Towards Reviving Electroweak Baryogenesis with a Fourth Generation

Abstract

Electroweak baryogenesis is an attractive scenario for explaining the baryon asymmetry of the universe. However, it does not work within the standard model due to two reasons: (1) the strength of CP violation from the Kobayashi-Maskawa mechanism with three generations is too small; (2) the electroweak phase transition is not first order for the experimentally allowed Higgs boson mass. We discuss possibilities to solve these problems by introducing a fourth generation of fermions and how electroweak baryogenesis might be revived. We also discuss briefly the recent observation of a Higgs-like boson with mass around 125 GeV, which puts the fourth generation in a difficult situation, and the possible way out.

1. Introduction

The origin of the baryon asymmetry of the universe (BAU) is not only a big mystery in particle physics and cosmology, but it is a core problem related to our very existence. The observational evidence is very compelling. There is no indication for any macroscopic objects made from antiparticles. The antiparticles observed in cosmic rays are thought to be of secondary origin. Thus, it is accepted that the universe contains negligible amount of antibaryons, as compared to the baryons that constitute usual matter (including ourselves!), or 𝑛_𝑏/𝑛_𝑏≪1. The standard big bang cosmology requires the baryon-to-entropy ratio of the universe to be [1] 𝑛_𝐵𝑠=𝑛_𝑏−𝑛_𝑏𝑠≃0.9×10⁻¹⁰. (1) If one does not take the view that this is merely an initial condition for the expanding universe, then the challenge is not so much the dominance of radiation, but that whether there is any matter left from the primordial matter-antimatter annihilation.

Many scenarios have been attempted to explain the origins of the minute amount of leftover matter based on a dynamical mechanism (baryogenesis). A common basis for such mechanisms is provided by the three Sakharov conditions [2, 3], which consist of (i) baryon number (𝐁) violation, (ii) 𝐶 and CP violation, and (iii) departure from equilibrium. In order to produce the BAU, these three conditions have to be satisfied simultaneously at some stage in the early universe.

Among many proposals, electroweak baryogenesis (EWBG) [4] is one of the well-motivated and appealing mechanisms to explain BAU, especially since its ingredients can be possibly probed by the Large Hadron Collider (LHC). This mechanism could operate during the electroweak phase transition (EWPT), occurring when the temperature of the universe is around 𝑇_𝑐∼100 GeV, and it is a possible baryogenesis mechanism within the standard model (SM) of particle physics. One should recall the truly remarkable fact that the SM carries all the necessary ingredients for the Sakharov conditions.(1)  𝐁 is violated due to the chiral anomaly induced by the SU(2)×U(1) gauge interaction. 𝐁-violating processes are nonperturbative phenomena mediated by topologically nontrivial SU(2) gauge field configurations. Although the instanton-mediated 𝐁 violation at zero temperature is highly suppressed, the rate of 𝐁 violation at finite temperature is not negligible via the so-called sphaleron transition, especially when the temperature is around or above 𝑇_𝑐. (2)  𝐶 is violated by the 𝑉−𝐴 structure of the weak interaction, while CP is violated by the Kobayashi-Maskawa (KM) mechanism [5] with three generations of quarks. A complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix provides the source of CP violation, which nicely describes all terrestrially observed CP-violating phenomena, that is, CP violation in the 𝐾- and 𝐵-meson (even the 𝐷-meson) systems. (3)  SU(2)×U(1) gauge symmetry is expected to be restored at high temperature above 𝑇_𝑐. Therefore, as the universe cools down to a temperature of around 𝑇_𝑐, the universe undergoes a phase transition (the EWPT) from symmetric phase of SU(2)×U(1) to the broken phase. If the EWPT is first order, the phase transition proceeds through bubble nucleation of the broken phase, where expansion of the bubbles fills up the entire universe. Thus, temporal departure from equilibrium is achieved near the surface of the expanding bubble.

Unfortunately, the measured parameters of the SM appear to be insufficient to bring about EWBG. The SM fails to explain the BAU due to two reasons. The first reason is that the KM mechanism with three quark generations cannot offer large enough CP violation (CPV) to produce the observed baryon-to-entropy ratio during the EWPT. The second reason is that the EWPT is not first order for a Higgs boson with 𝑚_ℎ>114 GeV, required already by direct search at the LEP [6] and now indicated to be around 125 GeV at the LHC [7, 8]. Therefore, physics beyond the SM is needed for EWBG, and various extensions of the SM have been investigated from this point of view. In particular, there are extensive studies (for a recent review on EWBG, see [9]; and references therein) for the two Higgs doublet model and supersymmetric extensions of the SM, such as the MSSM and the NMSSM. It should be noted, however, that MSSM itself is under stress from the LHC, and its case for EWBG is only marginal [9]. One of the biggest means of attraction to have EWBG is that the implied new physics should be accessible at the LHC, at least partially.

