text:
\section*{Introduction}

In
\cite{FP} it is shown that for a Fuchsian group the Selberg zeta function admits an analytic continuation for non-unitary twists.
If the group has cusps, an extra condition on the cusp-monodromy is needed.
In the papers
\cites{Spilioti,Spilioti2} the continuation is given for odd dimensional compact hyperbolic manifolds.
In this note, we use a different technique, which works well for arbitrary compact locally symmetric spaces. Using some metric estimates due to Lubotzky, Mozes and Raghunathan, we are able to invoke the calculus of Lefschetz formulas for torus actions developed earlier.
The present approach will also work for groups of higher splitrank.

\section{Notation and basic facts}\label{sec1}
\begin{definition}
Let $\Ga$ be a group acting on a set $X$.
A point $x_0\in X$ is a \e{free point}, if
 the stabilizer $\Ga_{x_0}=\operatorname{stab}_\Ga(x_0)$ is trivial.
\end{definition}

Recall that two metrics $d,d'$ on a space $X$ are called \e{Lipschitz equivalent}, if there is $C>0$ such that
$$
\frac1Cd\le d'\le Cd.
$$
Recall further that a metric $d$ on $X$ is called \e{proper}, if for every $x\in x$ and every $r>0$ the closed ball $\ol B_r(x)=\big\{ y\in X: d(x,y)\le r\big\}$ is compact.

\begin{definition}
\cite{Lub}.
A proper metric space $(X,d)$ is called a \e{path space}, if there exists $C>0$ such that for any $x,y\in X$ one has
$$
d(x,y)=\inf\left\{\sum_{j=0}^{n-1}d(x_{j},x_{j+1}):n\in\N, x=x_0, x_n=y, x_j\in X, d(x_j,x_{j+1})\le C\right\}.
$$
Any metric, which is Lipschitz equivalent to a path metric, is a path metric as well.
\end{definition}

Recall that a group $\Ga$ acts \e{properly discontinuously} on a space $X$, if every point $x\in X$ has a neighbourhood $U$ such that the set
$$
\big\{\ga\in\Ga: U\cap\ga U\ne\emptyset\big\}
$$
is finite.

\begin{proposition}[Proposition 3.2 of \cite{Lub}]
\label{prop1.3}
Let $\Ga$ be a finitely generated group acting properly discontinuously via isometries on a path space $(X,d_X)$ such that $\Ga\bs X$ is compact.
Let $S$ be a finite symmetric set of generators and let $d_S$ be the corresponding word metric on $\Ga$.
Let $x_0\in X$ be a $\Ga$-free point. Then the metric $d(\ga,\tau)=d(\ga x_0,\tau x_0)$ on $\Ga$ is Lipschitz equivalent to $d_S$.
\end{proposition}

\begin{proposition}
Let $G$ be a locally compact group which is second countable and compactly generated.
Fix a compact subgroup $K\subset G$.
Fix Haar measures on $G$ and $K$, such that $\operatorname{vol}(K)=1$.

Then the topology on $G$ is generated by a left $G$-invariant and right $K$-invariant path metric $d_G$.
The topology on the space $X=G/K$ is generated by a $G$-invariant path metric $d_X$.
\end{proposition}

\begin{proof}The proof os a variation of (3.3) in \cite{Lub}.
By the Theorem of Struble \cite{Struble}, the topology on $G$ is generated by a left-invariant proper metric $d$.
The metric
$$
d^K(x,y)=\int_Kd(xk,yk)\,dk
$$
has the same properties and is right $K$-invariant.
Let $\Om\subset G$ be a compact symmetric generating set of $G$.
Define
$$
d_G(x,y)=\inf\left\{\sum_{j=0}^{n-1}d^K(x_{j},x_{j+1}):n\in\N, x=x_0, x_n=y, x_j\in G, x_{j+1}^{-1}x_j\in\Om\right\}
$$
has the desired properties, where the properness is verified as in \cite{Lub}.
The same construction with $d^K$ replaced by
$$
d'(xK,yK)=\int_K\int_K d(xk,yl)\,dk\,dl
$$
yields the wanted metric $d_X$ on the space $X$.
\end{proof}

\begin{definition}
For $g\in G$
let
$$
\ell(g)=\inf_{x\in X}d_X(gx,x).
$$
\end{definition}

\begin{lemma}\label{lem1.3}
Let $G$ be as in Proposition \ref{prop1.3} and let $\Ga\subset G$ be a cocompact discrete subgroup with a free point in $X=G/K$ and let $\chi:\Ga\to\GL(V_\chi)$ be a group homomorphism, where $V_\chi$ is a finite-dimensional complex vector space.
Then there exist $A,B>0$ such that for every $\ga\in\Ga$ we have
$$
|\operatorname{tr}\chi(\ga)|\le A e^{B\ell(\ga)}\quad\text{and}\quad \norm{\chi(\ga)}_\mathrm{op}\le e^{Bd_G(\ga,e)},
$$
where $\norm._\mathrm{op}$ denotes the operator norm.
\end{lemma}

\begin{proof}
As $\Ga\bs G$ is compact and $\Ga$ is discrete, the group $\Ga$ is finitely generated.
Let $S$ be a finite symmetric set of generators and let $\ell_w$ denote the corresponding word length function on $\Ga$.
For a given minimal representation $\ga=s_1\cdots s_n$ of $\ga\in\Ga$ we have
$$
\norm{\chi(\ga)}_\mathrm{op}\le\norm{\chi(s_1)}_\mathrm{op}\cdots\norm{\chi(s_n)}_\mathrm{op}\le M^n=M^{\ell_w},
$$
where $M=\max_{s\in S}\norm {\chi(s)}_\mathrm{op}$.
By Proposition \ref{prop1.3}, this gives the estimate on $\norm{\chi(\ga)}_\mathrm{op}$.

For the first estimate, let $\ga\in\Ga$ and let $x_0=g_0K\in X$ with $\ell(\ga)\ge d_X(\ga x_0,x_0)-1$.
Then
\begin{align*}
\ell(\ga)&\ge \int_K\int_K d_G(\ga g_0k,g_0l)\,dk\,dl-1\\
&\ge \int_K\int_Kd_G(\ga g_0k,g_0)-d_G(g_0,g_0l)\,dk\,dl-1\\
&\ge\int_K\int_Kd_G(\ga g_0,g_0)-\underbrace{d_G(\ga g_0,\ga g_0k)}_{d_G(e,l)}\,dk\,dl-\int_K \underbrace{d_G(g_0,g_0k)}_{d_G(e,k)}\,dk-1\\
&\ge d_G(\ga g_0,g_0)-C
\end{align*}
For some constant $C$.
Fix a compact fundamental domain $F\subset G$ for $\Ga$ and $\tau$ such that $g_0=\tau y_0$ for some $y_0\in F$.
Then
\begin{align*}
d_G(\ga g_0,g_0)
&=d_G(\tau^{-1}\ga \tau y_0,y_0)\\
&\ge d_G(\tau^{-1}\ga \tau y_0,e)-d_G(e,y_0)\\
&\ge d_G(\tau^{-1}\ga \tau ,e)-\underbrace{d_G(\tau^{-1}\ga \tau ,\tau^{-1}\ga \tau y_0)}_{d_G(e,y_0)}-d_G(e,y_0)\\
&\ge d_G(\tau^{-1}\ga \tau ,e)-D
\end{align*}
for some constant $D>0$, which depends on $F$ only.
Therefore we have
\begin{align*}
|\operatorname{tr}\chi(\ga)|&=|\operatorname{tr}(\chi(\tau^{-1}\ga\tau)|\\
&\le\dim(\chi)\,\norm{\tau^{-1}\ga\tau}_\mathrm{op}\\
&\le \dim(\chi)\,e^{Bd_G(\tau^{-1}\ga\tau,e)}\\
&\le \dim(\chi)\,e^{B\ell(\ga)+C+D}\mqed
\end{align*}
\end{proof}

\begin{definition}
Let $E=E_\chi=\Ga\bs(G\times V_\chi)$, where $\Ga$ acts diagonally.
The projection onto the first factor makes $E$ a complex vector bundle over $\Ga\bs G$.
Choose a hermitian metric on $E$ to define the space $L^2(E)$ of $L^2$-sections.
This space can be identified with the space $L^2(\Ga\bs G,\chi)$ of all measurable functions $f:G\to V_\chi$ with $f(\ga x)=\chi(\ga)f(x)$ and $\int_{\Ga\bs G}\sp{f(x),f(x)}_x\,dx<\infty$ (modulo nullfunctions).
The group $G$ acts by right translations on the Hilbert space $L^2(\Ga\bs G,\chi)$.
This representation is continuous but in general not unitary.
\end{definition}

\begin{definition}
Let $\what G_\adm$ denote the admissible dual of $G$, which can be viewed as the set of isomorphy classes of irreducible Harish-Chandra modules.
By Casselman's Subrepresentation Theorem each $\pi\in\what G_\adm$ can be globalised to a Hilbert representation of $G$.
We shall occasionally identify $\pi$ with this globalisation.
\end{definition}

\begin{proposition}[The trace formula]
There exists a complete $G$-stable filtration of $L^2(\Ga\bs G,\chi)$ with irreducible quotients, such that each $\pi\in\what G_\adm$ occurs with finite multiplicity $N_{\Ga,\chi}(\pi)\in\N_0$ in the induced graded representation and such that for each $f\in C_c^\infty(G)$ the trace formula
$$
\sum_{\pi\in\what G_\adm}N_{\Ga,\chi}(\pi)\ \operatorname{tr}\pi(f)=\sum_{[\ga]}\operatorname{vol}(\Ga_\ga\bs G_\ga)\ \CO_\ga(f)\ \operatorname{tr}\chi(\ga)
$$
holds, where $G_\ga$ and $\Ga_\ga$ are the centralisers of $\ga$ in $G$ and $\Ga$ respectively.
Further, $\CO_\ga(f)$ is the \e{orbital integral}
$$
\CO_\ga(f)=\int_{G/G_\ga}f(x\ga x^{-1})\,dx.
$$
\end{proposition}

\begin{proof}
This is Theorem 3 in \cite{NonUnitaryTrace}.
\end{proof}

\section{Lie groups}
\begin{definition}
Let $G$ be a connected Lie group and $K$ a compact subgroup.
Let $\mathfrak g_\R$ denote its real Lie algebra and
let $\mathfrak g=\mathfrak g_\R\otimes_\R\C$ be its complex Lie algebra.
The universal envelope $U(\mathfrak g)$ will be identified with the algebra of all left-invariant differential operators on $G$.
For $k\in\N$, let $U_k(\mathfrak g)$ denote the finite-dimensional space of all $D\in U(\mathfrak g)$ of degree $\le k$.
\end{definition}

\begin{definition}
Fix a $G$-left-invariant and $K$-right-invariant Riemannian metric on $G$ and let $d_G$ denote the induced distance function on $G$.
For $r>0$ let $V(r)=\operatorname{vol}(B_r(e))$ be the volume of the open ball of radius $r$ around $e$.
As the sectional curvature of $G$ is bounded below, there are $A,\beta>0$ such that
$$
V(r)\le Ae^{\beta r}.
$$
For this, see \cite{BC}.
\end{definition}

\begin{lemma}\label{lem2.3}
For every $\eps>0$ one has $\sum_{\ga\in\Ga}e^{-(\beta+\eps)d(\ga,e)}<\infty$.
\end{lemma}

\begin{proof}
Let $U$ be a compact unit neighbourhood such that $U\cap\ga U=\emptyset$ for every $\ga\in\Ga\sm\{e\}$.
Then there exists $c_1>0$ with
\begin{align*}
\sum_{\ga\in \Ga}e^{-(\beta+\eps)d(\ga,e)}
&\le c_1\sum_{\ga\in\Ga}\int_Ue^{-(\beta+\eps)d(\ga u,e)}\,du\\
&\le c_1\int_Ge^{-(\beta+\eps)d(x,e)}\,dx<\infty.
\qedhere
\end{align*}
\end{proof}

\begin{definition}\label{def2.4}
Choose $B>0$  such that the conlcusions of Lemma \ref{lem1.3} are satisfied.
Fix some $C>2(\beta+B)$.
For $k\in\N$  let $C^{k,C}(G)$ denote the space
of all $k$-times continuously differentiable functions $f:G\to\C$ such that
$$
p_D(f)=\sup_{x\in G}e^{Cd_G(x,e)}|Df(x)|<\infty
$$
holds for every $D\in U_k(\mathfrak g)$.
Equip $C^{k,C}(G)$ with the topology given by the seminorms $p_D$, $D\in U_k(\mathfrak g)$.
Note that, as $C>\beta$, for $f\in C^{k,C}(G)$ we have $Df\in L^1(G)$ for every $D\in U_k(\mathfrak g)$.
\end{definition}

\begin{proposition}\label{prop2.4}
Let $N\in\N$ with $2N>\dim G$.
Then the trace formula is valid for every  $f\in C^{2N,C}(G)$.
Let $\operatorname{tr}_\Ga(f)$ denote either side of the trace formula. Then $\operatorname{tr}_\Ga$ is a continuous linear map $C^{2N,C}(G)\to\C$.
\end{proposition}

{\it Proof.} We need a lemma.

\begin{lemma}
Let $R_\chi$ denote the right translation representation of $G$ on $L^2(\Ga\bs G,\chi)$.
Equip the algebra of bounded operators $\CB\(L^2(\Ga\bs G,\chi)\)$ with the strong operator topology. Then for $f\in C^{0,C}(G)$ and $\phi\in L^2(\Ga\bs G,\chi)$, the  integral
$$
R_\chi(f)\phi(x)=\int_Gf(y)\phi(xy)\,dy
$$
exists almost everywhere for $x\in G$ and defines a Hilbert-Schmidt operator $R_\chi(f)\in\CB\(L^2(\Ga\bs G,\chi)\)$.
\end{lemma}

\begin{proof}
Let $F\subset G$ denote a compact fundamental domain for $\Ga$, i.e., $F$ is compact and contains a measurable set $V$ of representatives for $\Ga\bs G$ such that $F\sm V$ has measure zero.
Assume further that $e\in F$.
For $x,y\in F$ and $\ga\in\Ga$ one has
$
d_G(\ga,e)\le d_G(\ga,\ga x)+d_G(\ga x,y)+d_G(y,e)$ and
hence
\begin{align*}
d_G(x^{-1}\ga y,e)&=d_G(\ga y,x)\\
&\ge d_G(\ga,e)-d_G(e,x)-d_G(y,e)\\
&\ge d_G(\ga,e)-2\operatorname{diam}(F).
\end{align*}
It follows
\begin{align*}
|f(x^{-1}\ga y)|\norm{\chi(\ga)}_\mathrm{op}
&\le |f(x^{-1}\ga y)|\ e^{Bd_G(\ga,e)}\\
&\le T_1e^{-Cd_G(x^{-1}\ga y,e)}\ e^{Bd_G(\ga,e)}\\
&\le Te^{-2\beta d(\ga,e)}
\end{align*}
for some $T_1,T>0$.
By Lemma \ref{lem2.3}, the sum $k(x,y)=\sum_{\ga\in\Ga}f(x^{-1}\ga y)\chi(\ga)$
converges uniformly absolutely, therefore represents a continuous integral kernel $k$ on the compact manifold $\Ga\bs G$ for the operator $R_\chi(f)$. Having a continuous, hence square-integrable kernel, this operator is Hilbert-Schmidt.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop2.4}]
Let $\Delta$ denote the left-invariant group Laplacian given by the Riemannian metric on $G$.
 Since $\Delta\in U(\mathfrak g)$, the operator $R_\chi(\Delta)$ is well defined as a densely defined unbounded operator on $L^2(\Ga\bs G,\chi)$.
 Then, for $2N>\dim(G)$ and $c\in\C$, which is not in the spectrum on $\Delta$, the operator
$R(\Delta+c)^{-N}$ is of trace class.
This is folklore and can be deduced from \cite{Sh}.
One way to see it directly is to choose a smooth atlas $(U_i,\phi_i)_{i\in I}$ such that the bundle $E_\chi$ trivialises over each $U_i$, where the charts $\phi_i$ are maps taking values in $\R^n/\Z^n$. Then one uses a smooth partition of unity underlying $(U_i)_{i\in I}$ to translate the problem to $\R^n/\Z^n$ where it can be solved  using Fourier-series.

