A new insight into the phase transition in the early Universe with two Higgs doublets
PPrepared for submission to JHEP
A new insight into the phase transition in the earlyUniverse with two Higgs doublets
J´er´emy Bernon c , Ligong Bian a,b and Yun Jiang d a Department of Physics, Chongqing University, Chongqing 401331, China b Department of Physics, Chung-Ang University, Seoul 06974, Korea c Institute for Advanced Studies, The Hong Kong University of Science and Technology, ClearWater Bay, Kowloon, Hong Kong S.A.R, China d NBIA and Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17,DK-2100, Copenhagen, Denmark
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We study the electroweak phase transition in the alignment limit of the CP-conservingtwo-Higgs-doublet model (2HDM) of Type I and Type II. The effective potential is eval-uated at one-loop, where the thermal potential includes Daisy corrections and is reliablyapproximated by means of a sum of Bessel functions. Both 1-stage and 2-stage electroweakphase transitions are shown to be possible, depending on the pattern of the vacuum de-velopment as the Universe cools down. For the 1-stage case focused on in this paper, weanalyze the properties of phase transition and discover that the field value of the elec-troweak symmetry breaking vacuum at the critical temperature at which the first orderphase transition occurs is largely correlated with the vacuum depth of the 1-loop potentialat 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 testablesignatures 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 additionaldoublet. As a result, SFOEWPT might be able to be probed at the LHC and future leptoncolliders through Higgs pair production.
Keywords:
Electroweak phase transition, Beyond the Standard Model, Multi-scalar sec-tor
ArXiv ePrint: arXiv:1712.08430 a r X i v : . [ h e p - ph ] A p r ontents T c evaluation scheme 196 Properties of the first order EWPT 21 – 2 – Introduction
After the discovery of the 125 GeV Higgs boson [1, 2] and the accumulation of LHCdata, no evidence of the new physics has been observed yet. Therefore, it is time to inquirewhether the Standard Model (SM) of particle physics is actually complete to describe thephysics at the electroweak scale. In the meantime, the origin of the baryon asymmetryof the Universe (BAU) is still one of the important open puzzles in particle physics andcosmology. To explain the BAU, the three Sakharov conditions [3] must be fulfilled. Theelectroweak baryogenesis (EWBG) [4] is a possible mean to account for the generation ofan asymmetry (imbalance) between baryons and antibaryons produced in the very earlyUniverse. The success of EWBG requires two crucial ingredients: CP violation and strongfirst order phase transition (SFOPT), neither of which however can be addressed in theSM framework. First, the SM fails to produce a sufficiently large baryon number due toa shortage of CP violation in the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The othershortcoming of the SM is the absence of departure from thermal equilibrium which couldhave been realized by a SFOPT: for the observed value of the SM-like Higgs mass this isnot accomplished. The phase transition in the early Universe from the symmetric phase tothe electroweak symmetry breaking (EWSB) phase actually belongs to a smooth crossovertype [5]. It has been derived using lattice computation that the phase transition in the SMcan only be strong first order when the Higgs mass is around 70-80 GeV [6–9]. Therefore,a successful EWBG invokes new physics at the electroweak scale [10]. Theories that gobeyond the SM typically have an extended Higgs sector, which may contain the ingredientsfor a SFOPT as well as new CP-violating interactions as needed for EWBG, usually alsoproducing new signatures at colliders.The two-Higgs-doublet model (2HDM) is the simplest renormalizable framework torealize EWBG. In this model the scalar potential is extended by an additional SU (2) L doublet, where a charged Higgs together with two additional neutral scalars are introduced.Through their portal interactions with the SM-like Higgs, the finite temperature potentialcan develop a potential barrier during the Universe cooling down, leading to strengthen thephase transition at the critical temperature. On the other hand, the CP violation can existeither explicitly in these portal couplings or spontaneously via a relative phase between thevacuum expectation values (VEVs) of two doublets [12]. Interestingly, the CP violation(beyond the SM) can be detected indirectly at high precision electric dipole moments(EDMs) experiments. With the recent improvement of the EDMs measurements, the CPviolation phases that are needed for the baryon asymmetry are severely constrained [13].In order to evade the bound one may expect a cancellation arising from the differentcontributions of EDMs predictions, see Ref. [14–18].The electroweak phase transition (EWPT) in the 2HDM context has been extensivelystudied 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 insignificant in the EWPTprocess [19, 20, 23], the CP-violating phase at finite temperature is found to be important Though 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] – 3 –n a recent study [27] where the analysis was performed after taking into account the LHCrun-2 constraints. In general, none of the scalar states of the 2HDM resembles a SM-Higgsboson that was observed at the LHC. However, such a SM-like Higgs boson h can arise in thealignment limit, a particularly interesting limit of this model where only one Higgs doubletacquires the total electroweak vev, namely the couplings of H to gauge boson pairs vanishwhile h possesses SM-like couplings [28–30]. In terms of the model parameters, this limitcorresponds to sin( β − α ) (always positive in our convention) to be close to 1. Driven bythe LHC Higgs data, in this paper we focus on the alignment limit (here sin( β − α ) ≥ . h to be the 125 GeV SM-like Higgs observed at the LHC. To proceed thenumerical analysis we take the points passing all existing experimental bounds (by the timeof paper publication) generated from extensive scans in Ref. [28] and additionally employthe 1-loop improved theoretical constraints, the updated measurements coming from flavorphysics and the recent LHC run-2 bounds searching for heavy resonances. Our aim is toidentity the parameter space of the 2HDM that can lead to a SFOPT and investigate theimplications of a SFOPT required by baryogenesis on the LHC Higgs phenomenology.It is inspiring to note that the cosmological EWPT can leave signatures of gravitationalwaves (GW) after the nucleation of the true vacuum bubbles, with typical red-shifted spec-trum frequency around O (10 − − − ) Hz [31], which are detectable in the Evolved LaserInterferometer Space Antenna (eLISA) [32], DECi-hertz Interferometer Gravitational waveObservatory (DECIGO), UltimateDECIGO and Big Bang Observer (BBO) [33]. However,these two effects might be quite incompatible due to an opposite preference occurring in thebubble wall velocity. The baryon asymmetry generation process within EWBG demands arelatively low bubble wall velocity in order to have enough time for the chiral asymmetrygeneration process to take place, this will later be transformed to the baryon asymmetryby the sphelaron process [10]. Of course, when performing the computation of the BAU inthe EWBG mechanism, one should keep in mind that in addition to being subject to largetheoretical uncertainties, the detailed calculations of the baryon asymmetry rely on thewall velocity of the bubble generated during the EWPT, see Ref. [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 difficultin the 2HDM, to simultaneously accomplish the EWBG and produce the detectable GWsignals generated during the EWPT especially through acoustic waves [40, 41].This paper is organized as follows. In Sec. 2 we first briefly review the CP-conserving2HDMs of Type I and Type II and discuss the status in view of the existing experimentalbounds. Next, we describe in Sec. 3 the details of the finite temperature potential andprovide a fast numerical handle for the thermal potential. In Sec. 4, the one-stage andtwo-stage PT are demonstrated and classified. Subsequently, we present in Sec. 5 a usefulcomputational scheme used to single out the one-stage PT and, more importantly, toevaluate the critical temperature T c for the one-stage PT. Having studied the theoreticalissues of the model and built the computational tools, we then proceed with the numericalanalysis and investigate the properties of the phase transition which are presented in Sec. 6.In particular, the relations between T c and extra Higgs masses as well as the influence of– 4 –he effective potential at zero temperature on the field value of the electroweak symmetrybreaking vacuum are analyzed. In Sec. 7 benchmark scenarios leading to the SFOEWPT areestablished and their implications for future measurements at colliders are also discussed.Finally, Sec. 