Strong electroweak phase transition in a model with extended scalar sector
M. Saeedhoseini and A<EMAIL_ADDRESS>

Department of Physics, Faculty of Basic Sciences,
University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

In this paper we consider an extension of the Standard Model($SM$) with
additional gauge singlets which exhibits a strong first order phase
transition. Due to this first order phase transition in the early universe
gravitational waves are produced. We estimate the contributions such as the
sound wave, the bubble wall collision and the plasma turbulence to the
stochastic gravitational wave background, and we find that the strength at the
peak frequency is large enough to be detected at future gravitational
interferometers such as eLISA. Deviations in the various Higgs boson self
couplings are also evaluated.

PACS:12.60.Fr; 11.10.Wx; 95.35.+d
Key Words: Phase transition, gauge singlet model

##  1 Introduction

The discovery of a narrow resonance, with a mass near $125$ $GeV$, at the
Large Hadron Collider $(LHC)$ with properties similar to those of the Higgs
boson predicted by the Standard Model $(SM)$ $[1,2]$, sparked a lot of
excitements among high energy physicists. But in spite of this important
discovery the $(SM)$ is considered to be incomplete. For instance, in the
Standard Model of particle physics a strong first order phase transition
$(SFPT)$ does not occur $[3]$. However $SFPT$ is needed to justify the baryon
asymmetry of our universe $[4]$, moreover, the $SM$ does not have a candidate
for dark matter ($DM$).
Therefore, some new models are required to address these issues. A popular
model is to couple a singlet scalar to Higgs boson. In Ref. $[5]$ it is shown
that, it is possible to modify the standard theory by adding a scalar which
possesses a discrete $Z_{2}$ symmetry and to address the issue of the dark
matter of the universe, within the frame work of singlet extended $SM$, the
issue of dark matter has been studied in $[6-13]$, while electroweak phase
transition was studied in Refs. $[14-23]$ and in Refs. $[24-25]$ the authors
attempt to explain electroweak phase transition $EWPT$ and dark matter by
singlet extended $SM$.
Another class of models are the multi-singlet extensions of the $SM$ model
$[26-36]$. These models have a larger parameter space in comparison to the
singlet extended models, hence they can address several issues, in Refs.
$[26,27]$ cosmological implications of such models with classical conformal
invariance is presented. Electroweak phase transitions in two-Higgs doublet
model is analyzed in $[37,38]$ and within supersymmetric models in $[39-45]$.
A comprehensive review of $EWPT$ within various models has been given in
$[46]$.
In order to investigate the dynamics of the electroweak phase transition
$EWPT$ one has to utilize techniques from the domain of thermal field theory
$[47-50]$. The occurrence of a first order phase transition, requires that the
electroweak breaking and preserving minima to be degenerate, an event which
happens at a critical temperature $T_{c}$. Moreover, to prevent the washout of
any baryon asymmetry by electroweak sphalerons, the electroweak phase
transition must be strongly first order. Namely the ratio of vacuum
expectation value of the Higgs field to the critical temperature needs to be
greatar than unity. As we describe in the next section if a first order phase
transition occurs in the early universe, the dynamics of bubble collision and
subsequent turbulence of the plasma are expected to generate gravitational
wave ($GW$). If we detect these $GW$ then we can obtain information about
symmetry breaking in early universe. $GW$ signals from phase transitions has
been discussed in $[51-59]$.
In $[60]$ the authors study $EWPT$ within several exotic models and in $[61]$
an analysis of the $EWPT$ of a large number of minimal extensions of the $SM$
and their classically conformal limits is presented. Complex conformal singlet
extension of $SM$ with emphasis on the issue of dark matter and Higgs
phenomenology is studied in $[62]$. Recently baryogenesis within a
$\varphi^{6}$ model is addressed in $[63]$. An investigation of electroweak
phase transitions in a singlet extended model in the $100$ ($TeV)$ range is
given in $[64]$. The authors of $[65]$ study strong first order $EWPT$ in a
singlet scalar extension of the $SM$ where the singlet scalar is coupled non-
minimally to gravity. In this scheme the singlet field first derives inflation
and at a later time causes a strong $EWPT$, in a new study a first order
$EWPT$ in the $SM$ is obtained by varying Yukawas during phase transition
$[66]$.
In this work we propose a new model and we investigate the strength of $EWPT$
within this model. In this model which is a generalization of $[19]$, $N$ real
gauge singlet are coupled to Higgs boson via trilinear interactions. Previous
studies of multi-scalar singlet extension of $SM$ impose separate $Z_{2}$
symmetries on the singlets $[26,28,29,31,33,35,36]$, however in our model we
do not require such symmetry. In spite of it’s simple form , this model has a
very rich phenomenology. The main feature of the singlet extended $SM$ model
(without $Z_{2}$ symmetry) is that the potential barrier between the true and
the false vacua necessary for first order $EWPT$ can be formed mainly by tree-
level interactions. But in the singlet extended $SM$ (with $Z_{2}$ symmetry)
non-decoupling loop effects are needed for the occurrence of a strong first
order $EWPT$, however, these models have $DM$ candidate. In Ref. $[67]$ a non-
minimal composite model based on the coset $\frac{SO(7)}{SO(6)}$ has been
considered. At low energy the scalar sector of their model is composed of two
scalars, one with an unbroken $Z_{2}$ symmetry and another scalar with a
broken $Z_{2}$ symmetry.
The plan of this paper is as follows:
In section two we summarize the basic notions of $EWPT$. We describe the
finite temperature effective potential at one-loop. Then by emphasizing the
underlying physical mechanisms, we describe the basic quantities of interest
such as the strength of a phase transition, the rate of variation of bubble
nucleation rate per volume and the ratio of the latent heat released at the
phase transition to the radiation energy density. In section three we consider
a simple extension of the $SM$ by the addition of $N$ real scalar gauge
singlets with trilinear coupling to Higgs. We present the phenomenology of the
model for $N=2$ and due to lack of protective symmetry these gauge singlets
are not candidates for $DM$ and we discuss the issue of observability of
gravitational wave of our model. And finally in section four we present our
conclusions. Technical details are explained in the appendix.

