11institutetext: Department of Applied Physics and Astronomy, University of
Sharjah, P.O. Box 27272 Sharjah, UAE.22institutetext: Department of Physics,
Osaka University, Toyonaka, Osaka 560-0043, Japan.

# Gravitational Waves from Phase Transitions in Scale Invariant Models

Amine Ahriche${}^{\href https://orcid.org/0000-0003-0230-1774}$
<EMAIL_ADDRESS>2 Shinya Kanemura${}^{\href
https://orcid.org/0000-0002-1303-7043}$<EMAIL_ADDRESS>2
Masanori Tanaka${}^{\href https://orcid.org/0000-0002-1303-7043}$
<EMAIL_ADDRESS>
###### Abstract

We investigate the properties of the gravitational waves (GW) generated during
a strongly first order electroweak phase transition (EWPT) in models with the
classical scale invariance (CSI). Here, we distinguish two parameter space
regions that correspond to the cases of (1) light dilaton and (2) purely
radiative Higgs mass (PRHM). In the CSI models, the dilaton mass, or the Higgs
mass in the PRHM case, in addition to some triple scalar couplings are fully
triggered by the radiative corrections (RCs). In order to probe the RCs
effects on the EWPT strength and on the GW spectrum, we extend the standard
model by a real singlet to assist the electroweak symmetry breaking and an
additional scalar field $Q$ with multiplicity $N_{Q}$ and mass $m_{Q}$. After
imposing all theoretical and experimental constraints, we show that a strongly
first order EWPT with detectable GW spectra can be realized for the two cases
of light dilaton and PRHM. We also show the corresponding values of the
relative enhancement of the cross section for the di-Higgs production process,
which is related to the triple Higgs boson coupling. We obtain the region in
which the GW spectrum can be observed by different future experiments such as
LISA and DECIGO. We also show that the scenarios (1) and (2) can be
discriminated by future GW observations and measurements of the di-Higgs
productions at future colliders.

OU-HET-1201

##  1 Introduction

The standard model (SM) in particle physics is a successful theory to explain
results at the Large Hadron Collider (LHC) ATLAS:2012yve . On the other hand,
the SM has several experimental and theoretical problems. For the theoretical
aspect, radiative corrections (RCs) to the Higgs boson mass cause the
quadratic divergences in the SM, which is the origin of so-called the
hierarchy problem Bardeen:1995kv . Related to this problem, extended Higgs
models with the classical scale invariance (CSI) have been often considered
Lee:2012jn ; Englert:2013gz ; Guo:2014bha ; Endo:2015ifa ; Hashino:2015nxa ;
Hashino:2016rvx ; Ahriche:2016cio ; Ahriche:2016ixu ; Lane:2019dbc ;
Kanemura:2020yyr ; Braathen:2020vwo ; Ahriche:2021frb ; Soualah:2021xbn ;
Eichten:2022vys . The CSI requires that all mass terms in the Lagrangian are
forbidden. Therefore, the electroweak symmetry breaking (EWSB) does not occur
at the tree-level. However, the CSI is violated by quantum effects of new
particles. Then, the EWSB can be realized radiatively. This mechanism is often
called as the Coleman-Weinberg mechanism Coleman:1973jx . An extension of the
Coleman-Weinberg mechanism to the case with multi-scalar fields has been also
discussed by Gildener and Weinberg Gildener:1976ih . The phenomenology of CSI
extensions of the SM has been investigated in the literature, including the
testability at current and future collider experiments Lee:2012jn ;
Englert:2013gz ; Guo:2014bha ; Endo:2015ifa ; Hashino:2015nxa ;
Hashino:2016rvx ; Ahriche:2016cio ; Ahriche:2016ixu ; Lane:2019dbc ;
Kanemura:2020yyr ; Braathen:2020vwo ; Ahriche:2021frb ; Soualah:2021xbn .

Among the SM unsolved questions, the baryon asymmetry of the Universe is an
important problem for both particle physics and cosmology. The observed baryon
asymmetry is expressed by Planck:2018vyg

$\displaystyle\eta_{B}=\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}}=6.143\pm
0.190\times 10^{-10},$ (1)

where $n_{B}$, $n_{\bar{B}}$ and $n_{\gamma}$ refer to the number density of
baryons, antibaryons and the CMB photons, respectively. In order to explain
the ratio $\eta_{B}$, three conditions need to be satisfied simultaneously at
the early Universe, known as the Sakharov conditions Sakharov:1967dj . These
conditions can be summarized by the existence of such interactions that
violate the baryon number, violate C and CP symmetries; and occur out of the
thermal equilibrium. One of the most interesting scenarios for baryogenesis is
the electroweak baryogenesis (EWB), where the third Sakharov condition is
satisfied via a strongly first order phase transition at the electroweak scale
Kuzmin:1985mm . However, it has been shown that this scenario cannot be
realized in the SM since the CP violation source in Yukawa interactions is too
small to produce $\eta_{B}$ and the electroweak phase transition (EWPT) is not
first order. Therefore, going beyond the SM is mandatory. In order to realize
the EWB scenario, the sphaleron processes should decouple after the first
order EWPT. This condition can be approximately expressed by Bochkarev:1987wf
; Ahriche:2007jp ; Ahriche:2014jna ; Fuyuto:2014yia ; Kanemura:2022ozv

$\displaystyle\frac{\upsilon_{c}}{T_{c}}>1,$ (2)

where $T_{c}$ is the critical temperature at which the potential minima (the
false and true ones) are degenerate; and $\upsilon_{c}$ is the vacuum
expectation value (VEV) of the $SU(2)_{L}$ doublet scalar field at $T=T_{c}$.
Once the condition (2) is satisfied, the triple Higgs boson coupling $hhh$
significantly deviates from the SM prediction Kanemura:2004ch . Therefore,
precise measurements of the $hhh$ coupling at future colliders are important
to test the scenario of the EWB. In order to realize a large deviation in the
triple Higgs boson coupling, large quantum corrections from new particles play
an important role. Such large quantum corrections may be able to appear in the
other Higgs boson couplings like $h\gamma\gamma$ Shifman:1979eb ;
Kanemura:2002vm ; Kanemura:2004mg ; Arhrib:2012ia ; Kanemura:2019kjg ;
Kanemura:2019slf ; Degrassi:2023eii ; Aiko:2023nqj .

In addition to the $hhh$ coupling measurement, gravitational waves (GWs)
generated during a strongly first order EWPT can be used to explore new
physics models Kamionkowski:1993fg ; Kosowsky:1992rz ; Kosowsky:1991ua ;
Schwaller:2015tja . A first order phase transition at the early Universe
occurs via the nucleation and expansion of bubbles of the broken vacuum. When
the broken vacuum bubbles collide with each other, detectable GWs can be
produced. For a first order EWPT, the typical peak frequency of the GW
spectrum is around $10^{-3}-10^{-1}$ Hz Grojean:2006bp . Such GW spectrum can
be observed by future space-based interferometers like LISA LISA:2017pwj  and
DECIGO Kawamura:2006up . It means that new physics models can be explored by
using the GWs. Predictions on the GW spectrum in extended Higgs models with
the CSI have been discussed in the literature Kakizaki:2015wua ; Huang:2016cjm
; Hashino:2016rvx ; Hashino:2016xoj ; Hashino:2018zsi ; Brdar:2018num ;
Brdar:2019qut ; Salvio:2023qgb ; Salvio:2023ynn . The dynamics of the EWPT in
the CSI models has been also discussed in the literature Farzinnia:2014yqa ;
Mohamadnejad:2019vzg ; Kubo:2015joa ; Sannino:2015wka ; Hashino:2016rvx ;
Kierkla:2022odc .

