aainstitutetext: SGTB Khalsa College, University of Delhi, Delhi,
India.bbinstitutetext: Department of Physics $\&$ Astrophysics, University of
Delhi, Delhi, India.

# Lepto-philic 2-HDM + singlet scalar portal induced fermionic dark matter

Sukanta Dutta b, # Ashok Goyal a,b, $ Manvinder Pal Singh
<EMAIL_ADDRESS><EMAIL_ADDRESS>$^$$ Corresponding~Author:
<EMAIL_ADDRESS>
###### Abstract

We explore the possibility that the discrepancy in the observed anomalous
magnetic moment of the muon $\Delta a_{\mu}$ and the predicted relic abundance
of Dark Matter by Planck data, can be explained in a lepto-philic 2-HDM
augmented by a real SM singlet scalar of mass $\sim$ 10-80 GeV. We constrain
the model from the observed Higgs Decay width at LHC, LEP searches for low
mass exotic scalars and anomalous magnetic moment of an electron $\Delta
a_{e}$. This constrained light singlet scalar serves as a portal for the
fermionic Dark Matter, which contributes to the required relic density of the
universe. A large region of model parameter space is found to be consistent
with the present observations from the Direct and Indirect DM detection
experiments.

##  1 Introduction

Investigations into the nature of dark matter (DM) particles and their
interactions is an important field of research in Astro-particle physics. The
Atlas and CMS collaborations at the Large Hadron Collider (LHC) are searching
for the signature of DM particles involving missing energy
($\not\\!\\!\\!E_{T}$) Garny:2013ama ; GooDMan:2010yf accompanied by a single
or two jet events. Direct detection experiments measure the nuclear-recoil
energy and its spectrum in DM-Nucleon elastic scattering Hisano:2011um ;
Garny:2012eb . In addition, there are Indirect detection experiment
Ackermann:2013yva searching for the DM annihilation into photons and neutrinos
in cosmic rays. These experiments have now reached a level of sensitivity
where a significant part of parameter space required for the observed relic
density, if contributed by the dark matter composed of Weakly interacting
massive particles (WIMPs) that survive as thermal relics, has been excluded.
The null results of these direct and indirect experiments have given rise to
the consideration of ideas where the dark matter is restricted to couple
exclusively to either Standard Model (SM) leptons (lepto-philic) or only to
top quarks (top-philic). In these scenarios the DM-Nucleon scattering occurs
only at the loop level and the constraints from direct detection are weaker.

Extended Higgs sector have been studied in literature Dedes:2001nx ;
Abe:2015oca ; Chun:2015hsa to explain discrepancy in anomalous magnetic moment
of muon. Recently a simplified Dark Higgs portal model of the order of
$\lesssim$ few GeV, that couples predominantly to leptons with the coupling
constant $\sim\,m_{l}/v_{o}$ where $m_{l}$ is the lepton mass and $v_{o}$ is
the Higgs VEV, has been considered in the literature Chen:2015vqy . This model
induces large contribution to the anomalous magnetic moment of muon and can
explain the existing discrepancy between the experimental observation $a^{\rm
exp}_{\mu}=$ 11 659 209.1(5.4)(3.3)$\times 10^{-10}$ pdg2018-muon and
theoretical prediction $a^{\rm SM}_{\mu}=$ 116 591 823(1)(34)(26) $\times
10^{-11}$ Hagiwara:2011af ; pdg2018-muon of the muon anomalous magnetic moment
$\Delta a_{\mu}\equiv a_{\mu^{-}}^{\rm
exp}-a_{\mu^{-}}^{\text{SM}}\,=\,268(63)\,\times 10^{-11}$ pdg2018-muon
without compromising the experimental measurement of electron anomalous
magnetic moment $a^{\rm exp}_{e}=(1159.65218091\pm 0.00000026)\times 10^{-6}$
pdg2018-electron . It has been shown in the literature Batell:2016ove , that
with the inclusion of an additional singlet scalar below the electro-weak
scale to the lepto-philic 2-HDM makes the model UV complete. This UV complete
model with an extra singlet scalar $\sim\,\,<$ 10 GeV successfully explains
the existing 3 $\sigma$ discrepancy of muon anomalous magnetic moment and is
consistent with the constraints on the model parameters from muon and meson
decays Bird:2004ts ; OConnell:2006rsp ; Batell:2009jf ; Krnjaic:2015mbs .
These results have been analysed for 0.01 GeV < $m_{S^{0}}$ < 10 GeV when
compared with those for the singlet neutral vector $Z^{\prime}$ searches at
$B$ factories such as BaBar TheBABAR:2016rlg , from electron beam dump
experiments Battaglieri:2014hga and electroweak precision experiments
Lebedev:2000ix etc.

In reference Agrawal:2014ufa , the authors have explored the possibility of
explaining the anomalous magnetic moment of muon with an additional lepto-
philic light scalar mediator assuming the universal coupling of the scalar
with all leptons constrained from the LEP Schael:2013ita resonant production
and the BaBar experiments TheBABAR:2016rlg . These constraints were found to
exclude all of the scalar mediator mass range except between 10 MeV and 300
MeV.

In the current paper we consider fermionic dark matter that couples
predominately with SM leptons through the non-universal couplings with the
scalar portal in the UV complete lepto-philic 2-HDM model. We relax the
requirement of the very light scalar considered in Batell:2016ove and
investigate parameter space for a comparatively heavier scalar 10 GeV
$\lesssim\,m_{S^{0}}\,\lesssim$ 80 GeV. In section 2, we give a brief review
of this simplified model, using the full Lagrangian and couplings of the
Singlet scalar $S^{0}$ with all model particles. In section 3, we calculate
the contribution from scalars ($S^{0},H^{0},A^{0},H^{\pm}$) to the anomalous
magnetic moment of the muon and discuss bounds on the model parameters from
LEP-II, $\Delta a_{e}$ and upper bound on the observed total Higgs decay
width. Implications of the model contributions to the lepton couplings non-
universality and the oblique corrections are briefly discussed along-with
available constraints on them.

In the present study we are motivated to explore the possibility of
simultaneously explaining the discrepancy in the observed anomalous magnetic
moment of the muon on the one hand and the expected relic density contribution
from DM on the other. Accordingly, in section 4, we introduce the DM
contributing to the relic density through dark matter - SM particles
interactions induced by the additional scalar in the model and scan for the
allowed parameter space which is consistent with direct and indirect
experimental data as well as with the observed value of the $\Delta a_{\mu}$.
Section 5 is devoted to discussion and summary of results.

##  2 The Model

We consider a UV complete lepton specific 2-HDM with a singlet scalar portal
interacting with the fermionic DM. In this model the two Higgs doublets
$\Phi_{1}$ and $\Phi_{2}$ are so arranged that $\Phi_{1}$ couples exclusively
to leptons while $\Phi_{2}$ couples exclusively to quarks. The ratio of their
VEV’s
$\left\langle\Phi_{1}\right\rangle/\,\left\langle\Phi_{2}\right\rangle\equiv\,v_{2}/\,v_{1}$
= $\tan\beta$ is assumed to be large. In this model the scalars (other than
that identified with the CP even $h^{0}\,\sim$ 125 GeV) couple to leptons and
quarks with coupling enhanced and suppressed by $\tan\beta$ respectively. A
mixing term in the potential
$A_{12}\,\left[\Phi_{1}^{\dagger}\Phi_{2}+\Phi_{2}^{\dagger}\Phi_{1}\right]\,\varphi^{0}$
results in the physical scalar $S^{0}$ coupling to leptons with strength
proportional to $m_{l}/v_{o}$ where

$v_{o}\equiv\sqrt{v_{1}^{2}+v_{2}^{2}}\,=\,246\,{\rm GeV}.$ (1)

The full scalar potential is given by

$V(\Phi_{1},\Phi_{2},\varphi^{0})=V_{2-HDM}+V_{\varphi^{0}}+V_{\rm portal}$
(2)

where CP conserving $V_{2-HDM}$ is given as

$\displaystyle V_{2-HDM}(\Phi_{1},\Phi_{2})$ $\displaystyle=$ $\displaystyle
m^{2}_{11}\,\Phi_{1}^{\dagger}\Phi_{1}+m^{2}_{22}\,\Phi_{2}^{\dagger}\Phi_{2}-\left(m^{2}_{12}\,\Phi_{1}^{\dagger}\Phi_{2}+{\rm
h.c.}\right)+\frac{\lambda_{1}}{2}\,\left(\Phi_{1}^{\dagger}\Phi_{1}\right)^{2}+\frac{\lambda_{2}}{2}\,\left(\Phi_{2}^{\dagger}\Phi_{2}\right)^{2}$
(3)
$\displaystyle+\lambda_{3}\,\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\left(\Phi_{2}^{\dagger}\Phi_{2}\right)+\lambda_{4}\,\left(\Phi_{1}^{\dagger}\Phi_{2}\right)\left(\Phi_{2}^{\dagger}\Phi_{1}\right)+\left\\{\frac{\lambda_{5}}{2}\,\left(\Phi_{1}^{\dagger}\Phi_{2}\right)^{2}+{\rm
h.c.}\right\\}\,$

and $V_{\varphi^{0}}$ and $V_{\rm portal}$ is assumed to be

$\displaystyle V_{\varphi^{0}}$ $\displaystyle=$ $\displaystyle
B\varphi^{0}+\frac{1}{2}m_{0}^{2}(\varphi^{0})^{2}+\frac{A_{\varphi^{0}}}{2}(\varphi^{0})^{3}+\frac{\lambda_{\varphi^{0}}}{4}(\varphi^{0})^{4}.$
(4) $\displaystyle V_{\rm portal}$ $\displaystyle=$ $\displaystyle
A_{11}\,\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\,\varphi^{0}+A_{12}\,\left(\Phi_{1}^{\dagger}\Phi_{2}+\Phi_{2}^{\dagger}\Phi_{1}\right)\,\varphi^{0}+A_{22}\,\left(\Phi_{2}^{\dagger}\Phi_{2}\right)\,\varphi^{0}.$
(5)

where the scalar doublets

$\displaystyle\Phi_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{2}\omega_{1}^{+}\\\
\rho_{1}+v_{o}\,\cos\beta+i\,z_{1}\end{pmatrix};\,\,\,\,\,\Phi_{2}=\frac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{2}\omega_{2}^{+}\\\
\rho_{2}+v_{o}\,\sin\beta+i\,z_{2}\end{pmatrix}\,.$ (6)

are written in terms of the mass eigenstates $G^{0}$, $A^{0}$ , $G^{\pm}$ and
$H^{\mp}$ as

$\displaystyle\begin{pmatrix}z_{1}\\\
z_{2}\end{pmatrix}=\left(\begin{array}[]{ccc}\cos\beta&&-\sin\beta\\\
\sin\beta&&\cos\beta\end{array}\right)\begin{pmatrix}G^{0}\\\
A^{0}\end{pmatrix};$ $\displaystyle\,\,\,\begin{pmatrix}\omega_{1}\\\
\omega_{2}\end{pmatrix}=\left(\begin{array}[]{ccc}\cos\beta&&-\sin\beta\\\
\sin\beta&&\cos\beta\end{array}\right)\begin{pmatrix}G^{\pm}\\\
H^{\pm}\end{pmatrix}.$ (11)

Here $G^{0}$ & $G^{\pm}$ are Nambu-Goldstone Bosons absorbed by the $Z^{0}$
and W± vector Bosons, $A^{0}$ is the pseudo-scalar and $H^{\pm}$ are the
charged Higgs. The three CP even neutral scalar mass eigen-states mix among
themselves under small mixing angle approximations

$\displaystyle\sin\delta_{13}\sim\delta_{13}\simeq-\frac{v_{0}\,A_{12}}{m_{H^{0}}^{2}},\,\,{\rm
and}\,\,\sin\delta_{23}\sim\delta_{23}\simeq-\frac{v_{0}\,A_{12}}{m_{h^{0}}^{2}}\left[1+\xi^{h^{0}}_{\ell}\left(1-\frac{m_{h^{0}}^{2}}{m_{H^{0}}^{2}}\right)\right]\cot\beta$
(13)

to give three CP even neutral weak eigen-states as

$\displaystyle\left(\begin{array}[]{c}\rho_{1}\\\ \rho_{2}\\\
\varphi^{0}\end{array}\right)\simeq\left(\begin{array}[]{ccccc}-\sin\alpha&&\cos\alpha&&\delta_{13}\\\
\,\,\,\cos\alpha&&\sin\alpha&&\delta_{23}\\\
\delta_{13}\sin\alpha-\delta_{23}\cos\alpha&&\,\,-\delta_{13}\cos\alpha-\delta_{23}\sin\alpha&&1\end{array}\right)\left(\begin{array}[]{c}h^{0}\\\
H^{0}\\\ S^{0}\end{array}\right);$ (23)

The mixing matrix given in equation (23) validates the orthogonality condition
up to an order
$\lesssim\,\mathcal{O}\left(\delta_{13}^{2},\delta_{23}^{2},\delta_{13}\delta_{23}\right)$.
$\xi_{l}^{h^{0}}$ is chosen to be $\sim$ 1 in the alignment i.e.
($\beta-\alpha$) $\simeq\pi/2$.

