}$ increase, i.e., the distance between the source and the two relays increases. As explained earlier, this is because the farther away the two relay nodes are from the source node, the larger the experienced path-loss is, leading to less signal power to be received at $R_n$. Accordingly, the received signal power at the destinations node is poor, yielding inferior PEP performance. This result is in accordance with the majority of the research work in the literature of SWIPT relaying networks \cite{Nasir2013,Rabie1,Ojo,Rabie,Liu} and the references therein, where it is demonstrated that the best performance of the network was achieved when the relay nodes are located closer to the source node than the destination node. In our work, we demonstrate that this finding also holds when the network is operating under the impulsive noise. Moreover, we notice that in the case of low SNR regime (SNR=15dB), which is included in the flat region of the PEP performance under the HI noise, the PEP performance does not notably change with the change in the distance and that the performance is irrespective of the adopted relaying schemes. However, a rather more noticeable change is observed in the high SNR regime. This is in contrast to the NG noise environment case, where more rapid improvements can be seen at both low and high SNR regimes as the relays move closer to the source. Therefore, it turns out that moving the relays closer to the source is more rewarding in the NG noise environment. It can be further deduced from Fig. \ref{f6} that the performance gap between the four analyzed relaying schemes is more pronounced in the NG noise environment in the high SNR scenario. Finally, one can observe that the PEP performance does not notably change by increasing $d_{sr,1}$ and $d_{sr,2}$ beyond a certain value ($d_{sr,n}>0.8$), since as the relays get closer to the destination, smaller values of harvested energy are required to support the reliable communication through the $R_n\to D$ link. A similar conclusion can be drawn for all the presented relaying techniques for EH relays which are solely powered by the source. This suggests that the harvested energy at the relay nodes is the dominant performance limiting factor, rendering the $R_n\to D$ link to be the bottleneck of the system performance.
\begin{figure}[!t]
\centering
\includegraphics[width=3.7in]{PEP_Vs_dist.pdf}
\caption{PEP performance with respect to the normalized distances $d_{sr,n}, n\in \lbrace{1,2\rbrace}$ over NG and HI noise under Model I.}
\label{f6}
\vspace*{-0.3cm}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3.7in]{PEP_vs_theta.pdf}
\caption{PEP performance with respect to the PS factor $\theta_1$ at relay $R_1$ over NG and HI noise under Model I, where $\theta_2=\theta_1$.}
\label{f7a}
\vspace*{-0.3cm}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3.7in]{PEP_Vs_theta2.pdf}
\caption{PEP performance with respect to the PS factor $\theta_1$ at relay $R_1$ over NG and HI noise under Model I, where $\theta_2=1-\theta_1$.}
\label{f7b}
\vspace*{-0.6cm}
\end{figure}
\par In Figs. \ref{f7a} and \ref{f7b}, we investigate the impact of the PS factor $\theta_n$ at the relays on the associated PEP performance of the competing relaying techniques for NG and HI noise environments under Model I. The examination is carried out for low and high SNR regimes. Furthermore, in our work, we consider two scenarios for the PS factor of the two relays. The first scenario is depicted in Fig. \ref{f7a}, where we plot the PEP performance as a function of the PS factor of relay $R_1$, $\theta_1$, and we set the PS factor at the second relay $R_2$ to be $\theta_2=\theta_1$. In the second scenario, illustrated in Fig. \ref{f7b}, we set $\theta_2=1-\theta_1$. This is done to provide a deeper understanding on the behaviour of the system when equal or different power settings are imposed on the two relays. Interestingly, one can arrive at the same observation on the PEP performance from Fig. \ref{f7a} and Fig. \ref{f7b}. Specifically, it is noted that the PEP performance is insensitive to the change in the value of the PS factors at the two relays in the HI noise environment under the low SNR assumption due to the detrimental effects of the impulsive noise. On the contrary, it is demonstrated that for the other three scenarios (low SNR with NG noise and low and high SNR with NG and HI noise), there exists an optimal value for the PS factor that minimizes the PEP for the scenario in Fig. \ref{f7a}. This stems from the fact that when the value of $\theta_n, n\in\lbrace{1,2\rbrace}$ is smaller than the optimal, there is less power available for EH. Consequently, less transmission power is available at the two relay nodes causing the performance to deteriorate gradually. On the other hand, as the value of $\theta_n$ increases beyond the optimal value, more power is spent on EH at the expense of the power available for data transmission which considerably degrades the PEP performance. This phenomenon is expected, since the performance of dual-hop systems is constrained by the quality of the weakest hop \cite{Muhaidat2}. Comparing the two setups, we observe from Fig. \ref{f7a} that the minimum PEP performance is attained when $\theta_1=\theta_2=0.22$. However, when the PS factors are different, we observe from Fig. \ref{f7b} that the minimum PEP is achieved for $\theta_1=\theta_2=0.5$. This finding suggests that allocating equal PS factors displays a performance gain gap over the non-equal PS factors at the two relays. A final observation for both Fig. \ref{f7a} and \ref{f7b} is that when blind IEH-relaying is adopted, varying $\theta_n$ only makes a rather small change to the PEP performance. This trend is similar for all the examined noise and SNR scenarios. The aforementioned two scenarios imply that the PS factor for EH must be optimized for best performance.
\begin{figure}[!t]
\centering
\includegraphics[width=3.7in]{PEP_indp_dep_SNR.pdf}
\caption{PEP performance over NG and HI noise under Model I and Model II.}
\label{f8}
\end{figure}
\par To address the effect of the spatial independence, we plot in Fig. \ref{f8} the PEP performance for Models I and II under both NG and HI noise environments against the AWGN benchmark case. It is recalled that Model I refers to the case when the same set of interfering sources affects the relay and destination nodes together, while Model II refers to the case when different sets of interfering sources affect the relay and destination nodes. Fig. \ref{f8} illustrates that when the noise is HI, Model I outperforms Model II in the sufficiently low SNR regime (SNR $<22$dB). This behaviour is reversed in the higher SNR region and the performance over Model II becomes superior to that exhibited by Model I. On the other hand, both models exhibit a similar performance in the NG noise over the whole inspected SNR region. These results are in accordance with the ones reported in \cite{Al-Dharrab}.
\section{Conclusions}\label{sec:conc}
In this paper, we have investigated the performance of distributed Alamouti codes for SWIPT AF relaying systems in the presence of MCA noise. Assuming the PS receiver architecture, we have derived novel closed-form PEP expressions which are then exploited to provide detailed performance comparisons among the four relaying techniques under consideration. Besides the fact that our results are accurate and mathematically tractable, they provide efficient means for the design and evaluation of SWIPT relaying networks in practical scenarios where impulsive noise is present. In particular, the proposed analytical model is exploited to study the diversity gains of blind AF and CSI-assisted AF schemes considering AEH and IEH. In addition, we have illustrated that the performance of CSI-assisted AEH-relaying is superior to that exhibited by the other three relaying techniques, achieving the highest diversity order of 3. Furthermore, we have demonstrated that the performance loss incurred by the severity of noise impulsiveness increases with the diversity order and that the performance of the system in the low and medium SNR regions depends on the impulsive nature of the noise, resulting in different diversity orders to dominate the performance. Significant performance gains have been observed by locating the relays close to the source, offering a potential solution to mitigate the deleterious effect of MCA noise. Our results highlight the importance of accurately characterising the performance of the system for the successful implementation of SWIPT relay networks in the presence of impulsive noise.
\vspace*{-0.5cm}
\appendices
\newcounter{tempequationcounter8}
\begin{figure*}[b]
\normalsize
\hrulefill
\setcounter{equation}{48}
\begin{align}\label{I0}
I_0=-\frac{\xi_n}{B_n}\int_0^\infty\frac{\text{exp}\left(\frac{-t}{\xi_n}\right) dt}{\left(t+\left(\frac{\xi_n^2+\xi_n\sqrt{\xi_n^2-4B_n}}{2B_n}\right)\right)\left(t+\left(\frac{\xi_n^2-\xi_n\sqrt{\xi_n^2-4B_n}}{2B_n}\right)\right)} \nonumber \\
-\frac{\xi_n}{B_n}\int_0^\infty \frac{t\quad\text{exp}\left(\frac{-t}{\xi_n}\right) dt}{\left(t+\left(\frac{\xi_n^2+\xi_n\sqrt{\xi_n^2-4B_n}}{2B_n}\right)\right)\left(t+\left(\frac{\xi_n^2-\xi_n\sqrt{\xi_n^2-4B_n}}{2B_n}\right)\right)}
\end{align}
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\newcounter{tempequationcounter9}
\begin{figure*}[b]
\normalsize
\hrulefill
\setcounter{equation}{49}
\begin{align}\label{pepprop3}
P(\bold{s}\to\hat{\bold{s}}||h_{sr_1}|^2,|h_{sr_2}|^2)\leq\sum_{m=0}^{M}\alpha_m\left(\frac{\Delta_s}{4\beta_m N_0}+1\right)^{-1}\quad \quad\quad \quad\quad \quad\quad \quad\quad \quad\quad \quad\quad \quad\nonumber \\
\times \left(\frac{\epsilon_1\zeta_1/(4\beta_m N_0))|h_{sr_1}|^2}{\xi_1|h_{sr_1}|^2+1}+1\right)^{-1}\left(\frac{(\epsilon_2\zeta_2/(4\beta_m N_0))|h_{sr_2}|^2}{\xi_1|h_{sr_2}|^2+1}+1\right)^{-1}.
\end{align}
\setcounter{equation}{\value{equation}}
\vspace*{4pt}
\vspace*{-0.3cm}
\end{figure*}
\setcounter{equation}{42}
\section{Proof of Proposition 1}
\label{Appendix A}
Starting from the conditional PEP expression in \eqref{chernoff}, we take the expectation with respect to fading coefficients $|h_{sd}|^2$, $|h_{r_1d}|^2$, and $|h_{r_2d}|^2$, which follow an exponential distribution, resulting in
\begin{align}
P(\bold{S}\to\hat{\bold{S}}||h_{sr_1}|^4,|h_{sr_2}|^4)&\leq\sum_{m=0}^{M}\alpha_m\left(\frac{\Delta_s}{4\beta_m N_0}+1\right)^{-1}\nonumber \\
&\times\prod_{n=1}^2 \left(\frac{\epsilon_n\Phi_n^2 }{4\beta_m N_0}|h_{sr_n}|^4+1\right)^{-1}.
\end{align}
Performing an expectation with respect to the random variables $|h_{sr_1}|^4$, $|h_{sr_2}|^4$, which also follow an exponential distribution, yields the unconditional PEP, which is written as\\
\begin{align}\label{uncondPEP1}
P(\bold{s}\to\hat{\bold{s}})&\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1} \nonumber \\
&\times\prod_{n=1}^2\int_0^\infty\left(\frac{\epsilon_n \Phi_n^2 }{4\beta_m N_0}t^2+1\right)^{-1}\text{exp}(-t) dt,
\end{align}where $t$ is the integration variable and $n\in\lbrace{1,2\rbrace}$. Using the equality in \cite[Eq. (8.4.2.5)]{Prudnikov} to express the first integrand of \eqref{uncondPEP1} as \\
\begin{equation}
\left(\frac{\epsilon_n \Phi_n^2}{4\beta_m N_0}t^2+1\right)^{-1}=G_{1, 1}^{1, 1}\left[\frac{\epsilon_n \Phi_n^2}{4\beta_m N_0}t^2 \Big \vert \ {0 \atop 0} \right], \quad n\in\lbrace{1,2\rbrace},
\end{equation}\\then making use of the equality $e^{-t}=G_{0, 1}^{1, 0}\left[t\ \vert \ {- \atop 0} \right]$\cite[Eq. (8.4.3.1)]{Prudnikov} to rewrite the second integrand in \eqref{uncondPEP1}, the unconditional PEP can be derived in a closed-form as in \eqref{finalPEPmodIAPS} by exploiting the integral identity \cite[Eq. (2.24.1.2)]{Prudnikov}.\\
\section{Proof of Proposition 2}
\label{Appendix B}
In order to derive the PEP expression, we first obtain the exact PDF of the RV $Z_n$. It is recalled that RVs $X$ and $Y$\footnote{We drop in the proof the index $n$ for the convenience of analysis.} are independent RVs drawn from the exponential distribution. Therefore, their joint PDF is $f_{X,Y}=e^{-x-y}$ \cite{papoulis}. Expressing $X$ and $Y$ in terms of $U$ and $V$ as $X=(V-1)/\xi$ and $Y=U\xi^2/(V-1)^2$, then with the help of the Jacobian transformation method \cite{papoulis}, $(X,Y)$ are transformed to $(U,V)$. Consequently, the PDF of $(U,V)$ is obtained as \\
\begin{equation}\label{pdfuv}
f_{U,V}=J_d f_{X,Y}\left(\frac{(v-1)}{\xi},\frac{ub^2}{(v-1)^2} \right),
\end{equation}where $J_d=-\xi/(V-1)^2$ is the Jacobian of the transformation. Then using \cite[Eq. (6.60)]{papoulis} and \eqref{pdfuv}, and after some algebraic manipulations, the exact PDF of $Z$ is derived as\\ \\
\begin{align}\label{pdfz}
f(z)=\int_1^\infty v f_{U,V}(vz,v) dv \quad\quad\quad\quad\quad\quad\quad\quad \quad \quad \quad \quad \nonumber \\
=-\int_0^\infty \frac{\xi(t+1)}{t^2} f_{X,Y}\left(\frac{t}{\xi},\frac{(t+1)z\xi^2}{t^2}\right) dt\quad \quad \quad \quad \nonumber \\
=-\int_0^\infty \frac{\xi (t+1)}{t^{2}}\text{exp}\left(-\frac{t}{\xi}-\frac{\xi^2 (t+1) z}{t^2}\right)dt, \quad \quad\quad
\end{align}\\where the second equality in \eqref{pdfz} stems from the fact that $v>1$, as shown in \eqref{pdfuv}. To the best of the authors' knowledge, the integral in \eqref{pdfz} does not lend itself to a closed-form. However, we can obtain the exact PEP expression in a closed-form by substituting \eqref{distIPS} in the conditional PEP expression given in \eqref{chernoff}. Then, the desired unconditional PEP expression is deduced in \eqref{pepIEHIPS} by taking the expectation with respect to the RVs $|h_{sd}|^2, Z_1$ and $Z_2$, where we used the fact that the PDF of $|h_{sd}|^2$ follows the exponential distribution and that the PDF of each of $Z_1$ and $Z_2$ is computed using \eqref{pdfz}, yielding
\begin{align}\label{peppdfz}
P(\bold{s}\to\hat{\bold{s}})\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1}\quad\quad\quad\quad\quad\quad\quad\quad\nonumber\\
\times\prod_{n=1}^2\underbrace{\int_0^\infty -\xi_n (t+1) t^{-2}\left(\frac{\xi_n^2 (t+1)}{t^2}+B_n\right)^{-1}\text{exp}\left(\frac{-t}{\xi_n}\right) dt}_{I_0},
\end{align}where $B_n=(\epsilon_n\zeta_n)/(4\beta_m N_0)$. Finally, by rewriting $I_0$ as \eqref{I0} at the bottom of this page, followed by some algebraic manipulations, and invoking \cite[Eq. (3.354.3)]{Rizhik} and \cite[Eq. (3.354.4)]{Rizhik}, the integral in \eqref{peppdfz} is obtained in a closed-form as in \eqref{pepIEHIPS}.
\vspace*{-0.3cm}
\section{Proof of Proposition 3}
\label{AppendixC}
Substituting \eqref{distIPS2} in the conditional PEP expression in \eqref{chernoff}, then taking the expectation with respect to fading coefficients $|h_{sd}|^2$, $|h_{r_1d}|^2$, and $|h_{r_2d}|^2$, which follow an exponential distribution, to yield \eqref{pepprop3} at the bottom of this page. Performing an expectation with respect to the random variables $|h_{sr_1}|^2$, $|h_{sr_2}|^2$, which also follow an exponential distribution, yields the unconditional PEP which is written as
\setcounter{equation}{50}
\begin{align}\label{uncondPEP}
P(\bold{s}\to\hat{\bold{s}})\leq\sum_{m=0}^{M-1}\alpha_m\left(\frac{\Delta P_s}{4\beta_m N_0}+1\right)^{-1}\quad \quad\quad\quad \quad\quad\quad \quad\quad\nonumber \\
\times\prod_{n=1}^2\underbrace{\int_0^\infty\left(\frac{(\epsilon_1\zeta_1/(4\beta_m N_0))|h_{sr_1}|^2}{\xi_1|h_{sr_1}|^2+1}+1\right)^{-1}\text{exp}(-t) dt}_{\Upsilon},
\end{align}where $t$ is the integration variable and $n\in\lbrace{1,2\rbrace}$. To solve the integral $\Upsilon$, we perform simple algebraic manipulations to get
\begin{align}
\Upsilon&=\int_0^\infty\left(\xi_n t+1\right)\left(\left(\frac{\epsilon_n\zeta_n}{4\beta_m N_0}+\xi_n\right) t+1\right)^{-1}\text{exp}(-t) dt \nonumber \\
&=\underbrace{\int_0^\infty \left(\left(\frac{\epsilon_n\zeta_n}{4\beta_m N_0}+\xi_n\right) t+1\right)^{-1}\text{exp}(-t) dt}_{I_1}\nonumber \\
&+\xi_n\underbrace{\int_0^\infty t\left(\left(\frac{\epsilon_n\zeta_n}{4\beta_m N_0}+\xi_n\right) t+1\right)^{-1}\text{exp}(-t) dt}_{I_2} \label{47a}.
\end{align}
Then, with the aid of the equality in \cite[Eq. (8.4.2.5)]{Prudnikov}, followed by applying the transformation \cite[Eq. (8.2.2.14)]{Prudnikov}, the first and second integrands of $I_1$ and $I_2$, respectively, are expressed in terms of their Meijer G-function representations as\\
\begin{equation}
\left(\gamma_n t+1\right)^{-1}=G_{1, 1}^{1, 1}\left[\frac{1}{\gamma_n t } \ \Big \vert \ {1 \atop 1} \right].
\end{equation}where $\gamma_n=\left(\frac{\epsilon_n\zeta_n}{4\beta_m N_0}+\xi_n\right)$. Similarly, the second and third integrands of $I_1$ and $I_2$, respectively, are rewritten by making use of the equality $e^{-t}=G_{0, 1}^{1, 0}\left[t \ \vert \ {- \atop 0} \right]$\cite[Eq. (8.4.3.1)]{Prudnikov}, yielding
\begin{align}
\Upsilon&=\int_0^\infty G^{1,1}_{1,1}\left[\gamma_n t \ \Big\vert \ {0 \atop 0}\right]G^{1,0}_{0,1}\left[t \ \Big\vert \ {- \atop 0}\right]dt \nonumber \\
&+\xi_n\int_0^\infty t G^{1,1}_{1,1}\left[\gamma_n t \ \Big\vert \ {0 \atop 0}\right]G^{1,0}_{0,1}\left[t \ \Big\vert \ {- \atop 0}\right]dt. \label{b47}
\end{align}
Then, by exploiting the integral identity \cite[Eq. (3.356.4)]{Prudnikov}, followed by performing some algebraic manipulations, $\Upsilon$ can be derived in a closed-form as
\begin{align}
\Upsilon&=\gamma_n^{-1}G^{1,2}_{2,1}\left[\gamma_n \ \Big\vert \ {1, 1 \atop 1}\right] +\xi_n \gamma_n^{-2}G^{1,2}_{2,1}\left[\gamma_n \ \Big\vert \ {1, 2 \atop 2}\right] \label{c47}
\end{align}
Finally, after substituting \eqref{c47} in \eqref{uncondPEP}, the desired result in \eqref{PEPAEHIPS} is derived.
\balance
\bibliographystyle{IEEEtran}
\bstctlcite{BSTcontrol}
|
1,108,101,566,803 | arxiv | \section{Introduction}\label{sec:intro}
In the presence of a moving, accelerated mirror, the
electromagnetic field evolves from the vacuum to an excited state,
containing a non-vanishing number of photons. This `motion induced
radiation' or `Dynamical Casimir Effect' (DCE) has been the
subject of intense theoretical research since its discovery in the
seventies~\cite{moore,fv}. While this phenomenon was initially
regarded as being of just theoretical interest (for example as a
toy model for black hole evaporation), in recent years it has been
pointed out that the experimental verification of the DCE might
not be, after all, so far out of reach~\cite{job,Dodonov-rev}.
Indeed, taking advantage of parametric resonance amplification,
this effect could be dramatically increased \cite{varios}, since
the number of photons created within a cavity with a moving mirror
should grow exponentially at resonance (i.e., when the mirror's
oscillatory frequency doubles one of the eigenfrequencies of the
unperturbed cavity). For the case of microwave cavities, the
mechanical frequency of the mirror should, however, be extremely
high ($\sim$ 1GHz) for this to happen, and this poses the main
stumbling block for an experimental verification of the effect.
It has also been suggested that the DCE could be measured in
experiments in which the moving mirror is replaced by a
semiconductor slab which suddenly changes its conductivity due to
illumination with short laser pulses~\cite{braggio}.
Unfortunately, the unavoidable losses in the semiconductor could
put the viability of this proposal in jeopardy~\cite{dodloss}. Yet
another alternative that has been advanced~\cite{onofrioprl},
which amounts to consider an array of nanoresonators, moving
coherently at frequencies in the GHz range. The detection of the
created photons could, in this case, be performed using an
inverted population of Rydberg atoms.
From the theoretical point of view, the DCE has been analyzed for a variety
of geometries and using many different theoretical tools. A particularly
interesting functional approach has been proposed by Golestanian and
Kardar~\cite{GK}. They introduce auxiliary fields in the
functional integral for the quantum field, whose role is to impose
the boundary conditions on the mirrors. This method has been successfully
applied, for example, to the calculation of the Euclidean effective action
for one and two (slightly deformed) moving mirrors in $d+1$
dimensions~\cite{miri}, deriving also the effective equation of motion for
the mirror by analytic continuation of the Euclidean effective action.
In view of the possibility of detecting the DCE using
nanoresonators~\cite{onofrioprl}, it is of interest to extend this
formalism in several directions. On the one hand, it is important
to generalize the method, in order to be able to consider
dispersive mirrors, rather than just perfectly conducting ones. On
the other hand, since the nanoresonators could eventually show
quantum behaviour~\cite{quantumnano}, it is worthwhile to consider
their quantum to classical transition, and to describe their
effective dynamics in terms of a semiclassical Langevin equation.
This paper is a step in that direction~\cite{previousctp}.
Besides, to exhibit the quite general nature of the phenomenon, it is also
interesting to extend the formalism to consider mirrors coupled to
different fields, like the case of a moving wall that imposes bag
conditions on a Dirac field. In this article, we first show how to
generalize the functional approach of~\cite{GK}, to calculate the Euclidean
effective action for moving dispersive mirrors coupled to real scalar and
then to Dirac fields. We also show how the case of a relativistic mirror
also fits in the formalism, by performing minor modifications.
We then compute the Schwinger-Keldysh or Closed Time Path (CTP) effective
action for a mirror coupled to a scalar field. More realistic situations
(a cavity with two no-flat mirrors coupled to the electromagnetic field)
will be considered in a forthcoming publication.
This article is organized as follows: in section~\ref{sec:effect},
we use a path-integral approach to evaluate the Euclidean
effective action for a single, perfect or imperfect,
non-relativistic moving mirror in $1+1$ dimensions, both for the
real scalar and Dirac field cases. By `perfect mirror' we mean one
that imposes Dirichlet boundary conditions, when coupled to a
scalar field, or bag conditions in the Dirac field case. In both
cases, the boundary conditions due to the perfect mirror are
introduced by the coupling of the quantum field to a singular mass
term, localized on a region of codimension $1$, with a divergent
coupling constant $\lambda \to \infty$. The imperfect mirrors
that we shall consider here will be, on the other hand, described
by the same kind of interaction term, albeit with a {\em finite\/}
coupling constant $\lambda$.
The changes needed to cope with the relativistic mirror generalization are
also presented, taking the real scalar field case as a concrete example,
and evaluating the corresponding effective action.
In section~\ref{sec:dpo}, we consider the case of a (flat)
moving mirror in $d+1$ dimensions, coupled to a real scalar field,
evaluating explicitly the Euclidean effective action. In
section~\ref{sec:imaginary}, we evaluate and interpret the
imaginary part of the in-out effective action, obtained after Wick
rotating to real time, for the case of the real scalar field with
perfect boundary conditions.
In section~\ref{sec:eqs}, we evaluate the quantum corrections to the
mirrors' real-time equations of motion. In order to do this, we
compute the CTP effective action and obtain a semiclassical
stochastic equation for the mirror. Moreover, from the imaginary
part of the CTP effective action we provide an estimation of the
decoherence time for the mirror.
\section{Moving mirrors in $1+1$ dimensions}\label{sec:effect}
\subsection{Real scalar field}\label{ssec:scalar2}
We shall begin by considering a massive real scalar field $\varphi$
coupled to an imperfect mirror, whose position is described by a
function $q(x^0)$, so that the real-time Lagrangian density,
${\mathcal L}$ is:
\begin{equation}\label{eq:defls}
{\mathcal L} \;=\; \frac{1}{2} \partial_\mu\varphi \partial^\mu \varphi
-\frac{1}{2} m^2 \varphi^2 - \frac{1}{2} \, V(x^0,x^1) \,\varphi^2 \;,
\end{equation}
where $V$ is a $\delta$-like singular function:
\begin{equation}\label{eq:defsing}
V(x^0,x^1) \,=\, \lambda \, \delta(x^1 - q(x^0))\;,
\end{equation}
determined by the mirror's position. $\lambda$ is a positive
coupling constant. The coupling to the singular field has the
effect of introducing a perfect mirror (at $x^1 = q(x^0)$) when
$\lambda \to + \infty$, since it then enforces the condition
$\varphi = 0$ on the points of the spacetime curve ${\mathcal C}$
defined by the points $(x^0,\, q(x^0))$. On the other hand, the
imperfect mirror situation is simulated for $0 < \lambda < \infty$
\cite{barton1}.
Let us now perform a Wick rotation: $x^0 = - i \tau$, and
calculate the resulting Euclidean effective action
$\Gamma[q(\tau)]$ for the mirror, due to the scalar-field vacuum
fluctuations, in the functional-integral representation:
\begin{equation}\label{eq:defzsc}
e^{ - \Gamma[q(\tau)]} \,\equiv\, {\mathcal Z}[q(\tau)] \,=\,
\int {\mathcal D} \varphi \; e^{ - S[\varphi;q] } \;,
\end{equation}
where $S[\varphi;q]$ is the Euclidean action:
\begin{equation}
S[\varphi;q] \;=\; S_0[\varphi] \;+\; S_{\mathcal C}[\varphi; q]
\end{equation}
with $S_0$ denoting the free part
\begin{equation}
S_0[\varphi]\,=\, \frac{1}{2} \, \int d^2x \big( \partial_\mu
\varphi\partial_\mu
\varphi + m^2 \varphi^2 \big)
\end{equation}
and $S_{\mathcal C}$ the coupling to the mirror,
\begin{equation}
S_{\mathcal C}[\varphi; q] \,=\, \frac{\lambda}{2} \, \int d^2x
~\delta(x_1 - q(x_0)) \big[ \varphi(x) \big]^2 \,=\,
\frac{\lambda}{2}\, \int d\tau \big[\varphi(\tau,q(\tau))]^2 \;.
\end{equation}
Euclidean coordinates are denoted by $x_\mu$, where $x_0 \equiv
\tau$; the metric tensor is the $2\times 2$ identity matrix.
To proceed, we introduce an auxiliary field $\xi(\tau)$, living in $0+1$
dimensions, whose role is to linearize the term $S_{\mathcal C}$, which
couples the scalar field to the mirror. The resulting expression for
${\mathcal Z}[q(\tau)]$ is:
\begin{equation}
{\mathcal Z}[q(\tau)] \;=\;\int {\mathcal D}\xi \; e^{-\frac{1}{2
\lambda} \int d\tau \xi^2(\tau)} \; {\mathcal Z}_0 [J_\xi]
\label{auxfield}
\end{equation}
where ${\mathcal Z}_0$ is the free generating functional:
\begin{equation}
{\mathcal Z}_0[J] \,=\, e^{- W_0 [J]} \,=\, \int {\mathcal D} \varphi \;
e^{- S_0(\varphi) + i \int d^2 x J(x) \varphi(x)} \;,
\end{equation}
and $J_\xi$ is a current localized on the defect and proportional
to the auxiliary field: \mbox{$J_\xi(x_0,x_1) \equiv \xi(x_0)
\,\delta(x_1 - q(x_0))$}. Note that (\ref{auxfield}) reduces to
the approach of Golestanian and Kardar \cite{GK} when $\lambda \to
\infty$, i.e., when a perfect mirror is considered.
Since the integral over $\varphi$ is a Gaussian, we can
immediately write down the explicit form of $W_0$,
\begin{equation}
W_0[J] \;=\; \frac{1}{2} \int d^2x \int d^2 x' J(x) \Delta (x-x') J(x') \;,
\end{equation}
where $\Delta$ is the free Euclidean correlation function:
\begin{eqnarray}
\Delta(x-y) &=& \langle \varphi(x) \varphi(y)\rangle
\nonumber\\
&=& \int \frac{d^2k}{(2\pi)^2} e^{-i k \cdot (x-y)} \frac{1}{ k^2 + m^2} \;.
\end{eqnarray}
Thus we derive for ${\mathcal Z}[q(\tau)]$ a `dimensionally
reduced' path integral expression involving just the auxiliary
field $\xi$,
\begin{equation}
{\mathcal Z}[q(\tau)] \,=\, \int {\mathcal D}\xi \; e^{-\frac{1}{2} \int
d\tau \int d\tau' \xi(\tau) {\mathcal K}(\tau,\tau') \xi(\tau')} \;,
\end{equation}
where we have introduced the kernel $K(\tau,\tau')$
\begin{equation}
{\mathcal K}(\tau,\tau') \;=\; \frac{1}{\lambda} \delta(\tau-\tau') +
\Delta\big[\tau-\tau',q(\tau)-q(\tau')\big] \;.
\end{equation}
The $\xi$-integral, again a Gaussian, allows as to write down the
(formal) result for ${\mathcal Z}[q(\tau)]$ as follows:
\begin{equation}
{\mathcal Z}[q(\tau)] \;=\; \big( \det {\mathcal K} \big)^{-\frac{1}{2}}
\end{equation}
so that
\begin{equation}\label{eq:gqdet}
\Gamma[q(\tau)] \;=\; \frac{1}{2} {\rm Tr} \big[\ln {\mathcal K} \big] \;.
\end{equation}
Let us now approximate (\ref{eq:gqdet}) for small departures with
respect to the static mirror case. To that end, we first expand
${\mathcal K}$:
\begin{equation}
{\mathcal K} \;=\; {\mathcal K}_0 \,+\, {\mathcal K}_1 \,+\, {\mathcal
K}_2 \,+\, \ldots
\end{equation}
where the subscripts denote the order of the corresponding term.
To derive the linearized form of the equations of motion, it shall
be sufficient to keep terms of up to the quadratic order. It is
quite straightforward to see that
\begin{equation}
{\mathcal K}_0(\tau,\tau')\;=\;\int \frac{d\omega}{2\pi} e^{i \omega
(\tau-\tau')} {\widetilde K}_0 (\omega) \;,\;\;\;
{\widetilde{\mathcal K}}_0 (\omega) \;=\; \frac{1}{\lambda}
\,+\, \frac{1}{2\sqrt{\omega^2 + m^2}} \;,
\end{equation}
\begin{equation}
{\mathcal K}_2 (\tau ,\tau') \;=\; \frac{1}{4} \big(q(\tau) -
q(\tau')\big)^2 \, \int \frac{d\omega}{2\pi} \, e^{i \omega(\tau-\tau')}
\sqrt{\omega^2 + m^2} \;,
\end{equation}
and that ${\mathcal K}_1$ vanishes. It should be kept in mind that $q(\tau)$ is
the {\em departure\/} with respect to a constant (fixed to $0$ by a shift
of the axis, if necessary). This implies, in particular, that its Fourier
transform ${\tilde q}(\omega)$ will verify ${\tilde q}(0)=0$. Of course,
${\tilde q}(0)=0$ alone does not imply a small departure. Indeed, the
condition holds true for some motions that correspond to an unbounded
motion, like ${\tilde q}(\omega) \propto i \delta^{'''}(\omega)$, which
comes from $q(\tau) \propto \tau^3$. But in this case the quadratic
approximation fails, since $q(\tau)$ becomes large (and ${\tilde
q}(\omega)$ singular).
Coming back to the expression for $\Gamma[q(\tau)]$, expanding up to the
second order in the fluctuation, and discarding a $q(\tau)$-independent
term, we see that
\begin{equation}
\Gamma[q(\tau)] \,=\, \frac{1}{2} \,{\rm Tr} \ln \big[{\mathcal K}_0 + {\mathcal K}_2 \big]
\, \simeq \, \frac{1}{2} {\rm Tr} \big[{\mathcal K}_0^{-1} {\mathcal K}_2 \big]
\,\equiv\, \Gamma_2[q(\tau)]\;,
\end{equation}
where $\Gamma_2[q(\tau)]$ may be written more explicitly as follows:
\begin{equation}
\Gamma_2[q(\tau)] \,=\, \frac{1}{2} \int_{-\infty}^{+\infty} d\tau \int_{-\infty}^{+\infty} d\tau'
\big[{\mathcal K}_0^{-1}(\tau,\tau') {\mathcal K}_2(\tau',\tau)\big] \;.
\end{equation}
Using the explicit form for ${\mathcal K}_0$ and ${\mathcal K}_2$,
\begin{equation}
\Gamma_2[q(\tau)] \,=\, \frac{1}{2} \int_{-\infty}^{+\infty} d\tau \int_{-\infty}^{+\infty}
d\tau' ( q(\tau) - q(\tau') )^2 F(\tau -\tau') \;,
\end{equation}
where
\begin{equation}
F(\tau -\tau') \,=\, \int \frac{d\omega}{2\pi} e^{i \omega(\tau-\tau')}
{\tilde F}(\omega) \,,
\end{equation}
with
\begin{equation}\label{eq:defft}
{\tilde F}(\omega) \,=\,\frac{1}{4} \, \int \frac{d\nu}{2\pi} \,
\big[\frac{1}{\lambda} \,+\, \frac{1}{2 \sqrt{(\nu + \omega)^2 + m^2}}\big]^{-1}
\sqrt{\nu^2 + m^2}\;.
\end{equation}
It is clear that we may subtract from ${\tilde F(\omega)}$ its
value at zero-frequency, since any $\omega$-independent part would
give zero when inserted in $\Gamma_2[q(\tau)]$ (it would produce a
$\delta(\tau-\tau')$ contribution to $F$, multiplied by a
continuous function that vanishes when $\tau = \tau'$). Thus we
introduce
\begin{equation}
{\tilde F}_s(\omega) \,\equiv\, {\tilde F}(\omega) \,-\, {\tilde F}(0) \,,
\end{equation}
the subtracted version of ${\tilde F}$. Since ${\tilde F}_s(0) = 0$, we
obviously have \mbox{$\int d\tau F_s(\tau) = 0$}, and
\begin{eqnarray}
\Gamma_2[q(\tau)] &=& \frac{1}{2} \int_{-\infty}^{+\infty} d\tau \int_{-\infty}^{+\infty} d\tau'
( q(\tau) - q(\tau') )^2 F(\tau -\tau') \nonumber\\
&=& \frac{1}{2} \int_{-\infty}^{+\infty} d\tau \int_{-\infty}^{+\infty} d\tau'
( q(\tau) - q(\tau') )^2 F_s(\tau -\tau') \nonumber\\
&=& - \int_{-\infty}^{+\infty} d\tau \int_{-\infty}^{+\infty} d\tau'
q(\tau) q(\tau') \, F_s(\tau -\tau') \;. \label{eea}
\end{eqnarray}
Expression (\ref{eq:defft}) for ${\tilde F}$ is divergent; to
regulate it we introduce a symmetric frequency cutoff $\Xi$, such
that $|\nu | \leq \Xi$, and the regulated function ${\tilde
F}_s(\omega,\Xi)$ is:
\begin{eqnarray}\label{eq:deffto}
{\tilde F}_s(\omega,\Xi) &=&
\frac{1}{4} \, \int_{-\Xi}^\Xi \frac{d\nu}{2\pi} \,\Big\{
\big[\frac{1}{\lambda} \,+\, \frac{1}{2 \sqrt{(\nu + \omega)^2 + m^2}}\big]^{-1}
\sqrt{\nu^2 + m^2} \nonumber\\
&-& \big[\frac{1}{\lambda} \,+\, \frac{1}{2 \sqrt{\nu^2 + m^2}}\big]^{-1}
\sqrt{\nu^2 + m^2} \Big\} \;.
\end{eqnarray}
${\tilde F}_s(\omega,\Xi)$ in (\ref{eq:deffto}) is convergent for
$\Xi \to \infty$, so this `symmetric-limit' regularization yields
a finite value for \mbox{${\tilde F}_s(\omega)\equiv \lim_{\Xi \to
\infty}{\tilde F}_s(\omega,\Xi)$} when the regulator is removed.
Unfortunately, there seems to be no analytic expression for
${\tilde F}_s(\omega)$ which is valid for arbitrary values of the
constants $m$ and $\lambda$. We can, however, calculate it for
different relevant particular cases:
\begin{enumerate}
\item $m = 0$, $\lambda \to \infty$: in this case, we have
\begin{eqnarray}\label{eq:f0i}
{\tilde F}_s(\omega) &=& \lim_{\Xi \to \infty}
\frac{1}{2} \, \int_{-\Xi}^\Xi \frac{d\nu}{2\pi} \,
\Big[ |\nu + \omega| |\nu| \,-\,|\nu|^2 \Big] \nonumber\\
&=& \frac{1}{12\pi} |\omega|^3 \;.
\label{fsperf}
\end{eqnarray}
A cubic dependence in $\omega$ could have been guessed on
dimensional grounds. The numerical coefficient coincides with
previously obtained results \cite{LR}.
\item $m = 0$, $\lambda < \infty$: a property that we immediately see is that,
for any finite $\lambda$, the large-$\nu$ behaviour of the integral is
improved (by a power of $\nu$) with respect to the perfect mirror
($\lambda \to \infty$) case. As a consequence, the result obtained by taking the
$\Xi \to \infty$ and $\lambda \to \infty$ limits will depend on the order in
which they are taken.
The $\nu$ integral (for a finite $\lambda$) and its $\Xi \to
\infty$ limit can be evaluated exactly in this case, the result
being,
\begin{equation}
{\tilde F}_s(\omega) \,=\, \frac{\lambda^2}{16\pi^2} \Big[
2 |\omega| - \lambda \big( 1 + \frac{2}{\lambda} |\omega| \big)
\ln\big(1 + \frac{2}{\lambda} |\omega|\big)\Big]\;.
\end{equation}
Performing a large-$\lambda$ expansion in the previous expression;
we see that
\begin{equation}\label{eq:flexp}
{\tilde F}_s(\omega) \,=\, - \frac{\lambda}{8\pi} \omega^2
\,+\, \frac{1}{12\pi} |\omega|^3 \,+\,{\mathcal O}(\lambda^{-1})\;.
\end{equation}
The second term is independent of $\lambda$, and it coincides with
the result Eq.(\ref{fsperf}) obtained for $\lambda \to \infty$.
The first term was absent from the perfect mirror case, and is a
reflection of the fact that, as anticipated, the $\Xi \to \infty$
and $\lambda \to \infty$ limits do not commute. The reason for
that difference is that the finite-$\lambda$ system includes the
effect of more quantum fluctuations than in the infinite-$\lambda$
case. The resulting difference between the results obtained for
those different limits has, however, a simple physical
interpretation. Indeed, that difference $\delta{\tilde F}_s$ comes
from the ${\mathcal O}(\lambda)$ term in Eq.(\ref{eq:flexp}):
\begin{equation}
\delta{\tilde F}_s(\omega) \,\equiv \, - \frac{\lambda}{8\pi} \omega^2\,,
\end{equation}
a term which, when inserted into the expression for $\Gamma_2[q]$ yields
\begin{equation}
\delta\Gamma_2[q] \,=\, \int d\tau \frac{1}{2} \, \mu(\lambda)
\dot{q}^2(\tau) \;,
\end{equation}
where $\mu(\lambda) = \frac{\lambda}{4\pi}$. This term can, of
course, be regarded as a renormalization in the mirror's mass,
when the mirror has a non-relativistic kinetic-energy term, as it
is usually assumed. It shouldn't come as a surprise that the
outcome of the calculation is a non-relativistic invariant object:
the coupling between mirror and field, $S_{\mathcal C}$, does in
fact assume a non-relativistic description for the mirror, since
it is not a relativistic invariant. The covariant formulation of
this example is presented, for the sake of completeness, in
~\ref{ssec:cova}.
\item $m \neq 0$, $\lambda \to \infty$: the exact result for this
case can also be obtained, although the calculation is more
involved. As outlined in Appendix A, the final result is:
\begin{equation}
\tilde F_s(\omega)=\frac{1}{12\pi}\int_0^1\frac
{d\alpha}{\alpha(\alpha-1)}\left [\left [\alpha(\alpha
-1)\omega^2+m^2\right]^{3/2}-m^3\right]
\label{Fmperf}
\end{equation}
which reduces to the proper result Eq.(\ref{fsperf}) in the $m \to
0$ limit.
\end{enumerate}
\subsection{Real scalar field: relativistic mirror}\label{ssec:cova}
We present here a relativistically invariant formulation of the real
scalar field case.
The main reason, besides its intrinsic interest, is that
it makes it easier to understand the approximation incurred in the non relativistic
approach we (implicitly) used in the previous subsections. This
problem has been considered previously in Ref.~\cite{barton2} using
a canonical formalism.
An explicitly invariant coupling can be constructed, for example,
by considering a relativistic generalization of coupling term,
$S_{\mathcal C} \to S^{\rm rel}_{\mathcal C}$:
\begin{equation}
S^{\rm rel}_{\mathcal C}[\varphi,q] \;=\; \frac{\lambda}{2} \,
\int d^2x \int ds \sqrt{\dot{q}_\mu(s)\dot{q}_\mu(s)} \,
\delta^{(2)}[x - q(s)] \big[ \varphi(x) \big]^2
\end{equation}
where $q_\mu(s)$, $\mu=0,1$, is a suitable parametrization of the
worldline described by the mirror. We use the notation $\dot{q}_\mu=
\frac{dq_\mu}{ds}$.
When the parametrization is such that
$s$ coincides with the laboratory time, $x_0 \to (x_0, q(x_0))$, we obtain:
\begin{equation}
S^{\rm rel}_{\mathcal C}[\varphi,q] \;=\; \frac{\lambda}{2} \,
\int dx_0 \sqrt{1 + \dot{q}^2(x_0)} \,
\big[ \varphi(x_0, q(x_0)) \big]^2 \;,
\end{equation}
which indeed reduces to the non-relativistic term $S_{\mathcal C}$ for
\mbox{$|\dot{q}| << 1$}, and justifies {\em a posteriori\/} the non-relativistic
coupling when that condition is fulfilled.
Let us now write down the expressions for the (Euclidean) relativistic versions
of the objects we have considered before:
\begin{equation}
{\mathcal Z}^{\rm rel}[q(s)] \;=\;\int {\mathcal D}\xi \; e^{-\frac{1}{2 \lambda} \int
ds \xi^2(s)} \; {\mathcal Z}_0[J_\xi^{\rm rel}]
\end{equation}
where
\begin{equation}
J_\xi^{\rm rel} (x_0,x_1) \equiv \int ds \xi(s) \, |\dot{q}(s)|^{\frac{1}{2}}
\delta^{(2)}(x - q(s)) \;.
\end{equation}
Then
\begin{equation}
e^{-\Gamma^{\rm rel}[q(s)]} = {\mathcal Z}^{\rm rel}[q(s)] \;=\;
\big( \det {\mathcal K}^{\rm rel}\big)^{-\frac{1}{2}}
\end{equation}
where
\begin{equation}
{\mathcal K}^{\rm rel}(s,s')\;=\;\frac{1}{\lambda} \,\delta(s-s') \,+\,
|\dot{q}(s)|^{\frac{1}{2}} \,\Delta\big[q(s)-q(s')\big] \,|\dot{q}(s')|^{\frac{1}{2}} \,.
\end{equation}
The next step is to perform an expansion in powers of $f_\mu(s)$, the fluctuating
part of $q_\mu(s)$:
\begin{equation}
q_\mu(s) \;=\; q_\mu^{(0)}(s) \,+\, f_\mu(s) \;
\end{equation}
where $q_\mu(s)$ is analogous to the worldline for the `static' mirror.
It is in fact a linear function of $s$
\begin{equation}
q_\mu^{(0)} (s) \;=\; a_\mu \,+\, v_\mu \, s
\end{equation}
where $a_\mu$ and $v_\mu$ are constant vectors; $v_\mu$ is
time-like in the real time description. The fluctuating part,
$f_\mu$, will be assumed to be such that $\dot{f}_\mu$ is
orthogonal to $v_\mu$. The reason is that parallel components
amount to fluctuations in the parametrization, which are of course
irrelevant in a reparametrization-invariant theory.
In order to simplify matters, we use in what follows a specific convariant
parametrization; namely, we assume that $s$ is the mirror's proper time.
Then the $|\dot{q}|^{\frac{1}{2}}$ factors become both equal to $1$.
The calculations then proceed, for this parametrization, in a way that
mimics the non-relativistic ones. The quadratic part of the effective
action is given by,
\begin{equation}
\Gamma_2^{\rm rel}[q(s)] \,=\, \frac{1}{2} \int_{-\infty}^{+\infty} ds \int_{-\infty}^{+\infty} ds'
\big[({\mathcal K}^{\rm rel})_0^{-1}(s,s') {\mathcal K}^{\rm rel}_2(s',s)\big] \;.
\end{equation}
where
\begin{eqnarray}
{\mathcal K}^{\rm rel}_0(s,s')&=& \frac{1}{\lambda} \, \delta(s-s') \,+\,
\int \frac{d^2 k}{(2\pi)^2} e^{i k \cdot v (s-s')} \, \frac{1}{k^2 + m^2}
\nonumber\\
&=& \int \frac{d\omega}{2\pi}\, e^{i \omega (s-s')} \frac{1}{\lambda} \,+\,
\int \frac{d\omega}{2\pi}\, e^{i \omega |v|(s-s')} \frac{1}{2 \sqrt{\omega^2 + m^2}}
\nonumber\\
&=& \int \frac{d\omega}{2\pi}\, e^{i \omega (s-s')} \Big[ \frac{1}{\lambda} \,+\,
\frac{1}{2 |v| \, \sqrt{\omega^2 + m^2}} \Big]
\end{eqnarray}
and
\begin{equation}
{\mathcal K}^{\rm rel}_2 (s,s') \;=\; \frac{1}{4} \big[f(s) - f(s')\big]^2 \,
\frac{1}{|v|} \, \int \frac{d\omega}{2\pi} \, e^{i \omega (s-s')} \,\sqrt{\omega^2 + m^2} \;.
\end{equation}
Then
\begin{equation}
\Gamma_2^{\rm rel}[q(s)] \;=\; - \,\int_{-\infty}^{+\infty} ds \int_{-\infty}^{+\infty} ds'
q(s) \, F_s^{\rm rel} (s -s') \, q(s') \;.
\end{equation}
where:
\begin{eqnarray}
{\tilde F}^{\rm rel}_s(\omega) &=& \lim_{\Xi \to \infty}
\frac{1}{4} \, \int_{-\Xi}^\Xi \frac{d\nu}{2\pi} \,\Big\{
\big[\frac{|v|}{\lambda} \,+\, \frac{1}{2 \sqrt{(\nu + \omega)^2 + m^2}}\big]^{-1}
\sqrt{\nu^2 + m^2} \nonumber\\
&-& \big[\frac{|v|}{\lambda} \,+\, \frac{1}{2 \sqrt{\nu^2 + m^2}}\big]^{-1}
\sqrt{\nu^2 + m^2} \Big\} \;.
\end{eqnarray}
This implies, in particular, that in the $\lambda \to \infty$ limit, the
results for $\Gamma^{\rm rel}[q]$ coincide with the ones for the non-relativistic
case, if one uses the proper-time as the evolution parameter.
It is interesting to consider now the situation when $\lambda$ is large but
finite and $m =0$. It is quite straightforward to see that this contribution
generates, as in the non-relativistic case, an order-$\lambda$ term:
\begin{equation}
\delta\Gamma_2^{\rm rel}[q] \,=\, \frac{1}{|v|} \,\int ds \frac{1}{2} \, \mu(\lambda)
\dot{f}^2(s) \;,
\end{equation}
where $\mu(\lambda) = \frac{\lambda}{4\pi}$. On the other hand,
since \mbox{$\dot{q}^2(s)= v^2 + \dot{f}^2$} and we have an
equivalent expression for $\delta\Gamma_2^{\rm rel}$:
\begin{equation}
\delta\Gamma_2^{\rm rel}[q] \,=\, \mu^{\rm rel}(\lambda,v) \int ds \;,
\end{equation}
a term that has a form of a ($v$-dependent) mass counterterm for the relativistic mirror
action, with
\begin{equation}
\mu^{\rm rel}(\lambda,v) \;\equiv\;\frac{1}{2} \mu(\lambda) (\frac{1}{|v|}- |v|)
\;,
\end{equation}
since the natural relativistic action for a mirror with mass $M$ is:
\begin{equation}
S_{\rm mirror}^{\rm rel} \;=\; M \,\int ds \;.
\end{equation}
\subsection{Dirac field}\label{ssec:dirac}
Let us now consider the case of a Dirac field with bag-like
boundary conditions on the `mirror'. This kind of boundary
condition can also be introduced by means of an interaction with a
singular potential; indeed, as shown in~\cite{Sundberg:2003tc},
bag-like boundary conditions may be introduced by considering the
limit of a singular mass term. The real-time Lagrangian density is
then
\begin{equation}
{\mathcal L} \;=\; {\bar\psi} \big[ i \not \! \partial - m - V(x^0,x^1)
\big] \psi(x)
\end{equation}
where $V$ is the singular potential defined in (\ref{eq:defsing}).
As in the real scalar field case, we may pass to the Euclidean formulation,
to calculate $\Gamma[q(\tau)]$ for the mirror:
\begin{equation}
e^{-\Gamma[q(\tau)]}\;=\; {\mathcal Z}[q(\tau)] \;=\; \int {\mathcal D}\psi {\mathcal
D}{\bar\psi} \, e^{-S[{\bar\psi},\psi; q]} \;,
\end{equation}
where now:
\begin{equation}
S[{\bar\psi},\psi; q] \,=\, S_0[{\bar\psi},\psi] \,+\, S_{\mathcal C}[{\bar\psi},\psi; q]
\end{equation}
with
\begin{equation}
S_0[{\bar\psi},\psi] \,=\, \int d^2x ~{\bar\psi} ~ (\not \! \partial + m )
\psi
\end{equation}
and
\begin{equation}
S_{\mathcal C}[{\bar\psi},\psi;q] \,=\, \lambda \, \int d^2x ~{\bar\psi}(x)
~\delta[x_1 - q(x_0)] ~\psi(x) \;.
\end{equation}
Now to linearize the coupling we need two $2$-component (Grassmann) auxiliary fields,
${\xi}(\tau)$ and ${\bar\xi}(\tau)$, so that
\begin{equation}
{\mathcal Z}[q(\tau)] \;=\; \int {\mathcal D}\xi {\mathcal D}{\bar\xi} \,
e^{-\frac{1}{\lambda}\int d\tau {\bar\xi}(\tau) \xi(\tau)} \;
{\mathcal Z}_0[{\bar \eta}_\xi,\eta_\xi]
\;,
\end{equation}
with
\begin{equation}
{\mathcal Z}_0[{\bar\eta},\eta] \;=\;
e^{-W_0[{\bar\eta},\eta]}
\end{equation}
where
\begin{equation}
W_0[{\bar\eta},\eta] \;=\; \int d^2x \int d^2x' ~{\bar\eta}(x)
~{\mathcal S}_f (x-x') ~\eta(x')
\end{equation}
and ${\mathcal S}_f$ is the free Dirac propagator:
\begin{equation}
{\mathcal S}_f(x,x') \,=\, \langle \psi(x) {\bar\psi}(x') \rangle \,=\,
\int \frac{d^2p}{(2\pi)^2} e^{i p \cdot (x-x')} \frac{1}{i \not\! p + m} \;.
\end{equation}
We have introduced the sources:
\begin{equation}
\eta_\xi(x) \,=\, \xi (x_0) \, \delta[x_1 - q(x_0)] \;\;,\;\;\;
{\bar\eta}_{\bar\xi}(x) \,=\, {\bar\xi}(x_0) \, \delta[x_1 - q(x_0)] \,.
\end{equation}
Performing the (Grassmann) Gaussian integral over the auxiliary fields, we
see that
\begin{equation}
\Gamma[q] \;=\; - {\rm Tr} \ln\big[{\mathcal K}_f\big]\;
\end{equation}
where:
\begin{equation}
{\mathcal K}_f(\tau,\tau') \,=\, \frac{1}{\lambda} \delta(\tau-\tau')
\,+\, {\mathcal S}_f (\tau -\tau', q(\tau)- q(\tau') ) \;.
\end{equation}
We again expand in powers of the fluctuating $q(\tau)$,
\begin{equation}
{\mathcal K}_f \;=\; {\mathcal K}_f^{(0)} \,+\, {\mathcal K}_f^{(1)} \,+\,
{\mathcal K}_f^{(2)} \,+\,\ldots
\end{equation}
with
\begin{equation}
{\mathcal K}_f^{(0)}(\tau-\tau') \,=\, \int \frac{d\omega}{2\pi} e^{i
\omega (\tau-\tau')} {\widetilde{\mathcal K}}_f^{(0)}(\omega) \;,\;\;\;
{\widetilde{\mathcal K}}_f^{(0)}(\omega) \,=\, \Big[\frac{1}{\lambda} + \frac{-i
\gamma_0 \omega + m}{2 \sqrt{\omega^2 + m^2}} \Big]\;,
\end{equation}
\begin{equation}
{\mathcal K}_f^{(1)}(\tau-\tau') \,=\, - \gamma_1 \, [q(\tau) - q(\tau')] \,
\int \frac{d\omega}{2\pi} e^{i \omega (\tau-\tau')} \frac{1}{2} \sqrt{\omega^2 + m^2}
\end{equation}
and
\begin{equation}
{\mathcal K}_f^{(2)}(\tau-\tau') \,=\, \frac{1}{4} \, [q(\tau) -
q(\tau')]^2 \, \int \frac{d\omega}{2\pi} e^{i \omega (\tau-\tau')}
\sqrt{\omega^2 + m^2} \, ( - i \gamma_0 \omega + m) \;.
\end{equation}
Up to second order in the fluctuation,
\begin{eqnarray}
\Gamma[q(\tau)] &\simeq& \Gamma_2[q(\tau)] \nonumber\\
\Gamma_2[q] &=& -{\rm Tr}\Big[ ({\mathcal K}_f^{(0)})^{-1}{\mathcal K}_f^{(2)}\Big]
+ \frac{1}{2} {\rm Tr}\Big\{ \big[ ({\mathcal
K}_f^{(0)})^{-1}{\mathcal K}_f^{(1)}\Big]^2 \Big\} \nonumber\\
&\equiv& \Gamma_2^{(1)}[q(\tau)]\,+\,\Gamma_2^{(2)}[q(\tau)] \,
\end{eqnarray}
where
\begin{eqnarray}
\Gamma_2^{(1)}[q(\tau)]&=&
-{\rm Tr}\Big[ ({\mathcal K}_f^{(0)})^{-1}{\mathcal
K}_f^{(2)}\Big]\nonumber\\
&=& - \int d\tau \int d\tau' [q(\tau) -
q(\tau')]^2 \, F^{(1)}(\tau-\tau')
\end{eqnarray}
with
\begin{eqnarray}
{\tilde F}^{(1)}(\omega)&=&\int \frac{d\nu}{2\pi}\,
\frac{1}{(\lambda ^2+4) m^2+4 \lambda \sqrt{m^2+(\omega + \nu)^2} m+(\lambda^2+4)
(\omega + \nu)^2} \nonumber\\
&\times& \Big[ \lambda \sqrt{m^2+\nu^2} (2 m^3+\lambda
\sqrt{m^2+(\omega + \nu)^2} m^2 \nonumber\\
&+& 2 (\omega + \nu)^2 m+\lambda \nu (\omega +
\nu) \sqrt{m^2+(\omega + \nu)^2})\Big]
\end{eqnarray}
which for the $\lambda \to \infty$ and $m \to 0$ case reduces to:
\begin{equation}
{\tilde F}^{(1)}(\omega)\,=\,\int \frac{d\nu}{2\pi}\,
\frac{(\omega + \nu)}{|\omega + \nu|} \nu |\nu| \;.
\end{equation}
For the remaining term, ${\tilde F}^{(2)}$, a somewhat lengthy calculation
shows that it vanishes (for any value of $\lambda$ and $m$).
For the special case of $m=0$ and and $\lambda \to \infty$, the subtracted
version of ${\tilde F}^{(1)}(\omega)$ is
\begin{equation}
{\tilde F}^{(1)}_s(\omega) \,=\, \frac{2}{3} |\omega|^3 \;.
\end{equation}
\section{Plane mirror coupled to a real scalar field in $d+1$ dimensions}\label{sec:dpo}
We shall consider here the generalization of the calculations of
the previous section, in particular the ones for the real scalar
field, to the case of a flat moving mirror in $d+1$ dimensions.
The mirror's Euclidean world-volume is defined by the equation
\begin{equation}
x_d - q(x_0) \;=\; 0 \;;
\end{equation}
the coordinates $x_1,x_2,\ldots,x_{d-1}$ shall be denoted
collectively by $x_\parallel$, since they are parallel to the
mirror.
In a quite straightforward generalization of the derivation
implemented for the $1+1$ dimensional case, we introduce auxiliary
fields $\xi(\tau,x_\parallel)$, living in $d-1$ dimensions,
obtaining for the $d+1$ dimensional vacuum amplitude, ${\mathcal
Z}^{(d+1)}[q(\tau)]$, the expression:
\begin{equation}
{\mathcal Z}^{(d+1)}[q(\tau)] \,=\, \int {\mathcal D}\xi \; e^{-\frac{1}{2} \int
d\tau \int d\tau' \xi(\tau,x_\parallel) {\mathcal
K}(\tau,x_\parallel;\tau',x_\parallel') \xi(\tau',x_\parallel')} \;,
\end{equation}
where
\begin{equation}
{\mathcal K}(\tau,x_\parallel;\tau',x_\parallel')\,=\,
\frac{1}{\lambda} \delta(\tau-\tau') \delta(x_\parallel - x_\parallel') +
\Delta\big[\tau-\tau',x_\parallel - x_\parallel',q(\tau)-q(\tau')\big] \;.
\end{equation}
The $\xi$-integral is again Gaussian, and we take advantage of the
translation invariance along $x_\parallel$ to Fourier transform with
respect to those coordinates, obtaining
\begin{equation}
\Gamma^{(d+1)}[q(\tau)] \;=\; L^{d-1} \, \int
\frac{d^{d-1}k_\parallel}{(2\pi)^{d-1}} \,
\Gamma^{(1+1)}[q(\tau),m(p_\parallel)]
\end{equation}
where in the last expression we introduced $L^{d-1}$, the `area' of the plate,
and $\Gamma^{(1+1)}[q(\tau),m(k_\parallel)]$ denotes the effective action for
the $1+1$ dimensional case, calculated with a mass depending on the
parallel momentum, through the equation:
\begin{equation}
m^2(k_\parallel) \;=\; m^2 \,+\, k_\parallel^2 \;,
\end{equation}
with $m$ denoting the standard mass of the field.
The $L^{d+1}$ factor is divergent for an infinite plate. This
divergence is, however, harmless from the physical point of view,
since the natural object to calculate is not the force but rather
the {\em pressure\/} experienced by the mirror, hence the area
factor is divided out.
In the quadratic approximation we have
\begin{equation}
\frac{1}{L^{d-1}} \, \Gamma_2^{(d+1)}[q(\tau)] \;=\;
- \int_{-\infty}^{+\infty} d\tau \int_{-\infty}^{+\infty} d\tau'
q(\tau) q(\tau') \, F^{(d+1)}_s(\tau -\tau') \;,
\end{equation}
where
\begin{eqnarray}
{\tilde F}^{(d+1)}_s(\omega,\Xi) &=&
\frac{1}{4} \, \int \frac{d^{d-1}k_\parallel}{(2\pi)^{d-1}} \,
\int_{-\Xi}^\Xi \frac{d\nu}{2\pi} \,\Big\{
\big[\frac{1}{\lambda} \,+\, \frac{1}{2 \sqrt{(\nu + \omega)^2 + m^2 +
k_\parallel^2}}\big]^{-1} \nonumber\\
&-& \big[\frac{1}{\lambda} \,+\, \frac{1}{2 \sqrt{\nu^2 + m^2 +
k_\parallel^2}}\big]^{-1} \Big\} \, \sqrt{\nu^2 + m^2 + k_\parallel^2} \;.
\end{eqnarray}
As an example, we consider the particular case $m=0$ and $\lambda
\to \infty$:
\begin{eqnarray}
{\tilde F}^{(d+1)}_s(\omega,\Xi) &=&
\frac{1}{2} \, \int \frac{d^{d-1}k_\parallel}{(2\pi)^{d-1}} \,
\int_{-\Xi}^\Xi \frac{d\nu}{2\pi} \,\Big\{
\sqrt{(\nu + \omega)^2 + k_\parallel^2} \nonumber\\
&-& \sqrt{\nu^2 + k_\parallel^2}\Big\} \,
\sqrt{\nu^2 + k_\parallel^2} \;.
\end{eqnarray}
As already mentioned, the calculation in $d+1$ dimensions is
similar to the massive case in $1+1$ dimensions. Following the
steps described in Appendix A we find
\begin{equation}
\tilde F_s^{(d+1)}(\omega)=
\frac{\Gamma^2(\frac{1+d}{2})\Gamma(-1-(d/2))}{2^{d+3}\pi^{d/2+1}
\Gamma(d+1)}(\omega^2)^{1+\frac{d}{2}}
\label{Fperfd}
\end{equation}
While in an odd number of space dimensions the result could be
predicted by dimensional analysis, there is a subtle point in even
dimensions. As $\Gamma(-1-(d/2))$ is divergent in this case, it
is necessary to introduce in the Lagrangian a counterterm with
higher derivatives of the mirror's position. Once the divergence
is absorbed, a finite term remains, proportional to ${\rm
log}[\omega/\mu]$, where $\mu$ is an arbitrary constant,
determined by the renormalization point.
\section{Imaginary part of the in-out effective
action}\label{sec:imaginary}
One of the most distinctive signals of the dispersive effects
due to an accelerated mirror is a non-vanishing probability of
producing a particle pair out of the vacuum.
Indeed, the total probability of producing a particle
pair when the whole history of the mirror, from $t \to
-\infty$ to $t \to +\infty$, is taken into account, can be
obtained from the imaginary part of the `in-out' real time
effective action $\Gamma^{io}$:
\begin{equation}
P \;=\; 2 \, {\rm Im}[\Gamma^{io}] \;.
\end{equation}
This real-time effective action can, on the other hand, be
obtained by performing the inverse Wick rotation on the Euclidean
$\Gamma[q(\tau)]$ effective actions that we have just calculated,
back to real time.
Let us obtain the explicit form of $P$ for two illustrative
examples, the cases of the massless and massive real scalar fields
in $1+1$ dimensions, since they encode the main features of the
physical process we want to describe (other cases will indeed give
different results, but they will be kinematical in nature).
The quadratic approximation to the imaginary-time effective
action, $\Gamma_2[q(\tau)]$, whose general form in Fourier space
is:
\begin{equation}
\Gamma_2[q] \;=\; - \int_{-\infty}^{+\infty} \frac{d\omega}{2\pi} \;
{\tilde F}_s(\omega) \, |{\tilde q}(\omega)|^2 \;,
\end{equation}
leads to
\begin{equation}
\Gamma_2^{io}[q] \;=\; \int_{-\infty}^{+\infty} \frac{d\omega}{2\pi} \;
{\tilde F}_s(i \omega) \, |{\tilde q}(\omega)|^2 \;,
\end{equation}
where we kept the same notation for the rotated function $q$. Thus,
\begin{equation}\label{eq:P}
P \;=\; 2 \, \int_{-\infty}^{+\infty}
\frac{d\omega}{2\pi} \; {\rm Im}[ {\tilde F}_s(i \omega) ] \,
|{\tilde q}(\omega)|^2
\;.
\end{equation}
Let us now evaluate $P$ for the two cases mentioned above: in the
simplest case of a massless real scalar field with perfect
boundary conditions in $1+1$ dimensions, we have,
\begin{equation}
{\rm Im} {\tilde F}_s(i\omega) = \frac{1}{12\pi}
(-\omega^2)^{3/2}=\pm \, \frac{1}{12\pi} |\omega|^3\;.
\end{equation}
The two signs correspond to the two possible determinations of the
square root. Of course, only the positive one corresponds to the
right physical situation (Feynman conditions):
\begin{equation}\label{eq:P1}
P \;=\; \frac{1}{6\pi} \, \int_{-\infty}^{+\infty}
\frac{d\omega}{2\pi} \; |\omega|^3 \, |{\tilde q}(\omega)|^2 \;.
\end{equation}
In the case $m \neq 0$ and $\lambda \to \infty$, the imaginary
part of $\tilde F_s$ can also be computed. Indeed, from
(\ref{Fmperf}), we see that
\begin{equation}
{\rm Im} {\tilde F}_s(i\omega) = \frac{\theta(\omega^2-4
m^2)}{12\pi} \vert \omega^3\vert
\int_0^{1-\frac{4m^2}{\omega^2}}dx\,
\big[1-\frac{4m^2}{\omega^2(1-x^2)}\big]^{3/2}.
\end{equation}
The integral can be computed explicitly in terms of elliptic
functions, but we will not need that rather cumbersome expression
in what follows, since we want to pinpoint a rather interesting
physical phenomenon: the presence of a threshold in the imaginary
part of the effective action. This can be understood as follows:
if the mirror oscillates with a frequency $\omega$, the reflection
of a single field mode, with frequency $\omega_k$, will generate
frequency sidebands $\omega_k-\omega ,\, \omega_k+\omega$. In
order to create particles it is necessary to have a mix of
positive and negative frequencies, and the negative frequency
should be smaller than $-m$, i.e. $\omega_k-\omega < -m$. This
yields $\omega > 2 m$, as predicted by the previous equation.
\section{Real time dynamics: the Schwinger-Keldysh effective
action}\label{sec:eqs}
Up to now, we have considered the Euclidean effective action that
describes the dynamics of the mirror after integration of the
quantum fields, and applied it to study the probability of emitting a
particle pair during the whole evolution of the system, by performing
a Wick rotation back to Minkowski space. The last object is the
in-out effective action, which cannot be applied in a straightforward way
to the derivation of the equations of motion, since they would become
neither real nor causal.
As is well known, in order to get the correct effective equations
of motion, one should compute the in-in, Schwinger-Keldysh or
Closed Time Path Effective Action (CTPEA) \cite{ctp0}, which also has
information on the stochastic dynamics of the mirror \cite{ctp}.
The CTPEA is defined as
\begin{equation} e^{-i \Gamma_{\rm CTP} [q^+,q^-]} =
\int {\cal D} \phi^+{\cal D} \phi^- e^{i(S[q^+,\phi^+]-
S[q^-,\phi^-])}, \label{ctpeff}\end{equation} and the field
equations are obtained taking the variation of this action with
respect to the $q^+$, and then setting $q^+=q^-$. As in the
Euclidean case Eq.(\ref{auxfield}), one can introduce two
auxiliary fields $\xi_\pm(t)$ living in $0+1$ dimensions, in order
to linearize the coupling between the mirror and the field.
Instead of doing this, we will follow an alternative procedure.
Using a more concise notation, we can write the CTPEA as
\begin{equation} e^{- i \Gamma_{\rm CTP}[q]} =
\int {\cal D} \phi e^{i S^{{\cal
C}}[q,\phi]},\label{neweff}\end{equation} where we have introduced the CTP
complex temporal path ${\cal C}$, going from minus to plus infinity $\cal C_+$
and backwards $\cal C_-$, with a decreasing (infinitesimal) imaginary part.
Time integration over the contour ${\cal C}$ is defined by $\int_{{\cal C}} dt
=\int _{{\cal C_+}} dt -\int_{{\cal C_-}} dt$. The field $\phi$ appearing in
Eq.(\ref{neweff}) is related to those in Eq.(\ref{ctpeff}) by $\phi(t,\vec x) =
\phi_{\pm}(t,\vec x)$ if $t \in {\cal C}_{\pm}$. The same applies to
the mirror's position $q$.
The equation above is useful because it has the structure of the
usual in-out or the Euclidean effective action. Feynman rules are
therefore the ordinary ones, replacing Euclidean propagator by
\cite{ctprules}
\begin{eqnarray} G(x,y) = \left\{\begin{array}{ll}
G_F(x,y)=i \langle 0, in\vert T \phi (x) \phi(y)\vert 0, in\rangle,& ~t, t' ~
\mbox{both on} ~{\cal C}_+ \\ G_D(x,y)=-i \langle 0, in\vert {\tilde T}\phi (x)
\phi(y)\vert 0, in\rangle , & ~t, t' ~ \mbox{both on} ~{\cal C}_-\\ G_+(x,y)=-
i \langle 0, in\vert \phi (x) \phi(y)\vert 0, in\rangle, &~t ~\mbox{on}~
{\cal C}_-, t' ~\mbox{on} ~{\cal C}_+\\ G_-(x,y)=i \langle 0, in\vert \phi (y)
\phi(x)\vert 0, in\rangle, & ~ t ~\mbox{on} ~ {\cal C}_+, t'~ \mbox{on}~
{\cal C}_-\end{array}\right. \label{prop}
\end{eqnarray}
Explicitly
\begin{equation} G_F(x,y)= \int \frac{d^{d+1}p}{(2 \pi)^{d+1}}\frac {e^{ip(x-
y)}}{p^2+m^2-i\epsilon}= G_D^*(x,y),
\end{equation}
\begin{equation}
G_{\pm}(x,y)=\mp \int \frac{d^{d+1}p}{(2 \pi)^{d+1}} e^{ip(x-y)}2
\pi i
\delta(p^2-m^2)\theta(\pm p^0).
\label{gpm}
\end{equation}
The considerations above will allow us to compute the CTPEA using
the Euclidean results of the previous sections. To do this, we
will rewrite the Euclidean effective action given in Eq.(\ref{eea})
using a spectral decomposition for the form factor $F_s(\tau -
\tau ')$. For definiteness we will consider the concrete example
of a dispersive mirror (finite $\lambda$) coupled to a massless
scalar field.
The Euclidean effective action can be
rewritten as
$$\Gamma_2[q(\tau )]= -\frac{\lambda^2}{4\pi^3}\int d\tau\int d\tau'q(\tau )
q(\tau')\int_0^{\infty}dz \left(1 -
f(2z/\lambda)\right)\frac{d^2}{d\tau^2}G_{\rm E}(\tau -
\tau',z^2),$$ where $G_E(\tau,z^2)$ is 0+1 Euclidean propagator
with mass $z^2$ and
\begin{equation}
f(z) = \frac{\arctan z}{z}+ \frac{1}{2}\ln (1 + z^2)
\end{equation}
(see Appendix B for details).
The CTPEA can be obtained from the Euclidean one
considering the contour ${\cal C}$ and replacing $G_E(z,t)$
according to the rules given in Eq.(\ref{prop}). The result is
\begin{eqnarray}
\Gamma_{\rm CTP} &=& \frac{\lambda^2}{4\pi^3}\int dt\int dt' q^{\rm a}(t)
q^{\rm b}(t')\int_0^{\infty}dz(1 - f(2z/\lambda)) \psi_{\rm
ab}\nonumber \\
&=&\frac{\lambda^2}{4\pi^3}\int_0^{\infty}dz(1 -
f(2z/\lambda))\left[\int dt\int dt'q^+(t) q^+(t')
\psi_{++}
-\int dt\int dt'q^+(t) q^-(t') \psi_{+-}\right.\nonumber \\
&+&\int dt\int dt'q^-(t) q^+(t') \psi_{-+}
-\left.\int dt\int dt'q^-(t) q^-(t') \psi_{--}\right],
\end{eqnarray} where the CTP propagators are
\begin{eqnarray} \psi_{\pm\pm}(t,t')&=& \pm
\frac{d^2}{dt^2}\int\frac{d\omega}{2\pi}\frac{e^{i\omega(t -
t')}}{\omega^2 - z^2 \pm i\epsilon},\nonumber\\ \psi_{\pm\mp}(t,t')&=&
\mp \frac{d^2}{dt^2}\int\frac{d\omega}{2\pi}e^{i\omega(t - t')}
2\pi i \delta (\omega^2 - z^2)\theta (\pm \omega),
\end{eqnarray}
and satisfy the identities
\begin{eqnarray}
\psi_{++} &=&
\psi_{+-} \theta (t - t') - \psi_{-+}\theta
(t'-t),\nonumber\\
\psi_{--} &=& \psi_{-+} \theta (t - t') - \psi_{+-}\theta (t'-t).
\end{eqnarray}
Introducing the new variables $\Sigma = (q^++q^-)/2$ and $\Delta =
(q^+-q^-)/2 $, the CTP effective action can be written as
\begin{equation}\Gamma_{\rm CTP} = \int dt \int dt' \left[ \ddot{\Sigma}(t) \Sigma(t')
D(t'-t) - i \ddot{\Delta}(t) \Delta (t') N(t-t')\right],
\label{ctpeafin}
\end{equation}
where the dissipation $(D)$ and noise $(N)$ kernels are given by
\begin{eqnarray}D(t-t')&=& \frac{2\lambda^2}{\pi^2}\int_0^{\infty}dz~\left(1 -
f(2z/\lambda)\right) ~{\rm Re}\psi_{++}(t,t') ~\theta (t'-t),\nonumber\\
N(t-t')&=& \frac{\lambda^2}{\pi^2}\int_0^{\infty}dz~ \left(1 -
f(2z/\lambda)\right) ~{\rm Im}\psi_{++}(t,t'). \label{kernels}
\end{eqnarray}
Performing the integral in the spectral parameter $z$ we find
\begin{equation}
D(r) = \frac{\lambda^2}{2\pi}\left[(\frac{r \lambda}{2}+1) {\rm
ChI}(\frac{r\lambda}{2}) + \cos(\frac{r\lambda}{2})-
\sinh(\frac{r\lambda}{2}) -(\frac{r \lambda}{2}+1) {\rm
ShI}(\frac{r\lambda}{2})\right],
\end{equation}
and \begin{eqnarray} N(r) &=& \frac{-\lambda^2}
{48\pi^2}\left[-\pi^2 -24\gamma \ln (\frac{2}{\lambda}) + 12
\ln^2(2) - 12 \ln^2(\lambda) - 24
\ln(\frac{2}{\lambda})\left(\ln(r\lambda) - {\rm
ChI}(\frac{r\lambda}{2})\right)\right. \nonumber \\
&+& \left. 24 (\ln(\frac{2}{\lambda})-1) \left(\cosh
(\frac{r\lambda}{2}) - \frac{r\lambda}{2}{\rm
ShI}(\frac{r\lambda}{2})\right) \right]\end{eqnarray} where $r =
\vert t - t'\vert $, and ${\rm ChI}$, ${\rm ShI}$ are the
hyperbolic CosIntegral and SinIntegral, respectively.
From Eq.(\ref{ctpeafin}) we can also see that the dissipative force is given
by
\begin{equation}
F_{\rm diss}=\int_{-\infty}^t dt' \ddot q(t') D(t-t').
\end{equation}
Taking into account that dissipation kernel has the form $D(t-t')=
\lambda^2 g(\lambda (t-t'))$, we can rewrite the force as
\begin{equation}
F_{\rm diss}=\lambda \int_{0}^\infty dx \, D(x) \ddot
q(t-\frac{x}{\lambda}) \simeq \lambda \int_{0}^\infty dx \, D(x)
\left [\ddot q(t) -\frac{x}{\lambda}q^{(3)}(t)+....\right].
\end{equation}
A numerical evaluation of the remaining integrals gives the
correct perfect conductor limit: the term proportional to
$\lambda$ renormalizes the mass of the mirror, and the
$\lambda$-independent term gives a dissipation force proportional
to the third derivative of the mirror's position.
In order to derive the semiclassical Langevin equation that describes the
motion of the mirror, one can regard the imaginary part of $\Gamma_{\rm CTP}$
as coming from a noise source $\eta(t)$ with Gaussian functional
probability distribution given by
\begin{equation}
P[\eta (t)] = N_\eta \exp\left\{-\frac{1}{2}\int dt\int dt' \left[\eta (t) {\ddot N}^{-1}(t - t')
\eta (t')\right]\right\},
\end{equation}
where $N_\eta$ is a normalization factor. Indeed, we can write the
imaginary part of the CTP-effective action as a functional
integral over the Gaussian field $\eta(t)$
\begin{equation}
\int {\cal D}\eta (t) P[\eta ] ~e^{-i \Delta (t) \eta (t)} = e^{-i \int dt \int dt' \Delta (t)
{\ddot N}(t - t') \Delta (t')}.
\end{equation}
Therefore, the CTPEA can be rewritten as an average over the noise
of
\begin{equation}\Gamma_{\rm CTP}^{(\eta)} = \int dt \int dt' \left[ \ddot{\Sigma}(t) \Sigma(t')
D(t'-t)\right] - \int dt \Delta (t) \eta (t). \label{ctpeafin2}
\end{equation}
Thus, the associated Langevin equation comes from the variation
$\frac{\delta \Gamma_{\rm CPT}^{(\eta)}} {\delta
q_+}\vert_{q_+=q_-}=0$, obtaining
\begin{equation}
M\left( {\ddot q}(t) + \Omega^2 q^2(t)\right) + 2 \int dt' {\ddot
D}(t - t') q(t') = \eta (t), \label{langevineq}
\end{equation}
where $M$ is the mass of the mirror and the two-point correlation
function of the noise is given by
\begin{equation}
\langle \eta (t) \eta (t')\rangle = {\ddot N}(t - t').
\end{equation}
The Langevin equation describes the motion of the system (the
mirror) taking into account the main effects of the environment (the
quantum field): a dissipative force and a stochastic noise.
\subsection{Mirror's decoherence}
In the quantum open system approach that we have adopted here,
the imaginary part of the CTP-effective action (noise term) is
directly associated with the decoherence process of the mirror.
In fact, one can establish a direct link between the total number
of created particles and the decoherence functional
for a given classical (macroscopic) trajectory of the
mirror.
Decoherence means physically that the different coarse-graining
histories making up the full quantum evolution acquire individual
reality, and may therefore be assigned definite probabilities in
the classical sense. For our particular application, we wish to
consider as a single coarse-grained history all those fine-grained
ones where the trajectory $q(t)$ remains close to a prescribed
classical configuration $q_{\rm cl}$.
In principle, we can examine adjacent general classical solutions
for their consistency but, in practice, it is simplest to restrict
ourselves to particular solutions $q^{\pm}_{\rm cl}$, according to
the nature of the decoherence that we are studying. Therefore, we
evaluate the decoherence functional \cite{DF} for classical
trajectories such that the amplitude of one trajectory is $q^- =
q^+ - 2 \delta$, where $\delta$ is a small (constant) amplitude
difference. Thus, neglecting the dissipation we can write
$\Delta_{\rm cl}(t) = \delta \cos(\Omega t)$, and the decoherence
functional is formally given by
\begin{equation}
\vert{\cal D}(q_{\rm cl}^+,q_{\rm cl}^-)\vert =e^{-{\rm
Im}\Gamma_{\rm CTP}}=e^{-\delta^2 \Omega^2 \int dt \int dt' ~
\cos(\Omega t)N(t -t') \cos(\Omega t')}.
\end{equation}
Making the integration in the particular case $m=0$ and
$\lambda=\infty$, one can show that the decoherence time scales as
\begin{equation}
t_D \sim \frac{1}{\delta^2\omega^3}.
\end{equation}
This result is valid as long as the decoherence time is much shorter
than the dissipative time $t_{diss}$, which can be easily estimated
from Eq.(\ref{langevineq}) as $t_{\rm diss}\sim M/\Omega^2$. The
condition $t_D\ll t_{\rm diss}$ is satisfied as long as $\delta\gg
\sqrt{M\Omega}$, i.e the minimum uncertainty in the position of the
mirror. An alternative estimation based on the Fokker-Planck
equation for the Wigner function of the mirror gives the same order
of magnitude for the decoherence time $t_D$ \cite{paulodiego}.
\section{Final remarks}
In this paper we have extended, in several directions, the functional
approach to the dynamical Casimir effect introduced some years ago by
Golestanian and Kardar~\cite{GK} to consider different situations.
The main point in this approach, namely, the introduction of auxiliary
fields in the functional integral to impose the boundary conditions on the
quantum fields is retained, altough now they have an extra piece in the
action, to cope with the dispersive nature of the mirror.
After integration of the original quantum fields, the problem is
again reduced to the computation of a path integral over the
auxiliary fields. This is a kind of (non local) dimensionally reduced
theory, since the auxiliary field live on the boundary.
Firstly, we considered non-perfectly conducting
mirrors, by introducing a $\delta$-like potentials for the quantum
fields. As shown in Ref.\cite{barton1} for the scalar field, these
potentials serve as toy models to describe the interaction of the
electromagnetic field with a thin plasma sheet, and give rise to reflection
and transmission coefficients with a particular frequency dependence.
We believe that this generalization will be useful as a first step
towards solving more realistic situations. Indeed, our formalism can be
extended to include arbitrary reflection and transmission coefficients by
considering non local extensions of the singular potentials
considered here. We will describe this results in a forthcoming
publication.
We also considered a scalar field coupled to a relativistic mirror,
and also calculated the effective action for a mirror interacting with a
Dirac spinor, understanding here by mirror an object that reflects the
fermionic current.
Finally, we also extended the formulation to calculate the CTP effective
action. As an important by-product, we applied this effective action to
compute the semiclassical Langevin equation that describes the dynamics of
the mirror interacting with the vacuum fluctuations of the quantum fields,
and with the motion induced radiation produced by its accelerated motion.
\section*{Appendix A: Massive case}
We outline here the calculation of the Euclidean effective action
in the massive case, for perfectly conducting mirrors in $1+1$
dimensions.
In the definition of ${\tilde F}(\omega)$,
\begin{equation}
\tilde F(\omega)=\frac{1}{2}\int_ 0^\infty
\frac{d\nu}{2\pi}\sqrt{\nu^2+m^2}\sqrt{(\nu+\omega)^2+m^2} \;,
\end{equation}
we insert the representation
\begin{equation}
(\nu^2+m^2)^{\epsilon}=\frac{1}{\Gamma(-\epsilon)}\int_0^{\infty}\frac{d\beta}{\beta}
\beta^{-\epsilon}e^{-\beta(\nu^2+m^2)},
\end{equation}
and one with a shifted argument for the second factor, to obtain
the function
\begin{equation}
\tilde F_\epsilon(\omega)=
\frac{1}{2[\Gamma(-\epsilon)]^2}\int_0^\infty\frac{d\alpha_1}{\alpha_1}
\int_0^\infty\frac{d\alpha_2}{\alpha_2}\alpha_1^{-\epsilon}\alpha_2^{-\epsilon}
\int_ 0^\infty
\frac{d\nu}{2\pi}\exp\big[-\alpha_1(\nu^2+m^2)-\alpha_2((\nu+w)^2+m^2)\big]\;.
\end{equation}
The role of this representation is, to make it possible to
integrate over the frequency; in a way, it allows for the
introduction of Feynman-like parameters in the case of propagators
which have a non-standard form. Indeed, introducing the identity
\begin{equation}
1=\int_0^\infty d\rho\, \delta(\rho-\alpha_1-\alpha_2)\,,
\end{equation}
and after a rescaling of the $\alpha$'s plus some straightforward
calculations we see that
\begin{equation}
\tilde F_\epsilon(\omega)=\frac{\Gamma(-2\epsilon-1/2)}{4\sqrt{\pi}[\Gamma(-\epsilon)]^2}
\int_0^1d\alpha [\alpha(1-\alpha)]^{-1-\epsilon}[\alpha(1-\alpha)\omega^2+m^2]^{2\epsilon+1/2}
\end{equation}
The final result (\ref{Fmperf}) is obtained by subtracting $\tilde
F_\epsilon(0)$ and then taking the limit $\epsilon =1/2$. Note
that the previously used representation is not used as a
regularization, but as a device to do the integral. Indeed, one
could have worked with $\epsilon=1/2$ throughout in the subtracted
integral; no analytic extension to complex values of $\epsilon$
would be required.
The result for the massless case in $d+1$ dimensions,
(\ref{Fperfd}), can be derived using a similar procedure, but now
introducing a dimensional regularization for the momentum
integral. Now a regularization {\em is\/} required, since the
integral over the parallel momenta is divergent when $\epsilon =
1/2$.
\section*{Appendix B: Spectral decomposition}
In this Appendix we derive the spectral decomposition which allows us
to write the form factor $F(\tau - \tau')$ in terms of the Euclidean
propagator. Using the identities
\begin{equation}\vert\omega\vert =
\frac{2\omega^2}{\pi}\int_0^{+\infty}dz \frac{1}{\omega^2 +
z^2},\end{equation}
\begin{equation}\left(1 + \frac{2}{\lambda}\vert\omega\vert\right)\ln\left(1 +
\frac{2}{\lambda}\vert\omega\vert\right) =
\omega^2\frac{8}{\lambda^2\pi}\int_0^{+\infty}\frac{f(z)}{z^2 +
\frac{4\omega^2}{\lambda^2}}\,\, , \label{ident}
\end{equation}
where
\begin{equation}
f(z) = \frac{\arctan z}{z}+ \frac{1}{2}\ln (1 + z^2)\,\, ,
\end{equation}
we can write
\begin{equation}
F(\tau - \tau') = -\frac{d^2}{d\tau^2} G(\tau - \tau')
\end{equation}
with \begin{eqnarray} G(\tau) &=& \int
\frac{d\omega}{2\pi}e^{i\omega \tau} {\tilde G}(\omega
)\nonumber\\
{\tilde G}(\omega ) &=&
\frac{\lambda^2}{4\pi^3}\int_0^{+\infty}dz\left\{\frac{1}{\omega^2
+ z^2}\left[1 - f\left(\frac{2z}{\lambda}\right)\right]\right\}.
\end{eqnarray}
From these equations the form factor $G(\tau )$ can readily be
written in terms of the $0+1$ Euclidean propagator $G_{\rm
E}(\tau,z^2)$ with mass $z^2$
\begin{equation}
G(\tau )=\frac{\lambda^2}{4\pi^3}\int_0^{+\infty}dz\left[1 -
f\left(\frac{2z}{\lambda}\right)\right] G_{\rm E}(\tau, z^2)
\end{equation} where
\begin{equation}G_{\rm E}(\tau, z^2) = \int \frac{d\omega}{2\pi}
\frac{e^{i\omega\tau}}{\omega^2+z^2}.
\end{equation}
\acknowledgments
C.D.F. thanks CONICET and ANPCyT for financial support and to the FCEyN
(UBA) by the hospitality of its members.
F.D.M. acknowledges the warm hospitality of Centro At\'omico Bariloche,
where part of this work was done. The work of F.D.M. and F.C.L was
supported by UBA, CONICET and ANPCyT.
|
1,108,101,566,804 | arxiv | \section{Introduction}
As discussed in \citet{merz2010paid} the main task of reserving actuaries is to predict ultimate loss ratios and outstanding loss liabilities. In general such predictions are based on past information that comes from a variety of sources. Under a credibility based framework, the weighting of different data sources and their relative contribution to the estimated reserve is difficult to determine. Therefore, it is important to consider developing a unified prediction framework for the outstanding loss liabilities, known as the paid-incurred-claims (PIC) class of models. However, to date only simple dependence structures have been considered, with three parameters for the correlations which were not incorporated into the formal Bayesian estimation approach, and instead fixed deterministically \textit{a priori}. There are two technical difficulties in extending the current restrictive assumptions within a Bayesian framework. The first is being able to generate the positive definite matrices; the second is evaluating the joint likelihood of the mixture copula defined over the observed payments and incurred losses in each accident year row of the reserving matrix. Our article significantly extends the dependence structure of current PIC models by solving these two problems. The first problem is resolved through utilisation of a class of matrix-variate Inverse-Wishart priors coupled with an adaptive Markov chain sampler that restricts the proposed Markov chain states to remain on the manifold of such matrices. The second problem is solved by using a data augmentation strategy which treats the unobserved parts of the loss triangle as missing data so that one can perform evaluation of the copula based likelihood required for inference on the model parameters.
In order to ensure the financial security of an insurance company, it is important to predict future claims liabilities and obtain the corresponding prediction intervals which take into account parameter uncertainty. The PIC model is a claims reserving method which statistically combines information about claims payments and incurred losses. It allows actuaries to best utilise the available information for loss reserves. The Munich chain ladder method introduced by \citet{quarg2004munich} is one of the first claims reserving approaches in the actuarial literature to unify outstanding loss liability prediction based on both sources of information. This method aims to reduce the gap between the two chain ladder predictions that are based on claims payments and incurred losses data, respectively. It is achieved by adjusting the chain ladder factors with paid-incurred ratios to reduce the gap between the two predictions. The main drawback with the Munich chain ladder method is that it involves several parameter estimates whose precisions are difficult to quantify within a stochastic model framework.
\citet{merz2010paid} recently introduced a log-normal PIC chain model and used Bayesian methods to estimate the missing (future) part of the claims reserving triangles based on both payment and loss incurred information. Its major advantage is that the full predictive distribution of the outstanding loss liabilities can be quantified. One important limitation of the model of \citet{merz2010paid} is that it does not develop the dependence properties of the PIC model that will be applicable to loss reserving data observed in practice. Our article extends the proposed Bayesian PIC models of \citet{merz2010paid} to capture additional dependence structures.
\subsection{Brief Background}
Dependence within payment data, within incurred loss data, and even between payment and incurred loss data commonly exists due to the nature of the loss process. Payment and incurred loss ratios in the previous development period are likely to impact that of the next development period. Hence, correlation between development periods is practically appealing in claims reserving practice. Moreover, incurred loss is essentially payment data plus case estimates which are projections foreseen by case managers to estimate the remaining payments. Correlation between payment and incurred loss data is also found. \citet{happ2011paid} propose a fixed covariance structure to describe the correlation between payment and incurred loss, assuming that the correlations between different development periods are identical. In reality, correlations differ across development periods for various reasons, such as different stages of the life cycle for a claim and internal policy changes. In order to fully incorporate the actual correlations, we introduce a block covariance structure to allow for the variation between different development periods within payment and incurred losses. We also develop a second class of hierarchical mixture of copulas models.
The estimation challenge involves constructing and sampling from the resulting Bayesian models for PIC with flexible dependence structures. To specify the model, we vectorize the triangular random structures for payments and incurred loss and, applying appropriate permutations, we then assume a copula dependence structure on the vectorized data. We use a Gaussian copula with an unknown correlation matrix, which is restricted to be block diagonal for parsimony, or a mixture Archimedian copulas across development periods.
We estimate the Bayesian models by MCMC methods, using data augmentation to generate missing data values in the loss triangle and use an adaptive Metropolis algorithm to generate the unknown parameters. Bayesian simulation methodology is used to carry out inference on all aspects of the models considered and to obtain predictive distributions for reserves.
\subsection{Contributions}
We design a novel class of PIC models and illustrate it with two examples. The first involves a mixture of Clayton and Gumbel copulas for upper and lower tail dependence features in the development years for payments and incurred losses. The second example involves a Gaussian copula model in which the covariance structure is a telescoping block diagonal form representation which captures dependence between development lag years in the payments and incurred losses. By a telescoping block diagonal matrix we mean one in which the main diagonal is comprised of sub-blocks for which each incremental sub-block contains one less row and column compared to the previous. In constructing these models we consider hierarchical Bayesian models with hyperparameters on the priors for development factors and specially developed matrix-variate priors on the covariance structures which preserves the conjugacy properties of the independence models developed in \citet{merz2010paid} and \citet{merz2010estimation}.
For the independent and Gaussian copula based PIC models we develop a class of conjugate posterior models that can be efficiently estimated via an MCMC sampler known as a block Gibbs sampler. However, the extension to general copula dependence structures requires non-conjugate priors, making it necessary to develop adaptive MCMC algorithms. Adaptive sampling uses previous iterates of the Markov chain to form more efficient Metropolis proposals for the parameters, this class of MCMC algorithm has received growing attention in the statistics literature since it was recently developed and is now recognized as an important tool for Bayesian inference. There is an increasing interest in utilizing adaptive MCMC to facilitate more efficient sampling (\citet{andrieu2008tutorial}, \citet{atchade2005adaptive}). The adaptive techniques that we adopt in this paper fall within the general framework of adaptive Metropolis, and employ the optimal scale factors (\citet{roberts2009examples}) from the Single Component Adaptive Metropolis (SCAM) algorithm (\citet{haario2005componentwise}). There have been some initial utilisations of adaptive MCMC specifically in financial modelling such as \cite{peters2011calibration} and the references therein. In addition the adaption strategies we consider in this paper involve extensions of Euclidean space Adaptive Metropolis to the space of positive definite matrices, creating a class of matrix variate Markov chain adaptive proposals.
In the mixture copula based PIC models, we design data augmentation strategies which are a class of auxiliary variable methods. We modify these approaches to the PIC copula based models in order to circumvent the challenge of intractable likelihood evaluations which arise form the structure of the PIC reserving triangle. In particular we argue that the tail dependence features of the model should be consistent accross all development years for both payment and incurred loss data. This poses an evaluation challenge for the likelihood as it involves evaluation of marginal likelihood quantities given the observed data in accident year $i$, given by payment and incurred losses. The integral required when utilising mixture copula structures over the accident years is intractable, therefore we introduce auxiliary variables into the Bayesian model in a data-augmentation structure to overcome this dificulty.
\section{Review of the Merz-Wuethrich Independence Copula Paid-Incurred Claims Model}
This section introduces the PIC model which involves two sources of information. The first is the claims payment data, which involves payments made for reported claims. The second source of data incorporated into the statistical estimation are the incurred losses corresponding to the reported claim amounts. The differences between the incurred losses and the claim payments are known as the case estimates for the reported claims which should be equal once a claim is settled. This imposes a set of constraints on any statistical model developed to incorporate each of these sources of data into the parameter estimation. We use the constraints proposed in \citet{merz2010paid} which are used to specify a model based on a claims triangle constructed from vertical columns corresponding to development years of claims and rows corresponding to accident years. This structure for the observed data is summarized in triangular form which is utlised for both the claims payments and the incurred losses, as presented in Figure \ref{PICTriangle}.
Without loss of generality, we assume an equivalent number $J$ of accident years and development years. Furthermore, we assume that all claims are settled after the $J$-th development year. Let $P_{i,j}$ be the cumulative claims payments in accident year $i$ after $j$ development periods and $I_{i,j}$ the corresponding incurred losses. Moreover, for the ultimate loss we assume the constraint discussed on the case estimates corresponds to the observation that predicted claims should satisfy $P_{i,J}=I_{i,J}$ with probability 1, which means that ultimately (at time $J$) the claims reach the same value and therefore satisfy the required constraint.
We define (i) $P_{0:J,0:j} = \left\{P_{k,l}:0 \leq k \leq J, 0 \leq l \leq j\right\}$. (ii) Let $A$ and $B$ be square matrices. Then $diag(A,B)$ is the diagonal matrix, with the diagonal elements of A appearing topmost, then the diagonal elements of B. Let the matrices $A$ and $B$ be as in (ii). Then the direct sum of $A$ and $B$, written as $A \oplus B$ is the block diagonal matrix with $A$ in the top left corner and $B$ in the bottom right corner. It is clear that the definitions in (ii) and (iii) can be iterated. That is $diag(A,B,C) = diag(diag(A,B),C)$ and $A \oplus B \oplus C = (A \oplus B)\oplus C$. (iv) Define the $d \times d$ diagonal square identity matrix according to $\mathbb{I}_d$. (v) Define the indicator of an event by the dirac-delta function $\delta_{i}$. (vi) Define the vectorization operator on a $p \times n$ matrix $A$, denoted by $Vec(A)$, as the stacking of the columns to create a vector.
\begin{figure}[ht]
\includegraphics[height=6cm, width=\textwidth]{figures/Figure1PICTable.eps} \par \noindent
\caption{Claims triangle for payment data and incurred data, including constraint on zero case estimates at development period $J$ (source \citet{merz2010paid}).}
\label{PICTriangle}
\end{figure}
As in \citet{merz2010paid}, we consider a Log-Normal PIC model as this facilitates comparison between existing results and the results we derive based on different dependence frameworks in extensions to this model.
We now introduce the PIC model and the statistical assumptions for the independent case, followed by remarks on the resulting marginal posterior models for the payment and incurred losses.
\begin{model ass}[Independent PIC Log-Normal (\textsl{Model I})]\label{model ass}
The model assumptions for the independent model of \citet{merz2010paid} are:
\begin{itemize}
\item{The cumulative payments $P_{i,j}$ are given by the forward recursion
\begin{equation*}
P_{i,0} = \exp\left(\xi_{i,0}\right) \;\; \mathrm{and} \;\; \frac{P_{i,j}}{P_{i,j-1}} = \exp\left(\xi_{i,j}\right)\;\; \mathrm{for} \; j=1,\ldots,J
\end{equation*}}
\item{The incurred losses $I_{i,j}$ are given by the backward recursion
\begin{equation*}
I_{i,J} = P_{i,J} \;\; \mathrm{and} \;\; \frac{I_{i,j-1}}{I_{i,j}} = \exp\left(-\zeta_{i,j-1}\right).
\end{equation*}}
\item{The random vector
$(\xi_{0,0}, \ldots, \xi_{J,J}, \zeta_{0,0},\ldots, \zeta_{J,J-1})$
has independent components with
\begin{eqnarray*}
\xi_{i,j} & \sim & N \left( \Phi_j~,~ \sigma_j^{2} \right)
\qquad \text{ for $i\in \{0,\ldots, J\}$ and
$j\in \{0,\ldots, J\}$},\\
\zeta_{k,l} & \sim & N \left(\Psi_l~,~ \tau_l^{2} \right)
\qquad \text{ for $k\in \{0,\ldots, J\}$ and
$l\in \{0,\ldots, J-1\}$};
\end{eqnarray*}}
\item{ The parameter vector for the model is $\mathbf{\Theta}=\left(
\Phi_0,\ldots, \Phi_{J},\Psi_0,\ldots, \Psi_{J-1},
\sigma_0,\ldots, \sigma_{J},\tau_0,\ldots, \tau_{J-1}\right)$. It is assumed that the components of $\mathbf{\Theta}$ are independent apriori. The prior density for $\mathbf{\Theta}$ has independent components, with
$\sigma_j,\tau_j$ both positive for all $j$.
\item It follows that
\begin{equation}
\log\left(\frac{P_{i,j}}{P_{i,j-1}}\right) \sim N \left( \Phi_j~,~ \sigma_j^{2} \right) \;\; \mathrm{and} \;\; \log\left(\frac{I_{i,j}}{I_{i,j+1}}\right) \sim N \left(-\Psi_l~,~ \tau_l^{2} \right)
\end{equation}}
\end{itemize}
\end{model ass}
Let $\{\mathbf{P},\mathbf{I}\}=\{P_{i,j},I_{k,l};~ 0\le i,j,k, \le J,~0\le l \le J-1 \}$. Then, based on Model Assumptions 2.1 and the observed matrices $P$ and $I$, the likelihood for $\bm{\Theta}$ is given by three components, see derivation in \citet[Section 3.3, Equation 3.5]{merz2010paid}. The first and third components correspond to the payment and incurred data and the second component corresponds to the imposition of the restriction that ultimate claims for payments $P_{i,J}$ match incurred $I_{i,J}$ for all accident years, giving:
\begin{eqnarray} \label{LH_IndependentPIC}
f\left(
\left.\mathbf{P},\mathbf{I}\right|\boldsymbol{\Theta}\right)&=&
\underbrace{\prod^{J}_{j=0}\prod^{J-j}_{i=0}
\frac{1}{\sqrt{2\pi}\sigma_{j}P_{i,j}}~
\exp \left\{ -\frac{1}{2\sigma_j^{2}}(\Phi_{j}-\log(\frac{P_{i,j}}{P_{i,j-1}}))^2
\right\} \nonumber}_{\text{Component1: payment}}
\\&&\times ~
\underbrace{\prod^{J}_{i=1}
\frac{1}{\sqrt{2\pi(\upsilon_{J-i}^{2}-\omega_{J-i}^{2})}I_{i,J-i}}~
\exp \left\{ -\frac{1}{2(\upsilon_{J-i}^{2}-\omega_{J-i}^{2})}(\mu_{J-i}-\eta_{J-i}-\log(\frac{P_{i,J-i}}{P_{i,J-i}}))^2
\right\} \nonumber}_{\text{Component2: Discounted final development year restricted payment and incurred}}
\\&&\times ~
\underbrace{\prod^{J-1}_{j=0}\prod^{J-j-1}_{i=0}
\frac{1}{\sqrt{2\pi}\tau_{j}I_{i,j}}~
\exp \left\{ -\frac{1}{2\tau_j^{2}}(-\Psi_{j}+\log(\frac{I_{i,j}}{I_{i,j+1}}))^2
\right\}.}_{\text{Component3: incurred}}
\end{eqnarray}
where $\upsilon_{j}^{2} = \sum_{m=0}^{J}\sigma_{m}^{2}+\sum_{n=j}^{J-1}\tau_{n}^{2};$ \hspace{4mm}
$\omega_{j}^{2} = \sum_{m=0}^{j}\sigma_{m}^{2};$ \hspace{4mm} $\eta_{j} = \sum_{m=0}^{j}\Phi_{m};
$ \hspace{1mm} and $\mu_{j} = \sum_{m=0}^{J}\Phi_{m}-\Sigma_{n=j}^{J-1}\varphi_{n}.$
As noted in \citet{merz2010paid}, there are several consequences of the model assumptions made regarding the restriction $I_{i,J} = P_{i,J}$ which applies for all $i \in \left\{1,2\ldots,J\right\}$. The first is that this condition is sufficient to guarantee that the ultimate loss will coincide for both claims payments and incurred loss data. The second is that this model assumes that there is no tail development factor beyond the ultimate year $J$. However this could be incorporated into such models, see \citet{merz2010estimation}.
\citet{merz2010paid} discuss the relationship between the proposed Independent Log-Normal PIC model and existing models in the literature for payment loss based reserving and incurred loss based reserving. In particular, \citet{merz2010paid} [Section 2.1 and 2.2] show that the resulting cumulative payments $P_{i,j}$, conditional on model parameters $\boldsymbol{\Theta}$, will satisfy the model proposed in \citet{hertig85} and the incurred losses $I_{i,j}$, conditional on model parameters $\boldsymbol{\Theta}$, will satisfy the model proposed in \citet{gogol1993using}. Lemma \ref{LemmaHertigGogol} summarizes their results.
\begin{lemma} \label{LemmaHertigGogol} The relationships between consecutive payment development year losses in a given accident year is given conditionally according to
\begin{equation}
\left[\left.\log\left(\frac{P_{i,j}}{P_{i,j-1}}\right)\right|P_{0:J,0:j-1},\boldsymbol{\Theta}\right] \sim \mathcal{N}\left(\Phi_j,\sigma_j^2\right), \; \forall j \geq 0
\end{equation}
in agreement with Hertig's model. With conditional moments given according to the Chain Ladder property as in \citet[Lemma 5.2]{merz2010paid} by,
\begin{equation}
\mathbb{E}\left[P_{i,j}|P_{0:J,0:j-1},\boldsymbol{\Theta}\right] = P_{i,j-1}\exp\left(\Phi_j + \sigma_j^2/2\right).
\end{equation}
Furthermore, conditional upon the model parameters $\boldsymbol{\Theta}$, for all $0 \leq j < j+l \leq J$ the relationships between consecutive incurred losses in a given accident year are given in \citet{merz2010paid} [Proposition 2.2] according to
\begin{equation}
\left[\left.\log\left(I_{i,j+l}\right)\right|I_{0:J,0:j-1},I_{i,J},\boldsymbol{\Theta}\right] \sim \mathcal{N}\left( \mu_{j+1} + \frac{\nu_{j+1}^2}{\nu_j^2}\left(\log(I_{i,j}) - \mu_j \right), \nu_{j+1}^2(1 - \nu_{j+1}^2/\nu_j^2) \right),
\end{equation}
These results are consistent with the model assumptions of Gogol, and are derived using properties of multivariate normal distribution, see Lemma 2.1 in \citet{merz2010paid}.
\end{lemma}
Furthermore, for all accident years $i \in \left\{1,2,\ldots,J\right\}$, the resulting conditional expected ultimate payment loss equals the expected ultimate incurred loss, given the model parameters $\boldsymbol{\Theta}$, and is expanded in terms of the model parameters according to Equation (\ref{EqnCondExpectUltimateLoss}), which are given by \citet[Equation 1.1]{merz2010paid} as,
\begin{equation} \label{EqnCondExpectUltimateLoss}
E\left.\left[ P_{i,J} \right| \mathbf{\Theta} \right]
=E\left.\left[ I_{i,J} \right| \mathbf{\Theta} \right]
=\exp\left( \sum_{m=0}^J \Phi_m + \sigma_m^2/2 \right).
\end{equation}
\section{Incorporating the Gaussian Copula into Paid-Incurred-Claims Models}
This section discusses an important aspect of extending the original Log-Normal PIC model of \citet{merz2010paid}. In particular, when this model was developed in the independent setting it was observed by the authors that the assumption of conditional independence between $\xi_{i,j}$ and $\zeta_{k,l}$ for all $i,j,k,l \in \left\{1,2,\ldots,J\right\}$ was not necessarily consistent with observations. In particular, they note that \citet{quarg2004munich} discovered evidence for strong linear correlation between incurred and paid ratios. In Section 3.1 we explore in detail a different approach to incorporate dependence structures into the Log-Normal PIC model. Some aspects of the new approach have subsequently been proposed in the literature, while others are novel developments proposed in our article. We note that in developing the extended models, the convenient properties of conjugacy in the Bayesian framework, which aids estimation, is often lost. Hence, after presenting the models we develop efficient state of the art statistical estimation strategies based on adaptive MCMC.
\subsection{Dependence via Payment Loss Ratios and Incurred Loss Ratios (Model II)}
This section generalizes the model by \citet{happ2011paid}, which has a static covariance structure, see \citet[Figure 1.1]{happ2011paid}. We use a Bayesian approach, based on results in Lemma \ref{lemmaWishartDist} and Model Assumptions \ref{modelass2}, to estimate the extended models. We use properties of the matrix-variate Wishart and Inverse Wishart distributions to develop a Gaussian copula based statistical model. The relevant matrix-variate distributional assumptions and properties are provided in Lemma \ref{lemmaWishartDist} and Lemma \ref{lemmaInvWishartDist}.
\begin{model ass}[Dependent Payment-Incured Ratios: PIC Log-Normal (\textsl{Model II})]\label{modelass2}
The model assumptions for the Gaussian copula PIC Log-Normal model involve:
\begin{itemize}
\item The random matrix $\Sigma_i \in \mathbb{R}^{(2J+1)\times(2J+1)}$ representing the covariance structure for the random vector constructed from log payment ratios $\left(\xi_{i,j} = \log\left(\frac{P_{i,j}}{P_{i,j-1}}\right) \right)$ and log incurred loss ratios $\left(\zeta_{i,j} = \log\left(\frac{I_{i,j}}{I_{i,j+1}}\right) \right)$ in the $i$-th development year, denoted by $\Xi_i = \left(\xi_{i,0},\xi_{i,1},\zeta_{i,1},\xi_{i,2},\zeta_{i,2},\ldots,\xi_{i,J},\zeta_{i,J}\right)$, is assumed distributed according to an inverse Wishart distribution prior (see definition and properties in Lemma \ref{lemmaWishartDist} and Lemma \ref{lemmaInvWishartDist}),
\begin{equation}
\Sigma_i \sim \mathcal{IW}\left(\Lambda_i,k_i\right)
\end{equation}
where $\Lambda_i$ is a $\left((2J+1)\times(2J+1)\right)$ positive definite matrix and $k_i > 2J$.
\item Conditionally, given $\mathbf{\Theta}=\left(
\Phi_0,\ldots, \Phi_{J},\Psi_0,\ldots, \Psi_{J}\right)$ and the $(2J + 1) \times (2J + 1)$-dimensional covariance matrix $\Sigma$, we have:
\begin{itemize}
\item{The random matrix, constructed from log payment ratios $\left(\xi_{i,j} = \log\left(\frac{P_{i,j}}{P_{i,j-1}}\right) \right)$ and log incurred loss ratios $\left(\zeta_{i,j} = \log\left(\frac{I_{i,j}}{I_{i,j+1}}\right) \right)$, denoted by $\Xi$ and comprised of columns $\Xi_i = \left(\xi_{i,0},\xi_{i,1},\zeta_{i,1},\xi_{i,2},\zeta_{i,2},\ldots,\xi_{i,J},\zeta_{i,J}\right)$, is assumed distributed according to a matrix-variate Gaussian distribution $f_{\Xi}^{MVN}\left(\Xi|M,\Sigma,\Omega\right)$, see the definition and properties in Lemma \ref{lemmaMatrixGaussianDist}. The sufficient matrices are then the $\left((2J + 1) \times (J+1)\right)$ mean matrix $M = \left[\bm{\Theta}',\ldots,\bm{\Theta}'\right]$, column dependence given by $\left((2J + 1) \times (2J+1)\right)$ dimensional covariance matrix $\Sigma$ and row dependence given by $\left((J+1) \times (J+1)\right)$ dimensional matrix $\Omega$. If $\Omega = \mathbb{I}_{J+1}$, the covariance of the vectorization of $\widetilde{\Xi} = Vec(\Xi)$ is
\begin{align}
{\footnotesize{
\widetilde{\Sigma}=\mathbb{C}\text{ov}\left(\widetilde{\Xi}\right) = \bigoplus_{i=0}^J \Sigma_{i} = \left(
\begin{array}{cccc}
\Sigma_0 & \bm{0} & \ldots & \bm{0} \\
\bm{0} & \Sigma_1 & \ldots & \bm{0} \\
\vdots & \ddots & \ddots & \vdots \\
\bm{0} & \bm{0} & \bm{0} & \Sigma_J \end{array}
\right)}},
\end{align}
where it is assumed in the model in \citet{happ2011paid} that $\Sigma_{i} = \mathbb{C}ov(\Xi_i) = \Sigma$. However, this need not be the case and it is possible to consider two extensions, the first in which $\mathbb{C}ov(\Xi_i)$ varied as a function of $i \in \left\{0,1,\ldots,J\right\}$ and the second being the most general of these model structures given by the assumption
\begin{align}
{\footnotesize{
\mathbb{C}\text{ov}\left(\widetilde{\Xi}\right) = \Sigma \otimes \Omega.
}}
\end{align}
}
\item{For all accident years, $i \in \left\{0,1,\ldots,J\right\}$, the ultimate payment losses and incurred losses are equal a.s., $P_{i,J} = I_{i,J}.$}
\end{itemize}
\item{The matrix $\widetilde{\Sigma}$ is positive definite and the components of $\bm{\Theta}$ are independent with prior distributions
\begin{equation}
\Phi_i \sim \mathcal{N}\left(\phi_i,s_i^2\right) \; \text{ and } \; \Psi_j \sim \mathcal{N}\left(\psi_j,t^2_j\right),
\end{equation}
and hyper-prior distributions
\begin{equation}
s^2_i \sim \mathcal{IG}\left(\alpha_i,\beta_i\right) \; \text{ and } \; t^2_j \sim \mathcal{IG}\left(a_j,b_j\right),
\end{equation}
for all $i \in \left\{1,\ldots,J\right\}$ and $j \in \left\{0,\ldots,J\right\}$.
}
\end{itemize}
\end{model ass}
This model extends the model developed in \citet{happ2011paid} which assumes that $\Sigma$ is fixed and known with a tri-diagonal structure. The extension we consider generalizes the dependence structure to be unknown \textit{a priori} and given an inverse Wishart prior for matrix $\widetilde{\Sigma}$, so it forms part of the inference given the data, in the Bayesian inference. In addition, unlike in \citet{happ2011paid} where they assume $\Sigma = \Sigma_i, \forall i \in \left\{0,1,\ldots,J\right\}$, we also allow for variation in $\Sigma_i$ across development years.
Given these model assumptions, we now consider two consequences of the proposed model structures for the dependence between the log payment ratios and the log incurred loss ratios given in Lemma \ref{lemmaIndependentDevYr} and Lemma \ref{lemmaMultiVarGaussian}.
\begin{lemma} \label{lemmaIndependentDevYr} Conditional upon $\Lambda_i$ and $k_i$, for all i in $\left\{0,1,\ldots,J\right\}$, and given the marginal distributions for $\Sigma_i$ follow $\Sigma_i \sim \mathcal{IW}\left(\Lambda_i,k_i\right)$ with $\Lambda_i$ a $\left((2J+1)\times(2J+1)\right)$ positive definite matrix and $k_i > 2J$, the joint distribution for the $\left((2J^2 + 3J + 1) \times (2J^2 + 3J + 1)\right)$ covariance matrix $\widetilde{\Sigma}$ for the vectorized matrix for $\Xi$, given by $\widetilde{\Xi} = Vec(\Xi)$, under the assumption of independendence between development years,
\begin{align}
\widetilde{\Sigma}=\mathbb{C}\text{ov}\left(\widetilde{\Xi}\right) = \bigoplus_{i=0}^J \Sigma_{i} = (\Sigma_0 \oplus \cdots \oplus \Sigma_J),
\end{align}
results in a joint distribution given by:
\begin{equation}
\widetilde{\Sigma} \sim \mathcal{IW}\left(\widetilde{\Lambda},\widetilde{k}\right),
\end{equation}
with degrees of freedom $\widetilde{k} = \sum_{i=0}^J k_i > 2J^2 + 3J$ and scale matrix
\begin{equation}
\widetilde{\Lambda} = \bigoplus_{i=0}^J \Lambda_{i}.
\end{equation}
Furthermore, the joint prior mean and mode for the distribution of the random matrix $\widetilde{\Lambda}$ are
\begin{equation}
\begin{split}
\mathbb{E}\left[\widetilde{\Sigma}|\widetilde{\Lambda},\widetilde{k}\right] &= \frac{1}{\left(\sum_{i=0}^J k_i\right)-\left(2J^2 + 3J\right)}\widetilde{\Lambda}, \; \mathrm{and}\\
m\left(\widetilde{\Sigma}\right)&= \frac{1}{2J^2 + 3J + 1+\sum_{i=0}^J k_i}\widetilde{\Lambda}.
\end{split}
\end{equation}
\end{lemma}
{\textsl{The proof of this result is a consequence of the results in Lemma \ref{lemmaWishartDist}, the model assumptions and the properties of an inverse Wishart distributions; see \citet{gupta2000matrix}[Chapter 3].}}
{\begin{flushright}\vspace{-2mm}$\Box$\end{flushright}}
\begin{remark} We can demonstrate that under the proposed model assumptions the selection of the factorized covariance structure in Lemma \ref{lemmaIndependentDevYr} produces Bayesian conjugacy in the joint posterior of the model parameters given observed payment losses and incurred losses.
\end{remark}
\begin{remark} It is noted in \citet{happ2011paid} and Lemma \ref{lemmaIndependentDevYr} that in formulating the likelihood structure for this dependent model it is more convenient to work with the one-to-one (invertible) transformation for the observed data defined marginally for the $i$-th development year according to
\begin{equation}
\left[\bm{X}_i|\bm{\Theta}\right] = \left[B_i\Xi_i|\bm{\Theta}\right] \sim \mathcal{N}\left(B_iM_i,B_i\Sigma_iB_i'\right),
\end{equation}
where $M_i$ is the $i$-th column of matrix $M$ and $\bm{X}_i \in \mathbb{R}^{2J+1}$ defined by \\$\bm{X}_i = \left[\log I_{i,0},\log P_{i,0},\log I_{i,1},\log P_{i,1}, \ldots,\log I_{i,J-1},\log P_{i,J-1},\log I_{i,J}\right]$. This results in the joint matrix variate Normal distribution for random matrix $X = \left[\bm{X}_0',\bm{X}_1',\ldots,\bm{X}_J'\right]$ of all observed losses for payment and incurred data given after vectorisation $\widetilde{\bm{X}} = Vec\left(\bm{X}\right)$ by
\begin{equation} \label{EqnTransformedGaussianVect}
\left[\widetilde{\bm{X}}|\bm{\theta}\right] = \left[B\widetilde{\Xi}|\bm{\Theta}\right] \sim \mathcal{N}\left(BVec(M),B\left(\Sigma \otimes \Omega\right)B^T\right).
\end{equation}
\end{remark}
Furthermore, if we consider the property of multivariate Gaussian distributions given in Lemma \ref{lemmaMultiVarGaussian} we can find for the $i$-th accident year the required conditional distribution of the unobserved claims for payment and incurred loss data under the specified model. Furthermore, we can find the conditional distribution for unobserved claims for payment and incurred losses in the $i$-th accident year, given all observed claims triangles for payments and incurred losses data, see Lemma \ref{lemmaMultiVarGaussian} below. This is directly relevant for specifying the resulting likelihood model.
\begin{lemma} \label{lemmaMultiVarGaussian} Consider a $(n \times 1)$ random vector $\bm{Y}$ with multivariate Gaussian distribution, $\bm{Y} \sim \mathcal{N}\left(\bm{\mu},\Sigma\right)$, where $\bm{\mu} = \left[ \mu_1, \ldots, \mu_n\right]$ and $\mathbb{C}\text{ov}\left(\bm{Y}\right) = \Sigma$, and partition $Y = \left[Y^{(1)'},Y^{(2)'}\right]'$. Then the conditional distribution of $Y^{(1)}$ given $Y^{(2)}$ and the marginal distribution of $\bm{Y}^{(1)}$ is
\begin{equation}
\left[\bm{Y}^{(1)} | \bm{Y}^{(2)}\right] \sim \mathcal{N}\left(\bar{\bm{\mu}},\bar{\Sigma}\right),
\end{equation}
with $\bar{\bm{\mu}} = \bm{\mu}_1 + \Sigma_{1,2}\Sigma_{2,2}^{-1}\left(\bm{Y}^{(2)} - \bm{\mu}^{(2)}\right)$ and the Schur complement $\bar{\Sigma} = \Sigma_{1,1} - \Sigma_{1,2}\Sigma_{2,2}^{-1}\Sigma_{2,1}$ under the partitioning of the mean and covariance given by
\begin{equation}
\mu = \left[
\begin{array}{c}
\bm{\mu}_{1} \\
\bm{\mu}_{2}
\end{array}
\right] \; \text{ and } \;
\Sigma =
\left(
\begin{array}{cc}
\Sigma_{1,1} & \Sigma_{2,1} \\
\Sigma_{1,2} & \Sigma_{2,2}
\end{array}
\right).
\end{equation}
\end{lemma}
Definition \ref{DefnPermMatrices} below defines a family of permutation matrix operators. This permutation family allows the representation of the vectorization of the two loss triangles under different permutations that facilitate dependence specifications in the proposed models that admit conjugacy.
\begin{defi} \label{DefnPermMatrices} Let $Y$ be an $n \times n$ matrix, with $\widetilde{Y}= \left[Y_{1,1},Y_{1,2},\ldots \right]'$ and with $Vec(Y)$ defined as\\ $Vec(Y) = \left[Y_{1,1},Y_{1,2},\ldots,Y_{1,n},Y_{2,1},\ldots,Y_{2,n},\ldots,Y_{n,n}\right]'$. Define the family of permutation matrix operators, denoted by $\mathcal{P}^*_{\bm{i_{~}}}$ and indexed by $p \times 2, \; p\leq n^2$, indices matrix (vector of tuple elements) $\bm{i_{~}}$ with $j$-th element $\left[\bm{i_{~}}\right]_j = \left\{ \left(k,l\right); k,l \in \left\{1,2,\ldots,n\right\} \right\}$, and defined according to the mapping $\mathcal{P}^*_{\bm{i_{~}}}: Vec(Y) \mapsto Vec(Y)^*$ given by
\begin{equation}
\begin{split}
\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(Y)\right) &= P_{\bm{i_{~}}}^*Vec(Y)\\
&= \left[Y_{\left[\bm{i_{~}}\right]_1},Y_{\left[\bm{i_{~}}\right]_2},\ldots,Y_{\left[\bm{i_{~}}\right]_p},Vec(Y)_{\setminus \bm{i_{~}}}'\right]',
\end{split}
\end{equation}
where we define $Y_{\left[\bm{i_{~}}\right]_j}$ as the element of matrix $Y$ corresponding to the resulting tuple index location in the $j$-th element (column) of (tuple vector) $\bm{i_{~}}$, $P_i^*$ an $n^2 \times n^2$ permutation matrix defined by
\begin{equation}
P_{\bm{i_{~}}}^* = P_{\bm{i_{~}}} \oplus \mathbb{I}_{n^2 - p}
=\left[
\begin{array}{cc}
P_{\bm{i_{~}}} & 0_{n^2-p,n^2-p}\\
0_{n^2-p,n^2-p} & \mathbb{I}_{n^2-p}
\end{array}
\right],
\end{equation}
and $P_{\bm{i_{~}}}$ is a matrix with only non-zero identity elements at the $p$ locations in the indices matrix tuples in $\bm{i_{~}}$ corresponding to index elements.
\end{defi}
Using the property of the multivariate Gaussian distribution in Lemma \ref{lemmaMultiVarGaussian}, one can state the result in Proposition \ref{Propa} which is based on a generalization of the result in \citet{happ2011paid}[Lemma 2.1] to the model developed above. We consider two cases for the dependence structures in Proposition \ref{Propa} and Proposition \ref{Propb}.
\begin{prop} \label{Propa} Consider the $i$-th accident year. Conditional on the model parameters $\bm{\Theta}$ and the covariance matrix of the $i$-th accident year
\begin{equation}
\Sigma_i =
\left(
\begin{array}{cc}
\left[\Sigma_{i}\right]_{1,1} & \left[\Sigma_{i}\right]_{2,1} \\
\left[\Sigma_{i}\right]_{1,2} & \left[\Sigma_{i}\right]_{2,2}
\end{array}
\right),
\end{equation}
the dependence structure $\Omega = \mathbb{I}_{J+1}$ and the observed payment losses and incurred losses in the $i$-th accident year, denoted by $\bm{X}_i^{(1)} = \left[\log I_{i,0},\log P_{i,0},\log I_{i,1},\log P_{i,1}, \ldots,\log I_{i,J-i},\log P_{i,J-i}\right]$ with $\bm{X}_i \in \mathbb{R}^{q}$, the conditional distribution for the log of the unobserved payment losses and incurred losses \\
$(\bm{X}_i^{(2)} = \left[\log I_{i,J-i+1},\log P_{i,J-i+1},\ldots,\log I_{i,J-1},\log P_{i,J-1},\log I_{i,J}\right])$ is given by
\begin{equation}
\left[\bm{X}_i^{(2)}|\bm{X}_i^{(1)}, \bm{\Theta}\right] \sim \mathcal{N}\left(\bar{\bm{\mu}}^{(2)},\bar{\Sigma_i}^{(2)}\right)
\end{equation}
where $\bar{\bm{\mu}}_i^{(2)} = \bm{\mu}_i^{(2)} + \left[\Sigma_{i}\right]_{2,1}\left[\Sigma_{i}\right]_{1,1}^{-1} \left(\bm{X}_i^{(1)} - \bm{\mu}_i^{(1)}\right)$ and $\bar{\Sigma_i}^{(2)} = \left[\Sigma_{i}\right]_{22}$.
\end{prop}
\begin{prop}[Conditional Distribution of Unobserved Payment and Incurred Losses]{\label{Propb} Consider the $i$-th accident year and define indices for this year (vector of tuples), given by matrix \\
$\bm{i_{~}} = \left\{\left(k,j\right):\forall j \in \left\{J-k+1,\ldots,J\right\}\right\}\cup\left\{\left(k,j\right):\forall k \in \left\{0,1,\ldots,J\right\}, j \in \left\{0,\ldots,J-k\right\}\right\}$. Then consider the transformed vector of log payment and incurred losses $\mathcal{P}_{\bm{i_{~}}}^*\left(\widetilde{X}\right)$ defined by
\begin{equation}
\mathcal{P}_{\bm{i_{~}}}^*\left(\widetilde{X}\right) \sim \mathcal{N}\left(P_{\bm{i_{~}}}^* Vec(M), P_{\bm{i_{~}}}^*\left(\Sigma \otimes \Omega\right)(P_{\bm{i_{~}}}^*)'\right),
\end{equation}
for which the first $J-i$ elements of the permuted random vector $\left[\widetilde{X}^*\right]^{(1)} = \left[\mathcal{P}_{\bm{i_{~}}}^*\left(\widetilde{X}\right)\right]_{1:J-i-1}$ correspond to all un-observed payment and incurred loss random variables, and the remaining $J-i$ to $J-i+\left(\sum_{n=-1}^{J}(J-n)\right)$ elements are the observed payment and incurred data, denoted $\left[\widetilde{X}^*\right]^{(2)} = \left[\mathcal{P}_{\bm{i_{~}}}^*\left(\widetilde{X}\right)\right]_{J-i:J-i+\left(\sum_{n=-1}^{J}(J-n)\right)}$. Then, conditional on the model parameters $\bm{\Theta}$, the general dependence structre $\widetilde{\Sigma} = \Sigma\otimes\Omega$ with matrices $\Sigma$ and $\Omega$, and $\left[\widetilde{X}^*\right]^{(2)}$ the following results hold:
\begin{itemize}
\item{The conditional distribution for the log of the \underline{unobserved payment losses and incurred losses} in the $i$-th year, corresponding to the first $J-i$ elements of the permuted random vector $\left[\widetilde{X}^*\right]^{(1)} = \left[\mathcal{P}_{\bm{i_{~}}}^*\left(\widetilde{X}\right)\right]_{1:J-i-1}$ is given by
\begin{equation}
\left[\left[\widetilde{X}^*\right]^{(1)}|\left[\widetilde{X}^*\right]^{(2)}, \bm{\Theta}\right] \sim \mathcal{N}\left(\bar{\bm{\mu}}^{(1)},\bar{\Sigma_i}^{(1)}\right).
\end{equation}}
\item{The covariance matrix $\bar{\Sigma_i}^{(1)}$ is the postive definite \\ $\left(J-i+\left(\sum_{n=-1}^{J}(J-n)\right)\right)\times\left(J-i+\left(\sum_{n=-1}^{J}(J-n)\right)\right)$ sub-matrix denoted below by $\Gamma$ and defined by the top sublock of the permuted covariance matrix
\begin{equation}
P_{\bm{i_{~}}}^*\left(\Sigma \otimes \Omega\right)(P_{\bm{i_{~}}}^*)'
= \left[
\begin{array}{cc}
\Gamma & \left[P_{\bm{i_{~}}}^*\left(\Sigma \otimes \Omega\right)(P_{\bm{i_{~}}}^*)'\right]_{2,1} \\
\left[P_{\bm{i_{~}}}^*\left(\Sigma \otimes \Omega\right)(P_{\bm{i_{~}}}^*)'\right]_{1,2} & \left[P_{\bm{i_{~}}}^*\left(\Sigma \otimes \Omega\right)(P_{\bm{i_{~}}}^*)'\right]_{2,2}
\end{array}
\right].
\end{equation}
}
\item{Given, this covariance matrix one specifies the conditional mean vector, denoted by\\ $\bar{\bm{\mu}}^{(1)} = \bm{\mu}^{(1)} + \Gamma_{2,1}\Gamma_{1,1}^{-1} \left(\left[\widetilde{X}^*\right]^{(2)} - \bm{\mu}^{(2)}\right)$, according to the subblocks of the $\Gamma$ covariance matrix defined with respect to the first $J-i$ elements $\left[\widetilde{X}^*\right]^{(1)}$ and remaining elements of $\left[\widetilde{X}^*\right]^{(2)}$ as well as $\bm{\mu}^{(1)} = \left[P_{\bm{i_{~}}}^* Vec(M)\right]_{1:J-i}$ and the second $J-i$ to $J-i+\left(\sum_{n=-1}^{J}(J-n)\right)$ elements are given by $\bm{\mu}^{(2)} = \left[P_{\bm{i_{~}}}^* Vec(M)\right]_{J-i:J-i+\left(\sum_{n=-1}^{J}(J-n)\right)}$.}
\end{itemize}
}
\end{prop}
Having specified these statistical assumptions, we can formulate the joint likelihood from the observed data for both payments and incurred claims conditional upon the model parameters according to Equation (\ref{LH_DependentPICModI}). The joint data likelihood function in the dependent Log-Normal PIC Model I for the random vector of observations corresponding to the first $\sum_{n=-1}^{J}(J-n)$ elements of the permuted random vector, given by $\left[\widetilde{X}^*\right]^{(1)} = \left[\mathcal{P}_{\bm{i_{~}}}^*\left(\widetilde{X}\right)\right]_{1:\left(\sum_{n=-1}^{J}(J-n)\right)}$, where we define indices in this case by $\bm{i_{~}} = \left\{\left(i,j\right):\forall i \in \left\{0,1,\ldots,J\right\}, j \in \left\{0,\ldots,J-i\right\}\right\}$. The resulting likelihood is given by the matrix-variate Gaussian distribution in Equation (\ref{LH_DependentPICModI}).
{\small{
\begin{equation} \label{LH_DependentPICModI}
\begin{split}
&f\left(
\left.\left[\widetilde{X}^*\right]^{(1)} \right|\boldsymbol{\Theta},\Sigma, \Omega\right)=\\
&\frac{\exp\left[ \left(\left[\widetilde{X}^*\right]^{(1)}-\left[\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(M)\right)\right]^{(1)}\right) \left[ \left[P_{\bm{i_{~}}}^*\left(\Sigma \otimes \Omega\right)(P_{\bm{i_{~}}}^*)'\right]^{(1)} \right]^{-1} \left(\left[\widetilde{X}^*\right]^{(1)}-\left[\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(M)\right)\right]^{(1)}\right) \right]}{ \left(2\pi\right)^{\left(\sum_{n=-1}^{J}(J-n)\right)/2}\left|\left[P_{\bm{i_{~}}}^*\left(\Sigma \otimes \Omega\right)(P_{\bm{i_{~}}}^*)'\right]^{(1)} \right|^{\left(\sum_{n=-1}^{J}(J-n)\right)/2} }
\end{split}
\end{equation}
}}
We note that our proposed models also allow one to consider the dependence structures of \citet{happ2011paid} who assume that $\Sigma_i = \Sigma, \forall i \in \left\{0,1,\ldots,J\right\}$ and $\Omega = \mathbb{I}_{J+1}$, with the specific setting of $\Sigma$ via a tri-diagonal correlation matrix with three correlation parameters which are assumed either known \textit{a priori} or estimated prior to inference in the PIC model. Such an approach was motivated by the belief that a positive change in incurred loss results in an immediate payment in the same development period, and the remaining increased expectation is paid with some settlement delay. Therefore, the incurred losses increments $\zeta_{i}^{j}$ are assumed to be positively correlated to the claims payments increments $\xi_{i,j}$, $\xi_{i,j+1}$ and $\xi_{i,j+2}$ with positive correlations $\rho_{0},\rho_{1},\rho_{2}$, respectively. However, the argument for more general dependence structure that were introduced as extensions to the model of \citet{happ2011paid} are developed to account for the fact that these assumption may not be true, especially in long tail portfolios, such as compulsory third party. If the status of a claimant changes and requires long term medical treatment and rehabilitation, it might result in substantially high loss in the subsequent lengthy lag periods. The paper also assumes that the dependence is the same across different lag years, which is not always a realistic feature of such data. Our article aims to fill this gap and enhance the correlation structure in PIC models whilst maintaining a parsimonious model specification.
\subsection{Dependence Between Development Lag Years for Payment Losses and Incurred Losses (Model III)} \label{DevLagYrs}
This section considers an alternative dependence structure motivated by the fact that dependence between lag years is practically appealing in claims reserving practice. It affects the estimation of outstanding claims the most, and is widely recognized by actuaries in claims reserving. Lag is the measure of the difference between incurred month and paid month. Depending on the nature of the portfolio, many insurance claims often have lengthy settlement periods due to various reasons such as late reported claims, judicial proceedings, or schedules of benefits for employer's liability claims. During the lengthy lag periods, large payments in the previous lag period normally follow by small payments in the subsequent lag period. There may in fact be positive correlation if all periods are equally impacted by a change in claims status, e.g. if a claim becomes litigated, resulting in a huge increase in claims cost. There may also be negative correlation if a large settlement in one period replaces a stream of payments in later periods. The model developed in this section mainly focuses on capturing this feature of dependence between lag years. To achieve this we propose a block covariance structure for the covariance matrix, which will respect the dependence between each lag point. The model we propose is summarised in Model Assumptions \ref{modelass3} below.
\begin{model ass}[Dependent Development Lag Years: PIC Log-Normal (\textsl{Model III})]\label{modelass3}
The following statistical model assumptions are developed:
\begin{itemize}
\item Let $\Sigma^{P}_i \in \mathbb{S}\mathbb{D}^+(J-i)$ be the $(J-i)\times(J-i)$ random covariance matrix on the space $\mathbb{S}\mathbb{D}^+(J-i)$ of positive definite covariance matrices of dimension $(J-i)\times(J-i)$ corresponding to the observed payment data $\left[\log P_{i,0},\log P_{i,1},\ldots,\log P_{i,J-i}\right]$ in the $i$-th accident year and analogously for incurred loss data $\Sigma^{I}_i \in \mathbb{S}\mathbb{D}^+(J-i)$. When $i = 0$ we consider $\Sigma^{P}_0 \in \mathbb{S}\mathbb{D}^+(J+1)$ and for incurred loss data $\log I_{0,0:J-1}$ with $\Sigma^{I}_0 \in \mathbb{S}\mathbb{D}^+(J)$.
Assume an inverse Wishart distribution (see Lemma \ref{lemmaInvWishartDist} and Lemma \ref{lemmaWishartDist}) for each matrix defined according to
\begin{equation}
\Sigma^P_i \sim \mathcal{IW}\left(\Lambda^P_i,k^P_i\right) \; \text{ and } \; \Sigma^I_i \sim \mathcal{IW}\left(\Lambda^I_i,k^I_i\right),
\end{equation}
where $\Lambda^P_i$ and $\Lambda^I_i$ are the inverse scale matrices for the prior for the payment and incurred loss data covariance priors respectively. Hence, the joint covariance between all observed payment and incurred loss data satisfies the telescoping diagonal block size covariance structure:
\begin{equation} \label{EqnTelescopingDependence}
\begin{split}
&\widetilde{\Sigma} = \mathbb{C}\text{ov}\left(\left[\log P_{0,0},\ldots,\log P_{0,J} ,\log P_{1,0},\log P_{1,J-1},\ldots,\log P_{J,0},\log I_{0,0},\ldots,\log I_{0,J-1},\ldots,\log I_{J,0}\right]\right)\\
&\;\;\;\; =\left(\bigoplus_{i=0}^J \Sigma^{P}_0\right)\oplus\left(\bigoplus_{i=0}^J \Sigma^{I}_0\right) \sim \mathcal{IW}\left(\left(\bigoplus_{i=0}^J \Lambda^{P}_0\right)\oplus\left(\bigoplus_{i=0}^J \Lambda^{I}_0\right), \sum_{i=0}^J \left(k_i^{P}+ k_i^I\right) \right).
