# Impact of dark scalars on the radiative $H^{+}W^{-}Z$ vertex

Nabarun Chakrabarty<EMAIL_ADDRESS><EMAIL_ADDRESS>Centre
for High Energy Physics, Indian Institute of Science, C.V.Raman Avenue,
Bangalore 560012, India

###### Abstract

An $H^{+}W^{-}Z$ vertex, i.e, the interaction connecting a singly charged
physical scalar to the $W^{\pm},~{}Z$ gauge bosons is absent at the tree level
in a scalar sector comprising only $SU(2)_{L}$ doublets. However, the
interaction can be generated radiatively in such a case, an example of which
is a two-Higgs doublet model. In this study, we extend a two-Higgs doublet
scenario by additional $SU(2)_{L}$ scalar doublets. Upon endowing with
appropriate discrete symmetries, these additional doublets so introduced can
furnish successful candidates of dark matter. Interestingly, the same "dark"
scalars can also radiatively enhance the strength of the $H^{+}W^{-}Z$
interaction. We compute the $H^{+}W^{-}Z$ vertex at one-loop in a non-linear
gauge that eliminates certain unphysical interactions. While all possible
constraints are taken into account in doing so, particular emphasis is given
to the ones stemming from dark matter. Correlations between the dark matter
observables and the $H^{+}W^{-}Z$ interaction-strength are studied. The study
thus connects a radiatively generated $H^{+}W^{-}Z$ vertex to dark matter
phenomenology.

##  I Introduction

The Higgs-discovery at the Large Hadron Collider (LHC) Chatrchyan  _et al._
(2012); Aad  _et al._ (2012) completes the particle content of the Standard
Model (SM). Moreover, the interaction strengths of the discovered boson with
the SM fermions and gauge bosons are found to be in agreement with the
corresponding SM values. Despite such a success, the SM model cannot be a
complete theory since it is plagued with shortcomings both on theoretical as
well as experimental fronts. On the experimental side, observation of galactic
rotation curves, gravitational lensing and anisotropies in cosmic microwave
background (CMB) advocate the existence of cosmologically stable dark matter
(DM) in the present universe Spergel  _et al._ (2007); Aghanim  _et al._
(2020). Assuming DM has a particle character, no such particle candidate(s)
can be accommodated within the SM alone. This necessitates dynamics beyond the
SM (BSM). A thermal DM particle that thermalises with the visible sector with
a typical weak interaction strength is named a Weakly Interacting Massive
Particle (WIMP) (see Roszkowski  _et al._ (2018) for a review). It _freezes-
out_ from the thermal bath as its interaction rate is surpassed by the Hubble
expansion rate.

The lack of precise information on dark matter quantum numbers allows for the
possibility that DM consists of more than one type of particle. Such
multiparticle DM frameworks allow for DM-DM interaction (some recent studies
are Biswas  _et al._ (2013); Bhattacharya  _et al._ (2013); Bian  _et al._
(2014); Esch  _et al._ (2014); Bhattacharya  _et al._ (2017); Ahmed  _et al._
(2018); Poulin and Godfrey (2019); Herrero-Garcia  _et al._ (2017, 2019); Aoki
and Toma (2018); Aoki  _et al._ (2017); Elahi and Khatibi (2019); Bhattacharya
_et al._ (2020); Biswas  _et al._ (2019); Dutta Banik  _et al._ (2020)). While
such processes can contribute to the thermal relic, they do not have a role in
DM-nucluon scattering rates. Thus, a multipartite DM model can evade the ever
tightening bound on the direct detection (DD) cross section while successfully
accounting for the thermal relic. A multipartite DM framework comprising
scalar DM candidates is obtained by augmenting the SM by additional scalars
and introducing appropriate discrete symmetries in order to ensure DM
stability. A popular scalar dark matter scenario is the inert-doublet model
(IDM) in which case the SM is augmented by an additional $SU(2)_{L}$ scalar
doublet featuring a $\mathbb{Z}_{2}$ symmetry. In fact, an interesting
multipartite extension of the IDM was proposed recently where the SM was
augmented by two additional $SU(2)_{L}$ scalar doublets that were charged
under $\mathbb{Z}_{2}\times\mathbb{Z}^{\prime}_{2}$ Borah  _et al._ (2019).

A two-Higgs doublet model (2HDM) Branco  _et al._ (2012); Deshpande and Ma
(1978) is a widely studied extension of the SM Higgs sector. Motivated by the
minimal supersymmetric standard model (MSSM) in part, it potentially shuts off
the flavour changing neutral currents (FCNC), furnishes additional sources of
CP-violation that can eventually explain the observed matter-antimatter
imbalance, and, poses a solution to the strong CP problem. In fact, a 2HDM is
in fact the smallest $SU(2)_{L}$ multiplet predicting a singly charged Higgs
$H^{+}$. The more well known collider search channels of the same are its
fermionic decays. The charged Higgs is searched at the LHC through different
production and decay modes. The $H^{+}\to t\overline{b}$ decay is looked for
in the search of a heavy $H^{+}$ whereas the preferred search channel for a
light charged Higgs is the $H^{+}\to\overline{\tau}\nu_{\tau}$ channel.
However, such signals are generally swamped by a heavy QCD background. An
alternate strategy therefore is to search for its bosonic decays $H^{+}\to
W^{+}h,W^{+}Z,W^{+}\gamma$. The last two of the aforementioned modes are
_prima facie_ more interesting since they arise only at one-loop in multi-
Higgs doublet models111The $H^{+}W^{-}Z$ interaction is present at the tree
level itself in models with scalar triplets. Examples are the Georgi-Machacek
model Georgi and Machacek (1985); Chanowitz and Golden (1985); Gunion  _et
al._ (1990); Aoki and Kanemura (2008); Chiang and Yagyu (2013); Hartling  _et
al._ (2014, 2015); Chiang  _et al._ (2014); Logan and Rentala (2015); Blasi
_et al._ (2017); Cen  _et al._ (2019); Ghosh  _et al._ (2020); Banerjee  _et
al._ (2019) and scalar-triplet extensions of the MSSM Bandyopadhyay  _et al._
(2015, 2016); Bandyopadhyay and Costantini (2018). The absence of the
$H^{+}W^{-}Z$ coupling at the tree level is an artefact of the isospin
symmetry of the kinetic terms of the Higgs sector. This symmetry is broken at
one loop level through effects from other sectors that do not respect the
custodial invariance, these vertices are induced at loop level. Momentum
dependent interactions appear therein consequently. The $H^{+}W^{-}Z$ vertex
indeed has stirred some interest in the past. Its strength has been estimated
for the minimal supersymmetric standard model (MSSM) Raychaudhuri and
Raychaudhuri (1992); Arhrib  _et al._ (2007), a $\mathbb{Z}_{2}$ symmetric
2HDM Kanemura (2000a, b), an aligned 2HDM Abbas  _et al._ (2018) and a
particular version of 3HDM containing two active and one inert doublet Moretti
_et al._ (2015), and, the 2HDM in presence of a color-octet scalar isodoublet
Chakrabarty  _et al._ (2020).

In this work, we consider a scenario where a 2HDM is augmented by two inert
scalar doublets that are charged non-trivially under a
$\mathbb{Z}_{2}\times\mathbb{Z}^{\prime}_{2}$ symmetry. The ensuing DM
phenomenology is looked at and the parameter regions predicting the requisite
thermal relic are identified. In particular, the role of DM-DM conversions in
generating the observed relic in the IDM _desert region_ is illuminated. On
the other hand, we note that the $SU(2)_{L}$ nature of the inert scalars makes
them contribute radiatively to the $H^{+}W^{-}Z$ vertex. That is, the
contribution of the inert scalars adds to the one coming from the non-inert
scalars coming from the pure 2HDM. We subsequently estimate the strength of
the $H^{+}W^{-}Z$ vertex for the parameter regions allowed by the DM
constraints. In the process, we adopt the non-linear gauge to do away with
unphysical vertices involving goldstones. In all, as a major upshot, this
study correlates the $H^{+}W^{-}Z$ interaction strength with the observables
in the DM sector, the primary one being the thermal relic for this case.

The paper is organised as follows. We introduce the 2HDm + two inert doublet
scenario in section II. Section III elaborates on the calculation of the one-
loop $H^{+}W^{-}Z$ vertex in the chosen non-linear gauge. We discuss the
relevant constraints in section IV. Section V elucidates the key features of
the DM phenomenology and section VI aims to present results combining the DM
phenomenology and the one-loop results for the $H^{+}W^{-}Z$ vertex. Finally
we conclude in section VII. Various important formulae are relegated to the
Appendix.

