text:
\section{Introduction}

\noindent
Since the experimental observation of the discrepancy at the
Brookhaven National Laboratory (BNL)~\cite{Bennett:2004pv}, the anomalous
magnetic moment $g - 2$ of muon has served as a long-standing puzzle of
particle physics.
Recently, the Fermilab Muon $g-2$ collaboration has announced the new
measurement result~\cite{Abi:2021gix}, which has further strengthened the
significance of the BNL result on the muon $g-2$. Comparing to the
Standard Model (SM) prediction
from the Muon $g-2$ Theory Initiative~\cite{Aoyama:2020ynm},
the combined BNL and Fermilab result amounts be a $4.2\sigma$ discrepancy.
The deviation from the SM prediction is
\begin{equation}
  \Delta a_\mu = a_\mu^\text{exp} - a_\mu^\text{SM} = (25.1 \pm 5.9)
  \times 10^{-10} ,
\end{equation}
where $a_\mu \equiv (g_\mu - 2) / 2$.
Although the recent lattice calculation for the hadronic vacuum
polarization contribution to the muon $g-2$ has turned out to be in
accord with the measured value~\cite{Borsanyi:2020mff}, we take this
opportunity to examine the possibility of new physics accounting for the
muon $g-2$ anomaly.

Among various possible models that can explain the muon $g-2$ anomaly,
we consider the supersymmetric (SUSY) model as it is one of the most
promising candidates.
Ever since the announcement of the Fermilab result has come out, the
SUSY interpretations for the muon $g-2$ anomaly have already been
revisited or renewed in many works~\cite{Zhang:2021gun,
  Endo:2021zal, Baum:2021qzx, Ahmed:2021htr, Abdughani:2021pdc,
  Ibe:2021cvf, Cox:2021gqq, VanBeekveld:2021tgn, Yin:2021mls,
  Wang:2021bcx, Gu:2021mjd, Han:2021ify, Athron:2021iuf,
  Aboubrahim:2021rwz, Yang:2021duj,
  Baer:2021aax, Altmannshofer:2021hfu, Aboubrahim:2021phn,
  Chakraborti:2021squ, Zheng:2021wnu, Zhang:2021nzv}.
In the Minimal Supersymmetric Standard Model (MSSM), the contributions
to the muon $g-2$ can be generated by bino, winos, Higgsinos, smuons,
and sneutrino.
For sparticle masses of order $M_\text{SUSY}$, the leading SUSY
contributions have the generic behavior of
\begin{equation}
  \Delta a_\mu^\text{SUSY} \propto \frac{m_\mu^2 \, \mu
    M_a}{M_\text{SUSY}^4} \tan\beta ,
\end{equation}
where $\mu$ is the Higgsino mass, $M_a$ is the gaugino mass, and
$\tan\beta$ is the ratio of the Higgs vacuum expectation values~\cite{Stockinger:2006zn}.
Their relative contributions can differ by the
sparticle mass spectrum.
For example,   if the Higgsinos are very heavy,
only the bino-smuon loop contribution can become sizable.
Whereas the sfermion masses are highly model dependent,
the gaugino masses show a robust pattern
owing to the gauge coupling unification
at the grand unified theory (GUT) scale,  $M_{\rm GUT}$~\cite{Choi:2007ka}.
For instance,  in gravity mediation with universal gaugino masses at $M_{\rm GUT}$~\cite{Nilles:1982ik}
or in gauge mediation with messengers forming a GUT multiplet~\cite{Dine:1981za,Dimopoulos:1981au,Dine:1981gu},
the ratios of low energy gaugino masses read
\begin{equation}
  M_1 : M_2 : M_3 \simeq 1 : 2 : 6
\end{equation}
at the TeV scale,
regardless of the details of the model.
Anomaly mediation~\cite{Randall:1998uk,Giudice:1998xp},
which always exists in supergravity,  modifies
the above gaugino mass relation depending on its relative strength,
and may make the wino and/or bino much lighter than the gluino
as is required to explain the muon $g-2$ anomaly.
A natural framework for sizable anomaly mediation is provided by the KKLT string compactification~\cite{Kachru:2003aw}.
A remarkable feature of the KKLT moduli stabilization is that
the parameters of SUSY breaking are, in principle,    controlled by discrete numbers,
such as the winding number of D-branes, the number of fluxes that generate moduli potential, and so on.

In this article,
we point out that
mixed modulus-anomaly mediation~\cite{Choi:2004sx, Choi:2005ge, Endo:2005uy,
Choi:2005uz, Falkowski:2005ck}, which is realized in the KKLT setup,
can accommodate light electroweakinos (EWinos) and sleptons to explain
the muon $g-2$ anomaly and heavy colored sparticles to evade the lower
limits from the LHC\@.
To obtain the suitable sparticle mass spectra for the muon $g-2$, we
need to consider a generalized KKLT setup beyond the minimal one,   as
described in Sec.~\ref{sec:kklt}.
By imposing various conditions such as the Higgs boson
properties
and the vacuum stability of the scalar potential, we perform numerical
analysis in the parameter space to check the feasibility of the model.
We present our analysis result in Sec.~\ref{sec:g-2}.
Although our result is mostly safe from the lower limits set by the
search results on colored sparticles at the LHC, the bounds from the
searches for the chargino-neutralino and the slepton pair productions
may exclude the parameter points of light sleptons and gauginos.
We also find that,
in a large part of the parameter space for the muon $g-2$,
a slepton becomes lighter than the lightest neutralino.
In this case, we should consider alternative scenarios such as a
Peccei-Quinn (PQ) symmetric extension where the axino is the lightest
sparticle or R-parity violating (RPV) interactions to make the
lightest ordinary SUSY particle (LOSP), the lightest sparticle among
the MSSM sparticles, unstable.
The LHC limits and the phenomenological
scenarios with light sleptons are discussed in Sec.~\ref{sec:lhc}.
We summarize our study in the last section.
For reference,   we list the
benchmark sparticle mass spectra in Appendix~\ref{sec:benchmark}.

\section{\label{sec:kklt}\boldmath Mixed modulus-anomaly mediation}

\noindent
The sparticle mass spectrum crucially depends on how SUSY breaking in a hidden sector is
transmitted to the visible sector.
To be consistent with the experimental constraints, SUSY breaking
mediation should preserve flavor and CP symmetry with good accuracy
unless it makes the sparticles very heavy above $100$~TeV.
Indeed, many mediation schemes such as gravity mediation,
dilaton/moduli mediation, gaugino mediation, gauge mediation, anomaly
mediation, and their mixtures conserve flavor and CP symmetry and lead
to various patterns of sparticle spectra.
Among the sparticles, the gauginos are known to have a robust pattern of
masses under the condition of gauge coupling
unification~\cite{Choi:2007ka}.
The gaugino masses in mixed modulus-anomaly mediation are written as
\begin{equation}
M_a = M_0 \left( 1 + \frac{b_a g^2_{\rm GUT} }{4} \alpha \right),
\end{equation}
at the scale just below $M_{\rm GUT}$, where the gauge coupling constants have
the common value,   $g^2_a(M_{\mathrm{GUT}}) = g^2_{\rm GUT}$.
Here, $b_a=(33/5,\,1,\,-3)$ are the coefficients of the one-loop beta functions
at TeV,   and the $\alpha$ parameter represents the relative strength of
anomaly mediation:
\begin{equation}
\alpha \equiv \frac{m_{3/2}}{4\pi^2  M_0},
\end{equation}
with $m_{3/2}$ being the gravitino mass.
Note that anomaly mediation is a model-independent supergravity effect
proportional to the gravitino mass,
but it alone suffers from the tachyonic
slepton problem.
Because the combination $M_a/g^2_a$ is renormalization group (RG) invariant
at one loop, the gaugino masses at the TeV scale are found to
approximately obey the following relation:
\begin{equation}
\label{Ma-TeV}
M_1 : M_2 : M_3 \simeq (1 + 0.83 \alpha) :
(2 + 0.25 \alpha) : (6 -2.25 \alpha),
\end{equation}
where we have taken $g^2_{\rm GUT} =0.5$ and used the ratios of the
gauge couplings $g^2_1 : g^2_2: g^2_3 \simeq 1:2:6$ at the TeV scale.

As the first explicit realization of a de Sitter (dS) vacuum with all
string moduli stabilized, the KKLT mechanism~\cite{Kachru:2003aw} provides an interesting
framework to realize mixed modulus-anomaly mediation with $\alpha$ of
order unity.
In the minimal KKLT setup,  $\alpha$ is a positive rational
number~\cite{Choi:2004sx,Choi:2005ge}.
In the literature,
the phenomenology of positive $\alpha$ has been studied intensively
particularly to resolve the fine-tuning problem to realize the electroweak (EW) symmetry breaking scale~\cite{Kitano:2005wc,Lebedev:2005ge,Choi:2006xb,Abe:2014kla,Baer:2016hfa,Kawamura:2017qey,Jeong:2020wum}.
However,   it is known that the muon $g-2$ anomaly is hardly explained
in the cases with positive $\alpha$ due to relatively heavy winos and bino.
One can generalize the KKLT setup to obtain a negative $\alpha$ to get
a desired mass hierarchy in the gaugino spectrum.
For a concrete discussion, let us consider a model where the effective
moduli superpotential is  given by~\footnote{

  We take the reduced Planck mass unit, $M_{Pl}=1$, unless stated otherwise.
}
\begin{equation}
W = A_0 e^{-4\pi^2 \ell_0 S_0}
- A_1 e^{-4\pi^2(k_1 T + \ell_1 S_0)},
\end{equation}
with $A_0$ and $A_1$ being constants of order unity,
and the visible sector gauge kinetic function is written as
\begin{equation}
f_a = k T  + \ell S_0,
\end{equation}
after integrating out the heavy dilation $S$ and
complex structure moduli
fixed by fluxes at $S=S_0$.
Here, $\ell_0/\ell$,  $\ell_1/\ell$, and $k_1/k$ are rational numbers determined by topological or
group theoretical data of the underlying string compactification,
as can be dictated from the periodicities of ${\rm Im}(T)$ and ${\rm Im}(S)$.
A supersymmetric minimum is developed by the above superpotential
and is lifted to a dS vacuum by adding a SUSY breaking uplifting
potential that originates from a brane-localized source located at the
IR end
of the warped throat.
The uplifting potential is given by
\begin{equation}
V_{\rm lift}
= \frac{P e^{2K_0/3}}{(T+T^*)^{n_P} },
\end{equation}
for a rational number $n_P$.
$P$ is a positive constant fixed by the condition of vanishing cosmological constant.
Here, the modulus K\"ahler potential generally reads
\begin{equation}
K_0= -n_0 \ln(T+T^*),
\end{equation}
for a positive rational number $n_0$.
An extra-dimensional interpretation of the uplifting procedure
is possible for $n_P\geq 0$, because otherwise
the uplifting sector couples more strongly for a larger value of $T$~\cite{Aad:2015zhl}.
In the above model, one finds $\alpha$ to be~\cite{Choi:2006xb}
\begin{equation}
\alpha \simeq
\frac{2k_1}{k}
\left( 1 + \frac{3n_P}{2n_0} \right)^{-1},
\end{equation}
from the observations
that gauge coupling unification requires
$g^{-1}_{\rm GUT} = {\rm Re}(f_a)   \simeq 2$,
and that $T$ is stabilized at $k_1 T \simeq (\ell_0 -\ell_1) S_0$ with
$S_0$ written in terms of the gravitino mass as
$4\pi^2 \ell_0 {\rm Re}(S_0) \simeq \ln(M_{Pl}/m_{3/2})$.
A negative $\alpha$ is therefore obtained if either $k_1$ or $k$ is
negative.
See,  for instance,  Ref.~\cite{Abe:2005rx} for more discussion on the case with
$k_1<0$ and $k>0$.

Let us continue to examine the sfermion masses, which generally
possess a stronger model dependence compared to the gaugino masses.
In the mixed modulus-anomaly mediation under consideration, the
sfermion masses are determined by the modulus dependence of the matter
K\"ahler potential:
\begin{equation}
K = K_0 + \frac{ \Phi^*_i \Phi_i }{(T+T^*)^{n_i} }.
\end{equation}
We have taken into account a simple case where the matter K\"ahler
metric is not affected by the involved dilaton-modulus mixing.
Here,   the modular weight $n_i$ is a rational number of order unity fixed by
the location of the matter in extra dimensions.
The mixed mediation in the KKLT preserves CP and flavor symmetries respectively due to the axionic
shift symmetry associated with $T$ and flavor-universal modular weights.
The pure modulus-mediated (MM) trilinear $A$-parameters and soft
scalar masses are found to be
\begin{align}
A_{ijk}|_{\rm MM} &= - (a_i +a_j + a_k) M_0,
\nonumber \\
m^2_i|_{\rm MM} &= c_i M^2_0,
\end{align}
at $M_{\rm GUT}$,  where $a_i$ and $c_i$ are given by
\begin{align}
a_i &= \left(  \frac{n_0}{3} - n_i \right)
\left(
1 + \frac{k_1}{k} \frac{\ell}{\ell_0-\ell_1}
\right),
\nonumber \\
c_i &=
\left(
1 + \frac{k_1}{k} \frac{\ell}{\ell_0-\ell_1}
\right)
a_i.
\end{align}
Note that $a_i$ and $c_i$ are rational numbers of either sign depending on the choice
of the associated discrete numbers.