##  2 Electroweak Phase Transition

In this section we summarize basic notions and definitions of the electroweak
phase transitions.
The observable universe consists predominantly of matter. The asymmetry
between the matter and anti-matter content of the universe is expressed by the
baryon to photon ratio

$\rho=\frac{n_{b}-n_{\bar{b}}}{n_{\gamma}}\sim 10^{-9},$ (1)

where $n_{b}$, $n_{\bar{b}}$ and $n_{\gamma}$ are the number densities of
baryons, antibaryons and photons. In a symmetric universe one expects $\rho=0$
but experiments reveal that $\rho$ has a tiny but non-zero value. This paradox
can be resolved by requiring baryon number violation, C and CP violation and
departure from thermal equilibrium $[4]$.
Baryogenesis is the physical process which is responsible for this observed
baryon asymmetry of universe (BAU). In electroweak baryogenesis one assumes
that the physical mechanism is the occurrence of a strong first order
electroweak phase transition (EWPT) namely a smooth or a weak phase transition
can not explain BAU $[68,69]$.
A convenient tool for investigation of EWPT is effective potential. For any
quantum field theory if we replace the quantum field by it’s vacuum
expectation value in the presence of a source the result for the potential
energy to lowest order in perturbation theory is called the effective
potential. The physical meaning of the effective potential is that it
represent an energy density. In general effective potential contains other
terms (loop corrections) $[70,71]$. By using path integrals it is possible to
find effective potential. In an Euclidean space time at one loop order the
result is

$V^{eff}(\varphi_{c})=V_{0}(\varphi_{c})+\frac{i}{2}\int\frac{d^{4}p}{(2\pi)^{4}}log[\frac{p^{2}+m^{2}(\varphi_{c})}{p^{2}}]+O(\hbar^{2})+...,$
(2)

which is known as the Coleman-Weinberg potential. The first term in $eq.(2)$
is the classical tree-level potential.
While considering physical processes in a hot environment such as early
universe a proper method is finite temperature field theory $[47-50]$. The
expression for the one loop effective potential at finite temperature is
$[72]$

$V^{eff}_{1}(\varphi_{c},T)=\sum_{i}\frac{n_{i}T^{4}}{2\pi^{2}}J_{\mp}(\frac{m_{i}(\varphi_{c})}{T}),$
(3)

where $n_{i}$ is the number of degrees of freedom of the particle, $m_{i}$ is
the field dependent particle mass, and

$J_{\mp}=\pm\int^{\infty}_{0}dyy^{2}\log[1\mp exp(-\sqrt{x^{2}+y^{2}})],$ (4)

and $J_{-}(x)$, $J_{+}(x)$ denotes the contribution from bosons, fermions. In
addition there is another contribution to thermal effective potential from the
daisy subtraction $[73,74]$.
The best way for the evaluation of the finite temperature effective potential
is to use special packages (codes) developed for a specific model.
But when the temperature is much greater than the masses of the various
particles of the system under considerations it is possible to expand the
$J_{\mp}$ in a power series of $\frac{m}{T}$. In this work we use this high
temperature approximation. Recently, a new scheme for computation and
resummation of thermal masses beyond the high-temperature approximation in
general beyond SM scenarios has been proposed $[75]$.
Let us consider the shape of the effective potential. For a generic model, as
the universe cooled down at temperature above a critical temperature $T_{c}$
the effective potential had an absolute minima which was located at the
origin. At $T_{c}$ there were two degenerate minima, which were separated by
an energy barrier. At temperature below the critical temperature the second
minimum became the global one and presently there is no energy barrier.
Moreover the rate of expansion of the universe slowed, as the Hubble parameter
$H$ which characterizes the rate of expansion of the universe depends
quadratically on the temperature.
In $1976$ t Hooft discovered that baryon number is violated in the Standard
Model and it is due to transition between two topologically distinct
$SU(2)_{L}$ ground states $[76,77]$. While at zero temperature, the
probability for barrier penetration is vanishingly small, at non-zero
temperature the transition between two ground state differing by a unit of
topological charge can be achieved by a classical motion over the barrier.
Unstable static solutions of the field equations with energy equal to the
height of the barrier separating two topologically distinct $SU(2)_{L}$ ground
states (sphalerons) have been reported in $[78]$. Hence in the electroweak
baryogenesis , the baryon asymmetry of universe is generated through the
sphaleron process in the symmetric phase during the $EWPT$. At the symmetric
phase the rate of baryon violating process which we denote by
$\tilde{\Gamma}^{sph}$ is a quartic function of the temperature, hence in this
phase $\tilde{\Gamma}^{sph}>>H$.
But the third conditions to generate BAU is the departure from thermal
equilibrium. Hence, the baryon number changing sphaleron interaction must
quickly decouple in the broken phase, that is $\tilde{\Gamma}^{sph}<H$. But
the rate of sphaleron induced baryon violating process is suppressed by a
Boltzman factor

$\tilde{\Gamma}^{sph}\propto exp[-\frac{E_{sph}(T)}{T}],$ (5)

but at phase transition $E_{sph}(T)\propto\varphi_{c}$

$\frac{\varphi_{c}}{T_{c}}\geq 1,$ (6)

where $\varphi_{c}$ is the broken phase minimum at the critical temperature
$T_{c}$.
A first order phase transition proceeds by nucleation of bubbles of the broken
symmetry phase within the symmetric phase. The underlying mechanisms for
bubble nucleation are quantum tunneling and thermal fluctuations. These
bubbles then expand, merge and collide. Gravitational waves are produced due
to collision of bubble walls and turbulence in the plasma after the
collisions. In addition while these bubbles pass through the plasma sound
waves are created. These sound waves can provide additional sources of
gravitational waves.
A crucial parameter for the calculation of the gravitational wave spectrum is
the rate of variation of the bubble nucleation rate per volume, called
$\beta$. It is common to use a normalized dimensionless parameter which is
defined as

