Entropy Production Due to Electroweak Phase Transition in the Framework of Two Higgs Doublet Model

Abstract

We revisit the possibility of first order electroweak phase transition (EWPT) in one of the simplest extensions of the Standard Model scalar sector, namely the two-Higgs-doublet model (2HDM). We take into account the ensuing constraints from the electroweak precision tests, Higgs signal strengths and the recent LHC bounds from direct scalar searches. By studying the vacuum transition in 2HDM, we discuss in detail the entropy released in the first order EWPT in various parameter planes of a 2HDM.

1. Introduction

It is a well-established fact that electroweak phase transition (EWPT) is either a second order or a smooth crossover in the Standard Model (SM) of particle physics. It is also well-established that the entropy density in the early universe plasma is conserved in the course of the cosmological expansion if the plasma is in thermal equilibrium state with negligible chemical potential of every species [1,2]. The entropy conservation law is given by

$$\begin{array}{c}\hfill s=\frac{P+\rho }{T}{a}^3=\mathrm{const}.\end{array}$$

(1)

where s is the entropy density, $a\left(t\right)$ is the scale factor, $T\left(t\right)$ is the temperature of the fluid (or plasma) are function of time t, $\rho$ and P are the energy density and pressure of the plasma, respectively.

In the early universe, the state of matter is quite close to the equilibrium as the reaction rate $\Gamma \sim n\sigma v$, $n=1,2$, represents decays of one and two-body reactions, respectively, $\sigma v$ is the product of annihilation cross section, is much faster than the cosmological expansion rate, i.e., the Hubble parameter $\mathcal{H}=\dot{a}/a\propto T^2/m_{Pl}$, where $\dot{a}$ is the time derivative of the scale factor. The equilibrium condition $\Gamma >\mathcal{H}$ is always satisfied for at temperature $T<\alpha m_{\mathrm{Pl}}$. Here $\alpha$ is the coupling constant of the particle interaction of the order of $\sim {10}^{-2}$ and $m_{\mathrm{Pl}}$ is the Planck mass. Due to the large value of $m_{\mathrm{Pl}}$, thermal equilibrium exists in most of the history of the universe, if $\alpha$$\ll 1$.

As mentioned above, during thermal equilibrium, the entropy density in the comoving volume is conserved. However, there are scenarios where the entropy density is not conserved. For example, if the universe at a certain stage was dominated by primordial black holes [3], the entropy production can be very high, high enough to delete the pre-existing baryon asymmetry [4]. In the context of the modern cosmological paradigm of inflationary Universe with baryosynthesis and dark matter/energy, physics beyond the Standard Model (BSM) underlying these necessary elements of the modern cosmological model can provide many examples of various mechanisms of high entropy production (see e.g., [5] for review and references). Taking apart the wide range of various possibilities we consider here the problem of entropy production by minimal extension of SM and start the discussion from the SM predictions for the cosmological entropy production.

A large entropy production could take place during during quantum chromodynamical (QCD) phase phase transition at $T\sim$ 100–200 MeV. However, due to difficulty in numerical computation, QCD phase transition in early universe cosmology is not known in detail. For reference, see Ref. [6].

Few mechanisms of realistic though very weak entropy production could take place during the freeze-out of dark matter (DM) particles. However, usually, the fraction of DM density was quite low at the freezing out temperature and the effect is tiny.

An interesting effect, not covered in this paper, is the formation of bubble walls that can take place in the early universe. Their collision can lead to the formation of primordial black holes due to first order phase transition with background gravitational waves [7,8]. The entropy released in the process can wipe out the dark matter density that was present before EWPT. Recent results on muon $(g-2)$ experiment, see Refs. [9,10,11], have opened up a new window to study dark matter in the framework of extended SM physics which shall be done in our next work.

Most probably, the largest entropy release in the standard model took place in the process of the electroweak transition from symmetric to asymmetric electroweak phase in the course of the cosmological cooling down. In the SM with one Higgs field, the process is a mild crossover and the entropy production is about $13\%$ [12].

According to the electroweak (EW) theory at the temperature higher than a critical one, $T>T_c$, the expectation value of the Higgs field, $\varphi$, in the fluid (plasma) is zero and the universe is in electroweak symmetric phase [13].When the temperature drops below $T_c$, a nonzero expectation value is created, which gradually rises, with decreasing temperature, up to the vacuum expectation value $\eta$. Such a state does not satisfy the conditions necessary for the entropy conservation and an entropy production is expected.

A huge amount of entropy is released if EWPT is first order, which is the case even with the minimalist extension of standard model known as two-Higgs-doublet Model (2HDM). In what follows, we have considered a real 2HDM of type-I and are scanned over certain parameter spaces and used numerical analysis to calculate the entropy production for some interesting and unique benchmark points.

The paper is arranged as follows: in the next section details about 2HDM are given along with some some Large Hadron Collider (LHC) constraints, followed by the theoretical framework of the process. Due to cumbersome and very difficult analytical calculations, we did numerical analysis of certain parameters using the BSMPT package [14]. A generic discussion and conclusion is given there after. The paper closes by two appendices that give details about the metric being used here and also the masses of the scalar bosons generated by 2HDM.

2. 2HDM: A Brief Review and Current Constraints

There are two scalar doublets in the framework and they are defined as:

$${\phi }_I=\left(\begin{array}{c}{\varphi }_I^{+}\\ \frac{1}{\sqrt{2}}(v_I+{\rho }_I)+i{\eta }_I\end{array}\right),$$

(2)

with $I=1,2$. Here ${\varphi }_I^{\pm }$, ${\rho }_I$, ${\eta }_I$, and $v_I$ indicate the charged, neutral CP-even and neutral CP-odd degrees of freedom (d.o.f.) and the vacuum expectation value (VEV) of the I-th doublet, respectively.