The spectrum of the model at the electro-weak scale is dominated by
$V_{2-HDM}$. The $V_{\varphi^{0}}$ and $V_{\rm portal}$ interactions are
treated as perturbations. After diagonalization of the scalar mass matrix, the
masses of the physical neutral scalars are given by

$\displaystyle m_{S^{0}}^{2}$ $\displaystyle\simeq$ $\displaystyle
m_{0}^{2}+2\delta_{13}\,M_{13}^{2}+2\delta_{23}\,M_{23}^{2}$ (24)
$\displaystyle m_{h^{0},\,H^{0}}^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}\left[M_{11}^{2}+M_{22}^{2}\mp\sqrt{\left(M_{11}^{2}-M_{22}^{2}\right)^{2}+4\,M_{12}^{4}}\right]$
(25)

where

$\displaystyle
M_{11}^{2}=m_{12}^{2}\tan\beta+\lambda_{1}v_{0}^{2}\cos^{2}\beta;$
$\displaystyle
M_{12}^{2}=-\,m_{12}^{2}+\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)v_{0}^{2}\cos\beta\,\sin\beta;$
$\displaystyle
M_{22}^{2}=m_{22}^{2}\cot\beta+\lambda_{2}v_{0}^{2}\sin^{2}\beta;$
$\displaystyle
M_{13}^{2}=v_{0}\,A_{12}\,\sin\beta;\,\,\,M_{23}^{2}=v_{0}\,A_{12}\,\cos\beta;$
(26)

In the alignment limit, one of the neutral CP even scalar $h^{0}$ $\approx$
125 GeV is identified with the SM Higgs.

The coefficients $m_{0}^{2}$, $m_{11}^{2},\,m_{22}^{2}$ and $\lambda_{i}$ for
$i=1,\cdot\cdot\cdot,5$ are explicitly defined in terms of the physical scalar
masses, mixing angles $\alpha$ and $\beta$ and the free parameter $m_{12}^{2}$
and are given in the Appendix A. Terms associated with $A_{11}$ are
proportional to $\cot\beta$ and therefore can be neglected as they are highly
suppressed in the large $\tan\beta$ limit. Terms proportional to $A_{22}$ are
tightly constrained from the existing data at LHC on decay of a heavy exotic
scalar to di-higgs channel and therefore they are dropped. The coefficient $B$
is fixed by redefinition of the field $\varphi^{0}$ to avoid a non-zero VEV
for itself.

$\xi^{\phi}_{\psi}$/$\xi^{\phi}_{V}$ | $S^{0}$ | $h^{0}$ | $H^{0}$ | $A^{0}$ | $H^{\pm}$
---|---|---|---|---|---
$\ell$ | $\delta_{13}/c_{\beta}$ | $-s_{\alpha}/c_{\beta}$ | $c_{\alpha}/c_{\beta}$ | $-s_{\beta}/c_{\beta}$ | $-s_{\beta}/c_{\beta}$
$u_{q}$ | $\delta_{23}/s_{\beta}$ | $c_{\alpha}/s_{\beta}$ | $s_{\alpha}/s_{\beta}$ | $c_{\beta}/s_{\beta}$ | $c_{\beta}/s_{\beta}$
$d_{q}$ | $\delta_{23}/s_{\beta}$ | $c_{\alpha}/s_{\beta}$ | $s_{\alpha}/s_{\beta}$ | $-c_{\beta}/s_{\beta}$ | $c_{\beta}/s_{\beta}$
$Z^{0}$/$W^{\pm}$ | $\delta_{13}c_{\beta}+\delta_{23}s_{\beta}$ | $s_{\left(\beta-\alpha\right)}$ | $c_{\left(\beta-\alpha\right)}$ | - | -
Table 1: Values of $\xi^{\phi}_{\psi}$ and $\xi^{\phi}_{V}$ for $\phi=S^{0}$,
$h^{0}$, $H^{0}$, $A^{0}$ and $H^{\pm}$; $\psi=\ell$, $u_{q}$ and $d_{q}$;
$V$= $W^{\pm}$ and $Z^{0}$ in the lepto-philic 2-HDM+$S^{0}$ model. These
values coincide with couplings given in reference Batell:2016ove in the
alignment limit i.e. ($\beta-\alpha$) $\simeq\pi/2$. In the table $s$ and $c$
stands for $\sin$ and $\cos$ respectively.

The Yukawa couplings arising due to Higgs Doublets $\Phi_{1}$ and $\Phi_{2}$
in type-X 2-HDM is given by

$-{\cal
L}_{Y}=\bar{L}Y_{e}\Phi_{1}e_{R}+\bar{Q}Y_{d}\Phi_{2}d_{R}+\bar{Q}Y_{u}\tilde{\Phi}_{2}u_{R}+{\rm
h.c.},$ (27)
$m_{e}=\cos\beta\times\frac{Y_{e}v_{o}}{\sqrt{2}},~{}~{}m_{u(d)}=\sin\beta\times\frac{Y_{u(d)}v_{o}}{\sqrt{2}}.$
(28)

We re-write the Yukawa interactions of the physical neutral states as

$\displaystyle-{\cal L}_{Y}$ $\displaystyle\supset\sum_{\phi\equiv
S^{0},h^{0},H^{0}}\,\,\sum_{\psi=\ell,\,q}\xi^{\phi}_{\psi}\,\frac{m_{\psi}}{v_{o}}\,\,\phi\,\,\bar{\psi}\psi$
(29)

The couplings $\xi_{\psi}^{\phi}$ are given in the first three rows of table
1. It is important to mention here that the Yukawa couplings are proportional
to the fermion mass i.e. non-universal unlike the consideration in reference
Agrawal:2014ufa .

The interaction of the neutral scalar mass eigenstates with the weak gauge
Bosons are given by

$\displaystyle{\cal L}$ $\displaystyle\supset\sum_{\phi\equiv
S^{0},h^{0},H^{0}}\,\frac{\phi}{v_{o}}\,\,\left(2\,\,\xi^{\phi}_{W^{\pm}}\,\,m^{2}_{W^{\pm}}\,W^{+}_{\mu}{W^{-}}^{\mu}+\xi^{\phi}_{Z^{0}}\,\,\,\,m_{Z^{0}}^{2}\,{Z^{0}}_{\mu}{Z^{0}}^{\mu}\right).$
(30)

The couplings $\xi_{V}^{\phi}$ are given in the last row of table 1. It is to
be noted that for $m_{H^{0}}>>m_{h^{0}}$, the singlet scalar coupling with
$Z^{0}$ Boson can be fairly approximated as $\simeq\delta_{23}\sin\beta$.

The recent precision measurements at LHC constrains $\left|\kappa_{V}\right|$
= 1.06${}^{+0.10}_{-0.10}$ Aad:2015ona ; pdg2018Higgs (where $\kappa_{V}$ is
the scale factor for the SM Higgs Boson coupling) to the vector Bosons
restricts the generic 2HDM Models and its extension like the one in discussion
to comply with the alignment limit. In this model, the Higgs Vector Boson
coupling to gauge Bosons is identical to that of generic 2HDM model at tree
level as the additional singlet scalar contributes to $h^{0}VV$ couplings only
at the one loop level which is suppressed by
$\delta^{2}_{23}/\left(16\,\pi^{2}\right)$.

The triple scalar couplings of the mass eigen states are given in the Appendix
D, some of which can be constrained from the observed Higgs decay width and
exotic scalar Boson searches at LEP, TeVatron and LHC.

##  3 Electro-Weak Constraints

###  3.1 Anomalous Magnetic Moment of Muon

We begin our analysis by evaluating the parameter space allowed from 3
$\sigma$ discrepancy $a_{\mu^{-}}^{\rm
exp}-a_{\mu^{-}}^{\text{SM}}\,\equiv\Delta a_{\mu}=\,268(63)\,\times 10^{-11}$
pdg2018-muon . In lepto-philic 2-HDM + singlet scalar portal model all five
additional scalars $S^{0},\,H^{0},\,A^{0},\,H^{\pm}$ couple to leptons with
the coupling strengths given in table 1 and thus give contributions to $\Delta
a_{\mu}$ at the one-loop level and are expressed as:

$\displaystyle\Delta a_{\mu}$ $\displaystyle=$ $\displaystyle\Delta
a_{\mu}\arrowvert_{S^{0}}+\Delta a_{\mu}\arrowvert_{H^{0}}+\Delta
a_{\mu}\arrowvert_{A^{0}}+\Delta a_{\mu}\arrowvert_{H^{\pm}}$ (31)
$\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\frac{m^{2}_{\mu}}{v_{o}^{2}}\,\tan^{2}\beta\left[\delta_{13}^{2}\,\,I_{S^{0}}+I_{H^{0}}+I_{A^{0}}+\,I_{H^{\pm}}\right].$

Here $I_{i}$ are the integrals given as

$\displaystyle
I_{S^{0},\,H^{0}}=\int^{1}_{0}dz\frac{\left(1+z\right)\left(1-z\right)^{2}}{\left(1-z\right)^{2}+z\,r_{{S^{0}},\,H^{0}}^{-2}};\,\,\,$
$\displaystyle
I_{A^{0}}=-\,\int^{1}_{0}dz\frac{z^{3}}{r_{A^{0}}^{-2}\left(1-z\right)+z^{2}};\,\,\,{\rm
and}$ $\displaystyle
I_{H^{\pm}}=\int^{1}_{0}dz\frac{z(1-z)}{\left(1-z\right)-r_{H^{\pm}}^{-2}}\,\,\,$
$\displaystyle{\rm with}\,\,\,r_{i}\equiv\frac{m_{l}}{m_{i}}\,\,{\rm
for}\,\,i\equiv S^{0},\,H^{0},\,A^{0},\,H^{\pm}.$ (32)

(a)

(b)

(c)

(d)

Figure 1:  Figures 1(a), 1(b), 1(c) and 1(d) show contours on the
$\delta_{13}$ \- $\delta_{23}$ plane satisfying $\Delta
a_{\mu}=\,268(63)\,\times 10^{-11}$ corresponding to four different
combinations of $m_{H^{0}}$ and $m_{A^{0}}$ respectively. In each panel six
contours are depicted corresponding to six choices of singlet masses 10, 20,
30, 40, 50 and 60 GeV respectively.

We observe that in the limit $r_{i}<<1$ the charged scalar integral
$I_{H^{\pm}}$ is suppressed by 2-3 orders of magnitude in comparison to the
other integrals for the masses of the scalars varying between 150 GeV $\sim$
1.6 TeV. Since the present lower bound on the charged Higgs mass from its
searches at LHC is 600 GeV Aad:2015typ , we can neglect its contribution to
the $\Delta a_{\mu}$ in our calculations.

It is also important to note that the one loop contribution from the pseudo-
scalar integral $I_{A^{0}}$ is opposite in sign to that of the other neutral
scalars $I_{H^{0},\,h^{0},\,S^{0}}$, while at the level of two loops the Barr-
Zee diagrams Ilisie:2015tra , it gives positive contribution to $\Delta
a_{\mu}$ which may be sizable for low pseudo-scalar mass $m_{A^{0}}$ because
of large value of the coupling $\xi^{A^{0}}_{l}$. However, for heavy $A^{0}$
and $H^{0}$ considered here, we can safely neglect the two loop contributions.

In the alignment limit the mixing angle
$\delta_{23}\simeq\frac{\delta_{13}m^{2}_{H^{0}}}{m_{h^{0}}^{2}}\left[2-\frac{m_{h^{0}}^{2}}{m_{H^{0}}^{2}}\right]\cot\beta$
is fixed by constrains from $\Delta a_{\mu}$ and choice of $\delta_{13}$ and
neutral CP-even scalar masses. To understand the model we study the
correlation of the two mixing parameters $\delta_{13}$ and $\delta_{23}$
satisfying the $\Delta a_{\mu}$ for a given set of input masses of the
physical scalars and show four correlation plots in figures 1(a), 1(b), 1(c)
and 1(d) for varying $\delta_{13}$. We find that $\delta_{23}$ remains small
enough for all the parameter space in order to fulfill the small angle
approximation. We have chosen six singlet scalar masses 10, 20, 30, 40, 50,
and 60 GeV. In each panel $m_{H^{0}}$ and $m_{A^{0}}$ are kept fixed at
values, namely (a) $m_{H^{0}}$ = 400 GeV, $m_{A^{0}}$ = 200 GeV, (b)
$m_{H^{0}}$ = 400 GeV, $m_{A^{0}}$ = 400 GeV, (c) $m_{H^{0}}$ = 400 GeV,
$m_{A^{0}}$ = 600 GeV, and (d) $m_{H^{0}}$ = 600 GeV, $m_{A^{0}}$ = 400 GeV.
We find that relatively larger values of $\delta_{13}$ are required with the
increase in scalar mass $m_{S^{0}}$. Increase in the pseudo-scalar mass
$m_{A^{0}}$ for fixed $m_{H^{0}}$ results in the lower value of $\delta_{13}$
required to obtain the observed $\Delta a_{\mu}$.

