# Decays of Standard Model like Higgs boson $h\rightarrow\gamma\gamma,Z\gamma$
in a minimal left-right symmetric model

T.T. Hong<EMAIL_ADDRESS>An Giang University, VNU - HCM, Ung Van Khiem
Street, Long Xuyen, An Giang, Vietnam  V. K. Le
<EMAIL_ADDRESS>Binh Thuy Junior High School, Bui Huu Nghia
Street, Binh Thuy Ward, Binh Thuy District, Can Tho City, Vietnam  L. T. T.
Phuong<EMAIL_ADDRESS>An Giang University, VNU - HCM, Ung Van Khiem
Street, Long Xuyen, An Giang, Vietnam  N .C. Hoi<EMAIL_ADDRESS>An Giang
University, VNU - HCM, Ung Van Khiem Street, Long Xuyen, An Giang, Vietnam  N.
T. K. Ngan<EMAIL_ADDRESS>Department of Physics, Can Tho University, 3/2
Street, Can Tho, Vietnam  N. H. T. Nha 111Corresponding author
<EMAIL_ADDRESS>Subatomic Physics Research Group, Science and
Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam
Faculty of Applied Technology, School of Engineering and Technology, Van Lang
University, Ho Chi Minh City, Vietnam

###### Abstract

Two decay channels $h\rightarrow\gamma\gamma,Z\gamma$ of the Standard Model-
like Higgs in a left-right symmetry model are investigated under recent
experimental data. We will show there exist one-loop contributions that affect
the $h\rightarrow Z\gamma$ amplitude, but not the $h\rightarrow\gamma\gamma$
amplitude. From numerical investigations, we show that the signal strength
$\mu_{Z\gamma}$ of the decay $h\rightarrow Z\gamma$ is still constrained
strictly by that of $h\rightarrow\gamma\gamma$, namely
$|\Delta\mu_{\gamma\gamma}|<38\%$ results in max $|\Delta\mu_{Z\gamma}|<40\%$.
On the other hand, the future experimental sensitivity
$|\Delta\mu_{\gamma\gamma}|=4\%$ still allows $|\Delta\mu_{Z\gamma}|$ reaches
to values of $30\%$, which are larger than the expected sensitivity that
$|\Delta\mu_{Z\gamma}|=23\%$.

##  I  Introduction

The standard model-like (SM-like) Higgs decay $h\to Z\gamma$ is one of the
most important channels being searched at the experimental center CMS:2022dwd
. Meanwhile, the experimental evidence of this loop-induced decay relating to
the effective coupling $hZ\gamma$ has been reported by ATLAS and CMS recently
CMS:2022ahq ; CMS:2023mku , agreement with the SM prediction within 1.9
standard deviation. Experimental data shows that the effective coupling
$h\gamma\gamma$ derived from $h\to\gamma\gamma$ decay rates is constrained
very strictly CMS:2021kom . In contrast, the effective coupling $hZ\gamma$ in
many models beyond SM (BSM) might differ considerably from the SM prediction,
because the couplings of $Z$ to new particles are less strict than those of
the photon. Hence, studying the effective $hZ\gamma$ couplings will be a
indirect channel to determine properties of new particles. Controlled the by
strict experimental constraint of the decay $h\to\gamma\gamma$, constraints of
the SM-like Higgs decays $h\to Z\gamma$ affected by new fermions and charged
scalars were studied in several BSMs such as 3-3-1 models Yue:2013qba ;
Hung:2019jue , only Higgs extended SM versions Fontes:2014xva ;
Yildirim:2021kqs ; Benbrik:2022bol ; Hue:2023tdz , $U(1)$ gauge extensions
from SM Wang:2022fug ; Tran:2023vgk , supersymmetric models Archer-
Smith:2020gib ; Liu:2020nsm , $\dots$. Previous studies $h\to Z\gamma$ in
left-right symmetric models ignored one-loop contributions relating to the
diagrams consisting of both virtual Higgs and gauge particles in the loops
Martinez:1990ye ; Martinez:1989kr , where the $h$-Higgs-gauge boson couplings
were assumed to be suppressed.

The experimental results have been updated for loop-induced Higgs decays
$h\rightarrow\gamma\gamma$ Aaboud:2018ezd ; Sirunyan:2018ouh ; Aaboud:2018xdt
and $h\rightarrow Z\gamma$ Aaboud:2017uhw . In the future project, the
significant strength of the $h\rightarrow Z\gamma$, denoted as
$\mu_{Z\gamma}$, can reach $\Delta\mu_{Z\gamma}\equiv\mu_{Z\gamma}-1=\pm
0.23$, while that of the channel $h\rightarrow\gamma\gamma$ can reach around
$\Delta\mu_{\gamma\gamma}\equiv\mu_{\gamma\gamma}-1=\pm 0.04$ from two CMS and
ATLAS experiments Cepeda:2019klc . In addition, the ATLAS expected
significance at HL-LHC to the $h\rightarrow\,Z\gamma$ channel will be
$4.9~{}\sigma$ with $3000~{}\mathrm{fb}^{-1}$. Also, the Circular Electron
Positron Collider (CEPC) An:2018dwb  can reach a sensitivity of
$\mu_{Z\gamma}=1\pm 0.22$ Antonov:2022zwm .

One interesting extension of the beyond the SM models is an extension the
lepton sector. Namely, the minimal left-right version (MLRSM) is constructed
based on the parity symmetry $SU{(2)_{L}}\otimes SU(2)_{R}\otimes U(1)_{B-L}$
Pati:1974yy ; Mohapatra:1974gc ; Senjanovic:1975rk , which contains Higgs
fields included in two $SU(2)_{L}$ triplets denoted as $\Delta_{L,R}$, apart
from a bi-doublet field $\Phi$ playing the SM Higgs role. Therefore, the MLRSM
allows us to solve the parity problem of the SM as well as the neutrino
oscillation data through the seesaw mechanism. Besides, it contains extended
particles which may result in interesting consequences for rare decays such as
Higgs boson decay $h\to Z\gamma$.

This work is organized as follows. In section II, we present the overview of
the MLRSM, including the particle content and physical states. In Sec. III, We
present the necessary couplings that generate one-loop contributions to the
decays $h\rightarrow\gamma\gamma,Z\gamma$. We also collected analytic formulas
to determine the decay rates, reminding some new contributions which were not
discussed previously. Numerical results will be investigated in Sec. IV.
Namely, we will investigate the dependence of $\mu_{Z\gamma}^{\mathrm{MLRSM}}$
on several important parameters in this model. Finally, the summary is given
in the last Sec. IV.

##  II The minimal left-right symmetry model

###  II.1 The model review

All needed ingredients relevant to one-loop contributions to the decay
amplitudes $h\to Z\gamma,\gamma\gamma$ will be collected in this section. In
most general, the electric charge operator can be written as Zhang:2007da ;
Lee:2017mfg

$\displaystyle Q=T^{R}_{3}+T^{L}_{3}+\dfrac{B-L}{2}=T^{L,R}_{3}+\dfrac{Y}{2},$
(1)

where $T^{L,R}_{i}$ are the generators of the gauge groups $SU(2)_{L,R}$; $B$
($L$) is the baryon (lepton) number defining the $U(1)_{B-L}$ group in the
MLRSM. The baryon and lepton number of the fermions can be written in the
table 1.

$f$ | $e,\mu,\tau,\nu_{e},\nu_{\mu},\nu_{\tau}$ | $u,c,t,s$
---|---|---
$L$ | 1 | 0
$B$ | 0 | $\dfrac{1}{3}$
Table 1: The baryons and the leptons number of the fermions in MLRSM

With this information, we can write down the lepton representations as follows

${L^{\prime}_{Li}}=\begin{pmatrix}\nu^{\prime}_{Li}\\\ l^{\prime}_{Li}\\\
\end{pmatrix}\thicksim(2,1,-1),\
{L^{\prime}_{Ri}}=\begin{pmatrix}\nu^{\prime}_{Ri}\\\ l^{\prime}_{Ri}\\\
\end{pmatrix}\thicksim(1,2,-1),$ (2)
${Q^{\prime}_{Li}}=\begin{pmatrix}u^{\prime}_{Li}\\\ d^{\prime}_{Li}\\\
\end{pmatrix}\thicksim(2,1,\dfrac{1}{3}),\
{Q^{\prime}_{Ri}}=\begin{pmatrix}u^{\prime}_{Ri}\\\ d^{\prime}_{Ri}\\\
\end{pmatrix}\thicksim(1,2,\dfrac{1}{3}),$ (3)

where $i$ is the flavour index.

