# Exploring Direct Detection Suppressed Regions in a Simple 2-Scalar Mediator
Model of Scalar Dark Matter

Jérôme Claude111Email<EMAIL_ADDRESS>and Stephen Godfrey222Email:
<EMAIL_ADDRESS>Ottawa-Carleton Institute for Physics, Department
of Physics, Carleton University, Ottawa, Canada K1S 5B6

###### Abstract

We explore regions of parameter space that give rise to suppressed direct
detection cross sections in a simple model of scalar dark matter with a scalar
portal that mixes with the standard model Higgs. We found that even this
simple model allows considerable room in the parameter space that has not been
excluded by direct detection limits. A number of effects leading to this
result have been previously noted. Our main new result explores interference
effects between different contributions to DM annihilation when the DM mass is
larger than the scalar portal mass. New annihilation channels open up and the
parameters of the model need to compensate to give the correct DM relic
abundance, resulting in smaller direct detection cross sections. We find that
even in a very simple model of DM there are still sizeable regions of
parameter space that are not ruled out by experiment.

##  I Introduction

There is considerable evidence for Dark Matter (DM), a type of matter in the
universe which has so far only revealed itself through gravitational
interactions with normal matter Bergstrom:2000pn ; Bertone:2004pz ;
Bergstrom:2012fi . DM at most interacts very weakly with normal matter.
Various means of DM interacting with normal matter have been explored; Higgs
portals, e.g. Bertone:2004pz ; Silveira:1985rk ; McDonald:1993ex ; Patt:2006fw
; Baek:2011aa ; Djouadi:2011aa ; LopezHonorez:2012kv ; Baek:2012se ;
Walker:2013hka ; Esch:2014jpa ; Buchmueller:2014yoa ; Cheung:2015dta ;
Beniwal:2015sdl ; Han:2016gyy ; Arcadi:2016qoz ; Arcadi:2017kky ;
Bhattacharya:2017fid ; Gross:2017dan ; Azevedo:2018oxv ; Arcadi:2019lka ;
Cabrera:2019gaq ; Ghorbani:2014gka ; Ghosh:2017fmr ; Alanne:2020jwx , vector
portals, e.g. Beniwal:2015sdl ; Arcadi:2017kky ; Buckley:2011mm ;
Frandsen:2011cg ; Lebedev:2011iq ; Alves:2013tqa ; Arcadi:2013qia ;
Lebedev:2014bba ; Arcadi:2014lta ; Hooper:2014fda ; Alves:2015pea ;
Ghorbani:2015baa ; Jacques:2016dqz ; Duerr:2016tmh ; Ismail:2016tod ;
Escudero:2016gzx ; Kearney:2016rng ; Alves:2016cqf ; Dutra:2018gmv ;
Okada:2018ktp ; Blanco:2019hah ; Arcadi:2020jqf ; Okada:2020cue , and neutrino
portals, e.g. Beniwal:2015sdl ; Arcadi:2017kky ; Falkowski:2009yz ;
Cherry:2014xra ; Batell:2017cmf ; Cosme:2020mck . Higgs portal models have
been tightly constrained by experiment, leaving only small regions in the
parameter space viable Escudero:2016gzx . In particular, direct detection
experiments have tightly constrained the parameter space. However, there still
exist allowed regions, including regions referred to as blind spots which are
due to cancellations in the direct detection cross section amplitudes. This
has been explored in a number of papers, for example Baek:2012se ;
Gross:2017dan ; Azevedo:2018oxv ; Cabrera:2019gaq ; Cheung:2012qy ;
Cheung:2013dua ; Huang:2014xua ; Berlin:2015wwa ; Casas:2017jjg ;
Huitu:2018gbc . In addition to the blind spots mechanism, other mechanisms
exist that suppress direct detection cross sections which we discuss below.

Many Higgs portal models have a second scalar that mixes with the Standard
Model (SM) Higgs boson Baek:2011aa ; LopezHonorez:2012kv ; Baek:2012se ;
Esch:2014jpa ; Cheung:2015dta ; Arcadi:2016qoz ; Gross:2017dan ;
Azevedo:2018oxv ; Arcadi:2019lka ; Cabrera:2019gaq ; Ghosh:2017fmr ;
Alanne:2020jwx ; Duerr:2016tmh ; Huang:2014xua ; Barger:2007im ;
Falkowski:2015iwa ; Ko:2016ybp ; Arcadi:2016kmk ; Bell:2016ekl ; Bell:2017rgi
; Arcadi:2018pfo . The mechanism leading to blind spots in such models is the
destructive interference between the Higgs-like scalar and the second scalar
in the direct detection cross section amplitude LopezHonorez:2012kv ;
Baek:2012se ; Esch:2014jpa ; Arcadi:2016qoz ; Gross:2017dan ; Azevedo:2018oxv
; Cabrera:2019gaq ; Alanne:2020jwx ; Huang:2014xua ; Arcadi:2016kmk ;
Bell:2017rgi . Given that detecting dark matter is the focus of a broad
experimental program, we felt it useful to further explore regions of the
parameter space that give rise to suppressed direct detection cross sections.
Our preconceived bias was that the mixing angle between the two $t$-channel
exchange bosons could be tuned to create the direct detection blind spots
mentioned above. However, we found that values of the mixing angle that would
give rise to blind spots are for the most part ruled out by measurements of
Higgs boson properties — most generally by the Higgs signal strengths, but
also by the Higgs invisible width when Higgs decay to dark matter is
kinematically allowed. Another mechanism that can lead to suppressed direct
detection cross sections which has previously been pointed out Esch:2014jpa ;
Arcadi:2017kky  is the result of a resonance effect occurring when the dark
matter mass is roughly half the value of the scalar portal mass. However,
there is a third mechanism that suppresses the direct detection cross section
when the dark matter particle is more massive than either the Higgs boson or
the portal particle333Ref. Esch:2014jpa ; Azevedo:2018oxv ; Ghosh:2017fmr
noted a similar effect when $m_{S}>m_{h_{1},h_{2}}$ although the details
differ from those presented here.. In this case, a large region of the
parameter space has not been ruled out by any of the theory constraints, any
experimental constraints and, more to the point of this exercise, by direct
detection limits.

For the purposes of this study, we constructed a very simple toy model
consisting of scalar DM and an additional scalar portal that can mix with the
SM Higgs field to study direct detection suppressed regions. We use this toy
model to explore effects for the simplest possible case of a scalar dark
matter portal extension. There are, however, many possible variations of this
simple picture that can give rise to cancellations in the direct detection
cross section. An incomplete list of possibilities appearing in the literature
consists of the scalar portal being replaced with a pseudoscalar portal
Arcadi:2019lka ; Berlin:2015wwa ; Buckley:2014fba ; No:2015xqa ;
Goncalves:2016iyg ; Tunney:2017yfp ; Arcadi:2017wqi ; Ghosh:2020fdc ;
Butterworth:2020vnb ; Abe:2020pzp , or having a complex scalar which gives
rise to a second scalar portal Gross:2017dan ; Alanne:2020jwx ; Huitu:2018gbc
; Okada:2020zxo ; Alanne:2020jwx ; Zhang:2021alu , a two Higgs doublet model
Arcadi:2019lka ; Cabrera:2019gaq ; Berlin:2015wwa ; Bell:2017rgi ;
Arcadi:2018pfo ; Arcadi:2017wqi ; Ghosh:2020fdc ; Butterworth:2020vnb ;
Zhang:2021alu ; Bauer:2017ota ; Jiang:2019soj ; Arcadi:2020gge , higher Higgs
representations Arhrib:2011uy , or supersymmetric models Arcadi:2019lka ;
Cheung:2012qy ; Huang:2014xua ; Ellis:2005mb ; Carena:2006nv ; Hooper:2006wv ;
Cao:2010ph . Before proceeding, we note that, given that we simply want to
push the simplest of models as far as we could, we haven’t dealt with the
issue of UV completeness. However, Gross, Lebedev and Toma Gross:2017dan  and
Huitu et. al. Huitu:2018gbc  showed that they could make models very similar
to ours UV complete by assuming the system is invariant under a global $U(1)$
which is gauged in the UV-completeness. Other examples of similar UV complete
models are Ghosh:2017fmr ; Alanne:2020jwx .

Our simple model has eight parameters but two are fixed to their SM values,
one is fixed to give the correct DM relic abundance, and one is only weakly
constrained by DM self-interactions. This leaves four independent parameters
which we choose to be the scalar DM mass, the scalar portal mass, the scalar
singlet vacuum expectation value, and the mixing angle between the SM Higgs
scalar and the scalar portal. We scan through the parameter space and, by
transforming to the Lagrangian parameters, we test that perturbative unitarity
holds, that the potential is bounded from below, and that the parameters
result in a consistent set of parameters for the desired properties of the
model. We next fix the remaining parameter to give the correct relic
abundance. With these parameter values, we test that the parameters are
consistent with the Higgs boson invisible width, the Higgs signal strengths,
and DM self-interaction limits. Finally, we calculate the direct and indirect
detection cross sections and compare them to the experimental limits.

In Section II, we give the details of our model and examine the theoretical
constraints on its parameters. In Section III, we describe the details of
scanning the parameter space and the various experimental measurements we use
to constrain parameter points, starting by fitting the DM-portal coupling to
the DM relic abundance. The remaining experimental constraints are the Higgs
invisible width, the Higgs signal strength, the DM self-interaction, and the
DM indirect detection cross section. We then compare the points that pass all
these constraints to the direct detection limits and examine the various
mechanisms that lead to direct detection suppressed regions. Finally, in
Section IV, we summarize our conclusions.

##  II A 2-Scalar Mediator Model with Scalar DM

We consider an extension of the Standard Model that consists of two singlet
scalar fields $\varphi$ and $S$, with $\varphi$ a portal particle that mixes
with the SM Higgs field and $S$ the DM particle. We impose a $Z_{2}\times
Z_{2}$ symmetry on these fields so that they are odd under their respective
$Z_{2}$’s to ensure their stability and eliminate terms in the potential odd
in $\varphi$ and $S$ (see for example Ref. Arcadi:2017kky ; Arcadi:2019lka ).
We note that the $Z_{2}$ imposed on $\varphi$ is spontaneously broken when
$\varphi$ acquires a vev. The most general scalar potential with this symmetry
is then given by

$\displaystyle V(H,\varphi,S)$ $\displaystyle=$
$\displaystyle-\mu_{H}^{2}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}$ (1)
$\displaystyle-$ $\displaystyle{{\mu_{\varphi}^{2}}\over
2}\varphi^{2}+{{\lambda_{\varphi}}\over
4}\varphi^{4}+\lambda_{4}\varphi^{2}(H^{\dagger}H)$ $\displaystyle+$
$\displaystyle{{\mu_{S}^{2}}\over 2}S^{2}+{{\lambda_{S}}\over
4}S^{4}+{{\lambda_{\varphi\varphi SS}}\over 2}\varphi^{2}S^{2}$
$\displaystyle+$ $\displaystyle{{\lambda_{HHSS}}\over 2}(H^{\dagger}H)S^{2}.$

Following Ref. Arcadi:2017kky ; Arcadi:2019lka , we take $\lambda_{HHSS}=0$ so
that the Standard Model complex scalar doublet $H$ does not directly couple to
the dark matter candidate, $S$, at tree level. This choice does not affect our
conclusions, and we will discuss the consequences of not taking
$\lambda_{HHSS}=0$ in Section III.7 after we present our results. This term
can be generated via $\varphi$ loops and the natural size for the resulting
vertex would be the product of the couplings $\lambda_{\varphi\varphi
SS}\lambda_{4}/(16\pi^{2})$. We assume that the vertex can be made small
enough even if it requires some amount of tuning. Assuming this term is small
enough, and because the DM thermally averaged annihilation cross section is
typically dominated by the $s$-channel annihilation cross section and real
production of $h_{2}$, we will find that it will not have a big effect on the
relic abundance and that neglecting it will not qualitatively alter our
conclusions.

