# Loop induced $h\rightarrow\gamma\gamma,Z\gamma,gg$ decays in 3-Higgs doublet
models

Emine Yildirim<EMAIL_ADDRESS>Department of Physics, Karadeniz
Technical University, TR61080 Trabzon, Turkey

###### Abstract

## Abstract

In this paper, we analyze the loop induced decays of neutral Higgs boson $h$
into pairs of photons, gluons, and Z$\gamma$ in 3-Higgs Doublet Models
(3HDMs), where all the doublets have non-zero vacuum expectation values. In
the 3HDM, there are contributions to these decays from loops of charged
scalars $H^{\pm}_{1,2}$ that are not present in the Standard Model. We find an
enhancement in the channel $h\rightarrow Z\gamma$ with respect to the Standard
Model prediction in some parameter region while respecting constraints from
the measurements of $h\rightarrow\gamma\gamma$ at the LHC. The charged Higgs
bosons $H^{\pm}_{1,2}$ lead to the enhancement of $Z\gamma$ decay mode in some
part of the parameter space of the 3HDM. Then, we discuss the correlations
between the decay modes $h\rightarrow Z\gamma$ and $h\rightarrow\gamma\gamma$.
Moreover, we study Type-I and Type-Z of 3HDMs and find that these types of
3HDMs can be distinguished via only the gluonic decays, since the
contributions of the $H^{\pm}_{1,2}$ loops are dominant to those of fermion
loops. We also investigate the effects of the model parameters on these loop
induced decays. As an interesting phenomenological consequence, the 3HDM is
dominantly governed by the parameters: two ratios of vacuum expectation values
(VEVs); $\tan\gamma,~{}\tan\beta$ and the mixing angles of the CP-even Higgs
bosons, $\alpha_{1,2}~{}\text{and}~{}\theta$.

##  I Introduction

Recently a Higgs boson with a mass of 125 GeV has been discovered at the Large
Hadron Collider (LHC) Aad:2012tfa ; Chatrchyan:2012xdj  that resembles the
Standard Model (SM) Higgs boson. This discovery, however raises one important
question, whether it is indeed the SM Higgs boson or a Higgs boson of new
physics beyond the SM. In the class of the many beyond SM scenarios, an
interesting class of Higgs-sector extension is the n-Higgs Doublet Model
(nHDM). One of the simplest but important examples is the two-Higgs-doublet
model (2HDM) with one additional scalar doublet Branco:2011iw ;
Bhattacharyya:2015nca . In recent years, a lot of attention has been given to
the Three Higgs Doublet Model (3HDM) with additional two Higgs doublets my3hdm
; 3hdm ; 3hdm1 ; 3hdm2 ; 3hdm3 ; 3hdm4 ; 3hdm5 ; 3hdm6 ; Yagyu:2016whx ;
Ferreira:2008zy ; Machado:2010uc ; Ivanov:2012ry ; GonzalezFelipe:2013xok ;
GonzalezFelipe:2013yhh ; Keus:2013hya ; Ivanov:2014doa ; Maniatis:2014oza ;
3hdm-uni ; Das:2019yad . Extended Higgs sectors naturally give rise to Flavor-
Changing Neutral Currents (FCNCs) which are in conflict with various flavour
data at the tree level. This problem is usually avoided by implementing
discrete symmetries in the Yukawa interactions. For example, in the 3HDM, only
one of the three doublets couples to each type of fermion. The rich particle
spectrum of the 3HDM consists of three CP-even Higgs bosons, $h$, $H_{1}$ and
$H_{2}$, two CP-odd Higgs bosons, $A_{1}$ and $A_{2}$; and two pair of charged
Higgs bosons, $H^{\pm}_{1}$ and $H^{\pm}_{2}$. In this paper, we discuss the
possibility that the discovered Higgs boson at the LHC is the lightest CP-even
neutral Higgs boson $h$ of the 3HDM.

In this paper, we study the loop-induced decays $h\rightarrow\gamma\gamma$,
$Z\gamma$ and $gg$ in the 3HDM and their behavior over the model parameters.
The loop induced processes are sensitive to new physics contributions, which
would interfere with the SM contributions. Although a large number of works
have investigated the effects of new particles in the loop induced decay
widths of the Higgs 3hdm ; 3hdm3 ; loop ; loop2 ; loop3 ; Aranda:2013kq ;
Fontes:2014xva ; Arhrib:2011vc ; Hammad:2015eca , a systematic numerical study
of the 3HDM loop induced decay widths are still missing. In particular, we
will study under which circumstances, the loop induced decay widths could be
significantly affected. We consider the possibility of constructive
interference of the $H^{\pm}_{1}$ and $H^{\pm}_{2}$ contributions with the SM
particles contributions due to enhancement of the decay widths $h\rightarrow
Z\gamma$ in the 3HDM. Besides the two ratios of Vacuum Expectation Values
(VEVs), $\tan\beta$ and $\tan\gamma$, all loop induced decay widths are quite
sensitive to parameters $\alpha_{1,2}$ and $\theta$, which are the mixing
angles between the CP even Higgs bosons.

We will examine constraints on the parameter spaces of the Type-I and Type-Z-
3HDM by taking into account the measured value of $h\rightarrow\gamma\gamma$
decay at the LHC. Then, we will investigate whether an enhancement in the
$h\rightarrow Z\gamma$ decay mode exists while respecting these constraints.
We obtain an enhancement in the $h\rightarrow Z\gamma$ decay mode in part of
the parameter space. Archer-Smith et al. recently investigated the decay rate
of $h\rightarrow Z\gamma$ and found that simple models cannot lead to large
enhancements while still respecting other data, particularly measurements of
$h\rightarrow\gamma\gamma$ Archer-Smith:2020gib . According to Ref.
Chen:2013vi , the $h\rightarrow Z\gamma$ decay rate can enhance or reduce with
respect to its SM value: $0.76<\Gamma(h\rightarrow
Z\gamma)/\Gamma(h\rightarrow Z\gamma)_{SM}<2.05.$ In a study in the context of
an Inert Triplet Model (ITP) in Ref. Kanemura:2018esc , the $h\rightarrow
Z\gamma$ decay is suppressed by about 70 % from the SM prediction, while the
$h\rightarrow\gamma\gamma$ decay remains as in the SM. Wang et al.
investigated the $h\rightarrow\gamma\gamma$ decay in a 2HDM, but the partial
decay width $\Gamma(h\rightarrow Z\gamma)$ is generally less than the SM
expectation Wang:2017xml .

The correlation between $h\rightarrow\gamma\gamma$ and $h\rightarrow Z\gamma$
has been examined in some Refs.loop2 ; BhupalDev:2013xol ; Chen:2013dh ;
Chiang:2012qz ; Chen:2013vi ; Chabab:2018ert . An interesting thing is that
correlations between the $h\rightarrow\gamma\gamma$ and $h\rightarrow Z\gamma$
decays appear in some parameter space in our model by imposing the
$h\rightarrow\gamma\gamma$ decay measured by the LHC.

In particular, we study the Type-I and Type-Z 3HDM among the five Yukawa types
of 3HDMs. However, one can distinguish these Yukawa types in in gluonic decay
modes. Only heavy quark loops contribute to the gluonic decays, while
additional new charged Higgs bosons $H^{\pm}_{1,2}$ enter the loop in the
$\gamma\gamma$ and $Z\gamma$ decay modes as well as the SM particles. The
heavy quark contributions are small compared to new charged Higgs bosons
contributions in the $\gamma\gamma$ and $Z\gamma$ mode, hence Yukawa types in
the 3HDM could not be distinguished in these decay modes. In general, one of
the most important parameters to distinguish different Yukawa types in the
3HDM is denoted as $\tan\beta$ and $\tan\gamma$. The gluonic and photonic
decay widths are significant to investigate the Higgs production at hadron and
photon colliders, where the cross sections will be directly proportional to,
respectively, the gluonic and photonic decay widths.

Our work is organized as follows. In section II, we briefly describe the
theoretical structure of the 3HDM. In section III, we present the couplings of
the particles in the 3HDM relevant to our calculation. Analytic formulas
needed for calculating the decay widths of the
$h\rightarrow\gamma\gamma,~{}\gamma Z,~{}gg$ are presented in section IV, and
we discuss the numerical results of the partial decay widths and branching
ratios in the 3HDM. We finally summarize our results in section V. In the
appendix, we collect expressions for the loop functions used in our
calculations. Moreover, details on the analysis of the scalar potential and
the coefficients of the triple scalar couplings are given in the appendix.

##  II Three Higgs Doublet Model

Yukawa Types | $\Phi_{u}$ | $\Phi_{d}$ | $\Phi_{e}$
---|---|---|---
Type-I | $\Phi_{2}$ | $\Phi_{2}$ | $\Phi_{2}$
Type-II | $\Phi_{2}$ | $\Phi_{1}$ | $\Phi_{1}$
Type-X | $\Phi_{2}$ | $\Phi_{2}$ | $\Phi_{1}$
Type-Y | $\Phi_{2}$ | $\Phi_{1}$ | $\Phi_{2}$
Type-Z | $\Phi_{2}$ | $\Phi_{1}$ | $\Phi_{3}$
Table 1: The five physically-distinct types of Yukawa interactions under the $Z_{2}\times\tilde{Z}_{2}$ symmetries my3hdm .  | Factor for $\tilde{H}_{1}$, $\tilde{A}_{1}$ and $\tilde{H}_{1}^{\pm}$ | Factor for $\tilde{H}_{2}$, $\tilde{A}_{2}$ and $\tilde{H}_{2}^{\pm}$
---|---|---
| $R_{u2}/R_{u1}$ | $R_{d2}/R_{d1}$ | $R_{e2}/R_{e1}$ | $R_{u3}/R_{u1}$ | $R_{d3}/R_{d1}$ | $R_{e3}/R_{e1}$
Type-I | $\cot\beta$ | $\cot\beta$ | $\cot\beta$ | 0 | 0 | 0
Type-II | $\cot\beta$ | $-\tan\beta$ | $-\tan\beta$ | 0 | $-\tan\gamma/\cos\beta$ | $-\tan\gamma/\cos\beta$
Type-X | $\cot\beta$ | $\cot\beta$ | $-\tan\beta$ | 0 | 0 | $-\tan\gamma/\cos\beta$
Type-Y | $\cot\beta$ | $-\tan\beta$ | $\cot\beta$ | 0 | $-\tan\gamma/\cos\beta$ | 0
Type-Z | $\cot\beta$ | $-\tan\beta$ | $-\tan\beta$ | 0 | $-\tan\gamma/\cos\beta$ | $\cot\gamma/\cos\beta$
Table 2: Factors appearing in Eq. (15) for each type of Yukawa interaction.

The most general Higgs potential invariant under the $SU(2)_{L}\times
U(1)_{Y}\times Z_{2}\times\tilde{Z}_{2}$ symmetry is expressed by my3hdm

$\displaystyle V(\Phi_{1},\Phi_{2},\Phi_{3})$
$\displaystyle=\sum_{i=1}^{3}m_{i}^{2}\Phi_{i}^{\dagger}\Phi_{i}-(m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+m_{13}^{2}\Phi_{1}^{\dagger}\Phi_{3}+m_{23}^{2}\Phi_{2}^{\dagger}\Phi_{3}+\text{h.c.})$
$\displaystyle+\frac{1}{2}\sum_{i=1}^{3}\lambda_{i}(\Phi_{i}^{\dagger}\Phi_{i})^{2}+\rho_{1}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\rho_{2}|\Phi_{1}^{\dagger}\Phi_{2}|^{2}+\frac{1}{2}[\rho_{3}(\Phi_{1}^{\dagger}\Phi_{2})^{2}+\text{h.c.}]$
$\displaystyle+\sigma_{1}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{3}^{\dagger}\Phi_{3})+\sigma_{2}|\Phi_{1}^{\dagger}\Phi_{3}|^{2}+\frac{1}{2}[\sigma_{3}(\Phi_{1}^{\dagger}\Phi_{3})^{2}+\text{h.c.}]$
$\displaystyle+\kappa_{1}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\kappa_{2}|\Phi_{2}^{\dagger}\Phi_{3}|^{2}+\frac{1}{2}[\kappa_{3}(\Phi_{2}^{\dagger}\Phi_{3})^{2}+\text{h.c.}],$
(1)

where we allow the presence of the soft breaking terms $m_{12}^{2}$,
$m_{13}^{2}$ and $m_{23}^{2}$ for the discrete $Z_{2}$ symmetries.111We use
here the labelling $m_{12},~{}m_{13},~{}m_{23}$ in place of
$\mu_{12},~{}\mu_{13},~{}\mu_{23}$ in Ref. my3hdm , respectively. To avoid
explicit CP violation, we take all the parameters of the scalar potential to
be real.

In order to suppress FCNCs at tree level, we will enforce each type of fermion
to couple to only a single Higgs doublet in the Yukawa sector. In the 3HDM,
this condition leads to the five physically-distinct “types” given in Tab. 1.
The Yukawa Lagrangian of the 3HDM reads

$\displaystyle-{\cal
L}_{Y}=Y_{u}\bar{Q}_{L}(i\sigma_{2})\Phi_{u}^{*}u_{R}+Y_{d}\bar{Q}_{L}\Phi_{d}d_{R}+Y_{e}\bar{L}_{L}\Phi_{e}e_{R}+\text{h.c.},$
(2)

where $\Phi_{u,d,e}$ are either $\Phi_{1}$, $\Phi_{2}$ or $\Phi_{3}$.