Prior to spontaneous symmetry breaking (SSB), the tree-level 2HDM Lagrangian, comprising of 6-dimensional operators, assumes the form

$$\begin{array}{ccc}\hfill & & \mathcal{L}={\mathcal{L}}_{\mathrm{kin}}+{\mathcal{L}}_{\mathrm{Yuk}}-V({\phi }_1,{\phi }_2)+{\mathcal{L}}_6,\hfill \end{array}$$

(3)

where,

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mathrm{kin}}& =& -\frac{1}{4}\sum _{X=G^a,W^i,B}{X}_{\mu \nu }{X}^{\mu \nu }+\sum _{I=1,2}{\left|D_{\mu }{\phi }_I\right|}^2+\sum _{\psi =Q,L,u,d,l}\overline{\psi }i\cancel{D}\psi ,\hfill \\ \hfill {\mathcal{L}}_{\mathrm{Yuk}}& =& \sum _{I=1,2}{Y}_I^e\overline{l}e{\phi }_I+\sum _{I=1,2}{Y}_I^d\overline{q}d{\phi }_I+\sum _{I=1,2}{Y}_I^u\overline{q}u{\tilde{\phi }}_I,\hfill \\ \hfill V({\phi }_1,{\phi }_2)& =& m_{11}^2|{\phi }_1{|}^2+m_{22}^2|{\phi }_2{|}^2-({\mu }^2{\phi }_1^{†}{\phi }_2+\mathrm{h}.\mathrm{c}.)+{\lambda }_1|{\phi }_1{|}^4+{\lambda }_2|{\phi }_2{|}^4+{\lambda }_3|{\phi }_1{|}^2{|{\phi }_2|}^2\hfill \\ & & +{\lambda }_4{|{\phi }_1^{†}{\phi }_2|}^2+\left[\left(\frac{{\lambda }_5}{2}{\phi }_1^{†}{\phi }_2+{\lambda }_6|{\phi }_1{|}^2+{\lambda }_7{|{\phi }_2|}^2\right){\phi }_1^{†}{\phi }_2+\mathrm{h}.\mathrm{c}.\right],\hfill \\ \hfill {\mathcal{L}}_6& =& \sum _i{c}_i{O}_i/f^2.\hfill \end{array}$$

(4)

${\mathcal{L}}_{\mathrm{kin}}$ is the kinetic term of the Lagrangian, ${\mathcal{L}}_{\mathrm{Yuk}}$ is the Lagrangian originating from the Yukawa interaction. The indices $\mu ,\nu =0,1,2,3$ are the indexes for the time-space components. $W_i$, $i=1,2,3$, and B are the four gauge bosons, $G^a$ ($a=1,2$) are the Goldstone bosons and $Y^e,Y^d$ and $Y^u$ are the Yukawa coupling constants. D is the covariant derivative, see Ref. [15], and $q,u,d,e,l,q,m11,m22$ are defined in Appendix B. The term ‘h.c.’ and the symbol † stay for the Hermitian conjugate. $c_i$ is the Wilson coefficient of the 6-dimensinal operator $O_i$ and f. The terms proportional to ${\lambda }_{6,7}$ are known as ‘hard-$Z_2$ violating’ because, not only do they lead to a quadratically divergent amplitude for ${\phi }_1\leftrightarrow {\phi }_2$ transition [16] but also to CP-violation in the scalar sector for complex values [17]. However, it is possible to realize the CP-conserving limit with nonzero values of ${\lambda }_{6,7}$ as well [18]. In this paper, we contain our discussion to the CP-conserving 2HDM, and hence ${\lambda }_{6,7}=0$. The electroweak symmetry is broken by the VEVs, namely $v_1$ and $v_2$ corresponding to the two doublets ${\phi }_{1,2}$ respectively. This leads to the mixing of same types of degrees of freedom of ${\phi }_{1,2}$. For the case of CP-conservation, the mass matrices of the neutral CP-even and odd scalars and the charged scalars are diagonalized by the following operation:

$$\left(\begin{array}{c}H\\ h\end{array}\right)=R\left(\alpha \right)\left(\begin{array}{c}{\rho }_1\\ {\rho }_2\end{array}\right),\left(\begin{array}{c}{W}_L^{\pm }\\ H^{\pm }\end{array}\right)=R\left(\beta \right)\left(\begin{array}{c}{\varphi }_1^{\pm }\\ {\varphi }_2^{\pm }\end{array}\right),\left(\begin{array}{c}{Z}_L\\ A\end{array}\right)=R\left(\beta \right)\left(\begin{array}{c}{\eta }_1\\ {\eta }_2\end{array}\right).$$

(5)

Here,

$$\begin{array}{c}\hfill R\left(\theta \right)=\left[\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right],\end{array}$$

(6)

$h,H$ are the neutral CP-even physical d.o.f., and A and $H^{\pm }$ are the neutral CP-odd and charged d.o.f., respectively. $Z_L$ and A are the other scalars. From Equation (5) it can be seen that $\beta$ is the mixing angle of the charged and CP-odd sectors and is given by $\beta ={tan}^{-1}(v_2/v_1)$. $\Delta$ is the mixing angle of the CP-even neutral scalars and is expressed as

$$\begin{array}{ccc}\hfill \Delta & =& {sin}^{-1}\left[\frac{{\mathcal{M}}_{\rho 12}^2}{\sqrt{{\left({\mathcal{M}}_{\rho 12}^2\right)}^2+{({\mathcal{M}}_{\rho 11}^2-m_h^2)}^2}}\right],\hfill \end{array}$$

(7)

with ${\mathcal{M}}_{\rho }^2$ is the mass-squared matrix in the neutral CP-even sector and the repeating and non-repeating indices refers to the diagonal and off diagonal elements, respectively. Please see Appendix B for detailed values of the masses. Here, we have assumed that h is the SM-like Higgs with mass of $m_h\sim 125.09$ GeV and $m_H>m_h$. It has been shown that the tree-level Higgs-mediated flavor-changing neutral currents (FCNC) appear in models where more than one scalar doublet gives mass to the same kind of SM fermions [19,20]. Such a situation can be avoided under the framework of various discrete symmetries, for example, a $Z_2$ symmetry [19,20].

