aainstitutetext: School of Physics and State Key Laboratory of Nuclear Physics
and Technology, Peking University, Beijing 100871, Chinabbinstitutetext:
Center for High Energy Physics, Peking University, Beijing 100871,
Chinaccinstitutetext: School of Physics, Beihang University, Beijing 100083,
Chinaddinstitutetext: Beijing Key Laboratory of Advanced Nuclear Materials and
Physics, Beihang University, Beijing 100191, Chinaeeinstitutetext: Center for
Theoretical Physics, Department of Physics and Astronomy, Seoul National
University, Seoul 08826, Korea

# Searching for lepton portal dark matter with colliders and gravitational
waves

Jia Liu c,d Xiao-Ping Wang e Ke-Pan Xie<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the lepton portal dark matter (DM) model in which the relic abundance
is determined by the portal coupling among the Majorana fermion DM candidate
$\chi$, the singlet charged scalar mediator $S^{\pm}$ and the Standard Model
(SM) right-handed lepton. The direct and indirect searches are not sensitive
to this model. This article studies the lepton portal coupling as well as the
scalar portal coupling (between $S^{\pm}$ and SM Higgs boson), as the latter
is generally allowed in the Lagrangian. The inclusion of scalar portal
coupling not only significantly enhances the LHC reach via the $gg\to h^{*}\to
S^{+}S^{-}$ process, but also provides a few novel signal channels, such as
the exotic decays and coupling deviations of the Higgs boson, offering new
opportunities to probe the model. In addition, we also study the Drell-Yan
production of $S^{+}S^{-}$ at future lepton colliders, and find out that the
scenario where one $S^{\pm}$ is off-shell can be used to measure the lepton
portal coupling directly. In particular, we are interested in the possibility
that the scalar potential triggers a first-order phase transition and hence
provides the stochastic gravitational wave (GW) signals. In this case, the
terrestrial collider experiments and space-based GW detectors serve as
complementary approaches to probe the model.

##  1 Introduction

The Standard Model (SM) of particle physics has been a great triumph in
explaining and predicting the astrophysical and terrestrial experimental
phenomena, however there are still many unsolved problems remaining, such as
the dark matter (DM). Many astrophysical evidences support the existence of
DM, and the fitting result of Cosmic Microwave Background (CMB) to the
$\Lambda$CDM model yields a DM relic abundance of $\Omega_{\rm DM}h^{2}\approx
0.12$ Aghanim:2018eyx , which accounts for $\sim 27\%$ of the total universe
energy. However, we still know very little about the particle origin of DM,
except that none of the SM particles can be the DM candidate Bertone:2004pz .
Therefore, the existence of DM is a clear evidence for physics beyond the SM
(BSM).

Over the past several decades, the most popular particle explanation for DM
has been the freeze-out mechanism of the weakly interacting massive particles
(WIMPs) Lee:1977ua , as it naturally yields the observed DM relic density when
the coupling of DM to the SM particles is of the order of the electroweak (EW)
gauge couplings, and the DM mass is $\mathcal{O}(100~{}{\rm GeV})$. Although
the results from direct detection Schumann:2019eaa , indirect detection
Gaskins:2016cha  and collider searches Boveia:2018yeb  have been pushing more
and more stringent bounds on WIMPs, there is still room for this scenario.
There are many simplified models Alves:2011wf ; Abdallah:2014hon ;
Abdallah:2015ter ; Abercrombie:2015wmb  describe the interactions between DM
and SM particles. One category of them couples DM to SM fermions through
Yukawa interaction Boehm:2003hm , which is similar to neutralino-sfermion-
fermion vertex in supersymmetric (SUSY) models. The interactions can induce
$t$-channel annihilation diagrams for the DM pair. Such colored Goodman:2011jq
; Garny:2012eb ; Liu:2013gba ; An:2013xka ; DiFranzo:2013vra ; Bai:2013iqa ;
Papucci:2014iwa ; deSimone:2014pda ; Abdallah:2014hon ; Garny:2014waa ;
Gomez:2014lva ; Baker:2015qna  and uncolored Liu:2013gba ; Bai:2014osa ;
Chang:2014tea ; Agrawal:2014ufa ; Garny:2015wea  mediators have been studied
in literature.

The lepton portal DM model is proposed in Ref. Bai:2014osa , which assumes a
portal coupling among the SM lepton and two dark sector particles $S$ (scalar)
and $\chi$ (fermion), where “dark sector” means an odd $\mathbb{Z}_{2}$
symmetry is assigned to $S$ and $\chi$. Depending on the mass hierarchy, the
DM candidate could be $S$ or $\chi$, while the fermion DM case can be further
classified into Dirac or Majorana DM scenarios. This particular model has been
further studied in Refs. Yu:2014mfa ; Altmannshofer:2014cla ; Yu:2014pra ;
Garny:2015wea ; Agrawal:2015tfa ; Ibarra:2015fqa ; Cai:2015zza ; Baek:2015fma
; Chen:2015jkt ; Berlin:2015njh ; Agrawal:2015kje ; Mukherjee:2015axj ;
Evans:2016zau ; Chao:2016lqd ; Borah:2017dqx ; Kowalska:2017iqv ; Duan:2017pkq
; Yuan:2017ysv ; Tang:2017lfb ; Ge:2017tkd ; Ding:2017jdr ; Baker:2018uox ;
Hisano:2018bpz ; Gaviria:2018cwb ; Kavanagh:2018xeh ; Kawamura:2020qxo ;
Okada:2020oxh ; Ge:2020tdh ; Boehm:2020wbt ; Okawa:2020jea ; Kowalska:2020zve
; Verma:2021koo ; Alvarado:2021fbw ; Horigome:2021qof ; Bai:2021bau ;
Jueid:2020yfj ; Arcadi:2021glq ; Arcadi:2021cwg ; Calibbi:2018rzv . In this
article, we consider the model with right-handed lepton portal, taking $\chi$
as Majorana DM candidate and $S$ as the charged scalar mediator, which is
similar to the setup in Ref. Bai:2014osa .

This model has very small indirect search cross sections due to helicity
suppression. Because the DM candidate only couples to charged leptons, its
nuclear recoil cross section comes from loop diagrams. Thus, the direct search
signal is also suppressed. Therefore, collider experiments are crucial in
probing this model. The typical collider signal is the Drell-Yan pair
production of the $S^{\pm}$ mediator, and its subsequent decay to $\chi$ and a
charged lepton. In this article, we study this channel at the LHC and future
$e^{+}e^{-}$ colliders, and in the latter case we include the off-shell $S$
pair production $S^{\pm}S^{\mp(*)}$, which provides a direct probe for the
lepton portal coupling.

Different from previous studies, in addition to the lepton portal coupling, we
also consider the Higgs portal coupling $|S|^{2}|H|^{2}$, which is in general
allowed in the Lagrangian. The inclusion of this coupling leads to several
novel signals, such as the gluon-gluon fusion production of $S^{+}S^{-}$ at
the LHC, the Higgs exotic decay (e.g. $h\to\chi\chi$,
$h\to\ell^{+}\ell^{-}\chi\chi$), the Higgs coupling (e.g. $hZZ$,
$h\gamma\gamma$, $h\ell^{+}\ell^{-}$) deviations and the lepton $(g-2)$
corrections to the SM prediction. In particular, the scalar portal coupling
might be able to trigger a first-order phase transition (FOPT) in the early
universe, opening the window for detecting the model via the stochastic
gravitational wave (GW) signals. We finally consider the interplay between the
GW searches and collider searches and show their complementarity.

This paper is organized as follows. We describe the model and derive the
parameter space for the WIMP DM candidate in section 2. In section 3, various
terrestrial searches are investigated, including the $S^{+}S^{-}$ production,
the exotic decay and coupling deviations of the Higgs boson at the LHC and
future $e^{+}e^{-}$ colliders, together with lepton $(g-2)$ searches to
constrain the two portal couplings in this model. The scenario that this model
provides the right DM relic abundance and at the same time triggers a FOPT is
considered in section 4, in which we show that detectable GW signals suggest a
large Higgs portal coupling. In section 5, the interplay between GW detectors
and the collider experiments are discussed. Finally, we summarize and conclude
in section 6.

##  2 The model

The model contains two new fields: the Majorana DM candidate $\chi$, which is
a gauge singlet; and the complex scalar mediator $S$, which is an $SU(2)_{L}$
singlet with hypercharge $-1$. The relevant Lagrangian reads

$\displaystyle\mathcal{L}_{\chi}=$
$\displaystyle~{}\frac{1}{2}\bar{\chi}i\not{\partial}\chi-\frac{1}{2}m_{\chi}\bar{\chi}\chi+y_{\ell}\left(\bar{\chi}_{L}S^{\dagger}\ell_{R}+{\rm
h.c.}\right),$ (1) $\displaystyle\mathcal{L}_{S}=$
$\displaystyle~{}\left(D^{\mu}S\right)^{\dagger}D_{\mu}S-V(H,S),$ (2)
$\displaystyle V(H,S)=$
$\displaystyle~{}\mu_{H}^{2}|H|^{2}+\mu_{S}^{2}|S|^{2}+\lambda_{H}|H|^{4}+\lambda_{S}|S|^{4}+2\lambda_{HS}|H|^{2}|S|^{2},$
(3)

where $H$ is the SM Higgs doublet, and $\ell=e,\mu,\tau$ is the SM charged
lepton (mass eigenstate). We require $S$ couple to one generation of lepton at
a time to avoid lepton flavor violation. Such a flavor alignment typically
needs some specific underlying mechanism to realize, which however beyond the
scope of this work. For a more generic model that simultaneously involves all
three flavors, our results still apply, as long as suitable rescaling is
performed. The model contains a $Z_{2}$ symmetry for $\chi$ and $S$, that both
of them carry odd charges. Assuming $m_{\chi}<m_{S}$, the $\chi$ is stable and
thus can be the DM candidate.

Figure 1: The lepton portal coupling $y_{\ell}^{\rm th}$ as a function of
$m_{S}$ and $m_{\chi}$, which satisfies the DM relic abundance requirement.

The $\chi$ pair can annihilate into the lepton pair via the exchange of a
$t$-channel $S$. Due to the Majorana nature of $\chi$, the $s$-wave component
of $\chi\chi\to\ell^{+}\ell^{-}$ is suppressed by the lepton mass. Therefore,
the annihilation cross section is $p$-wave dominated Garny:2013ama ;
Luo:2013bua ,

$\sigma v_{\rm
rel}=\frac{y^{4}_{\ell}}{32\pi}\frac{m_{\ell}^{2}}{m_{S}^{4}}\frac{1}{(1+x)^{2}}+v_{\rm
rel}^{2}\frac{y^{4}_{\ell}}{48\pi
m_{S}^{2}}\frac{x(1+x^{2})}{(1+x)^{4}}\approx v_{\rm
rel}^{2}\frac{y^{4}_{\ell}}{48\pi m_{S}^{2}}\frac{x(1+x^{2})}{(1+x)^{4}},$ (4)

where $x\equiv m_{\chi}^{2}/m_{S}^{2}$, and we have applied the limit
$m_{\ell}\to 0$ in the second equality. We perform the thermal average of the
annihilation cross section according to Ref. Gondolo:1990dk . Given a set of
$(m_{S},m_{\chi})$, one can always tune $y_{\ell}$ to have the right
annihilation cross section at freeze-out to achieve the observed DM relic
abundance, and the corresponding $y_{\ell}$ is denoted as $y_{\ell}^{\rm th}$,
which is plotted in Fig. 1. One can see that for EW scale $m_{\chi}$ and
$m_{S}$, a Yukawa coupling $y_{\ell}\sim\mathcal{O}(1)$ can provide the
correct DM relic density. For $m_{\chi}<m_{S}$, smaller $m_{\chi}$ leads to
larger $y_{\ell}^{\rm th}$, because the annihilation cross section scales as
$m_{\chi}^{2}/m_{S}^{4}$.

As the DM annihilation signal $\chi\chi\to\ell^{+}\ell^{-}$ is helicity or
$p$-wave suppressed, it is hard to be probed by satellite experiments like
Fermi-LAT Ackermann:2015zua ; Fermi-LAT:2016uux , AMS-02 Accardo:2014lma ;
Aguilar:2014mma , or the CMB measurements from Planck Aghanim:2018eyx . For
direct detection, the scattering between $\chi$ and nucleons arises only at
one-loop level, which can be described by an effective operator. Since $\chi$
is a Majorana fermion, its dimension-five magnetic dipole operator vanishes.
It leaves the dimension-six operator as the leading contribution, which can be
matched to the electromagnetic anapole moment of DM. This receives additional
suppression from DM velocity square, so that it is difficult to detect from
the direct detection experiments Bai:2014osa .111The low energy electron
recoil cross section is
$(y_{\ell}^{4}/\pi)m_{e}^{2}/(m_{\phi}^{2}-m_{\chi}^{2})^{2}$, which is
typically $10^{-44}~{}{\rm cm}^{2}$ for
$m_{\phi},~{}m_{\chi}\sim\mathcal{O}(100)~{}{\rm GeV}$, well below the
constraint from LUX-ZEPLIN experiment Akerib:2021qbs . As a result, we
conclude that the lepton portal DM with Majorana DM has negligible signal in
indirect and direct searches, and is only subject to the constraints from the
thermal relic abundance and collider searches.