\end{split}
\end{equation}
\item Conditionally, given $\mathbf{\Theta}=\left(
\Phi_0,\ldots, \Phi_{J},\Psi_0,\ldots, \Psi_{J}\right)$ and the covariance matrix $\widetilde{\Sigma}$, we have the following results
\begin{itemize}
\item{Consider the marginal distribution of the first $\left(\sum_{n=-1}^{J}(J-n)\right)$ elements of the vectorized random matrix of observed payment and incurred losses, with $i$-th column $\bm{X}_i \in \mathbb{R}^{2J+1}$ given by $$\bm{X}_i = \left[\log I_{i,0},\log P_{i,0},\log I_{i,1},\log P_{i,1}, \ldots,\log I_{i,J-1},\log P_{i,J-1},\log I_{i,J}\right].$$
Then given the matrix of permutation indices $\bm{i_{~}} = \left[(1,2),(1,4),\ldots,(1,2(J-1))\right.$\\ $\left.,(2,2),(2,4),\ldots,(2,2J-4),\ldots,(J,1),(1,1),(1,3),\ldots(J-1,1),(J-1,2)\right]$ characterizing the elements of the marginal distribution for the observations, the transform $\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)$ has multivariate Gaussian distribution with covariance structure $\widetilde{\Sigma}$. Note, $\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right) = \left[\log P_{0,0},\log P_{0,1},\ldots,\log P_{0,J}\right.$\\
$\left.,\ldots,\log P_{J,0},\log I_{0,0},\ldots,\log I_{0,J-1},\log I_{1,0},\ldots,\log I_{J-1,0},\log I_{J-1,1}\right]$.}
\item{For all accident years, $i \in \left\{0,1,\ldots,J\right\}$, the ultimate payment losses and incurred losses are equal almost surely, $P_{i,J} = I_{i,J}.$}
\end{itemize}
\item{The matrix $\widetilde{\Sigma}$ is positive definite and the components of $\bm{\Theta}$ are independent with prior distributions
\begin{equation}
\Phi_i \sim \mathcal{N}\left(\phi_i,s_i^2\right) \; \text{ and } \; \Psi_j \sim \mathcal{N}\left(\psi_j,t_j^2\right)
\end{equation}
and hyper-prior distributions
\begin{equation}
s^2_i \sim \mathcal{IG}\left(\alpha_i,\beta_i\right) \; \text{ and } \; t^2_j \sim \mathcal{IG}\left(a_j,b_j\right)
\end{equation}
for all $i \in \left\{1,\ldots,J\right\}$ and $j \in \left\{0,\ldots,J\right\}$.
}
\end{itemize}
\end{model ass}
This proposed model is therefore another generalization of the dependence structure of the model structure proposed in \citet{happ2011paid}. As such, the likelihood structure is given by the multivariate Gaussian given in Equation (\ref{LH_DependentPICModI}) with the covariance matrix given by the telescoping diagonal block size covariance matrix structure in Equation (\ref{EqnTelescopingDependence}).
\subsection{Hierarchical Bayesian Conjugacy Under Gausian Copula Dependent PIC: Models I, II, III}
Under the Gaussian copula based dependence models, the ability to obtain the observed data likelihood in the form of a multivariate Gaussian distribution means that we obtain conjugacy properties. This makes the estimation of such models by MCMC more efficient because we can us Gibbs sampling in blocks. This section presents a generic set of such conjugate models for any of the dependence structures specified in Models I, II and III.
\begin{lemma} \label{LemmaTransformData} Conditional upon the parameters $\bm{\Theta}$ and the covariance matrix $\Sigma$, the permuted data $\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)$ can be transformed to produce the independent likelihood in Equation (\ref{LH_IndependentPIC}). This is achieved by considering the class of vector transformations $\mathcal{T}: \mathbb{R}^{(d \times 1)} \mapsto \mathbb{R}^{(d \times 1)}$, such that if the initial covariance structure of random vector $\bm{X}$ was given by $\Sigma = \mathbb{C}\text{ov}\left(\bm{X}\right)$, then the resulting covariance structure $\mathbb{C}\text{ov}\left(\mathcal{T}(\bm{X})\right) = \mathbb{I}_{d}$. The required rotation-dilation transformation is obtained by the spectral decomposition of the covariance according to a spectral decomposition (see \citet{stoica1997introduction})
$\Sigma = U\Lambda^{\frac{1}{2}}U'$ where $U$ is a $(d \times d)$ matrix of eigenvectors of $\Sigma$ and $\Lambda$ is a diagonal $d \times d$ matrix of the eigenvalues of $\Sigma$. Therefore the following holds for each of the models under a transform of the vector of permuted observations $\mathcal{T}\left(\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)\right)$:
\begin{enumerate}
\item{Model II - When $\tilde{\Sigma} = \Sigma \otimes \Omega$, with $\Omega = \mathbb{I}_{J+1}$ then, $\mathcal{T}\left(\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)\right) = \left(U\Lambda^{\frac{1}{2}} \otimes \mathbb{I}_{J+1}\right) \mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right),$ where the $\left((2J+1)\times(2J+1)\right)$ covariance $\Sigma$ is decomposed as $U\Lambda^{\frac{1}{2}}U'$.}
\item{Model II - When $\tilde{\Sigma} = \bigoplus_{i=0}^J \Sigma_i$, $\mathcal{T}\left(\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)\right) = \left(\bigoplus_{i=0}^J U_i\Lambda_i^{\frac{1}{2}}\right) \mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right),$ where each accident year's dependence between payments and incurred losses is given by the $(2J+1)\times(2J+1)$ matrix $\Sigma_i$ which is decomposed as $U_i\Lambda_i^{\frac{1}{2}}U_i'$.}
\item{Model III - When $\tilde{\Sigma} = \left(\bigoplus_{i=0}^J \Sigma^{P}_0\right)\oplus\left(\bigoplus_{i=0}^J \Sigma^{I}_0\right)$, $$\mathcal{T}\left(\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)\right) = \left(\bigoplus_{i=0}^J U^{P}_i\left(\Lambda^{P}_i \right)^{\frac{1}{2}}\right)\oplus\left(\bigoplus_{i=0}^J U^{I}_i\left(\Lambda^{I}_i \right)^{\frac{1}{2}}\right) \mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)$$ where each of the covariance matrices $\Sigma^P_i$ and $\Sigma^I_i$ decomposed to $U^P_i\left(\Lambda^P_i\right)^{\frac{1}{2}}(U^P_i)'$ and $U^I_i\left(\Lambda^I_i\right)^{\frac{1}{2}}(U^I_i)'$.}
\end{enumerate}
In each case, the resulting transformed random vector $\mathcal{T}\left(\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)\right)$, with elements $\widetilde{P}_{i,j}$ and $\widetilde{I}_{i,j}$, will produce a likelihood model given for the transformed data according to the independent Model I of \citet{merz2010paid} as defined in Equation (\ref{LH_IndependentPIC}). Of course this is defined now with respect to components in the likelihood corresponding to the transformed components, as detailed in Equation (\ref{EqnTransformedGaussianVect}).
\end{lemma}
\begin{remark}
The consequence is that results in Lemma \ref{LemmaTransformData} are that the conjugacy properties derived for the independent model in \citet{merz2010paid} can be directly applied post-transformation. This is of direct interest for MCMC based sampling schemes.
\end{remark}
In the models described so far, the following full conditional posterior distributions are now of relevance to the Bayesian MCMC estimation procedures developed for Models I, II and III.
\begin{lemma} The full conditional posterior distributions for sub-blocks of the model parameters can be decomposed under Model I, II and III into a conjugate model.
\begin{itemize}
\item{\textbf{Conjugate Posterior Distribution for Development Factors:} under the transformations $\mathcal{T}\left(\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)\right)$ on the data, described in Lemma \ref{LemmaTransformData}, the full conditional posterior distributions for sub-blocks of the transformed model parameters $\left(\widetilde{\Phi}_{0:J},\widetilde{\Psi}_{0:J}\right)$ are given by (see \citet{merz2010paid} [Theorem 3.4] for the independent case):
\begin{equation}
\left[\widetilde{\Phi}_{0:J},\widetilde{\Psi}_{0:J}|\Sigma,\Omega,\mathcal{T}\left(\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)\right)\right] \sim \mathcal{N}\left(\Pi,\Delta\right)
\end{equation}
with posterior mean $\Pi$ and posterior covariance $\Delta$, where the components of $\Delta^{-1} = \left(a_{n,m}\right)_{0 \geq n,m \leq 2J}$ are each given by
\begin{equation}
\begin{split}
a_{n,m} &= \left(s_n^{-2}+(J-n+1)\sigma_n^{-2}\right)\delta_{n=m} + \sum_{i=0}^{(n-1)\wedge(m-1)}\left(\nu^2_i - \omega^2_i\right)^{-1}, \; \text{ for } 0\leq n, m \leq J,\\
a_{J+1+n,J+1+m} &= \left(t_n^{-2} + (J-n)\tau_n^{-2} \right)\delta_{n=m} + \sum_{i=0}^{n\wedge m}\left(\nu^2_i - \omega^2_i\right)^{-1}, \; \text{ for } 0\leq n, m \leq J-1,\\
a_{n,J+1+m} &= \Delta_{n,J+1+m} = - \sum_{i=0}^{(n-1)\wedge m} \left(\nu^2_i - \omega^2_i\right)^{-1}, \; \text{ for } 0\leq n \leq J, 0 \leq m \leq J-1;\\
\end{split}
\end{equation}
where $\delta_{n=m}$ is the indicator of the event that index $m$ matches $n$, $m \wedge n$ is the minimum of $m$ and $n$ and the posterior mean is given on the transformed scale by,
\begin{equation}
\begin{split}
\left[\widetilde{\Phi}_{0:J},\widetilde{\Psi}_{0:J}\right] &= \Delta\left(\widetilde{c}_0,\widetilde{c}_1,\ldots,\widetilde{c}_J,\widetilde{b}_0,\ldots,\widetilde{b}_J\right),
\end{split}
\end{equation}
with
\begin{equation}
\begin{split}
\widetilde{c}_j &= s_j^{-2}\phi_j + \sigma^2_j \sum_{i=0}^{J-j} \log\left(\frac{\widetilde{P}_{i,j}}{\widetilde{P}_{i,j-1}}\right) + \sum_{i=J-j+1}^J \left(\nu_{J-i}^2 - \omega_{J-i}^2\right)^{-1}\log\left(\frac{\widetilde{I}_{i,J-i}}{\widetilde{P}_{i,J-i}}\right), \\
\widetilde{b}_j &= t_j^{-2}\psi_j + \tau^2_j \sum_{i=0}^{J-j-1} \log\left(\frac{\widetilde{I}_{i,j}}{\widetilde{I}_{i,j+1}}\right) - \sum_{i=J-j}^J \left(\nu_{J-i}^2 - \omega_{J-i}^2\right)^{-1}\log\left(\frac{\widetilde{I}_{i,J-i}}{\widetilde{P}_{i,J-i}}\right).
\end{split}
\end{equation}
Given the transform vector $\left[\widetilde{\Phi}_{0:J},\widetilde{\Psi}_{0:J}\right]$, the parameters on the orginal scale can be expressed according to the unique solution to the system of linear equations:
\begin{enumerate}
\item{Model II - On the untransformed scale, the solution is given by the following system of equations
\begin{equation}
\begin{split}
\left[\Phi_{0:J},\Psi_{0:J}\right]' = U^{-1}\Lambda^{-\frac{1}{2}} \left[\widetilde{\Phi}_{0:J},\widetilde{\Psi}_{0:J}\right].
\end{split}
\end{equation}
}
\item{Model II - On the untransformed scale, the solution is given by the following system of equations for each $i \in \left\{0,1,\ldots,J\right\}$, where we can randomly select $i$ or deterministically scan through $i$ for the results,
\begin{equation}
\begin{split}
\left[\Phi_{0:J},\Psi_{0:J}\right]' = U_i^{-1}\Lambda_i^{-\frac{1}{2}} \left[\widetilde{\Phi}_{0:J},\widetilde{\Psi}_{0:J}\right].
\end{split}
\end{equation}
}
\item{Model III - On the untransformed scale, the solution is given by the following system of equations,
\begin{equation*}
\begin{split}
\left[\Phi_{0:J},\Phi_{0:J-1},\Phi_{0:J-2},\ldots,\Phi_{J}\right]' &= \bigoplus_{i=0}^J \left(U^{P}_i\right)^{-1}\left(\Lambda^{P}_i \right)^{-\frac{1}{2}}\left[\widetilde{\Phi}_{0:J},\widetilde{\Phi}_{0:J-1},\widetilde{\Phi}_{0:J-2},\ldots,\widetilde{\Phi}_{J}\right],\\
\left[\Psi_{0:J},\Psi_{0:J-1},\Psi_{0:J-2},\ldots,\Psi_{J}\right]' &= \bigoplus_{i=0}^J \left(U^{I}_i\right)^{-1}\left(\Lambda^{I}_i \right)^{-\frac{1}{2}}\left[\widetilde{\Psi}_{0:J},\widetilde{\Psi}_{0:J-1},\widetilde{\Psi}_{0:J-2},\ldots,\widetilde{\Psi}_{J}\right].
\end{split}
\end{equation*}
}
\end{enumerate}
}
\item{\textbf{Conjugate Posterior Distribution for the Covariance Matrix:} Given the transformed observed payment and incurred losses have a multivariate Gaussian likelihood, as specified in Equation (\ref{LH_DependentPICModI}), with covaraince matrix $\widetilde{\Sigma} = \Sigma \otimes \Omega$ and mean vector $Vec\left(M\right)$. Then the posterior for the covariance matrix is the Inverse-Wishart-Gaussian distribution detailed in \citet{PetersBA1} [Section 3] and \citet{PetersBA2}
\begin{equation*} \label{EqnConjugateIW}
\begin{split}
\left[\widetilde{\Sigma}|\Phi_{0:J},\Psi_{0:J},\mathcal{T}\left(\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)\right)\right] \sim \mathcal{IW}\left(\Lambda + \mathcal{T}\left(\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)\right) \mathcal{T}\left(\mathcal{P}_{\bm{i_{~}}}^*\left(Vec(\bm{X})\right)\right)',dim\left(Vec(\bm{X})\right) + \widetilde{k}\right)
\end{split}
\end{equation*}
In cases in which the covariance matrix $\widetilde{\Sigma}$ takes any of the block diagonal forms presented in Models II and III, we may utilise Lemma \ref{lemmaWishartDist} and the result in Equation (\ref{EqnConjugateIW}) to further decompose the posterior covariance into blockwise components.
}
\item{\textbf{Conjugate Posterior Distribution for the Hyper-Parameters on Development Factors:} For all $i$ we have the following Inverse Gamma-Gaussian conjugacy for the hyper parameters in Models II and III,
\begin{equation*}
\begin{split}
\left[s^2_i|\Phi_{i}\right] \sim \mathcal{IG}\left(\alpha_i + \frac{1}{2}, \beta_i + \frac{\left(\Phi_i - \phi_i \right)^2}{2}\right)\; \; \text{ and } \; \; \left[t^2_i|\Psi_{i}\right] \sim \mathcal{IG}\left(a_i + \frac{1}{2}, b_i + \frac{\left(\Psi_i - \psi_i \right)^2}{2}\right).
\end{split}
\end{equation*}}
\end{itemize}
\end{lemma}
We next present alternative tail dependence structures for the PIC model. Previous studies on claims reserving that have incorporated copula based models, such as \citet{zhangpredicting} have done so through regression based frameworks. \citet{zhangpredicting} develop a parametric copula model to account for dependence between various lines of insurance claims. Their paper considers a bivariate Gaussian copula model with marginal generalized linear models to capture the positive correlation between the two insurance lines. Our article significantly extends the dependence modelling capability of the PIC model structure remaining in the frameworks presented above. However, to do so requires the introduction of auxiliary variables to enable computation. The approach developed involves modifying the posterior distribution by embeding the target posterior distribution for the model parameters into a much higher dimensional support comprised of the original model parameters and the additional auxiliary variables. The reason for this expansion of the posterior dimensions will be come clear below and is in general known in Bayesian statistics as an auxiliary variable framework.
\section{Incorporating Mixture-Archimedean Copula Dependence Structures into Paid-Incurred-Claims Models: Model IV}
This section presents an alternative parameteric approach to modelling and capturing dependence and tail dependence in the PIC model structure which involves considering copula based models within the PIC reserving framework. The dependence can be considered over the following combinations such as:
\begin{enumerate}
\item{Independent accident years and dependence between payment losses over the development years;}
\item{Independent accident years and dependence between incurred losses over the development years;}
\item{Independent accident years and dependence jointly between payment and incurred losses over the development years via a mixture copula, hierarchical copula (HAC) as in \citet{kurowicka2010dependence}, or a vine copula (d-vine, canononical vine) e.g. \citet{aas2009pair};}
\item{Dependent accident years and independent development years for payment, incurred or both sets of losses.}
\end{enumerate}
Our article concentrates on the mixture copula model which allows for combinations of upper and lower tail dependence of different strengths. We detail the class of auxiliary variable methods known in statistics as Data Augmentation and demonstrate how this class of models can be combined into our modelling framework to allow for consistent use of copula models in the PIC framework. There are many variations that can be explored in this approach. We give one such approach for Model IV, Assumptions \ref{modelass4}, that is directly comparable to that used for Model II in Assumption \ref{modelass2}.
We present fundamental properties of members of the Archimedean family of copula that we consider when constructing mixture copula models in the PIC framework in the Appendix, see Lemma \ref{LemmaArchCop} for the characteristics of the Archimedean family of copulas and Lemma \ref{lemmaArchCop} for the required distribution and densities for three members of this family. In addition references \citet{denuit2005actuarial},\citet{aas2009pair},\citet{embrechts2009copulas}, \citet{min2010bayesian} and \citet{patton2009copula} provide more detail.
In Lemma \ref{LemmaArchCop} the property of associativity of Archimedean copula models is particularly useful in the PIC model framework as it allows us to obtain analytic expressions for the likelihood structure of the matrix-variate PIC model. This is particularly useful if one specifies the model as a hierarchical Archimedean Copula (HAC) construction.
We consider the following popular members of the Archimedean family of copula models, due to their analytic tractability, their non-zero tail dependence properties and their parsimonious parameterizations. In addition, generating random variates from these class of models is trivial given the generator for the member of the Archimedean family of interest. Lemma \ref{lemmaArchCop} in the appendix presents the three Archimedean copulas for Clayton, Gumbel and Frank copulas that we consider and their properties. We use the following notation for copula densities we consider on $[0,1]^d$, see \citet[Section 4.3, Table 4.1]{nelsen2006introduction} and Lemma \ref{lemmaArchCop}:
the Clayton copula density is denoted by $c^C(u_1,...,u_n;\rho^C)$ with $\rho^C \in [0,\infty)$ the dependence parameter; the Gumbel copula density is denoted by $c^G(u_1,...,u_n;\rho^G)$ with $\rho^G \in [1,\infty)$ the dependence parameter; and the Frank copula density is denoted by $c^F(u_1,...,u_n;\rho^F)$ with $\rho^F \in \mathbb{R}/\{0\}$ the dependence parameter.
In addition, we also note that the properties of these copulas of interest include that the Clayton copula does not have upper tail dependence, however its lower tail dependence can be expressed as $\lambda_L = 2^{-1/\rho^C}$. The Gumbel copula does not have lower tail dependence, however its upper tail dependence of the Gumbel copula can be expressed as $\lambda_U = 2 - 2^{1/\rho^G}$. The Frank copula does not have upper or lower tail dependence.
In this class of copula dependence models we consider the marginal distribution of each log payment or log incurred loss as distributed according to a Gaussian distribution and the joint distribution vector is modelled via a mixture copula comprised of the above three components from the Archimedean family. Such a copula construction will still produce a copula as shown in Lemma \ref{lemmaCopMix}.
\begin{lemma}{ \label{lemmaCopMix} Consider copula distributional members $C_i\left(u_1,u_2,\ldots,u_n\right) \in \mathcal{A}^n$, where $\mathcal{A}^n$ defines the space of all possible n-variate distributional members of the Archimedean family of copula models, specified in Lemma \ref{lemmaArchCop}. Any finite mixture distribution constructed from such copula components that admit tractable density functions $c_i\left(u_1,u_2,\ldots,u_n\right)$, denoted by $\tilde{c}\left(u_1,u_2,\ldots,u_n\right) = \sum_{i=1}^m w_i c_i\left(u_1,u_2,\ldots,u_n\right)$, such that $\sum_{i=1}^m w_i = 1$, is also the density of a copula distribution.}
\end{lemma}
The proof of Lemma \ref{lemmaCopMix} is provided in Appendix \ref{AppendixProofCopMix}.
\subsection{Understanding Bayesian Data Augmentation}
The modeling framework of Data Augmentation in the Bayesian framework is typically invoked to deal with situations in which the likelihood evaluation is intractable to perform point-wise. This would make Bayesian inference in such a model also generally intractable. For example if one considers the generic likelihood $p\left(\bm{y}_{1:n}|\bm{\theta}\right)$ with observation random vectors $\bm{Y}_{1:n}$, which can be evaluate point-wise as a function of parameter vector $\bm{\theta}$ with respect to a realization of the observation process $\bm{y}_{1:n}$.
In the setting we encounter in the PIC models, we can generically consider the data random vector observation is partitioned into two vector sub-components $\bm{Y} = \left[\bm{Y}^{(1)},\bm{Y}^{(2)}\right]$, of which only one component, say $\bm{Y}^{(1)}$, is actually observed. Then evaluation of the likelihood pointwise for $\bm{\theta}$ given a realization of $\bm{Y}_{1:n}^{(1)}$ would require solving the integral in Equation \ref{EqnLHGeneric}
\begin{equation} \label{EqnLHGeneric}
p\left(\bm{Y}_{1:n}^{(1)}|\bm{\theta}\right) = \int p\left(\bm{Y}_{1:n}^{(1)}|\bm{\theta},\bm{Y}_{1:n}^{(2)}\right) p\left(\bm{Y}_{1:n}^{(2)}|\bm{\theta}\right) d\bm{Y}_{1:n}^{(2)}.
\end{equation}
Generally, this integral will not admit a closed form solution. Therefore, the Bayesian Data Augmentation approach involves extending the target posterior $p\left(\bm{\theta}|\bm{Y}_{1:n}^{(1)}\right)$ which is intractable due to the intractability of the likelihood to a new posterior model on a higher dimensional space, in which the target distribution is a marginal as given in Equation \ref{EqnDAGeneric}
\begin{equation} \label{EqnDAGeneric}
p\left(\bm{\theta},\bm{Y}_{1:n}^{(2)*}|\bm{Y}_{1:n}^{(1)}\right) = \frac{p\left(\bm{Y}_{1:n}^{(1)}|\bm{\theta},\bm{Y}_{1:n}^{(2)*}\right)p\left(\bm{Y}_{1:n}^{(2)*}|\bm{\theta}\right)p\left(\bm{\theta}\right)}{p\left(\bm{Y}_{1:n}^{(1)}\right)}
\end{equation}
where $\bm{Y}_{1:n}^{(2)*}$ are auxiliary random vectors with prior distribution $p\left(\bm{Y}_{1:n}^{(2)*}|\bm{\theta}\right)$, 'augmented' to the posterior parameter space to allow tractability of the posterior inference. This will be explained in detail for the PIC copula models below.
\subsection{Data Augmentation in the Bayesian PIC Copula Models}
Definition \ref{DefnLogAuxMod} gives some useful notation for the results that follow.
\begin{defi}[Auxiliary Data for Data Augmentation] \label{DefnLogAuxMod}
Consider the defined loss data under the one-to-one (invertible) transformation for the observed data
given by the joint matrix for all observations and auxiliary variables given by $X = \left[\bm{X}_0',\bm{X}_1',\ldots,\bm{X}_J'\right]$. In this framework, the $i$-th accident year is defined according to,
$\bm{X}_i = \left[\log I_{i,0},\log P_{i,0},\log I_{i,1},\log P_{i,1}, \ldots,\log I_{i,J-1},\log P_{i,J-1},\log I_{i,J}\right]$. Consider the permutation of each vector of log payments and log incurred losses given by\\
$\widetilde{\bm{X}}_i = \mathcal{P}^*_{\bm{i_{~}}}\left(\bm{X}_i\right) = \left[\log P_{i,0},\log P_{i,1},\ldots,\log P_{i,J},\log I_{i,0},\log I_{i,1},\ldots,\log I_{i,J-1}\right].$
Now consider the further partition by the decomposition of observed log payment losses and unobserved log payment losses as well as these quantities for the incurred losses defined for the $i$-th accident year by,
\begin{equation}
\begin{split}
\widetilde{\bm{X}}_i &= \left[\widetilde{\bm{X}}^P_{i,obs},\widetilde{\bm{X}}^P_{i,aux},\widetilde{\bm{X}}^I_{i,obs},\widetilde{\bm{X}}^I_{i,aux}\right] \\
&=\left[\widetilde{\bm{X}}^P_{0,i,obs},\ldots,\widetilde{\bm{X}}^P_{J-i,i,obs},\widetilde{\bm{X}}^P_{J-i+1,i,aux},\ldots,\widetilde{\bm{X}}^P_{J,i,aux},\widetilde{\bm{X}}^I_{0,i,obs},\ldots,\widetilde{\bm{X}}^I_{J-i,i,obs},\widetilde{\bm{X}}^I_{J-i+1,i,aux},\ldots,\widetilde{\bm{X}}^I_{J-1,i,aux}\right] \\
&=\left[ \underbrace{\log P_{i,0},\ldots,\log P_{i,J-i}}_{\text{observed Payments}},\underbrace{\log P_{i,J-i+1},\ldots,\log P_{i,J}}_{\text{unobserved Payments}},\underbrace{\log I_{i,0},\ldots,\log I_{i,J-i}}_{\text{observed Incurred}},\underbrace{\log I_{i,J-i+1},\ldots,\log I_{i,J-1}}_{\text{unobserved Incurred}} \right]'.\\
\end{split}
\end{equation}
Therefore the total data matrix of losses is given by $\widetilde{X} = \left[\widetilde{\bm{X}}_0,\ldots,\widetilde{\bm{X}}_J\right].$ Note, the introduction in this section of the notation subscripts $obs$ and $aux$ allows us to make explicit the fact that the upper triangle of log payment losses and the upper triangle of log incurred losses are un-observed quantities for these random variables, while the lower triangular regions for such losses are observed. We denote these random variables as auxiliary variables (augmented) to the observed data random variables to create a complete data set of all losses.
\end{defi}
By considering the unobserved data in the lower payment and incurred loss triangles as auxiliary variables to be jointly estimated along with the model parameters, we will demonstrate below that only under this approach is consistency ensured in the copula structure of the PIC model. However, we first make the following model assumptions about the statistical features of the PIC model.
The following assumptions illustrate a choice of copula models for the mixture from the Archimedean family. However, there are many related specifications and frameworks that can be explored in this context, be we leave that to future research.
\begin{model ass}[Data-Augmented Mixture Copula PIC (\textsl{Model IV})]\label{modelass4}
The model assumptions and specifications for the copula model we develop involve:
\begin{itemize}
\item Let the random matrix $\Sigma_i \in \mathbb{R}^{(2J+1)\times(2J+1)}$ be the covariance for $\widetilde{\bm{X}}_i = \left[\widetilde{\bm{X}}^P_{i,obs},\widetilde{\bm{X}}^P_{i,aux},\widetilde{\bm{X}}^I_{i,obs},\widetilde{\bm{X}}^I_{i,aux}\right]$ with $\widetilde{\bm{X}}_i \in \mathbb{R}^{2J+1}$ for all $i=0,\ldots,J$. We assume that $\Sigma$ is diagonal where
\begin{equation}
\Sigma_{i,i} \sim \mathcal{IG}\left(\alpha_i,\beta_i\right), \; \forall i \in \left\{0,\ldots,J\right\},
\end{equation}
where $\alpha_i$ and $\beta_i$ are the known hyper-parameters for shape and scale.
\item{\textbf{MARGINAL DISTRIBUTION:} given $\mathbf{\Theta}=\left(\Phi_0,\ldots, \Phi_{J},\Psi_0,\ldots, \Psi_{J}\right)$ and covariance matrices $\Sigma, \Omega \in \mathbb{R}^{(2J + 1) \times (2J + 1)}$ and $\bm{\rho}$, we assume the marginal distribution of the random matrix, of all log payments and log incurred losses $\widetilde{X}$, comprised of columns $\widetilde{\bm{X}}_i$ for the $i$-th accident year is matrix-variate Gaussian with density, defined as in Lemma \ref{lemmaMatrixGaussianDist}, with the $(2J + 1) \times (J+1)$ mean matrix $\widetilde{M} = \left[\bm{\Theta}',\ldots,\bm{\Theta}'\right]$, column dependence given by $(2J + 1) \times (2J+1)$ covariance matrix $\Sigma$ and row dependence given by $(J+1) \times (J+1)$ matrix $\Omega$. Here we only consider the case of $\Omega = \mathbb{I}_{J+1}$ for the marginal independent case.
}
\item{\textbf{DATA AUGMENTED PIC MIXTURE COPULA LIKELIHOOD:} Given $\widetilde{\bm{X}}^P_{0,aux},\widetilde{\bm{X}}^P_{1,aux},\ldots,\widetilde{\bm{X}}^P_{J-1,aux}$,
$\widetilde{\bm{X}}^I_{0,aux},\widetilde{\bm{X}}^I_{1,aux},\ldots,\widetilde{\bm{X}}^I_{J-1,aux}$,
$\mathbf{\Theta}=\left(\Phi_0,\ldots, \Phi_{J},\Psi_0,\ldots, \Psi_{J}\right)$, covariance matrices $\Sigma, \Omega \in \mathbb{R}^{(2J + 1) \times (2J + 1)}$ and $\bm{\rho}$, the joint distribution of the random matrix ($\widetilde{X}$) of all log permuted payment and incurred losses is assumed (in this example) to be independent between accident years. For the $i$-th column (corresponding to $i$-th accident year), the joint distribution of all losses $(\widetilde{\bm{X}}_i)$ is assumed to be hierarchical Archimedean Copula (HAC) mixture copula specified by distribution,
\begin{equation}
\begin{split}
\left[\widetilde{X}\right]_{\bullet, i} &\sim \tilde{C}_{\bm{\rho}_{i}}\left(F\left(\widetilde{\bm{X}}^P_{i,obs},\widetilde{\bm{X}}^P_{i,aux},\widetilde{\bm{X}}^I_{i,obs},\widetilde{\bm{X}}^I_{i,aux};\bm{[M]}_{\bullet, i},\Sigma\right)\right)\\
&=\tilde{C}^{P}_{\bm{\rho}^P_{i}}\left(F\left(\widetilde{\bm{X}}^P_{i,obs},\widetilde{\bm{X}}^P_{i,aux};\bm{[M]}^P_{\bullet i},\Sigma\right)\right) \tilde{C}^{I}_{\bm{\rho}^I_{i}}\left(F\left(\widetilde{\bm{X}}^I_{i,obs},\widetilde{\bm{X}}^I_{i,aux};\bm{[M]}^I_{\bullet i},\Sigma\right)\right),
\end{split}
\end{equation}
with supper script $P$ and $I$ denote the components for the log payments and log incurred losses in the $i$-th development year respectively and the density is given by
\begin{equation}
\begin{split}
&f\left(\widetilde{\bm{X}}^P_{i,obs},\widetilde{\bm{X}}^P_{i,aux},\widetilde{\bm{X}}^I_{i,obs},\widetilde{\bm{X}}^I_{i,aux}|\bm{[M]}_{\bullet i},\Sigma,\bm{\rho}^P_{i},\bm{\rho}^I_{i}\right) \\
&= \tilde{c}^P_{\bm{\rho}^P_{i}}\left(F\left(\widetilde{\bm{X}}^P_{i,obs},\widetilde{\bm{X}}^P_{i,aux};\bm{[M]}^P_{\bullet i},\Sigma\right)\right) \tilde{c}^I_{\bm{\rho}_{i}}\left(F\left(\widetilde{\bm{X}}^I_{i,obs},\widetilde{\bm{X}}^I_{i,aux};\bm{[M]}^I_{\bullet i},\Sigma\right)\right)\prod_{j=1}^{2J+1} \phi(\widetilde{X}_{j,i};M_{j,i},\Sigma_{i,i}),
\end{split}
\end{equation}
where
\begin{equation*}
\begin{split}
& \tilde{c}^{S}_{\bm{\rho}_{i}}\left(F\left(\widetilde{\bm{X}}^S_{i,obs},\widetilde{\bm{X}}^S_{i,aux};\bm{[M]}^{S}_{\bullet i},\Sigma\right)\right)
= w_1 c^{G}_{\rho^{(G,S)}_i}\left( F_{1,i}\left(\widetilde{\bm{X}}^S_{1,i,obs};\tilde{M}^S_{1,i},\Sigma_{1,1}\right),\ldots, F_{J,i}\left(\widetilde{\bm{X}}^S_{J,i,aux};\tilde{M}^S_{J,i},\Sigma_{J,J}\right) \right) \\
&\; \; + w_2 c^{F}_{\rho^{(F,S)}_i}\left( F_{1,i}\left(\widetilde{\bm{X}}^S_{1,i,obs};\tilde{M}^S_{1,i},\Sigma_{1,1}\right),\ldots, F_{J,i}\left(\widetilde{\bm{X}}^S_{J,i,aux};\tilde{M}^S_{J,i},\Sigma_{J,J}\right) \right) \\
& \;\;+ (1-w_1-w_2)
c^{C}_{\rho^{(C,S)}_i}\left( F_{1,i}\left(\widetilde{\bm{X}}^S_{1,i,obs};\tilde{M}^S_{1,i},\Sigma_{1,1}\right),\ldots, F_{J,i}\left(\widetilde{\bm{X}}^S_{J,i,aux};\tilde{M}^S_{J,i},\Sigma_{J,J}\right) \right), \; S \in \left\{P,I\right\},
\end{split}
\end{equation*}
and such that $w_1 + w_2 + (1-w_1-w_2) = 1$. This specifies a mixture of central, upper and lower tail dependence as denoted by the mixture of Archimedian copula models made up of Frank, Clayton and Gumbel members, such that for the source of data $S$, the copula parameters for each Archimedian family member is given by $\rho^{(G,S)}_i > 0$, $\rho^{(C,S)}_i > 1$ and $\rho^{(F,S)}_i \in \mathbb{R}/\left\{0\right\}$. Therefore the total conditional distribution corresponding to the likelihood model considered is given by,
\begin{equation} \label{EqnFullDataLH}
\begin{split}
f\left(\tilde{X}|M,\Sigma,\Omega,\bm{\rho}\right) &= \underbrace{\prod_{i=0}^J
\tilde{c}^P_{\bm{\rho}^P_{i}}\left(F\left(\widetilde{\bm{X}}^P_{i,obs},\widetilde{\bm{X}}^P_{i,aux};\bm{[M]}^P_{\bullet i},\Sigma\right)\right) \tilde{c}^I_{\bm{\rho}^I_{i}}\left(F\left(\widetilde{\bm{X}}^I_{i,obs},\widetilde{\bm{X}}^I_{i,aux};\bm{[M]}^I_{\bullet i},\Sigma\right)\right)}_{\text{Copula Dependence in Data Augmented PIC Likelihood}} \\
& \; \; \times \underbrace{\frac{\exp\left(-\frac{1}{2}\text{tr}\left[\Omega^{-1}\left(\widetilde{X} - M\right)'\Sigma^{-1}\left(\widetilde{X} - M\right)\right]\right)}{\left(2 \pi\right)^{(2J^2+3J+1)/2}\left|\Omega\right|^{(2J+1)/2}\left|\Sigma\right|^{(J+1)/2}}.
}_{\text{Marginal Distribution in Data Augmented Likelihood PIC Model}}
\end{split}
\end{equation}
}
\item{Assume that the tail dependence features of the Data-Augmented copula PIC model are such that the dependence structure is homogeneous accross accident years, $\bm{\rho}^P = \bm{\rho}^P_{i}$ and $\bm{\rho}^I = \bm{\rho}^I_{i}$ for all $i \in \left\{0,1,2,\ldots,J\right\}$.}
\item{Conditional on $\Sigma$, $\bm{\Phi} = \left[\Phi_0,\Phi_1,\ldots,\Phi_J\right]$ and $\bm{\Psi} = \left[\Psi_0,\Psi_1,\ldots,\Psi_J\right]$ the hierarchical prior distribution on the auxiliary payment data for the $i$-th accident year is given by a normal distribution, centered on the development year mean,
\begin{equation}
\begin{split}
&\widetilde{\bm{X}}^P_{i,aux} \sim \mathcal{N}\left(\left[\Phi_{J-i+1},\Phi_{J-i+2},\ldots,\Phi_J\right],\Sigma^P_2\right).
\end{split}
\end{equation}
The hierarchical prior distribution on the auxiliary incurred loss data for the $i$-th accident year is given by
\begin{equation}
\begin{split}
&\widetilde{\bm{X}}^I_{i,aux} \sim
\mathcal{N}\left(\left[\Psi_{J-i+1},\Psi_{J-i+2},\ldots,\Psi_J\right],\Sigma^I_2\right),
\end{split}
\end{equation}
with $\Sigma_2$ the lower portion of covariance $\Sigma$ corresponding to the lower triangle matrix from $(J-i+1)$ through to $J$ for all $i \in \left\{0,1,2,\ldots,J\right\}$.\\
}
\item{For all accident years, $i \in \left\{0,1,\ldots,J\right\}$, the ultimate payment losses and incurred losses are equal a.s., $P_{i,J} = I_{i,J}, \; \mathbb{P}-\text{a.s.}$}
\item{The matrix $\widetilde{\Sigma}$ is positive definite and components of $\bm{\Theta}$ are independent with prior distributions
\begin{equation}
\Phi_i \sim \mathcal{N}\left(\phi_i,s_i^2\right) \; \text{ and } \; \Psi_j \sim \mathcal{N}\left(\psi_j,t^2_j\right)
\end{equation}
and hyper-prior distributions
\begin{equation}
s^2_i \sim \mathcal{IG}\left(\alpha_i,\beta_i\right) \; \text{ and } \; t^2_j \sim \mathcal{IG}\left(a_j,b_j\right)
\end{equation}
for all $i \in \left\{1,\ldots,J\right\}$ and $j \in \left\{0,\ldots,J\right\}$.
}
\item{The matrix $\Sigma$ is distributed as $\Sigma \sim \mathcal{IW}\left(\Lambda,k\right)$ and the copula parameters are distributed as $\rho^{G,P} \sim \mathcal{IG}\left(\alpha^G,\beta^G\right)$, $\rho^{C,P} \sim \mathcal{IG}\left(\alpha^C,\beta^C\right)$ and $\rho^{F,P} \sim \mathcal{N}\left(0,\sigma^F\right)$}
\end{itemize}
\end{model ass}
Hence, we have made precise the auxilliary data scheme used in formulating the Data-Augmented-PIC model. In particular illustrating the importance of the role of the auxiliary data in evaluation of the model and estimation of the PIC claim development factors. Also we note we get indirectly via the data augmentation the distribution for the predicted payment and incurred Loss reserves.
\begin{remark} The following remarks provide motivation for the Data-Augmentation and resulting incorporation of auxiliary payment and incurred Losses data.
\begin{itemize}
\item{The use of data augmentation in the above model structure is critical in the PIC model formulation, since it allows one to ensure that the dependence structure considered (in this case a HAC-Mixture) is consistent both across accident years and across development years. \\
\textbf{Note:} In the case of a linear dependence structure such as with a covariance / correlation matrix under a Gaussian Copula or Independent Copula model, such as those presented previously under Models I,II, III, we have that conditional distributions and marginal distributions are Gaussian. This means that the evaluation of the likelihood is analytic without the need for auxiliary variables.
}
\item{In order to evaluate the likelihood one has two choices, to evaluate the observed data likelihood (Equation (\ref{EqnObsDataLH})) or to evaluate the full data likelihood (Equation (\ref{EqnFullDataLH})). }
\begin{itemize}
\item{The PIC copula model equivalent of Equation \ref{EqnDAGeneric} is the observed data likelihood is given for the $i$-th accident year by
\begin{equation} \label{EqnObsDataLH}
\begin{split}
&p\left(\widetilde{\bm{X}}^{P}_{i,obs},\widetilde{\bm{X}}^{I}_{i,obs}|\bm{\Theta},\Sigma,\Omega,\bm{\rho}\right) \\ \nonumber
& \; \; = \int\cdots \int p\left(\widetilde{\bm{X}}^{P}_{i,obs},\widetilde{\bm{X}}^{I}_{i,obs}|\bm{\Theta},\Sigma,\Omega,\bm{\rho},\widetilde{\bm{X}}^{P}_{i,aux},\widetilde{\bm{X}}^{I}_{i,aux}\right) p\left(\widetilde{\bm{X}}^{P}_{i,aux},\widetilde{\bm{X}}^{I}_{i,aux}|\bm{\Theta},\Sigma,\Omega,\bm{\rho}\right)
d\widetilde{\bm{X}}^{P}_{i,aux}d\widetilde{\bm{X}}^{I}_{i,aux}\\ \nonumber
& \; = \int\cdots \int \tilde{c}^P_{\bm{\rho}^P_{i}}\left(F\left(\widetilde{\bm{X}}^P_{i,obs},\widetilde{\bm{X}}^P_{i,aux};\bm{[M]}^P_{\bullet i},\Sigma\right)\right) \tilde{c}^I_{\bm{\rho}^I_{i}}\left(F\left(\widetilde{\bm{X}}^I_{i,obs},\widetilde{\bm{X}}^I_{i,aux};\bm{[M]}^I_{\bullet i},\Sigma\right)\right) \\
& \; \; \; \; \; \times
f_{\widetilde{X}_{i,aux}}^{MVN}\left(\widetilde{x}_{i,aux};M_{i,aux},\Sigma^P_2 \oplus \Sigma^I_2\right) f_{\widetilde{X}_{i}}^{MVN}\left(\widetilde{x}_{i};M_{i},\Sigma\right) \; d\widetilde{\bm{X}}^{P}_{i,aux} \; d\widetilde{\bm{X}}^{I}_{i,aux}
\end{split}
\end{equation}
where matrix-variate Gaussian distributions $f_X^{MVN}()$ and $f_X^{MVN}$ are as defined in Lemma \ref{lemmaMatrixGaussianDist}
with $\widetilde{X}_{i,aux} = Vec\left(\widetilde{\bm{X}}^P_{i,aux},\widetilde{\bm{X}}^I_{i,aux}\right)$, $M_{i,aux} = Vec\left([\Phi_{J-i+1:J}]',[\Psi_{J-i+1:J-1}]'\right)$, \\ $\widetilde{X}_i=[\widetilde{\bm{X}}^P_{i,obs},\widetilde{\bm{X}}^P_{i,aux},\widetilde{\bm{X}}^I_{i,obs},\widetilde{\bm{X}}^I_{i,aux}]$ and $M_i = [\Phi_0,\ldots,\Phi_J,\Psi_0,\ldots,\Psi_J]$ the equivalent mean.}
\item{Clearly, the marginalization required to evaluate the Observed data likelihood involves intractable integration, except in special cases in which the copula models are Gaussian or independence copulas.}
\end{itemize}
\item{The full data likelihood comprised of observed and auxiliary data involves incorporating auxiliary variables to represent the unobserved data in the lower reserve triangle for payment and incurred loss triangles. These become part of the inference procedure and are required to be estimated jointly with the model parameters in the estimation methodology.}
\end{itemize}
\end{remark}
\section{Estimation via Adaptive Data-Augmented MCMC for Claims Reserving PIC Models}
It has been shown for the Independent and Gaussian copula models that we can obtain the observed data likelihood analytically. Therefore the posterior distribution for all the model parameters can be sampled via a MCMC procedure comprised of block Gibbs sampler updates. In the case of a more general copula dependence model in which the observed data likelihood cannot be analytically evaluated pointwise, we must resort to a Data Augmentation scheme. In this case we will be able to perform sampling via a general MCMC Metropolis-Hastings sampler. In particular we will consider automating such a sampler using an adaptive MCMC scheme.
\subsubsection{Adaptive Metropolis within Data-Augmented Copula PIC Models}
This section presents the adaptive proposal we use to sample the parameters and the auxiliary variables. The advantage of an adaptive MCMC mechanism is that it automates the proposal design through consideration of a proposal distribution that learns the regions in which the posterior distribution for the static model parameters and auxiliary data has most mass. As such, the probability of acceptance under such an on-line adaptive proposal is likely to improve as the iterations progress and the generated MCMC samples will ideally have reduced autocorrelation. In such cases the variance of Monte Carlo estimators of integrals of smooth functionals formed from such samples will be reduced.
There are several classes of adaptive MCMC algorithms, see \citet{roberts2009examples}. The distinguishing feature of adaptive MCMC algorithms, compared to standard MCMC, is the generation of the Markov chain via a sequence of transition kernels. Adaptive algorithms utilize a combination of time or state inhomogeneous proposal kernels. Each proposal in the sequence is allowed to depend on the past history of the Markov chain generated, resulting in many possible variants.
\citet{haario2005componentwise} develop an adaptive Metropolis algorithm with proposal covariance adapted to the history of the Markov chain was developed. \citet{andrieu2008tutorial} is presenting a tutorial discussion of the proof of ergodicity of adaptive MCMC under simpler conditions known as \textit{Diminishing Adaptation} and \textit{Bounded Convergence}. We note that when using inhomogeneous Markov kernels it is particularly important to ensure that the generated Markov chain is ergodic, with the appropriate stationary distribution. Two conditions ensuring ergodicity of adaptive MCMC are known as \textit{Diminishing Adaptation} and \textit{Bounded Convergence}. These two conditions are summarised by the following two results for generic Adaptive MCMC strategies on a parameter vector $\bm{\theta}$. As in \citet{roberts2009examples}, we assume that each fixed MCMC kernel $Q_{\gamma}$, in the sequence of adaptions, has stationary distribution $P\left(\cdot\right)$ which corresponds to the marginal posterior of the static parameters. Define the convergence time for kernel $Q_{\gamma}$ when starting from a state $\bm{\theta} \in E$, as $M_{\epsilon}\left(\bm{\theta},\gamma\right) = \text{inf}\{s \geq 1 : \|Q^s_{\gamma}\left(\bm{\theta};\cdot\right) - P\left(\cdot\right)\| \leq \epsilon $. Under these assumptions, they give the following two conditions which are sufficient to guarantee that the sampler produces draws from the posterior distribution as the number of iterates tend to infinity. The two sufficient conditions are:
\begin{itemize}
\item{ \textit{Diminishing Adaptation:} $\text{lim}_{n\to\infty}\text{sup}_{\bm{\theta} \in E}\|Q_{\Gamma_{s+1}}\left(\bm{\theta},\cdot\right) - Q_{\Gamma_{s}}\left(\bm{\theta},\cdot\right)\|_{tv} = 0$ in probability. Note, $\Gamma_s$ are random indices.}
\item{ \textit{Bounded Convergence:} For $\epsilon > 0$, the sequence $\{M_{\epsilon}\left(\bm{\theta},\Gamma_j\right)\}^\infty_{j=0}$ is bounded in probability.}
\end{itemize}
The sampler converges asymptotically in two senses,
\begin{itemize}
\item{\textit{Asymptotic convergence:} $\text{lim}_{j\to\infty}\|\mathcal{L}\text{aw}\left(\bm{\theta} \right)-P\left(\bm{\theta}\right)\|_{tv}=0$ in probability.}
\item{\textit{Weak Law of Large Numbers}: $\text{lim}_{j\to\infty}\frac{1}{j}\sum^{j}_{i=1}\phi \left(\bm{\theta}\right)=\int \phi(\bm{\theta})P(d\bm{\theta}) $ for all bounded $\phi : E \to R$.}
\end{itemize}
In general, it is non-trivial to develop adaption schemes which can be verified to satisfy these two conditions. In this paper we use the adaptive MCMC algorithm to learn the proposal distribution for the static parameters in our posterior $\bm{\Phi}$. In particular we work with an adaptive Metropolis algorithm utilizing a mixture proposal kernel known to satisfy these two ergodicity conditions for unbounded state spaces and general classes of target posterior distribution, see \citet{roberts2009examples}
for details.
\subsubsection{Euclidean and Riemann-Manifold Adaptive Metropolis within Data-Augmented Copula PIC Models}
This section presents the specific details of the Adaptive Metropolis algorithm that we combine with Data-Augmentation to obtain an MCMC sampler for the Data Augmented Mixture Copula PIC Model proposed. This involves specifying the details of the proposal distribution in the AdMCMC algorithm which samples a new proposed update vector $\bm{\Upsilon}^*$ and matrix $\widetilde{\Sigma}^*$ from an existing Markov chain state $\bm{\Upsilon}$ with
$$\bm{\Upsilon} = \left[\bm{\Phi},\bm{\Psi}, s^2_{0:J}, t^2_{0:J},\bm{\rho}, \widetilde{\bm{X}}^P_{1,aux},\ldots,\widetilde{\bm{X}}^P_{J,aux},\widetilde{\bm{X}}^I_{1,aux}\ldots, \widetilde{\bm{X}}^I_{J,aux}\right]$$ and matrix $\widetilde{\Sigma}$. At the $j$-th iteration of the Markov chain we have existing state $\bm{\Upsilon}^{(j-1)}$ and $\widetilde{\Sigma}^{(j-1)}$ which is used to construct the proposal distribution $q\left(\bm{\Upsilon}^{(j-1)},\bm{\Upsilon}^{*}\right)q\left(\widetilde{\Sigma}^{(j-1)},\widetilde{\Sigma}^{*}\right)$. The choices we make for the two proposals will involve a novel development of a new adaptive proposal for positive definite matrices, required for the covariance matrix $\widetilde{\Sigma}$ should we choose not to specify it as diagonal.
\textbf{Euclidean Space Adaptive Metropolis for Static Parameters:}\\
We first detail the proposal for updating $\bm{\Upsilon}$ using a mixture of multivariate Gaussian distributions as specified for an Adaptive Metropolis algorithm which involves sampling from the proposal
\begin{equation} \label{AdapMetroEuclid}
q\left(\bm{\Upsilon}^{(t-1)},\cdot\right)=w_{1}\mathcal{N}\left(\bm{\Upsilon};\bm{\Upsilon}^{(t-1)},\frac{\left(2.38\right)^{2}}{d}\mathbb{C}\text{ov}\left(\left\{\bm{\Upsilon}^{(j)}\right\}_{0 \leq j \leq t-1}\right)\right)+\left(1-w_{1}\right)\mathcal{N}\left(\bm{\Upsilon};\bm{\Upsilon}^{(t-1)},\frac{\left(0.1\right)^{2}}{d}I_{d,d}\right),
\end{equation}
where we define the sample covariance for Markov chain past history by $\mathbb{C}\text{ov}\left(\left\{\bm{\Upsilon}^{(j)}\right\}_{0 \leq j \leq t-1}\right)$ and we note the following recursive evaluation, which significantly aids in algorithmic computational cost reduction
{\small{
\begin{equation}
\begin{split}
\mathbb{E}\left(\left\{\bm{\Upsilon}^{(j)}\right\}_{0 \leq j \leq t}\right) & =\mathbb{E}\left(\left\{\bm{\Upsilon}^{(j)}\right\}_{0 \leq j \leq t-2}\right)+\frac{1}{t}\left(\bm{\Upsilon}^{(t-1)}-\mathbb{E}\left(\left\{\bm{\Upsilon}^{(j)}\right\}_{0 \leq j \leq t-1}\right)\right)\\
\mathbb{C}\text{ov}\left(\left\{\bm{\Upsilon}^{(j)}\right\}_{0 \leq j \leq t}\right) &=\frac{1}{t+1}\left(\left(\bm{\Upsilon}^{(t-1)}-\mathbb{E}\left(\left\{\bm{\Upsilon}^{(j)}\right\}_{0 \leq j \leq t}\right)\right)\left(\bm{\Upsilon}^{(t-1)}-\mathbb{E}\left(\left\{\bm{\Upsilon}^{(j)}\right\}_{0 \leq j \leq t}\right)\right)'-\mathbb{C}\text{ov}\left(\left\{\bm{\Upsilon}^{(j)}\right\}_{0 \leq j \leq t-1}\right)\right)\\
&\;\; + \mathbb{C}\text{ov}\left(\left\{\bm{\Upsilon}^{(j)}\right\}_{0 \leq j \leq t-1}\right).
\end{split}
\end{equation}
}}
The theoretical motivation for the recommended choices of scale factors 2.38, 0.1 and dimension $d$ are provided in \citet{rosenthal2008optimal}.
\textbf{Riemannian Manifold Adaptive Metropolis for Covariance Matrices:}\\
Next we develop a novel proposal distribution for the sampling of the covariance matrix $\widetilde{\Sigma} \in \text{Sym}^+(d)$ in an adaptive MCMC proposal, restricted to the Riemann manifold of symmetric, postive definite $(d \times d)$ matrices, denoted by the space $\text{Sym}^+(d)$.
\begin{remark} First, we note two properties of the marginal posterior $p\left(\left. \widetilde{\Sigma}\right|\left\{ \widetilde{X}^P_{i,obs},\widetilde{X}^I_{i,obs} \right\}_{0\leq i \leq J}\right)$: its distribution is restricted to the Riemann-manifold of symmetric positive definite matrices, but in general will not be Inverse-Wishart; second, the Markov chain samples drawn from this marginal distribution at iteration $t$, $\left\{\widetilde{\Sigma}^{(s)}\right\}_{0\leq s \leq t}$, are not independent. The consequence of this is that we cannot simply apply the property of closure under convolution of independent Wishart distributed random matrices to find a suitable proposal.
\end{remark}
Therefore, we will adopt a strategy to perform adaptive moment matching of a distribution with support $\text{Sym}^+(d)$. We detail one possibility involving an inverse Wishart distribution fitted to the sample mean of the marginal posterior for the covariance. We note that future work could also consider specifying a distribution on the superset of the Riemannian manifold of symmetric positive definite matrices, given by the Riemannian manifold of symmetric matrices $\text{Sym}^+(d) \subset \text{Sym}(d)$.
\textbf{Adaptive Metropolis inverse Wishart Mixture:}
We note that one way to achieve this is a mixture of inverse Wishart distributions given by
\begin{equation}
q\left(\widetilde{\Sigma}^{(t-1)},\cdot\right)=w_{1}\mathcal{IW}\left(\widetilde{\Sigma};\Lambda^{adap}_t\left(\left\{\widetilde{\Sigma}^{(s)}\right\}_{0 \leq s \leq t-1}\right),p\right)+\left(1-w_{1}\right)\mathcal{IW}\left(\widetilde{\Sigma};\Lambda,p\right).
\end{equation}
Here, the adaptive proposal mixture component is specified through fixing the degrees of freedom $p$ and then selecting $\Lambda^{adap}_t$ with respect to the sample average of the covariance matrices $\left\{\widetilde{\Sigma}^{(s)}\right\}_{0\leq s \leq t-1}$ which are samples from the matrix-variate marginal posterior in the Markov chain, thereby adapting the proposal to the Markov chain history. To perform the moment matching (Equation (\ref{EqnMomentMatch})), we note that we need to ensure that the sample average considered is restricted to the Riemann-manifold of positive definite matrices.
\begin{equation} \label{EqnMomentMatch}
\Lambda^{adap}_t\left(\left\{\widetilde{\Sigma}^{(s)}\right\}_{0 \leq s \leq t-1}\right) = \widehat{\widetilde{\Sigma}}^{(t-1)} \left(p - dim(\widetilde{\Sigma}) -1 \right).
\end{equation}
This is satisfied through the choice of the estimator
\begin{equation}
\widehat{\widetilde{\Sigma}}^{(t-1)} = \frac{1}{t-1}\sum_{s=1}^{t-1} \widetilde{\Sigma}^{(s)}.
\end{equation}
To see this we observe that since we only form positive linear combinations of matrices on this manifold, with a scaling, such linear combinations will always remain on the manifold $\text{Sym}^+(d)$.
\FloatBarrier
\section{Real Data Analysis}
To illustrate the proposed models and compare with existing models and estimation methods in the actuarial literature we consider, as in \citet{merz2010paid}, the example presented in \citet{dahms2008loss} and \citet{dahms2009claims} (Tables 10 and 11). As in the second analysis framework in \citet{merz2010paid}, we treat the claim development factors, the likelihood dependence parameters and the hyperparameters on the claim development factor priors as parameters which we incorporate into the posterior inference.
We present two sets of results, the first studies the performance of the adaptive Markov chain Monte Carlo algorithms developed for the estimation and inference of the posterior distributions for the PIC-Copula models for Gaussian Copula (Models III) and the Data-Augmented-Mixture-Copula PIC (Models IV). The second stage of results assesses the estimation of predictive distributions and dependence features of the PIC claims reserving models compared to the independent PIC Model, the payment only model and the incurred only models. In particular, we focuss analysis on the data sets studied in \citet{merz2010paid} for comparison of the influence of dependence features in PIC models versus independence assumptions when performing PIC claims reserving.
\noindent \textbf{Convergence Analysis:} In all the Markov chain Monte Carlo simulations, for each model (payment, payment-incurred Gaussian copula Model III; and Data-Augmented hierarchical Archemdean mixture copula Model IV), we carried out convergence diagnostics. This included the Gelman-Rubin R-statistics (all less than 1.5), the ACF plots for each parameter were checked to ensure all parameters had ACF's which were less than 10\% by lag 20. Then the first 20\% of samples were discarded as burnin and the remaining samples were used in inference results presented below.
\subsection{Results: Euclidean and Riemann-Manifold Adaptive Metropolis for hierarchical Bayesian Copula PIC Models}
In the simualtion results, we consider a block Gibbs sampler with the following three stages performed at each iteration of the adaptive Metropolis-within-Gibbs sampler for the PIC Model III and Model IV:\\
\begin{enumerate}
\item[]{\textbf{Stage 1:} Perform exact sampling of the development factors and their hyperparameters under the conjugacy results developed.\\}
\item[]{\textbf{Stage 2:} Perform Euclidean space Adaptive Metropolis updates of the Augmented Data variables using proposal in Equation (\ref{AdapMetroEuclid}).\\}
\item[]{\textbf{Stage 3:} (Gaussian Copula Model III) - Perform Riemannian space Adaptive Metropolis updates of the covariance matrix in the Gaussian copula. Note, we consider the constrained specifications presented in the ``Dependent Lag Years'' model specification in Section \ref{DevLagYrs}, Equation (\ref{EqnTelescopingDependence}). Under this hierarchical Bayesian model, the joint covariance between all observed payment and incurred loss data under the dependent development years assumption, satisfies a telescoping diagonal block size form covariance matrix structure. Hence, the sampling of this structure can be performed blockwise on each covariance sub-block;\\
(Mixture Clayton-Gumbel Copula Model IV) - Perform Euclidean space Adaptive Metropolis updates of the mixture copula parameters.}
\end{enumerate}
\subsubsection{Hierarchical Bayesian Gaussian Copula (telescoping block covariance) PIC (Model III)}
This section presents the estimation results for the Gaussian Copula based PIC models (Model III) on the real data. Figure \ref{PIC_GausCopMod} summarizes the dependence structure by a heatmap for the posterior distribution of the Gaussian copula covariance matrix. As mentioned in the introduction, the telescoping block covariance refers to the fact that the covariance structure is reducing in rank by 1 on each diagonal block for the payment data and then the incurred data. This model has the joint covariance between all observed payment and incurred loss data under the assumption that the development years are dependent, satisfying a telescoping diagonal block size form covariance matrix structure. Summarising the information from such posterior samples for distributions of covariance matrices is non-trivial as discussed in \citet{tokuda2011visualizing}, where they develop a four layer approach. Our article adopts aspects of the ideas proposed in \citet{tokuda2011visualizing} to interpret the features of the posterior distribution samples for the dependence structures.
The posterior mean for estimated PIC covariance structure is obtained by using Monte Carlo samples from the Riemann-Manifold Adaptive Metropolis sampler and given by the estimator,
\begin{equation}
\mathbb{E}\left[\tilde{\Sigma}|\bm{P},\bm{I}\right] = \frac{1}{S}\sum_{s=1}^S \left\{\left(\bigoplus_{i=0}^J \Sigma^{P}_i\right)\oplus\left(\bigoplus_{i=0}^J \Sigma^{I}_i\right)\right\}^{(s)},
\end{equation}
where $\left\{\left(\bigoplus_{i=0}^J \Sigma^{P}_0\right)\oplus\left(\bigoplus_{i=0}^J \Sigma^{I}_0\right)\right\}^{(s)}$ is the $s$-th sample of the $J(J-1) \times J(J-1)$ covariance matrix. The estimated posterior mean covariance matrix is reported in a heatmap for the correlation matrix in Figure \ref{PIC_GausCopMod}. In addition, we present examples based on posterior mean covariance for covariance sub-blocks $p\left(\Sigma_4^P |\bm{P},\bm{I}\right)$ and then for $p\left(\Sigma_4^I |\bm{P},\bm{I}\right)$, where $\Sigma_4^P \in SP^{+}(6)$ and $\Sigma_4^I \in SP^{+}(5)$, again converted to heatmaps of the correlation. We see that although the priors selected for the dependence features in Model III in all cases favoured independence, since the scale matrices were all diagonal i.e. $\Lambda_5^P = \mathbb{I}_{6}$ and $\Lambda^I_4 = \mathbb{I}_{5}$, the resulting summaries of the marginal posteriors of the covariances clearly indicate non-trivial dependence patterns in the development years within the payments data and the incurred loss data. This is observed throughout each sub-block covariance matrix.
\begin{figure}[ht]
\includegraphics[height=6cm, width=\textwidth]{figures/PosteriorCorrelationMatrixF.eps}
\includegraphics[height=6cm, width=0.5\textwidth]{figures/PosterioSigma0Sigma1V2.eps}
\includegraphics[height=6cm, width=0.5\textwidth]{figures/PosterioSigma0ISigma1IV2.eps}
\caption{\textbf{Top panel:} Heatmap of the posterior distribution for the Gaussian copula covariance matrix $(100 \times 100)$, summarised by the heat map for the mean of correlation structure using samples from the Riemannian Manifold Adaptive Metropolis sampler under restriction to a telescoping diagonal block form. \textbf{Bottom Left Panel:} Heatmap for the posterior distribution sub-block covariance matrices $\Sigma_0^P$ and $\Sigma_1^P$ converted to correlation matrices. \textbf{Bottom Right Panel:} Heatmap for the posterior distribution sub-block covariance matrices $\Sigma_0^I$ and $\Sigma_1^I$ converted to correlation matrices. The color key is given at the top left.
}
\label{PIC_GausCopMod}
\end{figure}
Table \ref{Tab:CovPost} provides a second summary of the posterior for the covariance matrix which further demonstrates features of the dependence properties in the payment and incurred data per accident year and involves the estimates of the largest eigenvalue of each block diagonal matrix for the payment and incurred data as summary statistics. These estimates are given by
\begin{equation}
\widehat{\lambda}^{(s)}_i = \arg \max \left(\det(\Sigma_i^{(s)} - \lambda\mathbb{I})=0\right).\\
\end{equation}
The largest eigenvalue provides information on the posterior distribution of the magnitude of the first principal component of each development year, decomposed by accident year. That is, we can quantify in the PIC model, by accident year, the proportion of residual variation in the log payments for accident year $i$ currently unexplained by the development factors $\Phi_{0:J-i}$, which were jointly estimated in the PIC model and assumed constant accross each accident year (i.e. constant per development year) for parsimony. We can also repeat this for the incurred loss data. Suppose that a principal component analysis is performed, decomposing the variation in the payment and incurred data for each accident year $i$ with respect to the variation unexplained by the development factors in the PIC model. Then, up to proportionality, the distribution of the eigenvalues corresponds to the proportion of contribution from the leading eigenvector (principal component). When this is coupled with the fact that we can also easily obtain samples from the marginal posterior distribution of the leading eigenvector of the covariance matrix for the $i$-th accident year's payment of incurred loss data in the PIC model, then we get complete information per accident year on the ability of the development factors in the PIC model to explain variation in the observed loss data. Table \ref{Tab:CovPost} summarises the results for the average PCA weight (largest eigenvalue) and average posterior eigenvector.
\citet{tokuda2011visualizing} develops a framework which formalizes an approach to the summary of dependence structures. For the running example of results that we present for distributions $p\left(\Sigma_4^P |\bm{P},\bm{I}\right)$ and $p\left(\Sigma_4^I |\bm{P},\bm{I}\right)$, under such an approach the third and fourth layers of summary are presented in Figure \ref{PIC_GausCopModP4I4CovPosts}. This involves the presentation of contour maps of these marginal posteriors that are constructed using adaptive MCMC samples of these matrices.
In Figure \ref{PIC_GausCopModDevFactMargPosts}, the development factors for payment and incurred data marginal posterior distributions are presented along with the posteriors of the hyperparameters for the Gaussian Copula based PIC models (Model III). Finally, we also compare the estimated posterior marginal distributions of the development factors for the payment and incurred loss triangles for the models: payment only model; the incurred only model; the Gaussian Copula (Model III) dependent model; the PIC [Full] independent model and the PIC [Partial] independent model of \citet{merz2010paid}. The results of this comparison include the posterior mean estimates of $\mathbb{E}\left[\Phi_i|\bm{P},\bm{I}\right]$ and $\mathbb{E}\left[\Psi_i|\bm{P},\bm{I}\right]$, for all $i \in \left\{0,1,\ldots,J\right\}$ and the posterior quantiles for left and right tails as measured by the fifth and ninety-fifth percentiles, given in Table \ref{Tab:MMSEModels}. \textit{We note that the results in this section for the Gaussian copula models are obtained using the log ratio observational data and the restults for the Mixture Archimedian copula model are more conveniently obtained using the log observations (not ratio data).}
It is also worth noting other approaches that can be adopted in the case of the Gaussian copula model. One could also included a data-augmentation stage in the analysis as was utilised in the Mixture Archimedian copula example. In addition, the covariance matrices could have been specified under different structures with more or less parsimony. The examples utilised in this section were those which provided a reasonable trade-off between parsimonious model specification, while allowing a meaningful decomposition of the results.
The results of the comparison between the Gaussian copula PIC model and the independent PIC model illustrated that whilst the posterior marginal mean development factor estimates are not affected by the dependence feature included, the marginal posterior shape is affected. This is reflected by the comparison of the posterior confidence intervals for the Gaussian copula PIC model when compared to the payment or incurred individual models where there is a significant difference present in the shapes of the marginal posterior. It is expected that this will have implications for the estimation of reserves using these different will be quantified in the next section.
\begin{figure}[ht]
\includegraphics[height=6cm, width=\textwidth]{figures/Final_I4.eps}
\includegraphics[height=6cm, width=\textwidth]{figures/Final_P5.eps}
\caption{Heatmaps for the block diagonal covariance matrices $\Sigma_4^P$ ($2 \times 2$ sub-plot 1) and $\Sigma_4^I$ ($2 \times 2$ sub-plot 2). These are obtained using samples from the Riemannian Manifold Adaptive Metropolis sampler. Samples from the Posterior distribution of the telescoping diagonal block size form covariance matrix structures of the Gaussian copula under the hierarchical Bayesian model which has the joint covariance between all observed payment and incurred loss data under the dependent development years. Each set of $4 \times 4$ panels, starting from the top, summarizes the posterior distributions for the covariance matrices for $s \in \left\{P,I\right\}$ according to:
\textbf{\textit{Top Left Panel}}: contour map of posterior samples $\log\left[\Sigma_4^s\right]_{1,1}$ vs $\log\left[\Sigma_4^s\right]_{5,5}$.
\textbf{\textit{Top Right Panel}}: contour map of posterior samples $\log\left[\Sigma_4^s\right]_{1,1}$ vs $\left[\Sigma_4^s\right]_{1,5}$.
\textbf{\textit{Bottom Left Panel}}: kernel density estimator of the posterior distribution of the trace of the covariance matrix using samples $\left\{\log \text{tr}\left(\Sigma_4^s\right)\right\}$.
\textbf{\textit{Bottom Right Panel}}: scatter plot of posterior samples of the first, second and third largest eigenvalues scaled by total of the eigen valuse - (PCA weights - for linear combinations of the development factors when explaining variation in observed payment and incurred data for a given accident year).
}
\label{PIC_GausCopModP4I4CovPosts}
\end{figure}
\begin{figure}[ht]
\includegraphics[height=5cm, width=0.45\textwidth]{figures/Phi_MarginalPosteriors_ModelIII.eps}
\includegraphics[height=5cm, width=0.45\textwidth]{figures/Psi_MarginalPosteriors_ModelIII.eps} \\
\includegraphics[height=5cm, width=0.45\textwidth]{figures/PhiHP_MarginalPosteriors_ModelIII.eps}
\includegraphics[height=5cm, width=0.45\textwidth]{figures/PsiHP_MarginalPosteriors_ModelIII.eps}
\caption{Boxplot summaries of the marginal posterior distributions obtained using samples from the Riemannian Manifold Adaptive Metropolis sampler. Samples from the Posterior distribution under a telescoping diagonal block size form covariance matrix structures of the Gaussian copula under the hierarchical Bayesian model which has the joint covariance between all observed payment and incurred loss data under the dependent development years.
\textbf{\textit{Top Left Panel}}: box plots of marginal posterior distributions for $p\left(\Phi_i|\bm{P},\bm{I}\right)$.
\textbf{\textit{Top Right Panel}}: box plots of marginal posterior distributions for $p\left(\Psi_i|\bm{P},\bm{I}\right)$.
\textbf{\textit{Bottom Left Panel}}: box plots of marginal posterior distributions for $p\left(s_i|\bm{P},\bm{I}\right)$.
\textbf{\textit{Bottom Right Panel}}: box plots of marginal posterior distributions for $p\left(t_i|\bm{P},\bm{I}\right)$.
}
\label{PIC_GausCopModDevFactMargPosts}
\end{figure}
\subsubsection{Data-Augmented hierarchical Bayesian Mixture-Archimedian Copula PIC (Model IV)}
This section presents the estimation results for the mixture of Clayton and Gumbel Copula based PIC models (Model IV) on the real data are presented in this section. Figure \ref{PIC_ArchMixCopMod} presents a summary of the mixture copula dependence structure obtained from posterior samples of the copula parameters under the hierarchical Bayesian model. The results in this section are obtained using the log observational data, not ratio data. The figures summarise succinctly the estimated posterior dependence structure for the hierarchical Bayesian mixture Copula model, through plots of the dependence structure as captured by the estimatd mixture copula distribution, the scatter plots of copula parameter for the lower tail and rank correlation (Kendall's tau) and the upper tail copula parameter versus rank correlation. These results clearly demonstrate posterior evidence for non-trivial tail dependence features in the payment and incurred data, as well as potential for asymmetry in the upper and lower tail dependence. Note, uniformative prior choices were made on the copula parameters with uniform priors over $[0,50]$ and $[1,50]$ respectively, indicating these estimated copula parameters are data driven results.
\begin{figure}[ht]
\includegraphics[height=8cm, width=\textwidth]{figures/CopulaParamMixFit.eps} \par \noindent
\caption{Copula Dependence Parameter Posterior distributions estimated under the Data-Augmented Mixture Copula PIC Model IV. A mixture of Archimedean copula models is considered, with Clayton and Gumbel copula choices, allow for possible asymmetry in the tail dependence over development years.We chose uniformative uniform priors $U[0,20]$ for the copula parameters.
\textbf{\textit{Top Left Panel}}: Contour map of posterior estimated mixture copula dependence distribution between development years over paid and incurred loss data, with homogeneous dependence assumptions over accident years (estimated from posterior mean of $\rho_C^{MMSE}$ and $\rho_G^{MMSE}$.
\textbf{\textit{Top Right Panel}}: Surface plot of posterior estimated mixture copula dependence distribution between development years over paid and incurred loss data, with homogeneous dependence assumptions over accident years (estimated from posterior mean of $\rho_C^{MMSE}$ and $\rho_G^{MMSE}$.
\textbf{\textit{Bottom Left Panel}}: Scatter plot of posterior samples used to estimate Kendall's tau rank correlation versus copula parameter for the Clayton mixture component.
\textbf{\textit{Bottom Right Panel}}: Scatter plot of posterior samples used to estimate Kendall's tau rank correlation versus copula parameter for the Gumbel mixture component.
}
\label{PIC_ArchMixCopMod}
\end{figure}
Figure \ref{PIC_MixCopModDevFactMargPostss} presents the development factors for payment and incurred data marginal posterior distributions along with the hyperparameter marginal posteriors for the Data-Augmented Mixture Copula based PIC models (Model IV).
\begin{figure}[ht]
\includegraphics[height=5cm, width=0.45\textwidth]{figures/Phi_MarginalPosteriors_ModelIV.eps}
\includegraphics[height=5cm, width=0.45\textwidth]{figures/Psi_MarginalPosteriors_ModelIV.eps} \\
\includegraphics[height=5cm, width=0.45\textwidth]{figures/PhiHP_MarginalPosteriors_ModelIV.eps}
\includegraphics[height=5cm, width=0.45\textwidth]{figures/PsiHP_MarginalPosteriors_ModelIV.eps}
\caption{Boxplots of the marginal posterior distributions of the development factors and hyperparameters.
\textbf{\textit{Top Left Panel}}: box plots of marginal posterior distributions for $p\left(\Phi_i|\bm{P},\bm{I}\right)$.
\textbf{\textit{Top Right Panel}}: box plots of marginal posterior distributions for $p\left(\Psi_i|\bm{P},\bm{I}\right)$.
\textbf{\textit{Bottom Left Panel}}: box plots of marginal posterior distributions for $p\left(s_i|\bm{P},\bm{I}\right)$.
\textbf{\textit{Bottom Right Panel}}: box plots of marginal posterior distributions for $p\left(t_i|\bm{P},\bm{I}\right)$.
}
\label{PIC_MixCopModDevFactMargPostss}
\end{figure}
\FloatBarrier
\section{Comparison of PIC reserving with Gaussian Copula PIC and Mixture Archimedian Copula PIC Models}
This section discuss the effect of modelling the dependence structures on the reserving estimates. First we note two important details in calculating the reserves. We need to be able to draw samples from the predictive distributions for the payment and incurred data given below, for each accident year $i$, using
\begin{equation*} \label{Eqn:predictive}
p\left(P_{i,J}|\bm{P},\bm{I}\right) = \int p\left(P_{i,J}|P_{i,1:J-i},\bm{\Theta}\right)p\left(\bm{\Theta}|\bm{P},\bm{I}\right)d\bm{\Theta}\;\; \mathrm{and} \;\; p\left(I_{i,J}|\bm{P},\bm{I}\right) = \int p\left(I_{i,J}|I_{i,1:J-i},\bm{\Theta}\right)p\left(\bm{\Theta}|\bm{P},\bm{I}\right)d\bm{\Theta}.\\
\end{equation*}
In general it is not possible to solve these integrals analytically. Howerver, for the Gaussian copula models developed in this paper, under the results in Lemma \ref{LemmaTransformData}, one adopt the results of \citet{merz2010paid}[Theorem 2.4] to obtain analytic Gaussian predictive distributions. Alternatively, the predicitive distributions can be estimated as described in \citet[Section 3.3]{peters2010chain}. Although the results in Table \ref{Tab:MMSEModels} demonstrate that the incorporation of the dependence structures does not significantly alter the posterior mean of the development factors for the payment and incurred loss data, it is clearly possible for the predictive distribution to be altered, since the shape of the posterior distribution is altered by the dependence features. Second, regarding the hierarchical mixture Archimedian copula model, it does not admit an analytic solution for the predictive distribution. This does not matter if a data augmentation stage is set up in the joint posterior distribution to sample cumulative payments, since then we can use the MCMC sampler output for the ultimate cumulative payment and incurred losses in each accident year.
Finally, we also note that a simple Monte Carlo based approximation for the ultimate claim can be constructed. Take the samples from the MCMC output for the PIC model of interest (sampled from the complete PIC model with dependence features present) and then utilise these samples to construct a Laplace approximation to the predictive observation distribution for example $p\left(P_{i,J}|P_{i,1:J-i},\bm{\Theta}\right)$ which involves a normal approximation around the MAP or locally around each Monte carlo sample for the development factors, with precision given by the sampled observation covariance structure. Though this is not required, as we have shown for the Gaussian copula models independence models, it may be useful for alternative copula based models with simple data-augmentation approaches. In addition a second alternative would be to utilise in the predictive distribution the marginal distributions.
Figure \ref{PIC_UltimateRatio} presents the log posterior predictive distribution for the ultimate total claim given by the predictive distribution for the log of the cumulative payment over each accident year $\sum_{i=0}^JP_{i,J}$ for the full Bayesain models which incorporate priors on observation error, development factors and hyperpriors for precision of the development factors. We see that all three models are in good agreement with each other with the dependence parameters affecting the variance and tail behaviour of the distributions.
\begin{figure}[ht]
\includegraphics[height=8cm, width=\textwidth]{figures/UltimatePaymentLossTotal_EachModelFinal.eps} \par \noindent
\caption{Boxplots of the predictive distributions obtained from the MCMC samples. Ultimate Bayesian predictive distributions for log payment data from the payment only predictive distribution, the Full Independent PIC model, and the hierarchical PIC Mixture Copula model via Data Augmentation predictive distribution.
\textbf{\textit{Left Panel}}: Posterior predictive distribution box plots from samples.
\textbf{\textit{Right Panel}}: Kernel density estimates of the predictive distributions.
}
\label{PIC_UltimateRatio}
\end{figure}
Next we consider the distributions of the outstanding loss liabilities estimated using the S samples from the MCMC obtained for the posterior PIC model. We denoted these by random variables $\left\{R(\bm{P},\bm{I})^{(s)}\right\}_{s=1:S}$ where $R(\bm{P},\bm{I})^{(s)} = P_{i,J} - P_{i,J-i}$ and depending on whether payment, incurred, or both data is present we denoted by $R(\bm{P})^{(s)}$, $R(\bm{I})^{(s)}$ and $R(\bm{P},\bm{I})^{(s)}$ respectively. Figure \ref{PICReservesDist} presents the MCMC estimated claims reserve marginal posterior predictive distributions for each accident year per model developed.
\begin{figure}[ht]
\includegraphics[height=7cm, width=\textwidth]{figures/UltimateReserves.eps} \par \noindent
\caption{Boxplots of log ultimate Bayesian predictive reserve distributions for payment data per accident year, compared to (Partial) PIC Independent posterior mean estimates from \citet{merz2010paid} (karge unfilled black circles). \textbf{Top Row:} the (Full) hierarchical PIC Mixture Copula model via Data Augmentation; \textbf{Second Row from Top: } the (Full) hierarchical PIC Gaussian Copula model; \textbf{Third Row from Top: } the (Full) Independent PIC model; \textbf{Bottom Row: } the (Full) payment Only model.
}
\label{PICReservesDist}
\end{figure}
We compared our results to those obtained in \citet{merz2010paid} and find good agreement between the mean reserve per accident year and each proposed model. In addition, we note the possible differences between the distributions can be attributed to the utilisation of the full versus partial hierarchical Bayesian models in this paper and the different dependence structures considered. Additionally, we note that further analysis on comparisons to existing models in the literature can be obained for the models of \citet{mack1993distribution}, \citet{dahms2008loss} and \citet{quarg2004munich} for this data analysis in \citet{merz2010paid} [Table 4] and in the spreadsheet provided by Professor Mario Wuethrich at URL\begin{footnote}{URL:\url{http://www.math.ethz.ch/~wueth/claims_reserving3.html}}\end{footnote}.
\section{Conclusions}
This paper extends the class of PIC models to combine the two different channels of information as proposed in \citet{merz2010paid} by introducing several novel statistical models for the dependence features present within and between the payment and incurred loss data. This allows us to obtain a unified ultimate loss prediction which incorporates the potential for general dependence features. To achieve this we developed full hierarchical Bayesian models which incorporate several different potential forms of dependence, including generalized covariance matrix structure priors based on inverse Wishart distributions and conditional Bayesian conjugacy in the PIC independent log-normal model. This forms a general class of Gaussian copula models which extends the approach of \citet{happ2011paid}.
Second, we develop a class of hierarchical mixture Archimedian copula models to capture potential for tail dependence in the payment and incurred loss data, again developing and demonstrating how to appropriately construct a full Bayesian model incorporating hyperpriors. In this regard, we also develop a class of models in which data-augmentation is incorporated to both overcome challenging marginal likelihood evaluations required for the MCMC methodology to sample from the PIC Bayesian models. This had the additional feature that it also allowed for joint Bayesian inference of the reserves as part of the posterior inference.
Finally, to perform inference on these approaches we developed an adaptive Markov chain Monte Carlo sampling methodology incorporating novel adaptive Riemann-manifold proposals restricted to manifold spaces (postive definite symmetric matrices) to sample efficiently the covariance matrices in the posterior marginal for the Gaussian copula dependence. We make these advanced MCMC accessible to the actuarial audience to address challenging Bayesian inference problems in Claims Reserving modelling.
The consequence of these models for actuaries is that a new extended suite of flexible dependence structures have been incorporated into the recently proposed PIC models. These can now be extended and compared to existing chain ladder methods. We perform an analysis on real payment and incurred loss data discussed in \citet{merz2010paid} and compare our models with the analysis for the independent PIC model (partial) and the (full) Bayesian PIC model as well as several different dependent models and the payment only model. Furthermore, we provide reference on further comparisons to the alternative models of \citet{mack1993distribution}, \citet{dahms2008loss} and \citet{quarg2004munich} for this data.
\section{Acknowledgement}
We would like to thank Prof. Mario Wuethrich of ETH for his discussions on this topic and for introducing us to the family of models that we have explored. The research of Robert Kohn was partially supported by ARC Discovery Grant DP0667069.
\FloatBarrier
\bibliographystyle{plainnat}
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1,108,101,566,805 | arxiv |
\section{Introduction}
Glass formers incur a sensational dynamical slowdown as the temperature is decreased~\cite{debenedetti:01}. At high temperatures, this slowdown is well explained by \ac{mf} theory~\cite{cavagna:09,maimbourg:16}, which predicts a breaking of ergodicity at a temperature $T_\mathrm{d}$: at $T<T_\mathrm{d}$ the typical configurations are confined in local minima of the energy landscape. Since, in \ac{mf}, exiting these local minima requires overcoming barriers of height diverging with the system size $N$, in the thermodynamic limit the system remains stuck in a portion of phase space.
Because in non-\ac{mf} systems the energy barriers are not diverging, real low-dimensional glasses are able to escape local minima of the energy when $T<T_\mathrm{d}$, and ergodicity is restored. Nevertheless, the dynamics below $T_\mathrm{d}$ is extremely slow, since overcoming a barrier of height $\Delta$ requires an exponentially large time $\tau$, given by the Arrhenius law, $\tau\propto\exp(\Delta/k_\mathrm{B}T)$.
This extremely slow low-temperature dynamical regime, consisting in jumps over energy barriers, is often referred to as (thermal) \emph{activation} or activated dynamics.
Activated dynamics can be studied in \ac{mf} models by considering exponentially large times in $N$~\cite{crisanti:00,baityjesi:18,stariolo:19}, which restricts numerical simulations to very small system sizes. From the analytical point of view, the situation is even harder, because the thermodynamic limit cannot be taken, since once $N$ is sent to infinity it becomes impossible to treat times of order $\exp(N)$. A consequence is that, to our knowledge, there are very few exact analytical calculations for the activated regime in canonical glass models.
A successful alternative analytical framework for the description of activated dynamics is the \ac{tm}~\cite{dyre:87,bouchaud:92}, a toy model that describes the simplest case of activation, in the absence of any other dynamical mechanism, where local minima (traps) are exited by reaching a threshold energy ${E_\mathrm{th}}$. A detailed non-trivial analysis has been made for the \ac{tm}~\cite{bouchaud:92,bouchaud:95,dyre:95}, and a series of works culminated recently has shown that for large enough system sizes the \ac{tm} correctly describes the activated long-time dynamics of a paradigmatic glass system, the \ac{rem}~\cite{benarous:02,benarous:08,gayrard:16,gayrard:18,baityjesi:18}.
In this paper we go beyond these works in two main ways. Firstly, we thoroughly characterise the dynamics of the \ac{rem} at \emph{all} time scales. Secondly, we unveil an interesting behaviour of the maximum reached energy $E_\MAX(t)$, which can be related to static quantities such as the threshold energy ${E_\mathrm{th}}$. Our findings are then tested by numerical simulations in the $p$-spin model, which provides a more complex case of glassy dynamics.
Our approach is based on record statistics, which addresses questions on the extreme events arising in series of random variables~\cite{gumbel:58,schehr:13}. The simplest case of tracking the maximum in a series of
independent random variables is well understood, but
as soon as some dependence is added to the problem, the results become less trivial, and rigorous results
are only known for a limited number of problems, such as random walks \cite{schehr:13} (see \cite{godreche:17} for a review) and
eigenvalue statistics \cite{johnstone:08,tracy:94,tracy:96} (see \cite{majumdar:14} for a review).
We focus on three paradigmatic models of mean-field spin glasses, the \acf{tm}, the \acf{rem}, and the $p$-spin
model, with particular emphasis on the \ac{rem}. We study the evolution of the maximum energy, $E_\MAX(t)$, reached by time $t$ after an instant quench from an infinite to a small temperature $T$.
The \ac{tm} consists of a random walk in a fully-connected graph. The graph is equipped with a random potential and the transition rates do not depend on the destination energy. For this model we find that $E_\MAX(t)$ has a simple behaviour at all times.
In the \ac{rem}, the random walk is on a more sparse graph and the transition rates depend on the difference between the starting and destination energy. These features lead to a rich non-trivial behaviour of $E_\MAX(t)$, in which we are able to identify several different regimes for different lapses of time from the quench (see Sec.~\ref{subsec:summary} for a summary and Tab.~\ref{tab:summ} for more detailed results). Following $E_\MAX(t)$ opens a window on exponentially large time scales which are usually hard to study.
Finally, we make a digression to the $p$-spin model, whose landscape is more complex than the one of the \ac{rem}. This is due to an explicit correlation among the energy levels, which are no longer \ac{iid} random variables. We show numerically that even in this case the maximum energy reached grows
logarithmically. We also argue that the maximal energy $E_\MAX(t/2,t)$, reached in the time interval $[t/2,t]$, can be used to identify the threshold energy ${E_\mathrm{th}}$ (i.e. the highest energy at which local minima are found) in a finite size system. We further show numerically that this estimate converges smoothly to the analytical prediction in the thermodynamic limit.
In section \ref{sec:models} we define the models we study. In section \ref{sec:trap} we describe the evolution of the maximum energy reached in the \ac{tm}, whereas in section \ref{sec:rem} it is analysed in the \ac{rem}. In section \ref{sec:pspin}
we discuss the $p$-spin model and we show how the maximum energy reached can be used to locate the threshold energy. Finally, we give some concluding remarks in section \ref{sec:conclusions}.
In appendix~\ref{app:I} we remind the reader some standard extreme-value statistics results, and in appendix~\ref{app:finitecorr} we take into account a number of subleading corrections which allow for a quantitative comparison of our results with simulations on very small systems.
\section{Models}\label{sec:models}
In this section we describe the three models treated in this paper.
\subsection{Trap model}
In the \ac{tm}~\cite{dyre:87,bouchaud:92, monthus:96}, $M$ states lie on a fully connected graph, and their energies are drawn independently from
an exponential distribution
\begin{equation}\label{eq:rhoexp}
\rho_\mathrm{exp}(E) = \alpha e^{\alpha E} \Theta(-E)\,,
\end{equation}
where $\Theta(\cdot)$ is the Heaviside step function and $\alpha>0$ is a parameter that will be set to 1.
The transition rate, $q_{i,j}$, from a state $i$ to a state $j$ does not depend on the target state $j$:
\begin{equation}\label{eq:rateTrap}
q_{i,j}=\frac{1}{M}e^{\beta E(i)}\,,
\end{equation}
where $\beta=1/T$ is the inverse temperature and $E(i)$ is the energy of state $i$.
This represents a system in which leaving a state (or \emph{trap}) requires reaching a threshold energy ${E_\mathrm{th}}=0$. Once the threshold is reached, the whole space of states becomes accessible. This model has weak ergodicity breaking at temperatures $T\leq{T_\mathrm{c}}=\frac{1}{\alpha}$ \cite{bouchaud:92} (see also paragraph \ref{sec:emaxTM}).
Summarizing, in the \ac{tm} the energies of different states are independent,
the space of states is fully connected,
and the dynamics depends only on the state issuing the transition and not on the target state,
implying that there is a constant threshold energy ${E_\mathrm{th}}=0$ separating the configurations.
\paragraph{Gaussian Trap model}
The \ac{tm} can be defined analogously for a Gaussian energy distribution
\begin{equation}\label{eq:rhogauss}
{\rho_\mathrm{gauss}}(E) = \frac{1}{\sqrt{2\pi N}}e^{-\frac{E^2}{2N}}\,,
\end{equation}
setting $M=2^N$.
In this case, also positive energies can be reached, but since ${E_\mathrm{th}}=0$ the dynamics leaves them quickly with a rate given by equation \eqref{eq:rateTrap}.
\subsection{Random Energy Model}
The \ac{rem} describes a system of $N$ binary spins \cite{derrida:80,derrida:81}. As a consequence, the total number of states is $M=2^N$, and from
each one it is possible to reach $N$ new states by flipping a single spin. The energy change induced by flipping any of the spins is assumed to be so drastic that the energy levels are independent from each other. The energy $E$ of a state is drawn from
the probability distribution ${\rho_\mathrm{gauss}}$ (see Eq.~\eqref{eq:rhogauss}).
The statics of the \ac{rem} can be solved exactly. There is a disordered phase for temperature $T>{T_\mathrm{c}}=\frac{1}{\sqrt{2\ln(2)}}$,
and a glassy phase for $T<{T_\mathrm{c}}$~\cite{derrida:80,derrida:81}.~\footnote{
The critical temperature ${T_\mathrm{c}}$ is different from the one given in Refs.~\cite{derrida:80,derrida:81}, because we define the model with variance $\mathrm{Var}(E)=N$, instead
of $\mathrm{Var}(E)=N/2$.}
Concerning the dynamics of the \ac{rem}, we will focus on \ac{mc} Metropolis dynamics.
More precisely, we consider that spins flip one at a time and we set the transition rate from state $i$ to state $j$, differing by a single spin, to be
\begin{equation}\label{eq:mc}
q_{i,j}=\frac{1}{N}\min\left(1,e^{\beta (E(i)-E(j))}\right)\,.
\end{equation}
In a nutshell, the \ac{rem} can be seen as a \ac{tm} with (i) a hypercubic space of configurations (instead of fully connected), and (ii) with more physical dynamics to move in it.
In the limit of very long times and large system sizes, the dynamics of the \ac{rem} is qualitatively equivalent to the one of the \ac{tm}, with a threshold energy ${E_\mathrm{th}}=-\sqrt{2N\ln(N)}$ \cite{baityjesi:18, gayrard:18}.
\subsection[The \texorpdfstring{$p-$}-spin model]{The \texorpdfstring{$\boldsymbol{p}$}{p}-spin model} \label{sec:pspin-def}
The $p$-spin model represents a system of $N$ binary spins $\sigma_i=\pm1$ \cite{derrida:80}. At variance with the \ac{tm} and the \ac{rem}, where the energies are i.i.d.,
the energy depends on the microscopic configuration of the system through the Hamiltonian
\begin{equation}
\label{eq:pspin-H}
H = -\sum_{i_1<\ldots<i_p} \ J_{i_1,\ldots,i_p}\sigma_{i_1}\ldots\sigma_{i_p} \ ,
\end{equation}
in which the interactions couple all the possible groups of $p$ different spins.
The bonds $J_{i_1,\ldots,i_p}$, associated to a $p$-tuple of spins, are independent and usually follow a Gaussian or bimodal distribution. We choose the latter, with $J_{i_1,\ldots,i_p} = \pm \sqrt{p!/N^{p-1}}$
with probability one half.
An alternative way to define the previous Hamiltonian is as a Gaussian random field defined on the $N$-dimensional hypercube (in this case the couplings are Gaussian random variables). Its mean is zero, whereas its covariance in the large $N$ limit is {$\overline{H(\{\sigma_i\})H(\{\sigma_i'\})}\sim N q^p$} where $q=\sum_i \sigma_i\sigma_i'/N$.
When $p$ is finite, there is a general agreement that all models with finite $p\geq3$ fall in the same universality class \cite{billoire:05}.
Instead, if $p$ is diverging, the way limits are taken becomes important. On the one hand, if one studies the short-time limit, sending first $N\to\infty$, and only later $p,t\to\infty$, the $p$-spin reduces to the \ac{rem}~\cite{derrida:80} (which corresponds to naively saying that in the large $N$ limit the covariance tends to $\delta_{\{\sigma_i\}\{\sigma_i'\}}$).
On the other hand, when studying the long-time regime of activated dynamics, one needs to study large but finite $N$ at every time $t$ (recall that by definition $p\leq N$), so the REM cannot be recovered~\cite{baityjesi:18c}.
We will focus on $p=3$, in the limit of large but finite $N$, where activated processes become possible \cite{crisanti:00}.
As for the dynamics, we use the single-spin \ac{mc} Metropolis dynamics, described in Eq.~\eqref{eq:mc}.
\section{Energy maxima in the Trap Model}\label{sec:trap}
\subsection{Qualitative description of the dynamics in the TM}
\label{sec:emaxTM}
We consider a typical non-equilibrium protocol: an instant quench from infinite temperature to a target temperature $T$.
In the TM, the system is always stuck in a trap, where it will remain for a time $\tau\propto e^{-\beta E}$ (Eq.~\eqref{eq:rateTrap}), before emerging to the threshold to transition to the next trap.
We call $\tau$ the trapping time.
One can easily see that the distribution of trapping times is {heavy-tailed with density}
\begin{equation}
\psi(\tau)\propto\tau^{-(\alpha T+1)}\,,
\end{equation}
which identifies a critical temperature at ${T_\mathrm{c}}\equiv\frac{1}{\alpha}$~\cite{bouchaud:92}.
When $T<{T_\mathrm{c}}\equiv\frac{1}{\alpha}$ the average trapping time is infinite and the total waiting time is of the order of the maximal waiting time in a single state~\cite{bouchaud:92}.
Consequently, the process waits in the deepest trap it has seen for about as much time as the system has spent in the heat bath, implying
\begin{equation}\label{eq:et}
E(t)\approxE_\MIN(t)\sim-T\ln t\,.
\end{equation}
Eventually, the system is able to reach ${E_\mathrm{th}}$ and fall back into shallower traps, until it finds an even deeper trap than the one before, where it spends a time longer than previously elapsed time.
As a consequence, as time passes, the energy decreases gradually, and the dynamics becomes increasingly slower.
This dependence on how long the system has
been in the bath is called \emph{aging} and results in an increasing time correlation of the system.
In finite systems aging goes on until equilibrium is reached.
\subsection{Energy Maxima in the Exponential Trap Model}
For our consideration of the maximum energy, $E_\MAX(t)$, reached after a time $t$ we will restrict ourselves to the low-temperature \ac{tm}, where aging occurs. In other words, we assume that $T<{T_\mathrm{c}}$, so that \eqref{eq:et} holds. Then $E_\MAX(t)$ can be calculated through extremum statistics.
Recalling that each new trap is chosen independently, the number of traps, $n_t$, that the system will have visited satisfies
\begin{equation}
\label{eq:tm_positive}
n_t \int_{-\infty}^{{E_\mathrm{min}(t)}} \rho_\mathrm{exp}(E) dE \sim 1\,,
\end{equation}
where $E_\mathrm{min}(t)$ is the minimum energy reached until time $t$.
Therefore, $n_t\sim e^{-\alpha E_\mathrm{min}(t)}\sim e^{-\alpha E(t)}$, since in the \ac{tm} the current energy is almost always the lowest energy reached (i.e. $E(t)\approx E_\mathrm{min}(t)$).
The number of visited traps needs to satisfy an analogous relation for the highest energy reached
\begin{equation}
n_t \int_{E_\MAX(t)}^{0} \rho_\mathrm{exp}(E) dE \sim 1\,,
\end{equation}
that yields $n_t{\sim -1/E_\MAX(t)}$.
Equating the two expressions for $n_t$, and using \eqref{eq:et}, one obtains the behaviour
\begin{equation}
E_\MAX(t) \sim \ln\left(1-t^{-\frac{T}{{T_\mathrm{c}}}}\right)\sim -t^{-\frac{T}{{T_\mathrm{c}}}}\,,
\end{equation}
so in the exponential \ac{tm} the largest energy reached approaches its extremal value $E=0$ as a power law.
\footnote{{The high-temperature case, $T>{T_\mathrm{c}}$, can be treated similarly, yielding $E_\MAX(t)\sim -1/t$.}}
Note that with respect to all the models defined below, the exponential \ac{tm} is special (and a bit pathological) since
it has a cut-off in the energy distribution at the most numerous energies, i.e. it does not have any rare high energy.
In consequence, its behaviour is atypical with respect to all other cases considered below.
\subsection{Gaussian Trap Model}
In the Gaussian \ac{tm}, since $\rho(E)$ is symmetric, the
maximum and minimum records $E_\MAX(t)$ and $E_\MIN(t)$ follow the same law, with opposite signs.
This is because, from any configuration $i$, all the configurations are reached with equal probability,
and what changes is only the amount of time the system remains in them.
Therefore, a configuration with $E<E_\MIN(t)$ is reached with the same probability as a configuration with energy $E>-E_\MIN(t)$.
Since, by Eq.~\eqref{eq:et}, in the \ac{tm} $E(t)\simE_\MIN(t)\sim-T\ln t$, we can deduce that
\begin{equation}
\label{eq:gtm:aging}
E_\MAX(t)\sim T\ln t.\end{equation}
{Eq.~\eqref{eq:et} still holds for the Gaussian \ac{tm} in its aging regime, $t<\exp(\beta{\beta_\mathrm{i}} N)$, where ${\beta_\mathrm{i}}=\min(\beta,{\beta_\mathrm{c}})$ and ${\beta_\mathrm{c}}=\sqrt{2\ln 2}$. If $\beta<{\beta_\mathrm{c}}$ (i.e. in the high-temperature phase), a supplementary `post-equilibration' regime develops for $\exp(\beta^2 N)<t<\exp(N({\beta_\mathrm{c}}^2+\beta^2)/2)$, governed by
\begin{equation}
\label{eq:gtm:equilibrium}
E_\MAX(t)=\sqrt{2N(\ln t-\beta^2N/2)}.
\end{equation}
In both cases, the evolution of the maximum energy eventually ends when $E_\MAX$ reaches the maximum possible value
$N\sqrt{2 \ln 2}$.
As we shall discuss a similar argument for the \ac{rem} in Sec.\ref{subsec:verylongscales} and since there is no difference with the minimal energy, we omit the details here. However, let us note that depending on the temperature there are two different behaviours of $E_\MAX(t)/N$ as a function of $\ln t/N$. For $\beta>{\beta_\mathrm{c}}$ there is a single regime, linear in $\ln t$~\eqref{eq:gtm:aging}, while otherwise there are two different ones -- a linear law~\eqref{eq:gtm:aging} followed by a square-root dependency~\eqref{eq:gtm:equilibrium}.}
\section{Energy maxima in the REM}
\label{sec:rem}
We now turn to the Metropolis dynamics on the \ac{rem}. As in the previous section we examine quenches from infinite temperature to a target temperature $T$. Throughout this section we consider the thermodynamics large-$N$ limit and systematically discard subleading terms. Higher-order corrections for finite systems are discussed in App.\ref{app:finitecorr}.
\paragraph{Short summary of the maximum-energy records in the \ac{rem}.}
\label{subsec:summary}
We are able identify and study a large number of different regimes in the dynamics, which are summarised in quantitative terms in Tab.~\ref{tab:summ} at the end of Sec.~\ref{sec:rem}.
The energy of the initial state is typically of order $\sqrt{N}$ (Sec.~\ref{subsec:init}). In the initial stages of the evolution, the system quickly falls into a local minimum (Secs.~\ref{subsec:drift} and~\ref{subsec:firsttrap}). Since it is not necessary to reach energies of order $\sqrt{N}$ to escape the local minima, $E_\MAX(t)$ stays unchanged and only starts growing once a time of order $t_3=e^{\sqrt{2\beta^3 N\sqrt{2N\ln N}}}$ has passed (Eq.~\eqref{eq:e1}, Sec.~\ref{subsec:reachingcommonenergies}).
At larger times, and, in particular, those exponentially large in $N$, $E_\MAX(t)$ incurs a further slow down, because not only new records are hardly accepted, but they are also harder to find (Sec.~\ref{subsec:aging}). Even after equilibrium has been reached, $E_\MAX(t)$ keeps growing (Sec.~\ref{subsec:verylongscales}), and saturates only when the global maximum of the system has been visited (Sec.~\ref{subsec:saturation}).
\subsection{Basic features of the landscape and the dynamics of the REM}
In the \ac{rem}, most energy levels are concentrated around zero at a typical distance of order $\sqrt{N}$.
At variance with the \ac{tm}, the phase space has a structure, and each state has $N$ neighbours
whose energies,
as we show in Sec.~\ref{subsec:asymptoticbounds}, Eq.~\eqref{eq:Iasymp}, {typically}
lie in the interval
\begin{equation}
\label{eq:Ieqiv}
I\equiv\left[-\sqrt{2N\ln N},\sqrt{2N\ln N}\right]\,.
\end{equation}
The majority of states have at least one lower neighbour, and the dynamics spends little time there, since the Metropolis update rule [Eq.~\eqref{eq:mc}] privileges energy descent.
As a consequence, the states in which most of the time is spent are those without a lower neighbour,
which have energy $\inf I=-\sqrt{2N\ln N}$ or lower. Minima with energy $E\sim-N$ can be seen as the equivalent of the \ac{tm} traps,
since $N\gg\sqrt{2N\ln N}$ (the separation between these minima and the threshold diverges with $N$). Since the energy of such states is much lower than the typical energies, to leading order the time spent in these low-energy minima configurations scales as $\tau\sim e^{-\beta E}$ [Eq.~\eqref{eq:mc}].
\subsection{Energy maximum evolution}
In this section, as we did for the \ac{tm}, we discuss the evolution of the typical maximum energy $E_\MAX(t)$ in the \ac{rem} in the limit of very large system sizes. The final results will be summarised in Table \ref{tab:summ}.
When dealing with exponentially large time scales,
a central assumption of our computation is that, each time the dynamics leaves a state, the system becomes independent from its past. In other words, the neighbours are drawn anew, so that returns to a recently visited configuration are not taken into account. This hypothesis is supported by an argument in Ref.~\cite{baityjesi:18}, stating that even though, in the actual dynamics, some configurations are revisited, this happens with a sub-exponential rate. In consequence, this should not affect exponential time scales.
\subsubsection{Initial condition}\label{subsec:init}
Before the quench the temperature is infinite, so $E_\mathrm{max}(0)$ is drawn from ${\rho_\mathrm{gauss}}(E)$. Therefore, its intensive value is zero in the limit of large system sizes: $E_\mathrm{max}(0)/N\sim1/\sqrt{N}\stackrel{N\to\infty}{\longrightarrow} 0$.
\subsubsection{Drift towards the first trap}
\label{subsec:drift}
At the beginning of the evolution, the system quickly falls into a first local minimum of the energy. See Fig.~\ref{fig:beginning} (left part) for a schematic description of this and the following regimes.
In fact, with probability $1-\frac{1}{N+1}$, the initial state has at least one lower neighbour.
The rate $q_{i,j}$ [Eq.~\eqref{eq:mc}] privileges energy descent, and on time-scales of order one the energy will typically immediately decrease since climbing up moves will be discarded (their rate is exponentially small in $\sqrt N$). The transition toward the lowest neighbour, which is at energy $\inf I$, has a rate $q_{i,j}\sim 1/N$, hence a local minimum will be reached in a time of order $t_1=N$ (a configuration at energy $\inf I$ is a local minimum with probability of order $1$).
The number of transitions required to reach such state is of order $\ln(N)$. \footnote{At each transition, the number of lower neighbours is typically divided by two, so it will take $\mathcal{O}(\log_2(N))$ steps to reach a local minimum.}
During this regime, before the first trap is reached, $E_{\max}$ maintains its initial value of order $\sqrt N$, because the energy is decreasing.
\begin{figure}[!tbh]
\includegraphics[trim={0 13cm 0 0}, width=\columnwidth]{beginning-diagram.pdf}
\caption{A schematic representation of the dynamics of the \ac{rem} in the initial stages of the dynamics. In a time of order $t_1=N$ the system falls in the first local energy minimum, whose energy is typically $\inf I$ [Eq.~\eqref{eq:Ieqiv}]. It takes a time of order $t_2=\exp(\sqrt{N/\ln N})$ to leave that state. After that, the system will visit a large number, $n_{t_3}$, of minima before it can reach energies of order $\sqrt{N}$, after a time $t_3=e^{\beta E_1}$ [Eq.~\eqref{eq:e1}]. During all this time $E_\MAX(t)=E(0)$. The times and energies in the diagram are not drawn to scale.}
\label{fig:beginning}
\end{figure}
\subsubsection{Stay in the first trap}
\label{subsec:firsttrap}
Once the system reaches the first local minimum, which has energy $E\approx\inf I$ (that of a typical lowest neighbour), the energy difference with its neighbours is of order $\delta=\sqrt{\frac{N}{\ln N}}$ (see App.~\ref{subsec:lowestneighboursgap}).
Therefore, the system remains in the first trap and does not move at all for a time of order $t_2=\exp\left(\beta \sqrt{\frac{N}{\ln N}}\right)$. Consequently, $E_\MAX(t)$ does not change either.
\subsubsection{Reaching common energies}
\label{subsec:reachingcommonenergies}
We now estimate the time scale $t_3$ required, once the dynamics left the first trap, for the energy to become of order $\sqrt{N}$.
Such energy needs to be reached for a new record $E_\MAX(t)>E(0)\sim\sqrt{N}$ to be hit.
After $t_2$, the system starts visiting a series of local minima of the energy. When the dynamics leaves a local minimum, it does so by jumping to its lowest-energy neighbour, which, to leading order, also has energy $\inf I$. In order to reach energies of order $\sqrt{N}$, a large number of local minima needs to be visited.
The typical trapping time in such a minimum with energy $E$, is
\begin{equation}
\tau\sim Ne^{\beta\left(\inf I - E\right)}
\end{equation}
and the rate of jumping to a configuration of energy of order $\sim\sqrt{N}$ from the local minimum is
\begin{equation}
q_\mathrm{common}\sim e^{\beta\left(E-\mathcal{O}(\sqrt{N})\right)}\,,
\end{equation}
where $\mathcal{O}(\sqrt{N})$ indicates a quantity of order $\sqrt{N}$.
The probability of jumping in a time $\tau$ to a configuration of energy $\sim\sqrt{N}$ is then given by
\[
\tau q_\mathrm{com}\sim e^{\beta(\inf I-\mathcal{O}(\sqrt{N}))}\approx e^{\beta\inf I}\,.
\]
For such a transition to be likely, the number of visited traps $n_{t_3}$ needs to be of
order $e^{-\beta\inf I}$, since any site usually has neighbours at energy $\sim\sqrt{N}$. The lowest of the $n_{t}$ traps determines $E_\MIN(t)$. Thus, $E_\MIN(t)$ can be calculated as the minimum among $Nn_{t}$ Gaussians (the factor $N$ stems from the fact that we are dealing with minima of $N$ Gaussians, not arbitrary Gaussians) with variance $N$. Therefore, through known extremum statistics results (see App.~\ref{subsec:asymptoticbounds}, Eq.~\eqref{eq:mn}), one obtains
\begin{equation}\label{eq:e1}
-E_{\min}(t_3)\sim\sqrt{2N\ln {\left(Nn_{t_3}\right)}}\approx \sqrt{2\beta N\sqrt{2N\ln N}}=:E_1.
\end{equation}
By using Eq.~\eqref{eq:et}, which is valid also in the \ac{rem} \cite{baityjesi:18}, we obtain $t_3=e^{\beta E_1}$. Until $t_3$, $E_\MAX(t)$ typically still maintains its initial value.
\subsubsection{Aging}
\label{subsec:aging}
After $t_3$, $E_\MAX(t)$ starts increasing. We start by considering the aging stage, during which Eq.~\ref{eq:et} is valid.
We proceed analogously to Sec.~\ref{subsec:reachingcommonenergies} with an additional ingredient: high-energy records are harder to reach not only because transitions to them are rarely accepted -- as it was also previously -- but also because record-breaking states are rarely found among the neighbours of a given configuration.
The probability that a trap has a neighbour of energy at least $E$ with probability $p_{\mathrm{find}}(E)\sim e^{-\frac{E^2}{2N}}$. Thus, when exiting a trap, a neighbour of energy $E$ is chosen over the lowest one with probability
\begin{equation}
\label{eq:ptransit}
p_{\mathrm{find}}(E)p_{\mathrm{accept}}(E)\sim e^{\beta(\inf I - E)-\frac{E^2}{2N}}\,.
\end{equation}
Then, in order to hit a new record, the number of visited traps should be
\begin{equation}
\label{eq:ntptransit}
n_t\sim p_{\mathrm{find}}^{-1}(E)p_{\mathrm{accept}}^{-1}(E)\,.
\end{equation}
Again, by using extremum statistics (App.~\ref{subsec:asymptoticbounds}, Eq.~\eqref{eq:mn}), one has
\begin{equation}
\label{eq:nt}
n_t\sim e^{\frac{E_\MIN(t)^2}{2N}}
\end{equation}
Combining this with Eqs.~\eqref{eq:ptransit} and \eqref{eq:ntptransit} gives
\begin{equation}
\label{eq:firstfew1}
-E_\MIN(t)\approx\sqrt{2\beta N(E_\MAX(t)-\inf I)+E_\MAX(t)^2}\,.
\end{equation}
Plugging Eqs.~\eqref{eq:et} and \eqref{eq:Ieqiv} into Eq.~\eqref{eq:firstfew1}, we obtain
\begin{equation}
\label{eq:aging}
E_\MAX(t)=\beta N\left(-1+\sqrt{1-\frac{2}{\beta}\sqrt{\frac{2\ln N}{N}} +\left(\frac{\ln t}{\beta^2N}\right)^2}\right)\,.
\end{equation}
We can express Eq.~\eqref{eq:aging} in simpler forms by considering explicitly the different values of $\ln t$.
\paragraph{First records} We first consider the times for which $\ln t$ is of order $\beta E_1$ (but greater than it, so as to have $t>t_3$). Then, by substituting Eq.~\eqref{eq:e1}, Eq.~\eqref{eq:aging} becomes
\begin{equation}
\label{eq:firstfew2}
E_\MAX(t)=\sqrt{2N\ln N}\left(\left(\frac{\ln t}{\beta E_1}\right)^2-1\right)\,.
\end{equation}
\paragraph{Intermediate regime} We next assume that $\ln t$ is much larger than $\beta E_1$, but much smaller than $\beta^2 N$. Then Eq.~\eqref{eq:aging} takes the particularly simple form
\begin{equation}
\label{eq:intermediate}
E_\MAX(t)=\frac{\beta N}{2}\left(\frac{\ln t}{\beta^2 N}\right)^2\,.
\end{equation}
\paragraph{Exponential scales} Finally, we have $\ln t\sim \beta^2 N$. Then Eq.~\eqref{eq:aging} reduces to
\begin{equation}
\label{eq:longscales}
E_\MAX(t)=\beta N\left(-1 + \sqrt{1+\left(\frac{\ln t}{\beta^2 N}\right)^2}\right)\,.
\end{equation}
In order to determine time $t_4$ at which aging ends, we consider the mean energy, which is equal to $E_\MIN(t)$ as long as aging lasts. Then aging ends with $E_\mathrm{min}(t_4)=-\frac{\ln t_4}{\beta}$ equal to the equilibrium mean energy, itself given either by the global minimum
of the energy or by $\langle E\rangle=\frac{\int Ee^{-\beta E}{\rho_\mathrm{gauss}}(E)\mathrm{d} E}{\int e^{-\beta E}{\rho_\mathrm{gauss}}(E)\mathrm{d} E}=-\beta N\,.$
Hence, by App.~\ref{subsec:asymptoticbounds}, \eqref{eq:globalmax}
\begin{equation}
\label{eq:t5}
t_4=e^{\beta{\beta_\mathrm{i}} N}\,,
\end{equation}
where we introduced ${\beta_\mathrm{i}}\equiv\min(\beta,\beta_c)$.
We stress that this regime describes the aging occurring at exponential time scales, that are generally regarded as the relevant ones for activated dynamics.
In this regime, $E_\MAX(t)$ {grows} as $\ln t$ (and so does $-E_\MIN(t)$) until the system has eventually
equilibrated. Yet, since $E_\MAX(t)$ grows slower than $E_\MIN(t)$, it will keep increasing even after $E_\MIN(t)$ saturates.
\subsubsection{Equilibrium regime}
\label{subsec:verylongscales}
After $t_4$, $E_\MAX(t)$ continues to evolve even though the system has already reached macroscopic equilibrium. In this regime Eqs.~\eqref{eq:ptransit} and \eqref{eq:ntptransit} still hold, whereas Eq.~\eqref{eq:et} fails, making $E_\MIN(t)$ irrelevant for the calculation of $E_\MAX(t)$.
The calculation that led to Eq.~\eqref{eq:t5} also gives that the mean time spent per state $\frac{t}{n_t}$ stations at the end of aging (the system reaches macroscopic equilibrium) at the value $\frac{t_4}{n_{t_4}}$. In other words, $\frac{t}{n_{t}}=\frac{t_4}{n_{t_4}}$ for all $t>t_4$.
Using Eqs.~\eqref{eq:ptransit}, \eqref{eq:ntptransit} and \eqref{eq:t5} together with the fact that $\inf I$ is negligible compared to $E_\MAX(t)$, we get
\[t=n_t\frac{t_4}{n_{t_4}}=n_te^{\beta{\beta_\mathrm{i}} N}e^{-N\frac{{\beta_\mathrm{i}}^2}{2}}=e^{\beta{\beta_\mathrm{i}} N+\betaE_\MAX(t)-N\frac{{\beta_\mathrm{i}}^2}{2}+\frac{E_\MAX(t)^2}{2N}}\,.\] Thus,
\begin{equation}
\label{eq:verylongscales}
E_\MAX(t)=\beta N\left(-1 +\sqrt{\left(1-\frac{{\beta_\mathrm{i}}}{\beta}\right)^2+2\frac{\ln t}{\beta^2N}}\right)\,\,.
\end{equation}
Eq.~\eqref{eq:verylongscales} is valid until the global maximum $E_\mathrm{max}(\infty)=\beta_c N$ (see App.~\ref{subsec:asymptoticbounds}, Eq.~\eqref{eq:globalmax}) is reached. This means that it remains valid for
\begin{equation}\label{eq:verylongscales-ineq}
\ln t\leq (2\beta-{\beta_\mathrm{i}}+\beta_c)(\beta_c+{\beta_\mathrm{i}})\frac{N}{2}= \begin{cases} N\frac{(\beta+\beta_c)^2}{2} & \textrm{ if } \beta<\beta_c\\
2N\beta_c\beta & \textrm{ if } \beta>\beta_c\,.
\end{cases}
\end{equation}
Inequality \eqref{eq:verylongscales-ineq} defines the typical time $t_5$ required to reach the global maximum
\begin{equation}
\label{eq:microeq}
t_5=e^{(2\beta-{\beta_\mathrm{i}}+\beta_c)(\beta_c+{\beta_\mathrm{i}})\frac{N}{2}}\,.
\end{equation}
\subsubsection{Saturation}
\label{subsec:saturation}
Naturally, after $t_5$, that is after the global maximum has been attained, the maximum energy cannot change any more, so thereafter
\begin{equation}\label{eq:emax-tinfty}
E_\MAX(t) = \beta_cN\,.
\end{equation}
\subsection{Remarks}
\label{sec:comprehensive}
In our discussion, we treated the typical maximum of the energy $E_\MAX(t)$. We expect that when taking the thermodynamic limit there will be a concentration of the measure around a central value, so the same results should hold when replacing typical $E_\MAX(t)$ by the expectation $\mathbb{E}[{E_\MAX(t)}]$.
In App.~\ref{app:finitecorr}, we describe the subleading corrections that need to be taken into account in order to study finite systems, and tackle a few of them. The different regimes of $E(t)$ and $E_\MAX(t)$ (at leading order as $N\to \infty$) are summarised in Tab.~\ref{tab:summ}.
\begin{table}[tb]
\begin{center}
\begin{tabular}{c|c|c|c}
Regime & Time scale & $E(t)$/N & $E_\MAX(t)/N$\\\hline\hline
Initial & $t=0$ & $\sim1/\sqrt{N}$ & $E(0)\sim1/\sqrt{N}$\\\hline
Drift & $\ln t<\log_2 N$ & \small{decreasing} & $E(0)\sim1/\sqrt{N}$\\\hline
First trap & $\ln t<\beta\sqrt{\frac{N}{\ln N}}$ & $\sqrt{2\ln N/N}$ & $E(0)\sim1/\sqrt{N}$\\\hline
Common & $\ln t<\beta E_1$ & $-T\ln t/N$ & $E(0)\sim1/\sqrt{N}$\\\hline
First records & $\ln t\sim\beta E_1$ & $-T\ln t/N$ & $\sqrt{\frac{2\ln N}{N}}\left(\left(\frac{\ln t}{\beta E_1}\right)^2-1\right)$\\\hline
Intermediate & $\beta E_1\ll\ln t\ll\beta^2 N$ & $-T\ln t/N$ & $\frac{\beta}{2}\left(\frac{\ln t}{\beta^2N}\right)^2$\\\hline
Exponential & $\ln t<\beta{\beta_\mathrm{i}} N$ & $-T\ln t/N$ & $\beta\left(-1+\sqrt{1+\left(\frac{\ln t}{\beta^2 N}\right)^2}\right)$\\\hline
Equilibrium & $\ln t<(2\beta - {\beta_\mathrm{i}} + {\beta_\mathrm{c}})({\beta_\mathrm{c}}+{\beta_\mathrm{i}})\frac{N}{2}$ & $-{\beta_\mathrm{i}}$ & $\beta\left(-1+\sqrt{\left(1-\frac{{\beta_\mathrm{i}}}{\beta}\right)^2+2\frac{\ln t}{\beta^2 N}}\right)$\\\hline
Saturation & any larger time scale & $-{\beta_\mathrm{i}}$ & ${\beta_\mathrm{c}}$\\\hline
\end{tabular}
\end{center}
\caption{Summary of the regimes of the dynamics evolution in the \ac{rem} in chronological order.
For each regime (first column) we depict the time of validity, the typical intensive energy one finds at time $t$, $E(t)/N$,
and the typical intensive record of the maximum energy, $E_\MAX(t)/N$.
The energy $E_1=\sqrt{2\beta N\sqrt{2N\ln N}}$ is defined in Eq.~\eqref{eq:e1}, ${\beta_\mathrm{c}}=1/{T_\mathrm{c}}=\sqrt{2\ln 2}$ is the inverse critical temperature, and ${\beta_\mathrm{i}}=\min({\beta_\mathrm{c}},\beta)$.
}
\label{tab:summ}
\end{table}
Relevant time scales for the glassy activated dynamics stem naturally from our analysis. The longest time scale is exponentially large in the system size.
Thus, we can define a rescaled time $\theta=\ln t/N$, which is useful to express Eqs.~\eqref{eq:longscales}, \eqref{eq:verylongscales} and \eqref{eq:emax-tinfty} in a meaningful way. The intensive maximum energy for these time scales is plotted in Fig.~\ref{fig:Maxbeta}--left, with a comparison with numerics in small systems.
In the thermodynamic limit, the piecewise concatenation of these regimes is continuous.\footnote{For $\beta<{\beta_\mathrm{c}}$ its derivative is also continuous at $\theta=\beta{\beta_\mathrm{i}}$, corresponding to $t_4$.}
\begin{figure}[tb]\centering
\includegraphics[width=.45\textwidth]{Emax_inset}
\includegraphics[width=.45\textwidth]{Emax_biplot}
\caption{Intensive maximum energy, $E_\MAX$, in the \ac{rem} as a function of the rescaled time $\theta=\ln t/N$. \textbf{Left}: Plots for several values of the inverse temperature $\beta$. The $y$ axis is linear, in units of $\beta_c$. The curves are split in three, in order to stress the presence
of three subsequent regimes: Aging (Sec.~\ref{subsec:aging}), Equilibrium (Sec.~\ref{subsec:verylongscales}) and Saturation (Sec.~\ref{subsec:saturation}).
The tics on the $x$ axis show the beginning of these regimes for the highest $\beta$, corresponding to $T=0.25$.
We also show data from simulations at $\beta=1.13\beta_\mathrm{c}$, for systems of size $N=12$, which agrees well with our $N=\infty$ predictions. Finite-size corrections are discussed in App.~\ref{app:finitecorr}.
In the \textbf{inset} we show $E_\MAX$ for shorter time scales $\theta'$ (see main text).
\textbf{Right}: The \textbf{top} figure depicts the same curves shown in the left panel, for $\beta=1.13{\beta_\mathrm{c}}$. The solid line is the result of our calculation (in the thermodynamic limit), and the points are from runs with $N=12$ (error bars are smaller than the points).
The separation between the continuous lines emphasises the end of the aging regime according to our calculation [Eq.~\eqref{eq:t5}].
The \textbf{bottom} figure shows the energy as a function of time for comparison.\protect\\Numerical data here and in Fig.~\ref{fig:Emax_tw} are averaged over disorder realizations.
}
\label{fig:Maxbeta}
\end{figure}
In units of $\theta$, all the previous time regimes collapse to zero in the thermodynamic limit.
For {the regime at which the first records are observed,} the relevant rescaled time is $\theta'=\ln t/(\beta E_1(N))$, whereas the energy should be rescaled by a factor $1/\sqrt{2N\ln N}$ [see Eq.~\eqref{eq:firstfew2}]. The resulting rescaled curve is plotted in the inset of Fig.~\ref{fig:Maxbeta}--left.
In Fig.~\ref{fig:Maxbeta}--right, we show that $E_\MAX(t)$ keeps growing even after the system has thermalised, and that the equilibration time, $t_4$, extracted from $E_\MAX(t)$ (in the thermodynamic limit) corresponds roughly to the time at which $E(t)$ reaches its equilibrium value (in a system of size $N=12$).
\section{Energy maxima and threshold energy in the \texorpdfstring{$\boldsymbol{p}$}{p}-spin model}\label{sec:pspin}
A natural question is how the behaviour of $E_\MAX(t)$ extends to more complex glassy {systems}, and how it can be used to extract useful information on them.
We {consider} the $p$-spin model, since in the limit $p\to\infty$ the thermodynamics (though not the long-time activated dynamics~\cite{baityjesi:18c}) is the same as the one of the \ac{rem}.
In the $p$-spin model, there is a threshold energy ${E_\mathrm{th}}$ over which there are no energy minima~\cite{castellani:05},
as it also happens in the \ac{tm} and the \ac{rem}. Yet, at variance with the \ac{tm} and the \ac{rem}, where $\lim_{N\to\infty}({E_\mathrm{th}}/N)=0$, in the $p$-spin model
${E_\mathrm{th}}$ is extensively negative (i.e. $\lim_{N\to\infty}({E_\mathrm{th}}/N)<0$).
Let us now take into account the maximum energy $E_\MAX(t/2,t)$ reached in the time interval $[t/2,t]$.
Taking the maximum in a dilating time interval is useful to identify the separation of time scales that arises in the dynamics of glassy systems. In the thermodynamic limit, the dynamics of the $p$-spin model on time scales not diverging with $N$ is known \cite{cugliandolo:93}: the energy decreases monotonically towards the threshold energy. Moreover, in this regime, the energy as a function of time has fluctuations of order $\sqrt{N}$. In consequence, $E_\MAX(t/2,t)$ is equal to the energy at time $t/2$ and approaches ${E_\mathrm{th}}$ for $t$ large but not diverging with $N$. On times diverging with $N$ the dynamics becomes activated (on time-scales exponentially large in $N$), the intensive energy goes below ${E_\mathrm{th}}$, and the energy starts to have rare high excursions that increase the value of $E_\MAX(t/2,t)$ (see also Fig.~\ref{fig:sample} in App.~\ref{sec:sample}). Hence, one can expect that $E_\MAX(t/2,t)$ grows close to logarithmically, as in record-breaking dynamics for i.i.d. random variables (and as it happens for $E_\MAX(t)$ in the \ac{rem} during the aging regime, see table \ref{tab:summ}).
The minimum value reached by $E_\MAX(t/2,t)$ can be used to {define} the value of ${E_\mathrm{th}}$ in numerical simulations. This method adds to the usual procedures of calculating the threshold, and has the advantage of not requiring extrapolations to infinite times~\cite{cugliandolo:93}, nor the computation of the complexity of minima~\cite{crisanti:95}.
In Fig.~\ref{fig:Emax_tw} we show $E_\MAX(t/2,t)$ both in the \ac{rem} and in the $p$-spin model (with $p=3$), for different system sizes.
As expected, in both cases the curve reaches a minimum which we identify as ${E_\mathrm{th}}$. Note that, in the \ac{rem},
$\frac{{E_\mathrm{th}}}{N}$ grows as $N$ increases whereas in the $3$-spin model it decreases, in agreement with the fact that
the intensive threshold energy is zero in the former model and negative in the latter.
In the inset of Fig.~\ref{fig:Emax_tw}, we show that, in the $p$-spin model, the finite-size threshold energy obtained through this procedure is controlled by the $N^{-1/2}$ fluctuations, and converges to its analytical value in the thermodynamic limit~\cite{rizzo:13}.\footnote{This same scaling is not as clean in the REM, probably because the system sizes are too small. In App.~\ref{app:finitecorr} we argue that the asymptotic limit is reached for $N\sim10^3$.}
From the figure one can also see that, as argued in the previous paragraph, the relevant time scales for the growth of $E_\MAX(t/2,t)$ are exponential.
\begin{figure}[tb]
\centering
\includegraphics[width=0.49\columnwidth]{emaxHalfT075_gauss}
\resizebox{0.49\textwidth}{!}{\input{Emax_tw_tex}}
\caption{
Intensive maximum energy $E_\mathrm{max}(t/2,t)/N$ reached in the time interval $[t/2,t]$, for different system sizes $N$.
On the \textbf{left}, we show data for the \ac{rem} at $T=0.75$. Decreasing
the temperature (not shown), the bottom of the curve becomes progressively flatter,
but the lowest point stays approximately at the same height.
On the \textbf{right}, we show data for the $3$-spin model at $T=0.54$.
In both cases the curves seem to converge to a master curve proportional to $\ln t$. \textbf{Inset}: Scaling behaviour of the size-dependent threshold energy $E_{\text{th}}(N)$ (defined as the minimum of $E_\MAX(t)$) in the $3$-spin model.
The prediction from Ref.~\cite{rizzo:13}, valid for infinitely large systems, ${E_\mathrm{th}}=-0.762$, is also shown (red square).
The dotted line represents the fit $E_{\text{th}}(N)=aN^{-1/2}+b$.
The coefficients of the fit are $a=1.47(1)$ and $b=-0.758(1)$.
}
\label{fig:Emax_tw}
\end{figure}
\section{Conclusions}\label{sec:conclusions}
By counting the number of visited traps, and noting that in \ac{tm}-like models almost all the time is spent in the deepest trap, we were able to calculate the evolution of the maximum energy reached after a time $t$, $E_\MAX(t)$.
Our method, which applies record statistics to glasses, allowed us to add to the very reduced number of analytical calculations of activation in glasses (see also~\cite{bryngelson:89,gayrard:18}). It has the advantage of transparently treating the dynamics, providing new insight on the long-time behaviour of glassy systems. Further, we showed how the behaviour of the maximum energy record naturally reveals the different time scales involved in glassy relaxation, and how it can be used to identify a crucial static quantity, the threshold energy ${E_\mathrm{th}}$.
Calculating ${E_\mathrm{th}}$ by identifying the minimum of $E_\MAX(t/2,t)$ is arguably simpler than calculating the complexity of the minima~\cite{crisanti:95}, and does not require any kind of extrapolations to infinite times as required by traditional methods~\cite{cugliandolo:93}.
The relationship between ${E_\mathrm{th}}$ and $E_\MAX(t/2,t)$ relies on the assumption that it is necessary to reach ${E_\mathrm{th}}$ to connect low-lying energy minima. Therefore, its success in the $p$-spin model is a strong suggestion that there are no low-energy paths connecting minima in the $p$-spin.
Our results also connect to recent phenomenological works, suggesting to rationalise the dynamical slowdown of low-temperature
glasses as a record-breaking process over larger and larger domains~\cite{boettcher:05b, boettcher:11, robe:16}.
Our mean-field calculations could then account for this process in regions smaller than the correlation length, where the system is effectively fully-connected.
\section*{Acknowledgements}
We thank Valerio Astuti, Chiara Cammarota, Claude Godr\`eche and Satya Majumdar for interesting discussions.
This work was funded by the Simons Foundation for the collaboration ``Cracking the Glass Problem" (No. 454935 to G. Biroli and No. 454951 to D.R. Reichman).
M.B.-J. was partially supported through Grant No. FIS2015-65078-C2-1-P, jointly funded by MINECO (Spain) and
FEDER (European Union).
\clearpage
|
1,108,101,566,806 | arxiv | \section{Introduction}
\label{sec:intro}
Conditionally-solvable quantum-mechanical problems have been of great
interest during the last decades (see, for example, Turbiner's remarkable
review\cite{T16} and the references therein). However, the solutions to the
eigenvalue equations stemming from such models have been misinterpreted in a
wide variety of physical applications\cite{AF20,AF21,F21}.
In a recent paper, Mustafa\cite{M22a} derived apparently exact solutions to
an eigenvalue equation that is known to be conditionally solvable\cite
{AF20,AF21,F21}. For this reason, we deem it necessary to discuss Mustafa's
results in some detail. In section~\ref{sec:Phys_models} we outline
Mustafa's models. In section~\ref{sec:HO} we solve one of them, which is
actually exactly solvable, by means of the Frobenius (power-series) method.
In section~\ref{sec:CLH} we apply the Frobenius method to a
conditionally-solvable radial eigenvalue equation that contains Mustafa's
ones as particular cases. Finally, in section~\ref{sec:conclusions} we
summarize the main results and draw conclusions.
\section{Physical models}
\label{sec:Phys_models}
In this section we outline the eigenvalue equations for three models
discussed by Mustafa\cite{M22a}. The physical meaning of the parameters is
not relevant for present discussion and the interested reader is referred to
that paper.
For the ``KG-oscillator in cosmic string spacetime within KKT'' Mustafa\cite
{M22a} derived the radial equation
\begin{equation}
U^{\prime \prime }(r)+\left[ \tilde{\lambda}+\frac{1/4-\tilde{\gamma}^{2}}
r^{2}}-\tilde{\omega}^{2}r^{2}\right] U(r)=0, \label{eq:HO_Mus}
\end{equation}
and obtained the eigenvalues
\begin{equation}
\tilde{\lambda}^{M}=2\tilde{\omega}\left( 2n_{r}+|\tilde{\gamma}|+1\right) ,
\label{eq:HO_lambda^M}
\end{equation}
where $n_{r}=0,1,\ldots $ is the radial quantum number.
By means of the change of variables $\rho =\tilde{\omega}^{1/2}r$ we obtain
\begin{equation}
F^{\prime \prime }(\rho )+\left[ \frac{\tilde{\lambda}}{\tilde{\omega}}
\frac{1/4-\tilde{\gamma}^{2}}{\rho ^{2}}-\rho ^{2}\right] F(\rho )=0.
\label{eq:HO}
\end{equation}
For the ``pseudo-confined PDM KG-oscillator in cosmic string spacetime
within KKT'' Mustafa\cite{M22a} derived the radial equation
\begin{equation}
U^{\prime \prime }(r)+\left[ \mathcal{E}+\frac{1/4-\tilde{\beta}^{2}}{r^{2}}
\tilde{\omega}^{2}r^{2}-\eta \tilde{a}r-\frac{\tilde{b}}{r}\right] U(r)=0,
\label{eq:CLH1_Mus}
\end{equation}
and obtained the eigenvalues
\begin{equation}
\mathcal{E}^{M}=2\tilde{\omega}\left( 2n_{r}+|\tilde{\beta}|+1\right) -\frac
\tilde{a}^{2}\eta ^{2}}{4\tilde{\omega}^{2}}. \label{eq:CLH1_E^M}
\end{equation}
Curiously, these eigenvalues do not depend on $\tilde{b}$ in spite of the
fact that the eigenvalue equation already depends on this model parameter.
Well-known general theorems are useful for testing results; here, we can
resort to the celebrated Hellmann-Feynman theorem (HFT)\cite{G32,F39} that
in the present case states that
\begin{equation}
\frac{\partial \mathcal{E}}{\partial \tilde{b}}=\left\langle \frac{1}{r
\right\rangle >0,\;\frac{\partial \mathcal{E}}{\partial \eta }=\tilde{a
\left\langle r\right\rangle . \label{eq:CLH1_HFT}
\end{equation}
Clearly, Mustafa's result cannot be correct because
\begin{equation}
\frac{\partial \mathcal{E}^{M}}{\partial \tilde{b}}=0,\;\frac{\partial
\mathcal{E}^{M}}{\partial \eta }=-\frac{\tilde{a}^{2}\eta }{2\tilde{\omega
^{2}}. \label{eq:CLH1_HFT_M}
\end{equation}
Another unmistakable indication that equation (\ref{eq:CLH1_E^M}) is not
correct is that $\mathcal{E}^{M}$ does not yield the eigenvalues of the
Coulomb problem when $\tilde{\omega}=0$ and $\tilde{a}=0$.
The same change of variables indicated above yields
\begin{equation}
F^{\prime \prime }(\rho )+\left[ \frac{\mathcal{E}}{\tilde{\omega}}+\frac
1/4-\tilde{\beta}^{2}}{\rho ^{2}}-\rho ^{2}-\frac{\eta \tilde{a}}{\tilde
\omega}^{3/2}}\rho -\frac{\tilde{b}}{\tilde{\omega}^{1/2}\rho }\right]
F(\rho )=0. \label{eq:CLH1}
\end{equation}
For the ``confined PDM KG-oscillator-III in cosmic string spacetime within
KKT'' Mustafa\cite{M22a} derived the radial eigenvalue equation
\begin{equation}
U^{\prime \prime }(r)+\left[ \tilde{\lambda}_{1}+\frac{1/4-\tilde{\gamma
_{1}^{2}}{r^{2}}-\tilde{\omega}_{1}^{2}r^{2}-2mAr-\frac{2B}{r}\right] U(r)=0,
\label{eq:CLH2_Mus}
\end{equation}
and obtained the eigenvalues
\begin{equation}
\tilde{\lambda}_{1}^{M}=2\tilde{\omega}_{1}\left( 2n_{r}+|\tilde{\gamma
_{1}|+1\right) -\frac{m^{2}A^{2}}{\tilde{\omega}_{1}^{2}}.
\label{eq:CLH2_lambda_1^M}
\end{equation}
In this case, the HFT states that
\begin{equation}
\frac{\partial \tilde{\lambda}_{1}}{\partial B}=2\left\langle \frac{1}{r
\right\rangle >0,\;\frac{\partial \tilde{\lambda}_{1}}{\partial A
=2m\left\langle r\right\rangle , \label{eq:CLH2_HFT}
\end{equation}
which are not satisfied by Mustafa's $\tilde{\lambda}_{1}^{M}$. Besides,
\tilde{\lambda}_{1}^{M}$ does not yield the eigenvalues of the Coulomb
problem when $\tilde{\omega}_{1}=0$ and $A=0$.
The change of variables $\rho =\tilde{\omega}_{1}^{1/2}r$ yields
\begin{equation}
F^{\prime \prime }(\rho )+\left[ \frac{\tilde{\lambda}_{1}}{\tilde{\omega
_{1}}+\frac{1/4-\tilde{\gamma}_{1}^{2}}{\rho ^{2}}-\rho ^{2}-\frac{2mA}
\tilde{\omega}_{1}^{3/2}}\rho -\frac{2B}{\tilde{\omega}_{1}^{1/2}\rho
\right] F(\rho )=0. \label{eq:CLH2}
\end{equation}
It is clear from the analysis above that Mustafa's analytical expressions
for the energy in his equations (33) and (38) cannot be correct.
Consequently, all the physical conclusions derived from such equations, and
shown in Mustafa's figures 3 and 4, are based on wrong analytical
expressions for the eigenvalues $\mathcal{E}^{M}$ and $\tilde{\lambda
_{1}^{M}$.
The three radial eigenvalue equations outlined above are particular cases of
\begin{equation}
F^{\prime \prime }(\rho )+\left[ W+\frac{1/4-\gamma ^{2}}{\rho ^{2}}-\rho
^{2}-\frac{a}{\rho }-b\rho \right] F(\rho )=0, \label{eq:CLH}
\end{equation}
where $a$ and $b$ are arbitrary real parameters. We are interested in
square-integrable solutions $F(\rho )$ that vanish at origin. Such solutions
only take place for particular values of the eigenvalue $W$ that we may
label as $W_{\nu ,\gamma }(a,b)$, $\nu =0,1,\ldots $, in such a way that
W_{\nu ,\gamma }<W_{\nu +1,\gamma }$.
From the HFT we conclude that
\begin{equation}
\frac{\partial W}{\partial a}=\left\langle \frac{1}{\rho }\right\rangle >0,\
\frac{\partial W}{\partial b}=\left\langle \rho \right\rangle >0.