The general 3HDM contains three scalar $SU(2)_{L}$ doublet fields, denoted
here as

$\displaystyle\Phi_{i}=\left[\begin{array}[]{cc}\omega_{i}^{+}\\\
\frac{1}{\sqrt{2}}(h_{i}+v_{i}+iz_{i})\end{array}\right],~{}~{}(i=1,...3),$
(5)

where the $v_{i}$’s are the VEVs of the $\Phi_{i}$’s with the sum rule
$\sum_{i}v_{i}^{2}\equiv v^{2}=1/(\sqrt{2}G_{F})\simeq(246$ GeV$)^{2}$. The
VEVs may be introduced by the two ratios

$\displaystyle\tan\beta\equiv\frac{v_{2}}{v_{13}},\quad\tan\gamma\equiv\frac{v_{3}}{v_{1}},~{}~{}\text{with}~{}~{}v_{13}\equiv\sqrt{v_{1}^{2}+v_{3}^{2}}.$
(6)

Using this notation, each of the VEVs is parametrized by

$\displaystyle v_{1}=v_{13}\cos\gamma=v\cos\beta\cos\gamma,\quad
v_{2}=v\sin\beta,\quad v_{3}=v_{13}\sin\gamma=v\cos\beta\sin\gamma.$ (7)

Scalar mass eigenstates can be obtained by rotating the fields. The rotation
is performed in two stages. The first stage is to transform all scalar fields
from a weak basis with a generic choice of vacua to the so-called Higgs basis
higgsbasis ; higgsbasis2 , where only 1-Higgs doublet contains the SM VEV $v$
and the Nambu-Goldstone (NG) bosons. This can be done by rotating to the Higgs
basis using the orthogonal $3\times 3$ matrix $R$ as follows:

$\displaystyle\begin{pmatrix}h_{1}\\\ h_{2}\\\
h_{3}\end{pmatrix}=R\begin{pmatrix}\tilde{h}\\\ \tilde{H_{1}}\\\
\tilde{H_{2}}\end{pmatrix},~{}\begin{pmatrix}w_{1}^{\pm}\\\ w^{\pm}_{2}\\\
w^{\pm}_{3}\end{pmatrix}=R\begin{pmatrix}G^{\pm}\\\ \tilde{H^{\pm}_{1}}\\\
\tilde{H_{2}^{\pm}}\end{pmatrix},~{}\begin{pmatrix}z_{1}\\\ z_{2}\\\
z_{3}\end{pmatrix}=R\begin{pmatrix}G^{0}\\\ \tilde{A_{1}}\\\
\tilde{A_{2}}\end{pmatrix}.$ (8)

The $R$ matrix is expressed in terms of the three VEVs, following the notation
of Ref. my3hdm

$\displaystyle R$
$\displaystyle=\begin{pmatrix}\frac{v_{1}}{v}&-\frac{v_{2}v_{1}}{v_{13}v}&-\frac{v_{3}}{v_{13}}\\\
\frac{v_{2}}{v}&\frac{v_{13}}{v}&0\\\
\frac{v_{3}}{v}&-\frac{v_{2}v_{3}}{v_{13}v}&\frac{v_{1}}{v_{13}}\end{pmatrix}=\begin{pmatrix}\cos\gamma&0&-\sin\gamma\\\
0&1&0\\\ \sin\gamma&0&\cos\gamma\\\
\end{pmatrix}\begin{pmatrix}\cos\beta&-\sin\beta&0\\\ \sin\beta&\cos\beta&0\\\
0&0&1\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\cos\beta\cos\gamma&-\sin\beta\cos\gamma&-\sin\gamma\\\
\sin\beta&\cos\beta&0\\\
\cos\beta\sin\gamma&-\sin\beta\sin\gamma&\cos\gamma\end{pmatrix}.$ (9)

After spontaneous symmetry breaking (SSB), the NG bosons $G^{\pm}$ and $G^{0}$
are absorbed into the longitudinal components of the $W^{\pm}$ and $Z$
respectively. In Eq. (6), two singly-charged states $\tilde{H}_{a}^{\pm}$, two
CP-odd states $\tilde{A}_{a}$ and three CP-even states $\tilde{h}$ and
$\tilde{H}_{a}$ are not given in the mass eigenstates. The next stage is to
determine the scalar mass eigenstates through the orthogonal transformations

$\displaystyle\begin{pmatrix}G^{\pm}\\\ H^{\pm}_{1}\\\
H_{2}^{\pm}\end{pmatrix}=O_{c}^{\dagger}\begin{pmatrix}w^{\pm}_{1}\\\
w^{\pm}_{2}\\\
w^{\pm}_{3}\end{pmatrix},~{}O_{\theta_{2}}=\begin{pmatrix}1&0&0\\\
0&\cos(\theta_{2})&-\sin(\theta_{2})\\\ 0&\sin(\theta_{2})&\cos(\theta_{2})\\\
\end{pmatrix}$ (10) $\displaystyle\begin{pmatrix}G^{0}\\\ A_{1}\\\
A_{2}\end{pmatrix}=O^{\dagger}_{A}\begin{pmatrix}z_{1}\\\ z_{2}\\\
z_{3}\end{pmatrix},~{}O_{\theta_{1}}=\begin{pmatrix}1&0&0\\\
0&\cos(\theta_{1})&-\sin(\theta_{1})\\\ 0&\sin(\theta_{1})&\cos(\theta_{1})\\\
\end{pmatrix},$ (11)

where $\theta_{2}$ and $\theta_{1}$ are the mixing angles for the charged and
CP-odd states, respectively.
$O_{c}=O_{\theta_{2}}R,~{}~{}O_{A}=O_{\theta_{1}}R.$

For the CP-even states, we can obtain the mass eigenstates through

$\displaystyle\begin{pmatrix}\tilde{h}_{1}\\\ \tilde{H}_{1}\\\
\tilde{H}_{2}\end{pmatrix}=O^{h}\begin{pmatrix}h\\\ H_{1}\\\
H_{2}\end{pmatrix}.$ (12)

where $O^{h}=R^{T}O$ and

$\displaystyle O=R_{23}(\theta)R_{12}(\alpha_{1})R_{13}(\alpha_{2})$ (13)

where $\alpha_{1,2},~{}\theta$ mixing angles in the CP-even sector.

$\displaystyle
R_{13}(\alpha_{2})=\begin{pmatrix}\cos\alpha_{2}&-\sin\alpha_{2}&0\\\
\sin\alpha_{2}&\cos\alpha_{2}&0\\\
0&0&1\end{pmatrix},~{}R_{12}(\alpha_{1})=\begin{pmatrix}\cos\alpha_{1}&0&-\sin\alpha_{1}\\\
0&1&0\\\
\sin\alpha_{1}&0&\cos\alpha_{1}\end{pmatrix},~{}R_{23}(\theta)=\begin{pmatrix}1&0&0\\\
0&\cos(\theta)&-\sin(\theta)\\\ 0&\sin(\theta)&\cos(\theta)\\\ \end{pmatrix}.$
(14)

The masses of the three neutral physical states, $h,$ $H_{1}$ and $H_{2}$, are
evaluated in detail in Appendix A.3. Consequently, the model has nine scalar
physical particles: (i) three CP-even scalars $(h,H_{1,2}),$ (ii) two CP-odd
scalars $(A_{1,2}),$ and (iii) four charged scalars$(H^{\pm}_{1,2}).$ In this
paper, we focus on the physics related to the Higgs bosons
$h,~{}H_{1},~{}H_{2}$, $H_{1}^{\pm}$ and $H_{2}^{\pm}$, for which there are
eighteen significant parameters, namely

$\displaystyle
m_{h},~{}m_{H_{1}},~{}m_{H_{2}},~{}m_{H_{1}^{\pm}},~{}m_{H_{2}^{\pm}},~{}m_{A_{1}},~{}m_{A_{2}},~{}v,~{}\tan\beta,~{}\tan\gamma,~{}\theta_{1},~{}\theta_{2},~{}\alpha_{1,2},~{}\theta,m_{12},~{}m_{23},~{}\text{and}~{}m_{13}.$

###  II.1 The Yukawa sector

The Yukawa couplings of the Higgs bosons are given by my3hdm

$\displaystyle-{\cal
L}_{Y}=\sum_{f=u,d,e}\frac{m_{f^{i}}}{v}\bar{f}^{i}\left[\left(\tilde{h}+\frac{R_{f2}}{R_{f1}}\tilde{H}_{1}+\frac{R_{f3}}{R_{f1}}\tilde{H_{2}}\right)-2I_{f}\left(\frac{R_{f2}}{R_{f1}}\tilde{A}_{1}+\frac{R_{f3}}{R_{f1}}\tilde{A_{2}}\right)\gamma_{5}\right]f^{i}$
$\displaystyle+\frac{\sqrt{2}}{v}\left[\bar{u}^{j}V_{ji}m_{d^{i}}\left(\frac{R_{d2}}{R_{d1}}\tilde{H}_{1}^{+}+\frac{R_{d3}}{R_{d1}}\tilde{H}_{2}^{+}\right)P_{R}d^{i}-\bar{u}^{i}m_{u^{i}}V_{ij}\left(\frac{R_{u2}}{R_{u1}}\tilde{H}_{1}^{+}+\frac{R_{u3}}{R_{u1}}\tilde{H}_{2}^{+}\right)P_{L}d^{j}\right]+\text{h.c.}$
$\displaystyle+\frac{\sqrt{2}}{v}\bar{\nu}^{i}m_{e^{i}}\left(\frac{R_{e2}}{R_{e1}}\tilde{H}_{1}^{+}+\frac{R_{e3}}{R_{e1}}\tilde{H}_{2}^{+}\right)P_{R}e^{i}+\text{h.c.}$
(15)

where $I_{f}=+1/2\,(-1/2)$ for $f=u\,(d,e)$ and $V_{ij}$ is the Cabibbo-
Kobayashi-Maskawa (CKM) matrix element. The ratios of the matrix elements
$R_{f2}/R_{f1}$ and $R_{f3}/R_{f1}$ ($f=u,d,e$) are collected in Tab. 2 for
each of the five types of Yukawa interactions.

###  II.2 Kinetic Lagrangian

It is useful to take a closer look at kinetic Lagrangian to extract the
trilinear scalar-gauge-gauge couplings

$\displaystyle{\cal
L}_{kin}=\sum_{i=1}^{3}|D_{\mu}\Phi_{i}|^{2}\ni\dfrac{g^{2}}{2}W^{+}_{\mu}W^{\mu-}\left(\sum_{i=1}^{3}\nu_{i}h_{i}\right)$
(16)
$\displaystyle=\dfrac{g^{2}\nu}{2}W^{+}_{\mu}W^{-\mu}\left(\frac{1}{\nu}\sum_{i=1}^{3}\nu_{i}h_{i}\right)$

where g represents the $SU(2)_{L}$ gauge coupling.

##  III Relavant COUPLINGS AND ANALYTIC FORMULAS

###  III.1 Scalar couplings

The scalar, $hH^{+}_{i}H^{-}_{i}~{}(i=1,2)$, trilinear couplings, which are
relevant to the decay widths discussed in the next section, could be expressed
through $\beta,\gamma,\alpha_{1,2},\theta$ and other parameters as follows:

$\displaystyle\lambda_{hH_{1}^{+}H_{1}^{-}}=a~{}m_{h_{1}}^{2}+b~{}m_{H_{1}^{\pm}}^{2}+c~{}m_{H_{2}^{\pm}}^{2}+d~{}m_{12}^{2}+e~{}m_{13}^{2}+f~{}m_{23}^{2}$
(17)
$\displaystyle\lambda_{hH_{2}^{+}H_{2}^{-}}=a_{1}~{}m_{h_{1}}^{2}+b_{1}~{}m_{H_{1}^{\pm}}^{2}+c_{1}~{}m_{H_{2}^{\pm}}^{2}+d_{1}~{}m_{12}^{2}+e_{1}~{}m_{13}^{2}+f_{1}~{}m_{23}^{2}$
(18)

$a,b,c,d,e,f,a_{1},b_{1},c_{1},d_{1},e_{1},f_{1}$, which are coefficients of
$hH^{+}_{i}H^{-}_{i}$ terms in the potential, are expressed explicitly in
Appendix B.

