Hence $\sin \(\frac{\pi}{2}-\a\)$ tends to $0$ by \eqref{Eq:angle} and thus $\sin \a$ tends to $1$ exponentially fast as $|x| \to \infty$, respectively.
Noting that
\[
\frac{\|\rho(g^{-1}x)\|}{\|\rho(x)\|} \ge \frac{\|\rho(g^{-1}x) v_x\|}{\|\rho(x) v_x\|} \ge \frac{\|\rho(g^{-1}x)\|}{\|\rho(x)\|}\sin \a,
\]
where $\|\rho(x)v_x\|=\|\rho(x)\|$,
we conclude the claim.
\qed

Lemmas \ref{Lem:rep_RG} and \ref{Lem:rep_QE} show that for each dominated representation $\rho: \G \to \GL(m, \R)$, if we define
\[
\psi(x, y):=\log \|\rho(x^{-1}y)\| \quad \text{for $x, y \in \G$},
\]
then $\psi$ satisfies \eqref{Eq:QE} and \eqref{Eq:RG} relative to each fixed $d \in \Dc_\G$ by the Morse lemma (see Section \ref{Sec:PStempered}).
Moreover, since $\psi$ is $\G$-invariant,
the function $\psi:\G \times \G \to \R$ defines a $\G$-invariant tempered potential relative to a metric $d$ in $\Dc_\G$ (Definition \ref{Def:tempered}).

Let us define the contragredient representation $\check \rho:\G \to \GL(m,\R)$, i.e., $\check \rho(x)$ is the transpose of $\rho(x^{-1})$.
Note that
\[
\check \psi(x, y)=\log \|\rho(y^{-1}x)\|=\log \|\check \rho(x^{-1}y)\| \quad \text{for $x, y \in \G$},
\]
and the pair of cocycles $c_\rho(x, \x)$ and $c_{\check \rho}(x, \x)$ admits a Gromov product
\[
[\x | \y]_{\rho}:=-\log \llangle \check E_\rho(\x), E_\rho(\y)\rrangle \quad \text{for $(\x, \y) \in \partial^2 \G$}.
\]
Indeed, we have that
\begin{align*}
[x \x|x \y]_\rho-[\x|\y]_\rho
&=-\log \frac{|u_{x \x}(w_{x \y})|}{\|u_{x \x}\|\|w_{x \y}\|}+\log \frac{|u_{\x}(w_\y)|}{\|u_\x\|\|w_\y\|}\\
&=\log \frac{\|\rho(x)u_\x\|}{\|u_\x\|}+\log \frac{\|\rho(x)w_\y\|}{\|w_\y\|}
=c_{\check \rho}(x^{-1}, \x)+c_\rho(x^{-1}, \y),
\end{align*}
where $u_\x$ and $w_\y$ represent $\check E_\rho(\x)$ and $E_\rho(\y)$ respectively,
and $\rho(x)u_\x=u_{x \x}$ and $\rho(x)w_\y=w_{x \y}$,
further we have used $u_{x \x}(w_{x \y})=u_\x(w_\y)$ in the second equality.
Note that $[\x|\y]_\rho$ defines a locally H\"older continuous function on $\partial^2 \G$.
Given results through Sections \ref{Sec:pre}-\ref{Sec:TF},
we obtain the following:
Let us consider the series
\[
\sum_{x \in \G}\exp(-s\log\|\rho(x)\|) \quad \text{for $s \in \R$},
\]
and let $v_\rho$ be the divergence exponent in $s$.
Note that $v_\rho$ is positive and finite by \eqref{Eq:dom_QI}.
We call $v_\rho$ the {\it exponential growth rate} for $\rho$.
Furthermore, we have $v_{\check \rho}=v_\rho$.
Applying the above ideas to the cocycles $v_\rho c_\rho(x, \x)$ and $v_\rho c_{\check \rho}(x, \x)$,
we obtain finite measures $\m_\rho$ and $\m_{\check \rho}$ on $\partial \G$
satisfying \eqref{Eq:QC} by Proposition \ref{Prop:PSpotential},
where the exponent is $0$ for any metric $d$ in $\Dc_\G$.
there exists a $\G$-invariant Radon measure
\[
\L_\rho:=\exp(v_\rho [\x|\y]_\rho)\m_{\check \rho}\otimes \m_{\rho} \quad \text{on $\partial^2 \G$},
\]
by Proposition \ref{Prop:Radon}.
Note that $\L_{\rho}$ is ergodic with respect to the $\G$-action on $\partial^2 \G$ by Theorem \ref{Thm:Hopf}.

Given the above preparations, Theorem \ref{Thm:UMMD_ergodic-flip} implies that
\[
\dim_H(\L_\rho, q_\times)=v_\rho(\t(\check \rho/d:\L_{\check \rho})+\t(\rho/d:\L_\rho)),
\]
for $d \in \Dc_\G$ where $q_\times$ is the quasi-metric associated with $d$ since $\check \L_\rho=\L_{\check \rho}$, and
for any $\G$-invariant Radon measure $\L$ which is ergodic on $\partial^2 \G$,
\[
\dim_H(\L, q_\times)\ge \t(\check \rho/d : \check \L)+\t(\rho/d : \L),
\]
then $\L$ is a constant multiple of $\L_\rho$.
This concludes
Theorem \ref{Thm:UMMD_ergodic-flip_intro}.

\subsection{Intersections for pairs of dominated representations}

Fix an integer $m \ge 2$.
Let $\rho$ and $\rho_\ast$ be a pair of dominated representations from $\G$ to $\GL(m, \R)$.
We define the {\it Manhattan curve} for any such pair $(\rho, \rho_\ast)$
as the boundary of the domain where
\[
\Qc_{\rho, \rho_\ast}(s, t):=\sum_{[x] \in \conj_\G}\exp(-s \log \lambda(\rho_\ast(x))-t \log \lambda(\rho(x))) \quad \text{for $(s, t) \in \R^2$},
\]
is finite.
Note that the Manhattan curve is convex by the H\"older inequality.
\[
\Pc_{\rho, \rho_\ast}(s, t):=\sum_{x \in \G}\exp(-s \log \|\rho_\ast(x)\|-t \log \|\rho(x)\|) \quad \text{for $(s, t) \in \R^2$}.
\]
For each $s \in \R$, let $\th(s)$ be the divergence exponent of $\Pc_{\rho, \rho_\ast}(s, t)$ in $t$, and let us call $\th$ the {\it pressure curve} for the pair $(\rho ,\rho_\ast)$.
Analogous to the Manhattan curve for a pair of metrics \cite[Proposition 3.1]{CT},
we show that $\Qc_{\rho, \rho_\ast}(s, t)<\infty$ if and only if $\Pc_{\rho, \rho_\ast}(s, t)<\infty$ in Proposition \ref{Prop:Knieper} below.

\begin{lemma}\label{Lem:growth}
For large enough $L, R>0$ and
for each $s \in \R$, there exists a positive constant $C_{s, L, R}$ such that
\begin{equation*}\label{Eq:growth}
C_{s, L, R}^{-1}\,e^{\th(s)n} \le \sum_{x: \ |\log \|\rho(x)\|-n| \le L}e^{-s \log \|\rho_\ast(x)\|} \le C_{s, L, R}\,e^{\th(s)n},
\end{equation*}
for all positive integers $n$.
\end{lemma}
\proof
For each $s \in \R$, let $t:=\th(s)$.
Applying the cocycle $s c_{\rho_\ast}(x, \x)+tc_{\rho}(x, \x)$ to Proposition \ref{Prop:PSpotential},
we have a finite Borel measure $\m_{s, t}$ on $\partial \G$ satisfying \eqref{Eq:QC}
(where the exponent is $0$ relative to any metric in $\Dc_\G$) and
\[
C^{-1} \le \exp(s \log \|\rho_\ast(x)\|+t\log \|\rho(x)\|)\cdot \m_{s, t}(\Oc(x, R)) \le C,
\]
for all $x \in \G$ and for some positive constants $C, R>0$.

Fix a large enough $L>0$ and for all positive integer $n$,
the shadows $\Oc(x, R)$ for $x$ with $|\log \|\rho(x)\|-n|\le L$ cover the boundary $\partial \G$ with a uniformly bounded multiplicity $M$;
indeed, for each $\x \in \partial \G$, there exists a $C$-rough geodesic $\g$ from $o$ converging to $\x$, and \eqref{Eq:dom_QI} shows that for a large enough $L>0$, there exists $x$ on $\g$ such that $|\log \|\rho(x)\|-n| \le L$ and $\x \in \Oc(x, R)$.
Furthermore, if $\x \in \Oc(x_1, R)\cap \Oc(x_2, R)$ and $|\log \|\rho(x_i)\|-n|\le L$ for $i=1, 2$, then $\g$ passes through the balls $B(x_i, R)$ at some $y_i$ for $i=1, 2$, respectively.
Suppose that $|y_1| \le |y_2|$, otherwise we change the role $y_1$ and $y_2$ below.
By \eqref{Eq:RG} shown in Lemma \ref{Lem:rep_RG},
we have
\[
\log \|\rho(y_1)\|+\log \|\rho(y_1^{-1}y_2)\|=\log \|\rho(y_2)\|+O(1),
\]
and thus $\log \|\rho(y_1^{-1}y_2)\|=O(1)$, implying $|y_1^{-1}y_2|\le R_0$ for a constant $R_0$ by \eqref{Eq:dom_QI}.
Hence the multiplicity $M$ is at most $\#B(o, 2R+R_0)$.
This shows that for all $n$,
\begin{align*}
C^{-1}\m_{s, t}(\partial \G) &\le C^{-1}\sum_{x: \ |\log \|\rho(x)\|-n| \le L}\m_{s, t}(\Oc(x, R))
\le e^{|t|L}\sum_{x: \ |\log \|\rho(x)\|-n| \le L}\exp(-s \log \|\rho_\ast(x)\|-tn)\\
&\le C e^{2|t|L}\sum_{x: \ |\log \|\rho(x)\|-n| \le L}\m_{s, t}(\Oc(x, R))\le C e^{2|t|L} M\m_{s, t}(\partial \G),
\end{align*}
implying the claim.
\qed

\begin{proposition}\label{Prop:Knieper}
Let $\G$ be a non-elementary hyperbolic group.
If $\rho$ and $\rho_\ast$ are dominated representations from $\G$ to $\GL(m, \R)$ for $m \ge 2$,
then for each fixed $s \in \R$, and for a positive constant $L>0$,
\[
\Cc_{\rho, \rho_\ast, s}(n, L):=\sum_{[z]: \  |\log \lambda(\rho(z))-n| \le L}e^{-s \log \lambda(\rho_\ast(z))},
\]
we have
\[
C^{-1}n^{-2}e^{\th(s)n}\le\Cc_{\rho, \rho_\ast, s}(n, L)
\le
C e^{\th(s)n},
\]
for a constant $C>0$ and all large enough $n$.
In particular, for each $s \in \R$,
the divergence exponents in $t$ of $\Qc_{\rho, \rho_\ast}(s, t)$ and $\Pc_{\rho, \rho_\ast}(s, t)$ coincide.
\end{proposition}

\begin{remark}
We may take a pair of dominated representations into different spaces, namely,
$\rho: \G \to \GL(m_1, \R)$ and $\rho_\ast: \G \to \GL(m_2, \R)$ where $m_1$ and $m_2$ are integers at least $2$ and may not be identical.
The same result holds for this generality; see the proof of Proposition \ref{Prop:Knieper}.
However, we present the proof only when $m_1=m_2$ for the sake of brevity.
\end{remark}

