aainstitutetext: II. Institut für Theoretische Physik
Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg,
Germanybbinstitutetext: Deutsches Elektronen-Synchrotron DESY, Notkestr. 85,
22607 Hamburg, Germany

# Phenomenology of the Dark Matter sector in the Two Higgs Doublet Model with
Complex Scalar Singlet extension

Juhi Dutta a,b Gudrid Moortgat-Pick a and Merle Schreiber<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Extensions of the Two Higgs Doublet model with a complex scalar singlet
(2HDMS) can accommodate all current experimental constraints and are highly
motivated candidates for Beyond Standard Model Physics. It can successfully
provide a dark matter candidate as well as explain baryogenesis and provides
gravitational wave signals. In this work, we focus on the dark matter
phenomenology of the 2HDMS with the complex scalar singlet as the dark matter
candidate. We study variations of dark matter observables with respect to the
model parameters and present representative benchmark points in the light and
heavy dark matter mass regions allowed by existing experimental constraints
from dark matter, flavour physics and collider searches. We also compare real
and complex scalar dark matter in the context of 2HDMS. Further, we discuss
the discovery potential of such scenarios at the HL-LHC and at future
$e^{+}e^{-}$ colliders.

††preprint: DESY 21-200

##  1 Introduction

Dark Matter (DM) remains an unsolved mystery at the interface between particle
physics and cosmology. The particulate nature of the dark matter has been
under contemplation for several decades and particle physics models providing
a possible dark matter candidate are being actively explored at both
theoretical and experimental frontiers. A natural candidate for dark matter
can be provided by an additional singlet scalarBarger:2008jx ; Wu:2016mbe ;
Ghosh:2021qbo  added to the Standard Model (SM) with the SM-like 125 GeV Higgs
serving as the portal to the dark sector. However, such minimal Higgs portal
dark matter models are constrained stringently from current direct detection
data from Xenon-1TAprile:2018dbl . This is a strong motivation to look for
non-minimal Higgs portal sectors such as in extended Higgs sectors models.
Besides providing dark matter candidates, extended Higgs sectors are also
strongly motivated to address some of the still existing open questions in
particle physics such as the naturalness problem and baryogenesis. The Two
Higgs Doublet model(2HDM)Branco:2011iw  is a minimal extension of the Standard
Model (SM) of particle physics with an extra Higgs doublet. It provides a dark
matter candidate only in special cases such as the Inert Doublet model
Branco:2011iw . An extension by a scalar singlet, however, provides a natural
DM candidate and presents a portal to the dark matter from both the 125 GeV
SM-like Higgs as well as the heavy Higgs sector, enabling additional probes of
searching for dark matter at colliders. Singlet extensions can also provide
particle physics explanations for baryogenesis and gravitational
wavesDorsch:2016nrg . Real scalar singlet extensions of the 2HDM have been
exploited to address the dark matter issue with the heavy CP-even Higgses as
portal to DM with discovery potential in the upcoming HL-LHC run He:2008qm ;
Drozd:2014yla ; He:2016mls ; Muhlleitner:2016mzt ; Dey:2019lyr .

Complex scalar extensions to the 2HDM have also been studied in the context of
modified Higgs sectorsBaum:2018zhf ; Biekotter:2021ovi , gravitational wave
signals and pseudo-Nambu Goldstone dark matter together with a mixed Higgs
sector Kannike:2019mzk ; Biekotter:2021ovi . In this work, we investigate the
2HDMS model where the singlet scalar does not obtain a vacuum expectation
value(vev). This ensures that the Higgs sector is akin to the 2HDM, with five
physical Higgs bosons, while the complex scalar singlet is the DM candidate.
We study the dark matter phenomenology of this model consistent with all
experimental constraints including dark matter, flavour and collider
constraints. The presence of the dark matter leads to new decay modes for the
Higgs particles opening up into a pair of dark matter particles. We choose
representative benchmark points to study collider prospects at both the HL-LHC
as well as at future $e^{+}e^{-}$ colliders. Such decays would lead to
signatures involving missing energy in the final state at colliders.
Furthermore, associated production of the heavy Higgs would lead to signatures
involving visible SM quarks/leptons and missing energy signals. We also
discuss the viability of observing the channel 2$b$+$\not{E}_{T}$ channel at
the $e^{+}e^{-}$ collider at $\sqrt{s}=3$ TeV.

The important points of this work include

  * •

discussion of the parameter space allowed by dark matter constraints in 2HDMS;

  * •

comparison between the real and complex singlet models with regard to
observing the differences between the dark matter phenomenology in both cases;

  * •

identification of some representative benchmarks for 2HDMS allowed by all
experimental constraints including dark matter, flavour physics and collider
constraints from the SM-like Higgs as well as the heavy Higgs sector;

  * •

discussion of the collider analysis of 2HDMS in the upcoming HL-LHC and future
$e^{+}e^{-}$ collider.

The paper is organized as follows: we discuss the model followed by the dark
matter phenomenology and collider analyses in Sec. 2\- 8. We summarise our
results in Sec. 9.

##  2 The Model

We consider the CP-conserving Type II Two Higgs Doublet model (2HDM) augmented
with a complex scalar singlet (2HDMS)Baum:2018zhf  to avoid flavour changing
neutral currents (FCNCs) at tree-level. To consider a more general scenario,
one may imply soft $Z_{2}$ symmetry breaking consistent with FCNCs. It allows
for the presence of the mixing term between $\Phi_{1}$ and $\Phi_{2}$ , i.e,
$m^{2}_{12}$ while the explicit $Z_{2}$ breaking terms involving the couplings
$\lambda_{6}$ and $\lambda_{7}$ are set to zero. The complex scalar singlet
and the dark matter candidate $S$ is stabilised by a $Z^{\prime}_{2}$ symmetry
such that $S$ is odd under $Z_{2}^{\prime}$ while the SM fields are even under
the new $Z^{\prime}_{2}$ symmetry. Extensions of the 2HDM with a real scalar
singlet as the dark matter candidate have been studied e.g. in
Muhlleitner:2016mzt . The quantum numbers of the fields under $Z_{2}$ and
$Z_{2}^{\prime}$ are given in Table 1. The fields $\Phi_{1}$ and $S$ are even
under $Z_{2}$ while $\Phi_{2}$ is odd under $Z_{2}$. On the other hand, as
mentioned above, $\Phi_{1}$ and $\Phi_{2}$ are even under the new symmetry
$Z^{\prime}_{2}$ while $S$ is odd under $Z^{\prime}_{2}$. The $Z^{\prime}_{2}$
remains unbroken both explicitly and dynamically, i.e. the singlet doesn’t
obtain a vev. Therefore, the scalar potential $V$ with a softly broken
$Z_{2}$\- and a conserved $Z_{2}^{\prime}$-symmetry is given by:

$V=V_{2HDM}+V_{S}$

where, the softly broken $Z_{2}$-symmetric 2HDM potential is111The positive
sign convention is used for $m^{2}_{12}$ as used for Type II 2HDM
implementation in SARAH.

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

and the $Z_{2}^{\prime}$-symmetric singlet potential, $V_{S}$, is

$\displaystyle\begin{split}V_{S}&=m^{2}_{S}S^{\dagger}S+(\frac{m^{2\prime}_{S}}{2}S^{2}+h.c)+(\frac{\lambda^{\prime\prime}_{1}}{24}S^{4}+h.c)+(\frac{\lambda_{2}^{\prime\prime}}{6}(S^{2}S^{\dagger}S)+h.c)+\frac{\lambda_{3}^{\prime\prime}}{4}(S^{\dagger}S)^{2}\\\
&+S^{\dagger}S[\lambda_{1}^{\prime}\Phi_{1}^{\dagger}\Phi_{1}+\lambda_{2}^{\prime}\Phi_{2}^{\dagger}\Phi_{2}]+[S^{2}(\lambda^{\prime}_{4}\Phi_{1}^{\dagger}\Phi_{1}+\lambda_{5}^{\prime}\Phi_{2}^{\dagger}\Phi_{2})+h.c.].\end{split}$

For simplicity, we set $\lambda_{1}^{\prime\prime}=\lambda_{2}^{\prime\prime}$
without loss of generality throughout our study.222This simplifying assumption
was required to disentangle RGEs during implementing this model in SARAH.
Therefore, under the full symmetry $Z_{2}\times Z^{\prime}_{2}$ (including
softly broken $Z_{2}$), the scalar potential is,

Fields | $Z_{2}$ | $Z_{2}^{\prime}$
---|---|---
$\Phi_{1}$ | +1 | +1
$\Phi_{2}$ | -1 | +1
$S$ | +1 | -1
Table 1: The quantum numbers of the Higgs doublets and the singlet under the
$Z_{2}$\- and $Z_{2}^{\prime}$-symmetry.

$\displaystyle\begin{split}V&=m^{2}_{11}\Phi_{1}^{\dagger}\Phi_{1}+m^{2}_{22}\Phi_{2}^{\dagger}\Phi_{2}+(m^{2}_{12}\Phi_{1}^{\dagger}\Phi_{2}+h.c)+\frac{\lambda_{1}}{2}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\frac{\lambda_{2}}{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}\\\
&+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+[\frac{\lambda_{5}}{2}(\Phi_{1}^{\dagger}\Phi_{2})^{2}+h.c]\\\
&+m^{2}_{S}S^{\dagger}S+(\frac{m^{\prime}_{S}}{2}S^{2}+h.c)+(\frac{\lambda^{\prime\prime}_{1}}{24}S^{4}+h.c)+(\frac{\lambda_{1}^{\prime\prime}}{6}(S^{2}S^{\dagger}S)+h.c)+\frac{\lambda_{3}^{\prime\prime}}{4}(S^{\dagger}S)^{2}\\\
&+S^{\dagger}S[\lambda_{1}^{\prime}\Phi_{1}^{\dagger}\Phi_{1}+\lambda_{2}^{\prime}\Phi_{2}^{\dagger}\Phi_{2}]+[S^{2}(\lambda^{\prime}_{4}\Phi_{1}^{\dagger}\Phi_{1}+\lambda_{5}^{\prime}\Phi_{2}^{\dagger}\Phi_{2})+h.c.].\end{split}$
(1)

In terms of its components, the Higgs doublets and singlet S are,

$\displaystyle\Phi_{1}=(h_{1}^{+}\,\,\frac{1}{\sqrt{2}}(v_{1}+h_{1}+ia_{1}))^{T}$
$\displaystyle\Phi_{2}=(h_{2}^{+}\,\,\frac{1}{\sqrt{2}}(v_{2}+h_{2}+ia_{2}))^{T}$
$\displaystyle S=\frac{1}{\sqrt{2}}(h_{s}+ia_{s})$

where $v_{1},v_{2}$ are the vacuum expectation value(vev)’s obtained by the
neutral components of $\Phi_{1},\Phi_{2}$, respectively. Note that the singlet
scalar does not obtain a vev, therefore, the minimisation conditions are

$\displaystyle
m^{2}_{11}=-m^{2}_{12}\frac{v_{2}}{v_{1}}-\lambda_{1}v^{2}_{1}-\lambda_{345}v^{2}_{2}$
$\displaystyle
m^{2}_{12}=-m^{2}_{12}\frac{v_{1}}{v_{2}}-\lambda_{2}v^{2}_{2}-\lambda_{345}v^{2}_{1}$

After electroweak symmetry breaking (EWSB), there are fifteen free parameters
in the theory,

$\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$, $\lambda_{5}$,
$m^{2}_{12}$, $\tan\beta$, $\lambda_{1}^{\prime\prime}$,
$\lambda_{3}^{\prime\prime}$, $m^{2}_{S}$, $m^{2}_{S^{\prime}}$,
$\lambda_{1}^{\prime}$,$\lambda_{2}^{\prime}$, $\lambda_{3}^{\prime}$,
$\lambda_{4}^{\prime}$.