###  III.2 Yukawa couplings

To calculate the Yukawa couplings of the neutral Higgs bosons, we define the
coupling coefficients $B_{1,2,3}$ in terms of the elements of the matrix R
given in Tab. 2 my3hdm . In this case, the Yukawa couplings of the neutral
Higgs boson takes the following form:

$\displaystyle-{\cal
L}_{Y}=\sum_{f=u,d,e}\frac{m_{f}}{v}\bar{f}\left[\left(hB_{1}+H_{1}B_{2}+H_{2}B_{3}\right]\right)f,$
(19)

we find

$\displaystyle B_{1}$
$\displaystyle=c_{\alpha_{1}}c_{\alpha_{2}}c_{\beta}c_{\gamma}-\frac{c_{\alpha_{1}}c_{\alpha_{2}}c_{\gamma}R_{\text{f2}}s_{\beta}}{R_{\text{f1}}}-\frac{c_{\alpha_{1}}c_{\alpha_{2}}R_{\text{f3}}s_{\gamma}}{R_{\text{f1}}}+\frac{c_{\alpha_{2}}c_{\gamma}c_{\theta}R_{\text{f3}}s_{\alpha_{1}}}{R_{\text{f1}}}+c_{\alpha_{2}}c_{\beta}c_{\theta}s_{\alpha_{1}}s_{\gamma}-$
(20)
$\displaystyle\frac{c_{\alpha_{2}}c_{\theta}R_{\text{f2}}s_{\alpha_{1}}s_{\beta}s_{\gamma}}{R_{\text{f1}}}-\frac{c_{\alpha_{2}}c_{\beta}R_{\text{f2}}s_{\alpha_{1}}s_{\theta}}{R_{\text{f1}}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\beta}s_{\theta}+\frac{c_{\beta}c_{\theta}R_{\text{f2}}s_{\alpha_{2}}}{R_{\text{f1}}}+$
$\displaystyle\frac{c_{\gamma}R_{\text{f3}}s_{\alpha_{2}}s_{\theta}}{R_{\text{f1}}}+c_{\beta}s_{\alpha_{2}}s_{\gamma}s_{\theta}+c_{\theta}s_{\alpha_{2}}s_{\beta}-\frac{R_{\text{f2}}s_{\alpha_{2}}s_{\beta}s_{\gamma}s_{\theta}}{R_{\text{f1}}}$
$\displaystyle B_{2}$
$\displaystyle=\frac{c_{\alpha_{2}}c_{\beta}c_{\theta}R_{\text{f2}}}{R_{\text{f1}}}+c_{\alpha_{2}}c_{\theta}s_{\beta}+\frac{c_{\alpha_{2}}c_{\gamma}R_{\text{f3}}s_{\theta}}{R_{\text{f1}}}+c_{\alpha_{2}}c_{\beta}s_{\gamma}s_{\theta}-\frac{c_{\alpha_{2}}R_{\text{f2}}s_{\beta}s_{\gamma}s_{\theta}}{R_{\text{f1}}}-c_{\alpha_{1}}c_{\beta}c_{\gamma}s_{\alpha_{2}}+$
(21)
$\displaystyle\frac{c_{\alpha_{1}}c_{\gamma}R_{\text{f2}}s_{\alpha_{2}}s_{\beta}}{R_{\text{f1}}}+\frac{c_{\alpha_{1}}R_{\text{f3}}s_{\alpha_{2}}s_{\gamma}}{R_{\text{f1}}}-\frac{c_{\gamma}c_{\theta}R_{\text{f3}}s_{\alpha_{1}}s_{\alpha_{2}}}{R_{\text{f1}}}-$
$\displaystyle\frac{c_{\theta}R_{\text{f2}}s_{\alpha_{1}}s_{\alpha_{2}}s_{\beta}s_{\gamma}}{R_{\text{f1}}}+\frac{c_{\beta}R_{\text{f2}}s_{\alpha_{1}}s_{\alpha_{2}}s_{\theta}}{R_{\text{f1}}}+c_{\beta}c_{\theta}s_{\alpha_{1}}s_{\alpha_{2}}s_{\gamma}+s_{\alpha_{1}}s_{\alpha_{2}}s_{\beta}s_{\theta}$
$\displaystyle B_{3}$
$\displaystyle=\frac{c_{\alpha_{1}}c_{\gamma}c_{\theta}R_{\text{f3}}}{R_{\text{f1}}}+c_{\alpha_{1}}c_{\beta}c_{\theta}s_{\gamma}-\frac{c_{\alpha_{1}}c_{\theta}R_{\text{f2}}s_{\beta}s_{\gamma}}{R_{\text{f1}}}-\frac{c_{\alpha_{1}}c_{\beta}R_{\text{f2}}s_{\theta}}{R_{\text{f1}}}+$
(22)
$\displaystyle\frac{c_{\gamma}R_{\text{f2}}s_{\alpha_{1}}s_{\beta}}{R_{\text{f1}}}-c_{\beta}c_{\gamma}s_{\alpha_{1}}-c_{\alpha_{1}}s_{\beta}s_{\theta}+\frac{R_{\text{f3}}s_{\alpha_{1}}s_{\gamma}}{R_{\text{f1}}},$

where $c_{x}$, $s_{x}$ and $t_{x}$ are shorthand notations for $\cos x$, $\sin
x$ and $\tan x$, respectively.
These $B_{1,2,3}$ coefficients stand for the couplings
$g_{hff},~{}g_{H_{1}ff}$ and $g_{H_{2}ff}$, receptively.

###  III.3 Scalar-gauge-gauge couplings

The gauge$-$gauge$-$scalar type interactions are extracted from Eq.(14) as

$\displaystyle hW^{+}{}_{\mu}W^{\mu-}$
$\displaystyle=\frac{1}{2}g^{2}v\left(c_{\beta}\left(c_{\alpha_{1}}c_{\alpha_{2}}c_{\gamma}+s_{\gamma}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\right)+s_{\beta}\left(c_{\theta}s_{\alpha_{2}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}\right)\right)$
(23)

##  IV LOOP INDUCED DECAYS INTO $\gamma\gamma,~{}\gamma Z,~{}gg$

In the 3HDM, the decay widths of the $h\rightarrow\gamma\gamma,Z\gamma$
receive contributions from the new charged scalars $H^{\pm}_{1,2}$. The decay
widths of $h$ into $\gamma\gamma,~{}\gamma Z~{}\text{and}~{}gg$ are given
explicitly by djouadi ; djouadi1 ; hunter

$\displaystyle\Gamma(h\rightarrow\gamma\gamma)=$
$\displaystyle\frac{G_{F}\alpha^{2}m_{h}^{3}}{128\sqrt{2}\pi^{3}}\mid\sum_{f}N_{c}Q_{f}^{2}g_{hff}A_{1/2}(\tau_{f})+g_{hWW}A_{1}(\tau_{W})$
(24)
$\displaystyle+\frac{m_{W}^{2}\lambda_{hH^{+}_{i}H^{-}_{i}}}{2c_{w}^{2}m_{H^{\pm}_{i}}}A_{0}(\tau_{H^{\pm}_{i}})\mid^{2},$
$\displaystyle\Gamma(h\rightarrow Z\gamma)=$
$\displaystyle\frac{G_{F}^{2}m_{W}^{2}\alpha
m_{h}^{3}}{64\pi^{4}}\left(1-\frac{m_{Z}^{2}}{m_{h}^{2}}\right)\mid\sum_{f}g_{hff}\frac{N_{c}Q_{f}\hat{v}_{f}}{c_{w}}A_{1/2}(\tau_{f},\lambda_{f})$
(25)
$\displaystyle+g_{hWW}A_{1}(\tau_{W},\lambda_{W})+\frac{m_{W}^{2}v_{H^{\pm}_{i}}}{2c_{w}^{2}m_{H^{\pm}_{i}}}\lambda_{hH^{+}_{i}H^{-}_{i}}A_{0}(\tau_{H^{\pm}_{i}})\mid^{2}$
$\displaystyle\Gamma(h\rightarrow gg)$
$\displaystyle=\frac{G_{F}\alpha_{s}^{2}m^{3}_{h}}{36\sqrt{2}\pi^{3}}\mid\frac{3}{4}\sum_{f}g_{hff}A_{1/2}(\tau_{f})\mid^{2}$
(26)

with $\hat{v}_{f}=2I_{f}^{3}-4Q_{f}s^{2}_{W}$ and
$v_{H^{\pm}_{i}}=\frac{2c_{W}^{2}-1}{c_{W}}$.

Here $N_{c}$ is the color factor $N_{c}=3(1)$ for quarks(leptons) and $Q_{f}$
stands for the electric charge of a particle in the loop. $\alpha$ and
$\alpha_{s}$ are the fine-structure constant and strong coupling constant,
respectively. Dimensionless one-loop factors $A_{1}$ (for the W boson),
$A_{1/2}$ (for the fermions, $f=t,b,\tau$) and $A_{0}$ (for the charged
scalars, $H^{\pm}_{1,2}$ ) are written in Appendix C. The couplings $g_{hff}$
and $g_{hW^{+}W^{-}}$ and $\lambda_{hH^{+}_{i}H^{-}_{i}}$ are expressed in
Section III.

The decay $h\rightarrow Z\gamma$ in the 3HDM receives contributions from the
new charged scalars $H^{\pm}_{1,2}$, hence its decay width could be enhanced
in some parameter regions.

###  IV.1  $h\rightarrow\gamma\gamma$

We now consider the decay width of $h\rightarrow\gamma\gamma$ in the Type-I
3HDM. We first discuss constraints from the LHC measurements of the decay of
$h\rightarrow\gamma\gamma$ (Ref. ATLAS:2018hxb ) in our model parameter
spaces. By using the measured value of $h\rightarrow\gamma\gamma$
ATLAS:2018hxb , we perform an extensive scan within the parameter space of the
model. We then report the allowed values of the model parameters in Table 3 to
apply in all figures in subsection B.

| $\tan\gamma$ | $\tan\beta=1$ | $\tan\beta=2$ | $\tan\beta=3$
---|---|---|---
$\tan\gamma=1$ |  $\alpha_{1,2}=\theta=\frac{\pi}{3}$ $m_{H_{1}^{\pm}}=205,300~{}\text{GeV}$ |  $\alpha_{1,2}=\theta=\frac{\pi}{3}$ $m_{H_{1}^{\pm}}=319,500~{}\text{GeV}$ |  $\alpha_{1,2}=\theta=\frac{\pi}{3}$ $m_{H_{1}^{\pm}}=457,760~{}\text{GeV}$
$\tan\gamma=1$ |  $\alpha_{1,2}=\theta=\frac{13\pi}{45},\frac{14\pi}{45},\frac{\pi}{3},\frac{67\pi}{90}$ $m_{H_{1}^{\pm}}=300~{}\text{GeV}$ |  $\alpha_{1,2}=\theta=\frac{\pi}{3},\frac{4\pi}{9},\frac{41\pi}{90}$ $m_{H_{1}^{\pm}}=300~{}\text{GeV}$ |
$\tan\gamma=2$ |  $\alpha_{1,2}=\theta=\frac{\pi}{9},\frac{4\pi}{9}$ $m_{H_{1}^{\pm}}=290~{}\text{GeV}$ |  |
$\tan\gamma=3$ |  $\alpha_{1,2}=\theta=\frac{23\pi}{90},\frac{42\pi}{90},\frac{49\pi}{90}$ $m_{H_{1}^{\pm}}=320~{}\text{GeV}$ |  |

Table 3: Allowed values of the parameters in a Type-I 3HDM obtained by
performing the measured value of $h\rightarrow\gamma\gamma$ decay at the LHC.
Here we consider $m_{h}=125$ GeV, $\theta_{2}=\pi/6$,
$m_{H_{2}^{\pm}}=m_{H_{1}^{\pm}}+100$ GeV and $m_{12}=m_{23}=m_{13}=200$ GeV.

In Fig. 1, contour plots in the ($\tan\gamma,m_{H^{\pm}_{1}})$ and
$(m_{H^{\pm}_{1}},\alpha_{1,2}=\theta$ ) planes show the partial decay width
for $h\rightarrow\gamma\gamma$. In the plots, the allowed regions are bounded
by the contour lines by applying the measured values of Higgs to diphotons at
the LHC. We observed from the left panel of Fig. 1 that the experimental
constraint rules out the region of $\tan\gamma\geq 10$ for the cases of
$\alpha_{1,2}=\theta=\frac{\pi}{3}$ and $\tan\beta\geq 1$. The left panel
shows that there are allowed regions of the parameter space in the range of
$200~{}\text{GeV}\leq m_{H^{\pm}_{1,2}}\leq 750~{}\text{GeV}$ for the cases of
$\tan\gamma=1$ and $1\leq\tan\beta\leq 3$. It is clear from the center and
right panels of Fig. 1 that there are allowed parameter spaces for the cases
of $\frac{\pi}{10}\leq\alpha_{1,2},~{}\theta\leq\frac{13\pi}{30},$
$\tan\gamma=1,2$ and $1\leq\tan\beta\leq 3$. We can see that allowed regions
depend heavily upon the choice of
$\tan\beta,~{}\tan\gamma,~{}\text{and}~{}m_{H_{1,2}^{\pm}}$. In addition, our
scan shows that the curves in Fig. 1 are not much sensitive with the changes
of $\alpha_{1,2},\theta$ for the case of $\tan\gamma=1$. We have fixed the
mixing angle $\theta_{2}=\pi/6$ in all figures.

Figure 1:  The contour lines show the allowed points of parameter spaces from
the measurements of $h\rightarrow\gamma\gamma$ decay at the LHC in the left
plane $(\tan\gamma,~{}m_{H_{1}^{\pm}})$ and center and right planes
$(m_{H_{1}^{\pm}},~{}\alpha_{1,2}=\theta)$ in the Type-I 3HDM. The other
parameters are fixed to be: $m_{h}=125$ GeV, $\theta_{2}=\pi/6$,
$m_{H_{2}^{\pm}}=m_{H_{1}^{\pm}}+100$ GeV and $m_{12}=m_{23}=m_{13}=200$ GeV.

###  IV.2  $h\rightarrow Z\gamma$

In this section, we will examine the decay partial width of $h\rightarrow
Z\gamma$ while respecting the parameter spaces that are not ruled out by the
$\Gamma(h\rightarrow\gamma\gamma)$ analysis. Using the values of parameters in
Table 3, we first focus on the decay width of $\Gamma(h\rightarrow Z\gamma)$
in the Type I 3HDM. The dependence of $\Gamma(h\rightarrow Z\gamma)$ on
$m_{H_{1}^{\pm}}$ is shown on the upper panels of Fig. 2 for $\tan\gamma=1$
and different choice of $\tan\beta$. The curves show that the decay width is
sensitive to the charged Higgs bosons masses. One can see that the decay width
drops sharply with the charged Higgs masses $m_{H^{\pm}_{1,2}}$ increasing.
The decay width is sensitive to the choice of $\tan\beta$. On the lower panel
of Fig. 2, we interpret the dependence of the decay with on the CP-even mixing
angles $\alpha_{1,2},~{}\theta$ for various values of $\tan\gamma$ and
$m_{H_{1}^{\pm}}.$ We see that in order to have large values of $h\rightarrow
Z\gamma$ small values of the CP-even mixing angles are required.