$\tilde{\beta}=\frac{\beta}{H_{*}},$ (7)

where, $H_{*}$ denotes the Hubble parameter at the time of phase transition.
The density of latent heat released into the plasma is

$\epsilon_{*}=[-V^{eff}_{min}(T)+T\frac{d}{dT}V^{eff}_{min}(T)]_{T=T_{*}},$
(8)

where $V^{eff}_{min}(T)$ is the temperature-dependent true minimum of the
effective potential of the scalar fields which causes the phase transition
moreover, it’s value must be set to zero by adding a constant at each time.
Another dimensionless parameter for characterizing the spectrum of
gravitational wave is

$\alpha=\frac{\epsilon_{*}}{\rho_{rad}},$ (9)

where the radiation energy density of the plasma
$\rho_{rad}=\frac{\pi^{2}}{30}g_{*}T_{c}^{4}$. And the parameter $g_{*}$ is
the effective degrees of freedom in the thermal bath at the phase transition.
In this work we assume $g_{*}=106.75+N_{S}$ where $N_{S}$ denotes the number
of singlet scalars that facilitates the electroweak phase transition.

##  3  The Model

In Ref. $[19]$ the effects of a light scalar on the electroweak phase
transition has been considered. In their model the scalar sector has been
extended by an addition of a singlet, which has a trilinear interaction with
the Higgs boson. Recently an extension of the $SM$ with addition of $N$
isospin-singlet, which has a quartic interaction with the Higgs boson has been
considered $[57]$. Here we consider a generalization of model of $[19]$. At
zero temperature the effective potential of the scalar sector of our model is

$V_{0}=-DT^{2}_{0}\varphi^{2}+\frac{\lambda}{4}\varphi^{4}+\frac{1}{2}\sum^{N}_{i=1}m^{2}_{i}s_{i}^{2}+\varphi^{2}\sum^{N}_{i=1}\kappa_{i}{s_{i}}.$
(10)

Since we have not included quartic self coupling for the extra scalars , in
order to have a stable potential , we assume that the squared of the mass
parameters of the extra scalars ($m^{2}_{i}$)are positive definite (See
appendix for detail).
At high temperature we have

$V_{T}=D(T^{2}-T^{2}_{0})\varphi^{2}-ET\varphi^{3}+\frac{\lambda_{T}}{4}\varphi^{4}+\frac{1}{2}\sum^{N}_{i=1}m^{2}_{i}s_{i}^{2}+\varphi^{2}\sum^{N}_{i=1}\kappa_{i}{s_{i}}+\frac{T^{2}}{12}\sum^{N}_{i=1}\kappa_{i}s_{i}.$
(11)

The parameters of $eq.(11)$ are given by

$\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{1}{8v^{2}}(2m^{2}_{W}+m^{2}_{Z}+2m^{2}_{t}+2\lambda
v^{2})$ $\displaystyle E$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi
v^{3}}(4m^{3}_{W}+2m^{3}_{Z})$ $\displaystyle\lambda_{T}$ $\displaystyle=$
$\displaystyle\lambda-\frac{1}{16\pi^{2}v^{4}}(6m^{4}_{W}ln\frac{m^{2}_{w}}{a_{B}T^{2}}+3m^{4}_{Z}ln\frac{m^{2}_{Z}}{a_{B}T^{2}}-12m^{4}_{t}ln\frac{m^{2}_{t}}{a_{F}T^{2}})$
$\displaystyle lna_{B}$ $\displaystyle=$ $\displaystyle 3.91,\qquad
lna_{F}=1.14.$ (12)

The critical temperature of model is given by

$T_{c}=\frac{T_{0}}{\sqrt{1-\frac{E^{2}}{D(\lambda_{T}-2\zeta)}-\frac{\zeta}{12D}}},\qquad
where\qquad\zeta=\sum^{N}_{i=1}\frac{\kappa^{2}_{i}}{m^{2}_{i}}.$ (13)

The strength of the phase transition is denoted by $\xi$ and it is given by

$\xi=\frac{\varphi_{c}}{T_{c}}=\frac{2E}{\lambda_{T}-2\zeta}.$ (14)

Moreover, $s_{ic}$ is the $vev$ of the scalar field $s_{i}$ at the second
minimum of the effective potential at $T_{c}$ is

$s_{ic}=-\frac{\kappa_{i}}{m^{2}_{i}}(\varphi_{c}^{2}+\frac{T^{2}_{c}}{12}).$
(15)

###  3.1 Phenomenology of the models with N trilinear interactions

The structure of the scalar mass matrix is
$\left(\begin{array}[]{cccccc}2\lambda
v^{2}&2\kappa_{1}v&2\kappa_{2}v&2\kappa_{3}v&...&...\\\
2\kappa_{1}v&m^{2}_{1}&0&0&0&0\\\ 2\kappa_{2}v&0&m^{2}_{2}&0&0&0\\\
2\kappa_{3}v&&0&m^{2}_{3}&...&...\\\ ...&..&...&...&...&...\\\
...&...&...&...&...&...\\\ \end{array}\right).$
The physical mass squared of the scalar sector of our model are the
eigenvalues of this matrix. The characteristic equation is
$det\left(\begin{array}[]{cccccc}\omega-2\lambda
v^{2}&-2\kappa_{1}v&-2\kappa_{2}v&-2\kappa_{3}v&...&...\\\
-2\kappa_{1}v&\omega-m^{2}_{1}&0&0&0&0\\\
-2\kappa_{2}v&0&\omega-m^{2}_{2}&0&0&0\\\
-2\kappa_{3}v&&0&\omega-m^{2}_{3}&...&...\\\ ...&..&...&...&...&...\\\
...&...&...&...&...&...\\\ \end{array}\right)=0.$
Hence in general one should solve a polynomial of degree $(N+1)$ of $\omega$.
From the above equation we see that the coefficient of $\omega^{N+1}$ is
unity. If we designate the coefficient of $\omega^{N}$ by $\alpha$, then
$-\alpha$ is equal to the sum of physical masses of the model, namely