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. Let us start with a brief review of the tree-level 2HDM at zero temperature. Thegeneral 2HDM is obtained by doubling the scalar sector of the SM, two doublets withidentical quantum numbers are present. In general, CP violation may be present in thescalar sector and the Yukawa sector contains generic tree-level flavor changing neutralcurrents (FCNCs) mediated by the neutral scalar states. Here we consider a CP conservingHiggs sector and the absence of tree-level FCNCs. The first condition is obtained byimposing a reality condition on the parameters of the potential, and the second requirementis achieved by imposing a Type I or Type II structure on the Yukawa sector, this is achievedby imposing a softly-broken Z symmetry [42, 43].Denoting by Φ , Φ the two Higgs doublets, the tree-level potential of this model isexpressed as, V (Φ , Φ ) = m Φ † Φ + m Φ † Φ − (cid:104) m Φ † Φ + h.c. (cid:105) + λ † Φ ) + λ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ | Φ † Φ | + (cid:20) λ † Φ ) + h.c. (cid:21) . (2.1)In this basis, the Z symmetry under which Φ → − Φ is manifest in the quartic terms,while it is softly broken by the introduction of the m term. In general, m and λ arecomplex. We consider in this work a CP conserving Higgs sector and set all λ i and m as real parameters, see [27] for a CP violating study. In this basis, both Higgs doubletshave a non-zero vacuum expectation value (vev). We parametrize the degrees of freedomcontained in the Higgs doublets as,Φ i = (cid:32) φ + i ( v i + ρ i + iη i ) / √ (cid:33) , i = 1 , . (2.2)where v i are the vevs of the two Higgs doublets. At zero temperature one has the relation v + v = v (cid:39) (246 GeV) . For convenience, we use the shorthand notation v ≡ v fromnow on and define v = v cos β and v = v sin β , tan β is therefore the ratio of the two vevsat T = 0.The mass parameters m and m in the potential Eq. (2.1) are determined by thepotential minimization conditions, m = m t β − v (cid:0) λ c β + λ s β (cid:1) ,m = m /t β − v (cid:0) λ s β + λ c β (cid:1) , (2.3)– 5 –ere the shorthand notations s β ≡ sin β , c β ≡ cos β and t β ≡ tan β and λ ≡ λ + λ + λ are employed.Though tan β is a physical parameter here, it is still possible to redefine the twodoublets and go to a basis where the full vev resides entirely in one of the two doublets:the so-called Higgs basis ( H , H ) [44]. This change of basis is generally possible as observedby the invariance of the gauge kinetic terms of the two doublets under a U(2) Higgs flavortransformation. In the Higgs basis H has the full vev v and thus is precisely the SM Higgsdoublet. In general however both neutral components mix upon EWSB and a SM-like CP-even mass eigenstate is not automatic. On the contrary, if one of the CP-even eigenstatesis parallel to the neutral H 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 defined as the presence of a CP-even eigenstate in the vev directionin the scalar field space.In the electroweak vacuum, the squared mass matrices in the neutral CP-even, CP-oddand charged scalar sectors are respectively given by, M P = m t β + λ v c β − m + λ v s β − m + λ v m /t β + λ v s β , (2.4) M A = (cid:20) m − λ v s β (cid:21) (cid:32) t β − − /t β (cid:33) , (2.5) M ± = (cid:20) m −
14 ( λ + λ ) v s β (cid:21) (cid:32) t β − − /t β (cid:33) . (2.6)The CP-even mass eigenstates h and H , with m h ≤ m H , are obtained through the diago-nalization of M P , they are expressed in terms of the neutral components of the doubletsas, (cid:32) Hh (cid:33) = (cid:32) c α s α − s α c α (cid:33) (cid:32) ρ ρ (cid:33) , (2.7)where the mixing angle α is introduced and is expressed in terms of the entries of themass matrix. Diagonalization of M A leads to a massive CP-odd scalar A and a masslessGoldstone boson G , while M ± leads to a charged state H ± and a charged masslessGoldstone boson G ± . Their tree-level masses read m H,h = 12 (cid:104) M P, + M P, ± (cid:113) ( M P, − M P, ) + 4( M P, ) (cid:105) , (2.8) m A = m s β c β − λ v , (2.9) m H ± = m s β c β −
12 ( λ + λ ) v . (2.10) We point out that in the review article [46] a factor of 2 is missing in front of λ and λ + λ terms inthe formula of m A and m , respectively. – 6 – able 1 . Tree-level vector boson couplings C V ( V = W, Z ) and fermionic couplings C U and C D 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. Type I, II Type I Type IIHiggs C V C U C D C U C D h sin( β − α ) cos α/ sin β cos α/ sin β cos α/ sin β − sin α/ cos βH cos( β − α ) sin α/ sin β sin α/ sin β sin α/ sin β cos α/ cos βA β − cot β cot β tan β Using Eqs. (2.8)-(2.10) one can inversely solve the potential parameters, λ , . . . , λ interms of four physical Higgs masses and the CP-even Higgs mixing angle α , supplementedby the Z soft-breaking parameter m [45]. This means that the scalar potential can beentirely determined by these seven parameters and therefore allows us to choose them asa set of complete free inputs for the numerical analysis.As mentioned previously, we imposed a Z symmetry on the potential Eq. (2.1) inorder to forbid Higgs-mediated tree-level FCNCs. Out of the four independent realizationsof this symmetry in the fermion sector, we study two of them: the so-called Type I modelwhere only Φ couples to fermions and the Type II model where Φ couples to down-type fermions and Φ to up-type fermions, see [47] for details. These particular structuresredefine multiplicatively the Higgs couplings to fermions as compared to the SM predictions,we denote as C U,D,V theses multiplicative factors for the up-type fermions, down-typefermions and massive gauge bosons, respectively. The Higgs couplings to massive gaugebosons do not depend on the Z symmetry charges but are directly obtained from gaugesymmetry alone. In Table 1 we present these factors for the three physical scalar states ofthe theory. Important intuition can be gained by re-expressing these factors in terms of( β − α ) and β , in particular to understand their behavior in the alignment limit sin( β − α ) ≈ C h, I F = C h, II U = cos α/ sin β = sin( β − α ) + cos( β − α ) cot β, (2.11) C h, II D = − sin α/ cos β = sin( β − α ) − cos( β − α ) tan β, (2.12) C H, I F = C H, II U = sin α/ sin β = cos( β − α ) − sin( β − α ) cot β, (2.13) C H, II D = cos α/ cos β = cos( β − α ) + sin( β − α ) tan β. (2.14) For a viable 2HDM scenario, we require here tree-level stability of the potential, whichmeans that Eq. (2.1) has to be bounded from below, requiring λ , λ > , λ + λ − | λ | > −√ λ λ , λ > −√ λ λ . (2.15)In this work we improve the bounds supplemented by the radiative corrected potential,as will be shown in Sec. 3.2. Additional theoretical constraints from S-matrix unitarity– 7 –nd perturbativity are required. Tree-level unitarity imposes bounds on the size of thequartic couplings λ i or various combinations of them [49, 50]. Similarly (often less stringent)bounds on λ i may be obtained from perturbativity arguments. Next, we briefly describe the impact of the experimental bounds on the parameterspace of the model. First, electroweak precision data (EWPD), essentially the T parame-ter, constrains the mass difference between m H ± and m A or m H , one of the two neutralstates should indeed be approximatively paired with the charged state in order to restorea custodial symmetry of the Higgs sector [51, 52]. Second, the recent measurement on BR ( B → X s γ ) [53] excludes low values of m H ± (cid:46)
580 GeV in the Type II model [54]. Asa consequence, the preferred ranges for the scalar masses are pushed above ∼
400 GeV.Third, LHC measurements of the 125 GeV signal rates put large constraints on the 2HDMparameter space, in particular they tend to favor the alignment limit where the Higgs cou-plings 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 asperformed in [28]. A very important search for the Type II model is in the
A, H → τ τ channel, either through gluon-fusion or b ¯ b -associated production [56, 57]. The ATLAS Run-2 constraint is much stronger than the corresponding Run-1 searches, eliminating largerportion of the parameter space at large tan β in particular. For m A < ∼
350 GeV we only findfew scenarios compatible with the experimental constraints in the Type II model . This isboth coming from the aforementioned τ τ search, as well as the H → ZA searches for CP-odd state down to 60 GeV. This final state has been searched for by the CMS collaborationduring 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 theresulting constraints have little impact. In Fig. 1 we show the allowed spectra (red pluses)for the two types of models considered here. The points labeled ‘no-EWSB’ comes fromthe requirement of proper EWSB at the 1-loop level, which will be extensively discussedin Sec. 3.2. Due to the severe constraints on the mass spectrum of the extra Higgs bosons,these experimental constraints have significant influence on the requirement of a SFOPTas we will see in Sec. 6.We now move to investigate the possibility of having a first-order phase transitionfor the surviving sample points. The interesting question is whether the parameter spacethat LHC Higgs data favors, simultaneously satisfying both theoretical constraints andexperimental bounds, can lead to a favorable prediction for a strong first-order phasetransition. For a recent one-loop analysis, leading to slightly more stringent bounds, see [48]. This result is not fully consistent with Ref. [21] where the authors claimed the experimental constraintsare less severe for m A < ∼
120 GeV. – 8 –
Type I m H ± [ G e V ] m A [GeV] no-EWSB 20040060080010001200 0 200 400 600 800 1000 1200 Type II m H ± [ G e V ] m A [GeV] no-EWSB20040060080010001200 0 200 400 600 800 1000 1200 Type I m H [ G e V ] m A [GeV] no-EWSB 20040060080010001200 0 200 400 600 800 1000 1200 Type II m H [ G e V ] m A [GeV] no-EWSB Figure 1 . The allowed mass spectra (up to 1.2 TeV) of the BSM Higgs bosons confronting withLHC Run-2. In Type II (right), the B-physics constraint on the charged Higgs m H ± ≥
580 GeV [54]is imposed and nearly excludes the low m A points. Gray points indicate EWSB is not ensured atzero temperature when one-loop effect is included in the Higgs potential. To study the phase transition we consider the scalar potential of the model at finitetemperature. In the standard analysis, the effective potential V eff ( h , h , T ) is V eff ( h , h , T ) = V ( h , h ) + V CW ( h , h ) + V CT ( h , h ) + V th ( h , h , T ) , (3.1)which is composed of the tree-level potential at zero temperature V ( h , h ) derived inEq. (3.2), the Coleman-Weinberg one-loop effective potential V CW ( h , h ) at T = 0 givenin Eq. (3.3), the counter-terms V CT given in Eq. (3.11) being chosen to maintain the treelevel relations of the parameters in V , and the leading thermal corrections being denotedby V th ( h , h , T ). We discuss these terms separately now.