\label{eq:HFT}
\end{equation}
\section{The exactly-solvable case}
\label{sec:HO}
The eigenvalue equation (\ref{eq:CLH}) is exactly solvable \textit{only}
when $a=b=0$. In order to obtain exact solutions for this particularly
simple case we resort to the well known Frobenius (power-series) method. If
we insert the ansatz
\begin{equation}
F(\rho )=\rho ^{s}\exp \left( -\frac{\rho ^{2}}{2}\right) \sum_{j=0}^{\infty
}c_{j}\rho ^{2j},\;s=\left| \gamma \right| +\frac{1}{2},
\label{eq:HO_ansatz}
\end{equation}
into the eigenvalue equation we find that the expansion coefficients $c_{j}$
satisfy the two-term recurrence relation
\begin{equation}
c_{j+1}=\frac{4j+2s-W+1}{2\left( j+1\right) \left( 2j+2s+1\right)
c_{j},\;j=0,1,\ldots . \label{eq:HO_TTRR}
\end{equation}
For arbitrary values of $W$ the solution (\ref{eq:HO_ansatz}) is not square
integrable\cite{F21b}; however if we choose
\begin{equation}
W=W_{\nu ,\gamma }=2\left( 2\nu +|\gamma |+1\right) ,\;\nu =0,1,\ldots ,
\label{eq:HO_W}
\end{equation}
the series reduces to a polynomial and the resulting eigenfunctions are
square integrable. Note that Mustafa's result (\ref{eq:HO_lambda^M}) is
correct because $W=$ $\tilde{\lambda}^{M}/\tilde{\omega}$ when $\nu =n_{r}$.
The two-term recurrence relation thus takes the following simpler form
\begin{equation}
c_{j+1,\nu ,\gamma }=\frac{2\left( j-\nu \right) }{\left( j+1\right) \left(
2j+2s+1\right) }c_{j,\nu ,\gamma }, \label{eq:HO_TTRR_2}
\end{equation}
and the eigenfunctions become
\begin{equation}
F_{\nu ,\gamma }(\rho )=\rho ^{s}\exp \left( -\frac{\rho ^{2}}{2}\right)
\sum_{j=0}^{\nu }c_{j,\nu ,\gamma }\rho ^{2j}. \label{eq:HO_eigenf}
\end{equation}
\section{Conditionally-solvable models}
\label{sec:CLH}
When one of the model parameters $a$ or $b$ is nonzero the eigenvalue
equation (\ref{eq:CLH}) is not exactly solvable. It is an example of
conditionally-solvable quantum-mechanical problems\cite{T16}. Mustafa\cite
{M22a} obtained the results outlined above from a dubious analysis of the
biconfluent Heun function. In this section we resort to the Frobenius method
because it is not only simpler and clearer but leaves no room for doubts.
In the general case we resort to the ansatz
\begin{equation}
F(\rho )=\rho ^{s}\exp \left( -\frac{b}{2}\rho -\frac{\rho ^{2}}{2}\right)
\sum_{j=0}^{\infty }c_{j}\rho ^{j},\;s=\left| \gamma \right| +\frac{1}{2},
\label{eq:CLH_ansatz}
\end{equation}
that leads to the three-term recurrence relation
\begin{eqnarray}
c_{j+2} &=&A_{j}c_{j+1}+B_{j}c_{j}=0,\;j=-1,0,1,\ldots ,\;c_{-1}=0,\;c_{0}=1,
\nonumber \\
A_{j} &=&\frac{a+b\left( j+s+1\right) }{\left( j+2\right) \left(
j+2s+1\right) },\;B_{j}=\frac{4\left( 2j+2s-W+1\right) -b^{2}}{4\left(
j+2\right) \left( j+2s+1\right) }. \label{eq:CLH_TTRR}
\end{eqnarray}
In order to have a polynomial of degree $n$ we require that $c_{n}\neq 0$,
c_{n+1}=0$ and $c_{n+2}=0$, $n=0,1,\ldots $, that clearly leads to $c_{j}=0$
for all $j>n$. It follows from this condition that $B_{n}=0$ that yields
\begin{equation}
W=W_{\gamma }^{(n)}=2n+2s+1-\frac{b^{2}}{4}=2\left( n+|\gamma |+1\right)
\frac{b^{2}}{4}. \label{eq:CHL_W^n}
\end{equation}
When $W=W_{\gamma }^{(n)}$ $B_{j}$ takes the simpler form
\begin{equation}
B_{j}=\frac{2\left( j-n\right) }{\left( j+2\right) \left( j+2s+1\right) }.
\label{eq:CLH_B_j}
\end{equation}
Present expression for $W$ is not consistent with Mustafa's results (\ref
{eq:CLH1_E^M}) and (\ref{eq:CLH2_lambda_1^M}) unless $n=2n_{r}$. However,
this discrepancy is irrelevant because neither $\mathcal{E}^{M}$, $\tilde
\gamma}_{1}^{M}$ or $W^{(n)}_{\gamma}$ are the eigenvalues of the
corresponding radial equations\cite{AF20,AF21,F21}. Note that $W_{\gamma
}^{(n)}$ also fails to satisfy the HFT (\ref{eq:HFT}).
The most important point is that, in order to obtain such particular
polynomial solutions, we need a second condition $c_{n+1}(a,b)=0$ already
omitted by Mustafa\cite{M22a}. Since $c_{j}(a,b)$ is a polynomial function
of degree $j$ in each model parameter we conclude that we have $n+1$ roots
a^{(n,i)}(b)$, $i=1,2,\ldots ,n+1$, for a given value of $b$, or
b^{(n,i)}(a)$ for each value of $a$. It can be proved that all the roots are
real\cite{AF20,CDW00}. The exact polynomial solutions for a given value of $b
$ are
\begin{equation}
F_{\gamma }^{(n,i)}(\rho )=\rho ^{s}\exp \left( -\frac{b}{2}\rho -\frac{\rho
^{2}}{2}\right) \sum_{j=0}^{n}c_{j}^{(n,i)}\rho ^{j}. \label{eq:CLH_eigenf}
\end{equation}
The meaning of this kind of solutions has been discussed extensively in
recent papers\cite{AF20,AF21,F21}. However, in order to make present one
sufficiently self contained and convince the reader that our results are
correct we show some examples in what follows.
When $n=0$ we obtain
\begin{equation}
c_{1}(a,b)=\frac{a+bs}{2s}=0, \label{eq:c_1(a,b)=0}
\end{equation}
and the exact solution
\begin{equation}
F_{\gamma }^{(0)}(\rho )=\rho ^{s}\exp \left( -\frac{b}{2}\rho -\frac{\rho
^{2}}{2}\right) , \label{eq:F^(0)}
\end{equation}
of equation (\ref{eq:CLH}) with $W=W_{\gamma }^{(0)}$ and $a=a_{\gamma
}^{(0)}(b)=-bs$. Note that $F_{\gamma }^{(0)}(\rho )$ is the ground state of
a model given by $a=a_{\gamma }^{(0)}(b)$.
When $n=1$ the second condition reads
\begin{equation}
c_{2}(a,b)=\frac{a^{2}+ab\left( 2s+1\right) +b^{2}s\left( s+1\right) -4s}
4s\left( 2s+1\right) }=0, \label{eq:c_2(a,b)=0}
\end{equation}
with roots
\begin{equation}
a_{\gamma }^{(1,1)}(b)=\frac{\sqrt{b^{2}+16s}-b\left( 2s+1\right) }{2
,\;a_{\gamma }^{(1,2)}(b)=-\frac{\sqrt{b^{2}+16s}+b\left( 2s+1\right) }{2},
\label{eq:a^(1,i)}
\end{equation}
from which we obtain
\begin{eqnarray}
F_{\gamma }^{(1,1)}(\rho ) &=&\rho ^{s}\exp \left( -\frac{b}{2}\rho -\frac
\rho ^{2}}{2}\right) \left( 1+\frac{\sqrt{b^{2}+16s}-b}{4s}\rho \right) ,
\nonumber \\
F_{\gamma }^{(1,2)}(\rho ) &=&\rho ^{s}\exp \left( -\frac{b}{2}\rho -\frac
\rho ^{2}}{2}\right) \left( 1-\frac{\sqrt{b^{2}+16s}+b}{4s}\rho \right) .
\label{eq:F^(1,i)}
\end{eqnarray}
Note that $F_{\gamma }^{(1,1)}(\rho )$ is the ground state of a model with
a=a_{\gamma }^{(1,1)}(b)$ while $F_{\gamma }^{(1,2)}(\rho )$ is the first
excited state for a model given by $a=a_{\gamma }^{(1,2)}(b)$, both for
W=W_{\gamma }^{(1)}$. Anybody can easily verify that the functions in
equations (\ref{eq:F^(0)}) and (\ref{eq:F^(1,i)}) are solutions of equation
\ref{eq:CLH}) under the conditions just indicated.
As already stated above, this kind of solutions has been extensively
discussed in recent papers\cite{AF20,AF21,F21}. We just want to point out
that the occurrence of the exact polynomial solutions (\ref{eq:CLH_eigenf})
requires a second condition (overlooked by Mustafa\cite{M22a}) that
restricts the values of the model parameters and that such solutions are not
the only ones\cite{AF20,AF21,F21}. Therefore, any physical conclusion
derived from eigenvalues like (\ref{eq:CLH1_E^M}) and (\ref
{eq:CLH2_lambda_1^M}) are meaningless. In particular, Mustafa's expressions
for the energy in his equations (33) and (38) cannot be correct.
Consequently, all the physical conclusions derived from such equations, and
shown in Mustafa's figures 3 and 4, are based on wrong analytical
expressions for the eigenvalues $\mathcal{E}^{M}$ and $\tilde{\lambda
_{1}^{M}$.
\section{Conclusions}
\label{sec:conclusions}
We have already shown that the exact results obtained by Mustafa\cite{M22a}
for the eigenvalue equations (\ref{eq:CLH1_Mus}) and (\ref{eq:CLH2_Mus}) are
not correct because they are not exactly solvable. Those equations are
conditionally solvable and one obtains some particular polynomial solutions
for particular values of the model parameters determined by a second
condition overlooked by Mustafa. This conclusion also applies to another
paper by the same author where he apparently derived exact results for a
similar conditionally-solvable problem\cite{M22b}. Most physical conclusions
commonly derived from the polynomial solutions to conditionally-solvable
eigenvalue equations are meaningless\cite{AF20,AF21,F21}.
|
1,108,101,566,807 | arxiv | \section{Introduction}
\label{intro}
The topological He-McKellar-Wilkens (HMW) phase introduced in 1993 by X.G. He and B.H.J. McKellar \cite{HePRA93} and in 1994
by M.~Wilkens \cite{WilkensPRL94} was never tested since its theoretical discovery. We have recently published such a test
\cite{LepoutrePRL12} using our lithium atom interferometer. In a companion paper \cite{LepoutreXXX} quoted here as HMWI,
we have recalled the theory of this topological phase and its relations with the Aharonov-Bohm \cite{AharonovPR59} and
Aharonov-Casher phases \cite{AharonovPRL84}. We have also discussed the effects of phase dispersion on interferometer signals and we have considered in detail the phase shifts induced by electric and magnetic fields, namely the dynamical phase shifts due to the Stark and Zeeman Hamiltonian and the topological phase shift due to the Aharonov-Casher effect. The present paper is devoted to a
detailed presentation of the experiment, of its results and of the analysis of the stray effects which have complicated the test of the HMW phase.
In the following sections, we first describe the experiment, the data recording procedure and the interferometer signals (section \ref{expset}). Then, we discuss the effects of the electric fields (section \ref{elec}) and of the magnetic fields (section \ref{mag}). Section \ref{data} presents the data set for the HMW phase measurement and the raw results. The model describing the stray effects due to phase shift dispersion, introduced in HMWI and developed in the appendix (section \ref{App}), is tested thanks to numerous and sensitive measurements of the fringe phase and visibility (section \ref{test}). Thanks to this model, we have been able to reject most of the stray effects and to measure the HMW phase, as detailed in section \ref{HMWm}. A conclusion (section \ref{Conc}) summarizes what we have learnt from this experiment, recalls the open questions (in particular a phase shift presently not understood) and discusses how to improve this experiment.
\section{The experiment: the setup and the data recording procedure}
\label{expset}
In this part, we briefly describe our atom interferometer and, with greater details, the interaction region used to observe the HMW effect. We also describe the compensator coil used to produce a magnetic field gradient opposite to the one due to the HMW interaction region. Finally, we explain our data recording procedure which rejects the interferometer
phase drifts.
\subsection{Our atom interferometer}
\label{expset1}
\begin{figure}[t]
\begin{center}
\includegraphics[width= 8 cm]{figure1n.eps} \caption{(color online)
Schematic top views (not to scale) of our atom interferometer (panel a) and the interaction region for the observation of the HMW phase (panel b). Our atom interferometer is a Mach-Zehnder interferometer, with two entrances A and B (only A is used) and two exits C and D (C is detected). An atomic beam (dotted blue lines) entering by A is diffracted by three quasi-resonant laser standing waves produced by the mirrors M$_i$. The largest distance between interferometer arms, about $100$ $\mu$m, occurs just before the second laser standing wave, where we introduce the interaction region. The opposite electric fields necessary for the observation of the HMW phase are horizontal. They are produced by two plane capacitors (high voltage electrodes in red; grounded electrodes in black). The septum is a thin common electrode located between the two interferometer arms represented by dotted blue lines. Two rectangular coils (represented by the brown rectangle) produce the vertical magnetic field.
\label{fig1}}
\end{center}
\end{figure}
Our separated arm atom interferometer (see figure \ref{fig1}), has been previously described \cite{MiffreEPJD05,MiffrePhD,LepoutreEPJD11}. Here, we present only its main features and some recent improvements. The atomic source is a supersonic beam of lithium seeded in argon,
with a mean velocity $v_m \approx 1065$ m/s. Once collimated by two $18$ $\mu$m wide slits, this beam has a transverse velocity distribution with a width comparable to the recoil velocity of lithium, $v_r \approx 9$ cm/s. This beam is then
diffracted by three quasi-resonant laser standing waves in the Bragg regime: the present experiment uses first order diffraction, with only two diffracted beams of orders $0$ and $+1$ (or $-1$). We thus get a Mach-Zehnder atom interferometer with two output beams carrying complementary interference signals. One output beam is selected by a slit and its intensity $I$, measured by a surface ionization detector, is given by:
\begin{equation}
\label{a0} I=I_0 \left[1 + \mathcal{V}\cos\left(\varphi_d+\varphi_p\right)\right]
\end{equation}
\noindent $I_0$ is the mean intensity and $\mathcal{V}$ the fringe visibility. The phase is the sum of the phase $\varphi_p$ due to perturbations and the phase $\varphi_d $ due to the diffraction process: $\varphi_d =2k_L(x_1-2x_2+x_3)$, where $k_L$ is the laser wavevector and $x_i$ the $x$-position of mirror M$_i$. The choice of the laser frequency, at about $2$ GHz on the blue side of the $^2$S$_{1/2}$ - $^2$P$_{3/2}$ transition of $^7$Li, and the $92.5$\% natural abundance of $^7$Li ensure that the interferometer signal is almost purely due to this isotope \cite{MiffreEPJD05,JacqueyEPL07}. To record interference fringes, we sweep the phase $\varphi_d $ by varying $x_3$ with a piezoelectric actuator. We measure the variations of $x_3$ with a Michelson interferometer
\cite{LepoutreEPJD11}. Intense signals, with a mean intensity $I_0 \approx 60000$ atoms/s and a high fringe visibility $\mathcal{V} \approx 70$\% provide a large phase sensitivity, with a practically achieved value $\Delta\varphi_{min} \approx 25$ mrad$/\sqrt{ \mbox{Hz}}$.
\subsection{The HMW interaction region}
\label{expset2}
A HMW phase is induced when an atom propagates in crossed electric and magnetic fields, both transverse to the atom velocity. Our experimental arrangement is inspired by the ideas of H. Wei \textit{et al.} \cite{WeiPRL95}, the electric fields are horizontal, in the interferometer plane, and opposite on the two interferometer arms, while the common magnetic field is vertical and as homogeneous as possible.
The electric fields are produced by a double capacitor with a septum \cite{Ekstrom95} located between the interferometer
arms (see figure \ref{fig1}). Each of the two capacitors is similar to the one we used for the measurement of the electric
polarizability of lithium \cite{MiffreEPJD06}. Outer electrodes are made of polished $5$ mm-thick glass plates with evaporated aluminium electrodes. A central high-voltage electrode of length $2a \approx 48$ mm is separated from two $5$ mm-long grounded guard electrodes by $1$ mm-wide gaps: these gaps withstand a voltage larger than $1$ kV. The septum is a $30$ $\mu$m thick aluminium foil. The capacitors are assembled by gluing together the electrodes and the glass spacers (thickness $h\approx 1.10$ mm) with Epotex 301 glue (Epoxy Technologies). The septum must remain well stretched, even if the capacitor temperature varies. With some advice given by A. Cronin, we have acquired the know-how to glue a pre-stretched septum on the electrode-spacer assembly heated near $65^{\circ}$C and, due to differential thermal expansion, the septum is fully stretched when the assembly has cooled down \cite{LepoutrePhD11}. The capacitors are as symmetric as possible and they are powered by slightly different voltages issued from the same power supply, with an adjustable voltage ratio thanks to potentiometers. This arrangement minimizes Stark phase noise due to voltage fluctuations of the power supply. Figure \ref{fig2} presents the calculated $z$-variation of the electric field $E_x$-component, which is relevant for the HMW phase. $E_x$ is calculated at the septum surface whereas the atom-septum mean distance is near $40$ $\mu$m but the associated correction is very small \cite{MiffreEPJD06}.
The capacitors assembly is inserted in a brass support on which we have coiled $1.5$ mm-diameter enameled copper wires to produce the vertical magnetic field needed for the HMW phase. We use two rectangular coils, located below and above the interferometer plane, each coil being made of $2$ layers and each layer of $7$ windings, glued together and to the brass support with a high thermal conductivity glue (Stycast 2850 FT). A $2$-mm internal diameter copper pipe is also glued on the brass support at mid-distance between the two coils and with a $6$ cm$^3$/s flow of tap water (a low flow rate chosen to minimize vibrations),
the temperature rise is about $0.5$ K/W. Usually, we apply a current $I$ in the coils $50$\% of the time so that the maximum current $I = 25$ A induces a $20$ W mean Joule power and a temperature rise near $10$ K. In figure \ref{fig2}, we have plotted the calculated $z$-variation of the magnetic field $B_y$-component which is the one relevant for the HMW phase. As discussed in
HMWI, a Zeeman phase shift appears if the magnetic field modulus $B$ is different on the two interferometer arms, and we have minimized this difference by careful coiling and design of the connection wires geometry.
\begin{figure}
\begin{center}
\includegraphics[width= 8 cm]{figure2n.eps}
\caption{(color online) Calculated components $E_x$ (dashed line) and $B_y$ (full line) as a function of $z$ in the interaction region. For the electric field, the plotted quantity is $E_x/E_0$, where $E_0 = V/h$ is the field of a infinite plane capacitor, with a spacing $h$ and an applied voltage $V$. For the magnetic field, the plotted quantity $B_y/I$, where $I$ is the coil current, is in units of $10^{-4}$ T/A. \label{fig2}}
\end{center}
\end{figure}
The HMW interaction region is placed just ahead the second laser standing wave, where the distance between the center of the interferometer arms is largest, close to $100$ $\mu$m. In order not to induce vibrations of the standing wave mirrors, the interaction region is suspended from the top of the vacuum chamber. Initial adjustments of the rotation around the horizontal $z$-axis and the vertical $y$-axis are performed with optical methods. Rotation around $y$-axis as well as translation in the $x$-direction can be operated under vacuum, and the ultimate tunings are done with the atom interferometer running. After optimization of the interferometer signal, the mean intensity $I_0$ and the fringe visibility $\mathcal{V}$ are not modified by the presence of the septum between the two arms.
The magnetic field produced by the HMW coil was measured with a 3D Hall probe and compared to the field calculations, showing
a good agreement. Concerning the electric field, calibration measurements using the atom interferometer (described in part
\ref{elec1}) yield an accurate knowledge of each capacitor geometry needed for electric field calculations. With the electric and
magnetic field components $E_x$ and $B_y$ as a function of $z$, we can calculate the integral $\int E_x B_y dz$ and thanks to the very accurate knowledge of the electric polarizability of lithium atom \cite{MiffreEPJD06,PuchalskiPRA12}, we can predict the
slope of the HMW phase as a function of $VI$ product:
\begin{equation}
\label{a01} \varphi_{HMW} \left(V, I \right)/(VI) = - (1.28 \pm 0.03) \times 10^{-6} \mbox{ rad/VA}
\end{equation}
\noindent where the error bar is due to the uncertainty on the geometrical parameters of the capacitors and of the HMW coil.
\subsection{Compensator coil}
\label{expset3}
In spite of our efforts, the magnetic field of the HMW coil is slightly different on the two interferometer arms, with a mean
relative difference $\left|\Delta B \right|/ B \approx 10^{-4}$. This difference is most probably due to a bad centering of the septum in the HMW coil, with a distance between the coil symmetry plane and the septum of the order of $250$ $\mu$m. For $I=25$ A, the induced Zeeman phase shift is equal to $\varphi_{Z}(F,m_F)\approx \pm 11$ rad for the $F=2, m_F= \pm 2$ sublevels. We compensate these phase shifts thanks to a supplementary coil producing an opposite magnetic field gradient along the $x$-axis. This so-called compensator coil is made of $9$ turns of copper wire on a $30$ mm-diameter aluminium cylinder. It is located at mid-distance between the first and second laser standing waves, with a mean distance between the compensator coil and the interferometer arms near $10$ mm. This coil is cooled by conduction through its support and temperature rise limits its current $I_C$ to $5$ A, if applied only $50$\% of the time. Then, the magnetic field seen by the atoms is below $2\times 10^{-3}$ T, a range for which Zeeman effect is linear.
\subsection{Data recording and signals}
\label{expset4}
\begin{figure}
\begin{center}
\includegraphics[width= 8 cm]{figure3n.eps} \caption{ (color online) Recorded data with 6 ($V,I$) configurations during a single fringe scan: the number of detected atoms per second (the unit is $10^4$ detected atoms/second and the counting period is $0.1$ second) is plotted as a function of the phase of the optical Michelson interferometer which directly maps the position $x_3$ of mirror M$_3$. The signals corresponding to different field configurations are plotted with different symbols : black squares for $(0,0)$, red bullets for $(V,0)$, blue full triangle for $(V,I)$, open violet squares for $(0,I)$, open blue triangle for $(-V,I)$ and open red circles for $(-V,0)$). The top graph represents the signal as it is recorded and the signal corresponding to each one of the 6 ($V,I$) configurations is plotted separately below with its best fit. \label{fig3}}
\end{center}
\end{figure}
In our previous experiments \cite{MiffreEPJD06,JacqueyEPL07,JacqueyPRL07,LepoutreEPJD11}, we deduced the effect of a perturbation by comparing fringe signals successively recorded with and without this perturbation. The phase measured in the absence of perturbation, which should be constant, drifts with time, typically by several tens of mrad over the few minutes needed for recording a high-quality fringe signal. These drifts are not linear in time and they are due to minute distortions of the rail supporting the standing wave mirrors. Their magnitude is due to the high sensitivity of the diffraction phase $\varphi_d$ to the mirror positions, with $d\varphi_d/ dx_i \approx + 20$ rad/$\mu$m for M$_1$ or M$_3$ and $-40$ rad/$\mu$m for M$_2$. They were the main limitation of our phase shift measurements. For the present experiment, we have almost canceled the sensitivity to these drifts by applying several field configurations during each fringe recording: a field configuration is defined by the $(V,I)$
values, where $V$ is the capacitor mean voltage and $I$ the current in the HMW coil (this current is accompanied by a current $I_C$ in the compensator coil, as explained below). We have used either 4 configurations, $(0,0)$, $(V,0)$, $(V,I)$ and $(0,I)$ or 6 configurations, by adding $(-V,I)$ and ($-V,0)$ to this list. A typical fringe recording with $6$ configurations is shown in figure \ref{fig3}. A fit of the different fringe signal systems is made using equation (\ref{a0}), where the fringe systems of all configurations share the same value of the diffraction phase $\varphi_d$. We thus get the mean intensity $I_0(V,I)$, the fringe phase $\varphi(V,I)$ and the fringe visibility $\mathcal{V}(V,I)$ for each field configuration. In this way, we deduce the effects of the application of the electromagnetic field corresponding to each configuration. $I_0$ is independent of the field configuration, but the visibility and the phase are both modified. We define a relative visibility and a fringe phase shift for each field configuration by:
\begin{eqnarray}
\label{a1}
\mathcal{V}_E(V) &=& \mathcal{V}(V,0)/\mathcal{V}(0,0) \nonumber \\
\varphi_E(V) &=& \varphi(V,0) - \varphi(0,0)\nonumber \\
\mathcal{V}_B(I) &=& \mathcal{V}(0,I)/\mathcal{V}(0,0) \nonumber \\
\varphi_B(I) &=& \varphi(0,I) - \varphi(0,0)\nonumber \\
\mathcal{V}_{EB}(V,I) &=& \frac{\mathcal{V}_{E+B}(V,I)}{\mathcal{V}_E(V) \mathcal{V}_B(I)} =\frac{\mathcal{V}(V,I)\mathcal{V}(0,0)}{\mathcal{V}(V,0) \mathcal{V}(0,I)} \nonumber \\
\varphi_{EB}(V,I) &=& \varphi_{E+B}(V,I)- \varphi_E(V)- \varphi_B(I) \nonumber \\
&=& \varphi(V,I) - \varphi(V,0)-\varphi(0,I)+\varphi(0,0)\nonumber \\
\end{eqnarray}
\noindent
\noindent $\mathcal{V}_{E}(V)$ and $\varphi_E(V)$ are the fringe relative visibility and phase shift with the electric field only. $\mathcal{V}_{B}(I)$ and $\varphi_B(I)$ are the fringe relative visibility and phase shift with the magnetic field only and
$\mathcal{V}_{E+B}(V,I)$ and $\varphi_{E+B}(V,I)$ are the fringe relative visibility and phase shift with the electric and magnetic fields applied simultaneously. A fringe scan such as shown in figure \ref{fig3} lasts about $20$ s, a duration sufficiently small to ensure quasi-linearity of the interferometer phase drift with time (an exactly linear phase drift only alters $\varphi_d$ and leaves the results of eqs. (\ref{a1}) unchanged). The error bars are about $2$\% on the relative visibility and $30$ mrad on the induced phase shifts. We repeat about $100$ successive fringe scans, taking care that the fringe scan period and the field configuration period are not commensurate, in order to avoid any possible bias in the fits. The error bars on the averages of such a scan series are near $0.2$\% for the relative visibility and near $3$ mrad for the phase shifts, small enough to detect fine
perturbations of the interference fringe signals and to understand systematic effects.
\section{Effects of the electric fields on the fringe phase and visibility}
\label{elec}
\subsection{Experimental study of polarizability phase shifts}
\label{elec1}
During calibration measurements, we applied a voltage $V$ to one capacitor only, the other one being grounded. Figure \ref{fig4} presents typical results for the fringe visibility $\mathcal{V}_r = \mathcal{V}(V)/\mathcal{V}(0)$ and the induced phase shift $\varphi_S (V)$ as a function of $V^2$. These measurements were fitted using equations (19-20) and (31) of HMWI, which yields the
value of the parallel speed ratio $S_{\|}= 9.25 \pm 0.08$ and the values of the Stark phase shifts induced by each capacitor for the mean atom velocity: $\varphi_u/V^2 = (-4.830 \pm 0.005)$ rad/V$^2$ and $\varphi_l/V^2 = (4.760 \pm 0.007)$ rad/V$^2$. Using the very accurate theoretical value \cite{PuchalskiPRA12} of lithium atom electric polarizability $\alpha$, we may deduce the geometrical parameter $L_{eff}/h^2$ for both capacitors ($L_{eff}$ is the capacitor effective length and $h$ the plate spacing
\cite{MiffreEPJD06}). The effective length is the same for both capacitors with a good accuracy, $L_{eff}\approx 48\pm 0.5 $ mm, so that these experiments provide measurements of the mean values of the capacitor spacings $h_u = 1.101\pm 0.006$ mm and $h_l= 1.109\pm 0.006$ mm.
\begin{figure}
\begin{center}
\includegraphics[width = 8 cm]{figure4n.eps} \caption{ (color online) The relative visibility $\mathcal{V}_r$ (left scale, black squares and line) and the fringe phase shift $\varphi_{S}$ (rad) (right scale, blue bullets and line) are plotted as a function of $V^2$, where $V$ is the voltage applied to one capacitor only. The points are experimental and the curves are their best fits.
\label{fig4}}
\end{center}
\end{figure}
\subsection{Experiments with both electric fields on: phase measurements}
\label{elec2}
\begin{figure}
\begin{center}
\includegraphics[width = 8 cm]{figure5n.eps} \caption{(color online) Fringe phase shifts induced by electric fields applied to both capacitors: the measured phase shift $\varphi_E(V)$ is plotted as a function of $V^2$ where $V$ is the mean of the voltages $V_u$ and $V_l$ applied to the two capacitors. The points are experimental and the straight line is the best fit. \label{fig5}}
\end{center}
\end{figure}
When we apply electric fields to both capacitors, with the voltage ratio tuned to cancel the Stark phase shift $\varphi_E(V)$, we
observe a residual phase shift due to imperfect tuning: $\varphi_E(V)$ is small and approximately proportional to $V^2$, but with large fluctuations of the measured value (see figure \ref{fig5}). We have observed that $\varphi_E(V)$ drifts with time when the interaction region temperature varies: this behavior can be explained by a delay of the expansion of one capacitor with respect
to the other one, delay due to the low thermal conductivity of glass. For $V=800$ V, the Stark phase induced on each
interferometer arm can reach $\varphi_{S,u} \approx \varphi_{S,l} \approx 307$ rad. Then, a typical deviation for $\varphi_E(V)$ of $0.05$ rad from its mean value corresponds to a $1.7 \times 10^{-4}$ relative variation of the geometrical parameter
$L_{eff}/h^2$ of one capacitor with respect to the other one. This variation is somewhat larger than expected for a simple thermal expansion effect with a temperature variation smaller than $10$K. The conclusion is that, because of dispersion and drift,
the residual Stark phase shift $\varphi_E(V)$ does not carry much useful information.
\begin{figure}
\begin{center}
\includegraphics[width = 8 cm]{figure6n.eps} \caption{(color online) Fringe phase shifts induced by electric fields applied to both capacitors : the difference quantity $\left(\varphi_E(\left|V\right|)-\varphi_E(-\left|V\right|)\right)/2$ is plotted as a function of $\left|V\right|$. The points are experimental and the straight line is the best linear fit.
\label{fig6}}
\end{center}
\end{figure}
In the experiments with 6 field configurations, we can measure the difference of the Stark phase shifts for opposite $V$ values, with an error bar close to $1$ mrad. Figure \ref{fig6} plots the quantity $\left(\varphi_E(\left|V\right|)
-\varphi_E(-\left|V\right|)\right)/2$ as a function of $\left|V\right|$. Eq. (58) of HMWI predicts that the only $V$-odd
term in $\left<\varphi_S\right>$ is the contact potential phase $\left<\varphi_{Sc}\right> $ so that:
\begin{equation}
\label{ex0} \left(\varphi_E(\left|V\right|) -\varphi_E(-\left|V\right|)\right)/2 = 2 \varphi_{0} \frac{\left<\bar{V}_{c,u}\right> -\left<\bar{V}_{c,l}\right>}{V}
\end{equation}
\noindent We have fitted the measured values of $\left(\varphi_E(\left|V\right|) -\varphi_E(-\left|V\right|)\right)/2$ by a function $a+b\left|V\right|$. The fitted $a$-value, $a= 10 \pm 1$ mrad is not explained by eq. (\ref{ex0}) but its presence might be due to the use of different power supplies, one per polarity, to produce opposite voltages. The fitted slope $b= (-6 \pm 2) \times 10^{-3}$ mrad/V can be explained by a difference of the mean contact potentials $\left(\left<\bar{V}_{c,u}\right>
-\left<\bar{V}_{c,l}\right>\right) = 6\pm 2$ mV: we may conclude that contact potentials play a very minor role in our experiment and this idea will be supported by further results.
\subsection{Experiments with both electric fields on: visibility measurements}
\label{elec3}
\begin{figure}
\begin{center}
\includegraphics[width = 8 cm]{figure7n.eps} \caption{(color online) Fringe relative visibility $\mathcal{V}_E(V)$ as a function of the applied mean voltage $V$: the points are experimental, with different symbols (red squares and blue bullets) for two different alignments of the atom interferometer. The data points are fitted by eq. (\ref{ex1}) (red full line and blue dotted line). \label{fig7}}
\end{center}
\end{figure}
We now discuss the measurements of the relative visibility $\mathcal{V}_E(V)$. The residual Stark phase shift
$\varphi_E$ is sufficiently small to neglect any effect of the velocity dispersion on the visibility. The relative visibility showed highly dependent of the standing wave mirrors alignment. Therefore the collected data for $\mathcal{V}_E(V)$ was partitioned into 7 different sets: within a given set, this interferometer alignment is identical for all the data points. Fig. \ref{fig7} presents two of these data sets. All the sets exhibit different behaviors, but are well independently fitted using the following equation:
\begin{eqnarray}
\label{ex1}
\mathcal{V}_E(V)= 1 -\sum _{i=1,4} k_{Vi} V^i\nonumber \\
\end{eqnarray}
\noindent As illustrated by fig. \ref{fig7}, the relative visibility can become larger than $1$, a result apparently surprising. This happens when phase dispersions which exist when no interaction is applied, are partially canceled by the phase dispersion due to the application of the electric fields. The pre-existing phase dispersions originate from the Zeeman phase shifts due to the inhomogeneity of the laboratory magnetic field when $I= I_C = 0$ and from the diffraction phase which presents a spatial dispersion because of an imperfect alignment of the laser standing wave mirrors M$_i$. The application of the electric fields induces an Aharonov-Casher phase shift and a Stark phase shift: the Aharonov-Casher phase shift is dispersed because of its dependence with the $F,m_F$ sublevel and the Stark phase shift is dispersed because of capacitors defects. We must describe all these effects, in order to explain the behavior of $\mathcal{V}_E(V)$. Assuming a balanced hyperfine population i.e. $\chi=0$ ($\chi$ is defined in appendix A of HMWI), we use equations (23), (42) and (46-47) of HMWI to evaluate $\mathcal{V}(0,0)$:
\begin{eqnarray}
\label{ex2} \frac{\mathcal{V}(0,0)}{\mathcal{V}_{0}} &=& \left[1 -
\frac{\left< \left(\delta \varphi_d\right)^2\right>}{2}\right]
\frac{\mathcal{V}_{B0}}{\mathcal{V}_0} \nonumber \\
\mbox{with } \frac{\mathcal{V}_{B0}}{\mathcal{V}_0} &=& \frac{1 +
\cos\left(J_0\right) + 2 \cos\left(J_0/2\right)}{4}
\end{eqnarray}
\noindent $\delta \varphi_d (y)$ is the dispersion of the diffraction phase and $\mathcal{V}_{B0}$ is the visibility modified
by the inhomogeneity of the laboratory magnetic field. Because of these two effects, the observed visibility $\mathcal{V}(0,0)$ is smaller than its optimum value $\mathcal{V}_0$. With electric fields on both arms, the Stark phase shift $\varphi_S (y)$ is a function of $y$, and the Aharonov-Casher phase shift $\varphi_{AC}(F,m_F)$ is given by $\varphi_{AC}(2,m_F) = -\varphi_{AC}(1,m_F) = (m_F/2) \varphi_{AC}(2,2)$ (this formula is valid because Zeeman effect is linear in the laboratory field). We deduce the fringe visibility $\mathcal{V}(V,0)$:
\begin{eqnarray}
\label{ex3} \frac{\mathcal{V}(V,0)}{\mathcal{V}_0} &=&
\mathcal{V}_{B0}\left[1 - \frac{\left< \left(\delta \varphi_d+
\delta\varphi_S\right)^2\right>}{2}\right] \nonumber \\
&-& \frac{\varphi_{AC}(2,2)}{4} \left[\sin\left(J_0\right) +
\sin\left(\frac{J_0}{2}\right)\right]
\end{eqnarray}
\noindent The last term of the r.h.s. is a first-order Taylor expansion of the trigonometric functions of $\varphi_{AC}$, valid because $\left|\varphi_{AC} \right| \leq 70$ mrad in the present experiment. We get the relative visibility $\mathcal{V}_E(V)= \mathcal{V}(V,0)/ \mathcal{V}(0,0)$:
\begin{eqnarray}
\label{ex4} \mathcal{V}_E(V) &=& 1 - \frac{\left< \left(\delta
\varphi_S\right)^2+ 2 \delta
\varphi_d\delta\varphi_S\right>}{2}\nonumber \\
&& - \frac{\varphi_{AC}(2,2)}{4\mathcal{V}_{B0}}
\left[\sin\left(J_0\right) + \sin\left(\frac{J_0}{2}\right)\right]
\end{eqnarray}
\noindent The dispersion $\delta\varphi_S$ of the Stark phase shift is given by $\delta\varphi_S = \delta \varphi_{S,g}+\delta
\varphi_{S,c} $ with the geometrical defect term $\delta \varphi_{S,g} \propto V^2$ and the contact potential term $\delta
\varphi_{S,c} \propto V$. $\varphi_{AC}(2,2) \propto V$ while $\delta \varphi_d$ and $J_0$ are independent of $V$. We thus deduce the values of the $k_{Vi}$ coefficients:
\begin{eqnarray}
\label{ex5}
k_{V1} &=& \left< \delta \varphi_d\delta\varphi_{S,c}\right> + \frac{\varphi_{AC}(2,2)}{4\mathcal{V}_{B0}} \left[\sin\left(J_0\right) + \sin\left(\frac{J_0}{2}\right)\right] \nonumber \\
k_{V2} &=& \left< \left(\delta \varphi_{S,c}\right)^2/2\right> + \left<\delta \varphi_d \delta\varphi_{S,g}\right> \nonumber \\
k_{V3} &=& \left< \delta \varphi_{S,g}\delta \varphi_{S,c}\right> \nonumber \\
k_{V4} &=& \left< \left(\delta \varphi_{S,g}\right)^2/2\right>
\end{eqnarray}
\noindent Discussed below is a comparison of eqs. (\ref{ex5}) with the results of fits of $\mathcal{V}_E (V)$ for
the 7 available data sets: at the same time, we test the validity of our description of experimental defects and we get some insights on the nature of the systematic effects.
All $k_{V4}$ values are positive and compatible with their mean,
$k_{V4} = (6.0 \pm 0.5)\times 10^{-14}$ V$^{-4}$. This is in
agreement with eqs. (\ref{ex5}) which predicts that $k_{V4}$ is
positive and depends solely of the geometrical defects of the
capacitors and not of the interferometer alignment. From this result,
we may estimate the geometrical defects of the capacitors if we
assume that the spacing difference $\Delta h =h_u-h_l $ is the main
defect and that it varies linearly with $y$. We then find that
$\Delta h $ varies by about $1.4$ $\mu$m over the $y$-range sampled
by the atoms (about $2$ mm). This $\Delta h$ value appears to be quite small for capacitors
assembled by gluing parts together but, when $V = 800$ V, this small defect is sufficient to induce a total dispersion of the Stark phase shift along the atomic beam height equal to $0.8$ rad.
All $k_{V3}$ values (excepted one) are compatible with $0$, with a
very small mean value, $k_{V3}= (0.04 \pm 1.7)\times 10^{-12}$
V$^{-3}$, corresponding to $\left< \delta \varphi_{S,g}\delta
\varphi_{S,c}\right> < 10^{-6}$ rad$^2$ for $V=800$ V. The
dispersions $\delta \varphi_{S,g}$ and $\delta \varphi_{S,c}$ are not
correlated, in agreement with the idea that contact potentials
fluctuate on small scales and that geometrical defects are smooth
functions of $y$.
Each $k_{V2}$ value has a small error bar but $k_{V2}$ varies
strongly from one set of data to the next, covering the range from
$-5\times 10^{-8}$ to $+13\times 10^{-8}$ V$^{-2}$.
These large variations prove that the dominant contribution comes
from the interferometer alignment i.e. from the $\left<\delta \varphi_d
\delta\varphi_{S,g}\right>$ term. When $\delta \varphi_d$ and $\
\delta\varphi_{S,g}$ have opposite variations, the electric fields
increase the visibility, as observed in figure \ref{fig7}.
All $k_{V1}$ values are compatible with their mean, $k_{V1}=
\left(1.40 \pm 0.07\right) \times 10^{-5}$ V$^{-1}$. The first term
$\left< \delta \varphi_d\delta\varphi_{S,c}\right>$, which involves
the contact potential term, is expected to be very small for the same
reasons which explain the weakness of $k_{V3}$ and, if this term was
not negligible, $k_{V1}$ should vary with the interferometer
alignment like $k_{V2}$. The second term, which is due to the
Aharonov-Casher phase shift in the laboratory magnetic field, must be
dominant. Assuming that the laboratory magnetic field $\mathbf{B}_0$
is constant over the capacitor length, and that the electric fields
are equal to $E_0= V/h$ on a length $L_{eff}\approx 48$ mm, we
estimate the Aharonov-Casher phase shift given in
eqs. (2) and (49) of HMWI:
\begin{eqnarray}
\label{ex6}
\varphi_{AC}(2,2) = \frac{2\mu_B E_0 L_{eff}}{\hbar c^2}\mathbf{y}\cdot \mathbf{u}_0
\end{eqnarray}
\noindent where $\mathbf{u}_0 =\mathbf{B}_0/B_0$ points in the
direction of $\mathbf{B}_0$. The measurements presented in the next
section give access to $J_0 \approx -0.61$ rad and to
$\mathcal{V}_{B0}\approx 0.93$. We thus deduce $\mathbf{y}\cdot
\mathbf{u}_0 \approx -0.7$ i.e. $\mathbf{B}_0$ points downward, at
about $45^{\circ}$ from the vertical, in agreement with direct
measurements of the local laboratory field.
\section{Effects of the magnetic field on the fringe phase and visibility}
\label{mag}
\subsection{Experiments with the compensator coil only}
\label{mag1}
We have measured the relative visibility $\mathcal{V}_B(I_C)$ and the phase shift $\varphi_B(I_C)$ of the interference fringes as a function of the compensator coil current $I_C$. The results are plotted in fig. \ref{fig8} with fits based on eqs. (42) and (45) of HMWI, with $J_1 = A_{J1,C} \left|I_{C}-I_{0,C}\right| + J_{0,C}$ and assuming balanced sublevel populations (see appendix A of HMWI).
\begin{figure}
\begin{center}
\includegraphics[width = 8 cm]{figure8n.eps} \caption{Relative visibility (upper panel) and phase shift (lower panel) as a function of the compensator current $I_C$ (A). The points are experimental and the curves represent their best fit, with $A_{J1,C} = 1.52 \pm 0.02$ rad/A, $I_{0,C} = 0.09 \pm 0.02$ A and $J_{0,C} = -0.63 \pm 0.03$ rad. The minor deviations which appear when $I_C> 4$ A are
probably due to the arbitrary assumption of balanced sublevel populations ($\chi=0$) for this particular example. \label{fig8}}
\end{center}
\end{figure}
\subsection{Experiments with the HMW-coil only}
\label{mag2}
The relative visibility $\mathcal{V}_B(I)$ and the phase shift
$\varphi_B(I)$ were measured as a function of the HMW coil current
$I$. Some of the results are plotted in fig. \ref{fig9}.
Equation (41) of HMWI gives $\varphi_{Z}(F,m_F)$ as a function of
$J_1$, $J_2$ and $J_3$ and we use $J_1 = A_{J1}
\left|I-I_{0} \right| + J_{0,I}$, $J_2= A_{J2} I^2$, $J_3= A_{J3}
\left|I\right|^3$ to fit the data. The hyperfine
population unbalance parameter $\chi$ is also fitted, with a different
value for each data set corresponding to a slightly different laser frequency.
\begin{figure}
\begin{center}
\includegraphics[width = 8 cm]{figure9n.eps} \caption{(color online) Real and imaginary part of the complex fringe visibility as a function of the HMW coil current $I$ (note the expanded scale for Im$(\mathcal{V}_B(I)$). The points are experimental and the curves are the results of best fits, with 3 different $\chi$ values: $\chi= 0.077$ for $I= 0-4$ A; $\chi= -0.014$ for $I= 5-10$ A; $\chi= -0.062$ for $I= 11-15$ A. When these experiments were done, we had not understood that the laser frequency must be tightly controlled in order to keep $\chi$ very small and this explains why large $\chi$ values are observed. We get $A_{J1} = -0.46 \pm 0.02 $ rad/A; $A_{J2} = \left(-110 \pm 8\right)\times
10^{-4}$ rad/A$^2$; $A_{J3}= \left(-20 \pm 4 \right)\times 10^{-5}$
rad/A$^3$; $I_{0} = 0.32 \pm 0.05$ A and $J_{0,I} = -0.55 \pm 0.13$ rad. \label{fig9}}
\end{center}
\end{figure}
\subsection{Experiments with both coils and global fit}
\label{mag3}
With a HMW coil current $I$ and a compensator coil current $I_C$, optimum compensation of the linear part of the Zeeman phase shift is obtained with $I_C \approx \left|I\right|/3$. When $\left|I\right|> 15$ A, it is impossible to use $I_C>5$ because the compensator coil temperature rises too much and, then, we have used $I_C=5$ A. Fig. \ref{fig10} presents the relative visibility $\mathcal{V}_B$ as a function of $I$. Comparison with fig. \ref{fig9} proves the efficiency of the compensator: when $I_C=0$, $\mathcal{V}_B (I)$ vanishes for $I\approx 6$ A while, with the compensator in operation, it remains larger than $80$\% if $\left|I\right|\leq 12$ A and vanishes only for $\left|I\right|\approx 18$ A. The revival observed for $I=23$ A, with a relative visibility near $-70$\% and a phase shift close to $\pi$, is explained in \cite{LepoutrePhD11}.
In order to have the best estimate of the Zeeman phase
shifts induced during the HMW effect measurements, we performed a
single global fit of all the data recorded while testing the effects
of magnetic fields. This data set was collected during the HMW
effect measurements using both coils (with $I_C$ related to $I$ for optimum compensation) as well as during calibration
measurements using either both coils (with different relative tuning of $I$ and $I_C$) or only one coil. As introduced in HMWI, the Zeeman phase shifts are calculated using $J_1 = A_{J1} \left|I-I_{0}
\right|+ A_{J1,C} \left|I_{C}-I_{0,C} \right| + J_{0,I+C}$, with $J_{0,I+C} = J_0 - A_{J1} \left|I_{0}\right| - A_{J1,C}
\left|I_{0,C}\right|$. The data set for $\mathcal{V}_B \left(I,I_C \right)$ and
$\varphi_B \left(I,I_C \right)$ includes about 150 data points which belong to $31$ series corresponding to slightly different laser frequencies and a different $\chi$ value is fitted for each series. Here are the fitted values of $J_0$, $I_0$, $A_{Ji}$, $I_{0,C}$ and $A_{J1,C}$ provided by this global fit:
\begin{eqnarray}
\label{ex13}
J_0 &=& -0.61 \pm 0.01 \mbox{ rad} \nonumber \\
I_{0} &=& 0.31 \pm 0.03 \mbox{ A} \nonumber \\
A_{J1} &=& -0.430 \pm 0.005 \mbox{ rad/A} \nonumber \\
A_{J2} &=& \left(-662 \pm 5\right)\times 10^{-5} \mbox{ rad/A}^2 \nonumber \\
A_{J3} &=& \left(-180\pm 5 \right)\times 10^{-6} \mbox{ rad/A}^3 \nonumber \\
I_{0,C} &=& \left( 22 \pm 9\right) \times 10^{-3} \mbox{ A} \nonumber \\
A_{J1,C}&=& 1.43 \pm 0.015 \mbox{ rad/A}
\end{eqnarray}
\begin{figure}
\begin{center}
\includegraphics[width = 8 cm]{figure10n.eps} \caption{(color online) Real part of the complex relative visibility $Re\left(\underline{\mathcal{V}}_B\right)(I)$ plotted as a function of the HMW current $I$, the compensator current $I_C$ having the value described in the text. The points are experimental and the
curve is calculated using the global fit results, eqs.
(\ref{ex13}), with the population balance parameter fixed at $\chi
=0$. We have not represented the imaginary part
$Im\left(\underline{\mathcal{V}}_B\right)(I)$ which is very small ($<
0.03$) and very sensitive to $\chi$. \label{fig10}}
\end{center}
\end{figure}
\section{Data set for the HMW phase measurement and raw results}
\label{data}
\begin{figure}[t]
\begin{center}
\includegraphics[height= 6 cm]{figure11n.eps} \caption{Data set
collected for the measurement of the HMW phase. Each run is
represented in the $I$,$V$ plane by a triangle (respectively a
bullet) for a 4-field (respectively 6-field) experiment.
\label{fig11}}
\end{center}
\end{figure}
Fig. \ref{fig11} presents the data set collected for the HMW phase
measurement. As $\varphi_{HMW}$ is very small and proportional to the
$VI$ product, we have chosen to record data with large values either
of $V$ or of $I$, so that we have no data point near the origin.
The measured values of the phase shift $\varphi_{EB}(V,I)$ are
plotted as a function of the $VI$ product in fig. \ref{fig12}: these
results do not agree with the predicted variations of $\varphi_{HMW}$
and we explain this discrepancy by stray phase shifts which appear when the
electric and magnetic fields are simultaneously applied. The origin
of these stray phases have been explained on general grounds in HMWI
and the detailed calculation is presented in the appendix of the
present paper. We are going to test these calculations first on the
relative visibility $\mathcal{V}_{EB}(V,I)$ and afterwards on the
phase shift $\varphi_{EB}(V,I)$. The various stray effects differ by their symmetry with respect to the reversal of the electric and/or magnetic fields and, in order to test separately these effects, it is necessary to extract the even/odd
parts of these quantities with respect to field reversals by combining
measurements for opposite $V$ or $I$ values. For any quantity
$f(V,I)$, the mean $\mathcal{M}_Xf(V,I)$ and the half-difference
$\Delta_X f(V,I)$ for opposite values of $V$ (then $X=E$) or of $I$ (then
$X=B$) are equal to:
\begin{eqnarray}
\label{a4}
\mathcal{M}_Ef(V,I)&=& \left[f(V,I) + f(-V,I)\right]/2 \nonumber \\
\Delta_Ef(V,I)&=& \left[f(V,I) -f(-V,I)\right]/2 \nonumber \\
\mathcal{M}_Bf(V,I)&=& \left[f(V,I) + f(V,-I)\right]/2 \nonumber \\
\Delta_Bf(V,I)&=& \left[f(V,I) - f(V,-I)\right]/2
\end{eqnarray}
\noindent Most experiments were done with 6 field configurations and they provide simultaneous measurements of $\mathcal{V}_{EB}(V,I)$ and $\varphi_{EB}(V,I)$ for opposite voltages, with exactly the same current $I$ and the same value of the population unbalance parameter $\chi$: we thus have very sensitive tests of the effects of electric field reversal.
\begin{figure}[t]
\begin{center}
\includegraphics[width= 8 cm]{figure12n.eps}
\caption{(color online) Measured values of $\varphi_{EB}(V,I)$ given by eq. (\ref{a1}) as a function of the $VI$ product measured in VA. The (red) full line represents the expected value of the HMW phase, $\varphi_{HMW}(V,I)= - (1.28 \pm 0.03) \times 10^{-6} VI$ rad. \label{fig12}}
\end{center}
\end{figure}
\section{Some experimental tests of the effects of stray phases}
\label{test}
We are going to test the predictions of the calculations described in the appendix of the present paper.
\subsection{Tests involving the fringe visibility}
Following eq. (\ref{a6}), four combinations of
$\mathcal{V}_{EV} (V,I)$ separate the contributions of the four
$D_{\pm,\pm}(V,I)$ terms. However, as shown by eq.
(\ref{a14}), the quantity $\mathcal{V}_{EB}$ also
includes a contribution due to the Aharonov-Casher effect in the
corresponding $(V,0)$ field configuration. It is given by the term
$D_{AC,B0}(V)/D_{0,B0}$, with a value close to $1.1$\%
for $V=800$ V. Because it involves the AC phase, this effect is an
odd function of the voltage $V$. We eliminate this contribution by using the measured values of $\mathcal{V}_E (\pm V)$
to calculate $\Delta_E\mathcal{V}_{E}(V)$, from which we deduce a corrected fringe visibility given by
$\mathcal{V}_{EB}^{'}(V,I) = \mathcal{V}_{EB}(V,I)/\left( 1 -
\Delta_E\mathcal{V}_{E}(V)\right)$. This quantity is now simply expressed by eq. (\ref{a3}) (obviously, this correction is necessary only when studying $V$-odd terms).
The variations of $\mathcal{M}_B\Delta_E\mathcal{V}_{EB}^{'}(V,I)$
give a test of the $\sum D_{-,+}$ term. We do not plot these results
here because all the values are very small, in the $(-1\mbox{ to }
+5)\times 10^{-3}$ range, with error bars near $\pm 2\times 10^{-3}$,
with one exception, for $I=\pm 19$ A (the visibility is then very low
and some approximations of our calculations of the appendix are no more valid). The variations of
$\Delta_B\mathcal{M}_E\mathcal{V}_{EB}(V,I)$ give a test of the $\sum
D_{+,-}$ term. We do not plot these results here because all the
values are also very small, in the $(-2 \mbox{ to } +4)\times
10^{-3}$ range, with error bars near $\pm 2\times 10^{-3}$. These two
results prove that the $\sum D_{-,+}$ term and the $\sum D_{+,-}$
term are very small, in good agreement with our calculations which
predict that these terms should vanish if the contact potential terms
are negligible.
Having verified that the $D_{-,+}$ terms are negligible,
the quantity $\Delta_E \mathcal{V}_{EB}^{'}(V,I)$ reduces to $\sum
D_{-,-}/D_0$ (see eq. (\ref{a5})). The leading terms of $D_{-,-}$ given by eq. (\ref{Aa6}) are proportional to the Aharanov-Casher phase:
\begin{eqnarray}
\label{a15} \Delta_E \mathcal{V}_{EB}^{'}(V,I) \approx -\frac{\sum
\varphi_{AC} \sin \left( \phi_Z\right) }{\sum \cos \left( \phi_Z\right)
}
\end{eqnarray}
\noindent Thanks to our knowledge of the Zeeman phases (eqs. (\ref{ex13})), we can evaluate all the terms of
eq. (\ref{a15}) and we compare its prediction to our measurements in
fig. \ref{fig13} and fig. \ref{fig14}. The good agreement, obtained without any fitted parameter, proves that the
dominant $V$-odd effect is due to the AC phase shift, and
confirms the validity of our calculations.
\begin{figure}[t]
\begin{center}
\includegraphics[width= 8 cm]{figure13n.eps} \caption{Plot of
$\Delta_E\mathcal{V}_{EB}^{'}(V= 800 \mbox{ V},I)$ as a function of the current
$I$: the measured data points (squares) are compared to the result of our model (full line). The visibility, proportional to $D_0 = \sum \cos \left(\phi_Z\right)$ vanishes when $I\approx\pm 18$ A indicated by vertical
lines: this induces a divergence of the prediction of our model, which uses a first-order calculation in $D_{\pm,\pm}/D_0$.
Our model explains well the main variations of $\Delta_E \mathcal{V}_{EB}^{'}(V,I)$, even if some imperfections appear clearly.
\label{fig13}}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width= 8 cm]{figure14n.eps} \caption{Plot of
difference of visibility for opposite $V$ values,
$\Delta_E\mathcal{V}_{EB}^{'}(V,I=19\mbox{ A})$, as a function of $V$. The
current value $I=19$ A, chosen close to the cancelation of $D_0 =
\sum \cos \left( \phi_Z\right)$, enhances the sensitivity of the
visibility to the AC phase. The measured values (squares) are
well represented by a linear function of $V$, as predicted by our
model. \label{fig14}}
\end{center}
\end{figure}
\subsection{Tests involving the fringe phase}
We first discuss the combination
$\mathcal{M}_B\Delta_E\varphi_{EB}(V,I)$ given by:
\begin{eqnarray}
\label{a16} \mathcal{M}_B\Delta_E\varphi_{EB}(V,I)&=& -\frac{\sum
N_{-,+} }{D_0}
\end{eqnarray}
\noindent As $N_{-,+}$ is non-zero only if contact potentials are not
negligible, we expected this quantity to be negligible.
$\mathcal{M}_B\Delta_E\varphi_{EB}$ is plotted as a
function of $I$ on fig. \ref{fig15} and as a function of $V$ on fig.
\ref{fig16}. These experimental results are surprising:
$\mathcal{M}_B\Delta_E\varphi_{EB}(V,I)$ is almost independent of the
current $I$ and it rapidly increases with the voltage $V$. The
measured values are well fitted as the sum of two terms, one term in $V$ and
one in $V^3$ (odd powers of $V$ have been chosen because this quantity is $V$-odd).
Contact potentials can in principle explain non zero values of $\mathcal{M}_B\Delta_E\varphi_{EB}$ (the calculation is made in ref. \cite{LepoutrePhD11}), but the predicted effect depends of the current $I$ with divergences similar to those visible in fig. \ref{fig13}, in complete disagreement with the measurements plotted in fig. \ref{fig15}. Moreover, the observed
magnitude of $\mathcal{M}_B\Delta_E\varphi_{EB}(V,I)$ would require
values of contact potentials that are ruled out by the measurements of $\mathcal{V}_E (V)$ and $\mathcal{V}_{EB} (V,I)$
previously presented. This effect is strange because
$\varphi_{EB}(V,I)$ given by eq. (\ref{a1}) is already a
difference of phase shifts measured with and without the magnetic
field, so that $\mathcal{M}_B\Delta_E\varphi_{EB}(V,I)$ must vanish when
the applied magnetic field goes to zero: as a consequence, the
independence of $\mathcal{M}_B\Delta_E\varphi_{EB}(V,I)$ with the
current $I$ cannot extend to $I\rightarrow 0$. However, if the
transition occurs for instance when the laboratory field and the HMW field are
comparable in magnitudes, it should be observed with a
current $I$ of the order of $0.1$ A, a range of $I$-values we have
not studied.
We have investigated several possible explanations which revealed
unsatisfactory for different reasons: usually, either the symmetry
with respect to $V$ and $I$ reversals or the order of magnitude of
the observed phase are not in agreement with our observations.
Moreover, most explanations cannot explain why the
effect is sensitive to the presence of the magnetic field but
independent of its value in the studied range in figure \ref{fig15}.
We will not discuss here these failed explanations, for lack of
space. The origin of this phase shift remains mysterious but thanks
to its independence with regards to $I$, it can be
easily eliminated by combining data with opposite $I$-values.
\begin{figure}[t]
\begin{center}
\includegraphics[width= 8 cm]{figure15n.eps} \caption{Plot of the
quantity $\mathcal{M}_B\Delta_E\varphi_{EB}(V,I)$ in radians as a
function of the current $I$ for several values of the voltage $V$
applied to the capacitors. The dotted lines are simply connecting
values measured for the same $V$ values. The vertical line for
$I\approx 18$A indicates the place where the fringe visibility
vanishes. \label{fig15}}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width= 8 cm]{figure16n.eps} \caption{Plot of the
quantity $\mathcal{M}_B\Delta_E\varphi_{EB}(V,I)$ as a
function of the voltage $V$ applied to the capacitors for all the
values of the current $I$ in the HMW coil. The dotted line is a fit
with a single $V^3$ term while the full line is a fit with a term in $V$ and a term in $V^3$. \label{fig16}}
\end{center}
\end{figure}
We now discuss the quantity
$\Delta_B\mathcal{M}_E\varphi_{EB}(V,I)$ which vanishes if contact potentials are negligible. If they are taken
into account, this quantity is given by \cite{LepoutrePhD11}:
\begin{eqnarray}
\label{a17} \Delta_B\mathcal{M}_E\varphi_{EB}(V,I) \approx \frac{\sum
\varphi_{AC} \left< \delta \varphi_{S,c}\delta\varphi_Z\right>\cos
\left( \phi_Z\right) }{\sum \cos \left( \phi_Z\right) }
\end{eqnarray}
\noindent Because of the presence of a contact potential term $\delta \varphi_{S,c}$, $ \Delta_B\mathcal{M}_E\varphi_{EB}(V,I)$ is expected to be small, and we have not included higher order terms in eq. (\ref{a17}), because they should be even smaller. The measured values of $\Delta_B\mathcal{M}_E\varphi_{EB}(V,I)$ are plotted in fig.
\ref{fig17}, with different symbols for data points depending if
$\left|I\right|$ is smaller or larger than $12$ A. When
$\left|I\right|\leq 12 $ A, the measured values are very small and
compatible with $0$: in this range of $I$ values, the Zeeman phases
$\phi_Z$ are small thanks to the compensator, the systematic effects
are weak and the approximations done in our model are good. When
$\left|I\right|> 12 $ A, the Zeeman phases increase
rapidly with $\left|I \right|$, and several effects decrease
the accuracy of our model. First, the polynomial expansion of the Zeeman
phases in powers of the current $I$ is poorly convergent for some sublevels (see HMWI) while the systematic effects are very sensitive to the value of the Zeeman phases.
Secondly, with increasing Zeeman phases, the systematic effects which
involve the dispersion $\delta \varphi_Z$ increase (this point is discussed
below). Finally, increasing Zeeman phases induce a rapid decrease of the
visibility which cancels for $I\approx18$ A and higher
order terms in $N_i/D_0$ or $D_i/D_0$ are no more negligible.
\begin{figure}[t]
\begin{center}
\includegraphics[width= 8 cm]{figure17n.eps} \caption{Plot of the
quantity $\Delta_B\mathcal{M}_E\varphi_{EB}(V,I)$ as a
function of the $VI$ product: different symbols depending if
$\left|I\right|\leq 12 $ A or $\left|I\right|> 12 $ A. \label{fig17}}
\end{center}
\end{figure}
The tests on the fringe phase presented up to now have detected stray phase shifts not larger than $35$ mrad. We end this part by
considering the quantity $\mathcal{M}_E\varphi_{EB}$, which includes the largest stray phase shifts. $\mathcal{M}_E\varphi_{EB}(V,I)$ is an even function of the current $I$, because we have just shown that
$\Delta_B \mathcal{M}_E \varphi_{EB} (V,I) = \left[\mathcal{M}_E\varphi_{EB} (V,I) - \mathcal{M}_E\varphi_{EB}
(V,-I)\right]/2$ is negligibly small. $\mathcal{M}_E\varphi_{EB}$ is given by:
\begin{eqnarray}
\label{AjoutSL} \mathcal{M}_E \varphi_{EB} (V,I) &=& -\frac{\sum N_{+,+}}{D_0} \nonumber \\
&\approx& - \frac{\sum \left< \delta \varphi_{S,g} \delta \varphi_Z \right> \sin (\phi_Z)}{\sum \cos (\phi_Z)}
\end{eqnarray}
\noindent where we have neglected higher-order terms (see eq. (\ref{Aa6})). The measured values of
$\mathcal{M}_E\varphi_{EB}(V,I)$ for $V=800$ V are plotted in fig.
\ref{fig17SL18}. The Stark phase dispersion $\delta \varphi_{S,g}
(y)\propto V^2$ has been characterized thanks to the study of
$\mathcal{V}_E(V)$ (part \ref{elec3}). The evaluation of the
variations of $\delta \varphi_Z (y)$ with $I$ is done at the expense
of a supplementary approximation, assuming a rectangular profile for
the field of the HMW coil along the atom trajectory (compare fig.
\ref{fig2}). It then becomes possible to perform a fit of the measured values of $\mathcal{M}_E\varphi_{EB}$
(taking into account the terms neglected in eq.
(\ref{AjoutSL})). The result of this fit is also shown in figure
\ref{fig17SL18}: a good agreement is found for the behavior of this
quantity, and the fitted parameter values are compatible with the
expected dispersion of $\delta \varphi_Z(y)$ along the atomic beam
height according to the calculations of the magnetic field. This result confirms the importance of the spatial dispersion $\delta \varphi_Z (y)$ of the Zeeman phase shifts and it proves that the main systematic effects are due to these spatial phase dispersions.
\begin{figure}[t]
\begin{center}
\includegraphics[width= 8 cm]{figure18n.eps} \caption{\label{fig17SL18} Plot of the
quantity $\mathcal{M}_E\varphi_{EB}(V,I)$ for $V=800$ V, as a function of the
current $I$. The points are measured values, the line is a best fit
for all the measurements of $\mathcal{M}_E\varphi_{EB}$ (see
discussion). As for fig. \ref{fig13}, vertical gray lines indicate
cancelation of the visibility.}
\end{center}
\end{figure}
\subsection{ Conclusion concerning systematic effects}
Here are the main results of our study of these effects:
\begin{itemize}
\item the effects of the spatial dispersions $\delta\varphi_{S,g}$
and $\delta \varphi_Z$ of the Stark and Zeeman phase shifts
respectively are well identified;
\item the effects of the the dispersion $\delta\varphi_{S,c}$ of the
Stark phase due to contact potentials appear to be below our
experimental sensitivity;
\item our model provides a qualitative understanding of the systematic
effects for all values of the current $I$. The visibility decreases
rapidly and vanishes for $\left|I\right|\approx 18 $ A: this
circumstance has been used to enhance the sensitivity to certain
terms but clearly, as soon as in $\left|I\right|> 12 $ A, our
model describing the systematic effects is less accurate.
\item as the visibility presents a revival for $\left|I\right|\approx 23 $ A with the Zeeman phases $\phi_Z$ being close to $\pm \pi$, we have made several series of measurements in this range of $I$-values but we cannot expect our model to be accurate;
\item we have observed an unexpected phase shift which is independent
of the current $I$ in the range $8-23$ A and which is odd with
respect to $V$-reversal. We presently have no explanation
for this effect and we continue our investigations on
its possible sources. It may be either a systematic effect forgotten in our analysis or a fundamental physical effect,
for instance such as the effects discussed by J. Anandan
\cite{AnandanPRL00} but, as far as we can judge, these fundamental
effects are either too small or they have not the correct symmetry
with respect to $V$ and $I$.
\end{itemize}
\section{Measurement of the HMW phase}
\label{HMWm}
We now use our knowledge of the stray phase shifts in order to
eliminate their contributions to the measurement of the HMW phase.
The HMW phase $\varphi_{HMW}$ is proportional to the $VI$
product, i.e. it is odd with respect to $V$- and
$I$-reversals. The main contribution in the stray phase
shifts on the measurements of $\varphi_{EB}$ are even with respect to
$V$ and $I$, but because of the existence of a $V$-odd phase of
unknown origin, we choose to use the $I$-odd character of
$\varphi_{HMW}$ to cancel the maximum amount of systematic effects.
Accordingly, we plot the quantity $\Delta_B\varphi_{EB}(V,I)$ as a
function of the $VI$ product. We have used different symbols for the
measurements depending if $\left|I\right|$ is smaller or larger than
$12$ A and we have made separate fits of these two sets of data using
$\Delta_B\varphi_{EB}(V,I) = \alpha VI + \beta$.
\begin{eqnarray}
\label{sm4}
\alpha &=& \left( -1.94 \pm 0.06 \right)\times 10^{-6} \mbox{ rad/VA } \nonumber \\
\beta &=& \left( 7 \pm 4 \right) \times 10^{-4} \mbox{ rad } \nonumber \\
\mbox{ if} && \left|I\right|\leq 12 \mbox{ A}
\end{eqnarray}
\noindent and
\begin{eqnarray}
\label{sm5}
\alpha &=& \left( -2.16 \pm 0.14 \right)\times 10^{-6} \mbox{ rad/VA } \nonumber \\
\beta &=& \left( -26 \pm 19 \right) \times 10^{-4} \mbox{ rad } \nonumber \\
\mbox{ if} && \left|I\right|> 12 \mbox{ A}
\end{eqnarray}
\noindent In both fits, the intercept $\beta$ for $VI=0$ is
compatible with a vanishing value. The error bar on the slopes
$\alpha$ is substantially smaller when $\left|I\right|\leq 12$ A than
when $\left|I\right|> 12$ A: this is visible on the data which is
more dispersed when $\left|I\right|> 12$ A. For both
fits, the fitted slopes are larger (in modulus) than the predicted
value $\varphi_{HMW} \left(V, I \right)/(VI) = - (1.28 \pm 0.03)
\times 10^{-6} $ rad/VA. The discrepancy is equal $52$\% if
$\left|I\right|\leq 12$ A and $69$\% if $\left|I\right|> 12$ A. Our
model predicts that there are two contributions to
$\Delta_B\varphi_{EB}(V,I)$:
\begin{eqnarray}
\label{a18}
\Delta_B\varphi_{EB}(V,I) &=& \varphi_{HMW} - \frac{N_{-,-}}{D_0}
\end{eqnarray}
\noindent $N_{-,-}$ given by equation \ref{Aa6} is the
product of the AC phase by correlation terms. Thanks to the knowledge
of the experimental defects, it is possible to evaluate all the terms
involved in $N_{-,-}/D_0$. The only quantities which are not directly
measured are the correlations $ \left<\delta \varphi_d \delta
\varphi_Z\right>$ and $ \left<\delta \varphi_d \delta
\left(\varphi_Z\right)^2\right>$ which are evaluated from the
measurement the correlation with $\delta \varphi_S$ replacing $\delta
\varphi_d$, assuming that both effects are linear functions of $y$.
The calculated value of $N_{-,-}/D_0$ never exceeds $3$
mrad for the data set with $\left|I\right|\leq 12$ A,
and we have made this correction to get $\varphi_{final}(V,I)$ which is
plotted in fig. 3 of our letter \cite{LepoutrePRL12}. The fitted
slope $\varphi_{final}(V,I)/VI = (-1.68 \pm 0.07)\times 10^{-6}$ rad/V.A is
still too large but the discrepancy with the theoretical value is reduced to $31$\%.
\begin{figure}[h]
\begin{center}
\includegraphics[width= 8 cm]{figure19n.eps} \caption{(color online) Measured values
of $\Delta_B\varphi_{EB}(V,I)$ as a function of the $VI$ product. The
data points with $\left|I \right| \leq 12$ A are plotted in red as
well as their fit represented by a dotted line. The data points with
$\left|I \right| > 12$ A are plotted in black as well as their fit
represented by a dashed line. The expected dependence of
$\varphi_{HMW}(V,I)$ with the $VI$ product is represented by a black full line.
\label{FigDBPhiAEB}}
\end{center}
\end{figure}
\section{Conclusion}
\label{Conc}
\subsection{Some remarks on the present experiment}
We have described a measurement of the He-McKellar-Wilkens
topological phase by atom interferometry. This experiment was
feasible with our atom interferometer because the interferometer arms
are well separated in space and the interferometer signal is intense,
with a large fringe visibility, near $70$\%. The arm separation is
needed in order to insert a septum between the two interferometer
arms without any degradation of the signal. The signal intensity and
the large value of the fringe visibility both contribute to enhance
the phase sensitivity: its practically achieved value near $25$
mrad/$\sqrt{\mbox{Hz}}$ is needed for the present measurement. The
HMW phase shift is rather small, at most $27$
mrad under our experimental conditions, and appears as
the combination of four phase measurements for which
$2000$ s of data recording were needed to reduce the uncertainty
near $3$ mrad.
The analysis of the experiment revealed more complex than expected,
because of stray phases. The complexity of the signal is due to several
factors:
\begin{itemize}
\item the signal is the sum of the contributions of 8 sublevels which
are not exactly in phase because of the Zeeman phase shifts due to the
slightly different values of the magnetic field on the two interferometer arms;
\item we have built a compensator coil to produce an opposite gradient
of the magnetic field at another place in the interferometer. The use
of this compensator has been very fruitful as it has enabled us to
apply substantially higher fields with a limited loss of fringe
visibility. However the compensator produces a low field, so that it
can correct only the part of the phase shifts due to linear Zeeman
effect;
\item the weights of the various $F, m_F$ sublevels are functions of
the laser frequency and power density in the standing waves used for
atom diffraction. We had to control these parameters rather tightly
in order to keep these weights almost equal and constant;
\item the main phase shifts (diffraction phase shift, Stark and Zeeman
phase shifts) present a dispersion with the atomic trajectory
described in our calculations by the $y$-coordinate. In the presence
of several dispersed phase shifts, the visibility of the contribution
of a given sublevel to the total fringe signal is better or worse,
depending if the dispersions of the different phase shifts subtract or add their
effects.
\end{itemize}
We have developed a model taking into account all these effects and
this model has been very successful in explaining the variations of
the observed phase shifts and visibility with the capacitor voltage
$V$ and the HMW coil current $I$. However, an extra-phase has been
observed and characterized: this phase is odd with the
capacitor voltage $V$; it behaves roughly like $V^3$; it appears only when the
magnetic field is applied but its value is independent of the
magnetic field magnitude in a wide range. We continue our
investigations to understand the effect which produces this phase.
By combining measured phase shifts with opposite values of the current
$I$, we have eliminated this phase and we have obtained a first
measurement of the HMW phase shift. The observed effect is larger
than its expected value by $69$\% if we use all the collected data
points and only by $52$\% if we consider only the data points with
$\left|I \right|\leq12$ A. Finally, there is a small stray contribution of
the Aharonov-Casher phase to the measured phase shift, and
using our model, it was possible to evaluate this contribution and to
correct the measured values accordingly. The discrepancy
between our corrected measurements and the expected HMW phase shift
is then reduced to $31$\%.
\subsection{Possible improvements of this experiment}
It is necessary to improve this experiment in order to reduce the
uncertainty on the HMW phase-shift. Here are the main possibilities:
\begin{itemize}
\item reduction of stray effects by a better construction of the HMW
interaction region. The present construction has two main defects:
the difference of the capacitor thicknesses varies with the
$y$-coordinate and the septum does not coincide with the
symmetry plane of the HMW coils. The construction of capacitors with
a better controlled geometry is probably possible but quite
difficult, because of the need of using a stretched septum. A better
centering of the septum with respect to the HMW coils is probably
rather easy and this would reduce substantially the Zeeman phase
shifts which are the largest source of complication.
\item reduction of stray effects by optical pumping of the atomic
beam. If all the atoms are in one $F,m_F$ sublevel only, the signal
is no more an average on the hyperfine sublevels
populations. Moreover, the trajectory-averaged Zeeman phase shift
can be exactly canceled by the compensator if the pumping is done in
the $F=2$, $m_F=+2$ (or $-2$) sublevel for which Zeeman effect is
exactly linear. As a consequence, this arrangement, which should reduce most
of the stray phase shifts, is feasible with minor modifications
of our setup and experiments are in progress.
\item reduction of stray effects by using another atom: this requires
the development of a completely new atom interferometer with
separated arms. Most of the difficulties are due to the paramagnetic
character of lithium and an atom with a $^1$S$_0$ non-degenerate
ground state (i.e. with a zero nuclear spin) would be ideal
because there would be no Zeeman phase shift and no Aharonov-Casher
phase shift. We may consider either a thermal beam of a light atom or
a laser-cooled atomic source. In the case of a thermal beam, the most
obvious choice is ground state helium, with which a very nice
interferometer using diffraction by material gratings was developed
by J.P. Toennies and co-workers \cite{CroninRMP09}. Because helium
electric polarizability is small ($\alpha_{He} \approx
\alpha_{Li}/120$), larger electric and/or magnetic fields will be
needed. Among atoms which have been laser-cooled, magnesium, calcium,
strontium or ytterbium all have a $^1$S$_0$ ground state and at least one
stable isotope with a nuclear spin equal to $0$.
\end{itemize}
\acknowledgments
We thank the laboratory technical staff for their help, A. Cronin for fruitful discussions, G. Tr\'enec, A. Miffre and M. Jacquey for all the work done on our atom interferometer. We are greatly indebted toward CNRS INP, ANR (grants ANR-05-BLAN-0094 and ANR-11-BS04-016-01 HIPATI) and R\'egion Midi-Pyr\'en\'ees for supporting our research.
\section{Appendix: calculation of the fringe signal}
\label{App}
We describe here the main points of our calculation of the fringe
signal from which we deduce the stray phase and the fringe
visibility.
\subsection{Some simplifying assumptions}
\label{theory1}
The fringe phase $\varphi$ is the sum of the diffraction phase
$\varphi_d$, the Sagnac phase $\varphi_{Sagnac}$ due to Earth
rotation, the Stark phase $\varphi_S$, the Zeeman phase
$\varphi_{Z}(F,m_F)$, the HMW phase $\varphi_{HMW}$ and the
Aharonov-Casher phase $\varphi_{AC}(F,m_F)$:
\begin{eqnarray}
\label{t0} \varphi&=& \varphi_d + \varphi_{Sagnac} + \varphi_S +
\varphi_{Z}(F,m_F) \nonumber \\ && + \varphi_{HMW} + \varphi_{AC}(F,m_F)
\end{eqnarray}
\noindent $\varphi_d$, $\varphi_{HMW}$ and $ \varphi_{AC}(F,m_F)$ are
independent of the atom velocity $v$; $\varphi_{Sagnac}$ and
$\varphi_S$ vary like $1/v$ and $\varphi_{Z}(F,m_F)$ like $1/v^2$.
These velocity-dependent phases are small ($\varphi_{Sagnac}\approx
0.65$ rad, $\left| \varphi_S \right| \lesssim 0.2$ rad and
$\left|\varphi_{Z}(F,m_F)\right|<2$ rad when $\left|I\right|< 18$ A - only these data points will be retained for the final
analysis). As the parallel speed ratio $S_{\|}$ of the lithium beam
is large, $S_{\|}\approx 9$, we may forget the velocity
average and, as a consequence, the Sagnac phase $\varphi_{Sagnac}$
which is constant. We consider the spatial dispersion of $\varphi_d$,
$\varphi_S$ and $\varphi_{Z}(F,m_F)$ only and we neglect this
dispersion for $\varphi_{HMW}$ and $\varphi_{AC}(F,m_F)$, because they
are small, $\left| \varphi_{HMW} \right|< 27$ mrad and
$\left|\varphi_{AC}(F,m_F)\right|< 70$ mrad, for our largest fields.
The total phase dispersion $\delta \varphi$ is the sum of three terms
only:
\begin{eqnarray}
\label{t01}
\delta \varphi &=& \delta \varphi_d + \delta \varphi_S + \delta \varphi_Z
\end{eqnarray}
\noindent From now on, the $F,m_F$ dependence of $\varphi_Z$ and
$\varphi_{AC}$ is not explicit and, for $\varphi_d$, $\varphi_S$ and $\varphi_Z$, we note $\phi_X$ the spatial
average of $\varphi_X$ given by $\phi_X=\left< \varphi_X \right> =
\int dy P(y) \varphi_X(y) $. The average over the $F,m_F$ sublevels
is taken with equal weights, $P(F,m_F) = 1/8$. This is a
good approximation because in the experiments devoted to the HMW
phase measurement, we have kept $\chi$ small
($\left|\chi\right| <0.03$) and randomly distributed
around $0$ (its main effect is to induce a supplementary dispersion
of our phase measurements). With these approximations, following the
discussion of section IV of HMWI, the signal due to one $F,m_F$
sub-level is given by:
\begin{eqnarray}
\label{Aa1}
I(F,m_F) &=& I_0 \left[1 + \mathcal{V}_m\left<\cos\left(\varphi_m\right)\right>\right]/8 \nonumber \\
\mathcal{V}_m &=& \mathcal{V}_0 \left[ 1 -\left< {\left(\delta \varphi\right)}^2/2\right> \right] \nonumber \\
\varphi_m &=& \phi- \left< {\left(\delta \varphi \right)}^3/6\right>\nonumber \\
\mbox{with } \phi &=& \phi_d +\phi_S+\phi_{Z}(F,m_F)+ \varphi_{HMW}+ \varphi_{AC}(F,m_F) \nonumber \\
\mbox{and } \delta \varphi &=& \delta\varphi_d +\delta \varphi_S+ \delta\varphi_{Z}(F,m_F)
\end{eqnarray}
\noindent If we neglect nuclear magnetism, the $F,m_F$ sublevels form
4 pairs with exactly opposite Zeeman energy shifts: three pairs of
levels with the same $m_F$ value and the pair $F=2, m_F = \pm 2$. We
label these pairs by an $m_F$ value going from $-1$ to $+2$ and we
note $\varphi_{Z}$ the value of $\varphi_{Z}(F,m_F)$ for the sublevel
$F=2,m_F$.
\subsection{Tutorial calculation}
\noindent Because of numerous terms, these calculations are rather
complicated and we first present a tutorial calculation in which we
cancel $\delta\varphi_d$ and $\varphi_{AC}(F,m_F)$, and
we forget the cubic term in $\delta \varphi$. We first calculate the
signal $I_{pair}$ of a pair of sublevels:
\begin{eqnarray}
\label{Aa2} \left[\frac{I_{pair}}{I_{0}/4} -1\right]/ {\mathcal{V}}_0 & \approx & \left[1 -\frac{\left<\left(\delta \varphi_S\right)^2\right>+\left<\left(\delta \varphi_{Z}\right)^2\right>}{2}\right] \nonumber \\ &\times&
\cos\left(\phi_{Z}\right) \cos\left( \phi_d + \phi_S + \varphi_{HMW} -\theta \right) \nonumber \\
\mbox{with } \tan \theta &\approx & \theta \approx \left<\delta \varphi_S\delta \varphi_{Z}\right> \tan\left(\phi_{Z}\right)
\end{eqnarray}
\noindent The important point is the phase shift $\theta$
proportional to the correlation term $\left<\delta \varphi_S\delta
\varphi_{Z}\right>$ and this effect is due to the fact that the
contributions of the two levels of the pair have different
visibility: the term $\left[ 1 -\left<\left(\delta \varphi_{S}+
\delta \varphi_{Z}\right)^2\right>/2\right]$ modifies these
visibility in a different way because the dispersions $\delta
\varphi_S$ and $\delta\varphi_{Z}$ have the same sign for one level
of the pair and opposite signs for the other one. Because of the
$\tan\left(\phi_{Z}\right)$ factor, $\theta$ is very sensitive to
$\phi_{Z}$ value, especially when $\phi_{Z}$ is close to $\pi/2$.
\subsection{Complete calculation}
If we remove the approximations done in the tutorial example, we get
the signal $I_{tot}$ which has a form analogous to equation
(\ref{Aa2}):
\begin{eqnarray}
\label{Aa3}\left[\frac{I_{tot}}{I_{0}} -1\right]/ {\mathcal{V}}_0 &\approx& \frac{1}{4}\left[D \cos\left( \phi_d + \phi_S + \varphi_{HMW} \right) \right. \nonumber \\ &&+ \left. N \sin\left( \phi_d + \phi_S + \varphi_{HMW} \right)\right] \nonumber \\
&\approx & \frac{\sqrt{D^2 + N^2}}{4} \cos\left( \phi_d + \phi_S + \varphi_{HMW} -\theta \right) \nonumber \\
\mbox{with } \tan \theta &\approx & \theta \approx \frac{N}{D}
\end{eqnarray}
\noindent The numerator $N$ and the denominator $D$ of the fraction
giving $\theta $ are given by:
\begin{eqnarray}
\label{Aa4}D&=& \left[1 -\frac{\left<\left(\delta
\varphi_d + \delta \varphi_S\right)^2\right>}{2} \right] D_0 + D_Z
+ D_{+/-}\nonumber \\ N &=& \frac{\left<\left(\delta \varphi_d +
\delta \varphi_S\right)^3\right>}{6} D_0 + N_Z + N_{+/-}
\end{eqnarray}
\noindent with the following definitions:
\begin{eqnarray}
\label{Aa5}D_0&=& \sum \cos\left(\phi_{Z}\right) \nonumber \\ D_Z &=& \sum \left[-\frac{\left<\left(\delta \varphi_Z\right)^2\right>}{2} \cos\left(\phi_{Z}\right) \right. \nonumber \\ && + \left. \frac{\left<\left(\delta \varphi_Z\right)^3+ 3 \left(\delta \varphi_d\right)^2\delta \varphi_Z\right>}{6} \sin\left(\phi_{Z}\right)\right]\nonumber \\
D_{+/-} & =& \sum\left[ D_{+,+} + D_{-,+} + D_{+,-} + D_{-,-}\right]\nonumber \\
N_Z &=& \sum \left[\left<\delta \varphi_d \delta \varphi_Z\right> \sin\left(\phi_{Z}\right)+ \left<\delta \varphi_d\left( \delta \varphi_Z\right)^2\right> \cos\left(\phi_{Z}\right)\right] \nonumber \\
N_{+/-} & =& \sum \left[ N_{+,+} + N_{-,+} + N_{+,-} + N_{-,-}\right]
\end{eqnarray}
\noindent In these equations, $\sum$ is the sum over the 4 pair of
levels labeled by the $m_F$ value as defined after equation
(\ref{Aa1}) and this index is omitted everywhere. $D_0$ represents
the effect of the Zeeman phase shifts $\phi_{Z}$ on the visibility,
neglecting their spatial dispersion. $D_B$ and $N_B$ represent the
effects of the dispersions of the diffraction phase shift $\delta
\varphi_d$ and of the Zeeman phase shift $\delta\varphi_{Z}$. The
effects of $D_0$ and $D_B$ are independent of the application of the
electric field. $D_{+/-} $ and $N_{+/-} $ are the sum of four terms
which involve the simultaneous application of the electric and
magnetic field: the first index is the parity with respect to voltage
reversal and the second index is the parity with respect to current
reversal.
In ref. \cite{LepoutrePhD11}, we have developed the calculations of
the $D_{\pm,\pm}$ and $N_{\pm,\pm}$ terms including the contributions
of the dispersion $\delta \varphi_{S,c}$ due to contact potential
(see HMWI) and the presence of this $V$-odd phase largely increases
the number of terms in these equations. As the contact potential
terms appear to be extremely small, we do not take them into account
in the present discussion but we refer the reader to ref.
\cite{LepoutrePhD11} for a more complete discussion. With this
simplification, $\delta \varphi_{S}$ is reduced to the geometrical
defect term which is $V$-even and the $D_{\pm,\pm}$ and $N_{\pm,\pm}$
terms are given by:
\begin{eqnarray}
\label{Aa6} D_{+,+}&=& \left[\frac{\left<\left(\delta
\varphi_S\right)^2\delta \varphi_Z\right>}{2}+
\left<\delta \varphi_S\delta \varphi_d \delta \varphi_Z \right>
\right] \sin
\left( \phi_Z\right) \nonumber \\
D_{-,+} &=& D_{+,-} = 0\nonumber \\
D_{-,-} & =& \left[ -1 + \frac{\left<\left(\delta \varphi_d +
\delta \varphi_S \right)^2 \right>+ \left<\left(\delta
\varphi_Z\right)^2\right>}{2}\right]\nonumber \\ &\times & \varphi_{AC}
\sin \left( \phi_Z\right) + \left[\frac{\left<\left(\delta \varphi_Z
\right)^3 \right>}{6} + \left<\delta \varphi_d \delta \varphi_S
\delta \varphi_Z \right>\right. \nonumber \\
&+& \left.\frac{\left<\left(\delta \varphi_S\right)^2 \delta
\varphi_Z + \left(\delta \varphi_d\right)^2 \delta \varphi_Z
\right>}{2}
\right]\varphi_{AC} \cos \left( \phi_Z\right) \nonumber \\
N_{+,+} &=& \left<\delta \varphi_S \delta \varphi_Z\right> \sin \left( \phi_Z\right) + \frac{\left<\delta \varphi_S \left(\delta \varphi_Z\right)^2\right>}{2} \cos \left( \phi_Z\right) \nonumber \\
N_{-,+} &=& N_{+,-} = 0 \nonumber \\
N_{-,-} & =& \left[\left<\delta \varphi_S \delta \varphi_Z\right> + \left<\delta \varphi_d \delta \varphi_Z\right>\right] \varphi_{AC} \cos \left( \phi_Z\right) \nonumber \\
&-& \left[\frac{\left<\left(\delta \varphi_S +\delta \varphi_d \right)^3 \right>}{6} + \frac{\left<\left(\delta \varphi_S +\delta \varphi_d \right) \left( \delta \varphi_Z \right)^2 \right>}{2} \right]\nonumber \\
&&\times \varphi_{AC} \sin \left( \phi_Z\right)
\end{eqnarray}
\noindent From these equations, it is easy to deduce the relative
visibility and the phase shift of the interference fringes:
\begin{eqnarray}
\label{Aa9} \mathcal{V}_r &=& \frac{\mathcal{V}_m}{\mathcal{V}_0} =
\frac{\sqrt{D^2 + N^2}}{4} \approx \frac{D}{4} \nonumber \\
\phi_m &=& \phi_S + \varphi_{HMW} -\theta \nonumber \\
\theta &\approx & N/D
\end{eqnarray}
\noindent We have used a third-order approximation of the sine and
cosine function in eq. (23) of HMWI but we use only a first-order
approximation to get $\theta \approx N/D$ and $\mathcal{V}_r \approx
D/4$. This first order approximation is good if $N\ll D$. For a
practical use of these results, it will be necessary to assume that
$D_0$ is considerably larger that the other terms appearing in $D$
and that $N$ is small with respect to $D_0$ so that we will further
simplify the expression of $\theta \approx N/D_0$. We are going to
use the following equations for the analysis of our experimental
results:
\begin{eqnarray}
\label{a2} \frac{\mathcal{V}_m(V,I)}{\mathcal{V}_0} &=& \frac{1}{4}
\left[ \left(1 -\frac{\left<\left(\delta \varphi_S + \delta \varphi_d
\right)^2\right>}{2}\right)D_0 \right. \nonumber \\
&& \left. + D_Z + D_{+/-}\right] \nonumber \\
\phi_m (V,I)&=& \phi_S + \varphi_{HMW} - \frac{\left<\left(\delta \varphi_S + \delta \varphi_d \right)^3\right>}{6}\nonumber\\
&& - \frac{N_Z + N_{+/-}}{D_0}
\end{eqnarray}
\subsection{Calculation of the phase shift and the visibility neglecting the effects of the laboratory magnetic field}
\label{theory2}
To evaluate the visibility $\mathcal{V}_{EB}(V,I)$ and
the phase $\phi_{EB}(V,I)$ defined by eqs. (\ref{a1}), we
use eq. (\ref{a2}) to calculate the terms
corresponding to the different field configurations. As a first
simplified approach, we consider that the laboratory magnetic field
is homogeneous. In the case of fields configurations for which $I=0$,
apart from canceling the Zeeman phase shifts, this also enables
neglecting the effect of the Aharonov-Casher phase. At first order
in the stray terms, several cancelations appear:
\begin{eqnarray}
\label{a3}
\mathcal{V}_{EB}(V,I)&=& 1 + \frac{D_{+/-}(V,I)}{D_0(V,I)} \nonumber \\
\phi_{EB}(V,I) &=& \varphi_{HMW} (V,I) - \frac{N_{+/-}(V,I)}{D_0(V,I)}
\end{eqnarray}
\noindent Using the definitions of eqs. (\ref{a4}), we
separate the contributions in $D_{+/-}$ and in $N_{+/-}$
following their even/odd characters with respect to $V$
and $I$. Here are the results for the visibility:
\begin{eqnarray}
\label{a5}
\mathcal{M}_E\mathcal{V}_{EB}&=& 1 +\left[\sum\left(D_{+,+} + D_{+,-}\right)/D_0\right] \nonumber \\
\Delta_E\mathcal{V}_{EB}&=& \sum\left( D_{-,+} + D_{-,-}\right)/D_0 \nonumber \\
\mathcal{M}_B\mathcal{V}_{EB}&=& 1 + \left[\sum\left( D_{+,+} + D_{-,+}\right)/D_0 \right] \nonumber \\
\Delta_B\mathcal{V}_{EB}&=& \sum\left( D_{+,-} + D_{-,-}\right)/D_0
\end{eqnarray}
\noindent We fully separate the four $D_{\pm,\pm}$ terms by taking
means or half differences of the above quantities:
\begin{eqnarray}
\label{a6}
\mathcal{M}_B\mathcal{M}_E\mathcal{V}_{EB}&=& 1 +\left[\sum D_{+,+}/D_0\right] \nonumber \\
\mathcal{M}_B\Delta_E\mathcal{V}_{EB}&=& \sum D_{-,+}/D_0 \nonumber \\
\Delta_B\mathcal{M}_E\mathcal{V}_{EB}&=& \sum D_{+,-}/D_0 \nonumber \\
\Delta_B\Delta_E\mathcal{V}_{EB}&=& \sum D_{-,-}/D_0
\end{eqnarray}
\noindent Similar combinations with the phase $\phi_{EB}(V,I) $ also
enable the separation of the $N_{i,j}$ terms:
\begin{eqnarray}
\label{a7}
\mathcal{M}_B\mathcal{M}_E\phi_{EB}&=& - N_{+,+}/D_0 \nonumber \\
\mathcal{M}_B\Delta_E \phi_{EB}&=& - N_{-,+}/D_0 \nonumber \\
\Delta_B\mathcal{M}_E\phi_{EB}&=& - N_{+,-}/D_0 \nonumber \\
\Delta_B\Delta_E\phi_{EB}&=& \varphi_{HMW}- N_{-,-}/D_0
\end{eqnarray}
\subsection{Effect of the inhomogeneity of the laboratory magnetic field on the measurements}
We now take into account the inhomogeneity of the
laboratory magnetic field. Its main effect is to induce weak Zeeman
phase shifts, and we neglect their spatial dispersions ($\delta J_0
(y) = 0$). This brings corrections only to the $D(0,0)$ and $D(V,0)$
terms, i.e. to the visibility terms in the field configurations for
which $I=0$. It is straightforward to calculate $D(0,0)$:
\begin{eqnarray}
\label{a11}
D(0,0) &=& \left[1 -\frac{\left<\left( \delta \varphi_d \right)^2\right>}{2}\right] D_{0,B0}\nonumber \\
\mbox{with } D_{0,B0}&=& \left[1 + \cos\left(J_0\right) + 2
\cos\left(\frac{J_0}{2}\right)\right]
\end{eqnarray}
\noindent When the electric field is applied, the residual Zeeman
phase shifts are still present ($J_0 \neq 0$). With nonzero Zeeman
phase shifts, the AC effect modifies the visibility independently of
the presence of spatial phase dispersion $\delta \varphi (y)$: this
modification is described by the leading term $-\varphi_{AC} \sin
(\phi_Z)$ in the expressions of $D_{-,-}$, eqs. (\ref{Aa6}). We thus
obtain:
\begin{eqnarray}
\label{a11BisSL}
D(V,0) &=& \left[1 -\frac{\left<\left( \delta \varphi_S +\delta \varphi_d \right)^2\right>}{2}\right] D_{0,B0} + D_{AC,B0}\nonumber \\
\mbox{with } D_{AC,B0}&=& - \sum \varphi_{AC} \sin\left(\phi_Z\right)\nonumber \\
&=& - \varphi_{AC,B0} \left[ \sin\left(J_0\right) + \sin\left(\frac{J_0}{2}\right)\right]
\end{eqnarray}
\noindent Here $\varphi_{AC,B0}$ is the AC phase of the $F=2,m_F=2$
sub-level in the presence of the laboratory magnetic field: this
phase shift is proportional to the applied voltage $V$. In this way,
we regain the results of eq. (\ref{ex4}) for the relative
visibility $\mathcal{V}_{E}$, and we express the asymmetry $\Delta_E
\mathcal{V}_{E}$ of the visibility with voltage reversal:
\begin{eqnarray}
\label{a13}
\mathcal{V}_{E} &= & 1 -\frac{\left<\left( \delta \varphi_S \right)^2\right>}{2} - \left<\delta \varphi_S \delta \varphi_d \right> + \frac{D_{AC,B0}}{D_{0,B0}} \nonumber \\
\Delta_E \mathcal{V}_{E} &=& \frac{D_{AC,B0}(V)}{D_{0,B0}}
\end{eqnarray}
\noindent In the field configurations with $I \neq 0$, the
calculations of the type of eqs. (\ref{a2}) are not further modified.
Therefore, the presence of the laboratory magnetic field brings a
correction only to $\mathcal{V}_{EB}$ in equations (\ref{a3}), in the
following form :
\begin{eqnarray}
\label{a14}
\mathcal{V}_{EB}(V,I)&=& 1 + \frac{D_{+/-}(V,I)}{D_0(V,I)} - \frac{D_{AC,B0}(V)}{D_{0,B0}} \nonumber \\
\end{eqnarray}
|
1,108,101,566,808 | arxiv | \section{Introduction}
Self-testing is a process where an untrusted physical realization of states and measurement operations is certified to be equivalent to some ideal reference model.
Our lack of trust requires this certification to be based on a limited number of assumptions, and for this reason it is usually conditioned only on the observed measurement statistics.
The concept of self-testing was first introduced by Mayers and Yao \cite{MY}, who showed how particular statistics originating from a bipartite system can be reproduced only by performing a specific set of local measurements on a maximally entangled pair of qubits. This idea was later generalized to other states also composed by more than two parties, such as the three-qubit $W$ state \cite{stW} and graph states \cite{stGr}.
In this notes we follow the work in \cite{stW} and we generalize the protocol to self test (non trivial) Dicke states, \textit{i.e.} states of $N-k$ qubits in the ground state and $k>1$ qubits in the excited state, symmetric under particle exchange.
\section{Definition of self-testing}
Consider the scenario where $N$ physical devices (observers), labeled by $A_i$ with $i=1...N$, share a $N$-partite state $\vert\Psi\rangle$. Every device performs one out of $m_i$ possible local measurements $M_{j_i,A_i}$ (with $j_i=1...m_i$) on its share of the state, and obtains as outcome either $+ 1$ or $-1$. Moreover, the devices cannot communicate to each other during the measurement process. Our task is to certify, without assuming what has been measured, whether the physical realization of such experiment is equivalent to a reference model where the state and the measurements are known.
To be more precise, we formalize this concept by saying that a physical experiment and a reference experiment are equivalent if there exist a local isometry (\textit{i.e.} a map between Hilbert spaces)
\begin{equation}\label{lociso}
\Phi = \Phi_{A_1} \otimes ... \otimes \Phi_{A_N}
\end{equation}
and a state $\vert \text{junk}\rangle$, such that for every $j_i=1, ..., m_i$ and $A_i=1, ..., N$
\begin{align}
\Phi\left( \vert\Psi\rangle \right) &= \vert\text{junk}\rangle \otimes \vert\Psi^\star\rangle \label{isoTransfSt}\\
\Phi\left( M_{j_i,A_i} \vert\Psi\rangle \right) &= \vert\text{junk}\rangle \otimes M_{j_i,A_i}^\star \vert\Psi^\star\rangle \label{isoTransfMeas}
\end{align}
where $M_{j_i,A_i}^\star$ and $\vert\Psi^\star\rangle$ denote respectively the measurements and state in the reference experiment, and $\vert \text{junk}\rangle$ is in the same Hilbert space as $\vert\Psi\rangle$.
This definition of equivalence is motivated by the fact that performing local operations, as well as adding local degrees of freedom (ancillas), do not change the state: one can always exploit the arbitrariness of the reference system and neglect some degrees of freedom to go back to the original state. For this reason, two states mapped one into the other by an isometry are equivalent.
Self-testing consist in the claim that if the correlations observed in a physical experiment coincides with the one predicted by a particular reference experiment, namely
\begin{equation}
\langle\Psi\vert \bigotimes_{i=1}^N M_{j_i,A_i} \vert\Psi\rangle = \langle\Psi^\star\vert \bigotimes_{i=1}^N M_{j_i,A_i}^\star \vert\Psi^\star\rangle \;,
\end{equation}
for every choice of measurement settings, then the two experiments are equivalent. This means that physical realizations of states and measurements can be certified to be equivalent to a reference model only by looking at the statistics of the physical measurement outcomes.
For practical purposes, it is important that self-testing protocols require only few measurement settings per party, and that the correlators we want to measure involve only a subset of all possible combinations of measurements.
\section{Self-testing Dicke states}
Dicke state states are symmetric $N$-qubit states with $k$ qubits in $\vert 1 \rangle$ and $N-k$ qubits in $\vert 0\rangle$, in symbols
\begin{equation}
\vert D_N^k \rangle = {{N}\choose{k}}^{-\frac{1}{2}} \text{Sym} \left( \vert 0\rangle^{\otimes N-k} \vert 1 \rangle^{\otimes k} \right) \;,
\end{equation}
where $\text{Sym}(...)$ denotes the symmetrization by particle exchange. The reference experiment we consider is the one where the $N$ qubits of a Dicke state are shared among observers $A_{1}...A_{N}$, each allowed to perform local spin measurements $\sigma_x^{(A_i)}$ or $\sigma_z^{(A_i)}$, except for observer $A_N$ that can additionally measure $(\sigma_x^{(A_N)}+\sigma_z^{(A_N)})/\sqrt{2}$.
Consider now the physical realization of this scenario, where the $N$ observers share a state $\vert\Psi\rangle$ and perform local measurements $X_{A_i}$, $Z_{A_i}$ (for $i=1...N$) and $D_{A_N}$. Note that these measurements need not to be spin measurements, and nothing is assumed about them or about the state.
Our claim is that it is possible to conclude that the physical experiment is equivalent to the reference experiment if we observe the statistics
\begin{equation} \label{statP}
\langle\Psi\vert P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle = {{N}\choose{k}}^{-1} \quad\text{ for all $\vec{a}$ such that } \left\lVert\vec{a}\right\rVert_1 = k
\end{equation}
\begin{center}
and
\end{center}
\begin{eqnarray}
\langle\Psi\vert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{C_{N-1}} X_{A_{N}}\vert\Psi\rangle & = & 2 {{N}\choose{k}}^{-1} \label{PXX}\\
\langle\Psi\vert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} Z_{C_{N-1}} Z_{A_{N}}\vert\Psi\rangle & = & - 2 {{N}\choose{k}}^{-1} \label{PZZ}\\
\langle\Psi\vert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{C_{N-1}} Z_{A_{N}}\vert\Psi\rangle & = & 0 \label{PXZ}\\
\langle\Psi\vert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{C_{N-1}} D_{A_{N}}\vert\Psi\rangle & = & \sqrt{2} {{N}\choose{k}}^{-1} \label{PXD}\\
\langle\Psi\vert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} Z_{C_{N-1}} D_{A_{N}}\vert\Psi\rangle & = & -\sqrt{2} {{N}\choose{k}}^{-1} \label{PZD}
\end{eqnarray}
for all $\vec{a}$ such that $\sum_{i=1}^{N-2} a_i = k-1$ and all $\vec{C}=(C_1,...,C_{N-1})$ cyclic permutations of $(A_1,...,A_{N-1})$, and where $P_{A_{i}}^{a_i} = (1+(-1)^{a_i} Z_{A_i})/2 $ are projectors for the $Z_{A_i}$ measurement.
This claim is the main result of this work, and implies that Dicke states can be self-tested. To prove this statement, we propose an isometry and we use the experimentally observed statistics to show it certifies that the physical experiment is equivalent to the reference experiment. For this proof we will use a number of identities that we will derive in what follows.
\vspace{10mm}
\textbf{Projector identity.} Eq.\eqref{statP} implies
\begin{equation}\label{Proj}
\langle\Psi\vert \sum_{\vec{a}} \delta\left(\left\lVert\vec{a}\right\rVert_1 - k\right) P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle = 1 \;,
\end{equation}
where $\lVert\vec{a}\rVert_p = \left( \sum_{i=1}^N \vert a_i\vert^p \right)^{1/p}$ is the $p-$norm.
Note that since $\langle\psi\vert\phi\rangle = 1$ if and only if $\vert\psi\rangle = \vert\phi\rangle$, then $\langle\Psi\vert O \vert\Psi\rangle = 1$ implies that $\vert\Psi\rangle$ is an eigenstate of $O$ with eigenvalue $1$.
Using the fact that $P_{A_i}^{a_i}P_{A_i}^{b_i} = \delta(a_i-b_i) P_{A_i}^{a_i}$ we deduce
\begin{eqnarray}
\langle\Psi\vert \left(\sum_{\vec{a}} \delta\left(\left\lVert\vec{a}\right\rVert_1 - k\right) P_{A_1}^{a_1}...P_{A_N}^{a_N}\right)^2 \vert\Psi\rangle &=& \langle\Psi\vert \sum_{\vec{a},\vec{b}} \delta\left(\left\lVert\vec{a}\right\rVert_1 - k\right)\delta(\lVert\vec{b}\rVert_1 - k) P_{A_1}^{a_1}...P_{A_N}^{a_N} P_{A_1}^{b_1}...P_{A_N}^{b_N} \vert\Psi\rangle \nonumber\\
&=& \langle\Psi\vert \sum_{\vec{a},\vec{b}} \delta\left(\left\lVert\vec{a}\right\rVert_1 - k\right) \delta(\lVert\vec{b}\rVert_1 - k) \delta(\vec{a}-\vec{b}) P_{A_1}^{a_1}...P_{A_N}^{a_N}\vert\Psi\rangle \nonumber\\
&=& \langle\Psi\vert \sum_{\vec{a}} \delta\left(\left\lVert\vec{a}\right\rVert_1 - k\right) P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle = 1 \;,
\end{eqnarray}
which shows that the operator $\sum_{\vec{a}} \delta\left(\left\lVert\vec{a}\right\rVert_1 - k\right) P_{A_1}^{a_1}...P_{A_N}^{a_N}$ is a projector.
Eq.\eqref{statP} directly implies also
\begin{equation}\label{singleProj}
P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle = \begin{cases}
{{N}\choose{k}}^{-\frac{1}{2}}\vert\xi\rangle &\text{ if } \left\lVert\vec{a}\right\rVert_1 = k \\
0 &\text{ otherwise}
\end{cases}
\end{equation}
where $\vert\xi\rangle$ is some (normalized) state.
In a similar way we see that for all $\vec{a}$ such that $\sum_{i=1}^{N-2} a_i = k-1$ we have
\begin{equation}\label{eqPPP}
\langle\Psi\vert (P_{C_1}^{a_1})^2...(P_{C_{N-2}}^{a_{N-2}})^2 \vert\Psi\rangle = \langle\Psi\vert (P_{C_1}^{a_1})^2... (P_{C_{N-2}}^{a_{N-2}})^2(P_{C_{N-1}}^{0}+P_{A_{N-1}}^{1}) (P_{A_N}^{0}+P_{A_N}^{1})\vert\Psi\rangle = 2 {{N}\choose{k}}^{-1} \;,
\end{equation}
where $\vec{C}=(C_1,...,C_{N-1})$ is a cyclic permutations of $(A_1,...,A_{N-1})$. From Eq.\eqref{eqPPP} we define for later use
\begin{equation}\label{PHIdef}
P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} \vert\Psi\rangle = \sqrt{2 {{N}\choose{k}}^{-1}} \vert\phi\rangle \qquad\Rightarrow\qquad \vert\phi\rangle = \sqrt{\dfrac{1}{2} {{N}\choose{k}}} P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} \vert\Psi\rangle
\end{equation}
where $\vert\phi\rangle$ is some normalized state.
\vspace{10mm}
\textbf{Operators relabeling identities.} Using the definition of $\vert\phi\rangle$ given in Eq.\eqref{PHIdef}, and the observed measurement statistic Eq.\eqref{PXX}, we obtain
\begin{equation}
\langle \phi \vert X_{C_{N-1}} X_{A_N} \vert \phi \rangle = \dfrac{1}{2} {{N}\choose{k}} \langle\Psi\vert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{C_{N-1}} X_{A_N } \vert\Psi\rangle = 1
\end{equation}
which means that $X_{C_{N-1}}\vert \phi \rangle = X_{A_N} \vert \phi \rangle$ or equivalently
\begin{equation}\label{PXPX}
P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{C_{N-1}} \vert\Psi\rangle = P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{A_{N}} \vert\Psi\rangle \;.
\end{equation}
The statistics in Eq.\eqref{PZZ}, \eqref{PXZ}, \eqref{PXD} and \eqref{PZD} implies $\langle \phi \vert Z_{C_{N-1}} Z_{A_N} \vert \phi \rangle = -1$, $\langle \phi \vert X_{C_{N-1}} Z_{A_N} \vert \phi \rangle = 0$ and $\langle \phi \vert \left( \frac{X_{C_{N-1}} - Z_{C_{N-1}}}{2} \right) D_{A_N} \vert \phi \rangle = 1$, corresponding respectively to
\begin{equation}\label{PZPZ}
P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} Z_{C_{N-1}} \vert\Psi\rangle = -P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} Z_{A_{N}} \vert\Psi\rangle \;,
\end{equation}
\begin{equation}
P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{C_{N-1}} \vert\Psi\rangle \perp P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} Z_{A_{N}} \vert\Psi\rangle \;,
\end{equation}
\begin{equation}\label{PDPD}
P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} D_{A_{N}} \vert\Psi\rangle = P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} \left( \dfrac{X_{C_{N-1}} - Z_{C_{N-1}}}{2} \right) \vert\Psi\rangle = P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} \left( \dfrac{X_{A_{N}} + Z_{A_{N}}}{2} \right) \vert\Psi\rangle \;,
\end{equation}
where for the last equality we used Eq.\eqref{PXPX} and Eq.\eqref{PZPZ}. At this point we would like to emphasize that all the identities derived here are valid for every $\vec{a}$ such that $\sum_{i=1}^{N-2} a_i = k-1$, and for every cyclic permutation $\vec{C}$ of $(A_1,...,A_{N-1})$.
\vspace{10mm}
\textbf{Anticommutation identities.} Using Eq.\eqref{PDPD}, and the fact that $D_{A_N}^2=X_{A_i}^2=Z_{A_i}^2=\mathbb{I}$, we write
\begin{equation}
P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} \vert\Psi\rangle = P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} D_{A_{N}}^2 \vert\Psi\rangle = P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} \left( \mathbb{I} + X_{A_N} Z_{A_N} + Z_{A_N} X_{A_N} \right) \vert\Psi\rangle
\end{equation}
from which is derived the anticommutation relation
\begin{equation}\label{anticomm}
P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{A_N} Z_{A_N} \vert\Psi\rangle = -P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} Z_{A_N} X_{A_N} \vert\Psi\rangle \;.
\end{equation}
Again, this identity is valid for every $\vec{a}$ such that $\sum_{i=1}^{N-2} a_i = k-1$, and for every cyclic permutation $\vec{C}$ of $(A_1,...,A_{N-1})$.
\vspace{10mm}
\textbf{Swapping identity.} The relations derived until now allow us to obtain the identity
\begin{equation}\label{swapping}
P_{A_1}^{a_1} ...P_{A_i}^0...X_{A_j} P_{A_j}^1 ...P_{A_N}^{a_N} \vert\Psi\rangle = P_{A_1}^{a_1} ...X_{A_i} P_{A_i}^1... P_{A_j}^0 ...P_{A_N}^{a_N} \vert\Psi\rangle \;,
\end{equation}
which is the key tool to prove the main result of this work. To arrive at Eq.\eqref{swapping}, we make use of the fact that
\begin{align}
P_{A_1}^{a_1} ...P_{A_i}^0...X_{A_j} P_{A_j}^1 ...P_{A_N}^{a_N} \vert\Psi\rangle &= P_{A_1}^{a_1} ... P_{A_i}^0... P_{A_j}^0 ...P_{A_N}^{a_N} X_{A_j} \vert\Psi\rangle &\quad\qquad\text{from Eq.\eqref{anticomm}} \nonumber\\
&= P_{A_1}^{a_1} ... P_{A_i}^0... P_{A_j}^0 ...P_{A_N}^{a_N} X_{A_i} \vert\Psi\rangle &\quad\qquad\text{from Eq.\eqref{PXPX}} \nonumber\\
&= P_{A_1}^{a_1} ... X_{A_i} P_{A_i}^1... P_{A_j}^0 ...P_{A_N}^{a_N} \vert\Psi\rangle &\quad\qquad\text{from Eq.\eqref{anticomm}} \nonumber
\end{align}
which concludes the proof. Note that in the first and in the last step $\sum_{p=1}^N a_p = k$, but we exchanged $a_j \leftrightarrow a_i$.
\vspace{20mm}
\begin{figure}
\includegraphics[width=0.5\columnwidth]{selftesting_v2.pdf}
\caption{\label{localIso} Circuit representing the local isometry for party $A_i$. The upper state is an additional local degree of freedom (ancilla) initially in $\vert 0\rangle$, while the lower state is the partition of $\vert\Psi\rangle$ associated with $A_i$. Gate $H$ is the Hadamard transformation.}
\end{figure}
We now have all the tools to show that observing the statistics in Eqs.\eqref{statP}-\eqref{PZD} self-tests the Dicke state $\vert D_N^k \rangle$ and the spin measurements. Indeed, the isometry illustrated in Fig.\ref{localIso} applied on the initial state gives
\begin{align}\label{stState}
\Phi\left( \vert\Psi\rangle \vert 0\rangle^{\otimes N} \right) &= \sum_{\vec{a}} X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle \vert \vec{a} \rangle & \nonumber\\
&= \sum_{\left\lVert \vec{a}\right\rVert_1=k} X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle \vert \vec{a} \rangle & \quad\qquad\text{from Eq.\eqref{singleProj}} \nonumber\\
&= \sum_{\left\lVert \vec{a}\right\rVert_1=k} X_{A_1}...X_{A_k} P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \vert \vec{a} \rangle & \quad\qquad\text{from Eq.\eqref{swapping}} \nonumber\\
&= X_{A_1}...X_{A_k} P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \sum_{\left\lVert \vec{a}\right\rVert_1=k} \vert \vec{a} \rangle & \nonumber \\
&= \vert \text{junk}\rangle \vert D_N^k \rangle &
\end{align}
where $\vert \text{junk}\rangle$ is some state to be discarded. This concludes the proof that the Dicke state $\vert D_N^k \rangle$ is self-tested.
To see that the measurement operations are also self-tested, we evaluate
\begin{align}\label{stMeasX}
\Phi\left( X_{A_i} \vert\Psi\rangle \vert 0\rangle^{\otimes N} \right) &= \sum_{\vec{a}} X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_N}^{a_N} X_{A_i} \vert\Psi\rangle \vert \vec{a} \rangle & \nonumber\\
&= X_{A_i} \sum_{\vec{a}} X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_i}^{1-a_i}...P_{A_N}^{a_N} \vert\Psi\rangle \vert \vec{a} \rangle & \quad\qquad\text{from Eq.\eqref{anticomm}} \nonumber\\
&= \sum_{\vec{a}} X_{A_1}^{a_1}...X_{A_i}^{1-a_i}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_i}^{1-a_i}...P_{A_N}^{a_N} \vert\Psi\rangle \vert \vec{a} \rangle & \quad\qquad\text{from}\; X_{A_i}X_{A_i}^{a_i} = X_{A_i}^{1-a_i} \nonumber\\
&= \sum_{\vec{b}} X_{A_1}^{b_1}...X_{A_i}^{b_i}...X_{A_N}^{b_N} P_{A_1}^{b_1}...P_{A_i}^{b_i}...P_{A_N}^{b_N} \vert\Psi\rangle \vert b_1, ..., 1-b_i, ..., b_N \rangle & \quad\qquad\text{def.}\; {\scriptstyle b_l = \delta(l-i)+(-1)^{\delta(l-i)} a_l } \nonumber\\
&= \sum_{\left\lVert\vec{b}\right\rVert_1=k} X_{A_1}^{b_1}...X_{A_N}^{b_N} P_{A_1}^{b_1}...P_{A_N}^{b_N} \vert\Psi\rangle \vert b_1, ..., 1-b_i, ..., b_N \rangle & \quad\qquad\text{from Eq.\eqref{singleProj}} \nonumber\\
&= \sum_{\left\lVert \vec{b}\right\rVert_1=k} X_{A_1}...X_{A_k} P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \vert b_1, ..., 1-b_i, ..., b_N \rangle & \quad\qquad\text{from Eq.\eqref{swapping}} \nonumber\\
&= X_{A_1}...X_{A_k} P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \sum_{\left\lVert \vec{b}\right\rVert_1=k} \vert b_1, ..., 1-b_i, ..., b_N \rangle &\nonumber\\
&= \vert \text{junk}\rangle \sigma_x^{(A_i)} \vert D_N^k \rangle
\end{align}
which proves that the $X_{A_i}$ measurement act as the Pauli $x$ operator on party $A_i$.
For the $Z_{A_i}$ measurement note that $P_{A_i}^{a_i}Z_{A_i} = (-1)^{a_i} P_{A_i}^{a_i}$, and therefore
\begin{equation}\label{stMeasZ}
\Phi\left( Z_{A_i} \vert\Psi\rangle \vert 0\rangle^{\otimes N} \right) = \vert \text{junk}\rangle \sigma_z^{(A_i)} \vert D_N^k \rangle \;,
\end{equation}
proving that the $Z_{A_i}$ measurement act as the Pauli $z$ operator on party $A_i$.
To conclude, the linearity of the isometry allows us to show that
\begin{equation}\label{stMeasD}
\Phi\left( D_{A_N} \vert\Psi\rangle \vert 0\rangle^{\otimes N} \right) = \Phi\left( \left(\dfrac{X_{A_N}+Z_{A_N}}{\sqrt{2}}\right) \vert\Psi\rangle \vert 0\rangle^{\otimes N} \right) = \vert \text{junk}\rangle \left(\dfrac{\sigma_x^{(A_N)}+\sigma_z^{(A_N)}}{\sqrt{2}}\right) \vert D_N^k \rangle \;.
\end{equation}
Eqs.\eqref{stMeasX}, \eqref{stMeasZ} and \eqref{stMeasD} prove that the experimental measurement operations are also self-tested, meaning that $X_{A_i}$, $Z_{A_i}$ and $D_{A_N}$ are certified to be respectively equivalent to the spin measurements described by $\sigma_x^{(A_i)}$, $\sigma_z^{(A_i)}$ and $(\sigma_x^{(A_D)}+\sigma_z^{(A_N)})/\sqrt{2}$.
\newpage
\section{Robustness}
Inevitable experimental imperfections results in deviations from the ideal measurement statistics. Therefore, we would like to estimate the robustness of the proposed self-testing protocol.
We assume that the discrepancy between the measured and the ideal statistics is at most $\epsilon$, meaning that
\begin{equation}\label{epsilonSim}
\vert \langle\Psi\vert \bigotimes_{i=1}^N M_{j_i,A_i} \vert\Psi\rangle - \langle\Psi^\star\vert \bigotimes_{i=1}^N M_{j_i,A_i}^\star \vert\Psi^\star\rangle \vert \leq \epsilon \;,
\end{equation}
for every choice of measurement settings. In this situation we can not conclude that the physical experiment is (exactly) equivalent to the reference experiment, however we can bound tits deviation by saying that
\begin{align}
\left\lVert \Phi\left( \vert\Psi\rangle \right) - \vert\text{junk}\rangle \otimes \vert\Psi^\star\rangle \right\rVert_2 \leq \delta \label{StateRob}\\
\left\lVert \Phi\left( M_{j_i,A_i} \vert\Psi\rangle \right) - \vert\text{junk}\rangle \otimes M_{j_i,A_i}^\star \vert\Psi^\star\rangle \right\rVert_2 \leq \delta
\end{align}
where $\delta$ is a function of $\epsilon$. This means that physical realizations of states and measurements can be certified to be ``almost'' equivalent to a reference model, with some fidelity dependent on $\delta$.
In what follows we will give an expression for $\delta$, as a function of $\epsilon$, by bounding the norm in Eq.\eqref{StateRob}. Note that in the situation of Eq.\eqref{epsilonSim}, with $\epsilon > 0$, the isometry illustrated in Fig.\ref{localIso} might not be optimal, in the sense that there could be some other isometry giving a better (\textit{i.e.} lower) bound $\delta$, \cite{stW}. Moreover, adding measurements settings might also improve the bound. Here, for simplicity, we will not deal with such optimizations. Considering the same isometry as for the ideal case (Fig.\ref{localIso}), and the same measurement settings, we find a bound for Eq.\eqref{StateRob}.
We define, for compactness, the ideal ``output'' state of the isometry in Fig.\ref{localIso} as (see Eq.\eqref{stState}) $\vert \Theta \rangle = X_{A_1}...X_{A_k} P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \sum_{\left\lVert \vec{a}\right\rVert_1=k} \vert \vec{a} \rangle$, and to simplify our calculations we split Eq.\eqref{StateRob} into two terms
\begin{equation}\label{splitFidelity}
\left\lVert \Phi\left( \vert\Psi\rangle \right) - \vert\text{junk}\rangle \otimes \vert\Psi^\star\rangle \right\rVert_2 \quad\leq\quad \left\lVert \Phi\left( \vert\Psi\rangle \right) - \vert \Theta \rangle \right\rVert_2 + \left\lVert \vert \Theta \rangle - \vert\text{junk}\rangle \otimes \vert\Psi^\star\rangle \right\rVert_2 \;.
\end{equation}
Here, the first distance is the one between the non-ideal and the ideal output of the isometry considered, and the second distance is the one between the ideal output of the isometry and the ideal target state. In what follows we bound these two terms separately, to obtain an expression for $\delta$ of Eq.\eqref{StateRob}.
\vspace{10mm}
\textbf{First term of Eq.\eqref{splitFidelity}.} Remember that in the previous section we derived from the observed statistics a number of identities involving the measurement operators. Now, observing a deviation from the ideal statistics, Eq.\eqref{epsilonSim}, implies that the identities we derived might still hold approximately, \textit{i.e.} with some error. We start evaluating such errors, to then calculate their effect in the derivation of Eq.\eqref{stState}.
From Eq.\eqref{singleProj} and Eq.\eqref{eqPPP} we estimate
\begin{align}
\left\lVert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{C_{N-1}} \vert\Psi\rangle \right\rVert_2 &= \sqrt{\vert\langle\Psi\vert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{C_{N-1}} X_{C_{N-1}} P_{C_{N-2}}^{a_{N-2}} ... P_{C_1}^{a_1} \vert\Psi\rangle\vert} \nonumber\\
&= \sqrt{\vert\langle\Psi\vert (P_{C_1}^{a_1})^2...(P_{C_{N-2}}^{a_{N-2}})^2 \vert\Psi\rangle\vert} = \sqrt{\vert\langle\Psi\vert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} \vert\Psi\rangle\vert} \nonumber\\
& =\sqrt{\vert\langle\Psi\vert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} (P_{C_{N-1}}^{0}+P_{C_{N-1}}^{1}) (P_{A_N}^{0}+P_{A_N}^{1}) \vert\Psi\rangle\vert} \nonumber\\
& =\sqrt{ \left| 2{{N}\choose{k}}^{-1} - 4\epsilon \right| } \;,
\end{align}
and, following the same steps
\begin{align}
\left\lVert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} Z_{C_{N-1}} \vert\Psi\rangle \right\rVert_2 = \left\lVert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{A_N} \vert\Psi\rangle \right\rVert_2 &= \nonumber \\
= \left\lVert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} Z_{A_N} \vert\Psi\rangle \right\rVert_2 = \left\lVert P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} D_{A_N} \vert\Psi\rangle \right\rVert_2 & = \sqrt{ \left| 2{{N}\choose{k}}^{-1} - 4\epsilon \right| } \;.
\end{align}
Now we can estimate the error for Eq.\eqref{PXPX} by computing the norm
\begin{align}
& \left\lVert \left( P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{C_{N-1}} - P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{A_N} \right) \vert\Psi\rangle \right\rVert_2 = \nonumber\\
& = \sqrt{\vert \langle\Psi\vert P_{C_1}^{a_1}... X_{C_{N-1}}^2 P_{C_1}^{a_1}... \vert\Psi\rangle + \langle\Psi\vert P_{C_1}^{a_1}... X_{A_{N}}^2 P_{C_{1}}^{a_{1}} ... \vert\Psi\rangle - 2 \langle\Psi\vert P_{C_1}^{a_1}... X_{C_{N-1}} X_{A_{N}} P_{C_1}^{a_1}... \vert\Psi\rangle\vert} \nonumber\\
& = \sqrt{ \left| 2 \left(2 {{N}\choose{k}}^{-1} - 4\epsilon \right) - 2 \left( 2{{N}\choose{k}}^{-1} - \epsilon \right) \right| } = \sqrt{ 6 \vert \epsilon \vert } \;,
\end{align}
and, following the same steps, the error for Eq.\eqref{PZPZ} is
\begin{equation}
\left\lVert \left( P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} Z_{C_{N-1}} - P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} Z_{A_N} \right) \vert\Psi\rangle \right\rVert_2 = \sqrt{ 6 \vert \epsilon \vert} \;.
\end{equation}
For later use, we bound
\begin{align}
& \vert\langle\Psi\vert P_{C_1}^{a_1}... X_{C_{N-1}} Z_{C_{N-1}} \vert\Psi\rangle\vert = \vert\langle\Psi\vert P_{C_1}^{a_1}... X_{C_{N-1}} \left( Z_{C_{N-1}} + Z_{A_N} - Z_{A_N} \right) \vert\Psi\rangle\vert \leq \nonumber\\
& \leq \vert\langle\Psi\vert P_{C_1}^{a_1}... X_{C_{N-1}} Z_{A_N} \vert\Psi\rangle\vert + \vert\langle\Psi\vert P_{C_1}^{a_1}... X_{C_{N-1}} \left( Z_{C_{N-1}} - Z_{A_N}\right) \vert\Psi\rangle\vert = \nonumber\\
& = \vert\epsilon\vert + \vert\langle\Psi\vert P_{C_1}^{a_1}... X_{C_{N-1}} \left( Z_{C_{N-1}} - Z_{A_N}\right) \vert\Psi\rangle\vert \leq \nonumber\\
& \leq \vert\epsilon\vert + \left\lVert P_{C_1}^{a_1}... X_{C_{N-1}}\right\rVert_2 \; \left\lVert P_{C_1}^{a_1}... \left( Z_{C_{N-1}} - Z_{A_N}\right)\right\rVert_2 = \nonumber\\
& = \vert\epsilon\vert + \left\lVert P_{C_1}^{a_1}... X_{C_{N-1}} \vert\Psi\rangle \right\rVert_2 \; \left\lVert P_{C_1}^{a_1}... \left( Z_{C_{N-1}} - Z_{A_N}\right) \vert\Psi\rangle \right\rVert_2 = \nonumber\\
& = \vert\epsilon\vert + \sqrt{ \left| 2 {{N}\choose{k}}^{-1} - 4\epsilon \right|}\sqrt{ 6 \vert \epsilon \vert } \label{NPXZ}\;
\end{align}
where for the second line we used the triangle inequality $\vert a+b\vert \leq \vert a\vert + \vert b\vert$, and for the fourth the Cauchy-Schwarz inequality $\vert a b\vert \leq \vert a\vert \; \vert b\vert$.
Eq.\eqref{NPXZ} can now be used to estimate the error for Eq.\eqref{PDPD} as
\begin{align}
& \left\lVert \left( P_{C_1}^{a_1}...D_{A_N} - P_{C_1}^{a_1}...\dfrac{X_{C_{N-1}} + Z_{C_{N-1}}}{\sqrt{2}} \right)\vert\Psi\rangle \right\rVert_2 = \nonumber\\
& = \left( \vert \langle\Psi\vert P_{C_1}^{a_1}...D_{A_N}^2 P_{C_1}^{a_1}... + \dfrac{1}{2}P_{C_1}^{a_1}...X_{C_{N-1}}^2 ... + \dfrac{1}{2}P_{C_1}^{a_1}...Z_{C_{N-1}}^2 ... - P_{C_1}^{a_1}...X_{C_{N-1}}Z_{C_{N-1}} ... - \sqrt{2} P_{C_1}^{a_1}...D_{A_N}X_{C_{N-1}} ... + \right. \nonumber\\
& \phantom{= space} \left.+ \sqrt{2} P_{C_1}^{a_1}...D_{A_N}Z_{C_{N-1}} ... \vert\Psi\rangle \vert \right)^{\/2} = \nonumber \\
& = \sqrt{\left| 2 \left( 2 {{N}\choose{k}}^{-1} - 4\epsilon \right) - 2\sqrt{2}\left( \sqrt{2} {{N}\choose{k}}^{-1} - \epsilon \right) - \langle\Psi\vert P_{C_1}^{a_1}...X_{C_{N-1}}Z_{C_{N-1}} ... \vert\Psi\rangle \right|} = \nonumber \\
& = \sqrt{\vert 2(\sqrt{2}-4)\epsilon + \langle\Psi\vert P_{C_1}^{a_1}...X_{C_{N-1}}Z_{C_{N-1}} ... \vert\Psi\rangle \vert} \leq \nonumber \\
& \leq \sqrt{\vert 2(\sqrt{2}-4)\epsilon \vert + \vert\epsilon\vert + \sqrt{ \left| 2 {{N}\choose{k}}^{-1} - 4\epsilon \right|}\sqrt{ 6 \vert \epsilon \vert }} = \nonumber \\
& = \dfrac{\delta_1}{(2+2\sqrt{2})\sqrt{2}} \;,
\end{align}
where the symbol $\delta_1$ is for compactness, and the denominator is introduced for later convenience.
We now consider the error in commuting $X$ with $Z$.
From the observation that (see also Eqs.(B11) and (B12) of \cite{stW})
\begin{align}
&\left( P_{C_1}^{a_1}... X_{C_{N-1}} Z_{C_{N-1}} + P_{C_1}^{a_1}... Z_{C_{N-1}} X_{C_{N-1}} \right) \vert\Psi\rangle = \nonumber\\
= & \sqrt{2}\left( P_{C_1}^{a_1}... D_{A_N} + \dfrac{P_{C_1}^{a_1}... X_{C_{N-1}} - P_{C_1}^{a_1}... Z_{C_{N-1}}}{\sqrt{2}}\right)\left( P_{C_1}^{a_1}... D_{A_N} - \dfrac{P_{C_1}^{a_1}... X_{C_{N-1}} - P_{C_1}^{a_1}... Z_{C_{N-1}}}{\sqrt{2}}\right) \vert\Psi\rangle \;,
\end{align}
we estimate the error in the anticommutator between $X$ and $Z$, Eq.\eqref{anticomm}, by computing (see also Eqs.(B13) and (B14) of \cite{stW})
\begin{align}\label{Nanticomm}
& \left\lVert~\left( P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} X_{C_{N-1}} Z_{C_{N-1}} + P_{C_1}^{a_1}...P_{C_{N-2}}^{a_{N-2}} Z_{C_{N-1}} X_{C_{N-1}} \right) \vert\Psi\rangle \right\rVert_2 = \nonumber\\
=& \sqrt{2} \left\lVert \left( P_{C_1}^{a_1}... D_{A_N} + \dfrac{P_{C_1}^{a_1}... X_{C_{N-1}}}{\sqrt{2}} - \dfrac{P_{C_1}^{a_1}... Z_{C_{N-1}}}{\sqrt{2}} \right)\left( P_{C_1}^{a_1}... D_{A_N} - \dfrac{P_{C_1}^{a_1}... X_{C_{N-1}} - P_{C_1}^{a_1}... Z_{C_{N-1}}}{\sqrt{2}}\right) \vert\Psi\rangle \right\rVert_2 \nonumber\\
\leq & \sqrt{2} \left( \left\lVert P_{C_1}^{a_1}... D_{A_N}\right\rVert\infty + \left\lVert \dfrac{P_{C_1}^{a_1}... X_{C_{N-1}}}{\sqrt{2}} \right\rVert\infty + \left\lVert \dfrac{P_{C_1}^{a_1}... Z_{C_{N-1}}}{\sqrt{2}}\right\rVert\infty \right)\left\lVert \left( P_{C_1}^{a_1}... D_{A_N} - \dfrac{P_{C_1}^{a_1}... X_{C_{N-1}} - P_{C_1}^{a_1}... Z_{C_{N-1}}}{\sqrt{2}}\right) \vert\Psi\rangle \right\rVert_2 \nonumber\\
= & \sqrt{2}(2 +2\sqrt{2})\left( \dfrac{\delta_1}{(2+2\sqrt{2})\sqrt{2}} \right)= \delta_1 \;.
\end{align}
Finally, we estimate the error associated to the swapping identity Eq.\eqref{swapping} as
\begin{align}
&\left\lVert \left( P_{A_1}^{a_1} ...P_{A_i}^0...X_{A_j} P_{A_j}^1 ...P_{A_N}^{a_N} -P_{A_1}^{a_1} ... X_{A_i} P_{A_i}^1... P_{A_j}^0 ...P_{A_N}^{a_N} \right)\vert\Psi\rangle \right\rVert_2 = \nonumber\\
& = \left\lVert \left( P_{A_1}^{a_1} ...P_{A_i}^0...X_{A_j} P_{A_j}^1 ...P_{A_N}^{a_N} - P_{A_1}^{a_1} ... P_{A_i}^0 X_{A_i}... P_{A_j}^0 ...P_{A_N}^{a_N} + P_{A_1}^{a_1} ... P_{A_i}^0 X_{A_i}... P_{A_j}^0 - P_{A_1}^{a_1} ... X_{A_i} P_{A_i}^1... P_{A_j}^0 ...P_{A_N}^{a_N} \right)\vert\Psi\rangle \right\rVert_2 \nonumber\\
& = \left\lVert \left( P_{A_1}^{a_1} ...P_{A_i}^0...X_{A_j} P_{A_j}^1 ...P_{A_N}^{a_N} - P_{A_1}^{a_1} ...P_{A_i}^0... P_{A_j}^0 X_{A_j}...P_{A_N}^{a_N} + P_{A_1}^{a_1} ...P_{A_i}^0... P_{A_j}^0 X_{A_j}...P_{A_N}^{a_N} - P_{A_1}^{a_1} ... P_{A_i}^0 X_{A_i}... P_{A_j}^0 ...P_{A_N}^{a_N} + \right.\right.\nonumber\\
&\phantom{= space} \left.\left. + P_{A_1}^{a_1} ... P_{A_i}^0 X_{A_i}... P_{A_j}^0 - P_{A_1}^{a_1} ... X_{A_i} P_{A_i}^1... P_{A_j}^0 ...P_{A_N}^{a_N} \right)\vert\Psi\rangle \right\rVert_2 \nonumber\\
& \leq \left\lVert \left( P_{A_1}^{a_1} ...P_{A_i}^0...X_{A_j} P_{A_j}^1 ...P_{A_N}^{a_N} - P_{A_1}^{a_1} ...P_{A_i}^0... P_{A_j}^0 X_{A_j}...P_{A_N}^{a_N} \right)\vert\Psi\rangle \right\rVert_2 + \nonumber\\
&\phantom{+ space} \left\lVert \left( P_{A_1}^{a_1} ...P_{A_i}^0... P_{A_j}^0 X_{A_j}...P_{A_N}^{a_N} - P_{A_1}^{a_1} ... P_{A_i}^0 X_{A_i}... P_{A_j}^0 ...P_{A_N}^{a_N} \right)\vert\Psi\rangle \right\rVert_2 + \nonumber\\
&\phantom{+ space} \left\lVert \left( P_{A_1}^{a_1} ...P_{A_i}^0...X_{A_j} P_{A_j}^1 ...P_{A_N}^{a_N} - P_{A_1}^{a_1} ...P_{A_i}^0... P_{A_j}^0 X_{A_j}...P_{A_N}^{a_N} \right)\vert\Psi\rangle \right\rVert_2 \nonumber\\
& = 2\delta_1 + \sqrt{6 \vert\epsilon\vert} \label{errorSwap}\;.
\end{align}
In Eq.\eqref{stState} we used the swapping identity to transform every term of the form $X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_N}^{a_N} $, with $\left\lVert \vec{a}\right\rVert_1=k$, into $X_{A_1}...X_{A_k} P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0}$. Note here that every such term is unambiguously defined by the binary vector $\vec{a}$, with $\left\lVert \vec{a}\right\rVert_1=k$. This observation allows us to count how many times we need to apply the swapping identity in Eq.\eqref{stState}, by counting how many ``bit-flips'' are needed to transform one binary vector $\vec{a}$ into the vector $\vec{a}^\star=(1,...,1,0,...,0)$, having $k$ ones followed by $N-k$ zeros.
Recall that the for two strings $s_1$ and $s_2$, the Hamming distance $d_H(s_1,s_2)$ is defined as the number of symbols we need to change to transform $s_1$ into $s_2$. Therefore, a single application of the swapping identity changes a term defined by $\vec{a}$ into an other defined by $\vec{a}^\prime$, with Hamming distance $d_H(\vec{a},\vec{a}^\prime)=2$, since a $0$ is swapped with a $1$. In total, to transform every $\vec{a}$ with $\left\lVert \vec{a}\right\rVert_1=k$ into $\vec{a}^\star$, the number of times we need to apply the swapping identity is
\begin{equation}\label{countSwap}
\sum_{\left\lVert \vec{a}\right\rVert_1=k} \dfrac{d_H(\vec{a},\vec{a}^\star)}{2} = \frac{\Gamma (n)}{\Gamma (k) \Gamma (n-k)} \;.
\end{equation}
Note that the above number does not depend on which specific $\vec{a}^\star$ we chose among the vectors $\vec{a}$ with $\left\lVert \vec{a}\right\rVert_1=k$.
