Electroweak $\!$phase $\!$transition $\!$with $\!$spontaneous $\!$$Z_2$-breaking

This work investigates a simple, representative extension of the Standard Model with a real scalar singlet and spontaneous $Z_2$ breaking, which allows for a strongly first-order phase transition, as required by electroweak baryogenesis. We perform analytical and numerical calculations that systematically include one-loop thermal effects, Coleman-Weinberg corrections, and daisy resummation, as well as evaluation of bubble nucleation. We study the rich thermal history and identify the conditions for a strongly first-order electroweak phase transition with nearly degenerate extrema at zero temperature. This requires a light scalar with mass below 50 GeV. Exotic Higgs decays, as well as Higgs coupling precision measurements at the LHC and future collider facilities, will test this model. Additional information may be obtained from future collider constraints on the Higgs self-coupling. Gravitational-wave signals could potentially be probed by future gravitational wave experiments.


Introduction
The observed electroweak phase transition (EWPhT) in nature, together with sufficient C and CP violation, provides one of the most appealing opportunity to solve the puzzle of the matter-antimatter asymmetry of the universe, namely the mechanism of Electroweak Baryogenesis (EWBG) [1][2][3][4][5][6][7]. For a successful EWBG, the EWPhT needs to be strongly first-order to create an out-of-equilibrium condition, and to assure that the baryonic asymmetry generated during the bubble nucleation is not erased by the sphaleron processes [2,8]. In the Standard Model (SM) of particle physics, however, the EWPhT is a smooth crossover [6,9], and the amount of CP violation is insufficient [2], hence precluding the possibility of baryogenesis at the electroweak scale. Extensions to the SM that enhance the EWPhT and provide new sources of CP violation offer new possibilities for EWBG and have been studied extensively in the literature, e.g. see [10,11].
Singlet extensions of the SM provide a unique opportunity to generate a strongly first-order EWPhT [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], and are the subject of exploration in this work. These extensions, however, are relatively difficult to test, in comparison with other SM particle extensions with particles charged under the SM gauge groups. On the other hand, dark sector model building, involving a hidden sector with dark matter, often invokes spontaneously broken dark gauge symmetries. The simplest scalar sector charged under the dark symmetries would be a complex scalar, which is a singlet under the SM gauge groups. The effects from the dark Higgs, which obtains a vacuum expectation value (VEV), on the EWPhT can be approximated by a singlet extension of the SM with spontaneous Z 2 breaking, after rescaling the parameters by the corresponding degrees of freedoms. Given the above picture, in this work, we consider a comparative study of a real singlet extension of the SM and its impact on the strength of the EWPhT, in the presence of spontaneous Z2 breaking, through a detailed inclusion of various thermal and zero temperature quantum corrections to the tree-level potential.
Before moving on to details of this study, it is useful to review our current understanding of the EWPhT in singlet extensions of the SM. The strictly Z 2 -preserving version of this model has been studied to great detail in Ref. [12][13][14][15][16][17][18], presented as the so-called "nightmare scenario" for its challenges in testing it at future colliders. These scenarios generally enhance the EWPhT through loop effects of the singlet via the large quartic couplings (O(few)) between the singlet and Higgs pairs. This, however, occurs in the regime where perturbative unitarity is in question, where the one-loop corrections are large, and further studies are in need. A special mechanism, where the EWPhT is enhanced by tree-level effects through a two-step phase transition, can also be realized in these scenarios [14][15][16][17][18]. However, once the requirement of a non-relativistic bubble wall motion is imposed, solutions under this category only exist in a narrow region of parameter space. For general Z 2 -explicit breaking models, the large number of free parameters often requires numerical studies which can provide benchmark point solutions [14][15][16][19][20][21][22][23][24][25]. The solutions in these scenarios often invoke additional tree-level barriers from the explicit Z 2 breaking terms.
For the well-motivated scenario we are considering, where the Z 2 is spontaneously broken, it is a priori not clear if a sufficiently strong first-order EWPhT can be in place. First, a large mixing quartic coupling between the singlet pairs and the Higgs pairs is generically disfavored by Higgs precision tests, as this term will generate a sizable singlet-Higgs mixing when the singlet acquires a non-zero VEV. A small mixing quartic, instead, precludes a possible large loop effect from the singlet, which is one of the main mechanisms to enhance the EWPhT. Second, one might expect that the spontaneous Z 2 breaking singlet VEV could add additional trilinear terms and generate the |H † H| n or higher-order operators that could modify the Higgs potential directly via these tree-level couplings. Due to the relations among couplings in the spontaneous Z 2 breaking theory, it turns out that these operators are only generated at loop-level as if the Z 2 symmetry were not broken [31]. Hence this property prevents tree-level modifications to the Higgs potential that would be sizable enough to enhance the firstorder EWPhT strength.
The above considerations imply that it is far from trivial to anticipate the behavior of the EWPhT in singlet SM extensions with spontaneously discrete symmetry breaking. Understanding the situation and the possible region of allowed parameter space for a strongly first-order EWPhT demands a detailed study, which is the purpose of this work. 1 As we shall show, we obtain a particular type of solutions that enhances the EWPhT via engineering nearly degenerate zero temperature vacua in a very predictive manner. The paper is organized as follows: In Sec. 2 we introduce the SM extension and write down the expressions for the full one-loop potential and the daisy resummation. In Sec. 3 we classify the possible thermal histories, utilizing semi-analytic solutions that guide the understanding of our results. We also show the allowed region of parameter space for a strongly first-order EWPhT, and further check the robustness of our results against a nucleation calculation. An unavoidable, distinctive feature of our study is the prediction of a light singlet-like scalar. We present the phenomenological consequences of this model studying the implications for the Higgs exotic decays, Higgs precision measurements, double Higgs production, and gravitational wave signatures in Sec. 4. Finally, we reserve Sec. 5 to conclude and Appendices A-D to show some specific details of our analysis.
2 Singlet extension of the SM with spontaneous Z 2 -breaking 2.1 Tree-level potential We start with the tree-level Higgs boson potential with an additional real singlet s: The SM Higgs doublet φ is written as where χ 1 , χ 2 , χ 3 are three Goldstone bosons, and h is the Higgs boson. The tree-level potential of h and s in the unitary gauge reads At zero temperature, there are four non-degenerate extrema, with the possibility of the scalars having zero or non-zero VEVs. Amongst these four extrema, only two of them are consistent with the Higgs doublet obtaining a non-zero VEV. In this work, we are in particular interested in the case where the singlet also acquires a VEV. The VEVs of the Higgs doublet and the real singlet in terms of the bare parameters of the potential can be written as The physical scalar masses are obtained by diagonalizing the squared mass matrix evaluated at the physical VEV, .

