The Higgs Vacuum Uplifted: Revisiting the Electroweak Phase Transition with a Second Higgs Doublet

The existence of a second Higgs doublet in Nature could lead to a cosmological first order electroweak phase transition and explain the origin of the matter-antimatter asymmetry in the Universe. We explore the parameter space of such a two-Higgs-doublet-model and show that a first order electroweak phase transition strongly correlates with a significant uplifting of the Higgs vacuum w.r.t. its Standard Model value. We then obtain the spectrum and properties of the new scalars $H_0$, $A_0$ and $H^{\pm}$ that signal such a phase transition, showing that the decay $A_0 \rightarrow H_0 Z$ at the LHC and a sizable deviation in the Higgs self-coupling $\lambda_{hhh}$ from its SM value are sensitive indicators of a strongly first order electroweak phase transition in the 2HDM.


Introduction
In a cold, nearly empty Universe, spontaneous breaking of the electroweak (EW) symmetry takes place because the Higgs potential energy is minimized when the Higgs field(s) acquire non-vanishing vacuum expectation values (VEVs). But in the early Universe, when the scalar fields are surrounded by a thermal plasma of particles, the net free-energy of the entire system has further contributions stemming from interactions with this thermal bath, which yield a restoration of the EW symmetry for temperatures T 100 GeV. Tracing the thermal history of the Higgs field from the high temperature regime down to the T = 0 vacuum of today reveals the properties of the Electroweak Phase Transition (EWPT), the process of EW symmetry breaking in the early Universe.
The detailed dynamics of the EWPT is a crucial ingredient for a number of cosmological observables. One example is the baryon asymmetry of the Universe (BAU), which could be dynamically generated during a first order EWPT as long as the nucleation and expansion of vacuum bubbles provide a strong enough departure from thermal equilibrium as required by the Sakharov conditions [1]. For the minimal Higgs sector of the SM, a first order transition would only be achieved for a Higgs mass m h lower than the mass of the W boson, m h m W [2], and thus does not occur in the SM [3]. The BAU therefore constitutes concrete evidence of physics beyond the SM which can be connected to the EWPT and the precise nature of the Higgs sector. In addition, a first order EWPT would generate a stochastic background of gravitational waves, potentially observable with the upcoming space-based gravitational wave interferometer LISA (see [4] for a review). Since the properties of the EWPT are highly sensitive to the presence of new degrees of freedom at the EW scale coupling to the Higgs field, its study provides a tantalising research topic at the interface of particle physics and cosmology, shedding light on the ultimate structure of the sector responsible for EW symmetry breaking in Nature. This is a key goal of the physics programme of the LHC and future colliders.
To fully determine the nature of the EWPT one typically has to inspect the shape and evolution of the Higgs thermal effective potential with temperature, which faces various theoretical issues (see e.g. [5][6][7]). Furthermore, determining the phase transition strength is usually a computationally expensive algorithm. On the other hand, it has been recently pointed out that, in theories where a modified scalar sector acts as the main source of a strong phase transition, the EWPT strength is closely correlated with the zero temperature vacuum energy difference of the theory [8,9]. The amount by which the EW broken vacuum is "uplifted" with respect to the SM case constitutes a good indicator of the increase in the strength of the EWPT.
In this work we will investigate this correlation in the context of two-Higgs-doublet models (2HDMs) (see [10] for a review). Despite the minimality of the model, the existence of additional scalars can induce a strongly first order phase transition [11][12][13][14], as well as introduce new sources of Charge-Parity (CP) violation to enable the successful generation of the BAU via EW baryogenesis in some regions of its parameter space [15][16][17]. Ultimately, lattice calculations will provide a detailed map of the 2HDM parameter region in which a strong first order EWPT occurs, but perturbative calculations can already point to the main features of such a map. We show that the correlation between the EWPT strength and the zero temperature vacuum energy uplifting is a powerful analytic tool to explore the interplay between experimental/theoretical constraints and the strength of the EWPT in 2HDM scenarios.
Our analysis indicates that this interplay results in a strong EWPT favouring a hierarchical 2HDM scalar spectrum, with a preference for a heavy charged and pseudoscalar as compared to the neutral scalars (which includes the 125 GeV Higgs boson). This leads to a "smoking-gun" signature at the LHC [14] (see also [18,19]). We also show a significant deviation of the Higgs self-coupling from its SM value to be a collateral prediction of 2HDM scenarios with a strong EWPT [20,21]. Accessing the Higgs self-coupling is a key goal of the LHC and future colliders (see e.g. [22][23][24][25] for recent analyses), as it provides a direct probe of the nature of EW symmetry breaking. In the High-Luminosity LHC the sensitivity of such measurement is expected to be ∼ 50 % [26,27]. We will show that this could be enough to probe some scenarios with a strong EWPT in 2HDMs.