We have now everything we need to estimate the first term of Eq.\eqref{splitFidelity}. Since the binary vector $\vec{a}$ has $N$ digits, the sum in the first line of Eq.\eqref{stState} involves $2^N$ terms, out of which only ${{N}\choose{k}}$ have $\parallel\vec{a}\parallel_1=k$. We can therefore express
\begin{align}
\left\lVert \Phi\left( \vert\Psi\rangle \right) - \vert \Theta \rangle \right\rVert_2 &= \left\lVert \sum_{\vec{a}} X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle \vert \vec{a} \rangle - \vert \Theta \rangle \right\rVert_2 \nonumber\\
&\leq \left\lVert \sum_{\parallel\vec{a}\parallel_1\neq k} X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle \vert \vec{a} \rangle \right\rVert_2 + \left\lVert \sum_{\parallel\vec{a}\parallel_1= k} X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle \vert \vec{a} \rangle - \vert \Theta \rangle \right\rVert_2 \nonumber\\
&\leq \sum_{\parallel\vec{a}\parallel_1\neq k} \left\lVert X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle \vert \vec{a} \rangle \right\rVert_2 + \left\lVert \sum_{\parallel\vec{a}\parallel_1= k} X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle \vert \vec{a} \rangle - \vert \Theta \rangle \right\rVert_2 \nonumber\\
&= \left( 2^N - {{N}\choose{k}} \right) \vert\epsilon\vert + \left\lVert \sum_{\parallel\vec{a}\parallel_1= k} X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle \vert \vec{a} \rangle - \vert \Theta \rangle \right\rVert_2 \nonumber\\
&\leq \left( 2^N - {{N}\choose{k}} \right) \vert\epsilon\vert + \sum_{\parallel\vec{a}\parallel_1= k} \left\lVert X_{A_1}^{a_1}...X_{A_N}^{a_N} P_{A_1}^{a_1}...P_{A_N}^{a_N} \vert\Psi\rangle \vert \vec{a} \rangle - X_{A_1}...X_{A_k} P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \vert \vec{a} \rangle \right\rVert_2 \nonumber\\
&\leq \left( 2^N - {{N}\choose{k}} \right) \vert\epsilon\vert + \frac{\Gamma (N)}{\Gamma (k) \Gamma (N-k)} \left( 2\delta_1 + \sqrt{6 \vert\epsilon\vert} \right) \label{BoundFirstTerm} \;,
\end{align}
where in going from the second-to-last to the last row we used Eq.\eqref{countSwap} and Eq.\eqref{errorSwap}.
\vspace{10mm}
\textbf{Second term of Eq.\eqref{splitFidelity}.} To estimate this term, we first find
\begin{align}
\left(\left\lVert \Theta \right\rVert_2 \right)^2& = \left(\left\lVert X_{A_1}...X_{A_k} P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \sum_{\left\lVert \vec{a}\right\rVert_1=k} \vert \vec{a} \rangle \right\rVert_2\right)^2 \nonumber\\
& = {{N}\choose{k}} \left(\left\lVert X_{A_1}...X_{A_k} P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \right\rVert_2\right)^2 \nonumber\\
& = {{N}\choose{k}} \vert\langle\Psi\vert P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \vert \nonumber\\
& = {{N}\choose{k}} \left| {{N}\choose{k}}^{-1} + \epsilon \right| = \left| 1 + {{N}\choose{k}} \epsilon \right| \;,
\end{align}
which can be used to express
\begin{align}
\left\lVert \vert \Theta \rangle - \vert\text{junk}\rangle \otimes \vert\Psi^\star\rangle \right\rVert_2 & = \left\lVert \vert \Theta \rangle - \dfrac{\vert \Theta \rangle}{\left\lVert \vert \Theta \rangle \right\rVert_2} \right\rVert_2 \nonumber\\
& = \left\lVert \dfrac{\vert \Theta \rangle}{\left\lVert \vert \Theta \rangle \right\rVert_2} \left( \left\lVert \vert \Theta \rangle \right\rVert_2 - 1 \right) \right\rVert_2 \nonumber\\
& = \left| \sqrt{\left| 1 + {{N}\choose{k}} \epsilon \right|} - 1 \right| \label{BoundSecondTerm} \;.
\end{align}
\vspace{10mm}
\textbf{Bound for Eq.\eqref{splitFidelity}.}
The result Eq.\eqref{BoundFirstTerm}, together with Eq.\eqref{BoundSecondTerm}, is inserted into Eq.\eqref{splitFidelity} to get
\begin{equation}
\left\lVert \Phi\left( \vert\Psi\rangle \right) - \vert\text{junk}\rangle \otimes \vert\Psi^\star\rangle \right\rVert_2 \leq \left( 2^N - {{N}\choose{k}} \right) \vert\epsilon\vert + \frac{\Gamma (N)}{\Gamma (k) \Gamma (N-k)} \left( 2\delta_1 + \sqrt{6 \vert\epsilon\vert} \right) + \left| \sqrt{\left | 1 + {{N}\choose{k}} \epsilon \right| } - 1 \right| \;.
\end{equation}
This quantifies how ``close'' the physical state is to the ideal Dicke state.
\begin{comment}
\begin{align}
\left(\left\lVert \Theta \right\rVert_2 \right)^2& = \left(\left\lVert X_{A_1}...X_{A_k} P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \sum_{\left\lVert \vec{a}\right\rVert_1=k} \vert \vec{a} \rangle \right\rVert_2\right)^2 \nonumber\\
& = {{N}\choose{k}} \left(\left\lVert X_{A_1}...X_{A_k} P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \right\rVert_2\right)^2 \nonumber\\
& = {{N}\choose{k}} \vert\langle\Psi\vert P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \vert \nonumber\\
& = {{N}\choose{k}} \vert\langle\Psi\vert P_{A_1}^{1}...P_{A_{k-1}}^{1}X_{A_k}\left( X_{A_k} P_{A_k}^{1} - P_{A_k}^{0} X_{A_k} + P_{A_k}^{0} X_{A_k}\right) P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle \vert \nonumber\\
& \geq {{N}\choose{k}} \left( \vert\langle\Psi\vert X_{A_k}^2 P_{A_1}^{1}...P_{A_k}^{1}P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle\vert - \vert\langle\Psi\vert X_{A_k} P_{A_1}^{1}...\left( X_{A_k} P_{A_k}^{1} - P_{A_k}^{0} X_{A_k}\right) P_{A_{k+1}}^{0}...P_{A_N}^{0} \vert\Psi\rangle\vert \right)
\end{align}
\end{comment}
\section{conclusions}
We have shown that Dicke states can be self-tested. This conclusion is drawn by showing the existence of a local isometry which, together with a specific observed statistic, implies that the physical experiment is equivalent to a reference experiment where specific measurements are performed on a Dicke state. In the practical case where inevitable experimental imperfections cause deviations from the ideal reference statistics, we estimated the robustness of our protocol. We found that even in this case, the physical experiment can still be certified with high fidelity.
\vspace{10mm}
\textbf{Acknowledgements.} I am grateful to Jordi Tura for the useful discussions pointing to the possible extension of Ref.\cite{stW} to Dicke states.
|
1,108,101,566,809 | arxiv |
\section{Introduction}
Imagine a future where a doctor can write a few sentences describing a specialized drug for treating a patient and then receive the exact structure of the desired drug. Although this seems like science fiction now, with progress in integrating natural language and molecules, it might well be possible in the future.
Historically, drug creation has commonly been done by humans who design and build individual molecules. In fact, bringing a new drug to market can cost over a billion dollars and take over ten years \cite{gaudelet2021utilizing}. Recently, there has been considerable interest in using new deep learning tools to facilitate in silico drug design-- a field often called cheminformatics \cite{10.1093/bib/bby061}. Yet, many of these experiments still focus on molecules and their low-level properties, such as logP, the octanol/water partition, which helps measure the lipophilicity of a molecule \cite{bagal2021molgpt}. In the future, we foresee a need for a higher-level control over molecule design, which can easily be facilitated by natural language.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figures/page1_example_cropped.pdf}
\caption{An example output from our model for the molecule generation task. The left is the ground truth, and the right is a molecule generated from the given natural language caption. %
}
\label{fig:gen_task}
\end{figure}
In this work, we pursue an ambitious goal of translating between molecules and language by proposing two new tasks: molecule captioning and text-guided de novo molecule generation. In molecule captioning, we take a molecule (e.g., as a SMILES string) and generate a caption that describes it (Figure \ref{fig:task}). In text-guided molecule generation, the task is to create a molecule that matches a given natural language description (Figure \ref{fig:gen_task}). These new tasks would help to accelerate research in multiple scientific domains by enabling chemistry domain experts to generate new molecules and better understand them using natural language.
As a roadmap to multimodal molecule-language interactions, one can look to the intersection between natural language processing and computer vision (V+L), which has been studied extensively. Image captioning, for example, has made significant progress \cite{Herdade2019ImageCT,Pan2020XLinearAN}, and recent datasets for the task have also grown to large sizes \cite{Chen2015MicrosoftCC}. The benefits of incorporating natural language have become clear for enabling semantic-level control of images; as such, there is more and more interest in multimedia data and models \cite{chen2019uniter,li2020cross,radford2021learning,ramesh2021zero}.
While our proposed molecule-language tasks share some similarities with V+L tasks, they have several inherent difficulties that separate them from existing vision-language analogs: 1) creating annotations for molecules requires significant domain expertise, 2) thus, it is significantly more difficult to acquire large numbers of molecule-description pairs, 3) the same molecule can have many functions and thus be described in very different ways,
which causes 4) existing evaluation measures based on reference descriptions, such as BLEU, to fail to adequately evaluate these tasks.
To address the issue of data scarcity (i.e., difficulties 1 and 2), we propose a new self-supervised learning framework named MolT5 (\textbf{\underline{Mol}}ecular \textbf{\underline{T5}}) that is inspired by the recent progress in pretraining multilingual models \cite{devlin2019bert,Liu2020MultilingualDP}. MolT5 first pretrains a model on a vast amount of unlabeled natural language text and molecule strings using a simple denoising objective. After that, the pretrained model is finetuned on limited gold standard annotations. Furthermore, to adequately evaluate models for molecule captioning or generation, we consider various kinds of metrics and also propose a new metric named Text2Mol \cite{edwards2021text2mol}. Text2Mol repurposes a retrieval model for assessing the similarity between the ground truth molecule/description and the generated description/molecule, respectively.
To the best of our knowledge, there is no work yet on molecule captioning or text-guided molecule generation. The closest existing work attempts to convert images of molecules to their SMILES or InChi string formats \cite{sundaramoorthy2021end, campos2021img2smi}, to generate reaction steps in natural language \cite{vaucher2021inferring}, or to retrieve molecules from natural language \cite{edwards2021text2mol}. In molecule captioning, however, we take a molecule as input and generate a caption that describes it. The closest current approaches to this would fall within the scope of image captioning \cite{li2020oscar,hu2021scaling}. Current work in de novo molecule design primarily focuses on applying existing types of generative models to molecules \cite{olivecrona2017molecular,kell2020deep,krenn2020self,polykovskiy2020molecular}.
In summary, our main contributions are:
\begin{enumerate}[nolistsep]
\item We propose two new tasks: 1) molecule captioning, where a description is generated for a given molecule, and 2) text-based de novo molecule generation, where a molecule is generated to match a given text description.%
\item We consider multiple evaluation metrics for these new tasks, and we propose a new cross-modal retrieval similarity metric based on Text2Mol \cite{edwards2021text2mol}.
\item We propose \textbf{MolT5}: a self-supervised learning framework for jointly training a model on molecule string representations and natural language text, which is then finetuned on a cross-modal task.
\end{enumerate}
\section{Tasks}
With the ambitious goal of bi-directional translation between molecules and language, we propose two new novel tasks: molecule captioning (Section \ref{sec:task_molecule_captioning}) and text-based de novo molecule generation (Section \ref{sec:task_molecule_generation}).
\subsection{Molecule Captioning} \label{sec:task_molecule_captioning}
\begin{figure*}[!ht]
\centering
\includegraphics[width=\textwidth]{figures/example_figures_v2_horizontal.pdf}
\caption{An example of both the image captioning task \cite{Chen2015MicrosoftCC} and molecule captioning. Molecule captioning is considerably more difficult because of the increased linguistic variety in possible captions.} %
\label{fig:task}
\end{figure*}
For any given molecule, the goal of molecule captioning is to describe the molecule and what it does. At a high level, molecule captioning is very similar to image captioning. Figure~\ref{fig:task} compares an example of molecule captioning from the CheBI-20 dataset~\cite{edwards2021text2mol}, which we use for training, with an example of image captioning from COCO~\cite{Chen2015MicrosoftCC}. The molecule captioning case is considerably more difficult due to the increased linguistic variety in possible captions. For example, a molecule could be described with an IUPAC name, with one of many different synthetic routes from known precursor molecules, in terms of the properties (e.g. carcinogenic or lipophilic), with the applications of the molecule (e.g. a dye, an antipneumonic, or an antifungal), or in terms of its functional groups (e.g. ``substituted by hydroxy groups at positions 5 and 7 and a methyl group at position 8''), among other methods.
Unlike in the image domain, molecules are often represented as SMILES strings \cite{weininger1988smiles, weininger1989smiles}, a linearization of the molecular graph which can be interpreted as a language for molecules. Thus, this task can be considered an exotic translation task, and sequence to sequence models serve as excellent baselines. Existing work on image captioning cannot be easily applied to this task because molecules are discrete graphs while images are continuous vectors; instead existing work is more applicable to tasks such as converting images to SMILES strings \cite{campos2021img2smi}.
\subsection{Text-Based de Novo Molecule Generation}\label{sec:task_molecule_generation}
The goal of the de novo molecule generation task is to train a model which can generate a variety of possible new molecules. %
Existing work tends to focus on evaluating the model coverage of the chemical space~\cite{polykovskiy2020molecular}. Instead, we propose generating molecules based on a natural language description of the desired molecule--this is essentially swapping the input and output for the captioning task. An example of this task is shown in Figure \ref{fig:gen_task}. Recent work, such as DALL$\cdot$E \cite{ramesh2021zero}, which generates images from text, has shown the ability to seamlessly integrate multiple properties, such as chairs and avocados, in an image. This points towards exciting applications in the molecule generation domain via the usage of natural language.
\section{Data}
A primary difficulty of training joint molecule-text representations is the scarcity of training data pairs. Thus, to train MolT5 requires different sources of data, some of which are large quantities of single-modal data, in addition to the gold standard molecule-description pairs for finetuning.
For large-scale natural language data, we use the ``Colossal Clean Crawled Corpus'' (C4) \cite{raffel2019exploring}. For the molecule data, we use 100 million SMILES strings from the ZINC-15 dataset \cite{sterling2015zinc} which were used in Chemformer \cite{irwindimitriadishebjerrum2021}.
For our gold standard data, we use ChEBI-20, which was first presented in \cite{edwards2021text2mol}. It consists of 33,010 molecule-description pairs, which are separated into 80/10/10\% train/validation/test splits. In this paper, we work with SMILES strings, so we remove 2 molecules represented by the wildcard SMILES string: ``*''. This dataset is also used to train the baseline models. Further, many of the captions contain a name for the molecule at the start of the string. To force the molecule to focus on the semantics of the description, we replace the molecule's name with "The molecule is [...]".
\section{Evaluation Metrics}
\subsection{Text2Mol Metric}
Many evaluation metrics have been proposed which leverage pretrained models, such as Fréchet Inception Distance \cite{heusel2017gans} for images, BERTScore \cite{zhang2019bertscore} for natural language, and Fréchet ChemNet distance \cite{preuer2018frechet} for molecules. Cross-modal tasks have been evaluated using information retrieval metrics--\cite{ye2015eventnet} conducts event video retrieval given a natural language input. Since we are considering a new multimodal task between molecules and text, we introduce the Text2Mol metric. This is based off of \cite{edwards2021text2mol}, which trained a retrieval model to rank molecules given their text descriptions. However, the ranking function uses cosine similarity between embeddings, which allows this Text2Mol to be repurposed for evaluating the similarity between the ground truth molecule/description and the generated description/molecule, respectively. We train a base MLP model from Text2Mol for 320 epochs, which we will release for reproducibility. This model is used to generate similarities of the candidate molecule-description pairs, which can be compared to the average similarity of the ground truth molecule-description pairs. We also note that negative molecule-description pairs have an average similarity of roughly zero (i.e. the embeddings are perpendicular).
\subsection{Evaluating Molecule Captioning}
Traditionally, captioning tasks have been evaluated by natural language generation metrics such as BLEU \cite{papineni2002bleu}, ROUGE \cite{lin2004rouge}, and METEOR \cite{banerjee2005meteor}.
Unlike captioning tasks such as COCO \cite{Chen2015MicrosoftCC}, which has several captions per image, in our task we only have one reference caption. This makes these metrics less effective, especially because there are many non-overlapping ways to describe a molecule. Nonetheless, we still report these scores for comparison. We report aggregated sentence-level METEOR scores.
\subsection{Evaluating de Novo Molecule Generation from Text}
Over the last few years, considerable excitement has grown applying deep generative models to de novo molecule generation. Because of this, a number of metrics have been proposed, such as novelty and scaffold similarity \cite{polykovskiy2020molecular}. However, many of these metrics do not apply to our problem-- we want our generated molecule to match the input text instead of being generally diverse. Instead, we consider metrics which measure the distance to either the ground truth molecule, such as \cite{preuer2018frechet} and \cite{campos2021img2smi}, or the ground truth description, such as our proposed Text2Mol-based metric.
We employ three fingerprint metrics: MACCS FTS, RDK FTS, and Morgan FTS,
where FTS stands for fingerprint Tanimoto similarity \cite{tanimoto1958elementary}. MACCS \cite{durant2002reoptimization}, RDK \cite{schneider2015get}, and Morgan \cite{rogers2010extended} are each fingerprinting methods for molecules. The fingerprints of two molecules are compared using Tanimoto similarity (also commonly known as Jaccard index), and the average similarity over the evaluation dataset is reported. See \cite{campos2021img2smi} for more details. We also report exact SMILES
string matches, Levenshtein distance \cite{miller2009levenshtein}, and SMILES BLEU scores as in \cite{campos2021img2smi}.
\citet{preuer2018frechet} propose Fréchet ChemNet Distance (FCD), which is inspired by the Fréchet Inception Distance (FID) \cite{heusel2017gans}. FCD is based on the penultimate layer of a network called ``ChemNet'', which was trained to predict the activity of drug molecules. Thus, FCD takes into account chemical and biological information about molecules in order to compare them. This allows molecules to be compared based on the latent information required to predict useful properties rather than a string-based metric.
In the case of models which use SMILES strings, many generated molecules are syntactically invalid. Thus, we report validity as the percent of molecules which can be processed by RDKIT \cite{Landrum2021RDKit2021_03_2} as in \cite{polykovskiy2020molecular}. We report metrics on the successful molecules, as in \cite{campos2021img2smi}.
\section{MolT5 -- Multimodal Text-Molecule Representation Model} \label{sec:molt5}
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{figures/archi_cropped_v3.pdf}
\caption{A diagram of our framework. We first pre-train MolT5 on a large amount of data of both SMILES string and natural language using the ``replace corrupted spans'' objective \cite{raffel2019exploring}. After the pre-training stage, MolT5 can be easily fine-tuned for either the task of molecule captioning or generation (or both).}%
\label{fig:t5_task}
\end{figure*}
We can crawl a massive amount of natural language text from the Internet. For example, \newcite{raffel2019exploring} built a Common Crawl-based dataset that contains over 700 GB of reasonably clean and natural English text. On the other hand, over a billion molecules are also available from public databases such as ZINC-15 \cite{Sterling2015ZINC1}. Inspired by the recent progress in large-scale pretraining \cite{raffel2019exploring,ramesh2021zero}, we propose a new self-supervised learning framework named \textbf{MolT5} (\textbf{\underline{Mol}}ecular \textbf{\underline{T5}}) to leverage the vast amount of unlabeled natural language text and molecule strings.
Figure \ref{fig:t5_task} shows an overview of MolT5. We first initialize an encoder-decoder Transformer model \cite{vaswani2017attention} using one of the public checkpoints of T5.1.1\footnote{\url{https://tinyurl.com/t511-ckpts}}, an improved version of T5 \cite{raffel2019exploring}. After that, we pretrain the model using the ``replace corrupted spans'' objective \cite{raffel2019exploring}. More specifically, during each pretraining step, we sample a minibatch comprising both natural language sequences and SMILES sequences. For each sequence, some words in the sequence are randomly chosen for corruption. Each consecutive span of corrupted tokens is replaced by a sentinel token (shown as [X] and [Y] in Figure \ref{fig:t5_task}). Then the task is to predict the dropped-out spans.\footnote{For more explanation of the pretraining task, we refer the readers to the original T5 paper \cite{raffel2019exploring}.}
Molecules (e.g. represented as SMILES strings) can be thought of as a language with a very unique grammar. Then, intuitively, our pretraining stage essentially trains a single language model on two monolingual corpora from two different languages, and there is no explicit alignment between the two corpora. This approach is similar to how some multilingual language models such as mBERT \cite{devlin2019bert} and mBART \cite{Liu2020MultilingualDP} were pretrained. As models such as mBERT demonstrate excellent cross-lingual capabilities \cite{piresetal2019multilingual}, we also expect models pretrained using MolT5 to be useful for text-molecule translation tasks.
After the pretraining process, we can finetune the pretrained model for either molecule captioning or generation (depicted by the bottom half of Figure \ref{fig:t5_task}). In molecule generation, the input is a description, and the output is the SMILES representation of the target molecule. On the other hand, in molecule captioning, the input is the SMILES string of some molecule, and the output is a caption describing the input molecule.
\section{Experiments and Results}
\subsection{Data}
\paragraph{Pretraining Data} As described in Section \ref{sec:molt5}, the pretraining stage of MolT5 requires two monolingual corpora: one consisting of natural language text and the other consisting of molecule representations. We use the ``Colossal Clean Crawled Corpus'' (C4) \cite{raffel2019exploring} as the pretraining dataset for the textual modality. For the molecular modality, we directly utilize the 100 million SMILES strings used in Chemformer \cite{irwindimitriadishebjerrum2021}. As these strings were selected from the ZINC-15 dataset \cite{sterling2015zinc}, we refer to this pretraining dataset as ZINC from this point.
\paragraph{Finetuning and Evaluation Data} We use the ChEBI-20 dataset \cite{edwards2021text2mol} as our gold standard data for finetuning and evaluation. It consists of 33,010 molecule-description pairs, which are separated into 80/10/10\% train/validation/test splits. In this paper, we work with SMILES strings, so we remove two molecules represented by the wildcard SMILES string: ``*''. We use this dataset to finetune MolT5-based models and to train our baseline models. Furthermore, many captions in ChEBI-20 contain a name for the molecule at the start of the string (e.g., ``Rostratin D is an organic disulfide isolated from ...''). To force the models to focus on the semantics of the description, we replace the molecule's name with "The molecule is [...]" (e.g., ``The molecule is an organic disulfide isolated from ...'').
\subsection{Baselines}\label{sec:baselines}
Any sequence-to-sequence model is applicable to our new tasks (i.e., molecule captioning and generation). We implement the following baselines:
\begin{enumerate}
\item \textbf{RNN-GRU} \cite{cho2014properties}. We implement a 4-layer GRU recurrent neural network with a hidden size of 512. We use a learning rate of 1e-4 and a batch size of 128 for molecule generation. For caption generation, a batch size of 116 is used. The number of training epochs is 50. Additionally, the encoder is bidirectional. For training, teacher forcing is used 50\% of the time, and gradient clipping to 50 is applied.
\item \textbf{Transformer} \cite{vaswani2017attention}. We train a vanilla Transformer model consisting of six encoder and decoder layers. The number of training epochs is 40, the batch size is 16, and the learning rate is 1e-4. We use a linear decay with a warmup of 400 steps. %
\item \textbf{T5} \cite{raffel2019exploring}. We experiment with three public T5.1.1 checkpoints\footnote{\url{https://tinyurl.com/t511-ckpts}}: small, base, and large. We finetune each checkpoint for molecule captioning or molecule generation using the open-sourced t5x framework \cite{roberts2022t5x}. The number of training steps is set to be 50,000. The dropout rate is set to be 0.0 for the small and base models, and it is set to be 0.1 for the large model. For other hyperparameters, we use the default values provided by the t5x framework.
\end{enumerate}
We train the baseline models on the ChEBI-20 dataset using SMILES representations for the molecules. Molecule captioning and generation are trained with molecules as input/output and text as output/input. Sequences are limited to 512 tokens for input and output. During inference, a beam decoder with a beam size of 5 is used.
On the RNN and vanilla Transformer models, we use a character-split vocabulary for SMILES. For the text vocabulary, we use SciBERT's 31,090-token vocabulary \cite{beltagy2019scibert}.
\subsection{Pretraining Process and Hyperparameters}
Before the actual pretraining process, we initialize an encoder-decoder Transformer model using a public checkpoint of T5.1.1 (either \textit{t5.1.1.small}, \textit{t5.1.1.base}, or \textit{t5.1.1.large}). Next, we pretrain the model on the combined dataset of C4 and ZINC (i.e., C4+ZINC) for 1 million steps. Each step uses a batch size of 256 evenly split between the text and molecule sequences. After this, we finetune the pretrained model on ChEBI-20 for either molecule captioning or generation. Similar to training T5 baselines (Section \ref{sec:baselines}), we set the number of finetuning steps to be 50,000.
\begin{table*}[h!]
\resizebox{\textwidth}{!}{
\centering
\tiny
\begin{tabular}{ c|c|c|c|c|c|c|c }
\multicolumn{1}{c}{\textbf{Model}} & \multicolumn{1}{c}{BLEU-2} & \multicolumn{1}{c}{BLEU-4} & \multicolumn{1}{c}{ROUGE-1} & \multicolumn{1}{c}{ROUGE-2} & \multicolumn{1}{c}{ROUGE-L} & \multicolumn{1}{c}{METEOR} & \multicolumn{1}{c}{Text2Mol} \\
\noalign{\hrule height 2pt}
Ground Truth& & & & & & & 0.609 \\\hline
RNN & 0.303 & 0.213 & 0.347 & 0.191 & 0.303 & 0.337 & 0.426 \\
Transformer & 0.061 & 0.027 & 0.188 & 0.0597 & 0.165 & 0.126 & 0.0575 \\\hline
T5-Small & 0.525 & 0.414 & 0.612 & 0.457 & 0.568 & 0.533 & 0.526 \\
MolT5-Small & 0.520 & 0.436 & 0.624 & 0.475 & 0.581 & 0.549 & 0.540 \\\hline
T5-Base & 0.533 & 0.423 & 0.614 & 0.460 & 0.571 & 0.538 & 0.522 \\
MolT5-Base & 0.540 & 0.457 & 0.636 & 0.489 & 0.594 & 0.563 & 0.547 \\\hline
T5-Large & 0.558 & 0.467 & 0.631 & 0.482 & 0.584 & 0.570 & 0.563 \\
MolT5-Large & \textbf{0.594} & \textbf{0.508} & \textbf{0.650} & \textbf{0.509} & \textbf{0.605} & \textbf{0.591} & \textbf{0.582} \\
\end{tabular}
}
\caption{Molecule captioning results for the different baseline models on the test split of the CheBI-20 dataset. Rouge scores are F1 values.}
\label{tab:results_captioning}
\end{table*}
\begin{table*}[h!]
\resizebox{\textwidth}{!}{
\centering
\begin{tabular}{ c|c|c|c|c|c|c|c|c|c }
\multicolumn{1}{c}{\textbf{Model}} & \multicolumn{1}{c}{BLEU$\uparrow$} & \multicolumn{1}{c}{Exact$\uparrow$} & \multicolumn{1}{c}{Levenshtein$\downarrow$} & \multicolumn{1}{c}{MACCS FTS$\uparrow$} & \multicolumn{1}{c}{RDK FTS$\uparrow$} & \multicolumn{1}{c}{Morgan FTS$\uparrow$} & \multicolumn{1}{c}{FCD$\downarrow$} & \multicolumn{1}{c}{Text2Mol$\uparrow$} & \multicolumn{1}{c}{Validity$\uparrow$} \\
\noalign{\hrule height 2pt}
Ground Truth & 1.000 & 1.000 & 0.0 & 1.000 & 1.000 & 1.000 & 0.0 & 0.609 & 1.0 \\\hline
RNN & 0.652 & 0.004 & 38.09 & 0.591 & 0.400 & 0.362 & 0.223 & 0.409 & 0.542 \\
Transformer & 0.499 & 0.000 & 57.66 & 0.480 & 0.320 & 0.217 & 0.379 & 0.277 & \textbf{0.906} \\\hline
T5-Small & 0.741 & 0.063 & 27.7 & 0.704 & 0.578 & 0.525 & 0.213 & 0.479 & 0.608 \\
MolT5-Small & 0.755 & 0.076 & 25.99 & 0.704 & 0.568 & 0.517 & 0.198 & 0.482 & 0.721 \\\hline
T5-Base & 0.762 & 0.067 & 24.95 & 0.731 & 0.605 & 0.545 & 0.177 & 0.499 & 0.66 \\
MolT5-Base & 0.769 & 0.080 & 24.46 & 0.721 & 0.588 & 0.529 & 0.185 & 0.496 & 0.772 \\\hline
T5-Large & 0.854 & 0.272 & 16.721 & 0.823 & 0.731 & 0.670 & 0.117 & 0.552 & 0.902 \\
MolT5-Large & \textbf{0.854} & \textbf{0.302} & \textbf{16.07} & \textbf{0.834} & \textbf{0.746} & \textbf{0.684} & \textbf{0.116} & \textbf{0.554} & 0.905 \\
\end{tabular}
}
\caption{De novo molecule generation results for the different baseline models on the test split of the CheBI-20 dataset.
}
\label{tab:results_generation}
\end{table*}
\subsection{Molecule Captioning}
Table \ref{tab:results_captioning} shows reported results on the molecule captioning test. We find that the large pretrained models, either T5 or MolT5, are considerably better at generating realistic language to describe the molecule than the Transformer or RNN. We notice that the RNN model is more capable of extracting relevant properties from molecules than the Transformer, but it generally produces ungrammatical outputs. On the other hand, the Transformer baseline produces grammatical outputs, but they tend to repeat the same properties, such as carcinogenic, regardless of whether they apply. For this reason, the Text2Mol scores are much lower for the Transformer model, since its outputs match the given molecule much less frequently. We speculate that the ChEBI-20 dataset is too small to effectively train a transformer without large-scale natural language pretraining. We find that our additional pretraining of MolT5 results in a reasonable increase over T5 in captioning performance on both the traditional NLG metrics as well as our Text2Mol metric for each model size.
Several examples of different model outputs are shown in Figure \ref{fig:qual_caption}. In the first example (1), we see that MolT5's description matches best, identifying the molecule as a ``GDP-L-galactose''. We find that MolT5 is usually able to recognize what general class of molecule it is looking at (e.g. cyclohexanone, maleate salt, etc.). Generally, all models often look for the closest compound they know and base their caption on that. The argon atom, example (4) with SMILES `[39Ar]', is not present in the training dataset bonded to any other atoms (likely because it is an inert noble gas). All models recognize that (4) is a single atom, but they are unable to describe it. Interestingly, T5 comes closer to the actual atomic mass than MolT5 -- it may pick up on the number 39 in the SMILES string better without pretraining. In (7), the models try to caption a histological dye. MolT5 captions the molecule as an azure histological dye, which is very close to the ground truth ``brilliant cresyl blue'', while T5 does not.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/qual_mols1_cropped.pdf}
\caption{Examples of molecules generated by different models.}
\label{fig:qual}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/qual_caption_bigger_cropped.pdf}
\caption{Example captions generated by different models.}
\label{fig:qual_caption}
\end{figure*}
In the Transformer model, we see a phenomenon where many descriptions are reused. In the examples (2,7), the Transformer claims both molecules are deuterated and in (5,6) it claims that both are isotopes of oxygen, even though this is clearly not true. The RNN, on the other hand, tends to capture more properties of the molecule but has poor grammar. In (7), it recognizes the molecule as a dye, and it recognizes (6) as an antineoplastic agent. It recognizes (2) is an organaflourine compound, but it repeats this several times. Since it is extracting meaningful properties, the RNN achieves a much higher Text2Mol score than the Transformer.
\subsection{Text-Based de novo Molecule Generation}
As in the captioning task, we see that the RNN models tend to outperform the Transformer model in molecule generation. One interesting difference, however, is that the Transformer has a high validity; we speculate that the Transformer model is restricting itself to very confident outputs, much like in the captioning case where it repeats outputs. The large pretrained models perform much better than the RNN and Transformer models just as in the captioning task. Although it is well known that scaling model size and pretraining data leads to significant performance increases \cite{kaplan2020scaling}, it was still surprising to see the results on this task. For example, a default T5 model, which was only pretrained on text data, is capable of generating molecules which are much closer to the ground truth than the RNN and which are often valid. This trend also persists as language model size scales, since T5-large with 770M parameters outperforms the specifically pretrained MolT5-small with 60M parameters. Still, the pretraining in MolT5 slightly improves some molecule generation results, with especially large gains in validity.
We show results for the models in Figure \ref{fig:qual} and in Appendix \ref{appendix:examples} Figures \ref{fig:qual2} and \ref{fig:qual3}, which we number by input description. We find that compared to T5, MolT5 is better able to understand instructions for manipulating molecules, as shown in examples (1, 4, 5, 6, 7, 17, 18, 21). In many cases, MolT5 obtains exact matches with the ground truth (1, 4, 5, 6, 7, 8, 10, 12, 17, 20, 21). (6) is an interesting case, since it shows that MolT5 can understand crystalline solids like hydrates. (5) is another interesting example; it is the longest SMILES string, at 474 characters, which MolT5 is able to generate an exact match for. MolT5 understands peptides and can produce them from descriptions (5,15,17). Additionally, it also shows this ability for saccharides (4, 21) and enzymes (8,20). MolT5 is able to understand rare atoms such as Ruthenium (3). However, in this case it still misses the atom's charge. Some examples, such as (2), lack details so the molecules generated by MolT5 may be interesting to investigate.
As expected from its validity score, the RNN model has the most invalid matches. The Transformer, on the other hand, is rarely invalid. Like in the caption case, we note that the Transformer seems to be risk-averse. (4) is an example of this, where the RNN fails and the Transformer outputs a molecule which is very different than the ground truth instead of also failing. An example of an invalid SMILES string is (13)--T5 outputs ``\textit{C(C Cl)Cl}''. A more complicated example (19) from the RNN of a broken SMILES string is ``\textit{C1=CC2=C(C(=C1)S(=O)(=O)O)C3=CC(=C \\
(C=C3)S(=O)(=O)O)C4=CC=C(C=C4)S(=O) \\
(=O)O}'', which contains a broken ring.
\subsection{Ablations}
We perform ablations on MolT5-Small pretraining, which are shown in Appendix \ref{appendix:ablations}. If the model is only pretrained on the ZINC SMILES strings, then generated molecule validity increases significantly, as expected. Surprisingly, this leads to decreased similarity of the generated molecules to the ground truth. We speculate that more valid molecules are generated but that they occupy a larger portion of the chemical space, so they are farther from the molecules in ChEBI-20. While it would be expected that ZINC-only training would hurt natural language generation, we observe on the captioning task that performance differences are negligible. Joint training on C4+ZINC appears to effectively capture the benefits of each type of pretraining.
\section{Related Work}
\subsection{Multimedia Representation}
Much recent work on multimedia representations falls into vision-language models which leverage transformers \citep{chen2019uniter, su2019vl, lu2019vilbert}. More fine-grained multimedia embedding approaches have also been explored, such as \cite{li2020cross} which integrates events from images and their descriptions, or \cite{li2016event}, which studies multimodal pattern mining.
CLIP \citep{radford2021learning} trains a zero-shot image classifier by using natural language labels which can be easily applied to different datasets, and CLIP-Event \cite{li2022clip} extends CLIP to incorporate event-level alignment. A modification of CLIP's contrastive loss function, which follows
\cite{sohn2016improved}, is applied by Text2Mol \cite{edwards2021text2mol} for cross-modal retrieval between molecule and text pairs. \citet{edwards2021text2mol} also released the ChEBI-20 dataset of molecule-description pairs, which is used for training and evaluation in this paper. \citet{vallbioassayclr} leverage a contrastive loss between bioassay descriptions and molecules to predict activity between the two. \citet{zeng2022deep} pretrain a language model to learn a joint representation between molecules and biomedical text which they use for tasks such as relation extraction, molecule property prediction, and cross-modal retrieval like Text2Mol. Note that unlike our work, they do not explore generating text nor molecules. Due to the extremely large space of possible molecules, a retrieval approach can be intractable for finding the perfect (new) molecule for some task. \newcite{vaucher2020automated} create a dataset of chemical equations and associated action sequences in natural language. \citet{vaucher2021inferring} then leverage this dataset to learn a BART model which can plan chemical reaction steps, which is one of the most closely related attempts to ours. However, their natural language generation is constrained to the allowed reaction steps in their dataset-- the main purpose of their model is to list the steps for a reaction rather than describing molecules.
\subsection{Image Captioning and Text-Guided Image Generation}
Image captioning has been studied by the computer vision and NLP communities for several decades \cite{pan2004automatic, hossain2019comprehensive, stefanini2021show}. Recent state-of-the-art models tend to pretrain transformer-based models on massive text-image corpora \cite{li2020oscar,hu2021scaling}.
Work has also been done in the biomedical domain \cite{pavlopoulos2019survey}, a close cousin of the chemistry domain, where tasks tend to be focused on diagnosis of various image types, such as x-rays \cite{demner2016preparing} or gross lesions \cite{jing2017automatic}.
The reverse problem, text-guided image generation, has proven considerably more challenging \cite{khan2021transformers}. Several attempts have used GAN-based methods \cite{reed2016generative,zhang2017stackgan,xu2018attngan}. Recent work has shown remarkable results; DALL$\cdot$ E \cite{ramesh2021zero} and the recently released DALL $\cdot$ E 2 \cite{ramesh2022hierarchical} have made remarkable progress on this task--achieving realistic high resolution images which can seamlessly fuse multiple concepts together.
\subsection{Molecule Representation}
Molecule representation has been a long-standing problem in the field of cheminformatics. Traditionally, fingerprinting methods have been a preferred technique to featurize molecule structural representations \cite{rogers2010extended, cereto2015molecular}.
These approaches, however, do not allow representations to be learned from the data.
In recent years, advances in machine learning, and in particular natural language processing, have been applied to this problem. A popular input for these algorithms has been SMILES strings \cite{weininger1988smiles, weininger1989smiles}, which are a computer-readable linearization of molecule graphs. \citet{jaeger2018mol2vec} use the Morgan fingerprinting algorithm to convert each molecule into a `sentence' of its substructures, to which it applies the Word2vec algorithm \cite{mikolov2013efficient, mikolov2013distributed}. \cite{duvenaud2015convolutional} use neural methods to learn fingerprints.
Additionally, other recent advances such as BERT \citep{devlin2019bert} have been applied to the domain, such as MolBERT \citep{fabian2020molecular} and ChemBERTa \citep{chithrananda2020chemberta}, which use SMILES strings as inputs to pretrain a BERT-esque model.
Work has been done to use the molecule graph structure and known reactions for learning representations \cite{wang2021chemical}. \cite{schwaller2021mapping} trains a BERT model to learn representations of chemical reactions. \cite{schwaller2021extraction} leverages unsupervised representation learning with transformers to extract an organic chemistry grammar. Unlike existing work, our model MolT5 can represent both molecules and natural language, allowing us to translate between the two modalities
There has been particular interest in training generative models for de novo molecule discovery.
\cite{bagal2021molgpt} apply a GPT-style decoder for this task. \citet{lu2022unified} apply a T5 model to SMILES strings for multitask reaction prediction problems. MegaMolBART\footnote{https://tinyurl.com/megamolbart} trains a BART model on 500M SMILES strings from the ZINC-15 dataset \cite{sterling2015zinc}.
\subsection{Chemistry Text Representation}
Natural language representations for the chemistry domain have been less-well studied than molecule representations and biomedical text representations. Existing work, such as Text2Mol, has used more general representations such as SciBERT \cite{beltagy2019scibert}. Recently, a transformer-based encoder, ChemBERT, has been trained on a corpus of American Chemical Society papers, and it was extended as ChemRxnBERT for the purpose of extracting reactions \cite{doi:10.1021/acs.jcim.1c00284}.
\section{Conclusions and Future Work}
Chemists have collected a wealth of information over centuries of work, and a considerable portion of it is stored in natural language, which is incomprehensible to traditional computing approaches. By leveraging joint molecule-language representations, this resource can be applied to greatly augment chemistry research. In this work, we propose a new first-of-its-kind model, \textbf{MolT5}, which is trained on both natural language text and on molecule string representations. Further, we propose two new tasks: molecule captioning and text-guided molecule generation, for which we explore evaluation methods. Together, these tasks allow for translation between natural language and molecules. We find the results to be very exciting--we are able to obtain high Text2Mol and BLEU scores for both our proposed tasks. Notably, scaling model size is a driving factor in performance.
These new tasks are an exciting new direction within cheminformatics and natural language processing. They will allow an end-user with no knowledge of machine learning to generate a list of molecules according to their desired specifications. This has the potential to vastly improve the targeting of current molecule generation techniques and expand these tools to non-experts, allowing for significant improvements for creating new chemical techniques across a variety of industries.
\section{Broader Impacts}
Our proposed model and tasks will have the following broader impacts. 1) It will help to democratize molecular AI, allowing chemistry experts to take advantage of new AI technologies for discovering new life-changing drugs by interacting in the natural language they are experienced using. 2) Text-based molecule generation enables the ability to generate molecules with specific functions (such as taste) rather than properties, enabling the next generation of chemistry where custom molecules are used for each application. Specifically-designed molecular solutions have the potential to revolutionize fields such as medicine and material science. 3) Our model, \textbf{MolT5}, whose weights we will release, will allow further research in the NLP community on the applications of multimodal text-molecule models.
\section{Ablations}
\label{appendix:ablations}
\begin{table}[h!]
\resizebox{\textwidth}{!}{
\centering
\tiny
\begin{tabular}{ c|c|c|c|c|c|c|c }
\multicolumn{1}{c}{\textbf{Pretraining}} & \multicolumn{1}{c}{BLEU-2} & \multicolumn{1}{c}{BLEU-4} & \multicolumn{1}{c}{ROUGE-1} & \multicolumn{1}{c}{ROUGE-2} & \multicolumn{1}{c}{ROUGE-L} & \multicolumn{1}{c}{METEOR} & \multicolumn{1}{c}{Text2Mol} \\
\noalign{\hrule height 2pt}
Ground Truth & & & & & & & 0.609 \\\hline
C4-Only & 0.515 & 0.426 & 0.609 & 0.456 & 0.565 & 0.535 & 0.517 \\
ZINC-Only & 0.513 & 0.427 & 0.608 & 0.454 & 0.563 & 0.529 & 0.534 \\
C4+ZINC & 0.530 & 0.442 & 0.617 & 0.465 & 0.574 & 0.544 & 0.535 \\
\end{tabular}
}
\captionsetup{width=.8\textwidth,margin={0cm,-8cm}}
\caption{Pretraining ablation results for MolT5-Small on the validation split of the CheBI-20 dataset. Rouge scores are F1 values.}
\label{tab:ablation_results_captioning}
\end{table}
\begin{table}[h!]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{ c|c|c|c|c|c|c|c|c|c }
\multicolumn{1}{c}{\textbf{Pretraining}} & \multicolumn{1}{c}{BLEU$\uparrow$} & \multicolumn{1}{c}{Exact$\uparrow$} & \multicolumn{1}{c}{Levenshtein$\downarrow$} & \multicolumn{1}{c}{MACCS FTS$\uparrow$} & \multicolumn{1}{c}{RDK FTS$\uparrow$} & \multicolumn{1}{c}{Morgan FTS$\uparrow$} & \multicolumn{1}{c}{FCD$\downarrow$} & \multicolumn{1}{c}{Text2Mol$\uparrow$} & \multicolumn{1}{c}{Validity$\uparrow$} \\
\noalign{\hrule height 2pt}
Ground Truth & & & & & & & 0.0 & 0.609 & 1.0 \\\hline
C4-Only & 0.753 & 0.0769 & 29.21 & 0.720 & 0.591 & 0.543 & 0.191 & 0.490 & 0.599 \\
ZINC-Only & 0.712 & 0.0576 & 32.17 & 0.637 & 0.503 & 0.450 & 0.272 & 0.418 & 0.765 \\
C4+ZINC & 0.751 & 0.0836 & 29.19 & 0.705 & 0.564 & 0.516 & 0.197 & 0.483 & 0.693 \\
\end{tabular}
}
\captionsetup{width=.8\textwidth,margin={0cm,-8cm}}
\caption{Pretraining ablation de novo molecule generation results for MolT5-Small on the validation split of the CheBI-20 dataset.
}
\label{tab:ablation_results_generation}
\end{table}
\section{More Examples}
\label{appendix:examples}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/qual_mols2_cropped.pdf}
\caption{More examples of molecules generated by different models.}
\label{fig:qual2}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/qual_mols3_cropped.pdf}
\caption{More examples of molecules generated by different models.}
\label{fig:qual3}
\end{figure*}
\section*{Acknowledgements}
This work was supported by the Molecule Maker Lab Institute: An AI Research Institutes program supported by NSF under Award No. 2019897, and NSF No. 1705169. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect those of the National Science Foundation.
\clearpage
|
1,108,101,566,810 | arxiv |
\section{Hilbert basis for monopole operators}
\label{sec:general_idea}
\subsection{Preliminaries}
\label{subsec:preliminaries}
Let us recall some basic properties of Lie algebras,
c.f.~\cite{Humphreys:1972},
and combine them with the description of \emph{strongly convex rational
polyhedral cones} and \emph{affine semi-groups}, c.f.~\cite{Cox:2011}.
Moreover, we recapitulate the definition and properties of the GNO-dual group,
which can be found in~\cite{Goddard:1976qe,Kapustin:2005py}.
\paragraph{Root and weight lattices of $\gfrak$}
Let $\mathrm{G}$ be a Lie group with semi-simple Lie algebra $\gfrak$ and
$\mathrm{rk}(\mathrm{G})=r$. Moreover, $\widetilde{\mathrm{G}}$ is the universal covering group of
$\mathrm{G}$, i.e.\ the unique simply connected Lie group with Lie algebra $\gfrak$.
Choose a maximal torus $\mathrm{T} \subset \mathrm{G}$ and the corresponding Cartan subalgebra
$\mathfrak{t} \subset \gfrak$. Denote by
$\boldsymbol{\Phi}$ the set of all roots $\alpha \in \mathfrak{t}^*$.
By the choice of a
hyperplane, one divides the root space into positive $\boldsymbol{\Phi}_+$ and negative
roots $\boldsymbol{\Phi}_-$. In the half-space of positive roots one introduces the simple
positive roots as irreducible basis elements and denotes their set by
$\boldsymbol{\Phi}_s$.
The roots span a lattice $\Lambda_r(\gfrak)\subset \mathfrak{t}^*$, the \emph{root
lattice}, with basis $\boldsymbol{\Phi}_s$.
Besides roots, one can always choose a basis in the complexified Lie algebra
that gives rise to the notion of coroots $\alpha^\vee \in \mathfrak{t}$ which satisfy
$\alpha \left( \beta^\vee \right) \in \mathbb Z$ for any $\alpha,\beta\in
\mathbf{\Phi}$. Define $\alpha^\vee$ to be a simple
coroot if and only if $\alpha$ is a simple root.
Then the coroots span a lattice $\Lambda^\vee_r(\gfrak)$ in $\mathfrak{t}$ ---
called the \emph{coroot lattice} of $\gfrak$.
The dual lattice $\Lambda_w(\gfrak)$ of the coroot lattice is the set of points
$\mu \in \mathfrak{t}^*$ for which $\mu (\alpha^\vee) \in \mathbb Z$ for all $\alpha \in
\boldsymbol{\Phi}$. This lattice is called \emph{weight lattice} of $\gfrak$.
Choosing a basis $\boldsymbol{B}$ of simple coroots
\begin{equation}
\boldsymbol{B}\coloneqq \left\{ \alpha^{\vee} \ , \ \alpha\in \boldsymbol{\Phi}_s
\right\} \subset \mathfrak{t} \; ,
\end{equation}
one readily defines a basis for the dual space via
\begin{equation}
\boldsymbol{B^*} \coloneqq \left\{ \lambda_{\alpha} \ , \ \alpha \in \boldsymbol{\Phi}_s
\right\} \subset \mathfrak{t}^*\qquad\textrm{for}\quad \lambda_{\alpha} \left(\beta^{\vee}\right) =
\delta_{\alpha,\beta} \; , \; \forall \alpha,\beta \in \boldsymbol{\Phi}_s \; .
\end{equation}
The basis elements $\lambda_{\alpha} $ are precisely the fundamental weights of
$\gfrak$ (or $\widetilde{\mathrm{G}}$) and they are a basis for the weight lattice.
Analogous, the dual lattice $\Lambda_{mw}(\gfrak) \subset \mathfrak{t}$ of the root
lattice is the set of points $m \in \mathfrak{t}$ such that $\alpha(m) \in \mathbb Z$ for all
$\alpha \in \boldsymbol{\Phi}$. In particular, the coroot lattice is a
sublattice of $\Lambda_{mw}(\gfrak)$.
As a remark, the lattices defined so far solely depend on the Lie algebra
$\gfrak$, or equivalently on $\widetilde{G}$, but not on $\mathrm{G}$. Because any group
defined via $\widetilde{\mathrm{G}}\slash \Gamma$ for $\Gamma\subset \mathcal{Z}(\mathrm{G})$ has the
same Lie algebra.
\paragraph{Weight and coweight lattice of $\mathrm{G}$}
The weight lattice of the group $\mathrm{G}$ is the lattice of the infinitesimal
characters, i.e.\ a character $\chi: \mathrm{T} \to {{\rm U}(1)}$ is a homomorphism, which is
then uniquely determined by the derivative at the identity. Let $X \in \mathfrak{t}$
then $\chi(\exp{(X)}) = \exp{(i \mu(X))}$, wherein $\mu \in \mathfrak{t}^*$ is an
\emph{infinitesimal character} or weight of $\mathrm{G}$. The weights form then a
lattice $\Lambda_w(\mathrm{G})\subset \mathfrak{t}^*$, because the exponential map translates
the multiplicative structure of the character group into an additive structure.
Most importantly, the following inclusion of lattices holds:
\begin{equation}
\Lambda_r(\gfrak) \subset \Lambda_w(\mathrm{G}) \subset \Lambda_w(\gfrak) \;.
\label{eqn:inclusion_root_weight_weight}
\end{equation}
Note that the weight lattice $\Lambda_w$ of $\gfrak$ equals the weight lattice
of the universal cover $\widetilde{\mathrm{G}}$.
As before, the dual lattice for $\Lambda_w(\mathrm{G})$ in $\mathfrak{t}$ is readily defined
\begin{equation}
\Lambda_w^*(\mathrm{G}) \coloneqq \mathrm{Hom}\left( \Lambda_w(\mathrm{G}),\mathbb Z\right) = \ker
\left\{ \begin{matrix} \mathfrak{t} & \to & \mathrm{T} \\ X & \mapsto & \exp(2\pi \mathrm{i} X)
\end{matrix} \right\} \; .
\end{equation}
As we see, the \emph{coweight lattice} $\Lambda_w^*(\mathrm{G})$ is precisely the set of
solutions to the generalised Dirac quantisation
condition~\eqref{eqn:general_Dirac} for $\mathrm{G}$. In addition, an inclusion of
lattices holds
\begin{equation}
\Lambda_{r}^\vee(\gfrak) \subset \Lambda_w^*(\mathrm{G}) \subset \Lambda_{mw}(\gfrak)
\; ,
\end{equation}
which follows from dualising~\eqref{eqn:inclusion_root_weight_weight}.
\paragraph{GNO-dual group and algebra}
Following~\cite{Goddard:1976qe,Kapustin:2005py}, a Lie algebra $\widehat{\mathfrak{g}}$ is
the \emph{magnetic dual} of $\gfrak$ if its roots coincide with the coroots of
$\gfrak$. Hence, the Weyl groups of $\gfrak$ and $\widehat{\mathfrak{g}}$ agree. The
\emph{magnetic dual group} $\widehat{\G}$ is, by definition, the unique Lie
group with Lie algebra $\widehat{\mathfrak{g}}$ and weight lattice $\Lambda_w(\widehat{\G})$ equal to
$\Lambda_w^*(\mathrm{G})$. In physics, $\widehat{\G}$ is called the GNO-dual group; while in
mathematics, it is known under Langlands dual group.
\paragraph{Polyhedral cones}
A \emph{rational convex polyhedral cone} in $\mathfrak{t}$ is a set $\sigma_B$ of the
form
\begin{equation}
\sigma_B \equiv \mathrm{Cone}(\boldsymbol{B}) = \left\{ \sum_{\alpha^\vee \in
\boldsymbol{B}} f_{\alpha^\vee}\ \alpha^\vee \ | \ f_{\alpha^\vee} \geq
0 \right\} \subseteq \mathfrak{t}
\end{equation}
where $\boldsymbol{B} \subseteq \Lambda^\vee_r$, the basis of simple coroots,
is finite.
Moreover, we note that $\sigma_B$ is a \emph{strongly convex} cone, i.e.\
$\{0\}$ is a face of the cone, and of \emph{maximal dimension}, i.e.\ $\mathrm{dim}(
\sigma_B) =r$.
Following~\cite{Cox:2011}, such cones $\sigma_B$ are generated
by the \emph{ray generators} of their edges, where the ray generators in this
case are precisely the simple coroots of $\gfrak$.
For a polyhedral cone $\sigma_B\subseteq \mathfrak{t}$ one naturally defines the
\emph{dual cone}
\begin{equation}
\sigma_B^\vee = \left\{m \in \mathfrak{t}^* \ | \ m(u)\geq0 \ \text{for all} \ u \in
\sigma_B \right\} \subseteq \mathfrak{t}^* \; .
\end{equation}
One can prove that $\sigma_B^\vee$ equals the rational convex
polyhedral cone generated by $\boldsymbol{B^*}$, i.e.
\begin{equation}
\sigma_B^\vee = \sigma_{B^*} = \mathrm{Cone}(\boldsymbol{B^*}) = \left\{
\sum_{\lambda \in
\boldsymbol{B^*}} g_{\lambda}\ \lambda \ | \ g_{\lambda} \geq
0 \right\} \subseteq \mathfrak{t}^* \; ,
\end{equation}
which is well-known under the name \emph{(closed) principal Weyl chamber}. By
the very same arguments as above, the cone $ \sigma_{B^*}$ is generated by its
ray generators, which are the fundamental weights of $\gfrak$.
For any $m\in \mathfrak{t}$ and $d\geq0$, let us define an \emph{affine hyperplane}
$H_{m,d}$ and \emph{closed linear
half-spaces} $H_{m,d}^{\pm}$ in $\mathfrak{t}^*$ via
\begin{subequations}
\begin{align}
H_{m,d} &\coloneqq \left\{ \mu \in \mathfrak{t}^* \ | \ \mu(m)=d \right\} \subseteq
\mathfrak{t}^* \; ,\\
H_{m,d}^{\pm} &\coloneqq \left\{ \mu \in \mathfrak{t}^* \ | \ \mu(m)\geq \pm d
\right\} \subseteq
\mathfrak{t}^* \;.
\end{align}
\end{subequations}
If $d=0$ then $H_{m,0}$ is hyperplane through the origin, sometimes denoted as
\emph{central} affine hyperplane. A theorem~\cite{Ziegler:1995} then states: a
cone $\sigma\subset \mathbb R^n$ is finitely generated if and only if
it is the finite intersection of closed linear half spaces.
This result allows to make contact with the usual definition of the Weyl
chamber. Since we know that $\sigma_{B^*}$ is finitely generated by the
fundamental weights $\{ \lambda_{\alpha} \}$ and the dual basis is
$\{ \alpha^{\vee} \}$, one arrives at $\sigma_{B^*} = \cap_{\alpha \in
\boldsymbol{\Phi}_s} H_{\alpha^{\vee},0}^+ $; thus, the dominant Weyl chamber is
obtained by cutting the root space along the hyperplanes orthogonal to some root
and selecting the cone which has only positive entries.
\paragraph{Remark}
Consider the group ${{\rm SU}(2)}$, then the fundamental weight is simply $\tfrac{1}{2}$
such that $\Lambda_w^{{{\rm SU}(2)}}= \mathrm{Span}_{\Z}(\tfrac{1}{2})= \mathbb Z \cup \{\mathbb Z +\tfrac{1}{2}
\}$. Moreover, the corresponding cone (Weyl chamber) will be denoted by
$\sigma_{\boldsymbol{B^*}}^{{{\rm SU}(2)}}=\mathrm{Cone}(\tfrac{1}{2})$.
\subsection{Effect of conformal dimension}
\label{subsec:Hilbert basis}
Next, while considering the conformal dimension $\Delta(m)$ as map between two
Weyl chambers we will stumble across the notion of \emph{affine semi-groups},
which are known to constitute the combinatorial background for toric
varieties~\cite{Cox:2011}.
\paragraph{Conformal dimensions --- revisited}
Recalling the conformal dimension $\Delta$ to be interpreted as the highest
weight under ${{\rm SU}(2)}_R$, it can be understood as the following map
\begin{equation}
\Delta : \begin{matrix}
\sigma_{B^*}^{\widehat{\G}} \cap \Lambda_w(\widehat{\G}) & \to &
\sigma_{\boldsymbol{B^*}}^{{{\rm SU}(2)}} \cap \Lambda_w({{\rm SU}(2)}) \\
m & \mapsto & \Delta(m)
\end{matrix} \; .
\end{equation}
Where $\sigma_{B^*}^{\widehat{\G}}$ is the cone spanned by the fundamental
weights of $\widehat{\mathfrak{g}}$, i.e.\ the dual basis of the simple roots
$\boldsymbol{\Phi}_s$ of $\gfrak$. Likewise, $\sigma_{\boldsymbol{B^*}}^{{{\rm SU}(2)}}$
is the Weyl chamber for ${{\rm SU}(2)}_R$.
Upon continuation, $\Delta$ becomes a map between the dominant Weyl
chamber of $\widehat{\G}$ and ${{\rm SU}(2)}_R$
\begin{equation}
\Delta : \begin{matrix}
\sigma_{B^*}^{\widehat{\G}} & \to & \sigma_{\boldsymbol{B^*}}^{{{\rm SU}(2)}} \\
m & \mapsto & \Delta(m) \end{matrix} \; .
\end{equation}
By definition, the conformal dimension~\eqref{eqn:Def_ConfDim} has two types of
contributions: firstly, a
positive contribution $|\rho(m)|$ for a weight $\rho \in \Lambda_w(\mathrm{G})\subset
\mathfrak{t}^*$ and a magnetic weight $m \in \Lambda_w(\widehat{\G})\subset
\widehat{\mathfrak{t}}^*$. By definition $\Lambda_w(\widehat{\G})= \Lambda_w^*(\mathrm{G})$; thus,
$m$ is a coweight of $\mathrm{G}$ and $\rho(m)$ is the duality paring. Secondly, a
negative contribution $-|\alpha(m)|$ for a positive root $\alpha \in
\boldsymbol{\Phi}_+$ of $\gfrak$. By the same arguments, $\alpha(m)$ is the
duality pairing of weights and coweights. The paring is also well-defined on
the entire the cone.
\paragraph{Fan generated by conformal dimension}
The individual absolute values in $\Delta$ allow for another interpretation; we
use them to associate a collection of affine central hyperplanes and closed
linear half-spaces
\begin{align}
H_{\mu,0}^{\pm} = \left\{ m \in \mathfrak{t} \ \big| \ \pm \mu(m)
\geq 0 \right\} \subset \mathfrak{t} \quad\textrm{and}\quad H_{\mu,0} = \left\{ m \in
\mathfrak{t} \ \big| \ \mu(m) = 0 \right\} \subset \mathfrak{t}\; .
\end{align}
Here, $\mu$ ranges over all weights $\rho$ and all positive roots $\alpha$
appearing in the theory.
If two weights $\mu_1$, $\mu_2$ are (integer) multiples of each other,
then $ H_{\mu_1,0} = H_{\mu_2,0}$ and we can reduce the number of relevant
weights. From now on, denote by $\Gamma$ the set of weights $\rho$ and positive
roots $\alpha$ which are not multiples of one another. Then the conformal
dimension contains $Q \coloneqq |\Gamma| \in \NN$ distinct hyperplanes such
that there exist $2^Q$ different finitely generates cones
\begin{align}
\sigma_{\epsilon_1,\epsilon_2,\ldots,\epsilon_Q} \coloneqq
H_{\mu_1,0}^{\epsilon_1} \cap H_{\mu_2,0}^{\epsilon_2} \cap \cdots \cap
H_{\mu_Q,0}^{\epsilon_Q} \subset \mathfrak{t} \qquad\textrm{with}\quad \epsilon_i = \pm \qquad\textrm{for}\quad
i=1,\ldots, Q \; .
\end{align}
By construction, each cone $\sigma_{\epsilon_1,\epsilon_2,\ldots,\epsilon_Q}$
is a strongly convex rational polyhedral cone of dimension $r$, for
non-trivial cones, and $0$, for trivial intersections. Consequently,
each cone is generated by its ray generators and these can be chosen to be
lattice points of $\Lambda_w(\widehat{\G})$.
Moreover,
the restriction of $\Delta$ to any
$\sigma_{\epsilon_1,\epsilon_2,\ldots,\epsilon_Q}$ yields a linear function,
because we effectively resolved the absolute values by defining these cones.
It is, however, sufficient to restrict the considerations to the Weyl chamber
of $\widehat{\G}$;
hence, we simply intersect the cones with the hyperplanes defining
$\sigma_{B^*}^{\widehat{\G}}$, i.e.
\begin{equation}
C_p \equiv C_{\epsilon_1,\epsilon_2,\ldots,\epsilon_Q} \coloneqq
\sigma_{\epsilon_1,\epsilon_2,\ldots, \epsilon_Q} \cap \sigma_{B^*}^{\widehat{\G}}
\qquad\textrm{with}\quad p =(\epsilon_1,\epsilon_2,\ldots,\epsilon_Q) \; .
\end{equation}
Naturally, we would like to know for which $\mu \in \Lambda_w(\mathrm{G})$ the
hyperplane $H_{\mu,0}$ intersects the Weyl chamber $\sigma_{B^*}^{\widehat{\G}}$
non-trivially, i.e.\ not only in the origin. Let us emphasis the differences of
the Weyl chamber (and their dual cones) of $\mathrm{G}$ and $\widehat{\G}$:
\begin{subequations}
\begin{alignat}{3}
\sigma_{B^*}^{\mathrm{G}} &= \mathrm{Cone}\left( \lambda_\alpha \ | \ \lambda_\alpha
(\beta^\vee) =\delta_{\alpha,\beta} \ , \forall \alpha ,\beta \in \boldsymbol{\Phi}_s
\right) \subset \mathfrak{t}^* & \; &\xleftrightarrow{\ * \ } \; &
\sigma_{B}^{\mathrm{G}} &= \mathrm{Cone} \left( \alpha^\vee \ | \ \forall \alpha \in \boldsymbol{\Phi}_s
\right) \subset \mathfrak{t} \; ,
\\
\sigma_{B^*}^{\widehat{\G}} &= \mathrm{Cone} \left( m_\alpha \ | \ \beta(m_\alpha) =
\delta_{\alpha,\beta} \ , \forall \alpha ,\beta \in \boldsymbol{\Phi}_s \right) \subset
\mathfrak{t} & \; &\xleftrightarrow{\ * \ } \; &
\sigma_{B}^{\widehat{\G}} &= \mathrm{Cone} \left( \alpha \ | \ \forall \alpha \in \boldsymbol{\Phi}_s
\right) \subset \mathfrak{t}^* \; .
\end{alignat}
\end{subequations}
It is possible to prove the following statements:
\begin{enumerate}
\item If $\mu \in \mathrm{Int}\left( \sigma_{B}^{\widehat{\G}} \cup
(-\sigma_{B}^{\widehat{\G}}) \right) $, i.e.\ $\mu = \sum_{\alpha \in \boldsymbol{\Phi}_s}
g_\alpha \alpha $ where either all $g_\alpha >0$ \emph{or} all $g_\alpha < 0$ ,
then $H_{\mu,0} \cap \sigma_{B^*}^{\widehat{\G}} = \{0\}$.
\item If $\mu \in \partial \left( \sigma_{B}^{\widehat{\G}}\cup
(-\sigma_{B}^{\widehat{\G}}) \right)$ and
$\mu\neq0$, i.e.\ $\mu =
\sum_{\alpha \in \boldsymbol{\Phi}_s} g_\alpha \alpha $ where at least one $g_\alpha =0$,
then $H_{\mu,0}$ intersects $\sigma_{B^*}^{\widehat{\G}}$ at one of its boundary faces.
\item If $\mu \notin \sigma_{B}^{\widehat{\G}}\cup
(-\sigma_{B}^{\widehat{\G}}) $, i.e.\ $\mu =
\sum_{\alpha \in \boldsymbol{\Phi}_s} g_\alpha \alpha $ with at least one $g_\alpha >0$
\emph{and} at least one $g_\beta <0$, then $\left(H_{\mu,0} \cap
\sigma_{B^*}^{\widehat{\G}}\right) \setminus\{0\} \neq \emptyset $.
\end{enumerate}
\textbf{Consequently, a weight $\boldsymbol{\mu \in \Lambda_w(\mathrm{G})}$ appearing
in $\boldsymbol{\Delta}$ leads to a hyperplane intersecting the Weyl chamber of
$\widehat{\G}$ non-trivially if and only if neither $\boldsymbol{\mu}$ nor
$\boldsymbol{-\mu}$ lies in the rational cone spanned by the simple roots
$\boldsymbol{\Phi}_s$ of $\boldsymbol{\mathrm{G}}$.} \par \noindent
Therefore, the contributions $- |\alpha(m) |$, for $\alpha \in \boldsymbol{\Phi}_+$,
of the vector multiplet never yield a relevant hyperplane.
From now on,
assume that trivial cones $C_p$ are omitted in the index set $I$ for $p$. The
appropriate geometric object to consider is then the \emph{fan} $F_\Delta
\subset \mathfrak{t}$ defined by the family $F_\Delta= \left\{ C_{p}\ , \ p\in
I\right\}$ in $\mathfrak{t}$. A fan $F$ is a family of non-empty polyhedral
cones such that (i) every non-empty face of a cone in $F$ is a cone in
$F$ and (ii) the intersection of any two cones in $F$ is a face of
both. In addition, the fan $F_\Delta$ defined above is a \emph{pointed
fan}, because $\{0\}$ is a cone in $F_\Delta$ (called the trivial cone).
\paragraph{Semi-groups}
Although we already know the cone generators for the fan $F_\Delta$, we have to
distinguish them from the generators of $F_\Delta \cap \Lambda_w(\widehat{\G})$, i.e.\
we need to restrict to the weight lattice of $\widehat{\G}$.
The first observation is that
\begin{equation}
S_p\coloneqq C_p \cap \Lambda_w(\widehat{\G}) \qquad\textrm{for}\quad p\in I
\end{equation}
are semi-groups, i.e.\ sets with an associative binary operation. This is
because
the addition of elements is commutative, but there is no inverse
defined as ``subtraction'' would lead out of the cone. Moreover, the $S_p$
satisfy further properties, which we now simply collect, see for
instance~\cite{Ziegler:1995}. Firstly, the $S_p$ are affine semi-groups, which
are semi-groups that can be embedded in $\mathbb Z^n$ for some $n$. Secondly, every
$S_p$ possesses an identity element, here $m=0$, and such semi-groups are
called \emph{monoids}. Thirdly, the $S_p$ are \emph{positive} because the only
invertible element is $m=0$.
Now, according to \emph{Gordan's Lemma}~\cite{Ziegler:1995,Cox:2011}, we know
that every $S_p$ is finitely generated, because all $C_p$'s are finitely
generated, rational polyhedral cones. Even more is true, since the division into
the $C_p$ is realised via affine hyperplanes $H_{\mu_i,0}$ passing through
the origin, the $C_p$ are strongly convex rational cones of maximal dimension.
Then~\cite[Prop.~1.2.22.]{Cox:2011} holds and we know that there exist a
unique minimal generating set for $S_p$, which is called \emph{Hilbert basis}.
The Hilbert basis $\Hcal(S_p)$ is defined via
\begin{equation}
\Hcal(S_p) \coloneqq \left\{ m \in S_p \ | \ m \ \text{is
irreducible} \right\} \; ,
\end{equation}
where an element is called \emph{irreducible} if and only if $m = x
+ y$ for $x,y\in S_p$ implies $x=0$ or $y=0$.
The importance of the Hilbert basis is that it is a unique, finite,
minimal set of irreducible elements that generate
$S_p$. Moreover, $\Hcal(S_p)$ always contains the ray generators of the
edges of $C_p$. The elements of $\Hcal(S_p)$ are sometimes called \emph{minimal
generators}.
As a remark, there exist various algorithms for computing the Hilbert basis,
which are, for example, discussed in~\cite{Miller:2005,Sturmfels:1995}. For the
computations presented in this paper, we used the \texttt{Sage} module
\emph{Toric
varieties} programmed by A.~Novoseltsev and V.~Braun as well as the
\texttt{Macaulay2} package \emph{Polyhedra} written by René Birkner.
After the exposition of the idea to employ the conformal dimension to define a
fan in the Weyl chamber of $\widehat{\G}$, for which the intersection with the weight
lattice leads to affine semi-groups, we now state the main consequence: \par
\noindent
\textbf{The collection $\boldsymbol{\{\Hcal(S_p) \; ,
p \in I \}}$ of all Hilbert bases is the set of necessary (bare) monopole
operators for a theory with conformal dimension $\boldsymbol{\Delta}$.} \par
At this stage we did not include the Casimir invariance described by the
dressing factors $P_{G}(t,m)$. For a generic situation, the bare and dressed
monopole operators for a GNO-charge $m \in \Hcal(S_p)$ for some $p$ are all
necessary generators for the chiral ring $\mathbb C[\mathcal{M}_C]$. However, there will
be scenarios for which there exists a further reduction of the number of
generators. For those cases, we will comment and explain the cancellations.
\subsection{Dressing of monopole operators}
\label{subsec:Dressings_as_HS}
One crucial ingredient of the monopole formula of~\cite{Cremonesi:2013lqa}
are the dressing factors $P_{\mathrm{G}}(t,m)$ and this section provides an algebraic
understanding. We refer
to~\cite{Humphreys:1972,Varadarajan:1984,Humphreys:1990} for the exposition of
the
mathematical details used here.
It is known that in $\mathcal{N}=4$ the $\mathcal{N} =2$ BPS-monopole operator $V_{m}$ is
compatible with a constant background of the $\mathcal{N}=2$ adjoint complex scalar
$\Phi$, provided $\Phi$ takes values on the Lie algebra $\mathfrak{h}_m $ of the
residual gauge group $\mathrm{H}_m \subset \mathrm{G}$, i.e.\ the stabiliser of $m$ in $\mathrm{G}$.
Consequently, each bare monopole operator $V_{m}$ is compatible with \emph{any}
$\mathrm{H}_m$-invariant polynomial on $\mathfrak{h}_m$. We will now argue that the dressing
factors $P_{\mathrm{G}}(t,m)$ are to be understood as Hilbert (or Poincar\'{e}) series
for this so-called \emph{Casimir-invariance}.
\paragraph{Chevalley-Restriction Theorem}
Let $\mathrm{G}$ be a Lie group of rank $l$ with a semi-simple Lie algebra
$\gfrak$ over $\mathbb C$ and $\mathrm{G}$ acts via the adjoint representation on $\gfrak$.
Denote by $\mathfrak{P}(\gfrak)$ the algebra of all polynomial functions on
$\gfrak$. The action of $\mathrm{G}$ extends to $\mathfrak{P}(\gfrak)$ and
$\mathfrak{I}(\gfrak)^\mathrm{G}$ denotes the set of $\mathrm{G}$-invariant polynomials in
$\mathfrak{P}(\gfrak)$. In addition, denote by $\mathfrak{P}(\mathfrak{h})$ the
algebra of all polynomial functions on $\mathfrak{h}$. The Weyl group $\mathcal{W}_\mathrm{G}$,
which acts naturally on $\mathfrak{h}$, acts also on $\mathfrak{P}(\mathfrak{h})$ and
$\mathfrak{I}(\mathfrak{h})^{\mathcal{W}_\mathrm{G}}$ denotes the Weyl-invariant polynomials on
$\mathfrak{h}$. The \emph{Chevalley-Restriction Theorem} now states
\begin{equation}
\mathfrak{I}(\gfrak)^\mathrm{G} \cong \mathfrak{I}(\mathfrak{h})^{\mathcal{W}_\mathrm{G}} \; ,
\end{equation}
where the isomorphism is given by the restriction map $p \mapsto p |_{\mathfrak{h}}$
for $p \in \mathfrak{I}(\gfrak)^\mathrm{G}$.
Therefore, the study of $\mathrm{H}_m$-invariant polynomials on $\mathfrak{h}_m$ is reduced
to $\mathcal{W}_{\mathrm{H}_m}$-invariant polynomials on a Cartan subalgebra $\mathfrak{t}_m
\subset \mathfrak{h}_m$.
\paragraph{Finite reflection groups}
It is due to a theorem by Chevalley~\cite{Chevalley:1955}, in the context of
\emph{finite reflection groups}, that there exist $l$ algebraically independent
homogeneous elements $p_1,\ldots, p_l$ of positive degrees $d_i$, for
$i=1,\ldots,l$, such that
\begin{equation}
\mathfrak{I}(\mathfrak{h})^{\mathcal{W}_\mathrm{G}} = \mathbb C\left[p_1,\ldots, p_l\right] \; .
\end{equation}
In addition, the degrees $d_i$ satisfy
\begin{align}
\left| \mathcal{W}_\mathrm{G} \right| = \prod_{i=1}^l d_i \quad\textrm{and}\quad \sum_{i=1}^d (d_i -1) =
\text{number of reflections in $\mathcal{W}_\mathrm{G}$} \;.
\end{align}
The degrees $d_i$ are unique~\cite{Humphreys:1990} and tabulated for all Weyl
groups, see for instance~\cite[Sec. 3.7]{Humphreys:1990}. However, the
generators $p_i$ are themselves not uniquely determined.
\paragraph{Poincar\'{e} or Molien series}
On the one hand, the Poincar\'{e} series for the
$\mathfrak{I}(\mathfrak{h})^{\mathcal{W}_\mathrm{G}}$ is simply given by
\begin{equation}
P_{\mathfrak{I}(\mathfrak{h})^{\mathcal{W}_\mathrm{G}}}(t)= \prod_{i=1}^l \frac{1}{1-t^{d_i}} \; .
\end{equation}
On the other hand, since $\mathfrak{h}$ is a $l$-dimensional complex vector space and
$\mathcal{W}_\mathrm{G}$ a finite group, the generating function for the invariant
polynomials is known as Molien series~\cite{Molien:1897}
\begin{equation}
P_{\mathfrak{I}(\mathfrak{h})^{\mathcal{W}_\mathrm{G}}}(t)= \frac{1}{\left| \mathcal{W}_\mathrm{G} \right|}
\sum_{g \in \mathcal{W}_\mathrm{G} } \frac{1}{\mathrm{det}\left(\mathds{1}-t \ g\right)} \; .
\end{equation}
Therefore, the dressing factors $P_{\mathrm{G}}(t,m)$ in the Hilbert
series~\eqref{eqn:HS_refined} for the Coulomb branch are the Poincar\'{e}
series for graded algebra of $\mathrm{H}_m$-invariant polynomials on $\mathfrak{h}_m$.
\paragraph{Harish-Chandra isomorphism}
In~\cite{Cremonesi:2013lqa}, the construction of the $P_\mathrm{G}(t,m)$ is based on
Casimir invariants of $\mathrm{G}$ and $\mathrm{H}_m$; hence, we need to make contact with that
idea. Casimir invariants live in the centre $\mathcal{Z}(\mathfrak{U}(\gfrak))$ of the
universal enveloping algebra $\mathfrak{U}(\gfrak)$ of $\gfrak$. Fortunately, the
Harish-Candra isomorphism~\cite{Harish:Chandra} provides us with
\begin{equation}
\mathcal{Z}(\mathfrak{U}(\gfrak)) \cong \mathfrak{I}(\mathfrak{h})^{\mathcal{W}_\mathrm{G}} \; .
\end{equation}
Consequently, $\mathcal{Z}(\mathfrak{U}(\gfrak))$ is a polynomial algebra with $l$
algebraically independent homogeneous elements that have the same positive
degrees $d_i$ as the generators of $\mathfrak{I}(\mathfrak{h})^{\mathcal{W}_\mathrm{G}}$. It is
known that for semi-simple groups $\mathrm{G}$ these generators can be chosen to be the
$\mathrm{rk}(\mathrm{G})$ Casimir invariants; i.e.\ the space of Casimir-invariants
is freely generated by $l$ generators (together with the unity).
\paragraph{Conclusions}
So far, $\mathrm{G}$ (and $\mathrm{H}_m$) had been restricted to be semi-simple. However, in
most cases $\mathrm{H}_m$ is a direct product group of semi-simple Lie groups and
${{\rm U}(1)}$-factors. We proceed in two steps: firstly, ${{\rm U}(1)}$ acts trivially on its
Lie-algebra $\cong \mathbb R$, thus all polynomials are invariant and we obtain
\begin{equation}
\mathfrak{I}(\mathbb R)^{{{\rm U}(1)}} = \mathbb R[x] \quad\textrm{and}\quad P_{{{\rm U}(1)}}(t) =\frac{1}{1-t} \; .
\end{equation}
Secondly, each factor $\mathrm{G}_i$ of a direct product $\mathrm{G}_1 \times \cdots
\times \mathrm{G}_M$ acts via the adjoint representation on on its own Lie algebra
$\gfrak_i$ and trivially on all other $\gfrak_j$ for $j\neq i$. Hence, the space
of $\mathrm{G}_1 \times \cdots \times \mathrm{G}_M$-invariant polynomials on $\gfrak_1 \oplus
\cdots \oplus \gfrak_M$ factorises into the product of the
$\mathfrak{I}(\gfrak_i)^{\mathrm{G}_i}$ such that
\begin{equation}
\mathfrak{I}(\oplus_i \gfrak_i)^{\prod_i \mathrm{G}_i} = \prod_i
\mathfrak{I}(\gfrak_i)^{\mathrm{G}_i} \quad\textrm{and}\quad P_{\mathfrak{I}(\oplus_i \gfrak_i)^{\prod_i
\mathrm{G}_i} } (t) = \prod_i P_{\mathfrak{I}(\gfrak_i)^{\mathrm{G}_i} } (t) \; .
\end{equation}
For abelian groups $\mathrm{G}$, the Hilbert series for the Coulomb branch factorises
in the Poincar\'{e} series $\mathrm{G}$-invariant polynomials on $\gfrak$ times the
contribution of the (bare) monopole operators. In contrast, the Hilbert series
does not factorise for non-abelian groups $\mathrm{G}$ as the stabiliser $\mathrm{H}_m\subset
\mathrm{G}$ depends on $m$.
\subsection{Consequences for unrefined Hilbert series}
\label{subsec:summation_HS_unrefined}
The aforementioned dissection of the Weyl chamber $\sigma_{B^*}^{\widehat{\G}}$ into a
fan, induced by the conformal dimension $\Delta$, and the subsequent collection
of semi-groups in $\Lambda_w(\widehat{\G}) \slash \mathcal{W}_{\widehat{\G}} $ provides an
immediate consequence for the unrefined Hilbert series.
\begin{figure}[h]
\begin{subfigure}{0.485\textwidth}
\centering
\begin{tikzpicture}
\draw (0,0) node[circle,inner sep=0.8pt,fill,black] {};
\draw[black] (0,0) -- (1,3);
\draw[black] (0,0) -- (1.1*1.8,1.1*2.5);
\draw[black] (0,0) -- (3.2,2);
\draw[black] (0,0) -- (0.9*4,0.9*1.4);
\draw (3,0.8) node[circle,inner sep=0.4pt,fill,black] {};
\draw (3,0.6) node[circle,inner sep=0.4pt,fill,black] {};
\draw (3,0.4) node[circle,inner sep=0.4pt,fill,black] {};
\draw[black] (0,0) -- (3.5,0.2);
\draw[black] (0,0) -- (3.5,-1.5);
\draw[black] (0,0) -- (2.5,-2.5);
\draw (1.25,2.3) node {$C_1^{(2)}$};
\draw (0.9*2.3,0.9*2) node {$C_2^{(2)}$};
\draw (0.9*3,0.9*1.4) node {$C_3^{(2)}$};
\draw (0.9*3,-0.4*0.9) node {$C_{L-1}^{(2)}$};
\draw (0.85*2.4,-1.8*0.8) node {$C_L^{(2)}$};
\draw (1.25,3.3) node {$C_0^{(1)}$};
\draw (2.25,3.0) node {$C_1^{(1)}$};
\draw (3.3,2.3) node {$C_2^{(1)}$};
\draw (4.1,1.3) node {$C_3^{(1)}$};
\draw (4.1,0.2) node {$C_{L-2}^{(1)}$};
\draw (3.9,-1.6) node {$C_{L-1}^{(1)}$};
\draw (2.9,-2.6) node {$C_L^{(1)}$};
\draw (-0.5,0) node {$C^{(0)}$};
\end{tikzpicture}
\caption{}
\label{Fig:Rep_Fan}
\end{subfigure}
\begin{subfigure}{0.485\textwidth}
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (4.5,0);
\coordinate (YAxisMin) at (0,-3.5);
\coordinate (YAxisMax) at (0,2.5);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (4.7,-0.2) node {$m_1$};
\draw (-0.3,2.3) node {$m_2$};
\foreach \x in {0,1,...,8}{%
\foreach \y in {-6,-5,...,4}{%
\node[draw,circle,inner sep=0.5pt,fill,black] at (0.5*\x,0.5*\y) {};
}
}
\draw[black,dashed,thick] (0,0) -- (4,2) ;
\draw[black,dashed,thick] (0,0) -- (1.5,-3) ;
\filldraw[dotted,fill=red, fill opacity=0.2, draw=gray] (Origin) -- (1,1/2)
-- (3/2,-1/2) -- (1/2,-1) -- cycle ;
\draw[black,dashed] (1/2,-1) -- (8/2,2/2-1/4) ;
\draw[black,dashed] (1,1/2) -- (5.5/2,-6/2) ;
\draw[black,dashed] (1,-2) -- (4,-1/2) ;
\draw[black,dashed] (2,1) -- (4,-3) ;
\draw[black,dashed] (3/2,-3) -- (4,-7/4) ;
\draw[black,dashed] (3,3/2) -- (4,-1/2) ;
\draw[black,thick] (1,1/2) circle (4pt);
\draw[black,thick] (1/2,-1) circle (4pt);
\draw[blue,thick] (1/2,0) circle (4pt);
\draw[blue,thick] (1/2,-1/2) circle (4pt);
\end{tikzpicture}
\caption{}
\label{Fig:Rep_Cone}
\end{subfigure}
\caption{A representative fan, which is spanned by the
$2$-dim.\ cones $C_{p}^{(2)}$ for $p=1,\ldots,L$, is displayed
in~\ref{Fig:Rep_Fan}.
In addition, \ref{Fig:Rep_Cone} contains a $2$-dim.\ cone with a Hilbert basis
of the two ray generators (black) and two additional minimal generators (blue).
The ray generators span the fundamental parallelotope (red region).}
\label{Fig:Sketch}
\end{figure}
For simplicity, we illustrate the consequences for a rank two example. Assume
that the Weyl chamber is divided into a fan generated the $2$-dimensional cones
$C_p^{(2)}$ for $p=1,\ldots,L$, as sketched in
Fig.~\ref{Fig:Rep_Cone}. For each cone, one has two $1$-dimensional cones
$C_{p-1}^{(1)}$, $C_{p}^{(1)}$ and the trivial cone $C^{(0)}=\{0\}$ as boundary,
i.e.\ $\partial C_{p}^{(2)}= C_{p-1}^{(1)} \cup C_{p}^{(1)}$, where
$C_{p-1}^{(1)} \cap C_{p}^{(1)} = C^{(0)}$.
The Hilbert basis $\Hcal(S_p^{(2)})$ for $S_p^{(2)} \coloneqq C_p^{(2)} \cap
\Lambda_w^{\widehat{\G}}$ contains the ray generators $\{x_{p-1},x_p\}$, such that
$\Hcal(S_p^{(1)})= \{x_p\}$, and potentially other minimal generators
$u_{\kappa}^{p}$ for $\kappa$ in some \emph{finite} index set.
Although any element $s\in S_p^{(2)}$ can be generated by $\{x_{p-1},x_p,
\{u_{\kappa}^{p} \}_\kappa\}$, the representation $s= a_0 x_{p-1} + a_1 x_{p} +
\sum_\kappa b_\kappa u_{\kappa}^{p}$ is \emph{not} unique. Therefore, great
care needs to be taken if one would like to sum over all elements in
$S_p^{(2)}$. A possible realisation employs the \emph{fundamental
parallelotope}
\begin{equation}
\mathcal{P}(C_p^{(2)})\coloneqq \{ a_0 x_{p-1} + a_1 x_{p} \ | \ 0 \geq
a_0,a_1 \geq 1\} \; ,
\end{equation}
see also Fig.~\ref{Fig:Rep_Cone}. The number of points contained in
$\mathcal{P}(C_p^{(2)})$ is computed by the discriminant
\begin{equation}
d(C_p^{(2)}) \coloneqq |\mathrm{det}(x_{p-1},x_p)| \; .
\end{equation}
However, as known from solid state physics, the discriminant counts each of the
four boundary lattice points by $\tfrac{1}{4}$; thus, there are $d(C_p^{(2)})
-1$ points in the interior. Remarkably, each point $s \in \mathrm{Int}(
\mathcal{P}(C_p^{(2)}))$ is given by positive integer combinations of the
$\{u_{\kappa}^{p} \}_\kappa$ alone. A translation of $ \mathcal{P}(C_p^{(2)}) $
by non-negative integer combinations of the ray-generators $\{x_{p-1},x_p\}$
fills the entire semi-group $S_p^{(2)}$ and each point is only realised once.
Now, we employ this fact to evaluate the un-refined Hilbert series explicitly.
\begin{align}
\mathrm{HS}_{\mathrm{G}}(t) &= \sum_{m \in \Lambda_w(\widehat{\G}) \slash \mathcal{W}_{\widehat{\G}}}
t^{\Delta(m)} P_{\mathrm{G}}(t,m) \notag\\
&= P_{\mathrm{G}}(t,0)
+ \sum_{p=0}^{L} P_{\mathrm{G}}(t,x_p) \sum_{n_p>0} t^{n_p \Delta(x_p)}
+\sum_{p=1}^{L} \sum_{n_{p-1},n_p > 0} P_{\mathrm{G}}(t,x_{p-1}+x_p) t^{\Delta(n_{p-1}
x_{p-1} + n_{p} x_{p})} \notag \\
&\phantom{= P_{\mathrm{G}}(t,0) \ ; }
+\sum_{p=1}^{L} \sum_{s \in \mathrm{Int}(
\mathcal{P}(C_p^{(2)}))} \sum_{n_{p-1},n_p \geq0} P_{\mathrm{G}}(t,s) t^{\Delta(s +
n_{p-1} x_{p-1} + n_{p} x_{p})} \notag \\
&= P_{\mathrm{G}}(t,0) + \sum_{p=0}^{L} P_{\mathrm{G}}(t,x_p) \frac{t^{ \Delta(x_p)}}{1-
t^{\Delta(x_p)}} + \sum_{p=1}^{L} \frac{ P_{\mathrm{G}}(t,x_{p-1}+x_p) \
t^{\Delta(x_{p-1})
+\Delta(x_p)}}{\left(1-t^{\Delta(x_{p-1})}\right)
\left(1-t^{\Delta(x_{p})}\right)} \notag \\
&\phantom{= P_{\mathrm{G}}(t,0) \ ; }
+ \sum_{p=1}^{L} \sum_{s \in \mathrm{Int}(
\mathcal{P}(C_p^{(2)}))} \frac{ P_{\mathrm{G}}(t,s) \
t^{\Delta(s)}}{\left(1-t^{\Delta(x_{p-1})}\right)
\left(1-t^{\Delta(x_{p})}\right)} \notag \\
&= \frac{P_{\mathrm{G}}(t,0) }{\prod_{p=0}^{L} \left(1-t^{\Delta(x_{p})}\right)}
\Bigg\{ \prod_{q=0}^{L} \left(1-t^{\Delta(x_{q})}\right)
+ \sum_{q=0}^{L} \frac{P_{\mathrm{G}}(t,x_q)}{P_{\mathrm{G}}(t,0)} t^{\Delta(x_q)}
\prod_{r=0\atop r\neq q}^{L} \left(1-t^{\Delta(x_{r})}\right)
\label{eqn:HS_generic_solved}\\
&\qquad \qquad \qquad
+\sum_{q=1}^{L} \frac{P_{\mathrm{G}}(t,C_q^{(2)})}{P_{\mathrm{G}}(t,0)}
\bigg[t^{\Delta(x_{q-1}) +\Delta(x_q)}+ \sum_{s \in
\mathrm{Int}( \mathcal{P}(C_q^{(2)}))} t^{\Delta(s)}
\bigg]
\prod_{r=0\atop{r\neq q-1,q}}^L \left(1-t^{\Delta(x_{r})}\right)
\Bigg\} \notag \; .
\end{align}
Next, we utilise that the classical dressing factors, for rank two examples,
only have three different values: in the ($2$-dim.) interior of the Weyl
chamber $W$, the residual gauge group is the maximal torus $\mathrm{T}$ and
$P_{\mathrm{G}}(t,\mathrm{Int}W) \equiv P_2(t)= \prod_{i=1}^2 \tfrac{1}{(1-t)}$. Along
the $1$-dimensional boundaries, the residual gauge group is a non-abelian
subgroup $\mathrm{H}$ such that $\mathrm{T} \subset \mathrm{H} \subset \mathrm{G}$ and the $P_{\mathrm{G}}(t,\partial
W \setminus \{0\}) \equiv
P_1(t) = \prod_{i=1}^2 \tfrac{1}{(1-t^{b_i})} $, for the two degree $b_i$
Casimir invariants of $\mathrm{H}$. At the ($0$-dim.) boundary of the boundary, the
group is
unbroken and $P_{\mathrm{G}}(t, 0) \equiv P_0(t) = \prod_{i=1}^2 \tfrac{1}{(1-t^{d_i})}
$ contains the Casimir invariants of $\mathrm{G}$ of degree $d_i$.
Thus, there are a few observations to be addressed.
\begin{enumerate}
\item The numerator of~\eqref{eqn:HS_generic_solved}, which is everything in
the curly brackets $\{ \ldots \}$, starts with a one and is a polynomial with
integer coefficients, which is required for consistency.
\item The denominator of~\eqref{eqn:HS_generic_solved} is
given by $P_{\mathrm{G}}(t,0)\slash \prod_{p=0}^{L} (1-t^{\Delta(x_{p})})$ and
describes the poles due to the Casimir invariants of $\mathrm{G}$ and the bare monopole
$(x_p,\Delta(x_p))$ which originate from ray generators $x_p$.
\item The numerator has contributions $\sim t^{\Delta(x_p)}$ for the ray
generators with pre-factors $\tfrac{P_1(t)}{P_0(t)} -1$ for the two outermost
rays $p=0$, $p=L$ and pre-factors $\tfrac{P_2(t)}{P_0(t)} -1$ for the remaining
ray generators. None of the two pre-factors has a constant term as
$P_i(t\to0)=1$ for each $i=0,1,2$. Also $\mathrm{deg}(1\slash P_0(t))\geq \mathrm{deg}(1\slash
P_1(t))\geq \mathrm{deg}(1\slash P_2(t)) =2$ and
\begin{equation}
\frac{P_2(t)}{P_0(t)} = \frac{(1-t^{d_1})(1-t^{d_2})}{(1-t)(1-t)} =
\sum_{i=0}^{d_1 -1} \sum_{j=0}^{d_2-1}t^{i+j}
\end{equation}
is a polynomial for any rank two group. For the examples considered here, we
also obtain
\begin{equation}
\frac{P_1(t)}{P_0(t)} = \frac{(1-t^{d_1})(1-t^{d_2})}{(1-t^{b_1})(1-t^{b_2})}
= \frac{(1-t^{k_1 b_1})(1-t^{k_2 b_2})}{(1-t^{b_1})(1-t^{b_2})}
=
\sum_{i=0}^{b_1 -1} \sum_{j=0}^{b_2-1} t^{i \cdot k_1 + j \cdot k_2}
\end{equation}
for some $k_1,k_2 \in \NN$.
In summary, $(\tfrac{P_{\mathrm{G}}(t,x_p)}{P_{\mathrm{G}}(t,0)} -1)t^{\Delta(x_p)}$ describes
the dressed monopole operators corresponding to the ray generators $x_p$.
\item The finite sums $\sum_{s \in
\mathrm{Int}( \mathcal{P}(C_p^{(2)}))} t^{\Delta(s)}$ are entirely determined
by the conformal dimensions of the minimal generators $u_\kappa^p$.
\item The first contributions for the minimal generators $u_\kappa^p$ are
of the form
\begin{equation}
\tfrac{P_{2}(t)}{P_{0}(t)} t^{\Delta(u_\kappa^p)} = \sum_{i=0}^{d_1 -1}
\sum_{j=0}^{d_2-1}t^{i+j+ \Delta(u_\kappa^p)} \; ,
\end{equation}
which then comprise the bare and the dressed monopole operators simultaneously.
\item If $C_p^{(2)}$ is simplicial, i.e.\ $\Hcal(S_p^{(2)})=
\{x_{p-1},x_p\}$,
then the sum over $s \in \mathrm{Int}( \mathcal{P}(C_p^{(2)})) $
in~\eqref{eqn:HS_generic_solved} is zero, as the interior is empty. Also
indicated by $d(C_p^{(2)})=1$.
\end{enumerate}
In conclusion, the Hilbert series~\eqref{eqn:HS_generic_solved} suggests that
\emph{ray generators} are to be expected in the denominator, while other
minimal generators are manifest in the numerator. Moreover, the entire Hilbert
series is determined by a \emph{finite} set of numbers: the conformal
dimensions of the minimal generators $\{\Delta(x_p) \ | \ p=0,1,\ldots, L\} $
and $\{ \{ \Delta(u_\kappa^{(p)}) \ | \ \kappa=1,\ldots, d(C_p^{(2)})-1 \} \ |
\ p=1,\ldots,L \}$ as well as the classical dressing factors.
Moreover, the dressing behaviour, i.e.\ number and degree, of a minimal
generator $m$ is described by the quotient $P_{\mathrm{G}}(t,m) \slash P_{\mathrm{G}}(t,0)$.
Consolidating evidence for this statement comes from the analysis of the
plethystic logarithm, which we present in App.~\ref{app:PL}.
Together, the Hilbert series and the plethystic logarithm allow a better
understanding of the chiral ring.
We illustrate the formula~\eqref{eqn:HS_generic_solved} for the two simplest
cases in order to hint on the differences that arise if $d(C_p^{(2)})>1$ for
cones within the fan.
\paragraph{Example: one simplicial cone} Adapting the
result~\eqref{eqn:HS_generic_solved} to one cone $C_1^{(2)}$ with cone /
Hilbert basis $\{x_0,x_1\}$, we find
\begin{align}
\label{eqn:Example_HS_simplicial}
\mathrm{HS}= \frac{1+ \left( \frac{P_1(t)}{P_0(t)}-1\right)
\left(t^{\Delta(x_0)}+t^{\Delta(x_1)} \right) + \left(1-2\frac{P_1(t)}{P_0(t)}
+\frac{P_2(t)}{P_0(t)} \right) t^{\Delta(x_0)+\Delta(x_1)}}{\prod_{i=1}^2
\left( 1-t^{d_i} \right) \prod_{p=0}^{1} \left( 1-t^{\Delta(x_p)} \right)} \; .
\end{align}
Examples treated in this paper are as follows: firstly, the
representation $[2,0]$ for the quotients ${\rm Spin}(4)$, ${{\rm SO}}(3)\times {{\rm SU}(2)}$, ${{\rm SU}(2)}
\times {{\rm SO}}(3)$, $\mathrm{PSO}(4)$ of Sec.~\ref{subsec:A1xA1_Rep20}; secondly,
${\rm USp}(4)$ for the case $N_3=0$ of Sec.~\ref{subsec:USp4_N3=0}; thirdly, ${\rm G}_2$ in
the representations $[1,0]$, $[0,1]$ and $[2,0]$ of Sec.~\ref{subsec:G2_Cat1}.
The corresponding expression for the plethystic logarithm is provided
in~\eqref{eqn:Example_PL_simplicial}.
\paragraph{Example: one non-simplicial cone} Adapting the
result~\eqref{eqn:HS_generic_solved} to one cone $C_1^{(2)}$ with Hilbert basis
$\{x_0,x_1,\{u_\kappa \}\}$, fundamental parallelotope $\mathcal{P}$, and
discriminant $d>1$, we find
\begin{align}
\label{eqn:Example_HS_non-simplicial}
\mathrm{HS}= \frac{1+ \left( \frac{P_1(t)}{P_0(t)}-1\right)
\left(t^{\Delta(x_0)}+t^{\Delta(x_1)} \right) + \left(1-2\frac{P_1(t)}{P_0(t)}
+\frac{P_2(t)}{P_0(t)} \right)
t^{\Delta(x_0)+\Delta(x_1)} + \frac{P_2(t)}{P_0(t)} \sum_{s \in
\mathrm{Int}(\mathcal{P})} t^{\Delta(s)}
}{\prod_{i=1}^2 \left( 1-t^{d_i} \right)
\prod_{p=0}^{1} \left( 1-t^{\Delta(x_p)} \right)} \; .
\end{align}
An example for this case is ${{\rm SO}}(4)$ with representation $[2,0]$ treated
in Sec.~\ref{subsec:A1xA1_Rep20}. For the plethystic logarithm we refer
to~\eqref{eqn:Example_PL_non-simplicial}.
The difference between~\eqref{eqn:Example_HS_simplicial}
and~\eqref{eqn:Example_HS_non-simplicial} lies in the finite sum added in the
numerator which accounts for the minimal generators that are not ray generators.
\subsection{Consequences for refined Hilbert series}
\label{subsec:summation_HS_refined}
If the centre $\mathcal{Z}(\widehat{\G})$ of the GNO-dual group $\widehat{\G}$ is a non-trivial
Lie-group of rank $\mathrm{rk}(\mathcal{Z}(\widehat{\G}))=\rho$, one introduces additional
fugacities $\vec{z}\equiv (z_i)$ for $i=1,\ldots,\rho$ such that the Hilbert
series counts operators according to ${{\rm SU}(2)}_R$-spin $\Delta(m)$ and
topological charges $\vec{J}(m)\equiv(J_i(m))$ for
$i=1,\ldots,\rho$. Let us introduce the notation
\begin{equation}
\vec{z}^{\vec{J}(m)} \coloneqq \prod_{i=1}^{\rho} z_i^{J_i(m)} \quad\textrm{such that}\quad
\vec{z}^{\vec{J}(m_1+m_2)} =
\vec{z}^{\vec{J}(m_1)+\vec{J}(m_2)} = \vec{z}^{\vec{J}(m_1)}
\cdot \vec{z}^{\vec{J}(m_2)} \; ,
\end{equation}
where we \emph{assumed} each component $J_i(m)$ to be a linear function
in $m$.
By the very same arguments as in~\eqref{eqn:HS_generic_solved}, one can
evaluate the refined Hilbert series explicitly and obtains
\begin{align}
\mathrm{HS}_{\mathrm{G}}(t,\vec{z}) &= \sum_{m \in \Lambda_w^{\widehat{\G}} \slash
\mathcal{W}_{\widehat{\G}}} \vec{z}^{\vec{J}(m)}
t^{\Delta(m)} P_{\mathrm{G}}(t,m) \notag\\
&= \frac{P_{\mathrm{G}}(t,0) }{\prod_{p=0}^{L} \left(1-\vec{z}^{\vec{J}(x_p)}
t^{\Delta(x_{p})}\right)}
\Bigg\{
\prod_{q=0}^{L} \left( 1-\vec{z}^{\vec{J}(x_q)}t^{\Delta(x_{q})} \right)
\label{eqn:HS_generic_refined}\\
&\phantom{= \frac{P_{\mathrm{G}}(t,0) }{\prod_{p=0}^{L} \left(1-\vec{z}^{\vec{J}(x_p)}
t^{\Delta(x_{p})}\right)}}
+ \sum_{q=0}^{L} \frac{P_{\mathrm{G}}(t,x_q)}{P_{\mathrm{G}}(t,0)}
\vec{z}^{\vec{J}(x_q)} t^{\Delta(x_q)}
\prod_{r=0\atop r\neq q}^{L}
\left( 1-\vec{z}^{\vec{J}(x_r)} t^{\Delta(x_{r})} \right)
\notag \\
&\phantom{= \frac{P_{\mathrm{G}}(t,0) }{\prod_{p=0}^{L} \left(1-\vec{z}^{\vec{J}(x_p)}
t^{\Delta(x_{p})}\right)}}
+\sum_{q=1}^{L} \frac{P_{\mathrm{G}}(t,C_q^{(2)})}{P_{\mathrm{G}}(t,0)}
\bigg[\vec{z}^{\vec{J}(x_{q-1})+\vec{J}(x_q)} t^{\Delta(x_{q-1}) +\Delta(x_q)}
\notag\\*
&\phantom{= \frac{P_{\mathrm{G}}(t,0) }{\prod_{p=0}^{L} \left(1-\vec{z}^{\vec{J}(x_p)}
t^{\Delta(x_{p})}\right)}}
\qquad \qquad
+ \sum_{s \in \mathrm{Int}( \mathcal{P}(C_q^{(2)}))}
\vec{z}^{\vec{J}(s)} t^{\Delta(s)} \bigg] \prod_{r=0\atop{r\neq q-1,q}}^L
\left(1- \vec{z}^{\vec{J}(x_r)} t^{\Delta(x_{r})}\right)
\Bigg\} \notag \; .
\end{align}
The interpretation of the refined Hilbert
series~\eqref{eqn:HS_generic_refined} remains the same as before: the minimal
generators, i.e.\ their GNO-charge, ${{\rm SU}(2)}_R$-spin, topological charges
$\vec{J}$, and their dressing factors, completely determine the Hilbert series.
In principle, this data makes the (sometimes cumbersome) explicit
summation of~\eqref{eqn:HS_refined} obsolete.
\section{Case: \texorpdfstring{$\boldsymbol{A_1 \times A_1}$}{A1xA1}}
\label{sec:A1xA1}
This section concerns all Lie groups with Lie algebra $D_2$, which allows to
study products of the rank one gauge groups ${{\rm SO}}(3)$ and ${{\rm SU}(2)}$, but
also the proper rank two group ${{\rm SO}}(4)$.
\subsection{Set-up}
Let us consider the Lie algebra $D_2 \cong A_1\times A_1$.
Following~\cite{Goddard:1976qe}, there are five different groups with this
Lie algebra. The reason is that the universal covering
group $\widetilde{{{\rm SO}}}(4)$ of ${{\rm SO}}(4)$ has a
non-trivial centre $\mathcal{Z}(\widetilde{{{\rm SO}}}(4)) =\mathbb Z_2\times\mathbb Z_2$ of order
4. The quotient of $\widetilde{{{\rm SO}}}(4)$ by any of the five different
subgroups $\mathcal{Z}(\widetilde{{{\rm SO}}}(4))$ yields a Lie group with the same
Lie algebra. Fortunately, working with ${{\rm SO}}(4)$ allows to use the isomorphism
$\widetilde{{{\rm SO}}}(4) = {\rm Spin}(4)\cong {{\rm SU}(2)}\times {{\rm SU}(2)}$. We can
summarise the setting as displayed in Tab.~\ref{tab:A1xA1_quotients}.
\begin{table}[h]
\centering
\begin{doublespacing}
\begin{tabular}{c|c||c|c|c}
\toprule
Quotient & isomorphic group $\mathrm{G}$ & GNO-dual $\widehat{\G}$ &
$\mathcal{Z}(\widehat{\G}) $ & GNO-charges $(m_1,m_2)$ \\ \midrule
$\frac{\widetilde{{{\rm SO}}}(4)}{\{1\}}$ & ${{\rm SU}(2)}\times {{\rm SU}(2)}$ &
${{\rm SO}}(3)\times{{\rm SO}}(3)$ & $\{1\}$ & $ K^{[0]}$ \\
$\frac{\widetilde{{{\rm SO}}}(4)}{\mathbb Z_2\times\{1\}} $ & ${{\rm SO}}(3) \times {{\rm SU}(2)}$ & ${{\rm SU}(2)}
\times {{\rm SO}}(3)$ & $\mathbb Z_2 \times \{1\} $ & $ K^{[0]}\cup K^{[1]}$
\\
$\frac{\widetilde{{{\rm SO}}}(4)}{\mathrm{diag}(\mathbb Z_2)} $ & ${{\rm SO}}(4) $ & ${{\rm SO}}(4)$ &
$\mathbb Z_2$ & $ K^{[0]}\cup K^{[2]}$ \\
$\frac{\widetilde{{{\rm SO}}}(4)}{\{1\} \times \mathbb Z_2} $ & $ {{\rm SU}(2)}
\times {{\rm SO}}(3)$ & ${{\rm SO}}(3)\times {{\rm SU}(2)} $ & $\{1\} \times \mathbb Z_2 $ & $ K^{[0]}\cup
K^{[3]}$ \\
$\frac{\widetilde{{{\rm SO}}}(4)}{\mathbb Z_2 \times \mathbb Z_2} $ & $ {{\rm SO}}(3)
\times {{\rm SO}}(3)$ & ${{\rm SU}(2)} \times {{\rm SU}(2)} $ & $\mathbb Z_2 \times \mathbb Z_2 $ & $ K^{[0]}\cup
K^{[1]}\cup K^{[2]}\cup K^{[3]}$ \\
\bottomrule
\end{tabular}
\end{doublespacing}
\caption{All the Lie groups that arise taking the quotient of
$\widetilde{{{\rm SO}}}(4)$ by a subgroup of its centre; hence, their Lie algebra is
$D_2$. }
\label{tab:A1xA1_quotients}
\end{table}
Here, we employed $\widehat{{{\rm SU}(2)}}={{\rm SO}}(3)$ and that for semi-simple
groups $\mathrm{G}_1$, $\mathrm{G}_2$
\begin{equation}
\widehat{\mathrm{G}_1\times \mathrm{G}_2}=\widehat{\G}_1 \times \widehat{\G}_2
\end{equation}
holds~\cite{Goddard:1976qe}. Moreover, the GNO-charges are defined via the
following sublattices of the weight lattice of ${\rm Spin}(4)$ (see also
Fig.~\ref{fig:A1xA1_sublattices})
\begin{subequations}
\begin{align}
K^{[0]} &= \Big\{(m_1,m_2) \; | \; m_i = p_i \in \mathbb Z \, , \; p_1+p_2
=\text{even} \Big\} \; ,\\
K^{[1]} &= \Big\{(m_1,m_2) \; | \; m_i = p_i + \tfrac{1}{2} \, , \; p_i \in \mathbb Z
\, , \; p_1+p_2 =\text{even} \Big\} \; , \\
K^{[2]} &= \Big\{(m_1,m_2) \; | \; m_i = p_i \in \mathbb Z \, , \; p_1+p_2
=\text{odd} \Big\} \; , \\
K^{[3]} &= \Big\{(m_1,m_2) \; | \; m_i = p_i + \tfrac{1}{2} \, , \; p_i \in \mathbb Z
\, , \; p_1+p_2 =\text{odd} \Big\} \; .
\end{align}
\end{subequations}
\begin{figure}
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
%
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon,regular polygon sides=3,black,inner
sep=0.7pt] at (2*\x-1,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,regular polygon,regular polygon sides=3,black,inner
sep=0.7pt] at (2*\x,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,black,inner sep=0.8pt] at (2*\x +1/2,2*\y +1/2)
{};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,black,inner sep=0.8pt] at (2*\x-1
+1/2,2*\y-1+1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,cross out,thick,black,inner
sep=0.8pt] at (2*\x-1+1/2,2*\y+1/2) {};
%
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,cross out,thick,black,inner
sep=0.8pt] at (2*\x +1/2,2*\y-1 +1/2) {};
%
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\filldraw[dotted,fill=yellow, fill opacity=0.3, draw=gray] (Origin) --
(3.2,3.2) -- (4.6,3.2) -- (4.6,-3.2) -- (3.2,-3.2) -- cycle ;
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw (6,1) node[draw,diamond,inner sep=0.8pt,black] {};
\draw (7.2,1.05) node { $K^{[1]}$ lattice};
\draw (6,0) node[draw,regular polygon,regular polygon sides=3,inner
sep=0.8pt,black] {};
\draw (7.2,0.05) node { $K^{[2]}$ lattice};
\draw (6,-1 ) node[draw,cross out,thick,inner sep=0.8pt,black] {};
\draw (7.2,-0.95) node { $K^{[3]}$ lattice};
\draw (6,-2) node[rectangle,inner sep=8pt,fill,yellow,fill opacity=0.35]
{};
\draw (9,-2) node { Weyl chamber $m_1\geq |m_2| $};
\end{tikzpicture}%
\caption{The four different sublattices of the covering group of ${{\rm SO}}(4)$.
One recognises the root lattice $\Lambda_r^{\widetilde{{{\rm SO}}}(4)}=K^{[0]}$ and
the weight lattice $ \Lambda_w^{\widetilde{{{\rm SO}}}(4)} =K^{[0]}\cup K^{[1]}\cup
K^{[2]}\cup K^{[3]}$.}%
\label{fig:A1xA1_sublattices}%
\end{figure}%
The important consequence of this set-up is that the fan defined by the
conformal dimension will be the same for a given representation in each of the
five quotients, but the semi-groups will differ due to the different lattices
$\Lambda_w(\widehat{\G})$ used in the intersection. Hence, we will find different
Hilbert basis in each quotient group. Nevertheless, we are forced to consider
representations on the root lattice as we otherwise cannot compare all
quotients.
\paragraph{Dressings}
In addition, we have chosen to parametrise the principal Weyl chamber via
$m_1\geq |m_2|$ such that the classical dressing factors are given
by~\cite{Cremonesi:2013lqa}
\begin{equation}
P_{A_1\times A_1}(t,m_1,m_2)= \begin{cases} \frac{1}{(1-t^2)^2} \; ,& \qquad\textrm{for}\quad
m_1=m_2=0 \; , \\
\frac{1}{(1-t)(1-t^2)}\; , & \qquad\textrm{for}\quad m_1=|m_2|>0 \; , \\
\frac{1}{(1-t)^2} \; , & \qquad\textrm{for}\quad m_1>|m_2| \geq
0
\; .
\end{cases}
\end{equation}
Regardless of the quotient $\widetilde{{{\rm SO}}(4)} \slash \Gamma$, the space of
Casimir invariance is $2$-dimensional. We choose a basis such that the
two degree $2$ Casimir invariants stem either from ${{\rm SU}(2)}$ or ${{\rm SO}}(3)$,
i.e.~\footnote{In a different basis, the Casimir invariants for ${{\rm SO}}(4)$ are
the quadratic Casimir and the Pfaffian.}
\begin{equation}
\mathrm{diag} (\Phi) = (\phi_1,\phi_2) \quad \longrightarrow \quad \Casi{2}^{(i)}=
(\phi_i)^2 \; .
\end{equation}
Next, we can clarify all relevant bare and dressed monopole
operators for an $(m_1,m_2)$ that is a minimal generator. There are two cases:
On the one hand, for $m_2 = \pm m_1$, i.e. at the boundary of the Weyl
chamber, the residual gauge group is either ${{\rm U}(1)}_i\times {{\rm SU}(2)}_j$ or ${{\rm U}(1)}_i
\times {{\rm SO}}(3)_j$ (for $i,j=1,2$ and $i\neq j$), depending on the quotient
under consideration. Thus, only the degree $1$ Casimir invariant of the ${{\rm U}(1)}_i$
can be employed for a dressing, as the Casimir invariant of ${{\rm SU}(2)}_j$ or
${{\rm SO}}(3)_j$ belongs to the quotient $\widetilde{{{\rm SO}}(4)} \slash \Gamma$
itself. Hence, we get
\begin{subequations}
\label{eqn:A1xA1_dress}
\begin{equation}
V_{(m_1,\pm m_1)}^{\mathrm{dress},0} = (m_1,\pm m_1) \quad\textrm{and}\quad
V_{(m_1,\pm m_1)}^{\mathrm{dress},1} = \phi_i \ (m_1,\pm m_1) \; .
\end{equation}
Alternatively, we can apply the results of App.~\ref{app:PL} and deduce the
dressing behaviour at the boundary of the Weyl chamber to be $P_{A_1\times
A_1}(t,m_1,\pm m_1) \slash P_{A_1\times A_1}(t,0,0) = 1+t$, i.e.\ only one
dressed monopole arises.
On the other hand, for $m_1>|m_2|\geq0$, i.e. in the interior of the Weyl
chamber, the residual gauge group is ${{\rm U}(1)}^2$. From the resulting two degree $1$
Casimir invariants one constructs the following monopole operators:
\begin{equation}
V_{(m_1, m_2)}^{\mathrm{dress},0} = (m_1,m_2)
\quad \longrightarrow \quad
\left\{ \begin{matrix}
V_{(m_1,m_2)}^{\mathrm{dress},1,i} = \phi_i \ (m_1,m_2) \; ,& \qquad\textrm{for}\quad i=1,2
\\ V_{(m_1,m_2)}^{\mathrm{dress},2} = \phi_1 \phi_2 \ (m_1, m_2)\; .
\end{matrix} \right.
\end{equation}
\end{subequations}
Using App.~\ref{app:PL}, we obtain that monopole operator with GNO-charge
in the interior of the Weyl chamber exhibit the following
dressings $P_{A_1\times
A_1}(t,m_1,m_2) \slash P_{A_1\times A_1}(t,0,0) = 1+2t+t^2$, which agrees with
our discussion above.
\subsection{Representation \texorpdfstring{$[2,0]$}{[2,0]}}
\label{subsec:A1xA1_Rep20}
The conformal dimension for this case reads
\begin{equation}
\label{eqn:delta_A1xA1_Rep20}
\Delta(m_1,m_2)=(N-1)\left(|m_1 + m_2| + |m_1 - m_2|\right) \; .
\end{equation}
Following the ideas outlined earlier, the conformal
dimension~\eqref{eqn:delta_A1xA1_Rep20} defines a fan in the dominant Weyl
chamber. In this example, $\Delta$ is already a linear function on the entire
dominant Weyl chamber; thus, we generate a fan which just consists of one
$2$-dimensional rational cone
\begin{equation}
\label{eqn:fan_A1xA1_Rep20}
C^{(2)} = \Big\{ (m_1 \geq m_2) \wedge (m_1 \geq -m_2) \Big\} \; .
\end{equation}
\subsubsection{Quotient \texorpdfstring{${\rm Spin}(4)$}{Spin(4)}}
\paragraph{Hilbert basis}
Starting from the fan~\eqref{eqn:fan_A1xA1_Rep20} with the cone $C^{(2)}$, the
Hilbert basis for the semi-group $S^{(2)} \coloneqq C^{(2)} \cap K^{[0]}$ is
simply given by the ray generators
\begin{equation}
\Hcal(S^{(2)}) = \Big\{ (1,1), (1,-1) \Big\} \;,
\end{equation}
see for instance Fig.~\ref{Fig:Hilbert_basis_Spin4_Rep20}.
Both minimal generators exhibit a bare monopole operator and one dressed
operators, as explained in~\eqref{eqn:A1xA1_dress}.
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
%
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
%
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw[black,thick] (1,1) circle (4pt);
\draw[black,thick] (1,-1) circle (4pt);
\draw (4,1.3) node {$S^{(2)}$};
\end{tikzpicture}
\caption{The semi-group $S^{(2)}$ and its ray-generators (black circled points)
for the quotient ${\rm Spin}(4)$ and the representation $[2,0]$.}
\label{Fig:Hilbert_basis_Spin4_Rep20}
\end{figure}
\paragraph{Hilbert series}
We compute the Hilbert series to
\begin{equation}
\mathrm{HS}_{{\rm Spin}(4)}^{[2,0]}(t,N)= \frac{\left(1-t^{4
N-2}\right)^2}{\left(1-t^2\right)^2 \left(1-t^{2 N-2}\right)^2 \left(1-t^{2
N-1}\right)^2} \; ,
\label{eqn:HS_Spin4_Rep20}
\end{equation}
which is a complete intersection with $6$ generators and $2$ relations. The
generators are given in Tab.~\ref{tab:Ops_Spin4_Rep20}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings \\ \midrule
Casimirs & --- & --- & $2$ & --- & --- \\
monopole & $(1,\pm1)$ & $K^{[0]}$ & $2N-2 $ & ${{\rm U}(1)} \times {{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$\\
\bottomrule
\end{tabular}
\caption{Bare and dressed monopole generators for a ${\rm Spin}(4)$ gauge theory
with matter transforming in $[2,0]$.}
\label{tab:Ops_Spin4_Rep20}
\end{table}
\paragraph{Remark} The Hilbert series~\eqref{eqn:HS_Spin4_Rep20} can be
compared to the case of ${{\rm SU}(2)}$ with $n$ fundamentals and
$n_a$ adjoints such that $2N=n+2n_a$, c.f.~\cite{Cremonesi:2013lqa}. One
derives
at
\begin{equation}
\mathrm{HS}_{{\rm Spin}(4)}^{[2,0]}(t,N) = \mathrm{HS}_{{{\rm SU}(2)}}^{[1]+[2]}(t,n,n_a) \times
\mathrm{HS}_{{{\rm SU}(2)}}^{[1]+[2]}(t,n,n_a) \; ,
\end{equation}
which equals the product of two $D_{2N}$ singularities. As a consequence,
the minimal generator $(1,1)$ belongs to one ${{\rm SU}(2)}$ Hilbert series with adjoint
matter content, while $(1,-1)$ belongs to the other.
\subsubsection{Quotient \texorpdfstring{${{\rm SO}}(4)$}{SO(4)}}
The centre of the GNO-dual ${{\rm SO}}(4)$ is a $\mathbb Z_2$, which we choose to count if
$(m_1,m_2)$ belongs to $K^{[0]}$ or $K^{[2]}$. A realisation is given by
\begin{equation}
z^{m_1+m_2} = \begin{cases} z^{\text{even}}=1 & \qquad\textrm{for}\quad (m_1,m_2)\in K^{[0]} \;
,\\
z^{\text{odd}}=z & \qquad\textrm{for}\quad (m_1,m_2)\in K^{[2]} \; .
\end{cases}
\end{equation}
In other words, $z$ is a $\mathbb Z_2$-fugacity.
\paragraph{Hilbert basis}
The semi-group $S^{(2)}\coloneqq C^{(2)} \cap \left(K^{[0]} \cup K^{[2]}
\right)$ has a Hilbert basis as displayed in
Fig.~\ref{Fig:Hilbert_basis_SO4_Rep20} or explicitly
\begin{equation}
\Hcal(S^{(2)}) = \Big\{ (1,\pm1), (1,0) \Big\} \; .
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x-1,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x,2*\y-1) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw (6,1) node[draw, cross out,inner sep=0.8pt,thick,black] {};
\draw (7.2,1.05) node { $K^{[2]}$ lattice};
\draw[black,thick] (1,1) circle (4pt);
\draw[black,thick] (1,-1) circle (4pt);
\draw[red,thick] (1,0) circle (4pt);
\draw (4,1.3) node {$S^{(2)}$};
\end{tikzpicture}
\caption{The semi-group $S^{(2)}$ and its ray-generators (black circled
points) for the quotient ${{\rm SO}}(4)$ and the representation $[2,0]$. The red
circled lattice point completes the Hilbert basis for $S^{(2)}$.}
\label{Fig:Hilbert_basis_SO4_Rep20}
\end{figure}
\paragraph{Hilbert series}
The Hilbert series for ${{\rm SO}}(4)$ is given by
\begin{equation}
\label{eqn:HS_SO4_Rep20}
\mathrm{HS}_{{{\rm SO}}(4)}^{[2,0]}(t,z,N)=\frac{1
+t^{2 N-2}
+2 t^{2 N-1}
+z t^{2 N}
+2 z t^{2 N-1}
+z t^{4 N-2}
}{\left(1-t^2\right)^2
\left(1-t^{2 N-2}\right) \left(1-z t^{2 N-2}\right)} \; ,
\end{equation}
which is a rational function with a palindromic polynomial
of degree $4N-2$ as numerator, while the denominator is of degree $4N$. Hence,
the difference in degrees is $2$, i.e.\ the quaternionic dimension of the
moduli
space.
In addition, the denominator~\eqref{eqn:HS_SO4_Rep20} has a pole of order $4$
at $t\to1$, which equals the complex dimension of the moduli space.
\paragraph{Plethystic logarithm}
Analysing the PL yields for $N\geq3$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{{\rm SO}}(4)}^{[2,0]}) = 2t^2 &+ z t^{\Delta(1,0)}(1+2t^2+t^2)
+2t^{\Delta(1,\pm1)} (1+t) \\
&-t^{2\Delta(1,0)} (1+2(1+z)t + (6+4z)t^2 + 2(1+z)t^3 +t^4 ) +\ldots \notag
\end{align}
and for $N=2$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{{\rm SO}}(4)}^{[2,0]})= 2 t^2 + zt^2(1+2t+t^2) + 2t^2(1+t)
- t^4(1+ 2(1 + z) t + (6 +
4 z) t^2 ) + \ldots
\end{align}
such that we have generators as summarised in Tab.~\ref{tab:Ops_SO4_Rep20}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings
\\ \midrule
Casimirs & --- & --- & $2$ & --- & --- \\
monopole & $(1,0)$ & $K^{[2]}$ & $2N-2 $ & ${{\rm U}(1)} \times {{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$\\
monopole & $(1,\pm1)$ & $K^{[0]}$ & $2N-2 $ & ${{\rm U}(1)} \times {{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$
\\
\bottomrule
\end{tabular}
\caption{Bare and dressed monopole generators for a ${{\rm SO}}(4)$ gauge theory
with matter transforming in $[2,0]$.}
\label{tab:Ops_SO4_Rep20}
\end{table}
\paragraph{Gauging a $\boldsymbol{\mathbb Z_2}$}
Although the Hilbert series~\eqref{eqn:HS_SO4_Rep20} is not a complete
intersection, the gauging of the topological $\mathbb Z_2$ reproduces the ${\rm Spin}(4)$
result~\eqref{eqn:HS_Spin4_Rep20}, that is
\begin{equation}
\mathrm{HS}_{{\rm Spin}(4)}^{[2,0]}(t,N) = \frac{1}{2} \left(
\mathrm{HS}_{{{\rm SO}}(4)}^{[2,0]}(t,z{=}1,N) +
\mathrm{HS}_{{{\rm SO}}(4)}^{[2,0]}(t,z{=}-1,N)\right) \; .
\label{eqn:SO4_gauging_Rep20}
\end{equation}
\subsubsection{Quotient \texorpdfstring{${{\rm SO}}(3) \times {{\rm SU}(2)}$}{SO(3)xSU(2)}}
The dual group is ${{\rm SU}(2)} \times {{\rm SO}}(3)$ and the summation extends over
$(m_1,m_2) \in K^{[0]} \cup K^{[1]}$. The non-trivial centre $ \mathbb Z_2 \times \{1
\}$ gives rise to a $\mathbb Z_2$-action, which we choose to distinguish the two
lattices $K^{[0]}$ and $K^{[1]}$ as follows:
\begin{equation}
z_1^{m_1+m_2} =\begin{cases} z_1^{p_1+p_2} = z_1^{\text{even}} =1 & \qquad\textrm{for}\quad
(m_1,m_2) \in K^{[0]} \; , \\
z_1^{p_1+ \tfrac{1}{2}+p_2 + \tfrac{1}{2}} = z_1^{\text{even}+1} =z_1 & \qquad\textrm{for}\quad
(m_1,m_2) \in K^{[1]} \; .
\end{cases}
\end{equation}
\paragraph{Hilbert basis}
The semi-group $S^{(2)}\coloneqq C^{(2)} \cap \left(K^{[0]} \cup K^{[1]}
\right)$ has a Hilbert basis comprised of the ray generators. We refer to
Fig.~\ref{Fig:Hilbert_basis_SO3xSU2_Rep20} and provide
the minimal generators for completeness:
\begin{equation}
\Hcal(S^{(2)}) = \Big\{ (\tfrac{1}{2},\tfrac{1}{2}), (1,-1) \Big\} \; .
\label{eqn:Hilbert_basis_SO3xSU2_Rep20}
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x +1/2,2*\y +1/2)
{};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x-1 +1/2,2*\y-1+1/2)
{};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw (6,1) node[draw,diamond,inner sep=0.8pt,black] {};
\draw (7.2,1.05) node { $K^{[1]}$ lattice};
\draw[black,thick] (1/2,1/2) circle (4pt);
\draw[black,thick] (1,-1) circle (4pt);
\draw (4,1.3) node {$S^{(2)}$};
\end{tikzpicture}
\caption{The semi-group $S^{(2)}$ for the quotient ${{\rm SO}}(3)\times {{\rm SU}(2)}$ and the
representation $[2,0]$. The black circled points are the ray generators.}
\label{Fig:Hilbert_basis_SO3xSU2_Rep20}
\end{figure}
\paragraph{Hilbert series}
Computing the Hilbert series and using explicitly the $\mathbb Z_2$-properties of
$z_1$ yields
\begin{equation}
\mathrm{HS}_{{{\rm SO}}(3)\times {{\rm SU}(2)}}^{[2,0]}(t,z_1,N)= \frac{\left(1-t^{2 N}\right)
\left(1-t^{4 N-2}\right)}{\left(1-t^2\right)^2 \left(1-t^{2 N-2}\right)
\left(1-t^{2 N-1}\right) \left(1-z_1 t^{N-1}\right)
\left(1-z_1 t^{N}\right)} \; ,
\label{eqn:HS_SO3xSU2_Rep20}
\end{equation}
which is a complete intersection with $6$ generators and $2$ relations. The
generators are displayed in Tab.~\ref{tab:Ops_SO3xSU2_Rep20}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings
\\ \midrule
Casimirs & --- & --- & $2$ & --- & --- \\
monopole & $(\tfrac{1}{2},\tfrac{1}{2})$ & $K^{[1]}$ & $N-1 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$\\
monopole & $(1,-1)$ & $K^{[0]}$ & $2N-2 $ & ${{\rm U}(1)} \times {{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
\bottomrule
\end{tabular}
\caption{Bare and dressed monopole generators for a ${{\rm SO}}(3)\times {{\rm SU}(2)}$ gauge
theory with matter transforming in $[2,0]$.}
\label{tab:Ops_SO3xSU2_Rep20}
\end{table}
\paragraph{Remark}
Comparing to the case of ${{\rm SU}(2)}$ with $n_a$ adjoints and ${{\rm SO}}(3)$ with $n$
fundamentals presented in~\cite{Cremonesi:2013lqa}, we can re-express the
Hilbert series~\eqref{eqn:HS_SO3xSU2_Rep20} as the product
\begin{equation}
\mathrm{HS}_{{{\rm SO}}(3)\times {{\rm SU}(2)}}^{[2,0]}(t,z_1,N) = \mathrm{HS}_{{{\rm SO}}(3)}^{[1]}(t,z_1,n=N)
\times \mathrm{HS}_{{{\rm SU}(2)}}^{[2]}(t,n_a=N) \; ,
\end{equation}
where the $z_1$-grading belongs to ${{\rm SO}}(3)$ with $N$ fundamentals. The
minimal generator $(\tfrac{1}{2},\tfrac{1}{2})$ is the minimal generator for
${{\rm SO}}(3)$ with $N$ fundamentals, while $(1,-1)$ is the minimal generator for
${{\rm SU}(2)}$ with $N$ adjoints.
\subsubsection{Quotient \texorpdfstring{${{\rm SU}(2)}\times {{\rm SO}}(3)$}{SU(2)xSO(3)}}
The dual group is ${{\rm SO}}(3) \times {{\rm SU}(2)}$ and the summation extends over
$(m_1,m_2) \in K^{[0]} \cup K^{[3]}$. The non-trivial centre $ \{1\} \times
\mathbb Z_2 $ gives rise to a $\mathbb Z_2$-action, which we choose to distinguish the two
lattices $K^{[0]}$ and $K^{[3]}$ as follows:
\begin{equation}
z_2^{p_1+p_2} =\begin{cases} z_2^{\text{even}} =1 & \qquad\textrm{for}\quad
(m_1,m_2) \in K^{[0]} \; , \\
z_2^{\text{odd}} =z_2 & \qquad\textrm{for}\quad
(m_1,m_2) \in K^{[3]} \; . \end{cases}
\end{equation}
\paragraph{Hilbert basis}
The semi-group $S^{(2)}\coloneqq C^{(2)} \cap \left(K^{[0]} \cup K^{[3]}
\right)$ has as Hilbert basis the set of
ray generators
\begin{equation}
\Hcal(S^{(2)}) = \Big\{ (1,1), (\tfrac{1}{2},-\tfrac{1}{2}) \Big\} \; .
\label{eqn:Hilbert_basis_SU2xSO3_Rep20}
\end{equation}
Fig.~\ref{Fig:Hilbert_basis_SU2xSO3_Rep20} depicts the situation.
We observe that bases~\eqref{eqn:Hilbert_basis_SO3xSU2_Rep20}
and~\eqref{eqn:Hilbert_basis_SU2xSO3_Rep20} are related by reflection along the
$m_2=0$ axis, which in turn corresponds to the interchange of $K^{[1]}$ and
$K^{[3]}$.
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] at (2*\x-1+1/2,2*\y+1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] at
(2*\x +1/2,2*\y-1 +1/2) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw (6,1) node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] {};
\draw (7.2,1.05) node { $K^{[3]}$ lattice};
\draw[black,thick] (1/2,-1/2) circle (4pt);
\draw[black,thick] (1,1) circle (4pt);
\draw (4,1.3) node {$S^{(2)}$};
\end{tikzpicture}
\caption{The semi-group $S^{(2)}$ for the quotient ${{\rm SU}(2)} \times {{\rm SO}}(3)$ and
the representation $[2,2]$. The black circled points are the ray generators.}
\label{Fig:Hilbert_basis_SU2xSO3_Rep20}
\end{figure}
\paragraph{Hilbert series}
Similar to the previous case, employing the $\mathbb Z_2$-properties of $z_2$ we
obtain the following Hilbert series:
\begin{equation}
\mathrm{HS}_{{{\rm SU}(2)} \times {{\rm SO}}(3)}^{[2,0]}(t,z_2,N)= \frac{\left(1-t^{2 N}\right)
\left(1-t^{4 N-2}\right)}{\left(1-t^2\right)^2 \left(1-t^{2 N-2}\right)
\left(1-t^{2 N-1}\right) \left(1-z_2 t^{N-1}\right)
\left(1-z_2 t^{N}\right)} \; ,
\end{equation}
which is a complete intersection with $6$ generators and $2$ relations. We
summarise the generators in Tab.~\ref{tab:Ops_SU2xSO3_Rep20}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings \\ \midrule
Casimirs & --- & --- & $2$ & --- & --- \\
monopole & $(\tfrac{1}{2},-\tfrac{1}{2})$ & $K^{[3]}$ & $N-1 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$\\
monopole & $(1,1)$ & $K^{[0]}$ & $2N-2 $ & ${{\rm U}(1)} \times {{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
\bottomrule
\end{tabular}
\caption{Bare and dressed monopole generators for a ${{\rm SU}(2)}\times{{\rm SO}}(3)$ gauge
theory with matter transforming in $[2,0]$.}
\label{tab:Ops_SU2xSO3_Rep20}
\end{table}
\paragraph{Remark}
Also, the equivalence
\begin{equation}
\mathrm{HS}_{{{\rm SO}}(3)\times {{\rm SU}(2)}}^{[2,0]}(t,z_1,N) \xleftrightarrow{\quad z_1
\leftrightarrow z_2 \quad }
\mathrm{HS}_{{{\rm SU}(2)} \times {{\rm SO}}(3)}^{[2,0]}(t,z_2,N)
\label{eqn:Exchange_z1-z2_Rep20}
\end{equation}
holds, which then also implies
\begin{equation}
\mathrm{HS}_{{{\rm SU}(2)} \times {{\rm SO}}(3)}^{[2,0]}(t,z_2,N) = \mathrm{HS}_{{{\rm SO}}(3)}^{[1]}(t,z_2,n=N)
\times \mathrm{HS}_{{{\rm SU}(2)}}^{[2]}(t,n_a=N) \; .
\end{equation}
Thus, the moduli space is a product of two complete intersections.
\subsubsection{Quotient \texorpdfstring{$\mathrm{PSO}(4)$}{PSO(4)}}
Taking the quotient with respect to the entire centre of $\widetilde{{{\rm SO}}(4)}$
yields the projective group $\mathrm{PSO}(4)$, which has as GNO-dual
${\rm Spin}(4)\cong {{\rm SU}(2)}\times {{\rm SU}(2)}$. Consequently, the summation extends over the
whole weight lattice $K^{[0]}\cup K^{[1]}\cup K^{[2]}\cup K^{[3]}$ and there
is an
action of $\mathbb Z_2 \times \mathbb Z_2$ on this lattice, which is chosen as displayed in
Tab.~\ref{tab:PSO4_Z2-grading}.
\begin{table}[h]
\centering
\begin{tabular}{c|c||c}
\toprule
lattice & $\mathbb Z_2\times \mathbb Z_2$ & $\widetilde{\mathbb Z}_2\times \widetilde{\mathbb Z}_2$ \\
\midrule
$K^{[0]}$ & $(z_1)^0,(z_2)^0 $ & $(w_1)^0,(w_2)^0 $ \\
$K^{[1]}$ & $(z_1)^1,(z_2)^0 $ & $(w_1)^1,(w_2)^1 $ \\
$K^{[2]}$ & $(z_1)^0,(z_2)^1 $ & $(w_1)^0,(w_2)^1 $ \\
$K^{[3]}$ & $(z_1)^1,(z_2)^1 $ & $(w_1)^1,(w_2)^0 $ \\
\bottomrule
\end{tabular}
\caption{The $\mathbb Z_2\times \mathbb Z_2$ distinguishes the four different lattice
$K^{[j]}$, $j=0,1,2,3$. The choice of fugacities $z_1$, $z_2$ is used in the
computation, while the second choice $w_1$, $w_2$ is convenient for gauging
$\mathrm{PSO}(4)$ to ${{\rm SU}(2)}\times {{\rm SO}}(3)$.}
\label{tab:PSO4_Z2-grading}
\end{table}
\paragraph{Hilbert basis}
The semi-group $S^{(2)}\coloneqq C^{(2)} \cap \left(K^{[0]} \cup K^{[1]} \cup
K^{[2]} \cup K^{[3]} \right)$ has a Hilbert basis that is determined by the
ray generators. Fig.~\ref{Fig:Hilbert_basis_PSO4_Rep20} depicts the
situation and the Hilbert basis reads
\begin{equation}
\Hcal(S^{(2)}) = \Big\{ (\tfrac{1}{2},\tfrac{1}{2}) ,
(\tfrac{1}{2},-\tfrac{1}{2}) \Big\} \; .
\label{eqn:Hilbert_basis_PSO4_Rep20}
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x-1,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x +1/2,2*\y +1/2)
{};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x-1 +1/2,2*\y-1+1/2)
{};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon, regular polygon sides=3,inner
sep=0.7pt,black] at (2*\x-1+1/2,2*\y+1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon, regular polygon sides=3,inner
sep=0.7pt,black] at (2*\x +1/2,2*\y-1 +1/2) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw (6,3) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,3.05) node { $K^{[0]}$ lattice};
\draw (6,2) node[draw,diamond,inner sep=0.8pt,black] {};
\draw (7.2,2.05) node { $K^{[1]}$ lattice};
\draw (6,1) node[draw,cross out,inner sep=0.8pt,thick,black] {};
\draw (7.2,1.05) node { $K^{[2]}$ lattice};
\draw (6,0 ) node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] {};
\draw (7.2,0.05) node { $K^{[3]}$ lattice};
\draw[black,thick] (1/2,1/2) circle (4pt);
\draw[black,thick] (1/2,-1/2) circle (4pt);
\draw (4,1.3) node {$S^{(2)}$};
\end{tikzpicture}
\caption{The semi-group $S^{(2)}$ and its ray-generators (black circled
points) for the quotient $\mathrm{PSO}(4)$ and the representation $[2,0]$.}
\label{Fig:Hilbert_basis_PSO4_Rep20}
\end{figure}
\paragraph{Hilbert series}
An evaluation of the Hilbert series yields
\begin{equation}
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,0]}(t,z_1,z_2,N)= \frac{\left(1-t^{2
N}\right)^2}{\left(1-t^2\right)^2 \left(1-z_1 t^{N-1}\right) \left(1-z_1
t^{N}\right) \left(1-z_1 z_2 t^{N-1}\right) \left(1-z_1 z_2
t^{N}\right)} \; ,
\label{eqn:HS_PSO4_Rep20}
\end{equation}
which is a complete intersection with $6$ generators and $2$
relations. Tab.~\ref{tab:Ops_PSO4_Rep20} summarises the generators with their
properties.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings
\\ \midrule
Casimirs & --- & --- & $2$ & --- & --- \\
monopole & $(\tfrac{1}{2},\tfrac{1}{2})$ & $K^{[1]}$ & $N-1 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$\\
monopole & $(\tfrac{1}{2},-\tfrac{1}{2})$ & $K^{[3]}$ & $N-1 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
\bottomrule
\end{tabular}
\caption{Bare and dressed monopole generators for a $\mathrm{PSO}(4)$ gauge
theory with matter transforming in $[2,0]$.}
\label{tab:Ops_PSO4_Rep20}
\end{table}
\paragraph{Gauging a $\boldsymbol{\mathbb Z_2}$}
Now, we utilise the $\mathbb Z_2 \times \mathbb Z_2$ global symmetry to recover the Hilbert
series for \emph{all} five quotients solely from the $\mathrm{PSO}(4)$ result.
