# Electroweak phase transition confronted with dark matter detection
constraints

Cheng-Wei Chiang  , Da Huang  , and Bo-Qiang Lu

###### Abstract

We study the type-II first-order electroweak phase transition and dark matter
(DM) phenomenology in both real and complex singlet extensions of SM. In the
real singlet extension with a $\mathbb{Z}_{2}$ symmetry, we show that the
parameter regions favored by the phase transition suffer from strong
constraints from DM direct detection so that only a negligible fraction
($f_{X}\sim 10^{-4}-10^{-5}$) of DM composed of the real singlet scalar can
survive the LUX and XENON1T constraints. In the complex singlet $S$ case, we
impose a $CP$ symmetry $S\to S^{*}$ to the scalar potential. The real
component of $S$ can mix with SM Higgs boson while the imaginary component
becomes a DM candidate due to the protection of the $CP$ symmetry. By taking
into account the current experimental constraints of invisible Higgs decays,
Higgs signal strength measurements, and dark matter detections, we find that
there exists a large parameter space for the type-II electroweak phase
transition to occur while explaining all of the dark matter relic density. We
identify a subset of parameter space that is promising for future experiments,
including the di-Higgs and Higgs signal strength measurements at the HL-LHC
and the dark matter direct detection in the XENONnT project.

##  1 Introduction

Although the Standard Model (SM) of particle physics has been completed since
the discovery of the Higgs boson at the LHC in 2012 [1, 2], it is widely
believed that new physics is required to explain various phenomena beyond the
SM, including the existence of dark matter (DM) and the origin of matter-
antimatter asymmetry in the Universe [3]. The most popular class of DM
candidates is that of weakly interacting massive particles (WIMPs) [4, 5].
These particles decoupled from the thermal bath as the early Universe was
expanding and cooling, finally achieving the appropriate relic density. In
this scenario, the observed DM abundance is determined by the DM annihilation
cross section, provided that the DM particles are massive enough to become
non-relativistic at freeze-out. On the other hand, one of the widely accepted
mechanisms to generate matter-antimatter asymmetry in the Universe is
electroweak (EW) baryogenesis [6, 7, 8, 9], in which there are sufficiently
large $CP$ violation (CPV) sources and the electroweak phase transition (EWPT)
is sufficiently strong so that the washout of baryon number asymmetry through
sphalerons is suppressed. However, the only CPV source in the SM is provided
by the CKM phase, whose effect is too small to account for the observed
asymmetry. Moreover, the EWPT is found to be a crossover rather than a first-
order phase transition [10].

The simplest extension of the SM is to add a gauge singlet scalar. On the
theoretical side, the inclusion of a singlet can provide additional sources of
CP violation, and lead to a strong first-order EWPT. The singlet extension of
SM and the related DM and/or EWPT phenomenology have been widely studied in
the literature [11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 14, 15, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 41] (see refs.
[43, 44] for a recent review). On the experimental side, the process of the
strong first-order EWPT can leave a trace of stochastic background of
millihertz gravitational waves (GWs). In 2015, the GW generated by the merger
of binary black holes was first observed by the Advanced LIGO experiment [45].
In the near future, the space-based interferometers such as LISA [46], DECIGO
[47], and BBO [48] may be available to probe GWs in the range from millihertz
to decihertz, which can thus be used to test the first-order EWPT in the
singlet extension of SM. The singlet scenario may also give rise to detectable
signatures at the colliders, such as deviations in the triple Higgs coupling
and the Higgs signal strengths. In a recent work [49], we studied the two-step
phase transition $(0,~{}w)\to(v_{0},w_{0})$ (type-I EWPT) in a complex singlet
extension of SM with a $\mathbb{Z}_{3}$ symmetry and showed that the first-
order EWPT could occur if the mixing angle satisfied $|\theta|\gtrsim
11.5^{\circ}$. The measurements of the Higgs signal strengths at the LHC, on
the other hand, have restricted the mixing angle $|\theta|\lesssim
23^{\circ}$. Future precision Higgs measurements in collider experiments, such
as the high-luminosity LHC (HL-LHC) [50, 51], the International Linear
Collider (ILC) [52, 53], and the Circular Electron-Positron Collider (CEPC)
[54] could further probe the remaining parameter space in this model [55].

In the present work, we will continue our study in the singlet extensions of
the SM, but focus on the so-called type-II EWPT. In this scenario, the
Universe experiences a two-step phase transition. In the first step, the
singlet scalar obtains a vacuum expectation value (VEV) while the EW vacuum
remains symmetric. As the temperature decreases to the critical temperature
$T_{c}$ at which there exist two degenerate vacua, the second step of phase
transition takes place in which the EW gauge symmetries are spontaneously
broken by the Higgs doublet VEV $v_{0}$ while the singlet VEV vanishes. In
addition to providing a potential barrier between the first vacuum and the EW
vacuum, another attractive aspect of this scenario is that it is possible to
introduce a new CPV source. For example, we can implement a dimension-6
effective operator
$\mathcal{O}_{6}=\frac{S^{2}}{\Lambda^{2}}\bar{Q}_{3L}\tilde{H}t_{R}+{\rm
H.c.}$ in the real scalar case with the $\mathbb{Z}_{2}$ symmetry to implement
the additional CPV source. This can survive low-energy CPV experiment
constraints [34, 56] due to its effects on, e.g., the electric dipole moment
of an electron, are highly loop suppressed by a vanishing singlet VEV in the
present Universe [57].

A scalar DM candidate can naturally arise in the type-II EWPT for the widely
studied singlet extension with a $\mathbb{Z}_{2}$ [34, 17, 56, 41, 57, 58, 59,
33] or $U(1)$ [60, 28, 61] symmetry. If the scalar VEV vanishes, such an
unbroken symmetry would make the lightest neutral scalar in the dark sector
stable, so that it can be a DM candidate. In this paper, we shall firstly
revisit the $\mathbb{Z}_{2}$ model and perform a scan over the relevant
parameter space of physical interest. We show the updated constraints on this
model and find that the parameter space for the type-II EWPT in the singlet
extensions with a $\mathbb{Z}_{2}$ symmetry (or a $U(1)$ symmetry) is largely
excluded by the constraints form DM direct detections, such as LUX [62],
PandaX-II [63], and XENON1T [64], which strongly disfavor WIMP models with a
spin-independent (SI) DM annihilation cross section larger than about
$10^{-45}-10^{-46}~{}{\rm cm^{2}}$. We then propose a complex singlet
extension of the SM with a $CP$ symmetry in which the real component of the
complex scalar can mix with the SM Higgs boson while the imaginary component
can be a DM candidate, as protected by the $CP$ symmetry. We will show that
the type-II EWPT can proceed successfully while various constraints from
collider and DM detection experiments can be avoided.

This work is presented as follows. In Sec. 2, we study the type-II EWPT and
the DM phenomenology in the real singlet extension of SM with a
$\mathbb{Z}_{2}$ symmetry. In particular, we calculate the DM-nucleon
scattering cross section for DM direct detection experiments in this model. We
then perform a scan of model parameter space based on the obtained analytic
results. We comment on the model with a $U(1)$ symmetry toward the end of the
section. In Sec. 3, we introduce the model of a complex singlet extension with
a $CP$ symmetry and discuss in detail its phenomenology. Also, we explore the
type-II EWPT in this model, as well as various DM phenomena, including the
invisible Higgs decay, DM relic density, and indirect and direct DM
detections. Finally, we summarize our findings in Sec. 4.

##  2 The Real Singlet Scalar Extension with a $\mathbb{Z}_{2}$ Symmetry

###  2.1 The model

Let’s first revisit the SM extension with a real singlet scalar, denoted by
$S$, which interacts with SM particles through the Higgs portal. By imposing a
$\mathbb{Z}_{2}$ symmetry [34, 17, 56, 41] associated with the real scalar,
$S\to-S$, the most general renormalizable scalar potential is given by

$\displaystyle
V(H,S)=-\mu_{h}^{2}|H|^{2}+\lambda_{h}|H|^{4}+\frac{1}{2}\mu_{s}^{2}S^{2}+\frac{1}{4}\lambda_{s}S^{4}+\frac{1}{2}\lambda_{m}|H|^{2}S^{2},$
(2.1)

where $H$ denotes the $SU(2)$ Higgs doublet, and $\lambda_{m}$ is the coupling
between the real scalar and the Higgs and plays important roles in both EW
phase transition and DM phenomenology. We then expand the Higgs and real
scalar fields around their classical backgrounds as

$H=\left(\begin{array}[]{c}{G^{+}}\\\
{\frac{1}{\sqrt{2}}\left(h+iG^{0}\right)}\end{array}\right),~{}~{}S=s,$ (2.2)

where $G^{\pm}$ and $G^{0}$are the SM Goldstone bosons. In terms of the
classical fields $h$ and $s$ we have the following tree-level scalar potential
at zero temperature:

$V_{0}(h,s)=-\frac{1}{2}\mu_{h}^{2}h^{2}+\frac{1}{4}\lambda_{h}h^{4}+\frac{1}{2}\mu_{s}^{2}s^{2}+\frac{1}{4}\lambda_{s}s^{4}+\frac{1}{4}\lambda_{m}h^{2}s^{2}.$
(2.3)

As shown in ref. [56], the potential $V_{0}$ can give rise to a potential
barrier at tree level in the type-II EWPT of this model. In the following we
will neglect the one-loop Coleman-Weinberg potential, which just corrects the
details of the phase transition but does not qualitatively change the general
picture of this model. Here we also briefly comment that for the type-I EWPT,
the $\mathbb{Z}_{2}$ symmetry in this model enforces the existence of two
local minima at $s=\pm w$, which prohibits a local minimum at $h=0$ [65].
Thus, it is impossible to have a tree-level barrier for the type-I EWPT.

In order to study the phase transition at finite temperature, we need to
include the thermal contributions to the scalar potential. The leading terms
of one-loop finite-temperature corrections in the high-temperature expansion
are given by

$V_{\rm T}(h,s,T)=\frac{1}{2}(c_{h}h^{2}+c_{s}s^{2})T^{2},$ (2.4)

where the parameters $c_{h}$ and $c_{s}$ in this model are

$\displaystyle c_{h}$ $\displaystyle=$
$\displaystyle\frac{1}{48}(9g^{2}+3{g}^{\prime
2}+12y_{t}^{2}+24\lambda_{h}+2\lambda_{m}),$ $\displaystyle c_{s}$
$\displaystyle=$ $\displaystyle\frac{1}{12}(3\lambda_{s}+2\lambda_{m}),$ (2.5)

with $g$ and ${g}^{\prime}$ the $SU(2)_{L}$ and $U(1)_{Y}$ gauge couplings
while $y_{t}$ the top quark Yukawa coupling. Consequently, the total effective
potential is given by

$V_{\rm eff}(h,s,T)=V_{0}(h,s)+V_{\rm T}(h,s,T).$ (2.6)

###  2.2 Phase transition

Following the approach proposed in Refs. [65, 57], we add a constant term
(with respect to the temperature) to the potential (2.6) so that the total
potential takes the form of Eq. (2.3) at the critical temperature $T_{c}$. In
this way, we have

$V(h,s,T)=-\frac{1}{2}\mu_{h}^{2}h^{2}+\frac{1}{4}\lambda_{h}h^{4}+\frac{1}{2}\mu_{s}^{2}s^{2}+\frac{1}{4}\lambda_{s}s^{4}+\frac{1}{4}\lambda_{m}h^{2}s^{2}-\frac{1}{2}(c_{h}h^{2}+c_{s}s^{2})(T_{c}^{2}-T^{2}).$
(2.7)

This method makes the analysis of the phase transition at the critical
temperature much more convenient. It is worth mentioning that this finite-
temperature potential is gauge-independent [66, 67, 68, 69, 70]. We can absorb
the last term of Eq. (2.7) into the ones quadratic in h and s by defining the
T-dependent parameters

$\displaystyle\mu_{h}^{2}(T)$ $\displaystyle=$
$\displaystyle\mu_{h}^{2}-c_{h}(T^{2}-T_{c}^{2}),$ (2.8)
$\displaystyle\mu_{s}^{2}(T)$ $\displaystyle=$
$\displaystyle\mu_{s}^{2}+c_{s}(T^{2}-T_{c}^{2}).$ (2.9)

As a result, one can relate the T-dependent parameters defined at the critical
temperature to those at zero temperature.