It is worth noting that an anomalous U$(1)_A$ gauge symmetry allows
the $\alpha$ parameter and the
effective
modular weights to have various values
in a much wider range if $T$ transforms non-linearly to
implement the Green-Schwarz anomaly cancellation mechanism~\cite{Green:1984sg}.
Let us consider a simple case where the modulus-induced Fayet–Iliopoulos term is canceled by
a U$(1)_A$ charged but SM singlet matter field $X$.
After integrating out  the heavy U$(1)_A$ gauge superfield,  whose longitudinal component
comes mostly from $X$,
the effective low-energy theory is described by $\alpha$ and $n_i$
modified according to~\cite{Choi:2006bh}
\begin{align}
\alpha &\to
\alpha \left(
1 - \frac{4\pi^2}{\ln(M_{Pl}/m_{3/2}) } k_H
\right),
\nonumber  \\
n_i  &\to  n_i + ( 1-n_X ) \frac{q_i}{q_X},
\end{align}
where $n_\alpha$ and $q_\alpha$ are the modular weight and U$(1)_A$ charge of the corresponding
field,  and $k_H$ is a constant of order unity associated with the hidden gaugino condensation yielding
a non-perturbative superpotential term to stabilize $T$.
The above shows that a positive $k_H$ of order unity can  flip the sign of $\alpha$
in the effective theory,
implying that U$(1)_A$ not only enlarges the possible range of modular weights,
but also makes it possible to achieve a negative $\alpha$.
Meanwhile, the holomorphic Yukawa coupling changes as
\begin{equation}
\lambda_{ijk} \to \lambda_{ijk} \epsilon^{-(q_i+q_j + q_k)/q_X},
\end{equation}
because it arises from the superpotential,  $X^{n_{ijk}} \Phi_i \Phi_j \Phi_k$,  where
$n_{ijk} =-(q_i + q_j + q_k)/q_X$ should be a non-negative integer.
$\epsilon\sim 0.1$ represents the VEV of
$X$ relative to $M_{Pl}$.
It is clear that a
large Yukawa coupling $y_{ijk}$ of order unity apparently constrains
the U$(1)_A$ charges to be $n_{ijk} = 0$ or $1$.\footnote{
In the case where $n_X=1$, one can assign flavor-dependent U$(1)_A$
charges, for which U$(1)_A$ can account for the hierarchical Yukawa
couplings via the Froggatt-Nielsen mechanism~\cite{Froggatt:1978nt}.
}

To summarize,  the generalized KKLT setup leads to mixed
modulus-anomaly mediation where the sparticle masses are determined by
three types of dimensionless parameters:
\begin{equation}
\alpha, \quad a_i,  \quad c_i,
\end{equation}
with $c_i \propto a_i$.
The parameters can take various values of order unity with either
sign, leading to a variety of sparticle mass spectra.
The overall size of sparticle masses is fixed by $M_0$.
While achieving gauge coupling unification, the gauginos show a robust
mass relation given by Eq.~(\ref{Ma-TeV}) at the TeV scale, and
interestingly, they can have a large mass hierarchy for a negative
$\alpha$.
\begin{figure}[tb!]
  \begin{center}
    \includegraphics[width=0.6\textwidth]{muon-g-2-gauginos_gr1.pdf}
  \end{center}
  \caption{\label{fig:gaugino_mass}
    Gaugino masses
    at TeV as functions of the $\alpha$
    parameter.  The masses have been normalized
    by $M_0$,  which is the pure modulus-mediated contribution
    at $M_{\rm GUT}$.
    The red,  green,  and blue  colored lines correspond to the
    gluino, wino, and bino masses,  respectively.}
\end{figure}
In Fig.~\ref{fig:gaugino_mass}, we display the gaugino masses as
functions of the $\alpha$ parameter, including negative values.
It shows that gluinos can become much heavier than the others
for a largely negative $\alpha$.
As discussed in the next sections, this feature is essential for
evading the collider bounds while explaining the muon $g-2$ anomaly
with light EWinos.
Meanwhile, the sfermion soft parameters just below the
unification scale are given by\footnote{
We employ the convention in the SUSY Les Houches Accord (SLHA)
format~\cite{Skands:2003cj} with $M_3$ being positive for $\alpha
\lesssim 2.5$.
For example, the parameters in Ref.~\cite{Choi:2006xb} can be
obtained by redefining $M_a \to -M_a$, $A_{ijk} \to -A_{ijk}$ and $\mu
\to -\mu$.
It corresponds to the field redefinitions:
$\lambda_a \to i \lambda_a$, $\psi_i \to i \psi_i$ and $\phi_i \to -\phi_i$,
where $\lambda_a$, $\psi_i$ and $\phi_i$ are gauginos, Weyl fermions, and scalars,
respectively. The other coupling constants are unchanged under the field redefinition.
The signs of the gaugino mass terms in the SLHA format
are opposite from those in Ref.~\cite{Choi:2006xb}.
Here, $A_{ijk}$ is a trilinear coupling divided by a Yukawa coupling
constant $y_{ijk}$.
}
\begin{align}
 \frac{A_{ijk}}{M_0}
&=
-(a_i + a_j + a_k)
- \frac{\alpha}{4} (\gamma_i + \gamma_j  + \gamma_k),
\nonumber \\
\frac{m^2_i}{M^2_0}
&=
c_i
+ \Big(
\sum_{jk} (a_i +a_j + a_k)  |y_{ijk}|^2
- 4 k \sum_a g^2_a C_a(\Phi_i)
\Big)  \frac{\alpha}{4}
+   \dot\gamma_i
\left( \frac{\alpha}{4} \right)^2,
\end{align}
for $f_a = k T + \Delta f_a$
with $\Delta f_a$ depending on other moduli of the model.
The anomalous dimension $\gamma_i$ is given by
\begin{equation}
  16\pi^2 \gamma_i = \frac{1}{2} y^{imn}y_{imn}-2g_a^2 C_a(\Phi_i) ,
\end{equation}
and $\dot\gamma = 8\pi^2 d\gamma_i/d \ln Q$
with $Q$ being the RG scale.
$C_a(\Phi_i)$ is the quadratic Casimir invariant of $\Phi_i$.

\section{\label{sec:g-2}\boldmath The muon $g-2$ anomaly}

\noindent
We are now in a position to examine the possibility of explaining the
muon $g-2$ anomaly in mixed modulus-anomaly mediation
realized in the generalized KKLT setup.
Depending on sparticle mass spectrum,  various different SUSY
contributions can enhance (or reduce) the muon $g-2$.
For a recent review on the SUSY contributions in light of the muon $g-2$ anomaly,
we refer the reader to Ref.~\cite{Athron:2021iuf} and the references
therein.
If
the Higgsinos are significantly heavier than the other EWinos and smuons,
the SUSY contributions to the muon $g-2$ is dominated by the bino-smuon loop:
\begin{align}
    \label{eq:a_mu_BLR}
  \Delta a_\mu^\text{SUSY}
  &\approx \frac{g_1^2}{8\pi^2} \frac{m_\mu^2 \, \mu}{m_{\tilde\ell_+}^2 M_1}
    \tan\beta \times F_\mathrm{B}\left( \frac{m_{\tilde\ell_-}^2}{M_1^2} \right)
\\ \nonumber
  &\simeq
    2.5 \times 10^{-9}
    \left(\frac{500~{\rm GeV}}{m_{{\tilde\ell}_+}}\right)^2
    \left(\frac{250~{\rm GeV}}{M_1}\right)
    \left(\frac{\mu}{2000~{\rm GeV}}\right)
    \left(\frac{\tan\beta}{25}\right)
    \left(\frac{F_{\mathrm{B}}}{1/6}\right)
    ,
\end{align}
where
\begin{align}
  F_B(x) = \frac{-1+x^2-2x\ln x}{2(x-1)^3}
\end{align}
is the loop function.
In the expression above, $m_{\tilde\ell_- \,(\tilde\ell_+)}$ is the lighter
(heavier) smuon mass, and we have taken the limit where $m_Z \ll M_1$,
$m_{\tilde\ell_\pm}$ and $m_{\tilde\ell_-} \ll m_{\tilde\ell_+}$.
It shows that the sign of the Higgsino mass $\mu$ and the
bino mass $M_1$ must be matched to have a positive contribution to
$\Delta a_\mu^\text{SUSY}$.

As we have seen in Sec.~\ref{sec:kklt},
determining sparticle mass spectrum of mixed modulus-anomaly mediation
is governed by the three types of dimensionless parameters, $\alpha$, $a_i$,
$c_i$,  as well as
$M_0$.
For the sake of simplicity,  we focus on the case with
\begin{equation}
a_i = c_i,
\end{equation}
in our analysis.
The $M_0$ value,    which fixes the overall size of sparticle masses,   is not
an input but is determined by requiring the lighter CP-even Higgs mass
to be compatible with the observed Higgs boson
mass~\cite{Aad:2015zhl}.
Furthermore, we assume that the modular weights respect flavor
universality and follow the SU$(5)$ GUT relations for quarks and
leptons.
The assumption is motivated by
the flavor constraints
and the unification of gauge couplings.
Then,   the input parameters of the model are given as follows:
\begin{equation}
 \alpha,\quad \tan\beta, \quad \mathrm{sgn}(\mu), \quad
 c_5    \equiv c_L = c_D, \quad
 c_{10}  \equiv c_Q = c_U = c_E, \quad
 c_{H_u},\quad c_{H_d}.
\end{equation}
The mass splittings of squarks and sleptons are induced by the RG
effects, involved with gauginos, and anomaly-mediated
contributions. The size of $\mu$ is determined by the condition of
EW symmetry breaking,
and we take into account both signs of $\mu$ because $M_1$
can have either sign depending on the value of $\alpha$.

To investigate the parameter space of mixed modulus-anomaly
mediation,
we have added the boundary conditions of mixed mediation to the
\texttt{SOFTSUSY} program~\cite{Allanach:2001kg} and
calculated the sparticle and Higgs mass spectra using the program.
We require that the SM-like Higgs boson mass calculated with
\texttt{SOFTSUSY} is within $125.10 \pm 0.01$~GeV.
Then, for each parameter point,   we obtain the SUSY contributions to
$\Delta a_\mu$ by using \texttt{GM2Calc}, which can
compute the muon $g-2$ up to two-loop corrections~\cite{Athron:2015rva}.

Before looking into our analysis results, we should consider
some theoretical constraints.
The sleptons are required to be light to explain the muon $g-2$
anomaly, and the lightest stau can become tachyonic due to the large
values of $|\mu| \gtrsim 2$~TeV leading to a large mixing angle.
In our analysis, we discarded the parameter points with any tachyonic
sfermion, including sleptons, flagged by \texttt{SOFTSUSY}.
Furthermore, we should avoid the possibility of having a dangerous
charge-breaking minimum in the scalar potential deeper than the EW vacuum.
As it is not checked by \texttt{SOFTSUSY}, we impose the
vacuum stability condition given in Ref.~\cite{Endo:2013lva}: for
\begin{align}
  \label{eq:eta_l}
  \tilde{\eta}_\ell =
  &\  \abs{m^2_{\tilde{\ell}_{LR}}}\times
\left[ 101~\mathrm{GeV}\left( \sqrt{m_{\tilde{\ell}_L} m_{\tilde{\ell}_R}} + m_{\tilde{\ell}_L}
 + 1.03 m_{\tilde{\ell}_R} \right) \right.  \\ \nonumber
  &\ \left.  - 2.27\times 10^{4}~\mathrm{GeV}^2 +
  \frac{2.97\times10^6 \mathrm{GeV}^3}{m_{\tilde{\ell}_L}+m_{\tilde{\ell}_R}}
    - 1.14\times 10^8~\mathrm{GeV}^4 \left(\frac{1}{m_{\tilde{\ell}_L}^2}
    + \frac{0.983}{m_{\tilde{\ell}_R}^2} \right) \right]^{-1} ,
\end{align}
we require
\begin{equation}
  \tilde \eta_\ell < \eta_\ell ,
\end{equation}
where $\ell = \mu$, $\tau$.
In Eq.~(\ref{eq:eta_l}),
$m^2_{\tilde{\ell}_L}$, $m^2_{\tilde{\ell}_R}$, and $m^2_{\tilde{\ell}_{LR}}$
are the diagonal element of left-handed smuon,
that of right-handed smuon, and the off-diagonal element in the smuon
mass squared matrix, respectively.
Ignoring the small $\tan\beta$ dependence, we take $\eta_\tau$
($\eta_\mu$) $= 0.94$ ($0.88$)~\cite{Kitahara:2013lfa}.

\begin{figure}[tb!]
  \begin{center}
    \includegraphics[width=0.48\textwidth]{fig_alp-damu_scatter_0607.pdf}
    \includegraphics[width=0.48\textwidth]{fig_tnb-damu_scatter_0607.pdf}
  \end{center}
  \caption{\label{fig-scat_tbalp}
    Parameter points of $\alpha$ (left) and $\tan\beta$ (right)
    compatible with the muon $g-2$ anomaly. The yellow (orange) bands
    correspond to the $1\sigma$ ($2\sigma$) ranges of
    the measured $\Delta a_\mu$ value, and accordingly, the parameters
    points are colored in magenta ($1\sigma$), blue ($2\sigma$), and
    green ($3\sigma$). Points colored in gray violate the vacuum
    stability condition.}
\end{figure}
\begin{figure}[tb!]
  \begin{center}
    \includegraphics[width=0.48\textwidth]{fig_c5-c10_scatter_0607.pdf}
    \includegraphics[width=0.48\textwidth]{fig_cHu-cHd_scatter_0607.pdf}
  \end{center}
  \caption{\label{fig-scat_cs}
    Parameter points in the space of ($c_5$, $c_{10}$) (left) and ($c_{H_u}$,
    $c_{H_d}$) (right). The points colored in magenta, blue, and green
    can explain the muon $g-2$ anomaly within $1\sigma$, $2\sigma$,
    and $3\sigma$, respectively.}
\end{figure}

\begin{figure}[tb!]
   \centering
    \includegraphics[width=0.48\textwidth]{fig_bsg-bsm_scatter_0607.pdf}
  \caption{\label{fig-scat_bsgm}
    Scattering plot on ($\br{B_s}{\mu\mu}$, $\br{b}{s\gamma}$).

    The color coding of the points is the same as in the previous
    figures.

    The colored regions deviate from the experimental central values
    by more than $2\sigma$.}
\end{figure}

We now discuss our analysis results.
In Fig.~\ref{fig-scat_tbalp}, we display the parameter points of $\alpha$
and $\tan\beta$, which can explain the muon $g-2$ anomaly within
$3\sigma$, on the left and right panels, respectively.
The results show that the muon $g-2$ prefers the negative values of
$\alpha$, where bino is light: $M_1 \simeq 100$--$200$~GeV.
In particular, the $\Delta a_\mu$ value is the most sizable when
$\alpha \simeq -0.8$ or $-1.6$.
In the former case, $M_1$ is positive, whereas in the latter case, it
is negative as can be seen in Fig.~\ref{fig:gaugino_mass}.
There is a gap between the two cases, where $M_1$ becomes very
small. In the gap, either the $S = {\rm Tr}[Y_i m_{\phi_i}^2]$ or
the $A_\tau$ terms can drive the right-handed stau tachyonic in the
RG running.
The right panel of Fig.~\ref{fig-scat_tbalp} shows that a wide range
of $\tan\beta$ can be compatible with the measured $\Delta
a_\mu$.
However, when $\tan\beta$ is large ($\gtrsim 30$), a deeper
charge-breaking vacuum can be induced even if $\Delta a_\mu$ will be
enhanced by $\tan\beta$ as shown in Eq.~(\ref{eq:a_mu_BLR}).