We work in the unitarity gauge and shift the fields to the new minimum;
$H\to(0,(v+h)/\sqrt{2})^{T}$ and $\varphi\to(w+\phi)$, where $v$ and $w$ are
the vacuum expectation values (vevs) of the neutral component of $H$ and
$\phi$ respectively. We require that $S$ does not acquire a vev so that the
$Z_{2}$ symmetry remains unbroken and $S$ is stable. With this substitution,
we then minimize the resulting potential $V(h,\phi,S)$ with respect to the
scalar fields and obtain $\mu_{H}^{2}=\lambda_{H}v^{2}+\lambda_{4}w^{2}$ and
$\mu_{\varphi}^{2}=\lambda_{\varphi}w^{2}+\lambda_{4}v^{2}$. After
substituting these expressions into $V(h,\phi,S)$, we find the mass terms from
the resulting potential. Diagonalizing the mass matrix for the $h$ and $\phi$
fields leads to physical states that are linear combinations of the the $h$
and $\phi$ fields with mixing angle $\alpha$ given by:

$\displaystyle h_{1}$ $\displaystyle=$ $\displaystyle
h\cos{\alpha}-\phi\sin{\alpha}$ (2) $\displaystyle h_{2}$ $\displaystyle=$
$\displaystyle\phi\cos{\alpha}+h\sin{\alpha}$ (3)

with

$\displaystyle\sin(2\alpha)$ $\displaystyle=$
$\displaystyle\frac{2\lambda_{4}vw}{\sqrt{(\lambda_{H}v^{2}-\lambda_{\varphi}w^{2})^{2}+4\lambda_{4}^{2}v^{2}w^{2}}}$
(4) $\displaystyle\cos(2\alpha)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{\varphi}w^{2}-\lambda_{H}v^{2}}{\sqrt{(\lambda_{H}v^{2}-\lambda_{\varphi}w^{2})^{2}+4\lambda_{4}^{2}v^{2}w^{2}}},$
(5)

and the scalar masses given by

$\displaystyle m_{h_{1}}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{H}v^{2}+\lambda_{\varphi}w^{2}-\frac{\lambda_{\varphi}w^{2}-\lambda_{H}v^{2}}{\cos{(2\alpha)}}$
(6) $\displaystyle m_{h_{2}}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{H}v^{2}+\lambda_{\varphi}w^{2}+\frac{\lambda_{\varphi}w^{2}-\lambda_{H}v^{2}}{\cos{(2\alpha)}}$
(7) $\displaystyle m_{S}^{2}$ $\displaystyle=$
$\displaystyle\mu_{S}^{2}+\lambda_{\varphi\varphi SS}w^{2}.$ (8)

For small values of $\alpha$, we identify $h_{1}$ with the $125~{}\text{GeV}$
scalar associated with the Standard Model Higgs boson. Because of the mixing,
both $h_{1}$ and $h_{2}$ act as portals between the Standard Model and the
dark matter candidate $S$.

When we scan the parameter space, we will use the physical parameters
$m_{h_{1}}$, $m_{h_{2}}$, $\alpha$, $v$, and $w$, but the theoretical
constraints described below constrain the Lagrangian parameters. We will
therefore need the relationships between the physical and the Lagrangian
parameters, which are given by

$\displaystyle\lambda_{H}$ $\displaystyle=$
$\displaystyle\frac{1}{4v^{2}}\left(\left(m_{h_{1}}^{2}+m_{h_{2}}^{2}\right)-\left(m_{h_{2}}^{2}-m_{h_{1}}^{2}\right)\cos{2\alpha}\right)$
(9) $\displaystyle\lambda_{\varphi}$ $\displaystyle=$
$\displaystyle\frac{1}{4w^{2}}\left(\left(m_{h_{1}}^{2}+m_{h_{2}}^{2}\right)+\left(m_{h_{2}}^{2}-m_{h_{1}}^{2}\right)\cos{2\alpha}\right)$
(10) $\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle\frac{\sin{2\alpha}}{4vw}\left(m_{h_{2}}^{2}-m_{h_{1}}^{2}\right).$
(11)

In the following subsections, we examine the theoretical constraints on the
Lagrangian parameters.

###  II.1 Constraints from Partial Wave Unitarity

We start by using partial wave unitarity (PWU) of the $2\rightarrow 2$
scattering amplitudes to constrain the Lagrangian parameters. In the high
energy limit, only tree level diagrams involving four-point scalar
interactions contribute, as diagrams involving propagators are suppressed by
the square of the collision energy. Under these conditions, only the zeroth
partial wave amplitude $a_{0}$ contributes to the $2\rightarrow 2$ amplitudes
$\mathcal{M}$, so that the constraint $\left|a_{0}\right|<\frac{1}{2}$
corresponds to $\mathcal{M}<8\pi$. In the high energy limit, we can also use
the Goldstone equivalence theorem to replace the gauge bosons with the
Goldstone bosons.

There are therefore six fields to consider in the scattering amplitudes: $S$,
$\varphi$, and the four Goldstone bosons $\eta^{0}$, $\eta^{0*}$, $\eta^{+}$,
and $\eta^{-}$. The PWU condition must be applied to each of the eigenvalues
of the coupled-channel scattering matrix $\mathcal{M}$ for all pairs of
incoming and outgoing scalar fields. Because the scalar potential is invariant
under $SU(2)\times U(1)$ symmetry, the scattering processes conserve electric
charge and hypercharge, and can be classified by the total electric charge
($Q$) and hypercharge ($Y$) of the incoming and outgoing states. $S$ and
$\varphi$ are SM gauge singlets and the Goldstone bosons come from the
$SU(2)_{L}$ doublet with $Y=1$ (where $Q_{em}=T_{3}+Y/2$). A symmetry factor
of $1/\sqrt{2}$ is included for each pair of identical particles in the
initial and final states.

Starting with the $Q=2$ and $Y=2$ quantum numbers, there is only one
scattering channel, $\eta^{+}\eta^{+}\rightarrow\eta^{+}\eta^{+}$, which leads
to the constraint

$\left|\lambda_{H}\right|<4\pi.$ (12)

Likewise, the only scattering amplitude for $Q=1$ and $Y=0$ is
$\eta^{+}{\eta^{0}}^{*}\rightarrow\eta^{+}{\eta^{0}}^{*}$, which yields the
same constraint.

For $Q=0$ and $Y=1$, there is only the
$\eta^{0}\varphi\rightarrow\eta^{0}\varphi$ scattering amplitude, leading to
the constraint

$\left|\lambda_{4}\right|<4\pi.$ (13)

Likewise, the only scattering amplitude for $Q=1$ and $Y=1$ is
$\eta^{+}\varphi\rightarrow\eta^{+}\varphi$, which yields the same constraint.

For the $Q=0$ and $Y=0$ quantum numbers, there are five states:
$\eta^{0}\eta^{0*}$, $\eta^{+}\eta^{-}$, $\varphi\varphi$, $\varphi S$, and
$SS$. This results in a $5\times 5$ scattering matrix consisting of a $4\times
4$ block and the $\varphi S\rightarrow\varphi S$ channel. The $\varphi
S\rightarrow\varphi S$ channel leads to the constraint

$\left|\lambda_{\varphi\varphi SS}\right|<4\pi.$ (14)

We can partially diagonalize the $4\times 4$ matrix into a $3\times 3$ matrix
and a diagonal term. The diagonal term leads to the constraint
$\left|\lambda_{H}\right|<4\pi$. To find the remaining constraints, we
diagonalize the $3\times 3$ matrix by taking its determinant and imposing that
the roots of the resulting polynomial satisfy
$\left|\text{Roots}\left(p(x)\right)\right|<8\pi$, where

$\begin{split}p(x)=\left(x-3\lambda_{S}\right)\left(-4\lambda_{4}^{2}+\left(x-6\lambda_{H}\right)\left(x-3\lambda_{\varphi}\right)\right)\\\
-\left(x-6\lambda_{H}\right)\lambda_{\varphi\varphi SS}^{2}.\end{split}$ (15)

We follow the procedure of Ref. Campbell:2016zbp  to which we direct the
interested reader for details, and replace the bounds on the roots of $p(x)$
with the three equivalent conditions:

$\displaystyle 16\pi$ $\displaystyle>$
$\displaystyle\left|6\lambda_{H}+3\lambda_{\varphi}\pm\sqrt{\left(6\lambda_{H}-3\lambda_{\varphi}\right)^{2}+16\lambda_{4}^{2}}\right|$
(16) $\displaystyle\lambda_{S}$ $\displaystyle<$
$\displaystyle\frac{1}{3}\left[8\pi+\frac{\left(6\lambda_{H}-8\pi\right)\lambda_{\varphi\varphi
SS}^{2}}{\left(6\lambda_{H}-8\pi\right)\left(3\lambda_{\varphi}-8\pi\right)-4\lambda_{4}^{2}}\right]$
(17) $\displaystyle\lambda_{S}$ $\displaystyle>$
$\displaystyle\frac{1}{3}\left[-8\pi+\frac{\left(6\lambda_{H}+8\pi\right)\lambda_{\varphi\varphi
SS}^{2}}{\left(6\lambda_{H}+8\pi\right)\left(3\lambda_{\varphi}+8\pi\right)-4\lambda_{4}^{2}}\right].$
(18)

Thus, the constraints on the Lagrangian parameters from partial wave unitarity
are given by equations 12,13, 14, 16, 17 and 18.

###  II.2 Constraints from the Bounded from Below Requirement

We next include constraints on the Lagrangian parameters that ensure that the
scalar potential is bounded from below. Because the quartic terms dominate at
large field values, this constraint acts on the quartic terms in the
potential.