The local minima of the scalar potential at the critical temperature require
$\partial V_{c}/\partial h=0$ and $\partial V_{c}/\partial s=0$ where
$V_{c}\equiv V(T=T_{c})$. For the type-II EWPT with $(0,~{}w)\to(v,~{}0)$
111Here the notation refers to the VEVs of the two scalar fields: $(\langle
h\rangle,\langle s\rangle)$., we need the following solutions

$w^{2}=-\frac{\mu_{s}^{2}}{\lambda_{s}};~{}~{}v^{2}=\frac{\mu_{h}^{2}}{\lambda_{h}},$
(2.10)

where $v\equiv v(T=T_{c})$ and $w\equiv w(T=T_{c})$. In order to ensure the
vacuum stability, we demand

$\mu_{s}^{2}<0~{},~{}\lambda_{h}>0~{}{\rm and}~{}\lambda_{s}>0.$ (2.11)

Using Eqs. (2.8) and (2.10), the value of the Higgs doublet VEV at the
critical temperature $v$ is related to its counterpart at zero temperature
$v_{0}$ as follows

$v_{0}^{2}=v^{2}+\frac{c_{h}}{\lambda_{h}}T_{c}^{2}.$ (2.12)

where $v_{0}\equiv v(T=0)=246~{}{\rm GeV}$. We see that $v<v_{0}=246$ GeV is
aways established in this scenario because $c_{h},\lambda_{h}>0$.

The three elements of the scalar squared-mass matrix, evaluated at the EW
gauge symmetry broken minimum, are given by

$\left.M_{h}^{2}\equiv\frac{\partial^{2}V_{c}}{\partial h\partial
h}\right|_{b},~{}\left.M_{s}^{2}\equiv\frac{\partial^{2}V_{c}}{\partial
s\partial s}\right|_{b},~{}{\rm
and}~{}\left.M_{hs}^{2}\equiv\frac{\partial^{2}V_{c}}{\partial h\partial
s}\right|_{b},$ (2.13)

where the subscript b denotes the EW gauge symmetry broken vacuum $(v,~{}0)$.
At the critical temperature, there is another degenerate EW-symmetric vacuum
$(0,~{}w)$, which would be labelled by the subscript s in the following
discussions because only the $s$ field develops a nonzero VEV. For the model
considered here,

$\displaystyle
M_{h}^{2}=-\mu_{h}^{2}+3\lambda_{h}v^{2},~{}M_{s}^{2}=\mu_{s}^{2}+\frac{1}{2}\lambda_{m}v^{2},~{}{\rm
and}~{}M_{hs}^{2}=0\,,$ (2.14)

where $M_{h}^{2}>0$ and $M_{s}^{2}>0$ are required to ensure the existence of
the local minimum at the EW broken phase. Note that, with $\mu_{s}^{2}<0$, we
require the condition

$\lambda_{m}>0$ (2.15)

and sufficiently large to guarantee the positivity of $M_{s}^{2}$. We also
require

${\rm Det}\mathcal{M}^{2}\equiv M_{h}^{2}M_{s}^{2}-M_{hs}^{4}>0.$ (2.16)

so that both mass eigenvalues of the scalar squared-mass matrix are real and
positive. Similarly, $M_{h,s}^{2}\equiv\partial^{2}V_{c}/\partial
h^{2}|_{s}>0$ and $M_{s,s}^{2}\equiv\partial^{2}V_{c}/\partial s^{2}|_{s}>0$
are required to produce the local minimum at the EW symmetric vacuum. In order
for the vacuum $(0,~{}w)$ to be the deepest minimum along the $s$-axis, we
further impose the following condition [65]:

${\rm Det}\mathcal{M}^{2}>\frac{v^{2}}{w^{2}}M_{h}^{2}M_{h,s}^{2}.$ (2.17)

Finally, using Eqs. (2.8), (2.9), and (2.10), we obtain the T-dependent masses

$\displaystyle M_{h}^{2}(T)$ $\displaystyle=$ $\displaystyle
2\lambda_{h}\left[v^{2}-\frac{c_{h}}{\lambda_{h}}(T^{2}-T_{c}^{2})\right],$
(2.18) $\displaystyle M_{s}^{2}(T)$ $\displaystyle=$ $\displaystyle
M_{s}^{2}-\left(\frac{\lambda_{m}}{2\lambda_{h}}c_{h}-c_{s}\right)\left(T^{2}-T_{c}^{2}\right).$
(2.19)

The singlet mass at zero temperature is determined to be

$m_{s}^{2}\equiv
M_{s}^{2}(T=0)=M_{s}^{2}+\left(\frac{\lambda_{m}}{2\lambda_{h}}c_{h}-c_{s}\right)T_{c}^{2}.$
(2.20)

The degeneracy of the two vacua at the critical temperature implies
$V_{c}(0,~{}w)=V_{c}(v,~{}0)$. With the relations (2.10) and (2.14) and by
noting $\mu_{s}^{2}=M_{s}^{2}-\lambda_{m}v^{2}/2<0$, we obtain

$M_{s}^{2}=\frac{1}{2}v^{2}(\lambda_{m}-\sqrt{\lambda_{h}\lambda_{s}})$ (2.21)

By combining Eqs. (2.12) and (2.20), the singlet mass at zero temperature is
given by

$m_{s}^{2}=\frac{1}{2}\left(\lambda_{m}-2\sqrt{\lambda_{h}\lambda_{s}}\right)v_{0}^{2}+\left(c_{h}\sqrt{\frac{\lambda_{s}}{\lambda_{h}}}-c_{s}\right)T_{c}^{2}.$
(2.22)

As we shall show below, this relation requires $m_{s}$ to be smaller than
about $400$ GeV.

In order to have a correct direction of the EWPT, the EW broken minimum should
decrease faster than the symmetric one as the temperature drops. This can be
expressed as the following condition

$\left.\frac{d\Delta V_{bs}(T)}{dT^{2}}\right|_{T_{c}}>0,$ (2.23)

where $\Delta V_{bs}(T)=V(h,s,T)|_{b}-V(h,s,T)|_{s}$. This condition can be
transformed into the form

$\frac{c_{h}}{c_{s}}>\sqrt{\frac{\lambda_{h}}{\lambda_{s}}}$ (2.24)

in this model. One can easily confirm that this condition also ensures the EW
broken minimum to be the global minimum at zero temperature. The above general
considerations of the parameters are sufficient to ensure a successful type-II
EWPT in this scalar extension of the SM model.

Finally, we require the perturbative unitarity conditions [71]

$\lambda_{h}<4\pi,~{}\lambda_{s}<4\pi,~{}|\lambda_{m}|<8\pi,~{}3\lambda_{h}+2\lambda_{s}+\sqrt{\left(3\lambda_{h}-2\lambda_{s}\right)^{2}+2\lambda_{m}^{2}}<8\pi\,.$
(2.25)

By considering all relations discussed above, we are left with only three free
parameters in this real scalar model. Here we take
$\lambda_{s},~{}\lambda_{m},~{}{\rm and}~{}v/T_{c}$ as our input parameters
and make a random scan of the following parameter space:

$10^{-3}<\lambda_{s}<5,~{}~{}10^{-3}<\lambda_{m}<5,~{}~{}{\rm
and}~{}~{}1<\frac{v}{T_{c}}<10.$ (2.26)

Note that too large values of $\lambda_{s}$ and $\lambda_{m}$ would violate
the perturbative unitarity, and the upper limit on $v/T_{c}$ is to guarantee
the validity of high temperature expansions. Furthermore, we impose the
condition $v/T_{c}>1$ at the critical temperature in order to ensure a
sufficiently strong first-order EWPT [8]. We would like to mention that this
condition is necessary to generate a large enough baryon asymmetry through the
EW baryogenesis mechanism, since it helps suppress the washout effects caused
by the sphaleron process. Some discussions concerning the reliability of this
criterion can be found in Ref. [67].

Figure 1: Scatter plots of parameters in the $\lambda_{s}$-$m_{s}$ plane
(left) and the $\lambda_{m}$-$m_{s}$ plane (right) that can generate a
successful type-II EWPT. The colored bar indicates the corresponding critical
temperature.

In Fig. 1, the colored region shows the distributions of points in the
$\lambda_{s}-m_{s}$ plane (left plot) and the $\lambda_{m}-m_{s}$ plane (right
plot) that can trigger a sufficiently strong first-order type-II EWPT in the
model, with the color indicating the corresponding critical temperature for
the chosen parameters. The plot shows that the real scalar mass $m_{s}$ is
bounded to be less than $\sim 400$ GeV, as alluded to before. Furthermore, the
Higgs portal coupling $\lambda_{m}$ should be larger than $\sim 2\times
10^{-2}$. However, such large values of $\lambda_{m}$ would give rise to DM-
nucleon scatterings and DM annihilation signals. Thus, as discussed below,
this model suffers from strong constraints from DM detection experiments.

###  2.3 Dark matter phenomenology

After the EWPT, the VEV of the real singlet vanishes and the unbroken
$\mathbb{Z}_{2}$ symmetry protects it as a DM candidate. In the standard
freeze-out scenario, the DM particles are in chemical equilibrium with the
other SM particles via the annihilation-production reactions in the early
Universe. With the adiabatic expansion of the Universe, the DM population
becomes nonrelativistic and begins to decouple from the thermal bath at the
time with $m_{\rm DM}/T\sim 20$ when the annihilation rate falls behind the
cosmological expansion. The evolution of the DM number density is then
described by the following Boltzmann equation [72]:

$\frac{dY}{dT}=\sqrt{\frac{\pi g_{*}(T)}{45}}M_{\rm pl}\left\langle\sigma
v_{\mathrm{rel}}\right\rangle\left[Y(T)^{2}-Y_{\mathrm{eq}}(T)^{2}\right],$
(2.27)

where the abundance $Y(T)$ ($Y_{\rm eq}(T)$) denotes the ratio of the actual
(thermal equilibrium) DM number density to the entropy density,
$M_{\rm{pl}}=1.22\times 10^{19}~{}\mathrm{GeV}$ is the Planck mass, $g_{\ast}$
is the effective number of relativistic degrees of freedom, and
$\left\langle\sigma v_{\mathrm{rel}}\right\rangle$ is the thermally averaged
annihilation cross section times the relative velocity. The resulting DM relic
density is usually parametrized by

$h^{2}\Omega_{\rm DM}=2.742\times 10^{8}Y_{0}\frac{m_{\chi}}{\mathrm{GeV}},$
(2.28)

where $Y_{0}$ is the abundance of DM in the present Universe and $m_{\chi}$
denotes the DM mass. The analysis of the Planck satellite’s observations of
the CMB radiation [3] gives $h^{2}\Omega_{\mathrm{DM}}^{\rm obs}=0.12$. In our
numerical studies, we make use of the MicrOMEGAs 5.0.4 package [73] to
calculate the DM relic density.