##  3 The particle experiment searches

In this section, we discuss probing the lepton portal coupling $y_{\ell}$ and
Higgs portal coupling $\lambda_{HS}$ in particle experiments. First, we
consider the $S^{+}S^{-}\to\ell^{+}\chi\ell^{-}\chi$ channel at the LHC and
future lepton colliders. In the latter case, the off-shell production of
$S^{\pm}$ offers the opportunity to probe $y_{\ell}$ directly. Next, we study
the exotic decays of the Higgs and $Z$ bosons, including the three/four-body
decays $h/Z\to S^{\pm(*)}S^{\mp(*)}$ and the invisible decay $h\to\chi\chi$ at
the Higgs factory CEPC and FCC-ee, which probe the combination of couplings
$y_{\ell}$ and $\lambda_{HS}$. Then we turn to the correction to the Higgs
couplings $h\ell^{+}\ell^{-}$, $h\gamma\gamma$ and $hZZ$. Finally, the lepton
$(g-2)$ is discussed.

###  3.1 Pair production of $S^{\pm}$

####  3.1.1 $pp\to S^{+}S^{-}$ at the LHC

In the model, the lepton portal scalar $S$ carries one unit of hypercharge,
therefore it can be produced in pair via the EW Drell-Yan process $q\bar{q}\to
Z^{*}/\gamma^{*}\to S^{+}S^{-}$ mediated by off-shell $\gamma$ and $Z$ bosons.
However, in our model due to the large scalar sector coupling $\lambda_{HS}$,
one can also have the (off-shell) Higgs mediated process $pp\to h^{*}\to
S^{+}S^{-}$ which can significantly modify the total cross section of $pp\to
S^{+}S^{-}$. For example, with $\lambda_{HS}=1$ and $m_{S}=200$ GeV, the cross
section contributed by the Higgs mediation can be $30\%$ of the total cross
section. The production rates of the Drell-Yan and the gluon-gluon fusion
processes are shown in Fig. 2.

Figure 2: Left: the production rate of $S^{+}S^{-}$ at the 14 TeV LHC for the
Drell-Yan (red), gluon-gluon fusion (blue) channels and their combination
(black dashed). Right: the ratio of gluon-gluon fusion rate to the total cross
section, for $\lambda_{HS}=1$.

The produced $S^{\pm}$ exclusively decays to $\ell^{\pm}\chi$, leading to a
di-lepton plus missing transverse energy final state
($\ell^{+}\ell^{-}+\not{E}_{T}$) at the LHC. The LHC experiments have already
set constraints on such a final state in searches for the sleptons from SUSY
models Aad:2014vma ; Aad:2019vnb ; Aad:2019qnd . The LHC Run-I and Run-II data
from ATLAS Aad:2014vma ; Aad:2019vnb  have covered mass of $S^{\pm}$ up to 450
GeV for the exclusive decay channel $e^{\pm}\chi$ or $\mu^{\pm}\chi$. The
compressed parameter region when $m_{\chi}$ is close to $m_{S}$, has also been
studied by the ATLAS collaboration Aad:2019qnd . Earlier studies from LEP have
fully excludes such charged scalar $S$ with mass $m_{S}<100$ GeV LEPslepton .
We show the above existing constraints in Fig. 3 as colored regions.

To make future projections for our model, we write down the UFO model file
Degrande:2011ua  with the FeynRules package Alloul:2013bka  and use
MadGraph5_aMC@NLO Alwall:2014hca  to generate parton level events, and then
use Pythia8 Sjostrand:2007gs  and Delphes deFavereau:2013fsa  to implement
parton shower and fast detector simulation, respectively. Both the EW Drell-
Yan and the Higgs mediated $S^{+}S^{-}$ production processes are included in
our simulation. The signal events are selected using the cuts from two signal
regions (SRs) of the ATLAS study Aad:2019vnb  as follows,

  1. 1.

Exactly two opposite charged leptons with $p_{T}>25$ GeV and $|\eta|<2.47$;

  2. 2.

At most one light-flavor jet with $p_{T}>20$ GeV and $|\eta|<2.4$, and veto
the $b$-jets in such kinematic region;

  3. 3.

The invariant mass $m_{\ell\ell}>100$ GeV, and transverse missing momentum
$\not{E}_{T}>110$ GeV;

  4. 4.

$m_{T2}>100$ or 160 GeV.

Here the $m_{T2}$ observable is defined event-by-event as the minimum of the
function Lester:1999tx

$\max\left\\{m_{T}(\vec{p}_{T}^{\,\ell^{+}},\vec{p}_{T}^{\,a}),m_{T}(\vec{p}_{T}^{\,\ell^{-}},\vec{p}_{T}^{\,b})\right\\},$
(5)

subject to $\vec{p}^{\,a}+\vec{p}^{\,b}=\not{E}_{T}$, where
$\vec{p}_{T}^{\,\ell^{\pm}}$ are the transverse momentum of the two charged
leptons, $\vec{p}^{\,a}$ and $\vec{p}^{\,b}$ are the associated missing
momenta. The transverse mass $m_{T}$ is defined as

$m_{T}(\vec{p}_{T,1},\vec{p}_{T,2})=\sqrt{|\vec{p}_{T,1}||\vec{p}_{T,2}|(1-\cos\Delta\phi_{12})}.$
(6)

The $m_{T2}$ cut significantly suppresses the $W^{+}W^{-}$ and $t\bar{t}$
backgrounds as their $m_{T2}$ have an end point at the $m_{W}=80.4$ GeV.

Figure 3: The current constraints for right-handed selectron-like and right-
handed smuon-like scalar $S$ in the $m_{\chi}$-$m_{S}$ plane. The shaded
regions are exclusions from LEP LEPslepton , LHC Run-I ($20.3~{}{\rm
fb}^{-1}$) Aad:2014vma  and LHC Run-II ($139~{}{\rm fb}^{-1}$) Aad:2019vnb ;
Aad:2019qnd . The black lines are projections for the LHC reach at 300 ${\rm
fb}^{-1}$ with $\lambda_{HS}=0$ (solid) and $\lambda_{HS}=3$ (dashed).

According to the number (1 or 2) of light-flavor jets in the final state and
the different $m_{T2}$ cuts (100 or 120 GeV), we classify the events into 4
SRs, and adopt the simulated background event numbers from Ref. Aad:2019vnb .
We use $S/\sqrt{B}=2$ as the criterion of the LHC reach for a given integrated
luminosity, where $S$ and $B$ denote the signal and background event numbers,
respectively. The projections at the 14 TeV LHC with 300 fb-1 are plotted in
Fig. 3 as black lines, where we show both the pure Drell-Yan contribution
(means $\lambda_{HS}=0$, dashed) and the inclusion of gluon-gluon fusion
results for $\lambda_{HS}=3$ (dotted). For large $\lambda_{HS}$, the reach can
be visibly enhanced. Note that the enhancement is not significant in the low
$m_{S}$ region, although in that region the gluon-gluon fusion process has a
larger fraction, as shown in Fig. 2. That is because the $S^{\pm}$ from gluon-
gluon fusion typically have a softer $p_{T}$, and hence they are cut away by
the hard $m_{T2}$ cut in our simulation. Loosing $m_{T2}$ might help to probe
the low $m_{S}$ region, and we leave the detailed study for a future work.

####  3.1.2 $e^{+}e^{-}\to S^{\pm}S^{\mp(*)}$ at future $e^{+}e^{-}$colliders

As shown in Fig. 3, there is a gap between the LHC and LEP constraints for
$100~{}{\rm GeV}<m_{S}\lesssim 150$ GeV and $30~{}{\rm GeV}\lesssim
m_{\chi}\lesssim 100$ GeV. The future $e^{+}e^{-}$ colliders with a collision
energy of $\sim 250$ GeV can fill this gap. Moreover, an $e^{+}e^{-}$ machine
is able to probe the lepton portal coupling $y_{\ell}$ directly, provided one
$S^{\pm}$ is off-shell. For the on-shell production at LHC, since
$S^{\pm}\to\ell^{\pm}\chi$ decay branching ratio is $100\%$, the rate does not
depend on $y_{\ell}$. Therefore, the exclusion of slepton-like particle $S$ at
LHC is shown only in the $m_{S}$-$m_{\chi}$ plot, but can not constrain
$y_{\ell}$. However, for the $2\to 3$ process $e^{+}e^{-}\to
S^{\pm}\ell^{\mp}\chi$ mediated by an off-shell $S^{\mp}$, the rate does
depend on $y_{\ell}^{2}$, opening the window to directly probe the DM portal
coupling.

The analysis is carried out on the typical Higgs factory such as CEPC, ILC or
FCC-ee, with $\sqrt{s}=250$ GeV. We include the on-shell $2\to 2$ $S^{+}S^{-}$
pair production as well as the off-shell $2\to 3$ process. For $\ell=e$, the
$2\to 2$ process includes both Drell-Yan contribution and the $t$-channel
$\chi$ mediated contribution. Therefore, in this special case, the $2\to 2$
cross section already has the contribution of $y_{e}$. When
$\ell=\mu,~{}\tau$, the cross section does not depend on $y_{\ell}$, since
only Drell-Yan process contributes. Regarding $2\to 3$ process, there are two
advantages to include it. One is probing the $y_{\ell}$ coupling, and the
other is that we can probe $\sqrt{s}/2<m_{S}<\sqrt{s}$ region. Note that given
$m_{S}$ and $m_{\chi}$, $y_{\ell}$ is determined by the relic abundance of the
DM.

Figure 4: The current constraints for right-handed selectron-like and right-
handed smuon-like scalar S in the $m_{\rm DM}$-$m_{S}$ plane. The shaded
regions are exclusions from previous experiments, as stated in Fig 3. The
black lines are projections for the CEPC reach at 5 ${\rm ab}^{-1}$. The
zigzag shape of the projections at $m_{S}\sim 120$ GeV are due to the
inclusion of off-shell contributions.

We generate signal events using the packages mentioned in section 3.1.1.
Following Ref. Cao:2018ywk , we add a few cuts to select the signal events,
i.e.

  1. 1.

Exactly two opposite charged leptons with $p_{T}>5$ GeV and $|\eta|<3$;

  2. 2.

Veto any events with a jet within $p_{T}>5$ GeV and $|\eta|<3$;

  3. 3.

The transverse missing momentum $\not{E}_{T}>5$ GeV;

  4. 4.

$m_{T2}>20$ GeV;

  5. 5.

The polar angles of the leptons satisfy $\cos\theta_{\ell^{+}}<0.3$ and
$\cos\theta_{\ell^{-}}>-0.3$.

The background event numbers are adopted from the simulated results from Ref.
Cao:2018ywk . In Fig. 4, we can see that this search is complementary with the
LEP and LHC results. It can cover the region $100~{}{\rm GeV}<m_{S}\lesssim
150$ GeV region, which is not touched previously when $m_{\chi}$ mass is
moderately large, e.g. $30\sim 100$ GeV. For large $m_{\chi}$, the visible
energy shared in the leptons decreases, which makes it hard to compete the LHC
background from $W^{+}W^{-}$ pair. The cleaner environment at the lepton
collider makes it sensitive to softer leptons comparing with LHC. Including
$2\to 3$ process, we do see the sensitivity at future $e^{+}e^{-}$ collider
extends to $m_{S}\sim 170$ GeV and $150$ GeV for $y_{e}$ and $y_{\mu}$
respectively. For small $m_{\chi}$ there is higher reach for $m_{S}$, because
in this region a large $y_{\ell}$ is needed to get the correct DM abundance
(see Eq. (4)), which enhances the signal significance at the collider (through
the off-shell $S^{\pm}$).

###  3.2 Exotic decays from the Higgs and $Z$ bosons

####  3.2.1 Exotic decay: $h/Z\to
S^{\pm(*)}S^{\mp(*)}\to\ell^{+}\chi\ell^{\prime-}\chi$

A charged $S^{\pm}$ with $m_{S}<100$ GeV is already excluded by the LEP
experiment, forbidding the $h\to S^{+}S^{-}$ one-shell decay. However, for
$m_{\chi}<m_{h}/2\approx 62.5$ GeV, the $\lambda_{HS}$ portal coupling can
induce the exotic three- or four-body decays $h\to S^{\pm}\ell^{\mp}\chi$ or
$h\to\ell^{\pm}\chi\ell^{\mp}\chi$ mediated by one or two off-shell $S^{\pm}$,
depending on whether $m_{S}<m_{h}$ or not. The decay width is proportional to
$y_{\ell}^{2}\lambda_{HS}^{2}$ or $y_{\ell}^{4}\lambda_{HS}^{2}$, providing a
new way to probe $\lambda_{HS}$ and $y_{\ell}$. If we fix
$y_{\ell}=y_{\ell}^{\rm th}$ by the relic abundance requirement, it gives a
limit to $\lambda_{HS}$ for a given set of $(m_{S},m_{\chi})$.