$2\lambda v^{2}+\sum^{N}_{i=1}m^{2}_{i}=m_{H}^{2}+\sum^{N}_{i=1}\chi^{2}_{i}$
(16)

where the physical mass of the $ith$ scalar is denoted by $\chi_{i}$. From
this relation we find an important relation about the Higgs self coupling,
namely

$\lambda-\lambda_{SM}=\frac{1}{2v^{2}}\sum^{N}_{i=1}(\chi^{2}_{i}-m^{2}_{i}),$
(17)

where $\lambda_{SM}$ is the Higgs quartic self coupling of the standard model.
We see that if the mass parameters of the scalars as well as the physical
masses of the scalars are much smaller than $v$ then the the deviation of the
parameter $\lambda$ from the Higgs self-coupling of the standard model will be
very small.

###  3.2 The special case N=2

Here we consider the case $N=2$. The scalar mass matrix of the model at zero
temperature for this case is.

$\textsl{$M^{2}$}=\left(\begin{array}[]{ccc}2\lambda
v^{2}&2\kappa_{1}v&2\kappa_{2}v\\\ 2\kappa_{1}v&m^{2}_{1}&0\\\
2\kappa_{2}v&0&m^{2}_{2}\\\ \end{array}\right).$
The physical mass squared of the model can be obtained from

$\omega^{3}+A\omega^{2}+B\omega+C=0,$ (18)

where

$\displaystyle A$ $\displaystyle=$ $\displaystyle-(2\lambda
v^{2}+m^{2}_{1}+m^{2}_{2}),\qquad
B=2v^{2}[\lambda(m^{2}_{1}+m^{2}_{2})-2(\kappa_{1}^{2}+\kappa_{2}^{2})]+m^{2}_{1}m^{2}_{2}$
$\displaystyle C$ $\displaystyle=$ $\displaystyle
4v^{2}(\kappa^{2}_{1}m^{2}_{2}+\kappa^{2}_{2}m^{2}_{1})-2\lambda^{2}v^{2}m_{1}^{2}m_{2}^{2}.$
(19)

Now one of the eigenvalues is equal to $m^{2}_{H}$, hence we obtain

$m^{6}_{H}+Am_{H}^{4}+Bm_{H}^{2}+C=0,$ (20)

Moreover,by minimizing the potential with respect to variables $s_{1},s_{2}$
we obtain

$s_{i}=-\frac{\kappa_{i}v^{2}}{m^{2}_{i}},\qquad i=1,2.$ (21)

By minimizing the potential with respect to variable $\varphi$ we obtain

$-2DT^{2}_{0}+\lambda
v^{2}-2\sum^{2}_{i=1}\frac{\kappa_{i}^{2}v^{2}}{m^{2}_{i}}=0.$ (22)

The parameters of the model must satisfy eqs.(20,22). Hence the parameter
space of the model in this case contains four independent parameters
($\kappa_{1}$,$\kappa_{2}$, $m_{1}$,$m_{2}$).
If we subtract eq.(20) from eq.(18) we get

$\chi^{4}+(A+m_{H}^{2})\chi^{2}+m_{H}^{4}+m_{H}^{2}A+B=0,$ (23)

Therefore,the physical mass squared of the singlets are determined.
But models with extended Higgs sectors predicting strongly first order phase
transition simultaneously predict a significant deviation in the triple Higgs
boson coupling as well $[79]$. This deviation at the tree level in $[17]$ and
at loop level in $[33,79]$ has been studied, with

$\Delta_{hhh}=\frac{\lambda^{MSM}_{hhh}-\lambda^{SM}_{hhh}}{\lambda^{SM}_{hhh}},$
(24)

where $\lambda^{SM}_{hhh}$ is the Higgs triple coupling of the $SM$ and
$\lambda^{MSM}_{hhh}$ is the Higgs triple coupling of the multi-singlet
extension of the $SM$. Collider experiments could measure the Higgs triple
coupling. Here we want to explore the region of intermediate mass of the
singlets. But there are bounds on Higgs-Portal models from the LHC Higgs data
$[80,81,82]$. For instance the Higgs doublet is mixed with the extra singlet
scalars and the mixing element $cos(\varphi)$ between the CP-even component of
the doublet ($\varphi$) and the physical Higgs (whose $m_{H}$=125.09 GeV) is
not arbitrary and it is subject of a constraint coming from the Higgs coupling
to the W gauge bosons. Current data $[81]$ suggests $\cos(\varphi)>0.86$.
The results are presented in Table $1$ for the onset of a strong $EWPT$,
namely $\xi=1$. For each configuration in the table we present our results for
the deviation of Higgs triple coupling as well. Hence by adding one scalar to
the model we can have singlets in the intermediate mass region.

Set $\kappa_{1}$ $\kappa_{2}$ $m_{1}(GeV)$ $m_{2}(GeV)$ $\chi_{1}(GeV)$
$\chi_{2}(GeV)$ $cos(\varphi)$ $\Delta\lambda_{hhh}$ $\Delta\lambda_{hhhh}$ I
0.2 3.7 69.2 16.0 6.2 69.2 0.993 13.9 % 4.1% II 0.5 3.1 18.7 13.5 5.2 18.6
0.995 14.5 % 3.0% III 0.8 2.3 11.2 10.4 4.0 11.1 0.997 15.1 % 1.8% IV 1.2 0.7
18.1 3.2 17.5 1.3 0.999 5.7 % 0.6%

Table $1$: Different configurations associated with the onset of a strong
first order phase transition ($\xi=1$).