– 9 – .1 The tree level potential Since our model is CP conserving, the classical value of the CP-odd field A is zeroand so are the ones for the neutral Goldstone fields. We assume the charged fields do notget VEV during the EWPT process, by taking the classical values for the charged fieldsto be zero, to strictly respect the U(1) electromagnetic symmetry and therefore ensure thephoton massless [46]. The relevant tree level potential V in terms of their classical fields( h , h ) derived from Eq. (2.1) is V ( h , h ) = 12 m t β (cid:16) h − h t − β (cid:17) − v λ h + λ h t β t β − v λ ( h t β + h )1 + t β + 18 λ h + 18 λ h + 14 λ h h (3.2)here we have eliminated m and m by using the minimization conditions Eq. (2.3). To obtain the radiative corrections of the potential at one-loop level, we use Colemanand Weinberg method [70]. The Coleman-Weinberg (CW) potential in the MS scheme andLandau gauge at 1-loop level has the form: V CW ( h , h ) = (cid:88) i ( − s i n i ˆ m i ( h , h )64 π (cid:20) ln ˆ m i ( h , h ) Q − C i (cid:21) . (3.3)The sum i runs over the contributions from the top fermion, massive W ± , Z bosons, allHiggs bosons and Goldstone bosons ; in the sum s i and n i are the spin and the numbers ofdegree of freedom for the i -th particle listed in Table 2; Q is a renormalization scale whichwe fix to Q = v and C i are constants depending on the renormalization scheme. In theMS on-shell scheme employed, C i = ( ) for the transverse (longitudinal) polarizationsof gauge bosons and C i = 3 / The charge breaking vacuum in multi-Higgs doublet models has been studied in Refs. [61–67]. Oncethe U(1) electromagnetic symmetry is broken during the EWPT, the photon acquires mass, which maychange the thermal history of the Universe [67]. We leave it to future work. Also, we do not expect thepresence of color-breaking vacuum in the process of EWPT since the bosons which actively participate intothe evolution of Higgs scalar potential are color neutral. As of our knowledge, color-breaking baryogenesisis achievable in the model with the inclusion of colored bosons (i.e. scalar leptoquarks) [68, 69]. To avoid confusion we distinguish the dynamical fields and EW vev in this paper. The classical fields( h , h ) approach the EW vacuum ( v , v ) at zero temperature. As noted in [71], the VEVs are slightly different in various gauges and the recent study [26] find thiseffect to be numerically small in the physically interesting regions of parameter space. We ignore the light SM fermions because of the smallness of their masses. In contrast, the inclusionof Goldstone modes is necessary as their masses are non-vanishing for field configurations outside theelectroweak vacuum. The photon at zero temperature is strictly massless due to gauge invariance. In most literature C i = 5 / n Z C Z =2 × / × / × / n W C W = 2 n Z C Z . The mass difference between transverse and longitudinalmodes arises from thermal corrections as will see later. – 10 – able 2 . The number of d.o.f. from SM particles of different species contributing to the thermalpotential. The fermions except the top quark are neglected due to their small masses. i t W ± Z H = { h, H, A } H ± G , G ± n i × × × × m i for SM particles include ˆ m t = 12 y t h /s β , (3.4)ˆ m W ± = 14 g t (cid:0) h + h (cid:1) , ˆ m Z = 14 ( g + g (cid:48) ) (cid:0) h + h (cid:1) , ˆ m γ = 0 , (3.5)with the corresponding SM Yukawa and gauge couplings being defined g = 2 M W /v, g (cid:48) =2 (cid:113) M Z − M W /v , y t = √ m t /v and the ones for scalar bosons are given byˆ m h,H = eigenvalues( (cid:100) M ) , (3.6)ˆ m G,A = eigenvalues( (cid:100) M ) , (3.7)ˆ m G ± ,H ± = eigenvalues( (cid:100) M ± ) , (3.8)where the corresponding matrices (cid:99) M X ( X = P, A, ± ) are (cid:100) M X = λ h + m t β − λ v t β − λ v t β t β + Θ X − m + Θ X − m + Θ X λ h + m t − β − λ v t β t β − λ v t β + Θ X . (3.9)Here the Θ Xij terms listed below are different for X = P, A, ± Θ P = λ h + λ h , Θ A = ¯ λ h , Θ ± = λ h , Θ P = λ h h , Θ A = λ h h , Θ ± = ( λ + λ ) h h , Θ P = λ h + λ h , Θ A = ¯ λ h , Θ ± = λ h , (3.10)in which ¯ λ ≡ λ + λ − λ .With V CW being included in the potential, the minimum of the Higgs potential willbe slightly shifted, and hence the minimization conditions Eq. (2.3) no longer hold. Tomaintain these relations, we add the so-called “counter-terms” (CT)[24] , V CT = δm h + δm h + δλ h + δλ h h + δλ h , (3.11) We notice typos occurring in the thermal mass of SM fermions (c.f. Eqs. (A.19) and (A.20)) in Ref. [26]. In addition, we do not include more complicate terms to compensate the shift of mass matrix of h ,because these shift effects are estimated to be negligible in our scenario. – 11 –here the relevant coefficients are determined by, ∂V CT ∂h = − ∂V CW ∂h , ∂V CT ∂h = − ∂V CW ∂h , (3.12) ∂ V CT ∂h ∂h = − ∂ V CW ∂h ∂h , ∂ V CT ∂h ∂h = − ∂ V CW ∂h ∂h , ∂ V CT ∂h ∂h = − ∂ V CW ∂h ∂h , (3.13)which are evaluated at the EW minimum of { h = v , h = v , A = 0 } on both sides. As aresult, the vevs of h , h and the CP-even mass matrix will not be shifted.One technical difficulty involved at this step arises from the inclusion of the Goldstonebosons in the CW potential. Due to the variation of the scalar field configuration withtemperature (which we will see shortly), the Goldstone boson may acquire a non-zero massat finite temperature, enforcing the inclusion of Goldstone modes in the sum. Nonetheless,in the electroweak vacuum at zero temperature the masses of the Goldstone bosons arevanishing in the Landau gauge, which leads to an infrared (IR) divergence due to thesecond derivative present in our renormalization conditions Eq. (3.13). This means thatrenormalizing the Higgs mass at the IR limit is ill-defined [73]. To overcome this divergence,we take a straightforward treatment developed in [24] and impose for Goldstone bosons anIR cut-off at SM Higgs mass, m IR = m h . Although a rigorous prescription used to dealwith the Goldstone’s IR divergence was developed in [22], Ref. [24] argued that this simpleapproach 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 thesum and add instead the following Goldstone contribution to the right hand of Eq. (3.13)132 π ln m G ( h , h ) Q (cid:18) ∂ m G ( h , h ) ∂h ∂h , ∂ m G ( h , h ) ∂h ∂h , ∂ m G ( h , h ) ∂h ∂h (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) vev (3.14)with the replacement for the singular term m G ( h , h ) | VEV → m IR in the logarithm. Notethat the Goldstone bosons have a vanishing contribution to the first derivative evaluatedat the vev.Beyond tree level the true EW vacuum must be preserved when the one-loop correc-tions are taken into account. This demands that the potential after the inclusion of theCW and counter-terms still form a global minimum at the EW vacuum. As seen in Fig. 2,the CW term (green dotted) often lifts up the potential at the EW vacuum, resulting thelocal minimum shifting inward or even leading to a false vacuum. On the other hand, theCT effect (blue) drags down the potential at the EW vacuum and thus helps to accom-plish a true EW vacuum. As a result, the competition between these two opposite effectsdetermines the existence of a global minimum at the EW vacuum. We present in Fig. 2two examples where the left one accomplishes a true EW vacuum, while the potential inthe 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 Fig. 1. This is an additionalimportant constraint that excludes about 10% (5%) points in the Type I (II) model, inparticular for the points with m A ≤
300 GeV. In the m A < m h / m A scenario), it turns out that EWSB at zero temperature can be achieved as long as at least– 12 – [ GeV ]- - / [ GeV ] V V CW V + V CW V CT V + V CW + V CT
50 100 150 200 250 300 350 h [ GeV ]- - / [ GeV ] Figure 2 . Tree level and loop-correction contributions to the potential at zero temperature fortwo model points with tan β = 1 , sin( β − α ) = 1. The remaining parameters corresponding tothe point shown in the left (right) plot are λ = λ = λ = 2 . . , λ = − . − , λ =3 . − . , m = 315 ( −
70) GeV. Clearly, the point shown in the left plot has a true EW vacuumwhile the one on the right plot has only a local minimum at v .
100 200 300 400 h [ GeV ]- - ( h,T = ) / [ GeV ] ( m H = m H ± , m )( )( )( - )( )( )( - ) Figure 3 . The one-loop potential at zero temperature (including the CW potential and counter-terms). m H = m H ± is assumed at two common scales 300 (blue) and 600 (red) GeV, m A = 50 GeVis taken. The picture is negligibly modified for m A = 200 GeV. Three values of tan β , tan β = 1(solid and dashed lines), tan β = 10 (thick dashed lines) and tan β = 20 (thick dotdashed lines)are shown for which m is chosen such that none of quartic couplings exceeds the pertubativitybound. 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, wedisplay in Fig. 3 the one-loop potential at zero temperature (including the CW potentialand counter-terms). For simplicity, we assume m H = m H ± , which is typical mass spectrarequired by the T parameter . The authors of Ref. [74] have shown that low- m A scenariocan be phenomenologically alive in the parameter space where the SM-like Higgs h hasvery small coupling to AA , which leads to tan β (cid:46) β (cid:38)
12 for m H = 600 GeV inthe deep alignment limit. The low tan β solution requires a severe tuning in the parameter m , and in the allowed range m (cid:39) the zero temperature potential (c.f. the The lighter the CP-odd state A is, the stronger the degeneracy between H and H ± should be. – 13 –ed solid line) at EW vacuum v is higher than the one at the origin. Moreover, a proper EWvacuum can be developed as the symmetry soft-breaking parameter m increases. Thiscan be achievable for the case of m A ≥ m h / h → AA decay is kinematicallysuppressed. On the other hand, the large tan β solution, though possible in Type I model,strongly constrains m and tends to lift the potential. Hence, the importance of this classof solution is very marginal and no points were found in our numerical analysis. In addition,tan β (cid:38) τ τ final states during Run-1 [75]. As a comparison, we alsoexhibit the potential at a lower common scale m H = m H ± = 300 GeV. This example isonly applicable in Type I model. One can observe that the potential generically reachesa global minimum at the EW vacuum and the depth of this minimum is less sensitive totan β . This implies that when the new scalars introduced are not heavy, the loop effect isnot substantial and thus the potential is largely governed by the tree-level.We conclude that the requirement of proper EWSB at zero temperature, in synergywith m H ± ≥
580 GeV required by B -physics measurements [53], entirely exclude thescenario of existing a light pseudoscalar A in Type II model that was delicately studied inRef. [74]. As will show shortly, these theoretical constraints will play an important role inachieving a strong first-order phase transition. The finite temperature corrections to the effective potential at one-loop are givenby [76] V th ( h , h , T ) = T π (cid:88) i n i J B,F (cid:18) m i ( h , h ) T (cid:19) , (3.15)where the functions J B,F are J B,F ( y ) = ± (cid:90) ∞ dx x ln (cid:104) ∓ exp (cid:16) − (cid:112) x + y (cid:17)(cid:105) , (3.16)with y ≡ m i ( h , h ) /T and the upper (lower) sign corresponds to bosonic (fermionic) con-tributions. The numerical evaluation of this exact integral is very time-consuming (notablyfor the y < J B,F . At small y ( y (cid:28) , Eq. (3.16) can be approximatedby J y (cid:28) B ( y ) (cid:39) − π
45 + π y − π y / − y
32 ln ya B , (3.17) J y (cid:28) F ( y ) (cid:39) − π
360 + π y + y