In this contribution, we consider a four-generation (4G) extension of the SM and discuss a possibility to revive EWBG (beside a possible revival of EWBG, the 4G has another interesting implication on baryogenesis, if it is long-lived; see [10]) fully within the SM framework, as 4G does not introduce anything that was not already contained in SM. We focus on the quark sector, which includes fourth generation quarks 𝑡^′ and 𝑏^′. Compared to the three-generation case, the 4G quarks bring three additional mixing angles and two additional complex phases in the 4×4 CKM matrix, thus providing new sources of CP violation. One can argue intuitively for a possible effect of new CP violation on BAU by studying basis-independent invariants constructed from the quark mass matrices (or the Yukawa coupling matrices), which are extensions of the Jarlskog determinant in the SM. One of us [11] pointed out that such an invariant quantity can be highly enhanced compared to the three-generation case, mainly due to the large masses (or the large Yukawa couplings) of 𝑡^′ and 𝑏^′(see [12, 13] for an earlier discussion on 4G-enhanced CP violation for BAU). As for the issue of the EWPT, a one-loop analysis shows that the 4G quarks do not play a positive role towards a first-order phase transition [14]. However, given that bounds on the 4G quark masses from direct search at the LHC have entered 600 GeV level [15, 16], which is beyond the tree-level perturbative unitarity bound (UB) of 500–550 GeV [17], we have to reconsider the problem of EWPT beyond perturbation theory. In this regard, the nature of the EWPT with 4G quarks is still quite an open problem (for earlier works which address the issue of the EWPT within the four-generation framework (but not within the SM framework), see [14, 18] (supersymmetric model with the 4G fermions) and [19] (dynamical electroweak symmetry breaking model due to strong four-fermion interactions among the 4G fermions)).

The recent observation of a Higgs-like object with mass around 125 GeV poses a special impasse situation, since a naive application of heavy 𝑡^′ and 𝑏^′ quarks in the loop would enhance the gluon-gluon fusion production of the Higgs boson by about an order of magnitude, which is not supported by 𝑔𝑔→𝑉𝑉 data [7, 8], where 𝑉 is the 𝑊 or 𝑍 boson. Furthermore, quadratic corrections to the Higgs mass from heavy 𝑡^′ and 𝑏^′ quarks make the light Higgs mass rather difficult to sustain. With the deep conflict between a light Higgs and having 4G quarks beyond UB, it may seem cavalier to relegate this again to nonperturbative treatments. However, the aim of this work is very modest, which is just to show the nontrivial nature of 4G towards EWBG. Since the source of CPV arises from off-diagonal couplings, the Higgs boson does not directly enter EWBG computation. We will discuss towards the end the possibility that the observed 125 GeV boson could be a dilaton, rather than a bona fide SM Higgs boson, or it may be a pseudo-Goldstone Higgs boson.

This paper is organized as follows. In Section 2, we discuss the effect of CP violation from the fourth generation quark sector on BAU. Extending the EWBG mechanism in the case of the SM to the fourth generation case, we estimate the amount of the BAU produced during a first-order EWPT. In Section 3, we summarize the EWPT in the SM briefly and discuss possible mechanisms to induce a first-order phase transition within the four-generation framework. Finally, in Section 4, we offer some discussions and give our summary.

2. CP Violation

One of the reasons for the failure of EWBG based on SM is the insufficiency of CP violation from the KM phase. This situation can be naively understood as follows.

In the SM with three generations, a basis-independent invariant for CP violation is given by the well-known Jarlskog determinant [20], det𝑀_𝑢𝑀^†_𝑢,𝑀_𝑑𝑀^†_𝑑3−gen.≡𝑖𝐽_SM, (2) where 𝑀_𝑢,𝑑 are the up- and down-type quark mass matrices. In terms of quark masses and CKM matrix elements, one gets 𝐽_SM=2𝒥_SM𝑚²_𝑡−𝑚²_𝑐𝑚²_𝑡−𝑚²_𝑢𝑚²_𝑐−𝑚²_𝑢×𝑚²_𝑏−𝑚²_𝑠𝑚²_𝑏−𝑚²_𝑑𝑚²_𝑠−𝑚²_𝑑, (3) where 𝒥_SM is twice the area of any triangle defined by the unitarity condition of the CKM matrix, 𝑉^†𝑉=𝟏, with the experimentally determined value of 𝒥_SM≃3×10⁻⁵. One can estimate a naive strength of CP violation during the EWPT by constructing a dimensionless quantity from 𝐽_SM and the critical temperature, 𝑇_𝑐∼100 GeV. This gives 𝐽_SM/𝑇_𝑐¹²∼10⁻¹⁹, which is too small to account for BAU in (1).

