Strongly first-order electroweak phase transition by relatively heavy additional Higgs bosons

We discuss first-order electroweak phase transition in models with extended Higgs sectors for the case with relatively heavy additional scalar bosons. We first show that, by the combination of the sphaleron decoupling condition, perturbative unitarity and vacuum stability, mass upper bounds on additional scalar bosons can be obtained at the TeV scale even at the alignment limit where the lightest Higgs boson behaves exactly like the SM Higgs boson at tree level. We then discuss phenomenological impacts of the case with the additional scalar bosons with the mass near 1 TeV. Even when they are too heavy to be directly detected at current and future experiments at hadron colliders, the large deviation in the triple Higgs boson coupling can be a signature for first-order phase transition due to quantum effects of such heavy additional Higgs bosons. On the other hand, gravitational waves from the first-order phase transition are found to be weaker in this case as compared to that with lower masses of additional scalar bosons.


I. INTRODUCTION
Although the standard model (SM) has been successful being consistent with current data at LHC [1,2], there are phenomena that cannot be explained in the SM such as neutrino oscillation, dark matter and baryon asymmetry of the Universe (BAU). Therefore, new physics beyond the SM is absolutely necessary.
In order to explain the BAU, the idea of baryogenesis is the most promising. A new model is required to satisfy the Sakharov's conditions to realize baryogenesis [3]. It has turned out that the SM cannot satisfy these conditions [4][5][6][7]. Its extension has to be considered for successful baryogenesis. In particular, for the scenario of electroweak baryogenesis [8], extended Higgs sectors are often introduced to satisfy the Sakharov's conditions having the sufficient amount of CP violation and realizing strongly first-order electroweak phase transition.
The large quantum effect is often called the non-decoupling effect. New effective field theories describing the non-decoupling effects have recently been proposed [60][61][62][63]. Such large deviations in the triple Higgs boson coupling due to the quantum non-decoupling effects are often predicted in models of electroweak baryogenesis to satisfy the sphaleron decoupling condition (1) [13,64]. Namely, the strongly first-order electroweak phase transition can be tested by detecting a large deviation in the triple Higgs boson coupling from the SM prediction at future hadron colliders and lepton colliders such as the High Luminosity-LHC (HL-LHC) [65,66], Future Circular Collider (FCC-hh) [67], International Linear Collider (ILC) [68] and Compact LInear Collider (CLIC) [69]. For example, it has been shown that the electroweak baryogenesis can be realized in the framework of a CP-violating THDM without the constraints from the electric dipole moments [70,71]. In this model, the triple Higgs boson coupling should deviate from the SM prediction about 33-55 %.
It has also been known that gravitational waves (GWs) from the first-order phase transition can be used to explore such a scenario of electroweak baryogenesis [72] 1 . The spectrum takes a special shape with a peak around 10 −3 to 10 −1 Hz. Such GWs are expected to be observed at the LISA [73], DECIGO [74], BBO [75], TianQin [76] and Taiji [77]. From detailed measurements of the GWs, not only the nature of electroweak phase transition [72] but also the structure of the extended Higgs sector may be able to be determined [27,28,[78][79][80].
It has been known that from the unitarity argument [81] there are upper bounds on the masses of the additional Higgs bosons if the coupling constants of the lightest SM-like Higgs boson h deviate from the SM ones [34,82,83]. In the alignment limit where the lightest Higgs boson behaves exactly like the SM Higgs boson at the tree level, on the contrary, no such upper bound is obtained, and the masses of additional Higgs bosons can be very large.
In this letter, the first-order electroweak phase transition is discussed in models with extended Higgs sectors for the case with relatively heavy additional Higgs bosons. We here employ the more exact expression of the sphaleron decoupling condition. We then examine the parameter space of the THDM where the sphaleron decoupling condition is satisfied with perturbative unitarity [84][85][86] and vacuum stability [87]. A similar analyses are also performed in the model with N singlet scalar fields possessing a O(N ) global symmetry (Nscalar singlet model) [27] and the IDM [88]. We find that mass upper bounds on additional Higgs bosons are obtained to be at the TeV scale even in the alignment limit.
