A new insight into the phase transition in the early Universe with two Higgs doublets

Journal of High Energy Physics, May 2018

Abstract We study the electroweak phase transition in the alignment limit of the CP-conserving two-Higgs-doublet model (2HDM) of Type I and Type II. The effective potential is evaluated at one-loop, where the thermal potential includes Daisy corrections and is reliably approximated by means of a sum of Bessel functions. Both 1-stage and 2-stage electroweak phase transitions are shown to be possible, depending on the pattern of the vacuum development as the Universe cools down. For the 1-stage case focused on in this paper, we analyze the properties of phase transition and discover that the field value of the electroweak symmetry breaking vacuum at the critical temperature at which the first order phase transition occurs is largely correlated with the vacuum depth of the 1-loop potential at zero temperature. We demonstrate that a strong first order electroweak phase transition (SFOEWPT) in the 2HDM is achievable and establish benchmark scenarios leading to different testable signatures at colliders. In addition, we verify that an enhanced triple Higgs coupling (including loop corrections) is a typical feature of the SFOPT driven by the additional doublet. As a result, SFOEWPT might be able to be probed at the LHC and future lepton colliders through Higgs pair production.

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A new insight into the phase transition in the early Universe with two Higgs doublets

HJE new insight into the phase transition in the early Universe with two Higgs doublets Jeremy Bernon 0 2 4 Ligong Bian 0 2 3 5 Yun Jiang 0 1 2 Seoul 0 2 South Korea 0 2 Clear Water Bay 0 2 Kowloon 0 2 Hong Kong S.A.R. 0 2 China 0 2 0 Blegdamsvej 17 , DK-2100, Copenhagen , Denmark 1 NBIA and Discovery Center, Niels Bohr Institute, University of Copenhagen 2 Chongqing 401331 , China 3 Department of Physics, Chung-Ang University 4 Institute for Advanced Studies, The Hong Kong University of Science and Technology 5 Department of Physics, Chongqing University We study the electroweak phase transition in the alignment limit of the CPconserving two-Higgs-doublet model (2HDM) of Type I and Type II. The e ective potential is evaluated at one-loop, where the thermal potential includes Daisy corrections and is reliably approximated by means of a sum of Bessel functions. Both 1-stage and 2-stage electroweak phase transitions are shown to be possible, depending on the pattern of the vacuum development as the Universe cools down. For the 1-stage case focused on in this paper, we analyze the properties of phase transition and discover that the eld value of the electroweak symmetry breaking vacuum at the critical temperature at which the rst order phase transition occurs is largely correlated with the vacuum depth of the 1-loop potential at zero temperature. Beyond Standard Model; Thermal Field Theory; Higgs Physics; Cosmology - A 1 Introduction The two-Higgs-doublet model Theoretical constraints The experimental constraints The e ective potential at nite temperature The tree level potential The Coleman-Weinberg potential at zero temperature The thermal e ective potential Phase transition: classi cation Numerical procedures: Tc evaluation scheme Properties of the rst order EWPT First order vs. second order phase transition Properties of the rst order EWPT 2 3 4 5 6 7 2.1 2.2 3.1 3.2 3.3 6.1 6.2 7.1 7.2 Typical mass spectra and discovery channels at LHC Triple Higgs couplings and the implications of the future measurements 8 Conclusions and outlook A Thermal mass for SM gauge bosons 1 Introduction After the discovery of the 125 GeV Higgs boson [1, 2] and the accumulation of LHC data, no evidence of new physics has been observed yet. Therefore, it is time to inquire whether the Standard Model (SM) of particle physics is actually complete to describe the physics at the electroweak scale. In the meantime, the origin of the baryon asymmetry of the Universe (BAU) is still one of the important open puzzles in particle physics and cosmology. To explain the BAU, the three Sakharov conditions [3] must be ful lled. The electroweak baryogenesis (EWBG) [4] is a possible mean to account for the generation of an asymmetry (imbalance) between baryons and antibaryons produced in the very early Universe. The { 1 { success of EWBG requires two crucial ingredients: CP violation and strong rst order phase transition (SFOPT), neither of which however can be addressed in the SM framework. First, the SM fails to produce a su ciently large baryon number due to a shortage of CP violation in the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The other shortcoming of the SM is the absence of departure from thermal equilibrium which could have been realized by a SFOPT: for the observed value of the SM-like Higgs mass this is not accomplished. The phase transition in the early Universe from the symmetric phase to the electroweak symmetry breaking (EWSB) phase actually belongs to a smooth crossover type [5]. It has been derived using lattice computation that the phase transition in the SM can only be strong rst order when the Higgs mass is around 70-80 GeV [6{9]. Therefore, a successful studied for both the CP conserving case [19{21] and with the source of CP-violation [22{27]. While the CP phase at zero temperature is supposed to play an insigni cant in the EWPT process [19, 20, 23], the CP-violating phase at nite temperature is found to be important in a recent study [27] where the analysis was performed after taking into account the LHC Run-2 constraints. In general, none of the scalar states of the 2HDM resembles a SM-Higgs boson that was observed at the LHC. However, such a SM-like Higgs boson h can arise in the alignment limit, a particularly interesting limit of this model where only one Higgs doublet acquires the total electroweak vev, namely the couplings of H to gauge boson pairs vanish while h possesses SM-like couplings [28{30]. In terms of the model parameters, this limit corresponds to sin( ) (always positive in our convention) to be close to 1. Driven by the LHC Higgs data, in this paper we focus on the alignment limit (here sin( ) 0:99) of the CP-conserving 2HDMs of Type I and Type II models. We consider the lightest 1Though the electroweak phase transition has been extensively studied in the singlet extended model, the BAU generation cannot be addressed without extra CP-violation sources [11]. { 2 { CP-even state h to be the 125 GeV SM-like Higgs observed at the LHC. To proceed the numerical analysis we take the points passing all existing experimental bounds (by the time of paper publication) generated from extensive scans in ref. [28] and additionally employ the 1-loop improved theoretical constraints, the updated measurements coming from avor physics and the recent LHC Run-2 bounds searching for heavy resonances. Our aim is to identity the parameter space of the 2HDM that can lead to a SFOPT and investigate the implications of a SFOPT required by baryogenesis on the LHC Higgs phenomenology. It is inspiring to note that the cosmological EWPT can leave signatures of gravitational waves (GW) after the nucleation of the true vacuum bubbles, with typical red-shifted spectrum frequency around O(10 4 10 2) Hz [31], which are detectable in the Evolved Laser HJEP05(218) Interferometer Space Antenna (eLISA) [32], DECi-hertz Interferometer Gravitational wave Observatory (DECIGO), UltimateDECIGO and Big Bang Observer (BBO) [33]. However, these two e ects might be quite incompatible due to an opposite preference occurring in the bubble wall velocity. The baryon asymmetry generation process within EWBG demands a relatively low bubble wall velocity in order to have enough time for the chiral asymmetry generation process to take place, this will later be transformed to the baryon asymmetry by the sphelaron process [10]. Of course, when performing the computation of the BAU in the EWBG mechanism, one should keep in mind that in addition to being subject to large theoretical uncertainties, the detailed calculations of the baryon asymmetry rely on the wall velocity of the bubble generated during the EWPT, see refs. [34{39]. On the contrary, a testable GW signal requires a higher strength of the FOPT and a larger wall velocity. Very intriguingly, the recent development [25] shows that it is possible, although di cult in the 2HDM, to simultaneously accomplish the EWBG and produce the detectable GW signals generated during the EWPT especially through acoustic waves [40, 41]. This paper is organized as follows. In section 2 we rst brie y review the CP-conserving 2HDMs of Type I and Type II and discuss the status in view of the existing experimental bounds. Next, we describe in section 3 the details of the nite temperature potential and provide a fast numerical handle for the thermal potential. In section 4, the onestage and two-stage phase transitions are demonstrated and classi ed. Subsequently, we present in section 5 a useful computational scheme used to single out the one-stage phase transition and, more importantly, to evaluate the critical temperature Tc for the onestage phase transition. Having studied the theoretical issues of the model and built the computational tools, we then proceed with the numerical analysis and investigate the properties of the phase transition which are presented in section 6. In particular, the relations between Tc and extra Higgs masses as well as the in uence of the e ective potential at zero temperature on the eld value of the electroweak symmetry breaking vacuum are analyzed. In section 7 benchmark scenarios leading to the SFOEWPT are established and their implications for future measurements at colliders are also discussed. Finally, section 8 contains our conclusions and outlook for future studies. In appendix A, explicit formulas for the thermal mass corrections of the SM gauge bosons are given. { 3 { Let us start with a brief review of the tree-level 2HDM at zero temperature. The general 2HDM is obtained by doubling the scalar sector of the SM, two doublets with identical quantum numbers are present. In general, CP violation may be present in the scalar sector and the Yukawa sector contains generic