(2.5)
Electroweak Symmetry Breaking (EWSB) requires that the physical VEV (v EW , w EW ) is the deepest minimum of the potential. For (v EW , w EW ) to be a minimum, rendering 4λ h λ s − λ 2 m ≥ 0 a necessary condition for EWSB at tree level.
There are five bare parameters {µ 2 h , µ 2 s , λ h , λ s , λ m } in the tree-level potential. They can be traded by five physical parameters, two of which, the Higgs VEV and the Higgs mass m H , are fixed by boundary conditions v EW = 246 GeV, m H = 125 GeV. (2.7) The remaining three physical parameters are related to the singlet VEV, the singlet mass and the mixing angle of the mass eigenstates, and we defined tan β = w EW /v EW . Detailed discussion of the parametrization can be found in Appendix A.

One-loop effective potential at finite temperature
The one-loop effective potential at finite temperatures is calculated in the background of the Higgs and singlet fields. Effective masses of all degrees of freedom in the plasma dependent on the background fields are: where χ 1,2,3 are the Goldstone bosons and the particle degrees of freedom are: n W = 6, n Z = 3, n t = −12, n h = 1, n χ 1,2,3 = 1, n s = 1. (2.9) For the Higgs and singlet degrees of freedom, mass eigenvalues entering the effective potential are where ϕ 1 , ϕ 2 are the Higgs and singlet mass eigenstates with particle degrees of freedom n ϕ 1 ,ϕ 2 = 1.
In this study we work in the Landau gauge and the Goldstone modes contribute separately in addition to the massive bosons. There has been ample discussion in the literature on the issue of gauge dependence in perturbative calculations of the effective potential, both at zero and finite temperature [32][33][34][35][36][37][38][39][40][41]. In that sense, we understand that our treatment is not manifestly gauge invariant. We expect, however, that our analysis provides a realistic estimate of the EWPhT strength. 2 The temperature dependent part of the one-loop effective potential [42], referred in the following as (one-loop) thermal potential, reads where B includes all the bosonic degrees of freedom that couple directly to the Higgs boson, namely W, Z, χ i , ϕ 1 , ϕ 2 , and F stands for the top quark fermion only. The J B and J F functions for bosons and fermions can be evaluated by numerical integration or proper extrapolation. All the numerical study in this work is performed using a modified version of CosmoTransitions [43], where spline interpolation is implemented. The spline interpolation shows the best agreement with our full numerical integration results. For better analytical control, we use high-temperature expansion for analytical analyses in the next section. Appendix B provides a more detailed discussion of the thermal potential, including formalism of the J functions, numerical convergence, and the high-temperature expansion. The Coleman-Weinberg (CW) potential [44] is the temperature-independent part of the effective potential at one-loop order where B and F were defined above and c B = 3/2(5/6) for scalar (vector) bosons. The potential is calculated in the dimensional regularization and the MS renormalization scheme. Counterterms have been added to remove the UV divergences. Q is the renormalization scale that we have chosen to Q = 1000 GeV (see discussion in Appendix C). This part of the one-loop effective potential gives corrections to both the Higgs VEV and the Higgs mass at zero temperature; hence, the bare parameters deviate from their tree-level values to satisfy the boundary conditions (for more details see Appendix C).
In our numerical studies, we perform a 5-dimensional scan of the bare model parameters, selecting those consistent with the SM Higgs VEV v EW 246 GeV and the Higgs-like particle mass m ϕ i 125 GeV, with i = 1 or 2 depending on the mass hierarchy between mass eigenstates, where we allow for an uncertainty of ±2 GeV in the VEV and the mass value, respectively. Observe that adding the CW contributions is required to perform a consistent one-loop calculation, but significantly decreases the efficiency of the numerical scanning in comparison to the only one-loop thermal potential approximation, for which the number of scanning parameters is reduced to three.
Lastly, corrections from daisy resummation of ring diagrams need to be included in the full one-loop potential to ensure validity of the perturbative expansion. The leading order resummation results give thermal corrections of Π i = d i T 2 to effective masses, say where d i for different degrees of freedom in the plasma are [17] d L W ±,3 = (2.13) A truncated full dressing implementation corresponds to replacing all m 2 i (h, s) with m 2 i (h, s, T ) in the one-loop effective potential at finite temperatures [45,46].

Enhancing the Electroweak phase transitions
In this section, we analyze all possible electroweak phase transition patterns appearing in our real singlet scalar extension of the SM. The thermal history could be very rich, as depicted in Fig. 1. We highlight the cases for which a strongly first-order electroweak phase transition that is consistent with current SM EW and Higgs precision data is feasible.
Before proceeding with a more detailed analysis, we shall briefly described the possible thermal histories for the scalar potential defined in the previous section. The spontaneous Z 2 breaking singlet extension of the SM differs from the Z 2 -preserving case significantly through the allowed size of the mixing quartic coupling λ m . Large λ m certainly helps with enhancing the EWPhT by enhancing the thermal barrier term ET h 3 since the singlet is a new bosonic degree of freedom. However, in the spontaneous Z 2 breaking case, λ m is not an independent free parameter, but rather proportional to the Singlet-Higgs mixing angle sin θ, which in turn is constrained by LHC Higgs High T Figure 1: Schematic picture of the thermal histories with different phase transition patterns. Left: restoration scenarios, where the thermal history starts from the symmetric phase at (0, 0). Right: non-restoration scenarios, where the thermal history starts from the Z 2 non-restored phase at (0,w). Phases are represented by bubble areas: the high temperature phase by a black bubble, the zero temperature phase by a hatched bubble, and the intermedia phases by gray bubbles. Phase transition steps are represented by arrow lines with a color code depicting the different scenarios: scenario A, scenario A-NR, scenario B/B-NR, respectively. The dashed arrow line scenario is discussed in Appendix D.
precision data to be smaller than 0.4. Hence, in the spontaneous Z 2 breaking case, the smaller size of λ m implies that a sufficiently strong first-order EWPhT is only achievable via more subtle effects in the potential.
For scenarios of our interests, both the electroweak symmetry and the Z 2 symmetry are broken at zero temperature. At high temperatures, instead, the electroweak symmetry is preserved (high-temperature restoration of the EW symmetry), and the Z 2 symmetry can be either broken or restored. As a result, we will show how the path to the zero temperature electroweak physical vacuum can involve a one-or two-step phase transition.
In the following, we shall focus on the following four relevant scenarios: • Scenario A: Two-step phase transition (0,0)→(0,w)→(v,w) • Scenario B: One-step phase transition and their corresponding counterparts with Z 2 non-restoration (NR) at high temperatures: • Scenario A-NR: One-step phase transition (0,w)→(v,w) • Scenario B-NR: Two-step phase transition The correspondence between the restoration and non-restoration scenarios is defined by them sharing the same final path towards the electroweak physical vacuum. All the minima defined above are temperature dependent, and different VEVs are associated with different paths in the thermal history. There exist other possible scenarios, in which although the final step towards the true EW vacuum can involve a strongly first-order phase transition, it occurs when the sphalerons are already inactive. In such cases, the temperature at which the sphalerons are still active is associated with a previous step in which the EW symmetry breaking yields a false EW breaking vacuum and does not involve a sufficiently strongly firstorder phase transition. These scenarios are: For completeness, they are briefly discussed in Appendix D, however, they are not of interests to our study.