We explore this exotic Higgs decay at the Higgs factory FCC-ee and CEPC at
$\sqrt{s}=250$ GeV via the $e^{+}e^{-}\to Zh$ production channel, whose cross
section is 0.24 pb. We consider the following cascade decays
$Z\to\ell^{\prime\prime+}\ell^{\prime\prime-}$ and $h\to
S^{\pm(*)}S^{\mp(*)}\to\ell^{+}\chi\ell^{\prime-}\chi$,222Both $\ell$ and
$\ell^{\prime}$ denote the lepton $e$ and $\mu$. where $\chi$ plays the role
of missing energy. The main SM backgrounds include $ZW^{+}W^{-}$ and
$Z\tau^{+}\tau^{-}$, with all the particles decaying to leptonic final states,
and the $Zh$, with $h$ decaying to $W^{\pm}W^{\mp*}$, $ZZ^{*}$ and
$\tau^{+}\tau^{-}$. The total cross section for the backgrounds in the
$(\ell^{\prime\prime+}\ell^{\prime\prime-})\ell^{+}\ell^{\prime-}\not{E}_{T}$
final state is as small as 0.02 fb, where the leptons in the parentheses come
from $Z$ decay. The signal and background events are simulated by the packages
mentioned in section 3.1.1. We apply the following detailed requirements to
the events,

  1. 1.

At least four charged leptons with $p_{T}>10$ GeV and $|\eta|<2.47$;

  2. 2.

A pair of leptons with same-flavor and opposite-sign, and satisfies
$|m_{\ell^{\prime\prime+}\ell^{\prime\prime-}}-m_{Z}|<5~{}{\rm GeV}$;

  3. 3.

The missing energy $\not{E}_{T}>20$ GeV;

  4. 4.

The reconstructed Higgs resonance in the mass window Liu:2016zki

$120~{}{\rm
GeV}<\sqrt{\left(p_{e^{+}}+p_{e^{-}}-p_{\ell^{\prime\prime+}}-p_{\ell^{\prime\prime-}}\right)^{2}}<130~{}{\rm
GeV},$ (7)

which is equivalent to cut on the total energy of $Z$.

Figure 5: Left: The $95\%$ C.L. constraint on the coupling combination
$\lambda_{HS}$ as contours from the exotic decay $h\to
S^{\pm(*)}S^{\mp*}\to\ell^{+}\chi\ell^{\prime-}\chi$. The lepton portal
coupling $y_{\ell}$ is set to be $y_{\ell}^{\rm th}$, which is the value to
provide the correct DM relic abundance. Right: The $95\%$ C.L. constraint
contours (magenta) for $y_{\ell}$ for the exotic decay
$Z\to\ell^{+}\chi\ell^{\prime-}\chi$. The gray shaded region shows the
parameter space can be excluded if $y_{\ell}=y_{\ell}^{\rm th}$.

After the cuts, for a given integrated luminosity we are able to set bounds
for $\text{Br}(h\to S^{\pm(*)}S^{\mp(*)}\to\ell^{+}\chi\ell^{\prime-}\chi)$,
which in turn is translated into the upper limits for $\lambda_{HS}$ once
$y_{\ell}$ is fixed by the relic abundance requirement, as shown in left panel
of Fig. 5. The discontinuity of the curves around $m_{S}=125$ GeV is
originated from the phase space change from three-body to four-body decay. In
conclusion, the future $e^{+}e^{-}$ collider can significantly constrain the
couplings for light $m_{\chi}\lesssim 30$ GeV and $m_{S}$ of a few hundreds
GeV.

In addition to the exotic SM Higgs decay, another good target to probe new
physics is the $Z$ exotic decay Liu:2017zdh . To explore the SM parameters
with better precision, the future $e^{+}e^{-}$ colliders have the proposals to
run at $Z$-pole CEPCStudyGroup:2018ghi ; Abada:2019lih ; Abada:2019zxq , which
can provide Giga ($10^{9}$) or Tera ($10^{12}$) $Z$ boson. Interestingly, in
this model, there exists the exotic decay channel
$Z\to\ell^{+}\chi\ell^{\prime-}\chi$, which provides a di-lepton plus missing
energy final state. There are two types of diagrams responsible for this
channel, the first one involves two off-shell $S^{\pm}$ through $ZS^{+}S^{-}$
coupling and the second one involves one off-shell $S^{\pm}$ through
$Z\ell^{+}\ell^{-}$ coupling with $S$ attached to one of the charged lepton.
Since we are considering $S^{\pm}$ with mass larger than 100 GeV, it is
heavier than all other particles. Therefore, the decay width is dominated by
the latter diagram, which is proportional to $y_{\ell}^{4}m_{S}^{-4}$.

We explore this exotic $Z$ decay similar to the Higgs case. The dominant SM
background is $\bar{\nu}\nu\ell^{+}\ell^{\prime-}$ from off-shell gauge boson
pair production.333The background from tau pair provides softer leptons due to
more neutrinos in the final states, and can be further suppressed by requiring
large $m_{T2}$. The cut conditions are:

  1. 1.

At least two same-flavor opposite-sign leptons with $p_{T}>10$ GeV and
$\eta<2.5$;

  2. 2.

The missing energy should satisfy $\not{E}_{T}>10$ GeV.

The $95\%$ C.L. constraint on the exotic decay branching ratio is about
$\text{Br}(Z\to e^{-}e^{+}\chi\chi)\lesssim 10^{-9}$ for Tera $Z$ option,
while exact limit again depends on $m_{\chi}$ and $m_{S}$. In right panel of
Fig. 5, we show the $95\%$ C.L. constraint on $y_{\ell}$ presented as the
number above the magenta contours. Moreover, we compare this constraint with
the thermal relic requirement $y_{\ell}^{\rm th}$. It shows that for Tera $Z$
option, the thermal DM with mass $m_{\chi}\lesssim 13$ GeV can be excluded by
this exotic $Z$ search, because $y_{\ell}^{\rm th}$ exceeds the limit from the
exotic $Z$ decay $Z\to e^{-}e^{+}\chi\chi$. We plot such region in gray and
label as “thermal DM excluded”. This constraint provides a complimentary limit
for large $m_{S}$ comparing with LHC limits, because it does not require on-
shell $S$ production. Moreover, since the decay width and the DM annihilation
cross section are proportional to $y_{\ell}^{4}m_{S}^{-4}$, this exclusion
line can extend horizontally to very high $m_{S}$, thus is a powerful
constraint for this DM model.

####  3.2.2 Invisible decay: $h\to\chi\chi$

Figure 6: The one-loop induced Higgs invisible decay. The cross-diagrams for
Majorana fermion $\chi$ are not shown here, but are included in the
calculation.

The Higgs invisible decay $h\to\chi\chi$ is induced at one-loop level by the
two Feynman diagrams listed in Fig. 6.444This is similar to the Higgs to
neutralinos decay in the SUSY models Drees:1996pk ; Djouadi:2001kba ;
Eberl:2001vb ; Berlin:2015njh . The first diagram is negligible due to the
small lepton mass. We calculate the second diagram contribution to the exotic
Higgs decay using Package-X Patel:2015tea . The induced coupling is

$\mathcal{L}_{\rm
1-loop}^{\chi}\supset\frac{1}{2}g_{h\chi\chi}h\bar{\chi}\chi,$ (8)

where

$\displaystyle
g_{h\chi\chi}=-\frac{y_{\ell}^{2}\lambda_{HS}m_{\chi}v}{4\pi^{2}(4m_{\chi}^{2}-m_{h}^{2})}\times$
$\displaystyle\left[\text{DiscB}(m_{h}^{2},m_{S},m_{S})+\frac{m_{S}^{2}-m_{\chi}^{2}}{m_{\chi}^{2}}\left(\log\frac{m_{S}^{2}}{m_{S}^{2}-m_{\chi}^{2}}-m_{\chi}^{2}C_{0}(m_{h}^{2},m_{\chi}^{2},m_{\chi}^{2},m_{S},m_{S},0)\right)\right],$
$\displaystyle\approx-\frac{y_{\ell}^{2}\lambda_{HS}m_{\chi}v}{16\pi^{2}m_{S}^{2}}+\mathcal{O}\left(m_{S}^{-3}\right)$
(9)

where DiscB is the finite part of the Passarino-Veltman function $B_{0}$
defined in Package-X, $C_{0}$ is the Passarino-Veltman function following the
definition in Package-X and the lepton mass is taken to be zero. In the second
equality, we have expanded the result with large $m_{S}$. We have checked our
result with Ref. Djouadi:2001kba  and Ref. Berlin:2015njh , and found
agreement between each other. In numeric calculation, we use the full
expression from Eq. (3.2.2).

The $h\to\chi\chi$ partial width is given as

$\displaystyle\Gamma(h\to\chi\bar{\chi})=\frac{g_{h\chi\chi}^{2}m_{h}}{8\pi}\left(1-\frac{4m_{\chi}^{2}}{m_{h}^{2}}\right)^{\frac{3}{2}}\,.$
(10)

The current best limit for invisible Higgs decay is $\text{Br}(h\to{\rm
inv})<13\%$ at ATLAS Run-II with integrated luminosity $139~{}{\rm fb}^{-1}$
ATLAS-CONF-2020-008 . For future HL-LHC, the sensitivity for invisible Higgs
BR is $3.5\%$ from Bernaciak:2014pna . For future $e^{+}e^{-}$ collider such
as CEPC, the sensitivity can be increased to about $0.3\%$
CEPCStudyGroup:2018ghi .

The above projections from future colliders can set limits on the coupling
combination $y_{\ell}^{2}\lambda_{HS}$ as a function of masses $m_{S}$ and
$m_{\chi}$. In the left panel of Fig. 7, we show the contours of
$y_{\ell}^{2}\lambda_{HS}$ for LHC (brown), HL-LHC (blue) and CEPC (red)
sensitivities. The dashed and solid contours corresponds to
$y_{\ell}^{2}\lambda_{HS}=1,~{}10$ respectively. We clearly see that the
future $e^{+}e^{-}$ collider has a better sensitivity over the hadron
colliders. In the right panel of Fig. 7, $y_{\ell}$ is set to be
$y_{\ell}^{\rm th}$ to satisfy DM relic abundance requirement. Once fixing
$m_{S}$ and $m_{\chi}$, the future sensitivity on $\lambda_{HS}$ can be
calculated using CEPC sensitivity $\text{Br}(h\to{\rm inv})=0.3\%$ and the
contours are shown. One interesting feature is that for $m_{\chi}$ smaller
than 6 GeV, the sensitivity on $\lambda_{HS}$ goes down significantly, because
the leading order in the width for small $m_{\chi}$ expansion is linear in
$m_{\chi}$. When $m_{\chi}$ decreases further, $y_{\ell}^{\rm th}$ increases
to compensate the annihilation cross section, which makes the sensitivity on
$\lambda_{HS}$ becoming stronger again. Therefore, we see $\lambda_{HS}$
reaches its best sensitivity for small $m_{S}$ and moderate $m_{\chi}$.

Figure 7: Left: the Higgs invisible decay as a probe for the combination
$y_{\ell}^{2}\lambda_{HS}$ at different colliders. Right: the CEPC sensitivity
$\text{Br}(h\to{\rm inv})<0.3\%$ can set bounds on $\lambda_{HS}$ if we fix
$y_{\ell}=y_{\ell}^{\rm th}$ from the relic abundance requirement.

###  3.3 One-loop contributions to Higgs couplings

The $h\ell^{+}\ell^{-}$ vertex is modified by the one-loop diagram in Fig.
8,555Similar corrections have been studied in SUSY model already
Djouadi:2001kba ; Heinemeyer:2015pfa . and the induced interaction is
calculated using the Package-X

$\displaystyle\mathcal{L}_{\rm
1-loop}^{\ell}\supset-\frac{y_{\ell}^{2}\lambda_{HS}vm_{\ell}}{8\pi^{2}m_{h}^{2}}h\bar{\ell}\ell$
(11)
$\displaystyle\times\left[1+\text{DiscB}(m_{h}^{2},m_{S},m_{S})+\frac{m_{\chi}^{2}}{m_{S}^{2}-m_{\chi}^{2}}\log\frac{m_{S}^{2}}{m_{\chi}^{2}}-\left(m_{S}^{2}-m_{\chi}^{2}\right)C_{0}(0,0,m_{h}^{2},m_{S},m_{\chi},m_{S})\right],$

where we have taken the leading power in small lepton mass $m_{\ell}$. It is
obvious that the coupling is lepton mass suppressed, because the DM only
couples to right-handed lepton and one has to flip the helicity of lepton,
which induce this suppression. Of course, the SM Higgs couplings to leptons
are suppressed by the mass as well. Thus, the one-loop correction to the SM
coupling in fraction is proportional to $y_{\ell}^{2}\lambda_{HS}/(8\pi^{2})$.
In addition to vertex correction, the Higgs field renormalization and the
lepton field renormalization can also contribute to the coupling. The
experiments characterize the sensitivity of Higgs coupling measurements in the
$\kappa$-framework Cepeda:2019klc , which is calculated using the cross
section $\sigma(Zh)$ and $H$ decay branching ratios and hence the universal
Higgs field renormalization effect cancels out in the decay branching ratios.
Therefore, Higgs field renormalization does not affect the branching ratio to
leptons. For the lepton field renormalization, it only contributes to right-
handed lepton. We calculated that it is about $y_{\ell}^{2}/(128\pi^{2})$ for
large $m_{S}$ expansion, which is much smaller than Eq. (11). Together with
the fact that $\lambda_{HS}\gtrsim 1$, we can neglect the right-handed lepton
field renormalization effect and focus on the vertex correction.

Figure 8: The one-loop induced $h\ell^{+}\ell^{-}$ coupling modification.