Let $\om :[0,\infty )\to[0,1]$ be a monotonic $C^{2N}$-function
with compact support, $\om\equiv 1$ on $[0,1]$ and
$|\om^{(k)}(t)|\le 1$ for $k=1,\dots ,2N$. Let
$h_n(x):=\om\(\frac{d_G(x,e)}{n}\)$ for $x\in G$, $n\in \N$. Then
$|Dh_n(x)|\le \frac{C_D}{n}$ for any $D\in\mathfrak g U(\mathfrak g)$. Let
$f_n=h_nf$ then $f_n\to f$ locally uniformly.
As $f_n$ has compact support, we  get that $R_\chi(f_n)$ is a trace class operator.

Denoting the trace norm by $\norm{.}_{tr}$ we infer
\begin{align*}
\norm{R(f_n)-R(f)}_{tr}
&= \norm{R(\Delta +1)^{-N}(R((\Delta +1)^Nf_n)-R((\Delta +1)^Nf))}_{tr}\\
&
\le \norm{R(\Delta +1)^{-N}}_{tr}\norm{R((\Delta +1)^Nf_n)-R((\Delta
+1)^Nf)}_\mathrm{op}.
\end{align*}
{\it Claim.} For the operator norm  we have
$$
\parallel R(Df_n) -R(Df) \parallel_\mathrm{op} \to 0
$$
as $n\to \infty$ for any $D\in U(\mathfrak g)$ of degree $\le 2N$.

{\it Proof of the claim.} By the Poincar\'e-Birkhoff-Witt theorem
$D(f_n)=D(h_nf)$ is a sum of expressions of the type
$D_1(h_n)D_2(f)$ and $D_1$ can be chosen to be the identity
operator or in $\mathfrak g U(\mathfrak g)$. The first case gives the summand
$h_nD(f)$ and it is clear that $\norm{R(Df)-R(h_nDf)}_\mathrm{op}$ tends to zero
as $n$ tends to infinity. For the rest assume $D_1\in \mathfrak g U(\mathfrak g)$.
Then $|D_1(h_n)(x)|\le \frac{c}{n}$ hence
$\norm{R(D_1(h_n)D_2(f))}_\mathrm{op}$ tends to zero because $D_2(f)$ is in
$C^{0,C}(G)$. The claim follows.

We conclude that $\operatorname{tr} R(f_n)$ tends to $\operatorname{tr} R(f)$
as $n\to \infty$, which implies that the spectral side of the trace formula defines a continuous functional on $C^{2N,C}(G)$.
It remains to show that it coincides with the geometric side.
For this we go back to the situation where $f_n=h_nf$.
It follows
\begin{align*}
\sum_{\pi\in\what{G}_\adm} N_{\Ga ,\omega}(\pi)\operatorname{tr}\pi (f) &= \operatorname{tr} R(f)= \lim_n \operatorname{tr} R(f_n).\\
\end{align*}

Now suppose as an additional condition that $f\ge 0$ and $\chi =1$.
Then we are allowed to interchange the limit and the sum by monotone convergence and thus in this case
$$
\sum_{\pi\in\what{G}_\adm} N_{\Ga ,1}(\pi)\operatorname{tr}\pi (f)=
\sum_{[\ga]} \operatorname{vol}(\Ga_\ga\bs G_\ga)\ \CO_\ga(f).
$$
In particular, the right hand side is finite.
For general $\chi$, we use the bound on $\operatorname{tr}\chi(\ga)$ to justify the same interchange by dominated convergence using the result for $\chi =1$.
As both sides of the formula are linear in $f$, the claim follows generally.
\end{proof}

\section{The Lefschetz formula}\label{Sec3}
In this section, we adapt the contents of Section 1 in \cite{GAFA} to the situation of non-unitary twists.
For the most part, it consists of definitions.
At the end, we only give the changes necessary to adapt the proof of the main theorem of Section 1 in [loc.cit.] to our situation.

Let $G$ be a semi-simple, connected Lie group with finite center.
Let $H=AT$ be a maximally split Cartan subgroup, where $A$ is the  split component and $T$ is compact abelian.
Let $\a$ be real Lie algebra of $A$ and $\a^*=\Hom(\a,\R)$ be its dual space.
Fix a parabolic subgroup $P$ of $G$ with Langlands decomposition $P=MAN$.
The choice of $P$ induces a choice of positive roots $\Phi^+(\a,\mathfrak g)\subset \a^*$.
Let $\a^-=\big\{X\in \a: \al(a)<0\ \forall_{\al>0}\big\}$ and $A^-=\exp(\a^-)$ be the negative cones in $\a$ and $A$.

\begin{definition}
For $\mu\in\a^*$ and $j\in\N$, let $C^{\mu,j}(A^-)$
denote the space of $j$-times continuously differentiable functions $\phi$ on $A$ which vanish outside $A^-$ and satisfy
$$
N_D(\phi)=\sup_{a\in A}\,|D\phi(a)|a^{-\mu}<\infty
$$
for every $D\in U_j(A)$.
Equip $C^{\mu,j}(A^-)$ with the topology induced by the seminorms $N_D$, $D\in U_j(A)$.
Let $D_1,\dots,D_r$ be a basis of $U_j(A)$, then the norm
$$
\norm\phi=N_{D_1}(\phi)+\dots+N_{D_r}(\phi)
$$
induces the topology, hence $C^{\mu,j}(A^-)$ is a Banach space.
\end{definition}

\begin{definition}
Choose a maximal compact subgroup $K$ with
Cartan involution $\theta$, i.e., $K$ is the group of
fixed points of $\theta$.
After conjugation we can assume that $A$ and $M$ are stable under
$\theta$. Then $A$ is a maximal split torus of $G$ and $M$ is a subgroup of $K$. The centralizer of $A$
is $AM$. Let $W(A,G)$ be the \emph{Weyl group} of $A$, i.e. $W$ is the quotient of the
normalizer of $A$ by the centralizer. This is a finite group acting on $A$.
\end{definition}

\begin{definition}\label{def3.3}
We have to fix Haar measures. We use the normalization of
Harish-Chandra \cite{HC-HA1}. Note that this normalization
depends on the choice of an invariant bilinear form $B$ on
$\mathfrak g_\R$ which we keep at our disposal until later. Changing $B$ amounts to scaling the
metric of the symmetric space. Note further that in this normalization of Haar measures the
compact groups $K$ and $M$ have volume $1$.
\end{definition}

\begin{definition}
We write $\mathfrak g_\R, \mathfrak{k}_\R,\a_\R,\m_\R,\n_\R$ for the real Lie
algebras of the groups $G,K,A,M,N$ and $\mathfrak g,\mathfrak{k},\a,\m,\n$ for their
complexifications. $U(\mathfrak g)$ is the universal enveloping
algebra of $\mathfrak g$. This algebra is isomorphic to the algebra
of all left invariant differential operators on $G$ with
complex coefficients. Pick a Cartan subalgebra $\mathfrak{t}$ of $\m$. Then $\h=\a\oplus\mathfrak{t}$ is a
Cartan subalgebra of $\mathfrak g$. Let $W(\h,\mathfrak g)$ be the corresponding absolute Weyl group.

Let $\a^*$ denote the dual space of the complex vector
space $\a$. Let $\a_\R^*$ be the real dual of $\a_\R$. We identify $\a_\R^*$ with the real vector space of all $\la\in\a^*$ that map $\a_\R$ to $\R$. Let $\Phi\subset\a^*$ be the set of all roots
of the pair $(\a,\mathfrak g)$ and let $\Phi^+$ be the subset of
positive roots with respect to $P$. Let $\Delta\subset
\Phi^+$ be the set of simple roots. Then $\Delta$ is a basis of $\a^*$. The open {\it
negative Weyl chamber} $\a_\R^-\subset\a_\R$ is the cone of all
$X\in\a_\R$ with $\alpha(X)<0$ for every $\alpha\in\Delta$. Let $\overline{\a_\R^-}$ be the
closure of $\a_\R^-$. The $W(A,G)$-translates $w\a_\R^-$ of $\a_\R^-$ are pairwise disjoint
and their union equals $\a_\R$ minus a finite number of hyperplanes. This is called the
\emph{regular set},
$$
\a_\R^{reg}= \bigcup_{w\in W(A,G)} w\a_\R^-.
$$
Let $A^{reg}=\exp\left(\a_\R^{reg}\right)$ be the regular set in $A$. The elements are
called the regular elements of $A$. A given $a\in A$ lies in $A^{reg}$ if and only if the
centralizer of $a$ in $G$ equals the centralizer of $A$ in $G$ which is $AM$.
\end{definition}

The bilinear form $B$ is indefinite on $\mathfrak g_\R$, but the form
$$
\sp{X,Y}= -B(X,\theta(Y))
$$
is positive definite, ie an inner product on $\mathfrak g_\R$. We extend it to an inner product on
the complexification $\mathfrak g$. Let $\norm X=\sqrt{\sp{X,X}}$ be the corresponding norm.
The form $B$, being nondegenerate, identifies $\mathfrak g$ to its dual space $\mathfrak g^*$. In this way we
also define an inner product $\sp{.,.}$ and the corresponding norm on $\mathfrak g^*$.
Furthermore, if $V\subset\mathfrak g$ is any subspace on which $B$ is nondegenerate, then $B$ gives
an identification of $V^*$ with $V$ and so one gets an inner product and a norm on
$V^*$. This in particular applies to $V=\h$, a Cartan subalgebra of $\mathfrak g$, which is defines
over
$\R$.

\begin{definition}
Let $\Ga\subset G$ be a discrete, cocompact, torsion-free subgroup.
We are interested in the closed geodesics on the locally symmetric space $X_\Ga=\Ga\bs
X=\Ga\bs G/K$. Every such geodesic
$c$ lifts to a $\Ga$-orbit of geodesics on $X$ and gives a $\Ga$-conjugacy class $[\ga_c]$ of
elements closing the particular geodesics. This induces a bijection between the set of all
homotopy classes of closed geodesics in $X_\Ga$ and the set of all non-trivial conjugacy
classes in $\Ga$.
\end{definition}

\begin{definition}
An element $am$ of $AM$ is called {\it split regular} if
$a$ is regular in $A$. An element $\ga$ of $\Ga$
is called split regular if $\ga$ is in $G$ conjugate to a
split regular element $a_\ga m_\ga$ of $AM$. In that case
we may (and do) assume that $a_\ga$ lies in the negative
Weyl chamber $A^-=\exp(\a_\R^-)$ in $A$. Let $\CE(\Ga)$
denote the set of all conjugacy classes in $\Ga$ that
consist of split regular elements. Via the above correspondence the set $\CE(\Ga)$ can be
identified with the set of all homotopy classes of regular closed geodesics in $X_\Ga$, \cite{jost}.
\end{definition}

\begin{definition}\label{def3.7}
Let $[\ga]\in\CE(\Ga)$.
There is a closed geodesic $c$ in the Riemannian manifold $\Ga\bs G/K$ which gets closed by
$\ga$. This means that there is a lift $\tilde c$ to the universal covering $G/K$ which is
preserved by $\ga$ and $\ga$ acts on $\tilde c$ by a translation.
The closed geodesic $c$ is not unique in general but lies in a unique maximal flat
submanifold $F_\ga$ of $\Ga\bs G/K$. Let $\la_\ga$ be the volume of that flat,
$$
\la_\ga= \operatorname{vol}(F_\ga).
$$
Let $\Ga_\ga$ and $G_\ga$ denote the centralizers of $\ga$ in $\Ga$ and $G$ respectively.
The flat $F_\ga$ is the image of $\Ga_\ga\bs G_\ga /K_\ga$ in $\Ga\bs G/K$, where we assume
that $K$ is chosen so that $K_\ga=G_\ga\cap K$ is a maximal compact subgroup of $G_\ga$.
In Harish-Chandra's normalization $G_\ga$ is equipped with the Haar measure that satisfies
$\int_{G_\ga}=\int_{G_\ga/K_\ga}\int_{K_\ga}$, where $K_\ga$ has the Haar measure with
$\operatorname{vol}(K_\ga)=1$ and $G_\ga/K_\ga$ gets the measure induced by the metric of $G/K$. Therefore
$$
\la_\ga= \operatorname{vol}(\Ga_\ga\bs G_\ga /K_\ga)=\operatorname{vol}(\Ga_\ga\bs G_\ga).
$$
Note that for not maximally split elements $\ga$ the relation between $\la_\ga$ and
$\operatorname{vol}(\Ga_\ga\bs G_\ga)$ is more complicated \cites{geom,HRLef}.
\end{definition}

\begin{definition}
Let $\n$ denote the complexified Lie algebra of $N$. For
any $\n$-module $V$ let $H_q(\n,V)$ and $H^q(\n,V)$ for $q=0,\dots,\dim\n$
be the Lie algebra homology and cohomology \cite{BorWall}.
 If
$\pi\in\what G_\adm$, then $H_q(\n,\pi)$ and $H^q(\n,\pi)$ are finite
dimensional $AM$-modules \cite{HeSch}. Note that they a priori only are $(\a\oplus\m,M)$-modules, but since $A$ is isomorphic to its Lie algebra they are  $AM$-modules.
\end{definition}

Note that $AM$ acts on the Lie algebra $\n$ of $N$ by the
adjoint representation. Let $[\ga]\in\CE(\Ga)$. Since
$a_\ga\in A^-$ it follows that every eigenvalue of $a_\ga
m_\ga$ on $\n$ is of absolute value $<1$. Therefore
$\det(1-a_\ga m_\ga | \n)\ne 0$.