It is readily observed that the smallness of 𝐽_SM originates mainly from the powers of light quark masses, 𝑚²_𝑐𝑚⁴_𝑏𝑚²_𝑠/𝑇⁸_𝑐∼10⁻¹⁵, with milder suppression from the CKM factor 𝒥_S𝑀. Motivated by this observation and a possible hint of new physics in 𝑏→𝑠 transition, it was suggested in [11] to introduce the 4G quarks and replace (3) by an analogous quantity involving second to fourth generation quarks (or equivalently, treating the first two generations as degenerate), 𝐽₍₎^𝑠𝑏_2,3,4=2𝒥_𝑠𝑚²_𝑡′−𝑚²_𝑡𝑚²_𝑡′−𝑚²_𝑐𝑚²_𝑡−𝑚²_𝑐×𝑚²_𝑏′−𝑚²_𝑏𝑚²_𝑏′−𝑚²_𝑠𝑚²_𝑏−𝑚²_𝑠∼𝒥_𝑠𝒥_SM𝑚²_𝑡′𝑚²_𝑡𝑚−1^2𝑡′𝑚²_𝑐𝑚⁴_𝑏′𝑚²_𝑏𝑚²_𝑠𝐽_SM, (4) where 𝒥_𝑠=Im(𝑉^∗_𝑡𝑠𝑉_𝑡𝑏𝑉^∗_𝑡′𝑏𝑉_𝑡′𝑠). In fact, 𝐽^𝑠𝑏_(2,3,4) is a leading term of a basis-independent set of invariants for CP violation in the four-generation case and is given by Tr[𝑀_𝑢𝑀^†_𝑢,𝑀_𝑑𝑀^†_𝑑]³, up to an overall factor [20, 21]. Due to heaviness of 𝑡^′ and 𝑏^′, 𝐽^𝑠𝑏_(2,3,4) can be highly enhanced compared to 𝐽_SM in (3). For 𝑚_𝑡′∼𝑚_𝑏′∼600 GeV, for instance, the enhancement factor reaches ~10¹⁸ solely from the mass factors, thus, seemingly possible to overcome the smallness of 𝐽_SM.

In the following, after we briefly review EWBG in the SM case, we take one step further and examine the previous expectation of 4G-enhanced CP violation for the EWBG scenario.

A concrete mechanism of EWBG within SM was proposed by Farrar and Shaposhnikov (FS) [22–25]. In the FS mechanism, a baryon asymmetry is produced through CP violating scattering of quarks at the surface of an expanding bubble, that is, the bubble wall generated during a first-order EWPT. CP asymmetry in reflection of the quarks off the bubble wall is induced by thermal effects, that is, interactions of the quarks with the 𝑊 and charged-Higgs bosons in the cosmic plasma. FS included these effects employing a quasiparticle picture and treated the scattering problem in a quantum mechanical manner. Solving an effective Dirac equation for the quasiparticles, they found that CP violation from the KM phase is sufficient to explain the BAU under optimal conditions, contrary to the naive argument given previous.

The very attractive SM explanation for BAU by FS was, however, refuted by subsequent works by Gavela et al. [26–28]. These authors pointed out that the width of the quasiparticle (damping rate) was not included in the study of FS, and they found that inclusion of the width reduces the baryon asymmetry to a negligible amount. This result was also confirmed by Huet and Sather (HS) [29], who interpreted the damping as quantum decoherence phenomenon induced by plasma effects, leading to reduction of the CP asymmetry, in which quantum mechanical coherence plays an essential role. After the demonstration of these results, it was (re)accepted that CP violation from the KM phase is not sufficient to explain BAU. This conclusion is in accordance with the naive dimensional argument of 𝐽_SM given previous. We now estimate, by extending the work of HS, the baryon asymmetry generated by the FS mechanism in the four-generation case.

We assume the existence of a first-order phase transition. The baryon asymmetry generated during the first-order EWPT is given by [29] 𝑛_𝐵𝑠∼−10⁻²𝑇𝑑𝜔𝑛2𝜋⁰(𝜔)1−𝑛₀(𝜔)Δ𝐩⋅𝐯_𝑊𝑇Δ(𝜔), (5) where 𝑛₀(𝜔)=1/[exp(𝜔/𝑇)+1] is the Fermi-Dirac distribution, Δ𝐩≡𝐩_𝐿−𝐩_𝑅 is the difference between left-handed and right-handed quasiparticle momenta for a given energy 𝜔, and 𝐯_𝑊 is the velocity of the expanding bubble wall. We neglect 𝒪(𝐯²_𝑊) contribution by assuming |𝐯_𝑊|∼0.1. We take 𝑇∼𝑇_𝑐∼100 GeV in the previous formula. Δ(𝜔) is a reflection asymmetry defined by Δ(𝜔)≡Tr𝑅^†_𝐿𝑅𝑅_𝐿𝑅−𝑅^†_𝐿𝑅𝑅_𝐿𝑅, (6) where reflection coefficients 𝑅_𝐿𝑅 and 𝑅_𝐿𝑅 are matrices in flavor space; that is, 𝑅^𝑓𝑖_𝐿𝑅 is the reflection coefficient for 𝑞^𝑖_𝐿→𝑞^𝑓_𝑅, where 𝑖 and 𝑓 refer to quark flavors, and 𝑅^𝑓𝑖_𝐿𝑅 corresponds to the CP-conjugate process. Δ(𝜔) provides the CP asymmetry for the reflection rate of a left-handed quasiparticle, incident from the symmetric phase with an energy 𝜔, summed over all flavors. We further assume that the EWPT is strongly first order, so that the generated baryon asymmetry is not washed out by the sphaleron processes in the broke phase (see next section). We return to discuss the issue of order of phase transition in 4G context later.

