P3H–23–073, TTP23–047

Light mass window of inert doublet dark matter with lepton portal interaction

Ryo Higuchi1, Syuhei Iguro2,3, Shohei Okawa4, and Yuji Omura1

1 Department of Physics, Kindai University, Higashi-Osaka, Osaka 577-8502,
Japan

2 Institute for Theoretical Particle Physics (TTP), Karlsruhe Institute of
Technology (KIT), Engesserstraße 7, 76131 Karlsruhe, Germany
3 Institute for Astroparticle Physics (IAP), Karlsruhe Institute of Technology
(KIT), Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany

4 Departament de Física Quàntica i Astrofísica, Institut de Ciències del
Cosmos (ICCUB), Universitat de Barcelona, Martí i Franquès 1, E-08028
Barcelona, Spain

We study phenomenology of a light scalar dark matter (DM). In the model, there
are an inert doublet scalar and a singlet Dirac fermion $\psi$, both charged
under a global $Z_{2}$ symmetry. The mass of the lightest inert scalar $H$ can
be lighter than 10 GeV by imposing appropriate relations between three scalar
quartic couplings. The lightest $Z_{2}$ odd particle is stable and DM. In this
paper, focusing on the parameter space where $H$ is lighter than $\psi$ and is
DM, we discuss DM physics related to relic density, direct detection, indirect
detection, collider searches and other cosmological observations. We clarify
differences from the case where $\psi$ is instead DM, which has been focused
on in the previous works.

Keywords: Light scalar dark matter, Inert scalar doublet, Lepton portal
interactions

###### Contents

  1. 1 Introduction
  2. 2 Models
    1. 2.1 Minimal model
      1. 2.1.1 Scalar sector
      2. 2.1.2 Lepton portal couplings
      3. 2.1.3 Theoretical bounds
    2. 2.2 Singlet extension
  3. 3 Dark matter physics
    1. 3.1 Relic density
    2. 3.2 Direct detection
    3. 3.3 Indirect detection
    4. 3.4 $\Delta N_{{\rm eff}}$
    5. 3.5 Self-scattering
  4. 4 Other constraints
    1. 4.1 Constraints from the LHC experiments
    2. 4.2 Constraints from the LEP experiments
    3. 4.3 Constraints from SN 1987A
  5. 5 Results
  6. 6 Summary

##  1 Introduction

Dark matter (DM) with mass below the GeV scale has drawn significant attention
in recent years, given a continuous progress in DM direct detection
experiments using nuclear recoils. The experimental results put very strong
limits on the traditional Weakly Interacting Massive Particle (WIMP) DM above
10 GeV, while their detection techniques lose the sensitivity to lighter DM
mass range. To address DM detection in this light mass region, low-mass DM
searches have been designed making use of electron recoils [1, 2, 3, 4, 5, 6,
7], cryogenic detectors [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], CCDs
[20, 21, 22, 23, 24, 25, 26] and $p$-type point contact semiconductors [27,
28], which operate with much lower detection thresholds (see also Ref. [29]
for theoretical works). Other recent attempts to detect an irreducible
component of light DM boosted due to cosmic-ray up-scatterings or solar
reflection also provide new insights into the DM interactions with nucleons
[30, 31, 32, 33, 34, 35, 36], electrons [37, 38, 39, 40] and neutrinos [41].

Sub-GeV DM, if thermally produced, calls for light new mediator particles with
mass well below the electroweak (EW) scale [42, 43]. In a class of DM models,
such light mediators have no standard model (SM) gauge interactions, arising
from gauge singlet extensions of the SM. Well-studied singlet mediators
include dark photon [44, 45, 46, 47, 48], dark Higgs [49, 50, 51, 52, 53, 54,
55, 48], axion-like particles [56, 57, 58, 59, 60, 61, 62, 63, 64, 65] and
heavy neutral leptons [47, 66, 67, 68, 69, 70]. In other DM models, light
mediators can originate in EW multiplets. For example, a light neutral scalar
$H$ comes from a scalar doublet field and plays a role of $t$-channel
[71]#1#1#1In Ref. [72], a similar $t$-channel light scalar mediator has been
also discussed, based on a supersymmetric model. and $s$-channel mediators
[73] in DM annihilation. In Ref. [74], the discovery potential of such a light
scalar in high-energy collider experiments is studied. In Ref. [75], it is
shown that a particular linear combination of neutral scalars in two extra
scalar doublets can have sufficiently light mass without any conflict with EW
precision measurements and collier search bounds. In these models, a gauge
singlet field is added as a potential DM candidate, besides the extra scalar
doublets. In Refs. [71, 74], for instance, a Dirac fermion $\psi$ is
introduced as a DM candidate and the phenomenology is studied, assuming that
$\psi$ is the lightest in the models.

In this paper, we investigate another parameter space, where $H$ is lighter
than $\psi$ and is DM, which is not studied in the previous works [71, 74]. In
Sec. 2, we summarize the model setups proposed in Refs. [71, 74] and show that
the scalar quartic couplings are predicted to be ${\cal O}(1)$ and the charged
scalar mass range is limited from a perturbativity discussion. A simple
extension of that minimal realization to a less constrained setup is also
introduced. In Sec. 3, assuming $H$ is DM, we discuss the DM physics concerned
with the thermal production, direct detection, indirect detection, self-
scattering and a constraint from the effective number of neutrinos in
cosmology. After some comments on other constraints from high-energy collider
experiments and supernova 1987A in Sec. 4, we show in Sec. 5 viable parameter
space in the models as well as make a comparison with the case of $\psi$ being
DM. Sec. 6 is devoted to summary.

##  2 Models

In this section, we recapitulate the basic idea of realizing a light neutral
scalar from the inert doublet in a minimal way, following [71]. We also show a
simple extension that resolves a perturbativity problem in the minimal model.

Fields | spin |  $SU(3)$ |  $SU(2)_{L}$ |  $U(1)_{Y}$ |  $U(1)_{L}$ |  $Z_{2}$
---|---|---|---|---|---|---
$\ell^{i}_{L}$ | $1/2$ | ${\bf 1}$ | ${\bf 2}$ | $-\frac{1}{2}$ | 1 | $+$
$e^{i}_{R}$ | $1/2$ | ${\bf 1}$ | ${\bf 1}$ | $-1$ | 1 | $+$
$\psi_{L}$ | $1/2$ | ${\bf 1}$ | ${\bf 1}$ | $0$ | 1 | $-$
$\psi_{R}$ | $1/2$ | ${\bf 1}$ | ${\bf 1}$ | $0$ | 1 | $-$
$\Phi$ | $1$ | ${\bf 1}$ | ${\bf 2}$ | $\frac{1}{2}$ | 0 | $+$
$\Phi_{\nu}$ | $1$ | ${\bf 1}$ | ${\bf 2}$ | $\frac{1}{2}$ | 0 | $-$
Table 1: Charge assignment of the relevant particles. $i=1,\,2,\,3$ denotes
the lepton flavor. $U(1)_{L}$ and $Z_{2}$ are global symmetry.

###  2.1 Minimal model

The model features a SM gauge singlet Dirac fermion $\psi$ and an extra scalar
doublet $\Phi_{\nu}$ that has the same gauge quantum number as the SM Higgs
doublet field $\Phi$. The two scalar doublets are distinguished by an
additional global $Z_{2}$ symmetry which also guarantees the stability of DM.
The global $U(1)_{L}$ symmetry is also assigned to avoid radiative generation
of the active neutrino masses. The charge assignment is summarized in Table 1.

####  2.1.1 Scalar sector

The scalar potential of the model is given by

$\displaystyle V$
$\displaystyle=m_{11}^{2}(\Phi^{\dagger}\Phi)+m_{22}^{2}(\Phi_{\nu}^{\dagger}\Phi_{\nu})+\lambda_{1}(\Phi^{\dagger}\Phi)^{2}+\lambda_{2}(\Phi_{\nu}^{\dagger}\Phi_{\nu})^{2}$
$\displaystyle\quad+\lambda_{3}(\Phi^{\dagger}\Phi)(\Phi_{\nu}^{\dagger}\Phi_{\nu})+\lambda_{4}(\Phi^{\dagger}\Phi_{\nu})(\Phi_{\nu}^{\dagger}\Phi)+\frac{1}{2}\lambda_{5}[(\Phi^{\dagger}\Phi_{\nu})^{2}+\text{h.c.}],$
(2.1)

where all parameters are chosen to be real using the phase redefinition of
$\Phi$ and $\Phi_{\nu}$. We assume in this paper that only $\Phi$ develops a
non-zero vacuum expectation value (VEV) and $\Phi_{\nu}$ does not. The scalar
doublets are then expressed in terms of physical degrees of freedom and Nambu-
Goldstone bosons $G^{0,\pm}$ as

$\displaystyle\Phi=\begin{pmatrix}G^{+}\\\
\frac{1}{\sqrt{2}}(v+h+iG^{0})\end{pmatrix},\quad\Phi_{\nu}=\begin{pmatrix}H^{+}\\\
\frac{1}{\sqrt{2}}(H+iA)\end{pmatrix},$ (2.2)

where $v\simeq 246$ GeV. After the EW symmetry breaking, the scalar masses are
expressed as

$\displaystyle m_{h}^{2}$ $\displaystyle=2\lambda_{1}v^{2},$ (2.3)
$\displaystyle m_{H^{\pm}}^{2}$
$\displaystyle=m_{22}^{2}+\frac{\lambda_{3}v^{2}}{2},$ (2.4) $\displaystyle
m_{A}^{2}$
$\displaystyle=m_{H^{\pm}}^{2}+\frac{(\lambda_{4}-\lambda_{5})v^{2}}{2},$
(2.5) $\displaystyle m_{H}^{2}$
$\displaystyle=m_{H^{\pm}}^{2}+\frac{(\lambda_{4}+\lambda_{5})v^{2}}{2}.$
(2.6)

We see that the mass splittings between the extra scalars are generated only
by the quartic couplings.