Due to the rotation in the scalar sector following Equation (5), the couplings of the SM gauge bosons and fermions to the SM-like Higgs boson are required to be re-scaled compared to the corresponding SM values. After SSB, the Yukawa sector of the 2HDM can be written as:

$$\begin{array}{ccc}\hfill -{\mathcal{L}}_{\mathrm{Yuk}}& =& \frac{1}{\sqrt{2}}({\kappa }_D{s}_{\beta -\alpha }+{\rho }_D{c}_{\beta -\alpha })\overline{D}Dh+\frac{1}{\sqrt{2}}({\kappa }_D{c}_{\beta -\alpha }-{\rho }_D{s}_{\beta -\alpha })\overline{D}DH\hfill \\ & & +\frac{1}{\sqrt{2}}({\kappa }_U{s}_{\beta -\alpha }+{\rho }_U{c}_{\beta -\alpha })\overline{U}Uh+\frac{1}{\sqrt{2}}({\kappa }_U{c}_{\beta -\alpha }-{\rho }_U{s}_{\beta -\alpha })\overline{U}UH\hfill \\ & & +\frac{1}{\sqrt{2}}({\kappa }_L{s}_{\beta -\alpha }-{\rho }_L{c}_{\beta -\alpha })\overline{L}Lh+\frac{1}{\sqrt{2}}({\kappa }_L{c}_{\beta -\alpha }-{\rho }_L{s}_{\beta -\alpha })\overline{L}LH\hfill \\ & & -\frac{i}{\sqrt{2}}\overline{U}{\gamma }_5{\rho }_{U}UA+\frac{i}{\sqrt{2}}\overline{D}{\gamma }_5{\rho }_{D}DA+\frac{i}{\sqrt{2}}\overline{L}{\gamma }_5{\rho }_{L}LA\hfill \\ & & +\left(\overline{U}(V_{\mathrm{CKM}}{\rho }_D{P}_R-{\rho }_U{V}_{\mathrm{CKM}}{P}_L)D{H}^{+}+\overline{\nu }{\rho }_L{P}_{R}L{H}^{+}+\mathrm{h}.\mathrm{c}.\right).\hfill \end{array}$$

(8)

Here, ${\kappa }_f=\sqrt{2}{M}_f/v$ for $f=U,D$ and $L$. L and R are defined after Equation (11). For details of other parameters see Ref. [21]. The generation indices of the fermionic fields have been suppressed in Equation (8). As mentioned earlier, the measurement of the signal strengths of the SM-like Higgs at LHC demands that the properties of one of the neutral CP-even neutral scalars, here h, should closely resemble that of the SM Higgs. As Equation (8) indicates, this is satisfied at the vicinity of the so-called ‘alignment limit’, i.e., $cos(\beta -\alpha )\to 0$. The current measurement of Higgs signal strengths have pushed the 2HDMs close to the alignment limit [22,23,24,25]. The measurement of the Higgs signal strengths dictate that for type-II 2HDM, at $tan\beta \sim 1$, the constraint on $cos(\beta -\alpha )$ is given by $-0.05\lesssim cos(\beta -\alpha )\lesssim 0.15$ at $95\%$ CL. The allowed region becomes even smaller for higher values of $tan\beta$. The situation for type-III and -IV are quite similar to that of type-II 2HDM.

This constraint is comparably relaxed in type-I 2HDM, where the allowed range is $|cos(\beta -\alpha \left)\right|\lesssim 0.4$. Among the tree-level scalar-gauge couplings which are important for the cascade decays of the new scalars, $AZh$ and $H^{\pm }h{W}^{\mp }$ are proportional to $cos(\beta -\alpha )$, whereas $AZH$ and $H^{\pm }H{W}^{\mp }$ are proportional to $sin(\beta -\alpha )$. The tripple letters denote tripple coupling and Z is one of the SM guage bosons. It is possible to realize an exact alignment in the multi-Higgs-doublet models in the framework of certain additional symmetries of the 2HDM potential [26,27,28,29,30].

The impact of the measurement of the Higgs signal strengths in each individual search channels on the $cos(\beta -\alpha )$ vs. $tan\beta$ plane has been discussed in Ref. [31]. It should be mentioned that the coupling multipliers of the SM-like Higgs also becomes close to unity when $sin(\beta +\alpha )=1$, i.e., at the so-called ‘wrong-sign Yukawa’ limit [32] for type-II, type-III, and type-IV 2HDM. Though, with better measurement of the processes like $Vh\to b\overline{b}$, $h\to \gamma \gamma ,{\rm Y}\gamma$, (where b is the bottom quark), [33,34] the fate of the wrong-sign Yukawa region will be decided in near future.

It is possible to constrain the parameter space of the type-II 2HDM even at the alignment limit from the nonobservation of the heavier scalars [18] in processes like $gg/b\overline{b}\to H/A\to \tau \overline{\tau },\gamma \gamma$, $gg\to H\to hh$, etc. Here $g$ and $\tau$ represents gluon and tau and their respective interactions. Both ATLAS and CMS experiments are involved in numerous dedicated searches of these kinds, for instance Refs. [35,36,37,38,39], resulting in significant constraints on the 2HDM parameter space.