Scenario
We shall show that the electroweak phase transition can be strongly first-order if the transition occurs from a Z 2 breaking/EW preserving vacuum, (0,w), to the true EW physical vacuum with Z 2 breaking, (v, w). This behavior can develop in two different ways: the one discussed in this subsection, scenario A, that involves a two-step transition in which at high temperatures the system is in a symmetric vacuum (0, 0), and then evolves to a spontaneous Z 2 breaking/EW preserving vacuum at lower temperatures, to final transition to the true EW physical vacuum with Z 2 breaking. A different, one step phase transition path, that we call scenario A-NR, in which the system starts directly at a Z 2 breaking/EW preserving vacuum at high temperatures and then transitions to the true EW physical vacuum with Z 2 breaking, will be discussed in detail in Sec. 3.3.
First, we start considering a high-temperature expansion to show analytically the behavior. Under the high-temperature expansion, the finite temperature potential (without CW potential and daisy resummation) is given in Eq. (B.4). The complicated field-dependent term −E(h, s)T can enhance the trilinear coefficient E beyond the SM value used in Eq. (3.1) below, due to the effect of the additional quartic couplings. For simplicity, however, we shall neglect such subdominant effects in the following analytical considerations. Without such a term, the effective potential reads where relevant coefficients are given in the Appendix B.
For scenario A, the electroweak symmetry breaking proceeds through the second step from a Z 2 breaking/EW preserving vacuum, (0,w), to the true EW physical vacuum with Z 2 breaking, (v, w), at a critical temperature T c given by where both vacua coexist and are degenerate. In the Z 2 breaking/EW preserving vacuum, the singlet has a temperature dependent VEV that at T c reads while in the true EW physical vacuum with Z 2 breaking, both the Higgs and the singlet fields have non-zero temperature dependent VEVs which at T c respectively read The phase transition strength is determined by the ratio where a sufficiently strong first-order phase transition requires vc Tc 1. Accordingly, the EWPhT strength can be enhanced by having smaller singlet scalar mass m S compared to the Higgs boson mass m H . The lighter the singlet scalar and the larger the Higgssinglet mixing parameter, sin θ, the stronger the phase transition. In terms of the bare parameters, we observe that the strength of the phase transition is governed by the magnitude of the effective quartic coupling defined as Notice that the EWSB condition shown in Eq. (2.6) requiresλ h ≤ 0, which ensures vc Tc being positive definite, without constraining its absolute value.λ h 0 is the near criticality condition for EWSB, which at the same time yields maximal enhancement of the strength of the EWPhT. In the following we discuss the behavior of the potential at zero temperature, that will provide information of the potential energy difference between the true EW physical vacuum and the Z 2 breaking/EW preserving extremum at zero temperature, which in turn has information on the magnitude of the critical temperature, and hence on the strength of the EWPhT. Moreover, to better understand the EWPhT behavior, we shall further discuss the dependence of the relevant quantities at the critical temperature in terms of the model parameterλ h that governs them.
At zero temperature, the tree-level potential difference between the true vacuum, (v EW , w EW ), and the Z 2 breaking/EW preserving extremum, (0,w| T =0 ) is given by, This zero temperature potential energy difference reduces to the SM value, λ SM h , in the limit in which the singlet decouples. Eq.(3.7) depicts the proportionality between ∆V andλ h , and implies that near criticality, for whichλ h is small, ∆V is small as well. Given Eq. (3.5), we see that a small value of ∆V is naturally associated to a large value of vc Tc . When the Z 2 breaking/EW preserving extremum and the true vacuum have less potential energy difference at zero temperature, the critical temperature is lower. We consider now the specific dependence of the critical temperature on the model parameterλ h . The thermal evolution of the two zero temperature extrema is controlled by temperature dependent coefficients in the thermal potential. More specifically, we rewrite the critical temperature in Eq. (3.2) as 1 + λm λs . Numerically, the (E SM ) 2 term is negligible and we shall drop it. This corresponds to the fact that the We observe that near criticality, the critical temperature is very close to zero. Meanwhile, the Higgs VEV at the critical temperature is larger and closer to the zero temperature VEV of 246 GeV. More specifically, Notice that the E SM factor here, or else the trilinear term in the thermal potential, is required to give a non-zero value of v c . E SM is not essential to render a low critical temperature, but does ensure that the phase transition is first-order instead of secondorder.
In summary, we have determined all relevant quantities to the phase transition strength at the critical temperature in terms of the effective quartic couplingλ h , that controls our model behavior, as Within the mean field analysis considered, the effective quartic couplingλ h is bounded from above by the Higgs quartic coupling λ h , and from below at 0 by EWSB requirements. The near criticality condition, which corresponds to small values ofλ h , yields low values of the critical temperature and, therefore, a SFOPhT. Fig. 2 shows numerical results obtained with CosmoTransitions with full consideration of the one-loop thermal potential, as shown by the scattered black points. The dependence of v c , T c , and the transition strength v c /T c on the effective quartic couplingλ h , shows excellent agreement with our analytical results 3 derived within a high-temperature expansion of the one-loop thermal potential, as shown in Eq. (3.11). Fig. 2 also includes results for other scenarios that will be discussed below.
The enhancement of the phase transition strength due to the reduction of the potential depth at zero temperature has been discussed in the literature in other contexts triggered by loop effects [13,17,18,47]. However, when such a sizable reduction of the potential depth is due to loop effects, it requires sizable couplings, which in turn may break perturbativity, or it needs multiple singlets. In our scenarios, the potential depth reduction at zero temperature arises at tree level, similar to some other SM extensions [47][48][49], and relies on the spontaneous breaking of the Z 2 symmetry. These effects could be sizable even for sufficiently small coupling constants, which open a window to interesting Higgs phenomenology.
In Fig. 3, we show the numerical scan as in Fig. 2, but depicted in the c s − µ 2 s plane of model parameters, where c s ≡ 1 12 (2λ m + 3λ s ) is a parameter controlling the boundary between high temperature Z 2 restoration and non-restoration behaviors, as will be discussed in detail in Sec. 3.3. Scenario A is shown in burgundy, and regions rendering SFOPhT are shown with a burgundy darker shade. In this figure, we also show the approximated boundaries for SFOPhT, in burgundy solid lines, that are obtained from the mean field analysis withλ h ∼ 0.06, that is the value ofλ h at which v c /T c ≈ 1, as obtained from numerical estimation (see Fig. 2). The contours agree well with the dark region of SFOPhT from the numerical scanning. We shall discuss this figure in further detail when considering the other scenarios, including those with non-restoration of the Z 2 symmetry.