Figure 2: The decay width of $h\rightarrow Z\gamma$ as a function of
$m_{H^{\pm}_{1}}$(upper panel) and $\alpha_{1,2}=\theta$ (lower panel) in the
Type-I 3HDM. We take $m_{H_{2}^{\pm}}=m_{H_{1}^{\pm}}+100$ GeV, $m_{h}=125$
GeV, $m_{12}=m_{23}=m_{13}=200$ GeV, and $\theta_{2}=\pi/6.$

In Fig. 3, we present the deviation in the partial decay width of
$h\rightarrow Z\gamma$ relative to the SM prediction. On the left panel range
$319\leq m_{H_{1}^{\pm}}\leq 500$ is plotted, and on the right panel the range
$23\pi/90\leq\alpha_{1,2}=\theta\leq 49\pi/90$ is plotted. From these plots,
it is possible to see that the deviation in $\Gamma(h\rightarrow Z\gamma)$
increases with large values of $m_{H^{\pm}_{1,2}}$ and $\alpha_{1,2}=\theta$.
On the right panel, the partial decay width of $h\rightarrow Z\gamma$ with
respect to the SM one enhances about by a factor 2 for the case of
$\tan\gamma=3$ and $\alpha_{1,2}=\theta=23\pi/90.$ As can be seen from the
left panel, for $m_{H_{1}^{\pm}}=319$ GeV, the deviation from the SM
prediction is about 10 % when $\tan\beta=2$ and $\tan\gamma=1$.

Figure 3: Dependence of $\Gamma(h\rightarrow
Z\gamma)_{\text{3HDM}}/\Gamma(h\rightarrow Z\gamma)_{\text{SM}}$ on
$m_{H_{1}^{\pm}}$(left panel) and $\alpha_{1,2},~{}\theta$ (right panel) in
the Type-I 3HDM. We take $m_{H_{1}^{\pm}}=m_{H_{2}^{\pm}}=100$ GeV ,
$m_{h}=125$ GeV, $\theta_{2}=\pi/6$ and $m_{12}=m_{23}=m_{13}=200$ GeV.

To compare the branching ratio of BR$(h\rightarrow Z\gamma)$ in the Type I
3HDM to the branching ratio of BR$(h\rightarrow Z\gamma)$ in the SM, we define
the ratio of two branching ratios

$\displaystyle R_{\text{BR}}=\frac{BR(h\rightarrow
Z\gamma)^{\text{3HDM}}}{BR(h\rightarrow Z\gamma)^{\text{SM}}},$ (27)

where BR$(h\rightarrow Z\gamma)^{\text{SM}}$ represents the branching ratio in
the SM.

Figure 4: Dependence of $\text{BR}(h\rightarrow
Z\gamma)_{\text{3HDM}}/\text{BR}(h\rightarrow Z\gamma)_{\text{SM}}$ on
$m_{H_{1}^{\pm}}$ (left panel) and $\alpha_{1,2},~{}\theta$ (right panel) in
the Type-I 3HDM. We take $m_{H_{1}^{\pm}}=m_{H_{2}^{\pm}}+100$ GeV,
$m_{h}=125$ GeV, $\theta_{2}=\pi/6$ and $m_{12}=m_{23}=m_{13}=200$ GeV.

Fig. 4 shows the deviation in the branching ratio of BR$(h\rightarrow
Z\gamma)^{\text{3HDM}}$ with respect to the SM prediction as a function of
$m_{H_{1}^{\pm}}$ in the Type I-3HDM. $R_{BR}>1$ occurs due to the large
contribution of the $H^{\pm}_{1,2}$ loop. The left plot in Fig. 4 presents a
deviation from the SM prediction at small values of $m_{H^{\pm}_{1}}$ and
$\tan\beta=2.$ The right plot in Fig. 4 appears to be compatible with the SM
prediction for $\tan\beta=1,$ $\tan\gamma=3$ and the large values of
$\alpha_{1,2}=\theta$. As can be seen from the left plot, BR$(h\rightarrow
Z\gamma)^{\text{3HDM}}$ is close to the SM prediction at values of
$m_{H_{1}^{\pm}}=500$ GeV. The current searches at the LHC have a limit of
about 4 times the $h\rightarrow Z\gamma$ signal strength in the Standard Model
ATLAS:2020qcv . Thus, this decay would be observed in the high-luminosity (HL)
run of the LHC when $R_{BR}\simeq 1$. It is clear from the left and right
plots that the enhancement of the $h\rightarrow Z\gamma$ branching ratio is so
large in some regions of parameter space hence the HL-LHC could have the
potential to exclude this parameter region.

Note that the ratio of two branching ratios BR$(h\rightarrow
Z\gamma)^{\text{3HDM}}/\text{BR}(h\rightarrow Z\gamma)^{\text{SM}}$ is larger
than that of two partial widths $\Gamma(h\rightarrow
Z\gamma)^{\text{3HDM}}/\Gamma(h\rightarrow Z\gamma)^{\text{SM}}$. In the
Type-I 3HDM, the total decay width is smaller than the SM prediction because
the partial widths of $(h\rightarrow\bar{f}f,~{}gg)$ are significantly
suppressed with respect to their SM values. As a result, there is an
enhancement in the BR$(h\rightarrow Z\gamma)^{\text{3HDM}}$ relative to its SM
value.

We note that we have explicitly checked that the partial decay widths of
$(h\rightarrow Z\gamma(\gamma\gamma))$ in the Type-I 3HDM are nearly the same
as those in the Type-Z 3HDM.

$\displaystyle R_{Z\gamma(\gamma\gamma)}=\frac{\Gamma(h\rightarrow
Z\gamma(\gamma\gamma))^{\text{3HDM}}}{\Gamma(h\rightarrow
Z\gamma(\gamma\gamma))^{\text{SM}}},$ (28)

where $\Gamma(h\rightarrow Z\gamma)^{\text{SM}}$ represents the partial width
in the SM.

Before concluding, remarks on the correlations between the
$h\rightarrow\gamma\gamma$ and $h\rightarrow Z\gamma$ partial widths in Type-I
3HDM are in order. The left panel of Fig. 5 shows the contour plot of
$\Gamma^{\gamma\gamma}/\Gamma^{Z\gamma}$ in the plane of
$(m_{H^{\pm}_{1}},\tan\gamma)$ with the constraint from the measurement
$h\rightarrow\gamma\gamma$ decay. One can see that the allowed regions
increase with large value of $\Gamma^{\gamma\gamma}/\Gamma^{Z\gamma}$. The
left panel shows the regions where the partial width of $h\rightarrow Z\gamma$
could be either larger or smaller than that of $h\rightarrow\gamma\gamma$. On
the right panel of Fig. 5, we discuss the correlations between
$R_{\gamma\gamma}$ and $R_{Z\gamma},$ defined in Eq. (26), while we retain the
decay of $h\rightarrow\gamma\gamma$ as in its measured value at the LHC. The
right panel illustrates that the correlation is sensitive to the choice of
$m_{H^{\pm}_{1}}$. The plots in Fig. 5 show that correlation can exist between
the $h\rightarrow Z\gamma$ and $h\rightarrow\gamma\gamma$ decays at particular
parameter regions. This correlation could be tested in future experiments at
the LHC.

Figure 5: Left plane shows contour plot of
$\Gamma^{\gamma\gamma}/\Gamma^{Z\gamma}$ as a function of $m_{H^{\pm}_{1}}$
and $\tan\gamma$ in the Type-I 3HDM. Right plane shows correlation between
$R_{\gamma\gamma}$ and $R_{Z\gamma}$ in the Type-I 3HDM. We take $m_{h}=125$
GeV, $m_{H_{1}^{\pm}}=m_{H_{2}^{\pm}}+100$ GeV, $\theta_{2}=\pi/6$ and
$m_{12}=m_{23}=m_{13}=200$ GeV .

###  IV.3  $h\rightarrow gg$

In this section, we discuss the gluonic decay widths of the 125 GeV Higgs
boson $h$. First, we present the decay width $\Gamma(h\rightarrow gg)$ as a
function of $\tan\beta$ for various values of $\tan\gamma$ on the left plot of
Fig.6. The solid and dotted lines stand for the Type-I and Type-Z 3HDM
respectively. Hence one can use them to distinguish the Type-I and Type-Z
3HDM. The decay of $h$ into $gg$ increases with $\tan\gamma$ and $\tan\beta$
in the Type-Z 3HDM, while it remains the same at large $\tan\beta$ in the
Type-I 3HDM.

The right plot of Fig. 6 indicates the dependence of the CP-even mixing angles
on the decay of $h$ into $gg$. The decay width is very sensitive to the change
of $\alpha_{1,2}~{}\text{and}~{}\theta$. The solid and dotted lines stand for
the Type-I and Type-Z 3HDM, respectively, and thus one can disentangle the two
types in the gluonic decay.

Figure 6: The left plot indicates the decay width of $h\rightarrow gg$ as a
function of $\tan\beta$ in the Type-I (solid lines) and Type-Z (dotted lines)
3HDM. The right plot indicates the decay width of $h\rightarrow gg$ as a
function of the CP-even mixing angles in the Type-I (solid lines) and Type-Z
(dotted lines) 3HDM. We take $m_{h}=125$ GeV.

In general, some of the important parameters to distinguish different Yukawa
types of the model are denoted as $\tan\beta~{}\text{and}~{}\tan\gamma.$

##  V CONCLUSIONS

In this work, we have analyzed the new physics contributions predicted by the
3HDM to the loop induced decays $h\rightarrow\gamma\gamma,~{}\gamma Z,~{}gg$.
We have found that

  * •

The new charged Higgs bosons $H^{\pm}_{1,2}$ can significantly alter the decay
widths $h\rightarrow\gamma\gamma,~{}\gamma Z$ (where h is the lightest CP-even
scalar in the 3HDM), since their loop factor is the largest among others.

  * •

tan$\beta$, tan$\gamma$, $\alpha_{1,2},~{}\theta$, and $m_{H_{1,2}^{\pm}}$
parameters can have significant effects on these decays.

  * •

We can disentangle the Type-I and Type-Z 3HDM only in the gluonic decays,
since $\gamma\gamma$ and $Z\gamma$ decays are dominated by the $H^{\pm}_{1}$
and $H^{\pm}_{2}$ bosons loop contributions.

  * •

$h\rightarrow Z\gamma$ branching ratio increases considerably compared to its
SM value in the part of the parameter space while the
$h\rightarrow\gamma\gamma$ decay is compatible with the LHC measurement. Our
results on the BR$(h\rightarrow Z\gamma)$ can be tested at the LHC when
$R_{BR}\simeq 1$.

  * •

A correlation between the $h\rightarrow Z\gamma$ and
$h\rightarrow\gamma\gamma$ decays exist, which could be probed in future
experiments at the LHC.

In general, a large enhancement of $\Gamma(h\rightarrow Z\gamma)$ in the 3HDM
may be obtained with the contributions of the new charged Higgs bosons
$H^{\pm}_{1,2}$ and probed at the HL-LHC.

##  Appendix A Mass Matrices

Now, we determine the mass matrices in different sectors.

###  A.1 CP charged scalar sector

The analytic expressions for the mass matrices of the singly-charged (${\cal
M}_{\pm}^{2}$) scalar states in the Higgs basis of 3HDMs can be extracted from
the scalar potential as

$\displaystyle
V_{C}=\begin{pmatrix}w_{1}^{+}&w_{2}^{+}&w_{3}^{+}\end{pmatrix}M_{\pm}^{2}\begin{pmatrix}w_{1}^{-}\\\
w_{2}^{-}\\\ w_{3}^{-}\end{pmatrix}.$ (29)

The (${\cal M}_{\pm}^{2}$) $3\times 3$ mass matrix can be diagonalized as
follows:

$\displaystyle(M_{C})^{2}=R^{T}\cdot M_{\pm}^{2}\cdot
R=\begin{pmatrix}0&0&0\\\ 0&(M_{C})^{2}_{22}&(M_{C})^{2}_{23}\\\
0&(M_{C})^{2}_{23}&(M_{C})^{2}_{33}\end{pmatrix}.$ (30)

The matrix elements of ${\cal M}_{C}^{2}$ are given by

$\displaystyle({\cal M}_{C}^{2})_{22}$
$\displaystyle=-\frac{v^{2}}{2}\left[(\rho_{2}+\rho_{3})c_{\gamma}^{2}+(\kappa_{2}+\kappa_{3})s_{\gamma}^{2})\right]+m_{12}^{2}\frac{c_{\gamma}}{s_{\beta}c_{\beta}}+m_{23}^{2}\frac{s_{\gamma}}{s_{\beta}c_{\beta}},$
(31) $\displaystyle({\cal M}_{C}^{2})_{33}$
$\displaystyle=-\frac{v^{2}}{2}\left[(\rho_{2}+\rho_{3})s_{\beta}^{2}s_{\gamma}^{2}+(\sigma_{2}+\sigma_{3})c_{\beta}^{2}+(\kappa_{2}+\kappa_{3})s_{\beta}^{2}c_{\gamma}^{2}\right]$
$\displaystyle\quad+m_{12}^{2}t_{\beta}s_{\gamma}t_{\gamma}+\frac{m_{13}^{2}}{s_{\gamma}c_{\gamma}}+m_{23}^{2}\frac{t_{\beta}c_{\gamma}}{t_{\gamma}},$
(32) $\displaystyle({\cal M}_{C}^{2})_{23}$
$\displaystyle=-\frac{v^{2}}{4}(\rho_{2}+\rho_{3}-\kappa_{2}-\kappa_{3})s_{\beta}s_{2\gamma}+m_{12}^{2}\frac{s_{\gamma}}{c_{\beta}}-m_{23}^{2}\frac{c_{\gamma}}{c_{\beta}}.$
(33)

This matrix can be fully diagonalized by employing the orthogonal matrix
$O_{\theta_{2}}$ as follows:

$\displaystyle
O_{\theta_{2}}.(M_{C})^{2}.O_{\theta_{2}}^{T}=\begin{pmatrix}0&0&0\\\
0&m^{2}_{H^{\pm}_{1}}&0\\\
0&0&m^{2}_{H^{\pm}_{2}}\end{pmatrix}~{}~{}~{}~{}~{}~{}O_{\theta_{2}}=\begin{pmatrix}1&0&0\\\
0&\cos(\theta_{2})&-\sin(\theta_{2})\\\ 0&\sin(\theta_{2})&\cos(\theta_{2})\\\
\end{pmatrix}$ (34)