Recently, it is often discussed that the first order EWPT can be tested by
primordial black hole observations Hashino:2021qoq ; Hashino:2022tcs . When
the first order EWPT forms primordial black holes (PBHs), those masses are
around $10^{-5}$ times of the solar mass. It implies that we can test the
first order EWPT by observing such PBHs at current and future microlensing
experiments like Subaru HSC Subaru , OGLE OGLE , PRIME PRIME  and Roman
telescope Roman .

The CSI implies that a CP-even scalar should be massless at tree-level; and
becomes massive due to the quantum corrections that trigger the EWSB. In Ref.
Ahriche:2021frb , a generic model has been considered to investigate the RC
effects on the EWSB, dilaton mass, scalar mixing, as well other observables.
In this model, the SM is extended by a real singlet and an extra singlet $Q$
with multiplicity $N_{Q}$ and the couplings ($\alpha_{Q},\,\beta_{Q}$) to the
Higgs doublet and real singlet, respectively. Then, the RCs are quantified by
using $N_{Q}$, $\alpha_{Q}$ and $\beta_{Q}$ in addition to the singlet VEV. In
a case of large ($N_{Q}$, $\alpha_{Q}$, $\beta_{Q}$), the dilaton mass can be
125 GeV due to the large RCs. This scenario corresponds to the case of a
purely radiative Higgs mass (PRHM) Ahriche:2021frb . Although large RCs (with
large values for $N_{Q}$, $\alpha_{Q}$, and $\beta_{Q}$) are beneficial for
vacuum stability, they may introduce tension with the triviality bound
Alexander-Nunneley:2010tyr ; Kanemura:2012hr . This aspect needs to be
investigated within the viable parameter space of this model.

In this work, we will investigate the effect of quantum corrections on the
EWPT dynamics and on the corresponding GW spectrum using this generic CSI
model with a scalar mixing Ahriche:2021frb . Here, the triple scalar couplings
are very sensitive to the quantum corrections, which makes the di-Higgs
production cross section at both LHC and ILC useful observables to probe the
viable parameter space. According to the full LHC Run 2 data with $139\,{\rm
fb}^{-1}$, it is required that the cross section for the non-resonant di-Higgs
production process is lower than 3.4 times of the SM prediction CMS:2022dwd .
At the High Luminosity LHC (HL-LHC), the cross section will be measured by 0.7
times of the SM prediction CMS:2022dwd ; Cepeda:2019klc . At the International
Linear Collider (ILC), it is expected that the di-Higgs production cross
section is limited less than 2 or 3 times of the SM prediction Bambade:2019fyw
. The di-Higgs production processes in extended Higgs models have been
investigated in the literatures Arhrib:2009hc ; Dolan:2012ac ;
Kanemura:2016lkz ; Dawson:2017jja ; Carena:2018vpt ; Arco:2021bvf ;
Abouabid:2021yvw ; Iguro:2022fel . In this paper, we show that the light
dilaton and PRHM cases can be distinguished by using complementarity between
collider experiments and GW observations.

The structure of this paper is as follows. In Section 2, we introduce the CSI
models; and discuss the EWSB and the scalar mass spectrum. In Section 3, we
identify the theoretical and experimental constraints on these CSI models. In
Section 4, we define the parameters characterizing the first order EWPT and
the corresponding GW spectrum. Our numerical results for GW observations and
collider experiments are shown in Section 5. Our conclusion is given in
Section 6.

##  2 Models with Classical Scale Invariance

The CSI in quantum field theories is defined by
$x^{\mu}\to\kappa^{-1}x^{\mu},\,\psi_{i}\to\kappa^{a}\psi_{i}$ with
$a=1,\,3/2$ for bosons and fermions, respectively. This invariance implies
that the scalar quadratic terms vanish. Therefore, the general representation
of the potential with the CSI is given by Gildener:1976ih

$V(\Phi_{i})=\sum_{i,j,k,l}\lambda_{ijkl}\Phi_{i}\Phi_{j}\Phi_{k}\Phi_{l},$
(3)

where $\Phi_{i}$ represents all the scalar representations. Obviously, the
EWSB could not take place because the scalar potential in Eq. (3) contains
only quartic terms, which requires the contributions of the quantum
corrections that break both of the electroweak and CSI symmetries at the same
time. In order to achieve the EWSB in this class of models, the SM is extended
by a real singlet $S$; in addition to other bosonic and fermionic degrees of
freedom (dof) with $(n_{i},\,\alpha_{i},\,\beta_{i})$ as multiplicities and
couplings to the Higgs doublet and scalar $S$, respectively. Here, the Higgs
doublet field ${\cal H}$ and the singlet field $S$ are written as

${\cal H}=\left(\begin{array}[]{c}\chi^{+}\\\
\frac{H+i\,\chi^{0}}{\sqrt{2}}\end{array}\right),\quad S=s,$ (4)

with $\chi^{+}$ and $\chi^{0}$ are the Goldstones; and $\langle
H\rangle=\upsilon$ and $\langle s\rangle=\upsilon_{S}$ are the VEV of the
doublet and singlet, respectively. In this model, two CP-even eigenstates
$h_{1,2}$ (with $m_{2}>m_{1}$) are obtained by using the $2\times 2$ mixing
matrix with angle $\alpha$. We distinguish two possibilities where the
observed SM-like Higgs with the mass $m_{h}=125$ GeV corresponds to (1)
$h_{2}\equiv h$ and $h_{1}\equiv\eta$ is a light dilaton (light dilaton
scenario); and (2) $h_{1}\equiv h$ and $h_{2}\equiv\eta$ is a heavy scalar
(PRHM scenario) Ahriche:2021frb .

The full one-loop effective potential in terms of CP-even fields $(H,s)$ can
be written as

$V^{1\ell}(H,s)=\frac{1}{24}(\lambda_{H}+\delta\lambda_{H})H^{4}+\frac{1}{24}(\lambda_{S}+\delta\lambda_{S})s^{4}+\frac{1}{4}(\omega+\delta\omega)H^{2}s^{2}+\sum_{i}n_{i}G\big{(}m_{i}^{2}(H,s)\big{)},$
(5)

with $\delta\lambda_{H},\delta\lambda_{S},\delta\omega$ are the counter terms,
$n_{i}$ and $m_{i}^{2}(H,s)$ are the multiplicities and the field dependent
masses for particles running loops, respectively. Here, the function $G(r)$ is
defined a la $\overline{DR}$ scheme, i.e.,
$G(r)=\frac{r^{2}}{64\pi^{2}}\big{(}\log(r/\Lambda^{2})-3/2\big{)}$, with the
renormalization scale is taken to be $\Lambda=m_{h}=125.18\,\mathrm{GeV}$.