The most recent constraints from LHC for Higgs decay to leptons are
$\text{Br}(h\to\mu^{+}\mu^{-})<3.8\times 10^{-4}$ ATLAS-CONF-2019-028  and
coupling ratio $\kappa_{\tau}=1.05^{+0.16}_{-0.15}$ Aad:2019mbh . Moreover, we
can also consider the sensitivities at future $e^{+}e^{-}$ collider. For
example, the relative precision for coupling measurement from CEPC study in
10-parameter fit are $\delta\kappa_{\mu}<8.7\%$ and
$\delta\kappa_{\tau}<1.5\%$ An:2018dwb , similar to FCC-ee study Abada:2019zxq
. Therefore, we convert the above existing constraints and future
sensitivities on leptons into the combination of couplings
$y_{\ell}^{2}\lambda_{HS}$.

Figure 9: Left panel: The constraints on $\lambda_{HS}y_{\ell}^{2}$ from the
$h\to\ell\ell$ branching ratio and coupling strength $\kappa_{\tau}$ from
existing LHC constraints ATLAS-CONF-2019-028 ; Aad:2019mbh  with fixed DM mass
$m_{\chi}=10$ GeV. Middle and right panels: the constraint on $\lambda_{HS}$
from the projected precision $\delta\kappa_{\mu}<8.7\%$ and
$\delta\kappa_{\tau}<1.5\%$ from CEPC An:2018dwb , with
$y_{\ell}=y_{\ell}^{\rm th}$ by DM relic abundance.

The results are shown in Fig. 9. In the left panel, we show the limits for the
combination $y_{\ell}^{2}\lambda_{HS}$ from
$\text{Br}(h\to\mu^{+}\mu^{-},\tau^{+}\tau^{-}$) at LHC and CEPC. The shaded
regions are already excluded by current LHC limits, while the future
sensitivities from CEPC are plotted as dotted lines. The DM mass is fixed as
$m_{\chi}=10$ GeV and the limits get weakened linearly with increasing
$m_{S}$. In the middel and right panels, we fix $y_{\ell}=y_{\ell}^{\rm th}$
for the DM relic abundance requirement, and show the sensitivity contours for
$\lambda_{HS}$ as a function of $m_{S}$ and $m_{\chi}$. When taking
$m_{S}\to\infty$, the coupling in Eq. (11) is proportional to
$y_{\ell}^{2}m_{S}^{-2}$. Therefore, it has the similar dependence as the DM
relic abundance requirement which is $y_{\ell}^{4}m_{S}^{-4}$. This feature is
clearly shown that when $m_{S}$ increases, the contours for $\lambda_{HS}$ are
flat. It means that the sensitivity does not suffer for large $m_{S}$, because
the large mass $m_{S}$ is compensated by large $y_{\ell}^{\rm th}$. Since for
$\mu$ and $\tau$, this change of coupling ratio $\delta\kappa$ does not depend
on lepton mass, the constraint on $\lambda_{HS}$ is only linear depends on the
CEPC precision, which has better $\tau$ sensitivity. Therefore, the
sensitivity for $\tau$ lepton portal is better by a factor of $5.8$ compared
to $\mu$ lepton portal.

Figure 10: The shaded regions are constraints and projections from the
$h\gamma\gamma$ and $hZZ$ coupling measurements at the LHC.

Besides the one-loop modification to $h\ell^{+}\ell^{-}$ coupling, the charged
scalar loop can also modify the $h\gamma\gamma$ and the $hZZ$ couplings
Huang:2016cjm . The former one modifies $\text{Br}(h\to\gamma\gamma)$ via the
charged scalar triangle loop, and current fitting result for the signal
strength is $1.12\pm 0.09$ at the 13 TeV LHC with 137 fb-1 1851456 , while the
projection accuracy is 4% at the HL-LHC Cepeda:2019klc , better than the
projected sensitivity 6.9% at the 5.6 ab-1 CEPC CEPCStudyGroup:2018ghi . The
latter one is mostly contributed by Higgs field renormalization (via the
$S^{\pm}$ loop) if $\lambda_{HS}>e$ (the elementary charge). This modification
doesn’t rely only on $y_{\ell}$, therefore it provides a direct constraint on
the scalar interaction coupling $\lambda_{HS}$ independent of the Yukawa
coupling. Current constraint for $\delta\kappa_{Z}$ is 8% for the 13 TeV 139
fb-1 LHC Aad:2020mkp , and the projected result is 1.7% at the HL-LHC
Cepeda:2019klc . Therefore, the precision measurement on $Zh$ production cross
section can provide a limits on $\lambda_{HS}$. Following the procedure of
Ref. Huang:2016cjm , we calculate the sensitivity for $\lambda_{HS}$ from the
future limits on $\text{Br}(h\to\gamma\gamma$) and the coupling strength to
$Z$ gauge boson $\kappa_{Z}$. The results are presented in Fig. 10.

###  3.4 The lepton $g-2$

Another consequence of the lepton portal coupling $y_{\ell}$ is the magnetic
dipole moment for leptons at one-loop level. The $g-2$ contribution for lepton
is given as Moroi:1995yh ; Carena:1996qa

$\displaystyle\Delta a_{\ell}\equiv a_{\ell}^{\rm exp}-a_{\ell}^{\rm
SM}=-\frac{y_{\ell}^{2}}{16\pi^{2}}\frac{m_{\ell}^{2}}{m_{S}^{2}}\frac{1-6x+3x^{2}+2x^{3}-6x^{2}\log
x}{6(1-x)^{4}},$ (12)

where $x\equiv m_{\chi}^{2}/m_{S}^{2}$ and we keep the leading order result in
the limit of small $m_{\ell}$. The last term containing $x$ in Eq. (12) is a
monotonically decreasing function of $x$ and it equals to $1/6$ and $1/12$ in
the limit of $x\to 0$ and $x\to 1$. Moreover, $\Delta a_{\ell}$ is always
negative. The electron magnetic dipole moment has been directly measured in
Refs. Hanneke:2008tm ; Hanneke:2010au , and one can compare it with the SM
prediction Aoyama:2014sxa ; Mohr:2015ccw  once the fine structure constant is
given. The recent results are

$\Delta a_{e}=(-88\pm 36)\times 10^{-14},\quad\Delta\tilde{a}_{e}=(48\pm
30)\times 10^{-14},$ (13)

where $\Delta a_{e}$ comes from the $\alpha$ measurement in recent cesium
recoil experiments Parker:2018vye , while $\Delta\tilde{a}_{e}$ comes from a
new independent measurement using rubidium atom Morel:2020dww . For electron
$g-2$, the uncertainties from QED, EW and hadronic contributions are much
smaller than $\alpha$, while for muon $g-2$, the uncertainty in $\alpha$ is
less significant. $\Delta a_{e}$ has a mild anomaly at confidence level of
$2.4~{}\sigma$, which might need a negative contribution from new physics.
However, $\Delta\tilde{a}_{e}$ turns to be positive and the significance goes
down to $1.6~{}\sigma$. The measurement Morel:2020dww  placed $95\%$ C.L.
bounds to be $\Delta\tilde{a}_{e}\in[-34\times 10^{-14},98\times 10^{-14}]$
and we will adopt this value in the later calculation.

Figure 11: The interplay between $(g-2)_{e}$ and thermal DM requirements. In
blue (red) shaded region, we show the range for $y_{e}$ which satisfies
$\Delta a_{e}$ ($\Delta\tilde{a}_{e}$) at $95\%$ C.L with $m_{\chi}\ll m_{S}$.
The gray dotdashed lines shows the thermal $y_{e}^{\rm th}$ values with
different $m_{\chi}$. For both $\Delta a_{e}$ and $\Delta\tilde{a}_{e}$,
$m_{\chi}$ between 1 GeV and 2 GeV can simultaneously explain $(g-2)_{e}$ and
DM relic abundance.

In our model, the contribution is always negative for the electron $g-2$. In
Fig. 11, We consider the $y_{e}$ range which can fit to $\Delta a_{e}$ (blue
shaded region) and $\Delta\tilde{a}_{e}$ (red shaded region) at $95\%$ C.L.,
with a fixed $m_{\chi}$ which is much smaller than $m_{S}$. We fix
$m_{\chi}=1$ GeV for $(g-2)_{e}$ fit, which satisfies the condition
$m_{\chi}\ll m_{S}$ easily. For $g-2$ contribution in this limit, one can have
the following expansion with small $x$ as

$\displaystyle\Delta
a_{\ell}\approx-\frac{y_{\ell}^{2}}{16\pi^{2}}\frac{m_{\ell}^{2}}{m_{S}^{2}}\left(\frac{1}{6}-\frac{x}{3}+\mathcal{O}(x^{2})\right).$
(14)

Therefore, for $m_{\chi}$ considered in Fig. 11, we see that $\Delta a_{\ell}$
is determined by $m_{S}$ and $y_{\ell}$ most of the time. As a result, the
blue and red shaded regions will not shift when changing to other $m_{\chi}$.
On the other hand, we plot $y_{e}^{\rm th}$ with a given DM mass, which can
provide the correct DM relic abundance in Fig. 11. They are plotted as dot-
dashed gray lines with a fixed $m_{\chi}$. We can see that for $\Delta a_{e}$,
DM mass between $0.2\sim 2$ GeV is preferred and can satisfy DM relic
abundance at the same time, while for $\Delta\tilde{a}_{e}$, DM mass larger
than 1 GeV is required. The two measurements have a mutual region when DM mass
is between 1 and 2 GeV.

The muon $g-2$ also has a long-standing discrepancy Blum:2018mom ;
Bennett:2006fi . Combination of the newest Fermilab and previous BNL
measurements yields Abi:2021gix

$\Delta a_{\mu}=(2.51\pm 0.59)\times 10^{-9},$ (15)

corresponding to a significance of $4.2\sigma$, suggesting a positive
contribution from the new physics.666There are debates on this “excess”. A
lattice group shows that there is no significant tension between the SM
prediction and the recent FNAL experimental determination Borsanyi:2020mff .
Unfortunately, in this lepton portal model the contribution has a negative
sign that it can not explain the anomaly. To incorporate the $\Delta a_{\mu}$
result, one has to add new ingredients beyond current model.

##  4 Probing the model with gravitational waves

In this section we investigate the possibility of probing the scalar sector
via the GWs from a FOPT in the early universe.777The FOPT of such a singlet
charged scalar was also studied in Ref. Ahriche:2018rao . Also see Refs.
Jiang:2015cwa ; Chen:2019ebq ; Cheng:2018ajh  for the FOPT triggered by a
complex gauge singlet. The first subsection is devoted to the discussion of
FOPT, while the second subsection studies the GW detection limits.

###  4.1 First-order phase transition

The scalar potential in Eq. (1) is

$V(H,S)=\mu_{H}^{2}|H|^{2}+\mu_{S}^{2}|S|^{2}+\lambda_{H}|H|^{4}+\lambda_{S}|S|^{4}+2\lambda_{HS}|H|^{2}|S|^{2},$

where

$H=\frac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{2}G^{+}\\\
h+iG^{0}\end{pmatrix},\quad S=\frac{\phi+i\eta}{\sqrt{2}}.$ (16)

In terms of the real components, we get

$V(h,\phi)=\frac{\mu_{H}^{2}}{2}h^{2}+\frac{\mu_{S}^{2}}{2}\phi^{2}+\frac{\lambda_{H}}{4}h^{4}+\frac{\lambda_{S}}{4}\phi^{4}+\frac{\lambda_{HS}}{2}h^{2}\phi^{2}.$
(17)

Here the quartic coefficients should satisfy

$\lambda_{H}>0,\quad\lambda_{S}>0,\quad\sqrt{\lambda_{H}\lambda_{S}}+\lambda_{HS}>0,$
(18)

to ensure the potential is bounded below.

Since $S$ has electric charge number $-1$, it cannot develop a vacuum
expectation value (VEV) at zero temperature. Hence the vacuum configuration at
zero temperature is along the Higgs direction, i.e. $(\left\langle
h\right\rangle,\left\langle\phi\right\rangle)=(v,0)$. This means that the
Higgs-relevant coefficients have been fixed by the collider measurements,

$\mu_{H}^{2}=-\frac{m_{h}^{2}}{2},\quad\lambda_{H}=\frac{m_{h}^{2}}{2v^{2}},$
(19)

where $m_{h}=125$ GeV and $v=246$ GeV, leaving only three free parameters in
the scalar potential. If $\mu_{S}^{2}$ is positive, then the vacuum
configuration $(v,0)$ is trivially achieved, because along the $\phi$
direction there is no local minimum. However, if $\mu_{S}^{2}<0$, then the
$\phi$ direction also has a local minimum $w=\sqrt{-\mu_{S}^{2}/\lambda_{S}}$.
In this case, to make sure $(v,0)$ is a minimum but not a saddle point, the
coefficients must satisfy

$\lambda_{H}\mu_{S}^{2}>\lambda_{HS}\mu_{H}^{2},$ (20)

according to the Hessian matrix Bian:2019kmg . In case of
$\lambda_{S}\mu_{H}^{2}>\lambda_{HS}\mu_{S}^{2}$, $(0,w)$ is also a local
minimum, and a further condition

$V(v,0)=-\frac{\mu_{H}^{4}}{4\lambda_{H}}<V(0,w)=-\frac{\mu_{S}^{4}}{4\lambda_{S}},$
(21)

is required to ensure $(v,0)$ is the global minimum. The mass of $S$ is given
by $m_{S}^{2}=\mu_{S}^{2}+\lambda_{HS}v^{2}$. Although the collider
experiments have set a bound for $m_{S}$ (which also depends on $m_{\chi}$),
they cannot probe $\mu_{S}^{2}$ directly and neither the sign of
$\mu_{S}^{2}$. If $\mu_{S}^{2}<0$, the potential might trigger a FOPT in the
early universe, and this opens a new detection scenario for this kind of DM
models: the FOPT GWs.