##  II A framework with two inert doublets

This study extends a 2HDM (comprising two scalar $SU(2)_{L}$ doublets
$\phi_{1}$ and $\phi_{2}$) with two additional scalar $SU(2)_{L}$ doublets
$\eta_{1}$ and $\eta_{2}$. An additional
$\mathbb{Z}_{2}\times\mathbb{Z}^{\prime}_{2}$ is introduced under which the SM
fermions and $\phi_{1,2}$ transform trivially. The charges of $\eta_{1,2}$
under the same are given in Table 1.

Field | $\mathbb{Z}_{2}$ | $\mathbb{Z}^{\prime}_{2}$
---|---|---
$\eta_{1}$ | + | -
$\eta_{2}$ | - | +

Table 1: Discrete charges of $\eta_{1}$ and $\eta_{2}$.

The most general scalar potential consistent with the gauge and discrete
symmetries reads:

$\displaystyle V_{2}$ $\displaystyle=$ $\displaystyle-
m_{11}^{2}|\phi_{1}|^{2}-m_{22}^{2}|\phi_{2}|^{2}+m_{12}^{2}(\phi_{1}^{\dagger}\phi_{2}+\text{h.c.})+\mu_{1}^{2}|\eta_{1}|^{2}+\mu_{2}^{2}|\eta_{2}|^{2}$
(1a) $\displaystyle V_{4}^{a}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{1}}{2}|\phi_{1}|^{4}+\frac{\lambda_{2}}{2}|\phi_{2}|^{4}+\lambda_{3}|\phi_{1}|^{2}|\phi_{2}|^{2}+\lambda_{4}|\phi_{1}^{\dagger}\phi_{2}|^{2}+\frac{\lambda_{5}}{2}[(\phi_{1}^{\dagger}\phi_{2})^{2}+h.c.],$
(1b) $\displaystyle V_{4}^{b}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{1}}{2}|\eta_{1}|^{4}+\frac{\sigma_{2}}{2}|\eta_{2}|^{4}+\sigma_{3}|\eta_{1}|^{2}|\eta_{2}|^{2}+\sigma_{4}|\eta_{1}^{\dagger}\eta_{2}|^{2}+\frac{\sigma_{5}}{2}[(\eta_{1}^{\dagger}\eta_{2})^{2}+h.c.],$
(1c) $\displaystyle V_{4}^{c}$ $\displaystyle=$
$\displaystyle\sum_{i,j=1,2}\Big{[}\lambda_{3ij}|\phi_{i}|^{2}|\eta_{j}|^{2}+\lambda_{4ij}|\phi_{i}^{\dagger}\eta_{j}|^{2}+\frac{\lambda_{5ij}}{2}[(\phi_{i}^{\dagger}\eta_{j})^{2}+h.c.]\Big{]},$
(1d) $\displaystyle V$ $\displaystyle=$ $\displaystyle
V_{2}+V_{4}^{a}+V_{4}^{b}+V_{4}^{c}.$ (1e)

Here, $V_{2}$ and $V_{4}^{a,b,c}$ respectively contain the quadratic and
quartic terms of the scalar potential. All parameters in $V$ are chosen real
to avoid introducing CP-violation. The discrete
$\mathbb{Z}_{2}\times\mathbb{Z}^{\prime}_{2}$ symmetry prevents $\eta_{1,2}$
from picking up vacuum expectation values (VEVs). However, $\phi_{1,2}$ can
pick up VEVs $v_{1,2}$. The doublets can therefore be expressed as

$\displaystyle\phi_{i}=\begin{pmatrix}\phi_{i}^{+}\\\
\frac{1}{\sqrt{2}}(v_{i}+h_{i}+iz_{i})\end{pmatrix},~{}\eta_{i}=\begin{pmatrix}H_{i}^{+}\\\
\frac{1}{\sqrt{2}}(H_{i}+iA_{i})\end{pmatrix},~{}i=1,2.$ (2)

This particular assignment of the discrete charges ensures that $\eta_{1}$ and
$\eta_{2}$ neither mix with each other nor with $\phi_{1}$ and $\phi_{2}$.
Therefore, the component scalars of $\eta_{i}$, i.e. $H^{+}_{i},H_{i},A_{i}$,
are mass eigenstates themselves with masses $M_{i}^{+},M_{H_{i}}$ and
$M_{A_{i}}$ respectively. Moreover, the discrete symmetry renders the lightest
of each component scalar of $\eta_{1}$ and $\eta_{2}$ completely stable. The
CP-even components $H_{1,2}$ therefore are possible DM candidates. The
doublets $\eta_{1,2}$ are thus _inert_ having the following mass spectrum.

$\displaystyle M^{2}_{H_{j}}$ $\displaystyle=$
$\displaystyle\mu_{1}^{2}+\frac{1}{2}(\lambda_{31j}+\lambda_{41j}+\lambda_{51j})v_{1}^{2}+\frac{1}{2}(\lambda_{32j}+\lambda_{42j}+\lambda_{52j})v_{2}^{2}$
(3a) $\displaystyle M^{2}_{A_{j}}$ $\displaystyle=$
$\displaystyle\mu_{1}^{2}+\frac{1}{2}(\lambda_{31j}+\lambda_{41j}-\lambda_{51j})v_{1}^{2}+\frac{1}{2}(\lambda_{32j}+\lambda_{42j}-\lambda_{52j})v_{2}^{2}$
(3b) $\displaystyle M^{2}_{H^{+}_{j}}$ $\displaystyle=$
$\displaystyle\mu_{1}^{2}+\frac{1}{2}\lambda_{31j}v_{1}^{2}+\frac{1}{2}\lambda_{32j}v_{2}^{2}$
(3c)

On the contrary, the scalars coming from $\phi_{1}$ and $\phi_{2}$ as dictated
by $V_{1}$. The scalars in the "gauge basis" are rotated into the "mass basis"
by the action of two mixing angles $\alpha$ and $\beta$ where
tan$\beta=\frac{v_{2}}{v_{1}}$. The mass eigenstates are therefore the CP-even
scalars $h,H$, the CP-odd scalar $A$ and the charged scalar $H^{+}$. This part
is exactly same as in an ordinary 2HDM and we discuss no further on it for
brevity. We reiterate that $V_{4}^{a}$ describes interaction amongst the _non-
inert_ scalars coming from $\phi_{1,2}$, $V_{4}^{b}$ parametrises interaction
of the inert scalar sector with the non-inert one, and, $V_{4}^{c}$ encodes
interactions within the inert sector.

A counting of the independent model parameters is in order. In the active
sector, the relevant parameters are
$\\{m_{11},m_{22},m_{12},\lambda_{1-5},\text{tan}\beta\\}$. Of these, $m_{11}$
and $m_{22}$ can be eliminated by using the tadpole conditions. In addition,
the quartic couplings $\lambda_{1-5}$ can be traded off for the physical
masses $M_{h,H,A,H^{+}}$ and the mixing angle $\alpha$. The independent
parameters in the active sector are therefore
$\\{m_{12},M_{h},M_{H},M_{A},M_{H^{+}},\alpha,\text{tan}\beta\\}$. The inert
sector is more elaborate. A counting of parameters yields 19 parameters, i.e.,
$\\{\mu_{i},\lambda_{3ij},\lambda_{4ij},\lambda_{5ij},\sigma_{1-5}\\}$ for
$i,j=1,2$. However, some of these can be replaced by the physical masses of
the inert scalars and
$\lambda_{Lij}\equiv\lambda_{3ij}+\lambda_{4ij}+\lambda_{5ij}$. Here,
$\lambda_{Lij}$ parameterise the $H_{1}-H_{1}-h$, $H_{2}-H_{2}-h$,
$H_{1}-H_{1}-H$ and $H_{2}-H_{2}-H$ portal couplings and hence are of
paramount importance in a scalar DM scenario such as the present one. In all,
we deem the following parameters as our basis set:
$\\{M_{H_{1}},M_{A_{1}},M^{+}_{1},M_{H_{2}},M_{A_{2}},M^{+}_{2},\lambda_{L11},\lambda_{L12},\lambda_{L21},\lambda_{L22},\lambda_{411},\lambda_{412},\lambda_{511},\lambda_{512},\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4},\sigma_{5}\\}$.
The dimensionful parameters $\mu_{1,2}$ and the rest of the quartic couplings
can be expressed in terms of the basis parameters as