32 ln ya F , (3.18) The high/low T approximations do not necessarily lead to small/large y , which also depends on thefield-dependent mass in the numerator. – 14 – mallyapprox J F appr ( Bessel,n = ) largeyapprox J F appr ( Bessel,n = ) J F exact J F appr ( Bessel,n = ) - - - - J F [ y ] smallyapprox J B appr ( Bessel,n = ) largeyapprox J B appr ( Bessel,n = ) J B exact J B appr ( Bessel,n = ) - - - - J B [ y ] smallyapproxlargeyapprox J B exact J B appr ( Bessel,n = ) J B appr ( Bessel,n = ) J B appr ( Bessel,n = ) - -
20 0 20 40 - J B [ y ] Figure 4 . Thermal function for fermionic (left) and bosonic (middle) states for positive y . Forbosonic states, we additionally present the negative y range since their thermal mass can be negativeat T (cid:54) = 0. In each plot the result of the exact integral is shown in solid black curve. Red and bluecurves give the small and large y approximations, respectively. Three dashed lines illustrates theresult evaluated by summing over the Bessel functions at different order. where a F = π exp( − γ E ) and a B = 16 a F with the Euler constant γ E = 0 . y , J y (cid:29) B,F ( y ) (cid:39) − (cid:16) π (cid:17) / y / exp (cid:16) − y / (cid:17) (cid:18) y − / (cid:19) . (3.19)In order to make a quantitive assessment of the approximation precision we plot inFig. 4 the small/large y approximations as well as the direct numerical evaluation of theintegral. (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 inthe ranges y ∈ ( − ,
5) for bosons and y ∈ (0 ,
5) for fermions, while the large y expansion(blue curve) converges to the exact integral for y >
10 for both functions. A gap isthen present between the small and large y approximations in the transition range y ∈ (5 , T to know which approximation shouldbe applied, this largely increases the evaluation time. Second, the above approximationsEqs. (3.17)-(3.19) are only valid for y > If this happens, Ref. [21] suggests that only the real part of the integral J B should be chosen in the evaluation as the imaginary part is irrelevant in extracting theglobal minimum. The thermal integrals J B,F given by Eq. (3.16) can be expressed as an infinite sum of For instance, in the SM the field-dependent mass for Higgs field is m h = 3 λh − µ and turns negativeat low field configuration. Similarly for the Goldstone bosons. Tachyonic mass configurations generate a negative local curvature of the potential, leading to a localmaximum rather than a minimum. – 15 –odified Bessel functions of the second kind K n ( x ) with n = 2 [77], J B,F ( y ) = lim N → + ∞ ∓ N (cid:88) l =1 ( ± l yl K ( √ yl ) , (3.20)with the upper (lower) sign corresponds to bosonic (fermionic) contributions. Our numeri-cal results show that the leading order l = 1 does not provide a good approximation of thefull integrals. Instead, inclusion up to l = 5 order in the expansion can match the exactintegral 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). Fig. 4 also shows that thethermal function is negative for positive y thus dragging the potential down and leadingto the formation of two degenerate vacua. As expected, this dragging effect arising fromthe temperature corrections diminishes as y approaches to the infinity, which correspondsto zero temperature or the decoupling limit.Finally, there is another important part of the thermal corrections to the scalar massescoming from the resummation of ring (or daisy ) diagrams [79, 80], V daisy ( h , h , T ) = − T π (cid:88) i n i (cid:104)(cid:0) M i ( h , h , T ) (cid:1) − (cid:0) m i ( h , h ) (cid:1) (cid:105) , (3.21)where M i ( h , h , T ) are the thermal Debye masses of the bosons corresponding to theeigenvalues of the full mass matrix M i ( h , h , T ) = eigenvalues (cid:104) (cid:100) M X ( h , h ) + Π X ( T ) (cid:105) , (3.22)which consists of the field dependent mass matrices at T = 0 Eq. (3.9) and the finitetemperature correction to the mass function Π X , ( X = P, A, ± ) given byΠ X = (cid:32) Π X Π X Π X Π X (cid:33) T , (3.23)with the diagonal terms beingΠ P = Π A = Π ± = c SM − y t + 6 λ + 4 λ + 2 λ , Π P = Π A = Π ± = c SM + 6 λ + 4 λ + 2 λ , (3.24)here the subscripts { , } denote the states { h , h } and c SM = 92 g + 32 g (cid:48) + 6 y t , (3.25)is the known SM contribution from the SU(2) L and U(1) Y gauge fields and the topquark [79]. It is important to note that the temperature corrections are independentof λ where a possible CP phase can reside. On the other hand, the leading correction tooff-diagonal thermal mass is vanishingly small due to Z symmetry imposed in the scalar A similar numerical analysis taking N = 50 was performed in a recent study [78]. – 16 –ector. Moreover, it was argued by [81] that subleading thermal corrections to off-diagonalself-energies are suppressed by additional powers of coupling constants and EW vevs whichare usually neglected. Therefore, we shall treat the thermal mass correction Π i as diago-nal matrices in the following numerical analysis. The thermal mass corrections of the SMgauge bosons are given in Appendix A.Historically, there was an alternative algorithm proposed by Parwani in dealing withthe thermal corrections [82]. He included the effect of thermal correction from Daisy dia-grams by means of substituting m i ( h , h ) by M i ( h , h , T ) in the V th ( h , h , T ), Eq. (3.15).It is important to note that these two approaches are not physically equivalent and theresults produced are quantitively incompatible [21]. They differ in the organization of theperturbative expansion and consistent implementation of higher order terms. The methodformulated 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 ofthis dangerous artifact unrealistically large values of the phase transition strength ξ (de-fined in Eq. (6.1)) would be obtained. Therefore, we will adopt the former consistentmethod in the following analysis. In general, a system may transit from one symmetry phase to another one. Here theelectroweak symmetry is broken as the Universe cools down, this is singled as a change in thenature of the global 0-vacuum at high temperature that gets replaced by an electroweakbreaking global vacuum at lower temperature. At any given set of parameters, the fulleffective potential Eq. (3.1) can have several extrema. Our major interest is the globalminimum vacuum state, the deepest minimum of the potential. The other extrema can beeither saddle points or maxima or local minima of the potential. In studying the thermalphase transition, it is useful to trace the evolution of the extrema as well as calculate thedifference 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 effectivepotential has one global minimum, which tends towards the point ( h , h ) = (0 , h , h fields and thermal phase transition takes place.In general one can classify the thermal phase transition according to the behavior of thevacuum development during the cooling down. For instance, the phase transition may beof first or second order, one-stage or two-stage process. In Fig. 5 we show three examplesthat illustrate the different behavior of the vacuum development with temperature, wherethe temperature decreases from left to right and the true vacuum is marked as a redplus in each graph. The model parameters corresponding to each point are summarizedin Table 3. For the case shown in the top panel, the vacuum starts to depart from the It may have resulted not only in variation of the absolute values of particle masses, but also in rear-rangement of the particle mass spectrum, which can have interesting cosmological consequences. – 17 – � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = � + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = � + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = ��� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = �� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = �� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = �� ��� + � �� ��� ��� ��������������������� � � [ ��� ] � � [ � � � ] � = � Figure 5 . Temperature evolution of the Higgs potential on the ( h , h ) plane. As temperature coolsdown, 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 the2HDM: 1-stage second order (top panel), 1-stage first order (middle panel) and 2-stage transition(bottom panel). Red arrows indicate a jump between two degenerate vacuum in the first orderphase transition while the vacuum transitions smoothly in the second phase transition. origin at T = 163 GeV, and then moves closely along the yellow line until reaching theEW vacuum at zero temperature. This phase transition is called of second order, becausethe potential minimum shifts continuously while no potential barrier develops during thecooling down. In contrast, the vacuum of the potential displayed in the middle panel islocalized in the vicinity of the origin point at high temperature. When the temperaturedecreases to T (cid:39)
157 GeV an EW vacuum located away from the origin appears, formingtwo degenerate vacua separated by an energy barrier. This gives rise to the first-order phasetransition from the origin to the EW vacuum, which is indicated by the red arrow. In thesetwo examples, the phase transition is termed one-stage. In addition to experiencing onlyone standard EWSB phase transition, the 2HDM can undergo a two-stage phase transitionas the temperature falls as shown in the lower panel. In this mechanism the first stage is aconventional second order PT in which the symmetry is broken, shortly thereafter follows afirst order PT. Another remarkable thing is that the ratio of the classical value between thetwo fields h /h shown in the upper and middle panels has very little dependence on thetemperature. However, in general, the value of h ( T ) /h ( T ) is a temperature-dependentparameter 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 h ( T ) /h ( T ) astemperature decreases: at first it monotonically increases, resulting in a deviation of the– 18 – able 3 . Parameters for three benchmark points (with all mass parameters in units of GeV) thatlead to different types of EWPT.Points Properties tan β sin α λ λ λ λ λ m m H m A m H ± Top 2nd order PT 2.98 -0.24 3.99 0.29 0.86 -1.06 0.11 49 181 35 192Middle 1st order PT 1.30 -0.59 6.05 2.00 6.63 -8.27 -1.27 176 510 376 594Bottom 2 stage PT 40 0.06 0.16 0.27 4.25 1.04 1.80 59 375 176 232 vacuum from the yellow line, then jumps to its zero temperature value (that is tan β ) atthe transition point and maintains unchanged in the remaining process. T c evaluation scheme The dynamics of the EWPT is governed by the effective potential at finite temperatureEq. (3.1) in our model. For purposes of analyzing the temperature evolution of the potentialinvolving both h and h , it is convenient to work with a