As described in Sec.~\ref{sec:kklt}, the modular weights $c_i$ are
rational numbers in the KKLT setup.
In our analysis, we have taken them to be real positive or
negative numbers to find the viable ranges of $c_i$ that can explain
the muon $g-2$ anomaly.
The result of our parameter scan is shown in Fig.~\ref{fig-scat_cs}.
We find that
the favored regions are $0 \lesssim c_{10} \lesssim 0.5$,
and $-1 \lesssim c_5 \lesssim 0$,
$2 \lesssim c_{H_u} \lesssim 4$, and $-3\lesssim c_{H_d} \lesssim 0$,
which are compatible with the measured $\Delta a_\mu$ value within $2\sigma$.
Among them, $c_5$ and $c_{10}$ are important for
having sizable $\Delta a_\mu$ by the light smuons.
Even if the modulus-mediated contributions to the squarks are small,
the squarks can be heavy due to the RG effects of heavy gluinos.
The $c_{10}$ parameter is related to the mass and the mixing of top
squarks.
In our analysis result, it is mostly positive for achieving $m_h
\simeq 125$~GeV without having too high SUSY-breaking scale.
The $c_{H_u}$ parameter also plays an important role to have a stable
vacuum as it affects the $\mu$ value through the condition of EW
symmetry breaking.
The negative values of $c_{H_d}$ is favored because it lifts up the
slepton masses
through mixed anomaly-modulus mediation and RG running effects.
In Appendix~\ref{sec:benchmark}, we list benchmark sparticle mass
spectra.

Before closing this section, let us discuss the constraints from
flavor-violating processes.
We have calculated the flavor-violating observables by using
\texttt{SuperIso}~\cite{Mahmoudi:2007vz, Mahmoudi:2008tp, Mahmoudi:2009zz}.
Figure~\ref{fig-scat_bsgm} shows the scattering plot of
$\br{B_s}{\mu\mu}$ and $\br{b}{s\gamma}$.
The measured value
of $\br{b}{s\gamma}$ is $(3.32\pm 0.15)\times 10^{-4}$~\cite{Amhis:2019ckw},
and the SM prediction is $(3.36\pm 0.23)\times 10^{-4}$~\cite{Misiak:2015xwa}.
We refer to Ref.\cite{Altmannshofer:2021qrr}
for the combined measurements of $\br{B_s}{\mu\mu}$,
$(2.93\pm0.35)\times 10^{-9}$,
and the SM prediction, $(3.67\pm 0.15)\times 10^{-9}$.
The colored regions in Fig.~\ref{fig-scat_bsgm} are outside the
$2\sigma$ ranges from the experimental central values.
The uncertainties have been obtained by quadrature sums of the SM and
the experimental errors.
The points with larger $\Delta a_\mu$ tend to have smaller SUSY
contributions to the flavor-violating processes because the dangerous
points with light sleptons and large $\tan\beta$ have already been
excluded by requiring the vacuum stability condition.
We have also checked that all the other flavor-violating observables
calculated with \texttt{SuperIso} are consistent with the SM
predictions within current uncertainties.

\section{\label{sec:lhc}Collider signatures and LHC constraints}

\noindent
In this section, we discuss viable phenomenological scenarios of mixed
modulus-anomaly mediation motivated by the muon $g-2$ anomaly and the
relevant experimental constraints.
Among the sparticles,
in general,
the colored sparticles receive the most severe constraints from the
SUSY searches at hadron colliders.
The latest LHC Run 2 analysis results of the ATLAS~\cite{Aad:2020aze}
and CMS~\cite{Sirunyan:2019xwh} collaborations have excluded the
gluino mass below 2.3~TeV.
In our study, the $M_0$ value has been fixed by requiring the Higgs
mass to be compatible with the measured SM-like Higgs mass, given the
other model parameters.
It results in a large value of $M_0$ that leads to heavy gluinos.
Furthermore,   as seen in Sec.~\ref{sec:kklt},
a negative $\alpha$
can raise
the gluino mass up to multi-TeV scales
while
leaving the EWinos
around the weak scale.
The upper left panel of Fig.~\ref{fig-mass_spectrum} shows
that $m_{\tilde g} \gtrsim 2.5$~TeV in the parameter space that can
explain the muon $g-2$ anomaly within $2\sigma$.
On the other hand,
the sfermion masses
have a strong dependence on  $M_0$ and $c_i$.
In the parameter space for the muon $g-2$, the masses of the lighter
stop are close to or slightly above the current lower limit, which is
$m_{\tilde t_1} \gtrsim 1.2$~TeV~\cite{Aad:2020aob, Sirunyan:2021mrs}.
Consequently, we expect that the parameter points having stop
masses around or above 1~TeV will be tested by searches at the future
LHC Run 3 and the High-Luminosity LHC\@.
The other squark masses such as the lighter sbottom are well above the
current experimental bounds.
In Appendix~\ref{sec:benchmark}, we present the benchmark sparticle mass
spectra.

\begin{figure}[tb!]
  \begin{center}
    \includegraphics[width=0.48\textwidth]{fig_mt1-mgl_scatter_0607.pdf}
    \includegraphics[width=0.48\textwidth]{fig_mC1-mN1_scatter_0607.pdf}\\
    \includegraphics[width=0.48\textwidth]{fig_mstau1-mN1_scatter_0607.pdf}
    \includegraphics[width=0.48\textwidth]{fig_msmuR-mN1_scatter_0607.pdf}
  \end{center}
  \caption{\label{fig-mass_spectrum}
    The masses of $(\tilde t_1$, $\tilde g)$ (upper left),
    $(\tilde\chi_1^\pm$, $\tilde \chi_1^0)$ (upper right),
    $(\tilde\tau_1$, $\tilde \chi_1^0)$ (lower left),
    and
    $(\tilde \mu_R$, $\tilde \chi_1^0)$ (lower right) in GeV
    for the parameter points
    compatible with the muon $g-2$ anomaly. The color scheme is the same
    as in Fig.~\ref{fig-scat_cs}.}
\end{figure}

Contrary to the colored sparticles, the lighter chargino and
neutralinos, as well as the sleptons, have masses around the weak scale
to explain the muon $g-2$ anomaly.
Therefore, the search results on the direct productions of the
neutralinos/charginos and the sleptons can impose more serious limits
on the parameter space than the experimental bounds discussed above.
We exhibit the lighter chargino and slepton masses in
Fig.~\ref{fig-mass_spectrum}.
The lighter chargino is dominantly wino-like because the Higgsino is much
heavier, $\abs{\mu}\gtrsim 2$~TeV,
and hence the second lightest neutralino has degenerate mass with the chargino.
The left-handed sleptons tend to be heavier than the right-handed ones
due to the RG effects from relatively large wino mass.
The selectrons are nearly mass degenerate with the smuons,
whereas the stau can have different masses than the other
sleptons due to the left-right mixing terms and the RG effects.

Even though the muon $g-2$ anomaly hints at the existence of bino and
smuons around the weak scale, it can lead to various phenomenological
scenarios depending on the interactions and the mass spectrum
of the sparticles in the low-energy scale.
Classifying them by the property of the LOSP, we consider three
phenomenological scenarios:
\renewcommand*\labelenumi{(\theenumi)}
\begin{enumerate}
 \item {The neutralino is the LOSP and is stable.}
 \item {The charged slepton is the LOSP and is metastable.}
 \item {The LOSP is unstable due to the RPV.}
\end{enumerate}
Here, being stable means that the particle does not necessarily be
completely stable: it does not decay inside detectors at collider
experiments.

\subsection{Stable  neutralino LOSP}

In the first scenario, the LOSP is the lightest neutralino
$\tilde\chi_1^0$.
Because the Higgsino is very heavy, $|\mu| \gtrsim 2$~TeV, in the
parameter space for the muon $g-2$, $\tilde\chi_1^0$ is dominantly
bino-like or an admixture of wino and bino.
In this scenario, the most stringent limits come from the search
results on the direct productions of the neutralino-chargino
$\tilde\chi_2^0 \tilde\chi_1^\pm$ and the slepton pair $\tilde \ell
\tilde \ell$ at the LHC~\cite{Aad:2019vnb, CMS-PAS-SUS-19-012, Sirunyan:2020eab}.
Assuming mass-degenerate left-handed (right-handed) light flavor sleptons,
the search results for the slepton pair productions
have set the lower limit for the slepton mass
to be $m_{\tilde \ell_L} > 650$~GeV ($m_{\tilde \ell_R} > 500$~GeV)
for $m_{\tilde \chi_1^0}$ being up to 400~(200)~GeV~\cite{Aad:2019vnb}.
Searches for stau pair productions can also give constraints
because the stau is often the next-to-lightest SUSY particle
in a portion of the parameter space with the neutralino LOSP that can
explain the muon $g-2$ anomaly in our setup.
However, the current limits for the staus decaying into the
neutralino LOSP are not very stringent,   compared to those for  the
charginos and the other sleptons~\cite{CMS:2019eln, Aad:2019byo}.
A way out of the LHC constraints is to have sleptons
mass-degenerate with the neutralino LOSP,
$m_{\tilde \ell}-m_{\tilde \chi_1^0} \lesssim 80$~GeV.
The limits for such degenerate spectrum is quite restricted~\cite{Aad:2019qnd}.

The search results on the neutralino-chargino production at the LHC set
the stringent limits on the chargino mass~\cite{Aad:2019vnb, CMS-PAS-SUS-19-012}.
If the mass gaps among the sleptons and the winos are
sufficiently large, decays to all lepton flavors occur with almost
equal probability, i.e., flavor-democratic decays.
On the other hand, the wino-like states will dominantly decay into a
stau and a tau (neutrino) if the other decay modes are kinematically
forbidden.
In the flavor-democratic case,
the CMS analysis result for the integrated luminosity of 137~fb$^{-1}$
has excluded the wino-like chargino mass up to about 1.3~TeV when
$m_{\tilde{\chi}^0_1} \lesssim 800$~GeV~\cite{CMS-PAS-SUS-19-012}.
We see from Fig.~\ref{fig-mass_spectrum} that, in the
flavor-democratic case, all the parameter points are excluded by the
CMS search result except for the mass-degenerate region.
However, in the mass-degenerate region with $\alpha \lesssim -2.5$,
the bino becomes heavy, and thus the sleptons must be light to explain
$\Delta a_\mu$.
Consequently, the flavor-democratic decays of the wino-like states are
not achievable in the parameter region compatible with the muon $g-2$
anomaly.
Meanwhile,
if the wino-like states dominantly decay to a stau and a tau (neutrino),
the lower limit for the chargino mass is about
800~GeV for $m_{\tilde{\chi}^0_1} \lesssim 100$ GeV~\cite{CMS-PAS-SUS-19-012},
which is much weaker than that in the flavor-democratic case.
This can be realized
if the winos are lighter than the left-handed selectron and smuon.

We conclude that, in the case of neutralino LOSP, the current LHC
limits can be satisfied if the right-handed sleptons are nearly
degenerate with the neutralino LOSP, and the left-handed selectron and
smuon are sufficiently heavy so that the wino-like states dominantly
decay to the stau:\footnote{
  Here, we have used the ``$\lessapprox$'' symbol to indicate that the
  two particles are close in mass while evading the LHC limits by
  following the notation in Ref.~\cite{Athron:2021iuf}.
}
\begin{equation}
\label{eq-noLHCspec}
  m_{\tilde{\chi}^0_1} \lessapprox m_{\tilde{\ell}_R},
  \quad
  m_{\tilde{\chi}^0_1} \lesssim
  m_{\tilde \tau_1} \lesssim m_{\tilde\chi_1^\pm, \tilde\chi_2^0}
  \lesssim m_{\tilde e_L, \, \tilde \mu_L}, \quad
  m_{\tilde e_L, \, \tilde \mu_L} > 650~{\rm GeV} .
\end{equation}
In addition to the above mass hierarchy, one should ensure that the
right-handed slepton is heavier than about 100~GeV to avoid the lower
limit from LEP on the slepton masses.
The limit also applies indirectly to $m_{\tilde\chi_1^0}$ in the case
where the bino-like neutralino LOSP is degenerate in mass with the
slepton.
Furthermore, if $m_{\tilde\chi_1^0} \lesssim 100$~GeV, the lower
limit from CMS on the chargino mass, $m_{\tilde\chi_1^\pm} \gtrsim
800$~GeV, should also be taken into account.
The benchmark point A shown in Appendix~\ref{sec:benchmark}
corresponds to this scenario.
It is interesting that the neutralino LOSP
could serve as a good dark matter (DM) candidate through slepton co-annihilations
due to the mass degeneracy.\footnote{
The bino-like neutralino LOSP around the weak scale may be
overproduced via late-time decays of a modulus~\cite{Endo:2006zj}
unless the modulus is located quite close to the potential minimum
after the primordial inflation.
This fact also motivates us to consider the axino as the lightest SUSY
particle (LSP) or RPV scenarios, which will be discussed in the
following subsections.
}

\subsection{Metastable slepton LOSP}

If the slepton is the LOSP,
there should be lighter sparticle than it or R-parity should be violated
so that the slepton LOSP can decay. The latter scenario will be
discussed in the next subsection. In the former case, a scenario worth
considering is a PQ symmetric extension where the axion solves the
strong CP problem, and the axino $\tilde a$  contributes to the DM~\cite{Covi:1999ty}.
For instance,
if the saxion is radiatively stabilized~\cite{Nakamura:2008ey,Choi:2009qd},
the axino  naturally becomes the LSP because its mass is one-loop suppressed
compared to other sparticle masses.
The scenario is noteworthy because, in a majority of the
parameter space compatible with the muon $g-2$ anomaly, we find that
a slepton is lighter than the lightest neutralino:
the stau is the LOSP in more than half of the
parameter space and the selectron or the smuon is the LOSP in many
other parameter points.
Then, the slepton LOSP will mainly undergo the two-body decay,
$\tilde \ell \to \ell \tilde a$.

