Axionic Electroweak Baryogenesis

An axion can make the electroweak phase transition strongly first-order as required for electroweak baryogenesis even if it is weakly coupled to the Higgs sector. This is essentially because the axion periodicity naturally allows the structure of phase transition to be insensitive to the axion decay constant that determines the strength of axion interactions. Furthermore, the axion can serve as a CP phase relevant to electroweak baryogenesis if one introduces an effective axion coupling to the top quark Yukawa operator. Then, for $f$ between about TeV and order $10$~TeV, the observed baryon asymmetry can be explained while avoiding current experimental constraints. It will be possible to probe the axion window for baryogenesis in future lepton colliders and beam-dump experiments.

Electroweak baryogenesis (EWBG) is one of the most attractive ways to explain the observed baryon asymmetry of the Universe [1][2][3]. For successful EWBG, the Standard Model (SM) needs to be extended so that the electroweak phase transition (EWPT) is strongly firstorder and an extra source of CP violation is present [4][5][6][7][8][9][10]. The connection to EWPT has led to consider new physics near the electroweak scale with significant couplings to the Higgs sector. However, it would then be difficult to reconcile EWBG with the observed properties of the SM-like Higgs boson and LHC null results for new particle searches so far while avoiding the constraints from electric dipole moments (EDM) in association with CP violation relevant to baryogenesis.
In this paper we, present a scenario for EWBG where an axion φ weakly coupled to the Higgs sector induces a required strong first-order phase transition. The potential of the extended Higgs sector possesses a shift symmetry, φ → φ + 2πf , and is written where H is the Higgs doublet, and θ = φ/f . Let us consider a simple model where φ couples to the Higgs squared operator V 0 = λ|H| 4 + µ 2 |H| 2 − M 2 cos(θ + α)|H| 2 − Λ 4 cos θ, (2) and so φ plays a crucial role in EWPT for 0 < µ 2 < M 2 . 1 The potential shape is determined by the constant phase α and the ratio between M and Λ, but is insensitive to f as the dependence on it arises only radiatively. This explains why EWBG is viable even if φ weakly couples to 1 The potential terms depending φ can be generated nonperturbatively, for instance, if φ has an anomalous coupling to a hidden QCD, and H couples to additional vector-like lepton doublets and singlets via Yukawa interactions [11]. Here the leptons are charged under the hidden QCD and have Majorana masses such that only the singlet leptons are relevant light degrees of freedom at energy scales below the hidden confining scale.
the Higgs sector differently from other scenarios. Note also that a singlet-extended Higgs sector can generate a potential which allows a weakly coupled limit with a strong first-order phase transition as in our scenario. However, it is only when one adds a proper dimension-six or higher operator since otherwise, the potential becomes unstable.
How large can f be without spoiling EWBG? It turns out that a strong first-order phase transition is possible for f even larger than 10 6 GeV. A more stringent bound comes from the observed baryon asymmetry. To implement EWBG, one needs sizable CP violation at the phase interface, which can be provided for instance by an effective axion coupling to the top quark Yukawa operator. In such case, the axion itself corresponds to the CP phase and should have f lower than order 10 TeV to generate sufficient baryon asymmetry during phase transition. This is because the wall width increases with f , which reduces baryon asymmetry for a given source of CP violation. On the other hand, a lower bound on f comes from the Higgs searches and results because φ mixes with the Higgs boson, and also from the EDM searches associated with the CP violation for baryogenesis. Those constraints can be avoided for f above about 1 TeV. For f in the multi-TeV range, it would be possible to probe our scheme by future experiments for axion-like particle searches.
For large f above the electroweak scale, H is much heavier than φ. Thus, for a qualitative understanding of a phase transition, one can approximately explore EWPT using the effective potential of a single light field, φ, constructed by integrating out H via its equation of motion: which is parameterized by α ≥ 0 without loss of generality. To simplify our discussion, we further approximate thermal corrections to include only those to the Higgs mass squared. Then, ∆V is non-vanishing in the interval of θ where M 2 cos(θ − α) is larger than the thermal-corrected Higgs mass squared. The effective potential of φ is enough to discern which region of the parameter space leads to a strong firstorder phase transition. After a qualitative description of EWPT, we provide precise results taking a two-field dimensional analysis with full one-loop thermal corrections. Fig. 1 illustrates how a first-order phase transition is driven by the axion. At high temperature (blue curve), a large thermal mass for H leads to ∆V = 0, and the minimum ofV is located at θ = 0. An interval of θ where H is tachyonic appears and increases as the Universe cools down, and two degenerate minima are developed at θ = 0 and θ c at the critical temperature T c (orange curve). The electroweak minimum, θ = θ c , becomes the true vacuum at a temperature below T c because it is deeper than the symmetric one, θ = 0.