Fix a finite dimensional irreducible representation
$\sigma$ of $M$ and denote by $\breve\sigma$ the dual representation. A {\it quasi-character} of $A$ is a
continuous group homomorphism to $\C^\times$. Via
differentiation the set of quasi-characters can be
identified with the dual space $\a^*$. For $\la\in\a^*$ we write $a\mapsto a^\la$ for the
corresponding quasicharacter on $A$. We denote by
$\rho\in\a^*$ the modular shift with respect to $P$, i.e., for $a\in A$ we have
$\det(a|\n)=a^{2\rho}$.

\begin{definition}
For a complex vector space $V$ on which $A$ acts linearly and $\la\in\a^*$, let $(V)_\la$
denote the generalized $(\la+\rho)$-eigenspace, i.e.,
$$
(V)_\la= \{ v\in V\mid (a-a^{\la+\rho}Id)^n v=0\ \ {\rm for\ some\ } n\in\N\}.
$$
Since $H^p(\n,\pi)$ is finite dimensional, the Jordan Normal Form Theorem implies that
$$
H^p(\n,\pi) = \bigoplus_{\nu\in\a^*} H^p(\n,\pi)_\nu.
$$
Let $T$ be a Cartan subgroup of $M$ and let $\mathfrak{t}$ be its complex Lie algebra. Then $AT$ is a
Cartan subgroup of $G$. Let $\Lambda_\pi\in(\a\oplus\mathfrak{t})^*$ be a representative of the
infinitesimal character of $\pi$. By Corollary 3.32 of \cite{HeSch} it follows,
$$
H_p(\n,\pi)=\bigoplus_{\nu=w\Lambda_{\pi}|_\a}H_p(\n,\pi)_\nu,
$$
where $w$ ranges over $W(\mathfrak g,\h)$.
\end{definition}

\begin{lemma}\label{1.1}
For $0\le p\le d=\dim(\n)$ we have
$$
 H_p(\n,\pi)\ \cong\ H^{d-p}(\n,\pi)\otimes\det(\n),
$$
where the determinant of a
finite dimensional space is the top exterior power. So $\det(\n)$ is a one dimensional
$AM$-module on which $AM$ acts via the quasi-character $am\mapsto \det(am|\n)=a^{2\rho}$.
This in particular implies
$$
H^p(\n,\pi)=\bigoplus_{\nu= w\Lambda_\pi |_\a}H^p(\n,\pi)_{\nu-2\rho}.
$$
\end{lemma}

\begin{proof}
The first part follows straight from the definition of Lie algebra cohomology. The
second part by Corollary 3.32 of \cite{HeSch}.
\end{proof}

\begin{definition}
For $\la\in\a^*$
 and $\pi\in\what G_\adm$ let
$$
m_\la^\sigma(\pi)= \sum_{q=0}^{\dim \n} (-1)^{q+\dim\n} \dim\left(
H^q(\n,\pi)_\la\otimes\breve\sigma\right)^M,
$$
where the superscript $M$ indicates the subspace of $M$-invariants. Then
$m_\la^\sigma(\pi)$ is an integer and by the above, the set of $\la$ for which
$m_\la^\sigma(\pi)\ne 0$ for a given $\pi$ has at most $|W(\mathfrak g,\h)|$ many elements.
\end{definition}

\begin{definition}
For  $\mu\in\a^{*}$ and $j\in\N$ let $\CC^{j,\mu,-}(A)$ denote the space of functions
$\phi$ on $A$ which \nopagebreak
\begin{itemize}
  \item are $j$-times continuously differentiable on $A$,
  \item are zero outside $A^-$,
  \item are such that $a^{-\mu} D\phi(a)$ is bounded on $A$  for every invariant differential operator $D$ on $A$ of
degree $\le j$.
\end{itemize}

For every invariant differential operator $D$ of degree $\le j$ let $N_D(\phi)=\sup_{a\in A}\left| a^{-\mu}\, D\phi(a)\right|$.
Then $N_D$ is a seminorm.
Let $D_1,\dots,D_n$ be a basis of the space of invariant differential operators of degree $\le j$, then
$N(\phi)=\sum_{j=1}^n N_{D_j}(\phi)$ is a norm that makes $\CC^{j,\mu,-}(A)$ into a Banach space.
A different choice of basis will give an equivalent norm.
\end{definition}

\begin{definition}
For $[\ga]\in\CE(\Ga)$ let
$$
\ind(\ga)=\frac{\la_\ga }{\det(1-a_\ga m_\ga \mid \n)}\ >\ 0.
$$
\end{definition}

\begin{theorem}\label{lefschetz} (Lefschetz Formula)\\
There exists $j\in\N$ and $\mu\in\a^{*}$ such that
for any $\phi\in \CC^{j,\mu,-}(A)$ we have that
$$
\sum_{\pi\in\what G_\adm} N_{\Ga,\chi}(\pi)\sum_{\la\in\a^*} m_\la^\sigma(\pi) \int_{A^-}\phi(a)
a^{\la+\rho} da
$$
equals
$$
\sum_{[\ga]\in\CE(\Ga)}\ind(\ga)\,
\operatorname{tr}\sigma(m_\ga)\,\operatorname{tr}(\chi(\ga))\,\phi(a_\ga),
$$
where all sums and integrals converge absolutely. The inner sum on the left is always finite,
more precisely it has length $\le |W(\h,\mathfrak g)|$. The left hand side is called the
\emph{global side}, as opposed to the \emph{local side} on the right.
The map that sends $\phi$ to either side of the formula is continuous linear functional on the Banach space $\CC^{j,\mu,-}(A)$.
\end{theorem}

\begin{proof}
The proof is essentially the same as that of Theorem 1.2 in \cite{GAFA}, except for a missing estimate which justifies the factor $e^{C d_G(x,e)}$, needed in the non-unitary case as in Definition \ref{def2.4} and Proposition \ref{prop2.4}.
This is done as follows: By the definition of the test function $f$, one only needs to consider $x=kn\,am\,(kn)^{-1}$ for $am\in AM$, $k\in K$ and $n\in N$, where the latter stays in a fixed compact subset of $N$.
Using left $G$-and right$K$-invariance, one gets
\begin{align*}
d_G(kn\,am\,(kn)^{-1},e)&=d_G(n\,am\,n^{-1},e)\\
&=d_G(a,n^{am}n^{-1}m^{-1}),
\end{align*}
where the element $\om=kn\,am\,(kn)^{-1}$ stays in a fixed compact set $\Om\subset G$.
Then
$$
d_G(a,\om)\le d_G(a,e)+d_G(e,\om)\le d_G(a,e)+c
$$
for a constant $\al>0$.
Taking $e^{Cd_G(x,e)}$, the constant $\al$ contributes a multiplicative constant, which plays no role.
Next let $a=\exp(X)$ and $\ga(t)=\exp(tX)$.
then $d_G(a,e)=\int_0^1\norm{\ga'(t)}\,dt=\int_0^1\norm X\,dt=\norm X$.
Now finally $a^{-\mu}=\norm X^{-\mu}$, which shows that one achieves the boundedness of $e^{Cd_g(x,e)}|Df(x)|$ by increasing $\mu$.
\end{proof}

\section{Zeta functions of Ruelle and Selberg}\label{sec2}
From now on we assume that the splitrank of $G$ be one, i.e., $\dim A=1$.

In this case, $\CE(\Ga)$ coincides with the set of all non-trivial conjugacy classes in $\Ga$.

\begin{definition}
A class $[\ga]\in\CE(\Ga)$ is called \e{primitive}, if for every $\tau\in\Ga$ and every $n\in\N$ one has
$$
\ga=\tau^n\quad\Rightarrow\quad n=1.
$$
For every $\ga\in\Ga\sm\{1\}$ there exists a uniquely determined primitive $\ga_0\in\Ga$ such that
$$
\ga=\ga_0^m
$$
for some $m\in\N$.
we call $\ga_0$ the \e{underlying primitive} of $\ga$.
\end{definition}

\begin{definition}
For $a\in A$  we write $\ell(a)=|2\rho(\log a)|$.
Note that for $\ga\in\Ga\sm\{1\}$, the number $\ell(a_\ga)$ is the length of the closed geodesic in the locally symmetric space $\Ga\bs G/K$, whose free homotopy class corresponds to the $\Ga$-conjugacy class of $\ga$.
We also write
$$
\ell(\ga)
$$
for $\ell(a_\ga)$.
Note that
$$
\la_\ga=\ell(\ga_0),
$$
where $\ga_0$ is the underlying primitive of $\ga$ and $\la_\ga$ is as in Definition \ref{def3.7}.

\end{definition}

\begin{definition}
For $s\in\C$ with $\Re(s)>> 0$ the \e{Ruelle zeta function} is given by the  product
$$
R_\chi(s)=\prod_{\substack{[\ga]\\ \text{primitive}}}\(1- \chi(\ga)\,e^{-s\ell(\ga)}\)
$$
We shall show, that $R_\chi(s)$ converges locally uniformly in some half plane of the form $\{\Re(s)>A\}$ and that it admits an analytic continuation to a meromorphic function on $\C$.
The later will come about through its relation to the \e{Selberg zeta function}, which for a finite-dimensional representation $\sigma$ of the compact group $M$ is defined by
$$
Z_{\sigma,\chi}(s)=\prod_{\substack{[\ga]\\ \text{primitive}}}\prod_{k=0}^\infty\det\(1-\sigma(m_\ga)\otimes \chi(\ga)\otimes S^k\[\Ad(m_\ga a_\ga)|_{\n}\]\,e^{-s\ell(\ga)}\).
$$
\end{definition}

\begin{proposition}
Assuming convergence for the moment, we have the identity
$$
R_\chi(s)=\prod_{k=0}^{\dim\n}Z_{\bigwedge^k_\n,\chi}(s-j)^{(-1)^k}.
$$
Here $\bigwedge _\n^k$ stands for the representation of $M$ on the $k$-exterior power $\bigwedge^k\n$ of the space $\n$.
Further, the product defining $R_\chi$ converges, if the products of the $Z_{\bigwedge_\n^k,\chi}(s)$ converge absolutely for every $k=0,\dots,\dim\n$.
\end{proposition}

\begin{proof}
A straightforward computation shows that in the range of absolute convergence we have
\begin{align*}
\frac{Z_{\sigma,\chi}'}{Z_{\sigma,\chi}}(s)
&=\sum_{[\ga]}\ind(\ga)\ \operatorname{tr}(\chi(\ga))\ \operatorname{tr}\sigma(m_\ga)\ e^{-s\ell(\ga)}\\
&=\sum_{[\ga]}\frac{\ell(\ga_0)}{\det(1-a_\ga m_\ga|\n)}\ \operatorname{tr}(\chi(\ga))\ \operatorname{tr}\sigma(m_\ga)\ e^{-s\ell(\ga)}
\end{align*}
and
$$
\frac{R_\chi'}{R_\chi}(s)=\sum_{[\ga]}\ell(\ga_0)\ \operatorname{tr}\chi(\ga)\ e^{-s\ell(\ga)}.
$$
For any linear map $T$ on a finite-dimensional vector space one has $\det(1-T)=\sum_{k=0}^{\dim V}(-1)^k\operatorname{tr}\(\bigwedge^kT\)$.
Applying this to the action of $a_\ga m_\ga$ on $\n$ one gets
$$
\frac{R_\chi'}{R_\chi}(s)
=\sum_{k=0}^{\dim\n}(-1)^k\frac{Z_{\bigwedge^k_\n,\chi}'}
{Z_{\bigwedge^k_\n,\chi}}(s).
$$
Hence the logarithmic derivatives of the two sides of the claimed equation coincide.
Hence the two meromorphic functions coincide up to a multiplicative constant, which equals 1, since both functions tend to 1, as $s\to +\infty$.
\end{proof}

\begin{theorem}
The product of the Selberg zeta function converges in some half-plane and the so defined holomorphic function extends to a meromorphic function on $\C$.
The (vanishing) order at $s\in\C$ is given by the finite sum of integers
$$
\sum_{\pi\in\what G_\adm}N_{\Ga,\chi}(\pi)\ m_{(s-1)\rho}(\pi).
$$
\end{theorem}

\begin{proof}
We need to normalize Haar measures.
These depend on the choice of the form $B$ as in Definition \ref{def3.3}.
We normalise  $B$ such that the set
$$
\big\{ a\in A \mid 0\le 2\rho(\log a)\le 1\big\}
$$
has volume $1$.

For $s\in\C$ and $j\in\N$ define
$$
L^j(s)=\sum_{[\ga]\in\CE(\Ga)} \ind(\ga)\ \operatorname{tr}(\chi(\ga))\ \operatorname{tr}\sigma(m_\ga)\ \ell(\ga)^{j+1}\ e^{-s\ell(\ga)}.
$$
Then we have
$$
L^j(s)=D^{j+1}\frac{Z_{\sigma,\chi}'}{Z_{\sigma,\chi}}(s),
$$
where $D$ denotes the differential operator
$
D= -\frac \partial{\partial s}.
$
We shall show that this series converges absolutely for
$\Re(s)>A$ for some $A>0$.
As it is a (generalised) Dirichlet series, it then converges locally uniformly.

\begin{definition}
For given $\pi\in\what G_\adm$ let $\Lambda(\pi)$ denote the set of all $\la\in\a^*$ with
$m_{\la -\rho}(\pi)\ne 0$. Then $\Lambda(\pi)$ has at most $|W(\h,\mathfrak g)|$ elements.
\end{definition}

For given
$\la\in\a^*$ there exists a uniquely determined number $[\la]\in\C$ with $\la=[\la]\,2\rho$.

\begin{lemma}
There exists $A>0$, such that for $j\in\N$ large enough, the series $L^j(s)$ converges locally uniformly in the set
$$
\big\{ s\in\C : \Re(s)>A\big\}.
$$
The function $L^j(s)$ can be written as Mittag-Leffler series,
\begin{align*}
L^j(s)&=\sum_{\pi\in\what G_\adm} N_{\Ga,\chi}(\pi)
\sum_{\la\in\Lambda(\pi)} m_{\la-\rho}(\pi) D^{j+1}\frac 1{s+[\la]}.
\end{align*}
The double series converges locally uniformly on $\C$ minus its discrete set of poles.
These poles all lie in $\{\Re(s)\le A\}$.
\end{lemma}

\begin{proof}
We will show that the series $L^j(s)$ converges if $\Re(s_k)$ is
sufficiently large.
For $a\in A$ set
$$
\phi(a)= \ell(a)^{j+1}\ a^{2s\rho}.
$$
For $\Re(s)>>0$, the Lefschetz formula is valid for this test function. The local
side of the Lefschetz formula for $\sigma= triv$ equals
$$
\sum_{[\ga]\in\CE(\Ga)} \ind(\ga)\,\operatorname{tr}\(\chi(\ga)\)\, \ell(a_\ga)^{j+1}\ a_\ga^{2s\rho}=  L^j(s).
$$
The convergence assertion in the Lefschetz formula implies that the series  converges
absolutely  if
$\Re(s)>A$ for some $A\in\R$.
Since $L^j(s)$ is a Dirichlet series, it will
converge locally uniformly in that half-plane.
The theorem will follow, if we can show that the spectral side of the trace formula equals the Mittag-Leffler series above, since the trace formula grants convergence in the half plane and a Mittag-Leffler series, which converges at some point in $\C$, automatically converges locally uniformly outside its set of poles.