Here, $\tan\beta=\frac{v_{2}}{v_{1}}$ is the ratio of the vev’s of the up-type
and down-type Higgs doublet denoted by $v_{1}(=v\cos\beta)$ and
$v_{2}(=v\sin\beta)$ respectively where $v(=v^{2}_{1}+v^{2}_{2})\simeq 246$
GeV is the electroweak vev.

### Fermion and Gauge boson sector

For the Type II 2HDM, the up and down type quarks couple to the two different
Higgs doublets. The down-type quarks and leptons couple to $\Phi_{1}$ while
the up-type quarks couple to $\Phi_{2}$. Thus, the Yukawa Lagrangian
isBranco:2011iw 333The sign conventions for the Yukawa couplings are used as
in SARAH for Type II 2HDM.,

$\displaystyle\mathcal{L}_{Yukawa}=y_{u}^{ij}Q_{i}\Phi_{2}u_{j}-y_{d}^{ij}Q_{i}\Phi_{1}d_{j}-y_{l}^{ij}L_{i}\Phi_{1}^{ij}l_{j}$
(2)

where $i,j=1,2,3$ are the family indices of the fermions and $y_{f}$,
($f=u,d,l$) are the Yukawa coupling matrices for the quarks and leptons. The
couplings of the Higgses to the quarks and leptons (normalized to the SM) are
summarised in Table 2Branco:2011iw .

Particle | $h$ | $H$
---|---|---
$u$ | $\cos\alpha/\sin\beta$ | $\sin\alpha/\sin\beta$
$d$ | $-\sin\alpha/\sin\beta$ | $\cos\alpha/\cos\beta$
$l$ | $-\sin\alpha/\sin\beta$ | $\cos\alpha/\cos\beta$
Table 2: The fermion couplings in the Type II 2HDMBranco:2011iw .

For the couplings to the gauge bosons: the $HVV$ and $HHVV$ couplings are
suppressed by $\cos(\beta-\alpha)$ as compared to the SM couplings of the
Higgs boson while the couplings of the pseudoscalar to two gauge bosons vanish
in the CP-conserving caseBranco:2011iw .

### Higgs sector

We now discuss the Higgs sector in this model. The additional complex singlet
scalar doesnot develop a vev and therefore does not mix with the Higgses.
Therefore, the Higgs sector remains as in 2HDM consisting of two CP-even
neutral scalars Higgses $h$, $H$, a pseudoscalar Higgs $A$ and a pair of
charged Higgses $H^{\pm}$. The mass matrices for the charged, scalar and
pseudoscalar sector are Branco:2011iw ,

$M^{2}_{\pm}=[-m^{2}_{12}-(\lambda_{4}+\lambda_{5})v_{1}v_{2}]\begin{pmatrix}\frac{v_{2}}{v_{1}}&-1\\\
-1&\frac{v_{1}}{v_{2}}\\\ \end{pmatrix}$ (3)

$M^{2}_{S}=\begin{pmatrix}-m^{2}_{12}\frac{v_{2}}{v_{1}}+\lambda_{1}v_{1}^{2}&m^{2}_{12}+\lambda_{345}v_{1}v_{2}\\\
m^{2}_{12}+\lambda_{345}v_{1}v_{2}&-m^{2}_{12}\frac{v_{1}}{v_{2}}+\lambda_{2}v_{2}^{2}\\\
\end{pmatrix}$ (4)

$M^{2}_{PS}=\frac{m^{2}_{A}}{v^{2}_{1}+v^{2}_{2}}\begin{pmatrix}v^{2}_{2}&-v_{1}v_{2}\\\
-v_{1}v_{2}&v^{2}_{1}\\\ \end{pmatrix}$ (5)

where,
$m^{2}_{A}=[-\frac{m^{2}_{12}}{v_{1}v_{2}}-2\lambda_{5}](v^{2}_{1}+v^{2}_{2})$
and $\lambda_{345}=\lambda_{3}+\lambda_{4}+\lambda_{5}$. The squared mass of
the DM candidate, $m^{2}_{\chi}$, at tree-level, is

$\displaystyle\begin{split}m^{2}_{\chi}=m^{2}_{S}+\lambda^{\prime}_{1}\frac{v^{2}_{1}}{2}+\lambda^{\prime}_{2}\frac{v^{2}_{2}}{2}\end{split}$
(6)

### Relevant couplings

The Higgses couple to the dark matter giving rise to vertices $h_{i}SS$,
$h_{i}h_{i}SS$ where $i,j=1,2$. Trlinear couplings to the pseudoscalar are
absent at the tree-level due to CP-conservation therefore only $AASS$ vertices
are allowed.The trilinear couplings of the DM with the CP-even Higgses
(normalized to $v$) relevant for our discussions later in the section are,
while the rest are listed in the Appendix Appendix.

$\lambda_{hSS}=\frac{2i}{\sqrt{1+\tan^{2}\beta}}(\lambda_{4}^{\prime}\sin\alpha-\lambda_{5}^{\prime}\tan\beta\cos\alpha)$
(7)

$\lambda_{hS^{*}S}=\frac{i}{\sqrt{1+\tan^{2}\beta}}(\lambda_{1}^{\prime}\sin\alpha-\lambda_{2}^{\prime}\tan\beta\cos\alpha)$
(8)

$\lambda_{hS^{*}S^{*}}=\frac{2i}{\sqrt{1+\tan^{2}\beta}}(\lambda_{4}^{\prime}\sin\alpha-\lambda_{5}^{\prime}\tan\beta\cos\alpha)$
(9)

$\lambda_{HSS}=\frac{-2i}{\sqrt{1+\tan^{2}\beta}}(\lambda_{4}^{\prime}\cos\alpha+\lambda_{5}^{\prime}\tan\beta\sin\alpha)$
(10)

$\lambda_{HS^{*}S}=\frac{-i}{\sqrt{1+\tan^{2}\beta}}(\lambda_{1}^{\prime}\cos\alpha+\lambda_{2}^{\prime}\tan\beta\sin\alpha)$
(11)
$\lambda_{HS^{*}S^{*}}=\frac{-2i}{\sqrt{1+\tan^{2}\beta}}(\lambda_{4}^{\prime}\cos\alpha+\lambda_{5}^{\prime}\tan\beta\sin\alpha)$
(12)

##  3 Complex scalar singlet dark matter phenomenology

The complex scalar singlet dark matter candidate S couples to the Higgses via
the Higgs portal terms as in the scalar potential (eq. 2). In a CP conserving
scenario, no $SSA$ or $SS^{*}A$, $S^{*}S^{*}A$ vertices exist at tree-level.
The only interaction of the dark matter to the SM particles is via the Higgses
which act as scalar mediators. Therefore, there are no spin dependent
interactions for the dark matter candidate and the only stringent constraints
arise from the spin independent DM-nucleon direct detection searches.

### Relic density

For thermal dark matter, where the DM and SM particles are in thermal
equilibrium, the processes contributing to the relic density are the
annihilation and co-annihilation process of the DM with the SM particles and
itself in the thermal plasma. The processes contributing to the relic density
computation are summarised in Fig. 1-3. Since the only interaction of the DM
candidate to the SM particles is via the exchange of Higgses (at tree-level),
only s-channel Higgs $h(H)$ mediated channels as in Fig. 2 contribute besides
t-channel contributions mediated via $S(S^{*})$ as in Fig. 3. The relic
density therefore depends on the singlet portal couplings
$\lambda^{\prime}_{1},\lambda^{\prime}_{2},\lambda^{\prime}_{4},\lambda^{\prime}_{5}$
besides the self interactions of the dark matter
$\lambda^{\prime\prime}_{1},\lambda^{\prime\prime}_{3}$.

Figure 1: The four-point vertices contributing to computation of relic
density.

Figure 2: The vertices contributing to the $s$-channel Higgs mediated
processes contributing to computation of relic density. The Higgs decay to all
possible final states including fermions, gluons, gauge bosons as well as
lighter higgs bosons arising from decay of the heavier CP-even Higgs.

Figure 3: The $t$-channel mediated processes contributing to computation of
relic density.

### Direct detection

The interaction of the DM with nucleon are mediated by the t-channel neutral
scalar Higgses at tree-level. The relevant processes are summarised in Fig. 4.
The possible vertices involved in these processes are
$\lambda_{h_{i}SS^{*}}$,$\lambda_{h_{i}SS}$ and $\lambda_{h_{i}S^{*}S^{*}}$.
The latter two are DM number non-conserving while the former is DM number
conserving. For a $Z_{2}$ symmetric theory, there is apriori no reason for DM
number to be conserved since all of the vertices are allowed by the symmetry
arguments. However if a higher theory at ultra-violet (UV) scales lead to a
conserved quantum number which one can assign to DM Walker:2009en ;
Batell:2010bp , then assuming such a conservation the direct detection cross-
sections depend solely on the portal couplings $\lambda^{\prime}_{1}$ and
$\lambda^{\prime}_{2}$ since it is the coupling $\lambda_{h_{i}SS^{*}}$ which
contributes to the direct detection cross-section. This is solely the
consequence under the assumption of an underlying conserved charge
corresponding to the dark matter number conservation in presence of the
discrete $Z^{\prime}_{2}$ symmetry. For the $Z_{2}$ symmetry breaking, the
other vertices could give contributions of the same order to the cross-section
and hence strongly constrain the current parameter space. We leave this
analysis for a future work and proceed with the assumption of a conserved
quantum number for the DM.

Figure 4: Processes contributing to the direct detection of the dark matter
candidate S. Assuming DM number conservation, only the one involving the
vertex $hSS^{*}$ would contribute while the rest are DM number non-conserving.

The direct detection cross section rate for the proton and neutron scattering
are

$\sigma^{SI}_{p}=\frac{4\mu^{2}_{N}}{\pi}[f_{p}Z]^{2},$
$\sigma^{SI}_{n}=\frac{4\mu^{2}_{N}}{\pi}[f_{n}(A-Z)]^{2},$

where

$\mu_{N}=\frac{m_{N}m_{\chi}}{m_{N}+m_{\chi}}$ (13)

is the reduced mass of the DM-nucleon system where $N=p,n$ is the nucleon. The
couplings of the proton and neutron $f_{p}$ and $f_{n}$ to the DM are computed
from the DM-quark cross section corresponding to Fig. 2. The heavy Higgs
mediated processes would be relatively more suppressed because of the
propagator suppression due to $m_{H}>m_{h}$. The DM-quark amplitude
$\mathcal{M}$ is,

$\mathcal{M}=\sum\limits_{i}\frac{\lambda_{h_{i}SS^{*}}\lambda_{h_{i}f\bar{f}}}{t-m^{2}_{h_{i}}}$
(14)

where $i=1,2$ such that $h_{i}=h,H$ and $h_{i}f\bar{f}$ are the yukawa
couplings of the fermions to the Higgses. Assuming DM number conservation and
for zero momentum transfer, the propagator reduces to
$\frac{1}{-m^{2}_{h_{i}}}$. The cancellation in the amplitudes for scalar dark
matter is affected by the presence of dimension-4 terms U(1) breaking as
pointed out inGross:2017dan  (also Cai:2021evx ) where the terms are loop
suppressed arising from the loop corrections. For the $Z^{\prime}_{2}$
symmetry case, such terms are allowed at the tree-level and are not loop-
suppressed. Thus, the direct detection cross-section may be affected by the
presence of the portal terms at the tree-level and such cancellations between
the scalar contributions of the Higgses to the direct detection cross-section
may be spoiled.