Hereafter we consider the case where $m_{H}\leq m_{A}$#2#2#2This assumption
does not lose the generality of discussion since this is just a difference of
the $\Phi_{\nu}$ basis. Rephasing $\Phi_{\nu}\to i\Phi_{\nu}$ interchanges the
role of $H$ and $A$, which does not generate any physical difference. and
derive three basic conditions for $H$ to be lighter than 10 GeV. The LEP
result restricts the charged scalar mass to be
$m_{H^{\pm}}>70\text{--}90\,{\rm GeV}$ [76] and excludes the neutral scalar
mass region satisfying all three inequalities [77]:

$m_{H}<80\,{\rm GeV},\quad m_{A}<100\,{\rm GeV},\quad m_{A}-m_{H}>8\,{\rm
GeV}.$ (2.7)

In addition, the EW precision data, especially the oblique $T$ parameter,
restricts the mass difference between $A$ and $H^{\pm}$. The upper bound is
estimated as $|m_{H^{\pm}}-m_{A}|\lesssim{\cal O}(10)\,{\rm GeV}$, which
implies $\lambda_{4}\simeq\lambda_{5}$. Thus, the spectrum in our setup is
$m_{H}\ll m_{A}\simeq m_{H^{\pm}}={\cal O}(100)\,{\rm GeV}$. In Ref. [71], it
is shown that this scalar spectrum can be achieved if the following three
equations are satisfied:

$\displaystyle\lambda_{4}+\lambda_{5}$
$\displaystyle\approx-2m_{H^{\pm}}^{2}/v^{2}\,,$ (2.8)
$\displaystyle\lambda_{4}-\lambda_{5}$ $\displaystyle\approx 0\,,$ (2.9)
$\displaystyle\lambda_{3}+\lambda_{4}+\lambda_{5}$ $\displaystyle\approx 0\,.$
(2.10)

The first equation comes from Eq. (2.6) by using $m_{H}\ll m_{H^{\pm}}$ and
holds up to ${\cal O}(m_{H}^{2}/v^{2})$ corrections. The second equation is
necessary to respect the $T$ parameter constraint. The third equation requires
the consistency with the Higgs invisible decay bound,
$\mathrm{BR}(h\to\text{inv})\leq 0.13$[78, 79], which limits the $H$-$H$-$h$
coupling $|\lambda_{3}+\lambda_{4}+\lambda_{5}|\lesssim 0.01$. Eq. (2.10)
should be fulfilled with an accuracy better than 1%, since we have

$\lambda_{3}\simeq-2\lambda_{4}\simeq-2\lambda_{5}\simeq
2m_{H^{\pm}}^{2}/v^{2}={\cal O}(1)\,,$ (2.11)

from the three equations. The values of $\lambda_{3,4,5}$ are thus fixed once
a specific value of $m_{H^{\pm}}$ is provided.

####  2.1.2 Lepton portal couplings

The new scalars also have Yukawa couplings to $\psi$ and SM lepton doublets
$\ell_{L}$,

$\displaystyle-{\cal
L}_{\ell}=y^{i}_{\nu}\,\overline{\ell_{L}^{i}}\,\widetilde{\Phi_{\nu}}\psi_{R}+\text{h.c.}\,,$
(2.12)

where $\widetilde{\Phi_{\nu}}\equiv i\sigma_{2}\Phi_{\nu}^{*}$ and the Yukawa
couplings $y^{i}_{\nu}$ $(i=e,\,\mu,\,\tau)$ are lepton flavor dependent
complex parameters. For convenience, we choose the charged lepton mass basis,
i.e. $\ell_{L}^{i}=\left[(U\nu_{L})^{i},\,e^{i}_{L}\right]^{T}$ where
$e_{L}^{i}$ is a charged lepton in the mass basis and $U$ a neutrino mixing
matrix. As discussed in Sec. 3, the lepton portal couplings are responsible
for the thermal DM production via $HH\to\nu\bar{\nu}$ annihilation.

In general, all three Yukawa couplings are non-vanishing. However, if more
than one couplings have a comparably large magnitude, sizable charged lepton
flavor violation is induced and the model will be ruled out. In this paper,
therefore, we focus on three particular flavor structures:

  1. (i)

$|y^{e}_{\nu}|\gg|y^{\mu,\tau}_{\nu}|$ (electrophilic case).

  2. (ii)

$|y^{\mu}_{\nu}|\gg|y^{e,\tau}_{\nu}|$ (muonphilic case),

  3. (iii)

$|y^{\tau}_{\nu}|\gg|y^{e,\mu}_{\nu}|$ (tauphilic case).

These structures would be obtained by e.g. assigning a corresponding $U(1)$
lepton flavor charge to $\psi$, although in this paper we do not identify its
UV origin and we just take those structures for the phenomenological purpose.
In the following, we assume $y^{i}_{\nu}$ to be real for simplicity. The
relevant results would not change even if the Yukawa couplings are complex.

####  2.1.3 Theoretical bounds

As discussed above, the scalar quartic couplings are of ${\cal O}(1)$ to
realize a light enough $H$. These large couplings lead to Landau poles at low
energy scales. In Ref. [74] the authors study perturbativity bounds by
analyzing the one-loop renormalization group evolution of the quartic
couplings, and found that the charged scalar should be lighter than 350 GeV to
respect the perturbativity of the model up to 10 TeV. Note that other
theoretical constraints, such as the bounded-from-below condition and absence
of the charge-breaking vacuum, are satisfied in parameter space of our
interest.

###  2.2 Singlet extension

One might wonder if the model presented in Sec. 2.1 (referred to as the
minimal model) can be ameliorated in a simple way for the theory to be valid
at higher scales. Indeed we can consider an extended model with a new
$Z_{2}$-odd singlet scalar $S$ so that the fine-tuning between
$\mathcal{O}(1)$ quartic couplings and the perturbativity constraint are
significantly relaxed [71].

Let us add the extra terms,

$\Delta{\cal
L}=\frac{1}{2}(\partial_{\mu}S)^{2}-\frac{1}{2}m_{S}^{2}\,S^{2}-[A_{S}\,\Phi^{\dagger}\Phi_{\nu}S+\text{h.c.}]\,,$
(2.13)

to the minimal model. After the EW symmetry breaking, $H$ and $S$ are mixed as

$\displaystyle\begin{pmatrix}H\\\
S\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}h_{2}\\\ s\end{pmatrix}\,,$
(2.14)

where $\tan(2\theta)=2A_{S}v/(m_{H}^{2}-m_{S}^{2})$ and $h_{2}$ and $s$ are
the mass eigenstates. The scalar masses are given by

$m_{h_{2}}^{2}=\frac{\cos^{2}\theta\,m_{H}^{2}-\sin^{2}\theta\,m_{S}^{2}}{\cos(2\theta)}\,,\quad
m_{s}^{2}=\frac{\cos^{2}\theta\,m_{S}^{2}-\sin^{2}\theta\,m_{H}^{2}}{\cos(2\theta)}\,.$
(2.15)

One can see that $s$ can be arbitrarily light by setting
$m_{S}\simeq\tan\theta\,m_{H}$, while $h_{2}$ (and the other scalars $H^{\pm}$
and $A$) can reside above the EW scale without any strong restriction on the
quartic couplings $\lambda_{3,4,5}$. In this extension, $s$ is DM, if lighter
than $\psi$, and couples to $\psi$ through the scalar mixing $\theta$.

The $A_{S}$ term, which generates the scalar mixing, also induces a new decay
mode $h\to ss$. This contribution to the Higgs invisible decay is constrained
by experiments, and the current LHC bound corresponds to $\theta\leq{\cal
O}(0.1)$, which will be improved to $\theta\lesssim 0.03$ in future, combining
with direct searches for the charged scalar at the high luminosity (HL)-LHC
[74]. Note that, in principle, the Higgs invisible decay bound can be
significantly relaxed by adding a new quartic interaction
$S^{2}(\Phi^{\dagger}\Phi)$, which forms another contribution to the $h\to ss$
decay, at the cost of an additional 1% level tuning between this new coupling
and $A_{S}$. We do not pursue this possibility because only adding Eq. (2.13)
can resolve all issues present in the minimal model.

In the singlet extension, the interaction of DM to $\psi$ and neutrinos is
suppressed by a factor of $\theta$. This means that the DM annihilation into
neutrinos, which is responsible for the DM thermal production, is suppressed.
The result concerning the DM production is easily translated from that of the
minimal model by making $y_{\nu}^{i}\to y_{\nu}^{i}\sin\theta$ and $m_{H}\to
m_{s}$ replacement. Other DM constraints, such as those arising from the
direct detection, are very weak in the singlet extension, mainly because all
relevant processes are suppressed at least by a factor of $\theta^{2}$ at the
amplitude level. Therefore, in the rest of this paper, we only focus on the
minimal model unless otherwise noticed.

##  3 Dark matter physics

$H$ and $\psi$ are good DM candidates. In Refs. [71, 74], it is assumed that
$\psi$ is lighter than $H$ and is DM. Here, we study the DM physics in the
opposite case, i.e. $H$ is lighter than $\psi$ and is DM.

###  3.1 Relic density

We assume that DM is thermally produced, so that the DM abundance is
determined by its annihilation cross section. Our DM $H$ annihilates into
charged leptons, pions and photons in the sub-GeV region with its coupling to
the SM Higgs boson, namely the Higgs portal coupling,

${\cal L}\supset-\frac{\lambda_{345}v}{2}H^{2}h\,,$ (3.1)

where we define

$\lambda_{345}\equiv\lambda_{3}+\lambda_{4}+\lambda_{5}\,.$ (3.2)

It is known that the size of $\lambda_{345}$ required by the thermal DM
production is too large to be consistent with the Higgs invisible decay bound.
Thus, we do not consider this negligible contribution from $\lambda_{345}$ in
the DM production.

DM also annihilates into neutrinos with the lepton portal couplings
$y_{\nu}^{i}$. The cross section is given by [80, 81, 82]

$(\sigma v_{{\rm
rel}})_{HH\to\nu_{i}\bar{\nu}_{i}}\simeq\frac{(y_{\nu}^{i}/\sqrt{2})^{4}m_{H}^{6}}{60\pi(m_{H}^{2}+m_{\psi}^{2})^{4}}v_{{\rm
rel}}^{4}\,,$ (3.3)

where we ignore neutrino masses and show the leading term in the expansion of
the DM relative velocity $v_{{\rm rel}}$. Since the annihilation is $d$-wave
dominant, the coupling value required for the thermal production tends to be
larger than the fermion DM case [71, 74]. In our analysis, the DM abundance is
numerically calculated by employing micrOMEGAs_5_2_13 [83], which implements
calculations of all relevant coannihilation processes.

###  3.2 Direct detection

In this model, the lepton portal couplings do not generate measurable elastic
DM-nucleon scattering, since the first non-vanishing contribution arises from
two-loop diagrams. Instead, the Higgs portal coupling $\lambda_{345}$ (and
loop diagrams with the weak gauge couplings and scalar quartic couplings) can
make the sizable contribution.

When the contribution from the lepton portal couplings is ignored, the DM-
nucleon scattering in this model is induced the same way as in the inert
doublet DM model [84, 85, 86]. At the tree level, the Higgs portal process is
the only relevant contribution to the scattering. It is shown in Ref. [87]
that there are also irreducible loop contributions to the scattering, which
are non-vanishing even when $\lambda_{345}=0$ and can alter the tree-level
prediction. The loop corrections in the DM scattering are formally captured by
an effective shift of the Higgs portal coupling, i.e.
$\lambda_{345}\to\lambda_{345}+\delta\lambda$, where $\delta\lambda$ contains
not only one-loop vertex correction to $\lambda_{345}$ but also loop
contributions to $HH\bar{q}q$, $HHG_{\mu\nu}G^{\mu\nu}$ and quark twist-2
operators (see Eqs. (3.7) and (3.4) below).

The loop corrections $\delta\lambda$ depend on the nucleon species ($N=p,n$),
and following Ref. [87], we have#3#3#3In Ref. [87], they only show the neutron
coupling $\delta\lambda$ explicitly, which corresponds to $\delta\lambda_{n}$
in our notation, but it is straightforward to apply their results for the
proton case.