Gauge boson and fermion masses are originated from the following scalar
sector, consisting of a bi-doublet and two triplet scalar fields
$\Delta_{L,R}$ satisfying

$\Phi=\begin{pmatrix}\phi^{0}_{1}&\phi^{+}_{2}\\\ \phi^{-}_{1}&\phi^{0}_{2}\\\
\end{pmatrix},~{}\widetilde{\Phi}=\sigma_{2}\Phi^{*}\sigma_{2}\thicksim(2,2,0),\;\Delta_{L,R}=\begin{pmatrix}\dfrac{\delta^{+}_{L,R}}{\sqrt{2}}&\delta^{++}_{L,R}\\\
\delta^{0}_{L,R}&-\dfrac{\delta^{+}_{L,R}}{\sqrt{2}}\\\
\end{pmatrix}\thicksim(3,1,2).$ (4)

The Higgs components develop vacuum expectation values (VEV) defined as

$\langle\Phi\rangle=\begin{pmatrix}\langle\phi^{0}_{1}\rangle&0\\\
0&\langle\phi^{0}_{2}\rangle\\\ \end{pmatrix};\
\langle\Delta_{L,R}\rangle=\begin{pmatrix}0&0\\\
\langle\delta^{0}_{L,R}\rangle&0\\\ \end{pmatrix},$ (5)

where the neutral Higgs components are expanded as follows

$\displaystyle\phi^{0}_{i}$
$\displaystyle=\langle\phi^{0}_{i}\rangle+\frac{r_{i}+ia_{i}}{\sqrt{2}},\langle\phi^{0}_{1}\rangle=\frac{k_{1}}{\sqrt{2}},\;\langle\phi^{0}_{2}\rangle=\frac{k_{2}e^{i\alpha}}{\sqrt{2}},\;i=1,2;$
$\displaystyle\delta^{0}_{L,R}$
$\displaystyle=\langle\delta^{0}_{L,R}\rangle+\frac{v_{L,R}e^{i\theta_{L,R}}+r_{L,R}+ia_{L,R}}{\sqrt{2}},\;\langle\delta^{0}_{L}\rangle=\frac{v_{L}e^{i\theta_{L}}}{\sqrt{2}},\;\langle\delta^{0}_{R}\rangle=\frac{v_{R}}{\sqrt{2}}.$
(6)

The symmetry breaking pattern in MLRSM happens in two following steps:
$SU(2)_{L}\otimes SU(2)_{R}\otimes U(1)_{B-L}\xrightarrow{v_{R}\neq
0,k_{1},k_{2},v_{L}=0}SU(2)_{L}\otimes
U(1)_{Y}\xrightarrow{k_{1},k_{2},v_{L}\neq 0}U(1)_{Q}$, which corresponds to
the reasonable limits that $v_{R}\gg k_{1},k_{2}\gg v_{L}$. Ony new gauge
bosons will be massive after the first step. The second step is the SM
symmetry-breaking, which generate masses for the SM particles. When the
symmetry is broken to the step two, only $U(1)_{Q}$ remains unbroken, where
$Q$ is the quantifier. As a result, the photon $A_{\mu}$ has no mass. We
stress that the MLRSM contains not more than the three scalar multiplets
($\phi,\Delta_{L,R}$). Physical spectrum and masses of all particles in the
model under consideration are summarized as follows.

###  II.2 Fermions

Physical fermion state and their masses always relate to the Yukawa
interactions, which are included in the following Lagrangians for leptons and
quarks:

$\displaystyle\mathcal{L}_{lepton}^{Y}$
$\displaystyle=-\overline{L^{\prime}_{Li}}\left(f^{e}_{ij}\Phi+\widetilde{f}^{e}_{ij}\widetilde{\Phi}\right)L^{\prime}_{Rj}-\sum_{X=L,R}Y^{e}_{X,ij}\overline{L^{{}^{\prime}c}_{Xi}}i\sigma_{2}\Delta_{X}L^{\prime}_{Xj}+\text{h.c},$
$\displaystyle\mathcal{L}^{Y}_{\mathrm{quark}}$
$\displaystyle=-\overline{Q^{\prime}_{Li}}\left(f^{q}_{ij}\Phi+\widetilde{f}^{q}_{ij}\widetilde{\Phi}\right)Q^{\prime}_{Rj}+\text{h.c.}.$
(7)

Then, the mass terms for leptons and quarks are computed. We will use the
results for fermion masses and mixing presented in Refs. Zhang:2007da ;
Lee:2017mfg , i.e. all the original and the physical states of fermions are
the same. They are identified with the SM ones and will be denoted as
$e_{aL,R},u_{aL,R}$ and $d_{aL,R}$ in this work.

###  II.3 Gauge bosons

The covariant derivative corresponding to the symmetry of the MLRSM is defined
as Zhang:2007da

$D_{\mu}=\partial_{\mu}-ig_{L}\sum_{i=1}^{3}T^{a}_{L}W^{a}_{L\nu}+ig_{R}\sum_{i=1}^{3}T^{a}_{R}W^{a}_{R\nu}-ig^{\prime}\dfrac{B-L}{2}B_{\mu},$
(8)

where $g_{L,R}$ and $g^{\prime}$ are the $SU(2)_{L,R}$ and $U(1)_{B-L}$ gauge
couplings, respectively.

The Lagrangian for scalar kinetic is written as

$\displaystyle\mathcal{L}_{s}=$
$\displaystyle\sum_{S=\Phi,\Delta_{L,R}}\mathcal{L}_{S}$ $\displaystyle=$
$\displaystyle\sum_{S=\Phi,\Delta_{L,R}}\mathrm{Tr}\left[\left(D_{\mu}S\right)^{\dagger}\left(D^{\mu}S\right)\right]$
(9)

with the particular forms of covariant derivatives to the scalar multiplets

$\displaystyle D_{\mu}\Phi$ $\displaystyle=\partial_{\mu}\Phi-
ig_{L}\dfrac{\sigma^{b}}{2}W^{b}_{L\nu}\Phi+ig_{R}\Phi\dfrac{\sigma^{a}}{2}W^{a}_{R},$
$\displaystyle D_{\mu}\Delta_{X}$
$\displaystyle=\partial_{\mu}\Delta_{X}-i\frac{g_{X}}{2}\left(\sigma^{a}\Delta_{X}-\Delta_{X}\sigma^{a}\right)W^{a}_{X}-ig^{\prime}I_{2}B_{\mu}\Delta_{X},$
(10)

where $X=L,R$, $\sigma^{a}$ is the Pauli matrix corresponding to the $SU(2)$
doublet representation of $T^{L,R}_{a}$. Therefore, the mass terms of gauge
bosons are derived from the vev of Higgs components

$\displaystyle\langle\mathcal{L}_{s}\rangle=$
$\displaystyle\sum_{S=\Phi,\Delta_{L,R}}\langle\mathcal{L}_{S}\rangle$
$\displaystyle=$
$\displaystyle\dfrac{1}{8}\left(k^{2}+4v^{2}_{L}\right)g^{2}_{L}W^{3\mu}_{L}W^{3}_{L\mu}-\dfrac{1}{4}g_{L}g_{R}k^{2}W^{3\mu}_{L}W^{3}_{R\mu}$
$\displaystyle+\dfrac{1}{8}\left(k^{2}+4v^{2}_{R}\right)g^{2}_{R}W^{3\mu}_{R}W^{3}_{R\mu}-g_{L}g^{\prime}W^{3\mu}_{L}B^{\mu}-g_{R}g^{\prime}v^{2}_{R}W^{3\mu}_{R}B_{\mu}$
$\displaystyle+\dfrac{1}{2}g^{{}^{\prime}2}(v^{2}_{L}+v^{2}_{R})B^{\mu}B_{\mu}+\dfrac{1}{4}g^{2}_{L}\left(k^{2}+2v^{2}_{L}\right)W^{+\mu}_{L}W^{-}_{L\mu}-\dfrac{1}{2}g_{L}g_{R}k_{1}k_{2}e^{i\alpha}W^{+\mu}_{L}W^{-}_{R\mu}$
$\displaystyle-\dfrac{1}{2}g_{L}g_{R}k_{1}k_{2}e^{-i\alpha}W^{+\mu}_{L}W^{-}_{R\mu}+\dfrac{1}{4}g^{2}_{R}\left(k^{2}+2v^{2}_{R}\right)W^{+\mu}_{R}W^{-}_{R\mu},$
(11)

where $k^{2}\equiv k_{1}^{2}+k_{2}^{2}$. The mass terms of the neutral and
charged gauge bosons read:

$\displaystyle\mathcal{L}^{mass}_{g}$
$\displaystyle=\dfrac{1}{2}(W^{3\mu}_{L}W^{3\mu}_{R}B^{\mu})\begin{pmatrix}\dfrac{1}{4}\left(k_{1}^{2}+k_{2}^{2}+4v^{2}_{L}\right)g^{2}_{L}&-\dfrac{1}{4}g_{L}g_{R}(k^{2}_{1}+k^{2}_{2})&-g_{R}g^{\prime}v^{2}_{L}\\\
-\dfrac{1}{4}g_{L}g_{R}(k^{2}_{1}+k^{2}_{2})&\dfrac{1}{4}\left(k_{1}^{2}+k_{2}^{2}+4v^{2}_{R}\right)g^{2}_{R}&-g_{R}g^{\prime}v^{2}_{R}\\\
-g_{R}g^{\prime}v^{2}_{L}&-g_{R}g^{\prime}v^{2}_{R}&g^{{}^{\prime}2}(v^{2}_{L}+v^{2}_{R})\\\
\end{pmatrix}\begin{pmatrix}W^{3}_{L\mu}\\\ W^{3}_{R\mu}\\\ B_{\mu}\\\
\end{pmatrix}$
$\displaystyle+(W^{+\mu}_{L}W^{+\mu}_{R})\begin{pmatrix}\dfrac{1}{4}\left(k_{1}^{2}+k_{2}^{2}+2v^{2}_{L}\right)g^{2}_{L}&-\dfrac{1}{2}g_{L}g_{R}k_{1}k_{2}e^{i\alpha}\\\
-\dfrac{1}{2}g_{L}g_{R}k_{1}k_{2}e^{-i\alpha}&\dfrac{1}{4}\left(k_{1}^{2}+k_{2}^{2}+2v^{2}_{R}\right)g^{2}_{R}\\\
\end{pmatrix}\begin{pmatrix}W^{-}_{L\mu}\\\ W^{-}_{R\mu}\\\ \end{pmatrix},$
(12)

where $W^{\pm}_{X\mu}\equiv\dfrac{1}{\sqrt{2}}\left(W^{1}_{X\mu}\mp
iW^{2}_{X\mu}\right)$ with $X=L,R$. The mixing angle $\xi$ between two singly
charged gauge bosons $W^{\pm}_{L}$ and $W^{\pm}_{R}$ is determined by the
following formula

$\displaystyle\tan
2\xi=\dfrac{-4g_{L}g_{R}k_{1}k_{2}}{(g^{2}_{R}-g^{2}_{L})(k_{1}^{2}+k_{2}^{2})+2(g^{2}_{R}v^{2}_{R}-g^{2}_{L}v^{2}_{L})}.$
(13)

Using the approximation that $\tan 2\xi\ll 1\Rightarrow\tan 2\xi\approx\sin
2\xi\approx 2\sin\xi\approx 2\xi$, and $v_{L}\ll k_{1},k_{2}\ll v_{R}$, the
$W_{L}-W_{R}$ mixing angle $\xi$ is

$\displaystyle 2\xi$
$\displaystyle=\dfrac{-2g_{L}k_{1}k_{2}}{g_{R}v^{2}_{R}},\;\dfrac{k^{2}}{v^{2}_{R}}\approx
x\ll 1,\ \dfrac{v^{2}_{L}}{v^{2}_{R}}\approx 0,\
\dfrac{v^{2}_{L}}{k^{2}}\approx x_{L}\ll 1.$ (14)

The singly charged gauge bosons $W_{L,R}^{\pm}$ can be written as functions of
the mass basis ($W_{1}^{\pm},W_{2}^{\pm}$) as follows

$\displaystyle\begin{pmatrix}W^{\pm}_{L}\\\ W^{\pm}_{R}\\\
\end{pmatrix}=\begin{pmatrix}c_{\xi}&-s_{\xi}e^{i\alpha}\\\
s_{\xi}e^{-i\alpha}&c_{\xi}\\\ \end{pmatrix}\begin{pmatrix}W^{\pm}_{1}\\\
W^{\pm}_{2}\\\ \end{pmatrix},$ (15)

where $c_{\xi}\equiv\cos\xi$ and $s_{\xi}\equiv\sin\xi$. The respective
charged gauge boson masses are found to be

$\displaystyle m^{2}_{W_{1}}\approx\dfrac{1}{4}k^{2}g^{2}_{L},\quad
m^{2}_{W_{2}}\approx\dfrac{1}{2}g^{2}_{R}v^{2}_{R}.$ (16)

Identifying the $W^{\pm}_{1}\equiv W^{\pm}$ in the SM, we get
$k_{1}^{2}+k_{2}^{2}=k^{2}\equiv v^{2}=(246\;\mathrm{GeV})^{2}$.

The original neutral gauge basis $(W^{3}_{L\mu},W^{3}_{R\mu},B_{\mu})$ are
expressed in terms of the mass basis $(A_{\mu},Z_{1\mu},Z_{2\mu})$ as follows

$\displaystyle\begin{pmatrix}W_{L\mu}^{3}\\\ W_{R\mu}^{3}\\\ B_{\mu}\\\
\end{pmatrix}=C^{T}\begin{pmatrix}A_{\mu}\\\ Z_{1\mu}\\\ Z_{2\mu}\\\
\end{pmatrix},$ (17)

where

$\displaystyle C=\begin{pmatrix}s_{z_{2}}&c_{R}c_{z_{2}}&c_{z_{2}}s_{R}\\\
c_{z_{2}}c_{z_{3}}&-c_{R}c_{z_{3}}s_{z_{2}}-s_{R}s_{z_{3}}&-c_{z_{3}}s_{R}s_{z_{2}}+c_{R}s_{z_{3}}\\\
-c_{z_{2}}s_{z_{3}}&-c_{z_{3}}s_{R}+c_{R}s_{z_{2}}s_{z_{3}}&c_{R}c_{z_{3}}+s_{R}s_{z_{2}}s_{z_{3}}\\\
\end{pmatrix}$

and the mixing angles $t_{R},t_{z_{2}},t_{{z}_{3}}$ are given by

$\displaystyle t_{R}=\frac{g_{R}}{g^{\prime}},\quad
t_{z_{2}}=\frac{g_{R}g^{\prime}}{g_{L}\sqrt{g^{\prime 2}+g_{R}^{2}}},\quad
t_{{2z}_{3}}=\dfrac{-g_{R}^{2}k^{2}\sqrt{g_{L}^{2}g^{{}^{\prime}2}+g_{L}^{2}g_{R}^{2}+g^{{}^{\prime}2}g_{R}^{2}}}{2v_{R}^{2}(g_{R}^{2}+g^{{}^{\prime}2})}.$
(19)

State synchronization with the SM as follows: $W^{3}_{L\mu}\equiv
W^{3}_{\mu}$, $Z_{1\mu}\equiv Z_{\mu}$ in the limits $c_{z_{3}}\to 1$, then we
also have

$\displaystyle s_{z_{2}}\equiv s_{W},\;c_{z_{2}}\equiv c_{W}\Rightarrow
t_{W}=t_{z_{2}}=s_{R}t^{\prime}.$ (20)

The Weinberg angle $\theta_{W}$ is identified from the definition
$\cos\theta_{W}\equiv c_{W}\equiv\frac{m_{W_{1}}}{m_{Z_{1}}}\thickapprox
c_{z_{2}}$. Then, the neutral gauge boson masses of $Z_{1},Z_{2},$ and the
photon $A$ are given by

$\displaystyle m^{2}_{Z_{1}}$
$\displaystyle\thickapprox\dfrac{g^{2}_{L}v^{2}}{4c^{2}_{W}},~{}m^{2}_{Z_{2}}\thickapprox\dfrac{g^{2}_{R}v^{2}_{R}}{1-(g^{2}_{L}/g^{2}_{R})t^{2}_{W}},\text{and}~{}m_{A}^{2}=0.$
(21)

In addition, $Z_{1}\equiv Z$, and $Z_{2}\equiv Z^{\prime}$ are respectively SM
gauge boson $Z$ found experimentally, and the heavy one appearing in the
MLRSM.