In section 2 we provide a review of the 2HDM and establish our notation as well as the relevant theoretical constraints on the model parameters. Section 3 elaborates on the computation of the vacuum energy difference in the 2HDM. Section 4 presents the numerical scan of the 2HDM parameter space, establishing the correlation between the vacuum energy difference and the strength of the EWPT, well as highlighting a number of key features of 2HDMs that exhibit strongly first order EWPTs. We move to a more analytical treatment in section 5, using the vacuum energy difference as a proxy for the phase transition strength, delving deeper into the effects that establish the preferred regions of parameter space. Section 6 discusses the connection of the phase transition with the trilinear Higgs self-coupling before conclusions are drawn in section 7.

Reviewing Two Higgs Doublet Scenarios
Let us start with a brief review of the 2HDM, which also defines our notation in this work. We consider a 2HDM scalar potential with a softly broken Z 2 symmetry to inhibit tree-level flavour changing neutral currents (FCNC), and for simplicity we neglect effects from CP violation 1 . The scalar potential then reads where the two scalar SU (2) L doublets Φ j (j = 1, 2) may be written as The physical scalar sector of a 2HDM is comprised of two CP-even neutral scalars, h and H 0 (with m H 0 ≥ m h ), plus a neutral CP-odd scalar A 0 and a charged scalar H ± . In this work we identify h with the observed 125 GeV Higgs boson, but we stress that our main arguments can be easily extended to the flipped case where H 0 is the recently observed particle and h is a lighter and yet undetected scalar (experimental constraints on this scenario have been recently discussed in [29][30][31]). Apart from m h and v = 246 GeV, the scalar potential (2.1) may be parametrized in terms of , the angle α parametrising the mixing between the CP-even states, the scalar masses m H 0 , m A 0 , m H ± and the mass scale M , The relation between the physical states h, H 0 , A 0 , H ± and the states ϕ j , η j , φ ± j is given by with s β , c β , s α , c α ≡ sin β, cos β, sin α, cos α, respectively. Regarding the couplings of the two doublets Φ 1,2 to fermions, the Z 2 symmetry in (2.1), even when softly broken by µ 2 , may be used to forbid potentially dangerous tree-level FCNCs by requiring that each fermion type couple to one doublet only [32]. By convention, up-type quarks couple to Φ 2 . In Type I 2HDM all the other fermions also couple to Φ 2 , while for Type II down-type quarks and leptons couple to H 1 . There are two more possibilities (depending on the Z 2 parity assignment for leptons with respect to down-type quarks), but we focus here on Types I and II. The parameters t β ≡ tan β and c β−α ≡ cos (β − α) control the strength of the couplings of h, H 0 , A 0 and H ± to gauge bosons and fermions. In particular, one can identify the so-called alignment limit [33] c β−α = 0, for which h couples to SM particles exactly like the SM Higgs. The parameters in the scalar potential can be related to the masses and mixings in the scalar sector as shown in Appendix A.
In order to obtain a viable 2HDM scenario, theoretical constraints from unitarity, perturbativity and stability/boundedness from below of the scalar potential (2.1) need to be satisfied. These will play an important role in the following discussion. Tree-level boundedness from below of the potential (2.1) requires (2.4) At the same time, tree-level unitarity 2 imposes bounds on the size of various combinations of the quartic couplings λ i [35,36]. Similar (although generically less stringent) bounds on λ i may be obtained from perturbativity arguments. Finally, in order to guarantee absolute tree-level stability of the EW minimum (that is, the non-existence of a "panic vacuum" [37,38]), the couplings must satisfy which can be rewritten as Note that, in alignment, the condition that no panic-vacua exist at tree-level is satisfied for M 2 > 0.
In the following, it will prove convenient to use the Higgs basis of the 2HDM [33], given by the rotation from the doublet fields in (2.2) via (2.7) The two doublets in the Higgs basis read with the modified mass parametersμ 2 1 ,μ 2 2 ,μ 2 and quartic couplingsλ 1−7 being functions of m 2 . We also note that in the Higgs basis M precisely corresponds to the mass scale of the second doublet prior to EW symmetry breaking.