### Indirect detection

The dark matter self annihilation and via the CP-even Higgses to to final
state SM particles lead to the indirect detection signatures. Among these
signatures, stringent constraints arise from Fermi-LATFermi-LAT:2011vow  on
the channels $bb$ and $\tau\tau$ followed by $W^{+}W^{-}$. Among new
signatures specific to Higgs portal models are di-Higgs channels which are
relatively weakly constrained from experiments.

##  4 Experimental constraints

The following experimental constraints are imperative while moving ahead with
any BSM theory.

  * •

The lightest CP-even Higgs mass, $m_{h}=125$ GeV within the experimental
errorATLAS-CONF-2020-005 .

  * •

The invisible decay width of the light Higgs, is constrained by ATLAS and CMS
as below,

$\displaystyle\begin{split}BR(h\rightarrow\chi\chi)&\leq
0.11^{+0.04}_{-0.03}\text{ (ATLAS)}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ATLAS:2020kdi}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
&\leq 0.19\text{ (CMS)}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Sirunyan:2018owy}{\@@citephrase{(}}{\@@citephrase{)}}}}.\end{split}$
(15)

This limit is adhered to by choosing, $m_{DM}>62.5$ GeV.

  * •

Flavor physics constraints, namely $BR(b\rightarrow s\gamma$) =
(3.55$\pm$0.24$\pm$0.09)$\times 10^{-4}$ Lees:2012ym ,
$BR(B_{s}\rightarrow\mu^{+}\mu^{-}$)=(3.2${}^{+1.4\\\ +0.5}_{-1.2\\\
-0.3})\times 10^{-9}$Aaij:2013aka ; Chatrchyan:2013bka . The benchmark points
also are within the upper limit of $\Delta(g-2)_{\mu}(=261(63)(48)\times
10^{-11})$PDGgmu2 .

  * •

The benchmark points also satisfy the electroweak precision test constraints
on the STU parameters, where S = 0.02 $\pm$ 0.1, T =0.07$\pm$0.12, U =
0.00$\pm$0.09 10.1093/ptep/ptaa104 .

  * •

The relic density upper limit from PLANCK data, i.e, $\Omega
h^{2}=0.119$Aghanim:2018eyx  is adhered to.

  * •

DM-nucleon spin independent cross sections from XENON-1TAprile:2018dbl  and
indirect detection constraints from Fermi-LATFermi-LAT:2011vow .

  * •

Collider constraints from LEPAbbiendi:2013hk  and Run 2 ATLAS$/$CMS searches
on the heavy Higgs searcheshiggssumatlas ; higgssumcms  and the 125 GeV Higgs
signal strength measurements ATLAS-CONF-2020-027 .

The Higgs sector of 2HDMS is the same as in the 2HDM (for current status refer
to Eberhardt:2018lub ; Arbey:2017gmh ). For the 2HDMS, we compute the DM relic
and direct detection constraints using micrOMEGAsmco2 . Constraints on the
Higgs sector are checked using HiggsBounds and HiggsSignalsBechtle:2020uwn .
B-physics constraints are checked using SPheno-v4.0.4Porod:2003um .

##  5 Allowed parameter regions

In this section, we discuss the allowed parameter regions from the dark matter
constraints of relic density and spin independent direct detection cross-
section. We perform a scan using the parameters summarised in Table 3. The
model is implemented using SARAH-v4.14.3Staub:2013tta  and
SPheno-4.0.4Porod:2003um  is used for for generating the particle mass
spectrum and decays, and performing the parameter scans.

Parameters | BPA
---|---
$\lambda_{1}$ | 0.23
$\lambda_{2}$ | 0.25
$\lambda_{3}$ | 0.39
$\lambda_{4}$ | -0.17
$\lambda_{5}$ | 0.001
$m^{2}_{12}$ (GeV2) | -1.0$\times 10^{5}$
$\lambda_{1}^{\prime\prime}$ | 0.1
$\lambda_{3}^{\prime\prime}$ | 0.1
$\lambda_{1}^{\prime}$ | 0.042
$\lambda_{2}^{\prime}$ | 0.042
$\lambda_{4}^{\prime}$ | 0.1
$\lambda_{5}^{\prime}$ | 0.1
$m^{2\prime}_{S}$ (GeV2) | 1.13$\times 10^{5}$
$m_{h}$ (GeV) | 125.1
$m_{H}$(GeV) | 724.4
$m_{A}$(GeV) | 724.4
$m_{H^{\pm}}$(GeV) | 728.3
$m_{\chi}$(GeV) | 338.9
$\tan\beta$ | 5
Table 3: List of parameters kept fixed for the scans for relic density and direct detection cross-section along with the mass of the Higgses and dark matter candidate. Parameters |  $m^{2}_{S}$ (GeV2) | $\tan\beta$
---|---|---
Values | 100-400000 | 5
Table 4: Range of relevant parameters varied for the scans involving relic
density and direct detection cross-section versus dark matter mass,
$m_{\chi}$. The other parameters are fixed as in Table 3. Figure 5: Relic
density predicted by the model depending on the DM mass $m_{\chi}$. The
parameter $m^{2}_{S}$ is varied in this plot in the range given in Table 4.
The other input parameters are fixed as in Table 3.

### Relic Density

In Fig. 5, we observe that the relic density of the complex scalar dark matter
is satisfied at the resonance region corresponding to the lightest CP-even 125
GeV Higgs. One should note however no funnel regions corresponding to heavy
higgs annihilation are observed for the parameter values fixed for the scans.
However these regions are accessible for large values of the portal couplings
(close to the perturbativity limit). This is due to an interplay of the
couplings and mass of the heavy higgs while determining the cross-sections
during the relic density computation.

In Fig. 1 there is a rapid change observed in the relic density near the
$m_{\chi}\simeq 75$ GeV where the thermal relic density is achieved. We focus
on this region of light singlet dark matter region (defined as $m_{\chi}\leq
100$ GeV). In the plot, we observe a funnel region near $m_{\chi}\simeq
m_{h}/2$ where the dark matter resonant annihilation occurs via the lightest
CP-even Higgs with mass 125 GeV. The dominant processes contributing to the
relic density in this region are $b\bar{b}$ and sub dominant contributions
from $WW$. As one moves away from the funnel region, the $WW$ mode starts
dominating over $b\bar{b}$. This is due to the increasing phase space
available to the final state particles due to the increase in mass of the dark
matter candidate in the initial states. This continues upto masses close to
$m_{W}$ while near $m_{\chi}\simeq m_{W}$, the dominating mode is $WW$. As one
increases $m_{\chi}$ further the $ZZ$ mode is now allowed once the
$m_{\chi}\simeq m_{Z}$. In this region both the $WW$ and $ZZ$ modes contribute
to the relic density computation. As $m_{\chi}\simeq m_{h}$, the di-Higgs
channel opens up. Further, for $m_{\chi}\simeq m_{t}$ allows for $t\bar{t}$
channel to open up.

We also look into the high mass region ($m_{\chi}>100$ GeV) with a large
portion of the parameter space remains underabundant until $\simeq$ 480 GeV
after which the dark matter overcloses the universe. The processes
contributing to the relic density are summarised in the Fig. 1. We observe
that in these regions as discussed the dominant modes are $WW$, $t\bar{t}$,
$hh$, $ZZ$.

Figure 6: The different processes contributing to the relic density.

### Direct detection

The relevant constraint from direct detection cross section of dark matter
arises from the spin-independent (SI) interactions. The variation of the
direct detection cross section is plotted against the dark matter mass
$m_{\chi}$. We choose the parameters for the singlet and 2HDM parameters as in
BPA and vary $m^{2}_{S}$ to vary the dark matter mass for $\tan\beta=5$. We
observe the dark matter direct detection stringently rules out dark matter
masses less than 100 GeV while heavy dark matter remains allowed by current
data.

Figure 7: Variation of the direct detection cross-section against mass of the
dark matter candidate. The parameter $m^{2}_{S}$ is varied in this plot in the
range given in Table 4. The other input parameters are fixed as in Table 3.

### Case A: Light singlet dark matter

There is a rapid change observed in the relic density near the $Z$ boson
threshold where the thermal relic density is achieved. We focus on this region
of light singlet dark matter region (defined as $m_{\chi}\leq 100$ GeV). Fig.
8 shows the variation of the relic density with the dark matter mass in this
region.

Figure 8: Rapidly changing relic density in low DM mass region. The parameter
$m^{2}_{S}$ is varied in this plot in the range given in Table 4. The other
input parameters are fixed as in Table 3.

Figure 9: Variation of the direct detection cross-section with mass of the DM
for varying $\lambda_{2}^{\prime}$ for two values of $\tan\beta=5,20$. The
parameters varied in this plot is $\lambda^{\prime}_{2}$ as in the Table 5
while the other parameters are same as in Table 3.

In this Higgs funnel region the dominant processes contributing to the relic
density in this Higgs funnel region are $b\bar{b}$, $gg$, $\tau\bar{\tau}$. As
one moves to higher masses, the $b\bar{b}$ mode decreases while the $WW$ mode
opens up as seen in the top panel of Fig. 6. However from Fig. 7 such a region
is observed to be ruled out from the direct detection data. We now vary the
other singlet parameters to see their effect on the direct detection
parameters.

Figure 10: Variation of the direct detection cross-section with $m_{\chi}$ for $\lambda^{\prime}_{2}=0.001$ and $\tan\beta=12$. The scan range for $m^{2}_{S}$ is shown in Table 6 while the other parameters are fixed as in Table 3. Parameters | $\lambda^{\prime}_{2}$ | $\tan\beta$ |  $m^{2}_{S}$(GeV2)
---|---|---|---
Values | $10^{-4}-0.1$ | 5,20 | 4200
Table 5: List of parameters for the variation of relic density and direct
detection cross-section with $m_{\chi}$ for varying $\lambda^{\prime}_{2}$ for
two values of $\tan\beta$. The other input parameters are as in Table 3.

The strongest effect occurs of the portal coupling parameter
$\lambda^{\prime}_{2}$ and $\tan\beta$. This can be explained from the nature
of the coupling $\lambda_{hSS^{*}}$ and $\lambda_{HSS^{*}}$ (see eq. 8 and 11
respectively) which in the decoupling limit ($\sin(\beta-\alpha)\simeq 1$) are
functions of $\lambda^{\prime}_{1}$, $\lambda^{\prime}_{2}$ and $\tan\beta$.
We observe this behaviour in Fig. 9 demonstrates the effect of using a smaller
$\lambda^{\prime}_{2}$ on the direct detection plot for two values of
$\tan\beta=5,20$. As observed from the plot, a large part of the parameter
space remains allowed for such small values of $\lambda^{\prime}_{2}$. We now
scan over $m_{\chi}$ as in the Table 6 and demonstrate the allowed parameter
space in Fig. 10.

Parameters |  $m^{2}_{S}$ (GeV2)
---|---
Values | 1000-200000
Table 6: Scan range for the variation of direct detection cross-section
against the DM mass for $\lambda^{\prime}_{2}=0.001$ and $\tan\beta=12$. The
other parameters are fixed as in Table 3.

### Case B: heavy dark matter regions

From Fig. 6, we observe that the relic density remains underabundant upto
$m_{\chi}\leq 480$ GeV. From direct detection cross-section constraint in Fig.
7 and 10 is also weakly constrained for the heavy dark matter regions.

In summary, we observe that the relic density is satisfied only around
$m_{\chi}\simeq$ 75 GeV and $\simeq$ 480 GeV. Below 480 GeV, the DM relic
density is mostly underabundant with the Higgs resonance region opening up
near 62-63 GeV. In this region, the dominant annihilation process of the DM is
via the light 125 GeV Higgs with $b\bar{b}$ being the dominant annihilation
process. For heavier masses, the $WW$ process is the dominant process
especially near the 75 GeV mass peak. In the following sections, we look into
differences between real and complex scalar dark matter and some
representative benchmarks consistent with experimental constraints including
dark matter, flavour physics, Higgs sector and collider constraints on heavy
Higgses.