$\delta\lambda_{N}=\delta\Gamma_{h}(0)+\delta\lambda_{345}+\frac{m_{h}^{2}}{f_{N}}\sum_{q=u,d,s}\Gamma^{q}_{\rm
Box}f_{q}^{N}+\frac{2}{9}\frac{m_{h}^{2}}{f_{N}}\Gamma^{G}_{\rm
Box}f_{g}^{N}+\frac{3}{4}\frac{m_{h}^{2}}{f_{N}}\sum_{q}\left(\Gamma^{q}_{\rm
t2}+\Gamma^{\prime q}_{\rm t2}\right)\left(q^{N}(2)+\bar{q}^{N}(2)\right)\,,$
(3.4)

where the values for $f_{q}^{N}$, $q^{N}(2)$ and $\bar{q}^{N}(2)$ are shown in
Table 2, and $f_{N}$ and $f_{g}^{N}$ are defined as

$f_{N}=\frac{2}{9}+\frac{7}{9}\sum_{q=u,d,s}f_{q}^{N}\,,\quad
f_{g}^{N}=1-\sum_{q=u,d,s}f_{q}^{N}\,.$ (3.5)

$\delta\Gamma_{h}(q_{h}^{2})$ (with $q_{h}$ the Higgs four-momentum) and
$\delta\lambda_{345}$ are the vertex correction and counter term to
$\lambda_{345}$. We impose the renormalization condition as
$\delta\lambda_{345}=-\delta\Gamma_{h}(m_{h}^{2})$ so that the vertex
correction does not change the tree-level prediction for the Higgs invisible
decay: $\lambda_{345}$ in Eq. (3.7) is the renormalized value at
$q_{h}^{2}=m_{h}^{2}$. The last three terms in Eq. (3.4) are the loop
contributions, apart from the vertex correction, and we refer readers to Ref.
[87] for the definition and complete expressions, while noting that
$\delta\lambda_{N}$ depends on four free parameters $m_{H,A,{H^{\pm}}}$ and
$\lambda_{2}$, but is independent of $\lambda_{345}$. We have numerically
confirmed that the difference between $\delta\lambda_{n}$ and
$\delta\lambda_{p}$ is smaller than 1% in parameter space of our interest. We
thus focus only on the DM scattering to neutron in the following.

| $f_{u}^{n}$ | 0.0110 | $f_{u}^{p}$ | 0.0153
---|---|---|---
$f_{d}^{n}$ | 0.0273 | $f_{d}^{p}$ | 0.0191
$f_{s}^{n}$ | 0.0447 | $f_{s}^{p}$ | 0.0447
$u^{n}(2)$ 0.11 $\overline{u}^{n}(2)$ 0.034 $d^{n}(2)$ 0.22
$\overline{d}^{n}(2)$ 0.036 $s^{n}(2)$ 0.026 $\overline{s}^{n}(2)$ 0.026
$c^{n}(2)$ 0.019 $\overline{c}^{n}(2)$ 0.019 $b^{n}(2)$ 0.012
$\overline{b}^{n}(2)$ 0.012
Table 2:  Left: Nuclear matrix elements for light quarks. The values
correspond to the micrOMEGAs default. Right: The second moments of quark PDFs
for neutron, which are numerically evaluated at $\mu=m_{Z}$ [87] using the
CTEQ PDFs [88]. For proton, $u^{p}(2)=d^{n}(2)$ and $d^{p}(2)=u^{n}(2)$ and
the second moments of the other quarks and anti-quarks are the same as those
for neutron.

In Fig. 1, the black solid lines show the contours for the loop corrections
$\delta\lambda_{n}$ with $m_{H^{\pm}}=m_{A}$ and $m_{H}=10$ GeV. One can see
that $|\delta\lambda_{n}|$ is of ${\cal O}(10^{-3})$ in a large part of
parameter space and amounts to 40% of the possible maximal value of the tree-
level coupling, i.e. $|\lambda_{345}|_{\rm max}\simeq 0.01$. The loop
corrections can be dominant, depending on the value of $|\lambda_{345}|$ and
the parameter region. It may be illuminating to provide a good approximate
formula for $\delta\lambda_{n}$, which is given in the case of
$m_{H^{\pm}}=m_{A}$ by

$\displaystyle\delta\lambda_{n}$
$\displaystyle\simeq-0.00199+(1.18\,m_{H}-6.3\,m_{H^{\pm}}-4.46\times
10^{-3}\,m_{H^{\pm}}^{2})\times 10^{-6}$
$\displaystyle\quad+\lambda_{2}\left(0.00164+2.57/m_{H^{\pm}}^{2}+5.76\times
10^{3}/m_{H^{\pm}}^{4}\right)\,,$ (3.6)

where all masses are in the GeV unit.#4#4#4Our formula is different from the
one presented in Ref. [87] mainly because we consider a different mass region.
The formula in that paper predicts a non-vanishing value for
$\delta\lambda_{n}$ when taking $m_{H}\to 0$ and $\lambda_{2}\to 0$ limit, but
we figure out that $\delta\lambda_{n}$ is non-vanishing in that limit. We have
tried several different fitting functions for $\delta\lambda_{n}$ and found
that Eq. (3.6) better reproduces the numerical results in the parameter space
of our interest. The $\lambda_{2}$ independent and dependent terms each agree
with the numerical results with a precision better than 1$\%$, in the range of
$0.1\,{\rm GeV}<m_{H}<10\,{\rm GeV}$ and $100\,{\rm GeV}<m_{H^{\pm}}<350\,{\rm
GeV}$ and $0<\lambda_{2}<2$.#5#5#5$\lambda_{2}>0$ is required by the bounded-
from-below condition. The first line of the equation is always negative since
$m_{H}\ll m_{H^{\pm}}$, while the second line positive. The black dashed lines
in Fig. 1 represent the contours from the approximate formula Eq. (3.6), which
agree well with the numerical results evaluated using the full expression Eq.
(3.4).

Figure 1:  Contour lines for the loop corrections $\delta\lambda_{n}$ in unit
of $10^{-3}$, which are evaluated using the full expression Eq. (3.4) (solid)
and approximate formula Eq. (3.6) (dashed).

The spin-independent (SI) DM-neutron cross section is given by [87]

$\sigma_{\rm
SI}=\frac{|\lambda_{345}+\delta\lambda_{n}|^{2}m_{n}^{4}f_{n}^{2}}{4\pi(m_{H}+m_{n})^{2}m_{h}^{4}}\,,$
(3.7)

where $m_{n}$ is the neutron mass. The values of $\lambda_{345}$ are free in
our analysis as far as they satisfy the Higgs invisible decay
bound.#6#6#6$|\lambda_{345}|$ is fixed by the DM relic abundance in Ref.[87].
For reference, we show an estimate for $\sigma_{\rm SI}$,

$\sigma_{\rm SI}\simeq\frac{10^{-42}\,{\rm
cm}^{2}}{(1+m_{H}/m_{n})^{2}}\left(\frac{|\lambda_{345}+\delta\lambda_{n}|}{0.01}\right)^{2}\,.$
(3.8)

It follows from this estimate that for $m_{H}\simeq 10\,{\rm GeV}$,
$\sigma_{\rm SI}$ can be as large as $10^{-44}\,{\rm cm}^{2}$, which is large
enough to be excluded by the current experimental results. As $m_{H}$ gets
lighter, $\sigma_{\rm SI}$ approaches an asymptotic value for a fixed
$|\lambda_{345}+\delta\lambda_{n}|$. On the left panel in Fig. 2, we show the
predictions for $\sigma_{\rm SI}$ with several benchmark values of
$\lambda_{345}$ and $\lambda_{2}$ and with $m_{H^{\pm}}=m_{A}=220\,{\rm GeV}$.
The black solid line corresponds to almost the maximum cross section. The
shaded region is excluded by the direct detection bounds, for which we used
the current leading constraint in each mass region from DarkSide50
($m_{\psi}\leq 5\,{\rm GeV}$) [89], PandaX-4T ($5\,{\rm GeV}\leq m_{\psi}\leq
9\,{\rm GeV}$) [90] and LZ ($9\,{\rm GeV}\leq m_{\psi}$) [91].

Figure 2:  Left: Model predictions of $\sigma_{\rm SI}$ as a function of
$m_{H}$ with $m_{H^{\pm}}=m_{A}=220\,{\rm GeV}$ and several benchmark values
of $\lambda_{345}$ and $\lambda_{2}$. The shaded region is excluded by the
current direct detection experiments. Right: The DM direct detection bounds in
the parameter space, limited by the Higgs invisible decay (gray) and the
bounded-from-below condition $\lambda_{2}>0$. The red lines correspond to
$m_{H}=5\,{\rm GeV}$ with $m_{H^{\pm}}=150\,{\rm GeV}$ (dashed), 220 GeV
(solid) and 350 GeV (dotted), respectively. The green and blue solid lines are
for $m_{H}=7\,{\rm GeV}$ and 10 GeV with $m_{H^{\pm}}=220\,{\rm GeV}$. For
each set of $m_{H}$ and $m_{H^{\pm}}$ values, the region between the two lines
is allowed.

On the right panel in Fig. 2, we show the current direct detection constraints
for $\lambda_{345}$ and $\lambda_{2}$. The region of $|\lambda_{345}|\gtrsim
0.011$ is shaded in gray and excluded by the Higgs invisible decay at 95%
confidence level (CL). The other lines (red, blue, green) correspond to the
current sensitivity of the direct detection experiments for given sets of
$m_{H}$ and $m_{H^{\pm}}$. The regions between two lines with the same color
and style are allowed by the direct detection experiments. The red lines
represent the bounds for $m_{H}=5\,{\rm GeV}$ with $m_{H^{\pm}}=150\,{\rm
GeV}$ (dashed), 220 GeV (solid) and 350 GeV (dotted), respectively. We see
that the exclusion lines do not strongly depend on $m_{H^{\pm}}$. The green
and blue solid lines are for $m_{H}=7\,{\rm GeV}$ (green) and 10 GeV (blue)
with $m_{H^{\pm}}=220\,{\rm GeV}$ fixed. The allowed region for
$m_{H}=10\,{\rm GeV}$ is much smaller than the one for $m_{H}=5\,{\rm GeV}$,
mainly because the experimental limit on $\sigma_{\rm{SI}}$ weakens rapidly
below 10 GeV. A remarkable point is that $\lambda_{345}=0$, which is the
easiest choice just to evade the Higgs invisible decay bound, is not always
allowed by the direct detection results, taking the loop contributions into
account. This suggests that the interplay between those two different searches
provides an interesting window to test this model. Before closing this
subsection, we would like to repeat that the SI cross section discussed here
is independent of the thermal DM production in our study.

###  3.3 Indirect detection

DM indirect detection experiments search for signals originated from DM
annihilations into photon, charged particles, pions and neutrinos. Since some
experimental results limit the DM annihilation cross section to be far below
the canonical thermal relic value in the sub-GeV region, it might be useful to
look at the constraints in our case.

At the tree level, we have $HH\to\nu\bar{\nu}$ via the $t$-channel $\psi$
exchange. Given that the typical DM velocity in galaxies is so small as
$v_{\rm DM}\sim 10^{-3}$, this process is strongly suppressed by the velocity
(see Eq. (3.3)) and very weakly constrained. The annihilation to the other
particles occurs from the Higgs portal coupling and loop diagrams. Let us give
a rough estimate for those cross sections below, assuming DM is lighter than 1
GeV. After all, we will see that the predicted DM annihilation cross section
is too small for current and future planned experiments and indirect detection
is irrelevant in our study. This result clearly differs from that of the
fermion DM scenario [71, 74].