We also discuss phenomenological impacts of the case with the additional Higgs bosons with the mass near 1 TeV in these extended Higgs models. We find that even though they are too heavy to be directly detected at current and future experiments at hadron colliders, 1 It has been discussed that the observation of primordial black holes may be important as a new tool to verify the first-order electroweak phase transition via cosmological observations [127].
the large deviation in the triple Higgs boson coupling can be a signature for the first-order electroweak phase transition due to quantum effects of such heavy additional Higgs bosons, while GWs from the first-order phase transition is weaker in this case as compared to the case with light additional Higgs bosons.
The structure of this paper is as follows. In Section 2, we give a review of the THDM. In Section 3, we discuss the condition of the strongly first-order electroweak phase transition.
Then, in Section 4, we discuss the constraint on the THDM by utilizing the condition defined in Section 3. In Section 5, we consider three benchmark points to show phenomenological differences between the models with light and heavy additional Higgs bosons. We also discuss the triple Higgs boson coupling with relatively heavy additional Higgs bosons in the THDM, and show the GW spectra in each benchmark point. In Section 6, we show a similar discussion on the N -scalar singlet model and the IDM. Discussions and conclusions are given in Section 7.

II. THE TWO HIGGS DOUBLET MODEL
We here define the THDM, by using which we explain details of our analysis for the phase transition. For the results in the other models such as the N -scalar singlet model and the IDM, we only summarize them in Sec. VI.
We consider the CP-conserving THDM with a softly-broken Z 2 symmetry Φ 1 → Φ 1 , The symmetry plays a role to avoid flavor changing neutral currents at the tree level [89]. The Higgs potential in the model is given by Although m 2 12 and λ 5 are complex in general, we here assume that these are real for simplicity. The doublets Φ i (i = 1, 2) are parameterized as where tan β = v 2 /v 1 , v 1 = v cos β, v 2 = v sin β and v = v 2 1 + v 2 2 ( 246GeV). We reduce The THDM is classified by the Z 2 charge assignment for the quarks and charged leptons as shown in Tab. I. We especially focus on the Type-I and Type-II THDM in this paper.
(2) are constrained by perturbative unitarity. Unless the Higgs field h behaves like the SM Higgs boson, upper bounds on the masses of the additional Higgs bosons are obtained by perturbative unitarity [34,45,82,83]. On the contrary, if h is SM-like, no upper bound on the masses of the additional Higgs bosons is obtained. As we discuss later, the upper bound can be obtained even in such cases by imposing the sphaleron decoupling condition in addition to the unitarity bound. Another theoretical constraint comes from vacuum stability, which is expressed by [90,91] The direct searches at collider experiments also set the bound on the masses of the additional Higgs bosons. By the LEP experiments [92], the THDM with m H ± < 78 GeV 2 We utilize the definition described in Ref. [29].
is ruled out. The additional Higgs bosons are also explored by the LHC experiments. The lower bounds on the masses of the additional Higgs bosons are determined via the A → τ τ and A → tt processes [41]. For instance, in the Type-I THDM with tan β = 1, the mass regions m Φ < 600 GeV are excluded where Φ = H, A, H ± . In the Type-II THDM with tan β < 2 (tan β > 10), the mass regions m Φ < 350 GeV (m Φ < 400 GeV) are excluded.
The masses of the charged Higgs bosons are strongly constrained by flavor experiments [93]. In the Type-I THDM with tan β < 1.5, m H ± < 300 GeV is excluded via the measurement of the B s → µµ process. For the Type-II THDM, m H ± < 590 GeV is excluded independently of tan β via the measurement of the B → X s γ process.
The measurement of the Higgs boson couplings at the LHC is also important. For the Another parameter that is important when discussing constraints on the Higgs sector is the oblique parameters S, T and U [94]. The experimental constraints on these parameters are given by [93] S = 0.04 ± 0.11, T = 0.09 ± 0.14, U = −0.02 ± 0.11.
On the other hand, the two-point function of W and Z bosons in the THDM are calculated in Refs. [95][96][97]. The theoretical calculations and the measurements of the rho parameter indicate that the following condition should be satisfied approximately This condition is satisfied when the Higgs potential possesses a custodial symmetry [98][99][100].