Using Eqs. (A3)$-$ (A6), $\kappa_{2},\sigma_{2}~{}\text{and}~{}\rho_{2}$ can
be solved as

$\displaystyle\kappa_{2}$
$\displaystyle=\frac{2m_{A_{1}}^{2}c_{\theta_{1}}^{2}-\frac{2m_{A_{2}}^{2}c_{\theta_{1}}s_{\theta_{1}}}{s_{\beta}t_{\gamma}}+2m_{A_{2}}^{2}s_{\theta_{1}}^{2}+\frac{m_{A_{1}}^{2}s_{2\theta_{1}}}{s_{\beta}t_{\gamma}}-4c_{\theta_{2}}^{2}m_{H_{1}{}^{\pm}}^{2}+\frac{m_{H_{2}{}^{\pm}}^{2}s_{\theta_{2}}}{c_{\gamma}s_{\beta}s_{\gamma}}-4m_{H_{2}{}^{\pm}}^{2}s_{\theta_{2}}^{2}+\frac{m_{H_{2}{}^{\pm}}^{2}s_{\theta_{2}}}{s_{\beta}t_{\gamma}}}{2v^{2}}+$
(35)
$\displaystyle\frac{-\frac{2c_{\theta_{2}}s_{\theta_{2}}t_{\gamma}\left(\frac{m_{H_{1}{}^{\pm}}^{2}}{s_{\gamma}^{2}}+\frac{m_{H_{1}{}^{\pm}}^{2}}{t_{\gamma}^{2}}+m_{H_{2}{}^{\pm}}^{2}\right)}{s_{\beta}}+\frac{m_{12}^{2}c_{\gamma}}{c_{\beta}s_{\beta}}+\frac{m_{23}^{2}s_{\gamma}}{c_{\beta}s_{\beta}}+\frac{m_{23}^{2}}{c_{\beta}s_{\beta}s_{\gamma}}-\frac{m_{12}^{2}}{c_{\beta}c_{\gamma}s_{\beta}}+\frac{m_{12}^{2}s_{\gamma}t_{\gamma}}{c_{\beta}s_{\beta}}+\frac{m_{23}^{2}c_{\gamma}}{c_{\beta}s_{\beta}t_{\gamma}}+\frac{(m_{H_{1}{{}^{\pm}}}^{2})s_{\theta_{2}}t_{\gamma}}{s_{\beta}}}{2v^{2}}$
$\displaystyle\rho_{2}$
$\displaystyle=\frac{1}{2v^{2}}\left(2m_{A_{1}}^{2}c_{\theta_{1}}^{2}-4c_{\theta_{2}}^{2}m_{H_{1}{}^{\pm}}^{2}+2m_{A_{2}}^{2}s_{\theta_{1}}^{2}+\frac{m_{H_{1}{}^{\pm}}^{2}s_{\theta_{2}}}{c_{\gamma}s_{\beta}s_{\gamma}}-4m_{H_{2}{}^{\pm}}^{2}s_{\theta_{2}}^{2}+\frac{m_{H_{2}{}^{\pm}}^{2}s_{\theta_{2}}}{s_{\beta}t_{\gamma}}-\frac{m_{A_{1}}^{2}s_{2\theta_{1}}t_{\gamma}}{s_{\beta}}+\right.$
$\displaystyle\frac{m_{A_{2}}^{2}s_{2\theta_{1}}t_{\gamma}}{s_{\beta}}+\frac{m_{H_{1}{}^{\pm}}^{2}s_{\theta_{2}}t_{\gamma}}{s_{\beta}}-\frac{2c_{\theta_{2}}s_{\theta_{2}}t_{\gamma}\left(m_{H_{2}{}^{\pm}}^{2}\left(\frac{1}{s_{\gamma}^{2}}+1\right)+\frac{m_{H_{1}{}^{\pm}}^{2}}{t_{\gamma}^{2}}\right)}{s_{\beta}}+$
(36)
$\displaystyle\left.\frac{m_{12}^{2}c_{\gamma}}{c_{\beta}s_{\beta}}+\frac{m_{12}^{2}}{c_{\beta}c_{\gamma}s_{\beta}}+\frac{m_{23}^{2}s_{\gamma}}{c_{\beta}s_{\beta}}-\frac{m_{23}^{2}}{c_{\beta}s_{\beta}s_{\gamma}}+\frac{m_{12}^{2}s_{\gamma}t_{\gamma}}{c_{\beta}s_{\beta}}+\frac{m_{23}^{2}c_{\gamma}}{c_{\beta}s_{\beta}t_{\gamma}}+\frac{m_{H_{1}{}^{\pm}}^{2}s_{\theta_{2}}^{2}t_{\gamma}^{2}\left(\frac{m_{H_{1}{}^{\pm}}^{2}}{s_{\gamma}^{2}}+\frac{m_{H_{1}{}^{\pm}}^{2}}{t_{\gamma}^{2}}+m_{H_{2}{}^{\pm}}^{2}\right)}{s_{\beta}^{2}}\right)$
$\displaystyle\sigma_{2}$
$\displaystyle=\frac{1}{4v^{2}}\left(\frac{\left(m_{A_{1}}^{2}-m_{A_{2}}^{2}\right)\left(c_{2\beta}-3\right)c_{\theta_{1}}^{2}}{c_{\beta}^{2}}-\frac{2\left(m_{A_{1}}^{2}-m_{A_{2}}^{2}\right)c_{2\gamma}s_{2\theta_{1}}t_{\beta}}{c_{\beta}c_{\gamma}s_{\gamma}}+\frac{4c_{2\gamma}\left(m_{H_{1}{}^{\pm}}^{2}-m_{H_{2}{{}^{\pm}}}^{2}\right)s_{\theta_{2}}t_{\beta}}{c_{\beta}c_{\gamma}s_{\gamma}}+\right.$
$\displaystyle\left(t_{\beta}^{2}-1\right)\left(\left(m_{A_{1}}^{2}-m_{A_{2}}^{2}\right)s_{\theta_{1}}^{2}-m_{A_{1}}^{2}-m_{A_{2}}^{2}+c_{2\theta}\left(m_{H_{1}{}^{\pm}}^{2}-m_{H_{2}{}^{\pm}}^{2}\right)+m_{H_{1}{}^{\pm}}^{2}+3m_{H_{2}{}^{\pm}}^{2}\right)+$
(37)
$\displaystyle\frac{3m_{A_{1}}^{2}s_{\theta_{1}}^{2}+m_{A_{1}}^{2}+m_{A_{2}}^{2}-2\left(c_{2\beta}-3\right)c_{\theta_{2}}^{2}\left(m_{H_{1}{}^{\pm}}^{2}-m_{H_{2}{}^{\pm}}^{2}\right)-2m_{H_{1}{}^{\pm}}^{2}-2m_{H_{2}{}^{\pm}}^{2}}{c_{\beta}^{2}}$
$\displaystyle\frac{3m_{A_{2}}^{2}s_{\theta_{1}}^{2}-\frac{4m_{13}^{2}}{c_{\gamma}s_{\gamma}}-6s_{\theta_{2}}^{2}m_{H_{1}{}^{\pm}}^{2}+6s_{\theta_{2}}^{2}m_{H_{2}{}^{\pm}}^{2}+m_{13}^{2}s_{2\gamma}+2m_{13}^{2}s_{\gamma}^{2}t_{\gamma}+6m_{13}^{2}t_{\gamma}+\frac{8m_{13}^{2}}{t_{\gamma}}}{c_{\beta}^{2}}\Biggr{)}.$

###  A.2 CP odd scalar sector

The analytic expressions for the mass matrices of the CP-odd scalars $({\cal
M}_{P}^{2})$ in the Higgs basis of 3HDM can be extracted from the scalar
potential as

$\displaystyle
V_{C}=\begin{pmatrix}z_{1}&z_{2}&z_{3}\end{pmatrix}\frac{M_{P}^{2}}{2}\begin{pmatrix}z_{1}\\\
z_{2}\\\ z_{3}\end{pmatrix}.$ (38)

The CP-odd scalar mass matrix ${\cal M}_{P}^{2}$ can be diagonalized as
follows:

$\displaystyle(M_{A})^{2}=R^{T}\cdot M_{P}^{2}\cdot R=\begin{pmatrix}0&0&0\\\
0&(M_{A})^{2}_{22}&(M_{A})^{2}_{23}\\\
0&(M_{A})^{2}_{23}&(M_{A})^{2}_{33}\end{pmatrix}.$

The matrix elements of ${\cal M}_{A}^{2}$ is given by

$\displaystyle({\cal M}_{A}^{2})_{22}$
$\displaystyle=-v^{2}\left(\rho_{3}c_{\gamma}^{2}+\kappa_{3}s_{\gamma}^{2}\right)+m_{13}^{2}\frac{c_{\gamma}}{s_{\beta}c_{\beta}}+m_{23}^{2}\frac{s_{\gamma}}{s_{\beta}c_{\beta}},$
(39) $\displaystyle({\cal M}_{A}^{2})_{33}$
$\displaystyle=-v^{2}\left(\rho_{3}s_{\beta}^{2}s_{\gamma}^{2}+\sigma_{3}c_{\beta}^{2}+\kappa_{3}s_{\beta}^{2}c_{\gamma}^{2}\right)+m_{12}^{2}t_{\beta}s_{\gamma}t_{\gamma}+m_{13}^{2}\frac{s_{\gamma}}{s_{\gamma}c_{\gamma}}+m_{23}^{2}\frac{t_{\beta}c_{\gamma}}{t_{\gamma}},$
(40) $\displaystyle({\cal M}_{A}^{2})_{23}$
$\displaystyle=-\frac{v^{2}}{2}(\rho_{3}-\kappa_{3})s_{\beta}s_{2\gamma}+m_{12}^{2}\frac{s_{\gamma}}{c_{\beta}}-m_{23}^{2}\frac{c_{\gamma}}{c_{\beta}}.$
(41)

In order to fully diagonalise this matrix, an orthogonal matrix
$O_{\theta_{1}}$ can be employed as

$\displaystyle
O_{\theta_{1}}.(M_{A})^{2}.O_{\theta_{1}}^{T}=\begin{pmatrix}0&0&0\\\
0&m^{2}_{A_{1}}&0\\\
0&0&m^{2}_{A_{2}}\end{pmatrix}~{}~{}~{}~{}~{}~{}O_{\theta_{1}}=\begin{pmatrix}1&0&0\\\
0&\cos(\theta_{1})&-\sin(\theta_{1})\\\ 0&\sin(\theta_{1})&\cos(\theta_{1})\\\
\end{pmatrix}.$ (42)

Using Eqs. (A11)$-$(A14), $\kappa_{3},\sigma_{3}~{}\text{and}~{}\rho_{3}$ can
be solved as

$\displaystyle\kappa_{3}=$
$\displaystyle\frac{-2m_{A_{1}}^{2}c_{\theta_{1}}^{2}+\frac{2\left(m_{A_{2}}^{2}-m_{A_{1}}^{2}\right)c_{\theta_{1}}s_{\theta_{1}}}{s_{\beta}t_{\gamma}}-2m_{A_{2}}^{2}s_{\theta_{1}}^{2}+\frac{m_{23}^{2}s_{\gamma}}{c_{\beta}s_{\beta}}+\frac{m_{23}^{2}}{c_{\beta}s_{\beta}s_{\gamma}}-\frac{m_{12}^{2}}{c_{\beta}c_{\gamma}s_{\beta}}+\frac{m_{12}^{2}s_{\gamma}t_{\gamma}}{c_{\beta}s_{\beta}}+\frac{m_{23}^{2}c_{\gamma}+m_{12}^{2}s_{\gamma}}{c_{\beta}s_{\beta}t_{\gamma}}}{2v^{2}}$
(43) $\displaystyle\rho_{3}=$
$\displaystyle\frac{-2m_{A_{1}}^{2}c_{\theta_{1}}^{2}-2m_{A_{2}}^{2}s_{\theta_{1}}^{2}+\frac{\left(m_{A_{1}}^{2}-m_{A_{2}}^{2}\right)s_{2\theta_{1}}t_{\gamma}}{s_{\beta}}+\frac{m_{12}^{2}}{c_{\beta}c_{\gamma}s_{\beta}}+\frac{m_{23}^{2}s_{\gamma}}{c_{\beta}s_{\beta}}-\frac{m_{23}^{2}}{c_{\beta}s_{\beta}s_{\gamma}}+\frac{m_{12}^{2}s_{\gamma}t_{\gamma}}{c_{\beta}s_{\beta}}+\frac{m_{23}^{2}c_{\gamma}+m_{12}^{2}s_{\gamma}}{c_{\beta}s_{\beta}t_{\gamma}}}{2v^{2}}$
(44) $\displaystyle\sigma_{3}=$
$\displaystyle-\frac{\frac{c_{2\beta}\left(\left(m_{A_{1}}^{2}-m_{A_{2}}^{2}\right)c_{2\theta_{1}}+m_{A_{1}}^{2}+m_{A_{2}}^{2}\right)-3\left(m_{A_{1}}^{2}-m_{A_{2}}^{2}\right)c_{\theta_{1}}^{2}+3m_{A_{1}}^{2}s_{\theta_{1}}^{2}-3m_{A_{2}}^{2}s_{\theta_{1}}^{2}+m_{A_{1}}^{2}+m_{A_{2}}^{2}-\frac{4m_{13}^{2}}{c_{\gamma}s_{\gamma}}}{c_{\beta}}+\frac{2\left(m_{A_{2}}^{2}-m_{A_{1}}^{2}\right)c_{2\gamma}s_{2\theta_{1}}t_{\beta}}{c_{\gamma}s_{\gamma}}}{4v^{2}c_{\beta}}$
(45)