If $\rho:\G \to \GL(m, \R)$ is a dominated representation,
then by the above discussion it yields a continuous equivariant map
\[
\partial^2 \G \to \Pb({\R^m}^\ast)\times \Pb(\R^m)\setminus \D, \quad (\x, \y) \mapsto (\check E_\rho(\x), E_\rho(\y)),
\]
where $\D:=\{(u, v) \in \Pb({\R^m}^\ast)\times \Pb(\R^m) \ : \ \llangle u, v\rrangle=0\}$.
For any infinite order element $x \in \G$, let $x_+$ (resp.\ $x_-$) be the attracting  (resp.\ repelling) fixed point in $\partial \G$.

\begin{lemma}[cf.\ Lemma 5.7 in [42]]\label{Lem:Benoist}
Assume that $\G$ is torsion-free.
For every $0<\e<1/8$,
there exist a constant $c_\e\ge0$ and a finite set $F_\e$ in $\G$ such that
if $x \in \G \setminus F_\e$ and $\llangle \check E_\rho(x_-), E_\rho(x_+)\rrangle \ge 8 \e$,
then $\rho(x)$ is $\e$-proximal and
\[
\log \lambda(\rho(x)) \le \log \|\rho(x)\|  \le \log \lambda(\rho(x))+c_\e.
\]
If $\G$ is not torsion-free, then the same statement holds with $F_\e$ replaced by the union of a finite set and the set of finite order elements.
\end{lemma}

\proof
First we assume that $\G$ is torsion-free so that every element has infinite order.
For any $0<\e<1/8$, let
\[
P_{8\e}:=\{(u, v) \in \Pb({\R^m}^\ast)\times \Pb(\R^m)\setminus \D \ : \ \llangle u, v\rrangle \ge 8\e\}.
\]
Note that $P_{8\e}$ is compact, and thus there exists a finite collection of open sets $U \times V$ in $\Pb({\R^m}^\ast) \times \Pb(\R^m)$ such that $\llangle u, v\rrangle \ge 7\e$ for all $(u, v) \in U \times V$, and any two points in $U$ (resp.\ $V$) are $\e/2$-close.
For each such $U \times V$,
consider the sets $\check E_\rho^{-1}(U)$ and $E_\rho^{-1}(V)$ in the boundary,
cover each of them by finitely many shadows $\Oc(w_1, R)$ and $\Oc(w_2, R)$, respectively for some $w_1, w_2 \in \G$ and some $R>0$, so that if $\x \in \Oc(w_1, R)$ and $\y \in \Oc(w_2, R)$, then $\llangle \check E_\rho(\x), E_\rho(\y)\rrangle \ge 6\e$.
\[
U(w_1, w_2, R):=\{x \in \G \ : \ x_-\in \Oc(w_1, R), \ x_+ \in \Oc(w_2, R)\}.
\]
The $\d$-hyperbolicity implies that $(x_-|x_+)_o=(w_1|w_2)_o+O_{\d, R}(1)$ for all $x \in U(w_1, w_2, R)$,
and thus
\[
\ell[x]=|x|-2(x_-|x_+)_o=|x|+O_{\d, R, w_1, w_2}(1),
\]
(cf.\ \cite[Lemma 3.2]{CT}).
By \eqref{Eq:dom_sp}, we have that $\lambda_2(\rho(x))/\lambda_1(\rho(x)) \le e^{-c\ell[x]}$.
\[
\llangle \check E_\rho(x_-), E_\rho(x_+)\rrangle \ge 6\e
\]
and $\check E_\rho(x_-)$ and $E_\rho(x_+)$ are $6 \e$-apart by \eqref{Eq:angle}, there exists a constant $C_\ast(\e)$ depending on $\e, w_1, w_2, \d$ and $R$ such that
if $|x| \ge C_\ast(\e)$,
then $\lambda_2(\rho(x))/\lambda_1(\rho(x))$ is uniformly small and $\rho(x)$ is $\e$-Lipschitz outside the $\e$-neighborhood of the hyperplane $\check E_\rho(x_-)$ in $\Pb(\R^m)$ and it sends that outside into the $\e$-neighborhood of $E_\rho(x_+)$.
Hence $\rho(x)$ is $2\e$-proximal by appealing to \cite[Lemma 6.2]{Benoist}.
Furthermore, since the set of $2\e$-proximal elements with norm $1$ is compact in the space of (nonzero) linear transformation \cite[Corollaire 6.3]{Benoist},
there exists a constant $c_{\e, \d, R, w_1, w_2}$ such that for all $x \in U(w_1, w_2, R)$ and $|x| \ge C_\ast(\e)$,
\[
\log \lambda(\rho(x)) \le \log \|\rho(x)\| \le \log \lambda(\rho(x))+c_{\e, \d, R, w_1, w_2}.
\]
Let $c_\e$ and $C_{\ast\ast}(\e)$ be the maxima of such $c_{\e, \d, R, w_1, w_2}$ and $C_\ast(\e)$ over finitely many possibilities of $w_1, w_2$, respectively.
Letting $F_\e$ be the ball of radius $C_{\ast\ast}(\e)$ in the word metric, we conclude the claim.

Finally, we consider the case when $\G$ is not necessarily torsion-free.
Applying the above argument to the set of infinite order elements,
we obtain the second claim.
\qed

\medskip

For a large enough $R>0$ and for any $z, w \in \G$,
\[
O(z, w, R):=\{x \in \G \cup \partial \G \ : \ (z|x)_w \le R\},
\]
and for $x \in \G$,
\[
U(x^{-1}, x, R):=\{z \in \G \ : \ z^{-1}\in O(o, x^{-1}, R), \ z \in O(o, x, R)\}.
\]
Note that for an $R'>R$ depending on $x$, for any infinite order element $z$ in $U(x^{-1}, x, R)$,
we have $z_- \in O(o, x^{-1}, R')$ and $z_+ \in O(o, x, R')$.

\begin{lemma}\label{Lem:conj}
For each infinite order element $x$ in $\G$,
for all large enough $L>0$ there exists a constant $C_{x, R, L, \d}$ such that
for any positive integer $k$ and
any $z \in U(x^{-1}, x, R)$ with $|z|=k$,
the set
\[
C_k(z; x, L, R):=\{g\langle z\rangle \in \G/\langle z\rangle \ : \ g z g^{-1} \in U(x^{-1}, x, R), \ ||z|-k| \le L\}
\]
has cardinality at most $C_{x, R, L, \d}k$,
where $\langle z\rangle$ stands for the subgroup generated by $z$ in $\G$ and $\d$ is the hyperbolicity constant for a fixed word metric in $\G$.
\end{lemma}

\proof
The proof is found in the last paragraph in the proof of \cite[Proposition 3.1]{CT};
we include it here for the sake of convenience.
We write $[o, z]$ for a geodesic segment between $o$ and $z$ (with respect to a fixed word metric) and take $g\in\G$ with $g\langle z\rangle \in C_k(z; x, L, R)$. Since by assumption
\[
(z^{-1}|z)_o=(x^{-1}|x)_o+O_R \quad \text{and} \quad (gz^{-1}g^{-1}|gzg^{-1})_o=(x^{-1}|x)_o+O_R,
\]
we have that $\g(z):=\bigcup_{n \in \Z} z^n[o, z]$
is the image of a $(c_0, c_1)$-quasi-geodesic invariant under $z$ for some $c_0, c_1>0$
similarly $\g(gzg^{-1})$ is the image of a $(c_0, c_1)$-quasi-geodesic invariant under $g z g^{-1}$.
Note that $g \g(z)$ is also a $(c_0, c_1)$-quasi-geodesic invariant under $g z g^{-1}$,
and thus $g \g(z)$ and $\g(g z g^{-1})$ lie within a bounded Hausdorff distance.
Hence if $g\langle z\rangle \in C_k(z; x, L, R)$,
then $g\g(z)$ passes through near $o$ within a bounded distance $C_{x, R, \d}$ of $o$.
Here it is crucial that $c_0$, $c_1$ depend only on $x$, $R$ and the hyperbolicity constant.
Since for all such $g$, the inverse $g^{-1}$ lies in the $C_{x, R, \d}$-neighborhood of $[o, z]$ up to translation by $z$,
and $\g(z)$ is $z$-invariant, we obtain $\# C_k(z; x, L, R) \le C_{x, R, L, \d} k$,
where we have incorporated the thickness of the annulus $L$ in the counting.
\qed

\medskip

The following lemma is due to \cite[the proof of Lemma 5.2]{CoornaertKnieper}.

\begin{lemma}\label{Lem:CoornaertKnieper}
Fix a large enough $R>0$.
There exists a pair of infinite order elements $x, y$ in $\G$ such that
the following hold:
If we define
\[
U:=O(o, x^{-1}, R), \quad V:=O(o, x, R),
\]
$U':=yU$, $V':=yV$,  $U'':=y^{-1}U$,  $V'':=y^{-1}V$, and
\[
S(U, V):=\{z \in \G \ : \ U\cap z V =\emptyset\},
\]
and define $S(U', V')$, $S(U'', V'')$ analogously,
then $U, V, U', V', U''$ and $V''$ are disjoint,
\begin{equation}\label{Eq:x3}
x^3S(U, V)x^3 \subset U(x^{-1}, x, R),
\end{equation}
and $S(U, V)=y^{-1}S(U', V')y=y S(U'',V'')y^{-1}$.
there exists a finite (possibly empty) set $F_{x, y, R}$ such that
\begin{equation}\label{Eq:SUV}
\G \setminus F_{x, y, R} \subset S(U, V) \cup S(U', V') \cup S(U'', V'').
\end{equation}
\end{lemma}