III. CONDITION OF STRONGLY FIRST-ORDER PHASE TRANSITION
In this section, we discuss the sphaleron decoupling condition. In order to formulate the condition, we should consider the effective potential at finite temperatures. We follow the definition for the effective potential in the THDM in the Parwani scheme [101] discussed in Ref. [16]. We also utilize the definition of the nucleation temperature T n described in Ref. [72]. We use the public code CosmoTransitions to obtain T n for our numerical evaluation [102].
The key of electroweak baryogenesis is a sphaleron transition process. This process violates the baryon number via the chiral anomaly [103]. To generate the observed baryon asymmetry via the mechanism of electroweak baryogenesis, the sphaleron process must decouple in the broken phase. The transition rate of the sphaleron process is related to the energy of sphalerons at finite temperatures. In order to discuss the feasibility of electroweak baryogenesis, we should evaluate the sphaleron energy in extended Higgs models.
The sphaleron is a non-perturbative solution in field equations of the SU (2) gauge theory [103][104][105]. The sphaleron in extended Higgs models has been calculated [12,15,20,21,23,[106][107][108][109][110]. We propose a new ansatz for the configuration of the sphaleron, which is an extension of the ansatz proposed by Spannowsky and Tamarit [107] to the finite temperature where ξ = gv(T ) r/2, ξ = | ξ| and v(T ) = v 1 (T ) 2 + v 2 (T ) 2 . The profile functions R, S 1 and S 2 satisfy the following boundary conditions We take θ(ξ) = π and φ i (ξ) = π/2 (i = 1, 2) as taken in Ref. [107]. The sphaleron energy at finite temperatures E sph (T ) is given by where V eff is the effective potential at finite temperatures. The profile functions R, S 1 and S 2 are determined to realize the saddle point of the energy functional and satisfy the boundary conditions described in Eq. (12).
The condition for the suppression of the baryon number violating process in the broken phase is given by where N B is the baryon number, and H Hubble (T ) is the Hubble parameter at T . The prefactor A(T ) is a fluctuation determinant defined around the sphaleron configuration [111]. Using We take the above condition as the criterion for the strongly first-order phase transition. In the following, we discuss the constraint on the several extended Higgs models by utilizing the condition.
We comment on the thermal correction to the effective potential by new particles. For the THDM with the alignment, field dependent masses of additional Higgs bosons are given . φ is the order parameter, and λ Φ is the linear combination of λ i (i = 1, · · · , 5) in Eq. (2). On the other hand, the thermal correction to the effective potential has the Boltzmann suppression factor exp [−m 2 Φ (φ)/T 2 ] [16]. In the decoupling region (M 2 λ Φ v 2 ), the Boltzmann suppression is significant in the thermal correction. On the contrary, in the non-decoupling region (M 2 λ Φ v 2 ), the Boltzmann suppression factor is O(1) at φ = 0.

IV. BOUNDS ON MASSES OF THE ADDITIONAL HIGGS BOSONS
In this section, we discuss the constraint on the THDM by using the sphaleron decoupling condition (16) in the following several cases. because M is large. In such a case, the strongly first-order phase transition can be realized mainly by the radiative correction to the effective potential at the zero temperature [17]. The blue region is excluded by the condition for the completion of electroweak phase transition. 3 In the gray region, the unitarity bound is not satisfied. The green region indicates that the phase transition is two step where the first phase transition is a second-order phase In the alignment limit, there is no upper bound on the masses of the additional Higgs bosons without imposing the sphaleron decoupling condition [34,45,82,83]. Even in such a case, the upper bound is obtained by combining the sphaleron decoupling condition with the unitarity bound. only by the argument of perturbative unitarity. However, we have confirmed that the bound is weaker than the bound obtained by the combination of the sphaleron decoupling condition and perturbative unitarity in Scenario 2.
In Fig. 3  Before closing this section, we give two comments on our analysis. The strongly first-order electroweak phase transition in the THDM with the heavy additional Higgs bosons requires relatively large λ i (i = 1, · · · , 5). In such a case, the sub-leading terms neglected in our thermal mass calculation may be non-negligible due to additional contributions from super daisy diagrams [112]. Since there is no established method to systematically incorporate these effects in the THDM, we have only taken into account thermal masses, as often done so in the literature 4 .