The constraints from the decay of the new scalars into SM particles are significantly relaxed in the hierarchical scenario compared to the degenerate case [25]. However. for the hierarchical spectrum of new scalars, the channels like $H\left(A\right)\to ZA\left(H\right)$ dominates the total decay width of such states, leading to new bounds on the parameter space which are not applicable for the degenerate case. A hierarchical spectrum such as $m_A>m_H\sim m_{H^{\pm }}\sim v$ can lead to a first order electroweak phase transition providing an explanation for the matter-antimatter asymmetry, with $A\to ZH$ being its smoking gun signature at LHC [40,41]. In general, the importance of Higgs cascade decays as the possible probes of an extended scalar sector have been discussed in the literature [42,43,44,45,46,47] and $A\to ZH$ decay is dubbed as a ‘golden channel’ in this context [48].

3. EWPT Theory in 2HDM

The Lagrangian density of the electroweak theory (discussed in details in the previous section) in 2HDM can be expressed as [15]

$$\begin{array}{c}\hfill \mathcal{L}={\mathcal{L}}_f+{\mathcal{L}}_{\mathrm{Yuk}}+{\mathcal{L}}_{\mathrm{gauge},\mathrm{kin}}+{\mathcal{L}}_{\mathrm{Higgs}}.\end{array}$$

(9)

The first term on the right hand side, ${\mathcal{L}}_f$, is the kinetic term for the fermion fields:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_f& =& \sum _{j}i\left({\overline{\Psi }}_L^{\left(j\right)}\cancel{D}{\Psi }_L^{\left(j\right)}+{\overline{\Psi }}_R^{\left(j\right)}\cancel{D}{\Psi }_R^{\left(j\right)}\right)\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& =& i{\overline{\Psi }}_L{\gamma }^{\mu }({\partial }_{\mu }+ig{W}_{\mu }+i{g}^{\prime }{Y}_L{B}_{\mu }){\Psi }_L\hfill \\ & & +i{\overline{\Psi }}_R{\gamma }^{\mu }({\partial }_{\mu }+ig{W}_{\mu }+i{g}^{\prime }{Y}_R{B}_{\mu }){\Psi }_R,\hfill \end{array}$$

(11)

where L and R represents the left and right chiral field of that fermion and $\cancel{D}$ is the Feynman operator applied on the covariant derivative (it is defined as $\cancel{A}={\gamma }^{\mu }{A}_{\mu }$, where $\gamma$ are the gamma matrices) [15] and j runs over all fermionic species (the field ${\Psi }_j$) listed in

Table A1. Field-dependent mass of all fermions.

Fermions	${\mathit{n}}_{\mathit{i}}$	${\mathit{s}}_{\mathit{i}}$	${\mathit{m}}_{\mathit{f}}(\mathit{T}=0)$
e	4	$\frac{1}{2}$	$\frac{y_e}{\sqrt{2}}{v}_k$	lepton
$\mu$	4	$\frac{1}{2}$	$\frac{y_{\mu }}{\sqrt{2}}{v}_k$	lepton
$\tau$	4	$\frac{1}{2}$	$\frac{y_{\tau }}{\sqrt{2}}{v}_k$	lepton
u	12	$\frac{1}{2}$	$\frac{y_u}{\sqrt{2}}{v}_k$	quark
c	12	$\frac{1}{2}$	$\frac{y_c}{\sqrt{2}}{v}_k$	quark
t	12	$\frac{1}{2}$	$\frac{y_t}{\sqrt{2}}{v}_k$	quark
d	12	$\frac{1}{2}$	$\frac{y_d}{\sqrt{2}}{v}_k$	quark
s	12	$\frac{1}{2}$	$\frac{y_s}{\sqrt{2}}{v}_k$	quark
b	12	$\frac{1}{2}$	$\frac{y_b}{\sqrt{2}}{v}_k$	quark
Bosons	${\mathit{n}}_{\mathit{i}}$	${\mathit{s}}_{\mathit{i}}$	$\mathit{m}{\left(\mathit{v}\right)}^2$
h	1	1	eigenvalues of (A15)	Higgs
H	1	1	eigenvalues of (A15)	Higgs
A	1	1	eigenvalues of (A15)	Higgs
$G^0$	1	1	eigenvalues of (A15)	Goldstone
$H^{\pm }$	2	1	Equation (A7)	charged Higgs
$G^{\pm }$	2	1	Equation (A8)	charged Goldstone
$Z_L$	1	1	Equation (A5)	Higgs
$Z_T$	2	2	Equation (A5)	Higgs
$W_L$	2	1	Equation (A4)	Higgs
$W_T$	4	2	Equation (A4)	Higgs
${\gamma }_L$	1	2	Equation (A6)
${\gamma }_T$	2	2	Equation (A6)

. And g is the coupling constant. The partial derivative denotes derivative over space time, e.g., ${\partial }_0=d/dt$ and $\mu$ runs from 0 to 3.

The second term of Equation (9), Yukawa interaction term (for details, see previous section), ${\mathcal{L}}_{\mathrm{Yuk}}$ is [49]

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathrm{Yuk}}=-\left[y_e\overline{e_R}{\Phi }_a^{†}{L}_L+y_e^{\ast }\overline{L_L}{\Phi }_a^{†}{e}_R+\cdots \right],\end{array}$$

(12)

where $y_e$ is a complex dimensionless constant, ${\Phi }_a$ ($a=1,2$) is a $SU{\left(2\right)}_L$ doublet and for the Lagrangian to be gauge invariant it is coupled with another $SU{\left(2\right)}_L$ fermion $L_L$. $e_R$ is the right chiral electron field and the same goes for other fermions like quarks, neutrinos, etc.

The third term ${\mathcal{L}}_{\mathrm{gauge},\mathrm{kin}}$ represents $U\left(1\right)$ invariant kinetic term of four gauge bosons ($W^i,i=1,2,3$ and B). It can be written as

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathrm{gauge},\mathrm{kin}}=-\frac{1}{4}{G}_{\mu \nu }^i{G^i}^{\mu \nu }-\frac{1}{4}{F}_{\mu \nu }^B{F^B}^{\mu \nu },\end{array}$$

(13)

where $G_{\mu \nu }^i={\partial }_{\mu }{W}_{\nu }^i-{\partial }_{\nu }{W}_{\mu }^i-g{ϵ}^{ijk}{W}_{\mu }^j{W}_{\nu }^k$ and $F_{\mu \nu }^B={\partial }_{\mu }{B}_{\nu }-{\partial }_{\nu }{B}_{\mu }$.