Our DM annihilates into $e\bar{e},\mu\bar{\mu},\pi\pi,2\gamma$ through the
Higgs portal coupling. Given Eq. (3.1) and the low-energy Higgs couplings to
the fermions and pions,

${\cal
L}=-\frac{h}{v}\left(\sum_{f=e,\mu}m_{f}\bar{f}f+\sum_{a=1,2,3}\left[\frac{2}{9}(\partial_{\mu}\pi^{a})^{2}-\frac{5}{6}m_{\pi}^{2}(\pi^{a})^{2}\right]\right)\,,$
(3.9)

we find the cross sections,

$\displaystyle(\sigma v)_{f\bar{f}}^{\text{Higgs}}$
$\displaystyle\simeq\frac{1}{8\pi
s}\frac{|\lambda_{345}|^{2}m_{f}^{2}(s-4m_{f}^{2})}{m_{h}^{4}}\sqrt{1-\frac{4m_{\pi}^{2}}{s}}\,,$
(3.10) $\displaystyle(\sigma v)_{\pi^{a}\pi^{b}}^{\text{Higgs}}$
$\displaystyle\simeq\frac{\delta_{ab}}{16\pi
s}\frac{|\lambda_{345}|^{2}}{m_{h}^{4}}\left(\frac{2}{9}s+\frac{11}{9}m_{\pi}^{2}\right)^{2}\sqrt{1-\frac{4m_{\pi}^{2}}{s}}\,,$
(3.11) $\displaystyle(\sigma v)_{2\gamma}^{\text{Higgs}}$
$\displaystyle\sim\frac{1}{16\pi
s}\frac{|\lambda_{345}|^{2}\lambda_{h\gamma}^{2}s^{2}}{m_{h}^{4}}\,,$ (3.12)

where $a,b=1,2,3$ and $\lambda_{h\gamma}$ denotes the off-shell Higgs to
photon coupling evaluated at $p_{h}^{2}=s\ll m_{h}^{2}$ where $p_{h}$ is Higgs
four-momentum. Since $\lambda_{h\gamma}\sim\alpha/(4\pi)\ll 1$, the
annihilation into pions is the biggest. Taking the non-relativistic limit and
massless pions (i.e. $s\simeq 4m_{H}^{2}\gg 4m_{\pi}^{2}$) for reference, the
cross section is estimated as

$(\sigma v)_{\pi\pi}^{\text{Higgs}}\sim 2\times
10^{-33}\,\text{cm}^{3}/\text{sec}\times\left(\frac{|\lambda_{345}|}{10^{-2}}\right)^{2}\left(\frac{m_{H}}{350\,{\rm
MeV}}\right)^{2}\,.$ (3.13)

This is much smaller than the sensitivity of future gamma-ray telescopes – for
example, the projected 95% CL upper limits at future e-ASTROGAM experiments
are $10^{-29}\,\text{cm}^{3}/\text{sec}$ for $2\pi^{0}$ final state when DM
mass is 350 MeV (see Figure 4.5.2 in Ref. [92]). We note that this equation is
less valid when $m_{H}$ is close to $1$ GeV, because we apply the lowest order
chiral perturbation calculation for the process at $\sqrt{s}\simeq 2m_{H}$ and
higher order corrections could enhance or reduce the cross section. However,
given that the lowest order estimate Eq. (3.13) is four orders of magnitude
smaller than the future experimental reach, it is difficult to expect that our
model is tested in this process even if the cross section calculation is
refined.

Figure 3: Representative diagrams for one-loop DM annihilation.

Next, we discuss the processes induced by the loop corrections. $H$ can
annihilate into fermions and photon with $W$ and $H^{\pm}$ loop diagrams via
the weak gauge interactions (see Fig. 3). The cross sections are estimated as

$\displaystyle(\sigma v)_{f\bar{f}}^{\text{loop}}$
$\displaystyle\sim\frac{1}{8\pi
s}\left(\frac{g^{4}m_{f}}{16\pi^{2}m_{W}^{2}}\right)^{2}s\sim
10^{-34}\,\text{cm}^{3}/\text{sec}\times\left(\frac{m_{f}}{100\,{\rm
MeV}}\right)^{2},$ (3.14) $\displaystyle(\sigma v)_{2\gamma}^{\text{loop}}$
$\displaystyle\sim\frac{1}{16\pi
s}\left(\frac{g^{2}e^{2}}{16\pi^{2}m_{W}^{2}}\right)^{2}s^{2}\sim
10^{-33}\,\text{cm}^{3}/\text{sec}\times\left(\frac{m_{H}}{{\rm
GeV}}\right)^{2}\,,$ (3.15)

where the $s$-wave contribution are only kept and evaluated at $s\simeq
4m_{H}^{2}$. In Eq. (3.14), there is a helicity suppression factor, which is
understood by the fact that the fermions in the final state must form the
spin-0 state in the $s$-wave process. Clearly, both cross sections are too
small to be within reach of the future gamma-ray telescopes.

###  3.4 $\Delta N_{{\rm eff}}$

When $m_{H}\lesssim 20\,{\rm MeV}$, DM decouples from the thermal bath during
or after the neutrino decoupling, which changes the neutrino-photon
temperature ratio. This effect modifies the effective neutrino number $N_{{\rm
eff}}$ that is constrained by the BBN and CMB measurements.

In Refs. [72, 93, 94, 95, 96], the authors calculate a shift of $N_{{\rm
eff}}$ from the SM prediction $\Delta N_{{\rm eff}}\equiv N_{{\rm eff}}-N_{\rm
eff}^{\rm SM}$ as a function of dark sector particle mass, assuming that the
dark sector particle coupled to electron or neutrino is in the thermal bath
when becomes non-relativistic in the thermal history of the universe. This
result can be applied to the thermal relic DM case. For the real scalar DM
coupled exclusively to neutrinos, the latest BBN+Planck bound reads $m_{H}\geq
4.3\text{--}5.6\,{\rm MeV}$ at $2\sigma$ [96], depending on the choice of the
nuclear reaction rates for $d+d\to n+{{}^{3}{\rm He}}$ and $d+d\to
p+{{}^{3}{\rm H}}$. Since any set of the reaction rates in Refs. [97, 98, 99]
reproduces the observed primordial deuterium abundance D$/$H$|_{\rm P}^{\rm
obs}$ with good precision, we conservatively use the weakest limit $m_{H}\geq
4.3\,{\rm MeV}$ in this paper. Note that this low mass bound also applies to
the singlet extension.

###  3.5 Self-scattering

The quartic coupling $\lambda_{2}$ allows $H$ to scatter with itself at the
tree level. The cross section is given, with a negligible contribution from
the Higgs portal coupling, by

$\sigma_{\rm self}=\frac{9\lambda_{2}^{2}}{32\pi m_{H}^{2}}\,.$ (3.16)

The presence of this tree-level self-scattering is in sharp contrast to the
case of $\psi$ being DM, since $\psi$ has the self-scattering only at loop
levels.

The sizable DM self-interaction affects cosmological structures on different
scales, making a difference from the traditional collisionless DM paradigm and
bringing constraints on the self-scattering cross section from cosmological
observations (see e.g. Refs. [100, 101] for reviews and references therein).
The observations have been made on various cosmological scales, and on the
galaxy cluster scales, where the mean relative velocities of DM are 1000 –
2000 km/s, the most severe constraints come from measurements of the core
densities with strong gravitational lensing, providing $\sigma_{\rm
self}/m_{\rm DM}<0.13\,{\rm cm}^{2}/{\rm g}$ in Ref. [102] and $\sigma_{\rm
self}/m_{\rm DM}<0.35\,{\rm cm}^{2}/{\rm g}$ in Ref. [103], both at 95% CL.
The measurement on the galaxy group scales, with the average relative velocity
of 1150 km/s, gives a little weaker constraint $\sigma_{\rm self}/m_{\rm
DM}<1.1\,{\rm cm}^{2}/{\rm g}$ [103]. On the galaxy scales, where the DM
relative velocity is of order of 10 – 100 km/s, relatively large cross
sections $\sigma_{\rm self}/m_{\rm DM}\simeq\text{1\,--\,10}\,{\rm
cm}^{2}/{\rm g}$ are still allowed. Given that the DM self-scattering is
velocity independent in our model, we have an upper limit on $\lambda_{2}$,

$\lambda_{2}\lesssim 0.10\times\left(\frac{m_{H}}{10\,{\rm
MeV}}\right)^{3/2}\left(\frac{(\sigma_{\rm self}/m_{\rm DM})_{\rm
exp}}{0.2\,\text{cm}^{2}/\text{g}}\right)^{1/2}\,,$ (3.17)

where $(\sigma_{\rm self}/m_{\rm DM})_{\rm exp}$ is an experimental upper
bound we choose.

##  4 Other constraints

There are also experimental bounds on the heavy $Z_{2}$ odd particles $\psi$,
$H^{\pm}$ and $A$, which directly and indirectly limit the DM parameter space
when we assume the thermal production. In this section, we study the
constraints from the LHC and LEP experiments and SN 1987A.

###  4.1 Constraints from the LHC experiments

The extra scalars are produced only in pairs through the EW gauge interactions
in the proton collisions at the LHC. In Ref. [74], the authors studied the
searches for the extra scalar pair-productions at the LHC and showed that
relevant bounds come only from $pp\to H^{+}H^{-},\,AH$ productions followed by
$H^{\pm}\to\psi\ell^{\pm}$ and $A\to ZH$ decays. The bounds on the branching
ratios of the $H^{\pm}$ and $A$ decays were derived by reinterpreting the
results of mono-$Z$ [104, 105, 106] and slepton searches [107, 108, 109, 110,
111, 112] in our model. The constraints derived in Ref. [74] can be applied
directly to this model, because the analysis in that paper was done without
any assumption on the mass ordering of $H$ and $\psi$. Both particles were
reconstructed as missing momenta, and this situation keeps holding in this
paper. We do not repeat the analysis here and simply use the results of Ref.
[74].#7#7#7See Sec. 3 of Ref. [74] for further details. Recently the ATLAS
collaboration has reported a refined stau search [113]. This new search
considerably improves the bounds and closes a gap in the light stau region.
However, we do not use this latest ATLAS bound due to the lack of available
cross section data.