polar coordinate representationof the classical fields h ( T ) and h ( T ). To that end, we define h ( T ) and θ ( T ) via h ( T ) ≡ h ( T ) cos θ ( T ) , (5.1) h ( T ) ≡ h ( T ) sin θ ( T ) . (5.2)The tree-level potential Eq. (3.2) in the ( h, θ ) plane becomes V ( h, θ ) = 18 (cid:0) λ c θ + λ s θ + 2 λ s θ c θ (cid:1) h (5.3)+ h (cid:20) m ( t β − t θ ) t β (1 + t θ ) − v (cid:0) λ c θ c β + λ s θ s β + λ ( s θ c β + c θ s β ) (cid:1)(cid:21) , (5.4)and the remaining parts of the effective potential are much more involved and hence notshown here.When a first order PT takes place, a local minimum with (cid:104) h (cid:105) (cid:54) = 0 develops and becomesdegenerate with the symmetric minimum (cid:104) h (cid:105) = 0 as the temperature decreases, this definesthe critical temperature T c , and the two minima are separated by a potential barrier.Therefore, the evaluation of T c is of great importance in studying the EWPT and itscosmological consequences. A straightforward approach is to decrease the temperature bysmall steps and make a potential plot (like Fig. 5) at each step. Then the global minimumof the potential (starting from the EW vacuum at zero temperature) can be followed step-by-step and the critical point is found once the minimum displays a jump rather than asmooth transition. Obviously, this graphic method is feasible only for benchmark pointsbut is barely applicable for extensive scan due to its non-numerical nature. To date severalnumerical methods have been developed. In Ref. [19], going from zero temperature tohigher temperature, the critical point is taken to be the last one for which the minimumlied below the origin. This approach is no able to resolve the 2-step phase transitionswhere the potential experiences a second order PT prior to the first order PT, giving rise– 19 –o a vacuum shift from the origin at higher temperature. To overcome this problem, theauthors of [21] used advanced numerical algorithms to search for the global minimum of theeffective potential in the ( h, θ ) plane for each temperature. We employ a method consistingof the following procedures:First, we deal with points for which the ratio h /h is (approximately) temperature-independent, that is θ ( T ) = β . The effective potential Eq. (3.1) reduces to a function oftwo parameter – temperature T and T -dependent field norm h ( T ). In this case, one caneasily determine the critical temperature T c and the field norm v c at which the potentialreaches a minimum by solving the equations ∂∂h V eff ( h, T c ) (cid:12)(cid:12)(cid:12)(cid:12) h = v c = 0 , V eff ( h = v c , T c ) = V eff ( h = 0 , T c ) . (5.5)In searching for the solution of the above equations, we require a difference between thepotential at the minimum and its value at the origin smaller than 10 − GeV . As aconsequence, the solution for v c would be a value close to zero if there were no degenerateminima of the effective potential present in the process of temperature drop. This meansthat below a certain small value of v c we do not expect a decent probability of achieving afirst-order phase transition. Instead, very likely such points lead to a second-order phasetransition. For this reason we employ a technical cut v c > θ ( T ) = tan β is not obeyed at a certain temperature or within a smalltemperature interval. In this situation, ( v c , T c ) obtained as a solution of Eq. (5.5) is not thecritical vev and temperature where the phase transition occurs because the true vacuum isno longer located at the origin. Searching for the global minimum should be performed notalong the tan β line but on a two-dimensional ( h, θ ) space. We employ an algorithm whichuses the steepest descent method to find the global minimum of the effective potential.At T c the searched minimum is then compared with the value of the effective potentialevaluated at v c and the one with the lower value is chosen as the candidate for the globalminimum. For the general 2-stage PT, tracing the (temperature) evolution of the globalminimum on a 2D plane is inevitable. Here, we discard points leading to a 2-stage PTand focus on the scenarios featuring a 1-stage PT. In the following analysis, we only retainparameter points with T c ≤
300 GeV. It appears possible that the potential has a global minimum at large value of h . However, the probabilityof having a strong phase transition for these points is quite low, unless high scale phase transition isconsidered. – 20 – Properties of the first order EWPT
The strength of the phase transition is quantified as the ratio of the norm of the neutralfields to the temperature at the critical point, ξ = v c T c . (6.1)Here v c = (cid:112) (cid:104) h (cid:105) + (cid:104) h (cid:105) + (cid:104) A (cid:105) , in general, represents the value of the norm of allscalar fields involved at the broken vacuum at critical temperature T c . Note that wheninterpreting the ratio as the strength of the electroweak phase transition, one should beaware of its gauge dependence [76, 85–87]. In order to ensure that a baryon numbergenerated during the phase transition is not washed out, a strong first-order phase transitionis demanded and occurs if ξ ≥ Before presenting the main results, we discuss the specific features of the parameterspace compatible with the theoretical and experimental constraints and at the same timeleads to first order and second order phase transition. We will show results for bothType I and Type II models.
It has been shown in Fig. 5 that both first order and second order phase transition cantake place in the 2HDM. Whether first order or second order PT is developed depends onthe mass spectrum among the three extra Higgs bosons, which is directly related to the fivequartic couplings λ i and the soft symmetry breaking parameter m through Eqs. (2.8)-(2.10). Thus, it would be very interesting and useful if one can divide the entire modelparameter space into different sectors where distinct dynamics of vacuum evolution leadingto first order and second order PT take place. An initial attempt along this direction wasmade in [90] in accordance with the general geometric analysis of [91]. In [90] the authorsintroduced several discriminators in terms of certain combinations of λ i and succeeded individing into four sectors which do not overlap. However, the analysis conducted in [90] isoversimplified, only the effect of thermal mass corrections was included. When consideringthe full effective potential Eq. (3.1), such division may be highly difficult or even impossi-ble, which is reflected in Fig. 6 where we map our first order (red) and second order (blue)PT points in the 2D space of model parameters and none of the parameters exclusivelydistinguish the two types of PT points. As expected, λ and λ have marginal influ-ence since both of them only enter into the masses of two neutral CP-even scalars. Onthe contrary, λ , , can be potentially used as discriminators as they affect the masses of Since we restrict ourselves to a CP-conserving model, the global minimum has (cid:104) A (cid:105) = 0. In general theremay exist local minima that are CP-violating, while a recent study [21] found that it is always vanishes upto numerical fluctuations at both T = 0 and T = T c . The choice of the washout factor is subject to additional uncertainties. It was argued that the EWsphaleron is not affected much if extra degrees of freedom are SM-gauge singlets [89] but the situation inthe presence of an additional doublet is unclear yet. As a more conservative choice, other criterion such as ξ ≥ . We examined that none of the discriminators defined in [90] can effectively isolate the first order (red)and second order (blue) PT points when considering the full effective potential. – 21 –
Type I λ λ -10-50510 -10 -5 0 5 10 Type I λ λ -200-1000100200300400500600700800 0 10 20 30 40 50 60 Type I m tan β Type II λ λ -10-50510 -10 -5 0 5 10 Type II λ λ Type II m tan β Figure 6 . The mapping of the first order (red circles) and second order (blue boxes) PT points onthe 2D space of model parameters: λ vs. λ (left), λ vs. λ (middle) and m vs. tan β (right).Only T c ≤
300 GeV points are retained. Note that all the points with tan β >
25 in Type II modelhave been excluded by the
H, A → τ τ bounds [56, 57]. three extra Higgs bosons simultaneously. For instance, λ is bounded from -6 to 6 (2) inType I (II) model and a small value of λ tends to induce a second order PT unless thesum λ is negatively large.Another important observation is that for a given value of tan β , larger m , allowingfor larger m H , favours a second order PT. This points to the fact that the phase transitionin the theory degrades to the SM case when the new scalars reside in a decoupled sector, asexpected intuitively. In reverse, it has the implication that first order PT is more probablefor a small or modest value of m when m H is fixed. To illustrate this, we evaluate thephase transition properties in the process of slowly varying m , assuming the alignmentlimit and a common mass scale among the three BSM states M = 600 GeV for simplicity.The situation is shown in Fig. 7, where red and green curves represent tan β = 1 and 1 . v c : it startsfrom zero (in the second order PT stage) at large m to a non-zero value (in the firstorder PT stage). The jump from the second order PT to the first order PT is indicatedby a dashed line with an arrow. Notably, a severe fine-tuning on m is required for asuccessful first order PT and the v c value approaches the EW vacuum at smaller m . Thisinteresting behavior is explicitly illustrated in Fig 8 which gives, for tan β = 1, the 1-looppotential curve at zero temperature (left) and the finite temperature effective potentialevaluated at the critical temperature (right) for various values of m . As m decreases,thermal effects generate a higher potential barrier and simultaneously push the degenerate– 22 –
50 100 150 200 2500100200300400500 v c [ GeV ] m [ G e V ] Figure 7 . The evolution of the critical vev v c as function of m . The alignment limit and acommon mass scale among the three BSM states M = 600 GeV are assumed. The dashed linewith an arrow indicates the jump from the second order PT to the first order PT. Red and greencurves represent tan β = 1 and 1 .
5, respectively and terminate at which a proper EWSB at one looplevel does not happen at zero temperature. In the gray-shaded region v c exceeds the EW vacuum v = 246 GeV and in the green-shade region at least one of the λ ’s (mostly | λ | or | λ | ) exceeds theperturbativity bound (i.e. 4 π ) for the tan β = 1 .