Firstly, to obtain the ${{\rm SO}}(4)$ result, we need to average out the
contributions of $K^{[1]}$ and $K^{[3]}$, which is achieved for $z_1 \to \pm 1$
(we also relabel $z_2$ for consistence), see also
Tab.~\ref{tab:PSO4_Z2-grading}. This yields
\begin{subequations}
\label{eqn:PSO4_gauging_Rep20}
\begin{equation}
\mathrm{HS}_{{{\rm SO}}(4)}^{[2,0]}(t,z,N) = \frac{1}{2} \left(
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,0]}(t,z_1{=}1,z_2{=}z,N) +
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,0]}(t,z_1{=}-1,z_2{=}z,N)\right) \; .
\end{equation}
Secondly, a subsequent gauging leads to the ${\rm Spin}(4)$ result as demonstrated
in~\eqref{eqn:SO4_gauging_Rep20}, because one averages the $K^{[2]}$
contributions out.
Thirdly, one can gauge the other $\mathbb Z_2$-factor corresponding to $z_2 \to \pm1$,
which then eliminates the contributions of $K^{[2]}$ and $K^{[3]}$ due to the
choices of Tab.~\ref{tab:PSO4_Z2-grading}. The result then reads
\begin{equation}
\mathrm{HS}_{{{\rm SO}}(3)\times{{\rm SU}(2)}}^{[2,0]}(t,z_1,N) = \frac{1}{2} \left(
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,0]}(t,z_1,z_2{=}1,N) +
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,0]}(t,z_1,z_2{=}-1,N)\right) \; .
\end{equation}
Lastly, for obtaining the ${{\rm SU}(2)}\times {{\rm SO}}(3)$ Hilbert series one needs to
eliminate the $K^{[1]}$ and $K^{[2]}$ contributions. For that, we have to
redefine the $\mathbb Z_2$-fugacities conveniently. One choice is
\begin{equation}
z_1 \cdot z_2 \mapsto w_1 \; , \qquad z_1 \mapsto w_1 \cdot w_2 \; , \quad\textrm{and}\quad z_2
\mapsto w_2 \; ,
\label{eqn:redefine_Z2-grading_A1xA1}
\end{equation}
which is consistent in $\mathbb Z_2 \times \mathbb Z_2$. The effect on the lattices is
summarised in Tab.~\ref{tab:PSO4_Z2-grading}. Hence, $w_2\to \pm1$ has the
desired effect and leads to
\begin{equation}
\mathrm{HS}_{{{\rm SU}(2)}\times{{\rm SO}}(3)}^{[2,0]}(t,z_2{=} w_1,N) = \frac{1}{2} \left(
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,0]}(t,w_1,w_2{=}1,N) +
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,0]}(t,w_1,w_2{=}-1,N)\right) \; .
\end{equation}
\end{subequations}
Consequently, the Hilbert series for \emph{all} five quotients can be computed
from the $\mathrm{PSO}(4)$-result by gauging $\mathbb Z_2$-factors.
\paragraph{Remark}
As for most of the cases in this section, the Hilbert
series~\eqref{eqn:HS_PSO4_Rep20} can be written as a product of two complete
intersections. Employing the results of~\cite{Cremonesi:2013lqa} for ${{\rm SO}}(3)$
with $n$ fundamentals, we obtain
\begin{equation}
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,0]}(t,z_1,z_2,N) = \mathrm{HS}_{{{\rm SO}}(3)}^{[1]}(t,z_1,n=N)
\times \mathrm{HS}_{{{\rm SO}}(3)}^{[1]}(t,z_1 z_2,n=N) \; .
\end{equation}
\subsection{Representation \texorpdfstring{$[2,2]$}{[2,2]}}
Let us use the representation $[2,2]$ to further compare the results for the
five different quotient groups. The conformal dimension reads
\begin{equation}
\label{eqn:delta_A1xA1_Rep22}
\Delta(m_1,m_2)= N (\left| m_1-m_2\right| +\left| m_1+m_2\right|
+2 \left| m_1\right| +2 \left| m_2\right| )-\left|
m_1-m_2\right| -\left| m_1+m_2\right| \; .
\end{equation}
As described in the introduction, the conformal
dimension~\eqref{eqn:delta_A1xA1_Rep22} defines a fan in the dominant Weyl
chamber, which is spanned by two $2$-dimensional rational cones
\begin{equation}
\label{eqn:fan_A1xA1_Rep22}
C_{\pm}^{(2)} = \Big\{ (m_1 \geq \pm m_2) \wedge (m_2 \geq \pm 0) \Big\} \; .
\end{equation}
\subsubsection{Quotient \texorpdfstring{${\rm Spin}(4)$}{Spin(4)}}
\paragraph{Hilbert basis}
Starting from the fan~\eqref{eqn:fan_A1xA1_Rep22} with cones $C_{\pm}^{(2)}$,
the Hilbert bases for the semi-groups $S_{\pm}^{(2)} \coloneqq C_{\pm}^{(2)}
\cap K^{[0]}$ are simply given by the ray generators, see for instance
Fig.~\ref{Fig:Hilbert_basis_Spin4_Rep22}.
\begin{equation}
\Hcal(S_{\pm}^{(2)}) = \Big\{ (1,\pm1), (2,0) \Big\} \; .
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw[black,dashed,thick] (0,0) -- (4.5,0);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw (2,0) circle (4pt);
\draw (1,1) circle (4pt);
\draw (1,-1) circle (4pt);
\draw (4,1.3) node {$S_+^{(2)}$};
\draw (4,-1.3) node {$S_-^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups and their ray-generators (black circled points) for
the quotient ${\rm Spin}(4)$ and the representation $[2,2]$.}
\label{Fig:Hilbert_basis_Spin4_Rep22}
\end{figure}
\paragraph{Hilbert series}
The GNO-dual ${{\rm SO}}(3)\times {{\rm SO}}(3)$ has a trivial centre and the
Hilbert series reads
\begin{equation}
\label{eqn:HS_Spin4_Rep22}
\mathrm{HS}_{{\rm Spin}(4)}^{[2,2]}(t,N)=\frac{1+t^{6 N-2}+2 t^{6 N-1}+2 t^{8
N-3}+t^{8 N-2}+t^{14 N-4}}{\left(1-t^2\right)^2 \left(1-t^{6
N-2}\right) \left(1-t^{8 N-4}\right)} \; .
\end{equation}
The numerator of~\eqref{eqn:HS_Spin4_Rep22} is a palindromic polynomial of
degree $14N-4$; while the denominator is a polynomial of degree $14N-2$. Hence,
the difference in degree is two, which equals the quaternionic dimension of the
moduli space.
In addition, denominator of~\eqref{eqn:HS_Spin4_Rep22} has a pole of order four
at $t=1$, which equals the complex dimension of the moduli space.
\paragraph{Plethystic logarithm}
The plethystic logarithm takes the form
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm Spin}(4)}^{[2,2]})= 2t^2 &+ 2t^{\Delta(1,\pm1)} (1+t)
+ t^{\Delta(2,0)}(1+2t+t^2) \\
&-t^{2\Delta(1,\pm1)} (1 + 2t + 3t^2 + 2t^3 + 4t^4 + 2t^5 + 3t^6 + 2t^7
+t^8)+\ldots \notag
\end{align}
The appearing terms agree with the minimal generators of the Hilbert
bases~\eqref{Fig:Hilbert_basis_Spin4_Rep22}.
One has two independent degree two Casimir invariants. Further, there are
monopole operators of GNO-charge $(1,1)$ and $(1,-1)$ at conformal
dimension $6N-2$ with an independent dressed monopole generator of conformal
dimension $6N-1$ for both charges.
Moreover, there is a monopole operator of GNO-charge $(2,0)$ at dimension
$8N-4$ with two dressing operators at dimension $8N-3$ and
one at $8N-2$.
\subsubsection{Quotient \texorpdfstring{${{\rm SO}}(4)$}{SO(4)}}
\paragraph{Hilbert basis}
The semi-groups $S_{\pm}^{(2)}\coloneqq C_{\pm}^{(2)} \cap \left(K^{[0]} \cup
K^{[2]} \right)$ have Hilbert bases which again equal (the now different) ray
generators. The situation is depicted in Fig.~\ref{Fig:Hilbert_basis_SO4_Rep22}
and the Hilbert bases are as follows:
\begin{equation}
\Hcal(S_{\pm}^{(2)}) = \Big\{ (1,\pm1), (1,0) \Big\} \; .
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x-1,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x,2*\y-1) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw[black,dashed,thick] (0,0) -- (4.5,0);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw (6,1) node[draw,cross out,inner sep=0.8pt,thick,black] {};
\draw (7.2,1.05) node { $K^{[2]}$ lattice};
\draw (1,0) circle (4pt);
\draw (1,1) circle (4pt);
\draw (1,-1) circle (4pt);
\draw (4,1.5) node {$S_+^{(2)}$};
\draw (4,-1.5) node {$S_-^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups and their ray-generators (black circled points) for
the quotient ${{\rm SO}}(4)$ and the representation $[2,2]$.}
\label{Fig:Hilbert_basis_SO4_Rep22}
\end{figure}
\paragraph{Hilbert series}
The Hilbert series reads
\begin{equation}
\label{eqn:HS_SO4_Rep22}
\mathrm{HS}_{{{\rm SO}}(4)}^{[2,2]}(t,z,N)= \frac{1+ z t^{4 N}+2 z t^{4 N-1}+t^{6
N-2}+2 t^{6 N-1}+z t^{10 N-2}}{\left(1-t^2\right)^2 \left(1-z t^{4
N-2}\right)\left(1-t^{6 N-2}\right) } \; .
\end{equation}
The numerator of~\eqref{eqn:HS_SO4_Rep22} is a palindromic polynomial of degree
$10N-2$ (neglecting the dependence on $z$); while the
denominator is a polynomial of degree $10N$. Hence, the difference in degree
is two equals the quaternionic dimension of the moduli space.
Moreover, the denominator has a pole of order four at $t=1$, which equals the
complex dimension of the moduli space.
\paragraph{Plethystic logarithm}
Studying the PL, we observe
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{{\rm SO}}(4)}^{[2,2]})=2t^2 &+ z t^{\Delta(1,0)} (1+2t^2+t) + 2
t^{\Delta(1,\pm1)} (1+t) \\
&-t^{2\Delta(1,0)+2}(3+2t^2 + t^2 + 2t^3 + 4t^4 + 2t^5 + t^6 + 2 t^7 + 3 t^8)
+ \ldots \notag
\end{align}
such that we can associate the generators as follows: two degree two Casimir
invariants of ${{\rm SO}}(4)$, i.e.\ the quadratic Casimir and the Pfaffian;
A monopole of GNO-charge $(1,0)\in K^{[2]}$ at conformal dimension
$4N-2$ with two dressings at dimension
$4N-1$ and another dressing at $4N$; and two monopole operators of GNO-charges
$(1,1)$, $(1,-1) \in K^{[0]}$ at dimension $6N-2$ one dressed monopoles at
dimension $6N-1 $ each.
\paragraph{Gauging the $\mathbb Z_2$} In addition, one can gauge the topological
$\mathbb Z_2$ in~\eqref{eqn:HS_SO4_Rep22} and obtains
\begin{equation}
\label{eqn:SO4_gauging_Rep22}
\mathrm{HS}_{{\rm Spin}(4)}^{[2,2]}(t,N)= \frac{1}{2}
\left(\mathrm{HS}_{{{\rm SO}}(4)}^{[2,2]}(t,z{=}1,N)+\mathrm{HS}_{{{\rm SO}}(4)}^{[2,2]}(t,z{=}-1,N)
\right) \; .
\end{equation}
\subsubsection{Quotient \texorpdfstring{${{\rm SO}}(3)\times {{\rm SU}(2)} $}{SO(3)xSU(2)}}
\paragraph{Hilbert basis}
The semi-groups $S_{\pm}^{(2)}\coloneqq C_{\pm}^{(2)} \cap \left(K^{[0]} \cup
K^{[1]} \right)$ have Hilbert bases that go beyond the set of
ray generators. We refer to Fig.~\ref{Fig:Hilbert_basis_SO3xSU2_Rep22} and
the Hilbert bases are obtained as follows:
\begin{equation}
\Hcal(S_{+}^{(2)}) = \Big\{ (\tfrac{1}{2},\tfrac{1}{2}), (2,0) \Big\}
\quad\textrm{and}\quad
\Hcal(S_{-}^{(2)}) = \Big\{ (1,-1), (\tfrac{3}{2},-\tfrac{1}{2}) , (2,0)
\Big\} \; .
\label{eqn:Hilbert_basis_SO3xSU2_Rep22}
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x +1/2,2*\y +1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x-1
+1/2,2*\y-1+1/2) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw[black,dashed,thick] (0,0) -- (4.5,0);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw (6,1) node[draw,diamond,inner sep=0.8pt,black] {};
\draw (7.2,1.05) node { $K^{[1]}$ lattice};
\draw[black,thick] (1/2,1/2) circle (4pt);
\draw[black,thick] (1,-1) circle (4pt);
\draw[red,thick] (1+1/2,-1+1/2) circle (4pt);
\draw[black,thick] (2,0) circle (4pt);
\draw (4,1.5) node {$S_+^{(2)}$};
\draw (4,-1.5) node {$S_-^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups for the quotient ${{\rm SO}}(3)\times {{\rm SU}(2)}$ and the
representation $[2,2]$. The black circled points are the ray generators and the
red circled point completes the Hilbert basis for $S_{-}^{(2)}$.}
\label{Fig:Hilbert_basis_SO3xSU2_Rep22}
\end{figure}
\paragraph{Hilbert series}
The Hilbert series is computed to be
\begin{subequations}
\label{eqn:HS_SO3xSU2_Rep22}
\begin{align}
\mathrm{HS}_{{{\rm SO}}(3)\times {{\rm SU}(2)}}^{[2,2]}(t,z_1,N)&=
\frac{R(t,z_1,N)}{\left(1-t^2\right)^2 \left(1-t^{6 N-2}\right) \left(1-t^{8
N-4}\right)} \; , \\
R(t,z_1,N)&=1
+z_1 t^{3 N}
+z_1 t^{3 N-1}
+t^{6 N-2}
+2 t^{6 N-1}
+z_1 t^{7 N-3}
+2 z_1 t^{7 N-2}
+z_1 t^{7 N-1} \notag\\
&\qquad
+2 t^{8 N-3}
+t^{8 N-2}
+z_1 t^{11 N-4}
+z_1 t^{11 N-3}
+t^{14 N-4} \; .
\end{align}
\end{subequations}
Again, the numerator of~\eqref{eqn:HS_SO3xSU2_Rep22} is a palindromic
polynomial of degree $14N-4$; while the denominator is a polynomial of degree
$14N-2$. Hence, the difference in degree is two, which matches the quaternionic
dimension of the moduli space.
Also, the denominator has a pole of order four at $t=1$, which equals the
complex dimension of the moduli space.
\paragraph{Plethystic logarithm}
The inspection of the PL for $N\geq 2$ reveals
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{{\rm SO}}(3)\times {{\rm SU}(2)}}^{[2,2]}) = 2 t^2 &+ z_1
t^{\Delta(\frac{1}{2},\frac{1}{2})}(1 + t) +
t^{\Delta(1,\pm1)}(1+t-t^2) \\
&+z_1 t^{\Delta(1+\frac{1}{2},-1+\frac{1}{2})}
(1+2t+t^2)
+ t^{\Delta(2,0)}(1+2t+t^2) \notag \\
&-z_1 t^{3 \Delta(\frac{1}{2},\frac{1}{2})} (1+2t+t^2)+ \ldots \;. \notag
\end{align}
We summarise the generators in Tab.~\ref{tab:Ops_SO3xSU2_Rep22}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c}
\toprule
$(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ & dressings \\
\midrule
$(\tfrac{1}{2},\tfrac{1}{2})$ & $K^{[1]}$ & $3N-1$ & ${{\rm U}(1)}\times {{\rm SU}(2)}$ & $1$ by
${{\rm U}(1)}$\\
$(1,- 1)$ & $K^{[0]}$ & $6N-2$ & ${{\rm U}(1)}\times {{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
$(\tfrac{3}{2},-\tfrac{1}{2})$ & $K^{[1]}$ & $7N-3$ & ${{\rm U}(1)}\times {{\rm U}(1)}$ &
$3$ by ${{\rm U}(1)}^2$ \\
$(2,0)$ & $K^{[0]}$ & $8N-4$ & ${{\rm U}(1)}\times {{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$ \\
\bottomrule
\end{tabular}
\caption{The generators for the chiral ring of a ${{\rm SO}}(3)\times {{\rm SU}(2)}$ gauge
theory with matter in $[2,2]$.}
\label{tab:Ops_SO3xSU2_Rep22}
\end{table}
\subsubsection{Quotient \texorpdfstring{${{\rm SU}(2)}\times{{\rm SO}}(3) $}{SU(2)xSO(3)}}
\paragraph{Hilbert basis}
The semi-groups $S_{\pm}^{(2)}\coloneqq C_{\pm}^{(2)} \cap \left(K^{[0]} \cup
K^{[3]} \right)$ have Hilbert bases that go beyond the set
of ray generators. Fig.~\ref{Fig:Hilbert_basis_SU2xSO3_Rep22} depicts the
situation and the Hilbert bases are computed to be
\begin{equation}
\Hcal(S_{+}^{(2)}) = \Big\{ (1,1), (\tfrac{3}{2},\tfrac{1}{2}) , (2,0)
\Big\} \quad\textrm{and}\quad
\Hcal(S_{-}^{(2)}) = \Big\{ (\tfrac{1}{2},-\tfrac{1}{2}), (2,0) \Big\} \; .
\label{eqn:Hilbert_basis_SU2xSO3_Rep22}
\end{equation}
We observe that the bases~\eqref{eqn:Hilbert_basis_SO3xSU2_Rep22}
and~\eqref{eqn:Hilbert_basis_SU2xSO3_Rep22} are related by reflection along the
$m_2=0$ axis, which in turn corresponds to the interchange of $K^{[1]}$ and
$K^{[3]}$.
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] at (2*\x-1+1/2,2*\y+1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] at (2*\x +1/2,2*\y-1 +1/2) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw[black,dashed,thick] (0,0) -- (4.5,0);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw (6,1) node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] {};
\draw (7.2,1.05) node { $K^{[3]}$ lattice};
\draw[black,thick] (1/2,-1/2) circle (4pt);
\draw[black,thick] (1,1) circle (4pt);
\draw[red,thick] (3/2,1/2) circle (4pt);
\draw[black,thick] (2,0) circle (4pt);
\draw (4,1.5) node {$S_+^{(2)}$};
\draw (4,-1.5) node {$S_-^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups for the quotient ${{\rm SU}(2)} \times {{\rm SO}}(3)$ and the
representation $[2,2]$. The black circled points are the ray generators and the
red circled point completes the Hilbert basis for $S_{+}^{(2)}$.}
\label{Fig:Hilbert_basis_SU2xSO3_Rep22}
\end{figure}
\paragraph{Hilbert series}
The Hilbert series reads
\begin{subequations}
\label{eqn:HS_SU2xSO3_Rep22}
\begin{align}
\mathrm{HS}_{{{\rm SU}(2)} \times {{\rm SO}}(3)}^{[2,2]}(t,z_2,N)&=
\frac{R(t,z_2,N)}{\left(1-t^2\right)^2 \left(1-t^{6 N-2}\right) \left(1-t^{8
N-4}\right)} \; , \\
R(t,z_2,N)&=1
+z_2 t^{3 N}
+z_2 t^{3 N-1}
+t^{6 N-2}
+2 t^{6 N-1}
+z_2 t^{7 N-3}
+2 z_2 t^{7 N-2}
+z_2 t^{7 N-1} \notag\\
&\qquad
+2 t^{8 N-3}
+t^{8 N-2}
+z_2 t^{11 N-4}
+z_2 t^{11 N-3}
+t^{14 N-4} \; .
\end{align}
\end{subequations}
The numerator of~\eqref{eqn:HS_SU2xSO3_Rep22} is palindromic polynomial of
degree $14N-4$; while the denominator is a polynomial of degree $14N-2$. Hence,
the difference in degree is two, which equals the quaternionic dimension of the
moduli space.
In addition, the denominator has a pole of order four at $t=1$, which matches
the complex dimension of the moduli space.
As before, comparing the quotients ${{\rm SO}}(3)\times {{\rm SU}(2)}$ and ${{\rm SU}(2)} \times
{{\rm SO}}(3)$ as well as the symmetry of~\eqref{eqn:delta_A1xA1_Rep22}, it is
natural to expect the relationship
\begin{equation}
\mathrm{HS}_{{{\rm SO}}(3)\times {{\rm SU}(2)}}^{[2,2]}(t,z_1,N)
\xleftrightarrow{\quad z_1\leftrightarrow z_2 \quad }
\mathrm{HS}_{{{\rm SU}(2)} \times {{\rm SO}}(3)}^{[2,2]}(t,z_2,N) \; ,
\label{eqn:Exchange_z1-z2_Rep22}
\end{equation}
which is verified explicitly for~\eqref{eqn:HS_SO3xSU2_Rep22}
and~\eqref{eqn:HS_SU2xSO3_Rep22}.
\paragraph{Plethystic logarithm}
The equivalence to ${{\rm SO}}(3)\times {{\rm SU}(2)}$ is further confirmed by
the inspection of the PL for $N\geq 2$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{{\rm SU}(2)} \times {{\rm SO}}(3)}^{[2,2]})= 2 t^2 &+ z_2
t^{\Delta(\tfrac{1}{2},-\tfrac{1}{2})}(1 + t) + t^{\Delta(1,1)}(1+t-t^2) \\
&+z_2 t^{\Delta(\tfrac{3}{2},\tfrac{1}{2})} (1+2t+t^2)
+ t^{\Delta(2,0)}(1+2t+t^2)+ \ldots \notag
\end{align}
where we can summarise the monopole generators as in
Tab.~\ref{tab:Ops_SU2xSO3_Rep22}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c}
\toprule
$(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ & dressings \\
\midrule
$(\tfrac{1}{2},-\tfrac{1}{2})$ & $K^{[3]}$ & $3N-1$ & ${{\rm U}(1)}\times {{\rm SU}(2)}$ & $1$
by ${{\rm U}(1)}$ \\
$(1, 1)$ & $K^{[0]}$ & $6N-2$ & ${{\rm U}(1)}\times {{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
$(\tfrac{3}{2},\tfrac{1}{2})$ & $K^{[3]}$ & $7N-3$ & ${{\rm U}(1)}\times {{\rm U}(1)}$ &
$3$ by ${{\rm U}(1)}^2$ \\
$(2,0)$ & $K^{[0]}$ & $8N-4$ & ${{\rm U}(1)}\times {{\rm U}(1)}$ & $3 $ by ${{\rm U}(1)}^2$ \\
\bottomrule
\end{tabular}
\caption{The generators for the chiral ring of a ${{\rm SU}(2)}\times {{\rm SO}}(3)$ gauge
theory with matter in $[2,2]$.}
\label{tab:Ops_SU2xSO3_Rep22}
\end{table}
Note the change in GNO-charges in accordance with the use of $K^{[3]}$ instead
of $K^{[1]}$.
\subsubsection{Quotient \texorpdfstring{$\mathrm{PSO}(4) $}{PSO(4)}}
\paragraph{Hilbert basis}
The semi-groups $S_{\pm}^{(2)}\coloneqq C_{\pm}^{(2)} \cap \left(K^{[0]} \cup
K^{[1]} \cup K^{[2]}
\cup K^{[3]} \right)$ have Hilbert bases that are determined by the
ray generators. Fig.~\ref{Fig:Hilbert_basis_PSO4_Rep22} depicts the
situation and the Hilbert bases read
\begin{equation}
\label{eqn:Hilbert_basis_PSO4_Rep22}
\Hcal(S_{\pm}^{(2)}) = \Big\{ (\tfrac{1}{2},\pm\tfrac{1}{2}) , (1,0)
\Big\} \; .
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x-1,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x +1/2,2*\y +1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x-1
+1/2,2*\y-1+1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] at (2*\x-1+1/2,2*\y+1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] at (2*\x +1/2,2*\y-1 +1/2) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw[black,dashed,thick] (0,0) -- (4.5,0);
\draw (6,3) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,3.05) node { $K^{[0]}$ lattice};
\draw (6,2) node[draw,diamond,inner sep=0.8pt,black] {};
\draw (7.2,2.05) node { $K^{[1]}$ lattice};
\draw (6,1) node[draw,cross out,inner sep=0.8pt,thick,black] {};
\draw (7.2,1.05) node { $K^{[2]}$ lattice};
\draw (6,0) node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] {};
\draw (7.2,0.05) node { $K^{[3]}$ lattice};
\draw[black,thick] (1/2,1/2) circle (4pt);
\draw[black,thick] (1/2,-1/2) circle (4pt);
\draw[black,thick] (1,0) circle (4pt);
\draw (4,1.5) node {$S_+^{(2)}$};
\draw (4,-1.5) node {$S_-^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups and their ray-generators (black circled points) for
the quotient $\mathrm{PSO}(4)$ and the representation $[2,2]$.}
\label{Fig:Hilbert_basis_PSO4_Rep22}
\end{figure}
\paragraph{Hilbert series}
The Hilbert series reads
\begin{subequations}
\label{eqn:HS_PSO4_Rep22}
\begin{align}
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,2]}(t,z_1,z_2,N)&=
\frac{R(t,z_1,z_2,N)}{\left(1-t^2\right)^2 \left(1-t^{6 N-2}\right) \left(1-z_2
t^{4 N-2}\right)} \; ,\\
R(t,z_1,z_2,N)&=1
+z_1 t^{3 N}
+z_1 t^{3 N-1}
+z_1 z_2 t^{3 N}
+z_1 z_2 t^{3 N-1}
+z_2 t^{4 N}
+2 z_2 t^{4 N-1} \\
&\qquad
+t^{6 N-2}
+2 t^{6 N-1}
+z_1 z_2 t^{7 N-2}
+z_1 z_2 t^{7 N-1} \notag \\
&\qquad
+z_1 t^{7 N-2}
+z_1 t^{7 N-1}
+z_2 t^{10 N-2}\notag \; .
\end{align}
\end{subequations}
The numerator of~\eqref{eqn:HS_PSO4_Rep22} is palindromic polynomial of degree
$10N-2$; while the denominator is a polynomial of degree $10N$. Hence, the
difference in degree is two, which corresponds to the quaternionic dimension
of the moduli space.
Similarly to the previous cases, the denominator of~\eqref{eqn:HS_PSO4_Rep22}
has a pole of order four at $t=1$, which equals the complex dimension of the
moduli space.
\paragraph{Gauging a $\mathbb Z_2$}
As before, by gauging the $\mathbb Z_2$-factor corresponding to $z_1$ we recover the
${{\rm SO}}(4)$-result
\begin{subequations}
\label{eqn:PSO4_gauging_Rep22}
\begin{equation}
\mathrm{HS}_{{{\rm SO}}(4)}^{[2,2]}(t,z,N) = \frac{1}{2}
\left(\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,2]}(t,z_1{=}1,z_2{=}z,N) +
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,2]}(t,z_1{=}-1,z_2{=}z,N)\right) \; ,
\end{equation}
while gauging the $\mathbb Z_2$-factor with fugacity $z_2$ provides the
${{\rm SO}}(3)\times{{\rm SU}(2)}$-result
\begin{equation}
\mathrm{HS}_{{{\rm SO}}(3)\times{{\rm SU}(2)}}^{[2,2]}(t,z_1,N) = \frac{1}{2}
\left(\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,2]}(t,z_1,z_2{=}1,N) +
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,2]}(t,z_1,z_2{=}{-1},N)\right) \; .
\end{equation}
Furthermore, employing the redefined fugacities $w_1$, $w_2$
of~\eqref{eqn:redefine_Z2-grading_A1xA1} one reproduces the ${{\rm SU}(2)}\times
{{\rm SO}}(3)$
Hilbert series as follows:
\begin{equation}
\mathrm{HS}_{{{\rm SU}(2)}\times{{\rm SO}}(3)}^{[2,2]}(t,z_2{=}w_1,N) = \frac{1}{2}
\left(\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,2]}(t,w_1,w_2{=}1,N) +
\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,2]}(t,w_1,w_2{=}{-1},N)\right) \; .
\end{equation}
\end{subequations}
Therefore, one can obtain the Hilbert series for \emph{all} five quotients from
the $\mathrm{PSO}(4)$-result~\eqref{eqn:HS_PSO4_Rep22} by employing the
$\mathbb Z_2$-gaugings~\eqref{eqn:SO4_gauging_Rep22}
and~\eqref{eqn:PSO4_gauging_Rep22}.
\paragraph{Plethystic logarithm}
Inspecting the PL leads to
\begin{align}
\mathrm{PL}(\mathrm{HS}_{\mathrm{PSO}(4)}^{[2,2]})= 2 t^2 + z_1
t^{\Delta(\tfrac{1}{2},\tfrac{1}{2})}(1+t)
+ z_1 z_2 t^{\Delta(\tfrac{1}{2},-\tfrac{1}{2})}(1+t)
+z_2 t^{\Delta(1,0)} (1+2t+t^2) +\ldots
\end{align}
such that we can summarise the monopole generators as in
Tab.~\ref{tab:Ops_PSO4_Rep22}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c}
\toprule
$(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ & dressings \\
\midrule
$(\tfrac{1}{2},\tfrac{1}{2})$ & $K^{[1]}$ & $3N-1$ & ${{\rm U}(1)}\times {{\rm SU}(2)}$ & $1$
by ${{\rm U}(1)}$\\
$(\tfrac{1}{2},-\tfrac{1}{2})$ & $K^{[3]}$ & $3N-1$ & ${{\rm U}(1)}\times {{\rm SU}(2)}$ &
$1$ by ${{\rm U}(1)}$ \\
$(1, 0)$ & $K^{[2]}$ & $4N-2$ & ${{\rm U}(1)}\times {{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$ \\
\bottomrule
\end{tabular}
\caption{The generators for the chiral ring of a $\mathrm{PSO}(4)$ gauge
theory with matter in $[2,2]$.}
\label{tab:Ops_PSO4_Rep22}
\end{table}
\subsection{Representation \texorpdfstring{$[4,2]$}{[4,2]}}
The conformal dimension for this case reads
\begin{align}
\Delta(m_1,m_2)&=N\big(
\left| 3 m_1-m_2\right|
+\left| m_1-3 m_2\right|
+\left| m_1+m_2\right|
+3 \left| m_1-m_2\right|
+2 \left| m_1\right|
+2 \left| m_2\right|\big)
\notag \\*
&\qquad -|m_1 + m_2| - |m_1 - m_2|\; .
\label{eqn:delta_A1xA1_Rep42}
\end{align}
The interesting feature of this representation is its asymmetric behaviour
under exchange of $m_1$ and $m_2$.
As before, the conformal dimension~\eqref{eqn:delta_A1xA1_Rep42} defines a fan
in the dominant Weyl chamber of, which is spanned by three $2$-dimensional cones
\begin{subequations}
\label{eqn:fan_A1xA1_Rep42}
\begin{align}
C_{1}^{(2)} &= \Big\{ (m_1 \geq - m_2) \wedge (m_2 \leq 0) \Big\} \; , \\
C_{2}^{(2)} &= \Big\{ (m_1 \geq 3 m_2) \wedge (m_2 \geq 0) \Big\} \; , \\
C_{3}^{(2)} &= \Big\{ (m_1 \geq m_2) \wedge (m_1 \leq 3 m_2) \Big\} \; .
\end{align}
\end{subequations}
\subsubsection{Quotient \texorpdfstring{${\rm Spin}(4)$}{Spin(4)}}
\paragraph{Hilbert basis}
Starting from the fan~\eqref{eqn:fan_A1xA1_Rep42} with cones $C_p^{(2)}$
($p=1,2,3$), the Hilbert bases for the semi-groups $S_p^{(2)} \coloneqq
C_p^{(2)} \cap K^{[0]}$ are simply given by the ray generators, see for
instance Fig.~\ref{Fig:Hilbert_basis_Spin4_Rep42}.
\begin{equation}
\Hcal(S_{1}^{(2)}) = \Big\{ (2,0), (1,-1) \Big\} , \quad
\Hcal(S_{2}^{(2)}) = \Big\{ (3,1), (2,0) \Big\} , \quad
\Hcal(S_{3}^{(2)}) = \Big\{ (1,1), (3,1) \Big\} .
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3*1.5,1*1.5);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw[black,dashed,thick] (0,0) -- (4.5,0);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw[thick,black] (2,0) circle (4pt);
\draw[thick,black] (1,1) circle (4pt);
\draw[thick,black] (1,-1) circle (4pt);
\draw[thick,black] (3,1) circle (4pt);
\draw (3.8,2.3) node {$S_3^{(2)}$};
\draw (3.8,0.7) node {$S_2^{(2)}$};
\draw (4,-1.3) node {$S_1^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups and their ray-generators (black circled points) for
the quotient ${\rm Spin}(4)$ and the representation $[4,2]$.}
\label{Fig:Hilbert_basis_Spin4_Rep42}
\end{figure}
\paragraph{Hilbert series}
The Hilbert series reads
\begin{subequations}
\label{eqn:HS_Spin4_Rep42}
\begin{align}
\mathrm{HS}_{{\rm Spin}(4)}^{[4,2]}(t,N)&= \frac{R(t,N)}{\left(1-t^2\right)^2 \left(1-t^{18
N-2}\right) \left(1-t^{20 N-4}\right) \left(1-t^{26
N-6}\right)} \; , \\
R(t,N)&= 1
+t^{10 N-2}(1+t)
+t^{18 N-1}
+t^{20 N-4}(1+3t+t^2)
\\
&\qquad
+ t^{26 N-5}(2+t)
-t^{28 N-4}(1+t)
+t^{36 N-7}(1+t)
\notag \\
&\qquad
-t^{38 N-6}(1+2t)
-t^{44 N-8}(1+3t+t^2)
-t^{46 N-9} \notag \\
&\qquad
-t^{54 N-9}(1+t)
-t^{64 N-10} \notag \; .
\end{align}
\end{subequations}
The numerator of~\eqref{eqn:HS_Spin4_Rep42} is an anti-palindromic polynomial
of degree $64N-10$, while the denominator is of degree $64N-8$. Consequently,
the difference in degree is two.
Moreover, the rational function~\eqref{eqn:HS_Spin4_Rep42} has a pole of order
four as $t\to1$ because $R(t{=}1,N)=0$, but $\tfrac{\mathrm{d}}{\mathrm{d} t} R(t,N)
|_{t=1}\neq0$.
\paragraph{Plethystic logarithm}
Inspecting the PL yields for $N\geq 3$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm Spin}(4)}^{[4,2]}) = 2t^2 &+ t^{\Delta(1,1)} (1+t) +
t^{\Delta(1,-1)}
(1+t)
+ t^{\Delta(2,0)} (1+2t) + t^{\Delta(3,1)} (1+2t+t^2) \\
&-t^{\Delta(1,1)+\Delta(1,-1)} (1+2t+t^2)
-t^{\Delta(1,1)+\Delta(2,0)}(1+3t+3t^2+t^3) + \ldots \notag
\end{align}
leads to an identification of generators as in Tab.~\ref{tab:Ops_Spin4_Rep42}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ & \#
dressings \\ \midrule
Casimirs & --- & --- & $2$ & --- & --- \\
monopole & $(1,1)$ & $K^{[0]}$ & $10N-2 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
monopole & $(1,-1)$ & $K^{[0]}$ & $18N-2 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
monopole & $(2,0)$ & $K^{[0]}$ & $20N-4 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $2$ by ${{\rm U}(1)}^2$\\
monopole & $(3,1)$ & $K^{[0]}$ & $26N-6 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$ \\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a ${\rm Spin}(4)$ gauge
theory with matter transforming in $[4,2]$.}
\label{tab:Ops_Spin4_Rep42}
\end{table}
We observe that $(2,0)$ has only $2$ dressings, although we would
expect $3$. We know from other examples that there should be a relation at $2
\Delta(1,1) +2 =20N-2$ which is precisely the dimension of the second dressing
of $(2,0)$.
\subsubsection{Quotient \texorpdfstring{${{\rm SO}}(4)$}{SO(4)}}
\paragraph{Hilbert basis}
The semi-groups $S_p^{(2)}\coloneqq C_p^{(2)} \cap \left(K^{[0]} \cup K^{[2]}
\right)$ have Hilbert bases as shown in Fig.~\ref{Fig:Hilbert_basis_SO4_Rep42}
or explicitly:
\begin{subequations}
\begin{alignat}{2}
\Hcal(S_{1}^{(2)}) &= \Big\{ (1,0), (1,-1) \Big\} \; , \quad &
\Hcal(S_{2}^{(2)}) &= \Big\{ (3,1), (1,0) \Big\} \; , \\
\Hcal(S_{3}^{(2)}) &= \Big\{ (1,1), (2,1) , (3,0) \Big\} \; .
\end{alignat}
\end{subequations}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x-1,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x,2*\y-1) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3*1.5,1*1.5);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw[black,dashed,thick] (0,0) -- (4.5,0);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw (6,1) node[draw,cross out,inner sep=0.8pt,thick,black] {};
\draw (7.2,1.05) node { $K^{[2]}$ lattice};
\draw[black,thick] (1,0) circle (4pt);
\draw[black,thick] (1,1) circle (4pt);
\draw[black,thick] (1,-1) circle (4pt);
\draw[red,thick] (2,1) circle (4pt);
\draw[black,thick] (3,1) circle (4pt);
\draw (3.8,2.3) node {$S_3^{(2)}$};
\draw (3.6,0.6) node {$S_2^{(2)}$};
\draw (4.2,-1.4) node {$S_1^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups for the quotient ${{\rm SO}}(4)$ and the
representation $[4,2]$. The black circled points are the ray generators and the
red circled point completes the Hilbert basis for $S_{3}^{(2)}$.}
\label{Fig:Hilbert_basis_SO4_Rep42}
\end{figure}
\paragraph{Hilbert series}
\begin{subequations}
\label{eqn:HS_SO4_Rep42}
\begin{align}
\mathrm{HS}_{{{\rm SO}}(4)}^{[4,2]}(t,z,N) &= \frac{R(t,z,N)}{
\left(1-t^2\right)^2
\left(1-t^{10 N-2}\right)
\left(1-t^{18 N-2}\right)
\left(1-t^{26 N-6}\right)
\left(1-z t^{10 N-2}\right)} \; ,\\
R(t,z,N)&=1
+t^{10 N-1}
+ z t^{10 N-1} (2+t)
+z t^{18 N-4}(1+2t+t^3)
+t^{18 N-1} \notag \\
&\qquad
-z t^{20 N-4}(1+3t+t^2)
+2 t^{26 N-5}(2+t)
\label{eqn:HS_SO4_Rep42_Num}\\
&\qquad
-t^{28 N-6}(1+2t+2t^2+2t^3)
-z t^{28 N-3}\notag\\
&\qquad
-t^{36 N-7}
- z t^{36 N-7}(2+2t+2t^2+t^3)
+z t^{38 N-6}(1+2t)
\notag\\
&\qquad
-t^{44 N-8} (1+3t+t^2)
+z t^{46 N-9}
+t^{46 N-8} (1+2t+t^2)
\notag\\
&\qquad
+t^{54 N-10}(1+2t)
+z t^{54 N-9}
+z t^{64 N-10} \; . \notag
\end{align}
\end{subequations}
The numerator~\eqref{eqn:HS_SO4_Rep42_Num} is a palindromic polynomial of
degree
$64N-10 $, while the denominator is of degree $64N-8$. Consequently, the
difference of the degree is two.
Also, the Hilbert series~\eqref{eqn:HS_SO4_Rep42} has a pole of order four as
$t\to 1$, because $R(t{=}1,z,N)=0$ and $\tfrac{\mathrm{d}}{\mathrm{d} t} R(t,z,N) |_{t=1}
=0$, but $\tfrac{\mathrm{d}^2}{\mathrm{d} t^2} R(t,z,N) |_{t=1} \neq0$.
\paragraph{Plethystic logarithm}
Inspecting the PL reveals
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{{\rm SO}}(4)}^{[4,2]})= 2 t^2 &+ z t^{\Delta(1,0)} (1+2t +t^2)
+t^{\Delta(1,1)} (1+t)
+z t^{\Delta(2,1)} (1+ 2t + t^2) \\
&+ t^{\Delta(1,-1)} (1+t)
-z t^{2 \Delta(1,0)} (1+ 3t + 3t^2 + t^3)
-t^{2 \Delta(1,1) +2} (4+2t +t^2) \notag \\
&+t^{\Delta(3,1)} (1+2t+t^2) +\ldots \; , \notag
\end{align}
such that the monopole generators can be summarised as in
Tab.~\ref{tab:Ops_SO4_Rep42}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings
\\ \midrule
Casimirs & --- & --- & $2$ & --- & --- \\
monopole & $(1,0)$ & $K^{[2]}$ & $10N-2 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$\\
monopole & $(1,1)$ & $K^{[0]}$ & $10N-2 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
monopole & $(2,1)$ & $K^{[2]}$ & $18N-4 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$ \\
monopole & $(1,-1)$ & $K^{[0]}$ & $18N-2 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
monopole & $(3,1)$ & $K^{[0]}$ & $26N-6 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$ \\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a ${{\rm SO}}(4)$ gauge
theory with matter transforming in $[4,2]$.}
\label{tab:Ops_SO4_Rep42}
\end{table}
\paragraph{Gauging the $\boldsymbol{\mathbb Z_2}$}
Again, one can gauge the finite symmetry to recover the ${\rm Spin}(4)$ Hilbert
series
\begin{equation}
\mathrm{HS}_{{\rm Spin}(4)}^{[4,2]}(t,N) = \frac{1}{2} \left(
\mathrm{HS}_{{{\rm SO}}(4)}^{[4,2]}(t,z{=}1,N) +
\mathrm{HS}_{{{\rm SO}}(4)}^{[4,2]}(t,z{=}-1,N)\right) \; .
\label{eqn:SO4_gauging_Rep42}
\end{equation}
\subsubsection{Quotient \texorpdfstring{${{\rm SO}}(3) \times {{\rm SU}(2)}$}{SO(3)xSU(2)}}
\paragraph{Hilbert basis}
The semi-groups $S_p^{(2)}\coloneqq C_p^{(2)} \cap \left(K^{[0]} \cup K^{[1]}
\right)$ ($p=1,2,3$) have Hilbert bases that go beyond the set of ray
generators. We refer to Fig.~\ref{Fig:Hilbert_basis_SO3xSU2_Rep42} and the
Hilbert bases are obtained as follows:
\begin{subequations}
\label{eqn:Hilbert_basis_SO3xSU2_Rep42}
\begin{alignat}{2}
\Hcal(S_{1}^{(2)}) &= \Big\{ (2,1), (\tfrac{3}{2},-\tfrac{1}{2}), (1,-1)
\Big\} \; ,
\qquad &
\Hcal(S_{2}^{(2)}) &= \Big\{ (3,1), (\tfrac{5}{2},\tfrac{1}{2}) , (2,0)
\Big\} \; , \\
\Hcal(S_{3}^{(2)}) &= \Big\{ (1,1), (3,1) \Big\} \; .
\end{alignat}
\end{subequations}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x +1/2,2*\y +1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x-1 +1/2,2*\y-1+1/2)
{};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3*1.5,1*1.5);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw[black,dashed,thick] (0,0) -- (4.5,0);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw (6,1) node[draw,diamond,inner sep=0.8pt,black] {};
\draw (7.2,1.05) node { $K^{[1]}$ lattice};
\draw[thick,black] (1/2,1/2) circle (4pt);
\draw[thick,black] (1,-1) circle (4pt);
\draw[thick,green] (3/2,-1/2) circle (4pt);
\draw[thick,black] (2,0) circle (4pt);
\draw[thick,red] (5/2,1/2) circle (4pt);
\draw[thick,black] (3,1) circle (4pt);
\draw (3.8,2.3) node {$S_3^{(2)}$};
\draw (3.8,0.7) node {$S_2^{(2)}$};
\draw (4,-1.3) node {$S_1^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups for the quotient ${{\rm SO}}(3)\times {{\rm SU}(2)}$ and the
representation $[4,2]$. The black circled points are the ray generators, the
red circled point completes the Hilbert basis for $S_{2}^{(2)}$, while the
green circled point completes the Hilbert basis of $S_{1}^{(2)}$.}
\label{Fig:Hilbert_basis_SO3xSU2_Rep42}
\end{figure}
\paragraph{Hilbert series}
We compute the Hilbert series to
\begin{subequations}
\label{eqn:HS_SO3xSU2_Rep42}
\begin{align}
\mathrm{HS}_{{{\rm SO}}(3)\times {{\rm SU}(2)}}^{[4,2]}(t,z_1,N)&=
\frac{R(t,z_1,N)}{\left(1-t^2\right)^2
\left(1-t^{18 N-2}\right)
\left(1-t^{20 N-4}\right)
\left(1-t^{26 N-6}\right)} \; ,\\
R(t,z_1,N)&= 1
+z_1 t^{5 N-1}(1+t)
+t^{10 N-2}(1+t)
+z_1 t^{15 N-3}(1+t)
\\
&\qquad
+t^{18 N-1}
+z_1 t^{19 N-3}(1+2t+t^3)
\notag \\
&\qquad
+t^{20 N-4}(1+3t+t^2)
+z_1 t^{23 N-5}(1+2t-t^3)
\notag \\
&\qquad
+ t^{26 N-5}(2+t)
-t^{28 N-4}(1+t)
+z_1 t^{31 N-6}(1+t)
\notag \\
&\qquad
-z_1 t^{33 N-5}(1+t)
+t^{36 N-7}(1+t)
-t^{38 N-6}(1+2t)
\notag \\
&\qquad
+z_1 t^{41 N-8} (1-2t^2-t^3)
-t^{44 N-8}(1+3t+t^2)
\notag \\
&\qquad
-z_1 t^{45 N-9}(1+2t+t^2)
-t^{46 N-9}
-z_1 t^{49 N-8}(1+t)
\notag \\
&\qquad
-t^{54 N-9}(1+t)
-z_1 t^{59 N-10}(1+t)
-t^{64 N-10} \; . \notag
\end{align}
\end{subequations}
The numerator of~\eqref{eqn:HS_SO3xSU2_Rep42} is an anti-palindromic polynomial
of degree $64N-10$,
while the denominator is of degree $64N-8$. Thus, the difference in degrees is
again 2.
In addition, the Hilbert series~\eqref{eqn:HS_SO3xSU2_Rep42} has a pole of
order
4 as $t\to1$, because $R(t{=}1,z_1,N)=0$, but $\tfrac{\mathrm{d}}{\mathrm{d} t}
R(t,z_1,N)|_{t=1}\neq0$.
\paragraph{Plethystic logarithm}
Analysing the PL yields
\begin{align}
PL = 2t^2 &+ z_1 t^{\Delta(\tfrac{1}{2},\tfrac{1}{2})} (1+t) -
t^{\Delta(\tfrac{1}{2},\tfrac{1}{2})+2}
+t^{\Delta(1,-1)} (1+t) \\
&+z_1 t^{\Delta(\tfrac{3}{2},-\tfrac{1}{2})} (1+2t+t^2)
+t^{\Delta(2,0)} (1+2t+t^2) \notag \\
&+z_1 t^{\Delta(\tfrac{5}{2},\tfrac{1}{2})}(1+2t +\textcolor{red}{1})
-z_1 t^{\Delta(\tfrac{1}{2},\tfrac{1}{2})+\Delta(1,-1)} (\textcolor{red}{1} +
2t +t^2) \notag \\
&-t^{\Delta(\tfrac{1}{2},\tfrac{1}{2})+\Delta(\tfrac{3}{2},-\tfrac{1}{2})}
(1+3t+3t^2+t^3) \notag\\
&-z_1 t^{\Delta(\tfrac{1}{2},\tfrac{1}{2})+\Delta(2,0)} (1+3t+3t^2+t^3)\notag \\
&+t^{\Delta(3,1)} (1+2t+t^2) +\ldots \notag \; ,
\end{align}
verfies the set of generators as presented in
Tab.~\ref{tab:Ops_SO3xSU2_Rep42}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings
\\ \midrule
Casimirs & --- & --- & $2$ & --- & --- \\
monopole & $(\tfrac{1}{2},\tfrac{1}{2})$ & $K^{[1]}$ & $5N-1 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$\\
monopole & $(1,-1)$ & $K^{[0]}$ & $18N-2 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
monopole & $(\tfrac{3}{2},-\tfrac{1}{2})$ & $K^{[1]}$ & $19N-3 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$ \\
monopole & $(2,0)$ & $K^{[0]}$ & $20N-4 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}$ \\
monopole & $(\tfrac{5}{2},\tfrac{1}{2})$ & $K^{[1]}$ & $23N-5 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $\textcolor{red}{3} (2)$ by ${{\rm U}(1)}^2$ \\
monopole & $(3,1)$ & $K^{[0]}$ & $26N-6 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$ \\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a ${{\rm SO}}(3)\times {{\rm SU}(2)}$ gauge
theory with matter transforming in $[4,2]$.}
\label{tab:Ops_SO3xSU2_Rep42}
\end{table}
The coloured term indicates that we suspect a cancellation
between one dressing of $(\tfrac{5}{2},\tfrac{1}{2})$ and one relation because
$\Delta(\tfrac{5}{2},\tfrac{5}{2})+2= 23N-3 =
\Delta(\tfrac{1}{2},\tfrac{1}{2})+\Delta(1,-1)=5N-1+18N-2$.
\subsubsection{Quotient \texorpdfstring{${{\rm SU}(2)} \times {{\rm SO}}(3)$}{SU(2)xSO(3)}}
\paragraph{Hilbert basis}
The semi-groups $S_p^{(2)}\coloneqq C_p^{(2)} \cap \left(K^{[0]} \cup K^{[3]}
\right)$ (for $p=1,2,3$) have Hilbert bases consist of the ray generators as
shown
in Fig.~\ref{Fig:Hilbert_basis_SU2xSO3_Rep42} and we obtain explicitly
\begin{equation}
\label{eqn:Hilbert_basis_SU2xSO3_Rep42}
\Hcal(S_{1}^{(2)}) = \Big\{ (2,0), (\tfrac{1}{2},-\tfrac{1}{2}) \Big\} ,
\quad
\Hcal(S_{2}^{(2)}) = \Big\{ (\tfrac{3}{2},\tfrac{1}{2}),(2,0)
\Big\} , \quad
\Hcal(S_{3}^{(2)}) = \Big\{ (1,1), (\tfrac{3}{2},\tfrac{1}{2}) \Big\} .
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] at (2*\x-1+1/2,2*\y+1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] at (2*\x +1/2,2*\y-1 +1/2) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3*1.5,1*1.5);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw[black,dashed,thick] (0,0) -- (4.5,0);
\draw (6,2) node[circle,inner sep=0.8pt,fill,black] {};
\draw (7.2,2.05) node { $K^{[0]}$ lattice};
\draw (6,1) node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] {};
\draw (7.2,1.05) node { $K^{[3]}$ lattice};
\draw (1/2,-1/2) circle (4pt);
\draw (1,1) circle (4pt);
\draw (3/2,1/2) circle (4pt);
\draw (2,0) circle (4pt);
\draw (3.8,2.3) node {$S_3^{(2)}$};
\draw (3.8,0.7) node {$S_2^{(2)}$};
\draw (4,-1.3) node {$S_1^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups for the quotient ${{\rm SU}(2)} \times {{\rm SO}}(3)$ and the
representation $[4,2]$. The black circled points are the ray generators.}
\label{Fig:Hilbert_basis_SU2xSO3_Rep42}
\end{figure}
\paragraph{Hilbert series}
We compute the Hilbert series to
\begin{subequations}
\label{eqn:HS_SU2xSO3_Rep42}
\begin{align}
\mathrm{HS}_{{{\rm SU}(2)} \times {{\rm SO}}(3)}^{[4,2]}(t,z_2,N) &=
\frac{R(t,z_2,N)}{\left(1-t^2\right)^2
\left(1-t^{18 N-2}\right)
\left(1-t^{20 N-4}\right)
\left(1-t^{26 N-6}\right)} \; , \\
R(t,z_2,N)&=1
+z_2 t^{9 N-1} (1+t)
+t^{10 N-2}(1+t)
+z_2 t^{13 N-3} (1+2t+t^2)
\notag \\
&\qquad
+t^{18 N-1}
+t^{20 N-4}(1+3t+t^2)
+z_2 t^{23 N-5}(1+2t+t^2)
\notag \\
&\qquad
+ t^{26 N-5}(2+t)
-t^{28 N-4}(1+t)
+z_2 t^{29 N-4}(1+t)
\\
&\qquad
-z_2 t^{31 N-5}(1+2t+t^2)
+z_2 t^{33 N-7}(1+2t+t^2)
\notag \\
&\qquad
-z_2 t^{35 N-7}(1+t)
+t^{36 N-7}(1+t)
-t^{38 N-6}(1+2t)
\notag \\
&\qquad
-z_2 t^{41 N-7}(1+2t+t^2)
-t^{44 N-8}(1+3t+t^2)
\notag \\
&\qquad
-t^{46 N-9}
-z_2 t^{51 N-9}(1+2t+t^2)
-t^{54 N-9}(1+t)
\notag \\
&\qquad
-z_2 t^{55 N-10}(1+t)
-t^{64 N-10} \; . \notag
\end{align}
\end{subequations}
As before, we can try to compare the quotients ${{\rm SO}}(3)\times {{\rm SU}(2)}$
and ${{\rm SU}(2)} \times {{\rm SO}}(3)$. However, due to the asymmetry in $m_1$, $m_2$ or
the asymmetry of the fan in the Weyl chamber, the Hilbert series for the two
quotients are \emph{not} related by an exchange of $z_1$ and $z_2$.
\paragraph{Plethystic logarithm}
Upon analysing the PL we find
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{{\rm SU}(2)} \times {{\rm SO}}(3)}^{[4,2]}) = 2t^2 &+ z_2
t^{\Delta(\tfrac{1}{2},-\tfrac{1}{2})} (1+t)
+ t^{\Delta(1,1)} (1+t)
+ z_2 t^{\Delta(\tfrac{3}{2},\tfrac{1}{2})} (1+2t +t^2) \\
&-t^{2\Delta(\tfrac{1}{2},-\tfrac{1}{2}) +2} -
z_2 t^{\Delta(\tfrac{1}{2},-\tfrac{1}{2})+ \Delta(1,1)} (1+2t+t^2)
\notag \\
&+t^{\Delta(2,0)} (1+2t) -
t^{\Delta(\tfrac{1}{2},-\tfrac{1}{2})+\Delta(\tfrac{3}{2},\tfrac{1}{2})}
(1+3t+3t^2+t^3) +\dots \notag \; ,
\end{align}
through which one identifies the generators as given in
Tab.~\ref{tab:Ops_SU2xSO3_Rep42}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings
\\ \midrule
Casimirs & --- & --- & $2$ & --- & --- \\
monopole & $(\tfrac{1}{2},-\tfrac{1}{2})$ & $K^{[3]}$ & $9N-1 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$\\
monopole & $(1,1)$ & $K^{[0]}$ & $10N-2 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
monopole & $(\tfrac{3}{2},\tfrac{1}{2})$ & $K^{[3]}$ & $13N-3 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$ \\
monopole & $(2,0)$ & $K^{[0]}$ & $20N-4 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$ \\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a ${{\rm SU}(2)}\times {{\rm SO}}(3)$ gauge
theory with matter transforming in $[4,2]$.}
\label{tab:Ops_SU2xSO3_Rep42}
\end{table}
The terms in the denominator of the Hilbert series can be seen to reproduce
these generators
\begin{subequations}
\begin{align}
(1-t^{18N-2}) &= (1-z_2 t^{9N-1}) (1+z_2 t^{9N-1}) \; , \\
(1-t^{26N-6}) &= (1-z_2 t^{13N-3}) (1+z_2 t^{13N-3}) \; .
\end{align}
\end{subequations}
Unfortunately, we are unable to reduce the numerator accordingly.
\subsubsection{Quotient \texorpdfstring{$\mathrm{PSO}(4)$}{PSO(4)}}
\paragraph{Hilbert basis}
The semi-groups $S_p^{(2)}\coloneqq C_p^{(2)} \cap \left(K^{[0]} \cup K^{[1]}
\cup K^{[2]} \cup K^{[3]} \right)$ (for $p=1,2,3$) have Hilbert bases that are
determined by the ray generators. Fig.~\ref{Fig:Hilbert_basis_PSO4_Rep42}
depicts the situation and the Hilbert bases read:
\begin{equation}
\label{eqn:Hilbert_basis_PSO4_Rep42}
\Hcal(S_{1}^{(2)}) = \Big\{ (1,0), (\tfrac{1}{2},-\tfrac{1}{2}) \Big\} , \quad
\Hcal(S_{2}^{(2)}) = \Big\{ (\tfrac{3}{2},\tfrac{1}{2}), (1,0) \Big\}, \quad
\Hcal(S_{3}^{(2)}) = \Big\{ (\tfrac{1}{2},\tfrac{1}{2}),
(\tfrac{3}{2},\tfrac{1}{2}) \Big\} .
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-3);
\coordinate (YAxisMax) at (0,3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,3.2) node {$m_2$};
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (2*\x-1,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x-1,2*\y) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {0,1}{%
\node[draw,cross out,inner sep=0.8pt,thick,black] at (2*\x,2*\y-1) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x +1/2,2*\y +1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,diamond,inner sep=0.8pt,black] at (2*\x-1 +1/2,2*\y-1+1/2)
{};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] at (2*\x-1+1/2,2*\y+1/2) {};
}
}
\foreach \x in {0,1,...,2}{%
\foreach \y in {-1,0,...,1}{%
\node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] at (2*\x +1/2,2*\y-1 +1/2) {};
}
}
\draw[black,dashed,thick] (0,0) -- (3.2,3.2);
\draw[black,dashed,thick] (0,0) -- (3*1.5,1*1.5);
\draw[black,dashed,thick] (0,0) -- (3.2,-3.2);
\draw[black,dashed,thick] (0,0) -- (4.5,0);
\draw (6,3) node[circle,inner sep=0.8pt,fill,red] {};
\draw (7.2,3.05) node { $K^{[0]}$ lattice};
\draw (6,2) node[draw,diamond,inner sep=0.8pt,black] {};
\draw (7.2,2.05) node { $K^{[1]}$ lattice};
\draw (6,1) node[draw,cross out,inner sep=0.8pt,thick,black] {};
\draw (7.2,1.05) node { $K^{[2]}$ lattice};
\draw (6,0 ) node[draw,regular polygon,regular polygon sides=3,inner
sep=0.7pt,black] {};
\draw (7.2,0.05) node { $K^{[3]}$ lattice};
\draw[black,thick] (1/2,1/2) circle (4pt);
\draw[black,thick] (1/2,-1/2) circle (4pt);
\draw[black,thick] (1,0) circle (4pt);
\draw[black,thick] (3/2,1/2) circle (4pt);
\draw (3.8,2.3) node {$S_3^{(2)}$};
\draw (3.9,0.6) node {$S_2^{(2)}$};
\draw (4,-1.3) node {$S_1^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups and their ray-generators (black circled points) for
the quotient $\mathrm{PSO}(4)$ and the representation $[4,2]$.}
\label{Fig:Hilbert_basis_PSO4_Rep42}
\end{figure}
\paragraph{Hilbert series}
We obtain the following Hilbert series
\begin{subequations}
\label{eqn:HS_PSO4_Rep42}
\begin{align}
\mathrm{HS}_{\mathrm{PSO}(4)}^{[4,2]}(t,z_1,z_2,N)&=
\frac{R(t,z_1,z_2,N)}{\left(1-t^2\right)^2 \left(1-t^{10 N-2}\right)
\left(1-t^{18 N-2}\right) \left(1-t^{26 N-6}\right) \left(1-t^{10 N-2}
z_2\right)} \; , \\
R(t,z_1,z_2,N)&= 1
+z_1 t^{5 N-1}(1+t)
+z_1 z_2 t^{9 N-1}(1+t)
+z_1 z_2 t^{9 N}
+t^{10 N-1} \\
&\qquad
+z_2 t^{10 N-1} (2+t)
+z_1 z_2 t^{13 N-3} (1+2t+t^2)
-z_1 z_2 t^{15 N-3} (1+t)
\notag \\
&\qquad
+z_2 t^{18 N-4} (1+2t+t^2)
+t^{18 N-1}
-z_1 z_2 t^{19 N-3} (1+t)
\notag \\
&\qquad
+z_1 t^{19 N-2}(1+t)
-z_2 t^{20 N-4}(1+3t+t^2)
-z_1 t^{23 N-3}(1+t)
\notag \\
&\qquad
+ t^{26 N-5}(2+t)
-t^{28 N-6}(1+2t+2t^2+2t^3)
-z_2 t^{28 N-3}
\notag \\
&\qquad
-z_1 t^{29 N-4}(1+t)
+z_1 t^{31 N-6} (1+t)
-z_1 z_2 t^{31 N-5}(1+2t+t)
\notag \\
&\qquad
-z_1 t^{33 N-7}(1+2t+t^2)
+z_1 z_2 t^{33 N-5}(1+t)
\notag \\
&\qquad
-z_1 z_2 t^{35 N-7}(1+t)
- z_2 t^{36 N-7}(2+2t+2t^2+t^3)
-t^{36 N-7}
\notag \\
&\qquad
+z_2 t^{38 N-6}(1+2t)
-z_1 z_2 t^{41 N-8} (1+t)
-t^{44 N-8} (1+3t+t^2)\notag \\
&\qquad
+z_1 z_2 t^{45 N-9} (1+t)
-z_1 t^{45 N-8}(1+t)
\notag \\
&\qquad
+z_2 t^{46 N-9}
+t^{46 N-8} (1+2t+t^2)
-z_1 t^{49 N-8}(1+t)
\notag \\
&\qquad
+z_1 t^{51 N-9}(1+2t+t^2)
+t^{54 N-10}(1+2t)
+z_2 t^{54 N-9}
\notag \\
&\qquad
+z_1 t^{55 N-10}(1+t)
+z_1 z_2 t^{59 N-10}(1+t)
+z_2 t^{64 N-10} \notag \; .
\end{align}
\end{subequations}
The numerator of~\eqref{eqn:HS_PSO4_Rep42} is a palindromic polynomial of
degree
$64N-10$, while the denominator is of degree $64N-8 $. Hence, the difference in
degrees is again 2.
Moreover, the Hilbert series~\eqref{eqn:HS_PSO4_Rep42} has a pole of order 4 as
$t\to 1$ because $R(1,z_1,z_2,N)=0$ and $\tfrac{\mathrm{d}}{\mathrm{d} t}
R(t,z_1,z_2,N)|_{t\to1}=0$, while $\tfrac{\mathrm{d}^2}{\mathrm{d} t^2}
R(t,z_1,z_2,N)|_{t\to1}\neq0$.
\paragraph{Plethystic logarithm}
Working with the PL instead reveals further insights
\begin{align}
\mathrm{PL}(\mathrm{HS}_{\mathrm{PSO}(4)}^{[4,2]})= 2t^2 &+ z_1
t^{\Delta(\tfrac{1}{2},\tfrac{1}{2})} (1+t)
+ z_1 z_2 t^{\Delta(\tfrac{1}{2},-\tfrac{1}{2})} (1+t)
+z_2 t^{\Delta(1,0)} (1+2t +t^2) \\
&-t^{2\Delta(\tfrac{1}{2},\tfrac{1}{2}) +2}
+z_1 z_2 t^{\Delta(\tfrac{3}{2},\tfrac{1}{2})} (1+2t+t^2) \notag \\
&-z_2 t^{\Delta(\tfrac{1}{2},\tfrac{1}{2})+\Delta(\tfrac{1}{2},\tfrac{1}{2})}
(1+2t +t^2)
-z_1 z_2
t^{\Delta(\tfrac{1}{2},\tfrac{1}{2})+\Delta(1,0)}
(1+3t +3t^2+t^3) + \ldots \notag \; .
\end{align}
The list of generators, together with their properties, is provided in
Tab.~\ref{tab:Ops_PSO4_Rep42}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings
\\ \midrule
Casimirs & --- & --- & $2$ & --- & --- \\
monopole & $(\tfrac{1}{2},\tfrac{1}{2})$ & $K^{[1]}$ & $5N-1 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$\\
monopole & $(\tfrac{1}{2},-\tfrac{1}{2})$ & $K^{[3]}$ & $9N-1 $ & ${{\rm U}(1)} \times
{{\rm SU}(2)}$ & $1$ by ${{\rm U}(1)}$ \\
monopole & $(1,0)$ & $K^{[2]}$ & $10N-2 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$\\
monopole & $(\tfrac{3}{2},\tfrac{1}{2})$ & $K^{[3]}$ & $13N-3 $ & ${{\rm U}(1)} \times
{{\rm U}(1)}$ & $3$ by ${{\rm U}(1)}^2$ \\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a $\mathrm{PSO}(4)$ gauge
theory with matter transforming in $[4,2]$.}
\label{tab:Ops_PSO4_Rep42}
\end{table}
\paragraph{Gauging a $\mathbb Z_2$}
The global $\mathbb Z_2\times \mathbb Z_2$ symmetry allows us to compute the Hilbert series
for all five quotients from the $\mathrm{PSO}(4)$ result.
We start by gauging the $\mathbb Z_2$-factor with fugacity $z_1$ (and relabel $z_2$ as
$z$) and recover the ${{\rm SO}}(4)$-result
\begin{subequations}
\label{eqn:PSO4_gauging_Rep42}
\begin{equation}
\mathrm{HS}_{{{\rm SO}}(4)}^{[4,2]}(t,z,N) = \frac{1}{2}
\left(\mathrm{HS}_{\mathrm{PSO}(4)}^{[4,2]}(t,z_1{=}1,z_2{=}z,N) +
\mathrm{HS}_{\mathrm{PSO}(4)}^{[4,2]}(t,z_1{=}-1,z_2{=}z,N)\right) \; .
\end{equation}
In contrast, gauging the other $\mathbb Z_2$-factor with fugacity $z_1$ provides the
${{\rm SO}}(3)\times{{\rm SU}(2)}$-result
\begin{equation}
\mathrm{HS}_{{{\rm SO}}(3)\times{{\rm SU}(2)}}^{[4,2]}(t,z_1,N) = \frac{1}{2}
\left(\mathrm{HS}_{\mathrm{PSO}(4)}^{[4,2]}(t,z_1,z_2{=}1,N) +
\mathrm{HS}_{\mathrm{PSO}(4)}^{[4,2]}(t,z_1,z_2{=}{-1},N)\right) \; .
\end{equation}
Lastly, switching to $w_1$, $w_2$ fugacities as
in~\eqref{eqn:redefine_Z2-grading_A1xA1} allows to recover the Hilbert series
for ${{\rm SU}(2)}\times{{\rm SO}}(3)$ as follows:
\begin{equation}
\mathrm{HS}_{{{\rm SU}(2)}\times{{\rm SO}}(3)}^{[4,2]}(t,z_2{=}w_1,N) = \frac{1}{2}
\left(\mathrm{HS}_{\mathrm{PSO}(4)}^{[4,2]}(t,w_1,w_2{=}1,N) +
\mathrm{HS}_{\mathrm{PSO}(4)}^{[4,2]}(t,w_1,w_2{=}{-1},N)\right) \; .
\end{equation}
\end{subequations}
In conclusion, the $\mathrm{PSO}(4)$ result is sufficient to obtain the
remaining four quotients by gauging of various $\mathbb Z_2$ global symmetries as
in~\eqref{eqn:PSO4_gauging_Rep42} and~\eqref{eqn:SO4_gauging_Rep42}.
\subsection{Comparison to \texorpdfstring{${{\rm O}}(4)$}{O(4)}}
In this subsection we explore the orthogonal group ${{\rm O}}(4)$,
related to ${{\rm SO}}(4)$ by $\mathbb Z_2$. To begin with, we summarise the set-up as
presented in~\cite[App.~A]{Cremonesi:2014uva}. The dressing factor
$P_{{{\rm O}}(4)}(t)$ and the GNO lattice of ${{\rm O}}(4)$ equal those of ${{\rm SO}}(5)$.
Moreover, the dominant Weyl chamber is parametrised by $(m_1,m_2)$ subject to
$m_1 \geq m_2 \geq 0$. Graphically, the Weyl chamber is the upper half of the
yellow-shaded region in Fig.~\ref{fig:A1xA1_sublattices} with the lattices
$K^{[0]}\cup K^{[2]}$ present.
Consequently, the dressing function is given as
\begin{equation}
P_{{{\rm O}}(4)}(t,m_1,m_2) = \begin{cases} \frac{1}{\left(1-t^2 \right)
\left(1-t^4\right)} \; ,& m_1=m_2=0 \; ,\\
\frac{1}{\left(1-t \right) \left(1-t^2 \right)}\; , & m_1=m_2 >0 \; , \\
\frac{1}{\left(1-t \right) \left(1-t^2 \right)} \; , & m_1>0, \ m_2=0 \; ,\\
\frac{1}{\left(1-t \right)^2} \; , & m_1>m_2>0 \; .
\end{cases}
\end{equation}
It is apparent that ${{\rm O}}(4)$ has a different Casimir invariant as ${{\rm SO}}(4)$,
which comes about as the Levi-Civita tensor $\varepsilon$ is not an invariant
tensor under ${{\rm O}}(4)$. In other words, the Pfaffian of ${{\rm SO}}(4)$ is not an
invariant of ${{\rm O}}(4)$.
Now, we provide the Hilbert series for the three different representations
studied above.
\subsubsection{Representation \texorpdfstring{$[2,0]$}{[2,0]}}
The conformal dimension is the same as in~\eqref{eqn:delta_A1xA1_Rep20} and the
rational cone of the Weyl chamber is simply
\begin{equation}
C^{(2)} = \mathrm{Cone}\left((1,0),(1,1)\right) \; ,
\end{equation}
such that the cone generators and the Hilbert basis for $S^{(2)}\coloneqq
C^{(2)} \cap \left( K^{[0]}\cup K^{[2]} \right)$ coincide. The upper half-space
of Fig.~\ref{Fig:Hilbert_basis_SO4_Rep20} depicts the situation.
The Hilbert series is then computed to read
\begin{equation}
\mathrm{HS}_{{{\rm O}}(4)}^{[2,0]}(t,N)= \frac{1 +2 t^{2 N-1} + 2 t^{2 N} +2 t^{2 N+1} +
t^{4N}}{\left(1-t^2\right) \left(1-t^4\right) \left(1-t^{2 N-2}\right)^2}\; ,
\label{eqn:HS_O4_Rep20}
\end{equation}
which clearly displays the palindromic numerator, the order four pole for $t\to
1$,
and the order two pole for $t\to \infty$, i.e.\ the difference in degrees of
denominator and numerator is two.
By inspection of~\eqref{eqn:HS_O4_Rep20} and use of the plethystic logarithm
\begin{equation}
\mathrm{PL}(\mathrm{HS}_{{{\rm O}}(4)}^{[2,0]}) = t^2 +t^4 + t^{\Delta(1,0)}(1+t+t^2+t^3) +
t^{\Delta(1,1)}(1+t+t^2+t^3) - \mathcal{O}(t^{2\Delta(1,0) +2}) \; ,
\end{equation}
for $N\geq2$, we can summarise the generators as in Tab.~\ref{tab:Ops_O4_Rep20}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings
\\ \midrule
Casimirs & --- & --- & $2$, $4$ & --- & --- \\
monopole & $(1,0)$ & $K^{[2]}$ & $2N-2 $ & ${{\rm U}(2)}$ & $3$ \\
monopole & $(1,1)$ & $K^{[0]}$ & $2N-2 $ & ${{\rm U}(1)} \times {{\rm O}}(2)$ & $3$
\\
\bottomrule
\end{tabular}
\caption{Bare and dressed monopole generators for a ${{\rm O}}(4)$ gauge theory with
matter transforming in $[2,0]$.}
\label{tab:Ops_O4_Rep20}
\end{table}
The different dressing behaviour of the ${{\rm O}}(4)$ monopole generators $(1,0)$
and $(1,1)$ compared to their ${{\rm SO}}(4)$ counterparts can be deduced from
dividing the relevant dressing factor by the trivial one. In detail
\begin{equation}
\frac{P_{{{\rm O}}(4)}(t,\{(1,0) \text{ or } (1,1)\})}{P_{{{\rm O}}(4)}(t,0,0)} =
\frac{(1-t^2)(1-t^4)}{(1-t)(1-t^2)}= 1+t+t^2+t^3 \; .
\end{equation}
\subsubsection{Representation \texorpdfstring{$[2,2]$}{[2,2]}}
The conformal dimension is the same as in~\eqref{eqn:delta_A1xA1_Rep22} and
the
rational cone of the Weyl chamber is still
\begin{equation}
C^{(2)} = \mathrm{Cone}\left((1,0),(1,1)\right) \; ,
\end{equation}
such that the cone generators and the Hilbert basis for $S^{(2)}\coloneqq
C^{(2)} \cap \left( K^{[0]}\cup K^{[2]} \right)$ coincide. The upper half-space
of Fig.~\ref{Fig:Hilbert_basis_SO4_Rep22} depicts the situation. We note that
the Weyl chamber for ${{\rm SO}}(4)$ is already divided into a fan by two rational
cones, while the Weyl chamber for ${{\rm O}}(4)$ is not.
The computation of the Hilbert series then yields
\begin{equation}
\mathrm{HS}_{{{\rm O}}(4)}^{[2,2]}(t,N)=
\frac{1+t^{4 N-1}+ t^{4 N} +t^{4 N+1}
+t^{6 N-1} +t^{6 N} +t^{6 N+1} +t^{10 N}}{\left(1-t^2\right)
\left(1-t^4\right) \left(1-t^{4 N-2}\right) \left(1-t^{6 N-2}\right)}
\; .
\label{eqn:HS_O4_Rep22}
\end{equation}
Again, the rational function clearly displays a palindromic numerator, an order
four pole for $t\to 1$,
and an order two pole for $t\to \infty$, i.e.\ the difference in degrees of
denominator and numerator is two.
By inspection of~\eqref{eqn:HS_O4_Rep22} and use of the plethystic logarithm
\begin{equation}
\mathrm{PL}(\mathrm{HS}_{{{\rm O}}(4)}^{[2,2]}) = t^2 +t^4 + t^{\Delta(1,0)}(1+t+t^2+t^3) +
t^{\Delta(1,1)}(1+t+t^2+t^3) - \mathcal{O}(t^{2\Delta(1,0) +2}) \; ,
\end{equation}
for $N\geq2$, we can summarise the generators as in Tab.~\ref{tab:Ops_O4_Rep22}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings
\\ \midrule
Casimirs & --- & --- & $2$, $4$ & --- & --- \\
monopole & $(1,0)$ & $K^{[2]}$ & $4N-2 $ & ${{\rm U}(2)}$ & $3$ \\
monopole & $(1,1)$ & $K^{[0]}$ & $6N-2 $ & ${{\rm U}(1)} \times {{\rm O}}(2)$ & $3$
\\
\bottomrule
\end{tabular}
\caption{Bare and dressed monopole generators for a ${{\rm O}}(4)$ gauge theory with
matter transforming in $[2,2]$.}
\label{tab:Ops_O4_Rep22}
\end{table}
The dressings behave as discussed earlier.
\subsubsection{Representation \texorpdfstring{$[4,2]$}{[4,2]}}
The conformal dimension is given in~\eqref{eqn:delta_A1xA1_Rep42} and
the Weyl chamber is split into a fan generated by two rational cones
\begin{equation}
C_2^{(2)} = \mathrm{Cone}\left((1,0),(3,1)\right) \quad\textrm{and}\quad
C_3^{(2)} = \mathrm{Cone}\left((3,1),(1,1)\right) \; ,
\end{equation}
where we use the notation of the ${{\rm SO}}(4)$ setting, see the upper half plan of
Fig.~\ref{Fig:Hilbert_basis_SO4_Rep42}.
The Hilbert bases for $S_p^{(2)}\coloneqq
C_p^{(2)} \cap \left( K^{[0]}\cup K^{[2]} \right)$ differ from the cone
generators and are obtained as
\begin{equation}
\Hcal(S_2^{(2)}) = \left\{(1,0),(3,1) \right\} \quad\textrm{and}\quad
\Hcal(S_3^{(2)}) = \left\{(3,1),(2,1),(1,1) \right\} \; .
\end{equation}
The computation of the Hilbert series then yields
\begin{subequations}
\label{eqn:HS_O4_Rep42}
\begin{align}
\mathrm{HS}_{{{\rm O}}(4)}^{[4,2]}(t,N)&=
\frac{R(t,N)}{\left(1-t^2\right) \left(1-t^4\right) \left(1-t^{10 N-2}\right)
\left(1-t^{26 N-6}\right)}
\; , \\
R(t,N)&= 1 +t^{10 N-2}+2 t^{10 N-1} + 2 t^{10 N} +2 t^{10 N+1} \\
&\qquad +t^{18 N-4}+2 t^{18 N-3} +2 t^{18 N-2} + 2 t^{18 N-1} +t^{18 N} \notag
\\
&\qquad +2 t^{26 N-5}+2 t^{26 N-4}+2 t^{26 N-3}+t^{26 N-2}+t^{36 N-4} \notag
\end{align}
\end{subequations}
As before, the rational function~\eqref{eqn:HS_O4_Rep42} clearly displays a
palindromic numerator, an order
four pole for $t\to 1$,
and an order two pole for $t\to \infty$, i.e.\ the difference in degrees of
denominator and numerator is two.
By inspection of~\eqref{eqn:HS_O4_Rep42} and use of the plethystic logarithm
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{{\rm O}}(4)}^{[4,2]}) = t^2 +t^4 &+ t^{\Delta(1,0)}(1+t+t^2+t^3) +
t^{\Delta(1,1)}(1+t+t^2+t^3) \label{eqn:PL_O4_Rep42} \\
&+t^{\Delta(2,1)}(1+2(t+t^2+t^3)+t^4)\notag \\
&-t^{\Delta(1,0)+\Delta(1,1)}(1+2t+5t^2+6t^3+7t^4+4t^5+3t^6) \notag \\
&+t^{\Delta(3,1)}(1+2(t+t^2+t^3)+t^4)
- \mathcal{O}(t^{\Delta(1,0) +\Delta(2,1)}) \; , \notag
\end{align}
for $N\geq2$, we can summarise the generators as in Tab.~\ref{tab:Ops_O4_Rep42}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
\toprule
object & $(m_1,m_2)$ & lattice & $\Delta(m_1,m_2)$ & $\mathrm{H}_{(m_1,m_2)}$ &
dressings
\\ \midrule
Casimirs & --- & --- & $2$, $4$ & --- & --- \\
monopole & $(1,0)$ & $K^{[2]}$ & $10N-2 $ & ${{\rm U}(2)}$ & $3$ \\
monopole & $(1,1)$ & $K^{[0]}$ & $10N-2 $ & ${{\rm U}(1)} \times {{\rm O}}(2)$ & $3$
\\
monopole & $(2,1)$ & $K^{[2]}$ & $18N-4 $ & ${{\rm U}(1)}^2 $ & $7$
\\
monopole & $(3,1)$ & $K^{[0]}$ & $26N-6 $ & ${{\rm U}(1)}^2$ & $7$
\\
\bottomrule
\end{tabular}
\caption{Bare and dressed monopole generators for a ${{\rm O}}(4)$ gauge theory with
matter transforming in $[4,2]$.}
\label{tab:Ops_O4_Rep42}
\end{table}
The dressing behaviour of $(1,0)$, $(1,1)$ is as discussed earlier; however, we
need to describe the dressings of $(2,1)$ and $(3,1)$ as it differs from the
${{\rm SO}}(4)$ counterparts. Again, we compute the quotient of the dressing factor
of the maximal torus divided by the trivial one, i.e.\
\begin{equation}
\frac{P_{{{\rm O}}(4)}(t,m_1>m_2>0)}{P_{{{\rm O}}(4)}(t,0,0)}=
\frac{(1-t^2)(1-t^4)}{(1-t)^2}= 1+2(t+t^2+t^3)+t^4 \; .
\end{equation}
Consequently, each bare monopole $(2,1)$, $(3,1)$ is accompanied by seven
dressings, which is in agreement with~\eqref{eqn:PL_O4_Rep42}.
\section{Case: \texorpdfstring{$\boldsymbol{{\rm G}_2}$}{G2}}
\label{sec:G2}
Here, we study the Coulomb branch for the only exceptional simple Lie group of
rank two.
\subsection{Set-up}
The group ${\rm G}_2$ has irreducible representations labelled by two Dynkin labels
and the dimension formula reads
\begin{equation}
\mathrm{dim}[a,b]= \frac{1}{120} (a+1) (b+1) (a+b+2) (a+2 b+3) (a+3 b+4) (2 a+3 b+5) \;.
\end{equation}
In the following, we study the representations given in
Tab.~\ref{tab:G2_overview_reps}. The three categories defined are due to the
similar form of the conformal dimensions.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\toprule
Dynkin label & $[1,0]$ & $[0,1]$ & $[2,0]$ & $[1,1]$ & $[0,2]$ & $[3,0]$ &
$[4,0]$ & $[2,1]$ \\
\midrule
Dim. & $7$ & $14$ & $27$ & $64$ & $77$ & $77$ & $182$ & $189$ \\
& \multicolumn{3}{c|}{category 1} & \multicolumn{3}{c|}{category 2} &
\multicolumn{2}{c}{category 3} \\
\bottomrule
\end{tabular}
\caption{An overview of the ${\rm G}_2$-representations considered in this paper.}
\label{tab:G2_overview_reps}
\end{table}
The Weyl group of ${\rm G}_2$ is $D_6$ and the GNO-dual group is another ${\rm G}_2$.
Any element in the Cartan subalgebra $\mathfrak{h}=\mathrm{span}(H_1,H_2)$ can
be written as $H=n_1 H_1 + n_2 H_2$. Restriction to the principal Weyl chamber
is realised via $n_1,n_2\geq0$.
The group ${\rm G}_2$ has two Casimir invariants of degree $2$ and $6$. Therefore,
the classical dressing function is~\cite{Cremonesi:2013lqa}
\begin{align}
P_{{\rm G}_2}(t,n_1,n_2)= \left\{\begin{matrix} \frac{1}{(1-t^2)(1-t^6)} \; ,&
n_1=n_2=0 \; ,\\
\frac{1}{(1-t)(1-t^2)} \; , & {n_1>0,n_2=0 }\text{ or }{n_1=0,n_2>0} \; , \\
\frac{1}{(1-t)^2} \; ,& n_1,n_2>0 \; .
\end{matrix} \right.
\end{align}
\subsection{Category 1}
\label{subsec:G2_Cat1}
\paragraph{Hilbert basis}
The representations $[1,0]$, $[0,1]$, and $[2,0]$ have schematically conformal
dimensions of the form
\begin{equation}
\Delta(n_1,n_2)= \sum_j A_j | a_j n_1 + b_j n_2| + B_1 |n_1|+ B_2 |n_2|
\label{eqn:delta_G2_Cat1}
\end{equation}
for $a_j,b_j \in \NN$ and $ A_j,B_1,B_2 \in \mathbb Z$.
As a consequence, the usual fan within the Weyl chamber is simply one
$2$-dimensional rational polyhedral cone
\begin{equation}
C^{(2)}= \mathrm{Cone}((1,0),(0,1)) \; .
\end{equation}
The intersection with the weight lattice $\Lambda_w({\rm G}_2)$ yields the
relevant semi-group $S^{(2)}$, as depicted in
Fig.~\ref{Fig:Hilbert_basis_G2_Cat1}.
The Hilbert bases are trivially given by the ray generators
\begin{equation}
\Hcal(S^{(2)}) = \Big\{(1,0),(0,1) \Big\} \; .
\label{eqn:Hilbert_basis_G2_Cat1}
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (4.5,0);
\coordinate (YAxisMin) at (0,-0.5);
\coordinate (YAxisMax) at (0,4.5);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (4.7,-0.2) node {$n_1$};
\draw (-0.3,4.3) node {$n_2$};
\foreach \x in {0,1,...,4}{%
\foreach \y in {0,1,...,4}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
%
}
}
\draw[black,dashed,thick] (Origin) -- (0,4.2);
\draw[black,dashed,thick] (Origin) -- (4.2,0);
\draw[black,thick] (1,0) circle (4pt);
\draw[black,thick] (0,1) circle (4pt);
\draw (3.5,1.5) node {$S^{(2)}$};
\end{tikzpicture}
\caption{The semi-group $S^{(2)}$ for the representations $[1,0]$, $[0,1]$, and
$[2,0]$ obtained from the ${\rm G}_2$ Weyl chamber (considered as rational cone)
and its ray generators (black circled points).}
\label{Fig:Hilbert_basis_G2_Cat1}
\end{figure}
\paragraph{Dressings}
The two minimal generators lie at the boundary of the Weyl chamber and,
therefore, have residual gauge group $\mathrm{H}_{(1,0)}=\mathrm{H}_{(0,1)}={{\rm U}(2)}$. Recalling
that ${\rm G}_2$ has two Casimir invariants $\mathcal{C}_2$, $\mathcal{C}_6$ at
degree $2$ and $6$, one can analyse the dressed monopole operators
associated to $(1,0)$ and $(0,1)$.
The residual gauge group ${{\rm U}(2)} \subset {\rm G}_2$ has a degree one Casimir
$C_1\coloneqq \phi_1+\phi_2$ and a degree two Casimir
$C_2\coloneqq\phi_1^2+\phi_2^2$. Again, we employed the diagonal form of the
adjoint-valued scalar $\Phi$.
Consequently, the bare monopole $V_{(0,1)}^{\mathrm{dress},0}$ exhibits five
dressed monopoles $V_{(0,1)}^{\mathrm{dress},i}$ ($i=1,\ldots,5$)
of degrees $\Delta(0,1)+1, \ldots,\Delta(0,1)+5$. Since the
highest degree Casimir invariant is of order $6$ and the degree $2$ Casimir
invariant of ${\rm G}_2$ differs from the pure sum of squares~\cite{Okubo}, one can
build all dressings as follows:
\begin{align}
C_1 (0,1) \; , \quad C_2 (0,1)\; , \quad C_1 C_2 (0,1) \; , \quad C_1^2 C_2
(0,1)\; , \quad (C_1 C_2^2 + C_1^2 C_2) (0,1) \; .
\end{align}
The very same arguments applies for the bare and dressed monopole generators
associated to $(1,0)$. Thus, we expect six monopole operators: one bare
$V_{(1,0)}^{\mathrm{dress},0}$ and five dressed $V_{(1,0)}^{\mathrm{dress},i}$
($i=1,\ldots,5$).
Comparing with App.~\ref{app:PL}, we find that a magnetic weight at the
boundary of the dominant Weyl chamber has dressings given by
$P_{{\rm G}_2}(t,\{n_1=0 \text{ or }n_2=0\}) \slash
P_{{\rm G}_2}(t,0,0)=1+t+t^2+t^3+t^4+t^5 $, which is then consistent with the
exposition above.
We will now exemplify the three different representations.
\subsubsection{Representation \texorpdfstring{$[1,0]$}{[1,0]}}
The relevant computation has been presented in~\cite{Cremonesi:2013lqa} and the
conformal dimension reads
\begin{align}
\label{eqn:delta_G2_Rep10}
\Delta(n_1,n_2)= &N (\left| n_1+n_2\right| +\left| 2
n_1+n_2\right| +\left| n_1\right| ) \\
&-(\left|
n_1+n_2\right| +\left| 2 n_1+n_2\right| +\left| 3
n_1+n_2\right| +\left| 3 n_1+2 n_2\right| +\left|
n_1\right| +\left| n_2\right| ) \notag \; .
\end{align}
Evaluating the Hilbert series for $N>3$ yields
\begin{equation}
\label{eqn:HS_G2_Rep10}
\mathrm{HS}_{{\rm G}_2}^{[1,0]}(t,N) =
\frac{1+t^{2 N-4}+t^{2 N-3}+t^{2
N-2}+t^{2 N-1}+t^{4 N-5}}{\left(1-t^2\right) \left(1-t^6\right)
\left(1-t^{2 N-6}\right) \left(1-t^{2 N-5}\right)} \; .
\end{equation}
We observe that the numerator of~\eqref{eqn:HS_G2_Rep10} is a palindromic
polynomial of degree $4N-5$; while, the denominator has degree
$4N-3$. Hence, the difference in degree between denominator and numerator is
$2$, which equals the quaternionic dimension of moduli space.
In addition, the Hilbert series~\eqref{eqn:HS_G2_Rep10} has a pole of order
$4$
as $t\to 1$, which matches the complex dimension of the moduli space.
As discussed in~\cite{Cremonesi:2013lqa}, the plethystic logarithm has the
following behaviour:
\begin{equation}
\mathrm{PL}(\mathrm{HS}_{{\rm G}_2}^{[1,0]}(t,N)) = t^2 + t^6+
t^{2N-6}(1+t+t^2+t^3+t^4+t^5) -t^{4N-8} + \ldots \; .
\label{eqn:PL_G2_7dim}
\end{equation}
\paragraph{Hilbert basis}
According to~\cite{Cremonesi:2013lqa}, the monopole corresponding to GNO-charge
$(1,0)$, which has $\Delta(1,0)=4N-10$, can be generated. Again, this is due to
the specific form~\eqref{eqn:delta_G2_Rep10}.
\subsubsection{Representation \texorpdfstring{$[0,1]$}{[0,1]}}
\paragraph{Hilbert series}
For this representation, the conformal dimension is given as
\begin{align}
\Delta(n_1,n_2)=(N-1) (\left| n_1+n_2\right| +\left| 2
n_1+n_2\right| +\left| 3 n_1+n_2\right| +\left| 3
n_1+2 n_2\right| +\left| n_1\right| +\left| n_2\right| ) \; ,
\end{align}
and the computation of the Hilbert series for $N>1$ yields
\begin{equation}
\mathrm{HS}_{{\rm G}_2}^{[0,1]}(t,N)= \frac{1
+t^{6 N-5}(1+t+t^2+t^3+t^4)
+t^{10 N-9}(1+t+t^2+t^3+t^4)
+t^{16 N-10}}{\left(1-t^2\right) \left(1-t^6\right) \left(1-t^{6
(N-1)}\right) \left(1-t^{10 (N-1)}\right)} \; .
\label{eqn:HS_G2_Rep01}
\end{equation}
The numerator of~\eqref{eqn:HS_G2_Rep01} is a palindromic polynomial of
degree $16N-10$; while, the denominator is of degree $16N-8$. Hence, the
difference in degree between denominator and numerator is $2$, which matches
the quaternionic dimension of moduli space.
Moreover, the Hilbert series has a pole of order $4$ as $t\to 1$, i.e.\ it
equals complex dimension of the moduli space.
Employing the knowledge of the Hilbert basis~\eqref{eqn:Hilbert_basis_G2_Cat1},
the appearing objects in~\eqref{eqn:HS_G2_Rep01} can be interpreted as
in Tab.~\ref{tab:Ops_G2_Rep01}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c}
\toprule
object & & $\Delta(n_1,n_2)$ & $\mathrm{H}_{(n_1,n_2)}$\\ \midrule
Casimir & $\mathcal{C}_2$ & $2$ & ---\\
& $ \mathcal{C}_6$ & $6$ & ---\\ \midrule
bare monopole & $V_{(0,1)}^{\mathrm{dress},0}$ & $6(N-1)$ & ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(0,1)}^{\mathrm{dress},i}$ & $6(N-1)+i$ & ---
\\ \midrule
bare monopole & $V_{(1,0)}^{\mathrm{dress},0}$ & $10(N-1)$& ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(1,0)}^{\mathrm{dress},i}$ & $10(N-1)+i$
&---\\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a ${\rm G}_2$ gauge theory and matter
transforming in $[0,1]$.}
\label{tab:Ops_G2_Rep01}
\end{table}
\paragraph{Plethystic logarithm}
For $N\geq3$ the PL takes the form
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm G}_2}^{[0,1]}(t,N)) = t^2 + t^6&+
t^{6(N-1)}(1+t+t^2+t^3+t^4+t^5)\\
&+ t^{10(N-1)}(1+t+t^2+t^3+t^4+t^5) -t^{12N-10} + \ldots \notag
\end{align}
while for $N=2$ the PL is
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm G}_2}^{[0,1]}(t,2)) =
t^2 + t^6
+
t^{6}(1+t+t^2+t^3+t^4+t^5)
+ t^{10}(1+t+t^2+t^3) -2t^{16} + \ldots
\end{align}
In other words, the 4th and 5th dressing of $(1,0)$ are absent, because they
can be generated.
\subsubsection{Representation \texorpdfstring{$[2,0]$}{[2,0]}}
\paragraph{Hilbert series}
For this representation, the conformal dimensions is given by
\begin{align}
\Delta(n_1,n_2)=&N \Big( 2 \left|
n_1+n_2\right| +2 \left| 2 n_1+n_2\right| +\left| 3
n_1+n_2\right| +\left| 2 n_1+2 n_2\right| +\left| 3
n_1+2 n_2\right| \\
&\qquad +\left| 4 n_1+2 n_2\right| +2 \left|
n_1\right| +\left| 2 n_1\right| +\left| n_2\right| \Big) \notag \\
&-\left(\left|
n_1+n_2\right| +\left| 2 n_1+n_2\right| +\left| 3
n_1+n_2\right| +\left| 3 n_1+2 n_2\right| +\left|
n_1\right| +\left| n_2\right| \right) \notag \; .
\end{align}
The calculation for the Hilbert series is analogous to the previous cases and
we obtain
\begin{equation}
\mathrm{HS}_{{\rm G}_2}^{[2,0]}(t,N)= \frac{1
+t^{12 N-5}(1+t+t^2+t^3+t^4)
+t^{22 N-9}(1+t+t^2+t^3+t^4)
+t^{34 N-10}}{\left(1-t^2\right)
\left(1-t^6\right)
\left(1-t^{12 N-6}\right)
\left(1-t^{22 N-10}\right)} \; .
\label{eqn:HS_G2_Rep20}
\end{equation}
One readily observes, the numerator of~\eqref{eqn:HS_G2_Rep20} is a
palindromic polynomial of degree $34N-10$ and the denominator is of
degree $34N-8$. Hence, the difference in degree between denominator and
numerator is $2$, which is precisely the quaternionic dimension of moduli space.
Also, the Hilbert series has a pole of order $4$ as $t\to 1$, which equals the
complex dimension of the moduli space.
Having in mind the minimal generators~\eqref{eqn:Hilbert_basis_G2_Cat1}, the
appearing objects in~\eqref{eqn:HS_G2_Rep20} can be summarised as in
Tab.~\ref{tab:Ops_G2_Rep20}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c}
\toprule
object & & $\Delta(n_1,n_2)$ & $\mathrm{H}_{(n_1,n_2)}$\\ \midrule
Casimir & $\mathcal{C}_2$ & $2$ & ---\\
& $ \mathcal{C}_6$ & $6$ & ---\\ \midrule
bare monopole & $V_{(0,1)}^{\mathrm{dress},0}$ & $12N-6$ & ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(0,1)}^{\mathrm{dress},i}$ & $12N-6+i$ & ---
\\ \midrule
bare monopole & $V_{(1,0)}^{\mathrm{dress},0}$ & $22N-10$& ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(1,0)}^{\mathrm{dress},i}$ & $22N-10+i$ &---
\\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a ${\rm G}_2$ gauge theory and matter
transforming in $[2,0]$.}
\label{tab:Ops_G2_Rep20}
\end{table}
\paragraph{Plethystic Logarithm}
\begin{itemize}
\item For $N\geq3$ the PL takes the form
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm G}_2}^{[2,0]}(t,N)) = t^2 + t^6&+
t^{12N-6}(1+t+t^2+t^3+t^4+t^5)\\
&+ t^{22N-10}(1+t+t^2+t^3+t^4+t^5) -t^{12N-10} + \ldots \notag
\end{align}
\item While for $N=2$ the PL is
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm G}_2}^{[2,0]}(t,2)) = t^2 + t^6&+
t^{18}(1+t+t^2+t^3+t^4+t^5) \\
&+ t^{34}(1+t+t^2+t^3) -2t^{40} + \ldots \notag
\end{align}
By the very same reasoning as before, $V_{(1,0)}^{\mathrm{dress},4}$ and
$V_{(1,0)}^{\mathrm{dress},5}$ can be generated by monopoles associated
to $(0,1)$.
\item Moreover, for $N=1$ the PL looks as follows
\begin{equation}
\mathrm{PL}(\mathrm{HS}_{{\rm G}_2}^{[2,0]}(t,1)) = t^2 + t^6+
t^{6}(1+t+t^2+t^3+t^4+t^5)
+ t^{12}(1+t) -t^{16} + \ldots
\end{equation}
Looking at the conformal dimensions reveals that the missing dressed monopoles
of GNO-charge $(1,0)$ can be generated.
\end{itemize}
\subsection{Category 2}
\label{subsec:G2_Cat2}
\paragraph{Hilbert basis}
The representations $[1,1]$, $[0,2]$, and $[3,0]$ have schematically conformal
dimensions of the form
\begin{equation}
\label{eqn:delta_G2_Cat2}
\Delta(n_1,n_2)= \sum_j A_j | a_j n_1 + b_j n_2| + B_1 |n_1|+ B_2 |n_2| +
C|n_1-n_2|
\end{equation}
for $a_j,b_j \in \NN$ and $A_j,B_1,B_2,C\in \mathbb Z$.
The novelty of this conformal dimension, compared to~\eqref{eqn:delta_G2_Cat1},
is the difference $|n_1-n_2|$, i.e.\ a hyperplane that intersects the
Weyl chamber non-trivially.
As a consequence, there is a fan generated by two $2$-dimensional rational
polyhedral cones
\begin{equation}
C_1^{(2)}= \mathrm{Cone}((1,0),(1,1)) \quad\textrm{and}\quad C_2^{(2)}= \mathrm{Cone}((1,1),(0,1)) \; .
\end{equation}
The intersection with the weight lattice $\Lambda_w({\rm G}_2)$ yields the
relevant semi-groups $S_p$ ($p=1,2$), as depicted in
Fig.~\ref{Fig:Hilbert_basis_G2_Cat2}.
The Hilbert bases are again given by the ray generators
\begin{equation}
\label{eqn:Hilbert_basis_G2_Cat2}
\Hcal(S_1^{(2)}) = \Big\{(1,0),(1,1) \Big\} \quad\textrm{and}\quad \Hcal(S_2^{(2)}) =
\Big\{(1,1),(0,1) \Big\} \; .
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (4.5,0);
\coordinate (YAxisMin) at (0,-0.5);
\coordinate (YAxisMax) at (0,4.5);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (4.7,-0.2) node {$n_1$};
\draw (-0.3,4.3) node {$n_2$};
\foreach \x in {0,1,...,4}{%
\foreach \y in {0,1,...,4}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
%
}
}
\draw[black,dashed,thick] (Origin) -- (0,4.2);
\draw[black,dashed,thick] (Origin) -- (4.2,0);
\draw[black,dashed,thick] (Origin) -- (4.2,4.2);
\draw[black,thick] (1,0) circle (4pt);
\draw[black,thick] (0,1) circle (4pt);
\draw[black,thick] (1,1) circle (4pt);
\draw (1.5,3.5) node {$S_2^{(2)}$};
\draw (3.5,1.5) node {$S_1^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups $S_p^{(2)}$ ($p=1,2$) for the representations $[1,1]$,
$[0,2]$, and $[3,0]$ obtained from the ${\rm G}_2$ Weyl chamber (considered as
rational cone) and their ray generators (black circled points).}
\label{Fig:Hilbert_basis_G2_Cat2}
\end{figure}
\paragraph{Dressings}
The three minimal generators have different residual gauge groups, as two lie
on the boundary and one in the interior of the Weyl chamber. The GNO-charges
$(1,0)$ and $(0,1)$ are to be treated as in Subsec.~\ref{subsec:G2_Cat1}.
The novelty is the magnetic weight $(1,1)$ with $\mathrm{H}_{(1,1)}={{\rm U}(1)}^2$.
Thus, the dressing can be constructed with two independent ${{\rm U}(1)}$-Casimir
invariants,
proportional to $\phi_1$ and $\phi_2$. We choose a basis of dressed monopoles
\begin{subequations}
\begin{align}
V_{(1,1)}^{\mathrm{dress},j,\alpha} &= (1,1) (\phi_\alpha)^j \; , \qquad\textrm{for}\quad
j=1,\ldots 5 \; , \; \alpha=1,2 \; , \\
V_{(1,1)}^{\mathrm{dress},6} &= (1,1) \left( (\phi_1)^6 + (\phi_2)^6\right) \;
.
\end{align}
\end{subequations}
The reason behind the large number of dressings of the bare monopole $(1,1)$
lies in the delicate ${\rm G}_2$ structure~\cite{Okubo}, i.e.\ the degree two
Casimir $\mathcal{C}_2$ is not just the sum of the squares of $\phi_i$ and the
next ${\rm G}_2$-Casimir $\mathcal{C}_6$ is by four higher in degree and has a
complicated structure as well.
The number and degrees of the dressed monopole operators associated to $(1,1)$
can be confirmed by $P_{{\rm G}_2}(t,n_1>0,n_2>0)\slash
P_{{\rm G}_2}(t,0,0)=1+2t+2t^2+2t^3+2t^4+2t^5+t^6$, following App.~\ref{app:PL}.
We will now exemplify the three different representations.
\subsubsection{Representation \texorpdfstring{$[1,1]$}{[1,1]}}
\paragraph{Hilbert series}
The conformal dimension of the $64$-dimensional representation is given by
\begin{align}
\Delta(n_1,n_2)
&=N \Big(\left| n_1-n_2\right|
+8 \left| n_1+n_2\right|
+8 \left| 2 n_1+n_2\right|
+2 \left| 3 n_1+n_2\right|
+\left| 4 n_1+n_2\right| \\
&\qquad +\left| n_1+2 n_2\right|
+2 \left| 3 n_1+2 n_2\right|
+\left| 5 n_1+2 n_2\right|
+\left| 4 n_1+3 n_2\right|
+\left| 5 n_1+3 n_2\right| \notag \\
&\qquad
+8 \left| n_1\right|
+2 \left| n_2\right| \Big) \notag\\
&\qquad -\Big( \left| n_1+n_2\right|
+\left| 2 n_1+n_2\right|
+\left| 3 n_1+n_2\right|
+\left| 3 n_1+2 n_2\right|
+\left| n_1\right|
+\left| n_2\right| \Big) \notag \; .
\end{align}
Computing the Hilbert series provides the following expression
\begin{subequations}
\label{eqn:HS_G2_Rep11}
\begin{align}
\mathrm{HS}_{{\rm G}_2}^{[1,1]}(t,N)&= \frac{R(t,N)}{
\left(1-t^2\right) \left(1-t^6\right)
\left(1-t^{36 N-6}\right)
\left(1-t^{64 N-10}\right)
\left(1-t^{98 N-16}\right)} \; ,\\
R(t,N)&=1+
t^{36 N-5}(1+t+t^2+t^3+t^4)
+t^{64 N-9}(1+t+t^2+t^3+t^4)
\label{eqn:HS_G2_Rep11_Num}\\
&\qquad
+ t^{98 N-15}(2+2t+2t^2+2t^3+2t^4+t^5)
-t^{100 N-16} (1+2t+2t^2+2t^3+2t^4+2t^5)
\notag \\
&\qquad
-t^{134 N-21}(1+t+t^2+t^3+t^4)
-t^{162 N-25}(1+t+t^2+t^3+t^4)
-t^{198 N-26} \notag \; .
\end{align}
\end{subequations}
The numerator~\eqref{eqn:HS_G2_Rep11_Num} is a anti-palindromic
polynomial of degree $198N-26$; whereas the denominator is of degree
$198N-24$. Hence, the difference in degree
between denominator and numerator is $2$, which coincides
with the quaternionic dimension of moduli space.
The Hilbert series~\eqref{eqn:HS_G2_Rep11} has a pole of order $4$ as $t\to 1$,
which agrees with the complex dimension of the moduli space. (One can
explicitly show that $R(t=1,N)=0$, but $\tfrac{\mathrm{d}}{\mathrm{d}
t}R(t,N)|_{t=1}\neq0$.)
The appearing operators agree with the general setting outlined above and we
summarise them in Tab.~\ref{tab:Ops_G2_Rep11}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c}
\toprule
object & & $\Delta(n_1,n_2)$ & $\mathrm{H}_{(n_1,n_2)}$\\ \midrule
Casimir & $\mathcal{C}_2$ & $2$ & ---\\
& $ \mathcal{C}_6$ & $6$ & ---\\ \midrule
bare monopole & $V_{(0,1)}^{\mathrm{dress},0}$ & $36N-6$ & ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(0,1)}^{\mathrm{dress},i}$ & $36N-6+i$ & ---
\\ \midrule
bare monopole & $V_{(1,0)}^{\mathrm{dress},0}$ & $64N-10$& ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(1,0)}^{\mathrm{dress},i}$ & $64N-10+i$
&---\\ \midrule
bare monopole & $V_{(1,1)}^{\mathrm{dress},0}$ & $98N-16$& ${{\rm U}(1)}\times {{\rm U}(1)}$\\
dressings $(i=1,\ldots,5 ; \alpha=1,2)$ &
$V_{(1,1)}^{\mathrm{dress},i,\alpha}$ & $98N-16+i$
&--- \\
dressing &
$V_{(1,1)}^{\mathrm{dress},6}$ & $98N-16+6$
&--- \\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a ${\rm G}_2$ gauge theory and matter
transforming in $[1,1]$.}
\label{tab:Ops_G2_Rep11}
\end{table}
The new monopole corresponds to GNO-charge $(1,1)$ and displays a different
dressing behaviour than $(1,0)$ and $(0,1)$. The reason behind lies in the
residual gauge group being ${{\rm U}(1)}^2$.
\paragraph{Plethystic Logarithm}
Although the bare monopole $V_{(1,1)}^{\mathrm{dress},0}$ is generically a
necessary generator due to its origin as an ray generators
of~\eqref{eqn:Hilbert_basis_G2_Cat2}, not all dressings
$V_{(1,1)}^{\mathrm{dress}}$ might be independent.
\begin{itemize}
\item For $N\geq4$ the PL takes the form
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm G}_2}^{[1,1]}(t,N)) = t^2 + t^6&+
t^{36N-6}(1+t+t^2+t^3+t^4+t^5)\\
&+ t^{64N-10}(1+t+t^2+t^3+t^4+t^5) \notag\\
&-t^{2(36N-6)+2}(1+t+2t^2+2t^3+3t^4+2t^5+2t^6+t^7+t^8) \notag \\
&+t^{98N-16}(1+2t+2t^2+2t^3+2t^4+2t^5+t^6) -t^{100N-16} +\ldots \notag
\end{align}
\item For $N=3$ the PL is
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm G}_2}^{[1,1]}(t,N=3)) = t^2 + t^6&+
t^{102}(1+t+t^2+t^3+t^4+t^5)\\
&+ t^{182}(1+t+t^2+t^3+t^4+t^5) \notag\\
&-t^{206}(1+t+2t^2+2t^3+3t^4+2t^5+2t^6+t^7+t^8) \notag \\
&+t^{278}(1+2t+2t^2+2t^3+2t^4+2t^5) - 2t^{285} +\ldots \notag
\end{align}
Here, $\Delta(1,0)+\Delta(0,1)=284$ is precisely the conformal dimension of
$V_{(1,1)}^{\mathrm{dress},6}$; i.e.\ it is generated and absent from the PL.
\item For $N=2$ the PL is
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm G}_2}^{[1,1]}(t,N=2)) = t^2 + t^6&+
t^{66}(1+t+t^2+t^3+t^4+t^5)\\
&+ t^{118}(1+t+t^2+t^3+t^4+t^5) \notag\\
&-t^{134}(1+t+2t^2+2t^3+3t^4+2t^5+2t^6+t^7+t^8) \notag \\
&+t^{180}(1+2t+2t^2+2t^3+t^4) - 2t^{186} +\ldots \notag
\end{align}
Here, $\Delta(1,0)+\Delta(0,1)=184$ is precisely the conformal dimension of
$V_{(1,1)}^{\mathrm{dress},4,\alpha}$; i.e.\ only one of the dressings by the
4th power of ${{\rm U}(1)}$-Casimir is a generator. Consequently, the other one is
absent from the PL.
\item For $N=1$ the PL is
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm G}_2}^{[1,1]}(t,N=1)) = t^2 + t^6&+
t^{30}(1+t+t^2+t^3+t^4+t^5)\\
&+ t^{54}(1+t+t^2+t^3+t^4+t^5) \notag\\
&-t^{62}(1+t+2t^2+2t^3+3t^4+2t^5+2t^6+t^7+t^8) \notag \\
&+t^{82}(1+2t+t^2) - t^{62} +\ldots \notag
\end{align}
Here, $\Delta(1,0)+\Delta(0,1)=64$ is precisely the conformal dimension of
$V_{(1,1)}^{\mathrm{dress},2,\alpha}$; i.e.\ only one of the dressings by the
2th power of ${{\rm U}(1)}$-Casimir is a generator. Consequently, the other one is
absent from the PL.
\end{itemize}
\subsubsection{Representation \texorpdfstring{$[3,0]$}{[3,0]}}
\paragraph{Hilbert series}
The conformal dimension in this representation is given by
\begin{align}
\Delta(n_1,n_2)
&= N\Big( \left|5n_1+3n_2 \right|
+\left|5n_1+2n_2 \right|
+\left|4n_1 +3n_2 \right|
+\left|4n_1+n_2 \right|
+\left|n_1+2n_2 \right|
+\left|n_1-n_2 \right|\notag\\
&\qquad
+10\big(\left|2n_1+n_2 \right|
+\left|n_1+n_2 \right|
+\left|n_1 \right|\big)
+3\big(
\left|3n_1+2n_2 \right|
+\left|3n_1+n_2 \right|
+\left| n_2\right|\big)
\Big) \\
&\qquad -\Big(\left| n_1+n_2\right| +\left| 2 n_1+n_2\right| +\left| 3
n_1+n_2\right| +\left| 3 n_1+2 n_2\right| +\left|
n_1\right| +\left| n_2\right| \Big) \notag \; ,
\end{align}
such that we obtain for the Hilbert series
\begin{subequations}
\label{eqn:HS_G2_Rep30}
\begin{align}
\mathrm{HS}_{{\rm G}_2}{[3,0]}(t,N)&= \frac{R(t,N)}{\left(1-t^2\right) \left(1-t^6\right)
\left(1-t^{46 N-6}\right) \left(1-t^{82 N-10}\right)
\left(1-t^{126 N-16}\right)} \; ,\\
R(t,N)&=1
+t^{46 N-5}(1+t+t^2+t^3+t^4)
+t^{82 N-9}(1+t+t^2+t^3+t^4)
\label{eqn:HS_G2_Rep30_Num} \\
&\qquad
+ t^{126 N-15}(2+2t+2t^2+2t^3+2t^4+t^5)
\notag \\
&\qquad
-t^{128 N-16}(1+2t+2t^2+2t^3+2t^4+2t^5)
\notag \\
&\qquad
-t^{172 N-21}(1+t+t^2+t^3+t^4)
-t^{208 N-25}(1+t+t^2+t^3+t^4)
-t^{254 N-26} \notag \; .
\end{align}
\end{subequations}
The numerator~\eqref{eqn:HS_G2_Rep30_Num} is a anti-palindromic
polynomial of degree $254N-26$; while the denominator is of degree $254N-24$.
Hence, the difference in degree
between denominator and numerator is $2$, which coincides with the
quaternionic dimension of moduli space.
The Hilbert series~\eqref{eqn:HS_G2_Rep30} has a pole of order $4$ as $t\to 1$,
which equals the complex dimension of the moduli space. (One can
explicitly show that $R(t=1,N)=0$, but $\tfrac{\mathrm{d}}{\mathrm{d}
t}R(t,N)|_{t=1}\neq0$.)
Interpreting the appearing operators leads to a list of chiral ring generators
as presented in Tab.~\ref{tab:Ops_G2_Rep30}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c}
\toprule
object & & $\Delta(n_1,n_2)$ & $\mathrm{H}_{(n_1,n_2)}$\\ \midrule
Casimir & $\mathcal{C}_2$ & $2$ & ---\\
& $ \mathcal{C}_6$ & $6$ & ---\\ \midrule
bare monopole & $V_{(0,1)}^{\mathrm{dress},0}$ & $46N-6$ & ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(0,1)}^{\mathrm{dress},i}$ & $46N-6+i$ & ---
\\ \midrule
bare monopole & $V_{(1,0)}^{\mathrm{dress},0}$ & $82N-10$& ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(1,0)}^{\mathrm{dress},i}$ & $82N-10+i$
&---\\ \midrule
bare monopole & $V_{(1,1)}^{\mathrm{dress},0}$ & $126N-16$& ${{\rm U}(1)}\times {{\rm U}(1)}$\\
dressings $(i=1,\ldots,5 ; \alpha=1,2)$ &
$V_{(1,1)}^{\mathrm{dress},i,\alpha}$ & $126N-16+i$
&--- \\
dressing &
$V_{(1,1)}^{\mathrm{dress},6}$ & $126N-16+6$
&--- \\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a ${\rm G}_2$ gauge theory and matter
transforming in $[3,0]$.}
\label{tab:Ops_G2_Rep30}
\end{table}
The behaviour of the Hilbert series is absolutely identical to the case
$[1,1]$, because the conformal dimension is structurally identical. Therefore,
we do not provide further details.
\subsubsection{Representation \texorpdfstring{$[0,2]$}{[0,2]}}
\paragraph{Hilbert series}
The following conformal dimension reads
\begin{align}
\Delta(n_1,n_2)&=N\Big( \left|5n_1+3n_2 \right|
+ \left|5n_1+2n_2 \right|
+\left|4n_1+3n_2 \right|
+ \left|4n_1+n_2 \right|
+\left|n_1+2n_2 \right|
+\left|n_1-n_2 \right| \notag\\
&\qquad
+10\big( \left|2n_1+n_2 \right|
+\left|n_1+n_2 \right|
+\left|n_1 \right| \big)
+ 5\big( \left|3n_1+2n_2 \right|
+ \left|3n_1+n_2 \right|
+ \left|n_2 \right|\big)
\Big) \\
&\qquad -\Big(\left| n_1+n_2\right|
+\left| 2 n_1+n_2\right|
+\left| 3 n_1+n_2\right|
+\left| 3 n_1+2 n_2\right|
+\left| n_1\right|
+\left| n_2\right| \Big) \notag
\end{align}
The computation of the Hilbert series results in
\begin{subequations}
\label{eqn:HS_G2_Rep02}
\begin{align}
\mathrm{HS}_{{\rm G}_2}^{[0,2]}(t,N) &= \frac{R(t,N)}{
\left(1-t^2\right) \left(1-t^6\right)
\left(1-t^{52 N-6}\right)
\left(1-t^{90 N-10}\right)
\left(1-t^{140 N-16}\right)} \; ,\\
R(t,N)&=1
+t^{52 N-5}(1+t+t^2+t^3+t^4)
+t^{90 N-9}(1+t+t^2+t^3+t^4)
\label{eqn:HS_G2_Rep02_Num} \\
&\qquad
+t^{140 N-15}(2+2t+2t^2+2t^3+2t^4+t^5)
\notag \\
&\qquad
-t^{142 N-16}(1+2t+2t^2+2t^3+2t^4+2t^5)
\notag \\
&\qquad
-t^{192 N-21}(1+t+t^2+t^3+t^4)
-t^{230 N-25}(1+t+t^2+t^3+t^4)
-t^{282 N-26} \; . \notag
\end{align}
\end{subequations}
The numerator~\eqref{eqn:HS_G2_Rep02_Num} is a anti-palindromic
polynomial of degree $282N-26$; while, the denominator is of degree $282N-24$.
Hence, the difference in degree
between denominator and numerator is $2$, which agrees with the
quaternionic dimension of moduli space.
The Hilbert series~\eqref{eqn:HS_G2_Rep02} has a pole of order $4$ as $t\to 1$,
which equals complex dimension of the moduli space. (One can explicitly
show that $R(t=1,N)=0$, but $\tfrac{\mathrm{d}}{\mathrm{d} t}R(t,N)|_{t=1}\neq0$.)
Tab.~\ref{tab:Ops_G2_Rep02} summarises the appearing operators.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c}
\toprule
object & & $\Delta(n_1,n_2)$ & $\mathrm{H}_{(n_1,n_2)}$\\ \midrule
Casimir & $\mathcal{C}_2$ & $2$ & ---\\
& $ \mathcal{C}_6$ & $6$ & ---\\ \midrule
bare monopole & $V_{(0,1)}^{\mathrm{dress},0}$ & $52N-6$ & ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(0,1)}^{\mathrm{dress},i}$ & $52N-6+i$ & ---
\\ \midrule
bare monopole & $V_{(1,0)}^{\mathrm{dress},0}$ & $90N-10$& ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(1,0)}^{\mathrm{dress},i}$ & $90N-10+i$
&---\\ \midrule
bare monopole & $V_{(1,1)}^{\mathrm{dress},0}$ & $140N-16$& ${{\rm U}(1)}\times {{\rm U}(1)}$\\
dressings $(i=1,\ldots,5 ; \alpha=1,2)$ &
$V_{(1,1)}^{\mathrm{dress},i,\alpha}$ & $140N-16+i$
&--- \\
dressing &
$V_{(1,1)}^{\mathrm{dress},6}$ & $140N-16+6$
&--- \\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a ${\rm G}_2$ gauge theory and matter
transforming in $[0,2]$.}
\label{tab:Ops_G2_Rep02}
\end{table}
The behaviour of the Hilbert series is identical to the cases $[1,1]$ and
$[3,0]$, because the conformal dimension is structurally identical.
Again, we do not provide further details.
\subsection{Category 3}
\paragraph{Hilbert basis}
Investigating the representations $[2,1]$ and $[4,0]$, one recognises the
common structural form of the conformal dimensions
\begin{equation}
\Delta(n_1,n_2)= \sum_j A_j | a_j n_1 + b_j n_2| + B_1 |n_1|+ B_2 |n_2| +
C|n_1-n_2| + D|2n_1 -n_2|
\end{equation}
for $a_j,b_j \in \NN$ and $A_j,B_1,B_2,C,D\in \mathbb Z$.
The novelty of this conformal dimension, compared to~\eqref{eqn:delta_G2_Cat1}
and~\eqref{eqn:delta_G2_Cat2},
is the difference $|2n_1-n_2|$, i.e.\ a second hyperplane that intersects
the Weyl chamber non-trivially.
As a consequence, the Weyl chamber is decomposed into a fan
generated by three rational polyhedral cones of dimension 2. These are
\begin{align}
C_1^{(2)}= \mathrm{Cone}((1,0),(1,1))\; , \quad
C_2^{(2)}= \mathrm{Cone}((1,1),(1,2))
\quad\textrm{and}\quad C_3^{(2)}= \mathrm{Cone}((1,2),(0,1)) \; .
\end{align}
The intersection with the weight lattice $\Lambda_w({\rm G}_2)$ yields the
relevant semi-groups $S_p$ (for $p=1,2,3$), as depicted in
Fig.~\ref{Fig:Hilbert_basis_G2_Cat3}.
The Hilbert bases are again given by the ray generators
\begin{equation}
\label{eqn:Hilbert_basis_G2_Cat3}
\Hcal(S_1^{(2)}) = \Big\{(1,0),(1,1) \Big\} \; , \quad
\Hcal(S_2^{(2)}) = \Big\{(1,1),(1,2) \Big\} \quad\textrm{and}\quad
\Hcal(S_3^{(2)}) = \Big\{(1,2),(0,1) \Big\} \; .
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (4.5,0);
\coordinate (YAxisMin) at (0,-0.5);
\coordinate (YAxisMax) at (0,4.5);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (4.7,-0.2) node {$n_1$};
\draw (-0.3,4.3) node {$n_2$};
\foreach \x in {0,1,...,4}{%
\foreach \y in {0,1,...,4}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
%
}
}
\draw[black,dashed,thick] (Origin) -- (0,4.2);
\draw[black,dashed,thick] (Origin) -- (4.2,0);
\draw[black,dashed,thick] (Origin) -- (4.2,4.2);
\draw[black,dashed,thick] (Origin) -- (2.25*1,2.25*2);
\draw[black,thick] (1,0) circle (4pt);
\draw[black,thick] (0,1) circle (4pt);
\draw[black,thick] (1,1) circle (4pt);
\draw[black,thick] (1,2) circle (4pt);
\draw (0.9,3.5) node {$S_3^{(2)}$};
\draw (2.5,3.5) node {$S_2^{(2)}$};
\draw (3.5,1.5) node {$S_1^{(2)}$};
\end{tikzpicture}
\caption{The semi-groups $S_p^{(2)}$ (p=1,2,3) for the representations $[2,1]$
and $[4,0]$ obtained from the ${\rm G}_2$ Weyl chamber (considered as rational
cone) and their ray generators (black circled points).}
\label{Fig:Hilbert_basis_G2_Cat3}
\end{figure}
\paragraph{Dressings}
Compared to Subsec.~\ref{subsec:G2_Cat1} and~\ref{subsec:G2_Cat2}, the
additional magnetic weight $(1,2)$ has the same dressing behaviour as $(1,1)$,
because the residual gauge groups is ${{\rm U}(1)}^2$, too. Thus, the additional
necessary monopole operators are the bare operator
$V_{(1,2)}^{\mathrm{dress},0}$ and the dressed monopoles
$V_{(1,2)}^{\mathrm{dress},i,\alpha}$ for $i=1,\ldots,5$, $\alpha=1,2$ as well
as $V_{(1,2)}^{\mathrm{dress},6}$.
We will now exemplify the three different representations.
\subsubsection{Representation \texorpdfstring{$[4,0]$}{[4,0]}}
\paragraph{Hilbert series}
The conformal dimension reads
\begin{align}
\Delta(n_1,n_2)&= N \Big(
3 \left| n_1-n_2\right|
+\left| 2 n_1-n_2\right|
+27 \left| n_1+n_2\right|
+30 \left| 2 n_1+n_2\right|
+7 \left| 3 n_1+n_2\right|\\
&\qquad
+3 \left| 4 n_1+n_2\right|
+\left| 5 n_1+n_2\right|
+3 \left| n_1+2 n_2\right|
+7 \left| 3 n_1+2 n_2\right|
+3 \left| 5 n_1+2 n_2\right| \notag\\
&\qquad
+\left| 2 n_1+3 n_2\right|
+3 \left| 4 n_1+3 n_2\right|
+3 \left| 5 n_1+3 n_2\right|
+\left| 7 n_1+3 n_2\right|
+\left| 5 n_1+4 n_2\right| \notag\\
&\qquad
+\left| 7 n_1+4 n_2\right|
+27 \left| n_1\right|
+7 \left| n_2\right|
\Big) \notag \\
&\qquad -\Big(\left| n_1+n_2\right| +\left| 2
n_1+n_2\right| +\left| 3 n_1+n_2\right| +\left| 3 n_1+2 n_2\right|
+\left| n_1\right| +\left| n_2\right| \Big) \notag \; ,
\end{align}
from which we compute the Hilbert series to be
\begin{subequations}
\label{eqn:HS_G2_Rep40}
\begin{align}
\mathrm{HS}_{{\rm G}_2}^{[4,0]}(t,N)&=
\frac{R(t,N)}{\left(1-t^2\right)
\left(1-t^6\right) \left(1-t^{134 N-6}\right) \left(1-t^{238
N-10}\right) \left(1-t^{364 N-16}\right) \left(1-t^{496
N-22}\right)} , \\
R(t,N)&=1
+t^{134 N-5}(1+t+t^2+t^3+t^4)
+t^{238 N-9}(1+t+t^2+t^3+t^4)
\label{eqn:HS_G2_Rep40_Num}\\
&\qquad
+ t^{364 N-15}(2+2t+2t^2+2t^3+2t^4+t^5)
\notag\\
&\qquad
-t^{372 N-16}(1+2t+2t^2+2t^3+2t^4+2t^5)
\notag\\
&\qquad
+ t^{496 N-21}(2+2t+2t^2+2t^3+2t^4+t^5)
\notag\\
&\qquad
-t^{498 N-22}(1+3t+3t^2+3t^3+3t^4+3t^5+t^6)
\notag\\
&\qquad
-t^{602 N-25}(1+t+t^2+t^3+t^4)
-t^{630 N-27}(1+t+t^2+t^3+t^4)
\notag\\
&\qquad
-t^{734 N-32}(1+3t+3t^2+3t^3+3t^4+3t^5+t^6)
\notag\\
&\qquad
+t^{736 N-32}(1+2t+2t^2+2t^3+2t^4+2t^5)
\notag\\
&\qquad
- t^{860 N-37}(2+2t+2t^2+2t^3+2t^4+t^5)
\notag\\
&\qquad
+t^{868 N-38}(1+2t+2t^2+2t^3+2t^4+2t^5)
\notag\\
&\qquad
+t^{994 N-43}(1+t+t^2+t^3+t^4)
+t^{1098 N-47}(1+t+t^2+t^3+t^4)
+t^{1232 N-48} \notag \; .
\end{align}
\end{subequations}
The numerator~\eqref{eqn:HS_G2_Rep40_Num} is a palindromic polynomial of
degree $1232N-48$; while, the denominator is of degree $1232N-46$. Hence,
the difference in degree between denominator and numerator is $2$, which equals
the quaternionic dimension of moduli space.
The Hilbert series~\eqref{eqn:HS_G2_Rep40} has a pole of order $4$ as $t\to 1$,
which coincides with the complex dimension of the moduli space. (One can
explicitly show that $R(t=1,N)=0$ and $\tfrac{\mathrm{d}}{\mathrm{d} t}R(t,N)|_{t=1}=0$,
but $\tfrac{\mathrm{d}^2}{\mathrm{d} t^2}R(t,N)|_{t=1}\neq0$.)
The appearing operators can be summarised as in Tab.~\ref{tab:Ops_G2_Rep40}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c}
\toprule
object & & $\Delta(n_1,n_2)$ & $\mathrm{H}_{(n_1,n_2)}$\\ \midrule
Casimir & $\mathcal{C}_2$ & $2$ & ---\\
& $ \mathcal{C}_6$ & $6$ & ---\\ \midrule
bare monopole & $V_{(0,1)}^{\mathrm{dress},0}$ & $134N-6$ & ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(0,1)}^{\mathrm{dress},i}$ & $134N-6+i$ & ---
\\ \midrule
bare monopole & $V_{(1,0)}^{\mathrm{dress},0}$ & $238N-10$& ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(1,0)}^{\mathrm{dress},i}$ & $238N-10+i$
&---\\ \midrule
bare monopole & $V_{(1,1)}^{\mathrm{dress},0}$ & $364N-16$& ${{\rm U}(1)}\times {{\rm U}(1)}$\\
dressings $(i=1,\ldots,5 ; \alpha=1,2)$ &
$V_{(1,1)}^{\mathrm{dress},i,\alpha}$ & $364N-16+i$
&--- \\
dressing &
$V_{(1,1)}^{\mathrm{dress},6}$ & $364N-16+6$
&---\\ \midrule
bare monopole & $V_{(1,2)}^{\mathrm{dress},0}$ & $496N-22$& ${{\rm U}(1)}\times {{\rm U}(1)}$\\
dressings $(i=1,\ldots,5 ; \alpha=1,2)$ &
$V_{(1,2)}^{\mathrm{dress},i,\alpha}$ & $496N-22+i$
&--- \\
dressing &
$V_{(1,2)}^{\mathrm{dress},6}$ & $496N-22+6$
&--- \\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a ${\rm G}_2$ gauge theory and matter
transforming in $[4,0]$.}
\label{tab:Ops_G2_Rep40}
\end{table}
The new monopole corresponds to GNO-charge $(1,2)$ and displays the same
dressing behaviour as $(1,1)$.
Contrary to the cases $[1,1]$, $[3,0]$, and $[0,2]$, the bare and dressed
monopoles of GNO-charge $(1,1)$ are always independent generators as
\begin{equation}
\Delta(1,1)=364N-16 <372N-16=134N-6 + 238N-10= \Delta(0,1)+ \Delta(1,0)
\end{equation}
holds for all $N\geq1$.
\paragraph{Plethystic Logarithm}
By means of the minimal generators~\eqref{eqn:Hilbert_basis_G2_Cat3}, the bare
monopole $V_{(1,2)}^{\mathrm{dress},0}$ is a necessary generator.
Nevertheless, not all dressings $V_{(1,2)}^{\mathrm{dress}}$ need to be
independent.
For $N\geq1$ the PL takes the form
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm G}_2}^{[0,2]}(t,N)) = t^2 + t^6&+
t^{134N-6}(1+t+t^2+t^3+t^4+t^5)\\
&+ t^{238N-10}(1+t+t^2+t^3+t^4+t^5) \notag\\
&-t^{2(134N-6)+2}(1+t+2t^2+2t^3+3t^4+2t^5+2t^6+t^7+t^8) \notag \\
&+t^{364N-16}(1+2t+2t^2+2t^3+2t^4+2t^5+t^6) -t^{372N-16} +\ldots \notag
\end{align}
Based purely in conformal dimension and GNO-charge, we can argue the following:
\begin{itemize}
\item For $N=3$, $\Delta(1,1)+\Delta(0,1)=1472$ is precisely the conformal
dimension of
$V_{(1,2)}^{\mathrm{dress},6}$, i.e.\ it is generated.
\item For $N=2$, $\Delta(1,1)+\Delta(0,1)=974$ is precisely the conformal
dimension of
$V_{(1,2)}^{\mathrm{dress},4,\alpha}$, i.e.\ only one of the dressings by the
4th power of ${{\rm U}(1)}$-Casimir is a generator.
\item For $N=1$, $\Delta(1,1)+\Delta(0,1)=476$ is precisely the conformal
dimension of
$V_{(1,2)}^{\mathrm{dress},2,\alpha}$, i.e.\ only one of the dressings by the
2th power of ${{\rm U}(1)}$-Casimir is a generator.
\end{itemize}
\subsubsection{Representation \texorpdfstring{$[2,1]$}{[2,1]}}
\paragraph{Hilbert series}
The conformal dimension reads
\begin{align}
\Delta(n_1,n_2)&=
N \Big(
3 \left| n_1-n_2\right|
+\left| 2 n_1-n_2\right|
+24 \left| n_1+n_2\right|
+24 \left| 2 n_1+n_2\right|
+8 \left| 3 n_1+n_2\right| \\
&\qquad
+3 \left| 4 n_1+n_2\right|
+\left| 5 n_1+n_2\right|
+3 \left| n_1+2 n_2\right|
+8 \left| 3 n_1+2 n_2\right|
+3 \left| 5 n_1+2 n_2\right|\notag \\
&\qquad
+\left| 2 n_1+3 n_2\right|
+3 \left| 4 n_1+3 n_2\right|
+3 \left| 5 n_1+3 n_2\right|
+\left| 7 n_1+3 n_2\right|
+\left| 5 n_1+4 n_2\right| \notag \\
&\qquad
+\left| 7 n_1+4 n_2\right|
+24 \left| n_1\right|
+8 \left| n_2\right|
\Big) \notag \\
&\qquad
-\Big(\left|
n_1+n_2\right| +\left| 2 n_1+n_2\right| +\left| 3
n_1+n_2\right| +\left| 3 n_1+2 n_2\right| +\left|
n_1\right| +\left| n_2\right| \Big) \; , \notag
\end{align}
from which we compute the Hilbert series to be
\begin{subequations}
\label{eqn:HS_G2_Rep21}
\begin{align}
\mathrm{HS}_{{\rm G}_2}^{[2,1]}(t,N)&=
\frac{R(t,N)}{ \left(1-t^2\right)
\left(1-t^6\right)
\left(1-t^{132 N-6}\right)
\left(1-t^{232 N-10}\right)
\left(1-t^{356 N-16}\right)
\left(1-t^{486 N-22}\right)} , \\
R(t,N)&=1
+t^{132 N-5}(1+t+t^2+t^3+t^4)
+t^{232 N-9}(1+t+t^2+t^3+t^4)
\label{eqn:HS_G2_Rep21_Num}\\
&\qquad
+ t^{356 N-15}(2+2t+2t^2+2t^3+2t^4+t^5)
\notag\\
&\qquad
-t^{364 N-16}(1+2t+2t^2+2t^3+2t^4+2t^5)
\notag\\
&\qquad
+ t^{486 N-21}(2+2t+2t^2+2t^3+2t^4+t^5)
\notag\\
&\qquad
-t^{488 N-22}(1+3t+3t^2+3t^3+3t^4+3t^5+t^6)
\notag\\
&\qquad
-t^{588 N-25}(1+t+t^2+t^3+t^4)
-t^{618 N-27}(1+t+t^2+t^3+t^4)
\notag\\
&\qquad
-t^{718 N-32}(1+3t+3t^2+3t^3+3t^4+3t^5+t^6)
\notag\\
&\qquad
+t^{720 N-32}(1+2t+2t^2+2t^3+2t^4+2t^5)
\notag\\
&\qquad
- t^{842 N-37}(2+2t+2t^2+2t^3+2t^4+t^5)
\notag\\
&\qquad
+t^{850 N-38}(1+2t+2t^2+2t^3+2t^4+2t^5)
\notag\\
&\qquad
+t^{974 N-43}(1+t+t^2+t^3+t^4)
+t^{1074 N-47}(1+t+t^2+t^3+t^4)
+t^{1206 N-48} \notag .
\end{align}
\end{subequations}
The numerator~\eqref{eqn:HS_G2_Rep21_Num} is a palindromic polynomial of
degree $1206N-48$; whereas, the denominator is of degree $1206N-46$. Hence, the
difference in degree between denominator and numerator is $2$, which agrees
with the quaternionic dimension of moduli space.
The Hilbert series~\eqref{eqn:HS_G2_Rep21} has a pole of order $4$ as $t\to 1$,
which equals the complex dimension of the moduli space. (One can
explicitly show that $R(t=1,N)=0$ and $\tfrac{\mathrm{d}}{\mathrm{d} t}R(t,N)|_{t=1}=0$,
but $\tfrac{\mathrm{d}^2}{\mathrm{d} t^2}R(t,N)|_{t=1}\neq0$.)
The list of appearing operators is presented in Tab.~\ref{tab:Ops_G2_Rep21}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c}
\toprule
object & & $\Delta(n_1,n_2)$ & $\mathrm{H}_{(n_1,n_2)}$\\ \midrule
Casimir & $\mathcal{C}_2$ & $2$ & ---\\
& $ \mathcal{C}_6$ & $6$ & ---\\ \midrule
bare monopole & $V_{(0,1)}^{\mathrm{dress},0}$ & $132N-6$ & ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(0,1)}^{\mathrm{dress},i}$ & $132N-6+i$ & ---
\\ \midrule
bare monopole & $V_{(1,0)}^{\mathrm{dress},0}$ & $232N-10$& ${{\rm U}(2)}$\\
dressings $(i=1,\ldots,5)$ & $V_{(1,0)}^{\mathrm{dress},i}$ & $232N-10+i$
&---\\ \midrule
bare monopole & $V_{(1,1)}^{\mathrm{dress},0}$ & $356N-16$& ${{\rm U}(1)}\times {{\rm U}(1)}$\\
dressings $(i=1,\ldots,5 ; \alpha=1,2)$ &
$V_{(1,1)}^{\mathrm{dress},i,\alpha}$ & $356N-16+i$
&--- \\
dressing &
$V_{(1,1)}^{\mathrm{dress},6}$ & $356N-16+6$
&---\\ \midrule
bare monopole & $V_{(1,2)}^{\mathrm{dress},0}$ & $486N-22$& ${{\rm U}(1)}\times {{\rm U}(1)}$\\
dressings $(i=1,\ldots,5 ; \alpha=1,2)$ &
$V_{(1,2)}^{\mathrm{dress},i,\alpha}$ & $486N-22+i$
&--- \\
dressing &
$V_{(1,2)}^{\mathrm{dress},6}$ & $486N-22+6$
&--- \\
\bottomrule
\end{tabular}
\caption{The chiral ring generators for a ${\rm G}_2$ gauge theory and matter
transforming in $[2,1]$.}
\label{tab:Ops_G2_Rep21}
\end{table}
Due to the structure of the conformal dimension the behaviour of the $[2,1]$
representation is identical to that of $[4,0]$. Consequently, we do not discuss
further details.