 With our given test
function and the Haar measure chosen we compute
\begin{align*}
\int_{A^-}\phi(a) a^\la da &=(-1)^{j+1} \int_{A^-} (2\rho(\log a))^{j+1} a^{s\rho+\la} da\\
&= (-1)^{j+1} \int_0^\infty  {t}^{j+1}
e^{-(s+[\la])t} d t\\
&= D^{j+1}
\int_0^\infty  e^{-(s+[\la])t}\, d t\\
&= D^{j+1}  \frac
1{s+[\la]}.
\end{align*}
Writing $m_\la=m_\la^{triv}$ and performing a $\rho$-shift we see that the Lefschetz formula
gives
\begin{eqnarray*}
L^j(s)&=&\sum_{\pi\in\hat G}N_{\Ga,\chi}(\pi)\sum_{\la\in\a^*} m_{\la-\rho}(\pi)\, D^{j+1} \frac
1{s+[\la]}
\end{eqnarray*}
for $\Re(s)>A$.
The lemma follows.
\end{proof}

The convergence assertion in the theorem follows from the convergence of $L^j(s)$, since there are only finitely many classes $[\ga]$ with $\ell(\ga)<1$.
The rest is clear.
\end{proof}

\begin{bibdiv} \begin{biblist}

\bib{Alperin}{article}{
   author={Alperin, Roger C.},
   title={An elementary account of Selberg's lemma},
   journal={Enseign. Math. (2)},
   volume={33},
   date={1987},
   number={3-4},
   pages={269--273},
   issn={0013-8584},

}

\bib{ayoub}{book}{
   author={Ayoub, Raymond},
   title={An introduction to the analytic theory of numbers},
   series={Mathematical Surveys, No. 10},
   publisher={American Mathematical Society, Providence, R.I.},
   date={1963},
   pages={xiv+379},

}

\bib{BC}{book}{
   author={Bishop, Richard L.},
   author={Crittenden, Richard J.},
   title={Geometry of manifolds},
   note={Reprint of the 1964 original},
   publisher={AMS Chelsea Publishing, Providence, RI},
   date={2001},
   pages={xii+273},
   isbn={0-8218-2923-8},

   doi={10.1090/chel/344},
}

\bib{BorWall}{book}{
   author={Borel, A.},
   author={Wallach, N.},
   title={Continuous cohomology, discrete subgroups, and representations of
   reductive groups},
   series={Mathematical Surveys and Monographs},
   volume={67},
   edition={2},
   publisher={American Mathematical Society, Providence, RI},
   date={2000},
   pages={xviii+260},
   isbn={0-8218-0851-6},

   doi={10.1090/surv/067},
}

\bib{geom}{article}{
   author={Deitmar, Anton},
   title={Geometric zeta functions of locally symmetric spaces},
   journal={Amer. J. Math.},
   volume={122},
   date={2000},
   number={5},
   pages={887--926},
   issn={0002-9327},

}

\bib{class}{article}{
   author={Deitmar, Anton},
   title={Class numbers of orders in cubic fields},
   journal={J. Number Theory},
   volume={95},
   date={2002},
   number={2},
   pages={150--166},
   issn={0022-314X},

}

\bib{GAFA}{article}{
   author={Deitmar, A.},
   title={A prime geodesic theorem for higher rank spaces},
   journal={Geom. Funct. Anal.},
   volume={14},
   date={2004},
   number={6},
   pages={1238--1266},
   issn={1016-443X},

   doi={10.1007/s00039-004-0490-7},
}

\bib{HRLef}{article}{
   author={Deitmar, Anton},
   title={A higher rank Lefschetz formula},
   journal={J. Fixed Point Theory Appl.},
   volume={2},
   date={2007},
   number={1},
   pages={1--40},
   issn={1661-7738},

   doi={10.1007/s11784-007-0028-3},
}

\bib{NonUnitaryTrace}{article}{
   author={Deitmar, Anton},
   title={Spectral theory for non-unitary twists},
   journal={Hiroshima Math. J.},
   volume={49},
   date={2019},
   number={2},
   pages={235--249},
   issn={0018-2079},

   doi={10.32917/hmj/1564106546},
}

\bib{FP}{article}{
   author={Fedosova, Ksenia},
   author={Pohl, Anke},
   title={Meromorphic continuation of Selberg zeta functions with twists
   having non-expanding cusp monodromy},
   journal={Selecta Math. (N.S.)},
   volume={26},
   date={2020},
   number={1},
   pages={Paper No. 9, 55},
   issn={1022-1824},

   doi={10.1007/s00029-019-0534-3},
}

\bib{HC-DSII}{article}{
   author={Harish-Chandra},
   title={Discrete series for semisimple Lie groups. II. Explicit
   determination of the characters},
   journal={Acta Math.},
   volume={116},
   date={1966},
   pages={1--111},
   issn={0001-5962},

   doi={10.1007/BF02392813},
}

\bib{HeSch}{article}{
   author={Hecht, Henryk},
   author={Schmid, Wilfried},
   title={Characters, asymptotics and ${\germ n}$-homology of Harish-Chandra
   modules},
   journal={Acta Math.},
   volume={151},
   date={1983},
   number={1-2},
   pages={49--151},
   issn={0001-5962},

   doi={10.1007/BF02393204},
}

\bib{jost}{book}{
   author={Jost, J\"{u}rgen},
   title={Riemannian geometry and geometric analysis},
   series={Universitext},
   edition={7},
   publisher={Springer, Cham},
   date={2017},
   pages={xiv+697},
   isbn={978-3-319-61859-3},
   isbn={978-3-319-61860-9},

   doi={10.1007/978-3-319-61860-9},
}

\bib{Knapp}{book}{
   author={Knapp, Anthony W.},
   title={Representation theory of semisimple groups},
   series={Princeton Landmarks in Mathematics},
   note={An overview based on examples;
   Reprint of the 1986 original},
   publisher={Princeton University Press, Princeton, NJ},
   date={2001},
   pages={xx+773},
   isbn={0-691-09089-0},

}

\bib{Lub}{article}{
   author={Lubotzky, Alexander},
   author={Mozes, Shahar},
   author={Raghunathan, M. S.},
   title={The word and Riemannian metrics on lattices of semisimple groups},
   journal={Inst. Hautes \'{E}tudes Sci. Publ. Math.},
   number={91},
   date={2000},
   pages={5--53 (2001)},
   issn={0073-8301},

}

\bib{Selberg}{article}{
   author={Selberg, Atle},
   title={On discontinuous groups in higher-dimensional symmetric spaces},
   conference={
      title={Contributions to function theory},
      address={internat. Colloq. Function Theory, Bombay},
      date={1960},
   },
   book={
      publisher={Tata Institute of Fundamental Research, Bombay},
   },
   date={1960},
   pages={147--164},

}

\bib{Sh}{book}{
   author={Shubin, M. A.},
   title={Pseudodifferential operators and spectral theory},
   edition={2},
   note={Translated from the 1978 Russian original by Stig I. Andersson},
   publisher={Springer-Verlag, Berlin},
   date={2001},
   pages={xii+288},
   isbn={3-540-41195-X},
   review={\MR{1852334}},
   doi={10.1007/978-3-642-56579-3},
}

\bib{Spilioti}{article}{
   author={Spilioti, Polyxeni},
   title={Selberg and Ruelle zeta functions for non-unitary twists},
   journal={Ann. Global Anal. Geom.},
   volume={53},
   date={2018},
   number={2},
   pages={151--203},
   issn={0232-704X},

   doi={10.1007/s10455-017-9571-3},
}

\bib{Spilioti2}{article}{
   author={Spilioti, Polyxeni},
   title={Functional equations of Selberg and Ruelle zeta functions for
   non-unitary twists},
   journal={Ann. Global Anal. Geom.},
   volume={58},
   date={2020},
   number={1},
   pages={35--77},
   issn={0232-704X},

   doi={10.1007/s10455-020-09715-8},
}

\bib{Spilioti3}{article}{
   author={Frahm, Jan},
   author={Spilioti, Polyxeni},
   title={Twisted Ruelle zeta function at zero for compact hyperbolic surfaces
},
   eprint={https://arxiv.org/abs/2105.13321v1},
   date={2021},
}

\bib{HC-HA1}{article}{
   author={Harish-Chandra},
   title={Harmonic analysis on real reductive groups. I. The theory of the
   constant term},
   journal={J. Functional Analysis},
   volume={19},
   date={1975},
   pages={104--204},

   doi={10.1016/0022-1236(75)90034-8},
}

\bib{Struble}{article}{
   author={Struble, Raimond A.},
   title={Metrics in locally compact groups},
   journal={Compositio Math.},
   volume={28},
   date={1974},
   pages={217--222},
   issn={0010-437X},

}

\end{biblist} \end{bibdiv}

{\small Mathematisches Institut\\
Auf der Morgenstelle 10\\
72076 T\"ubingen\\
Germany\\
\tt deitmar@uni-tuebingen.de}

\today

\end{document}


text:
\section{Introduction}
The Standard Model (SM) has been more solidified by the Large Hadron Collider (LHC) data at
$\sqrt{s}=13{\;{\rm TeV}}$.
Supersymmetry models, composite Higgs models,
and other SM extensions are all strongly constrained.
Despite the absence of new signals,
we believe that new physics beyond the SM must exist
since the issue of naturalness and the existence of dark matter
cannot  be explained within the SM.
There are two kinds of strategies to confront no signals,
pushing
new particles out of the LHC reach~\cite{Kowalska:2016ent}
or introducing
a hidden sector~\cite{Patt:2006fw,Strassler:2006im}.
If either is the case,
we turn to radiative corrections mediated by new particles
or the linking to the hidden sector~\cite{Englert:2011yb} in order to find
a clue about new physics.

An interesting question
is how significantly the radiative correction or the linking changes tree level results.
They generally cause subleading corrections, but
there exist extreme cases also.
We may see entirely new signals
which are absent at tree level, such as the flavor changing neutral current processes through loops and
the invisible Higgs decay modes through the mixing with a hidden sector.
In this work, we investigate a possibility that such new signals are so significant that
they could  be useful in the search for a new heavy scalar boson.

Among various tree level results,
the decay of a \emph{heavy} scalar boson shows an intriguing and unique feature,
the longitudinal polarization enhancement in its decay into a massive gauge boson pair.
It is well-known that
 if the SM Higgs boson $h$ were much heavier,
it would have decayed dominantly into a heavy gauge boson pair, $VV$ ($V=W,Z$).
The $\ttop$ channel, the next dominant one, has
the maximum branching ratio about 19\%.
The extraordinarily large $\Gamma(h \to VV)$ when $m_h \gg m_V$
is due to its decay into the longitudinal modes, $V_L V_L$~\cite{Cahn:1983ip,Kane:1984bb}.
The longitudinal polarization vector of $V$ is proportional to
$p_V^\mu/m_V$ in the high energy limit,
which leads to $\Gamma(h \to V_L V_L) \propto m_h^3$.
The heavier the Higgs boson is, the larger the decay rate into $V_L V_L$ becomes.
For instance,
$\Gamma(h \to V_L V_L)$ is about 99\% of $\Gamma(h \to VV)$
when $m_h \simeq 440{\;{\rm GeV}}$.
Accordingly its total decay rate is also enhanced so that a heavy Higgs boson
becomes a broad resonance.

We wonder whether the same thing happens when a new heavy scalar boson  decays only radiatively.
To answer this question, we consider a simple extension of the SM
where
a  singlet scalar $S$~\cite{Schabinger:2005ei}
and vector-like quarks (VLQs)~\cite{Ellis:2015oso} are introduced.
This can be considered as a simplified model.
Theoretically, a singlet scalar has drawn a lot of interest
in the context of Higgs portal models \cite{Patt:2006fw}.
Its phenomenological signatures have been extensively studied~\cite{Barger:2007im,Dawson:2009yx,Fox:2011qc,Batell:2011pz,Bertolini:2012gu,Robens:2015gla}.
Heavy VLQs are also interesting as they
appear around the electroweak scale
in many new physics models~\cite{Lavoura:1992np,Aguilar-Saavedra:2013qpa}.
The VLQs
are compatible with the current experimental results
while new heavy SM-like chiral quarks are excluded
by the measurement of Higgs production rates~\cite{Anastasiou:2011qw,Anastasiou:2016cez}.
Moreover, the combination of a singlet scalar and vector-like quarks
is attractive:
it can shift
the metastability of the electroweak vacuum in the SM~\cite{Cao:2013cfa,Xiao:2014kba,Batell:2012zw,Degrassi:2012ry,Buttazzo:2013uya};
it is crucial to construct a model
where all of the gauge and Yukawa couplings remain asymptotically safe
up to infinite energy~\cite{Litim:2015iea,Pelaggi:2017wzr}.

In order to focus on the role of radiative corrections exclusively,
we consider a limiting scenario where $S$ does not couple to the SM particles at tree level.
The VLQs play the role of messengers
as connecting the SM particles and $S$ at loop level,
as interacting with the singlet scalar $S$, the SM Higgs boson, and the SM gauge bosons.
In order to allow the Yukawa interactions of VLQs with the Higgs boson,
we introduce three VLQ multiplets, an $SU(2)_L$ doublet, and two (up-type and down-type)
$SU(2)_L$ singlets.
The presence of multiple VLQs shall be shown crucial in the $S$ phenomenology.
The VLQ loops have two kinds of implications.
First $S$ decays into $gg$, $\rr$, $WW$, $ZZ$, and $hh$ through triangle VLQ loops.
The singlet scalar $S$
can be produced and probed at high energy colliders.
Secondly,
$S$ is radiatively mixed with the Higgs boson.
Naive expectation is that the heavier the VLQs are, the smaller the loop corrections
become.
We shall show that this is not true.
Large mass differences in the VLQ mass spectrum
induce the longitudinal polarization enhancement
and increase the $S$-$h$ radiative mixing.
The obtained condition for
the enhancement at loop level
shall help to study the physical properties of new
particles running in the loop.
These are our main results.

The paper is organized as follows.
In Sec.~\ref{sec:general},
we provide the general helicity amplitude framework for the decay of a scalar boson
into a massive gauge boson pair and into a
Higgs boson pair.
Section \ref{sec:model}
summarizes our new physics model with a singlet scalar $S$ and VLQs.
The gauge and Yukawa couplings of the VLQs in terms of the mass eigenstates are given.
In Sec.~\ref{sec:loop},
we present our main analytic results of loop calculations.
The radiatively generated $S$-$h$ mixing and the loop induced
decay rates of $S \to hh, VV$ are to be shown.
In particular, the asymptotic behaviors of the loop functions
are very useful to understand the enhancement of $\Gamma(S\to hh, VV)$
by large mass differences of the VLQs.
Section \ref{sec:results} is devoted to our numerical results
in a simple benchmark scenario.
The general physical properties of $S$ such as its branching ratio
and total decay rate are studied.
We also calculate the exclusion limits from the current LHC data
of the heavy scalar searches and the Higgs precision observation.
The future prospect at the $13{\;{\rm TeV}}$ LHC is also discussed.
Section \ref{sec:conclusions} contains our conclusions.

\section{Decays of a scalar boson into $VV$ and $hh$}
\label{sec:general}

We consider a $\mathcal{J^{PC}}=0^{++}$
scalar particle $S$ which has a  mass $m_S$ and
a momentum $p^\mu$.
In the CP-conserving framework,
the most general coupling of $S$
to a pair of gauge bosons and that to
a pair of the Higgs bosons can be parameterized by
\begin{eqnarray}
\label{eq:def:ABC}
S(p) V_{\mu} (p_1) V^{\prime}_{\nu}(p_2) & :&  m_S
\left[ \mathcal{A} \, g_{{\mu\nu}} + \mathcal{B} \, \frac{p_{2\mu} p_{1\nu}}{m_S^2} \right],
\\ \nonumber
S h h & : & m_S \, \mathcal{C},
\end{eqnarray}
where $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are
dimensionless.