At finite temperature, the scalar potential is modified by the thermal
correction. Taking the leading gauge invariant $T^{2}$ terms Dolan:1973qd ;
Braaten:1989kk , the thermal potential is

$V_{T}(h,\phi,T)\approx\frac{\mu_{H}^{2}+c_{h}T^{2}}{2}h^{2}+\frac{\mu_{S}^{2}+c_{\phi}T^{2}}{2}\phi^{2}+\frac{\lambda_{H}}{4}h^{4}+\frac{\lambda_{S}}{4}\phi^{4}+\frac{\lambda_{HS}}{2}h^{2}\phi^{2},$
(22)

where

$c_{h}=\frac{3g^{2}+g^{\prime
2}}{16}+\frac{y_{t}^{2}}{4}+\frac{\lambda_{H}}{2}+\frac{\lambda_{HS}}{6},\quad
c_{\phi}=\frac{g^{\prime 2}}{4}+\frac{\lambda_{S}}{3}+\frac{\lambda_{HS}}{3}.$
(23)

The necessary condition for a FOPT is the existence of a critical temperature
$T_{c}$ at which the system has two energetically degenerate vacua $(v_{c},0)$
and $(0,w_{c})$. For Eq. (22), this requires Bian:2019kmg

$\frac{c_{\phi}}{c_{h}}<\frac{\mu_{S}^{2}}{\mu_{H}^{2}}<\frac{\sqrt{\lambda_{S}}}{\sqrt{\lambda_{H}}}<\frac{\lambda_{HS}}{\lambda_{H}},$
(24)

and the critical temperature and VEVs are respectively

$T_{c}=\sqrt{\frac{\mu_{H}^{2}\sqrt{\lambda_{S}}-\mu_{S}^{2}\sqrt{\lambda_{H}}}{c_{\phi}\sqrt{\lambda_{H}}-c_{h}\sqrt{\lambda_{S}}}},\quad
v_{c}=\sqrt{\frac{c_{h}\mu_{S}^{2}-c_{\phi}\mu_{H}^{2}}{c_{\phi}\lambda_{H}-c_{h}\sqrt{\lambda_{H}\lambda_{S}}}}.$
(25)

Below $T_{c}$, the Higgs-direction minimum becomes the lower one and the
universe starts to decay to it from the $\phi$-direction minimum. The decay
rate per unit volume reads $\Gamma(T)\sim T^{4}e^{-S_{3}/T}$, where $S_{3}$ is
the classical action for the $O(3)$ symmetric bounce solution Linde:1981zj .
The nucleation temperature $T_{n}$ is defined by the equality of the
nucleation rate per Hubble volume and the universe expansion rate, i.e.
$\Gamma(T_{n})=H^{4}(T_{n})$. For a radiation-dominated universe and a FOPT
happening around the EW scale, $T_{n}$ can be solved by Quiros:1999jp

$S_{3}/T_{n}\sim 140.$ (26)

Eq. (26) is treated as the sufficient condition for a FOPT in this article.

Figure 12: The parameter space allowed by a FOPT. Different colors correspond
to different $\lambda_{S}$ values.

We use the Python package cosmoTransition Wainwright:2011kj  to calculate the
bounce solution and hence $T_{n}$ for the potential in Eq. (22). As discussed
before, after taking into account the Higgs mass and VEV, in the scalar
potential there are only three free parameters, which we choose to be
$\lambda_{S}$, $m_{S}$ and $\lambda_{HS}$. For a fixed $\lambda_{S}$, the
parameter space allowed by the FOPT can be projected to the
$m_{S}$-$\lambda_{HS}$ plane. We plot the parameter space for different
$\lambda_{S}$ as shaded areas in Fig. 12. The shapes of those areas can be
well understood by the analytical relation Eq. (24). A sizable $\lambda_{HS}$
is preferred by FOPT, because the phase transition is triggered by the
potential barrier induced by the $\lambda_{HS}h^{2}S^{2}/2$ term.

###  4.2 Gravitational waves

Stochastic GWs are produced during a FOPT via bubble collision Huber:2008hg ;
Di:2020nny , sound waves in the plasma Hindmarsh:2015qta  and the magneto-
hydrodynamics turbulence Binetruy:2012ze ; Caprini:2009yp . The expanding
bubble wall only accelerates for a short time before it reaches its final
velocity $v_{b}<1$. Therefore, only a tiny fraction of FOPT energy deposits in
the bubble shell, and the bubble collision contribution to the GWs is
negligible. Instead, most of the phase transition energy is pumped into the
surrounding fluid shells, making sound waves the dominant contribution to FOPT
GWs Ellis:2018mja . However, the sound wave period only lasts for a finite
time, after which the energy in the bulk fluid will cause the turbulence,
which is another source for GWs Ellis:2020awk . Consequently, the GW spectrum
today can be expressed as

$\Omega_{\rm GW}(f)=\Omega_{\rm sw}(f)+\Omega_{\rm turb}(f),$ (27)

where $f$ is the frequency, the subscripts “sw” and “turb” denote sound waves
and turbulence respectively, and $\Omega_{\rm GW}$ is the ratio of GW energy
density to the critical energy of the current universe, i.e.

$\Omega_{\rm GW}(f)=\frac{1}{\rho_{c}}\frac{\rho_{\rm GW}}{d\ln f},$ (28)

where $\rho_{c}=3H_{0}^{2}/(8\pi G)$, with $H_{0}$ being the Hubble constant
today.

The sound wave and turbulence spectra can be expressed as functions of two
FOPT parameters Grojean:2006bp ; Caprini:2015zlo ; Caprini:2019egz ,

$\alpha=\frac{1}{g_{*}\pi^{2}T_{n}^{4}/30}\left(T\frac{\partial\Delta
V_{T}}{\partial T}-\Delta
V_{T}\right)\Big{|}_{T_{n}};\quad\beta/H=T_{n}\frac{d(S_{3}/T)}{dT}\Big{|}_{T_{n}},$
(29)

where $\Delta V_{T}$ denotes the (negative) effective potential difference
between the true and false vacua, and $g_{*}\sim 100$ is the number of
relativistic degrees of freedom during FOPT.888It is suggested that the
$\alpha$ and $\beta/H$ parameters should be calculated at the percolation
temperature $T_{p}$ Megevand:2016lpr ; Kobakhidze:2017mru ; Ellis:2018mja ;
Ellis:2020awk ; Wang:2020jrd . However, we have checked that for our FOPT
scenario $\alpha\lesssim 1$, therefore the supercooling effect is not
prominent and $T_{n}\approx T_{p}$ is a good approximation. Namely, $\alpha$
is the ratio of FOPT latent heat to the radiation energy, while $\beta/H$ is
the inverse ratio of FOPT duration to the universe expansion time scale.
Numerically, the sound waves spectrum is Hindmarsh:2015qta

$\Omega_{\rm sw}(f)h^{2}=2.65\times
10^{-6}\frac{1}{\beta/H}\left(\frac{\kappa_{v}\alpha}{1+\alpha}\right)^{2}\left(\frac{g_{*}}{100}\right)^{-1/3}v_{b}\left(\frac{f}{f_{\rm
sw}}\right)^{3}\left(\frac{7}{4+3(f/f_{\rm sw})^{2}}\right)^{7/2},$ (30)

where

$f_{\rm sw}=1.9\times 10^{-2}~{}{\rm
mHz}\times\frac{\beta/H}{v_{b}}\left(\frac{T_{n}}{100~{}{\rm
GeV}}\right)\left(\frac{g_{*}}{100}\right)^{1/6}.$ (31)

To take into account the finite duration of the sound wave period, we make the
replacement $\Omega_{\rm sw}(f)\to\Omega_{\rm sw}(f)H(T_{n})\tau_{\rm sw}$ in
Eq. (30) Ellis:2020awk ; Wang:2020jrd , where

$\tau_{\rm sw}={\rm
min}\left\\{\frac{1}{H(T_{n})},\frac{v_{b}(8\pi)^{1/3}}{\beta\bar{U}_{f}}\right\\},\quad\bar{U}_{f}=\sqrt{\frac{3}{4}\frac{\kappa_{v}\alpha}{1+\alpha}}.$
(32)

A more accurate treatment for the sound wave cutoff factor can be found in
Ref. Guo:2020grp .

The turbulence spectrum is Binetruy:2012ze ; Caprini:2009yp

$\Omega_{\rm turb}(f)h^{2}=3.35\times
10^{-4}\frac{v_{b}}{\beta/H}\left(\frac{\kappa_{\rm
turb}\alpha}{1+\alpha}\right)^{3/2}\left(\frac{g_{*}}{100}\right)^{-1/3}S_{\rm
turb}(f),$ (33)

where

$S_{\rm turb}(f)=\frac{(f/f_{\rm turb})^{3}}{\left[1+(f/f_{\rm
turb})\right]^{11/3}(1+8\pi f/h_{*})},\quad h_{*}=16.5\times 10^{-3}~{}{\rm
mHz}\left(\frac{T_{n}}{100~{}{\rm
GeV}}\right)\left(\frac{g_{*}}{100}\right)^{1/6},$ (34)

and

$f_{\rm turb}=2.7\times 10^{-2}~{}{\rm
mHz}\times\frac{\beta/H}{v_{b}}\left(\frac{T_{n}}{100~{}{\rm
GeV}}\right)\left(\frac{g_{*}}{100}\right)^{1/6}.$ (35)

The factor $\kappa_{v}$ and $\kappa_{\rm turb}$ in Eq. (30) and Eq. (33) are
the fraction of FOPT energy that is transformed to bulk motion/turbulence,
respectively. We have adopted $\kappa_{\rm turb}=0.05\kappa_{v}$, and
$\kappa_{v}$ is extracted from the numerical function of Ref. Espinosa:2010hh
. $v_{b}=0.6$ is used as a benchmark in our study, although it might be
calculated using the hydrodynamics.999See Refs. Konstandin:2014zta ;
Cline:2021iff  for the $v_{b}$ determination in the real singlet extended SM.

Figure 13: Top: the parameter space that triggers FOPT and can be probed by
LISA, projected to different 2-dimension planes. Bottom: the GW spectrum from
two benchmark points which are labeled as stars in the top left panel.

The GWs from a FOPT around EW scale might be probed by the next generation
space-based laser interferometers such as LISA Audley:2017drz , BBO
Crowder:2005nr , TianQin Luo:2015ght ; Hu:2017yoc , Taiji Hu:2017mde ;
Guo:2018npi  and DECIGO Kawamura:2011zz ; Kawamura:2006up . The detectability
of GW signals can be characterized by the signal-to-noise ratio (SNR). Taking
the LISA detector as an example, the SNR is defined as Caprini:2015zlo

${\rm SNR}=\sqrt{\mathcal{T}\int_{f_{\rm min}}^{f_{\rm
max}}df\left(\frac{\Omega_{\rm GW}(f)}{\Omega_{\rm LISA}(f)}\right)^{2}},$
(36)

where $\Omega_{\rm LISA}$ is the sensitive curve of the LISA detector
Caprini:2015zlo  (in which we take the C1 configuration as a benchmark), and
$\mathcal{T}=9.46\times 10^{7}$ s is the data-taking duration Caprini:2019egz
. We adopt ${\rm SNR}=10$ as the detectable threshold, and found that for a
fixed $\lambda_{S}$, only a narrow band of parameter space in Fig. 12 can be
probed. The parameter space that can be probed by LISA is plotted in the top
panel of Fig. 13, where we show the results both in the $m_{S}$-$\lambda_{HS}$
plane (for fixed $\lambda_{S}$) and the $\lambda_{S}$-$\lambda_{HS}$ plane
(for fixed $m_{S}$). In the bottom panel of the same figure we plot the GW
spectrum for two benchmark points:

$\begin{split}{\rm BP1}:\quad&m_{S}=198~{}{\rm
GeV},~{}\lambda_{HS}=0.86,~{}\lambda_{S}=1,\\\ &T_{n}=76.6~{}{\rm
GeV},~{}v_{n}=189~{}{\rm GeV},~{}w_{n}=94.9~{}{\rm
GeV},~{}\alpha=0.0834,~{}\beta/H=228;\\\ {\rm BP2}:\quad&m_{S}=302~{}{\rm
GeV},~{}\lambda_{HS}=1.78,~{}\lambda_{S}=2,\\\ &T_{n}=91.8~{}{\rm
GeV},~{}v_{n}=123~{}{\rm GeV},~{}w_{n}=53.1~{}{\rm
GeV},~{}\alpha=0.0248,~{}\beta/H=177;\\\ \end{split}$ (37)

where $(0,w_{n})$ and $(v_{n},0)$ are respectively the old and new vacua at
the nucleation temperature $T_{n}$.

##  5 The interplay between phase transition and particle searches

In this section, we revisit the parameter space for $\lambda_{HS}$ from GW
study and the corresponding constraints from the collider studies. We focus on
how the two different types of experiments can be complimentary with each
other. For GW experiment, we focus on the LISA detectable parameter space in
the left panel of Fig. 13. Clearly, the scalar $S$ self-interaction coupling
$\lambda_{S}$ can affect the LISA detectable parameter space. Therefore, we
vary $\lambda_{S}$ between $0$ and $4\pi$ to obtain the entire GW detectable
region for a FOPT.