The Lagragian density for the doublet Higgs bosons is given by

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mathrm{Higgs}}& =& {\left(D^{\mu }{\Phi }_1\right)}^{†}\left(D_{\mu }{\Phi }_2\right)+{\left(D^{\mu }{\Phi }_1\right)}^{†}\left(D_{\mu }{\Phi }_2\right)-V_{\mathrm{tot}}({\Phi }_1,{\Phi }_2)\hfill \\ & =& {\left\{({\partial }_{\mu }+ig{T}^i{W}_{\mu }^i+i{g}^{\prime }Y{B}_{\mu }){\Phi }_1\right\}}^{†}\left\{({\partial }_{\mu }+ig{T}^i{W}_{\mu }^i+i{g}^{\prime }Y{B}_{\mu }){\Phi }_1\right\}\hfill \\ & & +{\left\{({\partial }_{\mu }+ig{T}^i{W}_{\mu }^i+i{g}^{\prime }Y{B}_{\mu }){\Phi }_2\right\}}^{†}\left\{({\partial }_{\mu }+ig{T}^i{W}_{\mu }^i+i{g}^{\prime }Y{B}_{\mu }){\Phi }_2\right\}-V_{\mathrm{tot}}({\Phi }_1,{\Phi }_2,T).\hfill \\ \hfill \end{array}$$

(14)

We define

$$\begin{array}{c}\hfill {\mathcal{W}}_{\mu }=g{T}^i{W}_{\mu }^i+g^{\prime }Y{B}_{\mu }.\end{array}$$

(15)

Thus, from Equation (14), we get:

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathrm{Higgs},\mathrm{kin}}=\left({\partial }^{\mu }{{\Phi }_a}^{†}\right)\left({\partial }_{\mu }{\Phi }_a\right)-i{\left({\mathcal{W}}^{\mu }{\Phi }_a\right)}^{†}\left({\partial }_{\mu }{\Phi }_a\right)+i\left({\partial }^{\mu }{{\Phi }_a}^{†}\right){\mathcal{W}}_{\mu }{\Phi }_a+{\left({\mathcal{W}}^{\mu }{\Phi }_a\right)}^{†}{\mathcal{W}}_{\mu }{\Phi }_a.\end{array}$$

(16)

The standard CP-conserving 2HDM potential $V_{\mathrm{tot}}({\Phi }_1,{\Phi }_2,T)$ consists of tree level potential $V_{\mathrm{tree}}({\Phi }_1,{\Phi }_2)$:

$$\begin{array}{ccc}\hfill V_{\mathrm{tree}}({\Phi }_1,{\Phi }_2)=& & m_{11}^2{\Phi }_1^{†}{\Phi }_1+m_{22}^2{\Phi }_2^{†}{\Phi }_2-\left[m_{12}^2{\Phi }_1^{†}{\Phi }_2+m_{12}^{\ast }{\Phi }_2^{†}{\Phi }_1\right]+\frac{1}{2}{\lambda }_1{\left({\Phi }_1^{†}{\Phi }_1\right)}^2\hfill \\ & & +\frac{1}{2}{\lambda }_2{\left({\Phi }_2^{†}{\Phi }_2\right)}^2+{\lambda }_3\left({\Phi }_1^{†}{\Phi }_1\right)\left({\Phi }_2^{†}{\Phi }_2\right)+{\lambda }_4\left({\Phi }_1^{†}{\Phi }_2\right)\left({\Phi }_2^{†}{\Phi }_1\right)\hfill \\ & & +\left[\frac{1}{2}{\lambda }_5{\left({\Phi }_1^{†}{\Phi }_2\right)}^2+\frac{1}{2}{\lambda }_5^{\ast }{\left({\Phi }_2^{†}{\Phi }_1\right)}^2\right]\hfill \end{array}$$

(17)

and other correction terms $V_{\mathrm{CW}}\left(v_1,v_2\right)$ and $V_T$. The correction terms are defined in Refs. [50,51]:

$$\begin{array}{c}\hfill V_{\mathrm{CW}}\left(v_1+v_2\right)=\sum _i\frac{n_i}{64{\pi }^2}{(-1)}^{2s_i}{m}_i^4\left(v_1,v_2\right)\left[log\left(\frac{m_i^2\left(v_1,v_2\right)}{{\mu }^2}\right)-c_i\right],\end{array}$$

(18)

$$\begin{array}{c}\hfill V_T=\frac{T^4}{2{\pi }^2}\left(\sum _{i=\mathrm{bosons}}{n}_i{J}_B\left[\frac{m_i^2(v_1,v_2)}{T^2}\right]+\sum _{i=\mathrm{fermions}}{n}_i{J}_F\left[\frac{m_i^2(v_1,v_2)}{T^2}\right]\right),\end{array}$$

(19)

where $\mu$ is the renormalisation scale which we take to be 246 GeV. For a detailed overview of 2HDM scenarios and other relevant aspects, see Ref. [21].

The potential-dependent mass of fermions and bosons $m_i\left(v_1+v_2\right)$ and the corresponding $n_i$, $s_i$, and $c_i$ are discussed in details in Appendix B.