Δ(𝜔) is obtained by solving an effective Dirac equation for the quasiparticles in the presence of space-dependent quark mass terms. Assuming planar wall with zero thickness, the effective Dirac equation is given by 2𝜔−Ω_𝐿1+𝑖𝛾+3𝑖𝝈⋅𝜕𝑀_𝑢,𝑑𝜃(𝑧)𝑀^†_𝑢,𝑑𝜃(𝑧)2𝜔−Ω_𝑅1+𝑖𝛾−3𝑖𝝈⋅𝜕×Ψ_𝑢,𝑑(𝑧)=0, (7) where Ψ_𝑢=(𝜓_𝑢,𝜓_𝑐,𝜓_𝑡,𝜓_𝑡′)^𝑇, Ψ_𝑑=(𝜓_𝑑,𝜓_𝑠,𝜓_𝑏,𝜓_𝑏′)^𝑇, and each 𝜓_𝑖 are four-component spinor wavefunctions. In (7), Ω_𝐿(𝑅) is a thermal mass matrix for the left- (right-) handed quasiparticles, 𝛾 is the quasiparticle width, and we use the one calculated in QCD [30], 𝛾≃0.15𝑔²_𝑠𝑇, neglecting possible flavor-dependent corrections from the Yukawa interactions.

Following HS, we solve the effective Dirac equations analytically based on the Green’s function method, which gives perturbative expansions of the reflection coefficients in terms of the quark mass matrices 𝑀_𝑢,𝑑. The leading-order contribution to Δ(𝜔) from the scattering of the 𝑑-type quarks is given by Δ_𝑑(𝜔)=4327𝜋𝛼_𝑊𝑇²64Ω₀𝑀²_𝑊³1+𝜔−Ω₀𝛾²⁻⁶×16𝛾⁹ImTr𝑀_𝑢𝑀^†_𝑢,𝑀_𝑑𝑀^†_𝑑³, (8) where Ω₀≃𝑔_𝑠𝑇/√6 is the dominant part in the thermal masses of quarks, coming from QCD. The 4G effects enter solely through the last factor in (8), which arises as one follows the scattering of an incoming 𝑑-type quark against the bubble wall. In the three-generation case, this factor is nothing but the Jarlskog determinant, discussed previous, ImTr𝑀_𝑢𝑀^†_𝑢,𝑀_𝑑𝑀^†_𝑑³_3−gen.=3Imdet𝑀_𝑢𝑀^†_𝑢,𝑀_𝑑𝑀^†_𝑑3−gen.=3𝐽_SM, (9) where 𝐽_SM is defined in (2) and (3). On the other hand, the four-generation counterpart is more complicated, but it can be simplified [21] by exploiting the hierarchical structure of the quark masses and the CKM matrix elements, with a mild assumption on new mixings, |𝑉_𝑡𝑑𝑉^∗_𝑡′𝑑|≪|𝑉_𝑡𝑖𝑉^∗_𝑡′𝑖| (𝑖=𝑠,𝑏,𝑏^′). Then, the last factor in (8) is approximated as ImTr𝑀_𝑢𝑀^†_𝑢,𝑀_𝑑𝑀^†_𝑑³≃−6𝒥_𝑠𝑚²_𝑡′−𝑚²_𝑡𝑚²_𝑡′𝑚²_𝑡𝑚⁴_𝑏′𝑚²_𝑏, (10) where 𝒥_𝑠=Im(𝑉^∗_𝑡𝑠𝑉^∗_𝑡′𝑏𝑉_𝑡𝑏𝑉_𝑡′𝑠). One notes that ImTr[𝑀_𝑢𝑀^†_𝑢,𝑀_𝑑𝑀^†_𝑑]³≃−3𝐽^𝑠𝑏_(2,3,4), where 𝐽^𝑠𝑏_(2,3,4) is defined in (4). Therefore, Δ_𝑑(𝜔) can be actually enhanced by large masses of 𝑡^′ and 𝑏^′ compared to the SM case, as anticipated in [11].

As discussed by HS, dimensionless perturbative expansion parameters in the Green’s function method are given by 𝑀_𝑢,𝑑/(6𝛾)≃𝑀_𝑢,𝑑/(1.4𝑇), which are of order unity or larger for the 𝑡, 𝑡^′, and 𝑏^′ quarks, given 𝑇∼100 GeV during the EWPT. Hence, the perturbative expansion breaks down for these heavy quarks, and the previous result for Δ_𝑑(𝜔) could be reduced due to large 𝑚_𝑏′ (equation (8) is obtained by the expansion in the 𝑑-type quark mass matrix 𝑀_𝑑 based on the Green’s function method. The 𝑢-type quark mass matrix 𝑀_𝑢 in (8) originates from the thermal mass matrix for the 𝑑-type quasiparticles, that is, the Yukawa interaction with the plasma. Thus, the heaviness of only the 𝑏^′ quark matters to Δ𝑑(𝜔)) if the perturbation is not used. Thus, the previous result might overestimate the CP asymmetry from the 𝑑-type quarks. The 𝑢-type quark contribution, at leading order, is the same as (8) except for the difference of the overall sign, hence, leading to a complete cancelation at this level. The previous mentioned remark is, however, also applied for the 𝑢-type quark contribution, and a possible amount of reduction would be even larger as it contains the contributions from the two heavy quarks, 𝑡 and 𝑡^′. We naively accept (8) as a dominant contribution to the reflection asymmetry Δ(𝜔), simply neglecting the 𝑢-type quark contribution (there is another type of major contribution to Δ(𝜔), called Δ₇ in HS, which arises when the finite quark masses in the broken phase are taken into account in the self-energy of quarks. While Δ₇ gives larger contribution to the BAU than (8) in the three-generation case, we confirmed that the contribution from (8) dominates in the four-generation case for experimentally allowed masses of 4G quarks).