###  II.4 Higgs bosons

The MLRSM scalar potential is written as Zhang:2007da

$\displaystyle V_{h}$
$\displaystyle=-\mu^{2}_{\Phi}\mathrm{Tr}[\Phi^{\dagger}\Phi]-\mu^{\prime
2}_{\Phi}(\mathrm{Tr}[\Phi^{\dagger}\widetilde{\Phi}]+\mathrm{Tr}[\widetilde{\Phi}^{\dagger}\Phi])-\mu^{2}_{\Delta}\sum_{X=L,R}\mathrm{Tr}[\Delta^{\dagger}_{X}\Delta_{X}]$
$\displaystyle{+\lambda_{1}(\mathrm{Tr}[\Phi^{\dagger}\Phi])^{2}+\lambda_{2}\Big{(}(\mathrm{Tr}[\Phi^{\dagger}\widetilde{\Phi}])^{2}+(\mathrm{Tr}[\widetilde{\Phi}^{\dagger}\Phi])^{2}\Big{)}}+\lambda_{3}(\mathrm{Tr}[\Phi^{\dagger}\widetilde{\Phi}]\mathrm{Tr}[\widetilde{\Phi}^{\dagger}\Phi])$
$\displaystyle+\lambda_{4}\mathrm{Tr}[\Phi^{\dagger}\Phi](\mathrm{Tr}[\Phi^{\dagger}\widetilde{\Phi}]+\mathrm{Tr}[\widetilde{\Phi}^{\dagger}\Phi])+\rho_{1}(\mathrm{Tr}[\Delta^{\dagger}_{L}\Delta_{L}]^{2}+\mathrm{Tr}[\Delta^{\dagger}_{R}\Delta_{R}]^{2})$
$\displaystyle+\rho_{2}\sum_{X=L,R}(\mathrm{Tr}[\Delta^{\dagger}_{X}\Delta^{\dagger}_{X}]{\mathrm{Tr}[\Delta_{X}\Delta_{X}]})+\rho_{3}(\mathrm{Tr}[\Delta^{\dagger}_{L}\Delta_{L}]\mathrm{Tr}[\Delta^{\dagger}_{R}\Delta_{R}])$
$\displaystyle+\sum_{X\neq
Y=L,R}\rho_{4}(\mathrm{Tr}[\Delta^{\dagger}_{X}\Delta^{\dagger}_{X}]\mathrm{Tr}[\Delta_{Y}\Delta_{Y}])+\alpha_{1}\mathrm{Tr}[\Phi^{\dagger}\Phi]\sum_{X=L,R}(\mathrm{Tr}[\Delta^{\dagger}_{X}\Delta_{X}])$
$\displaystyle+\left\\{\alpha_{2}e^{i\delta_{2}}\sum_{X=L,R}(\mathrm{Tr}[\Phi^{\dagger}\widetilde{\Phi}]\mathrm{Tr}[\Delta^{\dagger}_{X}\Delta_{X}])+\mathrm{h.c.}\right\\}+\sum_{X=L,R}\left(\alpha_{3}\mathrm{Tr}[\Phi\Phi^{\dagger}\Delta_{X}\Delta^{\dagger}_{X}]\right)$
$\displaystyle+\left\\{\alpha_{4}\mathrm{Tr}[\Phi^{\dagger}\Delta^{\dagger}_{L}\Phi\Delta_{R}]+\alpha_{5}\mathrm{Tr}[\Phi^{\dagger}\Delta^{\dagger}_{L}\widetilde{\Phi}\Delta_{R}]++{\alpha_{6}}\mathrm{Tr}[\widetilde{\Phi}^{\dagger}\Delta^{\dagger}_{L}\Phi\Delta_{R}]+\mathrm{h.c.}\right\\}.$
(22)

From the minimal conditions of the Higgs potential given in Eq. (II.4), three
parameters $\mu_{\Phi}^{2}$, $\mu^{\prime 2}_{\Phi}$, and $\mu_{\Delta}^{2}$
are expressed as functions of other independent parameters. Inserting them
into the Higgs potential (II.4), we can determine all Higgs boson masses and
physical states. Firstly, the original and the mass base of neutral CP-even
Higgs bosons are related to each other as follows

$\displaystyle\begin{pmatrix}h_{0}\\\ H^{0}_{1}\\\
\end{pmatrix}=\left(\begin{array}[]{cc}s_{\beta}&c_{\beta}\\\
-c_{\beta}&s_{\beta}\\\
\end{array}\right)\begin{pmatrix}\sqrt{2}Re(\phi^{0}_{1})\\\
\sqrt{2}Re(\phi^{0}_{2})\\\ \end{pmatrix}.$ (25)

In the limit $k_{1}\gg k_{2}$, therefore
$t_{\beta}=\dfrac{k_{1}}{k_{2}}\approx\dfrac{1}{\epsilon_{2}}\gg 1$,
$c_{\beta}=\dfrac{1}{\sqrt{t^{2}_{\beta}+1}}\approx\dfrac{1}{t_{\beta}}\approx\epsilon_{2}$,
$s_{\beta}\approx 1$, and $\epsilon_{1}\equiv k_{1}/v_{R}$,
$\epsilon_{2}\equiv k_{2}/k_{1}$. Besides that from (25) we get the same
result as in Ref. Zhang:2007da . In this study, the SM-like Higgs mass is
calculated approximately to the order $\epsilon^{2}=v^{2}/v_{R}^{2}$, namely

$\displaystyle m^{2}_{h^{0}}=$
$\displaystyle\left[2\lambda_{1}+8c_{\beta}^{2}s_{\beta}^{2}\left(2\lambda_{2}+\lambda_{3}\right)+8c_{\beta}s_{\beta}\lambda_{4}\right]v^{2}$
$\displaystyle-\frac{8\left(2c_{\beta}^{2}-1\right)^{3}v^{4}}{\alpha_{3}v_{R}^{2}}\left[4c_{\beta}^{4}(2\lambda_{2}+\lambda_{3})^{2}-4s_{\beta}c_{\beta}\lambda_{4}(2\lambda_{2}+\lambda_{3})-4c_{\beta}^{2}(2\lambda_{2}+\lambda_{3})^{2}-\lambda_{4}^{2}\right].$
(26)

The SM-like Higgs property appears in Eq. (II.4) as $m^{2}_{h^{0}}\simeq
v^{2}\times\mathcal{O}(\frac{v^{2}}{v_{R}^{2}})\sim v^{2}$ because
$\mathcal{O}(\epsilon^{2})\simeq 0$ when $v^{2}\ll v_{R}^{2}$. In this limit,
$h^{0}\equiv\,h$ can be identified with the SM-like Higgs boson with mass
$m_{h}=125.38$ GeV confirmed experimentally CMS:2022dwd . Then the Higgs self-
coupling $\lambda_{1}$ is expressed as follows

$\displaystyle\lambda_{1}$
$\displaystyle=\frac{1}{2}\left(\frac{m^{2}_{h}}{v^{2}}-8c_{\beta}\left[s_{\beta}^{2}c_{\beta}(2\lambda_{2}+\lambda_{3})+\lambda_{4}s_{\beta}\right]\right.$
$\displaystyle\left.\qquad\quad-\frac{8v^{2}\left(2c_{\beta}^{2}-1\right)^{3}\left[2c_{\beta}s_{\beta}(2\lambda_{2}+\lambda_{3})+\lambda_{4}\right]^{2}}{\alpha_{3}v_{R}^{2}}\right).$
(27)

Similarly, the original and mass base of the singly charged Higgs bosons have
the following relation

$\displaystyle\begin{pmatrix}\phi^{\pm}_{1}\\\ \phi^{\pm}_{2}\\\
\end{pmatrix}=\begin{pmatrix}-s_{\beta}&c_{\beta}\\\ c_{\beta}&s_{\beta}\\\
\end{pmatrix}\begin{pmatrix}G^{\pm}_{W}\\\ H^{\pm}_{1}\\\ \end{pmatrix},$
$\displaystyle
H^{\pm}_{2}\simeq\delta_{L}^{\pm},\;G_{W_{2}}^{\pm}\simeq\delta_{R}^{\pm},$
(28)

where $G_{W_{2}}^{\pm}$ is massless, corresponding to the Goldstone boson
eaten up by $W_{2}^{\pm}$, and the remaining singly charged gauge squared
masses are

$\displaystyle
m^{2}_{H^{\pm}_{1}}=\frac{\alpha_{3}v^{2}_{R}}{2\left(2s^{2}_{\beta}-1\right)},\;m^{2}_{H^{\pm}_{2}}=\frac{1}{2}\left(\rho_{3}-2\rho_{1}\right)v_{R}^{2}.$
(29)

Besides that, two components ($\delta^{++}_{L},\delta^{++}_{R}$) are also
physical states with the following masses

$\displaystyle
m^{2}_{H^{\pm\pm}_{1}}=\frac{1}{2}v^{2}_{R}(\rho_{3}-2\rho_{1}),\;\quad
m^{2}_{H^{\pm\pm}_{2}}=2v^{2}_{R}\rho_{2}.$ (30)

##  III Couplings and analytic formulas involved with loop-induced Higgs
decays

###  III.1 Couplings

From the above Higgs potential and the discussion on the masses and mixing of
Higgs bosons, all Higgs self-couplings of $h$ giving one-loop contributions to
the decays $h\rightarrow\gamma\gamma,Z\gamma$ can be derived analytically.
From the general notations in the interacting Lagrangian:
$-V_{h}\to\mathcal{L}_{hSS}=\sum_{S_{i},S_{j}=H^{\pm}_{1,2},H^{\pm\pm}_{1,2}}(-\lambda_{hS_{ij}}hS_{i}^{Q_{S}}S_{j}^{-Q_{S}}+\mathrm{h.c.})+\dots$,
the Feynman rule are $-i\lambda_{hSS}$ corresponds to the vertex $hSS$. All
non-zeros factors $\lambda_{hSS}$ are given in Table 2.