$J_B$ and $J_F$ are approximated Landau gauge up to leading orders:

$$\begin{array}{ccc}\hfill T^4{J}_B\left[\frac{m^2}{T}\right]& & =-\frac{{\pi }^4{T}^4}{45}+\frac{{\pi }^2}{12}{T}^2{m}^2-\frac{\pi }{6}T{\left(m^2\right)}^{3/2}-\frac{1}{32}{m}^{4}ln\frac{m^2}{a_b{T}^2}+\cdots ,\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill T^4{J}_F\left[\frac{m^2}{T}\right]& & =\frac{7{\pi }^4{T}^4}{360}-\frac{{\pi }^2}{24}{T}^2{m}^2-\frac{1}{32}{m}^{4}ln\frac{m^2}{a_f{T}^2}+\cdots ,\hfill \end{array}$$

(21)

where $a_b=16a_f=16{\pi }^{2}exp(3/2-2{\gamma }_E)$ with ${\gamma }_E$ being the Euler–Mascheroni constant.

When the temperature of the universe drops down to the critical temperature $T_c$, a second local minimum appears with the same height of the global minimum situated at $⟨{\Phi }_1⟩=⟨{\Phi }_2⟩=0$ [52]. The critical temperature can be obtained using the following expression:

$$\begin{array}{c}\hfill V_{\mathrm{tot}}\left({\Phi }_1=0,{\Phi }_2=0,T_c\right)=V_{\mathrm{tot}}\left({\Phi }_1=v_1,{\Phi }_2=v_2,T_c\right).\end{array}$$

(22)

During EWPT, if $\rho$ and P are, respectively, the energy density and pressure of the fluid determining the course of evolution of the early universe, then one gets [9]:

$$\begin{array}{ccc}\hfill \rho & =& {\rho }_f+{\rho }_{\mathrm{gauge},\mathrm{kin}}+{\rho }_{\mathrm{Higgs}}-g^{00}{\mathcal{L}}_{\mathrm{Yuk}},\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill P& =& P_f+P_{\mathrm{gauge},\mathrm{kin}}+P_{\mathrm{Higgs}}-\frac{1}{3}{g}^{ii}{\mathcal{L}}_{\mathrm{Yuk}}.\hfill \end{array}$$

(24)

where $\rho$ and P are the energy and the pressure density of the plasma and the the first three terms represents in the above equations the contribution from the fermionic, gauge kinetic and the Higgs sectors, respectively. We have assumed that dark matter and other components might have been present but they did not contribute much to the energy density of the universe during the particular epoch of EWPT which happened in radiation domination. The expressions for ${\rho }_f,P_f$ and ${\rho }_{\mathrm{gauge},\mathrm{kin}},P_{\mathrm{gauge},\mathrm{kin}}$ appear solely from fermionic and gauge sectors and their interactions (the stress–energy tensor for the above quantities is mentioned in Appendix A):

$$\begin{array}{ccc}\hfill {\rho }_{\mathrm{H}+\mathrm{F}+\mathrm{G}}& & =\left[\{{\partial }^0{\Phi }_a^{†}-i{\left({\mathcal{W}}^0{\Phi }_a\right)}^{†}\}{\partial }^0{\Phi }_a+\{{\partial }^0{\Phi }_a+i{\mathcal{W}}^0{\Phi }_a\}{\partial }^0{\Phi }_a^{†}\right],\hfill \\ & & -g^{00}\left[\left({\partial }^{\alpha }{{\Phi }_a}^{†}\right)\left({\partial }_{\alpha }{\Phi }_a\right)-i{\left({\mathcal{W}}^{\alpha }{\Phi }_a\right)}^{†}\left({\partial }_{\alpha }{\Phi }_a\right)+i\left({\partial }^{\alpha }{{\Phi }_a}^{†}\right){\mathcal{W}}_{\alpha }{\Phi }_a+{\left({\mathcal{W}}^{\alpha }{\Phi }_a\right)}^{†}{\mathcal{W}}_{\alpha }{\Phi }_a\right]\hfill \\ & & -g^{00}\left[V_{\mathrm{tot}}({\Phi }_1,{\Phi }_2,T)\right],\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill P_{\mathrm{H}+\mathrm{F}+\mathrm{G}}& & =\left[\{{\partial }^q{\Phi }_a^{†}-i{\left({\mathcal{W}}^q{\Phi }_a\right)}^{†}\}{\partial }^q{\Phi }_a+\{{\partial }^q{\Phi }_a+i{\mathcal{W}}^q{\Phi }_a\}{\partial }^q{\Phi }_a^{†}\right]\hfill \\ & & -g^{qq}\left[\left({\partial }^{\alpha }{{\Phi }_a}^{†}\right)\left({\partial }_{\alpha }{\Phi }_a\right)-i{\left({\mathcal{W}}^{\alpha }{\Phi }_a\right)}^{†}\left({\partial }_{\alpha }{\Phi }_a\right)+i\left({\partial }^{\alpha }{{\Phi }_a}^{†}\right){\mathcal{W}}_{\alpha }{\Phi }_a+{\left({\mathcal{W}}^{\alpha }{\Phi }_a\right)}^{†}{\mathcal{W}}_{\alpha }{\Phi }_a\right]\hfill \\ & & -g^{qq}\left[V_{\mathrm{tot}}({\Phi }_1,{\Phi }_2,T)\right].\hfill \end{array}$$

(26)

The subscript H+F+G stands for the total contribution that comes from the Higgs sector, the fermionic sector and the gauge sector. The index 00 means the first element of the metric tensor the repeated indices $qq$ means the diagonal elements.