For completeness the predictions of the model for the deviation of Higgs boson
quartic coupling $\Delta\lambda_{hhhh}$ are given in Table $1$.
Moreover, the existence of the extra singlet scalars could affect the total
Higgs decay if they are light enough, which becomes

$\Gamma_{total}(h)=cos^{2}(\varphi)\Gamma_{total}(h_{SM}),+\sum\Gamma(h\rightarrow
s_{i}+s_{k})$ (25)

where $s$ denote all the scalars. The deviation from the $SM$ value
$\Delta\Gamma_{total}<1.4(MeV)$.
In the $SM$, a total Higgs decay width around 4 $MeV$ is predicted. In this
work we assume the decay $h\rightarrow s_{i}+s_{k}$ is an invisible decay.
However, current analysis $[82]$ suggests that the Higgs invisible decay
branching ratio should be less than $17\%$. In Table $2$ we present Higgs
invisible decay branching ratio in various decay modes, as well the total
Higgs invisible branching ratio for all of Higgs invisible decay modes. The
invisible Higgs width , total Higgs width and the deviation of total Higgs
width of our model from that of the $SM$ are also shown. The unit for the
variuos width in this Table is ($MeV$), moreover in our calculation we have
assumed $\Gamma_{total}(h_{SM})=4(MeV)$.
It turns out that the critical temperature for this model $T_{c}<100(GeV)$.
Hence it is a good approximation to use $\lambda$ instead of $\lambda_{T}$ and
in Ref. $[19]$ this approximation is used to study $SFPT$ for the case $N=1$
but the mass of the light scalar is up to $12$ $GeV$, in Ref. $[20]$ a one
loop study of the same model has been presented but the mass of the light
scalar to catalyze a $SFPT$ is up to $20$ $GeV$. But for the model presented
in this work and using this approximation for the case $N=2$ the mass of the
scalar to catalyze a $SFPT$ is $69$ $GeV$. By adding more scalars we expect to
have heavy singlets.

Set | $B(h\rightarrow s_{1}s_{1})$ | $B(h\rightarrow s_{1}s_{2})$ | $B(h\rightarrow s_{2}s_{2})$ | $B_{tot}(h\rightarrow inv)$ | $\Gamma_{inv}(h)$ | $\Gamma_{tot}(h)$ | $\Delta\Gamma_{tot}$
---|---|---|---|---|---|---|---
I | 1.45 % | 0.31 % | 0.0 % | 1.49 % | 0.06 | 4.006 | 0.006
II | 0.75% | 0.04 % |  $3.6\times 10^{-5}$ % | 0.8 % | 0.03 | 3.992 | 0.008
III | 0.24 % | 0.004 % |  $1.3\times 10^{-6}$ % | .24 % | 0.01 | 3.99 | 0.01
IV | 0.02 % |  $4.6\times 10^{-7}$ % |  $6.3\times 10^{-12}$ % | 0.02 % | $5.8\times 10^{-4}$ | 3.993 | 0.007

Table $2$: Values of Higgs invisible branching ratios for various decay modes,
total value of Higgs invisible branching ratio, Higgs invisible width, Higgs
total width and deviation of the Higgs width from the SM value are shown for
various configurations. The unit for width in this table is $MeV$. All four
configurations are consistent with current experimental data.

###  3.3 Detection of gravitational waves

The direct detection of gravitational waves by LIGO $[83]$ generated a lot of
interest among researchers in cosmology, astrophysics and particle physics.
Major sources of gravitational waves are inflation, compact binary systems or
cosmological phase transitions.
In this section under phenomenological description we aim to make an estimate
for the observability of the gravitational waves produced by the model
presented in this work.
In all of previous multi-singlet models a $Z_{2}$ symmetry (either broken or
unbroken) is imposed on the fields. However, neither of the two fields of our
model has this symmetry. This is the main difference between our model and
previous studies.
The main parameters that are of value for obtaining the spectrum of
gravitational waves are the parameter $\alpha$ and $\beta$, which we discussed
in section two. And they are obtained from the thermal effective potential.
For the set $I$ of Table $1$ and from $eqs.(8,9)$ we obtain $\alpha=0.16$. But
the velocity of the bubble wall and the efficiency factor, the fraction of the
latent heat which is converted to the kinetic energy of the plasma are
determined from

$v_{b}=\frac{\frac{1}{\sqrt{3}}+\sqrt{\alpha^{2}+\frac{2\alpha}{3}}}{1+\alpha},\qquad\kappa=\frac{1}{1+0.715\alpha}(0.715\alpha+\frac{4}{27}\sqrt{\frac{3\alpha}{2}}),$
(26)

Thus for this configuration $v_{b}=0.81$ and $\kappa=0.17$. The The peak
frequency for bubble collision contribution is given by $[84]$

$f_{col}=16.5\times
10^{-6}\frac{0.62}{v^{2}_{b}-0.1v_{b}+1.8}\frac{\beta}{H_{*}}\frac{T_{*}}{100}(\frac{g_{*}}{100})^{\frac{1}{6}}.\qquad
Hz$ (27)

And the energy density at this frequency is given by

$\Omega h^{2}_{col}=1.67\times
10^{-5}(\frac{\beta}{H_{*}})^{2}\frac{0.11v_{b}^{3}}{0.42+v^{2}_{b}}(\frac{\kappa\alpha}{1+\alpha})^{2}(\frac{g_{*}}{100})^{\frac{-1}{3}}$
(28)

where $h$ is the reduced Hubble constant at present.
Another source of gravitational waves is the compression waves in the plasma
(sound waves) and the peak frequency is

$f_{sw}=1.9\times
10^{-5}\frac{\beta}{H_{*}}v^{-1}_{b}\frac{T_{*}}{100}(\frac{g_{*}}{100})^{\frac{1}{6}},\qquad
Hz$ (29)

the energy density at this peak frequency is obtained from

$\Omega h^{2}_{sw}=2.65\times
10^{-6}(\frac{\beta}{H_{*}})^{-1}(\frac{\kappa\alpha}{1+\alpha})^{2}(\frac{g_{*}}{100})^{\frac{-1}{3}}v_{b}.$
(30)

And finally the peak frequency for gravitational waves which are caused by the
turbulence of the plasma is