\section{Case: \texorpdfstring{$\boldsymbol{{{\rm SU}(3)}}$}{SU(3)}}
\label{sec:SU3}
The last rank two example we would like to cover is ${{\rm SU}(3)}$, for which the
computation takes a detour over the corresponding ${{\rm U}(3)}$ theory, similar
to~\cite{Cremonesi:2013lqa}. The advantage is that we can simultaneously
investigate the rank three example ${{\rm U}(3)}$ and demonstrate that the method of
Hilbert bases for semi-groups works equally well in higher rank cases.
\subsection{Set-up}
In the following, we systematically study a number of ${{\rm SU}(3)}$ representation,
where we understand a ${{\rm SU}(3)}$-representation $[a,b]$ as an ${{\rm U}(3)}$-representation
with a fixed ${{\rm U}(1)}$-charge.
\paragraph{Preliminaries for $\boldsymbol{{{\rm U}(3)}}$}
The GNO-dual group of ${{\rm U}(3)}$, which is again a ${{\rm U}(3)}$, has a weight lattice
characterised by $m_1,m_2,m_3\in \mathbb Z$
and the dominant Weyl chamber is given by the restriction $m_1 \geq m_2 \geq
m_3$, c.f.~\cite{Cremonesi:2013lqa}. The classical dressing factors associated
to the interior and boundaries of the dominant Weyl chamber are the following:
\begin{equation}
P_{{{\rm U}(3)}}(t^2,m_1,m_2,m_3) = \left\{
\begin{matrix} \frac{1}{(1-t^2)^3} \; , & m_1>m_2>m_3 \; , \\
\frac{1}{(1-t^2)^2(1-t^4)} \; , & (m_1=m_2 > m_3) \vee
(m_1>m_2 = m_3) \; ,\\
\frac{1}{(1-t^2)(1-t^4)(1-t^6)} \; , & m_1=m_2=m_3 \; .
\end{matrix} \right.
\label{eqn:dress_fct_U3}
\end{equation}
Note that we already introduced the fugacity $t^2$ instead of $t$.
Moreover, the GNO-dual ${{\rm U}(3)}$ has a non-trivial centre, i.e.\ $\mathcal{Z}({{\rm U}(3)})
=
{{\rm U}(1)}_J$; thus, the topological symmetry is a ${{\rm U}(1)}_J$ counted by
$z^{m_1+m_2+m_3}$.
The contributions of $N_{(a,b)}$
hypermultiplets transforming in $[a,b]$ to the conformal dimension are as
follows:
\begin{subequations}
\label{eqn:delta_h-plet_SU3}
\begin{align}
\Delta^{[1,0]}_{\mathrm{h-plet}} &= \frac{N_{(1,0)}}{2} \sum_i | m_i | \;,\\
\Delta^{[2,0]}_{\mathrm{h-plet}} &= \frac{3N_{(2,0)}}{2} \sum_i | m_i | \;,\\
\Delta^{[1,1]}_{\mathrm{h-plet}} &= N_{(1,1)} \sum_{i<j} | m_i -m_j| \;,\\
\Delta^{[3,0]}_{\mathrm{h-plet}} &= \frac{3N_{(3,0)}}{2} \sum_i | m_i |
+N_{[3,0]} \sum_{i<j} | m_i -m_j| \;,\\
\Delta^{[2,2]}_{\mathrm{h-plet}} &= 3N_{(2,2)} \sum_i | m_i |
+4N_{(2,2)} \sum_{i<j} | m_i -m_j| \; ,\\
\Delta^{[2,1]}_{\mathrm{h-plet}} &= 4N_{(2,1)} \sum_i | m_i |
+\frac{N_{(2,1)}}{2} \sum_{i<j} \left( |2 m_i -m_j| + | m_i - 2m_j| \right) \; ,
\end{align}
\end{subequations}
where $i,j=1,2,3$. In addition, the contribution of the vector-multiplets reads
as
\begin{equation}
\Delta_{\mathrm{v-plet}} = - \sum_{i<j} |m_i-m_j| \; .
\end{equation}
Consequently, one can study a pretty wild matter content if one considers the
conformal dimension to be of the form
\begin{equation}
\label{eqn:delta_U3_generic}
\Delta(m_1,m_2,m_3) = \frac{N_F}{2} \sum_i |m_i| + \left( N_A-1 \right)
\sum_{i<j} |m_i - m_j| + \frac{N_R}{2}\sum_{i<j} \left( |2m_i - m_j| + |m_i - 2
m_j|\right)
\end{equation}
and the relation to the various representations~\eqref{eqn:delta_h-plet_SU3} is
established via
\begin{subequations}
\begin{align}
N_F &= N_{(1,0)} + 3N_{(2,0)} + 3 N_{(3,0)} + 6 N_{(2,2)} + 4 N_{(2,1)} \; ,
\\
N_A &= N_{(1,1)} + N_{(3,0)} + 4 N_{(2,2)} \; , \\
N_R &= N_{(2,1)} \; .
\end{align}
\end{subequations}
\paragraph{Preliminaries for $\boldsymbol{{{\rm SU}(3)}}$}
As noted in~\cite{Cremonesi:2013lqa}, the reduction from ${{\rm U}(3)}$ to ${{\rm SU}(3)}$ (with
the same matter content) is realised by averaging over ${{\rm U}(1)}_J$, for the purpose
of setting $m_1+m_2+m_3=0$, and multiplying by $(1-t^2)$, such that
$\mathrm{Tr}(\Phi)=0$ for the adjoint scalar $\Phi$. In other words
\begin{equation}
\mathrm{HS}_{{{\rm SU}(3)}}^{[a,b]}(t^2)= (1-t^2) \oint_{|z|=1} \frac{\mathrm{d} z}{2 \pi \mathrm{i} z }
\mathrm{HS}_{{{\rm U}(3)}}^{[a,b]}(t^2,z) \; .
\label{eqn:Reduction_to_SU3}
\end{equation}
As a consequence, the conformal dimension for ${{\rm SU}(3)}$ itself is obtained
from~\eqref{eqn:delta_U3_generic} via
\begin{equation}
\Delta(m_1,m_2) \coloneqq \Delta(m_1,m_2,m_3) \big|_{m_3 =-m_1-m_2} \; .
\label{eqn:delta_SU3_generic}
\end{equation}
The Weyl chamber is now characterised by $m_1 \geq \max\{m_2,-2m_2\}$.
Multiplying~\eqref{eqn:dress_fct_U3} by $(1-t^2)$ and employing
$m_3=-m_1-m_2$ results in the classical dressing factors for ${{\rm SU}(3)}$
\begin{equation}
P_{{{\rm SU}(3)}}(t^2,m_1,m_2) = \left\{
\begin{matrix} \frac{1}{(1-t^2)^2} \; , & m_1 >\max\{m_2,-2m_2\} \; ,\\
\frac{1}{(1-t^2)(1-t^4)} \; , & (m_1=m_2 ) \vee
(m_1=-2m_2 ) \; , \\
\frac{1}{(1-t^4)(1-t^6)} \; , & m_1=m_2=0 \; .
\end{matrix} \right.
\label{eqn:dress_fct_SU3}
\end{equation}
\subsection{Hilbert basis}
\subsubsection{Fan and cones for \texorpdfstring{${{\rm U}(3)}$}{U(3)}}
Following the ideas outline previously,
$\Lambda_w(\widehat{{{\rm U}(3)}}) \slash
\mathcal{W}_{{{\rm U}(3)}}$ can be described as a collection of semi-groups
that originate from a fan.
Since this is our first $3$-dimensional example, we provide a detail
description on how to obtain the fan. Consider the absolute values $|a m_1 +b
m_2 +cm_2|$ in~\eqref{eqn:delta_SU3_generic} as \emph{Hesse normal form} for
the hyperplanes
\begin{equation}
\vec{n}\cdot \vec{m} \equiv \begin{pmatrix} a \\ b \\ c \end{pmatrix}\cdot
\begin{pmatrix} m_1 \\ m_2 \\ m_3 \end{pmatrix} =0
\end{equation}
which pass through the origin. Take all normal vectors $\vec{n}_j$, define the
matrices $M_{i,j} = (\vec{n}_i,\vec{n}_j)^T$ (for $i<j$) and compute the null
spaces (or kernel) $K_{i,j}\coloneqq\ker(M_{i,j})$. Linear algebra tell us
that $\mathrm{dim}(K_{i,j}) \geq1$, but by the specific form\footnote{$\Delta$ is
homogeneous and all hyperplanes pass through the origin; hence, no two
hyperplanes can be parallel. This implies that no two normal vectors can be
multiplies of each other.} of $\Delta$ we have the stronger condition
$\mathrm{rk}(M_{i,j})=2$ for all $i<j$; thus, we always have $\mathrm{dim}(K_{i,j})=1$. Next,
we select a basis vector $e_{i,j}$ of $K_{i,j}$ and check if $e_{i,j}$ or
$-e_{i,j}$ intersect the Weyl-chamber. If it does, then it is going to be an
edge for the fan and, more importantly, will turn out to be a ray generator
(provided one defines $e_{i,j}$ via the intersection with the corresponding
weight lattice).
Now, one has to define all $3$-dimensional cones, merge them into a fan, and,
lastly, compute the Hilbert bases. The programs
\texttt{Macaulay2} and \texttt{Sage} are convenient
tools for such tasks.
As two examples, we consider the conformal
dimension~\eqref{eqn:delta_SU3_generic} for $N_R=0$ and
$N_R\neq0$ and preform the entire procedure. That is: firstly, compute the
edges of the fan; secondly, define the all $3$-dimensional cones and; thirdly,
compute the Hilbert bases.
\paragraph{Case $N_R=0$:} In this circumstance, we deduce the following edges
\begin{align}
\begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix} \, , \quad
\begin{pmatrix}1 \\ 1 \\ 0 \end{pmatrix} \, , \quad
\begin{pmatrix}1 \\ 1 \\ 1 \end{pmatrix} \, , \quad
\begin{pmatrix}0 \\ 0 \\ -1 \end{pmatrix} \, , \quad
\begin{pmatrix}0 \\ -1 \\ -1 \end{pmatrix} \, , \quad
\begin{pmatrix}-1 \\ -1 \\ -1 \end{pmatrix} \, .
\end{align}
All these vectors are on the boundaries of the Weyl chamber.
The set of $3$-dimensional cones that generate the corresponding fan is given by
\begin{subequations}
\begin{alignat}{2}
C_1^{(3)} &= \mathrm{Cone}\left\{ \colvec{1}{0}{0}, \colvec{1}{1}{0},\colvec{1}{1}{1}
\right\} , \; &
C_2^{(3)} &= \mathrm{Cone}\left\{ \colvec{1}{0}{0},
\colvec{1}{1}{0},\colvec{0}{0}{-1}
\right\} ,\\
C_3^{(3)} &= \mathrm{Cone}\left\{ \colvec{1}{0}{0},
\colvec{0}{-1}{-1},\colvec{0}{0}{-1}
\right\} , \; &
C_4^{(3)} &= \mathrm{Cone}\left\{ \colvec{-1}{-1}{-1},
\colvec{0}{-1}{-1},\colvec{0}{0}{-1}
\right\} .
\end{alignat}
\end{subequations}
A computation shows that all four cones are strictly convex, smooth, and
simplicial. The Hilbert bases for the resulting semi-groups
comprise solely the ray generators
\begin{subequations}
\label{eqn:Hilbert_basis_U3_NR=0}
\begin{alignat}{2}
\Hcal(S_1^{(3)}) &=\left\{ \colvec{1}{0}{0},
\colvec{1}{1}{0},\colvec{1}{1}{1}
\right\} , \; &
\Hcal(S_2^{(3)}) &= \left\{ \colvec{1}{0}{0},
\colvec{1}{1}{0},\colvec{0}{0}{-1}
\right\} ,\\
\Hcal(S_3^{(3)}) &= \left\{ \colvec{1}{0}{0},
\colvec{0}{-1}{-1},\colvec{0}{0}{-1}
\right\} , \; &
\Hcal(S_4^{(3)}) &= \left\{ \colvec{-1}{-1}{-1},
\colvec{0}{-1}{-1},\colvec{0}{0}{-1}
\right\} .
\end{alignat}
\end{subequations}
From the above, we expect $6$ bare monopole operators plus their dressings for
a generic theory with $N_R=0$. Since all ray generators lie at the boundary of
the Weyl chamber, the residual gauge groups are ${{\rm U}(3)}$ for $\pm(1,1,1)$ and
${{\rm U}(2)} \times {{\rm U}(1)}$ for the other four GNO-charges.
\paragraph{Case $N_R\neq 0$:}
Here, we compute the following edges:
\begin{subequations}
\begin{alignat}{7}
&\begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix} , \; &
&\begin{pmatrix}1 \\ 1 \\ 0 \end{pmatrix} , \; &
&\begin{pmatrix}1 \\ 1 \\ 1 \end{pmatrix} , \; &
&\begin{pmatrix}2 \\ 1 \\ 0 \end{pmatrix} , \; &
&\begin{pmatrix}2 \\ 1 \\ 1 \end{pmatrix} , \; &
&\begin{pmatrix}2 \\ 2 \\ 1 \end{pmatrix}, \;&
&\begin{pmatrix}4 \\ 2 \\ 1 \end{pmatrix} , \\
&\begin{pmatrix}0 \\ 0 \\ -1 \end{pmatrix}, \;&
&\begin{pmatrix}0 \\ -1 \\ -1 \end{pmatrix} , \;&
&\begin{pmatrix}-1 \\ -1 \\ -1 \end{pmatrix} , \;&
&\begin{pmatrix}0 \\ -1 \\ -2 \end{pmatrix} , \; &
&\begin{pmatrix}-1 \\ -1 \\ -2 \end{pmatrix} , \; &
&\begin{pmatrix}-1 \\ -2 \\ -2 \end{pmatrix} , \; &
&\begin{pmatrix}-1 \\ -2 \\ -4 \end{pmatrix} .
\end{alignat}
\end{subequations}
Now, we need to proceed and define all $3$-dimensional cones that constitute
the fan and, in turn, will lead to the semi-groups that we wish to study.
\begin{subequations}
\label{eqs:Cones_U3_NR>0}
\begin{alignat}{2}
C_1^{(3)} &= \mathrm{Cone}\left\{ \colvec{1}{0}{0},\colvec{2}{1}{0},\colvec{4}{2}{1}
\right\} , \, &
C_2^{(3)} &= \mathrm{Cone}\left\{ \colvec{4}{2}{1},\colvec{1}{0}{0},\colvec{2}{1}{1}
\right\} , \\
C_3^{(3)} &= \mathrm{Cone}\left\{ \colvec{2}{2}{1},\colvec{1}{1}{0},\colvec{2}{1}{0}
\right\} , \, &
C_4^{(3)} &= \mathrm{Cone}\left\{ \colvec{2}{2}{1},\colvec{2}{1}{0},\colvec{4}{2}{1}
\right\} , \\
C_5^{(3)} &= \mathrm{Cone}\left\{ \colvec{2}{2}{1},\colvec{4}{2}{1},\colvec{2}{1}{1}
\right\} , \, &
C_6^{(3)} &= \mathrm{Cone}\left\{ \colvec{2}{2}{1},\colvec{2}{1}{1},\colvec{1}{1}{1}
\right\} , \\
C_7^{(3)} &= \mathrm{Cone}\left\{ \colvec{0}{0}{-1},\colvec{1}{0}{0},\colvec{2}{1}{0}
\right\} , \, &
C_8^{(3)} &= \mathrm{Cone}\left\{ \colvec{0}{0}{-1},\colvec{1}{1}{0},\colvec{2}{1}{0}
\right\} , \\
C_9^{(3)} &= \mathrm{Cone}\left\{
\colvec{0}{0}{-1},\colvec{0}{-1}{-2},\colvec{1}{0}{0}
\right\}, \, &
C_{10}^{(3)} &= \mathrm{Cone}\left\{
\colvec{0}{-1}{-2},\colvec{0}{-1}{-1},\colvec{1}{0}{0}
\right\} , \\
C_{11}^{(3)} &= \mathrm{Cone}\left\{
\colvec{0}{0}{-1},\colvec{0}{-1}{-2},\colvec{-1}{-2}{-4}
\right\} , \, &
C_{12}^{(3)} &= \mathrm{Cone}\left\{
\colvec{0}{0}{-1},\colvec{-1}{-2}{-4},\colvec{-1}{-1}{-2}
\right\} , \\
C_{13}^{(3)} &= \mathrm{Cone}\left\{
\colvec{0}{-1}{-2},\colvec{-1}{-2}{-4},\colvec{0}{-1}{-1}
\right\} , \, &
C_{14}^{(3)} &= \mathrm{Cone}\left\{
\colvec{0}{-1}{-1},\colvec{-1}{-2}{-4},\colvec{-1}{-2}{-2}
\right\} , \\
C_{15}^{(3)} &= \mathrm{Cone}\left\{
\colvec{-1}{-2}{-4},\colvec{-1}{-2}{-2},\colvec{-1}{-1}{-2}
\right\} , \, &
C_{16}^{(3)} &= \mathrm{Cone}\left\{
\colvec{-1}{-2}{-2},\colvec{-1}{-1}{-2},\colvec{-1}{-1}{-1}
\right\} .
\end{alignat}
\end{subequations}
All of the cones are strictly convex and simplicial, but only the cones $C_p$
for $p=1,2,3,6,\ldots,13,16$ are smooth.
Now, we compute the Hilbert bases for semi-groups $S_p^{(3)}$ for
$p=1,2,\dots,16$ and obtain
\begin{subequations}
\label{eqs:Hilbert_basis_U3_NR>0}
\begin{align}
\Hcal(S_1^{(3)}) &= \left\{
\colvec{1}{0}{0},\colvec{2}{1}{0},\colvec{4}{2}{1}
\right\} , \, &
\Hcal(S_2^{(3)}) &= \left\{
\colvec{4}{2}{1},\colvec{1}{0}{0},\colvec{2}{1}{1}
\right\} , \\
\Hcal(S_3^{(3)}) &= \left\{
\colvec{2}{2}{1},\colvec{1}{1}{0},\colvec{2}{1}{0}
\right\} , \, &
\Hcal(S_4^{(3)}) &= \left\{
\colvec{2}{2}{1},\colvec{2}{1}{0},\colvec{4}{2}{1},\colvec{3}{2}{1}
\right\} , \\
\Hcal(S_5^{(3)}) &= \left\{
\colvec{2}{2}{1},\colvec{4}{2}{1},\colvec{2}{1}{1},\colvec{3}{2}{1}
\right\} , \, &
\Hcal(S_6^{(3)}) &= \left\{
\colvec{2}{2}{1},\colvec{2}{1}{1},\colvec{1}{1}{1}
\right\} , \\
\Hcal(S_7^{(3)}) &= \left\{
\colvec{0}{0}{-1},\colvec{1}{0}{0},\colvec{2}{1}{0}
\right\} , \, &
\Hcal(S_8^{(3)}) &= \left\{
\colvec{0}{0}{-1},\colvec{1}{1}{0},\colvec{2}{1}{0}
\right\} , \\
\Hcal(S_9^{(3)}) &= \left\{
\colvec{0}{0}{-1},\colvec{0}{-1}{-2},\colvec{1}{0}{0}
\right\} , \, &
\Hcal(S_{10}^{(3)}) &= \left\{
\colvec{0}{-1}{-2},\colvec{0}{-1}{-1},\colvec{1}{0}{0}
\right\} , \\
\Hcal(S_{11}^{(3)}) &= \left\{
\colvec{0}{0}{-1},\colvec{0}{-1}{-2},\colvec{-1}{-2}{-4}
\right\} , \, &
\Hcal(S_{12}^{(3)}) &= \left\{
\colvec{0}{0}{-1},\colvec{-1}{-2}{-4},\colvec{-1}{-1}{-2}
\right\} , \\
\Hcal(S_{13}^{(3)}) &= \left\{
\colvec{0}{-1}{-2},\colvec{-1}{-2}{-4},\colvec{0}{-1}{-1}
\right\} , \, &
\Hcal(S_{14}^{(3)}) &= \left\{
\colvec{0}{-1}{-1},\colvec{-1}{-2}{-4},\colvec{-1}{-2}{-2} ,\colvec{-1}{-2}{-3}
\right\} , \\
\Hcal(S_{15}^{(3)}) &= \left\{
\colvec{-1}{-2}{-4},\colvec{-1}{-2}{-2},\colvec{-1}{-1}{-2},\colvec{-1}{-2}{-3}
\right\} , \, &
\Hcal(S_{16}^{(3)}) &= \left\{
\colvec{-1}{-2}{-2},\colvec{-1}{-1}{-2},\colvec{-1}{-1}{-1}
\right\} .
\end{align}
\end{subequations}
We observe that there are four semi-groups $S_p^{}$ for $p=4,5,14,15$ for which
the Hilbert bases exceeds the set of ray generators by an additional element.
Consequently, we expect $16$ bare monopoles plus their dressings for a generic
theory with $N_R \neq0$. However, the dressings exhibit a much richer structure
compared to $N_R=0$, because some minimal generators lie in the interior of
the Weyl chamber. The residual gauge groups are ${{\rm U}(3)}$ for $\pm(1,1,1)$; ${{\rm U}(2)}
\times {{\rm U}(1)}$ for $(1,0,0)$, $(0,0,-1)$, $(1,1,0)$, $(0,-1,-1)$, $(2,1,1)$,
$(-1,-1,-2)$, $(2,2,1)$, and $(-1,-2,-2)$; and ${{\rm U}(1)}^3$ for $(2,1,0)$,
$(0,-1,-2)$,$(4,2,1)$, $(-1,-2,-4)$, $(3,2,1)$, and $(-1,-2,-3)$.
\subsubsection{Fan and cones for \texorpdfstring{${{\rm SU}(3)}$}{SU(3)}}
The conformal dimension~\eqref{eqn:delta_SU3_generic} divides the Weyl chamber
of the GNO-dual into two different fans, depending on $N_R=0$ or $N_R\neq0$.
\paragraph{Case $\boldsymbol{N_R=0}$:}
For this situation, which is depicted in
Fig.~\ref{Fig:Hilbert_basis_SU3_NR=0},
there are three rays $~\sim |m_1|, |m_1-m_2|, |m_1+2m_2|$ present that
intersect the Weyl chamber non-trivially. The corresponding fan is generated
by two
$2$-dimensional cones
\begin{equation}
C_1^{(2)}=\mathrm{Cone}((2,-1),(1,0)) \quad\textrm{and}\quad C_2^{(2)}=\mathrm{Cone}((1,0),(1,1)) \; .
\end{equation}
The Hilbert bases for the semi-groups, obtained by intersecting the cones with
the weight lattice, are solely given by the ray generators, i.e.
\begin{equation}
\label{eqn:Hilbert_basis_SU3_NR=0}
\Hcal(S_1^{(2)}) = \Big\{(2,-1),(1,0) \Big\} \quad\textrm{and}\quad \Hcal(S_2^{(2)})= \Big\{
(1,0),(1,1) \Big\} \;.
\end{equation}
As a consequence, we expect three bare monopole operators (plus dressings) for
a generic $N_R=0$ theory. The residual gauge group is ${{\rm SU}(2)}\times {{\rm U}(1)}$ for
$(2,-1)$ and $(1,1)$, because these GNO-charges are at the boundary of the
Weyl-chamber. In contrast, $(1,0)$ has residual gauge group ${{\rm U}(1)}^2$ as it lies
in the interior of the dominant Weyl chamber.
\paragraph{Case $\boldsymbol{N_R\neq 0}$:}
For this circumstance, which is depicted in
Fig.~\ref{Fig:Hilbert_basis_SU3_NR>0},
there are two additional rays $~\sim |m_1 -2m_2 |, |m_1+3m_2|$
present, compared to $N_R=0$, that intersect the Weyl chamber non-trivially.
The corresponding fan is now generated by four $2$-dimensional cones
\begin{subequations}
\begin{alignat}{2}
C_{1-}^{(2)} &= \mathrm{Cone}((2,-1),(3,-1)) \; ,\qquad &
C_{1+}^{(2)} &= \mathrm{Cone}((3,-1),(1,0)) \; , \\
C_{2-}^{(2)} &= \mathrm{Cone}((1,0),(2,1)) \; , \qquad &
C_{2+}^{(2)} &= \mathrm{Cone}((2,1),(1,1)) \; .
\end{alignat}
\end{subequations}
The Hilbert bases for the resulting semi-groups are given by the ray
generators, i.e.
\begin{subequations}
\label{eqn:Hilbert_basis_SU3_NR>0}
\begin{alignat}{2}
\Hcal(S_{1-}^{(2)}) &= \Big\{(2,-1),(3,-1)\Big\} \; ,\qquad &
\Hcal(S_{1+}^{(2)}) &= \Big\{(3,-1),(1,0)\Big\} \; , \\
\Hcal(S_{2-}^{(2)}) &= \Big\{(1,0),(2,1)\Big\} \; , \qquad &
\Hcal(S_{2+}^{(2)}) &= \Big\{(2,1),(1,1)\Big\} \; .
\end{alignat}
\end{subequations}
Judging from the Hilbert bases, there are five bare monopole operators present
in the generic case. The residual gauge group for $(1,0)$, $(3,-1)$, and $(2,1)$
is ${{\rm U}(1)}^2$, as they lie in the interior. For $(1,1)$ and $(2,-1)$ the residual
gauge group is ${{\rm SU}(2)} \times {{\rm U}(1)}$, because these points lie at the boundary of
the Weyl chamber.
\begin{figure}[h]
\centering
\begin{subfigure}{0.485\textwidth}
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (4.5,0);
\coordinate (YAxisMin) at (0,-2.5);
\coordinate (YAxisMax) at (0,3.5);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax)
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax)
\draw (4.7,-0.2) node {$m_1$};
\draw (-0.3,3.3) node {$m_2$};
\foreach \x in {0,1,...,4}
\foreach \y in {-2,-1,...,3}
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
}
}
\draw[black,dashed,thick] (Origin) -- (3.8,3.8);
\draw[black,dashed,thick] (Origin) -- (1.2*4,-1.2*2);
\draw[black,dashed,thick] (Origin) -- (4.2,0);
\draw[black,thick] (1,0) circle (4pt);
\draw[black,thick] (1,1) circle (4pt);
\draw[black,thick] (2,-1) circle (4pt);
\end{tikzpicture}
\caption{$N_R=0$}
\label{Fig:Hilbert_basis_SU3_NR=0}
\end{subfigure}
\begin{subfigure}{0.485\textwidth}
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (4.5,0);
\coordinate (YAxisMin) at (0,-2.5);
\coordinate (YAxisMax) at (0,3.5);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax)
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax)
\draw (4.7,-0.2) node {$m_1$};
\draw (-0.3,3.3) node {$m_2$};
\foreach \x in {0,1,...,4}
\foreach \y in {-2,-1,...,3}
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
}
}
\draw[black,dashed,thick] (Origin) -- (3.8,3.8);
\draw[black,dashed,thick] (Origin) -- (1.2*4,-1.2*2);
\draw[black,dashed,thick] (Origin) -- (4.2,0);
\draw[black,dashed,thick] (Origin) -- (2.2*2,2.2*1);
\draw[black,dashed,thick] (Origin) -- (1.5*3,-1.5*1);
\draw[black,thick] (1,0) circle (4pt);
\draw[black,thick] (1,1) circle (4pt);
\draw[black,thick] (2,-1) circle (4pt);
\draw[black,thick] (3,-1) circle (4pt);
\draw[black,thick] (2,1) circle (4pt);
\end{tikzpicture}
\caption{$N_R\neq0$}
\label{Fig:Hilbert_basis_SU3_NR>0}
\end{subfigure}
\caption{The semi-groups for ${{\rm SU}(3)}$ and the corresponding ray generators
(black circled points).}
\label{Fig:Hilber_basis_SU3}
\end{figure}
\subsection{Casimir invariance}
\subsubsection{Dressings for \texorpdfstring{${{\rm U}(3)}$}{U(3)}}
Following the description of dressed monopole operators as
in~\cite{Cremonesi:2013lqa}, we diagonalise the adjoint-valued scalar $\Phi$
along the moduli space, i.e.
\begin{equation}
\mathrm{diag} \Phi = (\phi_1,\phi_2,\phi_3) \; .
\end{equation}
Moreover, the Casimir invariants of ${{\rm U}(3)}$ can then be written as $
\Casi{j}= \mathrm{Tr}(\Phi^j)= \sum_{l=1}^{3} (\phi_l)^j$ for $j=1,2,3$. We will now
elaborate on the possible dressed monopole operators by means of the insights
gained in Sec.~\ref{subsec:Dressings_as_HS} and App.~\ref{app:PL}.
To start with, for a monopole with GNO-charge such that $\mathrm{H}_{(m_1,m_2,m_3)}=
{{\rm U}(3)}$ the dressings are described by
\begin{equation}
\frac{P_{{{\rm U}(3)}}(t,m_1,m_1,m_1)}{P_{{{\rm U}(3)}}(t,0)} -1 = 0 \; ,
\end{equation}
i.e.\ there are no dressings, because the Casimir invariants of the
centraliser $\mathrm{H}_{(m_1,m_2,m_3)}$ are identical to those of $\mathrm{G}$, since the
groups coincide. Prominent examples are the (bare) monopoles of GNO-charge
$\pm(1,1,1)$.
Next, a monopole of GNO-charge such that $\mathrm{H}_{(m_1,m_2,m_3)}= {{\rm U}(1)} \times
{{\rm U}(2)}$ exhibit dressings governed by
\begin{equation}
\frac{P_{{{\rm U}(3)}}(t,m_1,m_2,m_3)}{P_{{{\rm U}(3)}}(t,0)} -1 =
\frac{(1-t^2)(1-t^4)(1-t^6)}{(1-t^2)^2(1-t^4)} -1 = t^2 + t^4 \; ,
\end{equation}
implying there to be exactly one dressing by a degree $2$ Casimir and one
dressing by a degree $4$ Casimir. The two degree $2$ Casimir invariants of
$\mathrm{H}_{(m_1,m_2,m_3)}$, one by ${{\rm U}(1)}$ and one by ${{\rm U}(2)}$, are not both
independent
because there is the overall Casimir $\Casi{1}$ of ${{\rm U}(3)}$. Therefore, only one
of them leads to an independent dressed monopole generator. The second dressing
is then due to the second Casimir of ${{\rm U}(2)}$. For example, the monopole of
GNO-charge $(1,1,0)$, $(0,-1,-1)$, $(2,1,1)$, $(-1,-2,-2)$, $(2,2,1)$, and
$(-1,-2,-2)$ exhibit these two dressings options.
Lastly, if the residual gauge group is $\mathrm{H}_{(m_1,m_2,m_3)} = {{\rm U}(1)}^3$ then the
dressings are determined via
\begin{equation}
\frac{P_{{{\rm U}(3)}}(t,m_1,m_2,m_3)}{P_{{{\rm U}(3)}}(t,0)} -1 =
\frac{(1-t^2)(1-t^4)(1-t^6)}{(1-t^2)^3} -1 = 2t^2 + 2t^4 +t^6 \; .
\end{equation}
Consequently, there are generically five dressings for each such bare monopole
operator. Examples for this instance are $(2,1,0)$, $(0,-1,-2)$, $(3,2,1)$,
$(-1,-2,-3)$, $(4,2,1)$, $(-1,-2,-4)$.
\subsubsection{Dressings for \texorpdfstring{${{\rm SU}(3)}$}{SU(3)}}
\label{subsec:Dress_SU3}
To determine the dressings, we take the adjoint scalar $\Phi$ and diagonalise
it, keeping in mind that it now belongs to ${{\rm SU}(3)}$, that is
\begin{equation}
\mathrm{diag} \Phi = (\phi_1,\phi_2,-(\phi_1+\phi_2)) \; .
\end{equation}
While keeping in mind that each $\phi_i$ has dimension one, we can write down
the dressings (in the dominant Weyl chamber):
$(1,0)$ can be dressed by two independent ${{\rm U}(1)}$-Casimir invariants, i.e.\
directly by $\phi_1$ and $\phi_2$
\begin{equation}
V_{(1,0)}^{\mathrm{dress},(0,0)}\equiv (1,0) \longrightarrow
\begin{cases} V_{(1,0)}^{\mathrm{dress},(1,0)} \equiv \phi_1 \ (1,0) \; ,\\
V_{(1,0)}^{\mathrm{dress},(0,1)} \equiv \phi_2 \ (1,0) \; ,\end{cases}
\end{equation}
such that the dressings have conformal dimension $\Delta(1,0)+1$.
Next, out of the three degree 2 combinations of $\phi_i$, only two of them are
independent and we choose them to be
\begin{equation}
V_{(1,0)}^{\mathrm{dress},(0,0)}\equiv (1,0) \longrightarrow
\begin{cases} V_{(1,0)}^{\mathrm{dress},(2,0)} \equiv \phi_1^2 \ (1,0) \;
,\\
V_{(1,0)}^{\mathrm{dress},(0,2)} \equiv \phi_2^2 \ (1,0) \; ,\end{cases}
\end{equation}
and these second order dressings have conformal dimension $\Delta(1,0)+2$.
Finally, one last dressing is possible
\begin{equation}
V_{(1,0)}^{\mathrm{dress},(0,0)}\equiv (1,0) \longrightarrow
V_{(1,0)}^{\mathrm{dress},(3,0)+(0,3)} \equiv (\phi_1^3+ \phi_2^3) \ (1,0)
\; ,
\end{equation}
having dimension $\Delta(1,0)+3$. Alternatively, we utilise App.~\ref{app:PL}
and compute the number and degrees of the dressed monopole operators of
magnetic charge $(1,0)$ via the quotient $ P_{{{\rm SU}(3)}}(t^2,1,0) \slash
P_{{{\rm SU}(3)}}(t^2,0,0)=1+2t^2+2t^4+t^6$.
For the two monopoles of GNO-charge $(1,1)$ and $(2,-1)$, the residual
gauge group is ${{\rm SU}(2)}\times {{\rm U}(1)}$,
i.e.\ the monopoles can be dressed by a degree one Casmir invariant of the
${{\rm U}(1)}$
and by a degree two Casimir invariant of the ${{\rm SU}(2)}$. These increase the
dimensions by one and two, respectively. Consequently, we obtain
\begin{align}
V_{(1,1)}^{\mathrm{dress},0}\equiv (1,1) \longrightarrow
\begin{cases} V_{(1,1)}^{\mathrm{dress},{{\rm U}(1)}} \equiv (\phi_1+\phi_2) \ (1,1)
\; ,\\
V_{(1,1)}^{\mathrm{dress},{{\rm SU}(2)}} \equiv (\phi_1^2+\phi_2^2 ) \ (1,1) \;
,\end{cases}
\end{align}
and similarly
\begin{align}
V_{(2,-1)}^{\mathrm{dress},0}\equiv (2,-1) \longrightarrow
\begin{cases} V_{(2,-1)}^{\mathrm{dress},{{\rm U}(1)}} \equiv (\phi_1+\phi_2) \
(2,-1) \; ,\\
V_{(2,-1)}^{\mathrm{dress},{{\rm SU}(2)}} \equiv (\phi_1^2+\phi_2^2 ) \ (2,-1)
\;. \end{cases}
\end{align}
Since the magnetic weights $(1,1)$, $(2,-1)$ lie at the boundary of the
dominant Weyl chamber, we can derive the dressing behaviour via $
P_{{{\rm SU}(3)}}(t^2,(1,1)\text{ or }(2,-1)) \slash P_{{{\rm SU}(3)}}(t^2,0,0)=1+t^2+t^4$ and
obtain agreement with our choice of generators.
The remaining monopoles of GNO-charge $(2,1)$ and $(3,-1)$ can be treated
analogously to $(1,0)$ and we obtain
\begin{align}
V_{(2,1)}^{\mathrm{dress},(0,0)}\equiv (2,1) &\longrightarrow
\begin{cases} V_{(2,1)}^{\mathrm{dress},(1,0)} \equiv \phi_1 \ (2,1) \; ,\\
V_{(2,1)}^{\mathrm{dress},(0,1)} \equiv \phi_2 \ (2,1) \; , \\
V_{(2,1)}^{\mathrm{dress},(2,0)} \equiv \phi_1^2 \ (2,1) \; ,\\
V_{(2,1)}^{\mathrm{dress},(0,2)} \equiv \phi_2^2 \ (2,1) \; , \\
V_{(2,1)}^{\mathrm{dress},(3,0)+(0,3)} \equiv (\phi_1^3+\phi_2^3) \ (2,1)
\; , \end{cases}\\
V_{(3,-1)}^{\mathrm{dress},(0,0)}\equiv (3,-1) &\longrightarrow
\begin{cases} V_{(3,-1)}^{\mathrm{dress},(1,0)} \equiv \phi_1 \ (3,-1) \;
,\\
V_{(3,-1)}^{\mathrm{dress},(0,1)} \equiv \phi_2 \ (3,-1) \; , \\
V_{(3,-1)}^{\mathrm{dress},(2,0)} \equiv \phi_1^2 \ (3,-1) \; ,\\
V_{(3,-1)}^{\mathrm{dress},(0,2)} \equiv \phi_2^2 \ (3,-1) \; , \\
V_{(3,-1)}^{\mathrm{dress},(3,0)+(0,3)} \equiv (\phi_1^3+\phi_2^3) \ (3,-1)
\; . \end{cases}
\end{align}
There can be circumstances in which not all dressings for the minimal
generators determined by the Hilbert bases~\eqref{eqn:Hilbert_basis_SU3_NR>0}
are truly independent. However, this will only occur for special
configurations of $(N_F,N_A,F_R)$ and, therefore, is considered as
``non-generic'' case.
\subsection{Category \texorpdfstring{$N_R =0$}{NR=0}}
\subsubsection{\texorpdfstring{$N_F$}{NF} hypermultiplets in
\texorpdfstring{$[1,0]$}{[1,0]} and \texorpdfstring{$N_A$}{NA} hypermultiplets
in \texorpdfstring{$[1,1]$}{[1,1]}}
\paragraph{Intermediate step at $\boldsymbol{{{\rm U}(3)}}$}
The conformal dimension~\eqref{eqn:delta_U3_generic} reduces for $N_R=0$ to the
following:
\begin{equation}
\Delta(m_1,m_2,m_3)=\frac{N_F}{2} \sum_i |m_i| + (N_A-1) \sum_{i<j} |m_i-m_j|
\; . \label{eqn:delta_U3_Rep10+11}
\end{equation}
The Hilbert series is then readily computed
\begin{subequations}
\label{eqn:HS_U3_Rep10+11}
\begin{align}
&\mathrm{HS}_{{{\rm U}(3)}}^{[1,0]+[1,1]}(N_F,N_A,t,z)= \frac{R(N_F,N_A,t,z)}{P(N_F,N_A,t,z)}
\;, \\
P(N_F,N_A,t,z) &= \prod_{j=1}^3 \left(1-t^{2j}\right)
\left(1-\tfrac{1}{z} t^{4 N_A+N_F-4}\right)
\left(1-z t^{4 N_A+N_F-4}\right)
\label{eqn:HS_U3_Rep10+11_Den}\\*
&\qquad \quad \times
\left(1-\tfrac{1}{z^2} t^{4 N_A+2 N_F-4}\right)
\left(1-z^2 t^{4 N_A+2 N_F-4}\right)
\left(1-\tfrac{1}{z^3} t^{3 N_F}\right)
\left(1-z^3 t^{3 N_F}\right) \; ,
\notag \\
R(N_F,N_A,t,z)&=
1+t^{8 N_A+2 N_F-2}
-t^{8 N_A+4 N_F-8} (1+2t^2+2t^4)
+2 t^{8 N_A+6 N_F-8}(1-t^6)
\label{eqn:HS_U3_Rep10+11_Num}\\
&\qquad
+ t^{8 N_A+8 N_F-6}(2+2t^2+t^4)
-t^{8 N_A+10 N_F-8}
+t^{16 N_A+6 N_F-10} \notag \\
&\qquad
-t^{16 N_A+12 N_F-10}
-t^{6 N_F} \notag \\
&+\left(z+\frac{1}{z}\right) \bigg(
t^{4 N_A+N_F-2} (1+t^2)
+t^{4 N_A+7 N_F-4}
-t^{4 N_A+5 N_F-4} (1+t^2+t^4)
\notag \\
&\phantom{+\left(z+\frac{1}{z}\right) \bigg( }
-t^{8 N_A+3 N_F-6}(1+t^2)
+t^{8 N_A+9 N_F-6}(1+t^2)
-t^{12 N_A + 5N_F-6}
\notag \\
&\phantom{+\left(z+\frac{1}{z}\right) \bigg( }
+t^{12 N_A+7 N_F -10}(1+t^2+t^4)
-t^{12 N_A+11 N_F-10}(1+t^2) \bigg)
\notag\\
&+\left(z^2+\frac{1}{z^2}\right) \bigg(
t^{4 N_A+2 N_F-2}
+t^{4 N_A+2 N_F}
-t^{4 N_A+4 N_F-4}(1+t^2+t^4)
+t^{4 N_A+8 N_F-4}
\notag \\
&\phantom{+\left(z^2+\frac{1}{z^2}\right)\bigg( }
-t^{12 N_A+4 N_F-6}
+t^{12 N_A+ 8 N_F-10}(1+t^2+t^4)
-t^{12 N_A+ 10 N_F-10}(1+t^2) \bigg)
\notag \\
&+\left(z^3+\frac{1}{z^3}\right) \bigg(
t^{8 N_A+3 N_F-2}
-t^{8 N_A+5 N_F-6}(1+t^2+t^4)
\notag \\
&\phantom{+\left(z^3+\frac{1}{z^3}\right)\bigg( }
+t^{8 N_A+7 N_F-8}(1+t^2+t^4)
-t^{8 N_A+9 N_F-8} \bigg) \; .
\notag
\end{align}
\end{subequations}
One can check that $R(N_F,N_A,t=1,z)=0$ and $\tfrac{\mathrm{d}^n}{\mathrm{d}
t^n}R(N_F,N_A,t,z)|_{t=1,z=1}=0$ for $n=1,2$.
Thus, the Hilbert series~\eqref{eqn:HS_U3_Rep10+11} has a pole of order $6$,
which matches the dimension of the moduli space. Moreover, one computes the
degree of the numerator~\eqref{eqn:HS_U3_Rep10+11_Num} to be
$12 N_F +16 N_A -10$ and the degree of the
denominator~\eqref{eqn:HS_U3_Rep10+11_Den} to be $12 N_F +16 N_A -4$,
such that their difference equals the dimension of the moduli space.
The interpretation follows the results~\eqref{eqn:Hilbert_basis_U3_NR=0}
obtained from the Hilbert bases and we summarise the minimal generators in
Tab.~\ref{tab:Ops_U3_NR=0}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c}
\toprule
\multicolumn{2}{c|}{$(m_1,m_2,m_3)$} &
$2\Delta(m_1,m_2,m_3)$ & $\mathrm{H}_{(m_1,m_2,m_3)}$ \\ \midrule
$(1,0,0)$ & $(0,0,-1)$ & $N_F +4N_A -4$ & ${{\rm U}(1)}\times {{\rm U}(2)}$ \\
$(1,1,0)$ & $(0,-1,-1)$ & $2N_F +4N_A -4$ & ${{\rm U}(1)}\times {{\rm U}(2)}$ \\
$(1,1,1)$ & $(-1,-1,-1)$ & $3N_F$ & ${{\rm U}(3)}$ \\
\bottomrule
\end{tabular}
\caption{The monopole generators for a ${{\rm U}(3)}$ gauge theory with $N_R=0$ that
together with the Casimir invariants generate the chiral ring.}
\label{tab:Ops_U3_NR=0}
\end{table}
\paragraph{Reduction to $\boldsymbol{{{\rm SU}(3)}}$}
Following the prescription~\eqref{eqn:Reduction_to_SU3}, we derive the
following Hilbert series:
\begin{subequations}
\label{eqn:HS_SU3_Rep10+11}
\begin{align}
\mathrm{HS}_{{{\rm SU}(3)}}^{[1,0]+[1,1]}(N_F,N_A,t)&= \frac{R(N_F,N_A,t)}{\left(1-t^4\right)
\left(1-t^6\right)\left(1-t^{8 N_A+2
N_F-8}\right) \left(1-t^{12 N_A+4 N_F-12}\right) } \; ,\\
R(N_F,N_A,t)&=1
+t^{8 N_A+2 N_F-6}(2+2t^2+t^4)
\label{eqn:HS_SU3_Rep10+11_Num}\\
& \qquad
+t^{12 N_A+4 N_F-12}(1+2t^2+2t^4)
+t^{20 N_A+6 N_F-14} \notag \;.
\end{align}
\end{subequations}
An inspection yields that the numerator~\eqref{eqn:HS_SU3_Rep10+11_Num} is
a palindromic polynomial of degree $20N_A+6N_F-14$; while the degree of the
denominator is $20N_A+6N_F-10$. Thus, the difference in the degrees is 4, which
equals the complex dimension of the moduli space.
In addition, the Hilbert series~\eqref{eqn:HS_SU3_Rep10+11} has a pole of order
four at $t\to 1$, which agrees with the dimension of Coulomb branch as well.
The minimal generators of~\eqref{eqn:Hilbert_basis_SU3_NR=0} are given by
$V_{(1,0)}^{\mathrm{dress},(0,0)} $ with
$2\Delta(1,0)=8 N_A+2 N_F-8$, and $V_{(1,1)}^{\mathrm{dress},0} $ and
$V_{(2,-1)}^{\mathrm{dress},0}$ with $2\Delta(2,-1)=2\Delta(1,1)=12 N_A+4
N_F-12 $. The dressed monopole operators are as described in
Subsec.~\ref{subsec:Dress_SU3}.
\subsubsection{\texorpdfstring{$N$}{N} hypermultiplets in
\texorpdfstring{$[1,0]$}{[1,0]}
representation}
Considering $N$ hypermultiplets in the fundamental representation is on extreme
case of~\eqref{eqn:delta_U3_generic}, as $N_A=0=N_R$. We recall the results
of~\cite{Cremonesi:2013lqa} and discuss them in the context of Hilbert bases
for semi-groups.
\paragraph{Intermediate step at $\boldsymbol{{{\rm U}(3)}}$}
The Hilbert series has been computed to read
\begin{equation}
\mathrm{HS}_{{{\rm U}(3)}}^{[1,0]}(N,t,z) = \prod_{j=1}^3
\frac{1-t^{2N+2-2j}}{(1-t^{2j})(1-zt^{N+2-2j})(1-\tfrac{t^{N+2-2j}}{z})} \; .
\end{equation}
Notably, it is a complete intersection in which the (bare and dressed) monopole
operators of GNO-charge $(1,0,0)$ and $(0,0,-1)$ generate all other monopole
operators. The to be expected minimal generators $(1,1,0)$, $(0,-1,-1)$,
$(1,1,1)$, and $(-1,-1,-1)$ are now generated because
\begin{subequations}
\begin{align}
V_{(1,1,0)}^{\mathrm{dress},0}&= V_{(1,0,0)}^{\mathrm{dress},1}
+V_{(0,1,0)}^{\mathrm{dress},1} \; ,\\
V_{(1,1,0)}^{\mathrm{dress},0}&= V_{(1,0,0)}^{\mathrm{dress},2}
+V_{(0,1,0)}^{\mathrm{dress},2} +V_{(0,0,1)}^{\mathrm{dress},2}\; .
\end{align}
\end{subequations}
\paragraph{Reduction to $\boldsymbol{{{\rm SU}(3)}}$}
The reduction leads to
\begin{equation}
\label{eqn:HS_SU3_Rep10_Hanany}
\mathrm{HS}_{{{\rm SU}(3)}}^{[1,0]}(N,t)= \frac{1+t^{2N-6} +2 t^{2N-4} +t^{2N-2} +t^{4N-8}
}{(1-t^4)(1-t^6) (1-t^{2N-6})(1-t^{2N-8})} \; .
\end{equation}
Although the form of the Hilbert series~\eqref{eqn:HS_SU3_Rep10_Hanany} is
suggestive: it has a pole of order $4$ for $t\to1$ and the numerator is
palindromic, there is one drawback: no monopole operator of conformal dimension
$(2N-6)$ exists. Therefore, we provide a equivalent rational function to
emphasis the minimal generators:
\begin{equation}
\label{eqn:HS_SU3_Rep10_mod}
\mathrm{HS}_{{{\rm SU}(3)}}^{[1,0]}(N,t)= \frac{1
+ t^{2 N-6}(2+2t^2+t^4)
+t^{4 N-12}(1+2t^2+2t^4)
+t^{6 N-14}}{(1-t^4)(1-t^6)
(1-t^{2N-8})(1-t^{4N-12})} \; .
\end{equation}
The equivalent form~\eqref{eqn:HS_SU3_Rep10_mod} still has a pole of order
$4$ and a palindromic numerator. Moreover, the monopole generators are clearly
visible, as we know the set of minimal
generators~\eqref{eqn:Hilbert_basis_SU3_NR=0}, and can be summarise for
completeness: $2\Delta(1,0)=2N-8$ and $2\Delta(1,1)=2\Delta(2,-1)=4N-12$.
\subsubsection{\texorpdfstring{$N$}{N} hypermultiplets in
\texorpdfstring{$[1,1]$}{[1,1]} representation}
Investigating $N$ hypermultiplets in the adjoint representation is another
extreme case of~\eqref{eqn:delta_U3_generic} as $N_F=0=N_R$.
The conformal dimension in this circumstance reduces to
\begin{equation}
\Delta(m_1,m_2,m_3)=(N-1) \sum_{i<j} | m_i-m_j| \; ,
\label{eqn:conformal_dim_11}
\end{equation}
and we notice that there is the shift symmetry $m_i \to m_i +a$ present. Due to
this, the naive calculation of the ${{\rm U}(3)}$ Hilbert series is divergent, which we
understand as follows: Define overall ${{\rm U}(1)}$-charge $M\coloneqq m_1+m_2+m_3$,
then the Hilbert series becomes
\begin{align}
\mathrm{HS}_{{{\rm U}(3)}}^{(1,1)}= \sum_{M\in \mathbb{Z}} \sum_{m_1,m_2 \atop m_1\geq \max{(
m_2 , M-2m_2 )} }
t^{2(N-1)(3m_1+3m_2-2M+|m_1-m_2|)} \ z^M \ P_{{{\rm U}(3)}}(t,m_1,m_2,m_3) \; .
\end{align}
Since we want to use the ${{\rm U}(3)}$-calculation as an intermediate step to derive
the ${{\rm SU}(3)}$-case, the only meaningful choice to fix the shift-symmetry is
$m_1+m_2+m_3=0$. But then
\begin{align}
\mathrm{HS}_{{{\rm U}(3)},\mathrm{fixed}}^{(1,1)}= \sum_{m_1,m_2 \atop
m_1\geq \max{(
m_2 ,-2m_2 )} }
t^{2(N-1)(3m_1+3m_2+|m_1-m_2|)} \ P_{{{\rm U}(3)}}(t,m_1,m_2,-m_1-m_2)
\end{align}
and the transition to ${{\rm SU}(3)}$ is simply
\begin{align}
\mathrm{HS}_{{{\rm SU}(3)}}^{(1,1)}&= (1-t^2) \int_{|z|=1} \frac{\mathrm{d} z}{2 \pi z}
\sum_{m_1,m_2 \atop m_1\geq \max{(
m_2 ,-2m_2 )} }
t^{2(N-1)(3m_1+3m_2+|m_1-m_2|)} \ P_{{{\rm U}(3)}}(t,m_1,m_2,-m_1-m_2) \notag \\
&= \sum_{m_1,m_2 \atop m_1\geq \max{(
m_2 ,-2m_2 )} }
t^{2(N-1)(3m_1+3m_2+|m_1-m_2|)} \ P_{{{\rm SU}(3)}}(t,m_1,m_2) \; .
\end{align}
The computation then yields
\begin{equation}
\mathrm{HS}_{{{\rm SU}(3)}}^{(1,1)}= \frac{1
+ t^{8 N-6} (2+2t^2+t^4)
+ t^{12 N-12} (1+2t^2+2t^4)
+ t^{20 N-14}}{\left(1-t^4\right) \left(1-t^6\right)
\left(1-t^{8 N-8}\right) \left(1-t^{12 N-12}\right) } \; .
\label{eqn:HS_U3_Rep11}
\end{equation}
We see that numerator of~\eqref{eqn:HS_U3_Rep11} is a palindromic
polynomial of degree $20N-14$; while the degree of the denominator is
$20N-10$. Hence, the difference in the degrees is 4, which coincides with the
complex dimension of the moduli space. The same holds for the order of the pole
of~\eqref{eqn:HS_U3_Rep11} at $t\to 1$.
The interpretation of the appearing monopole operators, and their dressings, is
completely analogous to~\eqref{eqn:HS_SU3_Rep10+11} and reproduces the picture
concluded from the Hilbert bases~\eqref{eqn:Hilbert_basis_U3_NR=0}. To be
specific, $2\Delta(1,0)=8N-8$ and $2\Delta(1,1)=2\Delta(2,-1)=12N-12$.
\subsubsection{\texorpdfstring{$N$}{N} hypers in
\texorpdfstring{$[3,0]$}{[3,0]}
representation}
\paragraph{Intermediate step at $\boldsymbol{{{\rm U}(3)}}$}
The conformal dimension reads
\begin{equation}
\Delta(m_1,m_2,m_3)= \frac{3}{2} N \sum_i |m_i| +(N-1) \sum_{i<j} |m_i-m_j| \; .
\label{eqn:delta_U3_Rep30}
\end{equation}
We then obtain for $N>2$ the Hilbert series:
\begin{subequations}
\label{eqn:HS_U3_Rep30}
\begin{equation}
\mathrm{HS}_{{{\rm U}(3)}}^{[3,0]}(t,z)= \frac{R(N,t,z)}{P(N,t,z)} \; ,
\end{equation}
\begin{align}
P(N,t,z) &=
\prod_{j=1}^3 \left(1-t^{2j}\right)
\left(1-\tfrac{1}{z}t^{7 N-4}\right)
\left(1-z t^{7 N-4}\right)
\left(1-\tfrac{1}{z^2} t^{10 N-4}\right)
\left(1-z^2 t^{10 N-4}\right)
\label{eqn:HS_U3_Rep30_Den}\\*
&\qquad \times
\left(1-\tfrac{1}{z^3} t^{9 N}\right)
\left(1-z^3 t^{9 N}\right) \notag \; ,\\
R(N,t,z)&= 1+t^{14 N-2}-t^{18 N}-t^{20 N-8}-2 t^{20 N-6}-2 t^{20 N-4}+2 t^{26
N-8}-2 t^{26 N-2}
\label{eqn:HS_U3_Rep30_Num}\\
&\qquad +2 t^{32 N-6} +2 t^{32 N-4}+t^{32 N-2}+t^{34 N-10}-t^{38 N-8}-t^{52
N-10} \notag \\
&+(z+\tfrac{1}{z})\Big(
t^{7 N-2}
+t^{7 N}
-t^{17 N-6}
-t^{17 N-4}
-t^{19 N-4}
-t^{19 N-2}
-t^{19 N}
+t^{25 N-4}\notag\\
&\phantom{(z+\tfrac{1}{z})}
-t^{27 N-6}
+t^{33 N-10}
+t^{33 N-8}
+t^{33 N-6}
+t^{35 N-6}
+t^{35 N-4}
-t^{45 N-10}
-t^{45 N-8} \Big) \notag \\
&+(z^2+\tfrac{1}{z^2})\Big(
t^{10 N-2}
+t^{10 N}
-t^{16 N-4}
-t^{16 N-2}
-t^{16 N}
-t^{24 N-6}
+t^{28 N-4}
+t^{36 N-10} \notag\\
&\phantom{(z^2+\tfrac{1}{z^2})}+t^{36 N-8}+t^{36 N-6}-t^{42 N-10}-t^{42 N-8}
\Big) \notag \\
&+(z^3+\tfrac{1}{z^3})\Big(
t^{17 N-2}-t^{23 N-6}-t^{23 N-4}-t^{23 N-2}+t^{29 N-8}+t^{29 N-6}+t^{29
N-4}-t^{35 N-8}\Big) \notag \; .
\end{align}
\end{subequations}
The Hilbert series~\eqref{eqn:HS_U3_Rep30} has a pole of order $6$ as $t\to1$,
because $R(N,t=1,z)=0$ and $\tfrac{\mathrm{d}^n}{\mathrm{d} t^n}R(N,t,z)|_{t=1}=0$ for
$n=1,2$. Therefore, the moduli space is $6$-dimensional. Also, the degree
of~\eqref{eqn:HS_U3_Rep30_Num} is $52N-10$, while the degree
of~\eqref{eqn:HS_U3_Rep30_Den} us $52N-4$; thus, the difference in degrees
equals the dimension of the moduli space.%
As this example is merely a special case of~\eqref{eqn:HS_U3_Rep10+11}, we just
summarise the minimal generators in Tab.~\ref{tab:Ops_U3_Rep30}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c}
\toprule
\multicolumn{2}{c|}{$(m_1,m_2,m_3)$} &
$2\Delta(m_1,m_2,m_3)$ & $\mathrm{H}_{(m_1,m_2,m_3)}$ \\ \midrule
$(1,0,0)$ & $(0,0,-1)$ & $7N-4$ & ${{\rm U}(1)}\times {{\rm U}(2)}$ \\
$(1,1,0)$ & $(0,-1,-1)$ & $10N-4$ & ${{\rm U}(1)}\times {{\rm U}(2)}$ \\
$(1,1,1)$ & $(-1,-1,-1)$ & $9N$ & ${{\rm U}(3)}$ \\
\bottomrule
\end{tabular}
\caption{The monopole generators for a ${{\rm U}(3)}$ gauge theory with matter
transforming in $[3,0]$ that together with the Casimir invariants generate the
chiral ring.}
\label{tab:Ops_U3_Rep30}
\end{table}
\paragraph{Reduction to $\boldsymbol{{{\rm SU}(3)}}$}
The Hilbert series reads
\begin{equation}
\label{eqn:HS_SU3_Rep30}
\mathrm{HS}_{{{\rm SU}(3)}}^{[3,0]}(t)=\frac{1+t^{14 N-6}(2+2t^2+t^4)
+t^{24N-12}(1+2t^2+2t^4)+t^{38 N-14}}{\left(1-t^4\right)
\left(1-t^6\right) \left(1-t^{14 N-8}\right) \left(1-t^{24 N-12}\right) } \; .
\end{equation}
It is apparent that the numerator of~\eqref{eqn:HS_SU3_Rep30} is a
palindromic polynomial of degree $38N-14$; while the degree of the denominator
is $38N-10$; hence, the difference in the degrees is $4$, which equals the
complex
dimension of the moduli space.
The structure of~\eqref{eqn:HS_SU3_Rep30} is merely a special case
of~\eqref{eqn:HS_SU3_Rep10+11}, and the conformal dimensions of the minimal
generators are $2\Delta(1,0)=14N-8$ and $2\Delta(1,1)=2\Delta(2,-1)=24N-12$.
\subsection{Category \texorpdfstring{$N_R\neq0$}{NR>0}}
\subsubsection{\texorpdfstring{$N_F$}{NF} hypers in
\texorpdfstring{$[2,1]$}{[2,1]}, \texorpdfstring{$N_A$}{NA} hypers in
\texorpdfstring{$[1,1]$}{[1,1]}, \texorpdfstring{$N_R$}{NR}
hypers in \texorpdfstring{$[2,1]$}{[2,1]} representation}
\paragraph{Intermediate step at $\boldsymbol{{{\rm U}(3)}}$}
The conformal dimension reads
\begin{align}
2\Delta(m_1,m_2,m_3)=(4 N_R+N_A )\sum_{i=1}^3 \left| m_i\right|
&+N_R \sum_{i<j}\left(\left| 2 m_i-m_j\right| +\left| m_i-2
m_j\right| \right) \\
&+2(N_A-1) \sum_{i<j}\left| m_i-m_j\right| \; . \notag
\end{align}
The Hilbert series reads
\begin{subequations}
\label{eqn:HS_U3_Rep10+11+21}
\begin{equation}
\mathrm{HS}_{{{\rm U}(3)}}^{[1,0]+[1,1]+[2,1]}(t,z)=
\frac{R(N_F,N_A,N_R,t,z)}{P(N_F,N_A,N_R,t,z)} \;,
\end{equation}
with
\begin{align}
P(N_F,N_A,N_R,t,z)=\prod_{j=1}^3 \left(1-t^{2j}\right)
&\left(1-\frac{t^{N_F+4 N_A+10 N_R-4}}{z}\right)
\left(1-z t^{N_F+4 N_A+10 N_R-4}\right)
\label{eqn:HS_U3_Rep10+11+21_Den}\\
&\times
\left(1-\frac{t^{2 N_F+4 N_A+16 N_R-4}}{z^2}\right)
\left(1-z^2 t^{2 N_F+4 N_A+16 N_R-4}\right) \notag \\
&\times
\left(1-\frac{t^{3 N_F+18 N_R}}{z^3}\right)
\left(1-z^3 t^{3 N_F+18 N_R}\right) \notag \\
&\times
\left(1-\frac{t^{3 N_F+8 N_A+24 N_R-8}}{z^3}\right)
\left(1-z^3 t^{3 N_F+8 N_A+24 N_R-8}\right) \notag \\
&\times
\left(1-\frac{t^{4 N_F+4 N_A+24 N_R-4}}{z^4}\right)
\left(1-z^4 t^{4 N_F+4 N_A+24 N_R-4}\right) \notag \\
&\times
\left(1-\frac{t^{5 N_F+4 N_A+30 N_R-4}}{z^5}\right)
\left(1-z^5 t^{5 N_F+4 N_A+30 N_R-4}\right) \notag \\
&\times
\left(1-\frac{t^{7 N_F+12 N_A+46 N_R-12}}{z^7}\right)
\left(1-z^7 t^{7 N_F+12 N_A+46 N_R-12}\right)\notag \; ,
\end{align}
\end{subequations}
and the numerator $R(N_F,N_A,N_R,t,z)$ is too long to be displayed,
because it contains $28650$ monomials. We checked
explicitly that $R(N_F,N_A,N_R,t=1,z) =0$ and $\tfrac{\mathrm{d}^n}{\mathrm{d} t^n}
R(N_F,N_A,N_R,t,z)|_{t=1,z=1}=0$ for all $n=1,2\ldots, 10$. Therefore, the
Hilbert series~\eqref{eqn:HS_U3_Rep10+11+21} has a pole of order $6$ at $t=1$,
which equals the dimension of the moduli space. In addition,
$R(N_F,N_A,N_R,t,z)$ is a polynomial of degree $50 N_F +72 N_A + 336 N_R -66$,
while the denominator~\eqref{eqn:HS_U3_Rep10+11+21_Den} is of degree $50 N_F
+72
N_A + 336 N_R -60$. The difference in degrees reflects the dimension of the
moduli space as well.
Following the analysis of the Hilbert bases~\eqref{eqn:Hilbert_basis_SU3_NR>0},
we identify the bare monopole operators and provide their conformal dimensions
in Tab.~\ref{tab:Ops_U3_Rep10+11+21}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c}
\toprule
\multicolumn{2}{c|}{$(m_1,m_2,m_3)$} & $2\Delta(m_1,m_2,m_3)$ &
$\mathrm{H}_{(m_1,m_2,m_3)}$ \\ \midrule
$(1,0,0)$ & $(0,0,-1)$ & $N_F+4 N_A+10 N_R-4$ & ${{\rm U}(1)}\times {{\rm U}(2)} $ \\
$(1,1,0)$ & $(0,-1,-1)$ & $2 N_F+4 N_A+16 N_R-4$ & ${{\rm U}(1)}\times {{\rm U}(2)} $
\\
$(1,1,1)$ & $(-1,-1,-1)$ & $3 N_F+18 N_R$ & ${{\rm U}(3)}$ \\
$(2,1,0)$ & $(0,-1,-2)$ & $3 N_F+8 N_A+24 N_R-8$ & ${{\rm U}(1)}^3$ \\
$(2,1,1)$ & $(-1,-1,-2)$ & $4 N_F+4 N_A+24 N_R-4$ & ${{\rm U}(1)} \times {{\rm U}(2)} $
\\
$(2,2,1)$ & $(-1,-2,-2)$ & $5 N_F+4 N_A+30 N_R-4$ & ${{\rm U}(1)} \times {{\rm U}(2)} $
\\
$(3,2,1)$ & $(-1,-2,-3)$ & $6 N_F+8 N_A+38 N_R-8$ & ${{\rm U}(1)}^3$ \\
$(4,2,1)$ & $(-1,-2,-4)$ & $7 N_F+12 N_A+46 N_R-12$ & ${{\rm U}(1)}^3$ \\
\bottomrule
\end{tabular}
\caption{The monopole generators for a ${{\rm U}(3)}$ gauge theory with a mixture of
matter transforming in $[1,0]$, $[1,1]$, and $[2,1]$.}
\label{tab:Ops_U3_Rep10+11+21}
\end{table}
The result~\eqref{eqn:HS_U3_Rep10+11+21} has been tested against the
independent calculations of the cases: $N$ hypermultiplets in $[1,0]$;
$N_F$ hypermultiplets in $[1,0]$ together with $N_A$ hypermultiplets in
$[1,1]$; and $N$ hypermultiplets in $[2,1]$. All the calculations agree.
\paragraph{Reduction to $\boldsymbol{{{\rm SU}(3)}}$}
The Hilbert series for the ${{\rm SU}(3)}$ theory reads
\begin{subequations}
\label{eqn:HS_SU3_Rep10+11+21}
\begin{equation}
\mathrm{HS}_{{{\rm SU}(3)}}^{[1,0]+[1,1]+[2,1]}(N_F,N_A,N_R,t)=
\frac{R(N_F,N_A,N_R,t)}{P(N_F,N_A,N_R,t)} \; ,
\end{equation}
\begin{align}
P(N_F,N_A,N_R,t)&=
\left(1-t^4\right) \left(1-t^6\right)
\left(1-t^{2 N_F+8 N_A+20 N_R-8}\right)
\label{eqn:HS_SU3_Rep10+11+21_Den}\\
&\qquad \times
\left(1-t^{4 N_F+12 N_A+36 N_R-12}\right)
\left(1-t^{6 N_F+20 N_A+54 N_R-20}\right) \; , \notag\\
R(N_F,N_A,N_R,t)&=1
+ t^{2 N_F+8 N_A+20 N_R-6} (2+2t^2+t^4)
\label{eqn:HS_SU3_Rep10+11+21_Num}
\\
&\qquad +t^{4 N_F+12 N_A+36 N_R-12}(1+2t^2+2t^4)
\notag \\
&\qquad +t^{6 N_F+20 N_A+54 N_R-20}(1+4t^2+4t^4+2t^6)
\notag \\
&\qquad - t^{6 N_F+20 N_A+56 N_R-20}(2+4t^2+4t^4+t^6)
\notag \\
&\qquad - t^{8 N_F+28 N_A+74 N_R-26}(2+2t^2+t^4)
\notag \\
&\qquad -t^{10 N_F+32 N_A+90 N_R-32}(1+2t^2+2t^4)
-t^{12 N_F+40 N_A+110 N_R-34}\notag \; .
\end{align}
\end{subequations}
Again, the numerator~\eqref{eqn:HS_SU3_Rep10+11+21_Num} is an anti-palindromic
polynomial of degree $12 N_F+40 N_A+110 N_R-34$; while the
denominator~\eqref{eqn:HS_SU3_Rep10+11+21_Den} is of degree $12 N_F+40
N_A+110 N_R-30$, such that the difference is again $4$.
The minimal generators from~\eqref{eqn:Hilbert_basis_SU3_NR>0} are now realised
with the following conformal dimensions: $2\Delta(1,0)=2 N_F+8 N_A+20
N_R-8$, $2\Delta(1,1)=2\Delta(2,-1)= 4 N_F+12 N_A+36 N_R-12 $ , and
$2\Delta(2,1)=2\Delta(3,-1)=6 N_F+20 N_A+54 N_R-20$. Moreover, the
appearing dressed monopoles are as described in
Subsec.~\ref{subsec:Dress_SU3}.
\paragraph{Remark}
The ${{\rm SU}(3)}$ result~\eqref{eqn:HS_SU3_Rep10+11+21} has been tested against the
independent calculations of the cases: $N$ hypermultiplets in $[1,0]$; $N$
hypermultiplets in $[1,1]$; $N_F$ hypermultiplets in $[1,0]$ together with
$N_A$ hypermultiplets in $[1,1]$; and $N$ hypermultiplets in $[2,1]$. All the
calculations agree.
\paragraph{Dressings of \texorpdfstring{$(2,1)$}{(2,1)} and
\texorpdfstring{$(3,-1)$}{(3,-1)}}
From the generic analysis~\eqref{eqn:Hilbert_basis_SU3_NR>0} the bare monopoles
of GNO-charges $(3,-1)$ and $(2,1)$ are necessary generators. However, not all
of their dressings need to be independent generators, c.f.\ App.~\ref{app:PL}.
\begin{itemize}
\item $N_R=0$: $(2,1)$ and $(3,-1)$ are generated by
$(1,0)$, $(1,1)$, and $(2,-1)$, which is the generic result
of~\eqref{eqn:Hilbert_basis_SU3_NR=0}.
\item $N_R=1$: Here, $(2,1)$ and $(3,-1)$ are independent, but not all
of their dressings, as we see
\begin{align}
(2,1) = (1,1) + (1,0) \quad\textrm{and}\quad \Delta(2,1) + 1 = \Delta(1,1) +
\Delta(1,0) \; .
\end{align}
Hence, \emph{only one} of the degree one dressings
$V_{(2,1)}^{\mathrm{dress},(1,0)}$, $V_{(2,1)}^{\mathrm{dress},(0,1)}$ is
independent, while the other can be generated. (Same holds for $(3,-1)$.)
\item $N_R=2$: Here, $(2,1)$ and $(3,-1)$ are independent, but not all
of their dressings, as we see
\begin{align}
(2,1) = (1,1) + (1,0) \quad\textrm{and}\quad \Delta(2,1) + 2 = \Delta(1,1) +
\Delta(1,0) \; .
\end{align}
Hence, \emph{only one} of the degree two dressings
$V_{(2,1)}^{\mathrm{dress},(2,0)}$, $V_{(2,1)}^{\mathrm{dress},(0,2)}$ is
independent, while the other can be generated. However, \emph{both} degree one
dressings $V_{(2,1)}^{\mathrm{dress},(1,0)}$,
$V_{(2,1)}^{\mathrm{dress},(0,1)}$ are independent. (Same holds for
$(3,-1)$.)
\item $N_R=3$: Here, $(2,1)$ and $(3,-1)$ are independent, but
still not all of their dressings, as we see
\begin{align}
(2,1) = (1,1) + (1,0) \quad\textrm{and}\quad \Delta(2,1) + 3 = \Delta(1,1) +
\Delta(1,0) \; .
\end{align}
Hence, the degree three dressing
$V_{(2,1)}^{\mathrm{dress},(3,0)+(0,3)}$ is not
independent. However, \emph{both} degree one
dressings $V_{(2,1)}^{\mathrm{dress},(1,0)}$,
$V_{(2,1)}^{\mathrm{dress},(0,1)}$ and \emph{both} degree two dressings
$V_{(2,1)}^{\mathrm{dress},(2,0)}$, $V_{(2,1)}^{\mathrm{dress},(0,2)}$
are
independent. (Same holds for $(3,-1)$.)
\item $N_R\geq4$: The bare and the all dressed monopoles corresponding to
$(2,1)$ and $(3,-1)$ are independent.
\end{itemize}
\subsubsection{\texorpdfstring{$N$}{N} hypers in
\texorpdfstring{$[2,1]$}{[2,1]}
representation}
\paragraph{Intermediate step at $\boldsymbol{{{\rm U}(3)}}$}
The conformal dimension reads
\begin{align}
2\Delta(m_1,m_2,m_3)=4 N \sum_{i=1}^3 \left| m_i\right|
+N \sum_{i<j}\left(\left| 2 m_i-m_j\right| +\left| m_i-2
m_j\right| \right) -2 \sum_{i<j}\left| m_i-m_j\right| \; .
\end{align}
From the calculations we obtain the Hilbert series
\begin{subequations}
\label{eqn:HS_U3_Rep21}
\begin{equation}
\mathrm{HS}_{{{\rm U}(3)}}^{[2,1]}(N,t,z)=\frac{R(N,t,z)}{P(N,t,z)} \; ,
\end{equation}
\begin{align}
P(N,t,z)= \prod_{j=1}^3 \left(1-t^{2j}\right)
&\left(1-\frac{t^{10 N-4}}{z}\right)
\left(1-z t^{10 N-4}\right)
\left(1-\frac{t^{16 N-4}}{z^2}\right)
\left(1-z^2 t^{16 N-4}\right)
\label{eqn:HS_U3_Rep21_Den}\\
&\times
\left(1-\frac{t^{18 N}}{z^3}\right)
\left(1-z^3 t^{18 N}\right)
\left(1-\frac{t^{24 N-8}}{z^3}\right)
\left(1-z^3 t^{24 N-8}\right) \notag \\
& \times
\left(1-\frac{t^{24 N-4}}{z^4}\right)
\left(1-z^4 t^{24 N-4}\right)
\left(1-\frac{t^{30 N-4}}{z^5}\right)
\left(1-z^5 t^{30 N-4}\right) \notag \\
& \times
\left(1-\frac{t^{46 N-12}}{z^7}\right)
\left(1-z^7 t^{46 N-12}\right) \notag \; ,
\end{align}
\end{subequations}
and the numerator $R(N,t,z)$ is with $13492$ monomials too long to be
displayed. Nevertheless, we checked
explicitly that $R(N,t=1,z) =0$ and $\tfrac{\mathrm{d}^n}{\mathrm{d} t^n}
R(N,t,z)|_{t=1,z=1}=0$ for all $n=1,2\ldots, 10$. Therefore, the
Hilbert series~\eqref{eqn:HS_U3_Rep21} has a pole of order $6$ at $t=1$,
which equals the dimension of the moduli space. In addition, the degree of
$R(N,t,z)$ is $296N-62$, while the denominator~\eqref{eqn:HS_U3_Rep21_Den} is
of degree $296N-56$; therefore, the difference in degrees is again equal to the
dimension of the moduli space.
The Hilbert series~\eqref{eqn:HS_U3_Rep21} appears as special case
of~\eqref{eqn:HS_U3_Rep10+11+21} and as such the appearing monopole operators
are the same. For completeness, we provide in Tab.~\ref{tab:Ops_U3_Rep21} the
conformal dimensions of all
minimal (bare) generators~\eqref{eqs:Hilbert_basis_U3_NR>0}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c}
\toprule
\multicolumn{2}{c|}{$(m_1,m_2,m_3)$} & $2\Delta(m_1,m_2,m_3)$ &
$\mathrm{H}_{(m_1,m_2,m_3)}$ \\ \midrule
$(1,0,0)$ & $(0,0,-1)$ & $10N-4$ & ${{\rm U}(1)}\times {{\rm U}(2)} $ \\
$(1,1,0)$ & $(0,-1,-1)$ & $16N-4$ & ${{\rm U}(1)}\times {{\rm U}(2)} $ \\
$(1,1,1)$ & $(-1,-1,-1)$ & $18N$ & ${{\rm U}(3)}$ \\
$(2,1,0)$ & $(0,-1,-2)$ & $24N-8$ & ${{\rm U}(1)}^3$ \\
$(2,1,1)$ & $(-1,-1,-2)$ & $24N-4 $ & ${{\rm U}(1)} \times {{\rm U}(2)} $ \\
$(2,2,1)$ & $(-1,-2,-2)$ & $30N-4 $ & ${{\rm U}(1)} \times {{\rm U}(2)} $ \\
$(3,2,1)$ & $(-1,-2,-3)$ & $38N-8 $ & ${{\rm U}(1)}^3$ \\
$(4,2,1)$ & $(-1,-2,-4)$ & $46N-12 $ & ${{\rm U}(1)}^3$ \\
\bottomrule
\end{tabular}
\caption{The monopole generators for a ${{\rm U}(3)}$ gauge theory with matter
transforming in $[2,1]$ that generate the chiral ring (together with the
Casimir invariants).}
\label{tab:Ops_U3_Rep21}
\end{table}
The GNO-charge $(3,2,1)$ is not apparent in the Hilbert series, but we
know it to be present due to the analysis of the Hilbert
bases~\eqref{eqs:Hilbert_basis_U3_NR>0}.
\paragraph{Reduction to $\boldsymbol{{{\rm SU}(3)}}$}
After reduction~\eqref{eqn:Reduction_to_SU3} of~\eqref{eqn:HS_U3_Rep21} to
${{\rm SU}(3)}$ we obtain the following Hilbert series:
\begin{subequations}
\label{eqn:HS_SU3_Rep21}
\begin{align}
HS_{{{\rm SU}(3)}}^{(2,1)}&=
\frac{R(N,t)}{\left(1-t^4\right)
\left(1-t^6\right) \left(1-t^{20 N-8}\right) \left(1-t^{36 N-12}\right)
\left(1-t^{54 N-20}\right)} \; ,
\\
R(N,t)&=1
+ t^{20 N-6}(2+2t^2+t^4)
+t^{36 N-12}(1+2t^2+2t^4)
\label{eqn:HS_SU3_Rep21_Num}\\
&\quad
+t^{54 N-20} (1+4t^2+4t^4+2t^6)
-t^{56 N-20}(2+4t^2+4t^4+t^6)
\notag \\
&\quad
- t^{74 N-26}(2+2t^2+t^4)
-t^{90 N-32}(1+2t^2+2t^4)
-t^{110 N-34} \notag \; .
\end{align}
\end{subequations}
The numerator of~\eqref{eqn:HS_SU3_Rep21_Num} is an
anti-palindromic polynomial of degree $110N-34$; while the numerator is of
degree $110N-30$. Consequently, the difference in degree reflects the complex
dimension of the moduli space.
The Hilbert series~\eqref{eqn:HS_SU3_Rep21} is merely a special case
of~\eqref{eqn:HS_SU3_Rep10+11+21} and, thus, the appearing (bare and dressed)
monopole operators are the same. For completeness we provide their conformal
dimensions: $2\Delta(1,0)=20N-8$, $2\Delta(1,1)=2\Delta(2,-1)=36N-12$, and
$2\Delta(2,1)=2\Delta(3,-1)= 54N-20$.
\section{Case: \texorpdfstring{$\boldsymbol{{{\rm U}(1)} \times {{\rm U}(1)}}$}{U(1)xU(1)}}
\label{sec:U1xU1}
In this section we analyse the abelian product ${{\rm U}(1)}\times {{\rm U}(1)}$. By
construction, the Hilbert series simplifies as the dressing factors are constant
throughout the lattice of magnetic weights. Consequently, abelian
theories do not exhibit dressed monopole operators.
\subsection{Set-up}
The weight lattice of the GNO-dual of ${{\rm U}(1)}$ is simply $\mathbb Z$ and no Weyl-group
exists due the abelian character; thus, $\Lambda_{w}(\widehat{{{\rm U}(1)}\times{{\rm U}(1)}})
= \mathbb Z^2$. Moreover, since ${{\rm U}(1)}\times {{\rm U}(1)}$ is abelian the classical dressing
factors are the same for any magnetic weight $(m_1,m_2)$, i.e.
\begin{equation}
P_{{{\rm U}(1)}\times {{\rm U}(1)}} (t,m_1,m_2) = \frac{1}{(1-t)^2} \; ,
\end{equation}
which reflects the two degree one Casimir invariants.
\subsection{Two types of hypermultiplets}
\paragraph{Set-up}
To consider a rank $2$ abelian gauge group of the form ${{\rm U}(1)}\times{{\rm U}(1)}$ requires
a delicate choice of matter content. If one considers $N_1$ hypermultiplets
with charges $(a_1,b_1)\in \mathbb{N}^2$ under ${{\rm U}(1)}\times{{\rm U}(1)}$, then the
conformal dimension reads
\begin{subequations}
\begin{equation}
\Delta_{1 \text{h-plet}}(m_1,m_2)= \frac{N_1}{2} \left|a_1 m_1 + b_1 m_2
\right| \qquad\textrm{for}\quad (m_1,m_2)\in \mathbb Z^2 \; .
\end{equation}
However, there exists an infinite number of points $\{m_1 =b_1 k, m_2=-a_1 k,
k\in \mathbb Z \} $ with zero conformal dimension, i.e.\ the Hilbert series does not
converge due to a decoupled ${{\rm U}(1)}$. Fixing this symmetry would reduce the
rank to one.
Fortunately, we can circumvent this problem by introducing a second set of
$N_2$ hypermultiplets with charges $(a_2,b_2)\in \mathbb{N}^2$, such that the
matrix
\begin{equation}
\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix}
\end{equation}
has maximal rank. The relevant conformal dimension then reads
\begin{equation}
\Delta_{2 \text{h-plet}}(m_1,m_2)= \sum_{j=1}^2 \frac{N_j}{2} \left|a_j m_1 +
b_j m_2 \right| \qquad\textrm{for}\quad (m_1,m_2)\in \mathbb Z^2 \; . \label{eqn:Delta:U1xU1_2hypers}
\end{equation}
Nevertheless, this set-up would introduce four charges and the summation of the
Hilbert series becomes tricky. We evade the difficulties by the
\emph{choice} $a_2=b_1$ and $b_2=-a_1$. Dealing with such a scenario leads to
summation
bounds such as
\begin{align}
a\, m_1 \geq b \, m_2 &\Leftrightarrow m_1 \geq \tfrac{b}{a} m_2
\Leftrightarrow m_1
\geq \lceil \tfrac{b}{a} m_2 \rceil \; ,\\
a\, m_1 < b\, m_2 &\Leftrightarrow m_1 < \tfrac{b}{a} m_2 \Leftrightarrow m_1
< \lceil \tfrac{b}{a} m_2 \rceil -1 \; .
\end{align}
Having the summation variable within a floor or ceiling function seems to be
an elaborate task with \texttt{Mathematica}. Therefore, we simplify the
setting by \emph{assuming} $\exists$ $k \in \mathbb{N}$ such that $b_1 = k a_1$.
Then we arrive at
\begin{equation}
\Delta_{2 \text{h-plet}}(m_1,m_2)=
\frac{ a_1}{2} \left( N_1 \left| m_1 + k m_2 \right|
+ N_2 \left|k m_1 - m_2 \right| \right) \qquad\textrm{for}\quad (m_1,m_2)\in \mathbb Z^2 \; .
\label{eqn:Delta_U1xU1_2hplet}
\end{equation}
\end{subequations}
For this conformal dimension, there exists exactly one point $(m_1,m_2)$ with
zero conformal dimension --- the trivial solution. Further, by a redefinition
of $N_1$ and $N_2$ we can consider $a_1=1$.
\paragraph{Hilbert basis}
Consider the conformal dimension~\eqref{eqn:Delta_U1xU1_2hplet} for $a_1=1$.
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-5,0);
\coordinate (XAxisMax) at (5,0);
\coordinate (YAxisMin) at (0,-5);
\coordinate (YAxisMax) at (0,5);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax)
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax)
\draw (5.2,-0.2) node {$m_1$};
\draw (-0.2,5.2) node {$m_2$};
\foreach \x in {-5,-4,...,5}
\foreach \y in {-5,-4,...,5}
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
}
}
\draw[black,dashed,thick] (-1.4,-5.6) -- (1.4,5.6);
\draw[black,dashed,thick] (-5.6,1.4) -- (5.6,-1.4);
\draw[black,thick] (1,4) circle (4pt);
\draw[black,thick] (-1,-4) circle (4pt);
\draw[black,thick] (4,-1) circle (4pt);
\draw[black,thick] (-4,1) circle (4pt);
\draw[blue,thick] (1,3) circle (4pt);
\draw[blue,thick] (1,2) circle (4pt);
\draw[blue,thick] (1,1) circle (4pt);
\draw[blue,thick] (1,0) circle (4pt);
\draw[red,thick] (3,-1) circle (4pt);
\draw[red,thick] (2,-1) circle (4pt);
\draw[red,thick] (1,-1) circle (4pt);
\draw[red,thick] (0,-1) circle (4pt);
\draw[green,thick] (-1,0) circle (4pt);
\draw[green,thick] (-1,-1) circle (4pt);
\draw[green,thick] (-1,-2) circle (4pt);
\draw[green,thick] (-1,-3) circle (4pt);
\draw[orange,thick] (0,1) circle (4pt);
\draw[orange,thick] (-1,1) circle (4pt);
\draw[orange,thick] (-2,1) circle (4pt);
\draw[orange,thick] (-3,1) circle (4pt);
\draw (2.5,2.5) node {$S_1^{(2)}$};
\draw (2.5,-2.5) node {$S_2^{(2)}$};
\draw (-2.5,-2.5) node {$S_3^{(2)}$};
\draw (-2.5,2.5) node {$S_4^{(2)}$};
\end{tikzpicture}
\caption{The dashed lines correspond the
$k\, m_1=m_2$ and $m_1=-k\,m_2$ and divide the lattice $\mathbb Z^2$ into four
semi-groups $S_j^{(2)}$ for $j=1,2,3,4$. The black
circles denote the ray generators, while the blue circles complete the Hilbert
basis for $S_1^{(2)}$, red circled points complete the basis for
$S_{2}^{(2)}$. Green circles correspond to the remaining minimal generators of
$S_{3}^{(2)}$ and orange circled points are the analogue for $S_{4}^{(2)}$.
(Here, the example is $k=4$.)}
\label{Fig:U1xU1}
\end{figure}
By resolving the absolute
values, we divide $\mathbb Z^2$ into four semi-groups
\begin{subequations}
\begin{align}
S_1^{(2)} &= \left\{ (m_1,m_2)\in \mathbb Z^2 | \left( km_1 \geq m_2 \right) \wedge
\left( m_1\geq -k m_2 \right) \right\} \; ,\\
S_2^{(2)} &= \left\{ (m_1,m_2)\in \mathbb Z^2 | \left( km_1 \geq m_2 \right) \wedge
\left( m_1 \leq -k m_2 \right) \right\} \; , \\
S_3^{(2)} &= \left\{ (m_1,m_2)\in \mathbb Z^2 | \left( km_1 \leq m_2 \right) \wedge
\left( m_1\geq -k m_2 \right) \right\} \; , \\
S_4^{(2)} &= \left\{ (m_1,m_2)\in \mathbb Z^2 | \left( km_1 \leq m_2 \right) \wedge
\left( m_1 \leq -k m_2 \right) \right\} \; ,
\end{align}
\end{subequations}
which all descend from $2$-dimensional rational polyhedral cones. The situation
is depicted in Fig.~\ref{Fig:U1xU1}.
Next, one needs to compute the Hilbert basis $\Hcal(S)$ for each
semi-group $S$. In this example, it follows from the drawing that
\begin{subequations}
\begin{align}
\Hcal(S_1^{(2)}) &= \Big\{ (k,-1), \big\{ (1,l) \; \big| \;
l=0,1, \ldots, k\big\} \Big\} \; ,\\
\Hcal(S_2^{(2)}) &=\Big\{ (-1,-k),\big\{ (l,-1) \; \big| \; l=0,1,
\ldots, k\big\} \Big\} \; ,\\
\Hcal(S_3^{(2)}) &= \Big\{ (-k,1),\{ (-1,-l) \; \big| \; l=0,1,
\ldots, k\big\} \Big\} \; ,\\
\Hcal(S_4^{(2)}) &= \Big\{ (1,k),\{ (-l,1) \; \big| \; l=0,1,
\ldots, k\big\} \Big\} \; .
\end{align}
\end{subequations}
For a fixed $k\geq1$ we obtain $4(k+1)$ basis elements.
\paragraph{Hilbert series}
We then compute the following Hilbert series
\begin{align}
\mathrm{HS}_{{{\rm U}(1)}\times {{\rm U}(1)}}^{k}(t,z_1,z_2) = \frac{1}{(1-t)^2} \sum_{m_1,m_2 \in \mathbb Z}
z_1^{m_1} z_2^{m_2} t^{\Delta_{2 \text{h-plet}}(m_1,m_2)} \; ,
\end{align}
for which we obtain
\begin{subequations}
\label{eqn:HS_U1xU1_2hplets}
\begin{equation}
\mathrm{HS}_{{{\rm U}(1)}\times {{\rm U}(1)}}^{k}(t,z_1,z_2) = \frac{R(t,z_1,z_2)}{P(t,z_1,z_2)} \; ,
\end{equation}
with denominator
\begin{align}
P(t,z_1,z_2)&=
(1-t)^2
\left(1-\frac{1}{z_1} t^{\frac{k N_2-N_1}{2}}\right)
\left(1-z_1 t^{\frac{k N_2-N_1}{2}}\right)
\left(1-\frac{1}{z_2} t^{\frac{k N_1-N_2}{2}}\right)
\left(1-z_2 t^{\frac{k N_1-N_2}{2}}\right) \notag \\
&\quad \times
\left(1-\frac{1}{z_1} t^{\frac{k N_2+N_1}{2}}\right)
\left(1-z_1 t^{\frac{k N_2+N_1}{2}}\right)
\left(1-\frac{1}{z_2} t^{\frac{k N_1+N_2}{2}}\right)
\left(1-z_2 t^{\frac{k N_1+N_2}{2}}\right) \\
&\quad \times
\left(1-\frac{1}{z_1 z_2^{k}} t^{\frac{1}{2} \left(k^2+1\right) N_1}\right)
\left(1-z_1 z_2^k t^{\frac{1}{2} \left(k^2+1\right) N_1}\right) \notag \\
&\quad \times
\left(1-\frac{z_1^k}{z_2} t^{\frac{1}{2} \left(k^2+1\right) N_2}\right)
\left(1-\frac{z_2}{ z_1^{k}} t^{\frac{1}{2} \left(k^2+1\right) N_2}\right)
\notag \; ,
\end{align}
\end{subequations}
while the numerator $R(t,z_1,z_2)$ is too long to be displayed, as it contains
$1936$ monomials.
Nonetheless, one can explicitly verify a few properties of the Hilbert series.
For example, the Hilbert series~\eqref{eqn:HS_U1xU1_2hplets} has a pole of
order
$4$ at $t\to 1$, because $R(1,z_1,z_2)=0$ and the derivatives $\tfrac{\mathrm{d}^n
}{\mathrm{d} t^n}R(t,z_1,z_2) |_{t=1}=0$ for
$n=1,2,\ldots 9$ (at least for $z_1=z_2=1$).
Moreover, the degrees of numerator and denominator depend on the relations
between $N_1$, $N_2$, and $k$; however, one can show that the difference in
degrees is precisely $2$, i.e.\ it matches the quaternionic dimension of the
moduli space.
\paragraph{Discussion}
Analysing the plethystic logarithm and the Hilbert series, the monopole
operators corresponding to the Hilbert basis can be identified as follows:
Eight poles of the Hilbert series~\eqref{eqn:HS_U1xU1_2hplets} can be
identified with monopole generators as shown in Tab.~\ref{tab:U1xU1_Ops_ray}.
\begin{table}[h]
\begin{subtable}{1\textwidth}
\centering
\begin{tabular}{c|c||c|c}
\toprule
$(m_1,m_2)$ & $\Delta(m_1,m_2)$ & $(m_1,m_2)$ & $\Delta(m_1,m_2)$ \\ \midrule
$(1,0)$, $(-1,0)$ & $\frac{1}{2} \left( N_1+k N_2 \right)$ & $(0,1)$,
$(0,-1)$ & $\frac{1}{2} \left( k N_1+ N_2 \right)$ \\
$(1,k)$, $(-1,-k)$ & $\frac{1}{2} \left(1 +k^2 \right) N_1$ & $(-k,1)$,
$(k,-1)$ & $\frac{1}{2} \left(1 +k^2 \right) N_2$ \\
\bottomrule
\end{tabular}
\caption{The minimal generators which are
ray generators or poles of the Hilbert series.}
\label{tab:U1xU1_Ops_ray}
\end{subtable}
\begin{subtable}{1\textwidth}
\centering
\begin{tabular}{c|c||c|c}
\toprule
$(m_1,m_2)$ & $\Delta(m_1,m_2)$ & $(m_1,m_2)$ & $\Delta(m_1,m_2)$ \\ \midrule
$(1,l)$, $(-1,-l)$ & $\frac{1}{2} N_1 (k l+1)+\frac{1}{2} N_2
(k-l)$ & $(-l,1)$,
$(l,-1)$ & $\frac{1}{2} N_1 (k-l)+\frac{1}{2} N_2 (k l+1)$ \\
\bottomrule
\end{tabular}
\caption{The minimal generators, labelled by $l= 1,2,\ldots , k-1$, which are
not ray generators.}
\label{tab:U1xU1_Ops_non-ray}
\end{subtable}
\caption{The set of bare monopole operators for a ${{\rm U}(1)}\times {{\rm U}(1)}$ theory with
conformal dimension~\eqref{eqn:Delta_U1xU1_2hplet}.}
\end{table}
Studying the plethystic logarithm clearly displays the remaining set, which is
displayed in Tab.~\ref{tab:U1xU1_Ops_non-ray}.
\paragraph{Remark}
A rather special case of~\eqref{eqn:Delta:U1xU1_2hypers} is $a_2=0=b_1$, for
which the theory becomes the product of two ${{\rm U}(1)}$-theories with $N_1$ or $N_2$
electrons of charge $a$ or $b$, respectively. In detail, the
conformal dimension is simply
\begin{equation}
\Delta_{2 \text{h-plet}}(m_1,m_2)\stackrel{a_2=0=b_1}{=}
\frac{N_1}{2} \left|a\, m_1 \right| +
\frac{N_2}{2} \left| b\, m_2 \right|\qquad\textrm{for}\quad (m_1,m_2)\in \mathbb Z^2 \; ,
\end{equation}
such that the Hilbert series becomes
\begin{align}
\mathrm{HS}_{{{\rm U}(1)}^2}^{a,b}(t,z_1,z_2) &=
\frac{1-t^{a N_1}}{ (1-t) \left(1-z_1 t^{\tfrac{a N_1}{2}}\right)
\left(1-\tfrac{1}{z_1} t^{\tfrac{a N_1}{2}} \right) } \times
\frac{1-t^{b N_2}}{ (1-t) \left(1-z_2 t^{\tfrac{b N_2}{2}}\right)
\left(1-\tfrac{1}{z_2} t^{\tfrac{b N_2}{2}}\right) } \notag \\
&= \mathrm{HS}_{{{\rm U}(1)}}^{a}(t,z_1,N_1) \times \mathrm{HS}_{{{\rm U}(1)}}^{b}(t,z_2,N_2) \; .
\end{align}
For the unrefined Hilbert series, that is $z_1=1=z_2$, the rational function
$\mathrm{HS}_{{{\rm U}(1)}}^{a}(t,N)$ equals the Hilbert series of the (abelian) ADE-orbifold
$\mathbb C^2 \slash \mathbb Z_{a\cdot N}$, see for instance~\cite{Cremonesi:2014xha}. Thus,
the ${{\rm U}(1)}{\times}{{\rm U}(1)}$ Coulomb branch is the product of two A-type singularities.
Quite intuitively, taking the corresponding limit $k\to0$
in~\eqref{eqn:HS_U1xU1_2hplets} yields the product
\begin{equation}
\lim_{k\to0}\mathrm{HS}_{{{\rm U}(1)}\times {{\rm U}(1)}}^{k}(t,z_1,z_2) = \mathrm{HS}_{{{\rm U}(1)}}(t,z_1,N_1)
\times \mathrm{HS}_{{{\rm U}(1)}}(t,z_2,N_2) \; ,
\end{equation}
which are ${{\rm U}(1)}$ theories with $N_1$ and $N_2$ electrons of unit charge. The
unrefined rational functions are the Hilbert series of $\mathbb Z_{N_1}$ and
$\mathbb Z_{N_2}$ singularities in the ADE-classification.
From Fig.~\ref{Fig:U1xU1} one observes that in the limit $k\to 0$ the relevant
rational cones coincide with the four quadrants of $\mathbb R^2$ and the
Hilbert basis reduces to the cone generators.
\subsection{Reduced moduli space of one
\texorpdfstring{${{\rm SO}}(5)$}{SO(5)}-instanton}
Consider the Coulomb branch of the quiver gauge theory depicted
in Fig.~\ref{Fig:Quiver_SO5_instanton} with
conformal dimension given by
\begin{equation}
\Delta(m_1,m_2)= \frac{1}{2} \left( |m_1| + | m_1 -2m_2| \right) \; .
\label{eqn:Delta_SO5_instanton}
\end{equation}
Instead of associating~\eqref{eqn:Delta_SO5_instanton} with the
quiver of Fig.~\ref{Fig:Quiver_SO5_instanton}, one could equally well
understand it as a special case of a ${{\rm U}(1)}^2$ theory with two different
hypermultiplets~\eqref{eqn:Delta:U1xU1_2hypers}.
\begin{figure}[h]
\centering
\begin{tikzpicture}
\draw[black,thick] (0,0) circle (10pt);
\draw[black,thick] (2,0) circle (10pt);
\draw[thick,double] (0.35,0) -- (1.65,0);
\draw[thick,black] (1.2,0.2) -- (1,0) -- (1.2,-0.2);
\draw[black,thick] (-0.3,1.7) rectangle (0.3,2.3);
\draw[thick,double] (0,0.35) -- (0,1.7);
\draw[thick,black] (0.2,1.2) -- (0,1) -- (-0.2,1.2);
\draw (0,-0.7) node {$U(1)$};
\draw (2,-0.7) node {$U(1)$};
\draw (-0.8,2) node {$U(1)$};
\end{tikzpicture}
\caption{Quiver gauge theory whose Coulomb branch is the reduced
moduli space of one ${{\rm SO}}(5)$-instanton.}
\label{Fig:Quiver_SO5_instanton}
\end{figure}
\paragraph{Hilbert basis}
Similar to the previous case, the conformal dimensions induces a fan which, in
this case, is generated by four $2$-dimensional cones
\begin{subequations}
\begin{alignat}{2}
C_1^{(2)}&=\mathrm{Cone} \left((2,1),(0,1) \right) \; , \qquad&
C_2^{(2)}&=\mathrm{Cone} \left((2,1),(0,-1) \right) \; , \\
C_3^{(2)}&=\mathrm{Cone} \left((-2,-1),(0,-1) \right) \; ,\qquad&
C_4^{(2)}&=\mathrm{Cone} \left((-2,-1),(0,1) \right) \; .
\end{alignat}
\end{subequations}
The intersection with the $\mathbb Z^2$ lattice defines the semi-groups $S_p^{(2)}
\coloneqq C_p^{(2)} \cap \mathbb Z^2$ for which we need to compute the Hilbert bases.
Fig.~\ref{Fig:Quiver_SO5} illustrates the situation and we obtain
\begin{subequations}
\label{eqn:Hilbert_basis_U1xU1_SO5}
\begin{alignat}{2}
\Hcal(S_1^{(2)}) &= \left\{(2,1),(1,1),(0,1) \right\} \; ,\qquad&
\Hcal(S_2^{(2)}) &= \left\{(2,1),(1,0),(0,-1) \right\} \; , \\
\Hcal(S_3^{(2)}) &= \left\{(-2,-1),(-1,-1),(0,-1) \right\} \; ,\qquad&
\Hcal(S_4^{(2)}) &= \left\{(-2,-1),(-1,0),(0,1) \right\} \; .
\end{alignat}
\end{subequations}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-3.5,0);
\coordinate (XAxisMax) at (3.5,0);
\coordinate (YAxisMin) at (0,-2.5);
\coordinate (YAxisMax) at (0,2.5);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax)
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax)
\draw (3.7,-0.2) node {$m_1$};
\draw (-0.2,2.7) node {$m_2$};
\foreach \x in {-3,-2,...,3}
\foreach \y in {-2,-1,...,2}
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
}
}
\draw[black,dashed,thick] (0,-2.3) -- (0,2.3);
\draw[black,dashed,thick] (-2*2,-1*2) -- (2*2,2*1);
\draw[black,thick] (0,1) circle (4pt);
\draw[black,thick] (0,-1) circle (4pt);
\draw[black,thick] (2,1) circle (4pt);
\draw[black,thick] (-2,-1) circle (4pt);
\draw[red,thick] (1,1) circle (4pt);
\draw[red,thick] (-1,-1) circle (4pt);
\draw[blue,thick] (1,0) circle (4pt);
\draw[blue,thick] (-1,0) circle (4pt);
\draw (1.5,1.5) node {$S_1^{(2)}$};
\draw (2.5,-1.5) node {$S_2^{(2)}$};
\draw (-1.5,-1.5) node {$S_3^{(2)}$};
\draw (-2.5,1.5) node {$S_4^{(2)}$};
\end{tikzpicture}
\caption{The dashed lines correspond the
$m_1=2m_2$ and $m_1=0$ and divide the lattice $\mathbb Z^2$ into four
semi-groups $S_j^{(2)}$ for $j=1,2,3,4$. The black
circles denote the ray generators, while the red circles complete the Hilbert
bases for $S_1^{(2)}$ and $S_3^{(2)}$. Blue circled lattice points complete the
bases for $S_2^{(2)}$ and $S_4^{(2)}$.}
\label{Fig:Quiver_SO5}
\end{figure}
\paragraph{Hilbert series}
The Hilbert series is evaluated to
\begin{subequations}
\label{eqn:HS_U1xU1_SO5}
\begin{align}
\mathrm{HS}_{{{\rm U}(1)}^2}^{{{\rm SO}}(5)}(t,z_1,z_2)&= \frac{R(t,z_1,z_2) }{
(1-t)^2
\left(1-\frac{t}{z_2}\right)
\left(1- z_2 t\right)
\left(1-\frac{t}{z_1^2 z_2}\right)
\left(1- z_1^2 z_2 t\right)} \; ,\\
R(t,z_1,z_2)&=1
+t \left(z_1+\frac{1}{z_1}+z_1 z_2+\frac{1}{z_1 z_2}\right) \\
&\qquad -2 t^{2} \left(1+ z_1+ \frac{1}{z_1} + z_1 z_2+\frac{1}{z_1
z_2}\right) \notag \\
&\qquad +t^{3} \left(z_1+\frac{1}{z_1}+ z_1 z_2+\frac{1}{z_1 z_2}\right)
+t^{4} \notag \; .
\end{align}
\end{subequations}
The Hilbert series~\eqref{eqn:HS_U1xU1_SO5} has a pole of order $4$ at $t=1$,
because one can explicitly verify that $R(t=1,z_1,z_2)=0$, $\tfrac{\mathrm{d}}{\mathrm{d}
t}R(t,z_1,z_2)|_{t=1}=0$,
but $\tfrac{\mathrm{d}^2}{\mathrm{d} t^2}R(t,z_1,z_2)|_{t=1}\neq0$. Thus, the complex
dimension of the moduli space is $4$. Moreover, the difference in degrees of
numerator and denominator is $2$, which equals the quaternionic dimension of
the
Coulomb branch.
\paragraph{Plethystic logarithm}
The plethystic logarithm for this scenario reads
\begin{align}
\label{eqn:PL_U1xU1_SO5}
\mathrm{PL}(\mathrm{HS}_{{{\rm U}(1)}^2}^{{{\rm SO}}(5)})= & \left(2+z_1^2 z_2+\frac{1}{z_1^2 z_2}+z_1
z_2+\frac{1}{z_1 z_2}+z_1+\frac{1}{z_1}+z_2+\frac{1}{z_2}\right)t \\
&- \bigg(4
+z_1^2 +\frac{1}{z_1^2}
+z_2 +\frac{1}{z_2}
+z_1^2 z_2^2 +\frac{1}{z_1^2 z_2^2}
+z_1^2 z_2 +\frac{1}{z_1^2 z_2} \notag\\*
&\qquad \qquad \quad +2 z_1 +\frac{2}{z_1}
+2 z_1 z_2+\frac{2}{z_1 z_2} \bigg)t^2
+\mathcal{O}(t^{3}) \; . \notag
\end{align}
\paragraph{Symmetry enhancement}
The information conveyed by the Hilbert
basis~\eqref{eqn:Hilbert_basis_U1xU1_SO5}, the Hilbert
series~\eqref{eqn:HS_U1xU1_SO5}, and the plethystic
logarithm~\eqref{eqn:PL_U1xU1_SO5} is that there are eight minimal generators
of conformal dimension one which, together with the two Casimir invariants,
span the adjoint representation of ${{\rm SO}}(5)$. It is
known~\cite{Cremonesi:2014xha,Hanany:2015hxa} that~\eqref{eqn:HS_U1xU1_SO5} is
the Hilbert series
for the reduced moduli space of one ${{\rm SO}}(5)$-instanton over $\mathbb C^2$.
\subsection{Reduced moduli space of one
\texorpdfstring{${{\rm SU}(3)}$}{SU(3)}-instanton}
The quiver gauge theories associated to the affine Dynkin diagram $\hat{A}_n$
have been studied in~\cite{Cremonesi:2013lqa}. Here, we consider the Coulomb
branch of the $\hat{A}_2$ quiver gauge theory as depicted in
Fig.~\eqref{Fig:Quiver_SU3_instanton} and with conformal dimension
given by
\begin{equation}
\Delta(m_1,m_2)= \frac{1}{2} \left( |m_1| + |m_2| + | m_1 -m_2| \right) \; .
\end{equation}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\draw[black,thick] (0,0) circle (10pt);
\draw[black,thick] (2,0) circle (10pt);
\draw[thick] (0.35,0) -- (1.65,0);
\draw[black,thick] (-0.3,1.7) rectangle (0.3,2.3);
\draw[thick] (0,0.35) -- (0,1.7);
\draw[black,thick] (1.7,1.7) rectangle (2.3,2.3);
\draw[thick] (2,0.35) -- (2,1.7);
\draw (0,-0.7) node {$U(1)$};
\draw (2,-0.7) node {$U(1)$};
\draw (-0.8,2) node {$U(1)$};
\draw (2.8,2) node {$U(1)$};
\end{tikzpicture}
\caption{Quiver gauge theory whose Coulomb branch is the reduced
moduli space of one ${{\rm SU}(3)}$-instanton.}
\label{Fig:Quiver_SU3_instanton}
\end{figure}
\paragraph{Hilbert basis}
Similar to the previous case, the conformal dimensions induces a fan which, in
this case, is generated by six $2$-dimensional cones
\begin{subequations}
\begin{alignat}{2}
C_1^{(2)}&=\mathrm{Cone} \left((0,1),(1,1) \right) \; , \qquad&
C_2^{(2)}&=\mathrm{Cone} \left((1,1),(1,0) \right) \; , \\
C_3^{(2)}&=\mathrm{Cone} \left((1,0),(0,-1) \right)\; , \qquad&
C_4^{(2)}&=\mathrm{Cone} \left((0,-1),(-1,-1) \right) \; , \\
C_5^{(2)}&=\mathrm{Cone} \left((-1,-1),(-1,0) \right) \; ,\qquad&
C_6^{(2)}&=\mathrm{Cone} \left((-1,0),(0,1) \right) \; .
\end{alignat}
\end{subequations}
The intersection with the $\mathbb Z^2$ lattice defines the semi-groups $S_p^{(2)}
\coloneqq C_p^{(2)} \cap \mathbb Z^2$ for which we need to compute the Hilbert bases.
Fig.~\ref{Fig:Quiver_SU3} illustrates the situation. We compute the Hilbert
bases to read
\begin{subequations}
\label{eqn:Hilbert_basis_U1xU1_SU3}
\begin{alignat}{2}
\Hcal(S_1^{(2)}) &= \left\{(0,1),(1,1) \right\} \qquad&
\Hcal(S_2^{(2)}) &= \left\{(1,1),(1,0)) \right\} \; , \\
\Hcal(S_3^{(2)}) &= \left\{(1,0),(0,-1) \right\} \qquad&
\Hcal(S_4^{(2)}) &= \left\{(0,-1),(-1,-1) \right\} \; ,\\
\Hcal(S_5^{(2)}) &= \left\{(-1,-1),(-1,0) \right\} \qquad&
\Hcal(S_6^{(2)}) &= \left\{(-1,0),(0,1) \right\} \; .
\end{alignat}
\end{subequations}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-3.5,0);
\coordinate (XAxisMax) at (3.5,0);
\coordinate (YAxisMin) at (0,-2.5);
\coordinate (YAxisMax) at (0,2.5);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax)
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax)
\draw (3.7,-0.2) node {$m_1$};
\draw (-0.2,2.7) node {$m_2$};
\foreach \x in {-3,-2,...,3}
\foreach \y in {-2,-1,...,2}
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
}
}
\draw[black,dashed,thick] (0,-2.3) -- (0,2.3);
\draw[black,dashed,thick] (-3.3,0) -- (3.3,0);
\draw[black,dashed,thick] (-3,-3) -- (3,3);
\draw[black,thick] (0,1) circle (4pt);
\draw[black,thick] (0,-1) circle (4pt);
\draw[black,thick] (1,1) circle (4pt);
\draw[black,thick] (-1,-1) circle (4pt);
\draw[black,thick] (1,0) circle (4pt);
\draw[black,thick] (-1,0) circle (4pt);
\draw (0.5,1.5) node {$S_1^{(2)}$};
\draw (2.5,1.5) node {$S_2^{(2)}$};
\draw (2.5,-1.5) node {$S_3^{(2)}$};
\draw (-0.5,-1.5) node {$S_4^{(2)}$};
\draw (-2.5,-1.5) node {$S_5^{(2)}$};
\draw (-2.5,1.5) node {$S_6^{(2)}$};
\end{tikzpicture}
\caption{The dashed lines correspond the
$m_1=m_2$, $m_1=0$, and $m_2=0$ and divide the lattice $\mathbb Z^2$ into six
semi-groups $S_j^{(2)}$ for $j=1,\ldots,6$. The black circled points denote the
ray generators, which coincide with the minimal generators.}
\label{Fig:Quiver_SU3}
\end{figure}
\paragraph{Hilbert series}
\begin{subequations}
\label{eqn:HS_U1xU1_SU3}
\begin{align}
\mathrm{HS}_{{{\rm U}(1)}^2 }^{{{\rm SU}(3)}}(t,z_1,z_2)&=\frac{R(t,z_1,z_2)}{
(1-t)^2
\left(1-\frac{t}{z_1}\right)
\left(1- z_1 t \right)
\left(1-\frac{t}{z_2}\right)
\left(1- z_2 t \right)
\left(1-\frac{t}{z_1 z_2}\right)
\left(1-z_1 z_2 t \right)}\\
R(t,z_1,z_2)&=1
- \left(3 +z_1 +\frac{1}{z_1} +z_2 +\frac{1}{z_2} +z_1 z_2 +\frac{1}{z_1 z_2}
\right) t^2 \\
&\qquad +2 \left(2+ z_1+\frac{1}{z_1} + z_2+\frac{1}{z_2} + z_1
z_2+\frac{1}{z_1 z_2}\right) t^3 \notag \\
&\qquad -\left(3 +z_1 +\frac{1}{z_1} +z_2 +\frac{1}{z_2} + z_1 z_2
+\frac{1}{z_1 z_2}\right)t^4 +t^6 \notag
\end{align}
\end{subequations}
The Hilbert series~\eqref{eqn:Hilbert_basis_U1xU1_SU3} has a pole of order $4$
as $t\to 1$, because $R(t=1,z_1,z_2)=0$ and $\tfrac{\mathrm{d}^n}{\mathrm{d} t^n}
R(t,z_1,z_2)|_{t=1,z_1=z_2=1}=0$ for $n=1,2,3$. Thus, the Coulomb branch is of
complex dimension $4$. In addition, the difference in degrees of numerator and
denominator is $2$, which equals the quaternionic dimension.
\paragraph{Plethystic logarithm}
\begin{align}
\label{eqn:PL_U1xU1_SU3}
\mathrm{PL}(\mathrm{HS}_{{{\rm U}(1)}^2 }^{{{\rm SU}(3)}})= &\left(2 +z_1 +\frac{1}{z_1} +z_2
+\frac{1}{z_2}+ z_1 z_2+\frac{1}{z_1 z_2} \right) t \\
&- \left(3 +z_1 +\frac{1}{z_1} +z_2 +\frac{1}{z_2} + z_1 z_2 +\frac{1}{z_1 z_2}
\right) t^2 + \mathcal{O}(t^3) \notag
\end{align}
\paragraph{Symmetry enhancement}
The information conveyed by the Hilbert
basis~\eqref{eqn:Hilbert_basis_U1xU1_SU3}, the Hilbert
series~\eqref{eqn:HS_U1xU1_SU3}, and the plethystic
logarithm~\eqref{eqn:PL_U1xU1_SU3} is that there are six minimal generators
of conformal dimension one which, together with the two Casimir invariants,
span the adjoint representation of ${{\rm SU}(3)}$. As proved
in~\cite{Cremonesi:2013lqa}, the Hilbert series~\eqref{eqn:HS_U1xU1_SU3} can be
resumed as
\begin{equation}
\label{eqn:HS_U1xU1_SU3_mod}
\mathrm{HS}_{{{\rm U}(1)}^2}^{{{\rm SU}(3)}}(t,z_1,z_2)= \sum_{k=0}^{\infty}
\chi_{[k,k]}
t^k
\end{equation}
with $\chi_{[k,k]}$ being the character of the ${{\rm SU}(3)}$-representation $[k,k]$.
Therefore, this theory has an explicit ${{\rm SU}(3)}$-enhancement in the Coulomb
branch. It is known~\cite{Benvenuti:2010pq}
that~\eqref{eqn:HS_U1xU1_SU3_mod} is the reduced instanton moduli space of one
${{\rm SU}(3)}$-instanton over $\mathbb C^2$.
\section{Case: \texorpdfstring{$\boldsymbol{{{\rm U}(2)}}$}{U(2)}}
\label{sec:U2}
In this section we aim to consider two classes of ${{\rm U}(2)}$ gauge theories
wherein ${{\rm U}(2)} \cong {{\rm SU}(2)} {\times} {{\rm U}(1)}$, i.e.\ this is effectively an ${{\rm SU}(2)}$
theory
with varying ${{\rm U}(1)}$-charge. As a unitary group, ${{\rm U}(2)}$ is self-dual under
GNO-duality.
\subsection{Set-up}
To start with, let consider the two view points and elucidate the relation
between them.
\paragraph{$\boldsymbol{{{\rm U}(2)}}$ view point}
The GNO-dual of ${{\rm U}(2)}$ is ${{\rm U}(2)}$ itself; hence, the weight
lattice is $\Lambda_w({{\rm U}(2)}) \cong \mathbb Z^2$. Moreover, the Weyl-group is $S_2$
and acts via permuting the two Cartan generators; consequently,
$\Lambda_w({{\rm U}(2)})\slash S_2 = \{ (m_1,m_2)\in \mathbb Z^2: m_1\geq m_2\}$.
\paragraph{$\boldsymbol{{{\rm U}(1)}\times{{\rm SU}(2)}}$ view point}
Considering ${{\rm U}(1)}\times {{\rm SU}(2)}$, we need to find the weight lattice of the
GNO-dual, i.e.\ find all solutions to the Dirac quantisation condition, see for
instance~\cite{Goddard:1976qe}. Since we consider the product, the
exponential in~\eqref{eqn:general_Dirac}
factorises in $\exp( 2\pi \mathrm{i} \ n \ T_{{{\rm U}(1)}}) $ and $ \exp(2\pi \mathrm{i} \ m \
T_{{{\rm SU}(2)}})$, where the $T$'s are the Cartan generators.
Besides the solution
\begin{subequations}
\begin{equation}
(n,m)\in H_0 \coloneqq \mathbb Z^2 = \mathbb Z
\times \Lambda_w({{\rm SO}}(3)) = \mathbb Z
\times \Lambda_r({{\rm SU}(2)})
\end{equation}
corresponding to the weight lattice of ${{\rm U}(1)} \times {{\rm SO}}(3)$, there exists also
the solution
\begin{equation}
(n,m)\in H_1 \coloneqq \mathbb Z^2 +(\tfrac{1}{2},\tfrac{1}{2}) =
\left( \mathbb Z+\tfrac{1}{2} \right) \times \left( \Lambda_w({{\rm SU}(2)})\setminus
\Lambda_r({{\rm SU}(2)}) \right) \; ,
\end{equation}
\end{subequations}
for which both factors are equal to $-1$. The action of the Weyl-group $S_2$
restricts then to non-negative $m$ i.e.\ $H_0^+ = H_0 \cap \{m\geq0\}$ and
$H_1^+ = H_1 \cap \{m\geq0\}$.
\paragraph{Relation between both}
To identify both views with one another, we select the ${{\rm U}(1)}$ as diagonally
embedded, i.e.\ identify the charges as follows:
\begin{equation}
\label{eqn:transf_U2_to_U1xSU2}
\begin{matrix} n\coloneqq \frac{m_1+m_2}{2} \\ m\coloneqq \frac{m_1 - m_2}{2}
\end{matrix} \Bigg\} \qquad \Leftrightarrow \qquad
\Bigg\{ \begin{matrix} m_1 =n+m \\ m_2=n-m \end{matrix} \; .
\end{equation}
The two classes of ${{\rm U}(2)}$-representations under consideration in this section
are
\begin{subequations}
\begin{align}
[1,a] &\qquad\textrm{with}\quad \chi_{[1,a]}^{{{\rm U}(2)}} = y_1^{a+1} y_2^a + y_1^a y_2^{a+1} \; ,\\
[2,a] &\qquad\textrm{with}\quad \chi_{[2,a]}^{{{\rm U}(2)}} = y_1^{a+2} y_2^a + y_1^{a+1} y_2^{a+1} +
y_1^a y_2^{a+2} \; ,
\end{align}
\end{subequations}
for $a \in \NN_0$.
Following~\eqref{eqn:transf_U2_to_U1xSU2}, we define the fugacities
\begin{equation}
q \coloneqq \sqrt{y_1 \ y_2} \qquad\textrm{for}\quad {{\rm U}(1)} \quad\textrm{and}\quad x\coloneqq \sqrt{\frac{y_1}{y_2}}
\qquad\textrm{for}\quad {{\rm SU}(2)},
\end{equation}
and consequently observe
\begin{subequations}
\begin{align}
\chi_{[1,a]}^{{{\rm U}(2)}} &= q^{2a+1} \left(x+ \tfrac{1}{x}\right) =
\chi_{2a+1}^{{{\rm U}(1)}} \cdot \chi_{[1]}^{{{\rm SU}(2)}} \; ,\\
\chi_{[2,a]}^{{{\rm U}(2)}} &= q^{2a+2} \left(x^2+ 1+ \tfrac{1}{x^2}\right) =
\chi_{2a+2}^{{{\rm U}(1)}} \cdot \chi_{[2]}^{{{\rm SU}(2)}} \; ,
\end{align}
where the ${{\rm SU}(2)}$-characters are defined via
\begin{equation}
\chi_{[L]}^{{{\rm SU}(2)}}=\sum_{r=-\tfrac{L}{2}}^{\tfrac{L}{2}} x^{2r} \; .
\end{equation}
\end{subequations}
Therefore, the family $[1,a]$ corresponds to the fundamental representation of
${{\rm SU}(2)}$ with \emph{odd} ${{\rm U}(1)}$-charge $2a+1$; while the family $[2,a]$
represents the adjoint representation of ${{\rm SU}(2)}$ with \emph{even}
${{\rm U}(1)}$-charge $2a+2$.
\paragraph{Dressing factors}
Lastly, the calculation employs the classical dressing function
\begin{equation}
P_{{{\rm U}(2)}}(t^2,m)\coloneqq \begin{cases} \frac{1}{(1-t^2)^2} & \; , m\neq 0 \\
\frac{1}{(1-t^2)(1-t^4)} &\; , m=0 \end{cases} \; ,
\label{eqn:Dressing_U2}
\end{equation}
as presented in~\cite{Cremonesi:2013lqa}. (Note that we rescaled $t$ to be
$t^2$ for later convenience.) Following the discussion of App.~\ref{app:PL},
monopoles with $m\neq0$ have precisely one dressing by a ${{\rm U}(1)}$ Casimir
invariant due
to $P_{{{\rm U}(2)}}(t^2,m) \slash P_{{{\rm U}(2)}}(t^2,0) = 1+t^2$. In contrast, there are
no dressed monopole operators for $m=0$.
\subsection{\texorpdfstring{$N$}{N} hypermultiplets in the
fundamental representation of \texorpdfstring{${{\rm SU}(2)}$}{SU(2)}}
\label{subsec:U2_fund_SU2}
The conformal dimension for a ${{\rm U}(2)}$ theory with $N$ hypermultiplets
transforming in $[1,a]$ is given as
\begin{equation}
\Delta(n,m)= \frac{N}{2} \big( \left| (2a+1) \cdot n +m \right| + \left|
(2a+1) \cdot n -m \right| \big) - 2 |m|\label{eqn:delta_U2_fundamental}
\end{equation}
such that the Hilbert series is computed via
\begin{equation}
\mathrm{HS}_{{{\rm U}(2)}}^{[1,a]}(t,z)= \sum_{n,m} P_{{{\rm U}(2)}}(t^2,m) \ t^{2 \Delta(n,m)}
z^{2n}
\, ,
\end{equation}
where the ranges of $n,m$ have been specified above. Here we use the fugacity
$t^2$ instead of $t$ to avoid half-integer powers.
\paragraph{Hilbert basis}
The conformal dimension~\eqref{eqn:delta_U2_fundamental} divides
$\Lambda_w({{\rm U}(2)}) \slash S_2$ into semi-groups
via the absolute values $|m|$,
$|(2a+1)n+m|$, and $|(2a+1)n-m|$. Thus, there are three semi-groups
\begin{subequations}
\begin{align}
S_+^{(2)} &= \left\{ (m,n)\in \Lambda_w^{{{\rm U}(2)}} \slash S_2 \ | \ (n \geq 0) \;
\wedge \; (0\leq m \leq (2a+1) n ) \right\} \; , \\
S_0^{(2)} &= \left\{ (m,n)\in \Lambda_w^{{{\rm U}(2)}} \slash S_2 \ | \ -(2a+1)n \leq
m \leq (2a+1)n \right\} \; , \\
S_-^{(2)} &= \left\{ (m,n)\in \Lambda_w^{{{\rm U}(2)}} \slash S_2 \ | \ ( n \leq 0) \;
\wedge \; ( 0\leq m \leq -(2a+1)n ) \right\}
\end{align}
\end{subequations}
originating from $2$-dimensional cones, see Fig.~\ref{Fig:U2_N-fund_a=odd}.
Since all these semi-groups $S_{\pm}^{(2)}$, $S_{0}^{(2)}$ are finitely
generated, one can compute the Hilbert basis $\Hcal(S_p)$ for each $p$ and
obtains
\begin{subequations}
\label{eqn:Hilbert_basis_U2_N-fund_a=odd}
\begin{align}
\Hcal(S_{\pm}^{(2)}) &= \Big\{ (0,\pm1), \{(l+\tfrac{1}{2},\pm\tfrac{1}{2})
\ | \ l=0,1,\ldots, a \} \Big\} \; , \\
\Hcal(S_0^{(2)}) &= \Big\{(a+\tfrac{1}{2},\tfrac{1}{2}), (1,0),
(a+\tfrac{1}{2}, -\tfrac{1}{2}) \Big\} \; .
\end{align}
\end{subequations}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (9.5,0);
\coordinate (YAxisMin) at (0,-3.1);
\coordinate (YAxisMax) at (0,3.3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (9.7,-0.2) node {$m$};
\draw (-0.2,3.2) node {$n$};
\foreach \x in {0,1,...,9}{%
\foreach \y in {-3,-2,...,3}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
%
}
}
\foreach \x in {0,1,...,8}{%
\foreach \y in {-3,-2,...,2}{%
\node[draw,diamond,inner sep=0.8pt,fill,gray] at (\x +1/2,\y +1/2) {};
%
}
}
\draw[black,dashed,thick] (Origin) -- (9*1.1,1.1);
\draw[black,dashed,thick] (Origin) -- (9*1.1,-1.1);
\draw[black,dashed,thick] (0,-2.8) -- (0,2.8);
\draw[black,thick] (4.5,0.5) circle (4pt);
\draw[black,thick] (4.5,-0.5) circle (4pt);
\draw[green,thick] (1,0) circle (4pt);
\draw[blue,thick] (3.5,0.5) circle (4pt);
\draw[blue,thick] (2.5,0.5) circle (4pt);
\draw[blue,thick] (1.5,0.5) circle (4pt);
\draw[blue,thick] (0.5,0.5) circle (4pt);
\draw[black,thick] (0,1) circle (4pt);
\draw[red,thick] (3.5,-0.5) circle (4pt);
\draw[red,thick] (2.5,-0.5) circle (4pt);
\draw[red,thick] (1.5,-0.5) circle (4pt);
\draw[red,thick] (0.5,-0.5) circle (4pt);
\draw[black,thick] (0,-1) circle (4pt);
\draw (3.95,2.5) node {$S_+^{(2)}$};
\draw (7.6,0.25) node {$S_0^{(2)}$};
\draw (3.95,-2.5) node {$S_-^{(2)}$};
\draw[black,thick,fill] (11,2) circle (1pt);
\draw (11.5,2) node {$H_0^+ $};
\draw (11,1) node[diamond,inner sep=0.8pt,fill,gray] {};
\draw (11.5,1) node {$H_1^+ $};
\end{tikzpicture}
\caption{The Weyl-chamber for the example $a=4$.
The black circled lattice points are the ray generators.
The blue circled lattice points complete the Hilbert basis (together with two
ray generators) for $S_+^{(2)}$; while the red circled points analogously
complete the Hilbert basis for $S_-^{(2)}$. The green circled point represents
the missing minimal generator for $S_0^{(2)}$.}
\label{Fig:U2_N-fund_a=odd}
\end{figure}
\paragraph{Hilbert series}
Computing the Hilbert series yields
\begin{subequations}
\label{eqn:HS_U2_fund_aOdd}
\begin{equation}
\mathrm{HS}_{{{\rm U}(2)}}^{[1,a]} (t,z,N)= \frac{R(t,z)}{P(t,z)} \; ,\\
\end{equation}
\begin{align}
P(t,z)&=
\left(1-t^2\right)^2
\left(1-t^4\right)
\left(1-t^{2 N-4}\right)
\left(1-\tfrac{1}{z^2} t^{(4 a+2) N}\right)
\left(1-z^2 t^{(4 a+2) N}\right)
\label{eqn:HS_U2_fund_aOdd_Den}\\*
&\phantom{=\left(1-t^2\right)^2 \left(1-t^4\right)}
\times
\left(1-\tfrac{1}{z} t^{(2a+1)(N-2)}\right)
\left(1-z t^{(2a+1)(N-2)}\right) \; ,\notag
\\
R(t,z)&=
1-t^2
+t^{2 N-2}
-t^{2 N}
+2 t^{4 a N-4 a+2 N}
-t^{4 a N-8 a+2 N-4}
-t^{4 a N-8 a+2 N-2} \\
&\qquad
-2 t^{4 a N-4 a+4 N-4}
+t^{4 a N-8 a+4 N-6}
+t^{4 a N-8 a+4 N-4}
+t^{8 a N+4 N}
+t^{8 a N+4 N+2} \notag \\
&\qquad
-2 t^{8 a N-4 a+4 N}
-t^{8 a N+6 N-2}
-t^{8 a N+6 N}
+2 t^{8 a N-4 a+6 N-4}
-t^{12 a N-8 a+6 N-4}\notag \\
&\qquad
+t^{12 a N-8 a+6 N-2}
-t^{12 a N-8 a+8 N-6}
+t^{12 a N-8 a+8 N-4}
\notag \\
&+\left(z + \tfrac{1}{z} \right)
\Big(
t^{2 a N-4 a+N}
-t^{2 a N+N+2}
+t^{2 a N+3 N-2}
-t^{2 a N-4 a+3 N-4}
+t^{6 a N+3 N+2}\notag \\
&\qquad
-t^{6 a N-8 a+3 N-2}
-t^{6 a N+5 N-2}
+t^{6 a N-8 a+5 N-6}
-t^{10 a N-4 a+5 N}
+t^{10 a N-8 a+5 N-2}\notag \\
&\qquad
+t^{10 a N-4 a+7 N-4}
-t^{10 a N-8 a+7 N-6}
\Big) \notag \\
&+\left(z^2 + \tfrac{1}{z^2} \right)
\big(
t^{4 a N-4 a+2 N}
-t^{4 a N+2 N}
+t^{4 a N+4 N}
-t^{4 a N-4 a+4 N-4}
-t^{8 a N-4 a+4 N}\notag \\
&\qquad
+t^{8 a N-8 a+4 N-4}
+t^{8 a N-4 a+6 N-4}
-t^{8 a N-8 a+6 N-4}
\big) \notag \; .
\end{align}
\end{subequations}
The Hilbert series~\eqref{eqn:HS_U2_fund_aOdd} has a pole of order $4$ at
$t\to1$, because $R(t=1,z)=0$ and $\tfrac{\mathrm{d}^n}{\mathrm{d} t^n} R(t,z)|_{t=1}=0$
for $n=1,2,3$. Hence, the moduli space is of (complex) dimension $4$.
As a comment, the additional $(1-t^2)$-term in the denominator can be cancelled
with a corresponding term in the numerator either explicitly for each
$a=\mathrm{fixed}$ or for any $a$, but the resulting expressions are not
particularly insightful.
\paragraph{Discussion}
The four poles of the Hilbert series~\eqref{eqn:HS_U2_fund_aOdd}, which are
graded as $z^{\pm2}$ and $z^{\pm1}$, can be
identified with the four ray generators $(0,\pm1) $ and
$(a+\tfrac{1}{2},\pm\tfrac{1}{2})$, i.e.\ they correspond to bare monopole
operators. In addition, the bare monopole operator for the minimal generator
$(1,0)$ is present in the denominator~\eqref{eqn:HS_U2_fund_aOdd_Den}, too.
In contrast, the family of monopoles $\{ ( l+\tfrac{1}{2} ,\pm\tfrac{1}{2})\,
, l=0,1,\ldots, a-1\} $ is not directly visible in
the Hilbert series, but can be deduced unambiguously from the plethystic
logarithm. These monopole operators correspond the \emph{minimal generators}
of $S_{\pm}^{(2)}$ which are not ray generators.
Tab.~\ref{tab:Ops_U2_fund_aOdd} provides as summary of the monopole
generators and their properties.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c}
\toprule
$(m,n)$ & $(m_1,m_2)$ & $2\Delta(m,n)$ & $H_{(m,n)}$ & dressings \\ \midrule
$(1,0)$ & $(1,-1)$ & $2N-4$ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\ \midrule
$(l+\tfrac{1}{2}, \tfrac{1}{2})$, for $l=0,1,\ldots,a$ & $(l+1,-l)$ &
$(2a+1)N -2(2l+1)$ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\
$(l+\tfrac{1}{2},- \tfrac{1}{2})$, for $l=0,1,\ldots,a$ & $(l,-(l+1))$ &
$(2a+1)N -2(2l+1)$ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\ \midrule
$(0,\pm1)$ & $\pm(1,1)$ & $(4a+2)N$ & ${{\rm U}(2)}$ & none \\
\bottomrule
\end{tabular}
\caption{Bare and dressed monopole operators for the family $[1,a]$ of
${{\rm U}(2)}$-representations.}
\label{tab:Ops_U2_fund_aOdd}
\end{table}
As a remark, the family of monopole operators $(l+\tfrac{1}{2},
\pm\tfrac{1}{2})$ is not always completely present in the plethystic logarithm.
We observe that $l$-th bare operator is a generator if $N\geq 2(a-l+1)$, while
the dressing of the $l$-th object is a generator if $N>2(a-l+1)$. The reason
for the disappearance lies in a relation at degree
$\Delta(1,0)+\Delta(a+\tfrac{1}{2},\pm\tfrac{1}{2})+2$, which coincides with
$\Delta(l+\tfrac{1}{2}, \pm\tfrac{1}{2})$ for $N-1=2(a-l+1)$, such that the
terms cancel in the PL. (See also App.~\ref{app:PL}.) Thus, for
large $N$ all above listed objects are
generators.
\subsubsection{Case: \texorpdfstring{$a=0$}{a=0}, complete
intersection}
For the choice $a=1$, we obtain the Hilbert series for the $2$-dimensional
fundamental representation $[1,0]$ of ${{\rm U}(2)}$ as
\begin{equation}
\mathrm{HS}_{{{\rm U}(2)}}^{[1,0]}(t,z,N)=\frac{ \left(1-t^{2 N}\right) \left(1-t^{2
N -2}\right) }{ \left(1- t^2\right) \left(1-t^4\right)
\left(1- \tfrac{1}{z} t^N\right) \left(1-z t^N\right)
\left(1-\tfrac{1}{z} t^{N-2}\right) \left(1-z t^{N-2}\right) }
\end{equation}
which agrees with the results of~\cite{Cremonesi:2013lqa}.
Let us comment on the reduction of generators compared to the Hilbert
basis~\eqref{eqn:Hilbert_basis_U2_N-fund_a=odd}. The minimal generators have
conformal dimensions $2\Delta(\tfrac{1}{2},\pm\tfrac{1}{2})=N-2$,
$2\Delta(1,0)=2N-4$, and $2\Delta(0,\pm1) =2N$. Thus, $(1,0)$ is generated by
$(\tfrac{1}{2},\pm\tfrac{1}{2})$ and $(0,\pm1)$ are generated by utilising the
dressed monopoles of $(\tfrac{1}{2},\pm\tfrac{1}{2})$ and suitable elements in
their Weyl-orbits.
\subsection{\texorpdfstring{$N$}{N} hypermultiplets in the adjoint
representation of \texorpdfstring{${{\rm SU}(2)}$}{SU(2)}}
\label{subsec:U2_adj_SU2}
The conformal dimension for a ${{\rm U}(2)}$-theory with $N$ hypermultiplets
transforming in the adjoint representation of ${{\rm SU}(2)}$ and arbitrary even
${{\rm U}(1)}$-charge is given by
\begin{equation}
\Delta(n,m)=\frac{N}{2} \big(
\left|(2a+2)n+2m\right|+\left|(2a+2)n\right|+\left|(2a+2)n-2m\right| \big)
-2|m| \; .
\label{eqn:delta_U2_adjoint}
\end{equation}
Already at this stage, one can define the four semi-groups induced by the
conformal dimension, which originate from $2$-dimensional cones
\begin{subequations}
\label{eqn:semi-groups_U2_adjoint}
\begin{align}
S_{2,\pm}^{(2)} &= \Big\{ (m,n) \in \Lambda_w^{{{\rm U}(2)}} \slash S_2 \ | \
(m\geq 0) \wedge (m \leq \pm (a+1) n) \wedge (\pm n \geq 0) \Big\} \; , \\
S_{1,\pm}^{(2)} &= \Big\{ (m,n) \in \Lambda_w^{{{\rm U}(2)}} \slash S_2 \ | \ (m\geq
0) \wedge (m \geq \pm (a+1) n) \wedge (\pm n \geq 0) \Big\} \; .
\end{align}
\end{subequations}
It turns out that the precise form of the Hilbert basis depends on the
divisibility of $a$ by $2$; thus, we split the considerations in
two cases: $a=2k-1$ and $a=2k$.
\subsubsection{Case: \texorpdfstring{$a= 1 \mod 2$}{a=1 mod 2}}
\paragraph{Hilbert basis}
The collection of semi-groups~\eqref{eqn:semi-groups_U2_adjoint} is
depicted in Fig.~\ref{Fig:U2_N-adj_a=0mod4}. As before, we compute the Hilbert
basis $\Hcal$ for each semi-group of the minimal generators.
\begin{subequations}
\begin{align}
\Hcal(S_{2,\pm}^{(2)}) &= \Big\{ (0,\pm1),(2k,\pm1) , \{
(j+\tfrac{1}{2},\pm\tfrac{1}{2}) \; | \; j=0,\ldots, k-1\} \Big\} \; , \\
\Hcal(S_{1,\pm}^{(2)}) &= \Big\{ (2k,\pm1), ( k+\tfrac{1}{2},\pm
\tfrac{1}{2}) ,(1,0) \Big\} \; .