The Electroweak Phase Transition with Two Higgs Doublets
The evolution of the Higgs vacuum in the early Universe, in thermal equilibrium, can be described by means of the finite temperature effective potential V T eff (φ, T ) for the Higgs (and possibly other scalar fields subject to evolution in the early Universe) with φ representing the set of relevant scalar fields including the Higgs, V 1 being the T = 0 radiative Coleman-Weinberg piece of the effective potential and V T the thermal contribution. The free-energy density difference F T between the SU (2) L × U (1) Y symmetric phase φ = 0 and the broken phase φ = v T = 0 at temperature T is then (3. 2) The first contribution, F 0 < 0, corresponds to the vacuum energy difference at T = 0, while the second contribution ∆V T ≥ 0 is monotonically increasing with T , vanishing as T vanishes. The critical temperature, T c , below which the EWPT can proceed in the early Universe is then defined by F Tc = 0. A first order EWPT is characterized by the presence of a potential barrier between the symmetric and broken phases as F T turns negative during the evolution of the Universe. Such a first order transition could be responsible for the generation of the matter-antimatter asymmetry of the Universe through EW baryogenesis, should the strength of the transition be sufficiently large (see [39][40][41] for reviews on the EWPT and baryogenesis). The details of the tunneling process [42][43][44] between symmetric and broken phases in a first order EWPT depend on the functional form of ∆V T in (3.2). Nevertheless, it has been recently shown that in a wide class of extensions of the SM potentially leading to a first order EWPT, the strength of the transition, which is the relevant quantity for EW baryogenesis, is dominantly controlled by the value of F 0 w.r.t. its corresponding value for the SM, F SM 0 [8,9]. In this work we show that this is indeed the case for the 2HDM. It is then possible to perform a systematic study of the 2HDM parameter space in which a strongly first order EWPT is favoured by analyzing the behaviour of ∆F 0 ≡ F 0 − F SM 0 . Moreover, we stress that ∆F 0 is renormalization scale independent and safe from potential gauge dependence issues [5,6], being manifestly gauge invariant. These highlight the advantage of using ∆F 0 to explore the regions of 2HDM parameter space where a strongly first order EWPT is possible, as well as its phenomenological implications.
Let us now discuss the vacuum energy at 1-loop in 2HDM scenarios. For the renormalization of the 2HDM 1-loop effective potential we use an on-shell scheme, imposing (among other conditions) that the value of the 1-loop vevs for the two doublets and the 1-loop physical masses m h , m H 0 , m A 0 and m H ± are equal to their tree-level values. The renormalized 1-loop effective potential in the Higgs basis reads with the counterterm potential being An immediate advantage of working in the Higgs basis is that, in order to obtain the vacuum energy F 0 , we only need to compute the on-shell renormalization conditions explicitly 3 for δμ 2 1 and δλ 1 −δμ 2 1 + The Higgs basis condition h 2 = 0 is maintained at 1-loop by the choice of δμ 2 and δλ 6 .
The 1-loop piece of the scalar potential V 1 in (3.3) is given in Landau gauge (see e.g. [15]) by The index α sums over W, Z gauge bosons, top quark and 2HDM scalars including Goldstone bosons 4 , with n α > 0 (n α < 0) for bosons (fermions). The various C α are constants which depend on the renormalization scheme, and may be disregarded as they drop out in the following analysis. The vacuum energy F 0 reads where ∆V 1 is to be understood as the difference of the Coleman-Weinberg terms (3.6) evaluated at the electroweak minimum and at the origin. As we are ultimately interested in ∆F 0 , we also need to compute F SM 0 using the same on-shell renormalization procedure (demanding the 1-loop Higgs vev and mass to match their tree level values), obtaining The first term in (3.7) and (3.8) corresponds to the tree-level vacuum energy difference for the SM. We also note that the contributions to ∆V 1 from the gauge bosons W and Z and the top quark are identical in the SM and 2HDM, and so drop out from ∆F 0 . Combining (3.7) and (3.8), we obtain with m 2 0 k the (possibly negative) squared scalar masses for k = H ± , A 0 , H 0 , h evaluated at the origin. Further details on the derivation of ∆F 0 including explicit expressions for I k and J k are given in Appendix B.
It is possible to show that the Q 2 dependence in (3.9) cancels out, so that ∆F 0 is renormalization scale independent. We also note that the first term in (3.9), which corresponds to the tree-level contribution to ∆F 0 , is negative definite and vanishes in the alignment limit c β−α → 0. In this limit, (3.9) simplifies considerably and reads

Vacuum Energy vs EW Phase Transition Strength: Numerical scan
In order to show explicitly the correlation between the vacuum energy difference ∆F 0 and the nature of the EW phase transition in 2HDMs, we perform a Monte-Carlo scan over an extensive region of the 2HDM parameter space. We vary mass parameters from 100 − 1000 GeV (but with m H 0 > m h ), and limit ourselves to the low tan β < 10 region, since very large tan β is uninteresting for practical applications such as the baryon asymmetry computation. Each scanned point is tested for: • Tree-level unitarity and perturbativity (by requiring the tree-level quartic selfcouplings among the physical scalars to be smaller than 2π) 5 .
• Stability of the electroweak vacuum at tree-level (c.f. eqs. (2.4) and (2.6)) and at 1-loop level by directly searching for lower secondary minima and/or unboundedness of the effective potential up to a cutoff Λ = 5 TeV 6 .
• Bounds from direct scalar searches using HiggsBounds [56], and agreement with measured properties of the m h = 125 GeV Higgs boson using HiggsSignals [57].
