Phase transition in multi-scalar-singlet extensions of the Standard Model

We propose a generalization of the Standard Model (SM) by adding two real gaugesinglets S1, S2. The field S1 will improve the strength of the electroweak phase transition (EWPT). Imposing a Z2 symmetry on the field S2 makes this field a possible candidate for dark matter. Both singlets interact with other observable fields through Higgs boson. They are allowed to interact with each other as well. We find that by introducing two different scalar fields, the model is less vulnerable to experimental constraints. In this paper, we consider the effects of a heavy scalar(M1>MH) on the electroweak phase transition. And we present configurations that produce a strong first order EWPT.


Introduction
A solution to explain the baryon asymmetry of our universe [1] is based on violation of baryon number, C and CP violation as well as a departure from thermal equilibrium. In a viable model of electroweak baryogenesis the departure from thermal equilibrium is realized via a strong first order phase transition [2]. But the SM of particle physics can not provide a strong first order phase transition [3]. In addition the SM does not have a candidate for dark matter (DM) as well. Moreover, from the three-loop β function for the Higgs self coupling it is found that Higgs vacuum is no longer stable beyond the scale 10 10 GeV . Hence we expect some new physics to appear before this scale [4]. Therefore, some new models are required to address these issues. A popular model is to couple a singlet scalar to Higgs boson [5 − 16]. In Ref. [17] a scheme for classifying models of the electroweak phase transition has been presented. One may associate the formation of gravitational waves to a strong first-order phase transition [18 − 21]. But models with an addition of one real singlet can not address all of the shortcomings of the SM. An interesting class of models are the multi-singlet extensions of the SM models [22 − 28]. In these models there are more opportunities to satisfy the constraints imposed by experimental findings [29]. In order to study the dynamics of the electroweak phase transition EW P T one has to resort to techniques from the domain of thermal field theory [30 − 33]. An essential element is finite-temperature effective potential, which is a measure of the free energy density of the system. Generally a loop-level analysis, in conjunction with a vigourous Monte-Carlo scan of the parameter space is needed to unravel the structure of EW P T . However in this work we study the dynamics of the EW P T at the tree level. The plan of this paper is as follows: In section two we propose a new model composed of two different gauge singlet scalar fields with coupling to Higgs boson, they can have mutual interaction as well. We impose a discrete symmetry on only one of them. Hence this field will be a candidate for dark matter. The other field will provide us the a strong first order phase transition. And we study the parameter space of the model. In section three we discuss the finite-temperature potential and explain the origin of the strongly first-order phase transition at the tree level. In section four we discuss the phenomenological implication of our model. And finally in section five we present our conclusions.

The Model
We propose a new extension of the SM by addition of two real gauge singlet scalars S 1 and S 2 . The Lagrangian of the scalar sector of our model is given by where H denotes the complex Higgs doublet, H T = (χ 1 + iχ 2 , ϕ + iχ 3 )/ √ 2. As pointed out in Ref. [11], the most general (renormalizable) tree-level potential for the SM Higgs field H and the singlet S 1 depends on 8 parameters. Three more parameters are needed for the field S 2 , which has a Z 2 symmetry. And another extra parameter as we allow interaction between the gauge singlets S 1 and S 1 . Hence, our potential depends on 12 parameters and it is given by However, one can always remove the linear term κ 0 S 1 by a redefinition of the field S 1 by a constant shift [11]. Therefore, we do not include this linear term in this work. After symmetry breaking the fields χ 1 , χ 2 , χ 2 becomes the longitudinal degrees of freedom of the weak gauge bosons. The field S 1 is to improve the strength of the phase transition. And the singlet S 2 has a Z 2 symmetry. Hence this singlet is a a dark matter candidate. Therefore at zero temperature the vacuum associated with this field must also be Z 2 symmetric. Hence vacuum expectation value (vev) of this field must vanish. At T = 0 we can parameterize the scalar fields of our model by with v ≃ 246(GeV ) and the parameter x is the vev of the gauge singlet S 1 . By expanding around the minimum we obtain the squared mass matrix At zero temperature the effective potential of the scalar sector of our model is It is convenient to express the parameters of the potential in terms of the physical masses M 1 and M 2 of the singlet scalars, the mixing angle θ of the singlet S 1 and the Higgs field. From the mass matrix squared we get and by minimizing the scalar potential we obtain Hence

The parameter space of the model
From previous section we know that the parameter space of the model consists of (θ, M 2 1 , M 2 2 , κ 1 , κ 2 κ 3 , κ 4 , κ 5 and λ 2 ). In order to have a stable potential we must have [11] λ > 0, But, if all the eigenvalues of the mass matrix eq. (4) are positive then the corresponding extremum point is a local minimum. Thus, From eqs. (9,10) and in terms of the parameters of our model we obtain In order to take the effect of mixing angle we notice that, for a light singlet (M 1 ≪ M H ) and the mixing angle cos(θ) = 0.95. From eq. (8) we obtain λ ≃ 0.12 and when cos(θ) = 0.99 λ ≃ 0.10. Here the Higgs boson self coupling is suppressed. For a heavy singlet (M 1 = 250GeV ) and the mixing angle cos(θ) = 0.95 we obtain λ ≃ 0.25 and when cos(θ) = 0.99 the value of Higgs boson self coupling λ ≃ 0.19, Hence the Higgs boson self coupling is enhanced in this region. Therefore, our model predicts variations of the Higgs boson quartic coupling from that of the SM to be tested by precision measurements. The triviality bound of the two singlets model is addressed in Ref. [34] and quartic coupling remain positive up to energy scalae of about 10T eV . In addition to the above theoretical bounds, there are constraints from experiments. Here we present ranges for the parameters of our model −250GeV < κ 1 < 250GeV, −333GeV < κ 4 < 333GeV −0.25 < κ 2 < 0.25, 0.0001 < κ 3 < 0.0025, 5GeV < M 1 < 650GeV, 0.95 < cos(θ) < 1, 0 < λ < 0.3, 0 < λ 1 < 4, 0 < λ 2 < 4.