Fig. 4 shows the constraints on the branching ratio of the charged scalar
BR($H^{\pm}\to\psi\ell^{\pm}$), for $m_{H}=m_{\psi}=1$ GeV (left) and 10 GeV
(right) and with $m_{H^{\pm}}=m_{A}$, which are derived from the LHC Run 2
data with 139 fb-1 luminosity. The shaded regions are excluded by the slepton
and mono-$Z$ searches. While the bound from slepton search depends on the
lepton flavor, the mono-$Z$ bound is common to all flavor structures. The
hatched region predicts the low cutoff scale: $\Lambda<10\,$TeV. When the
cutoff scale is required to be above 10 TeV, the charged scalar mass has to be
below $350\,{\rm GeV}$. The mono-$Z$ search indeed gives an upper bound on
$\mathrm{BR}(A\to HZ)$, which is translated into the lower bound on
$\mathrm{BR}(H^{\pm}\to\psi\ell^{\pm})$ in Fig. 4. The mono-$Z$ search
excludes the region where $m_{H^{\pm}}\leq 250\,$GeV, when
$\mathrm{BR}(H^{\pm}\to\psi\ell^{\pm})\approx 0$, i.e. $y^{i}_{\nu}\approx 0$.
One may notice that there is a narrow allowed region in $100\,{\rm
GeV}<m_{H^{\pm}}<110\,{\rm GeV}$ when $m_{H}=10\,{\rm GeV}$. This region is
not always consistent with the thermal production in this model. As we will
see later in Sec. 5, when $m_{H}=m_{\psi}=10\,{\rm GeV}$, the correct thermal
production requires $y_{\nu}^{i}\simeq 0.15$, which corresponds to
$\mathrm{BR}(H^{\pm}\to\psi\ell^{\pm})\simeq 0.56$ for $m_{H^{+}}=110\,{\rm
GeV}$. This branching ratio is compatible with the slepton searches in the
electro and tauphilic cases, but is excluded in the muonphilic case. This
conclusion might be changed by lifting $m_{A}=m_{H^{\pm}}$ and/or taking into
account off-shell $W$-mediated three-body decays $H^{\pm}\to HW^{*}\to
Hf\bar{f}^{\prime}$.#8#8#8We have confirmed that this process only mildly
reduces $\mathrm{BR}(H^{\pm}\to H\ell^{\pm})$ for the $m_{H^{\pm}}=m_{A}$
case. The slepton searches provide upper limits on
BR($H^{\pm}\to\psi\ell^{\pm}$). Combining the slepton and mono-$Z$ bounds,
$m_{H^{\pm}}\lesssim 200\,{\rm GeV}$ is excluded in the electro and muon-
philic scenarios, while a relatively large parameter space is still allowed in
the tauphilic case.

Figure 4:  The LHC bounds from the slepton searches (green, magenta, blue) and
mono-$Z$ search (gray) on the branching ratio of the charged scalar
$\mathrm{BR}(H^{\pm}\to\psi\ell^{\pm})$, with $m_{H^{\pm}}=m_{A}$ and for
$m_{H}=m_{\psi}=1\,{\rm GeV}$ (left) and $m_{H}=m_{\psi}=10\,{\rm GeV}$
(right). The shaded regions are excluded. The mono-$Z$ bound applies to all
flavor cases while the slepton bounds depend on the flavor structure chosen.
The hatched region predicts the low cutoff scales $\Lambda<10\,$TeV from the
perturbativity argument.

###  4.2 Constraints from the LEP experiments

In the electrophilic case, there is a Yukawa coupling,
$y^{e}_{\nu}\bar{\psi}_{R}H^{+}e_{L}$. This coupling leads the mono-photon
plus missing energy process, $e^{+}e^{-}\to\psi\bar{\psi}\gamma$, through the
charged scalar exchange in the LEP experiments. In Ref. [114], this mono-
photon search is studied in an effective field theory consisting of four-fermi
electron-DM operators (as well as in simplified single mediator models), and
gives upper limits on the Wilson coefficients. In Ref. [74], integrating out
the charged scalar and matching with their effective field theory, we
translate the limit in Ref. [114] into an upper limit on the electrophilic
lepton portal coupling in our model:

$|y_{\nu}^{e}|\leq\frac{m_{H^{\pm}}}{240\,{\rm GeV}}\,.$ (4.1)

Although we considered $m_{\psi}<m_{H}$ in that paper, this bound can be
applied to the present mass spectrum $m_{H}<m_{\psi}$, since $\psi$ decays
only to $H$ and $\nu$ and its decay does not affect the observed single photon
spectrum.

###  4.3 Constraints from SN 1987A

Light dark sector particles can be copiously produced in the core of
supernovae (SNe) through the interactions with the stellar medium and, when
escaping from the stars without absorption, carry the energy away. This extra
energy release modifies the cooling rate during the burst and brings
constraints on the interactions between the SM and dark sectors from the
observation of SN 1987A.

In this model, the mediator $\psi$ is produced via $e^{+}e^{-}$ and
$\mu^{+}\mu^{-}$ annihilation [115, 116, 117, 118, 119]. In the electrophilic
(muonphilic) case, the lepton portal coupling $y_{\nu}^{e}$ ($y_{\nu}^{\mu}$)
induces $e^{+}e^{-}\to\psi\bar{\psi}$ ($\mu^{+}\mu^{-}\to\psi\bar{\psi}$) with
the tree level $H^{\pm}$ exchange, while no such a process is induced at the
tree-level in the tauphilic case. In this paper, we cast SN bounds on Wilson
coefficients of electron-$\psi$ and muon-$\psi$ four-fermi operators, derived
in Ref. [119], onto constraints on $y_{\nu}^{e}$ and $y_{\nu}^{\mu}$. More
precisely we take as a benchmark constraint the solid lines in Fig. 3 of that
paper, which are obtained using numerical input from SFHo-18.8 SN simulation
[120]. It should noted that the full DM mass dependence of the SN bounds is
provided only for the $V\otimes V$ operators in Ref. [119], while the
$(V-A)\otimes(V+A)$ operator arises in our model after Fierz transformation.
We see, however, in Table 1 of that paper that this operator difference
affects the SN bound by 50% at most. This effect is smaller than the one
stemming from the uncertainty as to the SN simulation, which changes the
cooling bound by a factor of 2 (see Fig. 3 of that paper).

A remarkable difference from the literature is that in our model, $\psi$
produced in the core quickly decays into $H$ and $\nu$. The neutrino produced
from the decay easily loses the kinematic energy by scattering to the dense SN
medium, and gets reabsorbed. On the other hand, $H$ hardly scatters off the SN
medium and thus carries the energy away from the SN. We simply assume that
half of the $\psi$ energy is released from the SN.#9#9#9This assumption would
not be justified when $\psi$ is degenerate in $H$. We found, however, that in
the parameter space where the SN bound is relevant, $m_{H}\ll m_{\psi}$ is
satisfied. This half energy release is taken into account by rescaling the
upper boundary of the $\Lambda^{\rm eff}_{e,\mu}$ bound in Ref. [119] by a
factor of $2^{1/4}$ for a fixed $m_{\psi}$. The lower boundary of the
$\Lambda^{\rm eff}_{e,\mu}$ bound, which usually appears as a result of
absorption of the produced $\psi$ while passing through the SN, vanishes in
the present study. This is because $\psi$ decays away before it scatters, and
the decay product $H$ little scatters off the SN medium.

As another remark, we add that DM production from neutrino annihilation also
brings the SN 1987A bound into play. Recently Ref. [121] derives that bound on
a Majoron which is resonantly produced from annihilation of neutrinos abundant
in the SN. Ref. [119] also discusses the SN bound from the neutrino-initiated
DM production with an s-channel vector mediator $Z^{\prime}$ and in an
effective four-fermi theory, corresponding to the scenario with a mediator
heavy enough to be integrated out in the relevant cooling process. In our
model, $\psi$ ($H$) would be produced through
$\nu\overline{\nu}\to\psi\overline{\psi}\,(HH)$ via t-channel $H$ ($\psi$)
exchange, thereby increasing the DM emission rate. Although the increase of
the emission rate would strengthen the SN bound, any existing analysis cannot
apply to our scenario, because of completely different production and decay
channels for the DM and mediator. We do not analyze that process, which is
beyond the aim of this paper. The detailed study to implement the neutrino-
involved process will be done in future.

##  5 Results

In this section, we discuss the model predictions, based on the constraints
studied in Sec. 3 and Sec. 4. We also compare our results with those presented
in the previous works [71, 74] where $\psi$ is DM, i.e. $m_{\psi}<m_{H}$.
Hereafter, we refer to the case of $H$ being DM as the scalar DM case, and the
case of $\psi$ being DM as the fermion DM case for clarity.

In our model, the DM abundance is determined by three parameters, $m_{H}$,
$m_{\psi}$, and $y_{\nu}$, and independent of $m_{H^{\pm}}$ and the flavor
structures. Figure 5 illustrates how the observed DM abundance can be
thermally realized in the $(m_{H},y_{\nu}^{i})$ plane, with the red lines
representing values of $m_{\psi}=0.01,0.1,1$ and 10 GeV. Clearly, these red
lines exhibit a different behavior from those in the fermion DM case [74] (see
Fig. 6 in that paper). This is understood from the scaling of the DM
annihilation cross section for $m_{\rm DM}\ll m_{\rm med}$, where $m_{\rm DM}$
and $m_{\rm med}$ denote masses of the DM and mediator in each case. In this
limit, $(\sigma v_{\rm rel})\propto y_{\nu}^{4}m_{\rm DM}^{6}/m_{\rm med}^{8}$
in the scalar DM case, while $(\sigma v_{\rm rel})\propto y_{\nu}^{4}m_{\rm
DM}^{2}/m_{\rm med}^{4}$ in the fermion DM case. It follows from this scaling
that with $m_{\rm med}$ fixed, $y_{\nu}\propto m_{\rm DM}^{-3/2}$ in the
scalar DM case while $y_{\nu}\propto m_{\rm DM}^{-1/2}$ in the fermion DM
case, since the thermal freeze-out production requires a constant cross
section value, regardless of DM spin and mass. Further in the scalar DM case,
as $m_{H}\to m_{\psi}$, a coannihilation process
$\psi\bar{\psi}\to\nu\bar{\nu}$ efficiently produces the DM abundance, because
this process is not velocity suppressed. The efficient coannihilation
contribution allows DM to be produced with a smaller $y_{\nu}^{i}$, and this
is why in Fig. 5 the slope in the mass degenerate region ($m_{H}\simeq
m_{\psi}$) is steeper than the mass split region ($m_{H}\ll m_{\psi}$). In the
cyan hatched region, $H$ cannot be thermal relic DM, because any $m_{\psi}$
value does not reproduce the observed abundance in the thermal relic scenario,
unless $m_{H}>m_{\psi}$ and $\psi$ is DM.

Figure 5:  The experimental constraints from the $\Delta N_{\rm eff}$, LHC,
LEP, SN1987A with $m_{H^{\pm}}=220\,$GeV for the electrophilic (upper left),
muonphilic (upper right) and tauphilic (lower) cases. The shaded regions are
excluded. The red lines show the contours for the values of $m_{\psi}$ that
can thermally produce the observed DM abundance. In the cyan hatched region,
$H$ cannot be thermal relic DM, because any $m_{\psi}$ value does not
reproduce the observed abundance in the thermal relic scenario, unless
$m_{H}>m_{\psi}$, namely $\psi$ is DM, or non-thermal production is
considered.

