Magnetic mass effect on sphaleron energy

We elucidate a magnetic mass effect on a sphaleron energy that is crucial for baryon number preservation needed for successful electroweak baryogenesis. It is found that the sphaleron energy increases in response to the magnetic mass. As an application, we study the sphaleron energy and electroweak phase transition with the magnetic mass in a two-Higgs-doublet model. Although the magnetic mass can screen the gauge boson loops, it relaxes a baryon number preservation criterion more effectively, broadening the baryogenesis-possible region. Our findings would be universal in any new physics models as long as the gauge sector is common to the standard model.

Introduction.-Sphaleron [1,2] in electroweak (EW) theories plays a central role in the baryon and lepton number-violating process at high temperature and its rate in the EW-symmetry-broken phase is crucial to successful electroweak baryogenesis (EWBG) [3]. In particular, energy of the sphaleron, which is predominantly scaled by a vacuum expectation value (VEV) of the Higgs field, has to be sufficiently large in order to preserve the baryon asymmetry, calling for strong first-order electroweak phase transition (EWPT).
It is known that ordinary perturbative expansion at zero temperature would break down at high temperature due to uncontrollable temperature-dependent power corrections. Standard prescription for this problem is called thermal resummation that reorganizes the perturbative expansion in a consistent manner. It is common practice in the literature that finite-temperature effective potential is improved by resummation methods of Parwani [4] or Arnold-Espinosa [5]. With such a resummed effective potential, EWPT is studied in a plethora of models beyond the standard model (SM). Regarding the thermal resummation for the EW gauge sector, much attention has been given to electric masses despite a magnetic mass of the SU (2) gauge field can in principle enter the problems. It is thus legitimate to take both the resummation effects into consideration (for earlier studies in the SM, see Ref. [6,7]).
In evaluating the sphaleron rate using a WKB method, it is necessary for consistency to use the same resummed Lagrangian that is applied to the EWPT study. Due to the fact that the sphaleron is a magnetic configuration, the magnetic mass term appears at the lowest order in the equation of motion for the sphaleron, while the electric masses come into the gauge boson loops in the resummed effective potential, which are higher order. Regardless of the potential significance of the magnetic mass, devoted study has been missing so far. Since the required strength of the first-order EWPT is predominantly determined by the sphaleron energy, ramifications of the magnetic mass effect is of great importance for successful EWBG.
In this Letter, we investigate the magnetic mass effect on the sphaleron energy and quantify its impact on the baryon number preservation criterion (BNPC) needed for EWBG. For illustrative purpose, we first work on the SM at zero temperature and clarify to what extent the sphaleron energy can be affected by the presence of the magnetic mass. Given the fact that the magnetic mass is inherently nonperturbative and its robust estimate seems unavailable to date yet, we thus allow it to vary within a reasonable range studied in the literature. Then, as a realistic application, we consider a EWBG scenario in a general two Higgs doublet model (2HDM) (for a recent study, see, e.g., Ref. [8]) and evaluate the sphaleron energy and BNPC with the magnetic mass at critical temperature of EWPT. Our analysis shows that the sphaleron energy is the increasing function against the magnetic mass, which renders BNPC more relaxed and EWBG-possible regions get broadened accordingly. This conclusion would apply to any models beyond the SM as long as the gauge sector is common to the SM.
Gauge-invariant mass term.-We first delineate how the magnetic mass is implemented to our problem. Here we demonstrate two different approaches and show that the both lead to the same mass form under a certain condition.
In general, the bilinear term of the gauge field takes the form where Π µν is the self-energy and its general form in the momentum space is cast into the form [6] Π µν (p) = Π L (p)L µν (p) + Π T (p)T µν (p) + Π G (p)G µν (p) where L µν (p) = u T µ u T ν /(u T ) 2 , T µν (p) = g µν − p µ p ν /p 2 − L µν (p), G µν (p) = p µ p ν /p 2 , S µν (p) = (p µ u T ν + p ν u T µ )/ (p · u) 2 − p 2 with u T µ = u µ − p µ (u · p)/p 2 and u µ specifies thermal bath. We consider a static case in which p 0 = 0 in the rest frame of the thermal bath [u µ = (1, 0)]. In this case, under the condition of ∂ i A i = 0, which is satisfied by the sphaleron ansatz adopted here, the bilinear Lagrangian (1) is reduced to the local form Since the sphaleron is the magnetic object as mentioned above, only Π T is involved in the equations of motion for the sphaleron. It is well known that the computation of Π T requires a nonperturbative approach [9]. We do not attempt to do it here and treat it as an input parameter. We note that the nonabelian gauge invariance requires hight-order terms in A µ . In the case of hard thermal loop approximation in QCD, effective n-point (n ≥ 3) functions of A µ would be zero to the leading order and only Π L term in Eq. (3) survives in the static limit [10].