##  6 Distinguishing between real and complex scalar dark matter

In this section, we investigate the differences between a real and complex
singlet scalar dark matter in the context of 2HDMS. Similar comparison has
been previously studied for the SM with singlet extensionsWu:2016mbe  and for
the 2HDM $+$ scalar/pseudoscalar DM Arcadi:2020gge  in the context of collider
signals.

The real scalar singlet extended 2HDM potential is

$\displaystyle\begin{split}V_{RS}&=M^{2}_{RS}S^{2}+\frac{\lambda^{\prime\prime}_{R3}}{4}S^{4}+S^{2}[\lambda^{\prime}_{1R}\Phi_{1}^{\dagger}\Phi_{1}+\lambda^{\prime}_{2R}\Phi_{2}^{\dagger}\Phi_{2}]\end{split}$

In the real scalar limit of the complex scalar potential in Eq. 1, the
parameters are related as

$\displaystyle M^{2}_{RS}=m^{2}_{S}+m^{2\prime}_{S}$
$\displaystyle\lambda^{\prime\prime}_{3R}=\lambda^{\prime\prime}_{3}+\frac{5}{3}\lambda^{\prime\prime}_{1}$
$\displaystyle\lambda^{\prime}_{1R}=\lambda^{\prime}_{1}+2\lambda^{\prime}_{4}$
$\displaystyle\lambda^{\prime}_{2R}=\lambda^{\prime}_{2}+2\lambda^{\prime}_{5}$

In the limit of real $S$, the complex scalar potential also reduces to the
form of a real scalar potential for
$m^{2\prime}_{S},\lambda^{\prime}_{4},\lambda^{\prime}_{5},\lambda^{\prime\prime}_{1}=0$,

$\displaystyle M^{2}_{RS}=m^{2}_{S}$
$\displaystyle\lambda^{\prime\prime}_{3R}=\lambda^{\prime\prime}_{3}$
$\displaystyle\lambda^{\prime}_{1R}=\lambda^{\prime}_{1}$
$\displaystyle\lambda^{\prime}_{2R}=\lambda^{\prime}_{2}$

The results using the input parameters as in BPA is summarised in Table 7.

Parameters | BPB
---|---
$M^{2}_{RS}$ (GeV2) | 1.13e+05
$\lambda^{\prime\prime}_{3R}$ | 0.1
$\lambda^{\prime\prime}_{1R}$ | 0.042
$\lambda^{\prime\prime}_{2R}$ | 0.042
$m_{\chi}$ (GeV) | 338.9
$m_{h}$(GeV) | 124.99
$m_{H}$(GeV) | 724.4
$m_{A}$(GeV) | 724.4
$m_{H^{\pm}}$(GeV) | 728.3
$\Omega h^{2}$ | 1.2
$\sigma^{P}_{SI}$ (in pb) | 7.67$\times 10^{-11}$
$\sigma^{N}_{SI}$ (in pb) | 7.90$\times 10^{-11}$
Table 7: The benchmark point in the real singlet extended 2HDM.

The processes contributing to the relic density $WW$ (44$\%$), $hh$(24$\%$),
$ZZ$(22$\%$) and $t\bar{t}$ (10$\%$). In Table 8 we summarise all the cases
discussed so far. We observe that for the same value of input parameters, the
relic density in the real scalar case is larger than the complex case while
the direct detection cross-section remains the same.

We now perform a scan using the parameters listed in BPA for comparison
between the two cases. We fix the same mass for the dark matter candidate and
the Higgs spectrum in both cases. We also ensure the same portal couplings
$\lambda^{\prime}_{1}$, $\lambda^{\prime}_{2}$ and
$\lambda^{\prime\prime}_{3}$ in both real and complex cases.

Figure 11: Relic density (top) and SI (spin-independent) direct detection cross-section (bottom) for real and complex DM.  Benchmark | $\Omega h^{2}$ |  $\sigma^{SI}_{p}$ (in pb) |  $\sigma^{SI}_{n}$ (in pb)
---|---|---|---
BPA | 0.059 | 7.65e-11 | 7.88e-11
BPB | 1.20 | 7.67e-11 | 7.90e-11
Table 8: Comparison of the DM observables for BPA for both complex and real
scalar DM.

In Fig. 11, we observe the relic density for the same dark matter mass and
with same portal couplings is much larger for the real DM case than the
complex DM case. The extra contribution to the relic density arising from the
processes involving the extra portal and self couplings of the dark matter
leads to the different exclusion limits in the complex DM model. We also
compare the direct detection cross-section in the plot and observe no
differences for the real limit of the complex case. This can be attributed to
the fact that only the portal couplings $\lambda^{\prime}_{1}$ and
$\lambda^{\prime}_{2}$ contribute in both cases for conserved DM number.

##  7 Benchmarks

We now choose some representative benchmark regions for our model consistent
with all experimental constraints. We choose benchmark points in the light and
heavy dark matter regions and also the possibility of the decay of the Higgses
to the dark matter candidate as summarised in Table 9. The model is
implemented in SARAHStaub:2013tta  and SPheno-v4.0.4Porod:2003um  is utilised
to obtain the benchmarks for the study. The tree level unitarity constraints
are checked using SPheno-4.0.4. In the following analysis
micrOMEGAs-v5.2.4mco2  is used to compute the tree-level relic density and DM-
nucleon cross-sections. The constraints from the Higgs sector are checked
using HiggsBounds-v5Bechtle:2020pkv  and HiggsSignals-v2Bechtle:2013xfa  at
tree-level.

Parameters | BP1 | BP2 | BP3
---|---|---|---
$\lambda_{1}$ | 0.23 | 0.1 | 0.23
$\lambda_{2}$ | 0.25 | 0.26 | 0.26
$\lambda_{3}$ | 0.39 | 0.10 | 0.2
$\lambda_{4}$ | -0.17 | -0.10 | -0.14
$\lambda_{5}$ | 0.001 | 0.10 | 0.10
$m^{2}_{12}$(GeV2) | -1.0$\times 10^{5}$ | -1.0$\times 10^{5}$ | -1.0$\times 10^{5}$
$\lambda_{1}^{\prime\prime}$ | 0.1 | 0.1 | 0.1
$\lambda_{3}^{\prime\prime}$ | 0.1 | 0.1 | 0.1
$\lambda_{1}^{\prime}$ | 0.042 | 0.04 | 2.0
$\lambda_{2}^{\prime}$ | 0.042 | 0.001 | 0.01
$\lambda_{4}^{\prime}$ | 0.1 | 0.1 | 0.1
$\lambda_{5}^{\prime}$ | 0.1 | 0.1 | 0.1
$m_{h}$ (GeV) | 125.09 | 125.09 | 125.09
$m_{H}$(GeV) | 724.4 | 816.4 | 821.7
$m_{A}$(GeV) | 724.4 | 812.6 | 817.9
$m_{H^{\pm}}$ (GeV) | 728.3 | 816.3 | 822.2
$\tan\beta$ | 4.9 | 6.5 | 6.5
$m_{DM}$(GeV) | 338.0 | 76.7 | 323.6
$\Omega h^{2}$ | 0.058 | 0.119 | 0.05
$\sigma^{p}_{SI}\times 10^{10}$ (pb) | 0.76 | 0.052 | 2.9
$\sigma^{n}_{SI}\times 10^{10}$ (pb) | 0.78 | 0.054 | 3.1
Table 9: Relevant parameters of the benchmark used for the study. All mass
parameters have units GeV except for $m^{2}_{12}$ in GeV2. The decimal points
are rounded to the first decimal place for the masses and upto the third
decimal place for relic density, and direct detection cross-sections.

Benchmark BP1 with 125 GeV lightest CP-even Higgs as in Table 9. The relic
density $\Omega h^{2}=0.059$ and the direct detection cross section,
$\sigma_{DD}$ = 7.55 $\times 10^{-11}$ pb. The benchmark has an underabundant
relic density but is consistent with the DM constraints from PLANCK as well as
the XENON1T data for DM-nucleon scattering cross-sections. The dominating
annihilation channels are into $W^{+}W^{-}$ (43$\%$), $hh$ (22$\%$),
$ZZ(21\%)$ and $t\bar{t}$(10$\%$). This benchmark represents a heavy DM case
and with the Higgs sector which is 2HDM-like such that there are no
appreciable decays to the dark matter.

BP2 with $m_{\chi}=76.6$ GeV is another benchmark addressing the light dark
matter region which passes all the constraints from the Higgs signal strength
to ensure a phenomenologically viable benchmark point.

For both BP1 and BP2, the dominant decay modes of the Higgses are summarised
in Table 10. In this case, the invisible branching is suppressed and the
Higgses decay only with the 2HDM decay modes therefore is completely
indistinguishable with respect to the 2HDM. In presence of the dark matter,
there are additional decay channels opening up for the heavy Higgs,
$H\rightarrow\chi\bar{\chi}$. The presence of the invisible DM candidate in
the final state ensures the presence of missing energy in the final state
signal at colliders. Such signatures have been recently studied with the heavy
Higgs as a portal to dark matter and its collder signals of mono-jet and VBF
along with missing transverse energy at the LHCDey:2019lyr . We consider this
case in BP3 where the allowed invisible branching of the Higgses is
$\sim$4.8$\%$ and all experimental constraints are respected. BP3 is also a
heavy DM benchmark point where the dark matter mass$~{}\sim 324$ GeV and is
allowed by direct detection data at 90$\%$ confidence level (CL). We also
observe that the invisible decay branching of the Heavy Higgs in BP3 is
severely constrained from direct detection searches.

Decay Channels | Branching ratios for
---|---
| BP1 | BP2 | BP3
$H\rightarrow b\bar{b}$ | 0.14 | 0.29 | 0.24
$H\rightarrow t\bar{t}$ | 0.83 | 0.66 | 0.68
$H\rightarrow\tau\bar{\tau}$ | 0.02 | 0.45 | 0.04
$H\rightarrow\chi\bar{\chi}$ | 0.0 | 0.0 | 0.05
$A\rightarrow b\bar{b}$ | 0.12 | 0.27 | 0.27
$A\rightarrow t\bar{t}$ | 0.86 | 0.69 | 0.69
$A\rightarrow\tau\bar{\tau}$ | 0.02 | 0.04 | 0.04
$H^{\pm}\rightarrow t\bar{b}$ | 0.97 | 0.96 | 0.96
$H^{\pm}\rightarrow\tau\bar{\nu_{\tau}}$ | 0.022 | 0.03 | 0.03
Table 10: The branching ratios for the dominant decay modes of the heavy
Higgses for the benchmarks BP1, BP2 and BP3. The branching ratios are rounded
up to the second decimal place.

##  8 Collider Analysis

In this section, we discuss the potential signals of this model at HL-LHC and
future $e^{+}e^{-}$ colliders. As discussed in the previous section, the
presence of the invisible decay of the heavy Higgs to the dark matter
candidate is a source of missing energy at colliders. Therefore, direct
production of heavy Higgses and consequent decay of the Higgs to $\chi$ along
with visible SM particles can give rise to distinct signatures for this
scenario as opposed to the 2HDM like scenario. We investigate these
possibilities further in the following sub-sections and discuss their
prospects in the context of $\sqrt{s}=14$ TeV LHC at the targeted integrated
luminosity of 3-4 ab-1 and in future $e^{+}e^{-}$ colliders upto $\sqrt{s}=3$
TeV and integrated luminosities 5 ab-1. We also briefly summarise the
simulation set-up for our analysis below.