We use the approach described in Ref. Arhrib:2011uy  (see also Ref.
Campbell:2016zbp ) in which we use a hyperspherical coordinate system
replacing the scalar fields by the following parameters:

$\displaystyle r$ $\displaystyle=$
$\displaystyle\sqrt{\left|H\right|^{2}+\varphi^{2}+S^{2}}$ (19) $\displaystyle
r\sin{\beta}\cos{\gamma}$ $\displaystyle=$ $\displaystyle\left|H\right|^{2}~$
(20) $\displaystyle r\sin{\beta}\sin{\gamma}$ $\displaystyle=$
$\displaystyle\varphi^{2}~$ (21) $\displaystyle r\cos{\beta}$ $\displaystyle=$
$\displaystyle S^{2}.~$ (22)

The quartic part of the potential can be then be written as

$\frac{r^{4}}{(1+\tan^{2}{\beta})(1+\tan^{2}{\gamma})}\mathbf{x}^{\intercal}\mathbf{A}\mathbf{y}$
(24)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{1}{4}\begin{bmatrix}\lambda_{S}&2\lambda_{S}&\lambda_{S}\\\
0&2\lambda_{\varphi\varphi SS}&2\lambda_{\varphi\varphi SS}\\\
4\lambda_{H}&4\lambda_{4}&\lambda_{\varphi}\\\ \end{bmatrix}$ (25)
$\displaystyle x$ $\displaystyle=$ $\displaystyle\begin{bmatrix}1\\\
\tan{\beta}\\\ \tan^{2}{\beta}\end{bmatrix}$ (26) $\displaystyle y$
$\displaystyle=$ $\displaystyle\begin{bmatrix}1\\\ \tan{\gamma}\\\
\tan^{2}{\gamma}\end{bmatrix}.$ (27)

Since the prefactor is strictly positive, the requirement for the potential to
be bounded from below is that $\mathbf{x}^{\intercal}\mathbf{A}\mathbf{y}$ be
positive. This term can be written as a quadratic in $\tan^{2}{\beta}$ with
factors themselves quadratics in $\tan^{2}{\gamma}$, or vice-versa. Requiring
these expressions to be positive leads to the following constraints:

$\displaystyle\lambda_{H}$ $\displaystyle>$ $\displaystyle 0$ (28)
$\displaystyle\lambda_{\varphi}$ $\displaystyle>$ $\displaystyle 0~$ (29)
$\displaystyle\lambda_{S}$ $\displaystyle>$ $\displaystyle 0~$ (30)
$\displaystyle\lambda_{4}$ $\displaystyle>$
$\displaystyle-\sqrt{\lambda_{H}\lambda_{\varphi}}~$ (31)
$\displaystyle\lambda_{\varphi\varphi SS}$ $\displaystyle>$
$\displaystyle-\sqrt{\lambda_{\varphi}\lambda_{S}}.$ (32)

###  II.3 Constraints from Consistency of the Potential

With the sign conventions in our potential, for the $H$ and $\varphi$ fields
to obtain a vev and for $S$ to not obtain a vev we require $\mu_{H}^{2}>0$,
$\mu_{\varphi}^{2}>0$, and $\mu_{S}^{2}>0$. This leads to the following three
constraints:

$\displaystyle\mu_{H}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{H}v^{2}+\lambda_{4}w^{2}>0$ (33)
$\displaystyle\mu_{\varphi}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{\varphi}w^{2}+\lambda_{4}v^{2}>0$ (34)
$\displaystyle\mu^{2}_{S}$ $\displaystyle=$ $\displaystyle
m_{S}^{2}-\lambda_{\varphi\varphi SS}w^{2}>0.$ (35)

Imposing these constraints gives the only consistent set of parameters with a
DM candidate. Under these conditions, the potential and minima are unique.

##  III Parameter Scan and Relic Abundance

The model has eight independent parameters. At the Lagrangian level, these
parameters are $\lambda_{H}$, $\lambda_{\varphi}$, $\lambda_{4}$,
$\lambda_{S}$, $\lambda_{\varphi\varphi SS}$, $\mu_{H}$, $\mu_{\varphi}$, and
$\mu_{S}$. However, it is more transparent to use more physical parameters. We
take these to be $m_{h_{1}}$, $m_{h_{2}}$, $m_{S}$, the $h$-$\varphi$ mixing
angle $\alpha$, the two vacuum expectation values $v$ and $w$, and retain the
Lagrangian parameters $\lambda_{\varphi\varphi SS}$ and $\lambda_{S}$. The
relationship between these and the Lagrangian parameters was given by Eqns. 9,
10, 11, 33, 34 and 35.

We identify $v$ with the SM Higgs vacuum expectation value and $m_{h_{1}}$
with the observed 125 GeV scalar mass, leaving six parameters. Of these,
$\lambda_{S}$ is constrained by dark matter self-interaction and Eqns. 17 and
18. When these two constraints are not mutually exclusive, $\lambda_{S}$ can
be set to an arbitrary value that satisfies these constraints without
impacting any other quantity of interest. $\lambda_{\varphi\varphi SS}$
directly influences the dark matter annihilation cross section, and we fix its
value to give agreement with the measured relic abundance after all other
parameters have been fixed. This leaves $m_{h_{2}}$, $m_{S}$, $\alpha$, and
$w$ as free input parameters.

Our procedure is to randomly choose values for $m_{h_{2}}$, $m_{S}$, $\alpha$,
and $w$. We can limit the allowed range on $\alpha$ using the measured Higgs
boson signal strengths to constrain $|\cos\alpha|\gtrsim 0.97$. This will be
checked later by comparing the calculated and measured signal strengths. We
typically scan the four parameters by randomly varying $w$ and $m_{S}$ from
$1~{}\text{GeV}$ to $1~{}\text{TeV}$, $m_{h_{2}}$ from $100~{}\text{GeV}$ to
$1~{}\text{TeV}$, and $\alpha$ from $0.969<|\cos\alpha|<1.0$. We take
$\lambda_{S}=0.2$. We note that we find no qualitative difference in our
results or conclusions by increasing the scan range for $m_{S}$, $m_{h_{2}}$,
and $w$ to larger values so that scanning to 1 TeV is sufficient to reveal the
characteristics we are exploring.

We then check the resulting Lagrangian parameters against the relevant
theoretical constraints. For the parameter sets that pass this test, we use
the micrOMEGAs program Belanger:2018ccd  to search for values of
$\lambda_{\varphi\varphi SS}$ that agree with the measured value for the relic
abundance of $\Omega_{\text{DM}}=0.1200(12)\;h^{-2}$ Tanabashi:2018oca . We
then check the Lagrangian parameters against the remaining theoretical
constraints. For those that pass this test, we calculate and compare to
experimental measurements the Higgs boson invisible branching ratio, the Higgs
boson signal strength, and the dark matter self-interaction cross section. For
those parameter points that pass all these constraints, we calculate the
indirect detection cross sections for all possible final states and the direct
detection cross section using micrOMEGAs Belanger:2018ccd . The goal is to see
if parameter points that pass all these theoretical and experimental tests are
either ruled out or allowed by current limits on direct and indirect detection
cross section measurements.

In the following subsections, we describe the details of how we do these
calculations.

###  III.1 Fitting $\lambda_{\varphi\varphi SS}$ with the Relic Abundance

Figure 1: Dark matter relic abundance as a function of
$\lambda_{\varphi\varphi SS}$ for $\alpha=0.2$, $m_{h_{2}}=200~{}\text{GeV}$,
$m_{S}=300~{}\text{GeV}$, and for the fixed values of $w$ given in the legend.
The dashed line is for the measured value of
$\Omega_{\text{DM}}=0.1200(12)\;h^{-2}$ Tanabashi:2018oca .

We calculate the relic abundance and other DM properties using the micrOMEGAs
program Belanger:2018ccd . For each set of input parameters, we perform a
search by varying $\lambda_{\varphi\varphi SS}$ until we obtain agreement
between the calculated value for $\Omega_{\text{DM}}$ and the measured value.
However, when $m_{S}\gtrsim m_{h2}$, the relic abundance is no longer a
monotonic function of $\lambda_{\varphi\varphi SS}$, which complicates the
search and can lead to up to three solutions. For these cases, the relic
abundance starts by decreasing with increasing $\lambda_{\varphi\varphi SS}$
but then increases again due to a cancellation in the DM annihilation cross
sections. This is illustrated in Fig. 1.

The cancellation is due to interference between the diagrams contributing to
the $SS\rightarrow h_{2}h_{2}$ cross section that, for small $h_{1}$-$h_{2}$
mixing angles, occurs at $\lambda_{\varphi\varphi SS}\approx
m_{S}^{2}/2w^{2}$. This is a consequence of the Feynman rules for the various
vertices entering these matrix elements; the details are presented in Appendix
A. As is well known, when the annihilation cross section decreases, the relic
abundance increases due to earlier freeze-out. For finite values of the mixing
angle, this effect is also present in the $h_{1}h_{2}$ and $h_{1}h_{1}$ final
states, although it occurs at different values of $\lambda_{\varphi\varphi
SS}$ for each channel; this can be seen in Fig. 2. While the $h_{2}h_{2}$
final state generally dominates because the $h_{1}h_{2}$ and $h_{1}h_{1}$ are
suppressed by factors of $\sin\alpha$ and $\sin^{2}\alpha$ respectively, all
channels contribute to the relic abundance so that there is no simple formula
for the location of the maximum in $\Omega_{\text{DM}}$. As a consequence, we
use the small mixing angle formula given above to approximate the position of
the maxima. While the value of $m_{S}$ only affects the amplitude of the
maxima, $\alpha$ does influence their position, so this formula is not very
accurate for large values of $\alpha$. Nonetheless, the formula is an adequate
approximation for the local maximum in $\Omega_{\text{DM}}$ for the purposes
of searching for the values of $\lambda_{\varphi\varphi SS}$ that give the
correct relic abundance value $\Omega_{\text{DM}}=0.1200(12)\;h^{-2}$
Tanabashi:2018oca .

Figure 2: The dark matter annihilation cross section to scalar channels as a
function of $\lambda_{\varphi\varphi SS}$ for $\alpha=0.2$,
$w=300~{}\text{GeV}$, $m_{h_{2}}=200~{}\text{GeV}$, and
$m_{S}=300~{}\text{GeV}$, for a center of mass energy of $125~{}\text{GeV}$.
The value of $\lambda_{\varphi\varphi SS}$ where the cross section is a
minimum is different for each channel.

In general, as pointed out above, there can be up to three values of
$\lambda_{\varphi\varphi SS}$ that give $\Omega_{\text{DM}}=0.12\;h^{-2}$. We
must therefore take some care in our search so that we do not miss one of
these solutions. For $m_{S}<200$ GeV, the maximum is not high enough to yield
additional solutions for $\lambda_{\varphi\varphi SS}$. It is therefore
sufficient to perform a simple search procedure starting at
$\lambda_{\varphi\varphi SS}=0$. From this starting point, we increase
$\lambda_{\varphi\varphi SS}$ in small increments until $\Omega_{\text{DM}}$
falls below $0.12\;h^{-2}$, after which we perform a binary search between the
last two values of $\lambda_{\varphi\varphi SS}$ until we find a value of
$\lambda_{\varphi\varphi SS}$ that yields $\Omega_{\text{DM}}=0.12\;h^{-2}$.
If this does not occur before $\lambda_{\varphi\varphi SS}$ reaches $4\pi$,
the scan is aborted.

For larger values of $m_{S}$, we determine the position of the maximum in
$\Omega_{\text{DM}}$ using $\lambda_{\varphi\varphi
SS}^{\text{max}}=m_{S}^{2}/2w^{2}$. If $\Omega_{\text{DM}}<0.12\;h^{-2}$ for
$\lambda_{\varphi\varphi SS}^{\text{max}}$, there are no additional solutions
due to the maximum, and we follow the procedure described above starting at
$\lambda_{\varphi\varphi SS}=0$ to determine the unique solution, if it
exists.

If $\Omega_{\text{DM}}>0.12\;h^{-2}$, we follow the procedure described above
starting at $\lambda_{\varphi\varphi SS}^{\text{max}}$ to find a solution to
the right of the maximum. We repeat this procedure, this time decreasing
$\lambda_{\varphi\varphi SS}$ from $\lambda_{\varphi\varphi SS}^{\text{max}}$
to find a solution to the left of the maximum. If one is found, the procedure
is repeated starting from $\lambda_{\varphi\varphi SS}=0$ to find the final
solution.