Vertex | Coupling: $-i\lambda_{hSS}$
---|---
$\lambda_{hH^{+}_{1}H^{-}_{1}}$ | $2\left\\{c_{\beta}^{4}\lambda_{1}+2c_{\beta}^{2}s_{\beta}^{2}[\lambda_{1}-2(2\lambda_{2}+\lambda_{3})]+\lambda_{1}s_{\beta}^{4}\right\\}v$
$\lambda_{hH^{+}_{2}H^{-}_{2}}$ | $\frac{1}{2}(2\alpha_{1}+\alpha_{3}+8\alpha_{2}c_{\beta}s_{\beta})v$
$\lambda_{hH^{++}_{1}H^{--}_{1}}$ | $\left[\alpha_{1}+s_{\beta}(4\alpha_{2}c_{\beta}+\alpha_{3}s_{\beta})\right]v$
$\lambda_{hH^{++}_{2}H^{--}_{2}}$ | $\left[\alpha_{1}+s_{\beta}(4\alpha_{2}c_{\beta}+\alpha_{3}s_{\beta})\right]v$
$\lambda_{hH^{++}_{1}H^{--}_{2}}$ | $(\beta_{3}+\beta_{1}c_{\beta}s_{\beta})(-1+2s_{\beta}^{2})\frac{v}{s_{\beta}}$
$\lambda_{hH^{++}_{2}H^{--}_{1}}$ | $(\beta_{3}+\beta_{1}c_{\beta}s_{\beta})(-1+2s_{\beta}^{2})\frac{v}{s_{\beta}}$
Table 2: Feynman rules for the SM-like Higgs boson couplings with charged
Higgs bosons

We note that the vertex factors in Table 2 are derived following the general
notation defined in Ref. Hue:2017cph , so that we can use the analytic
formulas to compute the partial decay widths $h\to\gamma\gamma,Z\gamma$ in the
MLRSM mentioned in this work.

The couplings of $h$ with SM fermions can be determined using the Yukawa
Lagrangians given in Eqs. (7), where the Feynman rule is
$-i\left(Y_{h\bar{f}fL}P_{L}+Y_{h\bar{f}fR}P_{R}\right)$ for each vertex
$h\bar{f}f$. Because this model does not have exotic charged fermions and the
couplings of SM leptons to neutral Higgs/gauge bosons
($g_{h\bar{f}f},g_{Z\bar{f}f}$) are defined as in the SM Dedes:2019bew ;
Dawson:2018pyl , we will use the SM results for one-loop fermion contributions
to the decay amplitudes $h\to Z\gamma,\gamma\gamma$.

The Higgs-gauge boson couplings giving one-loop contribution to the decays
$h\rightarrow Z\gamma,\gamma\gamma$ are derived from the kinetic Lagrangian of
the Higgs bosons, namely

$\displaystyle\mathcal{L}^{H}_{\mathrm{kin}}$
$\displaystyle=\mathcal{L}_{\Phi}+\mathcal{L}_{\Delta_{L}}+\mathcal{L}_{\Delta_{R}}$
$\displaystyle=\sum_{i,j=1}^{2}g_{\mu\nu}g_{hW_{i}W_{j}}hW_{i}^{-\mu}W_{j}^{+\nu}$
$\displaystyle+$
$\displaystyle\sum_{S_{i},j}\left[-ig^{*}_{hSW_{j}}W_{j}^{-\mu}\left(S_{i}^{+Q}\partial_{\mu}h-h\partial_{\mu}S_{i}^{+Q}\right)+ig_{hSW_{j}}W_{j}^{+\mu}\left(S_{i}^{-Q}\partial_{\mu}h-h\partial_{\mu}S_{i}^{-Q}\right)\right]$
$\displaystyle+$
$\displaystyle\sum_{S_{i},S_{j}}ig_{ZS_{i}S_{j}}Z^{\mu}\left(S_{i}^{-Q}\partial_{\mu}S_{j}^{Q}-S_{j}^{Q}\partial_{\mu}S_{i}^{-Q}\right)$
$\displaystyle+$
$\displaystyle\sum_{S_{i},j}\left[ig_{ZW_{j}S_{i}}Z^{\mu}W_{j}^{+\nu}S_{i}^{-Q}g_{\mu\nu}+ig^{*}_{ZW_{j}S_{i}}Z^{\mu}W_{j}^{-\nu}S_{i}^{Q}g_{\mu\nu}\right]$
$\displaystyle+$
$\displaystyle\sum_{S_{i}}ieQA^{\mu}\left(S_{i}^{-Q}\partial_{\mu}S_{i}^{Q}-S_{i}^{Q}\partial_{\mu}S_{i}^{-Q}\right)+...,$
(31)

where $S_{i},S_{j}=H^{\pm}_{1,2},H^{\pm\pm}_{1,2}$ denote charged Higgs bosons
in the MLRSM. The Feynman rules for the $h$ couplings to at least one charged
gauge boson are shown in Table 3.

Vertex | Coupling
---|---
$g_{hW_{1}^{+}W_{1}^{-}}$ | $\frac{1}{2}\left\\{-4g_{L}g_{R}s_{\beta}c_{\beta}s_{\xi}c_{\xi}+(c_{\xi}^{2}g_{L}^{2}+g_{R}^{2}s_{\xi}^{2})\right\\}v$
$g_{hW_{2}^{+}W_{2}^{-}}$ | $\frac{1}{2}\left\\{4g_{L}g_{R}s_{\beta}c_{\beta}s_{\xi}c_{\xi}+(s_{\xi}^{2}g_{L}^{2}+g_{R}^{2}c_{\xi}^{2})\right\\}v$
$g_{hW_{1}^{+}W_{2}^{-}}$ | $\frac{1}{2}\left\\{s_{\xi}c_{\xi}(-g_{L}^{2}+g_{R}^{2})+2g_{L}g_{R}s_{\beta}c_{\beta}(-c_{\xi}^{2}+s_{\xi}^{2})\right\\}v$
$g_{hW_{2}^{+}W_{1}^{-}}$ | $\frac{1}{2}\left\\{s_{\xi}c_{\xi}(-g_{L}^{2}+g_{R}^{2})+2g_{L}g_{R}s_{\beta}c_{\beta}(-c_{\xi}^{2}+s_{\xi}^{2})\right\\}v$
$g_{hH_{1}^{-}W_{1}^{+}}$ | $\frac{1}{2}g_{R}s_{\xi}(c_{\beta}^{2}-s_{\beta}^{2})$
$g_{hH_{1}^{-}W_{2}^{+}}$ | $\frac{1}{2}g_{R}c_{\xi}(c_{\beta}^{2}-s_{\beta}^{2})$
Table 3: Feynman rules for couplings of the SM-like Higgs boson to charged
Higgs and gauge bosons.

The momenta appearing in the vertex factors are $\partial_{\mu}h\rightarrow-
ip_{0\mu}h$ and $\partial_{\mu}S_{i,j}\rightarrow-ip_{\mu}S_{i,j}$, where
$p_{0}$, $p_{\pm}$ are incoming momenta.

The Feynman rules for $Z$ couplings to charged Higgs and gauge bosons in Eq.
(III.1) are given in Table 4. The couplings
$g_{ZH_{i}^{+,++}H_{j}^{-,--}},g_{ZW_{2}^{+}H_{i,j}^{-}}$ are zero in the
MLRSM.

Vertex | Coupling
---|---
$g_{ZH_{1}^{+}H_{1}^{-}}$ | $\frac{1}{2}(p_{+}-p_{-})[c_{z_{2}}c_{z_{3}}g_{L}-g_{R}(c_{R}c_{z_{3}}s_{z_{2}}+s_{R}s_{z_{3}})]$
$g_{ZH_{2}^{+}H_{2}^{-}}$ | $g^{\prime}(p_{+}-p_{-})(-c_{z_{3}}s_{R}s_{z_{2}}+c_{R}s_{z_{3}})$
$g_{ZH_{1}^{++}H_{1}^{--}}$ | $(p_{++}-p_{--})(c_{z_{2}}c_{z_{3}}g_{L}-c_{z_{3}}g^{\prime}s_{R}s_{z_{2}}+c_{R}g^{\prime}s_{z_{3}})$
$g_{ZH_{2}^{++}H_{2}^{--}}$ | $(p_{++}-p_{--})(c_{R}c_{z_{3}}g_{R}s_{z_{2}}+c_{z_{3}}g^{\prime}s_{R}s_{z_{2}}-c_{R}g^{\prime}s_{z_{3}}+g_{R}s_{R}s_{z_{3}})$
$g_{ZW_{1}^{+}H_{1}^{-}}$ | $\frac{1}{2}c_{z_{2}}c_{z_{3}}g_{L}g_{R}(c_{\beta}^{2}-s_{\beta}^{2})s_{\xi}v$
$g_{ZW_{1}^{+}H_{2}^{-}}$ | $\frac{1}{2}c_{\xi}c_{z_{2}}c_{z_{3}}g_{L}g_{R}(c_{\beta}^{2}-s_{\beta}^{2})v$
Table 4: Feynman rules of couplings with $Z$ and photon. Notations $p_{+}$ and
$p_{-}$ are incoming momenta.