We also show in Fig. 5 the constraints from the slepton (blue) and mono-$Z$
(green) searches, $\Delta N_{\rm eff}$ (gray), LEP (purple) and SN 1987A
(magenta) for each flavor structure. The shaded regions are excluded. Of these
constraints, the $\Delta N_{\rm eff}$ bound depends only on $m_{H}$ and
restricts it to be $m_{H}\geq 4.3\,{\rm MeV}$, which forms another difference
from the fermion DM case in which the $\Delta N_{\rm eff}$ bound is $m_{H}\geq
10\,{\rm MeV}$. The other constraints are all described by four parameters,
$m_{H}$, $m_{\psi}$, $y_{\nu}$ and $m_{H^{\pm}}(=m_{A})$, except the DM direct
detection, which we do not consider in this section, since it depends on
$\lambda_{345}$ and $\lambda_{2}$ and can always be evaded by appropriately
choosing those two parameter values (see Fig. 2). Among the four parameters,
we fix the charged scalar mass at $m_{H^{\pm}}=m_{A}=220\,{\rm GeV}$ in Fig.
5, thereby making it easy to compare with the fermion DM results.

The LHC constraints exclude a large part of parameter space for this charged
scalar mass $m_{H^{\pm}}=220\,{\rm GeV}$. These constraints become
significantly weaker (stronger) as $H^{\pm}$ gets heavier (lighter).
Concretely, we see in Fig. 4 that in the electro and muonphilic cases and for
$m_{H^{\pm}}\lesssim 200\,{\rm GeV}$, the combination of the slepton and
mono-$Z$ searches rule out all possible
$\mathrm{BR}(H^{\pm}\to\psi\ell^{\pm})$ value, which in turn rules out any
$y_{\nu}^{i}$ value. Whereas, when $m_{H^{\pm}}\geq 250\,$GeV is chosen, the
mono-$Z$ search does not work and even
$\mathrm{BR}(H^{\pm}\to\psi\ell^{\pm})=0$ is allowed, and as a result the
mono-$Z$ exclusion region in Fig. 5 completely disappears from the plots. In
the tauphilic case, the constraint is relatively weak and charged scalar can
be a bit lighter, but only a limited parameter space around
$y_{\nu}^{\tau}\simeq 1$ is viable when $m_{H^{\pm}}\lesssim 200\,{\rm GeV}$.
All parameter spaces that are currently unexplored will be expected to be
within reach of the future HL-LHC experiment [74].

##  6 Summary

In this paper, we studied a light scalar DM model that features an inert
scalar doublet $\Phi_{\nu}$ and singlet Dirac fermion $\psi$, which are both
odd under a new global $Z_{2}$ symmetry while all SM fields are even. The DM
originates in the inert doublet, which couples to the SM sector with quartic
couplings to the SM Higgs doublet as well as Yukawa couplings to $\psi$ and
the left-handed leptons which are referred to as the lepton portal couplings.
We consider the case that the DM mass is below 10 GeV, which is realized by
adjusting three quartic couplings as shown in Eqs. (2.8)-(2.10). In this mass
region, the DM can be thermally produced by its pair annihilation into
neutrinos, which is mediated by $\psi$.

Focusing on the parameter space where the lightest inert scalar $H$ is DM, we
discussed the DM physics and experimental constraints on the extra scalars and
compared our results with those studied in the case of $\psi$ being DM [71,
74]. The differences are summarized as follows:

  * •

In the thermal production (Sec. 3.1), the velocity suppression and different
mass scaling in the DM annihilation result in a larger lepton portal coupling
than in the fermion DM case.

  * •

In the direct detection (Sec. 3.2), the lepton portal couplings do not
generate any relevant contribution to the elastic DM-nucleon scattering. This
makes a striking contrast to the fermion DM case, where the lepton portal
couplings are responsible for both the DM production and direct detection, and
thus correlate them closely. In the scalar DM case, the weak gauge and quartic
interactions instead make the leading contribution from the loop diagrams,
which depends on the Higgs portal coupling and DM self-coupling but decouples
from the DM production.

  * •

The indirect detection (Sec. 3.3) does not provide useful probes of the scalar
DM, while the monochromatic neutrino flux from the galactic DM annihilation is
a good probe in the fermion DM case.

  * •

The $\Delta N_{\rm eff}$ bound (Sec. 3.4) restricts the DM mass to be heavier
than 4.3 MeV, in contrast to 10 MeV for the fermion DM case.

  * •

The scalar DM has the self-scattering at the tree level (Sec. 3.5), while it
is loop induced in case $\psi$ is DM.

On the other hand, the collider bounds are basically the same as in the
fermion DM case, given that both $H$ and $\psi$ are supposed to be missing.

In closing, it is worth mentioning that adding the inert doublet with the
light $H$ triggers strong first order EW phase transition [122, 123, 124, 125,
126, 127], pointing to the EW baryogenesis as a promising baryogenesis
scenario. It is unfortunate that the EW baryogenesis mechanism does not
operate with the model considered in this paper, due to the absence of new CP
violating couplings to the Higgs field. This encourages us to extend the model
to incorporate other renormalizable CP-violating interactions to enable the EW
baryogenesis as discussed in Refs. [128, 129, 130].

## Acknowledgements

S.I. appreciates Robert Ziegler for the fruitful discussion on the SN bound.
S.O. is grateful to Tomohiro Abe for clarification of the loop effects on the
DM-nucleon scattering in the inert doublet model. S.I. would like to thank
support from a María de Maeztu grant for a visit to the Institute of Cosmos
Sciences, Barcelona University, where he stayed during the initial stage of
this project. S.I. enjoys the support from the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under grant 396021762-TRR 257. S.O.
acknowledges support from a Maria Zambrano fellowship, from the State Agency
for Research of the Spanish Ministry of Science and Innovation through the
“Unit of Excellence María de Maeztu 2020-2023” award to the Institute of
Cosmos Sciences (CEX2019-000918-M) and from PID2019-105614GB-C21 and
2017-SGR-929 grants. The work of Y.O. is supported by Grant-in-Aid for
Scientific research from the MEXT, Japan, No. 19K03867.