In this setup, we can eliminate the couplings $\lambda_{H},\lambda_{S}$ and
$\omega$ in favor of the tadpole conditions and the Higgs mass at tree-level,
and the corresponding counter-terms
$\delta\lambda_{H},\,\delta\lambda_{S},\,\delta\omega$ can be also eliminated
using the one-loop tadpole and Higgs mass conditions as shown in Refs.
Ahriche:2021frb ; Soualah:2021xbn . In case where all field dependent masses
can be written in the form111Indeed, all field dependent masses can be written
in this form except the eigenmasses of $h_{1,2}$. The contributions of
$h_{1,2}$ to the effective potential can be safely neglected.
$m_{i}^{2}(H,s)=\frac{1}{2}(\alpha_{i}H^{2}+\beta_{i}s^{2})$, the counter-
terms can be simplified as

$\displaystyle\delta\omega$
$\displaystyle=\frac{m_{h}^{2}}{\upsilon^{2}+\upsilon_{S}^{2}}\frac{(ab-c^{2})(\upsilon^{2}+\upsilon_{S}^{2})-a\upsilon_{S}^{2}-b\upsilon^{2}+2\,c\upsilon\upsilon_{S}}{a\upsilon^{2}+b\upsilon_{S}^{2}+2\,c\upsilon\upsilon_{S}-\upsilon^{2}-\upsilon_{S}^{2}},$
(6)

with

$\displaystyle\begin{aligned}
a&=\frac{1}{32\pi^{2}m_{h}^{2}}\sum_{i}n_{i}\alpha_{i}\Big{(}2m_{i}^{2}-\beta_{i}\upsilon_{S}^{2}\log\frac{m_{i}^{2}}{\Lambda^{2}}\Big{)},\\\
b&=\frac{1}{32\pi^{2}m_{h}^{2}}\sum_{i}n_{i}\beta_{i}\Big{(}2m_{i}^{2}-\alpha_{i}\upsilon^{2}\log\frac{m_{i}^{2}}{\Lambda^{2}}\Big{)},\\\
c&=\frac{\upsilon\upsilon_{S}}{32\pi^{2}m_{h}^{2}}\sum_{i}n_{i}\alpha_{i}\beta_{i}\log\frac{m_{i}^{2}}{\Lambda^{2}}.\end{aligned}$

The light dilaton and PRHM cases are identified via the conditions
Ahriche:2021frb ; Soualah:2021xbn

$\displaystyle\delta\omega(\upsilon^{2}+\upsilon_{S}^{2})/m_{h}^{2}<a+b<1+\delta\omega(\upsilon^{2}+\upsilon_{S}^{2})/m_{h}^{2}$
$\displaystyle\quad(\mbox{for light dilaton case}),$ (7) $\displaystyle
a+b>1+\delta\omega(\upsilon^{2}+\upsilon_{S}^{2})/m_{h}^{2}$
$\displaystyle\quad(\mbox{for PRHM case}),$ (8)

respectively. However, the condition
$a+b=1+\delta\omega(\upsilon^{2}+\upsilon_{S}^{2})/m_{h}^{2}$ corresponds to
the special case of degenerate eigenmasses $m_{1}=m_{2}=m_{h}$. This case is
of great interest that deserves an independent study.

In our analysis, we consider a generic model where the SM is extended by a
scalar singlet $S$ to assist the EWSB; and another boson $Q$ with multiplicity
$N_{Q}$ and the squared mass
$m_{Q}^{2}=\frac{1}{2}(\alpha_{Q}\upsilon^{2}+\beta_{Q}\upsilon_{S}^{2})$.
Clearly, the quantum correction from the boson $Q$ is proportional to the
field multiplicity $N_{Q}$, the couplings ($\alpha_{Q},\,\beta_{Q}$) to
$\mathcal{H}$ and $S$ and/or the singlet VEV $\upsilon_{S}$.

##  3 Constraints and Predictions

The different constraints on this model have been discussed in details in Ref.
Ahriche:2021frb . Here, we mention the constraints coming from the total Higgs
strength modifier $\mu_{{\rm
tot}}=c_{\alpha}^{2}\times(1-\mathcal{B}_{BSM})\geq 0.89$ at 95 % CL
ATLAS:2016neq , that implies the scalar mixing to be $s_{\alpha}^{2}\leq 0.11$
in the absence of invisible and undetermined Higgs decay
($\mathcal{B}_{BSM}=0$). Here, the RCs play a crucial role to satisfy this
bound. The mixing sine can be decomposed into a tree-level part and a one-loop
part as $s_{\alpha}=s_{\alpha}^{(0)}+s_{\alpha}^{(1)}$, where
$s_{\alpha}^{(0)}=\upsilon/\sqrt{\upsilon^{2}+\upsilon_{S}^{2}}$ in the light
dilaton case. Therefore, small RCs will keep the constraint
$s_{\alpha}^{2}\leq 0.11$ fulfilled for $\upsilon_{S}\gg\upsilon$. However, in
the PRHM case, we have
$s_{\alpha}^{(0)}=\upsilon_{S}/\sqrt{\upsilon^{2}+\upsilon_{S}^{2}}$ which
makes the constraint $s_{\alpha}^{2}\leq 0.11$ explicitly violated for
$\upsilon_{S}\gg\upsilon$. Interestingly, large RCs in the PRHM case that are
responsible to push the light CP-even eigenmass from zero tree-level value to
the measured 125 GeV; are also responsible to generate large negative
contributions to the mixing $s_{\alpha}^{(1)}$ that makes the condition
$s_{\alpha}^{2}\leq 0.11$ fulfilled. This point has been discussed in details
in Ref. Ahriche:2021frb  for this model; and in Ref. Soualah:2021xbn  for the
SI scotogenic model.

The counter-terms $\delta\lambda_{H},~{}\delta\lambda_{S},~{}\delta\omega$
cannot take any numerical values since they are constrained by the one-loop
perturbativity conditions
$\lambda_{H}^{1-\ell},~{}\lambda_{S}^{1-\ell},~{}|\omega^{1-\ell}|<4\pi$.
Here, the one-loop quartic couplings are defined as the $4^{th}$ derivatives
of the full one-loop scalar potential in Eq. (5) at the broken vacuum. In what
follows, we will consider the one-loop value for the Higgs-dilaton mixing
angle, since it has been shown that the role of RCs is crucial in driving the
mixing value towards the experimentally allowed region, both in the context of
a light dilaton and PRHM cases Ahriche:2021frb . In the CSI models, the
leading term in the one-loop scalar potential is $\varphi^{4}\log\varphi$
rather than $\varphi^{4}$, where $\varphi$ could be any direction in the
$H$-$s$ plane. Thus, the vacuum stability conditions differ from those used in
the literature, which are given by
$\sum_{i}n_{i}\alpha_{i}^{2}>0\,\wedge\,\sum_{i}n_{i}\beta_{i}^{2}>0$
Ahriche:2021frb ; Soualah:2021xbn .

In addition to the perturbativity and vacuum stability conditions, we discuss
here the triviality bound as a theoretical constraint on our model. The
triviality bound plays a significant role in discussing upper bounds on
couplings Lindner:1985uk . Especially, the triviality bound can give strong
constraints on models with the CSI or a strongly first order EWPT Alexander-
Nunneley:2010tyr ; Kanemura:2012hr . Generally, the Landau pole scale is
defined as the energy scale at which the perturbative description of the model
is broken down, which suggests the need for a more fundamental theory or
modifications at higher energy scales. In this work, we define the Landau pole
scale $(\Lambda_{\rm Lan})$ as the approximate scale where $\Lambda_{\rm
Lan}=\min(\\{\Lambda_{i}\\})$ with $\lambda_{i}(\mu=\Lambda_{i})=4\pi$, and
$\lambda_{i}(\mu)$’s are the couplings at the scale $\mu$ in our model. In
what follow, we consider the condition $\Lambda_{\rm Lan}<10\,{\rm TeV}$ as a
triviality bound.