The triple gauge couplings of $Z$ and photon to $W^{\pm}_{1,2}$ are derived
from the kinetic Lagrangian of the non-Abelian gauge bosons

$\displaystyle\mathcal{L}^{k}_{g}=-\frac{1}{4}\sum_{a=1}^{3}\left[F^{a}_{L\mu\nu}F^{a\mu\nu}_{L}+F^{a}_{R\mu\nu}F^{a\mu\nu}_{R}\right],$
(32)

where
$F^{a}_{L,R\mu\nu}=\partial_{\mu}W^{a}_{L,R\nu}-\partial_{\nu}W^{a}_{L,R\mu}+g_{L,R}\epsilon^{abc}W^{b}_{L,R\mu}W^{c}_{L,R\mu}$,
and $\epsilon^{abc}$ $(a,b,c=1,2,3)$ are the $SU(2)$ structure constants. The
respective $Z$couplings to $W^{\pm}_{1,2}$ are included in the following part:

$\displaystyle\mathcal{L}^{ZW^{+}W^{-}}\subset-
g_{L}\epsilon^{abc}(\partial_{\mu}W^{a}_{L\nu})W^{b\mu}_{L}W^{c\nu}_{L}-g_{R}\epsilon^{abc}(\partial_{\mu}W^{a}_{R\nu})W^{b\mu}_{R}W^{c\nu}_{R}.$
(33)

Then, the vertex factors corresponding to particular couplings are defined as

$\displaystyle\mathcal{L}^{g}_{D}\rightarrow$ $\displaystyle-
g_{ZW_{i}W_{j}}Z^{\mu}(p_{0})W^{+\nu}_{i}(p_{+})W^{-\lambda}_{j}(p_{-})\times\Gamma_{\mu\nu\lambda}(p_{0},p_{+},p_{-}),$
$\displaystyle-
eA^{\mu}(p_{0})W_{i}^{+\nu}(p_{+})W_{i}^{-\lambda}(p_{-})\times\Gamma_{\mu\nu\lambda}(p_{0},p_{+},p_{-}),$
(34)

where $\Gamma_{\mu\nu\lambda}(p_{0},p_{+},p_{-})\equiv
g_{\mu\nu}(p_{0}-p_{+})_{\lambda}+g_{\nu\lambda}(p_{+}-p_{-})_{\mu}+g_{\lambda\mu}(p_{-}-p_{0})_{\lambda}$,
and $i,j=1,2$. The photon always couples to two identical particles as the
consequence of the Ward Identity Hue:2023rks , see the second line of Eq.
(III.1). The non-zero factors for triple couplings of $Z$ with charged gauge
bosons are collected in Table 5.

Vertex | Coupling
---|---
$g_{ZW^{+\nu}_{1}W^{-\lambda}_{1}}$ | $\frac{-e}{s_{z_{2}}c_{R}c_{z_{2}}}\left[c_{R}c_{z_{3}}(-c^{2}_{z_{2}}+s^{2}_{\xi})+s_{R}s_{z_{3}}s_{z_{2}}s^{2}_{\xi}\right]$
$g_{ZW^{+\nu}_{2}W^{-\lambda}_{2}}$ | $\frac{e}{s_{z_{2}}c_{R}c_{z_{2}}}\left[c_{R}c_{z_{3}}(s^{2}_{\xi}-s^{2}_{z_{2}})-s_{R}s_{z_{3}}s_{z_{2}}c^{2}_{\xi}\right]$
$g_{ZW^{+\nu}_{1}W^{-\lambda}_{2}}$ | $-\frac{es_{\xi}c_{\xi}}{s_{z_{2}}c_{R}c_{z_{2}}}\left[c_{z_{3}}c_{R}+s_{R}s_{z_{3}}s_{z_{2}}\right]$
$g_{ZW^{+\nu}_{2}W^{-\lambda}_{1}}$ | $-\frac{es_{\xi}c_{\xi}}{s_{z_{2}}c_{R}c_{z_{2}}}\left[c_{z_{3}}c_{R}+s_{R}s_{z_{3}}s_{z_{2}}\right]$
Table 5: Feynman rules for triple gauge couplings relating with the decay
$h\rightarrow Z\gamma$.

###  III.2 Partial decay widths and signal strengths of the decays
$h\rightarrow Z\gamma,\gamma\gamma$

In the MLRSM framework, one-loop three-point Feynman diagrams giving
contributions to the decay amplitude $h\rightarrow\,Z\gamma$ are shown in Fig.
1,

Figure 1:  One-loop three-point Feynman diagrams contributing to the decay
$h\rightarrow\,Z\gamma$ in the unitary gauge, where $f_{i,j}$ are the SM
fermions, $s_{i,j}=H^{\pm}_{1,2},H^{\pm\pm}_{1,2}$,
$V_{i,j}=W_{1}^{\pm},W_{2}^{\pm}$.

where the unitary gauge is applied to determine the gauge boson contributions.
The fermion contributions to amplitude of the decay $h\to Z\gamma$ coincide
with the SM results calculated in Ref. Dedes:2019bew ; Dawson:2018pyl . Using
the general calculation introduced in Ref. Hue:2017cph , we can written these
contribution as follows

$F_{21,f}^{\mathrm{LR}}=F_{21,f}^{\mathrm{SM}}=\sum_{f_{i}=e_{i},u_{i},d_{i}}F_{21,f_{i}}^{\mathrm{SM}},$
(35)

where all form factors $F_{21,f_{i}}^{\mathrm{SM}}$ are written interms of the
Passrino-Veltman notations Passarino:1978jh .

Similarly, the contribution from the charged Higgs bosons can be given as

$\displaystyle F^{\mathrm{LR}}_{21,S}$
$\displaystyle=F_{21,H^{+}_{1}}+F_{21,H^{+}_{2}}+F_{21,H^{+}_{122}}+F_{21,H^{+}_{211}}$
$\displaystyle+F_{21,H^{++}_{1}}+F_{21,H^{++}_{2}}+F_{21,H^{++}_{122}}+F_{21,H^{++}_{211}}.$
(36)

The charged gauge boson contributions $W^{\pm}_{1,2}$ to the $h\to Z\gamma$
amplitude are

$\displaystyle F^{\mathrm{LR}}_{21,V}$
$\displaystyle=F^{\mathrm{LR}}_{21,W^{+}_{1}}+F^{\mathrm{LR}}_{21,W^{+}_{2}}+F^{\mathrm{LR}}_{21,W^{+}_{122}}+F^{\mathrm{LR}}_{21,W^{+}_{211}}.$
(37)

Similarly, the contribution from charged Higgs and gauge boson arising from
two diagrams 3 and 4 in Fig. 1 can be given as

$\displaystyle F^{\mathrm{LR}}_{21,VS}$
$\displaystyle=F_{21,WSS}^{\mathrm{LR}}+F_{21,SWW}^{\mathrm{LR}},$ (38)

where

$\displaystyle F^{\mathrm{LR}}_{21,WSS}$
$\displaystyle=F_{21,W^{+}_{1}H^{+}_{1}H^{+}_{1}}^{\mathrm{LR}}+F_{21,W^{+}_{1}H^{+}_{2}H^{+}_{2}}^{\mathrm{LR}}+F_{21,W^{+}_{2}H^{+}_{1}H^{+}_{1}}^{\mathrm{LR}}+F_{21,W^{+}_{2}H^{+}_{2}H^{+}_{2}}^{\mathrm{LR}},$
(39) $\displaystyle F^{\mathrm{LR}}_{21,SWW}$
$\displaystyle=F_{21,H^{+}_{1}W^{+}_{1}W^{+}_{1}}^{\mathrm{LR}}+F_{21,H^{+}_{1}W^{+}_{2}W^{+}_{2}}^{\mathrm{LR}}+F_{21,H^{+}_{2}W^{+}_{1}W^{+}_{1}}^{\mathrm{LR}}+F_{21,H^{+}_{2}W^{+}_{2}W^{+}_{2}}^{\mathrm{LR}}.$
(40)

Now, the $h\to Z\gamma$ partial decay width is Gunion:1989we ;
Degrande:2017naf

$\Gamma^{\mathrm{LR}}(h\rightarrow
Z\gamma)=\frac{m_{h}^{3}}{32\pi}\times\left(1-\frac{m_{Z}^{2}}{m_{h}^{2}}\right)^{3}|F^{\mathrm{LR}}_{21}|^{2},$
(41)

where the scalar factors $F^{\mathrm{LR}}_{21}$ is derived as follows
Hue:2017cph

$\displaystyle F^{\mathrm{LR}}_{21}$
$\displaystyle=F^{\mathrm{LR}}_{21,f}+F^{\mathrm{LR}}_{21,S}+F^{\mathrm{LR}}_{21,V}+F^{\mathrm{LR}}_{21,VS}.$
(42)

We note that $F^{\mathrm{LR}}_{21,VS}$ were omitted in some previous works
Yue:2013qba ; Martinez:1989kr ; Martinez:1990ye  because it was expected to be
smaller than the contributions from the SM and which is still far from the
sensitivity of the recent experiments. However, since collider sensitivity has
recently been improved and new scales have been established, their
contribution is necessary. The branching ratio
$\mathrm{Br}^{\mathrm{LR}}(h\rightarrow Z\gamma)$ in the MLRSM framework is