$f_{turbo}=2.7\times
10^{-5}\frac{\beta}{H_{*}}v^{-1}_{b}\frac{T_{*}}{100}(\frac{g_{*}}{100})^{\frac{1}{6}}\qquad
Hz,$ (31)

And the peak energy density of this part is given by

$\Omega h^{2}_{turbo}=3.35\times
10^{-4}(\frac{\beta}{H_{*}})^{-1}(\frac{\varepsilon\kappa\alpha}{1+\alpha})^{\frac{3}{2}}(\frac{g_{*}}{100})^{\frac{-1}{3}}v_{b}\frac{1}{2^{\frac{11}{3}}(1+8\pi\frac{f_{turbo}}{H_{*}})}$
(32)

where $\varepsilon$ denotes the fraction of latent heat that is transformed
into turbulent motion of the plasma. We choose $\varepsilon=0.05$.
Hence by knowing the values of the parameters $\alpha$, $\tilde{\beta}$ and
$T_{*}$ we can obtain the spectra of the $GW$. Even though the nucleation
temperature $T_{*}$ is lower than the critical temperature in this work we
assume $T_{*}\approx T_{c}$.
The standard method of calculation of the parameter $\tilde{\beta}$ from the
effective potential is to compute the Euclidean action of the model and it is
explained in $[36,85]$, but in an approximate scheme it is found that $[59]$,

$\tilde{\beta}\approx 170-4\ln(\frac{T_{*}}{1GeV})-2\ln g_{*},$ (33)

and in our case when the phase transition happens at weak scale,
$\tilde{\beta}\approx 144$.
Hence, in order to assess the implications of the model on the spectra of $GW$
we present results by varying this parameter in the interval
$50\leq\tilde{\beta}\leq 250$. In Table $3$ we present our results. The unit
for various frequencies is $mHz$.

$GWSpectra$ | $f_{sw}$ | $f_{col}$ | $f_{turb}$ | $\Omega h^{2}_{sw}$ | $\Omega h^{2}_{col}$ | $\Omega h^{2}_{turb}$ | $\Omega h^{2}_{total}$
---|---|---|---|---|---|---|---
$\tilde{\beta}=50$ | 0.65 | 0.12 | 1.85 | $2.34\times 10^{-11}$ | $1.98\times 10^{-13}$ | $1.67\times 10^{-15}$ | $2.36\times 10^{-11}$
$\tilde{\beta}=100$ | 1.30 | 0.24 | 3.69 | $1.17\times 10^{-11}$ | $9.9\times 10^{-14}$ | $8.35\times 10^{-16}$ | $1.18\times 10^{-11}$
$\tilde{\beta}=250$ | 3.25 | 0.60 | 9.23 | $4.67\times 10^{-12}$ | $3.96\times 10^{-14}$ | $3.34\times 10^{-16}$ | $4.71\times 10^{-12}$

Table $3$: The spectra of gravitational wave as predicted by our model. The
emitted $GW$ are within the reach of eLISA C1.

The results of Table $3$ shows that in our model the contribution from the
turbulent motion of always a few orders of magnitude smaller than the previous
two. Moreover, the contribution from the sound waves is the dominant source of
the total $GW$ spectrum. By studying the frequency dependent spectra $[84]$,
we find that the peak of energy density of the sound wave contribution and the
peak of energy density due to collision are well separated. A desirable
feature while detecting these waves. These waves are within the reach of
future gravitational wave interferometers (eLISA C1).

##  4 Conclusions

In this work we have presented a new extension of the $SM$. In this model we
amend the $SM$ by $N$ gauge singlets. In order to avoid proliferation of the
parameters we considered the most economical model. For each scalar we allowed
a mass term with a positive squared mass parameter to insure vacuum stability
in the direction of that scalar and a triple coupling with the Higgs field to
facilitate a strong electroweak phase transition, and for the special case
$N=2$ we find $SFPT$ with gauge singlets in the intermediate mass range. We
also investigated the deviations of the Higgs coupling constants from the $SM$
values. And we find that the deviation of the triple Higgs boson coupling can
be as large $15\%$.
Finally, we have obtained the gravitational wave spectrum from the electroweak
phase transition. We have shown that the gravitational wave signal can be
detected by eLISA. The present model has a large parameter space and as a
result a richer phenomenology in comparison to $[19]$. It would be of interest
to amend the model by inclusion of quartic self-coupling of the scalars as
well as the mixing term between scalars(see appendix for detail).
It would be of interest to consider higher values of $N$ for the model
proposed in this work. These and other related issues are presently under
considerations.

Acknowledgements:

We would like to thank the referees for their valuable comments and
suggestions.

Appendix : Vacuum stability conditions

For the special case $N=2$ of the model the weak condition for vacuum
stability is that if the value of the fields tend to infinity then the tree-
level potential is not unbounded-from-below directions. This leads to
$\lambda>0$. But if only the field $s_{1}$ tend to infinity, then the
condition vacuum stability will be $m_{1}^{2}>0$ and by similar argument for
the field $s_{2}$ the restriction will be $m_{2}^{2}>0$.
However, the strong condition for vacuum stability as stated in $[35,86,87]$is
that the masses of the particles of a particular model be positive.
The necessary conditions for a symmetric matrix A of order $3$ to have real
eigenvalues are;

$\displaystyle a_{11}>0,a_{22}>0,a_{33}>0,$
$\displaystyle\bar{a}_{12}=a_{12}+\sqrt{a_{11}a_{22}}>0,$
$\displaystyle\bar{a}_{13}=a_{13}+\sqrt{a_{11}a_{33}}>0,$
$\displaystyle\bar{a}_{23}=a_{23}+\sqrt{a_{22}a_{33}}>0,$ (34)

and

$\sqrt{a_{11}a_{22}a_{33}}+a_{12}\sqrt{a_{33}}+a_{13}\sqrt{a_{22}}+a_{23}\sqrt{a_{11}}+\sqrt{2\bar{a}_{12}\bar{a}_{13}\bar{a}_{23}}>0.$
(35)