Eventually, the baryon asymmetry generated during the EWPT is given by 𝑛_𝐵𝑠∼0.9×10⁻¹⁰𝒥_𝑠10⁻⁴𝑚_𝑡′650GeV⁴𝑚_𝑏′650GeV⁴, (11) where 𝑇=100 GeV and 𝑀_𝑊(𝑇)=50 GeV are adopted. 𝒥_𝑠 includes new CKM parameter 𝑉_𝑡′𝑠𝑉^∗_𝑡′𝑏 which can be measured via flavor observables related with 𝑏→𝑠 transition. We note that measurements for the CP violating phase 𝜙_𝑠 in 𝐵_𝑠−𝐵_𝑠 mixing, the forward-backward asymmetry for 𝐵→𝐾^∗𝜇⁺𝜇⁻, and the 𝐵_𝑠→𝜇⁺𝜇⁻ rate are making rapid progress at the LHC. Theoretical predictions for these observables are not subject to hadronic uncertainties so much; hence, they are useful to constrain 𝑉_𝑡′𝑠𝑉^∗_𝑡′𝑏 [31]. From the recent LHC data, 𝒥_𝑠 of around 10⁻⁴ with 𝑚_𝑡′=650 GeV seems to be at the border of the experimentally allowed region [32]. However, with rising 𝑚_𝑡′ (and 𝑚_𝑏′), there is a tendency that 𝒥_𝑠 drops, but the high powers of 𝑚_𝑡′𝑚_𝑏′ in (11) should easily compensate for it.

With reasonable 4G parameters, therefore, the FS mechanism with 4G quarks seems to generate the correct order of magnitude of the baryon-to-entropy ratio and, hence, can in principle explain BAU. We note that the previous estimate of BAU relies on the perturbative treatments of the Yukawa couplings/masses of the quarks including 𝑡^′ and 𝑏^′ in a few steps, while the LHC bounds on the 4G quark masses suggest a nonperturbative nature of the corresponding Yukawa couplings. For instance, the Green’s function method invokes the perturbation in the quark mass matrices as we already discussed. Besides this point, the thermal properties for the quasiparticles are obtained via perturbative calculations; for example, the thermal mass matrices are evaluated at one-loop level. Therefore, our result would be semiquantitative at most. Nevertheless, given remarkable enhancement of the CP asymmetry due to the 4G quarks and the agreement with the observed BAU as shown in (11), our finding should deserve further investigation. We further remark that the CP-violating Jarlskog-like invariants of (2) and (10) are of purely algebraic nature. Thus, we suspect that a fully dynamical calculation should still reflect this fact.

3. Electroweak Phase Transition

Besides CP violation, the other essential issue for EWBG is the nature of the EWPT. EWBG requires first-order EWPT, which proceeds through nucleation and growth of broken-phase bubbles. In addition, there is a stronger requirement [12, 13] on the EWPT as explained later.

In EWBG, the baryon asymmetry is generated at the EWPT utilizing the 𝐁-violating sphaleron transition. However, the sphaleron transition must decouple just after the phase transition, otherwise the generated baryon asymmetry would be washed out. In the broken phase, the sphaleron transition rate is proportional to the Boltzmann factor exp(−𝐸_sph/𝑇), where 𝐸_sph is the energy of a sphaleron configuration and is given by 𝐸_sph∼4𝜋𝜙/𝑔, where 𝜙 is the thermal average of the Higgs field, ⟨𝐻⁰⟩_𝑇=𝜙/√2, and 𝑔 is the SU(2) gauge coupling. In order to ensure the decoupling of the sphaleron transition in the broken phase, 𝜙 should be large enough to suppress the previous rate, leading to the condition 𝜙_𝑐/𝑇_𝑐≳1, where 𝜙_𝑐 is given by ⟨𝐻⁰⟩_𝑇𝑐=𝜙_𝑐/√2. Therefore, the first-order EWPT has to be strong enough to avoid a washout of the generated baryon asymmetry.

A basic tool to analyze the EWPT is the finite temperature effective potential (FTEP) ([33–35], and for a review on EWPT and FTEP, see [36] and references therein). In SM, the one-loop FTEP is given by 𝑉⁽⁰⁾𝜙=𝑉₀𝜙+𝑉₁⁽⁰⁾𝜙+𝑉₁^(𝑇)𝜙,𝑇, (12) where 𝑉₀ is the tree-level potential for the Higgs field, 𝑉₁⁽⁰⁾ is the temperature independent one-loop contribution, and 𝑉₁^(𝑇)(𝜙,𝑇) represents the finite temperature correction to the zero-temperature potential. 𝑉₁⁽⁰⁾ in Landau gauge and M𝑆 scheme is given by the Coleman-Weinberg potential 𝑉₁⁽⁰⁾𝜙=_{𝑖=ℎ,𝜒,𝑊,𝑍,𝑡}𝑛_𝑖𝑚⁴_𝑖𝜙64𝜋²𝑚ln^2𝑖𝜙𝜇²−𝐶_𝑖, (13) where 𝑚²_𝑖(𝜙) is the 𝜙-dependent squared mass for particle 𝑖 with 𝑛_𝑖 the corresponding number of degrees of freedom. See Table 1 for the definitions, where we note that fermions contribute with opposite sign.