## References

  * [1] R. Essig, A. Manalaysay, J. Mardon, P. Sorensen, and T. Volansky, “First Direct Detection Limits on sub-GeV Dark Matter from XENON10,” Phys. Rev. Lett. 109 (2012) 021301 [arXiv:1206.2644].
  * [2] R. Essig, T. Volansky, and T.-T. Yu, “New Constraints and Prospects for sub-GeV Dark Matter Scattering off Electrons in Xenon,” Phys. Rev. D 96 (2017) 043017 [arXiv:1703.00910].
  * [3] LUX Collaboration, “Results of a Search for Sub-GeV Dark Matter Using 2013 LUX Data,” Phys. Rev. Lett. 122 (2019) 131301 [arXiv:1811.11241].
  * [4] XENON Collaboration, “Light Dark Matter Search with Ionization Signals in XENON1T,” Phys. Rev. Lett. 123 (2019) 251801 [arXiv:1907.11485].
  * [5] XENON Collaboration, “Search for Light Dark Matter Interactions Enhanced by the Migdal Effect or Bremsstrahlung in XENON1T,” Phys. Rev. Lett. 123 (2019) 241803 [arXiv:1907.12771].
  * [6] PandaX-II Collaboration, “Search for Light Dark Matter-Electron Scatterings in the PandaX-II Experiment,” Phys. Rev. Lett. 126 (2021) 211803 [arXiv:2101.07479].
  * [7] LZ Collaboration, “A search for new physics in low-energy electron recoils from the first LZ exposure.” arXiv:2307.15753.
  * [8] SuperCDMS Collaboration, “Search for Low-Mass Weakly Interacting Massive Particles with SuperCDMS,” Phys. Rev. Lett. 112 (2014) 241302 [arXiv:1402.7137].
  * [9] SuperCDMS Collaboration, “New Results from the Search for Low-Mass Weakly Interacting Massive Particles with the CDMS Low Ionization Threshold Experiment,” Phys. Rev. Lett. 116 (2016) 071301 [arXiv:1509.02448].
  * [10] SuperCDMS Collaboration, “Low-mass dark matter search with CDMSlite,” Phys. Rev. D 97 (2018) 022002 [arXiv:1707.01632].
  * [11] SuperCDMS Collaboration, “First Dark Matter Constraints from a SuperCDMS Single-Charge Sensitive Detector,” Phys. Rev. Lett. 121 (2018) 051301 [arXiv:1804.10697]. [Erratum: Phys.Rev.Lett. 122, 069901 (2019)].
  * [12] SuperCDMS Collaboration, “Light Dark Matter Search with a High-Resolution Athermal Phonon Detector Operated Above Ground,” Phys. Rev. Lett. 127 (2021) 061801 [arXiv:2007.14289].
  * [13] CRESST Collaboration, “The CRESST dark matter search,” Astropart. Phys. 12 (1999) 107–114 [hep-ex/9904005].
  * [14] CRESST-II Collaboration, “Results on low mass WIMPs using an upgraded CRESST-II detector,” Eur. Phys. J. C 74 (2014) 3184 [arXiv:1407.3146].
  * [15] CRESST Collaboration, “Results on light dark matter particles with a low-threshold CRESST-II detector,” Eur. Phys. J. C 76 (2016) 25 [arXiv:1509.01515].
  * [16] CRESST Collaboration, “First results from the CRESST-III low-mass dark matter program,” Phys. Rev. D 100 (2019) 102002 [arXiv:1904.00498].
  * [17] CRESST Collaboration, “Results on sub-GeV dark matter from a 10 eV threshold CRESST-III silicon detector,” Phys. Rev. D 107 (2023) 122003 [arXiv:2212.12513].
  * [18] EDELWEISS Collaboration, “First germanium-based constraints on sub-MeV Dark Matter with the EDELWEISS experiment,” Phys. Rev. Lett. 125 (2020) 141301 [arXiv:2003.01046].
  * [19] EDELWEISS Collaboration, “Search for sub-GeV dark matter via the Migdal effect with an EDELWEISS germanium detector with NbSi transition-edge sensors,” Phys. Rev. D 106 (2022) 062004 [arXiv:2203.03993].
  * [20] SENSEI Collaboration, “SENSEI: First Direct-Detection Constraints on sub-GeV Dark Matter from a Surface Run,” Phys. Rev. Lett. 121 (2018) 061803 [arXiv:1804.00088].
  * [21] SENSEI Collaboration, “SENSEI: Direct-Detection Constraints on Sub-GeV Dark Matter from a Shallow Underground Run Using a Prototype Skipper-CCD,” Phys. Rev. Lett. 122 (2019) 161801 [arXiv:1901.10478].
  * [22] SENSEI Collaboration, “SENSEI: Direct-Detection Results on sub-GeV Dark Matter from a New Skipper-CCD,” Phys. Rev. Lett. 125 (2020) 171802 [arXiv:2004.11378].
  * [23] DAMIC Collaboration, “Direct Search for Low Mass Dark Matter Particles with CCDs,” Phys. Lett. B 711 (2012) 264–269 [arXiv:1105.5191].
  * [24] DAMIC Collaboration, “Search for low-mass WIMPs in a 0.6 kg day exposure of the DAMIC experiment at SNOLAB,” Phys. Rev. D 94 (2016) 082006 [arXiv:1607.07410].
  * [25] DAMIC Collaboration, “Constraints on Light Dark Matter Particles Interacting with Electrons from DAMIC at SNOLAB,” Phys. Rev. Lett. 123 (2019) 181802 [arXiv:1907.12628].
  * [26] DAMIC Collaboration, “Results on low-mass weakly interacting massive particles from a 11 kg-day target exposure of DAMIC at SNOLAB,” Phys. Rev. Lett. 125 (2020) 241803 [arXiv:2007.15622].
  * [27] CDEX Collaboration, “Limits on Light Weakly Interacting Massive Particles from the First 102.8 kg ${\times}$ day Data of the CDEX-10 Experiment,” Phys. Rev. Lett. 120 (2018) 241301 [arXiv:1802.09016].
  * [28] CDEX Collaboration, “First experimental constraints on WIMP couplings in the effective field theory framework from CDEX,” Sci. China Phys. Mech. Astron. 64 (2021) 281011 [arXiv:2007.15555].
  * [29] R. Essig, et al., “Direct Detection of sub-GeV Dark Matter with Semiconductor Targets,” JHEP 05 (2016) 046 [arXiv:1509.01598].
  * [30] T. Bringmann and M. Pospelov, “Novel direct detection constraints on light dark matter,” Phys. Rev. Lett. 122 (2019) 171801 [arXiv:1810.10543].
  * [31] C. Cappiello and J. F. Beacom, “Strong New Limits on Light Dark Matter from Neutrino Experiments,” Phys. Rev. D 100 (2019) 103011 [arXiv:1906.11283].
  * [32] K. Bondarenko, et al., “Direct detection and complementary constraints for sub-GeV dark matter,” JHEP 03 (2020) 118 [arXiv:1909.08632].
  * [33] G. Guo, Y.-L. S. Tsai, and M.-R. Wu, “Probing cosmic-ray accelerated light dark matter with IceCube,” JCAP 10 (2020) 049 [arXiv:2004.03161].
  * [34] Y. Ema, F. Sala, and R. Sato, “Neutrino experiments probe hadrophilic light dark matter,” SciPost Phys. 10 (2021) 072 [arXiv:2011.01939].
  * [35] J. Alvey, T. Bringmann, and H. Kolesova, “No room to hide: implications of cosmic-ray upscattering for GeV-scale dark matter,” JHEP 01 (2023) 123 [arXiv:2209.03360].
  * [36] C. Kouvaris, “Probing Light Dark Matter via Evaporation from the Sun,” Phys. Rev. D 92 (2015) 075001 [arXiv:1506.04316].
  * [37] Y. Ema, F. Sala, and R. Sato, “Light Dark Matter at Neutrino Experiments,” Phys. Rev. Lett. 122 (2019) 181802 [arXiv:1811.00520].
  * [38] J. B. Dent, B. Dutta, J. L. Newstead, I. M. Shoemaker, and N. T. Arellano, “Present and future status of light dark matter models from cosmic-ray electron upscattering,” Phys. Rev. D 103 (2021) 095015 [arXiv:2010.09749].
  * [39] C. Xia, Y.-H. Xu, and Y.-F. Zhou, “Azimuthal asymmetry in cosmic-ray boosted dark matter flux,” Phys. Rev. D 107 (2023) 055012 [arXiv:2206.11454].
  * [40] H. An, M. Pospelov, J. Pradler, and A. Ritz, “Directly Detecting MeV-scale Dark Matter via Solar Reflection,” Phys. Rev. Lett. 120 (2018) 141801 [arXiv:1708.03642]. [Erratum: Phys.Rev.Lett. 121, 259903 (2018)].
  * [41] Y. Jho, J.-C. Park, S. C. Park, and P.-Y. Tseng, “Cosmic-Neutrino-Boosted Dark Matter ($\nu$BDM).” arXiv:2101.11262.
  * [42] C. Boehm and P. Fayet, “Scalar dark matter candidates,” Nucl. Phys. B 683 (2004) 219–263 [hep-ph/0305261].
  * [43] P. Fayet, “Light spin 1/2 or spin 0 dark matter particles,” Phys. Rev. D 70 (2004) 023514 [hep-ph/0403226].
  * [44] P. Galison and A. Manohar, “TWO Z’s OR NOT TWO Z’s?” Phys. Lett. B 136 (1984) 279–283.
  * [45] B. Holdom, “Two U(1)’s and Epsilon Charge Shifts,” Phys. Lett. B 166 (1986) 196–198.
  * [46] R. Foot and X.-G. He, “Comment on Z Z-prime mixing in extended gauge theories,” Phys. Lett. B 267 (1991) 509–512.
  * [47] M. Pospelov, A. Ritz, and M. B. Voloshin, “Secluded WIMP Dark Matter,” Phys. Lett. B 662 (2008) 53–61 [arXiv:0711.4866].
  * [48] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, and N. Weiner, “A Theory of Dark Matter,” Phys. Rev. D 79 (2009) 015014 [arXiv:0810.0713].
  * [49] R. M. Schabinger and J. D. Wells, “A Minimal spontaneously broken hidden sector and its impact on Higgs boson physics at the large hadron collider,” Phys. Rev. D 72 (2005) 093007 [hep-ph/0509209].
  * [50] B. Patt and F. Wilczek, “Higgs-field portal into hidden sectors.” hep-ph/0605188.
  * [51] J. D. Wells, “How to Find a Hidden World at the Large Hadron Collider.” arXiv:0803.1243.
  * [52] B. Batell, M. Pospelov, and A. Ritz, “Probing a Secluded U(1) at B-factories,” Phys. Rev. D 79 (2009) 115008 [arXiv:0903.0363].
  * [53] E. Weihs and J. Zurita, “Dark Higgs Models at the 7 TeV LHC,” JHEP 02 (2012) 041 [arXiv:1110.5909].
  * [54] Y. G. Kim, K. Y. Lee, and S. Shin, “Singlet fermionic dark matter,” JHEP 05 (2008) 100 [arXiv:0803.2932].
  * [55] M. Pospelov and A. Ritz, “Higgs decays to dark matter: beyond the minimal model,” Phys. Rev. D 84 (2011) 113001 [arXiv:1109.4872].
  * [56] Y. Nomura and J. Thaler, “Dark Matter through the Axion Portal,” Phys. Rev. D 79 (2009) 075008 [arXiv:0810.5397].
  * [57] M. J. Dolan, F. Kahlhoefer, C. McCabe, and K. Schmidt-Hoberg, “A taste of dark matter: Flavour constraints on pseudoscalar mediators,” JHEP 03 (2015) 171 [arXiv:1412.5174]. [Erratum: JHEP 07, 103 (2015)].
  * [58] S. Gola, S. Mandal, and N. Sinha, “ALP-portal majorana dark matter,” Int. J. Mod. Phys. A 37 (2022) 2250131 [arXiv:2106.00547].
  * [59] P. J. Fitzpatrick, Y. Hochberg, E. Kuflik, R. Ovadia, and Y. Soreq, “Dark matter through the axion-gluon portal,” Phys. Rev. D 108 (2023) 075003 [arXiv:2306.03128].
  * [60] J. A. Dror, S. Gori, and P. Munbodh, “QCD axion-mediated dark matter,” JHEP 09 (2023) 128 [arXiv:2306.03145].
  * [61] L. Darmé, F. Giacchino, E. Nardi, and M. Raggi, “Invisible decays of axion-like particles: constraints and prospects,” JHEP 06 (2021) 009 [arXiv:2012.07894].
  * [62] A. Kamada, H. Kim, and T. Sekiguchi, “Axionlike particle assisted strongly interacting massive particle,” Phys. Rev. D 96 (2017) 016007 [arXiv:1704.04505].
  * [63] Y. Hochberg, E. Kuflik, R. Mcgehee, H. Murayama, and K. Schutz, “Strongly interacting massive particles through the axion portal,” Phys. Rev. D 98 (2018) 115031 [arXiv:1806.10139].
  * [64] A. Bharucha, F. Brümmer, N. Desai, and S. Mutzel, “Axion-like particles as mediators for dark matter: beyond freeze-out,” JHEP 02 (2023) 141 [arXiv:2209.03932].
  * [65] D. K. Ghosh, A. Ghoshal, and S. Jeesun, “Axion-like particle (ALP) portal freeze-in dark matter confronting ALP search experiments.” arXiv:2305.09188.
  * [66] B. Batell, T. Han, D. McKeen, and B. Shams Es Haghi, “Thermal Dark Matter Through the Dirac Neutrino Portal,” Phys. Rev. D 97 (2018) 075016 [arXiv:1709.07001].
  * [67] D. McKeen and N. Raj, “Monochromatic dark neutrinos and boosted dark matter in noble liquid direct detection,” Phys. Rev. D 99 (2019) 103003 [arXiv:1812.05102].
  * [68] M. Blennow, et al., “Neutrino Portals to Dark Matter,” Eur. Phys. J. C 79 (2019) 555 [arXiv:1903.00006].
  * [69] A. Biswas, D. Borah, and D. Nanda, “Light Dirac neutrino portal dark matter with observable $\Delta$Neff,” JCAP 10 (2021) 002 [arXiv:2103.05648].