$\displaystyle\mu_{1}^{2}$ $\displaystyle=$ $\displaystyle
M^{2}_{H_{1}}-\frac{1}{2}\lambda_{L11}v^{2}c^{2}_{\beta}-\frac{1}{2}\lambda_{L21}v^{2}s^{2}_{\beta},$
(4a) $\displaystyle\mu_{2}^{2}$ $\displaystyle=$ $\displaystyle
M^{2}_{H_{2}}-\frac{1}{2}\lambda_{L12}v^{2}c^{2}_{\beta}-\frac{1}{2}\lambda_{L22}v^{2}s^{2}_{\beta},$
(4b) $\displaystyle\lambda_{421}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{H_{1}}+M^{2}_{A_{1}}-2(M^{+}_{1})^{2}-\lambda_{411}v^{2}c^{2}_{\beta}}{v^{2}s^{2}_{\beta}},$
(4c) $\displaystyle\lambda_{521}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{H_{1}}-M^{2}_{A_{1}}-\lambda_{511}v^{2}c^{2}_{\beta}}{v^{2}s^{2}_{\beta}},$
(4d) $\displaystyle\lambda_{422}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{H_{2}}+M^{2}_{A_{2}}-2(M^{+}_{2})^{2}-\lambda_{412}v^{2}c^{2}_{\beta}}{v^{2}s^{2}_{\beta}},$
(4e) $\displaystyle\lambda_{522}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{H_{2}}-M^{2}_{A_{2}}-\lambda_{512}v^{2}c^{2}_{\beta}}{v^{2}s^{2}_{\beta}},$
(4f) $\displaystyle\lambda_{311}$ $\displaystyle=$
$\displaystyle\lambda_{L11}-\lambda_{411}-\lambda_{511},$ (4g)
$\displaystyle\lambda_{321}$ $\displaystyle=$
$\displaystyle\lambda_{L21}-\lambda_{421}-\lambda_{521},$ (4h)
$\displaystyle\lambda_{312}$ $\displaystyle=$
$\displaystyle\lambda_{L12}-\lambda_{412}-\lambda_{512},$ (4i)
$\displaystyle\lambda_{322}$ $\displaystyle=$
$\displaystyle\lambda_{L22}-\lambda_{422}-\lambda_{522}.$ (4j)

##  III One-loop form factors for $H^{+}\rightarrow W^{+}Z$

The amplitude for the $H^{+}\rightarrow W^{+}Z$ process can be written as

$\displaystyle i\mathcal{M}(H^{+}\rightarrow
W^{+}Z)=igm_{W}V_{Z}^{\mu\nu}\epsilon_{W\mu}^{*}(p_{W})\epsilon_{Z\nu}^{*}(p_{Z})\,.$
(5)

where,

$\displaystyle
V_{Z}^{\mu\nu}=g^{\mu\nu}F_{Z}+\frac{p_{Z}^{\mu}p_{W}^{\nu}}{m_{W}^{2}}G_{Z}+i\epsilon^{\mu\nu\rho\sigma}\frac{p_{Z\rho}p_{W\sigma}}{m_{W}^{2}}H_{Z}\,.$
(6)

Here $p_{W}^{\mu}p_{Z}^{\nu}$ denote the incoming momenta of $W^{\pm}$ and $Z$
and $F_{Z},~{}G_{Z}$ and $H_{Z}$ are the corresponding form-factors. Scalars
coming from both the _visible_ and _inert_ sectors contribute to
$H^{+}\rightarrow W^{+}Z$. A one-loop form factor is therefore a sum of the
contributions coming from the two sectors.

$\displaystyle X_{Z}$ $\displaystyle=$ $\displaystyle
X_{Z,\text{2HDM}}+X_{Z,\eta_{i}},$ (7)

for $X=F,~{}G,~{}H$. We adopt the $\beta-\alpha=\frac{\pi}{2}$ limit (known as
the _alignment limit_ in a 2HDM) where the interactions of $h$ to the fermions
and gauge bosons become identical to the corresponding SM values. At this
point, the following nonlinear gauge-fixing functions Fujikawa (1973); Bace
and Hari Dass (1975); Gavela  _et al._ (1981); Monyonko  _et al._ (1984);
Monyonko and Reid (1985); Hernandez  _et al._ (1999); Hernandez-Sanchez  _et
al._ (2004) are introduced:

$\displaystyle f^{+}$ $\displaystyle=$
$\displaystyle\Big{(}D^{e}_{\mu}+\frac{igs_{W}^{2}}{c_{W}}Z_{\mu}\Big{)}W^{+\mu}-i\xi
M_{W}G^{+},$ (8a) $\displaystyle f^{Z}$ $\displaystyle=$
$\displaystyle\partial_{\mu}Z^{\mu}-\xi M_{Z}G^{0},$ (8b) $\displaystyle
f^{A}$ $\displaystyle=$ $\displaystyle\partial_{\mu}A^{\mu}.$ (8c)

with $D^{e}_{\mu}$ the electromagnetic covariant derivative and $\xi$ the
gauge parameter. The corresponding gauge fixing Lagrangian reads:

$\displaystyle\mathcal{L}_{GF}$ $\displaystyle=$
$\displaystyle-\frac{1}{\xi}f^{+}f^{-}-\frac{1}{2\xi}(f^{Z})^{2}-\frac{1}{2\xi}(f^{A})^{2}$
(9a)

The advantage of using this particular gauge-fixing procedure is that it
eliminates the unphysical $G^{+}W^{-}V$ vertices that arise in the Higgs
kinetic-energy sector. In addition, for $\xi=1$, the
$Z_{\mu}(k)W^{+}_{\nu}(p)W^{-}_{\rho}(q)$ (all momenta incoming) triple-gauge
vertex have the following modified Feynman rules.

$\displaystyle\Gamma^{ZW^{+}W^{-}}_{\rho\nu\mu}(k,p,q)$ $\displaystyle=$
$\displaystyle-
igc_{W}\\{g_{\mu\nu}(k-p+\frac{s_{W}^{2}}{c_{W}^{2}}q)_{\rho}+g_{\nu\rho}(p-q)_{\mu}$
(10a)
$\displaystyle+g_{\rho\mu}(q-k-\frac{s_{W}^{2}}{c_{W}^{2}}p)_{\nu}\\}\,.$

With the $G^{+}W^{-}Z$ vertex now absent, the number of diagrams reduces. The
diagrams with inert scalars circulating in the loop can then be partitioned
into UV-finite sets, say, A and B. The diagrams corresponding to the sets A
and B are shown in Fig.1 and Fig.2 respectively. The contribution to $F_{Z}$
coming from $\eta_{i}$ for set A (set B) is denoted by $F^{A}_{Z,\eta_{i}}$
($F^{B}_{Z,\eta_{i}}$). Set B does not contribute to $G_{Z}$. We denote

$\displaystyle F_{Z,\text{inert}}$ $\displaystyle=$
$\displaystyle\sum_{i=1,2}F^{A}_{Z,\eta_{i}}+F^{B}_{Z,\eta_{i}},$ (11a)
$\displaystyle G_{Z,\text{inert}}$ $\displaystyle=$
$\displaystyle\sum_{i=1,2}G^{A}_{Z,\eta_{i}}.$ (11b)

Figure 1: One-loop diagrams in set A.

The decay width of $H^{+}\rightarrow W^{+}Z$ is given by

$\displaystyle\Gamma(H^{+}\rightarrow
W^{+}Z)=M_{H^{+}}\frac{\sqrt{\lambda(1,\omega,z)}}{16\pi}\sum_{i=L,T}|M_{ii}|^{2}\,,$
(12)

Figure 2: One-loop diagrams in set B.

where $i=L(T)$ represents the longitudinal and transverse polarization and,
$\lambda(a,b,c)=(a-b-c)^{2}-4abc,~{}\omega=(\frac{M_{W}}{M_{H^{+}}})^{2},z=(\frac{M_{Z}}{M_{H^{+}}})^{2}$.

The longitudinal and transverse contributions can be expressed in terms of
$F_{Z},G_{Z},H_{Z}$ as

$\displaystyle|M_{LL}|^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4z}|(1-\omega-z)F_{Z}+\frac{\lambda(1,\omega,z)}{2\omega}G_{Z}|^{2}\
,$ (13a) $\displaystyle|M_{TT}|^{2}$ $\displaystyle=$ $\displaystyle
g^{2}(2\omega|F_{Z}|^{2}+\frac{\lambda(1,\omega,z)}{2\omega}|H_{Z}|^{2})\,.$
(13b)

##  IV constraints

We discuss the relevant constraints on both the scenarios in this section

###  IV.1 Theoretical constraints

The scalar potential is subject to the theoretical requirements of
perturbativity, unitarity and vacuum stability. A perturbative theory demands
that all the quartic couplings remain $\leq 4\pi$. Therefore, for the more
general two-inert doublet case, maintaining
$|\lambda_{1-5}|,|\sigma_{1-5}|,|\lambda_{3ij}|,|\lambda_{4ij}|,|\lambda_{5ij}|\leq
4\pi$ ensures perturbativity.