\end{align}
\end{subequations}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (12.5,0);
\coordinate (YAxisMin) at (0,-3.1);
\coordinate (YAxisMax) at (0,3.3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (12.7,-0.2) node {$m$};
\draw (-0.2,3.2) node {$n$};
\foreach \x in {0,1,...,12}{%
\foreach \y in {-3,-2,...,3}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
%
}
}
\foreach \x in {0,1,...,11}{%
\foreach \y in {-3,-2,...,2}{%
\node[draw,diamond,inner sep=0.8pt,fill,gray] at (\x +1/2,\y +1/2) {};
%
}
}
\draw[black,dashed,thick] (Origin) -- (4*3.1,3.1);
\draw[black,dashed,thick] (Origin) -- (4*3.1,-3.1);
\draw[black,dashed,thick] (0,-2.8) -- (0,2.8);
\draw[black,dashed,thick] (Origin) -- (12.1,0);
\draw[black,thick] (4,1) circle (4pt);
\draw[black,thick] (4,-1) circle (4pt);
\draw[black,thick] (1,0) circle (4pt);
\draw[orange,thick] (2.5,0.5) circle (4pt);
\draw[green,thick] (2.5,-0.5) circle (4pt);
\draw[blue,thick] (1.5,0.5) circle (4pt);
\draw[blue,thick] (0.5,0.5) circle (4pt);
\draw[black,thick] (0,1) circle (4pt);
\draw[red,thick] (1.5,-0.5) circle (4pt);
\draw[red,thick] (0.5,-0.5) circle (4pt);
\draw[black,thick] (0,-1) circle (4pt);
\draw (3.95,2.5) node {$S_{2,+}^{(2)}$};
\draw (7.5,1.15) node {$S_{1,+}^{(2)}$};
\draw (7.5,-1.15) node {$S_{1,-}^{(2)}$};
\draw (3.95,-2.5) node {$S_{2,-}^{(2)}$};
\draw[black,thick,fill] (13,2) circle (1pt);
\draw (13.5,2) node {$H_0^+ $};
\draw (13,1) node[diamond,inner sep=0.8pt,fill,gray] {};
\draw (13.5,1) node {$H_1^+ $};
\end{tikzpicture}
\caption{The Weyl-chamber for odd $a$, here with the example $a=3$.
The black circled lattice points correspond to the ray generators originating
from the fan. The blue/red circled points are the remaining minimal
generators for $S_{2,\pm}^{(2)}$, respectively. Similarly, the orange/green
circled point are the generators that complete the Hilbert basis for
$S_{1,\pm}^{(2)}$.}
\label{Fig:U2_N-adj_a=0mod4}
\end{figure}
\paragraph{Hilbert series}
The computation of the Hilbert series yields
\begin{subequations}
\label{eqn:HS_U2_adj_0mod4}
\begin{equation}
\mathrm{HS}_{{{\rm U}(2)}}^{[2,2k-1]}(t,z,N)= \frac{R(t,z,N)}{P(t,z,N) } \; ,
\end{equation}
\begin{align}
P(t,z,N) &= \left(1-t^2\right)^2
\left(1-t^4\right)
\left(1-t^{4 N-4}\right)
\left(1-\tfrac{1}{z^2} t^{12 k N}\right)
\left(1-z^2 t^{12 k N}\right) \\*
&\phantom{= \left(1-t^2\right)^2 \left(1-t^4\right)}
\times
\left(1-\tfrac{1}{z^2} t^{12 k N-8 k}\right)
\left(1-z^2 t^{12 k N-8 k}\right) \; , \notag \\
R(t,z,N) &=
1-t^2+t^{4 N-2}-t^{4 N}
t^{24 k N}
+t^{24 k N+2}
-t^{24 k N-16 k}
-t^{24 k N-16 k+2}\\
&\qquad
-t^{24 k N+4 N-2}
-t^{24 k N+4 N}
+t^{24 k N-16 k+4 N}
+t^{24 k N-16 k+4 N-2}
-t^{48 k N-16 k}\notag \\
&\qquad
+t^{48 k N-16 k+2}
+t^{48 k N-16 k+4 N}
-t^{48 k N-16 k+4 N-2} \notag \\
&+ \left(z + \tfrac{1}{z} \right)
\bigg(
-t^{6 k N+2}
+t^{6 k N-4 k+2}
+t^{6 k N-4 k+2 N-2}
-t^{6 k N-4 k+2 N+2}
+t^{6 k N+4 N-2}\notag \\
&\qquad
-t^{6 k N-4 k+4 N-2}
+t^{18 k N+2}
-t^{18 k N-4 k+2}
+t^{18 k N-8 k+2}
-t^{18 k N-12 k+2}\notag\\
&\qquad
-t^{18 k N-4 k+2 N-2}
+t^{18 k N-4 k+2 N+2}
-t^{18 k N-12 k+2 N-2}
+t^{18 k N-12 k+2 N+2}\notag\\
&\qquad
-t^{18 k N+4 N-2}
+t^{18 k N-4 k+4 N-2}
-t^{18 k N-8 k+4 N-2}
+t^{18 k N-12 k+4 N-2}
+t^{30 k N-4 k+2}\notag\\
&\qquad
-t^{30 k N-8 k+2}
+t^{30 k N-12 k+2}
-t^{30 k N-16 k+2}
+t^{30 k N-4 k+2 N-2}
-t^{30 k N-4 k+2 N+2}\notag\\
&\qquad
+t^{30 k N-12 k+2 N-2}
-t^{30 k N-12 k+2 N+2}
-t^{30 k N-4 k+4 N-2}
+t^{30 k N-8 k+4 N-2}\notag\\
&\qquad
-t^{30 k N-12 k+4 N-2}
+t^{30 k N-16 k+4 N-2}
-t^{42 k N-12 k+2}
+t^{42 k N-16 k+2}
-t^{42 k N-12 k+2 N-2}\notag\\
&\qquad
+t^{42 k N-12 k+2 N+2}
+t^{42 k N-12 k+4 N-2}
-t^{42 k N-16 k+4 N-2}
\bigg) \notag \\
&+ \left(z^2 + \tfrac{1}{z^2} \right)
\bigg(
-t^{12 k N}
+t^{12 k N-8 k+2}
+t^{12 k N+4 N}
-t^{12 k N-8 k+4 N-2}
+t^{36 k N-16 k}\notag\\
&\qquad
-t^{36 k N-8 k+2}
-t^{36 k N-16 k+4 N}
+t^{36 k N-8 k+4 N-2}
\bigg) \notag \\
&+ \left(z^3 + \tfrac{1}{z^3} \right)
\bigg(
-t^{18 k N-4 k+2}
+t^{18 k N-8 k+2}
-t^{18 k N-4 k+2 N-2}
+t^{18 k N-4 k+2 N+2} \notag\\
&\qquad
+t^{18 k N-4 k+4 N-2}
-t^{18 k N-8 k+4 N-2}
-t^{30 k N-8 k+2}
+t^{30 k N-12 k+2} \notag\\
&\qquad
+t^{30 k N-12 k+2 N-2}
-t^{30 k N-12 k+2 N+2}
+t^{30 k N-8 k+4 N-2}
-t^{30 k N-12 k+4 N-2}
\bigg) \notag \; .
\end{align}
\end{subequations}
Inspection of the Hilbert series~\eqref{eqn:HS_U2_adj_0mod4} reveals that it
has
a pole of order $4$ as
$t\to1$ because one explicitly verifies $R(t=1,z,N)=0$, $\tfrac{\mathrm{d}}{\mathrm{d} t}
R(t,z,N)
|_{t=1}=0$, and $\tfrac{\mathrm{d}^n}{\mathrm{d} t^n} R(t,z,N)
|_{t=1,z=1}=0$ for $n=2,3$.
\paragraph{Discussion}
The denominator of the Hilbert series~\eqref{eqn:HS_U2_adj_0mod4} displays
poles for the five bare monopole operators $(0,\pm1)$, $(2k,\pm1)$, and
$(1,0)$, which are ray generators and charged under ${{\rm U}(1)}_J$ as $\pm2$, $\pm2$,
and $0$, respectively. The remaining operators, corresponding to the minimal
generators which are not ray generators, are apparent in the analysis of the
plethystic logarithm.
The relevant bare and dressed monopole operators are summarised in
Tab.~\ref{tab:Ops_U2_adj_0mod4}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c}
\toprule
$(m,n)$ & $(m_1,m_2)$ & $2\Delta(m,n)$ & $H_{(m,n)} $ & dressings \\
\midrule
$(1,0)$ & $(1,-1)$ & $4N-4$ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\\midrule
$(j+\tfrac{1}{2},\tfrac{1}{2})$, for $j=0,\ldots,k-1$ & $(j+1,-j)$ &
$6kN-4j-2$ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\
$(j+\tfrac{1}{2},-\tfrac{1}{2})$, for $j=0,\ldots,k-1$ & $(j,-(j+1))$ &
$6kN-4j-2$ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\
$(k+\tfrac{1}{2},\tfrac{1}{2})$ & $(k+1,-k)$ &
$6kN+2N-4k-2$ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\
$(k+\tfrac{1}{2},-\tfrac{1}{2})$ & $(k,-(k+1))$ &
$6kN+2N-4k-2$ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\ \midrule
$(0,\pm1)$ & $\pm(1,1)$ & $12kN$ & ${{\rm U}(2)}$ & none \\
$(2k,1)$ & $(2k+1,1-2k)$ & $12kN-8k$ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\
$(2k,-1)$ & $(2k-1,-(2k+1))$ & $12kN-8k$ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\
\bottomrule
\end{tabular}
\caption{Summary of the monopole operators for odd $a$.}
\label{tab:Ops_U2_adj_0mod4}
\end{table}
The plethystic logarithm, moreover, displays that not always all monopoles of
the family $(j+\tfrac{1}{2},\pm\tfrac{1}{2})$ are generators (in the sense of
the PL). The observation is: if $k-j < N$ then the $j$-th operator (bare as
well as dressed) is truely a generator in the PL. The reason behind lies in a
relation at degree $\Delta(k-\tfrac{1}{2},\pm \tfrac{1}{2}) + \Delta(1,0)$,
which coincides with $\Delta(j+\tfrac{1}{2},\pm\tfrac{1}{2})$ for $k-j=N$.
(See also App.~\ref{app:PL}.)
Hence, for large enough $N$ all above listed operators are generators.
\subsubsection{Case: \texorpdfstring{$a= 0 \mod 2$}{a=0 mod 2}}
\paragraph{Hilbert basis}
The diagram for the minimal generators is provided in
Fig.~\ref{Fig:U2_N-adj_a=2mod4}. Again, the appearing (bare) monopoles
correspond to the Hilbert basis of the semi-groups.
\begin{subequations}
\begin{align}
\Hcal(S_{2,\pm}^{(2)}) &= \Big\{ (0,\pm1), \{ (j+\tfrac{1}{2},\pm
\tfrac{1}{2}) \; , \; j=0,1,\ldots, k\} \Big\} \; ,
\\
\Hcal(S_{1,\pm}^{(2)}) &= \Big\{ ( k+\tfrac{1}{2},\pm
\tfrac{1}{2}),(1,0) \Big\} \; .
\end{align}
\end{subequations}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (12.5,0);
\coordinate (YAxisMin) at (0,-3.1);
\coordinate (YAxisMax) at (0,3.3);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax);%
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax);%
\draw (12.7,-0.2) node {$m$};
\draw (-0.2,3.2) node {$n$};
\foreach \x in {0,1,...,12}{%
\foreach \y in {-3,-2,...,3}{%
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
%
}
}
\foreach \x in {0,1,...,11}{%
\foreach \y in {-3,-2,...,2}{%
\node[draw,diamond,inner sep=0.8pt,fill,gray] at (\x +1/2,\y +1/2) {};
%
}
}
\draw[black,dashed,thick] (Origin) -- (5*2.5,2.5);
\draw[black,dashed,thick] (Origin) -- (5*2.5,-2.5);
\draw[black,dashed,thick] (0,-2.8) -- (0,2.8);
\draw[black,dashed,thick] (Origin) -- (12.1,0);
\draw[black,thick] (5/2,1/2) circle (4pt);
\draw[black,thick] (5/2,-1/2) circle (4pt);
\draw[black,thick] (1,0) circle (4pt);
\draw[blue,thick] (1.5,0.5) circle (4pt);
\draw[blue,thick] (0.5,0.5) circle (4pt);
\draw[black,thick] (0,1) circle (4pt);
\draw[red,thick] (1.5,-0.5) circle (4pt);
\draw[red,thick] (0.5,-0.5) circle (4pt);
\draw[black,thick] (0,-1) circle (4pt);
\draw (3.95,2.5) node {$S_{2,+}^{(2)}$};
\draw (8.5,1.15) node {$S_{1,+}^{(2)}$};
\draw (8.5,-1.15) node {$S_{1,-}^{(2)}$};
\draw (3.95,-2.5) node {$S_{2,-}^{(2)}$};
\draw[black,thick,fill] (13,2) circle (1pt);
\draw (13.5,2) node {$H_0^+ $};
\draw (13,1) node[diamond,inner sep=0.8pt,fill,gray] {};
\draw (13.5,1) node {$H_1^+ $};
\end{tikzpicture}
\caption{The Weyl-chamber for $a=0 \mod 2$, here with the example $a=4$.
The black circled lattice points correspond to the ray generators originating
from the fan. The blue/red circled points are the remaining minimal
generators for $S_{2,\pm}^{(2)}$, respectively.}
\label{Fig:U2_N-adj_a=2mod4}
\end{figure}
\paragraph{Hilbert series}
The computation of the Hilbert series for this case yields
\begin{subequations}
\label{eqn:HS_U2_adj_2mod4}
\begin{equation}
\mathrm{HS}_{{{\rm U}(2)}}^{[2,2k]}(t,z,N) = \frac{R(t,z,N)}{P(t,z,N)} \; ,
\end{equation}
\begin{align}
P(t,z,N)&=\left(1-t^2\right)^2
\left(1-t^4\right)
\left(1-t^{4 N-4}\right)
\left(1-\tfrac{1}{z} t^{6 k N-4 k+3 N-2}\right)
\left(1-z t^{6 k N-4 k+3 N-2}\right) \\*
&\qquad \times
\left(1-\tfrac{1}{z^2} t^{12 k N+6 N}\right)
\left(1-z^2 t^{12 k N+6 N}\right) \; , \notag\\
R(t,z,N) &=
1-t^2+t^{4 N-2}-t^{4 N}
+2 t^{12 k N-4 k+6 N}
-t^{12 k N-8 k+6 N-4}
-t^{12 k N-8 k+6 N-2} \\
&\qquad
-2 t^{12 k N-4 k+10 N-4}
+t^{12 k N-8 k+10 N-6}
+t^{12 k N-8 k+10 N-4}
+t^{24 k N+12 N}
+t^{24 k N+12 N+2}\notag \\
&\qquad
-2 t^{24 k N-4 k+12 N}
-t^{24 k N+16 N-2}
-t^{24 k N+16 N}
+2 t^{24 k N-4 k+16 N-4}\notag \\
&\qquad
-t^{36 k N-8 k+18 N-4}
+t^{36 k N-8 k+18 N-2}
-t^{36 k N-8 k+22 N-6}
+t^{36 k N-8 k+22 N-4}
\notag \\
&+\left(z +\tfrac{1}{z} \right) \bigg(
-t^{6 k N+3 N+2}
+t^{6 k N-4 k+3 N}
+t^{6 k N+7 N-2}
-t^{6 k N-4 k+7 N-4}
+t^{18 k N+9 N+2} \notag \\
&\qquad
-t^{18 k N-8 k+9 N-2}
-t^{18 k N+13 N-2}
+t^{18 k N-8 k+13 N-6}
-t^{30 k N-4 k+15 N}\notag \\
&\qquad
+t^{30 k N-8 k+15 N-2}
+t^{30 k N-4 k+19 N-4}
-t^{30 k N-8 k+19 N-6}
\bigg) \notag \\
&+\left(z^2 +\tfrac{1}{z^2} \right) \bigg(
-t^{12 k N+6 N}
+t^{12 k N-4 k+6 N}
+t^{12 k N+10 N}
-t^{12 k N-4 k+10 N-4}\notag \\
&\qquad
-t^{24 k N-4 k+12 N}
+t^{24 k N-8 k+12 N-4}
+t^{24 k N-4 k+16 N-4}
-t^{24 k N-8 k+16 N-4}
\bigg) \notag \; .
\end{align}
\end{subequations}
The Hilbert series~\eqref{eqn:HS_U2_adj_2mod4} has a pole of order $4$ as
$t\to1$ because one can explicitly verify that $R(t=1,z,N)=0$,
$\tfrac{\mathrm{d}}{\mathrm{d} t} R(t,z,N) |_{t=1}=0$, and
$\tfrac{\mathrm{d}^n}{\mathrm{d} t^n} R(t,z,N) |_{t=1,z=1}=0$ for $n=2,3$.
\paragraph{Discussion}
The five monopoles corresponding to the ray generators, i.e.\ $(0,\pm1)$,
$(k+\tfrac{1}{2},\pm \tfrac{1}{2})$, and $(1,0)$, appear as poles in
the Hilbert series~\eqref{eqn:HS_U2_adj_2mod4} and are charged under ${{\rm U}(1)}_J$ as
$\pm2$, $\pm1$, and $0$, respectively. The remaining minimal generator can be
deduced by inspecting the plethystic logarithm. We summarise the monopole
generators in Tab.~\ref{tab:Ops_U2_adj_2mod4}.
\begin{table}[h]
\centering
\begin{tabular}{c|c|c|c|c}
\toprule
$(m,n)$ & $(m_1,m_2)$ & $2\Delta(m,n) $ & $H_{(m,n)}$ & dressings \\ \midrule
$(1,0)$ & $(1,-1) $ & $4N-4$ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\ \midrule
$(j+\tfrac{1}{2},\tfrac{1}{2})$, for $j=0,1,\ldots,k$ & $(j+1,-j)$ &
$6kN+3N-4j-2 $ & ${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\
$(j+\tfrac{1}{2},- \tfrac{1}{2})$, for $j=0,1,\ldots,k$ & $(j,-(j+1)) $ &
$6kN+3N-4j-2 $ &
${{\rm U}(1)}^2$ & $1$ by ${{\rm U}(1)}$ \\ \midrule
$(0,\pm 1)$ & $\pm(1,1)$ & $12kN+6N $ & ${{\rm U}(2)}$ & none \\ \bottomrule
\end{tabular}
\caption{Summary of the monopole operators for even $a$.}
\label{tab:Ops_U2_adj_2mod4}
\end{table}
Similarly to the case of odd $a$, the plethystic logarithm displays
that not always all monopoles of
the family $(j+\tfrac{1}{2},\pm\tfrac{1}{2})$ are generators. The observation
is: if $k-j +1 \geq N$ then the $j$-th bare operator is a generator in
the PL, while for $k-j+2\geq N$ then also the dressing of the $j$-th monopole
is a generator. The reason behind lies, again, in a
relation at degree $\Delta(k-\tfrac{1}{2},\pm \tfrac{1}{2}) + \Delta(1,0)+2$,
which coincides with $\Delta(j+\tfrac{1}{2},\pm\tfrac{1}{2})$ for $k-j=N$. (See
also App.~\ref{app:PL}.)
Hence, for large enough $N$ all above listed operators are generators.
\subsection{Direct product of \texorpdfstring{${{\rm SU}(2)}$}{SU(2)} and
\texorpdfstring{${{\rm U}(1)}$}{U(1)}}
A rather simple example is obtained by considering the non-interacting product
of an ${{\rm SU}(2)}$ and a ${{\rm U}(1)}$ theory. Nonetheless, it illustrates how the rank two
Coulomb branches contain the product of rank one Coulomb branches as subclasses.
As first example, take $N_1$ fundamentals of ${{\rm SU}(2)}$ and $N_2$ hypermultiplets
charged under ${{\rm U}(1)}$ with charges $a \in \NN$. The conformal dimension is given
by
\begin{equation}
\Delta(m,n)= (N_1 -2) |m| + \frac{N_2 \cdot a}{2}|n| \qquad\textrm{for}\quad m\in \NN \quad\textrm{and}\quad n \in\mathbb Z
\end{equation}
and the dressing factor splits as
\begin{equation}
P_{{{\rm SU}(2)}}(t,m,n) = P_{{{\rm SU}(2)}}(t,m)\times P_{{{\rm U}(1)}}(t,n) \; ,
\end{equation}
such that the Hilbert series factorises
\begin{equation}
\mathrm{HS}_{{{\rm SU}(2)} \times {{\rm U}(1)}}^{[1],a}(t,N_1,N_2) = \mathrm{HS}_{{{\rm SU}(2)}}^{[1]}(t,N_1) \times
\mathrm{HS}_{{{\rm U}(1)}}^{a}(t,N_2) \; .
\end{equation}
The rank one Hilbert series have been presented in~\cite{Cremonesi:2013lqa}.
Moreover, $\mathrm{HS}_{{{\rm U}(1)}}^{a}(t,N_2)$ equals the $A_{a\cdot N_2 -1}$ singularity
$\mathbb C^2\slash \mathbb Z_{a\cdot N_2}$; whereas $\mathrm{HS}_{{{\rm SU}(2)}}^{[1]}(t,N_1)$ is precisely
the $D_{N_1}$ singularity.
The second, follow-up example is simply a theory comprise of $N_1$
hypermultiplets in the adjoint representation of ${{\rm SU}(2)}$ and $N_2$
hypermultiplets charged under ${{\rm U}(1)}$ as above. The conformal dimension is
modified to
\begin{equation}
\Delta(m,n)= 2(N_1 -1) |m| + \frac{N_2 \cdot a}{2}|n| \qquad\textrm{for}\quad m\in \NN \quad\textrm{and}\quad n
\in\mathbb Z
\end{equation}
and Hilbert series is obtained as
\begin{equation}
\mathrm{HS}_{{{\rm SU}(2)} \times {{\rm U}(1)}}^{[2],a}(t,N_1,N_2) = \mathrm{HS}_{{{\rm SU}(2)}}^{[2]}(t,N_1) \times
\mathrm{HS}_{{{\rm U}(1)}}^{a}(t,N_2) \; .
\end{equation}
Applying the results of~\cite{Cremonesi:2013lqa}, $\mathrm{HS}_{{{\rm SU}(2)}}^{[2]}(t,N_1)$ is
the Hilbert series of the $D_{2N_1}$-singularity on $\mathbb C^2$.
Summarising, the direct product of these ${{\rm SU}(2)}$-theories with ${{\rm U}(1)}$-theories
results in moduli spaces that are products of A and D type singularities,
which are complete intersections. Moreover, any non-trivial interactions
between these two gauge groups, as discussed in
Subsec.~\ref{subsec:U2_fund_SU2} and~\ref{subsec:U2_adj_SU2}, leads to a very
elaborate expression for the Hilbert series as rational functions. Also, the
Hilbert basis becomes an important concept for understanding the moduli space.
\section{Case: \texorpdfstring{$\boldsymbol{{\rm USp}(4)}$}{USp(4)}}
\label{sec:USp4}
This section is devoted to the study of the compact symplectic group ${\rm USp}(4)$
with corresponding Lie algebra $C_2$. GNO-duality relates them with the special
orthogonal group ${{\rm SO}}(5)$ and the Lie algebra $B_2$.
\subsection{Set-up}
For studying the non-abelian group ${\rm USp}(4)$, we start by providing the
contributions of $N_{a,b}$ hypermultiplets in various
representations $[a,b]$ of ${\rm USp}(4)$ to the conformal dimensions
\begin{subequations}
\begin{align}
\Delta_{\mathrm{h-plet}}^{[1,0]} &= N_{1,0} \sum_i |m_i| \; ,\\
\Delta_{\mathrm{h-plet}}^{[0,1]} &= N_{0,1} \Big( \sum_{i<j} |m_i-m_j| +
\sum_{i<j} |m_i+m_j| \Big) \; ,\\
\Delta_{\mathrm{h-plet}}^{[2,0]} &=2N_{2,0} \sum_i |m_i|
+ N_{2,0} \Big( \sum_{i<j} |m_i-m_j| + \sum_{i<j} |m_i+m_j| \Big) \; ,\\
\Delta_{\mathrm{h-plet}}^{[0,2]} &=2N_{0,2} \sum_i |m_i|
+ 3 N_{0,2} \Big( \sum_{i<j} |m_i-m_j| + \sum_{i<j} |m_i+m_j| \Big) \;
,\\
\Delta_{\mathrm{h-plet}}^{[1,1]} &=2N_{1,1} \sum_i |m_i|
+ N_{1,1} \Big( \sum_{i<j} \left( |2m_i-m_j| +|m_i-2m_j| \right) \\*
&\phantom{ =2N_{1,1} \sum_i |m_i| + N_{1,1} \Big( \sum_{i<j} }
+ \sum_{i<j} \left( |2m_i+m_j| + |m_i+2m_j| \right) \Big) \;
,\notag \\
\Delta_{\mathrm{h-plet}}^{[3,0]} &=5N_{3,0} \sum_i |m_i|
+ N_{3,0} \Big( \sum_{i<j} \left( |2m_i-m_j| +|m_i-2m_j| \right) \\*
&\phantom{ =5N_{3,0} \sum_i |m_i| + N_{3,0} \Big( \sum_{i<j} }
+ \sum_{i<j} \left( |2m_i+m_j| + |m_i+2m_j| \right) \Big) \; , \notag
\end{align}
wherein $i,j =1,2$, and the contribution of the vector multiplet is given by
\begin{equation}
\Delta_{\mathrm{V-plet}}=- 2 \sum_i |m_i|
- \Big( \sum_{i<j} |m_i-m_j| + \sum_{i<j} |m_i+m_j| \Big) \; .
\end{equation}
\end{subequations}
Such that we will consider the following conformal dimension
\begin{subequations}
\begin{align}
\label{eqn:delta_USp4_generic}
\Delta(m_1,m_2) &= (N_1 -2) (|m_1| +|m_2|) + (N_2-1) \left( |m_1-m_2| +
|m_1+m_2| \right) \\
&\qquad +N_3\left( |2m_1-m_2| +|m_1-2m_2| +|2m_1+m_2| + |m_1+2m_2| \right)
\notag
\end{align}
and we can vary the representation content via
\begin{align}
N_1 &= N_{1,0}+2N_{2,0}+2N_{0,2}+2N_{1,1}+5N_{3,0} \; , \\
N_2 &= N_{0,1} + N_{2,0}+3N_{0,2} \; ,\\
N_3 &= N_{1,1} + N_{3,0} \; .
\end{align}
\end{subequations}
The Hilbert series is computed as usual
\begin{equation}
\mathrm{HS}_{{\rm USp}(4)}(t,N) = \sum_{m_1\geq m_2 \geq 0} t^{\Delta(m_1,m_2)}
P_{{\rm USp}(4)}(t,m_1,m_2) \; ,
\end{equation}
where the summation for $m_1,m_2$ has been restricted to the principal Weyl
chamber of the GNO-dual group ${{\rm SO}}(5)$, whose Weyl group is $S_2\ltimes
(\mathbb Z_2)^2$. Thus, we use the reflections to restrict to non-negative
$m_i\geq0$ and the permutations to restrict to a ordering $m_1\geq m_2$.
The classical dressing factor takes the following form~\cite{Cremonesi:2013lqa}:
\begin{equation}
P_{{\rm USp}(4)}(t,m_1,m_2)= \begin{cases} \frac{1}{(1-t)^2}\; , & m_1 >m_2 >0 \; ,\\
\frac{1}{(1-t)(1-t^2)} \;, & (m_1>m_2=0) \vee (m_1=m_2>0) \; ,\\
\frac{1}{(1-t^2)(1-t^4)} \; , & m_1=m_2=0 \; .
\end{cases}
\end{equation}
\subsection{Hilbert basis}
The conformal dimension~\eqref{eqn:delta_USp4_generic} divides the dominant Weyl
chamber of ${{\rm SO}}(5)$ into a fan. The intersection with the corresponding weight
lattice $\Lambda_w({{\rm SO}}(5))$ introduces semi-groups $S_p$, which are sketched
in Fig.~\ref{Fig:Hilbert_basis_USp4_generic}.
As displayed, the set of semi-groups (and rational cones that constitute the
fan) differ if $N_3\neq0$. The Hilbert bases for both case are readily computed,
because they coincide with the set of ray generators.
\begin{itemize}
\item For $N_3\neq 0$, which is displayed in
Fig.~\ref{Fig:Hilbert_basis_USp4_N3>0}, there exists one hyperplane
$|m_1-2m_2|=0 $ which intersects the Weyl chamber non-trivially. Therefore,
$\Lambda_w({{\rm SO}}(5)) \slash \mathcal{W}_{{{\rm SO}}(5)}$ becomes a fan generated by
two $2$-dimensional cones. The Hilbert bases of the corresponding semi-groups
are computed to
\begin{equation}
\Hcal(S_{+}^{(2)}) = \Big\{ (1,1),(2,1) \Big\} \; , \qquad
\Hcal(S_{-}^{(2)}) = \Big\{ (2,1),(1,0)\Big\} \; .
\label{eqn:Hilbert_basis_USp4_N3>0}
\end{equation}
\item For $N_3 =0$, as shown in Fig.~\ref{Fig:Hilbert_basis_USp4_N3=0}, there
exists no hyperplane that intersects the dominant Weyl chamber non-trivially. As
a consequence, the $\Lambda_w({{\rm SO}}(5)) \slash \mathcal{W}_{{{\rm SO}}(5)}$ is
described by one rational polyhedral cone of dimension $2$. The Hilbert basis
for the semi-group is given by
\begin{equation}
\Hcal(S^{(2)}) = \Big\{ (1,1),(1,0) \Big\} \; .
\label{eqn:Hilbert_basis_USp4_N3=0}
\end{equation}
\end{itemize}
\begin{figure}
\begin{center}
\begin{subfigure}{0.485\textwidth}
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (4.5,0);
\coordinate (YAxisMin) at (0,-0.5);
\coordinate (YAxisMax) at (0,4.5);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax)
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax)
\draw (4.7,-0.2) node {$m_1$};
\draw (-0.3,4.3) node {$m_2$};
\foreach \x in {0,1,...,4}
\foreach \y in {0,1,...,4}
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
}
}
\draw[black,dashed,thick] (Origin) -- (4.2,4.2);
\draw[black,dashed,thick] (Origin) -- (4.2,0);
\draw[black,dashed,thick] (Origin) -- (2.2*2,2.2*1);
\draw[black,thick] (1,1) circle (4pt);
\draw[black,thick] (1,0) circle (4pt);
\draw[black,thick] (2,1) circle (4pt);
\draw (3.5,2.5) node {$S_+^{(2)}$};
\draw (3.5,1.3) node {$S_-^{(2)}$};
\end{tikzpicture}
\caption{$N_3\neq 0$}
\label{Fig:Hilbert_basis_USp4_N3>0}
\end{subfigure}
\begin{subfigure}{0.485\textwidth}
\centering
\begin{tikzpicture}
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-0.5,0);
\coordinate (XAxisMax) at (4.5,0);
\coordinate (YAxisMin) at (0,-0.5);
\coordinate (YAxisMax) at (0,4.5);
\draw [thin, gray,-latex] (XAxisMin) -- (XAxisMax)
\draw [thin, gray,-latex] (YAxisMin) -- (YAxisMax)
\draw (4.7,-0.2) node {$m_1$};
\draw (-0.3,4.3) node {$m_2$};
\foreach \x in {0,1,...,4}
\foreach \y in {0,1,...,4}
\node[draw,circle,inner sep=0.8pt,fill,black] at (\x,\y) {};
}
}
\draw[black,dashed,thick] (Origin) -- (4.2,4.2);
\draw[black,dashed,thick] (Origin) -- (4.2,0);
\draw[black,thick] (1,1) circle (4pt);
\draw[black,thick] (1,0) circle (4pt);
\draw (3.5,1.5) node {$S^{(2)}$};
\end{tikzpicture}
\caption{$N_3=0$}
\label{Fig:Hilbert_basis_USp4_N3=0}
\end{subfigure}
\end{center}
\caption{The various semi-groups for ${\rm USp}(4)$ depending on whether $N_3\neq0$
or $N_3=0$. For both cases the black circled points are the ray generators.}
\label{Fig:Hilbert_basis_USp4_generic}
\end{figure}
\subsection{Dressings}
Before evaluating the Hilbert series, let us analyse the classical dressing
factors for the minimal generators~\eqref{eqn:Hilbert_basis_USp4_N3>0}
or~\eqref{eqn:Hilbert_basis_USp4_N3=0}.
Firstly, the classical Lie group ${\rm USp}(4)$ has two Casimir invariants of
degree $2$ and $4$ and can they can be written as $\mathrm{Tr}(\Phi^2) = \sum_{i=1}^2
(\phi_i)^2$ and $\mathrm{Tr}(\Phi^4)= \sum_{i=1}^2 (\phi_i)^4$, respectively.
Again, we employ the diagonal form of the adjoint valued scalar field $\Phi$.
Secondly, the bare monopole operator corresponding to GNO-charge $(1,0)$ has
conformal dimension $N_1+ 2N_2 +6N_3 -4$ and the residual gauge group is
$\mathrm{H}_{(1,0)}={{\rm U}(1)}\times{{\rm SU}(2)}$, i.e.\ allowing for dressings by degree $1$ and $2$
Casimirs. The resulting set of bare and dressed monopoles is
\begin{subequations}
\begin{align}
V_{(1,0)}^{\mathrm{dress},0} &= (1,0) + (-1,0) + (0,1) + (0,-1) \; , \\
V_{(1,0)}^{\mathrm{dress},2} &=
\left( (1,0) + (-1,0) \right) (\phi_2)^2
+ \left((0,1) + (0,-1) \right) (\phi_1)^2 \; ,\\
V_{(1,0)}^{\mathrm{dress},1} &=
\left( (1,0) - (-1,0) \right) \phi_1
+ \left((0,1) - (0,-1) \right) \phi_2 \; ,\\
V_{(1,0)}^{\mathrm{dress},3} &=
\left( (1,0) - (-1,0) \right) (\phi_1)^3
+ \left((0,1) - (0,-1) \right) (\phi_2)^3 \; .
\end{align}
\end{subequations}
Thirdly, the bare monopole operators of GNO-charge $(1,1)$ has conformal
dimension
$2N_1+2N_2+8N_3-6$ and residual gauge group $\mathrm{H}_{(1,1)} = {{\rm U}(1)} \times {{\rm SU}(2)}$. The
bare and dressed monopole operators can be written as
\begin{subequations}
\begin{align}
V_{(1,1)}^{\mathrm{dress},0} &= (1,1) + (1,-1)+ (-1,1) +(-1,-1) \; , \\
V_{(1,1)}^{\mathrm{dress},2} &= ( (1,1) + (-1,-1)) ( (\phi_1)^2 + (\phi_2)^2 )
+ (1,-1) (\phi_2)^2 + (-1,1) (\phi_1)^2 \; ,\\
V_{(1,1)}^{\mathrm{dress},1} &= (1,1) (\phi_1+ \phi_2) + (-1,-1)
(-\phi_1-\phi_2) + (1,-1)(-\phi_2) + (-1,1)(-\phi_1) \; ,\\
V_{(1,1)}^{\mathrm{dress},3} &= (1,1) ((\phi_1)^3+ (\phi_2)^3) + (-1,-1)
(-(\phi_1)^3-(\phi_2)^3) \\
&\phantom{= (1,1) ((\phi_1)^3+ (\phi_2)^3) \; }
+ (1,-1)(-(\phi_2)^3) + (-1,1)(-(\phi_1)^3) \; .\notag
\end{align}
\end{subequations}
The two magnetic weights $(1,0)$, $(1,1)$ lie at the boundary of the dominant
Weyl chamber such that the dressing behaviour can be predicted by
$P_{{\rm USp}(4)}(t,m_1,m_2) \slash P_{{\rm USp}(4)}(t,0,0) = 1+t+t^2+t^3 $, following
App.~\ref{app:PL}. The above description of the bare and dressed monopole
operators is therefore a valid choice of generating elements for the chiral
ring.
Lastly, the bare monopole for $(2,1)$ has conformal dimension
$3N_1+4N_2+12N_3-10$ and residual gauge group $H_{(2,1)}={{\rm U}(1)}^2$. Thus, the
dressing proceeds by two independent degree $1$ Casimir invariants.
\begin{subequations}
\begin{align}
V_{(2,1)}^{\mathrm{dress},0} &= (2,1) + (2,-1) + (-2,1) + (1,2) + (1,-2)
+ (-1,2)+ (-1,-2) + (-2,-1) \notag \\
&\equiv (2,1) + (2,-1) + (-2,1) + (-2,-1)+\text{permutations} \; ,\\
V_{(2,1)}^{\mathrm{dress},2j-1,1}&=
(2,1) (\phi_1)^{2j-1}
+ (2,-1)(\phi_1)^{2j-1}
+ (-2,1)(-\phi_1)^{2j-1} \\
&\qquad + (-2,-1)(-\phi_1)^{2j-1}+\text{permutations} \qquad\textrm{for}\quad j=1,2 \; , \notag\\
V_{(2,1)}^{\mathrm{dress},2j-1,2}&=
(2,1) (\phi_2)^{2j-1}
+ (2,-1)(-\phi_2)^{2j-1}
+ (-2,1)(\phi_2)^{2j-1} \\
&\qquad + (-2,-1)(-\phi_2)^{2j-1}+\text{permutations} \qquad\textrm{for}\quad j=1,2 \; , \notag\\
V_{(2,1)}^{\mathrm{dress},2,1}&=
(2,1) (\phi_1)^2
+ (2,-1)(-(\phi_1)^2)
+ (-2,1)(-(\phi_1)^2) \\
&\qquad + (-2,-1)(\phi_1)^2+\text{permutations} \; , \notag \\
V_{(2,1)}^{\mathrm{dress},2,2}&=
(2,1) (\phi_1 \phi_2)
+ (2,-1)(-\phi_1 \phi_2)
+ (-2,1)(-\phi_1 \phi_2) \\
&\qquad
+ (-2,-1)(\phi_1 \phi_2)+\text{permutations} \; , \notag\\
V_{(2,1)}^{\mathrm{dress},4}&=
(2,1) (\phi_1^3 \phi_2)
+ (2,-1)(-(\phi_1)^3 \phi_2)
+ (-2,1)(-(\phi_1)^3 \phi_2) \\
&\qquad + (-2,-1)((\phi_1)^3 \phi_2)+\text{permutations} \; . \notag
\end{align}
\end{subequations}
The number and the degrees of dressed monopole operators of charge $(2,1)$ are
consistent with the quotient $P_{{\rm USp}(4)}(t,m_1>m_2>0) \slash P_{{\rm USp}(4)}(t,0,0) =
1+2t+2t^2+2t^3+t^4 $ of the dressing factors.
For ``generic'' values of $N_1$, $N_2$ and $N_3$ the Coulomb branch will be
generated by the two Casimir invariants together with the bare and dressed
monopole operators corresponding to the minimal generators of the Hilbert
bases. However, we will encounter choices of the three parameters such that
the set of monopole generators can be further reduced; for example, in the case
of complete intersections.
\subsection{Generic case}
The computation for arbitrary $N_1$, $N_2$, and $N_3$ yields
\begin{subequations}
\label{eqn:HS_USp4_generic}
\begin{equation}
\mathrm{HS}_{{\rm USp}(4)}(t,N_1,N_2,N_3)=\frac{R(t,N_1,N_2,N_3)}{P(t,N_1,N_2,N_3)} \; ,
\end{equation}
with
\begin{align}
P(t,N_1,N_2,N_3)&=\left(1-t^2\right)
\left(1-t^4\right)
\left(1-t^{N_1+2 N_2+6 N_3-4}\right)
\left(1-t^{2 N_1+2 N_2+8 N_3-6}\right)
\label{eqn:HS_USp4_Den}\\*
&\phantom{\left(1-t^2\right) \left(1-t^4\right)}
\times
\left(1-t^{3 N_1+4 N_2+12 N_3-10}\right) \; , \notag\\
R(t,N_1,N_2,N_3) &=1
+t^{N_1+2 N_2+6 N_3-3} ( 1+t+t^2)
+t^{2 N_1+2 N_2+8 N_3-5} (1+t+t^2)
\label{eqn:HS_USp4_Num} \\
&\qquad
+t^{3 N_1+4 N_2+12 N_3-9} (2+2t+2t^2+t^3)
-t^{3 N_1+4 N_2+14 N_3-10} (1+2t+2t^2+2t^3)
\notag \\
&\qquad
-t^{4 N_1+6 N_2+18 N_3-13}(1+t+t^2)
-t^{5 N_1+6 N_2+20 N_3-15}(1+t+t^2)
\notag \\
&\qquad
-t^{6 N_1+8 N_2+26 N_3-16} \; .
\notag
\end{align}
\end{subequations}
The numerator~\eqref{eqn:HS_USp4_Num} is an anti-palindromic polynomial of
degree $6N_1+8N_2+26 N_3-16$; while the denominator is of degree $6 N_1+8N_2+26
N_3-14$. The difference in degrees is $2$, which equals the quaternionic
dimension of the moduli space.
In addition, the pole of~\eqref{eqn:HS_USp4_generic} at $t\to1$ is of order
$4$, which matches the complex dimension of the
moduli space. For that, one verifies explicitly $R(t=1,N_1,N_2,N_3)=0$, but
$\tfrac{\mathrm{d}}{\mathrm{d} t} R(t,N_1,N_2,N_3) |_{t=1}\neq0$.
Consequently, the above interpretation of bare and dressed monopoles from the
Hilbert series~\eqref{eqn:HS_USp4_generic} is correct for ``generic'' choices
of $N_1$, $N_2$, and $N_3$. In particular, $N_3\neq 0$ for this arguments to
hold. Moreover, we will now exemplify the effects of the Casimir invariance in
various special case of~\eqref{eqn:HS_USp4_generic} explicitly. There are
cases for which the inclusion of the Casimir invariance, i.e.\ dressed
monopole operators, leads to a reduction of basis of monopole
generators.
\subsection{Category \texorpdfstring{$N_3=0$}{N3=0}}
\label{subsec:USp4_N3=0}
\subsubsection{Representation \texorpdfstring{$[1,0]$}{[1,0]}}
\paragraph{Hilbert series}
This choice is realised for $N_1=N$, $N_2=N_3=0$ and the Hilbert series
simplifies drastically to a complete intersection
\begin{equation}
\mathrm{HS}_{{\rm USp}(4)}^{[1,0]}(t,N)=
\frac{(1-t^{2N-4}) (1-t^{2N-2})}{(1-t^2) (1-t^4)
(1-t^{N-4}) (1-t^{N-3}) (1-t^{N-2}) (1-t^{N-1})} \; ,
\label{eqn:HS_USp4_Rep10}
\end{equation}
which was first obtained in~\cite{Cremonesi:2013lqa}.
Due to the complete intersection property, the plethystic logarithm terminates
and for $N>4$ we obtain
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[1,0]})=t^2+t^4&+t^{N-4}(1+t+t^2+t^3) -t^{2N-4} -
t^{2N-2} \; .
\end{align}
\paragraph{Hilbert basis}
Naively, the Hilbert series~\eqref{eqn:HS_USp4_Rep10} should be generated by
the Hilbert basis~\eqref{eqn:Hilbert_basis_USp4_N3=0} plus their dressings.
However, due to the particular form~\eqref{eqn:delta_USp4_generic} in
representation $[1,0]$ and the Casimir invariance, the bare monopole operator
of
GNO-charge $(1,1)$ can be generated by the dressings of $(1,0)$. To see this,
consider the Weyl-orbit $ \mathcal{O}_{\mathcal{W}} (1,0) = \big\{ (1,0),
(0,1),
(-1,0), (0,-1) \big\}$ and note the conformal dimensions align suitably, i.e.\
$\Delta(V_{(1,0)}^{\mathrm{dress},1})= N-3$, while
$\Delta(V_{(1,1)}^{\mathrm{dress},0})=2N-6$. Thus, we can symbolically write
\begin{equation}
V_{(1,1)}^{\mathrm{dress},0}=V_{(1,0)}^{\mathrm{dress},1}
+V_{(0,1)}^{\mathrm{dress},1} \; .
\end{equation}
The moduli space is then generated by the Casimir invariants and the bare and
dressed monopole operators corresponding to $(1,0)$, but this is to be
understood as a rather ``non-generic'' situation.
\subsubsection{Representation \texorpdfstring{$[0,1]$}{[0,1]}}
This choice is realised for $N_2=N$, and $N_1=N_3=0$ and the Hilbert series
simplifies to
\begin{equation}
\label{eqn:HS_USp4_Rep01}
\mathrm{HS}_{{\rm USp}(4)}^{[0,1]}(t,N)=
\frac{1+t^{2 N-5}+t^{2 N-4}+2 t^{2 N-3}+t^{2
N-2}+t^{2 N-1}+t^{4 N-6}}{\left(1-t^2\right)
\left(1-t^4\right) \left(1-t^{2 N-6}\right) \left(1-t^{2 N-4}\right)} \; .
\end{equation}
The Hilbert series~\eqref{eqn:HS_USp4_Rep01} has a pole of order $4$ at $t=1$
as well as a palindromic polynomial as numerator. Moreover, the
result~\eqref{eqn:HS_USp4_Rep01} reflects the expected basis of monopole
operators as given in the Hilbert basis~\eqref{eqn:Hilbert_basis_USp4_N3=0}.
\subsubsection{Representation \texorpdfstring{$[2,0]$}{[2,0]}}
This choice is realised for $N_1=2N$, $N_2=N$, and $N_3=0$ and the Hilbert
series reduces to
\begin{equation}
\label{eqn:HS_USp4_Rep20}
\mathrm{HS}_{{\rm USp}(4)}^{[2,0]}(t,N)=
\frac{1+t^{4 N-3}+t^{4 N-2}+t^{4 N-1}+t^{6
N-5}+t^{6 N-4}+t^{6 N-3}+t^{10
N-6}}{\left(1-t^2\right) \left(1-t^4\right) \left(1-t^{4
N-4}\right) \left(1-t^{6 N-6}\right)} \; .
\end{equation}
Also, the rational function~\eqref{eqn:HS_USp4_Rep20} has a pole of order $4$
for $t\to1$ and a palindromic numerator.
Evaluating the plethystic logarithm yields for all $N>1$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[2,0]})=t^2+t^4&+t^{4N-4}(1+t+t^2+t^3) \\
&+ t^{6N-6}(1+t+t^2+t^3) - t^{8N-6} +\mathcal{O}(t^{8N-5}) \notag \; .
\end{align}
This proves that bare monopole operators, corresponding to the
the minimal generators of~\eqref{eqn:Hilbert_basis_USp4_N3=0}, together with
their dressing generate all other monopole operators.
\subsubsection{Representation \texorpdfstring{$[0,2]$}{[0,2]}}
For $N_1=2N$, $N_2=3N$, and $N_3=0$ and the Hilbert
series is given by
\begin{equation}
\label{eqn:HS_USp4_Rep02}
\mathrm{HS}_{{\rm USp}(4)}^{[0,2]}(t,N)=
\frac{1+t^{8 N-3}+t^{8 N-2}+t^{8 N-1}+t^{10
N-5}+t^{10 N-4}+t^{10 N-3}+t^{18
N-6}}{\left(1-t^2\right) \left(1-t^4\right) \left(1-t^{8N-4}\right)
\left(1-t^{10 N-6}\right)} \; .
\end{equation}
Evaluating the plethystic logarithm yields for all $N>1$
\begin{subequations}
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[0,2]})=t^2+t^4&+t^{8N-4}(1+t+t^2+t^3) \\
&+ t^{10N-6}(1+t+t^2+t^3) - t^{16N-6} +\mathcal{O}(t^{16N-5}) \notag \; ,
\end{align}
and for $N=1$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[0,2]})=t^2+t^4&+t^{4}(1+t+t^2+t^3) \\
&+ t^{4}(1+t+t^2+t^3) - 3t^{10} +\mathcal{O}(t^{11}) \notag \; .
\end{align}
\end{subequations}
The inspection of the Hilbert series~\eqref{eqn:HS_USp4_Rep02}, together with
the PL, proves that Hilbert basis~\eqref{eqn:Hilbert_basis_USp4_N3=0},
alongside
all their dressings, are a sufficient set for all monopole operators.
\subsection{Category \texorpdfstring{$N_3 \neq0$}{N3>0}}
\subsubsection{Representation \texorpdfstring{$[1,1]$}{[1,1]}}
This choice corresponds to $N_1=2N$, $N_2=0$, and $N_3=N$ and we obtain the
Hilbert series to be
\begin{subequations}
\label{eqn:HS_USp4_Rep11}
\begin{align}
\mathrm{HS}_{{\rm USp}(4)}^{[1,1]}(t,N)&=
\frac{R(t,N)}{\left(1-t^2\right) \left(1-t^4\right)
\left(1-t^{8 N-4}\right) \left(1-t^{12 N-6}\right) \left(1-t^{18 N-10}\right)}
\; ,
\\
R(t,N)&=1+t^{8 N-3}(1+t+t^2) +t^{12 N-5}(1+t+t^2) \\
&\qquad+ t^{18 N-9}(2+2t+2t^2+t^3) -t^{20 N-10}(1+2t+2t^2+2t^3)\notag \\
&\qquad-t^{26 N-13}(1+t+t^2) -t^{30 N-15}(1+t+t^2)-t^{38 N-16} \notag \; .
\end{align}
\end{subequations}
Considering the plethystic logarithm, we observe the following behaviour:
\begin{subequations}
\begin{itemize}
\item For $N\geq5$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[1,1]})= t^2+t^4 &+ t^{8 N-4}(1+t+t^2+t^3)
+ t^{12 N-6}(1+t+t^2+t^3) \\
&-t^{2(8 N-4)+2}(1+t+2t^2+t^3+t^4) \notag \\
&+ t^{18 N-10}(1+2t+2t^2+2t^3+t^4) \notag \\
&-t^{20N-10}(1+2t+3t^2+4t^3+3t^4+2t^5+t^6)+\ldots \notag
\end{align}
\item For $N=4$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[1,1]})= t^2+t^4
&+ t^{28}(1+t+t^2+t^3)
+ t^{42}(1+t+t^2+t^3) \\
&-t^{58}(1+t+2t^2+t^3+\textcolor{red}{t^4}) \notag \\
&+ t^{62}(\textcolor{red}{1}+2t+2t^2+2t^3+t^4) \notag \\
&-t^{70}(1+2t+3t^2+4t^3+3t^4+2t^5+t^6)+\ldots \notag
\end{align}
We see, employing the previous results for $N>4$, that the bare monopole
$(2,1)$ and the last relation at $t^{62}$ coincide. Hence, the term $\sim
t^{62}$ disappears from the PL.
\item For $N=3$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[1,1]})= t^2+t^4
&+ t^{20}(1+t+t^2+t^3)
+ t^{30}(1+t+t^2+t^3) \\
&-t^{42}(1+t
+\textcolor{red}{2t^2}
+\textcolor{ForestGreen}{t^3}
+\textcolor{blue}{t^4}
) \notag \\
&+
t^{44}(\textcolor{red}{1}
+\textcolor{ForestGreen}{2t}+\textcolor{blue}{2t^2}
+2t^3+t^4) \notag \\
&-t^{70}(1+2t+3t^2+4t^3+3t^4+2t^5+t^6)+\ldots \notag
\end{align}
We see, employing again the previous results for $N>4$, that the some monopole
contributions of $(2,1)$ and the some of the relations coincide, c.f. the
coloured terms. Hence, there are, presumably, cancellations between generators
and relations.
\item For $N=2$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[1,1]})= t^2+t^4
&+ t^{12}(1+t+t^2+t^3)
+ t^{18}(1+t+t^2+t^3) \\
&-t^{26}(\textcolor{red}{1}
+\textcolor{ForestGreen}{t}
+\textcolor{blue}{2t^2}
+\textcolor{violet}{t^3}
+\textcolor{brown}{t^4}
) \notag \\
&+
t^{26}(\textcolor{red}{1}
+\textcolor{ForestGreen}{2t}
+\textcolor{blue}{2t^2}
+\textcolor{violet}{2t^3}
+\textcolor{brown}{t^4}) \notag \\
&-t^{30}(\textcolor{brown}{1}+2t+3t^2+4t^3+3t^4+2t^5+t^6)+\ldots \notag \\
= t^2+t^4
&+ t^{12}(1+t+t^2+t^3)
+ t^{18}(1+t+t^2+t^3) \\
&+
t^{26}(\textcolor{red}{0}
+\textcolor{ForestGreen}{t}
+\textcolor{blue}{0}
+\textcolor{violet}{t^3}
+\textcolor{brown}{0}) \notag \\
&-t^{30}(\textcolor{brown}{1}+2t+3t^2+4t^3+3t^4+2t^5+t^6)+\ldots \notag
\end{align}
\item For $N=1$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[1,1]})=
t^2 + 2t^4 + t^5 + 2t^6+2t^7+2t^8+3t^9-t^{11} + \ldots
\end{align}
\end{itemize}
\end{subequations}
Summarising, the Hilbert series~\eqref{eqn:HS_USp4_Rep11} and its plethystic
logarithm display that the minimal generators
of~\eqref{eqn:Hilbert_basis_USp4_N3>0} are indeed the basis for the bare
monopole operators, and the corresponding dressings generate the remaining
operators.
\subsubsection{Representation \texorpdfstring{$[3,0]$}{[3,0]}}
For the choice $N_1=5N$, $N_2=0$, and $N_3=N$ the Hilbert
series is given by
\begin{subequations}
\label{eqn:HS_USp4_Rep30}
\begin{align}
\mathrm{HS}_{{\rm USp}(4)}^{[3,0]}(t,N)&=
\frac{R(t,N)}{\left(1-t^2\right) \left(1-t^4\right)
\left(1-t^{11 N-4}\right) \left(1-t^{18 N-6}\right)
\left(1-t^{27 N-10}\right)} \; ,
\\
R(t,N)&=1+t^{11 N-3}(1+t+t^2)
+t^{18 N-5}(1+t+t^2) \\
&\qquad + t^{27 N-9}(2+2t+2t^2+t^3)
-t^{29 N-10}(1+2t+2t^2+2t^3) \notag \\
&\qquad
-t^{38 N-13}(1+t+t^2)
-t^{45 N-15}(1+t+t^2)
-t^{56N-16}\notag \; .
\end{align}
\end{subequations}
The inspection of the plethystic logarithm provides further insights:
\begin{subequations}
\begin{itemize}
\item For $N\geq3$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[3,0]})= t^2+t^4 &+ t^{11 N-4}(1+t+t^2+t^3)
+ t^{18 N-6}(1+t+t^2+t^3) \\
&-t^{2(11 N-4)+2}(1+t+2t^2+t^3+t^4) \notag \\
&+ t^{27 N-10}(1+2t+2t^2+2t^3+t^4) \notag \\
&-t^{(11 N-4)+(18N-6)}(1+2t+3t^2+4t^3+3t^4+2t^5+t^6)+\ldots \notag
\end{align}
\item For $N=2$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[3,0]})= t^2+t^4 &+ t^{18}(1+t+t^2+t^3)
+ t^{30}(1+t+t^2+t^3) \\
&-t^{38}(1+t+2t^2+t^3+t^4) \notag \\
&+ t^{44}(1+2t+2t^2+2t^3+\textcolor{red}{t^4}) \notag \\
&-t^{48}(\textcolor{red}{1} +2t+3t^2+4t^3+3t^4+2t^5+t^6)+\ldots \notag
\end{align}
We see that, presumably, one generator and one relation cancel at $~t^{48}$.
\item For $N=1$
\begin{align}
\mathrm{PL}(\mathrm{HS}_{{\rm USp}(4)}^{[3,0]})=
t^2+t^4+t^7(1+t+t^2+t^3)+t^{12}(1+t+t^2+t^3)-t^{16}-t^{20}+\ldots
\end{align}
\end{itemize}
\end{subequations}
Again, we confirm that the minimal generators of the Hilbert
basis~\eqref{eqn:Hilbert_basis_USp4_N3>0} are the relevant generators (together
with their dressings) for the moduli space.
\section{Plethystic Logarithm}
\label{app:PL}
In this appendix we summarise the main properties of the plethystic logarithm.
Starting with the definition, for a mulit-valued function $f(t_1,\ldots,t_m)$
with $f(0,\ldots,0)=1$, one defines
\begin{equation}
\mathrm{PL}[f]\coloneqq \sum_{k=1}^{\infty} \frac{\mu(k)}{k} \log\left(
f(t_1^k,\ldots,t_m^k) \right) \; ,
\end{equation}
where $\mu(k)$ denote the Möbius function~\cite{Benvenuti:2006qr}.
Some basic properties include
\begin{equation}
\mathrm{PL}[f\cdot g] = \mathrm{PL}[f] + \mathrm{PL}[g] \quad\textrm{and}\quad
\mathrm{PL}\left[ \frac{1}{\prod_n (1-t^n)^{a_n}} \right] = \sum_n a_n \ t^n \; .
\label{eqn:PL_properties}
\end{equation}
Now, we wish to compute the plethystic logarithm. Given a Hilbert series as
rational function, i.e.\ of the form~\eqref{eqn:HS_generic_solved}
or~\eqref{eqn:HS_generic_refined}, the denominator can be taken care of by
means of~\eqref{eqn:PL_properties}, while the numerator is a polynomial with
integer coefficients. In order to obtain an approximation of the PL, we employ
the following two equivalent transformations for the numerator:
\begin{subequations}
\begin{align}
\mathrm{PL}\left[ 1+ a t^n +\mathcal{O}(t^{n+1}) \right] &=
\mathrm{PL}\left[ \frac{(1-t^n)^a \ \left( 1+ a t^n +\mathcal{O}(t^{n+1}) \right)
}{(1-t^n)^a} \right] \notag \\
&= a t^n + \mathrm{PL}\left[1 +\mathcal{O}(t^{n+1}) \right] \; ,
\label{eqn:PL_mod_plus-term}\\
\mathrm{PL}\left[ 1- a t^n +\mathcal{O}(t^{n+1}) \right] &=
\mathrm{PL}\left[ \frac{(1-t^n)^a \ (1+t^n)^a \ \left( 1- a t^n +\mathcal{O}(t^{n+1})
\right)
}{(1-t^{2n})^a} \right] \notag \\
&= -a t^n + a t^{2n} + \mathrm{PL}\left[1 +\mathcal{O}(t^{n+1}) \right] \;
. \label{eqn:PL_mod_minus-term}
\end{align}
\end{subequations}
Now, we derive an approximation of the PL for a generic rank two
gauge group in terms of $t^\Delta$. More precisely, consider the Hilbert basis
$\{X_i\}$ then we provide an approximation of the PL up to second order, i.e.\
\begin{equation}
\mathrm{PL} = \text{Casimir inv.}
+ \left\{t^{\Delta(X_i)} \text{-terms} \right\}
+ \left\{t^{\Delta(X_i)+\Delta(X_j)} \text{-terms} \right\}
+\mathcal{O}\left(t^{\Delta(X_i)+\Delta(X_j)+\Delta(X_k)}\right)
\end{equation}
Considering~\eqref{eqn:HS_generic_solved}, the numerator is denoted by $R(t)$,
while the denominator $Q(t)$ is given by
\begin{equation}
Q(t) = \prod_{i=1}^2 (1-t^{d_i}) \ \prod_{p=0}^L \left(
1-t^{\Delta(x_p)}\right) \; ,
\end{equation}
with $d_i$ the degrees of the Casimir invariants. Then expand the numerator as
follows:
\begin{align}
R(t) = 1
&+ \sum_{q=0}^L \left( \frac{P_{\mathrm{G}}(t,x_q)}{P_{\mathrm{G}}(t,0)}-1\right)
t^{\Delta(x_q)}
+ \sum_{q=0}^L \sum_{s \in \mathrm{Int}(\mathcal{P}(C_q^{(2)}))}
\frac{P_{\mathrm{G}}(t,s)}{P_{\mathrm{G}}(t,0)} t^{\Delta(s)} \\
&-\sum_{{{q,p=0}\atop{q\neq p}}}^L
\left(\frac{P_{\mathrm{G}}(t,x_q)}{P_{\mathrm{G}}(t,0)}-\frac{1}{2} \right) t^{\Delta(x_p)
+\Delta(x_q)}
+\sum_{q=1}^L \frac{P_{\mathrm{G}}(t,C_q^{(2)})}{P_{\mathrm{G}}(t,0)} t^{\Delta(x_{q-1})
+\Delta(x_q)} \notag \\
&-\sum_{q=1}^L \sum_{s \in \mathrm{Int}(\mathcal{P}(C_q^{(2)}))}
\sum_{{{r=0}\atop{r\neq q{-}1,q}}}^L \frac{P_{\mathrm{G}}(t,s)}{P_{\mathrm{G}}(t,0)}
t^{\Delta(s) + \Delta{(x_r)}} \notag \; .
\end{align}
Note that the appearing factor $\tfrac{1}{2}$ avoids double counting when
changing summation $\sum_{q<p}$ to $\sum_{q\neq p}$. Still, the numerator is a
polynomial with integer coefficients.
The PL then reads
\begin{equation}
\mathrm{PL}\left[\mathrm{HS}_{\mathrm{G}}(t)\right] = \sum_{i=1}^2 t^{d_i} + \sum_{p=0}^L
t^{\Delta(x_p)}
+\mathrm{PL}\left[R(t)\right] \; .
\end{equation}
By step~\eqref{eqn:PL_mod_plus-term} we factor out the order $t^{\Delta(x_q)}$
and $t^{\Delta(s)}$ terms. However, this introduces further terms at order
$t^{\Delta(x_q) +\Delta(s)}$ and so forth, which are given by
\begin{equation}
-\left( \sum_{q=0}^L \left( \frac{P_{\mathrm{G}}(t,x_q)}{P_{\mathrm{G}}(t,0)}-1\right)
t^{\Delta(x_q)}
+ \sum_{q=1}^L \sum_{s \in \mathrm{Int}(\mathcal{P}(C_q^{(2)}))}
\frac{P_{\mathrm{G}}(t,s)}{P_{\mathrm{G}}(t,0)} t^{\Delta(s)} \right)^2 \; .
\end{equation}
Subsequently factoring the terms of this order by means
of~\eqref{eqn:PL_mod_minus-term}, one derives at the following expressing of the
PL
\begin{align}
\mathrm{PL}\left[\mathrm{HS}_{\mathrm{G}}(t)\right] = \sum_{i=1}^2 t^{d_i} &+ \sum_{q=0}^L
\frac{P_{\mathrm{G}}(t,x_q)}{P_{\mathrm{G}}(t,0)} \ t^{\Delta(x_q)}
+\sum_{q=1}^L \sum_{s \in \mathrm{Int}(\mathcal{P}(C_q^{(2)}))}
\frac{P_{\mathrm{G}}(t,s)}{P_{\mathrm{G}}(t,0)}\ t^{\Delta(s)}
\label{eqn:PL_truncated}\\
&-\sum_{{{q,p=0}\atop{q\neq p}}}^L
\left(\frac{P_{\mathrm{G}}(t,x_q)}{P_{\mathrm{G}}(t,0)}-\frac{1}{2} \right) t^{\Delta(x_p)
+\Delta(x_q)}
+\sum_{q=1}^L \frac{P_{\mathrm{G}}(t,C_q^{(2)})}{P_{\mathrm{G}}(t,0)} t^{\Delta(x_{q-1})
+\Delta(x_q)} \notag\\
&-\sum_{q=1}^L \sum_{s \in \mathrm{Int}(\mathcal{P}(C_q^{(2)}))}
\sum_{{{r=0}\atop{r\neq q{-}1,q}}}^L \frac{P_{\mathrm{G}}(t,s)}{P_{\mathrm{G}}(t,0)}
t^{\Delta(s) + \Delta{x_r}} \notag \\
&-\sum_{q,p=0}^{L}
\left( \frac{P_{\mathrm{G}}(t,x_q)}{P_{\mathrm{G}}(t,0)} -1 \right)
\left( \frac{P_{\mathrm{G}}(t,x_p)}{P_{\mathrm{G}}(t,0)} -1 \right) t^{\Delta(x_q)+\Delta(x_p)}
\notag \\
&-2 \sum_{p=0}^{L} \sum_{q=1}^L
\left( \frac{P_{\mathrm{G}}(t,x_p)}{P_{\mathrm{G}}(t,0)} -1 \right)
\sum_{s \in \mathrm{Int}(\mathcal{P}(C_q^{(2)}))}
\frac{P_{\mathrm{G}}(t,s)}{P_{\mathrm{G}}(t,0)} \ t^{\Delta(x_p)+\Delta(s)} \notag \\
&-\sum_{q,p=1}^{L} \sum_{s \in \mathrm{Int}(\mathcal{P}(C_p^{(2)}))} \sum_{s'
\in \mathrm{Int}(\mathcal{P}(C_q^{(2)}))}
\frac{P_{\mathrm{G}}(t,s)}{P_{\mathrm{G}}(t,0)} \frac{P_{\mathrm{G}}(t,s')}{P_{\mathrm{G}}(t,0)}
t^{\Delta(s)+\Delta(s')} \notag \\
&+\mathrm{PL}\left[1 +
\mathcal{O}\left(t^{\Delta(X_i)+\Delta(X_j)+\Delta(X_j)}\right)\right]\notag \;
.
\end{align}
Strictly speaking, the truncation~\eqref{eqn:PL_truncated} is
only meaningful if
\begin{equation}
\begin{aligned}
&\max\{\Delta(X) \} +\max\{d_i | i=1,2\} < \min\{\Delta(X)+\Delta(Y) \} = 2
\cdot \min\{ \Delta(X)\} \\
&\qquad\textrm{for}\quad X,Y= x_q \text{ or } s\; , s \in
\mathrm{Int}(\mathcal{P}(C_p^{(2)})) , q =0,1,\ldots,l
\end{aligned}
\label{eqn:PL_condition}
\end{equation}
holds. Only in this case do the positive contributions, i.e.\ the generators,
of the first line in~\eqref{eqn:PL_truncated} not mix with the negative
contributions, i.e.\ first syzygies or relations, of the remaining lines.
Moreover, the condition~\eqref{eqn:PL_condition} ensures that the remained
$\mathcal{O}\left(t^{\Delta(X_i)+\Delta(X_j)+\Delta(X_k)}\right)$ does not
spoil the truncation.
From the examples of Sec.~\ref{sec:U1xU1}-\ref{sec:SU3}, we see
that~\eqref{eqn:PL_condition} is at most satisfied for scenarios with just a
few generators, but not for elaborate cases. Nevertheless, there are some
observations we summarise as follows:
\begin{itemize}
\item The bare and dressed monopole operators associated to the GNO-charge $m$
are described by $\tfrac{P_\mathrm{G}(t,m)}{P_\mathrm{G}(t,0)} t^{\Delta(m)}$. In particular,
we emphasis that the quotient of dressing factors provides information on the
number and degrees of the dressed monopole operators.
\item The previous observation provides an upper bound on the number of dressed
monopole operators associated to a magnetic weight $m$. In detail, the value of
$\tfrac{P_\mathrm{G}(t,m)}{P_\mathrm{G}(t,0)}$ at $t=1$ equals the number of bare and
dressed monopole operators associated to $m$. Let $\{d_i\}$ and $\{b_i\}$, for
$i=1,\ldots,\mathrm{rk}(\mathrm{G})$ denote the degree of the Casimir invariants for $\mathrm{G}$ and
$\mathrm{H}_m$, respectively. Then
\begin{equation}
\begin{matrix} \text{\# dressed monopoles} \\
+1 \text{ bare monopole}
\end{matrix} =
\lim_{t \to 1} \frac{P_\mathrm{G}(t,m)}{P_\mathrm{G}(t,0)} = \lim_{t\to 1}
\frac{\prod_{i=1}^{\mathrm{rk}(\mathrm{G})} \left(1-t^{d_i} \right) }{\prod_{j=1}^{\mathrm{rk}(\mathrm{G})}
\left(1-t^{b_j} \right)} = \frac{\prod_{i=1}^{\mathrm{rk}(\mathrm{G})}
d_i}{\prod_{j=1}^{\mathrm{rk}(\mathrm{G})} b_j} = \frac{\left|\mathcal{W}_{\mathrm{G}}
\right|}{\left|\mathcal{W}_{\mathrm{H}_{m}}\right| } \; ,
\end{equation}
where the last equality holds because the order of the Weyl group equals
the product of the degrees of the Casimir invariants. Since $\mathcal{W}_{\mathrm{H}_{m}}
\subset \mathcal{W}_{\mathrm{G}} $ is a subgroup of the finite group $\mathcal{W}_{\mathrm{G}} $,
Lagrange's theorem implies that $\tfrac{\left|\mathcal{W}_{\mathrm{G}}
\right|}{\left|\mathcal{W}_{\mathrm{H}_{m}}\right| } \in \NN$ holds.
The situation becomes obvious whenever $m$ belongs to the interior of the Weyl
chamber, because $\mathrm{H}_m = \mathrm{T}$ and thus
\begin{equation}
\left. \begin{matrix} \text{\# dressed monopoles} \\
+1 \text{ bare monopole}
\end{matrix} \right|_{\text{interior of}\atop\text{Weyl chamber}}
=\left|\mathcal{W}_{\mathrm{G}} \right| \quad\textrm{and}\quad
\frac{P_\mathrm{G}(t,m)}{P_\mathrm{G}(t,0)} =\prod_{i=1}^{\mathrm{rk}(\mathrm{G})} \sum_{l_i=0}^{d_i -1}
t^{l_i} \; .
\end{equation}
\item The significance of the PL is limited, as, for instance, a positive
contribution $\sim t^{\Delta(X_1)}$ can coincide with a negative contribution
$\sim t^{\Delta(X_2)+\Delta(X_3)}$, but this does not necessarily imply that
the object of degree $\Delta(X_1)$ can be generated by others. The situation
becomes clearer if there exists an additional global symmetry $Z(\widehat{\G})$ on the
moduli space. The truncated PL for~\eqref{eqn:HS_generic_refined} is obtained
from~\eqref{eqn:PL_truncated} by the replacement
\begin{equation}
t^{\Delta(X)} \mapsto \vec{z}^{\vec{J}(X)} \ t^{\Delta(X)} \; .
\end{equation}
Then the ``syzygy'' $\vec{z}^{\vec{J}(X_2+X_3)}t^{\Delta(X_2)+\Delta(X_3)}$ can
cancel the ``generator'' $\vec{z}^{\vec{J}(X_1)}t^{\Delta(X_1)}$ only if the
symmetry charges agree $\vec{z}^{\vec{J}(X_1)}=\vec{z}^{\vec{J}(X_2+X_3)}$, in
addition to the ${{\rm SU}(2)}_R$ iso-spin.
\end{itemize}
Lastly, we illustrate the truncation with the two simplest examples:
\paragraph{Example: one simplicial cone}
For the Hilbert series~\eqref{eqn:Example_HS_simplicial} we obtain
\begin{align}
\label{eqn:Example_PL_simplicial}
\mathrm{PL}=\sum_{i=1}^2 t^{d_i} &+ \frac{P_1(t)}{P_0(t)} \left(t^{\Delta(x_0)} +
t^{\Delta(x_1)}\right)
-\left(2 \frac{P_1(t)}{P_0(t)} -1 - \frac{P_2(t)}{P_0(t)}
\right) t^{\Delta(x_0)+\Delta(x_1)} \\
&- \left(\frac{P_1(t)}{P_0(t)} \right)^2
\left(t^{2\Delta(x_0)}+t^{2\Delta(x_1)} +2t^{\Delta(x_0)+\Delta(x_1)} \right) +
\ldots \notag \; .
\end{align}
\paragraph{Example: one non-simplicial cone}
In contrast, for the Hilbert series~\eqref{eqn:Example_HS_non-simplicial} we
arrive at
\begin{align}
\label{eqn:Example_PL_non-simplicial}
\mathrm{PL}=\sum_{i=1}^2 t^{d_i} &+ \frac{P_1(t)}{P_0(t)} \left(t^{\Delta(x_0)} +
t^{\Delta(x_1)}\right)
+ \sum_{s \in
\mathrm{Int}\mathcal{P}} \frac{P_2(t)}{P_0(t)} t^{\Delta(s)} \\
&-\left(2 \frac{P_1(t)}{P_0(t)} -1 - \frac{P_2(t)}{P_0(t)}
\right) t^{\Delta(x_0)+\Delta(x_1)} \notag\\
&- \left(\frac{P_1(t)}{P_0(t)} \right)^2
\left(t^{2\Delta(x_0)}+t^{2\Delta(x_1)} +2t^{\Delta(x_0)+\Delta(x_1)} \right)
\notag \\
&-2\left(\frac{P_1(t)}{P_0(t)}-1 \right)\frac{P_2(t)}{P_0(t)} \sum_{s \in
\mathrm{Int}\mathcal{P}} \left( t^{\Delta(s)+\Delta(x_0)}
+t^{\Delta(s)+\Delta(x_1)} \right) \notag \\
&-\sum_{s \in
\mathrm{Int}\mathcal{P}} \sum_{s' \in
\mathrm{Int}\mathcal{P}} \left(\frac{P_2(t)}{P_0(t)} \right)^2
t^{\Delta(s)+\Delta(s')} + \ldots \notag \; .
\end{align}
\section{Conclusions}
\label{sec:conclusions}
In this paper we introduced a geometric concept to identify and compute the set
of bare and dressed monopole operators that are sufficient to describe the
entire chiral ring $\mathbb C[\mathcal{M}_C]$ of any $3$-dimensional $\mathcal{N}=4$ gauge
theory. The methods can be summarised as follows:
\begin{enumerate}
\item The matter content together with the positive roots of the gauge group
$\mathrm{G}$ define the conformal dimension, which in turn defines an arrangement of
hyperplanes that divide the dominant Weyl chamber of $\widehat{\G}$ into a fan.
\item The intersection of the fan with the weight lattice of the GNO-dual
group leads to a collection of affine semi-groups. All semi-groups are finitely
generated and the unique, finite basis is called Hilbert basis.
\item The knowledge of the minimal generators, together with their properties
${{\rm SU}(2)}_R$-spin, residual gauge group $\mathrm{H}_m$, and topological charges $J(m)$, is
sufficient to explicitly sum and determine the Hilbert series as rational
function.
\end{enumerate}
Utilising the fan and the Hilbert bases for each semi-group also allows to
deduce the dressing behaviour of monopole operators. The number of dressed
operators is determined by a ratio of orders of Weyl groups, while the degrees
are determined by the ratio of the dressing factors associated to the GNO-charge
$m$ divided by the dressing factor of the trivial monopole $m=0$.
Most importantly, the entire procedure works for any rank of the gauge group,
as indicated in Sec.~\ref{sec:SU3} for ${{\rm U}(3)}$. For the main part of the
paper, we, however, have chosen to provide a comprehensive collection of rank
two examples.
Before closing, let us outline and comment on the approach to higher rank cases.
\begin{enumerate}[(a)]
\item The gauge group $\mathrm{G}$ determines the GNO-dual group $\widehat{\G}$ and the
corresponding dominant Weyl chamber (or the product of several Weyl chambers).
The Weyl chamber is understood as finite intersection of positive half-spaces
$H^+_{\alpha} \subset \mathfrak{t}$, where $\alpha$ ranges over all simple roots of
$\mathrm{G}$. (If $\mathrm{G}$ is a product, then the roots of one factor have to be embedded
in a higher dimensional vector space.)
\item The \emph{relevant weights} $\mu_i$, as identified in
Sec.~\ref{subsec:Hilbert basis}, define a finite set of cones via the
intersection of all possible upper and lower half-spaces with the Weyl chamber.
This step can, for instance, be implemented by means of the package
\emph{Polyhedra} of \texttt{Macaulay2}.
\item Having defined all cones in \texttt{Macaulay2}, one computes the
dimension and the Hilbert basis for each cone. Identifying all cones
$C_p^{(\mathrm{rk}(G))}$ of the maximal dimension $\mathrm{rk}(\mathrm{G})$ can typically reduce
the number of cones one needs to consider.
\item Define the fan $F=
\{C_p^{(\mathrm{rk}(G))} | p=1,\ldots,L\}$ generated by all top-dimensional cones in
\texttt{Macaulay2}. This step is the computationally most demanding process so
far.
\item Next, one employs the \emph{inclusion-exclusion principle} for each cone
in the fan: that is the number of points in the (relative) interior
$\mathrm{Int}(S^{(p)})\coloneqq \mathrm{Relint}(C^{(p)}) \cap
\Lambda_w(\widehat{\G})$ is given by
\begin{subequations}
\begin{align}
\#|\mathrm{Int}(S^{(p)})| &= |S^{(p)}| - \Bigg( \sum_{j=1}^{\kappa_p}
|S_{j}^{(p-1)}|
-\sum_{1\leq i<j \leq \kappa_p} |S_{i}^{(p-1)} \cap S_{j}^{(p-1)}| \\*
&\phantom{= |S^{(p)}| -}
+\sum_{1\leq i<j<k \leq \kappa_p} |S_{i}^{(p-1)} \cap
S_{j}^{(p-1)}\cap S_{k}^{(p-1)}|
- \ldots
+(-1)^{\kappa_p -1} \left|\bigcap_{i=1}^{\kappa_p} S_{i}^{(p-1)} \right|
\Bigg) \notag \\
&\equiv|S^{(p)}| - |\partial S^{(p)} | \;,
\end{align}
\end{subequations}
where the $S_j^{(p-1)}$ for $j=1,\ldots,\kappa_p$ are the semi-groups resulting
from the facets of $C^{(p)}$. Note that the last term $\bigcap_{i=1}^{\kappa_p}
S_{i}^{(p-1)}$ equals the
trivial semi-group, while the intermediate intersections give rise to all lower
dimensional semi-groups contained in the boundary of $S^{(p)}$.
Then, the contribution for $\mathrm{Int}(S^{(p)})$ to the monopole formula is
computed as follows:
\begin{subequations}
\begin{align}
\mathrm{HS}(S^{(p)};t) &\coloneqq P_{\mathrm{G}}(t;S^{(p)}) \cdot \left[ \mathrm{H}_{S^{(p)}}(t)
-\mathrm{H}_{\partial S^{(p)}}(t) \right]\; , \\
\mathrm{H}_{S^{(p)}}(t) &\coloneqq \sum_{m\in S^{(p)}}
z^{J(m)} \, t^{\Delta(m)} \; ,\\
\mathrm{H}_{\partial S^{(p)}}(t) &\coloneqq
\sum_{j=1}^{\kappa_p}
\mathrm{H}_{S_{j}^{(p-1)}}(t)
-\sum_{1\leq i<j \leq \kappa_p} \mathrm{H}_{S_{i}^{(p-1)} \cap S_{j}^{(p-1)}}(t)
\\*
& \quad
+\sum_{1\leq i<j<k \leq \kappa_p} \mathrm{H}_{S_{i}^{(p-1)} \cap
S_{j}^{(p-1)}\cap S_{k}^{(p-1)}}(t)
- \ldots
+(-1)^{\kappa_p -1} \mathrm{H}_{\bigcap_{i=1}^{\kappa_p} S_{i}^{(p-1)} }(t) \; .
\notag
\end{align}
\end{subequations}
Each contribution $\mathrm{H}_{S^{(p)}}(t)$ is evaluated as
discussed in Sec.~\ref{subsec:summation_HS_unrefined} and
\ref{subsec:summation_HS_refined}. Although this step is algorithmically
simple, it can be computationally demanding. It is, however, crucial that the
fan $F$ has been defined, in order to work with the correct faces of each cone
and to sum over each cone in the fan only once.
\item Finally, one has to add all contributions
\begin{equation}
\mathrm{HS}(F;t)= \sum_{C \in F} \mathrm{HS}(S) \; .
\end{equation}
This last step is a simple sum, but to obtain the Hilbert series as a rational
function in a desirable form can be cumbersome.
\end{enumerate}
Equipped with this procedure, we hope to report on Coulomb branches for higher
rank gauge groups and quiver gauge theories in the future.
\section*{Acknowledgements}
We thank Roger Bielawski, Simon Brandhorst, Stefano Cremonesi, Giulia Ferlito,
Rudolph Kalveks, and Markus Röser for useful discussions.
A.~H.\ thanks the Institute für Theoretische Physik of the Leibniz Universität
Hannover for hospitality.
A.~H.\ is supported by STFC Consolidated Grant ST/J0003533/1, and EP- SRC
Programme Grant EP/K034456/1.
M.~S.\ thanks the Theoretical Physics Group of the Imperial College London for
hospitality.
M.~S.\ is supported by the DFG research training group GRK1463 ``Analysis,
Geometry, and String Theory''.
\section{Introduction}
The moduli spaces of supersymmetric gauge theories with $8$ supercharges have
generically two branches: the Higgs and the Coulomb branch. In this paper we
focus on $3$-dimensional $\mathcal{N}=4$ gauge theories, for which both branches are
hyper-Kähler spaces. Despite this fact, the branches are fundamentally
different.
The Higgs branch $\mathcal{M}_H$ is understood as hyper-Kähler quotient
\begin{equation}
\mathcal{M}_H = \mathbb R^{4N}/\!/\!/\,\mathrm{G}\, \; ,
\end{equation}
in which the vanishing locus of the $\mathcal{N}=4$ F-terms is quotient by the
complexified gauge group. The F-term equations play the role of complex
hyper-Kähler moment maps, while the transition to the complexified gauge group
eliminates the necessity to impose the D-term constraints. Moreover, this
classical description is sufficient as the Higgs branch is protected from
quantum corrections. The explicit quotient construction can be supplemented by
the study of the Hilbert series, which allows to gain further understanding of
$\mathcal{M}_H$ as a complex space.
Classically, the Coulomb branch $\mathcal{M}_C$ is the hyper-Kähler space
\begin{equation}
\mathcal{M}_C \approx (\mathbb R^3 \times S^1)^{\mathrm{rk}(\mathrm{G})}\slash \mathcal{W}_\mathrm{G} \; ,
\end{equation}
where $\mathcal{W}_\mathrm{G} $ is the Weyl group of $\mathrm{G}$ and $\mathrm{rk}(\mathrm{G})$ denotes the rank of
$\mathrm{G}$. However, the geometry and topology
of $\mathcal{M}_C$ are affected by quantum corrections. Recently, the
understanding of the Coulomb branch has been subject of active research
from various viewpoints: the authors
of~\cite{Bullimore:2015lsa} aim to provide a description for the
quantum-corrected Coulomb
branch of any $3d$ $\mathcal{N}=4$ gauge theory, with particular emphasis on the
full Poisson algebra of the chiral ring $\mathbb C[\mathcal{M}_C]$. In contrast, a
rigorous mathematical definition of the Coulomb branch itself lies at the heart
of the attempts presented
in~\cite{Nakajima:2015txa,Nakajima:2015gxa,Braverman:2016wma}.
In this paper, we take the perspective centred around the \emph{monopole
formula} proposed in~\cite{Cremonesi:2013lqa}; that is, the computation of the
Hilbert series for the Coulomb branch allows to gain information on $\mathcal{M}_C$
as a complex space.
Let us briefly recall the set-up. Select an $\mathcal{N} =2$ subalgebra in the
$\mathcal{N}=4$ algebra, which implies a decomposition of the $\mathcal{N}=4$ vector
multiplet into an $\mathcal{N}=2$ vector multiplet (containing a gauge field $A$ and
a real adjoint scalar $\sigma$) and an $\mathcal{N}=2$ chiral multiplet (containing a
complex adjoint scalar $\Phi$) which transforms in the adjoint representation
of the gauge group $\mathrm{G}$. In addition, the selection of an $\mathcal{N}=2$ subalgebra
is equivalent to the choice of a complex structure on $\mathcal{M}_C$ and
$\mathcal{M}_H$, which is the reason why one studies the branches only as complex
and not as hyper-Kähler spaces.
The description of the Coulomb branch relies on 't~Hooft monopole
operators~\cite{tHooft:1977hy}, which are local disorder
operators~\cite{Borokhov:2002ib} defined by specifying a Dirac monopole
singularity
\begin{equation}
A_{\pm} \sim \frac{m}{2} \left(\pm1 -\cos \theta \right) \mathrm{d} \varphi
\end{equation}
for the gauge field, where $m\in \gfrak=\mathrm{Lie}(\mathrm{G})$ and
$(\theta,\varphi)$ are coordinates on the $2$-sphere around the insertion point.
An important consequence is that the \emph{generalised Dirac quantisation
condition}~\cite{Englert:1976ng}
\begin{equation}
\exp\left(2 \pi \mathrm{i} m \right) = \mathds{1}_\mathrm{G}
\label{eqn:general_Dirac}
\end{equation}
has to hold. As proven in~\cite{Goddard:1976qe}, the set of solutions
to~\eqref{eqn:general_Dirac} equals the weight lattice $\Lambda_w(\widehat{\G})$ of
the GNO (or Langlands) dual group $\widehat{\G}$, which is uniquely associated to the
gauge group $\mathrm{G}$.
For Coulomb branches of supersymmetric gauge theories, the monopole operators
need to be supersymmetric as well, see for instance~\cite{Borokhov:2002cg}. In
a pure $\mathcal{N} =2$ theory, the supersymmetry condition amounts to the singular
boundary condition
\begin{equation}
\sigma \sim \frac{m}{2r} \qquad\textrm{for}\quad r \to \infty \; ,
\end{equation}
for the real adjoint scalar in the $\mathcal{N}=2 $ vector multiplet.
Moreover, an $\mathcal{N}=4$ theory also allows for a non-vanishing vacuum
expectation value of the complex adjoint scalar $\Phi$ of the adjoint-valued
chiral multiplet. Compatibility with supersymmetry requires $\Phi$ to take
values in the stabiliser $\mathrm{H}_m$ of the ``magnetic weight'' $m$ in $\mathrm{G}$. This
phenomenon gives rise to dressed monopole operators.
Dressed monopole operators and $\mathrm{G}$-invariant functions of $\Phi$ are believed
to generate the entire chiral ring $\mathbb C[\mathcal{M}_C]$. The corresponding Hilbert
series allows for two points of view: seen via the \emph{monopole formula},
each operator is precisely counted once in the Hilbert series --- no
over-counting appears. Evaluating the Hilbert series as rational function,
however, provides an over-complete set of generators that, in general, satisfies
relations. In order
to count polynomials in the chiral ring, a notion of degree or dimension is
required. Fortunately, in a CFT one employs the conformal dimension $\Delta$,
which for BPS states agrees with the ${{\rm SU}(2)}_R$ highest weight.
Following~\cite{Borokhov:2002cg,Gaiotto:2008ak,Benna:2009xd,Bashkirov:2010kz},
the conformal dimension of a BPS bare monopole operator of GNO-charge m is
given by
\begin{equation}
\Delta(m)= \frac{1}{2} \sum_{i=1}^{n} \sum_{\rho \in \mathcal{R}_i} \left|
\rho(m)\right| - \sum_{\alpha \in \Phi_+} \left|
\alpha(m)\right|\; ,
\label{eqn:Def_ConfDim}
\end{equation}
where $\mathcal{R}_i$ denotes the set of all weights $\rho$ of the
$\mathrm{G}$-representation in
which the $i$-th flavour of $\mathcal{N}=4$ hypermultiplets transform. Moreover,
$\Phi_+$ denotes the set of positive roots $\alpha$ of the Lie algebra $\gfrak$
and provides the contribution of the $\mathcal{N}=4$ vector multiplet. Bearing in
mind the proposed classification of $3d$ $\mathcal{N}=4$ theories
by~\cite{Gaiotto:2008ak}, we restrict ourselves to ``good'' theories (i.e.\
$\Delta > \tfrac{1}{2}$ for all BPS monopoles).
If the centre $\mathcal{Z}(\widehat{\G})$ is non-trivial, then the monopole operators can be
charged under this topological symmetry group and one can refine the counting
on the chiral ring.
Putting all the pieces together, the by now well-established \emph{monopole
formula} of~\cite{Cremonesi:2013lqa} reads
\begin{equation}
\mathrm{HS}_{\mathrm{G}}(t,z) = \sum_{m \in \Lambda_w(\widehat{\G}) \slash \mathcal{W}_{\widehat{\G}}}
z^{J(m)} t^{\Delta(m)} P_{\mathrm{G}}(t,m) \; .
\label{eqn:HS_refined}
\end{equation}
Here, the fugacity $t$ counts the ${{\rm SU}(2)}_R$-spin, while the (multi-)fugacity $z$
counts the quantum numbers $J(m)$ of the topological symmetry $\mathcal{Z}(\widehat{\G})$.
This paper serves three purposes: firstly, we provide a geometric derivation of
a sufficient set of monopole operators, called the \emph{Hilbert basis}, that
generates the entire chiral ring. Secondly, employing the Hilbert basis
allows an explicit summation of~\eqref{eqn:HS_refined}, which we demonstrate
for
$\mathrm{rk}(\mathrm{G})=2$ explicitly. Thirdly, we provide various examples for all rank two
gauge groups and display how the knowledge of the Hilbert basis completely
determines the Hilbert series.
The remainder of this paper is organised as follows:
Sec.~\ref{sec:general_idea} is devoted to the exposition of our main
points: after recapitulating basics on (root and weight) lattices and
rational polyhedral cones in Subsec.~\ref{subsec:preliminaries}, we explain in
Subsec.~\ref{subsec:Hilbert basis} how the conformal dimension decomposes the
Weyl chamber of $\widehat{\G}$ into a fan. Intersecting the fan with the weight lattice
$\Lambda_w(\widehat{\G})$ introduces affine semi-groups, which are finitely generated
by a unique set of irreducible elements --- called the Hilbert basis. Moving on
to Subsec.~\ref{subsec:Dressings_as_HS}, we collect mathematical results that
interpret the dressing factors $P_{\mathrm{G}}(t,m)$ as Poincar\'{e} series for the set
of $\mathrm{H}_m$-invariant polynomials on the Lie algebra $\mathfrak{h}_m$. Finally, we
explicitly sum the unrefined Hilbert series in
Subsec.~\ref{subsec:summation_HS_unrefined} and the refined Hilbert series
in~\ref{subsec:summation_HS_refined} utilising the knowledge about the Hilbert
basis.
After establishing the generic results, we provide a comprehensive collection
of examples for all rank two gauge groups in
Sec.~\ref{sec:U1xU1}-\ref{sec:SU3}. Lastly, Sec.~\ref{sec:conclusions}
concludes.
Before proceeding to the details, we present our main
result~\eqref{eqn:HS_generic_refined} already at
this stage: the refined Hilbert series for any rank two gauge group $\mathrm{G}$.
\begin{align}
\mathrm{HS}_{\mathrm{G}}(t,z)&= \frac{P_{\mathrm{G}}(t,0) }{\prod_{p=0}^{L}
\left(1-z^{J(x_p)}
t^{\Delta(x_{p})}\right)}
\Bigg\{
\prod_{q=0}^{L} \left( 1-z^{J(x_q)}t^{\Delta(x_{q})} \right)
\label{eqn:HS_generic_intro}\\
&\phantom{= \frac{P_{\mathrm{G}}(t,0) }{\prod_{p=0}^{L} \left(1-z^{J(x_p)}
t^{\Delta(x_{p})}\right)}}
+ \sum_{q=0}^{L} \frac{P_{\mathrm{G}}(t,x_q)}{P_{\mathrm{G}}(t,0)}
z^{J(x_q)} t^{\Delta(x_q)}
\prod_{r=0\atop r\neq q}^{L}
\left( 1-z^{J(x_r)} t^{\Delta(x_{r})} \right)
\notag \\
&\phantom{= \frac{P_{\mathrm{G}}(t,0) }{\prod_{p=0}^{L} \left(1-z^{J(x_p)}
t^{\Delta(x_{p})}\right)}}
+\sum_{q=1}^{L} \frac{P_{\mathrm{G}}(t,C_q^{(2)})}{P_{\mathrm{G}}(t,0)}
\bigg[z^{J(x_{q-1})+J(x_q)} t^{\Delta(x_{q-1}) +\Delta(x_q)}
\notag\\*
&\phantom{= \frac{P_{\mathrm{G}}(t,0) }{\prod_{p=0}^{L} \left(1-z^{J(x_p)}
t^{\Delta(x_{p})}\right)}}
\qquad \qquad
+ \sum_{s \in \mathrm{Int}( \mathcal{P}(C_q^{(2)}))}
z^{J(s)} t^{\Delta(s)} \bigg] \prod_{r=0\atop{r\neq q-1,q}}^L
\left(1- z^{J(x_r)} t^{\Delta(x_{r})}\right)
\Bigg\} \notag \; ,
\end{align}
where the ingredients can be summarised as follows:
\begin{itemize}
\item A fan $F_\Delta=\{C_p^{(2)} \, , \, p =1,\ldots,L\}$, and each
$2$-dimensional
cone satisfies $\partial C_p^{(2)} = C_{p-1}^{(1)} \cup C_p^{(1)}$ and
$C_{p-1}^{(1)} \cap C_p^{(1)} = \{0\} $.
\item The Hilbert basis for $C_p^{(2)}$ comprises the ray generators $x_{p-1}$,
$x_p$ as well as other minimal generators $\{u_{\kappa}^p\}$.
\item The $x_{p-1}$, $x_p$ generate a fundamental parallelotope
$\mathcal{P}(C_p^{(2)})$, where the discriminant counts the number of lattice
points in the interior $\mathrm{Int}( \mathcal{P}(C_p^{(2)}))$ via $d(C_p^{(2)})
-1 = \# \text{pts.}\ \left( \mathrm{Int}( \mathcal{P}(C_p^{(2)})\right)$.
\end{itemize}
The form of~\eqref{eqn:HS_generic_intro} is chosen to emphasis that the terms
within the curly bracket represent the numerator of the Hilbert series as
rational function, i.e.\ the curly bracket is a proper polynomial in $t$
without
poles. On the other hand, the first fraction represents the denominator of the
rational function, which is again a proper polynomial by construction.
|
1,108,101,566,811 | arxiv | \section{INTRODUCTION}
\label{sec:intro}
{Chondrites reserve information about the early stage of
the solar system that had been lost in the planets themselves.}
Spherical chondrules, grains 0.1 mm to 1 mm in size composed of the silicates found in meteorites, are considered to have been formed from precursor particles that were heated and melted in flash heating events.
These then cooled and resolidified in a short period of time ($\sim$ hours) in a protoplanetary disk (e.g., Jones et al. 2000).
It is believed that the efficiency of chondrule formation was high because chondrules are major constituents in chondritic meteorites.
So far, various mechanisms for chondrule formation have been proposed; however, it has not yet been concluded which process is predominant. One mechanism for chondrule formation that has received considerable attention is heating by shock waves (Hood \& Horanyi 1991, 1993; Boss 1996; Jones et al. 2000).
The heating process of chondrule precursors by shock waves has been investigated in detail in previous studies. It has been shown that the shock-wave heating model satisfies various constraints related to chondrule formation such as the peak temperature ($\sim 2000$ K) and short cooling time
(Hood \& Horanyi 1991, 1993; Iida et al. 2001; Ciesla \& Hood 2002; Miura, Nakamoto, \& Susa 2002; Desch \& Connolly 2002; Miura \& Nakamoto 2005, 2006).
As plausible sites for such shock waves to occur, highly eccentric planetesimals have been proposed (Hood 1998; Weidenschilling, Marzari, \& Hood 1998).
If the relative velocity between a planetesimal and the gas disk exceeds the speed of sound of the gas, a bow shock is produced.
Weidenschilling et al. (1998) suggested that Jovian mean-motion resonances excite planetesimals.
The formation of Jupiter in the gas disk induces resonances and strongly affects the motion of planetesimals around the asteroid belt ($\sim$ 2~AU--5~AU).
The evolution of such planetesimals under the influence of gas drag has also been studied in detail (Ida and Lin 1996; Marzari et al. 1997; Weidenschilling et al. 1998; Marzari \& Weidenschilling 2002).
The works showed that planetesimals migrate toward the sun due to gas drag even if their radii are on the order of 1000 km.
During the migration from 4~AU to 3~AU, the eccentricities of the planetesimals are excited by multiple Jovian mean-motion resonances.
The excited planetesimals can acquire further eccentricity up to about $e\sim 0.4$ ($\sim 6 {\rm km s^{-1}}$) during the trapping in the 2:1 resonance ($\sim$~3.3AU), provided Jupiter has an eccentricity larger than $e\ga 0.03$.
The excitation of eccentricity increases the gas drag, and the eccentricity and semi-major axis are quickly damped as a result.
The orbits of many planetesimals are circular before they reach 2~AU.
Marzari \& Weidenschilling (2002) showed that the velocity of 100 km--300 km-sized planetesimals relative to the gas disk reaches a maximum of 8 ${\rm km s^{-1}}$ ($e\sim$ 0.6 at 2~AU).
Planetesimals with such high velocities are rare and the period in which these velocities are achieved is limited to on the order of $10^4$ yr.
Simulations of gravitationally interacting planetesimals suggest that the process is not very efficient, i.e., the area swept by chondrule-forming shocks over a period of 1 Myr--2 Myr is just $\la$ 1\% and the planetesimals need to be about half the size of the Moon to accumulate the speed required for chondrule formation
(Hood \& Weidenschilling 2012).
\begin{figure*}\begin{center}
\includegraphics[width=13cm, bb=10 0 500 240]{fig1color.eps}
\caption{Evolution of $a$ versus $e$ for a 300 km planetesimal starting
from $a=4.1$ AU. Left: with disk potential. The location of the secular resonance is indicated by an arrow. Right: without disk potential.
\label{fig:wwodisk}}\end{center}
\end{figure*}
Chondrule formation induced by shock waves requires the relative velocity to be on the order of $\ga 7~{\rm kms^{-1} }$ for a partial melt of submillimeter-sized dusts in a gas disk with a density of $\rho \sim 10^{-9}$ ${\rm g cm^{-3}}$ (e.g., Hood 1998; Iida et al. 2001; Desch \& Connolly 2002).
Although the maximum velocities of planetesimals obtained in the previous simulations of planetesimal evolution in resonances suggest that chondrule formation by bow shocks is likely, the highest speeds obtained ($\la 8 $ ${\rm km s^{-1}}$) are rare and only marginally achieve efficient formation.
Furthermore, in the asteroid belt of the minimum-mass disk
($\rho \sim 10^{-10}$ ${\rm g cm^{-3}}$) where the resonances exist, a larger relative velocity $\ga 10$ ${\rm km s^{-1}}$ would be preferable to ensure complete melting of the 1 mm-sized dust.
If the planetesimals can achieve a relative velocity higher than
$10$ ${\rm km s^{-1}}$ more frequently during orbital migration, the ubiquitous existence of chondrules could be explained more satisfactorily.
In previous works regarding planetesimals in resonances, the gravity of the gas disk and planets other than Jupiter was neglected, i.e., the effect of secular resonances was neglected.
However, as we described in the previous paragraph, the effective shock-heating of chondrules requires a relatively dense gas disk, at least on the order of that of the minimum-mass disk.
Such a gas disk provides not only the drag force but also the gravitational force and causes secular resonance.
The gravitational potential of the disk precesses the Jovian pericenter.
When the precession rate coincides with the precession rate of the planetesimals, a secular resonance arises, which enhances the eccentricities of the planetesimals.
Such a secular resonance occurs between 2~AU and 4~AU in a disk of density
$\sim$ 0.1--5 times that of the minimum-mass disk (e.g., Heppenheimer 1980; Lecar \& Franklin 1997; Nagasawa, Tanaka, \& Ida 2000; Nagasawa, Ida, \& Tanaka 2001, 2002).
Even if the resonance is {\it not sweeping}, its existence causes a high-amplitude oscillation of the eccentricity and further excitation of the relative velocity in the vicinity of the secular resonance.
In chondrule formation induced by planetesimal bow shocks caused by Jovian perturbation, secular resonance inevitably occurs and plays an important role.
Planetesimal bow shocks may also contribute to the origin of the crystalline silicate in comets.
The presence of crystalline silicate in comets has been confirm though infrared observations of dust grains in a number of cases (Bregman et al. 1987; Molster et al. 1999; Hanner \& Bradley 2004).
It is thought that crystalline silicate is formed in the protoplanetary disk because the silicate dust in the interstellar medium is almost entirely amorphous (Li, Zhao, \& Li 2007).
Experimental studies on the thermal annealing of amorphous silicate show that the formation of crystalline silicates requires temperatures above 800~K (Hallenbeck, Nuth, \& Daukantas 1998).
In contrast, the composition of the gas in cometary comae indicates the preservation of interstellar ice in the cold outer nebula (Biermann, Giguere, \& Huebner 1982).
It is unclear why these two materials, which have contradicting heating records, co-exist in comets (see Yamamoto \& Chigai 2005; Tanaka, Yamamoto, \& Kimura 2010; Yamamoto et al. 2010).
It is possible that the amorphous silicates crystallize through shock heating. As shown later, a number of planetesimals are likely scattered by the Jupiter resonances. This mechanism may explain the incorporation of both high- and low-temperature materials in comets.
In this letter, we study the evolution of planetesimals in the gas disk, including the effect of secular resonances caused by the gas disk potential, and determine whether the relative velocity is sufficient for chondrule formation.
In Section \ref{sec:results}, we briefly describe the setup of the numerical simulations and present results.
We show that secular resonance excites most of the planetesimals up to $e \ga 0.6$ ($v_{\rm rel} \ga 12$ $ {\rm km s^{-1}})$.
Finally, we summarize and discuss our results in Section \ref{sec:conclusions}.
\section{NUMERICAL RESULTS}
\label{sec:results}
We investigate the orbital evolution of test particles perturbed by Jupiter and the disk using the {time-symmetric fourth-order Hermite code which has an advantage in the precise long-term calculations of pericenter evolution and detection of close encounters (e.g., Kokubo \& Makino 2004 and references therein). }
The protoplanetary disk provides a background gravitational field that induces secular resonance and gas drag.
We use a thin disk potential for the minimum-mass disk as described by Ward (1981).
The gap in the disk where Jupiter exists is neglected.
We perform simulations both with and without the disk potential and compare the results.
When we include the disk potential, the change in rotational speed of the gas disk due to its own gravity is taken into account.
We assume planetesimals 300~km in size with a material density of $\rho_{\rm mat} =3$ ${\rm g cm^{-3}}$ to calculate the gas drag.
{From the bow shock simulations of chondrule formation, it was shown that larger planetesimals ($\ga$ 1000 km) is required to account for the thermal history, particularly cooling rates (Ciesla, Hood, \& Weidenschilling 2004; Boley, Morris, \& Desch. 2013). It is shown by previous studies of the planetesimal evolution in the gas disk that the larger planetesimals obtain larger eccentricities because of the weakened gas drag. In this letter, we select 300~km planetesimals to show the required relative velocity for the chondrule melting is achieved under the gas drag. We tested other sizes (100~km, 300~km, 500~km, and 1000~km) and confirmed that our conclusion hardly changes.}
\begin{figure}
\includegraphics[width=.16\textwidth, bb=0 0 130 300]{fig2color.eps}
\caption{Evolution of the semi-major axis and eccentricity.
Red and blue lines show the simulation with
and without the disk potential, respectively.
\label{fig:tae}}
\end{figure}
We adopt the gas drag force given by Adachi, Hayashi, \& Nakazawa (1976).
The specific characteristics of the gas drag have little effect on the maximum eccentricity of the planetesimals (Marzari \& Weidenschilling 2002).
We set the drag coefficient, which varies with the Mach number and the Reynolds number (Tanigawa et al. in prep), but its choice is not essential in our simulations.
We use the current size, eccentricity, and semi-major axis of Jupiter, which are $1M_{\rm J}$, $e=0.048$ and $a=5.2$~AU, respectively.
With such parameters, the secular resonance caused by the minimum-mass disk occurs at around 3.2~AU.
We start our simulations putting planetesimals at 4.1~AU, which is just outside the 3:2 resonance region.
{When we start simulations from inside of 3AU, the eccentricity is kept smaller than 0.2 in the case of planetesimals larger than 100 km in size regardless of whether the disk potential is included.
Accordingly, the planetesimals hardly migrate since their migration timescales exceed the life-time of gas disk.}
The trajectories of 20 planetesimals 300~km in size with different initial orbital angles are plotted in Figure \ref{fig:wwodisk}.
The evolutions in which the disk's self-gravity is included are shown in the left panel and the evolutions without the disk potential are shown in the right panel.
The planetesimals migrate inwards due to the gas drag.
As already shown in previous works, the resonances between 3~AU and 4~AU increase the eccentricities of migrating planetesimals.
Since the 3:1 resonance at $\sim$ 2.5~AU is separated from the other resonances, the eccentricity is damped before the 3:1 resonance is reached as shown in the right panel.
On the other hand, in the left panel, the increase in eccentricity continues until the 3:1 resonance due to the extra excitation caused by the secular resonance.
The typical velocity of the planetesimals relative to the gas disk can be estimated using $v_{\rm rel} \sim e V_{\rm Kep}$, where $V_{\rm Kep}$ is the Kepler velocity at that semimajor axis $a$ and $e$ is its eccentricity (Adachi et al. 1976).
{ Note that $e V_{\rm Kep}$ is the redial velocity at the location of $r=a$ (where $r$ is the distance from the star).
In the case of low eccentricity, the relative velocity $e V_{\rm Kep}$ corresponds to the maximum value and the smallest relative velocity (1/2 $e V_{\rm Kep}$) occurs at its pericenter and apocenter.
In high eccentricity cases ($e \ga 0.5$), the actual maximum relative velocity is achieved between $r=a$ location and the pericenter and its magnitude is enlarged from $e V_{\rm Kep}$.}
In Figure \ref{fig:wwodisk}, the relative velocity ($e V_{\rm Kep}$) is shown by dotted gray lines. When the disk potential is not considered, the maximum speed rarely exceeds 10~${\rm km s^{-1}}$; when it is considered, however, the maximum speed exceeds 10~${\rm km s^{-1}}$ for all planetesimals.
Figure \ref{fig:tae} shows typical evolutions of the semi-major axes and eccentricities for two cases: including the disk potential (red lines) and excluding the disk potential (blue lines).
The planetesimals start migration when their eccentricities are pumped up by Jupiter.
Since the initial location of the planetesimals is near the chaotic resonance-overlapping region, the period for which the planetesimals remain near 4.1~AU depends on cases.
When they migrate to the 2:1 mean-motion resonance ($\sim 3.3$~AU), they become trapped in it.
In the case of Figure \ref{fig:tae}, the oscillations during 2.35 Myr--2.4 Myr (blue line) and 2.6~Myr--2.8~Myr (red line) correspond to such trapping.
In our 20 simulations, the time trapped in the resonance tends to be longer when the disk potential is considered.
When their eccentricities are excited to $e \sim 0.4$, the planetesimals become detached from the resonance as the discussed in previous papers (e.g., Marzari \& Weidenschilling 2002).
If there is no secular resonance between 2~AU and 3~AU, planetesimals continue rapid migration due to the gas drag until their orbits become circular.
If secular resonance is taken into account, however, the migration is again halted temporarily at the location of the resonance ($\sim$2.8~Myr).
The eccentricities are excited further, but with such high eccentricities, the gas drag drives away the planetesimals from the secular resonance at around $e \sim 0.7$.
The 3:1 resonance ($a=2.5$~AU) does not play as important a role as that of the 2:1 resonance or the 3:2 resonance, but with high eccentricity, it can further increase the eccentricity by a small amount ($\Delta e \la 0.1$).
The region where the maximum speed tends to be recorded is the location of the 3:1 resonance.
\begin{figure}
\includegraphics[width=.16\textwidth, bb=0 0 100 220]{fig3color.eps}
\caption{Histogram of maximum eccentricity. Red and blue lines
show the results for 20 simulations with and
without the disk potential, respectively.
Panel {\it a}: the maximum eccentricity for each planetesimal within 4.5 AU.
Panel {\it b}: the planetesimals that experienced a Jovian encounter are omitted.
\label{fig:hist}}
\end{figure}
Figure \ref{fig:hist} is a histogram of the achieved eccentricities ($e_{\rm max}$) of each planetesimal during the evolution.
The distributions obtained from the simulations with and without the disk potential are shown by red and blue lines, respectively.
Panel~{\it a} contains all 20 simulations for each case.
Seventeen planetesimals out of the 40 exhibit close encounters with Jupiter and enter the $a>4.5$ AU region at least one time.
These are omitted from Panel~{\it b}.
Since the scattered planetesimals obtain $e\ga 1$ values, we count $e_{\rm max}$ when the planetesimals are within 4.5 AU.
The groups of lower eccentricities in the bimodal distributions in Panel~{\it a} tend to correspond to the planetesimals that encountered Jupiter.
The planetesimals remaining in the asteroid region reach high velocities due to the secular resonance, not due to the Jovian encounters.
The figure reveals that the typical maximum eccentricity with secular resonance is $\Delta e \sim 0.15$ higher than that obtained when only mean-motion resonances are considered.
The evolution toward the 2:1 resonance basically follows a line of apocentral distance $a(1+e) \sim 4.5$ AU.
When the eccentricity excitation stops near the 2:1 resonance, $e_{\rm max}$ is $\sim 0.4$; the peak near $e_{\rm max} \sim 0.4$ (blue distribution in Figure \ref{fig:hist}) originates from this fact.
About 1/3 of the planetesimals drop out of the 2:1 resonance in the case without the disk potential.
On the other hand, in the case with the disk potential, all planetesimals that are not scattered by Jupiter continue on their trajectory until reaching the 3:1 resonance.
The peak at $e_{\rm max} \sim 0.7$ corresponds to this state.
At the location of the 3:1 resonance, $V_{\rm Kep} \sim 30 \ {\rm km s^{-1}}
(a {\rm /AU})^{-1/2} \sim 20~ {\rm km s^{-1}}$.
Thus, the typical maximum of the relative velocity is approximately given by
$20 \times e \ {\rm km s^{-1}}$.
The difference between the two cases is
$\sim 3$ ${\rm km s^{-1}}$ ($\Delta e \sim 0.15$).
{ Unlike the eccentricities, inclinations ($i$) remain lower than $10^{\circ}$ in most cases.
It is because a strong secular resonance of Jupiter which excites the inclination ($\nu_{15}$) hardly occurs in the gas disk within Jovian orbit (Nagasawa et al. 2000, 2001, 2002).
Although planetesimals with $i=10^{\circ}$ go out of the gas disk, they stay longer than 1/5 of their orbital period within one scale height of the disk, where the gas density is comparable to that at the disk midplane.
}
The planetesimals beyond the $a(1+e)\sim 4.5$ AU line enter into a region of resonance-overlapping and they are scattered by close encounters with Jupiter.
About half of the 300~km-sized planetesimals are scattered in both cases.
This fraction would be lower for smaller planetesimals due to the stronger gas drag.
One out of 15 scattered planetesimals returns to the region of $a<4.5$~AU, but the majority eventually reach $e \ga 1$ and leave orbit (Fig.~\ref{fig:scatter}).
\begin{figure}\epsscale{0.8}
\plotone{fig4color.eps}
\caption{Wide range $e$-$a$ diagram.
\label{fig:scatter}}
\end{figure}
\section{DISCUSSION AND CONCLUSIONS}
\label{sec:conclusions}
We studied the evolution of planetesimals in the gas disk, including the effect of secular resonance caused by the gas-disk potential.
We found that the planetesimals attain $e\sim 0.6$ with the help of this secular resonance.
The relative velocity of the planetesimals exceeds 12~${\rm km s^{-1}}$ around the 3:1 mean-motion resonance.
The high-eccentricity region is restricted to a relatively narrow band of 1.5 AU--4 AU.
In previous studies on the planetesimal evolution in the gas disk, the maximum velocity was found to be $\sim 8$ ${\rm km s^{-1}}$ ($e \sim 0.6$ at 2~AU) and such supersonic speeds were not frequent events (Marzari \& Weidenschilling 2002).
In our simulations with secular resonance, however, about the half of planetesimals reached $e \sim 0.6$.
Our results are supportive of the possibility of chondrule formation induced by planetesimal shock waves due to Jovian resonance.
The minimum relative speed required for melting 1 mm-sized dust at 1 AU is considered to be $\sim 7$ ${\rm kms^{-1}}$ in the minimum-mass disk (e.g., Hood 1998; Iida et al. 2001; Desch \& Connolly 2002).
The typical $e_{\rm max}$ of $\sim 0.65$ around 3~AU to 2~AU corresponds to a velocity of $11~{\rm km s^{-1}}$--$14~{\rm km s^{-1}}$ relative to chondrule precursors rotating with the disk.
The density of this region is $2 \times 10^{-10} {\rm g cm^{-3}}$--$7 \times 10^ {-11} {\rm g cm^{-3}}$.
With this disk gas density, the formation of 0.1 mm chondrules by shock waves requires $\sim 10$ ${\rm km s^{-1}}$--18 ${\rm km s^{-1}}$ (Iida et al. 2001), while $\sim 7 $ ${\rm km s^{-1}}$ is sufficient in 10 times denser regions.
If the speed exceeds $20$ ${\rm km s^{-1}}$, even a 1 cm precursor evaporates, but such speeds are not realized.
Note that the highly supersonic situation ($\ga$ 12$ {\rm km s^{-1}}$) is restricted to the 1~AU--3~AU region within the stable region of $a \ga 4.5 $AU.
That would suggest that the chemical or taxonomic evolution of planetesimals may depend on the semi-major axis via the gas-disk density and the maximum heating caused by bow shocks.
Our results relate to the origin of crystalline silicates observed in a number of comets.
In our simulations, about half of the planetesimals attain $e \ga 1$ and move toward the outer region of the disk due to Jovian resonances.
It was shown that such eccentric icy planetesimals with core-mantle structures are changed to rocky planetesimals due to the efficient evaporation of the icy mantle caused by shock heating, even in the region outside the snow line (Tanaka et al. 2013).
As long as the dry planetesimals are transported in the cometary region, there is no conflict with the fact that cold interstellar ice is preserved in the comae.
On the way to the outer region, the planetesimals are expected to accrete the silicate dust processed by the shock waves.
This process can explain the presence of crystalline silicates in comets, if the scattered planetesimals containing crystalline silicates become mixed with icy outer planetesimals.
In this letter, we considered 300 km-sized planetesimals in the minimum-mass gas disk.
If we considered smaller planetesimals ($\la 100$ km), the maximum velocities would have been smaller as a result of stronger gas drag.
On the other hand, the maximum velocity did not differ greatly to that for larger planetesimals ($\ga 100$ km); however, the ejection frequency would be enhanced due to weakened gas drag in such a case.
The maximum speed of planetesimals depends on their mass, the eccentricity, and the semi-major axis of Jupiter through the strength of the secular resonance.
The effect of the secular resonance starts to come into play when Jupiter exceeds $\sim 1/3$ of its current mass, and the effect continues until the disk is dissipated ($\sim$ 90\%).
Surveys changing these parameters will be conducted in subsequent work.
\acknowledgments
We are thankful for a careful and helpful review by an anonymous referee.
MN was supported in part by JSPS KAKENHI Grant Number 25610133.
KKT was supported in part by JSPS KAKENHI Grant Number 26108503 and 2540054.
This study was partly supported by the Grant for Joint Research Program of the Institute of Low Temperature Science, Hokkaido University.
|
1,108,101,566,812 | arxiv | \section{Introduction}
Turbulence is ubiquitous in natural phenomena and engineering
applications \cite{pope2001turbulent}. The understanding and
prediction of multiscale turbulent flow is one of the most difficult
problems for both mathematics and physical sciences. Direct
numerical simulation (DNS) solves the Navier-Stokes equations
directly, resolve all scales of the turbulent motion, and eliminate
modeling entirely \cite{kim1987turbulence}. With the advances of
high-order numerical methods and supercomputers, great success has
been achieved for the incompressible and compressible turbulent
flow, such as DNS in incompressible isotropic turbulence
\cite{chen1993far}, incompressible turbulent channel flow
\cite{kim1999turbulence, lee2015direct}, supersonic isotropic
turbulence \cite{wang2010hybrid,GKS-high-1}, compressible turbulent
channel flows \cite{coleman1995numerical, DNS-Li, yu2019genuine},
and compressible flat plate turbulence from the supersonic to
hypersonic regime \cite{pirozzoli2004direct,lixinliangma8}.
In the past decades, the gas-kinetic scheme (GKS) has been developed
systematically based on the Bhatnagar-Gross-Krook (BGK) model
\cite{BGK-1,BGK-2} under the finite volume framework, and applied
successfully in the computations from low speed flow to hypersonic
one \cite{GKS-Xu1,GKS-Xu2}. The gas-kinetic scheme presents a gas
evolution process from kinetic scale to hydrodynamic scale, where
both inviscid and viscous fluxes are recovered from a time-dependent
and genuinely multi-dimensional gas distribution function at a cell
interface. Starting from a time-dependent flux function, based on
the two-stage fourth-order formulation \cite{GRP-high-1,GRP-high-2},
the high-order gas-kinetic scheme (HGKS) has been constructed and
applied for the compressible flow simulation
\cite{GKS-high-1,GKS-high-2,GKS-high-3,GKS-high-4}. The high-order
can be achieved with the implementation of the traditional
second-order GKS flux solver. More importantly, the high-order GKS
is as robust as the second-order scheme and works perfectly from the
subsonic to hypersonic viscous heat conducting flows. Originally,
the parallel HGKS code was developed with central processing unit
(CPU) using open multi-processing (OpenMP) directives. However, due
to the limited shared memory, the computational scale is constrained
for numerical simulation of turbulence. To perform the large-scale
DNS, the domain decomposition and the message passing interface
(MPI) \cite{MPI-1} are used for parallel implementation
\cite{GKS-high-DNS}. Due to the explicit formulation of HGKS, the
CPU code with MPI scales properly with the number of processors
used. The numerical results demonstrates the capability of HGKS as a
powerful DNS tool from the low speed to supersonic turbulence study
\cite{GKS-high-DNS}.
Graphical processing unit (GPU) is a form of hardware acceleration,
which is originally developed for graphics manipulation and is
extremely efficient at processing large amounts of data in parallel.
Since these units have a parallel computation capability inherently,
they can provide fast and low cost solutions to high performance
computing (HPC). In recent years, GPUs have gained significant
popularity in computational fluid dynamics \cite{GPU-1,GPU-2,GPU-3}
as a cheaper, more efficient, and more accessible alternative to
large-scale HPC systems with CPUs. Especially, to numerically study
the turbulent physics in much higher Reynolds number turbulence, the
extreme-scale DNS in turbulent flows has been implemented using
multiple GPUs, i.e., incompressible isotropic turbulence up to
$18432^3$ resolution \cite{yeung2020advancing} and turbulent pipe
flow up to $Re_{\tau} \approx 6000$ \cite{pirozzoli2021one}.
Recently, the three-dimension discontinuous Galerkin based HGKS has
been implemented in single-GPU computation using compute unified
device architecture (CUDA) \cite{GKS-GPU}. Obtained results are
compared with those obtained by Intel i7-9700 CPU using OpenMP
directives. The GPU code achieves 6x-7x speedup with TITAN RTX, and
10x-11x speedup with Tesla V100. The numerical results confirm the
potential of HGKS for large-scale DNS in turbulence.
A major limitation in single-GPU computation is its available
memory, which leads to a bottleneck in the maximum number of
computational mesh. In this paper, to implement much larger scale
DNS and accelerate the efficiency, HGKS is implemented with multiple
GPUs using CUDA and MPI architecture (MPI + CUDA). It is not
straightforward for the multiple-GPU programming, since the memory
is not shared across the GPUs and their tasks need to be coordinated
appropriately. The multiple GPUs are distributed across multiple
CPUs at the cost of having to coordinate GPU-GPU communication via
MPI. For WENO-based HGKS in single-GPU using CUDA, compared with the
CPU code using Intel Core i7-9700, 7x speedup is achieved for TITAN
RTX and 16x speedup is achieved for Tesla V100. In terms of the
computation with multiple GPUs, the HGKS code scales properly with
the increasing number of GPU. Numerical performance shows that the
data communication crossing GPU through MPI costs the relative
little time, while the computation time for flow field is the
dominant one in the HGKS code. For the MPI parallel computation, the
computational time of CPU code with supercomputer using 1024 Intel
Xeon E5-2692 cores is approximately $3$ times longer than that of
GPU code using $8$ Tesla V100 GPUs. It can be inferred that the
efficiency of GPU code with $8$ Tesla V100 GPUs approximately equals
to that of MPI code using $3000$ CPU cores in supercomputer. To
reduce the loading and writing pressure of GPU memory, the benefits
can be achieved by using FP32 (single) precision compared with FP64
(double) precision for memory-intensive computing
\cite{lehmann2021accuracy, haidar2020mixed}. Thence, HGKS in GPUs is
compiled with both FP32 precision and FP64 precision to evaluate the
effect of precision on DNS of compressible turbulence. As expect,
the efficiency can be improved and the memory cost can be reduced
with FP32 precision. However, the difference in accuracy between
FP32 and FP64 precision appears for the instantaneous statistical quantities of
turbulent channel flows, which is not negligible for long time simulations.
This paper is organized as follows. In Section 2, the high-order
gas-kinetic scheme is briefly reviewed. The GPU architecture and
code design are introduced in Section 3. Section 4 includes
numerical simulation and discussions. The last section is the
conclusion.
\section{High-order gas-kinetic scheme}
The three-dimensional BGK equation \cite{BGK-1,BGK-2} can be
written as
\begin{equation}\label{bgk}
f_t+uf_x+vf_y+wf_z=\frac{g-f}{\tau},
\end{equation}
where $\boldsymbol{u}=(u,v,w)^T$ is the particle velocity, $f$ is the
gas distribution function, $g$ is the three-dimensional Maxwellian
distribution and $\tau$ is the collision time. The collision term
satisfies the compatibility condition
\begin{equation} \label{compatibility}
\int \frac{g-f}{\tau}\psi \text{d}\Xi=0,
\end{equation}
where
$\displaystyle\psi=(1,u,v,w,\frac{1}{2}(u^2+v^2+w^2+\xi^2))^T$,
$\xi^2=\xi_1^2+...+\xi_N^2$,
$\text{d}\Xi=\text{d}u\text{d}v\text{d}w\text{d}\xi_1,...,\text{d}\xi_{N}$,
$N=(5-3\gamma)/(\gamma-1)$ is the internal degree of freedom,
$\gamma$ is the specific heat ratio and $\gamma=1.4$ is used in the
computation.
In this section, the finite volume scheme on orthogonal structured
mesh is provided as example. Taking moments of the BGK equation
Eq.\eqref{bgk} and integrating with respect to the cell
$\Omega_{ijk}$, the finite volume scheme can be expressed as
\begin{align}\label{semi}
\frac{\text{d}(Q_{ijk})}{\text{d}t}=\mathcal{L}(Q_{ijk}),
\end{align}
where $Q_{ijk}$ is the cell averaged conservative variable over
$\Omega_{ijk}$, and the operator $\mathcal{L}$ reads
\begin{equation}\label{semi2}
\mathcal{L}(Q_{ijk})=-\frac{1}{|\Omega_{ijk}|}\sum_{p=1}^6\mathbb{F}_{p}(t).
\end{equation}
where $\Omega_{ijk}$ is defined as
$\Omega_{ijk}=\overline{x}_i\times\overline{y}_j\times
\overline{z}_k$ with $\overline{x}_i=[x_i-\Delta x/2,x_i+\Delta
x/2], \overline{y}_j=[y_j-\Delta y/2,y_j+\Delta y/2],
\overline{z}_k=[z_k-\Delta z/2,z_k+\Delta z/2]$, and
$\mathbb{F}_{p}(t)$ is the numerical flux across the cell interface
$\Sigma_{p}$. The numerical flux in $x$-direction is given as
example
\begin{align*}
\mathbb{F}_{p}(t)=\iint_{\Sigma_{p}}
F(Q)\cdot\boldsymbol{n}\text{d}\sigma=\sum_{m,n=1}^2\omega_{mn}
\int\psi u
f(\boldsymbol{x}_{i+1/2,j_m,k_n},t,\boldsymbol{u},\xi)\text{d}\Xi\Delta y\Delta z,
\end{align*}
where $\boldsymbol{n}$ is the outer normal direction. The Gaussian
quadrature is used over the cell interface, where $\omega_{mn}$ is
the quadrature weight,
$\boldsymbol{x}_{i+1/2,m,n}=(x_{i+1/2},y_{j_m},z_{k_n})$ and
$(y_{j_m},z_{k_n})$ is the Gauss quadrature point of cell interface
$\overline{y}_j\times\overline{z}_k$. Based on the integral solution
of BGK equation Eq.\eqref{bgk}, the gas distribution function
$f(\boldsymbol{x}_{i+1/2,j_m,k_n},t,\boldsymbol{u},\xi)$ in the
local coordinate can be given by
\begin{equation}
f(\boldsymbol{x}_{i+1/2,j_m,k_n},t,\boldsymbol{u},\xi)=\frac{1}{\tau}\int_0^t
g(\boldsymbol{x}',t',\boldsymbol{u}, \xi)e^{-(t-t')/\tau}\text{d}t'+e^{-t/\tau}f_0(-\boldsymbol{u}t,\xi),
\end{equation}
where
$\boldsymbol{x}'=\boldsymbol{x}_{i+1/2,j_m,k_n}-\boldsymbol{u}(t-t')$
is the trajectory of particles, $f_0$ is the initial gas
distribution function, and $g$ is the corresponding equilibrium
state. With the first order spatial derivatives, the second-order
gas distribution function at cell interface can be expressed as
\begin{align}\label{flux}
f(\boldsymbol{x}_{i+1/2,j_m,k_n},t,\boldsymbol{u},\xi)=&(1-e^{-t/\tau})g_0+
((t+\tau)e^{-t/\tau}-\tau)(\overline{a}_1u+\overline{a}_2v+\overline{a}_3w)g_0\nonumber\\
+&(t-\tau+\tau e^{-t/\tau}){\bar{A}} g_0\nonumber\\
+&e^{-t/\tau}g_r[1-(\tau+t)(a_{1}^{r}u+a_{2}^{r}v+a_{3}^{r}w)-\tau A^r)](1-H(u))\nonumber\\
+&e^{-t/\tau}g_l[1-(\tau+t)(a_{1}^{l}u+a_{2}^{l}v+a_{3}^{l}w)-\tau A^l)]H(u),
\end{align}
where the equilibrium state $g_{0}$ and the corresponding
conservative variables $Q_{0}$ can be determined by the
compatibility condition
\begin{align*}
\int\psi g_{0}\text{d}\Xi=Q_0=\int_{u>0}\psi
g_{l}\text{d}\Xi+\int_{u<0}\psi g_{r}\text{d}\Xi.
\end{align*}
The following numerical tests on the compressible turbulent flows
without discontinuities will be presented, thus the collision time
for the flow without discontinuities takes
\begin{align*}
\tau=\frac{\mu}{p},
\end{align*}
where $\mu$ is viscous coefficient and $p$ is the pressure at cell
interface determined by $Q_0$. With the reconstruction of
macroscopic variables, the coefficients in Eq.\eqref{flux} can be
fully determined by the reconstructed derivatives and compatibility
condition
\begin{equation*}
\begin{aligned}
\displaystyle \langle a_{1}^{k}\rangle=\frac{\partial
Q_{k}}{\partial x}, \langle
a_{2}^{k}\rangle=\frac{\partial Q_{k}}{\partial y},
\langle a_{3}^{k}\rangle&=\frac{\partial Q_{k}}{\partial
z}, \langle
a_{1}^{k}u+a_{2}^{k}v+a_{3}^{k}w+A^{k}\rangle=0,\\ \displaystyle
\langle\overline{a}_1\rangle=\frac{\partial Q_{0}}{\partial
x}, \langle\overline{a}_2\rangle=\frac{\partial
Q_{0}}{\partial y},
\langle\overline{a}_3\rangle&=\frac{\partial Q_{0}}{\partial
z},
\langle\overline{a}_1u+\overline{a}_2v+\overline{a}_3w+\overline{A}\rangle=0,
\end{aligned}
\end{equation*}
where $k=l,r$ and $\langle...\rangle$ are the moments of the
equilibrium $g$ and defined by
\begin{align*}
\langle...\rangle=\int g (...)\psi \text{d}\Xi.
\end{align*}
More details of the gas-kinetic scheme can be found in the
literature \cite{GKS-Xu1,GKS-Xu2}. Thus, the numerical flux can be
obtained by taking moments of the gas distribution function
Eq.\eqref{flux}, and the semi-discretized finite volume scheme
Eq.\eqref{semi} can be fully given.
Recently, based on the time-dependent flux function of the
generalized Riemann problem solver (GRP)
\cite{GRP-high-1,GRP-high-2} and gas-kinetic scheme
\cite{GKS-high-2,GKS-high-3,GKS-high-4}, a two-stage fourth-order
time-accurate discretization was recently developed for Lax-Wendroff
type flow solvers. A reliable framework was provided to construct
higher-order gas-kinetic scheme, and the high-order scheme is as
robust as the second-order one and works perfectly from the subsonic
to hypersonic flows. In this study, the two-stage method is used for
temporal accuracy. Consider the following time dependent equation
\begin{align*}
\frac{\partial Q}{\partial t}=\mathcal {L}(Q),
\end{align*}
with the initial condition at $t_n$, i.e., $Q(t=t_n)=Q^n$,
where $\mathcal {L}$ is an operator for spatial derivative of flux,
the state $Q^{n+1}$ at $t_{n+1}=t_n+\Delta t$ can be updated with
the following formula
\begin{equation}\label{two-stage}
\begin{split}
&Q^*=Q^n+\frac{1}{2}\Delta t\mathcal {L}(Q^n)+\frac{1}{8}\Delta
t^2\partial_t\mathcal{L}(Q^n), \\
Q^{n+1}=&Q^n+\Delta t\mathcal {L}(Q^n)+\frac{1}{6}\Delta
t^2\big(\partial_t\mathcal{L}(Q^n)+2\partial_t\mathcal{L}(Q^*)\big).
\end{split}
\end{equation}
It can be proved that for hyperbolic equations the above temporal
discretization Eq.\eqref{two-stage} provides a fourth-order time accurate solution for
$Q^{n+1}$. To implement two-stage fourth-order method for
Eq.\eqref{semi}, a linear function is used to approximate the time
dependent numerical flux
\begin{align}\label{expansion-1}
\mathbb{F}_{p}(t)\approx\mathbb{F}_{p}^n+ \partial_t
\mathbb{F}_{p}^n(t-t_n).
\end{align}
Integrating Eq.\eqref{expansion-1} over $[t_n, t_n+\Delta t/2]$ and
$[t_n, t_n+\Delta t]$, we have the following two equations
\begin{equation*}
\begin{aligned}
\mathbb{F}_{p}^n\Delta t&+\frac{1}{2}\partial_t
\mathbb{F}_{p}^n\Delta t^2 =\int_{t_n}^{t_n+\Delta t}\mathbb{F}_{p}(t)\text{d}t, \\
\frac{1}{2}\mathbb{F}_{p}^n\Delta t&+\frac{1}{8}\partial_t
\mathbb{F}_{p}^n\Delta t^2 =\int_{t_n}^{t_n+\Delta t/2}\mathbb{F}_{p}(t)\text{d}t.
\end{aligned}
\end{equation*}
The coefficients $\mathbb{F}_{p}^n$ and $\partial_t
\mathbb{F}_{p}^n$ at the initial stage can be determined by solving
the linear system. According to Eq.\eqref{semi2}, $\mathcal
{L}(Q_{i}^n)$ and the temporal derivative $\partial_t\mathcal
{L}(Q_{i}^n)$ at $t^n$ can be constructed by
\begin{align*}
\mathcal{L}(Q_{i}^n)&=-\frac{1}{|\Omega_{i}|}\sum_{p=1}^6\mathbb{F}_{p}^n,~~
\partial_t\mathcal{L}(Q_{i}^n)=-\frac{1}{|\Omega_{i}|}\sum_{p=1}^6\partial_t\mathbb{F}_{p}^n.
\end{align*}
The flow variables $Q^*$ at the intermediate stage can be updated.
Similarly, $\mathcal{L}(Q_{i}^*)$, $\partial_t\mathcal {L}(Q_{i}^*)$
at the intermediate state can be constructed and $Q^{n+1}$ can be
updated as well. For the high-order spatial accuracy, the
fifth-order WENO method \cite{WENO-JS, WENO-Z} is adopted. For the
three-dimensional computation, the dimension-by-dimension
reconstruction is used for HGKS \cite{GKS-high-3}.
\section{HGKS code design on GPU}
CPUs and GPUs are equipped with different architectures and are
built for different purposes. The CPU is suited to general
workloads, especially those for which per-core performance are
important. CPU is designed to execute complex logical tasks quickly,
but is limited in the number of threads. Meanwhile, GPU is a form of
hardware acceleration, which is originally developed for graphics
manipulation and is extremely efficient at processing large amounts
of data in parallel. Currently, GPU has gained significant
popularity in high performance scientific computing
\cite{GPU-1,GPU-2,GPU-3}. In this paper, to accelerate the
computation, the WENO based HGKS will be implemented with single GPU
using CUDA. To conduct the large-scale DNS in turbulence
efficiently, the HGKS also be further upgraded with multiple GPUs using MPI + CUDA.
\subsection{Single-GPU accelerated HGKS}
The CPU is regarded as host, and GPU is treated as device.
Data-parallel, compute-intensive operations running on the host are
transferred to device by using kernels, and kernels are executed on
the device by many different threads. For CUDA, these threads are
organized into thread blocks, and thread blocks constitute a grid.
Such computational structures build connection with Nvidia GPU
hardware architecture. The Nvidia GPU consists of multiple streaming
multiprocessors (SMs), and each SM contains streaming processors
(SPs). When invoking a kernel, the blocks of grid are distributed to
SMs, and the threads of each block are executed by SPs. In summary,
the correspondence between software CUDA and hardware Nvidia GPU is
simply shown in Fig.\ref{cuda-gpu}. In the following, the single-GPU
accelerated HGKS will be introduced briefly.
\begin{figure}[!h]
\centering
\includegraphics[width=0.85\textwidth]{schematic-cuda-gpu}
\caption{\label{cuda-gpu} Correspondence between software CUDA and hardware GPU.}
\end{figure}
\begin{algorithm}[!h]
\setstretch{1.25}
\begin{algorithmic}
\STATE
\textbf{\color{blue} Initialization}
\WHILE {TIME $ \leq $ TSTOP}
\STATE dimGrid $=$ dim3($N_z$, $N_x/\text{block}_x$, 1)
\STATE dimBlock $=$ dim3($\text{block}_x$,$N_y$,1)
\STATE \textbf{STEP 1} : \textbf{\color{blue} Calculation of time step}
\STATE \textbf{CALL} GETTIMESTEP$<<<$dimGrid,dimBlock$>>>$
\STATE istat=cudaDeviceSynchronize()
\STATE \textbf{STEP 2} : \textbf{\color{blue} WENO reconstruction}
\STATE \textbf{CALL} WENO-x$<<<$dimGrid,dimBlock$>>>$
\STATE istat=cudaDeviceSynchronize()
\STATE \textbf{STEP 3} : \textbf{\color{blue} Computation of flux}
\STATE \textbf{CALL} FLUX-x$<<<$dimGrid,dimBlock$>>>$ \\
\STATE istat=cudaDeviceSynchronize() \\
\quad \textcolor{SpringGreen4}{\% kernel for WENO reconstruction and flux calculation in $x$ direction as example.} \\
\quad \textcolor{SpringGreen4}{\% reconstruction and flux calculation in $y$ and $z$ directions can be also implemented.}
\STATE \textbf{STEP 4} : \textbf{\color{blue} Update of flow variables}
\ENDWHILE \\
\end{algorithmic}
\caption{\label{GPU-algorithm} Single-GPU HGKS code using CUDA}
\end{algorithm}
\begin{figure}[!htp]
\centering
\includegraphics[width=0.55\textwidth]{schematic-thread-element}
\caption{\label{thread-3d} Specifications of grid for the three-dimensional structured meshes.}
\end{figure}
Device implementation is handled automatically by the CUDA, while
the kernels and grids need to be specified for HGKS. One updating
step of single-GPU accelerated HGKS code can be divided into several
kernels, namely, initialization, calculation of time step, WENO
reconstruction, flux computation at cell interface in $x$, $y$, $z$,
and the update of flow variables. The main parts of the single-GPU
accelerated HGKS code are labeled in blue as
Algorithm.\ref{GPU-algorithm}. Due to the identical processes of
calculation for each kernel, it is natural to use one thread for one
computational cell when setting grids. Due to the limited number of
threads can be obtained in one block (i.e., the maximum $1024$ threads in one block), the whole computational
domain is required to be divided into several parts. Assume that the total number of cells for
computational mesh is $N_x\times N_y\times N_z$, which is
illustrated in Fig.\ref{thread-3d}. The computational domain is
divided into $N_x/\text{block}_x$ parts in $x$-direction, where the choice of $\text{block}_x$ for each kernel is an integer defined according to the requirement and experience. If $N_x$ is not divisible by $\text{block}_x$, an extra block is needed. When
setting grid, the variables "dimGrid" and "dimBlock" are defined as
\begin{align*}
{\rm dimGrid}&={\rm dim3}(N_z, N_x/\text{block}_x, 1), \\
{\rm dimBlock}&={\rm dim3}(\text{block}_x,N_y,1).
\end{align*}
As shown in Fig.\ref{thread-3d}, one thread is assigned to complete
the computations of a cell $\Omega_{ijk}$, and the one-to-one
correspondence of thread index $\text{threadidx}$, block index $\text{blockidx}$ and cell index $(i,j,k)$ is given by
\begin{align*}
i&=\text{threadidx}\%\text{x}+\text{block}_x*(\text{blockidx}\%\text{y}-1),\\
j&=\text{threadidx}\%\text{y}, \\
k&= \text{blockidx}\%\text{x}.