### Simulation details

We use MG5$\\_$aMC$\\_$v3$\\_1\\_1$ Alwall:2011uj ; Alwall:2014hca  to
generate the parton level processes for both signal and background samples.
For hadronisation and showering, Pythia-8Pythia8  is used while the detector
simulation is performed using Delphes-v3.4.1 deFavereau:2013fsa ;
Selvaggi:2014mya ; Mertens:2015kba using the default Delphes cards provided
for ATLAS for the LHC study and the default card for the ILD detector for the
$e^{+}e^{-}$ study. We performed the signal background analysis for LHC using
Delphes while the analysis for the electron-positron collider is performed
using Madanalysis-v5Dumont:2014tja ; Conte:2014zja ; Conte:2018vmg ;
Araz:2019otb ; Araz:2020lnp .

###  8.1 Prospects at LHC

The main processes contributing to neutral Higgs production are gluon fusion
(mediated by the top quark loop), vector boson fusion (VBF), associated Higgs
production ($Vh_{i}$), $b\bar{b}h_{i}$, $t\bar{t}h_{i}$Branco:2011iw . For the
charged Higgs pair, the possible production channels are $H^{+}H^{-}$ and
$W^{\pm}H^{\mp}$ Branco:2011iw . At LHC Run 3 at $\sqrt{s}=14$ TeV, all
possible Higgs production processes (including SM and BSM Higgses) are
summarised in Table 11 for BP1.

Processes | Cross section (in fb) at
---|---
|  $\sqrt{s}=14$ TeV
$h$ (ggF) | 29.3$\times 10^{3}$
$H$ | 22
$A$ | 35
$hjj$ ($VBF$) | 1.296$\times 10^{3}$
$Hjj$ | 1.843
$Ajj$ | 2.885
$Wh$ | 1.148$\times 10^{3}$
$WH$ | 1.195$\times 10^{-3}$
$WA$ | 4.3$\times 10^{-4}$
$Zh$ | 880.8
$ZH$ | 0.93
$ZA$ | 3.999
$bbh$ | 2.534
$bbH$ | 21.52
$bbA$ | 23.39
$t\bar{t}h$ | 478.3
$t\bar{t}H$ | 0.1988
$t\bar{t}A$ | 0.2552
$H^{+}H^{-}$ | 6.603$\times 10^{-2}$
$W^{\pm}H^{\mp}$ | 102.4
Table 11: The leading order (LO) cross-section (in fb) for dominant processes
for BP1 before analysis for $\sqrt{s}=14$ TeV LHC.

From the Table 11, we see HL-LHC running at 3-4 ab-1 is necessary to have a
through collider search for heavy Higgses close to a TeV range. The current
constraints on heavy Higgses are summarised inhiggssumcms ; higgssumatlas .
Projection studies of heavy Higgses predict mass reach of upto TeV for direct
detection ATLAS:2019mfr ; Adhikary:2018ise  and indirect detection
Bahl:2020kwe . The HL-LHC is important to achieve the required luminosity to
observe these channels and gain indirect insight into the BSM Higgs sector.
From the benchmark Table 9, we observe that for the case with the H decaying
to dark matter particle, di-jet+$\not{E}_{T}$, di-lepton+ $\not{E}_{T}$,
$b\bar{b}+\not{E}_{T}$ and mono-jet$+$ $\not{E}_{T}$ are the possible channels
to probe this model at hadron colliders. The VBF and gluon fusion channel have
been studied for the real singlet at LHC and are good discovery
probesDey:2019lyr . For the mass ranges of the heavy Higgses considered in our
study, the dominant production processes at $\sqrt{s}=$ 14 TeV LHC are:
$b\bar{b}H,H+jj(ggF),VBF,ZH$ and $t\bar{t}H$. involving $WH$ associated
prodcution are supressed due to the suppressed couplings of the heavy Higgses
with respect to the SM couplings to the gauge bosons.

In presence of the heavy Higgs $H$ decaying to two dark matter candidates, one
can obtain invisible momentum in the final state. Keeping this in mind, one
can look into the following final states:

  * •

$1j$ (ISR)+$\not{E}_{T}$ATLAS:2021kxv

  * •

$2j+\not{E}_{T}$ ATLAS-CONF-2020-008

We estimate the significance for the mono-jet and VBF channels using the cuts
from an existing cut-and-count analyses performed in Ref.Dey:2019lyr  for
$\sqrt{s}=14$ TeV LHC.

#### Pre-selection cuts

The cuts used in this paper are summarised below. We choose the Delphes ATLAS
card for reference.

  * •

The leptons are reconstructed with a minimum transverse momentum, $p_{T}>$ 10
GeV and pseudorapidity $|\eta|<2.5$ while excluding the transitional
pseudorapidity gap between the barrel and the end cap of the calorimeter
1.37$<|\eta|<$1.52.

  * •

Photons are reconstructed with $p_{T}>10$ GeV and $|\eta|<$2.5.

  * •

All jets are reconstructed using $\Delta R=0.4$ using the anti-$k_{T}$
algorithm and minimum $p_{T}>20$ GeV with pseudorapidity $|\eta|<2.5$.

#### Signal Region A: $1j+\not{E}_{T}$

In this subsection, we estimate the signal events for the monojet and missing
energy channel using the background estimates in Ref. Dey:2019lyr . Using the
cuts of $p_{T}(j)>250$ GeV and $\not{E}_{T}>250$ GeV, one obtains a cut
efficiency for the signal BP3 $\sim 18\%$. For the cut and count analysis, the
estimated significance we obtain a 0.111$\sigma$ excess at 3ab-1 using gluon
fusion production channel (at leading order (LO)).

#### Signal Region B: $2j+\not{E}_{T}$

We study the VBF topology for the signal region consisting of two forward jets
and missing energy. We follow the cuts in Ref.Dey:2019lyr  for estimating the
cut efficiency and use the background estimates from their paper. Using the
signal cross-section at LO, we get the a signal efficiency of 4.5$\%$ for BP3
and the signal significance is $\sim 0.2$ $\sigma$ at 3 ab-1. Therefore, we
observe that owing to the small invisible branching ratio and heavy Higgs
masses $\sim 820$ GeV (and hence small production cross-section) in BP3, the
final states are inaccessible at the upcoming HL-LHC run. We estimate the
prospects of observing such a benchmark at the $e^{+}e^{-}$ collider in the
following sub-section.

###  8.2 Prospects at $e^{+}e^{-}$ colliders

The cleaner environment and lesser background along the beam line compared to
hadron colliders make the electron positron linear colliders an attractive
choice for precision studies of new physics. The International Linear Collider
(ILC)Behnke:166034 , is a proposed $e^{+}e^{-}$ collider with center-of-mass
energies at the SM-like Higgs threshold ($\sqrt{s}=250$ GeV), top threshold
($\sqrt{s}=350$ GeV) and further upgrades such that the center of mass
energies are $\sqrt{s}=500$ GeV and upto $\sqrt{s}=1$ TeV with a maximum
target integrated luminosity of $\mathcal{L}=500$ fb-1. ILC gains advantage
over the LHC in the possibility of exploiting the polarisation of the electron
and positron beams leading to increased background
suppressionMoortgatPick:2005cw . Other proposed $e^{+}e^{-}$ colliders are
CLIC CLICdp:2018cto ; Roloff:2645352  with an energy upgrade upto
$\sqrt{s}=1.5,3$ TeV, FCC-ee Proceedings:2019vxr  and CEPC
CEPCStudyGroup:2018ghi with the latter having beam energies upto the
$t\bar{t}$ threshold. For an $e^{+}e^{-}$ collider, the main processes
contributing to neutral Higgs production are $Zh$ , $b\bar{b}h$,
$\nu\bar{\nu}h$Branco:2011iw . For the heavy scalar $H$ and pseudoscalar $A$,
the only relevant production channel which would be accessible upto
$\sqrt{s}=3$ TeV are $b\bar{b}H$, $b\bar{b}A$, $hA$, $HA$, $t\bar{t}H/A$ and
$\nu\bar{\nu}H/A$. For the charged Higgs pair, the possible production
channels are $H^{+}H^{-}$ and $W^{\pm}H^{\mp}$Branco:2011iw . One has the
possibility of accessing the heavy Higgses via the channels $b\bar{b}H/A$,
$HA$ and $t\bar{t}H/A$ at the CLIC with $\sqrt{s}=1.5(3)$ TeV with an
integrated luminosity up to 500 $fb^{-1}$ for ILC and up to 5 ab-1 for the
CLIC. As a representative study, we perform a signal-background analyses at a
generic $e^{+}e^{-}$ collider with $\sqrt{s}=3$ TeV and unpolarised electrons
and positrons to assess the observability of these channels after background
rejection for BP3. Presence of the invisible dark matter in the final state
manifests as missing energy in the final state along with visible SM-particles
such as $(b)$-jets, $e/\mu/\tau$ leptons from the pseudoscalar decay which we
investigate further in the collider analysis.

#### Signals and backgrounds

From Table 10, we observe that the dominant decay of the pseudoscalar are to a
pair of top quarks and $b$ quarks, and for BP3 a smaller fraction of the CP-
even heavy Higgs decays to the dark matter which manifests itself as missing
energy in the collider. In such cases of invisibly decaying heavy CP-even
Higgs, production processes such as $HA$, $b\bar{b}H$, $t\bar{t}H$ and $ZH$
lead to final states including (at least) two b-jets and missing energy while
$Hjj$ leads to a final state with two light jet associated with the missing
energy with the latter having a VBF topology. On the other hand, production
channel $\nu\bar{\nu}H$ state leads to a fully invisible state which in order
to probed requires an ISR photon against which the invisible system recoils.
Besides, note that the heavy Higgses ($A$) also decay to a pair of tau
leptons, therefore allowing final state of $2\tau+\not{E}_{T}$ from $HA$
production channel. Since the branching ratio of the $A$ are dominantly into
$t\bar{t}$ followed by $b\bar{b}$ one expects final states involving at least
$2b+\not{E}_{T}$ to be dominant over the tau final states. We look into the
prospects of observing signals including $2b+\not{E}_{T}$. We leave the study
of the tau and photon final states as a future study. Therefore the relevant
signal processes consists of the following production channels of the heavy
CP-even Higgs, $H$ such as

$e^{-}e^{+}\rightarrow HA,\,\,\,b\bar{b}H,\,\,\,t\bar{t}H$

Contributions from $HA,b\bar{b}H,t\bar{t}H$ lead to the final state of
$2b+\not{E}_{T}$ with the missing energy arising from
$H\rightarrow\chi\bar{\chi}$ while contributions from $ZH$ production cross-
section is considerably suppressed at $\sqrt{s}=3$ TeV upto integrated
luminosity of $\mathcal{L}=5$ ab-1.

#### Pre-selection cuts

The cuts used for identifying the reconstructed objects after reconstruction
with the default ILD card in Delphes are summarised below and denoted as C0,

  * •

Leptons are reconstructed with a minimum transverse momentum, $p_{T}>$ 10 GeV
and pseudorapidity $|\eta|<2.5$.

  * •

Photons are reconstructed with $p_{T}>10$ GeV and $|\eta|<$2.5.

  * •

All (b)-jets are identified with minimum $p_{T}>20$ GeV with pseudorapidity
$|\eta|<3.0$.

### Signal Region C: $2b+\not{E}_{T}$

We investigate the prospects of the final state consisting of two b-jets along
with missing energy. Dominant contributions arise from

  * •

$b\bar{b}$ (for misidentification of decay products of $b$-quark along with
missing energy from $b$ decays)

  * •

$b\bar{b}\nu\bar{\nu}$ (including both on-shell and off-shell contribution
from $Z$ boson decay as well as $\nu\bar{\nu}h$ contribution)

  * •

$Z(\rightarrow\ell^{+}\ell^{-})Z(\rightarrow\nu\bar{\nu})$

  * •

$hZ$ ($h\rightarrow b\bar{b},Z\rightarrow\nu\bar{\nu}$)

  * •

$t\bar{t}Z,(Z\rightarrow\nu\bar{\nu},t\rightarrow bW^{+})$ for misidentified
leptons/jets from $W$ bosons.