###  5.1 Interplay with $pp\to S^{+}S^{-}$ at the LHC

Figure 14: The interplay between gravitational wave detection and LHC
searches. The collider constrains and projections are the same as those in
Fig. 3. The light blue (orange) shaded region corresponds to $\lambda_{HS}=2$
(3), with the vertical boxed boundary regions being the LISA-detectable
parameter space, while the irregular boundary regions being enhanced part of
the LHC projections when including the $gg\to h^{*}\to S^{+}S^{-}$
contribution.

The $S^{\pm}$ particles can be pair produced at the LHC via Drell-Yan and
gluon-gluon fusion processes, and the current constrains and future
projections at the 14 TeV LHC has been studied in Section 3.1.1. As shown in
Fig. 3, a non-zero $\lambda_{HS}$ can visibly enhance the probe limit in
parameter space due to the $gg\to h^{*}\to S^{+}S^{-}$ contribution to the
signal events. On the other hand, a sizable $\lambda_{HS}$ may trigger a FOPT
and hence give detectable GW signals as well.

The interplay between the LHC and the future LISA experiments is plotted in
Fig. 14. The light blue (orange) shaded region corresponds to $\lambda_{HS}=2$
(3), with the vertical boxed boundary regions being the LISA-detectable
parameter space, while the irregular boundary regions being the enhanced LHC
projections when including the $gg\to h^{*}\to S^{+}S^{-}$ contribution. This
is because the GW signals is independent of the DM mass $m_{\chi}$. For a
given $\lambda_{HS}$, there is a set of upper and lower bounds for $m_{S}$ in
the LISA-detectable region. The enhanced parts of the LHC probe region due to
$\lambda_{HS}$ are also shown in the figure. We see that the LHC and GW
experiments mainly serve as complementary approaches to probe the DM parameter
space; while they also have some intersections, which can be used to identify
the origin of the excess (if found).

###  5.2 Higgs precision measurement at the future $e^{+}e^{-}$ colliders

Figure 15: The interplay between gravitational wave detection and future
$e^{+}e^{-}$ collider searches. The gray shaded region is the LISA detectable
parameter space, varying $\lambda_{S}$ from 0 to $4\pi$. From left to right,
we show the sensitivities for $\lambda_{HS}$ from future CEPC precision
measurements, based on invisible Higgs decay branching ratio
$\text{Br}(h\to{\rm inv})=0.3\%$, Higgs leptonic coupling precision reaches
$\delta\kappa_{\mu}<8.7\%$ and $\delta\kappa_{\tau}<1.5\%$.

The collider constraints on $\lambda_{HS}$ have to be related with SM Higgs.
The constraint from exotic Higgs decay is less sensitive compared to the Higgs
1-loop coupling as shown in the previous section. The 1-loop induced Higgs
couplings include the coupling to $\chi\chi$ and $\ell^{+}\ell^{-}$. The
former can be revealed by the Higgs invisible decay branching ratio, for
example we consider the future sensitivity from CEPC $\text{Br}(h\to{\rm
inv})=0.3\%$. The latter can be revealed by the Higgs precision measurements
at CEPC with relative precision of couplings $\delta\kappa_{\mu}<8.7\%$ and
$\delta\kappa_{\tau}<1.5\%$. In Fig. 15, we take the DM mass
$m_{\chi}=1,~{}5,~{}10,~{}30,~{}50,~{}60$ GeV respectively to show its effect
on the sensitivities for $\lambda_{HS}$. For a fixed DM mass, the
corresponding colored line shows the maximum allowed $\lambda_{HS}$ from the
future $e^{+}e^{-}$ collider searches. In general, the exclusion power is
better for light DM mass $m_{\chi}$.

In the left panel of Fig. 15, one can see that for $m_{\chi}<40$ GeV the
constraints on $\lambda_{HS}$ are quite similar. The reason is that the 1-loop
induced coupling is proportional to
$y^{2}_{\ell}\lambda_{HS}m_{\chi}/m_{S}^{2}$ for large $m_{S}$. At the same
time, the annihilation cross section is proportional to
$y^{4}_{\ell}m_{\chi}^{2}/m_{S}^{4}$ which requires this combination to be a
constant to satisfy the relic abundance. Therefore, the limits on
$\lambda_{HS}$ from Higgs invisible decay is a constant. The colored lines in
the left panel do show this feature, except when $m_{S}$ is too close to the
Higgs mass and the expansion on large $m_{S}$ is not valid anymore, the
sensitivity on $\lambda_{HS}$ changes slightly. For larger $m_{\chi}$, the
sensitivity of $\lambda_{HS}$ is downgraded because the phase space
suppression in the $h\to\chi\chi$ decay. It is worth to mention that the
sensitivity from Higgs invisible decay works equally good for all three
flavors of lepton portals. This search can test most of the LISA detectable
parameter regions for $m_{\chi}<m_{h}/2$.

In the right panel of Fig. 15, the limits from $\delta\kappa_{\tau}<1.5\%$ are
plotted for different DM mass. For large $m_{S}$, we can see that the
constraints on $\lambda_{HS}$ are proportional to $m_{\chi}$. The reason is
that the one-loop induced Higgs coupling is roughly proportional to
$y^{2}_{\ell}\lambda_{HS}m_{\ell}/m_{S}^{2}$ for large $m_{S}$ expansion.
Since the relic abundance fix the combination $y^{2}_{\ell}m_{\chi}/m_{S}^{2}$
to be constant, the sensitivity for $\lambda_{HS}$ from Higgs precision
measurement is proportional to $m_{\chi}$. Different from Higgs invisible
branching ratio, there is no phase space suppression for $m_{\chi}\sim
m_{h}/2$. One can see that for $m_{\chi}\lesssim 20$ GeV, the Higgs-tau
coupling precision measurement is the most sensitive among the three panels in
the figure, while for the intermediate mass $m_{\chi}$ between $20$ to $\sim
50$ GeV the Higgs invisible branching ratio measurement is better. For
$m_{\chi}$ close to $m_{h}/2$, the Higgs-tau coupling measurement becomes
better again due to the phase space suppression in the Higgs invisible decay.

In the middle panel of Fig. 15, we show the limits from
$\delta\kappa_{\mu}<8.7\%$. The results from muon coupling measurements are
fully analogous to tau coupling. The sensitivity is worse by a constant factor
from $\delta\kappa_{\tau}/\delta\kappa_{\mu}$, reflecting the fact that more
taus are produced due to larger Higgs-tau coupling.

Finally, it should be mentioned that the limits on $\lambda_{HS}$ from
$\text{Br}(h\to\gamma\gamma)$ and $\sigma(Zh)$ are also very powerful as shown
in section 3.3 and are able to exclude most of the parameter space for GW
detection Huang:2016cjm . Such constraints are independent of the DM Yukawa
coupling $y_{\ell}$ and therefore, are complementary with the limits from $h$
invisible and leptonic decays.

##  6 Conclusions

The GW detection opens a new window to the FOPT and the Higgs precision
measurement is an inevitable path after the Higgs discovery. In this paper, we
study their interplay in a specific DM model, namely lepton portal DM model.
We emphasize the Higgs portal coupling in this model, which is neglected in
the previous literature. The impact is investigated in two aspects. In the
cosmological aspect, we have studied the parameter space allowing a FOPT and
yielding detectable GW signals at the future detectors, taking LISA as an
example. In the particle aspect, we have considered various new channels to
further test this model:

  * •

$pp\to S^{+}S^{-}$ at the LHC, which mainly probes $m_{S}$ and $m_{\chi}$
since the production is dominated by the Drell-Yan process, can also test the
Higgs portal coupling between $h$ and $S^{\pm}$ (i.e. $\lambda_{HS}$) through
the gluon-gluon fusion process.

  * •

$e^{+}e^{-}\to S^{\pm}S^{\mp(*)}$ at the future lepton colliders, which can
fill in the gaps between the LEP and LHC constrains ($100~{}{\rm GeV}\lesssim
m_{S}\lesssim 150$ GeV), and probe the $y_{\ell}$ coupling via the off-shell
production of $S^{\pm}$.

  * •

Exotic decays of $h$ and $Z$ at the future lepton colliders, which probe the
couplings $\lambda_{HS}$ and/or $y_{\ell}$ for the low $m_{\chi}$ region.

  * •

Higgs precision measurements for invisible decay branching ratios and leptonic
coupling originated from one-loop contributions, which can provide the best
sensitivity for the combination $y_{\ell}^{2}\lambda_{HS}$ or $\lambda_{HS}$
assuming $y_{\ell}$ satisfies the DM relic abundance requirement.

  * •

Electron $g-2$ experiments have recently came up with two sets of results. For
$\Delta\tilde{a}_{e}$, DM mass should be larger than $1$ GeV, while for
$\Delta a_{e}$, DM mass between $0.2\sim 2$ GeV is preferred.

In summary, the future Higgs precision measurements can effectively interplay
with GW detection, since they both rely on the Higgs portal coupling. The
Higgs portal is allowed by this model and can contribute to the Higgs
couplings to DM and SM leptons at one-loop level. Therefore, most of the GW
detectable parameter space can be cross-checked by the Higgs precision
measurement. It shows the rigorous interplay between the future Higgs
precision measurement program and the GW detection program. Specific to the
lepton portal DM model, which is hard to probe through DM direct and indirect
detections, the Drell-Yan production of charged scalar pair is the useful way
to probe this model but only constrains the mass parameter of the scalar and
DM. We studied the Higgs mediated $S^{\pm}$ pair production, exotic decays of
$h/Z$ and electron $g-2$ experiment, which can help extending the constraints
on mass parameters and also providing useful constraints on the Yukawa and
scalar portal couplings.

## Acknowledgments

We would like to thank Caterina Doglioni, Manqi Ruan, Jian Wang and Lian-Tao
Wang for useful discussions and comments. The work of JL is supported by
National Science Foundation of China under Grant No. 12075005 and by Peking
University under startup Grant No. 7101502458. The work of XPW is supported by
National Science Foundation of China under Grant No. 12005009. KPX is
supported by the Grant Korea NRF-2019R1C1C1010050. KPX would like to thank the
hospitality of the University of Chicago where part of this work was
performed.