On imposing the perturbativity constraints on the Yukawa coupling
$\xi^{H^{0}}_{\tau}\equiv\tan\beta\,m_{\tau}/v_{0}$ involving the $\tau^{\pm}$
and $H^{0}$, we compute the upper bound on the model parameter
$\tan\beta\lesssim$ 485\. As a consequence, we observe that the values of
$\delta_{23}$ also gets restricted for each variation curve exhibited in
figures 1(a), 1(b), 1(c) and 1(d).

(a)

(b)

(c)

(d)

Figure 2:  Figures 2(a), 2(b), 2(c) and 2(d) show contours on $m_{S^{0}}$ \-
$\tan\beta$ plane satisfying $\Delta a_{\mu}=\,268(63)\,\times 10^{-11}$ for
fixed $m_{H^{\pm}}$ = 600 GeV and four different combinations of $m_{H^{0}}$
and $m_{A^{0}}$ as shown. In each panel, five contours along-with shaded one
$\sigma$ bands of $\Delta a_{\mu}$ are depicted corresponding to five choices
of $\delta_{13}$ 0.1, 0.2, 0.3, 0.4 and 0.5 respectively. The top horizontal
band (shaded in red) in each panel shows the forbidden region on $\tan\beta$
due to the perturbativity constraint on the upper limit of
$H^{0}\tau^{+}\tau^{-}$ coupling.

The contours satisfying $\Delta a_{\mu}=\,268(63)\,\times 10^{-11}$ on
$m_{S^{0}}$ \- $\tan\beta$ plane for fixed charged Higgs mass $m_{H^{\pm}}$ =
600 GeV are shown for four different combinations of heavy neutral Higgs mass
and pseudo-scalar Higgs mass namely (a) $m_{H^{0}}$ = 400 GeV, $m_{A^{0}}$ =
200 GeV, (b) $m_{H^{0}}$ = 400 GeV, $m_{A^{0}}$ = 400 GeV, (c) $m_{H^{0}}$ =
400 GeV, $m_{A^{0}}$ = 600 GeV, and (d) $m_{H^{0}}$ = 600 GeV, $m_{A^{0}}$ =
400 GeV respectively in figures 2(a), 2(b), 2(c) and 2(d). In each panel the
five shaded regions, correspond to five choices of mixing angle $\delta_{13}$
= 0.1, 02, 0.3, 0.4 and 0.5 respectively depict the 3 $\sigma$ allowed regions
for the discrepancy in $\Delta a_{\mu}$ around its central value shown by the
black lines. The horizontal band appearing at the top in all these panels
shows the forbidden region on account of the perturbativity constraint on the
upper limit of $H^{0}\tau^{+}\tau^{-}$ coupling as discussed above.

As expected the allowed value of $\tan\beta$ increases with the increasing
singlet scalar mass $m_{S^{0}}$ and decreasing mixing angle $\delta_{13}$. We
find that a very narrow region of the singlet scalar mass is allowed by
$\Delta a_{\mu}$ corresponding to $\delta_{13}\leq 0.1$.

###  3.2 LEP and $\Delta a_{e}$ Constraints

Searches for the light neutral Bosons were explored in the Higgs associated
vector Boson production channels at LEP Abdallah:2004wy . We consider the
s-channel bremsstrahlung process $e^{+}e^{-}\rightarrow
Z^{0}/\gamma^{0}+h^{0}\rightarrow\tau^{+}\tau^{-}\tau^{+}\tau^{-}$ whose
production cross-section can be expressed in terms of the SM $h^{0}Z^{0}$
production cross-section and given as

$\displaystyle\sigma_{e^{+}e^{-}\to S^{0}Z^{0}\to\tau^{+}\tau^{-}Z^{0}}$
$\displaystyle=$ $\displaystyle\sigma_{e^{+}e^{-}\to h^{0}Z^{0}}^{\rm
SM}\times\left|\frac{\xi^{S^{0}}_{Z^{0}}}{\xi^{h^{0}}_{Z^{0}}}\right|^{2}\times{\rm
BR}\left(S^{0}\to\tau^{+}\tau^{-}\right)$ (33) $\displaystyle\equiv$
$\displaystyle\sigma_{e^{+}e^{-}\to h^{0}Z^{0}}^{\rm
SM}\times\left|\frac{\delta_{13}c_{\beta}+\delta_{23}s_{\beta}}{\sin\left(\beta-\alpha\right)}\right|^{2}\times{\rm
BR}\left(S^{0}\to\tau^{+}\tau^{-}\right).$

Since the BR$\left(S^{0}\to\tau^{+}\tau^{-}\right)\simeq$ 1, we can compute
the exclusion limit on the upper bound on
$\left|\xi^{S^{0}}_{Z^{0}}\right|\equiv\left|\delta_{13}c_{\beta}+\delta_{23}s_{\beta}\right|$
from the LEP experimental data Abdallah:2004wy , which are shown in table 2
for some chosen values of singlet scalar masses in the alignment limit.

A light neutral Vector mediator ${Z^{\prime}}^{0}$ has also been extensively
searched at LEP Schael:2013ita . Vector mediator ${Z^{\prime}}^{0}$ of mass
$\leq$ 209 GeV is ruled out for coupling to muons $\gtrsim$ 0.01
Agrawal:2014ufa . Assuming the same production cross-section corresponding to
a light scalar mediator, the constraint on vector coupling can be translated
to scalar coupling by multiplying a factor of $\sqrt{2}$. For the case of non-
universal couplings where the scalar couples to the leptons with the strength
proportional to its mass as is the case in our model, a further factor of
$\sqrt{\frac{m_{\mu}}{m_{e}}}$ is multiplied. We therefore find the upper
limit on the Yukawa coupling for leptons to be
$\xi^{S^{0}}_{l}\frac{m_{l}}{v_{0}}\lesssim$ 0.2.

$m_{S^{0}}$(GeV) | 12 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65
---|---|---|---|---|---|---|---|---|---|---|---|---
$\left|\xi^{S^{0}}_{Z^{0}}\right|\lesssim$ | .285 | .316 | .398 | .530 | .751 | 1.132 | 1.028 | .457 | .260 | .199 | .169 | .093
Table 2: Upper limits on $\left|\xi^{S^{0}}_{Z^{0}}\right|$ from
bremsstrahlung process $e^{+}e^{-}\to
S^{0}Z^{0}\to\tau^{+}\tau^{-}\tau^{+}\tau^{-}$ LEP data Abdallah:2004wy

From the constrained parameter space of the model explaining the muon $\Delta
a_{\mu}$, we find that the total contribution to anomalous magnetic moment of
the electron comes out $\sim 10^{-15}$. This is two order smaller in the
magnitude than the error in the measurement of $a_{e}\simeq\pm 2.6\times
10^{-13}$ pdg2018-electron . The present model is thus capable of accounting
for the observed experimental discrepancy in the $\Delta a_{\mu}$ without
transgressing the allowed $\Delta a_{e}$.

(a) $m_{H^{0}}$ = 400 GeV; $m_{A^{0}}$ = 200 GeV

(b) $m_{H^{0}}$ = 400 GeV; $m_{A^{0}}$ = 400 GeV

(c) $m_{H^{0}}$ = 400 GeV; $m_{A^{0}}$ = 600 GeV

(d) $m_{H^{0}}$ = 600 GeV; $m_{A^{0}}$ = 400 GeV

Figure 3:  Figures 3(a), 3(b), 3(c) and 3(d) show $\Gamma_{h^{0}\to
S^{0}S^{0}}$ variation with the $m_{S^{0}}$ for $m_{H^{\pm}}$ = 600 GeV,
$\delta_{13}$ = 0.2 and four different combinations of $m_{H^{0}}$ and
$m_{A^{0}}$ respectively. In each panel we shade five regions corresponding to
$m_{12}^{2}$ = 10, 20, 30, 50, 100 GeV2 respectively. All points on the solid
curves satisfy the discrepancy $\Delta a_{\mu}=\,268(63)\,\times 10^{-11}$ and
their corresponding values of $\tan\beta$ are shown in the upper x-axis of all
the panels. We plot the contour corresponding to $m_{12}^{2}$ = 0 GeV2 in
black. The top horizontal band is forbidden from the measurement of the total
Higgs decay width at LHC. The red shaded region at the right in each panel is
forbidden due to non-perturbativity of $H^{0}\tau^{+}\tau^{-}$ coupling.

(a)

(b)

(c)

(d)

Figure 4:  Figures 4(a), 4(b), 4(c) and 4(d) show $\Gamma_{h^{0}\to
S^{0}S^{0}}$ variation with the $m_{S^{0}}$ for $m_{H^{\pm}}$ = 600 GeV,
$\delta_{13}$ = 0.4 and four different combinations of $m_{H^{0}}$ and
$m_{A^{0}}$ respectively. In each panel we shade five regions corresponding to
$m_{12}^{2}$ = 10, 20, 30, 50, 100 GeV2 respectively. All points on the solid
curves satisfy the discrepancy $\Delta a_{\mu}=\,268(63)\,\times 10^{-11}$ and
their corresponding values of $\tan\beta$ are shown in the upper x-axis of all
the panels. We plot the contour corresponding to $m_{12}^{2}$ = 0 GeV2 in
black. The top horizontal band is forbidden from the measurement of the total
Higgs decay width at LHC.

(a)

(b)

(c)

(d)

Figure 5:  Figures 5(a), 5(b), 5(c) and 5(d) show contours on the $m_{S^{0}}$
\- $m_{H^{0}}$ plane satisfying the 95 % C.L. upper limit on the total
observed Higgs decay width $\Gamma_{\rm obs.}^{h^{0}}\leq 2.4\times\Gamma_{\rm
SM}^{h^{0}}$ for fixed $m_{H^{\pm}}$ = 600 GeV, $\delta_{13}$ = 0.4 and four
different choices of $m_{12}^{2}$ = 20, 30, 50 and 100 GeV2. All points on the
contours satisfy the discrepancy $\Delta a_{\mu}=\,268(63)\,\times 10^{-11}$.
Each panel has three contours corresponding to $m_{A^{0}}$ = 200, 400 and 600
GeV respectively.

###  3.3 Constraints from Higgs decay-width

Recently CMS analysed the partial decay widths of the off-shell Higgs Boson
produced through gluon fusion decaying to $W^{+}W^{-}$ Bosons
Khachatryan:2016ctc and then combined the analysis with that for $Z\,Z$
Khachatryan:2015mma vector Bosons to obtain 95 % C.L. upper limit on the total
observed Higgs decay width of $2.4\times\Gamma_{\rm SM}^{h^{0}}$ pdg2018Higgs
; Khachatryan:2016ctc , where $\Gamma_{\rm SM}^{h^{0}}\simeq$ 4.1 MeV. The
authors have also investigated these decay channels for an off-shell Higgs
Boson produced from the vector Boson fusion channels and obtained the upper
bound on the total observed Higgs decay width of $19.3\times\Gamma_{\rm
SM}^{h^{0}}$ pdg2018Higgs ; Khachatryan:2016ctc . ATLAS also analysed the
Higgs decay width assuming that there are no anomalous couplings of the Higgs
boson to vector Bosons, and obtained 95% CL observed upper limit on the total
width of $6.7\times\Gamma_{\rm SM}^{h^{0}}$ Aad:2015xua . However, we have
used the conservative upper limit on the total observed decay width of Higgs
Boson of $2.4\times\Gamma_{\rm SM}^{h^{0}}$ for rest of the analysis in our
study.

However, in the present model, the scalar identified with SM Higgs Boson
$h^{0}$ is in addition likely to decay into two light singlet scalar portals
$h^{0}\to S^{0}\,S^{0}$ for $m_{S^{0}}\leq\frac{m_{h^{0}}}{2}$. The partial
decay width $\Gamma_{h^{0}\to S^{0}S^{0}}$ is given as

$\displaystyle\Gamma_{h^{0}\to
S^{0}S^{0}}=\frac{C_{h^{0}S^{0}S^{0}}^{2}}{32\,\pi\,m_{h^{0}}}\sqrt{1-\frac{4\,m_{S^{0}}^{2}}{m_{h^{0}}^{2}}}$
(34)

The tri-scalar coupling $C_{h^{0}S^{0}S^{0}}$ is given in equation
LABEL:Higgsdecay.