\proof

Since $\G$ is a non-elementary hyperbolic group,
there exists a pair of infinite order elements $x$ and $y$ such that
$n \mapsto x^n$ and $n \mapsto y^n$ for $n \in \Z$ give quasi-geodesics and their extremes are distinct in the boundary \cite[Section 3.1]{GromovHyperbolic}.
Replacing $x$ by its larger power if necessary,
we define
\[
\wt V:=O(x^{-3}, x^{-2}, R), \quad
U:=O(o, x^{-1}, R), \quad V:=O(o, x, R) \quad \text{and} \quad \wt U:=O(x^3, x^2, R),
\]
satisfying that
\[
U\cap V =\emptyset, \quad (\G \cup \partial \G)\setminus U \subset \wt V \quad \text{and} \quad (\G \cup \partial \G)\setminus V \subset \wt U.
\]
Furthermore, we define
\[
U':=yU, \quad V':=yV, \quad U'':=y^{-1}U \quad \text{and} \quad V'':=y^{-1}V,
\]
in such a way that they and together with $U$ and $V$ are all disjoint,
replacing $y$ by its large power if necessary.
Note that for every $z \in S(U, V)$,
\[
(x^3 z x^3)\wt V \subset V \quad \text{and} \quad (x^3 z x^3)^{-1}\wt U \subset U,
\]
since $(x^3zx^3)\wt V \subset (x^3z) V \subset x^3((\G \cup \partial \G) \setminus U) \subset x^3 \wt  V  = V$ and similarly for the second expression.
we have that
\begin{equation*}
x^3 S(U, V) x^3 \subset U(x^{-1}, x, R),
\end{equation*}
since $o \in \wt V$ and $o \in \wt U$.
Moreover, since $U \times U' \times U''$ and $V \times V' \times V''$ are disjoint relatively compact sets in the space of ordered distinct triples in $\G \cup \partial \G$, denoted by $(\G \cup \partial \G)^{(3)}$.
The diagonal action of $\G$ on $(\G \cup \partial \G)^{(3)}$ is properly discontinuous \cite[Lemma 1.2 and Proposition 1.12]{BowditchConvergence}.
Hence the set of $z$ in $\G$ such that $U \times U' \times U'' \cap z(V \times V' \times V'')\neq \emptyset$
is finite, implying the claim.
\qed

\proof[Proof of Proposition \ref{Prop:Knieper}]
For each $s \in \R$ and a large enough $L>0$,
\[
\Pc_{\rho, \rho_\ast, s}(n, L):=\sum_{x:\ |\log\|\rho(x)\|-n|\le L}\exp\(-s\log\|\rho_\ast(x)\|\),
\]
\[
\Cc_{\rho, \rho_\ast, s}(n, L):=\sum_{[x] :\ |\log \lambda(\rho(x))-n|\le L}\exp\(-s \log\lambda(\rho_\ast(x))\).
\]
First we show that for every $s \in \R$,
there exist positive constants $C, c>0$ such that
\begin{equation}\label{Eq:Knieper1}
\Cc_{\rho, \rho_\ast, s}(n, L)\le C \Pc_{\rho, \rho_\ast, s}(n, L+c),
\end{equation}
for all large enough $n$.
Observe that the left hand side depends only on the set of infinite elements in $\G$.
We apply Lemma \ref{Lem:Benoist} to both $\rho$ and $\rho_\ast$ simultaneously;
for any $0<\e<1/8$,
there exist a positive constant $c_\e>0$ and a set $\wt F_\e$ consisting of finite order elements together with some finite set in $\G$ such that
for all $x \in \G \setminus \wt F_\e$, if $\llangle \check E_\rho(x_-), E_\rho(x_+)\rrangle \ge 8 \e$, then we have
\[
\log \lambda(\rho(x)) \le \log \|\rho(x)\| \le \log \lambda(\rho(x))+c_\e,
\]
and analogous inequalities hold for $\rho_\ast(x)$.
\[
c_0:=\inf\{\llangle \check E_\rho(\x), E_\rho(\y)\rrangle \ : \ (\x|\y)_o \le 2\d\}>0,
\]
where $\d$ is a hyperbolic constant of $\Cay(\G, S)$ and
$c_0$ is positive since $(\x, \y)$ are uniformly away from the diagonal.
We fix $0<\e<c_0/8$.
Note that for any $[z] \in \conj_\G$ of an infinite order element,
there exists a representative $z$ such that $(z_-|z_+)_o \le 2\d$.
Therefore, for all large enough $n$, we have that
\[
\sum_{[z] :\ |\log \lambda(\rho(z))-n|\le L}\exp\(-s \log\lambda(\rho_\ast(z))\) \le e^{|s|c_\e}\sum_{z:\ |\log\|\rho(z)\|-n|\le L+c_\e}\exp\(-s\log\|\rho_\ast(z)\|\),
\]
implying \eqref{Eq:Knieper1}.

Second let us show that for every $s \in \R$,
there exist positive constants $C, c>0$ such that
\begin{equation}\label{Eq:Knieper2}
\Pc_{\rho, \rho_\ast, s}(n, L) \le C n^2\,\Cc_{\rho, \rho_\ast, s}(n, L+c),
\end{equation}
for all large enough $n$.
Observe that $\Pc_{\rho, \rho_\ast, s}(n, L)$ changes only by a multiplicative constant independent of $n$ under restricting the sum on a finite index subgroup of $\G$.
By the Selberg lemma applied to $\rho(\G)$, passing to a finite index subgroup in $\G$ if necessary,
we assume that the image $\rho(\G)$ consists of infinite order elements.

Let us take $x, y$ and define $U, V, U', V', U''$ and $V''$ as in Lemma \ref{Lem:CoornaertKnieper}.
Applying Lemma \ref{Lem:Benoist} to both $\rho$ and $\rho_\ast$ simultaneously, we have that
for each $0<\e<1/8$,
there exist a positive constant $c_\e>0$ and a finite set $F_\e$ in $\G$ such that
if $z \in \G \setminus F_\e$ and $\llangle \check E_\rho(z_-), E_\rho(z_+)\rrangle \ge 8 \e$,
\[
\log \lambda(\rho(z)) \le \log \|\rho(z)\| \le \log \lambda(\rho(z))+c_{\e}.
\]
and similarly for $\rho_\ast(z)$.
\[
r_{x,R'}:=\inf\{\llangle \check E_\rho(\x), E_\rho(\y) \rrangle \ : \ (\x, \y) \in O(o, x^{-1}, R') \times O(o, x, R') \}>0,
\]
for an $R'>R$ depending only on $x$,
we fix $0<\e<r_{x, R'}/8$.
Note that
if $z \in U(x^{-1}, x, R)$,
then $\llangle \check E_\rho(z_-), E_\rho(z_+)\rrangle \ge 8 \e$.

Given $z \in U(x^{-1}, x, R)$ with $|z|=k$,
the set
\[
C_k(z; x, L, R):=\{g\langle z\rangle \in \G/\langle z\rangle \ : \ g z g^{-1} \in U(x^{-1}, x, R), \ ||z|-k| \le L\}
\]
has cardinality at most $C_{x, R, L, \d}k$ by Lemma \ref{Lem:conj}.
Since $c_1^{-1}|z|-c_2 \le \log \|\rho(z)\| \le c_1|z|$ (see \eqref{Eq:dom_QI}) with constants $c_1, c_2$ depending only on the representation $\rho$ and the chosen word metric,
for $z \in U(x^{-1}, x, R)$ with $|\log \|\rho(z)\|-n|\le L$,
one has
\[
\sum_{c_1^{-1}(n-L)\le k \le c_1(n+L+c_2)}\#C_k(z; x, L, R) \le C_{x, L, R, \d}\cdot n^2.
\]
Therefore, we have for all large enough $n$,
\begin{align*}
&\sum_{z \in U(x^{-1}, x, R), \ |\log \|\rho(z)\|-n| \le L} \exp\(-s \log \|\rho_\ast(z)\|\) \\
&\qquad \qquad \qquad \qquad \qquad \qquad \le C_{x, L, R, \d}\cdot n^2\sum_{[z]: \ |\log\lambda(\rho(z))-n|\le L+c_{\e, x, R}}\exp\(-s \log \lambda(\rho_\ast(z))+|s|c_{\e}\),
\end{align*}
where $n$ is large enough so that $z \notin F_{x, y, R}\cup F_\e$ by the inequality $\log \|\rho(z)\| \le c_1 |z|$ in \eqref{Eq:dom_QI}.
By \eqref{Eq:x3} and \eqref{Eq:SUV} in Lemma \ref{Lem:CoornaertKnieper}, if we define
\[
\wt L:=L+6\log \|\rho(x)\|+\log \|\rho(y)\|+\log \|\rho(y^{-1})\|+c_{\e, x, R},
\]
in analogous inequalities for $S(U', V')$ and $S(U'', V'')$ and combine them,
then for all large enough $n$,
\[
\sum_{z: \ |\log\|\rho(z)\|-n|\le L}\exp\(-s \log \|\rho_\ast(z)\|\)\le C_{\e, x, y, L, R, \d}\cdot n^2\sum_{[z]: \  |\log \lambda(\rho(z))-n| \le \wt L}\exp\(-s \log \lambda(\rho_\ast(z))\),
\]
implying \eqref{Eq:Knieper2}, as required.
The claim follows \eqref{Eq:Knieper1} and \eqref{Eq:Knieper2} combined with Lemma \ref{Lem:growth}.
\qed

\begin{theorem}\label{Thm:Manhattan_rep}
Fix an integer $m \ge 2$.
For any pair of dominated representations $\rho$ and $\rho_\ast$ from a non-elementary hyperbolic group $\G$ to $\GL(m, \R)$,
the Manhattan curve for the pair $(\rho, \rho_\ast)$ is real analytic.
\end{theorem}

\proof
Let $c_{\rho_\ast}(x, \x)$ and $c_{\rho}(x, \x)$ be the cocycles associated to $\rho_\ast$ and $\rho$ respectively.
If we define
\[
\Psi_\ast(\o):=c_{\rho_\ast}(w(\o_0), w_\ast(\o)) \quad \text{and} \quad \Psi(\o):=c_{\rho}(w(\o_0), w_\ast(\o)) \quad \text{for $\o \in \SS_0$},
\]
then $\Psi_\ast$ and $\Psi$ define H\"older continuous potentials on the shift space $(\SS_0, \s)$ for an enhanced coding in Theorem \ref{Thm:enhanced}.
For each $s \in \R$, let $t:=\th(s)$ the value at $s$ of the pressure curve for the pair $(\rho, \rho_\ast)$.
Letting $\p_\ast(x, y):=\log \|\rho_\ast(x^{-1}y)\|$ and $\p(x, y):=\log \|\rho(x^{-1}y)\|$,
we define $\t_{s, t}$ as the local intersection number of $s \p_\ast+t\p$ relative to $\hat d$.
The rest follows as in the proof of Theorem \ref{Thm:Manhattan}.
\qed

\begin{proposition}\label{Prop:LDP}
Fix an integer $m \ge 2$.
Let $(\rho, \rho_\ast)$ be a pair of dominated representations from a non-elementary hyperbolic group $\G$ to $\GL(m, \R)$.
If we fix a large enough constant $L>0$ and define
\[
\Sc_{\rho, L}(n):=\{x\in \G:\ |\log \|\rho(x)\|-n|\le L\},
\]
then the large deviation principle holds for the sequence $(1/n)\log \|\rho_\ast(x)\|$
where $x$ is distributed according to the uniform counting measure on $\Sc_{\rho, L}(n)$ for $n=1, 2, \dots$.
More precisely, for any open set $U$ and any closed set $V$ in $\R$ such that $U \subset V$,
we have that
\begin{align*}
-\inf_{t \in U}I(t)
\liminf_{n \to \infty}\frac{1}{n}\log
\frac{1}{\#\Sc_{\rho, L}(n)}\#\left\{x \in \Sc_{\rho, L}(n) \ : \ \frac{1}{n}\log \|\rho_\ast(x)\| \in U\right\}\\
\limsup_{n \to \infty}\frac{1}{n}\log
\frac{1}{\#\Sc_{\rho, L}(n)}\#\left\{x \in \Sc_{\rho, L}(n) \ : \ \frac{1}{n}\log \|\rho_\ast(x)\| \in V\right\}
\le -\inf_{t \in V}I(t),
\end{align*}
\[
I(t):=\sup_{s \in \R}\{ts-\th(-s)\}+\th(0) \quad \text{for $t \in \R$},
\]
and $\th(s)$ is the pressure curve for the pair of dominated representations $(\rho, \rho_\ast)$.