tree-level avor changing neutral currents (FCNCs) mediated by the neutral scalar states. Here we consider a CP conserving Higgs sector and the absence of tree-level FCNCs. The rst condition is obtained by imposing a reality condition on the parameters of the potential, and the second requirement is achieved by imposing a Type I or Type II structure on the Yukawa sector, this is achieved by imposing ; 2 the two Higgs doublets, the tree-level potential of this model is hm212 y1 2 + h:c:i + 1 2 ( y1 1)2 + ( y2 2) 2 2 2 + 3( y1 1)( y2 2) + 4j y1 2j2 + ( y1 2)2 + h:c: : 2 ! In this basis, the Z2 symmetry under which 2 is manifest in the quartic terms, while it is softly broken by the introduction of the m212 term. In general, m212 and 5 are complex. We consider in this work a CP conserving Higgs sector and set all i and m212 as real parameters, see [27] for a CP violating study. In this basis, both Higgs doublets have a non-zero vacuum expectation value (vev). We parametrize the degrees of freedom contained in the Higgs doublets as, i = + i (vi + i + i i)=p2 ; i = 1; 2 : at T = 0. potential minimization conditions, where vi are the vevs of the two Higgs doublets. At zero temperature one has the relation v12 + v22 = v0T ' (246 GeV)2. For convenience, we use the shorthand notation v 2 v0T from now on and de ne v1 = v cos and v2 = v sin , tan is therefore the ratio of the two vevs The mass parameters m211 and m222 in the potential eq. (2.1) are determined by the 5 2 ! (2.1) (2.2) (2.3) is parallel to the neutral H1 direction, this realizes the alignment limit of the 2HDM [45] which the LHC Higgs data appears to favor. In general, in a basis-independent manner, the alignment limit is de ned as the presence of a CP-even eigenstate in the vev direction in the scalar eld space. In the electroweak vacuum, the squared mass matrices in the neutral CP-even, CP-odd and charged scalar sectors are respectively given by, HJEP05(218) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) H! h = c s c s ! The CP-even mass eigenstates h and H, with mh mH , are obtained through the diagonalization of M2P , they are expressed in terms of the neutral components of the doublets as, Goldstone boson G0, while M Goldstone boson G . Their tree-level masses read2 where the mixing angle is introduced and is expressed in terms of the entries of the mass matrix. Diagonalization of M2A leads to a massive CP-odd scalar A and a massless 2 leads to a charged state H and a charged massless 2 2 MP;11 + MP;22 q 2 (MP;11 MP;22)2 + 4(MP;12)2 i ; 2 2 m2H;h = m2A = m2H = 1 h 2 m212 s c m212 s c 5v2 ; 1 2 ( 4 + 5)v2 : Using eqs. (2.8){(2.10) one can inversely solve the potential parameters, 1; : : : ; 5 in terms of four physical Higgs masses and the CP-even Higgs mixing angle , supplemented by the Z2 soft-breaking parameter m212 [45]. This means that the scalar potential can be entirely determined by these seven parameters and therefore allows us to choose them as a set of complete free inputs for the numerical analysis. As mentioned previously, we imposed a Z2 symmetry on the potential eq. (2.1) in order to forbid Higgs-mediated tree-level FCNCs. Out of the four independent realizations of 2We point out that in the review article [46] a factor of 2 is missing in front of 5 and 4 + 5 terms in the formula of m2A and m2+, respectively. { 5 { Higgs to up-type and down-type fermions respectively, normalized to their SM values for the two scalars h; H and the pseudoscalar A in Type I and Type II models. this symmetry in the fermion sector, we study two of them: the so-called Type I model where only 1 couples to fermions and the Type II model where 1 couples to downtype fermions and 2 to up-type fermions, see [47] for details. These particular structures rede ne multiplicatively the Higgs couplings to fermions as compared to the SM predictions, we denote as CU;D;V theses multiplicative factors for the up-type fermions, down-type fermions and massive gauge bosons, respectively. The Higgs couplings to massive gauge bosons do not depend on the Z2 symmetry charges but are directly obtained from gauge symmetry alone. In table 1 we present these factors for the three physical scalar states of the theory. Important intuition can be gained by re-expressing these factors in terms of ( ) and , in particular to understand their behavior in the alignment limit sin( ) 1: HJEP05(218) F F U U Ch;I = Ch;II = cos =sin CH;I = CH;II = sin =sin CDh;II = sin =cos CDH;II = cos =cos = sin( = sin( = cos( = cos( ) + cos( ) ) cos( sin( ) + sin( ) cot ; ) tan ; ) cot ; ) tan : 2.1 Theoretical constraints For a viable 2HDM scenario, we require here tree-level stability of the potential, which means that eq. (2.1) has to be bounded from below, requiring 1; 2 > 0; perturbativity are required. Tree-level unitarity3 imposes bounds on the size of the quartic couplings i or various combinations of them [49, 50]. Similarly (often less stringent) bounds on i may be obtained from perturbativity arguments. 2.2 The experimental constraints Next, we brie y describe the impact of the experimental bounds on the parameter space of the model. First, electroweak precision data (EWPD), essentially the T parameter, 3For a recent one-loop analysis, leading to slightly more stringent bounds, see [48]. (2.11) (2.12) (2.13) (2.14) (2.15) { 6 { constrains the mass di erence between mH and mA or mH , one of the two neutral states should indeed be approximatively paired with the charged state in order to restore a custodial symmetry of the Higgs sector [51, 52]. Second, the recent measurement on BR(B ! Xs ) [53] excludes low values of mH . 580 GeV in the Type II model [54]. As a consequence, the preferred ranges for the scalar masses are pushed above Third, LHC measurements of the 125 GeV signal rates put large constraints on the 2HDM parameter space, in particular they tend to favor the alignment limit where the Higgs couplings are similar to the SM ones. To evaluate these constraints, we use Lilith-1.1.3 [55]. Finally, regarding direct searches, we implement the Run-1 and LEP constraints as performed in [28]. A very important search for the Type II model is in the A; H ! channel, either through gluon-fusion or bb-associated production [56, 57]. The ATLAS Run-2 constraint is much stronger than the corresponding Run-1 searches, eliminating larger portion of the parameter space at large tan in particular. For mA < 350 GeV we only nd few scenarios compatible with the experimental constraints in the Type II model.4 This is both coming from the aforementioned search, as well as the H ! ZA searches for CP-odd state down to 60 GeV. This nal state has been searched for by the CMS collaboration during Run-1 [58], and leads to severe constrains of the parameter region. The A ! Zh channel has been searched for during both LHC Run-1 [58, 59] and Run-2 [60] but the resulting constraints have little impact. In gure 1 we show the allowed spectra (red pluses) for the two types of models considered here. The points labeled `no-EWSB' comes from the requirement of proper EWSB at the 1-loop level, which will be extensively discussed in section 3.2. Due to the severe constraints on the mass spectrum of the extra Higgs bosons, these experimental constraints have signi cant in uence on the requirement of a SFOPT as we will see in section 6. We now move to investigate the possibility of having a rst-order phase transition for the surviving sample points. The interesting question is whether the parameter space that LHC Higgs data favors, simultaneously satisfying both theoretical constraints and experimental bounds, can lead to a favorable prediction for a strong rst-order phase transition. 3 The e ective potential at nite temperature To study the phase transition we consider the scalar potential of the model at nite temperature. In the standard analysis, the e ective potential Ve (h1; h2; T ) is Ve (h1; h2; T ) = V0(h1; h2) + VCW(h1; h2) + VCT(h1; h2) + Vth(h1; h2; T ) ; (3.1) which is composed of the tree-level potential at zero temperature V0(h1; h2) derived in eq. (3.2), the Coleman-Weinberg one-loop e ective potential VCW(h1; h2) at T = 0 given in eq. (3.3), the counter-terms VCT given in eq. (3.11) being chosen to maintain the tree level relations of the parameters in V0, and the leading thermal corrections being denoted by Vth(h1; h2; T ). We discuss these terms separately now. 4This result is not fully consistent with ref. [21] where the authors claimed the experimental constraints are less severe for mA < 120 GeV. { 7 { LHC Run-2. In Type II (right), the B-physics constraint on the charged Higgs mH 580 GeV [54] is imposed and nearly excludes the low mA points. Gray points indicate EWSB is not ensured at zero temperature when one-loop e ect is included in the Higgs potential. 3.1 The tree level potential Since our model is CP conserving, the classical value of the CP-odd eld A is zero and so are the ones for the neutral Goldstone elds. We assume the charged elds do not get VEV during the EWPT process, by taking the classical values for the charged elds to be zero, to strictly respect the U(1) electromagnetic symmetry and therefore ensure the photon massless [46].5 The relevant tree level potential V0 in terms of their classical elds 5The charge breaking vacuum in multi-Higgs doublet models has been studied in refs. [61{67]. Once the U(1) electromagnetic symmetry is broken during the EWPT, the photon acquires mass, which may change the thermal history of the Universe [67]. We leave it to future work. Also, we do not expect the presence of color-breaking vacuum in the process of EWPT since the bosons which actively participate into the evolution of Higgs scalar potential are color neutral. As of our knowledge, color-breaking baryogenesis is achievable in the model with the inclusion of colored bosons (i.e. scalar leptoquarks) [ 68, 69 ]. { 8 { (h1; h2),6 derived from eq. (2.1) is V0(h1; h2) = 1 2 + m122t 1 8 here we have eliminated m211 and m222 by using the minimization conditions eq. (2.3). The Coleman-Weinberg potential at zero temperature To obtain the radiative corrections of the potential at one-loop level, we use Coleman and Landau gauge7 at 1-loop level has the form: VCW(h1; h2) = X( 1)2si ni m^ i4(h1; h2) The sum i runs over the contributions from the top fermion, massive W Higgs bosons and Goldstone bosons;8 in the sum si and ni are the spin and the numbers of degree of freedom for the i-th particle listed in table 2; Q is a renormalization scale which we x to Q = v and Ci are constants depending on the renormalization scheme. In the MS onbosons9 and Ci = 3=2 for the particles of other species [72]. Finally, the shell scheme employed, Ci = 12 ( 3 ) for the transverse (longitudinal) polarizations of gauge 2 eld-dependent squared masses m^i2 for SM particles include10 m^ t2 = m^2W = 21 yt2h22=s2 ; 4 t with the corresponding SM Yukawa and gauge couplings being de ned g = 2MW =v; g0 = 2mt=v and the ones for scalar bosons are given by m^ 2h;H = eigenvalues(Md2P) ; m^2G;A = eigenvalues(Md2A) ; m^ 2G ;H = eigenvalues(Md2 ) ; (3.2) (3.4) (3.5) (3.6) (3.7) (3.8) 6To avoid confusion we distinguish the dynamical elds and EW vev in this paper. The classical elds (h1; h2) approach the EW vacuum (v1; v2) at zero temperature. 