50 100 150 200 250 300 350h [ GeV ]- V - loop / [ GeV ] [ GeV ] V eff1 - loop / [ GeV ] m [ GeV ] Figure 8 . The 1-loop potential curve at zero temperature (left) and finite temperature effectivepotential evaluated at the critical temperature (right) for tan β = 1 and various values of m givenin the legend. As Fig. 7, the mass of three BSM Higgs states are commonly fixed at 600 GeV andsin( β − α ) = 1 is assumed. vacuum towards the EW vev v , giving rise to a growth in ξ (owing to the small fluctuationon T c in the stage of the first order PT). On the other hand, a smaller contribution fromthe m term to the tree-level and 1-loop potential at zero temperature will remove thepotential barrier. For example, the SM potential V ∼ λh when the mass term µ → v c near v , aswe will also see in Fig. 9. Furthermore, the effect of increasing tan β on the phase propertiesis also visible by comparing the red and green curves. For a larger tan β and the same massspectrum, the first order PT is realised at a lower value of m and in the meanwhile the‘no-EWSB’ situation takes place at a smaller value of v c .As also seen in Fig. 6, most of our points have tan β close to one, which agree well– 23 – ξ T c [ T e V ] Type I ξ T c [ G e V ]
200 400 600 800 1000 1200 m H [ G e V ] ξ T c [ T e V ] Type II ξ T c [ G e V ]
200 400 600 800 1000 1200 m H [ G e V ] ξ T c [ T e V ] Type I ξ T c [ G e V ]
200 400 600 800 1000 1200 m H ± [ G e V ] ξ T c [ T e V ] Type II ξ T c [ G e V ]
200 400 600 800 1000 1200 m H ± [ G e V ] Figure 9 . Properties of first order PT in the 2HDM. Only T c (cid:46)
300 GeV points are retained. Threeblack contours from top to down correspond to v c = 250 , ,
50 GeV. The value of m H , m H ± iscolor coded as indicated by the scales on the right of the plots in the upper and lower panel,respectively. with the findings of previous studies [19]. Yet we would like to clarify that such preferenceis absolutely not the consequence of requiring a (strong) first order PT. The underlyingreason is that in the vicinity of tan β (cid:39)
1, a large range of m satisfying the theoreticalconstraints outlined in Sec. 2.1 is allowed. Oppositely, m is strongly constrained inthe high tan β region and a fine-tuning is required, which will greatly increase the difficultyof accumulating the points by means of random scan. Numerically, very limited range oftan β is allowed for large m . We now turn to discuss the general properties of the first-order PT accomplished inthe 2HDM. The crucial parameters of the phase transition include the critical temperature T c , the field value v c at T c and their ratio ξ = v c /T c which is used as a measure of thestrength of the EWPT. In Fig. 9 we display in the ( T c , ξ ) plane the points consistent withall theoretical constraints on the potential and up-to-date LHC limits at Run-2. Threeblack contours from top to down correspond to v c = 250 , ,
50 GeV. We first discuss the The correlation between tan β and m were discussed in details in Ref. [92]. – 24 –mpact of extra scalars in the spectrum. Suppose all extra scalars are heavy ( i.e , above800 GeV) and thus their masses are highly degenerate required by the EWPD (see Fig. 1),then the sector consisting of the new scalars decouple from the SM Higgs and the dynamicsof phase transition behaves like the SM. Of course, the strength of EWPT is not closelyrelated to the masses of any of additional Higgs bosons but more directly linked to themass splittings among them, which can be explicitly visualized in Fig. 14 presented later.A general tendency observed is that v c is more constrained as T c decreases. In theextreme case of T c (cid:46)
100 GeV, the thermal effect, while still playing the role of liftingthe effective potential and forming two degenerate minima, is too weak to compete withthe zero-temperature loop corrections to the potential. As a result, the critical classicalfield value is mostly localized around v , which makes it slowly vary with respect to thetemperature change. Nonetheless, v c shown in Fig. 9 does not exceed the zero temperatureEW vacuum value v owing to the EW vacuum run away (‘no-EWSB’ bound) as sketched inFig. 7, implying that the PT strength ξ necessarily improves at low T c . More quantitively,this leads to a maximum PT strength ξ (cid:39) T c = 50 GeV, and, on the other hand,implies an upper bound on T c at 250(350) GeV for ξ ≥ . . In addition, we observethat a lower bound on T c for each value of v c . For the value of the critical classical field v c being slightly away from the EW vacuum v , the lower bounds on the critical temperaturewould be around T c (cid:38)
100 GeV in Type I and the lower bound on T c in Type II model isslightly raised due to the lack of m A ≤
350 GeV points. We stress that this is an usefulfinding that one can utilize to greatly optimize the algorithm for the evaluation of T c .Last, we point out that the extremum, if coexisting in the vicinity of v c (cid:39)
135 GeV, oftendevelops to a local maximum (corresponding to a barrier) rather than a local minimum ofthe potential, which causes a narrow gap dividing the displayed points into two parts.An explicit dependence of the critical temperature T c on the mass spectrum of thethree extra Higgs bosons can be visualized in Fig. 10, where we display all points that passthe applied constraints as in Fig. 9 and additionally fulfill a strong first order EW phasetransition ( i.e. , ξ ≥ v c and T c ) from the effective potential at zero temperature. A recentprogress was reported in Ref. [93] (within the framework of the CP-conserving 2HDM) thatthe strength of the phase transition is dominantly controlled by the value of F , the depthof the 1-loop potential at zero temperature between the symmetry unbroken vacuum h = 0and the symmetry broken vacuum h = v which corresponds to, in our notation,∆ V ( v ) ≡ V ( v ) − V (0) (6.2)where V ( h ) = V ( h )+ V CW ( h )+ V CT ( h ) is the full 1-loop potential at zero temperature.Using the normalized depth ∆ F defined in [93] one can derive an upper bound that This result supports us to efficiently place a cut T c (cid:46)
300 GeV in the analysis. – 25 – ype I m A [GeV]020040060080010001200 m H [ G e V ]
50 100 150 200 250 300 T c [ G e V ] Type II m A [GeV]020040060080010001200 m H [ G e V ]
50 100 150 200 250 300 T c [ G e V ] Figure 10 . We present the critical temperature in the mass spectrum of the model. Only T c (cid:46)
300 GeV points obeying the strong first order EWPT condition ξ ≥ . definitely guarantees the PT to be strong, for example, ∆ F / F SM0 (cid:46) − .
34 necessarilyleads to ξ ≥ ξ itself, which is not an intrinsicproperty of the phase transition, but the characteristic quantities derived from the phasedynamics: v c and T c . Interestingly, we find that the magnitude of v c increases towards thezero temperature VEV with the decrease of the vacuum depth ∆ V ( v ) independent ofthe value of T c . This is illustrated in Fig 11 and is one of the nontrivial outcomes of thiswork. It is naively true that v c (cid:39) v when | ∆ V ( v ) | (cid:39)
0, which implies that the thermaleffects in the presence of extra scalars enhance the value of the effective potential at theSU(2) symmetry broken vacuum and almost do not shift the symmetry broken vacuumat the critical temperature. As expected, as the vacuum depth | ∆ V ( v ) | increases, v c decreases towards the classical field value of h = 0, which results in a smaller value of ξ for a given T c . In the meanwhile, we emphasize that the precise evaluation of v c (and T c ),of course, requires the inclusion of the temperature-dependent part in the potential. Thecritical temperature T c is supposed to be more related to the thermal corrections to theeffective potential, as demonstrated in the lower panels of Fig. 11.In addition to the non-thermal loop effect discussed above, the thermal effect in thepresence 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 contributionto the thermal mass through the quartic couplings ( λ , , , ), see Eq. (3.24). Thus, if– 26 – ξ T c [ T e V ] Type I ξ T c [ G e V ] ∆ V - l oop ( v ) / [ G e V ] ξ T c [ T e V ] Type II ξ T c [ G e V ] ∆ V - l oop ( v ) / [ G e V ] Type I ξ T c [ G e V ] -3-2.5-2-1.5-1-0.5 0 V T ( v c , T c ) / [ G e V ] Type II ξ T c [ G e V ] -3-2.5-2-1.5-1-0.5 0 V T ( v c , T c ) / [ G e V ] Figure 11 . Properties of first order PT in the 2HDM. The z-axis in the upper and lower plotsrepresents the vacuum depth of the zero temperature potential | ∆ V ( v ) | and the thermal poten-tial in the broken vacuum at the critical temperature V T ( v c , T c ), respectively. Only T c (cid:46)
300 GeVpoints are retained.
Type I m ( T c ) / m ξ AH ± H Type II m ( T c ) / m ξ AH ± H Figure 12 . The ratio of the thermal mass at the critical temperature to the zero-temperature massfor the three BSM states as a function of the PT strength ξ . Only T c (cid:46)
300 GeV points obeyingthe strong first order EWPT condition ξ ≥ . a proper cancellation between their masses and couplings is satisfied, an energy barriercan be generated so that the PT becomes strongly first order [94]. In order to see theimportance of the thermal effect, we estimate the thermal masses for three extra Higgsbosons Eq. (3.22) at critical temperature T c and present in Fig. 12 the ratio normalisingthe zero-temperature masses (the measured masses) as a function of the PT strength ξ .– 27 – ype I m H (T) / m H ± (T) m A ( T ) / m H ± ( T ) ξ Type II m H (T) / m H ± (T) m A ( T ) / m H ± ( T ) ξ Figure 13 . To examine the violation of the SU(2) custodial symmetry we normalize thefield-dependent mass for two neutral scalars A and H to the one for the charged Higgs H ± , m A ( T c ) /m H ± ( T c ) and m H ( T c ) /m H ± ( T c ) in the presentation. Only T c (cid:46)
300 GeV points obey-ing the strong first order EWPT condition ξ ≥ . Clearly, the ratio for the three states have a large variation around 1 on both sides, whichmeans their thermal corrections can be either constructive or destructive even for theSFOPT ( ξ ≥ A state in Type I model can beup to ∼
20 owing to the presence of the extremely light A . While the thermal correctiontends to suppress the m A and m H ± at T c , the preference over the enhancement on the H (relative to H ± ) is still visible. The importance of the thermal mass maximizes at ξ (cid:39) ξ further grows.Recall that the SU(2) custodial symmetry is not severely broken at zero temperaturedue to the T parameter in the EWPD which forces small mass difference for | m H ± − m A | or | m H ± − m H | or both. One may be curious whether this symmetry is broken at finitetemperature. This is especially interesting when such symmetry plays a crucial role inselecting a particular region of parameter space. In general, the thermal correction to thefield dependent masses might results in a shift of the symmetry of the model at finitetemperature. The particular case of interest is the Z symmetry cases studied in Refs. [95–97] where the Z symmetry is preserved at T = 0 but spontaneously broken at T (cid:54) = 0. Toexamine if the effect of thermal corrections leads to a shift of the SU(2) custodial symmetryin our model, we estimate the ratio of the thermal mass for two neutral states with respectto that for the charged state at critical temperature T c . The result is illustrated by Fig. 13where one can observe that the points displayed are well aligned either m A ( T c ) /m H ± ( T c ) (cid:39) m H ( T c ) /m H ± ( T c ) (cid:39) ype I m A [GeV]020040060080010001200 m H ± [ G e V ] ξ Type II m A [GeV]020040060080010001200 m H ± [ G e V ] ξ Type I m A [GeV]020040060080010001200 m H ± [ G e V ] | m H - m A | [ G e V ] Type II m A [GeV]020040060080010001200 m H ± [ G e V ] | m H - m A | [ G e V ] Figure 14 . The strength of first order EWPT shown on the Higgs mass plane. Only T c ≤ As seen from Fig. 9, a SFOPT is possible in both Type I and Type II models. Then onemay wonder what is the LHC Higgs phenomenology associated with a SFOPT. To answerthis question, in Fig. 14 we present in the m A versus m H ± (upper) and m H versus m H ± (lower) planes all points that pass the applied constraints as in Fig. 9 and additionallyrealize a SFOEWPT ( i.e. , ξ ≥ . ξ and the mass difference | m A − m H | are indicated in color scale in the upper and lower panels, respectively. We emphasizeagain that the EWPD, essentially the T parameter, force the mass differences between thecharged Higgs boson and at least one of the extra neutral Higgs bosons to be small andstrongly favor mass spectra where the masses of all new scalars are close to each other, inthe decoupling limit in particular. This severe constraint on the mass spectra for the non-SM Higgs bosons leads to five benchmark scenarios achieving a SFOPT in Type I model.– 29 – able 4 . Benchmark scenarios leading to the SFOPT. Mass spectra and the main decay modes ofthe heavier neutral Higgs boson ( H or A ) are given in each scenario.The numbers in the parenthesisfollowing each decay indicate an estimate on the branching ratios. Sce. m H [GeV] m A [GeV] m H ± [GeV] Type Main H/A decaysA 130 – 300 400 – 600 100 – 300 I A → W − H + (60%) , ZH (25%)B 400 – 600 10 – 200 400 – 600 I, II H → ZA (50 − A → ZH ( ∼ H → W − H + (60%) , ZA (25%)E 300 – 350 300 – 350 300 – 350 I A → Zh ( ∼ H → W + W − ( (cid:38) H or A ) with an estimate on the branching ratioare given in each scenario.We start with the analysis in Type I model. First, the most widely studied massconfiguration includes a pseudoscalar A with mass within the range of 400 −