Furthermore, for a large enough constant $L>0$,
if we define
\[
\Cc_{\rho, L}(n):=\{[x] \in \conj_\G \ : \ |\log \lambda(\rho(x))-n|\le L\},
\]
then the large deviation principle holds for the sequence $(1/n)\log \lambda(\rho_\ast(x))$ where $[x]$ is distributed according to the uniform counting measure on $\Cc_{\rho, L}(n)$ for $n=1, 2, \dots$ with the same rate function $I$.
\end{proposition}

\proof
By Lemma \ref{Lem:growth}, for each $s \in \R$, we have that
\[
\lim_{n \to \infty}\frac{1}{n}\log \frac{1}{\# \Sc_{\rho, L}(n)}\sum_{x: \ |\log \|\rho(x)\|-n|\le L}\exp\(s \log \|\rho_\ast(x)\|\)=\th(-s)-\th(0).
\]
Since $\th(-s)$ is continuously differentiable (in fact, real analytic) by Theorem \ref{Thm:Manhattan_rep},
the G\"artner-Ellis theorem (e.g., \cite[Theorem 2.3.6]{DemboZeitouni}) implies the first claim.

Concerning the second claim, by Proposition \ref{Prop:Knieper},
we have that for each $s \in \R$,
\[
C_{s, L, R}^{-1}\,n^{-2}e^{\th(s)n} \le \sum_{[x]: \ |\log \lambda(\rho(x))-n| \le L}\exp\(-s \log \lambda(\rho_\ast(x))\) \le C_{s, L, R}\,e^{\th(s)n},
\]
for all large enough $n$.
Hence for all large enough $L>0$ and for each $s \in \R$,
\[
\lim_{n \to \infty}\frac{1}{n}\log \frac{1}{\#\Cc_{\rho, L}(n)}\sum_{x: \ |\log \lambda(\rho(x))-n|\le L}\exp\(s \log \lambda(\rho_\ast(x))\)=\th(-s)-\th(0).
\]
The rest follows as in the proof of the first claim.
\qed

\begin{theorem}\label{Thm:intersection_norm}
Let $(\rho, \rho_\ast)$ be a pair of dominated representations from a non-elementary hyperbolic group $\G$ to $\GL(m, \R)$ for $m \ge 2$.
There exists a finite positive constant $\t(\rho_\ast/\rho)$ as the following limit:
\[
\t(\rho_\ast/\rho)=\lim_{T \to \infty}\frac{1}{\#\Bc_{\rho}(T)}\sum_{x \in \Bc_{\rho}(T)\setminus \Bc_\rho(0)}\frac{\log \|\rho_\ast(x)\|}{\log \|\rho(x)\|},
\]
where $\Bc_{\rho}(T):=\{x \in \G \ : \ \log \|\rho(x)\| \le T\}$ for positive real numbers $T$.

Moreover, we have that
\[
v_\rho \le \t(\rho_\ast/\rho)v_{\rho_\ast},
\]
where $v_\rho$ and $v_{\rho_\ast}$ are the exponential growth rates for $\rho$ and $\rho_\ast$, respectively.
\end{theorem}

We call the limit $\t(\rho_\ast/\rho)$ in Theorem \ref{Thm:intersection_norm} the {\it intersection number} for a pair of dominated representations $(\rho, \rho_\ast)$.

\proof[Proof of Theorem \ref{Thm:intersection_norm}]
Applying the cocycle $v_\rho c_\rho(x, \x)$ to Proposition \ref{Prop:PSpotential},
we obtain a finite Borel measure $\m_\rho$ satisfying \eqref{Eq:QC} and the exponent is $0$ relative to any metric $d$ in $\Dc_\G$ and for positive constants $C, L>0$,
\begin{equation}\label{Eq:mrho}
C^{-1} \le \exp(v_\rho \log \|\rho(x)\|)\cdot \m_\rho(\Oc(x, R)) \le C \quad \text{for all $x \in \G$},
\end{equation}
\begin{equation}\label{Eq:count}
C^{-1}\exp(v_\rho n) \le \#\{x \in \G \ : \ |\log \|\rho(x)\|-n| \le L\} \le C \exp(v_\rho n),
\end{equation}
for all positive integers $n$ by letting $s=0$ in Lemma \ref{Lem:growth}.
Fix a metric $d$ in $\Dc_\G$.
By Lemma \ref{Lem:localint} applied to the $\G$-invariant tempered potentials
\[
\psi_\ast(x, y):=\log \|\rho_\ast(x^{-1}y)\| \quad \text{and} \quad \psi(x, y):=\log \|\rho(x^{-1}y)\|,
\]
there exist constants $\t_\ast$ and $\t$ such that
\[
\t^{\psi_\ast}(\x)=\t_\ast \quad \text{and} \quad \t^{\psi}(\x)=\t \quad\text{for $\m_\rho$-almost every $\x \in \partial \G$}.
\]
Namely, for $\m_\rho$-almost every $\x \in \partial \G$,
we have that $\psi_\ast(o, \g_\x(n))/n \to \t_\ast$ as $n \to \infty$
where $\g_\x$ is any rough geodesic ray relative to $d$ issuing from $o$,
and similarly for $\psi(o, \g_\x(n))$.
This implies that
\[
\frac{\psi_\ast(o, \g_\x(n))}{\psi(o, \g_\x(n))} \to \frac{\t_\ast}{\t} \quad \text{for $\m_\rho$-almost every $\x \in \partial \G$ as $n \to \infty$}.
\]
\[
S_{n, \e}:=\left\{x \in \G \ : \ |\psi_\ast(o, x)-n (\t_\ast/\t)|>\e n \right\} \quad \text{for $n \ge 0$ and $\e>0$},
\]
and $S(n, L):=\{x \in \G \ : \ |\psi(o, x)-n|\le L\}$.
For a large enough $R>0$,
the shadows $\Oc(x, R)$ for $x \in S(n, L)$ cover the boundary $\partial \G$ with a uniformly bounded multiplicity.
Therefore we have by \eqref{Eq:mrho} and \eqref{Eq:count},
\begin{equation*}%\label{Eq:wLLN}
\frac{\#\(S_{n, \e}\cap S(n, L)\)}{\# S(n, L)} \le C'\sum_{x \in S_{n, \e}\cap S(n,L)}\m_\rho(\Oc(x, R))\le C''\m_\rho\left(\bigcup_{x \in S_{n,\e}\cap S(n, L)}\Oc(x, R)\right) \to 0,
\end{equation*}
as $n \to \infty$,
where the last term tends to $0$ as $n$ goes to infinity by the Fatou lemma; indeed
for $\x \in \Oc(x, R)$ and $x \in S_{n, \e}\cap S(n, L)$,
if $d(\g_\x(k), x) \le R$,
\[
|\psi_\ast(o, \g_\x(k))- (\t_\ast/\t)\psi(o, \g_\x(k))| \ge \e \psi(o, \g_\x(k))-R_0,
\]
where $R_0$ is a constant independent of $n$; it depends only on the constants in the condition \eqref{Eq:RG} for $\psi_\ast$ and $\psi$, the metric $d$, $\e$, $L$ and $R$.
This implies that for any given $\e>0$, for a uniformly distributed random element in $S(n, L)$,
we have $|\psi_\ast(o, x)-(\t_\ast/\t)\psi(o, x)| \le \e \psi(o, x)$ with probability at least $1-\e$ for all large enough $n$, and thus
\[
\frac{\t_\ast}{\t}=\lim_{n \to \infty}\frac{1}{\#S(n, L)}\sum_{x \in S(n, L)}\frac{\psi_\ast(o, x)}{n}.
\]
For any real $T>0$, let $n:=\lfloor T\rfloor$ be the largest integer at most $T$.
If $x_n$ is distributed according to the uniform counting measure on the set
$\Bc_{\rho}(T):=\{x \in \G \ : \ \log \|\rho(x)\| \le T\}$,
then since $x_n \notin S(n, L)$ with probability at most $O(e^{-v_\rho L})$ by \eqref{Eq:count} and the probability that $x_n \in S_{n, \e}\cap S(n, L)$ tends to $0$ as $n$ goes to infinity,
the event that $x_n \in S_{n, \e}$ has the probability at most $O(e^{-v_\rho L})$ for all large enough $n$.
Hence letting $T \to \infty$ and then $L \to \infty$,
we have that
\[
\frac{\t_\ast}{\t}=\lim_{T \to \infty}\frac{1}{\#\Bc_{\rho}(T)}\sum_{x \in \Bc_{\rho}(T)\setminus \Bc_\rho(0)}\frac{\psi_\ast(o, x)}{\psi(o, x)},
\]
since $\Bc_\rho(0)$ is finite by \eqref{Eq:dom_QI}.
Defining $\t(\rho_\ast/\rho):=\t_\ast/\t$, we obtain the first claim.

Moreover, the above discussion shows that for all small enough $\e>0$ and all large enough $L$ and $T$,
\[
(1-C e^{-v_\rho L})\# \Bc_\rho(T) \le \#\Bc_{\rho_\ast}((\t(\rho_\ast/\rho)+\e)T),
\]
where $C$ is independent of $L$ or $T$.
\[
c_1 \exp(v_\rho T) \le \#\Bc_\rho(T)\le c_2 \exp(v_\rho T),
\]
and similarly for $\#\Bc_{\rho_\ast}(T)$,
by taking logarithms, dividing by $T$ and letting $T \to \infty$,
we obtain the second claim.
\qed

\begin{corollary}\label{Cor:zero}
The rate function $I$ for the large deviation principles in Proposition \ref{Prop:LDP} has a unique zero at $\t(\rho_\ast/\rho)$.
\end{corollary}

\proof
We have $\th'(0)=-\t(\rho_\ast/\rho)$ by Theorem \ref{Thm:Manhattan_rep} and by the second claim of Theorem \ref{Thm:Manhattan} adapted to the pair of representations $(\rho, \rho_\ast)$
(cf.\ the proof of Theorem \ref{Thm:intersection_norm}).
Thus $I$ has a zero at $\t(\rho_\ast/\rho)$.
If $t \neq \t(\rho_\ast/\rho)$, then $I(t)>0$
since $\th$ is convex and differentiable at $0$.
\qed

\medskip

The following is shown in \cite[Theore 3.7 and (12) in p.1113]{BCLS}.
We provide another proof by using the large deviation principles in Proposition \ref{Prop:LDP}.