The constraints stated in eq. (34) applies to quantities with dimension of
squared mass. From our mass matrix of subsection $3-2$, the first constraint
is $2\lambda v^{2}>0$, which leads to $\lambda>0$. Hence, for the model
presented in this work, the criteria which provide the necessary and
sufficient vacuum stability conditions are given by,

$\displaystyle\lambda>0,\qquad m^{2}_{1}>0,\qquad m^{2}_{2}>0,$
$\displaystyle\kappa_{1}>-\sqrt{\frac{\lambda
m^{2}_{1}}{2}},\qquad\kappa_{2}>-\sqrt{\frac{\lambda m^{2}_{2}}{2}},$ (36)

and

$\sqrt{\lambda
m_{1}^{2}m_{2}^{2}}+\kappa_{1}\sqrt{2m^{2}_{2}}+\kappa_{2}\sqrt{2m^{2}_{1}}+\sqrt{(2\kappa_{1}+\sqrt{2\lambda
m_{1}^{2})}(2\kappa_{2}+\sqrt{2\lambda
m_{2}^{2})}\sqrt{m_{1}^{2}m_{2}^{2}}}>0,$ (37)

where the conditions stated in the first line of $eq.(36)$ has been obtained
by using weak conditions for vacuum stability.
In our model we have not included quartic terms for the gauge singles such as

$V_{quartic}=\sum^{2}_{i=1}\frac{\lambda_{i}s_{i}^{4}}{4}+\delta
s_{1}^{2}s_{2}^{2}$ (38)

in the effective potential, where $\lambda_{1}$, $\lambda_{2}$ and $\delta$
are dimensionless coupling parameters, in this case the vacuum stability in
the direction of the extra scalars is maintained if

$\lambda_{1}>0,\qquad\lambda_{2}>0,\qquad\sqrt{\lambda_{1}\lambda_{2}}>-2\delta,$
(39)

and the squared of the mass parameters $m^{2}_{1}$ and $m^{2}_{2}$ in
principle can assume any value (positive, zero or negative), for instance in
the multi-scalar model described in $[26]$ all of the extra scalars do not
have a mass term, while in the absence of quartic terms the squared mass
parameters of the model presented in section three must be positive as
required by vacuum stability.