As total Higgs decay width is known with a fair accuracy, any contribution
coming from other than SM particles should fit into the combined theoretical
and experimental uncertainty. Thus, using the LHC data on the total observed
Higgs decay-width, we can put an upper limit on the tri-scalar coupling
$C_{h^{0}S^{0}S^{0}}$. This upper limit is then used to constrain the
parameter space of the model.

Even restricting the parameter sets to satisfy the anomalous magnetic moment
and LEP observations, the model parameter $m_{12}^{2}$ remains unconstrained.
However, for a given choice of $\delta_{13}$, $m_{H^{0}}$, $m_{A^{0}}$ and
$m_{S^{0}}$ an upper limit on $\left|C_{h^{0}S^{0}S^{0}}\right|$ constrains
$m_{12}^{2}$ and thus fixes the model for further validation at colliders.

We study the partial decay-width $\Gamma_{h^{0}\to S^{0}S^{0}}$ w.r.t.
$m_{S^{0}}$ for five chosen values of the free parameter $m_{12}^{2}$ = 10,
20, 30, 50 and 100 GeV2. We depict the variation of the partial decay width
$\Gamma_{h^{0}\to S^{0}S^{0}}$ corresponding to four different combinations of
$\left(m_{H^{0}},\,m_{A^{0}}\right)$ in GeV: $\left(400,200\right)$,
$\left(400,400\right)$, $\left(400,600\right)$ and $\left(600,400\right)$ in
figures 3(a), 3(b), 3(c), 3(d) respectively for $\delta_{13}$ = 0.2 and in
figures 4(a), 4(b), 4(c), 4(d) respectively for $\delta_{13}$ = 0.4. The top
horizontal band in all the four panels in figures 3 and 4 corresponds to the
forbidden region arising from the observed total Higgs decay width at LHC. In
figure 3 the parameter region for $m_{S^{0}}$ $\geq 24$ GeV is forbidden by
non-perturbativity of $H^{0}\tau^{+}\tau^{-}$ couplings. We observe that the
constraints from the total Higgs decay width further shrinks the parameter
space allowed by $\Delta a_{\mu}$ between 10 GeV $\leq m_{S^{0}}\leq 62$ GeV
for $\delta_{13}$ = 0.4 corresponding to 100 GeV${}^{2}\geq m_{12}^{2}\geq$ 10
GeV2.

To have better insight of the bearings on the model from the observed total
Higgs decay width we plot the contours on the $m_{S^{0}}-m_{H^{0}}$ plane for
mixing angle $\delta_{13}$ = 0.4 in figures 5(a), 5(b), 5(c) and 5(d)
satisfying the upper bound of the total observed Higgs decay width obtained by
CMS Khachatryan:2016ctc . We have considered four choices of $m^{2}_{12}$
respectively. In each panel , three curves depict the upper limits on the
partial widths which are derived from the constraints on the total observed
decay width from LHC corresponding to three chosen values $m_{A^{0}}$ = 200,
400 and 600 GeV respectively. We note that with increasing $m^{2}_{12}$ the
allowed dark shaded region shrinks and remains confined towards a lighter
$m_{S^{0}}$.

###  3.4 Lepton non-Universality and Precision Constraints

Recently HFAG collaboration Amhis:2014hma provided stringent constraints on
the departure of SM predicted universal lepton-gauge couplings. Non
universality of the lepton-gauge couplings can be parameterized as deviation
from the ratio of the lepton-gauge couplings of any two different generations
from unity and is defined as
$\delta_{ll^{\prime}}\equiv\left(g_{l}/g_{l^{\prime}}\right)-1$. For example,
the said deviation for $\tau^{\pm}$ and $\mu^{\pm}$ can be extracted from the
measured respective pure leptonic decay modes and is defined as

$\displaystyle\delta_{\tau\mu}\equiv\left(g_{\tau^{-}}/g_{\mu^{-}}\right)-1=\frac{\sqrt{\Gamma\left(\tau^{-}\to
e^{-}\,\bar{\nu}_{e}\,\nu_{\tau}\right)}}{\sqrt{\Gamma\left(\mu^{-}\to
e^{-}\,\bar{\nu}_{e}\,\nu_{\mu}\right)}}-1.$ (35)

The measured deviations of the three different ratios are found to be
Amhis:2014hma

$\displaystyle\delta^{l}_{\tau\mu}=0.0011\pm 0.0015;\,\,\,\,\delta^{l}_{\tau
e}=0.0029\pm 0.0015,\quad{\rm and}\,\,\,\,\delta^{l}_{\mu e}=0.0018\pm
0.0014,$ (36)

out of which only two ratios are independent Chun:2015hsa .

The implication of these data on lepto-philic type X 2-HDM models have been
studied in great detail in reference Chun:2015hsa and are shown as contours in
$m_{H^{\pm}}-\tan\beta$ and $m_{A^{0}}-\tan\beta$ planes, based on $\chi^{2}$
analysis of non-SM additional tree $\delta_{\rm tree}$ and loop $\delta_{\rm
loop}$ contributions to the lepton decay process in the leptonic mode
Krawczyk:2004na . We find that the additional scalar in lepto-philic 2-HDM +
singlet scalar model contribute to $\delta_{\tau\mu}$, $\delta_{\tau e}$ and
$\delta_{\mu e}$ at the one loop level which is $\delta_{13}^{2}$ suppressed.
However, they make a negligibly small correction and render the $\delta_{\rm
loop}$ more negative.

Further we constraint the model from the experimental bound on the $S,\,T$ and
$U$ Peskin:1990zt oblique parameters. Constrains from these parameters for all
variants of 2-HDM models have been extensively studied in the literature
Funk:2011ad . We compute the additional contribution due to the singlet scalar
at one loop for $\Delta S$ and $\Delta T$ in 2-HDM + singlet scalar model and
find that they are suppressed by the square of the mixing angle
$\delta_{13}^{2}$ and are therefore consistent with the experimental
observations as long as $m_{H^{\pm}}$ is degenerate either with $m_{A^{0}}$ or
$m_{H^{0}}$ for large $\tan\beta$ region to a range within $\sim$ 50 GeV
Batell:2016ove .

##  4 Dark matter Phenomenology

We introduce a spin 1/2 fermionic dark matter particle $\chi$ which is taken
to be a SM singlet with zero-hyper-charge and is odd under a discrete $Z_{2}$
symmetry. The DM $\chi$ interacts with the SM particle through the scalar
portal $S^{0}$. The interaction Lagrangian ${\cal L}_{DM}$ is given as

${\cal L}_{DM}=i\bar{\chi}\gamma^{\mu}\partial_{\mu}\chi-
m_{\chi}\bar{\chi}\chi+g_{\chi S^{0}}\bar{\chi}\chi S^{0}$ (37)

We are now equipped to compute the relic density of the DM, the scattering
cross-section of such DM with the nucleon and its indirect detection
annihilation cross-section.

###  4.1 Computation of the Relic Density

In early universe, when the temperature of the thermal bath was much greater
than the corresponding mass of the particle species, the particles were in
thermal equilibrium with the background. This equilibrium was maintained
through interactions such as annihilation and scattering with other SM
particles, such that the interaction rate remained greater than the expansion
rate of the universe. As the Universe cooled, massive particles such as our DM
candidate $\chi$, became non-relativistic and the interaction rate with other
particles became lower than the expansion rate of the universe, hence
decoupling the DM and giving us the relic abundance 0.119 Ade:2015xua ;
Komatsu:2014ioa we observe today. Evolution of the number density of the DM
$n_{\chi}$ is governed by the Boltzmann equation:

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6: Figures 6(a) to 6(f) show contours on the $m_{\chi}$ \- $g_{\chi
S^{0}}$ plane satisfying the relic density 0.119 Ade:2015xua ; Komatsu:2014ioa
for fixed $m_{H^{\pm}}$ = 600 GeV, $\delta_{13}$ = 0.2 and different choices
of $m_{12}^{2}$ and $m_{A^{0}}$. All points on the contours satisfy the
discrepancy $\Delta a_{\mu}=\,268(63)\,\times 10^{-11}$. In the left and right
panels, we show allowed (shaded) regions for four and five combinations of
$m_{S^{0}}$, $m_{H^{0}}$ respectively.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: Figures 7(a) to 7(f) show contours on the $m_{\chi}$ \- $g_{\chi
S^{0}}$ plane satisfying the relic density 0.119 Ade:2015xua ; Komatsu:2014ioa
for fixed $m_{H^{\pm}}$ = 600 GeV, $\delta_{13}$ = 0.4 and different choices
of $m_{12}^{2}$ and $m_{A^{0}}$. All points on the contours satisfy the
discrepancy $\Delta a_{\mu}=\,268(63)\,\times 10^{-11}$. In the left and right
panels, we show allowed (shaded) regions for four and five combinations of
$m_{S^{0}}$, $m_{H^{0}}$ respectively.

$\frac{dn_{\chi}}{dt}+3\frac{\dot{a}}{a}\,n_{\chi}=-\langle\sigma\left|\vec{v}\right|\rangle\left(n^{2}_{\chi}-n^{2}_{\chi
eq}\right)$ (38)

where $\frac{\dot{a}}{a}=\sqrt{\frac{8\pi\rho}{3\,M_{Pl}}}$ ,
$\langle\sigma\left|\vec{v}\right|\rangle$ is thermally averaged cross-section
and

$n^{2}_{\chi
eq}={\mathbf{g}}\left(\frac{m_{\chi}T}{2\pi}\right)^{\frac{3}{2}}\exp\left[\frac{-m_{\chi}}{T}\right]$
where ${\mathbf{g}}$ is the degrees of freedom, and it is 2 for fermions. As
for a massive thermal relics, freeze-out occurs when the species is non-
relativistic $\left|\vec{v}\right|<<c$. Therefore, we expand
$\langle\sigma\left|\vec{v}\right|\rangle$ as
$\langle\sigma\left|\vec{v}\right|\rangle=a+b\left|\vec{v}\right|^{2}+\mathcal{O}(\left|\vec{v}\right|^{4})$.
The Boltzmann equation can be solved to give the thermal relic density
F.Tanedo

$\Omega_{\chi}{\mathbf{h}}^{2}\simeq\frac{1.07\times
10^{9}x_{F}}{M_{Pl}\,\,\sqrt{{\mathbf{g}^{*}}(x_{F})}(a+\frac{6b}{x_{F}})}$
(39)

where ${\mathbf{h}}$ is dimensionless Hubble parameter,
${\mathbf{g}^{*}}(x_{F})$ is total number of dynamic degrees of freedom near
freeze-out temperature $T_{F}$ and $x_{F}=\frac{m_{\chi}}{T_{F}}$ is given by

$x_{F}=\ln\left[c\,(c+2)\sqrt{\frac{45}{8}}\,\frac{{\mathbf{g}}\,M_{Pl}\,\,m_{\chi}\left(a+\frac{6\,b}{x_{F}}\right)}{2\pi^{3}\sqrt{{\mathbf{g}^{*}}\left(x_{F}\right)}\sqrt{x_{F}}}\right]$
(40)

where $c$ is of the order 1. The thermal-averaged scattering cross-sections as
a function of DM mass $m_{\chi}$ are given in the Appendix C.

To compute relic density numerically, we have used MadDM Ambrogi:2018jqj and
MadGraph Alwall:2014hca . We have generated the input model file required by
MadGraph using FeynRules Alloul:2013bka , which calculates all the required
couplings and Feynman rules by using the full Lagrangian.