In Appendix A, we present the $\beta$-functions for the renormalization group
equations (RGEs) in our model using the ordinary scheme. However, these
functions do not decouple the effects of heavy new particles in the RGEs flow,
even when our focus is on physics at a low energy scale. These non-decoupling
effects are commonly known as threshold effects. Recently, it has been
confirmed that such threshold effects can be naturally taken into account by
employing mass-dependent $\beta$-functions Kanemura:2023wap . In the
following, we will utilize the mass-dependent $\beta$-function when discussing
the RGEs analysis.

As well known, the triple Higgs boson coupling,
$\lambda_{hhh}=\lambda_{hhh}^{{\rm SM}}(1+\Delta_{hhh})$, is an important
quantity to probe the strongly first order EWPT Grojean:2004xa ;
Kanemura:2004ch ; Noble:2007kk . In order to obtain information about the
relative triple Higgs coupling enhancement ($\Delta_{hhh}$), it is necessary
to measure the di-Higgs production processes at the LHC or the ILC. However,
in models where the SM Higgs doublet mixes with a singlet, as in our model,
the di-Higgs production process involves an extra Feynman triangle diagram
mediated by the extra scalar $\eta$. In this case, the di-Higgs production
cross section has three independent contributions that come from: (1) the
Feynman diagrams involving only the triple scalar couplings $\lambda_{hhh}$
and $\lambda_{hh\eta}$ ($\sigma_{\lambda}$), (2) the diagrams with only pure
gauge couplings ($\sigma_{G}$); and (3) the interference contribution
($\sigma_{G\lambda}$) Ahriche:2014cpa ; Ahriche:2021frb . In our model, the
di-Higgs production cross section scaled by its SM value can be expressed as

$\displaystyle R(hh+X)=\frac{\sigma(hh+X)}{\sigma_{{\rm
SM}}(hh+X)}=\frac{\xi_{1}\sigma_{G}+\xi_{2}\sigma_{\lambda}+\xi_{3}\sigma_{G\lambda}}{\sigma_{G}+\sigma_{\lambda}+\sigma_{G\lambda}}.$
(9)

The coefficients $\xi_{i}~{}(i=1,2,3)$ are defined at the CM energy $\sqrt{s}$
as Ahriche:2014cpa ; Ahriche:2021frb

$\displaystyle\xi_{1}=c_{\alpha}^{4},\quad\xi_{2}=|\mathcal{P}|^{2},\quad\xi_{3}=c_{\alpha}^{2}\Re(\mathcal{P}),$
(10)

with

$\displaystyle\mathcal{P}=c_{\alpha}\frac{\lambda_{hhh}}{\lambda_{hhh}^{{\rm
SM}}}+s_{\alpha}\frac{\lambda_{hh\eta}}{\lambda_{hhh}^{{\rm
SM}}}\frac{s-m_{h}^{2}+im_{h}\Gamma_{h}}{s-m_{\eta}^{2}+im_{\eta}\Gamma_{\eta}},$
(11)

where $\Gamma_{h}$ is the measured Higgs total decay width, $\Gamma_{\eta}$ is
the estimated heavy scalar total decay width; and $\lambda_{hhh}^{{\rm SM}}$
is the triple Higgs boson coupling in the SM. We take the value of
$\lambda_{hhh}^{{\rm SM}}$ as in Refs. Kanemura:2002vm ; Kanemura:2004mg .

Here, we consider the di-Higgs production processes $pp\to hh$ and
$e^{+}e^{-}\to Zhh$ at the LHC with 14 TeV and ILC with 500 GeV, respectively.
In the following, we use the notations $R_{{\rm LHC}}$ and $R_{{\rm ILC}}$ for
the processes $R(pp\to hh{\rm@LHC14\,TeV})$ and $R(e^{+}e^{-}\to
Zhh{\rm@ILC500\,GeV})$, respectively. The values
$\sigma_{\lambda},~{}\sigma_{G}$ and $\sigma_{G\lambda}$ for the process
$pp\to hh{\rm@LHC14\,TeV}$ and $e^{+}e^{-}\to Zhh{\rm@ILC500\,GeV}$ are given
in Table. 1.

|  $\sigma_{\lambda}$ [fb] |  $\sigma_{G}$ [fb] |  $\sigma_{G\lambda}$ [fb] | Reference
---|---|---|---|---
$pp\to hh\,@{\rm LHC}\,14$TeV |  70.1 |  9.66 |  $-49.9$ | Spira:1995mt
$e^{+}e^{-}\to Zhh\,@{\rm ILC}\,500$GeV |  0.0837 |  0.01565 |  0.05685 | Ahriche:2021frb
Table 1:  The cross section contributions in Eq. (9) for each di-Higgs
production process at the LHC and the ILC.

One has to notice that, in the PRHM scenario, the heavy scalar $\eta$ decays
into all SM Higgs final states in addition to a di-Higgs channel. This means
that the negative searches for a heavy resonance at both ATLAS and CMS can be
used to constraint our model, like: (1) the heavy CP-even resonance in the
channels of pair of leptons, jets or gauge bosons $pp\to
H\to\ell\ell,\,jj,\,VV$ ATLAS:2020zms ; ATLAS:2020tlo ; CMS:2021klu ; and (2)
the resonance in the di-Higgs production $pp\to H\to hh$ ATLAS:2021fet ;
ATLAS:2021ulo ; ATLAS:2021jki . It has been shown in Ref. Ahriche:2021frb
that almost the full parameter space region is allowed even if we consider the
heavy CP-even resonance. Thus, these constraints will not be considered in our
work.

##  4 Gravitational Waves from a First Order Phase Transition

In order to analyze the GW spectrum generated during a strongly first order
EWPT, we need to define the parameters $\alpha$ and $\beta$, which
characterize the GWs from the dynamics of vacuum bubbles Grojean:2006bp .
These parameters represent the latent heat and the inverse of the EWPT time
duration, respectively. However, an accurate description of the EWPT dynamics
is mandatory, which requires the exact knowledge of a key physical quantity:
the full one-loop effective potential at finite temperature222Recently, there
has been a discussion on methods to decrease the uncertainty associated with
the thermal effective potential, considering factors such as renormalization
scale and gauge dependencies Croon:2020cgk ; Lofgren:2021ogg ; Schicho:2022wty
. , that is given by Dolan:1973qd