## References

  * [1] ATLAS Collaboration, G. Aad et al., Phys. Lett. B 716 (2012) 1.
  * [2] CMS Collaboration, S. Chatrchyan et al., Phys. Lett. B 716 (2012) 30.
  * [3] G. W. Anderson and L. J. Hall, Phys. Rev. D 45 (1992) 2685.
  * [4] A. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32.
  * [5] V. Silveira and A. Zee, Phys. Lett. B 161 (1985) 136.
  * [6] J. McDonald, Phys. Rev. D 50  (1994) 3637.
  * [7] C. P. Burgess, M. Pospelov and T. ter Veldhuis, Nucl. Phys. B 619 (2001) 709.
  * [8] J. J. van der Bij, Phys. Lett. B 636 (2006) 56.
  * [9] V. Barger, P. Langacker, M. McCaskey, M. J. Ramsy-Musolf and G. Shaughnessy, Phys. Rev. D 77 (2008) 035005.
  * [10] W. -L. Guo, Y. -L. Wu, JHEP 1010 (2010) 083.
  * [11] C. E. Yaguna, JHEP 1108 (2011) 060.
  * [12] R. Coimbra, M. O. P. Sampaio and R. Santos, Eur. Phys. J. C 73 (2013) 2428.
  * [13] L. Feng, A. S. Profumo and B. L. Ubaldi, JHEP 03 (2015) 045.
  * [14] J. R. Espinosa and M. Quiros, Phys. Lett. B305 (1993) 98.
  * [15] J. Choi and R. R. Volkas, Phys. Lett. B317 (1993) 385.
  * [16] S. Profumo, M. J. Ramsey-Musolf, G. Shaughnessy, JHEP 0708 (2007) 010.
  * [17] A. Noble and M. Perelstein, Phys. Rev. D 78 (2008) 063518.
  * [18] A. Ashoorioon and T. Konstandin, JHEP 0907 (2009) 086.
  * [19] S. Das, P. J. Fox, A. Kumar and N. Weiner, JHEP 1011 (2010) 108\.
  * [20] J. R. Espinosa, T. Konstandin, and F. Riva, Nucl. Phys. B 854 (2012) 592.
  * [21] T. Alanne, K. Tuominen, and V. Vaskonen, Nucl. Phys. B 889 (2014) 692.
  * [22] D. Curtin, P. Meade and C. T. Yu, JHEP 1411 (2014) 127.
  * [23] S. Profumo, M J. Ramsey-Musolf, C. L. Wainwright and P. Winslow, Phys. Rev. D 91 (2015) 035018.
  * [24] J. M. Cline, K. Kainulainen, P. Scott and C. Weniger, Phys. Rev. D 88 (2013) 055025.
  * [25] J. M. Cline, K. Kainulainen, JCAP 1301 (2013) 012.
  * [26] J. R. Espinosa and M. Quiros, Phys. Rev. D 76 (2007) 07600.
  * [27] J. R. Espinosa, T. Konstandin, J. M. No and M. Quiros, Phys. Rev. D 78 (2008) 123528.
  * [28] A. Drozd, B. Grzadkowski and J. Wudka, Acta Phys. Polon. B 42 (2011) 2255.
  * [29] A. Abada, S. Nasri and D. Ghaffor, Phys. Rev. D 83 (2011) 095021\.
  * [30] A. Abada and S. Nasri, Phys. Rev. D 85 (2012) 075009.
  * [31] A. Drozd, B. Grzadkowski and J. Wudka, JHEP 1204 (2012) 006.
  * [32] A. Ahriche and S. Nasri, Phys. Rev. D 85 (2012) 093007.
  * [33] A. Ahriche, A. Arhrib and S. Nasri, JHEP 02 (2014) 042.
  * [34] A. Tofighi, O. N. Ghodsi and M. Saeedhoseini, Phys. Lett. B 748 (2015) 208.
  * [35] K. P. Modak, D. Majumdar and S. Rakshit, JCAP 1503 (2015) 011.
  * [36] R. Jinno, K. Nakayama and M. Takimoto, Phys. Rev. D 93 (2016) 045024\.
  * [37] J. M. Cline and P.-A. Lemieux, Phys.Rev. D 55 (1997) 3873.
  * [38] L. Fromme, S. J. Huber, and M. Seniuch, JHEP 0611 (2006) 038.
  * [39] S. J. Huber and M. G. Schmidt, Eur. Phys. J. C 10 (1999) 473.
  * [40] K. Funakubo and E. Senaha, Phys. Rev. D 79 (2009) 115024.
  * [41] D. J. H. Chung and A. J. Long, Phys. Rev. D 81 (2010) 123531.
  * [42] J. Kozaczuk, S. Profumo, L. S. Haskins and C. L. Weinwright, JHEP 1501 (2015) 144.
  * [43] C. Balazs, A. Mazumdar, E. Pukartas and G. White, JHEP 1401 (2014) 073.
  * [44] W. Huang, Z. Kang, J. Shu and J. M. Yang, Phys. Rev. D 91 (2015) 025006.
  * [45] C. -W. Chiang and E. Senaha, JHEP 1006 (2010) 030.
  * [46] D. J. Chung, A. J. Long, and L.-T. Wang, Phys. Rev. D 87 (2013) 023509.
  * [47] L. Dolan and R. Jackiw, Phys. Rev. D 9 (1974) 3320.
  * [48] E. W. Kolb and M. S. Turner, The Early Universe, Front. Phys. 69 (1990) 1 547.
  * [49] J. I. Kapusta, Finite emperature field theory, Cambridge University Press, (1989).
  * [50] M. Le-Bellac, Thermal field theory, Cambridge University Press, (2000).
  * [51] A. Kosowsky, A. Mack and T. Kahniashvili, Phys. Rev. D 66 (2002) 024030.
  * [52] A. Kosowsky, M. S. Turner and R. Watkins, Phys. Rev. D 45 (1992) 4514.
  * [53] M. Kamionkowski, A. Kosowsky and M. S. Turner, Phys. Rev. D 49 (1994) 2837\.
  * [54] C. Caprini and R. Durrer, Phys. Rev. D 74 (2006) 063521.
  * [55] L. Leitao and A. Megevand, arxiv:1512.08962.
  * [56] M. Artymowski, M. Lewicki and J. D. Wells, arxiv:1609.07143.
  * [57] M. Kakizaki, S. Kanemura and T. Matsui, Phys. Rev. D 92 (2015) 115007\.
  * [58] P. Huang, A. J. Long, L.-T. Wang, Phys. Rev. D 95 (2016) 075008\.
  * [59] C. Balazs, A. Fowlie, A. Mazumdar and G. White, Phys. Rev. D 95 (2017) 043505.
  * [60] A. Katz and M. Perelstein, JHEP 1407 (2014) 108.
  * [61] F. Sannino and J. Virkaja$\ddot{a}$rvi, Phys. Rev. D 92 (2015) 045015\.
  * [62] Z. -W. Wang, T. G. Steele, T. Hanif and R. B. Mann, JHEP 08(2016) 065.
  * [63] M. Lewicki, T. Rindler-Daller and J. D. Wells, JHEP 06 (2016) 055\.
  * [64] A. V. Kotwal, M. J. Ramsey-Musolf, J. M. No and P. Winslow, Phys. Rev. D 94 (2016) 035022.
  * [65] I. Baldes, T. Konstandin and J. Servant, arxiv:1604.04526.
  * [66] T. Tenkanen, K. Tuominen and V. Vaskonen, arxiv:1606.06063.
  * [67] M. Chala, G. Nardinib and I. Sobolevc, Phys. Rev. D 94 (2016) 055006\.
  * [68] V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155 (1985) 36.
  * [69] D. E. Morrissey and M. J. Ramsey-Musolf, New J. Phys. 14 (2012) 125003.
  * [70] S. Coleman, Aspects of symmetry: Selected Erice lectures, Cambridge University Press, Cambridge, (1988).
  * [71] A. Zee, Quantum field theory in a nutshell, Princeton University Press, Princeton, (2010).
  * [72] M. Carena, A. Megevand, M. Quiros and C. E. M. Wagner, Nucl. Phys. B 716 (2005) 319.
  * [73] P. B. Arnold and O. Espinosa, Phys. Rev. D 47 (1993) 3546.
  * [74] Z. Fodor and A. Hebecker, Nucl. Phys. B 432 (1994) 127.
  * [75] D. Curtin, P. Meade and H. Ramani, arxiv:1612.00466.
  * [76] G. ${}^{\prime}t$ Hooft, Phys. Rev. Lett. 37 (1976) 8.
  * [77] G. ${}^{\prime}t$ Hooft, Phys. Rev. D 14 (1976) 3432.
  * [78] N. S. Manton, Phys. Rev. D 28 (1983) 2019.
  * [79] S. Kanemura, M. Kikuchia and K. Yagyub, Nucl. Phys. B 917 (2017) 154.
  * [80] T. Robens and T. Stefaniak, Eur. Phys. J. C 76 (2016) 268.
  * [81] K. Cheung, P. Ko, J. S. Lee and P. Y. Tseng, JHEP 10 (2015) 057\.
  * [82] P. Bechtle, S. Heinemeyer, O. Stål, T. Stefaniak, and G. Weiglein, JHEP 11 (2014) 039.
  * [83] Virgo, LIGO Scientific collaboration, B. P. Abbott et al., Phys. Rev. Lett. 116 (2016) 061102.
  * [84] C. Caprini, et al., JCAP 1604 (2016) 001.
  * [85] V. Vaskonen, arxiv:1611.02073.
  * [86] K. Kannike, Eur. Phys. J. C 72, 2093 (2012) 2093.
  * [87] J. Chakrabortty, P. Konar and T. Mondal, arXiv:1311.5666.