A strongly first order phase transition and experimental constraints
At high temperature the effective potential is The parameters of eq.(14) are given by Let us consider the shape of the potential. As the universe cools down at temperature above a critical temperature T c the potential has an absolute minima. At T c we have two degenerate minima. The symmetric vacuum is denoted by (0, s 1T , 0) and the true vacuum is designated by (h c , s 1c , 0), where we assumed that the field S 2 develops a vev at a temperature above T c [13]. The critical temperature is the on-set of the electroweak symmetry breaking, and a first order EW P T occurs from the symmetric vacuum to the true vacuum. A transition is considered as strong if ξ = hc Tc > 0.6 − 1.6, but this ratio for the SM is 2E λ SM = 0.23. In order to have a symmetric vacua The conditions for the existence of the broken vacua are and m 2 1 s 1c + λ 1 s 3 1c + 3κ 4 s 2 1c + κ 1 h 2 c + 2κ 2 s 1c h 2 c + 1 12 [(3λ 1 + 8κ 2 )s 1c + (4κ 1 + 3κ 4 + 4κ 5 )]T 2 c = 0.
Finally, to have a pair of degenerate vacua the following expression must holds By solving eqs. (16)(17)(18)(19) one can determine the variables s 1T , s 1c , T c and h c . The model has a rich parameter space and it is possible to generate a strong first order EW P T with a critical temperature varying from twenty GeV up to few hundred GeV .

Phenomenological implications
In order to investigate the physical implications of the model, we consider two different cases.

Gauge singlets without mutual interactions
In this case κ 5 = 0. In table 1

Strong first order EWPT
Since our numerical results are obtained by using the high temperature-expanded potential, we explore models which yields critical temperature which are higher than the mass of the top quark. Hence we expect that high temperature approximation to be quantitatively reasonable. The value of the parameter κ 3 is chosen from Refs. [15,35], in order to incorporate the most recent experimental and theoretical constraints. Due to the small value of κ 3 , the dark sector has little influence on the dynamics which leads to a strong EW P T . We also selected the value of the parameter cos(θ) from Ref. [36].
The results of table 1 shows that for lower values of M 1 , the electroweak broken vacuum at zero temperature lies much deeper than the electroweak broken vacuum at the critical temperature. In the last configuration of table 1, T c = 196. Within the range of validity of high temperature expansion, we compute the temperature evolution of the vevs of the doublet and the singlet field up to T c . We also evaluate the temperature evolution of the electroweak broken vacuum. The results are presented in table 2. The parameter β 1 = |V min (T =0)| |V min (T =T )| and for this configuration V min (T = 0) = −5.568654133 × 10 9 (GeV ).

Dark matter considerations
As far as the issue of dark matter is concerned we notice that, the bounds of the parameter κ 3 expressed in eq. (11) are from the stability of the potential and from the positivity of the mass squared matrix. At present a region of interest for the mass of dark matter is the region 56.8(GeV ) < M 2 < 63(GeV ) [35]. We see that, a severe bound of this parameter in this region can be obtained from the decay width of H → S 2 S 2 . In Ref. [13] it is found that With M 2 = 60(GeV ) we get κ 3 ≤ 0.0096. Yet a more restrictive bound on this parameter (see Ref. [35]) based on the relic density of dark matter, which comes from experiments is 0.0000625 < κ 3 < 0.00125. While in Refs. [13,15] these results, which prevent the occurrence of a strong first order (EW P T ) is catastrophic, in our model the small values of κ 3 are acceptable. In fact from eq.(15) we see that smaller values of κ 3 are preferred as in this case the scalar S 2 will have a minor role in the occurrence of a strong first order EW P T . Moreover, we see that in this region the variation of the parameter λ due to singlet scalar S 1 does not have influence on the bounds for κ 3 . Hence as far as dark matter is concerned, we find that the situation is very similar to that of Ref. [15].

Gauge singlets with mutual interactions
In this case the gauge singlets are allowed to interact with each other directly. By considering the annihilation of DM into SM particles one must include the effect of a s channel reaction mediated by the gauge singlet S 1 , which has an amplitude proportional to The parameter κ 5 must chosen in a way that the extra s channel reaction does not alter the dark matter cross section significantly, hence we defer an investigation of this matter to a future work.

Conclusions
There have been some attempts to address the issues of dark matter and occurrence of a strong first order electroweak phase transition in a single unified model [13]. However, we find that it is difficult to solve both problems with a single scalar. Hence, in this paper we have presented a new two singlet scalar model. In our model, when the dark matter field S 2 is only coupled to Higgs boson, we find that the phenomenology of the dark sector of the model is similar to that of a singlet scalar dark matter. We have explored the characteristics of a strong first order EW P T in the mass range 150GeV < M 1 < 550GeV .  Table 1 : The values of the critical temperature and the strength of a strong first order EW P T for several configurations. The values of κ 3 and cos(θ)are within the current experimental bounds. Table 2 : Temperature evolution of the vevs of the doublet (h) and the singlet field (s 1 ) up to T c , for the case of M 1 = 550(GeV ). The parameter β 1 is a measure of the location of the electroweak broken vacuum at temperature T .