Unitarity dictates that the tree-level $2\to 2$ scattering matrix computed
between various two particle states consisting of charged and neutral scalars
Lee  _et al._ (1977) must have eigenvalues bounded from above at $\leq 8\pi$
in magnitude. These eigenvalues are combinations of the quartic couplings.
Therefore, demanding unitarity is tantamount to deriving upper bounds on the
quartic couplings.

In addition, the following conditions for the two-inert doublet case ensure
that the scalar potential remains bounded from below (BFB) along various
directions in field space:

$\displaystyle\lambda_{1}>0,~{}\lambda_{2}>0,~{}\lambda_{3}+\sqrt{\lambda_{1}\lambda_{2}}>0,~{}\lambda_{3}+\lambda_{4}-|\lambda_{5}|+\sqrt{\lambda_{1}\lambda_{2}}>0,$
(14a)
$\displaystyle\sigma_{1}>0,~{}\sigma_{2}>0,~{}\sigma_{3}+\sqrt{\sigma_{1}\sigma_{2}}>0,~{}\sigma_{3}+\sigma_{4}-|\sigma_{5}|+\sqrt{\sigma_{1}\sigma_{2}}>0.$
(14b)
$\displaystyle\lambda_{311}+\sqrt{\lambda_{1}\sigma_{1}}>0,~{}\lambda_{311}+\lambda_{411}-|\lambda_{511}|+\sqrt{\lambda_{1}\sigma_{1}}>0,$
(14c)
$\displaystyle\lambda_{312}+\sqrt{\lambda_{1}\sigma_{2}}>0,~{}\lambda_{312}+\lambda_{412}-|\lambda_{512}|+\sqrt{\lambda_{1}\sigma_{2}}>0,$
(14d)
$\displaystyle\lambda_{321}+\sqrt{\lambda_{2}\sigma_{1}}>0,~{}\lambda_{321}+\lambda_{421}-|\lambda_{521}|+\sqrt{\lambda_{2}\sigma_{1}}>0,$
(14e)
$\displaystyle\lambda_{322}+\sqrt{\lambda_{2}\sigma_{2}}>0,~{}\lambda_{322}+\lambda_{422}-|\lambda_{522}|+\sqrt{\lambda_{2}\sigma_{2}}>0.$
(14f)

The corresponding conditions for the single inert doublet scenario are
obtained by omitting the inequalities involving
$\sigma_{2-5},\lambda_{312},\lambda_{412},\lambda_{512},\lambda_{322},\lambda_{422},\lambda_{522}$.

###  IV.2 Oblique parameters

The additional scalars modify the oblique parameters Peskin and Takeuchi
(1992) with respect to the SM value. Moreover in multi-doublet scenarios
Grimus  _et al._ (2008), the $T$-parameter is the most constraining among the
oblique parameters that restricts scalar mass splittings. The total
$T$-parameter can be written in terms of the SM and BSM contribution ($\Delta
T$) as:

$\displaystyle T$ $\displaystyle=$ $\displaystyle T_{\rm SM}+\Delta T\,.$ (15)

The most updated bound on the New Physics contribution to $T$-parameter is
Tanabashi  _et al._ (2018):

$\displaystyle\Delta T$ $\displaystyle=$ $\displaystyle 0.07\pm 0.12.$ (16)

In our case, the source of BSM contribution is two-fold, i.e., contribution
arising from 2HDM and the scalar octet. Thus,

$\displaystyle\Delta T$ $\displaystyle=$ $\displaystyle T_{\rm
2HDM}+T_{\eta_{1}}+T_{\eta_{2}},$ (17a) $\displaystyle T_{\rm 2HDM}$
$\displaystyle=$ $\displaystyle\frac{1}{16\pi
s^{2}_{W}M^{2}_{W}}\Big{[}F(M^{2}_{H^{+}},M^{2}_{A})+s^{2}_{\beta-\alpha}\Big{(}F(M^{2}_{H^{+}},M^{2}_{H})-F(M^{2}_{H},M^{2}_{A})\Big{)}$
$\displaystyle+c^{2}_{\beta-\alpha}\Big{(}F(M^{2}_{H^{+}},M^{2}_{h})-F(M^{2}_{A},M^{2}_{h})+F(M^{2}_{W},M^{2}_{H})-F(M^{2}_{W},M^{2}_{h})$
$\displaystyle+F(M^{2}_{Z},M^{2}_{h})-F(M^{2}_{Z},M^{2}_{H})+4M^{2}_{Z}\overline{B_{0}}(M^{2}_{Z},M^{2}_{H},M^{2}_{h})-4M^{2}_{W}\overline{B_{0}}(M^{2}_{W},M^{2}_{H},M^{2}_{h})\Big{)}\Big{]}\,,$
$\displaystyle T_{\eta_{i}}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
s^{2}_{W}M^{2}_{W}}\Big{[}F((M^{+}_{i})^{2},M^{2}_{H_{i}})+F((M^{+}_{i})^{2},M^{2}_{A_{i}})-F(M^{2}_{H_{i}},M^{2}_{A_{i}})\Big{]}\,,$
(17b)

where,

$\displaystyle F(x,y)$ $\displaystyle=$
$\displaystyle\frac{x+y}{2}-\frac{xy}{x-y}~{}{\rm
ln}\bigg{(}\frac{x}{y}\bigg{)}~{}~{}~{}{\rm for}~{}~{}~{}x\neq y\,,$ (18a)
$\displaystyle=$ $\displaystyle 0~{}~{}~{}{\rm for}~{}~{}~{}x=y\,$
$\displaystyle\overline{B_{0}}(m_{1}^{2},m_{2}^{2},m_{3}^{2})$
$\displaystyle=$
$\displaystyle\frac{m_{1}^{2}\text{log}(m_{1}^{2})-m_{3}^{2}\text{log}(m_{3}^{2})}{m_{1}^{2}-m_{3}^{2}}-\frac{m_{1}^{2}\text{log}(m_{1}^{2})-m_{2}^{2}\text{log}(m_{2}^{2})}{m_{1}^{2}-m_{2}^{2}}.$
(18b)

###  IV.3 $h\to\Gamma\Gamma$ signal strength

For $\beta-\alpha=\frac{\pi}{2}$, the only signal strength to get modified
_w.r.t._ the SM value is for the $h\to\Gamma\Gamma$ channel. The physical
charged scalars $H^{+},H^{+}_{1,2}$ modify the diphoton amplitude as

$\displaystyle\mathcal{M}_{h\to\gamma\gamma}$ $\displaystyle=$
$\displaystyle\mathcal{M}^{\text{SM}}_{h\to\gamma\gamma}+\mathcal{M}^{\prime}_{h\to\gamma\gamma}$
(19a) $\displaystyle\mathcal{M}^{\text{SM}}_{h\to\gamma\gamma}$
$\displaystyle=$
$\displaystyle\sum_{f}N_{f}Q_{f}^{2}A_{1/2}\Big{(}\frac{M^{2}_{\phi}}{4M^{2}_{f}}\Big{)}+A_{1}\Big{(}\frac{M^{2}_{h}}{4M^{2}_{W}}\Big{)},$
(19b) $\displaystyle\mathcal{M}^{\prime}_{h\to\gamma\gamma}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{hH^{+}H^{-}}v}{2M^{2}_{H^{+}}}A_{0}\Big{(}\frac{M^{2}_{h}}{4M^{2}_{H^{+}}}\Big{)}+\sum_{i=1,2}\frac{\lambda_{hH_{i}^{+}H_{i}^{-}}v}{2M^{2}_{H_{i}^{+}}}A_{0}\Big{(}\frac{M^{2}_{\phi}}{4M^{2}_{H_{i}^{+}}}\Big{)}\,,$
(19c) $\displaystyle\Gamma_{h\to\gamma\gamma}$ $\displaystyle=$
$\displaystyle\frac{G_{F}\alpha^{2}M_{h}^{3}}{128\sqrt{2}\pi^{3}}|\mathcal{M}_{h\to\gamma\gamma}|^{2}.$
(19d)

where $G_{F}$ and $\alpha$ denote respectively the Fermi constant and the QED
fine-structure constant. The loop functions are listed below.