  * [70] S.-P. Li and X.-J. Xu, “Dark matter produced from right-handed neutrinos,” JCAP 06 (2023) 047 [arXiv:2212.09109].
  * [71] S. Okawa and Y. Omura, “Light mass window of lepton portal dark matter,” JHEP 02 (2021) 231 [arXiv:2011.04788].
  * [72] C. Boehm, M. J. Dolan, and C. McCabe, “A Lower Bound on the Mass of Cold Thermal Dark Matter from Planck,” JCAP 08 (2013) 041 [arXiv:1303.6270].
  * [73] J. Herms, S. Jana, V. P. K., and S. Saad, “Minimal Realization of Light Thermal Dark Matter,” Phys. Rev. Lett. 129 (2022) 091803 [arXiv:2203.05579].
  * [74] S. Iguro, S. Okawa, and Y. Omura, “Light lepton portal dark matter meets the LHC,” JHEP 03 (2023) 010 [arXiv:2208.05487].
  * [75] J. Herms, S. Jana, V. P. K., and S. Saad, “Light neutrinophilic dark matter from a scotogenic model,” Phys. Lett. B 845 (2023) 138167 [arXiv:2307.15760].
  * [76] A. Pierce and J. Thaler, “Natural Dark Matter from an Unnatural Higgs Boson and New Colored Particles at the TeV Scale,” JHEP 08 (2007) 026 [hep-ph/0703056].
  * [77] E. Lundstrom, M. Gustafsson, and J. Edsjo, “The Inert Doublet Model and LEP II Limits,” Phys. Rev. D 79 (2009) 035013 [arXiv:0810.3924].
  * [78] ATLAS Collaboration, “Search for invisible Higgs boson decays with vector boson fusion signatures with the ATLAS detector using an integrated luminosity of 139 fb-1.”.
  * [79] CMS Collaboration, “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].
  * [80] T. Toma, “Internal Bremsstrahlung Signature of Real Scalar Dark Matter and Consistency with Thermal Relic Density,” Phys. Rev. Lett. 111 (2013) 091301 [arXiv:1307.6181].
  * [81] F. Giacchino, L. Lopez-Honorez, and M. H. G. Tytgat, “Scalar Dark Matter Models with Significant Internal Bremsstrahlung,” JCAP 10 (2013) 025 [arXiv:1307.6480].
  * [82] J. Kawamura, S. Okawa, and Y. Omura, “Current status and muon $g-2$ explanation of lepton portal dark matter,” JHEP 08 (2020) 042 [arXiv:2002.12534].
  * [83] G. Bélanger, F. Boudjema, A. Goudelis, A. Pukhov, and B. Zaldivar, “micrOMEGAs5.0 : Freeze-in,” Comput. Phys. Commun. 231 (2018) 173–186 [arXiv:1801.03509].
  * [84] E. Ma, “Verifiable radiative seesaw mechanism of neutrino mass and dark matter,” Phys. Rev. D 73 (2006) 077301 [hep-ph/0601225].
  * [85] R. Barbieri, L. J. Hall, and V. S. Rychkov, “Improved naturalness with a heavy Higgs: An Alternative road to LHC physics,” Phys. Rev. D 74 (2006) 015007 [hep-ph/0603188].
  * [86] L. Lopez Honorez, E. Nezri, J. F. Oliver, and M. H. G. Tytgat, “The Inert Doublet Model: An Archetype for Dark Matter,” JCAP 02 (2007) 028 [hep-ph/0612275].
  * [87] T. Abe and R. Sato, “Quantum corrections to the spin-independent cross section of the inert doublet dark matter,” JHEP 03 (2015) 109 [arXiv:1501.04161].
  * [88] J. Pumplin, et al., “New generation of parton distributions with uncertainties from global QCD analysis,” JHEP 07 (2002) 012 [hep-ph/0201195].
  * [89] DarkSide-50 Collaboration, “Search for low mass dark matter in DarkSide-50: the bayesian network approach,” Eur. Phys. J. C 83 (2023) 322 [arXiv:2302.01830].
  * [90] PandaX-4T Collaboration, “Dark Matter Search Results from the PandaX-4T Commissioning Run,” Phys. Rev. Lett. 127 (2021) 261802 [arXiv:2107.13438].
  * [91] LZ Collaboration, “First Dark Matter Search Results from the LUX-ZEPLIN (LZ) Experiment,” Phys. Rev. Lett. 131 (2023) 041002 [arXiv:2207.03764].
  * [92] A. De Angelis, V. Tatischeff, I. A. Grenier, J. McEnery, and M. Mallamaci, eds., “Science with e-ASTROGAM: A space mission for MeV–GeV gamma-ray astrophysics,” JHEAp 19 (2018) 1–106 [arXiv:1711.01265].
  * [93] K. M. Nollett and G. Steigman, “BBN And The CMB Constrain Neutrino Coupled Light WIMPs,” Phys. Rev. D 91 (2015) 083505 [arXiv:1411.6005].
  * [94] J. H. Heo and C. S. Kim, “Light Dark Matter and Dark Radiation,” J. Korean Phys. Soc. 68 (2016) 715–721 [arXiv:1504.00773].
  * [95] N. Sabti, J. Alvey, M. Escudero, M. Fairbairn, and D. Blas, “Refined Bounds on MeV-scale Thermal Dark Sectors from BBN and the CMB,” JCAP 01 (2020) 004 [arXiv:1910.01649].
  * [96] N. Sabti, J. Alvey, M. Escudero, M. Fairbairn, and D. Blas, “Addendum: Refined bounds on MeV-scale thermal dark sectors from BBN and the CMB,” JCAP 08 (2021) A01 [arXiv:2107.11232].
  * [97] C. Pitrou, A. Coc, J.-P. Uzan, and E. Vangioni, “A new tension in the cosmological model from primordial deuterium?” Mon. Not. Roy. Astron. Soc. 502 (2021) 2474–2481 [arXiv:2011.11320].
  * [98] O. Pisanti, G. Mangano, G. Miele, and P. Mazzella, “Primordial Deuterium after LUNA: concordances and error budget,” JCAP 04 (2021) 020 [arXiv:2011.11537].
  * [99] T.-H. Yeh, K. A. Olive, and B. D. Fields, “The impact of new $d(p,\gamma)$3 rates on Big Bang Nucleosynthesis,” JCAP 03 (2021) 046 [arXiv:2011.13874].
  * [100] S. Tulin and H.-B. Yu, “Dark Matter Self-interactions and Small Scale Structure,” Phys. Rept. 730 (2018) 1–57 [arXiv:1705.02358].
  * [101] S. Adhikari et al., “Astrophysical Tests of Dark Matter Self-Interactions.” arXiv:2207.10638.
  * [102] K. E. Andrade, et al., “A stringent upper limit on dark matter self-interaction cross-section from cluster strong lensing,” Mon. Not. Roy. Astron. Soc. 510 (2021) 54–81 [arXiv:2012.06611].
  * [103] L. Sagunski, S. Gad-Nasr, B. Colquhoun, A. Robertson, and S. Tulin, “Velocity-dependent Self-interacting Dark Matter from Groups and Clusters of Galaxies,” JCAP 01 (2021) 024 [arXiv:2006.12515].
  * [104] ATLAS Collaboration, “Search for an invisibly decaying Higgs boson or dark matter candidates produced in association with a $Z$ boson in $pp$ collisions at $\sqrt{s}=$ 13 TeV with the ATLAS detector,” Phys. Lett. B 776 (2018) 318–337 [arXiv:1708.09624].
  * [105] CMS Collaboration, “Search for dark matter produced in association with a leptonically decaying Z boson in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” Eur. Phys. J. C 81 (2021) 13 [arXiv:2008.04735]. [Erratum: Eur.Phys.J.C 81, 333 (2021)].
  * [106] ATLAS Collaboration, “Search for associated production of a $Z$ boson with an invisibly decaying Higgs boson or dark matter candidates at $\sqrt{s}=13$ TeV with the ATLAS detector.” arXiv:2111.08372.
  * [107] CMS Collaboration, “Search for supersymmetric partners of electrons and muons in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” Phys. Lett. B 790 (2019) 140–166 [arXiv:1806.05264].
  * [108] CMS Collaboration, “Search for supersymmetry in events with a $\tau$ lepton pair and missing transverse momentum in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” JHEP 11 (2018) 151 [arXiv:1807.02048].
  * [109] CMS Collaboration, “Search for direct pair production of supersymmetric partners to the $\tau$ lepton in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” Eur. Phys. J. C 80 (2020) 189 [arXiv:1907.13179].
  * [110] ATLAS Collaboration, “Search for electroweak production of charginos and sleptons decaying into final states with two leptons and missing transverse momentum in $\sqrt{s}=13$ TeV $pp$ collisions using the ATLAS detector,” Eur. Phys. J. C 80 (2020) 123 [arXiv:1908.08215].
  * [111] ATLAS Collaboration, “Searches for electroweak production of supersymmetric particles with compressed mass spectra in $\sqrt{s}=$ 13 TeV $pp$ collisions with the ATLAS detector,” Phys. Rev. D 101 (2020) 052005 [arXiv:1911.12606].
  * [112] CMS Collaboration, “Search for direct pair production of supersymmetric partners of $\tau$ leptons in the final state with two hadronically decaying $\tau$ leptons and missing transverse momentum in proton-proton collisions at $\sqrt{s}$ = 13 TeV,” Phys. Rev. D 108 (2023) 012011 [arXiv:2207.02254].
  * [113] ATLAS Collaboration, “Search for electroweak SUSY production in final states with two $\tau$-leptons in $\sqrt{s}=13$ TeV $pp$ collisions with the ATLAS detector.”.
  * [114] P. J. Fox, R. Harnik, J. Kopp, and Y. Tsai, “LEP Shines Light on Dark Matter,” Phys. Rev. D 84 (2011) 014028 [arXiv:1103.0240].
  * [115] H. K. Dreiner, C. Hanhart, U. Langenfeld, and D. R. Phillips, “Supernovae and light neutralinos: SN1987A bounds on supersymmetry revisited,” Phys. Rev. D 68 (2003) 055004 [hep-ph/0304289].
  * [116] H. K. Dreiner, J.-F. Fortin, C. Hanhart, and L. Ubaldi, “Supernova constraints on MeV dark sectors from $e^{+}e^{-}$ annihilations,” Phys. Rev. D 89 (2014) 105015 [arXiv:1310.3826].
  * [117] K. Kadota and J. Silk, “Constraints on Light Magnetic Dipole Dark Matter from the ILC and SN 1987A,” Phys. Rev. D 89 (2014) 103528 [arXiv:1402.7295].
  * [118] A. Guha, P. S. B. Dev, and P. K. Das, “Model-independent Astrophysical Constraints on Leptophilic Dark Matter in the Framework of Tsallis Statistics,” JCAP 02 (2019) 032 [arXiv:1810.00399].
  * [119] C. A. Manzari, J. Martin Camalich, J. Spinner, and R. Ziegler, “Supernova Limits on Muonic Dark Forces.” arXiv:2307.03143.
  * [120] R. Bollig, W. DeRocco, P. W. Graham, and H.-T. Janka, “Muons in Supernovae: Implications for the Axion-Muon Coupling,” Phys. Rev. Lett. 125 (2020) 051104 [arXiv:2005.07141]. [Erratum: Phys.Rev.Lett. 126, 189901 (2021)].
  * [121] D. F. G. Fiorillo, G. G. Raffelt, and E. Vitagliano, “Strong Supernova 1987A Constraints on Bosons Decaying to Neutrinos,” Phys. Rev. Lett. 131 (2023) 021001 [arXiv:2209.11773].
  * [122] T. A. Chowdhury, M. Nemevsek, G. Senjanovic, and Y. Zhang, “Dark Matter as the Trigger of Strong Electroweak Phase Transition,” JCAP 02 (2012) 029 [arXiv:1110.5334].
  * [123] D. Borah and J. M. Cline, “Inert Doublet Dark Matter with Strong Electroweak Phase Transition,” Phys. Rev. D 86 (2012) 055001 [arXiv:1204.4722].
  * [124] G. Gil, P. Chankowski, and M. Krawczyk, “Inert Dark Matter and Strong Electroweak Phase Transition,” Phys. Lett. B 717 (2012) 396–402 [arXiv:1207.0084].
  * [125] J. M. Cline and K. Kainulainen, “Improved Electroweak Phase Transition with Subdominant Inert Doublet Dark Matter,” Phys. Rev. D 87 (2013) 071701 [arXiv:1302.2614].
  * [126] N. Blinov, J. Kozaczuk, D. E. Morrissey, and C. Tamarit, “Electroweak Baryogenesis from Exotic Electroweak Symmetry Breaking,” Phys. Rev. D 92 (2015) 035012 [arXiv:1504.05195].
  * [127] N. Blinov, S. Profumo, and T. Stefaniak, “The Electroweak Phase Transition in the Inert Doublet Model,” JCAP 07 (2015) 028 [arXiv:1504.05949].
  * [128] J. M. Cline, K. Kainulainen, and D. Tucker-Smith, “Electroweak baryogenesis from a dark sector,” Phys. Rev. D 95 (2017) 115006 [arXiv:1702.08909].
  * [129] E. Fernández-Martínez, J. López-Pavón, T. Ota, and S. Rosauro-Alcaraz, “$\nu$ electroweak baryogenesis,” JHEP 10 (2020) 063 [arXiv:2007.11008].
  * [130] E. Fernández-Martínez, J. López-Pavón, J. M. No, T. Ota, and S. Rosauro-Alcaraz, “$\nu$ Electroweak baryogenesis: the scalar singlet strikes back,” Eur. Phys. J. C 83 (2023) 715 [arXiv:2210.16279].