###  A.3 CP even scalar sector

In the general weak basis, the CP-even scalar mass matrix $M^{2}_{H}$ can be
extracted from the potential as

$\displaystyle
V_{S}=\begin{pmatrix}h_{1}&h_{2}&h_{3}\end{pmatrix}\frac{M_{H}^{2}}{2}\begin{pmatrix}h_{1}\\\
h_{2}\\\ h_{3}\end{pmatrix}.$ (46)

The matrix elements of ${\cal M}_{H}^{2}$ are given by

$\displaystyle({\cal M}_{H}^{2})_{11}$
$\displaystyle=\frac{m_{12}^{2}t_{\beta}}{c_{\gamma}}+\lambda_{1}v^{2}c_{\beta}^{2}c_{\gamma}^{2}+m_{13}^{2}t_{\gamma},$
(47) $\displaystyle({\cal M}_{H}^{2})_{22}$
$\displaystyle=\frac{m_{12}^{2}c_{\gamma}}{t_{\beta}}+\frac{m_{23}^{2}s_{\gamma}}{t_{\beta}}+\lambda_{2}v^{2}s_{\beta}^{2},$
(48) $\displaystyle({\cal M}_{H}^{2})_{33}$
$\displaystyle=\lambda_{3}v^{2}c_{\beta}^{2}s_{\gamma}^{2}+\frac{m_{23}^{2}t_{\beta}}{s_{\gamma}}+\frac{m_{13}^{2}}{t_{\gamma}},$
(49) $\displaystyle({\cal M}_{H}^{2})_{12}$
$\displaystyle=\left(\rho_{1}+\rho_{2}+\rho_{3}\right)v^{2}c_{\beta}c_{\gamma}s_{\beta}-m_{12}^{2},$
(50) $\displaystyle({\cal M}_{H}^{2})_{13}$
$\displaystyle=\left(\sigma_{1}+\sigma_{2}+\sigma_{3}\right)v^{2}c_{\beta}^{2}c_{\gamma}s_{\gamma}-m_{13}^{2},$
(51) $\displaystyle({\cal M}_{H}^{2})_{23}$
$\displaystyle=\left(\kappa_{1}+\kappa_{2}+\kappa_{3}\right)v^{2}c_{\beta}s_{\beta}s_{\gamma}-m_{23}^{2}.$
(52)

In the above expressions, we used the shorthand notations $c_{X}=\cos X$,
$s_{X}=\sin X$ and $t_{X}=\tan X$.

For the masses of the three neutral physical states, $h,$ $H_{1}$ and $H_{2}$,
we need to diagonalise the $3\times 3$ mass matrix $M^{2}_{H}$, defined in the
general weak basis, by employing the orthogonal matrix $O$ given in Eq. (13)
as

$\displaystyle O^{T}\cdot M_{H}^{2}\cdot O=\begin{pmatrix}m_{h}^{2}&0&0\\\
0&m_{H_{1}}^{2}&0\\\ 0&0&m_{H_{2}}^{2}\end{pmatrix}.$ (53)

Finally, the remaining parameters
$\lambda_{1},~{}\lambda_{2},~{}\lambda_{3},~{}\kappa_{1},\sigma_{1}~{}\text{and}~{}\rho_{1}$
can be obtained from Eqs. (A19)$-$(A25) as follows:

$\displaystyle\lambda_{1}$
$\displaystyle=-\frac{c_{\alpha_{1}}^{2}\left(-\left(c_{\alpha_{2}}^{2}m_{h}^{2}+m_{H_{1}}^{2}s_{\alpha_{2}}^{2}\right)\right)+\frac{m_{12}^{2}t_{\beta}}{c_{\gamma}}-m_{H_{2}}^{2}s_{\alpha_{1}}^{2}+m_{13}^{2}t_{\gamma}}{v^{2}c_{\beta}^{2}c_{\gamma}^{2}}$
(54) $\displaystyle\lambda_{2}$
$\displaystyle=\frac{c_{\alpha_{2}}^{2}\left(c_{\theta}^{2}m_{H_{1}}^{2}+m_{h}^{2}s_{\alpha_{1}}^{2}s_{\theta}^{2}\right)-c_{\alpha_{2}}s_{\alpha_{1}}s_{\alpha_{2}}s_{2\theta}\left(m_{h}^{2}-m_{H_{1}}^{2}\right)+c_{\alpha_{1}}^{2}m_{H_{2}}^{2}s_{\theta}^{2}}{v^{2}s_{\beta}^{2}}+$
(55)
$\displaystyle\frac{c_{\theta}^{2}m_{h}^{2}s_{\alpha_{2}}^{2}-\frac{m_{12}^{2}c_{\gamma}}{t_{\beta}}+m_{H_{1}}^{2}s_{\alpha_{1}}^{2}s_{\alpha_{2}}^{2}s_{\theta}^{2}-\frac{m_{23}^{2}s_{\gamma}}{t_{\beta}}}{v^{2}s_{\beta}^{2}}$
$\displaystyle\lambda_{3}$
$\displaystyle=\frac{-c_{\alpha_{2}}^{2}c_{\theta}^{2}m_{h}^{2}s_{\alpha_{1}}^{2}-2c_{\alpha_{2}}c_{\theta}m_{h}^{2}s_{\alpha_{1}}s_{\alpha_{2}}s_{\theta}+c_{\alpha_{1}}^{2}\left(-c_{\theta}^{2}\right)m_{H_{2}}^{2}-c_{\alpha_{2}}^{2}m_{H_{1}}^{2}s_{\theta}^{2}}{v^{2}c_{\beta}^{2}s_{\gamma}^{2}}+$
(56)
$\displaystyle-\frac{-c_{\theta}^{2}m_{H_{1}}^{2}s_{\alpha_{1}}^{2}s_{\alpha_{2}}^{2}+2c_{\alpha_{2}}c_{\theta}m_{H_{1}}^{2}s_{\alpha_{1}}s_{\alpha_{2}}s_{\theta}-m_{h}^{2}s_{\alpha_{2}}^{2}s_{\theta}^{2}+\frac{m_{23}^{2}t_{\beta}}{s_{\gamma}}+\frac{m_{13}^{2}}{t_{\gamma}}}{v^{2}c_{\beta}^{2}s_{\gamma}^{2}}$
$\displaystyle\kappa_{1}$
$\displaystyle=\frac{1}{2v^{2}s_{\gamma}}\left(4c_{\theta_{2}}^{2}m_{H_{1}{}^{\pm}}^{2}s_{\gamma}-\frac{2c_{\alpha_{1}}^{2}c_{\theta}m_{H_{2}}^{2}s_{\theta}}{c_{\beta}s_{\beta}}+\frac{c_{\alpha_{2}}^{2}s_{2\theta}\left(c_{2\alpha_{1}}m_{h}^{2}-m_{h}^{2}+2m_{H_{1}}^{2}\right)}{2c_{\beta}s_{\beta}}-\right.$
(57)
$\displaystyle\frac{c_{\alpha_{2}}s_{\alpha_{1}}s_{\alpha_{2}}\left(c_{2\theta}\left(m_{H_{1}}^{2}-m_{h}^{2}\right)+m_{h}^{2}+m_{H_{1}}^{2}\right)}{c_{\beta}s_{\beta}}+\frac{m_{h}^{2}s_{\alpha_{2}}^{2}s_{2\theta}}{c_{\beta}s_{\beta}}-\frac{2c_{\theta}m_{H_{1}}^{2}s_{\alpha_{1}}^{2}s_{\alpha_{2}}^{2}s_{\theta}}{c_{\beta}s_{\beta}}+\frac{c_{\theta}^{2}m_{h}^{2}s_{\alpha_{1}}s_{2\alpha_{2}}}{c_{\beta}s_{\beta}}+$
$\displaystyle\frac{m_{H_{1}}^{2}s_{\alpha_{1}}s_{2\alpha_{2}}s_{\theta}^{2}}{c_{\beta}s_{\beta}}+\frac{m_{H_{1}{}^{\pm}}^{2}s_{\theta_{2}}}{c_{\gamma}s_{\beta}}-\frac{m_{H_{2}{}^{\pm}}^{2}s_{\theta_{2}}}{c_{\gamma}s_{\beta}}+4m_{H_{2}{}^{\pm}}^{2}s_{\gamma}s_{\theta_{2}}^{2}-\frac{m_{H_{1}{}^{\pm}}^{2}s_{\gamma}s_{\theta_{2}}t_{\gamma}}{s_{\beta}}+$
$\displaystyle\left.\frac{c_{\gamma}\left(\left(m_{H_{1}{}^{\pm}}^{2}-m_{H_{2}{}^{\pm}}^{2}\right)s_{\theta_{2}}-\frac{2m_{12}^{2}s_{\gamma}}{c_{\beta}}\right)}{s_{\beta}}-\frac{2m_{23}^{2}c_{\gamma}^{2}}{c_{\beta}s_{\beta}}-\frac{2m_{23}^{2}s_{\gamma}^{2}}{c_{\beta}s_{\beta}}+\frac{2m_{12}^{2}t_{\gamma}}{c_{\beta}s_{\beta}}-\frac{2m_{12}^{2}s_{\gamma}^{2}t_{\gamma}}{c_{\beta}s_{\beta}}+\frac{m_{H_{2}{}^{\pm}}^{2}s_{\gamma}s_{\theta_{2}}t_{\gamma}}{s_{\beta}}\right)$
$\displaystyle\rho_{1}=\frac{1}{2v^{2}}\left(4c_{\theta_{2}}^{2}m_{H_{1}{}^{\pm}}^{2}-\frac{2c_{\alpha_{1}}c_{\alpha_{2}}^{2}m_{h}^{2}s_{\alpha_{1}}s_{\theta}}{c_{\beta}c_{\gamma}s_{\beta}}+\frac{m_{H_{2}}^{2}s_{2\alpha_{1}}s_{\theta}}{c_{\beta}c_{\gamma}s_{\beta}}-\frac{2c_{\alpha_{1}}c_{\alpha_{2}}c_{\theta}m_{H_{1}}^{2}s_{\alpha_{2}}}{c_{\beta}c_{\gamma}s_{\beta}}-\frac{2c_{\alpha_{1}}m_{H_{1}}^{2}s_{\alpha_{1}}s_{\alpha_{2}}^{2}s_{\theta}}{c_{\beta}c_{\gamma}s_{\beta}}+\right.$
(58)
$\displaystyle\frac{c_{\alpha_{1}}c_{\theta}m_{h}^{2}s_{2\alpha_{2}}}{c_{\beta}c_{\gamma}s_{\beta}}-\frac{m_{H_{1}{}^{\pm}}^{2}s_{\theta_{2}}}{c_{\gamma}s_{\beta}s_{\gamma}}+\frac{m_{H_{2}{}^{\pm}}^{2}s_{\theta_{2}}}{c_{\gamma}s_{\beta}s_{\gamma}}+4m_{H_{2}{}^{\pm}}^{2}s_{\theta_{2}}^{2}+\frac{m_{H_{1}{}^{\pm}}^{2}s_{\theta_{2}}}{s_{\beta}t_{\gamma}}-\frac{m_{H_{2}{}^{\pm}}^{2}s_{\theta_{2}}}{s_{\beta}t_{\gamma}}-$
$\displaystyle\left.\frac{2m_{23}^{2}}{c_{\beta}s_{\beta}s_{\gamma}}-\frac{2m_{23}^{2}s_{\gamma}}{c_{\beta}s_{\beta}}-\frac{2m_{12}^{2}s_{\gamma}t_{\gamma}}{c_{\beta}s_{\beta}}-\frac{2\left(m_{23}^{2}c_{\gamma}+m_{12}^{2}s_{\gamma}\right)}{c_{\beta}s_{\beta}t_{\gamma}}+\frac{m_{\left(H_{1}\right){}^{+}}^{2}s_{\theta_{2}}t_{\gamma}}{s_{\beta}}+\frac{m_{\left(H_{2}\right){}^{+}}^{2}s_{\theta_{2}}t_{\gamma}}{s_{\beta}}\right)$
$\displaystyle\sigma_{1}$
$\displaystyle=\frac{1}{8v^{2}c_{\beta}}\left(\frac{6\left(c_{2\beta}-2\right)c_{\theta_{2}}^{2}\left(m_{H_{1}{}^{\pm}}^{2}-m_{H_{2}{}^{\pm}}^{2}\right)}{c_{\beta}}-\frac{8c_{2\gamma}\left(m_{H_{1}{}^{\pm}}^{2}-m_{H_{2}{}^{\pm}}^{2}\right)s_{\theta_{2}}t_{\beta}}{c_{\gamma}s_{\gamma}}-\right.$
(59)
$\displaystyle\frac{1}{c_{\beta}c_{\gamma}s_{\gamma}}\left(-c_{\gamma}s_{\gamma}\left(c_{2\beta}\left(c_{2\theta}\left(m_{H_{1}{}^{\pm}}^{2}-m_{H_{2}{}^{\pm}}^{2}\right)+m_{\left(H_{1}\right){}^{+}}^{2}+7m_{H_{2}{}^{\pm}}^{2}\right)+4\left(m_{\left(H_{1}\right){}^{+}}^{2}+m_{H_{2}{}^{\pm}}^{2}\right)\right)+\right.$
$\displaystyle
8c_{\alpha_{1}}\left(c_{\alpha_{2}}s_{\alpha_{2}}s_{\theta}\left(m_{H_{1}}^{2}-m_{h}^{2}\right)+c_{\theta}s_{\alpha_{1}}\left(-c_{\alpha_{2}}^{2}m_{h}^{2}-m_{H_{1}}^{2}s_{\alpha_{2}}^{2}+m_{H_{2}}^{2}\right)\right)-6m_{H_{1}{}^{\pm}}^{2}s_{2\gamma}s_{\theta_{2}}^{2}+$
$\displaystyle\left.\left.16m_{13}^{2}c_{\gamma}^{2}+6m_{H_{2}{}^{\pm}}^{2}s_{2\gamma}s_{\theta_{2}}^{2}-8m_{13}^{2}+4m_{13}^{2}s_{\gamma}^{4}+12m_{13}^{2}s_{\gamma}^{2}+m_{13}^{2}s_{2\gamma}^{2}\right)\right)$