$\displaystyle A_{1/2}(x)$ $\displaystyle=$
$\displaystyle\frac{2}{x^{2}}\big{(}(x+(x-1)f(x)\big{)},$ (20a) $\displaystyle
A_{1}(x)$ $\displaystyle=$
$\displaystyle-\frac{1}{x^{2}}\big{(}(2x^{2}+3x+3(2x-1)f(x)\big{)},$ (20b)
$\displaystyle A_{0}(x)$ $\displaystyle=$
$\displaystyle-\frac{1}{x^{2}}\big{(}x-f(x)\big{)},$ (20c)
$\displaystyle\text{with}~{}~{}f(x)$ $\displaystyle=$
$\displaystyle\text{arcsin}^{2}(\sqrt{x});~{}~{}~{}x\leq 1$ (20d)
$\displaystyle=-\frac{1}{4}\Bigg{[}\text{log}\frac{1+\sqrt{1-x^{-1}}}{1-\sqrt{1-x^{-1}}}-i\pi\Bigg{]}^{2};~{}~{}~{}x>1.$

where $A_{1/2}(x),A_{1}(x)$ and $A_{0}(x)$ are the respective amplitudes for
the spin-$\frac{1}{2}$, spin-1 and spin-0 particles in the loop. The latest 13
TeV results on the diphoton signal strength from ATLAS Aaboud  _et al._ (2018)
and CMS Sirunyan  _et al._ (2018) read

$\displaystyle\mu_{\Gamma\Gamma}$ $\displaystyle=$ $\displaystyle
0.99^{+0.14}_{-0.14},$ (21a) $\displaystyle=$ $\displaystyle
1.18^{+0.17}_{-0.14}.$ (21b)

Upon using the standard combination of signal strengths and uncertainties, we
obtain $\mu_{\Gamma\Gamma}\simeq 1.06\pm 0.1$ and impose this constraint at
2$\sigma$.

###  IV.4 Dark matter constraints

In concurrence with the observation on the relic abundance of the dark matter
provided the PLANCK experiment, we demand for our model

$\displaystyle 0.117\leq\Omega h^{2}\leq 0.123.$ (22)

Further, the dark matter parameter space is constrained significantly by the
direct detection experiments such as LUX Akerib  _et al._ (2017), PandaX-II
Zhang  _et al._ (2019) and Xenon-1T Aprile  _et al._ (2018). We abide by the
most stringent bound that comes from Xenon-1T, in the present analysis.

##  V Dark matter phenomenology

The present framework comprises two different dark sectors endowed by two
different discrete symmetries. Two different DM candidates $H_{1},~{}H_{2}$,
one from each sector, thus emerge. The total relic density is thus the sum of
the individual components, i.e., $\Omega
h^{2}=\Omega_{1}h^{2}+\Omega_{2}h^{2}$. Here, $h$ refers to the reduced Hubble
parameter. The individual relic densities are determined by solving the
Boltzmann equations. Before writing the equations themselves, we define the
parameter $x=\frac{\mu}{T}$, where $\mu$ is the reduced mass defined through
$\mu=\frac{M_{H_{1}}M_{H_{2}}}{M_{H_{1}}+M_{H_{2}}}$. The comoving number
density for $H_{1,2}$ at a temperature $T$ is denoted by $Y_{1,2}(x)$. In
terms of these, the Boltzmann equations are given by

$\displaystyle\frac{dy_{1}}{dx}$ $\displaystyle=$
$\displaystyle-\frac{1}{x^{2}}\Bigg{[}\sum_{B}\langle\sigma
v\rangle_{H_{1}H_{1}\to
BB}\big{(}y^{2}_{1}-(y_{1}^{\text{eq}})^{2}\big{)}\Theta(M_{H_{1}}-M_{\phi})$
(23a) $\displaystyle+\langle\sigma v\rangle_{H_{1}H_{1}\to
H_{2}H_{2}}\Bigg{(}y_{1}^{2}-\frac{(y_{1}^{\text{eq}})^{2}}{(y_{2}^{\text{eq}})^{2}}y_{2}^{2}\Bigg{)}\Theta(M_{H_{1}}-M_{H_{2}})$
$\displaystyle-\langle\sigma v\rangle_{H_{2}H_{2}\to
H_{1}H_{1}}\Bigg{(}y_{2}^{2}-\frac{(y_{2}^{\text{eq}})^{2}}{(y_{1}^{\text{eq}})^{2}}y_{1}^{2}\Bigg{)}\Theta(M_{H_{2}}-M_{H_{1}})\Bigg{]},$
$\displaystyle\frac{dy_{2}}{dx}$ $\displaystyle=$
$\displaystyle-\frac{1}{x^{2}}\Bigg{[}\sum_{B}\langle\sigma
v\rangle_{H_{2}H_{2}\to
BB}\big{(}y^{2}_{2}-(y_{2}^{\text{eq}})^{2}\big{)}\Theta(M_{H_{2}}-M_{\phi})$
(23b) $\displaystyle+\langle\sigma v\rangle_{H_{2}H_{2}\to
H_{1}H_{1}}\Bigg{(}y_{2}^{2}-\frac{(y_{2}^{\text{eq}})^{2}}{(y_{1}^{\text{eq}})^{2}}y_{1}^{2}\Bigg{)}\Theta(M_{H_{1}}-M_{H_{2}})$
$\displaystyle-\langle\sigma v\rangle_{H_{1}H_{1}\to
H_{2}H_{2}}\Bigg{(}y_{1}^{2}-\frac{(y_{1}^{\text{eq}})^{2}}{(y_{2}^{\text{eq}})^{2}}y_{2}^{2}\Bigg{)}\Theta(M_{H_{1}}-M_{H_{2}})\Bigg{]}.$

In the above, $B$ denotes a bath particle and $y_{i}\equiv
0.26M_{\text{Pl}}\sqrt{g_{*}}\mu Y_{i}$. The equilibrium density
$Y_{i}^{\text{eq}}$ can now be expressed as

$\displaystyle Y_{i}^{eq}(x)$ $\displaystyle=$ $\displaystyle
0.145\frac{g}{g_{*}}x^{3/2}\Bigg{(}\frac{M_{H_{i}}}{\mu}\Bigg{)}^{3/2}e^{-x\Big{(}\frac{M_{H_{i}}}{\mu}\Big{)}}$
(24)

Figure 3: DM annihilation diagrams relevant for the [$M_{W}$,500 GeV] mass
range.

Apart from the annihilation processes $H_{i}H_{i}\to BB$ (see Fig.3), assuming
$M_{H_{2}}>M_{H_{1}}$, the $H_{2}H_{2}\to H_{1}H_{1}$ conversion process (see
Fig.4) forms a key aspect of the analysis as will be elaborated later.
Finally, the relic abundance $\Omega_{i}h^{2}$ is obtained using

$\displaystyle\Omega_{i}h^{2}$ $\displaystyle=$ $\displaystyle\frac{8.54\times
10^{-13}}{\sqrt{g_{*}}}\frac{M_{H_{i}}}{\mu}y_{i}\Big{(}\frac{\mu}{M_{H_{i}}}x_{\infty}\Big{)},$
(25)

where $x_{\infty}$ implies a large enough post-decoupling value.

Figure 4: DM conversion diagrams for $M_{H_{2}}>M_{H_{1}}$.

On another part, SI elastic scatterings involving the DM particles and the
nucleons are triggered by t-channel exchanges of $h$ and $H$. The expression
for the SI-DD cross section corresponding to $H_{i}$ reads:

$\displaystyle\sigma_{i}^{\text{SI}}$ $\displaystyle=$
$\displaystyle\frac{f_{n}^{2}\mu^{2}_{i}m^{2}_{n}}{4\pi
M^{2}_{H_{i}}}\Bigg{(}\frac{\lambda_{L1i}}{M_{h}^{2}}+\frac{\lambda_{L2i}}{M_{H}^{2}}\Bigg{)}^{2},i=1,2.$
(26)

In the above, $\mu_{i}=\frac{m_{n}M_{H_{i}}}{m_{n}+M_{H_{i}}}$ is the DM-
nucleon reduced mass and $f_{n}$ refers to the nuclear form factor. Moreover,
In this two-component DM framework, the effective SI-DD cross sections
relevant for each of the candidates can be expressed by scaling the individual
DM-nucleon cross-section by the relative abundance of that particular
component ($\Omega_{i}h^{2}$) in total DM relic density ($\Omega h^{2}$). That
is,

$\displaystyle\sigma^{SI}_{i,~{}\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{i}}{\Omega}\sigma^{SI}_{i}.$ (27)

The effective SI-DD cross sections corresponding to the two DM species must
satisfy the XENON 1T bound.