$\displaystyle V_{{\rm eff}}^{T}(H,s,T)$
$\displaystyle=V^{1\ell}(H,s)+T^{4}\sum_{i={\rm
all}}n_{i}J_{B,F}\big{(}m_{i}^{2}/T^{2}\big{)}+V_{{\rm daisy}}(H,s,T),$ (12)
$\displaystyle J_{B,F}\left(\alpha\right)$
$\displaystyle=\frac{1}{2\pi^{2}}\int_{0}^{\infty}drr^{2}\log\left[1\mp\exp(-\sqrt{r^{2}+\alpha})\right],$
(13)

where the last contribution in Eq. (12), that is called the daisy (or ring)
part, represents the leading term of higher order thermal corrections
Carrington:1991hz . This contribution can be taken into account by replacing
the scalar and longitudinal gauge field-dependent masses in the first and
second terms of Eq. (12) by their thermally corrected values, i.e.,
$m_{i}^{2}\rightarrow\widetilde{m}_{i}^{2}=m_{i}^{2}+\Pi_{i}$ Gross:1980br .
The thermal self-energies are given by Ahriche:2018rao

$\displaystyle\begin{aligned}
&\Pi_{H}=\Pi_{\chi}=T^{2}\left(\frac{1}{2}\lambda_{H}+\frac{3}{16}g^{2}+\frac{1}{16}g^{\prime
2}+\frac{1}{4}y_{t}^{2}+\frac{\omega}{24}+\frac{\alpha_{Q}N_{Q}}{24}\right),\\\
&\Pi_{S}=\Big{(}\frac{\omega}{6}+\frac{\lambda_{S}}{24}+\frac{\beta_{Q}N_{Q}}{24}\Big{)}\,T^{2},\\\
&\Pi_{Q}=T^{2}\left(\frac{\alpha_{Q}}{4}+\frac{\beta_{Q}}{12}+\frac{\lambda_{Q}N_{Q}}{24}\right),\\\
&\Pi_{W_{L}}=\frac{11}{6}g^{2}T^{2},\quad\Pi_{B_{L}}=\frac{11}{6}g^{\prime
2}T^{2},\quad\Pi_{W_{L},B_{L}}\simeq 0,\end{aligned}$ (14)

with $g$ and $g^{\prime}$ are the $SU(2)_{L}$ and $U(1)_{Y}$ gauge couplings,
respectively. Here, $\lambda_{Q}$ is the self-coupling constant of boson $Q$.
Since the thermal corrections related to $\lambda_{Q}$ only come from the
thermal mass of the new boson $Q$, this correction can be negligible. Thus,
for reasons of simplicity, we take $\lambda_{Q}=0$ in our analysis.

The EWSB takes place during the transition from the symmetric vacuum ($\langle
H\rangle=0$) to the broken one ($\langle H\rangle\neq 0$), at the nucleation
temperature $T_{n}\leq T_{c}$. A strongly first order EWPT occurs through the
tunneling between the symmetric and broken vacuum, which corresponds to the
vacuum bubble nucleation at random points in the space. These bubbles expand
and collide with each other until filling the Universe by the broken vacuum
($\langle H\rangle\neq 0$). When a bubble wall passes through an unbroken
symmetry region (where $\langle H\rangle=0$) at which a net baryon asymmetry
is generated by the $B+L$ and CP violating processes, the thermal equilibrium
will washout this asymmetry unless the $B$ number violating process is
suppressed inside the bubble (broken vacuum $\langle H\rangle\neq 0$). This
condition is often called as the sphaleron decoupling condition and
approximately expressed by Eq. (2). It has been shown that the singlet plays
an important role in the EWPT dynamics even though its VEV $\langle
S\rangle_{c}$ is absent in the condition in Eq. (2) Ahriche:2007jp ;
Ahriche:2014jna . The precise evaluation of the sphaleron decoupling condition
in extended Higgs models has been also discussed in Refs. Fuyuto:2014yia ;
Kanemura:2022ozv .

In what follows, we analyze the GW spectrum from a first order EWPT in this
model by estimating the above mentioned parameters $\alpha$ and $\beta$
Grojean:2006bp . The parameter $\alpha$ is the latent heat normalized by the
radiation energy density, which is given by

$\alpha=\frac{\epsilon(T_{n})}{\rho_{{\rm rad}}(T_{n})},$ (15)

with $\rho_{{\rm rad}}(T)=(\pi^{2}/30)g_{\ast}T^{4}$ is the radiation energy
density, $g_{\ast}$ is relativistic degrees of freedom in the thermal plasma;
and

$\epsilon(T)=-V_{{\rm eff}}^{T}(H(T),s(T),T)+T\frac{\partial V_{{\rm
eff}}^{T}(H(T),s(T),T)}{\partial T},$ (16)

is the released energy density, where the configuration $(H(T),s(T))$ is the
broken vacuum at temperature $T$.

The parameter $\beta$ that describes approximately the inverse of time
duration of the EWPT is defined as

$\beta=-\left.\frac{dS_{E}}{dt}\right|_{t=t_{n}}\simeq\left.\frac{1}{\Gamma}\frac{d\Gamma}{dt}\right|_{t=t_{n}},$
(17)

where $S_{E}$ and $\Gamma$ are the 4d Euclidean action of a critical bubble
and vacuum bubble nucleation rate per unit volume per unit time at the time of
the EWPT $t_{n}$, respectively. In the following, we use the normalized
parameter $\beta$ by Hubble parameter $H_{T}$

$\widetilde{\beta}=\frac{\beta}{H_{T}}=T_{n}\frac{d}{dT}\left(\frac{S_{3}(T)}{T}\right)\Bigg{|}_{T=T_{n}},$
(18)

where $S_{3}(T)$ is the 3d action for the bounce solution. The transition
temperature $T_{n}$ is defined by

$\left.\frac{\Gamma}{H_{T}^{4}}\right|_{T=T_{n}}=1,$ (19)

with the bubble nucleation rate Linde:1981zj

$\Gamma(T)\simeq T^{4}\left(\frac{S_{3}(T)}{2\pi
T}\right)^{3/2}\exp\left[-\frac{S_{3}(T)}{T}\right].$ (20)

If $\Gamma/H_{T}^{4}$ cannot be larger than the unity ($\Gamma/H_{T}^{4}\ll
1$), the first order phase transition does not complete by today. Thus, this
condition is used as another theoretical constraint on our model. Here, we use
the public code CosmoTransitions to obtain the bounce solutions
Wainwright:2011kj .

The GWs from a first order phase transition can be produced via three
mechanisms: (1) collisions of bubble walls and shocks in the plasma
$\Omega_{\phi}h^{2}$ Kosowsky:1992rz , (2) the compressional waves (sound
waves) $\Omega_{{\rm sw}}h^{2}$ Huber:2008hg ; and (3) magnetohydrodynamic
(MHD) turbulence in the plasma $\Omega_{{\rm tur}}h^{2}$ Caprini:2006jb .
Therefore, the stochastic GW background can be approximately expressed by

$\Omega_{{\rm GW}}h^{2}\simeq\Omega_{\phi}h^{2}+\Omega_{{\rm
sw}}h^{2}+\Omega_{{\rm tur}}h^{2}.$ (21)

The importance of each contribution depends on the EWPT dynamics, especially
on the bubble wall velocity $v_{b}$. In this work, we take $v_{b}=0.95$ as a
free parameter; and focus on the contribution to GWs from the sound waves in
the plasma, which is the dominant contributions among the other GW sources.
According to the numerical simulations, a fitting function for the GW spectrum
from the sound waves is given by Caprini:2015zlo

$\Omega_{{\rm sw}}(f)h^{2}=\widetilde{\Omega}_{{\rm
sw}}h^{2}\times(f/\tilde{f}_{{\rm
sw}})^{3}\left(\frac{7}{4+3(f/\tilde{f}_{{\rm sw}})^{2}}\right)^{7/2},$ (22)

where the peak of the amplitude is given by

$\widetilde{\Omega}_{{\rm sw}}h^{2}\simeq 2.65\times
10^{-6}v_{b}\widetilde{\beta}^{-1}\left(\frac{\kappa\alpha}{1+\alpha}\right)^{2}\left(\frac{100}{g_{\ast}}\right)^{1/3};$
(23)

and the peak frequency is expressed as

$\widetilde{f}_{{\rm sw}}\simeq 1.9\times
10^{-5}\mbox{Hz}\frac{1}{v_{b}}\widetilde{\beta}\left(\frac{T_{n}}{100{\rm
GeV}}\right)\left(\frac{g_{\ast}}{100}\right)^{1/6},$ (24)

with $\kappa$ is the efficiency factor which characterize how much of the
vacuum energy is converted into the fluid motion Espinosa:2010hh .