Scenario B: (0,0)→(v,w)
A direct one-step phase transition from a fully symmetric phase to the physical vacuum could be realized in restricted regions of parameter space, while allowing for a strong first-order EWPhT. As we shall discuss in the following, such a one-step transition requires a comparable critical temperature for the (0, 0) → (ṽ, 0) and (0, 0) → (0,w).
In Fig. 3, we show in green a scan of points for scenario B (and its non-restoration counterpart, scenario B-NR to be discussed later on), whereas regions rendering SFOPhT are shown in a darker green shade. As we observe in Fig. 3, the scenario B lies within a narrow restricted region where −µ 2 s cs ∼ 140 GeV (shown as a black line in the figure). This can be understood in the sense that −µ 2 s cs features the temperature of Z 2 breaking, while 140 GeV features the temperature of the electroweak breaking in the limit of decoupling the singlet. When these two temperatures are comparable, the Z 2 symmetry and the electroweak symmetry may break simultaneously, which is realized in scenario B through the phase transition step (0, 0) → (v, w). Observe that, given our knowledge of EWPhT in the SM, we would need the actual temperature of simultaneous Z 2 /EW breaking to be below −µ 2 s cs ∼ 140 GeV, if we expect this scenario to allow for a sufficiently strong first-order EWPhT. We shall study this in the following.
Using the high temperature expansion of the effective potential, Eq.(3.1), we can compute analytically, for scenario B (and similarly for scenario B-NR), the strength of (3.12) In the above,λ h is defined as in Eq. (3.7) and T c is the critical temperature at which the Z 2 /EW symmetric vacuum, (0, 0), is degenerate with the physical vacuum, (v, w).
Since both terms in the denominator are positive definite (without one-loop Coleman-Weinberg correction), they must be sufficiently small for the transition to be strongly first-order. Indeed, the second term in the denominator, (µ 2 , is numerically small for scenario B, and one can then approximate the v c /T c ratio by showing identical behavior, mainly controlled by the parameterλ h as in scenario A above. The numerical results shown in Fig. 2, highlight scattered points for scenario B (and scenario B-NR) in green. According to our discussion above, the quantity v c /T c (upper panel) follows closely the expected behavior as a function ofλ h , in a very good agreement with Eq. (3.13). We observed that the data are scattered more downward compared with scenario A, and this is due to the small correction from the additional second term in the denominator of Eq. (3.12).

Z 2 Non-restoration scenarios
In scenarios A and B discussed above, the phase transition, either one-step or twosteps, starts from the trivial phase (0, 0) at high temperatures. Interestingly, it is also possible to consider that the Z 2 symmetry is not restored at high temperatures.
Using the same high temperature approximation as in Eq. (3.1), when the coefficient c h is negative, h will acquire a non-zero VEV at high temperatures, which has been recently discussed in [50,51]. For c h to be negative, a relevant negative contribution to it from λ m is required (see Eq. (B.5)), and this can be in general achieved in models with multiple singlets. However, since, in our case, we only have one singlet, such large negative contributions will require a large value of λ m . Thus, the electroweak symmetry is always restored at high temperatures h hT = 0 in our one-singlet extension of the SM.
With h = 0 at high temperatures, the singlet phase reads For c s ≥ 0, with µ 2 s ≥ 0, the phase (0,w) does not exist throughout the thermal history; while with µ 2 s < 0, the finite temperature phase (0,w) can undergo Z 2 symmetry restoration into the trivial phase (0, 0) at a higher temperature The case c s ≥ 0 and µ 2 s < 0 is what drives Scenario A. Observe that for Scenario B we also consider c s ≥ 0 (see Eq. (3.14)), and because the transition is from (0, 0) to (v, w), it also requires a positive defined T Z 2 r which in turn needs to be of same order of the T EW r ≈ 140GeV. Hence scenario B also requires µ 2 s < 0 as clearly shown in Fig. 3. For c s < 0, which can be achieved with negative λ m , Eq. (3.14) shows that the Z 2 symmetry remains non-restored at very high temperatures. This allows for thermal histories that start from a (0,w) phase and can lead to extending scenarios A and B to their Z 2 non-restoration corresponding cases. For both signs of µ 2 s , depending on its magnitude and the one of c s , one obtains the one-step phase transition that leads to scenario A-NR. If, however, µ 2 s ≥ 0, the Z 2 symmetry is temporarily restored at the temperature T Z 2 r , given in Eq. (3.15), and it is broken again to a different vacuum state, (v, w), during a later phase transition at a yet lower temperature. This is the path for scenario B-NR.
In summary, these are the possibilities we are considering as a function of the model parameters controlling the singlet sector, The correspondence between the restoration and non-restoration scenarios is defined by them sharing the same final path towards the electroweak physical vacuum. Thus, the enhancement effects on the transition strength from the singlet contribution can be described in the same manner. This implies that the ratio of v c /T c for scenario A-NR is described by the same Eq. (3.5) as in the scenario A. Analogously, v c /T c for scenario B-NR is described by Eq. (3.12), that after simplification becomes Eq. (3.13) as in scenario B, and therefore the same result as for scenario A. The separation between the Z 2 restoration and non-restoration cases is clear in Fig. 3, corresponding to the positive and negative c s regions, respectively. We have already described the restrictive region of scenario B. For scenario B-NR, −µ 2 s cs is the temperature scale where Z 2 is temporarily restored from the high temperature Z 2 nonrestoration phase, provided µ 2 s > 0. For a strong electroweak phase transition to happen in the step of (0, 0) → (v, w) in scenario B-NR, this temperature needs to be below the 140 GeV scale, i.e. −µ 2 s cs < 140 GeV, otherwise after Z 2 symmetry restoration to the trivial phase, the transition to an electroweak breaking vacuum (ṽ, 0) will develop at a temperature around 140 GeV, which will imply a small perturbation to the SM situation that we already know does not produce a SFOPhT. In addition, we expect this will result in scenario B-NR transitioning from (0, 0) → (v, w) at a temperature significantly below 140 GeV, rendering a SFOPhT. In Fig. 3, this can be seen in the dark green shade points with negative c s . Also observe from Fig. 3 that there is no SFOPhT points for the Z 2 restored scenario B.
In Fig. 4, we show our numerical results for all the scenarios, projected in the λ s -λ m plane of the quartic couplings, zoomed into the small λ s region. As the Higgs quartic λ h varies within a small numerical range, the EWSB conditionλ h ≥ 0 corresponds to the outer parabolic boundary of the dark region, and the SFOPhT conditionλ h 0.06 corresponds to the inner parabolic boundary of the dark region. Different scenarios are coded by color in the same way as in Fig. 3 with dark shaded points corresponding to a SFOPhT. The points inside the rectangle are compatible with current bounds on the Higgs exotic decays, as will be discussed in Sec. 4.2.