In this work, we probe the IDM _desert region_ i.e., $M_{W}<M_{\text{DM}}<500$
GeV, where a $\simeq$ 0.1 relic is not obtainable in presence of a single
inert doublet. Our objective is to find out if the desert region can be
revived upon introduction of the second doublet. We first show the variation
of the individual relics $\Omega_{1,2}h^{2}$ versus $M_{H_{1}}$ in Fig.5. We
fix $M_{H_{2}}-M_{H_{1}}$ = 10 GeV in order to kinematically allow for the
$H_{2}H_{2}\to H_{1}H_{1}$ conversion process. The conversion amplitude is
sensitive to the combination $\sigma_{3}+\sigma_{4}+\sigma_{5}$. We set
$\sigma_{3}=\sigma_{4}=\sigma_{5}$ for simplicity and subsequently take
$\sigma_{3}$ = 0.1, 0.5. The rest of the parameters in the IDM sector are
chosen to be: $M_{A_{i}}-M^{+}_{i}=M^{+}_{i}-M_{H_{i}}$ = 1 GeV;
$\lambda_{Lij}=0.01$ for $i,j$ = 1,2;
$\lambda_{411}=\lambda_{511}=\lambda_{412}=\lambda_{512}$ = 0.01, 0.1. The
narrow mass splittings in a particular inert are motivated from the IDM itself
wherein co-annihilations are triggered for small mass gaps. The small values
for $\lambda_{Lij}$ help mitigate the DD bound. The 2HDM spectrum taken is
reads $M_{H}=M_{H^{+}}=500$ GeV, $M_{A}$ = 510 GeV. The left (right) panel of
Fig.5 corresponds to $\sigma_{3}$ = 0.01 (0.1). Upon inspecting the left
panel, it is seen that the total relic density attains the requisite value for
$M_{H_{1}}\simeq 385$ GeV in case of $\sigma_{3}$ = 0.1. More specifically,
$\Omega_{1}h^{2}\simeq 0.62$ and $\Omega_{2}h^{2}\simeq 0.55$ at this mass
point. It is therefore demonstrated that the current framework can revive the
IDM desert region. Even more striking is the observation for $\sigma_{3}$ =
0.5. In this case, $H_{1}$ alone having mass $\simeq$ 465 GeV can account for
the observed relic. This enhanced relic of an individual DM component is a
direct consequence of an enhanced DM-DM conversion rate. That is, the
$H_{2}H_{2}\to H_{1}H_{1}$ rate is higher for $\sigma_{3}$ = 0.5 than it is
for $\sigma_{3}$ = 0.1 and the abundance of $H_{1}$ is higher in case of the
former. The abundance of $H_{2}$ becomes accordingly lower.

Figure 5: Variations of $\Omega_{1,2}h^{2}$ with DM mass for tan$\beta$ = 2.
The left (right) panel corresponds to $\lambda_{411}$ = 0.01 (0.1). Values for
the other parameters are given in the text.

On another part, an important difference between the current framework and the
one in REF is that the annihilation final states in the current scenario can
involve 2HDM scalars. An example worthy of elucidation here is $H_{i}H_{i}\to
W^{\pm}H^{\mp},ZA$ for $i$ = 1, 2. Since, The edges seen at $M_{H_{1}}\simeq$
240 GeV can be understood as the kinematical thresholds for these two final
states. Now, the $H_{1}H_{1}\to W^{\pm}H^{\pm}$ channel involves both s- and
t-channel amplitudes. The s-channel amplitude involves the
$\lambda_{hH_{1}H_{1}}$ trilinear coupling which can further be expressed in
terms of $\lambda_{Lij}$ and tan$\beta$ for $i,j$ = 1, 2. On the other hand,
the t-channel amplitude involves the $\lambda_{H^{+}H_{1}^{-}H_{1}}$ coupling
that in turn depends on the values of $\lambda_{411}$ and $\lambda_{511}$.
Increasing the value to $\lambda_{411}$ = $\lambda_{411}$ = 0.1 enhances the
$H_{1}H_{1}\to W^{\pm}H^{\mp}$ annihilation rate thereby leading to a reduced
relic. This is concurred by an inspection of the right panel. The edges at the
kinematical threshold is expectedly sharper in this case.

Figure 6: Variations of $\Omega_{1,2}h^{2}$ with DM mass for tan$\beta$ = 10.
The left (right) panel corresponds to $\lambda_{411}$ = 0.01 (0.1). Values for
the other parameters are given in the text.

Fig.6 displays the corresponding curves for tan$\beta$ = 10. The behaviour is
qualitatively the same as the case of tan$\beta$ = 2. However, the magnitude
of $\lambda_{H^{+}H_{i}^{-}H_{i}}$ reduces accordingly. The edges
corresponding to the kinematical thresholds are consequently softened.

Figure 7: Allowed parameter regions in $M_{H_{1}}-M_{H_{2}}$ plane.

We show the parameter region allowed the relic and DD constraints in the
$M_{H_{1}}-M_{H_{2}}$ plane for tan$\beta$ = 2, 10 in Fig.7. The role of the
_conversion couplings_ $\sigma_{3},\sigma_{4}$ and $\sigma_{5}$ is clarified
here. The kinematics whenever $M_{H_{1}}=M_{H_{2}}$ implies little to no
conversion regardless of the value of the conversion couplings. This is
precisely why the two parameter bands for tan$\beta$ = 2 and 10 meet at the
$M_{H_{1}}\simeq M_{H_{2}}$ corner. Increasing $M_{H_{2}}-M_{H_{1}}$ and
keeping the conversion couplings fixed increases the conversion rate. In other
words, upon increasing the conversion couplings, demanding the same conversion
rate is tantamount to reducing the mass gap between $H_{1}$ and $H_{2}$. This
explains why the parameter band in the $M_{H_{1}}-M_{H_{2}}$ plane shifts
towards right upon increasing $\sigma_{3},\sigma_{4}$ and $\sigma_{5}$ from
0.1 to 0.5.

##  VI Combined results

We aim to correlate DM phenomenology with the strength of the $H^{+}W^{-}Z$
vertex in this section. Similarly as in the previous section, we take
tan$\beta$ = 2, 10; $\sigma_{3}=\sigma_{4}=\sigma_{5}$ = 0.1, 0.5;
$M_{i}^{+}-M_{i}$ = 1 GeV and $\lambda_{Lij}$ = 0.01 for $i,j$ = 1, 2. We
further choose $M_{H_{1}}$ = 380 GeV and $M_{H_{2}}$ = 420 GeV, values typical
of the desert region. In addition, the following variation is made.

$\displaystyle|\lambda_{411}|=|\lambda_{412}|\leq
4\pi;~{}|\lambda_{511}|=|\lambda_{512}|\leq 4\pi;$ (28a)
$\displaystyle~{}0\leq\Delta M\equiv
M_{A_{1}}-M^{+}_{1}=M_{A_{2}}-M^{+}_{2}\leq 200~{}\text{GeV}.$ (28b)

The 2HDM contribution is computed taking $M_{H}=M_{H^{+}}$ = 200 GeV and
$M_{A}$ = 500 GeV. Such a large $M_{A}-M_{H^{+}}$ is motivated by the fact
that the 2HDM contribution increases with increasing this mass-splitting
Kanemura (2000a); Moretti  _et al._ (2015); Chakrabarty  _et al._ (2020). We
skip the details of the 2HDM contribution for brevity and focus on the one
from the inert doublets.

Figure 8: Heat plot showing the variation of relic density and
$F_{Z,\text{inert}}$ with $\Delta M$ for tan$\beta$ = 2.

Fig.8 is a scatter plot in the $\Delta M-F_{Z,\text{inert}}$ plane for
tan$\beta$ = 2. It is seen that the form factor increases with increasing
$\Delta M$. This can be explained as follows: Increasing $\Delta M$ leads to
an increase in the magnitudes of
$\lambda_{421},\lambda_{521},\lambda_{422},\lambda_{522}$. This enhances
$\lambda_{H^{+}H_{i}^{-}H_{i}}$ and $\lambda_{H^{+}H_{i}^{-}A_{i}}$, and,
ultimately $F_{Z,\text{inert}}$. At the highest mass splitting taken, i.e.,
$\Delta M$ = 200 GeV, $F_{Z,\text{inert}}\simeq 2.4\times 10^{-3}$ while
$F_{Z,2HDM}\simeq 1.7\times 10^{-2}$ thereby indicating $\simeq 15\%$
enhancement due to the inert sector. The contribution of the inert scalars to
$G_{Z}$ is relatively more suppressed with the inert scalars accounting only
for $\simeq 1\%$ of the total contribution even for large mass splittings.
Fig.9 corresponds to tan$\beta$ = 10. The form factor $F_{Z,2HDM}$ diminishes
since it is proportional to tan$\beta$. In fact, it is $\simeq 3.4\times
10^{-3}$ for $\Delta M$ = 200 GeV. The contribution from inert scalars does
not shrink much though, it is $\simeq 1.9\times 10^{-3}$. The relative
enhancement coming from the inert sector is therefore $\simeq 3.6\%$.

Figure 9: Heat plot showing the variation of relic density and
$F_{Z,\text{inert}}$ with $\Delta M$ for tan$\beta$ = 10.