##  5 Numerical Results

In our analysis, we consider the following values for the multiplicity
$N_{Q}=6,\,12,\,24$; and the singlet VEV
$\upsilon_{S}=500\,\mathrm{GeV},~{}1\,\mathrm{TeV},~{}3\,\mathrm{TeV}$, while
allowing the couplings to lie within the perturbative regime
$|\alpha_{Q}|,|\beta_{Q}|\leq 4\pi$. In addition, we consider the constraints:
(1) vacuum stability, (2) the mixing angle $s_{\alpha}^{2}\leq 0.11$, (3) the
perturbativity one-loop constraints
$\lambda_{H,S}^{1-\ell},\,|\omega^{1-\ell}|<4\,\pi$; and (4) the completion
condition of the phase transition. As well known, the GWs from a first order
EWPT is useful to probe the structure of the Higgs potential Kakizaki:2015wua
; Hashino:2016rvx . Since the RCs have the key role in this model, one expects
that future space-based interferometers would be able to detect the GWs
generated during the first order EWPT. In order to understand the impact on
the EWPT strength from the quantum corrections, we show the ratio
$\upsilon_{c}/T_{c}$ in the palette in Fig. 1 for different values of
$(\alpha_{Q},~{}\beta_{Q})$, $N_{Q}$ and $\upsilon_{S}$.

Figure 1: The dependence of the EWPT strength parameter $\upsilon_{c}/T_{c}$
on the couplings ($\alpha_{Q},~{}\beta_{Q}$) with different values of the
multiplicity $N_{Q}$ and the singlet VEV $\upsilon_{S}$. The green and red
stars are the benchmark points (BPs) that correspond to the PRHM and light
dilaton scenarios, respectively, which will be used in Fig. 3. The area
colored in cyan represents the parameter space region that could be tested by
DECIGO, while the magenta region is eliminated by the completion condition for
the first order EWPT (19). The gray region is constrained by the triviality
bound with $\Lambda_{\rm Lan}<10\,{\rm TeV}$.

In all panels of Fig. 1, the upper parameter space region corresponds to the
PRHM scenario; and the lower region is the light dilaton case. The regions
colored in cyan represent the parameter region where the GW spectra from the
first order EWPT may be detected at DECIGO Kawamura:2006up . Here, one notices
that not only the dilaton case, but also the PRHM scenario predicts testable
GW spectra. The magenta region in Fig. 1 is constrained by the completion
condition of the first order EWPT. Interestingly, in Fig. 1, there is a
parameter space region where the GW spectra can be tested at future space-
based interferometers even if the EWPT is not strongly first order, i.e., the
condition in Eq. (2) is not satisfied. One remarks that satisfying the
condition in Eq. (2) requires small $\alpha_{Q}$ and $\beta_{Q}$. Furthermore,
the triviality bound ($\Lambda_{\rm Lan}<10~{}\rm{TeV}$) excludes a portion of
the parameter space corresponding to large values of $\alpha_{Q}$ and
$\beta_{Q}$ in the PRHM case. This observation has also been noted in extended
scalar sector models, where more parameter space is excluded when the
triviality bound is considered at a higher scale Khojali:2022squ ;
Baouche:2021wwa . On the other hand, as we will explain later, the di-Higgs
production cross section can be large for relatively large $\beta_{Q}$ value.
It indicates a complementarity between the GW observations and collider
experiments to test our model. The details for the di-Higgs production will be
discussed later.

According to Fig. 1, as $N_{Q}$ is getting large, the parameter space region
satisfying the condition in (2) becomes large in the PRHM scenario. Whereas,
in the light dilaton case, such parameter space region is narrowed down, which
means that the PRHM scenario is preferred in order to realize the EWBG
scenario within CSI models. From the results in Fig. 1, one learns that the
scenario with degenerate masses $m_{1}=m_{2}=m_{h}$ is possible; and it
corresponds to the interference region between the upper and lower islands.
This option cannot be realized in cases with the large $\upsilon_{S}$.

In order to understand more about the EWPT dynamics, we show the relative
difference between the critical and the nucleation temperature values
$(T_{c}-T_{n})/T_{n}$ in Fig. 2 for different values of the model free
parameters.

Figure 2: The relative difference between the critical and the nucleation
temperature values in function of the couplings $\alpha_{Q}$ (left)
$\beta_{Q}$ (middle) and the singlet VEV $\upsilon_{S}$ (right). The phase
transition in our model is clearly supercooling.

Since this difference is significantly large, the first order EWPT in our
analysis can be supercooling. In Fig. 2, it is shown that the relative
difference $(T_{c}-T_{n})/T_{n}$ is large for larger singlet VEV
$\upsilon_{S}$, and/or smaller couplings ($\alpha_{Q},\,\beta_{Q}$). As a
consequence, the testable GWs can be produced even if the EWPT strength
parameter $\upsilon_{c}/T_{c}$ is smaller than unity. This is the reason why
the wide parameter space region in Fig. 1 can be probed by GW experiments.

By considering the two BPs that correspond to the green (PRHM case) and red
(light dilaton case) stars in Fig. 1, we show the GW amplitude $\Omega_{{\rm
GW}}h^{2}$ as functions of the frequency values.

Figure 3: The predicted GW spectra for the two BPs with different values of
the multiplicity $N_{Q}$ and the singlet VEV $\upsilon_{S}$. The red and green
lines correspond to the light dilaton scenario and the PRHM scenario,
respectively, as shown as the green and red starts in Fig. 1. We also show the
sensitivity curves of LISA LISA:2017pwj , DECIGO Kawamura:2006up , TianQin
TianQin:2015yph  and Taiji Ruan:2018tsw .

We conclude from Fig. 3 that our model may be tested at future space-based
interferometers such as LISA LISA:2017pwj , DECIGO Kawamura:2006up , TianQin
TianQin:2015yph  and Taiji Ruan:2018tsw . Even if the predicted GW spectrum
cannot reach the sensitivity curves of the future GW observations, we may be
able to extract the information of the GWs by using improved analysis methods
Hashino:2018wee ; Cline:2021iff . However, we do not consider these methods in
our analysis for the sake of simplicity.