This yields a list of points in the parameter space that give the correct
relic abundance. We then check to see that the values of
$\lambda_{\varphi\varphi SS}$ satisfy the remaining theoretical constraints
given by Eqns. 17,18, 32 and 35.

Once we have a set of parameters that give the correct relic abundance and
satisfy the theoretical constraints, we test them against the experimental
constraints.

###  III.2 Constraints from the Higgs Invisible Width

The current limits on the invisible width of the $H^{0}$ boson at 125 GeV is
$\text{BR}_{\text{invis}}<0.26$ at 95% C.L. (ATLAS Aaboud:2019rtt ) and
$\text{BR}_{\text{invis}}<0.19$ at 95% C.L. (CMS Sirunyan:2018owy ). We use
the less constraining limit of $\text{BR}_{\text{invis}}<0.26$ but this has
very little effect on our results. Identifying $h_{1}$ with the $H^{0}$, the
$h_{1}$ invisible BR is given by

$\text{BR}_{\text{inv}}=\frac{\Gamma_{\text{inv}}}{\Gamma_{\text{total}}}=\frac{\Gamma_{\text{inv}}}{\Gamma_{\text{SM}}\cos^{2}\alpha+\Gamma_{\text{inv}}}$
(36)

where $\Gamma_{\text{SM}}=4.07$ GeV Tanabashi:2018oca  (see also HDECAY
Djouadi:1997yw ), which is modified by the $h_{1}$-$h_{2}$ mixing,
$\cos\alpha$. The $h_{1}SS$ vertex is $2i\lambda_{\varphi\varphi
SS}w\sin{\alpha}$, so that the invisible width is given by

$\Gamma_{\text{inv}}=\frac{\lambda_{\varphi\varphi
SS}^{2}w^{2}\sin^{2}{\alpha}}{8\pi
m_{h_{1}}}\sqrt{1-4\frac{m_{S}^{2}}{m_{h_{1}}^{2}}}.$ (37)

This constraint eliminates points for $m_{S}\lesssim m_{h_{1}}/2$, where the
kinematically allowed decay $h_{1}\to SS$ results in a large
$\Gamma_{\text{inv}}$.

###  III.3 Constraints from the Higgs Signal Strength

The Higgs signal strength $\mu$ is given by

$\mu=\sum_{i}c_{i}\omega_{i},$ (38)

where the sum runs over all channels, and where the channel signal strength
$c_{i}$ and the SM channel weight $\omega_{i}$ are given by

$\displaystyle c_{i}$ $\displaystyle=$
$\displaystyle\frac{\left[\sigma\times\text{BR}\right]_{i}}{\left[\sigma_{\text{SM}}\times\text{BR}_{\text{SM}}\right]_{i}}$
(39) $\displaystyle\omega_{i}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{i}\left[\sigma_{\text{SM}}\times\text{BR}_{\text{SM}}\right]_{i}}{\sum_{j}\epsilon_{j}\left[\sigma_{\text{SM}}\times\text{BR}_{\text{SM}}\right]_{j}}$
(40)

for channel $i$ with cross section $\sigma$ ($\sigma_{\text{SM}}$) and
branching ratio BR ($\text{BR}_{\text{SM}}$) in the BSM (SM) model and
$\epsilon_{i}$ the experimental efficiency for that channel Bechtle:2013xfa .
For the Standard Model, the Higgs signal strength parameter is $\mu=1$. The
current PDG quoted average for the signal strength is $\mu=1.13\pm.06$
Tanabashi:2018oca . In our model, $\mu\leq 1$. As such, relative to this best
fit point, the 95% C.L. limit is $\mu>0.94$.

For our model, all production channels are modified equally by the
$h_{1}$-$h_{2}$ mixing, $\cos\alpha$. This leads to a factor of
$\sigma_{i}/\sigma_{\text{SM}i}=\cos^{2}{\alpha}$ for the production channels.
The decay channels are slightly different as one needs to include the
modification of the invisible width in the total width, so that

$\frac{\text{BR}_{i}}{\text{BR}_{\text{SM}i}}=\frac{\Gamma_{\text{SM}}}{\Gamma_{\text{SM}i}}\frac{\Gamma_{i}}{\Gamma}=\cos^{2}{\alpha}\frac{\Gamma_{\text{SM}}}{\Gamma}$
(41)

where $\Gamma=\Gamma_{\text{SM}}\cos^{2}\alpha+\Gamma_{\text{inv}}$. Putting
it together we obtain

$\mu=\cos^{4}{\alpha}\frac{\Gamma_{\text{SM}}}{\Gamma},$ (42)

which can be used to apply the constraint $\mu>0.94$. This constraint
eliminates parameter points for which $\cos\alpha\lesssim 0.97$, as
anticipated. Additionally, when $h_{1}$ is kinematically allowed to decay to
$SS$, the $h_{1}$ width is significantly larger than the Standard Model value
so that the signal strength is altered, also eliminating parameter points.

###  III.4 Constraints from Dark Matter Self-Interaction

At tree level, the strength of dark matter self-interaction is determined by
$\lambda_{S}$ from the quartic coupling and $\lambda_{\varphi\varphi SS}$ from
$t$-channel and $s$-channel processes. Once $\lambda_{\varphi\varphi SS}$ is
set by the relic abundance, we compare the predicted self-interaction cross
section to current limits on $\sigma_{\text{DM}}$. Constraints from the Bullet
Cluster give a limit of $\sigma_{\text{SIDM}}/m_{S}<1.25$ cm2/g
Robertson:2016xjh , while constraints from colliding galaxies clusters give
$\sigma_{\text{DM}}/m<0.47$ cm2/g (95% CL) Harvey:2015hha . We use the tighter
constraint of $\sigma_{\text{SIDM}}/m_{S}<0.47$ cm2/g $\approx 2.2\times
10^{3}$ GeV-3. However, $\sigma_{\text{SIDM}}$ only constrains $\lambda_{S}$,
and we have chosen a value that avoids this limit.

###  III.5 Constraints from Indirect Detection

Dwarf spheroidal satellite galaxies (dSphs) are typically DM dominated, and so
are a good place to study dark matter. We calculated cross-sections for our
model using the micrOMEGAs program Belanger:2018ccd  which outputs
$\sigma_{\text{ID}}v$ at rest. We compared our results to a global analysis by
Hoof et al Hoof:2018hyn  of DM signals from 27 dwarf spheroidal galaxies using
11 years of observations by Fermi-LAT Atwood:2009ez .

In Fig. 3, we show our results along with the Fermi-LAT limits for the
$b\bar{b}$, $\tau^{+}\tau^{-}$, and $W^{+}W^{-}$ final states. Because
$\sigma_{\text{ID}}v$ is evaluated at threshold, the lower bound is dictated
by the kinematic threshold and each plot has a different lower bound. Below
$m_{S}\approx m_{h_{1}}/2$, the cross sections are relatively insensitive to
$m_{S}$. In this region, the $h_{1,2}SS$ vertices are proportional to
$\lambda_{\varphi\varphi SS}w$, which appears in both $\sigma_{\text{ID}}$ and
$\langle\sigma v\rangle$ (which feeds into the relic abundance via the
Boltzmann equation Kolb:1990vq ). As a consequence, any change in $w$ leads to
a corresponding change in the value for $\lambda_{\varphi\varphi SS}$ to give
the correct relic abundance so that the product $\lambda_{\varphi\varphi SS}w$
remains constant for a given value of $m_{S}$. In any case, the points for
$m_{S}\lesssim m_{h_{1}}/2$ are almost always ruled out by
$\text{BR}_{\text{inv}}$ when the decay $h_{1}\to SS$ is kinematically allowed
because of the resulting large $\Gamma_{\text{inv}}$. The dip at $m_{S}\approx
m_{h_{1}}/2$ is due to the Higgs resonance in the annihilation cross section
entering in the calculation of the relic abundance, forcing
$\lambda_{\varphi\varphi SS}$ to be small to give the correct relic abundance
and resulting in a dip in the indirect detection cross section.

While the indirect detection limits do reject some parameter points for
$b\bar{b}$ and $\tau^{-}\tau^{+}$ final states, most of these were already
rejected by previous constraints. Only a few points are rejected for the
$W^{+}W^{-}$ final state, but modest improvements in experimental sensitivity
will start ruling out regions of the parameter space allowed by other
constraints.

Figure 3: Product of dark matter annihilation cross section and velocity at
$v\approx 0$ as a function of the mass of the dark matter candidate $S$ for
the $b\bar{b}$, $\tau^{+}\tau^{-}$, and $W^{+}W^{-}$ final states for the
theoretically available points in our scan. Points labeled as “rejected" are
points that do not satisfy at least one of the invisible width, Higgs signal
strength, or self-interaction constraints.

###  III.6 Constraints from Direct Detection

Now that all the theoretical constraints and various experimental constraints
have been applied to the parameter scan, we turn to our original purpose of
confronting the surviving points with the direct detection experimental
limits. In this section we compare our parameter points to the limits from the
XENON1T experiment Aprile:2017iyp . We want to see if patterns emerge with
respect to regions in the parameter space where the direct detection cross
section is suppressed. We start with an overview of the direct detection cross
sections ($\sigma_{\text{DD}}$) for the scan of parameter points and then
examine specific characteristics of the results.

Figure 4: Feynman diagrams for the $t$-channel exchange involved in direct
detection, where $N$ is the nucleon.

In our model, the Higgs boson $t$-channel exchange from a Higgs portal is
replaced with $t$-channel exchange of the $h_{1}$ and $h_{2}$ which is shown
in Fig. 4. The direct detection cross section for scalar DM with a Higgs
portal is given by Cline:2013gha

$\sigma_{\text{DD}}={1\over{4\pi}}{{M_{N}^{2}}\over{(m_{S}+M_{N})^{2}}}{{f_{N}^{2}M_{N}^{2}}\over{v^{2}}}\left({\lambda_{hs}\over{m_{h}^{2}}}\right)^{2}$
(43)

with $m_{h}$ the Higgs boson mass, $\lambda_{hs}$ the Higgs-scalar DM
coupling, $M_{N}=938.95$ MeV the nucleon mass, and $f_{N}=0.30$ the Higgs
nucleon coupling so that $h_{1}$ and $h_{2}$ exchange results in the following
substitution

$\displaystyle\left({\lambda_{hs}\over{m_{h}^{2}}}\right)^{2}$
$\displaystyle\to$
$\displaystyle\left({{g_{{h_{1}}SS}\cos\alpha}\over{m_{h_{1}}^{2}}}+{{g_{{h_{2}}SS}\sin\alpha}\over{m_{h_{1}}^{2}}}\right)^{2}$
$\displaystyle=$ $\displaystyle
4\cos^{2}\alpha\sin^{2}\alpha\lambda_{\varphi\varphi
SS}^{2}w^{2}\left({{1}\over{m_{h_{1}}^{2}}}-{{1}\over{m_{h_{1}}^{2}}}\right)^{2},$

where we used the relations from Eqns. 49 and 50. One notes the cancellation
between the two $t$-channel exchanges and, more importantly, that the direct
detection cross section is proportional to $\lambda_{\varphi\varphi SS}^{2}$
which, as pointed out above, is fitted to give the correct relic abundance.