600 GeVaccompanied with m H ≈ m H ± (cid:39)
200 GeV [19, 20]. In this case, m must be relativelysmall since large m tends to reduce the strength of the phase transition. This leads toa special relation among the quartic couplings λ , , (cid:39) λ (cid:39) − λ (cid:39)
5, meaningthat the strength of the phase transition is mainly governed by λ and λ , see also [19].Dictated by symmetry argument, one can image that the mass spectrum consisting of alight CP-odd state and two highly mass degenerate H and H ± can also lead to a SFOPT,which is reflected by the existence of a bulk of red points at the upper left corner ( i.e , m H ± (cid:39) −
600 GeV, m A (cid:46)
200 GeV) in Fig. 14. The situation of the model parametersis opposite due to the flip of mass hierarchy among the three BSM Higgs states. To bespecific, m is large as a consequence of large m H , and λ , , (cid:39) λ (cid:39) − λ (cid:39) − ξ ≥
1) can be also realized provided that m A and m H ± are closeto each other, while both having a large gap relative to m H . Strictly speaking, suchcondition provides two possibilities for the mass spectra: i) m A (cid:39) m H ± (cid:39)
600 GeV and m H (cid:39)
200 GeV and ii) m A (cid:39) m H ± (cid:39)
200 GeV and m H (cid:39)
600 GeV, which correspondto two isolated red-orange points densely distributed along the diagonal line in the lowerpanel plot. Deduced from Eqs (2.9) and (2.10) the mass degeneracy between A and H ± states in this scenario restrict λ (cid:39) λ , while an additional coupling λ participates into thepotential evolution and influences the phase transition. Apart from these four scenariosthat are visible in the low panel plot, the upper left plot in Fig. 14 demonstrates anadditional possible scenario that is compatible with ξ ≥ −
350 GeV. This scenario was unfortunatelyignored [19, 20] or paid less attention [21]. It is also worth noting that in this highlydegenerate scenario none of λ i couplings can be close to zero if the first order PT takes One might indeed have believed that large mass splitting among the non-SM Higgs bosons are anecessary condition for the requirement of a strong first-order EWPT. – 30 –lace.On the other hand, the allowed mass spectrum that is compatible with ξ ≥ B -physics observables and EWPD pushes m H ± (cid:38)
580 GeV and simultaneously raises the massscale for at least one of the extra Higgs bosons. Consequently, many scenarios available inType I model are eliminated, resulting in an allowed mass spectrum that leads to a strongfirst-order EWPT being quite restrained: m A (cid:39) m H ± ≈
600 GeV and a large positive massgap between m H ± and m H : m H ± − m H > ∼
300 GeV.Generally speaking, requiring a SFOPT forces down the mass scale for the new scalarsand the preferred ranges for all the scalar masses below 600 GeV, which coincidentlyapproaches to the current lower bound on the charged Higgs mass strongly constrained bythe latest measurement of B → X s γ . This means that future improvement on B -physicsobservables may decisively rule out the success of SFOPT in the Type II 2HDM. Of course,Fig. 14 also informs us that weak first order PT (under the criterion of ξ (cid:39) .
7) would stillbe possible even if no additional Higgs bosons were discovered below 1 TeV.Finally, we briefly discuss the prospects of testing the EWPT at the colliders in ac-cordance with the mass spectrum provided above. In the alignment limit sin( β − α ) ≈ g hAZ is vanishingly small but the coupling g HAZ ∝ sin( β − α ) isenhanced. Hence, the branching ratios for A → ZH and H → ZA as long as kinematicallyallowed can be substantially large depending on the mass spectrum in the model. Theseresults point towards the observation of the A → ZH and/or H → ZA decay channelswould be “smoking gun” signatures of 2HDMs with a SFOEWPT. LHC search prospectsfor the former decay have been analyzed and proposed as a promising EWPT benchmarkscenario in [20], while the collider analysis looking at both decays was performed in Ref. [98]but not specifically aiming at the EWPT. In Fig. 15 we show the 13 TeV cross sections atthe LHC for these two channels in the gluon-fusion production mode. In all cases, a crosssection above the pb level can be achieved for the scenarios realizing a SFOPT. Althoughthese signatures are characteristic ones in most of the 2HDM scenarios discussed above (seeTable 4), no strong correlation in these channels is found between ξ and the correspondingcross sections, which means that there is no guarantee to observe these decays in colliders.We leave a detailed collider analysis to future studies.Searching for a new scalar resonance is performed at the LHC mostly through itsdecay into SM particles. These decay channels include H → ZZ → (cid:96) , H, A → γγ, τ τ, t ¯ t .For the purpose of testing the EWPT, it would be very useful to find channels with strongcorrelation to the ξ value. The one served as an example here is the gluon-fusion productioncross section of A and H in the τ τ decay channel, which is shown in Fig. 16. In general,the gluon-fusion cross section in Type I model is considerably small, so there is very littlehope 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, m A ≤
60 GeV, a few points with large ξ values are observed, which could be excluded by Although our results confirm the results in the earlier literature [20], more importantly, we clarify thatthe 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]. – 31 – ype I
200 400 600 800 1000 1200m A [GeV]10 -6 -5 -4 -3 -2 -1 σ ( gg → A → Z H ) [ pb ] ξ Type II
200 400 600 800 1000 1200m A [GeV]10 -6 -5 -4 -3 -2 -1 σ ( gg → A → Z H ) [ pb ] ξ Type I
200 400 600 800 1000 1200m H [GeV]10 -6 -5 -4 -3 -2 -1 σ ( gg → H → Z A ) [ pb ] ξ Type II
200 400 600 800 1000 1200m H [GeV]10 -6 -5 -4 -3 -2 -1 σ ( gg → H → Z A ) [ pb ] ξ Figure 15 . 13 TeV cross sections at the LHC as a function of the relevant mass scale, for the gg → A → ZH (upper panels) and gg → H → ZA (bottom panels) channels in Type I (left panels)and Type II (right panels). the upcoming experimental searches in that channel. In Type II the situation is different,for a given scalar mass the achievable cross-sections have a lower bound. The large ξ pointsare located at low m H (cid:46)
350 GeV and have reasonably large cross-sections just below thecurrent experimental upper limit. In short, we estimate that a factor of 4 improvement inthe search sensitivity, which is very likely to be reached, would either see an exciting signalor eliminate these points, as a result, the first order PT with strength ξ >
The scenarios that lead to the first order PT in the model have a mass spectrumbelow the TeV scale, as can be seen in Fig. 14 and Table 4. The presence of additionalscalars that couple to the SM-like Higgs h can modify the triple Higgs coupling hhh atboth tree-level and loop-level and thus leads to the deviation with respect to its SM value g SM tree hhh . Moreover, such deviation can be significant near the alignment limit providedbeing away from the decoupling limit [28, 99]. We examine both the tree-level couplingand the one after the inclusion of the one-loop corrections. They are computed by takingthe third derivative of the tree level potential V and the one-loop potential V + V CW + V CT with respect to h , respectively and shown in the upper and lower panels of Fig. 17 (afternormalizing the SM value g SM tree hhh = 3 m h /v ). Focusing on the tree-level results, one can– 32 – ype I
200 400 600 800 1000 1200m A [GeV]10 -6 -5 -4 -3 -2 -1 σ ( gg → A → ττ ) [ pb ] ξ Type II
200 400 600 800 1000 1200m A [GeV]10 -6 -5 -4 -3 -2 -1 σ ( gg → A → ττ ) [ pb ] ξ Type I
200 400 600 800 1000 1200m H [GeV]10 -6 -5 -4 -3 -2 -1 σ ( gg → H → ττ ) [ pb ] ξ Type II
200 400 600 800 1000 1200m H [GeV]10 -6 -5 -4 -3 -2 -1 σ ( gg → H → ττ ) [ pb ] ξ Figure 16 . 13 TeV cross-sections at the LHC as a function of the relevant mass scale, for the gg → A → τ τ (upper panels) and gg → H → τ τ (bottom panels) channels in Type I (left panels)and Type II (right panels). observe that the triple SM-like Higgs self-coupling g hhh in favor of the highly strong PT( i.e , ξ (cid:38)
3) is close to its SM value g SM tree hhh , while large deviation (mostly suppression) of g hhh from g SM tree hhh is possible for the weakly strong PT ( i.e , ξ (cid:46) . hhh coupling at tree-level cannot be enhanced in Type I (for m H (cid:38)
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-loopcorrections to the hhh coupling are taken into account. As shown in the lower panel plots,the coupling g hhh at one-loop level are absolutely enhanced in both models and the largestnormalized coupling g hhh /g SM tree hhh can be about 2.5, corresponding to ∼ ξ ≥
1) in the 2HDM typically induces theenhancement on the hhh coupling. Next, we would like to quantitatively explore therelation between the phase transition strength and the content of the derivation the tripleHiggs coupling. In general, the loop-level hhh coupling exhibits a larger deviation withincreased strength of the phase transition. Whereas, the tree-level hhh coupling shownin the upper panel plots does not display such a proportionality behavior. This dramaticchange implies that the loop corrections coming from the CW potential and counter-termsare important in general when the phase transition is of strong first order and can evenbe dominant over the tree-level contribution in the case of the extremely strong phasetransition. It is also apparent in Fig. 17 that the highly strong PT induces a substantial– 33 – ype I
200 400 600 800 1000 1200 m H [GeV]0.70.750.80.850.90.9511.051.11.15 g t r eehhh / g S M t r eehhh ξ Type II
200 400 600 800 1000 1200 m H [GeV]0.550.60.650.70.750.80.850.90.9511.05 g t r eehhh / g S M t r eehhh ξ Type I
200 400 600 800 1000 1200 m H [GeV]11.21.41.61.822.22.42.62.83 g hhh / g S M t r eehhh ξ Type II
200 400 600 800 1000 1200 m H [GeV]11.21.41.61.822.22.42.62.8 g hhh / g S M t r eehhh ξ Figure 17 . Triple Higgs coupling hhh at tree-level (upper) and at one-loop level (lower) normalizedto the SM tree value g SM tree hhh = 3 m h /v . Note that tree-level Higgs self-coupling is not enhanced inType II. To have a better visualization only ξ ≥ enhancement on the hhh coupling. In contrast, the hhh coupling normalized to the SMtree value can vary from ∼ ξ = 1 −
2. This means thatlarge triple Higgs coupling hhh is a necessary but not sufficient condition of realizing thehighly strong PT. For instance, if the deviation is smaller than 100%, then possibility ofthe highly strong PT ( ξ (cid:38) .