We  Fig. 1.  As we mentioned in the introduction, the triple Higgs boson coupling is the key to verify the first-order electroweak phase transition. The triple Higgs boson coupling is defined by using the effective potential V eff as We define the deviation in the triple Higgs boson coupling from the SM prediction as where λ SM hhh is the value in the SM. ∆λ hhh /λ SM hhh can be significant even in extended Higgs models with heavy additional Higgs bosons due to their quantum effects [29,39,40]. We have confirmed that a large ∆λ hhh /λ SM hhh is required to satisfy the sphaleron decoupling condition in the heavy scenario such as BM2. In order to satisfy the sphaleron decoupling condition in the THDM with m Φ > 700 GeV (Φ = H, A, H ± ), ∆λ hhh /λ SM hhh > 60% is required at the one-loop level. The results indicate that the THDM with relatively heavy additional Higgs bosons can be tested by the measurement of the triple Higgs boson coupling at future collider experiments.
Two-loop corrections to the triple Higgs boson coupling in the THDM have been calculated in Refs. [39,40]. Including the scalar two-loop corrections, the deviation in the triple Higgs boson coupling is larger than the one-loop result. We have evaluated the constraint on the triple Higgs boson coupling from the sphaleron decoupling condition including the two-loop corrections. Then, we have obtained that ∆λ hhh /λ SM hhh > 80% is required to satisfy the sphaleron decoupling condition in the THDM with m Φ > 700 GeV (Φ = H, A, H ± ) at the two-loop level. The lower bound on the triple Higgs boson coupling is larger by including the two-loop corrections.
On the other hand, we do not take into account the two-loop corrections to the strength of the phase transition. According to Refs. [113,114], the strength of the phase transition is weakened by about 10% due to the two-loop corrections in the IDM. Since the twoloop corrections to the effective potential at finite temperatures in THDM have not been calculated completely, we only consider the effective potential with the one-loop corrections h 2 GW T i a n Q i n T a i j i L I S A and daisy resummation.
We also discuss the GWs from the first-order electroweak phase transition. The spectrum of the GWs from the strongly first-order electroweak phase transition is characterized by α GW andβ GW . These parameters are defined by [72] where is the bounce solutions for the vacuum bubbles. S 3 (T ) is the free energy of the vacuum bubbles.
We focus on the three benchmarks in Tab. II. In Fig. 5 the GW spectra in each benchmark scenario are shown for the different wall velocity (v b ). The sensitivity curves at each future GW observation are also shown. The solid (dashed) lines correspond to the cases that the wall velocity is 95% (40%) of the light speed 6 . Interestingly, although magnitudes of the deviation in the triple Higgs boson coupling at the one-loop level are similar between BM1 and BM2, the peak height of the GW spectrum is lower when the additional Higgs bosons are heavy. If both large ∆λ hhh /λ SM hhh and the peaked GW spectrum are determined at future experiments, the additional Higgs bosons are expected to be relatively light. On the other hand, if large ∆λ hhh /λ SM hhh is found but no GW spectrum is observed, the scenario with relatively heavy additional Higgs bosons may be plausible.
We note that a detailed analysis for the detectability of the GWs is required in order to determine the mass scale of the additional Higgs bosons. We may be able to guess the mass scale of additional Higgs bosons by using the correlation between the GW spectrum and the triple Higgs boson coupling. The analysis for the detectability of the GW spectrum is beyond the scope of this paper. Although the GW spectrum in BM1 is lower than the sensitivity curves of the LISA, Taiji and DECIGO, we may be able to detect the signal by investigating the sensitivity of these interferometers in details [79]. For simplicity, we consider the model with N additional singlet real scalar fields S i which have a global O(N ) symmetry [25], where ( S) T = (S 1 , ..., S N ) is a vector under the O(N ) symmetry. We also assume µ 2 S > 0. 6 There are previous studies that have clarified a relation between the wall velocity and the Higgs potential by using quasiclassical calculation methods [132]. In this paper, however, the wall velocity is treated as a free parameter.