We write the helicity amplitudes for the decay $S\to V V'$
as
\begin{eqnarray}
\langle V_{\mu} (p_1,\lambda_1) V^{\prime}_{\nu}(p_2, \lambda_2) |S(P) \rangle
&\equiv&
m_S \mathcal{T}_{\lambda_1 \lambda_2},
\end{eqnarray}
where $\lambda_{1}$ and $\lambda_2$ are the helicities of the outgoing gauge bosons.
The dimensionless amplitudes $\mathcal{T}_{\lambda_1 \lambda_2}$'s are then
written in terms of $\mathcal{A}$ and $\mathcal{B}$ in Eq.~(\ref{eq:def:ABC}) as~\cite{Choi:2002jk}
\begin{eqnarray}
\label{eq:hel:amp}
\mathcal{T}_{++} &=&  \mathcal{T}_{--} = -\mathcal{A},
\\ \nonumber
\mathcal{T}_{00} &=&
\left\{
\begin{array}{ll}
\dfrac{m_S^2}{4 m_V^2}(2 \mathcal{A} + \mathcal{B}) - (\mathcal{A}+\mathcal{B}), &
\hbox{ if } m_V \equiv m_{V_1}=m_{V_2}\neq 0;
\\[5pt]
0, & \hbox{ if } m_{V_1}=0 \hbox{ or } m_{V_2}= 0,
\end{array}
\right.
\end{eqnarray}
and the other helicity amplitudes are zero.
The partial decay rates are
\begin{eqnarray}
\label{eq:decay:rate}
\Gamma (S \to VV') &=& \frac{1}{\mathcal{S}}\frac{\beta_{VV'}}{16 \pi } m_S
\sum_{\lambda_1,\lambda_2} \left| \mathcal{T}_{\lambda_1\lambda_2}\right|^2,
\\[3pt] \nonumber
\Gamma (S \to hh) &=&\frac{\beta_{hh}}{32\pi } m_S \left| \mathcal{C} \right|^2,
\end{eqnarray}
where the symmetric factor $\mathcal{S}$ is $1/2$ for two identical outgoing particles,
and
$\beta_{ij} = \sqrt{1 -  2(m_i^2+m_j^2)/m_S^2 + (m_i^2-m_j^2)^2/m_S^4}$.

When a scalar particle is heavy enough,
its decay into a massive gauge boson pair $VV$ ($V=W^\pm,Z$)
has a special feature.
The condition $m_S \gg m_V$ makes $\mathcal{T}_{00}$
greatly enhanced if $(2\mathcal{A} + \mathcal{B}) \neq 0$.
The SM Higgs boson, if its mass is greater than $2m_V$,
has
\begin{eqnarray}
\label{eq:A:B:SM}
\mathcal{A}^{h_{\rm SM}} =
\frac{2m_V^2}{v m_h},
\quad \mathcal{B}^{h_{\rm SM}} =0.
\end{eqnarray}
The partial
decay rate of $h_{\rm SM} \to V_L V_L$ is proportional to the cube of $m_h$
while that of $h_{\rm SM} \to V_T V_T$ is inversely proportional to $m_h$.
The heavier the Higgs boson is, the more dominant $h \to V_L V_L$
will become.
Another significant decay rate $\Gamma(h\to \ttop)$ is linearly proportional to $m_h$.
The Higgs boson decay into $ V_L V_L$ is dominant in the total decay rate.
This is called the longitudinal polarization enhancement.

The partial decay rate of $S$ into a pair of SM Higgs bosons
is sizable if $\mathcal{C} \sim \mathcal{O} (1)$.
In the MSSM,  an obvious scalar boson candidate which decays into $hh$ is
the heavy CP-even Higgs boson $H$.
However,
the decay into a pair of light Higgs bosons is suppressed in the alignment limit
since $\mathcal{C}$  is ~\cite{Craig:2013hca}
\begin{eqnarray}
\label{eq:C:MSSM}
\mathcal{C}^{H_{\rm MSSM}} &=&
-
\frac{3 g_Z^2 \sin{4\beta}}{8} \frac{v}{M_H}
\left[
1+\mathcal{O} (\cos(\beta-\alpha))
\right],
\end{eqnarray}
where
$g_Z = g/\cos\theta_W$ and $\theta_W$ is the weak mixing angle.
The partial decay rate $\Gamma(H \to hh)$ is inversely proportional to the
heavy Higgs mass: there is no
enhancement in the $hh$ decay channel.

\section{Model with a singlet scalar and vector-like quarks}
\label{sec:model}

We consider a simple extension of the SM
by introducing a CP-even singlet scalar boson $S_0$,
a VLQ doublet $\mathcal{Q}_{L/R}$, two
VLQ singlets $\mathcal{U}_{L/R}$ and $\mathcal{D}_{L/R}$:
\begin{eqnarray}
\mathcal{Q}_{L/R}=\left( \begin{array}{c}
              \mathcal{U}^{\prime} \\
              \mathcal{D}^{\prime} \end{array} \right)_{L/R},
\quad
\mathcal{U}_{L/R}\,, \quad \mathcal{D}_{L,R}.
\end{eqnarray}
The $SU(3)_c \times SU(2)_L \times U(1)_Y$ quantum numbers of
$\mathcal{Q}_{L/R}$, $\mathcal{U}_{L/R}$, $D_{L/R}$ are
$(\mathbf{3}, \mathbf{2}, 1/3)$,  $(\mathbf{3}, \mathbf{1},4/3)$,
and $ (\mathbf{3}, \mathbf{1},  -2/3)$, respectively.
The hypercharges of VLQs are set to be the same as the SM quarks.
Different assignment
shall affect the decays of $S$ into $ZZ$ and $\rr$.

The most general scalar potential of the SM Higgs doublet $H$
and a real singlet scalar $S_0$
is \cite{Chen:2014ask}
\begin{eqnarray}
\label{eq:V:HS}
V(H,S_0) &=& -\mu^2 H^\dagger H +\lambda( H^\dagger H)^2
+\frac{a_1}{2} S_0 H^\dagger H + \frac{a_2}{2} S_0^2 H^\dagger H
\\ \nonumber
&& + b_1   S_0 + \frac{b_2}{2}  S_0^2 + \frac{b_3}{3} S_0^3+ \frac{b_4}{4} S_0^4.
\end{eqnarray}
Note that we do not assume any discrete $Z_2$ symmetry for $S_0$.
When defining the neutral component of $H$ as
$\phi_0 = (v_0 + h_0)/\sqrt{2}$ and
the VEV of the singlet field as
$\langle S_0 \rangle = x$, the extrema of the potential satisfy
\begin{eqnarray}
\label{Eq:min:condition}
\frac{\partial V(v_0/\sqrt{2},x)}{\partial v_0} =0, \quad \frac{\partial V(v_0/\sqrt{2},x)}{\partial x} =0.
\end{eqnarray}
Although there exist many possible extrema, the
true minimum of $H$ should generate proper EWSB, i.e., $v_0=v=246{\;{\rm GeV}}$.
On the other hand, the VEV of $S$
is free to choose
since the shift of the singlet field, $S \to S +\dts$,
corresponds to redefining the parameters of $a_{1,2}$ and $b_{1,\cdots,4}$.
There is no change in physics.
Without loss of generality we take $(v_0,x) = (v, 0)$.
Note that the choice of vanishing VEV for $S_0$ eliminates the tadpole
term of $S_0$.
The minimization conditions in Eq.~(\ref{Eq:min:condition}) become
\begin{eqnarray}
\label{eq:min:condition:0Svev}
\mu^2 &=& \lambda v^2,
\quad
b_1 = -\frac{v^2}{4}a_1.
\end{eqnarray}

The Yukawa terms of VLQs with the singlet $S_0$ and the SM Higgs doublet $H$
as well as their mass terms are
\begin{eqnarray}
\label{eq:L:Yukawa}
-\lg_Y&=&
S_0 \left[
y_\mathcal{Q} \bar{\mathcal{Q}}\mathcal{Q} + y_\mathcal{U} \bar{\mathcal{U}}\mathcal{U} + y_\mathcal{D} \bar{\mathcal{D}}\mathcal{D}
\right]
+M_\mathcal{Q}  \bar{\mathcal{Q}}\mathcal{Q} + M_\mathcal{U} \bar{\mathcal{U}}\mathcal{U} + M_\mathcal{D} \bar{\mathcal{D}}\mathcal{D}
\\ \nonumber &&
+\left[
Y_\mathcal{D} \bar{\mathcal{Q}}_L H \mathcal{D}_R + Y'_\mathcal{D} \bar{\mathcal{Q}}_R H \mathcal{D}_L
+
Y_\mathcal{U} \bar{\mathcal{Q}}_L \widetilde{H} \mathcal{U}_R
+
Y'_\mathcal{U} \bar{\mathcal{Q}}_R \widetilde{H} \mathcal{U}_L + H.c.
\right],
\end{eqnarray}
where $\widetilde{H} = i \tau_2 H^*$.
For simplicity, we assume
$y_\mathcal{Q} = y_\mathcal{U} = y_\mathcal{D} \equiv y_S$, $Y_{\mathcal{U}}=Y_{\mathcal{U}'}$, and $Y_{\mathcal{D}}=Y_{\mathcal{D}'}$
in what follows.

The VLQ mass matrix
$\mathbb{M}_{F}$  in the basis of $(F', F)$ where $F=\mathcal{U},\mathcal{D}$
is
\begin{eqnarray}
\mathbb{M}_F =
\left(
\begin{array}{cc}
~M_\mathcal{Q}~ & ~\frac{Y_{F} v}{\sqrt{2}}~ \\[5pt] \frac{Y_{F} v}{\sqrt{2}} & M_F
\end{array}
\right),
\end{eqnarray}
which is diagonalized by
the mixing matrix of
\begin{eqnarray}
\label{eq:mixing:R}
\mathbb{R}_{\theta_F}
=
\left(
\begin{array}{cc}
\cthf & -\sthf \\ \sthf & \cthf
\end{array}
\right).
\end{eqnarray}
Here we adopt simplifying notations of $c_x =\cos x$ and $s_x =\sin x$.
The $Y_\mathcal{U}$ and $Y_\mathcal{D}$ terms
generate the mixings between VLQ doublet and VLQ singlets.
If $Y_F \sim \mathcal{O}(1)$,
VLQ mixing angles are small
since VLQs are expected to be heavy.
The mass eigenvalues and the mixing angle  are then
\begin{eqnarray}
\label{eq:MF:sinF}
M_{{F_1},{F_2}}&=& \frac{1}{2} \left[M_\mathcal{Q}+M_F \mp
\sqrt{ ( M_F-M_\mathcal{Q} )^2+ 2 Y_F^2 v^2 }
\right],
\\ \nonumber
\stthf &=&  \,\frac{ \sqrt{2} Y_F v }{M_{F_2} - M_{F_1}},
\end{eqnarray}
where $M_{F_1}<M_{F_2}$.

The Yukawa terms of the VLQ mass eigenstates become
\begin{eqnarray}
-\lg_{\rm Yukawa}
&=& y_S S_0 \sum_{i}
\left[
\overline{\mathcal{U}}_i \mathcal{U}_i + \overline{\mathcal{D}}_i \mathcal{D}_i
\right]
+
h_0 \sum_{F} \sum_{i,j}
 y_{_{h F_i F_j} }
\bar{F}_i F_j
,
\end{eqnarray}
where $F=\mathcal{U},\mathcal{D}$, $i,j=1,2$, and
$y_{_{h F_i F_j} }$ are
\begin{eqnarray}
\label{eq:y:hFF}
y_{_{h F_1 F_1}} &=& - y_{_{h F_2 F_2}}
=  -\frac{Y_F}{\sqrt{2}}\,s_{2\theta_F},
\quad
y_{_{h F_1F_2}} = y_{_{h F_2 F_1}} = -\frac{Y_F}{\sqrt{2}}\,c_{2 \theta_F}.
\end{eqnarray}

The gauge interaction Lagrangian in terms of the VLQ mass eigenstates is
\begin{eqnarray}
\lg_{\rm gauge} &=&
e  A_\mu \sum_F \sum_{i} Q_F \bar{F}_i \gamma^\mu F_i
+ g_Z Z_\mu \sum_{F} \sum_{i,j} \hat{g}_{_{Z F_i F_j}}
\bar{F}_i \gamma^\mu F_j
\\ \nonumber
&& +
\frac{g}{\sqrt{2}}\,
\Big[
W^{+\mu} \sum_{i,j} \hat{g}_{_{W \mathcal{U}_i \mathcal{D}_j}}
\bar{\mathcal{U}}_i \gamma_\mu \mathcal{D}_j
+ H.c. \Big].
\end{eqnarray}
Here $Q_F$ is the electric charge of the fermion $F$
and the effective gauge couplings $\hat{g}_{_{VFF'}}$ are
\begin{eqnarray}
\label{eq:gh:VFF}
\hat{g}_{_{Z F_1 F_1 }} &=& \bar{g}_{\mathcal{Q}}^{v} \cthf^2 + \bar{g}_{F}^{v} \sthf^2, \quad
\hat{g}_{_{ZF_2F_2}}= \bar{g}_{\mathcal{Q}}^{v}\sthf^2 + \bar{g}_{ F}^v \cthf^2,
\\ \nonumber
\hat{g}_{_{Z F_1 F_2}} &=&
\left( \bar{g}_{\mathcal{Q}}^{v} - \bar{g}_{F}^{v} \right) \sthf\cthf,
\\ \nonumber
\hat{g}_{_{W\mathcal{U}_1 \mathcal{D}_1}} &=&
\cthu \cthd ,\quad
\hat{g}_{_{W \mathcal{U}_1 \mathcal{D}_2}} =
 \cthu \sthd,
\\ \nonumber
\hat{g}_{_{W \mathcal{U}_2 \mathcal{D}_1}} &=&
\sthu \cthd , \quad
\hat{g}_{_{W \mathcal{U}_2 \mathcal{D}_2} } =
  \sthu\sthd ,
\end{eqnarray}
where $\hat{g}_{_{VFF'}}=\hat{g}_{_{VF'F}}$ and
$\bar{g}_{\mathcal{F}}^v = \tfrac{1}{2}T_3^{\mathcal{F}} - s_{W}^2 Q_\mathcal{F}$
for $\mathcal{F}=\mathcal{Q},\mathcal{U},\mathcal{D}$.
There is a big difference between $h$-$F$-$F'$ couplings and $V$-$F$-$F'$ couplings.
In the limit of $\theta_{\mathcal{U},\mathcal{D}}\ll 1$,
the gauge couplings to different mass eigenstates of VLQs
(e.g. $\hat{g}_{_{V F_1 F'_2}}$)
are suppressed by $s_{\theta_F}$.
On the contrary, the VLQ couplings to the Higgs boson
are suppressed for the same mass eigenstates.

Without the $Z_2$ symmetry,
the $S_0$ field can couple to the SM particles at tree level.
Since the singlet scalar $S_0$ is neutral under all quantum numbers of the SM
gauge group,
the only possible renormalizable couplings of $S_0$ to the SM particles
at tree level
are to the Higgs boson
through $a_1$ and $a_2$ terms in Eq.~(\ref{eq:V:HS}).
However, a nonvanishing $a_1$ term
will generate the $S$-$h$ mixing with the mixing angle $\eta$,
which shall change the Higgs coupling modifiers of $\kappa_V$ and $\kappa_f$ into $c_\eta$.
According to the global fit analysis of the LHC Higgs precision data~\cite{Dolan:2016eki,kappa:2015,Cheung:2015dta},
$c_\eta$ is very close to 1.
Nonzero $a_1$ builds up
some tension with the Higgs boson constraints.
Moreover, our main question is whether the unique characteristic of a heavy scalar boson
such as the longitudinal polarization enhancement
remains even at loop level.
Therefore,
we consider a limiting scenario in which
the singlet scalar
has no tree level couplings with the Higgs boson:
\begin{eqnarray}
\label{eq:no:tree}
a_1^{\rm tree}=0=a_2^{\rm tree}.
\end{eqnarray}

\section{The effects of the VLQ loops}
\label{sec:loop}
In the previous section,
we suggested a rather  extreme scenario
where $S_0$
does not interact with the SM Higgs boson at tree level.
The singlet field $S_0$ could be considered as a field in a hidden sector.
In the model,
the visible sector and the hidden sector are connected via VLQ loops:
the VLQs play the role of messengers.
There are two phenomenological implications: (i)
the singlet-Higgs mixing and (ii)
the radiative decays of $S$  into SM particles.
We study the effects at one loop level.