A point passing all these tests is considered physical. For each of these, the strength of the phase transition is computed by increasing the temperature, starting at T = 0, and following the electroweak minimum (whose norm at temperature T is denoted v T ), until we reach the critical temperature T c for which F Tc = 0. The phase transition is Clearly, the larger ∆F 0 is, the smaller the temperature corrections required in order to reach F Tc = 0. Since v T also grows as T decreases, the overall result is that the strength of the phase transition should be directly related to ∆F 0 . This is illustrated in Fig. 1. Here, the filled green contours indicate the number of physical points in a given region of the parameter space. In any such region we also define whose contours are shown in the empty curves indicating the percentage of points in the encircled region for which the phase transition is strong (e.g. in Fig. 1 (top), 95% of points inside the black solid curve have ξ ≥ 1). Note that the latter curves, being the ratio of density distributions in a certain region, are less sensitive to the priors of the scan than the actual distribution of points alone, and therefore offer a more meaningful physical picture in that they can be interpreted as a posterior probability density for requiring a strongly first-order EWPT given the existing constraints on the model. For convenience, we normalize the vacuum energy by the SM value at 1-loop 7 F SM 0 ≈ −1.25 × 10 8 GeV 4 . It is clear from Fig. 1 (top) that as ∆F 0 /F SM 0 decreases both T c and the likelihood of having a strong phase transition increase. Notice, furthermore, that the phase transition is guaranteed to be strong if ∆F 0 /F SM 0 −0.34 for the sample generated in our scan. This can be used as an efficient criterion to judge the nature of the phase transition, as it does not require the evaluation of the thermal potential (although it is not used in what follows). We however emphasize that the details of the temperature-dependent part of the effective potential are obviously important for the thermal evolution of the system, and oftentimes one cannot precisely judge the nature of the phase transition by the vacuum energy alone. E.g. for ∆F 0 = 0 in Type I, the EWPT can be weak or strong, as shown in Fig. 1 (bottom, left).
Yet, a direct correlation certainly exists between these quantities, from which one can understand and predict the favoured corners of the parameter space for a strong EWPT. Eq. (3.9) shows that the vacuum energy difference receives a negative treelevel contribution away from alignment, which increases with m H 0 . We thus expect a strong EWPT to favour the alignment limit, and the more so the heavier H 0 is. These expectations are confirmed by the data, as shown in Fig. 2. In both Type I and II scenarios the probability contours increasingly favour alignment for a strong EWPT as m H 0 grows. For Type I, even though the distribution of physical points already narrows around alignment for m H 0 550 GeV, the narrowing of the P ξ>1 bands is significantly more drastic and does not merely follow that of the physical distribution. It is also worth noticing that, while for Type I the low-mass region is the mostly populated, for Type II the lower bound m H ± > 480 GeV from flavour constraints tends to shift the masses of the additional scalars towards rather large values, which is why the physical points are mostly concentrated in the region of m H 0 ∼ 500 GeV. For Type II we 7 As F SM 0 is negative, larger values of ∆F 0 will correspond to more negative values of ∆F 0 /F SM 0 .
-11 - Number of physical points Type I Type II Figure 2. Distribution of physical points, as in Figure 1, and P ξ>1 contours in the (m H 0 , c β−α ) plane. As H 0 gets heavier, a strong first order EWPT increasingly favours alignment. In Type II the wrong-sign scenario, albeit less populated, can also lead to a strong EWPT. Number of physical points Type I Type II Figure 3. Distribution of physical points, as in Figure 1, and P ξ>1 contours in the also note the physical region for c β−α 0.4, corresponding to the 2HDM wrong-sign scenario [58]. Both in Type I and II scenarios one sees that away from the alignment limit there is a tension between a strong EWPT and a heavy H 0 .
The dependence of the vacuum uplifting with the overall mass scale M is determined mostly by stability, perturbativity and unitarity constraints. Indeed, close to the alignment limit the quartic couplings λ 1,2 read where the parameter has been introduced for its usefulness in the analysis of the stability and unitarity requirements. Recalling eq. (2.4), both couplings λ 1,2 must be positive and it follows that so that as M 2 grows larger, m 2 H 0 has to follow it closely. In addition eq. (2.4) shows that cannot grow too negative either, from which it follows that m 2 H ± and m 2 A 0 cannot be much smaller than a large M 2 . In summary, for M 2 m 2 h , stability enforces m 2 H 0 , m 2 A 0 , m 2 H ± ∼ M 2 , for which the decoupling limit is approached and ∆F 0 → 0, as can be verified by setting . Therefore, a significant uplifting of the vacuum energy can only be achieved for M ∼ v, which is confirmed by Fig. 3. We note that again in Type II the distribution of physical points is peaked around larger values due to the lower bound on m H ± from theB → X s γ constraint. However, in both types a moderate uplifting of the vacuum energy is achieved only for M 500 GeV.