$\mathrm{Br}^{\mathrm{LR}}(h\rightarrow
Z\gamma)=\frac{\Gamma^{\mathrm{LR}}(h\rightarrow
Z\gamma)}{\Gamma^{\mathrm{LR}}_{h}},$ (43)

where $\Gamma^{\mathrm{LR}}_{h}$ is the total decay width of the SM-like Higgs
boson $h$ Gunion:1989we ; Degrande:2017naf . Although there are available
experimental measurements of the SM-like Higgs boson productions and decays
Khachatryan:2016vau , we focus only on the Higgs production through the gluon
fusion process $ggF$ at LHC, in which the respective signal strength predicted
by two models SM and MLSM are equal. Then the signal strength corresponding to
the decay mode $h\to Z\gamma$ predicted by the MLRSM is:

$\mu_{Z\gamma}^{\mathrm{LR}}\equiv\frac{\mathrm{Br}^{\mathrm{LR}}(h\rightarrow
Z\gamma)}{\mathrm{Br}^{\mathrm{SM}}(h\rightarrow Z\gamma)},$ (44)

where $\mathrm{Br}^{\mathrm{SM}}(h\rightarrow Z\gamma)$ is the SM branching
ratio of the decay $h\to Z\gamma$. The recent $ggF\to h\to Z\gamma$ singal
strength is $\mu_{Z\gamma}=2.4\pm 0.9$ at $2.7\sigma$ (standard deviation)
CMS:2022ahq ; CMS:2023mku .

Similarly, the partial decay width and signal strength of the decay
$h\rightarrow\gamma\gamma$ can be calculated as Degrande:2017naf ; Hue:2017cph

$\displaystyle\Gamma^{\mathrm{LR}}(h\rightarrow\gamma\gamma)=$
$\displaystyle\frac{m_{h}^{3}}{64\pi}\times|F^{\mathrm{LR}}_{\gamma\gamma}|^{2},$
$\displaystyle\mu_{\gamma\gamma}^{\mathrm{LR}}\equiv$
$\displaystyle\frac{\Gamma^{\mathrm{LR}}(h\rightarrow\gamma\gamma)}{\Gamma^{\mathrm{SM}}(h\rightarrow\gamma\gamma)},$
(45)

where

$\displaystyle F^{\mathrm{LR}}_{\gamma\gamma}$
$\displaystyle=\sum_{f}F^{\mathrm{LR}}_{\gamma\gamma,f}+\sum_{s}F^{\mathrm{LR}}_{\gamma\gamma,s}+\sum_{v}F^{\mathrm{LR}}_{\gamma\gamma,v},$
(46)

and

$\displaystyle F_{\gamma\gamma,f}^{\mathrm{LR}}=$ $\displaystyle
F_{\gamma\gamma,f}^{\mathrm{SM}}=\sum_{f_{i}=e_{i},u_{i},d_{i}}F_{\gamma\gamma,f_{i}}^{\mathrm{SM}},$
$\displaystyle F^{\mathrm{LR}}_{\gamma\gamma,S}=$ $\displaystyle
F_{\gamma\gamma,H^{+}_{1}}+F_{\gamma\gamma,H^{+}_{2}}+F_{\gamma\gamma,H^{++}_{1}}+F_{\gamma\gamma,H^{++}_{2}},$
$\displaystyle F^{\mathrm{LR}}_{\gamma\gamma,V}=$ $\displaystyle
F_{\gamma\gamma,W^{+}_{1}}+F_{\gamma\gamma,W^{+}_{2}}.$ (47)

Here we have used the notations that Hung:2019jue

$\displaystyle F^{\mathrm{SM}}_{\gamma\gamma,f_{i}}$
$\displaystyle=-\frac{e^{2}\,Q^{2}_{f_{i}}\,N_{c}}{2\pi^{2}}\left(m_{f}Y_{h\bar{f}fL}\right)\left[4X_{2}+C_{0}\right],$
$\displaystyle F_{\gamma\gamma,s}$
$\displaystyle=\frac{e^{2}\,Q^{2}_{s}\lambda_{hss}}{2\pi^{2}}X_{2},$
$\displaystyle F_{\gamma\gamma,v}$
$\displaystyle=\frac{e^{2}\,Q^{2}_{V}\,g_{hvv}}{4\pi^{2}}\times\left\\{\left(6+\frac{m_{h}^{2}}{m_{V}^{2}}\right)X_{2}+4C_{0}\right\\},$
(48)

where $X_{2}=C_{12}+C_{22}+C_{2}$ and $C_{0,i,ij}\equiv
C_{0,i,ij}(0,0,m_{h}^{2};m_{x}^{2},m_{x}^{2},m_{x}^{2})$ are Passarino-Veltman
functions Passarino:1978jh  with $x=f,s,v$ implying fermions, charged Higgs
and gauge bosons, respectively. Particular forms given in Eq. (III.2) are
defined precisely in Ref. Hung:2019jue . In the following section, the
numerical results will be evaluated using LoopTools Hahn:1998yk .

##  IV  Numerical discussions

###  IV.1  Setup parameters

In this section, there are following quantities fixed from experiments
Tanabashi:2018oca : $m_{h}=125.38$ GeV, $m_{W}$, $m_{Z}$, well-known fermion
masses, $v\simeq 246$ GeV, the $SU(2)_{L}$ gauge coupling $g_{2}\simeq 0.651$,
$\alpha_{\mathrm{em}}=1/137$, $e=\sqrt{4\pi\alpha_{\mathrm{em}}}$,
$s^{2}_{W}=0.231$.

The unknown independent parameters of the MLRSM used as inputs are
$\beta_{1,3}$, $\rho_{1,2,3,4}$, $\alpha_{1,2,3}$, $\lambda_{1,2,3,4}$. The
dependent parameters include $\lambda_{1}$ calculated by the Eq. (II.4),
$\alpha_{1,2}$ and $\alpha_{4}$ calculated by the Eq. (49)

$\alpha_{2}=\frac{\alpha_{3}t_{\beta}}{2(1-t_{\beta}^{2})},\alpha_{1}=\frac{\alpha_{3}}{t_{\beta}^{2}-1},\alpha_{4}=-\frac{2\alpha_{6}}{t_{\beta}}.$
(49)

Higgs self-couplings are calculated through the heavy Higgs boson masses in
the MLRSM, namely

$\displaystyle m^{2}_{H^{0}_{1}}$
$\displaystyle=m^{2}_{A_{2}}=m^{2}_{H^{\pm}_{1}}=\frac{\alpha_{3}v_{R}^{2}}{2\left(2s_{\beta}^{2}-1\right)},$
$\displaystyle m^{2}_{H^{0}_{2}}$
$\displaystyle=m^{2}_{H^{\pm}_{2}}=m^{2}_{H^{\pm\pm}_{1}}=m^{2}_{A_{2}}=\frac{v_{R}^{2}}{2}\left(-2\rho_{1}+\rho_{3}\right),\;$
$\displaystyle m^{2}_{H^{0}_{3}}$
$\displaystyle=2\rho_{1}v_{R}^{2},\;m^{2}_{H^{\pm\pm}_{2}}=2\rho_{2}v_{R}^{2}.$
(50)

Choosing the mass of $m_{H^{+}_{1}}$, $m_{H^{+}_{2}}$, and $m_{H^{++}_{1}}$ as
free parameters we get

$\displaystyle\alpha_{1}$
$\displaystyle=\frac{2m_{H_{2}^{+}}^{2}}{\left(t_{\beta}^{2}+1\right)v_{R}^{2}},\;\alpha_{2}=-\frac{t_{\beta}m_{H_{2}^{+}}^{2}}{\left(t_{\beta}^{2}+1\right)v_{R}^{2}},\;\alpha_{3}=\frac{2\left(t_{\beta}^{2}-1\right)m_{H_{2}^{+}}^{2}}{\left(t_{\beta}^{2}+1\right)v_{R}^{2}},\;$
$\displaystyle\alpha_{4}$
$\displaystyle=-\frac{2\alpha_{6}}{t_{\beta}},\;\rho_{2}=\frac{m_{H_{2}^{\text{++}}}^{2}}{2v_{R}^{2}},\;\rho_{3}=2\rho_{1}+\frac{2m_{H_{1}^{+}}^{2}}{v_{R}^{2}}.$
(51)

The other free parameters are $\lambda_{2,3,4}$,
$\rho_{1}=m^{2}_{H^{0}_{3}}/(2v_{R}^{2})>0$, the mixing angle and the gauge
bosons masses will be at the orders of $\mathcal{O}(v^{2}/v_{R}^{2})$

$\displaystyle g_{R}$
$\displaystyle=g_{L}=g_{2}=e/s_{W},\;g^{\prime}=\frac{e}{\sqrt{1-2s^{2}_{W}}},$
$\displaystyle s_{\xi}$
$\displaystyle=-\frac{s_{\beta}c_{\beta}v^{2}}{v_{R}^{2}}\ll
1,\;c_{\xi}=1+\mathcal{O}\left(\frac{v^{4}}{v_{R}^{4}}\right)\simeq 1,$
$\displaystyle s_{R}$
$\displaystyle=\frac{\sqrt{1-2s_{W}^{2}}}{c_{W}},\;c_{R}=t_{W},$
$\displaystyle s_{z_{2}}$ $\displaystyle=s_{W},\;c_{z_{2}}=c_{W},$
$\displaystyle s_{z_{3}}$
$\displaystyle=\frac{t_{W}^{2}\sqrt{1-2s_{W}^{2}}v^{2}}{4c_{W}^{2}v_{R}^{2}}\ll
1,\;c_{z_{3}}=1+\mathcal{O}\left(\frac{v^{4}}{v_{R}^{4}}\right)\simeq 1.$ (52)

We note here that the relations given in Eq. (IV.1) are consistent with the SM
because the two couplings $hW^{+}W^{-}$ and $ZW^{+}W^{-}$ are consistent with
the SM predictions.