On the other hand, the larger is $\Delta M$, the smaller is the relic density.
This can be attributed primarily to enhanced annihilations to the
$H^{\pm}W^{\mp},AZ$ final states due to the enhanced
$\lambda_{H^{+}H_{i}^{-}H_{i}}$ trilinear coupling. A secondary reason is the
reduction in the co-annihilation rates due to the large mass gap. This anti-
correlation between $F_{Z,\text{inert}}$ and $\Omega h^{2}$ is clear from the
heat plots. Expectedly, there is no appreciable change in relic with change in
tan$\beta$. The conversion couplings $\sigma_{3},\sigma_{4}$ and $\sigma_{5}$
leave an impact on the heat plots.

##  VII Conclusions

In this work, we extended the 2HDM by two additional inert scalar doublets
that are charged non-trivially under a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\prime}$ discrete symmetry. A key part of
the ensuing phenomenology is the two-component DM scenario originating from
the two inert doublets. We explored the DM phenomenology in detail with
emphasis on the $M_{W}\leq M_{\text{DM}}<500$ GeV mass range, the well-known
desert region where the IDM fails to score the requisite relic. We showed in
this study that the desert region can be revived with the introduction of the
second doublet, and, highlighted the role of DM-DM conversions in the process.
Moreover, the sensitivity of the thermal relic to the 2HDM spectrum and the
parameter tan$\beta$ was also illustrated.

On another part, we computed the strength of the radiatively generated
$H^{+}W^{-}Z$ interaction was in the non-linear gauge for the present
framework. Particularly, the contribution coming from the inert doublets was
detailed. The correlation of the $H^{+}W^{-}Z$ interaction strength with the
relic density in the desert region was studied. It was shown that the
interaction strength decreases with increasing relic.

###### Acknowledgements.

NC acknowledges financial support from IISc (Indian Institute of Science)
through the C.V.Raman postdoctoral fellowship. NC also sincerely thanks
Indrani Chakraborty and Rishav Roshan for their help with some of the figures.

##  Appendix A Appendix

This section contains analytical expressions of relevant scalar couplings and
form factors.

###  A.1 Scalar trilinear couplings

$\displaystyle\lambda_{hH^{+}H^{-}}$ $\displaystyle=$ $\displaystyle
v\\{(-\lambda_{3}c^{3}_{\beta}+(-\lambda_{1}+\lambda_{4}+\lambda_{5})s^{2}_{\beta}c_{\beta})s_{\alpha}$
(29a)
$\displaystyle+(\lambda_{3}s^{3}_{\beta}+(\lambda_{2}-\lambda_{4}-\lambda_{5})c^{2}_{\beta}s_{\beta})c_{\alpha}\\}\,,$
$\displaystyle\lambda_{HH^{+}H^{-}}$ $\displaystyle=$
$\displaystyle\cos\alpha\\{v\sin^{2}\beta\cos~{}\beta(\lambda_{1}-\lambda_{4}-\lambda_{5})+\lambda_{3}v\cos^{3}\beta\\}$
(29b)
$\displaystyle+\sin\alpha~{}\sin\beta\\{v\cos^{2}\beta(\lambda_{2}-\lambda_{4}-\lambda_{5})+\lambda_{3}v\sin^{2}\beta\\}$
$\displaystyle\lambda_{hH^{+}G^{-}}$ $\displaystyle=$
$\displaystyle\frac{v}{4}\\{(\lambda_{2}-\lambda_{3}+\lambda_{4}+\lambda_{5})c_{\beta}c_{\alpha}+(-\lambda_{2}+\lambda_{3}+\lambda_{4}+\lambda_{5})c_{3\beta}c_{\alpha}$
(29c)
$\displaystyle+(\lambda_{1}-\lambda_{3}+\lambda_{4}+\lambda_{5})s_{\beta}s_{\alpha}-(-\lambda_{1}+\lambda_{3}+\lambda_{4}+\lambda_{5})s_{3\beta}s_{\alpha}\\}$
$\displaystyle\lambda_{HH^{+}G^{-}}$ $\displaystyle=$
$\displaystyle\frac{v}{4}\\{(\lambda_{2}-\lambda_{3}+\lambda_{4}+\lambda_{5})c_{\beta}s_{\alpha}+(-\lambda_{2}+\lambda_{3}+\lambda_{4}+\lambda_{5})c_{3\beta}s_{\alpha}$
(29d)
$\displaystyle-(\lambda_{1}-\lambda_{3}+\lambda_{4}+\lambda_{5})s_{\beta}c_{\alpha}+(-\lambda_{1}+\lambda_{3}+\lambda_{4}+\lambda_{5})s_{3\beta}c_{\alpha}\\}$
$\displaystyle\lambda_{hH_{1}H_{1}}$ $\displaystyle=$ $\displaystyle
v\big{(}\lambda_{L11}c_{\beta}s_{\alpha}-\lambda_{L21}s_{\beta}c_{\alpha}\big{)},$
(29e) $\displaystyle\lambda_{hH_{2}H_{2}}$ $\displaystyle=$ $\displaystyle
v\big{(}\lambda_{L12}c_{\beta}s_{\alpha}-\lambda_{L22}s_{\beta}c_{\alpha}\big{)},$
(29f) $\displaystyle\lambda_{HH_{1}H_{1}}$ $\displaystyle=$
$\displaystyle-v\big{(}\lambda_{L11}c_{\beta}c_{\alpha}+\lambda_{L21}s_{\beta}s_{\alpha}\big{)},$
(29g) $\displaystyle\lambda_{HH_{2}H_{2}}$ $\displaystyle=$
$\displaystyle-v\big{(}\lambda_{L12}c_{\beta}c_{\alpha}+\lambda_{L22}s_{\beta}s_{\alpha}\big{)},$
(29h) $\displaystyle\lambda_{hH^{+}_{1}H^{-}_{1}}$ $\displaystyle=$
$\displaystyle
v\big{(}\lambda_{311}c_{\beta}s_{\alpha}-\lambda_{321}s_{\beta}c_{\alpha}\big{)}$
(29i) $\displaystyle\lambda_{hH^{+}_{2}H^{-}_{2}}$ $\displaystyle=$
$\displaystyle
v\big{(}\lambda_{312}c_{\beta}s_{\alpha}-\lambda_{322}s_{\beta}c_{\alpha}\big{)}$
(29j) $\displaystyle\lambda_{H^{+}H^{-}_{1}H_{1}}$ $\displaystyle=$
$\displaystyle\frac{v}{4}\Big{(}\lambda_{411}+\lambda_{511}-\lambda_{421}-\lambda_{521}\Big{)}\sin
2\beta$ (29k) $\displaystyle\lambda_{H^{+}H^{-}_{1}A_{1}}$ $\displaystyle=$
$\displaystyle\frac{v}{4}\Big{(}\lambda_{411}-\lambda_{511}-\lambda_{421}+\lambda_{521}\Big{)}\sin
2\beta$ (29l) $\displaystyle\lambda_{H^{+}H^{-}_{2}H_{2}}$ $\displaystyle=$
$\displaystyle\frac{v}{4}\Big{(}\lambda_{412}+\lambda_{512}-\lambda_{422}-\lambda_{522}\Big{)}\sin
2\beta$ (29m) $\displaystyle\lambda_{H^{+}H^{-}_{2}A_{2}}$ $\displaystyle=$
$\displaystyle\frac{v}{4}\Big{(}\lambda_{412}-\lambda_{512}-\lambda_{422}+\lambda_{522}\Big{)}\sin
2\beta$ (29n)

###  A.2 Form factors

The loop functions have been expressed in terms of Passarino-Veltman functions
throughout the analysis Passarino and Veltman (1979). The form-factors for the
inert doublet $\eta_{i}$ ($i$ = 1, 2) read

$\displaystyle F_{Z,\eta_{i}}^{A}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}vc_{W}}\Big{[}\lambda_{H^{+}H_{i}^{-}H_{i}}[(2-4s_{W}^{2})C_{24}(H_{i},H_{i}^{+},H_{i}^{+})-2C_{24}(H_{i}^{+},A_{i},H_{i})+s_{W}^{2}B_{0}(q^{2};H_{i}^{+},H_{i})]$
(30a)
$\displaystyle+\lambda_{H^{+}H_{i}^{+}A_{i}}[(2-4s_{W}^{2})C_{24}(A_{i},H_{i}^{+},H_{i}^{+})-2C_{24}(H_{i}^{+},H_{i},A_{i})+s_{W}^{2}B_{0}(q^{2};H_{i}^{+},A_{i})]\Big{]}\,,$
$\displaystyle F_{Z,\eta_{i}}^{B}$ $\displaystyle=$
$\displaystyle\frac{s^{2}_{W}}{16\pi^{2}vc_{W}}\Big{[}\lambda_{H^{+}H_{i}^{-}H_{i}}\big{(}B_{0}(q^{2},H_{i}^{+},H_{i})+2B_{1}(q^{2},H_{i}^{+},H_{i})\big{)}$
(30b)
$\displaystyle+\lambda_{H^{+}H_{i}^{-}A_{i}}\big{(}B_{0}(q^{2},H_{i}^{+},A_{i})+2B_{1}(q^{2},H_{i}^{+},A_{i})\big{)}\Big{]}\,,$
$\displaystyle G_{Z,\eta_{i}}^{A}$ $\displaystyle=$
$\displaystyle\frac{M_{W}^{2}}{16\pi^{2}vc_{W}}\Big{[}\lambda_{H^{+}H_{i}^{-}H_{i}}[(2-4s_{W}^{2})(C_{12}+C_{23})(H_{i},H_{i}^{+},H_{i}^{+})-2(C_{12}+C_{23})(H_{i}^{+},A_{i},H_{i})]$
(30c)
$\displaystyle+\lambda_{H^{+}H_{i}^{-}A_{i}}[(2-4s_{W}^{2})(C_{12}+C_{23})(A_{i},H_{i}^{+},H_{i}^{+})-2(C_{12}+C_{23})(H_{i}^{+},H_{i},A_{i})]\Big{]}\,,$