The early universe was flat, hence the metric $g_{\mu \nu }=(+,-,-,-)$. Furthermore, hence ${\rho }_{\mathrm{H}+\mathrm{F}+\mathrm{G}}+P_{\mathrm{H}+\mathrm{F}+\mathrm{G}}$ become:

$$\begin{array}{ccc}\hfill {\rho }_{\mathrm{H}+\mathrm{F}+\mathrm{G}}+P_{\mathrm{H}+\mathrm{F}+\mathrm{G}}& & ={\partial }^0{\Phi }_a{\partial }^0{\Phi }_a^{†}+2\left[{\left({\mathcal{W}}^{\alpha }{\Phi }_a\right)}^{†}{\mathcal{W}}_{\alpha }{\Phi }_a\right],\hfill \\ & & +\left[\left({\partial }^0{{\Phi }_a}^{†}\right)\left({\partial }_0{\Phi }_a\right)-i{\left({\mathcal{W}}^0{\Phi }_a\right)}^{†}\left({\partial }_0{\Phi }_a\right)+i\left({\partial }^0{{\Phi }_a}^{†}\right){\mathcal{W}}_0{\Phi }_a\right]\hfill \\ \hfill \end{array}$$

(27)

where the explicit expression for ${\rho }_{\mathrm{H}+\mathrm{F}+\mathrm{G}}$ is given as

$$\begin{array}{ccc}\hfill {\rho }_{\mathrm{H}+\mathrm{F}+\mathrm{G}}& =& {\partial }^0{\Phi }_a{\partial }^0{\Phi }_a^{†}+\left[{\left({\mathcal{W}}^{\alpha }{\Phi }_a\right)}^{†}{\mathcal{W}}_{\alpha }{\Phi }_a\right]\hfill \\ & & -\left[V_{\mathrm{tot}}({\Phi }_1,{\Phi }_2,T)+{\mathcal{L}}_{\mathrm{Yuk}}\right].\hfill \end{array}$$

(28)

The oscillations of the Higgs fields around minimum after it appeared in the course of the phase transition, are damped due to particle production by the oscillating field. The characteristic time is equal to the decay width of the Higgses and it is large in comparison with the expansion rate and the universe cooling rate. So we may assume that Higgses essentially live in the minimum, of the potential. In principle, it can be calculated numerically by the solution of the corresponding Klein–Gordon equation with damping induced by the particle production [12].

With the above assumption,

$$\begin{array}{ccc}\hfill \rho & =& {\dot{\Phi }}_{a,\min }^2+V_{\mathrm{tot}}({\Phi }_1,{\Phi }_2,T)+\frac{g_{\ast }{\pi }^2}{30}{T}^4.\hfill \end{array}$$

(29)

where $g_{\ast }$ is the effective degrees of freedom.

The last term in Equation (29) arises from the Yukawa interaction between fermions and Higgs bosons and from the energy density of the fermions, the gauge bosons and the interaction between the Higgs and gauge bosons. This is the energy density of the relativistic particles which have not gained mass till the moment of EWPT. $g_{\ast }$ is the effective degrees of freedom. We have assumed that the energy density consists of the two parts: the energy density of the fields $\varphi \left(t\right)$s sitting at the minima of the potential and the contribution from relativistic matter.

Since for relativistic species $P=(1/3)\rho$, we can write

$$\begin{array}{c}\hfill P=\frac{{\dot{\Phi }}_{a,\min }^2}{2}+\frac{1}{3}\frac{g_{\ast }{\pi }^2}{30}{T}^4.\end{array}$$

(30)

The oscillations of scalar fields around their minima are quickly damped, so we take the time derivative of the fields equivalent to the their derivative around the minima and neglect higher order terms of their time derivative ${\dot{\varphi }}^2$ and so on. Furthermore, as a result the evolution of the minima induced by the expansion of the universe is very slow.

The entropy conservation law holds when the plasma (assumed to be an ideal fluid) was in thermal equilibrium with negligible chemical potential. However, as the temperature went below $T_c$, EWPT happened and the universe went into a thermally nonequilibrium state. It is to be noted that one of the main consequences of EWPT is electroweak baryogenesis and, following Sakharov’s principle [53], the out-of-equilibrium process is a necessary condition for successful baryogenesis.

As a result of this deviation from thermal equilibrium, the entropy conservation law is no longer valid during EWPT and, hence, a rise in the entropy production can be noticed significantly during this process.

To calculate this production, it is necessary to solve the evolution equation for energy density conservation,

$$\begin{array}{c}\hfill \dot{\rho }=-3\mathcal{H}(\rho +P).\end{array}$$

(31)

From henceforward computational analysis was used for further calculations which are discussed in the next section.

4. Entropy Release in 2HDM Scenarios

In the early universe when the temperature of universe $T\gg T_c$, the universe was in thermal equilibrium and also was dominated by relativistic species. Almost all of the fermions and bosons were massless and the contribution from those who were already massive (e.g., DM) to the total energy density of the universe was insignificant. During that epoch, the chemical potential of the massless bosons was zero, and with the assumption that chemical potential of the fermions was negligible, the entropy density per comoving volume was conserved and given by

$$\begin{array}{c}\hfill s\equiv \frac{{\rho }_r+P_r}{T}{a}^3=\mathrm{const}.,\end{array}$$

(32)

where the subscript r is used to indicate relativistic components. For our scenario, generalising Equation (29),

$$\begin{array}{c}\hfill {\rho }_r+P_r\sim g_{\ast }{T}^4,\end{array}$$

(33)

$g_{🟉}$ is not constant over time; it depends on the components of the hot primordial hot soup. These two equations, Equation (32) and Equation (33), imply

$$\begin{array}{c}\hfill T\sim a^{-1}.\end{array}$$

(34)

As long as the thermal equilibrium was maintained, s remained constant. If the thermal equilibrium was not maintained at some epoch at later time, the value of s and thus $g_{🟉}\left(T\right)a^3{T}^3$, might have increased as entropy can only either increase or remain constant. The contribution to entropy is dominated by a single particle species, namely by that with the largest mass in the temperature range $m\left(T\right)<T$. So for these temperatures $g_{\ast }\left(T\right)T=const.$ and the relative entropy rise is given just by $a^3{T}^3$. Since the final temperature $T_f=m\left(T_f\right)$, below which new particle species start to dominate, does not depend upon $g_{\ast }$, the relatives entropy rise is determined by $T^3{a}^3$. In other words, we calculate below the entropy rise over background of the constant $g_{\ast }{a}^3{T}^2$.