In this scenario, the slepton LOSP becomes a heavy stable
charged particle (HSCP), mostly decaying outside the detector, because
it can have a lifetime longer than $10^{4}$~ns to a few hundred
seconds, depending on the axion decay constant and the masses of
the involved sparticle masses~\cite{Brandenburg:2005he}.
Due to the long lifetime, the scenario is not constrained by
the search results for displaced leptons because it is sensitive to
the particles with lifetime shorter than 1~ns~\cite{Aad:2020bay}.
At LEP2, the null detection of the HSCPs set the lower mass limit of
about 100 GeV~\cite{Abbiendi:2003yd}.
In recent years, the constraint has been updated further by the
searches for HSCPs at the LHC\@.
In particular, the CMS collaboration performed model-independent
analyses for various possible HSCPs and excluded stau masses
below 360~GeV~\cite{CMS-PAS-EXO-16-036}.
In the scenario with the axino LSP,
the CMS limit can impose a serious impact on our analysis result
because the lighter stau is lighter than about 350~GeV in the
parameter points with the stau LOSP for the muon $g-2$ anomaly within
$3\sigma$, as can be seen in the lower left panel of Fig.~\ref{fig-mass_spectrum}.

\subsection{Unstable LOSP}

The LOSP, either neutralino or slepton,
decays to SM particles if the RPV interactions are allowed.
In the RPV scenario, the axion can serve as a candidate for the DM~\cite{Baer:2014eja}.
The relevant RPV terms in the superpotential are given as follows:
\begin{equation}
  W \supset \frac{1}{2} \lambda_{ijk} L_i L_j \bar E_k +
  \lambda_{ijk}^\prime L_i Q_j \bar D_k + \frac{1}{2}
  \lambda_{ijk}^{\prime \prime} \bar U_i \bar D_j \bar D_k ,
\end{equation}
where $i$, $j$, $k$ are flavor indices. The $\lambda_{ijk}$ and
$\lambda_{ijk}^\prime$ terms violate lepton number while the
$\lambda_{ijk}^{\prime \prime}$ terms violate baryon number.
Either of the lepton or baryon number conservation should hold with good accuracy
to avoid too fast proton decay.

If the terms with the $\lambda_{ijk}$ couplings are dominant among
the others, the slepton LOSP will mainly decay into a charged lepton and a
neutrino, $\tilde \ell \to \ell_j \nu_{\ell_k}$.
The slepton pair
production then gives rise to the signature of $2\ell+E_T^\mathrm{miss}$.
The signature is similar to that of the R-parity conserving case,
and it receives the bounds from the aforementioned SUSY searches for
multi-lepton final states.
Recasting of the LHC search results has revealed that the
lower limit of the stau LOSP is about 225~GeV in the case where the
stau is right-handed~\cite{Dreiner:2020lbz}.
A large portion of our parameter space could be excluded by the limit.
Meanwhile, if either $\lambda_{ijk}^\prime$ or $\lambda_{ijk}^{\prime
  \prime}$ term is dominant, the final states of the slepton decays
are quite different. In the case where the $\lambda_{ijk}^{\prime}$
operators are dominant, the slepton LOSP can decay into
the final state of two leptons $+$ two quark-jets via
four-body processes. For example, the stau LOSP can decay as
\begin{equation}
  \tilde \tau_1 \to \tau + \tilde\chi_1^{0 \ast} \to \tau + \mu\,
  u \, \bar d
\end{equation}
via the $\lambda_{211}^{\prime}$ operator.
See Refs.~\cite{Desch:2010gi, Dercks:2017lfq}
for a list of possible LHC signatures.
The decay length of the stau LOSP can be $\mathcal{O}(10^{-6})$~m
for $\lambda^\prime \simeq 10^{-3}$ and $m_{\tilde \tau_1} \simeq
m_{\tilde\chi_1^0} \simeq 100$~GeV,
resulting in displaced vertices~\cite{Allanach:2003eb}.
As the gluino and squark masses are beyond the current experimental
limits, the slepton LOSP could be produced via the neutralino-chargino or
the direct slepton pair processes.
The signatures have not yet been covered by the LHC searches so far.
Therefore, we conclude that the slepton LOSP scenario with RPV would
be viable unless the $\lambda_{ijk}$ operator is the dominant RPV
interaction.

In the case of neutralino LOSP with RPV, we can reach a similar conclusion.
If R-parity is violated dominantly by the $\lambda_{ijk}$
coupling, there are strong constraints due to the signatures of
high-multiplicity leptons~\cite{Aad:2021qct,Aad:2020cqu}.
For instance, the limits for sleptons and charginos are
about 800 and 1000~GeV, respectively, in the scenario of nonzero
$\lambda_{i33}$~\cite{Aad:2021qct}.
In the other cases where either $\lambda_{ijk}^\prime$ or
$\lambda_{ijk}^{\prime \prime}$ is dominant, the limits are much
weaker or absent.

\section{\label{sec:summary}Summary}

\noindent
Since the new measurement of the muon $g-2$ at the Fermilab
experiment,
physicists have regained attention on the existence of new
physics in the lepton sector.
If new physics responsible for the muon $g-2$ anomaly is
supersymmetric, one should consider how to obtain light
EWinos and sleptons in a systematic way.
Combined with the gauge coupling unification,
the gaugino masses exhibit a robust pattern controlled by a single parameter $\alpha$
that  represents the size of anomaly mediation.
The EWinos can be much lighter than the gluino if $\alpha$ is negative and of order unity,
as is required to explain the muon $g-2$ anomaly while avoiding experimental constraints.
The KKLT provides a natural and interesting framework for such mixed mediation,  where
the pattern of gaugino masses is determined by $\alpha$,  while that of sfermion masses
depends on how the corresponding matter field couples to the string moduli sector.

We have performed a numerical analysis to explore the parameter space
of mixed modulus-anomaly mediation
realized in the generalized KKLT setup
and identified the parameter region
compatible with the muon $g-2$ anomaly.
To have light EWinos,
it is essential to construct a setup of KKLT moduli stabilization
yielding a negative $\alpha$.
As a byproduct,  it can make
the gluino
heavier than a few TeV, thus we can easily evade the lower limit of
gluino at the LHC\@.
On the other hand, due to light sleptons, imposing the condition of
vacuum stability of the scalar potential is crucial,    and it excludes
the parameter space of large $\tan\beta \gtrsim 30$.

In the viable parameter region,  we find that the LOSP can be either
bino-like neutralino or slepton.
However, in the case of the neutralino LOSP,
the slepton and chargino-neutralino searches at the LHC
exclude
a vast parameter space of the R-parity conservation.
The current LHC limits can be satisfied only when the mass spectrum
of Eq.~\eqref{eq-noLHCspec} is realized.
In most cases, the wino cannot be sufficiently heavy or degenerate with the bino
as far as the sleptons are sufficiently light due to the gaugino mass relations
predicted in the mixed modulus-anomaly mediation.
To avoid this difficulty, one may consider a more general case with
$c_i \neq a_i$. Another way is to add gauge-mediated contributions so
that the deflection of sparticle masses occurs at the gauge-messenger
scale~\cite{Everett:2008qy,Everett:2008ey}.
Meanwhile,  when a slepton is  lighter than the neutralinos,  we should consider
alternative scenarios such as axino LSP or RPV interactions.
In the former case with axino LSP, the lightest slepton becomes
long-lived and will decay outside the detector.
The recent CMS result on long-lived charged particles has excluded
such possibility.
On the other side, the RPV interactions with either lepton or baryon
number violation can be a viable option because of unexplored
signatures with the final states of multi-jets and -leptons with small
or no missing energy at the LHC\@.

\section*{Acknowledgments}

\noindent
This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korean government NRF-2018R1C1B6006061 (K.S.J.),
the Institute for Basic Science (IBS) under the project code
IBS-R018-D1 (J.K. and C.B.P.), and
the Grant-in-Aid for Scientific Research from the Ministry of
Education, Science, Sports and Culture (MEXT), Japan No.\ 18K13534
(J.K.).


text:
\section{Introduction}\label{sec1-bpw2}

One of the most interesting questions in the theory of univalent
functions is to address the region of variability of the $n$-th
Taylor (Laurent resp.) coefficient for functions $f$
that are analytic (meromorphic resp.) and univalent in the
unit disk $\ID =\{z:\,|z|<1\}$. The leading example is the
Bieberbach conjecture settled by de Branges in 1985 for the class
$\mathcal S$ of normalized analytic univalent functions $f$ in $\ID$ although
corresponding results for important subclasses of $\mathcal S$ are
relatively easy and were settled positively much earlier. In this paper,
we consider the family $Co(p)$ of
 functions $f:\ID\to \overline{\IC}$ that satisfy the following conditions:
\bee
\item[(i)] $f$ is meromorphic in $\ID$ and has a simple pole at the point
$p\in (0,1)$ with the standard normalization $f(0)=f'(0)-1=0$.
\item[(ii)] $f$ maps $\ID$ conformally onto a set whose complement with respect to
$\overline{\IC}$ is convex.
\eee
Each $f\in Co(p)$ has the power series expansion of the form
\be\label{p1eq0}
f(z)=z+\sum_{n=2}^{\infty} A_n(f)z^n, \quad |z|<p.
\ee
For our investigation, we  consider the Laurent expansion of $f\in Co(p)$
about the pole $z=p$:
\be\label{p1eq1}
f(z)=\sum_{n=-1}^{\infty} a_n(z-p)^n, \quad z\in \Delta_p,
\ee
where $\Delta_p=\{z\in \IC:\, |z-p|<1-p\}$.
Motivated by the works of Pfaltzgraff  and Pinchuk \cite{Pfa-Pin-90},
Miller \cite{mill-80}, and Livingston \cite{Living-94}, the class $Co(p)$ has been
investigated recently in \cite{A-wirths06,Avk-Wir-93,Avk-Wir-92,Wir-91,Wirths-pre}.
A necessary and sufficient condition
for a function $f$ to be in $Co(p)$  (\cite{Living-94}) is
that ${\rm Re}\,\phi(z,f) >0$ for all $z\in \ID$, where
$$
\phi(z,f)=-(1+p^2)+2pz-\frac{(z-p)(1-pz)f''(z)}{f'(z)}, \quad z\in \ID.
$$
Livingston \cite{Living-94} determined some estimates regarding the real part
of $A_n(f)$ for $n=2,3$ when $f\in Co(p)$ has
the expansion (\ref{p1eq0}). In the same article he conjectured an
estimate for the real part of the general coefficient $A_n(f)$ $(n\geq 2$)
for $ f\in Co(p)$. After a long gap of ten years,  positive developments have
occurred in this line of work. For example, the recent work of
Avkhadiev and Wirths \cite{A-wirths06} settles
the conjecture of Avkhadiev, Pommerenke and Wirths
\cite{Avk-Wir-93} which, in particular, provides a proof of the Livingston conjecture.
For a ready reference, we now recall it here.

\bigskip
\noindent
{\bf Theorem A.} {\rm \cite{A-wirths06}}
{\em Let $n\geq 2$ and $p\in (0,1).$ For each $f \in Co(p)$ with the expansion
$(1.1)$ the inequality
\be\label{p1eq21}
\left|A_n(f)-\frac{1-p^{2n+2}}{p^{n-1}(1-p^4)}\right|\leq
\frac{p^2(1-p^{2n+2})}{p^{n-1}(1-p^4)}
\ee
is valid. Equality in {\rm (\ref{p1eq21})} is attained if and only if $f$
is one of the functions $f_{\theta}$, $\theta \in [0, 2\pi)$, where
\be\label{p1eq18aa}
f_{\theta}(z)=\frac{z-\frac{p}{1+p^2}(1+e^{i\theta})z^2}{(1-\frac{z}{p})(1-zp)}.
\ee
For each complex number in the disk described in {\rm (\ref{p1eq21})} there
exists a function $f\in Co(p)$ such that this number occurs as the
$n$-th Taylor coefficient of $f.$
}
\bigskip

Interestingly,  Wirths  \cite{Wirths-pre} established the following
representation formula for functions in $Co(p)$.

\bigskip
\noindent
{\bf Theorem B.} {\rm \cite{Wirths-pre}}
{\em
For each $f\in Co(p)$, there exists a function $\omega$
holomorphic in $\ID$ such that $\omega(\ID)\subset \overline{\ID}$ and
\be\label{p1eq17}
f(z)=\frac{z-\frac{p}{1+p^2}(1+\omega(z))z^2}{(1-\frac{z}{p})(1-zp)},
\quad \mbox{$ z\in \ID $}.
\ee
}
\bigskip

The above representation formula
has been used by the authors in \cite{BPW1} to obtain some other
kind of coefficient estimates for functions in the class $Co(p)$
with the Laurent expansion of the form (\ref{p1eq1}).

In the present article, we first obtain certain coefficient estimates
for functions in $Co(p)$  but this time for the expansion of the form (\ref{p1eq1}).
Next we move on to discuss a related class of meromorphically
starlike functions, namely, the class $\Sigma^s(p, w_0$ and obtain
a simple and easily applicable
representation formula for this class. Using this formula, we also
obtain some sharp coefficient estimates for functions in this class.
As a consequence of our investigation, we rectify a mistake that appeared in
the work of Livingston in \cite[Theorem 9]{Living-94}.

Now, we state our first result.

\bthm\label{1th1}
Let $p\in(0,\frac{\sqrt 5-1}{2}]$ and $f \in Co(p)$ have the expansion $(\ref{p1eq1})$.
Then
\be\label{p1eq2}
\left |p-(1-p^2)\frac{a_0}{a_{-1}}
\right |\leq \frac{p}{|a_{-1}|} ,
\quad \mbox{i.e. }\quad
\left |a_{-1}-\frac{1-p^2}{p}a_0
\right |\leq 1.
\ee
The inequality is sharp.
\ethm

\br
In \cite{Wirths-pre} Wirths  has obtained the region of variability
for $a_{-1}(f)$, namely, the inequality
$$
\left |a_{-1}+ \frac{p^2}{1-p^4}\right |\leq \frac{p^4}{1-p^4}
\quad \mbox{for $0<p<1$}.
$$
In \cite{BPW1}, the domain of variability of $a_0(f)$
is determined by the inequality
$$\left|\frac{(1-p^2)a_0}{p}+\frac{1-p^2+p^4}{1-p^4}\right|
\leq \frac{p^2(2-p^2)}{1-p^4} \quad \mbox{for $p\in (0,\sqrt{3}-1]$}.
$$
Equality in each of the above two inequalities is attained if and only if
$f$ is one of the functions given in (\ref{p1eq18aa}).
\hfill $\Box$\er

Next result presents sharp coefficient estimates for all $n\geq 3$
if $f\in Co(p)$ has the expansion (\ref{p1eq1}).