In the left plot of Fig. 2, we show the obtained DM relic density of the
sample points that can trigger a successful type-II EWPT. For $m_{s}\lesssim
m_{h}/2$ where $m_{h}=125$ GeV is the mass of the observed Higgs boson, a good
portion of the parameter points are excluded due to their extremely large DM
relic density, indicating that the corresponding DM annihilation rates are too
small. The resonant DM annihilation occurs at $m_{s}\sim m_{h}/2$, which
results in a sharp decrease of the DM relic abundance, as shown by the dip in
the plot. When $m_{s}\gtrsim m_{h}$, the real scalar DM has a negligible relic
density with $h^{2}\Omega_{\mathrm{DM}}\sim 10^{-6}-10^{-5}$, which can be
explained by two reasons. For one thing, as the DM mass increases, the DM
annihilation channels to the massive gauge bosons $W^{\pm}$ and $Z$, as well
as the Higgs boson $h$ open up, leading to a decrease in the DM relic density.
The other reason lies in the fact that as $m_{s}$ increases a larger Higgs
portal coupling is required in order to induce the type-II EWPT, as clearly
shown in the right plot of Fig. 1. In particular, when $m_{s}\gtrsim m_{h}$
and $\lambda_{m}\gtrsim 1.0$, it would lead to an efficient DM annihilation to
reduce the DM density.

Figure 2: Left: DM thermal relic density as a function of the DM mass. The
black line denotes the DM relic density derived from the Planck satellite’s
observationCMB radiation [3]. Right: effective SI DM-nucleon elastic
scattering cross section as a function of the DM mass. The blue and red curves
denote the upper limits from LUX [62] and XENON1T [64] experiments,
respectively. The black dashed curve represents the potential constraints from
the future XENONnT project [78]. In both plots, the green scatter points
represent the samples that can produce a successful type-II EWPT.

We now turn to the DM direct detection constraint. For the present Higgs-
portal DM models, the cross section of the SI DM-nucleon elastic scattering is
given by

$\sigma_{\rm{SI}}=\frac{\lambda_{m}^{2}m_{N}^{4}f^{2}}{\pi
m_{s}^{2}m_{h}^{4}},$ (2.29)

where $m_{N}$ is the nucleon mass and $f\simeq 0.3$ is the form factor [74,
75, 20, 76, 77]. Here we have used the MicrOMEGAs package [73] to compute
$\sigma_{\rm SI}$. Furthermore, in order to compare with the experimental
upper bounds, we have to scale the obtained scattering cross section as [56]

$\tilde{\sigma}_{\rm SI}=f_{X}\sigma_{\rm SI},$ (2.30)

where the dark matter fraction

$f_{X}\equiv\frac{h^{2}\Omega_{\rm DM}}{h^{2}\Omega_{\rm DM}^{\rm obs}}.$
(2.31)

Theoretically, the dark matter fraction should be $f_{X}\leq 1$; otherwise,
the Universe would be over-closed. In our numerical calculations, however, we
do not impose this restriction in the right plot of Fig. 2 and always apply
the definition in Eq. (2.31), since it does not affect our final numerical
results by simultaneously considering the constraints from the DM relic
density and DM direct detections.

The prediction of the DM-nucleon scattering cross sections for the parameter
points surviving from the requirement of the type-II EWPT are displayed in the
right plot of Fig. 2. In the same plot, we also show the most stringent
constraints from the DM direct detection experiments up to date, with the blue
curve denoting the latest upper limit from the LUX experiment [62] and the red
curve from the XENON1T experiment [64]. It is seen that there are two relevant
parameter regions which are still compatible with the strong constraints from
the DM direct detections: one is the Higgs resonant region and the other is
the region with a higher DM mass $100~{}{\rm GeV}\lesssim m_{s}\lesssim
400~{}{\rm GeV}$. Note that both allowed regions predict that the real singlet
scalar has a negligible contribution to the DM relic density with a fraction
of $f_{X}<10^{-4}$, which suppresses the DM-nucleon signal to a level beyond
the probe of current direct detection experiments. Moreover, as evident in the
plot, most parameter points in these two regions could be further probed or
excluded by the future direct detection experiment XENONnT [78], an upgraded
version of XENON1T, as indicated by the black dashed curve.

Before closing this section, we note that the complex singlet scalar extension
of SM with a global or local $U(1)$ symmetry (in which case the scalar carries
a $U(1)$ charge of $q=1$) can have a potential structure much like the real
singlet model with a $\mathbb{Z}_{2}$ symmetry. After the type-II EWPT, this
$U(1)$ symmetry would be recovered, thus protecting the complex scalar from
decaying. Therefore, the complex scalar can be a DM candidate. In this case,
the only way for the DM particle to communicate with the SM particle is via
the Higgs portal coupling, which would also suffer from the strong constraints
of DM experiments. Therefore, we expect that the final parameter space allowed
by the constraints from the EWPT and the DM phenomenology would be similar to
the real singlet case. We have confirmed this expectation with almost the same
calculations as those performed in this section.

##  3 The Complex Singlet Scalar Extension with a CP Symmetry

###  3.1 The model

From the above discussions we know that in the singlet extension with a
$\mathbb{Z}_{2}$ symmetry or a $U(1)$ symmetry, the type-II EWPT favors larger
values of the Higgs portal coupling $\lambda_{m}$, which, however, suffers
from the strong constraints of the DM relic density and direct detections. In
order to get out of this dilemma, we need a scalar potential with a different
structure. For this purpose, we consider the complex singlet extension of the
SM with a “CP symmetry” $S\to S^{*}$, under which the most general
renormalizable scalar potential can be written as

$\displaystyle V(H,S)=$ $\displaystyle-$
$\displaystyle\mu_{h}^{2}|H|^{2}+\lambda_{h}|H|^{4}-\mu_{1}^{2}(S^{*}S)-\frac{1}{2}\mu_{2}^{2}(S^{2}+S^{*2})+\lambda_{1}(S^{*}S)^{2}+\frac{1}{4}\lambda_{2}(S^{2}+S^{*2})^{2}$
(3.1) $\displaystyle+$
$\displaystyle\frac{1}{2}\lambda_{3}(S^{*}S)(S^{2}+S^{*2})+\kappa_{1}|H|^{2}(S^{*}S)+\frac{1}{2}\kappa_{2}|H|^{2}(S^{2}+S^{*2})+\frac{1}{\sqrt{2}}a_{1}^{3}(S+S^{*})$
$\displaystyle+$
$\displaystyle\frac{1}{2\sqrt{2}}b_{m}|H|^{2}(S+S^{*})+\frac{\sqrt{2}}{3}c_{1}(S^{*}S)(S+S^{*})+\frac{\sqrt{2}}{3}c_{2}(S^{3}+S^{*3}),$

where all the parameters in the potential are assumed to be real. Note that
the operator $(S^{3}+S^{*3})(S+S^{*})$ has already been contained in the sixth
and seventh term of Eq. (3.1). By expanding the complex scalar $S$ in terms of
the background fields, $S=(s+i\chi)/\sqrt{2}$, and the doublet field as in Eq.
(2.2), the scalar potential can be rewritten as

$\displaystyle V(h,s,\chi)=$ $\displaystyle-$
$\displaystyle\frac{1}{2}\mu_{h}^{2}h^{2}-\frac{1}{2}\mu_{s}^{2}s^{2}-\frac{1}{2}\mu_{\chi}^{2}\chi^{2}+\frac{1}{4}\lambda_{h}h^{4}+\frac{1}{4}\lambda_{s}s^{4}+\frac{1}{4}\lambda_{\chi}\chi^{4}+\frac{1}{2}\lambda_{a}s^{2}\chi^{2}$
(3.2) $\displaystyle+$
$\displaystyle\frac{1}{4}\kappa_{s}h^{2}s^{2}+\frac{1}{4}\kappa_{\chi}h^{2}\chi^{2}+a_{1}^{3}s+\frac{1}{4}b_{m}h^{2}s+\frac{1}{3}c_{s}s^{3}+\frac{1}{3}c_{\chi}s\chi^{2},$

where

$\displaystyle\mu_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\mu_{s}^{2}+\mu_{\chi}^{2}),~{}~{}\mu_{2}^{2}=\frac{1}{2}(\mu_{s}^{2}-\mu_{\chi}^{2}),$
$\displaystyle\kappa_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\kappa_{s}+\kappa_{\chi}),~{}~{}\kappa_{2}=\frac{1}{2}(\kappa_{s}-\kappa_{\chi}),$
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(3c_{s}+c_{\chi}),~{}~{}~{}c_{2}=\frac{1}{4}(c_{s}-c_{\chi}),$
(3.3)

and

$\displaystyle\lambda_{1}=\frac{1}{2}\left[\frac{1}{2}(\lambda_{s}+\lambda_{\chi})+\lambda_{a}\right],~{}\lambda_{2}=\frac{1}{2}\left[\frac{1}{2}(\lambda_{s}+\lambda_{\chi})-\lambda_{a}\right],~{}\lambda_{3}=\frac{1}{2}(\lambda_{s}-\lambda_{\chi}).$
(3.4)

We assume that the pseudoscalar $\chi$ does not develop a VEV during the whole
process of the type-II EWPT. Note that we can get rid of the parameter
$a_{1}^{3}$ or $c_{s}$ by shifting the singlet $s$ with a constant. However,
we refrain from doing so for later convenience [65].

###  3.2 Phase transition

Following the analysis procedure in section 2, we rewrite the total effective
potential at finite temperature as

$\displaystyle V_{\rm
eff}(h,s,\chi,T)=V(h,s,\chi)-\frac{1}{2}\left(g_{h}h^{2}+g_{s}s^{2}+g_{\chi}{\chi}^{2}+2m_{3}s\right)(T_{c}^{2}-T^{2}),$
(3.5)

where

$\displaystyle g_{h}$ $\displaystyle=$
$\displaystyle\frac{3}{16}g^{2}+\frac{1}{16}{g}^{\prime
2}+\frac{1}{4}y_{t}^{2}+\frac{1}{2}\lambda_{h}+\frac{1}{24}(\kappa_{s}+\kappa_{\chi}),$
$\displaystyle g_{s}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[\frac{1}{2}(\lambda_{s}+\lambda_{\chi})+\lambda_{a}+\kappa_{s}\right]+\frac{1}{8}(\lambda_{s}-\lambda_{\chi}),$
$\displaystyle g_{\chi}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[\frac{1}{2}(\lambda_{s}+\lambda_{\chi})+\lambda_{a}+\kappa_{\chi}\right]-\frac{1}{8}(\lambda_{s}-\lambda_{\chi}),$
$\displaystyle m_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{12}(b_{m}+c_{s}+c_{\chi}).$ (3.6)

As before, it is more convenient to combine the two parts in $V_{\rm
eff}(h,s,\chi,T)$ and define four temperature-dependent parameters:

$\displaystyle\mu_{h}^{2}(T)$ $\displaystyle=$
$\displaystyle\mu_{h}^{2}-g_{h}(T^{2}-T_{c}^{2}),~{}~{}\mu_{s}^{2}(T)=\mu_{s}^{2}-g_{s}(T^{2}-T_{c}^{2}),$
$\displaystyle\mu_{\chi}^{2}(T)$ $\displaystyle=$
$\displaystyle\mu_{\chi}^{2}-g_{\chi}(T^{2}-T_{c}^{2}),~{}~{}a_{1}^{3}(T)=a_{1}^{3}+m_{3}(T^{2}-T_{c}^{2}).$
(3.7)

With the above relations we have

$\displaystyle\mu_{h}^{2}\equiv\mu_{h}^{2}(T_{c}),~{}\mu_{s}^{2}\equiv\mu_{s}^{2}(T_{c}),~{}\mu_{\chi}^{2}\equiv\mu_{\chi}^{2}(T_{c}),~{}{\rm
and~{}}a_{1}^{3}\equiv a_{1}^{3}(T_{c}).$ (3.8)

From now on, all parameters in Eq. (3.2) are re-defined to be those at the
critical temperature $T_{c}$.