\begin{theorem}\label{Thm:intersection_rep}
Let $(\rho, \rho_\ast)$ be a pair of dominated representations from a non-elementary hyperbolic group $\G$ to $\GL(m, \R)$ for $m \ge 2$.
There exists a finite positive constant $\t(\rho_\ast/\rho)$ as the following limit:
\[
\t(\rho_\ast/\rho)=\lim_{T \to \infty}\frac{1}{\#\Cc_{\rho}(T)}\sum_{[x] \in \Cc_{\rho}(T)\setminus \Cc_\rho(0)}\frac{\log \lambda(\rho_\ast(x))}{\log \lambda(\rho(x))},
\]
where $\Cc_{\rho}(T):=\{[x] \in \conj_\G \ : \ \log \lambda(\rho(x)) \le T\}$ for positive real numbers $T$.
\end{theorem}

\proof
We recall that for a fixed large enough $L>0$ and for positive integers $n$,
\[
\Cc_{\rho, L}(n)=\{[x] \in \conj_\G \ : \ |\log \lambda(\rho(x))-n|\le L\},
\]
in the second claim in Proposition \ref{Prop:LDP}; it shows that
since the rate function $I$ has a unique zero at $\t(\rho_\ast/\rho)$ by Corollary \ref{Cor:zero},
for each $\e>0$ there exists $0<c_\e<v_\rho$ such that for all large enough $n$,
\[
\#\{[x] \in \Cc_{\rho, L}(n) \ : \ |\log \lambda(\rho_\ast(x))-n\t(\rho_\ast/\rho)| \ge \e n\} \le \#\Cc_{\rho, L}(n)\cdot e^{-c_\e n} \le C e^{(v_\rho-c_\e)n},
\]
where in the last inequality we have used $\#\Cc_{\rho, L}(n) \le C e^{v_\rho n}$ by Proposition \ref{Prop:Knieper} at $s=0$.
\begin{align*}
\sum_{n=0}^{\lceil T\rceil}\#\{[x] \in \Cc_{\rho, L}(n) \ : \ |\log \lambda(\rho_\ast(x))-n\t(\rho_\ast/\rho)| \ge \e n\}
\le \sum_{n=0}^{\lceil T\rceil}C e^{(v_\rho-c_\e)n} \le C' e^{(v_\rho-c_\e)T},
\end{align*}
for all $T>0$,
where $\lceil T\rceil$ is the smallest integer greater than or equal to $T$.
we have a positive constant $C>0$ such that for all large enough $T$ and $n:=\lfloor T\rfloor$,
\[
\#\Cc_\rho(T) \ge \#\Cc_{\rho, L}(n-\lceil L \rceil) \ge \frac{1}{Cn^2}e^{v_\rho n},
\]
which follows by letting $s=0$ in Proposition \ref{Prop:Knieper}.
Therefore we obtain for each $\e>0$,
\begin{align*}
&\frac{1}{\#\Cc_\rho(T)}\#\{[x] \in \Cc_\rho(T) \ : \ |\log \lambda(\rho_\ast(x))-\t(\rho_\ast/\rho)\log \lambda(\rho(x))| \ge \e \log \lambda(\rho(x))\}\\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \le \frac{C n^2}{e^{v_\rho n}}C' e^{(v_\rho-c_\e)n}
\le C'' n^2 e^{-c_\e n} \to 0 \quad \text{as $n \to \infty$},
\end{align*}
implying that
\[
\frac{1}{\#\Cc_\rho(T)}\sum_{[x] \in \Cc_\rho(T)\setminus\Cc_\rho(0)}\frac{\log \lambda(\rho_\ast(x))}{\log \lambda(\rho(x))} \to \t(\rho_\ast/\rho) \quad \text{as $T \to \infty$},
\]
since $\Cc_\rho(0)$ is finite; \eqref{Eq:dom_QI_sp} and there are only finitely many conjugacy class of torsion elements (e.g., \cite[Lemma 2.3.6]{Calegari}),
as claimed.
\qed

\subsection{Real analytic variations of dominated representations}

Let $\Uc$ be a real analytic manifold.
We say that a family of representation $\rho_u: \G \to \GL(m, \R)$ parameterized by $u \in \Uc$ is a \textit{real analytic family} if for each $x \in \G$, the map $u \mapsto \rho_u(x)$ yields a real analytic map with values in $\GL(m, \R)$.
Recall that associated with any dominated representation $\rho$,
there exists a pair of H\"older continuous $\rho$-equivariant maps,
\[
\check E_\rho: \partial \G\to \Pb({\R^m}^\ast) \quad \text{and} \quad E_\rho:\partial \G \to \Pb(\R^m).
\]
Let us consider the spaces $C^\a(\partial \G, \Pb(\R^m))$ and $C^\a(\partial \G, \Pb({\R^m}^\ast))$ of $\a$-H\"older maps for some $\a>0$ as Banach manifolds.

\begin{theorem}[Theorem 6.1 in [2] and Theorem 6.1 in [11]]\label{Thm:RA}
Let $\rho_u:\G \to \GL(m, \R)$ for $u \in \Uc$ be a real analytic family of representations.
If $\rho_{u_0}$ is dominated at $u_0 \in \Uc$,
then there exists a neighborhood $\Uc'$ of $u_0$ in $\Uc$ such that
for every $u \in \Uc'$,
the representations $\rho_u$ are dominated.
Moreover, for some $\a>0$, the map from $\Uc'$ to $C^\a(\partial \G, \Pb(\R^m))$, $u \mapsto E_{\rho_u}$ and the map from $\Uc'$ to $C^\a(\partial \G, \Pb({\R^m}^\ast))$, $u \mapsto \check E_{\rho_u}$ are real analytic, respectively.
\end{theorem}

The following is shown in \cite[Propositions 3.12 and 8.1]{BCLS}.
We give a proof by using Theorem \ref{Thm:enhanced}.

\begin{theorem}
Let $\rho_{u}$ for $u \in \Uc$ and $\rho_{\ast v}$ for $v \in \Vc$ be two real analytic families of dominated representations from $\G$ to $\GL(m, \R)$ for $m \ge 2$.
The exponential growth rate $v_{\rho_u}$ is a real analytic function on $\Uc$, and
the intersection number $\t(\rho_{\ast v}/\rho_u)$ is a real analytic function on $\Uc \times \Vc$.
\end{theorem}

\proof
We consider $(\SS_0, \s)$ in Theorem \ref{Thm:enhanced}.
Letting $c_{\rho_u}(x, \x)$ be the cocycle associated with $\rho_u$ for $u \in \Uc$,
we define
\[
\Psi_u(\o):=c_{\rho_u}(w(\o_0), w_\ast(\o)) \quad \text{for $\o \in \SS_0$}.
\]
Note that $\Psi_u$ is a H\"older continuous potential for each $u \in \Uc$.
Since $v_{\rho_u}c_{\rho_u}(x, \x)$ has the exponent $0$ with respect to any metric in $\Dc_\G$,
by Lemma \ref{Lem:Pr}, we have $\Pr(v_{\rho_u}\Psi_u)=0$, and thus
$\Pr_{\Cc_0}(v_{\rho_u}\Psi_u)\le 0$.
For each $u \in \Uc$, there exists a $\G$-invariant Radon measure $\L_u$ on $\partial^2 \G$ equivalent to $\m_{\check \rho_u}\otimes \m_{\rho_u}$.
By Proposition \ref{Prop:code-measure}(2),
there exists a $\s$-invariant Borel probability measure $\lambda_u$ on $\SS_0$ such that
\[
w_\ast \lambda_u \otimes \Leb_{[0, T_0)}\le c \L_u \otimes dt \qquad \text{on $\partial^2 \G \times \R$},
\]
for some positive constants $T_0, c>0$, further $w_\ast \lambda_u \le C \m_{\check \rho_u}\otimes \m_{\rho_u}$ on $\partial^2 \G$ for some positive constant $C>0$.
For the fixed strongly hyperbolic metric $\hat d$ used to define $\Sus(\SS_0, r_0)$ in Theorem \ref{Thm:enhanced},
let $\hat q$ be the corresponding quasi-metric in $\partial \G$.
We then have that
\[
D_{\hat q}(\x_-, \x_+; \L_u)=\hat \t_u \qquad \text{for $\L_u$-almost all $(\x_-, \x_+) \in \partial^2 \G$},
\]
where $\hat \t_u$ is the local intersection number of $\log \|\rho_u(x^{-1}y)\|$ relative to $\hat d$ for $\m_{\rho_u}$-almost every point in the boundary by Lemma \ref{Lem:localint}.
Furthermore, Lemma \ref{Lem:localint_potential} shows that
\[
\int_{\SS_0}\Psi_u\,d\lambda_u=\hat \t_u\int_{\SS_0}\Psi_B\,d\lambda_u.
\]
Lemma \ref{Lem:dim-ent} shows that
\[
h(\s, \lambda_u)+\int_{\SS_0}\Psi_u\,d\lambda_u=h(\s, \lambda_u)+\hat \t_u\int_{\SS_0}\Psi_B\,d\lambda_u \ge 0,
\]
and by the variational principle (Proposition \ref{Prop:VP}),
we have $\Pr_{\Cc_0}(v_{\rho_u}\Psi_u)\ge 0$.
Hence $\Pr_{\Cc_0}(v_{\rho_u}\Psi_u)=0$.
Proposition \ref{Prop:VP} shows that $\lambda_u$ is the unique equilibrium state for $\Psi_u$.
Note that $\Uc\times \R \to \R$, $(u, s)\mapsto \Pr_{\Cc_0}(s\Psi_u)$ is real analytic by Theorems \ref{Thm:TF}(2) and \ref{Thm:RA}.
Furthermore, we have that
\[
\frac{\partial}{\partial s}\Big|_{s=v_{\rho_u}}\Pr_{\Cc_0}(s\Psi_u)=\int_{\SS_0}\Psi_u\,d\lambda_u <0,
\]
where the last strict inequality holds since $S_n \Psi_u<0$ for all large enough $n$ by Lemmas \ref{Lem:rep_RG} and \ref{Lem:rep_QE} together with \eqref{Eq:dom_QI}.
The implicit function theorem shows that on an open neighborhood $\Uc_0$ of $u$ in $\Uc$, the function $w \mapsto v_{\rho_w}$ for $w \in \Uc_0$ is real analytic.
Since this holds for each $u \in \Uc$, the function $u \mapsto v_{\rho_u}$ for $u \in \Uc$ is real analytic,
concluding the first claim.