## References

  * (1) Planck collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” _Astron. Astrophys._ 641 (2020) A6, [1807.06209].
  * (2) G. Bertone, D. Hooper and J. Silk, “Particle dark matter: Evidence, candidates and constraints,” _Phys. Rept._ 405 (2005) 279–390, [hep-ph/0404175].
  * (3) B. W. Lee and S. Weinberg, “Cosmological Lower Bound on Heavy Neutrino Masses,” _Phys. Rev. Lett._ 39 (1977) 165–168.
  * (4) M. Schumann, “Direct Detection of WIMP Dark Matter: Concepts and Status,” _J. Phys. G_ 46 (2019) 103003, [1903.03026].
  * (5) J. M. Gaskins, “A review of indirect searches for particle dark matter,” _Contemp. Phys._ 57 (2016) 496–525, [1604.00014].
  * (6) A. Boveia and C. Doglioni, “Dark Matter Searches at Colliders,” _Ann. Rev. Nucl. Part. Sci._ 68 (2018) 429–459, [1810.12238].
  * (7) LHC New Physics Working Group collaboration, D. Alves, “Simplified Models for LHC New Physics Searches,” _J. Phys. G_ 39 (2012) 105005, [1105.2838].
  * (8) J. Abdallah et al., “Simplified Models for Dark Matter and Missing Energy Searches at the LHC,” 1409.2893.
  * (9) J. Abdallah et al., “Simplified Models for Dark Matter Searches at the LHC,” _Phys. Dark Univ._ 9-10 (2015) 8–23, [1506.03116].
  * (10) D. Abercrombie et al., “Dark Matter Benchmark Models for Early LHC Run-2 Searches: Report of the ATLAS/CMS Dark Matter Forum,” _Phys. Dark Univ._ 27 (2020) 100371, [1507.00966].
  * (11) C. Boehm and P. Fayet, “Scalar dark matter candidates,” _Nucl. Phys._ B683 (2004) 219–263, [hep-ph/0305261].
  * (12) J. Goodman and W. Shepherd, “LHC Bounds on UV-Complete Models of Dark Matter,” 1111.2359.
  * (13) M. Garny, A. Ibarra, M. Pato and S. Vogl, “Closing in on mass-degenerate dark matter scenarios with antiprotons and direct detection,” _JCAP_ 11 (2012) 017, [1207.1431].
  * (14) J. Liu, B. Shuve, N. Weiner and I. Yavin, “Looking for new charged states at the LHC: Signatures of Magnetic and Rayleigh Dark Matter,” _JHEP_ 07 (2013) 144, [1303.4404].
  * (15) H. An, L.-T. Wang and H. Zhang, “Dark matter with $t$-channel mediator: a simple step beyond contact interaction,” _Phys. Rev. D_ 89 (2014) 115014, [1308.0592].
  * (16) A. DiFranzo, K. I. Nagao, A. Rajaraman and T. M. P. Tait, “Simplified Models for Dark Matter Interacting with Quarks,” _JHEP_ 11 (2013) 014, [1308.2679].
  * (17) Y. Bai and J. Berger, “Fermion Portal Dark Matter,” _JHEP_ 11 (2013) 171, [1308.0612].
  * (18) M. Papucci, A. Vichi and K. M. Zurek, “Monojet versus the rest of the world I: t-channel models,” _JHEP_ 11 (2014) 024, [1402.2285].
  * (19) A. De Simone, G. F. Giudice and A. Strumia, “Benchmarks for Dark Matter Searches at the LHC,” _JHEP_ 06 (2014) 081, [1402.6287].
  * (20) M. Garny, A. Ibarra, S. Rydbeck and S. Vogl, “Majorana Dark Matter with a Coloured Mediator: Collider vs Direct and Indirect Searches,” _JHEP_ 06 (2014) 169, [1403.4634].
  * (21) M. A. Gomez, C. B. Jackson and G. Shaughnessy, “Dark Matter on Top,” _JCAP_ 12 (2014) 025, [1404.1918].
  * (22) M. J. Baker et al., “The Coannihilation Codex,” _JHEP_ 12 (2015) 120, [1510.03434].
  * (23) Y. Bai and J. Berger, “Lepton Portal Dark Matter,” _JHEP_ 08 (2014) 153, [1402.6696].
  * (24) S. Chang, R. Edezhath, J. Hutchinson and M. Luty, “Leptophilic Effective WIMPs,” _Phys. Rev. D_ 90 (2014) 015011, [1402.7358].
  * (25) P. Agrawal, Z. Chacko and C. B. Verhaaren, “Leptophilic Dark Matter and the Anomalous Magnetic Moment of the Muon,” _JHEP_ 08 (2014) 147, [1402.7369].
  * (26) M. Garny, A. Ibarra and S. Vogl, “Signatures of Majorana dark matter with t-channel mediators,” _Int. J. Mod. Phys. D_ 24 (2015) 1530019, [1503.01500].
  * (27) Z.-H. Yu, X.-J. Bi, Q.-S. Yan and P.-F. Yin, “Tau Portal Dark Matter models at the LHC,” _Phys. Rev. D_ 91 (2015) 035008, [1410.3347].
  * (28) W. Altmannshofer, P. J. Fox, R. Harnik, G. D. Kribs and N. Raj, “Dark Matter Signals in Dilepton Production at Hadron Colliders,” _Phys. Rev. D_ 91 (2015) 115006, [1411.6743].
  * (29) J.-H. Yu, “Vector Fermion-Portal Dark Matter: Direct Detection and Galactic Center Gamma-Ray Excess,” _Phys. Rev. D_ 90 (2014) 095010, [1409.3227].
  * (30) P. Agrawal, Z. Chacko, C. Kilic and C. B. Verhaaren, “A Couplet from Flavored Dark Matter,” _JHEP_ 08 (2015) 072, [1503.03057].
  * (31) A. Ibarra and S. Wild, “Dirac dark matter with a charged mediator: a comprehensive one-loop analysis of the direct detection phenomenology,” _JCAP_ 05 (2015) 047, [1503.03382].
  * (32) Y. Cai and A. P. Spray, “Fermionic Semi-Annihilating Dark Matter,” _JHEP_ 01 (2016) 087, [1509.08481].
  * (33) S. Baek and Z.-F. Kang, “Naturally Large Radiative Lepton Flavor Violating Higgs Decay Mediated by Lepton-flavored Dark Matter,” _JHEP_ 03 (2016) 106, [1510.00100].
  * (34) M.-C. Chen, J. Huang and V. Takhistov, “Beyond Minimal Lepton Flavored Dark Matter,” _JHEP_ 02 (2016) 060, [1510.04694].
  * (35) A. Berlin, D. S. Robertson, M. P. Solon and K. M. Zurek, “Bino variations: Effective field theory methods for dark matter direct detection,” _Phys. Rev. D_ 93 (2016) 095008, [1511.05964].
  * (36) P. Agrawal, Z. Chacko, E. C. F. S. Fortes and C. Kilic, “Skew-Flavored Dark Matter,” _Phys. Rev. D_ 93 (2016) 103510, [1511.06293].
  * (37) A. Mukherjee and M. K. Das, “Neutrino phenomenology and scalar Dark Matter with $A_{4}$ flavor symmetry in Inverse and type II seesaw,” _Nucl. Phys. B_ 913 (2016) 643–663, [1512.02384].
  * (38) J. A. Evans and J. Shelton, “Long-Lived Staus and Displaced Leptons at the LHC,” _JHEP_ 04 (2016) 056, [1601.01326].
  * (39) W. Chao, H.-K. Guo and H.-L. Li, “Tau flavored dark matter and its impact on tau Yukawa coupling,” _JCAP_ 02 (2017) 002, [1606.07174].
  * (40) D. Borah, S. Sadhukhan and S. Sahoo, “Lepton Portal Limit of Inert Higgs Doublet Dark Matter with Radiative Neutrino Mass,” _Phys. Lett. B_ 771 (2017) 624–632, [1703.08674].
  * (41) K. Kowalska and E. M. Sessolo, “Expectations for the muon g-2 in simplified models with dark matter,” _JHEP_ 09 (2017) 112, [1707.00753].
  * (42) G. H. Duan, L. Feng, F. Wang, L. Wu, J. M. Yang and R. Zheng, “Simplified TeV leptophilic dark matter in light of DAMPE data,” _JHEP_ 02 (2018) 107, [1711.11012].
  * (43) Q. Yuan et al., “Interpretations of the DAMPE electron data,” 1711.10989.
  * (44) Y.-L. Tang, L. Wu, M. Zhang and R. Zheng, “Lepton-portal Dark Matter in Hidden Valley model and the DAMPE recent results,” _Sci. China Phys. Mech. Astron._ 61 (2018) 101003, [1711.11058].
  * (45) S.-F. Ge, H.-J. He and Y.-C. Wang, “Flavor Structure of the Cosmic-Ray Electron/Positron Excesses at DAMPE,” _Phys. Lett. B_ 781 (2018) 88–94, [1712.02744].
  * (46) R. Ding, Z.-L. Han, L. Feng and B. Zhu, “Confronting the DAMPE Excess with the Scotogenic Type-II Seesaw Model,” _Chin. Phys. C_ 42 (2018) 083104, [1712.02021].
  * (47) M. J. Baker and A. Thamm, “Leptonic WIMP Coannihilation and the Current Dark Matter Search Strategy,” _JHEP_ 10 (2018) 187, [1806.07896].
  * (48) J. Hisano, R. Nagai and N. Nagata, “Singlet Dirac Fermion Dark Matter with Mediators at Loop,” _JHEP_ 12 (2018) 059, [1808.06301].
  * (49) A. Gaviria, R. Longas and O. Zapata, “Charged lepton flavor violation and electric dipole moments in the inert Zee model,” _JHEP_ 10 (2018) 188, [1809.00655].
  * (50) B. J. Kavanagh, P. Panci and R. Ziegler, “Faint Light from Dark Matter: Classifying and Constraining Dark Matter-Photon Effective Operators,” _JHEP_ 04 (2019) 089, [1810.00033].
  * (51) J. Kawamura, S. Okawa and Y. Omura, “Current status and muon $g-2$ explanation of lepton portal dark matter,” _JHEP_ 08 (2020) 042, [2002.12534].
  * (52) H. Okada and Y. Shoji, “Dirac dark matter in a radiative neutrino model,” _Phys. Dark Univ._ 31 (2021) 100742, [2003.11396].
  * (53) S.-F. Ge, H.-J. He, Y.-C. Wang and Q. Yuan, “Probing flavor structure of cosmic ray $e^{\mp}$ spectrum and implications for dark matter indirect searches,” _Nucl. Phys. B_ 959 (2020) 115140, [2004.10683].
  * (54) C. Bœhm, X. Chu, J.-L. Kuo and J. Pradler, “Scalar Dark Matter Candidates \- Revisited,” 2010.02954.
  * (55) S. Okawa and Y. Omura, “Light mass window of lepton portal dark matter,” 2011.04788.
  * (56) K. Kowalska and E. M. Sessolo, “Minimal models for $g-2$ and dark matter confront asymptotic safety,” 2012.15200.
  * (57) R. Verma, M. Kashav, S. Verma and B. C. Chauhan, “Scalar Dark Matter in an Inverse Seesaw Model with A_4 Discrete Flavor Symmetry,” 2102.03074.
  * (58) C. Alvarado, C. Bonilla, J. Leite and J. W. F. Valle, “Phenomenology of fermion dark matter as neutrino mass mediator with gauged B-L,” 2102.07216.
  * (59) S.-i. Horigome, T. Katayose, S. Matsumoto and I. Saha, “Leptophilic fermion WIMP ~ Role of future lepton colliders,” 2102.08645.
  * (60) Y. Bai and J. Berger, “Muon g-2 in Lepton Portal Dark Matter,” 2104.03301.
  * (61) A. Jueid, S. Nasri and R. Soualah, “Searching for GeV-scale Majorana Dark Matter: inter spem et metum,” _JHEP_ 04 (2021) 012, [2006.01348].
  * (62) G. Arcadi, L. Calibbi, M. Fedele and F. Mescia, “Systematic approach to $B$-physics anomalies and $t$-channel dark matter,” 2103.09835.
  * (63) G. Arcadi, L. Calibbi, M. Fedele and F. Mescia, “Muon $g-2$ and $B$-anomalies from Dark Matter,” 2104.03228.
  * (64) L. Calibbi, R. Ziegler and J. Zupan, “Minimal models for dark matter and the muon g$-$2 anomaly,” _JHEP_ 07 (2018) 046, [1804.00009].
  * (65) M. Garny, A. Ibarra, M. Pato and S. Vogl, “Internal bremsstrahlung signatures in light of direct dark matter searches,” _JCAP_ 1312 (2013) 046, [1306.6342].
  * (66) F. Luo and T. You, “Enhancement of Majorana Dark Matter Annihilation Through Higgs Bremsstrahlung,” _JCAP_ 1312 (2013) 024, [1310.5129].
  * (67) P. Gondolo and G. Gelmini, “Cosmic abundances of stable particles: Improved analysis,” _Nucl. Phys._ B360 (1991) 145–179.
  * (68) Fermi-LAT collaboration, M. Ackermann et al., “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, [1503.02641].
  * (69) Fermi-LAT, DES collaboration, A. Albert et al., “Searching for Dark Matter Annihilation in Recently Discovered Milky Way Satellites with Fermi-LAT,” _Astrophys. J._ 834 (2017) 110, [1611.03184].
  * (70) AMS collaboration, L. Accardo et al., “High Statistics Measurement of the Positron Fraction in Primary Cosmic Rays of 0.5–500 GeV with the Alpha Magnetic Spectrometer on the International Space Station,” _Phys. Rev. Lett._ 113 (2014) 121101.
  * (71) AMS collaboration, M. Aguilar et al., “Electron and Positron Fluxes in Primary Cosmic Rays Measured with the Alpha Magnetic Spectrometer on the International Space Station,” _Phys. Rev. Lett._ 113 (2014) 121102.
  * (72) LZ collaboration, D. S. Akerib et al., “Projected sensitivities of the LUX-ZEPLIN (LZ) experiment to new physics via low-energy electron recoils,” 2102.11740.
  * (73) ATLAS collaboration, G. Aad et al., “Search for direct production of charginos, neutralinos and sleptons in final states with two leptons and missing transverse momentum in $pp$ collisions at $\sqrt{s}=$ 8 TeV with the ATLAS detector,” _JHEP_ 05 (2014) 071, [1403.5294].
  * (74) ATLAS collaboration, G. Aad et al., “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._ C80 (2020) 123, [1908.08215].
  * (75) ATLAS collaboration, G. Aad et al., “Searches for electroweak production of supersymmetric particles with compressed mass spectra in $\sqrt{s}=$ 13 TeV $pp$ collisions with the ATLAS detector,” _Phys. Rev._ D101 (2020) 052005, [1911.12606].
  * (76) ALEPH, DELPHI, L3, OPAL Experiments, “Combined LEP Chargino Results, up to 208 GeV for low DM,” 2002.
  * (77) C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter, “UFO - The Universal FeynRules Output,” _Comput. Phys. Commun._ 183 (2012) 1201–1214, [1108.2040].
  * (78) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, “FeynRules 2.0 - A complete toolbox for tree-level phenomenology,” _Comput. Phys. Commun._ 185 (2014) 2250–2300, [1310.1921].
  * (79) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations,” _JHEP_ 07 (2014) 079, [1405.0301].