\subsection{$S$-$h$ mixing and Higgs Modifiers}

\begin{figure}[h] \centering
\begin{center}
\includegraphics[width=.35\textwidth]{fig-Feyn-Sh-mixing.pdf}
\end{center}
\caption{\label{fig-Feyn-Sh-mixing}
\baselineskip 3.0ex
Feynman diagrams for the loop induced $S$-$h$ mixing.
}
\end{figure}

First, the $S$-$h$ mixing is radiatively generated through
the VLQ loops as shown
in Fig.~\ref{fig-Feyn-Sh-mixing}.
The scalar-mass-squared matrix in the basis
of $(h_0,S_0)$
becomes
\begin{eqnarray}
\mathbb{M}_{hS}^2
\equiv
\left(
\begin{array}{cc}
2 \lambda v^2 & \delta M_{Sh}^2 \\
\delta M_{Sh}^2 & M_{SS}^2
\end{array}
\right)\,,
\end{eqnarray}
where $M_{SS}^2= b_2$
since we have used the conditions in
Eq.~(\ref{eq:min:condition:0Svev})
for our choice of the vacuum $(v_0,x) = (v,0)$.
At one loop level, we have
\begin{eqnarray}
\delta M_{Sh}^2 &=& -
\frac{y_S N_c}{4\pi^2}
\sum_F \sum_i
y_{_{hF_i F_i }} M_{F_i}^2
\bigg[4(\tau^S_{F_i}-1)g(\tau^S_{F_i}) -4\tau^S_{F_i}+5
\bigg]\,,
\end{eqnarray}
where $N_c=3$ is the color factor of the VLQ,
$F=\mathcal{U},\mathcal{D}$, $i=1,2$,
$\tau^i_{j} = m_i^2/(4m_j^2)$,  and
$y_{_{h F F'}}$'s are in Eq.~(\ref{eq:y:hFF})\footnote{
There is UV divergence in the one loop calculation of $\delta M_{Sh}^2$ which must be properly renormalized. Detailed description on the renormalization of the whole model is in preparation~\cite{future}.
}.
The loop function $g(\tau)$ is given by
\begin{eqnarray}
g(\tau) &=& \left\{ \begin{array}{ll}
\sqrt{\tau^{-1}-1}\arcsin\sqrt{\tau} & \hbox{ if } \tau \le 1; \\[3pt]
\frac{\sqrt{1-\tau^{-1}}}{2}\left[
\log\frac{1+\sqrt{1-\tau^{-1}}}{1-\sqrt{1-\tau^{-1}}}-i\pi \right]
& \hbox{ if } \tau > 1\,. \end{array} \right.
\end{eqnarray}
Note that $\delta M_{Sh}^2$ vanishes if $M_{F_1}=M_{F_2}$
since $y_{h F_1 F_1} = - y_{h F_2 F_2}$: see Eq.~(\ref{eq:y:hFF}).
Significant $S$-$h$ mixing requires sizable mass differences of $F_1$ and $F_2$.

The mass eigenvalues and the $S$-$h$ mixing angle $\eta$ are
\begin{eqnarray}
m_{h,S}^2&=&
\frac{1}{2}
\left(
M_{hh}^2+M_{SS}^2 \mp \sqrt{(M_{SS}^2-M_{hh}^2)^2+ 4 (\delta M_{Sh}^2)^2 }
\right),
\\ \nonumber
s_{2\eta} &=& -
\frac{ 2 \,\delta M_{Sh}^2}{m_S^2-m_h^2 }
,
\end{eqnarray}
where we use the $S$-$h$ mixing matrix $\mathbb{R}_\eta$
in Eq.~(\ref{eq:mixing:R}).
Since $\delta M_{Sh}^2$ is radiatively generated,
we expect $s_\eta \ll 1$.
We take the mass eigenstate $h = c_\eta h_0 - s_\eta S_0$
to be the observed Higgs boson with a mass of $125{\;{\rm GeV}}$, and assume
$S$ to be heavy such as  $m_S \gtrsim 500{\;{\rm GeV}}$.

The nonzero $S$-$h$ mixing changes
the Higgs coupling modifiers
of $\kappa_Z$, $\kappa_W$, $\kappa_t$, $\kappa_\tau$, and $\kappa_b$
to be $c_\eta$.
The loop induced decays of the Higgs boson into $gg$ and $\rr$
have additional loop contributions from VLQs.
We define $\kappa_g$ and $\kappa_\gamma$ as
\begin{eqnarray}
\lg_{\rm Higgs}
&=& \kappa_g c_g^{\rm SM}
\,\frac{h}{v}\, G^{a{\mu\nu}} G^a_{\mu\nu}
+ \kappa_\gamma c_\gamma^{\rm SM}
\,\frac{h}{v}\, F^{{\mu\nu}} F_{\mu\nu}\,.
\end{eqnarray}
The SM values $c_g^{\rm SM}$ and $c_\gamma^{\rm SM}$ are
\begin{eqnarray}
c_g^{\rm SM} \equiv
\frac{\alpha_s}{16\pi}
 A^{\rm SM}_{hgg}
, \quad
c_\gamma^{\rm SM} \equiv
\frac{\alpha_e}{8 \pi}
 A^{\rm SM}_{h\rr},
\end{eqnarray}
where $A^{\rm SM}_{hgg} = \sum_{f=t,b} A_{1/2}(\tau^h_f)$,
$A^{\rm SM}_{h\rr}
=
A_1(\tau^h_W) +
\sum_{f=t,b,\tau} N_C^f Q_f^2 A_{1/2}(\tau^h_f)$, and $ A_{1/2}(\tau)$ and $A_1(\tau)$
are referred to Ref.~\cite{Djouadi:2005gi}.
The modifiers
$\kappa_g$ and $\kappa_\gamma$ receive two kinds of new contributions.
One is from the modified couplings of $h$ to the SM particles through the $S$-$h$ mixing.
The other is from the triangle VLQ loops, parameterized by
\begin{eqnarray}
\label{eq:Ahgg}
\mathcal{A}^{\rm VLQ}_{hgg}
&=&
\sum_{F}\sum_i
y_{_{h F_i F_i}}  \frac{v}{M_{F_i}} A_{1/2}(\tau^h_{F_i})
,
\\ \nonumber
\mathcal{A}^{\rm VLQ}_{h\rr}
&=&
\sum_{F}\sum_i
N_C
Q_{F_i}^2 y_{_{h F_i F_i}}
\frac{v}{M_{F_i}} A_{1/2}(\tau^h_{F_i})
,
\end{eqnarray}
where $F=\mathcal{U},\mathcal{D}$,
$i=1,2$,
$\tau^i_j=m_i^2/(4 m_j^2)$.
Then $\kappa_g$ and $\kappa_\gamma$ are
\begin{eqnarray}
\kappa_{g,\gamma} &=&
\frac{ c_\eta A^{\rm SM}_{hgg,h\rr}
+
\mathcal{A}^{\rm VLQ}_{hgg,h\rr} }
{A^{\rm SM}_{hgg,h\rr}
}
.
\end{eqnarray}
Since $y_{_{h F_1 F_1}}$ and $y_{_{h F_2 F_2}}$
in Eq.~(\ref{eq:y:hFF}) have opposite signs,
both $\delta M_{Sh}^2$ as well as $\mathcal{A}^{\rm VLQ}_{hgg,h\rr}$ are suppressed
when $M_{F_1} \simeq M_{F_2}$.

\begin{figure*}[t!]
    \centering
    \begin{subfigure}[b]{0.5\textwidth}
        \centering
        \includegraphics[width=0.8\textwidth]{fig-Feyn-Shh}
        \caption{}
    \end{subfigure}%
    ~
    \begin{subfigure}[b]{0.5\textwidth}
        \centering
        \includegraphics[width=0.8\textwidth]{fig-Feyn-SWW}
        \caption{}
    \end{subfigure}
    \caption{\label{fig-Feyn-SWW-Shh}
Feynman diagrams of  $S \to hh$ and $S \to WW$
from the VLQ loops.
}
\end{figure*}

\subsection{Radiative Decays of $S$}

Another important effect of the VLQ loops
is the radiative decay of $S$
into the SM particles,
which occurs through the $S$-$h$ mixing as in Fig.~\ref{fig-Feyn-Sh-mixing}
and/or through the triangle VLQ loops into a gauge boson pair
or a Higgs boson pair as in Fig.~\ref{fig-Feyn-SWW-Shh}.
Since we consider the case of $m_S \gtrsim 500{\;{\rm GeV}}$,
the main decay modes are into $\ttop$, $gg$, $\rr$, $WW$,  $ZZ$, and $hh$.

The decay of $S$ into a top quark pair is only through the $S$-$h$ mixing.
The partial decay rate is
\begin{eqnarray}
\Gamma (S \to \ttop ) = s_\eta^2 \, \Gamma ({h_{\rm SM}}\to \ttop) \Big|_{m_{h_{\rm SM}} = m_S}.
\end{eqnarray}
Another important decay channel is $S \to hh$
shown in Fig.~\ref{fig-Feyn-SWW-Shh}(a).
The vertex $\mathcal{C}$
in Eq.~(\ref{eq:def:ABC}) at one loop level
is
\begin{eqnarray}
\label{eq:Ctot}
\mathcal{C} &=&  \frac{y_S N_c}{4\pi^2}\sum_{F} \;\sum_{i,j}
y_{_{h F_i F_j}}^2  \,
\mathcal{C}_T(m_h,m_S, M_{F_i}, M_{F_j})
 + \frac{3 m_h^2}{v m_S} s_\eta
 \, ,
\end{eqnarray}
where $y_{_{hFF^\prime}}$
are given in Eq.~(\ref{eq:y:hFF}).
The first term is due to the triangle diagrams while the second one is from the $S$-$h$ mixing.
The asymptotic expression\footnote{Full expressions of form factors such as
$\mathcal{A}_T$, $\mathcal{B}_T$, and $\mathcal{C}_T$
are to be presented in Ref.~\cite{future}.} of
$\mathcal{C}_T(m_h,m_S,M_{F_i},M_{F_j})$
when $m_h \ll m_S$ and $\Delta_F
\ll M_\mathcal{F}$, where $\Delta_F
= M_{F_j}-M_{F_i}$ and
$M_\mathcal{F} =
(M_{F_i} + M_{F_j})/2)
$,
is very useful to understand the enhancement of $\Gamma (S\to hh)$ in some parameter space:
\begin{eqnarray}
\label{eq:CM:asymptotic}
\sqrt{\tau}\,{\cal C}_T &=& 2 + (1-2\tau^{-1})f(\tau) -2g(\tau)
\nonumber\\
&& ~
+\Big(\frac{\Delta_F^2}{M_{\cal F}^2}\Big)\bigg[
\frac{8\tau^2+49\tau-48}{12\tau(1-\tau)} + \frac{(\tau^3+12\tau^2-26\tau+16)g(\tau)}{4\tau(1-\tau)^2}
\bigg]\nonumber\\
&& ~
+\Big(\frac{m_h^2}{m_S^2}\Big)
\Big[\frac{2(6-\tau)}{3}+\frac{2(\tau-2)f(\tau)}{\tau}\Big]
+{\cal O}\Big(\frac{\Delta_F^4}{M_{\cal F}^4}\Big)
+{\cal O}\Big(\frac{m_h^4}{m_S^4}\Big)
\,.
\end{eqnarray}
where $\tau =m_S^2/(4 M_\mathcal{F}^2)$ and $f(\tau)$ is referred to Ref.~\cite{Djouadi:2005gi}.
Note that the odd power terms in $(\Delta_F/M_{\cal F})$ are
neglected since they cancel each other after the summation in Eq. (\ref{eq:Ctot}).
If $y_S, Y_{\mathcal{U},\mathcal{D}} \sim \mathcal{O}(1)$,
$\mathcal{C}$ is not suppressed by large $m_S$,
contrary to the case of a heavy CP-even scalar $H$ of the MSSM in Eq.~(\ref{eq:C:MSSM}).
Another important result is that
the partial decay rate $\Gamma(S\to hh)$ increases with $\Delta_F$,
the mass difference between
$M_{F_i}$ and $M_{F_j}$.
Since $\Delta_F$ is proportional to the Higgs VEV from the SM-like Yukawa couplings of the VLQs to the Higgs boson,
the enhancement of $S \to h h $ can be considered as non-decoupling effects.

The VLQ loops also allow the decay of $S$
into a massive gauge boson pair $VV$ ($V=W,Z$)
as shown in Fig.~\ref{fig-Feyn-SWW-Shh}(b).
The dimensionless parameters $\mathcal{A}$ and $\mathcal{B}$
in Eq.~(\ref{eq:def:ABC}) are
\begin{eqnarray}
\nonumber
{\cal A}_{WW} &=&
\frac{g^2 y_S N_c }{8\pi^2}\sum_{i,j}
\left[
   \hat{g}_{_{W\mathcal{U}_i \mathcal{D}_j}}^2
{\cal A}_T(m_W,m_S, M_{\mathcal{U}_i}, M_{\mathcal{D}_j})
+ \{ \mathcal{U} \leftrightarrow \mathcal{D} \}
\right] +
\frac{2 m_W^2}{v m_S} s_\eta
, \\ \nonumber
 {\cal B}_{WW} &=& \frac{g^2 y_S N_c}{8\pi^2}\sum_{i,j}
\left[
  \hat{g}_{_{W\mathcal{U}_i \mathcal{D}_j}}^2
{\cal B}_T(m_W,m_S, M_{\mathcal{U}_i}, M_{\mathcal{D}_j})
+ \{ \mathcal{U} \leftrightarrow \mathcal{D} \}
\right]\,,
\\ \nonumber
{\cal A}_{ZZ} &=&
 \frac{g_Z^2 y_S N_c}{4\pi^2}\sum_{i,j}\left[
  \hat{g}_{_{Z {\cal U}_i {\cal U}_j}}^2
 {\cal A}_T(m_Z,m_S, M_{{\cal U}_i}, M_{{\cal U}_j})
 + \{ \mathcal{U} \leftrightarrow \mathcal{D} \}
\right]\,
 +\frac{ 2m_Z^2}{v m_S} s_\eta
,\\ \label{eq:AVV:BVV}
{\cal B}_{ZZ} &=&
\frac{g_Z^2 y_S N_c}{4\pi^2}\sum_{i,j}
\left[
 \hat{g}_{_{Z{\cal U}_i {\cal U}_j}}^2\,
 {\cal B}_T(m_Z,m_S, M_{{\cal U}_i}, M_{{\cal U}_j})\,
 + \{ \mathcal{U} \leftrightarrow \mathcal{D} \}
\right]\,,
\end{eqnarray}
where
$i,j=1,2$.
$\mathcal{A}_{VV}$ consists of two parts,
one from the triangle VLQ loops and the other from the $S$-$h$ mixing,
while
$\mathcal{B}_{VV}$ is only from the triangle loops.