A strongly first order EWPT generally relies on the existence of sizable couplings between the symmetry breaking scalar field (the Higgs) and the particles in the plasma, which means that one or more of the additional scalars must be significantly heavier than the overall mass scale M , as the mass splitting would be controlled by these large couplings. We have already established that a large m H 0 becomes disadvantageous for a strong EWPT away (even if only slightly) from alignment. Furthermore, for t β = 1 a large Ω 2 quickly violates perturbativity bounds. On the other hand, EW precision observables constrain the charged scalar H ± to be close in mass to either m H 0 or m A 0 . This leaves A 0 as the only scalar whose mass is free to be large 8 , and Fig. 4 confirms that a rather heavy 9 A 0 is indeed the most favoured scenario, with > 94% of strong phase transition points lying above the lower bound m A 0 300 GeV. 8 H ± may also be significantly heavier than M if paired to A 0 , but not on its own. 9 We note that a heavy pseudoscalar (m 2  These results are put together in Fig. 5, illustrating how the likelihood of a strong EWPT varies with m H 0 and m A 0 . In both Type I and II 2HDM scenarios a strong transition favoures a large splitting m A 0 > m H 0 + m Z , pointing to the A 0 → ZH 0 decay as a smoking gun signature of a 2HDM with a strongly first order EWPT. The detection prospects of this channel, and its importance as complementary to searches into SM final states, have been discussed in [14,18,19,65].

Analytic results
We now turn to an analytic exploration of the 2HDM vacuum uplifting as computed from eq. (3.9). Given the large dimensionality of the 2HDM parameter space, we perform the study in various limits which allow us to explicitly investigate the relevant parameter dependences. In the following section we focus on the alignment limit, pair m H ± exactly with either m H 0 or m A 0 , and work out the dependence of the vacuum energy and phase transition strength with the splitting ∆m AH ≡ m A 0 − m H 0 and Ω ≡ |Ω 2 |×sign(Ω 2 ) for different fixed values of m H 0 . Then, in section 5.2 we allow for deviations from the alignment limit, fixing a degenerate spectrum (m H 0 = m A 0 = m H ± ) for simplicity. Finally we devote section 5.3 to the special case of the Inert 2HDM where only one double takes a vev and the Z 2 symmetry is exact.

5.1
The Alignment Limit c β−α = 0 We start by considering the alignment limit c β−α = 0, where h behaves exactly as the SM Higgs boson. In this case, ∆F 0 is given by (3.10). Since measurements of EW precision observables (in particular the T -parameter) require an approximate degeneracy m H ± ∼ m H 0 or m H ± ∼ m A 0 , we set for simplicity this pairing as exact, analysing both possibilities. With these parameters fixed, ∆F 0 is then solely dependent on m H 0 , m A 0 , and Ω 2 . We first fix m H ± = m A 0 and show in Fig. 6 the parameter space regions of constant ∆F 0 /F SM 0 in the (Ω, ∆m AH ) plane, respectively for m H 0 = 200, 500 GeV (Left to Right) and t β = 1.5, 3, 5 (Top to Bottom). In each case we show the constraints from treelevel unitarity, boundedness from below of the scalar potential and non-existence of a panic vacuum. We note that as opposed to unitarity and stability, ∆F 0 /F SM 0 and the existence of a panic vacuum do not depend on t β (this last one for c β−α = 0). To estimate the breakdown of perturbativity, we show the region for which any quartic coupling grows larger than 4π at a cutoff µ = 5 TeV from 2-loop running [63], starting from µ 0 = max(m H 0 , m H ± , m A 0 ) to ensure that the heavy degrees of freedom will only contribute above their threshold. While this is not a hard limit on the model compared to the others presented, it provides an idea of the UV scale of new physics that would be required in such a picture. Finally, we also show the lines of a constant strength of the EWPT ξ in the (Ω, ∆m AH ) plane, obtained numerically. These smoothly track the lines of constant ∆F 0 /F SM 0 , confirming the observations in section 4 regarding the tight correlation between the strength of the EWPT and ∆F 0 in 2HDM scenarios.