As shown in Fig. 3, the peak height of the GW amplitude lowers as the value of
$\beta_{Q}$ gets smaller in general. Here, one has to mention that for small
$\upsilon_{S}$ values, the light dilaton scenario is most likely to be
detectable than the PRHM one. The qualitatively reason is as follows. The
strength of the first order phase transition is generally related to the
potential difference at the zero temperature between the symmetric vacuum
$(H,s)=(0,0)$ and the broken vacuum $(H,s)=(\upsilon,\upsilon_{S})$. The
potential difference is approximately given by

$\displaystyle V_{{\rm eff}}(0,0)-V_{{\rm
eff}}(\upsilon,\upsilon_{S})\simeq\frac{N_{Q}m_{Q}^{4}}{128\pi^{2}},$ (25)

where the loop corrections from the SM particles are neglected here. According
to Eq. (25), the parameters $N_{Q},\,\alpha_{Q}$ and $\beta_{Q}$ should be
small to make a high potential barrier, which means that the EWPT is strongly
first order in the parameter space region with small $N_{Q},\,\alpha_{Q}$ and
$\beta_{Q}$.

Generally, in models with singlet scalar fields a non-trivial vacuum ($0,s\neq
0$) can be preferred rather than the origin ($0,0$). If such vacuum exists,
the strongly first order EWPT can be easily realized Ahriche:2007jp . In order
to avoid the existence of such non-trivial vacuum, the following condition
should be satisfied

$\lambda_{S}+\delta\lambda_{S}+\frac{3}{32\pi^{2}}N_{Q}\beta_{Q}^{2}\left[\log\left(\frac{T^{2}}{\Lambda^{2}}\right)+0.99\right]>0.$
(26)

Since our model has a $Z_{2}$ symmetry $s\to-s$ even at the quantum level, the
condition in Eq. (26) is always satisfied. Therefore, the approximation in Eq.
(25) is meaningful in our model.

We note that the typical values of $\alpha$ and $\widetilde{\beta}$ we found
are around $0.1-1$ and $100-10000$, respectively. Is is well known that the
bubble collision contribution can be dominant when $\alpha\gg 1$
Caprini:2015zlo . Our results indicate that the dominant source of the GWs in
our scenario is the sound wave contribution.

As noted above, the parameter space region satisfying the condition in Eq. (2)
may imply a large deviation in the cross section for the di-Higgs production
from the SM prediction. To confirm this, we present the di-Higgs production
cross section at the LHC and ILC in Fig. 4 and Fig. 5, respectively. These
figures show the ratio of $R_{{\rm LHC}}$ and $R_{{\rm ILC}}$ as defined in
Eq. (9) for different values of $N_{Q}$ and $\upsilon_{S}$. These figures are
produced by taking into account the different theoretical and experimental
constraints that are mentioned above.

Figure 4: The parameter dependence for the value of $R_{{\rm LHC}}$. The cyan
region can be explored by the GW observations at the DECIGO. The magenta
region is excluded by the completion condition for the phase transition in Eq
(19). The large deviation in the cross section for the di-Higgs production
prefers the large $\alpha_{Q}$. The black region is constrained by the
triviality bound with $\Lambda_{\rm Lan}<10\,{\rm TeV}$. Figure 5: The
parameter dependence for the value of $R_{{\rm ILC}}$. The cyan region can be
explored by the GW observations at the DECIGO. The magenta region is excluded
by the completion condition for the phase transition in Eq. (19). The black
shaded region is the current constraint on the di-Higgs production from the
LHC results. The black region is constrained by the triviality bound with
$\Lambda_{\rm Lan}<10\,{\rm TeV}$.

According to Fig. 4 and Fig. 5, the large $\alpha_{Q}$ values are preferred to
realize a large deviation in the cross section of the di-Higgs production from
the SM prediction. As was mentioned earlier, for the light dilaton case, the
$R$ ratios cannot be large due to the constraints from the mixing angle and
the completion condition for the phase transition. This means that we can
distinguish the PRHM and the light dilaton scenarios by combining the
measurement of the $R_{{\rm LHC}}$ and $R_{{\rm ILC}}$ with the GW
observations. By comparing Fig. 4 with Fig. 5, the value of $R_{{\rm ILC}}$ is
larger than $R_{{\rm LHC}}$. One has to mention that the recent LHC negative
searches for the di-Higgs signal established the upper bound on the di-Higgs
production cross section $\sigma_{hh}^{{\rm LHC}}<112\,{\rm fb}$ CMS:2022dwd ,
which excludes significant regions in the parameter space, especially in the
case with $\upsilon_{s}=3\,{\rm TeV}$. These regions are presented in Fig. 5
as the black dashed regions. Thus, it is expected that the ILC may impose more
severe constraints on the parameter space region with a large deviation in the
$R$ ratios.

Figure 6: The parameter space region explored by the LHC, HL-LHC and ILC. The
gray, orange and green regions are explored by HL-LHC, LHC and ILC,
respectively. DECIGO can explore the cyan shaded region. The magenta region is
constrained by the completion condition for the phase transition. The blue
regions are the unexplored parameter space regions. The black region is
constrained by the triviality bound with $\Lambda_{\rm Lan}<10\,{\rm TeV}$.

In Fig. 6, we show that the constraints on the model from current and future
collider experiments such as the LHC, HL-LHC and ILC. The orange and gray
regions can be explored by the LHC CMS:2022dwd  and HL-LHC Cepeda:2019klc ,
respectively. The yellow regions are within the reach of the ILC with
$\sqrt{s}=500\,{\rm GeV}$ Bambade:2019fyw , while the blue regions cannot be
explored by the di-Higgs production measurement at the LHC, HL-LHC and ILC. On
the other hand, the GW observations can be used to probe the models within the
blue region.

Even if the PRHM scenario cannot be distinguished from the dilaton scenario by
the di-Higgs production measurements, we can determine which scenario is
preferred by combining collider and GW results. For instance, we focus on the
two BPs with $N_{Q}=12$ and $v_{S}=3\,{\rm TeV}$ shown in Fig. 2. The
predicted GW spectra of these BPs has been shown in the panel with $N_{Q}=12$
and $v_{S}=3\,{\rm TeV}$ in Fig. 3. One remarks that the positions of peak of
the GWs are close. However, the GW peak position in the PRHM scenario is
different from that in the light dilaton scenario. This implies that we are
able to determine which scenario is preferred by observing the GW spectra even
if a large cross section di-Higgs signal would not be observed at future
collider experiments. Therefore, one can determine whether the PRHM scenario
or the light dilaton scenario are realized by utilizing complementarity of
collider experiments and GW observations.

We provide a commentary on the Landau pole scale in our scenario. It has been
numerically confirmed that the typical Landau pole scale in our model is
around $O(1)$ TeV when using the beta functions outlined in Appendix A. In
Table 2, the Landau pole scales for the two BPs considered in Fig. 3, with
mass-dependent beta functions, are presented. The results in Table 2 indicate
that the typical values of $\Lambda_{\text{\rm Lan}}$ in our model are around
$O(10)$ TeV. This finding suggests that our phenomenological analyses satisfy
perturbativity.

For the PRHM scenarios with $v_{s}=500,{\rm GeV}$, the Landau pole scale can
be in the $O(1)$ TeV range. However, such BPs are constrained by the
triviality bound with $\Lambda_{\text{\rm Lan}}<10,{\rm TeV}$.