7As noted in [71], the VEVs are slightly di erent in various gauges and the recent study [26] nd this e ect to be numerically small in the physically interesting regions of parameter space. 8We ignore the light SM fermions because of the smallness of their masses. In contrast, the inclusion of Goldstone modes is necessary as their masses are non-vanishing for eld con gurations outside the electroweak vacuum. The photon at zero temperature is strictly massless due to gauge invariance. 9In most literature Ci = 5=6 is taken for gauge bosons without the distinction between transverse and longitudinal modes. In fact, these two ways of counting are equivalent as the eld-dependent mass are identical for both transverse and longitudinal modes at zero temperature. For instance, nZ CZ = 2 1=2 + 1 3=2 = 3 5=6 and nW CW = 2nZ CZ . The mass di erence between transverse and longitudinal modes arises from thermal corrections as will see later. 10We notice typos occurring in the thermal mass of SM fermions (cf. eqs. (A.19) and (A.20)) in ref. [26]. { 9 { i ni t 2 2 3 1+2 potential. The fermions except the top quark are neglected due to their small masses. where the corresponding matrices Mc2X (X = P; A; ) are Md2X = B 0 21 h12 + m212t 1 v2 iXj terms listed below are di erent for X = P; A; . With VCW being included in the potential, the minimum of the Higgs potential will be slightly shifted, and hence the minimization conditions eq. (2.3) no longer hold. To maintain these relations, we add the so-called \counter-terms" (CT) [24],11 VCT = m12h21 + m22h22 + where the relevant coe cients are determined by, which are evaluated at the EW minimum of fh1 = v1; h2 = v2; A = 0g on both sides. As a result, the vevs of h1, h2 and the CP-even mass matrix will not be shifted. One technical di culty involved at this step arises from the inclusion of the Goldstone bosons in the CW potential. Due to the variation of the scalar eld con guration with temperature (which we will see shortly), the Goldstone boson may acquire a non-zero mass at nite temperature, enforcing the inclusion of Goldstone modes in the sum. Nonetheless, in the electroweak vacuum at zero temperature the masses of the Goldstone bosons are vanishing in the Landau gauge, which leads to an infrared (IR) divergence due to the second derivative present in our renormalization conditions eq. (3.13). This means that 11In addition, we do not include more complicate terms to compensate the shift of mass matrix of h, because these shift e ects are estimated to be negligible in our scenario. Tree level and loop-correction contributions to the potential at zero temperature for two model points with tan = 1; sin( ) = 1. The remaining parameters corresponding to the point shown in the left (right) plot are 1 = 3:3 ( 0:2); m12 = 315 ( 70) GeV. Clearly, the point shown in the left plot has a true EW vacuum while the one on the right plot has only a local minimum at v. renormalizing the Higgs mass at the IR limit is ill-de ned [73]. To overcome this divergence, we take a straightforward treatment developed in [24] and impose for Goldstone bosons an IR cut-o at SM Higgs mass, mI2R = m2h. Although a rigorous prescription used to deal with the Goldstone's IR divergence was developed in [22], ref. [24] argued that this simple approach can give a good approximation to the exact on-shell renormalization. Practically, in evaluating the derivatives for the CW potential, we remove the Goldstone modes from the sum and add instead the following Goldstone contribution to the right hand of eq. (3.13) 1 32 2 ln m2G(h1; h2) Q2 vev (3.14) with the replacement for the singular term m2G(h1; h2)jVEV ! mI2R in the logarithm. Note that the Goldstone bosons have a vanishing contribution to the rst derivative evaluated at the vev. Beyond tree level the true EW vacuum must be preserved when the one-loop corrections are taken into account. This demands that the potential after the inclusion of the CW and counter-terms still form a global minimum at the EW vacuum. As seen in gure 2, the CW term (green dotted) often lifts up the potential at the EW vacuum, resulting the local minimum shifting inward or even leading to a false vacuum. On the other hand, the CT e ect (blue) drags down the potential at the EW vacuum and thus helps to accomplish a true EW vacuum. As a result, the competition between these two opposite e ects determines the existence of a global minimum at the EW vacuum. We present in gure 2 two examples where the left one accomplishes a true EW vacuum, while the potential in the right plot has only a local minimum at v. The latter example is phrased `no-EWSB' in our terminology and such type of points are displayed in gure 1. This is an additional important constraint that excludes about 10% (5%) points in the Type I (II) model, in particular for the points with mA 300 GeV. In the mA < mh=2 regime (termed low-mA scenario), it turns out that EWSB at zero temperature can be achieved as long as at least one lighter H or H is present in the spectrum. For the case where both H and H are heavier than 550 GeV, EWSB would be hardly successful. To understand this, we is assumed at two common scales 300 (blue) and 600 (red) GeV, mA = 50 GeV is taken. The picture is negligibly modi ed for mA = 200 GeV. Three values of tan , tan = 1 (solid and dashed lines), tan = 10 (thick dashed lines) and tan = 20 (thick dotdashed lines) are shown for which m212 is chosen such that none of quartic couplings exceeds the pertubativity bound. display in gure 3 the one-loop potential at zero temperature (including the CW potential and counter-terms). For simplicity, we assume mH = mH , which is typical mass spectra required by the T parameter.12 The authors of ref. [74] have shown that low-mA scenario can be phenomenologically alive in the parameter space where the SM-like Higgs h has very small coupling to AA, which leads to tan . 2 or tan & 12 for mH = 600 GeV in the deep alignment limit. The low tan solution requires a severe tuning in the parameter m212, and in the allowed range m212 ' 5000 GeV2 the zero temperature potential (cf. the red solid line) at EW vacuum v is higher than the one at the origin. Moreover, a proper EW vacuum can be developed as the symmetry soft-breaking parameter m212 increases. This can be achievable for the case of mA mh=2 where the h ! AA decay is kinematically suppressed. On the other hand, the large tan solution, though possible in Type I model, strongly constrains m212 and tends to lift the potential. Hence, the importance of this class of solution is very marginal and no points were found in our numerical analysis. In addition, tan & 5 in Type II model was already excluded by the CMS bound searching for a light pseudoscalar scalar in the mass range of 20{80 GeV through the bottom-quark associated production and decaying into nal states during Run-1 [75]. As a comparison, we also exhibit the potential at a lower common scale mH = mH = 300 GeV. This example is only applicable in Type I model. One can observe that the potential generically reaches a global minimum at the EW vacuum and the depth of this minimum is less sensitive to tan . This implies that when the new scalars introduced are not heavy, the loop e ect is not substantial and thus the potential is largely governed by the tree-level. We conclude that the requirement of proper EWSB at zero temperature, in synergy 580 GeV required by B-physics measurements [53], entirely exclude the scenario of existing a light pseudoscalar A in Type II model that was delicately studied in ref. [74]. As will show shortly, these theoretical constraints will play an important role in achieving a strong rst-order phase transition. 12The lighter the CP-odd state A is, the stronger the degeneracy between H and H should be. where the functions JB;F are Vth(h1; h2; T ) = mi2(h1; h2)=T 2 and the upper (lower) sign corresponds to bosonic (fermionic) contributions. The numerical evaluation of this exact integral is very time-consuming (notably for the y < 0 case present for the bosonic degrees of freedom). Thus, computational techniques to reduce the computation time are welcome. A widely used solution is to consider the asymptotic expansions of JB;F . At small y (y 1),13 eq. (3.16) can be approximated by The nite temperature corrections to the e ective potential at one-loop are given by [76] J J y 1(y) ' B y 1(y) ' F 4 45 2 E) and aB = 16aF with the Euler constant E = 0:5772156649. where aF = Whereas at large y, 2 exp( 32 J By;F1(y) ' 2 1=2 y3=4 exp y1=2 1 + y 1=2 : 15 8 In order to make a quantitive assessment of the approximation precision we plot in gure 4 the small/large y approximations as well as the direct numerical evaluation of the integral. (For the evaluation of the latter one we use the NIntegrate function built in Mathematica.) It is clearly seen that the small y approximation (red curve) is valid in the ranges y 2 ( 5; 5) for bosons and y 2 (0; 5) for fermions, while the large y expansion (blue curve) converges to the exact integral for y > 10 for both functions. A gap is then present between the small and large y approximations in the transition range y 2 (5; 10). In this situation an interpolation can be introduced to connect smoothly the two approximations. Even though this reduces the deviation of the approximate