  * •

Leptonic $t\bar{t}$ for misidentified leptons or semi-leptonic $t\bar{t}$
decays with missing energy arising from the $b$ decaying leptonically.

  * •

$WWZ$

  * •

$ZZZ$

Process | Production cross-section (in fb)
---|---
$HA$ | 0.0341
$b\bar{b}H$ | 0.0231
$t\bar{t}H$ | 0.0202
$ZZ$ | 1.54
$WW$ | 22.12
$b\bar{b}\nu\bar{\nu}$ | 40.84
$b\bar{b}$ | 10.17
$t\bar{t}$ (leptonic) | 0.87
$t\bar{t}$(semi-leptonic) | 5.2
$t\bar{t}Z$ | 0.0149
$hZ$ | 0.23
$WWZ$ | 0.23
$ZZZ$ | 0.0151
Table 12: Production cross sections (in fb) for the signal processes at
$\sqrt{s}=3$ TeV for BP3 (with $H\rightarrow\chi\bar{\chi}$) and SM
backgrounds (with leptonic decay for tops, $W$ boson, invisible decay of $Z$,
$Z\rightarrow b\bar{b}$ and $h\rightarrow bb~{}$).

For the backgrounds such as $t\bar{t},t\bar{t}Z$ we generate the leptonic
mode. For the other SM backgrounds: $Z(\rightarrow
bb~{})Z(\rightarrow\nu\bar{\nu}),h(\rightarrow
bb~{})Z(\rightarrow\nu\bar{\nu})$, $W(l\nu)W(l\nu)Z(\rightarrow b\bar{b})$
modes are generated.Since the signal benchmark BP3 has a large dark matter
mass ($m_{\chi}\simeq 323$ GeV) , with the DM candidate originating from the
decay of the heavy Higgs $H$ and a large mass gap between the parent and
daughter particles, the signal sample has a large missing transverse energy as
seen in Fig. 12 for the 2 b-jet final state. In order to generate the
irreducible background final state $b\bar{b}\nu\bar{\nu}$, a large
$\not{E}_{T}(>350\text{ GeV})$ to tame the large the cross-section of this
background. The signal and background production cross-section in the missing
energy final state are summarised in Table 12. We now perform a signal
background analyses using the cuts C1-C6 as follows,

  * •

C1: The final state consists of two b-jets and no leptons or photons.

  * •

C2: The leading b-jet has transverse momentum $p_{T}>100$ GeV and sub-leading
b-jet has $p_{T}>80$ GeV. The hard $p_{T}$ cuts on the b-jets help reduce
backgrounds from SM backgrounds from $Z$ and $h$ bosons.

  * •

C3: The invariant mass of the two b-jets within the mass window
80$<M_{b_{1}\,b_{2}}<$130 is rejected to remove contributions from $Z$ and $h$
bosons.

  * •

C4: Since the dark matter is heavy, we demand a large cut on the effective
mass $M_{eff}>1.2$ TeV where $M_{eff}=\sum_{i}(p_{T_{i}})+\not{E}_{T}$.

  * •

C5: Further, the large mass-gap between the heavy Higgs and the dark matter
allows for a large missing energy. We demand $\not{E}_{T}>650$ GeV on the
final state which reduces the dominant SM backgrounds.

  * •

C6: The $\Delta\Phi$ between two b-jets is significantly different for the
signal and background from $b\bar{b}$ where the b-jets are mostly back-to-
back. We demand $\Delta\Phi(b_{1},b_{2})<1.60$. This also reduces the
backgrounds from $b\bar{b}$ as well as from $t\bar{t}$ and $t\bar{t}Z$
sharply.

Figure 12: Normalized distribution of missing transverse energy($\not{E}_{T}$)
and effective mass($M_{eff}$) for signal vs. some backgrounds after cut C1.
The $t\bar{t}$ background here is the leptonic $t\bar{t}$ contribution.

The number of signal and background events after applying the cuts C1-C6 (at
$\mathcal{L}=5$ ab-1) are summarised in Table 13. We observe that the large
$M_{eff}$ and $\not{E}_{T}$ cut are instrumental in reducing SM backgrounds
sharply. Further, the variable $\Delta\Phi(b_{1},b_{2})$ also reduces
contributions from $b\bar{b}$ owing to the fact that the jets in the
background are back to back while in the signal they are more collimated. It
also reduces the backgrounds from $t\bar{t}$ and $t\bar{t}Z$ sharply which are
also peaked towards higher values of $\Delta\Phi$ as compared to the signal.
We also plot the invariant mass of the two b-jets in Fig. 13 and observe that
the SM backgrounds from $Z$ and $h$ are peaked at the resonance masses but for
the signal the peak is more broad since the parent particle is heavy. However
the irreducible background $b\bar{b}\nu\bar{\nu}$ has a similar shape compared
to the signal.

Figure 13: Normalized distribution of the invariant mass of the two b-jets for
signal vs. background after cut C1. The $t\bar{t}$ background here is the
leptonic $t\bar{t}$ contribution.

Therefore, excluding the mass window 80$<M_{b_{1}\,b_{2}}<$130 removes the
resonant backgrounds from $Z$ and $h$ bosons however retaining backgrounds
involving b-jets associated with neutrinos and also from $t\bar{t}$ associated
backgrounds.

Figure 14: Normalized distribution of $\Delta\Phi$ separation between the two
final state b-jets after cut C1. The $t\bar{t}$ background here is the
leptonic $t\bar{t}$ contribution.

We discuss the signal significance for the unpolarised electron and positron
beams. The statistical significance (${\mathcal{S}}$) of the signal ($s$) over
the total SM background ($b$) is calculated using Cowan:2010js ; pdg

$\mathcal{S}=\sqrt{2\times\left[(s+b){\rm ln}(1+\frac{s}{b})-s\right]}.$ (16)

where $s$ and $b$ are the total signal and background event numbers after the
cuts C1-C6. We use this expression for the statistical significance as is
relevant in this case since the background events are not overwhelmingly large
compared to the signal and the limit $b>>s$ is not fully accurate. 444In the
limit $b>>s$, Eq. 16 reduces to $S=\frac{s}{\sqrt{b}}\simeq$ 4.21 $\sigma$.
Also, for $\mathcal{S}=\frac{s}{\sqrt{s+b}}\simeq 3.63$. We observe that for
BP3, although the invisible decay branching ratio of the heavy Higgs is small,
i.e, $H\rightarrow\chi\bar{\chi}\sim 4.8\%$, one can obtain $\sim 4\sigma$
signal at the integrated luminosity of 5 ab-1.

Process | C0 | C1 | C2 | C3 | C4 | C5 | C6
---|---|---|---|---|---|---|---
$b\bar{b}H$ | 115 | 32 | 27 | 26 | 26 | 25 | 21
$t\bar{t}H$ | 101 | 22 | 13 | 12 | 12 | 11 | 10
$HA$ | 170 | 38 | 28 | 24 | 24 | 22 | 20
BP3 | 51
$b\bar{b}\nu\bar{\nu}$ | 204200 | 35159 | 15738 | 2040.9 | 330.3 | 147.6 | 124.3
$b\bar{b}$ | 50850 | 9514.5 | 8432.5 | 8387.2 | 6697.5 | 65.6 | 4.07
$ZZZ$ | 75 | 10.89 | 3.75 | 3.07 | 1.5 | 0.51 | 0.28
$WWZ$ | 1163 | 42.8 | 3.14 | 1.1 | 0.14 | 0.02 | -
$t\bar{t}Z$ | 74 | 7.1 | 5.68 | 5.6 | 4.04 | 0.71 | 0.35
$t\bar{t}$ (semi-leptonic) | 25955 | 3846.1 | 2843.9 | 2818.8 | 2500.6 | 338.5 | 16.61
$t\bar{t}$ (leptonic) | 4328 | 565.4 | 481.5 | 478.3 | 401.9 | 29.65 | 1.13
$WW$ | 110620 | 0.55 | 0.28 | 0.28 | - | - | -
$hZ$ | 1131 | 6.8 | 1.26 | 0.023 | - | - | -
$ZZ$ | 7699 | 167.9 | 42.81 | 13.0 | - | - | -
Total background | 146.4
Significance | 3.99
Table 13: The cut-flow table showing the change in the number of events for
for benchmark BP3 for the unpolarised electron and positron at $\sqrt{s}=3$
TeV at $\mathcal{L}=5$ ab-1. The cuts (C0-C6) are defined in the text in sec.
8.2. The ’-’ represent less than 1 event at integrated luminosity
$\mathcal{L}=5$ ab-1.

Note that we have used unpolarised incoming beams for this study. It has been
observed that the use of a right-handed electron and left-handed positron beam
polarisation can significantly suppress SM backgrounds MoortgatPick:2005cw
and can be effectively used to improve the signal significance as proposed in
the upcoming ILC with the possibility of achieving beam polarisations ($\pm
80\%,\mp 30\%)$) for $(e^{-},e^{+})$. With the CLIC experiment potentially
targetting higher center-of-mass energies along with electron polarisation,
one may hope to improve upon the signal significance further.

##  9 Summary and Conclusions

In this work, we consider the Two Higgs Doublet model extended with a complex
singlet scalar (2HDMS) which is motivated from the possibility of addressing
multiple open issues in Nature such as dark matter, baryogenesis and CP-
violation. In this work, we focus on the dark matter aspect with the complex
scalar singlet as the dark matter candidate. Since the scalar doesnot develop
a vacuum expectation value, the Higgs spectrum is the same as in 2HDM. We
focus on a Type II softly broken $Z_{2}$ symmetric 2HDM scalar potential
augmented with a complex scalar symmetric under $Z^{\prime}_{2}$ stabilizing
the dark matter candidate. In the CP-conserving scenario, the CP-even Higgses
act as a portal to the dark matter. We explore the allowed parameter space
allowed by current experimental data from dark matter, flavour physics and
collider constraints from the SM-like Higgs as well as searches for the heavy
Higgses and choose some representative benchmarks for light and heavy dark
matter mass region are chosen. We observe that direct detection results
stringently constrain the parameter space and therefore requiring low values
of $\lambda^{\prime}_{2}$. Further, we compare our results to the real singlet
extended 2HDM and observe that the relic density is much larger for the
complex scalar DM over real scalar DM while the direct detection cross-section
remains unchanged.

Presence of the singlet also leads to new decay modes for the Higgses with the
Higgses decaying to a pair of dark matter particles. Such a final state will
lead to the presence of missing energy at colliders. We choose a benchmark BP3
with $m_{H}\simeq$ 820 GeV consistent with all experimental data in order to
demonstrate the prospects of observing such a signal at LHC and future
$e^{+}e^{-}$ colliders. In this case, the constraints from direct detection
data as well as the competing fermionic decays of the heavy Higgses
stringently constrain the invisible decay $H\rightarrow\chi\bar{\chi}\simeq
4.8\%$. Owing to the small branching fraction and heavy mass of $H$, it is
difficult to observe such a scenario due to the low production cross-section
times branching ratio. We perform a signal-background analyses at the
$e^{+}e^{-}$ collider with $\sqrt{s}=3$ TeV with unpolarised beams and observe
that the $2b+\not{E}_{T}$ channel is observable with a $=3.99\sigma$
significance at integrated luminosity $\mathcal{L}=5$ ab-1.

## Acknowledgements

JD and GMP acknowledge support by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy EXC 2121
”Quantum Universe”- 390833306. The authors thank G.Belanger, A.Pukhov from the
micrOMEGAs team, J.Heisig, O.Mattalear and C.Arina from the maDDM team for
helpful correspondence. The authors also thank H.Bahl, S.Heinemeyer, C.Li, S.
Paasch, T.Stefaniak and G.Weiglein for helpful discussions and help with
HiggsBounds and HiggsSignals.