$N_{Q}$ |  $v_{S}$ = 500 GeV |  $v_{S}=$1 TeV |  $v_{S}=$3 TeV
---|---|---|---
6 | (R) 31.0 TeV (G) 7.05 TeV | (R) 19.8 TeV (G) 26.0 TeV | (R) 14.9 TeV (G) 53.0 TeV
12 | (R) 34.0 TeV (G) 6.66 TeV | (R) 37.6 TeV (G) 40.3 TeV | (R) 17.4 TeV (G) 92.8 TeV
24 | (R) 27.5 TeV (G) 6.19 TeV | (R) 29.7 TeV (G) 46.6 TeV | (R) 19.3 TeV (G) 96.9 TeV
Table 2: The Landau pole scale values with the mass dependent beta functions.
Here, (R) and (G) denote the values for the BPs considered in Fig. 3.

Before closing this section, we would like to comment on the possibility of
projecting our results to other renormalizable new physics models. In this
work, we considered a generic CSI model, where the quantum effects that
trigger the EWSB lead to the EWPT dynamics and GWs discussed above. These
quantum corrections lead to the same behavior whether they are a consequence
of single/multiple field contributions that are just bosonic or bosonic and
fermionic contributions. However, in renormalizable new physics models,
additional model-dependent theoretical and experimental constraints should be
considered; and the viable parameter space is significantly affected. For
instance, if such a new physics model suggests a DM candidate; and/or proposes
a solution to the neutrino mass smallness problem, the new masses and
couplings in the model would be severely constrained not only by the
motivation requirements (DM relic density and/or neutrino mass smallness); but
also by other constraints like lepton flavor violating processes, Higgs
invisible and di-photon decays. For instance, in the current generic model, if
the singlet $Q$ is considered to be $U(1)_{Y}$ charged, a significant part of
the parameter space $\alpha_{Q},\,\beta_{Q},\,N_{Q}$ and $\upsilon_{S}$ would
be excluded by the bound from $h\to\gamma\gamma$. Therefore, one expects that
the parameter space regions where the light dilation or PRHM scenarios can be
realized are in general narrowed down when specific new physics models are
considered.

##  6 Conclusion

In this work, we have investigated the possibility of detectable GWs that are
produced during a strongly first order EWPT, within a class of models with the
CSI. In order to get model independent results, we have considered the SM
extended by a real scalar singlet to assist the EWSB, and an additional scalar
field with multiplicity $N_{Q}$, and couplings $\alpha_{Q}$ and $\beta_{Q}$
that accounts for the RCs. By scanning over the model free parameters
($N_{Q}$, $\alpha_{Q}$, $\beta_{Q}$ and $\upsilon_{S}$), the impact of the RCs
on the EWPT and GW observables have been estimated. Especially, we have
focused on the difference between the light dilaton and PRHM scenarios. We
have analyzed the properties of the GWs from a strongly first order EWPT by
considering a generic model with the CSI. We have shown that peaked GW
spectrum might be detected at future space-based interferometers such as the
DECIGO and other GW observations even if the condition $\upsilon_{c}/T_{c}>1$
is not satisfied. As a result, the wide parameter space region in the CSI
model may be tested by utilizing the GW observations.

Also, we have analyzed the cross section in the di-Higgs production processes
at the LHC and ILC. As we have shown, the cross section can be more than three
times larger than the SM prediction in the PRHM scenario. As shown in Fig. 6,
the parameter regions with such large deviation in the di-Higgs production
process can be explored at the LHC, HL-LHC and ILC. In addition, we have shown
that GW observations are useful to distinguish the light dilaton scenario from
the PRHM scenario even if collider experiments cannot observe the di-Higgs
production signal. This fact indicates that we may be able to determine
whether the light dilaton scenario or the PRHM scenario is preferred by
utilizing complementary of collider experiments and GW observations.

## Acknowledgements

A. A. was funded by the University of Sharjah under the research projects No
21021430107 “Hunting for New Physics at Colliders” and No 23021430135
“Terascale Physics: Colliders vs Cosmology”. S. K. was supported by the JSPS
KAKENHI Grant No. 20H00160, No. 202F21324 and No. 203K17691. M. T. was
supported by JSPS KAKENHI Grant No. JP21J10645. M. T. would like to thank
Sharjah university (IAESTE exchange program, HEP group, RISE and VCRGS) for
the financial support and the warm hospitality.

##  Appendix A Renormalization Group Equations

In this appendix, the renormalization group equations for the couplings in our
model are discussed. The beta functions for each coupling in our model at one-
loop level are given by

$\displaystyle 16\pi^{2}\beta_{g^{\prime}}=\frac{41}{6}g^{\prime
3},16\pi^{2}\beta_{g}=-\frac{19}{6}g^{3},16\pi^{2}\beta_{g_{3}}=-7g_{3}^{3},$
$\displaystyle
16\pi^{2}\beta_{y_{t}}=y_{t}\left(-8g_{3}^{2}-\frac{9}{4}g^{2}-\frac{17}{12}g^{\prime
2}+\frac{9}{2}y_{t}^{2}\right),$ $\displaystyle
16\pi^{2}\beta_{\lambda_{h}}=4\lambda_{h}^{2}+3N_{Q}\alpha_{Q}^{2}+3\omega^{2}+\frac{27}{4}g^{4}+\frac{9}{2}g^{2}g^{\prime
2}+\frac{9}{4}g^{\prime 4}-36y_{t}^{4}-\lambda_{h}(9g^{2}+3g^{\prime
2}-12y_{t}^{2}),$ $\displaystyle
16\pi^{2}\beta_{\lambda_{s}}=3\lambda_{s}^{2}+3N_{Q}\beta_{Q}^{2}+12\omega^{2},16\pi^{2}\beta_{\lambda_{Q}}=\frac{N_{Q}+8}{3}\lambda_{Q}^{2}+3\beta_{Q}^{2}+12\alpha_{Q}^{2},$
$\displaystyle
16\pi^{2}\beta_{\omega}=4\omega^{2}+\omega\lambda_{s}+2\lambda_{h}\omega+N_{Q}\alpha_{Q}\beta_{Q}+6\omega
y_{t}^{2}-\frac{9}{2}\omega g^{2}-\frac{3}{2}\omega g^{\prime 2},$
$\displaystyle
16\pi^{2}\beta_{\alpha_{Q}}=4\alpha_{Q}^{2}+\frac{N_{Q}+2}{3}\alpha_{Q}\lambda_{Q}+2\alpha_{Q}\lambda_{h}+\omega\beta_{Q}+6\alpha_{Q}y_{t}^{2}-\frac{9}{2}\alpha_{Q}g^{2}-\frac{3}{2}\alpha_{Q}g^{\prime
2},$ $\displaystyle
16\pi^{2}\beta_{\beta_{Q}}=4\beta_{Q}^{2}+\frac{N_{Q}+2}{3}\lambda_{Q}\beta_{Q}+\lambda_{s}\beta_{Q}+4\omega\alpha_{Q},$
(27)

where $g^{\prime}$, $g$ and $g_{3}$ are gauge couplings for $U(1)_{Y}$,
$SU(2)_{L}$ and $SU(3)_{C}$ gauge groups, respectively. Also, $y_{t}$ is the
Yukawa coupling for top quarks. It is confirmed that these equations are
consistent with the previous work if we take $N_{Q}=1$ Abada:2013pca .

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