In contrast, we are not aware of the counterpart in effective theories of the magnetic mass.
In contrast to this approach, the second one is manifestly gauge invariant by construction. As discussed in Ref. [11], one of the gauge-invariant operator with a mass dimension of two can be constructed by minimizing the volume integral of Tr[A 2 µ ] along gauge orbits, i.e., where U is the gauge transformation function, that is, with g being gauge coupling. As proven, A 2 min can be expressed by an infinite series of nonlocal and gauge-invariant terms constructed by the covariant derivative and field strength of A µ such as Tr[F µν (D 2 ) −1 F µν ] [11] (for a dedicated study of Yang-Mill theories with this nonlocal term, see Ref. [12]). It is shown that A 2 min can be local if ∂ µ A µ = 0 is satisfied [11,12], which greatly simplifies our calculation as delineated below.
Sphaleron with the magnetic mass.-Now we work out the sphaleron energy in the presence of the magnetic mass in the SM. As demonstrated in Ref. [13], U (1) Y contribution is rather minor so that we do not include it in the estimate of the sphaleron energy.
The original ansatz of the sphaleron adopted in Ref. [2] causes a divergence in the A 2 i term in the energy functional of the sphaleron. To avoid this, we perform a SU (2) gauge transformation as [2]), the ansatz of A i is cast into the form where ǫ ija is the Levi-Civita symbol, , τ a are the Pauli matrices and f (r) denotes the profile function. Unlike the original sphaleron ansatz, A i is proportional to (1 − f ) that is damped exponentially at r → ∞, and thus the A 2 i term becomes finite. Most importantly, the above ansatz satisfies ∂ i A i = 0. Exploiting this property as well as the A 0 = 0 gauge, which is often adopted for finding a static classical solution, Eq. (4) is reduced to the local form, thereby the magnetic mass contribution to the energy function of the sphaleron takes the form Adding this energy shift into the ordinary energy functional of the sphaleron in the SM [2], one arrives at where ξ = gvr with v being the VEV of the Higgs field, V 0 (h) = λv 4 (h 2 − 1) 2 /4 and r m = m T /(gv/2) = m T /m W . From the above energy functional, it follows that Since the magnetic mass correction is positive definite, we expect that the resulting sphaleron would be more massive than the usual one. Moreover, r 2 m could exceed unity unlike the h 2 term.
In Fig. 1, a dimensionless sphaleron energy defined by E sph = E sph (g/4πv) is plotted as a function of r m . We use the values of m W = 80.4 GeV, m h = 125 GeV and v = 246 GeV to fix the input parameters. One can see that E sph increases from 1.92 to 5.68 as r m varies from 0 to 2. This drastic change stands in stark contrast to the dependence of the Higgs mass on E sph . As first clarified in Ref. [2], E sph is an increasing function of the Higgs mass, which would be at most around 2.7 even when the Higgs mass goes to infinity. The question to be answered is how large r m could be in a realistic case. As widely studied, the magnetic mass is given by m T = cg 2 T with c being a coefficient that is determined by nonperturbative methods. For instance, in early studies [6,7] it is found in SU (2) gauge-Higgs model that c = 1/(3π) ≃ 0.11 by solving a gap equation in the high-T limit. However, this result turns out to be gauge dependent. Estimates using the gauge-invariant gap equations provide somewhat bigger values: c = 0.28 [14], c = 0.38 [15], and c = 0.35 [16]. On the other hand, a lattice calculation finds even bigger value c = 0.46 [17] with a caveat [18]. Given the lack of the convincing estimate, we treat c as the varying parameter and take the maximal value of c = 0.45 in our analysis. It should be noted that the degree to which E sph is enhanced depends not only on the value of m T but that of m W (T ) = gv(T )/2 at finite temperature T . Generally, r m gets bigger for weaker first-order EWPT because of r m = 2cg(T C /v C ). For c = 0.45 and v C /T C = 0.3, one would get r m ≃ 2, where g = 0.65 is used. Therefore, E sph could be three times as large as the case without the magnetic mass effect. Now we discuss implications of the enhanced E sph for EWBG. As alluded above, the baryon and lepton number-changing process has to shut off in the EW broken phase, which yields BNPC as [19,20] v T > g 4πE sph 44.35 + log corrections ≡ ζ sph .