The stationary points of the effective potential lie on the curves along which
$\partial V_{\rm eff}/\partial\phi_{i}=0$ where $\phi_{i}=h,~{}s,~{}{\rm
and}~{}\chi$. For the type-II EWPT, $(0,w(T),0)\to(v(T),0,0)$, we have the
following conditions

$\mu_{h}^{2}(T)=\lambda_{h}v(T)^{2},~{}b_{m}=-\frac{4a_{1}^{3}(T)}{v(T)^{2}},~{}{\rm
and}~{}c_{s}=-\frac{1}{w(T)^{2}}(a_{1}^{3}(T)-\mu_{s}^{2}(T)w(T)+\lambda_{s}w(T)^{3}).$
(3.9)

The VEV of the EW Higgs doublet at the critical temperature $v\equiv v(T_{c})$
is related to its value at zero temperature $v_{0}$ by

$v^{2}=v_{0}^{2}-\frac{g_{h}}{\lambda_{h}}T_{c}^{2}.$ (3.10)

Using the condition of vacua degeneracy at the critical temperature, i.e.,
$V_{\rm eff}(0,w,0,T_{c})=V_{\rm eff}(v,0,0,T_{c})$ where $w\equiv w(T_{c})$,
we derive

$a_{1}^{3}=\frac{1}{8w}\left(\lambda_{s}w^{4}-3\lambda_{h}v^{4}+2\mu_{s}^{2}w^{2}\right).$
(3.11)

The field-dependent scalar mass matrix elements are determined by
${M}_{\phi_{i}\phi_{j}}^{2}=\partial^{2}V_{\rm
eff}/\partial\phi_{i}\partial\phi_{j}$. Due to the $CP$-odd nature of the
pseudoscalar $\chi$, it cannot mix with other two scalars $h$ and $s$. On the
other hand, $h$ and $s$ fields can mix with each other, as induced by the term
proportional to $b_{m}$ in the potential. Through an orthogonal rotation, we
can diagonalize the mass matrix and define the following mass eigenstates:

$\displaystyle\begin{pmatrix}\mathcal{H}\\\
\mathcal{S}\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}h\\\ s\end{pmatrix}.$ (3.12)

At zero temperature, the mass matrix elements of $h$ and $s$ can be related to
physical parameters, such as the masses of the two scalars
$m_{\mathcal{H}}=125$ GeV (the SM Higgs mass) and $m_{\mathcal{S}}$, as well
as the mixing angle $\theta$, as given by the following relations:

$\displaystyle\begin{split}M_{hh}^{2}&\equiv\cos^{2}\theta
m_{\mathcal{H}}^{2}+\sin^{2}\theta
m_{\mathcal{S}}^{2}=2\lambda_{h}v_{0}^{2},\\\ M_{ss}^{2}&\equiv\sin^{2}\theta
m_{\mathcal{H}}^{2}+\cos^{2}\theta
m_{\mathcal{S}}^{2}=-\mu_{s}^{2}(0)+\frac{1}{2}\kappa_{s}v_{0}^{2},\\\
M_{hs}^{2}&\equiv\cos\theta\sin\theta(m_{\mathcal{S}}^{2}-m_{\mathcal{H}}^{2})=\frac{1}{2}b_{m}v_{0},\end{split}$
(3.13)

where $M_{\phi_{i}\phi_{j}}^{2}$ take values at the broken phase
$(v_{0},0,0)$. As a result of the mixing, the physical real scalar
$\mathcal{S}$ is unstable and can decay into SM particles. On the other hand,
the pseudoscalar $\chi$ is still stable and becomes a DM candidate since it is
protected by the imposed CP symmetry.

In summary, to trigger a type-II EWPT in this complex scalar model with a CP
symmetry, the model parameters should satisfy the following relations:

$\displaystyle\lambda_{h}$ $\displaystyle=$
$\displaystyle\frac{M_{hh}^{2}}{2v_{0}^{2}},~{}~{}\mu_{h}^{2}=\lambda_{h}v^{2},~{}~{}b_{m}=\frac{2M_{hs}^{2}}{v_{0}},~{}~{}a_{1}^{3}=-\frac{b_{m}v^{2}}{4},$
$\displaystyle\lambda_{s}$ $\displaystyle=$
$\displaystyle\frac{36\lambda_{h}v^{4}+w[96a_{1}^{3}+(24M_{ss}^{2}+4\kappa_{s}T_{c}^{2}+4\lambda_{a}T_{c}^{2}-\lambda_{\chi}T_{c}^{2}-12\kappa_{s}v_{0}^{2})w]}{12w^{4}-5T_{c}^{2}w^{2}},$
$\displaystyle\mu_{s}^{2}$ $\displaystyle=$
$\displaystyle\frac{15\lambda_{h}T_{c}^{2}v^{4}+40a_{1}^{3}T_{c}^{2}w+(24M_{ss}^{2}+4\kappa_{s}T_{c}^{2}+4\lambda_{a}T_{c}^{2}-\lambda_{\chi}T_{c}^{2}-12\kappa_{s}v_{0}^{2})w^{4}}{10T_{c}^{2}w^{2}-24w^{4}},$
$\displaystyle c_{s}$ $\displaystyle=$
$\displaystyle-\frac{1}{w^{2}}(a_{1}^{3}-\mu_{s}^{2}w+\lambda_{s}w^{3}).$
(3.14)

The mass of the pseudoscalar $\chi$ is given by

$m_{\chi}^{2}=-\mu_{\chi}^{2}(0)+\frac{1}{2}\kappa_{\chi}v_{0}^{2}.$ (3.15)

From the potential given in Eq. (3.2), we see that the parameters
$\kappa_{\chi}$, $\lambda_{a}$, $c_{\chi}$ and $\lambda_{\chi}$ are directly
related to the properties of the DM candidate $\chi$ so that they play
important roles in the DM phenomenology. On the other hand, the same
parameters would also affect significantly the type-II EWPT via their thermal
contributions to the potential. Therefore, the EWPT and the DM physics are
closely related to each other in the present model.

There are various theoretical constraints on the model parameters. The
condition corresponding to Eq. (2.23) ensures a correct direction of the EWPT
and requires

$g_{h}v^{2}-w(g_{s}w+2m_{3})>0\,.$ (3.16)

Again, the same condition also warrants that the EW broken minimum at zero
temperature is a global minimum. Other bounds on the parameters have been
discussed in Sec. 2, a summary of which can be found in Table 1 of Ref. [65].

By taking into account the above relations and conditions, we take

$\\{w,~{}m_{\mathcal{S}},~{}\theta,~{}v/T_{c},~{}\lambda_{a},~{}\lambda_{\chi},~{}\kappa_{a},~{}\kappa_{\chi},~{}c_{\chi},~{}m_{\chi}\\}.$
(3.17)

as our input parameter set. When searching for the parameter space of the
type-II EWPT, we fix

$\lambda_{\chi}=0.1,~{}~{}{\rm and}~{}~{}\kappa_{\chi}=0,$ (3.18)

as a concrete example and make a random scan of the other eight parameters in
the following ranges:

$\displaystyle 1<\frac{w}{\rm GeV}<2\times
10^{3},~{}1<\frac{m_{\mathcal{S}}}{\rm GeV}<2\times
10^{3},~{}1<\frac{m_{\chi}}{\rm GeV}<2\times 10^{3},~{}-0.4<\theta<0.4,$
$\displaystyle
1<\frac{v}{T_{c}}<10,~{}10^{-3}<\kappa_{s}<1,~{}10^{-3}<\lambda_{a}<1,~{}-10^{3}<\frac{c_{\chi}}{\rm
GeV}<10^{3}.$ (3.19)

Figure 3: Distributions of parameters that can generate a sufficiently strong
type-II EWPT in the $CP$-symmetric complex singlet extension of the SM.

The model parameter space is well constrained by the current LHC measurements
on various Higgs decay channels, including
$\gamma\gamma,WW^{*},ZZ^{*},b\bar{b}$, and $\tau^{+}\tau^{-}$. It is
remarkable [25, 23] that in the Higgs portal model all these channels have
only one common Higgs signal strength:

$\mu_{\mathcal{H}}=\frac{\Gamma_{\mathcal{H}}^{\mathrm{SM}}\cos^{4}\theta}{\Gamma_{\mathcal{H}}^{\mathrm{SM}}\cos^{2}\theta+\Gamma_{\mathcal{H}\rightarrow\chi\chi}+\Gamma_{\mathcal{H}\rightarrow\mathcal{S}\mathcal{S}}}~{},$
(3.20)

where $\Gamma_{\mathcal{H}}^{\rm SM}$ is the total Higgs decay width in the SM
and $\Gamma_{\mathcal{H}\rightarrow\chi\chi}$ and
$\Gamma_{\mathcal{H}\rightarrow\mathcal{S}\mathcal{S}}$ refer to those of the
exclusive channels for the SM-like Higgs decays into particles in the dark
sector. It is obvious that the Higgs signal strength is suppressed by two
factors: $\cos^{2}\theta$ and the presence of new decay channels. In our
previous work [49], we have shown that the Higgs signal strength measurements
at the LHC restrict the mixing angle $|\theta|\lesssim 0.4$. In this section,
we have restricted the scan range of $\theta$ in accordance with this bound.

In the numerical scan, we generate one million random parameter points
uniformly distributed in the range of Eq. (3.2). About $\sim 3.5\%$ of the
points are found to be able to trigger a strong type-II EWPT while fulfilling
other theoretical and experimental constraints. Furthermore, we show the
distributions of the input and derived parameters relevant to the type-II EWPT
in Fig. 3. The distribution of the real scalar VEV $w$ defined at the critical
temperature peaks around 400 GeV and can extend up to 2 TeV. The type-II EWPT
favors a relatively light $\cal{S}$ whose mass concentrates at $m_{\cal S}\sim
200$ GeV but can be as large as $\sim 1$ TeV, in comparison with the real
singlet model with a $\mathbb{Z}_{2}$ symmetry in Sec. 2. Also contrary to the
real scalar case where the type-II phase transition requires large values of
the Higgs portal coupling $\lambda_{m}$, most values of $\kappa_{s}$ in the
current model are small and located in the bin of $0-0.2$. The critical
temperature $T_{c}$ falls in the range of $\sim 25-125$ GeV, and the triple
Higgs coupling $\lambda_{h}$ falls mainly in the range of $0.1-0.3$. Note that
a common feature for models to generate a strong first-order EWPT is the
prediction of large deviations in the cubic and quartic Higgs couplings from
the SM values. Therefore, precision measurements of the Higgs self-couplings
via di-Higgs production at the future colliders, including HL-LHC, CEPC, and
ILC, can be used to reconstruct the Higgs potential so as to confirm the
nature of the EWPT [30, 79, 80, 81, 82]. Finally, the allowed ranges of some
other parameters relevant to the EWPT are summarized as follows:

$\displaystyle 0\lesssim\lambda_{a}\lesssim
0.3,~{}0\lesssim\lambda_{s}\lesssim 5,~{}-300\lesssim\frac{b_{m}}{\rm
GeV}\lesssim 1000,$ $\displaystyle-2500\lesssim\frac{c_{s}}{\rm GeV}\lesssim
0,~{}-800\lesssim\frac{c_{\chi}}{\rm GeV}\lesssim 200.$ (3.21)

###  3.3 Dark matter phenomenology

In our model, the imaginary component $\chi$ of the complex scalar is found to
be a DM candidate owing to the CP symmetry of the scalar potential. From the
potential in Eq. (3.2), it is clear that there always exists a
$\mathbb{Z}_{2}$ symmetry acting on $\chi$ as $\chi\to-\chi$, which is just
another representation of the $CP$ symmetry. As noted above, the real
component $s$ can mix with the Higgs boson $h$ even though it has a vanishing
VEV at the final step of the type-II EWPT. Thus, the DM candidate $\chi$ can
interact with SM particles via the mediation of the mixture of both particles,
leading to observable phenomena in the DM experiments.