Full one-Loop study and nucleation
In this section, we shall show the results of the numerical scanning after implementing the CW and daisy resummation corrections introduced in Sec. 2.2. All scanning results satisfy the Higgs mass and Higgs VEV boundary conditions. Other bounds will be introduced and shown in the following discussions. In Appendix C, we will expand on details of the CW implementation, including discussions on the renormalization schemes, scale dependence, and Renormalization Group Equation (RGE) running. Fig. 5 shows the parameter space rendering SFOPhT after implementation of the full one-loop effective potential, including the one-loop thermal and CW potential, and the daisy resummation, projected on the physical parameter space of the singlet mass m S and the mixing angle sin θ. Our study shows that the valid parameter region rendering SFOPhT has been reduced after including the full one-loop results and features smaller singlet mass values. Importantly, including the full one-loop effective potential with daisy resummation still allows for all types of solutions that existed in the thermal only analysis.
The CW correction to the scalar potential effectively accounts for the one-loop running of the tree-level potential parameters [40,41,[52][53][54][55]. The top quark Yukawa coupling yields the most relevant contribution in the running of the quartic couplings, with the possibility of rendering them negative at large scales. Furthermore, as we have discussed in detail in Sec. 3.1, the effective quarticλ h (v EW ), which is directly related to the phase transition strength, is required to be small to yield a SFOPhT. Hence the stronger the first-order phase transition, the smaller the effective quarticλ h (v EW ) and the most likely it is to be rendered negative at large scales, through the effects of the top Yukawa coupling in its running. This implies that after including the CW potential in the analysis, the points with stronger first-order phase transition strength in the thermal only analysis will be more likely to become unstable (acquire a negative effective quartic coupling) and will be discarded from the accepted solutions. If instead, one would implement a RGE improvement of the CW potential, this will include the effects of running of the top quark Yukawa coupling itself, diminishing its value at large scales and, hence, also its impact in rendering the effective quartic coupling unstable. As a result, the inclusion of the one-loop CW without the RGE improvement has the effect of reducing the parameter space of SFOPhT as shown in Fig. 5, beyond what would be the case with a more comprehensive analysis. In this sense the results presented in Fig. 5 are conservative. We shall postpone a full study of the RGE-improved effective one-loop scalar potential, as well as exploration of gauge dependence effects, for future work.
It is crucial to check that our results are robust against the nucleation calculation. Fig. 6 shows the nucleation calculation results including the full one-loop effective potential and the daisy resummation correction, we observe that the actual transition strength at nucleation temperatures is stronger than the strength evaluated at the critical temperatures. For computational efficiency, all the previous calculations have been done at the critical temperature that gives a good indication of the actual transition strength at the nucleation temperature. Therefore, Fig. 6 indicates that it is sufficient to require v c /T c 0.8 as criteria for a SFOPhT. Parameter space for SFOPhT in the m S -sin θ plane, after including the full potential up to one loop order ( tree-level potential, one-loop thermal potential and one-loop zero temperature CW potential) plus finite temperature daisy resummation (darker shaded points in green and black for the B-NR and A/A-NR cases, respectively). Also shown are the points with SFOPhT when only the tree-level with one-loop thermal potential is considered (gray scattered points).

Phenomenology
The analysis of the thermal history of the spontaneous Z 2 breaking singlet extension of the SM leads to a firm prediction of a light singlet-like scalar mass eigenstate. The viable parameter space can be tested through various phenomenological probes. First of all, the spontaneous Z 2 breaking will result in mixing between the singlet scalar and the doublet Higgs boson. The Higgs precision measurements and electroweak precision measurements constrain the mixing angle sin θ to be smaller than 0.4 for light singlets. 4 This constraint has been applied directly to our numerical scans. Furthermore, the precision Higgs program will improve with the full HL-LHC dataset [56,57], and even more with data from future colliders [58][59][60][61][62][63][64][65].
In this section, we discuss three leading observational aspects of the model in regions of parameter space compatible with a strongly first-order electroweak phase transitions (SFOEWPT). First of all, the 125 GeV Higgs-like boson can decay to a pair of singlet-like scalars that can be directly searched for at the HL-LHC and/or at a future collider Higgs factory. Second, the Higgs trilinear coupling is modified when compared with the SM one. Third, the strongly first-order phase transition can be potentially probed by the next generation of gravitational wave detectors.