Our main question is whether the longitudinal polarization enhancement
in $S\to VV$ remains significant at loop level, which happens
when $2\mathcal{A} + \mathcal{B} \neq 0$ as shown in Eq.~(\ref{eq:hel:amp}).
The $S$-$h$ mixing induced terms,
proportional to $s_\eta$ in Eq.~(\ref{eq:AVV:BVV}),
appear only in $\mathcal{A}$ and thus
generate the longitudinal polarization enhancement.
The condition that the triangle VLQ loops induce the enhancement
is easy to see through
the asymptotic behaviors of $\mathcal{A}_T$ and $\mathcal{B}_T$
in the limit of $\Delta_F \ll M_\mathcal{F}$
and $m_{V} \ll m_S$, given by
\begin{eqnarray}
\label{eq:AM}
\sqrt{\tau}\,{\cal A}_T &=& 1+(1-\tau^{-1}) f(\tau)
\\ \nonumber && ~
+\Big(\frac{\Delta_F^2}{M^2}\Big)\Big[
-\frac{1}{4} + \frac{(3\tau-4)f(\tau)}{4\tau^2}
+\frac{(\tau^2+4\tau-8)g(\tau)}{4\tau(\tau-1)}
\Big]
\\ \nonumber && ~
+2\Big(\frac{m_V^2}{m_S^2}\Big)
\Big[ 3-\tau -\tau^{-1} f(\tau) - 2g(\tau) \Big]
 +{\cal O}\Big(\frac{\Delta_F^4}{M^4}\Big)
 +{\cal O}\Big(\frac{m_V^4}{m_S^4}\Big)
\,,
\\
\label{eq:BM}
\sqrt{\tau}\,{\cal B}_T &=& -2-2(1-\tau^{-1}) f(\tau)
\\ \nonumber && ~
+\Big(\frac{\Delta_F^2}{M^2}\Big)\bigg[
\frac{5}{2} + \frac{(8-5\tau)f(\tau)}{2\tau^2}
-\frac{(\tau^2+12\tau-16)g(\tau)}{2\tau(\tau-1)}
\bigg]\\ \nonumber
&& ~
+4\Big(\frac{m_V^2}{m_S^2}\Big)
\Big[\tau-4 +(2-\tau)\tau^{-1} f(\tau) + 2g(\tau) \Big]
+{\cal O}\Big(\frac{\Delta_F^4}{M^4}\Big)
+{\cal O}\Big(\frac{m_V^4}{m_S^4}\Big)\,,
\end{eqnarray}
where $\tau =m_S^2/(4 M_\mathcal{F}^2)$.
Equations (\ref{eq:AM}) and (\ref{eq:BM}) show
that $2\mathcal{A} + \mathcal{B} \sim \mathcal{O}\left(m_V^2/m_S^2 \right) $ if $\Delta_F=0$.
Sizable mass differences of VLQs
are crucial for the longitudinal polarization enhancement
through the triangle VLQ loops.

The last category of the radiative decays of $S$ is into $gg$, $\rr$, and
$Z\gamma$.
When at least one of the outgoing gauge bosons is massless,
there is no longitudinal polarization mode as shown in Eq.~(\ref{eq:hel:amp}).
The
$\mathcal{A}$'s are
\begin{eqnarray}
{\cal A}_{\gamma\gamma} &=& \frac{ e^2 y_S N_c}{4\pi^2}
\sum_{F} \sum_i Q_{F_i}^2 \frac{1}{\sqrt{\tau_{F_i}}}
\left[1+(1-\tau_{F_i}^{-1})f(\tau_{F_i}) \right],
\\ \nonumber
{\cal A}_{gg} &=& \
\delta^{ab}
\frac{ g_s^2 y_S }{8\pi^2}
\sum_{F} \sum_i
\frac{1}{\sqrt{\tau_{F_i}}}
\left[
1+(1-\tau_{F_i}^{-1})f(\tau_{F_i}) \right]\,,
\\ \nonumber
{\cal A}_{Z\gamma} &=&
\frac{ e \, g_Z\, y_S N_c}{2\pi^2}
\sum_F \sum_i Q_{F_i} \hat{g}_{_{Z F_i F_i}}
\frac{1}{\sqrt{\tau_{F_i}}}
\left[
-1-(1-\tau_{F_i}^{-1})f(\tau_{F_i})
+ \mathcal{O} \left( \frac{m_Z^2}{m_S^2} \right) \right],
\end{eqnarray}
where
$a,b$ are color indices of the outgoing gluons,
$F=\mathcal{U},\mathcal{D}$, $i=1,2$, and $\tau_{F_i} = m_S^2/(4 M_{F_i}^2)$.
The ${\cal B}$'s can be obtained by using Ward identity as follows
\begin{eqnarray}
{\cal B}_{\rr} = -2 {\cal A}_{\rr},~~
{\cal B}_{gg} = -2 {\cal A}_{gg},~~
{\cal B}_{Z\gamma} = -2 \Big( 1- \frac{m_Z^2}{m_S^2}\Big)^{-1}{\cal A}_{Z\gamma}\,.
\end{eqnarray}

The final comment in this section is the importance of the VLQ Yukawa couplings
with the \emph{Higgs boson}
in enhancing the radiative decay rates of $S$.
If we do not allow the $Y_\mathcal{U}$ and $Y_\mathcal{D}$ terms,
which happens for example when we introduce only one VLQ multiplet,
the $S$-$h$ mixing and the $S \to hh$ decay will be absent.
In addition, the VLQs running in the triangle VLQ loops for the decay of $S\to WW,ZZ$
have the same masses because of no VLQ mixing.
There is no longitudinal polarization enhancement
and thus the signal rates of the radiative decays have typical loop suppression~\cite{Cao:2009ah}.
In summary, the presence of the VLQ doublet \emph{and} the VLQ singlets
are crucial for the enhanced radiative decays of $S$.

\section{Numerical Results}
\label{sec:results}

The phenomenological characteristics of the singlet scalar $S$
depend on the model parameters of $y_S$, $m_S$,
$Y_{\mathcal{U},\mathcal{D}}$,
$M_\mathcal{Q}$,
$M_{\mathcal{U}}$ and $M_{\mathcal{D}}$.
The $y_S$ contributes equally to all
of the partial decay rates of $S$ by the common factor of $y_S^2$
since $S$ decays only radiatively through VLQ loops in our model.
The branching ratios of $S$ are independent of $y_S$.
The $m_S$ dependence on the branching ratios is also weak for the heavy $S$.
The $Y_{\mathcal{U},\mathcal{D}}$,
$M_\mathcal{Q}$, and
$M_{\mathcal{U},\mathcal{D}}$
specify the VLQ mass matrices and thus the mass difference $\Delta_F$.
Since
$Y_{\mathcal{U}}$ and $Y_{\mathcal{D}}$ also quantify the VLQ couplings with the Higgs boson,
they are the most crucial parameters.

Therefore,  we consider a simple benchmark parameter line, given by
\begin{eqnarray}
\label{eq:benchmark}
M_\mathcal{Q} = M_\mathcal{U}=M_\mathcal{D}, \quad Y_\mathcal{U}=0, \quad Y_\mathcal{D} \hbox{ varies}.
\end{eqnarray}
We found that the results in this simple case
display the main characteristic features of the radiative decays of $S$.
The VLQ mass spectra become
\begin{eqnarray}
\label{eq:mass:benchmark}
M_{\mathcal{U}_1}&=&M_{\mathcal{U}_2}=M_\mathcal{Q},\quad
M_{\mathcal{D}_{1,2}} = M_\mathcal{Q} \mp \frac{1}{\sqrt{2}} |Y_\mathcal{D}| v.
\end{eqnarray}
Note that $\mathcal{D}_1$ becomes the lightest VLQ
and $\Delta M_{\mathcal{U}_1 \mathcal{D}_1} = \Delta M_{\mathcal{D}_2 \mathcal{U}_1} = (1/2) \Delta M_{\mathcal{D}_2 \mathcal{D}_1}$
where $\Delta M_{ij} \equiv M_i - M_j $.
Our setting of $Y_\mathcal{U} \neq Y_\mathcal{D}$
generates a sizable mass difference $\Delta M_{\mathcal{U}_1 \mathcal{D}_1}$
which is essencial for the longitudinal polarization enhancement of
$S \to WW$.

Brief comments on the VLQ masses are in order here.
The mass bounds on the VLQs from the direct searches at the Tevatron and the LHC
depend sensitively on the decay channels of the VLQs.
If the main decay mode
includes the third generation quarks,
the bounds
are strong: $M_{\rm VLQ}\gtrsim 400-600$ GeV~\cite{Okada:2012gy}.
If VLQs mix only with lighter generations, the mass bounds become much less than
$400{\;{\rm GeV}}$ \cite{Okada:2012gy}, which is adopted here.

\begin{figure*}[t!]
    \centering
    \begin{subfigure}[b]{0.5\textwidth}
        \centering
        \includegraphics[width=0.95\textwidth]{fig-500-BR}
        \caption{$m_S=500{\;{\rm GeV}}$}
    \end{subfigure}%
    ~
    \begin{subfigure}[b]{0.5\textwidth}
        \centering
        \includegraphics[width=0.95\textwidth]{fig-750-BR}
        \caption{$m_S =750{\;{\rm GeV}}$}
    \end{subfigure}
    \caption{\label{fig-BR}
    Branching ratios of the radiative decays of the singlet scalar $S$
with mass $m_S=500,750{\;{\rm GeV}}$
as functions of $\Delta M_{\mathcal{U}_1 \mathcal{D}_1}( \equiv M_{\mathcal{U}_1} - M_{\mathcal{D}_1})$.
For the VLQ masses we set the lightest VLQ mass as $M_{\mathcal{D}_1}=0.6 \, m_S$ and assume
$M_\mathcal{Q} = M_\mathcal{U}=M_\mathcal{D}$ and $Y_\mathcal{U}=0$
with varying $Y_\mathcal{D}$.}
\end{figure*}

In Fig.~\ref{fig-BR}, we present the branching ratios
of the singlet scalar $S$
as functions of $\Delta M_{\mathcal{U}_1 \mathcal{D}_1}$, or equivalently of $Y_\mathcal{D}$,
along the benchmark parameter line.
We consider two cases, $m_S=500{\;{\rm GeV}}$ and
$m_S = 750{\;{\rm GeV}}$
with $M_{\mathcal{D}_1}= 0.6\, m_S$.
When $Y_\mathcal{D} =0$ ($Y_\mathcal{U}=0$ by setting),
the dominant decay mode is into $gg$ with almost 100\% branching ratio.
The radiative decay of $S$ into $hh$ is certainly prohibited.
In addition there is no radiatively generated $S$-$h$ mixing,
i.e., $s_\eta = 0$,
which forbids the decay of $S \to \ttop$.
The mixing induced decays in $S \to WW,ZZ$
are closed and only the triangle VLQ loop contributions
become relevant.
The next dominant decay mode is into $WW$ with very small branching ratio
of the order of $10^{-3}$.
This is because of the suppression of the longitudinal polarization enhancement
since the $Y_\mathcal{D}=0$ condition makes
all of the VLQ masses degenerate
and thus $2 \mathcal{A}+ \mathcal{B} \sim \mathcal{O}(m_V^2/m_S^2)$:
see Eqs.~(\ref{eq:CM:asymptotic}),
(\ref{eq:AM}), and (\ref{eq:BM}).
The reason why ${\rm B}(S \to WW)$ is much larger than
${\rm B}(S \to ZZ)$ when $Y_{\mathcal{U},\mathcal{D}}=0$ is that the gauge couplings of VLQs
to the $Z$ boson
are smaller than those to the $W$ boson
with our choice of the electric charges of VLQs.
Note that $\Gamma(S \to WW) \gg \Gamma(S \to \rr,ZZ)$
is generic in the view of high dimensional operators
in the effective field theory~\cite{Franceschini:2015kwy}.

As $Y_\mathcal{D}$ increases,
the decay modes into $hh$, $WW$, $ZZ$ and $\ttop$
all become significant.
For both $m_S=500{\;{\rm GeV}}$ and $m_S=750{\;{\rm GeV}}$ cases,
the $hh$ mode is as important as the $gg$ mode when $Y_\mathcal{D} \simeq 0.8$,
and dominant when $Y_\mathcal{D} \gtrsim 0.9$,
followed by the $WW$, $ZZ$, and $\ttop$ modes.
We found that the little hierarchy among $hh$, $WW$, $ZZ$ and $\ttop$ modes
is quite generic with more general parameter setup other than
our benchmark scenario.
In some extreme corners
 of the parameter space such as small $Y_{\mathcal{U},\mathcal{D}}$
but large $\Delta_F$,
the $WW$ decay mode is dominant.

\begin{figure}[t]
\begin{center}
\includegraphics[width=.6\textwidth]{fig-gamtot.pdf}
\end{center}
\caption{\label{fig-gamtot}
\baselineskip 3.0ex
Total decay rate of the singlet scalar $S$
as a function of $\Delta M_{\mathcal{U}_1 \mathcal{D}_1}$ or $Y_\mathcal{D}$
for $m_S=500{\;{\rm GeV}}$ and $m_S=750{\;{\rm GeV}}$.
We take the benchmark parameter line in Eq.~(\ref{eq:benchmark}).
}
\end{figure}

In Fig.~\ref{fig-gamtot},
we show the total decay rate of $S$ as a function of $\Delta  M_{\mathcal{U}_1 \mathcal{D}_1}$
for $m_S=500{\;{\rm GeV}}$ and $m_S=750{\;{\rm GeV}}$.
When $\Delta  M_{\mathcal{U}_1 \mathcal{D}_1}=0$,
$\Gamma^{\rm tot}_S \sim 0.1 {\;{\rm GeV}}$ for both mass cases.
The singlet scalar is a very narrow resonance.
With increasing $\Delta  M_{\mathcal{U}_1 \mathcal{D}_1}$,
$\Gamma^{\rm tot}_S$ starts decreasing,
which is expected since $\mathcal{U}_{1,2}$ and $\mathcal{D}_2$
become heavier with
the fixed $M_{\mathcal{D}_1}$
and thus make smaller loop corrections.
When $\Delta  M_{\mathcal{U}_1 \mathcal{D}_1}$ is large enough, however,
$\Gamma^{\rm tot}_S$ turns to increase,
reaching about 10 GeV when $\Delta  M_{\mathcal{U}_1 \mathcal{D}_1}=300{\;{\rm GeV}}$.
The enhancement
compared to the $\Delta  M_{\mathcal{U}_1 \mathcal{D}_1}=0$ case
is almost by two orders of magnitude.
This is unexpected since the VLQ masses for $\Delta  M_{\mathcal{U}_1 \mathcal{D}_1}=300{\;{\rm GeV}}$
are much heavier than those for $\Delta  M_{\mathcal{U}_1 \mathcal{D}_1}=0$.
This shows how dramatical the enhancement of the radiative decays of $S$
can be when there exist sizable mass differences of the VLQs.

\begin{figure*}[t!]
    \centering
    \begin{subfigure}[b]{0.5\textwidth}
        \centering
        \includegraphics[width=0.95\textwidth]{fig-500-Constraints}
        \caption{$m_S=500{\;{\rm GeV}}$}
    \end{subfigure}%
    ~
    \begin{subfigure}[b]{0.48\textwidth}
        \centering
        \includegraphics[width=0.95\textwidth]{fig-750-Constraints}
        \caption{$m_S =750{\;{\rm GeV}}$}
    \end{subfigure}
    \caption{\label{fig-constraints}
The constraints
in the parameter space of $(\Delta M_{\mathcal{U}_1\mathcal{D}_1}, y_S)$
from the current LHC Higgs data
as well as the $\sqrt{s}=8{\;{\rm TeV}}$ searches for a heavy Higgs
decaying into $WW$,
$ZZ$, and $hh$:
(a) is for $m_S=500{\;{\rm GeV}}$ and (b) is for $m_S=750{\;{\rm GeV}}$.
    }
\end{figure*}

Figure \ref{fig-constraints} presents the 95\% C.L. exclusion region
in the $(\Delta M_{\mathcal{U}_1\mathcal{D}_1}, y_S)$ parameter plane
by the LHC Higgs precision data as well as
the heavy Higgs search results in the $WW$, $ZZ$, and $hh$ channels.
We also show the contours for $s_\eta$ by dashed (orange) lines.
For the Higgs precision data,
we adopt the global fit results from the ATLAS/CMS combined analysis
for $\kappa_V \leq 1$~\cite{kappa:2015}:
$\kappa_V = 0.97 \pm 0.060$, $\kappa_g = 0.81^{+0.13}_{-0.10}$,
and $\kappa_\gamma = 0.90^{+0.10}_{-0.09}$.
Note that $\kappa_\tau = 0.87^{+0.12}_{-0.11}$
and $\kappa_b = 0.57^{+0.16}_{-0.16}$ are consistent within $2\sigma$ but $\kappa_t = 1.42^{+0.23}_{-0.22}$
shows some deviation.
For heavy scalar boson searches
with mass $m_S=500$ $(750){\;{\rm GeV}}$,
the observed 95\% C.L.
upper bounds on $\sigma\cdot {\rm B}$
at $\sqrt{s}=8{\;{\rm TeV}}$
are
 $200{\,{\rm fb}}$ ($40{\,{\rm fb}}$) for $WW$~\cite{Aad:2015agg,Khachatryan:2015cwa},
$43{\,{\rm fb}}$ ($12{\,{\rm fb}}$) for $ZZ$~\cite{Aad:2015kna},
and $107.6{\,{\rm fb}}$ ($34{\,{\rm fb}}$)
for $hh$~\cite{Aad:2014yja,Aad:2015uka,Aad:2015xja}.
We
found that the heavy scalar search channels of dijet~\cite{Aad:2014aqa,CMS:2015neg}
and $W\gamma/ Z\gamma$~\cite{Aad:2014fha} provide weaker
 constraints.
We do not consider the $\ttop$ channel~\cite{Aad:2015fna,Khachatryan:2015sma}
because the current bound ignores the interference with the
continuum background,
which can be very significant~\cite{Jung:2015gta,Carena:2016npr,Barcelo:2010bm}.