Table 1 

The 𝜙-dependent squared masses 𝑚²_𝑖(𝜙) and the number of degrees of freedom 𝑛_𝑖 for particle 𝑖 in SM. 𝑚²_𝑖(𝜙=𝑣) gives the physical masses for 𝑣=246 GeV, which is the vacuum expectation value of the Higgs field. 𝐶_𝑖 are given for MS scheme. We include the contributions coming from the Higgs boson ℎ, the would-be Nambu-Goldstone bosons 𝜒, the gauge bosons 𝑊 and 𝑍, and the top quark 𝑡.

The one-loop finite temperature contribution, 𝑉₁^(𝑇), is given by 𝑉₁^(𝑇)=𝑇𝜙,𝑇₄2𝜋²_{𝑖=ℎ,𝜒,𝑊,𝑍}𝑛_𝑖𝐽_𝐵𝑚²_𝑖𝜙𝑇²+𝑛_𝑡𝐽_𝐹𝑚²_𝑡𝜙𝑇², (14) where 𝐽_𝐵 and 𝐽_𝐹 are defined by 𝐽_𝐵(𝑎)=^∞₀𝑑𝑥𝑥²ln1−𝑒^{−√𝑥2+𝑎},𝐽_𝐹(𝑎)=^∞₀𝑑𝑥𝑥²ln1+𝑒^{−√𝑥2+𝑎}. (15) In the high temperature limit where 𝑚(𝜙)≪𝑇, 𝐽_𝐵,𝐹 can be expanded in terms of 𝑚(𝜙)/𝑇 (high-temperature expansion) as 𝐽_𝐵𝑚²𝑇²𝜋=−⁴+𝜋45²𝑚12𝑇²²−𝜋6𝑚²𝑇^23/2−1𝑚32𝑇⁴⁴𝑚ln²𝑎_𝑏𝑇²𝑚+𝒪⁶𝑇⁶,𝐽_𝐹𝑚²𝑇²=7𝜋⁴−𝜋360²𝑚24𝑇²²−1𝑚32𝑇⁴⁴𝑚ln²𝑎_𝑓𝑇²𝑚+𝒪⁶𝑇⁶, (16) where 𝑎_𝑏=16𝜋²exp(3/2−2𝛾_𝐸), and 𝑎_𝑓=𝜋²exp(3/2−2𝛾_𝐸), that is, ln𝑎_𝑏≈5.4076 and ln𝑎_𝑓≈2.6351.

Using high temperature expansion, the one-loop FTEP can be written as 𝑉𝜙,𝑇≃𝐷𝑇²−𝑇²_𝑜𝜙²−𝐸𝑇𝜙³+𝜆(𝑇)4𝜙⁴, (17) where 𝐷=2𝑚²_𝑊+𝑚²_𝑍+2𝑚²_𝑡8𝑣²,𝑇²_𝑜=𝑚²_ℎ,4𝐷𝐸=2𝑚³_𝑊+𝑚³_𝑍4𝜋𝑣³,𝜆(𝑇)3=𝜆−16𝜋²𝑣⁴×2𝑚⁴_𝑊𝜇ln²𝐴_𝐵𝑇²+𝑚⁴_𝑍𝜇ln²𝐴_𝐵𝑇²−4𝑚⁴_𝑡𝜇ln²𝐴_𝐹𝑇². (18) Here, 𝑚²_𝑖≡𝑚²_𝑖(𝑣) with 𝑣=246 GeV are physical masses for particle 𝑖, and ln𝐴_𝐵=ln𝑎_𝑏−3/2, as well as ln𝐴_𝐹=ln𝑎_𝑓−3/2. In (17), terms which do not have 𝜙 dependence are omitted. For sake of illustration, we also neglect contributions from ℎ and 𝜒 by assuming that ℎ is lighter than 𝑊 and 𝑍, though this is phenomenologically not correct. One sees that only weak bosons contribute to the coefficient 𝐸 of the cubic term in (17); hence, 𝐸 is rather small in SM.

With this FTEP, one can examine the behavior of the EWPT analytically. For 𝑇>𝑇₁≡8𝜆(𝑇₁)𝐷𝑇²_𝑜/(8𝜆(𝑇₁)𝐷−9𝐸²), the only minimum of the effective potential, (17), is 𝜙=0. Hence, the electroweak symmetry is restored. As the electroweak symmetry is broken at 𝑇=0, the phase transition must occur at a temperature 𝑇 that satisfies 𝑇₁>𝑇>0. Inspection of (17), one clearly sees that the phase transition is first order for 𝐸≠0, taking the form depicted in Figure 1, and is second order for 𝐸=0. When 𝐸≠0, the critical temperature 𝑇_𝑐 and 𝜙_𝑐 are given by 𝑇²_𝑐=𝜆𝑇_𝑐𝐷𝑇²₀𝜆𝑇_𝑐𝐷−𝐸²,𝜙_𝑐=2𝐸𝑇_𝑐𝜆𝑇_𝑐. (19) Then, the strength of the first-order phase transition is given by 𝜙_𝑐𝑇_𝑐=2𝐸𝜆𝑇_𝑐∼4𝐸𝑣²𝑚²_ℎ. (20) One notes that a lighter Higgs boson is preferred to maintain a stronger first-order phase transition.