In investigating the $h\to Z\gamma$ decays, the values of unknown independent
parameters we choose here will be scanned in the following ranges:

$\displaystyle
m_{H^{+}_{1}},\;m_{H^{+}_{2}},m_{H^{++}_{2}}\in[100,\;1000]\;\mathrm{GeV},\;v_{R}\in[1,\;10]\;\mathrm{TeV},\;t_{\beta}\in[5,50],$
$\displaystyle\lambda_{2,3,4},\alpha_{3}\in[-10,10],$ (53)

where the Higgs self-couplings satisfy the perturbative limits, $v_{R}$ is
large enough to generate the heavy gauge bosons masses satisfying the current
experimental constraints, $t_{\beta}>1$ due to model survey conditions.

###  IV.2 Results and discussions

To express the differences of the prediction between the SM and the MLRSM, we
define a quantity $\Delta\mu_{Z\gamma}$ as in Ref. Hung:2019jue

$\displaystyle\Delta{\mu}^{\mathrm{LR}}_{Z\gamma}\equiv\left(\mu^{\mathrm{LR}}_{Z\gamma}-1\right)\times
100\%,$ (54)

which is constrained by recent experiments $\Delta\mu_{Z\gamma}=1.4\pm 0.9$
CMS:2022ahq ; CMS:2023mku , implying the $1\sigma$ deviation is

$50\%\leq\Delta{\mu}^{\mathrm{LR}}_{Z\gamma}\leq 230\%.$ (55)

The $1\sigma$ constraint from $h\to\gamma\gamma$ decay originating from $ggF$
fusion is defined as
$\Delta\mu^{\mathrm{LR}}_{\gamma\gamma}\equiv(\mu^{\mathrm{LR}}_{\gamma\gamma}-1)\times
100\%$, leading to the respective $2\sigma$ deviation is

$\displaystyle-12\%<\Delta\mu^{\mathrm{LR}}_{\gamma\gamma}<38\%.$ (56)

The numerical results we discuss in the following will always satisfy this
constraint. We will focus in the region satisfying
$|\Delta\mu^{\mathrm{LR}}_{Z\gamma}|\geq 10\%$ to collect interesting points
that may support the $1\sigma$ range given in Eq. (55). We have checked
numerically that the MLRSM always contains regions of the parameter space that
both values of $\Delta{\mu}_{Z\gamma},\Delta{\mu}_{\gamma\gamma}\to 0$,
implying the consistency with the SM results. Firstly, we discuss on the
dependence of $\Delta\mu^{\mathrm{LR}}_{Z\gamma}$ on
$\Delta\mu^{\mathrm{LR}}_{\gamma\gamma}$, which is illustrated in Fig. 2.

Figure 2: Correlations between $\Delta\mu_{Z\gamma}^{\mathrm{LR}}$ and
$\Delta\mu_{\gamma\gamma}^{\mathrm{LR}}$.

It can be seen that $\Delta\mu_{Z\gamma}^{\mathrm{LR}}$ is constrained
strictly by $\Delta\mu_{\gamma\gamma}^{\mathrm{LR}}$, i.e.,
$\Delta\mu_{Z\gamma}^{\mathrm{LR}}<40\%$ in the range of $2\sigma$ deviation
given in “Eq. (56)”. It is noted that negative values of
$\Delta\mu_{\gamma\gamma}^{\mathrm{LR}}<0$ can give large
$\Delta\mu_{Z\gamma}^{\mathrm{LR}}$ than the positive ones. But largest values
of $\Delta\mu_{Z\gamma}^{\mathrm{LR}}$ is still smaller than the $1\sigma$
deviation given by recent experimental data. Therefore, in the future, if
large $\Delta\mu_{Z\gamma}^{\mathrm{LR}}$ and small
$\Delta\mu_{\gamma\gamma}^{\mathrm{LR}}$ are confirmed experimentally, the
assumption given in Eq. (IV.1) must be changed.

We comment here a point that the future sensitivities are
$|\Delta\mu_{\gamma\gamma}|\leq 4\%$, and $|\Delta\mu_{Z\gamma}|\leq 23\%$,
respectively Cepeda:2019klc . In the model under consideration, large values
of $\Delta\mu_{Z\gamma}>23\%$ are still allowed in the very strict constraint
from $\Delta\mu_{\gamma\gamma}$.

Now, we discuss on the dependence of $\Delta\mu^{\mathrm{LR}}_{Z\gamma}$ on
$t_{\beta}$ and $v_{R}$, which are shown in Fig. 3.

Figure 3:  Correlations between $\Delta\mu_{Z\gamma}^{\mathrm{LR}}$ with
different values of $t_{\beta}$ and $v_{R}$.

We can see that $\Delta\mu_{Z\gamma}^{\mathrm{LR}}$ depends weakly on these
two parameters. Namely, all values of $t_{\beta}$ and $v_{R}$ can give large
$\Delta\mu_{Z\gamma}^{\mathrm{LR}}>35\%$.

The correlations between $\Delta\mu_{Z\gamma}^{\mathrm{LR}}$ and charged Higgs
boson masses are shown in Fig. 4.

Figure 4: Correlations between $\Delta\mu_{Z\gamma}^{\mathrm{LR}}$ and charged
Higgs bosons masses $m_{H^{\pm\pm}_{1}}$ and $m_{H^{\pm}_{1,2}}$.

The results show that only $m_{H^{\pm}_{1}}$ gives large
$\Delta\mu^{\mathrm{LR}}_{Z\gamma}$, i.e. $m_{H^{\pm}_{1}}<200$ GeV for
$\Delta\mu^{\mathrm{LR}}_{Z\gamma}>37\%$. In contrast, the other masses do not
affect strongly on values of $\Delta\mu^{\mathrm{LR}}_{Z\gamma}$.

##  V Conclusions

We have studied all one-loop contributions to the SM-like Higgs decays
$h\to\gamma\gamma,Z\gamma$ in the MLRSM framework. Interesting properties of
the new gauge and Higgs bosons were explored. Namely, the SM-like Higgs
couplings were identified with the SM prediction and experimental data. All
masses, physical states of gauge and Higgs bosons, and their mixing were
presented clearly so that all couplings relate to one-loop contributions the
decay amplitudes $h\to\gamma\gamma,Z\gamma$ are derived analytically. From
this, decays $h\rightarrow\gamma\gamma,Z\gamma$ in MLRSM have been discussed
using the relevant recent experimental results. The one-loop contributions
from the diagrams containing both gauge and Higgs mediation were included in
the decay amplitude $h\rightarrow Z\gamma$. These contributions were ignored
in previous studies, although they may enhance the $h\rightarrow Z\gamma$
amplitude, but do not affect the $h\rightarrow\gamma\gamma$ one, leading to
the possibility that large $\Delta\mu_{Z\gamma}$ may be allowed under the
strict experimental constraint of $\Delta\mu_{\gamma\gamma}$. We have shown
that the mentioned $h$ decay rates depend weakly on $t_{\beta}$, the
$SU(2)_{R}$ vacuum scale $v_{R}$. The $2\sigma$ deviation of
$\mu_{\gamma\gamma}$ results in a rather strict constraint
$\left|\Delta\mu_{Z\gamma}\right|<40\%$. On the other hand, the large values
of $\Delta\mu_{Z\gamma}>23\%$ can appear under the very strict constraint of
$|\Delta\mu_{\gamma\gamma}|\leq 4\%$ corresponding to the future experimental
sensitivities. Therefore, the future experimental searches of the two decays
mentioned in this work will be important to constrain the parameter space of
the MLRSM.

## Acknowledgments

This research is funded by Vietnam National University HoChiMinh City (VNU-
HCM) under grant number “C2022-16-06”.

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