There are thirteen parameters in the potential Eq. (3.2), some of which can be
related to the physical parameters $v_{0}$, $m_{\mathcal{H}}$, $\theta$,
$m_{\mathcal{S}}$, and $m_{\chi}$ with the following relations:

$\displaystyle\lambda_{h}$ $\displaystyle=$
$\displaystyle\frac{M_{hh}^{2}}{v_{0}^{2}},~{}\mu_{h}^{2}(0)=\lambda_{h}v_{0}^{2},~{}b_{m}=\frac{2M_{hs}^{2}}{v_{0}},$
$\displaystyle\mu_{s}^{2}(0)$ $\displaystyle=$ $\displaystyle-
M_{ss}^{2}+\frac{1}{2}\kappa_{s}v_{0}^{2},~{}\mu_{\chi}^{2}(0)=-m_{\chi}^{2}+\frac{1}{2}\kappa_{\chi}v_{0}^{2},$
$\displaystyle a_{1}^{3}(0)$ $\displaystyle=$
$\displaystyle-\frac{1}{4}b_{m}v_{0}^{2}.$ (3.22)

The equations in the first and second lines of Eq. (3.3) are obtained by
requiring $\partial V_{\rm eff}/\partial h=0$ in the EW symmetry broken phase
$(v_{0},0,0)$ of the present Universe, while the last equation of Eq. (3.3)
represents the condition $\partial V_{\rm eff}/\partial s=0$ in the same
phase.

we shall focus on five benchmark models with parameters summarized in Table 1
in order to highlight the prominent roles played by these parameters in the
dark matter phenomenology. In addition, other physical parameters are fixed to
the following values:

$\lambda_{\chi}=0.1,~{}\kappa_{\chi}=0,~{}\lambda_{s}=0.2,~{}c_{s}=-100~{}{\rm
GeV},~{}{\rm and}~{}\kappa_{s}=0.1.$ (3.23)

Note that the mixing angle $\theta$ and $c_{\chi}$ are directly related to the
interaction between the DM particle $\chi$ and SM particles via $\mathcal{H}$
and $\mathcal{S}$. Hence, direct detection experiments are sensitive to these
two parameters. Also, the coupling $\lambda_{a}$ will play a significant role
in the DM annihilation when $m_{\chi}>m_{\mathcal{S}}$. The choices of the
parameters here fall in the ranges that are preferred by the type-II EWPT, as
shown in Fig. 3. Our calculations below are again based upon the MicrOMEGAs
package [73].

Table 1:  A summary of parameters for the five benchmark models. Model | $\theta$ | $\lambda_{a}$ |  $c_{\chi}$ [GeV]
---|---|---|---
A | 0.1 | 0.5 | $-100$
B | 0.1 | 0.1 | $-100$
C | 0.3 | 0.5 | $-100$
D | 0.1 | 0.5 | $-300$
E | 0.2 | 0.3 | $-200$

####  3.3.1 Invisible Higgs decay

Figure 4: Invisible Higgs decay branching ratio as a function of $\theta$. The
blue solid line, pink dotted line, and yellow dashed line correspond
respectively to $c_{\chi}=-1$ GeV, $c_{\chi}=-5$ GeV, and $c_{\chi}=-40$ GeV.
In this plot, we fix the DM mass $m_{\chi}=20$ GeV, the real scalar mass
$m_{\mathcal{S}}=400$ GeV, and the coupling $\lambda_{a}=0.5$, with the other
parameters given in Eq. (3.23). The black solid, dotted, and dashed lines
correspond respectively to the upper limit on the invisible Higgs decay
branching ratio from the current LHC analysis [84], the future HL-LHC [85] and
CEPC [86].

For $m_{\chi}<m_{\mathcal{H}}/2$, the $\mathcal{H}\to\chi\chi$ decay is
kinematically allowed. Its partial decay width is

$\Gamma_{\mathcal{H}\to\chi\chi}=\frac{\lambda_{H\chi\chi}^{2}}{32\pi
m_{\mathcal{H}}}\sqrt{1-\frac{4m_{\chi}^{2}}{m_{\mathcal{H}}^{2}}},$ (3.24)

where

$\lambda_{\mathcal{H}\chi\chi}=-\kappa_{\chi}v_{0}\cos\theta+\frac{2}{3}c_{\chi}\sin\theta.$
(3.25)

In Eq. (3.25), the first term proportional to $\kappa_{\chi}$ represents the
widely studied Higgs portal originated from the $h^{2}\chi^{2}$ operator in
the potential (3.2), and the second term stems from the mixing between $h$ and
$s$ and hence the suppression factor of mixing angle. For simplicity, we focus
on the case with $\kappa_{\chi}=0$, i.e., we assume $\kappa_{1}=\kappa_{2}$
for the initial potential in Eq. (3.1). Fig. 4 shows the
$\mathcal{H}\to\chi\chi$ branching ratio as a function of $\theta$. In this
plot, we have assumed the DM mass $m_{\chi}=20$ GeV, the real scalar mass
$m_{\mathcal{S}}=400$ GeV, and the coupling $\lambda_{a}=0.5$. The other
parameters are given in Eq. (3.23). In the SM, the dominant Higgs decay
channel is $\mathcal{H}\to b\bar{b}$. However, when the invisible Higgs decay
opens up, the ratio of the invisible decay width to the $\mathcal{H}\to
b\bar{b}$ decay width is estimated to be
$R\simeq\lambda_{\mathcal{H}\chi\chi}^{2}/(\lambda_{f}^{2}m_{\mathcal{H}}^{2})=1.7c_{\chi}^{2}\sin^{2}\theta/(m_{b}^{2}\cos^{2}\theta)$,
where $\lambda_{f}=m_{b}\cos\theta/v_{0}$ and $m_{b}\simeq 4$ GeV is the
bottom quark mass GeV [83]. Thus, for $\theta\sim 0.1$ and $|c_{\chi}|\sim 40$
GeV, the invisible decay becomes competitive with the $b\bar{b}$ channel,
which can be seen in Fig. 4. Currently, the upper limit on the branching ratio
of the invisible Higgs decay is 19% by the LHC [84] (black solid line), and
this limit can be further improved to 5.6% at HL-LHC [85] (black dotted line)
and 0.24% at CEPC [86] (black dashed line).

Figure 5: Branching ratios of $\mathcal{S}$ decays (lower plot) and its total
decay width (upper plot) as a function of $m_{\mathcal{S}}$. Here DM mass
$m_{\chi}=400$ GeV and the other parameters are given in Eq. (3.23) and model
A.

When $m_{\mathcal{S}}<m_{\mathcal{H}}/2$, the
$\mathcal{H}\to\mathcal{S}\mathcal{S}$ decay process can also contribute to
the invisible Higgs decay if $\mathcal{S}\to\chi\chi$ is kinematically
allowed. In this case, the total invisible Higgs decay branching ratio is
given by

$\displaystyle BR({\rm
inv})=BR(\mathcal{H}\to\chi\chi)+BR(\mathcal{H}\to\mathcal{S}\mathcal{S})BR(\mathcal{S}\to\chi\chi)^{2}.$
(3.26)

As in Eq. (3.24), the partial decay width
$\Gamma_{\mathcal{H}\to\mathcal{S}\mathcal{S}}$ is proportional to the square
of the coupling

$\displaystyle\lambda_{\mathcal{H}\mathcal{S}\mathcal{S}}$ $\displaystyle=$
$\displaystyle-\kappa_{s}v_{0}\cos^{3}\theta-(b_{m}-2c_{s})\cos^{2}\theta\sin\theta$
(3.27)
$\displaystyle+\frac{1}{2}\sin^{2}\theta[4(\kappa_{s}-3\lambda_{h})v_{0}\cos\theta+b_{m}\sin\theta].$

With $m_{\chi}=20$ GeV, $m_{\cal{S}}=50$ GeV, $c_{\chi}=-5$ GeV, and other
parameters given in Eq. (3.23), we find the branching ratio of invisible Higgs
decays in the range $\sim 10^{-5}-10^{-3}$ for $0.01\lesssim\theta\lesssim
0.1$.

Fig. 5 shows the total decay width of $\cal{S}$ in the upper plot, and its
respective branching ratios to SM particles and to $\chi$ in the lower plot.
Here we fix the DM mass to be $m_{\chi}=400$ GeV, and other parameters are
given in Eq. (3.23) and model A in Table 1. We see that when
$m_{\mathcal{S}}<2m_{W}$, the process $\mathcal{S}\to b\bar{b}$ is the
dominant channel of $\mathcal{S}$ decays. As $m_{\mathcal{S}}$ increases, the
$W^{+}W^{-}$, $ZZ$, $\cal HH$, and $t\bar{t}$ channels open up successively
and dominate over the $\mathcal{S}$ decays. When $m_{\mathcal{S}}>2m_{\chi}$,
the channel $\mathcal{S}\to\chi\chi$ becomes available. But, by comparing with
$\mathcal{S}$ decays to SM particles, this process only contributes a small
fraction of $\simeq 5\times 10^{-3}$ and thus plays a subdominant role. For
$m_{\mathcal{S}}$ around $\sim 100$ GeV, the total decay width of
$\mathcal{S}$ is estimated to be $\Gamma_{S}\simeq 6\times 10^{-5}~{}{\rm
GeV}\simeq 10^{-2}\Gamma_{H}$, where $\Gamma_{H}$ is the total decay width of
the SM Higgs boson. This is because the real scalar $\mathcal{S}$ decaying to
SM particles is suppressed by a factor of $\sin^{2}\theta$ in comparison with
the SM Higgs decays. The $\mathcal{S}$ total decay width can go up to $\sim
10$ GeV with its mass increasing to $\sim 1$ TeV. Note again that due to the
mixing, $\mathcal{S}$ could induce observable effects sensitive to $\theta$ at
the LHC. Here we have set $\theta=0.1$, which is allowed by the Higgs signal
strength restriction $|\theta|\lesssim 0.4$. The future hadron and lepton
colliders, such as HL-LHC and ILC, could probe the mixing angle $\theta$ in
the range of $0.1-0.4$ and provide more opportunities to test our scenarios.

####  3.3.2 Relic density

Figure 6: DM relic density in the five benchmark models as a function of
$m_{\chi}$.