As the universe dropped down to $T_c$ the Higgs potential gets degenerate minima. If the temperature drops to the mass of any component of the relativistic plasma, that component gained mass and became non-relativistic and decoupled from relativistic fluid. We are assuming this process was instantaneous and the universe was not in thermal equilibrium.There is change in $g_{🟉}$ of the relativistic plasma. This led to increase of s.

Suppose at $T_c$, the scale factor was $a_c$ and $g_{🟉}\equiv g_{🟉,\mathrm{tot}}=110.75$ for our 2HDM model and thus at that moment $s_c\sim g_{🟉,\mathrm{tot}}{a}_c^3{T}_c^3$. As the temperature drops down to $T\sim T_x$, the component ‘x’ would decoupled and thus $g_{🟉}$ would decrease. If $g_{🟉,+}$ and $g_{🟉,-}$ be the $g_{🟉}$-factor before and after the decoupling of the ‘x’, then change in s relative to the time of critical temperature is:

$$\begin{array}{c}\hfill \frac{\delta s}{s_c}=\frac{{\left(g_{🟉,+}{a}_x{T}_x\right)}^3-{\left(g_{🟉,\mathrm{tot}}{a}_c{T}_c\right)}^3}{{\left(g_{🟉,\mathrm{tot}}{a}_c{T}_c\right)}^3}.\end{array}$$

(35)

In what follows, the BSMPT C++ package, which deals with various properties and features related to 2HDM and baryon asymmetry is applied. The package was used to calculate the critical temperature for $T_c$ and the vacuum expectation value VEV and $V_{\mathrm{eff}}\left(T\right)$ for each benchmark points. The calculation was repeated for 4 parameter space, the first one being the benchmark points provided in the BSMPT manual. The parameters are chosen in such a way that they satisfy the limiting conditions for type-I real 2HDM. It should also be noted that we have assumed that VEV/$T_c>1$. If this limiting assumption is varied, more benchmark points would arise, which is beyond the scope of this paper.

The differential equation Equation (31) was solved numerically by interpolating the data for $V_{\mathrm{eff}}\left(T\right)$ for all the benchmark points and the entropy release was calculated for the same. For four different benchmark points, as mentioned in

Table 1. 2HDM benchmark points for entropy production.

	${\mathit{m}}_{\mathit{h}}$ [GeV]		${\mathit{m}}_{\mathit{H}}$ [GeV]	${\mathit{m}}_{{\mathit{H}}^{\pm }}={\mathit{m}}_{\mathit{A}}$ [GeV]		$tan\mathit{\beta }$
Benchmark-I	125		500	500		10
Benchmark-II	125		400	500		10
Benchmark-III	125		90	400		10
Benchmark-IV	125.09		228.17	233		6.94
	${\mathbf{\lambda }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{2}}$	${\mathbf{\lambda }}_{\mathbf{3}}$	${\mathbf{\lambda }}_{\mathbf{4}}$	${\mathbf{\lambda }}_{\mathbf{5}}$	${\mathbf{T}}_{\mathbf{c}}$[GeV]
Benchmark-I	4.13	0.29	4.15	0	0	255.5
Benchmark-II	0.25	0.25	12.65	−1.48	−1.48	173.5
Benchmark-III	0.133	0.259	5.02	−2.51	−2.51	151
Benchmark-IV	1.22	0.29	−0.51	4.07	−3.86	139.5

, the entropy release has been calculated with the assumption of $a_c{T}_c\sim 1$ and and the results are presented in Figure 1.

As it is seen from Figure 1, the amount of entropy released increases as the critical temperature for EWPT increases. For example, the entropy production for $T_c=139.5$ GeV is $\sim 41\%$, for $T_c=151$ GeV is ∼52 %, for $T_c=173.5$ GeV is ∼63 % and for $T_c=255.5$ GeV is ∼73 %. All these results are way higher than the entropy release by EWPT in the SM which is ∼13% [4].

The main reason for this excess in the production of entropy is the extra scalar bosons produced in the 2HDM which contributes the most to the process. It is to be noted that the contributions from lighter particles like electrons and neutrinos are similar to that of the SM.

5. Conclusions

In this paper, it is shown that the total release of entropy due to electroweak phase transition (EWPT) is very large even in the framework of minimal extension of the Standard Model (SM) of particle physics namely type-I two-Higgs-doublet model (2HDM) compared to the SM. It is a proven fact that unlike the SM where electroweak phase transition EWPT is of second order, in the mere extension of the SM, EWPT becomes a first order phase transition. An interesting fact is that as $g_{\ast }$, which is the effective degrees of freedom, decreases as the temperature falls down. However, as we go to a very low temperature scale, the minimum temperature takes the value of the particle mass and their contribution remains the same as that of SM.

There are two points which should be noted. First, the benchmark points are unique and they were calculated using the BSMPT package. The limiting condition of the BSMPT was set so that the vacuum expectation value (VEV) exceeds the critical temperature, $T_c$: VEV/$T_c>1$. All the benchmark points used here satisfy this condition. Second, in this paper, we have considered only the real sector of 2HDM. If other extensions of 2HDM such as the complex 2HDM is considered, there might be considerable change in the entropy production.

In addition, one needs to mention two effects even though they are beyond the scope of this paper. First, the entropy release due to the EWPT can considerably reduce the abundance of dark matter present in the universe before EWPT. Detailed calculation of this dilution factor for SM scenario is done in Ref. [4]. Second, bubble walls that were formed might collide and may produce primordial black holes and might lead to a sufficient entropy production. The bubble collisions are also the source of primordial gravitational wave background. These will be studied in the subsequent papers.