For a given charged Higgs mass of 600 GeV we depict the contours of constant
relic density $\simeq$ 0.119 Ade:2015xua ; Komatsu:2014ioa in $g_{\chi S^{0}}$
(DM coupling) and $m_{\chi}$ (DM mass) plane in figure 6 corresponding to two
choices of singlet scalar masses of 10 and 20 GeV for $\delta_{13}$ = 0.2 and
in figure 7 corresponding to three choices of singlet scalar masses of 10, 30
and 50 GeV for $\delta_{13}$ = 0.4. The six different panels in figures 6 and
7 correspond to the following six different combinations of
$\left(m_{A^{0}},m_{12}^{2}\right):$

$\displaystyle\left(200\,{\rm GeV},\,30\,{\rm
GeV}^{2}\right),\,\left(200\,{\rm GeV},\,50\,{\rm
GeV}^{2}\right),\,\left(400\,{\rm GeV},\,30\,{\rm GeV}^{2}\right),$
$\displaystyle\left(400\,{\rm GeV},\,50\,{\rm
GeV}^{2}\right),\,\left(600\,{\rm GeV},\,30\,{\rm GeV}^{2}\right)\,\,{\rm
and}\,\,\left(600\,{\rm GeV},\,50\,{\rm GeV}^{2}\right).$

The un-shaded regions in $g_{\chi S^{0}}-m_{\chi}$ plane in figures
corresponding to over closing of the Universe by DM relic density
contribution. The successive dips in the relic density contours arise due to
opening up of additional DM annihilation channel with the increasing DM mass.
Initial dip is caused by s-channel propagator. Dip observed around 0.2 TeV and
0.4 TeV are caused by opening of $\bar{\chi}\chi\rightarrow S^{0}H^{0}$ and
$\bar{\chi}\chi\rightarrow H^{0}H^{0}\,(A^{0}A^{0})$ channels. The parameter
sets chosen for the calculation of the relic density are consistent with the
observed value of $\Delta a_{\mu}$ and measured total Higgs decay width.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8: Figures 8(a) to 8(f) show the spin-independent DM-Nucleon cross-
section variation with the $m_{\chi}$ for fixed $m_{H^{\pm}}$ = 600 GeV ,
$\delta_{13}$ = 0.2 and different choices of $m_{12}^{2}$ and $m_{A^{0}}$. All
points on the contours satisfy the relic density 0.119 and also explain the
discrepancy $\Delta a_{\mu}=\,268(63)\,\times 10^{-11}$. In the left and right
panels, we plot the variation curves (bold lines) and allowed (shaded) regions
for five combinations of $m_{S^{0}}$ and $m_{H^{0}}$. The upper limit from
PANDA 2X-II 2017 Cui:2017nnn and XENON-1T Aprile:2015uzo ; Aprile:2017aty are
also shown along with the forbidden region shaded in red.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9: Figures 9(a) to 9(f) show the spin-independent DM-Nucleon cross-
section variation with the $m_{\chi}$ for fixed $m_{H^{\pm}}$ = 600 GeV ,
$\delta_{13}$ = 0.4 and different choices of $m_{12}^{2}$ and $m_{A^{0}}$. All
points on the contours satisfy the relic density 0.119 and also explain the
discrepancy $\Delta a_{\mu}=\,268(63)\,\times 10^{-11}$. In the left and right
panels, we plot the variation curves (bold lines) and allowed (shaded) regions
for five combinations of $m_{S^{0}}$ and $m_{H^{0}}$. The upper limit from
PANDA 2X-II 2017 Cui:2017nnn and XENON-1T Aprile:2015uzo ; Aprile:2017aty are
also shown along with the forbidden region shaded in red.

###  4.2 Direct Detection

Direct detection of DM measures the recoil generated by DM interaction with
matter. For the case of lepto-philic DM, we have tree level DM-Electron
interaction, where DM can scatter with electron in-elastically, leading to
ionization of the atom to which it is bound or elastically, where excitation
of atom is succeeded by de-excitation, releasing a photon. The DM-Nucleon
scattering in this model occurs at the loop level and though suppressed by one
or two powers of respective coupling strengths and the loop factor, it vastly
dominates over the DM-Electron and DM-Atom scattering Kopp:2009et ;
Kopp:2014tsa ; DEramo:2017zqw .

The scalar spin-independent DM-Nucleon scattering are induced through the
effective DM-photon, DM-quark and DM-gluon interactions which are mediated by
the singlet scalar portal of the model. Following reference Kopp:2009et , we
approximate the DM-Nucleon scattering cross-section through two photons by
integrating out the contributions of heavier fermions running in the loop. The
total cross-section Spin-Independent DM-Nucleon in this case is given as

$\displaystyle\sigma^{\gamma\gamma}_{N}=\left(\frac{\alpha_{em}Z}{\pi}\right)^{2}\left[\frac{\mu^{2}_{N}}{\pi}\left(\frac{\alpha_{em}Z}{\pi
m^{2}_{S^{0}}}\right)^{2}\right]\left(\frac{\pi^{2}}{12}\right)^{2}\left(\frac{\mu_{N}v}{m_{\tau}}\right)^{2}2\left(g_{\chi
S^{0}}\xi^{S^{0}}_{l}\frac{m_{\tau}}{v_{0}}\right)^{2}$ (41)

where $Z$ is the atomic number of the detector material, $\mu_{N}$ is the
reduced mass of the DM-Nucleon system and $v$ is the DM velocity of the order
of 10-3.

The effective DM-gluon interactions are induced through a quark triangle loop,
where, the negligible contribution of light quarks $u$, $d$ and $s$ to the
loop integral can be dropped. In this approximation, the effective Lagrangian
for singlet scalar-gluon interactions can be derived by integrating out
contributions from heavy quarks $c$, $b$ and $t$ in the triangle loop and can
be written as

${\cal L}_{\rm
eff.}^{S^{0}gg}=-\frac{\xi^{S^{0}}_{q}}{12\pi}\frac{\alpha_{s}}{v_{o}}\left\\{\sum_{q=c,b,t}I_{q}\right\\}G_{\mu\nu}^{a}G^{\mu\nu
a}S^{0}$ (42)

where the loop integral $I_{q}$ is given in Appendix B. The DM-gluon effective
Lagrangian is the given as

${\cal L}_{\rm eff.}^{\chi\chi
gg}=\frac{\alpha_{s}(m_{S^{0}})}{12\pi}\frac{\xi^{S^{0}}_{q}g_{\chi
S^{0}}}{v_{o}m^{2}_{S^{0}}}\left\\{\sum_{q=c,b,t}I_{q}\right\\}\bar{\chi}\chi
G_{\mu\nu}^{a}G^{\mu\nu a}.$ (43)

Using (43), the DM-gluon scattering cross-section can be computed and given
as:

$\displaystyle\sigma^{gg}_{N}=\left(\frac{2\xi^{S^{0}}_{q}g_{\chi
S^{0}}m_{N}}{m^{2}_{S^{0}}27v_{o}}\right)^{2}\left|\sum_{q=c,b,t}I_{q}\right|^{2}\frac{2}{\pi(m_{\chi}+m_{N})^{2}}m_{N}^{2}m^{2}_{\chi}$
(44)

To compare the cross-sections given in (41) and (44), we evaluate the ratio

$\frac{\sigma^{\gamma\gamma}_{N}}{\sigma^{gg}_{N}}\simeq(\alpha_{em})^{4}\frac{\mu^{2}_{N}}{m^{2}_{N}}\left(\frac{\xi^{S^{0}}_{\tau}}{\xi^{S^{0}}_{q}}\right)^{2}\left(\frac{9}{8}\right)^{2}\frac{v^{2}}{c^{2}}\simeq
10^{-6}-10^{-10}.$ (45)

Thus even though the effective DM-quark coupling is suppressed by
$\tan^{2}\beta$ w.r.t that of DM-lepton coupling, the scattering cross-
sections induced via the singlet coupled to the quark-loop dominates over the
$\sigma^{\gamma\gamma}_{N}$ due to suppression resulting from the fourth power
of the electromagnetic coupling.

We convolute the DM-quark and DM-gluon scattering cross-sections with the
quark form factor $F^{q_{i}/N}(q^{2})$ and gluon form factor $F^{g/N}(q^{2})$
respectively to compute nuclear recoil energy observed in the experiment.
However, this form factor is extracted at low $q^{2}\ll m^{2}_{N}$
Bishara:2017nnn ; DelNobile:2013sia ; Dutta:2017jfj . The form factors are
defined as

$\displaystyle\left\langle
N^{\prime}\left|\frac{\alpha_{s}}{12\pi}{G^{a}}^{\mu\nu}G^{a}_{\mu\nu}\right|N\right\rangle$
$\displaystyle=$ $\displaystyle F^{g/N}(q^{2})\bar{u}^{\prime}_{N}u_{N}$ (46a)
$\displaystyle\left\langle
N^{\prime}\left|m_{q_{i}}\bar{q_{i}}q_{i}\right|N\right\rangle$
$\displaystyle=$ $\displaystyle F^{q_{i}/N}(q^{2})\bar{u}^{\prime}_{N}u_{N}$
(46b)

Since, $m_{N}\equiv\sum_{u,d,s}\left\langle
N\left|m_{q}\bar{q}q\right|N\right\rangle-\frac{9\alpha_{S}}{8\pi}\left\langle
N\left|{G^{a}}^{\mu\nu}{G^{a}}_{\mu\nu}\right|N\right\rangle$, the gluon form
factor can be expressed as

$\displaystyle
F^{g/N}=1-\sum_{u,d,s}\frac{F^{q_{i}/N}_{S}(q^{2})}{m_{N}}=-\frac{1}{m_{N}}\frac{9\alpha_{s}}{8\pi}\left\langle
N\left|{G^{a}}^{\mu\nu}{G^{a}}_{\mu\nu}\right|N\right\rangle$ (47)

The $F^{g/N}$ is found to be $\approx$ 0.92 using the values for
$F^{q_{i}/N}_{S}(q^{2})$ as quoted in the literature DelNobile:2013sia . Thus,
at the low momentum transfer the quartic DM-gluon $\left(\chi\chi gg\right)$
effective interaction induced through relatively heavy quarks dominates over
the quartic DM-quark $\left(\chi\chi qq\right)$ effective interactions for
light quarks in the direct-detection experiments.

Using the expression 44 we have plotted the spin-independent DM-Nucleon
scattering cross-section as a function of the DM mass $m_{\chi}$. Figures 8
and 9 corresponding to mixing angle $\delta_{13}$=0.2 and $\delta_{13}$=0.4
respectively. The parameter sets used in the computation of direct detection
cross-section are consistent with the observed relic density as given in
figures 6 and 7. Different panels in figures 8 and 9 show combinations of
$m_{A^{0}}$ and $m^{2}_{12}$. In each panel different combinations of
$m_{S^{0}}$ and $m_{H^{0}}$ are used as shown. Current bounds on spin-
independent interactions from experiments like PANDA 2X-II 2017 Cui:2017nnn
and XENON-1T Aprile:2015uzo ; Aprile:2017aty are also shown. It can be seen
that most of the parameter space for $m_{S^{0}}$ less than 10 GeV is ruled out
by the current bounds.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10: Figures 10(a) to 10(f) show the velocity-averaged scattering cross-
section $<\sigma v>_{\tau^{+}\tau^{-}}$ variation with the $m_{\chi}$ for
fixed $m_{H^{\pm}}$ = 600 GeV, $\delta_{13}$ = 0.2 and different choices of
$m_{12}^{2}$ and $m_{A^{0}}$. All points on the contours satisfy the relic
density 0.119 and also explain the discrepancy $\Delta
a_{\mu}=\,268(63)\,\times 10^{-11}$. In the left and right panels, we plot the
variation curves (bold lines) and allowed (shaded) regions for four five
combinations of $m_{S^{0}}$ and $m_{H^{0}}$. The upper limit on velocity-
averaged annihilation cross-section observed from Fermi-LAT Ackermann:2015zua
is shown.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11: Figures 11(a) to 11(f) show the velocity-averaged scattering cross-
section $<\sigma v>_{\tau^{+}\tau^{-}}$ variation with the $m_{\chi}$ for
fixed $m_{H^{\pm}}$ = 600 GeV, $\delta_{13}$ = 0.4 and different choices of
$m_{12}^{2}$ and $m_{A^{0}}$. All points on the contours satisfy the relic
density 0.119 and also explain the discrepancy $\Delta
a_{\mu}=\,268(63)\,\times 10^{-11}$. In the left and right panels, we plot the
variation curves (bold lines) and allowed (shaded) regions for four five
combinations of $m_{S^{0}}$ and $m_{H^{0}}$. The upper limit on velocity-
averaged annihilation cross-section observed from Fermi-LAT Ackermann:2015zua
is shown.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 12: Figures 12(a) to 12(f) show the velocity-averaged scattering cross-
section $<\sigma v>_{W^{+}W^{-}}$ variation with the $m_{\chi}$ for fixed
$m_{H^{\pm}}$ = 600 GeV, $\delta_{13}$ = 0.2 and different choices of
$m_{12}^{2}$ and $m_{A^{0}}$. All points on the contours satisfy the relic
density 0.119 and also explain the discrepancy $\Delta
a_{\mu}=\,268(63)\,\times 10^{-11}$. In the left and right panels, we plot the
variation curves (bold lines) and allowed (shaded) regions for five
combinations of $m_{S^{0}}$ and $m_{H^{0}}$ . The upper limit on velocity-
averaged annihilation cross-section observed from Fermi-LAT Ackermann:2015zua
is shown.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 13: Figures 13(a) to 13(f) show the velocity-averaged scattering cross-
section $<\sigma v>_{W^{+}W^{-}}$ variation with the $m_{\chi}$ for fixed
$m_{H^{\pm}}$ = 600 GeV, $\delta_{13}$ = 0.4 and different choices of
$m_{12}^{2}$ and $m_{A^{0}}$. All points on the contours satisfy the relic
density 0.119 and also explain the discrepancy $\Delta
a_{\mu}=\,268(63)\,\times 10^{-11}$. In the left and right panels, we plot the
variation curves (bold lines) and allowed (shaded) regions for five
combinations of $m_{S^{0}}$ and $m_{H^{0}}$ . The upper limit on velocity-
averaged annihilation cross-section observed from Fermi-LAT Ackermann:2015zua
is shown.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 14: Figures 14(a) to 14(f) show the velocity-averaged scattering cross-
section $<\sigma v>_{S^{0}S^{0}}$ variation with the $m_{\chi}$ for fixed
$m_{H^{\pm}}$ = 600 GeV, $\delta_{13}$ = 0.2 and and different choices of
$m_{12}^{2}$ and $m_{A^{0}}$. All points on the contours satisfy the relic
density 0.119 and also explain the discrepancy $\Delta
a_{\mu}=\,268(63)\,\times 10^{-11}$. In the left and right panels, we plot the
variation curves (bold lines) and allowed (shaded) regions for five
combinations of $m_{S^{0}}$ and $m_{H^{0}}$. The upper limit on velocity-
averaged annihilation cross-section for the process $\chi\chi\rightarrow
S^{0}S^{0}$ computed from 4$\tau$ final states from Fermi-LAT
dataProfumo:2017obk is shown.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 15: Figures 15(a) to 15(f) show the velocity-averaged scattering cross-
section $<\sigma v>_{S^{0}S^{0}}$ variation with the $m_{\chi}$ for fixed
$m_{H^{\pm}}$ = 600 GeV, $\delta_{13}$ = 0.4 and and different choices of
$m_{12}^{2}$ and $m_{A^{0}}$. All points on the contours satisfy the relic
density 0.119 and also explain the discrepancy $\Delta
a_{\mu}=\,268(63)\,\times 10^{-11}$. In the left and right panels, we plot the
variation curves (bold lines) and allowed (shaded) regions for five
combinations of $m_{S^{0}}$ and $m_{H^{0}}$. The upper limit on velocity-
averaged annihilation cross-section for the process $\chi\chi\rightarrow
S^{0}S^{0}$ computed from 4$\tau$ final states from Fermi-LAT data
Profumo:2017obk is shown.