From Fig. 6 we see that a strongly first order EWPT is achieved by increasing ∆m AH in all cases. For m H 0 v (m H 0 = 500 GeV in Fig. 6) and t β ∼ 1 it is also possible to achieve such a strongly first order transition by increasing Ω (with Ω < m H 0 ) for  ∆m AH < 0, but this possibility is forbidden by unitarity as t β departs significantly from 1. We repeat the analysis, now for m H ± = m H 0 , and show the results in Fig. 7. These are qualitatively similar to those from Fig. 6 for the m H ± = m A 0 scenario. Together,  these show that a strongly first order EWPT within the 2HDM generically favours m A 0 − m H 0 100 GeV, leading to the landmark signature A 0 → H 0 Z at colliders. Before continuing, let us note that in our analytical study of the 2HDM vacuum energy we haven't imposed several experimental constraints that would further restrict the allowed parameter space within the 2HDM, briefly outlined in section 4. The reason for not doing so is that these constraints depend significantly on the Type of 2HDM, while our analysis of the EWPT and the bounds from stability, unitarity, perturbativity and existence of a panic vacuum do not. However, it is important to briefly discuss these experimental constraints so that the reader is well informed of their potential impact on the 2HDM parameter space: (i) LEP searches yield the limit m H ± > 72 GeV (80 GeV) for 2HDM Type I (II) [60] as well as the bound m H 0 + m A 0 209 GeV [61]. (ii) LHC measurements of Higgs signal strengths constrain the allowed value of c β−α as a function of t β (see e.g. [62][63][64][65]). These do not provide a constraint in the alignment limit c β−α = 0 (since the 125 GeV Higgs behaves as the SM one in this case), but do constrain significant deviations from the alignment limit, and thus will be relevant for the analysis of section 5.2. In addition, Higgs signal strength measurements constrain the size of the h → A 0 A 0 partial width for m A 0 < 62 GeV, which in alignment translates into the strong constraint Ω 2 m 2 H 0 − m 2 A 0 − m 2 h /2 on the allowed range of Ω in this region [66]. (iii) LHC searches for H 0 , A 0 and H ± constrain the masses of the new scalars as a function of c β−α and t β (and Ω in certain regions of parameter space). In the alignment limit, and for the parameters considered in Figs. 6 and 7, relevant limits come from A 0 → ZH 0 (H 0 → ZA 0 ) 8 TeV CMS searches [67] in the region ∆m AH > 0 (∆m AH < 0), as discussed in [65]. Searches for H ± are also relevant for m H ± < m t (see e.g. [68]). (iv) Flavour constraints, particularly fromB → X s γ B-meson decays, yield strong limits on the (m H ± , t β ) parameter space both for Type I [54] and Type II [54,55] 2HDM (see also [69]).
In order to shed some more light on the impact of the quartic coupling values from the 2HDM potential (2.1) on the strength of the EWPT, we now analyze the interplay between ∆F 0 and the theoretical constraints using a different choice of independent parameters: c β−α , t β , M 2 , λ 3 , λ 4 , λ 5 . Together with v = 246 GeV and m h = 125 GeV, these completely determine the parameters in (2.1). We fix c β−α = 0, and note that ∆F 0 in this limit, given by (3.10), is symmetric under m A 0 ↔ m H 0 . Fixing m H ± to be close to either m A 0 or m H 0 breaks this symmetry. However, there is still a symmetry between the scenario m H ± = m A 0 with ∆m AH > 0 and the scenario m H ± = m H 0 with ∆m AH < 0. Using the relations from Appendix A.1 we find that in the former scenario λ 4 = λ 5 while in the latter Choosing M = 246 GeV as an illustrative example, we compare in Fig. 8  with both m A 0 − m H 0 > m Z and m H 0 − m A 0 > m Z (and even m H 0 = m A 0 ). This is the case for both the m H ± = m A 0 (Fig. 8 Left) and m H ± = m H 0 (Fig. 8 Right) scenarios. However, as t β increases, the region m H 0 > m A 0 becomes progressively excluded by unitarity, and already for t β = 3 a vacuum uplifting ∆F 0 /F SM 0 = −0.2 demands m A 0 − m H 0 > m Z , as can also be inferred from Figs. 6 and 7.

Away from the Alignment Limit: Degenerate 2HDM Spectrum
We now investigate the effect of departing from the alignment limit, setting for simplicity m H 0 = m A 0 = m H ± = m φ . In this approximation the vacuum energy difference can be expressed in terms of c β−α , t β , m 2 φ and Ω 2 (see Appendix B for details). We show in Fig. 9 the behaviour of the vacuum energy difference in the (Ω, c β−α ) plane for m φ = 200, 500 GeV (Left to Right) and t β = 1.5, 3, 5 (Top to Bottom). In all cases a sizable vacuum uplifting demands Ω v (the only exception corresponds to m φ = 500 GeV, t β = 5 and c β−α 0.4, excluded by vacuum stability). As shown in Fig. 9 (Left), for light m φ uplifting of the vacuum is in conflict with the panic vacuum constraint (and also excluded by unitarity for t β 1). In contrast, Fig. 9 (Right) shows that sufficient vacuum uplifting is possible for m φ = 500 GeV and v Ω m H 0 , provided that t β ∼ 1. Again, as t β increases the parameter space region where the 2HDM Higgs vacuum is uplifted compared to the SM one becomes excluded by unitarity.

An Inert Second Doublet
The inert doublet model [70][71][72] (IDM) is a special case of 2HDM scenario in which the second doublet is protected by a Z 2 symmetry and does not develop a vev. This Z 2 symmetry leads to the lightest state of the second doublet being stable, yielding a viable dark matter (DM) candidate if this corresponds to either A 0 or H 0 . This scenario has been widely studied in the literature (see e.g. [73,74] for updated analyses, and references therein), including its impact on the EWPT [75][76][77][78].
The scalar potential for the IDM is given by (2.1) with µ = 0, and due to the unbroken Z 2 symmetry the dictionaries from Appendix A.1 -A.2 do not apply in any particular limit, and instead the relations among parameters are given in A.3 (note however that some of the parameter relations are identical to those of the Higgs basis with c β−α = 0 and M 2 = 0). The relevant IDM parameters can be conveniently chosen to be m H 0 , m A 0 , m H ± , λ 345 ≡ λ 3 + λ 4 + λ 5 and λ 2 . In the following we consider DM to be H 0 (both choices are physically equivalent in the IDM), which amounts to requiring ∆m AH > 0, and we also consider m H ± = m A 0 as a simplifying assumption to satisfy EW precision constraints.