results from the exact integral to less than 2%, there are still two serious shortcomings. First, this requires a conditional judgement for each state at temperature T to know which approximation should be applied, this largely increases the evaluation time. Second, the above approximations eqs. (3.17){(3.19) are only valid for y > 0 as shown in gure 4. However, the eigenvalues of the mass matrix of the neutral scalar states can become negative depending on the eld 13The high/low T approximations do not necessarily lead to small/large y, which also depends on the eld-dependent mass in the numerator. (3.15) (3.16) (3.17) (3.18) (3.19) bosonic states, we additionally present the negative y range since their thermal mass can be negative at T 6= 0. In each plot the result of the exact integral is shown in solid black curve. Red and blue curves give the small and large y approximations, respectively. Three dashed lines illustrates the result evaluated by summing over the Bessel functions at di erent order. con guration.14 If this happens, ref. [21] suggests that only the real part of the integral JB should be chosen in the evaluation as the imaginary part is irrelevant in extracting the global minimum.15 The thermal integrals JB;F given by eq. (3.16) can be expressed as an in nite sum of modi ed Bessel functions of the second kind Kn(x) with n = 2 [77], with the upper (lower) sign corresponds to bosonic (fermionic) contributions. Our numerical results show that the leading order l = 1 does not provide a good approximation of the full integrals. Instead, inclusion up to l = 5 order in the expansion can match the exact integral very well for both positive and negative y values. Therefore, in this work we take N = 5 in the evaluation of the thermal integrals eq. (3.20).16 Figure 4 also shows that the thermal function is negative for positive y thus dragging the potential down and leading to the formation of two degenerate vacua. As expected, this dragging e ect arising from the temperature corrections diminishes as y approaches to the in nity, which corresponds to zero temperature or the decoupling limit. Finally, there is another important part of the thermal corrections to the scalar masses coming from the resummation of ring (or daisy ) diagrams [79, 80], T 12 i Vdaisy (h1; h2; T ) = X ni h Mi2 (h1; h2; T ) 2 3 mi2 (h1; h2) 2 ; 3 i where Mi2 (h1; h2; T ) are the thermal Debye masses of the bosons corresponding to the eigenvalues of the full mass matrix Mi2 (h1; h2; T ) = eigenvalues h Md2X (h1; h2) + X (T )i ; 14For instance, in the SM the eld-dependent mass for Higgs eld is m2h = 3 h2 2 and turns negative at low eld con guration. Similarly for the Goldstone bosons. 15Tachyonic mass con gurations generate a negative local curvature of the potential, leading to a local maximum rather than a minimum. 16A similar numerical analysis taking N = 50 was performed in a recent study [78]. (3.20) (3.21) (3.22) (3.23) (3.24) (3.25) which consists of the eld dependent mass matrices at T = 0 eq. (3.9) and the nite temperature correction to the mass function X ; (X = P; A; ) given by with the diagonal terms being X = X 11 X 12 X ! T 2 12 X 22 24 ; 1P1 = 2P2 = 1A1 = 2A2 = 11 = cSM 6yt2 + 6 1 + 4 3 + 2 4; 22 = cSM + 6 2 + 4 3 + 2 4 ; here the subscripts f1; 2g denote the states fh1; h2g and cSM = 2 is the known SM contribution from the SU(2)L and U(1)Y gauge elds and the top quark [79]. It is important to note that the temperature corrections are independent of 5 where a possible CP phase can reside. On the other hand, the leading correction to o -diagonal thermal mass is vanishingly small due to Z2 symmetry imposed in the scalar sector. Moreover, it was argued by [81] that subleading thermal corrections to o -diagonal self-energies are suppressed by additional powers of coupling constants and EW vevs which are usually neglected. Therefore, we shall treat the thermal mass correction i as diagonal matrices in the following numerical analysis. The thermal mass corrections of the SM gauge bosons are given in appendix A. Historically, there was an alternative algorithm proposed by Parwani in dealing with the thermal corrections [82]. He included the e ect of thermal correction from Daisy diagrams by means of substituting mi2(h1; h2) by Mi2(h1; h2; T ) in the Vth(h1; h2; T ), eq. (3.15). It is important to note that these two approaches are not physically equivalent and the results produced are quantitively incompatible [21]. They di er in the organization of the perturbative expansion and consistent implementation of higher order terms. The method formulated in eq. (3.21) restricts the corrections to the thermal masses at one-loop level, whereas Parwani's method inconsistently blends higher-order contributions. Because of this dangerous artifact unrealistically large values of the phase transition strength (de ned in eq. (6.1)) would be obtained. Therefore, we will adopt the former consistent method in the following analysis. 4 Phase transition: classi cation In general, a system may transit from one symmetry phase to another one. Here the electroweak symmetry is broken as the Universe cools down, this is singled as a change in the nature of the global 0-vacuum at high temperature that gets replaced by an electroweak breaking global vacuum at lower temperature. At any given set of parameters, the full e ective potential eq. (3.1) can have several extrema. Our major interest is the global minimum vacuum state, the deepest minimum of the potential. The other extrema can be down, the EW vacuum shifts away from the 0-vacuum. Depending on the way in which the vacuum (marked by the red plus) develops, three types of phase transition presented are possible in the 2HDM: 1-stage second order (top panel), 1-stage rst order (middle panel) and 2-stage transition (bottom panel). Red arrows indicate a jump between two degenerate vacuum in the rst order phase transition while the vacuum transitions smoothly in the second phase transition. either saddle points or maxima or local minima of the potential. In studying the thermal phase transition, it is useful to trace the evolution of the extrema as well as calculate the di erence in potential depth between the global minimum (called true electroweak (EW) vacuum) and a secondary local minimum. First, since at very high temperatures electroweak symmetry is not broken, the e ective potential has one global minimum, which tends towards the point (h1; h2) = (0; 0). We refer to this minimum as the 0-vacuum. As the Universe is cooling down, the parameters that characterize the thermal e ects of the model evolve with temperature. This leads to a change of the classical values of h1; h2 elds17 and thermal phase transition takes place. In general one can classify the thermal phase transition according to the behavior of the vacuum development during the cooling down. For instance, the phase transition may be of rst or second order, one-stage or two-stage process. In gure 5 we show three examples that illustrate the di erent behavior of the vacuum development with temperature, where the temperature decreases from left to right and the true vacuum is marked as a red plus in each graph. The model parameters corresponding to each point are summarized in table 3. For the case shown in the top panel, the vacuum starts to depart from the origin at T = 163 GeV, and then moves closely along the yellow line until reaching the EW vacuum at zero temperature. This phase transition is called of second order, because 17It may have resulted not only in variation of the absolute values of particle masses, but also in rearrangement of the particle mass spectrum, which can have interesting cosmological consequences. Points Top Middle Bottom Properties tan sin 2nd order PT 2.98 -0.24 3.99 0.29 0.86 -1.06 1st order PT 2 stage PT 594 232 lead to di erent types of EWPT. the potential minimum shifts continuously while no potential barrier develops during the cooling down. In contrast, the vacuum of the potential displayed in the middle panel is localized in the vicinity of the origin point at high temperature. When the temperature decreases to T ' 157 GeV an EW vacuum located away from the origin appears, forming two degenerate vacua separated by an energy barrier. This gives rise to the rst-order phase transition from the origin to the EW vacuum, which is indicated by the red arrow. In these two examples, the phase transition is termed one-stage. In addition to experiencing only one standard EWSB phase transition, the 2HDM can undergo a two-stage phase transition as the temperature falls as shown in the lower panel. In this mechanism the rst stage is a conventional second order PT in which the symmetry is broken, shortly thereafter follows a rst order PT. Another remarkable thing is that the ratio of the classical value between the two elds h2=h1 shown in the upper and middle panels has very little dependence on the temperature. However, in general, the value of h2(T )=h1(T ) is a temperature-dependent parameter and the change in the temperature growth can even be large in magnitude. In particular, the lower panel displays a peculiar behavior of the ratio h2(T )=h1(T ) as temperature decreases: at rst it monotonically increases, resulting in a deviation of the vacuum from the yellow line, then jumps to its zero temperature value (that is tan ) at the transition point and maintains unchanged in the remaining process. 