###  4.3 Indirect detection

Observations of diffused gamma rays from the regions of our Galaxy, such as
Galactic Center (GC) and dwarf spheroidal galaxies (dsphs), where DM density
appears to be high, impose bounds on DM annihilation to SM particles.
Experiments like Fermi-LAT TheFermi-LAT:2015kwa ; Ackermann:2015zua and
H.E.S.S. Moulin:2007qw have investigated DM annihilation as a possible source
of the incoming photon-flux. These experiments provide us with an upper-limit
to velocity-averaged scattering cross-section for various channels, which can
attribute to the observed photon-flux.

DM annihilations contribute to the photon-flux through Final State Radiation
(FSR) and radiative decays Mazziotta:2014ada ; Ackermann:2013yva from leptonic
channels in lepto-philic models. FSR contributions are important in
understanding the photon-spectra from DM annihilations to charged final states
and therefore are instrumental in calculation of the observed bounds by
experiments like Fermi-LAT Ackermann:2015zua ; Ackermann:2013yva ;
Essig:2009jx ; Bringmann:2007nk . The radiation emitted by the charged
relativistic final state fermions $f$ in the annihilation process
$\bar{\chi}+\chi\rightarrow f+\bar{f}+\gamma$ are approximately collinear with
the charged fermions. In this regime, the differential cross-section for the
real emission process can be factorized into the a collinear factor and cross-
section $\sigma(\chi\chi\to f\bar{f})$ as discussed in the reference
Birkedal:2005ep .

$\begin{array}[]{rcl}\frac{d\sigma(\chi\chi\,\to\,f\bar{f}\gamma)}{dx}\approx\frac{\alpha_{em}Q_{f}^{2}}{\pi}\,{\cal
F}_{f}(x)\,\log\left(\frac{s(1-x)}{m_{f}^{2}}\right)\sigma(\chi\chi\to
f\bar{f}),\end{array}$ (48)

where $Q_{f}$ and $m_{f}$ are the electric charge and the mass of the $f$
particle, $s$ is the center-of-mass energy, and $x=2E_{\gamma}/\sqrt{s}$. For
fermion final states, the splitting function ${\cal F}$ is given by

$\begin{array}[]{rcl}{\cal
F}_{f}(x)=\frac{1+\left(1-x\right)^{2}}{x}\end{array}$ (49)

The suppression factor of $p$-wave suppressed thermal averaged cross-section
$\left\langle\sigma(\chi\chi\to f\bar{f})\,v\right\rangle$ is mitigated in the
thermal averaged cross-section of the real emission process
$\left\langle\sigma(\chi\chi\to f\bar{f}\gamma)\,v\right\rangle$ by the virtue
of collinear factor given in equation (48).

In the present model, the fermionic DM can annihilate to SM particles via
$s-channnel$ through the scalar portal as well as to a pair of singlet scalars
through $t-channel$ diagrams. Recently authors of the reference
Siqueira:2019wdg ; Queiroz:2019acr explored the discovery potential of the
pair production of such lepto-philic scalars which pre-dominantly decay into
pairs of charged leptons at Cherenkov Telescope Array (CTA). Given the
spectrum of pair produced SM particles through single scalar mediator and into
two pairs of charged leptons through scalar pair production, we should be able
to simulate the expected DM fluxes which will enable us to get the upper
limits on the annihilation cross section for a given mediator mass in the
model.

We calculate the velocity averaged cross-sections analytically for the
annihilation processes $\chi\,\bar{\chi}\to f\bar{f}$, $\chi\,\bar{\chi}\to
Z^{0}Z^{0}$, $\chi\,\bar{\chi}\to W^{+}W^{-}$,
$\chi\,\bar{\chi}\to\gamma\gamma$ and $\chi\,\bar{\chi}\to H_{i}\,H_{j}$ and
are given in equations (LABEL:thermalfermion), (66), (67), (69) and (70)
respectively where $H_{i}\equiv h^{0},\,H^{0},\,S^{0}$ and $A^{0}$ are the
scalars of the model. In addition, the velocity averaged annihilation cross-
section for $\chi\,\bar{\chi}\to S^{0}S^{0}$ through the $t$ and $u$ channel
diagrams are given in (71). We observe that the velocity averaged scattering
cross-sections for all these processes are $p$-wave suppressed and are,
therefore, sensitive to the choice of velocity distribution of the DM in the
galaxy.

The annihilation channels to fermions are proportional to the Yukawa coupling
of the fermions with $S^{0}$. We present the analysis for the most dominant
$s-channel$ annihilation process $\chi\,\bar{\chi}\to\tau^{+}\tau^{-}$, which
is enhanced due to its coupling strength being proportional to
$m_{\tau}\tan\beta$ and plot the variation of the velocity averaged scattering
cross-section $\left\langle\sigma
v\right\rangle(\chi\bar{\chi}\rightarrow\tau^{+}\tau^{-})$ as a function of
the DM mass in figures 10 and 11 for mixing angle $\delta_{13}$ = 0.2 and 0.4
respectively. The coupling $g_{\chi S^{0}}$ for a given DM mass and all other
parameters are chosen to satisfy the observed relic density and electro-weak
constraints as shown in the figures 6 and 7. Annihilation of DM pairs to gauge
Bosons are proportional to the square of their masses and therefore it is the
second dominant process followed by the annihilation to $\tau\pm$ pairs.
Similarly, we show the variation of the velocity averaged scattering cross-
section $\left\langle\sigma v\right\rangle(\chi\bar{\chi}\rightarrow
W^{+}W^{-})$ as a function of the DM mass in figures 12 and 13 for
$\delta_{13}$ = 0.2 and 0.4 respectively. The DM pair annihilation to photons
is loop suppressed and is not discussed further. The $s$ channel mediated DM
pair annihilation to pair of scalars in the theory involve the triple scalar
couplings, which are experimentally constrained and are therefore suppressed.

As mentioned above, the t-channel pair production of singlet scalars dominates
over the other channels. The $S^{0}$ pair production through its decay to
dominant $\tau$ pairs will modify the $\gamma$ ray spectrum that one would
have expected from the two body decay processes. We plot the velocity averaged
scattering cross-section $\left\langle\sigma
v\right\rangle(\chi\bar{\chi}\rightarrow S^{0}S^{0})$ as a function of the DM
mass which satisfies the relic density constraint in figures 14 and 15 for
$\delta_{13}$ = 0.2 and 0.4 respectively with all the other parameters fixed
from the observed relic density and electro-weak constraints. The experimental
upper limit on velocity-averaged annihilation cross-section for the process
$\chi\chi\rightarrow S^{0}S^{0}$ for the varying DM mass are derived from the
upper limits on the events contributed to 4$\tau$ final states at Fermi-LAT
Profumo:2017obk and shown in figures 14 and 15.

We find that the annihilation cross-sections for all these processes are three
or more orders of magnitude smaller than the current upper-bounds from Fermi-
Lat data Ackermann:2015zua ; Profumo:2017obk .

##  5 Summary

In this article we have made an attempt to address the observed discrepancy in
anomalous magnetic moment of muon by considering a lepto-philic type X 2-HDM
and a singlet scalar portal for fermionic DM. We have presented the model in
such a manner where most of it’s scalar sector parameters can be constrained
in terms of the lower bound on the physical neutral and charged scalar’s
masses derived from the direct and indirect searches at LEP and LHC.

The model is analysed in the alignment limit, where one of its scalar is
identified with the Higgs Boson of SM and the Yukawa couplings of fermions
with the singlet scalar are found to be proportional to mass of the fermions
i.e. non-universal. It is then validated with low energy constraints. We have
illustrated the constraints from anomalous magnetic moment in figures 1 and 2
and fixed the parameters $\tan\beta$ and $\delta_{23}$ for a given
$\delta_{13}$. We have considered two choices 0.2 and 0.4 respectively for the
mixing angle $\delta_{13}$. Contrary to the results obtained in reference
Agrawal:2014ufa for the singlet scalar with mass lying between 10 - 300 MeV
with universal couplings to leptons, this study establishes the acceptability
of the model to explain the discrepancy $\Delta a_{\mu}$ for singlet scalar
mass lying between 10 GeV $\leq$ $m_{S^{0}}$ $\leq$ 80 GeV with couplings to
leptons being non-universal. The requirement of the Yukawa coupling
$H^{0}\,\tau^{+}\tau^{-}$ to remain perturbative further imposes an upper
limit $\tan\beta\leq 485$ which in turn provides the upper bound on the
allowed mass range of singlet scalars to be $\sim 80$ GeV.

Exclusion limits on the couplings of SM gauge Bosons with the singlet scalars
are obtained from the process $e^{+}e^{-}\to Z^{0}S^{0}$ at LEP-II experiment
and have been displayed in table 2 for some chosen singlet scalar masses.

Validation of the model is further subjected to the observed total Higgs decay
width at LHC pdg2018Higgs ; Khachatryan:2016ctc . It is shown that the
parameter $m_{12}^{2}$, which has no bearing on the $\Delta a_{\mu}$, can now
be constrained from the from upper bound on the triple scalar coupling
involved in the decay of SM like Higgs to a pair of singlet scalars $h^{0}\to
S^{0}S^{0}$. The observed total decay width of SM like Higgs $h^{0}$ restricts
this additional channel and put a upper limit on the partial decay width,
which has been shown in figures 3 for $\delta_{13}$ = 0.2 and in figures 4 and
5 $\delta_{13}$ = 0.4 respectively. We have found that in the probed region of
interest for singlet scalar mass, $m_{12}^{2}$ greater than 100 GeV2 and less
than 0 GeV2 are forbidden.

We have addressed reasons for which there can be a deviation from SM predicted
universality in lepton-gauge Boson couplings. The precision constraints are
also discussed for our model and found that corrections are suppressed due to
the smallness of mixing angle.

We augment our analysis by including a fermionic DM candidate $\chi$ and
compute the relic density which are depicted in figures 6 & 7 for
$\delta_{13}$ = 0.2 and 0.4 respectively. The parameter sets chosen
corresponding to points lying on contours satisfying relic density of 0.119
also fulfill the $\Delta a_{\mu}$ discrepancy and are consistent with the
total Higgs decay width observed at LHC and LEP data.

The scalar portal induced DM interactions are now probed in the Direct-
detection experiment by the DM-nucleon scattering propelled through the
gluons. The variation of spin-independent scattering cross-sections with the
DM mass are shown in figures 8 and 9 for $\delta_{13}$ = 0.2 and 0.4
respectively. It can be seen that most of the parameter space for $m_{S^{0}}$
lighter than 10 GeV is excluded by current Direct-detection constraints from
PANDA 2X-II and XENON-1T experiments.