Let us explicitly examine the parameter region where EWPT is strongly first-order for the case with Λ = 130 GeV. In the white region of Fig. 2, the axion-Higgs coupling induces a strong first-order phase transition, i.e. v c /T c > 1, where v c is the Higgs vacuum expectation value at T c . The blue region leads to no phase transition since the symmetric minimum is deeper than the electroweak minimum, while the green region leads to the same situation as in the SM phase transition. In the red region, a first-order phase transition takes place, but not strong. The orange region is excluded because there is a potential barrier between two minima at T = 0. If there EWPT for Λ = 130 GeV. A strong first-order phase transition occurs in the white region. The symmetric minimum is deeper than the electroweak minimum in the blue region, while it is metastable in the orange region. The green and red regions are excluded because the transition is not first-order as in the SM, and is weak first-order, respectively. remains a barrier at T = 0, the vacuum transition rate is significantly suppressed for f above TeV, making the symmetric minimum metastable. For concreteness in the later discussion of baryogenesis, we choose a benchmark parameter point as marked by a star in the figure. The benchmark case leads to a strong phase transition, v c /T c 3, with T c 68 GeV and T 2 54 GeV. One also finds that the axion mass reads m φ 17 GeV × (f /TeV) −1 , and the axion-Higgs mixing angle is given by δ 0.1 × (f /TeV) −1 , which makes the axion detectable at colliders. Here T 2 denotes the temperature at which the barrier disappears.
The estimated values are slightly changed depending on f if one includes radiative and other thermal contributions in the potential. Now we move on to analyzing the phase transition in detail and examining the constraints on f . Electroweak bubbles of the broken phase are nucleated at a temperature below T c and expand. The bubble nucleation rate per unit volume is roughly given by T 4 e −S3/T , and it exceeds the Hubble rate when S 3 /T ≈ 130, which defines the nucleation temperature T n . Here S 3 is the Euclidean action of an O(3)-symmetric critical bubble [12,13]. Note that S 3 = 0 below T 2 because there is no potential barrier between two minima. the cosmoTransition python package to find a bubble solution through the overshooting-undershooting method [14]. The dashed lines are obtained simply by the potential V 0 (h, φ) including only the thermal corrections to the Higgs mass squared, for which T 2 54 GeV regardless of f as one can see in the figure. On the other hand, the solid lines show the bubble action evaluated by taking the full one-loop potential of h and φ at finite temperature [15], V (h, φ), which depends radiatively on f . The nucleation temperature is determined by the intersection of each solid line with the dotted horizontal line, and it approaches to T 2 as f increases. Here the barrier disappearing temperature is given by T 2 57 GeV.
It is interesting to notice that the Higgs contributions to the bubble action can be approximated by integrating out the Higgs field since the importance of its kinetic term gets suppressed as f increases. Consequently, S 3 is approximately given by f 3 times some function of Λ, α, and T . This feature can be understood because the action is written for the Higgs field h, showing that the Higgs contributions are suppressed by f 2 . Here V includes radiative and thermal corrections, and the prime denotes a derivative with respect to x ≡ r/f with r being the radial distance from the center of the bubble. The approximate scaling behavior, S 3 ∝ f 3 , reflects the fact that the effective potentialV (φ) can describe well the phase transition and the tunneling occurs mainly along the axion direction if φ is much lighter than H, that is, for large f . Our scenario leads to T n rather close to T 2 compared to other scenarios, but the duration of phase transition, ∆t PT , is long enough to complete the first-order phase transition. In fact, a more relevant requirement is that the time scale of generating baryon asymmetry, ∆t BG , should satisfy where we have used that it takes time ∼ m −1 φ for φ to settle down to the true vacuum after quantum tunneling. Since baryogenesis takes place outside bubbles via electroweak sphaleron processes, its time scale is the inverse sphaleron rate in the symmetric phase, ∆t BG ∼ T 3 /(Γ sph /V ), where Γ sph /V ∝ T 4 is the sphaleron rate per unit volume [16,17]. For large f , the bubble action shows the scaling behavior, S 3 ∝ f 3 , and S 3 /T is approximately linear in T during the phase transition. It thus follows that the duration of phase transition, which corresponds to the inverse of the time derivative of S 3 /T [18], is roughly given by ∆t PT ∝ 1/(T 2 f 3 ) assuming radiation domination. Note that the above condition (7) puts an upper bound on f because both m φ and ∆t PT are sensitive to f . For the benchmark case, baryogenesis requires f below or around 10 6 GeV.
To evaluate baryon asymmetry generated during the phase transition, one needs to specify the source of CP violation. In this paper, we consider an axion coupling to the top quark Yukawa operator, e iθq L3 t R3 H, so that the effective top quark Yukawa coupling is written for positive constants y t and x t , taking an appropriate field redefinition. Thus, the axion plays the role of a CP phase, and the top quark acquires a mass whose phase varies across the wall depending on the bubble profile. Note that the above axion coupling is subject to the EDM constraints if there is a relative phase between the Higgs and axion couplings to the top quark, that is, between y t + x t e iθ0+β and ix t e iθ0+β with θ 0 = φ /f at T = 0 [19][20][21]. It is clear that the EDM constraints depend on β, but EWBG is insensitive to it for x t 1 because what matters in baryogenesis is the phase difference of the top quark mass across the wall.