5 Numerical procedures: Tc evaluation scheme The dynamics of the EWPT is governed by the e ective potential at nite temperature eq. (3.1) in our model. For purposes of analyzing the temperature evolution of the potential involving both h1 and h2, it is convenient to work with a polar coordinate representation of the classical elds h1(T ) and h2(T ). To that end, we de ne h(T ) and (T ) via h1(T ) h2(T ) h(T ) cos (T ); h(T ) sin (T ): The tree-level potential eq. (3.2) in the (h; ) plane becomes V0(h; ) = 1c4 + 2s4 + 2 345s2c2 h 4 1 8 + h 2 2m212(t 4 t (1 + t2)2 t )2 v 2 1c2c2 + 2s2s2 + 345(s2c2 + c2s2 ) ; (5.1) (5.2) (5.3) (5.4) and the remaining parts of the e ective potential are much more involved and hence not shown here. When a rst order PT takes place, a local minimum with hhi 6= 0 develops and becomes degenerate with the symmetric minimum hhi = 0 as the temperature decreases, this de nes the critical temperature Tc, and the two minima are separated by a potential barrier. Therefore, the evaluation of Tc is of great importance in studying the EWPT and its cosmological consequences. A straightforward approach is to decrease the temperature by small steps and make a potential plot (like gure 5) at each step. Then the global minimum of the potential (starting from the EW vacuum at zero temperature) can be followed stepby-step and the critical point is found once the minimum displays a jump rather than a smooth transition. Obviously, this graphic method is feasible only for benchmark points but is barely applicable for extensive scan due to its non-numerical nature. To date several numerical methods have been developed. In ref. [19], going from zero temperature to higher temperature, the critical point is taken to be the last one for which the minimum lied below the origin. This approach is no able to resolve the 2-step phase transitions where the potential experiences a second order PT prior to the rst order PT, giving rise to a vacuum shift from the origin at higher temperature. To overcome this problem, the authors of [21] used advanced numerical algorithms to search for the global minimum of the e ective potential in the (h; ) plane for each temperature. We employ a method consisting of the following procedures: First, we deal with points for which the ratio h2=h1 is (approximately) temperatureindependent, that is (T ) = . The e ective potential eq. (3.1) reduces to a function of two parameter | temperature T and T -dependent eld norm h(T ). In this case, one can easily determine the critical temperature Tc and the eld norm vc at which the potential reaches a minimum by solving the equations Ve (h = vc; Tc) = Ve (h = 0; Tc): (5.5) In searching for the solution of the above equations, we require a di erence between the potential at the minimum and its value at the origin smaller than 10 10GeV4. As a consequence, the solution for vc would be a value close to zero if there were no degenerate minima of the e ective potential present in the process of temperature drop. This means that below a certain small value of vc we do not expect a decent probability of achieving a rst-order phase transition. Instead, very likely such points lead to a second-order phase transition. For this reason we employ a technical cut vc > 1 GeV in order to remove these points. Next, we are going to deal with the points that exhibit an explicit temperature dependence for the ratio of two elds. This type of points often lead to a 2-stage phase transition [83, 84], as illustrated in the last row of gure 5. There must exist a global minimum for which tan (T ) = tan is not obeyed at a certain temperature or within a small temperature interval. In this situation, (vc; Tc) obtained as a solution of eq. (5.5) is not the critical vev and temperature where the phase transition occurs because the true vacuum is no longer located at the origin. Searching for the global minimum should be performed not along the tan line but on a two-dimensional (h; ) space. We employ an algorithm which uses the steepest descent method to nd the global minimum of the e ective potential. At Tc the searched minimum is then compared with the value of the e ective potential evaluated at vc and the one with the lower value is chosen as the candidate for the global minimum. For the general 2-stage PT, tracing the (temperature) evolution of the global minimum on a 2D plane is inevitable. Here, we discard points leading to a 2-stage PT and focus on the scenarios featuring a 1-stage PT. In the following analysis, we only retain parameter points with Tc The strength of the phase transition is quanti ed as the ratio of the norm of the neutral elds to the temperature at the critical point, = vc Tc : (6.1) Here vc = phh1i2 + hh2i2 + hAi2,19 in general, represents the value of the norm of all scalar elds involved at the broken vacuum at critical temperature Tc. Note that when interpreting the ratio as the strength of the electroweak phase transition, one should be aware of its gauge dependence [76, 85{87]. In order to ensure that a baryon number generated during the phase transition is not washed out, a strong rst-order phase transition is demanded and occurs if Before presenting the main results, we discuss the speci c features of the parameter space compatible with the theoretical and experimental constraints and at the same time rst order and second order phase transition. We will show results for both Type I and Type II models. 6.1 First order vs. second order phase transition It has been shown in gure 5 that both rst order and second order phase transition can take place in the 2HDM. Whether rst order or second order PT is developed depends on the mass spectrum among the three extra Higgs bosons, which is directly related to the ve quartic couplings i and the soft symmetry breaking parameter m212 through eqs. (2.8){(2.10). Thus, it would be very interesting and useful if one can divide the entire model parameter space into di erent sectors where distinct dynamics of vacuum evolution leading to rst order and second order PT take place. An initial attempt along this direction was made in [90] in accordance with the general geometric analysis of [91]. In [90] the 18It appears possible that the potential has a global minimum at large value of h. However, the probability of having a strong phase transition for these points is quite low, unless high scale phase transition is considered. 19Since we restrict ourselves to a CP-conserving model, the global minimum has hAi = 0. In general there may exist local minima that are CP-violating, while a recent study [21] found that it is always vanishes up to numerical uctuations at both T = 0 and T = Tc. 20The choice of the washout factor is subject to additional uncertainties. It was argued that the EW sphaleron is not a ected much if extra degrees of freedom are SM-gauge singlets [89] but the situation in the presence of an additional doublet is unclear yet. As a more conservative choice, other criterion such as 0:7 was also taken in other works. common mass scale among the three BSM states M = 600 GeV are assumed. The dashed line with an arrow indicates the jump from the second order PT to the rst order PT. Red and green curves represent tan = 1 and 1:5, respectively and terminate at which a proper EWSB at one loop level does not happen at zero temperature. In the gray-shaded region vc exceeds the EW vacuum v = 246 GeV and in the green-shade region at least one of the 's (mostly j 1j or j 2j) exceeds the perturbativity bound (i.e. 4 ) for the tan = 1:5 case. for a small or modest value of m12 when mH is xed. To illustrate this, we evaluate the phase transition properties in the process of slowly varying m12, assuming the alignment limit and a common mass scale among the three BSM states M = 600 GeV for simplicity. The situation is shown in gure 7, where red and green curves represent tan = 1 and 1:5, respectively. This plot can be used to track the evolution of the critical vev vc: it starts from zero (in the second order PT stage) at large m12 to a non-zero value (in the rst order PT stage). The jump from the second order PT to the rst order PT is indicated by a dashed line with an arrow. Notably, a severe ne-tuning on m12 is required for a successful rst order PT and the vc value approaches the EW vacuum at smaller m12. This interesting behavior is explicitly illustrated in gure 8 which gives, for tan = 1, the 1-loop potential curve at zero temperature (left) and the nite temperature e ective potential evaluated at the critical temperature (right) for various values of m12. As m12 decreases, thermal e ects generate a higher potential barrier and simultaneously push the degenerate vacuum towards the EW vev v, giving rise to a growth in (owing to the small uctuation on Tc in the stage of the rst order PT). On the other hand, a smaller contribution from the m212 term to the tree-level and 1-loop potential at zero temperature will remove the potential barrier. For example, the SM potential V h 4 when the mass term 2 ! 0. Consequently, the desired vacuum disappears, resulting in a terminal value of vc near v, as we will also see in gure 9. Furthermore, the e ect of increasing tan on the phase properties is also visible by comparing the red and green curves. For a larger tan and the same mass spectrum, the rst order PT is realised at a lower value of m12 and in the meanwhile the `no-EWSB' situation takes place at a smaller value of vc. As also seen in gure 6, most of our points have tan close to one, which agree well with the ndings of previous studies [19]. Yet we would like to clarify that such preference potential evaluated at the critical temperature (right) for tan = 1 and various values of m12 given in the legend. As gure 7, the mass of three BSM Higgs states are commonly xed at 600 GeV and sin( ) = 1 is assumed. is absolutely not the consequence of requiring a (strong) rst order PT. The underlying reason is that in the vicinity of tan ' 1, a large range of m212 satisfying the theoretical constraints outlined in section 2.1 is allowed.22 Oppositely, m212 is strongly constrained in the high tan region and a ne-tuning is required, which will greatly increase the di culty of accumulating the points by means of random scan. Numerically, very limited range of tan is allowed for large m212. Properties of the rst order EWPT We now turn to discuss the general properties of the rst-order PT accomplished in the 2HDM. The crucial parameters of the phase transition include the critical temperature Tc, the eld value vc at Tc and their ratio = vc=Tc which is used as a measure of the strength of the EWPT. In gure 9 we display in the (Tc; ) plane the points consistent with all theoretical constraints on the potential and up-to-date LHC limits at Run-2. Three black contours from top to down correspond to vc = 250; 135; 50 GeV. We rst discuss the impact of extra scalars in the spectrum. Suppose all extra scalars are heavy (i.e., above 800 GeV) and thus their masses are highly degenerate required by the EWPD (see gure 1), then the sector consisting of the new scalars decouple from the SM Higgs and the dynamics of phase transition behaves like the SM. Of course, the strength of EWPT is not closely related to the masses of any of additional Higgs bosons but more directly linked to the mass splittings among them, which can be explicitly visualized in gure 14 presented later. A general tendency observed is that vc is more constrained as Tc decreases. In the extreme case of Tc . 