The velocity averaged cross-sections for dominant DM pair annihilation
channels like $\tau^{+}\tau^{-}$, $W^{+}W^{-}$, $S^{0}S^{0}$ and
$\gamma\gamma$ are analytically derived, analysed and compared with the
available space borne indirect-detection experiments. The velocity averaged
cross-sections variation w.r.t DM mass are shown for $\delta_{13}$ = 0.2 and
0.4 in figures 10 and 11 respectively for $\chi\bar{\chi}\to\tau^{+}\tau^{-}$,
in figures 12 and 13 respectively for $\chi\bar{\chi}\to W^{+}W^{-}$, in
figures 14 and 15 respectively for $\chi\bar{\chi}\to S^{0}S^{0}$. We find
that the contribution to the gamma ray spectrum from the most dominant
annihilation channel to $\tau^{\pm}$ pairs is at least three orders of
magnitude lower than the current reach for the DM mass varying between 5 GeV -
8 TeV.

In conclusion the lepton-specific type X 2-HDM model with a singlet scalar
portal for fermionic dark matter is capable of explaining both the observed
discrepancy in the anomalous magnetic moment of the muon and the observed
relic density. This model with the shrunk parameter space after being
constrained by low energy experiments, LEP Data, observed total decay width of
Higgs at LHC and constrained by dark matter detection experiments can now be
tested at the ongoing and upcoming collider searches.

###### Acknowledgements.

Authors acknowledge the fruitful discussions with Mamta. SD and MPS
acknowledge the partial financial support from the CSIR grant No.
03(1340)/15/EMR-II. MPS acknowledges the CSIR JRF fellowship for the partial
financial support. SD and MPS thank IUCAA, Pune for providing the hospitality
and facilities where this work was initiated.  Appendix

##  Appendix A Model Parameters

The parameters used in the Lagrangian for lepto-philic 2-HDM and dark matter
portal singlet scalar given in equation 3 are expressed in terms of the
physical scalar masses, mixing angles $\alpha$ and $\beta$ and the model
parameter $m_{12}^{2}$.

$\displaystyle m^{2}_{11}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[m_{H^{0}}^{2}\cos^{2}\alpha+m_{h^{0}}^{2}\sin^{2}\alpha+\left\\{\sin\alpha\cos\alpha\left(m_{H^{0}}^{2}-m_{h^{0}}^{2}\right)-2\,m^{2}_{12}\right\\}\tan\beta\right]$
(50) $\displaystyle m^{2}_{22}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[m_{h^{0}}^{2}\cos^{2}\alpha+m_{H^{0}}^{2}\sin^{2}\alpha+\left\\{\sin\alpha\cos\alpha\left(m_{H^{0}}^{2}-m_{h^{0}}^{2}\right)-2\,m^{2}_{12}\right\\}\cot\beta\right]$
(51) $\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{v_{o}^{2}\cos^{2}\beta}\left[m_{H^{0}}^{2}\cos^{2}\alpha+m_{h^{0}}^{2}\sin^{2}\alpha-m^{2}_{12}\tan\beta\right]$
(52) $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{v_{o}^{2}\sin^{2}\beta}\left[m_{h^{0}}^{2}\cos^{2}\alpha+m_{H^{0}}^{2}\sin^{2}\alpha-m^{2}_{12}\cot\beta\right]$
(53) $\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle\frac{2}{v_{o}^{2}\,\sin\left(2\,\beta\right)}\left[\sin\alpha\cos\alpha\left(m_{H^{0}}^{2}-m_{h^{0}}^{2}\right)-m^{2}_{12}+\,m_{H^{+}}^{2}\sin\left(2\,\beta\right)\right]$
(54) $\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{v_{o}^{2}\sin\left(2\,\beta\right)}\left[2m^{2}_{12}+\left(m_{A^{0}}^{2}-2\,m_{H^{+}}^{2}\right)\sin\left(2\,\beta\right)\right]$
(55) $\displaystyle\lambda_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{v_{o}^{2}\sin\left(2\,\beta\right)}\left[2\,m^{2}_{12}-m_{A^{0}}^{2}\,\sin\left(2\,\beta\right)\right]$
(56)

##  Appendix B Decay widths of the singlet scalar $S^{0}$

The tree level partial decay widths of the scalar mediator are computed and
are given by:

$\Gamma(S^{0}\rightarrow
f\,\bar{f})=\frac{N_{c}}{8\,\pi}\left(\frac{\xi^{S}_{f}m_{f}}{v_{o}}\right)^{2}\,\,\,m_{S^{0}}\,\,\,\left(1-\frac{4m_{f}^{2}}{m_{S^{0}}^{2}}\right)^{3/2}\,\,\,\theta\left(m_{S^{0}}-2\,mf\right)$
(57)

where $N_{c}$ = 1 for leptons and 3 for quarks

$\displaystyle\Gamma(S^{0}\rightarrow W^{+}\,W^{-})$ $\displaystyle=$
$\displaystyle\frac{(\xi^{S^{0}}_{V})^{2}}{16\,\pi\,v_{o}^{2}}\,m_{S^{0}}^{3}\,\left(1-\frac{4m_{W}^{2}}{m_{S^{0}}^{2}}\right)^{1/2}\,\left[12\,\left(\frac{m_{W}}{m_{S^{0}}}\right)^{4}-4\,\left(\frac{m_{W}}{m_{S^{0}}}\right)^{2}+1\right]$
(58) $\displaystyle\times\theta\left(m_{S^{0}}-2\,m_{W}\right)$
$\displaystyle\Gamma(S^{0}\rightarrow Z^{0}\,Z^{0})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\,\Gamma(S^{0}\rightarrow W^{+}W^{-})\,\,\,{\rm
with}\,\,m_{W}\rightarrow m_{Z}$ (59)
$\displaystyle\Gamma(S^{0}\rightarrow\chi\,\bar{\chi})$ $\displaystyle=$
$\displaystyle\frac{g_{\chi
S^{0}}^{2}}{8\pi}\,\,\,m_{S^{0}}\,\,\,\left(1-\frac{4m_{\chi}^{2}}{m_{S^{0}}^{2}}\right)^{3/2}\,\,\,\theta\left(m_{S^{0}}-2\,m_{\chi}\right)$
(60)

The one loop induced partial decay width of the scalar to gluons in this model
arises mainly from relatively heavy quarks and is given by

$\Gamma(S^{0}\rightarrow
gg)=\left(\frac{m_{t}\xi^{S^{0}}_{q}}{v_{o}}\right)^{2}\,\,\frac{\alpha_{s}^{2}}{72\,\pi^{3}}\,\,\,\frac{m_{S^{0}}^{3}}{m_{t}^{2}}\,\,\,\,\left|\sum_{q=c,b,t}I_{q}\right|^{2}$
(61)

For the case of photons it is given by

$\Gamma(S^{0}\rightarrow\gamma\gamma)=\frac{m_{S^{0}}^{3}}{16\pi
v^{2}_{0}}\,\,\left(\frac{\alpha_{em}}{\pi}\right)^{2}\left|\sum_{q}\xi^{S^{0}}_{q}Q^{2}_{q}I_{q}+\sum_{l}\xi^{S^{0}}_{l}Q^{2}_{q}I_{q}-\xi^{S^{0}}_{W^{\pm}}I_{W^{\pm}}+C_{SH^{+}H^{-}}\frac{v_{o}}{2m^{2}_{H^{\pm}}}I_{H^{\pm}}\right|^{2}$
(62)

The integrals are given as

$\displaystyle
I_{q}=3[2\lambda_{q}+\lambda_{q}(4\lambda_{q}-1)f(\lambda_{q})];$
$\displaystyle I_{l}=2\lambda_{q}+\lambda_{q}(4\lambda_{q}-1)f(\lambda_{q});$
$\displaystyle
I_{W}=3\lambda_{W}(1-2\lambda_{W})f(\lambda_{W})-\lambda_{W}-\frac{1}{2};$
$\displaystyle
I_{H^{\pm}}=-\lambda_{H^{\pm}}[1+2\lambda_{H^{\pm}}f({H^{\pm}})].$ (63)

The integrals are defined in terms of dimensionless parameter
$\lambda_{i}=m^{2}_{i}/m^{2}_{S^{0}}$ and its function $f(\lambda)$ as

$\displaystyle f(\lambda)$ $\displaystyle=$
$\displaystyle-2\left(sin^{-1}\frac{1}{2\sqrt{\lambda}}\right)^{2},for\,\lambda>\frac{1}{4}$
(64) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\ln\frac{\eta^{+}}{\eta^{-}}\right)^{2}-\frac{\pi^{2}}{2}-i\pi\frac{\eta^{+}}{\eta^{-}},\,for\,\lambda<\frac{1}{4}$

with $\eta^{\pm}=\frac{1}{2}\pm\sqrt{\frac{1}{4}-\lambda}$.

##  Appendix C Thermally averaged scattering cross-sections

We compute the thermal averaged annihilation cross-section of the fermionic DM
via the singlet scalar portal $S^{0}$ to the SM final states. These processes
contributes to the relic density of the universe and are directly used in
computing the annihilation cross-section for indirect detection of the DM.

$\displaystyle\left\langle\sigma(\chi\bar{\chi}\rightarrow
f\bar{f})\,v\right\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{m_{f}\xi^{S^{0}}_{f}}{v_{o}}\right)^{2}g_{\chi
S^{0}}^{2}\,\frac{1}{4\pi}\,\left(1-\frac{m_{l}^{2}}{m_{\chi}^{2}}\right)^{\frac{3}{2}}\frac{m_{\chi}^{2}}{(4m_{\chi}^{2}-m_{\phi}^{2})^{2}}\left(\frac{3}{x_{f}}\right)\theta\left(m_{\chi}-m_{f}\right)$
$\displaystyle\left\langle\sigma\left(\chi\bar{\chi}\rightarrow
Z^{0}Z^{0}\right)\,v\right\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{\xi_{V}^{S^{0}}}{v_{o}}\right)^{2}\frac{g_{\chi
S^{0}}^{2}}{8\pi}\sqrt{1-\frac{m_{Z}^{2}}{m_{\chi}^{2}}}\frac{(16m_{\chi}^{4}+12m_{Z}^{4}-16m_{\chi}^{2}m_{Z}^{2})}{(4m_{\chi}^{2}-m_{S^{0}}^{2})^{2}}\left(\
\frac{3}{x_{f}}\right)$ (66)
$\displaystyle\times\,\,\theta\left(m_{\chi}-m_{Z^{0}}\right)$
$\displaystyle\left\langle\sigma\left(\chi\bar{\chi}\rightarrow
W^{+}W^{-}\right)\,v\right\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{\xi_{V}^{S^{0}}}{v_{o}}\right)^{2}\frac{g_{\chi
S^{0}}^{2}}{16\pi}\sqrt{1-\frac{m_{W}^{2}}{m_{\chi}^{2}}}\frac{(16m_{\chi}^{4}+12m_{W}^{4}-16m_{\chi}^{2}m_{W}^{2})}{(4m_{\chi}^{2}-m_{S^{0}}^{2})^{2}}\left(\
\frac{3}{x_{f}}\right)$ (67)
$\displaystyle\times\,\,\theta\left(m_{\chi}-m_{W^{\pm}}\right)$
$\displaystyle\left\langle\sigma(\chi\bar{\chi}\rightarrow
gg)\,v\right\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{\xi^{S^{0}}_{q}\alpha_{s}\,g_{\chi
S^{0}}}{3\pi^{3/2}v_{o}}\right)^{2}\frac{m_{\chi}^{4}}{(4m^{2}_{\chi}-m_{S^{0}}^{2})^{2}+m_{S^{0}}^{2}\Gamma}\left(\frac{3}{2x_{f}}\right)\left|\sum_{q=c,b,t}I_{q}\right|^{2}$
(68)

$\displaystyle\left\langle\sigma\left(\chi\bar{\chi}\to\gamma\gamma\right)\,v\right\rangle=$
$\displaystyle\frac{g_{\chi
S^{0}}^{2}\alpha^{2}_{em}}{2\pi^{3}v^{2}_{o}}\frac{m^{4}_{\chi}\left|\sum_{q}\xi^{S^{0}}_{q}Q^{2}_{q}I_{q}+\sum_{l}\xi^{S^{0}}_{l}Q^{2}_{q}I_{q}-\xi^{S^{0}}_{W^{\pm}}I_{W^{\pm}}+C_{SH^{+}H^{-}}\frac{v_{o}}{2m^{2}_{H^{\pm}}}I_{H^{\pm}}\right|^{2}}{(4m_{\chi}-m^{2}_{S^{0}})^{2}+m^{2}_{S^{0}}\Gamma^{2}}$
(69) $\displaystyle\times\left(\frac{3}{x_{f}}\right)$

$\displaystyle\left\langle\sigma(\chi\bar{\chi}\rightarrow
H^{i}\,H^{j})\,v\right\rangle$ $\displaystyle=$ $\displaystyle
c_{0}\sum_{H^{i};H^{j}\equiv
H^{0},\,A^{0},\,H^{\pm}}C_{S^{0}H^{i}H^{j}}^{2}\frac{g_{\chi
S^{0}}^{2}}{16\pi}\lambda^{\frac{1}{2}}\left(1,\frac{m^{2}_{H^{i}}}{4m^{2}_{\chi}},\frac{m^{2}_{H^{j}}}{4m^{2}_{\chi}}\right)$
(70)
$\displaystyle\times\frac{1}{(4m^{2}_{\chi}-m^{2}_{\phi})^{2}}\left(\frac{3}{x_{f}}\right)$

where $C_{S^{0}H^{i}H^{j}}$ are the tri-linear scalar couplings given in the
Appendix D; $c_{0}=\frac{1}{2}$ for i=j and $c_{0}=1$ for i$\neq$j ;
$\lambda(X,a,b)=X^{2}+a^{2}+b^{2}-2ab-2aX-2bX$.