In Fig. 6, we show the DM relic density in the five benchmark models as a
function of $m_{\chi}$. For $m_{\chi}<m_{\mathcal{H}}/2$, the processes
contributing to the DM annihilation are those with the SM fermions $f$ in the
final states, $\chi\chi\to f\bar{f}$, mediated by $\mathcal{H}$ and
$\mathcal{S}$. Still, the relevant parameters are $\theta$ and $c_{\chi}$ (see
Eq. (3.25)) and the DM annihilation cross sections are proportional to the
product $|\sin\theta\times c_{\chi}|^{2}$. As $m_{\chi}$ increases, the DM
relic density can be reduced greatly due to the allowance of annihilation into
pairs of the SM weak gauge bosons and Higgs bosons, as well as the one into
$\mathcal{H}\mathcal{S}$. The process $\chi\chi\to\mathcal{H}\mathcal{S}$ is
kinematically allowed when $2m_{\chi}\gtrsim m_{\mathcal{H}}+m_{\mathcal{S}}$.
The resonant DM annihilations occur at $m_{\chi}\simeq m_{\mathcal{H}}/2$ and
$m_{\mathcal{S}}/2$, thus sharp decrease of the DM relic density at the two
dips in the curves. The final decrease in the relic density takes place when
$m_{\chi}>m_{\mathcal{S}}=400$ GeV, since $\chi\chi\to\mathcal{S}\mathcal{S}$
dominates over the other annihilation processes. In this case, $\lambda_{a}$
controls the DM freeze-out process in the five benchmark models. In
particular, models A, C, and D share the same large coupling
$\lambda_{a}=0.5$, which can boost the cross section of
$\chi\chi\to\mathcal{S}\mathcal{S}$ so that it leads to a DM relic density
consistent with the current cosmological observations in the DM mass range
between 400 GeV and 1.6 TeV. Model E with $\lambda_{a}=0.3$ can also generate
a DM relic density $\lesssim 0.12$ in the DM mass range of $400-900$ GeV. On
the other hand, in model B, $\lambda_{a}$ is fixed at $0.1$ that is
insufficient to suppress the relic density.

Figure 7: Upper plot: Total DM annihilation cross section as a function of DM
mass in the present Universe. Lower plot: Branching ratios DM annihilation to
various final states as a function of DM mass. The real scalar mass
$m_{\mathcal{S}}=400$ GeV, and the other parameters are given in Eq. (3.23)
and model A. Figure 8: Effective annihilation cross section in the five
benchmark models as function of DM mass. The red and blue curves represent
respectively the constraints on the DM annihilation to $W^{+}W^{-}$ and
$b\bar{b}$ final states from a combined analysis of Fermi-LAT [87] and MAGIC
[88] observations of dwarf satellite galaxies.

####  3.3.3 Indirect detection

We now turn to the study of DM indirect detections via the measurements of
cosmic rays. For this purpose, in Fig. 7 we calculate the total DM
annihilation cross section in the present Universe (upper plot) and its
branching ratios to various final states (lower plot). Here, we take
parameters in model A as an example. In the calculation of the total
annihilation cross section, the polarizations of the $W$ and $Z$ bosons and
the photon radiation from 3-body final states are also considered. As we have
observed in the DM relic density computations, there are two resonant
annihilation regions at around $m_{\chi}=m_{\mathcal{H}}/2$ and
$m_{\chi}=m_{\mathcal{S}}/2$. Moreover, the annihilation cross section becomes
nearly a constant $\approx 5\times 10^{-26}~{}{\rm cm^{-3}/s}$ when
$m_{\chi}>m_{\mathcal{S}}=400$ GeV. From the annihilation branching ratios, we
observe that for $m_{\chi}<(m_{\mathcal{H}}+m_{\mathcal{S}})/2$ the
annihilation to SM final states $b\bar{b}$, $W^{+}W^{-}$, $ZZ$, $HH$, and
$t\bar{t}$ are the dominant processes. When
$m_{\chi}>(m_{\mathcal{H}}+m_{\mathcal{S}})/2$, the process
$\chi\chi\to\mathcal{H}\mathcal{S}$ assumes the main contribution to
annihilation. Finally, the $\mathcal{S}\mathcal{S}$ final state dominates the
annihilation when $m_{\chi}>m_{\mathcal{S}}$.

In order to compare with the experimental upper limits, we define the
effective annihilation cross section as follows:

$\left\langle\widetilde{\sigma v}\right\rangle=f_{X}^{2}\left\langle\sigma
v\right\rangle,$ (3.28)

which is plotted in Fig. 8 for the five benchmark models. In the same plot,
the solid red and blue lines represent respectively the constraints on the DM
annihilations to the $W^{+}W^{-}$ and $b\bar{b}$ final states from a combined
analysis of Fermi-LAT [87] and MAGIC [88] observations of dwarf satellite
galaxies. It is seen that model B has been restricted severely by these gamma-
ray observations due to its relatively large value of DM relic density, while
the other models can survive them. Note that such considerations of DM
indirect detection limits are conservative since the cross section of a
specific DM annihilation channel is always less than the total one. We also
note that the reason why we have applied the constraints from the $b\bar{b}$
and $W^{+}W^{-}$ final states is that the former channel dominates the DM
annihilation when $m_{\chi}<m_{W}$ while the latter decay mode becomes
relevant in the large DM mass regime when the annihilation
$\chi\chi\to\mathcal{S}\mathcal{S}$ opens up, with ${\cal S}$ mostly decaying
into $W^{+}W^{-}$ for the present parameter choice, as seen in Fig. 5. In
fact, a more appropriate upper limit for the $m_{\chi}>m_{\cal S}$ case should
be given by the DM annihilation process $\chi\chi\to 2W^{+}+2W^{-}$. However,
we cannot find the DM indirect detection constraints on this $4W$ final state
in the literature, and have thus applied the bounds on the $W^{+}W^{-}$
channel instead. Nevertheless, one can expect that the two constraints can
differ at most by a factor of $2$ [89, 90].

####  3.3.4 Direct detection

Figure 9: Effective SI DM-nucleon elastic scattering cross section in the five
benchmark models as a function of the DM mass. The blue and red curves denote
the upper limits on scattering cross section from LUX [62] and XENON1T [64]
experiments, respectively. The black dashed curve represents the potential
constraints from the future XENONnT project [78].

The DM direct detection tries to measure the DM-nucleon scatterings in the
deep underground laboratories. In the present model, the DM-nucleon
interaction can proceed via the $\mathcal{H}$ and $\mathcal{S}$ portals, which
are, however, suppressed by the mixing angle $\theta$. In Fig. 9, we plot the
rescaled cross section of DM-nucleon scatterings $\tilde{\sigma}_{\rm SI}$
defined in Eq. (2.30) as a function of DM mass. As shown in the figure, for
$m_{\chi}\lesssim m_{\mathcal{S}}=400$ GeV, all the benchmark models are
constrained by LUX and XENON1T, except for the DM mass sitting at the
$\mathcal{H}$ and $\mathcal{S}$ resonances. As for the heavy DM regime with
$m_{\chi}>m_{\mathcal{S}}$, the effective DM-nucleon scattering cross section
of all models is suppressed to the level that can evade the current direct
detection constraints. In order to probe the large DM mass region, it is
widely believed that the near-future XENONnT project can play an important
role with its expected sensitivity shown as the black dashed curve in Fig. 9.
As a result, the XENONnT experiment can test and constrain the entire DM mass
range in the plot for models B and E, a correlated result of the large DM
relic density predicted in the model. In contrast, the XENONnT constraint on
model A is the weakest. This can be understood as follows: the DM relic
density in the large mass region is controlled by the coupling $\lambda_{a}$,
while the DM direct detection signal is induced via the mediation of $\cal{H}$
and $\cal{S}$ and can only place limits on $|\sin\theta\times c_{\chi}|$.
Therefore, models A, C, and D predict the same DM relic density due to the
same value of $\lambda_{a}$, but, due to their different values of
$|\sin\theta\times c_{\chi}|$, model A can totally avoid the probe of XENONnT,
while models C and D can be tested when the DM is lighter than 1 TeV.
Interestingly, with the parameter choices in Table 1, the pseudoscalar $\chi$
with $m_{\chi}\sim 1.6$ TeV in models A, C, and D, and $m_{\chi}\sim 0.9$ TeV
in model E can constitute all the DM density in our Universe, while being
consistent with all the current DM experimental constraints.

Figure 10: Effective SI DM-nucleon elastic scattering cross section as a
function of the DM mass. The light green scatter points represent the samples
that can trigger a type-II EWPT, and the red scatter points are the samples
further having a relic density in the range of $0.10-0.12$.

In Fig. 10, we plot our numerical scan result in the
$m_{\chi}$-$\tilde{\sigma}_{\rm SI}$ plane in order to show the importance of
the DM direct detection constraints on the parameter space of this
$CP$-symmetric complex scalar model. The light green points represent the
samples that can generate a sufficiently strong type-II EWPT, and the red
points are those predicting a DM relic density in the range of $\Omega_{\rm
DM}h^{2}=0.10-0.12$, close to the value of observed DM relic density from the
Planck experiment. From this plot, we see that there still exists a large
parameter space in the model that have predictions below the most stringent
constraint from DM direct detections. As interpreted above, this is because
the DM scattering with the nucleon in the direct detection via $\mathcal{H}$
and $\mathcal{S}$ portals are suppressed by the mixing angle $\theta$, while
the observed DM relic density can be achieved via the annihilation process
$\chi\chi\to\mathcal{S}\mathcal{S}$, in which the coupling $\lambda_{a}$ plays
the leading role.

###  3.4 Final remarks

Table 2:  A summary of parameters for model G1 and G2. Model |  $m_{\mathcal{S}}$ [GeV] | $\theta$ | $\kappa_{s}$ |  $c_{\chi}$ [GeV]
---|---|---|---|---
G1 | 95.5 | $-0.086$ | 0.068 | $-501.2$
G2 | 141.3 | 0.234 | 0.191 | $-871.0$
Table 3:  A summary of phase transition parameters for model G1 and G2. Model |  $T_{c}$ [GeV] |  $T_{n}$ [GeV] | $v/T_{c}$ | $\alpha$ | $\beta/H_{n}$
---|---|---|---|---|---
G1 | 113.6 | 65.3 | 1.24 | 0.23 | 1044.2
G2 | 105.7 | 57.3 | 1.51 | 0.39 | 849.6

Baryogenesis and gravitational waves resulted from the electroweak phase
transition are two attractive topics that, however, are beyond the scope of
this work. Nevertheless, here we show two concrete $CP$-symmetric models, G1
and G2, in which laudable gravitational waves could be produced. We fix the
following parameters:

$w=500~{}{\rm GeV},~{}~{}m_{\chi}=300~{}{\rm GeV},~{}~{}{\rm
and}~{}~{}\lambda_{a}=0.2~{}.$ (3.29)

The other parameters are given in Table 2. We follow Ref. [49] to calculate
the bubble nucleation temperature $T_{n}$ and the gravitational wave
parameters $\alpha$ and $\beta/H_{n}$ at the nucleation temperature. A summary
of the results for models G1 and G2 are given in Table 3. In figure 12 of Ref.
[49], we show that the gravitational wave parameter region of $\alpha\gtrsim
0.1$ and $\beta/H_{n}\lesssim 10^{4}$ are within the sensitivity of LISA
experiment [46], which thus could provide another test of our model. The BBO
interferometer [48] could further detect the gravitational wave parameter
regions that are beyond the ability of LISA experiment. In Table 3, we also
show the values of $v/T_{c}$, which is an important parameter for successful
baryogenesis. It has been shown that, for successful baryogenesis, the first-
order EWPT should be strong enough so that the sphaleron process in the broken
phase is sufficiently suppressed to avoid baryon asymmetry washout [8]. This
leads to the conventional criterion $v/T_{c}\gtrsim 1$, which is satisfied in
both models G1 and G2 (note that the phase transition parameter scan performed
in section 3.2 has been restricted by this criterion). We leave the extensive
study on the sphaleron process and gravitational waves in the $CP$-symmetric
models for a future work.