\end{align*}
Thus, the single-GPU accelerated HGKS code can be implemented after
specifying kernels and grids.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\textwidth]{schematic-mpi-com}
\caption{\label{total-separate} The domain decomposition and GPU-GPU communications for multiple-GPU implementation.}
\end{figure}
\subsection{Multiple-GPU accelerated HGKS}
Due to the limited computational resources of single-GPU, it is
natural to develop the multiple-GPU accelerated HGKS for large-scale
DNS of turbulence. The larger computational scale can be achieved
with the increased available device memories. It is not
straightforward for the programming with multiple-GPUs, because the
device memories are not shared across GPUs and the tasks need to be
coordinated appropriately. The computation with multiple GPUs is
implemented using multiple CPUs and multiple GPUs, where the GPU-GPU
communication is coordinated via the MPI. CUDA-aware MPI library is
chosen \cite{CUDA-MPI}, where GPU data can be directly passed by MPI
function and transferred in the efficient way automatically.
\begin{algorithm}[!h]
\setstretch{1.25}
\begin{algorithmic}
\IF{($i =0$ ~or ~$i = N-1$)}
\STATE \textbf{\color{blue} Boundary conditions} for $P_0$ and $P_{N-1}$\\
\quad \textcolor{SpringGreen4}{\% for wall boundary, boundary condition can be given directly.} \\
\quad \textcolor{SpringGreen4}{\% for periodic boundary, \textbf{\color{blue}MPI$\_$SEND} and \textbf{\color{blue}MPI$\_$RECV} are also needed.}
\ENDIF \\
\IF{($i =$ even number)}
\STATE \textbf{\color{blue}MPI$\_$SEND} data from $P_{i}$ to $P_{i-1}$,
\STATE \textbf{\color{blue}MPI$\_$RECV} data from $P_{i-1}$ to $P_{i}$.
\ELSE
\STATE \textbf{\color{blue}MPI$\_$RECV} data from $P_{i+1}$ to $P_{i}$,
\STATE \textbf{\color{blue}MPI$\_$SEND} data from $P_{i}$ to $P_{i+1}$
\ENDIF
\IF{($i =$ odd number)}
\STATE \textbf{\color{blue}MPI$\_$SEND} data from $P_{i}$ to $P_{i-1}$,
\STATE \textbf{\color{blue}MPI$\_$RECV} data from $P_{i-1}$ to $P_{i}$.
\ELSE
\STATE \textbf{\color{blue}MPI$\_$RECV} data from $P_{i+1}$ to $P_{i}$,
\STATE \textbf{\color{blue}MPI$\_$SEND} data from $P_{i}$ to $P_{i+1}$
\ENDIF
\end{algorithmic}
\caption{\label{Boundary-algorithm} Data communication for MPI+CUDA}
\end{algorithm}
\begin{figure}[htp]
\centering
\includegraphics[width=0.9\textwidth]{schematic-code-frame-2}
\caption{\label{code-frame} Multiple-GPU accelerated HGKS code using MPI+CUDA.}
\end{figure}
For the parallel computation with MPI, the domain decomposition are
required for both CPU and GPU, where both one-dimensional (slab) and
two-dimensional (pencil) decomposition are allowed. For CPU
computation with MPI, hundreds or thousands of CPU cores are usually
used in the computation, and the pencil decomposition can be used to
improve the efficiency of data communications \cite{GKS-high-DNS}. In terms of GPU
computation with MPI, the slab decomposition is used, which reduces
the frequency of GPU-GPU communications and improves performances.
The domain decomposition and GPU-GPU communications are presented in
Fig.\ref{total-separate}, where the computational domain is divided
into $N$ parts in $x$-direction and $N$ GPUs are used. For
simplicity, $P_i$ denotes the $i$-th MPI process, which deals with
the tasks of $i$-th decomposed computational domain in the $i$-th
GPU. To implement the WENO reconstruction, the processor $P_i$
exchanges the data with $P_{i-1}$ and $P_{i+1}$ using MPI$\_$RECV
and MPI$\_$SEND. The wall boundary condition of $P_0$ and $P_{N-1}$
can be implemented directly. For the periodic boundary, the data is
exchanged between $P_0$ and $P_{N-1}$ using MPI$\_$RECV and
MPI$\_$SEND as well. In summary, the detailed data communication for
GPU computation with MPI are shown in
Algorithm.\ref{Boundary-algorithm}. Fig.\ref{code-frame} shows the
HGKS code frame with multiple-GPUs. The initial data will be divided
into $N$ parts according to domain decomposition and sent to
corresponding process $P_1$ to $P_{N-1}$ using MPI$\_$SEND. As for
input and output, the MPI process $P_0$ is responsible for
distributing data to other processors and collecting data from other
processors, using MPI$\_$SEND and MPI$\_$RECV. For each decomposed
domain, the computation is executed with the prescribed GPU. The
identical code is running for each GPU, where kernels and grids are
similarly set as single-GPU code \cite{GKS-GPU}.
\subsection{Detailed parameters of CPUs and GPUs}
In following studies, the CPU codes run on the desktop with Intel
Core i7-9700 CPU using OpenMP directives and TianHe-II supercomputer
system with MPI and Intel mpiifort compiler, The HGKS codes with CPU
are all compiled with FP64 precision. For the GPU computations, the
Nvidia TITAN RTX GPU and Nvidia Tesla V100 GPU are used with Nvidia
CUDA and Nvidia HPC SDK. The detailed parameters of CPU and GPU
are given in Table.\ref{GPU-CPU-TG-A} and Table.\ref{GPU-CPU-TG-B},
respectively. For TITAN RTX GPU, the GPU-GPU communication is
achieved by connection traversing PCIe, and there are $2$ TITAN TRTX
GPUs in one node. For Tesla V100 GPU, there are $8$ GPUs inside one
GPU node, and more nodes are needed for more than $8$ GPUs. The
GPU-GPU communication in one GPU node is achieved by Nvidia NVLink.
The communication across GPU nodes can be achieved by Remote Direct
Memory Access (RDMA) technique, including iWARP, RDMA over Converged
Ethernet (RoCE) and InfiniBand. In this paper, RoCE is used for
communication across GPU nodes.
As shown in Table.\ref{GPU-CPU-TG-B}, Tesla V100 is equipped with
more FP64 precision cores and much stronger FP64 precision
performance than TITAN RTX. For FP32 precision performance, two GPUs
are comparable. Nvidia TITAN RTX is powered by Turing architecture,
while Nvidia Tesla V100 is built to HPC by advanced Ampere
architecture. Additionally, Tesla V100 outweighs TITAN RTX in GPU
memory size and memory bandwidth. Thence, much excellent
performance is excepted in FP64 precision simulation with Tesla
V100. To validate the performance with FP32 and FP64 precision,
numerical examples will be presented.
\begin{table}[!h]
\centering
\begin{tabular}{c|c|c|c}
\hline
\hline
~ & CPU & Clock rate & Memory size \\
\hline
Desktop & Intel Core i7-9700 &$3.0$ GHz &$16$ GB/$8$ cores \\
\hline
TianHe-II & Intel Xeon E5-2692 v2 &$2.2$ GHz &$64$ GB/$24$ cores\\
\hline
\hline
\end{tabular}
\caption{\label{GPU-CPU-TG-A} The detailed parameters of CPUs.}
~\\~\\
\centering
\begin{tabular}{c|c|c}
\hline
\hline
~ & Nvidia TITAN RTX & Nvidia Tesla V100 \\
\hline
Clock rate & 1.77 GHz &1.53 GHz \\
\hline
Stream multiprocessor & 72 &80 \\
\hline
FP64 core per SM & 2 &32 \\
\hline
FP32 precision performance & 16.3 Tflops &15.7 Tflops\\
\hline
FP64 precision performance & 509.8 Gflops &7834 Gflops \\
\hline
GPU memory size & 24 GB &32 GB \\
\hline
Memory bandwidth & 672 GB/s &897 GB/s \\
\hline
\hline
\end{tabular}
\caption{\label{GPU-CPU-TG-B} The detailed parameters of GPUs.}
\end{table}
\section{Numerical tests and discussions}
Benchmarks for compressible turbulent flows, including Taylor-Green
vortex and turbulent channel flows, are presented to validate the
performance of multiple-GPU accelerated HGKS. In this section, we
mainly concentrate on the computation with single-GPU and
multiple-GPUs. The GPU-CPU comparison in terms of computational
efficiency will be presented firstly, and the scalability of
parallel multiple-GPU code will also be studied. Subsequently, HGKS
implemented with GPUs is compiled with both FP32 and FP64 precision
to evaluate the effect of precision on DNS of compressible
turbulence. The detailed comparison on numerical efficiency and
accuracy of statistical turbulent quantities with FP32 precision and
FP64 precision is also presented.
\subsection{Taylor-Green vortex for GPU-CPU comparison}
Taylor-Green vortex (TGV) is a classical problem in fluid dynamics
developed to study vortex dynamics, turbulent decay and energy
dissipation process \cite{Case-Brachet,Case-Debonis,Case-Bull}. In
this case, the detailed efficiency comparisons of HGKS running on
the CPU and GPU will be given. The flow is computed within a
periodic square box defined as $-\pi L\leq x, y, z\leq \pi L$. With
a uniform temperature, the initial velocity and pressure are given
by
\begin{equation*}
\begin{aligned}
U=&V_0\sin(\frac{x}{L})\cos(\frac{y}{L})\cos(\frac{z}{L}),\\
V=&-V_0\cos(\frac{x}{L})\sin(\frac{y}{L})\cos(\frac{z}{L}),\\
W=&0,\\
p=&p_0+\frac{\rho_0V_0^2}{16}(\cos(\frac{2x}{L})+\cos(\frac{2y}{L}))(\cos(\frac{2z}{L})+2).
\end{aligned}
\end{equation*}
In the computation, $L=1, V_0=1, \rho_0=1$, and the Mach number
takes $Ma = V_0/c_0=0.1$, where $c_0$ is the sound speed. The fluid
is a perfect gas with $\gamma=1.4$, Prandtl number $Pr=1.0$, and
Reynolds number $Re=1600$. The characteristic convective time $t_c =
L/V_0$.
\begin{table}[!h]
\centering
\begin{tabular}{c|c|c|c|c|c|c}
\hline
\hline
Mesh size & Time step & i7-9700 & TITAN RTX & $S_{gc}$ & Tesla V100 & $S_{gc}$\\
\hline
$64^3$ & $3.571\times10^{-3}$ &0.757 &0.126 &6.0 &0.047& 16.1 \\
\hline
$128^3$ & $1.785\times10^{-3}$ & 11.421 &1.496 &7.6 &0.714& 16.0\\
\hline
$256^3$ & $8.925\times10^{-4}$ &182.720 &24.256 &7.4 &11.697&15.4\\
\hline
\hline
\end{tabular}
\caption{\label{GPU-CPU-TG-C} Taylor-Green vortex: the total execution times and speedup for
single-GPU versus CPU.}
\end{table}
In this section, GPU codes are all compiled with FP64 precision. To
test the performance of single-GPU code, this case is run with
Nvidia TITAN RTX and Nvidia Tesla V100 GPUs using Nvidia CUDA. As
comparison, the CPU code running on the Intel Core i7-9700 is also
tested. The execution times with different meshes are shown in
Table.\ref{GPU-CPU-TG-C}, where the total execution time for CPU and
GPUs are given in terms of hours and $20$ characteristic convective
time are simulated. Due to the limitation of single GPU memory size,
the uniform meshes with $64^3$, $128^3$ and $256^3$ cells are used.
For the CPU computation, $8$ cores are utilized with OpenMP parallel
computation, and the corresponding execution time
$T_{8,CPU}^{total}$ is used as the base for following comparisons.
The speedups $S_{gc}$ are also given in Table.\ref{GPU-CPU-TG-C},
which is defined as
\begin{equation*}
\displaystyle {S_{gc}}=\frac{T_{8, CPU}^{total}}{T_{1, GPU}^{total}},
\end{equation*}
where $T_{8, CPU}^{total}$ and $T_{1, GPU}^{total}$ denote the total
execution time of $8$ CPU cores and $1$ GPU, respectively. Compared
with the CPU code, 7x speedup is achieved by single TITAN RTX GPU
and 16x speedup is achieved by single Tesla V100 GPU. Even though
the Tesla V100 is approximately $15$ times faster than the TITAN RTX
in FP64 precision computation ability, Tesla V100 only performs $2$
times faster than TITAN RTX. It indicates the memory bandwidth is
the bottleneck for current memory-intensive simulations, and the
detailed technique for GPU programming still required to be
investigated.
\begin{table}[!h]
\centering
\begin{tabular}{c|c|c|c}
\hline
\hline
Mesh size & Time step &No. CPUs & $T_{n,CPU}^{total}$ \\
\hline
$128^3$ & $1.785\times10^{-3}$ & 16 & 13.3 \\
\hline
$256^3$ & $8.925\times10^{-4}$ & 256 & 13.5 \\
\hline
$512^3$ &$4.462\times10^{-4}$ & 1024 & 66 \\
\hline
\hline
\end{tabular}
\caption{\label{time_table2} Taylor-Green vortex: the detailed
computational parameters in TianHe-II supercomputers.}
\end{table}
\begin{table}[!h]
\centering
\begin{tabular}{c|c|c|c|c}
\hline
\hline
Mesh size & No. GPUs & $T_{n,GPU}^{total}$ & $T_{n,GPU}^{flow}$ & $T_{n,GPU}^{com}$ \\
\hline
$256^3$ & 1 &11.697 & ~ & ~ \\
\hline
$256^3$ & 2 & 6.201 &6.122 & 0.079 \\
\hline
$256^3$ & 4 & 3.223 &2.997 & 0.226 \\
\hline
$256^3$ & 8 & 1.987 &1.548 & 0.439 \\
\hline
\hline
$512^3$ & 8 & 22.252 &21.383 & 0.869 \\
\hline
$512^3$ & 12 & 19.055 & 17.250 & 1.805 \\
\hline
$512^3$ & 16 & 13.466 & 11.483 & 1.983 \\
\hline
\hline
\end{tabular}
\caption{\label{GPU-CPU-TG-D} Taylor-Green vortex: the detailed
computational parameters with Tesla V100.}
\end{table}
For single TITAN TRX with $24$ GB memory size, the maximum
computation scale is $256^3$ cells. To enlarge the computational
scale, the multiple-GPU accelerated HGKS are designed. Meanwhile,
the parallel CPU code has been run on the TianHe-II supercomputer
\cite{GKS-high-DNS}, and the execution times with respect to the
number of CPU cores are presented in Table.\ref{time_table2} for
comparison. For the case with $256^3$ cells, the computational time
of single Tesla V100 GPU is comparable with the MPI code with
supercomputer using approximately $300$ Intel Xeon E5-2692 cores.
For the case with $512^3$ cells, the computational time of CPU code
with supercomputer using 1024 Intel Xeon E5-2692 cores is
approximately $3$ times longer than that of GPU code using $8$ Tesla
V100 GPUs. It can be inferred that the efficiency of GPU code with
$8$ Tesla V100 GPUs approximately equals to that of MPI code with
$3000$ supercomputer cores, which agree well with the multiple-GPU
accelerated finite difference code for multiphase flows
\cite{GPU-3}. These comparisons shows the excellent performance of
multiple-GPU accelerated HGKS for large-scale turbulence simulation.
To further show the performance of GPU computation, the scalability
is defined as
\begin{align}
S_{n}=\frac{T_{n,GPU}^{total}}{T_{1,GPU}^{total}}.
\end{align}
Fig.\ref{mpi-efficience} shows the log-log plot for $n$ and $S_n$,
where $2$, $4$ and $8$ GPUs are used for the case with $256^3$
cells, while $8$, $12$ and $16$ GPUs are used for the case with
$512^3$ cells. The ideal scalability of parallel computations would
be equal to $n$. With the log-log plot for $n$ and
$T_{n,GPU}^{total}$, an ideal scalability would follow $-1$ slope.
However, such idea scalability is not possible due to the
communication delay among the computational cores and the idle time
of computational nodes associated with load balancing. As expected,
the explicit formulation of HGKS scales properly with the increasing
number of GPU. Conceptually, the total computation amount increases
with a factor of $16$, when the number of cells doubles in every
direction. Taking the communication delay into account, the
execution time of $256^3$ cells with 1 GPU approximately equals to
that of $512^3$ cells with 16 GPUs, which also indicates the
scalability of GPU code. When GPU code using more than $8$ GPUs, the
communication across GPU nodes with RoCE is required, which accounts
for the worse scalability using $12$ GPUs and $16$ GPUs.
Table.\ref{GPU-CPU-TG-D} shows the execution time in flow solver
$T_{n,GPU}^{flow}$ and the communication delay between CPU and GPU
$T_{n,GPU}^{com}$. Specifically, $T_{n,GPU}^{flow}$ is consist of
the time of WENO reconstruction, flux calculation at cell interface
and update of flow variables, while $T_{n,GPU}^{com}$ concludes time
for MPI$\_$RECV and MPI$\_$SEND for initialization and boundary
exchange, and MPI$\_$REDUCE for global time step. The histogram of
$T_{n,GPU}^{com}$ and $T_{n,GPU}^{flow}$ with different number of
GPU is shown in Fig.\ref{mpi-efficience-2} with $256^3$ and $512^3$
cells. With the increase of GPU, more time for communication is
consumed and the parallel efficiency is reduced accounting for the
practical scalability in Fig.\ref{mpi-efficience}. Especially, the
communication across GPU nodes with RoCE is needed for the GPU code
using more than $8$ GPUs, which consumes longer time than the
communication in single GPU node with NVLink. The performance of
communication across GPU nodes with InfiniBand will be tested, which
is designed for HPC center to achieve larger throughout among GPU
nodes.
\begin{figure}[!h]
\centering
\includegraphics[width=0.55\textwidth]{tg-comm-comp-1}
\caption{\label{mpi-efficience} Taylor-Green vortex: speedup and
efficiency of Tesla V100 GPUs with $256^3$ and $512^3$ uniform
cells.}
~\\
\includegraphics[width=0.495\textwidth]{tg-comm-comp-2A}
\includegraphics[width=0.495\textwidth]{tg-comm-comp-2B}
\caption{\label{mpi-efficience-2} Taylor-Green vortex: the histogram
of $T_{com}^n$ and $T_{flow}^n$ using different number of Tesla V100
GPUs with $256^3$ (left) and $512^3$ (right) uniform cells.}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=0.45\textwidth]{tg-vortex-1}
\includegraphics[width=0.45\textwidth]{tg-vortex-2}
\includegraphics[width=0.45\textwidth]{tg-vortex-3}
\includegraphics[width=0.45\textwidth]{tg-vortex-4}
\caption{\label{tg-vortex-q} Taylor-Green vortex: top view of
Q-criterion with Mach number $Ma = 0.25$, $0.5$, $0.75$ and $1.0$ at
$t =10$ using $256^3$ cells.}
\end{figure}
The time history of kinetic energy $E_k$, dissipation rate
$\varepsilon(E_k)$ and enstrophy dissipation rate
$\varepsilon(\zeta)$ of the above nearly incompressible case from
GPU code is identical with the solution from previous CPU code, and
more details can be found in \cite{GKS-high-DNS}. With the efficient
multiple-GPU accelerated platform, the compressible Taylor-Green
vortexes with fixed Reynolds number $Re = 1600$ Mach number $Ma =
0.25$, $0.5$, $0.75$ and $1.0$ are also tested. Top view of
Q-criterion with Mach number $Ma = 0.25$, $0.5$, $0.75$ and $1.0$ at
$t =10$ using $256^3$ cells are presented in Fig.\ref{tg-vortex-q}.
The volume-averaged kinetic energy is given by
\begin{align*}
E_k=\frac{1}{\rho_0\Omega}\int_\Omega\frac{1}{2}\rho\boldsymbol{U}\cdot\boldsymbol{U} \text{d} \Omega,
\end{align*}
where $\Omega$ is the volume of the computational domain.
The total viscous dissipation is defined as
\begin{align*}
\varepsilon_{com}=\frac{\mu}{\rho_0\Omega}\int_\Omega\boldsymbol{\omega}\cdot\boldsymbol{\omega} \text{d} \Omega
+\frac{4}{3}\frac{\mu}{\rho_0\Omega}\int_\Omega(\nabla\cdot\boldsymbol{U})^2 \text{d} \Omega,
\end{align*}
where $\mu$ is the coefficient of viscosity and
$\boldsymbol{\omega}=\nabla\times \boldsymbol{U}$. The time
histories of kinetic energy and total viscous dissipation are given
in Fig.\ref{tg-vortex-compressible}. Compared with the low Mach
number cases, the transonic and supersonic cases with $Ma = 0.75$
and $Ma = 1.0$ have regions of increasing kinetic energy rather than
a monotone decline. This is consistent with the compressible TGV
simulations \cite{Case-Peng}, where increases in the kinetic energy
were reported during the time interval $2 \leq t \leq 4$. A large
spread of total viscous dissipation curves is observed in
Fig.\ref{tg-vortex-compressible}, with the main trends being a delay
and a flattening of the peak dissipation rate for increasing Mach
numbers. The peak viscus dissipation rate for the $Ma=0.25$ curve
matches closely to that of $Ma=0.1$. Meanwhile, for the higher Mach
number cases, the viscous dissipation rate is significantly higher
in the late evolution region (i.e., $t \ge 11$) than that from near
incompressible case.
\begin{figure}[!h]
\centering
\includegraphics[width=0.495\textwidth]{tg-compressible-ener}
\includegraphics[width=0.495\textwidth]{tg-compressible-rate}
\caption{\label{tg-vortex-compressible} Taylor-Green vortex: the
time history of kinetic energy $E_k$ and viscous dissipation
$\varepsilon_{com}$ with Mach number $0.1$, $0.25$, $0.5$, $0.75$
and $1.0$ using $256^3$ cells.}
\end{figure}
\subsection{Turbulent channel flows for FP32-FP64 comparison}
For most GPUs, the performance of FP32 precision is stronger than
FP64 precision. In this case, to address the effect of FP32 and FP64
precision in terms of efficiency, memory costs and simulation
accuracy, both the near incompressible and compressible turbulent
channel flows are tested. Incompressible and compressible turbulent
channel flows have been studied to understand the mechanism of
wall-bounded turbulent flows
\cite{kim1987turbulence,kim1999turbulence,coleman1995numerical}. In
the computation, the physical domain is $(x,y,z)\in[0,2 \pi H]
\times [-H,H] \times[0,\pi H]$, and the computational mesh is given
by the coordinate transformation
\begin{align*}
\begin{cases}
\displaystyle x=\xi,\\
\displaystyle y=\tanh(b_g(\frac{\eta}{1.5\pi}-1))/\tanh(b_g),\\
\displaystyle z=\zeta,
\end{cases}
\end{align*}
where the computational domain takes $(\xi,\eta,\zeta)\in[0, 2\pi
H]\times[0, 3\pi H]\times[0, \pi H]$ and $b_g=2$. The fluid is
initiated with density $\rho = 1$ and the initial streamwise
velocity $U(y)$ profile is given by the perturbed Poiseuille flow
profile, where the white noise is added with $10\%$ amplitude of
local streamwise velocity. The spanwise and wall-normal velocity is
initiated with white noise. The periodic boundary conditions are
used in streamwise $x$-direction and spanwise $z$-directions, and
the non-slip and isothermal boundary conditions are used in vertical
$y$-direction. The constant moment flux in the streamwise direction
is used to determine the external force \cite{GKS-high-DNS}. In
current study, the nearly incompressible turbulent channel flow with
friction Reynolds number $Re_\tau=395$ and Mach number $Ma=0.1$, and
the compressible turbulent channel flow with bulk Reynolds number
$Re=3000$ and bulk Mach number $Ma=0.8$ are simulated with the same
set-up as previous studies \cite{GKS-high-DNS,GKS-high-cao-iles}.
For compressible case, the viscosity $\mu$ is determined by the
power law as $\mu= \mu_w(T/T_w)^{0.7}$, and the Prandtl number $Pr
=0.7$ and $\gamma=1.4$ are used. For the nearly incompressible
turbulent channel flow, two cases $G_1$ and $G_2$ are tested, where
$256\times128\times128$ and $256\times256\times128$ cells are
distributed uniformly in the computational space as
Table.\ref{channel_grids-1}. For the compressible flow, two cases
$H_1$ and $H_2$ are tested as well, where $128\times128\times128$
and $128\times256\times128$ cells are distributed uniformly in the
computational space as Table.\ref{channel_grids-2}. In the
computation, the cases $G_1$ and $H_1$ are implemented with single
GPU, while the cases $G_2$ and $H_2$ are implemented with double
GPUs. Specifically, $\Delta y^{+}_{min}$ and $\Delta y^{+}_{max}$
are the minimum and maximum mesh space in the $y$-direction. The
equidistant meshes are used in $x$ and $z$ directions, and $\Delta
x^{+}$ and $\Delta z^{+}$ are the equivalent plus unit,
respectively. The fine meshes arrangement in case $G_2$ and $H_2$
can meet the mesh resolution for DNS in turbulent channel flows. The
$Q$-criterion iso-surfaces for $G_2$ and $H_2$ are given
Fig.\ref{channel_iso_surface} in which both cases can well resolve
the vortex structures. With the mesh refinement, the high-accuracy
scheme corresponds to the resolution of abundant turbulent
structures.
\begin{table}[!htp]
\centering
\begin{tabular}{c|c|c|c|c}
\hline
\hline
Case &Mesh size &$\Delta y^{+}_{min}$/$\Delta y^{+}_{max}$ &$\Delta x^{+}$ &$\Delta z^{+} $\\
\hline
$G_1$ &$256\times128\times128$ &0.93/12.80 &9.69 &9.69 \\
\hline
$G_2$ &$256\times256\times128$ &0.45/6.40 &9.69 &9.69 \\
\hline
\hline
\end{tabular}
\caption{\label{channel_grids-1} Turbulent channel flow: the details
of mesh for nearly incompressible flow with friction Reynolds number
$Re_\tau=395$ and $Ma=0.1$.}
~\\
\centering
\begin{tabular}{c|c|c|c|c}
\hline
\hline
Case &Mesh size &$\Delta y^{+}_{min}$/$\Delta y^{+}_{max}$ &$\Delta x^{+}$ &$\Delta z^{+} $\\
\hline
$H_1$ &$128\times128\times128$ &0.43/5.80 &8.76 &4.38 \\
\hline
$H_2$ &$128\times256\times128$ &0.22/2.90 &8.76 &4.38 \\
\hline
\hline
\end{tabular}
\caption{\label{channel_grids-2} Turbulent channel flow: the
details of mesh for compressible flow with bulk Reynolds number
$Re=3000$ and bulk Mach number $Ma=0.8$.}
\end{table}
\begin{figure}[!h]
\centering
\includegraphics[width=0.475\textwidth]{channel-vor-ma01-B}
\includegraphics[width=0.475\textwidth]{channel-vor-ma08-B}
\caption{\label{channel_iso_surface} Turbulent channel flow: the
instantaneous $Q$-criterion iso-surfaces with $G_2$ (left) and $H_2$
(right).}
\end{figure}
\begin{table}[!h]
\centering
\begin{tabular}{c|c|c|c}
\hline
\hline
TITAN RTX & $T_{2,GPU}^{total}$ with FP32 & $T_{2,GPU}^{total}$ with FP64 & $S_{fp}$ \\
\hline
$G_1$ &0.414 &1.152 & 2.783 \\
\hline
$G_2$ &1.635 &4.366 & 2.670 \\
\hline
TITAN RTX & memory cost with FP32 & memory cost with FP64 & $R_{fp}$ \\
\hline
$G_1$ &2.761 &5.355 & 1.940 \\
\hline
$G_2$ &5.087 &10.009 & 1.968 \\
\hline
\hline
Tesla V100 & $T_{2,GPU}^{total}$ with FP32 & $T_{2,GPU}^{total}$ with FP64 & $S_{fp}$ \\
\hline
$G_1$ &0.377 & 0.642 & 1.703 \\
\hline
$G_2$ &1.484 &2.607 & 1.757 \\
\hline
Tesla V100 & memory cost with FP32 & memory cost with FP64 & $R_{fp}$ \\
\hline
$G_1$ &3.064 &5.906 & 1.928 \\
\hline
$G_2$ &5.390 &10.559 & 1.959 \\
\hline
\hline
\end{tabular}
\caption{\label{channel-precision} Turbulent channel flow: the
comparison of computational times and memory for FP32 and FP64
precision.}
\end{table}
Because of the reduction in device memory and improvement of
arithmetic capabilities on GPUs, the benefits can be achieved by
using FP32 precision compared to FP64 precision. In view of these
strength, FP32-based and mixed-precision-based high-performance
computing start to be explored \cite{lehmann2021accuracy,
haidar2020mixed}. However, whether FP32 precision is sufficient for
the DNS of turbulent flows still needs to be discussed. It is
necessary to evaluate the effect of precision on DNS of compressible
turbulence. Thus, the GPU-accelerated HGKS is compiled with FP32
precision and FP64 precision, and both nearly incompressible and
compressible turbulent channel flows are used to test the
performance in different precision. For simplicity, only the memory
cost and computational time for the nearly incompressible cases are
provided in Table.\ref{channel-precision}, where the execution times
$T_{2,GPU}^{total}$ are given in terms of hours and the memory cost
is in GB. In the computation, two GPUs are used and $10H/U_c$
statistical period is simulated. In Table.\ref{channel-precision},
$S_{fp}$ represents the speed up of $T_{2,GPU}^{total}$ with FP32 to
$T_{2,GPU}^{total}$ with FP64, and $R_{fp}$ denotes the ratio of
memory cost with FP32 to memory cost with FP64. As expected, the
memory of FP32 precision is about half of that of FP64 precision
for both TITAN RTX and Tesla V100 GPUs. Compared with FP64-based
simulation, 2.7x speedup is achieved for TITAN RTX GPU and 1.7x
speedup is achieved for Tesla V100 GPU. As shown in
Table.\ref{channel-precision}, the comparable FP32 precision
performance between TITAN RTX and Tesla V100 also provides
comparable $T_{2,GPU}^{total}$ for these two GPUs. We conclude that
Turing architecture and Ampere architecture perform comparably for
memory-intensive computing tasks in FP32 precision.
\begin{figure}[!h]
\centering
\includegraphics[width=0.495\textwidth]{channel-precision-ave}
\caption{\label{channel_ave_1} Turbulent channel flow: the mean
velocity for nearly incompressible turbulent flow. The reference
data is given in \cite{kim1999turbulence}.} \centering
\includegraphics[width=0.495\textwidth]{channel-ma08-u1}
\includegraphics[width=0.495\textwidth]{channel-ma08-u2}
\caption{\label{channel_ave_2} Turbulent channel flow: the mean
velocity $\langle U \rangle^+$ and velocity with VD transformation
$\langle U \rangle_{VD}^+$ for compressible turbulent flow. The
reference data is given in \cite{DNS-Li}.}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=0.45\textwidth]{channel-precision-rms-1}
\includegraphics[width=0.45\textwidth]{channel-precision-rms-2}
\includegraphics[width=0.45\textwidth]{channel-precision-rms-3}
\includegraphics[width=0.45\textwidth]{channel-precision-rms-4}
\caption{\label{channel_fluctuation_1} Turbulent channel flow: the
root-mean-square fluctuation velocity $U_{rms}^+$, $V_{rms}^+$,
$W_{rms}^+$ and Reynolds stress $-<U'V'>$ profiles for nearly
incompressible turbulent flow. The reference data is given in
\cite{kim1999turbulence}.}
\end{figure}
\begin{figure}[!htp]
\centering
\includegraphics[width=0.45\textwidth]{channel-ma08-1}
\includegraphics[width=0.45\textwidth]{channel-ma08-2}
\includegraphics[width=0.45\textwidth]{channel-ma08-3}
\includegraphics[width=0.45\textwidth]{channel-ma08-4}
\includegraphics[width=0.45\textwidth]{channel-ma08-5}
\includegraphics[width=0.45\textwidth]{channel-ma08-6}
\caption{\label{channel_fluctuation_2} Turbulent channel flow: the
root-mean-square fluctuation velocity $U_{rms}^+$, $V_{rms}^+$,
$W_{rms}^+$, Reynolds stress $\langle-\rho U^{'} V^{'}
\rangle/\langle \tau_{w} \rangle$, the root-mean-square of Mach
number $M_{rms}^+$ and turbulent Mach number $M_t$ profiles for
compressible turbulent flow. The reference data is given in
\cite{DNS-Li}.}
\end{figure}
To validate the accuracy of HGKS and address the effect with
different precision, the statistical turbulent quantities are
provided for the nearly incompressible and compressible turbulent
channel flows. For FP32-FP64 comparison, all cases restarted from
the same flow fields, and $200 H/U_c$ statistical periods are used
to average the computational results for all cases. For the nearly
incompressible channel flow, the logarithmic formulation is given by
\begin{align}\label{log-law}
U^+ = \frac{1}{\kappa}\ln Y^+ + B,
\end{align}
where the von Karman constant $\kappa=0.40$ and $B=5.5$ is given for
the low Reynolds number turbulent channel flow
\cite{kim1987turbulence}. The mean streamwise velocity profiles with
a log-linear plot are given in Fig.\ref{channel_ave_1}, where the
HGKS result is in reasonable agreement with the spectral results
\cite{kim1999turbulence}. In order to account for the mean property
of variations caused by compressibility, the Van Driest (VD)
transformation \cite{vdtransformation} for the density-weighted
velocity is considered
\begin{align}\label{log-law-vd}
{\left\langle U \right\rangle}_{VD}^+ = \int_{0}^{{\left\langle U \right\rangle}^+} \large \big(\frac{\left\langle \rho \right\rangle}{\left\langle \rho_w \right\rangle} \big)^{1/2} \text{d} {\left\langle U \right\rangle}^+,
\end{align}
where $\langle\cdot\rangle$ represents the mean average over
statistical periods and the X- and Z-directions. For the
compressible flow, the transformed velocity is expected to satisfy
the incompressible log law Eq.\eqref{log-law}. The mean streamwise
velocity profiles $\langle U \rangle^+$ and $\langle U
\rangle_{VD}^+$ with VD transformation as Eq.\eqref{log-law-vd} are
given in Fig.\ref{channel_ave_2} in log-linear plot. HGKS result is
in reasonable agreement with the reference DNS solutions with the
mesh refinements \cite{DNS-Li}. The mean velocity with FP32 and FP64
precision are also given in Fig.\ref{channel_ave_1} and
Fig.\ref{channel_ave_2}. It can be observed that the qualitative
differences between different precision are negligible for
statistical turbulent quantities with long time averaging.
For the nearly incompressible turbulent channel flow, the time
averaged normalized root-mean-square fluctuation velocity profile
$U_{rms}^+$, $V_{rms}^+$, $W_{rms}^+$ and normalized Reynolds stress
profiles $-<U'V'>$ are given in Fig.\ref{channel_fluctuation_1} for
the cases $G_1$ and $G_2$. The root mean square is defined as
$\phi_{rms}^{+} = \sqrt{(\phi -\langle \phi \rangle)^2}$ and
$\phi'=\phi -\langle \phi \rangle$, where $\phi$ represents the flow
variables. With the refinement of mesh, the HGKS results are in
reasonable agreement with the reference date, which given by the
spectral method with $256 \times 193 \times 192$ cells
\cite{kim1999turbulence}. It should also be noted that spectral
method is for the exact incompressible flow, which provides accurate
incompressible solution. The HGKS is more general for both nearly
incompressible flows and compressible flows. For the compressible
flow, the time averaged normalized root-mean-square fluctuation
velocity profiles $U_{rms}^+$, $V_{rms}^+$, $W_{rms}^+$, Reynolds
stress $\langle-\rho U^{'} V^{'} \rangle/\langle \tau_{w} \rangle$,
the root-mean-square of Mach number $M_{rms}^+$ and turbulent Mach
number $M_t$ are presented in Fig.\ref{channel_fluctuation_2} for
the cases $H_1$ and $H_2$. The turbulent Mach number is defined as
$M_t = q/\left\langle c \right \rangle$, where $q^2 =\langle
(U^{'})^2+(V^{'})^2+(W^{'})^2\rangle$ and $c$ is the local sound
speed. The results with $H_2$ agree well with the refereed DNS
solutions, confirming the high-accuracy of HGKS for DNS in
compressible wall-bounded turbulent flows. The statistical
fluctuation quantities profiles with FP32 precision and FP64
precision are also given in Fig.\ref{channel_fluctuation_1} and
Fig.\ref{channel_fluctuation_2}. Despite the deviation of
$U_{rms}^+$ for different precision, the qualitative differences in
accuracy between FP32 and FP64 precision are acceptable for
statistical turbulent quantities with such a long time average.
\begin{figure}[!h]
\centering
\includegraphics[width=0.45\textwidth]{channel-inst-a1}
\includegraphics[width=0.45\textwidth]{channel-inst-b1}
\includegraphics[width=0.45\textwidth]{channel-inst-a2}
\includegraphics[width=0.45\textwidth]{channel-inst-b2}
\caption{\label{channel_fluctuation_3} Turbulent channel flow: the
instantaneous profiles of $U^+$, $V^+$, $W^+$ for $G_2$ and $H_2$ at
$t=20H/U_c$ (top) and $200H/U_c$ (bottom) with FP32 and FP64
precision.}
\end{figure}
The instantaneous profiles of $U^+_{rms}$, $V^+_{rms}$, $W^+_{rms}$
at $t=20H/U_c$ and $200H/U_c$ are shown in
Fig.\ref{channel_fluctuation_3} for the cases $G_2$ and $H_2$ with
FP32 and FP64 precision, where cases restarted with the same initial
flow field for each case. Compared with the time averaged profiles,
the obvious deviations are observed for the instantaneous profiles.
The deviations are caused by the round-off error of FP32 precision,
which is approximately equals to or larger than the errors of
numerical scheme. Since detailed effect of the round-off error is
not easy to be analyzed, FP32 precision may not be safe for DNS in
turbulence especially for time-evolutionary turbulent flows. In
terms of the problems without very strict requirements in accuracy,
such as large eddy simulation (LES) and Reynolds-averaged
Navier–Stokes (RANS) simulation in turbulence, FP32 precision may
be used due to its improvement of efficiency and reduction of
memory. It also strongly suggests that the FP64 precision
performance of GPU still requires to be improved to accommodate the
increasing requirements of GPU-based HPC.
\section{Conclusion}
Based on the multi-scale physical transport and the coupled
temporal-spatial gas evolution, the HGKS provides a workable tool
for the numerical study of compressible turbulent flows. In this
paper, to efficiently conduct large-scale DNS of turbulence, the
HGKS code is developed with single GPU using CUDA architecture, and
multiple GPUs using MPI and CUDA. The multiple GPUs are distributed
across multiple CPUs at the cost of having to coordinate network
communication via MPI. The Taylor-Green vortex problems and
turbulent channel flows are presented to validate the performance of
HGKS with multiple Nvidia TITAN RTX and Nvidia Tesla V100 GPUs. We
mainly concentrate on the computational efficiency with single GPU
and multiple GPUs, and the comparisons between FP32 precision and
FP64 precision of GPU. For single-GPU computation, compared with the
OpenMP CPU code using Intel Core i7-9700, 7x speedup is achieved by
TITAN RTX and 16x speedup is achieved with Tesla V100. The
computational time of single Tesla V100 GPU is comparable with the
MPI code using $300$ supercomputer cores with Intel Xeon E5-2692.
For multiple GPUs, the HGKS code scales properly with the number of
GPU. It can be inferred that the efficiency of GPU code with $8$
Tesla V100 GPUs approximately equals to that of MPI code with $3000$
CPU cores. Compared with FP64 precision simulation, the efficiency of HGKS can be
improved and the memory is reduced with FP32 precision. However,
with the long time computation in compressible turbulent channels
flows, the differences in accuracy appear. Especially, the
instantaneous statistical turbulent quantities is not acceptable
using FP32 precision. The choice of precision should depend on the
requirement of accuracy and the available computational resources.
In the future, more challenging compressible flow problems, such as
the supersonic turbulent boundary layer and the interaction of shock
and turbulent boundary layer \cite{adams2000direct,wu2007direct},
will be investigated with efficient multiple-GPU accelerated HGKS.
\section*{Ackonwledgement}
This research is supported by National Natural Science Foundation of
China (11701038), the Fundamental Research Funds for the Central
Universities. We would thank Prof. Shucheng Pan of Northwestern
Polytechnical University, Dr. Jun Peng and Dr. Shan Tang of Biren
Technology for insightful discussions in GPU-based HPC.
|
1,108,101,566,813 | arxiv | \section{Introduction}
The study of atomic coherence effects in multilevel atoms is one of the most active area in atomic spectroscopy~\cite{ao76,ari90,fs}. Especially, the theory of coherent population trapping (CPT) in a three-level $\Lambda$-type atom has been extensively studied and the phenomenon has been observed experimentally in a sodium vapor~\cite{alz76,gra78}, photoassociation systems~\cite{dumke}, BEC~\cite{win05} and solids~\cite{kol05}. The CPT results from the formation of a coherent superposition of the ground atomic states that is decoupled from the external fields and hence referred to as a dark state. The particular interest of this phenomenon consists of the possibility of storage and coherent manipulation of the population in a coherent superposition of the ground states of the atoms~\cite{fl00,lu03}. These phenomena have received greatly increased experimental attention in recent years and experimental techniques have been developed which allow a reversible transfer of quantum information from light to the dark state of the atoms~\cite{jsc04}. The coherent population trapping has also been investigated in the context of lasing without inversion~\cite{ha97}, subrecoil laser cooling~\cite{as88} and a search for materials that display a high index of refraction accompanied by vanishing absorption~\cite{mos,sz92,zms}.
The atomic coherence effects are sensitive to decoherence. In the CPT effect, one source of decoherence is fluctuations of the laser fields used to create the coherent superposition of the atomic ground states~\cite{dk82}. The fluctuations redistribute the population among the atomic states including the excited atomic states from which it can be spontaneously emitted resulting in optical losses.
Recent investigations of decoherence processes in atomic systems have demonstrated that CPT and quantum storage in an ensemble of noninteracting atoms are limited primarily by different decoherence processes such as atomic collisions, atom loss and motion of atoms~\cite{dumke,mf05}. The results show an interesting property that in the limit of the total number of excitations much smaller than the number of atoms, the decoherence rate of the multiatom system is of the same order of magnitude as in the single atom, i.e. is independent of the number of atoms in the sample. In an earlier study, Jyotsna and Agarwal~\cite{ja96} showed that the CPT effect in a dense atomic medium is unaffected by local-field effects.
It is well known that the dominant contribution to the decoherence processes in the interaction of atoms with the electromagnetic field stems from the thermal fluctuations. They are present in a non-zero temperature reservoir to which the atoms are coupled. The fluctuations cause a pumping of the population stored in the dark state into the excited states of the atoms from which it can be spontaneously emitted resulting in an increase in decoherence. The magnitude of thermal fluctuations depends on temperature of the reservoir and determines the minimum level of thermal decoherence.
In this paper we propose a method to suppress the decoherences that occur due to the thermal fluctuations of the environmental electromagnetic reservoir at temperature $T$. Essentially, we examine the CPT effect in three-level $\Lambda$ systems by addressing a practical question: How can one increase the efficiency of trapping and storage of the population in the presence of thermal decoherence. In particular, we will investigate limits to the efficiency of the CPT effect in a single atom and next will explore the role of multiatom collective behavior in the reduction of the single-atom decoherence rate induced by the thermal field. The dipole-dipole interactions between the atoms will not be taken into account here assuming lower atomic densities, so that the collective behavior we consider stems entirely from the mutual coupling of all the atoms with the common radiation field \cite{ex_L}. Employing the analytic solution for the density operator of the system, we find that in general the single-atom coherent population trapping effect, reduced by thermal fluctuations, can be significantly improved or even completely restored when the atoms interact collectively with the thermal modes of the reservoir. We are particularly interested in the manner in which multiatom effects can lead to a suppression of thermal decoherence. With appropriate selection of atomic parameters, we will find cases of almost perfect coherent population trapping in the presence of the thermal decoherence. Our physical interpretation of the results is based on the semiclassical dressed atom model of the collective atomic system. The collective dressed states of the system are identified, and the effect of suppression of the thermal decoherence is explained in terms of the increased capacity of these states. This is shown to arise from correlation-enhanced transition rates among the multiatom dressed states, in particular those entering the trapped state. Hence, the effects of decoherence by thermal fields may by reverted more rapidly.
\section{Approach}
The system we consider is an ensemble of $N$ identical three-level $\Lambda$-type atoms each with excited
state $|1\rangle$ and two nondegenerate ground states $|2\rangle$ and $|3\rangle$. The atoms are driven by two single-mode
cw laser fields of Rabi frequencies $2\Omega_{2}$ and $2\Omega_{3}$ and angular frequencies $\omega_{L2}$ and
$\omega_{L3}$ significantly different from each other, so that each laser is coupled only to one of the allowed transitions, as shown in~Fig.~\ref{fig-1}.
\begin{figure}[b]
\includegraphics[height=3cm]{mfkfig1.eps}
\caption{\label{fig-1} Energy-level diagram of a three-level $\Lambda$-type atom driven by two laser
fields of Rabi frequencies $2\Omega_{2}$ and $2\Omega_{3}$.}
\end{figure}
The transitions are associated with nonzero dipole moments $\vec{\mu}_{12}$ and $\vec{\mu}_{13}$, and the laser fields
are detuned from the atomic transition frequencies, such that there is a nonzero two-photon detuning
$\Delta = (\omega_{13} -\omega_{12}+\omega_{L2} -\omega_{L3})/2$.
The transition $\ket 2 \rightarrow \ket 3$ is forbidden in the electric dipole approximation
$(\vec{\mu}_{23}=0)$. The exited atoms may decay spontaneously due to the zero point fluctuations of the electromagnetic
field from the state $|1\rangle$ to both ground states $|2\rangle$ and $|3\rangle$ with the decay rates $2\gamma_{2}$ and
$2\gamma_{3}$, respectively. We assume that the atoms are contained in a volume with linear dimensions that are small
compared with the radiation wavelengths, the Dicke model~\cite{dik}. Thus, all atoms experience the same Rabi frequencies
of the driving fields including their phases, and propagation effects are negligible due to the small size of the sample.
In addition, we assume that the atomic transitions are driven by a thermal field of the mean photon numbers $\bar n_{2}$ and
$\bar n_{3}$ at the atomic transition frequencies $\omega_{12}$ and $\omega_{13}$, respectively.
The system is described by the reduced density operator, which in the interaction picture and under the usual Born-Markov
and rotating-wave approximations satisfies the master equation
\begin{eqnarray}
\frac{\partial \rho}{\partial t} = -\frac{i}{\hbar}[H_{0},\rho]
+\gamma _{2}{\cal L}_{2}\rho +\gamma _{3}{\cal L}_{3}\rho ,\label{master}
\end{eqnarray}
where
\begin{eqnarray}
H_{0} &=& \hbar\Delta (S_{22}-S_{33})+\hbar\sum_{\alpha \in \{2,3\}}\Omega_{\alpha} (S_{1\alpha} + S_{\alpha 1}), \nonumber \\
{\cal L}_{2}\rho &=& (1+\bar{n}_{2})[S_{21}\rho ,S_{12}] + \bar{n}_{2} [S_{12}\rho ,S_{21}] + {\rm H.c.} , \nonumber \\
{\cal L}_{3}\rho &=& (1+\bar{n}_{3})[S_{31}\rho ,S_{13}] + \bar{n}_{3} [S_{13}\rho ,S_{31}] + {\rm H.c.} .
\end{eqnarray}
Here ${\cal L}_{2}\rho$ and ${\cal L}_{3}\rho$ are operators representing the damping of the atoms via spontaneous emission
and $H_{0}$ is the Hamiltonian describing the coupling of the atoms to the laser fields. The operators $S_{\alpha \beta }$ are the collective atomic operators
\begin{eqnarray}
S_{\alpha \beta } = \sum_{j=1}^{N}S_{\alpha \beta }^{(j)} = \sum_{j=1}^{N}|\alpha \rangle_{j}{}_{j}\langle \beta | ,
\quad \alpha,\beta = 1,2,3 ,
\end{eqnarray}
which obey the usual commutation relations
\begin{eqnarray}
[S_{\alpha \beta },S_{\alpha ^{^{\prime }}\beta^{^{\prime }}}] =
\delta _{\beta \alpha ^{^{\prime }}}S_{\alpha \beta ^{^{\prime }}} -
\delta _{\beta ^{^{\prime }}\alpha }S_{\alpha ^{^{\prime }}\beta } .\label{com}
\end{eqnarray}
The master equation (\ref{master}) allows to obtain equations of motion for the expectation value of an arbitrary
combination of the atomic operators. The calculations can be performed without much troubles for the simple case of
a single atom $(N=1)$ and arbitrary $\Delta$. However, for $N>1$ the calculation of the expectation value is not an
easy task. In even the simplest cases of small numbers of atoms, the calculations are prohibitively difficult due to the enormity of the number of coupled equation of motion. Fortunately, for the $\Delta =0$ case and high field strengths, $\Omega_{k}\gg N\gamma_{k}$, an approximation technique has been developed, which greatly simplifies the master equation (\ref{master}) and thus ables to perform analytical calculations of the expectation value of an arbitrary combination of the atomic operators. The restriction to the $\Delta =0$ case stems from the difficulty in obtaining a closed set of equations when the two-photon detuning is present~\cite{cor82}. A full discussion of the technique is
given in Refs.~\cite{symm,ag73,at76,ag78,pu94,law,bog,mek}. In the interest of brevity only the key results will be given here.
The technique is implemented by introducing dressed states of a single atom, which are obtained by a diagonalization
of the single-atom interaction Hamiltonian
\begin{eqnarray}
H_{0j} = \Omega_{2} (S_{12}^{(j)}+S_{21}^{(j)}) + \Omega_{3} (S_{13}^{(j)}+S_{31}^{(j)}) .
\end{eqnarray}
The single-atom dressed states are of the form
\begin{eqnarray}
\ket{\Psi_{1}}_{j} &=& \frac{1}{\Omega}\left(\Omega_{2}\ket 3_{j} -\Omega_{3}\ket 2_{j}\right) , \nonumber \\
\ket{\Psi_{2}}_{j} &=& \frac{1}{\sqrt{2}}\ket 1_{j} +\frac{1}{\sqrt{2}\Omega}\left(\Omega_{2}\ket{2}_{j} +\Omega_{3}\ket{3}_{j}\right) , \nonumber \\
\ket{\Psi_{3}}_{j} &=& \frac{1}{\sqrt{2}}\ket 1_{j} -\frac{1}{\sqrt{2}\Omega}\left(\Omega_{2}\ket{2}_{j} +\Omega_{3}\ket{3}_{j}\right) , \label{DS}
\end{eqnarray}
where $\Omega =\sqrt{\Omega^{2}_{2}+\Omega^{3}_{3}}$ is the generalized Rabi frequency.
The idea of the approximate technique is now to replace the collective operators $S_{\alpha \beta }$ by the
collective dressed-atom operators
\begin{eqnarray}
R_{\alpha \beta} = \sum^{N}_{j=1}R^{(j)}_{\alpha \beta} = \sum^{N}_{j=1}|\Psi_{\alpha}
\rangle_{j}{}_{j}\langle \Psi_{\beta}| , \quad \alpha,\beta = 1,2,3 ,\label{rab}
\end{eqnarray}
and then substitute for $S_{\alpha \beta }$ into the damping terms of the master equation (\ref{master}). Next, we make
the unitary transformation of the density operator
\begin{eqnarray}
\tilde{\rho} = \exp\left(\frac{i}{\hbar}\tilde{H}_{0}t\right)\rho \exp\left(-\frac{i}{\hbar}\tilde{H}_{0}t\right) ,
\end{eqnarray}
where
\begin{eqnarray}
\tilde{H}_{0} = \hbar\Omega \left(R_{22} - R_{33}\right) = \hbar\Omega R_{z}
\end{eqnarray}
and on carrying out this procedure it is found that certain terms in the transformed master equation are slowly varying
while the others are rapidly oscillating at frequencies $\Omega$ and $2\Omega$. The approximation then consists of dropping
these rapidly oscillating terms. The master equation (\ref{master}) in the dressed state basis reduces to
\begin{eqnarray}
\frac{\partial \tilde{\rho}}{\partial t} &=& -i\Omega[R_{z},\tilde{\rho}] + \{ \Gamma _{0}([R_{z}\tilde{\rho}, R_{z}] + [R_{32}\tilde{\rho},R_{23}] \nonumber \\
&+&[R_{23}\tilde{\rho}, R_{32}]) + \Gamma_{1}([R_{12}\tilde{\rho}, R_{21}]+[R_{13}\tilde{\rho}, R_{31}]) \nonumber \\
&+& \Gamma_{2}([R_{21}\tilde{\rho},R_{12}] + [R_{31}\tilde{\rho}, R_{13}]) + {\rm H.c.} \}, \label{masterd}
\end{eqnarray}
where
\begin{eqnarray}
\Gamma_{0} &=&\frac{1}{2}\{\gamma_{2}(1 + 2\bar n_{2})[\frac{\Omega_{2}}{\sqrt{2}\Omega}]^{2} + \gamma_{3}(1 + 2\bar n_{3})[\frac{\Omega_{3}}{\sqrt{2}\Omega}]^{2}\},\nonumber \\
\Gamma_{1} &=& \frac{1}{2}\{\gamma_{2}(1 + \bar n_{2})[\Omega_{3}/\Omega]^{2} + \gamma_{3}(1 + \bar n_{3})[\Omega_{2}/\Omega]^{2}\},\nonumber \\
\Gamma_{2} &=& \frac{1}{2}\{\gamma_{2}\bar n_{2}[\Omega_{3}/\Omega]^{2} + \gamma_{3}\bar n_{3}[\Omega_{2}/\Omega]^{2}\},\label{trd}
\end{eqnarray}
are the transition rates between the single-atom dressed states.
Using the approximate master equation, it is straightforward to obtain a simple analytical solution for the steady-state
density operator of the system. The solution can be written in the form
\begin{eqnarray}
\rho_{s} = Z^{-1}\exp[-\xi R_{11}] , \label{SS}
\end{eqnarray}
where
\begin{eqnarray}
\xi = \ln\left[\frac{\Omega^{2}_{3}\bar n_{2}
+ \eta \Omega^{2}_{2}\bar n_{3}}{\Omega^{2}_{3}(1+\bar n_{2})
+ \eta \Omega^{2}_{2}(1+\bar n_{3})}\right] ,\label{xi}
\end{eqnarray}
and $\eta =\gamma_{3}/\gamma_{2}$. The parameter $Z$ is the normalization constant such that Tr$\{\rho_{s}\}=1$. It is easily verified that~$\xi$ is always negative independent of the parameters used and approaches zero when $\bar n_{2}$ and/or $\bar n_{3}$ go to infinity. The solution~(\ref{SS}) was obtained in Refs.~\cite{bog,law,mek}, and some applications are discussed there in details. In Ref.~\cite{mek}, the solution has been used to investigate different control schemes for collective systems of three-level atoms. In this paper, we focus on the competition between thermal fluctuations and the collective effects that can lead to collective population trapping.
The steady-state solution (\ref{SS}) enables to calculate any statistical moment of the diagonal elements
$R_{\alpha \alpha}$, and thus population distributions between atomic states.
In particular, an $k$-th order moment of $R_{11}$ (expectation value of a product of $k$
operators $R_{11}$), is of the form
\begin{eqnarray}
\langle R^{k}_{11}\rangle_{s} = (-1)^{k}Z^{-1}\frac{\partial^{k}}{\partial \xi^{k}}Z , \quad k=1,2,\ldots ,\label{r1sol}
\end{eqnarray}
and the first order statistical moments of $R_{22}$ and $R_{33}$~are
\begin{eqnarray}
\langle R_{22}\rangle_{s} = \langle R_{33}\rangle_{s} = [N - \langle R_{11}\rangle_{s}]/2 ,\label{rsol}
\end{eqnarray}
where
\begin{eqnarray}
Z=\frac{N+2-(N+1){\rm e}^{\xi}-{\rm e}^{-\xi(N+1)}}{(1-{\rm e}^{\xi})(1-{\rm e}^{-\xi})} . \label{Z}
\end{eqnarray}
One can easily show from (\ref{SS}) that the steady-state off-diagonal elements $R_{\alpha \beta}\ (\alpha\neq \beta)$
equal zero. Note from Eq.~(\ref{rsol}) that all the non-zero expectation values can be represented in terms of
$\langle R^{k}_{11}\rangle_{s}$. The steady-state solutions are to be used in the forthcoming treatment of the coherent
population trapping in a multiatom system.
\section{Coherent Population Trapping}
Before we proceed to the detailed analysis of the multiatom trapping effect, we briefly investigate the trapping behavior of single atoms in the presence of thermal fluctuations. In this way we may see what restrictions are brought
by the thermal fluctuations for the trapping phenomenon and how they are related to the coherent driving process.
Coherent population trapping effect in a system of three-level atoms may be monitored experimentally in terms of the
intensity of the fluorescence light emitted~\cite{alz76,gra78,dumke,win05,kol05}. It is manifested by the disappearance of the fluorescence which, on the other side, manifests the vanishing of the population of the upper atomic states~$\ket 1_{j}$. Therefore, we will consider first the effect of the thermal field on the so-called transparency window, i.e. the dependence of the stationary population $\rho_{11}^{s}$ on the two-photon detuning. Next, using the stationary solution (\ref{SS}), we will find the analytical expression for the population at the two-photon resonance, $\Delta =0$, and will analyze how one could reduce the destructive effect of the thermal field on the minimum of the population at $\Delta =0$.
\begin{figure}[t]
\includegraphics[height=4cm]{mfkfig2.eps}
\caption{\label{fig-2} Stationary population of the upper state $\ket 1_{j}$ as a function of the two-photon detuning
$\Delta$ for $\gamma_{2}=\gamma_{3}=\gamma $, $\Omega_{2}=\Omega_{3}=5\gamma$ and different $\bar{n}$: $\bar{n}=0$ (solid line), $\bar{n}=0.5$ (dashed line), $\bar{n}=2$ (short dashed line).}
\end{figure}
Fig.~(\ref{fig-2}) illustrates the stationary population $\rho_{11}^{s}$ as a function of the two-photon detuning $\Delta$. We have obtained the population by solving numerically the master equation (\ref{master}) for $N=1$.
It is seen that in the absence of the thermal field, $\bar{n}=0$, there is perfect CPT observed at $\Delta =0$. When the atom is in the thermal field equally affecting both transitions, the CPT effect is reduced and the thermal field washes out the transparency window as $\bar{n}\gg 1$. Thus, the thermal field has a destructive effect on the CPT, because the thermal field is an incoherent field with random fluctuations that destroy the coherent process induced by the laser fields.
The variation with $\bar{n}$ of the minimum of the upper state population at $\Delta =0$ can be analyzed explicitly
using Eq.~(\ref{DS}) which for~$N=1$ and together with the steady-state solution (\ref{SS}) gives a simple analytical
expression for $\rho^{s}_{11}$ in the form
\begin{eqnarray}
\rho_{11}^{s} = \frac{1}{2}\left(\langle R_{22}\rangle_{s} + \langle R_{33}\rangle_{s}\right)
= \frac{{\rm e}^{\xi}}{1+2{\rm e}^{\xi}} . \label{UP}
\end{eqnarray}
First, we note from Eq.~(\ref{UP}) that the population distribution between the atomic states
is determined solely by the parameter~$\xi$. Clearly, the population distribution and consequently the trapping effect depend on several
parameters such as the laser intensity, spontaneous emission rates, and mean number of thermal photons.
We can call the parameter $\xi$ as a measure of efficiency of the CPT effect.
Here the efficiency of the CPT is examined in various intensity regimes of the coherent fields for equal and also unequal
average numbers of thermal photons. The average numbers can be made unequal by a suitable choice of bandwidth of the thermal field.
The selective excitation of the atomic transitions can be realized in practice by applying a finite bandwidth multimode thermal field whose
bandwidth is much smaller than the splitting of the lower atomic levels, but large compared to the natural linewidths of the
atomic transitions to satisfy the Markov approximation used in the derivation of the master equation.
In the limit of $\bar n_{2}=0$ that the thermal fluctuations affect only the $\ket 1 \rightarrow \ket 3$ transition, the
parameter $\xi$ reduces to
\begin{eqnarray}
\xi = \ln\left[\frac{\eta\Omega^{2}_{2}\bar n_{3}}{\Omega^{2}_{3}
+ \eta \Omega^{2}_{2}(1+\bar n_{3})}\right] .\label{xi1}
\end{eqnarray}
The parameter $\xi$ does not change substantially with the Rabi frequencies unless $\Omega_{3}$ is much larger than the
Rabi frequency $\Omega_{2}$ of the other transition. In the very strong-field
regime of $\Omega^{2}_{3}\gg \eta \Omega^{2}_{2}(1+\bar n_{3})$, the parameter $\xi$ approaches the limit of
$\xi\rightarrow -\infty$. This minimum value is that one which leads to vanishing of the population of the
upper atomic state, because $\lim_{\xi \to -\infty}\rho^{s}_{11} =0 $. This predicts that perfect coherent population
trapping can be observed even in the presence of thermal decoherence on one of the two atomic transitions, which is in contrast to the result of~\cite{bla97}. However, it requires that the transition influenced by the decoherence is simultaneously driven by a strong laser field. It can be understood rather easily. For a large Rabi frequency $\Omega_{3}$, the coherent processes on the $\ket 1 \rightarrow \ket 3$ transition dominate over the incoherent thermal processes resulting in perfect transparency.
Various other intensity regimes can also be distinguished. If $\bar n_{2}\neq \bar n_{3}$, the parameter $\xi$
can depend entirely on $\bar n_{2}$ or $\bar n_{3}$ depending on the ratio $\Omega_{3}/\Omega_{2}$. For instance,
when $\bar{n}_{2}\Omega^{2}_{3}\gg \eta \bar{n}_{3}\Omega^{2}_{2}$, we find that
\begin{eqnarray}
\xi = \ln\left(\frac{\bar n_{2}}{1+\bar n_{2}}\right) .\label{xi2}
\end{eqnarray}
This predicts that the coherent population trapping depends entirely on the thermal fluctuations at the weakly driven
$\ket 1 \rightarrow \ket 2$ transition. In the opposite limit of $\eta\bar{n}_{3}\Omega^{2}_{2}\gg \bar{n}_{2}\Omega^{2}_{3}$, the parameter $\xi$ now depends entirely on $\bar n_{3}$. Thus, the driving fields are relatively efficient in controlling decoherence in a single atom. Again, it can be interpreted as caused by coherent processes that dominate
incoherent thermal processes on the strongly driven transition. This also shows that the suppression of the thermal
decoherence in a single atom is limited to the level set by the lowest thermal fluctuations affecting the atomic transitions.
In the case when the thermal field equally contributes to both atomic transitions, $\bar n_{2}=\bar n_{3}\equiv\bar n$,
we have
\begin{eqnarray}
\xi = \ln\left(\frac{\bar n}{1+\bar n}\right) ,\label{xi3}
\end{eqnarray}
independent of the Rabi frequencies and the spontaneous emission rates. Obviously, the trapping effect is reduced
regardless of how strong are the Rabi frequencies of the laser fields relative to the thermal fluctuations.
In other words, there is no possibility of obtaining perfect population trapping or a control of the decoherence level
in a single atom when both transitions are equally affected by the thermal field.
A qualitative understanding of this effect can be obtained in terms of the transition rates (\ref{trd}). Figure~\ref{fig-3} shows the single-atom dressed states and the transition rates $\Gamma$. One can see from the figure that the population flows into the state $\ket{\Psi_{1}}_{j}$ with the rate $\Gamma_{1}$, and is removed from this state with the rate $\Gamma_{2}$. The state $\ket{\Psi_{1}}_{j}$ is a linear superposition of only the ground states of the atom that it is the trapping (dark) state. Therefore, we can call the rate $\Gamma_{2}$ a decoherence rate, as it transfers the population from the dark state to the upper state $\ket 1$ from which it can be spontaneously radiated resulting in an increase in decoherence and optical losses. Only in the absence of the thermal field, $\bar n_{2}=\bar n_{3}=0$, the transition rate $\Gamma_{2}=0$.
Evidently, the CPT effect depends crucially on $\Gamma_{2}$, and therefore the key to maintain a large efficiency of the CPT is to make $\Gamma_{2}$ as small as possible.
It can be done when the thermal field unequally affects the atomic transitions, i.e. when the number of thermal photons affecting one of the transitions is different than on the other transition.
\begin{figure}[t]
\includegraphics[height=4cm]{mfkfig3.eps}
\caption{\label{fig-3} Single atom dressed states and possible transitions with the rates $\Gamma_{0}, \Gamma_{1}$
and $\Gamma_{2}$.}
\end{figure}
For example, when $\bar n_{2}\ll \bar n_{3}$, the rate $\Gamma_{2}$ can be made small, proportional to $\bar n_{2}$, by changing the ratio $\Omega_{3}/\Omega_{2}$. It is easily to see from (\ref{trd}) that in the case of $\Omega_{3}\gg \Omega_{2}$, the rate $\Gamma_{2}$ is only of the order of $\bar n_{2}$ despite the fact that there is a large number of thermal photons present on the $\ket 1 -\ket 3$ transition. When the thermal field equally contributes to both atomic transitions, $\gamma_{2}\bar n_{2}=\gamma_{3}\bar n_{3}=\gamma\bar{n}$, and from Eq.~(\ref{trd}) we find that $\Gamma_{2}=\bar{n}\gamma/2$ independent of the Rabi frequencies of the laser fields. This is the smallest decoherence rate one can achieve in the single atom interacting with a thermal field that equally affects the atomic transitions. The limit is set by the number of photons $\bar{n}$ that, on the other hand, depends on temperature of the reservoir. An improvement of the CPT effect in the $\Lambda-$ type system with asymmetric spontaneous decay rates has been predicted in the absence of the thermal field \cite{PRA52}, but in this case the transparency window shows a strong sensitivity to the Rabi frequencies and is observed only in the limit of very weak driving fields.
The limit set in single atoms by temperature of the reservoir can be circumvented to improve the efficiency of the CPT effect if one considers multiatom collective systems in which interatomic interactions can create collective states of a significantly enhanced storage capacity compared with the capacity of the corresponding states of individual atoms.
\section{Collective trapping states}
The effects described in Section III can be seen in dilute atomic gases where the interatomic interactions are not important.
However, a more interesting situation emerges as we have considered here atomic samples where radiative interactions between
the atoms can lead to a collective (entangled) behavior of the atoms. Here, we include the multiatom effects and calculate the
population $\rho_{11}^{s}$ as a function of the number of atoms and the number of thermal photons.
The upper state population $\rho^{s}_{11}$ can be evaluated using Eq.~(\ref{DS}) which, together with the steady-state solution
(\ref{SS}), gives the analytical expression for $\rho^{s}_{11}$ in terms of $\xi$ and $N$ as
\begin{eqnarray}
\rho_{11}^{s} &=& \frac{Z^{-1}}{\left(1-{\rm e}^{-\xi}\right)}[\frac{1}{2}N(N + 1) \nonumber \\
&-& \frac{{\rm e}^{-\xi N} + N{\rm e}^{\xi} - N - 1}{(1 - {\rm e}^{\xi})^{2}}]. \label{CUP}
\end{eqnarray}
In Fig.~\ref{fig-4}, we present a three-dimensional plot which shows that in the absence of the thermal field, i.e.
$\bar n =0$, the stationary population $\rho_{11}^{s}$ is equal to zero independent of the number of atoms.
\begin{figure}[b]
\includegraphics[width = 7cm]{mfkfig4.eps}
\caption{\label{fig-4} The upper-state population $\rho_{11}^{s}/N$ as a function of $\bar n$ and $N$
for $\bar n_{2} =\bar n_{3} =\bar n$ and moderate numbers of atoms.}
\end{figure}
Thus, for $\bar n =0$ the collective behavior of the atoms does not affect the trapping effect. The presence of the
thermal field has a destructive effect on the trapping phenomenon that the population in the upper state is no longer
zero and increases with increasing number of thermal photons $\bar n$. However, the rate of the increase of the population
decreases with increasing number of atoms $N$ such that for a suitably large $N$ the population $\rho_{11}^{s}$
may remain very small even for large $\bar n$. In other words, the thermal decoherence decreases with increasing number of atoms.
Thus, the collective interactions are relatively efficient in suppression of thermal decoherence such that the atoms may remain
in their ground states even in the presence of the thermal decoherence. This is a suprising result as one might expect that
decoherence should increase with the increasing number of atoms.
Figure~\ref{fig-5} shows the population $\rho_{11}^{s}$ as a function of $\bar n_{2}=\bar n_{3} \equiv \bar n$ for different numbers of atoms.
Here we see that the rate of the increase of the population decreases with $N$. For a small number of atoms, the population saturates quickly
with $\bar n$. But, for a large number of atoms, a much stronger thermal field is required to reach saturation. In other words, the collective
population stored in the ground states is less affected by the thermal fluctuations than for the case of independent atoms. As a consequence, one has a
practical scheme to reduce thermal decoherence and preserve CPT in the thermal field.
\begin{figure}[t]
\includegraphics[height=4cm]{mfkfig5.eps}
\caption{\label{fig-5} The upper-state population $\rho_{11}^{s}/N$ as a function of $\bar n_{2}=\bar n_{3}=\bar n$ for different
numbers of atoms: $N=10$ (solid line), $N=100$ (dashed line), $N=1000$ (short dashed line).}
\end{figure}
In order to obtain an insight into the physical origin of the reduction of thermal decoherence and the improvement of the population trapping,
we examine the energy structure of the collective system. In general, in the absence of the driving fields, the system can be represented in
terms of collective symmetric and antisymmetric states. However, in the case of $N$ identical atoms contained in a volume with linear dimensions
that are small compared with the radiation wavelengths, only the symmetric states couples to external driving fields. The dipole-dipole interactions
between the atoms lead to a shift of these states from the laser resonance \cite{fs}. Thus, here the Rabi frequencies should be larger than the shift caused by
the dipole-dipole effects, i.e. the latter can be neglected. The antisymmetric states do not participate in the dynamics of the small sample system~\cite{dik}.
Therefore, we may limit the dynamics to only those involving the symmetric states. Moreover, only the lowest in energy symmetric states are of interest in the
analysis of the collective population trapping. We therefore consider the lowest energy states defined as
\begin{widetext}
\begin{eqnarray}
\ket 3 &=& \left(
\begin{array}{c}
N\\
0
\end{array}
\right)^{-\frac{1}{2}}
\ket{3_{1},3_{2},\ldots,3_{N}} ,\nonumber \\
\ket 2 &=& \left(
\begin{array}{c}
N\\
1
\end{array}
\right)^{-\frac{1}{2}}
\sum_{i=1}^{N} \ket{3_{1},\ldots,2_{i},\ldots,3_{N}} ,\nonumber \\
\ket 1 &=& \left(
\begin{array}{c}
N\\
1
\end{array}
\right)^{-\frac{1}{2}}
\sum_{i=1}^{N}
\ket{3_{1},\ldots,1_{i},\ldots,3_{N}} ,\nonumber \\
\ket{2^{2}} &\equiv& \ket{22} = \left(
\begin{array}{c}
N\\
2
\end{array}
\right)^{-\frac{1}{2}} \sum_{i<j=1}^{N}
\ket{3_{1},\ldots,2_{i},\ldots,2_{j},\ldots,3_{N}} ,\nonumber \\
\ket{12} &=& \frac{1}{\sqrt{2}}\left(
\begin{array}{c}
N\\
2
\end{array}
\right)^{-\frac{1}{2}}
\sum_{i\neq j=1}^{N}
\ket{3_{1},\ldots,1_{i},\ldots,2_{j},\ldots,3_{N}} ,\nonumber \\
\ket{2^{3}} &\equiv& \ket{222} = \left(
\begin{array}{c}
N\\
3
\end{array}
\right)^{-\frac{1}{2}} \sum_{i<j<k =1}^{N}
\ket{3_{1},\ldots,2_{i},\ldots,2_{j},\ldots,2_{k}\ldots,3_{N}} ,\nonumber \\
\mathrm{etc.} ,&& \label{4.1}
\end{eqnarray}
\end{widetext}
where the binomial coefficients are the normalization constants. The states (\ref{4.1}) are superpositions of single-atom product
states $\ket{m}_{i}\otimes \ket{n}_{j}\otimes\ldots\otimes\ket{k}_{l}$ that are symmetric under the exchange of any pair of atoms.
For example, the state $\ket 2$ is a linear superposition of the product states in which atom $i$ is in the state $\ket 2_{i}$ and
the remaining $N-1$ atoms are in their states $\ket 3_{j}$.
If we now allow the atoms to interact with the laser fields, each state $\ket{2^{k}}$ couples to the first excited states
$\ket{12^{k-1}}$ and $\ket{2^{k}}$ with the Rabi frequencies $\Omega_{2}$ and $\Omega_{3}$, respectively. Figure~\ref{fig-6} shows
the collective symmetric states and possible couplings of the two laser fields. As we have already mentioned, we limit the presentation
to the lowest energy levels which will be mixed together by the interaction leading to a ground collective dressed state, which is of
the main interest here. The lowest energy state is the product state $\ket 3$=$\ket{3_{1},3_{2},\ldots,3_{N}}$. Each succeeding state
$\ket{2^{k}}$ is of energy higher by successive increments of $\delta = \omega_{13}-\omega_{12}$. Similarly, each succeeding state
$\ket{12^{k}}$ is of energy higher by successive increments of $\delta$. It is interesting to note that the rotating-wave approximation,
which we are assuming to be valid, ignores coupling of states which differ in excitation by two and higher. In other words, the laser
fields couple only the neighboring ground states through the first excited states. It forms a two-dimensional chain of $\Lambda$
configurations.
\begin{figure}[t]
\includegraphics[width=8cm]{mfkfig6.eps}
\caption{\label{fig-6} Energy-level structure of noninteracting collective states of the $N$ atom system.}
\end{figure}
With the state ordering $\ket 3, \ket 1, \ket 2, \ket{12}, \ket{22}, \ldots ,\ket{2^{N}}$, corresponding to
the path of successive excitations of the states~$\ket{2^{k}}$ by the laser fields, the interaction Hamiltonian
$H_{0}$ can be expressed as an infinite tridiagonal matrix
\begin{widetext}
\begin{eqnarray}
H_{0}/\hbar = \left(
\begin{array}{cccccc}
-N\Delta & \Omega_{3}\sqrt{N} & 0 & 0 & 0 & \cdots\\
\Omega_{3}\sqrt{N} & -(N-1)\Delta & \Omega_{2}\sqrt{1} & 0 & 0 & \cdots\\
0 & \Omega_{2}\sqrt{1} & -N\Delta & \Omega_{3}\sqrt{N-1} & 0 & \cdots\\
0 & 0 & \Omega_{3}\sqrt{N-1} & -(N-1)\Delta & \Omega_{2}\sqrt{2} & \cdots\\
0 & 0 & 0 & \Omega_{2}\sqrt{2} & -N\Delta & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots
\end{array}
\right) .\label{4.2a}
\end{eqnarray}
\end{widetext}
It is interesting to note that the matrix element of the $\ket 3 -\ket 1$ transition coupled by the Rabi
frequency $\Omega_{3}$ is enhanced by a factor $\sqrt{N}$ and the magnitude of the matrix elements of the
successive transitions coupled by the same field decreases along the path to the state $\ket{2^{N}}$.
On the other hand, the matrix element of the $\ket 2 -\ket 1$ transition coupled by the Rabi frequency $\Omega_{2}$
is the same as in the single atom case, but the the magnitude of the matrix elements of the successive transitions
coupled by the same field increases as $\sqrt{k}$ when one moves along the excitation path to the state $\ket{2^{N}}$.
Thus, the coupling strength of the lasers to the atoms is transferred from one field to the other as one moves
along the path of excitations from $\ket 3$ to $\ket{2^{N}}$.
We now proceed to diagonalize the matrix (\ref{4.2a}) which will result in collective dressed states.
The diagonalization is performed by solving Schr\"{o}dinger's time-independent equation in the form
\begin{eqnarray}
\left(H_{0} -\lambda_{n}I\right)\ket{D_{n}}_{N} = 0 ,\label{4.2b}
\end{eqnarray}
where $I$ is the identity matrix and $\ket{D_{n}}_{N}$ is an eigenvector.
Substituting Eq.~(\ref{4.2a}) into Eq.~(\ref{4.2b}) yields the eigenvalue equation
\begin{eqnarray}
&&(\lambda_{n} + N\Delta)\{[\lambda_{n} + (N - 1)\Delta ]( \cdots ) + \Omega_{2}^{2}[\lambda_{n} \nonumber \\
&& + (N - 1)\Delta ] ( \cdots ) \} + \Omega_{3}^{2}N [(\lambda_{n} + N\Delta) (\cdots ) \nonumber \\
&& + \Omega_{3}^{2}(N - 1 )( \cdots )] = 0 \,
\end{eqnarray}
where the $\left(\cdots\right)$ refers to terms of which the explicit form is not needed apart from that those are polynomial functions
of $\lambda_{n}$. It is easily to show that in the case of $\Delta =0$, the eigenvalue equation reduces to
\begin{eqnarray}
\lambda_{n}\left[\lambda_{n}\left(\cdots\right) + \Omega_{2}^{2}\lambda_{n}\left(\cdots\right)
+\Omega_{3}^{2}N\left(\cdots\right)\right] = 0 \,
\end{eqnarray}
from which we see that $\lambda_{n} =0$ is one eigenvalue of $H_{0}$.
In the single atom case the dressed state $\ket{\Psi_{1}}_{j}$ corresponding to the zero eigenvalue was of very
special significance as corresponding to a trapping (dark) state completely decoupled from the fields~\cite{zms,mf05,usb}.
Let us investigate this possibility in the multiatom case.
If we represent the eigenvector $\ket{D_{n}}_{N}$ by the column vector
\begin{eqnarray}
\ket{D_{n}}_{N} =
\left(
\begin{array}{c}
c_{1}\\
c_{2}\\
\vdots\\
c_{i}\\
\vdots\\
c_{n}
\end{array}
\right) ,\label{4.2c}
\end{eqnarray}
we find by substituting into Eq.~(\ref{4.2b}) that for $\lambda_{n}=0$ the coefficients $c_{n}$ for even $n$
are all zero, whereas for odd $n$ the coefficients are given by the recurrence relation
\begin{eqnarray}
c_{2k+1}=\left(-\frac{\Omega_{3}}{\Omega_{2}}\right)^{k}\sqrt{\frac{N!}{k!(N-k)!}}\ c_{1}, ~~k = 1,2,3, \ldots
\end{eqnarray}
and $c_{1}$ is found from the normalization condition.
The dressed state corresponding to the eigenvalue $\lambda_{n}=0$ can thus be written as
\begin{eqnarray}
\ket{D}_{N} & \equiv & \ket{D_{0}}_{N} \nonumber \\
&=& \left(\cos\theta\right)^{N} \sum_{k=0}^{N} \left(
\begin{array}{c}
N\\
k
\end{array}
\right)^{\frac{1}{2}} \left(-\tan\theta\right)^{k} \ket{2^{k}} ,\label{4.3}
\end{eqnarray}
where $\ket{2^{0}} = \ket 3$, and
\begin{eqnarray}
\tan\theta =\frac{\Omega_{3}}{\Omega_{2} } .\label{4.4}
\end{eqnarray}
The collective dressed state (\ref{4.3}) is a linear combination of the state $\ket 3$ and $N$ of the states $\ket{2^{k}}$.
The important feature of the state is that it does not contain the excited states of the atoms
and thus does not radiate. The dressed state is a stationary state of the Hamiltonian $H_{0}$ describing
the atoms driven by two coherent fields. Therefore, if nothing else is allowed to interact with this system, the state (\ref{4.3})
will never change in time.
We can write the multi-atom dressed state (\ref{4.3}) in the basis of the single-atom dressed states (\ref{DS}). Surprisely, we
find that the state is of the form
\begin{eqnarray}
\ket{D}_{N} = \ket{\Psi_{1}}_{1}\otimes\ket{\Psi_{1}}_{2}\otimes\cdots\otimes\ket{\Psi_{1}}_{N}, \label{prod}
\end{eqnarray}
which is a product of the single-atom trapping states $\ket{\Psi_{1}}_{j}$. Obviously, the state~(\ref{prod}) is not entangled, which
shows that trapping of the population in all of the atomic ground states is equally effective in destroying collective (entangled)
properties of the system. Thus, the improvement of the CPT in the collective multiatom system, seen in Figs.~\ref{fig-4} and~\ref{fig-5},
does not arise from collective excitations of the dark state~$\ket{D}_{N}$.
We note in passing that the state in Eq.~(\ref{4.3}) is similar in form to that found by Mewes and Fleischhauer~\cite{mf05},
see also~\cite{fl00,lu03}, who considered collective quantum memories in three-level atoms driven by a classical field and a
single-mode quantum field. The results of their work demonstrate that the dark states of the multiatom system are highly entangled states.
However, the state $\ket{D}_{N}$ which is the analog of the dark states found in~\cite{mf05}, is not entangled. Thus, a question arises:
Why does the dark state $\ket{D}_{N}$ is not entangled? The reason is that the state $\ket{D}_{N}$ is a linear superposition of all the
collective ground states, whereas the dark states considered in Ref.~\cite{mf05} is restricted to having involved a small number of the
ground states corresponding to a small number of excitations $k\ll N$. It is easily verified that if we limit the number of the states
involved in the superposition (\ref{4.3}) to $k<N$, then the resulting state cannot be written as a product of the single atomic states.
Clearly, entanglement properties of the dark state $\ket{D}_{N}$ depend on the number of atoms involved in the interaction with the laser
fields, that only for $k<N$ the interaction can produce a dark state which is an entangled state.
To find the explanation why the CPT in the collective system interacting with the thermal field decoheres slower than the system of independent
atoms, we introduce the interaction between the collective dressed states and the thermal field. This interaction leads to a distribution of
the population, initially trapped in the dark state $\ket{D}_{N}$, among the collective dressed states. Let us look at the evolution of the
population of the state $\ket{D}_{N}$. Using the master equation~(\ref{masterd}), we obtain the following equation of motion for the population
of the state $\ket{D}_{N}$:
\begin{eqnarray}
\dot{\rho}_{DD} = -4N\Gamma_{2}\rho_{DD} + 2N\Gamma_{1}\left(\rho_{D2} + \rho_{D3}\right), \label{eqm}
\end{eqnarray}
where $\rho_{DD}$ is the population of the state $\ket{D}_{N}$ and $\rho_{D2}$ are $\rho_{D3}$ are populations of the following superposition states
\begin{eqnarray}
\ket{D2}_{N} \equiv \frac{1}{\sqrt{N}}\sum^{N}_{j=1} \ket{\Psi_{1}^{(1)},\Psi_{1}^{(2)},\ldots ,\Psi_{2}^{(j)},\ldots ,\Psi_{1}^{(N)}}, \nonumber \\
\ket{D3}_{N} \equiv \frac{1}{\sqrt{N}}\sum^{N}_{j=1} \ket{\Psi_{1}^{(1)},\Psi_{1}^{(2)},\ldots ,\Psi_{3}^{(j)},\ldots ,\Psi_{1}^{(N)}}, \label{6.1}
\end{eqnarray}
which differ in energy from the state $\ket{D}_{N}$ by $+\Omega$ and $-\Omega$, respectively.
The states $\ket{D2}_{N}$ and $\ket{D3}_{N}$ are linear superpositions of the product states in which $N-1$ atoms are in state $\ket{\Psi_{1}}_{j}$
and one atom is in the state $\ket{\Psi_{2}}_{j}$ and $\ket{\Psi_{3}}_{j}$, respectively.
It is interesting to note from Eq.~(\ref{eqm}) that both spontaneous emission and the thermal field couple the state $\ket{D}_{N}$ to only those states
which differ in the excitation by one. Moreover, the transition rates between these states are $N$ times larger than that for single atoms. This shows
that the system is superradiant despite the fact that the state $\ket{D}_{N}$ is the product state of the single-atom trapping states. In addition, the
collective decay rate of the radiators entering the state $\ket{D}_{N}$ is larger than that describing the atoms escaping from it. Thus, the collective
properties of the system are preserved due to the presence of the superposition states involving the single-atom states $\ket{\Psi_{2}}_{j}$ and
$\ket{\Psi_{3}}_{j}$.
In fact, the master equation (\ref{masterd}) leads to a set of $(N+1)(N+2)/2$ coupled equations of motion for the populations of the collective dressed
states. Fortunately, however, an explanation of the enhancement of the CPT effect, seen in Figs.~\ref{fig-4} and \ref{fig-5}, does not require a complete
solution for the populations of the dressed states. It is enough to consider only the population of the state $\ket{D2}_{N}$ or $\ket{D3}_{N}$.
Thus, using Eqs.~(\ref{DS}) and (\ref{rab}), we can show that for $N=1$ the stationary population of the state $\ket{D2}_{N}$ is simply equal to
$\langle R_{22}\rangle$, and for $N>1$ is given by the following expectation value
\begin{eqnarray}
\rho_{D2} &=& \frac{1}{N!}\langle (R_{12}R_{21} + R_{22} - R_{11})R_{11}(R_{11} - 1) \nonumber \\
&\times& (R_{11} - 2)\ldots (R_{11} - N + 2)\rangle, \label{4.5}
\end{eqnarray}
which can be evaluated using the steady-state solution (\ref{SS}).
We also calculate the population of the state $\ket{D2}_{N}$ in the case of independent atoms and find
\begin{eqnarray}
\rho_{D2}^{in}=\frac{\Gamma_{2}\left(\Gamma_{1}\right)^{N-1}}{\left(\Gamma_{1} + 2\Gamma_{2}\right)^{N}}. \label{4.6}
\end{eqnarray}
To see the difference between the populations (\ref{4.5}) and (\ref{4.6}), we study the ratio $\rho_{D2}^{in}/\rho_{D2}$.
Figure \ref{fig-7} shows the ratio for different numbers of atoms. For $N=1$, the ratio is equal to one, but for $N>1$ the ratio is smaller than one
and decreases with $N$. This shows that the population of the collective states is larger than the population of the equivalent states of independent atoms.
In other words, we may say that the capacity of the collective states is larger than the capacity of the equivalent states of independent atoms.
\begin{figure}[t]
\includegraphics[height=4cm]{mfkfig7.eps}
\caption{\label{fig-7} The ratio $\rho_{D2}^{in}/\rho_{D2}$ as a function of $\bar n_{2} = \bar n_{3} =\bar{n}$ for different numbers
of atoms: $N=2$ (solid line), $N=4$ (dashed line), $N=20$ (short dashed line).}
\end{figure}
The above analysis give us a simple physical interpretation of the collective trapping effect. We may conclude that the improvement of trapping effect by
multiatom system is due simply to the increased storage capacity of the collective (entangled) states compared with the storage capacity of the equivalent
states of independent atoms.
\section{Summary}
We have investigated the coherent population trapping effect in a collective system of three level atoms driven by two coherent laser fields and
simultaneously coupled to the reservoir of a non-zero temperature. The thermal reservoir causes thermal decoherence which affects the trapping effect.
We have shown that in a single atom there is no perfect CPT when both atomic transitions are affected by thermal decoherence. The perfect CPT may occur
when only one of the two atomic transitions is affected by thermal decoherence. Extending the analysis to multi-atom systems, we have shown that the
destructive effect of the thermal decoherence on the CPT can be circumvented by the collective behavior of the atoms. Unlike the case of noninteracting
atoms in which decoherence processes are independent of the number of atoms, we have found that the collective behavior of the atoms can substantially
improve the trapping effect destroyed by the thermal decoherence. In the collective atomic system the trapping effect increases with increasing number
of atoms. If number of atoms is large enough, an almost complete CPT is observed even at high temperatures of the reservoir.
This feature is explained in terms of the semiclassical dressed atom model. We have shown that the improvement of the CPT trapping in the multiatom system
arises from the presence of collective (entangled) states whose capacity of storage of the atomic population is larger than the corresponding states of
independent atoms.
\section*{ACKNOWLEDGMENTS}
ZF would like to thank The Max-Planck Institute for hospitality and The University of Queensland for
the travel support.
{\small $^\star$ On leave from \it{Technical University of Moldova, Physics Department,
\c{S}tefan Cel Mare Av. 168, MD-2004 Chi\c{s}in\u{a}u, Moldova.}}
|
1,108,101,566,814 | arxiv | \section{Introduction}
Generative models for text generation is of vital importance in a wide range of real world applications such as dialog system \cite{young2013pomdp} and machine translation \citep{brown1993mathematics}. Impressive progress has been achieved with the development of neural generative models \cite{serban2016building, zhao2017learning, zhao2018unsupervised, zhang-etal-2016-variational-neural, li2017deep, gupta2018deep, zhao2018adversarially} . However, most of prior methods focus on the improvement of text generation quality such as fluency and diversity. Besides the quality, the interpretability or controllability of text generation process is also critical for real world applications. Several recent papers recruit deep latent variable models for interpretable text generation where the latent space is learned to capture interpretable structures such as topics and dialog actions which are then used to guide text generation \citep{wang2019topic, zhao2018unsupervised}.
Deep latent variable models map a latent vector to the observed example such as a piece of text. Earlier methods \cite{kingma2013auto, rezende2014stochastic, bowman-etal-2016-generating} utilize a continuous latent space. Although it is able to generate text of high quality, it is not suitable for modeling interpretable discrete attributes such as topics and dialog actions. A recent paper \citep{zhao2018unsupervised} proposes to use a discrete latent space in order to capture dialog actions and has shown promising interpretability of dialog utterance generation. A discrete latent space nevertheless encodes limited information and thus might limit the expressiveness of the generative model. To address this issue, \citet{shi2020dispersed} proposes to use Gaussian mixture VAE (variational auto-encoder) which has a latent space with both continuous and discrete latent variables. By including a dispersion term to avoid the modes of the Gaussian mixture to collapse into a single mode, the model produces promising results on interpretable generation of dialog utterances.
To improve the expressivity of the latent space and the generative model as a whole, \citet{pang2020learning} recently proposes to learn an energy-based model (EBM) in the latent space, where the EBM serves as a prior model for the latent vector. Both the EBM prior and the generator network are learned jointly by maximum likelihood or its approximate variants. The latent space EBM has been applied to text modeling, image modeling, and molecule generation, and significantly improves over VAEs with Gaussian prior, mixture prior and other flexible priors. \citet{aneja2020ncpvae} generalizes this model to a multi-layer latent variable model with a large-scale generator network and achieves state-of-the-art generation performance on images.
Moving EBM from data space to latent space allows the EBM to stand on an already expressive generator model, and the EBM prior can be considered a correction of the non-informative uniform or isotropic Gaussian prior of the generative model. Due to the low dimensionality of the latent space, the EBM can be parametrized by a very small network, and yet it can capture regularities and rules in the data effectively (and implicitly).
In this work, we attempt to leverage the high expressivity of EBM prior for text modeling and learn a well-structured latent space for both interpretable generation and text classification. Thus, we formulate a new prior distribution which couples continuous latent variables (i.e., vector) for generation and discrete latent variables (i.e., symbol) for structure induction. We call our model \lsebmfull (\lsebm).
Two key differences of our work from \citet{pang2020learning} enable incorporation of information bottleneck \citep{tishby2000information}, which encourages the continuous latent vector to extract information from the observed example that is informative of the underlying structure. First, unlike \citet{pang2020learning} where the posterior inference is done with short-run MCMC sampling, we learn an amortized inference network which can be conveniently optimized. Second, due to the coupling formulation of the continuous latent vector and the symbolic one-hot vector, given the inferred continuous vector, the symbol or category can be inferred from it via a standard softmax classifier (see Section \ref{sec:svc} for more details).
The model can be learned in unsupervised setting where no category labels are provided. The symbol-vector coupling, the generator network, and the inference network are learned jointly by maximizing the variational lower bound of the log-likelihood.
The model can also be learned in semi-supervised setting where the category labels are provided for a subset of training examples. The coupled symbol-vector allows the learned model to generate text from the latent vector controlled by the symbol. Moreover, text classification can be accomplished by inferring the symbol based on the continuous vector that is inferred from the observed text.
\textbf{Contributions. \quad} (1) We propose a symbol-vector coupling EBM in the latent space, which is capable of both unsupervised and semi-supervised learning. (2) We develop a regularization of the model based on the information bottleneck principle. (3) Our experiments demonstrate that the proposed model learns well-structured and meaningful latent space, allowing for interpretable text generation and effective text classification.
\begin{figure}[tb]
\centering
\includegraphics[width=.52\linewidth]{figure/fig.pdf}\\
\caption{Graphical illustration of \lsebmfull (\lsebm). $y$ is a symbolic one-hot vector, and $z$ is a dense continuous vector. $x$ is the observed example. $y$ and $z$ are coupled together through an EBM, $p_\alpha(y, z)$, in the latent space. Given $z$, $y$ and $x$ are independent, i.e., $z$ is sufficient for $y$, hence giving the generator model $p_\beta(x|z)$. The intractable posterior, $p_\theta(z|x)$ with $\theta = (\alpha, \beta)$, is approximated by a variational inference model, $q_\phi(z|x)$.}
\label{fig:ibebm}
\end{figure}
\section{Model and learning}
\subsection{Model: symbol-vector coupling}
\label{sec:svc}
Let $x$ be the observed text sequence. Let $z \in \mathbb{R}^d$ be the continuous latent vector. Let $y$ be the symbolic one-hot vector indicating one of $K$ categories. Our generative model is defined by
\begin{align}
p_\theta(y, z, x) = p_\alpha(y, z) p_\beta(x | z), \label{eq:whole}
\end{align}
where $p_{\alpha}(y, z)$ is the prior model with parameters $\alpha$, $p_\beta(x|z)$ is the top-down generation model with parameters $\beta$, and $\theta = (\alpha, \beta)$. Given $z$, $y$ and $x$ are independent, i.e., $z$ is sufficient for $y$.
The prior model $p_{\alpha}(y, z)$ is formulated as an energy-based model,
\begin{align}
p_{\alpha}(y, z) = \frac{1}{Z_\alpha} \exp(\langle y, f_\alpha(z)\rangle) p_0(z), \label{eq:prior}
\end{align}
where $p_0(z)$ is a reference distribution, assumed to be isotropic Gaussian (or uniform) non-informative prior of the conventional generator model. $f_{\alpha}(z) \in \mathbb{R}^K$ is parameterized by a small multi-layer perceptron. $Z_\alpha$ is the normalizing constant or partition function.
The energy term $ \langle y, f_\alpha(z)\rangle$ in Equation (\ref{eq:prior}) forms an associative memory that couples the symbol $y$ and the dense vector $z$. Given $z$,
\begin{align}
p_\alpha(y|z) \propto \exp(\langle y, f_\alpha(z)\rangle), \label{eq:softmax}
\end{align}
i.e., a softmax classifier, where $f_\alpha(z)$ provides the $K$ logit scores for the $K$ categories. Marginally,
\begin{align}
p_\alpha(z) = \frac{1}{Z_\alpha} \exp(F_\alpha(z))p_0(z),
\end{align}
where the marginal energy term
\begin{align}
F_\alpha(z) = \log \sum_y \exp(\langle y, f_\alpha(z)\rangle), \label{eq:lse}
\end{align}
i.e., the so-called log-sum-exponential form. The summation can be easily computed because we only need to sum over $K$ different values of the one-hot $y$.
The above prior model $p_\alpha(y, z)$ stands on a generation model $p_\beta(x|z)$. For text modeling, let $x = (x^{(t)}, t=1,...,T)$ where $x^{(t)}$ is the $t$-th token. Following previous text VAE model \cite{bowman-etal-2016-generating}, we define $p_{\beta}(x|z)$ as a conditional autoregressive model,
\begin{align}
p_\beta(x|z) = \prod_{t=1}^T p_\beta(x^{(t)}|x^{(1)}, ..., x^{(t-1)}, z)
\end{align}
which is parameterized by a recurrent network with parameters $\beta$. See Figure~\ref{fig:ibebm} for a graphical illustration of our model.
\subsection{Prior and posterior sampling: symbol-aware continuous vector computation}
Sampling from the prior $p_\alpha(z)$ and the posterior $p_\theta(z|x)$ can be accomplished by Langevin dynamics. For prior sampling from $p_\alpha(z)$, Langevin dynamics iterates
\begin{align}
z_{t+1} = z_t + s \nabla_z \log p_\alpha(z_t) + \sqrt{2s} e_t,
\end{align}
where $e_t \sim \mathcal{N}(0, I_d)$, $s$ is the step size, and the gradient is computed by
\begin{align}
\nabla_z \log p_\alpha(z) &= \E_{p_\alpha(y|z)}[\nabla_z \log p_\alpha(y, z)] \nonumber \\
&= \E_{p_\alpha(y|z)}[\langle y, \nabla_z f_\alpha(z)\rangle ] ,
\end{align}
where the gradient computation involves averaging $\nabla_z f_\alpha(z)$ over the softmax classification probabilities $p_\alpha(y|z)$ in Equation (\ref{eq:softmax}). Thus the sampling of the continuous dense vector $z$ is aware of the symbolic $y$.
Posterior sampling from $p_\theta(z|x)$ follows a similar scheme, where
\begin{align}
\nabla_z \log p_\theta(z|x) &= \E_{p_\alpha(y|z)}[\langle y, \nabla_z f_\alpha(z)\rangle ] \nonumber \\
&+ \nabla_z \log p_\beta(x|z).
\end{align}
When the dynamics is reasoning about $x$ by sampling the dense continuous vector $z$ from $p_\theta(z|x)$, it is aware of the symbolic $y$ via the softmax $p_\alpha(y|z)$.
Thus $(y, z)$ forms a coupling between symbol and dense vector, which gives the name of our model, Symbol-Vector Coupling Energy-Based Model (SVEBM).
\citet{pang2020learning} proposes to use prior and posterior sampling for maximum likelihood learning. Due to the low-dimensionality of the latent space, MCMC sampling is affordable and mixes well.
\subsection{Amortizing posterior sampling and variational learning}
Comparing prior and posterior sampling, prior sampling is particularly affordable, because $f_\alpha(z)$ is a small network. In comparison, $\nabla_z \log p_\beta(x|z)$ in the posterior sampling requires back-propagation through the generator network, which can be more expensive. Therefore we shall amortize the posterior sampling from $p_\theta(z|x)$ by an inference network, and we continue to use MCMC for prior sampling.
Specifically, following VAE \cite{kingma2013auto}, we recruit an inference network $q_\phi(z|x)$ to approximate the true posterior $p_\theta(z|x)$, in order to amortize posterior sampling. Following VAE, we learn the inference model $q_\phi(z|x)$ and the top-down model $p_\theta(y, z, x)$ in Equation (\ref{eq:whole}) jointly.
For unlabeled $x$, the log-likelihood $\log p_\theta(x)$ is lower bounded by the evidence lower bound (ELBO),
\begin{align}
&\ELBO(x|\theta,\phi) = \log p_\theta(x) - \KL(q_\phi(z|x)\|p_\theta(z|x)) \nonumber \\
&= \E_{q_\phi(z|x)} \left[\log p_\beta(x|z)\right] - \KL(q_\phi(z|x) \| p_\alpha(z)),
\end{align}
where $\KL$ denotes the Kullback-Leibler divergence.
For the prior model, the learning gradient is
\begin{align}
\nabla_\alpha \ELBO = \E_{q_\phi(z|x)} [\nabla_\alpha F_{\alpha}(z)]
- \E_{p_\alpha(z)} [\nabla_\alpha F_{\alpha}(z)],
\end{align}
where $F_\alpha(z)$ is defined by (\ref{eq:lse}), $\E_{q_\phi(z|x)}$ is approximated by samples from the inference network, and $\E_{p_\alpha(z)}$ is approximated by persistent MCMC samples from the prior.
Let $\psi = \{\beta, \phi\}$ collect the parameters of the inference (encoder) and generator (decoder) models. The learning gradients for the two models are
\begin{align}
&\nabla_\psi \ELBO = \nabla_\psi \E_{q_\phi(z|x)} [\log p_\beta(x|z)] \nonumber\\
&- \nabla_\psi \KL(q_\phi(z|x) \| p_0(z)) + \nabla_\psi \E_{q_{\phi(z|x)}} [F_{\alpha}(z)], \label{eq:extra}
\end{align}
where $p_0(z)$ is the reference distribution in Equation (\ref{eq:prior}), and $\KL(q_\phi(z|x) \| p_0(z))$ is tractable. The expectations in the other two terms are approximated by samples from the inference network $q_\phi(z|x)$ with reparametrization trick \cite{kingma2013auto}. Compared to the original VAE, we only need to include the extra $F_\alpha(z)$ term in Equation (\ref{eq:extra}), while $\log Z_\alpha$ is a constant that can be discarded. This expands the scope of VAE where the top-down model is a latent EBM.
As mentioned above, we shall not amortize the prior sampling from $p_\alpha(z)$ due to its simplicity. Sampling $p_\alpha(z)$ is only needed in the training stage, but is not required in the testing stage.
\subsection{Two joint distributions}
Let $q_{\rm data}(x)$ be the data distribution that generates $x$. For variational learning, we maximize the averaged ELBO: $\E_{q_{\rm data}(x)}[\ELBO(x|\theta,\phi)]$, where $\E_{q_{\rm data}(x)}$ can be approximated by averaging over the training examples. Maximizing $\E_{q_{\rm data}(x)}[\ELBO(x|\theta,\phi)]$ over $(\theta, \phi)$ is equivalent to minimizing the following objective function over $(\theta, \phi)$
\begin{align}
& \KL(q_{\rm data}(x) \| p_\theta(x))
+ \E_{q_{\rm data}(x)} [\KL(q_\phi(z|x) \| p_\theta(z|x))] \nonumber \\
&= \KL(q_{\rm data}(x) q_\phi(z|x)\| p_\alpha(z) p_\beta(x | z) ). \label{eq:joint}
\end{align}
The right hand side is the KL-divergence between two joint distributions: $Q_\phi(x, z) = q_{\rm data}(x) q_\phi(z|x)$, and $P_\theta(x, z) = p_\alpha(z) p_\beta(x|z)$.
The reason we use notation $q$ for the data distribution $q_{\rm data}(x)$ is for notation consistency. Thus VAE can be considered as joint minimization of $\KL(Q_\phi\|P_\theta)$ over $(\theta, \phi)$. Treating $(x, z)$ as the complete data, $Q_\phi$ can be considered the complete data distribution, while $P_\theta$ is the model distribution of the complete data.
For the distribution $Q_\phi(x, z)$, we can define the following quantities.
\begin{align}
q_\phi(z) = \E_{q_{\rm data}(x)} [q_\phi(z|x)] = \int Q_\phi(x, z) dx
\end{align}
is the aggregated posterior distribution and the marginal distribution of $z$ under $Q_\phi$. $\HH(z)=-\E_{q_\phi(z)}[ \log q_\phi(z)]$ is the entropy of the aggregated posterior $q_\phi(z)$.
$\HH(z|x) = - \E_{Q_\phi(x, z)}[ \log q_\phi(z|x)]$ is the conditional entropy of $z$ given $x$ under the variational inference distribution $q_{\phi}(z|x)$.
\begin{align}
&\II(x, z) = \HH(z) - \HH(z|x) \nonumber \\
& = -\E_{q_\phi(z)} [\log q_\phi(z)] + \E_{Q_\phi(x, z)}[ \log q_\phi(z|x)]
\end{align}
is the mutual information between $x$ and $z$ under $Q_\phi$.
It can be shown that the VAE objective in Equation (\ref{eq:joint}) can be written as
\begin{align}
\KL&(Q_\phi(x, z) \| P_\theta(x, z)) \nonumber \\
&=-\HH(x) - \E_{Q_\phi(x, z)} [\log p_\beta(x|z)] \nonumber \\
& \quad +\II(x, z) + \KL(q_\phi(z) \| p_\alpha(z)), \label{eq:joint1}
\end{align}
where
$\HH(x) = - \E_{q_{\rm data}(x)} [\log q_{\rm data}(x)]$ is the entropy of the data distribution and is fixed.
\subsection{Information bottleneck}
Due to the coupling of $y$ and $z$ (see Equations (\ref{eq:prior}) and (\ref{eq:softmax})), a learning objective with information bottleneck can be naturally developed as a simple modification of the VAE objective in Equations (\ref{eq:joint}) and (\ref{eq:joint1}):
\begin{align}
\LL(\theta, \phi) &= \KL(Q_\phi(x, z)\| P_\theta(x, z)) - \lambda \II(z, y) \label{eq:ib}\\
&=-\HH(x) - \underbrace{\E_{Q_\phi(x, z)}[ \log p_\beta(x|z)]}_\text{reconstruction} \label{eq:recon}\\
&\quad + \underbrace{\KL(q_\phi(z) \| p_\alpha(z))}_\text{EBM learning}\\
&\quad + \underbrace{\II(x, z) - \lambda \II(z, y)}_\text{information bottleneck},
\end{align}
where $\lambda \ge 0$ controls the trade-off between the compressivity of $z$ about $x$ and its expressivity to $y$. The mutual information between $z$ and $y$, $\II(z, y)$, is defined as:
\begin{align}
\II(z, y) &= \HH(y) - \HH(y|z) \nonumber\\
& = -\sum_y q(y) \log q(y) \nonumber\\
&\quad + \E_{q_\phi(z)} \sum_y p_\alpha(y|z) \log p_\alpha(y|z), \label{eq:mi}
\end{align}
where $q(y) = \E_{q_\phi(z)} [p_\alpha(y|z)]$. $\II(z, y)$, $\HH(y)$, and $\HH(y|z)$ are defined based on $Q(x, y, z) = q_{\rm data}(x) q_\phi(z|x) p_\alpha(y|z)$, where $p_\alpha(y|z)$ is softmax probability over $K$ categories in Equation (\ref{eq:softmax}).
In computing $\II(z, y)$, we need to take expectation over $z$ under $q_\phi(z) = \E_{q_{\rm data}(x)} [q_\phi(z|x)]$, which is approximated with a mini-batch of $x$ from $q_{\rm data}(x)$ and multiple samples of $z$ from $q_\phi(z|x)$ given each $x$.
The Lagrangian form of the classical information bottleneck objective \citep{tishby2000information} is,
\begin{align}
\min_{p_\theta(z|x)} [\II(x, z|\theta) -\lambda \II(z, y|\theta)] \label{eq:cib}.
\end{align}
Thus minimizing $\LL(\theta, \phi)$ (Equation (\ref{eq:ib})) includes minimizing a variational version (variational information bottleneck or VIB; \citealt{alemi2016deep}) of Equation (\ref{eq:cib}). We do not exactly minimize VIB due to the reconstruction term in Equation (\ref{eq:recon}) that drives unsupervised learning, in contrast to supervised learning of VIB in \citet{alemi2016deep}.
We call the \lsebm learned with the objective incorporating information bottleneck (Equation (\ref{eq:ib})) as \ibebm.
\subsection{Labeled data}
\label{sec:ssl}
For a labeled example $(x, y)$, the log-likelihood can be decomposed into $\log p_\theta(x, y) = \log p_\theta(x) + \log p_\theta(y |x)$. The gradient of $\log p_\theta(x)$ and its ELBO can be computed in the same way as the unlabeled data described above.
\begin{align}
p_\theta(y|x) = \E_{p_\theta(z|x)} [p_\alpha(y|z)] \approx \E_{q_\phi(z|x)} [p_\alpha(y|z)] \label{eq:yx},
\end{align}
where $p_\alpha(y|z)$ is the softmax classifier defined by Equation (\ref{eq:softmax}), and $q_\phi(z|x)$ is the learned inference network. In practice, $\E_{q_\phi(z|x)} [p_\alpha(y|z)]$ is further approximated by $p_\alpha(y | z = \mu_\phi(x))$ where $\mu_\phi(x)$ is the posterior mean of $q_\phi(z|x)$. We found using $\mu_\phi(x)$ gave better empirical performance than using multiple posterior samples.
For semi-supervised learning, we can combine the learning gradients from both unlabeled and labeled data.
\subsection{Algorithm}
The learning and sampling algorithm for \lsebm is described in Algorithm \ref{alg:lsebm}. Adding the respective gradients of $\II(z, y)$ (Equation (\ref{eq:mi})) to Step 4 and Step 5 allows for learning \ibebm.
\begin{algorithm}[tb]
\caption{Unsupervised and Semi-supervised Learning of \lsebmfull.}
\label{alg:lsebm}
\begin{algorithmic}
\STATE {\bfseries Input:} Learning iterations~$T$, learning rates~$(\eta_0,\eta_1,\eta_2)$, initial parameters~$(\alpha_0, \beta_0, \phi_0)$, observed unlabelled examples~$\{x_i\}_{i=1}^M$, observed labelled examples~$\{(x_i, y_i)\}_{i=M+1}^{M+N}$ (optional, needed only in semi-supervised learning), unlabelled and labelled batch sizes $(m,n)$, initializations of persistent chains~$\{z_i^{-}\sim p_0(z)\}_{i=1}^{L}$, and number of Langevin dynamics steps $T_{LD}$.
\STATE {\bfseries Output:} $(\alpha_{T}, \beta_{T}, \phi_{T})$.
\FOR{$t=0$ {\bfseries to} $T-1$}
\STATE {\bfseries 1. mini-batch:} Sample unlabelled $\{ x_i \}_{i=1}^m$ and labelled observed examples $\{ x_i, y_i \}_{i=m+1}^{m+n}$.
\STATE {\bfseries 2. prior sampling:} For each unlabelled $x_i$, randomly pick and update a persistent chain $z_i^{-}$ by Langevin dynamics with target distribution $p_\alpha(z)$ for $T_{LD}$ steps.
\STATE {\bfseries 3. posterior sampling:} For each $x_i$, sample $z_i^{+} \sim q_\phi(z|x_i)$ using the inference network and reparameterization trick.
\STATE {\bfseries 4. unsupervised learning of prior model:} $\alpha_{t+1} = \alpha_t + \eta_0 \frac{1}{m}\sum_{i=1}^{m} [\nabla_\alpha F_{\alpha_t}(z_i^{+}) - \nabla_\alpha F_{\alpha_t}(z_i^{-})]$.
\STATE {\bfseries 5. unsupervised learning of inference and generator models:} \\$\psi_{t+1} = \psi_t + \eta_1 \frac{1}{m}\sum_{i=1}^{m}[\nabla_\psi \log p_{\beta_t}(x_i|z_i^{+}) - \nabla_\psi \KL(q_{\phi_t}(z|x_i) \| p_0(z)) + \nabla_\psi F_{\alpha_t}(z_i^{+})]$, with backpropagation through $z_i^{+}$ via reparametrization trick.\\
\IF{labeled examples $(x, y)$ are available}
\STATE {\bfseries 6. supervised learning of prior and inference models:} Let $\gamma = (\alpha, \phi)$.
$\gamma_{t+1} = \gamma_t + \eta_2 \frac{1}{n}\sum_{i=m+1}^{m+n}$ \\$\nabla_\gamma \log p_{\alpha_t}(y_i| z_i= \mu_{\phi_t}(x_i))$.
\ENDIF
\ENDFOR
\end{algorithmic}
\end{algorithm}
\section{Experiments}
We present a set of experiments to assess (1) the quality of text generation, (2) the interpretability of text generation, and (3) semi-supervised classification of our proposed models, \lsebm and \ibebm, on standard benchmarks. The proposed \lsebm is highly expressive for text modeling and demonstrate superior text generation quality and is able to discover meaningful latent labels when some supervision signal is available, as evidenced by good semi-supervised classification performance. \ibebm not only enjoys the expressivity of \lsebm but also is able to discover meaningful labels in an unsupervised manner since the information bottleneck objective encourages the continuous latent variable, $z$, to keep sufficient information of the observed $x$ for the emergence of the label, $y$. Its advantage is still evident when supervised signal is provided.
\begin{figure}
\centering
\begin{subfigure}[b]{0.35\textwidth}
\centering
\includegraphics[width=\textwidth]{figure/8_gaussians_combined.png}
\end{subfigure}
\hfill
\hfill
\hfill
\begin{subfigure}[b]{0.35\textwidth}
\centering
\includegraphics[width=\textwidth]{figure/pinwell_combined.png}
\end{subfigure}
\caption{Evaluation on 2D synthetic data: a mixture of eight Gaussians (upper panel) and a pinwheel-shaped distribution (lower panel). In each panel, the first, second, and third row display densities learned by \ibebm, \lsebm, and \dgmvae, respectively.}
\label{fig:8gaussians}
\end{figure}
\subsection{Experiment settings}
Generation quality is evaluated on the Penn Treebanks (\citealt{marcus1993building}, PTB) as pre-processed by \citet{mikolov2010recurrent}. Interpretability is first assessed on two dialog datasets, the Daily Dialog dataset \citep{li2017dailydialog} and the Stanford Multi-Domain Dialog (SMD) dataset \citep{eric2017key-value}. DD is a chat-oriented dataset and consists of $13,118$ daily conversations for English learner in a daily life. It provides human-annotated dialog actions and emotions for the utterances. SMD has $3,031$ human-Woz, task-oriented dialogues collected from three different domains (navigation, weather, and scheduling). We also evaluate generation interpretability of our models on sentiment control with Yelp reviews, as preprocessed by \citet{li-etal-2018-delete}. It is on a larger scale than the aforementioned datasets, and contains $180,000$ negative reviews and $270,000$ positive reviews.
Our model is compared with the following baselines: (1) RNNLM \citep{mikolov2010recurrent}, language model implemented with GRU \citep{cho2014learning}; (2) AE \citep{vincent2010stacked}, deterministic autoencoder which has no regularization to the latent space; (3) DAE, autoencoder with a discrete latent space; (4) VAE \citep{kingma2013auto}, the vanilla VAE with a continuous latent space and a Gaussian noise prior; (5) DVAE, VAE with a discrete latent space; (6) DI-VAE \citep{zhao2018unsupervised}, a DVAE variant with a mutual information term between $x$ and $z$; (7) semi-VAE \citep{kingma2014semi}, semi-supervised VAE model with independent discrete and continuous latent variables; (8) GM-VAE, VAE with discrete and continuous latent variables following a Gaussian mixture; (9) DGM-VAE \citep{shi2020dispersed}, GM-VAE with a dispersion term which regularizes the modes of Gaussian mixture to avoid them collapsing into a single mode; (10) semi-VAE $+ \II(x,y)$, GM-VAE $+ \II(x,y)$, DGM-VAE $+ \II(x,y)$, are the same models as (7), (8), and (9) respectively, but with an mutual information term between $x$ and $y$ which can be computed since they all learn two separate inference networks for $y$ and $z$. To train these models involving discrete latent variables, one needs to deal with the non-differentiability of them in order to learn the inference network for $y$. In our models, we do not need a separate inference network for $y$, which can conveniently be inferred from $z$ given the inferred $z$ (see Equation \ref{eq:softmax}), and have no need to sample from the discrete variable in training.
The encoder and decoder in all models are implemented with a single-layer GRU with hidden size 512. The dimensions for the continuous vector are $40$, $32$, $32$, and $40$ for PTB, DD, SMD and Yelp, respectively. The dimensions for the discrete variable are $20$ for PTB, $125$ for DD, $125$ for SMD, and $2$ for Yelp. $\lambda$ in information bottleneck (see Equation \ref{eq:ib}) that controls the trade-off between compressivity of $z$ about $x$ and its expressivity to $y$ is not heavily tuned and set to $50$ for all experiments. Our implementation is available at \url{https://github.com/bpucla/ibebm.git}.
\subsection{2D synthetic data}
We first evaluate our models on 2-dimensional synthetic datasets for direct visual inspection. They are compared to the best performing baseline in prior works, \dgmvae $+ \II(x,y)$ \citep{shi2020dispersed}. The results are displayed in Figure~\ref{fig:8gaussians}. In each row, \textit{true x} indicates the true data distribution $q_{\rm data}(x)$; \textit{posterior x} indicates the KDE (kernel density estimation) distribution of $x$ based on $z$ samples from its posterior $q_\phi(z|x)$; \textit{prior x} indicates the KDE of $p_\theta(x) = \int p_\beta(x|z) p_\alpha(z) dz$, based on $z$ samples from the learned EBM prior, $p_\alpha(z)$; \textit{posterior z} indicates the KDE of the aggregate posterior, $q_\phi(z) = \int q_{\rm data}(x)q_\phi(z|x) dx$; \textit{prior z} indicates the KDE of the learned EBM prior, $p_\alpha(z)$.
It is clear that our proposed models, \lsebm and \ibebm model the data well in terms of both \textit{posterior x} and \textit{prior x}. In contrast, although \dgmvae reconstructs the data well but the learned generator $p_\theta(x)$ tend to miss some modes. The learned prior $p_\theta(z)$ in \lsebm and \ibebm shows the same number of modes as the data distribution and manifests a clear structure. Thus, the well-structured latent space is able to guide the generation of $x$. By comparison, although \dgmvae shows some structure in the latent space, the structure is less clear than that of our model. It is also worth noting that \lsebm performs similarly as \ibebm, and thus the symbol-vector coupling \textit{per se}, without the information bottleneck, is able to capture the latent space structure of relatively simple synthetic data.
\subsection{Language generation}
We evaluate the quality of text generation on PTB and report four metrics to assess the generation performance: reverse perplexity (rPPL; \citealt{zhao2018adversarially}), BELU \cite{papineni2002bleu}, word-level KL divergence (wKL), and negative log-likelihood (NLL). Reverse perplexity is the perplexity of ground-truth test set computed under a language model trained with generated data. Lower rPPL indicates that the generated sentences have higher diversity and fluency. We recruit ASGD Weight-Dropped LSTM \cite{merity2018regularizing}, a well-performed and popular language model, to compute rPPL. The synthesized sentences are sampled with $z$ samples from the learned latent space EBM prior, $p_\alpha(z)$. The BLEU score is computed between the input and reconstructed sentences and measures the reconstruction quality. Word-level KL divergence between the word frequencies of training data and synthesized data reflects the generation quality. Negative log-likelihood~\footnote{It is computed with importance sampling \cite{burda2015importance} with 500 importance samples.} measures the general model fit to the data. These metrics are evaluated on the test set of PTB, except wKL, which is evaluated on the training set.
The results are summarised in Table~\ref{tab:generation_quality}. Compared to previous models with (1) only continuous latent variables, (2) only discrete latent variables, and (3) both discrete and continuous latent variables, the coupling of discrete and continuous latent variables in our models through an EBM is more expressive. The proposed models, \lsebm and \ibebm, demonstrate better reconstruction (higher BLEU) and higher model fit (lower NLL) than all baseline models except AE. Its sole objective is to reconstruct the input and thus it can reconstruct sentences well but cannot generate diverse sentences.
The expressivity of our models not only allows for capturing the data distribution well but also enables them to generate sentences of high-quality. As indicated by the lowest rPPL, our models improve over these strong baselines on fluency and diversity of generated text. Moreover, the lowest wKL of our models indicate that the word distribution of the generated sentences by our models is most consistent with that of the data.
It is worth noting that \lsebm and \ibebm have close performance on language modeling and text generation. Thus the mutual information term does not lessen the model expressivity.
\begin{table}[!htbp]
\scriptsize
\centering
\begin{tabular}{l c c c c}
\toprule
{\bf Model} & {\bf rPPL$^\downarrow$} & {\bf BLEU$^\uparrow$} & {\bf wKL$^\downarrow$} & {\bf NLL$^\downarrow$} \\
\midrule
{ Test Set} & - & 100.0 & \textbf{0.14} & -\\
\midrule
{ RNN-LM} & - & - & - & 101.21 \\
\midrule
{ AE} & 730.81 & {\bf 10.88} & 0.58 & - \\
{ VAE} & 686.18 & 3.12 & 0.50 & 100.85 \\
\midrule
{ DAE } & 797.17 & 3.93 & 0.58 & - \\
{ \textsc{DVAE}} & 744.07 & 1.56 & 0.55 & 101.07 \\
{ \textsc{DI-VAE}} & 310.29 & 4.53 & 0.24 & 108.90 \\
\midrule
{ semi-VAE} & 494.52 & 2.71 & 0.43 & 100.67 \\
{ semi-VAE $+ \II(x,y)$} & 260.28 & 5.08 & 0.20 & 107.30 \\
{ \textsc{GM-VAE}} & 983.50 & 2.34 & 0.72 & 99.44 \\
{ \textsc{GM-VAE} $+ \II(x,y)$} & 287.07 & 6.26 & 0.25 & 103.16 \\
{ \dgmvae } & 257.68 & 8.17 & 0.19 & 104.26 \\
{ \dgmvae $+ \II(x,y)$} & 247.37 & 8.67 & 0.18 & 105.73 \\
\midrule
{ \lsebm } & 180.71 & \textbf{9.54} & 0.17 & 95.02 \\
{ \ibebm } & \textbf{177.59} & 9.47 & \textbf{0.16} & \textbf{94.68} \\
\bottomrule
\end{tabular}
\caption{ Results of language generation on PTB.}\label{tab:generation_quality}
\end{table}
\subsection{Interpretable generation}
We next turn to evaluate our models on the interpretabiliy of text generation.
\textbf{Unconditional text generation.\quad} The dialogues are flattened for unconditional modeling. Utterances in DD are annotated with action and emotion labels. The generation interpretability is assessed through the ability to unsupervisedly capture the utterance attributes of DD. The label, $y$, of an utterance, $x$, is inferred from the posterior distribution, $p_\theta(y|x)$ (see Equation \ref{eq:yx}). In particular, we take $y = {\rm argmax}_k p_\theta(y=k|x)$ as the inferred label. As in \citet{zhao2018unsupervised} and \citet{shi2020dispersed}, we recruit homogeneity to evaluate the consistency between groud-truth action and emotion labels and those inferred from our models. Table~\ref{tab:dd} displays the results of our models and baselines.
Without the mutual information term to encourage $z$ to retain sufficient information for label emergence, the continuous latent variables in \lsebm appears to mostly encode information for reconstructing $x$ and performs the best on sentence reconstruction. However, the encoded information in $z$ is not sufficient for the model to discover interpretable labels and demonstrates low homogeneity scores. In contrast, \ibebm is designed to encourage $z$ to encode information for an interpretable latent space and greatly improve the interpretability of text generation over \lsebm and models from prior works, as evidenced in the highest homogeneity scores on action and emotion labels.
\begin{table}[!htbp]
\scriptsize
\centering
\hspace{-2 pt}
\begin{tabular}{l c c c c}
\toprule
{\bf Model} & {\bf MI$^\uparrow$} & {\bf BLEU$^\uparrow$} & {\bf Action$^\uparrow$} & {\bf Emotion$^\uparrow$} \\
\midrule
{ \text{DI-VAE}} & 1.20 & 3.05 & 0.18 & 0.09\\
\midrule
{ semi-VAE} & 0.03 & 4.06 & 0.02 & 0.08\\
{ semi-VAE $+ \II(x,y)$} & 1.21 & 3.69 & 0.21 & 0.14\\
{ \textsc{GM-VAE}} & 0.00 & 2.03 & 0.08 & 0.02\\
{ \textsc{GM-VAE} $+ \II(x,y)$} & 1.41 & 2.96 & 0.19 & 0.09\\
{ \dgmvae } & 0.53 & 7.63 & 0.11 & 0.09\\
{ \dgmvae $+ \II(x,y)$} & 1.32 & 7.39 & 0.23 & 0.16 \\
\midrule
{ \lsebm} & 0.01 & {\bf 11.16} & 0.03 & 0.01 \\
{ \ibebm} & {\bf 2.42} & 10.04 & {\bf 0.59} & {\bf 0.56} \\
\bottomrule
\end{tabular}
\caption{ Results of interpretable language generation on DD. Mutual information (MI), BLEU and homogeneity with actions and emotions are shown.
}\label{tab:dd}
\end{table}
\textbf{Conditional text generation.\quad} We then evaluate \ibebm on dialog generation with SMD. BELU and three word-embedding-based topic similarity metrics, embedding average, embedding extrema and embedding greedy \citep{mitchell2008vector, forgues2014bootstrapping, rus2012comparison}, are employed to evaluate the quality of generated responses. The evaluation results are summarized in Table~\ref{tab:smd_quality}. \ibebm outperforms all baselines on all metrics, indicating the high-quality of the generated dialog utterances.
SMD does not have human annotated action labels. We thus assess \ibebm qualitatively. Table~\ref{tab:case-actions} shows dialog actions discovered by it and their corresponding utterances. The utterances with the same action are assigned with the same latent code ($y$) by our model. Table~\ref{tab:case-response} displays dialog responses generated with different values of $y$ given the same context. It shows that \ibebm is able to generate interpretable utterances given the context.
\begin{table}[!htbp]
\scriptsize
\centering
\begin{tabular}{lllll}
\toprule
{\bf Model} & {\bf BLEU$^\uparrow$} & {\bf Average$^\uparrow$} & {\bf Extrema$^\uparrow$} & {\bf Greedy$^\uparrow$} \\
\midrule
DI-VAE & 7.06 & 76.17 & 43.98 & 60.92 \\
\dgmvae $+ \II(x,y)$ & 10.16 & 78.93 & 48.14 & 64.87 \\
\ibebm & \bf 12.01 & \bf 80.88 & \bf \bf 51.35 & \bf 67.12 \\
\bottomrule
\end{tabular}
\caption{ Dialog evaluation results on SMD with four metrics: BLEU, average, extrema and greedy word embedding based similarity. }\label{tab:smd_quality}
\end{table}
\begin{table}[!htbp]
\centering
\scriptsize
\begin{tabular}{ll}
\toprule
{\bf Action} & Inform-weather \\
\midrule
\multirow{3}{*}{\bf Utterance} & Next week it will rain on Saturday in Los Angeles \\
& It will be between 20-30F in Alhambra on Friday. \\
& It won't be overcast or cloudy at all this week in Carson \\
\bottomrule
\toprule
{\bf Action} & Request-traffic/route \\
\midrule
\multirow{3}{*}{\bf Utterance} & Which one is the quickest, is there any traffic? \\
& Is that route avoiding heavy traffic? \\
& Is there an alternate route with no traffic? \\
\bottomrule
\end{tabular}
\caption{Sample actions and corresponding utterances discovered by \ibebm on SMD. }\label{tab:case-actions}
\end{table}
\begin{table}[t]
\centering
\scriptsize
\begin{tabular}{ll}
\toprule
\multirow{2}{*}{\bf Context} & \textit{Sys:} What city do you want to hear the forecast for?\\
&\textit{User:} Mountain View\\
\midrule
\multirow{4}{*}{\bf Predict} & Today in Mountain View is gonna be overcast, with low of 60F \\
&and high of 80F. \\
\\
& What would you like to know about the weather for Mountain View? \\
\bottomrule
\toprule
\multirow{3}{*}{\bf Context} & \textit{User:} Where is the closest tea house? \\
&\textit{Sys:} Peets Coffee also serves tea. They are 2 miles away \\
&at 9981 Archuleta Ave.\\
\midrule
\multirow{3}{*}{\bf Predict} & OK, please give me an address and directions via the shortest distance. \\
\\
& Thanks! \\
\bottomrule
\end{tabular}
\caption{ Dialog cases on SMD, which are generated by sampling dialog utterance $x$ with different values of $y$.}\label{tab:case-response}
\end{table}
\textbf{Sentence attribute control. \quad} We evaluate our model's ability to control sentence attribute. In particular, it is measured by the accuracy of generating sentences with a designated sentiment. This experiment is conducted with the Yelp reviews. Sentences are generated given the discrete latent code $y$. A pre-trained classifier is used to determine which sentiment the generated sentence has. The pre-trained classifier has an accuracy of $98.5\%$ on the testing data, and thus is able to accurately evaluate a sentence's sentiment. There are multiple ways to cluster the reviews into two categories or in other words the sentiment attribute is not identifiable. Thus the models are trained with sentiment supervision. In addition to \dgmvae $+ \II(x,y)$, we also compare our model to text conditional GAN \cite{subramanian2018towards}.
The quantitative results are summarized in Table~\ref{tab:yelp-accuracy}. All models have similar high accuracies of generating positive reviews. The accuracies of generating negative reviews are however lower. This might be because of the unbalanced proportions of positive and negative reviews in the training data. Our model is able to generate negative reviews with a much higher accuracy than the baselines, and has the highest overall accuracy of sentiment control. Some generated samples with a given sentiment are displayed in Table~\ref{tab:yelp-samples}.
\begin{table}[!htbp]
\scriptsize
\centering
\hspace{-2 pt}
\begin{tabular}{l c c c}
\toprule
{\bf Model} & {\bf Overall$^\uparrow$} & {\bf Positive$^\uparrow$} & {\bf Negative$^\uparrow$} \\
\midrule
{ \dgmvae $+ \II(x,y)$} & 64.7\% & 95.3\% & 34.0\% \\
{ CGAN} & 76.8\% & 94.9\% & 58.6\% \\
{ \ibebm} & {\bf 90.1\%} & 95.1\% & {\bf 85.2\%} \\
\bottomrule
\end{tabular}
\caption{Accuracy of sentence attribute control on Yelp.
}\label{tab:yelp-accuracy}
\end{table}
\begin{table}[t]
\centering
\scriptsize
\begin{tabular}{ll}
\toprule
\multirow{5}{*}{\bf Positive} & The staff is very friendly and the food is great.\\
& The best breakfast burritos in the valley.\\
& So I just had a great experience at this hotel.\\
& It's a great place to get the food and service.\\
& I would definitely recommend this place for your customers.\\
\midrule
\multirow{5}{*}{\bf Negative} & I have never had such a bad experience.\\
& The service was very poor. \\
& I wouldn't be returning to this place.\\
& Slowest service I've ever experienced. \\
& The food isn't worth the price. \\
\bottomrule
\end{tabular}
\caption{Generated positive and negative reviews with \ibebm trained on Yelp.}\label{tab:yelp-samples}
\end{table}
\subsection{Semi-supervised classification}
We next evaluate our models with supervised signal partially given to see if they can effectively use provided labels. Due to the flexible formulation of our model, they can be naturally extended to semi-supervised settings (Section \ref{sec:ssl}).
In this experiment, we switch from neural sequence models used in previous experiments to neural document models \cite{miao2016neural, card2018neural} to validate the wide applicability of our proposed models. Neural document models use bag-of-words representations. Each document is a vector of vocabulary size and each element represents a word's occurring frequency in the document, modeled by a multinominal distribution. Due to the non-autoregressive nature of neural document model, it involves lower time complexity and is more suitable for low resources settings than neural sequence model.
We compare our models to VAMPIRE \cite{gururangan2019variational}, a recent VAE-based semi-supervised learning model for text, and its more recent variants (Hard EM and CatVAE in Table~\ref{tab:ssl}) \cite{jin-etal-2020-discrete} that improve over VAMPIRE. Other baselines are (1) supervised learning with randomly initialized embedding; (2) supervised learning with Glove embedding pretrained on $840$ billion words (Glove-OD); (3) supervised learning with Glove embedding trained on in-domain unlabeled data (Glove-ID); (4) self-training where a model is trained with labeled data and the predicted labels with high confidence is added to the labeled training set. The models are evaluated on AGNews \cite{zhang2015character} with varied number of labeled data. It is a popular benchmark for text classification and contains $127,600$ documents from $4$ classes.
The results are summarized in Table~\ref{tab:ssl}. \lsebm has reasonable performance in the semi-supervised setting where partial supervision signal is available. \lsebm performs better or on par with Glove-OD, which has access to a large amount of out-of-domain data, and VAMPIRE, the model specifically designed for text semi-supervised learning. It suggests that \lsebm is effective in using labeled data. These results support the validity of the proposed symbol-vector coupling formation for learning a well-structured latent space. \ibebm outperforms all baselines especially when the number of labels is limited ($200$ or $500$ labels), clearly indicating the effectiveness of the information bottleneck for inducing structured latent space.
\begin{table}[!htbp]
\scriptsize
\centering
\begin{tabular}{l c c c c}
\toprule
{\bf Model} & {200} & {500} & {2500} & {10000} \\
\midrule
{ \text{Supervised}} & 68.8 & 77.3 & 84.4 & 87.5\\
{ Self-training} & 77.3 & 81.3 & 84.8 & 87.7\\
{ Glove-ID} & 70.4 & 78.0 & 84.1 & 87.1\\
{ Glove-OD} & 68.8 & 78.8 & 85.3 & 88.0\\
{ VAMPIRE} & 82.9 & 84.5 & 85.8 & 87.7\\
{ Hard EM} & 83.9 & 84.6 & 85.1 & 86.9\\
{ CatVAE} & 84.6 & 85.7 & 86.3 & 87.5\\
\midrule
{ \lsebm} & 84.5 & 84.7 & 86.0 & 88.1 \\
{ \ibebm} & {\bf 86.4} & {\bf 87.4} & {\bf 87.9} & {\bf 88.6} \\
\bottomrule
\end{tabular}
\caption{ Semi-supervised classification accuracy on AGNews with varied number of labeled data.
}\label{tab:ssl}
\end{table}
\section{Related work and discussions}
\textbf{Text generation.\quad} VAE is a prominent generative model \cite{kingma2013auto, rezende2014stochastic}. It is first applied to text modeling by \citet{bowman-etal-2016-generating}. Following works apply VAE to a wide variety of challenging text generation problems such as dialog generation \cite{serban2016building, serban2017hierarchical, zhao2017learning, zhao2018unsupervised}, machine translation \cite{zhang-etal-2016-variational-neural}, text summarization \cite{li2017deep}, and paraphrase generation \cite{gupta2018deep}. Also, a large number of following works have endeavored to improve language modeling and text generation with VAE by addressing issues like posterior collapse \cite{zhao2018adversarially, li-etal-2019-surprisingly, fu-etal-2019-cyclical, he2018lagging}.
Recently, \citet{zhao2018unsupervised} and \citet{shi2020dispersed} explore the interpretability of text generation with VAEs. While the model in \citet{zhao2018unsupervised} has a discrete latent space, in \citet{shi2020dispersed} the model contains both discrete ($y$) and continuous ($z$) variables which follow Gaussian mixture. Similarly, we use both discrete and continuous variables. But they are coupled together through an EBM which is more expressive than Gaussian mixture as a prior model, as illustrated in our experiments where both \lsebm and \ibebm outperform the models from \citet{shi2020dispersed} on language modeling and text generation. Moreover, our coupling formulation makes the mutual information between $z$ and $y$ can be easily computed without the need to train and tune an additional auxiliary inference network for $y$ or deal with the non-diffierentibility with regard to it, while \citet{shi2020dispersed} recruits an auxiliary network to infer $y$ conditional on $x$ to compute their mutual information \footnote{Unlike our model which maximizes the mutual information between $z$ and $y$ following the information bottleneck principle \cite{tishby2000information}, they maximizes the mutual information between the observed data $x$ and the label $y$.}. \citet{kingma2014semi} also proposes a VAE with both discrete and continuous latent variables but they are independent and $z$ follows an non-informative prior. These designs make it less powerful than ours in both generation quality and interpretability as evidenced in our experiments.
\textbf{Energy-based model.\quad} Recent works \cite{xie2016theory, nijkamp2019learning, Han2020CVPR} demonstrate the effectiveness of EBMs in modeling complex dependency. \citet{pang2020learning} proposes to learn an EBM in the latent space as a prior model for the continuous latent vector, which greatly improves the model expressivity and demonstrates strong performance on text, image, molecule generation, and trajectory generation \cite{pang2020molecule, pang2021trajectory}. We also recruit an EBM as the prior model but this EBM couples a continuous vector and a discrete one, allowing for learning a more structured latent space, rendering generation interpretable, and admitting classification. In addition, the prior work uses MCMC for posterior inference but we recruits an inference network, $q_\phi(z|x)$, so that we can efficiently optimize over it, which is necessary for learning with the information bottleneck principle. Thus, this design admits a natural extension based on information bottleneck.
\citet{grathwohl2019your} proposes the joint energy-based model (JEM) which is a classifier based EBM. Our model moves JEM to latent space. This brings two benefits. (1) Learning EBM in the data space usually involves expensive MCMC sampling. Our EBM is built in the latent space which has a much lower dimension and thus the sampling is much faster and has better mixing. (2) It is not straightforward to apply JEM to text data since it uses gradient-based sampling while the data space of text is non-differentiable.
\textbf{Information bottleneck.\quad} Information bottleneck proposed by \citet{tishby2000information} is an appealing principle to find good representations that trade-offs between the minimality of the representation and its sufficiency for predicting labels. Computing mutual information involved in applying this principle is however often computationally challenging. \citet{alemi2016deep} proposes a variational approach to reduce the computation complexity and uses it train supervised classifiers. In contrast, the information bottleneck in our model is embedded in a generative model and learned in an unsupervised manner.
\section{Conclusion}
In this work, we formulate a latent space EBM which couples a dense vector for generation and a symbolic vector for interpretability and classification. The symbol or category can be inferred from the observed example based on the dense vector. The latent space EBM is used as the prior model for text generation model. The symbol-vector coupling, the generator network, and the inference network are learned jointly by maximizing the variational lower bound of the log-likelihood. Our model can be learned in unsupervised setting and the learning can be naturally extended to semi-supervised setting. The coupling formulation and the variational learning together naturally admit an incorporation of information bottleneck which encourages the continuous latent vector to extract information from the observed example that is informative of the underlying symbol. Our experiments demonstrate that the proposed model learns a well-structured and meaningful latent space, which (1) guides the top-down generator to generate text with high quality and interpretability, and (2) can be leveraged to effectively and accurately classify text.
\section*{Acknowledgments}
The work is partially supported by NSF DMS 2015577 and DARPA XAI project N66001-17-2-4029. We thank Erik Nijkamp and Tian Han for earlier collaborations.
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