100 GeV, the thermal e ect, while still playing the role of lifting the e ective potential and forming two degenerate minima, is too weak to compete with the zero-temperature loop corrections to the potential. As a result, the critical classical eld value is mostly localized around v, which makes it slowly vary with respect to the temperature change. Nonetheless, vc shown in gure 9 does not exceed the zero temperature EW vacuum value v owing to the EW vacuum run away (`no-EWSB' bound) as sketched in gure 7, implying that the PT strength necessarily improves at low Tc. More quantitively, this leads to a maximum PT strength ' 5 at Tc = 50 GeV, and, on the other hand, implies 22The correlation between tan and m122 were discussed in details in ref. [92]. black contours from top to down correspond to vc = 250; 135; 50 GeV. The value of mH ; mH is color coded as indicated by the scales on the right of the plots in the upper and lower panel, respectively. an upper bound on Tc at 250(350) GeV for 1(0:7).23 In addition, we observe that a lower bound on Tc for each value of vc. For the value of the critical classical eld vc being slightly away from the EW vacuum v, the lower bounds on the critical temperature would be around Tc & 100 GeV in Type I and the lower bound on Tc in Type II model is slightly raised due to the lack of mA 350 GeV points. We stress that this is an useful nding that one can utilize to greatly optimize the algorithm for the evaluation of Tc. Last, we point out that the extremum, if coexisting in the vicinity of vc ' 135 GeV, often develops to a local maximum (corresponding to a barrier) rather than a local minimum of the potential, which causes a narrow gap dividing the displayed points into two parts. An explicit dependence of the critical temperature Tc on the mass spectrum of the three extra Higgs bosons can be visualized in gure 10, where we display all points that pass the applied constraints as in gure 9 and additionally ful ll a strong rst order EW phase transition (i.e., 1). Having explored these SFOPT behaviors, we shall investigate the relation between critical classical eld values, critical temperatures and di erent contributions to the e ective potential in the model. While the thermal contribution is crucial in controlling the process of vacuum tunneling, lots of attempts have been made to describe the properties of the phase (i.e. vc and Tc) from the e ective potential at zero temperature. A recent progress was reported in ref. [93] (within the framework of the CP-conserving 2HDM) that 23This result supports us to e ciently place a cut Tc . 300 GeV in the analysis. 300 GeV points obeying the strong rst order EWPT condition the strength of the phase transition is dominantly controlled by the value of F0, the depth of the 1-loop potential at zero temperature between the symmetry unbroken vacuum h = 0 and the symmetry broken vacuum h = v which corresponds to, in our notation, 0 V 1-loop(v) 0 V 1-loop(v) 0 V 1-loop(0) ; (6.2) where V01-loop(h) = V0(h)+VCW(h)+VCT(h) is the full 1-loop potential at zero temperature. Using the normalized depth F0 de ned in [93] one can derive an upper bound that de nitely guarantees the PT to be strong, for example, F0=F0SM . 0:34 necessarily 1 in the 2HDM. This, of course, can be used as an empirical test to assess the strength of the phase transition. However, a strong rst order PT is still possible even though this upper bound is over owed, in this situation the thermal potential plays a more important role for the thermal evolution of the system. Therefore, while appreciating the advantage of this approach in simplifying the phase transition study, which allows to nd regions of the parameter space where a SFOPT could be achieved, we expect a deeper comprehension by investigating not the strength itself, which is not an intrinsic property of the phase transition, but the characteristic quantities derived from the phase dynamics: vc and Tc. Interestingly, we nd that the magnitude of vc increases towards the zero temperature VEV with the decrease of the vacuum depth 0 V 1-loop(v) independent of the value of Tc. This is illustrated in gure 11 and is one of the nontrivial outcomes of this work. It is naively true that vc ' v when j V01-loop(v)j ' 0, which implies that the thermal e ects in the presence of extra scalars enhance the value of the e ective potential at the SU(2) symmetry broken vacuum and almost do not shift the symmetry broken vacuum at the critical temperature. As expected, as the vacuum depth j V01-loop(v)j increases, vc decreases towards the classical eld value of h = 0, which results in a smaller value of for a given Tc. In the meanwhile, we emphasize that the precise evaluation of vc (and Tc), of course, requires the inclusion of the temperature-dependent part in the potential. The critical temperature Tc is supposed to be more related to the thermal corrections to the e ective potential, as demonstrated in the lower panels of gure 11. represents the vacuum depth of the zero temperature potential j V01-loop(v)j and the thermal potential in the broken vacuum at the critical temperature VT (vc; Tc), respectively. Only Tc . 300 GeV points are retained. In addition to the non-thermal loop e ect discussed above, the thermal e ect in the presence of extra scalars is another promising source driving the SFOPT. In the 2HDM, extra BSM bosonic states are present in the plasma and induce the additional contribution to the thermal mass through the quartic couplings ( 1;2;3;4), see eq. (3.24). Thus, if a proper cancellation between their masses and couplings is satis ed, an energy barrier can be generated so that the PT becomes strongly rst order [94]. In order to see the importance of the thermal e ect, we estimate the thermal masses for three extra Higgs bosons eq. (3.22) at critical temperature Tc and present in gure 12 the ratio normalising the zero-temperature masses (the measured masses) as a function of the PT strength . Clearly, the ratio for the three states have a large variation around 1 on both sides, which means their thermal corrections can be either constructive or destructive even for the SFOPT ( particular, this ratio for the CP-odd A state in Type I model can be up to 1). In 20 owing to the presence of the extremely light A. While the thermal correction tends to suppress the mA and mH at Tc, the preference over the enhancement on the H (relative to H ) is still visible. The importance of the thermal mass maximizes at signi cant as further grows. ' 1 and becomes less Recall that the SU(2) custodial symmetry is not severely broken at zero temperature due to the T parameter in the EWPD which forces small mass di erence for jmH mAj or mH j or both. One may be curious whether this symmetry is broken at nite temfor the three BSM states as a function of the PT strength . Only Tc . 300 GeV points obeying the strong rst order EWPT condition 0:7 are shown. To examine the violation of the SU(2) custodial symmetry we normalize the eld-dependent mass for two neutral scalars A and H to the one for the charged Higgs H , mA(Tc)=mH (Tc) and mH (Tc)=mH (Tc) in the presentation. Only Tc . 300 GeV points obeying the strong rst order EWPT condition 0:7 are shown. perature. This is especially interesting when such symmetry plays a crucial role in selecting a particular region of parameter space. In general, the thermal correction to the eld dependent masses might results in a shift of the symmetry of the model at nite temperature. The particular case of interest is the Z2 symmetry cases studied in refs. [95{97] where the Z2 symmetry is preserved at T = 0 but spontaneously broken at T 6= 0. To examine if the e ect of thermal corrections leads to a shift of the SU(2) custodial symmetry in our model, we estimate the ratio of the thermal mass for two neutral states with respect to that for the charged state at critical temperature Tc. The result is illustrated by gure 13 where one can observe that the points displayed are well aligned either mA(Tc)=mH (Tc) ' 1 or mH (Tc)=mH (Tc) ' 1 with about 10{20% departure, indicating a large violation of the SU(2) custodial symmetry is not possible at nite temperature during the SFOEWPT in the 2HDM. Sce. mH [GeV] mA [GeV] mH [GeV] Type A B C D E 130{300 400{600 130{200 400{600 300{350 400{600 10{200 450{800 10{250 300{350 100{300 400{600 450{800 100{250 300{350 I, II I, II I I I Main H=A decays A ! W H+(60%); ZH(25%) H ! ZA(50 75%) A ! ZH( 100%) H ! W H+(60%); ZA(25%) A ! Zh( 100%), H ! W +W (& 40%) the heavier neutral Higgs boson (H or A) are given in each scenario.The numbers in the parenthesis following each decay indicate an estimate on the branching ratios. rst order EWPT and the implications for future measurements at colliders Typical mass spectra and discovery channels at LHC As seen from gure 9, a SFOPT is possible in both Type I and Type II models. Then one may wonder what is the LHC Higgs phenomenology associated with a SFOPT. To answer this question, in gure 14 we present in the mA versus mH (upper) and mH versus mH (lower) planes all points that pass the applied constraints as in gure 9 and additionally realize a SFOEWPT (i.e., 0:7). The values of and the mass di erence jmA mH j are indicated in color scale in the upper and lower panels, respectively. We emphasize again that the EWPD, essentially the T parameter, force the mass di erences between the charged Higgs boson and at least one of the extra neutral Higgs bosons to be small and strongly favor mass spectra where the masses of all new scalars are close to each other, in the decoupling limit in particular. This severe constraint on the mass spectra for the nonSM Higgs bosons leads to ve benchmark scenarios achieving a SFOPT in Type I model. They are summarized in table 4 where the characteristic mass spectra and the main decay modes of the heavier neutral Higgs boson (H or A) with an estimate on the branching ratio are given in each scenario. We start with the analysis in Type I model. First, the most widely studied mass con guration includes a pseudoscalar A with mass within the range of 400{600 GeV accompanied ' 200 GeV [19, 20]. In this case, m212 must be relatively small since large m212 tends to reduce the strength of the phase transition. This leads to a special relation among the quartic couplings 1;2;3 ' 0 and 4 ' 5 ' 5, meaning that the strength of the phase transition is mainly governed by 4 and 5, see also [19]. Dictated by symmetry argument, one can image that the mass spectrum consisting of a light CP-odd state and two highly mass degenerate H and H can also lead to a SFOPT, which is re ected by the existence of a bulk of red points at the upper left corner (i.e., mH ' 400 600 GeV, mA . 