Let $\Psi_{\ast v}(\o):=c_{\rho_{\ast v}}(w(\o_0), w_\ast(\o))$ for $\o \in \SS_0$ a family of H\"older continuous potentials associated with $\rho_{\ast v}$ for $v \in \Vc$.
the function $\Uc \times \R \to \R$ and the function $\Uc\times \Vc \times \R \to \R$ defined by
\[
(u, s) \mapsto \Pr_{\Cc_0}(s\Psi_u), \quad \text{and} \quad (u, v, s) \mapsto \Pr_{\Cc_0}(v_{\rho_u}\Psi_u+s \Psi_{\ast v}),
\]
respectively, are real analytic by Theorems \ref{Thm:TF}(2) and \ref{Thm:RA} combined with the real analyticity of $v_{\rho_u}$ in $u \in \Uc$ in the first claim.
Hence for $u \in \Uc$ and for $(u, v) \in \Uc \times \Vc$, the following values depend real analytically in $u$ and in $(u, v)$, respectively,
\[
\frac{\partial}{\partial s}\Big|_{s=v_{\rho_u}}\Pr_{\Cc_0}(s \Psi_u)=\int_{\SS_0}\Psi_u\,d\lambda_u
\quad \text{and} \quad
\frac{\partial}{\partial s}\Big|_{s=0}\Pr_{\Cc_0}(v_{\rho_u}\Psi_u+s\Psi_{\ast v})=\int_{\SS_0}\Psi_{\ast v}\,d\lambda_u.
\]
Noting that
\[
\t(\rho_{\ast v}/\rho_u)=\frac{\int_{\SS_0}\Psi_{\ast v}\,d\lambda_u}{\int_{\SS_0}\Psi_u\,d\lambda_u},
\]
by Lemma \ref{Lem:localint_potential} (cf.\ the proof of Theorem \ref{Thm:intersection_norm}).
Therefore, we conclude that $\t(\rho_{\rho_{\ast v}}/\rho_u)$ is a real analytic function on $\Uc \times \Vc$ and thus the second claim.
\qed

\section{Examples}\label{Sec:example}

\subsection{A free product group}\label{Sec:free}

Consider the following group with presentation
\[
\G:=\langle s_1, s_2 \ | \ s_1^4, s_2^6\rangle,
\]
which is isomorphic to the free product $(\Z/4)\ast(\Z/6)$.
Let $S:=\{s_1, s_1^{-1}, s_2, s_2^{-1}\}$.
We define a family of presentations $\rho_u$ of $\G$ into $\SL(2, \R)$ by setting
\[
\rho_u(s_1):=A_{-u}R(\pi/2)A_u \quad \text{and} \quad \rho_u(s_2):=R(\pi/3),
\]
\[
\begin{pmatrix}
e^u & 0\\
0 & e^{-u}
\end{pmatrix}
\quad \text{and} \quad
\begin{pmatrix}
\cos {t} & -\sin{t}\\
\sin {t} & \cos {t}
\end{pmatrix}
\quad \text{where $u \in \R$ and $t\in [0, \pi)$}.
\]
It indeed defines a representation since $\rho_u(s_1)^2=\rho_u(s_2)^3=-I$,
and further it descends to $\PSL(2, \R)$.
Note that $\{\rho_u\}_{u \in \R}$ is a real analytic family.
If $|u|$ is large enough, then $\rho_u$ defines a dominated representation (cf.\ \cite[Section 5.5]{BochiPotrieSambarino}).
For $u \in \R$ such that $\rho_u$ is dominated,
let $\m_{\rho_u}$ be an associated finite Borel measure on $\partial \G$ (see the end of Section \ref{Sec:dom_rep}), and $\t(\rho_u/d_S)$ be the local intersection number of the associated tempered potential with $\rho_u$ over the word metric $d_S$.
By construction, if $u$ is large, then $\t(\rho_u/d_S)$ is large.
Since the local intersection number is constant $\m_{\rho_u}$-almost everywhere on $\partial \G$ (see Lemma \ref{Lem:localint}),
if $\t(\rho_u/d_S) \neq \t(\rho_{u'}/d_S)$,
then $\m_{\rho_u}$ and $\m_{\rho_{u'}}$ are mutually singular for $u, u' \in \R$.

\subsection{A surface group with the translation generators}\label{Sec:surface}
Let us consider the fundamental group of closed Riemann surface of genus $2$ with the set of generators $S:=\{s_i, s_i^{-1}, i=1,2,3,4\}$ arising from identifying pairs of opposite sides of a regular octagon in the hyperbolic plane, i.e.,
\[
\G:=\langle s_1, s_2, s_3, s_4 \ | \ s_4^{-1}s_3^{-1}s_2^{-1}s_1^{-1}s_4 s_3 s_2 s_1\rangle.
\]
We will refer to $S$ as the set of {\it translation generators}.
We define a representation of $\G$ into $\SL(2, \R)$ which acts via $\PSL(2, \R)$ and admits a hyperbolic regular octagon as a fundamental polygon.
\[
t:=\frac{\pi}{8} \quad \text{and} \quad u:=\log\frac{\sqrt{\cos 2t}+\cos t}{\sin t},
\]
we define a homomorphism
\[
\rho: \G \to \SL(2, \R)
\]
\[
\rho(s_1):=A_u,
\quad
\rho(s_2):=R(3t)A_u R(-3t),
\]
\[
\rho(s_3):=R(6t)A_u R(-6t),
\quad \text{and} \quad
\rho(s_4):=R(t)A_u R(-t),
\]
where we have used the same notations in the previous Section \ref{Sec:free}.
One can check that $\rho$ is indeed a homomorphism since the group generated by $\rho(s_i)$, $i \in \{1, 2, 3, 4\}$ admits a regular octagon as a Dirichlet fundamental domain and they satisfy the required relation.
Moreover, the representation $\rho$ is dominated;
this follows since the associated topological flow is H\"older orbit equivalent to the classical geodesic flow on $\rho(\G)\backslash \SL(2, \R)/{\{\pm I\}}$ (cf.\ \cite[Theorem A]{BochiGourmelon}).
More concretely, one can describe a strictly invariant family of multicones of index $1$ (Figure \ref{Fig:multicones}; see \cite[Section 5]{BochiPotrieSambarino} for the details).

\begin{figure}
\centering
\includegraphics[width=120mm, bb=0 0 595 842]{FigM.pdf}
\caption{A part of a strictly invariant family of multicones of index $1$ for $(\G, S)$, where $\G$ is a genus two surface group and $S$ is the translation generators.
The arrows indicate the destinations of intervals after translation by a single generator $s \in S$ (bottom), and an expanded picture shows a small portion of the disk bounded by the thick arc (top).}
\label{Fig:multicones}
\end{figure}

The group $\G$ with the set of generators $S$ admits a deterministic automaton on $8$ letters with $37$ states [26].
The underlying directed graph has a unique strongly connected component with $36$ states.
We have performed a Markov chain $\{w_n\}_{n=0, 1, \dots}$ on the strongly connected component where the transition probability is given by the Parry measure; the extreme point of sample path is distributed according to a Patterson-Sullivan measure $\m_S$ defined in terms of the word metric for $S$ (cf.\ \cite[Section 4.2]{CalegariFujiwara2010} and Figure \ref{Fig:histogram_translation} (left)).
A numerical experiment shows that
\[
\t(\rho/d_S)=\lim_{n \to \infty}\frac{1}{n}\Eb\log \s_1(\rho(w_n)) \quad \text{is approximately}\quad
\]
(where $\Eb$ stands for the expectation-----we compute an average over the number of samples $10^4$ with the number of steps $n=10^3$).
The exponential growth rate of $(\G, S)$ is
\[
\]
and $v_\rho=2$ (since the exponential growth rate for the hyperbolic metric is $1$ and $2\log \s_1(\rho(w))$ coincides with the displacement by $\rho(w)$ in the hyperbolic plane).
Computations suggest that
\[
\frac{v_S}{v_\rho}<1.13837....
\]
In fact, we can actually show that a finite Borel measure $\m_\rho$ associated with $\rho$ (where we note that $\m_\rho$ is in the Lebesgue measure class if it is realized on the boundary of the hyperbolic plane) and a Patterson-Sullivan measure $\m_S$ associated with the word metric for $S$ are mutually singular.
The following is a sketch of the proof:
Suppose that two measures $\m_\rho$ and $\m_S$ are mutually absolutely continuous,
then the logarithm of spectral radius for $\rho$ and the stable length function are proportional on the set of conjugacy classes of $\G$; this follows from \cite[Theorem 1.2]{CT} for the pair of the hyperbolic metric and the word metric.
However, the values for the one associated with $\rho$ is not arithmetic, i.e., not included in a discrete subgroup in $\R$ \cite[Proposition 2.1]{Dalbo} whereas the values for the one associated with the word metric are contained in a discrete set of rational numbers $\Z[1/N]$ for some $N \ge 1$ \cite[Theorem 3.17, Chapter III.$\Gamma$]{BridsonHaefliger}, yielding a contradiction.
%(See Figure \ref{Fig:histogram_translation} (left)).

Concerning the harmonic measure, we have performed the simple random walks $\{w_n\}_{n=0, 1, \dots}$ on the Cayley graph for $(\G, S)$ (Figure \ref{Fig:histogram_translation} (right)).
Let $\n$ be the associated harmonic measure on the boundary $\partial \G$, and $\L_{\rm harm}$ be a $\G$-invariant Radon measure in the class of $\check \n\otimes \n$ on $\partial^2 \G$ (see Section \ref{Sec:RW}).
A numerical experiment provides that
\[
\text{$\t(\rho/d_S: \L_{\rm harm})$ is approximately $1.12909...$.}
\]
Note that the local intersection number for $\t(\rho/d_S:\L)$ relative to an ergodic measure $\L$ is constant $\L$-almost everywhere,
and $\t(\rho/d_S)=\t(\rho/d_S: \L_S)$ where $\L_S$ is a $\G$-invariant Radon measure in the class of $\m_S\otimes \m_S$ on $\partial^2 \G$.
Therefore if $\t(\rho/d_S)\neq \t(\rho/d_S: \L_{\rm harm})$,
then a Patterson-Sullivan measure $\m_S$ and the harmonic measure $\n$ are mutually singular on $\partial \G$;
the fact itself has been shown as a special case of \cite[Theorems 1.2 and 1.3]{GMM2018}.
Moreover, we note that $\m_\rho$ and $\n$ are mutually singular \cite[Theorem 1]{KosenkoTiozzo};
see also the discussion in Remark \ref{Rem:multifractal}.

\begin{figure}
\begin{minipage}[t]{0.5\columnwidth}
\centering
\includegraphics[clip, width=65mm, bb=0 0 360 214]{FigPS.pdf}
\subcaption{A Patterson-Sullivan measure}
\end{minipage}%
\begin{minipage}[t]{0.5\columnwidth}
\centering
\includegraphics[clip, width=65mm, bb=0 0 360 214]{FigRW.pdf}
\subcaption{The harmonic measure}
\end{minipage}
\caption{The histograms (the number of steps $n=10^3$ and the number of samples $10^4$) for
the distributions of angles in $[-3.14\dots, 3.14\dots]$ of $\rho(w_n).o$ where $o$ is the center of the unit disk:
$\{w_n\}_{n=0, 1, \dots}$ is associated with the Markov chain on the strongly connected component in the underlying directed graph in an automatic structure for $(\G, S)$  (left), and $\{w_n\}_{n=0, 1, \dots}$ is the simple random walk on the Cayley graph for $(\G, S)$ (right).}
\label{Fig:histogram_translation}
\end{figure}

\subsection*{Acknowledgment}
We thank Eduardo Oreg\'on-Reyes for his helpful comments and the referee for a careful review that helped us to improve the presentation of the paper.
R.T.\ is partially supported by JSPS Grant-in-Aid for Scientific Research JP20K03602.
This work was supported by the Research Institute for Mathematical
Sciences, an International Joint Usage/Research Center located
in Kyoto University.