\bthm\label{1th2}
If $f\in Co(p)$ with $p\in (0,1)$ and has the expansion $(\ref{p1eq1})$, then we have for $(n\geq 3)$\\

$\ds \left|a_{n-2}-\frac{(1-p^2)a_{n-1}}{p}\right|$
\begin{equation}\label{p1eq18}
\leq
\frac{p}{(1-p^4)(1-p)^{n-1}}
\left[1-\left (\frac{1-p^4}{p^4}\right )^2\,
\left|a_{-1}+\frac{p^2}{1-p^4}\right|^2\right].
\end{equation}
The equality holds in the above inequality for the functions $f_\theta
~~(0\leq\theta\leq 2\pi)$ of the form $(\ref{p1eq18aa})$.
\ethm

\section{Proofs of Theorems \ref{1th1} and \ref{1th2}}\label{1proofs}
\begin{nonsec}Proof of Theorem \ref{1th1}\end{nonsec}
Let $f\in Co(p)$. Then, by Theorem B,
there exists a function $\omega$ holomorphic in $\ID$ such that
$\omega(\ID)\subset \overline{\ID}$ satisfying
the representation formula (\ref{p1eq17}).

Now, let $f\in Co(p)$ have the Laurent expansion (\ref{p1eq1}) and let
$\omega$ have the Taylor expansion
\be\label{p1eq4a}
\omega(z)=\sum_{n=0}^{\infty}c_n(z-p)^n, \quad z\in\Delta_p.
\ee
Using these two expansions, the series formulation of (\ref{p1eq17}) takes
the form
\beq \label{p1eq5}
(z-p)\left ((z-p)-\frac{1-p^2}{p}\right )\sum_{n=-1}^{\infty}a_n(z-p)^n \hspace{3cm}\\
= p+(z-p)-\frac{p}{1+p^2}\left (1+\sum_{n=0}^{\infty}c_n(z-p)^n
\right)\left((z-p)^2+2p(z-p)+p^2\right ).\nonumber
\eeq
Comparing  the coefficient of $(z-p)$  on both sides of (\ref{p1eq5}), we see that
\begin{equation}\label{p1eq6}
a_{-1}-\frac{1-p^2}{p}a_0 = \frac{1-p^2}{1+p^2}-\frac{p^2}{1+p^2}(2c_0+pc_1).
\end{equation}
Using the classical Schwarz-Pick lemma, it follows that
$$
|\omega'(p)|\leq \frac{1-|\omega (p)|^2}{1-p^2},\quad \mbox{i.e. }~|c_1|\leq \frac{1-|c_0|^2}{1-p^2}.
$$
In view of this observation, we have the estimate
$$
|2c_0+pc_1|\leq \frac{p(1-|c_0|^2)+2(1-p^2)|c_0|}{1-p^2}.
$$
For convenience, we set $x=|c_0|$ and consider
$$R_p(x)=p(1-x^2)+2(1-p^2)x.
$$
We see that $R_p(x)$ attains local maximum at $x_m=\frac{1-p^2}{p}$. Since
$x_m \geq 1$ for $p\in \left(0,\frac{\sqrt 5-1}{2}\right]$,
we see that
$$
|R_p(x)|\leq R_p(1)= 2(1-p^2)\quad, \, x\in[0,1],\quad p\in\left(0,\frac{\sqrt 5-1}{2}\right],
$$
and therefore, we have the estimate $|2c_0+pc_1|\leq 2$ for those $p$ in the said
interval. Now using this we get
from (\ref{p1eq6}) the  estimate (\ref{p1eq2}).
It is a simple exercise to see that the
equality is attained in (\ref{p1eq2}) for the following function
$$
f(z)=\frac{-zp}{(z-p)(1-zp)}.
$$
\hfill $\Box$

\begin{nonsec}Proof of Theorem \ref{1th2}\end{nonsec}
Let $f\in Co(p)$, with the expansion (\ref{p1eq1}).
Next, following the notation of the proof of Theorem \ref{1th1},
we compare the coefficients of $(z-p)^n$ $(n\geq 3$) on both side of
the equation (\ref{p1eq5}). This gives
$$a_{n-2}-\frac{1-p^2}{p}a_{n-1}=-\frac{p}{1+p^2}(c_{n-2}+2p c_{n-1}+p^2c_n)
\quad \mbox{$(n\geq 3)$}.
$$
Now, for a unimodular bounded analytic function $\omega$ in the unit disk $\ID$
having the expansion (\ref{p1eq4a}) in $\Delta_p$, we recall the following
result due to Ruscheweyh \cite[Theorem 2]{Rus-89}
$$
(1-p)^n(1+p)|c_n|\leq 1-|c_0|^2  \quad (n\geq 1)
$$
where the equality holds for $\omega(z)=e^{i\theta}, \theta\in[0,2\pi)$.
Using this, we easily obtain that
$$
\left|a_{n-2}-\frac{(1-p^2)a_{n-1}}{p}\right|
\leq \frac{p(1-|c_0|^2)}{(1+p^2)(1+p)(1-p)^n} \quad (n\geq 3).
$$
Consequently, (\ref{p1eq18}) follows since
$$
c_0=\frac{1-p^4}{p^4}a_{-1}+\frac{1}{p^2},
$$
by comparing  the constant terms on both sides of (\ref{p1eq5}).
Now, the equality holds in (\ref{p1eq18}) for the functions $f_\theta$,
$\theta\in [0, 2\pi)$,
in (\ref{p1eq18aa}), since both sides of the inequality are zero.
\hfill $\Box$

\section{Meromorphically starlike functions}\label{sec1-bpw3}
Let $\Sigma^s(p,w_0)$ denote the class of
meromorphic and univalent functions $f$ in $\ID$ (with the standard
normalization $f(0)=f'(0)-1=0$) having a simple pole
at $p\in (0,1)$ with the expansion (\ref{p1eq1}) such that $f$ is starlike
with respect to a fixed
$w_0\in \IC, w_0\neq 0$ (i.e.  $\overline\IC\backslash f(\ID)$ is a starlike
set with respect to $w_0$).
A well-known fact is that \cite{Living-94} $f\in\Sigma^s(p,w_0)$
if and only if ${\rm Re}\,\psi(z,f) >0$ for all $z\in \ID$, where
\be\label{p1eq20}
\psi(z,f) =\frac{-(z-p)(1-pz)f'(z)}{f(z)-w_0},\quad z\in \ID.
\ee

We now state and prove a useful representation formula for functions
in the class $\Sigma^s(p, w_0)$.

\bthm\label{1th5}
For $0<p<1$, let $f\in \Sigma^s(p, w_0)$. Then there exists a function
$\omega$ holomorphic in $\ID$ such that
$\omega(\ID)\subset \overline \ID,\, \omega(0)= -\frac{1}{2} (\frac{1}{w_0}+p+\frac{1}{p})$
and
\be\label{p1eq22}
f(z)= w_0 + \frac{pw_0(1+z\omega(z))^2}{(z-p)(1-zp)},\quad z\in \ID.
\ee
\ethm\bpf
The proof of this theorem is indeed a direct consequence of \cite[Corollary 2]{Zhang-92}
in which they use the notation $\sigma^*(p, w_0)$ in place of $\Sigma^s(p, w_0)$.
By this corollary we get if $f\in \Sigma^s(p, w_0)$, then
$$\left|\left\{\frac{f(z)-w_0}{pw_0}(z-p)(1-pz)\right\}^{1/2}-1\right|\leq |z| ,\quad z\in \ID.
$$
Now writing
$$
\omega(z)= \frac{1}{z}\left\{\frac{f(z)-w_0}{pw_0}(z-p)(1-pz)\right\}^{1/2}-\frac{1}{z}
$$
and simplifying the above expression for $f$ we get the desired
representation formula for functions in the class $\Sigma^s(p, w_0)$. Here we note
that $ \omega$ is holomorphic in $\ID$ and $| \omega(z)|\leq 1$. Also since $f'(0)=1$
we get $\omega(0)= -\frac{1}{2} (\frac{1}{w_0}+p+\frac{1}{p})$.
\epf

As a consequence of Theorem \ref{1th5}, we have the following result
which has been proved in \cite{Chang-88} but by using a different method of
proof.

\bcor
For $0<p<1$, let  $f\in \Sigma^s(p, w_0)$. Then, we have
\be\label{p1eq25}
\left |w_0+\frac{p(1+p^2)}{(1-p^2)^2}\right |\leq \frac{2p^2}{(1-p^2)^2} .
\ee
In particular, one has
$$\frac{p}{(1+p)^2}\leq |w_0|\leq \frac{p}{(1-p)^2}.
$$
\ecor \bpf
As $|\omega(0)|\leq 1$ and $\omega(0)= -\frac{1}{2} (\frac{1}{w_0}+p+\frac{1}{p})$,
it follows that
$$\left | \frac{1}{w_0}+\frac{1+p^2}{p}\right |\leq 2
$$
which is easily seen to be equivalent to the inequality (\ref{p1eq25}). The second inequality is a simple
consequence of  (\ref{p1eq25}).
\epf

\bthm\label{1th6}
Let $f\in\Sigma^s(p, w_0)$ have the Laurent expansion $(\ref{p1eq1})$. Then
\begin{enumerate}
\item[(i)] $\displaystyle
\left|a_{-1}-\frac{pw_0}{1-p^2}\right|\leq \frac{p|w_0|}{1-p^2}
\frac{|\frac{p}{w_0}+p^2+1|+2p^2}{2+|\frac{p}{w_0}+p^2+1|}
\left(\frac{|\frac{p}{w_0}+p^2+1|+2p^2}{2+|\frac{p}{w_0}+p^2+1|}
+2\right)$ ~, $p\in (0,1)$
\item[(ii)] $\displaystyle
 \left |a_0-\frac{1-p^2+p^4}{(1-p^2)^2}w_0\right|\leq \frac{p(2+2p-p^3)}{(1-p^2)^2}|w_0|
$~, $ p\in \left(0,\frac{\sqrt5 -1}{2}\right]$.
\end{enumerate}
Both inequalities are sharp for
$$
f(z)=\frac{-zp}{(z-p)(1-pz)}= w_0+\frac{pw_0}{(z-p)(1-pz)}(1-z)^2\in \Sigma^{s}(p, w_0)
$$
where $w_0=\frac{-p}{(1+p)^2}$.
\ethm\bpf
Let $f\in\Sigma^s(p, w_0)$ have the Laurent expansion (\ref{p1eq1})
and  consider the Taylor expansion  for $\omega$:
$$\omega(z)=\sum_{n=0}^{\infty}c_n(z-p)^n, \quad z\in\Delta_p.
$$
Now substituting (\ref{p1eq1}) and (\ref{p1eq4a}) in the representation formula
(\ref{p1eq22}) we get the following  series formulation of  (\ref{p1eq22}) valid in $\Delta_p$:\\

$\ds \sum_{n=-1}^{\infty} a_n(z-p)^n-w_0$
\beq\label{p1eq23}
&=& \frac{pw_0}{1-p^2}\sum_{n\geq 0}\left(\frac{p}{1-p^2}\right)^n (z-p)^{n-1}\\
\nonumber && \left[1+\{(z-p)^2+p^2+2p(z-p)\}\sum_{n\geq 0}\right.\left(\sum_{k=0}^{n}c_k c_{n-k}\right)(z-p)^n\\
\nonumber && \left.+\{2p+2(z-p)\}\sum_{n\geq 0}c_n(z-p)^n\right].
\eeq
While obtaining the above series form, we make use of the following relation
\beqq
(z-p)(1-pz) &=& (1-p^2)(z-p)\left (1- \frac{p}{1-p^2}(z-p) \right )\\
(1+z\omega(z))^2 &=& 1+ 2(z-p+p)w(z)+ ((z-p)^2+2p(z-p) +p^2)w(z)w(z).
\eeqq
Now, we proceed to prove (i).
Comparing the coefficients of $1/(z-p)$ on both sides of $(\ref{p1eq23})$ we get
$$a_{-1}=\frac{pw_0}{1-p^2}[1+p^2c_0^2+2pc_0].
$$
Finally, Schwarz Pick lemma applied to $\omega$ shows that
$$|c_0|=|\omega(p)|\leq \frac{|\omega(0)|+p}{1+p|\omega(0)|},
$$
where
$$|\omega(0)|=\frac{1}{2}\left|\frac{1}{w_0}+p+\frac{1}{p}\right| .
$$
Using this, we now get the desired estimate for $a_{-1}$.
It is also easy to check that the estimate stated in (i)
is sharp for the function mentioned in the statement of the theorem.\\

(ii) Comparing constant terms on  both sides of (\ref{p1eq23}) we get
$$a_0-w_0
= \frac{p^2 w_0}{(1-p^2)^2}(1+p^2c_0^2+2pc_0)+
\frac{2pw_0}{1-p^2}(p^2c_0c_1+p^2c_0^2+pc_1+c_0)
$$
or equivalently,
$$
a_0-\frac{1-p^2+p^4}{(1-p^2)^2}w_0
= \frac{p^2 w_0}{(1-p^2)^2}(p^2c_0^2+2pc_0)+
\frac{2pw_0}{1-p^2}(p^2c_0c_1+p^2c_0^2+pc_1+c_0).
$$
Now, we recall the well known estimates from the classical Schwarz Pick lemma:
$$|c_0|\leq 1, \quad |c_1|\leq \frac{1-|c_0|^2}{1-p^2}.
$$
For convenience, we use the notation $x= |c_0|$.
Using the above estimates, it is easy to see that the last equality
implies that
$$\left |a_0-\frac{1-p^2+p^4}{(1-p^2)^2}w_0\right|\leq
\frac{p|w_0|}{(1-p^2)^2}(2p+2x+2p^2x-2p^2x^3-p^3x^2).
$$
Next, we introduce
$$Q_p(x)= 2p+2x+2p^2x-2p^2x^3-p^3x^2, \quad 0\leq x\leq 1.
$$
It follows that $Q_p$ attains local maximum at
$$x_m= (-p^2+\sqrt{p^4+12(1+p^2)}\,)/(6p).
$$
Since $x_m\geq 1$ for
$p\in \left(0,\frac{\sqrt5 -1}{2}\right]$, we have
$$
\max\{Q_p(x):\, x\in[0,1]\}= Q_p(1)= 2+2p-p^3.
$$
This proves the inequality (ii) and the sharpness part can easily be
verified for the function given in the statement of
the theorem.
\epf

\br
It is a simple exercise to see that
\be\label{p1eq26}
\frac{\left |\frac{p}{w_0}+p^2+1\right |+2p^2}{2+\left |\frac{p}{w_0}+p^2+1
\right |}\leq p
\ee
is equivalent to
$$\frac{1}{2}\left |\frac{p}{w_0}+p^2+1\right |\leq p,
~\mbox{ i.e. }~ |\omega (0)|\leq 1 .
$$
Thus, (\ref{p1eq26}) holds.
If we use the  inequality (\ref{p1eq26}) then the inequality (i)
in Theorem \ref{1th6} turns out to be
$$
\left|a_{-1}-\frac{pw_0}{1-p^2}\right|\leq \frac{p^2}{1-p^2}(p+2)|w_0|,\quad p\in(0,1).
$$
Applying the triangle inequality in the above inequality and
the inequality (ii) in Theorem \ref{1th6} we get
\be\label{p1eq24}
|a_{-1}|\leq \frac{p(1+p)}{1-p}|w_0|, \quad p\in (0,1)
\ee
and
$$
|a_0|\leq \frac{1}{(1-p)^2}|w_0|, \quad p\in \left(0,\frac{\sqrt5 -1}{2}\right],
$$
respectively. Both the above estimates are sharp for the function stated in
Theorem \ref{1th6}.
\hfill $\Box$\er

In view of the estimate (\ref{p1eq24}) we observe that there was a minor error in
one of the results of Livingston, namely Theorem 9 in  \cite{Living-94}.
Indeed a counterexample is given by the function
$$g(z)=\frac{-zp}{(z-p)(1-pz)}\in \Sigma^{s}\left(p, \frac{-p}{1+p^2}\right).
$$
Here we note that
$$a_{-1}(g)= \frac{-p^2}{1-p^2}
$$
does not belong to the disk stated in Theorem 9 of \cite{Living-94}.
Moreover, the error is actually occurred in  \cite[p.~290]{Living-94} where the
inequality in the 6th line needs to be reversed, since $\xi-p\leq 0$.
We may now formulate a corrected version of \cite[Theorem 9]{Living-94}
for future use.