The Higgs precision data
exclude large $\Delta M_{\mathcal{U}_1 \mathcal{D}_1}$,
almost independently of $y_S$:
$\Delta M_{\mathcal{U}_1 \mathcal{D}_1} \lesssim 200 \,(300){\;{\rm GeV}}$ for $m_S=500\,(750){\;{\rm GeV}}$ is allowed.
This exclusion mainly comes from the constraint on $\kappa_g$
of which the deviation from the SM value is generated from the $S$-$h$ mixing
or the triangle VLQ loops.
When $\Delta M_{\mathcal{U}_1 \mathcal{D}_1}$ is small,
or equivalently when all of the VLQ masses are almost degenerate,
the opposite signs
between $y_{_{h F_1 F_1}}$ and $y_{_{h F_2 F_2}}$
cause significant cancellation of the $F_1$ and $F_2$ contributions.
Therefore, $\kappa_g$ is within the allowed value.
As the VLQ mass difference increases,
the VLQ loop corrections become more important.
The Higgs precision data put an upper bound on $\Delta M_{\mathcal{U}_1 \mathcal{D}_1}$.
The $\kappa_\gamma$ is less sensitive since
the dominant contribution to $\kappa_\gamma$
comes from the $W$ loop.
The $S$-$h$ mixing effect, mainly on $\kappa_V$,
is minor since we adopt the Higgs precision data at $2\sigma$ level
such that $s_\eta\lesssim 0.5$~\cite{Cheung:2015dta}.

Figure \ref{fig-constraints} shows that
the $ZZ$ channel in the heavy scalar searches puts the strongest bound
for both mass cases.
This is attributed to compatible branching ratios of $WW$, $hh$, and $ZZ$ modes
but
much smaller LHC upper bounds on $\sigma \cdot {\rm B}$ for the $ZZ$ mode
because of its clean signal.
The parameter space with large $y_S$ and large $Y_\mathcal{D}$
is excluded.
We also present the contours of $s_\eta$
by dashed (orange) lines.
It is clear to see that
the current heavy Higgs searches
put stronger bounds on the $S$-$h$ mixing angle
than the Higgs precision data.
In most parameter space, $s_\eta$ should be less than about 0.01\,(0.05)
for $m_S = 500\,(750){\;{\rm GeV}}$.
The radiatively generated $S$-$h$ mixing
is significantly constrained by the current LHC data.

\begin{figure}[h] \centering
\begin{center}
\includegraphics[width=.6\textwidth]{fig-500-XS13.pdf}
\end{center}
\caption{\label{fig-500-XS13}
\baselineskip 3.0ex
Cross sections of production and decay of $S$ for the main decay channels
with $m_S = 500\,{\rm GeV}$ and $\sqrt{s}=13\,{\rm TeV}$ at the LHC.
The cross sections are normalized by $y_S^2$.
}
\end{figure}

Finally, we show in Fig.~\ref{fig-500-XS13} the cross section times branching ratio
 $\sigma(pp \to S) \times B(S \to XY)$
as a function of $\Delta M_{\mathcal{U}_1\mathcal{D}_1}$ with $m_S=500{\;{\rm GeV}}$
at the $13\,{\rm TeV}$ LHC.
The decay of $S$ into $gg$ is not considered because of the overwhelming QCD background.
We normalize $\sigma \cdot {\rm B}$ by $y_S^2$.
Incorporating the current Higgs precision constraint on $\Delta M _{\mathcal{U}_1 \mathcal{D}_1}$,
we present the results for $\Delta M _{\mathcal{U}_1 \mathcal{D}_1}$ up to 200 GeV.
In the whole parameter space,
the $WW$ mode is leading or next-to-leading,
having $\sigma\cdot {\rm B} \sim {\cal O}(100-1000){\,{\rm fb}}$.
The cleanest search mode, the $ZZ$ one, also has sizable signal rate
about 100 fb
if $\Delta M _{\mathcal{U}_1 \mathcal{D}_1} \gtrsim 50{\;{\rm GeV}}$.
The $hh$ channel is also promising with sufficient VLQ mass differences.

\section{Conclusions}
\label{sec:conclusions}

In a simple extension of the SM with an additional singlet scalar field $S$,
we answer the question
whether a unique feature of a heavy scalar boson,
the longitudinal polarization enhancement in its decay into a massive gauge boson pair,
remains at loop level.
In order to focus on the loop induced effects,
we consider a limiting scenario where $S$ does not interact with the SM Higgs boson
at tree level.
Since $S$ decouples from the SM world at tree level,
we introduced vector-like quarks (VLQs) as messengers between $S$ and the SM particles.
In order for the Higgs boson to interact also with the VLQs,
one VLQ doublet and two VLQ singlets are suggested.
There are two up-type VLQs and two down-type VLQs,
$\mathcal{U}_{1,2}$ and $\mathcal{D}_{1,2}$.
Through the Yukawa couplings of VLQs with $S$ and the Higgs boson,
the VLQs
generate radiatively the $S$-$h$ mixing as well as
the decays of $S$ into $gg$, $WW$, $ZZ$, and $hh$.

We found that the most required condition
for enhancing the radiative decay rates of $S$
into $WW$, $ZZ$ and $hh$ is the large mass differences of VLQs.
This is contrary to the common expectation
since large mass differences with the fixed lightest VLQ mass
mean heavy VLQs and thus smaller loop corrections.
First the radiatively generated $S$-$h$ mixing is
proportional to the coupling of $h$-$F_i$-$F_i$ ($F=\mathcal{U},\mathcal{D}$ and $i=1,2$).
When $M_{F_1} = M_{F_2}$, the opposite signs between
$h$-$F_1$-$F_1$ and $h$-$F_2$-$F_2$ couplings
cancel the contributions of $F_1$ and $F_2$.
As $\Delta M_{F_2 F_1}(\equiv M_{F_2}-M_{F_1})$ increases,
the $S$-$h$ mixing angle is enhanced.
The mixing induced decays of $S$ into $ WW$, $ZZ$, $hh$, and $\ttop$ become significant.
Another kind of the VLQ contribution to the radiative decay of $S$ is
through the triangle VLQ loops.
We showed that the longitudinal polarization enhancement in $S \to WW,ZZ$
through the triangular VLQ loops
happens also when the mass differences of the VLQs become large.

In order to illustrate the phenomenological features,
we considered a simple benchmark scenario
where $Y_\mathcal{D}$ controls the VLQ mass differences
with the fixed lightest VLQ mass.
Two cases of $m_S=500{\;{\rm GeV}}$ and $m_S=750{\;{\rm GeV}}$
are studied.
When $\Delta M_{F F'}=0$, both the $S$-$h$ mixing and the
longitudinal polarization enhancement in $S\to VV$
vanish, which
makes $S\to gg$ dominant.
The total decay rate is of the order of 0.1 GeV for $m_S\sim 500{\;{\rm GeV}}$.
If $\Delta M_{F F'}$ is sizable such as $Y_\mathcal{D} \simeq 0.8$,
the decay of $S$ into $hh$ becomes as important as that into $gg$.
For $Y_\mathcal{D} \gtrsim 0.8$,
${\rm B}(S \to gg)$ drops rapidly,
and the decays into $hh$, $WW$, $ZZ$, and $\ttop$ become
similarly dominant.
The enhancement of the total decay rate of $S$
is huge, by one order of magnitude when $Y_\mathcal{D} =1$.
This is contrary to the naive prediction
that heavier VLQs  running in the loop would cause
smaller loop corrections.

We also presented the 95\% C.L. exclusion regions of $(Y_\mathcal{D}, y_S)$
from the current LHC bounds
including the Higgs precision data and the heavy scalar searches
in the channel of $WW$, $ZZ$, and $hh$.
Among various Higgs precision data, $\kappa_g$
puts the strongest bound on $Y_\mathcal{D}$: $Y_\mathcal{D} \gtrsim 1.1$ for $m_S=500{\;{\rm GeV}}$
and $Y_\mathcal{D} \gtrsim 1.7$ for $m_S=750{\;{\rm GeV}}$ are excluded.
The heavy scalar searches also put additional constraints.
In particular the $ZZ$ channel data severely limit the $S$-$h$ mixing angle $\eta$,
more than the Higgs precision data:
$s_\eta \lesssim 0.05$  for $m_S=500{\;{\rm GeV}}$
and $s_\eta \lesssim 0.1$ for $m_S=750{\;{\rm GeV}}$ are allowed.
In conclusions,
our loop calculation in a UV model
with a singlet scalar and three VLQ multiplets shows that
the radiative decays of $S$
can be very enhanced when the mass spectrum of the VLQs shows diversity.
Note that the presence of multiple VLQs is crucial for the enhanced radiative decays of $S$
since sizable mass differences among VLQs are required.
Therefore, the persistent searches for a heavy scalar boson at the future LHC
are of great importance in constraining new particles that appear at loop level.

\acknowledgments
K.C. was supported by the MoST of Taiwan under
Grants No. MOST-105-2112-M-007-028-MY3.
S.K. was supported by the National Research Foundation of Korea,
2015R1D1A1A01058726.
The work of J.S. and Y.W.Y. was supported by the National Research Foundation of Korea, NRF-2016R1D1A1B03932102.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

In unified AGN models, powerful radio sources are believed to have a
central source emitting in the optical, UV and X-rays. The dusty torus that
surrounds the nuclear region absorbs this light and re-emits it in the
infrared. This mechanism makes it difficult to observe the objects viewed
through the torus directly in the optical, UV and soft X-rays. Willott et
al. (2000) define the quasar fraction $f_{q}$ to be the ratio of unobscured to
obscured quasars. In \cite{eleniv06} we expressed the need to study radio
sources in the mid-infrared in order to investigate the existence of any
hidden  accretion activity in radio sources. Spitzer observations at 24
$\mu \rm m$ are ideal for this task, since they trace hot dust emission
which can be obscured only by extreme columns (\cite{ogle06}).

Our sample is the 36 brightest radio sources from the VLA survey of the SXDF
(\cite{simpson06}) with flux densities greater than 2 mJy at 1.4 GHz. Most of
our objects are not as yet spectroscopically confirmed, so we use photometric
redshifts for our analysis. We use a radio spectral index $\alpha$ = 0.8
($S_{\nu} \propto \nu^{-\alpha}$). We assume throughout a low-density,
$\Lambda$-dominated Universe in which
$H_{0}=70~ {\rm km~s^{-1}Mpc^{-1}}$, $\Omega_{\rm M}=0.3$ and
$\Omega_{\Lambda}=0.7$.

More details of this work will appear in Vardoulaki et al. (in prep.).

\section{Discussion}

We have constructed spectral energy distributions (SEDs) for our SXDF sample
and fitted Bruzual-Charlot galaxy templates using the HyperZ photometric code
(\cite{hyperz}) to estimate photometric redshifts. An example is shown on
the left side of Fig. \ref{fig1}. This object is spectroscopically confirmed
at redshift $z$ = 1.095. From its SED we can see excess emission at 24
$\mu \rm m$, indicating that this object has hidden accretion activity. This
object is not a spectroscopically confirmed quasar but has high-excitation
narrow lines in its optical spectrum; its radio map shows an unresolved source.

We define objects to be `quasar-like' when their rest-frame luminosity at
$\lambda =$ 24 $\mu \rm m$ is $\lambda L_{24 \mu \rm m}>10^{37.3}$ ${\rm W}$,
since this corresponds to  $\lambda L_{24 \mu \rm m} > 10^{-1.8} L_{Edd}$ for
a quasar with $M_{\rm BH}$ $\mbox{{\raisebox{-0.4ex}{$\stackrel{>}{{\scriptstyle\sim}}$}}}$ $10^{8}$ $\rm M_{\odot}$
(see \cite{mclure04}; \cite{mclureetal04}). The luminosity at 24 $\mu \rm m$
is calculated using the spectral index measured from 24 $\mu \rm m$ to the
nearest lower-$\lambda$ detection in the observed frame SED. The quasar-like
fraction $q_{\rm l}$ in the region between the RLF break and the FRI/II break
is high, $q_{\rm l} \sim 0.5-0.9$, depending on how many of the 24
$\mu \rm m$ upper limits turn out to be close to detections. [The most radio
luminous of the two non-quasar-like objects, is an FRI at $z \sim 0.7$].
Recall that $f_{q} \sim 0.2$ in this region (\cite{willott00}), i.e. the
24 $\mu \rm m$ data have revealed at least some hidden accretion activity.

The quasar-like fraction drops significantly below the FRI/II break, and if
one excludes compact ($D <$ 100 kpc) sources as potentially part of a separate
(beamed) population, we see that there are no definite quasar-like objects.

We conclude that Spitzer 24-$\mu \rm m$ data on the SXDF radio sources are
consistent with the hypothesis that quasar-like objects are almost always
associated with powerful (above the FRI/II break) radio sources, but are
rarely connected to less luminous sources, unless they are part of the compact
population.

\begin{figure}
\plottwo{vardoulaki_e_fig1.ps}{vardoulaki_e_fig2.ps}
\caption{{\itshape Left:\/} Observed frame SED of sxds\_0033 with a redshift
estimated using HyperZ and data from Spitzer, UKIDSS and SXDF surveys.
{\itshape Right:\/} $\log_{10}( L_{1.4{\rm GHz}})$ versus largest projected
linear size $D$: possible quasar-like objects have upper limits on
$\lambda L_{24 \mu \rm m}$ consistent with
$\lambda L_{24 \mu \rm m}>10^{37.3}$ ${\rm W}$.}
\label{fig1}
\end{figure}