Fig. 5 shows the direct detection cross sections calculated using micrOMEGAs
Belanger:2018ccd  for the 8148 points of the original 10,000 points that
passed the theoretical constraints in our parameter scan. The red points were
rejected by at least one of the invisible width, Higgs signal strength, dark
matter self-interaction, or indirect detection constraints. We remind the
reader that, for a given value of $m_{S}$, we vary $m_{h_{2}}$, $\cos\alpha$,
and $w$. We fit $\lambda_{\varphi\varphi SS}$ to give the correct relic
abundance, and since $\lambda_{S}$ is mainly constrained by the self-
interaction cross section, we chose a value that passes this constraint.

In the region below $m_{S}\approx m_{h_{1}}/2$, $\sigma_{\text{DD}}$ is mainly
determined by $m_{S}$ as can be seen from Eqn. 43 with slight variations due
to the value of $\alpha$, and is largely independent of the other parameters.
This is for the same reason as with indirect detection as discussed in section
III.5: the $h_{1,2}SS$ vertices are proportional to $\lambda_{\varphi\varphi
SS}w$, which appears in both $\sigma_{\text{DD}}$ and $\langle\sigma
v\rangle$, so that any change in $w$ leads to a corresponding change in the
value for $\lambda_{\varphi\varphi SS}$ to give the correct relic abundance
and the product $\lambda_{\varphi\varphi SS}w$ remains constant for a given
value of $m_{S}$. Likewise, as also discussed in section III.5, the dip in
$\sigma_{\text{DD}}$ around $m_{S}\approx m_{h_{1}}/2$ is due to the Higgs
resonance where $\lambda_{\varphi\varphi SS}$ needs to be small to compensate
for the enhancement in the $SS$ annihilation cross section to obtain the
correct relic abundance.

Figure 5: The dark matter direct detection cross section as a function of
$m_{S}$ for the 8148 theoretically allowed points from our scan of 10,000
points. Points labeled as “rejected" are points that do not satisfy at least
one of the invisible width, Higgs signal strength, self-interaction, or
indirect detection constraints.

The region for $m_{S}\gtrsim 80$ GeV shows numerous parameter points not ruled
out by direct detection limits. There are two effects contributing to this.
The first is due to the resonance effect of the portal scalar when
$m_{S}\approx m_{h_{2}}/2$, which is analogous to the Higgs resonance effect
described above LopezHonorez:2012kv ; Arcadi:2017kky ; Azevedo:2018oxv . Near
the $h_{2}$ resonance, the $SS$ annihilation cross section increases,
requiring a smaller value for $\lambda_{\varphi\varphi SS}$ to obtain the
correct relic abundance, resulting in a small direct detection cross section.
This is illustrated in Fig. 6 which shows, in addition to the Higgs/$h_{1}$
resonance, dips in the direct detection cross section at $m_{S}=100$, 200 and
300 GeV corresponding to $m_{h_{2}}=200$, 400 and 600 GeV respectively. The
linear relationship corresponding to $m_{h_{2}}\approx 2m_{S}$ shows up
clearly as a cluster of points along the diagonal in Fig. 7, which plots the
parameter points allowed by direct detection on a plot of $m_{h_{2}}$ vs
$m_{S}$. The cluster of points in the vertical band at $m_{S}\approx 62.5$ GeV
corresponds to the Higgs resonance, and the cluster of points below the
diagonal in the bottom right portion of the plot reflects a second effect
which we discuss next. The lack of points along $m_{S}=m_{h_{2}}$ simply
reflects the fact that there are no similar effects in that region.

Figure 6: The dark matter direct detection cross section as a function of
$m_{S}$ for random theoretically allowed points with fixed values of
$m_{h_{2}}$. Points labeled as “rejected" are points that do not satisfy at
least one of the invisible width, Higgs signal strength, self-interaction, or
indirect detection constraints. Figure 7: All points allowed by invisible
width, Higgs signal strength, self-interaction, indirect detection, and direct
detection constraints plotted on the $m_{h_{2}}$-$m_{S}$ plane. The dotted
line indicates $m_{h_{2}}=2m_{S}$.

This second effect results in a big spread of the direct detection cross
section and the allowed parameter points, and is more interesting due to non-
trivial relationships between the parameters $w$ and $\lambda_{\varphi\varphi
SS}$ and how this influences the annihilation cross sections as described in
Section III. We refer to Fig. 8 to examine the details of this behavior. For
$m_{S}<m_{h_{i}}$ where $i=1$ or 2, the annihilation cross section is
dominated by $SS\to W^{+}W^{-}$ and $ZZ$, while for $m_{S}>m_{h_{i}}$ the
annihilation cross sections into $h_{1}$ and $h_{2}$ become important for
achieving the correct relic abundance. In Fig. 8, we see that the resulting
direct detection cross section drops at $m_{S}=m_{H}=125$ GeV and again at
$m_{S}=m_{h_{2}}=200$ GeV, the value used for $m_{h_{2}}$ in this figure.
These points correspond to where the $SS\to h_{i}h_{i}$ annihilation channels
open up so that a smaller value of $\lambda_{\varphi\varphi SS}$ is needed to
achieve the correct relic abundance.

When $m_{S}>m_{h_{i}}$, the direct detection cross section in Fig. 8 depends
on the value of $w$ because it is the product $w\lambda_{\varphi\varphi SS}$
that enters the expressions for the $s$-channel annihilation cross sections
for $SS\to h_{i}h_{j}$, where $i,j=1$ or 2. In this situation, as seen in Fig.
1, there can be multiple values of $\lambda_{\varphi\varphi SS}$ that give the
correct relic abundance for a given set of the free parameters, $m_{h_{2}}$,
$m_{S}$, $\alpha$, and $w$, due to the peak in $\Omega_{DM}$ at
$\lambda_{\varphi\varphi SS}\approx m_{s}^{2}/2w^{2}$. This results in the
multiple values for the direct detection cross section seen in Fig. 8.
Referring to Fig. 1, we can see that this situation only arises for
intermediate values of $w$. This is because for small values of $w$ the peak
shifts to large values of $\lambda_{\varphi\varphi SS}$ where $\Omega_{DM}$
falls below the observed value, while for large values of $w$ the calculated
value of $\Omega_{DM}$ sits above the measured value until after the peak. As
such, for small and large values of $w$, there is only one solution for
$\lambda_{\varphi\varphi SS}$. The multiple values of $\lambda_{\varphi\varphi
SS}$ for intermediate values of $w$ result in multiple values for the direct
detection cross section, although it should be noted that the additional
points with large values of $\lambda_{\varphi\varphi SS}$ are more likely to
be inconsistent with direct detection limits.

We can see how the solutions evolve with $w$ from a different perspective in
Fig. 9, where we plot $\sigma_{DD}$ versus $w$ while keeping the other
parameters fixed and as usual fitting $\lambda_{\varphi\varphi SS}$ to give
the correct relic abundance. The horizontal lines are the XENON1T limits, so
points below the lines are allowed and points above are ruled out. The regions
of parameter space at both small and large values of $w$ are allowed by the
direct detection limits. In the intermediate region, starting with small
values of $w$, there are multiple values for the direct detection cross
sections reflecting the multiple solutions for $\lambda_{\varphi\varphi SS}$
that give the correct relic abundance. In this region, some solutions give
rise to large direct detection cross sections that are ruled out by
experimental limits while others are allowed. As $w$ increases further, we
leave the region of multiple solutions and the remaining solutions are ruled
out by direct detection limits until eventually they fall below the XENON1T
limits. The size of the ruled out region depends on the cancellations of the
dark matter annihilation cross sections for the available scalar channels. In
our model, when kinematically allowed, the $SS\rightarrow h_{2}h_{2}$ channel
dominates. However, this region could be larger for cases where multiple
scalar channels are comparable in importance.

Figure 8: The dark matter direct detection cross section as a function of
$m_{S}$ for theoretically allowed points with $\alpha=0.2$,
$m_{h_{2}}=200~{}\text{GeV}$, and the fixed values of $w$ given in the legend.
Figure 9: Dark matter direct detection cross section as a function of $w$ for
theoretically available points with $\alpha=0.1$,
$m_{h_{2}}=200~{}\text{GeV}$, and the fixed values of $m_{S}$ given in the
legend. The dotted lines correspond to the XENON1T limit for each
corresponding value of $m_{S}$. For each value of $m_{S}$ shown, there is a
intermediate range of $w$ values that have no points below the direct
detection cross section limit.

###  III.7 Effect of taking $\lambda_{HHSS}\neq 0$

We end this section with some comments on the consequences of not setting
$\lambda_{HHSS}$ to zero in Eqn. 1. We chose $\lambda_{HHSS}=0$ to highlight
the interplay between parameters of the model, and altering this choice will
not affect our conclusions. Allowing $\lambda_{HHSS}\neq 0$ introduces an
additional parameter so that for this case it is a linear combination of
$\lambda_{\varphi\varphi SS}$ and $\lambda_{HHSS}$ that is fitted to reproduce
the observed relic abundance. This gives a family of solutions for these two
Lagrangian parameters when keeping the rest of the parameters fixed. This is
illustrated in Fig. 10 where the relic abundance is plotted as a function of
$\lambda_{\varphi\varphi SS}$ and $\lambda_{HHSS}$ with the other parameters
fixed to the same values given in Fig. 1; $\alpha=0.2$, $m_{h_{2}}=200$ GeV,
$m_{S}=300$ GeV, with the choice $w=300$ GeV. We see that there is now a
continuum of solutions, with our choice in this paper corresponding to
solutions where $\lambda_{HHSS}=0$. Rotating from the $\lambda_{HHSS}=0$ axis
to the $\lambda_{\varphi\varphi SS}=0$ axis simply corresponds to another
choice of parameters. For the parameters used in Fig. 10, we can obtain the
correct relic abundance for a continuum of $\lambda_{HHSS}$ and
$\lambda_{\varphi\varphi SS}$ values, but the multi-valueness of $\Omega_{DM}$
is only present near the $\lambda_{HHSS}=0$ axis, so that when
$\lambda_{HHSS}\neq 0$ the prediction for the direct detection cross section
is more straightforward. Thus, taking $\lambda_{HHSS}\neq 0$ does not
qualitatively change our results but misses the subtleties and richness of the
effects that are discussed in this paper.

Figure 10:  Dark matter relic abundance as a function of
$\lambda_{\varphi\varphi SS}$ and $\lambda_{HHSS}$ for $\alpha=0.2$,
$m_{h_{2}}=200~{}\text{GeV}$, $m_{S}=300~{}\text{GeV}$, and
$w=300~{}\text{GeV}$. The grey plane is for the measured value of
$\Omega_{\text{DM}}=0.1200(12)\;h^{-2}$ Tanabashi:2018oca .

##  IV Conclusions

We studied a simple model of scalar DM with a scalar portal that can mix with
the SM Higgs. Our purpose was to explore regions of parameter space with a
suppressed direct detection cross section for a Higgs portal model. We found
that even in this simple model there remains significant regions of parameter
space that are not ruled out by direct detection measurements, with many
points lying below the neutrino floor. Three of the mechanisms leading to
these regions have been discussed previously; a small Higgs-portal mixing
angle leading to a small coupling with the DM, the Higgs resonance effect
which requires a small DM-portal coupling to compensate for the enhanced DM
annihilation cross section due to the Higgs resonance, and the similar effect
as a result of the portal resonance.