##  Appendix B Coefficients of $hH^{+}_{i}H^{-}_{i}$ Terms in the Potential

The coefficients of triple scalar couplings in the potential are explicitly
given as

$\displaystyle a=$
$\displaystyle\frac{-1}{2v}\left(c_{\alpha_{1}}c_{\alpha_{2}}s_{\beta}\left(c_{\alpha_{1}}c_{\theta}s_{2\alpha_{2}}-c_{\alpha_{2}}^{2}s_{2\alpha_{1}}s_{\theta}\right)\left(\frac{c_{\theta_{2}}}{t_{\beta}}+s_{\gamma}s_{\theta_{2}}\right){}^{2}-\right.$
(60) $\displaystyle
s_{\beta}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\left(c_{\alpha_{2}}^{2}s_{\alpha_{1}}^{2}s_{2\theta}-c_{2\theta}s_{2\alpha_{2}}s_{\alpha_{1}}-s_{\alpha_{2}}^{2}s_{2\theta}\right)\left(\frac{c_{\theta_{2}}}{t_{\beta}}+s_{\gamma}s_{\theta_{2}}\right){}^{2}-$
$\displaystyle
2s_{\beta}\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(c_{\alpha_{2}}^{2}s_{\alpha_{1}}^{2}s_{\theta}^{2}+c_{\theta}\left(c_{\theta}s_{\alpha_{2}}^{2}-s_{\alpha_{1}}s_{2\alpha_{2}}s_{\theta}\right)\right)\left(\frac{c_{\theta_{2}}}{t_{\beta}}+s_{\gamma}s_{\theta_{2}}\right){}^{2}+$
$\displaystyle
2c_{\alpha_{1}}^{3}c_{\alpha_{2}}^{3}c_{\gamma}s_{\beta}t_{\beta}+c_{\gamma}s_{\beta}t_{\beta}\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(c_{\alpha_{2}}^{2}s_{2\alpha_{1}}s_{\theta}-c_{\alpha_{1}}c_{\theta}s_{2\alpha_{2}}\right)+$
$\displaystyle
c_{\gamma}s_{\beta}t_{\beta}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\left(c_{\alpha_{2}}^{2}c_{\theta}s_{2\alpha_{1}}+c_{\alpha_{1}}s_{2\alpha_{2}}s_{\theta}\right)+\frac{c_{\alpha_{1}}c_{\alpha_{2}}c_{\beta}\left(c_{\alpha_{2}}^{2}c_{\theta}s_{2\alpha_{1}}+c_{\alpha_{1}}s_{2\alpha_{2}}s_{\theta}\right)\left(s_{\theta_{2}}-c_{\theta_{2}}s_{\gamma}t_{\beta}\right){}^{2}}{s_{\gamma}}+$
$\displaystyle\frac{c_{\beta}\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(c_{\alpha_{2}}^{2}s_{\alpha_{1}}^{2}s_{2\theta}-c_{2\theta}s_{2\alpha_{2}}s_{\alpha_{1}}-s_{\alpha_{2}}^{2}s_{2\theta}\right)\left(s_{\theta_{2}}-c_{\theta_{2}}s_{\gamma}t_{\beta}\right){}^{2}}{s_{\gamma}}+$
$\displaystyle\left.\frac{2c_{\beta}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\left(c_{\alpha_{2}}^{2}c_{\theta}^{2}s_{\alpha_{1}}^{2}+c_{\theta}s_{2\alpha_{2}}s_{\alpha_{1}}s_{\theta}+s_{\alpha_{2}}^{2}s_{\theta}^{2}\right)\left(s_{\theta_{2}}-c_{\theta_{2}}s_{\gamma}t_{\beta}\right){}^{2}}{s_{\gamma}}\right)$
$\displaystyle b=$
$\displaystyle\frac{1}{4v}\left(-\frac{4\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\left(c_{\beta}c_{\theta_{2}}+s_{\beta}s_{\gamma}s_{\theta_{2}}\right){}^{2}\left(2c_{\theta_{2}}^{2}s_{\beta}s_{\gamma}+c_{\gamma}s_{2\theta_{2}}\right)}{t_{\beta}}+\right.$
(61) $\displaystyle
c_{\gamma}^{2}s_{\beta}^{2}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\left(\frac{4c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}}-\frac{2s_{\gamma}\left(\left(2c_{2\beta}-3\right)c_{\theta_{2}}^{2}+3s_{\theta_{2}}^{2}+1\right)}{c_{\beta}}\right)+$
$\displaystyle
c_{\gamma}s_{\beta}\left(c_{\gamma}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)+c_{\alpha_{1}}c_{\alpha_{2}}s_{\gamma}\right)\left(c_{\beta}s_{\theta_{2}}-c_{\theta_{2}}s_{\beta}s_{\gamma}\right)\left(\frac{\left(6-4c_{2\beta}\right)c_{\theta_{2}}^{2}+3c_{2\theta_{2}}-5}{c_{\beta}}+\frac{4c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}s_{\gamma}}\right)+$
$\displaystyle
4\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(c_{\theta_{2}}s_{\beta}s_{\gamma}-c_{\beta}s_{\theta_{2}}\right){}^{2}\left(2c_{\theta_{2}}^{2}s_{\beta}+\frac{s_{2\theta_{2}}}{t_{\gamma}}\right)+4\left(c_{\beta}s_{\gamma}\left(c_{\theta}s_{\alpha_{2}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}\right)+\right.$
$\displaystyle\left.+s_{\beta}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\right)(c_{\beta}s_{\theta_{2}}-c_{\theta_{2}}s_{\beta}s_{\gamma})\left(c_{\beta}c_{\theta_{2}}+s_{\beta}s_{\gamma}s_{\theta_{2}}\right)\left(2c_{\theta_{2}}^{2}+\frac{s_{2\theta_{2}}}{s_{\beta}t_{\gamma}}\right)+$
$\displaystyle\frac{c_{\alpha_{1}}c_{\alpha_{2}}\left(c_{\theta_{2}}s_{\beta}s_{\gamma}-c_{\beta}s_{\theta_{2}}\right){}^{2}\left(\frac{4c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}}-\frac{2s_{\gamma}\left(\left(2c_{2\beta}-3\right)c_{\theta_{2}}^{2}+3s_{\theta_{2}}^{2}+1\right)}{c_{\beta}}\right)}{t_{\gamma}}+$
$\displaystyle
4c_{\gamma}^{2}s_{\beta}^{2}\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(2c_{\theta_{2}}^{2}s_{\beta}-s_{2\theta_{2}}t_{\gamma}\right)-\frac{4c_{\alpha_{1}}c_{\alpha_{2}}c_{\gamma}\left(c_{\beta}c_{\theta_{2}}+s_{\beta}s_{\gamma}s_{\theta_{2}}\right){}^{2}\left(2c_{\theta_{2}}^{2}s_{\beta}-s_{2\theta_{2}}t_{\gamma}\right)}{t_{\beta}}+$
$\displaystyle\left.c_{\gamma}s_{\beta}\left(c_{\beta}c_{\theta_{2}}+s_{\beta}s_{\gamma}s_{\theta_{2}}\right)\left(c_{\beta}c_{\gamma}\left(c_{\theta}s_{\alpha_{2}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}\right)+c_{\alpha_{1}}c_{\alpha_{2}}s_{\beta}\right)\left(-8c_{\theta_{2}}^{2}-\frac{4c_{\theta_{2}}s_{\theta_{2}}}{s_{\beta}t_{\gamma}}+\frac{2s_{2\theta_{2}}\left(\frac{1}{c_{\gamma}s_{\gamma}}+t_{\gamma}\right)}{s_{\beta}}\right)\right)$
$\displaystyle c=$
$\displaystyle\frac{1}{8v}\left(-\frac{8\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\left(c_{\beta}c_{\theta_{2}}+s_{\beta}s_{\gamma}s_{\theta_{2}}\right){}^{2}\left(2s_{\beta}s_{\gamma}s_{\theta_{2}}^{2}-c_{\gamma}s_{2\theta_{2}}\right)}{t_{\beta}}+\right.$
(62) $\displaystyle
2c_{\gamma}^{2}s_{\beta}^{2}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\left(-\frac{s_{\gamma}\left(6c_{\theta_{2}}^{2}+\left(4c_{2\beta}-6\right)s_{\theta_{2}}^{2}+2\right)}{c_{\beta}}-\frac{4c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}}\right)+$
$\displaystyle
c_{\gamma}s_{\beta}\left(c_{\gamma}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)+c_{\alpha_{1}}c_{\alpha_{2}}s_{\gamma}\right)(c_{\beta}s_{\theta_{2}}-c_{\theta_{2}}s_{\beta}s_{\gamma})\left(\frac{2\left(c_{\beta-\theta_{2}}+c_{\beta+\theta_{2}}-2c_{2\beta}-6c_{2\theta_{2}}-2\right)}{c_{\beta}}-\right.$
$\displaystyle\left.\frac{8c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}s_{\gamma}}\right)+8\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(c_{\theta_{2}}s_{\beta}s_{\gamma}-c_{\beta}s_{\theta_{2}}\right){}^{2}\left(2s_{\beta}s_{\theta_{2}}^{2}-\frac{s_{2\theta_{2}}}{t_{\gamma}}\right)+$
$\displaystyle
4s_{\beta}^{2}\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(c_{\gamma}s_{\gamma}s_{2\theta_{2}}+c_{\gamma}^{2}\left(4s_{\beta}s_{\theta_{2}}^{2}-\frac{s_{2\theta_{2}}}{t_{\gamma}}\right)+\frac{s_{2\theta_{2}}}{t_{\gamma}}\right)+$
$\displaystyle\frac{2c_{\alpha_{1}}c_{\alpha_{2}}\left(c_{\theta_{2}}s_{\beta}s_{\gamma}-c_{\beta}s_{\theta_{2}}\right){}^{2}\left(-\frac{s_{\gamma}\left(6c_{\theta_{2}}^{2}+\left(4c_{2\beta}-6\right)s_{\theta_{2}}^{2}+2\right)}{c_{\beta}}-\frac{4c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}}\right)}{t_{\gamma}}-$
$\displaystyle
2\left(c_{\beta}s_{\gamma}\left(c_{\theta}s_{\alpha_{2}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}\right)+\right.\left.s_{\beta}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\right)(c_{\beta}s_{\theta_{2}}-c_{\theta_{2}}s_{\beta}s_{\gamma})(c_{\beta}c_{\theta_{2}}+s_{\beta}s_{\gamma}s_{\theta_{2}})$
$\displaystyle\left(-8s_{\theta_{2}}^{2}+\frac{2s_{2\theta_{2}}\left(\frac{1}{c_{\gamma}s_{\gamma}}+\frac{1}{t_{\gamma}}\right)}{s_{\beta}}-\frac{4c_{\theta_{2}}s_{\theta_{2}}t_{\gamma}}{s_{\beta}}\right)+2c_{\gamma}s_{\beta}\left(c_{\beta}c_{\gamma}\left(c_{\theta}s_{\alpha_{2}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}\right)+c_{\alpha_{1}}c_{\alpha_{2}}s_{\beta}\right)$
$\displaystyle\left(c_{\beta}c_{\theta_{2}}+s_{\beta}s_{\gamma}s_{\theta_{2}}\right)\left(-\frac{4c_{\theta_{2}}\left(\frac{1}{s_{\gamma}^{2}}+1\right)s_{\theta_{2}}t_{\gamma}}{s_{\beta}}-8s_{\theta_{2}}^{2}+\frac{2s_{2\theta_{2}}}{s_{\beta}t_{\gamma}}\right)-$
$\displaystyle\left.\frac{8c_{\alpha_{1}}c_{\alpha_{2}}c_{\gamma}\left(c_{\beta}c_{\theta_{2}}+s_{\beta}s_{\gamma}s_{\theta_{2}}\right){}^{2}\left(2s_{\beta}s_{\theta_{2}}^{2}+s_{2\theta_{2}}t_{\gamma}\right)}{t_{\beta}}\right)$
$\displaystyle d=$
$\displaystyle-\frac{-c_{\alpha_{1}}c_{\alpha_{2}}s_{\beta}\left(\frac{c_{\theta_{2}}}{t_{\beta}}+s_{\gamma}s_{\theta_{2}}\right){}^{2}+c_{\beta}c_{\gamma}\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(\frac{c_{\theta_{2}}}{t_{\beta}}+s_{\gamma}s_{\theta_{2}}\right){}^{2}+c_{\gamma}s_{\beta}t_{\beta}\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)}{v}$
(63)
$\displaystyle+\frac{c_{\alpha_{1}}c_{\alpha_{2}}s_{\beta}t_{\beta}^{2}+2c_{\beta}\left(c_{\theta_{2}}+s_{\gamma}s_{\theta_{2}}t_{\beta}\right)\left(c_{\gamma}\left(c_{\theta}s_{\alpha_{2}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}\right)+c_{\alpha_{1}}c_{\alpha_{2}}t_{\beta}\right)}{v}$
$\displaystyle e=$
$\displaystyle\frac{c_{\beta}\left(s_{\theta_{2}}+2s_{\gamma}s_{\frac{\theta_{2}}{2}}^{2}t_{\beta}\right){}^{2}\left(c_{\alpha_{1}}c_{\alpha_{2}}+\frac{c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}}{t_{\gamma}}\right)}{vs_{\gamma}}$
(64) $\displaystyle f=$
$\displaystyle\frac{\left(\frac{c_{\theta_{2}}}{c_{\beta}}-\frac{c_{\gamma}s_{\beta}s_{\theta_{2}}}{t_{\gamma}}\right){}^{2}\left(c_{\theta}\left(c_{\alpha_{2}}s_{\alpha_{1}}+\frac{s_{\alpha_{2}}s_{\gamma}}{t_{\beta}}\right)+s_{\theta}\left(s_{\alpha_{2}}-\frac{c_{\alpha_{2}}s_{\alpha_{1}}s_{\gamma}}{t_{\beta}}\right)\right)}{vs_{\beta}}$
(65) $\displaystyle a_{1}=$
$\displaystyle-\frac{\frac{c_{\gamma}^{2}s_{\theta_{2}}^{2}\left(c_{\theta}s_{\alpha_{2}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}\right)}{s_{\beta}}+\frac{c_{\gamma}c_{\theta_{2}}^{2}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)}{c_{\beta}t_{\gamma}}+\frac{c_{\alpha_{1}}c_{\alpha_{2}}s_{\gamma}t_{\gamma}}{c_{\beta}}}{v}$
(66)