200 GeV) in gure 14. The situation of the model parameters is opposite due to the ip of mass hierarchy among the three BSM Higgs states. To be speci c, m12 is large as a consequence of large mH , and 1;2;3 ' 0 and 4 ' 5 ' 5. Likewise, a SFOPT 1) can be also realized provided that mA and mH are close to each other, while both having a large gap relative to mH . Strictly speaking, such condition provides two 300 GeV points are retained. possibilities for the mass spectra: i) mA ' mH ' 600 GeV and mH ' 200 GeV and ii) mA ' mH ' 200 GeV and mH ' 600 GeV, which correspond to two isolated red-orange points densely distributed along the diagonal line in the lower panel plot. Deduced from eqs. (2.9) and (2.10) the mass degeneracy between A and H states in this scenario restrict 4 ' 5, while an additional coupling 3 participates into the potential evolution and in uences the phase transition. Apart from these four scenarios that are visible in the low panel plot, the upper left plot in gure 14 demonstrates an additional possible scenario that is compatible with 1 where all three non-SM-like Higgs bosons have similar mass scales at 300{350 GeV. This scenario was unfortunately ignored [19, 20] or paid less attention [21].24 It is also worth noting that in this highly degenerate scenario none of i couplings can be close to zero if the rst order PT takes place. On the other hand, the allowed mass spectrum that is compatible with 1 in Type II model is quite simple. As explained in section 2.2, the combination e ect from Bphysics observables and EWPD pushes mH & 580 GeV and simultaneously raises the mass 24One might indeed have believed that large mass splitting among the non-SM Higgs bosons are a necessary condition for the requirement of a strong rst-order EWPT. scale for at least one of the extra Higgs bosons. Consequently, many scenarios available in Type I model are eliminated, resulting in an allowed mass spectrum that leads to a strong rst-order EWPT being quite restrained: mA ' mH gap between mH and mH : mH mH > 300 GeV. 600 GeV and a large positive mass ' 0:7) would Generally speaking, requiring a SFOPT forces down the mass scale for the new scalars and the preferred ranges for all the scalar masses below 600 GeV, which coincidently approaches to the current lower bound on the charged Higgs mass strongly constrained by the latest measurement of B ! Xs . This means that future improvement on B-physics observables may decisively rule out the success of SFOPT in the Type II 2HDM. Of course, gure 14 also informs us that weak rst order PT (under the criterion of still be possible even if no additional Higgs bosons were discovered below 1 TeV. Finally, we brie y discuss the prospects of testing the EWPT at the colliders in accordance with the mass spectrum provided above. In the alignment limit sin( we consider, the coupling ghAZ is vanishingly small but the coupling gHAZ / sin( enhanced. Hence, the branching ratios for A ! ZH and H ! ZA as long as kinematically allowed can be substantially large depending on the mass spectrum in the model. These results point towards the observation of the A ! ZH and/or H ! ZA decay channels would be \smoking gun" signatures of 2HDMs with a SFOEWPT.25 LHC search prospects for the former decay have been analyzed and proposed as a promising EWPT benchmark scenario in [20], while the collider analysis looking at both decays was performed in ref. [98] but not speci cally aiming at the EWPT. In gure 15 we show the 13 TeV cross sections at the LHC for these two channels in the gluon-fusion production mode. In all cases, a cross section above the pb level can be achieved for the scenarios realizing a SFOPT. Although these signatures are characteristic ones in most of the 2HDM scenarios discussed above (see table 4), no strong correlation in these channels is found between and the corresponding cross sections, which means that there is no guarantee to observe these decays in colliders. ) is We leave a detailed collider analysis to future studies. Searching for a new scalar resonance is performed at the LHC mostly through its decay into SM particles. These decay channels include H ! ZZ ! 4`, H; A ! ; ; tt. For the purpose of testing the EWPT, it would be very useful to nd channels with strong correlation to the value. The one served as an example here is the gluon-fusion production cross section of A and H in the decay channel, which is shown in gure 16. In general, the gluon-fusion cross section in Type I model is considerably small, so there is very little hope to ever observe A or H in this channel. An exception occurs in the very light CP-odd A region with cross-section as large as the level of 10 100 pb [74]. Moreover in this region, mA 60 GeV, a few points with large values are observed, which could be excluded by the upcoming experimental searches in that channel. In Type II the situation is di erent, for a given scalar mass the achievable cross-sections have a lower bound. The large points are located at low mH . 350 GeV and have reasonably large cross-sections just below the current experimental upper limit. In short, we estimate that a factor of 4 improvement in 25Although our results con rm the results in the earlier literature [20], more importantly, we clarify that the decay A ! ZH is not a unique \smoking gun" signature of SFOPT in the 2HDM of Type I model. This conclusion is also supported by another recent study [21]. gg ! A ! ZH (upper panels) and gg ! H ! ZA (bottom panels) channels in Type I (left panels) and Type II (right panels). gg ! A ! (upper panels) and gg ! H ! and Type II (right panels). (bottom panels) channels in Type I (left panels) the search sensitivity, which is very likely to be reached, would either see an exciting signal or eliminate these points, as a result, the rst order PT with strength > 3 can be fully tested at the LHC. to the SM tree value ghShMhtree = 3m2h=v. Note that tree-level Higgs self-coupling is not enhanced in Type II. To have a better visualization only 1 points are shown. Triple Higgs couplings and the implications of the future measurements The scenarios that lead to the rst order PT in the model have a mass spectrum below the TeV scale, as can be seen in gure 14 and table 4. The presence of additional scalars that couple to the SM-like Higgs h can modify the triple Higgs coupling hhh at both tree-level and loop-level and thus leads to the deviation with respect to its SM value ghShMh tree. Moreover, such deviation can be signi cant near the alignment limit provided being away from the decoupling limit [28, 99]. We examine both the tree-level coupling and the one after the inclusion of the one-loop corrections. They are computed by taking the third derivative of the tree level potential V0 and the one-loop potential V0 + VCW + VCT with respect to h, respectively and shown in the upper and lower panels of gure 17 (after normalizing the SM value ghShMh tree = 3m2h=v). Focusing on the tree-level results, one can observe that the triple SM-like Higgs self-coupling ghhh in favor of the highly strong PT (i.e., & 3) is close to its SM value ghShMh tree, while large deviation (mostly suppression) of ghhh from ghShMh tree is possible for the weakly strong PT (i.e., . 1:5). Another transparent observation is that the hhh coupling at tree-level cannot be enhanced in Type I (for mH & 600 GeV) and Type II models, see ref. [28] for analytical understanding of these features. However, we stress that this conclusion will be dramatically changed when the one-loop corrections to the hhh coupling are taken into account. As shown in the lower panel plots, the coupling ghhh at one-loop level are absolutely enhanced in both models and the largest normalized coupling ghhh=ghShMh tree can be about 2.5, corresponding to 150% enhancement. This allow us to conclude that the strong PT ( 1) in the 2HDM typically induces the enhancement on the hhh coupling. Next, we would like to quantitatively explore the relation between the phase transition strength and the content of the derivation the triple Higgs coupling. In general, the loop-level hhh coupling exhibits a larger deviation with increased strength of the phase transition. Whereas, the tree-level hhh coupling shown in the upper panel plots does not display such a proportionality behavior. This dramatic change implies that the loop corrections coming from the CW potential and counter-terms are important in general when the phase transition is of strong rst order and can even be dominant over the tree-level contribution in the case of the extremely strong phase transition. It is also apparent in gure 17 that the highly strong PT induces a substantial enhancement on the hhh coupling. In contrast, the hhh coupling normalized to the SM tree value can vary from 1 to 2.5 for the weakly strong PT of = 1 large triple Higgs coupling hhh is a necessary but not su cient condition of realizing the highly strong PT. For instance, if the deviation is smaller than 100%, then possibility of the highly strong PT ( & 3 (2:5) in Type I (II)) will be eliminated. As a result, the size of the triple Higgs coupling hhh derives an upper bound on the achievable value of . In some sense, this is phenomenologically useful because we have built a connection between the phase transition involving the thermal contribution and a measurable observable at zero temperature. Therefore, the measurement of the triple Higgs coupling could be an indirect approach of probing the phase