Figure 1 

Typical forms of the effective potential in a case of the first-order phase transition for 𝑇=0, 𝑇=𝑇𝑐, and 𝑇>𝑇1.

At high temperature, a certain class of higher-order diagrams, the so-called ring diagrams (or daisy diagrams) [36], give significant contributions to the FTEP. Dominant contributions from the ring-diagrams can be resummed, and it amounts to shifting the masses 𝑚²_𝑖(𝜙) for bosons in the one-loop contributions, given in (13) and (14), to thermal masses given by ℳ²_𝑖(𝜙,𝑇)≡𝑚²_𝑖(𝜙)+Π_𝑖(𝑇), where Π_𝑖(𝑇) is the one-loop self-energy of particle 𝑖 in the infrared limit. As for gauge bosons, only the longitudinal modes receive corrections to the masses when one-loop self-energies are adopted for Π_𝑖(𝑇). Basically, this forbids the longitudinal modes of 𝑊 and 𝑍 to contribute to the cubic term in (17). As a consequence, the coefficient of the cubic term is reduced as 𝐸=2𝑚³_𝑊+𝑚³_𝑍4𝜋𝑣³→232𝑚³_𝑊+𝑚³_𝑍4𝜋𝑣³∼9.5×10⁻³, (21) leading to the reduction of the strength of the EWPT.

With these results, the sphaleron decoupling condition, 𝜙_𝑐/𝑇_𝑐≳1, reads 𝑚_ℎ≲√4𝐸𝑣∼42GeV. (22) However, this condition conflicts with the mass bound from direct search at the LEP, 𝑚_ℎ>114 GeV [6]. Furthermore, recent results from the LHC indicate a Higgs-like particle at 125 GeV. So, the EWPT is not strongly first order in the SM, and the baryon asymmetry generated during EWPT cannot be retained in the broken phase.

For a heavier Higgs boson, 𝑚_ℎ≳𝑚_𝑊, the approximations used in the previous analysis are no longer correct in the SM. In particular, higher-order diagrams beyond the ring diagrams also become nonnegligible at critical temperature, and a nonperturbative analysis is required. Lattice studies [37–40] suggest, however, that there is an endpoint of first-order EWPT around 𝑚_ℎ∼70 GeV, above which the transition turns into a continuous crossover. Therefore, there is no EWPT for the experimentally preferred Higgs boson mass.

For successful EWBG, then, physics beyond the SM is also required. It is clear that the existence of the cubic term in the FTEP of (17) is essential for a first-order phase transition, and the cubic term arises from (𝑚²)^3/2 term in the high temperature expansion for the bosonic loop function 𝐽_𝐵, given in (16), while there is no such term from fermionic contributions. So, if a new boson with strong coupling to the Higgs boson is introduced, their thermal loop can enhance the cubic term, and the EWPT can be strengthened. On the other hand, introducing a new fermion does not affect the cubic term and, hence, does not improve the strength of the EWPT at one-loop level.

Based on the one-loop result, therefore, the introduction of 4G fermions seems useless from the viewpoint of making a first-order EWPT. However, given that bounds on the 4G quark masses from direct search at the LHC have reached beyond the 600 GeV level, that is, beyond the perturbative unitarity bound of 500–550 GeV, we have to reconsider the problem of the EWPT beyond perturbative level. Interestingly, several studies (for zero temperature case) suggest that strong Yukawa couplings of 4G quarks can induce new bound states of 4G quarks [41–43]. If such Yukawa bound states are bound tightly enough so that they do not dissolve around the critical temperature of the EWPT, bosonic bound states may contribute to the FTEP via loop effect and may induce the cubic term, leading to strongly first-order phase transition. Beside this possibility, if the 4G quarks 𝑄 form a pair condensate, ⟨𝑄𝑄⟩≠0, due to strong Yukawa couplings, an effective description of the theory would be given by a three Higgs doublet model (1 elementary + 2 composite Higgs doublet) [41] (the EWPT in multicomposite Higgs doublet model with the 4G fermions was studied in [19], where the composite Higgs fields appear as bound states of the 4G fermions, formed by strong four-fermion interactions). Therefore, the potential for the Higgs field would be modified even at tree level, and dynamics of the EWPT could be drastically changed.

4. Discussion and Summary

Let us continue the discussion from the previous section. Bosonization is only one aspect of very heavy 4G quarks that could change the landscape for EWBG. After all, the bosonization described previous is due to strong Yukawa coupling. A special feature of the SM is that fermion, in fact all, masses reflect a dynamical coupling. But fermions are special in that there is no theory of these Yukawa couplings, and, neglecting neutrinos, they span a range of six orders of magnitude! If a fourth generation exists above the unitarity bound, the strong Yukawa coupling could induce ⟨𝑄𝑄⟩ condensation, which in principle could replace the usual condensation of the Higgs field as the electroweak symmetry breaking mechanism. A “bootstrap” dynamical symmetry breaking (DSB) equation was recently formulated [44] and studied [45], and it was found numerically that DSB can occur for Yukawa coupling 𝜆_𝑄≳4𝜋. For such strong Yukawa coupling, our traditional notions for EWBG may have to be reconsidered.