Higgs exotic decays
Since the singlet consistent with SFOEWPT should have a mass well below half of the SM-like Higgs boson one, the Higgs boson will decay into a pair of the new singlet scalars, H → SS. The singlet-like scalar S will then decay back to SM particles, dominantly into a bb final state, if m S is greater than 10 GeV, and into other fermions and hadrons for lower singlet-like scalar masses [67]. The partial width of the SM Higgs decaying to the light singlet-like scalar S is  The current LHC Higgs exotic decay searches constrain the BR(H → SS) to be smaller than around 25% from a global fit [68][69][70][71] and 30-50% from direct searches [72,73]. This translates into a constraint on the HSS coupling Λ HSS to be smaller than about 3 GeV. Given that for a large part of the parameter space, the size of this coupling reaches values up to O(100) GeV, the Higgs exotic decay bounds provide an important constraint on this model. Fig. 7 shows the allowed values in the log 10 BR(H → SS) − m S parameter space for different calculations of the SFOEWPT, with v c /T c 0.8. The gray region includes only the tree-level and one-loop thermal contributions to the scalar potential. The full one-loop results, including the CW corrections as well as the daisy resummation, are shown as the red, blue, and green regions for different requirements on the value of the Higgs-singlet mixing angle θ. The HL-LHC Higgs precision measurements will be able to probe deviations of the Higgs boson couplings at the 5% level, and this is shown by the blue and green regions. A future Higgs precision program at a prospective Higgs factory will measure the Higgs couplings at the 0.5% level, which would limit the Higgs-singlet mixing angle cos θ to be greater than 0.995, and is shown by the green region. 5 Above the dashed lines in Fig. 7 are regions constrained by direct searches of the Higgs decaying to a singlet scalar pair: from top to bottom, the dashed lines represent the current LHC coverage, the corresponding HL-LHC coverage, and projections for a future electron-positron collider [66], respectively. As shown in Fig. 7, imposing the future Higgs precision bounds implies a strong preference towards low singlet masses, however, we expect that a more intense numerical scan targeted to specific mass regions may expand the mass values allowed. 6 The boundary is also affected by the renormalization scale choice of the CW potential (see discussion in Appendix C). We argue that the HL-LHC will be able to actively probe a significant region of the SFOEWPT parameter space in a spontaneous Z 2 breaking singlet extension of the SM and that a future Higgs factory could compellingly test this model.

Higgs pair production
The Higgs pair production process provides a unique handle in exploring the vacuum structure of the Higgs potential [31]. The HL-LHC program can probe the Higgs trilinear coupling through double Higgs boson production with an accuracy of 50% [56], whereas it could be measured at the 40% level at a low energy lepton collider [74], and at the 5-7% level at the FCC-hh [64] as well as at CLIC [58].
The Higgs pair production receives three contributions: the triangle diagram of an s-channel off-shell singlet S through a SHH vertex, the triangle diagram of an schannel off-shell H through a HHH vertex, and a top-quark box diagram with double top Yukawa insertions. The first contribution from the s-channel off-shell scalar S is additional to the other SM ones, while the SM diagrams in turn are modified by mixing 5 Note that the colored regions show allowed solutions without implying any assumptions on the density of such solutions, since this would be correlated to the density of scanned points, implying a highly prior dependent result. The same consideration is valid for Fig. 8. 6 Note that for these results on a five-dimensional parameter space, we performed scans with approximately 10 5 CPU hours. We have a total of 10 7 points, of which 10 5 are compatible with SFOEWPT, and 10 4 satisfy the current Higgs precision and exotic decay constraints. A further simplification can be made due to the fact that m S is much smaller than twice the Higgs mass. For the double Higgs production at hadron colliders such as the LHC and FCC-hh, within a good approximation, one can define an effective trilinear coupling that combines the two triangle diagrams via sin θΛ SHH + cos θΛ HHH . (4.5) The determination and measurement of the trilinear Higgs coupling uses the differential information of the process as a result of the different diagrams and the interferences between the SM di-Higgs box diagram and the effective triangle diagram. Indeed, given the smallness of the singlet mass, the double Higgs production is far off-shell and can be absorbed into the above effective Higgs trilinear redefinition, which is valid at the differential cross-section level. We show the contributing trilinear couplings, Λ HHH and Λ SHH , in the mass basis in the left panel of Fig. 8. The modified Higgs trilinear coupling Λ HHH varies broadly between 0.08 to 0.20. There is, in general, a positive correlation between Λ HHH and the singlet scalar-di-Higgs trilinear coupling Λ SHH . Such a positive correlation follows from Eq. (4.3) for a subdominant contribution of the negative sin 3 θ term in Λ HHH , which corresponds to the mixing quartic coupling contribution. The case of negative correlation, instead, follows from the dominance of the negative sin 3 θ term over the positive second term in Λ HHH . The shading and color choices are the same as in Fig. 7. We can see that as we restrict the Higgs-singlet mixing parameter sin θ to be smaller, the Higgs trilinear coupling is also reduced to be closer to the SM value (which is shown as a gray reference line). The right panel of Fig. 8 shows the departure of the effective trilinear coupling Λ Eff HHH from its SM value as a function of the mixing parameter sin θ. We have defined the ratio with Λ Ef f HHH defined in Eq. (4.5) and, again, the color code is the same as in Fig. 7. We observe that for negative values of the mixing parameter sin θ, the effective Higgs trilinear coupling can be suppressed as much as 30%, while for positive values, the suppression is at most of the order 10%. These changes in the Higgs trilinear coupling are beyond the current reach of colliders and set a compelling challenge for the di-Higgs boson search program and related precision measurements at future colliders.