5) in Type I (II)) will be eliminated. As a result, the size ofthe triple Higgs coupling hhh derives an upper bound on the achievable value of ξ . In somesense, this is phenomenologically useful because we have built a connection between thephase transition involving the thermal contribution and a measurable observable at zerotemperature. Therefore, the measurement of the triple Higgs coupling could be an indirectapproach of probing the phase transition at colliders.Experimentally, the deviation of the triple Higgs coupling can be detected at bothlepton colliders (i.e., ILC [100], CEPC [101] and FCC-ee[102] ) and hadron colliders such asLHC and SppC [103]. At hadron colliders, the resonant Higgs pair production is promisingwhile special attention needs to be paid when the heavier CP-even state H produces adestructive interference with the SM top box diagram process [104]. Upon the sensitivityof 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 thedi-Higgs production into b ¯ bγγ and b ¯ bW + W − channels in the ultimate operation of LHCRun-2 [105–107]. In our case, g hhh has the same sign as the SM value and hence resultsin the destructive interference between the s -channel h -mediator triangle diagram and thetop box diagrams of the gg → hh production process. This implies increasing g hhh will– 34 – ype I
200 400 600 800 1000 1200 m H [GeV]-2-1.5-1-0.500.511.52 g H hh / g S M t r eehhh ξ Type II
200 400 600 800 1000 1200 m H [GeV]-1.5-1-0.500.511.52 g H hh / g S M t r eehhh ξ Figure 18 . Triple Higgs coupling
Hhh at one-loop level normalized to the SM tree value g SM tree hhh = 3 m h /v . In contrast to the hhh coupling, the one-loop corrections to the Hhh cou-pling are vanishingly small near the alignment limit. To have a better visualization only ξ ≥ decrease the production cross section [108]. Previous studies demonstrated that when g hhh (cid:39) . g SM tree hhh an exact cancellation between these two diagrams is accomplished atthe threshold of the di-Higgs invariant mass m hh = 2 m t [104, 109, 110]. Due to the lowacceptance at LHC for large g hhh , a cut m hh < m t is imposed [104, 109]. MVA analysis ofRef. [109] shows that, for the parameter space leading to the SFOPT (presented in Fig. 17),the observation significance in the b ¯ bγγ channel with the integrated luminosity of 3 ab − at 14 TeV would decrease from 10 to 4 in both Type I and Type II models. In measuringthe triple Higgs coupling hhh the lepton machines are typically more powerful, using theHiggs associated process e + e − → Z ∗ → Zh ∗ ( hh ). The better designed sensitivities at theCEPC [101], FCC-ee[102] and ILC1000 are roughly 20-30%. This indicates that almostthe full parameter spaces that are compatible with ξ ≥
1, particularly for m H (cid:38)
500 GeV,are within the future detection reach.The other Higgs self-coupling of interest is the
Hhh coupling g Hhh , which is alsorelevant to the Higgs pair production through the s-channel H mediator triangle diagram.The Hhh coupling at one-loop as a function of m H is depicted in Fig. 18. In contrast tothe hhh coupling, the one-loop corrections to Hhh coupling are vanishingly small near thealignment limit. We thus do not show the tree-level result. It is important to mention thatthe
Hhh coupling can be significant even in the alignment limit, which can be observedin Fig. 18. For instance, the
Hhh coupling is about ± (30 − hhh coupling for the highly strong PT ( ξ ≥
3) and can be even comparable with orlarger than g SM tree hhh as the PT is weakly strong ( ξ ≈ − Hhh coupling g Hhh in the successful SFOEWPT scenarios can have either the same sign as orthe opposite sign to the coupling g hhh . The consequence of the sign flip of the g Hhh willaffect the s-channel H -resonant triangle diagram contribution to the gg → hh process,whose amplitude is proportional to the product of g Hhh and g Ht ¯ t = C HU y t , resulting in achange on the m hh lineshape due to the interference between the triangle diagram of thesignal and the continuum top box diagram. When the interference is destructive, specialattention needs to be paid [104]. The study of Ref. [111] indicates that most of our SFOPT– 35 –oints can be detected at 5 σ significance provided that g Hhh × g Ht ¯ t >
300 GeV at 14 TeVwith integrated luminosity of 3 ab − using the b ¯ bγγ channel. Taking into account theoretical and up-to-date experimental constraints, we stud-ied the electroweak phase transition in the framework of the CP-conserving 2HDM ofType I and Type II models near the alignment limit. The thermal potential was expressedin terms of modified Bessel functions, which allows for a fast numerical evaluation andhigh precision compared to the simpler high/low temperature approximations. While both1-stage and 2-stage phase transitions were shown to be realized within the 2HDM, in thispaper we focused on scenarios leading to 1-stage phase transitions at electroweak scale, forwhich first order and the second order phase transitions are distinguished.We analyzed the properties of the first order phase transition, observing that the fieldvalue of the electroweak symmetry breaking vacuum at the critical temperature is stronglyrelated to the vacuum depth of the 1-loop potential at zero temperature, while the criticaltemperature reflecting the size of the thermal effect is characterized by the temperature-dependent potential. In general, the critical temperature T c tends to be higher as the BSMstates becomes heavier, and on the other hand T c can be down to ∼
100 GeV when atleast one light BSM Higgs bosons present in the spectrum. We have also observed that thethermal correction to the mass is important in driving a SFOPT.The strength of the transition, a key property for the electroweak baryogenesis mecha-nism, depends largely on the allowed mass spectrum. Requiring a SFOPT with ξ ≥ i.e. , ξ ≥
1) required for baryogenesis ispossible in both Type I and Type II models. In Type I model, SFOPT is achievable in theparameter space where a large mass splitting is present between two neutral Higgs bosonssuch as m H (cid:29) m A and m A (cid:29) m H . In either case, the charged Higgs mass is close to either m H or m A required by the EWPD. The mass spectrum among the extra Higgs bosons inthe Type II model is, on the contrary, strongly constrained due to flavor observables, whichpush the mass of the charged Higgs above ∼
600 GeV. As a result, scenarios leading to aSFOPT in Type II are m H ± (cid:39) m H (cid:29) m A and m H ± (cid:39) m A (cid:29) m H . In view of large masssplitting between H and A , both pp → H → ZA and pp → A → ZH can be “smokinggun” 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 ∼ A , H and H ± ) are degenerate around350 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 ofa SFOEWPT on the LHC Higgs phenomenology. Various characteristic collider signaturesat the 13 TeV LHC have been identified, among which the gluon-fusion production crosssection of A and H in the τ τ decay channel displays a correlation with the PT strength ξ .– 36 –t turns out that new physics searches at collider machines can provide an indirect channelto examine the EWPT scenarios. Finally, we verify that an enhancement on the tripleHiggs coupling hhh (including loop corrections) is a typical signature of the SFOPT drivenby the additional doublet. The PT with larger strength is associated with larger deviationof the loop-level triple Higgs coupling hhh with respect to the SM value, which can helpto enhance an energy barrier. Meanwhile, we notice that the other triple Higgs coupling g Hhh can also be comparable with the triple Higgs coupling in the SM for SFOPT so thatthe search for the heavy neutral Higgs H through the gg → hh process is possible for smalltan β since the top Yukawa coupling of the H is proportional to cot β .We leave for future work the interplay of gravitational waves signals and testablecolliders signatures for SFOPT benchmark scenarios presented in this paper. This successwould build a link between early Universe cosmology and collider detection, which couldprovide additional constraints in the allowed parameter space of the 2HDM. We believethat such connection will have a significant physical value and serves as a useful guide forcollider 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 sup-ported 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 ofthis work was performed. The work of LGB is partially supported by the National NaturalScience Foundation of China (under Grant No. 11605016), Basic Science Research Pro-gram through the National Research Foundation of Korea (NRF) funded by the Ministryof Education, Science and Technology (Grant No. NRF- 2016R1A2B4008759), and KoreaResearch 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 financial support provided by Chongqing University for multiple visits atdifferent stages during the completion of this paper.– 37 –
Thermal mass for SM gauge bosons
The thermal masses of the gauge bosons are more complicated. Only the longitudinalcomponents receive corrections. The expressions for these in the SM can be found inRef. [79], Π LW ± = 116 g T , Π TW ± = 0Π LW = 116 g T , Π TW = 0 (A.1)Π LA = 116 g (cid:48) T where the script L ( T ) denotes the longitudinal (transversal) mode. Their contributionsfrom the extra Higgs doublet are easy to be included∆Π LW ± = ∆Π LW = 16 g T , (A.2)∆Π LA = 16 g (cid:48) T , (A.3)Adding them together, for the longitudinally polarized W boson, the result is M W ± L = 14 g ( h + h ) + 2 g T . (A.4)This includes contributions from gauge boson self-interactions, two Higgs doublets andall three fermion families. The masses of the longitudinal Z and A are determined bydiagonalizing the matrix14 ( h + h ) (cid:32) g − gg (cid:48) − gg (cid:48) g (cid:48) (cid:33) + (cid:32) g T
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