where the logarithmic corrections come from the fluctuation about the sphaleron such as zero mode factors, which could be 10% correction in the minimal supersymmetric SM [20]. Since ζ sph is predominantly affected by E sph , we do not consider the subdominant logarithmic corrections hereafter. Without large supercooling, T can be set to a critical temperature T C at which the Higgs potential has two degenerate minima. As seen from Eq. (12), the larger E sph lowers ζ sph . One can find ζ sph = 1.20−0.41 for E sph obtained in Fig. 1. Therefore, the commonly used BNPC in the literature, v C /T C > 1, does not always give the reasonable criterion in the presence of the magnetic mass. One remark here is that as emphasized in Ref. [20], E sph (T = T C ) tends to be smaller than E sph (T = 0), implying that the above values of ζ sph could be underestimated to some extent. We therefore evaluate E sph (T C ) explicitly in the case of 2HDM, which is one of the simplest extended models that accommodates EWBG possibility. Application to 2HDM.-Here, we consider a general 2HDM in which an ad hoc Z 2 symmetry is not imposed. For the sake of simplicity, we take an alignment limit in which sin(β − α) = 1, where α and β are the mixing angles between two neutral Higgs bosons and two Higgs VEVs, respectively. In this case, the Higgs couplings to gauge bosons and fermions are reduced to the SM values at tree level. In addition, we put another simplifying assumption that tan β = 1 and Z 2 -breaking Higgs quartic couplings are taken zero to first approximation, while flavor-changing neutral Higgs couplings in the Yukawa sector are allowed to exist in order to provide CP violation for baryogenesis [8]. In this setup, the effective potential at tree-level is SM like. We emphasize here that the above simplification do not spoil the core of our discussion, and main findings would not be affected by it.
Following the resummation method of Rarwani [4], we construct the resummed Lagrangian as L = (L R − ∆m 2 (T )Ψ 2 ) + (L CT + ∆m 2 (T )Ψ 2 ), where L R and L CT denote the renormalized Lagrangian and corresponding counterterms, respectively. The thermal mass terms of the particles in question are collectively denoted as m 2 (T )Ψ 2 (for explicit forms of m 2 (T ), see, e.g., Ref. [21]). The thermal masses in the first parenthesis are treated as zeroth-order in the resummed perturbation theory while those in the second one as the part of the counterterms. For the magnetic mass, we expand the gauge-invariant operator d 4 xA 2 min as where the ellipsis denotes higher-order terms in A µ . We retain only the quadratic term that is needed for the oneloop effective potential calculation. Taking the Landau gauge and regularizing the resummed effective potential using the MS scheme, one finds wherem . Their numerical evaluations are made using fitting functions described in Ref. [20].
For the renormalization of the vacuum and the mass of the SM-like Higgs (h) at one-loop level, we employ a scheme in which the tree-level relations are not altered by the one-loop contributions [22]. A thorny problem in this scheme is that the massless NG bosons cause an infrared divergence that calls for resummation of higherorder corrections [23]. It is demonstrated in Ref. [24] that such a NG resummation has little effect on EWPT so that we do not take the NG bosons into account in our numerical study. Moreover, we do not include the SM-like Higgs boson loop contribution either since its treatment is somewhat technical in the case ofm 2 h < 0 despite its numerical impact is rather minor. With that in mind, we calculate v C /T C ≡ R C using Eq. (14) plus the tree-level Higgs potential. Likewise, E sph is evaluated by using the same resummed effective potential at T C .
For illustration, we consider a case in which all the heavy Higgs boson masses are degenerate in masses, m Φ ≡ m H = m A = m H ± , and M 3 = m 2 3 /(sin β cos β) = (300 GeV) 2 with m 2 3 being the squared mixing mass between the two Higgs doublets in a generic basis where the both Higgs doublets develop the VEVs.
In Fig. 2, R C ≡ v C /T C and ζ sph are shown as functions of m Φ varying from 450 GeV to 500 GeV. The solid curves represent R C in the cases of c = 0.0 (red), 0.3 (blue), and 0.45 (black). In any case, R C gets enhanced when m Φ is considerably bigger than M , which is due to the fact that thermal loops of the heavy Higgs bosons enhance the potential barrier. The differences among the three cases become more pronounced in the lower R C region, where the gauge boson thermal loops are main contributor to the potential barrier and thus the screening effect by the magnetic mass is more influential. The three dashed curves display ζ sph for c = 0.0, 0.3, 0.45 with the same color scheme as R C . Without the magnetic mass effect, it is found that ζ sph = 1.38 − 1.32 in which E sph (T C ) = 1.67 − 1.75, which is smaller than E sph = 1.92 found in the case of the SM at T = 0 discussed above. The nonzero magnetic mass cases show more dramatic effects on ζ sph compared to R C , which is attributed to the substantial enhancement of E sph as expected from the analysis in the SM. As is the case of R C , the magnetic mass effect on ζ sph gets more pronounced in the lower R C region and its change is even more drastic, which is due to the enhancement of r m = 2cg/(πR C ). For c = 0.45 and m Φ = 450 GeV, it is found that r m = 2. Conclusion.-Our analysis has unveiled that the sphaleron energy can get increased considerably for the nonzero magnetic mass, which relaxes BNPC and broadens the parameter space for successful EWBG. The findings here would apply for any other new physics models as long as the gauge sector is common to the SM.