Gravitational wave signature
Discussions on Gravitational Wave (GW) signatures associated with a SFOEWPT in singlet extensions of the SM have been carried out in recent studies, see, e.g., Refs. [75][76][77][78][79][80]. Here we study for the first time the potential for detectability of gravitational waves in a singlet extension of the SM with spontaneous Z 2 breaking. We provide a rough estimate of the GW signatures of the various underlying thermal histories and evaluate the opportunities to observe them at current and future GW detection experiments.
Following Ref. [81], we estimate the GW signature spectrum from our model parameter points. The phase transition process induces the GW through bubble collision, GW energy: Figure 9: The gravitational wave power spectra as a function of frequency for all of our model points with a strong first-order EWPhT. The green and orange curves correspond to scenarios A and B, respectively, including the tree-level and one-loop thermal calculation, respectively. The dark red curves correspond to the full 1-loop evaluation plus the daisy resummation. Also shown are the power-law integrated sensitivities of the LIGO, LISA and BBO projections in brown, red, and blue, respectively. dubbed as Ω φ , propagation of the sound wave, dubbed Ω sw , and the decay of magnetohydrodynamic (MHD) turbulence, dubbed Ω M HD , respectively. The stochastic GW background power spectrum is the summation of these three sources, whose relative strengths differ depending on the given model. The detailed spectral and parametric dependences of the GW signature from the different sources are given in Ref. [81,82]. In this section, we describe the key parameters and show the numerical results of our study.
The inverse duration of the phase transition is characterized by β Γ /Γ with Γ the bubble nucleation rate. In turn, the relevant parameter for the GW signal is where S 3 /T is the O(3)-symmetric bounce Euclidean action, T * denotes the temperature of the thermal bath when the GW was generated, and H * is the corresponding Hubble parameter at temperature T * . For a strongly first-order phase transition without significant reheating, T * is approximately the nucleation temperature T n . The bubble collision generated power spectrum is suppressed by two powers of the duration of the phase transition, H * /β, while the corresponding spectra generated by the sound waves and turbulences last longer and are suppressed by only one power of the duration of the phase transition. Another crucial characteristic parameter is the fraction of vacuum energy released during the transition with respect to the radiation bath. Specifically, where ρṽ ,w and ρ v,w are the zero temperature vacuum energy densities before and after the phase transition, evaluating the VEVs at the phase transition temperature T * . The radiation energy density, ρ rad , is approximately given by g * π 2 T 4 * /30, where g * is the number of relativistic degrees of freedom in the plasma at T * . Note that as mentioned in Ref. [81,82], α also approximately coincides with the latent heat of the PT in the limit of a strong PT and large supercooling.
Another important parameter is the bubble wall velocity. If the wall velocity is small, then the GW spectrum is suppressed and hence less detectable. Detailed understanding of bubble wall velocity is, however, difficult, although one generically expects the plasma and matter reflection effects to let the bubble reach a relativistic terminal velocity [82]. In this study, we assumed it to be 0.5 of the speed of light, corresponding to non-runaway bubbles in the plasma. In such a case, the energy from the scalar field is negligible, and the sound wave contribution to the GW signal dominates. Moreover, we conservatively assume that the contribution from the MHD turbulence represents about 5% of the total sound wave energy.
In Fig. 9 we show the GW spectral density associated to the strong first-order phase transition for various scenarios in comparison to the corresponding LIGO [83], LISA [84] and BBO [85] projected power-law integrated sensitivities [86,87]. Other gravitational wave observatories such as Taiji [88] and TianQin [89] have similar sensitivities to LISA but different frequency bands, and future ones can further extend the gravitational wave sensitivities [90][91][92]. The green and orange curves correspond to scenarios A and B, respectively, in the tree-level plus one-loop thermal calculation. We observe that future GW wave experiments such as LISA and BBO would have the sensitivity to these scenarios. However, the dark red curves correspond to the full one-loop with daisy resummation study, and this renders a much weaker GW signal. This is related to the loss of points with very strong first-order phase transition strength due to zero temperature one-loop radiative corrections, as has been discussed in Sec. 3.4. The RGE-improved CW potential could potentially relax this reduction of electroweak phase transition strength and hence recover in part the some strong gravitational wave signals. We postpone such a study for future work.

Summary and outlook
The electroweak phase transition provides an appealing avenue for baryogenesis. In the SM, however, the electroweak phase transition turns to be a crossover instead of a firstorder phase transition, as would be required to preserve a possibly generated baryon asymmetry in the presence of sufficient CP violation. In this paper, we study the simplest extension of the SM to render the EWPhT to be strongly first-order. The SM singlet extension has been studied to great detail in the past, both in the Z 2 preserving scenario as well as for the case of a generic potential where Z 2 is explicitly broken. In our systematic study, we consider the unique scenario of spontaneous Z 2 breaking, including one-loop thermal effects with daisy resummation and the Coleman-Weinberg potential corrections. We identify several very distinctive features of the spontaneous Z 2 breaking model: • A variety of thermal histories can be generically achieved. We classify them according to the number of steps to achieve the EWPhT. We define scenario A (Sec. 3.1) for (0, 0) → (0,w) → (v, w) (two steps) and scenario B (Sec. 3.2) for (0, 0) → (v, w) (one step). We also consider the possibility that at high temperatures there is non-restoration of the Z 2 symmetry and define scenario A-NR for (0,w) → (v, w) (one step), and scenario B-NR for (0,w) → (0, 0) → (v, w) (two steps) (Sec. 3.3). The relation between the restoration and non-restoration scenarios is defined by them sharing the same final path towards the electroweak physical vacuum; • Our study shows that scenario A, A-NR and B-NR lead to solutions with strong first-order EWPhT; • We derive simple analytical relations for such scenarios and perform detailed numerical simulations. Our study has the potential to be generalized to other scalar extensions of the SM with novel phenomenology, e.g., in the limit of EW symmetry non-restoration; • We find that in the spontaneous Z 2 breaking singlet extension of the SM, due to an upper bound on the singlet Higgs mixing quartic λ m , the enhanced EWPhT can only be achieved via a particular scenario of nearly degenerate extrema. As shown in detail in this paper, having an extremum close in vacuum energy to the global minimum at zero temperature yields a low critical temperature and a large critical EW VEV, enabling strong first-order EWPhT. Furthermore, we check our results performing a nucleation calculation and found v n /T n larger than v c /T c for these solutions, further validating our results; • The realization of a strong first-order EWPhT in this model predicts a light singlet-like scalar with a mass smaller than 50 GeV, which allows for a rich phenomenology. Special properties of the model can be tested through Higgs exotic decays and via Higgs coupling precision measurements at current and future collider facilities. The trilinear Higgs boson coupling is modified and can be enhanced or suppressed with respect to its SM value. Future constraints on the Higgs boson self-coupling could shed light on the physics of the EWPhT. In addition, the strongly first-order EWPhT transition can generate gravitational-wave signals, which are a challenging target for future gravitational wave experiments such as LISA and BBO.
The above points summarize distinctive aspects of the spontaneous Z 2 -breaking, singlet extension of the SM. The existence of a light scalar with accessible collider signatures as well as possible GW signals are common features that can also be present in more general models connecting the SM to a plausible dark sector via a Higgs portal.

A Parameterization
There are five bare parameters in the tree-level potential, whereas the remaining three parameters are the mass of the singlet-like eigenstate, the ratio of the singlet field VEV to the Higgs field VEV and the mixing between the mass eigenstates, respectively: The tree-level relations between these two sets of parameters are given by These tree-level relations provide a guidance for the understanding of the parametric dependence of the EWPhT strength, although such relations are modified after considering the CW corrections.