\bthm\label{1th7}
If $f\in\Sigma^{s}(p, w_0)$ and has the Laurent expansion
$(\ref{p1eq1})$, then we have
$$|a_{-1}|\geq \frac{p(1-p)}{1+p}|w_0|.
$$
The inequality is sharp for the function
$$g(z)=\frac{-zp}{(z-p)(1-pz)}
= w_0+\frac{pw_0}{(z-p)(1-pz)}(1-z)^2\in \Sigma^{s}(p, w_0)
$$
where $w_0=\frac{-p}{(1-p)^2}$.
\ethm

Here we also note that
$${\rm Re}\, \left(\frac{(z-p)(1-zp)g'(z)}{f(z)+\frac{p}{(1-p)^2}}\right)
=-(1-p)^2 {\rm Re}\left(\frac{1+z}{1-z}\right)<0
$$
for all $z\in \ID$ and $g$ satisfies the normalization condition $g(0)=0=g'(0)-1$
whenever $w_0=\frac{-p}{(1-p)^2}$.

\br
In view of the last theorem, the corollary that follows from Theorem 9 in \cite{Living-94}
is also not true since it uses the incorrect estimate for $|a_{-1}|$.
\hfill $\Box$
\er

{\bf Acknowledgement:} The authors thank Prof. K.-J. Wirths for his
suggestions on various stages of the paper and for his
continuous encouragement.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

 One of the central problems in the theory of smooth representations of reductive groups over nonarchimdean local fields is to determine when a representation of a group $G$ admits a linear functional invariant under a closed subgroup $H$ (an $H$-invariant period).

 The Gross--Prasad conjectures \cite{grossprasad92} give a very precise and elegant description of when such periods exist, for many natural pairs $(G, H)$, in terms of $\varepsilon$-factors. However, the original formulation of these conjectures applies to members of \emph{generic} $L$-packets for $G$; and the analogous picture for representations in non-generic $L$-packets is rather more complex. Although the $\varepsilon$-factor is still well-defined for all such $L$-packets, the conjecture formulated in \cite{gangrossprasad20} only applies when the $L$-parameters satisfy an additional ``relevance'' condition, raising the natural question of whether the $\varepsilon$-factors for non-relevant $L$-packets have any signficance in terms of invariant periods.

 In this short note, we describe some computations of branching laws in the following simple case: $G$ is $\GL_2(A)$, where $A/F$ is a cubic \'etale algebra, and $H$ is the subgroup $\GL_2(F)$. Our computations suggest an alternative approach to the theory: rather than studying branching laws for non-generic irreducible representations, we focus on representations which are possibly reducible, but satisfy a certain ``Whittaker type'' condition. We show that $H$-invariant periods on these representations are unique if they exist, and that their existence is governed by $\varepsilon$-factors, extending the results of Prasad \cite{prasad90,prasad92} for irreducible generic representations, and Harris--Scholl \cite{harrisscholl01} for $A$ the split algebra (in which case the $\varepsilon$-factor is always $+1$). In this optic, the ``relevance'' condition appears as a criterion for the $H$-invariant period to factor through the unique irreducible quotient.

 We conclude with an application of this result to global arithmetic. For $\pi$ a Hilbert modular form over a real quadratic field, the constructions of \cite{Kings-Higher-Regulators, leiloefflerzerbes18, grossi19} give rise to a family of cohomology classes taking values in the 4-dimensional Asai Galois representation associated to $\pi$. We show that if $\pi$ is not of CM type and not a base-change from $\mathbf{Q}$, then these elements all lie in a 1-dimensional subspace. This is the analogue for quadratic Hilbert modular forms of the result proved in \cite{harrisscholl01} for Beilinson's elements attached to the Rankin convolution of two modular forms.

 \subsubsection*{Acknowledgements} I am grateful to Giada Grossi and Dipendra Prasad for interesting conversations in connection with this paper, and especially to Nadir Matringe for his answer to a question of mine on MathOverflow, which provided the key to Theorem \ref{thm:asaiA}.

\section{Statements}

 Throughout this paper, $F$ denotes a non-archimedean local field of characteristic 0. If $G$ is a reductive group over $F$, then a ``representation'' of $G(F)$ shall mean a smooth linear representation on a complex vector space.

 \subsection{Epsilon-factors}

  We choose a nontrivial additive character $\psi$ of $F$. For Weil--Deligne representations $\rho$ of $F$, we define epsilon-factors $\varepsilon(\rho) = \varepsilon(\rho, \psi)$ following Langlands (the ``$\varepsilon_L$'' convention in \cite[\S 3.6]{tate79}), so that $\varepsilon(\rho)$ is independent of $\psi$ if $\det(\rho) = 1$. We note that
  \begin{align*}
   \varepsilon(\rho_1 \oplus \rho_2) &= \varepsilon(\rho_1) \varepsilon(\rho_2), &
   \varepsilon(\rho) \varepsilon(\rho^\vee) &= (\det \rho)(-1)
  \end{align*}
  where $\det(\rho)$ is identified with a character of $F^\times$ via class field theory.
  \renewcommand{\sp}{\operatorname{sp}}

  We write $\sp(n)$ for the $n$-dimensional Weil--Deligne representation given by the $(n-1)$-st symmetric power of the Langlands parameter of the Steinberg representation, so that the eigenvalues of the Frobenius element on $\sp(n)$ are $q^{\tfrac{1-n}{2}}, q^{\tfrac{3-n}{2}}, \dots, q^{\tfrac{n-1}{2}}$ where $q$ is the size of the residue field.

 \subsection{The generic Langlands correspondence for $\GL_2$}

  The classical local Langlands correspondence for $\GL_2$ is a bijection between irreducible smooth representations of $\GL_2(F)$, and 2-dimensional Frobenius-semisimple representations of the Weil--Deligne group of $F$.

  In this paper, we will use the following modification of the correspondence. A representation of $\GL_2(F)$ is said to be \emph{of Whittaker type} if it is either irreducible and generic, or a reducible principal series representation with 1-dimensional quotient. (These are precisely the representations of $\GL_2(F)$ which have well-defined Whittaker models.) The \emph{generic Langlands correspondence} is a bijection between Whittaker-type representations of $\GL_2(F)$ and 2-dimensional Frobenius-semisimple Weil--Deligne representations; it agrees with the classical Langlands correspondence on irreducible generic representations, and maps a reducible Whittaker-type principal series to the classical Langlands parameter of its 1-dimensional quotient.\footnote{This correspondence was introduced in \cite{breuilschneider07}; but our conventions differ from \emph{op.cit.} by a power of the norm character, in order that our generic Langlands correspondence extend the classical one.}

  In particular, the unramified Weil--Deligne representation with Frobenius acting as $\stbt{q^{1/2}}{}{}{q^{-1/2}}$ corresponds to the reducible principal series $\Sigma_F$ containing the Steinberg representation $\mathrm{St}_F$ as subrepresentation and trivial 1-dimensional quotient. (We omit the subscript $F$ if it is clear from context.)

 \subsection{Statement of the theorem}

  We now state our main theorem, which is an extension of the main results of \cite{prasad90,prasad92} to possibly reducible representations of Whittaker type.

  Let $A / F$ be a separable cubic algebra, so $A$ is a product of field extensions of $F$ of total degree 3. Let $\omega_A$ be the quadratic character of $F^\times$ determined by the class of $\operatorname{disc}(A)$ in $F^\times / F^{\times 2}$. We let $G = \GL_2(A)$, and $H = \GL_2(F)$, embedded in $G$ in the obvious way.

  The Langlands dual group of $\GL_2 / A$ has a natural 8-dimensional \emph{Asai}, or \emph{multiplicative induction}, representation; in the case $A = F^3$ this is simply the tensor product of the defining representations of the factors. We use this representation, and the generic Langlands correspondence for $\GL_2$ above, to define Asai epsilon-factors $\varepsilon(\operatorname{As}(\Pi))$ for Whittaker-type representations of $\GL_2(A)$.

  Finally, we consider Jacquet--Langlands transfers. Let $H' = D^\times$ where $D/F$ is the unique non-split quaternion algebra. Let $G' = (D \otimes_F A)^\times$, and let $\Pi'$ be the Jacquet--Langlands transfer of $\Pi$ to $G'$ if this exists, and 0 otherwise.

  \begin{remark}
   Note that if $A = E \times F$, then $D^\times$ is split over $E$, and hence $G' = \GL_2(E) \times D^\times(F)$. Thus if $\Pi = \pi \boxtimes \sigma$, for $\pi$, $\sigma$ representations of $\GL_2(E)$ and $\GL_2(F)$ respectively, we have $\Pi' = \pi \boxtimes \sigma'$. In particular, $\Pi' \ne 0$ whenever $\sigma'$ is discrete series (even if $\pi$ is principal series, possibly reducible).
  \end{remark}

  \begin{mainthm}
   Let $\Pi$ be a representation of $\GL_2(A)$ of Whittaker type, whose central character is trivial on $F^\times$. Then we have
   \[ \dim \Hom_{H}(\Pi, \mathbb{1}) = \begin{cases}
    1 & \text{if $\varepsilon(\operatorname{As}(\Pi)) \omega_A(-1) = 1$}, \\
    0 & \text{if $\varepsilon(\operatorname{As}(\Pi)) \omega_A(-1) = -1$};
    \end{cases}
   \]
   and
   \[
    \dim \Hom_{H}(\Pi, \mathbb{1}) + \dim \Hom_{H'}(\Pi', \mathbb{1}) = 1. \]
  \end{mainthm}

  If $\Pi$ is an irreducible generic representation, then this is the main result of \cite{prasad90} for $A$ the split algebra, and \cite{prasad92} for non-split $A$ (modulo the case of supercuspidal representations of cubic fields, completed in \cite{prasadSP08}). The new content of the above theorem is that this also holds for reducible Whittaker-type $\Pi$.

  \begin{remark}
   Any such $\Pi$ can be written as the specialisation at $s = 0$ of an analytic family of Whittaker-type representations $\Pi(s)$ indexed by a complex parameter $s$, which are irreducible for generic $s$ and all have central character trivial on $F^\times$. For such families, the $\varepsilon$-factors $\varepsilon(\operatorname{As} \Pi(s))$ are locally constant as a function of $s$; so, given the results of \cite{prasad92, prasadSP08} in the irreducible case, our theorem is equivalent to the assertion that $\dim \Hom_H(\Pi(s), \mathbb{1})$ and $\dim \Hom_{H'}(\Pi(s)', \mathbb{1})$ are locally constant in $s$.
  \end{remark}

\section{Split triple products}

 We first put $A = F \times F \times F$.

 \begin{theorem}[Prasad, Harris--Scholl] Let $\pi_1$, $\pi_2$, $\pi_3$ be representations of $\GL_2(F)$ of Whittaker type, with central characters $\omega_i$ such that $\omega_1 \omega_2 \omega_3 = 1$. Then we have
 \[ \dim \Hom_{\GL_2(F)}(\pi_1 \otimes \pi_2 \otimes \pi_3, \mathbb{1}) =
   \begin{cases}
    1 & \text{if $\varepsilon(\pi_1 \times \pi_2 \times \pi_3) = +1$},\\
    0 & \text{if $\varepsilon(\pi_1 \times \pi_2 \times \pi_3) = -1$},\\
   \end{cases} \]
  and
  \[ \dim \Hom_{\GL_2(F)}(\pi_1 \otimes \pi_2 \otimes \pi_3, \mathbb{1}) + \dim \Hom_{D^\times(F)}(\pi'_1 \otimes \pi'_2 \otimes \pi'_3, \mathbb{1}) = 1.\]
 \end{theorem}

 If the $\pi_i$ are all irreducible, then the above is the main result of \cite{prasad90}. If one or more of the $\pi_i$ is isomorphic to a twist of $\Sigma_F$, then the $\varepsilon$-factor is automatically $+1$, and $\pi'_1 \otimes \pi'_2 \otimes \pi'_3$ is the zero representation. So all that remains to be shown is that in this case we have $\dim \Hom_{\GL_2(F)}(\pi_1 \otimes \pi_2 \otimes \pi_3, \mathbb{1})= 1$. This is estabished in Propositions 1.5, 1.6 and 1.7 of \cite{harrisscholl01} (except for one specific case which we consider below): in the majority of cases, an $\varepsilon$-factor computation using Prasad's results shows that only one of the composition factors of $\pi_1 \otimes \pi_2 \otimes \pi_3$ admits an $H$-invariant period. The remaining awkward cases are all principal-series representations, which can be handled explicitly using Mackey theory.