$\displaystyle b_{1}$
$\displaystyle=-\frac{1}{4v}\left(4c_{\beta}c_{\gamma}^{2}s_{\theta_{2}}^{2}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\left(\frac{c_{\gamma}s_{2\theta_{2}}}{s_{\beta}}+2c_{\theta_{2}}^{2}s_{\gamma}\right)-\right.$
(67) $\displaystyle
s_{\gamma}^{2}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\left(\frac{4c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}}-\frac{2s_{\gamma}\left(\left(2c_{2\beta}-3\right)c_{\theta_{2}}^{2}+3s_{\theta_{2}}^{2}+1\right)}{c_{\beta}}\right)-$
$\displaystyle
c_{\gamma}c_{\theta_{2}}s_{\gamma}\left(c_{\gamma}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)+c_{\alpha_{1}}c_{\alpha_{2}}s_{\gamma}\right)\left(\frac{\left(6-4c_{2\beta}\right)c_{\theta_{2}}^{2}+3c_{2\theta_{2}}-5}{c_{\beta}}+\frac{4c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}s_{\gamma}}\right)-$
$\displaystyle
2c_{\theta_{2}}^{2}\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(-2c_{\gamma}c_{\theta_{2}}s_{\gamma}s_{\theta_{2}}+c_{\gamma}^{2}\left(4c_{\theta_{2}}^{2}s_{\beta}+\frac{s_{2\theta_{2}}}{t_{\gamma}}\right)+\frac{s_{2\theta_{2}}}{t_{\gamma}}\right)+$
$\displaystyle\frac{4c_{\gamma}c_{\theta_{2}}s_{\theta_{2}}\left(2c_{\theta_{2}}^{2}s_{\beta}s_{\gamma}+c_{\gamma}s_{2\theta_{2}}\right)\left(s_{\theta}\left(s_{\alpha_{2}}s_{\beta}-c_{\alpha_{2}}c_{\beta}s_{\alpha_{1}}s_{\gamma}\right)+c_{\theta}\left(c_{\beta}s_{\alpha_{2}}s_{\gamma}+c_{\alpha_{2}}s_{\alpha_{1}}s_{\beta}\right)\right)}{s_{\beta}t_{\gamma}}-$
$\displaystyle\frac{c_{\alpha_{1}}c_{\alpha_{2}}c_{\gamma}^{2}c_{\theta_{2}}^{2}\left(\frac{4c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}}-\frac{2s_{\gamma}\left(\left(2c_{2\beta}-3\right)c_{\theta_{2}}^{2}+3s_{\theta_{2}}^{2}+1\right)}{c_{\beta}}\right)}{t_{\gamma}}+2\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(-4c_{\gamma}c_{\theta_{2}}^{2}s_{\beta}s_{\gamma}-\right.$
$\displaystyle
c_{\gamma}^{2}s_{2\theta_{2}}+\left.\left(s_{\gamma}^{2}+1\right)s_{2\theta_{2}}\right)t_{\gamma}+4c_{\alpha_{1}}c_{\alpha_{2}}c_{\beta}c_{\gamma}^{3}s_{\theta_{2}}^{2}\left(2c_{\theta_{2}}^{2}-\frac{s_{2\theta_{2}}t_{\gamma}}{s_{\beta}}\right)+c_{\gamma}s_{\gamma}\left(c_{\alpha_{1}}c_{\alpha_{2}}s_{\beta}+\right.$
$\displaystyle\left.\left.c_{\beta}c_{\gamma}\left(c_{\theta}s_{\alpha_{2}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}\right)\right)s_{\theta_{2}}\left(-8c_{\theta_{2}}^{2}-\frac{4c_{\theta_{2}}s_{\theta_{2}}}{s_{\beta}t_{\gamma}}+\frac{2s_{2\theta_{2}}\left(\frac{1}{c_{\gamma}s_{\gamma}}+t_{\gamma}\right)}{s_{\beta}}\right)\right)$
$\displaystyle c_{1}=$
$\displaystyle\frac{1}{8v}\left(8c_{\beta}c_{\gamma}^{2}s_{\theta_{2}}^{2}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\left(\frac{c_{\gamma}s_{2\theta_{2}}}{s_{\beta}}-2s_{\gamma}s_{\theta_{2}}^{2}\right)+\right.$
(68) $\displaystyle
2s_{\gamma}^{2}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\left(-\frac{s_{\gamma}\left(6c_{\theta_{2}}^{2}+\left(4c_{2\beta}-6\right)s_{\theta_{2}}^{2}+2\right)}{c_{\beta}}-\frac{4c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}}\right)+$
$\displaystyle
c_{\gamma}c_{\theta_{2}}s_{\gamma}\left(c_{\gamma}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)+c_{\alpha_{1}}c_{\alpha_{2}}s_{\gamma}\right)\left(\frac{2\left(c_{\beta-\theta_{2}}+c_{\beta+\theta_{2}}-2c_{2\beta}-6c_{2\theta_{2}}-2\right)}{c_{\beta}}-\frac{8c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}s_{\gamma}}\right)-$
$\displaystyle
4c_{\theta_{2}}^{2}\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(-2c_{\gamma}c_{\theta_{2}}s_{\gamma}s_{\theta_{2}}+c_{\gamma}^{2}\left(\frac{s_{2\theta_{2}}}{t_{\gamma}}-4s_{\beta}s_{\theta_{2}}^{2}\right)+\frac{s_{2\theta_{2}}}{t_{\gamma}}\right)+$
$\displaystyle\frac{2c_{\alpha_{1}}c_{\alpha_{2}}c_{\gamma}^{2}c_{\theta_{2}}^{2}\left(-\frac{s_{\gamma}\left(6c_{\theta_{2}}^{2}+\left(4c_{2\beta}-6\right)s_{\theta_{2}}^{2}+2\right)}{c_{\beta}}-\frac{4c_{2\gamma}s_{2\theta_{2}}t_{\beta}}{c_{\gamma}}\right)}{t_{\gamma}}+4\left(c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}-c_{\theta}s_{\alpha_{2}}\right)\left(4c_{\gamma}s_{\beta}s_{\gamma}s_{\theta_{2}}^{2}-c_{\gamma}^{2}s_{2\theta_{2}}+\right.$
$\displaystyle\left.\left(s_{\gamma}^{2}+1\right)s_{2\theta_{2}}\right)t_{\gamma}+2c_{\gamma}^{2}c_{\theta_{2}}\left(c_{\beta}s_{\gamma}\left(c_{\theta}s_{\alpha_{2}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}\right)+\right.\left.s_{\beta}\left(c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}\right)\right)s_{\theta_{2}}$
$\displaystyle\left(-\frac{4c_{\theta_{2}}s_{\theta_{2}}t_{\gamma}}{s_{\beta}}+\frac{2s_{2\theta_{2}}\left(\frac{1}{c_{\gamma}s_{\gamma}}+\frac{1}{t_{\gamma}}\right)}{s_{\beta}}-8s_{\theta_{2}}^{2}\right)$
$\displaystyle-2c_{\gamma}s_{\gamma}s_{\theta_{2}}\left(c_{\beta}c_{\gamma}\left(c_{\theta}s_{\alpha_{2}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}\right)+c_{\alpha_{1}}c_{\alpha_{2}}s_{\beta}\right)\left(-\frac{4c_{\theta_{2}}\left(\frac{1}{s_{\gamma}^{2}}+1\right)s_{\theta_{2}}t_{\gamma}}{s_{\beta}}-8s_{\theta_{2}}^{2}+\frac{2s_{2\theta_{2}}}{s_{\beta}t_{\gamma}}\right)$
$\displaystyle\left.-8c_{\alpha_{1}}c_{\alpha_{2}}c_{\beta}c_{\gamma}^{3}s_{\theta_{2}}^{2}\left(2s_{\theta_{2}}^{2}+\frac{s_{2\theta_{2}}t_{\gamma}}{s_{\beta}}\right)\right)$

$\displaystyle d_{1}=$
$\displaystyle\frac{\left(c_{\beta}c_{\gamma}\left(c_{\theta}s_{\alpha_{2}}-c_{\alpha_{2}}s_{\alpha_{1}}s_{\theta}\right)+c_{\alpha_{1}}c_{\alpha_{2}}s_{\beta}\right)\left(\frac{8c_{\gamma}s_{\theta_{2}}}{s_{\beta}}-\frac{8t_{\gamma}}{c_{\beta}}\right){}^{2}}{64v}$
(69) $\displaystyle e_{1}=$
$\displaystyle\frac{s_{\gamma}\left(\frac{1}{c_{\gamma}s_{\gamma}}+\frac{2c_{\theta_{2}}-1}{t_{\gamma}}+t_{\gamma}\right){}^{2}\left(c_{\alpha_{1}}c_{\alpha_{2}}+\frac{c_{\alpha_{2}}c_{\theta}s_{\alpha_{1}}+s_{\alpha_{2}}s_{\theta}}{t_{\gamma}}\right)}{4vc_{\beta}}$
(70) $\displaystyle f_{1}=$
$\displaystyle\frac{\left(s_{\theta}\left(s_{\alpha_{2}}s_{\beta}-c_{\alpha_{2}}c_{\beta}s_{\alpha_{1}}s_{\gamma}\right)+c_{\theta}\left(c_{\beta}s_{\alpha_{2}}s_{\gamma}+c_{\alpha_{2}}s_{\alpha_{1}}s_{\beta}\right)\right)\left(\frac{c_{\gamma}s_{\theta_{2}}}{s_{\beta}}+\frac{c_{\theta_{2}}}{c_{\beta}t_{\gamma}}\right){}^{2}}{v}$
(71)

##  Appendix C Definitions of the One-Loop Factors

The detailed analytic formulas of the one-loop factors are given as follows
djouadi ; djouadi1 :

$\displaystyle A_{1/2}(\tau)$ $\displaystyle=2[\tau+(\tau-1)f(\tau)]\tau^{-2}$
(72) $\displaystyle A_{1}(\tau)$
$\displaystyle=-[2\tau^{2}+3\tau+3(2\tau-1)f(\tau)]\tau^{-2}$ $\displaystyle
A_{0}(\tau)$ $\displaystyle=-[\tau-f(\tau)]\tau^{-2}$

The function $f(\tau)$ is given by

$\displaystyle f(\tau)=\begin{cases}\text{arcsin}^{2}\sqrt{\tau}&\tau\leq 1\\\
-\frac{1}{4}\left[\text{log}\frac{1+\sqrt{1-\tau^{-1}}}{1-\sqrt{1-\tau^{-1}}}-i\pi\right]&\tau>1.\end{cases}$
(73)

We used

$\displaystyle\tau_{i}=\frac{m_{\phi}^{2}}{4m_{i}^{2}}~{}~{}~{}\text{with}~{}~{}i=f,W^{\pm},H^{\pm}_{1,2}.$

Now $\tau_{i}=\frac{4m_{i}^{2}}{m_{\phi}^{2}}$,
$\lambda_{i}=\frac{4m_{i}^{2}}{m_{Z}^{2}}$ and other loop factors

$\displaystyle A_{1/2}(\tau,\lambda)=$
$\displaystyle[I_{1}(\tau,\lambda)-I_{2}(\tau,\lambda)]$ (74) $\displaystyle
A_{1}(\tau,\lambda)=$ $\displaystyle
c_{W}\left\\{4\left(3-\frac{s_{W}^{2}}{c_{W}^{2}}\right)I_{2}(\tau,\lambda)+\left[\left(1+\frac{2}{\tau}\right)\frac{s_{W}^{2}}{c_{W}^{2}}-\left(5+\frac{2}{\tau}\right)\right]I_{1}(\tau,\lambda)\right\\}$
(75)

The functions $I_{1}$ and $I_{2}$ are

$\displaystyle I_{1}(\tau,\lambda)=$
$\displaystyle\frac{\tau\lambda}{2(\tau-\lambda)}+\frac{\tau^{2}\lambda^{2}}{2(\tau-\lambda)^{2}}[f(\tau^{-1})-f(\lambda^{-1})]+\frac{\tau^{2}\lambda}{(\tau-\lambda)^{2}}[g(\tau^{-1})-g(\lambda^{-1})]$
(76) $\displaystyle I_{2}(\tau,\lambda)=$
$\displaystyle-\frac{\tau\lambda}{2(\tau-\lambda)}[f(\tau^{-1})-f(\lambda^{-1})]$
(77)

The function $g(\tau)$ is written as

$\displaystyle
g(\tau)=\begin{cases}\sqrt{\tau^{-1}-1}~{}\text{arcsin}\sqrt{\tau}&\tau\geq
1\\\
\frac{\sqrt{1-\tau^{-1}}}{2}\left[\text{log}\frac{1+\sqrt{1-\tau^{-1}}}{1-\sqrt{1-\tau^{-1}}}-i\pi\right]&\tau<1.\end{cases}$
(78)

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