B Aspects of the thermal potential
The one-loop thermal potential reads where B includes all the bosonic degrees of freedom that couple directly to the Higgs boson, namely W, Z, χ i , ϕ 1 , ϕ 2 and F stands for the fermionic particles that couple to the Higgs boson, but we are only considering the top quark. The J B and J F functions for bosons and fermions are defined as and for positive values of α they have real values and are well defined. For negative α values, instead, the J B and J F functions become complex. As the effective squared masses in Eq. (B.1) can be negative for some field values, we regulate the functions by taking their real parts [93]. After taking the real part, it occurs that for large values of |α|, the J functions have an oscillatory behavior around a central value. Numerically CosmoTransitions deals with such an oscillatory behavior by assigning constant values to the functions once |α| becomes larger than a certain large cut-off value, as it occurs when the system is close to zero temperature. All numerical studies in this work are based on results obtained with a modified version of CosmoTransitions [43], that appropriately accounts for our scenarios as well as to improve on instabilities under certain conditions. Our version of the CosmoTransitions code is available at CosmoTransitions-Z2SB. For the numerical evaluation of the thermal potential with CosmoTransitions, we chose to utilize the spline interpolation. We have compared the results obtained by performing the spline interpolation with those obtained by performing the exact integration of the J functions. For all the benchmark points considered, we obtained a reduction of at most 10% in the strength of the phase transition. This difference is well within the limitations in accuracy due to the effective potential approximation we consider, namely one loop effective potential with daisy resummation. Since the calculation with the spline interpolation is much more efficient than that one with the exact integration, we used the former throughout our study.
For the analytic evaluations we expand the J B and J F functions in terms of small α, which yields the high-temperature approximation of the thermal potential. Under the high temperature expansion, the J functions read Based on the expansion (up to leading order in T), without the Coleman-Weinberg potential and daisy resummation contributions, the field-dependent part of the one-loop effective potential at finite temperature reads where m 2 ϕ 1,2 is given in Eq. (2.10).

C Aspects of the Coleman-Weinberg potential
As it is well known, it is possible to consider different schemes to evaluate the CW potential and each scheme has its own subtleties. In particular, the so called on-shell scheme is defined as the one that includes counter-terms such that the relations between the bare parameters in the Lagrangian and the physical parameters are not affected at one loop level. This scheme has the advantage that in principle it has a fixed prescription for the renormalization scales for each particle and that it is computationally more efficient, since the boundary conditions for the Higgs mass and VEV fixed at tree level remain valid at one loop level for the same set of bare parameters. The CW potential in the on-shell scheme reads (C.1) The on-shell scheme, however, has a subtlety related to the existence of massless particles at zero temperature. Indeed, the Goldstone fields χ i are massless at zero temperature and this yields a CW potential in field space that is infrared divergent and ill-defined. Proper resummations should be employed to render the on-shell scheme consistent [94].
In this study, we have chosen to work in the MS scheme, as has been introduced in Sec. 2.2, to avoid the zero mass infrared divergency. However, in the MS scheme the potential at the one loop level depends explicitly on the choice of the renormalization scale Q. In our model, the singlet scalar could acquire a TeV scale VEV at zero temperature and hence we have chosen Q = 1 TeV throughout the study. To reduce the scale-dependence, an RGE improvement should be performed for the CW potential [40,41,52,55]. We leave the implementation of the RGE improvement for future studies.
Note that, similar to the thermal potential, the CW potential is regulated by taking its real part in our study when the field-dependent squared masses become negative at some field values [93].

D Other phase transition patterns
Another type of phase transitions that could occur in the thermal history is (0,0)→(ṽ,0)→(v,w) (0,w)→(0,0)→(ṽ,0)→(v,w), where the two scenarios differ from each other by the fact that the Z 2 symmetry is restored or non-restored at high temperatures. Otherwise, they share the same final path towards the electroweak physical vacuum. In both scenarios, the electroweak symmetry is first broken through the step (0, 0) → (ṽ, 0), (D .1) where (ṽ, 0) is an intermediate phase at which the electroweak symmetry is broken while the Z 2 symmetry remains preserved. Since the singlet does not acquire a VEV, it plays no major role in perturbing the potential depth at tree-level. Therefore the phase transition strength in this step is not largely affected by the existence of the extended singlet sector. Solving the finite temperature effective potential under the high temperature approximation, given in Eq. (3.1), the strength of such a step is The transition is enhanced when m S < m H . However, the enhancement is bounded from above by constraints from Higgs precision measurements, which roughly set the mixing angle | sin θ| < ∼ 0.4. Accordingly, the transition strength is bounded as  Figure 10: Higgs VEV to temperature ratios of the high temperature phase and low temperature phase for the transition step (ṽ, 0) → (v, w) at a critical temperature T c . Results are obtained from numerical scanning with effective potential including treelevel and one-loop thermal potential. The sub figure shows phase transition strength of the previous step (0, 0) → (ṽ, 0) at a critical temperature T c , where the electroweak symmetry is first broken and the Higgs obtains its non-zero VEVṽ.
This upper bound on the transition strength is far below the requirement of SFOPhT. After including the CW potential and the daisy resummation corrections, such a step still yields small values ofṽ (Tc) Tc , provided the couplings still fulfill perturbative unitarity conditions.
From the temperature T c , at which the (ṽ, 0) is present, the thermal history proceeds to the next phase transition step, (ṽ, 0) → (v, w), at a lower temperature T c , which breaks Z 2 and may further change the value of the electroweak symmetry breaking vacuum. Such a step is either a smooth cross over, or a first-order phase transition. If it is of first-order nature, the singlet field can play a role in rendering the strength of the phase transition strongly first-order. As shown in Fig. 10, such is the case for v(T c ) T c > ∼ 1 at T c , for which the sphaleron rate inside the bubble is suppressed. However, we observe that the sphaleron rate outside the bubble is also suppressed during the bubble nucleation whenever the ratio of the high temperature phaseṽ (T c ) T c > ∼ 1. Therefore although the step (ṽ, 0) → (v, w) can evolve a strongly first-order phase transition, no net baryon asymmetry can be created during the bubble nucleation.
In summary, although this type of thermal history occupies a sizable parameter space, it is not of special interests for modeling electroweak baryogenesis. The first electroweak breaking step (0, 0) → (ṽ, 0) is weakly first-order, and any baryon asym-metry created is to be erased. For the following step (ṽ, 0) → (v, w), although the phase transition can be strongly first-order, the sphaleron process is suppressed both inside and outside the bubble through the transition, therefore, no baryon asymmetry can be sourced.