  * (80) T. Sjostrand, S. Mrenna and P. Z. Skands, “A Brief Introduction to PYTHIA 8.1,” _Comput. Phys. Commun._ 178 (2008) 852–867, [0710.3820].
  * (81) DELPHES 3 collaboration, J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens et al., “DELPHES 3, A modular framework for fast simulation of a generic collider experiment,” _JHEP_ 02 (2014) 057, [1307.6346].
  * (82) C. Lester and D. Summers, “Measuring masses of semiinvisibly decaying particles pair produced at hadron colliders,” _Phys. Lett. B_ 463 (1999) 99–103, [hep-ph/9906349].
  * (83) Q.-H. Cao, G. Li, K.-P. Xie and J. Zhang, “Searching for weak singlet charged scalars at lepton colliders,” _Phys. Rev. D_ 99 (2019) 015027, [1810.07659].
  * (84) Z. Liu, L.-T. Wang and H. Zhang, “Exotic decays of the 125 GeV Higgs boson at future $e^{+}e^{-}$ lepton colliders,” _Chin. Phys._ C41 (2017) 063102, [1612.09284].
  * (85) J. Liu, L.-T. Wang, X.-P. Wang and W. Xue, “Exposing the dark sector with future Z factories,” _Phys. Rev._ D97 (2018) 095044, [1712.07237].
  * (86) CEPC Study Group collaboration, M. Dong et al., “CEPC Conceptual Design Report: Volume 2 - Physics & Detector,” 1811.10545.
  * (87) FCC collaboration, A. Abada et al., “FCC Physics Opportunities,” _Eur. Phys. J._ C79 (2019) 474.
  * (88) FCC collaboration, A. Abada et al., “FCC-ee: The Lepton Collider,” _Eur. Phys. J. ST_ 228 (2019) 261–623.
  * (89) M. Drees, M. M. Nojiri, D. P. Roy and Y. Yamada, “Light Higgsino dark matter,” _Phys. Rev._ D56 (1997) 276–290, [hep-ph/9701219].
  * (90) A. Djouadi, M. Drees, P. Fileviez Perez and M. Muhlleitner, “Loop induced Higgs and Z boson couplings to neutralinos and implications for collider and dark matter searches,” _Phys. Rev._ D65 (2002) 075016, [hep-ph/0109283].
  * (91) H. Eberl, M. Kincel, W. Majerotto and Y. Yamada, “One loop corrections to neutral Higgs boson decays into neutralinos,” _Nucl. Phys._ B625 (2002) 372–388, [hep-ph/0111303].
  * (92) H. H. Patel, “Package-X: A Mathematica package for the analytic calculation of one-loop integrals,” _Comput. Phys. Commun._ 197 (2015) 276–290, [1503.01469].
  * (93) ATLAS Collaboration collaboration, “Search for invisible Higgs boson decays with vector boson fusion signatures with the ATLAS detector using an integrated luminosity of 139 fb-1,” Tech. Rep. ATLAS-CONF-2020-008, CERN, Geneva, Apr, 2020.
  * (94) C. Bernaciak, T. Plehn, P. Schichtel and J. Tattersall, “Spying an invisible Higgs boson,” _Phys. Rev._ D91 (2015) 035024, [1411.7699].
  * (95) S. Heinemeyer and C. Schappacher, “Higgs Decays into Charginos and Neutralinos in the Complex MSSM: A Full One-Loop Analysis,” _Eur. Phys. J._ C75 (2015) 230, [1503.02996].
  * (96) M. Cepeda et al., “Report from Working Group 2: Higgs Physics at the HL-LHC and HE-LHC,” _CERN Yellow Rep. Monogr._ 7 (2019) 221–584, [1902.00134].
  * (97) ATLAS Collaboration collaboration, “A search for the dimuon decay of the Standard Model Higgs boson in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS Detector,” Tech. Rep. ATLAS-CONF-2019-028, CERN, Geneva, Jul, 2019.
  * (98) ATLAS collaboration, G. Aad et al., “Combined measurements of Higgs boson production and decay using up to $80$ fb-1 of proton-proton collision data at $\sqrt{s}=$ 13 TeV collected with the ATLAS experiment,” _Phys. Rev. D_ 101 (2020) 012002, [1909.02845].
  * (99) F. An et al., “Precision Higgs physics at the CEPC,” _Chin. Phys. C_ 43 (2019) 043002, [1810.09037].
  * (100) P. Huang, A. J. Long and L.-T. Wang, “Probing the Electroweak Phase Transition with Higgs Factories and Gravitational Waves,” _Phys. Rev._ D94 (2016) 075008, [1608.06619].
  * (101) CMS collaboration, A. M. Sirunyan et al., “Measurements of Higgs boson production cross sections and couplings in the diphoton decay channel at $\sqrt{s}=$ 13 TeV,” 2103.06956.
  * (102) ATLAS collaboration, G. Aad et al., “Higgs boson production cross-section measurements and their EFT interpretation in the $4\ell$ decay channel at $\sqrt{s}=$13 TeV with the ATLAS detector,” _Eur. Phys. J. C_ 80 (2020) 957, [2004.03447].
  * (103) T. Moroi, “The Muon anomalous magnetic dipole moment in the minimal supersymmetric standard model,” _Phys. Rev._ D53 (1996) 6565–6575, [hep-ph/9512396].
  * (104) M. Carena, G. F. Giudice and C. E. M. Wagner, “Constraints on supersymmetric models from the muon anomalous magnetic moment,” _Phys. Lett._ B390 (1997) 234–242, [hep-ph/9610233].
  * (105) D. Hanneke, S. Fogwell and G. Gabrielse, “New Measurement of the Electron Magnetic Moment and the Fine Structure Constant,” _Phys. Rev. Lett._ 100 (2008) 120801, [0801.1134].
  * (106) D. Hanneke, S. F. Hoogerheide and G. Gabrielse, “Cavity Control of a Single-Electron Quantum Cyclotron: Measuring the Electron Magnetic Moment,” _Phys. Rev. A_ 83 (2011) 052122, [1009.4831].
  * (107) T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, “Tenth-Order Electron Anomalous Magnetic Moment — Contribution of Diagrams without Closed Lepton Loops,” _Phys. Rev. D_ 91 (2015) 033006, [1412.8284].
  * (108) P. J. Mohr, D. B. Newell and B. N. Taylor, “CODATA Recommended Values of the Fundamental Physical Constants: 2014,” _Rev. Mod. Phys._ 88 (2016) 035009, [1507.07956].
  * (109) R. H. Parker, C. Yu, W. Zhong, B. Estey and H. Müller, “Measurement of the fine-structure constant as a test of the Standard Model,” _Science_ 360 (2018) 191, [1812.04130].
  * (110) L. Morel, Z. Yao, P. Cladé and S. Guellati-Khélifa, “Determination of the fine-structure constant with an accuracy of 81 parts per trillion,” _Nature_ 588 (2020) 61–65.
  * (111) RBC, UKQCD collaboration, T. Blum, P. A. Boyle, V. Gülpers, T. Izubuchi, L. Jin, C. Jung et al., “Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment,” _Phys. Rev. Lett._ 121 (2018) 022003, [1801.07224].
  * (112) Muon g-2 collaboration, G. W. Bennett et al., “Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL,” _Phys. Rev._ D73 (2006) 072003, [hep-ex/0602035].
  * (113) Muon g-2 collaboration, B. Abi et al., “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm,” _Phys. Rev. Lett._ 126 (2021) 141801, [2104.03281].
  * (114) S. Borsanyi et al., “Leading hadronic contribution to the muon 2 magnetic moment from lattice QCD,” _Nature_ 593 (2021) 51–55, [2002.12347].
  * (115) A. Ahriche, K. Hashino, S. Kanemura and S. Nasri, “Gravitational Waves from Phase Transitions in Models with Charged Singlets,” _Phys. Lett. B_ 789 (2019) 119–126, [1809.09883].
  * (116) 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, [1502.07574].
  * (117) N. Chen, T. Li, Y. Wu and L. Bian, “Complementarity of the future $e^{+}e^{-}$ colliders and gravitational waves in the probe of complex singlet extension to the standard model,” _Phys. Rev. D_ 101 (2020) 075047, [1911.05579].
  * (118) W. Cheng and L. Bian, “From inflation to cosmological electroweak phase transition with a complex scalar singlet,” _Phys. Rev. D_ 98 (2018) 023524, [1801.00662].
  * (119) L. Bian, Y. Wu and K.-P. Xie, “Electroweak phase transition with composite Higgs models: calculability, gravitational waves and collider searches,” _JHEP_ 12 (2019) 028, [1909.02014].
  * (120) L. Dolan and R. Jackiw, “Symmetry Behavior at Finite Temperature,” _Phys. Rev. D_ 9 (1974) 3320–3341.
  * (121) E. Braaten and R. D. Pisarski, “Resummation and Gauge Invariance of the Gluon Damping Rate in Hot QCD,” _Phys. Rev. Lett._ 64 (1990) 1338.
  * (122) A. D. Linde, “Decay of the False Vacuum at Finite Temperature,” _Nucl. Phys. B_ 216 (1983) 421.
  * (123) M. Quiros, “Finite temperature field theory and phase transitions,” in _ICTP Summer School in High-Energy Physics and Cosmology_ , pp. 187–259, 1, 1999.  hep-ph/9901312.
  * (124) C. L. Wainwright, “CosmoTransitions: Computing Cosmological Phase Transition Temperatures and Bubble Profiles with Multiple Fields,” _Comput. Phys. Commun._ 183 (2012) 2006–2013, [1109.4189].
  * (125) S. J. Huber and T. Konstandin, “Gravitational Wave Production by Collisions: More Bubbles,” _JCAP_ 09 (2008) 022, [0806.1828].
  * (126) Y. Di, J. Wang, R. Zhou, L. Bian, R.-G. Cai and J. Liu, “Magnetic field and gravitational waves from the first-order Phase Transition,” 2012.15625.
  * (127) M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, “Numerical simulations of acoustically generated gravitational waves at a first order phase transition,” _Phys. Rev. D_ 92 (2015) 123009, [1504.03291].
  * (128) P. Binetruy, A. Bohe, C. Caprini and J.-F. Dufaux, “Cosmological Backgrounds of Gravitational Waves and eLISA/NGO: Phase Transitions, Cosmic Strings and Other Sources,” _JCAP_ 06 (2012) 027, [1201.0983].
  * (129) C. Caprini, R. Durrer and G. Servant, “The stochastic gravitational wave background from turbulence and magnetic fields generated by a first-order phase transition,” _JCAP_ 12 (2009) 024, [0909.0622].
  * (130) J. Ellis, M. Lewicki and J. M. No, “On the Maximal Strength of a First-Order Electroweak Phase Transition and its Gravitational Wave Signal,” _JCAP_ 04 (2019) 003, [1809.08242].
  * (131) J. Ellis, M. Lewicki and J. M. No, “Gravitational waves from first-order cosmological phase transitions: lifetime of the sound wave source,” _JCAP_ 07 (2020) 050, [2003.07360].
  * (132) C. Grojean and G. Servant, “Gravitational Waves from Phase Transitions at the Electroweak Scale and Beyond,” _Phys. Rev. D_ 75 (2007) 043507, [hep-ph/0607107].
  * (133) C. Caprini et al., “Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions,” _JCAP_ 04 (2016) 001, [1512.06239].
  * (134) C. Caprini et al., “Detecting gravitational waves from cosmological phase transitions with LISA: an update,” _JCAP_ 03 (2020) 024, [1910.13125].
  * (135) A. Megevand and S. Ramirez, “Bubble nucleation and growth in very strong cosmological phase transitions,” _Nucl. Phys. B_ 919 (2017) 74–109, [1611.05853].
  * (136) A. Kobakhidze, C. Lagger, A. Manning and J. Yue, “Gravitational waves from a supercooled electroweak phase transition and their detection with pulsar timing arrays,” _Eur. Phys. J. C_ 77 (2017) 570, [1703.06552].
  * (137) X. Wang, F. P. Huang and X. Zhang, “Phase transition dynamics and gravitational wave spectra of strong first-order phase transition in supercooled universe,” _JCAP_ 05 (2020) 045, [2003.08892].
  * (138) H.-K. Guo, K. Sinha, D. Vagie and G. White, “Phase Transitions in an Expanding Universe: Stochastic Gravitational Waves in Standard and Non-Standard Histories,” _JCAP_ 01 (2021) 001, [2007.08537].
  * (139) J. R. Espinosa, T. Konstandin, J. M. No and G. Servant, “Energy Budget of Cosmological First-order Phase Transitions,” _JCAP_ 06 (2010) 028, [1004.4187].
  * (140) T. Konstandin, G. Nardini and I. Rues, “From Boltzmann equations to steady wall velocities,” _JCAP_ 09 (2014) 028, [1407.3132].
  * (141) J. M. Cline, A. Friedlander, D.-M. He, K. Kainulainen, B. Laurent and D. Tucker-Smith, “Baryogenesis and gravity waves from a UV-completed electroweak phase transition,” 2102.12490.
  * (142) LISA collaboration, P. Amaro-Seoane et al., “Laser Interferometer Space Antenna,” 1702.00786.
  * (143) J. Crowder and N. J. Cornish, “Beyond LISA: Exploring future gravitational wave missions,” _Phys. Rev. D_ 72 (2005) 083005, [gr-qc/0506015].
  * (144) TianQin collaboration, J. Luo et al., “TianQin: a space-borne gravitational wave detector,” _Class. Quant. Grav._ 33 (2016) 035010, [1512.02076].
  * (145) Y.-M. Hu, J. Mei and J. Luo, “Science prospects for space-borne gravitational-wave missions,” _Natl. Sci. Rev._ 4 (2017) 683–684.
  * (146) 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) 685–686.
  * (147) W.-H. Ruan, Z.-K. Guo, R.-G. Cai and Y.-Z. Zhang, “Taiji program: Gravitational-wave sources,” _Int. J. Mod. Phys. A_ 35 (2020) 2050075, [1807.09495].
  * (148) S. Kawamura et al., “The Japanese space gravitational wave antenna: DECIGO,” _Class. Quant. Grav._ 28 (2011) 094011.
  * (149) S. Kawamura et al., “The Japanese space gravitational wave antenna DECIGO,” _Class. Quant. Grav._ 23 (2006) S125–S132.