An additional effect is the result of a heavy DM particle with a lighter
portal. This opens up new DM annihilation channels so that the parameters
controlling this annihilation need to compensate, resulting in a smaller
direct detection cross section. For certain regions of the parameter space,
destructive interference between diagrams leads to multiple solutions for the
DM-portal couplings, resulting in a spread of allowed parameter points. We
therefore find, contrary to common lore, that even in a very simple model of
DM there are sizeable regions of parameter space that are still allowed by
direct detection limits.

###### Acknowledgements.

The authors thank Thomas Gregoire, Heather Logan, Alex Poulin and Daniel
Stolarski for helpful conversations. This work was supported by the Natural
Sciences and Engineering Research Council of Canada under grant number
SAPIN-2016-00041.

##  Appendix A Interference in the dark matter annihilation amplitude

The model presented in section II features a dark matter candidate $S$ which
couples only to the other two physical scalars $h_{1}$ and $h_{2}$, giving
three annihilation channels: $SS\rightarrow h_{1}h_{1}$, $SS\rightarrow
h_{1}h_{2}$, and $SS\rightarrow h_{2}h_{2}$. At tree-level, each of these
channels features the five diagrams shown in Fig. 11: two $s$-channels with
$h_{1}$ and $h_{2}$ mediators, $t$\- and $u$-channels with an $S$ mediator,
and a quartic vertex.

Figure 11: Feynman diagrams contributing to the $SS\rightarrow h_{2}h_{2}$
process.

The relevant vertices are given by

$\displaystyle g_{h_{1}h_{1}h_{1}}$ $\displaystyle=$ $\displaystyle
6i(\lambda_{H}vc_{\alpha}^{3}-\lambda_{4}wc_{\alpha}^{2}s_{\alpha}+\lambda_{4}vc_{\alpha}s_{\alpha}^{2}$
(45) $\displaystyle-\lambda_{\varphi}ws_{\alpha}^{3})$ $\displaystyle
g_{h_{1}h_{1}h_{2}}$ $\displaystyle=$ $\displaystyle
2i(\lambda_{4}wc_{\alpha}^{3}-\left(2\lambda_{4}-3\lambda_{H}\right)vc_{\alpha}^{2}s_{\alpha}$
(46)
$\displaystyle-\left(2\lambda_{4}-3\lambda_{\varphi}\right)wc_{\alpha}s_{\alpha}^{2}+\lambda_{4}vs_{\alpha}^{3})$
$\displaystyle g_{h_{1}h_{2}h_{2}}$ $\displaystyle=$ $\displaystyle
2i(\lambda_{4}vc_{\alpha}^{3}+\left(2\lambda_{4}-3\lambda_{\varphi}\right)wc_{\alpha}^{2}s_{\alpha}$
(47)
$\displaystyle-\left(2\lambda_{4}-3\lambda_{h}\right)vc_{\alpha}s_{\alpha}^{2}-\lambda_{4}ws_{\alpha}^{3})$
$\displaystyle g_{h_{2}h_{2}h_{2}}$ $\displaystyle=$ $\displaystyle
6i(\lambda_{\varphi}wc_{\alpha}^{3}+\lambda_{4}vc_{\alpha}^{2}s_{\alpha}+\lambda_{4}wc_{\alpha}s_{\alpha}^{2}$
(48) $\displaystyle+\lambda_{H}vs_{\alpha}^{3})$ $\displaystyle g_{h_{1}SS}$
$\displaystyle=$ $\displaystyle-2i\lambda_{\varphi\varphi SS}ws_{\alpha}$ (49)
$\displaystyle g_{h_{2}SS}$ $\displaystyle=$ $\displaystyle
2i\lambda_{\varphi\varphi SS}wc_{\alpha}$ (50) $\displaystyle
g_{h_{1}h_{1}SS}$ $\displaystyle=$ $\displaystyle 2i\lambda_{\varphi\varphi
SS}s_{\alpha}^{2}$ (51) $\displaystyle g_{h_{1}h_{2}SS}$ $\displaystyle=$
$\displaystyle-2i\lambda_{\varphi\varphi SS}c_{\alpha}s_{\alpha}$ (52)
$\displaystyle g_{h_{2}h_{2}SS}$ $\displaystyle=$ $\displaystyle
2i\lambda_{\varphi\varphi SS}c_{\alpha}^{2},$ (53)

where $c_{\alpha}=\cos{\alpha}$ and $s_{\alpha}=\sin{\alpha}$. In the limit
$\alpha\rightarrow 0$, using Eqns. 10 and 9, these vertices become

$\displaystyle g_{h_{1}h_{1}h_{1}}$ $\displaystyle=$ $\displaystyle
3i\frac{m_{h_{1}}^{2}}{v}$ (54) $\displaystyle g_{h_{2}h_{2}h_{2}}$
$\displaystyle=$ $\displaystyle 3i\frac{m_{h_{2}}^{2}}{w}$ (55) $\displaystyle
g_{h_{2}SS}$ $\displaystyle=$ $\displaystyle 2i\lambda_{\varphi\varphi SS}w$
(56) $\displaystyle g_{h_{2}h_{2}SS}$ $\displaystyle=$ $\displaystyle
2i\lambda_{\varphi\varphi SS},$ (57)

with all other couplings going to 0, effectively decoupling $h_{1}$ from the
other scalars.

Under this approximation, the amplitudes of $SS\rightarrow h_{1}h_{1}$ and
$SS\rightarrow h_{1}h_{2}$ vanish, and the amplitude of $SS\rightarrow
h_{2}h_{2}$ is given by

$\displaystyle\mathcal{M}_{SS\rightarrow h_{2}h_{2}}$ $\displaystyle=$
$\displaystyle
g_{h_{2}h_{2}SS}-\frac{ig_{h_{2}h_{2}h_{2}}g_{h_{2}SS}}{s-m_{h_{2}}^{2}}$ (58)
$\displaystyle-\frac{ig_{h_{2}SS}^{2}}{t-m_{S}^{2}}-\frac{ig_{h_{2}SS}^{2}}{u-m_{S}^{2}}$
$\displaystyle\approx$ $\displaystyle 4i\lambda_{\varphi\varphi
SS}-8i\frac{w^{2}\lambda_{\varphi\varphi SS}^{2}}{m_{S}^{2}},$ (59)

where we used a theshold approximation to set the Mandelstam variables to
$s=\left(2m_{h_{2}}\right)^{2}$ and $t=u=0$. The resulting amplitude is zero
at both $\lambda_{\varphi\varphi SS}=0$ and $\lambda_{\varphi\varphi
SS}=m_{S}^{2}/2w^{2}$.