 For completeness, we note that the case where all three of the $\pi_i$ are twists of $\Sigma$ by characters is not considered in \cite{harrisscholl01}. By twisting, we may assume $\pi_2 = \pi_3 = \Sigma$ and $\pi_1 = \Sigma \otimes \eta$ where $\eta$ is a character of $F^\times$ with $\eta^2 = 1$. The case $\eta = 1$ is covered by Proposition 1.7 of \emph{op.cit.}, so we assume $\eta$ is a nontrivial quadratic character. In this case $\Hom_H( \eta \otimes \Sigma_F \otimes \Sigma_F, \mathbb{1}) = \Hom_H(\Sigma_F, \Sigma_F^\vee \otimes \eta) = 0$, so $\Hom_H( \eta \otimes \Sigma_F \otimes \Sigma_F, \mathbb{1})$ injects into $\Hom_H( \eta \mathrm{St}_F \otimes \Sigma_F \otimes \Sigma_F, \mathbb{1})$, and the 1-dimensionality of this space is established in \emph{op.cit.}.

\section{Quadratic fields}

 We now suppose $A = E \times F$ with $E/F$ quadratic, so $\Pi = \pi \boxtimes \sigma$ for Whittaker-type representations $\pi$ of $\GL_2(E)$ and $\sigma$ of $\GL_2(F)$ such that $\omega_{\pi}|_{F^\times} \cdot \omega_{\sigma} = 1$. Since the case of $\pi$, $\sigma$ irreducible is proved in \cite{prasad92}, it suffices to consider the following cases:
 \begin{enumerate}[(a)]
 \item $\pi$ is irreducible and $\sigma = \Sigma_F$.
 \item $\sigma$ is irreducible and $\pi = \Sigma_E$;
 \item $\pi = \Sigma_E$ and $\sigma = \Sigma_F \otimes \varepsilon$, where $\varepsilon$ is a quadratic character.
 \end{enumerate}

 In cases (a) and (c), we always have $\varepsilon(\operatorname{As}(\pi) \times \sigma)\varepsilon_{E/F}(-1) = 1$, and $\sigma' = \{0\}$, so the Main Theorem amounts to the assertion that $\dim \Hom_H(\pi \boxtimes \sigma, \mathbb{1}) = 1$. In case (b), both signs can occur.

 \addtocounter{theorem}{1}
 \begin{subtheorem}
  \label{thm:asaiA}
  Let $\pi$ be an irreducible generic representation of $\GL_2(E)$ such that $\omega_{\pi} |_{F^\times} = 1$. Then we have $\dim \Hom_H(\pi \boxtimes \Sigma_F, \mathbb{1}) = 1$.
 \end{subtheorem}

 \begin{remark}
  Note that the case when $E / F$ is unramified, and $\pi$ is unramified and tempered, is part of \cite[Theorem 4.1.1]{grossi19}. However, the proof of this statement \emph{loc.cit.}~has a minor error which means the argument does not work when $\pi$ is the normalised induction of the trivial character of $B_E$. So the argument below fixes this small gap.
 \end{remark}

 \begin{proof}

  We first observe that $\Hom_H(\pi \boxtimes \Sigma_F, \mathbb{1})$ is non-zero. Since $\pi$ is generic, it has a Whittaker model $\mathcal{W}(\pi)$ with respect to any non-trivial additive character of $E$. We may suppose that this additive character is trivial on $F$, so that we may define the Asai zeta-integral
  \[ Z(W, \Phi, s) = \int_{N_H \backslash H} W(h)\Phi( (0, 1) h) |\det h|^{s}\ \mathrm{d}h,\]
  for $W \in \mathcal{W}(\pi)$ and $\Phi \in \mathcal{S}(F^2)$ (the space of Schwartz functions on $F$). Here $N_H$ is the upper-triangular unipotent subgroup of $H$.

  It is well known that this integral converges for $\Re(s) \gg 0$ and has meromorphic continuation to the whole complex plane; and the values of $Z(-, -, s)$ span a nonzero fractional ideal of $\mathbf{C}[q^s, q^{-s}]$, generated by an $L$-factor independent of $\Phi$ and $W$, which is the Asai $L$-factor $L(\operatorname{As}(\pi), s)$. Thus the map
  \[ (W, \Phi) \mapsto \lim_{s \to 0} \frac{Z(W, \Phi, s)}{L(\operatorname{As}(\pi), s)} \]
  defines a non-zero, $H$-invariant bilinear form $W(\pi) \otimes \mathcal{S}(F^2) \to \mathbf{C}$. Since the maximal quotient of $\mathcal{S}(F^2)$ on which $F^\times$ acts trivially is isomorphic to $\Sigma_F$ (see e.g.~\cite[Proposition 3.3(b)]{loeffler-zeta1}), this shows that $\Hom_H(\pi \boxtimes \Sigma_F, \mathbb{1}) \ne 0$ as claimed.

  So, to prove \cref{thm:asaiA}, it suffices to show that $\dim \Hom_H(\pi \boxtimes \Sigma_F, \mathbb{1}) \le 1$. As $\pi$ has unitary central character, it is either a discrete-series representation, in which case it is automatically tempered, or an irreducible principal series, which may or may not be tempered. We shall consider these cases separately.

  Theorem 1.1 of \cite{AKT04} states that if $\pi$ is an irreducible tempered representation of $\GL_2(E)$, then we have $\dim \Hom_{M(F)}(\pi, \mathbb{1}) = 1$, where $M(F) = \{ \stbt{\star}{\star}{0}{1}\}$ is the mirabolic subgroup of $\GL_2(F)$. If we assume $\omega_{\pi} |_{F^\times} = 1$, then since $F^\times \cdot M(F) = B(F)$ is the Borel subgroup of $\GL_2(F)$, we have
  \[ \Hom_{M(F)}(\pi, \mathbb{1}) = \Hom_{B(F)}(\pi, \mathbb{1}) = \Hom_{H}(\pi, \operatorname{Ind}_{B(F)}^{H}(\mathbb{1})).\]
  As $\operatorname{Ind}_{B(F)}^{H}(\mathbb{1}) = \Sigma_F^\vee$, this proves \cref{thm:asaiA} for tempered $\pi$.

  We now consider the principal-series case. For $\alpha, \beta$ smooth characters of $E^\times$, we write $I_E(\alpha, \beta)$ for the normalised induction to $\GL_2(E)$ of the character $\alpha \boxtimes \beta$ of $B(E)$.
  Note that this representation is tempered if and only if $\alpha$ and $\beta$ are unitary. We suppose $\alpha/\beta \ne |\cdot|_E^{\pm 1}$ and $\alpha\beta|_{F^\times} = 1$. Then we have the following results:
  \begin{itemize}
  \item $\Hom_H(\pi \boxtimes \mathrm{St}_F, \mathbb{1})$ is zero if $\alpha \beta^c = 1$, and one-dimensional otherwise, where $\beta^c$ denotes the character $x \mapsto \beta(x^c)$. See \cite[Remark 4.1.1]{prasad92}.
  \item $\Hom_H(\pi \boxtimes \mathbb{1}, \mathbb{1})$ is one-dimensional if $\alpha \beta^c = 1$, or if $\alpha|_{F^\times} = \beta|_{F^\times} = 1$; otherwise it is zero. See \cite[Theorem 5.2]{matringe11}.
  \end{itemize}

  We conclude that exactly one of $\Hom_H(\pi \boxtimes \mathrm{St}_F, \mathbb{1})$ and $\Hom_H(\pi \boxtimes \mathbb{1}, \mathbb{1})$ is nonzero (and \cref{thm:asaiA} therefore follows), \emph{unless} $\pi$ is of the form $I_E(\alpha, \beta)$ with $\alpha|_{F^\times} = \beta|_{F^\times} = 1$ and $\alpha\beta^c \ne 1$. However, in this exceptional case $\alpha$ and $\beta$ are unitary, and thus $\pi$ is tempered, so \cref{thm:asaiA} has already been established for $\pi$ above. This completes the proof of \cref{thm:asaiA}.\end{proof}

  For case (b) we need the following lemma:

  \begin{lemma}
   Let $\sigma$ be an irreducible generic representation of $\GL_2(F)$ with $\omega_{\sigma} = 1$. Then $\varepsilon(\operatorname{As}(\Sigma_E) \times \sigma) = \varepsilon(\sigma) \varepsilon(\sigma \times \omega_{E/F})$. Moreover, if $\sigma \ne \mathrm{St}_F$, then we have
   \[ \varepsilon(\sigma) \varepsilon(\sigma \times \omega_{E/F}) = \varepsilon(\operatorname{As}(\mathrm{St}_E) \times \sigma), \]
   while for $\sigma = \mathrm{St}_F$ we have $\varepsilon(\operatorname{As}(\mathrm{St}_E) \times \mathrm{St}_F)\omega_{E/F}(-1) = 1$ and $\varepsilon(\operatorname{As}(\Sigma_E) \times \mathrm{St}_F)\omega_{E/F}(-1) = -1$.
  \end{lemma}

  \begin{proof}
   If $\sigma$ is not a twist of Steinberg, then its Weil--Deligne representation has trivial monodromy action, so we compute that
   \[ \varepsilon(\operatorname{As}(\mathrm{St}_E) \times \sigma) = \varepsilon( (\sp(3) \oplus \omega_{E/F}) \times \sigma) = \varepsilon(\sigma \times \omega_{E/F}) \varepsilon(\sigma)^3 \det(-\operatorname{Frob}: \rho_{\sigma}^{I_F})^2.\]
   Since $\sigma$ has trivial central character, $\varepsilon(\sigma) = \pm 1$. If $\sigma$ is supercuspidal we are done, since in this case $\rho_{\sigma}^{I_F} = 0$. If $\sigma$ is principal series, then $\rho_{\sigma}^{I_F}$ must be either 0, or all of $\rho_\sigma$, since $\rho_{\sigma}$ has determinant 1. Thus $\det(-\operatorname{Frob}: \rho_{\sigma}^{I_F}) = 1$, so $\varepsilon(\operatorname{As}(\mathrm{St}_E) \times \sigma) = \varepsilon(\sigma) \varepsilon(\sigma \times \omega_{E/F})$, proving the claim in this case. The case when $\sigma$ is a twist of the Steinberg by a nontrivial (necessarily quadratic) character can be computed similarly.
  \end{proof}
  \addtocounter{theorem}{-2}
  \addtocounter{subtheorem}{1}
  \begin{subtheorem}
   \label{thm:asaiB}
   Let $\sigma$ be an irreducible generic representation of $\GL_2(F)$ with $\omega_{\sigma} = 1$. Then
   \begin{enumerate}[(i)]
    \item If $\varepsilon(\sigma) \varepsilon(\sigma \times \omega_{E/F}) = \omega_{E/F}(-1)$, then $\dim \Hom_H(\Sigma_E \boxtimes \sigma, \mathbb{1}) = 1$ and $\Hom_{H'}(\Sigma_E \boxtimes \sigma', \mathbb{1}) = 0$.
    \item If $\varepsilon(\sigma) \varepsilon(\sigma \times \omega_{E/F}) = -\omega_{E/F}(-1)$, then $\Hom_H(\Sigma_E \boxtimes \sigma, \mathbb{1}) = 0$ and $\dim \Hom_{H'}(\Sigma_E \boxtimes \sigma', \mathbb{1}) = 1$.
   \end{enumerate}
  \end{subtheorem}

  \begin{proof}
   We first consider the situation for $H'$. This case is relatively simple, since $H'$ acts transtively on $\mathbf{P}^1(E)$, with stabiliser isomorphic to $E^\times$. Consider the representation $I(s) \coloneqq I_E( |\cdot|_E^{s + 1/2}, |\cdot|_E^{-s-1/2})$ for a parameter $s$. Thus $I(0) = \Sigma_E$. For a generic $s$ these representations are irreducible, and we have $\varepsilon(\operatorname{As} I(s) \times \sigma) = \varepsilon(\sigma) \varepsilon(\sigma \times \omega_{E/F})$ (independently of $s$). On the other hand, using Mackey theory we see that the isomorphism class of $I(s)|_{H'}$ is independent of $s$ (it is isomorphic to $\operatorname{Ind}_{E^\times}^{H'}(\mathbb{1})$ for all $s$). So the dimension of the $\Hom$-space $\Hom_{H'}(I(s) \boxtimes \sigma, \mathbb{1})$ is independent of $s$. Hence, using the results of \cite{prasad92} for a generic non-zero $s$, we obtain the required formula for $s = 0$ also.

   We now consider $H$-invariant periods. We first prove the following claim:

   \noindent\emph{Claim}: If $\varepsilon(\sigma) \varepsilon(\sigma \times \omega_{E/F}) = \omega_{E/F}(-1)$, then $\Hom_H(\Sigma_E \times \sigma, \mathbb{1}) \ne 0$. \medskip

   If $\sigma$ is a principal-series representation, then we can construct an element of $\Hom_H(\Sigma_E \times \sigma, \mathbb{1})$ using zeta-integrals as before (using the fact that $\Sigma_E$, although reducible, has a well-defined Whittaker model).

   If $\sigma = \eta \cdot \mathrm{St}_F$ for a trivial or quadratic character $\eta$, the same zeta-integral argument shows that at least one of the spaces
   \[ \Hom_H(\Sigma_E \boxtimes \eta, \mathbb{1})\quad \text{and}\quad \Hom_H(\Sigma_E \boxtimes (\eta \cdot \mathrm{St}_F), \mathbb{1})\]
   must be non-zero. There is an exact sequence of $H$-representations
   \[ 0 \to \operatorname{cInd}_{E^\times}^{\GL_2(F)}(\mathbb{1}) \to \Sigma_E \to I_F(|\cdot|_F, |\cdot|_F^{-1}) \to 0,\]
   and $\Hom_H(\operatorname{cInd}_{E^\times}^{\GL_2(F)}(\mathbb{1}), \eta) = \Hom_{E^\times}(\mathbb{1}, \eta \circ \operatorname{Nm}_{E/F})$, so if $\eta$ is not $\mathbb{1}$ or $\omega_{E/F}$, then we must have $\Hom_H(\Sigma_E \boxtimes \eta, \mathbb{1}) = 0$ and hence $\Hom_H(\Sigma_E \boxtimes (\eta \cdot \mathrm{St}_F), \mathbb{1}) \ne 0$. On the other hand, if $\eta = \mathbb{1}$ or $\eta = \omega_{E/F}$, then $\varepsilon(\sigma) \varepsilon(\sigma \times \omega_{E/F}) = -\omega_{E/F}(-1)$ so there is nothing to prove.