Using (3.5), (3.6) and the results from Appendix B we can easily obtain the vacuum energy difference ∆F 0 for the IDM, which reads and we investigate its interplay with theoretical constraints: stability, unitarity and the requirement that the Z 2 symmetry is preserved in the EW broken vacuum, which leads to the condition µ 2 1 / λ 1 < µ 2 2 / λ 2 .
We also include in our analysis the constraint on the IDM parameter space from the latest LUX bounds on the spin-independent DM-nucleon scattering cross section [79], as well as the IDM parameter space region for which the H 0 relic abundance through thermal freeze-out Ω H 0 does not exceed the observed DM relic density Ω DM = 0.1199 ± 0.0022 [80]. The H 0 relic abundance and the spin-independent H 0 -nucleon scattering cross section are both obtained with micrOMEGAs 4.3 [81], and we note that the nucleon scattering cross section has to be weighted by Ω H 0 /Ω DM when comparing with the LUX limits (as these assume Ω H 0 = Ω DM ).
In Fig. 10 we show the vacuum energy difference in the plane (λ 345 , ∆m AH ) for benchmark values m H 0 = 70 GeV (left) and m H 0 = 150 GeV (right), as well as the theoretical constraints for λ 2 = 1, 0.1. We also show the contours of constant Ω H 0 /Ω DM = 1, 0.1, 0.02, 0.01 and the bound from LUX. For m H 0 = 70 GeV the LUX bound combined with Ω H 0 /Ω DM ≤ 1 exclude the entire parameter space except for the small island ∆m AH 10 GeV and −0.05 λ 345 0.05. As shown in Fig. 10 significant vacuum uplifting requires ∆m AH v and is thus not possible in this case 10 . In contrast for m H 0 = 150 GeV, sizable uplifting and thus a strongly first order EWPT is possible, requiring ∆m AH 200 GeV. We emphasize that while previous works have already identified a large mass splitting ∆m AH in the IDM as providing a strong EWPT (see e.g. [78]), the dominant strengthening effect was attributed to the thermal contributions of H 0 , A 0 , H ± to V T eff . While these do play an important role, we show here that the most important effect is due to the uplifting of the T = 0 vacuum.

Trilinear Higgs self-coupling
Finally, it is useful to discuss the behaviour of the trilinear Higgs self-coupling λ hhh in the (Ω, ∆m AH ) plane, w.r.t. its value in the SM λ SM hhh . It has been suggested that a strong first order EWPT in the 2HDM is tightly correlated with sizable deviation in the value of λ hhh w.r.t. the SM value [20,21]. In the alignment limit, we note that λ hhh = λ SM hhh at tree-level (as was also noted in [20,21]). However, in the 2HDM 1-loop corrections may lead to sizable deviations from the SM value. The Higgs self-coupling λ hhh in the 2HDM is approximately given at 1-loop by where λ SM hhh includes the SM 1-loop corrections due to the top quark, Higgs and gauge bosons. Our result agrees with [20,21] and includes some sub-leading pieces that become relevant when the new scalar states are not so heavy with respect to the 125 GeV Higgs boson. Given the tight correlation between the vacuum energy difference and the strength of the EWPT, one would also expect a relationship to exist between the former and the Higgs self-coupling. Defining κ hhh ≡ λ hhh /λ SM hhh , the region |1 − κ hhh | ≥ 0.5 is of particular interest, since such a deviation in λ hhh from its SM value could be probed at the HL-LHC [26,27]. In Fig. 11 we show contours of κ hhh , for m H 0 = 200 GeV and m H 0 = 500 GeV in both m H ± = m H 0 and m H ± = m A 0 scenarios. We also superimpose the normalized vacuum energy difference ∆F 0 /F SM 0 , highlighting (in red/green) the values 0 and -1. The latter case corresponds to the limit above which the EW vacuum is lifted above the trivial one (F 0 > 0), preventing EWSB from ever occurring, while the former denotes a vacuum energy difference equal to that of the SM. Interestingly, we see that the region of unchanged vacuum energy difference with respect to the SM coincides almost exactly with the region where the Higgs self-coupling does not deviate from the SM prediction. Furthermore, the self-coupling grows as the EW vacuum is uplifted, reaching values of 2-4 times the SM prediction in the regions shown in Figs. 6 and 7 where a strong EWPT is expected to occur.