## References

  * (1) L. Bergström, Nonbaryonic dark matter: Observational evidence and detection methods, Rept. Prog. Phys. 63, 793 (2000) doi:10.1088/0034-4885/63/5/2r3 [arXiv:hep-ph/0002126 [hep-ph]].
  * (2) G. Bertone, D. Hooper and J. Silk, Particle dark matter: Evidence, candidates and constraints, Phys. Rept. 405, 279-390 (2005) doi:10.1016/j.physrep.2004.08.031 [arXiv:hep-ph/0404175 [hep-ph]].
  * (3) L. Bergstrom, Dark Matter Evidence, Particle Physics Candidates and Detection Methods, Annalen Phys. 524, 479-496 (2012) doi:10.1002/andp.201200116 [arXiv:1205.4882 [astro-ph.HE]].
  * (4) V. Silveira and A. Zee, SCALAR PHANTOMS, Phys. Lett. B 161, 136-140 (1985) doi:10.1016/0370-2693(85)90624-0
  * (5) J. McDonald, Gauge singlet scalars as cold dark matter, Phys. Rev. D 50, 3637-3649 (1994) doi:10.1103/PhysRevD.50.3637 [arXiv:hep-ph/0702143 [hep-ph]].
  * (6) B. Patt and F. Wilczek, Higgs-field portal into hidden sectors, [arXiv:hep-ph/0605188 [hep-ph]].
  * (7) S. Baek, P. Ko and W. I. Park, Search for the Higgs portal to a singlet fermionic dark matter at the LHC, JHEP 02, 047 (2012) doi:10.1007/JHEP02(2012)047 [arXiv:1112.1847 [hep-ph]].
  * (8) A. Djouadi, O. Lebedev, Y. Mambrini and J. Quevillon, Implications of LHC searches for Higgs–portal dark matter, Phys. Lett. B 709, 65-69 (2012) doi:10.1016/j.physletb.2012.01.062 [arXiv:1112.3299 [hep-ph]].
  * (9) L. Lopez-Honorez, T. Schwetz and J. Zupan, Higgs portal, fermionic dark matter, and a Standard Model like Higgs at 125 GeV, Phys. Lett. B 716, 179-185 (2012) doi:10.1016/j.physletb.2012.07.017 [arXiv:1203.2064 [hep-ph]].
  * (10) S. Baek, P. Ko, W. I. Park and E. Senaha, Higgs Portal Vector Dark Matter : Revisited, JHEP 05, 036 (2013) doi:10.1007/JHEP05(2013)036 [arXiv:1212.2131 [hep-ph]].
  * (11) D. G. E. Walker, Unitarity Constraints on Higgs Portals, [arXiv:1310.1083 [hep-ph]].
  * (12) S. Esch, M. Klasen and C. E. Yaguna, A minimal model for two-component dark matter, JHEP 09, 108 (2014) doi:10.1007/JHEP09(2014)108 [arXiv:1406.0617 [hep-ph]].
  * (13) O. Buchmueller, M. J. Dolan, S. A. Malik and C. McCabe, Characterising dark matter searches at colliders and direct detection experiments: Vector mediators, JHEP 01, 037 (2015) doi:10.1007/JHEP01(2015)037 [arXiv:1407.8257 [hep-ph]].
  * (14) K. Cheung, P. Ko, J. S. Lee and P. Y. Tseng, Bounds on Higgs-Portal models from the LHC Higgs data, JHEP 10, 057 (2015) doi:10.1007/JHEP10(2015)057 [arXiv:1507.06158 [hep-ph]].
  * (15) A. Beniwal, F. Rajec, C. Savage, P. Scott, C. Weniger, M. White and A. G. Williams, Combined analysis of effective Higgs portal dark matter models, Phys. Rev. D 93, no.11, 115016 (2016) doi:10.1103/PhysRevD.93.115016 [arXiv:1512.06458 [hep-ph]].
  * (16) H. Han, J. M. Yang, Y. Zhang and S. Zheng, Collider Signatures of Higgs-portal Scalar Dark Matter, Phys. Lett. B 756, 109-112 (2016) doi:10.1016/j.physletb.2016.03.010 [arXiv:1601.06232 [hep-ph]].
  * (17) G. Arcadi, C. Gross, O. Lebedev, S. Pokorski and T. Toma, Evading Direct Dark Matter Detection in Higgs Portal Models, Phys. Lett. B 769, 129-133 (2017) doi:10.1016/j.physletb.2017.03.044 [arXiv:1611.09675 [hep-ph]].
  * (18) G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, M. Pierre, S. Profumo and F. S. Queiroz, The waning of the WIMP? A review of models, searches, and constraints, Eur. Phys. J. C 78, no.3, 203 (2018) doi:10.1140/epjc/s10052-018-5662-y [arXiv:1703.07364 [hep-ph]].
  * (19) S. Bhattacharya, P. Ghosh, T. N. Maity and T. S. Ray, Mitigating Direct Detection Bounds in Non-minimal Higgs Portal Scalar Dark Matter Models, JHEP 10, 088 (2017) doi:10.1007/JHEP10(2017)088 [arXiv:1706.04699 [hep-ph]].
  * (20) C. Gross, O. Lebedev and T. Toma, Cancellation Mechanism for Dark-Matter-Nucleon Interaction, Phys. Rev. Lett. 119, no.19, 191801 (2017) doi:10.1103/PhysRevLett.119.191801 [arXiv:1708.02253 [hep-ph]].
  * (21) D. Azevedo, M. Duch, B. Grzadkowski, D. Huang, M. Iglicki and R. Santos, Testing scalar versus vector dark matter, Phys. Rev. D 99, no.1, 015017 (2019) doi:10.1103/PhysRevD.99.015017 [arXiv:1808.01598 [hep-ph]].
  * (22) G. Arcadi, A. Djouadi and M. Raidal, Dark Matter through the Higgs portal,’ Phys. Rept. 842, 1-180 (2020) doi:10.1016/j.physrep.2019.11.003 [arXiv:1903.03616 [hep-ph]].
  * (23) M. Cabrera, J. Casas, A. Delgado and S. Robles, Generalized Blind Spots for Dark Matter Direct Detection in the 2HDM, JHEP 02, 166 (2020) doi:10.1007/JHEP02(2020)166 [arXiv:1912.01758 [hep-ph]].
  * (24) K. Ghorbani and H. Ghorbani, Scalar split WIMPs in future direct detection experiments, Phys. Rev. D 93, no.5, 055012 (2016) doi:10.1103/PhysRevD.93.055012 [arXiv:1501.00206 [hep-ph]].
  * (25) P. Ghosh, A. K. Saha and A. Sil, Study of Electroweak Vacuum Stability from Extended Higgs Portal of Dark Matter and Neutrinos, Phys. Rev. D 97, no.7, 075034 (2018) doi:10.1103/PhysRevD.97.075034 [arXiv:1706.04931 [hep-ph]].
  * (26) T. Alanne, N. Benincasa, M. Heikinheimo, K. Kannike, V. Keus, N. Koivunen and K. Tuominen, Pseudo-Goldstone dark matter: gravitational waves and direct-detection blind spots, JHEP 10, 080 (2020) doi:10.1007/JHEP10(2020)080 [arXiv:2008.09605 [hep-ph]].
  * (27) M. R. Buckley, D. Hooper and J. L. Rosner, A Leptophobic Z’ And Dark Matter From Grand Unification, Phys. Lett. B 703, 343-347 (2011) doi:10.1016/j.physletb.2011.08.014 [arXiv:1106.3583 [hep-ph]].
  * (28) M. T. Frandsen, F. Kahlhoefer, S. Sarkar and K. Schmidt-Hoberg, Direct detection of dark matter in models with a light Z’, JHEP 09, 128 (2011) doi:10.1007/JHEP09(2011)128 [arXiv:1107.2118 [hep-ph]].
  * (29) O. Lebedev, H. M. Lee and Y. Mambrini, Vector Higgs-portal dark matter and the invisible Higgs,’ Phys. Lett. B 707, 570-576 (2012) doi:10.1016/j.physletb.2012.01.029 [arXiv:1111.4482 [hep-ph]].
  * (30) A. Alves, S. Profumo and F. S. Queiroz, The dark $Z^{{}^{\prime}}$ portal: direct, indirect and collider searches,’ JHEP 04, 063 (2014) doi:10.1007/JHEP04(2014)063 [arXiv:1312.5281 [hep-ph]].
  * (31) G. Arcadi, Y. Mambrini, M. H. G. Tytgat and B. Zaldivar, Invisible $Z^{\prime}$ and dark matter: LHC vs LUX constraints, JHEP 03, 134 (2014) doi:10.1007/JHEP03(2014)134 [arXiv:1401.0221 [hep-ph]].
  * (32) O. Lebedev and Y. Mambrini, Axial dark matter: The case for an invisible $Z^{\prime}$, Phys. Lett. B 734, 350-353 (2014) doi:10.1016/j.physletb.2014.05.025 [arXiv:1403.4837 [hep-ph]].
  * (33) G. Arcadi, Y. Mambrini and F. Richard, Z-portal dark matter,’ JCAP 03, 018 (2015) doi:10.1088/1475-7516/2015/03/018 [arXiv:1411.2985 [hep-ph]].
  * (34) D. Hooper, $Z^{\prime}$ Mediated Dark Matter Models for the Galactic Center Gamma-Ray Excess, Phys. Rev. D 91, 035025 (2015) doi:10.1103/PhysRevD.91.035025 [arXiv:1411.4079 [hep-ph]].
  * (35) A. Alves, A. Berlin, S. Profumo and F. S. Queiroz, Dark Matter Complementarity and the Z′ Portal, Phys. Rev. D 92, no.8, 083004 (2015) doi:10.1103/PhysRevD.92.083004 [arXiv:1501.03490 [hep-ph]].
  * (36) K. Ghorbani and H. Ghorbani, Two-portal Dark Matter, Phys. Rev. D 91, no.12, 123541 (2015) doi:10.1103/PhysRevD.91.123541 [arXiv:1504.03610 [hep-ph]].
  * (37) T. Jacques, A. Katz, E. Morgante, D. Racco, M. Rameez and A. Riotto, Complementarity of DM searches in a consistent simplified model: the case of $Z^{\prime}$’, JHEP 10, 071 (2016) [erratum: JHEP 01, 127 (2019)] doi:10.1007/JHEP10(2016)071 [arXiv:1605.06513 [hep-ph]].
  * (38) M. Duerr, F. Kahlhoefer, K. Schmidt-Hoberg, T. Schwetz and S. Vogl, How to save the WIMP: global analysis of a dark matter model with two s-channel mediators, JHEP 09, 042 (2016) doi:10.1007/JHEP09(2016)042 [arXiv:1606.07609 [hep-ph]].
  * (39) A. Ismail, W. Y. Keung, K. H. Tsao and J. Unwin, Axial vector $Z^{\prime}$ and anomaly cancellation, Nucl. Phys. B 918, 220-244 (2017) doi:10.1016/j.nuclphysb.2017.03.001 [arXiv:1609.02188 [hep-ph]].
  * (40) M. Escudero, A. Berlin, D. Hooper and M. X. Lin, Toward (Finally!) Ruling Out Z and Higgs Mediated Dark Matter Models, JCAP 12, 029 (2016) doi:10.1088/1475-7516/2016/12/029 [arXiv:1609.09079 [hep-ph]].
  * (41) J. Kearney, N. Orlofsky and A. Pierce, $Z$ boson mediated dark matter beyond the effective theory, Phys. Rev. D 95, no.3, 035020 (2017) doi:10.1103/PhysRevD.95.035020 [arXiv:1611.05048 [hep-ph]].
  * (42) A. Alves, G. Arcadi, Y. Mambrini, S. Profumo and F. S. Queiroz, Augury of darkness: the low-mass dark $Z^{\prime}$ portal, JHEP 04, 164 (2017) doi:10.1007/JHEP04(2017)164 [arXiv:1612.07282 [hep-ph]].
  * (43) M. Dutra, M. Lindner, S. Profumo, F. S. Queiroz, W. Rodejohann and C. Siqueira, MeV Dark Matter Complementarity and the Dark Photon Portal, JCAP 03, 037 (2018) doi:10.1088/1475-7516/2018/03/037 [arXiv:1801.05447 [hep-ph]].
  * (44) S. Okada, $Z^{\prime}$ Portal Dark Matter in the Minimal $B-L$ Model, Adv. High Energy Phys. 2018, 5340935 (2018) doi:10.1155/2018/5340935 [arXiv:1803.06793 [hep-ph]].
  * (45) C. Blanco, M. Escudero, D. Hooper and S. J. Witte, Z’ mediated WIMPs: dead, dying, or soon to be detected?, JCAP 11, 024 (2019) doi:10.1088/1475-7516/2019/11/024 [arXiv:1907.05893 [hep-ph]].
  * (46) G. Arcadi, A. Djouadi and M. Kado, The Higgs-portal for vector Dark Matter and the Effective Field Theory approach: a reappraisal, Phys. Lett. B 805, 135427 (2020) doi:10.1016/j.physletb.2020.135427 [arXiv:2001.10750 [hep-ph]].
  * (47) N. Okada, S. Okada and Q. Shafi, Light $Z^{\prime}$ and dark matter from U(1)X gauge symmetry, Phys. Lett. B 810, 135845 (2020) doi:10.1016/j.physletb.2020.135845 [arXiv:2003.02667 [hep-ph]].
  * (48) A. Falkowski, J. Juknevich and J. Shelton, Dark Matter Through the Neutrino Portal, [arXiv:0908.1790 [hep-ph]].
  * (49) J. F. Cherry, A. Friedland and I. M. Shoemaker, Neutrino Portal Dark Matter: From Dwarf Galaxies to IceCube, [arXiv:1411.1071 [hep-ph]].