The difficulty for the bootstrap DSB scenario is again the newly observed [7, 8] 125 GeV boson at the LHC. However, the observed state could be a dilaton of scale invariance violation, rather than the genuine SM Higgs boson. The couplings of the dilaton to vector bosons and fermions are suppressed by 𝑣/𝑓 compared to the SM Higgs boson case, where 𝑓 is the dilaton decay constant, while 𝛾𝛾 and 𝑔𝑔 couplings of the dilaton are essentially free parameters depending on the details of scale invariance violation [46]. Therefore, if the observed signal arises mostly from the gluon fusion, the dilaton could mimic the SM Higgs boson. Discrimination is provided by the detection of the Higgs production through the vector boson fusion (VBF), or the bremsstrahlung off a vector boson (VH). In particular, important modes are the VBF-produced 𝛾𝛾 mode and the 𝑏𝑏 and 𝜏𝜏 modes in this regard. As VBF and VH production are subdominant compared with gluon fusion, these modes are not yet firmly established by the LHC experiments. It is interesting that the bootstrap DSB equation is scale invariant by construction, and at the present level of study, the scale is introduced heuristically as a physical condition for the bootstrap [45]; hence, a dilaton is in principle allowed. Whether the 125 GeV state is the SM-like Higgs or a dilaton with rather modified couplings can be checked by ATLAS and CMS.

Even if the 125 GeV object is verified as SM-like Higgs boson, the strong Yukawa coupling of 4G quarks may still be relevant. The lightness of the Higgs boson may be because it is a pseudo-Goldstone boson from a TeV scale strongly coupled theory, which indeed the previous 𝜆_𝑄≳4𝜋 situation seems to qualify it as a candidate. Pseudo-Goldstone Baryogenesis (PGBG) has been advocated [47] as a possible mechanism, where strong coupling brings about parametric enhancement of effective dimension-six interactions that loosen the relation between Higgs self-coupling and the Higgs mass; it is 𝜆(𝑇_𝑐) that really appears in (20), and Higgs mass enters only through the standard relation of 𝑚²_ℎ=2𝜆𝑣². Thus, whether one could have PGBG at work or not has to be tested by checking the Higgs boson self-coupling, which likely can be done only at an 𝑒⁺𝑒⁻ Linear Collider with energy above 500 GeV, which will take several decades.

Of course, the formulation of the bootstrap DSB equation is not yet at the level to demonstrate a possible PGBG, and much more work needs to be done. The formulation of the bootstrap DSB equation itself may offer a different path towards the study of order of phase transition. The equation is a coupled set of two integral equations of loop momentum in the ladder approximation [45]. At finite temperature, these would become four equations, since the temporal integration would be replaced by a summation. One could check the temperature dependence of DSB, both in finding the critical 𝑇_𝑐 when symmetry is restored, but also check what is “𝜙_𝑐,” the critical Higgs field expectation value, and whether its strength allows a strongly first-order transition. Of course, this is not yet done, but it may offer further insight that is along a different path than the usual approach of [36] discussed in the previous section.

To summarize, EWBG is an attractive scenario to address the BAU puzzle, especially because this scenario is based on particle physics models that can be tested at the LHC. Although the EWBG scenario fails within the minimal SM with three quark generations, introduction of the fourth generation may revive the scenario, offering hope to solve the problems in the SM case. CP violation coming from the 4×4 CKM matrix would be highly enhanced compared to the SM, mainly through the large masses (or the large Yukawa couplings) of 𝑡^′ and 𝑏^′ quarks. Naive extension of the EWBG mechanism of SM to the 4G case shows that this CP violating effect can be large enough to explain BAU with reasonable 4G parameters. The issue of whether the EWPT becomes strongly first order with 4G quarks is still an open problem, especially since the bounds on 4G quark masses from direct search at the LHC have reached the 600 GeV level, which is beyond the perturbative unitarity bound of 500–550 GeV. A mechanism of the strongly first-order phase transition may be accommodated by new bound states and/or 𝑄𝑄 condensation of 4G quarks, both induced by strong Yukawa couplings, which could modify the FTEP at loop/tree level. Higgs boson search at the LHC has uncovered a new boson with SM Higgs features. The study is still ongoing, and its nature is important to obtain better understanding for the EWPT. All in all, the possibility of providing both sufficient CP violation as well as perhaps a strongly first-order phase transition (which we did not demonstrate), all rooted in large Yukawa couplings of fourth generation quarks, makes this a worthy pursuit.

Acknowledgments

The authors are grateful to Y. Kikukawa for collaboration on the FS mechanism in the four-generation case, presented in Section 2. M. Kohda is supported by the NTU Grant no. 10R40044 and the Laurel Program, and W.-S. Hou by the Academic Summit Grant NSC 100-2745-M-002-002-ASP of the National Science Council of Taiwan and various NTU grants under the Excellence Program of the Ministry of Education of Taiwan.