In order to obtain upper bounds on the masses of the additional Higgs bosons, we utilize the bound from perturbative unitarity [116] and the sphaleron decoupling condition given in Eq. (16). In this model, we obtain the upper bounds on the masses of the additional singlet fields as 2 TeV (1.4 TeV) when N = 1 (N = 4). As N is larger, this upper bound is more stringent.
Next, we show the results in the IDM. The Higgs potential is given by We note that we have obtained the above lower bound by using the sphaleron decoupling condition given in Eq. (16). It means that our result is the improvement of the previous work [118][119][120].

VII. DISCUSSIONS AND CONCLUSIONS
We give comments on several issues. We have treated the CP-conserving THDM with softly-broken Z 2 symmetry. As confirmed in Refs. [14,121], due to the inclusion of non-zero CP-violating phases, strength of the first-order phase transition tends to be weakened. In this case, the constraints on the THDM from the sphaleron decoupling condition might be more stringent than our results.
We have analyzed the constraint on the extended Higgs models by using the sphaleron decoupling condition, the completion condition of electroweak phase transition, perturbative unitarity and vacuum stability. In addition to these theoretical constraints, if we include the bound from the triviality [122], the allowed parameter region can be narrowed down in general [90,91,123,124]. Thus, we expect that the upper bounds on the additional Higgs boson masses are lower. However, the mass upper bounds determined by the triviality include a cutoff scale dependence. Therefore, we have not taken into account the triviality as a theoretical constraint.
We have utilized perturbative unitarity at the tree level to discuss the constraints on the extended Higgs models. When we consider the unitarity bound at the one-loop level, the extended Higgs models might be more strongly constrained [125]. But, the unitarity bounds at the one-loop level are inherently energy dependent. In our paper, to obtain the conservative mass upper bounds on the additional Higgs bosons, we have only considered the constraint from perturbative unitarity at the tree level.
We mention the relation between our results and the predictions in the effective field theories. In the Standard Model Effective Field Theory (SMEFT) with a dimension-six operator |Φ| 6 /Λ 2 where Λ is the cutoff scale, the sphaleron decoupling condition requires Λ < 750GeV as shown in Refs. [64,126]. On the other hand, we have shown that the strongly first-order electroweak phase transitions are possible in the renormalizable extended Higgs models such as the THDM even in the masses of the additional Higgs bosons are above 750 GeV. It indicates that the strongly first-order electroweak phase transition cannot be comprehensively explored by the SMEFT framework. Instead, the non-linear form of the effective field theory (Higgs EFT) would well describe the strongly first-order phase transition [60][61][62][63].
In this paper, in addition to the unitarity bound, we have evaluated the constraint on the extended Higgs models by using the sphaleron decoupling condition given in Eq. (16).
In the THDM, we have obtained the new result that the upper bounds on the masses of additional Higgs bosons exist around 1.6-2 TeV even when h is SM-like. This indicates that even if the THDM with relatively heavy Higgs bosons whose masses are TeV scale, the strongly first-order electroweak phase transition can be realized. Since light additional Higgs bosons will soon be strongly constrained by future flavor and collider experiments, it might be important to clarify the possibility of the strongly first-order phase transition due to the quantum effects of heavy additional Higgs bosons.
We have found that in order to realize the strongly first-order phase transition in the THDM with m Φ > 700 GeV (Φ = H, A, H ± ), the triple Higgs boson coupling must deviate from the SM prediction at least 80% at the two-loop level. This result is important to verify such scenarios at near future collider experiments such as the HL-LHC and the ILC where the deviation in the triple Higgs boson coupling can be measured.
We have also confirmed that the peak height of the GW spectrum is lower as the masses of the additional Higgs bosons are larger even when the deviation in the triple Higgs boson coupling is similar. If the large deviation in the triple Higgs boson coupling and the peaked GW spectrum are found, we can expect that the additional Higgs bosons are relatively light.
On the contrary, if the large deviation is found in the triple Higgs boson coupling but no GW spectrum is observed, it would be plausible that the additional Higgs bosons are relatively heavy. We may be able to guess the scale of masses of the additional Higgs bosons even if these additional fields are not discovered by direct searches at future collider experiments.