  * (50) B. Batell, T. Han, D. McKeen and B. Shams Es Haghi, Thermal Dark Matter Through the Dirac Neutrino Portal, Phys. Rev. D 97, no.7, 075016 (2018) doi:10.1103/PhysRevD.97.075016 [arXiv:1709.07001 [hep-ph]].
  * (51) C. Cosme, M. Dutra, T. Ma, Y. Wu and L. Yang, Neutrino Portal to FIMP Dark Matter with an Early Matter Era, JHEP 21, 026 (2020) doi:10.1007/JHEP03(2021)026 [arXiv:2003.01723 [hep-ph]].
  * (52) C. Cheung, L. J. Hall, D. Pinner and J. T. Ruderman, Prospects and Blind Spots for Neutralino Dark Matter, JHEP 05, 100 (2013) doi:10.1007/JHEP05(2013)100 [arXiv:1211.4873 [hep-ph]].
  * (53) C. Cheung and D. Sanford, Simplified Models of Mixed Dark Matter, JCAP 02, 011 (2014) doi:10.1088/1475-7516/2014/02/011 [arXiv:1311.5896 [hep-ph]].
  * (54) P. Huang and C. E. M. Wagner, Blind Spots for neutralino Dark Matter in the MSSM with an intermediate$m_{A}$, Phys. Rev. D 90, no.1, 015018 (2014) doi:10.1103/PhysRevD.90.015018 [arXiv:1404.0392 [hep-ph]].
  * (55) A. Berlin, S. Gori, T. Lin and L. T. Wang, Pseudoscalar Portal Dark Matter, Phys. Rev. D 92, 015005 (2015) doi:10.1103/PhysRevD.92.015005 [arXiv:1502.06000 [hep-ph]].
  * (56) J. A. Casas, D. G. Cerdeño, J. M. Moreno and J. Quilis, Reopening the Higgs portal for single scalar dark matter, JHEP 05, 036 (2017) doi:10.1007/JHEP05(2017)036 [arXiv:1701.08134 [hep-ph]].
  * (57) K. Huitu, N. Koivunen, O. Lebedev, S. Mondal and T. Toma, Probing pseudo-Goldstone dark matter at the LHC,’ Phys. Rev. D 100, no.1, 015009 (2019) doi:10.1103/PhysRevD.100.015009 [arXiv:1812.05952 [hep-ph]].
  * (58) V. Barger, P. Langacker, M. McCaskey, M. J. Ramsey-Musolf and G. Shaughnessy, LHC Phenomenology of an Extended Standard Model with a Real Scalar Singlet, Phys. Rev. D 77, 035005 (2008) doi:10.1103/PhysRevD.77.035005 [arXiv:0706.4311 [hep-ph]].
  * (59) A. Falkowski, C. Gross and O. Lebedev, A second Higgs from the Higgs portal, JHEP 05, 057 (2015) doi:10.1007/JHEP05(2015)057 [arXiv:1502.01361 [hep-ph]].
  * (60) P. Ko and J. Li, Interference effects of two scalar boson propagators on the LHC search for the singlet fermion DM, Phys. Lett. B 765, 53-61 (2017) doi:10.1016/j.physletb.2016.11.056 [arXiv:1610.03997 [hep-ph]].
  * (61) G. Arcadi, C. Gross, O. Lebedev, Y. Mambrini, S. Pokorski and T. Toma, Multicomponent Dark Matter from Gauge Symmetry, JHEP 12, 081 (2016) doi:10.1007/JHEP12(2016)081 [arXiv:1611.00365 [hep-ph]].
  * (62) N. F. Bell, G. Busoni and I. W. Sanderson, Self-consistent Dark Matter Simplified Models with an s-channel scalar mediator, JCAP 03, 015 (2017) doi:10.1088/1475-7516/2017/03/015 [arXiv:1612.03475 [hep-ph]].
  * (63) N. F. Bell, G. Busoni and I. W. Sanderson, Two Higgs Doublet Dark Matter Portal, JCAP 01, 015 (2018) doi:10.1088/1475-7516/2018/01/015 [arXiv:1710.10764 [hep-ph]].
  * (64) G. Arcadi, 2HDM portal for Singlet-Doublet Dark Matter, Eur. Phys. J. C 78, no.10, 864 (2018) doi:10.1140/epjc/s10052-018-6327-6 [arXiv:1804.04930 [hep-ph]].
  * (65) M. R. Buckley, D. Feld and D. Goncalves, Scalar Simplified Models for Dark Matter,’ Phys. Rev. D 91, 015017 (2015) doi:10.1103/PhysRevD.91.015017 [arXiv:1410.6497 [hep-ph]].
  * (66) J. M. No, Looking through the pseudoscalar portal into dark matter: Novel mono-Higgs and mono-Z signatures at the LHC, Phys. Rev. D 93, no.3, 031701 (2016) doi:10.1103/PhysRevD.93.031701 [arXiv:1509.01110 [hep-ph]].
  * (67) D. Goncalves, P. A. N. Machado and J. M. No, Simplified Models for Dark Matter Face their Consistent Completions, Phys. Rev. D 95, no.5, 055027 (2017) doi:10.1103/PhysRevD.95.055027 [arXiv:1611.04593 [hep-ph]].
  * (68) P. Tunney, J. M. No and M. Fairbairn, Probing the pseudoscalar portal to dark matter via $\bar{b}bZ(\to\ell\ell)+\not{E}_{T}$ : From the LHC to the Galactic Center excess, Phys. Rev. D 96, no.9, 095020 (2017) doi:10.1103/PhysRevD.96.095020 [arXiv:1705.09670 [hep-ph]].
  * (69) G. Arcadi, M. Lindner, F. S. Queiroz, W. Rodejohann and S. Vogl, Pseudoscalar Mediators: A WIMP model at the Neutrino Floor, JCAP 03, 042 (2018) doi:10.1088/1475-7516/2018/03/042 [arXiv:1711.02110 [hep-ph]].
  * (70) S. Ghosh, A. Dutta Banik, E. J. Chun and D. Majumdar, Pseudoscalar-portal Dark Matter in the Light of AMS-02 positron excess,’ [arXiv:2003.07675 [hep-ph]].
  * (71) J. M. Butterworth, M. Habedank, P. Pani and A. Vaitkus, A study of collider signatures for two Higgs doublet models with a Pseudoscalar mediator to Dark Matter,’ [arXiv:2009.02220 [hep-ph]].
  * (72) T. Abe, M. Fujiwara, J. Hisano and Y. Shoji, Future detectability of a pseudoscalar mediator dark matter model,’ J. Phys. Conf. Ser. 1468, no.1, 012012 (2020) doi:10.1088/1742-6596/1468/1/012012
  * (73) N. Okada, D. Raut and Q. Shafi, Pseudo-Goldstone Dark Matter in gauged $B-L$ extended Standard Model,’ [arXiv:2001.05910 [hep-ph]].
  * (74) Z. Zhang, C. Cai, X. M. Jiang, Y. L. Tang, Z. H. Yu and H. H. Zhang, Phase transition gravitational waves from pseudo-Nambu-Goldstone dark matter and two Higgs doublets,’ [arXiv:2102.01588 [hep-ph]].
  * (75) M. Bauer, U. Haisch and F. Kahlhoefer, Simplified dark matter models with two Higgs doublets: I. Pseudoscalar mediators,’ JHEP 05, 138 (2017) doi:10.1007/JHEP05(2017)138 [arXiv:1701.07427 [hep-ph]].
  * (76) X. M. Jiang, C. Cai, Z. H. Yu, Y. P. Zeng and H. H. Zhang, Pseudo-Nambu-Goldstone dark matter and two-Higgs-doublet models,’ Phys. Rev. D 100, no.7, 075011 (2019) doi:10.1103/PhysRevD.100.075011 [arXiv:1907.09684 [hep-ph]].
  * (77) G. Arcadi, G. Busoni, T. Hugle and V. T. Tenorth, Comparing 2HDM $+$ Scalar and Pseudoscalar Simplified Models at LHC,’ JHEP 06, 098 (2020) doi:10.1007/JHEP06(2020)098 [arXiv:2001.10540 [hep-ph]].
  * (78) A. Arhrib, R. Benbrik, M. Chabab, G. Moultaka, M. C. Peyranere, L. Rahili and J. Ramadan, The Higgs Potential in the Type II Seesaw Model, Phys. Rev. D 84, 095005 (2011) doi:10.1103/PhysRevD.84.095005 [arXiv:1105.1925 [hep-ph]].
  * (79) J. R. Ellis, K. A. Olive, Y. Santoso and V. C. Spanos, Update on the direct detection of supersymmetric dark matter, Phys. Rev. D 71, 095007 (2005) doi:10.1103/PhysRevD.71.095007 [arXiv:hep-ph/0502001 [hep-ph]].
  * (80) M. Carena, D. Hooper and A. Vallinotto, The Interplay Between Collider Searches For Supersymmetric Higgs Bosons and Direct Dark Matter Experiments, Phys. Rev. D 75, 055010 (2007) doi:10.1103/PhysRevD.75.055010 [arXiv:hep-ph/0611065 [hep-ph]].
  * (81) D. Hooper and A. M. Taylor, Determining Supersymmetric Parameters With Dark Matter Experiments, JCAP 03, 017 (2007) doi:10.1088/1475-7516/2007/03/017 [arXiv:hep-ph/0607086 [hep-ph]].
  * (82) J. Cao, K. i. Hikasa, W. Wang, J. M. Yang and L. X. Yu, Constraints of dark matter direct detection experiments on the MSSM and implications on LHC Higgs search, Phys. Rev. D 82, 051701 (2010) doi:10.1103/PhysRevD.82.051701 [arXiv:1006.4811 [hep-ph]].
  * (83) R. Campbell, S. Godfrey, H. E. Logan and A. Poulin, Real singlet scalar dark matter extension of the Georgi-Machacek model, Phys. Rev. D 95, no.1, 016005 (2017) doi:10.1103/PhysRevD.95.016005 [arXiv:1610.08097 [hep-ph]].
  * (84) G. Bélanger, F. Boudjema, A. Goudelis, A. Pukhov and B. Zaldivar, micrOMEGAs5.0 : Freeze-in, Comput. Phys. Commun. 231, 173-186 (2018) doi:10.1016/j.cpc.2018.04.027 [arXiv:1801.03509 [hep-ph]].
  * (85) P.A. Zyla et al. [Particle Data Group], Review of Particle Physics, Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * (86) M. Aaboud et al. [ATLAS], Combination of searches for invisible Higgs boson decays with the ATLAS experiment, Phys. Rev. Lett. 122, no.23, 231801 (2019) doi:10.1103/PhysRevLett.122.231801 [arXiv:1904.05105 [hep-ex]].
  * (87) A. M. Sirunyan et al. [CMS], 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, 520-551 (2019) doi:10.1016/j.physletb.2019.04.025 [arXiv:1809.05937 [hep-ex]].
  * (88) A. Djouadi, J. Kalinowski and M. Spira, HDECAY: A Program for Higgs boson decays in the standard model and its supersymmetric extension, Comput. Phys. Commun. 108, 56-74 (1998) doi:10.1016/S0010-4655(97)00123-9 [arXiv:hep-ph/9704448 [hep-ph]].
  * (89) P. Bechtle, S. Heinemeyer, O. Stål, T. Stefaniak and G. Weiglein, $HiggsSignals$: Confronting arbitrary Higgs sectors with measurements at the Tevatron and the LHC, Eur. Phys. J. C 74, no.2, 2711 (2014) doi:10.1140/epjc/s10052-013-2711-4 [arXiv:1305.1933 [hep-ph]].
  * (90) A. Robertson, R. Massey and V. Eke, What does the Bullet Cluster tell us about self-interacting dark matter?, Mon. Not. Roy. Astron. Soc. 465, no.1, 569-587 (2017) doi:10.1093/mnras/stw2670 [arXiv:1605.04307 [astro-ph.CO]].
  * (91) D. Harvey, R. Massey, T. Kitching, A. Taylor and E. Tittley, The non-gravitational interactions of dark matter in colliding galaxy clusters, Science 347, 1462-1465 (2015) doi:10.1126/science.1261381 [arXiv:1503.07675 [astro-ph.CO]].
  * (92) S. Hoof, A. Geringer-Sameth and R. Trotta, A Global Analysis of Dark Matter Signals from 27 Dwarf Spheroidal Galaxies using 11 Years of Fermi-LAT Observations, JCAP 02, 012 (2020) doi:10.1088/1475-7516/2020/02/012 [arXiv:1812.06986 [astro-ph.CO]].
  * (93) W. B. Atwood et al. [Fermi-LAT], The Large Area Telescope on the Fermi Gamma-ray Space Telescope Mission, Astrophys. J. 697, 1071-1102 (2009) doi:10.1088/0004-637X/697/2/1071 [arXiv:0902.1089 [astro-ph.IM]].
  * (94) E. W. Kolb and M. S. Turner, The Early Universe, Front. Phys. 69, 1-547 (1990)
  * (95) E. Aprile et al. [XENON], First Dark Matter Search Results from the XENON1T Experiment, Phys. Rev. Lett. 119, no.18, 181301 (2017) doi:10.1103/PhysRevLett.119.181301 [arXiv:1705.06655 [astro-ph.CO]].
  * (96) J. M. Cline, K. Kainulainen, P. Scott and C. Weniger, Update on scalar singlet dark matter, Phys. Rev. D 88, 055025 (2013) [erratum: Phys. Rev. D 92, no.3, 039906 (2015)] doi:10.1103/PhysRevD.88.055025 [arXiv:1306.4710 [hep-ph]].