##  4 Summary

In this work, we focus on studying the first-order type-II EWPT and the
related DM phenomenology in the real and complex singlet extensions of SM,
amended with $\mathbb{Z}_{2}$ and $CP$ symmetries, respectively. Note that the
type-II EWPT is attractive because it can provide a natural mechanism to
generate a sufficiently strong first-order EWPT, a necessary condition for
successful electroweak baryogenesis. We first study the real singlet extension
of SM with a $\mathbb{Z}_{2}$ symmetry. We notice that the real singlet
becomes a DM candidate after the first-order type-II EWPT. By scanning the
entire parameter space, we have found that the model samples with the type-II
EWPT suffer severely from the constraints of DM direct detection experiments
by LUX and XENON1T. As a result, only a negligible fraction of ${\cal
O}(10^{-4}-10^{-5})$ of DM relic density can be explained by the real singlet
scalar.

We then turn to the complex singlet scalar extension of the SM with a CP
symmetry, where we still examine the interplay between the first-order type-II
EWPT and DM physics. In this model, due to the protection of the $CP$
symmetry, the imaginary component of $S$ can become a DM candidate. On the
other hand, its real component can mix with the remaining real component in
the Higgs doublet to form two physical states, ${\cal H}$ and ${\cal S}$, with
the former much similar to the SM Higgs boson and the latter more singlet-
like. We first perform a large-scale scan over the parameter space of physical
interest, and find that there are ample parameter samples that can induce
sufficiently strong type-II EWPT while satisfying the Higgs signal strength
measurements at the LHC. For the selected model parameters, we then explore
their DM phenomenology, including the DM relic density and Higgs invisible
decays, as well as DM direct and indirect detections. It turns out that there
are still a large number of samples which can survive all of these DM
constraints. In particular, we find parameters that can generate both strong
type-II EWPT and correct DM relic abundance without any conflict with the
current experimental data. This clearly shows that the complex scalar model
with the $CP$ symmetry is superior to the real scalar model in this regard.
Especially, when the DM $\chi$ is heavier than the mediator $\mathcal{S}$, the
dominance of DM annihilation channel $\chi\chi\to\mathcal{S}\mathcal{S}$
during the DM freeze-out can help avoid the strong upper bounds of DM direct
detections, since the two processes involve different parameter dependences,
which is impossible for the real scalar model.

This $CP$-symmetric complex singlet model also predicts some new physics
phenomena that can be further tested and constrained by future experiments.
The mixing between the Higgs doublet field and the singlet field can lead to
deviations in the cubic and quartic Higgs couplings, which in turn can
significantly modify the di-Higgs production at colliders. In fact, our large-
scale scan over the parameter space has already given some hint to the triple-
Higgs coupling $\lambda_{h}v_{0}$ in this model with $\lambda_{h}$ restricted
in the range of $0.1-0.3$, as shown in Fig. 3. Also, the mixing angle could be
further restricted by the Higgs signal strength measurements and the DM direct
detections. Moreover, the pair annihilation of the pseudoscalar DM $\chi$ may
give rise to observable anomalies in the cosmic rays or $\gamma$-ray spectra.
Finally, the strong EWPT can potentially generate a significant stochastic
gravitational wave background, which can be probed by the future space-based
gravitational wave detectors, such as LISA [46], BBO [48], and Taiji [91, 92].

## Acknowledgments

This work was supported in part by the Ministry of Science and Technology
(MOST) of Taiwan under Grant Nos. MOST-108-2112-M-002-005-MY3,
108-2811-M-002-548 and 109-2811-M-002-550. DH is supported in part by the
Chinese Academy of Sciences (CAS) Hundred-Talent Program and by National
Science Foundation of China (NSFC) under Grant No. 12005254.

Note added: While preparing this manuscript, we found that the $CP$ symmetric
model was also considered in a recent work by T. Alanne et al., [93]. Our work
emphasizes on the type-II EWPT and the related DM phenomenology, while their
study focuses on the type-I phase transition and the possible gravitational
wave signals.