For the benchmark case, we show in Fig. 4 the parameter region where the correct amount of baryon asymmetry is obtained while avoiding the electron EDM constraint. Here we have taken β −θ 0 so that the EDM contribution from the axion is maximized. The observed baryon asymmetry, η B = (8.2 -9.4) × 10 −11 , can be explained for f below around 10 TeV if one takes x t < 0.3. One can also see that the EDM constraint is important only for f around or below TeV. If one takes a different value of β, the EDM constraint gets weaker, and the generated baryon asymmetry is slightly modified by a factor of order unity for x t above 0.1. It should be noted that η B is sensitive to the bubble dynamics, and our results are obtained under the assumptions summarized below. A more careful study will be necessary to estimate the relic baryon asymmetry precisely. The analysis assumes that the bubble profiles for h and φ are approximated by a kink, 1 − tanh(z/L w ), where z is the distance from the wall. The wall width L w is numerically obtained by examining the S 4 critical bubble configuration [12] , and the relation between f and L w is shown in the figure. For a thick wall with L w T n > 10, the numerical computation suffers from unstability when finding an inhomogeneous solution of the transport equations [22][23][24], and so we estimate the baryon asymmetry by performing an extrapolation, η B ∝ x(L w /L 0 ) n1+n2 ln(Lw/L0) , with constants n i and L 0 . The baryon asymmetry also replies on the wall velocity v w at a stationary situation, which can be computed based on the fluctuation-dissipation theorem using that the pressure on the wall [25] is determined by the potential difference between the broken and symmetric phases. We find that the benchmark case leads to v w 0.007.
Let us continue to discuss the experimental constraints on the axion properties and the future testability of our scenario. The LHC measurements [26], which require the Higgs signal strength relative to the SM prediction to be above 0.8, constrain axion-Higgs mixing and Higgs decay into axions. For the benchmark case, one needs f above 340 GeV, and the lower bound on f would increase to 1.4 TeV if the Higgs signal rate is measured at a sub-percent level in future lepton colliders [27]. A more stringent constraint comes from the Higgs searches at LEP because axion-Higgs mixing leads to the pro-cess, e + e − → Zφ [28]. In the benchmark case, the axion should be lighter than about 20 GeV to avoid the LEP bound, implying f > 850 GeV. For the case that φ changes the phase of the top quark mass, there is a constraint coming from cosmology and beam-dump experiments searching for axion-like-particles if the axion has an anomalous coupling to photons, φFF [29]. Those constraints rule out m φ smaller than 0.5 GeV, from which we find the upper bound on f to be about 35 TeV in the benchmark case. It is thus weaker than the bound imposed by the observed baryon asymmetry.
Taking into account the constraints discussed above, we find that the viable range of the axion mass is approximately between 1.3 GeV and 20 GeV in the benchmark case, which is obtained for the axion with f between 0.85 TeV and 13 TeV. This parameter window can be tested at future lepton colliders [30].
Finally, we comment on how the viable parameter range changes when one considers a case different from the benchmark case. If one moves away from the orange region in Fig. 2, T n and T 2 get higher, and thus the Higgs vacuum expectation value at T n decreases and the phase transition is weakened. As a result, the size of CP violation associated with the top quark is reduced during phase transition, lowering the upper bound on f . On the other hand, if one approaches to the orange region, the upper bound on f can be released, but instead, successful baryogenesis would require tuning of the parameters. The benchmark case has been chosen to implement baryogenesis without severe tuning while allowing large f as possible. On the other hand, the lower bound on m φ becomes weaker if one takes smaller since the axion-Higgs mixing is proportional to it. In this case, the LHC constraints on the Higgs properties get stronger than the LEP bound, restricting severely the parameter space to give, for instance, m φ O(40) GeV for ∼ 0.2.
In this paper we have explored an axion-extended Higgs sector where EWPT is strengthened enough to implement EWBG even when the axion is weakly coupled to the Higgs sector. This is essentially owing to the axion periodicity, which helps to avoid an instability of the potential independently of the axion decay constant, f . Interestingly, the axion can also serve as a CP phase for EWBG if one considers an effective axion coupling to the top quark Yukawa operator. In such case, axion-induced EWBG can account for the observed baryon asymmetry of the Universe while evading the current experimental constraints if f lies in the range between about TeV and order 10 TeV. Note that roughly the axion mass reads m φ ∼ v 2 /f , and the axion-Higgs mixing angle is given by δ ∼ αv/f . Therefore it will be possible to probe our scenario by future lepton colliders and beam-dump experiments.