Similarly, the form factors for the 2HDM can be expressed as

$\displaystyle F^{A}_{Z,{\rm 2HDM}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}v}[-\lambda_{HH^{+}H^{-}}\\{\frac{s_{W}^{2}}{c_{W}}~{}B_{0}(q^{2},H,H^{+})-\frac{2}{c_{W}}~{}C_{24}(H^{+},A,H)+\frac{2~{}c_{2W}}{c_{W}}~{}C_{24}(H,H^{+},H^{+})\\}$
(31a)
$\displaystyle+\lambda_{hH^{+}G^{-}}\\{\frac{s_{W}^{2}}{c_{W}}~{}B_{0}(q^{2},h,G^{+})-\frac{2}{c_{W}}~{}C_{24}(G^{+},G_{0},h)+\frac{2~{}c_{2W}}{c_{W}}~{}C_{24}(h,G^{+},G^{+})\\}]$
$\displaystyle F^{B}_{Z,{\rm 2HDM}}$ $\displaystyle=$
$\displaystyle\frac{s^{2}_{W}}{16\pi^{2}vc_{W}}~{}[\lambda_{HH^{+}H^{-}}\\{B_{0}(q^{2},H^{+},H)+2B_{1}(q^{2},H^{+},H)\\}$
$\displaystyle-\lambda_{hH^{+}G^{-}}\\{B_{0}(q^{2},G^{+},h)+2B_{1}(q^{2},G^{+},h)\\}]\,,$
$\displaystyle G^{A}_{Z,{\rm 2HDM}}$ $\displaystyle=$
$\displaystyle\frac{M_{W}^{2}}{16\pi^{2}v}[\lambda_{HH^{+}H^{-}}\\{-\frac{2}{c_{W}}~{}(C_{12}+C_{23})(H^{+},A,H)+\frac{2c_{2W}}{c_{W}}~{}(C_{12}+C_{23})(H,H^{+},H^{+})\\}$
(31c)
$\displaystyle+\lambda_{hH^{+}G^{-}}\\{-\frac{2}{c_{W}}~{}(C_{12}+C_{23})(G^{+},G^{0},h)+\frac{2c_{2W}}{c_{W}}~{}(C_{12}+C_{23})(h,G^{+},G^{+})\\}]\,,$

The total form factor is the sum of the individual components.

$\displaystyle F^{A,F}_{Z,{\rm 2HDM}}$ $\displaystyle=$
$\displaystyle\frac{2N_{t}}{16\pi^{2}v^{2}c_{W}}[$ (32a) $\displaystyle
M_{t}^{2}\zeta_{t}~{}(v_{b}+a_{b})~{}\\{4C_{24}(t,b,b)-B_{0}(q^{2},t,b)-B_{0}(p_{W}^{2},b,t)-(2M_{b}^{2}-M_{Z}^{2})~{}C_{0}(t,b,b)\\}$
$\displaystyle-
M_{b}^{2}\zeta_{b}~{}(v_{b}+a_{b})~{}\\{4C_{24}(t,b,b)-B_{0}(p_{Z}^{2},b,b)-B_{0}(q^{2},t,b)-(M_{t}^{2}+M_{b}^{2}-M_{W}^{2})~{}C_{0}(t,b,b)\\}$
$\displaystyle-
M_{b}^{2}\zeta_{b}~{}(v_{b}-a_{b})~{}\\{B_{0}(p_{Z}^{2},b,b)+B_{0}(p_{W}^{2},t,b)+(M_{t}^{2}+M_{b}^{2}-q^{2})~{}C_{0}(t,b,b)\\}$
$\displaystyle+2M_{t}^{2}M_{b}^{2}\zeta_{t}~{}(v_{b}-a_{b})~{}C_{0}(t,b,b)]+(M_{t},\zeta_{t},v_{b},a_{b})\leftrightarrow(M_{b},-\zeta_{b},v_{t},a_{t})\,,$
$\displaystyle F^{B,F}_{Z,{\rm 2HDM}}$ $\displaystyle=$
$\displaystyle\frac{4s_{W}^{2}N_{t}}{16\pi^{2}v^{2}c_{W}}[M_{t}^{2}\zeta_{t}~{}(B_{0}+B_{1})-M_{b}^{2}\zeta_{b}~{}B_{1}](q^{2},t,b)\,,$
(32b) $\displaystyle G^{A,F}_{Z,{\rm 2HDM}}$ $\displaystyle=$
$\displaystyle\frac{4N_{c}M_{W}^{2}}{16\pi^{2}v^{2}c_{W}}[M_{t}^{2}\zeta_{t}~{}(v_{b}+a_{b})~{}(2C_{23}+2C_{12}+C_{11}+C_{0})$
(32c) $\displaystyle-
M_{b}^{2}\zeta_{b}~{}(v_{b}+a_{b})~{}(2C_{23}+C_{12})-M_{b}^{2}\zeta_{b}~{}(v_{b}-a_{b})~{}(C_{12}-C_{11})](t,b,b)$
$\displaystyle+(M_{t},\zeta_{t},v_{b},a_{b})\leftrightarrow(M_{b},-\zeta_{b},v_{t},a_{t})\,,$

$\displaystyle H^{\rm 1PI}_{Z,F}$ $\displaystyle=$
$\displaystyle\frac{4N_{c}M_{W}^{2}}{16\pi^{2}v^{2}c_{W}}\times$ (33)
$\displaystyle[M_{t}^{2}\zeta_{t}(v_{b}+a_{b})(C_{0}+C_{11})-M_{b}^{2}\zeta_{b}(v_{b}+a_{b})C_{12}+M_{b}^{2}\zeta_{b}(v_{b}-a_{b})(C_{12}-C_{11})](t,b,b)$
$\displaystyle+(M_{t},\zeta_{t},v_{b},a_{b})\leftrightarrow(M_{b},+\zeta_{b},v_{t},a_{t})\,.$

where,

$\displaystyle v_{f}=I_{f}-s_{W}^{2}Q_{f},~{}a_{f}=I_{f}.$ (34)

$\displaystyle F_{Z}$ $\displaystyle=$ $\displaystyle
F_{Z,\eta_{i}}^{A}+F_{Z,\eta_{i}}^{B}+F_{Z,{\rm 2HDM}}^{A}+F_{Z,{\rm
2HDM}}^{B}+F_{Z,{\rm 2HDM}}^{A,F}\,,$ (35a) $\displaystyle G_{Z}$
$\displaystyle=$ $\displaystyle G_{Z,\eta_{i}}^{A}+G_{Z,{\rm
2HDM}}^{A}+G_{Z,{\rm 2HDM}}^{A,F}\,,$ (35b) $\displaystyle H_{Z}$
$\displaystyle=$ $\displaystyle H^{\rm 1PI}_{Z,F}\,,$ (35c) $\displaystyle
F_{\gamma}$ $\displaystyle=$ $\displaystyle
F_{\gamma,\eta_{i}}^{A}+F_{\gamma,\eta_{i}}^{B}+F_{\gamma,{\rm
2HDM}}^{A}+F_{\gamma,{\rm 2HDM}}^{B}+F_{\gamma,{\rm 2HDM}}^{A,F}\,,$ (35d)
$\displaystyle G_{\gamma}$ $\displaystyle=$ $\displaystyle
G_{\gamma,\eta_{i}}^{A}+G_{\gamma,{\rm 2HDM}}^{A}+G_{\gamma,{\rm
2HDM}}^{A,F}\,,$ (35e) $\displaystyle H_{\gamma}$ $\displaystyle=$
$\displaystyle H^{\rm 1PI}_{\gamma,F}\,.$ (35f)

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