In addition to s-channel processes considered above, we also have
contributions to the relic density from t-channel process
$\chi\bar{\chi}\rightarrow S^{0}S^{0}$, given by

$\displaystyle\left\langle\sigma\left(\chi\bar{\chi}\rightarrow
S^{0}S^{0}\right)\,v\right\rangle$ $\displaystyle=$
$\displaystyle\frac{3g_{\chi S^{0}}^{4}}{64\pi
m^{2}_{\chi}}\left(\frac{3}{x_{f}}\right)\,\theta\left(m_{\chi}-m_{S^{0}}\right)$
(71)

##  Appendix D  Triple scalar coupling

Here, we extract the triple scalar coupling from the 2-HDM + singlet scalar
Lagrangian in the alignment limit. Some of these scalars can be directly
constrained from the ongoing experiments at the Colliders. We define
dimensionless ratios $r_{0}=\frac{m^{2}_{H^{0}}}{m^{2}_{h^{0}}}$ and
$s_{0}=\frac{m^{2}_{12}}{m^{2}_{h^{0}}}$. All triple scalar couplings are
defined in terms of $r_{0}$, $s_{0}$ and $\tan\beta$.

$\displaystyle C_{h^{0}S^{0}S^{0}}$ $\displaystyle=$
$\displaystyle\frac{\delta^{2}_{13}m_{h^{0}}^{2}}{v_{o}}\left(-\frac{4\delta_{13}^{2}r_{0}^{2}}{\tan\beta}+\frac{2\delta_{13}^{2}r_{0}}{\tan\beta}-\frac{8r_{0}^{3}}{\tan^{4}\beta}+\frac{8r_{0}^{2}s_{0}}{\tan^{3}\beta}+\frac{4r_{0}^{2}}{\tan^{4}\beta}-\frac{8r_{0}s_{0}}{\tan^{3}\beta}-\frac{8r_{0}s_{0}}{\tan\beta}\right.$
$\displaystyle+\left.\frac{2r_{0}}{\tan^{4}\beta}-\frac{2r_{0}}{\tan^{2}\beta}+\frac{2s_{0}}{\tan^{3}\beta}+2s_{0}\tan\beta+\frac{4\delta_{13}^{2}s_{0}}{\tan\beta}-\frac{1}{\tan^{4}\beta}-\frac{2}{\tan^{2}\beta}-1\right)$
$\displaystyle C_{S^{0}h^{0}h^{0}}$ $\displaystyle=$
$\displaystyle\frac{\delta_{13}m_{h^{0}}^{2}}{v_{o}}\left(-\frac{4\delta_{13}^{2}r_{0}^{2}}{\tan^{2}\beta}+\frac{2\delta_{13}^{2}r_{0}}{\tan^{2}\beta}-2\delta_{13}^{2}r_{0}-\frac{6r_{0}}{\tan^{3}\beta}-\frac{4r_{0}}{\tan\beta}\right)$
(73) $\displaystyle C_{S^{0}H^{0}H^{0}}$ $\displaystyle=$
$\displaystyle\frac{\delta_{13}m_{h^{0}}^{2}}{v_{o}}\left(-\frac{8\delta_{13}^{2}r_{0}^{3}}{\tan^{2}\beta}+\frac{12\delta_{13}^{2}r_{0}^{2}}{\tan^{2}\beta}-\frac{4\delta_{13}^{2}r_{0}}{\tan^{2}\beta}-\frac{6r_{0}^{2}}{\tan^{5}\beta}-\frac{4r_{0}^{2}}{\tan^{3}\beta}+\frac{2r_{0}^{2}}{\tan\beta}+\frac{6r_{0}s_{0}}{\tan^{4}\beta}+\frac{4r_{0}s_{0}}{\tan^{2}\beta}\right.$
$\displaystyle\left.-2r_{0}s_{0}+\frac{3r_{0}}{\tan^{5}\beta}+\frac{r_{0}}{\tan^{3}\beta}-3r_{0}\tan\beta-\frac{7r_{0}}{\tan\beta}-\frac{3s_{0}}{\tan^{4}\beta}+3s_{0}\tan^{2}\beta-\frac{3s_{0}}{\tan^{2}\beta}+3s_{0}\right)$
$\displaystyle C_{S^{0}S^{0}H^{0}}$ $\displaystyle=$
$\displaystyle\frac{\delta^{2}_{13}m_{h^{0}}^{2}}{v_{o}}\left(-\frac{8\delta_{13}^{2}r_{0}^{3}}{\tan^{2}\beta}+\frac{8\delta_{13}^{2}r_{0}^{2}}{\tan^{2}\beta}-\frac{2\delta_{13}^{2}r_{0}}{\tan^{2}\beta}+\frac{12r_{0}^{3}}{\tan^{5}\beta}+\frac{4r_{0}^{3}}{\tan^{3}\beta}-\frac{12r_{0}^{2}s_{0}}{\tan^{4}\beta}-\frac{4r_{0}^{2}s_{0}}{\tan^{2}\beta}-\frac{12r_{0}^{2}}{\tan^{5}\beta}\right.$
(75)
$\displaystyle\left.+\frac{8r_{0}^{2}}{\tan\beta}+\frac{12r_{0}s_{0}}{\tan^{4}\beta}+\frac{8r_{0}s_{0}}{\tan^{2}\beta}-4r_{0}s_{0}+\frac{3r_{0}}{\tan^{5}\beta}-\frac{r_{0}}{\tan^{3}\beta}-3r_{0}\tan\beta-\frac{11r_{0}}{\tan\beta}\right.$
$\displaystyle\left.-\frac{3s_{0}}{\tan^{4}\beta}+3s_{0}\tan^{2}\beta-\frac{3s_{0}}{\tan^{2}\beta}+3s_{0}\right)$
$\displaystyle C_{S^{0}H^{0}h^{0}}$
$\displaystyle=\frac{\delta_{13}m_{h^{0}}^{2}}{v_{o}}\left(-\frac{4\delta_{13}^{2}r_{0}^{3}}{\tan^{3}\beta}+\frac{4\delta_{13}^{2}r_{0}^{2}}{\tan^{3}\beta}-\frac{4\delta_{13}^{2}r_{0}^{2}}{\tan\beta}-\frac{\delta_{13}^{2}r_{0}}{\tan^{3}\beta}+\frac{3\delta_{13}^{2}r_{0}}{\tan\beta}+\frac{4r_{0}^{2}}{\tan^{4}\beta}+\frac{4r_{0}^{2}}{\tan^{2}\beta}-\frac{4r_{0}s_{0}}{\tan^{3}\beta}\right.$
(76)
$\displaystyle\left.-\frac{4r_{0}s_{0}}{\tan\beta}-\frac{3r_{0}}{\tan^{2}\beta}-r_{0}+\frac{2s_{0}}{\tan^{3}\beta}+2s_{0}\tan\beta+\frac{4s_{0}}{\tan\beta}-\frac{1}{\tan^{4}\beta}-\frac{2}{\tan^{2}\beta}-1\right)$
$\displaystyle C_{S^{0}H^{+}H^{-}}$
$\displaystyle=\frac{\delta_{13}m_{h^{0}}^{2}}{v_{o}}\left(-\frac{2r_{0}^{2}}{\tan^{5}\beta}+\frac{2r_{0}^{2}}{\tan\beta}+\frac{2r_{0}s_{0}}{\tan^{4}\beta}+\frac{4r_{0}s_{0}}{\tan^{2}\beta}+2r_{0}s_{0}+\frac{r_{0}}{\tan^{5}\beta}-\frac{r_{0}}{\tan^{3}\beta}-r_{0}\tan\beta\right.$
(77)
$\displaystyle\left.-\frac{5r_{0}}{\tan\beta}-\frac{s_{0}}{\tan^{4}\beta}+s_{0}\tan^{2}\beta-\frac{s_{0}}{\tan^{2}\beta}+s_{0}-\frac{4m_{H^{\pm}}^{2}r_{0}}{\tan^{3}\beta
m^{2}_{h^{0}}}-\frac{4m_{H^{\pm}}^{2}r_{0}}{\tan\beta m^{2}_{h^{0}}}\right)$
$\displaystyle C_{S^{0}A^{0}A^{0}}$
$\displaystyle=\frac{\delta_{13}m_{h^{0}}^{2}}{v_{o}}\left(-\frac{4m_{A^{0}}^{2}r_{0}}{\tan^{3}\beta
m_{h^{0}}^{2}}-\frac{4m_{A^{0}}^{2}r_{0}}{\tan\beta
m_{h^{0}}^{2}}-\frac{2r_{0}^{2}}{\tan^{5}\beta}+\frac{2r_{0}^{2}}{\tan\beta}+\frac{2r_{0}s_{0}}{\tan^{4}\beta}+\frac{4r_{0}s_{0}}{\tan^{2}\beta}+2r_{0}s_{0}\right.$
(78)
$\displaystyle\left.+\frac{r_{0}}{\tan^{5}\beta}-\frac{r_{0}}{\tan^{3}\beta}-r_{0}\tan\beta-\frac{5r_{0}}{\tan\beta}-\frac{s_{0}}{\tan^{4}\beta}+s_{0}\tan^{2}\beta-\frac{s_{0}}{\tan^{2}\beta}+s_{0}\right)$
$\displaystyle C_{S^{0}S^{0}S^{0}}$
$\displaystyle=\frac{\delta^{3}_{13}m_{h^{0}}^{2}}{v_{o}}\left(-\frac{24r_{0}^{4}}{\tan^{5}\beta}+\frac{24r_{0}^{3}s_{0}}{\tan^{4}\beta}+\frac{36r_{0}^{3}}{\tan^{5}\beta}-\frac{12r_{0}^{3}}{\tan^{3}\beta}-\frac{36r_{0}^{2}s_{0}}{\tan^{4}\beta}-\frac{12r_{0}^{2}s_{0}}{\tan^{2}\beta}-\frac{18r_{0}^{2}}{\tan^{5}\beta}+\frac{12r_{0}^{2}}{\tan^{3}\beta}\right.$
(79)
$\displaystyle+\frac{18r_{0}^{2}}{\tan\beta}+\frac{18r_{0}s_{0}}{\tan^{4}\beta}+\frac{12r_{0}s_{0}}{\tan^{2}\beta}-6r_{0}s_{0}+\frac{3r_{0}}{\tan^{5}\beta}-\frac{3r_{0}}{\tan^{3}\beta}-3r_{0}\tan\beta-\frac{15r_{0}}{\tan\beta}$
$\displaystyle\left.-\frac{3s_{0}}{\tan^{4}\beta}+3s_{0}\tan^{2}\beta-\frac{3s_{0}}{\tan^{2}\beta}+3s_{0}\right)$
$\displaystyle C_{h^{0}H^{+}H^{-}}$ $\displaystyle=$
$\displaystyle\frac{m^{2}_{h^{0}}}{v_{o}}\left(\frac{2\delta^{2}_{13}r_{0}}{\tan\beta}+\frac{2s_{0}}{\tan^{3}\beta}+2s_{0}\tan\beta+\frac{4s_{0}}{\tan\beta}-\frac{1}{\tan^{4}\beta}-\frac{2}{\tan^{2}\beta}-1-\frac{2m_{H^{\pm}}^{2}}{\tan^{4}\beta
m^{2}_{h^{0}}}\right.$ (80)
$\displaystyle\left.-\frac{4m_{H^{\pm}}^{2}}{\tan^{2}\beta
m^{2}_{h^{0}}}-\frac{2m_{H^{\pm}}^{2}}{m^{2}_{h^{0}}}\right).$

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