The strong correlation between the vacuum energy and the trilinear Higgs coupling shown in Fig. 11 can qualitatively be understood in terms of an effective potential for the SM Higgs. The extra Higgs states induce higher dimensional operators, with the leading one being of mass dimension six. When only keeping the mass term, the quartic coupling and the dimension-6 operator in the Higgs potential, we can vary the vacuum energy independently of the Higgs mass and trade the coefficient of the dimension-6 operator for the vacuum energy to parametrize this effective potential [9]. We can then compute the third derivative of this potential to obtain the trilinear Higgs coupling. Setting this in ratio to the SM result, which corresponds to a vanishing dimension-6 operator, we obtain means κ hhh > 1. Quantitatively, we find that this estimate falls short of the full result in Fig. 11 up to about 30%. This is not surprising, as the Higgs states integrated out are not very much heavier than v. So we expect operators of higher mass dimension to play a role, which, however, do not spoil the overall qualitative picture.
In fact, the contribution to the dimension-6 operator affecting the Higgs potential from integrating out the new states in the 2HDM is known [82]. Only one operator plays a role here. Its effect of the vacuum energy difference and the Higgs trilinear coupling is as follows In the alignment limit, the Wilson coefficient of interest has been calculated as .
Being positive definite, we see that it contributes both to an uplifting of the EW vacuum and an increase in the Higgs trilinear coupling. Furthermore, since EW precision tests constrain the charged Higgs mass to be near one or the other neutral state (m H ± ∼ H 0 or m H ± ∼ A 0 ), we are left with precisely the aforementioned mass splitting between the two, new neutral states controlling the effects of interest, lending further support to our previous findings.

Conclusions
In this work we have established a correlation between the strength of the electroweak phase transition and the zero-temperature free-energy of the broken minimum in two-Higgs-doublet models. Considering similar statements made previously in the literature in the context of other SM scalar sector extensions [8,9], we claim this is a general effect of any model where the modified scalar sector acts as the main source of strong phase transition. Because working with the zero-temperature vacuum energy is analytically much simpler than with the full thermal potential, this correlation can be used to better predict the behaviour of a certain model concerning the nature of the EWPT, as well as to better understand the impact of parameter space constraints on the strength of the phase transition predicted by the model. In particular, we have in this way clarified the preferred hierarchy in the scalar sector from the requirement of a strong EWPT, with a heavier pseudoscalar and charged scalar.
We have further investigated the relation between the triple Higgs self-coupling and the vacuum energy uplifting in the model. Large deviations from the SM predictions of these couplings are expected as a collateral effect of a model with a strong EWPT, and we have shown that these deviations can be measurable at the HL-LHC in some scenarios here presented. A measurement of the Higgs self-couplings is a key goal in any future collider experiment as a probe of the ultimate structure of the Higgs potential. Results such as the ones we present here show that this measurement would also serve as an indirect probe for the nature of the nature of the electroweak phase transition, and of the viability of electroweak baryogenesis as an explanation for the baryon asymmetry of the Universe.
B On-Shell Renormalization of the 2HDM: F 0 in the Higgs basis We recall the scalar contribution to the zero-temperature 2HDM vacuum energy in the basis of 2.9 (eq. (3.7)) The first two terms correspond to the tree-level piece, −λ 1 v 4 /8, translated with eq. (A.2). The second half of the expression is the 1-loop correction, comprising of the difference between the Coleman Weinberg potential evaluated at the EW minimum and the origin as well as the relevant counterterms. The latter are chosen to preserve the tree-level minimum and scalar masses at 1-loop, which fixes their value to Note that there is a caveat in carrying out the condition in eq. (B.4). For the Goldstone bosons, the first term in eq. (B.4) is infrared divergent, so that trying to define the physical mass by taking derivatives of V eff actually yields unphysical results. This happens because, by definition, the effective potential takes into account only diagrams with vanishing external momenta, whereas the physical mass must be evaluated on-shell, with p 2 = m 2 . A rigorous solution to the problem has been developed in [11], and also in [83,84] via resummation of the Goldstone contributions. Here we choose to adopt the more straightforward approach of replacing the vanishing Goldstone masses in the logarithmic divergent term by an IR cutoff at m 2 IR = m 2 h 0 , which gives a good approximation to the exact procedure of on-shell renormalization, as argued in [15].
Plugging eqs. (B.2) and (B.4) into (B.1), one finds the contribution of the counterterms to the effective potential at the electroweak minimum, Finally, putting everything together back into eq. (B.1), including the explicit contributions to ∆V 1 , we find where the SM vacuum energy of eq. (3.8) has been reintroduced and the contribution to the vacuum energy from loops of the SM Higgs and Goldstones, which also occur in ∆V 1 , are explicitly subtracted to avoid double counting these terms. Here, m 2 k,0 denotes a field dependent mass squared evaluated at the origin. This defines the vacuum energy difference of eq. (3.9).
What remains is to compute the derivatives of the field dependent masses with respect to h 1 via the general relations [85]  v I H ± = 2m 2 (B.10) It is easy to show that eq. (B.6) simplifies to eq. (3.10) in alignment. Through a laborious computation one can also show that the Q 2 dependence always cancels out, so that F 0 is indeed renormalization scale independent.