## Appendix

### DM interaction vertices

We discuss the relevant interaction vertices of the DM and Higgses.

#### DM-scalar Higgs cubic interactions

$\lambda_{hSS}=2i\frac{v}{\sqrt{1+\tan^{2}\beta}}(\lambda_{4}^{\prime}\sin\alpha-\lambda_{5}^{\prime}\tan\beta\cos\alpha)$
(17)

$\lambda_{hSS^{*}}=i\frac{v}{\sqrt{1+\tan^{2}\beta}}(\lambda_{1}^{\prime}\sin\alpha-\lambda_{2}^{\prime}\tan\beta\cos\alpha)$
(18)

$\lambda_{hS^{*}S^{*}}=2i\frac{v}{\sqrt{1+\tan^{2}\beta}}(\lambda_{4}^{\prime}\sin\alpha-\lambda_{5}^{\prime}\tan\beta\cos\alpha)$
(19)
$\lambda_{HSS}=-2i\frac{v}{\sqrt{1+\tan^{2}\beta}}(\lambda_{4}^{\prime}\cos\alpha+\lambda_{5}^{\prime}\tan\beta\sin\alpha)$
(20)

$\lambda_{HSS^{*}}=-i\frac{v}{\sqrt{1+\tan^{2}\beta}}(\lambda_{1}^{\prime}\cos\alpha+\lambda_{2}^{\prime}\tan\beta\sin\alpha)$
(21)

$\lambda_{HS^{*}S^{*}}=-2i\frac{v}{\sqrt{1+\tan^{2}\beta}}(\lambda_{4}^{\prime}\cos\alpha+\lambda_{5}^{\prime}\tan\beta\sin\alpha)$
(22)

#### DM-Higgs scalars quartic interactions

$\lambda_{hhSS}=-2i(\lambda_{4}^{\prime}\sin^{2}\alpha+\lambda_{5}^{\prime}\cos^{2}\alpha)$
(23)

$\lambda_{hhS^{*}S^{*}}=-2i(\lambda_{4}^{\prime}\sin^{2}\alpha+\lambda_{5}^{\prime}\cos^{2}\alpha)$
(24)

$\lambda_{hhSS^{*}}=-i(\lambda_{1}^{\prime}\sin^{2}\alpha+\lambda_{2}^{\prime}\cos^{2}\alpha)$
(25)

$\lambda_{hHSS}=i(\lambda_{4}^{\prime}-\lambda_{5}^{\prime})\sin 2\alpha$ (26)

$\lambda_{hHS^{*}S^{*}}=2i(\lambda_{4}^{\prime}-\lambda_{5}^{\prime})\sin
2\alpha$ (27)

$\lambda_{HHSS}=-2i(\lambda_{4}^{\prime}\cos^{2}\alpha+\lambda_{5}^{\prime}\sin^{2}\alpha)$
(28)

$\lambda_{HHS^{*}S^{*}}=-2i(\lambda_{4}^{\prime}\cos^{2}\alpha+\lambda_{5}^{\prime}\sin^{2}\alpha)$
(29)

### Pseudoscalar interactions with DM

$\\\
\lambda_{AASS}=-2i(\frac{\lambda_{4}^{\prime}\tan^{2}\beta}{1+\tan^{2}\beta}+\frac{\lambda_{5}^{\prime}}{1+\tan^{2}\beta})$
(30)

$\lambda_{AASS^{*}}=i(\frac{\lambda_{1}^{\prime}\tan^{2}\beta}{1+\tan^{2}\beta}-\frac{\lambda_{2}^{\prime}}{1+\tan^{2}\beta})$
(31)
$\lambda_{AAS^{*}S^{*}}=-2i(\frac{\lambda_{4}^{\prime}\tan^{2}\beta}{1+\tan^{2}\beta}+\frac{\lambda_{5}^{\prime}}{1+\tan^{2}\beta})$
(32)

#### Charged Higgs interactions with DM

$\lambda_{H^{+}H^{-}SS}=-2i(\lambda_{4}^{\prime}\frac{\tan^{2}\beta}{1+\tan^{2}\beta}+\lambda_{5}^{\prime}\frac{1}{1+\tan^{2}\beta})$
(33)

$\lambda_{H^{+}H^{-}SS^{*}}=-i(\lambda_{1}^{\prime}\frac{\tan^{2}\beta}{1+\tan^{2}\beta}+\lambda_{2}^{\prime}\frac{1}{1+\tan^{2}\beta})$
(34)

$\lambda_{H^{+}H^{-}S^{*}S^{*}}=-2i(\lambda_{4}^{\prime}\frac{\tan^{2}\beta}{1+\tan^{2}\beta}+\lambda_{5}^{\prime}\frac{1}{1+\tan^{2}\beta})$
(35)

#### DM self interactions

$\lambda_{SSSS}=-i\lambda_{1}^{\prime\prime}$ (36)

$\lambda_{SSS^{*}S^{*}}=-i\lambda_{3}^{\prime\prime}$ (37)

$\lambda_{SSSS^{*}}=-i\lambda_{1}^{\prime\prime}$ (38)

$\lambda_{S^{*}S^{*}S^{*}S}=-i\lambda_{1}^{\prime\prime}$ (39)

$\lambda_{S^{*}S^{*}S^{*}S^{*}}=-i\lambda_{1}^{\prime\prime}$ (40)

#### Analytical computation of direct detection cross-section

We compute the direct detection cross-section. The DM-quark amplitude of the
tree-level Feynman diagram Fig. 4 for zero momentum transfer is

$\mathcal{M}=\sum\limits_{h_{i}}\lambda_{h_{i}SS^{*}}\frac{1}{-m^{2}_{h_{i}}}\lambda_{h_{i}q\bar{q}}$
(41)

where $h_{i}=h,H$. Recall,

$\lambda_{hSS^{*}}=\frac{i}{\sqrt{1+\tan^{2}\beta}}(\lambda_{1}^{\prime}\sin\alpha-\lambda^{\prime}_{2}\tan\beta\cos\alpha)$
$\lambda_{HSS^{*}}=-\frac{i}{\sqrt{1+\tan^{2}\beta}}(\lambda_{1}^{\prime}\cos\alpha-\lambda^{\prime}_{2}\tan\beta\sin\alpha),$

and

$\lambda_{h_{i}q\bar{q}}=\frac{m_{q}}{v}C^{h_{i}}$

is the Yukawa coupling of the quark to the Higgses. For Type II
2HDMBranco:2011iw  the Higgs couplings to the fermions are summarised in Table
14.

Higgses | $C_{u}$ | $C_{d}$ | $C_{s}$
---|---|---|---
$h$ | $\cos\alpha/\sin\beta$ | -$\sin\alpha/\cos\beta$ | -$\sin\alpha/\cos\beta$
$H$ | $\sin\alpha/\sin\beta$ | $\cos\alpha/\cos\beta$ | $\cos\alpha/\cos\beta$
Table 14: The couplings of the fermions in Type II 2HDMBranco:2011iw .

In order to compute the DM-nucleon amplitude, the nucleon form factors $f_{N}$
need to folded into the quark-DM amplitude $\mathcal{M}$. The form factors
$f^{N}$ for $N=p,n$ areDrozd:2014yla

$f_{N}=\frac{m_{N}}{2m_{\chi}}(\sum\limits_{q=u,d,s}f^{N}_{Tq}\frac{\mathcal{M}}{m_{q}}+\frac{2}{27}f^{N}_{TG}\sum\limits_{q=c,b,t}\,\frac{\mathcal{M}}{m_{q}})$
(42)

where the first term is due to the contribution of the light valence quarks
and the second term due to the gluonic form factor $f^{N}_{TG}$ defined as,

$f^{N}_{TG}=1-\sum\limits_{q=u,d,s}f^{N}_{Tq}$ (43)

Folding in the form factors above and the phase space of the DM-nucleon
scattering, the DM-nucleon cross-section for the proton and neutron
areJungman:1995df ; Drozd:2014yla ,

$\displaystyle\sigma^{p}_{SI}=\frac{4\mu^{2}_{N}}{\pi}(Zf_{p})^{2}$ (44)
$\displaystyle\sigma^{n}_{SI}=\frac{4\mu^{2}_{N}}{\pi}((A-Z)f_{n})^{2}$ (45)

where $\mu_{N}=\frac{m_{\chi}m_{N}}{(m_{\chi}+m_{N})}$ for nucleon $N=p,n$.