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MESON2023

# Deciphering $\Xi^{-}$ capture events in light emulsion nuclei

Eliahu Friedman<EMAIL_ADDRESS>Institute of Physics,
The Hebrew University, Jerusalem 9190401, Israel1 Avraham Gal
<EMAIL_ADDRESS>
###### Abstract

We recently showed that all five KEK and J-PARC uniquely assigned two-body
$\Xi^{-}+{{{}^{A}}Z}\to{{}_{\Lambda}^{A^{\prime}}}{Z^{\prime}}+{{}_{\Lambda}^{A^{\prime\prime}}}{Z^{\prime\prime}}$
capture events in CNO light emulsion nuclei are consistent with Coulomb-
assisted $1p_{\Xi^{-}}$ nuclear states in a $\Xi$-nuclear potential of
nuclear-matter depth $V_{\Xi}\gtrsim 20$ MeV FG21 . Here we argue that the
recently reported 14N capture events named KINKA and IRRAWADDY are more likely
$\Xi^{0}_{1p}-{{}^{14}}$C nuclear states FG23  than $\Xi^{-}_{1s}-{{}^{14}}$N
states, the latter assignment implying considerably smaller values of
$V_{\Xi}$.

##  1 Introduction

Nuclear configurations of $\Xi$ hyperons are poorly known GHM16 ; HN18 .
Because of the large momentum transfer in the standard $(K^{-},K^{+})$
production reaction, induced by the two-body $K^{-}p\to K^{+}\Xi^{-}$
strangeness exchange reaction, $\Xi^{-}$ hyperons are produced dominantly in
the quasi-free continuum region. No $\Xi^{-}$ nor $\Lambda\Lambda$ nuclear
bound states have ever been observed unambiguously in such experiments E224 ;
E885 ; E906 . Nevertheless, an attractive $\Xi$-nuclear Woods-Saxon (WS)
potential of depth $V_{\Xi}=17\pm 6$ MeV HH21  was deduced recently from the
9Be($K^{-},K^{+}$) quasi-free $\Xi^{-}$ spectrum shape E906  shown in Fig. 1.

Figure 1: WS fit of the BNL-AGS E906 9Be($K^{-},K^{+}$) spectrum. Figure
adapted from Ref. HH21 .

A potential depth $V_{\Xi}=17\pm 6$ MeV is considerably larger than
$V_{\Xi}\lesssim 10$ MeV deduced from strong-interaction models, e.g., HALQCD
HALQCD20  (confirmed in $p\Xi^{-}$ femtoscopy ALICE19 ), EFT@NLO HM19 ; K19
and RMF Gaitanos21 . A notable exception is provided by versions ESC16*(A,B)
of the latest Nijmegen model in which values of $V_{\Xi}$ higher than 20 MeV
arise ESC16 . However, these $V_{\Xi}$ values get reduced by substantial $\Xi
NN$ three-body contributions within the same ESC16* model.

Here we focus on $\Xi$ nuclear constraints derived by observing $\Xi^{-}$
capture events in exposures of light-emulsion CNO nuclei to the
$(K^{-},K^{+})$ reaction. A small fraction of the produced high-energy
$\Xi^{-}$ hyperons slows down in the emulsion, undergoing an Auger process to
form high-$n$ atomic states, and cascades down radiatively. Strong-interaction
capture takes over atomic radiative cascade in a $3D$ atomic orbit bound by
126, 175, 231 keV in C, N, O, respectively, affected to less than 1 keV by the
strong interaction BFG99 . Nevertheless, captures from a lower orbit have also
been observed, as follows.

Table 1: Twin-$\Lambda$ two-body $\Xi^{-}$ capture events from KEK and J-PARC emulsion work. Deduced $\Xi^{-}$-nuclear binding energies $B_{\Xi^{-}}$ are contrasted with purely Coulomb $2P$ atomic binding energies $B_{\Xi^{-}}^{2P}$. Experiment | Event | ${}^{A}Z$ | ${{}_{\Lambda}^{A^{\prime}}}{Z^{\prime}}+{{}_{\Lambda}^{A^{\prime\prime}}}{Z^{\prime\prime}}$ |  $B_{\Xi^{-}}$ (MeV) |  $B_{\Xi^{-}}^{2P}$ (MeV)
---|---|---|---|---|---
KEK E176 E176  | 10-09-06 |  12C |  ${}^{4}_{\Lambda}$H+${}^{9}_{\Lambda}$Be | 0.82$\pm$0.17 | 0.285
KEK E176 E176  | 13-11-14 |  12C |  ${}^{4}_{\Lambda}$H+${}^{9}_{\Lambda}$Be∗ | 0.82$\pm$0.14 | 0.285
KEK E176 E176  | 14-03-35 |  14N |  ${}^{3}_{\Lambda}$H+${}^{12}_{\leavevmode\nobreak\ \Lambda}$B | 1.18$\pm$0.22 | 0.393
KEK E373 E373b  | KISO |  14N |  ${}^{5}_{\Lambda}$He+${}^{10}_{\leavevmode\nobreak\ \Lambda}$Be∗ | 1.03$\pm$0.18 | 0.393
J-PARC E07 E07a  | IBUKI |  14N |  ${}^{5}_{\Lambda}$He+${}^{10}_{\leavevmode\nobreak\ \Lambda}$Be | 1.27$\pm$0.21 | 0.393

Listed in Table 1 are all two-body $\Xi^{-}$ capture events
$\Xi^{-}+{{}^{A}Z}\to{{}_{\Lambda}^{A^{\prime}}}{Z^{\prime}}+{{}_{\Lambda}^{A^{\prime\prime}}}{Z^{\prime\prime}}$
to twin single-$\Lambda$ hypernuclei uniquely identified in KEK and J-PARC
light-nuclei emulsion $K^{-}$ exposures E176 ; E373b ; E07a ; E07b . Expecting
$\Lambda$ hyperons in $\Xi^{-}p\to\Lambda\Lambda$ capture to form a spin
$S=0\,\,1s_{\Lambda}^{2}$ configuration, the initial $\Xi^{-}$ hyperon and the
proton on which it is captured must satisfy $l_{\Xi^{-}}=l_{p}$ Zhu91 , which
for $p$-shell nuclear targets favors the choice $l_{\Xi^{-}}=1$. Indeed, all
the listed events are consistent with $\Xi^{-}$ capture from Coulomb-assisted
$1p_{\Xi^{-}}$ nuclear states, with $B_{\Xi^{-}}^{1p}$ larger by about 0.5 MeV
than the corresponding $2P$ atomic binding energies $B_{\Xi^{-}}^{2P}$. Not
listed in the table are multi-body capture events requiring undetected capture
products, mostly neutrons, besides a pair of single-$\Lambda$ hypernuclei. Two
such events E07b , KINKA (KEK-E373) and IRRAWADDY (J-PARC E07), correspond to
a few MeV $\Xi^{-}$ binding each, suggesting $\Xi^{-}$ capture from
$1s_{\Xi^{-}}$ nuclear states. Given that $1s_{\Xi^{-}}$ capture rates are of
order 1% of the $1p_{\Xi^{-}}$ capture rates Zhu91 ; Koike17 , this poses a
problem. Its likely resolution is discussed below.

##  2 $\Xi$ nuclear optical potential

$\Xi^{-}$ atomic and nuclear bound states in $N=Z$ nuclei such as 12C and 14N
are calculated using a finite-size Coulomb potential $V_{c}^{\Xi^{-}}$,
including vacuum-polarization terms, plus a ‘$t\rho$’ optical potential
$V_{\rm opt}^{\Xi}$ FG21  where $t$ is a spin-isospin averaged in-medium $\Xi
N$ $t$-matrix and $\rho=\rho_{n}+\rho_{p}$ is a nuclear density normalized to
the number of nucleons $A$. For $V_{\rm opt}^{\Xi}$ we adopt a form applied in
Ref. FG23a ; FG23b  to $V_{\rm opt}^{\Lambda}$:

$V_{\rm
opt}^{\Xi}(r)=-\frac{2\pi}{\mu_{\Xi}}\,b_{0}^{A}(\rho)\,\rho(r),\,\,\,\,\,\,\,\,\,b_{0}^{A}(\rho)=\frac{{\rm
Re}\,b_{0}^{A}}{1+(3k_{F}/2\pi){\rm Re}\,b_{0}^{A}}+{\rm Im}\,b_{0}.$ (1)

Here $\mu_{\Xi}$ is the $\Xi^{-}$-nucleus reduced mass, $b_{0}^{A}(\rho)$ is
an effective density-dependent $\Xi N$ isoscalar c.m. scattering amplitude,
$b_{0}^{A}=(1+\frac{A-1}{A}\frac{\mu_{\Xi}}{m_{N}})b_{0}$ transforms $b_{0}$
from the $\Xi N$ c.m. frame to the $\Xi$-nucleus c.m. frame and $k_{F}$ is the
Fermi momentum associated with density $\rho$, $k_{F}^{3}=3{\pi}^{2}\rho/2$.
This form of $V_{\rm opt}^{\Xi}$ accounts for long-range Pauli correlations in
$\Xi N$ in-medium multiple scatterings, starting at $\rho^{4/3}$ when
$b_{0}^{A}(\rho)$ is expanded in powers of the density $\rho$ DHL71 ; WRW97 .
Shorter-range correlation terms, arising in the present context from three-
body $\Xi NN$ interactions, start at $\rho^{2}$ and are briefly discussed in
the concluding section.

For $N=Z$ nuclear densities we assumed $\rho_{n}=\rho_{p}$ and identified the
r.m.s. radius of $\rho_{p}$ with that of the nuclear charge density. Folding
reasonably chosen $\Xi N$ interaction ranges other than corresponding to the
proton charge radius, varying the spatial form of the charge density, or
introducing realistic differences between neutron and proton r.m.s. radii,
made little difference: in 12C, for example, all such calculated binding
energies varied within 20% of the $\pm$0.15 MeV measurement uncertainty of
$B_{\Xi^{-}}^{1p}(^{12}$C) in Table 1.

For a given absorptivity of Im$\,b_{0}=0.01$ fm in Eq. (1),
$B_{\Xi^{-}}^{1p}(^{12}$C)=0.82$\pm$0.15 MeV was fitted by Re$\,b_{0}=0.495\pm
0.030$ fm which in the limit $A\to\infty$ and $\rho(r)\to\rho_{0}=0.17$ fm-3
leads to a depth value $V_{\Xi}=21.2\pm 0.7$ MeV in nuclear matter. This value
is compatible with that derived from AGS-E906 as shown in Fig. 1 and is in
agreement with values 21–24 MeV extracted from old emulsion events DG83 .
Disregarding Pauli correlations by setting $k_{F}=0$ leads to almost 15%
increase of the depth, whereas doubling Im$\,b_{0}$ increases the fitted
Re$\,b_{0}$ by only 1% FG21 .