transition at colliders. Experimentally, the deviation of the triple Higgs coupling can be detected at both lepton colliders (i.e., ILC [100], CEPC [101] and FCC-ee [102] ) and hadron colliders such as LHC and SppC [103]. At hadron colliders, the resonant Higgs pair production is promising while special attention needs to be paid when the heavier CP-even state H produces a destructive interference with the SM top box diagram process [104]. Upon the sensitivity of 50% supposed to be achieved at the HL-LHC, a large amount of the (nearly entire) parameter space in Type I (II) model leading to strong PT can be probed through the di-Higgs production into bb and bbW +W channels in the ultimate operation of LHC Run-2 [105{107]. In our case, ghhh has the same sign as the SM value and hence results in the destructive interference between the s-channel h-mediator triangle diagram and the top box diagrams of the gg ! hh production process. This implies increasing ghhh will decrease the production cross section [108]. Previous studies demonstrated that when ghhh ' 2:45ghShMh tree an exact cancellation between these two diagrams is accomplished at the threshold of the di-Higgs invariant mass mhh = 2mt [104, 109, 110]. Due to the low acceptance at LHC for large ghhh, a cut mhh < 2mt is imposed [104, 109]. MVA analysis of ref. [109] shows that, for the parameter space leading to the SFOPT (presented in gure 17), the observation signi cance in the bb channel with the integrated luminosity of 3 ab 1 at 14 TeV would decrease from 10 to 4 in both Type I and Type II models. In measuring the triple Higgs coupling hhh the lepton machines are typically more powerful, using the Higgs associated process e+e ! Z ! Zh (hh). The better designed sensitivities at the CEPC [101], FCC-ee [102] and ILC1000 are roughly 20-30%. This indicates that almost the full parameter spaces that are compatible with 1, particularly for mH & 500 GeV, are within the future detection reach. The other Higgs self-coupling of interest is the Hhh coupling gHhh, which is also relevant to the Higgs pair production through the s-channel H mediator triangle diagram. The Hhh coupling at one-loop as a function of mH is depicted in gure 18. In contrast to the hhh coupling, the one-loop corrections to Hhh coupling are vanishingly small near the alignment limit. We thus do not show the tree-level result. It is important to mention that the Hhh coupling can be signi cant even in the alignment limit, which can be observed ghShMhtree = 3m2h=v. In contrast to the hhh coupling, the one-loop corrections to the Hhh coupling are vanishingly small near the alignment limit. To have a better visualization only HJEP05(218) points are shown. in gure 18. For instance, the Hhh coupling is about (30 50)% of the tree-level SM hhh coupling for the highly strong PT ( 3) and can be even comparable with or larger than ghShMh tree as the PT is weakly strong ( 1 2). Notably, the obtained Hhh coupling gHhh in the successful SFOEWPT scenarios can have either the same sign as or the opposite sign to the coupling ghhh. The consequence of the sign ip of the gHhh will a ect the s-channel H-resonant triangle diagram contribution to the gg ! hh process, whose amplitude is proportional to the product of gHhh and gHtt = CUH yt, resulting in a change on the mhh lineshape due to the interference between the triangle diagram of the signal and the continuum top box diagram. When the interference is destructive, special attention needs to be paid [104]. The study of ref. [111] indicates that most of our SFOPT points can be detected at 5 signi cance provided that gHhh with integrated luminosity of 3 ab 1 using the bb channel. gHtt > 300 GeV at 14 TeV 8 Conclusions and outlook Taking into account theoretical and up-to-date experimental constraints, we studied the electroweak phase transition in the framework of the CP-conserving 2HDM of Type I and Type II models near the alignment limit. The thermal potential was expressed in terms of modi ed Bessel functions, which allows for a fast numerical evaluation and high precision compared to the simpler high/low temperature approximations. While both 1-stage and 2-stage phase transitions were shown to be realized within the 2HDM, in this paper we focused on scenarios leading to 1-stage phase transitions at electroweak scale, for which rst order and the second order phase transitions are distinguished. We analyzed the properties of the rst order phase transition, observing that the eld value of the electroweak symmetry breaking vacuum at the critical temperature is strongly related to the vacuum depth of the 1-loop potential at zero temperature, while the critical temperature re ecting the size of the thermal e ect is characterized by the temperaturedependent potential. In general, the critical temperature Tc tends to be higher as the BSM states becomes heavier, and on the other hand Tc can be down to 100 GeV when at least one light BSM Higgs bosons present in the spectrum. We have also observed that the thermal correction to the mass is important in driving a SFOPT. The strength of the transition, a key property for the electroweak baryogenesis mechanism, depends largely on the allowed mass spectrum. Requiring a SFOPT with 1 forces down the mass scale for the new scalars and the preferred ranges for all the scalar masses below 600 GeV. We demonstrate that SFOPT (i.e., 1) required for baryogenesis is possible in both Type I and Type II models. In Type I model, SFOPT is achievable in the parameter space where a large mass splitting is present between two neutral Higgs bosons such as mH mA and mA mH . In either case, the charged Higgs mass is close to either mH or mA required by the EWPD. The mass spectrum among the extra Higgs bosons in the Type II model is, on the contrary, strongly constrained due to avor observables, which push the mass of the charged Higgs above 600 GeV. As a result, scenarios leading to ' mH mA and mH ' mA mH . In view of large mass splitting between H and A, both pp ! H ! ZA and pp ! A ! ZH can be \smoking gun" collider signatures related to a SFOPT in the 2HDMs as the cross sections via gluon-fusion production in these two channels predicted for SFOPT points are typically up to 1 pb. In addition to large mass splitting, SFOPT can also take place in Type I even if all the masses of the three extra Higgs bosons (A, H and H ) are degenerate around 350 GeV. Such scenario leads to potentially testable consequences through the A ! Zh decay channel at colliders. Following the analysis of the benchmark scenarios, we investigated the implications of a SFOEWPT on the LHC Higgs phenomenology. Various characteristic collider signatures at the 13 TeV LHC have been identi ed, among which the gluon-fusion production cross section of A and H in the decay channel displays a correlation with the PT strength . It turns out that new physics searches at collider machines can provide an indirect channel to examine the EWPT scenarios. Finally, we verify that an enhancement on the triple Higgs coupling hhh (including loop corrections) is a typical signature of the SFOPT driven by the additional doublet. The PT with larger strength is associated with larger deviation of the loop-level triple Higgs coupling hhh with respect to the SM value, which can help to enhance an energy barrier. Meanwhile, we notice that the other triple Higgs coupling gHhh can also be comparable with the triple Higgs coupling in the SM for SFOPT so that the search for the heavy neutral Higgs H through the gg ! hh process is possible for small tan since the top Yukawa coupling of the H is proportional to cot . We leave for future work the interplay of gravitational waves signals and testable colliders signatures for SFOPT benchmark scenarios presented in this paper. This success would build a link between early Universe cosmology and collider detection, which could provide additional constraints in the allowed parameter space of the 2HDM. We believe that such connection will have a signi cant physical value and serves as a useful guide for collider search strategies. Acknowledgments We would like to thank M. Trott and J. Cline for useful discussions and communication. We also appreciate N. Chen for careful reading and comments on the manuscript. JB is supported by the Collaborative Research Fund (CRF) under Grant No. HUKST4/CRF/13G. He also thanks the LPSC Grenoble for support for a research stay during which part of this work was performed. The work of LGB is partially supported by the National Natural Science Foundation of China (under Grant No. 11605016), Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. NRF- 2016R1A2B4008759), and Korea Research Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, and I. C. T (Grant No. 2017H1D3A1A01014046). Y.J. acknowledges generous support by the Villum Fonden and the Discovery center. Y.J. also thanks for nancial support provided by Chongqing University for multiple visits at di erent stages during the completion of this paper. A Thermal mass for SM gauge bosons The thermal masses of the gauge bosons are more complicated. Only the longitudinal components receive corrections. The expressions for these in the SM can be found in ref. [79], where the script L (T ) denotes the longitudinal (transversal) mode. Their contributions from the extra Higgs doublet are easy to be included (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) L W L W 3 = LA = 6 6 6 11 g2T 2; 11 g2T 2; 11 g02T 2 ; T W = 0 T W 3 = 0 L W LA = 1 g02T 2 : 6 W 3 = 1 g2T 2; L 6 M W2L 4 = 1 g2(h21 + h22) + 2g2T 2: Adding them together, for the longitudinally polarized W boson, the result is This includes contributions from gauge boson self-interactions, two Higgs doublets and all three fermion families. The masses of the longitudinal Z and A are determined by diagonalizing the matrix The eigenvalues can be written as 41 (h12 + h22) g 2g02T 2 : where 2 = 1 64 (g2 + g02)2(h12 + h22 + 8T 2)2 g2g02T 2(h21 + h22 + 4T 2): Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 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Jérémy Bernon, Ligong Bian, Yun Jiang. A new insight into the phase transition in the early Universe with two Higgs doublets, Journal of High Energy Physics, 2018, 151, DOI: 10.1007/JHEP05(2018)151