## References

  * [1] ATLAS collaboration, _Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC_ , _Phys. Lett._ B716 (2012) 1 [arXiv:1207.7214].
  * [2] CMS collaboration, _Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC_ , _Phys. Lett._ B716 (2012) 30 [arXiv:1207.7235].
  * [3] Planck collaboration, _Planck 2015 results. XIII. Cosmological parameters_ , _Astron. Astrophys._ 594 (2016) A13 [arXiv:1502.01589].
  * [4] B. W. Lee and S. Weinberg, _Cosmological lower bound on heavy-neutrino masses_ , _Phys. Rev. Lett._ 39 (1977) 165.
  * [5] P. Hut, _Limits on masses and number of neutral weakly interacting particles_ _Phys. Lett._ 69B (1977) 85.
  * [6] V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, _On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe_ , _Phys. Lett._ B155 (1985) 36.
  * [7] A. G. Cohen, D. B. Kaplan and A. E. Nelson, _Weak scale baryogenesis_ , _Phys. Lett._ B245 (1990) 561.
  * [8] J. M. Cline, _Status of electroweak phase transition and baryogenesis_ , _Pramana_ 55 (2000) 33 [hep-ph/0003029].
  * [9] D. E. Morrissey and M. J. Ramsey-Musolf, _Electroweak baryogenesis_ , _New J. Phys._ 14 (2012) 125003 [arXiv:1206.2942].
  * [10] M. D’Onofrio, K. Rummukainen, and A. Tranberg, _Sphaleron Rate in the Minimal Standard Model_ , _Phys. Rev. Lett._ 113, (2014) 141602. [arXiv:1404.3565].
  * [11] V. Barger, P. Langacker, M. McCaskey, M. Ramsey-Musolf, and G. Shaughnessy, _Complex singlet extension of the standard model_ , _Phys. Rev. D_ 79 (2009) 015018. [arXiv:0811.0393].
  * [12] X. G. He, T. Li, X. Q. Li, J. Tandean and H.-C. Tsai, _Constraints on scalar dark matter from direct experimental searches_ , _Phys. Rev. D_ 79 (2009) 023521. [arXiv:0811.0658].
  * [13] M. Gonderinger, H. Lim and M. J. Ramsey-Musolf, _Complex scalar singlet dark matter: Vacuum stability and phenomenology_ , _Phys. Rev. D_ 86 (2012) 043511. [arXiv:1202.1316].
  * [14] M. Jiang, L. Bian, W. Huang and J. Shu, _Impact of a complex singlet: Electroweak baryogenesis and dark matter_ , _Phys. Rev. D_ 93 (2016) 065032. [arXiv:1502.07574].
  * [15] C. W. Chiang, M. J. Ramsey-Musolf and E. Senaha, _Standard model with a complex scalar singlet: Cosmological implications and theoretical considerations_ , _Phys. Rev. D_ 97 (2018) 015005. [arXiv:1707.09960].
  * [16] C. E. Yaguna, _Gamma rays from the annihilation of singlet scalar dark matter_ , _JCAP_ 3 (2009) 003 [arXiv:0810.4267].
  * [17] W. L. Guo and Y. L. Wu, _The real singlet scalar dark matter model_ , _JHEP_ 10 (2010) 083 [arXiv:1006.2518].
  * [18] S. Profumo, L. Ubaldi and C. Wainwright, _Singlet scalar dark matter: monochromatic gamma rays and metastable vacua_ , _Phys. Rev. D_ 82 (2010) 1–10 [arXiv:1009.5377].
  * [19] Y. Mambrini, _Higgs searches and singlet scalar dark matter: combined constraints from XENON 100 and the LHC_ , _Phys. Rev. D_ 84 (2011) 115017 [arXiv:1108.0671].
  * [20] J. M. Cline, P. Scott, K. Kainulainen and C. Weniger, _Update on scalar singlet dark matter_ , _Phys. Rev. D_ 88 (2013) 055025. [arXiv:1306.4710].
  * [21] J. A. Casas, D. G. Cerdea$\acute{\rm a}$o, J. M. Moreno and J. Quilis, _Reopening the Higgs portal for single scalar dark matter_ , _JHEP_ 05 (2017) 036 [arXiv:1701.08134].
  * [22] GAMBIT Collaboration P. Athron et al., _Status of the scalar singlet dark matter model_ , _Eur. Phys. J. C_ 77 (2017) 568. [arXiv:1705.07931].
  * [23] S. Baek, P. Ko and W. I. Park, _Search for the Higgs portal to a singlet fermionic dark matter at the LHC_ , _JHEP_ 02 (2012) 047 [arXiv:1112.1847].
  * [24] M. Fairbairn and R. Hogan, _Singlet fermionic dark matter and the electroweak phase transition_ , _JHEP_ 09 (2013) 022 [arXiv:1305.3452].
  * [25] T. Li and Y. F. Zhou, _Strongly first order phase transition in the singlet fermionic dark matter model after LUX_ , _JHEP_ 07 (2014) 006 [arXiv:1402.3087].
  * [26] A Beniwal, M Lewicki, JD Wells, M White and A. G. Williams, _Gravitational wave, collider and dark matter signals from a scalar singlet electroweak baryogenesis_ , _JHEP_ 08 (2017) 108 [arXiv:1702.06124].
  * [27] A. Beniwal, M. Lewicki, M. White and A. G. Williams, _Gravitational waves and electroweak baryogenesis in a global study of the extended scalar singlet model_ , _JHEP_ 02 (2019) 183. [arXiv:1810.02380].
  * [28] K Hashino, M Kakizaki, S Kanemura, P. Ko and T. Matsui, _Gravitational waves from first order electroweak phase transition in models with the $U(1)_{X}$ gauge symmetry,_, _JHEP_ 06 (2018) 088 [arXiv:1802.02947].
  * [29] A. Ahriche, K. Hashino, S Kanemura and S. Nasri, _Gravitational waves from phase transitions in models with charged singlets_ , _Phys. Lett._ B789 (2019) 119–126. [arXiv:1809.09883].
  * [30] A. Alves, T. Ghosh, H. K. Guo, K. Sinha and D. Vagie _Collider and gravitational wave complementarity in exploring the singlet extension of the standard model_ , _JHEP_ 04 (2019) 052. [arXiv:1812.09333].
  * [31] M. Duch, B. Grzadkowski and M. McGarrie, _A stable Higgs portal with vector dark matter_ , _JHEP_ 09 (2015) 162 [arXiv:1506.08805].
  * [32] X. G. He and J. Tandean, _New LUX and PandaX-II results illuminating the simplest Higgs-portal dark matter models_ , _JHEP_ 12 (2016) 074 [arXiv:1609.03551].
  * [33] W. Chao, H. K. Guo and J. Shu, _Gravitational wave signals of electroweak phase transition triggered by dark matter_ , _JCAP_ 09 (2017) 009 [arXiv:1702.02698].
  * [34] B. Grzadkowski and D. Huang, _Spontaneous CP-violating electroweak baryogenesis and dark matter from a complex singlet scalar_ , _JHEP_ 08 (2018) 135. [arXiv:1807.06987].
  * [35] Z. Kang, P. Ko and T. Matsui, _Strong first order EWPT $\&$ strong gravitational waves in $Z_{3}$-symmetric singlet scalar extension_, _JHEP_ 02 (2018) 115 [arXiV:1706.09721].
  * [36] W. Cheng and L. Bian, _From inflation to cosmological electroweak phase transition with a complex scalar singlet_ , _Phys. Rev. D_ 98 (2018) 023524 [arXiV:1801.00662].
  * [37] K. Kannike, K. Loos and M. Raidal, _Gravitational Wave Signals of Pseudo-Goldstone Dark Matter in the $Z_{3}$ Complex Singlet Model_, _Phys. Rev. D_ 101 (2020) 035001 [arXiv:1907.13136].
  * [38] K. Ghorbani, P. H. Ghorbani, _A Simultaneous Study of Dark Matter and Phase Transition: Two-Scalar Scenario_ , _JHEP_ 12 (2019) 077. [arXiv:1906.01823].
  * [39] K. Ghorbani, P. H. Ghorbani, _Strongly First-Order Phase Transition in Real Singlet Scalar Dark Matter Model_ , _J. Phys. G_ 47 (2020) 015201. [arXiv:1804.05798].
  * [40] S. Profumo, M. J. Ramsey-Musolf and G. Shaughnessy, _Singlet higgs phenomenology and the electroweak phase transition_ , _JHEP_ 08 (2007) 010. [arXiv:0705.2425].
  * [41] M.J. Ramsey-Musolf, _The Electroweak Phase Transition: A Collider Target_ , [arXiv:1912.07189].
  * [42] J. Kozaczuk, M.J. Ramsey-Musolf and J. Shelton, _Exotic Higgs Decays and the Electroweak Phase Transition,_ _Phys. Rev. D_ 101 101 (2020) 115035 [arXiv:1911.10210].
  * [43] G. Arcadi et al, _The waning of the WIMP? A review of models, searches, and constraints_ , _Eur. Phys. J. C_ 78 (2018) 203 [arXiv:1703.07364].
  * [44] G Arcadi, A Djouadi and M Raidal, _Dark matter through the Higgs portal_ , _Phys. Rep._ 842 (2020) 1-180. [arXiv:1903.03616].
  * [45] B. P. Abbott et al. (LIGO and Virgo Collaborations), _Observation of Gravitational Waves from a Binary Black Hole Merger_ _Phys. Rev. Lett._ 116 (2016) 061102.
  * [46] LISA Collaboration, _Laser Interferometer Space Antenna_ , [arXiv:1702.00786].
  * [47] S. Sato et al., _The status of DECIGO_ , _J. Phys. Conf. Ser._ 840 (2017), 012010.
  * [48] J. Crowder and N. J. Cornish, _Beyond LISA: Exploring future gravitational wave missions_ , _Phys. Rev. D_ 72 (2005) 083005 [gr-qc/0506015].
  * [49] C. W. Chiang and B. Q. Lu, _First-order electroweak phase transition in a complex singlet model with $\mathbb{Z}_{3}$ symmetry_, _JHEP_ 07 (2020) 082 [arXiv:1912.12634].
  * [50] ATLAS Collaboration, Report No. ATL-PHYS-PUB-2013-014.
  * [51] CMS Collaboration, _Projected Performance of an Upgraded CMS Detector at the LHC and HL-LHC: Contribution to the Snowmass Process_ , [arXiv:1307.7135].
  * [52] D.M. Asner et al, _ILC Higgs White Paper_ , [arXiv:1310.0763].
  * [53] CLICdp, ILD concept group collaboration, _On the physics potential of ILC and CLIC_ , _PoS_ CORFU2019 (2020) 037 [arXiv:2004.14628].
  * [54] D. Yu et al, _The measurement of the $H\to\tau\tau$ signal strength in the future $e^{+}e^{-}$ Higgs factories_, _Eur. Phys. J. C_ 80 (2020) 7.
  * [55] S. Profumo, M.J. Ramsey-Musolf, C.L. Wainwright and P. Winslow, _Singlet-catalyzed electroweak phase transitions and precision Higgs boson studies_ , _Phys. Rev. D_ 91 (2015) 035018 [arXiv:1407.5342].
  * [56] J. M. Cline and K. Kainulainen, _Electroweak baryogenesis and dark matter from a singlet Higgs_ , _JCAP_ 01 (2013) 012 [arXiv:1210.4196].
  * [57] J.R. Espinosa, B. Gripaios, T. Konstandin and F. Riva, _Electroweak baryogenesis in non-minimal composite Higgs models_ , _JCAP_ 01 (2012) 012 [arXiv:1110.2876].
  * [58] D. Curtin, P. Meade and C. T. Yu, _Testing electroweak baryogenesis with future colliders_ , _JHEP_ 11 (2014) 127 [arXiv:1409.0005].
  * [59] V. Vaskonen, _Electroweak baryogenesis and gravitational waves from a real scalar singlet_ , _Phys. Rev. D_ 95 (2017) 123515 [arXiv:1611.02073].
  * [60] C. Gross, O. Lebedev and T. Toma, _Cancellation Mechanism for Dark-Matter-Nucleon Interaction_ , _Phys. Rev. Lett._ 119 (2017) 191801 [arXiv:1708.02253].
  * [61] W. Chao, _First order electroweak phase transition triggered by the Higgs portal vector dark matter_ , _Phys. Rev. D_ 92 (2015) 015025 [arXiv:1412.3823].
  * [62] LUX Collaboration, _Results from a Search for Dark Matter in the Complete LUX Exposure_ , _Phys. Rev. Lett._ 118 (2017) 021303. [arXiv:1608.07648].
  * [63] PandaX-II Collaboration, _Dark Matter Results from 54-Ton-Day Exposure of PandaX-II Experiment_ , _Phys. Rev. Lett._ 119 (2017) 181302. [arXiv:1708.06917].
  * [64] XENON Collaboration, _Dark Matter Search Results from a One Ton-Year Exposure of XENON1T_ , _Phys. Rev. Lett._ 121 (2018) 111302. [arXiv:1805.12562].
  * [65] J. R. Espinosa, T. Konstandin and F. Riva, _Strong electroweak phase transitions in the Standard Model with a singlet_ , _Nucl. Phys._ B854 (2012) 592–630. [arXiv:1107.5441].
  * [66] R. Jackiw, _Functional evaluation of the effective potential_ , _Phys. Rev. D_ 9 (1974) 1686.
  * [67] H. H. Patel and M. J. Ramsey-Musolf, _Baryon Washout, Electroweak Phase Transition, and Perturbation Theory_ , _JHEP_ 1107 (2011) 029 [arXiv:1101.4665].
  * [68] C. Wainwright, S. Profumo and M. J. Ramsey-Musolf, 84 (2011) . _Phase transitions and gauge artifacts in an Abelian Higgs boson plus singlet model_ , _Phys. Rev. D_ 84 (2011) 023521. [arXiv:1204.5464].
  * [69] A. Katz, M. Perelstein, M. J. Ramsey-Musolf and P. Winslow, _Stop-Catalyzed Baryogenesis Beyond the MSSM_ , _Phys. Rev. D_ 92 (2015) 095019 [arXiv:1509.02934].
  * [70] C. W. Chiang and E. Senaha, _On gauge dependence of gravitational waves from a first-order phase transition in classical scale-invariant $U(1)^{\prime}$ models_, _Phys. Lett._ B774 (2017) 489–493. [arXiv:1707.06765].
  * [71] S. Kanemura, M. Kikuchi and K. Yagyu, _Radiative corrections to the Higgs boson couplings in the model with an additional real singlet scalar field_ , _Nucl. Phys. B_ 907 (2016) 286 [arXiv:1511.06211].
  * [72] P. Gondolo and G. Gelmini, _Cosmic abundances of stable particles: Improved analysis_ , _Nucl. Phys. B_ 360 (1991) 145.
  * [73] D. Barducci et al., _Collider limits on new physics within MicrOMEGAs_4.3_ , _Comput. Phys. Commun._ 222 (2018) 327 [arXiv:1606.03834].
  * [74] J.M. Alarcon, J. Martin Camalich and J.A. Oller, _The chiral representation of the $\pi N$ scattering amplitude and the pion-nucleon sigma term_, _Phys. Rev. D_ 85 (2012) 051503 [arXiv:1110.3797].
  * [75] H. Y. Cheng and C. W. Chiang, _Revisiting Scalar and Pseudoscalar Couplings with Nucleons_ , _JHEP_ 07, 009 (2012) [arXiv:1202.1292].
  * [76] J.M. Alarcon, L.S. Geng, J. Martin Camalich and J.A. Oller, _The strangeness content of the nucleon from effective field theory and phenomenology_ , _Phys. Lett. B_ 730 (2014) 342 [arXiv:1209.2870].
  * [77] X.L. Ren, L.S. Geng, J. Martin Camalich, J. Meng and H. Toki, _Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory_ , _JHEP_ 12 (2012) 073 [arXiv:1209.3641].
  * [78] XENON Collaboration, _Physics reach of the XENON1T DM experiment_ , _JCAP_ 04 (2016) 027 [arXiv:1512.07501].
  * [79] S. Di Vita et al., _A global view on the Higgs self-coupling at lepton colliders_ , _JHEP_ 02 (2018) 178 [arXiv:1711.03978].
  * [80] A. Adhikary et al., _Revisiting the non-resonant Higgs pair production at the HL-LHC_ , _JHEP_ 07 (2018) 116 [arXiv:1712.05346].
  * [81] F. Maltoni, D. Pagani and X. Zhao, _Constraining the Higgs self-couplings at $e^{+}e^{-}$ colliders_, _JHEP_ 07 (2018) 087 [arXiv:1802.07616].
  * [82] S. Borowka et al., _Probing the scalar potential via double Higgs boson production at hadron colliders_ , _JHEP_ 04 (2019) 016 [arXiv:1811.12366].
  * [83] M. Tanabashi et al. (Particle Data Group), _Review of Particle Physics_ , _Phys. Rev. D_ 98 (2018) 030001.
  * [84] A. M. Sirunyan et al. (CMS Collaboration), _Search for invisible decays of a Higgs boson produced through vector boson fusion in proton-proton collisions at $\sqrt{13}$ TeV_, _Phys. Lett. B_ 793 (2019) 520. [arXiv:1809.05937].
  * [85] Z. Liu, L. T. Wang and H. Zhang, _Exotic decays of the 125 GeV Higgs boson at future $e^{+}e^{-}$ colliders_, _Chin. Phys. C_ 41 (2017) 063102 [arXiv:1612.09284].
  * [86] Y. Tan et al, _Search for invisible decay of a Higgs boson produced at the CEPC_ , [arXiv:2001.05912].
  * [87] M. Ackermann et al. Fermi-LAT collaboration, _Searching for Dark Matter Annihilation from Milky Way Dwarf Spheroidal Galaxies with Six Years of Fermi Large Area Telescope Data,_ _Phys. Rev. Lett._ 115 (2015) 231301 [arXiv:1503.02641].
  * [88] M.L. Ahnen et al., _Limits to dark matter annihilation cross-section from a combined analysis of MAGIC and Fermi-LAT observations of dwarf satellite galaxies_ , _JCAP_ 02 (2016) 039 [arXiv:1601.06590].
  * [89] H. B. Jin, Y. L. Wu and Y. F. Zhou, _Implications of the first AMS-02 measurement for dark matter annihilation and decay_ , _JCAP_ 11 (2013) 026 [arXiv:1304.1997].
  * [90] S.J. Clark, B. Dutta and L.E. Strigari, _Dark matter annihilation into four-body final states and implications for the AMS antiproton excess_ , _Phys. Rev. D_ 97 (2018) 023003. [arXiv:1709.07410].
  * [91] W. R. Hu and Y. L. Wu, _The Taiji Program in Space for gravitational wave physics and the nature of gravity_ , _Natl. Sci. Rev._ 4 (2017) 685.
  * [92] W. H. Ruan, C. Liu, Z. K. Guo, Y. L. Wu and R. G. Cai, _The LISA–Taiji network_ , _Nature Astronomy_ 4 (2020) 108–109. [arXiv:2008.2002.03603].
  * [93] T. Alanne et al, _Pseudo-Goldstone dark matter: gravitational waves and direct-detection blind spots_ , [arXiv:2008.09605].