## References

  * (1) V. Barger, P. Langacker, M. McCaskey, M. Ramsey-Musolf, and G. Shaughnessy, Complex Singlet Extension of the Standard Model, Phys. Rev. D 79 (2009) 015018, [arXiv:0811.0393].
  * (2) H. Wu and S. Zheng, Scalar Dark Matter: Real vs Complex, JHEP 03 (2017) 142, [arXiv:1610.06292].
  * (3) A. Ghosh, D. Ghosh, and S. Mukhopadhyay, Cosmology of complex scalar dark matter: Interplay of self-scattering and annihilation, Phys. Rev. D 104 (2021), no. 12 123543, [arXiv:2103.14009].
  * (4) XENON Collaboration, E. Aprile et al., Dark Matter Search Results from a One Ton-Year Exposure of XENON1T, Phys. Rev. Lett. 121 (2018), no. 11 111302, [arXiv:1805.12562].
  * (5) G. Branco, P. Ferreira, L. Lavoura, M. Rebelo, M. Sher, and J. P. Silva, Theory and phenomenology of two-Higgs-doublet models, Phys. Rept. 516 (2012) 1–102, [arXiv:1106.0034].
  * (6) G. C. Dorsch, S. J. Huber, T. Konstandin, and J. M. No, A Second Higgs Doublet in the Early Universe: Baryogenesis and Gravitational Waves, JCAP 05 (2017) 052, [arXiv:1611.05874].
  * (7) X.-G. He, T. Li, X.-Q. Li, J. Tandean, and H.-C. Tsai, Constraints on Scalar Dark Matter from Direct Experimental Searches, Phys. Rev. D 79 (2009) 023521, [arXiv:0811.0658].
  * (8) A. Drozd, B. Grzadkowski, J. F. Gunion, and Y. Jiang, Extending two-Higgs-doublet models by a singlet scalar field - the Case for Dark Matter, JHEP 11 (2014) 105, [arXiv:1408.2106].
  * (9) X.-G. He and J. Tandean, New LUX and PandaX-II Results Illuminating the Simplest Higgs-Portal Dark Matter Models, JHEP 12 (2016) 074, [arXiv:1609.03551].
  * (10) M. Muhlleitner, M. O. P. Sampaio, R. Santos, and J. Wittbrodt, The N2HDM under Theoretical and Experimental Scrutiny, JHEP [arXiv:1612.01309].
  * (11) A. Dey, J. Lahiri, and B. Mukhopadhyaya, LHC signals of a heavy doublet Higgs as dark matter portal: cut-based approach and improvement with gradient boosting and neural networks, JHEP 09 (2019) 004, [arXiv:1905.02242].
  * (12) S. Baum and N. R. Shah, Two Higgs Doublets and a Complex Singlet: Disentangling the Decay Topologies and Associated Phenomenology, arXiv:1808.02667. [JHEP12,044(2018)].
  * (13) T. Biekötter and M. O. Olea-Romacho, Reconciling Higgs physics and pseudo-Nambu-Goldstone dark matter in the S2HDM using a genetic algorithm, JHEP 10 (2021) 215, [arXiv:2108.10864].
  * (14) K. Kannike, K. Loos, and M. Raidal, Gravitational wave signals of pseudo-Goldstone dark matter in the $\mathbb{Z}_{3}$ complex singlet model, Phys. Rev. D 101 (2020), no. 3 035001, [arXiv:1907.13136].
  * (15) D. G. E. Walker, Dark Matter Stabilization Symmetries from Spontaneous Symmetry Breaking, arXiv:0907.3146.
  * (16) B. Batell, Dark Discrete Gauge Symmetries, Phys. Rev. D 83 (2011) 035006, [arXiv:1007.0045].
  * (17) C. Gross, O. Lebedev, and T. Toma, Cancellation Mechanism for Dark-Matter–Nucleon Interaction, Phys. Rev. Lett. 119 (2017), no. 19 191801, [arXiv:1708.02253].
  * (18) C. Cai, Y.-P. Zeng, and H.-H. Zhang, Cancellation mechanism of dark matter direct detection in Higgs-portal and vector-portal models, arXiv:2109.11499.
  * (19) Fermi-LAT Collaboration, M. Ackermann et al., Constraining Dark Matter Models from a Combined Analysis of Milky Way Satellites with the Fermi Large Area Telescope, Phys. Rev. Lett. 107 (2011) 241302, [arXiv:1108.3546].
  * (20) ATLAS Collaboration Collaboration, Measurement of the Higgs boson mass in the $H\rightarrow ZZ^{*}\rightarrow 4\ell$ decay channel with $\sqrt{s}=13$ TeV $pp$ collisions using the ATLAS detector at the LHC, Tech. Rep. ATLAS-CONF-2020-005, CERN, Geneva, Apr, 2020.
  * (21) ATLAS Collaboration, Combination of searches for invisible Higgs boson decays with the ATLAS experiment, .
  * (22) CMS Collaboration, A. M. Sirunyan et al., Search for invisible decays of a Higgs boson produced through vector boson fusion in proton-proton collisions at $\sqrt{s}=$ 13 TeV, Phys. Lett. B 793 (2019) 520–551, [arXiv:1809.05937].
  * (23) BaBar Collaboration Collaboration, J. Lees et al., Precision Measurement of the $B\to X_{s}\gamma$ Photon Energy Spectrum, Branching Fraction, and Direct CP Asymmetry $A_{CP}(B\to X_{s+d}\gamma)$, Phys.Rev.Lett. 109 (2012) 191801, [arXiv:1207.2690].
  * (24) LHCb collaboration Collaboration, R. Aaij et al., Measurement of the $B^{0}_{s}\to\mu^{+}\mu^{-}$ branching fraction and search for $B^{0}\to\mu^{+}\mu^{-}$ decays at the LHCb experiment, Phys.Rev.Lett. 111 (2013) 101805, [arXiv:1307.5024].
  * (25) CMS Collaboration Collaboration, S. Chatrchyan et al., Measurement of the B(s) to mu+ mu- branching fraction and search for B0 to mu+ mu- with the CMS Experiment, Phys.Rev.Lett. 111 (2013) 101804, [arXiv:1307.5025].
  * (26) https://pdg.lbl.gov/2020/reviews/rpp2020-rev-g-2-muon-anom-mag-moment.pdf.
  * (27) P. D. Group and P. A. e. Zyla, Review of Particle Physics, Progress of Theoretical and Experimental Physics 2020 (08, 2020) [https://academic.oup.com/ptep/article-pdf/2020/8/083C01/33653179/ptaa104.pdf]. 083C01.
  * (28) Planck Collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6, [arXiv:1807.06209].
  * (29) ALEPH, DELPHI, L3, OPAL, LEP Collaboration, G. Abbiendi et al., Search for Charged Higgs bosons: Combined Results Using LEP Data, Eur. Phys. J. C 73 (2013) 2463, [arXiv:1301.6065].
  * (30) ATLAS Collaboration. https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CombinedSummaryPlots/HDBS/.
  * (31) CMS Collaboration. https://twiki.cern.ch/twiki/bin/view/CMSPublic/Summary2HDMSRun2.
  * (32) ATLAS Collaboration Collaboration, A combination of measurements of Higgs boson production and decay using up to $139$ fb-1 of proton–proton collision data at $\sqrt{s}=$ 13 TeV collected with the ATLAS experiment, Tech. Rep. ATLAS-CONF-2020-027, CERN, Geneva, Aug, 2020.
  * (33) O. Eberhardt, Current status of Two-Higgs-Doublet models with a softly broken $\mathbb{Z}_{2}$ symmetry, PoS ICHEP2018 (2019) 457, [arXiv:1809.04851].
  * (34) A. Arbey, F. Mahmoudi, O. Stal, and T. Stefaniak, Status of the Charged Higgs Boson in Two Higgs Doublet Models, Eur. Phys. J. C 78 (2018), no. 3 182, [arXiv:1706.07414].
  * (35) G. Belanger, F. Boudjema, A. Pukhov, A. Semenov, and et al., The user’s manual, version 5.2,http://lapth.cnrs.fr/micromegas/v5.2/manual_5.2.pdf, .
  * (36) P. Bechtle, S. Heinemeyer, T. Klingl, T. Stefaniak, G. Weiglein, and J. Wittbrodt, HiggsSignals-2: Probing new physics with precision Higgs measurements in the LHC 13 TeV era, Eur. Phys. J. C 81 (2021), no. 2 145, [arXiv:2012.09197].
  * (37) W. Porod, SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e- colliders, Comput. Phys. Commun. 153 (2003) 275–315, [hep-ph/0301101].
  * (38) F. Staub, SARAH 4 : A tool for (not only SUSY) model builders, Comput. Phys. Commun. 185 (2014) 1773–1790, [arXiv:1309.7223].
  * (39) G. Arcadi, G. Busoni, T. Hugle, and V. T. Tenorth, Comparing 2HDM $+$ Scalar and Pseudoscalar Simplified Models at LHC, JHEP 06 (2020) 098, [arXiv:2001.10540].
  * (40) P. Bechtle, D. Dercks, S. Heinemeyer, T. Klingl, T. Stefaniak, G. Weiglein, and J. Wittbrodt, HiggsBounds-5: Testing Higgs Sectors in the LHC 13 TeV Era, arXiv:2006.06007.
  * (41) P. Bechtle, S. Heinemeyer, O. St$\dot{a}$l, T. Stefaniak, and G. Weiglein, $HiggsSignals$: Confronting arbitrary Higgs sectors with measurements at the Tevatron and the LHC, Eur. Phys. J. C 74 (2014), no. 2 2711, [arXiv:1305.1933].
  * (42) J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, and T. Stelzer, MadGraph 5 : Going Beyond, JHEP 06 (2011) 128, [arXiv:1106.0522].
  * (43) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079, [arXiv:1405.0301].
  * (44) Pythia. http://home.thep.lu.se/~torbjorn/pythia81html/Welcome.html.
  * (45) DELPHES 3 Collaboration, J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi, DELPHES 3, A modular framework for fast simulation of a generic collider experiment, JHEP 02 (2014) 057, [arXiv:1307.6346].
  * (46) M. Selvaggi, DELPHES 3: A modular framework for fast-simulation of generic collider experiments, J. Phys. Conf. Ser. 523 (2014) 012033\.
  * (47) A. Mertens, New features in Delphes 3, J. Phys. Conf. Ser. 608 (2015), no. 1 012045.
  * (48) B. Dumont, B. Fuks, S. Kraml, S. Bein, G. Chalons, E. Conte, S. Kulkarni, D. Sengupta, and C. Wymant, Toward a public analysis database for LHC new physics searches using MADANALYSIS 5, Eur. Phys. J. C 75 (2015), no. 2 56, [arXiv:1407.3278].
  * (49) E. Conte, B. Dumont, B. Fuks, and C. Wymant, Designing and recasting LHC analyses with MadAnalysis 5, Eur. Phys. J. C 74 (2014), no. 10 3103, [arXiv:1405.3982].
  * (50) E. Conte and B. Fuks, Confronting new physics theories to LHC data with MADANALYSIS 5, Int. J. Mod. Phys. A 33 (2018), no. 28 1830027, [arXiv:1808.00480].
  * (51) J. Y. Araz, M. Frank, and B. Fuks, Reinterpreting the results of the LHC with MadAnalysis 5: uncertainties and higher-luminosity estimates, Eur. Phys. J. C 80 (2020), no. 6 531, [arXiv:1910.11418].
  * (52) J. Y. Araz, B. Fuks, and G. Polykratis, Simplified fast detector simulation in MADANALYSIS 5, Eur. Phys. J. C 81 (2021), no. 4 329, [arXiv:2006.09387].
  * (53) ATLAS, CMS Collaboration, Addendum to the report on the physics at the HL-LHC, and perspectives for the HE-LHC: Collection of notes from ATLAS and CMS, CERN Yellow Rep. Monogr. 7 (2019) Addendum, [arXiv:1902.10229].
  * (54) A. Adhikary, S. Banerjee, R. Kumar Barman, and B. Bhattacherjee, Resonant heavy Higgs searches at the HL-LHC, JHEP 09 (2019) 068, [arXiv:1812.05640].
  * (55) H. Bahl, P. Bechtle, S. Heinemeyer, S. Liebler, T. Stefaniak, and G. Weiglein, HL-LHC and ILC sensitivities in the hunt for heavy Higgs bosons, Eur. Phys. J. C 80 (2020), no. 10 916, [arXiv:2005.14536].
  * (56) ATLAS Collaboration, G. Aad et al., Search for new phenomena in events with an energetic jet and missing transverse momentum in $pp$ collisions at $\sqrt{s}$ =13 TeV with the ATLAS detector, Phys. Rev. D 103 (2021), no. 11 112006, [arXiv:2102.10874].
  * (57) ATLAS Collaboration Collaboration, Search for invisible Higgs boson decays with vector boson fusion signatures with the ATLAS detector using an integrated luminosity of 139 fb-1, tech. rep., CERN, Geneva, Apr, 2020\.  All figures including auxiliary figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2020-008.
  * (58) T. Behnke, J. E. Brau, B. Foster, J. Fuster, M. Harrison, J. M. Paterson, M. Peskin, M. Stanitzki, N. Walker, and H. Yamamoto, The International Linear Collider Technical Design Report - Volume 1: Executive Summary, arXiv:1306.6327. See also http://www.linearcollider.org/ILC/TDR . The full list of contributing institutes is inside the Report.
  * (59) G. Moortgat-Pick et al., The Role of polarized positrons and electrons in revealing fundamental interactions at the linear collider, Phys. Rept. 460 (2008) 131–243, [hep-ph/0507011].
  * (60) CLICdp, CLIC Collaboration, T. K. Charles et al., The Compact Linear Collider (CLIC) - 2018 Summary Report, arXiv:1812.06018.
  * (61) P. G. Roloff and A. Robson, Updated CLIC luminosity staging baseline and Higgs coupling prospects, tech. rep., CERN, Geneva, Oct, 2018.  9 pages, 6 figures.
  * (62) A. Blondel, J. Gluza, S. Jadach, P. Janot, and T. Riemann, eds., Theory for the FCC-ee: Report on the 11th FCC-ee Workshop Theory and Experiments, vol. 3/2020 of CERN Yellow Reports: Monographs, (Geneva), CERN, 5, 2019.
  * (63) G. Cowan, K. Cranmer, E. Gross, and O. Vitells, Asymptotic formulae for likelihood-based tests of new physics, Eur. Phys. J. C 71 (2011) 1554, [arXiv:1007.1727]. [Erratum: Eur.Phys.J.C 73, 2501 (2013)].
  * (64) https://pdg.lbl.gov/2019/reviews/rpp2018-rev-statistics.pdf, .
  * (65) G. Jungman, M. Kamionkowski, and K. Griest, Supersymmetric dark matter, Phys. Rept. 267 (1996) 195–373, [hep-ph/9506380].