   Finally, if $\eta$ is supercuspidal, then we obtain a long exact sequence
   \[ 0 \to \Hom_H(I_F(|\cdot|_F, |\cdot|_F^{-1}), \sigma^\vee) \to \Hom_H(\Sigma_E, \sigma^\vee) \to \Hom_{E^\times}(\sigma, \mathbb{1}) \to \operatorname{Ext}^1_H(I_F(|\cdot|_F, |\cdot|_F^{-1}), \sigma^\vee) \to \dots.\]
   and the first term is clearly 0. Under our hypotheses, $\Hom_{E^\times}(\sigma, \mathbb{1})$ is 1-dimensional; and for a supercuspidal $\sigma$, we have $\operatorname{Ext}^1_H(I_F(|\cdot|_F, |\cdot|_F^{-1}), \sigma^\vee) = 0$, since $\sigma^\vee$ and the principal-series representation lie in distinct Bernstein components of the category of smooth $H$-representations. Hence $\Hom_H(\Sigma_E, \sigma^\vee) = \Hom_{E^\times}(\sigma, \mathbb{1})$. This completes the proof of the claim.

   So it remains to prove the upper bound for the various spaces. Since $\sigma$ is irreducible and non-trivial, $\Hom_H(\Sigma_E \boxtimes \sigma, \mathbb{1})$ injects into $\Hom_H(\mathrm{St}_E \boxtimes \sigma, \mathbb{1})$; and similarly, if $\sigma \ne \mathrm{St}_F$, $\sigma'$ is either zero or non-trivial, and hence $\Hom_{H'}(\Sigma_E \boxtimes \sigma', \mathbb{1})$ injects into $\Hom_{H'}(\mathrm{St}_E \boxtimes \sigma', \mathbb{1})$. So we have
   \begin{multline*}
    \dim \Hom_H(\Sigma_E \boxtimes \sigma, \mathbb{1}) + \dim \Hom_{H'}(\Sigma_E \boxtimes \sigma', \mathbb{1}) \\
    \le \dim \Hom_H(\mathrm{St}_E \boxtimes \sigma, \mathbb{1}) + \dim \Hom_{H'}(\mathrm{St}_E \boxtimes \sigma', \mathbb{1}) = 1,
   \end{multline*}
   where the final equality comes from the irreducible case of the Main Theorem proved in \cite{prasad92}. This completes the proof of Theorem \ref{thm:asaiB} for $\sigma \ne \mathrm{St}_F$.

   It remains to consider the case $\sigma = \mathrm{St}_F$. We must show that the non-trivial $H$-invariant period of $\mathrm{St}_E \times \mathrm{St}_F$ does not extend to $\Sigma_E \times \mathrm{St}_F$. However, we can compute $\Sigma_E |_{\GL_2(F)}$ via Mackey theory, using the two orbits of $H$ on $\mathbf{P}^1(E)$: it is given by the exact sequence
    \[ 0 \to \operatorname{cInd}_{E^\times}^{\GL_2(F)}(\mathbb{1}) \to \Sigma_E \to I_F(|\cdot|_F, |\cdot|_F^{-1}) \to 0.\]
   The latter representation is irreducible and has no homomorphisms to $\mathrm{St}_F$; and we saw in the proof of Theorem \ref{thm:asaiA} that
   \[ \Hom_H(\operatorname{cInd}_{E^\times}^{H}(\mathbb{1}) \otimes \mathrm{St}_F, \mathbb{1}) = \Hom_{E^\times}(\mathrm{St}_F, \mathbb{1}) = 0.\]
   This shows that $\Hom_H(\Sigma_E \boxtimes \mathrm{St}_F, \mathbb{1}) = 0$, completing the proof.
  \end{proof}

  \begin{subtheorem}
   Let $\eta$ be a quadratic character of $F^\times$ (possibly trivial). Then we have $\dim \Hom_H(\Sigma_E \boxtimes \Sigma_F, \eta) = 1$.
  \end{subtheorem}

  \begin{proof}[Proof]
   The computation of the epsilon-factor is immediate; and by a zeta-integral argument as before, we can show that $\Hom_H(\Sigma_E \boxtimes \Sigma_F, \eta) \ne 0$ (since the representation $\Sigma_E$, despite being reducible, has a well-defined Whittaker model). So it suffices to show that the hom-space has dimension $\le 1$.

   If $\eta$ is not the trivial character, then $\Hom_H(\mathbb{1} \boxtimes \Sigma_F, \eta) = 0$, so the desired Hom-space injects into $\Hom_H(\mathrm{St}_E \boxtimes \Sigma_F, \eta)$, which is 1-dimensional by Theorem \ref{thm:asaiA}.

   If $\eta$ is trivial, then we have seen above that $\Hom_H(\Sigma_E \boxtimes \mathrm{St}_F, \mathbb{1})$ is zero. So $\Hom_H(\Sigma_E \boxtimes \Sigma_F, \mathbb{1}) = \Hom_H(\Sigma_E, \mathbb{1})$. From the Mackey decomposition of $\Sigma_E |_{\GL_2(F)}$ above, one sees easily that this space is 1-dimensional.
  \end{proof}

\section{Cubic fields}

 We briefly discuss the case where $A$ is a cubic extension of $F$.

 \begin{theorem}
  Let $\pi$ be a Whittaker-type representation of $\GL_2(E)$. Then the space $\Hom_H(\pi, \mathbb{1})$ has dimension 1 if $\varepsilon(\operatorname{As}(\pi))\omega_{A}(-1) = 1$ and is zero otherwise.
 \end{theorem}

 \begin{proof}The case of irreducible generic $\pi$ is proved in \cite{prasad92} assuming $\pi$ non-supercuspidal, and the supercuspidal case is filled in by \cite{prasadSP08}. In this case, the only example of a reducible Whittaker-type representation of $G$ is $\Sigma_E \otimes \eta$, where $\eta$ is a character of $E^\times$; and the central-character condition implies that $\lambda = \eta|_{F^\times}$ must be trivial or quadratic.

 The $\varepsilon$-factors $\varepsilon(\operatorname{As}(\mathrm{St}_E) \times \lambda)$ are computed in \cite[\S 8]{prasad92}. We find that $\varepsilon(\operatorname{As}(\Sigma_E) \times \lambda)\omega_{E/F}(-1)$ is always $+1$. On the other hand, $\varepsilon(\operatorname{As}(\mathrm{St}_E) \times \lambda) \omega_{E/F}(-1)$ is $+1$ if $\lambda$ is nontrivial quadratic, and $-1$ if $\lambda = 1$. So it follows that exactly one of $\Hom_H(\mathbb{1}, \lambda)$ and $\Hom_H(\mathrm{St}_E, \lambda)$ is non-zero, implying that $\dim \Hom_H(\Sigma_E \otimes \eta, \mathbb{1}) \le 1$.

 To complete the proof, we must show that when $\lambda \ne 1$, the $H$-invariant homomorphism $\Hom_H(\mathrm{St}_E, \lambda)$ extends to $\Sigma_E$. However, this is clear since the obstruction lies in $\operatorname{Ext}^1_H(\mathbb{1}, \lambda)$, which is zero.
 \end{proof}

 This completes the proof of the Main Theorem.

\section{An application to Euler systems}

 \renewcommand{\AA}{\mathbf{A}}
 \newcommand{\mathcal{I}}{\mathcal{I}}

 We now give a global application, a strengthening of some results from \cite{leiloefflerzerbes18} and \cite{grossi19} on Euler systems for quadratic Hilbert modular forms. Let $K / \mathbf{Q}$ be a real quadratic field and write $G = \operatorname{Res}_{K/\mathbf{Q}}(\GL_2)$ and $H = \GL_{2/\mathbf{Q}} \subset G$; set $G_f = G(\AA_{f}) = \GL_2(\AA_{K, f})$ and $H_f$ similarly.

 \subsection{Adelic representations}

 Let $\chi$ be a finite-order character of $\AA_f^\times$ and define a representation of $H_f$ by $\mathcal{I}(\chi) = \sideset{}{'}\bigotimes_\ell \mathcal{I}_\ell(\chi_\ell)$, where $\mathcal{I}_{\ell}(\chi_{\ell})$ denotes the representation of $H_\ell$ given by normalised induction of the character $\chi_{\ell} |\cdot|^{1/2} \boxtimes |\cdot|^{-1/2}$ of the Borel subgroup. For $\chi = 1$, we let $\mathcal{I}^0(1)$ denote the codimension 1 subrepresentation of $\mathcal{I}(1)$. Exactly as in \cite[\S 2]{harrisscholl01}, the local results above imply the following branching law for $G_f$-representations:

 \begin{proposition}
 \label{prop:semilocal}
  Let $\pi$ be an irreducible admissible representation of $G_f$, all of whose local factors are generic, with $\omega_{\pi} |_{\AA_f^\times} = \chi^{-1}$.
  \begin{itemize}
   \item We have $\dim \Hom_{H_f}(\pi \otimes \mathcal{I}(\chi), \mathbb{1}) = 1$.
   \item If $\chi = 1$ and there exists some $\ell$ such that $\Hom_{H_\ell}(\pi_\ell, \mathbb{1}) = 0$, then $\dim \Hom_{H_f}(\pi \otimes \mathcal{I}^0(1), \mathbb{1}) = 1$ and the natural restriction map $\Hom_{H_f}(\pi \otimes \mathcal{I}(1), \mathbb{1}) \to \Hom_{H_f}(\pi \otimes \mathcal{I}^0(1), \mathbb{1})$ is a bijection.
   \item If $\chi = 1$ and $\Hom_{H_\ell}(\pi_\ell, \mathbb{1}) \ne 0$ for all $\ell$, then $\dim \Hom_{H_f}(\pi \otimes \mathcal{I}^0(1), \mathbb{1}) = \infty$.
  \end{itemize}
 \end{proposition}

 \subsection{Hilbert modular forms}

  Suppose now that $\pi$ is (the finite part of) a cuspidal automorphic representation, arising from a Hilbert modular cusp form of parallel weight $k+2 \ge 2$, normalised so that $\omega_{\pi}$ has finite order.

  \begin{proposition}\label{prop:giada}
   Suppose $\pi$ is not a twist of a base-change from $\GL_{2/\mathbf{Q}}$. Then, for any Dirichlet character $\tau$, there exist infinitely many primes $\ell$ such that $\Hom_{H_\ell}\left(\pi_\ell \otimes \tau_\ell, \mathbb{1}\right) = 0$.
  \end{proposition}

  \begin{proof}
   See \cite[Proposition 7.2.5]{grossi19}.
  \end{proof}

  There is a natural $H_f$-representation $\mathcal{O}^\times(Y)_{\mathbf{C}}$ of \emph{modular units}, where $Y$ is the infinite-level modular curve (the Shimura variety for $\GL_2$). Note that this representation is smooth, but not admissible. It fits into a long exact sequence
  \[ 0 \to (\mathbf{Q}^{\mathrm{ab}})^\times \otimes \mathbf{C} \to \mathcal{O}^\times(Y)_{\mathbf{C}} \to \mathcal{I}^0(1) \oplus \bigoplus_{\eta \ne 1}\mathcal{I}(\eta) \to 0,\]
  with $H_f$ acting on $(\mathbf{Q}^{\mathrm{ab}})^\times$ via the Artin reciprocity map of class field theory, and the sum is over all even Dirichlet characters $\eta$.

  There is a canonical homorphism, the \emph{Asai--Flach map}, constructed in \cite{leiloefflerzerbes18} (building on several earlier works such as \cite{Kings-Higher-Regulators}):
  \[ \mathcal{AF}^{[\pi, k]}: \left( \pi \otimes \mathcal{O}^\times(Y)_{\mathbf{C}} \right)_{H_f} \to H^1(\mathbf{Q}, V^{\mathrm{As}}(\pi)^*(-k)),  \]
  where $V^{\mathrm{As}}(\pi)$ is the Asai Galois representation attached to $\pi$, and we have fixed an isomorphism $\overline{\mathbf{Q}}_p \cong \mathbf{C}$. The subscript $H_f$ indicates $H_f$-coinvariants.

  \begin{theorem}
   Suppose $\pi$ is not a twist of a base-change from $\mathbf{Q}$. Then the Asai--Flach map factors through $\pi \otimes \mathcal{I}(\chi)$, and its image is contained in a 1-dimensional subspace of $H^1(\mathbf{Q}, V^{\mathrm{As}}(\pi)^*(-k))$.
  \end{theorem}

 \begin{proof}
  Using \cref{prop:giada}, we see that $\mathcal{AF}^{[\pi, k]}$ must vanish on $(\mathbf{Q}^{\mathrm{ab}})^\times \otimes \mathbf{C}$, so it factors through $\pi \otimes \mathcal{I}(\chi)$ if $\chi \ne 1$, or $\pi \otimes \mathcal{I}^0(\chi)$ if $\chi = 1$, where $\chi = (\omega_{\pi} |_{\AA_f^\times})^{-1}$ as above. Using \cref{prop:semilocal}, combined with a second application of \cref{prop:giada} if $\omega_{\pi}$ is trivial on $\mathbf{Q}$, the result follows.
 \end{proof}

 As in the $\GSp_4$ case described in \cite[\S 6.6]{LZ20b-regulator}, one can remove the dependency on the test data entirely: using zeta-integrals, we can construct a canonical basis vector $Z_{\mathrm{can}} \in \Hom(\pi_f \otimes \mathcal{I}(\chi), \mathbb{1})$, and define $\mathcal{AF}_{\mathrm{can}}^{[\pi, k]} \in H^1\left(\mathbf{Q}, V^{\mathrm{As}}(\pi)^*(-k)\right)$ as the unique class such that $\mathcal{AF}^{[\pi, k]} = Z_{\mathrm{can}} \cdot \mathcal{AF}_{\mathrm{can}}^{[\pi, k]}$. We hope that this perspective may be useful in formulating and proving explicit reciprocity laws in the Asai setting.

 \begin{remark}
  The constructions of \cite{leiloefflerzerbes18} also apply to other twists of $V^{\mathrm{As}}(\pi)$, and to Hilbert modular forms of non-parallel weight; but in these other cases the input data for the Asai--Flach map lies in an irreducible principal series representation of $H_f$, so the necessary multiplicity-one results are standard. (The delicate cases are those which correspond to near-central values of $L$-series.)
 \end{remark}

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