##  3 $1s_{\Xi^{-}}$ states in 14N?

Figure 2: Binding energies of $\Xi^{-}$ nuclear states in 14N deduced from
$\Xi^{-}$ capture events identified by their twin-$\Lambda$ hypernuclear
decays in KEK-E373 and J-PARC E07 emulsion experiments. Figure provided by Dr.
Kazuma Nakazawa, based on recent results from Refs. E373b ; E07a ; E07b .

In addition to the KISO and IBUKI $\Xi^{-}_{1p}$ twin-$\Lambda$ capture events
listed in Table 1, two new 14N twin-$\Lambda$ capture events were reported
recently E07b , KINKA from KEK E373 and IRRAWADDY from J-PARC E07, both
assigned as $\Xi^{-}_{1s}$ in Fig. 2. We note that $2P\to 1S$ radiative decay
rates are of order 1% of $3D\to 2P$ radiative decay rates Zhu91 ; Koike17
suggesting that $\Xi^{-}$ capture from a nuclear $\Xi^{-}_{1s}$–14N state is
suppressed to this order relative to capture from a nuclear $\Xi^{-}_{1p}$–14N
state. Assigning a $\Xi^{-}_{1s}$–14N bound state to IRRAWADDY or to KINKA is
therefore questionable.

Figure 3: Level diagram of $\Xi^{-}$–14N. Shown on the right are a $3D$ atomic
state and $1p,1s$ nuclear states assigned, respectively, to E07 $\Xi^{-}$
capture events IBUKI E07a  and IRRAWADDY E07b . The $\Xi^{0}$+14C threshold at
$-$6.18 MeV is marked on the left. Electromagnetic $E1$ transitions deexciting
the $\Xi^{-}_{3D}$ atomic state to lower, slightly mixed together
$\Xi^{-}$–14N and $\Xi^{0}_{1p}$–14C nuclear states, are marked by red and
green arrowed lines, respectively. A near-threshold $\Xi^{0}_{1p}$–14C state
on the left provides an alternative interpretation of IRRAWADDY FG23 .

It has been suggested by us recently FG23  that IRRAWADDY is a near-threshold
$\Xi^{0}_{1p}$–14C bound state that has nothing to do with a
$\Xi^{-}_{1s}$–14N bound state claimed by E07. The Coulomb potential’s role in
forming a Coulomb assisted $\Xi^{-}_{1p}$–14N nuclear bound state is replaced
for $\Xi^{0}_{1p}$–14C($T=1$) by adding a strong-interaction Lane term for
isospin $T\neq 0$ nuclear cores in $V_{\rm opt}$ of Eq. (1). The sign and
strength of this Lane term relative to Re $b_{0}$ were estimated in Ref. FG23
from the sign and strength of $V_{\tau}$ relative to $V_{0}$ in the $\Xi N$
$s$-wave HALQCD underlying interaction HALQCD20

$V_{\Xi
N}=V_{0}+V_{\sigma}{\vec{\sigma}}_{\Xi}\cdot{\vec{\sigma}}_{N}+V_{\tau}{\vec{\tau}}_{\Xi}\cdot{\vec{\tau}}_{N}+V_{\sigma\tau}{\vec{\sigma}}_{\Xi}\cdot{\vec{\sigma}}_{N}\,{\vec{\tau}}_{\Xi}\cdot{\vec{\tau}}_{N}\,,$
(2)

and were found sufficient to bind $\Xi^{0}_{1p}$–14C near threshold, within
the J-PARC E07 experimental uncertainty of IRRAWADDY, as shown in Fig. 3.
Here, the introduction of $V_{\sigma\tau}$ causes the newly considered
$\Xi^{0}_{1p}$–14C state to get slightly mixed with IBUKI’s $\Xi^{-}_{1p}$–14N
bound state, sufficiently to make the $\sim$6 MeV $E1$ radiative deexcitation
of the $\Xi^{-}_{3D}$–14N atomic state to the dominantly $\Xi^{0}_{1p}$–14C
nuclear state as strong as to the $\sim$1 MeV deexcitation to the IBUKI
$\Xi^{-}_{1p}-{{}^{14}}$N nuclear state. Assigning a $\Xi^{0}_{1p}$–14C bound
state structure to IRRAWADDY contrasts with viewing it as a $\Xi^{-}_{1s}$–14N
state motivated largely by IRRAWADDY’s binding energy of a few MeV.

It is worth recalling that in spite of limiting discussion to the fairly
narrow IRRAWADDY, given that a KINKA+IRRAWADDY weighted average of
$B_{\Xi^{-}}$=6.13$\pm$0.25 MeV or 6.46$\pm$0.25 MeV differs little from
IRRAWADDY’s own value of $B_{\Xi^{-}}$=6.27$\pm$0.27 MeV, our arguments apply
equally well to either one of KINKA’s considerably broader versions.

##  4 Discussion

Table 2: Input (underlined) and calculated mean values of $\Xi^{-}$ binding energies in optical potential fits, plus resulting $\Xi N$ and $\Xi NN$ induced $\Xi$ nuclear potential depths $V^{(2)}_{\Xi}$ and $V^{(3)}_{\Xi}$, respectively, at nuclear-matter density $\rho_{0}=0.17\,$fm-3, and their correlated sum $V_{\Xi}$; see text. Potentials and energies are given in MeV. $B_{\Xi^{-}}^{1p}(^{12}$C) |  $B_{\Xi^{-}}^{1s}(^{14}$N) |  $B_{\Xi^{-}}^{1p}(^{14}$N) | $V_{\Xi}^{(2)}$ | $V_{\Xi}^{(3)}$ | $V_{\Xi}$ |  $B_{\Xi^{-}}^{1s}(^{11}$B)
---|---|---|---|---|---|---
$\underline{0.82}$ | 11.79 | 1.94 | 21.2$\pm$0.7 | – | 21.2$\pm$0.7 | 9.00
0.32 | $\underline{6.27}$ | 0.52 | 13.6$\pm$0.4 | – | 13.6$\pm$0.4 | 4.20
$\underline{0.82}$ | $\underline{8.00}$ | $\underline{1.27}$ | 26.4$\pm$2.6 |  $-$15.4$\pm$5.7 | 11.0$\pm$3.1 | 6.29
$\underline{0.82}$ | $\underline{6.27}$ | $\underline{1.27}$ | 30.6$\pm$1.7 |  $-$28.2$\pm$3.9 | 2.4$\pm$2.2 | 5.15

Some $\Xi$-nuclear scenarios are outlined in Table 2. Choosing
$B_{\Xi^{-}}^{1p}$=0.82$\pm$0.15 MeV for the two KEK-E176 12C events listed in
Table 1 to fit the strength $b_{0}$ of the $\Xi N$ induced $\Xi$-nuclear
attractive optical potential $V_{\rm opt}^{\Xi}$ in Eq. (1) results in several
other $\Xi^{-}$ nuclear binding energies listed in the first row. Choosing
instead the J-PARC E07 14N IRRAWADDY event E07b  with
$B_{\Xi^{-}}^{1s}$=6.27$\pm$0.27 MeV as input results in values listed in the
second row of the table. Clearly, these two sets of results differ strongly
for the $\Xi^{-}_{1s}$–11B binding energies discussed below and for the
$\Xi$-nuclear attractive potential depths $V_{\Xi}^{(2)}$. As for the Coulomb
assisted $\Xi^{-}_{1p}$–12C nuclear state, we note that it lies deeper by
merely 40 keV than the Coulomb $2P$ atomic state in 12C
($B_{\Xi^{-}}^{2P}=283$ keV) when constrained by IRRAWADDY in the second row,
strongly disagreeing with the 540$\pm$150 keV extra strong-interaction binding
deduced from the KEK E176 events E176  underlined in the first row.

The next two rows in Table 2 report on fitting two $\Xi$-nucleus interaction
parameters, $b_{0}$ for the $\Xi N$ induced attractive $V_{\rm opt}^{\Xi}$,
Eq. (1), and $B_{0}$ for a $\Xi NN$ induced potential term

$\delta V^{\Xi}_{\rm
opt}(r)=\frac{2\pi}{\mu_{\Xi}}\,(1+\frac{A-2}{A}\frac{\mu_{\Xi}}{2m_{N}})\,B_{0}\,\frac{\rho^{2}(r)}{\rho_{0}}$
(3)

introduced in our recent work on the content of the $\Lambda$-nuclear optical
potential FG23a ; FG23b . Both 12C and 14N $\Xi^{-}_{1p}$ bound states are
used for input, along with KINKA’s higher-binding option in the third row or
IRRAWADDY in the fourth row for $B_{\Xi^{-}}^{1s}(^{14}$N) to allow for some
variation. Both fits are acceptable, $\chi^{2}<1$, with a substantial $\Xi NN$
induced repulsive $\delta V^{\Xi}_{\rm opt}$ almost doubling its strength as
$B_{\Xi^{-}}^{1s}(^{14}$N) input is decreased from 8.00$\pm$0.77 MeV in the
third row to 6.27$\pm$0.27 MeV in the fourth row. We note that the $\Xi N$
induced potential $V_{\Xi}^{(2)}$ depth values obtained when $\delta
V^{\Xi}_{\rm opt}$ is introduced increase farther away from the considerably
smaller values of $V_{\Xi}^{(2)}$ obtained in recent theoretical models
HALQCD20 ; HM19 , while the total depth $V_{\Xi}$ decreases farther away from
$V_{\Xi}=17\pm 6$ MeV suggested by the 9Be($K^{-},K^{+}$) spectrum in Fig. 1.

The solution proposed here to the difficulty of interpreting IRRAWADDY as a
$\Xi^{-}_{1s}$ bound state in 14N is by pointing out that it could correspond
to a $\Xi^{0}_{1p}$–14C bound state, something that cannot occur kinematically
in the other light-emulsion nuclei 12C and 16O. Given that in this nuclear
mass range capture rates from $1s_{\Xi^{-}}$ states are estimated to be two
orders of magnitude below capture rates from $1p_{\Xi^{-}}$ states Zhu91 ;
Koike17 , our $\Xi^{0}_{1p}$–14C assignment addresses satisfactorily the
capture rate hierarchy.

Regarding $1s_{\Xi^{-}}$ states, J-PARC E70
12C$(K^{-},K^{+})^{12}_{\leavevmode\nobreak\ \Xi}$Be experiment with record 2
MeV FWHM simulated resolution Go22 , following an earlier experiment E05 with
5.4 MeV FWHM resolution Go20 , aims particularly to observe
$\Xi^{-}_{1s}-{{}^{11}}$B signals. The last column in Table 2 lists a wide
range of predicted $B_{\Xi^{-}}^{1s}(^{11}$B) values depending on which
$\Xi^{-}$-capture data are accepted. Corrections of order 0.5 MeV are likely
from the three spin-isospin $\Xi N$ terms in Eq. (2).

## Acknowledgments

One of us (A.G.) thanks the organizers of MESON2023 in Krakow for support and
hospitality. This work is part of a project funded by the EU Horizon 2020
Research & Innovation Programme under grant agreement 824093.

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